1 Self-drainage of viscous liquids in vertical and inclined pipes Ali, A., Underwood, A., Lee, Y-R. and D.I. Wilson* Department of Chemical Engineering and Biotechnology, New Museums Site, Pembroke Street, Cambridge, CB2 3RA, UK Food & Bioproducts Processing © AA, AU, YRL & DIW March 2016 *Corresponding author D. Ian Wilson Department of Chemical Engineering and Biotechnology New Museums Site Pembroke Street Cambridge CB2 3RA, UK E-mail diw11@cam.ac.uk Tel +44 1223 334791 FAX +44 1223 334796 2 Self-drainage of viscous liquids in vertical and inclined pipes Ali, A., Underwood, A., Lee, Y-R. and D.I. Wilson* Department of Chemical Engineering and Biotechnology, New Museums Site, Pembroke Street, Cambridge, CB2 3RA, UK Abstract The rate of drainage of a viscous liquid from initially full cylindrical tubes inclined at various angles to the vertical (0, 30, 45 and 60) was studied in glass and polymethylmethacrylate (Perspex) tubes of various lengths and diameters using three food materials: honey (Newtonian) and two variants of Marmite spread (both exhibiting complex rheological behaviour, including shear-thinning and thixotropy). The behaviour was marked by an initially steady rate of drainage in which an air slug descended the tube, followed by slower drainage from an annular film remaining on the wall. Eventually the liquid stopped draining as a filament and entered a dripping regime. Drainage was insensitive to the tube material, whereas the stages of drainage were influenced by the geometry and angle of inclination. Quantitative models are presented for the rate and extent of the initial drainage stage, the rate in a second linear stage (where it existed), and the rate of drainage in the third, falling rate stage. In the fourth and final stage, characterised by drop formation, was not modelled. The initial rate can be predicted with reasonable accuracy, allowing the time to remove approximately 50% of the material in a short waiting phase to be calculated, e.g. t = 8L/R2g for a Newtonian liquid with kinematic viscosity in a vertical pipe of radius R and length L. The agreement with the other models is less exact but they capture the general trends reasonably. Keywords: cleaning, drainage, fluid mechanics, laminar, shear-thinning, thixotropy 3 1. Introduction Viscous liquids are widely used in food processing. Products such as sauces and spreads (e.g. Marmite TM , White et al., 2008) are manufactured and sold as viscous, non-Newtonian liquids. Others such as ice cream and chocolate (e.g. Taylor et al., 2009) are processed as non- Newtonian fluids but sold as solids, while other products employ viscous liquids as components in their assembly, e.g. chocolate for coating, jams and syrups for filling. Food processing operations regularly require the lines carrying these viscous liquids to be cleared, either as part of shutdown for maintenance, changeover to a different batch, or for cleaning and disinfection. This is often achieved by recirculating water as part of a cleaning-in-place cycle. The water initially pushes out a central core of material and the surrounding annulus is subsequently eroded by the shear stress imposed by the water flow (Mickaily and Middleman, 1993; Palabiyik et al., 2014; Fan et al., 2015), aided by dissolution if the material is soluble. Water flushing can be fast but causes product loss and generation of large volumes of contaminated water, which must be treated. An alternative approach is to allow the material to drain back to a reservoir under the action of gravity. This will extend the time required to clear the line and will still need to be followed by a water flush to complete the operation, but will reduce product loss and water consumption. There will be an optimal time to start flushing, which will be determined by the amount of material remaining in a line over time. This is a transient fluid flow problem which does not appear to have received much attention in the literature, and particularly for viscous, non-Newtonian liquids which are commonly encountered in the food sector. The amount of product wasted can be considerable: for example, Cragnell et al. (2014) reported that 5-10% of fermented milk products are left behind on packaging surfaces when the consumer decants the contents from a carton. We report a series of experiments where a transparent tube, initially full of viscous liquid and open at the top, is allowed to discharge from the open lower end under the action of gravity. As liquid drains, air enters from the top in the form of a long slug which descends at velocity Us, leaving an annular film behind on the tube wall (see Figure 1). Drainage behaviour was studied for a series of glass and polymethylmethacrylate (Perspex) pipes of different lengths and diameters. Three liquids were studied: a commercial honey and two varieties of Marmite TM , all of which are viscous food fluids. Whereas the former is Newtonian, 4 Marmite TM is a complex fluid, being a concentrated suspension of protein fragments from brewer’s yeast in a highly saline solution. It exhibits shear-thinning behaviour and thixotropy, with an apparent memory of recent shear history (White et al., 2008). The two varieties studied here differed in solids content and rheology. Related studies Taylor (1961) described an elegant experimental investigation of the flushing of a viscous liquid from a horizontal pipe by air. The pipe diameters were large enough (2 and 3 mm diameter) so that there were no capillary effects arising from the pressure drop across the meniscus. The slug of displacing fluid (air) left an annulus of the initial liquid in its wake. The fraction of initial liquid remaining in the pipe, which we denote m*, was found to be determined by the dimensionless group Us/, where is the dynamic viscosity of the Newtonian liquid being flushed, Us the slug velocity and  the surface tension. Cox (1962) continued this work, again with Newtonian liquids, and showed that m* initially increased with (Us/) 0.5 and approached an asymptote of 0.62 at high Us/. Taylor also reported that the flow pattern in the liquid immediately ahead of the finger changed significantly around m* = 0.5, as shown in Figure 2. He stated that at m* = 0.5 the flow velocity in the liquid at points ahead of the meniscus is identical to that at the meniscus, i.e. it is in plug flow. For m* < 0.5 there is recirculation in the liquid ahead of the slug nose. We report this result because we observe in our experiments that the initial phase of drainage under gravity is associated with m*  0.4-0.5. For m* > 0.5, liquid ahead of the meniscus flows into the film at the side of the descending slug. We present a model, derived for Newtonian liquids such as those studied by Taylor and by Cox, which predicts the effect of experimental parameters on m* and compare the model predictions with the results obtained with the Newtonian fluid (honey) and the non- Newtonian ones. Following the passage of the slug of rinsing fluid, drainage involves the gravity-driven flow of the liquid film remaining on the inner wall of the tube. Self-drainage from a plane wall has been studied at length, starting with the work of Jeffreys (1930) on the dynamics of the film remaining on a flat plate as it is pulled upwards from a bath of liquid. Jeffreys’ stated 5 motivation was the drainage of liquid from the walls of cylindrical vessels and considered cases of low curvature, where the wall could be considered as a flat plate; the current work considers cases where curvature is important. These flows underpin many coating operations and has been studied for various geometries and fluid rheologies (e.g. White and Tallmadge, 1966; de Kee et al., 1988). More recently, Sherwood has considered the draining of fluid from the walls of process vessels of various curved shapes subject to gravity (2009) and centrifugal (2013) body forces. A model based on the approach reported by Van Rossum (1958) is shown here to give a reasonable description of drainage of the annular film formed in the later stages of the experiments with vertical tubes. 2. Modelling The liquid leaving the tube is collected on a balance. Figure 3 shows an example of the mass of material remaining in the tube, calculated by difference, for a typical experiment with honey. Four stages are evident, labelled as: I Plug flow The air slug moves downwards and the tube empties at a constant rate. Videos indicated that the nose of the slug travelled at a constant velocity, leaving an annular film of liquid behind. This stage ends when the slug reaches the bottom of the tube, at time tI. The mass in the tube at this point is mI and the ratio of mI to the initial mass is denoted m*, for comparison with the Taylor (1961) results. II Second linear stage In several cases, stage I was followed by a shorter period in which the drainage rate was constant. This stage ended at time tII when the mass remaining in the tube was mII. III, IV Decreasing rate stages After tII, or tI in cases where a second linear stage was not evident, the rate of drainage decreased with time. At some point the liquid ceased to drain as a steady filament and changed to a dripping regime, labelled IV. The steps in m in Figure 3 are the result of droplet formation. Quantitative models, based on steady state flows, are presented to describe stages I to III. 6 2.1 Stage I, plug flow Consider the steady flow of liquid of density  along a tube of internal diameter R inclined at angle  to the vertical. The wall shear stress, w, matches the matches the component of the weight of the fluid in the direction of the tube axis, giving  cos 2 1 gRw  [1] where g is the acceleration due to gravity. For steady laminar flow of a Newtonian fluid with viscosity  along a tube, the wall shear stress is given by 2Re 16 2U w    [2] where Re is the Reynolds number, and U is the mean velocity of the liquid. This gives explicit results for U,   cos 8 2 gR U  , [3] and the apparent wall shear rate, app    cos 2 1 4 gR R U app  [4] Equations [2] and [4] are useful for determining the range of shear rates and/or shear stresses that need to be considered when determining the rheological behaviour of a non-Newtonian fluid. For the test in Figure 3, R ~ 0.01 m,  ~ 1415 kg m-3, g = 9.81 m s-2 and  ~ 7.1 Pa s, giving w = 70 Pa, app ~ 10 s -1 , U = 24 mm s-1 and Re = 0.005, indicating that the flows are expected to be laminar. The volumetric flow rate, QI = R 2 U, is also used as a reference flow rate:   cos 8 4 gR QI  [5] For a power law fluid which exhibits Ostwald-van de Waele behaviour, viz. nK  [6] where n is the power law index and K the consistency, the corresponding result is:   n n n n n PLI K g n n DQ 1 1 13 , 32 cos 8 1 13                       [7] 7 where D is the tube diameter. These predictions for the volumetric flow rate are compared with the measured (mass) flow rates. The finding that m*  0.5 suggests that this quantity can be predicted by building on the work by Taylor (1961) and Cox (1962). For steady flow in stage I, consider the control volume drawn round the slug nose shown by the dashed box in Figure 1. Equating volumetric flows in and out gives ISiA QUrQ  2 [8] where QA is the flow rate in an annular film with inner radius ri. The experimental results show that the ratio of the mass remaining in the tube at the end of stage I to the initial mass, m * , was around 0.5, giving ri ~ R/2. It can be shown that the local velocity, u, at radial position r in such an annulus of liquid flowing downwards under gravity is given by   xxxgRu ln1 2 22 2 1 2    [9] where x = r/R. The volumetric flow rate in the annulus, QA, is given by  iiiiIiA xxxxQxQ ln4341)( 242  [10] For the case where m * = 0.5, xi = 1/2 and QII = 0.0966 QI, or QII ~ QI/10. Combining [8] and [10] gives    ii iiii i s xxU xxxx r R UU ln434 ln434 2 242 2 2     [11] Taylor (1961) presented data relating the fraction of mass left by an air slug to the group Us/. His data were replotted in the form Us/, = f(xi), and a third order polynomial fitted to the data (see Appendix) over the range of interest (m* > 0.4), giving 29.3868.14737.18612.797)( 23  iiii s xxxxf U   [12] Substituting for U from Equation [3] yields        cos 8ln434 2 2 gR xF xx xf i ii i   [13] Or   cos 8 Eo xF i  [14] where Eo is the Eötvös number. It is notable that the viscosity does not appear in this relationship. The expression in Equation [10] is unlikely to be accurate for the non- 8 Newtonian materials, and it is of interest to compare the agreement obtained for these liquids with that for the honey, which is Newtonian. For the parameters in this study (0.0088 m  R  0.0217 m,  = 1330 or 1415 kg m-3;  values in Table 2), Eo ranges from 4 to 69 for honey and 5 to 100 for the Marmite fluids. Equation [14] was solved numerically for the range of values of 1 /8 Eo cos  arising in this work and m* was then calculated using 21* ixm  [15] The results are compared with the experimental values of m* in Figure 6. 2.2 Second linear stage, stage II Passage of the air slug leaves an annulus of liquid behind. Measurements of the thickness of the annular film, achieved by placing a draining tube at stage II promptly in a freezer, indicated that the film was quite uniform. Drainage in the second linear stage for vertical tubes was modelled as the steady flow of an annulus of liquid with outer radius R and inner radius ri, where ri is the radius corresponding to the fraction of material remaining at the end of the plug flow stage, using Equation [10]. The result for QA for a vertical tube ( = 0) can be compared with the flow rate estimated using the Nusselt film result (Nusselt, 1916) for a steady flow of liquid of thickness  down a vertical wall. The flow rate per unit width of a Nusselt film is g3/3: the flow in the annulus, QN, is then approximated as  3 3 3 2 3 2 iN rR gRg RQ       [16] and  31 3 16 i I N x Q Q  [17] Figure 4 compares QA and QN. The latter overpredicts the flow rate but the two expressions converge to the same result as xi approaches unity and the effect of curvature diminishes. At xi ~ 1/2, QA/QI = 0.097 and QN/ QI = 0.134. Drainage at a constant rate could be expected to continue until the thinning of the annulus became significant. The time for this to occur was estimated from L/ui, where L is the length of the tube and ui is the velocity of the fluid at the interface when xi = 1/2. This gives tII-tI = 9 3.26L/U. Inspection of the data where a second linear region was observed indicated that there was no consistent trend in the duration of this second stage compared to the first: the above estimate provided an upper bound for the values of tII-tI, but there was considerable variation in this value (data not reported). 2.3 Drainage with a shrinking annulus, stage III The viscous draining film model presented by Van Rossum (1958) was adapted to the annular geometry. The thickness of the annular film, , varies with axial position along the tube, z. Liquid enters a control volume drawn between z and z + dz at flow rate QA: a volume balance gives 0      t V z Q AA [18] where VA is the volume of liquid in the annulus between z and z+dz. The approximation dVA = 2Rdz is employed to give an analytical solution. The difference between this and the correct result (2R-)dz is about 15% for the widest annulus considered here. Equation [18] becomes 0 2 1       tz Q R A   [19] The result for QA, Equation [10], does not have a simple dependency on  so the function was fitted to a power law expression, QA/QI = a1( /R)  over the range of  values of interest (0 < /R < 0.707), giving 8644.2 3936.3        RQ Q I A  R2 = 0.9997 [20] For comparison, the Nusselt film result is QN/QI = 5.33(/R) 3 . Substituting [20] into [19] gives 0 2 3936.3 8644.2 8644.3       tzR Q k I     [21] Writing 2.8644 = , one solution, based on scaling and similarity, is   1 1 1 1            t z kk [22] The measured quantity is the flow rate at the tube exit, QIII. Setting QIII = QA(z = L) gives 10   1 2 1                     t L a R Lz QaQ IIII [23] Writing Equation [23] as QIII = a3t  , where = -1.536, the drainage rate in stage III is given by  ta dt dm 3 [24] Integrating from m = mII (or mI if there is no second linear stage) at time tII (or tI) gives          13 1 IIII tt a mm [25] With  = -1.536, Equation [25] predicts mII – m(t)  t’ -0.536 , where t’ is elapsed time (t’ = t-tII) It is more convenient to present the experimental results in the form (mII – m(t)) -1.836 versus t’. A result similar to Equation [25] is obtained for draining of a thin annular film of thickness  and volume 2RL. If the rate of drainage per unit width is given by the Nusselt film result, i.e. Q 2R3, the solution is of the form -2 (and thus m-2)  t’. Methods and Materials Drainage tests Perspex or borosilicate glass tubes were obtained with three different internal diameters and cut to give comparable L/D ratios, summarised in Table 1. The tubes were rinsed out with water, cleaned thoroughly with detergent solution, rinsed in hot water then dried before each test. The mass of each tube before and after filling was measured. The bottom end of the tube was stoppered and the fluid added slowly to avoid entraining air bubbles. The tube was then mounted at the desired angle to the vertical, determined using an electronic spirit level. Liquid drained into a dish located on an electronic balance connected to a datalogging PC. The response time for measuring the mass was short for steady flows. The fluids tested were not strongly viscoelastic so negative internal stresses, which could give rise to a fluid siphon effect and reduce the weight of the filament, were not expected to occur. Surface tension contributions were estimated to generate an upward force in the filament equivalent to less 11 than 0.1 g (and would decrease steadily with filament diameter): this was considered to be a small effect. Test fluids Honey The honey was a clear variety purchased from a local supermarket. Its rheology was studied using cone and plate tools on a Bohlin CVO 120 controlled stress device over the temperature range 15-25C likely to be encountered in the laboratory. The honey was Newtonian with a viscosity of approximately 8 Pa s at 21C. The temperature dependency fitted an Andrade relationship, viz.         T 8900 exp101.1 13 [26] where T is in Kelvin. The honey density was measured as 1415 10 kg m-3. Marmite Two varieties of Marmite were obtained: DExtract, an intermediate from the factory line, and Squeezy, a product with a lower apparent viscosity marketed in squeezable plastic containers. The solids content of the two materials were determined by heating in an oven at 90C to constant residual mass, giving solids fractions of 0.731 and 0.712 for DExtract and Squeezy, respectively. The density of the materials was similar, at 133010 kg m-3. The rheology of both varieties was studied on the Bohlin device using roughened parallel plates (maximum peak height 63 ±10 µm; Malvern Instruments, 2016) with a 1 mm gap. Cone and plate tools were not used owing to the high solids content. The solids were chiefly protein aggregates with sizes less than 1 m. A small number salt crystals were present, with particle sizes up to 50 m (White et al., 2008). Increasing then decreasing shear stress ramps were imposed from (i) 10 to 100 Pa; (ii) 10 to 300 Pa; and (iii) 10 to 1000 Pa, to determine the influence of thixotropy. Each step lasted 3 s: the apparent viscosity was recorded when the strain rate reached a steady value, which took less than 3 s. The samples were left to rest for approximately 5 minutes after loading. Pre- shear was not applied. 12 The results obtained at 19C are summarised in Figure 5. Both varieties show an initial increase in apparent viscosity (on the upward leg) until the shear stress reaches about 20 Pa, after which the material exhibits shear thinning. The extent of shear thinning increases with the applied shear stress. On the return (decreasing shear stress) ramp there are noticeable differences from the behaviour on the upward sweep. These differences are particularly large for samples which had been subjected to shear stresses above 100 Pa. These data confirm that both materials are thixotropic. For both materials one of the series shows a different profile for the initial ramp, even though the sample was subjected to the same loading, preshear and stress-time history. This variation illustrates the challenges in studying these complex food fluids. For the samples sheared up to 100 Pa the difference for the Squeezy material is smaller: for both materials the return leg data could be fitted to Equation [6] and the power law parameters thus generated are reported in Table 2. The DExtract exhibited less Newtonian behaviour (smaller n), with a larger consistency. These parameters were used to estimate the steady drainage rate using Equation [7]. Also plotted on Figure 5 is the largest shear stress expected to be generated in the drainage tests. At approximately 70 Pa, this lies below the range at which strong thixotropic effects were observed in the rheological tests, discussed above. Loading the sample also subjects the sample to some shear history and the influence of the loading stage was assessed by a simple draining test. In these, the tube was half-filled with sample and stoppered at both ends. It was then inverted twice, allowing the bulk of the fluid to flow to the other end each time, then left to rest. After a rest period ranging from 30 min to 24 h, a standard draining test was conducted and the initial draining rates compared. There was no significant difference between the flow rates for tubes left for 1 h or longer after filling, so a standard waiting time of 2 h was used. The surface tension and contact angles on each substrate were measured for each fluid using a Krüss DSA 100 goniometer. The results in Table 3 show that the difference in contact angles between the fluids, compared to those for water, is small. 13 Results and Discussion Figure 3 shows the mass in the tube, m, versus time for a test in which all four stages are evident. In first two stages, I and II, m decreases linearly with time and these are labelled the linear regions. The first linear stage was observed in all tests. The third stage, labelled III, is where annular drainage occurs at a falling rate. The final stage, IV, is where the liquid collects into droplets before falling. Cases where stage II was observed are summarised in Tables 4-6. Almost all tests showed two linear stages for a vertical pipe θ = 0˚, with exceptions at smaller diameters. For an inclination of 30˚, two linear stages tended to be observed for longer tubes (higher L/D) – particularly with DExtract – which is attributed to the longer time for the thinning of the annular region to reach the end of the tube (see the estimate above for tII -tI). For tests with DExtract and honey in the narrowest tubes, stage I was followed directly by stage IV (dropping), indicating that the rate of drainage from the annulus was insufficient to maintain a steady filament. The criterion for the filament to dropping transition represents an area for further work. Stage II was rarely observed at inclinations of 45 and 60, which is attributed to the absence of an annular flow pattern: the drainage flow is unlikely to be axisymmetric and the shape of the interface with changes with time (see Sherwood, 2009; Ng et al., 2001). For food processing applications, tubes are likely to be vertical or horizontal: drainage from a horizontal tube with a vapour cavity into a vertical leg represents an area for future investigation. The nature of the substrate had little influence on the observed drainage patterns. Subsequent results will show no quantitatively significant influence of substrate on drainage rates. This finding is expected, particularly for smaller angles of inclination, as dewetting (formation of dry patches) was not observed over the timescales of these tests. The following discussion focuses on drainage rates. Stage I – plug flow Figure 6 shows that the mass remaining in the tube at the end of stage I, m*, lay consistently around 0.5 ( 0.13), for all three liquids. There is noticeable scatter but there was no clear influence of θ and D on the m* values. High m* values were observed with the shortest tubes 14 when not vertical (L/D = 10, θ > 0; marked on the plot) indicating that steady flow conditions may not have been achieved in these cases. There was no noticeable effect of L/D for longer tubes. This result indicates that at least half the product remaining in the tube can be recovered by waiting for an appropriate period of time, tI. Discounting the outliers, the m* values range from 0.4-0.52, suggesting that the flow pattern at the slug nose is expected to resemble that in Figure 2(b). The data for honey are plotted against the angle of inclination in Figure 6(a) and against the dimensionless group 1 /8 Eo cos  in Figure 6(b). There is a weak decrease in m* with increasing  in Figure 6(a), with noticeable scatter. This feature is also predicted by the model: the results for D = 8.8 mm describe the overall trend in Figure 6(a) but overestimate the absolute value of m*. However, the systematic increase in m* with increasing tube diameter predicted by the model is not present in the experimental data. When the data are plotted in the form suggested by Equation [14], see Figure 6(b), the model systematically overpredicts m* for all values of 1 /8 Eo cos < 1 /2 (for L/D >10). The data distributions for Squeezy in Figure 6(c)) and DExtract in Figure 6(d) exhibit very similar patterns: if plotted together the data overlap to a large extent (see Appendix Figure A2). Shorter pipe lengths inclined to the vertical tend to give larger m* values. There is little effect of the non-Newtonian nature of the draining fluid on m*. The data suggest that a value of 0.5 could be used to estimate the amount of product recovered, and m*= 0.4 could be used to estimate tI, the time required for this to be achieved. Figure 7 shows the reliability of the model to predict tI, via the drainage rate. The plots compare the measured mass flow rate, ?̇?, measured over the stage I linear portions of the m-t profiles (see Figure 3) to the value predicted by equations [5] and [7], QI and QI,PL, respectively, using the measured rheological parameters. There is good agreement between the measured and predicted flow rates for honey and Squeezy, i.e. the Newtonian and weakly shear-thinning liquid, for all values of D, L/D and  tested. The differences that arise between measured and predicted values for Squeezy (Figure 7(b)) are likely due to the selection of rheological model and its parameters. The DExtract results in Figure 7(c) show a systematic difference between the model and measured flow rates. Equation [7] tends to underpredict the experimental values and indicates that a more detailed rheological model, such as the 15 Carreau-Yasuda model employed by White et al. (2008), should be used to estimate QI. The choice of rheological parameters, however, remains a challenge, as the thixotropy evident in Figure 5 indicates that the parameters will be determined by the recent shear history of the material, particularly the rate at which it was being pumped before flow was stopped, and the length of any delay before emptying. In the absence of a reliable a priori prediction of mass flow rate for rheologically complex materials such as DExtract, the effect of pipe inclination was tested by comparing the ratio of the mass flow rates of the vertical and inclined cases suggested by Equation [7], namely   n I I m m /1 cos )0( )(       [27] The DExtract data in Figure 7(c) are plotted in this form in Figure 8, using the n values in Table 2. The data exhibit the expected trend, with noticeable scatter. This result indicates that the dependency of the flow rate on the wall shear stress (which is proportional to gcos, Equation [1]) is not modelled reliably by Equation [7]: the trend is captured but the absolute value of QI,PL. Stage II – second linear stage The analysis of a falling annular Newtonian film, Equation [10], predicts that for vertical tubes ṁII/ṁI ~ 0.1. For vertical tubes ( = 0), the ratio was around 0.1 for all three fluids, despite the differences in rheology. Figure 9 presents the ratio of the two flow rates for most of the tests where a second linear stage was observed (see Tables 4-6): the angle is plotted as cosθ, to capture the influence of gravity. In all cases, ṁII < ṁI. The ratio decreases with increasing cosθ, (decreasing angle of inclination), which reflects the increase in ṁI with cosθ. The variation in the data increases with angle of inclination (smaller cosθ). This is likely to arise from the change in flow pattern in stage II from a concentric annulus to a stratified flow as the angle of inclination increases: the flow pattern will be determined by surface tension (and Bond number) as well as the rheology of the fluid. Stage III – falling rate regime Examples of data collected in stage III are presented in the form suggested by Equation [25], m-1.8644 versus t’, in Figure 10. For honey there is good agreement with the model trend over the first 100 s, shown by the inset on the Figure, by which point m-1.8644 is 16 approximately 60 g -1.8644 . Similarly linear behaviour is evident up to 60 g -1.8644 for the Squeezy and DExtract cases, but these fluids require longer times, of around 2000 and 8000 s, respectively. The longer times required for DExtract and Squeezy is consistent with their higher apparent viscosity as well as their shear thinning nature: their apparent viscosity is expected to increase as the film thickness (and shear stress) decreases. Stage IV – drop regime At the end of this period, at which drainage switched to drop behaviour (stage IV), the mass fraction in the tube (m/m0) ranged from 0.11 to 0.13. Tests with the more viscous liquids were often curtailed before this point was reached. Most of the experiments gave good agreement with the model to xi values around 0.9. Drop formation was observed at 0.91 < xi < 0.93 for all but the narrowest tubes. The above results suggest that the model captures the drainage behaviour. The agreement found for Squeezy and DExtract was surprising as the model assumes Newtonian behaviour, which was not observed in the rheometry testing and the estimation of the flow rate in stage I. The degree to which these fluids can be treated as pseudo-Newtonian in this thin film drainage regime represents a topic for further investigation. Further evidence suggesting that Equation [20] should be treated as a semi-empirical result is that the values of a3 (Equation [24]) obtained from fitting the experimental data did not agree with the value calculated using the properties for honey and the test geometry. The mismatch ranged from a factor of 2 to 500 across the configurations studied. This is not unexpected as there are several approximations made in the model, including the estimate of the film volume. The boundary conditions are unlikely to match those encountered in practice in moving from Stages I to III: this is only likely to be overcome by a detailed numerical model which calculates the flow (and evolution of film thickness) at every location. Application The aim of this investigation was to determine the feasibility of including a self-draining step into a cleaning protocol in order to increase the amount of product recovered and reduce subsequent contamination of the cleaning solutions. The results show that around 50% of the material is removed in stage I, with the model giving a reasonable estimate of the waiting time. More material can be recovered by waiting longer, but the rate decreases significantly after tI. The existence of a second linear stage is related to the angle of inclination and length, 17 with more vertical and longer pipes favouring this behaviour. The accuracy of the models to predict the flow rate in stage II is reasonable for vertical pipes but has not been explored further here. Likewise, a model for stage III drainage has been proposed, which describes the observed behaviour up to 90% removal. Its predictive accuracy, even for the Newtonian fluid tested, is poor. Figure 11 puts these results into perspective. After time tI around 50% of the fluid has been removed from the pipe: tI depends on its configuration and the fluid rheology. Where stage II is observed, a further 10% or so drains off after waiting until tII, which is several times tI: clearly, there is a diminishing return. This is confirmed by the tIII values, which range from 10-100 tI in achieving 80-90% removal. In a processing unit, waiting for over an hour may be acceptable but this will depend on the application and nature of the product. A priori prediction of tIII is not achievable with the models presented here. Evaluating a delay stage For a vertical pipe of length L, the time taken for 50% of the fluid to drain, t50, can be estimated from t50  L/U. From Equation [3], gR L t   250 8  [28] Consider a 10 m length of 50 mm i.d. pipe initially filled with the honey used in this work ( = 8 Pa s,  = 1415 kg m-3). Equation [28] gives t50 = 74 s, suggesting that 50% of the product could be recovered by waiting for 2 minutes, say, before starting the cleaning-in-place (CIP) system. This is short compared to standard food industry cleaning cycle times. The Marmite TM varieties would require longer periods. The prospective financial return could be estimated by comparing the cost of extra equipment required to add the step (valves, tankage, pump and time spent reprogramming the control system) against the savings incurred. The latter would include (i) The value of product recovered rather than being purged with the initial CIP rinse. (ii) The reduction in energy and chemicals required for cleaning, related to there being less product to remove from the pipe; (iii) The reduction in volume of aqueous effluent sent for waste treatment. For the fluids studied here, the volumes of water can be considerable. The honey, with a 18 high sugar content, generates waste with a high biological oxygen demand. Likewise, Marmite TM has a high salt content and the CIP waste water must be diluted or treated in order to reach discharge limits. These costs are all likely to be site specific and all depend on the frequency with which the line is cleaned. Non-Newtonian fluids Where the process is able to accommodate long drainage times, the fluid is likely to be viscous and non-Newtonian, like the DExtract and Squeezy fluids employed in this work. Both DExtract and Squeezy demonstrated thixotropy. This introduces challenges into modelling, some of which have been mentioned above. The rheological results (Figure 5) indicated that this would have noticeable effects when the wall shear stress exceeded 100 Pa, which was not encountered in these tests. Commercial lines are likely to employ larger pipes: for example, a vertical 50 mm i.d. line would give a wall shear stress of 160 Pa for DExtract. The apparent viscosity is then expected to be smaller and the drainage times shorter. Moreover, the liquid is likely to have been pumped in the period prior to drainage, subjecting it to an even higher wall shear stress, and will exhibit, again, a lower apparent viscosity. Selection of the rheological model and parameters for use in the drainage calculation in this case requires careful consideration. Conclusions The self-drainage of three food-related viscous liquids from circular pipes was investigated in experiments featuring different pipe diameters, lengths and angles of inclination to the vertical. The mass of fluid remaining in the pipe was measured. Drainage exhibited an initial stage characterised by a constant drainage rate, during which the volume of material in the pipe decreased by about 50%. Thereafter drainage could follow a second constant rate regime, a falling rate regime and one characterised by drop formation, depending on the pipe configuration and nature of the fluid. The nature of the pipe wall did not have a significant effect on drainage behaviour. Quantitative models for the rate and extent of drainage in the initial stage were developed. The former gave reasonable estimates of the drainage rate while the latter tended to overpredict the fraction of material remaining in the tube at the end of the initial stage. 19 Whereas the rate was strongly affected by the rheology of the fluid, the rheology had little influence on the fraction remaining: further work is required to allow this to be predicted reliably. Similarly, models for the drainage rate in the second and third stages offered insight into the behaviour but were not able to predict the rates reliably. The DExtract and Squeezy materials studied are complex fluids. They exhibited noticeable thixotropy but the models developed, particularly for the initial stage, gave reasonable estimates of their behaviour. There is significant difference in the times required to remove 50% and 90% of the product. This suggests that partial recovery of material by self-drainage is feasible: the extended period required to remove 90% of the product may not be practicable. Acknowledgements Samples of Marmite TM were provided by Unilever. The data in Figure A1 of the Appendix were extracted by Ole Mathis Magens. An EPSRC studentship for AA, supported by Procter & Gamble, is gratefully acknowledged, as are helpful discussions with Dr David Scott. An Open Data Statement will be added if the manuscript is accepted for publication. 20 References Cox, B.G. (1962) On driving a viscous fluid out of a tube, J. Fluid Mech., 14, 81-96. Cragnell, C., Hansson, K., Andersson, T, Jonsson, B. and Skepo, M. (2014) Underlying mechanisms behind adhesion of fermented milk to packaging surfaces, J. Food Engineering, 130, 52-59. de Kee, D, Schlesinger, M and Godo M (1988) Postwithdrawal drainage of different types of fluids, Chem. Eng. Sci., 43(7) 1603-1614. Fan, M., Phinney, D.M. and Heldman, D.R. (2015) Effectiveness of rinse water during in- place cleaning of stainless steel pipe lines, J. Food Sci., 80(7), 1490-1497. Irons, F.E. (1995) An interferometric measurement of the wall thickness of a cylindrical glass tube with application to a draining liquid film, Meas. Sci. Tech., 6, 1356-1361. Jeffreys, H. (1930) The draining of a vertical plate, Proc. Camb. Phil. Soc., 26, 204. Lister, J.R., Rallison, J.M., King, A.A., Cummings, L.J. and Jensen, O.E. (2006) Capillary drainage of an annular film: the dynamics of collars and lobes, J. Fluid Mech., 552, 311-343. Loibl, F., Schmidt, M.C. Auer-Seidl, A., Kirchner, C., Holtz, C., Miiller, K., Stramm, C. and Langowski, H.-C.(2012) The emptying behaviour of highly viscous liquids. Part II: development of test methods and evaluation of untreated and coated film, J. Adhes Sci Tech, 26, 2469-2503. Malvern Instruments (2016) Personal communication, 15 February 2016. Mickaily E.S. and Middleman, S. (1993) Hydrodynamic cleaning of a viscous film from the inside of a long tube, AIChE J, 39(5), 885-893. Ng, T.S., Lawrence, C.J. and Hewitt, G.F. (2001) Gravity-driven laminar flow in a partially- filled pipe, Chem. Eng. Res. Des., 79, 499-511. Nusselt, W. (1916) Oberflachen kondensation des Wasserdampfes. Zeitschrift Verein Deutscher Ingenieure, 60, 541-546 and 569-575. Palabiyik, I., Olunloyo, B., Fryer, P.J. and Robbins, P.T. (2014) Flow regimes in the emptying of pipes filled with a Herschel-Bulkley fluid, Chem. Eng. Res. Des., 92, 2201–2212. Sherwood, J.D. (2009) Optimal shapes for best draining, Phys. Fluids, 21, 113102. Sherwood, J.D. (2013) Optimal shapes for best centrifugal draining, Eur. J. Mech. B Fluids, 37, 124-128. Taylor, G.I. (1961) Deposition of a viscous fluid on the wall of a tube, J. Fluid Mech., 10, 161-165. Taylor, J.E., Van Damme, I., Johns, M.L., Routh, A.F. and Wilson, D.I. (2009) Shear rheology of molten crumb chocolate, J. Food Sci., 74(2) E55-61. Van Rossum, J.J. (1958) Viscous lifting and drainage of liquids, J. Appl. Res., A7, 121-144 White, D.A. and Tallmadge, J.A. (1966) A theory of withdrawal of cylinders from liquid baths, AIChE J, 12(2), 333-339. White, D.E, Moggridge, G.D. and Wilson, D.I. (2008) Solid-liquid transitions in the rheology of a structured yeast extract paste, Marmite TM , J. Food Eng, 88, 353-363. 21 Nomenclature Roman a1 constant, after equation [19] m 2- s -1 a2 constant, equation [23] m 2+1/ s -1/ a3 constant, equation [24] m 3 s -(1+ D tube diameter m Eo Eötvös number - g acceleration due to gravity m s -2 k constant, equation [21] m 2- s -1 K consistency, power law fluid Pa s n L tube length m ?̇? mass flow rate g s-1 ?̇?pred predicted mass flow rate g s -1 m Mass of liquid in tube g m* Fraction of liquid remaining in tube at end of stage I - m0, mi Mass of liquid initially, at time ti g n Power law index, Equation [6] - QA Flow rate in annulus m 3 s -1 Qi Flow rate, stage i m 3 s -1 QN Flow rate in annulus, Nusselt approximation m 3 s -1 QI,PL Flow rate in stage I, power law fluid m 3 s -1 R tube radius m r radial co-ordinate m ri radial position of annulus interface m Re Reynolds number - T temperature K t time s 𝑡′ elapsed time in stage III s ti time at end of stage i s u, ui velocity in annulus, velocity at interface m s -1 U mean velocity of liquid m s -1 Us velocity of slug front m s -1 22 VA volume of annulus m 3 x dimensionless radius, x = r/R - z axial co-ordinate m Greek  power law index, Equation [20] -  power law index, Equation [24] -  film thickness m app apparent shear rate s -1  dynamic viscosity Pa s  angle of inclination from vertical -  density kg m -3  surface tension N m -1  shear stress Pa w wall shear stress Pa 23 Tables Captions Table 1 Dimensions of tubes used in drainage tests Table 2 Rheological power law model parameters for Marmite™ fluids extracted from return sweeps for w < 100 Pa (see Figure 5) Table 3 Surface tension and advancing contact angles on test substrates Table 4: Summary of honey drainage tests: observation of one or two linear regions. A dash indicates that this configuration was not tested. Table 5: Summary of Squeezy drainage tests: observation of one or two linear regions. A dash indicates that this configuration was not tested. Table 6 Summary of DExtract drainage tests: observation of one or two linear regions. A dash indicates that this configuration was not tested. 24 Table 1 Dimensions of tubes used in drainage tests Material D /mm L1 /mm L2 /mm L3 /mm L1/D - L2/D - L3/D - Glass 8.8 100 200 500 11.4 22.7 56.8 15.3 158 317 791 10.3 20.7 51.7 21.7 217 433 1083 10.0 20.0 49.9 Perspex 7.9 100 200 500 12.7 25.3 63.3 15.0 158 317 791 10.5 21.1 52.7 22.0 217 433 1083 9.9 19.7 49.2 Table 2 Rheological power law model parameters for Marmite™ fluids extracted from return sweeps for w < 100 Pa (see Figure 5) Temperature Parameter 17°C 19°C 21°C DExtract n 0.85 0.82 0.75 K /Pa s n 108 100 95 Squeezy n 0.91 0.89 0.89 K /Pa s n 45 43 38 Table 3 Surface tension and advancing contact angles on test substrates Fluid Honey DExtract Squeezy Water Surface tension /mN m -1 72  4.0 46  2.1 46.9  0.5 73 Contact angle Borosilicate glass Perspex 81 5.0 49  2.4 45  4.3 51  1 81  4.0 55  3.1 54  4.0 71 1 25 Table 4: Summary of honey drainage tests: observation of one or two linear regions. A dash indicates that this configuration was not tested. D L m0 L/D Number of linear regions Material (mm) (mm) (g) (-)  = 0  = 30  = 45  = 60 Perspex 15 158 40 10.5 1 - - - 15 317 80 21.1 2 1 1 1 Glass 8.8 100 8 11.4 2 2 1 1 8.8 200 17 22.7 1 2 2 1 8.8 500 43 56.8 2 2 2 1 15.3 158 40 10.3 2 - 2 1 15.3 317 81 20.7 2 1 1 1 15.3 791 207 51.7 2 - 2 - 21.7 217 107 10.0 2 1 1 1 21.7 433 225 20.0 2 1 1 1 21.7 1083 576 49.9 2 2 1 1 Table 5: Summary of Squeezy drainage tests: observation of one or two linear regions. A dash indicates that this configuration was not tested. D L m0 L/D Number of linear regions Material (mm) (mm) (g) (-)  = 0  = 30  = 45  = 60 Perspex 7.9 100 6.0 12.7 1 - 2 - 15 158 37.0 10.5 2 2 1 1 15 317 76.0 21.1 2 2 1 1 22 217 105 9.9 2 1 1 1 Glass 8.8 100 8.0 11.4 1 - 1 - 15.3 158 38.6 10.3 2 1 1 1 15.3 317 78.0 20.7 2 2 2 1 15.3 791 198.0 51.7 2 - - - 21.7 217 105 10 2 2 1 1 26 Table 6 Summary of DExtract drainage tests: observation of one or two linear regions. A dash indicates that this configuration was not tested. D L m0 L/D Number of linear regions Material (mm) (mm) (g) (-)  = 0  = 30  = 45  = 60 Perspex 7.9 100 6 12.7 1 1 - - 15 158 38 10.5 2 2 1 1 15 317 76 21.1 2 2 2 1 22 217 106 9.9 2 2 1 - Glass 8.8 100 8 11.4 1 - 1 - 15.3 158 38 10.3 2 2 1 1 15.3 317 77 20.7 2 2 1 1 15.3 791 196 51.7 2 - - - 21.7 217 107 10 2 2 1 1 27 Figures Captions Figure 1 Schematic of flow from a draining tube in region I. A slug of air moves downwards at velocity Us, while liquid drains from the bottom as a filament. Dot-dash box indicates the control volume used to derive Equation [1]: dashed box indicates the control volume used to derive Equation [8]. Figure 2 Sketches of liquid motion near nose of a slug, based on sketches in Taylor (1961). Arrows indicate motion of fluid displaced by the slug, relative to the slug nose. Figure 3 Mass of liquid remaining in tube (calculated by difference) for honey in glass tube (L = 433 mm, D = 21.7 mm; 22.5 °C). The initial mass, m0, was measured as 225.85 g. Dashed lines show boundaries between stages. Inset shows data with time plotted on logarithmic scale. Figure 4 Comparison of flow rate in viscous Newtonian annular film calculated by Equation [10], QA, and that estimated using the Nusselt film result, QN, Equation [16]. QI is the flow rate in a full tube, Equation [5]. Figure 5 Apparent viscosity of Marmite™ fluids obtained from steady state shear stress sweeps at 19C. (a) DExtract, (b) Squeezy. Vertical dashed line shows upper limit of wall shear stresses calculated using Equation [1] for the drainage tests. Figure 6: Mass fraction remaining at end of stage I, m*. (a) honey, showing effect of angle of inclination; (b) honey data, plotted against 1 /8 Eo cos ; (c) Squeezy; (d) DExtract. Open symbols – Perspex; solid symbols - glass. Error bars are smaller than symbols. Loci show model predictions for each case. Figure 7 Agreement between measured and predicted mass flow rates in region I for (a) honey (QI, Equation [5]); (b) Squeezy, (c) DExtract (both QI,PL Equation [7]). Open symbols – Perspex; solid symbols - glass. Symbol shape indicates angle of inclination: ○ – 0°; △ - 30°; ☐ - 45°; ♢ - 60°. Error bars represent 95% confidence intervals. Dashed (diagonal) locus shows the line of equality (y = x). Figure 8 Effect of angle of inclination on drainage rate of DExtract in stage I. DExtract data in Figure 7(c) expressed as the ratio of the drainage rate at angle  to the vertical case ( = 0). Open and solid symbols denote tests using perspex and glass, respectively. Figure 9: Effect of angle of inclination, expressed as cos, on ratio of measured mass flow rates in linear regions I and II, ṁII/ṁI. Open symbols – honey; grey symbols – Squeezy; black symbols - Dextract. Symbol shape indicates diameter (see legend). 28 Figure 10. Evolution of mass remaining in tube during Stage III. L/D = 433/21.5, = 0. Grey symbols – mass fraction; black symbols, data plotted in the form suggested by Equation [25]. Circles – honey, inset shows detail of first 200 s where Equation [20] describes the honey data; triangles - Squeezy; squares – DExtract. Dashed horizontal line indicates limit of linear behaviour for honey. Figure 11 Comparison of drainage times. Symbols: open – honey; grey – Squeezy; black – DExtract: circles – glass, triangles – Perspex. Dashed line shows locus for t = tI. Figure A1 Experimental data from Taylor (1961, Figure 2) replotted in the form Us/ = f(xi). Dashed locus shows line of best fit obtained by linear regression to a third order polynomial using Microsoft Excel TM . Figure A2 Composite plot of data from Figure 6(b)-(d) for L/D > 10. 29 Figure 1 Schematic of flow from a draining tube in region I. A slug of air moves downwards at velocity Us, while liquid drains from the bottom as a filament. Dot-dash box indicates the control volume used to derive Equation [1]: dashed box indicates the control volume used to derive Equation [8]. 30 (a) (b) Figure 2 Sketches of liquid motion near nose of a slug, based on sketches in Taylor (1961). Arrows indicate motion of fluid displaced by the slug, relative to the slug nose. 31 Figure 3 Mass of liquid remaining in tube (calculated by difference) for honey in glass tube (L = 433 mm, D = 21.7 mm; 22.5 °C). The initial mass, m0, was measured as 225.85 g. Dashed lines show boundaries between stages. Inset shows data with time plotted on logarithmic scale. 0 50 100 150 200 0 50 100 150 200 m ( g ) t (s) I II III IV 0 50 100 150 200 1 10 100 m ( g ) t (s) 32 Figure 4 Comparison of flow rate in viscous Newtonian annular film calculated by Equation [10], QA, and that estimated using the Nusselt film result, QN, Equation [16]. QI is the flow rate in a full tube, Equation [5]. 33 (a) Figure 5 Apparent viscosity of Marmite™ fluids obtained from steady state shear stress sweeps at 19C. (a) DExtract, (b) Squeezy. Vertical dashed line shows upper limit of wall shear stresses calculated using Equation [1] for the drainage tests. 34 (b) Figure 5 Apparent viscosity of Marmite™ fluids obtained from steady state shear stress sweeps at 19C. (a) DExtract, (b) Squeezy. Vertical dashed line shows upper limit of wall shear stresses calculated using Equation [1] for the drainage tests. 35 (a) Figure 6: Mass fraction remaining at end of stage I, m*. (a) honey, showing effect of angle of inclination; (b) honey data, plotted against 1 /8 Eo cos ; (c) Squeezy; (d) DExtract. Open symbols – Perspex; solid symbols - glass. Error bars are smaller than symbols. Loci show model predictions for each case. 36 (b) Figure 6: Mass fraction remaining at end of stage I, m*. (a) honey, showing effect of angle of inclination; (b) honey data, plotted against 1 /8 Eo cos ; (c) Squeezy; (d) DExtract. Open symbols – Perspex; solid symbols - glass. Error bars are smaller than symbols. Loci show model predictions for each case. 37 (c) Figure 6: Mass fraction remaining at end of stage I, m*. (a) honey, showing effect of angle of inclination; (b) honey data, plotted against 1 /8 Eo cos ; (c) Squeezy; (d) DExtract. Open symbols – Perspex; solid symbols - glass. Error bars are smaller than symbols. Loci show model predictions for each case. 38 (d) Figure 6: Mass fraction remaining at end of stage I, m*. (a) honey, showing effect of angle of inclination; (b) honey data, plotted against 1 /8 Eo cos ; (c) Squeezy; (d) DExtract. Open symbols – Perspex; solid symbols - glass. Error bars are smaller than symbols. Loci show model predictions for each case. 39 (a) (b) 0.001 0.01 0.1 1 10 100 0.001 0.01 0.1 1 10 100  Q I (g /s ) (g/s) 0.001 0.01 0.1 1 10 100 0.001 0.01 0.1 1 10 100  Q I, P L ( g /s ) (g/s) m 40 (c) Figure 7 Agreement between measured and predicted mass flow rates in region I for (a) honey (QI, Equation [5]); (b) Squeezy, (c) DExtract (both QI,PL Equation [7]). Open symbols – Perspex; solid symbols - glass. Symbol shape indicates angle of inclination: ○ – 0°; △ - 30°; ☐ - 45°; ♢ - 60°. Error bars represent 95% confidence intervals. Dashed (diagonal) locus shows the line of equality (y = x). 0.001 0.01 0.1 1 10 100 0.001 0.01 0.1 1 10 100  Q I, P L ( g /s ) (g/s) 41 Figure 8 Effect of angle of inclination on drainage rate of DExtract in stage I. DExtract data in Figure 7(c) expressed as the ratio of the drainage rate at angle  to the vertical case ( = 0). Open and solid symbols denote tests using perspex and glass, respectively.  = 45  = 0 42 Figure 9: Effect of angle of inclination, expressed as cos, on ratio of measured mass flow rates in linear regions I and II, ṁII/ṁI. Open symbols – honey; grey symbols – Squeezy; black symbols - DExtract. Symbol shape indicates diameter (see legend). ṁ II /ṁ I 43 Figure 10. Evolution of mass remaining in tube during Stage III. L/D = 433/21.5, = 0. Grey symbols – mass fraction; black symbols, data plotted in the form suggested by Equation [25]. Circles – honey, inset shows detail of first 200 s where Equation [20] describes the honey data; triangles - Squeezy; squares – DExtract. Dashed horizontal line indicates limit of linear behaviour for honey. m /m 0 44 Figure 11 Comparison of drainage times. Symbols: open – honey; grey – Squeezy; black – DExtract: circles – glass, triangles – Perspex. Dashed line shows locus for t = tI. tII tIII 45 Appendix In the experiments reported by Taylor (1961), air was forced through tubes filled with viscous liquid at set flow rates. Us and m were measured. His Figure 2 presented measurements of m* against Us/: m* was used to calculate xi using Equation [15] and the data are replotted in Figure A1 as Us/ = f(xi). Figure A1 Experimental data from Taylor (1961, Figure 2) replotted in the form Us/ = f(xi). Dashed locus shows line of best fit obtained by linear regression to a third order polynomial using Microsoft Excel TM . 46 Figure A2 Composite plot of data from Figure 6(b)-(d) for L/D > 10.