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Prediction of deposit depth and transmembrane pressure during crossflow microfiltration
Journal of Membrane Science 154 (1999) 229±237
Prediction of deposit depth and transmembrane pressure during cross¯ow micro®ltration I.W. Cumming*, R.G. Holdich, B. Ismail Department of Chemical Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK Received 14 July 1997; received in revised form 28 July 1998; accepted 11 September 1998
Abstract Most cross¯ow micro®ltrations of suspensions of signi®cant solids volume concentration exhibit non-Newtonian ¯ow behaviour, and may lead to the deposition of appreciable cake depths. The cake depth may even extend to the full ®lter tube diameter, thus blocking any further ®ltration. Cross¯ow ®ltration modelling also requires knowledge of the cake depth, and the transmembrane pressure. Experiments have been performed for both the cross¯ow micro®ltration of non-Newtonian talc suspensions, and an investigation of the pressures and ¯ow rates in impermeable tubes of diameters similar to the cross¯ow ®lters. The investigation also illustrates how the transmembrane pressure drop can be corrected to take account of differences in cross-sectional area between where the pressure measurements are recorded and the ®lter ¯ow channel. A method is demonstrated for the estimation of the depth of deposit on a fouled ®lter tube from pressure and ¯ow measurements, when combined with a knowledge of the suspension rheology. The pseudo-equilibrium steady-state ¯ux has been shown to correlate with shear stress at the ®lter or deposit surface, after the above corrections have been performed. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Membrane preparation and structure; Macrovoid; Formation mechanism
1. Introduction Predicting the deposition of solid particles on a ®lter operating in cross¯ow is important because it has a major impact on the ®ltration ¯ux. In extreme circumstances, it will result in the ®lter ¯ow path becoming completely blocked preventing any further ¯ow. Many different authors have proposed models for the mechanisms of deposition on a ®ltration surface in an attempt to predict the ¯ux which can be maintained with little further decline, the so-called steady-state ¯ux. The prediction of this ¯ux is important as it sets *Corresponding author. Tel.: +44-1509150922; fax: +44-1509223923.
the ¯ux at which a cross¯ow ®ltration unit may be operated. The models have used different mechanisms to explain particle behaviour but, due to the wide range of operating conditions in cross¯ow ®ltration, it has proved dif®cult to explain all the observations with one single model. The models described in the literature have been reviewed by a number of authors [1±3]. The model illustrated in this paper is based on a force balance on a particle at the ®ltration surface [4,5]; using this approach it has been shown that the steady-state ¯ux is proportional to the shear at the ®ltration surface for Newtonian and magnesia suspensions, respectively. It is assumed that the particles depositing on the ®lter are too large to enter the ®lter structure.
0376-7388/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S0376-7388(98)00293-2
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The cross¯ow ®ltration of non-Newtonian slurries has practical applications in the thickening of foods, metallic hydroxide slurries, clays, ®ne chemicals and biochemical products. It has been found that thickening is normally best carried out using ®lter tubes, rather than capillaries or hollow ®bres, as these are less prone to the development of tube blockages. However, if the ®lter is operated at extreme conditions such as high solids concentration, the ®lter tube may still block so the determining conditions of ®lter blockage are important. In order to employ the shear model, or most other alternatives, the true channel diameter open to ¯ow (rather than the original channel diameter) and the transmembrane pressure are required. If the pressures are measured outside the ®lter channel, and in pipes of a different cross-sectional area to the ®lter ¯ow channel, the pressures within the ®lter will be different to those recorded in the surrounding pipework. In most cases, the ®lter channels are narrower in diameter than the surrounding pipework and the slurry accelerates to a higher velocity in the channel compared to the entry and exit manifolds. An energy balance on the slurry provides the solution that if the kinetic energy increases, then the pressure energy or head must decrease, in order to conform to the conservation of energy principle. In Newtonian ¯ow, the well-known Bernoulli equation can be used to predict this reduction in pressure head. Thus the pressures inside the ®lter channel may be signi®cantly below those registered in the ®lter manifolds, and estimates of the transmembrane pressure based on the latter will be in error. The deposition of cake on the surface of the ®lter will reduce the ¯ow channel area, further contributing to the error in transmembrane pressure prediction based on the manifold values. This last factor will also limit the effectiveness of positioning the pressure transducers within the ®lter ¯ow channel, i.e. using the same diameter for both the pressure measurements and ®lter channel; as ®lter cake will not deposit on the transducers but will deposit on the surrounding membrane ®lter. Thus the ¯ow channel will again have a different cross-sectional area. Clearly, the signi®cance of this effect depends on the amount of deposit formed, but is important in thickening operations if complete blockage of the ¯ow channel is to be avoided. The suspension used in the test programme was talc dispersed in water. Talc was used as it is a fairly
incompressible material, which also tends to agglomerate. It was assumed that these properties would encourage the formation of a stable cake on the ®lter, resisting size segregation of the deposit under differing conditions of shear. 2. Analytical procedure The model for particle deposition is based on a semi-empirical approach to the prediction of the steady-state ¯ux when ®ltering a slurry chie¯y consisting of particles larger than one micron. This approach uses a force balance on a particle at the ®lter surface, which gives the relation that the ®ltration ¯ux should be linearly dependent on the shear stress at the ®ltration surface [4]. This surface could be either the surface of the ®lter medium or the surface of particles deposited on the ®lter medium. The model was demonstrated for Newtonian slurries made up of single size particles. In the case of a slurry consisting of particles with a range of sizes the relation between ¯ux (J) and shear stress ( w) is J k1 w k0 ;
(1)
where k1 and k0 are the empirical constants. The shear stress at the wall of a cylindrical tube of length L and diameter d is related to the pressure drop (P) between the inlet and outlet of the tube by w
Pd : 4L
(2)
So, if the pressure loss along the tube and the tube dimensions are known, then the wall shear stress can be calculated. In a cross¯ow ®lter, a layer of deposited particles forms at the ®ltration surface, so the diameter open to ¯ow is not known and must be determined. To calculate the diameter open to ¯ow requires the use of standard ¯uid mechanics equations for pressure drop with ¯ow rate. 2.1. Pressure drop calculation The non-Newtonian talc slurry used in this work was found to ®t a power law expression: w _ n ;
(3)
where _ is the shear rate, and n are the empirically
I.W. Cumming et al. / Journal of Membrane Science 154 (1999) 229±237
derived constants known, respectively, as the consistency coef®cient and the ¯ow index. Using this relation, the ¯ow of a power law ¯uid in a channel can be characterised by a modi®ed Reynolds number [6]: Re
8mv2ÿn dn ; 6 2=nn
(4)
where v is the average tube velocity and m is the slurry density. If the friction factor (f) is known, then the shear stress at the tube wall can be calculated using the Fanning equation: 2 f 4Q ; (5) w m 2 d2 where Q is the volumetric ¯ow in the tube. The pressure drop in the tube can then be calculated using Eq. (2). Methods for estimating the friction factor are available for both laminar and turbulent ¯ows. In the case of laminar ¯ow, the friction factor can be calculated from an analytical expression [6] as f
16 : Re
(6)
For turbulent ¯ow, the friction factor can be calculated from the empirical relation [6]: 1 4 0:4 p 0:75 log Re f 1ÿn=2 ÿ 1:2 : n f n
(7)
So the tube diameter can be calculated for both laminar and turbulent ¯ows from measured values of: volumetric ¯ow, slurry rheological properties, tube pressure drop and length. In the case of laminar ¯ow, this requires the use of Eqs. (3)±(6), whilst for turbulent ¯ow, Eqs. (3)±(5) and (7) are required. The
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laminar case can be solved algebraically, but for turbulent ¯ow, an iterative calculation is necessary. This model does assume that the depth of deposit over the surface of the membrane is similar over all its surface. This has previously been shown to be a reasonable assumption, except close to the tube entrance, where the velocity pro®le is changing rapidly [3]. 2.2. Pressure measurement A complication in applying this technique to calculate the deposit depth is that the pressure is not normally measured at the ends of a membrane tube, but in a manifold beyond either end of the tube. In the case of the experiments where a single ®lter tube was used, the slurry will undergo a signi®cant acceleration on entering the tube and a corresponding deceleration on leaving the tube. Deposition of solids on the inside tube surface will further reduce the diameter, resulting in a further increase in velocity. There will be pressure changes associated with these velocity changes which can signi®cantly alter the pressure drop across the ®ltration surface (transmembrane pressure), and there will also be energy losses due to eddies formed on entry and exit. Fig. 1 shows the pressure variations that occur as the slurry enters and leaves a ®lter tube. Often it is assumed that the average transmembrane pressure (TMP) can be calculated as TMP
P1 P 4 ÿ Pp ; 2
(8)
where P1 and P4 are the module inlet and outlet
Fig. 1. Pressure changes occurring inside the filter module.
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pressures, respectively, and Pp is the permeate pressure at the outside of the ®lter tube. This approximation assumes that the pressure loss along the ®lter is linear with tube length, which is reasonable if the velocity pro®le develops rapidly, and the tube maintains a constant area open to ¯ow. The mean transmembrane pressure is obviously actually nearer the value calculated as TMP
P2 P3 ÿ Pp ; 2
(9)
where P2 and P3 are the ®lter tube inlet and outlet pressures. If the pressure loss on entering the ®lter tube is Pin and the pressure recovery on the leaving the ®lter tube is Prise, then in terms of the module inlet and outlet pressures, the transmembrane pressure becomes TMP
P1 P4 ÿ Pin ÿ Prise ÿ Pp : 2
(10)
This will give a lower value for the average transmembrane pressure than Eq. (8) and in this paper, it will be shown that not using Eq. (10) can lead to a signi®cant error, particularly at high velocities and pressure drops in tubes. In addition to the pressure change between the manifold and the ®lter channel due to ¯uid acceleration, and hence pressure reduction ± i.e. the Bernoulli effect, there will be energy losses due to the eddies formed on entry and exit from the ®lter channel. These energy losses will give rise to further pressure changes from those indicated by transducers mounted in the ®lter manifold. The estimation of inlet and outlet pressure changes are not well documented for nonNewtonian ¯ow. This is particularly true for the inlet loss as there is no information for the vena contracta. Fortunately, the inlet energy loss is less signi®cant as there tends to be little cake deposited at the actual ®lter tube inlet, so that the cake formed a funnel shape at the beginning of the ®lter tube which will reduce energy loss. In this work, it has been assumed that the inlet pressure loss is only due to Bernoulli effects and that frictional loss along the ®lter tube can be calculated as already described. The pressure rise on leaving the ®lter tube and passing into the header can be calculated for laminar ¯ow by the classical energy and momentum balance to give the following equation [7]:
Prise m
3n 1 2n 1
Q A1
2 " 2 # A1 A1 : ÿ A2 A2 (11)
Here A1 and A2 are the upstream and downstream cross-section areas, in this work, the ratio between A1 and A2 was varied from 0.022 to 0.31. The energy loss due to eddy formation as the slurry passes through the tube expansion can be similarly calculated in terms of pressure change as 2 2 " 3n 1 Q n3 A1 Peddy m 2n 1 A1 2 5n 3 A2 # A1 3 3n 1 ÿ : (12) A2 2 5n 3 These equations were also used for the case of turbulent ¯ow pressure losses. The velocity pro®le for laminar ¯ow of non-Newtonian power law ¯uids is ¯atter than for Newtonian ¯ow, so Eqs. (11) and (12) may be valid at higher Reynolds numbers. The goodness of ®t of this calculation procedure is discussed later. 3. Experimental A conventional cross¯ow loop was used for all the tests and is shown schematically in Fig. 2. The feed tank containing talc dispersed in water was stirred by a slowly rotating large paddle to ensure that the talc was kept well mixed and prevented from settling. The talc was pumped from the feed tank by a moving cavity Monopump. The pressure and ¯ows at the ®lter/tube module were controlled using the by-pass valve and the two in-line valves. The pressures in and out of the module were monitored by two stainless steel pressure transducers mounted a little upstream and downstream of the module, these were ¯ush mounted in the 20 mm nominal bore PVC pipe used to recirculate the slurry. To measure the slurry ¯ow a magnetic inductive ¯ow meter was ®tted in-line with the ®lter. The three in-line heat exchangers had the same internal diameter as the PVC pipes and with the use of cooling water, the slurry temperature was maintained at 2028C. The pressures and ¯ows were continuously logged by a microcomputer.
I.W. Cumming et al. / Journal of Membrane Science 154 (1999) 229±237
233
Fig. 2. Schematic diagram of crossflow microfiltration equipment.
Details of the ®lters used in the experiments are given in Table 1. Two stainless steel ®lters had internal diameters of 10 and 14 mm, with nominal pore sizes of 2 and 3 mm, respectively. Two ceramic ®lters were also used, these were supplied in the form of seven channel monoliths but in these tests only the central channel was used with the others being blocked at their ends. The ceramic channels had internal diameters of 4.3 and 5.8 mm with respective nominal pore sizes of 0.3 and 0.1 mm. The experiments to determine whether the internal diameter of a tube could be predicted from pressure and ¯ow measurements used different materials such as PVC, acrylic, glass and metal. A large number of different tubes were tested with internal diameters of 3.5, 3.7, 3.9, 4.2, 4.5, 5.2, 5.8, 6.2, 9.5 and 13 mm. The
lengths of these tubes were similar to that of the different ®lters used. The slurry used was a suspension of talc which varied only slightly in composition and properties throughout the experimental programme. The slurry composition varied from 26.3 to 28.3 wt%. The size distribution was routinely measured using laser diffraction, a Coulter LS 130. There was a slight reduction in the Sauter mean particle diameter from 8.9 to 8.3 mm throughout the programme of work. The slurry rheological properties were measured using a Haake RV2 viscometer, with a concentric cylinder measuring head ®tted. Each time a new series of experiments were undertaken, the slurry rheological properties were measured using a sample taken from the slurry feed tank. The shear stress/rate graph is shown in
Table 1 Dimensions of crossflow filters Dimensions
Internal diameter (mm) External diameter (mm) Total length (mm) Filtration length (mm) Pore size (mm) Number of tubes
Membrane type Metal tubes Stainless steel 316
Ceramic tubes
14 20 380 376 3 1 tube
5.8 20 380 340 0.1 7 tubes (with 6 closed in both cases)
10 13 298 260 2 1 tube
4.3 19 360 320 0.3 7 tubes
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I.W. Cumming et al. / Journal of Membrane Science 154 (1999) 229±237
Fig. 4. Comparison between measured and predicted pressure drop at various flow rates inside tubes of different diameters. Fig. 3. Rheogram of talc suspension used during the tests ± illustrating power-law fit.
Fig. 3, where it is plotted using logarithmic scales. This shows how well the measured data ®ts a power law expression. The ®ltration experiments were carried out recycling the talc slurry through the tubes until the permeate ¯ow became constant, it usually took between 15 and 70 min for a pseudo-steady-state ¯ux to be achieved. At this point, the data for the system pressures and ¯ow were measured as cake deposition had essentially ceased. 4. Results and discussion 4.1. Prediction of tube pressure drop and diameter The tests of the applicability of the calculation procedure for predicting the pressure drops for different diameter tubes were carried out over a wide range of conditions. The tube diameters varied from 3.5 to 13 mm, whilst the velocity in the tubes was between 1.9 and 11.1 m sÿ1 with Reynolds numbers in the range 1300±14 000. The value of n in the powerlaw expression varied from 0.548 to 0.601 in this set of experiments. The ¯ow is expected to vary from laminar up to a reasonable degree of turbulence. Using Eqs. (3)±(7), it was found that the pressure drop due to friction within the tube varied between 50% and 80% of the total pressure change measured by the transducers located in the main loop pipework. The pressure loss due to the formation of an eddy at the tube outlet was calculated using Eq. (11) to represent between
50% and 20% of the total pressure loss. Adding these two pressure losses together gives the pressure change between the pressure transducers in the experimental loop. A comparison between the measured and predicted pressure drop for four of the tubes tested, covering a wide range of diameters, is shown in Fig. 4. It appears from this ®gure that the prediction of pressure loss was best for the larger 13 mm pipe and poorest for the much smaller 3.9 mm pipe. All the experimental data are presented in Fig. 5, where the predicted pressure drop is plotted against the measured values. For the prediction to ®t the data, all the points would have to fall on the line shown in the graph. As can be seen at typical ®ltration pressure drops of less than 1 bar, the ®t is good. As the pressure drop increases, the ®t declines until at almost 2 bar, for the 3.5 mm diameter tube, the error in predicted pressure loss is up to 23%, however, most of the data had an error of between 0% and 15%. The reasons for
Fig. 5. Comparison between measured and predicted pressure drops for all tubes.
I.W. Cumming et al. / Journal of Membrane Science 154 (1999) 229±237
Fig. 6. Comparison between measured and predicted tube diameters for all tubes.
this overprediction of pressure drop may be due to errors in the exit loss equation, as the ¯ow becomes increasingly turbulent, the laminar exit loss equation is likely to become less valid. The prediction of tube diameter from slurry ¯ow and measured pressure drop was very good. This is shown in Fig. 6, where the measured diameter is plotted against the calculated diameter using the measured pressure drop and the ¯ow equations to infer the tube diameter. The agreement can be seen to be close over the complete range of tubes tested. The reason for the better ®t for diameter than pressure drop can be seen by examining Eqs. (5)±(7) and (11), which indicate that a small change in measured diameter will have a much larger effect on the calculated pressure drop. From these tests, it can be seen that the pressure drop in a cylindrical restriction can be predicted quite well particularly at lower pressure drops. The diameter can be predicted over the complete range of tubes tested. Thus it should be possible to accurately infer the deposit thickness from the pressure readings taken either side of the ®lter tube, having corrected for pressure changes due to the Bernoulli effect and eddies on expansion. 4.2. Mean transmembrane pressure As has been described, the mean transmembrane pressure has often been taken as the arithmetic mean of the module inlet and outlet pressures. This is not a very good approximation when there are signi®cant pressure changes associated with the acceleration and deceleration of the slurry as it enters and leaves a ®lter
235
Fig. 7. Effect of entrance and exit pressure changes on mean transmembrane pressure.
tube. The method used here is to estimate the ®lter tube outlet pressure by subtracting the pressure recovery term from the outlet transducer reading and estimating the ®lter tube inlet pressure by adding the calculated pressure drop in the tube to the ®lter tube outlet pressure. Fig. 7 shows how the mean transmembrane pressure calculated by this technique, Eq. (10), compares with the mean transmembrane pressure calculated by the simple averaging technique, Eq. (8), and is called the `super®cial average pressure'. These data have been generated for the tube diameters and ¯ows used in the experimental programme to test the non-Newtonian ¯ow calculations already described. The line on the graph indicates agreement between the two calculation procedures. At the highest mean transmembrane pressure, the error introduced by the simple averaging technique can be as great as 100%. This error does decrease as the cross¯ow velocity is reduced and the entrance and exit losses become less signi®cant. 4.3. Flux/shear stress correlation Experiments ®ltering at constant operating conditions of ¯ow showed that the ¯ux declined fairly rapidly with time until a pseudo-constant permeate ¯ow was achieved. An example of this is shown in Fig. 8, where talc was ®ltered using the 4.2 mm inside diameter ceramic ®lter, at a cross¯ow of 2.5 l minÿ1 and four different transmembrane pressures of 0.06, 0.15, 0.67 and 1.28 bar. The permeate ¯ow declined to an apparently constant value within 20 min for the lowest transmembrane pressure, whilst at the highest
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I.W. Cumming et al. / Journal of Membrane Science 154 (1999) 229±237
Fig. 8. Example of permeate decline during the experiments ± at constant crossflow rate.
pressure, the time to reach a steady ¯ow was about 70 min. All the experiments using talc slurry showed similar behaviour. In all these tests, the ®ltrate was clear from the start of ®ltration, this was believed to be due to initial rapid deposition of talc particles on to the ®lter at the beginning of each experiment so that the talc deposit gave good ®ltration as a form of precoat. This meant that the ®lter medium did not suffer from particle penetration and the performance of the process was mainly controlled by the talc slurry interaction. The experiments were carried out for the different ®lter media used at the range of conditions shown in Table 2. The minimum cross¯ow rate was set by the limitation that any lower ¯ow would result in complete blockage of the ®lter tubes by the talc deposit. The Reynolds number for all the experiments can be seen to be in the transition regime of turbulence. The method of calculating the deposit thickness was by using Eqs. (3)±(7) and (11), and solving for the diameter open to ¯ow using the measured data for ®lter inlet and outlet pressures for different cross¯ows. The calculated thickness of the cake deposit showed that for the larger diameter metal ®lters, the thickness
Fig. 9. Permeate flux rate as a function of wall shear stress for talc suspensions.
varied from 1.8 to 3.6 mm, whilst for the ceramic ®lters, the depth varied from 0.04 to 1.5 mm. Where the calculated deposit thickness was of the order of 100 microns and below, the cake may not be very many particle diameters thick. The ¯ux rates with shear are shown in Fig. 9, where the ®lter tube diameters remaining open to ¯ow were used in Eq. (2) together with the pressure drop within the ®lter module (P2 to P3 in Fig. 1), to give the wall shear stress. The least squares error straight line ®t through these data points gives an intercept quite close to the origin, and conforms reasonably well to Eq. (1). 5. Conclusions When ®ltering viscous slurries, the deposition of solids at the surface of the ®lter can be substantial. To test the shear stress model, Eq. (1), the diameter open to ¯ow in a ®lter tube was ®rstly deduced from measurement of the pressure loss as the slurry ¯ows through a tube. Using tubes of known internal diameter, it was demonstrated that, by applying standard non-Newtonian ¯ow theory, the internal diameter can
Table 2 Talc slurry flow regimes in filtration tests Filter
Mean velocity in clean tube (m sÿ1)
Mean velocity in fouled tube (m sÿ1)
Reynolds number in clean tube
Reynolds number in fouled tube
Metal 14 mm Metal 10 mm Ceramic 5.8 mm Ceramic 4.3 mm
1.2±3.3 1.2±3.2 2.2±5.4 2.7±9.2
2.1±5.8 1.8±5.5 3.8±9.2 4.8±16.2
685±2900 721±3500 1500±5500 2300±12 700
2500Ð6800 5200±11 900 2800Ð7200 3000±15 000
I.W. Cumming et al. / Journal of Membrane Science 154 (1999) 229±237
be predicted from measurement of the pressure at either end of the tube and the ¯ow rate through the tube. It has also been shown that using the pressure transducer measurements in the larger diameter manifolds at either end of a tube can lead to substantial inaccuracy in establishing the pressure drop across the ®ltration surface (the transmembrane pressure). The method to determine the open tube diameter, or cake thickness, was then employed to the ®lter tubes of diameters ranging from 14 to 4.3 mm, and for ¯ows of Reynolds numbers from 685 to 15 000. These values encompass those of practical use during micro®ltration of concentrated dispersions. For the pseudoplastic talc suspensions, the steady-state permeate ¯ux gave a reasonable correlation with the wall shear stress, in accordance with the shear stress model ± Eq. (1). 6. Nomenclature A1 A2 d f J k 0, k 1 L n P Q Re* TMP v
upstream area (m2) downstream area (m2) tube diameter (m) friction factor flux (m sÿ1) constants tube length (m) flow index pressure (N mÿ2) volumetric flow (m3 sÿ1) Reynolds number transmembrane filtration pressure average velocity (m sÿ1)
237
Greek letters
_ m w
shear rate (sÿ1) consistency coefficient suspension density (kg mÿ3) wall shear stress (N mÿ2)
Subscripts p
permeate
Acknowledgements The authors wish to thank Tech Sep for the provision of the ceramic ®lters used in this work. References [1] M.H. Lojkine, R.W. Field, J.A. Howell, Crossflow microfiltration of cell suspensions: A review of models with emphasis on particle size, Trans. I Chem. E, Part C 70 (1992) 149±164. [2] C. Smith, R.J. Wakeman, S. Tarleton, The relevance of fouling models to crossflow microfiltration, Proceedings of the Filtech 1991 Conference, Karlesruhe, Germany (Filtration Society, Horsham, UK), pp. 51±63. [3] M.R. Mackley, N.E. Sherman, Crossflow cake filtration mechanisms and kinetics, Chem. Eng. Sci. 47 (1992) 3067± 3084. [4] N.J. Blake, I.W. Cumming, M. Streat, Prediction of steady state crossflow filtration using a force balance model, J. Membr. Sci. 68 (1992) 205±216. [5] R.G. Holdich, I.W. Cumming, B. Ismail, The variation of crossflow filtration rate with wall shear stress and the effect of deposit thickness, Trans. I Chem. E, Part A 73 (1995) 20±25. [6] A.B. Metzner, J.C. Reed, Flow of non-Newtonian fluids ± correlation of the laminar, transition and turbulent flow regimes, AIChE J. 1 (1955) 434±440. [7] W.L. Wilkinson, Non-Newtonian Fluids, Pergamon Press, Oxford, 1960.
Prediction of deposit depth and transmembrane pressure during cross¯ow micro®ltration I.W. Cumming*, R.G. Holdich, B. Ismail Department of Chemical Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK Received 14 July 1997; received in revised form 28 July 1998; accepted 11 September 1998
Abstract Most cross¯ow micro®ltrations of suspensions of signi®cant solids volume concentration exhibit non-Newtonian ¯ow behaviour, and may lead to the deposition of appreciable cake depths. The cake depth may even extend to the full ®lter tube diameter, thus blocking any further ®ltration. Cross¯ow ®ltration modelling also requires knowledge of the cake depth, and the transmembrane pressure. Experiments have been performed for both the cross¯ow micro®ltration of non-Newtonian talc suspensions, and an investigation of the pressures and ¯ow rates in impermeable tubes of diameters similar to the cross¯ow ®lters. The investigation also illustrates how the transmembrane pressure drop can be corrected to take account of differences in cross-sectional area between where the pressure measurements are recorded and the ®lter ¯ow channel. A method is demonstrated for the estimation of the depth of deposit on a fouled ®lter tube from pressure and ¯ow measurements, when combined with a knowledge of the suspension rheology. The pseudo-equilibrium steady-state ¯ux has been shown to correlate with shear stress at the ®lter or deposit surface, after the above corrections have been performed. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Membrane preparation and structure; Macrovoid; Formation mechanism
1. Introduction Predicting the deposition of solid particles on a ®lter operating in cross¯ow is important because it has a major impact on the ®ltration ¯ux. In extreme circumstances, it will result in the ®lter ¯ow path becoming completely blocked preventing any further ¯ow. Many different authors have proposed models for the mechanisms of deposition on a ®ltration surface in an attempt to predict the ¯ux which can be maintained with little further decline, the so-called steady-state ¯ux. The prediction of this ¯ux is important as it sets *Corresponding author. Tel.: +44-1509150922; fax: +44-1509223923.
the ¯ux at which a cross¯ow ®ltration unit may be operated. The models have used different mechanisms to explain particle behaviour but, due to the wide range of operating conditions in cross¯ow ®ltration, it has proved dif®cult to explain all the observations with one single model. The models described in the literature have been reviewed by a number of authors [1±3]. The model illustrated in this paper is based on a force balance on a particle at the ®ltration surface [4,5]; using this approach it has been shown that the steady-state ¯ux is proportional to the shear at the ®ltration surface for Newtonian and magnesia suspensions, respectively. It is assumed that the particles depositing on the ®lter are too large to enter the ®lter structure.
0376-7388/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S0376-7388(98)00293-2
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The cross¯ow ®ltration of non-Newtonian slurries has practical applications in the thickening of foods, metallic hydroxide slurries, clays, ®ne chemicals and biochemical products. It has been found that thickening is normally best carried out using ®lter tubes, rather than capillaries or hollow ®bres, as these are less prone to the development of tube blockages. However, if the ®lter is operated at extreme conditions such as high solids concentration, the ®lter tube may still block so the determining conditions of ®lter blockage are important. In order to employ the shear model, or most other alternatives, the true channel diameter open to ¯ow (rather than the original channel diameter) and the transmembrane pressure are required. If the pressures are measured outside the ®lter channel, and in pipes of a different cross-sectional area to the ®lter ¯ow channel, the pressures within the ®lter will be different to those recorded in the surrounding pipework. In most cases, the ®lter channels are narrower in diameter than the surrounding pipework and the slurry accelerates to a higher velocity in the channel compared to the entry and exit manifolds. An energy balance on the slurry provides the solution that if the kinetic energy increases, then the pressure energy or head must decrease, in order to conform to the conservation of energy principle. In Newtonian ¯ow, the well-known Bernoulli equation can be used to predict this reduction in pressure head. Thus the pressures inside the ®lter channel may be signi®cantly below those registered in the ®lter manifolds, and estimates of the transmembrane pressure based on the latter will be in error. The deposition of cake on the surface of the ®lter will reduce the ¯ow channel area, further contributing to the error in transmembrane pressure prediction based on the manifold values. This last factor will also limit the effectiveness of positioning the pressure transducers within the ®lter ¯ow channel, i.e. using the same diameter for both the pressure measurements and ®lter channel; as ®lter cake will not deposit on the transducers but will deposit on the surrounding membrane ®lter. Thus the ¯ow channel will again have a different cross-sectional area. Clearly, the signi®cance of this effect depends on the amount of deposit formed, but is important in thickening operations if complete blockage of the ¯ow channel is to be avoided. The suspension used in the test programme was talc dispersed in water. Talc was used as it is a fairly
incompressible material, which also tends to agglomerate. It was assumed that these properties would encourage the formation of a stable cake on the ®lter, resisting size segregation of the deposit under differing conditions of shear. 2. Analytical procedure The model for particle deposition is based on a semi-empirical approach to the prediction of the steady-state ¯ux when ®ltering a slurry chie¯y consisting of particles larger than one micron. This approach uses a force balance on a particle at the ®lter surface, which gives the relation that the ®ltration ¯ux should be linearly dependent on the shear stress at the ®ltration surface [4]. This surface could be either the surface of the ®lter medium or the surface of particles deposited on the ®lter medium. The model was demonstrated for Newtonian slurries made up of single size particles. In the case of a slurry consisting of particles with a range of sizes the relation between ¯ux (J) and shear stress ( w) is J k1 w k0 ;
(1)
where k1 and k0 are the empirical constants. The shear stress at the wall of a cylindrical tube of length L and diameter d is related to the pressure drop (P) between the inlet and outlet of the tube by w
Pd : 4L
(2)
So, if the pressure loss along the tube and the tube dimensions are known, then the wall shear stress can be calculated. In a cross¯ow ®lter, a layer of deposited particles forms at the ®ltration surface, so the diameter open to ¯ow is not known and must be determined. To calculate the diameter open to ¯ow requires the use of standard ¯uid mechanics equations for pressure drop with ¯ow rate. 2.1. Pressure drop calculation The non-Newtonian talc slurry used in this work was found to ®t a power law expression: w _ n ;
(3)
where _ is the shear rate, and n are the empirically
I.W. Cumming et al. / Journal of Membrane Science 154 (1999) 229±237
derived constants known, respectively, as the consistency coef®cient and the ¯ow index. Using this relation, the ¯ow of a power law ¯uid in a channel can be characterised by a modi®ed Reynolds number [6]: Re
8mv2ÿn dn ; 6 2=nn
(4)
where v is the average tube velocity and m is the slurry density. If the friction factor (f) is known, then the shear stress at the tube wall can be calculated using the Fanning equation: 2 f 4Q ; (5) w m 2 d2 where Q is the volumetric ¯ow in the tube. The pressure drop in the tube can then be calculated using Eq. (2). Methods for estimating the friction factor are available for both laminar and turbulent ¯ows. In the case of laminar ¯ow, the friction factor can be calculated from an analytical expression [6] as f
16 : Re
(6)
For turbulent ¯ow, the friction factor can be calculated from the empirical relation [6]: 1 4 0:4 p 0:75 log Re f 1ÿn=2 ÿ 1:2 : n f n
(7)
So the tube diameter can be calculated for both laminar and turbulent ¯ows from measured values of: volumetric ¯ow, slurry rheological properties, tube pressure drop and length. In the case of laminar ¯ow, this requires the use of Eqs. (3)±(6), whilst for turbulent ¯ow, Eqs. (3)±(5) and (7) are required. The
231
laminar case can be solved algebraically, but for turbulent ¯ow, an iterative calculation is necessary. This model does assume that the depth of deposit over the surface of the membrane is similar over all its surface. This has previously been shown to be a reasonable assumption, except close to the tube entrance, where the velocity pro®le is changing rapidly [3]. 2.2. Pressure measurement A complication in applying this technique to calculate the deposit depth is that the pressure is not normally measured at the ends of a membrane tube, but in a manifold beyond either end of the tube. In the case of the experiments where a single ®lter tube was used, the slurry will undergo a signi®cant acceleration on entering the tube and a corresponding deceleration on leaving the tube. Deposition of solids on the inside tube surface will further reduce the diameter, resulting in a further increase in velocity. There will be pressure changes associated with these velocity changes which can signi®cantly alter the pressure drop across the ®ltration surface (transmembrane pressure), and there will also be energy losses due to eddies formed on entry and exit. Fig. 1 shows the pressure variations that occur as the slurry enters and leaves a ®lter tube. Often it is assumed that the average transmembrane pressure (TMP) can be calculated as TMP
P1 P 4 ÿ Pp ; 2
(8)
where P1 and P4 are the module inlet and outlet
Fig. 1. Pressure changes occurring inside the filter module.
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pressures, respectively, and Pp is the permeate pressure at the outside of the ®lter tube. This approximation assumes that the pressure loss along the ®lter is linear with tube length, which is reasonable if the velocity pro®le develops rapidly, and the tube maintains a constant area open to ¯ow. The mean transmembrane pressure is obviously actually nearer the value calculated as TMP
P2 P3 ÿ Pp ; 2
(9)
where P2 and P3 are the ®lter tube inlet and outlet pressures. If the pressure loss on entering the ®lter tube is Pin and the pressure recovery on the leaving the ®lter tube is Prise, then in terms of the module inlet and outlet pressures, the transmembrane pressure becomes TMP
P1 P4 ÿ Pin ÿ Prise ÿ Pp : 2
(10)
This will give a lower value for the average transmembrane pressure than Eq. (8) and in this paper, it will be shown that not using Eq. (10) can lead to a signi®cant error, particularly at high velocities and pressure drops in tubes. In addition to the pressure change between the manifold and the ®lter channel due to ¯uid acceleration, and hence pressure reduction ± i.e. the Bernoulli effect, there will be energy losses due to the eddies formed on entry and exit from the ®lter channel. These energy losses will give rise to further pressure changes from those indicated by transducers mounted in the ®lter manifold. The estimation of inlet and outlet pressure changes are not well documented for nonNewtonian ¯ow. This is particularly true for the inlet loss as there is no information for the vena contracta. Fortunately, the inlet energy loss is less signi®cant as there tends to be little cake deposited at the actual ®lter tube inlet, so that the cake formed a funnel shape at the beginning of the ®lter tube which will reduce energy loss. In this work, it has been assumed that the inlet pressure loss is only due to Bernoulli effects and that frictional loss along the ®lter tube can be calculated as already described. The pressure rise on leaving the ®lter tube and passing into the header can be calculated for laminar ¯ow by the classical energy and momentum balance to give the following equation [7]:
Prise m
3n 1 2n 1
Q A1
2 " 2 # A1 A1 : ÿ A2 A2 (11)
Here A1 and A2 are the upstream and downstream cross-section areas, in this work, the ratio between A1 and A2 was varied from 0.022 to 0.31. The energy loss due to eddy formation as the slurry passes through the tube expansion can be similarly calculated in terms of pressure change as 2 2 " 3n 1 Q n3 A1 Peddy m 2n 1 A1 2 5n 3 A2 # A1 3 3n 1 ÿ : (12) A2 2 5n 3 These equations were also used for the case of turbulent ¯ow pressure losses. The velocity pro®le for laminar ¯ow of non-Newtonian power law ¯uids is ¯atter than for Newtonian ¯ow, so Eqs. (11) and (12) may be valid at higher Reynolds numbers. The goodness of ®t of this calculation procedure is discussed later. 3. Experimental A conventional cross¯ow loop was used for all the tests and is shown schematically in Fig. 2. The feed tank containing talc dispersed in water was stirred by a slowly rotating large paddle to ensure that the talc was kept well mixed and prevented from settling. The talc was pumped from the feed tank by a moving cavity Monopump. The pressure and ¯ows at the ®lter/tube module were controlled using the by-pass valve and the two in-line valves. The pressures in and out of the module were monitored by two stainless steel pressure transducers mounted a little upstream and downstream of the module, these were ¯ush mounted in the 20 mm nominal bore PVC pipe used to recirculate the slurry. To measure the slurry ¯ow a magnetic inductive ¯ow meter was ®tted in-line with the ®lter. The three in-line heat exchangers had the same internal diameter as the PVC pipes and with the use of cooling water, the slurry temperature was maintained at 2028C. The pressures and ¯ows were continuously logged by a microcomputer.
I.W. Cumming et al. / Journal of Membrane Science 154 (1999) 229±237
233
Fig. 2. Schematic diagram of crossflow microfiltration equipment.
Details of the ®lters used in the experiments are given in Table 1. Two stainless steel ®lters had internal diameters of 10 and 14 mm, with nominal pore sizes of 2 and 3 mm, respectively. Two ceramic ®lters were also used, these were supplied in the form of seven channel monoliths but in these tests only the central channel was used with the others being blocked at their ends. The ceramic channels had internal diameters of 4.3 and 5.8 mm with respective nominal pore sizes of 0.3 and 0.1 mm. The experiments to determine whether the internal diameter of a tube could be predicted from pressure and ¯ow measurements used different materials such as PVC, acrylic, glass and metal. A large number of different tubes were tested with internal diameters of 3.5, 3.7, 3.9, 4.2, 4.5, 5.2, 5.8, 6.2, 9.5 and 13 mm. The
lengths of these tubes were similar to that of the different ®lters used. The slurry used was a suspension of talc which varied only slightly in composition and properties throughout the experimental programme. The slurry composition varied from 26.3 to 28.3 wt%. The size distribution was routinely measured using laser diffraction, a Coulter LS 130. There was a slight reduction in the Sauter mean particle diameter from 8.9 to 8.3 mm throughout the programme of work. The slurry rheological properties were measured using a Haake RV2 viscometer, with a concentric cylinder measuring head ®tted. Each time a new series of experiments were undertaken, the slurry rheological properties were measured using a sample taken from the slurry feed tank. The shear stress/rate graph is shown in
Table 1 Dimensions of crossflow filters Dimensions
Internal diameter (mm) External diameter (mm) Total length (mm) Filtration length (mm) Pore size (mm) Number of tubes
Membrane type Metal tubes Stainless steel 316
Ceramic tubes
14 20 380 376 3 1 tube
5.8 20 380 340 0.1 7 tubes (with 6 closed in both cases)
10 13 298 260 2 1 tube
4.3 19 360 320 0.3 7 tubes
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Fig. 4. Comparison between measured and predicted pressure drop at various flow rates inside tubes of different diameters. Fig. 3. Rheogram of talc suspension used during the tests ± illustrating power-law fit.
Fig. 3, where it is plotted using logarithmic scales. This shows how well the measured data ®ts a power law expression. The ®ltration experiments were carried out recycling the talc slurry through the tubes until the permeate ¯ow became constant, it usually took between 15 and 70 min for a pseudo-steady-state ¯ux to be achieved. At this point, the data for the system pressures and ¯ow were measured as cake deposition had essentially ceased. 4. Results and discussion 4.1. Prediction of tube pressure drop and diameter The tests of the applicability of the calculation procedure for predicting the pressure drops for different diameter tubes were carried out over a wide range of conditions. The tube diameters varied from 3.5 to 13 mm, whilst the velocity in the tubes was between 1.9 and 11.1 m sÿ1 with Reynolds numbers in the range 1300±14 000. The value of n in the powerlaw expression varied from 0.548 to 0.601 in this set of experiments. The ¯ow is expected to vary from laminar up to a reasonable degree of turbulence. Using Eqs. (3)±(7), it was found that the pressure drop due to friction within the tube varied between 50% and 80% of the total pressure change measured by the transducers located in the main loop pipework. The pressure loss due to the formation of an eddy at the tube outlet was calculated using Eq. (11) to represent between
50% and 20% of the total pressure loss. Adding these two pressure losses together gives the pressure change between the pressure transducers in the experimental loop. A comparison between the measured and predicted pressure drop for four of the tubes tested, covering a wide range of diameters, is shown in Fig. 4. It appears from this ®gure that the prediction of pressure loss was best for the larger 13 mm pipe and poorest for the much smaller 3.9 mm pipe. All the experimental data are presented in Fig. 5, where the predicted pressure drop is plotted against the measured values. For the prediction to ®t the data, all the points would have to fall on the line shown in the graph. As can be seen at typical ®ltration pressure drops of less than 1 bar, the ®t is good. As the pressure drop increases, the ®t declines until at almost 2 bar, for the 3.5 mm diameter tube, the error in predicted pressure loss is up to 23%, however, most of the data had an error of between 0% and 15%. The reasons for
Fig. 5. Comparison between measured and predicted pressure drops for all tubes.
I.W. Cumming et al. / Journal of Membrane Science 154 (1999) 229±237
Fig. 6. Comparison between measured and predicted tube diameters for all tubes.
this overprediction of pressure drop may be due to errors in the exit loss equation, as the ¯ow becomes increasingly turbulent, the laminar exit loss equation is likely to become less valid. The prediction of tube diameter from slurry ¯ow and measured pressure drop was very good. This is shown in Fig. 6, where the measured diameter is plotted against the calculated diameter using the measured pressure drop and the ¯ow equations to infer the tube diameter. The agreement can be seen to be close over the complete range of tubes tested. The reason for the better ®t for diameter than pressure drop can be seen by examining Eqs. (5)±(7) and (11), which indicate that a small change in measured diameter will have a much larger effect on the calculated pressure drop. From these tests, it can be seen that the pressure drop in a cylindrical restriction can be predicted quite well particularly at lower pressure drops. The diameter can be predicted over the complete range of tubes tested. Thus it should be possible to accurately infer the deposit thickness from the pressure readings taken either side of the ®lter tube, having corrected for pressure changes due to the Bernoulli effect and eddies on expansion. 4.2. Mean transmembrane pressure As has been described, the mean transmembrane pressure has often been taken as the arithmetic mean of the module inlet and outlet pressures. This is not a very good approximation when there are signi®cant pressure changes associated with the acceleration and deceleration of the slurry as it enters and leaves a ®lter
235
Fig. 7. Effect of entrance and exit pressure changes on mean transmembrane pressure.
tube. The method used here is to estimate the ®lter tube outlet pressure by subtracting the pressure recovery term from the outlet transducer reading and estimating the ®lter tube inlet pressure by adding the calculated pressure drop in the tube to the ®lter tube outlet pressure. Fig. 7 shows how the mean transmembrane pressure calculated by this technique, Eq. (10), compares with the mean transmembrane pressure calculated by the simple averaging technique, Eq. (8), and is called the `super®cial average pressure'. These data have been generated for the tube diameters and ¯ows used in the experimental programme to test the non-Newtonian ¯ow calculations already described. The line on the graph indicates agreement between the two calculation procedures. At the highest mean transmembrane pressure, the error introduced by the simple averaging technique can be as great as 100%. This error does decrease as the cross¯ow velocity is reduced and the entrance and exit losses become less signi®cant. 4.3. Flux/shear stress correlation Experiments ®ltering at constant operating conditions of ¯ow showed that the ¯ux declined fairly rapidly with time until a pseudo-constant permeate ¯ow was achieved. An example of this is shown in Fig. 8, where talc was ®ltered using the 4.2 mm inside diameter ceramic ®lter, at a cross¯ow of 2.5 l minÿ1 and four different transmembrane pressures of 0.06, 0.15, 0.67 and 1.28 bar. The permeate ¯ow declined to an apparently constant value within 20 min for the lowest transmembrane pressure, whilst at the highest
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Fig. 8. Example of permeate decline during the experiments ± at constant crossflow rate.
pressure, the time to reach a steady ¯ow was about 70 min. All the experiments using talc slurry showed similar behaviour. In all these tests, the ®ltrate was clear from the start of ®ltration, this was believed to be due to initial rapid deposition of talc particles on to the ®lter at the beginning of each experiment so that the talc deposit gave good ®ltration as a form of precoat. This meant that the ®lter medium did not suffer from particle penetration and the performance of the process was mainly controlled by the talc slurry interaction. The experiments were carried out for the different ®lter media used at the range of conditions shown in Table 2. The minimum cross¯ow rate was set by the limitation that any lower ¯ow would result in complete blockage of the ®lter tubes by the talc deposit. The Reynolds number for all the experiments can be seen to be in the transition regime of turbulence. The method of calculating the deposit thickness was by using Eqs. (3)±(7) and (11), and solving for the diameter open to ¯ow using the measured data for ®lter inlet and outlet pressures for different cross¯ows. The calculated thickness of the cake deposit showed that for the larger diameter metal ®lters, the thickness
Fig. 9. Permeate flux rate as a function of wall shear stress for talc suspensions.
varied from 1.8 to 3.6 mm, whilst for the ceramic ®lters, the depth varied from 0.04 to 1.5 mm. Where the calculated deposit thickness was of the order of 100 microns and below, the cake may not be very many particle diameters thick. The ¯ux rates with shear are shown in Fig. 9, where the ®lter tube diameters remaining open to ¯ow were used in Eq. (2) together with the pressure drop within the ®lter module (P2 to P3 in Fig. 1), to give the wall shear stress. The least squares error straight line ®t through these data points gives an intercept quite close to the origin, and conforms reasonably well to Eq. (1). 5. Conclusions When ®ltering viscous slurries, the deposition of solids at the surface of the ®lter can be substantial. To test the shear stress model, Eq. (1), the diameter open to ¯ow in a ®lter tube was ®rstly deduced from measurement of the pressure loss as the slurry ¯ows through a tube. Using tubes of known internal diameter, it was demonstrated that, by applying standard non-Newtonian ¯ow theory, the internal diameter can
Table 2 Talc slurry flow regimes in filtration tests Filter
Mean velocity in clean tube (m sÿ1)
Mean velocity in fouled tube (m sÿ1)
Reynolds number in clean tube
Reynolds number in fouled tube
Metal 14 mm Metal 10 mm Ceramic 5.8 mm Ceramic 4.3 mm
1.2±3.3 1.2±3.2 2.2±5.4 2.7±9.2
2.1±5.8 1.8±5.5 3.8±9.2 4.8±16.2
685±2900 721±3500 1500±5500 2300±12 700
2500Ð6800 5200±11 900 2800Ð7200 3000±15 000
I.W. Cumming et al. / Journal of Membrane Science 154 (1999) 229±237
be predicted from measurement of the pressure at either end of the tube and the ¯ow rate through the tube. It has also been shown that using the pressure transducer measurements in the larger diameter manifolds at either end of a tube can lead to substantial inaccuracy in establishing the pressure drop across the ®ltration surface (the transmembrane pressure). The method to determine the open tube diameter, or cake thickness, was then employed to the ®lter tubes of diameters ranging from 14 to 4.3 mm, and for ¯ows of Reynolds numbers from 685 to 15 000. These values encompass those of practical use during micro®ltration of concentrated dispersions. For the pseudoplastic talc suspensions, the steady-state permeate ¯ux gave a reasonable correlation with the wall shear stress, in accordance with the shear stress model ± Eq. (1). 6. Nomenclature A1 A2 d f J k 0, k 1 L n P Q Re* TMP v
upstream area (m2) downstream area (m2) tube diameter (m) friction factor flux (m sÿ1) constants tube length (m) flow index pressure (N mÿ2) volumetric flow (m3 sÿ1) Reynolds number transmembrane filtration pressure average velocity (m sÿ1)
237
Greek letters
_ m w
shear rate (sÿ1) consistency coefficient suspension density (kg mÿ3) wall shear stress (N mÿ2)
Subscripts p
permeate
Acknowledgements The authors wish to thank Tech Sep for the provision of the ceramic ®lters used in this work. References [1] M.H. Lojkine, R.W. Field, J.A. Howell, Crossflow microfiltration of cell suspensions: A review of models with emphasis on particle size, Trans. I Chem. E, Part C 70 (1992) 149±164. [2] C. Smith, R.J. Wakeman, S. Tarleton, The relevance of fouling models to crossflow microfiltration, Proceedings of the Filtech 1991 Conference, Karlesruhe, Germany (Filtration Society, Horsham, UK), pp. 51±63. [3] M.R. Mackley, N.E. Sherman, Crossflow cake filtration mechanisms and kinetics, Chem. Eng. Sci. 47 (1992) 3067± 3084. [4] N.J. Blake, I.W. Cumming, M. Streat, Prediction of steady state crossflow filtration using a force balance model, J. Membr. Sci. 68 (1992) 205±216. [5] R.G. Holdich, I.W. Cumming, B. Ismail, The variation of crossflow filtration rate with wall shear stress and the effect of deposit thickness, Trans. I Chem. E, Part A 73 (1995) 20±25. [6] A.B. Metzner, J.C. Reed, Flow of non-Newtonian fluids ± correlation of the laminar, transition and turbulent flow regimes, AIChE J. 1 (1955) 434±440. [7] W.L. Wilkinson, Non-Newtonian Fluids, Pergamon Press, Oxford, 1960.