Sep Purif-2002

Separation and Purification Technology 26 (2002) 51 – 59 www.elsevier.com/locate/seppur Sieve mechanism of microfiltration separation V. Starov a,*, D. Lloyd b, A. Filippov c, S. Glaser c a Department of Chemical Engineering, Loughborough Uni6ersity, Loughborough, Leicestershire LE11 3TU, UK b Department of Chemical Engineering, Uni6ersity of Texas at Austin, Austin, TX 78712 -1062, USA c Moscow State Uni6ersity of Food Industry, 11 Volokolamskoe sh., Moscow 125080, Russia Abstract A theoretical model of dead-end microfiltration purification (MFP) of dilute suspensions is suggested. The model is based on a sieve mechanism of MFP and takes into account the probability of membrane pore blocking during MFP of dilute colloidal suspensions. An integro-differential equation (IDE) that includes both the membrane pore size and particle size distribution functions is deduced. According to the suggested model a similarity property is deduced, which allows one to predict the flux through the membrane as a function of time for any pressure and dilute concentration based on one experiment at a single pressure and concentration. For a narrow pore size distribution in which one pore diameter predominates (track-etched membranes), the IDE is solved analytically and the derived equation is in a good agreement with the measurements on four different track-etched membranes. A simple approximate solution of the IDE is deduced and that approximate solution as well as the similarity property of MFP process is in a good agreement with the measurements on a commercial Teflon microfiltration membranes. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Microfiltration; Theory; Similarity property Nomenclature D d f t c l S p n particle diameter pore diameter probability distribution function time concentration characteristic scale of pore and particle diameters membrane area probability number of particles * Corresponding author. Fax: + 44-1509-223-923. E -mail addresses: [email protected] (V. Starov), [email protected] (D. Lloyd), a.fi[email protected] (A. Filippov). 1383-5866/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 3 - 5 8 6 6 ( 0 1 ) 0 0 1 1 6 - 2 (4) permeability Greek letters i parameter to characterise membrane pore influence v viscosity l Dirac delta function … see definitions after Eq.52 V. (11) € rejection coefficient Subscripts p m min max d 0 tr . Staro6 et al. / Separation /Purification Technology 26 (2002) 51 – 59 N J p L h m V a A k number of pores flux pressure length of the pore membrane thickness porosity volume of liquid in the permeate see definition after Eq. (8) see definition after Eq. (8) ~ tortuosity k see definitions after Eq. * − particle membrane minimum value maximum value particles with diameter D B d initial value track-etched membrane at time tends to . Distilled water often contains up to 100 000 such particles cm − 3. (10) according to mean value theorem 1. specific value dimensional value Superscripts * characteristic value ** after Eq. During MF.1 – 20 mm [1]. but for microelectronics processing this concentration should be less than 500 and in some cases not more than 2 particles cm − 3 [2]. MF is useful for purification. Introduction Microfiltration (MF) is widely used for the purification of colloid solutions having particles in the range 0. Thus. the hydrodynamic resistance of the MF membrane increases (and water flux decreases) with time due to mem- . Consequently. Hubble [10] con- siders MF separation of protein solutions in the framework of a two-parameter model of blocked and partially blocked membrane pores. the log-normal distribution of membrane pore diameters and the influence of membrane fouling on that distribution are considered [11. the particle is retarded by the membrane. In addition. particle adheres to the membrane surface. which describe a flux decline with time. particle blocking a small pore. 2. It was assumed that the ratio of number of partially blocked pores and free pores is a linear function of the solution flux at the moment t. and there have been attempts to fit selected functions to the time dependency of experimental flux data [7]. if a particular particle is outside that cylinder. .17]. Existing experimental data and theoretical estimations show osmotic pressure influences on MF are negligible compared with pore blocking and membrane fouling [3]. where i (defined below) \ 1. but at low concentrations it is possible to neglect this foulant layer formation and to consider only membrane pore blocking. A number of models [13. [15] a model is proposed for a description of volume on time dependencies in microfiltration of suspensions. 1). In a number of recent publications. the pore is blocked. and the hydrodynamic resistance of the membrane is increased. etc.9]. it is possible to assume a sieve mechanism for MF of dilute solutions and to neglect particles deposition inside the membrane pores. In the following analysis. At high enough particle concentrations a layer of particles forms on the MF membrane [4]. and 4. In such a procedure the physico-chemical meaning of fitted parameters is not disclosed and there is no possibility of applying the resulting empirical equations to any other membrane or for different experimental conditions. but again flux decline with time and the dynamics of the pore size distribution in time were not considered. a theoretical model is presented.14] were proposed for cake layer (or gel-layer) formation on the surface of membranes. either go through or block this particular pore depending on their diameter. then all particles that are in the cylinder above the membrane with the bottom area iyd 2/4. time series were predicted and coefficients were calculated and measured. a similar model is used for MF with the additional consideration of taking into account both pores and particles probability distribution functions.12]. 1. The flux decline in time was not considered.) is presented elsewhere [8. if the pore diameter is d with a cross section yd 2/4. the pore under consideration does not influence that particular particle’s behaviour (Fig. small particle goes through a pore. In Ref. Staro6 et al. In this sense. The phrase ‘approaching the pore’. A stochastic model of deep bed filtration has been suggested and elaborated [16. foulant layer formation. / Separation /Purification Technology 26 (2002) 51 – 59 53 brane pore blocking by particles from the feed solution and the formation of a foulant layer [1]. If the particle diameter is bigger than the pore diameter. the particle goes through the pore. a probability sieve model of dead-end MF proposed earlier [18] is extended and tested. Photomicrograph investigations show that even in the case of ultrafiltration of protein solutions it is possible to neglect protein molecules entering into membrane pores [5]. First. particle will block the pore. 3. an integro-differential equation (IDE) is deduced for Fig. The sieve mechanism can be described as follows: if the particle approaching the pore has a diameter less than the pore diameter. The sieve mechanism of membrane pore blocking was introduced more than a decade ago [6]. Schematic presentation of the microfiltration process: 1.V. The material below is organised as follows. Analyses of different membrane fouling mechanisms (sieve mechanism. is defined in terms of ‘a region of pore influence’. Staro6 et al. Dmin. the concentration cd of particles with diameters D \ d is determined. Diffusive flux is neglected. i is assumed below to be independent of the pore and particle sizes. t ) represent the number of particles of diameter D \ d that have approached the membrane surface during time t. the probability becomes iy (ld )2/4S. Hence. which are neglected below. fm(d ) [11. 1). a similarity property is deduced and a simple approximate solution of the IDE is obtained. y (ld )2/4S. It is assumed below. that is.54 V. therefore. Usually for MF purposes log-normal. That is n (d. dmin and dmax are minimum and maximum particle diameters (in the feed solution) and pore diameters. t ). Then the probability that the centre of a random particle far from the membrane is projected into the pore area is the ratio of the pore area to the total membrane area. y (ld )2/4  S . Then the probability P (d. experimental measurements on tracketched membranes and a commercial Teflon membrane are presented and compared with the theoretical predictions. All particle diameters D and membrane pore diameters d are made dimensionless as follows: D = D /l and d = d /l. the particle goes through the pore into the permeate. bilog-normal. respectively. t ) is much greater than 1. the flow of the colloidal feed solution is normal to the membrane surface. from Eqs. that the probability of a single pore with diameter d being blocked by an approaching particle of diameter D \ d is equal to the ratio of the pore influence area y (ld )2/4 to the total membrane area S (Fig. hydrodynamic. / Separation /Purification Technology 26 (2002) 51 – 59 the flux-time dependency. t ) is an integral over time from that flux. that can be interpret as an increase of the effective pore area. that is. for example). Assuming that hydrodynamic and specific surface forces of interaction between the particle and the membrane pore result in an increase of the above probability by a factor i \ 1. respectively. and an exact solution of that IDE is obtained for the case of tracketched membranes. Theoretical model In dead-end MF. and Gaussian functions are used for the membrane pore diameter distribution function.12]. t ) = [1 − iy (ld )2/4S ]n(d. t ) of the event the fixed pore with diameter d being not blocked at the time t is P (d. t ) = exp − (yil 2d 2/4)  & t J (u ) du & Dmax fp(x ) dx 0 d n (3) . and (ii) the number of particles approaching the membrane surface n (d. according to the previous consideration and [18]. particles are transferred by convection only. Let n (d. t ) because (i) the cross section of a single pore is much smaller than the membrane area. It is possible to conclude cd = c & Dmax fm(x ) dx d where c is the particle concentration (number per cm3) in the feed solution. 2. To calculate the i value it is necessary to solve the hydrodynamic equation taking into account a local interaction in the vicinity of the membrane pores (electrostatic. t ) = cS & t J (u ) du & Dmax fp(x ) dx (2) 0 d Eqs.t) (1) To calculate the number of particles n (d. It is easy to show a low influence of particle diffusion and concentration polarisation. Second. From examination of IDE. or other forces) for particles and membrane pores with all diameters under investigation. If the particle diameter D B d. (1) and (2) P (d. (1) and (2) allow one to simplify substantially the probability of pore blocking P (d. molecular. The probability distribution functions of the particle diameters and membrane pore diameters are fp(D ) and fm(d ). Dmax. To derive the probability of pore blocking. This is beyond of the scope of the present investigation. where l is a characteristic length scale (Dmin or dmin. The flux of particles with diameter D \ d is J (t )cd (where J (t ) is the solution flux) and the total number of particles n (d. consider a model membrane with only one pore with diameter ld. trS /… Dp. but for other membranes L = ~h. where the tortuosity ~ accounts for deviation of the pore length from the membrane thickness. Particles that adhere to the membrane surface and particles that block membrane pores are rejected. the solution flux through the clean membrane (no pores blocked) J0 is obtained (this flux is referred to below as the initial flux at the moment t = 0): J0 = yNml 4 Dp /(128vLSA ) (4) where Nm/S is the number of pores per unit membrane area and A is defined as A= & The upper limit of integration is chosen as min(dmax. This probability is equal to ratio of the area of non-blocked pores (corrected by the factor i ) having diameters greater than the particle diameter D to the total membrane area.Dmax) & & t Jtr(u ) du . It is necessary to note that according to the suggested model the mechanism of particle rejection in MF is drastically different from that is reverse osmosis [19] and ultrafiltration. they can only block membrane pores. After averaging all local fluxes. (6). 1 attempts to clarify the mechanism of particle rejection in MF. It is now possible to calculate the probability that a particle with diameter D goes through the membrane at the moment t. For track-etched membranes L is equal to the membrane thickness h. but it is seen below that the results obtained are for the most part independent of that assumption. is known. t ) dx (5) d min n 0 x (7) It is very important to notice that the expression fm(d)P (d. where L is the length of the membrane pore. It is necessary to stress that the use of the Poiseuille equation is justified for track-etched membranes only. an v is the viscosity. To deduce an equation for J (t ). Staro6 et al. Eqs. can not go through the membrane pores. no pore blocking takes place). dmax 0 x 2fm(x )P (x. (3). (7) yields J (t ) = J0. (7) can be solved in the case of track-etched membranes. (5) to yield the following integro-differential equation (IDE): J (t ) = J0A dmax x fm(x ) dx 4 d min n −1 . (3) was substituted into Eq. is J (t ) = J0A & × exp − (yil 2c /4)x 2 ×fp(D ) dD dx  x 4fm(x ) & t J (u ) du & Dmax dmax x 4fm(x )P (x. . Eq. (5) and (6) show that MF can be fully described if the flux as a function of time. the local flux through the single pore with diameter d is yl 4d 4 Dp /(128vL ). In that case all membrane pores have the same diameter dtr and fm(d ) is the Dirac delta function (in this particular case it is reasonable to choose l = dtr). & dmax d min The flux through the membrane at time t. that is. t ) is the probability distribution function of non-blocked pores at time t. (7) in this case results in the following dependency of the permeate volume on time Vtr(t ) = a ln(1 + …t Dp ) (8) where Vtr(t ) represent the volume of the solution in the permeate at time t Vtr(t ) = S & min(dmax. Eq. Dmax) because particles with diameters bigger than the maximum pore diameter dmax. then and Eq. Dp is the applied pressure drop across the membrane. / Separation /Purification Technology 26 (2002) 51 – 59 55 It is assumed below that the flow of the solution in each membrane pore is simple Poiseuille flow. J (t ). Fig. Averaging of this probability over all particles that can go through the membrane pores results in the following expression for the rejection coefficient dependency on time: € (t ) = 1 − (Nmiyl 2/4S ) × If Dmax B dmin (that is. taking into account pore blocking. All those events are taken into account in Eq. only particles in a region influenced by a pore go through the pores if the particle diameter is smaller than the pore diameter. Solution of Eq. t ) (6) D min D × dx fp(D ) dD a and … are constants. which are obviously independent of time and applied pressure difference: a = J0.V. Experimental determination of the pure water flux through the track-etched membrane. According to a mean value theorem for integrals Eq. in these special co-ordinates the dimensionless flux j (t ) should be valid for any pressure difference and any concentration. Staro6 et al.tr is not difficult. Now an important similarity property of MF will be deduced from Eq. “ If we suppose now that d ** value is independent of time the latter equation can be immediately integrated and the solution is as follows k. This means. 1 In all experiments reported below. Eq. it is desirable to determine a and … simultaneously in the same experiment. J (t ) can be calculated with the help of J (t* ) shifting the latter dependency along both axes: J (t ) = (Dp /Dp )J [t  Dp c /(c Dp )] (10) Eq. if an experimental curve J (t* ) is generated for fixed values of Dp* and c*. (7) can be rewritten as j (t ) = A Hence. J0. For this purpose a dimensionless flux j = J /J0 and a dimensionless time t = t /t * are introduced. Thus.56 … = (J0. It is necessary to stress that time scale t * is inversely proportional to concentration and pressure difference (because J0 in the denominator is proportional to Dp ). (8). and J (t ) is desired for any other values of Dp and c. / Separation /Purification Technology 26 (2002) 51 – 59 Dmax fp(D ) dD. consequently.tryid 2 trc /4 Dp ) & V. the concentration dependence is not specified in Eq. With the help of the dimensionless variables j and t. however. the following steps are required: “ according to the definition of the dimensionless time: t  Dp c  = t Dp c. (7). a and … are considered below as fitting parameters. where t * = 4/ (iyl 2cJ0). Therefore. (9) does not include any parameters connected with applied pressure difference or concentration of particles in the feed solution. (7) can be found in the following way. (10) is important with regard to the similarity property of MF processes. the concentration of particles in the feed solution is kept constant. An approximate solution of Eq. (7) can be rewritten as J (t ) = J0 exp − (yil 2c / 4)d** 2 &  t J (u ) du 0 dx = J0 exp − (yil 2c /4)d** 2 × &  & Dmax fp(D ) dD A d** n & n dmax x 4fm(x ) d min t J (u ) du & Dmax fp(D ) dD 0 d** & dmax d min × exp −  & x 4fm(x ) t j (u ) du & Dmax fp(y ) dy dx (9) 0 x n Eq. Dp J (t ) = (11) k. 1− 1− exp( − k. kc Dp t ) k0 where k. = J. k0 = J0/Dp is the initial membrane permeability.   J./Dp is the membrane permeability at the final stage of the process. or J = J0(J /J0) = J (Dp /Dp ) Eq. (11) includes three parameters: k. = J0A & dmax x 4fm(x ) dx . D max k = (yid** 2/4) & Dmax fp(D ) dD d** or t = t  Dp c /(c Dp ) according to the definition of the dimensionless flux: j = J /J0 = J /J0. Permeabilities k.. k0 and k. and k0 can be determined independently and k can be determined by fitting only one experimental dependency of flux on time. Direct numerical calculations according to Eq. Eq. J (t ) can be easily calculated according to similarity property. J* (t ). at a fixed pressure difference Dp* and concentration c*. at any pressure difference and concentration. (10). After these three parameters are determined. (11) gives in most cases a reasonable approximation of the exact solution of the IDE. . (7) showed that the approximate solution according to Eq. Russian Federation). / Separation /Purification Technology 26 (2002) 51 – 59 57 Fig. 3. . values a and … were determined from separate experiments with different applied pressures. which served as a dilute colloidal solution. According to the theory presented above.9 atm (that is.03. parameters a and … should be independent of the operating pressure drop (a manifestation of the similarity property in the case of track-etched membranes). particle distribution and their concentration in water might be different for different membranes. 3. Experiments Two series of experiments were conducted. For each membrane two experimental curves were constructed at different pressures with the same water (dilute colloid solution). and 2 mm were used. and (ii) a commercially available Teflon membrane was used in the second series to verify the similarity property of MF processes according to Eq.V. a typical pressure range for MF) and the permeate volume of water as a function of time was measured. (8) are in excellent agreement with the experimental results. To test this.2 to 0. Four track-etched membranes of pore diameters 0.2 to 0. Moscow 117333. Leninski pr. and (b) dtr = 0.1. (11).45. the pressure ranged from 0. 59. Experiments with different membranes were conducted on different days: consequently. In Fig. 3 two typical plots out of a series of eight cases are presented in all eight cases (two experiments with each membrane) the theoretical curves according to Eq. Mtchedlishvily (Track Etched Membranes Laboratory. ( ) experimental points. 2. The cell was pressurised to operating pressure in the rage from 0. (8). The feed was City of Austin tap water. 2. As mentioned above.9 atm. Track-etched poly(ethylene terephthalate) membranes (commercial name LAVSAN) were supplied by Dr B. As mentioned above. Particle concentration and size distribution in the water were determined using a Coulter Multisizer Counter.17 cm2. Staro6 et al. (i) Track-etched membranes were used in the first series to check Eq.45 Fig. Comparison of experimental dependencies of the water volume in the permeate on time and fitted theoretical dependencies (solid lines) in two cases: track-etched membrane with (a) dtr = 2 mm. 0. A typical particle distribution is shown in Fig. Example of particle size distribution of the water used (dilute colloid solution) for MF experiments on track-etched membranes.1 mm. but in some cases d = 0. 0.l and d = 0. over this pressure range a and … varied by less than 10% in all cases. (10) and the applicability of the approximate solution according to Eq. The experiments were conducted in a standard dead-end MF setup with a membrane area of 32. Moscow Institute of Crystallography. parameters a and … were determined using a fitting procedure. Russian Academy of Sciences. / Separation /Purification Technology 26 (2002) 51 – 59 4.55 atm. The model is based on a sieve mechanism and takes into account the probability of the membrane pores blocking in the course of the MF process. Conclusions A theoretical model of the dead-end microfiltration (MF) process of dilute colloid suspensions is suggested. Staro6 et al. According to the suggested model a similarity property of MF processes is deduced. ( + ) Dp3 = 1. 4. an integro-differential equation (IDE) that includes both the membrane pore size and particle size probability distribution functions is deduced. An approximate equation for flux dependency on time is derived from the IDE and this solution is in good agreement with the experimental results on a commercial MF Teflon membrane.58 V. The similarity property is in good agreement with the experimental data on both track-etched and commercial Teflon membranes.8 atm. Experimental and theoretical dependencies of the dimensionless flux on time for the MF Teflon membrane MMF: ( × ) Dp1 = 0. This property allows one to predict the flux as a function of time for any pressure and concentration (dilute) based on one experiment at a single pressure and concentration. (“) Dp2 = 1.05 atm. Solid line corresponds to the fitted theoretical dependency j (t1) = j. Fig. − {1 − (1 − j. ) exp( − k j. t1)} − 1 with j. 87 ml cm2 min kPa. 4) confirms the similarity property of the MF process.03. The universal curve (Fig. The solid curve in Fig. 4 experimental data for flux versus time for three different pressures are presented. k0. Pressure DP3 = 0. Vladimir. The second series of experiments was performed on a commercially available Teflon MF membrane having an average pore diameters of 0. Concentration of colloid particles in the feed solution was fixed and according to Eq. = 0. (10) experimental curves J1(Dp1t1/Dp3)/J01. The good fit of the data and pressure independence of a and … support the theory presented here as a description of the MF process for track-etched membranes. was 0.35 mm (commercial name MMF. (11). thus.18 min − 1. the rejection coefficient was close to 1. Russian Federation).55 atm was chosen as a basis. p . and J2(Dp2t2/Dp3)/J02 should lie on the experimental curve J3(t3)/J03. Concentration of colloid particles in the feed solution was 55 000 particles per cm3 and the concentration in the permeate was 55 – 100 particles per cm3. 4 presents the fitted curve according to the approximate solution Eq. mm) those variations were less than 1%. k = 0. In Fig. rewritten in the following form j (t3) = j. In each experiment the initial water permeability. supplied by VNIISS. − {1 − (1 − j. ) exp( − k j. t3)} − 1. Chakravarti for help in experimental measurements.). R.D. Springer-Verlag. Madison. 442. New York. to Dr V. 1987. Starov expresses his gratitude to the Royal Society for support (Grant ESEP/JP/JEB/ 11159). Kemmer (Ed. Science Technology. Cohen. . 381. Test. second ed. [2] F. The NALCO Water Handbook. Probstein. V. 114 (1986) 194 – 207. in: F. Gorsky for his assistance in computer programming and to Mr S. where k = kcJ03. p. Membrane Filtration. Colloid Interface Sci.. Brock. p. WI. Colloidal fouling of reverse osmosis membranes. [3] R. McGraw-Hill. 1983. J. References [1] T.F.D.N. Acknowledgements The authors express their gratitude to the Texas Advanced Technology Program and the University of Texas Separations Research Program for their financial support. Membr. Germany. Churaev. Hwang. Persson. [13] D. Technol. 12 (1990) 483 – 488 (English Translation). P. S.A. Suki. [11] P. Wakeman. R. N. Staro6 et al. 54 (1990) 321 – 328.T. N. Yasminov. S. Microhydrodynamic effects in membrane filtration of high pure chemicals.D. Analysis of deep bed filtration data: modelling as a birth-death process. Herrera-Zeppelin. V.H. Velicangil.M. J. Koenders.V. Adv. Howell. [7] A. S. Prien. Le. Nilsson. Filippov.A. Ser. Modelling of protein rejection by microfiltration membranes. Birth-death modelling of deep bed filtration secondary analysis. [15] M. Reactives and Pure Chemicals.T. Acad.V. [19] V.V. Petsev. Koenders. 43 (1993) 145 – 167. S.L. Meireles.T. Kessler. Trans. Desalination 80 (1991) 123 – 138.A.J. Yasminov. J. Concentration polarisation in crossflow microfiltration. V.A. M.A. Schulz. Filter cake formation from structured suspensions. 3rd Int. p. Membr. [17] L. Aspects 81 (1993) 65 – 81. Fouling and Cleaning in Food Processing.A. AIChE J. Colloid Interface Sci. D. Fan. Membr. Fell. [6] J. IChemE 75 (Part A) (1997) 309 – 320.M. Fane. Wakeman. R. Sci. 65 (in Russian). Eng. . 369 (1981) 355. A. Sanches. Colloids Surfaces A: Physicochem. Moscow. A. 43 (4) (1997) 946 – 958. AICHE J. V. Aimar. [18] A. Mathematical modelling of microfiltration processes with the help of probability sieve mechanism. Concentrated dispersions of charged colloidal particles: sedimentation. Fouling and Cleaning in Food Processing. [14] M. Hubble. J.H. Ann. 239. Sanches. M. Yasminov. Nassar. [5] A. Starov. Glaser. [10] J. O. Conf. 1989. Fouling resistance models in microfiltration and ultrafiltration. ultrafiltration and diffusion. Membr. Volodin. R. 40 (1989) 173 – 187. A.J. 56 (1991) 13 – 28. Membrane microfiltration in purification technology of chemicals. in: Information Review. Eng. B. Ivanov.S. Visoko Tchistie Veshestva 1 (1988) 110 (in Russian). 59 [12] M. I. Starov.L. Sov. Effect of protein fouling on the apparent pore size distribution of sieving membranes. Hwang. A.J. A contribution to the transformation of retention curves into pore size distribution for sieving membranes. Gaydukova. Chou. 1987. J. S. Sci. Chou. [9] K. J. Fan. V. Water Chem.G. Sci.G. / Separation /Purification Technology 26 (2002) 51 – 59 [4] G. Ripperger. Nassar. 31 (1985) 1781 – 1790.M. S. Commun. Sci. Aimar. [8] A. Separation of electrolyte solutions by reverse osmosis. Grekov. S. Meireles.B. J. in: H.). Y.V. 27 (1986) 181 – 193. C. Initial deposition of interactive particles by filtration of dilute suspensions. Lurid (Eds. 35 (1985) 101 – 121.T. Glaser. Modelling of fouling mechanism in protein ultrafiltration. Sci.V. Chem. Starov. I. [16] L. NIITECHIM.V. p.N. R.