sieve mechanism of microfiltration

-

journal of

MEiEEE

Journal of Membrane Science 89 ( 1994) 199-2 13 ELSEVIER

Sieve mechanism of microfiltration

A. Filippov”, V.M. Starovb,*, D.R. Lloydb, S. Chakravartib, S. Glaser” “Moscow Institute of Food Technology, Moscow 125080, Russian Federation

bDepartment of ChemicaI Engineering, The University of Texas at Austin, Austin, TX 78712-1062, USA

(Received June 29, 1993; accepted in revised form November 1, 1993)

Abstract

A mathematical model for dead-end microfiltration (MF) of dilute suspensions is suggested. The model is based on a sieve mechanism and takes into account the probability of membrane pore blocking during MF of dilute colloid suspensions. An integro-differential equation (IDE) that includes both the membrane pore size and particle size probability distribution functions is deduced. According to the suggested model a similarity property is deduced, which allows one to predict the flux through the MF membrane as a function of time for any pressure and dilute concentration based on one experiment at a single pressure and concentration. The model enables one to calculate all the necessary physico-chemical parameters that are relevant for the determination of the dependency of flux on time, 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 good agreement with the measurements on four different track-etched membranes. A condition is found for which the MF membrane transforms into an ultrafiltration membrane. A simple approximate solution of the IDE is deduced and that approximate solution as well as the similarity property of MF processes is in good agreement with the measurements on a commercial teflon MF membrane.

Key words: Microfiltration; Theory

1. Introduction

Microfiltration (MF) is widely used for the purification of colloid solutions having particles in the range 0.1-20 pm [ 11. Distilled water often contains up to 100,000 such particles/cm3, but for microelectronics processing this concentration should be less than 500 and in some cases not more than 2 particles/cm3 [ 21. Thus, MF is useful for purification. During MF, the hydrodynamic resistance of the MF membrane increases (and water flux decreases) with time due

*Corresponding author.

to membrane pore blocking by particles from the feed solution and the formation of a foulant layer [ I 1. Existing experimental data and theoretical estimations show osmotic pressure influences on MF are negligible compared with pore blocking and membrane fouling [ 3 1. At high enough particle concentrations a layer of particles forms on the MF membrane [ 41, but at low concentrations it is possible to neglect this foulant layer formation and to consider only membrane pore blocking. Photomicrographic investigations show that even in the case of ultrafiltration of protein solutions it is possible to neglect protein mole- cules entering into membrane pores [ 5 1. Conse-

0376-7388/94/%07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI0376-7388(93)EO195-P

200 A. Filippov et al. /Journal of Membrane Science 89 (1994) 199-213

quently, it is possible to assume a sieve mechanism for MF of dilute solutions and to neglect particles deposition inside the membrane pores.

The sieve mechanism can be described as follows: if the particle approaching the pore has a diameter less than the pore diameter, the particle goes through the pore. If the particle diameter is bigger than the pore diameter, the particle is re- tarded by the membrane, the pore is blocked, and the hydrodynamic resistance of the membrane is increased. The phrase “approaching the pore”, is defined in terms of “a region of pore influence”. In this sense, if the pore diameter is dwith a cross section 7#/4, then all particles that are in the cylinder above the membrane with the bottom area /?zc?*/~, where p (defined below) > 1, either go through or block this particular pore depend- ing on their diameter; if a particular particle is outside that cylinder, the pore under consideration does not influence that particular particle’s behavior (Fig. 1).

The sieve mechanism of membrane pore blocking was introduced more than a decade ago [ 61, and there have been attempts to fit selected functions to the time dependency of experimen-

flow

I I 4 a

P 10 cj /

Fig. 1. Schematic presentation of the microfiltration process: ( 1) particle adheres to the membrane surface; (2 ) particle blocking a small pore; ( 3 ) small particle goes through a big pore; (4) particle will block the pore. The circle around the pore represents the bottom of the region of the pore influence with the area jIn(ld)‘/4.

tal flux data [ 7 1. In such a procedure the phys- ice-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.

Analyses of different membrane fouling mech- anisms (sieve mechanism, foulant layer formation, etc. ) is presented elsewhere [ 8,9]. Hubble [ lo] considers MF separation of protein solutions in the framework of a two-parameter model of blocked and partially blocked membrane pores. 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 L The flux decline in time was not considered. In a number of recent publications, the log-normal distribution of membrane pore diameters and the influence of membrane fouling on that distribution are considered [ 11,121, but again flux decline with time and the dynamics of the pore size distribution in time were not considered.

A stochastic model of deep bed filtration has been suggested and elaborated [ 13,141. In the following analysis, a similar model is used for MF with the additional consideration of taking into account both pores and particles probability distribution functions. In addition, a probability- sieve model of dead-end MF proposed earlier [ 15 ] is extended and tested.

The paper is organized as follows. First, a theoretical model is presented, an in-

tegro-differential equation (IDE) is deduced for the flux-time dependency, and an exact solution of that IDE is obtained for the case of track- etched membranes. From examination of this solution, a similarity property is deduced and a simple approximate solution of the IDE is obtained.

Second, numerical simulations of the flux decline in time according to the IDE are presented and a condition is found whereby the MF membrane is transformed into an ultrafiltration membrane.

Third, experimental measurements on track- etched membranes and a commercial Teflon membrane are presented and compared with the theoretical predictions.

A. Filippov et al. /Journal ofMembrane Science 89 (1994) 199-213 201

2. Theoretical model

In dead-end MF, the flow of the colloidal feed solution is normal to the membrane surface. All particle diameters d and membrane pore diameters dare made dimensionless as follows: D = D/ I and d=d/l, where 1 is a characteristic scale (Dmin or &in, for example ) . The probability distribution functions of the particle diameters and membrane pore diameters are fp (D) and f, (d) , respectively. Usually for MF purposes log-normal, bilog-normal, and gaussian functions are used for the membrane pore diameter distribution function, f,(d) [ 11,121. Dmin, D,,, dmi,y and d,,, are minimum and maximum particle diameters (in the feed solution) and pore diameters, respectively. It is assumed

dmax

s An(x) h= 1

dmin

and

The particle diffusion coefficient can be esti- mated as N k T/ (3 K p D) (where k is the Boltz- man constant and T the temperature in K) and for particles with a characteristic diameter _ 1 ,um the diffusion coefficient is H 10m9 cm*/s. Ac- tually, the diffusion coefficient must be even less because of a considerable decrease of particle hydrodynamic mobility near the wall [ 16- 18 1. A characteristic distance between pores can be es- timated as (S/N,)“‘- (x 62/4 m)‘/*. For the track-etched membranes used below, m N 0.03 and d- 1 pm; hence, the characteristic distance is N 10 -’ cm. Consequently, for a characteristic filtration velocity N 10m3 cm/s and a Peclet number N lo3 * 1, particle diffusion and concentration polarization can be neglected below. In that point MF differs from ultrafiltration, where concentration polarization is of great im- portance [ 19,201 and actual membrane rejection is different from the observed module rejection [21,22].

To derive the probability of pore blocking, consider a model membrane with only one pore with diameter 1 d. Then the probability that the center 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; that is, n( I d)*/4S. 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 @- 1, that can be interpret as an increase of the effec- tive pore area, the probability becomes /3 a( Id)*/ 4s. To calculate the B value it is necessary to solve the hydrodynamic equation taking into account a local interaction in the vicinity of the membrane pores for particles and membrane pores with all diameters under investigation. This is beyond of the scope of the present paper.

It is assumed below, according to the previous consideration and ref. 15, 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 j?nl*d*/4 to the total membrane area S (Fig. 1) . If the particle diameter D< d, the particle goes through the pore into the permeate. As before B ( > 1) is a coefficient that accounts for the membrane pore influence, but the nature of that influence is not considered; it can be caused by electrostatic, mo- lecular, hydrodynamic, or other forces. B is assumed below to be independent of the pore and particle sizes.

Let n(d, f) represent the number of particles of diameter D>d that have approached the membrane surface during time K Then the probability P(d, ij of the event “the fixed pore with diameter d being not blocked at the time ?’ is

P( d, F) = ( 1 - nj312d2/4S)“(d~fi (I)

To calculate the number of particles n (d, f), the concentration cd of particles with diameters D> d is determined. It is possible to conclude

Dmax

cd =C I

f,(D) do

d

where c is the particle concentration (number per cm3) in the feed solution. Diffusive flux is ne-


glected; therefore, particles are transferred by convection only. The flux of particles with diameter D > d is J( F) c, (where J( F) is the solution flux) and the total number of particles n(d, F) is an integral over time from that flux. That is

n(d, T)=cS jJ(u)duq^f,(D) dD (2) 0 d

Eqs. ( 1) and (2 ) allow one to simplify substan- tially the probability of pore blocking P( d, F) because ( i ) the cross section of a single pore is much smaller than the membrane area, that is, nl*d*/ 4 c S; and (ii) the number of particles approaching the membrane surface n (d, ij is much greater than 1. Hence, from Eqs. ( 1) and (2 )

P(d, T) =exp - (njI12d2c/4)

x J(u)du J J f,(W) (3) 0 d

-!

It is assumed below that the flow of the solution in each membrane pore is simple Poiseuille flow; that is, the local flux through the single pore with diameter d is n14d4Ap/ ( 128 pL), where L is the length of the membrane pore, Ap is the applied pressure drop across the membrane, and p is the viscosity. For track-etched membranes L is equal to the membrane thickness h, but for other membranes L= oh, where the tortuosity r accounts for deviation of the pore length from the membrane thickness. It is necessary to stress that the use of the Poiseuille equation is justified for track- etched membranes only, but it is seen below that the results obtained are for the most part independent of that assumption.

After averaging all local fluxes, the solution flux through the clean membrane (no pores blocked) Jo is obtained (this flux is referred to below as the initial flux at the moment t= 0) :

J,,=zN,,,14Ap/(128pLSA)

where A is defined by

(4)

1 -I

A= x4fm(x)b

and N,,.,/S is the number of pores per unit membrane area. The flux through the membrane at time 6 taking into account pore blocking, is

dmar

J(+J,A I

x4fm(x)fYx, 0 d-x (5) dmn

where the expression f, (d) P( d, F) is the probability distribution function of non-blocked pores at time t:

It is now possible to calculate the probability that a particle with diameter D goes through the membrane at the moment L This probability is equal to ratio of the area of non-blocked pores (corrected by the factor 8) having diameters greater than the particle diameter D to the total membrane area:

&ax

(Nn,M2/4S) 1 x*fm(x)P(x, f) d.x b

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:

q(F) = 1 - (N,,&,*/4S) (6)

min ( dmax, Dmax ) d-x

x s s x2&, (xY(x, f> ck(p(~) dD

&I,” D

The upper limit of integration is chosen as min ( &,,, D,,,) because particles with diameters bigger than the maximum pore diameter D *ax cannot go through the membrane pores, they can only block membrane pores.

It is necessary to note that the mechanism of particle rejection in MF is different from that is reverse osmosis [ 23 ] and ultrafiltration. Fig. 1 attempts to clarify the mechanism of particle rejection in MF. Particles that adhere to the mem-

A. Filippov et al. /Journal ofMembrane Science 89 (I 994) 199-Z 13 203

brane surface and particles that block membrane pores are rejected; only particles in a region in- fluenced by a pore go through the pores if the particle diameter is smaller than the pore diameter. All those events are taken into account in Eq. (6).

Eqs. (3), (5), and (6) show that MF can be fully described if the flux as a function of time, J(F) is known. To deduce an equation for J(T), Eq. (3) was substituted into Eq. (5) to yield the following integro-differential equation (IDE) :

dmax

J(f) =JOA S x4f,(x)exp[ - (7~fi/~c/4)x~ dmin

X J=&,(D) dD jl(u)d+x (7) X 0

If D,,, < dmin (that is, no pore blocking), then

Dmu

I s,(DW=O X

and Eq. (7) yields J( F) =Jo. Eq. ( 7) can be solved in the case of track-

etched membranes. In that case all membrane pores have the same diameter a,, and f,(d) is the Dirac delta function (in this particular case it is reasonable to choose I= a,, ). In this case Eq. ( 7 ) becomes

&(O=Jo,,, exp [

- (7@CX4)

X Dj=A,(D)~ f JtrO4du-j (8) I 0

Let

t

v,,C~l =s Jt,(uW, s 0

represent the volume of the solution in the permeate at time E Then

&(F)=(W) dK,(ti/dt

and from Eq. (8 )

d V,,( F) /dr=&, S exp - ( xpGr c/4 S)

Dmax

x s,(W dD K,(f) I 1

1 with boundary condition V,, (0) = 0.

Solution of this equation results in the following dependency of the permeate volume on time

&(C) =a ln( 1 +oCfp) (9)

where

DIllax

1

and a=&, S/o Ap. Jo,,r is proportional to Ap; thus, a and o are independent of pressure difference.

In all experiments reported below, the concentration of particles in the feed solution is kept constant; consequently, the concentration de- pendence is not specified in Eq. (9). Experimen- tal determination of the pure water flux through the track-etched membrane, &, is not difficult; however, it is desirable to determine a and o si- multaneously in the same experiment. Thus, a and o are considered below as fitting parameters.

From Eqs. (6) and (9), the flux and rejection coefficient time dependencies are:

Jtr(~)=(awAp/S)/(l+wAp~).

Assuming 4, < Dmax,

where m = N,,, 7ra$/4S is the track-etched membrane porosity. If &, 2 D,,,, rejection remains constant in time. It is easy to conclude from these equations that the solution flux Jt,( f) tends to zero and selectivity qtr(?) tends to 1 as time increases.

Now an important similarity property of MF will be deduced from Eq. ( 7 ) . For this purpose a


dimensionless flux j= J/J0 and a dimensionless time t= f/P are introduced, where f*= 4/ (p K l2 c Jo). It is necessary to stress that time scale ?* is inversely proportional to concentration and pressure difference (because J,, in the denomi- nator is proportional to the pressure). With the help of the dimensionless variables j and t, Eq. (7 ) can be rewritten as

dmax *

j(t) =A s j(WC

dm.

Dmax

xx+ s

.fw)~ dx 1 (10) x Eq. ( 10) does not include any parameters con- nected with applied pressure difference or concentration of particles in the feed solution. Therefore, in these special coordinates the dimensionless flux j(t) should be valid for any pressure difference and any concentration. This means, if an experimental curve J, ( & ) is gener- ated for fixed values of Ap, and c,, and J(i) is desired for any other values of Ap and c, the following steps are required:

according to the definition of the dimensionless time:

according to the definition of the dimensionless flux:

j= J,/J o~=J/Jo,o~J=Jo(J,lJ~)=J,(~~l~~~)

Hence, J(i) can be calculated with the help of J, ( t, ) shifting the latter dependency along both axes:

J(~)=(AP/AP,) J*[~;,AP~(cAP)I (11)

Eq. ( 11) is important with regard to the similarity property of MF processes.

Consider Eq. ( 7 ) in the case &in c D,, < d,,; that is, all membrane pores with diameters larger than D,, cannot be blocked. In this case, Eq. (7 ) gives

&ax

J(i)=JoA s x4fm(x)~

&lax

Dmax

+J,,A s x4fx-(x)

dm.

x exp [

- ( a/U2c/4) x2

X ?‘f,(D)dD j J(u)du] dx (12) X 0

Introducing

dmax

J ,=J,A I x4fm(xW

Dmax

it can be shown J(i) -+ J, as i/i*+ co. From Eq. ( 12), J(i) > J,. Integration of this inequality gives

r

s J(u)du> J, t; 0

but J, i+oo as i/i*+m and hence the left-hand side of the inequality tends to infinity. Conse- quently, the exponential in Eq. ( 12) tends to 0 as i/ i*+m and the second term on the right- hand side of Eq. ( 12) approaches 0. Hence, J(i)+J, as i/i*400 and Eq. (12) can be rewritten as

&lax

J(i)=J,+J,A s x4fm(x) dmin

xexp [

- (rr/Z2c/4)x2

X q,&,(D)0 1 J(u)du]dr (13) X 0

From the definition of J,, J,= 0 if all membrane pores are blocked (that is, at D,,, > d,,,)

A. FiIippov et al. /Journal ofMembrane Science 89 (1994) 199-213 205

and J, = Jo if no pore blockage takes place (that is, at Dmax < &in).

An approximate solution of Eq. ( 13 ) can be found as follows. In Eq. ( 13 ) the expression

exp [ - (rr/312c,4)x2 Tf,(o)dD 1 J(u)du] X 0

depends on the variable x, but according to the mean value theorem for definite integrals it is possible to find a special value of x (that value is denoted below as d**) such that the value of the previous exponential is independent of x but still the value of the right-hand side of the Eq. ( 13 ) is unchanged. Hence, Eq. ( 13 ) can be rewritten in the following form

J(r) = J, + Jo B exp [

- (7@~/4)d**~

X D~&,(D)dD j J(u)du] (14) d- 0

where 6** is a pore diameter from the mean value theorem for definite integrals and

Dmax

B=A s x”f,(x)k.

dmin

According to definitions, Jo = J, + Jo B. Although d* depends on time, to obtain an

approximate solution of Eq. ( 13) d** is assumed to be independent of time. In this case, the solution of Eq. ( 14) is found using the method given in the Appendix, and the result is

J(f)=J,[l-(I-J,/J,)exp(-J,ycij]-’

(15)

where

Y= (7@d**2/4) qf,(D)dD d"

Unfortunately, the exact value of y cannot be

precisely determined because of the arbitrary assumption that d** is independent of time. It is easy to conclude from Eq. ( 15) that J( ij decreases from Jo at f=O to J, as time increases. Eq. ( 15 ) can be rewritten as

xexp(-k,ycf&)]-’ (16)

where k, = J,/Ap is the membrane permeability at the final stage of the process and b = JolAp is the initial membrane permeability. The parameters k,, b, and y are independent of both pressure difference and concentration. In the case of k,=O, Eq. ( 16) transforms into

J(F)=koAp (1 +k,yctip)-’ (17)

Eq. ( 16 ) includes three parameters: k,, k,, and y. Permeabilities k, and b can be determined independently and y can be determined by tit- ting experimental flux data, J,( & ), at a fixed pressure difference Ap, and concentration c,. After these three parameters are determined, J(F) can be easily calculated according to Eq. (16) at any pressure difference and concentration.

Eq. ( 16) is checked below to see if it gives a reasonable approximation of the exact solution of the IDE given by Eq. (7 ) . For this purpose Eq. (7) [rewritten in the dimensionless form, Eq. ( 10) ] is numerically solved for different particle and pore distributions&(D) and_&(d).

3. Numerical results

Eq. ( 10) was solved numerically for membranes with pore diameters represented by the log-normal probability distribution fm (d) = exp[ - (In d-e,)*/2a~]/[J(2n)o,d], with average pore diameter E, = exp ( af/2 + e,) and pore diameter dispersion C,= exp (of + 2e,) [ exp (a& - 11. The log-normal probability distribution function suitably approximates globular, librillar, and globular-fibrillar (Sarto- rius SM- 11806, Germany ) membranes [ 7,10,11]. In the case of membranes with globular-cellular structure (Vladipore, Russian Fed-


eration; Durapore: GVHP, GVWP and Sarto- rius: SM-11905-11907, Germany) the so-called bilog-normal probability distribution function is more suitable [ 6,7 1. In this case the probability distribution function has two maxima: one for small pores with average diameter E,,, and the second for big pores with average diameter Em,b.

Thwf&O= 4,&O +(1-4.&,WLwhere a is a fraction of pores having smaller diameters. The bilog-normal distribution is also used in the calculations below.

The log-normal distribution is adopted for particle diameters f,(O) =exp[ - (In O-e,)*/

20; Il[Jcm~, Dl, with average particle diameter E, = exp (cri /2 + e,) and particle diameter dispersion C, = exp ( tri + 2e,) (exp r_$ - 1) . This probability distribution is in agreement with physical notions about particle size distributions in nature [ 24 1.

Numerical simulations were undertaken to generate dependencies of the dimensionless flux and the rejection coefficient v, [according to Eq. (6) ] as well as the evolution of the non-blocked pore distribution in time. Use of the approximate solution according to Eq. ( 16 ) [or ( 17) ] instead of the exact solution of the IDE given by Eq. ( 10) was also analyzed. Eq. ( 10) was solved for three log-normal membrane pore and particle size distributions:

(i) e,=O.OOl, a,=O.l; e,= 1.04, a;,=O.2; (ii) e,=O.Ol, o,=O.l; e,=O.Ol, a,=O.l;

(iii) e,= 1.04, a,=O.2; ep= 1.04, cr =0.2. Eq. (9) was also solved for combinathns of the bilog-normal distribution for membrane pores:

(i) e,,,=O.Ol, e +=0.95; cTm,s=o.l, f&&=0.5; (ii) e,,,=O.Ol, e,,b=0.79; 0~,~=0.1, um,b=0.3;

and the following log-normal distributions for particles:

(i) e,= 1.04, q=O.2; (ii) e,=O.44, a,=O.2.

In all cases, the exact solution was compared with the approximate solution according to Eq. ( 16 ) or ( 17 ) rewritten in the dimensionless form

i(t)=joo[l-_(l-_jm)exp(Xi,t)l-’

ifj, #O (18)

and

j(t)=(l+xt)-’ ifj,=O (19)

where j,- -.JJJ,, and x=&t* c. In the case j, # 0, parameters j, and x (or x only in the case j,=O) were found by comparison with the corresponding exact solution using a least-squares fit procedure. For all but one combination of pore and particle diameter distributions mentioned above, reasonable agreement was obtained between the exact and approximate solutions. One example of these comparisons is presented in Fig. 2 (a). In Fig. 2 (a), the average particle size E,, is bigger than the average pore size Em; therefore,

‘i-... 04.

-t\+_

03 --------w 02

i

(b)

Fig. 2. Comparison of the exact solution of Eq. ( 10) (solid line) for dimensionless dependency of flux on time and the approximate dependency according to Eq. ( 16) or ( 17 ) (dotted line). (a) E,= 1.015, &,=O.Ol; E,=2.886, +0.34; (b) E,,= 1.015, E_=2.305, z&,=0.01, .&,=OS; E,=2.886, &=0.34. (m) Exact solution; (+ ) according to the approximate solution.

A. Filippov et al. /Journal of Membrane Science 89 (I 994) 199-213 207

i, should be 0. After fitting of the exact solution of Eq. (10) using Eq. (18) or (19) the fitted value j, was found to be equal 0 in accordance with the theory prediction. This confirms that the approximate solution works in this case.

Agreement was unsatisfactory only in the case of the bilog-normal pore diameter distribution combined with the log-normal particle size distribution in which the average particle size, Ep,

is bigger than the average value of the larger membrane pores [see Fig. 2 (b) 1. In this case, there was good agreement at short times, but not at longer times. As shown below, this case is a peculiar one and that the peculiarity probably is a source of the failure of the approximate solution in this particular case.

presented. The curves in Fig. 3 show that increasing the average particle size (from curve 1 to curve 5 ) results in a more pronounced flux decrease and rejection coefficient increase with increased time. These phenomena take place because the increased fraction of the bigger particles results in an increased number of pores being blocked, which improves rejection but decreases permeability.

As mentioned above, the bilog-normal distribution is a good approximation for pore size distribution in globular-cellular MF membranes. It is interesting in this case to calculate the time dependency of the non-blocked pore size distribution. Numerical solutions of Eq. (10) are presented for two cases:

In Fig. 3 results of numerical calculations of first, the average particle size is between the rejection coefficient and dimensionless flux ac- two maxima of the initial membrane pore distri- cording to Eqs. ( 6) and ( lo), respectively, are bution (Fig. 4); and

9 I I I 0.9

- 0.6

0 100 200 300 400

1. min

Fig. 3. Calculated time dependencies of dimensionless flux [ Eq. ( IO) ] and rejection coefficient (p [ Eq. (6) 1 for the MF membrane with the log-normal pore size distribution (E,= 10, &,,= 5, &in= 1, d,,,,= 20) and the following particle size distributions: (l)E,=S,C,=3,D,i,=l,D,,=ll; (2)E,=lO,~~:,=3,0,,=5,0,,= 15; (3)E,=ll,~~=3,D,i,=6,0,,=16; (4) E,=13,~~=3,D,,,=S,D,,=lS; (5) E,,=16,CP=3,D,,,=10,D,,=20.

208 A. Filippov et al. /Journal of Membrane Science 89 (I 994) 199-213

0.25

0.20

0.15 a

-E

0.10

0.05

0.00

a

b

+ c

0 5 10 15 20 25 30 35

diameter, pm

Fig. 4. Particle size distribution and change of the non-blocked membrane pore size distribution with time. E,,,,<: E,<E,,,. The initial pore size distribution is bilog-normal ((u=O.6, Em,=& Z,,,,=5, E,,=22, &,=7, &in= 1, d,,,=35, solid curve 2). The particle size distribution is log-normal (I&,= 16, &, = 3, D,,, = 11, D,, = 2 1, solid curve 1). Pore size distributions after: (a) 25; (b) 250; (c) 500 min.

second, the average particle size is bigger than both maxima of the initial membrane pore distribution (Fig. 5).

In both cases the total number of non-blocked pores (both large and small) decreases with time. The membrane pore blocking process is different for these two cases because the rate of decrease is different for small and large pores and depends on the particle size distribution.

In the first case, the fraction of non-blocked small pores decreases in time and the fraction of non-blocked large pores increases in time. Con- sequently, with increasing time, the initial bilog- normal pore distribution transforms into a log- normal distribution with the average pore diameter approximately equal to the larger diameter of the initial pore distribution.

In the second case, the large pores are blocked

faster than the small pores and the fraction of non-blocked small pores increases in time. If the small pore diameters are in the range of ultralil- tration pores (which is sometimes the case), the MF membrane transforms into an ultrafiltration membrane. This is the case that showed the worst agreement between the exact solution and the approximate one [Fig. 2 (b) 1.

The phenomena of transformation of an MF membrane into an ultrafiltration membrane has been observed experimentally [ 25 1. The theory presented here predicts for the first time this transformation. This transformation can be avoided by using MF membranes with bilog- normal pore distribution for purification of colloid solutions having an average particle size less than the larger membrane pore maxima.

A. Filippov et al. /Journal ofMembrane Science 89 (1994) 199-213 209

0.25

0.20

0.15 P +

+-E

0.10

0.05

0.00 1 0 5 10 15 20 25 30 35

diameter, pm

--.--- a

b

+ c

Fig. 5. Particle size distribution and change of the non-blocked membrane pore size distribution with time. Ep z E,,,b. The initial pore size distribution is bilog-normal (cr = 0.7, E,,,, = 8,~~,=5,E,,=22,C,,=7,d,i.=1,d,,=35,solidcurve2).Theparticle size distribution is log-normal (I&, = 26, Z,, = 3, Dmin = 2 1, D,, = 31, solid curve 1). Pore size distributions after: (a) 25; (b) 250; (c) 500 min.

4. Experiments

Two series of experiments were conducted. Track-etched membranes were used in the first series to check Eqn. (9) and a commercially available teflon membrane was used in the second series to verify the similarity property of MF processes according to Eq. ( 11) and the appli- cability of the approximate solution Eq. ( 15 ) .

Track-etched poly (ethylene terephthalate) membranes (commercial name LAVSAN) were supplied by Dr. B. Mtchedlishvily (Track Etched Membranes Laboratory, Moscow Institute of Crystallography, Russian Academy of Sciences, Leninski pr. 59, Moscow 117333, Russian Fed- eration ) . The experiments were conducted in a standard dead-end MF setup with a membrane area of 32.17 cm*. The cell was pressurized to

f,(D)

Fig. 6. Example of particle size distribution of the water used (dilute colloid solution) for MF experiments on track-etched membranes.


V, ml

V, ml

Fig. 7. Comparison of experimental dependencies of the water volume in the permeate on time and fitted theoretical dependencies (solid lines) in two cases: track-etched mem- branewith (a) d=2pmand (b) d=O.l pm; (w) experimental points.

operating pressures in the range from 0.2 to 0.9 atm (that is, a typical pressure range for microfiltration) and the permeate volume of water collected as a function of time was measured. The feed was City of Austin tap water, which served as a dilute colloidal solution. Particle concentration and size distribution in the water were determined using a Coulter Multisizer Counter. A typical particle distribution is shown in Fig. 6. Four track-etched membranes of pore diameters 0.03, 0.1, 0.45, and 2 pm were used. For each membrane two experimental curves were con- structed at different pressures with the same water (dilute colloid solution). Experiments with different membranes were conducted on different days; consequently, particle distribution and their concentration in water might be different for different membranes. As mentioned above,

j(4)

1 1

50 100 t, min

Fig. 8. Experimental and theoretical dependencies of the dimensionless flux on time for the MF teflon membrane MM& (~)dp,=0.55atm;(O)dp~=1.8atm;(+)dp,=1.05atm. Solid line corresponds to the fitted theoretical dependency j(&)=j,{l- (1 -j,)exp( -7 j, 6 )}-I, with j,=O.O3, g=O.18 min-‘.

parameters a and o were determined using a lit- ting procedure. In Fig. 7 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. (9 ) are in the excellent agreement with the experimental results.

According to the theory presented above, parameters a and o should be independent of the applied pressure drop (a manifestation of the similarity property in the case of track-etched membranes). To test this, values a and o were determined from separate experiments with different applied pressures. As mentioned above, the pressure ranged from 0.2 to 0.9 atm; over this pressure range a and o varied by less than 10% in all cases, but in some cases d=O. 1 pm and d=0.45 pm) those variations were less than 1%. The good lit of the data and pressure independence of a and o support the theory presented here as a description of the MF process for track- etched membranes.

The pressure independence of a and o suggested by the data is not the only means of con- firming the theory. A stronger test would be to determine whether the calculated values of j3 are reasonable. One would expect, for example, B

A. Filippov et al. /Journal of Membrane Science 89 (1994) 199-213 211

should not exceed the ratio of membrane area to pore area. It should be possible to calculateb and compare it to this ratio for the track-etched membranes. To do this, pore diameters and pore number density need to be determined for the membranes.

The second series of experiments was per- formed on a commercially available teflon MF membrane having an average pore diameters of 0.35 brn (commercial name MM@, supplied by VNIISS, Vladimir, Russian Federation). Con- centration of colloid particles in the feed solution was 55,000 particles per cm3 and the concentration in the permeate was 55 to 100 particles per cm3; thus, the rejection coefficient was close to 1. In each experiment the initial water permeability, b, was 0.87 ,ul cm* min kPa. In Fig. 8 experimental data for flux versus time for three different pressures are presented. Pressure dp, = 0.5 5 atm was chosen as a basis, p*. Concen- tration of colloid particles in the feed solution was fixed and according to Eq. ( 11) experimental curves J1 (Apd,lA~~)lJ~, and JA&~zIAP~ )/Jo2 should lie on the experimental curve J3 ( t3) /Jo3. The universal curve (Fig. 8 ) confirms the similarity property of the MF process. The solid curve in Fig. 8 presents the fitted curve according to the approximate solution Eq. ( 16)) rewritten in the following form

j(t,)=i&-(1-L) exp(-%t;)Y

where y= yc J,,.

5. Conclusions

A mathematical model of the dead-end microfiltration (MF) process of dilute colloid suspensions is suggested. 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; an integro-differential equation (IDE) that includes both the membrane pore size and particle size probability distribution functions is deduced. According to the suggested model a similarity property of MF processes is deduced. This property allows one to predict the flwr as a function of time for any

pressure and concentration (dilute) based on one experiment at a single pressure and concentration. The similarity property is in good agreement with the experimental data on both track- etched and commercial teflon membranes. Based on the deduced IDE for flux a condition is found where an MF membrane transforms into an ultrafiltration membrane in the course of the MF process and, hence, that undesirable transformation can be prevented. An approximate equation for flux including two fitting parameters is derived from the IDE and is in good agreement with the experimental results on a commercial MF teflon membrane.

6. Appendix

Solution of Eq. (7) for track-etched membranes with two kinds of pores: some with small diameter d, and some with large diameter d,, is presented. In this case the probability distribution function of membrane pore diameters has the form

fm(d)=a6(d-d,) + (1 -a)b(d-db) (Al)

where cx is the fraction of smaller pores and 6(x) is the Dirac delta function. For ordinary track- etched membranes, (Y = 1.

For the case &,, <a,, the MF membrane has a constant rejection. In the case a, < &,< &, particles go through larger pores and block smaller pores. In this case, Eq. ( 13 ) gives

J(f)=J,+J,B exp -(a/?dIfc/4) [

X Df=f.odo 1 JOdu] (A21 1 0

where J,=[nIV,,,Apd~(l-a)]/(128 pL S), Ja= (x&,Apd~a)/( 128&T). Eq. (A2) can be rewritten as

(dV/dF)/S=J,+J,B


x f,ww V(t) s I

1

Solution of the latter equation with boundary condition V( 0) = 0 results in the following equation for the flux on time dependency, J( F) :

J(~)==J,{l-(l-J,/J,)exp(-vJ,~)}-’

(A3)

where

1

From Eq. (A3 ), if I&,,, > &,, (that is, .I, = 0) ,

the flux dependency on time is as follows:

J(+J,{l+vJ,~-’ (A4)

7. List of symbols

f:

f t

;

s P

k B J

P L h M V

z k

particle diameter pore diameter probability distribution function time concentration characteristic scale of pore and particle diameters membrane area probability number of particles number of pores see definition after Eq. ( 13) flux pressure length of the pore membrane thickness porosity volume of liquid in the permeate see definition after Eq. (9 ) see definition after Eq. (4 ) permeability

e, E parameters in log-normal distribution function

7.1. Greek letters

B

T

Y X 9

parameter to characterize membrane pore influence viscosity Dirac delta function fraction of small pores see definitions after Eq. (9) parameters in log-normal probability distribution functions tortuosity see definitions after Eq. ( 15 ) see definition after Eq. ( 19) rejection coefficient

7.2. Subcripts

P m min max d

s, b tr 00

*

particle membrane minimum value maximum value particles with diameter D < d smaller initial value bigger track-etched membrane at time tends to 00 specific value

7.3. Superscripts

* - **

characteristic value dimension time or diameter in Eq. ( 14) according to mean value theorem

8. Acknowledgments

The authors express their gratitude to the Texas Advanced Technology Program and the Univer- sity of Texas Separations Research Program for their financial support and to Dr. V. Gorsky for his assistance in computer programming.

A. Filippov et al. /Journal of Membrane Science 89 (I 994) 199-213 213

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