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A combined model for the prediction of the permeation flux during the cross-flow ultrafiltration of a whey suspension
Journal of Membrane Science 361 (2010) 71–77
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Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
A combined model for the prediction of the permeation flux during the cross-flow ultrafiltration of a whey suspension Kensuke Karasu a , Shiro Yoshikawa a,∗ , Shinichi Ookawara a,b , Kohei Ogawa a , Sandra E. Kentish c , Geoffrey W. Stevens c a
Department of Chemical Engineering, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8550, Japan Department of Energy Resources and Environmental Engineering, Egypt-Japan University of Science and Technology, MuCSAT, Postal Code 21934, New Borg El-Arab, Alexandria, Egypt c Department of Chemical and Biomolecular Engineering, School of Engineering, The University of Melbourne, Victoria 3010, Australia b
a r t i c l e
i n f o
Article history: Received 18 January 2010 Received in revised form 30 May 2010 Accepted 2 June 2010 Available online 9 June 2010 Keywords: Ultrafiltration Cross-flow Whey suspension Modeling
a b s t r a c t In the dairy industry, whey protein is concentrated by means of cross-flow ultrafiltration. In this membrane process, the thickness of the cake layer is assumed to be controlled by the shear stress exerted by the feed flow. In our prior work, we developed a model for the cross-flow ultrafiltration of a whey suspension that incorporated the effect of this shear stress. In this study, the model is extended to consider the permeation process as two periods determined by the dominant factor controlling the filtration resistance. In the first period, pore blockage dominates while in the latter, cake deposition is dominant. The models for each period were combined using the results of the model for the first period as the initial conditions for the second period. The combined model was compared with experimental results of crossflow ultrafiltration. Results show that the permeation process was well described under the conditions of this study by the combined model and the possibility of the application of the model to the cross-flow ultrafiltration system was indicated. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The ultrafiltration (UF) process has been utilized widely in the dairy industry. It is most often used for the separation and concentration of the protein in dairy whey solutions. In such processes, a cake layer is formed on the surface of membrane. Therefore, the processes are regarded as a form of cake filtration. A range of models has been developed to describe such processes. The resistance-inseries model sums all factors for the decline of permeate flux as resistance terms; membrane, adsorption, concentration polarization and fouling resistances [1]. Choi et al. [2] further subdivided the fouling resistance into those of reversible and irreversible fouling and experimentally determined the values of these resistances in the cross-flow UF of biological suspensions. In UF, it is considered that the concentration of solute in the liquid phase above the cake varies in the direction perpendicular to the membrane or the cake. The range where the concentration is distributed is regarded as a film. The resistance to permeation due to this distribution was defined in the concentration polarization model by Kimura and Sourirajan [3]. Hermans and Bredee [4] presented the resistance of pore blockage of the membrane by solid
∗ Corresponding author. Tel.: +81 3 5734 3278; fax: +81 3 5734 3278. E-mail address: [email protected] (S. Yoshikawa). 0376-7388/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2010.06.008
particles in suspension as the pore blockage model. They investigated the possible types of blockage such as intermediate, standard and complete pore blockage. The relationships between permeate flux and resistance are expressed as equations of the same form, but with different exponents on the permeate flux term for the different types of blockage mechanism. Iritani et al. [5,6] reported that in the UF of a whey suspension the protein cake was often compressible. In such a compressible cake layer, an increase of the filtration pressure or the permeate flux results in a more compacted cake. Iritani et al. [5] presented a method for evaluating the internal structure in the cake based on a compressible cake resistance model. This work involved the determination of the local specific flow resistance and the relation between the porosity of the cake and the compressive pressure on the solute phase based on the results of ultracentrifugation experiments. They revealed that a compressible cake resistance model can accurately describe the UF behavior of whey systems. Buscall and White [7] studied the interactions of a network of particles and the compressibility in flocculated suspensions, based on the balance between driving forces, drag forces and the stresses exerted on the network during sedimentation. They reported methods for the estimation of the compressive yield stress. The compressive yield stress model was further developed by Landman et al. [8,9]. In this model, the con-
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centration distribution and the resistance to the permeation in cake layer were considered based on the pressure distribution and the compressive yield stress. Landman and White [10] applied this model to optimize the UF operation in a dead-end mode. In our previous work [11], the use of cross-flow UF for the separation of proteins from a whey protein concentrate (WPC) suspension obtained in the manufacturing process of cheese was considered. The compressive yield stress model for dead-end UF was modified and applied to a cross-flow UF system of WPC suspension. In this case, it was assumed that the surface elements of the cake layer were continuously removed by the effect of the shear stress exerted by the feed flow. The rheology of the concentrated WPC suspension was measured in order to describe the behavior of the surface elements of the cake when a shear stress was applied. The removal rate of the cake was introduced into the model and the volume fraction of the surface element was determined based on the results of the rheology measurements and the shear stress exerted by feed flow. The resulting model was able to effectively represent experimental results in the later period of the process, which was regarded as a steady state. However, the initial period of the permeation process was not effectively represented. In an industrial process, it is important to estimate the time course of the permeate flux in the initial period more accurately in order to explain the entire permeation process of cross-flow UF. Models for the prediction of the permeation process have been developed by considering a single factor of the filtration resistance such as a cake filtration model or several factors of the filtration resistance which are coupled in series such as a resistance-in-series model [1]. Bolton et al. [12] proposed a combined model which considered several factors of the filtration resistance such as pore blocking, pore constriction and caking. They reported that several factors simultaneously affected on the permeation process. Yee et al. [13] proposed a unified model for the long-term cross-flow ultrafiltration of whey. They divided the permeation process into several periods and the dominant factor of the filtration resistance in each period was considered. In this model, the compressibility of the cake, the flow effect of the feed liquid and the connection of the models between periods were not considered enough. The purpose of this study is to further develop our previous model by considering the transition of the dominant factors controlling the filtration resistance from the initial period to the later period based on the permeation process and the behavior of solid particles and cake layer. In order to estimate the permeation process, the process is divided into two periods. The first period is based on the pore blockage resistance model. As the pore blockage is developed over time, it is assumed that the cake layer begins to form on the surface of the membrane and the permeation process is moved into the region where the resistance of the cake layer is dominant. Then the second period is based on the compressive yield stress model as developed in our previous work. The models for two periods are combined and a new model for the estimation of the permeation processes including both the first and the second periods are developed. The possibility of the application of the new model is evaluated by comparison to experimental results.
2. A combined permeation model In this study, the permeation process was divided into two periods according to the dominant factor controlling the filtration resistance. It is considered that the dominant factor in the initial period of the permeation process is pore blockage and that it is the compressible cake deposition in the second period. Detail of the models applied to each period and the estimation procedure using these two models are described below.
2.1. Pore blockage model In the initial period of the permeation process, protein aggregates in the suspension block or cover the pores. Consequently, the membrane area available for permeation is reduced. This pore blockage is the dominant factor controlling the filtration resistance in the first period. At the same time, the protein aggregates depositing on the blocked or covered pores begin to form a cake. The compressibility of this cake is assumed to be negligible because the cake is thin at this period. In such processes, Ho and Zydney [14] proposed the combined pore blockage and cake filtration model. In this model, it is considered that the volumetric permeate flow rate through the membrane can be expressed as the sum of the flow rates through the open and blocked pores: Q = Qopen + Qblocked
(1)
Qopen and Qblocked are derived as the following equations by Ho and Zydney:
Qopen = Q0 exp
− t
Qblocked = Q0 0
˛PCf Rm
t
˛PCf
Rm + Rp
(2)
exp
−
˛PCf Rm
t
dt
(3)
Therefore the permeate flow rate is derived by combining Eqs. (2) and (3). The equation of the permeate flow rate is assumed as a much simpler analytical solution by Ho and Zydney:
Q = Q0 exp
−
˛PCf Rm
t
+
Rm Rm + Rp
1 − exp
−
˛PCf Rm
t
(4)
where Rp means the resistance of the protein resistance and is given as a function of t by the following equation:
Rp = Rm
1+
2f R PCf 2 Rm
t − Rm
(5)
The first term on the right hand side of Eq. (4) is equivalent to the classical pore blockage model and gives a simple exponential decay in the volumetric flow rate. In the initial period, this term is assumed to be dominant in Eq. (4) to first approximation. Eq. (5) can be used to evaluate Rp in Eq. (4), and these equations give a simple approximate solution for the time course of the volumetric flow rate. The rate of pore blockage, ˛, and the rate of increase of the protein layer resistance with time, f R , are unknown parameters in Eqs. (4) and (5). 2.2. Compressive yield stress model 2.2.1. Details of the model As the pore blockage is developed over time, it is assumed that the cake layer begins to form on the surface of the membrane and the permeation process is moved into the region where the resistance of the cake layer is dominant. The model for the region is described below. The period where the dominant factor controlling the resistance is the cake deposition is defined as the second period of the model. In compressible cake filtration, Landman et al. [8] proposed a compressive yield stress model for dead-end UF, and we applied the model for cross-flow UF in our previous work [11]. A concentrated particle suspension, the volume fraction of which becomes larger than the gel point fraction, gel , forms networks and the suspension is transformed to a gel. The cake layer consists of the gel and shows a normal stress resisting compression. This stress is called the compressive yield stress, Py . Buscall and White [7] experimentally measured the relationship between volume fraction of the gel, , and Py . It is expressed by the following
K. Karasu et al. / Journal of Membrane Science 361 (2010) 71–77
equation:
Py () = p
gel
q
Py () = 0
−1
gel <
0 < ≤ gel
(6)
where p and q are constants. The values depend on the properties of the suspension. The value of gel determined for a WPC suspension in our study is 0.142 (0.184 weight fraction on dry WPC base). The compressive yield stress model developed in our previous work [11] is based on the pressure and concentration distribution in a compressible cake, the compressive yield stress and the rheology of the cake. When the cake layer shows compressibility ( > gel ), the volume fraction of the cake layer at the surface of the membrane (m ) varies from t to ∞ with time. Here, t is equal to or larger than gel , and Py (∞ ) is equal to P. When m = ∞ , the permeate flux is considered to be zero. Therefore, m cannot increase to a larger value than ∞ . Rather than using an incremental time step, we divide the volume fraction m from t to ∞ into n steps of size = (∞ − t )/n. Then the time course of the permeate flux and the distribution of the volume fraction in the cake layer are estimated for every step by the model. To estimate the volume fraction distribution and the height of the cake layer at the ith step, the range of the volume fraction from the top of the layer, t , to the membrane surface, mi , is divided into m stages of size ij = (mi − t )/m. The distribution of the volume fraction in the cake layer is also obtained based on the relation between the height of each stage and the volume fraction of each stage. Based on this distribution, the volume of the solid in the cake layer at the ith step is calculated by the integration of the volume fraction over the height of the cake layer. As a result, a material balance of the solid in the cake between the ith and (i + 1)th step is expressed by the following equation:
li
dz +
0
0 Vi − v · ti = 1 − 0
li+1
dz
(7)
0
where v is the volume of the solid removed from the cake layer by the feed liquid flow per unit membrane area and unit time. In Eq. (7), the first term on the left hand side is the solid volume in the cake at the ith step, the second term is the solid volume which is transferred to the cake layer by convection, the third term is the solid volume which is removed from the cake layer during the ith step and the right side is the solid volume in the cake at the (i + 1)th step. In cross-flow UF, the cake layer is predicted to be so thin compared with the height of the channel on the membrane that the thickness of the layer does not affect the shear stress on the cake layer. In addition, since the thickness of the concentration polarization layer is much thinner than the height of the channel on the membrane due to the small diffusion coefficient, it can also be assumed that the concentration polarization layer does not affect the shear stress on the cake layer and the shear rate and the shear stress are dominated by the effect of the feed flow which is assumed as laminar flow. It is, therefore, assumed that the shear stress on the cake layer is constant throughout the permeation process and that the removal rate of the cake layer, v, is also constant. In this study, the distribution of the volume fraction is expressed not as a continuous function. The solid volume in the cake is thus represented as a discrete summation shown in the following equation:
li
dz ≈
0
ij zij
(8)
j
ti is determined so as to satisfy Eq. (7). In addition, Vi is calculated by the following equation: Vi = ti ·
dV dt i
73
(9)
Fig. 1. Relationship between volume fraction and yield shear stress. Reproduced from [11].
In this manner, ti and Vi are calculated from the first step to the nth step. Consequently, the relation between filtration time, t, and filtrate volume, V, is calculated by the summation of ti and Vi .
2.2.2. Rheology of the cake layer When the compressive yield stress model is used, the volume fraction at the top surface of the cake layer, t , is required as a boundary condition. In our prior work, the rheology of a range of WPC suspensions [11] was evaluated with a rheometer (Physica MCR-300 Anton Paar GmbH, Austria). The results showed that dilute suspensions ( ≤ 0.15) could be regarded as a Newtonian fluid. On the other hand, more concentrated suspensions ( ≥ 0.4) typical of the concentration within the cake layer exhibited a yield shear stress and could be characterized as a plastic fluid. The relationship between the WPC concentration and the yield shear stress, 0 , is shown in Fig. 1. It is assumed that the surface of the cake layer can be removed by the shear stress even if the volume fraction is higher than gel when the shear stress is greater than the yield stress. On the other hand, the surface of the cake layer is immobile and is not removed by the shear stress when the shear stress is lower than the yield shear stress. Based on these considerations, the volume fraction at the top surface of the cake layer in cross-flow UF is predicted to be equal to the volume fraction at which the yield shear stress is the same as the shear stress exerted by the feed flow. This shear stress on the surface of the cake layer is calculated as the product of the viscosity of the feed liquid and the velocity gradient at the wall of the channel of the membrane module, measured by an electrochemical method [15]. The volume fraction at the top surface of the cake layer, t , is estimated from the shear stress by means of the relationship shown in Fig. 1.
3. Experiments The model outcomes should be compared with experimental results to confirm its validity. Although we conducted cross-flow UF experiments in our previous work [11], the measured permeate flux data were scattered especially in the initial periods of UF due to errors caused by the measurement methods. The experimental apparatus has been improved in order to carry out much more accurate measurements of the time course of the permeate flux, particularly in the initial period. The details of these experiments are given below.
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Fig. 3. Cross-sectional drawing of the flat membrane module.
times larger than that of the permeate liquid removed during the experiment. 3.4. Cross-flow membrane module
Fig. 2. Schematic diagram of the experimental apparatus.
Fig. 3 shows a cross-sectional image of the flat sheet membrane module used for the cross-flow experiments. The membrane holder was made of acrylic resin. The flow channel was rectangular in shape with length 60 mm, width 40 mm and height 7 mm. There was a supporting plate under the membrane which had 24 holes for permeate flow. Since permeation occurred only at the place of contact between the membrane and the holes, the effective membrane area was calculated as the summation of the area of the holes as 0.00068 m2 .
3.1. Materials 3.5. Experimental conditions Re-constituted whey protein concentrate suspension was used as the test liquid in this work. WPC powder, ALACENTM (NZMP Ltd.), was suspended into pure water. The volume fraction of all WPC suspensions was standardized to 0.1 (0.13 weight fraction of the dry WPC base). 3.2. Membrane A regenerated cellulose membrane (MILLIPORE) was used in all experiments with diameter 90 mm and molecular weight cutoff (MWCO) of 5000. As the smallest molecular weight among the protein components of WPC is ␣-lactalbumin at 14,200 g/mol, it was considered that all proteins were fully rejected by the membrane. 3.3. Cross-flow UF apparatus The cross-flow UF experiments were conducted using the apparatus shown in Fig. 2. The apparatus consisted of separate circuits for water and for the WPC suspension. Before the UF operation with the WPC suspension, water is fed to the membrane filter module and the permeate flux measured in order to determine the membrane resistance. After the measurement, three-way valves installed at each end of the module were switched over and the WPC suspension pumped from the suspension tank to the module through the suspension line. The time course of the permeate flow rate was measured with an ASL1600 Liquid Mass Flow Meter (CENSIRION AG, Switzerland). Because the feed flow passed through the membrane module continuously and permeate flow also passed through the permeate line continuously in this method, the permeate flux could be accurately measured in the initial period of the experiments. The temperature of the feed liquid was kept constant by circulating the coolant through a copper coil, which was installed in the reservoir. The feed flow rate and the trans-membrane pressure (TMP) were adjusted with the valves installed in the retentate line and the bypass. The concentration of the feed liquid was assumed to be constant because the volume of the feed liquid was about 100
It was considered that the shear stress was a function of the feed velocity across the surface of the membrane and that the volume fraction of the surface of the cake layer is related to the shear stress in the model. On the other hand, the cake compressibility is influenced by the TMP. In order to investigate the relationship between the operational conditions and the permeate flux, the feed velocity and the TMP were varied systematically, as shown in Table 1. 4. Results and discussion 4.1. Results of the experiments The relation between filtrate volume, V, and filtration time, t, was obtained as the results of the experiments. The results under some conditions are shown in Fig. 4. The abscissa is VJ0 and the ordinate is tJ0 /V. The relationship between them for cake filtration is expressed by the following equation based on Ruth’s filtration equation. The water flux of each membrane (J0 = P/Rm ) is introduced to remove the effect of membrane resistance: tJ0 = V
ac 2P
VJ0 + 1
(10)
From the results, it is clear that the value of tJ0 /V increases sharply at the beginning of the experiments. After a period, the Table 1 Experimental conditions. No.
TMP (kPa)
Feed velocity (cm/s)
Temperature (◦ C)
Time (h)
1 2 3 4 5 6 7 8 9
220 200 180 220 200 180 220 200 180
6.0 6.0 6.0 4.8 4.8 4.8 3.0 3.0 3.0
25 25 25 25 25 25 25 25 25
10 10 10 10 10 10 10 10 10
K. Karasu et al. / Journal of Membrane Science 361 (2010) 71–77
Fig. 4. Results of cross-flow UF experiments under some conditions.
Fig. 5. Results of cross-flow UF experiments in the initial period in Fig. 4.
Fig. 6. Results of comparison of estimation and experimental data.
gradient becomes more gradual. In this period, it is considered that the increase of the filtration resistance became more gradual due to the decrease of the permeate flux. When the permeate flux declined, amount of the deposition of the solid in the suspension decreased. Consequently, the growth rate of the cake layer declined. In the experiments, the reproducibility was broadly confirmed especially in the later period of the permeation process and the error between the experiments under the same conditions was approximately 20%. It is considered that the difference was caused by the properties of membrane filtration based on the differences of the formation process of the cake layer, the condition of the depositing aggregates and so on. Fig. 5 shows the relation between tJ0 /V and VJ0 in the initial period. The intercept of each result is approximately equal to unity and it appears that the permeate flux at t = 0 is almost equal to J0 . This result coincides with Eq. (10) at t = V = 0. From the comparison of the values of tJ0 /V in the region of the more gradual gradient, tJ0 /V increases as the TMP increases at a constant feed velocity and as the feed velocity decreases at constant
75
Fig. 7. The magnified drawing of the result of comparison of estimation and experimental data in the conditions of 220 kPa and 6.0 cm/s around the point where the models is combined.
Fig. 8. Relation between the pore blockage factor, ˛, determined in the estimation and the operating conditions.
Fig. 9. Relation between the cake growth factor, f R , determined in the estimation and the operating conditions.
TMP. Here, an increase of the value of tJ0 /V means a decrease of the specific permeate flux, J/J0 . 4.2. Comparison of the model with the experimental results To estimate the time course of the permeate flux, the models for the two periods described above were fitted to the data. In the initial period, the time course of the volumetric flow rate or the flux of the permeation liquid was fitted to Eqs. (4) and (5). In this calculation, the pore blockage parameter, ˛, and the rate of increase of the cake layer resistance, f R , were set as fitting parameters and the values of the parameters were determined simultaneously based on the multi-parameter regression. At the time when the model and experimental results diverged, the calculation of the model was changed from the first period to the second period, and the time was determined as the transition time from the first period to the second period. In this study, the transition time is assumed as the time when the difference of the gradient of tJ0 /V vs. VJ0 between
76
K. Karasu et al. / Journal of Membrane Science 361 (2010) 71–77
Fig. 10. Relation between the removal rate, v, determined in the estimation and the operating conditions.
the estimation result and the experimental result becomes 20%. The compressive yield stress model for the second period was calculated by using the permeate flux, the filtrate volume and the filtration time at the end of the first period as the initial condition of the second period. The membrane resistance in the second period was calculated from the permeate flux at the end of the first period. This resistance term includes the resistance from pore blockage and from the cake, which was deposited during the first period. The removal rate of the surface elements of the cake, v, was determined by trial and error until the simulated t/V data best fitted the experimental values. The results of the comparison between the estimation and experimental data are shown in Fig. 6. The errors of values between the estimation results and the experimental results were smaller than 5% under all the operating conditions. Therefore, it is considered that the model agrees well with the experimental results during the whole period of the permeation process for all conditions. Fig. 7 shows a magnified drawing of Fig. 6 under the conditions of 220 kPa and 6.0 cm/s around the point where the models are combined. The estimation result based on the models is a smooth curve and agrees well with the experimental result in the whole region as shown in Fig. 7. Therefore, it is considered that the estimation based on the models described above is suitable for explaining the whole permeation process, including the very initial period, of the cross-flow UF of WPC suspension. The relations between fitting parameters, ˛ and f R for the first period and v for the second period, and the operating conditions, TMP and feed velocity, are shown in Figs. 8–10 respectively. The pore blockage factor, ˛, which is equal to the membrane area blocked per unit mass of protein aggregates transferred to the membrane surface, seems to decrease with an increase of TMP or a decrease of feed velocity. The cake growth factor, f R , which is equal to the rate of increase of the protein layer resistance, seems to be hardly influenced by the operating conditions. The removal rate, v, which means the volume of the cake removed by the feed flow per unit membrane area and unit time, has a slight tendency to decrease with an increase of TMP and tends to slightly increase with an increase of feed velocity. The validity of the model based on the physics and numerical analysis on these relations should be discussed in detail in the next stage. 5. Conclusion In this study, the permeation of a WPC suspension in crossflow ultrafiltration was modeled by dividing the process into two periods according to the dominant factor controlling the filtration resistance. The first factor was a pore blockage, and the second was the deposition of compressible cake. We applied a pore blockage model for the first period and a compressive yield stress model for the second period. In the estimation, the models were fitted by
optimization of the pore blockage factor, ˛, and the cake growth factor, f R in the first period and the removal rate, v, in the second period. The two models were combined by using the values at the end of the first period as the initial conditions of the second period. The estimated results based on the models were compared to the experimental results. Since the errors of values between the estimation results and the experimental results were smaller than 5%, we considered that the models agreed well with the experimental results across the entire cross-flow ultrafiltration process. Based on the results of the simulation, relations between the parameters, ˛, f R and v, and the operating conditions, trans-membrane pressure and feed velocity, were presented. The validity of the model based on the physics and numerical analysis on the relations should be discussed in detail in the next stage. Based on the results presented in this study, it was concluded that the combined use of both models was able to explain the entire permeation process.
Nomenclature a A0 Ablocked Aopen c Cf f J0 Jblocked l mp p P Pf Ps Py q Q Q0 Qblocked Qopen r R Rm Rp t uf
v V Vp z
hydraulic resistance of compressible cake per unit mass (m/kg) total membrane area (m2 ) membrane area blocked by protein aggregates (m2 ) uncovered membrane area (m2 ) mass of cake per unit filtrate volume (kg/m3 ) protein concentration of feed liquid (kg/m3 ) fraction of the proteins that contribute to the growth of the cake permeate flux of pure water (m/s) permeate flux through the blocked pores (m/s) total height of cake layer (m) mass of protein cake per unit membrane area (kg/m2 ) constant number of compressive yield stress (Pa) trans-membrane pressure (Pa) pressure on the liquid phase in cake layer (Pa) pressure on the solid phase in cake layer (Pa) compressive yield stress (Pa) constant number of compressive yield stress volumetric permeate flow rate (m3 /s) initial volumetric permeate flow rate through the clean membrane (m3 /s) volumetric permeate flow rate through the blocked pores (m3 /s) volumetric permeate flow rate through the open pores (m3 /s) hindered settling factor specific resistance of cake layer (m/kg) membrane resistance (m−1 ) resistance of protein deposition (m−1 ) filtration time (s) feed velocity on membrane (cm/s) removal rate of cake layer (m/s) volume of permeate liquid per unit membrane area (m) volume of solid particle (m3 ) height of cake layer in a stage (m)
Greek letters ˛ pore blockage factor (m2 /kg) volume fraction volume fraction of feed liquid 0 maximum volume fraction at the surface of mem∞ brane
K. Karasu et al. / Journal of Membrane Science 361 (2010) 71–77
gel m t 0
volume fraction at gel point volume fraction at the surface of membrane volume fraction at the top surface of cake layer viscosity of permeate liquid (Pa s) stokes drag coefficient (Pa s m) yield shear stress (Pa)
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Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
A combined model for the prediction of the permeation flux during the cross-flow ultrafiltration of a whey suspension Kensuke Karasu a , Shiro Yoshikawa a,∗ , Shinichi Ookawara a,b , Kohei Ogawa a , Sandra E. Kentish c , Geoffrey W. Stevens c a
Department of Chemical Engineering, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8550, Japan Department of Energy Resources and Environmental Engineering, Egypt-Japan University of Science and Technology, MuCSAT, Postal Code 21934, New Borg El-Arab, Alexandria, Egypt c Department of Chemical and Biomolecular Engineering, School of Engineering, The University of Melbourne, Victoria 3010, Australia b
a r t i c l e
i n f o
Article history: Received 18 January 2010 Received in revised form 30 May 2010 Accepted 2 June 2010 Available online 9 June 2010 Keywords: Ultrafiltration Cross-flow Whey suspension Modeling
a b s t r a c t In the dairy industry, whey protein is concentrated by means of cross-flow ultrafiltration. In this membrane process, the thickness of the cake layer is assumed to be controlled by the shear stress exerted by the feed flow. In our prior work, we developed a model for the cross-flow ultrafiltration of a whey suspension that incorporated the effect of this shear stress. In this study, the model is extended to consider the permeation process as two periods determined by the dominant factor controlling the filtration resistance. In the first period, pore blockage dominates while in the latter, cake deposition is dominant. The models for each period were combined using the results of the model for the first period as the initial conditions for the second period. The combined model was compared with experimental results of crossflow ultrafiltration. Results show that the permeation process was well described under the conditions of this study by the combined model and the possibility of the application of the model to the cross-flow ultrafiltration system was indicated. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The ultrafiltration (UF) process has been utilized widely in the dairy industry. It is most often used for the separation and concentration of the protein in dairy whey solutions. In such processes, a cake layer is formed on the surface of membrane. Therefore, the processes are regarded as a form of cake filtration. A range of models has been developed to describe such processes. The resistance-inseries model sums all factors for the decline of permeate flux as resistance terms; membrane, adsorption, concentration polarization and fouling resistances [1]. Choi et al. [2] further subdivided the fouling resistance into those of reversible and irreversible fouling and experimentally determined the values of these resistances in the cross-flow UF of biological suspensions. In UF, it is considered that the concentration of solute in the liquid phase above the cake varies in the direction perpendicular to the membrane or the cake. The range where the concentration is distributed is regarded as a film. The resistance to permeation due to this distribution was defined in the concentration polarization model by Kimura and Sourirajan [3]. Hermans and Bredee [4] presented the resistance of pore blockage of the membrane by solid
∗ Corresponding author. Tel.: +81 3 5734 3278; fax: +81 3 5734 3278. E-mail address: [email protected] (S. Yoshikawa). 0376-7388/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2010.06.008
particles in suspension as the pore blockage model. They investigated the possible types of blockage such as intermediate, standard and complete pore blockage. The relationships between permeate flux and resistance are expressed as equations of the same form, but with different exponents on the permeate flux term for the different types of blockage mechanism. Iritani et al. [5,6] reported that in the UF of a whey suspension the protein cake was often compressible. In such a compressible cake layer, an increase of the filtration pressure or the permeate flux results in a more compacted cake. Iritani et al. [5] presented a method for evaluating the internal structure in the cake based on a compressible cake resistance model. This work involved the determination of the local specific flow resistance and the relation between the porosity of the cake and the compressive pressure on the solute phase based on the results of ultracentrifugation experiments. They revealed that a compressible cake resistance model can accurately describe the UF behavior of whey systems. Buscall and White [7] studied the interactions of a network of particles and the compressibility in flocculated suspensions, based on the balance between driving forces, drag forces and the stresses exerted on the network during sedimentation. They reported methods for the estimation of the compressive yield stress. The compressive yield stress model was further developed by Landman et al. [8,9]. In this model, the con-
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centration distribution and the resistance to the permeation in cake layer were considered based on the pressure distribution and the compressive yield stress. Landman and White [10] applied this model to optimize the UF operation in a dead-end mode. In our previous work [11], the use of cross-flow UF for the separation of proteins from a whey protein concentrate (WPC) suspension obtained in the manufacturing process of cheese was considered. The compressive yield stress model for dead-end UF was modified and applied to a cross-flow UF system of WPC suspension. In this case, it was assumed that the surface elements of the cake layer were continuously removed by the effect of the shear stress exerted by the feed flow. The rheology of the concentrated WPC suspension was measured in order to describe the behavior of the surface elements of the cake when a shear stress was applied. The removal rate of the cake was introduced into the model and the volume fraction of the surface element was determined based on the results of the rheology measurements and the shear stress exerted by feed flow. The resulting model was able to effectively represent experimental results in the later period of the process, which was regarded as a steady state. However, the initial period of the permeation process was not effectively represented. In an industrial process, it is important to estimate the time course of the permeate flux in the initial period more accurately in order to explain the entire permeation process of cross-flow UF. Models for the prediction of the permeation process have been developed by considering a single factor of the filtration resistance such as a cake filtration model or several factors of the filtration resistance which are coupled in series such as a resistance-in-series model [1]. Bolton et al. [12] proposed a combined model which considered several factors of the filtration resistance such as pore blocking, pore constriction and caking. They reported that several factors simultaneously affected on the permeation process. Yee et al. [13] proposed a unified model for the long-term cross-flow ultrafiltration of whey. They divided the permeation process into several periods and the dominant factor of the filtration resistance in each period was considered. In this model, the compressibility of the cake, the flow effect of the feed liquid and the connection of the models between periods were not considered enough. The purpose of this study is to further develop our previous model by considering the transition of the dominant factors controlling the filtration resistance from the initial period to the later period based on the permeation process and the behavior of solid particles and cake layer. In order to estimate the permeation process, the process is divided into two periods. The first period is based on the pore blockage resistance model. As the pore blockage is developed over time, it is assumed that the cake layer begins to form on the surface of the membrane and the permeation process is moved into the region where the resistance of the cake layer is dominant. Then the second period is based on the compressive yield stress model as developed in our previous work. The models for two periods are combined and a new model for the estimation of the permeation processes including both the first and the second periods are developed. The possibility of the application of the new model is evaluated by comparison to experimental results.
2. A combined permeation model In this study, the permeation process was divided into two periods according to the dominant factor controlling the filtration resistance. It is considered that the dominant factor in the initial period of the permeation process is pore blockage and that it is the compressible cake deposition in the second period. Detail of the models applied to each period and the estimation procedure using these two models are described below.
2.1. Pore blockage model In the initial period of the permeation process, protein aggregates in the suspension block or cover the pores. Consequently, the membrane area available for permeation is reduced. This pore blockage is the dominant factor controlling the filtration resistance in the first period. At the same time, the protein aggregates depositing on the blocked or covered pores begin to form a cake. The compressibility of this cake is assumed to be negligible because the cake is thin at this period. In such processes, Ho and Zydney [14] proposed the combined pore blockage and cake filtration model. In this model, it is considered that the volumetric permeate flow rate through the membrane can be expressed as the sum of the flow rates through the open and blocked pores: Q = Qopen + Qblocked
(1)
Qopen and Qblocked are derived as the following equations by Ho and Zydney:
Qopen = Q0 exp
− t
Qblocked = Q0 0
˛PCf Rm
t
˛PCf
Rm + Rp
(2)
exp
−
˛PCf Rm
t
dt
(3)
Therefore the permeate flow rate is derived by combining Eqs. (2) and (3). The equation of the permeate flow rate is assumed as a much simpler analytical solution by Ho and Zydney:
Q = Q0 exp
−
˛PCf Rm
t
+
Rm Rm + Rp
1 − exp
−
˛PCf Rm
t
(4)
where Rp means the resistance of the protein resistance and is given as a function of t by the following equation:
Rp = Rm
1+
2f R PCf 2 Rm
t − Rm
(5)
The first term on the right hand side of Eq. (4) is equivalent to the classical pore blockage model and gives a simple exponential decay in the volumetric flow rate. In the initial period, this term is assumed to be dominant in Eq. (4) to first approximation. Eq. (5) can be used to evaluate Rp in Eq. (4), and these equations give a simple approximate solution for the time course of the volumetric flow rate. The rate of pore blockage, ˛, and the rate of increase of the protein layer resistance with time, f R , are unknown parameters in Eqs. (4) and (5). 2.2. Compressive yield stress model 2.2.1. Details of the model As the pore blockage is developed over time, it is assumed that the cake layer begins to form on the surface of the membrane and the permeation process is moved into the region where the resistance of the cake layer is dominant. The model for the region is described below. The period where the dominant factor controlling the resistance is the cake deposition is defined as the second period of the model. In compressible cake filtration, Landman et al. [8] proposed a compressive yield stress model for dead-end UF, and we applied the model for cross-flow UF in our previous work [11]. A concentrated particle suspension, the volume fraction of which becomes larger than the gel point fraction, gel , forms networks and the suspension is transformed to a gel. The cake layer consists of the gel and shows a normal stress resisting compression. This stress is called the compressive yield stress, Py . Buscall and White [7] experimentally measured the relationship between volume fraction of the gel, , and Py . It is expressed by the following
K. Karasu et al. / Journal of Membrane Science 361 (2010) 71–77
equation:
Py () = p
gel
q
Py () = 0
−1
gel <
0 < ≤ gel
(6)
where p and q are constants. The values depend on the properties of the suspension. The value of gel determined for a WPC suspension in our study is 0.142 (0.184 weight fraction on dry WPC base). The compressive yield stress model developed in our previous work [11] is based on the pressure and concentration distribution in a compressible cake, the compressive yield stress and the rheology of the cake. When the cake layer shows compressibility ( > gel ), the volume fraction of the cake layer at the surface of the membrane (m ) varies from t to ∞ with time. Here, t is equal to or larger than gel , and Py (∞ ) is equal to P. When m = ∞ , the permeate flux is considered to be zero. Therefore, m cannot increase to a larger value than ∞ . Rather than using an incremental time step, we divide the volume fraction m from t to ∞ into n steps of size = (∞ − t )/n. Then the time course of the permeate flux and the distribution of the volume fraction in the cake layer are estimated for every step by the model. To estimate the volume fraction distribution and the height of the cake layer at the ith step, the range of the volume fraction from the top of the layer, t , to the membrane surface, mi , is divided into m stages of size ij = (mi − t )/m. The distribution of the volume fraction in the cake layer is also obtained based on the relation between the height of each stage and the volume fraction of each stage. Based on this distribution, the volume of the solid in the cake layer at the ith step is calculated by the integration of the volume fraction over the height of the cake layer. As a result, a material balance of the solid in the cake between the ith and (i + 1)th step is expressed by the following equation:
li
dz +
0
0 Vi − v · ti = 1 − 0
li+1
dz
(7)
0
where v is the volume of the solid removed from the cake layer by the feed liquid flow per unit membrane area and unit time. In Eq. (7), the first term on the left hand side is the solid volume in the cake at the ith step, the second term is the solid volume which is transferred to the cake layer by convection, the third term is the solid volume which is removed from the cake layer during the ith step and the right side is the solid volume in the cake at the (i + 1)th step. In cross-flow UF, the cake layer is predicted to be so thin compared with the height of the channel on the membrane that the thickness of the layer does not affect the shear stress on the cake layer. In addition, since the thickness of the concentration polarization layer is much thinner than the height of the channel on the membrane due to the small diffusion coefficient, it can also be assumed that the concentration polarization layer does not affect the shear stress on the cake layer and the shear rate and the shear stress are dominated by the effect of the feed flow which is assumed as laminar flow. It is, therefore, assumed that the shear stress on the cake layer is constant throughout the permeation process and that the removal rate of the cake layer, v, is also constant. In this study, the distribution of the volume fraction is expressed not as a continuous function. The solid volume in the cake is thus represented as a discrete summation shown in the following equation:
li
dz ≈
0
ij zij
(8)
j
ti is determined so as to satisfy Eq. (7). In addition, Vi is calculated by the following equation: Vi = ti ·
dV dt i
73
(9)
Fig. 1. Relationship between volume fraction and yield shear stress. Reproduced from [11].
In this manner, ti and Vi are calculated from the first step to the nth step. Consequently, the relation between filtration time, t, and filtrate volume, V, is calculated by the summation of ti and Vi .
2.2.2. Rheology of the cake layer When the compressive yield stress model is used, the volume fraction at the top surface of the cake layer, t , is required as a boundary condition. In our prior work, the rheology of a range of WPC suspensions [11] was evaluated with a rheometer (Physica MCR-300 Anton Paar GmbH, Austria). The results showed that dilute suspensions ( ≤ 0.15) could be regarded as a Newtonian fluid. On the other hand, more concentrated suspensions ( ≥ 0.4) typical of the concentration within the cake layer exhibited a yield shear stress and could be characterized as a plastic fluid. The relationship between the WPC concentration and the yield shear stress, 0 , is shown in Fig. 1. It is assumed that the surface of the cake layer can be removed by the shear stress even if the volume fraction is higher than gel when the shear stress is greater than the yield stress. On the other hand, the surface of the cake layer is immobile and is not removed by the shear stress when the shear stress is lower than the yield shear stress. Based on these considerations, the volume fraction at the top surface of the cake layer in cross-flow UF is predicted to be equal to the volume fraction at which the yield shear stress is the same as the shear stress exerted by the feed flow. This shear stress on the surface of the cake layer is calculated as the product of the viscosity of the feed liquid and the velocity gradient at the wall of the channel of the membrane module, measured by an electrochemical method [15]. The volume fraction at the top surface of the cake layer, t , is estimated from the shear stress by means of the relationship shown in Fig. 1.
3. Experiments The model outcomes should be compared with experimental results to confirm its validity. Although we conducted cross-flow UF experiments in our previous work [11], the measured permeate flux data were scattered especially in the initial periods of UF due to errors caused by the measurement methods. The experimental apparatus has been improved in order to carry out much more accurate measurements of the time course of the permeate flux, particularly in the initial period. The details of these experiments are given below.
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Fig. 3. Cross-sectional drawing of the flat membrane module.
times larger than that of the permeate liquid removed during the experiment. 3.4. Cross-flow membrane module
Fig. 2. Schematic diagram of the experimental apparatus.
Fig. 3 shows a cross-sectional image of the flat sheet membrane module used for the cross-flow experiments. The membrane holder was made of acrylic resin. The flow channel was rectangular in shape with length 60 mm, width 40 mm and height 7 mm. There was a supporting plate under the membrane which had 24 holes for permeate flow. Since permeation occurred only at the place of contact between the membrane and the holes, the effective membrane area was calculated as the summation of the area of the holes as 0.00068 m2 .
3.1. Materials 3.5. Experimental conditions Re-constituted whey protein concentrate suspension was used as the test liquid in this work. WPC powder, ALACENTM (NZMP Ltd.), was suspended into pure water. The volume fraction of all WPC suspensions was standardized to 0.1 (0.13 weight fraction of the dry WPC base). 3.2. Membrane A regenerated cellulose membrane (MILLIPORE) was used in all experiments with diameter 90 mm and molecular weight cutoff (MWCO) of 5000. As the smallest molecular weight among the protein components of WPC is ␣-lactalbumin at 14,200 g/mol, it was considered that all proteins were fully rejected by the membrane. 3.3. Cross-flow UF apparatus The cross-flow UF experiments were conducted using the apparatus shown in Fig. 2. The apparatus consisted of separate circuits for water and for the WPC suspension. Before the UF operation with the WPC suspension, water is fed to the membrane filter module and the permeate flux measured in order to determine the membrane resistance. After the measurement, three-way valves installed at each end of the module were switched over and the WPC suspension pumped from the suspension tank to the module through the suspension line. The time course of the permeate flow rate was measured with an ASL1600 Liquid Mass Flow Meter (CENSIRION AG, Switzerland). Because the feed flow passed through the membrane module continuously and permeate flow also passed through the permeate line continuously in this method, the permeate flux could be accurately measured in the initial period of the experiments. The temperature of the feed liquid was kept constant by circulating the coolant through a copper coil, which was installed in the reservoir. The feed flow rate and the trans-membrane pressure (TMP) were adjusted with the valves installed in the retentate line and the bypass. The concentration of the feed liquid was assumed to be constant because the volume of the feed liquid was about 100
It was considered that the shear stress was a function of the feed velocity across the surface of the membrane and that the volume fraction of the surface of the cake layer is related to the shear stress in the model. On the other hand, the cake compressibility is influenced by the TMP. In order to investigate the relationship between the operational conditions and the permeate flux, the feed velocity and the TMP were varied systematically, as shown in Table 1. 4. Results and discussion 4.1. Results of the experiments The relation between filtrate volume, V, and filtration time, t, was obtained as the results of the experiments. The results under some conditions are shown in Fig. 4. The abscissa is VJ0 and the ordinate is tJ0 /V. The relationship between them for cake filtration is expressed by the following equation based on Ruth’s filtration equation. The water flux of each membrane (J0 = P/Rm ) is introduced to remove the effect of membrane resistance: tJ0 = V
ac 2P
VJ0 + 1
(10)
From the results, it is clear that the value of tJ0 /V increases sharply at the beginning of the experiments. After a period, the Table 1 Experimental conditions. No.
TMP (kPa)
Feed velocity (cm/s)
Temperature (◦ C)
Time (h)
1 2 3 4 5 6 7 8 9
220 200 180 220 200 180 220 200 180
6.0 6.0 6.0 4.8 4.8 4.8 3.0 3.0 3.0
25 25 25 25 25 25 25 25 25
10 10 10 10 10 10 10 10 10
K. Karasu et al. / Journal of Membrane Science 361 (2010) 71–77
Fig. 4. Results of cross-flow UF experiments under some conditions.
Fig. 5. Results of cross-flow UF experiments in the initial period in Fig. 4.
Fig. 6. Results of comparison of estimation and experimental data.
gradient becomes more gradual. In this period, it is considered that the increase of the filtration resistance became more gradual due to the decrease of the permeate flux. When the permeate flux declined, amount of the deposition of the solid in the suspension decreased. Consequently, the growth rate of the cake layer declined. In the experiments, the reproducibility was broadly confirmed especially in the later period of the permeation process and the error between the experiments under the same conditions was approximately 20%. It is considered that the difference was caused by the properties of membrane filtration based on the differences of the formation process of the cake layer, the condition of the depositing aggregates and so on. Fig. 5 shows the relation between tJ0 /V and VJ0 in the initial period. The intercept of each result is approximately equal to unity and it appears that the permeate flux at t = 0 is almost equal to J0 . This result coincides with Eq. (10) at t = V = 0. From the comparison of the values of tJ0 /V in the region of the more gradual gradient, tJ0 /V increases as the TMP increases at a constant feed velocity and as the feed velocity decreases at constant
75
Fig. 7. The magnified drawing of the result of comparison of estimation and experimental data in the conditions of 220 kPa and 6.0 cm/s around the point where the models is combined.
Fig. 8. Relation between the pore blockage factor, ˛, determined in the estimation and the operating conditions.
Fig. 9. Relation between the cake growth factor, f R , determined in the estimation and the operating conditions.
TMP. Here, an increase of the value of tJ0 /V means a decrease of the specific permeate flux, J/J0 . 4.2. Comparison of the model with the experimental results To estimate the time course of the permeate flux, the models for the two periods described above were fitted to the data. In the initial period, the time course of the volumetric flow rate or the flux of the permeation liquid was fitted to Eqs. (4) and (5). In this calculation, the pore blockage parameter, ˛, and the rate of increase of the cake layer resistance, f R , were set as fitting parameters and the values of the parameters were determined simultaneously based on the multi-parameter regression. At the time when the model and experimental results diverged, the calculation of the model was changed from the first period to the second period, and the time was determined as the transition time from the first period to the second period. In this study, the transition time is assumed as the time when the difference of the gradient of tJ0 /V vs. VJ0 between
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K. Karasu et al. / Journal of Membrane Science 361 (2010) 71–77
Fig. 10. Relation between the removal rate, v, determined in the estimation and the operating conditions.
the estimation result and the experimental result becomes 20%. The compressive yield stress model for the second period was calculated by using the permeate flux, the filtrate volume and the filtration time at the end of the first period as the initial condition of the second period. The membrane resistance in the second period was calculated from the permeate flux at the end of the first period. This resistance term includes the resistance from pore blockage and from the cake, which was deposited during the first period. The removal rate of the surface elements of the cake, v, was determined by trial and error until the simulated t/V data best fitted the experimental values. The results of the comparison between the estimation and experimental data are shown in Fig. 6. The errors of values between the estimation results and the experimental results were smaller than 5% under all the operating conditions. Therefore, it is considered that the model agrees well with the experimental results during the whole period of the permeation process for all conditions. Fig. 7 shows a magnified drawing of Fig. 6 under the conditions of 220 kPa and 6.0 cm/s around the point where the models are combined. The estimation result based on the models is a smooth curve and agrees well with the experimental result in the whole region as shown in Fig. 7. Therefore, it is considered that the estimation based on the models described above is suitable for explaining the whole permeation process, including the very initial period, of the cross-flow UF of WPC suspension. The relations between fitting parameters, ˛ and f R for the first period and v for the second period, and the operating conditions, TMP and feed velocity, are shown in Figs. 8–10 respectively. The pore blockage factor, ˛, which is equal to the membrane area blocked per unit mass of protein aggregates transferred to the membrane surface, seems to decrease with an increase of TMP or a decrease of feed velocity. The cake growth factor, f R , which is equal to the rate of increase of the protein layer resistance, seems to be hardly influenced by the operating conditions. The removal rate, v, which means the volume of the cake removed by the feed flow per unit membrane area and unit time, has a slight tendency to decrease with an increase of TMP and tends to slightly increase with an increase of feed velocity. The validity of the model based on the physics and numerical analysis on these relations should be discussed in detail in the next stage. 5. Conclusion In this study, the permeation of a WPC suspension in crossflow ultrafiltration was modeled by dividing the process into two periods according to the dominant factor controlling the filtration resistance. The first factor was a pore blockage, and the second was the deposition of compressible cake. We applied a pore blockage model for the first period and a compressive yield stress model for the second period. In the estimation, the models were fitted by
optimization of the pore blockage factor, ˛, and the cake growth factor, f R in the first period and the removal rate, v, in the second period. The two models were combined by using the values at the end of the first period as the initial conditions of the second period. The estimated results based on the models were compared to the experimental results. Since the errors of values between the estimation results and the experimental results were smaller than 5%, we considered that the models agreed well with the experimental results across the entire cross-flow ultrafiltration process. Based on the results of the simulation, relations between the parameters, ˛, f R and v, and the operating conditions, trans-membrane pressure and feed velocity, were presented. The validity of the model based on the physics and numerical analysis on the relations should be discussed in detail in the next stage. Based on the results presented in this study, it was concluded that the combined use of both models was able to explain the entire permeation process.
Nomenclature a A0 Ablocked Aopen c Cf f J0 Jblocked l mp p P Pf Ps Py q Q Q0 Qblocked Qopen r R Rm Rp t uf
v V Vp z
hydraulic resistance of compressible cake per unit mass (m/kg) total membrane area (m2 ) membrane area blocked by protein aggregates (m2 ) uncovered membrane area (m2 ) mass of cake per unit filtrate volume (kg/m3 ) protein concentration of feed liquid (kg/m3 ) fraction of the proteins that contribute to the growth of the cake permeate flux of pure water (m/s) permeate flux through the blocked pores (m/s) total height of cake layer (m) mass of protein cake per unit membrane area (kg/m2 ) constant number of compressive yield stress (Pa) trans-membrane pressure (Pa) pressure on the liquid phase in cake layer (Pa) pressure on the solid phase in cake layer (Pa) compressive yield stress (Pa) constant number of compressive yield stress volumetric permeate flow rate (m3 /s) initial volumetric permeate flow rate through the clean membrane (m3 /s) volumetric permeate flow rate through the blocked pores (m3 /s) volumetric permeate flow rate through the open pores (m3 /s) hindered settling factor specific resistance of cake layer (m/kg) membrane resistance (m−1 ) resistance of protein deposition (m−1 ) filtration time (s) feed velocity on membrane (cm/s) removal rate of cake layer (m/s) volume of permeate liquid per unit membrane area (m) volume of solid particle (m3 ) height of cake layer in a stage (m)
Greek letters ˛ pore blockage factor (m2 /kg) volume fraction volume fraction of feed liquid 0 maximum volume fraction at the surface of mem∞ brane
K. Karasu et al. / Journal of Membrane Science 361 (2010) 71–77
gel m t 0
volume fraction at gel point volume fraction at the surface of membrane volume fraction at the top surface of cake layer viscosity of permeate liquid (Pa s) stokes drag coefficient (Pa s m) yield shear stress (Pa)
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