ANALYSIS AND OPTIMISATION OF
THERMAL ENERGY STORAGE
JOSHUA MCTIGUE
St. Catharine’s College
Cambridge University Engineering Department
September 29, 2016
A DISSERTATION SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
Declaration
This dissertation is the result of my own work and includes nothing which is the outcome
of work done in collaboration except as declared in the Preface and specified in the text.
It is not substantially the same as any that I have submitted, or, is being concurrently sub-
mitted for a degree or diploma or other qualification at the University of Cambridge or any
other University or similar institution except as declared in the Preface and specified in the
text. I further state that no substantial part of my dissertation has already been submitted, or,
is being concurrently submitted for any such degree, diploma or other qualification at the
University of Cambridge or any other University of similar institution except as declared
in the Preface and specified in the text.
The following does not exceed 65 000 words or 150 figures as set out by the Department
of Engineering degree committee. This thesis contains 56 661 words and 84 figures.
i
Abstract
The focus of this project is the storage of thermal energy in packed beds for bulk electricity
storage applications. Packed beds are composed of pebbles through which a heat transfer
fluid passes, and a thermodynamic model of the heat transfer processes within the store
is described. The packed beds are investigated using second law analysis which reveals
trade-offs between several heat transfer processes and the importance of various design
parameters. Parametric studies of the reservoir behaviour informs the design process and
leads to a set of design guidelines. Two innovative design features are proposed and investi-
gated. These features are segmented packed beds and radial-flow packed beds respectively.
Thermal reservoirs are an integral component in a storage system known as Pumped Ther-
mal Energy Storage (PTES). To charge, PTES uses a heat pump to create a difference in
internal energy between two thermal stores; one hot and one cold. The cycle reverses dur-
ing discharge with PTES operating as a heat engine. The heat pumps/engines require com-
pression and expansion devices, for which simple models are described and are integrated
with the packed bed models. The PTES system behaviour is investigated with parametric
studies, and alternative design configurations are explored.
A multi-objective genetic algorithm is used to undertake thermo-economic optimisations
of packed-bed thermal reservoirs and PTES systems. The algorithm generates a set of
optimal designs that illustrate the trade-off between capital cost and round-trip efficiency.
Segmentation is found to be particularly beneficial in cold stores, and can add up to 1% to
the round-trip efficiency of a PTES system. On the basis of the assumptions made, PTES
can achieve efficiencies and energy densities comparable with other bulk electricity stor-
age systems. However, the round-trip efficiency is very sensitive to the efficiency of the
compression–expansion system. For designs that utilised bespoke reciprocating compres-
sors and expanders, PTES might be expected to achieve electricity-to-electricity efficien-
cies of 64%. However, using compression and expansion efficiencies typical of off-the-
shelf devices the round-trip efficiency is around 45%.
ii
Contents
Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Journal papers resulting from this thesis . . . . . . . . . . . . . . . . . . . . . . xv
Conference proceedings resulting from this thesis . . . . . . . . . . . . . . . . . xv
Funding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
1 Introduction 1
1.1 Context: current and future trends in energy supply . . . . . . . . . . . . . 2
1.2 Background and applications of energy storage . . . . . . . . . . . . . . . 4
1.3 Solutions to the intermittency of renewable power; how much energy stor-
age is required? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Prominent energy storage types . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.1 PHES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.2 CAES, I-CAES and AA-CAES . . . . . . . . . . . . . . . . . . . 15
1.4.3 Thermal energy storage . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Pumped Thermal Energy Storage . . . . . . . . . . . . . . . . . . . . . . . 22
1.5.1 Energy and power densities . . . . . . . . . . . . . . . . . . . . . 26
1.5.2 Round-trip efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5.3 The Joule-Brayton cycle PTES . . . . . . . . . . . . . . . . . . . . 32
1.5.4 The Rankine cycle based PTES . . . . . . . . . . . . . . . . . . . 36
1.6 Aims of the project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2 Modelling of packed-bed thermal reservoirs 41
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
iii
2.2 Description of packed bed reservoirs . . . . . . . . . . . . . . . . . . . . . 42
2.3 Heat transfer in packed bed reservoirs . . . . . . . . . . . . . . . . . . . . 44
2.4 Numerical solutions of the energy equations . . . . . . . . . . . . . . . . . 52
2.5 Availability destruction in packed beds . . . . . . . . . . . . . . . . . . . . 58
2.5.1 Thermal losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5.2 Pressure losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.5.3 Conductive losses . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.5.4 Heat leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.5.5 Exit losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.5.6 Storage losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.5.7 Radiative losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3 Behaviour of packed-bed thermal reservoirs 86
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.2 Parametric studies of reservoir thermodynamics . . . . . . . . . . . . . . . 88
3.2.1 Design parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.2.2 Operational parameters . . . . . . . . . . . . . . . . . . . . . . . . 97
3.3 Segmented packed beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.3.1 Segment control and spacing requirements . . . . . . . . . . . . . 102
3.3.2 Availability losses in segmented reservoirs . . . . . . . . . . . . . 104
3.4 Radial-flow packed beds . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.4.1 Radial-flow packed bed governing equations . . . . . . . . . . . . 115
3.4.2 Size of the empty column . . . . . . . . . . . . . . . . . . . . . . 119
3.4.3 Parametric studies of radial-flow packed beds . . . . . . . . . . . . 120
3.4.4 Comparison of radial-flow and axial-flow packed beds . . . . . . . 123
3.4.5 Storage losses in radial-flow stores . . . . . . . . . . . . . . . . . . 126
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4 Pumped Thermal Energy Storage (PTES) 130
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.1.1 Metrics of PTES performance . . . . . . . . . . . . . . . . . . . . 134
4.2 Modelling of reciprocating devices . . . . . . . . . . . . . . . . . . . . . . 135
iv
4.2.1 Pressure loss estimation . . . . . . . . . . . . . . . . . . . . . . . 140
4.2.2 Heat transfer losses . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.3 Other components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.3.1 Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.3.2 Electric heater . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.3.3 Pipe work and valves . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.3.4 Buffer tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.4 Behaviour of PTES systems . . . . . . . . . . . . . . . . . . . . . . . . . 157
4.4.1 Sensitivity to loss factors . . . . . . . . . . . . . . . . . . . . . . . 158
4.4.2 Further parametric studies . . . . . . . . . . . . . . . . . . . . . . 161
4.5 Avoiding electric heat addition . . . . . . . . . . . . . . . . . . . . . . . . 167
4.5.1 Neglecting heat addition . . . . . . . . . . . . . . . . . . . . . . . 168
4.5.2 Recuperated cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.5.3 Variable pressure ratios . . . . . . . . . . . . . . . . . . . . . . . . 171
4.5.4 Variable cold store outlet temperature . . . . . . . . . . . . . . . . 175
4.5.5 Liquid stores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5 Thermo-economic optimisation 181
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
5.2 Economic modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.2.1 Thermal reservoir costs . . . . . . . . . . . . . . . . . . . . . . . . 189
5.2.2 Reciprocating device costs . . . . . . . . . . . . . . . . . . . . . . 192
5.2.3 PTES capital cost . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.3 Multi-objective optimisation techniques . . . . . . . . . . . . . . . . . . . 197
5.3.1 Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . 198
5.3.2 Multi-objective Optimisation and Pareto Fronts . . . . . . . . . . . 200
5.3.3 Diversity in Objective and Solution Spaces . . . . . . . . . . . . . 201
5.4 Optimisation of packed-bed thermal reservoirs . . . . . . . . . . . . . . . . 202
5.4.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . 202
5.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.5 Optimisation of PTES systems . . . . . . . . . . . . . . . . . . . . . . . . 213
5.5.1 Optimisation of normal PTES configurations . . . . . . . . . . . . 214
v
5.5.2 Optimisation of alternative PTES configurations . . . . . . . . . . 222
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
6 Conclusions and future work 228
Nomenclature 240
Bibliography 245
A Estimation of thermal front lengths A1
A.1 Front lengths in axial-flow thermal stores . . . . . . . . . . . . . . . . . . A2
A.2 Front lengths in radial-flow thermal stores . . . . . . . . . . . . . . . . . . A4
vi
List of Figures
1.1 Installed capacity of various renewable energy sources in the UK. . . . . . 3
1.2 Instantaneous wind generation and electricity demand in the United Kingdom 7
1.3 Instantaneous wind generation during the first week of January 2013 in
three European countries. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Distribution of the duration of lulls in wind power in the UK for 2015. . . . 11
1.5 Optimal energy storage capacity as calculated by Strbac [1] for various
storage costs and duration of storage. . . . . . . . . . . . . . . . . . . . . . 12
1.6 Schematic of an AA-CAES system . . . . . . . . . . . . . . . . . . . . . . 16
1.7 (Left) PTES during charge acting as a heat pump. (Right) PTES during
discharge acting as a heat engine. . . . . . . . . . . . . . . . . . . . . . . . 23
1.8 Ideal T–s diagram for PHES systems during charge for systems based on
(a) the Joule-Brayton cycle and (b) a transcritical CO2 Rankine cycle. . . . 24
1.9 Variation in PTES round-trip efficiency with operating temperatures and
work ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.10 Variation of work ratio R with pressure ratio and T3/T1. . . . . . . . . . . . 30
1.11 T–s diagrams for Joule-Brayton cycle based PHES systems proposed by
Isentropic Ltd. and SAIPEM . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1 Schematic layout of a packed bed . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Schematic view of the packed bed structure and the equivalent control vol-
ume which displays heat transfer processes . . . . . . . . . . . . . . . . . 44
2.3 Schematic of an ideal thermal front between temperatures T1 and T2 trav-
elling at velocity Vf (see equation 2.3) . . . . . . . . . . . . . . . . . . . . 45
2.4 Experimental results showing how thermal fronts and void fraction depend
on the distance from the packed bed wall. . . . . . . . . . . . . . . . . . . 47
vii
2.5 Comparison of theoretical models and experimental results showing ther-
mal fronts in packed beds. Theoretical models both use one dimensional
energy equations similar to those developed in section 2.3. (a) Figure from
Zanganeh et al. [2]. (b) Figure from Anderson et al. [3] . . . . . . . . . . . 48
2.6 A stencil of the computational grid for the predictor-corrector numerical
scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.7 Temperature profiles generated using the predictor-corrector scheme at three
different times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.8 A stencil of the computational grid for the semi-implicit method . . . . . . 55
2.9 Temperature profiles generated using the semi-implicit scheme at three dif-
ferent times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.10 Heat capacity per unit volume for various materials . . . . . . . . . . . . . 61
2.11 Effect of variable cs on thermal front formation in cold stores . . . . . . . . 62
2.12 Control volume for heat conduction along a packed bed . . . . . . . . . . . 65
2.13 Packing arrangements giving maximum and minimum axial heat conduction 70
2.14 Comparison of experiments and theoretical correlations for the effective
conductivity. Figure taken from van Antwerpen et al. [4]. . . . . . . . . . . 71
2.15 Idealised depiction of successive layers of spherical particles in a packed bed 72
2.16 Electrical analogy for radiative heat transfer between successive layers of
particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.17 Correlations for axial effective conductivity kxeff and its impact on thermal
front shape and availability losses in hot stores. . . . . . . . . . . . . . . . 74
2.18 Reversible heat engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.1 Effect of θc and θd on the returned energy and utilisation of a hot store . . . 90
3.2 Parametric study of aspect ratio and particle diameter on hot and cold
packed beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.3 Parametric study of aspect ratio and particle diameter on hot packed beds. . 94
3.4 Distribution of losses for different packing materials computed at the nom-
inal conditions of table 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.5 Effect of reservoir sizing on availability losses in packed beds for Π = 0.75 98
3.6 Parametric study of θc and θd on hot and cold packed beds . . . . . . . . . 99
3.7 Parametric study of operating temperatures in hot and cold packed beds. . . 101
viii
3.8 Possible segmented packed bed arrangements . . . . . . . . . . . . . . . . 103
3.9 Behaviour of an unsegmented packed bed and a segmented packed bed with
eight segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.10 Availability losses of cold segmented packed beds. . . . . . . . . . . . . . 107
3.11 Conductive effects in the storage phase of segmented reservoirs. . . . . . . 109
3.12 Combined effect of conduction and heat leakage on loss coefficients during
the storage phase of segmented packed beds. . . . . . . . . . . . . . . . . . 112
3.13 Combined effect of conduction and heat leakage during the storage phase
of segmented packed beds. . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.14 Schematic of a radial-flow packed bed during charge. . . . . . . . . . . . . 114
3.15 Control volume of a radial-flow packed bed during charge. . . . . . . . . . 116
3.16 Thermal front steepening due to decreasing mass flow rate per unit area in
radial packed beds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.17 Parametric study on aspect ratio and particle diameter on hot and cold
radial-flow packed beds. . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.18 Comparison of dimensionally similar axial-flow packed beds and radial-
flow packed beds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.19 Comparison of thermal fronts formed in dimensionally similar axial-flow
and radial-flow packed beds for Π = 0.75. . . . . . . . . . . . . . . . . . . 125
3.20 Conductive and leakage losses during the storage phase of a cold radial-
flow packed bed reservoir with a linear thermal front. . . . . . . . . . . . . 127
4.1 Layout of PTES system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.2 Schematic and p–V diagram for an ideal reciprocating piston. . . . . . . . . 138
4.3 Mass flow rate–pressure ratio characteristic for reciprocating devices. . . . 140
4.4 p–V diagram from a prototype piston developed by Isentropic Ltd. which
shows a comparable shape to the ideal p–V diagram in figure 4.2(b). Figure
is reproduced from [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.5 Prototype piston developed by Isentropic Ltd. [5] which illustrates the valve
arrangement that consists of many rectangular orifices. . . . . . . . . . . . 142
4.6 Schematic diagram of gas at velocity vp passing through a valve of area Av
into the piston cylinder of area Ap . . . . . . . . . . . . . . . . . . . . . . 142
ix
4.7 Simplified model of the irreversibility associated with preheating during
induction and cooling during delivery, shown for a compression process.
The overall process 1–2 is assumed to be adiabatic. . . . . . . . . . . . . . 146
4.8 Schematic of heat transfer in a reciprocating device during compression. . . 148
4.9 Experimental and theoretical results for heat transfer losses in gas springs.
Figure is taken from Lee [6]. The figure shows the analytical expression
developed by Lee [7] as well as isothermal and adiabatic limits, and exper-
imental results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.10 Loss in efficiency due to heat transfer irreversibilities in gas springs . . . . 149
4.11 Experimental and theoretical results for heat transfer in reciprocating de-
vices. Figure is from Lawton [8] and shows the cylinder wall heat flux
against the crank angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.12 Layout of PTES system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.13 Sensitivity of thermodynamic PTES round-trip efficiency to loss factors . . 159
4.14 Sensitivity of thermodynamic round-trip efficiency to loss factors . . . . . . 160
4.15 Sensitivity of availability losses in each PTES component to variations in
(a) polytropic efficiency η (b) heat leakage factor ν . . . . . . . . . . . . . 161
4.16 Impact of heat exchanger loss factor ∗ on PTES performance. . . . . . . . 162
4.17 Variation of thermodynamic round-trip efficiency with packed-bed geomet-
ric parameters. Minimum, nominal and maximum values are given in ta-
ble 4.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
4.18 Distribution of available energy losses in a PTES system as the number of
layers in the cold store is varied. . . . . . . . . . . . . . . . . . . . . . . . 164
4.19 Variation of round-trip efficiency and energy density with PTES opera-
tional parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.20 Distribution of availability losses in PTES components as βchg and βdis are
varied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.21 T–s diagrams and temperature traces for the nominal PTES design. . . . . . 169
4.22 Simplified schematic of a PTES system using recuperation. . . . . . . . . . 170
4.23 T–s diagrams and temperature traces for a PTES design with T1 > T3. . . . 172
4.24 Charging profiles of a PTES scheme where the charging pressure ratio βchg
is increased such that T2 is constant . . . . . . . . . . . . . . . . . . . . . 173
x
4.25 Discharging profiles of a PTES scheme where the discharging pressure ra-
tio βdis is varied such that approximately equal amounts of heat are rejected
in each heat exchanger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.26 T–s diagrams and temperature traces for a PTES design with a cold store
discharging temperature of T ′1. . . . . . . . . . . . . . . . . . . . . . . . . 176
4.27 Thermal profiles in the cold store at (A) the end of discharging and (B) the
end of charging. Profiles are shown for the nominal design and for a PTES
design with a cold store discharging temperature of T ′1. . . . . . . . . . . . 177
4.28 Schematic of an alternative PTES design utilising liquid during the charg-
ing phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
5.1 Variation in capital cost per stored available energy for a 16 MWh ther-
mal reservoir using different storage materials. Costs factors and relevant
material data are given in table 5.2. Shaded areas indicate the cost ± one
standard deviation and represent the uncertainty of the cost factors . . . . . 190
5.2 Optimisation results for cold and hot packed-bed thermal reservoirs. . . . . 206
5.3 Distribution of capital costs in optimal cold and hot stores. . . . . . . . . . 212
5.4 Pareto fronts of the optimised results of PTES systems with the basic con-
figuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
5.5 Distribution of availability losses in the nominal PTES design. . . . . . . . 217
5.6 Distribution of availability losses in optimal PTES designs for points P1 –
P4 which are illustrated on figure 5.4. The optimisation was carried out on
a PTES system with the basic configuration. (a) P1 (b) P2 (c) P3 (d) P4 . . 219
5.7 Distribution of availability losses in optimal PTES designs for points P5
and P6 which are illustrated on figure 5.4. This optimisation was carried out
on a PTES system with the basic configuration, but with the compressor–
expander polytropic efficiency set to η = 0.90. (a) P5 (b) P6 . . . . . . . . 220
5.8 Distribution of capital costs amongst the system components for optimal
PTES systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
5.9 Pareto fronts of the optimised results of PTES systems with alternative con-
figurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
xi
5.10 Distribution of availability losses in optimal PTES designs for points P7
and P8 which are illustrated on figure 5.9. This optimisation was carried out
on a PTES system with the ‘SAIPEM’ configuration, in that temperature
T2 was constrained to 1100°C rather than 600°C. (a) P7 (b) P8 . . . . . . . 225
5.11 Distribution of availability losses in optimal PTES designs for points P9
and P10 which are illustrated on figure 5.9. This optimisation was carried
out on a PTES system where heat addition was avoided as described in
section 4.5. (a) P9 (b) P10 . . . . . . . . . . . . . . . . . . . . . . . . . . 226
A.1 Idealised thermal fronts at the end of charge (1 – green) and the start of
charge (2 – purple). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A3
A.2 Wave speed coefficients calculated with either the average thermal front
length or a fluctuating thermal front length. Compare with equations A.2, A.6
and A.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A5
A.3 Idealised thermal fronts in a radial-flow thermal store at the end of charge
(green) and the start of charge (purple). . . . . . . . . . . . . . . . . . . . A5
A.4 Fractional front lengths calculated for axial-flow stores λx and radial-flow
stores λ1 and λ2, see equation A.17. . . . . . . . . . . . . . . . . . . . . . A7
xii
List of Tables
1.1 Comparison of energy and power densities and efficiencies of several promi-
nent energy storage systems and media. . . . . . . . . . . . . . . . . . . . 21
1.2 Operating conditions for Joule-Brayton based PHES systems suggested by
patents and literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.1 Comparison of CPU times for each numerical scheme. . . . . . . . . . . . 57
3.1 Nominal PTES and packed bed designs . . . . . . . . . . . . . . . . . . . 89
3.2 Efficiency and losses in nominal packed bed designs . . . . . . . . . . . . 90
3.3 Physical properties of materials for hot and cold thermal stores. . . . . . . . 95
3.4 Nominal radial-flow packed bed designs . . . . . . . . . . . . . . . . . . . 120
3.5 Geometry of dimensionally similar axial-flow and radial-flow cold packed-
beds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.1 Hot and cold reservoir details for a nominal 16 MWh PTES system. . . . . 133
4.2 Details of the compression-expansion devices for a nominal power rating
of 2 MW. The nominal designs have a clearance ratio of ε = 5% and a
cylinder aspect ratio of Ls/d = 0.25. . . . . . . . . . . . . . . . . . . . . . 140
4.3 Pressure loss factors in reciprocating devices. . . . . . . . . . . . . . . . . 144
4.4 Approximate effectiveness and pressure loss factors for several heat ex-
changers of different sizes . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.5 Pressure loss factors in pipes. Each pipe section corresponds to a point on
the PTES schematic in figure 4.1. The diameters are those which give a
pressure loss factor of 1%. ∗ The final column is the pressure loss factor
for a pipe of diameter 0.55 m. . . . . . . . . . . . . . . . . . . . . . . . . . 156
xiii
4.6 Nominal values of loss factors, and minimum and maximum values used
in parametric studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.7 PTES variables that are used in the parametric study. Minimum and maxi-
mum values indicate the range through which the parameters are varied. . . 161
5.1 Costs factors k used in the economic analysis . . . . . . . . . . . . . . . . 189
5.2 Material properties and cost factors k used in the economic analysis. . . . . 191
5.3 Data from the literature regarding diesel engine rated power, geometry, and
capital costs. Further information about the literature sources is given in
table 5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.4 Further details about the literature sources for data on reciprocating diesel
engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
5.5 Capital costs of each component in a nominal 16 MWh, 2MW PTES sys-
tem described in table 4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.6 Capital costs of several prominent energy storage systems. . . . . . . . . . 197
5.7 Lower and upper bounds for parameters used in optimisation of hot stores. . 204
5.8 Lower and upper bounds for parameters used in optimisation of cold stores. 204
5.9 Design variables and results for selected points from the optimisation of a
cold store. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.10 Design variables and results for selected points from the optimisation of a
hot store. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
5.11 Typical efficiencies and capital costs of optimal packed-bed thermal reser-
voirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
5.12 Lower and upper bounds for parameters used in optimisation of PTES sys-
tems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5.13 Design variables and results of PTES optimal points P1–P6 (see figure 5.4). 216
5.14 Design variables and results of PTES optimal points P7–P10 (see figure 5.9).223
xiv
Journal papers resulting from this thesis
“Analysis and optimisation of segmented packed-bed thermal reservoirs for electricity stor-
age applications”
A. J. White, J. D. McTigue, C. N. Markides
Manuscript accepted by the IMechE Journal of Power and Engineering
“Segmented packed beds for improved thermal energy storage performance”
J. D. McTigue, A. J. White
Manuscript accepted by IET Renewable Power Generation, June 2016
“Parametric studies and optimisation of pumped thermal electricity storage”
J. D. McTigue, A. J. White, C. N. Markides
Applied Energy, vol. 137, pp. 800–811, Sept. 2015
“Wave propagation and thermodynamic losses in packed-bed thermal reservoirs for energy
storage”
A. J. White, J. D. McTigue, C. N. Markides
Applied Energy, vol. 130, pp. 648–657, 2014
Conference proceedings resulting from this thesis
Presentations
“A comparison of radial-flow and axial-flow packed beds for thermal energy storage”
International Conference on Applied Energy (ICAE 2016), 8–11/10/2016, Beijing, China
“Analysis of radial-flow packed beds for thermal energy storage”
UK Thermal Energy Storage (UKTES) Conference, 8–9/06/2016, London, UK
xv
“Segmented packed beds for improved thermal energy storage performance”
FETE (Fluids, Energy and Turbomachinery Expo, Department of Engineering Division A
Graduate Conference), 16/07/2015, Imperial War Museum, Duxford, UK
“Segmented packed beds for improved thermal energy storage performance.”
Offshore Energy and Storage (OSES) Conference, 1–3/07/2015, Edinburgh, UK
“A parametric study of losses in packed-bed reservoirs for electrical energy storage.”
13th UK Heat Transfer Conference, 2-3/09/2013, London, UK
Posters
“Optimisation of Pumped Thermal Energy Storage.”
P. Farres-Antunez, C. Willich, J. D. McTigue, A. J. White
SolarStor, London UK, 14/04/2016
“Operational management and electrical grid integration of Pumped Thermal Electricity
Storage”
C. Bordin, A. Ibrahim, J. D. McTigue, C. Dent, B. Kazemtabrizi, A. J. White
UK Energy Storage Conference, Birmingham, UK
“The role of energy storage in future energy systems”
UK Energy Research Centre International Summer School, 5-10/07/2015, Cambridge, UK
“Thermo-economic optimisation of a thermal energy storage system”
9th International Renewable Energy Storage (IRES) Conference, 9-11/03/2015, Dussel-
dorf, Germany
xvi
Funding
I gratefully acknowledge the following organisations for their financial support:
The UK Engineering and Physical Sciences Research Council (EPSRC) for providing a
Doctoral Training Grant
St. Catharine’s College, University of Cambridge for providing travel grants.
Cambridge University Engineering Department for providing travel grants.
The Energy Storage Research Network (ESRN) for providing a travel grant.
Isentropic Ltd. who provided a maintenance bursary.
Acknowledgements
I’m grateful to my supervisor, Alex White, for his guidance and help throughout this inter-
esting and rewarding project. I have also appreciated the useful discussions and enjoyable
conferences that I’ve shared with Alex and the rest of the group.
I am very thankful to my parents and to the Docherty clan for their generous and unerring
support and hospitality. And to Fiona Docherty, I am more grateful than I could express
here. I am also thankful to all those people who have made life in Cambridge a wonderful
and memorable experience. In particular, (and in chronological order) I thank BR, MB,
CS, FS, GM, RG, BL, LP, SS, AC, FD, AR, SS, SW, RS, SH.
I would also like to convey my thanks to the staff and academics at St. Catharine’s College,
Cambridge, and Cambridge University Engineering Department, who have been a pleasure
to know and have provided invaluable assistance to me throughout my time in Cambridge.
Finally, I acknowledge the people and organisations who have used their time and resources
to develop open-source software. Without the use of the following programs and databases,
this project would have been very different: Unix, Linux, GNU Project (specifically Gnu-
plot, gfortran, GIMP, GNU Emacs), GSView, LATEX, TeXstudio, Open office, xfig, Cool-
props, Mendeley Desktop, Notepad++, CES Edupack
xvii
We are like tenant farmers chopping down the fence around our house for fuel when we
should be using Nature’s inexhaustible sources of energy sun, wind and tide. . . . I’d put my
money on the sun and solar energy. What a source of power! I hope we don’t have to wait
until oil and coal run out before we tackle that.
Thomas Edison
Humanity had been strong, energetic, and intelligent, and had used all its abundant vitality
to alter the conditions under which it lived. And now came the reaction of the altered
conditions.
“The Time Traveller”, H. G. Wells
xviii
Chapter 1
Introduction
1
1.1 Context: current and future trends in energy supply
Since the late 20th century there has been a surge in the deployment of renewable energy
technologies (see figure 1.1) driven by concerns about anthropogenic climate change, the
health impacts of particulate pollution, and diminishing fossil fuel reserves. Although cli-
mate change is a natural process, recent human activities have contributed to changes in the
global climate. Carbon dioxide levels have increased from 284 ppm in 1832 to 397 ppm in
2013 [9] and ice cores have determined that the CO2 levels in 2005 (379 ppm) exceed the
natural range for the past 650 000 years (180 - 300 ppm) [10]. These increases are primarily
due to the burning of fossil fuels and changing land use. Scientific consensus suggests that
climate change could severely impact a large proportion of the world’s population through
a number of mechanisms including an increased rate of severe weather patterns and risks
to the security of supply of food, water and energy [11–14].
A number of policies have been developed in response to the above risks. In 1997 the
Kyoto Protocol set internationally binding emissions reduction targets for participating in-
dustrialised countries. The UK’s target is to reduce greenhouse gas (GHG) emissions to
8% below 1990 levels in the period 2008-2012 [21]. The Doha Amendment to the Ky-
oto Protocol extended this value to 18% below 1990 levels in the period 2013-2020 [22].
The 2008 Climate Change Act committed the UK to reducing GHG emissions by 80% by
2050 [23], and the UK had achieved a 35% reduction by 2014 [24].
The 2009 EU Renewable Directive aims to increase the renewable energy share to 15% of
the final energy consumption by 2020 for EU member states. Whilst there has been good
progress so far, it is predicted that many countries may not meet these targets unless further
changes to policy are made [25]. In 2015, 7.0% of the UK’s energy consumption came
from renewable sources (up from 5.2% in 2011) [15]. To meet the Renewable Directive
target, the UK will have to increase renewable energy deployment from around 64 TWh to
approximately 230 TWh (for heat, transport and electricity) [16,26]. Figure 1.1 shows pre-
dicted capacities of several renewable sources in 2020 and illustrates the large investment
that the UK is preparing to make into renewable energy. By implementing these changes,
it is forecast that around 30% of UK electricity will come from renewable sources [16].
There is much discussion about how to tackle these pressing issues and a variety of views
2
0.0
10.0
20.0
30.0
40.0
50.0
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
In
st
al
le
d 
ca
pa
ci
ty
 (G
W
)
Year
Onshore wind
Offshore wind
Photovoltaics
Biomass
Figure 1.1: Installed capacity of various renewable energy sources in the UK. Data from 2008–
2014 from Department for Energy and Climate Change (DECC) DUKES 2015 report [15, 16].
2015 values are from DECC Energy trends reports [17]. 2016 values are estimated based on in-
stalled capacity plus capacity under construction, data for which can be found from various sources,
including DECC [18–20]. Values at 2020 are predicted capacities taken from [15] and are subject
to uncertainty (for instance the value for solar photovoltaics could vary between 7 and 20 GW)
on technologies, economics and politics have been presented [27–34]. However, it is clear
that regardless of the route taken, the UK will become increasingly dependent on renew-
able sources, especially onshore and offshore wind and solar power. The intermittent nature
of renewable technologies creates problems for the electrical grid such as congestion, fre-
quency and voltage control, and balancing of supply and demand. Several solutions exist,
as discussed in section 1.3, and this thesis focusses on energy storage. As well as facil-
itating the deployment of renewable technologies, energy storage provides a number of
further benefits to the electrical grid. The next section provides some historical context
and applications of energy storage, before the intermittency problem is discussed in more
detail. This thesis is concerned with a particular energy storage technology, known here as
3
Pumped Thermal Energy Storage (PTES), which utilises packed-bed thermal reservoirs as
the energy stores. Several prominent energy storage technologies are summarised before
PTES is described in detail.
1.2 Background and applications of energy storage
Technology based on the accumulation of energy has a long history. 30 000 years ago,
rocks were employed as thermal stores and were used to cook food and boil water, thereby
leading to more nutritious and digestible meals [35]. Somewhat more recently, the Romans
developed a mining method known as hushing where powerful torrents of water were re-
leased from reservoirs with capacities of over 10 million litres in order to reveal mineral
veins [36].
The development of the electrical grid in the 19th and 20th centuries provided opportunities
for bulk electricity storage. At the beginning of the 20th century generating stations were
shut down at night, leaving the residual loads to be supplied by lead-acid accumulators [37].
While electrical batteries that power portable devices are currently a ubiquitous technology,
bulk electricity storage has not been widely deployed leading to a system where electricity
is, for the most part, consumed as it is produced. Generating plants must be capable of
satisfying demand that varies on daily and seasonal timescales, with maximum demand that
may only last for a few hours each year [38]. Chen et al. [39] suggests that the resulting
plants are inefficient, over-designed and expensive.
There is a wide array of energy storage technologies which have a diverse range of appli-
cations [39]. These can be categorised roughly according to the time scales at which they
discharge and the applications they are therefore suitable for.
Milliseconds – Seconds
Energy storage can combat random short-term fluctuations and improve power quality and
reliability by providing voltage and frequency control [39, 40]. Appropriate technologies
typically have high power ratings and low energy capacity, for example capacitors, batter-
ies, fly wheels and superconducting magnetic energy storage [39].
4
Seconds – Minutes
Energy storage can provide continuity of service by allowing ride through of momentary
shortages. Appropriate technologies may overlap with the above category, since a fast
response is required, although a larger energy storage capacity is beneficial. Batteries and
fuel cells are perhaps the most suitable since they have fast response (less than a second)
and appropriate discharge durations [39].
Hours – Days
Energy storage can be used in ‘peak shaving’: unexpected surges in demand are matched
by using energy that was stored during a period of surplus supply. This may be done on a
large scale by storing energy generated at off-peak times in order to supply it during periods
of higher demand, and is known as load levelling. Load levelling also allows for arbitrage
of the electricity price. Another application is to integrate energy storage with intermittent
renewable energy generators to help match fluctuating demand and supply.
Appropriate technologies have large energy storage capacities. Chen et al. [39] suggests
that for large scale management (e.g. applications greater than 100 MW) Pumped Hydro
(PHES) and Compressed Air Energy Storage (CAES) are most suitable, whilst thermal
energy storage, large scale batteries, fuel cells and solar fuels are more suitable for smaller
capacities of 10 – 100 MW. It has been suggested that increasing proportions of renewable
energy will move the energy mix away from large scale generating plants, towards smaller
scale more widely distributed plants [40]. Therefore a distributed network of medium scale
energy storage may be an appropriate way of integrating with distributed supply.
Storage can decouple energy supply from demand, and thereby provide benefits to all as-
pects of electricity generation, transmission and supply. For instance, energy storage could
be used to ease electrical grid congestion; installing storage may be more affordable than
upgrading the grid [41]. Ref. [41] suggests that other benefits include providing back up
and the ability to shift load to off-peak hours for both home users and commercial and
industrial use.
Strbac et al. [1] carried out a detailed analysis of a number of mechanisms by which stor-
5
age can provide value. For instance, energy storage could be used to offset investment in
additional generation plants, investment in the transmission network and interconnection,
and reduce the need for distribution network reinforcement. As energy systems continue to
develop and evolve, other applications for energy storage systems may emerge.
Strbac’s analysis suggested that energy storage provides greatest value when there is a
high proportion of renewable energy technologies. Furthermore, he suggested that only a
few hours of storage are necessary to reduce peak demand and capture significant value.
Surprisingly, Strbac’s analysis suggested that the value of storage was quite insensitive to
the efficiency of the storage system: for instance, the value of storage only increased by
10% when the efficiency was increased from 50% to 90%. However, it was also noted that
the round-trip efficiency became much more important in applications which required large
storage capacity.
A recent report by the National Infrastructure Commission [42], a UK government backed
organisation, recognised the benefits that storage could provide and recommended regu-
latory change to encourage investment in storage. In response, the UK government has
committed £50 million towards “innovation in energy storage, demand-side response and
other smart technologies” [43].
The above discussion demonstrates there are a range of applications for energy storage
systems, and that energy storage could provide value in a number of ways. Each of these
applications has different requirements in terms of, for instance, power and energy capacity,
storage duration, depth of discharge, and frequency of charge-discharge. Since different
technologies will be more or less suited to different applications, it is likely that a portfolio
of diverse technologies will be necessary.
1.3 Solutions to the intermittency of renewable power; how
much energy storage is required?
Renewable energy sources are intermittent, and electricity demand fluctuates throughout
the day. Figure 1.2 illustrates the variability in wind power and electricity demand in the
United Kingdom for the week at the beginning of January and July 2013. Electricity de-
6
 0
 1
 2
 3
 4
 0  1  2  3  4  5  6  7
 20
 25
 30
 35
 40
 45
 50
 55
W
in
d 
po
w
er
 (G
W
)
D
em
an
d 
(G
W
)
Time (days) - January 2013
Wind power
Demand
(a)
 0
 1
 2
 3
 4
 0  1  2  3  4  5  6  7
 20
 25
 30
 35
 40
 45
 50
 55
W
in
d 
po
w
er
 (G
W
)
D
em
an
d 
(G
W
)
Time (days) - July 2013
Wind power
Demand
(b)
Figure 1.2: Instantaneous wind generation (left axis) and electricity demand (right axis) in the
United Kingdom during the first week in (a) January 2013 and (b) July 2013. Data from Grid-
watch [44].
7
mand clearly follows a diurnal pattern and peaks at around 55 GW in the winter and 40 GW
in the summer. Wind power follows seasonal patterns, with the average wind power output
being greater in the winter than the summer. Variability in the weather (which includes
a degree of randomness) leads to variations in the supply of electricity which can include
lulls lasting hours to days [28]. Consequently, demand and supply of renewable electricity
are unmatched, which leads to difficulties in scenarios with large proportions of renewable
energy. At shorter time scales, intermittency may cause problems with frequency control
as demand and supply can change very quickly.
There are a number of solutions to these problems. One suggestion is that increasing the
land area over which wind is installed will average out the variability [45–47]. Figure 1.3
shows the normalised wind power output in the first week of January 2013 for three coun-
tries: Ireland, the UK and Denmark. The curves show the sum of wind generation across
each country and the output is clearly still intermittent. In addition, wind generation fol-
lows a similar pattern in each country: the UK lags behind Ireland by a few hours (as the
prevailing wind is from the south-west), and Denmark lags this by another few hours1.
Combining the wind output of these countries would reduce the intermittency to an extent,
but variability would still exist. This solution would require greater interconnection and
collaboration between countries on a continent-wide scale (which may provoke concern
over energy security). However, interconnection is thought to have a number of advantages
and there are plans to expand interconnection between the UK and Europe from the current
4 GW to around 10–11 GW [42].
A second option is demand side response, whereby energy consumers use energy in a more
flexible manner to accommodate the current supply of energy. For instance, the national
grid may ask large industrial consumers to switch off equipment in periods of low supply
and high demand. Smaller, more distributed options involve controlling when household
systems such as washing machines and dishwashers are switched on and off, requiring a
degree of flexibility from the user. Electric car batteries have also been suggested as a tool
for a demand-side management system [28].
Other options include maintaining peaking power plants, developing more flexible gener-
ation plants and over-sizing renewable energy capacity, all of which may not be economi-
1Denmark’s data looks a little different as it is reported hourly, whereas Irish data was reported every 15
minutes, and UK data every 5 minutes.
8
 0
 0.5
 1
 1.5
 2
 2.5
 0  1  2  3  4  5  6  7
W
in
d 
po
w
er
 / 
A
ve
ra
ge
 w
in
d 
po
w
er
Time (days)
Ireland
UK
Denmark
Figure 1.3: Instantaneous wind generation during the first week of January 2013 in three European
countries. Wind power is normalised by the average wind power in that country for that week.
Sources: Ireland (EirGrid Group) [48], UK (Gridwatch) [44] and Denmark (Energinet.dk) [49].
cally feasible.
Another solution to the ‘intermittency problem’ is bulk electricity storage, where surplus
electricity is stored until a later period when electricity demand increases above power
generation. A wide array of storage technologies has been proposed and an overview is
given in the next section.
Estimating the energy storage capacity that would address the intermittency problem de-
pends on a number of uncertain factors, such as the proportion of renewables, the influence
of other schemes (interconnection and demand side management), and economic consid-
erations. Nevertheless, several authors have made estimates based on simplified scenarios.
For instance, MacKay [28] calculated the storage capacity required if storage were to com-
pensate for a lull in wind power that lasted for five days. For an installed wind capacity
of 33 GW2 delivering 10 GW on average, Mackay proposed a storage capacity of 1200
GWh. This amounts to a storage capacity of 36 GWh for every GW of installed wind ca-
pacity, which can deliver power at 30% of installed capacity. This is an overestimate, since
during the five day lull the wind output is not zero, and energy could be obtained from
233 GW of installed wind is in line with the UK targets for 2020, see figure 1.1.
9
other sources. In addition, the storage would be under-utilised as five-day lulls are fairly
infrequent.
Carbajales-Dale et al. [50, 51] carried out the same analysis and provided mathematical
expressions for the required energy storage capacity. Their analysis therefore provides the
same result to MacKay for the example of a five-day lull.
Another simple estimate of required energy storage capacity is presented here. Wind gen-
eration data for 2015 was collected [44] and the average wind output was found to be 2.7
GW for a total installed capacity of around 13 GW3. Like the above analyses, it was as-
sumed that wind output would be kept constant at the average value of 2.7 GW. Any time
that the wind output dropped below this average value was classified as a lull. The lull
continued until the power increased above the average value. Lulls that occurred within
four hours of one another were counted as one lull (as there probably would not have been
sufficient time to recharge the storage). Figure 1.4 demonstrates that the majority of the
lulls (>50%) are less than 20 hours, and over 75% of the lulls are less than 50 hours long.
The histogram indicates that lulls lasting for over 5 days do occasionally occur, although
it probably would not be economical to build energy storage to provide for all possible
scenarios although a solution is required. The storage required to account for 75% of the
lulls at the average wind output would require 2.7× 50 = 135 GWh, which corresponds to
a rule of thumb of 15 GWh of storage per GW of installed wind capacity. Scaling this up
to MacKay’s scenario with 33 GW of installed wind suggests a storage energy capacity of
340 GWh, which is ten times the current UK storage capacity. Note, it is possible that the
storage requirement may not scale linearly with the installed capacity of renewables, and
that these values may underestimate the required storage.
These analyses are somewhat approximate as many of the factors are arbitrary, and a num-
ber of complex factors that may significantly affect the problem have been neglected. For
instance, buffering wind to provide a constant output is not realistic: supply needs to be
variable in order to be able to match demand. These studies also consider only one appli-
cation of energy storage and do not include other intermittent energy sources such as solar
power. However, the above analyses give some indication as to the storage energy capacity
3This suggests quite a low capacity factor of 21%. However, wind data is only available for metered
farms, and many smaller farms are not metered. Assuming a capacity factor of 30% suggests that the installed
capacity of metered farms is around 9 GW
10
 0
 5
 10
 15
 20
 25
 30
 0  50  100  150  200  250
 0
 20
 40
 60
 80
 100
Fr
eq
ue
nc
y 
(%
)
Cu
m
ul
at
iv
e 
fre
qu
en
cy
 (%
)
Duration of lulls (hours) 
Figure 1.4: Distribution of the duration of lulls in wind power in the UK for 2015. Lulls are
considered to be any time wind power fell below the average wind power that year (2.67 GW). If
lulls occur within four hours of one another, they are considered to be the same lull.
that may be required, and suggest that the required energy capacity is not insignificant.
For instance, the UK currently has around 30 GWh of storage installed (predominantly as
pumped hydro) whereas the above estimates suggested an order of magnitude more may
be required if intermittency is to be addressed in 2020. It is also interesting to note that all
the UK’s pumped hydro plants (which provide 26.7 GWh) were opened between 1963 and
1984, which indicates the low levels of investment in storage in the past three decades.
Other authors have undertaken different types of analyses, and frequently focus on the
economic aspects of storage. Gru¨newald et al. [52] took into account energy prices to
calculate the net present value of several storage technologies under scenerios with different
proportions of renewable energy. For one case with 60 GW of renewables (42 GW wind
and 18 GW solar) and a mean demand of 36 GW, Gru¨newald found that the optimal energy
capacity of a Compressed Air Energy Storage (CAES) system was 69 GW h. Another
analysis [53] considered de-carbonisation of the US electricity grid, and how the cost of
storage (per kW and per KWh) would affect storage capacity. Ref. [53] found that the
optimal energy capacity of the cheapest storage systems modelled (100 $/kW and 5 $/kWh)
11
would store two days worth of average electricity demand. In 2015 the average UK demand
was 33 GW suggesting that complete de-carbonisation would require 1580 GWh storage.
 0
 50
 100
 150
 200
 250
 300
 350
 400
 450
 50  100  150  200  250  300  350  400
En
er
gy
 st
or
ag
e 
ca
pa
ci
ty
 (G
W
h)
Cost of storage (£ / kW.year)
Storage duration =   6 hours
Storage duration = 24 hours
Storage duration = 48 hours
Figure 1.5: Optimal energy storage capacity as calculated by Strbac [1] for various storage costs
and duration of storage.
Strbac et al. [1] carried out a detailed investigation into the benefit of energy storage to
the electrical grid. The analysis considered the economic value that storage would provide
in future scenarios. They used the DECC 2050 Pathways as a guide for future energy
mixes, and particularly focussed on a pathway known as Grassroots where, of a generation
capacity of 190 GW, approximately 75 GW were renewable (of which 70 GW were wind
power) in 2030. Strbac developed a system model incorporating generation, transmission
and distribution systems. The model aims to satisfy the short-term supply and demand
balance in an economically optimal way by considering long term investment decisions.
Strbac investigated the optimal storage capacity for various costs of storage and durations of
storage. Unsurprisingly, Strbac noted that cheaper storage corresponded to larger optimal
storage capacities. Some of these results are reproduced in figure 1.5. Further analysis by
Strbac suggested that only a few hours of storage were sufficient to reduce peak demand.
For storage durations beyond six hours, the marginal value of storage decreased sharply.
By considering the six hours storage duration curve in figure 1.5 for optimistic storage costs
12
of 75–150 / kW·year suggests storage capacities of 25–60 GW h.
A wide range of energy storage capacities have been predicted, and there are a number of
methods for analysing the problem. These studies have many different assumptions, and
calculate storage energy capacity for different objectives and scenarios. Thus it is hard to
compare the results, in which a large degree of variation exists. While there may not be
much agreement on the required quantity of storage, the above authors all identified that
storage would provide value in a number of ways.
1.4 Prominent energy storage types
Chen et al. [39] provides a review of a large number of prominent energy storage technolo-
gies, and compares their technical and economic characteristics. Technologies are typically
categorised into broad groups, such as mechanical, electrical, chemical and thermal stor-
age. These categories are then further divided and sub-divided due to the breadth and
diversity of technologies that exist. Mechanical storage technologies include fly-wheels,
Compressed Air Energy Storage, and pumped hydro; electrical storage includes capacitors,
supercapacitors and magnetic storage; chemical storage includes fuel cells, conventional
battery technologies (such as lead-acid, lithium-ion) and flow cells (such as zinc-bromine
and vanadium redox); thermal storage may include cryogenic storage, sensible heat stor-
age or latent heat storage such as phase change materials. In some cases technologies from
one category may be integrated with those of another category, for instance thermal stores
can be incorporated in CAES technology. The range of storage systems makes a compre-
hensive review beyond the scope of this thesis and the reader is directed to Chen’s review
for further details. However, a brief discussion of Pumped Hydro Energy Storage (PHES),
Compressed Air Energy Storage (CAES) and thermal energy storage systems is now pro-
vided in order to provide context, background and comparison for the technology that is
the focus of this work: Pumped Thermal Energy Storage (PTES).
Metrics are required to provide a comparison between different storage schemes, and a
number of these factors have been developed. The energy density ρE and power density ρP
are important performance metrics for energy storage systems and give an indication of the
scale and economic feasibility of the system; capital cost per unit energy storage ($ / kWh)
13
and per unit power capacity ($ / kW) vary approximately inversely with ρE and ρP . The
precise definition of energy and power density varies from author to author and depends
on the technology. Densities may be volumetric or gravimetric, depending on whether the
energy stored is divided by the volume of the system or its weight, respectively. In this
thesis, densities are volumetric, and ρE is defined as the stored available energy divided by
the volume of the storage media. (Energy densities which incorporate the full footprint of
the plant will necessarily be lower than this measure, but this measure is not used here due
to uncertainty over the plant size.)
The round-trip of energy storage systems is defined as the ratio of net work output during
discharge to the net work input during charge, such that
χ =
Net work output
Net work input
=
W netdis
W netchg
(1.1)
1.4.1 PHES
Pumped Hydro Energy Storage (PHES) is the most widely implemented energy storage
technology [39] for electricity, with over 129 GW installed at more than 200 sites globally
[54]. PHES operates using two water reservoirs at different elevations and energy is stored
in the gravitational potential of water. The quantity of energy stored is proportional to the
height difference and volume of the reservoirs. The available energy density ρE is given by
ρE = ρg∆H (1.2)
Where ρ is the density of water, g is the acceleration due to gravity, and ∆H is the height
difference between the upper and lower reservoirs. The largest PHES scheme in the UK is
Dinorwig power station in Gwynedd, Wales which has ∆H of roughly 500 m [28]. The
three other PHES schemes in Wales have height differences of 170–350 m. Dinorwig’s
energy density is roughly 1.4 kWh m–3, while Chen et al. [39] suggests PHES schemes
have energy densities in the range 0.5–1.5 kWh m–3. A comparison with other technologies
is shown in table 1.1 on page 21.
PHES is a proven technology with a high efficiency, fast response and long storage time.
14
For instance, Dinorwig power station has an efficiency of 74–75% and can attain full load
in less than two minutes from standstill, and less than 20 seconds if spinning [55]. Fur-
thermore, it has a low risk investment with a low levelised cost of energy4 [39, 41]. It
has numerous drawbacks, however, including limited suitable sites, long construction time,
high capital cost and low energy density. Moreover, there is a significant environmental im-
pact associated with clearing and flooding a large area, and the moving of large quantities
of earth to build the dam [56].
Various alternatives to the standard design have been proposed with the aim of reducing
the geographical limitations of PHES. Underground pumped hydro uses an underground
reservoir which exchanges water with a surface reservoir [57]. Since the two reservoirs can
be vertically aligned, frictional losses are reduced. Excavating the lower reservoir includes
several challenges, such as finding geologically suitable sites, technical requirements and
cost [58]. Disused mine shafts and underground caverns may provide suitable locations,
which would then lead underground pumped hydro to having similar geographical con-
straints to CAES.
1.4.2 CAES, I-CAES and AA-CAES
Compressed Air Energy Storage has garnered much interest due to its low capital costs,
high energy densities and efficiencies. There are currently two CAES plants in operation,
although several more are planned or under construction [39]. These existing plants are
known as Diabatic CAES since they are similar to conventional gas turbines, except that
the compression and expansion stages are decoupled. After compression, the high pressure
gas is stored in a large underground cavern. Energy is discharged by taking the compressed
air, heating it and expanding it through a high pressure turbine. Fuel is then added to the
exhaust, which combusted and passed through a low pressure turbine.
The first Diabatic CAES plant was installed at Huntorf, Germany in 1978. The storage
comprises two salt domes with a volume of 300 000 m3 which operated at a maximum
pressure of 100 bar. The plant takes eight hours to charge and can discharge at 290 MW for
two hours. The second operational CAES system is the McIntosh plant in Alabama, USA
4The levelised cost of energy is the price of energy from a certain source that will allow that venture to
break even over its lifetime.
15
and has a cavern of 500 000 m3 which operates at 75 bar. The McIntosh plant can produce
110 MW for 26 hours and uses a recuperator to reuse heat from the gas turbine exhaust in
order to reduce fuel consumption [39, 59].
air in
comp. exp.
air out
work outwork in
cavern
thermal store
Figure 1.6: Schematic of an AA-CAES system
The available energy density for a constant volume container is given by [60]
ρE = p
{
ln
(
p
p0
)
−
(
1− p0
p
)}
(1.3)
where p0 is ambient pressure and p is the charged pressure which is typically 40–80 bar [39],
leading to energy densities of around 7.5 kWh m–3. However, in practice, CAES systems
tend not to be discharged to p0 and energy densities are therefore lower than this.
CAES round-trip efficiency can be improved with isothermal compression and expansion,
and such systems are known as Isothermal CAES (I-CAES). Thermal energy is continu-
ously removed during compression, and continuously added during expansion [61]. The
gas is therefore kept at ambient temperature. Isothermal compression and expansion are
difficult to achieve, although Grazzini and Milazzo [61] point out that some schemes have
been proposed.
A more practicable approach involves a number of compression and expansion stages, be-
16
tween each of which heat can be removed or added. As the number of stages increases,
the process gets closer to being isothermal, although with increasing cost. Such a system
is known as Advanced Adiabatic Compressed Energy Storage (AA-CAES), illustrated in
figure 1.6. Furthermore, AA-CAES avoids the need for fossil fuels by storing the thermal
energy generated by compression and returning it to the gas before the expansion [62]. In
a multi-stage AA-CAES system, it is conceivable that different storage materials could be
used at each stage, depending on what is most appropriate. For instance, a packed bed
could be used after the first stage of compression where pressures are reasonably low, lead-
ing to lower cost pressure vessels. At later stages, heat could be transferred to water or
molten salts with a heat exchanger.
The main advantage of CAES systems is that they are based on well established technolo-
gies, have a low capital cost, and AA-CAES plants are predicted to achieve efficiencies of
60-80% [59]. The performance of these systems is mainly governed by the efficiency of the
turbo-machinery [61], however, the thermal energy store may also have significant impact.
For instance, Pickard et al. [63] suggests that low heat exchanger effectiveness could limit
the quantity of energy stored in the thermal store, therefore leading to lower round-trip ef-
ficiencies (around 50%). Of course, these estimates depend on the exact type of thermal
store employed and the manner in which heat is transferred to these stores (i.e. directly
or via several heat exchangers which serve to intercool the compression process [61, 62]).
CAES and AA-CAES have geographical limitations and are constrained by the location of
suitable caverns. CAES is likely to have a lower environmental impact than PHES [64].
1.4.3 Thermal energy storage
Thermal energy storage has applications in cooling and refrigeration [39], solar power
plants [65], solar cooking [66] and seasonal energy storage [67]. It can be divided into
two categories: latent and sensible heat. Latent heat systems store energy by changing the
phase of the material. This has several advantages, such as high energy densities and the
ability to provide heat at a constant temperature. However, problems include a change in
volume associated with the phase change, pinch point constraints and identifying suitable
materials [68].
17
A simple example of a latent heat system is the conversion of water to ice for cooling
purposes although there are a wide range of other materials (paraffins, waxes etc.) that can
be used. Cryogens (that is, liquefied gases) such as air, nitrogen, or hydrogen are also used
as thermal stores [69]. Cryogens have the advantage of high exergy densities (table 1.1) and
Li et al. [69] notes that the second law of thermodynamics indicates that a cold reservoir
is a “higher quality” energy source than a hot store. In addition, cryogens can be readily
stored and transported.
Liquid-vapour transition PCMs require a large volume for the gas storage (unless air is
used) and generally solid–liquid transitions are preferred [70]. There are a wide range of
such materials, although finding a material that satisfies the necessary requirements (such
as chemical and thermal stability, phase change at the appropriate temperature etc.) is
challenging [70]. Moreover, PCMs can be expensive. There have been many attempts to
improve the properties of phase change materials, such as increasing the thermal conduc-
tivity by encapsulating the material in graphite or some other material [70].
Sensible thermal energy
Sensible heat systems store energy by virtue of the change of temperature of a material5.
One example from the UK in the 1970s is the storage heater: a domestic heating system
where a material (brick or clay) was heated during the night on low-cost tariffs. This heat
was then delivered through the day when it was required [71]. Sensible heat technologies
are relatively cheap and simple to manufacture although they have a lower energy density
than latent heat systems.
Water is a popular choice for sensible heat storage, due to its high specific heat capacity
and density, chemical inertness and the fact it can be used as both a heat transfer fluid and
a storage medium. Water is limited to a certain temperature range unless it is pressurised,
therefore for high temperature operation solids (or liquids with a low vapour pressure) are
generally preferred. Heat resistant oils are available that can operate in a broader range
of temperatures without pressurisation than water. These oils tend to have lower specific
heat capacities than water [72]. Other suitable liquids include molten salts, which have a
5Note that availability can be stored by cooling a material, since there is still the potential to extract work
with the use of a heat engine.
18
wide range of operating temperatures (despite having high freezing points) and are non-
toxic, non-flammable and are widely used in solar plants [70]. Rocks are a good alternative
to liquids since they are cheap, readily available and can operate at high temperatures.
However, the effect of high temperatures on the mechanical performance of solid stores
should be considered; the compressive strength of concrete reduces by about 20% at 400oC,
for example [73].
Comprehensive reviews of thermal energy storage materials and technologies can be found
in [68, 70, 73]. Thermal energy storage systems may be comprised solely of the storage
medium as is the case in some heating applications, or as part of a wider system. Systems
which contain a thermal energy store include AA-CAES (as above) and PTES (below).
The available energy density for a thermal store operating between states 1 (charged) and
2 (discharged) is given by
ρE = ρs [h1 − h2 − T0(s1 − s2)] (1.4)
Where ρs is the density of the storage material, h is its specific enthalpy, s is its specific
entropy, and T0 is the dead state (or environment) temperature. For a sensible thermal
energy store consisting of a packed bed of solid pebbles this expression becomes
ρE = (1− ε) ρs
[∫ T2
T1
cs(T )dT − T0
∫ T2
T1
cs(T )
dT
T
]
(1.5)
where cs is the specific heat capacity which is a function of temperature, (1 − ε) is the
fraction of volume taken up by the storage, T1 is the discharged temperature and T2 is the
charged temperature. The energy density of several materials is given in table 1.1.
A number of innovative design features for solid thermal stores have been suggested.
Zanganeh et al. [2] developed a conical store to reduce the effect of thermal ratcheting.
Daschner et al. [74] suggested that pressure losses could be reduced by using packed beds
where the gas flowed radially rather than axially. Refs. [75–77] investigated incorporating
phase change materials into packed beds. Ref. [77] found that PCMs increased the energy
density, although the thermal front would de-stratify more rapidly. One disadvantage of
solid stores is that the exit temperature changes as the thermal front approaches the end of
the store. Generally this energy has to be rejected in heat exchangers, although it could be
19
used in waste heat applications or domestic heating systems. Utilising the waste heat in this
way is complicated by the fact the outlet temperature varies as the thermal front progresses
out of the reservoir. Placing PCMs at the end of the reservoirs was also seen to stabilise the
outlet temperature.
Crandall and Thatcher [78] developed another design feature called segmentation as a way
to maintain thermal stratification in packed beds for solar air heating systems. Segmented
stores were being developed by Isentropic Ltd. for PTES systems [79] and are discussed
in detail in section 3.3. White et al. [80] showed that by reducing pressure losses, segmen-
tation allowed smaller particles to be used, thereby reducing thermal losses and leading to
larger efficiencies. Furthermore, segmentation reduces conductive losses during the storage
phase as described by McTigue and White [81].
20
Storage system χ (%) ρE (kWh m−3) ρP (kW m−3)
Hot watera - 10–100 -
Hot gravelb - 130 -
Cold gravelb - 60 -
Cryogen (nitrogen)c - 170 -
Cryogen (hydrogen)c - 236 -
PHESd 70–85 0.5–1.5 5000
CAESd 70–80 3–6 330e
CAES - Huntorff 42 1.9 -
CAES - McIntoshf 54 5.7 -
PTESg 50–70 100 260
Table 1.1: Comparison of energy and power densities and efficiencies of several prominent energy
storage systems and media.
a For hot water ρE = ρH2O [h1 − h2 − T0(s1 − s2)] c.f. equation 1.5. Properties are calcu-
lated with CoolProps [82] for water between 15°C and the saturation temperature at 1 bar (lower
bound) and 100 bar (upper bound).
b Gravel is taken to be magnetite with ρs = 5175 kg m−3, ε = 0.4 and c¯s = 800 kJ/kgK. For hot
gravel T1 = 500°C and for cold gravel T1 = −150°C.
c Values taken from Li et al. [69] for the thermal exergy density of the cryogen. This value does
not include the chemical exergy which is due to the compositional difference between the chemical
and the reference environment (air at standard conditions). Nitrogen’s chemical exergy is very low
whilst that of hydrogen is very large (∼2300 kWh m−3). It is not included as the cryogens are
assumed to be part of a closed cycle.
d χ and ρE for PHES and CAES are estimates from Chen et al. [39] (energy densities show good
agreement with equations 1.2 and 1.3) ρP are taken from White et al. [83].
e White et al. [83] uses the power density of a low specification gas turbine to provide an
approximate comparison with other storage technologies
f This data is for operational diabatic CAES plants and data has been taken from [39, 59].
g PTES data is from equations 1.7 and 1.8 and the efficiency is from [84].
For comparison, the energy density of this thesis is roughly 3 kWh m−3, based on a calorific value
of 13.5 MJ kg−1 and a density of 800 kg m−3
21
1.5 Pumped Thermal Energy Storage
This thesis is concerned primarily with a thermal energy storage system based on heat
pumps/engines. A number of variants exist and perhaps the most widely used name is
Pumped Heat Energy Storage (PHES) [5, 85, 86]. Other names include Electrothermal
Energy Storage (ETES) [87], Thermo-electrical energy storage (TEES) [88], Compressed
Heat Energy Storage (CHEST) [89] or Stockage d’E´lectricite´ par Pompage Thermique
(SEPT) [90, 91]. Liquid Air Energy Storage (LAES) [69, 92–95] can also be included in
this class of thermal energy storage technologies. Stamatiou noted that the abundance of
names led to difficulties with identifying relevant work before introducing a further variant
known as the Dual Energy Storage and Converter (DESC) [96]. Following the group based
in Cambridge, and to avoid confusion with pumped hydro (PHES), this thesis uses the
nomenclature Pumped Thermal Energy Storage (PTES) [60, 80, 81, 83, 84, 97].
The concept that unifies these variants is energy storage by means of ‘reversible’ heat
pumps and engines. In this context, reversible means that the direction of gas flow through
the heat pump is reversed to begin operation as a heat engine. PTES systems typically
utilise two thermal stores (one hot and one cold) although this is not necessarily the case.
The modern study of PTES systems has concentrated on two main variants, which are heat
pumps/engines based on either the Joule-Brayton cycle or the Rankine cycle, as illustrated
in figure 1.8. During charge the system operates as a heat pump, and uses electrical energy
to move heat from a cooler region to a hotter region, thereby storing available energy, as in
figure 1.7. The T–s diagrams are traversed anti-clockwise during charge, with compression
occurring from 1–2 and expansion from 3–4. The flow reverses direction during discharge
and operates as a heat engine to generate electricity.
PTES systems have comparable energy densities, power densities, and efficiencies to other
large scale energy storage technologies as illustrated in table 1.1 on p. 21. It is also con-
sidered to have a number of advantages over these systems. For instance, unlike PHES or
CAES, it is not constrained geographically. Unlike batteries, the energy and power capaci-
ties are decoupled and can be varied independently by varying the size of the thermal stores
and the size of the compression–expansion system (or mass flow rate), respectively. Fur-
thermore, PTES systems are likely to have a low impact on the environment as the storage
22
Q4
Q2
T2
T4
Q4
Q2
T2
T4
Win WoutHEHP
Figure 1.7: (Left) PTES during charge acting as a heat pump. (Right) PTES during discharge acting
as a heat engine.
materials are relatively abundant, inert and non-toxic. Finally, it is anticipated that PTES
systems will be relatively affordable, since the system could be constructed from off-the-
shelf components. However, PTES requires four compression-expansion processes and is
therefore sensitive to the performance of these devices.
This section proceeds by providing a historical background to the development of PTES-
type systems. The basic theory of PTES systems is then developed with the results being
generally applicable to all its variants. The modern development of these systems is then
categorised into those based on the Joule-Brayton cycle and those based on the Rankine
cycle and an overview of the academic and industrial work that has been undertaken is
presented.
Historical background
The concept of using a heat pump/engine to store energy has been re-invented a number of
times. Marguerre published perhaps the first example of a Rankine cycle based system in
1924 [98]. This system used a steam turbine and compressor, and stored energy in steam
23
-200
-100
 0
 100
 200
 300
 400
 500
 600
-0.4 -0.2  0
T
em
p
er
at
u
re
, 
(°
C
)
Specific entropy, s (kJ / kgK)
T1
T2
T3
T4
(a)
-20
 0
 20
 40
 60
 80
 100
 120
 0.8  1  1.2  1.4  1.6  1.8  2
T
em
p
er
at
u
re
, 
T
 (
°
C
)
Specific entropy, s (kJ / kgK)
T1
T2
T3
T4
(b)
Figure 1.8: Ideal T–s diagram for PHES systems during charge for systems based on (a) the Joule-
Brayton cycle and (b) a transcritical CO2 Rankine cycle.
accumulators. During charge, low pressure steam from the low pressure accumulator was
drawn into the compressor and its temperature increased. The high pressure steam was then
condensed in the high pressure steam accumulator. In discharge, the high pressure steam
was drawn from the high pressure accumulator and used to power the steam turbine, before
being condensed in the low pressure accumulator [99]. Descriptions of Marguerre’s system
in English can be found in two patents filed in the 1930s [100, 101].
In 1972 Babcock [102] patented a system that utilised superheated steam drawn from a
nuclear power plant. This steam was compressed and stored in an accumulator composed
of ceramic refractory material at a temperature of around 450°C and pressure of 200 bar.
In 1978 Cahn [103,104] developed a similar system also based on the use of steam. He en-
visioned a system that was independent of the power plant and could use waste heat [104].
In Cahn’s system energy was transferred to the hot and cold stores in heat exchangers. The
cold store consisted of water that was cycled between approximately 40°C and 99°C. The
hot store was described as a ‘low vapour pressure thermal energy retention material’, which
Cahn specified to be some kind of hydrocarbon oil. When charged the hot store would have
24
a temperature of 230–320°C, and when discharged it would be 65–150°C. Each store there-
fore consisted of two tanks, with the storage media passing from one tank to the other via
the heat exchanger. Cahn mentioned that this had the advantage over a solid store such as
the Babcock design, since there are no thermal fronts and conductive losses would therefore
not occur.
Another early heat pump system was described by Smith in 1977 [92]. During charge,
air is compressed and subsequently cooled and liquefied. The energy extracted during
cooling is stored in a regenerator or packed bed and the resulting cryogen is stored in a
tank. Discharge of the system uses an open super-critical Rankine cycle; the liquid air is
compressed and reheated in the regenerator before being expanded through a turbine to
produce work. The resulting cool air is stored in a cool store which is used in the liquefac-
tion process during charge. This system was the precursor of Liquid Air Energy Systems
(LAES) described by Ameel et al. [94] and Morgan et al. [95] and developed by Highview
Power [93]. While these systems are broadly similar, there are a number of differences,
such as the liquefaction process (Linde cycle or Claude cycle), the compression–expansion
processes (isothermal or adiabatic), the number of compression/expansion stages, and the
details of how energy is recovered in the regenerator. The prototype recently developed by
Highview Power achieved efficiencies of just 8%, but Morgan et al. [95] attributed this low
value to the small scale of the plant and the fact that the regenerator was not used to its
full extent (i.e. not all the energy produced during expansion was stored). Morgan identi-
fied that pinch point constraints in the Highview design restricted the maximum round-trip
efficiency to 36%. However, it was noted that optimisation of the system (particularly the
regenerator and expansion processes) should enable efficiencies above 50%. Therefore,
LAES could achieve similar efficiencies to other PTES systems, and shares the advantages
of using well established technologies, abundant storage media, and geographical flexibil-
ity. Unlike PTES systems, LAES is not a cyclic process and it consequently has fewer
compressions and expansions than PTES.
Recent research has focussed on PTES systems based on Rankine cycles and Joule-Brayton
cycles. Isentropic Ltd. began construction of a Joule-Brayton cycle based prototype in
2015, and ABB expressed interest in constructing a Rankine cycle based system, and a
number of papers have been published on both systems. Before carrying out a literature
review of this work, the basic thermodynamic theory of these PTES systems is set out.
25
1.5.1 Energy and power densities
Simple expressions for the energy and power density are obtained by considering a fully
reversible PTES system, as represented by the T–s diagram in figure 1.8. Compression and
expansion are isentropic and heat addition and rejection are isobaric.
The energy density is defined as the available energy stored per unit volume of storage
medium. The stored availability is the difference between the stored enthalpies of the two
reservoirs, such that
ρE =
(H2 −H3)− (H1 −H4)
V H + V C
(1.6)
Where H is the enthalpy, V is the volume of the store, and superscripts H and C refer to
the hot and cold store respectively. The above equation is written in such a way that it can
be applied to systems that may have different storage materials for the hot and cold store,
and includes enthalpy changes associated with phase change. A simplified expression was
developed by White et al. [83] for a system with the same solid packing material in both
reservoirs. The enthalpies therefore become ∆H = ρsV cs(1− ε)∆T , where ρs is the solid
density, V is the volume of the reservoir, cs is the specific heat capacity of the solid (which
varies with the temperature of the store), and ε is the void fraction. By noting that charging
the stores in the same amount of time introduces the constraint that (ρscsV )H = (ρscsV )C
(see [60] and equation 2.5 for the nominal charging time of a thermal reservoir) where
superscripts H , C indicate the hot and cold store. The energy density then becomes
ρE = ρs(1− ε)
(
cHs c
C
s
cHs + c
C
s
)
[(T2 − T3)− (T1 − T4)] (1.7)
A more accurate estimation could include the volume required for insulation and contain-
ment, or even the full footprint of the plant. However, as a first estimation the above
expression is satisfactory and useful.
The power density is defined here as the power capacity divided by the volumetric flow
rate of working fluid. Other definitions include normalising by the volume required by the
compression and expansion devices or considering the maximum exergetic flux per unit
26
power capacity [84]. However, the above definition again leads to a simple expression:
ρP = ρg,1cp [(T2 − T1)− (T3 − T4)] (1.8)
where ρg,1 is the fluid density at point 1 on figure 1.8, and cp is the specific isobaric heat ca-
pacity for the working fluid. White et al. [83] further manipulated equations 1.7 and 1.8 to
reveal how these factors depend on PTES design for Joule–Brayton type systems. By intro-
ducing the compressor/expander temperature ratio τ = T2/T1 = T3/T4 and the work ratio
R = T1/T4 = T2/T3 which is approximately the ratio of compression work to expansion
work during charge, the above equations become
ρE = 1/2ρscsT1 (τ − 1)
(
1− 1
R
)
(1.9)
ρP =
γ
γ − 1p1 (τ − 1)
(
1− 1
R
)
(1.10)
where T1 and p1 are the temperature and pressure at point 1 and γ is the ratio of specific
heats. These expressions indicate that the energy and power densities may be increased by
using larger temperature ratios τ , and therefore larger pressure ratios. Increasing the work
ratio R leads to larger energy and power densities, and this can be achieved by increasing
T1 or decreasing T3. Furthermore, the power density is increased by raising the system
pressure p1. White et al. [83] additionally point out the importance of the isentropic index
on the power density, noting that γ/(γ − 1) takes a value of 5/2 for monatomic gases and
7/2 for diatomic gases.
Table 1.1 provides a comparison of these ideal energy and power densities with other es-
tablished energy storage systems, and indicates that PTES has competitive values.
1.5.2 Round-trip efficiency
The round-trip efficiency of energy storage systems is defined as the ratio of electricity out-
put during discharge to electricity input during charge. For PTES systems a thermodynamic
round-trip efficiency is used, which neglects the efficiency of the electrical machines. This
metric is defined as the ratio of net work output during discharge to the net work input
27
during charge, such that
χ =
Net work output
Net work input
=
W netdis
W netchg
(1.11)
For an ideal PTES cycle with reversible compression and expansion stages evaluation of
the round-trip efficiency is trivial and is given by the COP of a heat pump multiplied by the
Carnot efficiency of a heat engine.
χ =
(
1
1− T4
T2
)
×
(
1− T4
T2
)
= 1 (1.12)
Where T2 is the maximum temperature of the hot store, and T4 is the minimum temperature
in the cold store. Thus, the round-trip efficiency of PTES is limited only by irreversibilities
within the cycle. Several approximations to the efficiency have emerged in the literature,
and these take the form of either classical cycle calculations [83, 90] or endo-reversible
thermodynamics [85, 87].
The endo-reversible approach assumes that irreversibilities are the result of heat transfer
between the thermal reservoirs and the thermodynamic cycles (i.e. heat pump or heat en-
gine) that interact with them. By assuming that a finite temperature difference ∆T exists
between the thermal reservoirs and the cycles, Mercango¨z et al. [87] developed the follow-
ing expression:
χ =
(
T2 − T4 − 2∆T
T2 −∆T
)
×
(
T2 + ∆T
T2 − T4 + 2∆T
)
(1.13)
Thess [85] used a similar methodology to develop an alternative expression, which he then
further manipulated to remove the variable ∆T . Thess’s expression has been widely cited,
however Guo et al. [105] noted that a mistake had been made, and found the correct ex-
pression to be:
χ =
(T2/T4)
1/2 − 1/2
(T2/T4)1/2 + 1/2
(1.14)
(This expression has the same form as Thess’s equation, which slightly over-predicts the
efficiency). This analysis involved maximising the power output of the heat engine which
is an unrealistic assumption as PTES is unlikely to be operated in this way. The heat pump
is not treated in an equivalent way since the heat pump power cannot be optimised. Fur-
thermore, as shown on figure 1.9(a), this expression predicts an efficiency of 33% when
28
 20
 40
 60
 80
 100
 2  4  6  8
R
o
u
n
d
-t
ri
p
 e
ff
ic
ie
n
cy
, 
χ 
(%
)
T2 / T4
T4 = 120 K
T4 = 200 K
Guo
A
B
C
DE
F
(a)
 20
 40
 60
 80
 100
 0  2  4  6
R
o
u
n
d
-t
ri
p
 e
ff
ic
ie
n
cy
, 
χ 
(%
)
Work ratio, R
White, η = 0.90
White, η = 0.99
A
B
C
DE
(b)
Figure 1.9: Variation in PTES round-trip efficiency with operating temperatures and work ratio.
Data points are from numerical evaluations of PTES systems: A. McTigue et al. [84] with η =
0.90; B. McTigue et al. [84] with η = 0.99; C. Desrues et al. [90]; D. Mercango¨z et al. [87]; E.
Morandin et al. [88]; F. Steinmann [89]. (a) PTES round-trip efficiency calculated using endo-
reversible thermodynamics. Purple and green lines are calculated using equation 1.13. Solid lines
have ∆T = 5 K, dotted lines have ∆T = 20 K. Guo’s expression is from equation 1.14. (b) PTES
round-trip efficiency calculated using classical cycle analysis from equation 1.15.
T2/T4 = 1. At such a condition the power input and output are zero, and any irreversibil-
ities should lead to a negative efficiency. As a result, equation 1.14 is not applicable at
T2/T4 = 1, and may in fact be unreliable at a larger range of values.
Mercango¨z’s and Guo’s expressions are plotted in figure 1.9(a) against T2/T4. It is clear
that the efficiency is maximised by increasing the hot reservoir storage temperature, and
decreasing the cold store storage temperature. Equation 1.13 implies that the finite temper-
ature difference should be minimised, which could be achieved by increasing the area for
heat transfer, or reducing the rate of heat transfer. Equation 1.13 allows the impact of the
low temperature store at a fixed T2/T4 to be studied. Figure 1.9(a) illustrates that increasing
T4 at fixed T2/T4 leads to higher efficiencies.
Classical cycle analysis allows the irreversibility of the compression and expansion devices
29
 0
 5
 10
 15
 20
 25
 30
 0  2  4  6  8  10
W
o
rk
 r
a
ti
o
, 
R
Pressure ratio, β
CO2 - T3 / T1 = 0.4
CO2 - T3 / T1 = 1.0
Ar - T3 / T1 = 0.4
Ar - T3 / T1 = 1.0
Figure 1.10: Variation of work ratio R with pressure ratio and T3/T1. Results are shown for
Rankine cycles with carbon dioxide as the working fluid, and for Joule-Brayton cycles with argon
as the working fluid. Working fluid data taken from CoolProps [82].
to be considered and includes the influence of the parameters T1 and T3. White et al. [83]
developed an approximate expression for the efficiency based on the irreversibility in the
compressors and expanders and assuming ideal heat transfer in the thermal reservoirs. By
scaling the ideal compression work by 1/ηs and the ideal expansion work by ηs, where ηs
is the isentropic efficiency, they obtained
χ ≈ Rη
2
s − 1
R− η2s
(1.15)
The work ratio R was introduced above as the ratio of compression work to expansion
work during charge. This expression was also independently developed by Kim et al. [106]
shortly after. As depicted in figure 1.9(b), equation 1.15 implies that the efficiency is a
monotonically increasing function of R; a larger work ratio ensures that losses will have a
less significant effect on the net work of the system. For Joule-Brayton cycles, the work
ratio is given by R = τT1/T3 and can be increased by a) increasing the temperature ratio
(and hence pressure ratio) b) increasing T1 c) decreasing T3 as shown in figure 1.10. For
30
Rankine cycle based PTES systems, equation 1.15 is also true, although the expression for
R is no longer so simple due to the phase change between point 4 and 1. In this case, the
work ratio is given by
R =
cpT1
p4/ρ4
β
γ−1
γ
−1
β − 1 (1.16)
where cp is the isobaric specific heat capacity of the gas, p4 and ρ4 are the liquid pressure
and density at point 4, and β is the pressure ratio. It is less clear how to maximise the
work ratio from this expression, and instead some results are shown on figure 1.10 with
carbon dioxide as the working fluid. For these calculations, the pressure at point 4 is
fixed at the saturation pressure at -5°C. The pressure ratio is varied and the properties are
calculated around the Rankine cycle (assuming isentropic compression/expansion) from
which the work ratio may be evaluated. A trade-off now occurs in Rankine cycles: reducing
the pressure ratio increases the work ratio, albeit at the expense of the energy and power
density. In addition, the results show that the ratio T3/T1 should be minimised, although
this may not be practicable: Rankine based cycles typically use ice as the cold storage
medium, and to match the temperature profiles of the ice and working fluid T1 and T3
should ideally be almost equal. However, it is conceivable that the cold storage could be
made up of more than one thermal reservoir. For instance, if T1 is greater than the saturation
temperature at that pressure (denoted say by T ′1), then a sensible heat reservoir could store
the energy between T1 and T ′1, while ice storage operates between points 1
′ and 3.
PTES systems require two compressions, and two expansions for one complete charge-
discharge cycle. Consequently, the isentropic efficiency has a significant impact on the
round-trip efficiency as shown in figure 1.9(b). White et al. developed a more detailed
analysis based on the polytropic efficiency and found that the round-trip efficiency was
most sensitive to the polytropic efficiency at low temperature ratios τ (and therefore pres-
sure ratios, β) and that reducing T3/T1 reduced the sensitivity [83]. Numerical results by
McTigue et al. [84] indicated that for the Isentropic Ltd. design with a value of τ = 2.5 the
efficiency would drop around 2 percentage points for every percentage drop in polytropic
efficiency.
While the equations above are simplifications of the true behaviour of PTES systems, to-
gether these two approaches provide a useful insight and are a first step in design optimi-
sation. More accurate estimations of the round-trip efficiency, energy density and power
31
density require numerical simulation which is discussed in the following sections.
1.5.3 The Joule-Brayton cycle PTES
In this scheme, the charging phase consists of heat pumping by a reverse Joule-Brayton
cycle which establishes a temperature difference between two thermal stores. Energy is
recovered by reversing the cycle into heat engine mode (forward Joule-Brayton cycle),
and returning heat from the hot to the cold store. Two independent patents seem to have
emerged almost simultaneously in 2008–2009 for similar schemes which are referred to
here as the Saipem approach [107] and the Isentropic approach [108], after the companies
that filed the patents. A further scheme was patented much earlier in 1979 by Weissenbach
of Messerschmitt-Bo¨lkow-Blohm (MBB) [109]. T–s diagrams of the ideal Isentropic and
Saipem schemes are shown in figure 1.11. These variants will be described briefly before
discussing the Joule-Brayton PTES system in more detail.
MBB system
Weissenbach’s system is an open cycle using atmospheric air as the working fluid. As such
the ‘cold’ store consists of a regenerator at atmospheric temperature and pressure. The hot
store was specified to be around 800–900°C, with thermal energy stored in ceramic balls
contained in steel tubes. Little further information is available to the knowledge of the
author, so it is unclear what temperatures T2 and T3 were and therefore what the pressure
ratio was. Weissenbach anticipated efficiencies in the region of 65–75%.
Saipem system
The Saipem scheme was the first to suggest using argon as the working fluid, due to its
high ratio of heat capacities. The scheme therefore uses a closed cycle. The thermal stores
consisted of porous refractory material, or clay with a high content of magnesia, alumina
and lime. This material was formed into bricks and perforated with cylindrical holes for the
working fluid to pass through. Compression and expansion was undertaken with turboma-
chinery meaning that two compressors and two expanders are required (one for charging
32
-200
 0
 200
 400
 600
 800
 1000
 1200
-0.6 -0.4 -0.2  0  0.2  0.4  0.6
T
em
p
er
at
u
re
, 
(°
C
)
Specific entropy, s (kJ / kgK)
1.
2.
3.
4.
1.
2.
3.
4.
Isentropic Ltd.
Saipem
Figure 1.11: T–s diagrams for Joule-Brayton cycle based PHES systems proposed by Isentropic
Ltd. and SAIPEM
and one for discharging).
The T–s diagram for a typical Saipem scheme is shown in figure 1.11, and further details are
given in table 1.2. The original design uses a low pressure ratio of around 4, but achieves
a high work ratio by increasing the value of T1 to around 480°C and fixing T3 to ambient
temperature. The low pressure ratio should reduce the cost of the system, although the use
of four turbo-machines would be expensive.
The Saipem scheme has been modelled in the literature by Desrues et al. [90, 91] and Ni
and Caram [86]. Desrues et al. [91] modelled the heat transfer processes within the thermal
stores with a finite volume scheme, and included axial conductivity and pressure losses.
The turbomachines were modelled using polytropic efficiencies of around 90%, although
off-design behaviour was also modelled by developing characteristics from commercial
data as described in Desrues’ PhD thesis [91]. On the other hand, Ni and Caram mod-
elled the turbomachines with polytropic efficiencies. For the thermal reservoirs, they used
a technique known as exponential matrix solutions to solve the governing thermal reser-
voir energy equations (which are known as the Schumann equations and are described in
33
section 2.3). The authors suggested that this has the advantage of quickly obtaining the con-
verged steady state solution. However, numerical experimentation has indicated that this
scheme is slower than the numerical schemes that are described in section 2.4. In addition,
the variation of physical properties with temperature cannot be included and Ni’s analysis
did not include conduction or heat leakage as it was based on the Schumann model.
Isentropic system
The Isentropic system6 also utilises a closed cycle with argon as the working fluid. Com-
pression and expansion is achieved with reciprocating piston engines. By changing the
valve timings, a reciprocating compressor can also work as an expander. Consequently,
only two devices are required: one on the ‘hot’ side between points 1 and 2 to carry out
compression during charge, and expansion during discharge; and another on the ‘cold’
side to expand during charging and compress during discharge. Isentropic efficiencies of
reciprocating devices may be in the range of 75-85% [110], but much of the loss is due
to valve pressure losses. By developing bespoke reciprocating devices with a new valve
system [5], Isentropic Ltd. aimed to reduce these losses significantly and (perhaps opti-
mistically) quoted isentropic efficiencies of 95-99%. Modelling of reciprocating devices is
discussed in more detail in section 4.2.
The original patent suggested that the thermal stores would consist of particles or fibres ran-
domly packed to allow gas to flow through. These particles would be metallic or possibly
a mineral or ceramic. Subsequent work [97] suggested that iron oxides such as magnetite
(Fe3O4) or hematite (Fe2O3) would be suitable materials, as they have reasonably high spe-
cific heat capacities, a wide temperature range of operation, and are relatively cheap and
abundant. Investigations typically assume these particles are uniformly sized and spherical,
although this is unlikely to be the case in practise.
The T–s diagram for a typical Isentropic scheme is shown in figure 1.11, and further details
are given in table 1.2. The original design uses a pressure ratio of around 12, and has
T3/T1 ≈ 1 meaning that the Isentropic scheme has a lower work ratio than the Saipem
scheme. Achieving high round-trip efficiencies therefore depends heavily upon developing
6Note that Isentropic is capitalised to emphasise this scheme is named after the company Isentropic Ltd.,
and does not imply that any process in the system is isentropic.
34
reciprocating devices with high isentropic efficiencies.
The majority of the literature on the Isentropic scheme originated at Cambridge Univer-
sity, and has focussed predominantly on the design of the packed-bed thermal reservoirs
and simple cycle modelling. These papers include a semi-analytical packed bed model
based on Schumann’s equations (therefore neglecting conduction and heat leakage) and
defined a number of exergetic loss coefficients and the effect of certain design parameters
upon them [60]. The impact of the variation of specific heat capacity with temperature
was investigated in detail [97]. An early paper by White et al. [83] that focussed on the
Isentropic scheme developed equation 1.15 and explored the sensitivity of the efficiency
to various uncertain parameters. This paper suggested the importance of the reciprocating
device isentropic efficiency as well as the irreversible heat transfer that occurs in these de-
vices. This analysis was extended by McTigue et al. [84] who integrated the packed bed
model with compression-expansion devices that were modelled by polytropic efficiencies,
pressure loss factors and heat leakage factors. They undertook a parametric study of the
main parameters and used a genetic algorithm to optimise the design. The optimisation
converged on solutions similar to the Isentropic design, and some results are shown in ta-
ble 1.2. The top temperature T2 was constrained to be less than 600°C by material limits.
If this constraint is relaxed, then it is possible that the optimiser would have converged
on Saipem-like designs. Again, this study highlighted the importance of the compression-
expansion efficiency. Ref. [84] also described the importance of the charging frequency
or utilisation (the duration of the charging/discharging phases) in controlling the trade-off
between efficiency and energy density: high utilisations leads to high energy densities but
low efficiencies also observed by Ni and Caram [86]. It was also noted that the optimal
discharging pressure ratio was slightly less than the charging pressure ratio.
More recent papers have again focussed on the thermal reservoir model, which was updated
to include axial conduction and heat leakage, behaviour during storage phases, and to model
segmentation of the packed beds [80,81]. In these works, the packed beds were considered
in isolation, but were sized for bulk electricity storage. It was found that segmentation
(i.e., dividing the bed into layers) could improve performance in both hot and cold stores,
although only the cold store benefit was significant enough to suggest that it would be a
worthwhile investment. These results are also applicable to other thermal energy storage
systems which utilise hot or cold packed beds.
35
Saipem Isentropic Desrues McTigue 1. McTigue 2.
[107] [5] [90] [84] [84]
Polytropic efficiency 0.90 0.90 0.90 0.90 0.99
Pressure ratio 2–4 12.0 4.6 10.9 7.6
T1 (°C) 400–1000 45 477 35 117
T2 (°C) 1000–1500 500 995 600 598
T3 (°C) 10–50 45 25 35 37
T4 (°C) -80 – -20 -166 -73 -164 -134
Work ratio, R ∼4 2.4 4.2 2.8 2.8
Efficiency∗, χ (%) 70 61 71 64 96
Efficiency+, χ (%) >60 - 67 59 87
Energy density, - - 100 65 94
ρE (MJ m−3)
Table 1.2: Operating conditions for Joule-Brayton based PHES systems suggested by patents and
literature. ∗ refers to efficiency calculated with equation 1.15, and + refers to the efficiency cal-
culated in the source. McTigue 1. and 2. refer to optimised results where the points with highest
efficiency have been selected for the given polytropic efficiency.
1.5.4 The Rankine cycle based PTES
The first PTES systems that were described by Maguerre and Cahn were based on Rankine
cycles. Modern research has taken place predominantly in Switzerland with input from the
company ABB.
Mercango¨z et al. [87] described a transcritical CO2 cycle which used water as the hot store
and ice slurry as the cold store. Most of the subsequent research has developed this type
of cycle. Mercango¨z et al. emphasised the importance of matching the temperatures of
the cycle working fluid and the storage fluid. For instance, the water storage is a sensible
heat store, thus its temperature increases as energy is transferred to it. Irreversibilities
due to heat transfer can be minimised by reducing the temperature difference between the
working fluid and the storage material. If the working fluid changes phase (condenses)
it would do so at a constant temperature, leading to large temperature differences and,
consequently, exergetic losses. During charge, the CO2 is therefore compressed to a super-
critical state, before following a trans-critical path as it releases its energy - i.e. the fluid
remains above the critical point as illustrated on figure 1.8(b). The temperatures of the
working fluid and the storage media are therefore ‘matched’ for the high pressure part of
36
the cycle. However, as figure 1.8(b) shows, the CO2 must undergo a phase change in the
low pressure part of the cycle. Matching the temperatures of the working fluid and storage
therefore requires a storage medium that also undergoes a phase change, and ice slurry is
the suggested medium.
The requirement to match temperatures therefore places constraints upon the Rankine-cycle
system that do not exist for the Joule–Brayton system. Materials with suitable thermody-
namic properties are required. Matching the CO2 and ice slurry temperatures requires the
CO2 to be at high pressure: around 32 bar on the low pressure side and 140 bar on the high
pressure side. The high pressures add to the cost, but the cycle benefits from not need-
ing to pressurise the thermal stores. In addition, hot water has a high heat capacity but a
narrow temperature range of operation unless it is pressurised. Mercango¨z and subsequent
researchers therefore employ several hot water tanks that operate over different temperature
ranges. In addition, there is the possibility of using molten salts or heat transfer oils for the
appropriate temperature ranges.
The use of liquid pumps or expanders means that the Rankine-cycle system can achieve
higher work ratios than the Joule-Brayton cycle based system. Mercango¨z et al. [87] opti-
mised a 50 MW system and achieved a work ratio of roughly 6 and an efficiency of 65%,
as shown in figure 1.9. Morandin et al. [111] developed the work by Mercango¨z et al. and
described a modelling method based on pinch analysis. An optimisation study gave an
efficiency of 60% at a work ratio of 5. Morandin also undertook a more detailed thermo-
economic optimisation of the Rankine-cycle system [88]. The study optimised the system
topology (i.e. the number of storage tanks and their layout). Morandin also investigated
intercooling between the discharging expanders in this study.
Another Rankine-cycle system developed by Steinmann [89] is known as CHEST: Com-
pressed Heat Energy Storage. The basic CHEST configuration system uses steam as the
working fluid, and the atmosphere as the cold store. The hot store comprises both sensible
heat and latent storage, which enables the temperature of the storage media to match the
temperature of the condensing steam. Steinmann also considered “cascaded cycles”: a low
pressure ammonia cycle is used as it has lower volumetric flow rates compared to water.
The hot store of the ammonia cycle then acts as the cold reservoir for the high pressure
steam cycle. In a simplified analysis, Steinmann estimated the round-trip efficiency to be
37
72.8%. The analysis did not include pressure losses or heat leakage losses in any of the
compressors, expanders, pipes or storage systems. Analysis in section 4.4.1 indicates that
inclusion of these factors (at least in the compression-expansion system) may be expected
to cause a further loss in efficiency of several percentage points.
Figure 1.9 compares the results of more detailed studies on PTES systems against the sim-
plified expressions for efficiency that were derived in section 1.5.2. A direct comparison
between different PTES configurations is challenging as authors typically make different
assumptions in the modelling process.
1.6 Aims of the project
This thesis focusses on the Joule-Brayton cycle variant of Pumped Thermal Energy Storage
(PTES). The ultimate objective of the project is to develop a detailed thermo-economic
model of the full system, and to use optimisation techniques to determine optimum designs.
Ideally these methods should be combined with other approaches such as experiments and
computational fluid dynamics in order to validate the results. However, these technologies
are at an early stage of development. Consequently, thermodynamic models are a useful
tool which can be used to quickly investigate the potential performance of energy systems.
The models will help to develop an understanding of the system behaviour, generate design
guidelines, and also suggest promising designs or areas for further work.
The bulk of the thesis concentrates on thermodynamic modelling of the system and its
subcomponents. Of particular interest are the packed-bed thermal reservoirs to which ther-
mal energy is transferred and stored. A number of heat transfer processes occur within
the packed beds. In chapter 2 the packed bed theory is discussed with the aim of charac-
terising the most significant phenomena, and developing a robust thermodynamic model.
Having developed a thermodynamic model, individual processes that cause a reduction in
efficiency (or ‘loss generating mechanisms’) are quantified.
The objective of chapter 3 is to take the theory developed in chapter 2 and improve the
understanding of the behaviour and performance of packed-bed thermal reservoirs for bulk
electricity storage. This is achieved by undertaking parametric studies of a number of de-
38
sign variables to uncover their influence on the various loss generating mechanisms. These
studies inform design, and two innovative design features are investigated. The first is
segmentation of the packed beds, and the second is radial-flow packed beds. A rigorous
investigation of these design features is undertaken using the theoretical framework de-
veloped in chapter 2 and in such a way that the investigations are consistent and allow
comparison between different features.
The aim of chapter 4 is to develop a thermodynamic model of the PTES system. This in-
volves developing models of the compression and expansion system and the heat exchang-
ers, and integrating them with the thermal reservoir model. Like the preceding chapters, a
framework is established that allows the various processes that reduce the efficiency to be
accounted for. Having developed a model of the PTES system, its behaviour is investigated
with parametric studies. By doing so, and by considering alternative modes of operation,
an understanding of the system can be developed that is necessary for design optimisation.
Realistically, the system design cannot be optimised by considering only technical factors.
The ‘best’ design may not be the most efficient, but may be the one that is most affordable or
most flexible. Holistic design optimisation therefore requires the consideration of several,
possibly conflicting objectives. Since externalities can be uncertain or difficult to quantify
chapter 5 begins by developing a simplified economic model of the PTES system. The aim
is to keep the model relatively simple whilst capturing the key parameters that impact on
the cost of the system.
The objective of chapter 5 is to find optimum system designs for thermal energy storage.
Optimisation theory is discussed and an appropriate multi-objective optimisation algorithm
is described. The optimisation scheme is applied with the aim of finding a range of optimal
solutions: some designs will have better thermodynamic performance, some will have bet-
ter economic performance, and others will be a compromise between these objectives. Op-
timisation of hot and cold packed-bed thermal reservoirs is first undertaken. This is carried
out on a range of different scenarios with the aim of testing some of the designs developed
in earlier chapters. The overall PTES system is then optimised, and several alternative con-
figurations are tested. Conclusions and a discussion of areas for future research are given
in chapter 6.
39
Contribution of the author
At the beginning of this project, a thermal reservoir model had already been developed by
Alex White of Cambridge University. The author extended this model to include other heat
transfer processes, such as heat leakage and conduction, and to model other behaviour, such
as that of segmented stores. The thermal reservoir codes were then integrated with models
of compressors, expanders and heat exchangers and thereby developed a model of PTES.
The author undertook a number of investigations using these models, such as parametric
studies and thermo-economic optimisation. For the latter, the author programmed a multi-
objective optimisation algorithm, and developed simple cost models of the storage systems.
Much of the work presented in this thesis is either an extension of earlier published work
[84, 97], or work that is currently being published [80, 81], and has also contributed to a
number of conference proceedings, see page xv.
40
Chapter 2
Modelling of packed-bed thermal reservoirs
41
2.1 Introduction
Packed-bed thermal reservoirs are an integral component of the Pumped Thermal Energy
Storage (PTES) system described in chapter 1, and are also used in other storage systems.
Investigating the performance of PTES requires accurate modelling of the heat transfer
processes in the packed beds. This chapter begins with a discussion of the assumptions that
underlie the thermodynamic theory of packed-bed thermal reservoirs. These assumptions
lead to a set of governing energy equations known as the Schumann model which are
described in section 2.3.
A numerical scheme based on that developed by White et al. [80] is used to solve the
governing equations. As part of this project, several modifications and extensions have been
made, such as including axial conduction, modelling of segmented stores, and integrating
the numerical scheme into a model of PTES. For comparison, the equations are also solved
with a widely used, but less efficient, numerical scheme: the predictor-corrector (or Adams-
Bashforth-Moulton) method. These schemes are validated against the existing analytical
solution to the Schumann equations.
Capturing the packed bed characteristics allows assessment and comparison of packed bed
designs and is therefore a key requirement of the thermodynamic model. In section 2.5 a
Second Law approach to packed bed analysis is presented and various entropy generating
processes are discussed in detail. These loss-generating mechanisms are quantified by so-
called ‘loss coefficients’ as defined by White [60]. Solid and gas properties that affect the
loss coefficients are discussed. By simplifying the expressions for the loss coefficients, the
influence of the packed bed design on performance is revealed. Together this approach
provides a basis for the more detailed analysis that is presented in chapter 3
2.2 Description of packed bed reservoirs
Packed-bed thermal reservoirs typically contain a solid storage material composed of spheres,
irregularly shaped pebbles or gravel, or some other internal structure (such as a concrete
matrix) through which the heat transfer fluid passes. The packing is encased in one or more
42
layers of insulation and a steel containment vessel which may be pressurised (as is the case
for the hot store in PTES). PTES pressure vessels are cylindrical (with closed ends), but
some packed beds that have been investigated are conical [2] or other shapes.
During charge, the heat transfer fluid (which is argon in PTES) enters the packed bed at
temperature T1 and, having transferred its energy to the packing material, exits the packed
bed at temperature T2. The flow reverses direction during discharge. A schematic diagram
for a hot store is shown in figure 2.1. The packing material has a length L and a diameterD,
although the store itself will be larger than this due to the insulation and steel containment.
The particles have a diameter of dp and occupy a fraction of the store equal to (1 − ε)V ,
where ε is known as the void fraction.
leakage
Heat
loss
Exit
loss






















































































































































































































































































































































































































































































































































































































































































































































loss
In
su
la
tio
nConductive












































heat transfer
loss
Pressure
Steel
pressure
vessel
loss
Gas−particle
T1
T2
L
D
Figure 2.1: Schematic layout of a packed bed. The hot thermal reservoir is orientated vertically dur-
ing charge. Gas enters at T1 and exits at T2. Several entropy generating mechanisms are illustrated
and are discussed in section 2.5.
43
2.3 Heat transfer in packed bed reservoirs
Main concepts and thermal fronts
A number of different models exist for describing heat transfer in packed beds and are
discussed comprehensively in [112, 113]. Figure 2.2 illustrates the main heat transfer pro-
cesses that occur in an infinitesimal control volume for a hot reservoir, in which it is as-
sumed that the flow is one-dimensional, and that the particles have a uniform temperature.
Gas flows from top to bottom to avoid buoyancy issues. Convective heat transfer between
the gas and solid results in a temperature change across the packed bed. This region is
known as the ‘thermal front’ or ‘thermal wave’.
(a) Basic structure (b) Equivalent control volume
Figure 2.2: Schematic view of the packed bed structure and the equivalent control volume which
displays heat transfer processes: where Qx, Ql and Qc are gas-solid heat transfer, heat leakage
through the walls and conduction along the bed, respectively.
The velocity of the thermal front Vf can be found by assuming the front remains perfectly
abrupt (i.e. a step change from T1 to T2) as shown in figure 2.3. Applying the first law to
the thermal front yields
dEs
dt
= m˙cp(T1 − T2) (2.1)
(1− ε)ρsc¯sA(T1 − T2)dx
dt
= ρguscpA(T1 − T2) (2.2)
Where us = m˙/ρgA is the superficial (or open tube) gas velocity, cp is the gas specific heat
capacity, c¯s is the average solid heat capacity and ρs and ρg are the solid and gas densities.
44
The thermal front velocity Vf = dx/dt is therefore
Vf =
ρgcp
(1− ε)ρsc¯sus (2.3)
T
T1
T2
Vf
δx
x
Figure 2.3: Schematic of an ideal thermal front between temperatures T1 and T2 travelling at ve-
locity Vf (see equation 2.3)
The solid heat capacity cs is a function of temperature (see figure 2.10 on page 61), and
thereby influences the shape of the thermal front as cold regions travel more rapidly that
hot regions. The nominal thermal front velocity in equation 2.3 is defined in terms of the
average solid heat capacity over the temperature range of interest c¯s which is given by
c¯s =
Es,1 − Es,2
T1 − T2 (2.4)
where Es =
∫ T
Tref
csdT + Eref is the energy of the solid. An abrupt thermal front would
require there to be no thermal resistance between the gas and solid, so in practice, the front
tends to vary between T1 and T2 with a finite gradient that changes. However, Vf can still
be interpreted as the nominal speed of the front, and the nominal charging time is then
given by
tN =
L
Vf
=
Msc¯s
m˙cp
(2.5)
where L is the reservoir length, and Ms is the mass of the solid packing material.
45
Other considerations
Thermal gradients within particles are generally ignored, since the internal thermal resis-
tance of particles is small for the pebble sizes considered here. Thus the Biot number is
effectively zero and heat transfer to and from the solid is limited by the thermal resistance
at the surface. For instance, the average Biot number is 0.07 in a hot store and 0.05 in a
cold store [60].
In certain cases, radial variations in temperature and velocity can be significant. For in-
stance, Beasley and Clark [114] demonstrated the different thermal fronts that occur along
the wall and the centreline of a packed bed, as reproduced in figure 2.4(a). Radial effects
are predominantly due to two factors: heat leakage from the walls of the store, and vari-
ations in the packing density. According to [4] there are damped oscillations in the void
fraction ε with radius: at the wall ε increases to 1 and then decreases to around 0.3–0.4 (de-
pending on the packing structure) as distance from the wall increases. The lower frictional
resistance at the wall results in a ‘by-pass’ flow [114]. Van Antwerpen et al. [4] collected
several correlations for the void fraction variation, and shows that the oscillations are fully
damped after around five particle diameters, as shown in figure 2.4(b).
The radial variations shown in figure 2.4(a) are particularly pronounced because the store
was not insulated leading to more heat leakage than a typical insulated thermal store. Fur-
thermore, the ratio of bed diameter to particle diameter was D/dp ≈ 30 meaning that
roughly 56% of the bed area was affected by variations in the packing density and there-
fore the velocity. These factors are less significant in the packed beds that are studied in this
work. For instance, the stores are well insulated to prevent energy loss which reduces radial
temperature variations. Moreover, thermal stores typically have much larger volumes than
those on which experiments are performed. The lower surface-area-to-volume ratios will
also reduce heat leakage.
Furthermore, the large stores investigated in the present study have values of D/dp of the
order of 1000. As a result, the area of the bed that is affected by void fraction variations
is typically around 1–2%. Meier et al. [116] suggest that radial non-uniformities can be
neglected for D/dp > 40.
Consequently, radial effects are not considered and a set of one-dimensional governing
46
(a) (b)
Figure 2.4: (a) Experimental and numerical results showing thermal fronts in a packed bed during
charge. The curves show the fronts at two radial positions: on the centreline (R/R0 = 0.0) and
along the wall (R/R0 = 1.0). Figure from Beasley and Clark [114]. (b) Experimental results
showing the variation in void fraction with distance from the wall for uniform spheres in a packed
bed with D/dp =∞. Figure from Benenati and Brosilow [115].
equations is developed. Furthermore, it is assumed that the particle heat transfer character-
istics can be represented by an average equivalent diameter – i.e. the particles are spherical
and uniformly sized. Numerous studies of this type have been presented in the literature
but only a few have been experimentally validated. Figure 2.5 provides two examples and
demonstrates that one-dimensional models can provide a good fit to experimental data.
However, in most cases (including the two shown) only the centreline temperatures are
measured so it is unclear if what the radial effects may be. In particular, heat leakage is ap-
parent in the thermal profiles of figure 2.5(b) which suggests that a two-dimensional model
may have been more appropriate.
To the best of the authors knowledge there have been no experimental studies into the sig-
nificance of radial variations in insulated stores with very high D/dp. Such a study would
clarify when it is appropriate to use one-dimensional models and when two-dimensional
models are necessary. However, under these conditions it is likely that radial effects will be
reduced. One-dimensional models therefore provide a comparatively efficient and straight-
forward way to gain an understanding of the major parameters that affect packed bed be-
haviour.
47
(a) (b)
Figure 2.5: Comparison of theoretical models and experimental results showing thermal fronts
in packed beds. Theoretical models both use one dimensional energy equations similar to those
developed in section 2.3. (a) Figure from Zanganeh et al. [2]. (b) Figure from Anderson et al. [3]
Models typically have a temperature for the solid Ts and a temperature for the gas Tg.
However, ‘single-phase’ models use a single temperature [113]. In deriving the single-
phase model, Ref. [117] showed that it was not necessary to assume the gas and solid
temperatures were equivalent, but only that their second derivatives were equal. Anderson
et al. [3] also developed a one-phase model coupled with the Navier-Stokes equation in
order to investigate energy storage applications. This model showed reasonable agreement
to experimental results. However ‘single-phase’ methods are not sufficiently detailed for
evaluation of energy storage systems, since the gas-solid temperature difference is required
to calculate the availability destruction due to heat transfer across the finite temperature
difference, as detailed in section 2.5.1. A ‘two-phase’, one dimensional model is now
described in detail.
Governing equations
The governing equations may be derived by considering the energy flows in the control
volume of figure 2.2. The convective heat transfer rate between gas and solid is given by
δQ˙x = (1− ε)ASvh(Tg − Ts)δx (2.6)
48
where A is the cross-sectional area, h is the heat transfer coefficient, and Sv is the particle
surface-to-volume ratio. The heat flux due to conduction into the control volume is
Q˙inc = −keffA
∂Ts
∂x
(2.7)
where keff is the effective conductivity, which takes into account various physical processes
as discussed in more detail in section 2.5.3. The heat flux out of the control volume due to
conduction is given by
Q˙outc = −keffA
∂Ts
∂x
− ∂
∂x
(
keffA
∂Ts
∂x
)
δx (2.8)
The heat flux resulting from heat leaking through the wall of the packed bed is
δQ˙L = UwAw(Ts − T0)δx (2.9)
where Uw is the overall heat transfer coefficient, Aw is the surface area of the wall, and T0
is the ambient temperature.
The gas and solid equations are then given by applying the unsteady energy equation to the
gas and the First Law to the solid:
ε
∂
∂t
(ρgeg) +
∂
∂x
(Ghg) = (1− ε)Svh(Ts − Tg) (2.10)
ρscs(1− ε)∂Ts
∂t
= Svh(1− ε)(Tg − Ts) + keff∂
2Ts
∂x2
− UwAw
A
(Ts − T0) (2.11)
Where G = m˙/A. The continuity and momentum equations are given by
ε
∂ρg
∂t
+
∂G
∂x
= 0 (2.12)
∂G
∂t
+
∂
∂z
(
G2
ρg
+ ε2(p− ρggx)
)
= −ε(1− ε)τsSv (2.13)
where τs is the effective shear stress on the particle. These equations are coupled, and the
numerical scheme should take this into account. A numerical scheme along these lines
would require small time-steps and be computationally expensive. However, numerical
49
experimentation has shown that gas unsteady terms are usually small and that the energy
and momentum equations are effectively ‘one-way coupled’ since the mass flow rate is
approximately constant. (For example, the main cause of density change in the reservoir is
due to the progression of the thermal front, which moves slowly).
This may be illustrated by considering the fractional change of mass across an ideal thermal
front. The thermal front is vertical as illustrated in figure 2.3 travelling a distance of δx.
The change in mass of the control volume is
∆m = (ρ2 − ρ1)Aδx = p
R
(
1
T2
− 1
T1
)
Aδx (2.14)
The mass that enters the control volume is m˙1δt where δt is the time taken for the front to
move δx which is given by
δt =
δx
Vf
(2.15)
Therefore the fractional change in mass is
∆m
min
=
1
(1− ε)ρscs
pcp
R
(
T1 − T2
T1T2
)
(2.16)
For a nominal hot reservoir operating between 500°C and 25°C at 10 bar with argon, the
fractional change in mass is around 0.2% suggesting that the mass flow rate is approxi-
mately constant.
Employing these simplifications, and using hg = eg + p/ρg = cpTg and introducing the
Stanton number St = h/cpG allows the energy equations to be written as
∂Tg
∂x
=
Ts − Tg
`
+ C (2.17)
∂Ts
∂t
=
Tg − Ts
τ
+ α
∂2Ts
∂x2
− β(Ts − T0) (2.18)
where the length scale (`), time scale (τ ), diffusivity (α) and other factors are given by
` =
1
(1− ε)SvSt (2.19)
τ =
ρscs
cpGSvSt
(2.20)
50
α =
keff
ρscs(1− ε) (2.21)
β =
UwAw
ρscs(1− ε)A (2.22)
C = ερg
G
∂
∂t
(
p
ρgcp
− Tg
)
(2.23)
This formulation of the energy equations is often known as the Schumann model [118], or
a Schumann-style model. The quantity C is a small correction to the steady-flow energy
equation to account for the unsteady accumulation of energy within the control volume.
For the special case where α = 0, β = 0, ` and τ are constant, and pressure losses are
ignored, an analytical solution can be developed by non-dimensionalising and transforming
equations 2.17 and 2.18 in the following way [112]:
∂θg
∂ξ
= θs − θg (2.24)
∂θs
∂η
= θg − θs (2.25)
Where θg,s = (Tg,s − T2)/∆T , ∆T = T1 − T2, and ξ and η are dimensionless length and
time variables given by
ξ =
x
`
; η =
t
τ
− εx
usτ
(2.26)
The dependence of η on x arises from the unsteady accumulation of internal energy within
the gas. For the application considered here (argon flowing through a rock bed), this term
can be ignored, due to the low heat capacity of gas per unit volume compared with that of
the solid. The analytical solution was first presented by Anzelius in 1926 [119] and is given
by
θg − θs = exp
[
−(ξ + η)I0
(
2
√
ξη
)]
(2.27)
Where I0 is the zero-th order modified Bessel function of the first kind. Full equations for
Ts and Tg of a similar form are presented in [118] but these expressions require evaluation
of an infinite sum of Bessel functions making calculation of the analytical solution cum-
bersome. In addition, the analytical result assumes that thermal constants are independent
of temperature, whereas this is not a necessary requirement for numerical approaches. Fur-
thermore, this solution does not include heat leakage, axial conduction, or pressure losses.
51
As a result, numerical solutions generally prove to be a comparatively straightforward,
flexible and efficient alternative, and the analytical solution may be used for validation.
2.4 Numerical solutions of the energy equations
The Schumann-style model presented above can be solved with a variety of numerical
schemes, several of which will be described and validated now. Firstly, a correlation for the
Stanton number is required. There is significant variation in the available experimental data
as a result of difficulties in obtaining reliable measurements of heat transfer coefficients in
packed beds. For the range of Reynolds numbers of interest, the following correlation was
developed by Wakao et al. [120]
St =
2
RePr
+
1.1
Re2/5Pr2/3
(2.28)
where Re = ρgusdp/µg is the Reynolds number based on the particle size. The most
straightforward approach to solving the gas and solid equations 2.17 and 2.18 is by ex-
plicit finite differences as used by Zanganeh et al. [2]. The stability of this method can
be improved substantially by using a predictor-corrector method, also known as Adams-
Bashforth-Moulton. The equations are solved explicitly in the predictor step, and the result
is then refined in the corrector step which typically uses an implicit formulation of the equa-
tions [121]. Firstly, equations 2.17 and 2.18 are re-formulated with the time differential on
the left-hand side
∂Tg
∂t
=
G
ερ
(
Ts − Tg
`
− ∂Tg
∂x
)
+
1
ρcp
∂p
∂t
(2.29)
= A
∂Ts
∂t
=
Tg − Ts
τ
+ α
∂2Ts
∂x2
− β(Ts − T0) (2.30)
= B
The stencil for the predictor-corrector scheme is shown in figure 2.6. Information is known
at all nodes at time (n − 1). An initial guess for the temperatures Tˆg,s at time n and node
52
∆ t
time n−1
time n
∆x
ii −1 i +1
Figure 2.6: A stencil of the computational grid for the predictor-corrector numerical scheme. In-
formation is known at all filled nodes, and unknown at the open nodes.
i is found in the predictor step by solving the equations using central differences and the
explicit Euler method such that
Tˆ ng,i − T n−1g,i
∆t
=
Gni
ερni
(
T n−1s,i − T n−1g,i
`n−1i
− T
n−1
g,i+1 − T n−1g,i−1
2∆x
)
+
1
ρn−1i cp
(
pˆni − pn−1i
∆t
)
= An−1i (2.31)
Tˆ ns,i − T n−1s,i
∆t
=
T n−1g,i − T n−1s,i
τni
+ α
(
T n−1s,i+1 − 2T n−1s,i + T n−1s,i−1
∆x2
)
− β (T n−1s,i − T0)
= Bn−1i (2.32)
In the corrector step, the initial guesses are corrected by re-evaluating Aˆn and Bˆn using the
predicted values of Tˆ ns,g. The updated gas and solid temperatures at time n are then given
by
T ng,i − T n−1g,i
∆t
=
1
2
(
An−1i + Aˆni
)
(2.33)
T ns,i − T n−1s,i
∆t
=
1
2
(
Bn−1i + Bˆni
)
(2.34)
53
In this method, other variable parameters such as pn, Gn and `n can be updated after the
predictor step. Strictly, the mass flow rate per unit area Gn should be calculated in a similar
manner using a predictor-corrector scheme. However, G only varies slightly as a result of
the change in mass of stored gas brought about by the progression of the thermal front. The
pressure is calculated from Ergun’s equation (equation 2.53) from which the gas density
can be updated from ρg = p/RTg. Finally, the mass flow rate per unit area is found from
the continuity equation:
Gni = G
n
i−1 + ε
∆x
∆t
(
ρn−1g,i − ρng,i
)
(2.35)
The predictor-corrector method is verified against the analytical solution for single blow
operation of a charging hot packed bed in figure 2.7 from which it is clear there is good
agreement between the solutions. In accordance with the derivation of the analytical solu-
tion, there are no pressure losses, conduction, or heat leakage, and factors such as ` are kept
constant. Even in this simplified scenario, extremely small time-steps are required; in this
case ∆t/τ ≤ 0.5 × 10−3 in order match the analytical results as well as obtain converged
estimates for the thermal loss. Grid steps can be somewhat larger at ∆x/` ≤ 0.5. As a
result, the predictor-corrector method runs slowly and developing an alternative scheme is
worthwhile.
An alternative method devised by White et al. [80] achieves stable results with larger grid
and time-steps. This semi-implicit method is similar to that in [60] and [97], although it is
extended to include conduction and heat leakage. Similar integration schemes are presented
in Willmott [112].
The Schumann-style gas and solid equations are marched forward in space and time, re-
spectively, along the paths indicated in figure 2.8. All derivatives are represented semi-
implicitly (with the exception of the conduction term in the solid equation) such that quan-
tities in the derivatives take on their average value along the path of integration. The equa-
tions are therefore
54
 0
 0.2
 0.4
 0.6
 0.8
 1
 0  20  40  60  80  100
 0
 0.002
 0.004
 0.006
 0.008
 0.01
G
as
 t
em
p
er
at
u
re
, 
θ g
(θ
g
 -
 θ
s)
2
  
  
  
  
  
  
  
  
  
  
 
Normalised distance, ξ 
η = 8
η = 32
η = 64
Numerical
Analytical
Figure 2.7: Temperature profiles generated using the predictor-corrector scheme at three different
times. (θg − θs)2 are plotted on the right-hand axis, and are verified against the analytical solution
of equation 2.27
∆ t
∆x
time n−1
time n
ii −1 i +1
A
B
Figure 2.8: A stencil of the computational grid for the semi-implicit method. Properties are known
at nodes that are filled, and unknown at open points. The two energy equations are integrated along
paths A and B to obtain Ts and Tg at node (i, n)
55
T ng,i − T ng,i−1
∆x
=
T¯As − T¯Ag
`
+ C (2.36)
T ns,i − T n−1s,i
∆t
=
T¯Bg − T¯Bs
τ
+ α
T n−1s,i+1 − 2T n−1s,i + T n−1s,i−1
∆x2
− β (T¯Bs − T0) (2.37)
where superscripts A and B represent that the quantity is evaluated at its average value
along that path (see figure 2.8)
T¯A =
1
2
(
T ni−1 + T
n
i
)
(2.38)
T¯B =
1
2
(
T ni + T
n−1
i
)
(2.39)
The unsteady gas accumulation term is accounted for by C which is small and may be eval-
uated with sufficient accuracy at the upstream location (i−1). Expanding these expressions
and rearranging, allows the resulting pair of equations to be represented in matrix form as[
(1 + φ) −φ
−ψ1 (1 + ψ1 + ψ2)
]{
T ng,i
T ns,i
}
=
{
K1
K2
}
(2.40)
where
K1 = (1− φ)T ng,i−1 + φT ns,i−1 + Cni−1∆x (2.41)
K2 = (1− ψ1 − ψ2 − 2ψ3)T n−1s,i
+ ψ3
(
T n−1s,i+1 + T
n−1
s,i−1
)
+ ψ1T
n−1
g,i + 2ψ2T0 (2.42)
and
φ =
∆x
2`
ψ1 =
∆t
2τ
; ψ2 =
β∆t
2
; ψ3 =
α∆t
∆x2
The gas and solid temperatures at node (i, n) can then be found by inverting this 2 x 2
matrix at each node. The pressure, density and mass flow rate per unit area are then up-
dated as in the predictor-corrector method. The semi-implicit scheme is compared with
56
the predictor-corrector method in figure 2.9 for a hot reservoir which includes conduction,
heat leakage and pressure losses. (The semi-implicit scheme was also verified against the
analytical solution for the most simplified scenario as in figure 2.9). It is possible to take
much larger time-steps than the predictor-corrector scheme, and for convergence of the
thermal losses ∆t/τ,∆x/` ≤ 1/2 is sufficient. These results demonstrate that the numeri-
cal results are consistant with the theory, although validation against experimental results is
also required. (Note, these equations are commonly used in the literature and can produce
comparable results to experimental data, as shown in figure 2.5 on page 48).
A further stability constraint is α∆t/∆x2 ≤ 1/2 which arises from the explicit coding of
the conduction term. The conduction terms can be represented implicitly by separating the
solid equation into two separate steps. In the first step, ψ2 = ψ3 = 0 are substituted into
the above equations to obtain the gas and solid temperatures. In the second step, the solid
temperature is updated such that
∂Ts
∂t
= α
∂2Ts
∂x2
− β(Ts − T0) (2.43)
This is simply the one-dimensional heat diffusion equation which can be solved implicitly
using the Crank-Nicolson method [121]. This method was tested and has very similar
performance to the full semi-implicit method described above. A comparison of the CPU
times taken by each algorithm for the case described above is given in table 2.1.
Predictor-corrector Semi-implicit
∆t/τ = 0.1 ∆t/τ = 0.4
CPU times (s) 128 2.4 0.40
Table 2.1: Comparison of CPU times for one cycle for each numerical scheme. Calculations were
undertaken on a typical laptop (Intel i5 -2430M, 2.40 GHz, 6 GB RAM)
A wide range of other techniques have been developed in the literature. Beasley and
Clark [114] describes a range of approaches that have been used, as well as developing
a two-dimensional solver in which the energy equations were formulated such that the con-
ductive terms were included in the gas energy equation. The solid equation could then be
discretised implicitly and substituted into the finite-difference form of the gas equation.
The resulting set of equations could be solved using matrix decomposition approaches for
banded matrices. This method was deemed to be quite stable, although small time-steps
57
 0
 0.2
 0.4
 0.6
 0.8
 1
 0  20  40  60  80  100
 0
 0.002
 0.004
 0.006
G
as
 t
em
p
er
at
u
re
, 
θ g
(θ
g
 -
 θ
s)
2
  
  
  
  
  
 
Normalised distance, ξ 
η = 8
η = 32
η = 64
Semi-implicit
Predictor-corrector
Figure 2.9: Temperature profiles generated using the semi-implicit scheme at three different times.
(θg − θs)2 are plotted on the right-hand axis, and are verified against the predictor-corrector method.
Conduction (keff = 2 W/mK), heat leakage (Uw = 0.15 W m−2K−1) and pressure losses are
included.
were required in order to achieve convergence. Another approach by Ni and Caram [86] de-
velops a closed form matrix solution to the steady-state cyclic operation problem. However,
this method requires at least as much computational expense as typical numerical schemes,
and is not easily extendible to treat the temperature-dependent properties described in [97].
2.5 Availability destruction in packed beds
Various physical processes lead to a reduction in the available energy stored in the reser-
voir. Each loss mechanism may be affected in different ways by the packed bed design
parameters. Accurate modelling therefore provides a way to understand the impact of each
variable on performance, and aid the design of optimal systems. In this section the main
mechanisms that cause a reduction in availability are described.
For the reservoir alone (i.e. excluding the power conversion machinery) the round-trip
storage efficiency χ may be defined as the available energy recovered from the reservoir
during discharge as a fraction of that which enters during charge. A loss coefficient ζ may
58
be derived for each of the mechanisms that causes a loss in availability. Each of these
loss coefficients is defined as the fraction of the inlet available energy that is lost such that
ζ = Blost/Bin and
χ =
Bout
Bin
= 1−
∑
ζ (2.44)
where B is the total amount of available energy added and Bin is given by
Bin = m˙cptchg
[
(T1 − T2 − T0 ln
(
T1
T2
)]
= m˙cptchgβ (2.45)
where tchg is the charging duration.
Note that in the present work the term availability or available energy is used. The specific
availability function is defined as b = h− T0s, and as such only changes in availability are
meaningful. Exergy is a commonly used quantity and is the difference between the actual
value of b and b0, which is the availability at a point where the fluid has zero potential to
do work. This point is called the dead state and is typically at atmospheric pressure and
temperature. As such, the specific exergy may be defined as ξ = (b − bo) = (h − T0s)−
(h0 − T0s0), and it can be seen that ∆b = ∆ξ. In this work, availability or exergy could be
used interchangeably. However, the use of availability is more natural because the thermal
stores operate between T1 and T2 and are not necessarily discharged to the dead state.
2.5.1 Thermal losses
Availability losses are caused by the irreversible heat transfer that occurs across the finite
temperature difference between gas and solid. Following [60, 97] this loss is referred to
here as the “thermal loss”. The instantaneous entropy generation rate is given by,
S˙irr =
∫ (
1
Ts
− 1
Tg
)
dQ˙ (2.46)
and dQ˙ = hASv(1 − ε)(Tg − Ts)dx. By incorporating the definition of `, equation 2.46
becomes
S˙irr = m˙cp
∫ λ
0
(Tg − Ts)2
TgTs
dx
`
(2.47)
59
where λ is the length of the thermal front. (Equation 2.47 could equally be integrated over
the length of the packed bed L, since Tg−Ts = 0 outside of the thermal front). The thermal
loss coefficient ζt is found by integrating equation 2.47 over the duration of operation t and
normalising by the net flux of inlet availability Bin
ζt =
T0
βtchg
∫ tchg
0
∫ λ
0
(Tg − Ts)2
TgTs
dx
`
dt (2.48)
By approximating the thermal fronts as linear profiles of length λ the effect of various de-
sign parameters on ζt may be revealed. Ignoring the small contribution of C in equation 2.17
the thermal gradient may be estimated as
∂Tg
∂x
≈ ∆T
λ
=
Ts − Tg
`
(2.49)
where ∆T = T1 − T2. Substituting this into the expression for ζt and integrating along the
linear profile gives the simplified expression
ζt ≈ T0
β
(
∆T 2
T1T2
)
`
λ
(2.50)
Minimising thermal losses clearly requires a long thermal front and small value of ` which
may be achieved by increasing the Stanton number or reducing the particle size. As will be
seen, this conflicts with minimising pressure losses and a compromise must be sought.
Factors affecting thermal losses
Various factors affect the length of the thermal front, including the time which the packed
bed charges and discharges for during cyclic operation (discussed in more detail in chap-
ter 3) and the dependence of specific heat capacity cs on temperature. The heat capacity
per unit volume tends to increase with temperature as illustrated for various materials in
figure 2.10. Since the thermal front speed varies inversely with cs (equation 2.3), cold
regions of the front travel more quickly than hot regions. As a result, the thermal fronts
change shape. For instance, during charge thermal fronts in a hot reservoir will become
less steep, whereas those cold reservoirs become steeper, as shown in figure 2.11(a). As the
60
 0
 1000
 2000
 3000
 4000
 5000
 6000
 7000
 0  200  400  600  800  1000
H
ea
t 
ca
p
ac
it
y
 p
er
 u
n
it
 v
o
lu
m
e,
 c
s 
(k
J 
/ 
m
3
)
Temperature (K)
discontinuities 
indicate a phase 
change
Al2O3
Fe2O3
Fe3O4
SiO2
TiO2
Figure 2.10: Heat capacity per unit volume for various materials [97]
cold thermal front travels along the reservoir, its length λ decreases, and thermal losses thus
increase. Eventually, the losses reach a constant value and White et al. [97] explains that
this is the result of the wave catch-up effect being balanced by the dissipative effects caused
by irreversible heat transfer, as shown in figure 2.11(a). The sudden change in properties
across the thermal front, as well as the losses becoming independent of the detailed dissi-
pative processes (i.e. `) means that the front can be thought of as a thermal “shock-wave”.
The shape of the thermal front also remains constant.
Realistic operation of the thermal stores will probably involve regular charging and dis-
charging phases. During discharge, any effects of wave steepening in the cold store will
be reversed and thermal shock-waves are unlikely to form during such cyclic operation.
However, as figure 2.11(b) illustrates, thermal fronts still steepen during charge compared
to the case where cs is constant. The average thermal gradients that occur for constant
and variable cs are actually approximately equal. However, for variable cs the fluctuation
of these gradients between charge and discharge generates larger losses because ζt varies
61
-150
-100
-50
 0
 50
 0  1  2  3  4  5
S
o
li
d
 t
em
p
er
at
u
re
, 
T
s 
(o
C
)
Distance along reservoir, x (m)
1% charge 30% 60%
(a)
-150
-100
-50
 0
 50
 0  1  2  3  4  5
S
o
li
d
 t
em
p
er
at
u
re
, 
T
s 
(o
C
)
Distance along reservoir, x (m)
1% tc 40% 85%
Variable cs
Constant cs
(b)
 0
 0.05
 0.1
 0.15
 0.1  0.3  0.5  0.7  0.9
A
v
er
ag
e 
th
er
m
al
 l
o
ss
 c
o
ef
fi
ci
en
t,
 ζ t 
(%
)
Normalised charge period, tc / tN
Fe3O4 constant cs
Fe3O4 variable cs
Fe2O3 constant cs
Fe2O3 variable cs
(c)
 0
 100
 200
 300
 400
 500
 0  0.25  0.5  0.75  1
G
as
 t
em
p
er
at
u
re
, 
T
g
 (
°
C
)
Normalised distance
tc / tN = 0.30
tc / tN = 0.75
(d)
Figure 2.11: Effect of variable cs on thermal front formation in cold stores (a) Temperature profiles
in a cold store during the charging phase. Variable cs causes wave catch-up and shock wave for-
mation. (b) Temperature profiles in a cold store under cyclic operation. Profiles are shown at 1%,
40% and 85% through the charge phase. (c) Thermal losses for 20 mm diameter particles in a cold
reservoir undergoing cyclic operation. (d) Effect of cycle period tc/tN on thermal front shape in a
hot store.
62
inversely with the front length λ. For instance, the thermal loss is given by
ζt ≈ T0
βtchg
(
∆T 2
T1T2
)
`
∫ tchg
0
1
λ
dt (2.51)
If the average front length λ¯ is substituted into this rather than the fluctuating front length
(which is a function of time λ = λ(t)) then it is the case that∫ tchg
0
1
λ(t)
dt >
tchg
λ¯
(2.52)
This result is demonstrated in Appendix A. As a result, the fluctuation of the thermal front
shape due to variable cs leads to slightly greater thermal losses than when cs is constant.
Furthermore, larger fractional changes in cs (given by ∆cs/c¯s) will lead to greater losses
[97]. This is supported by figure 2.11(c) which shows thermal losses in a cold reservoir
during cyclic operation for two materials (hematite and magnetite) which have different
fractional changes in cs. It is notable that ζt for magnetite and hematite with constant cs
are almost identical since the average heat capacity for these materials is similar. However,
when cs was allowed to vary losses were greater for hematite as this material has a higher
fractional change in cs.
As a result, thermal losses can be reduced by selecting materials and operating temperatures
that minimise the fractional change in heat capacity ∆cs/c¯s where c¯s is the average heat
capacity over the temperature range of interest [97].
Figure 2.11(c) shows that thermal losses increase as the length of the cycle period tc/tN
is increased. This is due to the steeper gradients that are formed during longer cycles, as
shown in figure 2.11(d). During cyclic operation, the thermal fronts are the same from one
cycle to another, and the loss in available energy is the same for every cycle. In order for the
reservoir to return to its initial condition at the end of each charge-discharge cycle, the total
flux of entropy out of the reservoir must equal the entropy generation within the reservoir.
To achieve this, ‘information’ must be propagated from one end of the packed bed to the
other in the time of one charge cycle. For short cycle periods, the thermal front will only
travel a small fraction of the reservoir length (roughly given by Vf tc/L). Therefore, the
thermal fronts are longer and less steep so that entropy can be ‘flushed’ out of the packed
bed, resulting in lower thermal losses.
63
2.5.2 Pressure losses
The momentum flux is approximately constant along the reservoir, so the pressure drop
is computed in a manner analogous to that for fully-developed pipe flow. The pressure
gradient is given by [60],
∂p
∂x
= −Sv(1− ε)G
2Cf
2ε3ρg
(2.53)
Where Cf is the friction coefficient which depends on the Reynolds number and is given
here by the Carman correlation [122]
Cf =
10
Rem
+
8
10Re
1
10
m
(2.54)
Rem is the “modified” Reynolds number given by Rem = ρgus/(1− ε)Svµg. The instanta-
neous entropy generation rate due to the pressure drop is
S˙irr = −m˙R ln
(
Pout
Pin
)
(2.55)
Equation 2.53 is integrated along the length of the reservoir to give the overall pressure
drop which is then converted to an entropy increase and hence a loss coefficient,
ζp =
To
m˙cpβtchg
∫ tchg
0
S˙irrdt =
To
βtchg
γ − 1
γ
∫ tchg
0
ln
(
Pout
Pin
)
dt (2.56)
Pressure losses are typically larger in the cold reservoir compared to hot reservoirs since
fractional pressure changes are more significant, and gas densities are relatively low. Min-
imisation of pressure losses requires a reduction in frictional effects which is achieved by
increasing particle size and reducing the length of the reservoir, which is directly at odds
with minimising thermal losses. This conflict motivated the concept of segmented stores,
whereby gas is diverted around areas of the packed bed where no heat exchange is taking
place in order to reduce frictional effects. Segmented stores are discussed in more detail in
section 3.3.
64
2.5.3 Conductive losses
Entropy generation due to axial conduction
The dissipative nature of conduction causes a loss in availability. This loss is generated
during the charging and discharging phases but is also significant during storage periods
since spreading of the thermal front results in a loss of available energy. The instantaneous
entropy generation due to axial conduction is found by considering the first and second
laws applied to a control volume, as in figure 2.12.
Ts Ts +
∂Ts
∂x
δx
Q˙ Q˙+ ∂Q˙
∂x
δx
δx
Figure 2.12: Control volume for heat conduction along a packed bed
Applying the first law of thermodynamics to the elemental control volume yields
csδMs
∂Ts
∂t
= −∂Q˙
∂x
δx (2.57)
where cs and δMs are the specific heat capacity and mass of solid within the element. The
rate of entropy generation due to irreversibility is found by applying the second law to give
∆S˙irr =
∂
∂t
(δSCV )−
∑( Q˙
Ts
)
=
∂
∂t
(δSCV ) +
∂
∂x
(
Q˙
Ts
)
δx
= csδMs
1
Ts
∂Ts
∂t
+
1
Ts
∂Q˙
∂x
δx− Q˙
T 2s
∂Ts
∂x
δx (2.58)
where δSCV is the entropy of elemental mass δMs. Substituting in equation 2.57 eliminates
65
the first two terms on the right-hand-side. Fourier’s law of conduction is
Q˙ = −kxeffA
∂Ts
∂x
(2.59)
The instantaneous entropy generation rate is then given by
S˙irr =
∫ L
0
kxeffA
(
1
Ts
∂Ts
∂x
)2
dx (2.60)
where kxeff is the effective conductivity in the axial direction, the value of which will be dis-
cussed in section 2.5.3. The loss coefficient due to conduction ζc is defined in a similar way
to equation 2.48. The expression for ζc can also be simplified so that it is comparable to the
thermal loss ζt by assuming that the gas and solid temperature gradients are approximately
equal and using a simplified version of equation 2.17 to give
∂Ts
∂x
≈ ∂Tg
∂x
≈ Ts − Tg
`
(2.61)
This simplification assumes that conduction will not significantly affect thermal front shapes.
Substituting this expression into equation 2.60 produces
S˙irr =
∫ L
0
kxeffA
(
Ts − Tg
`Ts
)2
dx (2.62)
By comparing this expression to the instantaneous entropy generation caused by tempera-
ture differences in equation 2.47, and by assuming T 2s ≈ TsTg in the denominator, the ratio
of the two entropy generation rates simplifies to
ζc
ζt
=
kxeff
cpG`
(2.63)
This equation is most appropriate for low values of kxeff and figure 2.17(d) on p. 74 illustrates
its accuracy compared to the full expression above.
66
Evaluation of the effective axial conductivity kxeff
The effective axial conductivity is subject to much uncertainty and a large number of papers
have been published on the subject. There are a several mechanisms that contribute to the
effective conductivity and Yagi and Kunii [123] divided these into two groups:
Heat transfer processes that are independent of the fluid flow:
1. Thermal conduction through the bulk of the solid particles
2. Thermal conduction through the contact surfaces of the particles
3. Radiative heat transfer between the faces of two particles
4. Radiative heat transfer between neighbouring voids
Heat transfer processes that are dependent on the fluid flow:
1. Thermal conduction through the fluid film near the contact surface of the particles
2. Convection between the solid and fluid
3. Heat transfer by lateral mixing of the fluid (turbulent transport in eddies)
Many empirical and numerical correlations have been developed, and several reviews ex-
ist [4, 124]. These correlations are typically empirical and require estimation of several
unknown parameters. For instance, several models require details of the number of points
of contact (the coordination number). This can be estimated for idealised packings with
uniformly sized spherical particles, but it is unlikely to be reliable for more realistic packed
beds. Various other correlations, which incorporate factors such as surface roughness and
contact area, have been developed more recently (see for example, [4]). However, these
models contain additional parameters that are difficult to obtain experimentally. Further-
more, experimental results from which the correlations are developed are often very specific
to certain types and shapes of materials, and the results may not be generally applicable.
Balakrishnan and Pei [124] suggests that the difficulty of obtaining a general correlation
is due to the interaction of the major modes of heat transfer; gas-to-particle heat transfer
convection and effective conduction through the bed. This may be illustrated by combining
equations 2.18 and 2.17 (with β = C = 0) which gives an expression for the progress of
67
the thermal front:
∂Ts
∂t
+ Vf
∂Tg
∂x
=
α`
τ
∂2Ts
∂x2
(2.64)
where Vf = `/τ is the thermal wave speed, as defined in equation 2.3. Substituting Tg =
Ts+(Tg−Ts) into this equation, and then replacing (Tg−Ts) using equation 2.17 produces
∂Ts
∂t
+ Vf
∂Ts
∂x
=
`
τ
∂
∂x
(
`
∂Tg
∂x
)
+ α
∂2Ts
∂x2
(2.65)
The gas-solid equation above shows wave propagation at speed Vf combined with a dis-
sipative process [97] (on the right-hand-side). Therefore, both axial conduction and gas-
particle heat transfer manifest themselves as diffusion of the thermal front. When α = 0
the effective diffusivity is
D = `
2
τ
=
Vf
St(1− ε)Sv (2.66)
This equation is consistent with the analysis in section 2.5.1 and equation 2.50 and in
particular it suggests that thermal losses associated with irreversible heat transfer can be
minimised by reducing `. In the case of finite thermal conductivity kxeff there is an additional
dissipative term due to axial conduction. The relative importance of the two sources of
dissipation is given by the ratio ατ/`2 = kxeff/(cpG`), which is the ratio between conductive
and thermal losses as shown above. Thus equation 2.65 suggests that there are two main
sources of dissipation due to the two different heat transfer processes.
The combination of dissipative processes makes designing experiments that quantify the
individual effects a challenging task. In the following section, several correlations are
compared with the aim of establishing the correct order of magnitude for the effective
conductivity.
Thermal conductivities of magnetite (Fe3O4) and hematite (Fe2O3)
Lis and Kellard [125] report that experimental data on magnetite conductivity range be-
tween 0.5 W/mK and 8 W/mK for experiments carried out at the same temperature. They
found the following correlation for thin oxide films (∼0.1 mm thick) in terms of porosity
68
P (expressed as a percentage) in the temperature range 35–80oC
kmag = 1.24± 0.18− 9.15× 10−3P (2.67)
For a single crystal of magnetite, Slack [126] measured thermal conductivities in the range
of 5–7 W/mK at 300K, and noted that the conductivity was independent of temperature in
the range 119–300K. In addition, Slack noted that the orientation of the crystal affected the
conductivity.
Mølgaard [127] carried out experiments on a range of magnetite and hematite samples
across a range of temperatures. Regression analysis gave the following correlations for
thermal conductivity in terms of temperature T :
kmag = 4.23− 1.37× 10−3T (2.68)
khem = 8.39− 6.63× 10−3T (2.69)
These different measurements indicate that there is considerable uncertainty for the con-
ductivity of the solid material, even without considering other factors that contribute to the
effective conductivity.
Correlations for effective conductivities
Many well established models for effective conductivity were developed for packed beds
with stagnant fluids, presumably since it is more straightforward to obtain correlations
without having to consider turbulent transport of heat. These results are applicable for the
storage phases (i.e. between charge and discharge) which are discussed in more detail in
section 2.5.6.
Upper and lower estimates of the effective thermal conductivity can be found by placing
the solid and fluid in series or in parallel, as in [128]. The series arrangement has succes-
sive layers of solid and gas which are perpendicular to the flow direction, as illustrated in
69
a) minimum conductivity b) maximum conductivity
so
lid ga
s
so
lid ga
s
so
lid
flow direction
solid
gas
solid
gas
solid
flow direction
Figure 2.13: Packing arrangements giving maximum and minimum axial heat conduction
figure 2.13 and gives the minimum conductivity,
kmineff
kf
=
1
ε+ (1− ε)kf/ks (2.70)
The parallel arrangement has successive layers of solid and gas parallel to the flow direction
and gives an upper bound on the thermal conductivity:
kmaxeff
kf
= ε+ (1− ε)ks
kf
(2.71)
where kf/s is the fluid/solid thermal conductivity and ε is the void fraction. Using kAr =
17× 10−3 W/mK, kmag = 4.23 W/mK (from Mølgaard’s correlation above), and ε = 0.40
this suggests that the effective conductivity will lie in the range of 0.042 W/mK < keff <
2.54 W/mK.
A number of more complex models for effective conductivities are reviewed in [4]. These
typically also consider factors such as the contact number – i.e., the number of other parti-
cles in contact with the particle in question. Kunii and Smith [129] developed correlations
with and without radiation including parameters that describe the packing density and the
contact number, but not the contact area. On the basis of data from Prasad et al. [130]
Ref. [4] shows that the correlation without radiation fits experimental data well provided
70
Figure 2.14: Comparison of experiments and theoretical correlations for the effective conductivity.
Figure taken from van Antwerpen et al. [4]. Experimental data is from Prasad et al. [130]. The
correlation developed by Kunii and Smith [129] is shown, as are the results using the minimum and
maximum effective conductivity of equations 2.70 and 2.71 which were developed by Deissler and
Boegli [128].
that ks/kf < 103, as shown in figure 2.14. Above this value, additional factors such as the
contact area become important. (For the particular case given above with argon and mag-
netite ks/kf ∼ 250). These experiments were carried out over quite a limited temperature
range of 50 K.
Impact of radiation on effective conductivity
Zanganeh et al. [2] argues that radiation makes an important contribution to the effective
conductivity, especially at high temperatures. Using Kunii and Smith’s correlation, Zan-
ganeh suggests that at around 900 K, radiation could contribute to 70% of the value of keff,
giving keff a value of ∼2.0 W/mK. (For this analysis, Zanganeh used concrete as the solid
medium, and air as the fluid). However, the inclusion of radiation in Ref. [2] only has a
slight impact on the thermal front shapes.
A simplified model of radiation can be developed to estimate the dependence of the effec-
tive conductivity on various parameters. Particles are assumed to be arranged in layers,
71
flow direction
s = λ dp
Figure 2.15: Idealised depiction of successive layers of spherical particles in a packed bed
and radiation occurs between the surfaces of one layer and the next, as illustrated in fig-
ure 2.15. The surfaces are uneven due to the curvature of the particles, but it is assumed
that the surfaces are flat. The centres of successive layers are a distance s = λdp apart,
where λ is approximately unity. (For example, for close packed spheres of a uniform size,
λ =
√
2/3). The heat transfer is modelled as a resistor network as shown in figure 2.16.
With a view factor of F = 1 the rate of heat transfer Q˙ is
Q˙ =
εσA (T 41 − T 42 )
2− ε (2.72)
Assuming that
T2 ≈ T1 + λdp∂T1
∂x
(2.73)
and comparing the resulting heat transfer rate to Fourier’s law of conduction leads to an
effective conductivity due to radiation of
keff ≈ 4σλdpT 31 (2.74)
72
1 ε
εA
1 ε
εA
1
FA
T1
4 T2
4
σσ
Figure 2.16: Electrical analogy for radiative heat transfer between successive layers of particles
where it has been assumed that the emissivity is one. If ε < 1 then keff is reduced slightly,
but the correction can be absorbed into λ which is treated as an empirical factor. Equa-
tion 2.74 shows that larger particles increase the effective conductivity. keff also depends
strongly on the temperature.
Figure 2.17(a) illustrates the correlations described using magnetite and argon. Radiation
has a significant effect on the value of the effective conductivity, especially for high temper-
atures and large particles and this is consistent with [2,129]. Particle size is unlikely to have
much of an effect on the effective conductivity in cold stores, where radiative effects are
small. Hot stores may also see reasonably low values of keff despite the high temperatures,
because particle sizes will be small due to the low pressure losses. The simplified model
of equation 2.74 is shown for dp = 10 mm and under-predicts keff compared to Kunii’s
expression. However, equation 2.74 does show the correct trends.
Impact of gas flow on effective conductivity
During charging and discharging phases, the turbulent transport of heat can have a signifi-
cant effect on the effective conductivity, although quantifying this is difficult as explained
above. Ref. [124] describes how the effective conductivity due to mixing of fluid could be
represented as
keff,t
kf
= cRePr (2.75)
where c is an empirical factor depending on the packing geometry, Re is the Reynolds num-
ber and Pr is the Prandtl number. Yagi and Kunii [123] argued that the physical processes
that influenced keff for stagnant fluid and flowing fluid were independent of one another,
73
 0
 0.5
 1
 1.5
 2
 2.5
 3
 3.5
-100  0  100  200  300  400  500
k
ef
f 
(W
/m
K
)
Temperature (
o
C)
Series packing
Parallel packing
Kunii (no radiation)
Kunii (radiation) dp = 2 mm
Kunii (radiation) dp = 10 mm
Eq. (2.74)
(a)
 0
 1
 2
 3
 4
 0  200  400  600  800
k
ef
f 
(W
/m
K
)
Reynolds number, Re
dp =   2 mm, T = -150
o
C
dp = 10 mm, T = -150
o
C
dp =   2 mm, T =    25
o
C
dp = 10 mm, T =    25
o
C
dp =   2 mm, T =  500
o
C
dp = 10 mm, T =  500
o
C
(b)
 0
 100
 200
 300
 400
 500
 0  1  2  3  4
T
em
p
er
at
u
re
 (
o
C
)
Distance along reservoir (m)
keff = 0 W/mK
keff = 2 W/mK
(c)
 0
 0.25
 0.5
 0.75
 1
 1.25
 1.5
 0  5  10  15  20  25
L
o
ss
 c
o
ef
fi
ci
en
t,
 ζ c
 /
 ζ t
Particle diameter, dp (mm)
keff = 0.5 W/mK
              (approx)
keff = 1.0 W/mK
              (approx)
keff = 2.0 W/mK
              (approx)
(d)
Figure 2.17: Correlations for axial effective conductivity kxeff and its impact on thermal front shape
and availability losses in hot stores. (a) Correlations for effective conductivity keff in a stagnant
fluid. Calculations for equation 2.74 are undertaken with dp = 10 mm and λ = 0.8. (b) Correlations
for effective conductivity keff in a flowing fluid. (c) Effect of conductivity on thermal front shapes
in a hot reservoir with dp = 5mm. (d) Conductive loss coefficient normalised by thermal loss
coefficient for a hot reservoir (nominal design). Approximate values are given by equation 2.63.
74
and that a general correlation could therefore be given by
keff
kf
=
koeff
kf
+ cRePr (2.76)
where koeff is the effective conductivity with stagnant fluid as given by Kunii and Smith [129].
Clearly, the presence of fluid flow increases the effective conductivity and this is shown in
figure 2.17(b). For typical thermal reservoir designs, the Reynolds number varies between
10–200 for hot stores, and 50–500 for cold stores, depending on particle diameter and
geometry. While these correlations are somewhat uncertain, they give approximate val-
ues that are useful for understanding the significance that axial conductivity may have in
packed beds. On the basis of the above references, values of keff in the range of 0.5–1.5
W/mK seem realistic. The effective conductivity changes along the length of the bed since
it depends on temperature. However, for realistic packed bed designs (where hot stores
have small particles less than 2 mm and cold stores operate at low temperatures), these
variations are small. Furthermore, the effective conductivities are sufficiently low to have
an almost negligible impact on the thermal front shape (figure 2.17(c)). Given the uncer-
tainty, it seems satisfactory to use an average value of keff rather than recalculating it at
every time-step at each axial node along the reservoir.
Impact of conduction on losses
Figure 2.17(c) shows that axial conductivity does not have a significant effect on thermal
front shape in a hot reservoir even when quite a large value of keff is chosen. Loss co-
efficients are shown for a range of particle diameters in figure 2.17(d). The conductive
losses are shown as a fraction of thermal losses ζt and numerical results are compared to
the simplified expression in equation 2.63. The figure shows good agreement between the
numerical and approximate solutions, particularly for large dp and low keff i.e. the cases
when Tg ≈ Ts and conductivity does not significantly effect thermal profiles.
As would be expected from equation 2.62, conductive losses increase almost linearly with
effective conductivity. In addition, figure 2.17(d) illustrates that ζc drops rapidly with
particle diameter, since larger particles form thermal fronts with less steep gradients (i.e.
∂Ts/∂x is reduced). Figure 2.17(d) shows that conductive losses have a similar order of
75
magnitude to thermal losses, especially for small particles when thermal losses are lowest.
Consequently, axial conduction is an important process to include in modelling of packed
beds for thermal energy storage. Thermal losses were also seen to decrease by a small
amount as the effective conductivity was increased, as conduction acts to spread out the
thermal fronts. However, these effects were hardly significant, as conduction only has a
small impact on the thermal gradient.
While conductive losses are not insignificant during charging and discharging phases, they
will probably have the greatest impact during storage periods, although the extent of the
losses will depend on the duration of storage and geometry of the system.
2.5.4 Heat leakage
Ts
T0
WLδ
Q Lδ
Figure 2.18: Reversible heat engine
The rate of available energy loss due to heat
leakage is given by multiplying the rate of heat
loss by the efficiency with which it could have
been converted to work via a reversible heat en-
gine, see figure 2.18. Thus,
W˙L =
∫ (
1− T0
Ts
)
dQ˙L (2.77)
Heat leakage may occur from the side walls
and from the top or bottom of the store. For
side wall leakage dQ˙L,w = piDUw(Ts − T0)dx,
whilst heat transfer from an end wall is Q˙L,t = piD
2
4
Ut(Ts − T0) where U is an overall heat
transfer coefficient. (Note that this expression assumes flat end walls. It is likely that the
end walls will in fact be a domed shape, meaning that the heat transfer area would be greater
than piD2/4). As with other loss coefficients, these lost work terms are multiplied by T0 and
integrated over time, noting that leakage losses occur during charge, discharge and storage
phases, and normalised by the inlet available energy input during charge. Therefore,
ζL,w =
piD
m˙cptchgβ
∫ tchg
0
∫ L
0
Uw
(Ts − T0)2
Tg
dxdt (2.78)
76
ζL,t =
piD2
4m˙cptchgβ
∫ tchg
0
Ut
(Ts − T0)2
Tg
dt (2.79)
To a first approximation, the overall heat transfer process can be estimated by considering
a simple lumped heat capacity model for the packing and calculating the fractional energy
loss per second, giving
dEs
dt
= −ULAL(Ts − T0) = −ULAL
Msc¯s
Es = −Es
τL
(2.80)
where Es is the energy stored in the solid, UL is the overall heat transfer coefficient, AL
is the surface area at the packing-insulation interface, and τL = Msc¯s/ULAL is the heat
leakage time constant. For instance, setting τL to 200 days corresponds to an internal energy
loss rate of approximately 0.5% per day. (The fractional available energy loss will be
somewhat higher, depending on Ts). Nominal designs for hot and cold stores are developed
in chapter 3 and quantified in table 3.2. For the hot nominal design, with L/D = 1,
AL = 8piR
2 (including hemispherical ends), and setting τ = 200 days, then the overall
heat transfer coefficient is approximately
UL =
ρsc¯sR
4τL
≈ 0.13Wm−2K−1 (2.81)
However, the lumped capacity model is only valid if the temperature distributions are ra-
dially uniform. White et al. [80] shows that, as would be expected, low Biot numbers are
required in order to obtain uniform temperature profiles1. For an infinite cylinder the Biot
number is given by Bi = ULR/keff where keff is the effective conductivity of packing. The
requirement of a low Biot number thus places an upper limit on the overall heat transfer
coefficient UL: for a nominal design with Bi = 0.01 then UL ≈ 0.11 W m−2K−1.
The overall heat transfer coefficient UL can also be calculated by considering the various
thermal resistances between the solid particles and the atmosphere as described below.
1Radial temperature variations could be permitted, although it is not possible to asses the full impact that
this would have on radial performance without undertaking two-dimensional calculations.
77
Thermal resistance of casing
Each material in the casing will contribute to the thermal resistance depending on its thick-
ness and thermal conductivity. The insulation thickness tI has the greatest influence, and
should be set so as to reduce the overall heat transfer coefficient to a sufficiently low num-
ber, such as that described above. In addition, the insulation thickness may have to be
increased further so that the steel temperature does not rise excessively (to prevent creep).
The thermal conductivities of insulating materials vary by a factor of over 10, but a typical
value for glass foam (a cheap and widely used insulator) is ki ≈ 0.05 Wm−1K−1 [131].
The steel thickness ts can be estimated from substituting the longitudinal σl and hoop σh
stresses of a thin walled cylinder into the von Mises yield criterion, which is given by
(σh − σl)2 + σ2h + σ2l = 2Y 2 (2.82)
where Y is the yield stress of steel (Y ≈ 200 MPa). The steel thickness is therefore given
by,
ts =
Σ
√
3∆PrI
2Y
= σ¯rI (2.83)
where Σ is a safety factor, rI is the outer radius of the insulation, and ∆P = Pin+ρsgL−P0
is the pressure difference across the casing walls (including the hydrostatic pressure due to
the packing material’s weight). For designs that are typical for the present application,
the steel thickness is roughly 10–20 mm for a hot store. Combining the high thermal
conductivity the steel casing ks ≈ 25 Wm−1K−1 [131] with the low thickness means that
steel only makes a small contribution to the total thermal resistance.
Wall convective heat transfer coefficient
There are numerous correlations for the convective heat transfer coefficient at the packing-
insulation interface hw and a summary is presented by Ref. [132]. Evaluation of hw is not
straightforward due to radial variations in temperature, and the increased flow rate that may
occur at the walls as a result of reduced particle packing density. Yagi and Wakao [133]
developed a correlation which assumes that the temperature does not vary in the radial
direction. Li and Finlayson [134] extended the analysis to consider radial temperature
78
Figure 2.19: Comparison of experimental results and the correlation for the wall heat transfer
coefficient developed by Li and Finlayson [134] in equation 2.84. Figure is taken from [134].
variations and an abrupt temperature change at the wall. This study also incorporated data
from a range of sources (including Yagi) and developed the following correlation which fits
experimental data well, as shown in figure 2.19.
Nuw = 0.17Re
0.79
p ; 20 ≤ Rep ≤ 7600 (2.84)
where Nuw is the Nusselt number at the wall and Rep is the Reynolds number based on
the particle diameter. Li and Finlayson found that this correlation fitted the data better
than another widely used correlation developed by Beek [135]. Note that this correlation
is not suitable for storage periods, since, as Ref. [136] points out, the correlation implies
that Nuw = 0 when Rep = 0. Equation 2.84 yields a wall heat transfer coefficient in the
region of hw ≈ 14 W m−2K−1 for a hot store, and hw ≈ 25 W m−2K−1 for a cold store,
for nominal designs with average Reynolds numbers. The large value of hw suggests that
it may be reasonable to neglect these terms.
79
External convective heat transfer coefficient
Free convection on the outer surface of the reservoir can lead to the formation of a vertical
plume that grows with increasing height. Again, various correlations exist for the external
convective heat transfer coefficient hx and are discussed in [137] and [138]. Ref. [138]
compares various correlations for flat plates and isothermal cylinders, and flat plate cor-
relations provide a good approximation for aspect ratios close to or less than 1 (which is
likely in thermal stores). A common flat plate correlation was developed by Churchill and
Chu [139] and is given by
Nux = 0.68 +
0.67(GrPr)0.25{
1 +
(
0.492
Pr
) 9
16
} 4
9
(2.85)
where Gr is the Grashoff number, which for ideal gases is given by
Gr =
gβ(Tx − T0)L3
ν2
≈ g(Tx − T0)L
3
T¯ ν2
(2.86)
where T¯ = 1/2(Tx + T0). As the stores charge and discharge the surface temperature
changes with position and time and consequently a steady-state plume is not formed. One
approximation is to take L = R which corresponds to a half charged reservoir. For lower
states of charge, the effective height over which buoyancy driven flow can develop is re-
duced and leads to lower Grashoff and Nusselt numbers. However, convective currents are
then likely to be dominated by heat transfer at the top surface of the vessel. The surface
temperature is required to calculate Gr so that knowledge of the thermal resistance of the
casing is required before Tx can be calculated iteratively. A constraint is placed on the
maximum surface temperature and the insulation thickness is recalculated if this value is
exceeded. For a maximum value of |Tx − T0| = 50 K, the external heat transfer coefficient
is approximately hx ≈ 2.75 W m−2K−1 for a nominal hot reservoir.
Overall heat transfer coefficient – illustrative example
Each of the above processes contributes to the overall thermal resistance. The relative
magnitude of these processes may be compared by considering an illustrative example
80
with a hot packed bed radius of R = 2.5 m and insulation thickness of tI = 0.5 m. The
overall heat transfer coefficient for the side walls is given by
1
Uw
=
1
hw
+
R ln(1 + tI/R)
kI
+
R ln(1 + σ¯)
ks
+
R + tI(1 + σ¯)
Rhx
1
Uw
= 0.07 + 9.12 + 0.001 + 0.44
Internal Insulation Steel External
convection convection
Each term has been estimated by using the approximate values developed above. The
thermal resistance due to the insulation clearly dominates, and the other factors may be
neglected.
The analysis on page 77 suggested a value for the overall heat transfer coefficient of
UL = 0.11 W m−2K−1. The insulation thickness may be calculated by only considering
the insulation term in the above equation, yielding
tI = R
[
exp
(
kI
RUw
)
− 1
]
≈ 0.5 m (2.87)
(In fact, this value slightly over-estimates the required insulation thickness.) The surface
temperature are then calculated by the inclusion of the hx term and, as discussed above, if
this exceeds the desired surface temperature, the insulation thickness is increased accord-
ingly (although the surface temperature is rarely found to exceed 50°C. In this instance it
is 45°C for the hot store).
Strictly speaking, for accurate evaluation of heat leakage and the associated availability
loss, the overall heat transfer coefficient should be re-calculated at every x position and
time-step in order to take into account variations in hw and hx. However, since the vast
majority of thermal resistance is provided by the insulation, this is unnecessary and has a
negligible effect on the results. It is more efficient and straightforward to assume that Uw
remains at a constant value determined by either the heat leakage time constant τ , the Biot
number constraints, or the steel surface temperature requirement; whichever provides the
lowest value of Uw.
81
2.5.5 Exit losses
Exit losses ζx occur as the thermal fronts approach each end of the reservoir, as shown by
the 85% charged curve in figure 2.11(b). When the ‘nose’ of the thermal front emerges
from the reservoir at a temperature greater (or less) than ambient temperature a loss in
available energy is incurred. These exit losses serve to ‘flush’ entropy out of the reservoir.
Since irreversible processes occur internally in the reservoir, for the reservoir to reach a
cyclic state then the entropy of the system must be reduced and thus exit losses cannot be
avoided. Consequently, exit losses are closely related to thermal losses ζt.
The loss coefficient is defined as the excess availability energy leaving the store above that
which occurs if the gas exited at the design temperature and pressure. For instance, during
the charge of a hot reservoir the gas initially leaves the reservoir at temperature T2 and an
exit loss occurs when the exit temperature Tx > T2. Thus, the loss coefficient is
ζx =
1
Bin
∫ tchg
0
{
B˙(Tx, p1)− B˙(T2, p1)
}
dt (2.88)
Note, that the availability fluxes are calculated using the reservoir inlet pressure, because
pressure losses have already been accounted for in the pressure loss coefficient.
In principle, the available energy in the exit flow could be recovered and utilised. There are
many uses for excess industrial heat [140, 141], and a commonly discussed application is
domestic heating. As such, it is unclear whether exit losses should be included in definitions
of the round-trip efficiency. Whilst exit losses are a loss of availability from the reservoir,
it is conceivable that it should not be counted as a loss if that energy is used elsewhere.
Furthermore, for reservoirs that are part of a thermodynamic cycle such as PTES, this
energy could be recovered in other parts of the cycle.
2.5.6 Storage losses
Storage losses ζs are the availability losses that are incurred during storage periods and
are comparable to self-discharge of batteries. Axial conduction causes thermal gradients
to decay. In the case where there is no heat leakage (perfectly insulated store) and the
total energy within the store remains constant, the degradation of the thermal gradients
82
results in a loss of availability. Furthermore, heat leakage reduces the energy content of
the reservoir and contributes to the storage loss. While insulation thickness and effective
conductivity are important parameters, it is obvious that storage losses depend heavily on
the duration of storage. Innovative reservoir design (such as segmenting the stores) can
reduce these effects significantly, but cannot eradicate them completely. Thermal stores are
consequently more appropriate for storage durations of hours to weeks rather than months.
The storage loss is calculated as the difference in availability within the reservoir between
the beginning and end of storage
ζs =
ρsA(1− ε)
m˙cpβtchg
∫ L
0
cs
(
Tb − Te − T0 ln
(
Tb
Te
))
dx (2.89)
≈ 1
Vfβtchg
∫ L
0
Tb − Te − T0 ln
(
Tb
Te
)
dx
where Tb and Te are the temperatures in the reservoir at the beginning and end of storage.
The maximum storage loss occurs after an infinite time when the thermal front has com-
pletely equilibrated. For an adiabatic store, this is achieved once the storage medium has
reached a uniform temperature given by its average value
T¯ =
1
L
∫ L
0
Ts(x)dx (2.90)
2.5.7 Radiative losses
Radiative losses are a special case of availability loss, and occur when a packed bed has
been segmented. Segmentation is a design feature where the packed bed may be divided
into separate layers. Segmentation is discussed comprehensively in section 3.3 and a
schematic is given in figure 3.8. Each segment or layer is separated by a small gap. Dur-
ing charge and discharge, gas flows out of one segment, through the gap and into the next
segment, such that the gap may be assumed not to be present. On the other hand, there is
no gas flow between segments that are inactive or during storage phases. Heat transfer still
occurs between segments by conduction through the gas (which is small), convection and
radiation. Convective processes are neglected by noting that the packed-beds are orientated
such that hot regions are at the top in order to reduce buoyant forces.
83
Radiation between two successive layers is modelled with an electrical analogy similar to
that in figure 2.16. For segments of area A = pir2 with a view factor of F = 1 between
them, the rate of heat transfer is
Q˙ =
εσA
2− ε
(
T 41 − T 42
)
(2.91)
where ε is the emissivity of the particles in the segment which may be assumed to be close
to 1 for pebbles due to their low reflectivity. The irreversible entropy generation is given
by
S˙irr =
(
1
T2
− 1
T2
)
Q˙ =
εσA
2− ε
(T1 − T2)
T1T2
(
T 41 − T 42
)
(2.92)
Simplifications can be made to this expression if T1 ≈ T2 but this is not necessarily the
case between segments. As is discussed further in section 3.3.2, heat transfer between
layers reduces the benefits that segmentation is thought to provide.
2.6 Summary
In this chapter, the underlying theory of packed-bed thermal reservoirs has been described.
The governing equations have been derived, appropriate methods of solution have been dis-
cussed, and a number of processes that reduce the available energy have been introduced.
Whilst much of the behaviour is complex, employing a number of simplifications develops
an understanding of these processes. For instance, thermal losses were seen to depend on
the particle diameter, the duration of charging in cyclic operation and the variation of the
specific heat capacity with temperature. cs was also seen to have a significant impact on
the shape of the thermal fronts. The value of the axial effective conductivity is a source of
uncertainty, but a consideration of various correlations suggests that for the current appli-
cation kxeff is sufficiently small that it does not significantly affect thermal front shapes, and
that conductive losses only have a comparable magnitude to thermal losses at small parti-
cle sizes. Conduction is therefore not negligible and it is especially important in storage
phases.
Other simplifications include assuming that effective conductivities and overall heat trans-
fer coefficients remain constant throughout operation of the bed. These adjustments were
84
justified based on the realistic magnitudes of each process. Some confidential experimental
results were made available by Isentropic Ltd., from which it has been possible to verify
that the reported figures have realistic magnitudes. Moreover, a survey of the literature
demonstrated that these kind of theoretical analyses were comparable to experimental re-
sults. Further experimental work is clearly required in order to justify these claims. Fur-
thermore, there are some assumptions that may not be particularly realistic. For instance,
particles are unlikely to be spherical and uniformly sized. An understanding of the impact
of rough, unevenly shaped particles could be achieved through experiments or detailed
CFD simulations.
85
Chapter 3
Behaviour of packed-bed thermal reservoirs
86
3.1 Introduction
In the previous chapter, a thermodynamic model of packed-bed thermal reservoirs for en-
ergy storage applications was described. Availability loss coefficients were defined and
these are used to assess the performance and behaviour of thermal reservoirs in this chap-
ter.
Packed bed behaviour is reasonably well understood [112,113]. Thermal and pressure loss
coefficients have been investigated by White [60] but this analysis is extended in this chap-
ter to include conductive losses, heat leakage losses, and losses during the storage phase.
Parametric studies are presented in section 3.2 and involve varying each design parameter
individually, and quantifying its effect on each loss coefficient. These results provide an
understanding of reservoir behaviour from a second law perspective that is necessary for
design and optimisation of PTES systems.
Establishing the major factors that affect performance enables innovative design features
to be developed and two such features are investigated. The first is ‘segmentation’ of the
packed bed into discrete layers that the gas may be diverted around or directed into. In the
context of energy storage, this idea was developed by Isentropic Ltd. [108], and papers have
been written on the subject by White and McTigue [80, 81]. The present author extended
the thermal reservoir model to include segmented stores, and the behaviour of these stores
during charge and discharge is presented in section 3.3. The performance of segmented
stores during storage phases is also of interest, and an account is given in section 3.3.2.
A second innovative design feature is radial-flow packed beds. These stores have been
patented [142, 143] and some numerical and experimental results are given in [74]. In
section 3.4, the governing equations are derived and a method of solution is described based
on that presented in section 2.4. The behaviour of radial-flow packed beds is described in
terms of loss coefficients and compared with the performance of axial-flow stores.
87
3.2 Parametric studies of reservoir thermodynamics
The impact of key design variables is investigated through parametric studies: each pa-
rameter is varied, whilst the others are held constant at nominal design values given in
table 3.1. This approach develops an understanding of reservoir behaviour, and informs
decision making during optimisation. The variables are divided into two categories: de-
sign parameters that are fixed once the packed bed has been constructed, and operational
parameters that can be varied during operation.
The nominal packed bed designs in table 3.1 are chosen such that if the two reservoirs
were integrated into a PTES system as described in section 1.5.3, the device would have a
nominal energy storage capacity of ∆Emax = 16 MWh which could be delivered at W˙max =
2 MW. The nominal charging and discharging durations are therefore 8 hours1. Two storage
phases of four hours (after each charge and discharge period) allow the system to operate
on a 24 hour cycle. Typical behaviour of the nominal reservoirs is tabulated in table 3.2
which gives the round-trip efficiency χ as defined by equation 2.44, loss coefficients as
defined in section 2.5 and the available energy that is returned during discharge as given by
Bret = χBin (3.1)
where Bin is the net flux of available energy during charge.
Control of the reservoir charging period
The normalised cycle period Π is defined as the charging duration as a fraction of the nom-
inal charging time, Π = tchg/tN . This may also be interpreted as the reservoir utilisation
because it corresponds to the extent to which the reservoir is charged. The reservoir charge
and discharge durations can be controlled in a number of ways, and the most obvious
method is to switch between charge and discharge after fixed periods of time have elapsed.
However, in practice the utilisation is likely to be controlled by ending the current opera-
tion phase once the exit temperature of the packed bed reaches a certain threshold. This
1This is the time required to fully charge the reservoirs, in accord with equation 2.5. In practice the actual
charging and discharging duration will be less than this in order to avoid large exit losses.
88
System properties PTES
Energy capacity ∆Emax 16.0 MW h
Power W˙max 2.0 MW
Nominal charge time tN 8 hours
Exit temperature fraction θc,d 0.25
Charging time tchg 5.4 hours
Working fluid Argon (Ar)
Gas specific heat capacity cp 520 J/kgK
Heat capacity ratio γ 1.667
Mass flow rate m˙ 13.7 kg/s
Packing material Magnetite (Fe3O4)
Particle diameter dp 20 mm
Void fraction ε 0.40
Hot reservoir Cold reservoir
Charging temperature T1 505°C -150°C
Discharging temperature T2 37°C 37°C
Pressure P 10.5 bar 1.05 bar
Effective thermal conductivity keff 0.5 W m−1 K−1 0.5 W m−1 K−1
Overall heat transfer coefficient Uw 0.16 W m−2K−1 0.09 W m−2K−1
Length L 4.58 m 5.45 m
Diameter D 4.58 m 5.45 m
Packing volume Vpack 75.5 m3 127.1 m3
Insulation thickness tI 238 mm 342 mm
Steel thickness ts 16 mm 2 mm
Total volume Vtot 128 m3 222 m3
Table 3.1: Nominal PTES and packed bed designs
method is straightforward to monitor and would prevent excessive heat rejection. Tempera-
ture thresholds for the charging and discharging periods may be varied independently. The
thresholds are defined as a fraction of the temperature difference in the reservoir:
θc =
T cx − T2
T1 − T2
θd =
T dx − T1
T1 − T2 (3.2)
where θc,d is the charging or discharge temperature threshold, and T c,dx is the exit tempera-
ture of the gas during charge or discharge. Larger values of θ correspond to higher utilisa-
tions and energy densities, as shown in figure 3.1. For the case where θc 6= θd charging and
89
χ (%) ζt (%) ζp (%) ζc (%) ζL (%) ζs (%) Bret (MW h)
Hot store 94.97 4.47 0.03 0.07 0.18 0.28 7.13
Cold store 86.55 10.90 1.47 0.20 0.29 0.59 2.96
Table 3.2: Efficiency and losses in nominal packed bed designs
 3
 4
 5
 6
 7
 8
 9
 10
 11
 0.1  0.2  0.3  0.4  0.5
 0
 0.2
 0.4
 0.6
 0.8
 1
R
et
u
rn
ed
 e
n
er
g
y
 (
M
W
h
)
U
ti
li
sa
ti
o
n
 Π
 =
 t
ch
g
 /
 t
N
θc
θd = 0.05, 0.14, 0.26,
        0.38, 0.50
Returned energy
Π
Figure 3.1: Effect of θc and θd on the returned energy and utilisation of a hot store
discharging durations are roughly equal during cyclic operation.
The following studies use θc,d to control the cycle period, and these values are fixed at 0.25
unless otherwise specified. It is worth noting that the thermal gradient is affected by the
choice of design and operational variables. As a result, when θc,d are constant the utilisation
may change if other parameters are varied.
Packed beds can be operated in a number of ways. ‘Single-blow’ operation involves a single
charge-discharge cycle. On the other hand, in ‘cyclic’ operation the stores are repeatedly
charged and discharged for the same duration each cycle. In practice, the stores may be
required to operate more flexibly in order to meet constraints set by economic factors or the
electrical grid. As a result, charge and discharge periods may change from cycle to cycle,
while mass flow rates may be varied in order to change to power output. Understanding the
impact of these erratic load cycles requires a comprehensive study and a knowledge of the
90
typical load cycles that may occur. The subsequent investigations use cyclic operation, as
this mode could be used to smooth daily demand fluctuations.
3.2.1 Design parameters
Aspect ratio and particle diameter
The particle diameter and aspect ratio are varied whilst holding the reservoir volumes, mass
flow rate and operating temperatures constant, thereby ensuring that the nominal storage
capacity and power also remain constant. Figure 3.2 shows how each loss coefficient varies
with dp and L/D for hot and cold stores. As with all heat exchange processes there is an in-
herent trade-off between efficient heat transfer and low pressure loss, which manifests itself
as an optimal particle diameter and aspect ratio. Increasing the aspect ratio and decreasing
the particle size both increase frictional effects, thereby increasing ζp, but cause a decrease
in ζt since the area for heat transfer is increased and dissipative effects are reduced.
The optimal shape of cold reservoirs is shorter and fatter than that of the hot stores due to
the dominant effect of pressure losses: cold reservoirs have lower gas densities, larger gas
velocities, and a lower pressure meaning that a given pressure drop has a greater impact.
The large pressure losses in cold reservoirs also lead to larger optimal particle sizes than in
hot stores. Pressure losses only become significant in hot reservoirs for rather large aspect
ratios and very small particles.
Storage losses
Storage losses are less significant than thermal or pressure losses. Storage losses are com-
posed of conductive and leakage losses during the storage phase, hence ζs follows a similar
trend to ζc (which itself is closely related to the thermal gradient and therefore to ζt, as
shown in equation 2.62). Conductive losses drop continuously as dp is increased but stor-
age losses show a slight increase at large particle sizes. Figure 3.3(a) shows that large
particles lead to less steep thermal fronts which consequently reach the exit of the packed
bed sooner. Since the full cycle is fixed at 24 hours, the impact of large particles is to reduce
charging duration and increase storage duration.
91
 0
 1
 2
 3
 4
 5
 0  2  4  6  8  10  12
L
o
ss
es
, 
ζ (
%
)
Aspect ratio, L / D
ζt
ζp
ζL
ζc
ζs
(a)
 0
 2
 4
 6
 8
 10
 12
 14
 16
 0.5  1  1.5  2
L
o
ss
es
, 
ζ (
%
)
Aspect ratio, L / D
ζt
ζp
ζL
ζc
ζs
(b)
 0
 2
 4
 6
 8
 10
 10  20  30  40  50
L
o
ss
es
, 
ζ (
%
)
Particle diameter, dp (mm)
ζt
ζp
ζL
ζc
ζs
(c)
 0
 5
 10
 15
 20
 25
 30
 0  10  20  30  40  50
L
o
ss
es
, 
ζ (
%
)
Particle diameter, dp (mm)
ζt
ζp
ζL
ζc
ζs
(d)
Figure 3.2: Parametric study of aspect ratio and particle diameter on hot and cold packed beds (a)
Effect of aspect ratio on losses in a hot store (dp = 20 mm) (b) Effect of aspect ratio on losses in a
cold store (dp = 20 mm) (c) Effect of particle diameter on losses in a hot store (L / D = 1) (d) Effect
of particle diameter on losses in a cold store (L / D = 1)
92
The shape of the thermal fronts has an impact on the energy that is stored in the packed
bed. Figure 3.3(b) displays the round-trip efficiency χ against the available energy that is
returned during the discharge phaseBret. Decreasing the particle size leads to steeper fronts
and generally increases the quantity of energy that is returned during discharge. However,
very small particles and large aspect ratios lead to large pressure losses, which reduces
the round-trip efficiency and therefore also reduces the returned energy (consider the open
point on the dp = 5mm curve).
Figure 3.3(c) indicates that smaller aspect ratios lead to thermal fronts that take up a larger
fractional length of the reservoir. As these fronts are ‘less steep’ a smaller quantity of
energy is returned. Increasing the aspect ratio increases the returned energy, until pressure
losses become significantly large and reduce the efficiency in a similar way to that shown
in figure 3.3(b).
Packing material
The choice of storage media is an important consideration in the design of the PTES sys-
tem. The material must fulfil several requirements, such as good thermal and mechanical
properties – e.g. it should be able to resist the compressive forces due to its own self weight.
Ideally, the material should also be cheap, widely available, environmentally friendly and
non-toxic. In order to make detailed comparisons between materials, a mechanical model
of the particles should be developed (e.g. particle packing and arrangement, how the par-
ticles compact or interact with one another, and how this changes with time) so that the
interactions between mechanical stresses and thermodynamic behaviour can be considered
such as in [144, 145]. The development of a mechanical model is outside the scope of this
thesis which concentrates on the thermodynamic performance.
The availability density provides a useful metric to compare different materials and is given
by
ρa = ρs(1− ε) [(echg − edis)− T0(schg − sdis)] (3.3)
Data for several suitable materials are given in table 3.3, where the values are calculated
for the nominal design. By comparison, PHES with a height difference of 500 m has a
density of 5 MJ m−3 of water, whilst CAES at 100 bar has a value of around 36 MJ m−3
93
 0
 100
 200
 300
 400
 500
 0  0.2  0.4  0.6  0.8  1
G
as
 t
em
p
er
at
u
re
, 
T
g
 (
°
C
)
Distance / Length, x / L
1% tchg 30% 60% 95%
dp = 5 mm
dp = 20 mm
(a)
 80
 85
 90
 95
 100
 6  7  8  9  10
E
ff
ic
ie
n
cy
 (
%
)
Returned Available Energy (MWh)
dp = 5mm
10mm
15mm
20mm
25mm
(b)
 0
 100
 200
 300
 400
 500
 0  0.2  0.4  0.6  0.8  1
G
as
 t
em
p
er
at
u
re
, 
T
g
 (
°
C
)
Distance / Length, x / L
1% tchg 30% 60% 95%
L / D = 0.5
L / D = 4.0
(c)
Figure 3.3: Parametric study of aspect ratio and particle diameter on hot packed beds. (a) Effect
of particle diameter on thermal front shape in hot stores (b) Effect of geometric parameters on the
trade-off between efficiency and returned available energy in a hot store. Lines of constant dp are
plotted and L/D is varied along the lines. Filled and open points mark the minimum and maximum
L/D, respectively. (c) Effect of aspect ratio on thermal front shapes in hot stores
94
Al2O3 Fe2O3 Fe3O4 SiO2 TiO2
Hot reservoir
ρs (kg m−3) 4040 5242 5175 2650 4230
ρa (MJ m−3) 509 534 549 327 425
c¯s (J/kg·K) 1070 850 860 1020 855
∆cs/c¯s 0.347 0.367 0.468 0.433 0.242
k¯s (W/m·K) 25.50 4.83 3.50 8.20 7.50
Cold reservoir
ρs (kg m−3) 4040 5242 5175 2650 4230
ρa (MJ m−3) 85 112 121 65 100
c¯s (J/kg·K) 540 500 520 570 550
∆cs/c¯s 1.048 0.774 0.643a 0.740 0.708
k¯s (W/m·K) 56.80 6.95 3.94 22.50 11.0
Table 3.3: Physical properties of materials for hot and cold thermal stores. Data for densities and
heat capacities taken from [146] and [147], thermal conductivities are from [148] and [127].
a ∆cs/c¯s for magnetite has been calculated as if there were no spike in cs at 120 K (see figure 2.10),
in order to give a more representative value. The true value is 0.20 but this is distorted by the
presence of the spike which is apparently due to a magnetic effect. While not included in the
calculation of ∆cs/c¯s this feature has been included in the calculation of losses.
of compressed air, demonstrating that thermal stores achieve competitive energy densities
[97].
The solid specific heat capacity cs varies with temperature, as shown in figure 2.10 on
page 61. This causes the length of the thermal front to fluctuate, which subsequently affects
the thermal losses since the area for heat transfer is changed. The extent to which fronts
steepen or spread is dependent on the range of heat capacities over which the reservoir
operates. Table 3.3 gives the fractional change in heat capacity ∆cs/c¯s for each material
over the range of operating temperatures, where c¯s is the average heat capacity (defined by
equation 2.4). In cyclic operation, the greater the value of ∆cs/c¯s, the more the length of
the thermal front changes about its average gradient, and therefore the greater the thermal
losses, as discussed in section 2.5.1 and appendix A.
Figure 3.4 shows the distribution of losses for hot and cold reservoirs in cyclic operation for
different materials. Pressure and leakage losses are insensitive to the choice of material.
Thermal losses in the hot reservoirs do not vary significantly due to smaller variations
in heat capacity at high temperatures (figure 2.10). The choice of material has a more
significant effect in cold reservoirs, however, since the heat capacity curves are steeper at
low temperatures. Magnetite (Fe3O4) is a promising material due to its low variation in
95
0.0
2.0
4.0
6.0
Al2O3 Fe2O3 Fe3O4 SiO2 TiO2
Lo
ss
 c
oe
ffi
ci
en
ts,
 ζ 
(%
)
ζt ζp ζL ζc ζs
(a) Hot store
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
Al2O3 Fe2O3 Fe3O4 SiO2 TiO2
Lo
ss
 c
oe
ffi
ci
en
ts,
 ζ 
(%
)
ζt ζp ζL ζc ζs
(b) Cold store
Figure 3.4: Distribution of losses for different packing materials computed at the nominal condi-
tions of table 3.1
heat capacity.
Conductive losses are fairly similar in each of the materials. Table 3.3 indicates that each
material has quite different values of average thermal conductivity k¯s. However, it was
found that the solid conductivities did not significantly affect the effective conductivity
keff which was calculated using the correlations of section 2.5.3 (i.e. keff is dominated by
contributions from radiation and the turbulent transport of heat). Aluminium oxide has the
largest conductive and storage loss as a result of the large solid conductivity.
These results suggest that the choice of material has an important effect on the thermal
losses, particularly in cold stores. For the temperature ranges considered, magnetite appears
to be the most suitable material, and aluminium oxide the least suitable. However, this
choice may change if different temperature ranges are chosen. It is conceivable that hot
and cold reservoirs may use different materials in order to exploit the material with the best
heat capacity profile for that temperature range. Furthermore, mechanical properties and
cost will play an important role in material selection.
96
Reservoir sizing
Reservoirs of different volumes are compared in this section. The operating temperatures
of the packed beds are fixed and the availability capacity of the stores therefore varies in
proportion with the volume. The nominal charge time tN of the reservoirs is held constant
at 8 hours, and the stores are cycled with a period of Π = tc/tN = 0.75 (6 hours). The
mass flow rate is varied in order to keep tN constant and is thus
m˙ =
ρscs(1− ε)Vpack
cptN
(3.4)
where Vpack is the volume of the packing material.
The mass flow rate scales with the cube of the packed-bed length scale, such that the mass
flow rate per unit area is proportional to the length scale,
G =
ρscs(1− ε)
cptN
L (3.5)
The variation in mass flow rate per unit area is directly responsible for an increase of ζp with
volume, and for a decrease in ζt and ζc as the volume increases as shown in figure 3.5(b).
Leakage losses also decrease as the volume is increased at constant aspect ratio since the
surface area to volume ratio is given by 4/D.
The trade-off between thermal losses and pressure losses leads to an optimal packing vol-
ume. As the aspect ratio is increased, this optimal point occurs at smaller volumes due to
the increased pressure losses that occur. Comparable behaviour was observed in the cold
store.
3.2.2 Operational parameters
Cycle frequency
Once the packed bed has been constructed the geometric parameters are fixed. Operational
parameters, such as the cycle frequency and the operating temperatures influence the ef-
ficiency and energy density. Previous work [60, 84] has indicated that one of the most
97
 80
 82
 84
 86
 88
 90
 92
 94
 96
 0  200  400  600  800  1000
E
ff
ic
ie
n
cy
 (
%
)
Packing volume (m
3
)
L / D = 1
L / D = 3
L / D = 5
L / D = 7
(a)
 0
 1
 2
 3
 4
 5
 6
 0  100  200  300  400  500  600  700
L
o
ss
 c
o
ef
fi
ci
en
ts
, 
ζ (
%
)
Packing volume (m
3
)
ζt
ζp
ζL
ζc
ζs
(b)
Figure 3.5: Effect of reservoir sizing on availability losses in packed beds for Π = 0.75 (a) Round-
trip efficiency of hot stores for different volumes and aspect ratios (b) Breakdown of availability
losses in a hot store for L/D = 3
significant parameters is the normalised cycle period, Π.
During cyclic operation, instantaneous temperature profiles are identical from one cycle to
the next and the packed bed returns to its original state at the end of the cycle. Information
must be propagated along the entire length of the reservoir in order to flush out entropy
that was generated irreversibly within the packed bed. When θc,d (or Π) is low, the thermal
front moves only a small fraction of the reservoir length. The thermal front must therefore
become longer and less steep, as shown in figure 3.6(a). Heat transfer then occurs over
a larger surface area reducing thermal losses, although the energy density of the storage
is compromised, as shown in figure 3.1. The utilisation therefore controls the trade-off
between efficiency and energy density.
Figures 3.6(b) and 3.6(c) indicate that lower θc,d lead to larger efficiencies. However, the
orientation of the constant efficiency lines indicate that these parameters do not have an
equal impact. The efficiency is more sensitive to θc than θd in hot packed beds, and the
opposite is true in cold packed beds. Calculating the correlation coefficients between θ and
98
 0
 100
 200
 300
 400
 500
 0  0.25  0.5  0.75  1
G
as
 t
em
p
er
at
u
re
, 
T
g
 (
°
C
)
Normalised distance
θc = 0.10
θc = 0.25
(a)
 0.1
 0.2
 0.3
 0.4
 0.5
 0.1  0.2  0.3  0.4  0.5
θ d
θ
c
98
97.5
97
96.5 96 95.5
95
(b)
 0.1
 0.2
 0.3
 0.4
 0.5
 0.1  0.2  0.3  0.4  0.5
θ d
θ
c
89
88
87
86
85
84
83
82
81
(c)
Figure 3.6: Parametric study of θc and θd on hot and cold packed beds (a) Effect of θc and θd on
thermal front shapes in a hot store (b) Contours of constant efficiency in a hot store. (c) Contours
of constant efficiency in a cold store.
99
the round-trip efficiency χ for this data emphasises this result, since corr(θc, χ) = −0.95
and corr(θd, χ) = −0.30 for a hot store, and corr(θc, χ) = −0.21 and corr(θd, χ) = −0.96
for a cold store.
Operating temperatures
For PTES systems reservoir temperatures are not independent as setting the temperatures
of one reservoir impacts on those in the other reservoir. For instance, during charge the
inlet temperature of the hot reservoir cannot be varied without changing either the pres-
sure ratio or the outlet temperature of the cold reservoir. Changing these quantities also
affects the rated power and energy density of the system. Furthermore, these factors affect
the operating point on the mass flow rate–pressure ratio characteristic of the reciprocating
devices, see section 4.2.
A parametric study of the operating temperatures and pressure ratios within the context
of the full PTES system is presented in chapter 4. The above constraints are relaxed in
this analysis which considers the thermal stores in isolation, and is applicable to storage
systems that use a single reservoir.
The charging and discharging temperatures of hot and cold reservoirs are varied indepen-
dently. The reservoir volume and geometry is fixed, as is the mass flow rate such that the
nominal storage time is constant. Contours of constant ∆T = T1 − T2 are plotted in fig-
ures 3.7(a) and 3.7(b) and are approximately proportional to the internal energy density of
the system. For hot stores, increasing the charging temperature T1 at a constant ∆T leads
to higher efficiencies since thermal losses decrease (compare with equation 2.50 where
ζt ∼ ∆T 2/T1T2). On the other hand, T1 should be minimised in cold reservoirs.
In figures 3.7(c) and 3.7(d) the charging temperature is varied while the discharging tem-
perature is held constant (i.e. ∆T varies). As ∆T increases, thermal losses increase and
pressure losses are reduced due to lower gas densities. Heat leakage losses also reduce
which seems counter-intuitive. However, whilst the real heat leakage rate does increase
with T1, the inlet available energy Bin (the denominator in the loss coefficients) increases
at a greater rate than the heat leakage, so that the leakage loss coefficient decreases.
These results suggest that the charging temperature should be increased (or decreased for
100
 92
 93
 94
 95
 96
 97
 98
 99
 200  300  400  500  600
E
ff
ic
ie
n
cy
 (
%
)
Charging temperature, T1 (°C)
T1 - T2 = 100 °C
200 °C
300 °C
400 °C
500 °C
(a)
 65
 70
 75
 80
 85
 90
 95
-160 -140 -120 -100 -80 -60 -40 -20
E
ff
ic
ie
n
cy
 (
%
)
Charging temperature, T1 (°C)
T1 - T2 = -60 °C
-80 °C
-100 °C
-120 °C
-140 °C-160 °C
(b)
 0.5
 1
 1.5
 2
 200  300  400  500  600
A
v
ai
la
b
il
it
y
 l
o
ss
 c
o
ef
fi
ci
en
ts
 ζ 
(%
)
Charging temperature, T1 (°C)
ζt
ζp
ζL
ζc
ζs
(c)
 0
 1
 2
 3
 4
 5
 6
 7
 8
-150 -140 -130 -120 -110 -100 -90 -80
A
v
ai
la
b
il
it
y
 l
o
ss
 c
o
ef
fi
ci
en
ts
 ζ 
(%
)
Charging temperature, T1 (°C)
ζt
ζp
ζL
ζc
ζs
(d)
Figure 3.7: Parametric study of operating temperatures in hot and cold packed beds. (a) Contours
of constant T1 − T2 in a hot store. (b) Contours of constant T1 − T2 in a cold store. (c) Loss
coefficients in a hot store as the charging temperature T1 is varied. The discharging temperature T2
is fixed at 185°C (d) Loss coefficients in a cold store as the charging temperature T1 is varied. The
discharging temperature T2 is fixed at -30°C
101
a cold store) as much as possible, and that the discharging temperature should be as close
as possible to T1 for high efficiencies. As a result, there is a trade-off with maximising the
energy density which requires large values of T1 − T2.
3.3 Segmented packed beds
The previous section emphasised the conflict between thermal and pressure losses and their
impact on the optimal value of variables such as the particle diameter and aspect ratio.
Pressure losses occur over the full length of the packed bed whilst heat exchange only
occurs in the fraction of the reservoir where the thermal front is present. The magnitude of
the pressure losses could be reduced by dividing the packed bed into several segments and
only directing flow through those segments where the thermal front is present. By diverting
the flow around the remaining segments the pressure losses are reduced.
Segmentation of thermal stores has been proposed in the literature for different applica-
tions. Crandall and Thatcher [78] suggested that it could be used to maintain thermal
stratification in solar air heating applications. Similarly, Dickinson et al. [149] employed
multiple hot water tanks for a solar hot water scheme. These authors noted that segmenta-
tion could reduce the extent of de-stratification of the temperature gradients. This would be
beneficial during the storage periods of packed beds where conduction erodes the thermal
fronts. Segmentation has recently been patented for thermal energy storage applications
by Isentropic Ltd. [108] who cite greater control over the thermal front in addition to the
above advantages.
3.3.1 Segment control and spacing requirements
Segment control is the process by which segments are activated (gas flow is directed into the
segment) and de-activated (gas flow is diverted around the segment). Various possibilities
exist for activation/deactivation and two schemes suggested by Isentropic Ltd. [108] are
illustrated in figure 3.8. In 3.8(a) the central valve/baffle system moves down through the
reservoir during charge, tracking the thermal front. Most of the flow takes the path of least
resistance so that, as shown, only layers B, C and D are active. As shown in Ref. [80] this
102
(a) (b)
Figure 3.8: Possible segmented packed bed arrangements based on [108]. (a) A fixed number of
layers are maintained active by means of the baffle/valve arrangement (b) Independent valving of
each segment allows greater flexibility.
leads to impractical temperature distributions. Alternatively, in the design of 3.8(b) each
layer is individually gated (with gates B, C and D shut, as shown). This allows each segment
to be controlled independently. In practice, some flow will leak through the bypassed layers
(as indicated by the dashed lines through layer E), depending on the relative resistance of
the layers and the bypass channels.
For segmented beds, additional space is required between each segment and in the case of
design 3.8(b) in an annular ring around the segments. The space required may be estimated
by setting the pressure drop through the spaces between and around the segments to be
a small fraction (denoted by φ) of the pressure drop that would occur if the flow passed
through an optimised unsegmented reservoir2. The pressure drop of the bypass flow takes
into account the loss due to turning the flow and is estimated from
∆pb =
1
2
kρu2 =
k
2ρ
(
m˙
Ab
)2
(3.6)
where Ab is the area through which the bypass flow travels and k is an empirical factor
which is of the order of unity [150]. The pressure drop along an unsegmented packed bed
2The objective of segmentation is to reduce pressure losses. Therefore, the losses in the bypass areas
should be small compared to that in an optimal unsegmented store.
103
is given by
∆pp =
1
2ρε3
Cf
St
(
m˙
A
)2
L
`∗
(3.7)
where A is the cross sectional area of the packed bed, L is the total length of the bed, and
`∗ is the optimal value of ` for an unsegmented store. Note that the density ρ, and the other
thermodynamic properties (Cf , St, and `∗) are taken at their average values along the bed.
Consequently this leads to
Ab
A
=
√
kε3
φ
`∗
L
St
Cf
(3.8)
The spacing s between segments must also have the same area as the bypass flow ratio and
is given by
Ab
A
=
2piRs
piR2
⇒ s
D
=
1
4
Ab
A
(3.9)
As a result, the space between segments is approximately equal to the size of the gap
outside the segments. Estimates using the nominal reservoir design suggest a spacing of 11
mm in hot stores (an increase in cross-sectional area A of 1%) and 21 mm in cold stores
(an increase in A of 1.5%).
Further additional space may be required to provide room for the valve mechanisms, other
control systems, and structural support. Although segmented beds could lead to an in-
creased utilisation of the packed beds, the extra space required may in fact decrease the
energy density of the stores.
3.3.2 Availability losses in segmented reservoirs
Operational losses
The evolution of the thermal front within a segmented store is affected by the way the
individual segments are activated and deactivated. Figure 3.9(a) illustrates the temperature
profiles at the end of charge and discharge for packed beds with one segment and eight
segments. During cyclic operation using the valve arrangement shown in figure 3.8(b) new
segments are activated at the leading edge of the thermal front when the normalised exit
temperature from the previous segment exceeds θl = 0.1, where θl = (To − T2)/(T1 − T2)
104
 0
 0.2
 0.4
 0.6
 0.8
 1
 0  0.2  0.4  0.6  0.8  1
G
as
 t
em
p
er
at
u
re
, 
T
g
 (
°
C
)
Normalised distance, x / L 
θl
θh
C
D
(a)
 0
 200
 400
 0  1  2  3
 0
 0.05
 0.1
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
E
x
it
 t
em
p
er
at
u
re
, 
T
g
 (
°
C
)
T
h
er
m
al
 l
o
ss
es
 (
%
) 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
 
Normalised time, t / tN
charge discharge charge discharge
(b)
Figure 3.9: Behaviour of an unsegmented packed bed (black) and a segmented packed bed (blue)
with eight segments. Cyclic operation with Π = 0.75, dp = 20 mm. (a) Temperature profiles at the
end of charge (C) and discharge (D) (b) Exit temperatures and instantaneous thermal losses for the
first two cycles of operation
and To is its outlet temperature. The segments at the trailing edge of the thermal front are
then deactivated when their exit temperature exceeds θh = 0.9 where θh is defined in the
same way as θl.
Truncation of the nose and tail of the thermal front in segmented stores leads to steeper
temperature profiles, but also generates a ‘sawtooth wake’ in the deactivated segments that
gradually decays as a result of axial conduction. Figure 3.9(b) (top) shows how the exit
temperature from the packed bed varies for the first two cycles of operation. The exit
temperature displays the same sawtooth behaviour and implies that there are exit losses
through much of the duration of the cycle. This counteracts the increased utilisation of
the reservoir due to the steeper fronts, and as a result the packed beds do not store more
available energy per cycle. On the other hand, the exit flow does not reach such high
temperatures as the unsegmented case.
Figure 3.9(b) (bottom) shows the importance of the switching temperatures θl,h on thermal
losses. During the first charging cycle, whenever a segment is activated a temperature
105
difference exists between the exit gas of the previous segment (which is T = T2 + θl(T1 −
T2)) and the solid temperature near the inlet of the new segment (which is T2). This causes
spikes in the instantaneous thermal losses, the extent of which will clearly depend on the
size of θl. Notably, figure 3.9(b) shows that the spikes are substantially smaller during the
second charging period compared to the first. This is because the remnants of the sawtooth
wake from the previous cycle reduce the temperature difference between the inlet gas and
the existing solid temperature.
When a segment is deactivated, flow is diverted around this segment and is directed into the
next segment. Consequently, there is also a large temperature difference at the trailing edge
of the front. For instance, during charge the inlet gas is at T1 whereas the solid temperature
near the inlet of the first activated segment is T = T2 + θh(T1−T2). During the first charge
cycle these losses are observed in figure 3.9(b) (bottom) as small spikes between the larger
spikes. (The instantaneous losses are smaller than those at the leading edge of the front due
to the higher average temperature). The large spikes shown during discharge are trailing
edge losses. These are much larger during the discharging phases where temperatures are
low. Unlike the leading edge losses which are reduced in subsequent cycles, the trailing
edge losses cannot be avoided, and the loss spikes are particularly pronounced throughout
all discharge cycles. Note that the charging losses gradually decrease, and discharge losses
increase due to the spreading and steepening of the thermal front.
The impact of particle diameter
Figure 3.10 illustrates the impact of segmentation on availability losses in cold reservoirs.
As expected, segmenting the bed significantly reduces the pressure losses, especially at
small particle diameters. On the other hand, thermal losses increase slightly due to the
losses caused when activating and deactivating segments. There is a trade-off in the value
of θl/h; smaller values reduce thermal losses with a penalty for pressure losses.
The round-trip efficiency is maximised at smaller particle diameters when segmented, as
illustrated by the filled in dots in figure 3.10(a). This has an additional benefit: smaller
particles form steeper fronts and therefore more available energy is stored. Figure 3.10(b)
compares the performance of three cold packed beds which use the optimal particle size
for the given number of segments, with θc,d = 0.25. The figure is created by varying θc,d
106
 55
 60
 65
 70
 75
 80
 85
 90
 95
 0  5  10  15  20
 0
 5
 10
 15
 20
 25
 30
 35
 40
E
ff
ic
ie
n
cy
 (
%
)
L
o
ss
es
, 
ζ (
%
)
Particle diameter, dp (mm)
ζtζp
Efficiency
(a)
 80
 82
 84
 86
 88
 90
 92
 94
 1  1.5  2  2.5  3  3.5  4
E
ff
ic
ie
n
cy
 (
%
)
Returned available energy (MWh)
1 segment
8 segments
32 segments
(b)
Figure 3.10: Availability losses of cold segmented packed beds. (a) Efficiency and availability
losses in cyclic operation of packed beds with one segment (solid lines) and 32 segments (dashed
lines). Solid dots represent ‘optimal’ points at θc,d = 0.25 (b) Particle diameter is set at the optimal
value for efficiency (figure 3.10(a)) which is 10 mm for 1 segment, 8 mm for 8 segments and 7mm
for 32 segments. θc is varied to generate the plot.
and it therefore displays ‘off-design’ performance in the sense that the optimal particle
size also depends on the utilisation: longer charging periods imply steeper, shorter thermal
fronts (see section 3.2.2). The benefits derived from segmentation are clear and may be
interpreted as either an increase in efficiency at high utilisation (by up to 4 percentage
points) or a substantial improvement in returned energy at fixed efficiency (e.g. an increase
of 30% at an efficiency of 92%). Note that this improvement will be offset to some degree
by the additional space required for bypass flows etc. In addition, it is clear that there are
diminishing returns to installing a greater number of segments, and that the majority of the
potential benefit can be achieved with just eight segments.
107
Storage losses
Segmentation can reduce the losses that occur during the storage phase, when conduction
and heat leakage cause equilibration of the solid temperature. Conductive losses in par-
ticular can be reduced and figures 3.11 and 3.12(a) illustrate a simplified scenario where
conduction acts on thermal fronts that are assumed to be linear. In this scenario, the gas
and solid are assumed to have the same temperature, and the front behaviour is governed
by the heat equation
∂T
∂t
= α
∂2T
∂x2
(3.10)
where α = keff/ρscs(1− ε). The reservoir inlet and outlet are assumed fully insulated (i.e.
adiabatic) as are the boundaries between segments so that the boundary conditions are
∂T (0, t)
∂x
=
∂T (L, t)
∂x
= 0 (3.11)
For a linear front which varies from T1 at the inlet to T2 at the exit, the initial condition is
given by
T (x, 0) = T1 − (T1 − T2)x
L
(3.12)
Separation of variables and use of the Fourier Series gives the analytical solution:
T (x, t) =
(
T1 + T2
2
)
(3.13)
+
2(T1 − T2)
pi2
∞∑
n=1
1− (−1)n
n2
cos
(npix
L
)
exp
{
−α
(npi
L
)2
t
}
Figures 3.11(a) and 3.11(b) show how the thermal front equilibrates in reservoirs with one
and four segments, respectively. In the segmented case, the thermal front is constrained
to equilibrate within its respective segment, rather than level out over the full length of
the reservoir. Consequently, the thermal profile is preserved, albeit, with a stepped profile.
For figures 3.11(a) and 3.11(b) the temperature profiles have the same energy content.
However, the stepped profile of the segmented store has a greater available energy content.
The instantaneous conductive loss coefficient at each x position for the above cases is
shown in figures 3.11(c) and 3.11(d). (In this section loss coefficients are defined as the
108
-150
-100
-50
 0
 0  0.25  0.5  0.75  1
T
em
p
er
at
u
re
 (
°
C
)
Normalised distance, x / L
t = 0 days
t = 15 days
(a)
-150
-100
-50
 0
 0  0.25  0.5  0.75  1
T
em
p
er
at
u
re
 (
°
C
)
Normalised distance, x / L
(b)
 0
 2
 4
 6
 8
 10
 12
 0  0.25  0.5  0.75  1
C
o
n
d
u
ct
iv
e 
lo
ss
es
 ζ c
 x
 1
0
-9
(%
)
Normalised distance, x / L
t = 0 days
t = 15 days
(c)
 0
 2
 4
 6
 8
 10
 12
 0  0.25  0.5  0.75  1
C
o
n
d
u
c
ti
v
e
 l
o
ss
e
s 
ζ c 
x
 1
0
-9
 (
%
)
Normalised distance, x / L
(d)
Figure 3.11: Conductive effects in the storage phase of segmented reservoirs. Boundaries of the
reservoir and each segment are adiabatic. Each line shows the profile at intervals of 3 days with the
darkest line being at 0 days. (a) Equilibration of a linear temperature profile during the storage phase
of an unsegmented cold reservoir (b) Equilibration of a linear temperature profile during storage in
a cold reservoir with four segments (c) Instantaneous conductive losses in an unsegmented cold
reservoir (d) Instantaneous conductive losses in a cold reservoir with four segments
109
available energy lost divided by the total available energy in the packed bed at the begin-
ning of storage.) By comparing these figures with equation 2.60 on page 66 it is clear that
the largest conductive losses are generated at points where the thermal gradient is steepest
and where the temperature is lowest. Conductive losses decrease at the boundaries first,
where the thermal gradient reduces most rapidly. The segmented store has the same con-
ductive loss generation as the unsegmented store at zero days. However, the larger number
of boundaries means that the thermal gradient is reduced at more points in the segmented
reservoir. As a result, the total conductive loss at each time-step is less than in the unseg-
mented reservoir. (This can be seen by observing that the total area under each curve for the
segmented reservoir is always equal to or less than the corresponding unsegmented curve).
Conductive and leakage losses during the storage phase
Segmented reservoirs are compared by considering designs that would minimise the pres-
sure losses. Thus, it is assumed that the thermal front length is equal to the length of the
segments. At the end of charge, the thermal profile is given by a linear gradient in the
final segment, while all the other segments are fully charged. Figure 3.12(a) shows how
the conductive losses vary with time for different numbers of segments with keff = 0.5 W
m−1K−1. Conductive losses increase more quickly in packed beds with more segments due
to the steeper thermal gradient in the final layer. However, over a longer period of time the
benefit of segmenting the reservoirs is obvious, for instance conductive losses in a reservoir
with 32 layers are around one fifth of those in an unsegmented store.
Heat leakage occurs during storage periods and its behaviour is included by adding an
additional term to the heat equation, such that
∂T
∂t
= α
∂2T
∂x2
+ β(T − T0) (3.14)
where β = 4Us/((1 − ε)ρscsD). For insulated boundaries, the analytical solution can be
obtained in the same manner as equation 3.14 and is given by
T (x, t) = Θ(x, t) exp (−βt) (3.15)
where Θ(x, t) is the solution given in equation 3.14. When the boundaries are no longer
110
insulated and heat transfer occurs between segments and from the top and bottom of the
stores, the heat equation can be most easily and accurately solved using the Crank-Nicolson
method [121].
The impact of leakage in a segmented cold store with a linear temperature profile in the
final layer is investigated. The overall heat transfer coefficient is Us = Ut = Ub = 0.16
W/mK which is consistent with UL for the nominal designs. Leakage occurs from the side
walls and from the top and bottom of the store. Radiative heat transfer also occurs between
the layers and generates thermal gradients which are then eroded by conduction as shown
in figure 3.13. Thermal gradients also develop in the first layer as a result of heat leakage
to the environment through the bottom of the reservoir.
The variation of each loss with time is shown in figure 3.12(b) for a 16 segment reservoir
(radiative losses as defined in section 2.5.7 are included in ζL,t). Leakage losses are a
similar order of magnitude to conductive losses in this example, but in practice could be
reduced by using more insulation. The effect of heat transfer between the segments and
from the reservoir ends is to substantially increase the conductive losses compared to the
adiabatic case. Furthermore, the conductive losses continually increase rather than levelling
off.
Once leakage and conduction are both included in the analysis, the benefit of segmented
reservoirs is significantly reduced as shown in figure 3.12(c). This analysis highlights the
importance of several factors when designing the packed beds. If the reservoir is likely to
have storage phases that last several days, then segmentation would significantly reduce
conductive losses. However, the packed bed should be sufficiently insulated to reduce
leakage losses to a small fraction of the conduction losses. This is particularly important
since leakage losses exacerbate conductive losses. On the other hand, if the storage duration
is short (e.g. a number of hours) then conductive losses are likely to be small compared
to other losses in the system, and the system should be designed with an emphasis on the
operational periods.
The analysis in this section suggests that segmentation may have reasonable benefits during
operational periods. Furthermore, segmented reservoirs could reduce losses during storage
phases that last several days.
111
 0
 0.5
 1
 1.5
 2
 2.5
 3
 0  1  2  3  4  5
C
o
n
d
u
ct
iv
e 
lo
ss
es
, 
ζ c 
(%
)
Time (days)
 1 segment 
 8 segments
16 segments
32 segments
(a)
 0
 1
 2
 3
 4
 5
 0  1  2  3  4  5
L
o
ss
es
, 
ζ (
%
)
Time (days)
ζc
ζL
ζL,b
ζL,t
(b)
 0
 2
 4
 6
 8
 10
 12
 0  1  2  3  4  5
T
o
ta
l 
lo
ss
es
, 
ζ (
%
)
Time (days)
 1 segment 
 8 segments
16 segments
32 segments
(c)
Figure 3.12: Combined effect of conduction and heat leakage on loss coefficients during the storage
phase of segmented packed beds. (a) Conductive losses in cold segmented packed beds with no heat
leakage and with adiabatic boundaries. (b) Loss coefficients in a cold packed beds with 16 segments
and with heat leakage (c) Total losses from cold segmented packed beds with the thermal front in
the final segment
112
-150
-100
-50
 0
 0  0.25  0.5  0.75  1
T
em
p
er
at
u
re
 (
°
C
)
Normalised distance, x / L
(a)
 0
 1
 2
 3
 4
 5
 6
 0  0.25  0.5  0.75  1
C
o
n
d
u
c
ti
v
e
 l
o
ss
e
s 
ζ c 
x
 1
0
-8
 (
%
)
Normalised distance, x / L
(b)
Figure 3.13: Combined effect of conduction and heat leakage during the storage phase of packed
beds with four segments. Segments do not have adiabatic boundaries (there is heat leakage to the
atmosphere and radiation between the segments). Profiles plotted every 3 days with the darkest line
being 0 days. (a) Conduction and heat leakage acting on a profile with the thermal front in the final
segment of a segmented cold store. (b) Instantaneous conductive loss coefficients for a reservoir
with the thermal front in the final segment of a segmented cold store.
3.4 Radial-flow packed beds
Radial-flow packed beds are one alternative to the packed beds that have been described so
far. Figure 3.14 is a schematic for a radial-flow packed bed, where gas enters a chamber in
the centre of the cylindrical container, before flowing through the bed along the radial co-
ordinate and exiting through an outer plenum. The packing material is contained by a grid
through which the gas can flow. Various aspects of these packed beds have been studied,
including the effective conductivity [151, 152], flow distributions [153] and applications
in air filters [154, 155] and the synthesis of ammonia [156, 157]. These models typically
include reaction terms and mass transfer, and the emphasis of these papers tends to be on
the chemical conversion efficiency. For electrical energy storage applications particular
consideration must be given the heat transfer processes and available energy losses.
113










































































































































































































































































































































































































































































































































































































































































































































































































































Ri
Ro
steel
casing
insulation
flow     paths
outer
plenum
inner
plenum






















































































































































Figure 3.14: Schematic of a radial-flow packed bed during charge. Hot gas enters the inner plenum,
and flows radially through the bed from Ri to Ro and into the outer plenum.
Radial-flow packed beds were first patented for thermal energy storage in 1942 by Bradley
[158]. In this patent, the thermal store heated input air to 1030°C for use in a blast furnace
for smelting iron ore. Bradley suggested that particles of different sizes could be used. The
mass flow rate decreases along the radius, such that pressure losses are lowest at the outer
plenum. Smaller particles can then be placed near the outer plenum in order to reduce
thermal losses. (The decreasing mass flow rate alters the trade-off between thermal losses
and pressure losses, such that the optimal particle size also varies with radius). Grids
would have to be installed at intervals along the radius to keep particles in their respective
positions.
Two further patents [142, 143] in the 1990s aimed to make Bradley’s concept more practi-
cable. It was thought that the packing material might flow with the gas into the inner and
outer plena. Thermal expansion could cause disintegration of the rocks, damage to the con-
tainment, or non-uniformities in the packing structure. Fassbinder’s [142] and Emmel et
al.’s [143] patents provided specific design features to tackle these problems. A subsequent
patent [159] described a system for converting thermal energy into mechanical work which
specifically used radial-flow thermal stores.
Daschner et al. [74] carried out experimental work on Emmel et al.’s design and described
114
results from a 80 kW h, 235 kW packed bed that was operated between 710°C and 50°C.
The store recorded a first law thermal efficiency of 92%. Daschner noted that because the
hot gas enters the inner plenum, the exit flow will always be close to ambient temperature
so that smaller quantities of insulation will be required. This may result in a cheaper design
and lower heat leakage losses. However, the outer surface of the packed bed is not designed
for high temperatures which may compromise flexibility of operation. Furthermore, the
need for an inner and outer plenum means that the radial-flow design may have a lower
energy density than an axial-flow device of the same energy capacity.
The governing equations for radial-flow packed beds are now described, and their be-
haviour is investigated. The thermal front length is found to vary with the radial position
and this has consequences for thermal losses and energy stored. In section 3.4.4 the per-
formance of radial-flow stores is compared to axial-flow stores. This will establish, for
example, which design results in lower pressure losses and higher energy densities once
factors such as aspect ratio and segmentation are included.
3.4.1 Radial-flow packed bed governing equations
Daschner et al. [74] employed a one-temperature model of the thermal store to generate
numerical results, but a two-temperature model is required here so that the thermal loss
coefficient can be calculated. The analysis and derivation of governing equations follows
the same methodology as for axial-flow packed beds that was described in section 2.3.
Figure 3.15 shows a control volume for the radial-flow packed bed. The mass continuity
equation for radial-flow is given by
ε
∂ρ
∂t
= −1
r
∂ (rG)
∂r
(3.16)
When mass accumulation is small and the mass flow rate is constant, the mass flow rate per
unit area decreases with radius
G =
GiRi
r
(3.17)
where Gi is the mass flow rate per unit area at the inlet radius Ri. As a result, the thermal
wave speed (which can be found from an energy balance like equation 2.3) is also a function
115
Figure 3.15: Control volume of a radial-flow packed bed during charge, where Tg is the gas tem-
perature, Ts is the solid temperature, Q˙x is the gas-solid heat transfer, Q˙c is heat transfer due to
conduction along the bed and m˙ is the mass flow rate.
of radius
V rf =
cpGiRi
ρscs(1− ε)r (3.18)
The wave speed thus depends on the radial position of the front; for gas flow from the
inner surface to outer surface, points in the front that are more advanced travel more slowly
and the front shape will be affected. For instance, figure 3.16 shows how the shape of
an initially linear front in hot and cold reservoirs changes during the first charging period.
When the specific heat capacity cs is held constant the front steepens. As discussed in
section 2.5.1, the dependence of the wave speed on cs effects the thermal front shape. For
instance, it causes spreading of the front in hot stores, thereby counteracting the steepening
caused by radial position. A similar effect is seen in the cold reservoir. Variable cs in the
cold store contributes to the wave steepening during charge and causes thermal shocks to
form. If the gas were to pass in the opposite direction (i.e. from the outer plenum towards
the inner plenum) during charge then the steepening impact of variable cs may be reduced
by the wave slackening effect of the mass flow rate. Consequently, thermal losses may be
116
 0
 100
 200
 300
 400
 500
 0  0.5  1  1.5  2  2.5
G
as
 te
m
pe
ra
tu
re
, T
g 
(°C
)
Distance (m)
Constant cs
Variable cs
(a)
37
-150
-100
-50
 0
 0  0.5  1  1.5  2  2.5  3
G
as
 te
m
pe
ra
tu
re
, T
g 
(°C
)
Distance (m)
Constant cs
Variable cs
(b)
Figure 3.16: Thermal front steepening due to decreasing mass flow rate per unit area in radial
packed beds during the first charging period. (a) Hot radial packed beds. (b) Cold radial packed
beds.
reduced as the thermal gradients are reduced. A disadvantage of this mode of operation is
that the cold gas would be in direct contact with the outer plenum, and the ‘self-insulating’
benefits of the radial store would not occur. Also, since the thermal front shape changes
are undone during the discharging phase, wave steepening is not as prominent in cyclic
operation as it is in the single blow case. As a result, there may not be very much to gain
by charging cold reservoirs from the outer plenum.
The governing equations can be derived in a similar way to section 2.3 by considering an
infinitesimal control volume, see figure 3.15. The heat transfer rate between gas and solid
is given by
δQ˙x = (1− ε)Svh(Tg − Ts)rδrδθδz (3.19)
The net heat flux due to conduction through the control volume is given by
Q˙netc = −
∂
∂r
(
rkeff
∂Ts
∂r
δθδz
)
δr (3.20)
117
The energy equations are then given by
ε
∂
∂t
(ρgeg) +
1
r
∂
∂r
(rGhg) = (1− ε)Svh(Ts − Tg) (3.21)
ρscs (1− ε) ∂Ts
∂t
= Svh (1− ε) (Tg − Ts) + 1
r
∂
∂r
(
rkeff
∂Ts
∂r
)
(3.22)
The gas equation can be simplified for the case where mass accumulation is zero since the
continuity equation gives
∂(rG)
∂r
≈ 0 (3.23)
meaning that
1
r
∂
∂r
(rGhg) ≈ G∂h
∂r
(3.24)
Using these simplifications, and re-writing the equations to highlight the dependence on
radius r and be comparable with the axial-flow equations of section 2.3 produces
∂Tg
∂r
=
Ts − Tg
`
+
r
Ri
Ci (3.25)
∂Ts
∂t
=
Ri
r
Tg − Ts
τi
+
α
r
∂
∂r
(
r
∂Tg
∂r
)
(3.26)
where `, τi, α and Ci have the same form as in section 2.3. However, these factors are all
functions of the radius through the mass flow rate per unit area G, and must be integrated
appropriately.
These equations are solved using the semi-implicit method described for axial packed beds
in section 2.4 using the stencil in figure 2.8. The conductive term on the right-hand-side of
the solid equation is expanded and the ∂Ts/∂r term is incorporated into this method. The
second derivative is then evaluated in a second step after the main integration as described
in section 2.4. Heat leakage is also included in this second step and only occurs at the final
node. This approach therefore generates the following matrix for each grid-step i at each
time-step n: [
(1 + φ) −φ
−ψ1 (1 + ψ1 + ψ2)
]{
T ng,i
T ns,i
}
=
{
K1
K2
}
(3.27)
118
where
φ =
∆r
2`
(3.28)
ψ1 =
∆t
2τ
(3.29)
ψ2 =
α∆t
r¯∆r
(3.30)
where r¯ is the average radius between grid points i and i− 1. The constants are given by
K1 = (1− φ)T ng,i−1 + φT ns,i−1 + ∆rCn−1i (3.31)
K2 = (1− ψ1)T n−1s,i − ψ2T ns,i−1 + ψ1T n−1g,i (3.32)
3.4.2 Size of the empty column
The size of the inner plenum can be estimated by requiring the pressure loss down the
central column ∆pc to be small compared to the pressure loss along the radius ∆pr in order
to maintain uniform flow. The inner plenum is similar to a pipe, so that ∆pc is given by
∆pc =
4CpfL
2Ri
1/2ρV 2 =
(
m˙
pi
)2 CpfL
ρR5i
(3.33)
where Cpf is the coefficient of friction in a pipe which is approximately 0.02.
The pressure drop through the packed bed varies with radius as the mass flow rate per unit
area varies. Assuming dimensionless numbers do not vary significantly, and integrating
gives
∆pr =
1
2ρε3
Cf
St
1
`
∫ Ro
Ri
(
m˙
2piRL
)
dR (3.34)
=
1
2ρε3
Cf
St
1
`
(
m˙
2piL
)2(
Ro −Ri
RoRi
)
(3.35)
Setting ∆pc as a fraction φ = 0.1 of ∆pr and solving for Ri iteratively suggests that the
inner column should have a radius of around Ri/Ro ≈ 0.25. The column then occupies
around 5% of the volume of the packed bed (not including insulation and steel volume).
119
The increase in volume compared to a reservoir which has the same volume of packing
material but no inner column is approximately 7%.
Radial-flow packed beds also require an outer plenum (see figure 3.14) through which the
gas can exit the reservoir. The space occupied can be found by the same method as above
and the required area is the same as for the inner plenum (although the additional increase
to the radius is much smaller).
For comparison, the patent by Bradley [158] suggested a system with Ri = 5 feet (1.52 m),
Ro = 10 feet (3.05 m), meaning that Ri/Ro = 0.5. Bradley’s store had a height of 50 feet
(15.24 m) such that the aspect ratio is 5. The prototype developed by Daschner et al. [74]
usedRi = 200 mm andRo = 850 mm such thatRi/Ro = 0.24. The prototype had a height
of 900 mm so that the aspect ratio is 0.82.
3.4.3 Parametric studies of radial-flow packed beds
The nominal bed design is chosen to be similar to the axial-flow nominal designs: reser-
voirs have the same energy capacity and the same power rating and thereby have the same
nominal charging time tN . Magnetite is used as the packing material, and the operating
temperatures, pressures, mass flow rates and packing volumes are the same. The nominal
particle diameter is 20 mm. The main parameters free to be varied are the particle diameter
and the aspect ratio which is defined as H/2(Ro − Ri) for radial-flow packed beds. The
geometry of a nominal design for a radial packed bed with an aspect ratio of 1 is described
in table 3.4.
Geometry Hot reservoir Cold reservoir
H 4.0 m 4.76 m
Ri/Ro 0.20 0.20
Ri 0.50 m 0.60 m
Ro 2.50 m 2.98 m
Packing volume 75.5 m3 127.1 m3
Outer plenum thickness 50 mm 60 mm
tI 224 mm 321 mm
ts 17 mm 2 mm
Total volume 189 m3 328 m3
Table 3.4: Nominal radial-flow packed bed designs
120
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.4  0.8  1.2  1.6  2
A
va
ila
bi
lit
y 
lo
ss
es
, ζ
 
(%
)
Aspect ratio, H / (Ro - Ri)
ζt x 10ζpζLζc
(a)
 0
 0.2
 0.4
 0.6
 0.8
 1
 1.2
 1.4
 1.6
 1.8
 2
 0.4  0.8  1.2  1.6  2
A
va
ila
bi
lit
y 
lo
ss
es
, ζ
 
(%
)
Aspect ratio, H / (Ro - Ri)
ζt x 10ζpζLζc
(b)
 0
 0.5
 1
 1.5
 2
 2.5
 3
 3.5
 4
 0  2  4  6  8  10
A
va
ila
bi
lit
y 
lo
ss
es
, ζ
 
(%
)
Particle diameter, dp (mm)
ζtζpζLζc
(c)
 0
 2
 4
 6
 8
 10
 12
 14
 16
 18
 0  2  4  6  8  10
A
va
ila
bi
lit
y 
lo
ss
es
, ζ
 
(%
)
Particle diameter, dp (mm)
ζtζpζLζc
(d)
Figure 3.17: Parametric study on aspect ratio and particle diameter on hot and cold radial-flow
packed beds. (a) Hot store (dp = 20 mm. Solid lines have Ri/Ro = 0.2 and dotted points are
Ri/Ro = 0.5) (b) Cold store (dp = 20 mm. Solid lines have Ri/Ro = 0.2 and dotted points are
Ri/Ro = 0.5) (c) Hot store (H/2(Ro−Ri) = 1,Ri/Ro = 0.2) (d) Cold store (H/2(Ro−Ri) = 1,
Ri/Ro = 0.2)
121
In this section availability losses during the storage phase are not considered. A simple
treatment of storage losses in radial-flow stores will be discussed in section 3.4.5.
Parametric studies of aspect ratio and particle diameter for hot and cold radial-flow reser-
voirs are shown in figure 3.17. The trends in each loss are very similar to those in axial-flow
packed beds and as expected a trade-off between thermal and pressure losses is observed.
However, it is apparent that pressure losses are significantly smaller than in axial-flow
reservoirs as shown in figure 3.2 on page 92 3. This suggests that it may be possible to use
smaller particles thereby further reducing thermal and conductive losses.
Varying the size of the central empty column only changes the losses by a small amount. As
expected, figures 3.17(a) and 3.17(b) show that pressure losses decrease slightly as Ri/Ro
is increased since the mass flow rate undergoes a commensurate reduction. Increasing
Ri/Ro reduces the distance the gas passes through Ro − Ri for a fixed volume and aspect
ratio. Steeper thermal gradients are then formed which causes the slight increase in thermal
losses.
3Note that the axial-flow results include storage losses whereas the radial-flow results do not. Inclusion
of storage phases does not significantly effect the size of the pressure losses, nor the optimal particle size
122
3.4.4 Comparison of radial-flow and axial-flow packed beds
To make a comparison of radial-flow and axial-flow reservoirs, the stores are set up so that
they are dimensionally similar. This requires matching of the dimensionless length scale
Λ = L/` and the dimensionless time scale Γ = tc/τ . The reservoirs have the same volume,
mass flow rates, temperatures and pressures as the nominal designs that have been discussed
so far. The axial-flow store has the nominal design geometry as in table 3.1. The radial-
flow geometry (aspect ratio and particle diameter) are then chosen such that Λ and Γ are
the same for both reservoir types. Since the dimensionless numbers vary with temperature
and mass flow rate and the numbers that are matched correspond to the average reservoir
conditions. Table 3.5 provides details of the selected geometry for a cold reservoir.
Axial-flow store Radial-flow store
Length La (m) 5.45 3.00
Aspect ratiob 1.0 0.5
dp (mm) 20 16
V (m3) 127 127
tc (h) 6 6
` (mm) 50.6 28.4
τ (s) 280 280
Λ 108 106
Γ 77 77
Table 3.5: Geometry of dimensionally similar axial-flow and radial-flow cold packed-beds.
a L is the length of the flow path through the store. For an axial-flow reservoir it is equal to the
height of the store. For a radial-flow store it is given by ro − ri.
b The aspect ratio is defined as L/D for an axial-flow store, and as H/2(ro − ri) in a radial-flow
store.
Comparison of pressure losses
The cycle period Π = tc/tN = Γ/Λ was varied and results are shown in figure 3.18. As
indicated by the parametric studies, pressure losses in the radial-flow store are smaller than
those in the axial-flow store. This is predominantly the result of smaller mass flow rates per
unit area. For instance, the average G in the radial-flow store is 0.39 kg m−2s−1, whereas
in an axial-flow store, G = 0.59 kg m−2s−1. Furthermore, the pressure drop depends on
123
 0
 0.5
 1
 1.5
 2
 0  0.25  0.5  0.75  1
L
o
ss
 c
o
ef
fi
ci
en
t,
 ζ 
(%
)
Utilisation, Π = tc / tN
ζt / 10
ζp
ζc
(a)
 78
 80
 82
 84
 86
 88
 90
 92
 94
 0  1  2  3  4
E
ff
ic
ie
n
cy
 (
%
)
Returned available energy (MWh)
Radial
Axial - 1 segment
Axial - 32 segments
(b)
Figure 3.18: Comparison of dimensionally similar axial-flow packed beds and radial-flow packed
beds as the cycle frequency Π = tc/tN is varied. Curves generated by varying θc,d between 5%
and 50%. (a) Loss coefficients. Lines correspond to radial-flow stores. Dotted lines correspond to
axial-flow stores. Note that thermal losses are 10 times larger than those displayed. (b) Trade-off
between efficiency and returned available energy.
the square of G. The pressure gradient in equation 2.53 can be reformulated as
∂p
∂x
= − G
2
2ρgε3`
Cf
St
(3.36)
Integrating this expression over the length of the reservoir to give the pressure drop and for
an axial-flow reservoir this is
∆px = − m˙
2
2pi2ε3ρ¯g
Cf
St
L
`
1
R4x
(3.37)
where ρ¯g is the average gas density and Rx is the radius. For a radial-flow reservoir the
total pressure drop is
∆pr = − m˙
2
8pi2ε3ρ¯g
Cf
St
Ro −Ri
`
1
RoRiH2
(3.38)
124
-150
-100
-50
 0
 50
 0  0.25  0.5  0.75  1
T
em
p
er
at
u
re
, 
T
 (
°
C
)
Normalised length
Axial
Radial
(a)
 0
 0.5
 1
 1.5
 0  0.25  0.5  0.75  1
In
st
an
ta
n
eo
u
s 
lo
ss
 c
o
ef
fi
ci
en
t,
 ζ t 
x
 1
0
3
 (
%
)
Normalised length
Axial
Radial
(b)
Figure 3.19: Comparison of thermal fronts formed in dimensionally similar axial-flow and radial-
flow packed beds for Π = 0.75. Profiles are plotted at t/tc = 10%, 40% and 70% during charge.
‘Normalised length’ is x/L in axial-flow stores and (r − ri)/(ro − ri) in radial-flow stores. (a)
Thermal front shapes (b) Instantaneous thermal loss coefficient at each x position
where H is the height of the radial reservoir. Cf/St ≈ 4 and L/` = (Ro − Ri)/` = Λ for
both radial and axial packed beds. Consequently, the ratio of the pressure losses is given
by
∆pr
∆px
=
R4x
4RiRoH2
(3.39)
which is 0.5 for the dimensionally similar designs in table 3.5. This simplified analysis
supports the results above, although it does not take into account the variation of parameters
such as ` which depends on the mass flow rate per unit area.
Comparison of thermal and conductive losses
Figure 3.18(a) indicates that thermal and conductive losses are larger in radial-flow stores.
For the same Π a radial-flow store forms a steeper thermal front on average than in the
equivalent axial-flow store. As discussed in section 2.5.1 shorter thermal fronts lead to
larger thermal losses as there is a smaller area for heat transfer. In addition, the front length
125
changes due to the variation in the mass flow rate per unit area with radius. Figure 3.19
compares the thermal fronts in dimensionally similar axial-flow and radial-flow stores. At
the start of charge the radial-flow front is slightly longer than the axial flow front. However,
as the front progresses it becomes steeper than the axial-flow front. A simplified analysis
in Appendix A.2 indicates that the average radial-flow front length is less than the average
axial-flow front length. Moreover, the fluctuations in front length lead to larger thermal
losses than if the front was kept at its average value, as described in section 2.5.1 and Ap-
pendix A. The thermal loss coefficient is calculated at each x position along the thermal
fronts and is plotted in figure 3.19(b). This illustrates that larger thermal losses are gen-
erated in radial-flow stores. Furthermore, steeper thermal gradients increase conductive
losses.
Figure 3.18(b) illustrates the trade-off between efficiency and returned available energy as
the charging frequency Π is varied. Axial-flow and radial-flow packed beds that are dimen-
sionally similar perform equally well. Radial-flow stores attain slightly higher efficiencies,
but axial-flow reservoirs can store slightly more available energy. The figure also shows
results for an axial-flow store with 32 segments. The segmented store does not attain high
efficiencies since its design has not been optimised.
The above studies indicate the radial-flow stores have a comparable performance to axial-
flow stores. An optimisation algorithm can be used to compare the best possible designs
of radial-flow and axial-flow stores. Additional factors and objectives such as economic
considerations and/or space requirements can be included. For instance, it is important
to determine the impact of the volume requirements of the inner and outer plenum on the
energy density. Optimisation studies are carried out in chapter 5.
3.4.5 Storage losses in radial-flow stores
The impact of conductive and leakage losses in radial packed beds during the storage phase
is evaluated in the same way as described in section 3.3.2. The radial heat equation is
∂T
∂t
=
α
r
∂
∂r
(
r
∂T
∂r
)
(3.40)
126
where α = keff/ρscs(1− ε). The heat equation is solved using the Crank-Nicolson numer-
ical scheme.
The development of conductive losses with time is shown in figure 3.20. As in section 3.3.2
the thermal front is initially linear and either crosses the full radius of the reservoir (“Full
profile”) or only the final fifth of the reservoir (“Realistic profile”). The “realistic” profile
is similar to the thermal profiles that were observed at the end of the charging phase. Con-
ductive losses are initially larger in the realistic profile due to steeper thermal gradients, but
the long-term losses are lower.
 0
 2
 4
 6
 8
 10
 12
 14
 16
 18
 0  1  2  3  4  5
A
va
ila
bi
lit
y 
lo
ss
es
, ζ
 
(%
)
Time (days)
Full profile - ζc
Realistic profile - ζc
Realistic profile - ζc + ζL
(a)
-150
-100
-50
 0
 0  0.25  0.5  0.75  1
Te
m
pe
ra
tu
re
 (°
C)
Normalised distance, x / L
(b)
Figure 3.20: Conductive and leakage losses during the storage phase of a cold radial-flow packed
bed reservoir with a linear thermal front. (a) Conductive losses during the storage phase of a cold
radial reservoir. The thermal front crosses either the full reservoir, or only the final fifth. The final
curve includes heat leakage losses and the profile crosses the final fifth of the reservoir. (b) Thermal
fronts in a cold radial reservoir. The front initially covers the final fifth of the reservoir. Profiles are
plotted every 3 days with the darkest line being 0 days.
It is clear that the conductive losses are substantially larger than those that occur in ax-
ial reservoirs (see figure 3.12). This is partly because the thermal front is initially much
steeper. In addition, conduction occurs across the full radius of the radial store unlike
segmented stores.
127
Heat leakage was included in a radial reservoir with a realistic profile in figure 3.20(a).
Leakage does not add significantly to the total availability losses due to the ‘self-insulating’
behaviour: unlike axial stores, the external surface temperature of the radial stores is never
significantly different from ambient temperature. However, as section 3.3.2 noted, suf-
ficient levels of insulation in axial-flow stores could reduce the leakage losses and also
conductive losses.
3.5 Summary
In this chapter, the detailed thermodynamic behaviour of packed-bed thermal reservoirs
was investigated. Nominal designs for hot and cold reservoirs were developed, and key
parameters were varied about these nominal values. These investigations improve under-
standing of packed bed thermal store behaviour. The uncertainties in the modelling and
the correlations used should be borne in mind when considering the quantitative results.
Experiments would provide useful validation for these methods. Nonetheless, the trends
that emerge are useful and can be thought of as design guidelines. The expected trade-off
between thermal losses and pressure losses was observed when geometric parameters such
as aspect ratio and particle diameter were varied. Pressure losses are less significant in hot
stores, meaning that optimal particle diameters are smaller, and optimal aspect ratios are
larger. The optimal value of these parameters typically depends on other variables, such as
the charging time, however.
Several different packing materials were compared, and it was found that the variation
in specific heat capacity over the range of operation was a key factor in determining the
thermal losses. The physical size of the packed beds was investigated by comparing stores
with different volumes but the same charging period. Again, a trade-off between thermal
and pressure losses led to there being an optimal volume. It was also found that larger
packed beds would benefit from having a lower aspect ratio, L/D and that the heat leakage
losses decreased as the reservoir volume was increased.
Parametric studies of the charging period revealed further conflicts between the round-trip
efficiency and the available energy that was returned during discharge: longer charging
periods created steeper fronts, leading to lower efficiencies and higher utilisation of the
128
packed beds.
The operating temperatures were varied and it was seen that the charging temperature
should be as large as possible for hot stores and as small as possible for cold stores to
increase efficiency. The study also suggested that the difference between charging and dis-
charging temperature should be minimised, although this is not practical as it would min-
imise the storage density. The operating temperature results are only applicable to packed
beds considered in isolation, since in PTES schemes the temperatures of the hot and cold
reservoirs are dependent on one another.
The concept of segmenting the packed beds in order to reduce the pressure losses was in-
troduced. By reducing pressure losses, segmentation allows smaller particles to be used,
thereby reducing thermal losses and creating steeper thermal gradients. As a result, seg-
mented packed beds were found to increase the efficiency and returned available energy
during operational periods. Segmentation also significantly reduced conductive losses dur-
ing storage periods. However, heat leakage during storage exacerbates the conductive
losses and the packed bed should be sufficiently insulated to avoid this.
Radial-flow packed beds were introduced. The governing equations were derived and some
simple parametric studies were undertaken. The radial-flow causes the mass flow rate per
unit area to decrease with radius which can affect the shape of the thermal front, and most
notably, significantly reduces the pressure losses. Radial-flow packed beds were found
to have comparable performance to segmented and unsegmented packed beds, and more
detailed investigations are recommended.
129
Chapter 4
Pumped Thermal Energy Storage (PTES)
130
4.1 Introduction
Pumped Thermal Energy Storage (PTES) is a novel energy storage system. Several vari-
ations exist and an overview is given in the literature review. This chapter is mainly con-
cerned with the PTES system that was under development by Isentropic Ltd. Energy is
stored in two packed-bed thermal reservoirs (one hot, one cold). During charge, the system
operates as a heat pump, moving energy from the cold store to the hot store. The process
is reversed in discharge, where the system behaves as a heat engine.
Figure 4.1 shows a schematic of the PTES layout, and a T–s diagram of ideal cycles pro-
posed by Isentropic Ltd. and Saipem. The main system components are two compression-
expansion devices (CE and EC) and two thermal stores (one hot, HS, and one cold, CS).
The discharged state of the reservoirs is set close to ambient temperature: 310 K in the
present case. With argon as the working fluid (as proposed in [107]), and with a pressure
ratio of 10:1, the nominal hot and cold storage temperatures (based on isentropic compres-
sion and expansion) are then 778 K and 123 K.
Following previous work, investigations centre on a nominal design with an energy storage
capacity of 16 MW h, and a power rating on 2 MW. For a reversible, adiabatic system, the
stored energy that can be converted back to useful work (i.e. the available energy) is the
difference between the stored internal energies of the two reservoirs
E = Mhs c
h
s (T2 − T3)−M csccs(T1 − T4) (4.1)
where Ms is the mass of storage material, cs is its average specific heat capacity over the
relevant temperature range, and the superscripts h and c refer to the hot and cold reservoirs
respectively. The mass of storage material in a reservoir is given by Ms = ρs(1 − ε)Vs
where ε is the void fraction. The reservoirs are sized by specifying that they charge in the
same period of time. Since the nominal charging time is given by tN = Mscs/m˙cp,g where
m˙ is the mass flow rate and cp,g is the specific heat capacity of the gas, if the reservoirs are
to charge in the same time then Mhs c
h
s = M
c
sc
c
s. Consequently, the stored energy is
E = Mhs c
h
s (T2 − T3 − T1 + T4) = M csccs(T2 − T3 − T1 + T4) (4.2)
131
-200
 0
 200
 400
 600
 800
 1000
 1200
-0.6 -0.4 -0.2  0  0.2  0.4  0.6
T
em
p
er
at
u
re
, 
(°
C
)
Specific entropy, s (kJ / kgK)
1.
2.
3.
4.
1.
2.
3.
4.
Isentropic Ltd.
Saipem
(a)
(1)(2)
(3) (4)
discharge
charge
cold front
hot front
HS CS
CE
EC
(b)
Figure 4.1: (a) T–s diagrams of ideal Joule-Brayton based PTES systems as proposed by Isentropic
Ltd. and SAIPEM. During charge the diagrams are traversed in the anti-clockwise direction. (b)
Schematic layout of an ideal PTES system during charge. Key: HS/CS hot/cold store; CE/EC
reversible compressor–expanders.
Therefore the hot reservoir volume can be calculated given a specific energy storage capac-
ity, and the cold store volume can be calculated from the equality of heat capacities.
As discussed in the previous chapters, magnetite (Fe3O4) provides a suitable storage ma-
terial due to its high heat capacity per unit volume, and its low fractional variation of heat
capacity over the temperature ranges of interest. Data for magnetite are given in table 4.1,
together with the required storage masses computed from equation 4.2 for 16 MWh of
storage. The reservoir volumes are also given in the table, calculated with an average void
fraction of 0.40. The nominal stores have an aspect ratio of L/D = 1, such that the internal
diameters are 4.5 m and 5.3 m for the hot and cold stores respectively.
The power output for a reversible, adiabatic PTES system is given by
W˙x = m˙cp,g [(T2 − T1)− (T3 − T4)] (4.3)
Using the temperatures listed above and data for argon (cp,g = 520 J kg−1K−1), the mass
flow rate required for a 2 MW device is 13.7 kg s−1.
132
P (bar) T (K) ρg (kg m−3) Mg (kg) c¯s (J kg−1K−1) Ms (tonne) V (m3)
Hot
Charge 10.5 778 6.5 196
860 234 75.5
Discharge 10.5 310 16.3 492
Cold
Charge 1.05 123 4.1 208
520 395 127.1
Discharge 1.05 310 1.63 83
Table 4.1: Hot and cold reservoir details for a nominal 16 MWh PTES system. The storage material
is Fe3O4 (density 5.175 tonne m−3) in the form of a packed bed with an assumed void fraction of
ε = 0.40. Further reservoir details are given in table 3.1.
Isentropic Ltd.’s design uses reciprocating machines as the compression and expansion
devices. A similar thermal energy storage scheme developed in France (SEPT) uses turbo-
machinery as described in chapter 1 and displayed in figure 4.1(a). The Isentropic design
has the advantage that only two machines are required: by changing the valve timings a
reciprocating device may operate as a compressor or an expander. This could potentially
lead to lower system costs, and as a result reciprocating devices are investigated rather than
turbomachinery.
Simple models of reciprocating devices are developed in this chapter which allow the re-
quired geometry to be calculated, and describe their behaviour. Developing realistic mod-
els of these components is important because preliminary studies have suggested that the
round-trip efficiency depends heavily on the compression and expansion efficiency, partic-
ularly since a full charge-discharge cycle involves four compression-expansion processes.
Other components, such as heat exchangers, buffer vessels, and electrical heaters are also
discussed. The previous chapter suggested that high efficiencies could be achieved in the
packed-beds and optimisation studies in chapter 5 suggest that efficiencies could reach
99.5% in hot stores and 95% in cold stores. Therefore, careful design of the reciprocating
devices is a necessity for PTES to be a competitive option.
133
4.1.1 Metrics of PTES performance
Generally, the round-trip efficiency of energy storage systems is defined as the ratio of
net work output during discharge to the net work input during charge1. In the present
study, only the “thermodynamic” efficiency χ is considered – i.e. excluding losses due to
electrical and mechanical processes. χ is given by
χ =
Net work output
Net work input
=
W netdis
W netchg
(4.4)
For a 2 MW machine using an induction motor-generator, electrical efficiencies of 97%
in each direction are commonplace, but mechanical losses for a custom-built reciprocating
devices are less easy to estimate. Howes [5] suggests a mechanical efficiency of 92%
might be possible in each direction. Together with the electrical losses, the electricity-to-
electricity round-trip efficiency is (ηmechηelec)2χ ≈ 0.8χ.
In chapter 1 simplified expressions for χ were developed and revealed the impact of design
parameters on the system performance. In this chapter, χ is evaluated by integrating the
power input and output over the duration of the charge-discharge cycle. This is necessary
because the power output changes as the hot and cold fronts reach the exits of the reservoirs.
Other useful performance metrics are the energy density ρE and power density ρP which
are defined as:
ρE =
Net work output
Reservoir volume
=
W netdis
VCR + VHR
(4.5)
ρP =
Net work output
Hot cylinder volume
=
W netdis
V CEs
(4.6)
where Vs indicates the swept volume which is discussed in more detail in section 4.2. The
power density typically increases monotonically with χ [84]. Maximising energy density
typically leads to a trade-off with efficiency and therefore ρE is related to cost. This chapter
focuses on thermodynamic metrics as economic factors tend to be uncertain (see chapter 5).
1Some storage systems, such as CAES, have a dual role of storage and generation, and a different defini-
tion of the round-trip efficiency is required.
134
4.2 Modelling of reciprocating devices
The theory of reciprocating compressors and expanders is less fully developed than for
packed beds. Their performance is compromised by mechanical irreversibilities (such as
friction and pressure losses) and thermal irreversibilities caused by heat transfer between
the fluid and the cylinder walls. A simplified model of reciprocating devices was developed
by White [160]. Heat losses are quantified by a fractional heat loss factor ν defined as
dqc = νdwc (4.7)
where dqc is the heat loss and dwc is the work input during an infinitesimal compression
process. The corresponding reversible work input is dp/ρ, regardless of whether the pro-
cess is adiabatic. Irreversibility may then be defined by introducing the (compression)
polytropic efficiency ηc, where
dwc =
dp
ρηc
(4.8)
As shown in Ref. [160] this leads to a polytropic expression of the form
τc = β
φc
c (4.9)
where τc and βc are the compression temperature and pressure ratios, and φc is the com-
pression polytropic exponent,
φc =
γ − 1
γ
(
1− νc
ηc
)
(4.10)
Similar expressions exist for the expander. In the polytropic model heat leakage losses
and irreversibilities are accounted for independently. On the other hand, isentropic models
are only suitable for adiabatic processes and cannot resolve the individual influences of
heat transfer and other irreversibilities. For example, during compression the effects of
irreversibility and heat transfer could ‘cancel’ one another out thereby leading to a process
which is isentropic but not ideal.
Pressure losses occur in pipework and valves and are included by specifying a fractional
pressure loss factor fp = ∆p/p. Losses occur at the inlet and outlet valves of the reciprocat-
135
ing devices such that each component has two pressure loss factors. For example, for the in-
let valve with an inlet p1 and outlet p′1 the pressure loss factor is given by fp = (p1−p′1)/p1.
This is a convenient definition, since the pressure loss factors are closely related to the ex-
ergetic pressure loss ζp (see equation 2.55). Thus, a given absolute pressure loss ∆p has a
greater impact on performance if it occurs in the low pressure part of the cycle.
Availability loss coefficients ζ
Second law analysis of the PTES system requires the loss in available energy in each com-
ponent to be calculated. Loss coefficients for various loss generating mechanisms in the
packed-bed thermal reservoirs were discussed in section 2.5. In reciprocating devices avail-
ability loss coefficients are given by the loss in available energy divided by the net work
input during charge, such that
ζ =
Blost
W netchg
(4.11)
whereas the availability loss coefficients derived in section 2.5 were divided by the net
inlet available energy into the thermal store. Note the terminology loss coefficient refers
to a fractional loss in available energy, whilst loss factor refers to an empirical factor that
determines the magnitude of a loss generating mechanism (η, ν or fp).
The irreversible entropy generation due to the pressure loss at each valve is given by:
S˙irr = −m˙R ln(1− fp) (4.12)
so that the pressure loss coefficient in the reciprocating devices is
ζp = −m˙RT0 ln(1− fp)
W netchg
(4.13)
In a compressor, the available energy loss coefficient corresponding to the polytropic effi-
ciency η is found by substituting νc = 0 into the expression for the polytropic exponent φc
and calculating the increase in entropy which leads to
ζt =
(
1− η
η
)
m˙RT0 ln β
W netchg
(4.14)
136
The availability loss coefficient associated with heat addition ζQ can then by found by
taking the increase in availability for the compression process (after the inlet valve and
before the outlet valve) and subtracting from it the thermal loss ζt and the work input wc
which leads to
ζQ =
m˙
W netchg
[
νccpT1
(
βφc − 1)
νc − 1 + 2
(
η − 1
η
)
RT0 ln β
]
(4.15)
Similar expressions can be derived for the expander.
The general model described above is equally applicable to turbomacinery and recipro-
cating devices but the present work focuses on the latter. Evaluating the availability loss
coefficients requires values for each of the loss factors η, ν and fp. In the next section a
simplified model of reciprocating device geometry is used to obtain order of magnitude
estimates of ν and fp. More detailed models of turbomachinery or reciprocating devices
could have been employed. However, the aim is to develop simple expressions that re-
veal the influence of design parameters on the loss factors. Due to the uncertainty of these
results, the sensitivity of PTES performance to each loss factor is explored in section. 4.4.1.
Simplified model of a reciprocating compressor
A schematic diagram of a reciprocating compressor is shown in figure 4.2(a). The com-
pressor has a diameter d and a stroke length Ls = 2r (where r is the crank length), and
rotates at an angular velocity ω. The pressure-volume diagram for the ideal compression
process is shown in figure 4.2(b) (ideal expansion processes are represented by the same
curve traversed in the opposite direction). After compression, the valves are opened at (2)
and the flow is discharged until the piston reaches top dead centre (TDC) at (3). At this
point the valves are closed and the piston reverses direction, thereby expanding the gas.
The valves are then re-opened at (4) for the suction phase, and closed again at bottom dead
centre (BDC) at (1). In the absence of heat leakage and irreversibility, processes (1)–(2)
and (3)–(4) are isentropic. The volumetric compression ratio is therefore
rv =
V1
V2
=
V4
V3
= β1/γ (4.16)
137
V1V3
Lc
Ls = 2r
ω
TDC BDC
r
d
(a)
    
P
re
ss
u
re
 (
b
ar
)
Volume (m
3
)
V1V2V3 V4
1.
2.3.
4.
C
om
pression
E
x
p
an
sio
n
Discharge
Suction
(b)
-15
-10
-5
 0
 5
 10
 15
pi/2 pi 3pi/2
P
is
to
n
 v
el
o
ci
ty
 (
m
/s
)
Crank angle (radians)
Velocity
Approx. vel.
(c)
Figure 4.2: (a) Schematic diagram of reciprocating piston. (b) p-V diagram for an ideal recipro-
cating compressor. The lines are followed anti-clockwise. The direction is reversed (but otherwise
identical) for an expander (c) Velocity variation in an ideal reciprocating compressor given by 4.23.
Approximate velocities given by vp = wr sin θ. Filled circles indicate where valves close and open
circles indicate valves opening for a cycle with a pressure ratio of 10.
138
where β is the pressure ratio. If the inlet valve pressure losses are small then the mass of
gas entering the cylinder during the suction stroke (4–1) is
min = ρin(V1 − V4) = ρinV1(1− εrv) (4.17)
where ρin is the inlet density and ε is the clearance (or dead) volume ratio given by ε =
V3/V1. The volumetric efficiency is therefore given by
ηv =
m
ρinVs
=
1− εrv
1− ε (4.18)
where Vs = V1 − V3 is the swept volume. The same expression for volumetric efficiency
may be applied to expanders but with the density evaluated at the outlet rather than the inlet.
(For instance, for the PTES scheme illustrated in figure 4.1, for charging compressors use
ρ1 and for charging expanders use ρ4).
The mass flow rate versus pressure ratio characteristic curve for the compressor is devel-
oped by considering the mass flow rate in and is thus,
m˙ =
ρ1Vsω
2pi
ηv =
ρ1Vsω
2pi
(
1− εβ1/γ)
(1− ε) (4.19)
This model is used to size the reciprocating devices for the PTES system. The mass flow
rate is fixed by the power rating (equation 4.3) and the swept volumes are then found
from equation 4.19. Nominal reciprocating device details are tabulated in table 4.2. These
designs have 6 cylinders, a nominal dead space ratio of 0.05, a speed of 1200 RPM and an
aspect ratio (stroke / diameter) of 0.25. This low aspect ratio is proposed in preliminary
designs by Isentropic Ltd. [5] on the grounds that the resulting low piston velocity leads
to low valve pressure losses and low inertial loading. Figure 4.3 illustrates the mass flow
characteristics of these nominal designs for different dead volume ratios. (Characteristics
for the compressor and expander are coincident, since the devices are sized to have the
same mass flow rate at the same pressure ratio). At low dead volume ratios the pressure
ratio can be varied without significantly effecting the mass flow rate. Small ε also increases
the volumetric efficiency of the device. However, in practice it may be challenging to
reduce ε too much below 5%.
139
Speed Vs (total) Ncyl Bore d Stroke Ls Clearance
(RPM) (m3) (m) (m) (mm)
Hot cylinders 1200 0.50 6 0.75 0.19 10
Cold cylinders 1200 0.20 6 0.55 0.14 7.4
Table 4.2: Details of the compression-expansion devices for a nominal power rating of 2 MW. The
nominal designs have a clearance ratio of ε = 5% and a cylinder aspect ratio of Ls/d = 0.25.
 0
 5
 10
 15
 20
 5  10  15  20
M
as
s 
fl
o
w
 r
at
e,
 m˙
 (
k
g
/s
)
Pressure ratio, β
ε = 1%
ε = 5%
ε = 10%
Figure 4.3: Mass flow rate–pressure ratio characteristic for reciprocating devices calculated using
equation 4.19 and data from table 4.2. Characteristics for the hot and cold cylinders are coincident.
4.2.1 Pressure loss estimation
Pressure losses occur in the reciprocating devices as the gas enters and exits the piston
chamber during the suction and discharge steps, respectively. These losses can be min-
imised by actuating the valves when there is no pressure difference across them and if the
valves open and close rapidly. An indicator diagram for a prototype device developed by
Isentropic Ltd [5] is shown in figure 4.4 and shows behaviour that is reasonably similar
to the ideal p–V diagram of figure 4.2(b). This suggests that the objective of significantly
reducing pressure losses may be achievable in practice. Isentropic Ltd. designed valves
that comprise two thin sheets of metal which are perforated with rectangular openings, as
140
shown in figure 4.5. When the valves are closed, the sheets lie over one another such that
there are no openings for the gas to flow through. The pressure difference keeps the sheets
in place until the pressure equilibrates across both sides and the opening force causes one
sheet to slide over the other, and allows the gas to flow through the perforations. This design
allows careful control of valve opening and closing times, and maximises the valve open
area which Isentropic Ltd. estimates is around 30% of the bore area [5]. Reportedly, this
design configuration also helps to reduce large scale turbulence after the valve. Pressure
Figure 4.4: p–V diagram from a prototype piston developed by Isentropic Ltd. which shows a
comparable shape to the ideal p–V diagram in figure 4.2(b). Figure is reproduced from [5].
losses are included in computational models by specifying a pressure loss factor fp for each
valve. These factors can be estimated by considering the schematic diagram in figure 4.6
whereby a single orifice represents the rectangular grid that makes up Isentropic’s valves.
Miller [150] suggests that this is a reasonable assumption, especially if the plate is thin in
comparison to the hole diameter. The orifice has an area Av such that it equals the open
area of the real valve (therefore σ = Av/Ap ≈ 0.30). A vena contracta (or narrowing of
the fluid jet due to streamline curvature ) occurs after the valve and the contraction area is
typically given by σvc = Avc/Av ≈ 0.64.
Assuming that the gas travels at the piston velocity, making use of the mass continuity
equation, and by applying the steady flow momentum equation between B–C and the steady
141
Figure 4.5: Prototype piston developed by Isentropic Ltd. [5] which illustrates the valve arrange-
ment that consists of many rectangular orifices.
vp
A B C
ApAv
Figure 4.6: Schematic diagram of gas at velocity vp passing through a valve of area Av into the
piston cylinder of area Ap
flow energy equation between A–B, the pressure loss is thus
∆p = pA − pC =
ρv2p
2
(
σσvc − 1
σσvc
)2
(4.20)
Where the density is approximately constant across the control volume at this instant in
time. The pressure loss factor is given by
fp =
∆p
p
=
v2p
2RT
(
σσvc − 1
σσvc
)2
(4.21)
Evaluation of fp requires the piston velocity vp, which can be derived by considering fig-
142
ure 4.2(a). The piston position xp is given by
xp =
√
L2c − r2 sin2 θ − r cos θ (4.22)
where Lc is the length of the connecting rod, r is the length of the crank arm, and θ is the
crank angle. Differentiating this expression gives the piston velocity as
vp = ω
r sin θ − r2 sin θ cos θ(
L2c − r2 sin2 θ
) 1
2
 (4.23)
A conservative pressure loss factor fp is estimated by taking the piston velocity at its max-
imum value vp = ωr [84] to give
fp =
ω2L2s
8RT
(
σσvc − 1
σσvc
)2
(4.24)
where Ls is the stroke length. Estimated values for the loss factors at inlet and outlet of the
compressor and expander are given in table 4.3. A more accurate estimate is obtained by
considering how the instantaneous pressure loss factor varies with piston velocity. An av-
erage pressure loss factor f¯p is given by the value which produces the same irreversible en-
tropy generation as that calculated by integrating instantaneous values of fp. For instance,
the irreversible entropy generation caused by an instantaneous pressure loss is given by
(compare with equation 2.55)
S˙irr = −m˙R ln (1− fp) (4.25)
The total irreversible entropy generation is found by integrating over the time that the valves
are open
Sirr = −R
∫
ln (1− fp) dm
= −ρrefRAp
∫
ln (1− fp) dx
= −ρrefRAp
ω
∫ θ2
θ1
vp ln (1− fp) dθ (4.26)
143
Ls/d = 0.25 Ls/d = 1.00
vp,max fp f¯p vp,max fp f¯p
(m/s) (%) (%) (m/s) (%) (%)
Hot cylinder inlet
11.9
1.96 1.43
29.5
12.2 6.92
Hot cylinder outlet 0.78 0.33 4.84 1.49
Cold cylinder inlet
8.8
1.06 0.45
21.9
6.60 2.90
Cold cylinder outlet 2.67 1.96 16.6 12.54
Table 4.3: Pressure loss factors in reciprocating devices. fp is calculated using the maximum
velocity (as in [84]) whereas f¯p is calculated from equation 4.28. Values are calculated for charging
compressors (hot cylinder) and charging expanders (cold cylinder).
In these expressions ρref is the gas density for the particular process. For instance, in the
charging compressor, ρref = ρ1 for the suction pressure loss factor, and ρref = ρ2 for the
discharge pressure loss factor. θ1 and θ2 represent the crank angles at which the valves
open and close respectively. The crank angle at which the charge compressor opens its
valves during suction can be found by considering the volume (and thereby xp) at which
the correct pressure ratio is achieved. In this case, the valve closing crank angle is simply
given by θ2 = 2pi.
Setting equation 4.26 equal to the entropy generation that would occur for some average
value of pressure loss factor f¯p gives
− ρrefApLR ln
(
1− f¯p
)
= −ρrefRAp
ω
∫ θ2
θ1
vp ln (1− fp) dθ (4.27)
where L is the distance that the piston travels whilst the valves are open. In the charge
compressor, during suction L = x1− x4 and during discharge L = x2− x3. Consequently,
the entropy averaged value of pressure loss factor is given by
f¯p = 1− exp
{
1
ωL
∫ θ2
θ1
vp ln (1− fp) dθ
}
(4.28)
Calculated values for f¯p are given in table 4.3. As expected, these values are lower than
the pressure loss factors using the maximum gas velocity, and indicate that for the given
design, values of fp of around 1% are reasonable. Table 4.3 also indicates that the lower
the aspect ratio Ls/d, the lower the valve pressure losses.
144
4.2.2 Heat transfer losses
In theory, the time-mean heat leakage can be reduced to very small levels with sufficient
insulation. Heat leakage factors have therefore been set to 2% for baseline calculations.
However, even if the processes are globally adiabatic, losses are still incurred due to time-
varying heat transfer to and from the cylinder walls. To estimate the magnitude of this loss,
the irreversibility is divided into three parts: heat transfer between the cylinder walls and the
inlet (or outlet) gas, heat transfer between the gas and the walls during the compression and
expansion strokes, and a mixing loss between the residual gas and the inlet gas. The loss
in efficiency due to these three processes is estimated for compression, and it is assumed a
similar loss factor applies to expanders.
Heat transfer during induction and delivery
The walls of the compressor have thermal inertia and are likely to be at a temperature
greater than the cool inlet gas, and lower than the compressed outlet gas. The gas that en-
ters the compressor is preheated by the walls prior to the compression stroke. Similarly, the
gas is cooled by the walls during the delivery stroke. The net effect is that heat is transferred
from the high exit temperature down to the low inlet temperature, as illustrated schemati-
cally in figure 4.7. Analysis of this problem was carried out in the appendix of [84]. To
simplify the problem, the complete process (1–2) is assumed adiabatic and the compressor
(1’–2’) is isentropic. Since the wall temperature is between the inlet and outlet tempera-
tures it is written as Tw = T1 + θ(T2 − T1) where 0 < θ < 1. The heat transferred to the
inlet gas can be written as
Q = hS(Tw − T1)τ = hSθ(T2 − T1)τ (4.29)
where h is an average heat transfer coefficient, S is the average internal surface area and τ
is the duration of the induction stroke. The heat transfer increases the enthalpy of the inlet
mass according to
Q = mcp(T
′
1 − T1) = ρ1vpAcp(T ′1 − T1)τ (4.30)
145
21
1’
2’
s
T
1
2
1’
2’
C
Q
Q
Figure 4.7: Simplified model of the irreversibility associated with preheating during induction and
cooling during delivery, shown for a compression process. The overall process 1–2 is assumed to
be adiabatic.
where vp is the average piston speed during induction and A is the piston face area. Equat-
ing the two expressions for Q gives
T ′1 − T1 = Stθ
S
A
(T2 − T1) = T ′2 − T2 (4.31)
where St is the Stanton number (based on average piston speed). The equality on the right-
hand side is a result of the full process being adiabatic. By inspection of the T–s diagram
of figure 4.7 and using the above equations, the isentropic efficiency is given by
η =
T ′2 − T ′1
T2 − T1 (4.32)
= 1− κ [β(γ−1)/γ − 1] (4.33)
where κ = StθS/A which represents the heat transferred during induction as a fraction of
the work transfer. As expected, this fraction is independent of the pressure ratio β although
efficiency losses are larger at high pressure ratios due to the larger temperature difference
across which heat is transferred. The fraction κ is difficult to accurately determine. A
rough estimate involves setting θ = 1/2 and approximating the Stanton number from a
turbulent pipe flow correlation, such as St = 0.023Re−1/5Pr−3/5. For the nominal design
conditions given in table 4.2 this gives κ = 0.0025 which corresponds to an efficiency loss
of approximately 0.4% at a pressure ratio of 10:1.
146
Heat transfer during compression and expansion strokes
Heat transfer occurs during the compression and expansion strokes (1–2 and 3–4 on fig-
ure 4.2(b), respectively). It is no longer appropriate to use a simple heat transfer coeffi-
cient due to the interaction between heat and work transfers, and the inherently unsteady
behaviour of these two processes. Lawton [8] explains this behaviour and notes that in gen-
eral the heat flux is not in phase with the temperature difference. For instance, figure 4.8
is a schematic of the temperature variation across a cylinder during compression which
illustrates the interaction between the heat (diffusion in the boundary layer) and work (adi-
abatic compression) transfers. At the beginning of compression, the wall temperature is
greater than the core temperature, and heat transfer occurs from the wall to the gas. During
compression the temperature of the gas increases. In the absence of conduction, the tem-
perature at point 1 which is located in the thermal boundary layer increases above the wall
temperature. Consequently, the wall is heated by the gas despite the core gas temperature
being lower than the wall temperature. During expansion, the reverse effect occurs, and the
wall is cooled even though the core is hotter than the wall. As a result, the heat flux is not
always in phase with the temperature difference and this can be modelled with a complex
Nusselt number.
A number of studies have attempted to quantify the above effect in the context of gas
springs. These comprise a fixed mass of gas enclosed in a cylinder and a piston but with no
flow of gas into or out of the system. Ideal gas springs are fully reversible, such that com-
pression and expansion lines on the p–V diagram are coincident. However, heat transfer
irreversibilities result in a hysteresis loss such that there is a net work input into the system.
Early work by Lee [7] led to the development of a simple analytical expression for this loss
which fits experimental data reasonably well, as reported by Kornhauser and Smith [6] and
reproduced in figure 4.9. Minor modifications were made to the theory in [6] to improve
the fit. In more recent work by Mathie et al. [161] an extended semi-analytical model was
proposed for the full conjugate (solid-gas) thermal problem in gas-springs.
Kornhauser et al.’s results are shown in figure 4.10. The loss is expressed as a reduction in
the polytropic efficiency where half of the hysteresis loss is attributed to the compression
stroke. The horizontal axis in this figure is the Peclet number, Pe = ωD2h/2αt, where
Dh is the cylinder mean hydraulic diameter, and αt is the mean thermal diffusivity of the
147






Twall
Tcore
Tcore
Piston
Piston
A
A
Figure 4.8: Schematic of heat transfer in a reciprocating device during compression. The figure
on the left shows the beginning of compression where the wall temperature is greater than the core
fluid temperature. The right-hand figure shows the compression at a later stage, where at some
points (e.g. A) the boundary layer gas temperature is greater than the wall temperature.
Figure 4.9: Experimental and theoretical results for heat transfer losses in gas springs. Figure is
taken from Lee [6]. The figure shows the analytical expression developed by Lee [7] as well as
isothermal and adiabatic limits, and experimental results.
148
 0
 2
 4
 6
 8
 10
 12
 0.1  1  10  100  1000  10000
L
o
ss
 i
n
 e
ff
ic
ie
n
cy
 (
%
)
Peclet number, Pe
Increasing pressure ratio
from 2.5 to 10.0
Gas spring theory
Laminar diffusivity
Turbulent diffusivity
Figure 4.10: Loss in efficiency due to heat transfer irreversibilities in gas springs. Gas spring theory
uses the expression by Kornhauser [6].
gas. The Peclet number for the nominal design of hot cylinder is 34 000, suggesting an
efficiency loss of around 0.3%. However, the thermal dissipation loss in a real compressor
(as opposed to a gas spring) may be considerably larger because the eddying and turbulent
motion generated by the inlet valve is likely to enhance the effective diffusivity. In this
context, Lawton [8] found that good agreement between theory and experiment for heat
transfer rates in a motored piston engine (albeit with a different valve design) could be
obtained by multiplying the gas conductivity by 25 as shown in figure 4.11. This would
reduce the Peclet number to 1360 and thereby increase the loss to 1.3%. Other models
(such as that in [161]) predict higher losses than this.
Mixing losses
Due to irreversibility and heat transfer during processes 1→4, the residual gas in the cylin-
der may end up at a different temperature to that entering the cylinder, thereby giving rise
to an additional mixing loss. The mixing loss can be estimated in the following way. The
149
Figure 4.11: Experimental and theoretical results for heat transfer in reciprocating devices. Figure
is from Lawton [8] and shows the cylinder wall heat flux against the crank angle.
gas that enters the cylinder has temperature T1 and its mass mi is
mi = ρ1V1
(
1− εβ1/γ) (4.34)
The residual gas has temperature T4 which can be estimated from polytropic relationships.
For a polytropic efficiency of 97.5%, T4 ≈ 325 K. The mass of the residual gas mr is
approximately given by
mr = ρ4εV1
(
β1/γ − 1) (4.35)
The temperature at which the inlet and residual gas equilibrate Tf is found by equating the
heat flux, for example
mrcp(T4 − Tf ) = micp(Tf − T1) (4.36)
For the current case, the final temperature is approximately 314 K. The entropy increase
due to mixing is given by
∆Sirr = (mi +mr)cp ln
(
Tf
T0
)
−micp ln
(
T1
T0
)
−mrcp ln
(
T4
T0
)
(4.37)
The loss factor is then given the loss of available energy normalised by the work input in
the adiabatic case Wad
ζ =
T0∆Sirr
Wad
(4.38)
150
The adiabatic work is the work input during compression for a gas-spring (rather than a
compressor) so that the mixing loss factors are calculated in a consistent manner with the
other heat transfer irreversibilities. For a gas spring where compression occurs between V1
and V3 the adiabatic work input is
Wad =
∫ 3
1
pdV =
p1V1
1− γ
(
ε1−γ − 1) (4.39)
For the case described above, the mixing loss factor is around 0.01%. When the residual
gas is above the expected temperature, the residual mass is small compared to the inlet
mass thereby giving rise to the small mixing losses.
Concluding remarks on heat transfer losses
The simplified analysis above is somewhat uncertain due to the interplay between a number
of heat transfer processes within the reciprocating devices. In addition, it is unclear exactly
what impact the turbulent eddies of the gas flow through the valves will have. As a result,
accurate estimates of efficiency require detailed experimental investigations.
4.3 Other components
4.3.1 Heat Exchangers
Heat exchangers reject heat and are required for steady state cyclic operation to be achieved.
They are placed at points 1 and 3 in figure 4.12 where they are labelled HX1 and HX3.
These locations are closest to ambient temperature and are therefore the most efficient
locations for heat rejection to occur. Figure 4.12(b) illustrates the T–s diagram for a non-
ideal PTES system. Irreversibilities in the compressors and expanders, plus the progression
of the thermal fronts out of the stores leads to a need for heat rejection.
Heat exchanger design should consider a wide range of factors such as the required inlet
and outlet temperatures of gas flow, the heat exchanger geometry (shell and tube, cross
flow, multiple passes, fin shapes and sizes etc.), the gas or liquid to which heat is rejected
151
HX1
HX3
CE
EC
(1)(2)
(3) (4)
discharge
charge
cold front
hot front
HS CS
(a)
-200
-100
 0
 100
 200
 300
 400
 500
 600
-0.5 -0.4 -0.3 -0.2 -0.1  0  0.1
T
em
p
er
at
u
re
, 
°
C
Specific entropy, KJ/kgK
1.
2.
3.
4.
heat exchange
Charge
Discharge
(b)
Figure 4.12: (a) Schematic layout of a PTES system during charge. Key: HS/CS hot/cold store;
CE/EC reversible compressor–expanders; HX1/HX3 heat exchangers. (b) T–s diagrams of a PTES
system during charge and discharge. During charge the diagrams are traversed in the anti-clockwise
direction. In this example, the discharging cycle operates at a lower pressure ratio and heat rejection
occurs at both HX1 and HX3.
(and its mass flow rate), the desired performance and the economic cost of these design
requirements [162]. The modelling is simplified here by considering two parameters: the
effectiveness  governs the efficiency of the heat transfer processes, and a pressure loss
factor fp which accounts for frictional pressure drops.
The effectiveness is defined as the ratio between the actual rate of heat transfer and the
maximum possible rate of heat transfer. For a heat exchanger where gas enters at a high
temperature Tg,i and is cooled to Tg,o by water which enters at Tw,i, the effectiveness is
given by
 =
m˙gcp,g(Tg,i − Tg,o)
(m˙cp)min (Tg,i − Tw,i)
(4.40)
where (m˙cp)min is the minimum heat capacity rate of the two flows. For the outlet gas flow
to be as close to the water inlet temperature as possible (i.e. to maximise the heat transfer)
the minimum heat capacity rate should equal that of the gas. The outlet gas temperature is
152
thus
Tg,o = Tg,i −  (Tg,i − Tw,i) (4.41)
The lost work associated with heat rejection may be calculated by finding the change in
available energy between the inlet and oulet gas, which, in the case of zero pressure losses
is given by
S˙irr = m˙(bi − bo) = m˙cp
(
Tg,i − Tg,o − T0 ln
(
Tg,i
Tg,o
))
(4.42)
The effectiveness may be calculated using approaches such as the NTU- method. How-
ever, this requires geometric data, and it is more revealing to choose a value of  and con-
sider its impact on the behaviour of the cycle. Heat exchanger design can begin in earnest
once its importance relative to other component parameters has been determined. Nonethe-
less, it is useful to estimate the magnitude of  and, following [160], data from a compact
heat exchanger of surface type 8.0 − 3/8T has been employed. This design has finned
cylindrical tubes which are perpendicular to the flow direction. The purpose of the fins is
to enhance the heat transfer rate and provide structural support. The fin geometry means
that the exchanger can be used in scenarios where one stream is at a higher pressure than
the other, as may be the case in PTES systems [162]. Data for an 8.0 − 3/8T exchanger
is available in Kays and London [163]. Heat transfer results correlate the Colburn j factor
jH = StPr
2/3 against Reynolds number from which it is possible to calculate the overall
heat transfer coefficient. An approximate value of the effectiveness can be obtained by
using the standard result for a cross-flow heat exchanger [137]. The effectiveness is eval-
uated for several heat exchangers (HX3) of different volumes in table 4.4. These results
use 8.0 − 3/8T exchanger data and assume the heat capacity flow rates of water and gas
are equal (resulting in a water mass-flow-rate of ∼1.7 kg s−1). An average inlet gas tem-
perature is 50°C is also assumed. An effectiveness of 80% is achieved using a reasonably
compact heat exchanger that has a total volume of 1 m3. To achieve an effectiveness close
to 90% requires a much larger heat exchanger of 5m3. Sensitivity of PTES performance to
the heat exchanger effectiveness is investigated in section 4.4.1.
Pressure losses are the result of friction along the heat exchanger, and losses at the in-
let/outlet manifolds and in the inlet/outlet ducts or valves. Like the reciprocating devices,
losses are defined using pressure loss factors where fp = ∆p/p. Losses in the core of
the heat exchanger are estimated in this section, whilst those that arise in the connecting
153
HX3 volume (m3) 0.5 1.0 5.0
Side wall length, Lhx (m) 0.80 1.0 1.70
Heat exchange area, Ax (m2) 290 590 2900
Reynolds number, Re 5400 3400 1200
Overall heat transfer coefficient, U (W m−2K−1) 145 116 63
Effectiveness,  0.77 0.81 0.87
Friction factor, fv 0.022 0.025 0.03
Pressure loss factor, fp = ∆p/p (%) 0.11 0.06 0.014
Table 4.4: Approximate effectiveness and pressure loss factors for several heat exchangers of dif-
ferent sizes for HX3 with a 8.0− 3/8T surface [160, 162, 163].
pipes/manifolds are discussed in 4.3.3.
Losses are estimated for the 8.0−3/8T exchanger described above. Kays and London [163]
provide correlations for the friction factor fv and Reynolds number. The pressure drop
across the cylinders is given by [160, 162]
∆P = 1/2ρv2xfv
Ax
Aff
(4.43)
where vx is the maximum gas velocity in the heat exchanger which occurs at the minimum
free flow area Aff , and Ax is the area over which heat transfer takes place. Pressure losses
for several heat exchangers are tabulated in table 4.4 from which it is clear that within the
heat exchanger they are very small (around 0.1%). Losses in additional pipework are more
significant and should be included.
The above discussion gives an indication of the typical performance and size of the heat
exchangers in a PTES system. There are a number of additional practical considerations.
For instance, the inlet water temperature Tw,i may typically be around 15°C, whereas for
the nominal PTES design the desired gas outlet temperature from the heat exchanger is
37°C. Consequently, the heat exchanger would not seek to cool the inlet gas down to Tw,i,
but rather increase the water temperature to that of the inlet gas flow. Furthermore, as the
thermal front emerges from the packed bed, the inlet gas temperature changes. To maintain
a constant outlet gas temperature, the mass flow rate of water should be varied. As the
operating point changes, the performance of the heat exchanger may also be affected.
For the cycle calculations presented in the rest of this chapter, the heat exchanger behaviour
154
is simplified and its performance is governed by a factor ∗ that corresponds to the change
in temperature of the gas as a fraction of the total required change, that is:
∗ =
Tg,i − Tg,x
Tg,i − Tg,o (4.44)
where Tg,o and Tg,x are the ideal scenario and obtained outlet gas temperatures respectively.
This definition simplifies the integration of heat exchangers into the PTES system, and
gives a sufficient indication of their influence on PTES behaviour.
4.3.2 Electric heater
In some cases, the outlet temperature of a packed bed may fall below the desired value.
For instance, during charge the thermal front in the cold reservoir will eventually emerge
from the exit. The exit temperature will then be less than T1 and heat addition is required to
maintain cyclic operation. Heat addition may be provided by an electric heater, for which
the work input is given by
W˙h = m˙cp,g (Tg,o − Tg,i) (4.45)
If the heater does not heat the gas back up to the required temperature, a ‘heater effective-
ness’ h can be defined in the same way as ∗ for the heat exchangers. Availability losses
are given by the work input minus the increase in availability of the gas. In this chapter,
heater losses are combined with the heat exchanger loss.
These additional work inputs (plus the expense of installing a heater) should be avoided,
and section 4.5 investigates designs that achieve this.
4.3.3 Pipe work and valves
Pressure losses occur in the pipes that connect the packed beds, reciprocating devices and
heat exchangers. Without detailed knowledge of the geometry it is difficult to accurately
calculate these losses. Factors that should be considered are the length, diameter and rough-
ness of the pipes, the number of bends, expansions and contractions and the precise geome-
try of these individual features [164]. For instance, a sudden expansion will cause a greater
155
Pipe location Pressure (bar) Temperature (K) Diameter (m) ∆P/P ∗ (%)
1. 1.05 310 0.55 1.0
2. 10.5 778 0.22 0.026
3. 10.5 310 0.17 0.010
4. 1.05 123 0.43 0.41
Table 4.5: Pressure loss factors in pipes. Each pipe section corresponds to a point on the PTES
schematic in figure 4.1. The diameters are those which give a pressure loss factor of 1%. ∗ The final
column is the pressure loss factor for a pipe of diameter 0.55 m.
pressure loss than a gradual expansion, and a tight bend could lead to flow separation and
secondary flows [165].
The pressure loss ∆P can be written as
∆P = 1/2ρV 2K (4.46)
where K is an empirical factor that depends on the flow conditions and geometry. Values
of K are given in several textbooks [164, 165], and Miller [150] provides data for a wide
range of pipe types. Use of this data requires detailed geometric information, which at this
early design stage is unspecified. Instead, the pressure losses are assumed to be of the order
of 1/2ρV 2, as generally K ∼ 1. By keeping these losses within 1% of the local pressure
it is possible to calculate the required pipe diameters at each point around the cycle as
in [84], and these are shown in table 4.5. The largest pipe diameter occurs in section 1 and
is around 0.55 m. The pressure loss factors are also evaluated for the case where all the
pipes are identical with diameter 0.55 m.
In the PTES model pressure loss factors are incorporated with the compressor and expander
valve pressure losses to reduce the number of variables. This is appropriate because these
loss factors are additive if they are reasonably small, as is the case here: the new pressure
P2 after a number of loss factors fi have been applied is P2 = P1
∏
(1 − fi). If the fi are
small then the expression becomes P2 ≈ P1(1−
∑
fi).
156
4.3.4 Buffer tank
A buffer tank is required because the total mass of gas within the packed beds changes
during charge. The change in mass of gas is given by
∆m =
εV P
R
(
1
Tchg
− 1
Tdis
)
(4.47)
where V is the volume of the reservoir, and Tchg and Tdis are the charging and the dis-
charging temperatures. Table 4.1 shows the masses of charged and discharged hot and cold
reservoirs, and indicates that the total change in mass between charge and discharge is 171
kg. Locating the buffer tank at point 3 on diagram 4.1 minimises the required volume which
is then 10.5 m3, but this does require the vessel to be pressurised, and insulation may also
be necessary. Pressure losses occur in the valves and pipes for both filling and emptying
of the tanks. These losses are taken into account by the other pressure loss factors in the
cycle.
4.4 Behaviour of PTES systems
The behaviour of PTES systems is investigated using parametric studies. A nominal 16
MWh, 2 MW design has been chosen as described at the beginning of chapter 3. The
nominal design has a thermodynamic efficiency of χ = 67.4%, an energy density of ρE =
34.3 kWh m−3 (123.5 MJ m−3) and a power density of ρP = 2.9 MW m−3. Values for each
of the nominal design parameters are given in table 3.1. These parameters are individually
varied through a range of values to investigate the impact on PTES performance. The
nominal design uses a polytropic efficiency of 98% and values in this range have been used
in the literature before [160, 166]. Detailed simulation results suggest that reciprocating
devices could achieve indicated efficiencies of 87–93% if leakage flow and pressure losses
are reduced [167].
157
fp 1− η ν fp,hx 1− ∗
Nominal 0.02 0.02 0.02 0.02 0.20
Minimum 0.00 0.00 0.00 0.00 0.00
Maximum 0.04 0.04 0.04 0.04 0.40
Table 4.6: Nominal values of loss factors, and minimum and maximum values used in parametric
studies
4.4.1 Sensitivity to loss factors
As discussed above, the availability losses that occur in the reciprocating devices, heat
exchangers and pipe work are defined in terms of loss factors. The sensitivity of the round-
trip efficiency to each of the loss factors was investigated by White et al. [83] as part of a
simplified analysis of a PTES system. Ref. [83] developed an expression for the round-trip
efficiency χ in terms of the temperature ratios τ = T2/T1 and θ = T3/T1 and included the
effect of pressure loss fp, heat leakage ν and irreversibilities η. Partial derivatives of χ with
each loss factor indicate the sensitivity to these factors. For instance, the sensitivity to the
compression/expansion heat leakage factor is defined as
Sν = −∂χ
∂ν
(4.48)
According to [83] Sν = 2.0 and is independent of τ which means that there will be a 2%
increase in losses for each 1% of heat leakage. Results from [83] are shown in figure 4.13
for various values of the isentropic temperature ratio τ . For the nominal design, τ = 2.5,
and from the figure 4.13 it can be seen that Sη = 2.0, Sν = 2.0 and Sfp = 0.9. This anal-
ysis suggests that the PTES round-trip efficiency is susceptible to polytropic efficiencies,
especially at low pressure ratios.
Sensitivity to the loss factors was explored using the full PTES thermodynamic model.
Nominal values for the loss factors are given in table 4.6, as are the maximum and mini-
mum values. Figure 4.14 shows how the round-trip efficiency changes as the loss factors
are varied and the results support the simplified analysis of Ref. [83]. For instance, the effi-
ciency drops around 2% for every 1% decrease in the polytropic efficiency. Figure 4.15(a)
shows the distribution of availability losses in each component, from which it can be seen
that the polytropic efficiency significantly affects the losses in the reciprocating devices.
158
 0
 1
 2
 3
 4
 5
 6
 1  1.5  2  2.5  3  3.5  4
S
en
si
ti
v
it
y
 t
o
 l
o
ss
 f
ac
to
rs
Isentropic temperature ratio, τ
Sν 
Sfp
Sη 
Figure 4.13: Sensitivity of thermodynamic PTES round-trip efficiency to loss factors. Results are
from a simplified model of PTES in [83]. Open circles indicate the sensitivities at τ = 2.5, the
nominal PTES design.
Heat exchanger losses are also affected, although the net impact remains fairly constant.
The heat loss factor ν has a slightly larger impact than that predicted by Ref. [83]. Fig-
ure 4.15(b) indicates that the reciprocating device losses significantly affected by ν as ex-
pected. In addition, heat exchanger losses increase to a similar extent because of increased
exit losses from the packed beds. Heat exchanger behaviour was not included in the analy-
sis of Ref. [83] which accounts for the difference in the results.
These results highlight the importance of minimising heat leakage from the reciprocat-
ing devices, and for more thoroughly understanding the thermal irreversibility that occurs
within them.
Note that these results are also applicable to compression and expansion with turboma-
chinery. Heat leakage losses may be less significant in turbomachines but the polytropic
efficiency will be just as influential.
The heat exchanger loss factor 1 − ∗ has quite a small effect on the round-trip efficiency.
As figure 4.16(a) shows, losses in the thermal reservoirs increase if the heat exchanger ef-
159
Figure 4.14: Sensitivity of thermodynamic round-trip efficiency to loss factors
fectiveness is increased. This is surprising, but is due to changes in the inlet temperature of
the reservoirs when ∗ < 1, as shown in figure 4.16(b). For instance, during discharge, the
inlet temperature to the discharge compressor increases above T4 as the cold front emerges,
and this leads to a compressor outlet temperature above T3. As HX3 does not reject all the
required heat, the hot reservoir inlet temperature during discharge is greater than the design
value of T3.
Figure 4.16(b) illustrates how hot store thermal profiles are affected by the heat exchanger
effectiveness. When ∗ < 1, the thermal gradients occur across a greater proportion of
the store. However, the thermal front is typically less steep and as a result, thermal losses
actually reduce as indicated by considering the area under the curve of (Tg − Ts)2 /TgTs in
figure 4.16(b). (This expression scales with the thermal losses, see section 2.5.1.)
Overall, reducing the heat exchanger performance ∗ has a negative effect on the round-
160
 0
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 4
 6
 8
 10
 0  0.01  0.02  0.03  0.04
A
v
ai
la
b
il
it
y
 l
o
ss
es
 (
%
)
1 - η
Compressor
Expander
Hot store
Cold store
HX1
HX3
(a)
 3
 4
 5
 6
 7
 8
 9
 10
 0  0.01  0.02  0.03  0.04
A
v
ai
la
b
il
it
y
 l
o
ss
es
 (
%
)
ν
Compressor
Expander
Hot store
Cold store
HX1
HX3
(b)
Figure 4.15: Sensitivity of availability losses in each PTES component to variations in (a) poly-
tropic efficiency η (b) heat leakage factor ν
trip efficiency, which is due to an increase in losses in the heat exchangers themselves:
the irregular thermal fronts mean that the packed-bed outlet temperatures are never at the
design value such that heat is rejected over the entire duration of the cycle.
4.4.2 Further parametric studies
Parametric studies of the main PTES design variables are now carried out. Parameters are
varied about their nominal values as given in table 4.7, whilst the remaining variables are
fixed at their nominal values.
Geometric parameters Operating conditions
(L/D)HS (L/D)CS dp (mm) Nseg T1 (K) T3 (K) βchg βdis Π
Nominal 1.00 1.00 20.0 1 310 310 10.0 10.0 0.75
Minimum 0.10 0.10 2.0 1 273 273 5.0 5.0 0.10
Maximum 10.0 2.0 50.0 32 347 347 15.0 15.0 1.00
Table 4.7: PTES variables that are used in the parametric study. Minimum and maximum values
indicate the range through which the parameters are varied.
161
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 8
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 0  0.1  0.2  0.3  0.4
A
v
ai
la
b
il
it
y
 l
o
ss
es
 (
%
)
1 - ε*
Compressor
Expander
Hot store
Cold store
HX1
HX3
(a)
 0
 100
 200
 300
 400
 500
 0  1  2  3  4
 0
 2
 4
 6
 8
 10
G
as
 t
em
p
er
at
u
re
, 
T
g
 (
°
C
)
[(
T
g
 -
 T
s)
2
 /
 T
g
T
s]
  
 x
 1
0
-4
Distance along reservoir, x (m)
A B
ε* = 0.6
ε* = 1.0
(b)
Figure 4.16: Impact of heat exchanger loss factor ∗ on PTES performance (a) Distribution of
availability losses in various components (b) Influence of ∗ on thermal profiles in a hot packed bed.
Profiles are shown at the start of charge (A) and the end of charge (B). Dashed lines show thermal
losses of (A) on the right-hand axis.
Geometric parameters
The geometric parameters that are varied are the hot and cold reservoir aspect ratio (L/D),
particle diameter (dp) and number of segments (Nseg). The maximum in efficiency that is
evident in figure 4.17 reflects the trade-off between heat transfer losses and pressure losses
as discussed in detail in chapter 3. For example, small particles give large heat exchange
area but increase the frictional effects. Likewise, long reservoirs give lower thermal losses,
but the associated reduction in cross-sectional area implies higher fluid velocities and hence
higher pressure drop. Due to the lower average gas density, pressure losses are more sig-
nificant in the cold reservoir and consequently the optimal aspect ratio is lower and the
optimal particle size larger than for the hot reservoir. However, it should be recalled that
these results have been obtained with all the other parameters fixed at their nominal values,
and the picture changes when overall system optimisation is considered.
In chapter 3 it was found that segmentation of the packed-beds could have a number of
162
 60
 62
 64
 66
 68
 70
 72
-0.4 -0.3 -0.2 -0.1  0  0.1  0.2  0.3  0.4
E
ff
ic
ie
n
cy
, 
χ 
(%
)
Normalised parameter (x - xlo) / (xhi - xlo)
(L / D)HS
(L / D)CS
(dp)HS
(dp)CS
Figure 4.17: Variation of thermodynamic round-trip efficiency with packed-bed geometric param-
eters. Minimum, nominal and maximum values are given in table 4.7.
advantages, such as reduced pressure and storage losses, increased utilisation of the reser-
voirs, and more uniform power output. However, these benefits could only be achieved
under certain conditions. For instance, storage losses were only reduced if heat leakage
was small compared to axial conduction. Furthermore, the main benefit of segmentation
was that it allows smaller particles to be used thereby reducing thermal losses without in-
curring significant pressure losses. As a result, segmentation is most beneficial in cold
packed beds where pressure losses are reasonably large.
Figure 4.18 illustrates the impact of a layered cold store on the round-trip efficiency of a
PTES system. Calculations are carried out using particle diameters in the cold store of 4
mm and 10 mm, since there is no benefit to segmenting the packed bed at larger particle
sizes. For instance, losses can be reduced by 1.4% when dp = 4 mm, but only by 0.4%
for dp = 10 mm. Furthermore, most of the loss reduction can be achieved with only eight
segments, and the benefit of additional segments diminishes. Figure 4.18 shows that seg-
mentation has no impact on the performance of the reciprocating devices or the hot store.
However, heat exchanger losses do increase by a small amount as a result of the presence
of the saw-tooth wake discussed in section 3.3. For the particular case when (dp)CS = 4
163
 0
 5
 10
 15
 20
 25
 30
1 8 16 32
A
v
ai
la
b
il
it
y
 l
o
ss
es
 (
%
)
Number of segments
CE
EC
HS
CS
HX1
HX3
7.0 7.0 7.0 7.0
6.7 6.7 6.7 6.7
3.5 3.5 3.5 3.5
4.5 3.5 3.3 3.1
3.8 4.1 4.2 4.3
4.3 4.2 4.2 4.2
(a)
 0
 5
 10
 15
 20
 25
 30
1 8 16 32
A
v
ai
la
b
il
it
y
 l
o
ss
es
 (
%
)
Number of segments
CE
EC
HS
CS
HX1
HX3
7.0 7.0 7.0 7.0
6.7 6.7 6.7 6.7
3.5 3.5 3.5 3.5
3.0 2.7 2.7 2.6
5.0 5.1 5.1 5.2
4.6 4.6 4.5 4.5
(b)
Figure 4.18: Distribution of available energy losses in a PTES system as the number of layers in
the cold store is varied. In (a) dp = 4 mm and (b) dp = 10 mm
mm, the thermodynamic round-trip efficiency (i.e. excluding mechanical and electrical
losses) is improved from 70.2% to 71.2% when the number of segments is increased from
1 to 32. Detailed economic analysis is required to determine whether the additional cost
of installing the segments, valving and control systems is covered by the 1% increase in
efficiency.
Operational parameters
Figure 4.19 shows the effect of various cycle operating conditions on the efficiency and
energy density. In varying these parameters, the nominal power and storage capacity have
been held constant by adjusting the size of the reservoirs and the mass flow rate.
At fixed pressure ratio, the efficiency and energy density are both improved by either in-
creasing T1 or decreasing T3. Increasing the pressure ratio at fixed T1 and T3 has a similar
effect as shown in figure 4.19(a). These trends are consistent with the findings in refer-
164
 56
 58
 60
 62
 64
 66
 68
 70
 72
 60  80  100  120  140  160  180
E
ff
ic
ie
n
cy
, 
χ 
(%
)
Energy density, ρE (MJ/m
3
)
βchg
T1
T3
(a)
 56
 58
 60
 62
 64
 66
 68
 70
 72
 60  80  100  120  140  160  180
E
ff
ic
ie
n
cy
, 
χ 
(%
)
Energy density, ρE (MJ/m
3
)
βchg
βdis
Π
(b)
Figure 4.19: Variation of round-trip efficiency and energy density with PTES operational parame-
ters. For each curve the open and solid square symbols are at the minimum and maximum values of
each parameter in accord with table 4.7. (a) Variation with T1, T3 and charging pressure ratio βchg
(with T1 and T3 fixed). (b) Variation with βchg (with T2 and T4 fixed), discharge pressure ratio βdis
and utilisation factor Π.
ence [83] where it was argued that, if compression and expansion losses dominate, then the
efficiency is governed mainly by the ratio T2/T3 since this determines the ratio between
compression and expansion work. Increasing this ratio (as all of the above variations in
T1, T3 and βchg will achieve) thus makes the cycle less susceptible to compression and
expansion losses. However, improvements along these lines will almost certainly result
in increased costs and technical challenges since they imply higher top temperatures or
pressures, or lower bottom temperatures. A more detailed examination of the distribution
of losses also reveals that the benefits of higher work ratio are partly offset by increased
thermal losses in the reservoirs, as indicated in figure 4.20(a).
If the maximum and minimum cycle temperatures (T2 and T4) are fixed, instead of fixing
T1 and T3, then variation of the pressure ratio gives trends similar to those observed for gas
turbine cycles for which there are optimum pressure ratios, as shown in figure 4.19(b). The
optimum pressure ratios for energy density and efficiency are the same, and in this case
are 8.8, but the corresponding value of T3 (10.6°C) lies below ambient temperature. This
165
 2
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 4
 5
 6
 7
 8
 9
 10
 4  6  8  10  12  14  16
A
v
ai
la
b
il
it
y
 l
o
ss
es
 (
%
)
βchg
Compressor
Expander
Hot store
Cold store
HX1
HX3
(a)
 0
 5
 10
 15
 20
 25
 4  6  8  10  12  14  16
A
v
ai
la
b
il
it
y
 l
o
ss
es
 (
%
)
βdis
Compressor
Expander
Hot store
Cold store
HX1
HX3
(b)
Figure 4.20: Distribution of availability losses in PTES components as βchg and βdis are varied. (a)
Variation with charging pressure ratio with T1 and T3 held constant (b) Variation with the discharg-
ing pressure ratio
precludes the possibility of rejecting heat in HX3, making it difficult to manage the thermal
fronts as they emerge from the reservoirs.
Figure 4.19(b) also shows the influence of the ‘utilisation factor’ Π and the discharge pres-
sure ratio. Increasing Π (i.e. longer charge and discharge period) obviously increases the
energy stored per cycle, but this is at the expense of lower efficiency. This is due to steeper
temperature gradients within the reservoirs, which imply that gas-solid heat exchange oc-
curs over a smaller interfacial area and hence with a larger temperature difference.
Varying βdis effectively controls the balance of heat rejection between HX1 and HX3, as
can be seen by examining the T–s diagram shown in figure 4.12(b). A sharp optimum in this
pressure ratio is observed in figure 4.19(b) and occurs at around βdis = 7.60. Inspection
of figure 4.20(b) indicates that this optimum corresponds to when heat rejection is roughly
evenly divided between the two heat exchangers.
166
4.5 Avoiding electric heat addition
When the exit temperatures of the thermal reservoirs change heat addition and rejection
from the cycle is necessary. Heat rejection is unavoidable due to irreversibilities within the
cycle. On the other hand, it should be possible to avoid heat addition which increases the
work input and requires an electric heater as described in section 4.3.2.
Heat addition is apparent in the T–s diagram of the nominal PTES cycle, as shown in
figure 4.21. During charge, temperature T1 decreases as the thermal front emerges from
the cold store. Figure 4.21(c) shows that T1 drops below the required compressor inlet
temperature of 37°C, and a heat input is required. Heat addition is also required during
discharge: as the thermal front reaches the end of the discharging hot store T2 reduces
below the design value of 505°C. Consequently, after expansion the temperature T1 drops
below 37°C. Furthermore, changes in the exit temperature lead to reduced power output
and a system that has constant power output is preferable.
In this section, alternative methods of operating the PTES cycle to avoid heat addition are
described. Solutions include neglecting the need for heat addition, placing a recuperator
between T3 and T1 or varying pressure ratios throughout operation of the cycle. A number
of other solutions exist, and more detailed investigations could compare these as well as
taking into account other factors such as reciprocating device behaviour.
Calculations are carried out using the nominal PTES design as described in section 4.4 and
table 3.1, albeit with all pressure loss factors set to fp = 0.0 and heat exchanger effec-
tiveness set to ∗ = 1.0. These modifications are made in order to simplify the analysis
and ensure that the effect of any modifications can be observed more clearly. Under these
conditions the thermodynamic efficiency is χ = 72.6%. When electrical heating is used,
the electrical work is added to the work input during charge, and subtracted from the work
output during discharge. During charge 3.1% of the work input is heat addition. Dur-
ing discharge the electrical work is 3.4% of the work output. By avoiding electrical heat
addition the efficiency could therefore be increased by
χnew =
(1 + 0.034)W netdis
(1− 0.031)W netchg
=
(1 + 0.034)
(1− 0.031)χ = 1.07χ (4.49)
167
Therefore, the efficiency could potentially rise by 7% to 77.5%.
4.5.1 Neglecting heat addition
The simplest way to avoid heat addition is to ignore the need for it. When the temperature
at T1 or T3 falls below the required value of 37°C, the gas is simply fed through to the next
component with no heat addition. Varying the inlet temperatures to the thermal stores leads
to unusual thermal fronts shapes, and cyclic operation cannot be achieved. After 20 cycles
the round-trip efficiency is 37.0% compared to 72.6% of the nominal cycle above.
The stores develop profiles without a clearly defined thermal front which leads to the
packed beds storing less energy, while thermal and conductive losses are exacerbated by
the increased rate of heat transfer. Ideally the power should be constant, or at least easy to
predict so that the energy store can be a useful asset to the electrical grid. However, the
large temperature variation across the compressors and expanders also causes significant
variation in the input and output power.
This option would be simple and cheap to implement. However, it significantly compro-
mises the efficiency and energy density of the PTES plant and is not a sensible solution.
4.5.2 Recuperated cycles
Figure 4.21 indicates that heat is rejected at 3. and added at 1. in both charge and discharge
cycles. The heat that is rejected at 3. could be used as a heat input at 1. as illustrated in
the schematic layout in figure 4.22. The temperature traces in figure 4.21(c) and 4.21(d)
suggest that T3 is always high enough to be able to heat the gas at T3 up to 37°C, even
considering heat exchanger inefficiencies. For instance, at the end of charging T3 = 168°C
while T1 = −25°C. Using a heat exchanger with an effectiveness of  = 0.7, both outlet
streams could reach 37°C if the ratio of mass flow rates was m˙3/m˙1 = 0.46. The remaining
54% of the T3 would pass through HX3 as usual and could conceivably be used in other
waste heat purposes. The true behaviour would be more complicated than this since the
inlet temperatures to the recuperator would vary as the thermal fronts emerge from the
stores. Consequently, m˙3/m˙1 may be required to change (although preliminary analysis
168
-200
-100
 0
 100
 200
 300
 400
 500
 600
-0.5 -0.4 -0.3 -0.2 -0.1  0  0.1
T
em
p
er
at
u
re
, 
°
C
Specific entropy, KJ/kgK
1.
2.
3.
4.
1’.
3’.
heat 
 rejection
heat 
 addition
t / tN = 0.0
t / tN = 0.7
(a)
-200
-100
 0
 100
 200
 300
 400
 500
 600
-0.5 -0.4 -0.3 -0.2 -0.1  0  0.1
T
em
p
er
at
u
re
, 
°
C
Specific entropy, KJ/kgK
t / tN = 0.0
t / tN = 0.7
(b)
37
-200
-100
 0
 100
 200
 300
 400
 500
 600
 0  0.25  0.5  0.75
T
em
p
er
at
u
re
, 
°
C
Time / Nominal charge time, t / tN
T1
T2
T3
T4
(c)
37
-200
-100
 0
 100
 200
 300
 400
 500
 600
 0  0.25  0.5  0.75
T
em
p
er
at
u
re
, 
°
C
Time / Nominal charge time, t / tN
T1
T2
T3
T4
(d)
Figure 4.21: T–s diagrams and temperature traces for the nominal PTES design. (a) T–s diagram
during charge. Points 1’. and 3’. indicate how the exit temperature of the stores changes towards
the end of the charging phase and leads to heat being rejected and added. (b) T–s diagram dur-
ing discharge (c) Reservoir inlet and exit temperatures during charge (d) Reservoir inlet and exit
temperatures during discharge
169
(1)(2)
(3) (4)
cold front
hot front
HS CS
Recuperator
EC
CE
Figure 4.22: Simplified schematic of a PTES system using recuperation. Heat exchangers are not
shown for clarity. As the thermal fronts exit the stores, the hot store gas at 3. is used to reheat the
cold store gas at 1.
suggests only by a small amount). Integration of a recuperator in a PTES system therefore
requires a thorough analysis of the off-design operating behaviour.
The benefit of recuperation is difficult to evaluate without detailed calculations. However,
as noted above, if heat addition is avoided the efficiency could potentially increase from
72.6% to 77.5% which provides a strong case for including a recuperator in the PTES
system. Furthermore, it may be possible to avoid any increase in capital cost by re-directing
water between the existing heat exchangers to achieve the recuperation.
If different operating temperatures are used then recuperation may not be possible. Para-
metric studies in section 4.4.2 and [84] showed that the round-trip efficiency could be im-
proved by decreasing T3 and increasing T1. Furthermore, optimisation studies of a PTES
system [84] found that the cheapest designs often had lower pressure ratios and values of
T1 greater than T3. Figure 4.23 shows T–s plots and temperature traces for a system similar
to the nominal design albeit with T1 = 117°C, T3 = 37°C, βchg = 7.6 and βdis = 7.2.
These values have been taken from the optimised results in [84]. In both charge and dis-
charge, T1 drops below the design value of 117°C for some time before T3 becomes high
170
enough for recuperation to be possible. One option would be to avoid heat addition during
the times it is required which would create irregular thermal front shapes and reduce the
efficiency. However, T1 decreases only a small amount below 117°C before recuperation
would be possible (during charge T1 drops by around 30°C, and in discharge it drops by
around 5°C). Therefore, the impact of neglecting heat addition may not be as dramatic as
in the previous section.
Recuperation could be a promising technique to avoid heat addition. However, further
analysis is required to quantify the benefit, assess the economic cost, and to investigate the
feasibility of recuperation in designs with alternative operating conditions.
4.5.3 Variable pressure ratios
Heat addition can be avoided by varying the pressure ratio throughout the charge and dis-
charge cycles. The wider impacts are currently unclear since changing β requires either the
mass flow rate or rotational speed of the compressors and expanders to change according
to the characteristic equation derived in section 4.2. Changing the pressure ratio therefore
moves the devices away from their design point. In this section it is assumed that the rota-
tional speed changes so that the mass flow rate remains constant. If such a system were to
be used, then a variable speed motor-generator would be required.
Variable pressure ratio during charging
Heat addition is required during charge when the thermal front reaches the end of the
cold store causing a drop in the temperature T1. Passing the reduced value of T1 into
the compressor leads to lower T2. Increasing the charging pressure ratio βchg to keep T2
constant, that is, βchg = (T2/T1)
1/φc . Figure 4.24 shows the T–s plot and temperature traces
for this scheme. It is notable that the pressure ratio increases from 10 to 17 which will
significantly increase the cost of the stores. However, the round-trip efficiency increases
from 72.6% to 75.2% if the nominal discharging cycle is used (i.e. heat addition still occurs
during discharge).
Deployment of this scheme requires flexible operation of HX3: if the output of the heat
171
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-100
 0
 100
 200
 300
 400
 500
 600
-0.5 -0.4 -0.3 -0.2 -0.1  0  0.1  0.2
T
em
p
er
at
u
re
, 
°
C
Specific entropy, KJ/kgK
recuperation
t / tN = 0.0
t / tN = 0.7
(a)
-200
-100
 0
 100
 200
 300
 400
 500
 600
-0.5 -0.4 -0.3 -0.2 -0.1  0  0.1  0.2
T
em
p
er
at
u
re
, 
°
C
Specific entropy, KJ/kgK
Charging
t / tN = 0.0
t / tN = 0.7
(b)
-200
-100
0
117
 200
 300
 400
 500
 600
 0  0.25  0.5  0.75
T
em
p
er
at
u
re
, 
°
C
Time / Nominal charge time, t / tN
T1
T2
T3
T4
(c)
-200
-100
0
117
 200
 300
 400
 500
 600
 0  0.25  0.5  0.75
T
em
p
er
at
u
re
, 
°
C
Time / Nominal charge time, t / tN
T1
T2
T3
T4
(d)
Figure 4.23: T–s diagrams and temperature traces for a PTES design with T1 > T3. (a) T–s
diagram during charge. For the t/tN = 0.7 curve recuperation can occur between the hot store exit
and the cold store exit as indicated. (b) T–s diagram during discharge (c) Reservoir inlet and exit
temperatures during charge (d) Reservoir inlet and exit temperatures during discharge
172
-200
-100
 0
 100
 200
 300
 400
 500
 600
-0.5 -0.4 -0.3 -0.2 -0.1  0  0.1
T
em
p
er
at
u
re
, 
°
C
Specific entropy, KJ/kgK
t / tN = 0.0
t / tN = 0.7
(a)
37
-200
-100
 0
 100
 200
 300
 400
 500
 600
 0  0.25  0.5  0.75
 10
 12
 14
 16
T
em
p
er
at
u
re
, 
°
C
C
h
ar
g
in
g
 p
re
ss
u
re
 r
at
io
, 
β ch
g
Time / Nominal charge time, t / tN
T1
T2
T3
T4βchg
(b)
Figure 4.24: Charging profiles of a PTES scheme where the charging pressure ratio βchg is in-
creased such that T2 is constant (a) T–s diagram during charge (b) Temperatures at the reservoir
inlets and outlets during charge
exchanger was fixed at T3 = 37°C then the increased pressure ratio would reduce T4. This
would create problems with the thermal fronts in the discharging phase. Keeping T4 con-
stant sets the outlet temperature from HX3 to be T3 = T4β
φe
chg. For instance, figure 4.24(a)
shows that HX3 exit needs to rise to 100°C, so that T4 is constant at -150°C. Operating
HX3 in this way should reduce the quantity of heat rejected, although the heat is rejected
at a higher average temperature.
Variable pressure ratio during discharging
Heat addition occurs in the discharging phase when the thermal front exits the hot store.
The reduction in T2 leads to a decrease in the expander exit temperature T1. The pressure
ratio can be varied in a number of ways. For instance βdis could be varied such that the
exit of the compressor remains constant at T3 = 37°C. As a result, heat rejection would
occur exclusively in HX1. As noted in section 4.4.2, an optimal value of βdis occurs when
heat rejection is evenly divided between the two heat exchangers. This can be achieved
173
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-100
 0
 100
 200
 300
 400
 500
 600
-0.5 -0.4 -0.3 -0.2 -0.1  0  0.1
T
em
p
er
at
u
re
, 
°
C
Specific entropy, KJ/kgK
Charging
t / tN = 0.0
t / tN = 0.7
(a)
37
-200
-100
 0
 100
 200
 300
 400
 500
 600
 0  0.25  0.5  0.75
 5
 6
 7
 8
 9
 10
T
em
p
er
at
u
re
, 
°
C
D
is
ch
ar
g
in
g
 p
re
ss
u
re
 r
at
io
, 
β di
s
Time / Nominal charge time, t / tN
T1
T2
T3
T4βdis
(b)
Figure 4.25: Discharging profiles of a PTES scheme where the discharging pressure ratio βdis is
varied such that approximately equal amounts of heat are rejected in each heat exchanger. (a) T–s
diagram during discharge (b) Temperatures at the reservoir inlets and outlets during discharge
by choosing a pressure ratio that leads to T1 = T3. By noting that T2/T1 = β
φe
dis and
T3/T4 = β
φc
dis, rearranging such that T1 = T3 leads to
βdis =
(
T2
T4
) 1
φc+φd
(4.50)
Figure 4.25 illustrates the T–s diagram and temperature traces. It is clear that heat addition
is not required and that T1 and T3 are equal. The pressure ratio reduces from 9.1 to 5.1.
Varying only βdis (i.e. βchg is constant) increases the round-trip efficiency from 72.6% to
76.9%.
Varying the pressure ratio in both the charging and discharging cycle successfully avoids
undesirable heat addition and leads to a 7% increase in efficiency. There are two major
disadvantages to this approach. Firstly, the pressure ratio varies between 5 and 17 which
will require reciprocating devices with good off-design performance. Such wide variation
could lead to low polytropic efficiencies thereby compromising the increase in efficiency.
Secondly, the components have to be able to withstand a maximum pressure of 17 bar rather
174
than 10 bar which will significantly increase the cost of the reciprocating devices and hot
store.
Rather than operating the reciprocating devices at fixed mass flow rate as above, they could
be operated at a fixed speed. Thus, as the pressure ratio decreases the mass flow rate would
increase (see figure 4.3) which would help to maintain constant power output.
4.5.4 Variable cold store outlet temperature
Varying the operating temperatures of the cold store is another way to avoid electric heat
addition. The cold store is operated between T ′1 and T4 where T
′
1 is higher than the nominal
temperature T1 (i.e. the compressor inlet temperature during charge). During charge, gas
exits the cold store at T ′1 and is cooled in HX1 to T1 before continuing into the charge
compressor. As the thermal front emerges from the cold store T ′1 decreases until at the end
of charge T ′1 = T1. As a result, heat is continually rejected between T
′
1 and T1 thereby
avoiding heat addition during charge. The position of T ′1 is shown on figure 4.26(a) and
heat rejection from T ′1 to T1 is shaded in on figure 4.26(c).
The discharging phase needs to be carefully controlled to ensure that T ′1 > T1. In this
example, βdis is varied so that T3 is held constant. During discharge, this leads to an increase
in the discharge expander exit temperature T ′1, as shown on figure 4.26(c). This gas is fed
directly into the cold store rather than HX1. As a result, the cold reservoir has a temperature
above the design value of T1 = 37°C at this end of the store. During the next charging cycle,
the gas that exits the reservoir is therefore also above 37°C, and heat addition is avoided.
During discharge the inlet temperature to the cold store T ′1 is not constant. The varying
discharging pressure ratio causes T ′1 to steadily increase to a maximum, and then reduce
slightly towards the end of discharge. Consequently, thermal fronts in the cold store have ir-
regular shapes, as shown in figure 4.27. Rather than being a smooth curve, the thermal front
actually has a maximum temperature of about 120°C at the warm end. This impacts the
charging cycle where the output temperature of the cold reservoir increases before decreas-
ing. Figure 4.27 demonstrates that the thermal profiles are quite different to the nominal
case.
Generating unusual thermal fronts leads to increased heat transfer and conduction losses,
175
-200
-100
 0
 100
 200
 300
 400
 500
 600
-0.5 -0.4 -0.3 -0.2 -0.1  0  0.1  0.2
T
em
p
er
at
u
re
, 
°
C
Specific entropy, KJ/kgK
1.
1’.
2.
3.
4.
t / tN = 0.0
t / tN = 0.7
(a)
-200
-100
 0
 100
 200
 300
 400
 500
 600
-0.5 -0.4 -0.3 -0.2 -0.1  0  0.1  0.2
T
em
p
er
at
u
re
, 
°
C
Specific entropy, KJ/kgK
t / tN = 0.0
t / tN = 0.7
(b)
-200
-100
0
37
 100
 200
 300
 400
 500
 600
 0  0.25  0.5  0.75
T
em
p
er
at
u
re
, 
°
C
Time / Nominal charge time, t / tN
T1’
T2 
T3 
T4 
(c)
-200
-100
0
37
 100
 200
 300
 400
 500
 600
 0  0.25  0.5  0.75
 4
 6
 8
T
em
p
er
at
u
re
, 
°
C
D
is
ch
ar
g
in
g
 p
re
ss
u
re
 r
at
io
, 
β di
s
Time / Nominal charge time, t / tN
T1’
T2 
T3 
T4 βdis
(d)
Figure 4.26: T–s diagrams and temperature traces for a PTES design with a cold store discharging
temperature of T ′1. (a) T–s diagram during charge (b) T–s diagram during discharge (c) Reservoir
inlet and exit temperatures during charge. Shaded area illustrates heat rejection that occurs (d)
Reservoir inlet and exit temperatures during discharge
176
-150
-100
-50
 0
 50
 100
 150
 0  1  2  3  4  5
G
as
 t
em
p
er
at
u
re
, 
T
g
 (
°
C
)
Distance along reservoir, x (m)
A
B
T1’ scheme
Nominal
Figure 4.27: Thermal profiles in the cold store at (A) the end of discharging and (B) the end
of charging. Profiles are shown for the nominal design and for a PTES design with a cold store
discharging temperature of T ′1.
but this cycle does successfully avoid heat addition and achieves a round-trip efficiency of
78.4%. Furthermore, this approach does not require additional components (such as recu-
perators) or excessively high pressure ratios. However, the variable discharging pressure
ratio has implications for the reciprocating devices and means the motor-generator may
have to be variable speed.
A final consideration is that no heat is rejected during discharge. Therefore, to maintain
cyclic operation all of the irreversibilities must be ‘flushed’ out of the system during charge.
The T–s diagram indicates that most of the heat is rejected in HX1. It is possible to balance
heat rejection more evenly between the two heat exchangers and also reduce the average
temperature of heat rejection, by using a scheme where the discharging pressure ratio keeps
T1 = T3. This may also reduce the maximum value of T ′1.
177
4.5.5 Liquid stores
Heat has to be added to the PTES system as a result of the thermal gradients in the packed
beds. During charge and discharge the thermal fronts travel out of the reservoirs and heat
rejection or addition is required to maintain cyclic operation. The preceding sections have
discussed a number of options that avoid this problem. An alternative solution uses liquid
stores rather than packed-bed thermal reservoirs.
Each liquid store comprises two tanks: one contains fluid at the discharged conditions and
the other is for fluid at the charged conditions. Between the tanks is a heat exchanger that
integrates the store with the heat pump/engine cycle, as illustrated in figure 4.28. As an
example, consider the hot store during charge. Tank H3 is full of fluid at the discharged
temperature T3. Fluid leaves tank H3 and passes through the heat exchanger where it is
heated to T2 before entering tank H2. The reverse process occurs during discharge.
CE
EC
(4)(3)
HX2 HX1
(1)(2)
CS1HS2
HS3 CS4
charge
discharge
Figure 4.28: Schematic of an alternative PTES design utilising liquid during the charging phase.
The advantage of this method is that heat exchange always occurs between fixed temper-
atures. There are no thermal fronts so heat addition is not required (heat rejection will
still occur, however). Furthermore, the thermal stores can be fully charged and discharged
unlike the packed beds where the thermal front prevents this.
178
These systems require a liquid medium with suitable melting and boiling points since phase
change should ideally be avoided. Potential candidates include water, molten salts and
cryogens. The hot and cold stores would probably use different liquids. Although liquid
based PTES systems have four storage vessels rather than two, they may not need to be
pressurised (unless water is used at high temperatures) which would reduce the cost. Liq-
uids such as water typically have higher specific heat capacities than rocks, but have smaller
operating temperature ranges. Without detailed calculations it is difficult to determine how
the energy density of liquid stores compare to the nominal PTES system. Further work is
therefore required, although thermodynamic models of such systems have already indicated
that the round-trip efficiency is heavily dependent upon heat exchanger efficiency [168].
4.6 Summary
In this chapter, a thermodynamic model of Pumped Thermal Energy Storage (PTES) sys-
tems has been developed, and the system’s behaviour has been investigated in detail.
Reciprocating devices were characterised by simple loss factors such as the pressure loss
factor fp, the heat loss factor ν and the polytropic efficiency η. In section 4.2 the ther-
modynamic processes that governed availability losses were investigated which allowed
estimates to be made. For instance, fp is approximately 1% while ν is of the order of 2%.
Simplified models of heat exchangers, electric heaters, pipes, valves and the buffer tank
were then discussed. The estimated loss factors were only approximations, so the sensitiv-
ity of the system to each loss factor was studied. It was found that the round-trip efficiency
decreased for around 2% for every 1% drop in η or ν. Pressure losses were less significant
with each percentage decrease causing an 1% loss.
Parametric studies of geometric variables and operating variables were undertaken in sec-
tion 4.4.2. Varying the geometric parameters illustrated the trade-off between thermal
losses and pressure losses that was described in chapters 2 and 3. For instance, longer
stores and smaller particles caused larger pressure losses and lower thermal losses. It was
found that segmenting the cold store could increase the efficiency and that the effect was
most significant when small particles were used. The parametric studies confirmed the
simplified analysis of PTES systems described in section 1.5. One important parameter
179
is the utilisation factor Π which controls the trade-off between round-trip efficiency and
the energy density. Increasing the charging pressure ratio, increasing T1 and decreasing
T3 were all found to increase the round-trip efficiency. The discharging pressure ratio was
found to have a significant effect on the round-trip efficiency, and primarily controlled the
distribution of heat rejection between the two heat exchangers.
The objective of section 4.5 was to avoid the need for an electric heater, since the work input
for the heater significantly reduces the round-trip efficiency. Several methods of operating
the PTES system were considered and it was found that the thermodynamic round-trip
efficiency could be increased from 72.6% to 78.4%. Further research is required in this
area, since a large number of configurations exist.
180
Chapter 5
Thermo-economic optimisation
181
5.1 Introduction
Previous chapters have developed detailed thermodynamic models of packed-bed thermal
reservoirs and Pumped Thermal Energy Storage (PTES) systems. These models include a
large number of parameters that can make optimisation of the system by simple parameter
variation a challenging task. Furthermore, optimisation of engineering systems typically
requires consideration of several objectives: most commonly efficiency and cost. This
chapter begins with a discussion of economic modelling of energy storage systems and a
number of economic metrics are described. Most economic metrics are limited by uncer-
tainty, and the approach of this chapter is to develop an economic model that is as simple
as possible while still providing meaningful information.
The capital cost model of PTES systems is described in sections 5.2.1, 5.2.2 and 5.2.3 and
is based on the material requirements of each component. The objective of this chapter is
to compare PTES designs against one another, so the relative magnitude of the capital cost
is of greater interest than its absolute magnitude.
Thermo-economic optimisation is carried out on packed-bed thermal reservoirs and on
PTES systems. The optimisations investigate the benefit of various design features such
as segmentation, radial-flow stores, and the avoidance of heat addition. In addition, the
sensitivity of the optimal designs to the reciprocating device polytropic efficiency is inves-
tigated. Stochastic optimisation techniques are used as these are suitable for these types
of problem, as described in section 5.3. Both the thermodynamic and economic models
involve a number of simplifications and estimations and the outputs are therefore uncer-
tain. Consequently, the uncertainties in the modelling must be taken into account when
comparing the results with other technologies or when considering the values of efficiency
and cost. By undertaking multi-objective optimisation it is possible to see the full range of
optimal designs, i.e. designs that achieve high efficiency at high cost through to designs
that achieve low costs at lower efficiencies. In this way, it is possible to compare one design
to another and make informed judgements. As a result, the following optimisation studies
are most appropriate for identifying promising designs.
Thermo-economic optimisation of Rankine-cycle based PTES systems has been carried
out by Morandin et al. [88]. Optimisation of Joule-Brayton cycle based PTES systems
182
have been carried out by the Cambridge group and are presented in [84] and the results in
this chapter are extensions of this work.
5.2 Economic modelling
Holistic optimisation of the PTES system should consider economic factors as well as
technical issues. Economic factors are often incorporated into thermodynamic studies by
a method known as ‘thermo-economics’, ‘techno-economics’, or, if the analysis is based
on second-law considerations, ‘exergo-economics’. The theory of exergo-economics was
developed in the 1980s and early 1990s, and several introductory texts exist [169–171]. In
1994 an example problem (the ‘CGAM’ problem) was formulated to demonstrate a range
of techniques [172–175]. The theory has since been widely applied to a number of energy
systems [176] and several methodologies have been developed [177]. Exergo-economics
considers not only the exergetic losses within components, but also the exergy that was
required to produce those components and any fuel sources. Such analyses requires a set
of equations to relate the flows of exergy between and within components. A further set of
‘auxiliary equations’ are also necessary to relate monetary costs to the exergetic flows. This
approach requires the development of its own model rather than incorporating economic
factors into an existing thermodynamic model.
These approaches consolidated several aims into a single objective. Reducing the dimen-
sionality of the problem was advantageous as complex problems could be efficiently stud-
ied using the optimisation techniques and computing power that were available at the time.
However, multi-objective optimisation theory was also being developed in the 1980s, and
since then computational power has increased significantly. Multi-objective optimisation
problems can now be easily tackled, and it is beneficial to consider technical and economic
factors with separate objectives. Multi-objective optimisation theory is discussed in sec-
tion 5.3.
The economics of energy storage is a complex field, riddled with uncertainty and specula-
tion. A sophisticated economic model would take into account capital costs, operation and
maintenance costs and future energy markets; electricity prices, subsidies, taxes and the
influence of other technologies. Energy storage has several potential revenue streams, and
183
it is unclear how profitable these streams may be, how payments would be organised, and
how the market would be regulated. The most commonly discussed revenue stream is ar-
bitrage, which some authors believe could lead to ‘self-cannibalisation’ of its own market.
The various economic benefits of storage are discussed in more detail in the chapter 1 and
in [1, 41]. Inclusion of these details would provide information about the feasibility and/or
profitability of the system, and the design and mode of operation that would achieve this.
The inherent uncertainty of economic factors (and other externalities) suggests that any
judgements about the profitability of an energy storage system are meaningless without
undertaking comprehensive sensitivity analysis. The uncertainty and complexity can be
reduced by developing simpler economic models although these may be unable to capture
the true nature of the problem as well as still being quite uncertain.
It is not the aim of this thesis to evaluate how economically feasible a PTES system may
be. Instead, economic factors are used to compare different designs and to evaluate which
may be the most feasible design. Consequently, it is the relative magnitude of the economic
factors that are of importance, rather than the absolute values themselves.
The simplest, and least uncertain, economic metric is the capital cost. Optimisation of the
PTES system therefore involves minimisation of the capital cost and maximisation of the
round-trip efficiency. A number of slightly more sophisticated economic metrics have been
applied in the literature and are summarised below.
Net Present Value (NPV)
The Net Present Value (NPV) considers the capital cost and revenue streams from the sys-
tem. Future revenue streams are discounted to the present value, thus giving an indication
of whether investing the capital cost in the system would be more or less profitable than
investing the money. An NPV greater than zero indicates a worthwhile investment. The
simplest case to consider is the system earning revenue through arbitrage i.e. buying elec-
tricity at a low price, and selling it at a higher price. Rastler [41] and Strbac et al. [1] suggest
that this is unlikely to be the most profitable scheme. However, its simplicity means it is
184
the most commonly discussed in the literature. The NPV is given by
NPV =
N∑
i
piouti W
out
i − piini W ini
(1 + r)i
− Ccap (5.1)
Where i is the time period in consideration, r is the discount rate, and Ccap is the capital
cost. Superscripts ‘in’ and ‘out’ indicate whether energy (or work) W is being bought or
sold, respectively, and the price pi at which the transaction occurs in that period. These
parameters (which vary with time) are then summed over the period of operation; typically
assumed to be around 20 years. Clearly this model introduces a great degree of uncertainty,
requiring estimates of interest rates, capital costs, and how electricity prices will vary over
the next 20 years at quite a fine resolution (e.g. on an hourly basis). An operational strategy
is also required for the energy storage system – i.e. the conditions under which it will buy
and sell energy – which also requires an understanding of the impact on this mode of
operation on the system efficiency and level of storage.
The equation above can be simplified by assuming that the system operates by charging and
discharging for regular periods, and that the efficiency is given by χ = Wout/Win. Thus,
NPV = W in
[
N∑
i
χpiouti − piini
(1 + r)i
− Ccap
W in
]
(5.2)
For cyclic operation Win is the same for every period, and no knowledge of pi is required
to see that maximisation of the NPV can be simply achieved by maximising the round-
trip efficiency and minimising the capital cost. Therefore, it is clear that optimisation of
this economic model is equivalent to the proposed optimisation scheme. The proposed ap-
proach has the additional benefit that electricity price markets do not have to be considered.
In the energy storage literature, DeForest et al. [178] used the NPV to compare the costs
and operating behaviour of cold water thermal energy storage systems that were used for
domestic cooling under a range of different economic and climatic conditions. Grunewald
et al. [52] assessed several energy storage technologies by investigating the integration of
storage into the grid. The economic benefit of CAES, compressed hydrogen, and flow bat-
teries were compared as the proportion of renewables in the energy mix was varied. It was
found that for low renewables penetration none of the storage systems were economically
185
viable. As the proportion of renewables was increased CAES was the first to become viable
and were found to be more affordable than the other systems for all proportions of renew-
ables. As the renewables proportion became very large, the NPV decreased: there was an
abundance of excess renewable energy to be stored, but rarely an opportunity for storage to
discharge to the grid.
Levelised cost of energy (LCOE)
The Levelised Cost of Energy (LCOE) is the total cost of the system throughout its lifetime
(including construction, operation and maintenance) divided by its useful energy output.
The costs are levelised to the current value using a discount rate r: the LCOE can be
thought of as the Net Present Value of the unit cost of energy over the lifetime of the
system [41].
Rastler [41] carried out a comparison of PHES, CAES and batteries using LCOE as eco-
nomic metric, as did Schoenung and Hassenzahl [179]. When storage earned revenue from
transmission and distribution grid support, and renewable integration/time shifting capa-
bilities Rastler found that pumped hydro had the lowest LCOE followed by CAES, while
batteries were more expensive.
Schoenung and Hassenzahl [179] considered three broad applications for energy storage:
bulk storage, distributed generation and power quality. Different energy storage systems
were placed in each category according to which application they were most suited. For
bulk storage, Schoenung found similar results to Rastler, with CAES being the most afford-
able technology, closely followed by PHES, while batteries were more expensive. Both
reports placed the LCOE for CAES and PHES in the range of 10–20 c/kWh (USD cents
per kilowatt hour) while lead acid batteries were in the range of 40–60 c/kWh.
Life cycle analysis (LCA)
As the name suggests, life cycle analysis (LCA) or life cycle energy analysis (LCEA) is a
method that takes into account the full life cycle of the system: from construction and oper-
ation, through to decommissioning. This can be evaluated in terms of economic costs, or of
the energy that is required for each stage. It is also possible to incorporate externalities into
186
LCA and thereby give an indication of the wider impact of the system [180]. For instance,
Denholm and Kulcinski [56] assigned an economic cost to the greenhouse gas emissions
from different storage technologies. The analysis of pumped hydro, for example, includes
emissions generated when constructing the dam, from clearing biomass and from the de-
cay of biomass in the area that is flooded. Denholm investigated PHES, CAES and battery
energy storage using LCEA, and the use of this technique provided a subtle comparison
of the technologies: it was shown that batteries had much greater energy requirements for
construction than PHES or CAES when normalised by the energy capacity since battery
components are relatively complicated, and the energy storage media requires intensive
mining and processing. On the other hand, CAES produced the most greenhouse gases
throughout its lifetime, which is unsurprising given the required gas input. However, these
results only occurred when the input energy to the three technologies came from renew-
ables or nuclear. If the input energy was produced by a greenhouse-gas producing source
such as coal, it was shown that CAES had lower greenhouse gas emissions than the other
technologies.
Oro et al. [181] used LCA to compare three different thermal energy storage materials for
applications in solar power plants. The materials considered were solid media (concrete),
phase change materials and molten salts. The analysis also included the ecological impact
and the effect on human health. These factors were combined into a single metric known
as Eco-Indicator 99 (EI99) and the data was extracted from a dedicated database. For each
of the stores it was found that the storage media contributed the most to the environmental
impact. However, since the application is for solar power plants the stores were not pres-
surised and it is therefore unclear how significant the pressure vessel contribution would
be. The study also found that concrete had the lowest environmental impact out of the
three storage media. The disadvantage of such approaches is that it is somewhat arbitrary
to assign an economic cost to external factors. Furthermore, this method generally depends
on databases having reliably collected the required data.
Energy returned on energy invested (EROI)
The energy returned on energy invested (EROI) is defined as the ratio of the energy deliv-
ered by an energy system to the total energy required to produce the system over its entire
187
life-cycle. As a result, the EROI metric has similarities to Life Cycle Analysis and Net
Energy Analysis. The EROI approach has been applied to energy storage systems by Barn-
hart et al. [182, 183] who defined another metric specifically for storage technologies: the
‘energy stored on energy invested’ (ESOI). The ESOI is the electrical energy stored over
the lifetime of the storage divided by the electrical energy equivalent required to build the
system.
Barnhart et al. calculated the ESOI and EROI for a number of storage technologies and
renewable energy sources. Barnhart’s results supports Denholm’s LCA analysis which
suggested that batteries are energetically expensive compared to bulk energy storage sys-
tems such as CAES and PHES. The analysis suggests that over their lifetime, batteries may
only provide 5–30 times the amount of energy required to construct them, while bulk en-
ergy storage systems are of the order of 1000. The disadvantage of this approach is that
it considers only the energetic cost rather than the full economic cost and involves many
assumptions. Nevertheless, this methodology provides an interesting way to consider the
energy system and how energy storage may best be integrated into it.
Evaluation of economic approaches
The range of economic approaches makes it difficult to compare technologies that are de-
scribed in the literature. Even if the same economic metric is used, it is necessary to
carefully consider the assumptions made, the factors that are included, and the data sources
and values in order to fairly compare results. When a comparison is made using the same
metric in the same piece of literature, it should be remembered that the quality of infor-
mation and data available may vary substantially between technologies, especially if some
systems are well established and others are only in the developmental stage.
The above discussion described a number of economic approaches that have been applied
to energy storage systems. Despite the differences in methodologies and data, it is notable
that most studies agreed that bulk storage systems such as PHES and CAES were less
expensive and less energy intensive than battery systems. PTES potentially has similar
economic characteristics to CAES because it uses similar components and is predicted to
have similar technical performance.
188
5.2.1 Thermal reservoir costs
The capital cost of the thermal reservoirs is modelled by considering the material require-
ments of each component and is given by
Cbed = kpVp + kiVi + kPVVPV (P0 + ∆P + ρsgL) (5.3)
where V refers to the volume of a material, and k is the material cost per unit volume. The
subscripts p, i and PV refer to the packing material (magnetite), insulation, and pressure
vessel, respectively. The pressure vessel cost is assumed to be proportional to the volume
and the pressure difference across the walls, ∆P . The additional term P0 allows unpres-
surised vessels to be modelled. The term ρsgL takes into account the weight of the packing
material, where ρs is the solid density, L is the height of the store, and g = 9.81 ms−2
is gravitational acceleration. The packing material will probably be supported by grids
throughout the pressure vessel, to reduce the stress on the particles and the pressure vessel
but such detail is not included in the present estimations.
The pressure vessel volume includes the volume required for the packing material, insula-
tion, and end-caps that are similar to ‘squashed’ hemispheres. The volume of each end-cap
is given by apiD3/12 where a is taken as 1/2 to reflect the fact that the dome ends are
somewhat flatter than hemispheres (a = 1). For segmented stores, VPV includes the ad-
ditional volume which is required for bypass flow and inter-segment gaps as described in
section 3.3.1. However, the extra costs of valves, sensing and control which are necessary
in layered stores are not included. Values for k and its standard deviation are estimated
using the CES Materials Selector database [131], and are shown in table 5.1. A value of
kPV was chosen following discussions with Isentropic Ltd [184].
Cost factors Value Standard deviation Source
kp (£m−3) 500 100 [131]
ki (£m−3) 1400 500 [131]
kPV (£m−3bar−1) 200 50 [184]
kr (£m−3s) 8000 2000 (see table 5.3)
Table 5.1: Costs factors k used in the economic analysis
The cost of pressure vessels has also been estimated by Morandin et al. [88] who undertook
189
Figure 5.1: Variation in capital cost per stored available energy for a 16 MWh thermal reservoir
using different storage materials. Costs factors and relevant material data are given in table 5.2.
Shaded areas indicate the cost ± one standard deviation and represent the uncertainty of the cost
factors
a thermo-economic optimisation of a Rankine-cycle based PTES system. Morandin fitted
linear trends to data from vendors to suggest that the pressure vessel cost is
CPV = Z
[
4500V 0.6PV + 10000
]
(5.4)
where V is the pressure vessel volume and Z ≈ 0.77 is a factor that converts Morandin’s
cost (in US dollars in 2009) into January 2016 pounds sterling. The pressure vessels in
Ref. [88] stored water at 10 bar. For an empty pressure vessel of volume 100 m3 at 10 bar,
equation 5.4 suggests a cost of £63 000 whereas the estimate using the above value of kPV
is £200 000. The difference emphasises the inherent uncertainty in these analyses.
Equation 5.3 has been used to calculate the cost of a 16 MWh thermal store for various
storage materials, data for which are given in table 5.2. Figure 5.1 shows that increasing
the storage temperature generally leads to a decrease in the cost per stored availability as
the reservoir volume is reduced. For packed-beds, the pressure is increased in accordance
190
with T2/T1 = βφc as if the heat transfer fluid is supplied by a compressor. As a result, as the
pressure increases with temperature, and the value of Cbed/B reduces despite the increased
pressure before plateauing. For liquid storage heat is transferred in a heat exchanger and
the pressure thus remains at 1 bar. Consequently the cost drops continually as the temper-
ature increases. These materials have a limited working temperature range: sunflower oil
due to a maximum ‘smoke temperature’ [185]; and molten salts due to their freezing and
evaporating points. Heat is also transferred to water in a heat exchanger, and the pressure
is fixed at 1 bar until the water begins to boil at 100°C, at which point the pressure is set to
the saturation pressure at that temperature. The large increase in pressure that is required to
keep water in a liquid form is reflected in the steep increase in cost for temperatures above
100°C.
Material ρs λs c¯s Temperature Cost factor Standard
(kg m−3) (W/kg K) (J/kg K) range (°C) k (£m−3) deviation of k
Magnetite 5175 4.0 850 -200 – 800 500 100
Concrete 2300 - 920 -200 – 800 100 25
Sunflower oil 918 - 2400 177 550 100
Soybean oil 925 - 1900 191 410 100
Molten salt 1680 - 1560 112 – 459 1000 500
Glass foam 135 0.04 780 - 1200 200
Graphite foam 100 0.08 775 - 1800 500
Lightweight 1100 0.20 950 - 50 10
concrete
Pressure vessel 200 50
(£m−3 bar−1) (£m−3 bar−1)
Table 5.2: Material properties and cost factors k used in the economic analysis. Data comes from
CES Edupack Materials Selector database [131]. Standard deviations have been estimated from the
range of available data. The cost factor for the pressure vessel arose from discussions with Isentropic
Ltd. Solid densities ρs, thermal conductivities λs and specific heat capacities are averages over the
temperature range of operation.
Magnetite is a competitive option since it can be operated at high temperatures and thereby
reach low values of cost per stored energy. Despite the high capital costs of molten salts,
they are also a competitive option since the container does not need to be pressurised, al-
though these estimates do not include the heat exchanger cost. For comparison, Strasser
and Selvam [186] performed a cost analysis of concrete thermal storage in a solar farm.
The storage had an extremely large capacity of 2165 MWh and Strasser obtained economic
data from an EPRI study and by scaling certain parameters. The unpressurised concrete
191
storage operated at a temperature of 565°C and was found to have a cost of 30 USD $/kWh
which converted to current units is 20 £/kWh. This price included a number of additional
factors such as construction costs, tax and additional components. For just the storage me-
dia, insulation and tank Strasser’s results suggested a cost of approximately 8 £/kWh. For
a magnetite system of the equivalent size at the same temperature, the cost given by equa-
tion 5.3 is 36.6 ± 8 £/kWh, and this drops to 17.3 ± 4 £/kWh for an unpressurised vessel.
This is a similar order of magnitude to Strasser’s analysis and indicates that pressurising
the storage adds significantly to the cost.
5.2.2 Reciprocating device costs
Isentropic Ltd. were developing bespoke compressors and expanders specifically for the
PTES system, and estimating their cost is not straightforward. Correlations for turboma-
chinery have been presented frequently in the literature. These correlations tend to com-
bine thermodynamic modelling with uncertain constants that are calculated by a regression
through manufacturer’s data [172, 187]. This approach has been widely used [172, 188–
190], such as in Ref. [191] to evaluate the cost of an underwater compressed air energy
storage system. However, the expressions are technically only applicable to turbomachin-
ery. Furthermore, the data and regression analysis needs to have been obtained recently so
that the resulting equation represents the current value of the components.
In the present study, reciprocating device cost is estimated in a similar way to the pressure
vessel. Compressor–expander cost is likely to increase with the pressure ratio β, speed
of rotation ω, swept volume Vs and mass flow rate m˙. These factors are all related by
equation 4.17. The capital cost is thus assumed to be proportional to the pressure ratio and
volumetric flow rate V˙ = ωVs as
Cr = krβV˙ (5.5)
Where kr is the cost factor. kr is estimated by considering economic analyses of systems in-
volving diesel generators. Although there are obviously some differences in design, diesel
generators contain reciprocating components and can therefore provide a rough estimate of
cost. There are a number of papers which present economic or thermo-economic studies of
systems containing reciprocating devices or diesel engines, but meaningful data can only
192
be extracted from a subset of these. This data is shown in table 5.3.
The literature generally provided the diesel engine rated power, and either the cost of the
engine or the cost per kilowatt of rated power, Cr/W˙ . Some sources also gave cylinder
geometry making it possible to calculate kr. The data in table 5.3 show that kr is typically
of the order of 10 000 £m−3s. A representative value of kr = 8000 £m−3s has been cho-
sen. Capital costs were calculated by inflating the cost to the current value (January 2016)
for that currency, and then converting to pounds sterling using the current exchange rate.
Further information about the data that was used is shown in table 5.4.
For a nominal design (see table 4.2) with ω = 1200rpm = 20revs−1, β = 10, and Vs ≈
0.5m3, and using kr = 8000 m−3s gives a compressor cost of approximately £800 000.
193
Power Cr Ncyl β Vs ω Cr/W˙ kr = Cr/βV˙
W˙ (kW) (£× 103) (m3 × 10−3) (rpm) (£kW−1) (£m−3s)
Bonnet et al. [192] 24 6.4 1 6.0 7.0 1000 271 9100
Herrera et al. [193] 44 8.4 4 17.0 3.9 1800 191 4200
Ehyaei et al. [194] 145 29.9 6 10.5 10.5 1800 206 9050
Acikkalp et al. [195] 5900 2200.0 - - - - 371 -
Balli et al. [196] 6500 2340.0 18 ≈30 580.0 750 360 10800
Abusoglu et al. [176] 28800 4000.0 18 - - - 139 -
Table 5.3: Data from the literature regarding diesel engine rated power, geometry, and capital costs.
Further information about the literature sources is given in table 5.4.
Notes
Bonnet et al. [192] Investigates an Ericsson engine which uses reciprocating compressors
and expanders. Compression occurs in two stages and there is a lack
of data concerning the operating points of these two compressors. Ex-
pansion occurs in a single stage and data in table 5.3 is based on the
expander which cost 7 200 euros in 2005.
Herrera et al. [193] Diesel engine as part of a co-generation plant. Economic data originates
in Herrera’s PhD thesis (in Portuguese) and suggests a capital cost of
USD$12 555.
Ehyaei et al. [194] Internal combustion engine based on the Caterpillar G3306B engine for
co-generation applications. Assumes cost of installation is 300 USD
$/kW in 2012.
Acikkalp et al. [195] Diesel engine for use in a tri-generation plant. Engine has a Purchasing
Equipment Cost (PEC) of USD$3 280 000 in 2014.
Balli et al. [196] Engine for use in a tri-generation plant, based on a Wartsilla 18V32GD
diesel engine. Purchasing Equipment Cost is given as USD$3 250 000
in 2010. Engine geometry data in table 5.3 is from the Wartsila web-
site [197].
Abusoglu et al. [176] Data is for a diesel engine as part of a co-generation system. The diesel
engine has a cost of USD$5 400 000 in 2009 and this data was provided
by Sanko energy
Table 5.4: Further details about the literature sources for data on reciprocating diesel engines
194
5.2.3 PTES capital cost
Evaluation of the full PTES cost also requires estimation of the cost of heat exchangers,
the motor-generator, pipework, valving and control systems, as well as construction costs.
Thermo-economic optimisation of a Rankine-cycle based PTES system was undertaken
in [88]. The costs for several components were estimated by fitting linear trends to data
from vendors. The heat exchanger cost was given by
CHX = Z
[
450A0.82HX + 5000
]
(5.6)
where AHX is the heat transfer area, and Z ≈ 0.77 is a factor that converts the cost (in US
dollars in 2009) into January 2016 pounds sterling. The compact heat exchanger of surface
type 8.0−3/8T described in section 4.3.1 has a geometry of AHX/V = 587 m2 / m3 [163].
Therefore, a heat exchanger of volume 1 m3 would have a capital cost of CHX = £68 000.
Ref. [88] proposed that the cost of a variable speed motor-generator would vary as
Cmg = Z [110P + 5000] (5.7)
where P is the electric power (in kW). Therefore, a 2 MW system would have a capital
cost of Cmg = £173 000.
The total capital cost is thus,
Ccap = C
hot
bed + C
cold
bed + 2Cr + 2CHX + Cmg (5.8)
The expression for capital cost contains four rather uncertain cost factors k (not including
the variables in the expressions of Ref. [88]). Table 5.1 shows these factors as derived from
data in the literature. The standard deviation of the cost factors has also been estimated
given the range of data found and assuming the costs follow a normal distribution.
The capital cost of the nominal 16 MW h, 2 MW PTES system of table 4.1 is estimated
as £1.4 million, or 160 £/kWh. The distribution of cost between each component is given
in table 5.5. The standard deviation is calculated using the values given in table 5.1 and is
195
CE EC HS CS HX MG Total
Ccap (£× 1000) 245 101 417 310 136 173 1380
Ccap/Wdis (£/kWh) 28 12 48 36 16 20 160
standard deviation (£/kWh) 7 3 10 6 - - 14
Ccap/W˙dis (£/kW) 170 70 290 215 94 120 958
standard deviation (£/kW) 43 18 61 37 - - 85
Table 5.5: Capital costs of each component in a nominal 16 MWh, 2MW PTES system described in
table 4.1. Standard deviations are calculated assuming that the cost factors k are normally distributed
with the values given in table 5.1. Key: CE/EC Reversible compressor-expander; HS/CS hot/cold
store; HX heat exchangers; MG motor-generator.
roughly 14 £/kWh1. The standard deviation indicates the uncertainty in the cost factors but
it does not reflect the uncertainty of the estimation method, which may be considerable and
could vary by as much as a factor of two or three.
For comparison, table 5.6 gives the costs of several other prominent energy storage types.
The data was obtained from Chen et al. [39] and adjusted to the current value. The present
study suggests that PTES is significantly more expensive that PHES or CAES in terms of
cost per work output, and has a value more comparable to lead-acid batteries. It is notable
that PTES is an order of magnitude more expensive than other TES systems. PTES is
comparable to PHES and CAES in terms of cost per power output, and is more expensive
than most of the batteries by this metric.
The actual cost of PTES may be quite different to this value since optimisation has not been
undertaken. In addition, these costs do not include the capital cost of control systems or
construction costs. Morandin et al.’s [88] study of a 50 MW, 100 MWh PTES system sug-
gested optimal designs would cost in the range of 210–290 £/kWh or 420–580 £/kW which
is a similar order of magnitude to the estimates presented here. Alternatively Howes [5]
suggested that Isentropic Ltd. could build a PTES system (including installation) for 10
£/kWh and 280 £/kW. Howes’ estimation is significantly cheaper than the values suggested
here, and seems to be optimistic. Nevertheless, it emphasises how uncertain even a rela-
1The overall standard deviation is found by assuming that each cost factor varies with a normal distribution
such that kj ∼ N
(
µj , σ
2
j
)
. Since the capital cost is given by a weighted sum of the cost factors such as C =∑
j kjWj , then the capital cost also has a normal distribution given by C ∼ N
(∑
j µjWj ,
∑
j (σjWj)
2
)
from which the standard deviation is found.
196
System Capital cost per work
output (£/kWh)
Capital cost per power
output (£/kW)
PHES 4 - 75 440 - 1500
CAES 1 - 40 300 - 600
Flywheel 700 - 4000 190 - 260
TES 20 - 50 -
Lead-acid 150 - 300 200 - 400
Li-ion 400 - 1800 900 - 3000
Super-capacitor 200 - 1500 75 - 200
PTES 150 - 200 850 - 1050
Table 5.6: Capital costs of several prominent energy storage systems, data originally from
Chen [39] has been inflated and converted to current value of GBP (£). Capital costs for the nominal
PTES design are shown. N.B. These do not represent the cost that the optimal design may be able
to achieve.
tively simple measure such as the capital cost can be. As a result, the capital cost is used
here to compare one PTES design with another, rather than making comparisons with other
technologies.
Due to the uncertainty of the economic cost, a comprehensive sensitivity analysis should
be carried out to explore the impact of each of the economic factors. Preliminary analysis
suggests that the capital cost is dominated by the pressure vessel and the reciprocating
devices. Without more detailed studies, it is unreliable to compare capital costs to other
systems.
5.3 Multi-objective optimisation techniques
Optimisation can assist designers and engineers to find the best solution to a problem, by al-
lowing them to maximise the potential of their work whilst minimising the expense. There
are a wide range of optimisation algorithms each with its own strengths and weaknesses,
and each suitable for certain types of problem.
The simplest technique is to differentiate the objective function (i.e. the function that is
to be minimised or maximised) with respect to the parameters of interest. This method
can be extended to constrained problems with the technique of Lagrange multipliers and
Kuhn-Tucker multipliers. Iterative algorithms that search through potential solutions are
197
also popular, and are appropriate in certain circumstances. Common examples are the
golden section method, method of steepest descent, the Newton-Rhapson method and the
conjugate gradient method.
These techniques generally rely on the function being defined analytically and differen-
tiable. Moreover, approaches such as the Newton-Rhapson method work most efficiently
if the function is approximately quadratic, while other techniques may be computationally
intensive, for instance, they may involve the inversion of matrices at each iteration.
5.3.1 Evolutionary Algorithms
The optimisation techniques mentioned above are limited to tackling certain types of prob-
lems. In practical scenarios, optimisation problems can become extremely complex for
several reasons:
1. The objective function and its derivatives may not be obtainable analytically, for
example, it may be the output of a complex computer model
2. The objective function may be a function of many design variables so that the search
space is large and multi-dimensional
3. There may be several competing objectives. A common example in engineering
problems is to maximise performance and minimise cost
4. The problem may be highly constrained, so that the search space is not only large
and multi-dimensional, but also disjoint
5. The problem may be multi-modal, so that many local optima may exist that could
easily trap an optimisation routine
6. The problem may be dynamic and the solutions may change with time
An appropriate approach to difficult optimisation problems such as these is to use a stochas-
tic search method such as Monte Carlo sampling, which explores the design space in a
random manner. This approach is suited to complex problems, since it can traverse the
entire search space without the risk of becoming trapped in local optima. Monte Carlo
198
sampling can be improved if the random search is directed in some way, and there are
several methods of doing this which are discussed below.
Simulated Annealing is inspired by the cooling of metal during which the structure formed
is such that the energy of the configuration is minimised. The algorithm therefore seeks
to minimise the internal energy of the system, that is, the objective function. Solutions are
stochastically chosen and are accepted with a given probability. (The probability is one for
solutions that are better than the previous solution, and less than one otherwise). Cooling
is mimicked by reducing the probability of accepting worse solutions with time. Allowing
worse solutions (with diminishing probabilities) prevents the algorithm becoming trapped
in local minima.
Simulating the movements of large groups of organisms, such as schools of fish, or colonies
of ants, led to the development of Particle Swarm Optimisation. Each particle in the swarm
moves about the search space, and its direction and velocity is influenced by the movements
and discoveries of other members of the swarm.
Tabu search [198, 199] uses a memory archiving scheme to store the most recently visited
points and record them as tabu, thereby preventing them from being revisited for a certain
period. Tabu search methods are well suited to highly constrained problems and make rapid
initial progress.
Evolutionary algorithms are a popular branch of stochastic optimisation. These algorithms
are inspired by the Darwinian theory of evolution, and aim to evolve solutions to prob-
lems [200]. This typically involves generating an initial population of solutions whose
members breed and mutate, with only the ‘fittest’ members surviving. Fitness implies how
good the member is in terms of the objective function, however, other factors can be in-
corporated, see the discussion on diversity below. Each activity (recombination, mutation,
selection) is subject to much variation, depending on the exact implementation of the algo-
rithm. This is a very broad field and many diverse approaches have been suggested; popular
methods are genetic algorithms [201] and evolution strategies [202, 203].
One reason for the diversity, even within sub-disciplines, of optimisation algorithms is the
“no free lunch” theorems [204] which imply that every optimisation algorithm performs
equally well when its performance is averaged across all possible problems.
199
5.3.2 Multi-objective Optimisation and Pareto Fronts
A particularly relevant issue is how to handle problems in which multiple objectives require
optimisation, especially when the improvement of one objective may come at the expense
of another: that is, when trade-offs exist between the objectives. One solution to this is to
combine each of the N objectives fi into one composite objective, f¯ , by assigning a weight
wi to each objective depending on its perceived importance.
f¯ =
N∑
i=1
wifi (5.9)
1 =
N∑
i=1
wi (5.10)
This approach therefore allows the application of single-objective algorithms such as those
discussed in section 5.3.1, however, it clearly has the disadvantage that the designer has
to choose the weightings prior to running the optimisation leaving the results at risk of
the designer’s preconceptions. Ideally, the designer would like to be able to see the entire
trade-off surface, in order to make a more informed decision.
Game theory and the Nash equilibrium provides the foundation for an alternative approach
whereby each objective is treated as a ‘player’ in a game where every player aims to max-
imise their utility [205–207]. Every player seeks the strategy that maximises their pay-off
given the strategy of every other player, where the strategy is the vector of decision vari-
ables. Whilst an advantage of this approach is its speed and the fact that weightings are
not required, it also suffers from the fact that only one point on the trade-off surface is
found. Game theoretic optimisation has been incorporated into algorithms which generate
the full trade-off surfaces, however, as the game theoretic solution can be used to ‘seed’ an
algorithm which generates the full trade-off front [208].
Algorithms which find points on the entire trade-off surface (also known as Pareto fronts)
do so by the concept of dominance. Each of the solutions on the Pareto front is deemed
as good as every other solution: one solution may be better in terms of the first objective
function, but it will be worse in terms of the second. A Pareto front is therefore composed
of solutions which are not dominated by any other solution. Formally, a solution x (where
200
x is the vector of design variables) is dominated by a solution y if y is at least as good in
all objectives and is better in at least one:
fi(y) ≤ fi(x) ∀i = 1, N and fi(y) < fi(x) for some i (5.11)
where N is the number of objectives, and the optimiser is seeking to minimise the objec-
tives.
A Pareto dominance based approach thus allows the designer to see the entire range of
potential solutions without having to specify a weighting for each objective and can po-
tentially suggest solutions that the designer would not have previously thought of or con-
sidered. This diversity in the solution space can also prevent ‘inbreeding’ – i.e. where the
algorithm converges on a single, non-optimal point.
5.3.3 Diversity in Objective and Solution Spaces
There are many methods of achieving diversity which generally focus on diversity in the
objective function space [209–211]. For example, the Non-dominated Sorting Genetic
Algorithm (NSGA-II) [212] encourages diversity along the Pareto front by calculating a
measure of distance between each member on the front known as the crowding distance.
This measure of the density can be used to determine which solutions to keep and which to
discard. In an alternative approach known as the Strength Pareto Evolutionary Algorithm
(SPEA) [213] a distance metric is not required, and instead a ranking procedure based on
the number of solutions that a given solution dominates is used. The fewer points a solution
dominates, the more this solution is preferred since it is located in a less-explored area of
the objective space. Comparisons of various algorithms (bearing in mind the ‘no free lunch
theorems’) have determined that SPEA and NSGA-II generally show the best performance
[212, 213], although SPEA has the disadvantage of greater computational complexity.
It is also important to maintain diversity in the design variable space. Members of a diverse
population will generate equally diverse offspring therefore helping the design space to
be more fully explored. The objectives of achieving convergence to a Pareto front and
maximising diversity of the population may be at odds. The Genetic Diversity Evolutionary
Algorithm (GDEA) [214] explores this conflict by making diversity an additional objective
201
in the problem.
5.4 Optimisation of packed-bed thermal reservoirs
5.4.1 Problem formulation
The multi-objective genetic algorithm (MOGA) is first applied to packed-bed thermal reser-
voirs. Thermal reservoirs can be integrated into a variety of storage systems including
PTES, CAES, and LAES. While holistic optimisation should consider the full system and
its requirements, an interesting preliminary study is to discover the optimum performance
of packed beds.
Both hot and cold reservoirs are optimised independently. Furthermore, optimisations are
carried out on unsegmented, segmented and radial-flow stores, which were introduced in
chapter 3. Since the intended application of the packed beds is in bulk electricity storage,
the internal energy storage capacity and nominal charging time of the stores is held con-
stant. For consistency, the reservoirs have the same energy capacity as the nominal PTES
design, see table 3.1. The nominal charging time is 8 hours, and the maximum energy
capacities of the hot and cold store are 26.5 MWh (95.3 GJ) and 10.7 MWh (38.5 GJ),
respectively.
The objective functions of the thermo-economic optimisation are the round-trip efficiency
and the capital cost per unit returned available energy. For the thermal stores, the round-trip
efficiency is defined as in equation 2.44, as the ratio between the recovered available energy
flux, and the net available energy flux during charge
χ =
Bout
Bin
(5.12)
The capital cost per unit returned available energy is given by
CˆresE =
Cbed
Bout
(5.13)
Where Cbed is the capital cost of the packed bed as defined in equation 5.3. The MOGA is
202
a minimisation routine so that the objective functions are expressed as
f1 = 1− χ (5.14)
f2 = Cˆ
res
E (5.15)
The design variables of interest are the aspect ratio L/D, the particle diameter dp, the
exit temperature fractions which indicate the end of charge or discharge θc and θd, the
charging and discharging temperatures T1 and T2, the number of segments Nseg and the
temperature thresholds which switch segments on and off θl and θh. The nominal values of
these parameters, and the lower and upper bounds are shown in tables 5.7 and 5.8 for hot
and cold stores respectively.
The operating temperatures are design variables, and the reservoir volume is calculated to
keep the energy capacity of the storesEres constant. For instance, for a hot store the volume
is given approximately by
Vpack =
Eres
ρs(1− ε)c¯s (T1 − T2) (5.16)
(A more exact value would integrate over the temperatures to take into account the varying
specific heat capacity cs). In addition, the nominal charging time tN (equation 2.5) is held
constant which determines the mass flow rate:
m˙ =
ρs(1− ε)Vpackc¯s
cp,gtN
(5.17)
Constraints are placed on the temperatures so that T1 > T2 + 50 for a hot store and T1 <
T2 − 50 for a cold store. The constraint is imposed such that the temperatures must be at
least 50°C apart (an infinitesimal difference would be the most efficient, but would require
an infinitely large store which is clearly not practicable).
As well as the objective functions, a number of other metrics are of interest:
• The charging duration ∆tchg and storage duration ∆tstore.
• The volume of the packing media Vpack
• The mass flow rate, m˙
203
L/D dp (mm) θc (%) θd (%) T1 (°C) T2 (°C) Nseg θl (%) θh (%)
Nominal 1.0 20.0 25.0 25.0 505.0 37.0 1 1.0 99.0
Lower bound 0.5∗ 1.5 5.0 5.0 37.0 37.0 1 1.0 95.0
Upper bound 2.0 20.0 50.0 50.0 600.0 600.0 32 5.0 99.0
Table 5.7: Lower and upper bounds for parameters used in optimisation of hot stores. For radial
stores and unsegmented stores θl and θh are not variables andNseg = 1. ∗ for radial stores the lower
bound of L/D is 0.1.
L/D dp (mm) θc (%) θd (%) T1 (°C) T2 (°C) Nseg θl (%) θh (%)
Nominal 1.0 20.0 25.0 25.0 -150.0 37.0 1 1.0 99.0
Lower bound 0.5∗ 1.5 5.0 5.0 -170.0 -170.0 1 1.0 95.0
Upper bound 2.0 20.0 50.0 50.0 37.0 37.0 32 5.0 99.0
Table 5.8: Lower and upper bounds for parameters used in optimisation of cold stores. For radial
stores and unsegmented stores θl and θh are not variables andNseg = 1. ∗ for radial stores the lower
bound of L/D is 0.1.
• The returned available energy Bout
• The average power output Pave = Bout/∆tchg
• The energy density ρE = Bout/Vpack
• The power density ρP = Pave/Vpack
• The capital cost of the packed bed Cbed
• The capital cost per unit average power output CˆresP = Cbed/Pave
5.4.2 Results
Thermo-economic optimisation results for hot and cold stores are presented below. Pareto
fronts for cold and hot thermal stores are shown in figures 5.2(a) and 5.2(c), respec-
tively. The Pareto front visualises the optimisation results by plotting one objective function
against the other. The Pareto front effectively displays the trade-off between efficiency and
capital cost: each point on the Pareto front is better than any other point in at least one
objective. For instance, on figure 5.2(a), the point marked C1 is cheaper than C2, although
C2 is more efficient than C1.
The design variables, and several other metrics are tabulated in tables 5.9 and 5.10 for
204
selected optimal points (as indicated on the Pareto fronts). A comprehensive understanding
of the optimal designs is best achieved by studying the full range of designs along the
Pareto front. For brevity, two points are selected from each Pareto front with the aim of
being representative of the type of designs that occur at each end of the trade-off surface.
A thermo-economic optimisation with fixed operating temperatures was also carried out,
using the same objective functions and thermodynamic model. Results are presented in [80]
and are not replicated here as the results of both optimisations followed the same trends.
The main differences were that when the temperatures were fixed, the optimal stores were
found to be less efficient than when the temperatures could vary. For instance, the most
efficient hot and cold stores were 99.7% and 96.8% respectively, compared to 98.7% and
94.2%. This is not unduly surprising, but indicates the importance of operating tempera-
tures on the design. It also indicates that optimising a full PTES system cannot be achieved
by optimising individual components independently.
Pareto front behaviour
The Pareto fronts for hot and cold stores show the expected trade-off between capital cost
and efficiency: high efficiency stores have high costs. The Pareto front of the cold store is
particularly flat at high efficiencies meaning that it is possible to significantly reduce the
capital cost without unduly compromising the efficiency. However, the efficiency drops
very suddenly once the cheapest designs are sought. The best compromise exists around
the ‘knee’ of the curve where the gradient of the trade-off surface changes most rapidly.
Table 5.11 below indicates the likely performance of points in this region for hot and cold
segmented stores. Comparing these results with the performance of other established tech-
nologies in table 5.6 on page 197 suggests that packed-bed thermal reservoirs are slightly
more expensive per work output than other bulk technologies such as PHES and CAES.
Thermal storage systems that utilise packed-beds will be more expensive than the indicated
values, as this study does not include the cost of the rest of the storage system. However,
these estimates are very approximate and the true values could vary significantly. These
results indicate that hot stores are more efficient and cost effective than cold stores, which
are unlikely to be used unless the full energy storage system specifically requires it.
205
 82
 84
 86
 88
 90
 92
 94
 96
 98
 25  50  75  100  125  150  175  200  225
E
ff
ic
ie
n
cy
 (
%
)
Capital cost / returned available energy (£ / kWh)
C1
C2
C3
C4 C5
C6
Nominal
Unsegmented
Segmented
Radial flow
(a)
 0
 5
 10
 15
 20
Nom. C1 C2 C3 C4 C5 C6
A
va
ila
bi
lit
y 
lo
ss
es
 (%
)
ζt ζp ζL ζc ζs
(b)
 97
 98
 99
 100
 25  50  75  100  125  150
E
ff
ic
ie
n
cy
 (
%
)
Capital cost / returned available energy (£ / kWh)
H1
H2
H3
H4
H5
H6
Unsegmented
Segmented
Radial flow
(c)
 0
 1
 2
 3
 4
 5
Nom. H1 H2 H3 H4 H5 H6
A
va
ila
bi
lit
y 
lo
ss
es
 (%
)
ζt ζp ζL ζc ζs
(d)
Figure 5.2: Optimisation results for cold and hot packed-bed thermal reservoirs. (a) Cold store
Pareto fronts (b) Cold store distribution of availability losses for the selected points in figure (a)
(c) Hot store Pareto fronts (d) Hot store distribution of availability losses for the selected points in
figure (c)
206
Nom. C1 C2s C3r C4 C5s C6r
L/D 1.00 0.50 0.51 0.43 0.51 0.51 0.45
dp (mm) 20.0 6.0 5.6 5.9 7.5 8.1 10.7
θc (%) 25.0 50.0 44.0 47.0 17.0 24.0 8.5
θd (%) 25.0 50.0 49.0 50.0 9.7 18.1 17.5
T1 (°C) -150.0 -170.0 -170.0 -170.0 -169.7 -167.7 -165.5
T2 (°C) 37.0 15.0 17.6 -5.1 -76.9 -114.1 -104.9
Nseg 1 1 9 - 1 14 -
θl (%) - - 3.2 - - 3.9 -
θh (%) - - 98.7 - - 98.5 -
∆tchg (h) 5.5 7.3 7.2 7.3 4.5 6.2 4.1
∆tstore (h) 6.5 6.5 4.7 4.8 4.7 5.8 7.9
m˙ (kg/s) 13.7 13.9 13.7 15.6 27.7 47.9 42.4
Vpack (m3) 127.1 139.4 136.8 162.5 342.9 667.5 564.8
Bout (MWh) 3.0 5.4 5.3 5.9 6.1 10.2 6.4
Pave (MW) 0.55 0.74 0.74 0.81 1.36 1.66 1.53
ρE (kWh m−3) 23.5 38.7 39.0 36.3 17.7 15.3 11.2
ρP (kW m−3) 4.3 5.3 5.4 5.0 4.0 2.5 2.7
Ccap (£×1000) 320.0 304 305 365 717 1510 1300
CˆresP (£/ kW) 586 409 414 452 529 909 845
χ (%) 86.7 85 85.3 83.3 96.1 96.8 96.0
CˆresE (£/ kWh) 107 56.4 57.2 62.0 118.3 147.4 204.1
Table 5.9: Design variables and results for selected points from the optimisation of a cold store.
Points correspond to those in figure 5.2(a). Note that those marked with a s are segmented beds
and those with r are radial-flow stores. The first data block provides the design variables for these
points. The second and third data block gives some other useful information, while the final data
block provides the objective functions (efficiency and cost per unit returned energy).
General trends in decision variables
Reductions in CˆresE are achieved by either reducing the capital cost or increasing the avail-
able energy that is returned during discharge. The capital cost is generally minimised by
reducing the pressure vessel volume. This is achieved by the charging temperature T1 con-
verging to its extreme limit (maximum for hot stores and minimum for cold stores) which
also increases the efficiency. Conversely, T2 varies along the Pareto front as the least ex-
pensive designs have a large ∆T = |T1−T2|whereas a small ∆T maximises the efficiency,
as in section 3.2.2. However, it should be noted that operating temperatures may in fact be
207
Nom. H1 H2s H3r H4 H5s H6r
L/D 1.00 0.85 0.93 1.07 0.50 0.51 1.90
dp (mm) 20.0 2.9 4.1 2.5 8.0 4.0 1.7
θc (%) 25.0 0.50 0.50 0.50 7.0 12.0 33.8
θd (%) 25.0 0.15 0.35 0.39 17.3 10.9 38.3
T1 (°C) 505.0 600.0 600.0 600.0 595.0 600.0 600.0
T2 (°C) 37.0 110.3 133.4 112.4 308.8 350.7 516.2
Nseg 1 1 4 - 1 15 -
θl (%) - - 2.6 - - 1.2 -
θh (%) - - 98.2 - - 97.7 -
∆tchg (h) 5.5 7.9 7.9 8.0 4.3 5.8 6.6
∆tstore (h) 6.5 4.1 4.1 4.0 7.7 6.2 5.4
m˙ (kg/s) 13.7 13.0 13.1 13.3 22.2 25.5 76.3
Vpack (m3) 75.5 66.5 66.9 66.9 104.4 117.5 329.0
Bout (MWh) 7.1 12.6 12.7 12.8 8.3 11.5 14.0
Pave (MW) 1.31 1.60 1.61 1.60 1.92 1.97 2.1
ρE (kWh m−3) 94.6 190.0 190.6 190.9 79.4 97.7 42.5
ρP (kW m−3) 17.4 24.1 24.2 24.0 18.4 16.8 6.4
Ccap (£×1000) 422 374 378 405 592 698 1784
CˆresP (£/ kW) 322.3 233.5 233.9 252.9 308.3 353.7 844.9
χ (%) 95.0 97.3 97.4 97.0 99.5 99.7 99.6
CˆresE (£/ kWh) 59.1 29.6 29.6 31.8 74.5 60.8 127.7
Table 5.10: Design variables and results for selected points from the optimisation of a hot store.
Points correspond to those in figure 5.2(c). Note that those marked with a s are segmented beds
and those with r are radial-flow stores. The first data block provides the design variables for these
points. The second and third data block gives some other useful information, while the final data
block provides the objective functions (efficiency and cost per unit returned energy).
constrained by the whole system design.
The trade-off between efficiency and cost may be illustrated by considering the shape of
the thermal fronts that are formed. Choosing values of decision variables that lead to steep
thermal fronts leads to lower efficiencies due to larger thermal and conductive losses. On
the other hand, steeper fronts travel further along the packed bed by the end of the charging
phase and therefore increase the returned available energy.
As a result, the position of a point on the Pareto front is primarily controlled by the charging
duration, which is determined by the gas exit temperature fractions θc and θd. As discussed
in chapter 3 longer charging times lead to steeper thermal fronts. The optimal particle size
208
χ (%) CˆresE (£/kWh) Cˆ
res
P (£/kW)
Hot store 99.5 35–50 400–500
Cold store 95.5 60–90 250–350
Table 5.11: Typical efficiencies and capital costs of optimal packed-bed thermal reservoirs
depends on the charging periods. When a steep front is formed, smaller particles increase
the heat transfer area and reduce the thermal losses. As a result, lower cost stores tend to
use smaller particles.
Available energy loss mechanisms for the selected points are illustrated in histograms in
figures 5.2(b) and 5.2(d). The most efficient points (C4–C6 and H4–H6) have designs
where thermal losses are approximately equal to pressure losses which is a result of the
optimiser finding a trade-off between these two conflicting loss mechanisms. On the other
hand, thermal losses dominate in the high energy density designs as a result of the steep
thermal fronts. The steep fronts (and small particles) in designs C1–C3 and H1–H3 also
lead to large conductive losses during operating and storage.
Benefit of segmentation
Inspection of the Pareto fronts suggests that segmented thermal stores out-perform un-
segmented stores. As expected, segmentation provides the greatest improvement to cold
stores. For example, the highest efficiency unsegmented cold store is 96.2% whilst the best
segmented store is 96.8%. At a fixed efficiency, cold segmented stores offer a significant
saving. For example, the unsegmented store C4 has CˆresE = 118 £/ kWh but a correspond-
ing segmented point costs around 93 £/ kWh which is roughly 20% cheaper. However,
this does not include the cost of control systems or valving, so this saving represents the
amount of money that is available to be spent on additional equipment before segmented
stores stop being economically competitive with unsegmented stores. (Note, that this study
does in fact include a simple economic penalty for segmentation; bypass flow requires ex-
tra volume resulting in larger containment vessels and higher pressure vessel costs.) The
optimiser finds value in layering hot stores, but the benefit is quite small as pressure losses
are low.
The reduced pressure losses in segmented stores typically lead to larger aspect ratios (par-
209
ticularly in the most efficient hot stores) and smaller particles. Consequently, the thermal
fronts are steeper, and the utilisation of the reservoirs increases.
θh and θl affect the truncation of the thermal front that occurs when segments are activated
and de-activated as discussed in section 3.3.2. Front truncation leads to thermal losses that
are most significant at the trailing edge during discharge. The optimiser therefore converges
towards the upper limit of θh in cold stores and the lower limit of θl in hot stores.
Optimisation of radial-flow stores
Radial-flow stores are outperformed by segmented and unsegmented axial-flow packed
beds. Although radial-flow stores can achieve similar efficiencies to the axial-flow stores,
it is always more expensive to do so. These high costs stem from the additional space that
is required for the inner and outer plena. A good example is to compare H1 (unsegmented)
and H3 (radial-flow). These points have similar operating temperatures, require similar
volumes of packing material and achieve similar efficiencies and available energy outputs.
However, the extra volume required for the inner and outer plena leads to a pressure vessel
that is 6% larger than the unsegmented store which increases CˆresE .
The optimisation was carried out with the assumption that the radius of the inner plenum
should be 20% of the packing radius. The outer plenum has an equal area to the inner
plenum. Reducing the size of the inner plenum would reduce the required volume and
therefore the cost. However, detailed studies are necessary to determine how small the
inner plenum may be and to realise the full implications of this design feature. The potential
improvements in radial-store performance could be checked by undertaking an optimisation
with Ri/Ro = 0.
Economic considerations
A breakdown of the capital cost between the various components for selected points is
given in figure 5.3. Both the capital cost Ccap and the capital cost per unit returned avail-
able energy CˆresE are shown. Error bars are plotted to indicate the uncertainty in the cost
factors and show the total cost plus or minus one standard deviation, which is calculated
as described in a footnote on page 196. A cursory glance at the figures suggests that the
210
uncertainty is quite high. Furthermore, the uncertainty is likely to be higher than the error
bars indicate, as they do not reflect the uncertainty in the estimation method. The capital
cost is therefore best used to compare one design against another rather than determining
the feasibility of packed-bed thermal reservoirs.
The cost of the steel containment vessel dominates the capital cost. There is also a large
difference in cost between the most efficient designs and the least efficient designs. This
is primarily due to the variation in volume due to the variable operating temperatures.
For instance, points C4–C6 have more than double the packing volume of points C1–C3.
This leads to much larger reservoirs with larger insulation and containment requirements.
Hot stores are more expensive (despite their lower volumes) due to the higher pressure of
operation. Radial stores are typically more expensive due to the larger volumes that are
required as a result of the inner and outer plena. H6 (radial) has a much higher capital cost
Ccap as a result of a small difference between T1 and T2.
The results above are consistent with the theory and parametric studies of packed beds
presented in chapters 2 and 3. The optimisation algorithm is nevertheless a useful tool as
it identifies the surface of optimal points which would have been difficult to achieve using
only the knowledge developed in the opening chapters.
211
 0
 250
 500
 750
 1000
 1250
 1500
 1750
Nom. C1 C2 C3 C4 C5 C6
Ca
pi
ta
l c
os
t (
£ x
 10
3 )
Steel Packing Insulation
(a)
 0
 25
 50
 75
 100
 125
 150
 175
 200
 225
 250
Nom. C1 C2 C3 C4 C5 C6
Ca
pi
ta
l c
os
t /
 re
tu
rn
ed
 a
va
ila
bl
e 
en
er
gy
 (£
 / k
W
h)
Steel Packing Insulation
(b)
 0
 500
 1000
 1500
 2000
Nom. H1 H2 H3 H4 H5 H6
Ca
pi
ta
l c
os
t (
£ x
 10
3 )
Steel Packing Insulation
(c)
 0
 25
 50
 75
 100
 125
 150
 175
Nom. H1 H2 H3 H4 H5 H6
Ca
pi
ta
l c
os
t /
 re
tu
rn
ed
 a
va
ila
bl
e 
en
er
gy
 (£
 / k
W
h)
Steel Packing Insulation
(d)
Figure 5.3: Distribution of capital costs in optimal cold and hot stores. Indicated points are shown
on figure 5.2. (a) Capital costs (cold) (b) Capital costs normalised by the returned available energy
(cold) (c) Capital costs (hot) (d) Capital costs normalised by the returned available energy (hot)
212
5.5 Optimisation of PTES systems
The optimisation method has been applied to several complete PTES systems. Optimisation
is first carried out on the nominal PTES design described in section 5.5.1, before also
considering systems with layered stores, and with lower values of reciprocating device
polytropic efficiency. In addition, constraints on the highest temperature T2 are relaxed so
that systems more similar to the SAIPEM design can be considered. A PTES design that
does not require an electric heater, as described in section 4.5, is also investigated.
Each optimisation study has the same decision variables, objective functions and con-
straints. The decision variables of interest are the aspect ratio L/D, particle diameter dp,
and number of segments Nseg of both reservoirs, as well as the charging and discharging
pressure ratios βchg and βdis, the utilisation Π, and the discharged reservoir temperatures,
T1 and T3. The nominal values, as well as upper and lower bounds are given in table 5.12.
This table also illustrates the constraints that are placed on temperatures T2 and T4.
L/D dp (mm) Nseg βchg,dis T1 (°C) T2 (°C) T3 (°C) T4 (°C) Π
Nominal 1.0 20.0 1 10.0 37.0 505.0 37.0 -150.0 0.75
Lower bound 0.5 2.0 1 5.0 37.0 T3 + 50 37.0 -160.0 0.10
Upper bound 2.0 20.0 32 15.0 600.0 600.0 600.0 T1 − 50 0.99
Table 5.12: Lower and upper bounds for parameters used in optimisation of PTES systems. For
unsegmented stores Nseg = 1, and in all cases θl = 1% and θh = 99%. Note that T2 and T4 are not
decision variables, but are constrained as indicated.
The objective functions are the thermodynamic round-trip efficiency χ and the capital cost
per unit net work during discharge CˆE . To recap, the thermodynamic efficiency does not
include mechanical or electrical losses, and is given by the net work during discharge di-
vided by the net work input during charge. There are a number of other metrics of interest
and including:
• The charging duration ∆tchg
• The mass flow rate m˙
• The work ratio R, defined as the ratio of compression to expansion work during
charge. For simplicity, and to allow comparison with the use of the work ratio in
section 1.5.2, the work ratio is calculated as R = T2/T3
213
• The average power output W˙ave = W netdis /∆tchg
• The energy density ρE = W netdis /(VCR + VHR), where VCR,HR are the volumes of
packing material in the cold and hot reservoirs, respectively
• The average power density ρP = W˙dis/Vcyl where Vcyl is the total volume of the
largest compression–expansion device
• The capital cost per unit power CˆP = Ccap/W˙ave
Two previous optimisation studies of PTES systems have been undertaken by the author
and are presented in [81, 84]. Ref. [84] was a thermodynamic optimisation with three
objective functions: the round-trip efficiency, the energy density, and the power density.
This initial study did not consider layered stores or include conductive losses, leakage
losses or storage periods. This investigation found that there was no trade-off between
power density and efficiency: designs that increased one objective would increase the other.
A trade-off did occur between efficiency and energy density and the compromise between
the two objectives was mainly controlled by the utilisation.
In Ref. [81] conduction, leakage, and storage losses were included, and segmented reser-
voirs were allowed. The objective functions were round-trip efficiency and capital cost
per unit work output. The paper carried out three optimisations with different values of
kr to investigate what impact the economic factors may have on the optimal designs. It
was found that while kr significantly affected the capital cost, it had little influence over
the designs that the optimisation routine converged on. This suggests that the PTES design
may be robust to variations in component costs, and that these factors have little impact on
the final design. Further sensitivity analysis is ideally required.
5.5.1 Optimisation of normal PTES configurations
The ‘normal’ PTES configuration is investigated in this section, and results are presented in
figure 5.4. Data for several key points are tabulated in table 5.13 and loss distributions are
shown in figure 5.6. The ‘Normal’ design configuration is that which has been discussed
throughout this report and is based on the Isentropic Ltd. scheme. Details of the nominal
design are given in tables 3.1 and 4.1. In this optimisation the number of segments Nseg is
214
a decision variable, but the temperatures at which segments are turned on and off are fixed
to θl = 1% and θh = 99%. PTES systems with unsegmented stores are also optimised
for comparison. The impact of the compression-expression polytropic efficiency is investi-
gated by optimising a system with η = 90% as well as the nominal value of η = 98%.
 45
 50
 55
 60
 65
 70
 75
 80
 100  125  150  175  200
R
o
u
n
d
-t
ri
p
 e
ff
ic
ie
n
cy
, 
χ 
(%
)
Capital cost / returned available energy (£ / kWh)
Nominal
P1
P2
P3
P4
P5
P6
Normal
Unsegmented
η = 0.90
Figure 5.4: Pareto fronts of the optimised results of PTES systems with the basic configuration.
‘Normal’ refers to a system with the design variables shown in table 5.12; ‘Unsegmented’ has the
same design variables although the thermal stores are constrained to have a single segment;‘η =
0.90’ indicates the polytropic efficiency that the compression–expansion system used.
215
Nom. P1 P2 P3 P4 P5 P6
Hot L/D 1.00 0.95 0.99 0.88 0.71 0.61 1.08
Hot dp (mm) 20.00 2.71 2.11 4.05 3.22 2.65 3.44
Hot Nseg 1 4 8 1 1 5 7
Cold L/D 7.00 0.53 0.51 0.51 0.51 0.50 0.52
Cold dp (mm) 20.00 4.39 3.16 4.80 4.83 4.80 4.05
Cold Nseg 1 14 29 1 1 10 3
βchg 10.00 9.57 9.98 9.25 9.75 8.44 10.90
βdis 10.00 7.28 9.89 7.73 9.20 5.95 6.97
Π 0.75 0.86 0.55 0.86 0.49 0.89 0.72
T1 (°C) 37.0 77.4 72.9 84.3 75.7 70.1 35.0
T2 (°C) 505.9 591.9 595.2 597.3 594.3 596.0 599.1
T3 (°C) 37.0 37.6 35.4 39.9 36.3 35.3 36.2
T4 (°C) -145.1 -142.7 -145.6 -140.0 -144.1 -127.8 -140.0
m˙ (kg/s) 13.7 11.5 11.3 11.5 11.4 10.6 9.9
R 2.51 2.78 2.81 2.78 2.80 2.82 2.82
W netdis (MWh) 8.64 10.79 7.41 10.78 6.51 7.64 6.94
∆tchg (h) 6.01 6.88 4.42 6.92 3.88 7.15 5.77
W˙ave (MW) 1.44 1.57 1.68 1.56 1.68 1.07 1.20
Ccap (£× 1000) 1382 1191 1205 1166 1184 1065 1133
ρE (kWh m−3) 43.78 62.26 41.28 65.44 39.38 48.24 46.94
ρP (MW m−3) 2.94 3.76 4.10 3.71 4.08 2.85 3.31
CˆP (£/ kW) 960 759 717 747 705 995 944
CˆE (£/ kWh) 160 110 163 108 182 139 163
χ (%) 67.48 73.70 79.77 73.12 79.65 50.22 56.12
Table 5.13: Design variables and results of PTES optimal points P1–P6 (see figure 5.4). Objective
functions (efficiency and capital cost per work output) are shown in the final two rows..
216
 0
 2
 4
 6
 8
Hot cyl.
Cold cyl.
Hot store
Cold store
HX 1.
HX 3.
Heater
A
va
ila
bi
lit
y 
lo
ss
es
 (%
)
Nominal
ζt ζp ζL ζc ζs
Figure 5.5: Distribution of availability losses in the nominal PTES design.
Optimal design characteristics
Comparing the optimal design results with the nominal design indicates that the improve-
ment in round-trip efficiency is mainly due to a reduction in the packed bed losses, which
cause a corresponding reduction in heat rejection. The reciprocating device losses do not
change significantly which may be an artefact of the simplified modelling of the compression–
expansion system. For instance, the polytropic efficiency is likely to be a function of the
operating conditions which would cause greater variation in the reciprocating device losses.
Like the optimisation of the thermal stores, the cycle period (or utilisation Π) has the great-
est influence over a design’s position on the Pareto front. High utilisations correspond to
low efficiencies but large work outputs and therefore low CˆE . The operating pressures and
temperatures have a smaller impact and the cheapest designs typically have slightly lower
pressure ratios and higher values of T1. Lower pressure ratios reduce the cost of the com-
pressor, expander and hot store. The higher value of T1 then maintains T2 at close to its
maximum value of 600°C.
Operational parameters therefore have a significant influence, and this may have important
consequences for off-design behaviour. For instance, the electrical grid may require PTES
to operate flexibly, with irregular charge durations or power requirements. Further research
into realistic load cycles is required to quantify this effect.
217
Benefits of segmented packed beds
For costs in the range of 115–150 £/kWh the calculations predict that segmentation in-
creases the efficiency by around 1%. Moreover, segmented systems are able to achieve
high efficiencies at lower costs than unsegmented systems. This may be illustrated by con-
sidering systems with an efficiency of 79% in which case segmented systems cost around
125 £/kWh and unsegmented systems cost around 140 £/kWh, saving 11%. (This margin
will decrease once the cost for the valves and control systems required for segmentation
have been included). Layering the stores typically leads to longer reservoirs and smaller
particles, which is consistent with the packed bed optimisation results in section 5.4.
Influence of the polytropic efficiency
The round-trip efficiency is sensitive to the polytropic efficiency, with the results suggest-
ing that reducing η from 0.98 to 0.90 reduces the maximum achievable efficiency from
79.8% to 56.1%. Reducing the polytropic efficiency increases the availability losses in the
reciprocating devices from approximately 6% to 15%. In addition, heat exchanger losses
increase (particularly during discharge). The optimal designs of the η = 0.90 case are
reasonably similar to the segmented results. The main difference is that the discharging
pressure ratios are typically lower in order to reduce heat rejection.
The capital costs are quite similar to those of the η = 0.98 designs, since the systems have
similar designs. However, the capital cost per unit returned work is larger due to the low
round-trip efficiency which reduces the work output. This effect is most significant for the
lowest efficiency designs. For instance, the cheapest η = 0.90 design (P5) is approximately
21% more expensive than a cheap segmented design (P1).
218
 0
 2
 4
 6
Hot cyl.
Cold cyl.
Hot st.
Cold st.
HX 1.
HX 3.
Heater
A
va
ila
bi
lit
y 
lo
ss
es
 (%
)
P1
ζt ζp ζL ζc ζs
(a)
 0
 2
 4
 6
Hot cyl.
Cold cyl.
Hot st.
Cold st.
HX 1.
HX 3.
Heater
A
va
ila
bi
lit
y 
lo
ss
es
 (%
)
P2
ζt ζp ζL ζc ζs
(b)
 0
 2
 4
 6
Hot cyl.
Cold cyl.
Hot st.
Cold st.
HX 1.
HX 3.
Heater
A
va
ila
bi
lit
y 
lo
ss
es
 (%
)
P3
ζt ζp ζL ζc ζs
(c)
 0
 2
 4
 6
Hot cyl.
Cold cyl.
Hot st.
Cold st.
HX 1.
HX 3.
Heater
A
va
ila
bi
lit
y 
lo
ss
es
 (%
)
P4
ζt ζp ζL ζc ζs
(d)
Figure 5.6: Distribution of availability losses in optimal PTES designs for points P1 – P4 which
are illustrated on figure 5.4. The optimisation was carried out on a PTES system with the basic
configuration. (a) P1 (b) P2 (c) P3 (d) P4
219
 0
 5
 10
 15
Hot cyl.
Cold cyl.
Hot st.
Cold st.
HX 1.
HX 3.
Heater
A
va
ila
bi
lit
y 
lo
ss
es
 (%
)
P5
ζt ζp ζL ζc ζs
(a)
 0
 5
 10
 15
Hot cyl.
Cold cyl.
Hot st.
Cold st.
HX 1.
HX 3.
Heater
A
va
ila
bi
lit
y 
lo
ss
es
 (%
)
P6
ζt ζp ζL ζc ζs
(b)
Figure 5.7: Distribution of availability losses in optimal PTES designs for points P5 and P6 which
are illustrated on figure 5.4. This optimisation was carried out on a PTES system with the basic
configuration, but with the compressor–expander polytropic efficiency set to η = 0.90. (a) P5 (b)
P6
Economic considerations
Figure 5.8 illustrates the distribution of costs between each component for the points P1–
P10. Reciprocating devices are generally around a quarter of the total cost per unit work
output, while the reservoirs are around half. The hot store is more expensive than the cold
store due to higher pressures. Hot cylinders are more expensive than the cold cylinders
since they work at lower densities and thus have larger volumes.
Error bars are shown on the cost distributions and indicate the total cost plus or minus one
standard deviation which is around ±10–20% for most designs. The cheapest designs are
approximately 100 £/kWh while the most efficient designs are around 175 £/kWh. The cost
of additional components such as the electrical heater, pipes/valves, control systems or land
costs is not included and it should be emphasised that these estimates are very approximate.
However, these results do provide an indication of the minimum capital cost.
220
 0
 250
 500
 750
 1000
 1250
 1500
Nom. P1 P2 P3 P4 P5 P6 P7 P8 P9 P10
Ca
pi
ta
l c
os
t (
£ x
 10
00
)
Heat exchangers
Motor-generator
Hot cylinder
Cold cylinder
Hot store
Cold store
(a)
 0
 50
 100
 150
 200
Nom. P1 P2 P3 P4 P5 P6 P7 P8 P9 P10
Ca
pi
ta
l c
os
t /
 re
tu
rn
ed
 a
va
ila
bl
e 
en
er
gy
 (£
 / k
W
h)
Heat exchangers
Motor-generator
Hot cylinder
Cold cylinder
Hot store
Cold store
(b)
Figure 5.8: Distribution of capital costs amongst the system components for optimal PTES systems.
(a) Capital costs (b) Capital costs normalised per unit returned available energy
221
5.5.2 Optimisation of alternative PTES configurations
Optimisations have been carried out on two alternative PTES configurations, and Pareto
fronts are presented in figure 5.9. The first alternative relaxes the top temperature con-
straint, and T2 can increase up to 1100°C. This option investigates systems similar to the
Saipem design which uses a top temperature of T2 = 1200°C, small pressure ratios of
β = 4, and results in high work ratios (see section 1.5.3).
The second alternative design avoids electric heat addition as discussed in section 4.5 but is
otherwise similar to the nominal design. The configuration that is optimised has a variable
temperature T ′1 which is greater than T1. During charge, heat is continually rejected at
the exit of the cold store between T ′1 and T1. As the cold thermal front emerges from the
reservoir T ′1 decreases, but does not drop below T1. During discharge, the pressure ratio is
varied and HX1 is bypassed to ensure that T ′1 > T1. A more detailed explanation is given
in section 4.5.
 65
 70
 75
 80
 85
 75  100  125  150  175
R
o
u
n
d
-t
ri
p
 e
ff
ic
ie
n
cy
, 
χ 
(%
)
Capital cost / returned available energy (£ / kWh)
Nominal
P1
P2
P7
P8
P9
P10
Normal
SAIPEM
No heat in
Figure 5.9: Pareto fronts of the optimised results of PTES systems with alternative configurations.
‘Normal’ refers to a system with the design variables shown in table 5.12; ‘SAIPEM’ is a scheme
similar to that proposed by Desrues [91] and the constraint on top temperature is set to 1100°C rather
than 600°C; ‘No heat in’ is a configuration that avoids heat addition as discussed in section 4.5.
222
Nom. P7 P8 P9 P10
Hot L/D 1.00 0.93 1.15 1.00 0.83
Hot dp (mm) 20.00 2.69 2.00 3.99 2.04
Hot Nseg 1 7 13 6 2
Cold L/D 7.00 0.53 0.53 0.52 0.51
Cold dp (mm) 20.00 2.48 2.53 2.69 4.23
Cold Nseg 1 24 29 3 1
βchg 10.00 11.30 13.95 9.73 12.97
βdis 10.00 10.49 13.93 7.07 9.26
Π 0.75 0.80 0.52 0.96 0.65
T1 (°C) 37.0 241.9 189.2 77.3 36.3
T2 (°C) 505.9 1085.1 1053.7 597.7 589.6
T3 (°C) 37.0 42.7 43.3 36.1 37.6
T4 (°C) -145.1 -148.7 -158.2 -144.1 -157.0
m˙ (kg/s) 13.7 5.9 5.8 11.3 10.7
R 2.51 4.30 4.19 2.82 2.78
W netdis (MWh) 8.64 10.80 7.71 11.12 8.84
∆tchg (h) 6.01 6.69 4.35 7.64 5.22
W˙ave (MW) 1.44 1.61 1.77 1.45 1.69
Ccap (£× 1000) 1382 891 969 1181 1300
ρE (kWh m−3) 43.78 130.47 91.15 67.70 54.11
ρP (MW m−3) 2.94 7.24 7.77 3.56 4.16
CˆP (£/ kW) 960 553 547 814 769
CˆE (£/ kWh) 160 83 126 106 147
χ (%) 67.48 76.63 84.47 69.80 80.88
Table 5.14: Design variables and results of PTES optimal points P7–P10 (see figure 5.9). Objective
functions (efficiency and capital cost per work output) are shown in the final two rows.
Optimisation of the Saipem design
This optimisation converges on solutions which maximise the temperature T2 to 1100°C.
These designs have large values of T1 (∼200°C rather than 40°C) and larger pressure ra-
tios (11–14 rather than 10) which is somewhat different to the original Saipem design of
Ref. [90]. These differences increase the maximum efficiency from 79.8% to 84.5% as
well as reducing the capital cost. For instance, higher temperatures lead to smaller thermal
stores, which reduces the cost despite the increase in pressure, as can be seen on the cost
distributions in figure 5.8. The cost reductions are significant: the most efficient Saipem
design (P8) is roughly 30% cheaper than the most efficient segmented design (P2). At a
223
fixed efficiency of 80% the Saipem scheme is almost half the cost of a segmented system.
However, this may be a result of the simplified economic models which are independent
of temperature. The high temperatures are in fact likely to increase the component costs
significantly.
The benefit of increasing T2 as much as possible is clear but further work is required to de-
termine the maximum operating temperatures for the storage media and other components,
as well as the economic implications.
The distribution of availability losses (figure 5.10) reveals more information about how
these designs achieve high efficiencies. The largest gains occur in the reciprocating de-
vices. The analysis of PTES systems in section 1.5 suggested that the efficiency could be
increased by maximising the work ratio R = T1/T4 = T2/T3 – i.e. maximising T1 and T2
and minimising T3 and T4. The optimal temperatures for the Saipem design achieve this:
the work ratio of P8 is 4.2 compared to 2.8 for P2. As a result, each reciprocating device
availability loss is around 4% for P8 compared to 6% for P2.
These optimal operational temperatures increase packed bed losses. Thermal losses in the
hot store are slightly larger in P8 than P2 as a result of the increased temperature difference.
The high temperatures also lead to large conductive and leakage losses, particularly during
the storage phase. On the other hand, pressure losses decrease slightly due to reduced mass
flow rates.
The benefit of avoiding heat addition
The optimal designs of normal PTES configurations have low levels of heat addition. For
instance, heat addition losses in P2 are 0.28% suggesting there is limited scope to increase
the efficiency by avoiding heat addition. However, the No Heat Addition Configuration
(NHAC) achieves a higher maximum efficiency (80.9%) than the normal design (79.8%),
and has comparable performance along the entire Pareto front, as shown in figure 5.9. The
expression for CˆE does not include the cost of the electric heater, meaning that the NHAC
is slightly more attractive than the Pareto fronts would suggest. On the other hand, NHACs
require a variable discharging pressure ratio e.g. 5.8 < βdis < 9.9 for P10 which may
compromise off-design performance of the reciprocating devices. However, this may be
224
 0
 1
 2
 3
 4
 5
Hot cyl.
Cold cyl.
Hot st.
Cold st.
HX 1.
HX 3.
Heater
A
va
ila
bi
lit
y 
lo
ss
es
 (%
)
P7
ζt ζp ζL ζc ζs
(a)
 0
 1
 2
 3
 4
 5
Hot cyl.
Cold cyl.
Hot st.
Cold st.
HX 1.
HX 3.
Heater
A
va
ila
bi
lit
y 
lo
ss
es
 (%
)
P8
ζt ζp ζL ζc ζs
(b)
Figure 5.10: Distribution of availability losses in optimal PTES designs for points P7 and P8 which
are illustrated on figure 5.9. This optimisation was carried out on a PTES system with the ‘SAIPEM’
configuration, in that temperature T2 was constrained to 1100°C rather than 600°C. (a) P7 (b) P8
outweighed by better power variation.
Availability losses in the cold store of design P9 are large compared to other optimal de-
signs. Section 4.5 described how irregularly shaped thermal fronts were formed in the cold
reservoir of the NHAC, as illustrated in figure 4.27. When the cold store is not segmented,
the irregular section of the front is contained in the top of the store - i.e. at the ‘warm’ end.
However, point P9 has three segments, and as segments at the ‘warm’ end are deactivated
during discharge irregular front shapes penetrate further towards the ‘cold’ end and take up
a larger proportion of the reservoir. Consequently, thermal and conductive losses are sig-
nificantly higher than comparative designs such as P1. The cold store in the most efficient
NHAC design P10 has only one segment and avoids these high losses.
These optimisation results indicate that the NHAC could achieve high efficiencies without
the expense of segmenting the cold packed bed or installing an electric heater. Further
work is necessary to quantify the magnitude of this saving and to investigate the off-design
behaviour of the reciprocating devices.
225
 0
 2
 4
 6
 8
Hot cyl.
Cold cyl.
Hot st.
Cold st.
HX 1.
HX 3.
Heater
A
va
ila
bi
lit
y 
lo
ss
es
 (%
)
P9
ζt ζp ζL ζc ζs
(a)
 0
 2
 4
 6
 8
Hot cyl.
Cold cyl.
Hot st.
Cold st.
HX 1.
HX 3.
Heater
A
va
ila
bi
lit
y 
lo
ss
es
 (%
)
P10
ζt ζp ζL ζc ζs
(b)
Figure 5.11: Distribution of availability losses in optimal PTES designs for points P9 and P10
which are illustrated on figure 5.9. This optimisation was carried out on a PTES system where heat
addition was avoided as described in section 4.5. (a) P9 (b) P10
5.6 Summary
In this chapter, optimisation studies were carried out on the thermal energy storage sys-
tems that have been described throughout this thesis: packed-bed thermal reservoirs and
Pumped Thermal Energy Storage. The optimisations considered technical and economic
metrics separately. This chapter began by describing a number of economic metrics and
their application to energy storage systems. A simple model of capital cost was developed
for packed beds and PTES systems and was found to be quite uncertain. As a result, it
is most appropriate to use the capital cost to compare different designs. This provides an
insight into which designs may be the most economically feasible, but does not provide a
reliable indication of how feasible the system is.
Multi-objective optimisation is a useful tool for identifying promising designs and it allows
the full trade-off between optimal designs to be examined. However, due to uncertain-
ties in the thermodynamic and economic modelling quantitative results should be viewed
226
cautiously.
Multi-objective optimisation was first applied to hot and cold packed-bed thermal reser-
voirs. The observed behaviour was consistent with the parametric studies of chapter 3.
The cycle period was the main factor that controlled the trade-off between efficiency and
energy density. Segmented thermal stores were found to outperform unsegmented stores,
particularly in cold packed beds. Radial-flow packed beds had a similar performance to
axial-flow stores but were typically more expensive due to larger volume requirements.
PTES systems were optimised in section 5.5 by maximising efficiency and minimising the
capital cost per work output. Again, the cycle frequency was found to be the main param-
eter that controlled the trade off between efficiency and cost. By segmenting the thermal
stores, the round-trip efficiency could be increased by around 1%. PTES performance was
very sensitive to the polytropic efficiency of the reciprocating devices: reducing η by 8%
led to a 23% drop in round-trip efficiency. Further research on reciprocating devices is re-
quired in order to analyse and maximise their performance. If these devices can be designed
to achieve high polytropic efficiencies, then the optimisation suggested that thermodynamic
round-trip efficiencies of around 80% may be achievable. Including mechanical and elec-
trical efficiencies of 80% [84], suggests that PTES systems could achieve efficiencies of up
to 64%.
Two variants on the standard PTES design were optimised. A Saipem-like scheme where
the maximum hot store charging temperature was increased to 1100°C reached round-trip
efficiencies of up to 85%. The optimisation results therefore suggest that there is a signifi-
cant benefit to this type of design. However, further work is required to establish the impact
of high temperatures on the components, as well as including temperature dependence in
the economic models. An alternative configuration which avoided heat addition was also
optimised and achieved efficiencies of up to 81%. This configuration has the benefit of not
requiring an electric heater to be installed, although it does require flexible operation of the
reciprocating devices. Further research is require to determine how off-design performance
of the compressors and expanders may affect the PTES performance.
227
Chapter 6
Conclusions and future work
228
Energy storage solutions
Chapter 1 introduced energy storage technologies and described how they could be used to
improve the electrical grid and the integration of greater proportions of renewables. This
project set out to investigate a novel energy storage system known as Pumped Thermal
Energy Storage (PTES). The ultimate objective of the project was to develop a detailed
thermo-economic model of the PTES system, and to use optimisation techniques to deter-
mine the best designs.
Modelling of thermal stores
PTES systems are comprised of a number of components, the most important of which are
the thermal stores and the compression-expansion system. Chapter 2 concentrated on the
thermodynamic modelling of the packed-bed thermal stores. Heat transfer processes were
described in detail, and a one-dimensional system of energy equations was developed. An
efficient numerical scheme was described, and it was verified against the analytical solution
of the equations and against a less efficient numerical scheme.
The packed-bed thermodynamic model captured the most influential heat transfer processes
and provided the necessary starting point to investigate their behaviour. However, future
work could relax some of the assumptions. For instance, the stores could be modelled
using two-dimensional energy equations which would allow the impact of radial variations
in temperature and velocity to be studied. In particular, the packing density of particles
decreases near the walls, creating by-pass flows due to the reduced frictional resistance. It
would be interesting to study the influence of this on the behaviour of the store, particularly
with regard to heat leakage losses.
Improvements to thermal store modelling by CFD and experiments
More detailed studies of packed beds could be performed using Computational Fluid Dy-
namics (CFD) analysis. CFD has been used to study packed beds, such as in [215, 216]
which presented analysis of the flow distributions and heat transfer coefficient. Extending
this work to consider second-law factors would provide a more detailed understanding of
229
the loss coefficients. The model described in chapter 2 assumed that heat transfer charac-
teristics are represented by an average equivalent diameter. It may be possible to use CFD
to investigate particles that are non-spherical, have a range of shapes and sizes and are not
uniformly distributed.
Experimental studies of packed beds would also be a beneficial area for further research.
Experimental studies could provide data to validate the existing numerical models (or CFD
schemes). In particular, values of the effective conductivity and heat transfer coefficient are
quite uncertain, and experiments could provide correlations or estimates that are specific to
the applications of interest. Furthermore, experiments could be used to investigate thermal
store behaviour from a more practical perspective.
Investigations have concentrated on the thermodynamics of the problem. However, further
work is required to establish the mechanical properties of storage material. For instance,
it is necessary to understand the conditions under which particles may be crushed, fuse
together, or otherwise be damaged.
Improvements to packed bed understanding and design
The thermodynamic model developed in chapter 2 provided a way of rapidly studying a
wide range of packed bed behaviour. A detailed account was provided in chapter 3 which
illustrated the impact of a number of design variables on the performance of the store.
Studying the packed beds from a second-law perspective with the use of loss coefficients
revealed areas where the thermal store behaviour could be improved. These investigations
resulted in a set of design guidelines for packed beds which include:
• Hot stores should have higher aspect ratios and lower particle diameters than cold
stores due to lower pressure losses.
• Larger volumes reduce heat leakage and lead to lower optimal aspect ratios.
• The charging period is the main factor that controls the trade off between efficiency
and energy density, with low utilisations providing the highest efficiency.
• The efficiency can be increased by increasing the charging temperature of the hot
store and decreasing that of the cold store.
230
• The energy density can be maximised by increasing the difference between charging
and discharging temperature, although this reduces the efficiency.
• Using a packing material with a low fractional change in specific heat capacity re-
duces the thermal losses.
As a result of these investigations, two innovative design solutions were investigated in
more detail. In section 3.3, segmentation of the packed beds was introduced. It was found
that segmentation reduced pressure losses. As a result, smaller particles could be used and
led to the formation of steeper thermal fronts which increases the efficiency and the energy
density. Further work on layered stores should concentrate on the heat transfer that occurs
between successive segments. In addition, it would be beneficial to understand in more
detail the process by which the segments are switched on or off and the impact of gas trav-
elling where it is not intended to (for instance, through inactive segments). Furthermore,
understanding the space requirements of the control system would improve estimates of the
energy density of segmented stores.
The second innovative design feature that was investigated was radial-flow packed beds.
The thermodynamic model was designed to be comparable to the axial-flow packed bed
model. Comparison of dimensionally similar radial-flow and axial-flow packed beds in-
dicated that radial-flow packed beds typically had lower pressure losses. However, these
stores also had steeper thermal fronts which led to larger thermal and conductive losses.
Further work could investigate whether decreasing the particle size along the radius of the
bed would improve performance, as suggested by [158].
Modelling of reciprocating devices
Having established the fundamental characteristics of packed-bed thermal reservoirs, a
thermodynamic model of the PTES system was developed in chapter 4. Simplified models
of the reciprocating compressors and expanders were first developed. Performance was
characterised in terms of average loss factors: pressure loss factors, heat leakage factors,
and polytropic efficiencies, and the likely magnitude of these factors was investigated. Fur-
ther work is required to model reciprocating devices in comparable detail to the packed
beds. Currently, experimental work is being carried out at Imperial College, London, and
231
CFD analysis is being undertaken at Cambridge University. This work is focussing on gas
springs, which are similar to reciprocating compressors except that there is no through flow
of gas. The CFD analysis at Cambridge has modelled gas springs with a baffle near top
dead centre in order to represent the valves and the associated pressure loss. This work
provides the necessary background for developing models of reciprocating devices.
Understanding PTES behaviour
In chapter 4 the influence of a number of design variables on PTES behaviour was in-
vestigated using parametric studies. This provided an understanding of the importance of
certain variables, and how they may affect the optimum design. These investigations add
to the packed bed design guidelines. In particular, the cycle period was found to control
the trade-off between efficiency and energy density. The polytropic efficiency also had a
significant impact on the round-trip efficiency as a result of there being four compression or
expansion processes. The round-trip efficiency can be improved by increasing the charging
pressure ratio and the temperature T1 and by reducing temperature T3. The discharging
pressure ratio controlled the distribution of heat rejection between the two heat exchangers.
Avoiding the electric heater
The nominal design required a heat input due to the behaviour of thermal fronts in the
packed beds, which is costly and inefficient. Several configurations that avoided the heater
were investigated in section 4.5. An effective method involved varying the discharging
pressure ratio, and bypassing HX1 during discharge. As a result, during charge gas always
leaves the cold store at a temperature greater than the design temperature T1, so that heat is
rejected rather than added. The potential success of this scheme depends on the off-design
performance of the reciprocating devices and emphasises the need for further research in
this area.
Other methods for avoiding heat addition were discussed. For instance, recuperation or
the use of liquid stores also provide promising alternatives. Liquid store PTES is currently
being investigated at Cambridge University, and results have so far indicated the need for
high heat exchanger effectiveness, and the importance of the temperature limits of the liquid
232
storage media. Another avenue for further research is to consider the integration of PTES
into other industrial systems. Waste heat from industrial applications could be used as a
heat input to the PTES system. Furthermore, the heat that is rejected by PTES could be
used in waste heat applications such as domestic heating. Such arrangements may place
additional constraints or requirements upon the system, and future work could investigate
the impact and potential benefits.
Irregular load cycles
Once a PTES system is integrated into the electrical grid it may not necessarily charge and
discharge with regular periods, as has been assumed in this report. Chapter 4 emphasised
the importance of the cycle period on the round-trip efficiency, and the load cycle is likely
to be equally influential. Furthermore, PTES performance depends on the historical opera-
tion: previous load cycles will affect the thermal front shape and thereby impact on future
performance.
Load cycles may be affected by a number of factors such as instantaneous electricity de-
mand and generation, the position of the storage on the grid, and the current state of charge
of the storage (and of other storage systems). Network analysis tools have been developed
by other research groups in order to analyse the behaviour of storage on the grid. These
models typically assume very simple characteristics and efficiency and ‘self-discharge’ typ-
ically have fixed values. This project has shown that PTES efficiency depends on many
factors, such as the state of charge, the position and shape of the thermal fronts, the power
input or output to the system, and historical operation.
There are several options for future research on the impact on load cycles. For instance,
research could focus on the influence of irregular load cycles on PTES behaviour. The
existing model could be executed using a large number of typical load cycles. This would
provide information about how a given load input may affect future performance. Alter-
natively, the storage characteristics in network analysis models could be improved by inte-
grating the PTES model. This would suggest how PTES would behave in the electrical grid
and how it may best be operated. Furthermore, a range of scenarios could be investigated,
with different types of generation, or different objectives.
233
Such investigations have been initiated in a collaboration between Durham University and
Cambridge University. It was found that integrating the PTES model and network analysis
model was not straightforward. For instance, the network analysis tool solved a set of linear
equations in a dedicated software package, whereas the PTES model is coded in Fortran
and solves a set of differential equations. One option that is being investigated is to reduce
the complexity of the PTES model by representing its behaviour with regression equations
which could be generated from the existing model and should capture the main behaviour.
However, whether these equations are able to represent the complexity of the thermal fronts
and the dependency on historical behaviour remains to be seen.
Application of thermo-economic optimisation techniques
Having developed a model of the PTES system and investigated its behaviour, chapter 5
undertook thermo-economic optimisation of a number of systems that had been introduced
throughout this thesis. Since the ‘best’ design may not be the one with the best techni-
cal characteristics, it is important to consider other objectives such as the cost. Thermo-
economic optimisation was therefore an suitable method to study a range of optimal de-
signs.
Chapter 5 described several economic metrics and their application to energy storage tech-
nologies. It was recognised that economic metrics could be uncertain and difficult to quan-
tify. Of the available economic metrics capital cost was the most straightforward and least
uncertain to model and a simplified economic model of the PTES capital cost was de-
veloped. The objective was to keep the model relatively simple whilst capturing the key
parameters that would impact the economic cost. Despite this, the absolute values of PTES
capital cost are likely to be inaccurate and it was most appropriate to use the capital cost as
a metric to compare different PTES designs.
Optimisation of packed-bed thermal reservoirs
Multi-objective optimisation was first applied to hot and cold packed-bed thermal reser-
voirs. A thermo-economic optimisation of the efficiency and energy density provided an
insight into optimal packed bed designs. The observed behaviour was consistent with the
234
parametric studies of chapter 3. Segmented thermal stores were found to outperform un-
segmented stores, particularly in cold packed beds. Radial-flow packed beds had a similar
performance to axial-flow stores but were typically more expensive due to larger volume
requirements. Further research into radial-flow stores should investigate the volume re-
quirements of the inner and outer plena, as these empty spaces are the main factor that
increases the cost.
Optimisation of PTES
PTES systems were optimised in section 5.5 by maximising efficiency and minimising the
capital cost per work output. Cycle frequency was found to be the main parameter that
controlled the trade-off between efficiency and cost. By segmenting the thermal stores,
the round-trip efficiency could be increased by around 1%. PTES performance was very
sensitive to the polytropic efficiency of the reciprocating devices: reducing the polytropic
efficiency by 8% led to a 23% drop in round-trip efficiency. If the reciprocating devices
can be designed to achieve high polytropic efficiencies, then the optimisation suggested
that it may be possible to achieve thermodynamic round-trip efficiencies of around 80%.
Including mechanical and electrical losses, which may have an efficiency of 80% [84],
suggests that PTES systems could achieve efficiencies of up to 64%. However, if a less op-
timistic polytropic efficiency of 90% was used for the compressor and expander, PTES had
a maximum thermodyanmic round-trip efficiency of 56% - corresponding to an electricity-
to-electricity round-trip efficiency of 45%.
Two variants on the standard PTES design were optimised. A Saipem-like scheme where
the maximum hot store charging temperature was increased to 1100°C reached round-trip
efficiencies of up to 85%. However, further work should determine the maximum operating
temperatures of the components and how this may affect the cost. An alternative config-
uration which avoided heat addition was also optimised and achieved efficiencies of up to
81%. This configuration has the benefit of not requiring an electric heater to be installed,
although it does require flexible operation of the reciprocating devices.
235
Improving economic models
Improved estimates of capital cost could be made by collaboration with an energy economist
or commercial partner. In addition, more complex economic models that incorporate a
range of revenue streams could be developed. This type of investigation could feasibly
occur as part of a network analysis model. However, comprehensive sensitivity analysis is
required in order to quantify the uncertainty of the results.
Improving thermo-economic optimisation techniques
Thermo-economic optimisation has shown itself to be a useful tool to compare different
systems, whether they are packed beds or the PTES scheme. As a result, further research
could compare different energy storage types using this methodology. It would be im-
portant to develop comparable technical and economic models in order to provide useful
information.
One limitation of the optimisation algorithm was that it was time consuming to obtain
results. For instance, the optimum designs typically had small particle diameters which re-
quired the model to take very small grid- and time-steps. The model was executed several
hundred or even thousand times before the results converged. As a result, it may be bene-
ficial to investigate alternative optimisation routines or approaches. A promising option is
known as ‘multi-fidelity’ optimisation. In this approach, a number of models of the system
exist, with varying degrees of complexity. Undertaking the optimisation on the simplest
model would quickly provide information about regions where the optimal solutions may
exist. The search space is then constrained to this region and a more complex (higher fi-
delity) model is optimised. For this approach to be successful, the low fidelity models need
to be able to capture the main characteristics of the problem. Low fidelity models could
be obtained either from simplified analysis such as that in section 1.5 or by developing
regression equations.
236
Main conclusions:
1. Energy storage (along with interconnection and demand-side response) will have an
integral role in future energy systems. Storage provides value in a number of ways,
such as through arbitrage, easing grid congestion or upgrade deferrals. However,
current regulations and uncertainty about future electricity markets has prevented
investment.
2. Thermodynamic models of packed-bed thermal stores have been developed and have
provided a set of design guidelines, which are summarised on page 230.
3. Optimisation studies suggest that packed-bed thermal reservoirs can effectively store
thermal energy. Hot stores can achieve round-trip efficiencies of 99.6% and energy
densities of 190 kWh m−3. Cold stores can achieve round-trip efficiencies of 96.5%
and energy densities of 38 kWh m−3. However, there is some uncertainty to these
values and experimental studies are required.
4. A number of variants of a storage scheme known as Pumped Thermal Energy Stor-
age (PTES) exist. The behaviour of a variant based on the Joule cycle which employs
packed-bed thermal reservoirs has been investigated and design guidelines devel-
oped, as summarised on page 232.
5. Under the assumptions made thermo-economic optimisation suggested that PTES
could achieve thermodynamic round-trip efficiencies of up to 79.8%. Once mechan-
ical and electrical inefficiencies are included the efficiency is 64%.
6. The PTES system is very sensitive to the polytropic efficiency of the compressors
and expanders. A 1% decrease in polytropic efficiency reduces the thermodynamic
round-trip efficiency by 2%.
7. Several innovative design features were investigated and found that:
(a) Segmentation of the packed beds could improve the performance of the thermal
stores by reducing pressure losses and enabling smaller particles to be used.
When included in PTES systems, segmentation increased the round-trip effi-
ciency by 1% at a fixed cost, or could reduce costs by 10% at fixed efficiency.
237
(b) Radial-flow packed beds were effective energy stores and are comparable to
axial flow stores. The mass flow rate decreases with radius, leading to lower
pressure losses. However, radial flow stores are expensive due to large volume
requirements.
(c) Heat addition to PTES systems can be avoided by controlling the discharging
pressure ratio and the thermal front in the cold store. This removes the need
for an electric heater, but requires flexible operation of the compressors and
expanders.
Promising areas for future research:
1. Detailed modelling (including CFD) and experimental investigations on the thermal
stores
2. Modelling of the reciprocating devices, particularly off-design behaviour
3. Investigations into the impact of realistic load cycles on PTES behaviour and its
integration into electrical networks
4. More reliable modelling and sensitivity analysis of PTES economics.
238
Nomenclature
239
Abbreviations
CAES Compressed Air Energy Storage
CE Reversible compressor/expander (hot cylinders)
EC Reversible expander/compressor (cold cylinders)
HS, CS Hot/cold store
HX Heat exchanger
MOGA Multi-objective Genetic Algorithm
PCM Phase Change Materials
PHES Pumped Hydroelectric Energy Storage
PTES Pumped Thermal Energy Storage
TES Thermal Energy Storage
Greek symbols
α Packed bed diffusivity, see equation 2.21 (m2 / s)
β Compression or expansion pressure ratio
β Non-dimensional availability
β Packed bed heat leakage time constant (s-1)
Γ Dimensionless packed bed charge period, tc/τ
γ Ratio of specific heat capacities
∆ A difference
δ An infinitesimal difference
 Heat exchanger effectiveness
ε Clearance (dead) volume ratio of reciprocating devices, see equation 4.19
ε Emissivity
ε Packed bed void fraction
ζ Loss coefficient, see section 2.5 (%)
η Non-dimensional time in the Schumann equations, see equation 2.27
η Polytropic efficiency of reciprocating devices
ηs Isentropic efficiency of reciprocating devices
ηv Reciprocating device volumetric efficiency
θ Dimensionless temperature
Λ Dimensionless packed bed length L/`
λ Thermal front length (m)
240
ν Kinematic viscosity (m2 / s)
νc,d Compression or expansion heat leakage factor, see section 4.2
ξ Non-dimensional length in the Schumann equations, see equation 2.27
ξ Specific exergy (J / kg)
φc,d Compression or expansion polytropic exponent, see section 4.2
Π Dimensionless cycle period (or utilisation) Π = tc/tN = Γ/Λ
ρ Density (kg / m3)
ρE Energy density (kWh / m3)
ρP Power density (kW / m3)
σ Stefan-Boltzmann constant (W / m2·K 4)
τ Compression or expansion temperature ratio
τ Packed bed time scale, see equation 2.20 (s)
τL Heat leakage time constant, see section 2.5.4 (s)
χ Round-trip efficiency (%)
ω Reciprocating device angular velocity (rad / s)
Roman symbols
A Packed bed area (m2)
B Availability (J)
b Specific availability (J / kg)
Bi Biot number
C Unsteady gas terms, see equation 2.23 (K / m)
C Capital cost (£)
c Specific heat capacity (J / kg·K)
CˆE Capital cost per work output (£/kWh)
Cf Friction coefficient, see equation 2.54
CˆP Capital cost per power output (£/kW)
D Diameter of the packed bed (m)
dp Particle diameter (m)
e Specific internal energy (J / kg)
∆Emax Maximum energy capacity of a packed bed or PTES system (MW h)
F View factor
fp Pressure loss factor, see section 4.2
241
G Mass flow rate per unit area (kg / m2·s)
g Acceleration due to gravity (m / s2)
Gr Grashoff number
h Heat transfer coefficient (W / m2·K)
h Specific enthalpy (J / kg)
k Conductivity (W / m·K)
k Cost factors, see section 5.2.2
keff Effective conductivity of the packed bed (W / m·K)
` Packed bed length scale, see equation 2.19 (m)
L Length of the packed bed (m)
m˙ Mass flow rate (kg / s)
M Mass (kg)
Ncyl Number of cylinders in the reciprocating devices
Nseg Number of segments in a packed bed
Nu Nusselt number
p Pressure (bar)
Pr Prandtl number
Q Heat transfer (J)
R Specific gas constant (J / kg·K)
R Work ratio = T1/T4 = T2/T3
r Radius of the packed bed (m)
Re Reynolds number
ri,o Inner or outer radius in radial-flow stores (m)
rv Reciprocating device volumetric compression ratio
s Specific entropy (J/ kg·K)
S˙irr Rate of irreversible entropy generation (W / K)
St Stanton number St = h/Gcp
Sv Particle surface-area-to-volume ratio, Sv = 6/dp, (m-1)
T Temperature (K)
t Thickness (m)
t Time (s)
tchg Charging duration (s)
242
tN Nominal charging time of the packed bed, see equation 2.5 (s)
U Overall heat transfer coefficient (W / m2·K)
u Gas velocity (m /s)
ui Interstitial gas velocity = us/ε (m /s)
us Superficial gas velocity = m˙/ρgA (m /s)
V Volume (m3)
Vf Velocity of the thermal front, see equation 2.3 (m / s)
Vs Reciprocating device swept volume
W Work (J)
W˙max Maximum power capacity of a PTES system (MW)
Subscripts
0 Ambient conditions
chg, dis Charging, discharging conditions
c Availability losses due to conduction
c, e Compression, expansion
c, h Cold, hot reservoir
L Availability losses due to heat leakage
p Availability losses due to pressure drops
s, g Solid, gas
t Availability losses due to convective heat transfer
x Availability losses or temperatures due to exit flow from a packed bed
1, 2 Charging, discharging conditions
243
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Appendix A
Estimation of thermal front lengths
A1
A.1 Front lengths in axial-flow thermal stores
In section 2.5.1 thermal losses ζt in the packed-bed thermal reservoirs were discussed. It
was noted that the specific heat capacity of solid cs varies with temperature. The speed
of the thermal front depends inversely on cs, and consequently the shape of the thermal
front changes. Numerical results demonstrated that variable cs leads to slightly increased
thermal losses, and claimed that this was the result of the thermal front length changing. A
simple model is now developed to demonstrate this behaviour.
The thermal loss was shown in section 2.5.1 to be approximately
ζt ≈ T0
βtchg
(
∆T 2
T1T2
)
`
∫ tchg
0
1
λ
dt (A.1)
where λ is the front length and is a function of time t. In order for changing front lengths
to cause larger losses than the average front length λ¯ then it should be the case that∫ tchg
0
1
λ(t)
dt >
tchg
λ¯
(A.2)
If λ(t) is a function that varies linearly with time some thought leads to the conclusion that
this expression is true. This may be demonstrated for the case where variable cs causes
the front lengths to fluctuate. Expressions are developed for the left and right hand side of
equation A.2 and these terms are referred to as ‘wave-speed coefficients’.
Figure A.1 illustrates idealised thermal fronts at the beginning and end of a charging phase
during cyclic operation of a hot store. The thermal fronts are assumed to have a linear
shape. The hot store is charged at temperature T1 and discharged at temperature T2. The
cycle period is given by Π = tc/tN where tc is the charging duration, and tN is the nominal
time for a wave at temperature T1 to travel the length of the store, where
tN =
ρscs1AL(1− ε)
m˙cp
(A.3)
(c.f equation 2.5). cs1 is the solid heat capacity at temperature T1. At the beginning of
charge, the front length is λ1 = L− Vf2tc where Vf2 is the speed of the thermal front at T2
A2
T (1)
T2
1
(2)
λ1 x2
L
λ 2x1
T
x
Figure A.1: Idealised thermal fronts at the end of charge (1 – green) and the start of charge (2 –
purple). cs is assumed to vary with temperature leading to the front changing shape. Charge and
discharge periods are estimated by assuming the period ends when the leading edge of the front
reaches the exit.
– i.e. Vf2tc is the distance point (2) on figure A.1 travels during charge. Rearranging gives
λ1 = L
(
1− Πcs1
cs2
)
(A.4)
A similar expression can be found for the front length at the end of charge, λ2. The average
front length is then given by
λ¯ = 1/2(λ1 + λ2) = L
[
1− Π
2
(
cs1 + cs2
cs2
)]
(A.5)
The wave-speed coefficient for constant front length (L.H.S of equation A.2) is then
tc
λ¯
=
Π
Vf
[
1− Π
2
(
cs1 + cs2
cs2
)]−1
(A.6)
An expression now needs to be developed for the variable front length λ(t). The change in
A3
front length δλ is found by noting that in time δt point (1) will have moved a distance
δx1 =
m˙cp
(1− ε)ρAcs1 δt (A.7)
A similar expression is found for δx2. The change in front length is then δλ = δx2 − δx1
δλ =
m˙cp
(1− ε)ρsA
(
1
cs2
− 1
cs1
)
δt (A.8)
The front length therefore varies linearly and can be written as
λ = λ1 +
dλ
dt
t (A.9)
λ =
(
1− Πcs1
cs2
)
L+
t
tN
(
cs1 − cs2
cs2
)
L (A.10)
An expression for the wave-speed coefficient for fluctuating front lengths is then given by∫ tchg
0
1
λ(t)
dt =
1
Vf
(
cs2
cs1 − cs2
)
ln
[
1− Π
(
cs1 − cs2
Πcs1 − cs2
)]
(A.11)
The two wave-speed coefficients are plotted on figure A.2 for a range of charging periods
Π. It is clear that the fluctuating wave-speed coefficient is always slightly larger than when
the average thermal front length is used. As a result, this analysis demonstrates the fact that
variations in cs with temperature lead to slightly larger thermal losses.
A.2 Front lengths in radial-flow thermal stores
Front lengths in radial-flow thermal stores can be estimated in a similar way to those in
axial-flow stores. Figure A.3 illustrates linear thermal fronts in a radial-flow store. As
described in section 3.4 the mass flow rate per unit area decreases along the radius. Con-
sequently the thermal wave speed is inversely proportional to the radius (equation 3.18),
thereby leading the thermal front to steepen up during charge.
The nominal charging time of an axial-flow store with outer radius ro and inner radius ri is
A4
 0
 2
 4
 6
 8
 10
 0  0.1  0.2  0.3  0.4  0.5  0.6
W
av
e-
sp
ee
d
 c
o
ef
fi
ci
en
t 
x
 1
 /
 V
f 
(s
/m
)
Π = tc / tN
Average λ
Fluctuating λ
Figure A.2: Wave speed coefficients calculated with either the average thermal front length or a
fluctuating thermal front length. Compare with equations A.2, A.6 and A.11
rori
r1
r2
1λ x2
λ 2x1
T
T2
1
T
r
Figure A.3: Idealised thermal fronts in a radial-flow thermal store at the end of charge (green) and
the start of charge (purple). cs is assumed to be constant but the varying mass flow rate per unit area
changes the front shape. Charge and discharge periods are estimated by assuming the period ends
when the leading edge of the front reaches the exit.
A5
given by
tN =
ρsc¯s(1− ε)
2Giricp
(
r2o − r2i
)
(A.12)
where it is assumed that the specific heat capacity is a constant. The thermal fronts are
assumed to have a linear shape and to just reach the ends of the reservoir at the end of
charge and discharge. By inspection of figure A.3 the charging time tc may be written as
Π =
tc
tN
=
r21 − r2i
r2o − r2i
=
r2o − r22
r2o − r2i
(A.13)
Furthermore the front lengths λ1 and λ2 are given by
λ1 = ro − ri − x2 = r2 − ri (A.14)
λ2 = ro − ri − x2 = ro − r1 (A.15)
By substituting the utilisation Π into these expressions, normalising by the flow length
ro − ri and writing ρ = ri/ro the fractional front lengths are given by
λ1
ro − ri =
ρ
1− ρ
{
1
ρ
[
1− Π (1− ρ2)]1/2 − 1} (A.16)
λ2
ro − ri =
1
1− ρ
{
1− ρ
[
1 + Π
1− ρ2
ρ2
]1/2}
(A.17)
Figure A.4 compares the radial-flow front lengths to the fractional front length in axial
reservoirs (which is given by λx/L = 1 − Π when cs is constant). As expected λ1 > λ2.
However, the axial front length lies between these values, and this is supported by the
numerical results in figure 3.19. The figure also illustrates the average radial front length
which is less than the axial front length for Π < 0.5. (For larger values of Π this model
becomes increasingly inaccurate since the thermal fronts protrude further from the store).
These results have particular significance for the thermal loss. An approximate expression
for the thermal loss was derived in equation 2.50 as
ζt ≈ T0
β
(
∆T 2
T1T2
) ¯`
λ
(A.18)
This expression also applies to radial-flow stores if the length scale ¯` is calculated at the
A6
 0
 0.2
 0.4
 0.6
 0.8
 1
 0  0.25  0.5  0.75  1
 F
ra
ct
io
n
al
 f
ro
n
t 
le
n
g
th
, 
λ 
/ 
L
Π = tc / tN
λx
λ1
λ2
1/2(λ1 + λ2)
Figure A.4: Fractional front lengths calculated for axial-flow stores λx and radial-flow stores λ1
and λ2, see equation A.17.
average radius r¯ = 1/2(ri + ro). The above analysis has shown that the front length may
be replaced by λ = Lf(Π), where L is the length of the flow path (ro − ri for radial-flow
stores) and f(Π) is a function of the utilisation. The thermal loss coefficient is then
ζt ≈ T0
β
(
∆T 2
T1T2
) ¯`
Lf(Π)
(A.19)
Axial-flow and radial-flow stores are approximately equivalent if they have the same di-
mensionless length Λ = L/` = (ro− ri)/¯`. For such reservoirs, the store with the smallest
value of f(Π) will generate the largest thermal loss. Therefore, these results suggest that
thermal losses are larger in radial-flow stores than in the equivalent axial-flow store, par-
ticularly at low utilisations. In addition, the fluctuation of the radial front length between
λ1 and λ2 will lead to greater losses, as described in the previous section. However, the
variable cs will counteract these fluctuations to some extent, and the effect may not be as
large as anticipated.
A7