Ultrafiltration membrane performance:
Effects of pore blockage / constriction
Yuriy S. Polyakova*, Andrew L. Zydneyb
a
b
USPolyResearch, Ashland, PA 17921, USA
Department of Chemical Engineering, The Pennsylvania State University,
University Park, PA 16802, USA
Abstract
Several recent studies have quantified the performance characteristics of ultrafiltration
membranes in terms of the inherent trade-off between the membrane selectivity and permeability.
However, none of these studies have accounted for the effects of membrane fouling on the evolution of
the selectivity and permeability during typical ultrafiltration processes. This review paper examines a
range of available fouling models, including the classical pore blockage / pore constriction models as
well as newer models based on depth filtration and solute adsorption, with a particular focus on
understanding the effects of these fouling phenomena on the permeability – selectivity tradeoff.
Although fouling always causes a reduction in permeability, the selectivity can actually increase, e.g.,
if the larger (less selective) pores are preferentially blocked during ultrafiltration or if the pores are
constricted by the foulants. The evolution of the permeability – selectivity tradeoff can be quite
complex, depending on both the underlying fouling mechanism as well as the distribution of pore and
solute sizes. These results provide new insights into the behavior of ultrafiltration processes.
Keywords: Ultrafiltration, fouling, pore blockage, permeability, membrane transport
Highlights:

Analysis of effects of pore blockage on ultrafiltration membrane performance

Consideration of multiple fouling models / mechanisms

Review of previous work on permeability – selectivity tradeoff in ultrafiltration

Development of new equations describing effects of fouling on flux and retention
*
Corresponding author. Tel.: +1-347-673-7747.
E-mail address: ypolyakov@uspolyresearch.com (Yu.S. Polyakov).
1
1. Introduction
Ultrafiltration (UF) membranes provided by different manufacturers and made from different
polymeric or ceramic materials are normally characterized by the membrane’s nominal molecular
weight cut-off, which is typically defined as the molecular weight of a solute that has a rejection of
90% [1–3]. The instantaneous value of the rejection coefficient R is evaluated experimentally as:
R 1  C p / C f ,
(1)
where C p and C f are the solute concentrations in the permeate and bulk feed solution, respectively, at
any given instant during the filtration process. Some investigators report results in terms of the solute
sieving coefficient S :
S  Cp / C f ,
(2)
which is simply equal to one minus the rejection coefficient.
Eqs. (1) and (2) define the observed rejection / sieving coefficients, which are functions of the
intrinsic (or actual) properties of the membrane as well as the extent of concentration polarization (CP)
in the membrane module, i.e., the accumulation of rejected solutes at the upstream membrane surface.
The actual rejection and sieving coefficients can be expressed in terms of the solute concentration at
the upstream surface of the membrane ( Cw ) as:
Ra 1  C p / Cw  1  C p /  C f ,
(3)
S a  C p / Cw = S /  ,
(4)
where  = Cw/Cf is the CP factor and is related to the magnitude of the filtrate flux and bulk mass
transfer coefficient in the membrane module [1]. The measured rejection coefficient approaches Ra in
the limit of low filtrate flux, i.e., under conditions where concentration polarization effects are
minimal.
Recent studies [4,5] have demonstrated that a more appropriate framework for describing the
performance characteristics of different ultrafiltration membranes is the trade-off between the
membrane selectivity and membrane permeability. For traditional ultrafiltration processes used for
protein concentration, the selectivity  is defined as the ratio of the flux of a small solute (e.g., a
buffer component) to that of the product / protein of interest and is thus equal to the reciprocal of the
protein sieving coefficient (assuming that the sieving coefficient of the small solute is equal to one):
1
S
 .
(5)
The membrane permeability is defined as:
2
L p  J / P ,
(6)
where J is the permeate flux through the membrane at a transmembrane hydraulic pressure P .
Fig. 1 shows the selectivity – permeability tradeoff for a variety of cellulosic and polysulfone
ultrafiltration membranes using bovine serum albumin as a model protein. Increasing the membrane
pore size leads to an increase in the membrane permeability but with a corresponding reduction in the
selectivity [3]. The data for all of the membranes examined in Fig. 1 appear to fall along the same
curve; none of these membranes has the combination of high permeability and high selectivity that
would be desired for high performance ultrafiltration membranes / processes. A number of recent
studies have used the selectivity – permeability tradeoff to analyze the performance of surface
modified membranes [6] or membranes made with novel pore structures [7,8].
Fig. 1. Selectivity–permeability tradeoff (here,  1/ Sa ) for a variety of cellulosic and polysulfone
ultrafiltration membranes using bovine serum albumin as a model protein. Solid black curve is
a model calculation using a log-normal pore size distribution. Thin blue curve is given by Eq.
(18). Adapted with permission from [4].
The actual filtrate flux during protein ultrafiltration will be lower than the pure water flux due
to both osmotic pressure effects and membrane fouling:
J
P   os 
rm  rc
(7)
3
where ∆ is the osmotic pressure difference across the membrane, rm is the hydraulic resistance of the
membrane (equal to 1/L0 in the absence of any pore fouling, where L0 is the pure water membrane
permeability), and rc is the hydraulic resistance of any cake layer formed on the membrane surface.
The parameter os is the osmotic or Staverman reflection coefficient and is equal to unity for a
perfectly retentive membrane (Ra = 1) and zero for a totally non-retentive membrane. Note that Eq. (7)
ignores the hydraulic resistance of the CP boundary layer, which is usually negligible compared to rm
and rc in most ultrafiltration systems. Zydney [3] has examined the effects of concentration
polarization on the selectivity – permeability tradeoff in the absence of membrane fouling, i.e., under
conditions where rc = 0 and the membrane resistance remains constant.
One of the major limitations of the selectivity – permeability analysis presented in Fig. 1 is that
it does not consider any of the fouling phenomena that occur during almost every ultrafiltration
process. Numerous studies [e.g., 9–24] have shown that both surface and pore fouling can dramatically
affect the permeate flux and retention characteristics of ultrafiltration (UF) membranes. The objective
of this review is to examine the current theories of complete blocking and pore constriction and
evaluate the effects of these fouling phenomena on membrane transport and, in turn, the selectivitypermeability tradeoff for UF membranes. The first section of this review examines underlying
transport theory, with an emphasis on the relationship between the intrinsic membrane transport
properties (both permeability and sieving coefficient) and the underlying pore size characteristics of
the membrane.
2. Transport Theory
Expressions for solute and solvent transport through UF membranes are typically developed
from steady-state hydrodynamic models of hindered diffusion and convection [25–28]. In these
models, the solute flux is evaluated by directly solving the governing set of hydrodynamic equations
for the motion of a single solute particle in a well-defined (usually cylindrical) pore. The hindrances to
diffusion and convection, which are due to solute steric restriction at the pore entrance and frictional
drag caused by hydrodynamic interactions with the pore wall, are accounted for by simple hindrance
factors, which are functions of the ratio of the solute to pore radius. The hydrodynamic interactions
depend on how close the particle is to the wall and are thus a function of any force that influences the
equilibrium particle position within the pore. Even if long-range (e.g., van der Waals or electrostatic)
forces are negligible, the finite size of the solute restricts its access to the region near the pore wall and
4
therefore affects its flux. The solute concentrations in the pore interior are related to those in the
solution external to the membrane pores by the solute equilibrium partition coefficient. The resulting
transport and partition coefficients can be substituted into the differential equations (extended NernstPlanck equation, Spiegler-Kedem equation, etc.) obtained by equating the gradient in the chemical
potential of the solute to the hydrodynamic drag force acting on the solute in the pore [28–34].
2.1 Extended steady-state Nernst-Planck equation
The extended differential Nernst-Planck equation accounts for the solute flux due to hindered
diffusion, convection, and electrophoretic motion driven by either an applied or induced electric field
[30,31]. The electrostatic potential inside the pore can be found by solving the Poisson-Boltzmann
equation for a charged particle inside a charged pore [30,34–37].
Generally, the extended differential Nernst-Planck equation can be solved only by numerical
methods [30]. However, the linearized Poisson-Boltzmann equation allows for analytical solutions [35,
36], which can be used to directly write simple expressions for the electrostatic potential energy for a
charged hard-sphere solute in a long pore with charged walls, with the resulting expression used to
evaluate the solute equilibrium partition coefficient. This enables one to use simple analytical
expressions for the solute flux to evaluate the protein sieving coefficient [3,4,5,29,32,36–40].
For an uncharged solute in an uncharged pore, the extended Nernst-Planck equation reduces to
the well-familiar Spiegler-Kedem differential equation for solute transport through a membrane.
Integration of the Spiegler-Kedem equation over the pore length gives a simple analytical expression
for the actual sieving coefficient [29,38]:
Sa 
 K c exp  Pem 
 K c  exp  Pem  1
,
(8)
where  is the solute partition coefficient, Kc is the hindrance factor for solute convective transport,
and Pem is the membrane pore Peclet number:
 K  J l p 
Pem   c 
,
K
D
 d   
(9)
where K d is the hindrance factor for solute diffusive transport, l p is the membrane thickness (pore
length), and D is the solute diffusion coefficient in the free solution outside the pore.
5
For an uncharged hard sphere in an uncharged cylindrical pore, the partition coefficient
accounts for the steric exclusion of the solute from the region within one solute radius of the pore wall
[28]:
  1    ,
2
(10)
where   rs / rp , and rs and rp are the solute and pore radii, respectively. When 0    0.8 , the
hindrance factors for convection ( Kc ) and diffusion ( K d ) evaluated using the centerline approximation
are given as [28,33,41]:
K c  1  2   2 1.0  0.054  0.988 2  0.441 3  ,
(11)
K d 1.0  2.30  1.154 2  0.224 3 .
(12)
More detailed expressions for the hindrance factors at larger  are available in the literature [28].
The partition coefficient for a charged hard sphere in a cylindrical pore with charged walls can
be expressed as [28,30,36,38]:
  e 
,
k
T
 B 
  1    exp 
2
(13)
where  e / kBT is the dimensionless electrostatic energy of interaction, k B is the Boltzmann constant,
and T is the absolute temperature. The electrostatic energy can be evaluated by solving the linearized
Poisson-Boltzmann equation with the final result given as [35,42]:
e
k BT

As s2  Ap p2  Asp s p
Aden
,
(14)
where As , Ap , Asp , and Aden are all positive coefficients that depend on the solution ionic strength,
pore radius, and solute radius;  s and  p are the dimensionless surface charge densities of the solute
and pore wall, respectively. Detailed expressions for these coefficients are given elsewhere
[35,36,42,43]. The first term in the numerator of Eq. (14) is associated with the distortion of the
electrical double layer around the solute caused by the presence of the pore boundary, leading to a
repulsive interaction even when the membrane has no net electrical charge. The second term accounts
for the increase in free energy associated with the deformation of the double layer adjacent to the
charged pore walls associated with the presence of the solute and is repulsive even when the solute
bears a net neutral charge. The last term accounts for the direct charge-charge interactions between the
solute and the pore and is positive when the solute and pore have like polarity.
6
The contribution of the electrophoretic term in the Nernst-Planck equation (in the absence of an
applied external electric field) is typically assessed using an approximate analytical solution developed
by expressing the electrostatic potential in the cylindrical pore as the pairwise summation of the
potential energies arising from (1) the interaction between the solute and pore in the absence of any
flow and thus in the absence of a streaming potential, and (2) the interaction between the solute and the
streaming potential in an unbounded system with the streaming potential assumed to be unaffected by
the presence of the solute [30]. This approximation thus neglects the possible effects of the streaming
potential on the structure of the double layer surrounding the solute, as well as any effects of the solute
on the streaming potential. Additionally, the derivative of the potential energy of interaction associated
with the streaming potential is typically assumed to be that of an equivalent electric field acting on an
isolated solute in an unbounded system. This means that it is proportional to the streaming potential
and the solute electrophoretic mobility ue . The streaming potential in this system is proportional to the
average solution velocity with the proportionality constant  being a complex function of the pore
radius, the Debye length, and the solution conductivity. Under these conditions, the actual membrane
sieving coefficient can be expressed as [30]:
Sa 
 K c 1    exp  Pem 1    
 K c 1     exp  Pem 1     1
,
(15)
where    Kd / Kc  ue is the electrophoretic ratio describing the relative importance of
electrophoretic transport compared to convection inside the membrane.
In another approach [31], the apolar (Lifshitz-van der Waals), electrostatic, and polar (acidbase) interactions between spherical particles and pore walls were estimated by an original surface
element integration method and used to calculate Kc , K d , and  in addition to the osmotic reflection
coefficient (os).
The hydraulic permeability of a membrane with uncharged uniform cylindrical pores can be
evaluated from the Hagen-Poiseuille equation as:
Lp 
J
  rp2 /  8 l p  .
P
(16)
where  is the membrane porosity (pore area per unit membrane cross-sectional area), µ is the solution
viscosity, and lp is the pore thickness. Eq. (16) can also be used to evaluate the equivalent pore radius
from the measured value of the membrane permeability. Note that this definition of the equivalent pore
radius ignores the effects of a pore size distribution and tortuosity on the measured value of the
permeability.
7
The solvent flux through a charged membrane is reduced compared to that through a membrane
with uncharged pores due to counter-electroosmosis arising from the induced streaming potential.
Mehta and Zydney [32] evaluated the effects of counter-electroosmosis on the effective membrane
permeability as a function of solution ionic strength and membrane zeta potential by solving the
Navier-Stokes equation including the electrical stress term.
2.2 Pore size distribution effects
The effect of a pore size distribution on the membrane permeability and sieving coefficient has
been examined by a number of investigators [3–5,32,36,44]. These studies have typically employed the
log-normal pore size distribution, which is defined only for positive values of the pore radius (in
contrast to the standard Gaussian) and more accurately captures the longer tail of the distribution at
very large pore radii [45]. The solid curve in Fig. 1 represents the model calculations developed using
Eqs. (8) and (18) but with the expressions for the solute flux integrated over the log-normal pore size
density using a coefficient of variation (the ratio of the standard deviation to the mean pore radius)
equal to 0.2. The model is in good agreement with the experimental data, although it should be
recognized that the selectivity is plotted on a logarithmic scale. The deviation between the model and
data could be due to differences in pore size distribution, thickness, porosity, and / or tortuosity
between the different membranes. It is also possible that the lower selectivities seen with some of the
polysulfone membranes could be due to protein sorption on these more hydrophobic membranes [46].
Several authors have noted that the pore size distribution in existing UF membranes has a fairly
small effect on the permeability–selectivity tradeoff [3,44]. For example, current UF membranes have
a permeability of approximately 32 L/  m 2 h psi  at a sieving coefficient of 0.01 [44]. Completely
eliminating the pore size distribution, that is, having a membrane with perfectly uniform pores of a
single size, only provides a 60% increase in permeability [44]. The pore size distribution would have a
much larger effect on the behavior for highly selective separations, e.g., for membranes with
selectivities greater than 104, or for separations between solutes with similar size.
Mochizuki and Zydney [38] developed an approximate expression for the sieving coefficient
using a simple theoretical expression for the partition coefficient in a porous media formed by the
intersection of a random array of planes:
 r
Sa  exp   s
 s

,

(17)
8
where s is the specific area of the pores, equal to the total pore volume divided by the pore surface
area. Opong and Zydney [29] evaluated the specific pore area in terms of the membrane permeability,
giving a simple analytical expression for the sieving coefficient in terms of the permeability:

rs
Sa  exp  2 1/2

 Lp l p


.


(18)
Eqs. (17) and (18) neglect the effects of the hindrance factor for convection; this has a relatively small
effect on the sieving coefficient since Kc only varies between 1.0 and 1.47 for an uncharged sphere in
an uncharged cylindrical pore. Eq. (18) has been shown to be in good agreement with experimental
data for BSA and cytochrome c [3]. The thin blue curve in Fig. 1 represents the predicted selectivity –
permeability tradeoff given by Eq. (18) with rs = 3.6 nm,  = 0.25, and lp = 1 µm.
3. Complete blocking models
The mechanism of complete blocking is illustrated in Fig. 2. Solutes of various sizes are
dragged by liquid flow to the mouths of pores of various diameters. Solutes with sizes less than the
pore diameter can enter and pass through the pore, whereas solutes that are equal to or larger than the
pore diameter completely plug the pore mouth.
Fig.2. Schematic diagram of complete blocking for solutes and pores of various sizes.
3.1 Classical complete blocking
The initial volumetric permeate flow rate G0 through one cylindrical pore can be written
according to the Hagen-Poiseuille equation [47]:
G0 
 rp4 P
.
8  lp
(19)
9
The initial permeate flux through a membrane consisting of an array of uniformly sized pores is thus
given as
J 0  G0 N p ,
(20)
where N p is the number of pores per m2.
If all solutes are the same size with rs  rp , the solute rejection coefficient will be R  1 and the
selectivity relative to a small solute will be infinite. The probability of pore plugging is proportional to
the solute concentration and the total cumulative permeate volume. Under these conditions, the
permeate flux J at any time t is given as:
t


J  t   G0  N p  C0  J  u  du  ,
0


(21)
where C0 is the number concentration of solutes in the feed solution and t is the time.
Differentiating Eq. (21) with respect to t gives the differential equation:
dJ
 G0C0 J  0 ,
dt
(22)
which can be easily solved using the initial condition given by Eq. (20), yielding:
J  t   J 0 exp  kG t  ,
(23)
where kG  G0C0 is an empirical fouling coefficient that is typically determined experimentally from a
plot of the logarithm of the filtrate flux as a function of time. Since the classical complete blocking
model assumes that all solutes are retained by the membrane, the selectivity remains infinite while the
permeate flux declines throughout the ultrafiltration process.
3.2. Complete blocking with pore- and solute-size distributions
The theoretical expressions developed in this section are based on the general equations
formulated by Santos et al. [48,49], which were originally presented for filtration of a particulate
suspension through porous rock.
When a membrane has a pore-size distribution with minimum pore radius rpmin and maximum
radius rpmax , the permeate flux is given by an integral over the pore size distribution [49]:
P
J t  
8  lp
rpmax
 n  r , t  r dr
p
4
p
p
,
(24)
rpmin
where n  rp , t  is the pore concentration density satisfying the equality:
10
n  rp , t  drp  N  t  f n  rp , t  drp .
(25)
Here, N  t  is the total number of open pores per m2 at time t and f n  rp , t  is the probability density
function:
rpmax
 f  r , t  dr
n
p
p
 1.
(26)
rpmin
For cross-flow ultrafiltration, rpmax may be replaced by the effective radius rpeff , which corresponds to
the threshold value of the solute size when shear forces start preventing solutes from plugging
membrane pores [46], although this effect is likely to be important primarily for larger (micron-size)
particles.
The pore plugging kinetics are assumed to be given as [48,49]:
dn  rp , t 
dt
J  t  n  rp , t  rp4
  r max
 n  r , t  r dr
p
p
4
p
rpmax
 c  r  dr
s
,
s
(27)
rp
p
rpmin
where c  rs  is the solute concentration density satisfying the equality
c  rs  drs  C f s  rs  drs .
(28)
Here, C is the total number of solutes per m3 of solution (assumed to be constant) and f s  rs  is the
solute probability density function obeying the relation:
rsmax
 f  r  dr
c
s
s
1.
(29)
rsmin
Eq. (27) states that the pore-plugging rate for pores of radius rp is proportional to the permeate flux
through those pores multiplied by the number concentration of hard-sphere solutes with radii equal to
or larger than rp , which is a generalization of the results for the classical pore blockage model.
The permeate flux can be eliminated from Eq. (27) using Eq. (24), yielding
dn  rp , t 
dt
P

n  rp , t  rp4
8  lp
rpmax
 c  r  dr
s
s
.
(30)
rp
Eq. (30) can be integrated to evaluate n  rp , t  :
11
rp

4
n  rp , t   n  rp , 0  exp   rp t  c  rs  drs

rp

max
where  

,


(31)
P
. Eq. (31) can be substituted into Eq. (24) to evaluate the permeate flux as a function
8  lp
of time:
rp

4
J  t     n  rp , 0  exp   rp t  c  rs  drs

rp
rpmin

rpmax
max

 rp4 drp .


(32)
It can easily be shown that Eq. (32) reduces to Eq. (23) for uniform pore- and solute-size distributions.
The selectivity depends on both the total rejection of solutes with radii equal to or larger than
rp and the exclusion of smaller solutes due to steric and/or electrostatic interactions. For a specific
value of rs , the permeate solute concentration density c p  rs , t  at time t can be evaluated as:
rpmax
c p  rs , t   c  rs 

rs
r
Sa  s
r
 p

4
 n  rp , t  rp drp

,
rpmax
(33)
 n  r , t  r dr
p
4
p
p
rpmin
where S a is the actual sieving coefficient estimated from transport theory by Eq. (8) or (15). The
contribution for pore sizes from
min
rp
to rs is excluded since these solutes are completely retained by the
pores.
The integral sieving coefficient S  for a solute size range from rs to rs  rs can be calculated
as
rs rs
S  t  

c p  r , t  dr
rs
rs rs
,
(34)
 c  r  dr
rs
with the corresponding rejection coefficient written as
R  t   1  S .
(35)
The effects of pore plugging on the membrane selectivity and permeability are illustrated in the
following subsections by examining the behavior of three simple cases showing the most important
features of the process.
12
3.2.1 Two pore sizes and a single solute size
In the case of two pore sizes rp ,1 and rp ,2 and one solute size rs ,1 , the initial pore and solute
concentration distributions can be written as [49]:
n  rp , 0   n1  0    rp  rp ,1   n2  0    rp  rp ,2  ,
(36)
c  rs   C0   rs  rs ,1  ,
(37)
where   x  is the Dirac delta function for the argument x.
Consider the case when rp ,1  rs ,1  rp ,2 . After substituting Eqs. (36) and (37) into Eq. (32) we
obtain


J  t    n1  0  rp4,1 exp  rp4,1 C0 t   n2  0  rp4,2 .
(38)
The dimensionless flux j  t   J  t  / J  0 can then be written as
n1  0  rp4,1 exp   rp4,1 C0 t   n2  0  rp4,2
j t  
.
n1  0  rp4,1  n2  0  rp4,2
(39)
Note that the dimensionless flux j  t   J  t  / J  0 is equal to the dimensionless permeability
 p  Lp / L0 , which will be used to plot the selectivity-permeability trade-offs below. Eq. (39) indicates
that the permeate flux decreases only by plugging the pores of smaller size rp ,1 .
The permeate solute concentration density c p  rs ,1 , t  in this case is given as
r 
C0 Sa  s ,1  n2  0  rp4,2
r 
 p ,2 
.
c p  rs ,1 , t  
4
n1  0  rp ,1 exp   rp4,1 C0 t   n2  0  rp4,2
(40)
As there is only one solute size, the rejection coefficient can be evaluated by
R  R  1 
n1  0  rp4,1
r 
Sa  s ,1  n2  0  rp4,2
r 
 p ,2 
.
exp   rp4,1 C0 t   n2  0  rp4,2
(41)
In terms of dimensionless variables, Eqs. (39) and (41) can be rewritten as:
1  0   p4,1 exp   b   2  0   p4,2
,
 p  b   j  b  
1  0   p4,1  2  0   p4,2
(42)
13
 1 
Sa 
 0 4
   2   p ,2
 p ,2 
R  b   R  1 
,
4
1  0   p ,1 exp   b    2  0   p4 ,2
where 1  0  
(43)
r
r
n1  0 
n2  0 
, 2  0  
,  p ,1  p ,1 ,  p ,2  p ,2 ,  b   rp4,1 C0 t .
n1  0   n2  0 
n1  0   n2  0 
rs
rs
Pore plugging decreases the flow of pure solvent passing through the smaller pores, which
leads to an increase in the particle concentration in the permeate. Ultimately, every pore of smaller size
is plugged and only the pores of larger size are functional, with R approaching the retention coefficient
for the transport of solutes through the larger pores. This behavior is shown in Fig. 3 for a model
system in which rp,1 = 2.5 nm and rp,2 = 5 nm for a protein with rs = 4 nm. Increasing the relative
number of small pores leads to a reduction in the initial permeation flux and the rate of flux decline,
but causes a corresponding increase in the protein rejection coefficient. Note that the permeability in
Figure 3(b) has been normalized by the initial permeability through the small pores; thus,  p is greater
than 1 throughout the process.
The effect of this pore blockage on the selectivity-permeability plot is shown in Fig. 3(c). The
initial performance of the membrane is a function of the ratio n1  0 : n2  0 , with the membrane having
the largest number of small pores having the highest selectivity but lowest permeability. Blockage of
the small pores causes a reduction in both the selectivity and permeability, leading to a linear evolution
of the membrane performance on a plot of the selectivity versus permeability (plotted with the
selectivity on a linear scale, in contrast to the logarithmic scale used in Figure 1).
14
15
Fig. 3. (a) Rejection, (b) dimensionless permeability, and (c) selectivity-permeability plots for
complete blocking with two pore sizes and a single solute size: n1  0 : n2  0  0.5: 0.5 (1),
0.7 : 0.3 (2); and 0.9 : 0.1 (3). R is calculated by Eq. (43) with   1/ 1  R  .  p is evaluated
with (42) but with the permeability normalized by the initial flux through the small pores, i.e.,
with n2  0  0 . Arrows denote the evolution of the selectivity-permeability tradeoff during
ultrafiltration.
3.2.2 One pore size and two solute sizes
In the case of one pore size rp ,1 and two solute sizes rs ,1 and rs ,2 , the initial pore and solute
concentration distributions can be written as
n  rp , 0   n1  0    rp  rp ,1  ,
(44)
c  rs   C01   rs  rs ,1   C02   rs  rs ,2  .
(45)
For the case where rs ,1  rp ,1  rs ,2 , Eq. (32) becomes
J  t    n1  0  rp4,1 exp   rp4,1 C02 t  .
(46)
The dimensionless permeability and flux can then be written as
 p  j  t   exp   rp4,1 C02 t  .
(47)
In this case, the permeate flux declines with time due to the plugging of pores by the larger solute,
eventually declining to zero at infinitely long time.
16
The permeate solute concentration densities c p  rs ,1 , t  and c p  rs ,2 , t  in this case are written as
r 
c p  rs ,1 , t   C01 S a  s ,1  ,
r 
 p ,1 
(48)
c p  rs ,2 , t   0 .
(49)
The rejection coefficients R1 and R2 for solutes with sizes rs ,1 and rs ,2 , respectively, are given as
r 
R1  t   1  S a  s ,1  ,
r 
 p ,1 
(50)
R2  t   1 .
(51)
Under these conditions, the process path on the selectivity – permeability plot for the smaller solute
would correspond to a horizontal line, with the permeability decreasing with time while the selectivity
remains constant. The selectivity of the larger solute is infinite throughout the process.
3.2.3 Two pore sizes and two solute sizes
In the case of two pore sizes rp ,1 , rp ,2 and two solute sizes rs ,1 , rs ,2 , the initial pore and solute
concentration distributions take the form:
n  rp , 0   n1  0    rp  rp ,1   n2  0    rp  rp ,2  ,
(52)
c  rs   C01   rs  rs ,1   C02   rs  rs ,2  .
(53)
Here we consider the case when rp ,1  rs ,1  rp ,2  rs ,2 . Derivations similar to those in Sections 3.2.1 and
3.2.2 give the permeate flux equation:
J  t   n1  0  rp4,1 exp   rp4,1 C0t   n2  0  rp4,2  exp   rp4,2 C02t  .
(54)
Eq. (54) implies that the permeate flux declines with time and ultimately vanishes when all of the pores
are plugged.
The dimensionless permeability and flux can then be written as
 p t   j t  
n1  0  exp   rp4,1 C0t  rp4,1  n2  0  exp   rp4,2 C02t  rp4,2
n1  0  rp4,1  n2  0  rp4,2
,
(55)
where the rejection coefficients R1 and R2 for solutes with sizes rs ,1 and rs ,2 , respectively, are given by
R1  t   1 ,
(56)
17
r 
Sa  s ,1  n  rp ,2 , 0  exp   rp4,2 C02t  rp4,2
r 
 p ,2 
.
R2  t   1 
n  rp ,1 , 0  exp   rp4,1 C0t  rp4,1  n  rp ,2 , 0  exp   rp4,2 C02t  rp4,2
(57)
In dimensionless form, Eqs. (55) and (57) can be written as:
 p  b   j  b  
1  0  exp   b   p4,1  2  0  exp   b   p4,2
,
1  0   p4,1  2  0   p4,2
(58)
 1 
Sa 
 0 exp   b   p4 ,2
   2  
 p ,2 
R2  b   1 
,
1  0  exp   b   p4 ,1  2  0  exp   b   p4,2
where 1  0  
 
rp4,2 C02
rp4,1 C0
n1  0 
,
n1  0   n2  0 
2  0  
n2  0 
,
n1  0   n2  0 
 p ,1 
(59)
rp ,1
rs ,1
,
 p ,2 
rp ,2
rs ,1
,
 b   rp4,1 C0 t ,
.
In this case, the pure size-exclusion rejection shows a complex behavior depending on the
interplay between the parameters for both pores and solutes. With C02  0 , Eqs. (54) and (57) reduce to
Eqs. (38) and (41), respectively, with the rejection coefficient decreasing with time. Eq. (57) predicts
that the rejection coefficient decreases with time only when rp4,2 C02  rp4,1 C0   1 , in which case the
solvent flow through the smaller pores declines faster than the flow through the larger pores. In
contrast, when rp4,2 C02  rp4,1 C0   1 , the rejection coefficient increases with time because the larger
pores are plugged faster than the smaller ones. This behavior is illustrated in Fig. 4, which shows the
predicted rejection coefficient for the smaller solute as a function of time for several values of C02 :C01 .
The initial rejection coefficient is independent of the relative amount of the small and large solutes.
The rejection coefficient decreases with time for the smaller ratios of C02 :C01 , with the reverse
behavior seen for large ratios of C02 :C01 .
18
Fig. 4. Rejection of small solutes as a function of time for complete blocking with two pore sizes and
two solute sizes: C02 :C01 = (1) 0:1.0, (2) 0.05:0.95, (3) 0.065:0.935, (4) 0.15:0.85, (5) 0.5:0.5;
R2 evaluated by Eq. (59); rs ,1  4 nm; rs ,2  6 nm; rp ,1  2.5 nm; rp ,2  5 nm .
3.2.4 Concluding remarks
The above three cases clearly demonstrate that complete pore blocking can cause the solute
rejection coefficient to decrease, increase, or remain constant during an ultrafiltration process
depending on the relative size and concentration of the solutes and pores. At the same time, the
permeate flux decreases with time and can ultimately vanish or come to a steady state value, with the
latter occurring only when the maximum solute size is smaller than the maximum pore diameter. In
this case, the flux decline is characterized by the plugging of the smaller pores on the membrane
surface while the larger pores remain open and determine the steady state value of the flux. The fouling
caused by complete blocking reduces the selectivity as the filtrate flow is directed towards the larger
(less selective) pores. The greater the ratio of the number of small (to be plugged) to large (to remain
open) pores, the greater the reduction in both the selectivity and permeation flux. The permeate flux
and rejection coefficients for more complex pore- and solute-size distributions (e.g., log-normal
distribution) can be determined by the integrals in Eqs. (32) and (35), respectively.
It should also be noted that of some theoretical interest is the fouling model in which the
probability of pore plugging is assumed to be proportional to the pore influence area (pore area
multiplied by some factor) rather than to the volumetric flow rate through the pore [50–53]. This
19
theoretical approach was tested for deadend microfiltration using a limited set of experimental data
obtained with track-etched membranes having very narrow pore size distributions. The probability of
pore plugging in this “pore area” approach is proportional to rp2 , which is in contrast to the rp4 function
on which the conventional “volumetric” approach is based. There is no independent evidence showing
the applicability of the pore-area fouling model to UF pore blocking, while numerous studies have
suggested that the volumetric approach provides an accurate description of the flux decline data during
UF [1,10–24].
4. Pore constriction models
When the size of a solute is smaller than the hydraulic diameter of a pore, the solute can enter
the pore. In this case, it can pass into the permeate or approach the pore wall and get deposited on the
pore surface. The deposition of solutes on the pore wall leads to pore constriction, reducing the
hydraulic (flow) diameter, which causes a decline in the permeate flux and a change in solute rejection.
In addition, the deposition of solutes on the pore wall can change the effective surface charge density
of the pore (especially if the solute is charged), which can also alter the retention characteristics. Thus,
pore constriction can change both the steric and electrostatic interactions. For simplicity, we will
consider cylindrical pores and hard-sphere solutes. Pore- and solute-size distribution effects and solute
diffusion inside pores are ignored.
4.1. Classical standard blocking
Like the complete blocking model, the classical standard blocking (or pore constriction) model
is based on the Hagen-Poiseuille equation for flow through a single cylindrical pore. The permeate flux
produced by 1 m2 of membrane area is given as [1]:
J  Np G 
N p  P rp4
8  lp
,
(60)
where rp is the pore radius at time t and N p is the number of uniform pores per m2 of membrane area.
The volume occupied by the solutes uniformly deposited over the whole pore wall is assumed to be
proportional to the produced permeate volume and the feed concentration (Fig. 5) [54]:
G Sa  rp  Rc dt
c0

 2 rp l p drp ,
(61)
where c0 is the volume fraction of suspended solutes in the feed,  is the porosity of the layer of
deposited solutes, Rc is the fraction of solutes deposited on the pore wall (equal to the average solute
20
rejection coefficient due to pore constriction; discussed in more detail subsequently), and S a is the
solute sieving coefficient due to steric/electrostatic exclusion, which can be estimated from transport
theory by Eq. (8) or (15) for a given value of the solute radius rs .
Fig. 5. Schematic diagram of intrapore solute deposition for standard blocking model.
Eqs. (60)-(61) can be combined and solved to give the radius rp as an implicit function of time:
t
where G0 
2  l p r04
rp
dr
 r S r  ,
3
Rc c0G0
r0
(62)
a
 P r04
is the initial permeate flow rate and r0 is the initial pore radius. In dimensionless
8  lp
form, Eqs. (60) and (62) can be rewritten as:
 p   p   j   p    p4 ,
2
 
Rc
where  p 
rp
r0
p
d
  S  ,
3
1
, s 
(63)
(64)
a
rs
cG
,   0 02 t . The total rejection coefficient, based on the combination of
r0
 l p r0 
steric exclusion and solute capture by pore wall, is given as
Rt   p   1  Sa   p  1  Rc  .
(65)
21
4.2. m-Model
The m-model assumes a stepwise deposit profile over the pore wall caused by the decrease in
the solute concentration as the fluid moves through the pore (Fig. 6). This model was developed in an
attempt to build a more realistic model for solute fouling than the classical standard blocking model
[55]. In this case, the permeate flow rate through a single pore is given as [55,56]:
G
 P
 m 1 m 
8  lp  4  4 
r
r0 
 p

G0
 m 1 m 
r04  4  4 
r
r0 
 p
,
(66)
where m is the ratio of the length of the inlet pore portion with deposited solutes to the entire length of
the pore (Fig. 6). The value of m can be determined empirically by fitting to experimental data or
calculated by the formula given by Polyakov [57,58].
Fig. 6. Schematic diagram of intrapore solute deposition for m-model.
The material balance equation takes the form:
G dt Sa  rp  Rc
c0

 2 m rp l p drp .
(67)
Combining Eqs. (66) and (67) and solving for rp gives the radius as an implicit function of time:
2  ml p r04 p  m 1  m  r dr
t
.
  4 
Rc c0G0 r0  r 4
r0  S a  r 
r
(68)
In dimensionless form, Eqs. (66) and (68) can be written as:
22
p p  j p 
1
m
 p4
2m
 
Rc
p
m
  
1
4
,
(69)
1 m
  d
.
1 m 
 Sa   
(70)
The total rejection coefficient is determined by the size of the inlet region and can thus be evaluated by
Eq. (65), just as in the standard blocking case.
Fig. 7(a) shows ultrafiltration selectivity – permeability plots calculated using the m-model for
several values of m. The initial dimensionless permeability is 1.0 with an initial selectivity of Rt
independent of the value of m. Particle deposition causes a reduction in the permeability with a
corresponding increase in the selectivity due to the constriction of the membrane pores. The highest
selectivity (at a given value of the permeability) occurs with the smallest value of m since the
selectivity is determined by the pore size at the pore entrance while the permeability is determined by
the sum of the resistances in the fouled and clean portions of the pore. Thus, a membrane with small m
and a thick deposit (high rejection) will have the same permeability as a membrane with larger m and a
thinner deposit (and lower rejection). Corresponding results for m = 1 (classical standard blocking
model) but with different values of Rc are shown in Fig. 7(b). In this case, the initial selectivity
increases with increasing Rc due to the greater solute capture by the pore walls. The fouling again
causes a reduction in the permeability and an increase in the selectivity.
23
24
Fig. 7. Selectivity – permeability plots calculated by the m-model: (a) m = (1) 1.0, (2) 0.66, (3) 0.33;
Rc  0.5 ; (b) Rc = (1) 0.5; (2) 0.7; (3) 0.9; m  1 . Here,   1/ 1  Rt  ; Rt evaluated using Eq.
(65); dimensionless permeability  p calculated by (69); rs  4 nm; rp  20 nm ; arrows denote
the evolution of selectivity-permeability tradeoff during ultrafiltration.
4.3 Depth filtration model
A common drawback of both the standard blocking approach and the m-model is that they do
not provide a physical mechanism for the deposition of solutes on the pore wall, nor do they account
for the gradual decrease in the local concentration of suspended solutes over the length of the pore. As
a result, these models are typically unable to accurately describe the change in solute retention with
time due to solute capture by the pore walls (Fig. 8).
25
Fig. 8. Schematic diagram of nonuniform solute deposition [57].
The non-uniform deposition of solutes within the membrane pores can be described by depth
filtration models [57,58], which are based on the macroscopic theory of filtration across deep granular
beds of particle collectors [59–61]. According to this approach, suspended solutes are deposited on the
pore wall, which makes the local concentration of suspended solutes decrease with the axial coordinate
as one moves through the membrane / filter. The solute deposition (collection) rate depends on the
filter (deposition) coefficient, the surface area available for the deposition of solutes inside the pore,
and the local concentration of suspended solutes in the pore [57]. The reduction in local solute
concentration causes the thickness of the solute deposit to decrease over the length of the pore as the
solution moves from the pore inlet to outlet (Fig. 8). The model described below assumes that the
membrane is composed of a parallel array of uniform circular cylindrical pores of radius rp and length
lp, the feed consists of a suspension of uniform hard-sphere solutes of single radius rs. In addition,
solute diffusion inside the pore is ignored and the solution is assumed to be perfectly mixed over the
pore cross-section (i.e., no radial concentration gradients).
The length of the pore is divided into a sequence of circular sections with lengths equal to the
diameter of the solute. The solutes can deposit on the clean pore wall or on the surface of the layer of
solutes previously deposited on the pore wall. The flow in each circular section is described by the
Hagen-Poiseuille formula, which neglects any entrance and exit effects. Since the pore length is much
26
larger than the solute diameter, the sum over the very large number of circular sections can be
transformed to the corresponding integral, yielding [57]:

G  t    P  8 


rp ( )
r0
 1
1
lp

0

dz1
 ,
4
 rp    

(71)

,
0
(72)
where  0 is the initial membrane porosity, and the specific deposit σ (volume fraction occupied by
deposited particles) is a function of time and axial coordinate.
The phenomenological theory of suspension flow across deep granular beds [59,61] is then
used to formulate the macroscopic mass balance equations and initial conditions for flow through a
membrane with uniform cylindrical pores [57]:
   cs  

,

 cs J   
t
z
t
(73)
d
     J cs ,
dt
(74)
cs  c0 ,
when t  0, z  0 ,
(75)
cs  0,   0 ,
when t  0, z  0 .
(76)
where J  N p G is the permeate flux, and c0 is the solute concentration immediately outside the pore
inlet. The effect of the porosity of the layer of deposited solutes is implicitly accounted for by the filter
coefficient  f , which is assumed to depend only on the surface area available for the deposition of
solutes inside the pore:
 f    0
rp
r0
 0 1 

.
0
(77)
The effect of steric/electrostatic exclusion at the pore entrance can be incorporated into Eqs.
(73)-(77) by adjusting the boundary condition (75):
cs  S a  rp  0  c0 ,
when t  0, z  0 ,
(78)
where  0 is the specific deposit at pore entrance ( z  0 ), which can be related to the pore radius at the
pore entrance using Eq. (72).
The system of equations given by Eqs. (71) - (74), (76), and (78) can be transformed to two
uncoupled ordinary differential equations [57]:
27

  N      ,
Z
(79)
   0 at Z  0,
(80)
d 0
  0 j ( * ) N    0  S a  0  ,
d *
(81)
 0  0 at  *  0 ,
(82)
where  * 
c0G0
z
t , Z  , N  0 l p ,      f   / 0 , and  0 is the specific deposit at the pore
2
 l p r0
lp
inlet ( Z  0 ).
It can be seen from Eqs. (79)-(80) that the profile of specific deposit  depends only on its
value at the pore entrance and does not directly depend on time. This implies that the permeate flux is
always the same for a given value of the entrance pore radius (related to  0 by Eq. (72)). Formally,
this can be written as

1

j  0    
0


1


dZ1

,
2
   0 , Z1   
1 
 
0

 
(83)
where j ( 0 )  J ( 0 ) / J 0 . Eq. (83) is a dimensionless form of Eq. (71).
An expression for the specific deposit   0 , Z  can be derived by integrating Eq. (79) using
the boundary condition given by Eq. (80). The resulting algebraic equation can be solved for  :
2

 N Z

 

0
  0 , Z    0 1  tanh 
 arctanh  1 
   .

2


0

  


(84)
Substituting Eq. (84) into Eq. (83) and evaluating the integral gives the following analytical expression
for the dimensionless permeability, or dimensionless permeate flux, as a function of the specific
deposit (pore radius) at the pore entrance:
 2 1  3 p2  0 
 p  0   j  0   3 3 

N   3p  0 

2
2
 N

 N
  
coth 
 arctanh   p  0    4  csch 
 arctanh   p  0   
N
 2
 
 2
  
1
,
(85)
28
 p  0   1 
0
,
0
(86)
where  p    rp   / r0 is the dimensionless pore radius.
The dependence of  0 on time can be calculated from the equation:
* 
0
d *
0  0 j ( * ) N   *  Sa  *  ,
(87)
which was obtained by solving Eqs. (81)-(82). The integral on the right hand side cannot be evaluated
analytically due to the complex dependence of j and S a on  0 , but it can be easily evaluated by
numerical integration. Therefore, Eqs. (85)-(87) describe the dependence of the permeate flux on time
in parametric form.
The total rejection coefficient can be written as
Rt  1 
cl
 1  Sa  0  1  Rc  ,
c0
(88)
where
Rc  1 
cl
Sa  0  c0
.
(89)
Here, cl is the solute concentration at the pore outlet and S a , which is a function of  0 , accounts for
the contribution of steric/electrostatic rejection at the pore entrance (intrapore concentration is
decreased by a factor S a ).
According to the derivations on pages 32-33 of Ref. [56] and Eqs. (50)-(53) of Ref. [57], we
have the following identity:
cl
Sa  0  c0

l
,
0
(90)
where  l is the specific deposit at the pore outlet. In view of Eqs. (84) and (90), Eq. (88) transforms to
 N

 0 Sa  0  
 0   
Rt  0   1 
1  tanh 
 arctanh  1 
 .

0
2
 0   





2
(91)
Eqs. (87) and (91) describe the dependence of the rejection coefficient on time in parametric form. For
small values of t, it is easy to show that
Rt  1  Sa  0  exp   N  ,
(92)
which can be used to estimate the values of N  from experimental data at very short times.
29
Fig. 9 shows the selectivity – permeability and rejection plots calculated using the depth
filtration model for several values of N. The deposition of solutes on the pore wall increases with
increasing N, which is a function of the filter coefficient 0 and accounts for the ability of the pore
wall, or the surface of the layer of deposited particles, to collect suspended solutes. This deposition
reduces the current pore diameter, which causes a decrease in the permeation flux and, hence, in the
mass flux of solutes entering the pore. On the other hand, the deposition of solutes on the pore wall
reduces the area available for solute deposition, which decreases the ability of the membrane pores to
catch solutes. At the same time, the reduction in the pore diameter causes an increase in the purely
steric rejection by the membrane. The interplay between the above three processes determines the
behavior of the solute rejection coefficient and the membrane selectivity. For the parameter values
used in Fig. 9, the reduction of the area available for solute deposition inside the pore initially
dominates, which causes an initial reduction in the solute rejection coefficient. At longer times, the
increase in steric rejection of solutes at the pore mouth begins to dominate, giving rise to an increase in
solute rejection. Thus, the solute rejection coefficient (and selectivity) goes through a minimum before
increasing rapidly at long times as the pore radius approaches the solute radius (Fig. 9). It should be
noted that this qualitative behavior of the rejection coefficient was experimentally observed in
unstirred ultrafiltration of aqueous solutions of BSA and myoglobin [46; see Fig. 6], which suggests
that the protein adsorption/deposition on pore walls may be the dominant mechanism in this
ultrafiltration process.
Interestingly, the value of N does not noticeably affect the kinetics of the permeation flux over
this range of parameter values. Therefore, the best combination of selectivity and permeability is
obtained for the system with the highest value of N since this provides the greatest protein capture and
thus the largest values of the protein rejection. Note that this analysis does not account for the loss of
protein (product) due to capture within the membrane pores – the high selectivity (or rejection) in this
system would not be desirable if large quantities of the desired product are lost within the membrane.
30
Fig. 9. (a) Selectivity – permeability and (b) rejection curves calculated by the depth filtration model:
N  (1) 2; (2) 3; (3) 4;   1/ 1  Rt  ; Rt evaluated by (91); dimensionless permeability  p
calculated by (85); rs  4 nm; rp  20 nm . Arrows denote the evolution of the selectivitypermeability tradeoff during ultrafiltration.
31
4.4 Monolayer coverage model (Langmuir adsorption)
In general, the deposition rate for the first layer of solutes on the clean pore wall may be very
different than the deposition rates for the second and further layers because of the difference between
the solute-pore and solute-solute surface interactions. For stable solutions, i.e., for conditions without
coagulation and aggregation, the formation of a second layer is often suppressed by the same doublelayer electrostatic repulsion forces that prevent coagulation [62]. In this case, the monolayer coverage
model (Fig. 10), i.e., Langmuir adsorption, is more appropriate than the continuous multilayer model
discussed in section 4.3.
Monolayer solute deposition in the membrane pore can be described by Eqs. (71) - (74), (76),
and (78). In this case, the relationship between the filtration coefficient  f and specific deposit  is
written as [57]:

 f    0 1 

 
,
 max 
(93)
where
 max 
4 0
 r0  rs  rs ,
r02
(94)
 max is the value of the specific deposit corresponding to the completed single layer.
Fig. 10. Schematic diagram of monolayer coverage model for steady-state conditions.
32
The system of equations given by Eqs. (71) - (74), (76), and (93) can be re-written as two
uncoupled ordinary differential equations, which are mathematically of the same form as Eqs. (79)(82) but with    given as:

.
 max
    1 
(95)
It can be easily shown that the specific deposit   0 , Z1  in this case is given by
  0 , Z  
 max 0
.
 0   0   max  exp  N Z 
(96)
The dimensionless permeability, or dimensionless permeate flux, as a function of  0 is written as
2

 2  0 max  0   max 
 0 max
 0   max 
1
 p  0   j  0   
0 


2 

N  0   0 
N   0 exp  N    max   0    0   0   max 
   0   max  
 max
N
 
ln   0   0   max   ln  0  0  exp  N   0  exp  N   max    0 max   
 
 2 0   max  

1
(97)
The total rejection coefficient takes the form:
Rt  0   1 
Sa  0  max
 0   0   max  exp  N  
.
(98)
Eqs. (97), (98), and (87) give the dependence of the permeate flux and rejection coefficient on time in
parametric form.
At small values of   (  0   0 ), Eq. (98) reduces to Eq. (92). The rejection coefficient reaches
a steady-state value as soon as the specific deposit at the pore entrance  0 reaches  max (complete
monolayer):
Rt  0   1  Sa  max  .
(99)
The corresponding dimensionless steady-state permeability  s , or dimensionless permeate flux js , is
written as:
2
  
 s  js   1  max  ,
0 

(100)
which can be derived directly from Eq. (97) by substituting  0   max or from the Hagen-Poiseuille
equation for the case where the pore radius is reduced by 2 rs (one solute diameter).
33
The selectivity – permeability plots for monolayer adsorption are shown in Fig. 11. As in the
depth filtration case, the overall behavior is governed by the interplay between three processes: the
reduction in solute mass flux due to the narrowed pore diameter, the reduction in intrapore area
available for solute deposition, and the increase in steric rejection due to the reduced pore mouth flow
diameter. This is illustrated by curve 1 plotted in Fig 11(a) for the case where the solute radius is only
slightly smaller than the radius of the pore. In contrast, the results for rp  rs (curve 2) are very
different than that predicted by the depth filtration model. In this case, the steric rejection is small at
the beginning and remains as such until the monolayer is completed. Fig. 11(a) shows that the initially
high solute rejection caused by solute deposition on the pore walls starts declining as soon as a
significant area of the pore walls is covered by the deposited solutes. Solute rejection decreases until
the monolayer is complete, at which point the rejection coefficient is equal to that for solute transport
through a pore with radius  rp  2rs  . As in the depth filtration case, the deposition of solutes on the
pore wall increases with increasing N or 0 , which leads to a higher value of the initial solute
rejection coefficient (Fig. 11(b)).


34
Fig. 11. Selectivity – permeability curves calculated by the monolayer adsorption model: (a) rp = (1)
12, (2) 20 nm; N  3; (b) N  (1) 2, (2) 3, (3) 4; rp  20 nm . Here, rs  4 nm ,
  1/ 1  Rt  , Rt evaluated by Eq. (98), dimensionless permeability  p calculated by Eq.
(97), and arrows denote the evolution of the selectivity-permeability tradeoff during
ultrafiltration.
It should be noted that the deposition rate under monolayer restriction is initially higher in the
pore entrance region as compared to the rest of the pore because the solute concentration at the pore
entrance is higher. However, at longer times, the probability of solute capture in the pore entrance
region becomes lower than that deeper in the pore (monolayer exclusion) as described by Eq. (95).
This results in a leveling of the profile of deposited solutes along the length of the pore, leading to a
virtually uniform monolayer deposition as the process approaches steady-state conditions (Fig. 10).
4.2.5 Concluding remarks
The fouling caused by pore constriction reduces the permeation flux of the membrane and
increases the solute rejection coefficient and the selectivity. The increase in selectivity due to pore
constriction (intrapore solute deposition) can be substantial right from the very beginning of the
ultrafiltration process. The overall performance of the ultrafiltration process in the presence of pore
constriction depends on the magnitude of the surface interaction forces, which in the monolayer and
35
multilayer models of this study are indirectly accounted for by the value of the filtration (deposition)
coefficient 0 and its dimensionless analog N  .
The evolution of the selectivity during UF depends on the interplay between three simultaneous
processes: (1) the increase in steric/electrostatic rejection due to the reduced pore mouth flow diameter
caused by particle deposition on the pore wall; (2) the reduction in the permeation flux and, hence,
solute mass flux due to the narrowed pore flow diameter; and (3) the reduction in intrapore area
available for solute deposition (mostly determined by electrostatic phenomena). The net result leads to
the complex behavior illustrated in Figs. 9 and 11, with an initial decline in selectivity followed by an
increase at longer times when the exclusion effects at the pore entrance begin to dominate.
When rs is relatively close to rp , the pore constriction is best described by the monolayer
adsorption model. In this case, the selectivity first decreases due to the reduction of the wall surface
area available for solute deposition, but then rapidly increases due to the reduction in the pore diameter
at the pore entrance and the corresponding increase in steric rejection. In contrast, when rs is much
less than rp , the fouling might be best described by either the monolayer adsorption model, when the
long-time selectivity is low, or the depth filtration (multilayer) model, when it is high. In the depth
filtration model, multiple layers of deposited solutes build up until the pore mouth flow diameter
approaches the solute diameter; that is, the pore becomes almost perfectly retentive to the solute at
long filtration times.
It is important to note that the calculations presented in this section do not directly incorporate
electrostatic effects, which can play a major role in both the initial selectivity-permeability tradeoff and
the pore blockage/constriction fouling dynamics (indirectly included in the intrapore depth filtration
and monolayer adsorption models via the filtration coefficient  f [63-65]). Moreover, these effects
can largely control the magnitude of the critical flux and the onset of cake layer formation in protein
ultrafiltration [66,67], both of which are of great practical importance in membrane ultrafiltration
processes. A more advanced, integrated theory is needed that incorporates the effects of key
electrostatic parameters, such as solution ionic strength, pH, and membrane and solute surface
potentials [67], on the ultrafiltration performance. In this regard, the analytical mathematical
expressions presented in this paper can provide an initial framework to analyze the effects of
electrostatic interactions by incorporating more complex expressions for the solute sieving coefficient
(as discussed in section 2).
36
5. Summary
The results presented in this review demonstrate that the selectivity-permeability relationship
during UF processes can be mathematically described by coupling existing models for
steric/electrostatic solute rejection with appropriate models for solute fouling based on different
fouling mechanisms, such as pore blockage (complete blocking) and pore constriction. The resulting
algebraic and differential equations can be solved analytically for many simplified situations, e.g., for
membranes with a small number of discrete pore and solute sizes. Pore blockage typically decreases
the membrane selectivity relative to its fouling-free value (determined by transport theory alone) due to
shunting of the fluid flow to the larger (less selective) pores. In contrast, pore constriction typically
increases the selectivity due to the reduction in the effective pore size. However, the detailed evolution
of the selectivity during ultrafiltration can be quite complex, with the selectivity displaying a distinct
minimum under some conditions.
Our results also demonstrate that the selectivity-permeability tradeoff for the clean membrane
is insufficient for characterizing the performance during actual UF processes. In particular, the tradeoff
analysis for the clean membrane ignores the changes in selectivity and permeation flux due to protein
fouling during the membrane process. The evolution of the selectivity-time or selectivity-permeability
curves strongly depends on the underlying fouling mechanism as well as the ratio between the solute
and pore radii and the underlying solute and pore size distributions. In addition, the overall behavior is
a function of the membrane’s ability to capture solutes by the pore walls, i.e., the strength and kinetics
of protein adsorption. This suggests that experimental data for the variation of the membrane
selectivity and permeation flux with time can provide important information about the properties of the
membrane and feed solution as well as the mechanisms governing fouling behavior. This information
could enable the design of more efficient ultrafiltration membranes and processes for specific
applications.
37
Nomenclature
c
solute number concentration density (m-4)
c0
volume fraction of suspended solutes in the feed
cl
volume fraction of suspended solutes at the pore outlet
cp
permeate solute number concentration density (m-4)
cs
volume fraction of suspended solids
C
total number of solutes per unit of solution volume (m-3)
C0
number concentration of single-size solutes in the feed solution (m-3)
Cf
solute concentration in the bulk feed solution (kg m-3)
Cp
solute concentration in the permeate (kg m-3)
Cw
solute concentration at the upstream surface of the membrane (kg m-3)
fn
pore probability density function (m-1)
fs
solute probability density function (m-1)
D
solute diffusion coefficient in the free solution outside the pore (m2 s-1)
G
volumetric permeate flow rate through one cylindrical pore (m3 s-1)
G0
initial volumetric permeate flow rate through one cylindrical pore (m3 s-1)
j
dimensionless permeate flux
js
steady-state dimensionless permeate flux (monolayer adsorption model)
J
permeate flux (m s-1)
J0
initial permeate flux (m s-1)
kB
Boltzmann constant (J K-1)
Kc
hindrance factor for solute convective transport
Kd
hindrance factor for solute diffusive transport
lp
membrane thickness (pore length) (m)
L0
initial membrane permeability (m s-1 Pa-1)
Lp
membrane permeability (m s-1 Pa-1)
m
ratio of the length of the inlet pore portion with deposited solutes to the entire length of the pore
n
pore concentration density (m-3)
38
N
total number of open pores per square unit of membrane (m-2)
Np
number of pores per square unit of membrane (m-2)
N   0 l p
dimensionless factor accounting for the efficiency of solute capture by pore walls
P
transmembrane pressure (Pa)
Pem
membrane pore Peclet number
r0
initial pore radius (m)
rc
hydraulic resistance of cake layer (Pa s m-1)
rm
hydraulic resistance of the membrane (Pa s m-1)
rp
pore radius (m)
rpmin
minimum pore radius (m)
rpmax
maximum pore radius (m)
rs
solute radius (m)
R
rejection coefficient
Ra
actual rejection coefficient (at the upstream surface of the membrane)
Rc
fraction of solutes deposited on the pore wall
Rt
total rejection coefficient (includes both steric/electrostatic and pore constriction contributions)
R
integral rejection coefficient for a solute size range from rs to rs  rs
s
specific area of the pores (m)
S
solute sieving coefficient
Sa
actual solute sieving coefficient (at the upstream surface of the membrane)
S
integral sieving coefficient for a solute size range from rs to rs  rs
t
time (s)
T
absolute temperature (K)
z
axial coordinate (m)
Z
dimensionless axial coordinate
Greek letters
 = Cw/Cf

concentration polarization factor
dimensionless filter coefficient
39
p
dimensionless permeability
s
steady-state dimensionless permeability (monolayer adsorption model)
0 
initial membrane porosity

membrane porosity

porosity of the layer of deposited solutes
  rs / rp
ratio of solute radius to pore radius
0
initial filter coefficient (m-1)
f
filter coefficient (m-1)

P
8  lp
pore-size-independent factor in Hagen-Poiseuille equation (m-1 s-1)

fluid viscosity (Pa s)
∆ 
osmotic pressure difference (Pa)
p
dimensionless pore radius
s
dimensionless solute radius

specific deposit (volume fraction occupied by deposited particles)
0
specific deposit at the pore inlet
 max
specific deposit corresponding to the completed single layer
l
specific deposit at the pore outlet
os
osmotic (Staverman) reflection coefficient
p
dimensionless surface charge density of the pore wall
s
dimensionless surface charge density of the solute

dimensionless time (in standard blocking models)
*
dimensionless time (in depth filtration and monolayer coverage models)
b
dimensionless time (in complete blocking models)

solute partition coefficient

selectivity
e
electrostatic energy of interaction (J)
40

electrophoretic ratio describing the relative importance of electrophoretic transport compared to
convection inside the membrane
41
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46
Figure Captions:
Fig. 1. Selectivity–permeability tradeoff (here,  1/ Sa ) for a variety of cellulosic and polysulfone
ultrafiltration membranes using bovine serum albumin as a model protein. Solid black curve is
a model calculation using a log-normal pore size distribution. Thin blue curve is given by Eq.
(18). Adapted with permission from [4].
Fig.2. Schematic diagram of complete blocking for solutes and pores of various sizes.
Fig. 3. (a) Rejection, (b) dimensionless permeability, and (c) selectivity-permeability plots for
complete blocking with two pore sizes and a single solute size: n1  0 : n2  0  0.5: 0.5 (1),
0.7 : 0.3 (2); and 0.9 : 0.1 (3). R is calculated by Eq. (43) with   1/ 1  R  .  p is evaluated
with (42) but with the permeability normalized by the initial flux through the small pores, i.e.,
with n2  0  0 . Arrows denote the evolution of the selectivity-permeability tradeoff during
ultrafiltration.
Fig. 4. Rejection of small solutes as a function of time for complete blocking with two pore sizes and
two solute sizes: C02 :C01 = (1) 0:1.0, (2) 0.05:0.95, (3) 0.065:0.935, (4) 0.15:0.85, (5) 0.5:0.5;
R2 evaluated by Eq. (59); rs ,1  4 nm; rs ,2  6 nm; rp ,1  2.5 nm; rp ,2  5 nm .
Fig. 5. Schematic diagram of intrapore solute deposition for standard blocking model.
Fig. 6. Schematic diagram of intrapore solute deposition for m-model.
Fig. 7. Selectivity – permeability plots calculated by the m-model: (a) m = (1) 1.0, (2) 0.66, (3) 0.33;
Rc  0.5 ; (b) Rc = (1) 0.5; (2) 0.7; (3) 0.9; m  1 . Here,   1/ 1  Rt  ; Rt evaluated using Eq.
(65); dimensionless permeability  p calculated by (69); rs  4 nm; rp  20 nm ; arrows denote
the evolution of selectivity-permeability tradeoff during ultrafiltration.
Fig. 8. Schematic diagram of nonuniform solute deposition [57].
Fig. 9. (a) Selectivity – permeability and (b) rejection curves calculated by the depth filtration model:
N  (1) 2; (2) 3; (3) 4;   1/ 1  Rt  ; Rt evaluated by (91); dimensionless permeability  p
calculated by (85); rs  4 nm; rp  20 nm . Arrows denote the evolution of the selectivitypermeability tradeoff during ultrafiltration.
Fig. 10. Schematic diagram of monolayer coverage model for steady-state conditions.
Fig. 11. Selectivity – permeability curves calculated by the monolayer adsorption model: (a) rp = (1)
12, (2) 20 nm; N  3; (b) N  (1) 2, (2) 3, (3) 4; rp  20 nm . Here, rs  4 nm ,
  1/ 1  Rt  , Rt evaluated by Eq. (98), dimensionless permeability  p calculated by Eq.
(97), and arrows denote the evolution of the selectivity-permeability tradeoff during
ultrafiltration.
47