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Modelling and simulation of membrane fouling in batch ultrafiltration on pilot plant
Pergamon PII:
Computers them. Engng Vol. 22, Suppl., pp. S901-S904, 1998 0 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0098-1354/98 $19.00 + 0.00 SOO98-1354(98)00176-8
Modelling and simulation of membrane fouling in batch ultrafiltration on pilot plant F. Rouvetl, K. Fiaty”, I, P. Laurent’, J.K. Liou2 I Laboratoire d’Automatique et de Genie des pro&d&s, UPRES-A CNRS QSOO7 Universite Claude Bernard Lyon-l et CPE Lyon, Bat 308G, 43 Bd du 11 Novembre 1918 69626 Villeurbanne C&x, France 2 Tech&p,
BP 347,01703 Miribel C&x, France
Abstract In order to control membrane fouling, a batch ultrafiltration was used to study the reduction in membrane permeability in turbulent flow. A model of the process was developped on pilot plant as function of time, taking into account the non uniformities in flow, cake deposits along the length of the channel. The model predictions compared to the experimental results for ultrafiltration of bentonite solutes shows a good agreement. The effect of transmembrane pressure on the filtration has been studied. 0 1998 Elsevier Science Ltd. All rights reserved. Keywords: Mcdelling, Membrane fouling, Cross-flow filtration, Pilot plant IlltHJdUCtiOEl
Cross-flow ultrafiltration or microfiltration is an of some for clarification efficient method the performance of suspensions. However microfiltration in ultrat%.ration and many applications is limited by membrane fouling causing a decay in filtrate flux, and involving mainly high cost of membrane filtration process (Takizawa et al,1996). This phenomenon refers to the deposit of rejected particles of the feed components on the top surface of the membrane (external fouling) or to adsorption of small particles such as protein and polysaccharides molecules, or macromolecules within the membranes pores (internal fouling). Membrane fouling owing to specific physical or chemical interactions between solutes particles and understood membrane, remains a PoarlY phenomenon as has been confirmed the recent work of Chen et al (1997). Various authors explained differently the continuously decline of the flux showing the difficulty in modelling the process. To explain the flux decline, the mechanisms proposed by Tracey and Davis in the study of protein fouling consist of two consecutive steps: the internal fouling characterised by particles adsorption on the pores walls and mouths, and the external fouling due to the deposition and the growth of particles aggregates on the membrane surface leading to a cake layer building (Gttell et al., 1996). In order to prevent this phenomena a good understanding of the mechanism is necessary. This can be achieved by a good mcdelling step. There are two different models used to analyse internal fouling: the standard blocking model which assumed pore radius reduction due to particles adsorption on the pores walls, and the pore * Corresponding author e-mail: [email protected]
blocking model which assumed that the flux decline is due to complete blocking of some pores while the rest are unaffected by the particles (Giiell et al., 1996). In the case of external fouling the gel polarisation model can be combined with the osmotic pressure model. The gel polarisation or cake filtration model considered that the cake layer provides additional resistance due to deposition of rejected particles on the membrane surlbce and therefore increases witb time. The permeate flux in the osmotic pressure model is determined by difference between the applied transmembrane pressure and the osmotic pressure difference associated with retained particles. Hydraulic resistance provided by cake layers are assumed negligible. In fact the osmotic pressure must be taken into account if the macromolecules solutes like dextmn are present in the solution to be ultrafiltered, on the other hand the permeate rate will be controlled by cake formation if the feed is colloidal suspension such as bentonite. The purpose of this work is to develop a mathematical model of ultrafiltration cross-flow process of colloidal suspension on industrial scale pilot in order to study the e&ct of sensitive parameters on the fouling and to control membrane fouling. Modelling results have been compared with experimental data. Experimental apparatus The pilot (figure 1) is made with two vertical monolithic tubular modules. Each monolith contains seven 4.5 mm internal diameter tubes of 0.856 m length. The inorganic membrane used (type: Kerasep 300kD made by Tech-&p) has a Molecular Weight
SYO2
European Symposium on Computer Aided Process Engineering-S
Cut Off (MWCO) of 300,000 Dalton. Feed flows through the bore while permeate travels through the porous walls of the element into the surrounding shell for withdrawal. The permeate is collected in a reservoir placed on an electronic balance interfaced to a computer to collect and record time and mass data. The two monolithic tubular modules are fed with solution/suspension to be clarified contained in a tank which capacity is 80.10” m3. The entire retentate stream is recycled back to the feed tank. The feed suspension flow rate as well as the inlet and outlet pressures and the pressure in permeate side are measured continuously with a sample period of 60 s. Flux data were obtained by numerical differentiation of the permeate mass versus time data. Retentate and permeate concentrations were determined by weighting the dried extract. Experiments were conducted at different inlet transmembrane nressure.
resistance R, is constituted of the membrane resistance before fouling %, and the additional hydraulic resistance provided by particle cake. This latest is proportional to the cake mass per unit membrane area I$ constituting the fouling layer : RtW = R, +Q&&t)&(‘4t) (5) Choe et al (1986) have shown that the bentonite cake is compressible and one way of representing the specific resistance in terms of applied pressure as mentioned by Coulson and Richardson, is by means of a simple power law relationship: crde(z,t) = agAP’O(z,t) (6) Different expressions are presented in literature to describe particles removed by erosion. Xu et al (1994) in their theoretical analysis on microfiltration have written it as a function of inlet tangential flow rate. In this paper we considered the removal mechanism to be depending of the wall shear stress. In contrast to previous work, this concept integrated the variation of the mass removed according to the length of the membrane. The rate of particle deposit is related to the mass balance between particle convection to the surface and particle removed by erosion which is defined as power law of wall shear stress (figure 2).
Figure 1: Schematic diagram of pilot plant Process modelling The most common model used to describe particles accumulation on membrane surface is the cake filtration model, where the membrane total resistance is rewritten in terms of a serial resistance. Most of the model are in steady-state and do not take into account the effect of state parameters changing along the membrane length particularly for long module. The concentration of the retained solutes or particles in the feed tank increases with time as permeate is removed causing a continual decline in filtrate flux due to the increase in the bulk solute concentration. The model is developed assuming that the feed flow rate is the same in each tube. The variation of the feed volume V and bulk concentration C,, with time is given as: J$ = 2S(U(LJ) - U(OJ))
(1)
Figure 2: Schematic of membrane surface rearrangement the equations After some characterizing the dynamic of cake layer growth and the bulk concentration in membrane tube at transient state are: aMd(z,t) = (CW-CpW))# dt ’ (7) @(z, t)
vJ$zsu(L,t)(c
(2)
ac(2,t) -=---_--_
(L,t)-Cb)
The fluid velocity in each membrane tube and the axial transmembrane pressure are obtained f?om the Darcy’s law and the equation of motion as: 4 AP(z,t) au(z,t) -(3) az -7 pq(Z,t) ~am&t)
az
= -p(z,t)
(zy 1-m
(4)
where f the Fanning friction factor is obtained from experiments done with pure water as f = 0.1 14Rea.25 similar to Blasius correlation. The membrane total
az
4 Cp(z,t)
L\P(z,t)
d W,t> IrR,(z,t) +$K,(p~)nu2n-1(z,1) (')
The effects of bulk concentration on erosion constante have been shown in the literature (Xu et al, 1994). This latest varies linearly according to the bulk concentration up to a limit value of the bulk concentration and does not change beyond this limit value. On the basis of this observation the cake erosion constant can finally be related qualitatively to the solute concentration:
European Symposium on Computer Aided Process Engineering-S
where & and p are parameters to be estimated. The mean permeate flux and membrane mean resistance are then calculated as:
The set of partial differential equations (l)-(9) with the following initial and boundary conditions: att=O V=V,,,Cb=Co,andforw &=O., at z = 0 U(O,t) = UO, AP(O,t) = Po(O,t)-P,, C(O,t) = Cb(f), are solved using IMSL routine DASSL after orthogonal collocation discretisation giving a system of algebraic and difFerential equations. Unknown parameters F&, Q, a~, p and n of the model were estimated using Marquardt algorithm by minimizing at each time the error between the mean permeate flux and the experimental one. Results The optimal values listed in table 1 were used to simulate the model. Figure 2 shows the comparison between the experimental data and the predicted value of bulk concentration in the feed tank. In figure 3 is depicted the membrane mean resistance in transient state. It can be seen a good agreement between the model and the corresponding experiment points. The analysis of the process in terms of cake erosion works quite well.
o-
i
300
1500
2400
3000
3600
4200
4500
5400
6300
The(a)
Figure 3: Evolution of bulk concentration in feed tank versus time ”
&OE+lZ
$
7.ou+12 --
,
I
6.OE+12 --
2.OE+12 -I!
l.OE+12 -.
2
O.OE+OO 240
1140 2040 2940 3540 4740 5640 6540 Time(s)
Figure 4: Evolution of membrane mean resistance versus time
s903
Simulntion In this section are presented the results obtained from a simulation with the following input parameters: U0 = 4 m.s*‘, V0 = 60.10-’ m3, Co = 1 lkg.m”, and T = 293% and the optimal values of table 1. The temperature is used to calculate the solution dynamic viscosity p. Figures 5 to 7 are obtained for APO= 2.10’ Pa. In figure 5 is shown the dimensionless transmembrane pressure as function of the module dimensionless length and time. It can be seen that the transmembrane pressure decreases linearly along the tube and this variation is not modified with time.
Figure 5: transmembrane pressure along the module length versus time. The dimensionless total resistance of membrane is depicted in figure 6. We observe that the cake deposit is more important at the module inlet than at the module outlet, the non-uniformity of the cake resistance is due to the non-uniformity of the mass deposed at the membrane surface, this leads to a non-uniformity in the permeate flux as shown in figure 7. This phenomenon on the flux is less important with time and the permeate flux becomes uniform along the module. A limiting flux effect has not been found in this system, this almost is due to the concentration of the retained particles in the feed tank which increases with time as the permeate is removed, causing the increase in the bulk solute concentration and of course a continual decline in permeate flwc. Taking into account the transmembrane pressure curve profile and the permeate flux one, it appears that the flux decay particularly at initial time is due to pressure drop along the membrane, and due to particles accumulation with time. The et&t of transmembrane pressure has been also studied. In figure 8, the evolution of permeate flux for inlet transmembrane pressure APO= 0.510’ Pa is shown. We observe that the flux decline is only due to pressure drop and thus fouling is limited. In fact at low transmembrane pressure drops the membrane surf&e is free of any particle deposit, thus the permeate flux remains equal to that for the pure solvent through the clean membrane. This result has been confirmed by experiments and is in agreement with Chen et al (1997) results.
s904
European
Symposmm
on Computer
Aided Process
Engineering-8
and membrane fouling. In fact an increasing transmembrane pressure leads to a greater resistance owing to cake formation or fouling. Table 1: Table of ammeters used in simulation
~a” Figure 6: Evolution of membrane resistance
Figure 7: Permeate flux decay as fimction of time and membrane length (APO= 2.10’ Pa)
Figure 8: Permeate flux decay as 8mction of time and membrane length (APO= 0.510’ Pa) Conclusion A mathematical model of ultrafiltration of colloidal suspension on pilot plant is developed. Contrary to the previous works this model takes into account the variation of the hydraulic resistance and the cake mass according to the membrane length, and the effect of the wall shear stress at each position in the tube on the particle cake erosion. The model is valid on bentonite suspension ultrafiltration and can be used for the process optimisation. The simulation done shows that the process non-linearity is limited within the time. A good choice of the transmembrane pressure can prevent cake growing
Notation C: monolith tube bulk concentration (kg.m”) Cb: tank bulk concentration (kg.m”) Co: tank initial concentration (kg.mw3) d: tube diameter (m) f friction factor g: acceleration of gravity (m.s-*) Jr,: local permeate flux (m3.m”.s*‘) K: erosion constante L: tube length (m) &: local mass of cake per unit membrane area (kg.mw2) Pr: permeate pressure (Pa) PO:tube inlet pressure (Pa) AP: local transmembrane pressure (Pa) Re: Reynolds number R: membrane total resistance (rn-‘) &: membrane resistance before fouling (m-r) z: axial coordinate (m) S: monolith axial cross section (m*) t: time (s) U: retentate tangential flow rate (m.s-‘) UO:inlet tangential flow rate (m.s~‘) V: tank volume (m3) VO:tank initial volume (m3) cock: specific cake resistance (m.kg-‘) p: density of solution (kg.m”) u: dynamic viscosity of solution (Pa.s) References Chen, V., Fane, A. G., Madaeni, S. and Wenten, I. G., ” Particle deposition during membrane filtration of colloids: transition between concentration polarization and cake formation “, J. Membrane Sci.,125(1997)109-122. Choe, T.B., Masse, P. and Verdier A., “Flux decline in batch ultrafiltration: concentration polarization and cake formation “, J. Membrane Sci., 26(1986)115. GUel, C. and Davis, R. H., ” Membrane fouling during microfiltration of protein mixtures I’, J. Membrane Sci., 119(1996)269-284. Takizawa, S., Fujita, K. and Soo, K. H., ” Membrane fouling decrease by microfiltration with ozone scrubbing “, Desalination, 106(1996)423-426. Xu, Y., Dodds, J. and Leclerc, D., ” Analyse theorique de la microfiltration tangentielle dans les canaux poreux en tenant compte de la nonuniformite de l’ecoulemem ‘1, Entropie, 182(1994)43-50.
Computers them. Engng Vol. 22, Suppl., pp. S901-S904, 1998 0 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0098-1354/98 $19.00 + 0.00 SOO98-1354(98)00176-8
Modelling and simulation of membrane fouling in batch ultrafiltration on pilot plant F. Rouvetl, K. Fiaty”, I, P. Laurent’, J.K. Liou2 I Laboratoire d’Automatique et de Genie des pro&d&s, UPRES-A CNRS QSOO7 Universite Claude Bernard Lyon-l et CPE Lyon, Bat 308G, 43 Bd du 11 Novembre 1918 69626 Villeurbanne C&x, France 2 Tech&p,
BP 347,01703 Miribel C&x, France
Abstract In order to control membrane fouling, a batch ultrafiltration was used to study the reduction in membrane permeability in turbulent flow. A model of the process was developped on pilot plant as function of time, taking into account the non uniformities in flow, cake deposits along the length of the channel. The model predictions compared to the experimental results for ultrafiltration of bentonite solutes shows a good agreement. The effect of transmembrane pressure on the filtration has been studied. 0 1998 Elsevier Science Ltd. All rights reserved. Keywords: Mcdelling, Membrane fouling, Cross-flow filtration, Pilot plant IlltHJdUCtiOEl
Cross-flow ultrafiltration or microfiltration is an of some for clarification efficient method the performance of suspensions. However microfiltration in ultrat%.ration and many applications is limited by membrane fouling causing a decay in filtrate flux, and involving mainly high cost of membrane filtration process (Takizawa et al,1996). This phenomenon refers to the deposit of rejected particles of the feed components on the top surface of the membrane (external fouling) or to adsorption of small particles such as protein and polysaccharides molecules, or macromolecules within the membranes pores (internal fouling). Membrane fouling owing to specific physical or chemical interactions between solutes particles and understood membrane, remains a PoarlY phenomenon as has been confirmed the recent work of Chen et al (1997). Various authors explained differently the continuously decline of the flux showing the difficulty in modelling the process. To explain the flux decline, the mechanisms proposed by Tracey and Davis in the study of protein fouling consist of two consecutive steps: the internal fouling characterised by particles adsorption on the pores walls and mouths, and the external fouling due to the deposition and the growth of particles aggregates on the membrane surface leading to a cake layer building (Gttell et al., 1996). In order to prevent this phenomena a good understanding of the mechanism is necessary. This can be achieved by a good mcdelling step. There are two different models used to analyse internal fouling: the standard blocking model which assumed pore radius reduction due to particles adsorption on the pores walls, and the pore * Corresponding author e-mail: [email protected]
blocking model which assumed that the flux decline is due to complete blocking of some pores while the rest are unaffected by the particles (Giiell et al., 1996). In the case of external fouling the gel polarisation model can be combined with the osmotic pressure model. The gel polarisation or cake filtration model considered that the cake layer provides additional resistance due to deposition of rejected particles on the membrane surlbce and therefore increases witb time. The permeate flux in the osmotic pressure model is determined by difference between the applied transmembrane pressure and the osmotic pressure difference associated with retained particles. Hydraulic resistance provided by cake layers are assumed negligible. In fact the osmotic pressure must be taken into account if the macromolecules solutes like dextmn are present in the solution to be ultrafiltered, on the other hand the permeate rate will be controlled by cake formation if the feed is colloidal suspension such as bentonite. The purpose of this work is to develop a mathematical model of ultrafiltration cross-flow process of colloidal suspension on industrial scale pilot in order to study the e&ct of sensitive parameters on the fouling and to control membrane fouling. Modelling results have been compared with experimental data. Experimental apparatus The pilot (figure 1) is made with two vertical monolithic tubular modules. Each monolith contains seven 4.5 mm internal diameter tubes of 0.856 m length. The inorganic membrane used (type: Kerasep 300kD made by Tech-&p) has a Molecular Weight
SYO2
European Symposium on Computer Aided Process Engineering-S
Cut Off (MWCO) of 300,000 Dalton. Feed flows through the bore while permeate travels through the porous walls of the element into the surrounding shell for withdrawal. The permeate is collected in a reservoir placed on an electronic balance interfaced to a computer to collect and record time and mass data. The two monolithic tubular modules are fed with solution/suspension to be clarified contained in a tank which capacity is 80.10” m3. The entire retentate stream is recycled back to the feed tank. The feed suspension flow rate as well as the inlet and outlet pressures and the pressure in permeate side are measured continuously with a sample period of 60 s. Flux data were obtained by numerical differentiation of the permeate mass versus time data. Retentate and permeate concentrations were determined by weighting the dried extract. Experiments were conducted at different inlet transmembrane nressure.
resistance R, is constituted of the membrane resistance before fouling %, and the additional hydraulic resistance provided by particle cake. This latest is proportional to the cake mass per unit membrane area I$ constituting the fouling layer : RtW = R, +Q&&t)&(‘4t) (5) Choe et al (1986) have shown that the bentonite cake is compressible and one way of representing the specific resistance in terms of applied pressure as mentioned by Coulson and Richardson, is by means of a simple power law relationship: crde(z,t) = agAP’O(z,t) (6) Different expressions are presented in literature to describe particles removed by erosion. Xu et al (1994) in their theoretical analysis on microfiltration have written it as a function of inlet tangential flow rate. In this paper we considered the removal mechanism to be depending of the wall shear stress. In contrast to previous work, this concept integrated the variation of the mass removed according to the length of the membrane. The rate of particle deposit is related to the mass balance between particle convection to the surface and particle removed by erosion which is defined as power law of wall shear stress (figure 2).
Figure 1: Schematic diagram of pilot plant Process modelling The most common model used to describe particles accumulation on membrane surface is the cake filtration model, where the membrane total resistance is rewritten in terms of a serial resistance. Most of the model are in steady-state and do not take into account the effect of state parameters changing along the membrane length particularly for long module. The concentration of the retained solutes or particles in the feed tank increases with time as permeate is removed causing a continual decline in filtrate flux due to the increase in the bulk solute concentration. The model is developed assuming that the feed flow rate is the same in each tube. The variation of the feed volume V and bulk concentration C,, with time is given as: J$ = 2S(U(LJ) - U(OJ))
(1)
Figure 2: Schematic of membrane surface rearrangement the equations After some characterizing the dynamic of cake layer growth and the bulk concentration in membrane tube at transient state are: aMd(z,t) = (CW-CpW))# dt ’ (7) @(z, t)
vJ$zsu(L,t)(c
(2)
ac(2,t) -=---_--_
(L,t)-Cb)
The fluid velocity in each membrane tube and the axial transmembrane pressure are obtained f?om the Darcy’s law and the equation of motion as: 4 AP(z,t) au(z,t) -(3) az -7 pq(Z,t) ~am&t)
az
= -p(z,t)
(zy 1-m
(4)
where f the Fanning friction factor is obtained from experiments done with pure water as f = 0.1 14Rea.25 similar to Blasius correlation. The membrane total
az
4 Cp(z,t)
L\P(z,t)
d W,t> IrR,(z,t) +$K,(p~)nu2n-1(z,1) (')
The effects of bulk concentration on erosion constante have been shown in the literature (Xu et al, 1994). This latest varies linearly according to the bulk concentration up to a limit value of the bulk concentration and does not change beyond this limit value. On the basis of this observation the cake erosion constant can finally be related qualitatively to the solute concentration:
European Symposium on Computer Aided Process Engineering-S
where & and p are parameters to be estimated. The mean permeate flux and membrane mean resistance are then calculated as:
The set of partial differential equations (l)-(9) with the following initial and boundary conditions: att=O V=V,,,Cb=Co,andforw &=O., at z = 0 U(O,t) = UO, AP(O,t) = Po(O,t)-P,, C(O,t) = Cb(f), are solved using IMSL routine DASSL after orthogonal collocation discretisation giving a system of algebraic and difFerential equations. Unknown parameters F&, Q, a~, p and n of the model were estimated using Marquardt algorithm by minimizing at each time the error between the mean permeate flux and the experimental one. Results The optimal values listed in table 1 were used to simulate the model. Figure 2 shows the comparison between the experimental data and the predicted value of bulk concentration in the feed tank. In figure 3 is depicted the membrane mean resistance in transient state. It can be seen a good agreement between the model and the corresponding experiment points. The analysis of the process in terms of cake erosion works quite well.
o-
i
300
1500
2400
3000
3600
4200
4500
5400
6300
The(a)
Figure 3: Evolution of bulk concentration in feed tank versus time ”
&OE+lZ
$
7.ou+12 --
,
I
6.OE+12 --
2.OE+12 -I!
l.OE+12 -.
2
O.OE+OO 240
1140 2040 2940 3540 4740 5640 6540 Time(s)
Figure 4: Evolution of membrane mean resistance versus time
s903
Simulntion In this section are presented the results obtained from a simulation with the following input parameters: U0 = 4 m.s*‘, V0 = 60.10-’ m3, Co = 1 lkg.m”, and T = 293% and the optimal values of table 1. The temperature is used to calculate the solution dynamic viscosity p. Figures 5 to 7 are obtained for APO= 2.10’ Pa. In figure 5 is shown the dimensionless transmembrane pressure as function of the module dimensionless length and time. It can be seen that the transmembrane pressure decreases linearly along the tube and this variation is not modified with time.
Figure 5: transmembrane pressure along the module length versus time. The dimensionless total resistance of membrane is depicted in figure 6. We observe that the cake deposit is more important at the module inlet than at the module outlet, the non-uniformity of the cake resistance is due to the non-uniformity of the mass deposed at the membrane surface, this leads to a non-uniformity in the permeate flux as shown in figure 7. This phenomenon on the flux is less important with time and the permeate flux becomes uniform along the module. A limiting flux effect has not been found in this system, this almost is due to the concentration of the retained particles in the feed tank which increases with time as the permeate is removed, causing the increase in the bulk solute concentration and of course a continual decline in permeate flwc. Taking into account the transmembrane pressure curve profile and the permeate flux one, it appears that the flux decay particularly at initial time is due to pressure drop along the membrane, and due to particles accumulation with time. The et&t of transmembrane pressure has been also studied. In figure 8, the evolution of permeate flux for inlet transmembrane pressure APO= 0.510’ Pa is shown. We observe that the flux decline is only due to pressure drop and thus fouling is limited. In fact at low transmembrane pressure drops the membrane surf&e is free of any particle deposit, thus the permeate flux remains equal to that for the pure solvent through the clean membrane. This result has been confirmed by experiments and is in agreement with Chen et al (1997) results.
s904
European
Symposmm
on Computer
Aided Process
Engineering-8
and membrane fouling. In fact an increasing transmembrane pressure leads to a greater resistance owing to cake formation or fouling. Table 1: Table of ammeters used in simulation
~a” Figure 6: Evolution of membrane resistance
Figure 7: Permeate flux decay as fimction of time and membrane length (APO= 2.10’ Pa)
Figure 8: Permeate flux decay as 8mction of time and membrane length (APO= 0.510’ Pa) Conclusion A mathematical model of ultrafiltration of colloidal suspension on pilot plant is developed. Contrary to the previous works this model takes into account the variation of the hydraulic resistance and the cake mass according to the membrane length, and the effect of the wall shear stress at each position in the tube on the particle cake erosion. The model is valid on bentonite suspension ultrafiltration and can be used for the process optimisation. The simulation done shows that the process non-linearity is limited within the time. A good choice of the transmembrane pressure can prevent cake growing
Notation C: monolith tube bulk concentration (kg.m”) Cb: tank bulk concentration (kg.m”) Co: tank initial concentration (kg.mw3) d: tube diameter (m) f friction factor g: acceleration of gravity (m.s-*) Jr,: local permeate flux (m3.m”.s*‘) K: erosion constante L: tube length (m) &: local mass of cake per unit membrane area (kg.mw2) Pr: permeate pressure (Pa) PO:tube inlet pressure (Pa) AP: local transmembrane pressure (Pa) Re: Reynolds number R: membrane total resistance (rn-‘) &: membrane resistance before fouling (m-r) z: axial coordinate (m) S: monolith axial cross section (m*) t: time (s) U: retentate tangential flow rate (m.s-‘) UO:inlet tangential flow rate (m.s~‘) V: tank volume (m3) VO:tank initial volume (m3) cock: specific cake resistance (m.kg-‘) p: density of solution (kg.m”) u: dynamic viscosity of solution (Pa.s) References Chen, V., Fane, A. G., Madaeni, S. and Wenten, I. G., ” Particle deposition during membrane filtration of colloids: transition between concentration polarization and cake formation “, J. Membrane Sci.,125(1997)109-122. Choe, T.B., Masse, P. and Verdier A., “Flux decline in batch ultrafiltration: concentration polarization and cake formation “, J. Membrane Sci., 26(1986)115. GUel, C. and Davis, R. H., ” Membrane fouling during microfiltration of protein mixtures I’, J. Membrane Sci., 119(1996)269-284. Takizawa, S., Fujita, K. and Soo, K. H., ” Membrane fouling decrease by microfiltration with ozone scrubbing “, Desalination, 106(1996)423-426. Xu, Y., Dodds, J. and Leclerc, D., ” Analyse theorique de la microfiltration tangentielle dans les canaux poreux en tenant compte de la nonuniformite de l’ecoulemem ‘1, Entropie, 182(1994)43-50.