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HUNGARIAN JOURNAL
OF INDUSTRIAL CHEMISTRY

VESZPRÉM
Vol. 37(2) pp. 159-164 (2009)

MATHEMATICAL MODELING OF DIAFILTRATION

Z. KOVÁCS1 , M. FIKAR2, P. CZERMAK1,3

1Institute of Biopharmaceutical Technology, University of Applied Sciences, Giessen-Friedberg, Giessen, GERMANY
E-mail: kovacs.zoltan@tg.fh-giessen.de

2Department of Information Engineering and Process Control FCFT, Slovak University of Technology, SLOVAKIA
3Department of Chemical Engineering, Kansas State University, Manhattan, Kansas, USA

The main objective of this study is to provide a general mathematical model in a compact form for batch diafiltration
techniques. The presented mathematical framework gives a rich representation of diafiltration processes due to the
employment of concentration-dependent solute rejections. It unifies the existing models for constant-volume dilution
mode, variable-volume dilution mode, and concentration mode operations. The use of such a mathematical framework
allows the optimization of the overall diafiltration process. The provided methodology is particularly applicable for
decision makers to choose an appropriate diafiltration technique for the given separation design problem.

Keywords: membrane separations, diafiltration, mathematical modeling, optimization

Introduction

The objective of industrial purification processes is
usually dual: (1) to separate certain solutes from the
process liquor and (2) to concentrate the purified solution
in order to obtain a final product. In this work we
examine a batch diafiltration process that is designed to
fulfill these objectives simultaneously.

In the following we consider a binary aqueous
solution consisting of two solutes, namely a macrosolute
and a microsolute. Diafiltration is known as a conventional
process technique to achieve high purification of
macrosolutes with an economically acceptable flux [1].
The requirement for an effective separation is the
utilization of a membrane which has a high rejection for
the macrosolute and a low rejection for the microsolute.
The terms macrosolute and microsolute are widely-used
in the literature dealing with membrane diafiltration. In
order to eliminate ambiguity, we would like to point out,
that the separation is not necessarily based on solely size
exclusion as it might be suggested by this nomenclature.
Membrane filtration also allows separation of solutes of
similar molecular weights but having different charges
as reported in many studies, for example in [2, 3].

There have been many published works on batch
diafiltration. However, there is no exact and uniform
definition for the term diafiltration. Indeed, the
terminology currently being used is somewhat conflicting.
In this paper, we use the term diafiltration in its broad
sense referring to the actual technological goal. Thus,
diafiltration is a membrane-assisted process that can be
used to achieve the twin-objectives of concentrating a
solution of a macrosolute, and removing a microsolute

by the utilization of a diluant. In this context, batch
diafiltration is a complex process that may involve a
sequence of consecutive operational steps. We consider
three frequently used operational modes. These are the
concentration mode (C), the constant-volume dilution
mode (CVD), and the variable-volume dilution mode
(VVD). They differ from each other in the utilization of
wash-water as it is discussed in more details later in this
paper. Note that an operation mode does operate with
fixed operational settings. A diafiltration process, in
contrast, is usually constructed by changing the settings
of wash-water addition (i.e. switching to another
operational mode) according to a pre-defined schedule.
In the following, we examine two frequently used
diafiltration techniques: the traditional diafiltration
(TD) and the pre-concentration combined with variable-
volume dilution (PVVD).

The most commonly used concept of diafiltration is
the TD process that involves three consecutive steps
(i.e. operational modes). First, a pre-concentration is
used to reduce the fluid volume and remove some of the
microsolute. Then, a constant-volume dilution step is
employed to “wash out” the micro-solute by adding a
washing solution (e.g. diluant) into the system at a rate
equal to the permeate flow rate. Thus, the volume of the
solution in the feed tank is kept constant during this
operational mode. Finally, a post-concentration is used to
obtain the final volume and concentrate the macrosolute
to the final concentration due to the specific technological
demands.

The VVD is an operation mode in which fresh water
is continuously added to the feed tank at a rate that is
proportional but less than the permeate flow. This causes
a simultaneous concentration of macrosolute and removal

160

of microsolute. This operation has been proposed by
Jaffrin and Charrier [4], analyzed in some detail by
Tekić et. al and Krstić et. al [5, 6], and recently revised
by Foley [7]. A modification of VVD is PVVD, i.e., a
two step process in which the solution is first pre-
concentrated to an intermediate macrosolute concentration
and then subjected to VVD to reach the final desired
concentrations of both solutes. This concept is credited
to Foley [8].

Several studies have examined the different types of
diafiltration techniques in terms of process time and
wash-water requirement [1, 4-11]. However, only a few
works have considered concentration-dependent rejections
in the optimization procedure [12]. Assuming constant
rejections might lead to inaccurate simulation and
subsequent optimization results under conditions where the
rejections of solutes are strongly vary depending on their
feed concentrations and a considerably interdependence in
their permeation occurs.

In this work, we attempt to enlarge our perspective
on how engineers in general should cope with the
complexity of a diafiltration design problem. We present
a general mathematical model in a compact form for
batch diafiltration techniques. From this perspective we
discuss the model limitations when simplifying
assumptions on solute rejections are being used. We
consider a common separation objective and through a
specific example we demonstrate the power of the
presented modeling methodology. Finally, we present
some specific ideas of how optimization should support
decision makers in finding the best wash-water utilizing
profile for the given engineering design problem.

Theory

Configuration of diafiltration

The schematic representation of membrane diafiltration
setting is shown in Fig. 1.

Figure 1: Schematic representation of diafiltration

settings

In a batch operation, the retentate stream is

recirculated to the feed tank, and the permeate stream
q(t) is collected separately. During the operation, fresh
solute-free diluant stream u(t) (i.e. wash-water) can be
added into the feed tank to replace solvent losses.

General mathematical framework

In this section we derive the governing differential
equations for diafiltration. The proportionality factor
α(t) is defined as the ratio of diluant flow u(t) to
permeate flow q(t):

)(
)(

)(
tq
tu

t =α (1)

where the diluant flow u(t) is given as a product of the
membrane area A and the permeate flux J(t). The
change in the volume in the permeate tank Vp is given
by the permeate volumetric flow-rate q:

)(
)(

tq
dt

tdV p = (2)

The change in the feed volume during the operation is
given as

)()(
)(

tqtu
dt

tdV f −= (3)

Considering two solutes and assuming that the diluant
consists of no solutes, the mass balance for the solute
concentrations yields

2,1)()()()( ,, =−= itctqtctVdt
d

ipiff (4)

where cp,i(t) denotes the permeate concentration of
solute i at time t. Equation (4) can be rewritten in the
following way:

2,1)()(
)(

)()(
)(

,
,

, =−=+ itctqdt
tdc

tVtc
dt

tdV
ip

if
fif

Using Eq.(3) and recalling that cp,i(t) = cf,i(t)(1–Ri(t)),
where Ri(t) is the rejection of solute i at time t, we
obtain, for i = 1, 2, ...

[ ])()()()(
)(

)( ,
, tutRtqtc

dt
tdc

tV iif
if

f −=

Thus, we have the following initial-value problems:

⎪
⎩

⎪
⎨

⎧

−=

0)0(

)()(
)(

tqtu
tdt

(5)

and, for i = 1, 2, ...

[ ]

⎪
⎩

⎪
⎨

⎧

−=

0
,,

,
,

)0(

)()()()(
)(

)(

ifif

iif
if

tutRtqtc
dt

tdc
tV

(6)

which describe the evolution in time of the volume in
the feed tank Vf and of the feed concentration cf,i. Vf

0 and
cf

0
,i denote respectively the initial feed volume and the

initial feed concentration of the solute i.
In the next two sections, we briefly describe discuss

the possible strategies to determine flux and rejection.

161

Later we formulate an optimization problem that
represents a frequent industrial separation flask. Then,
to examine and compare the TD and PVVD processes,
we make a use of the filtration data from our earlier
work [14].

Rejection and permeate flow

The separation behaviour of the membrane can be
characterized in terms of permeate flux and solute
rejections. The estimation of the flow q(t) and of the
rejection Ri(t) can be carried out separately using the
most convenient approach for the problem at hand.
Possible strategies to determine flux and rejection are
presented in our previous study [13]. In brief, either
mechanism-driven or data-driven models can be
employed. Mechanism-driven models are based on a
physical understanding of the transport phenomenon. In
contrast with that, data-driven models make a direct use
of the experimental data obtained from filtration tests
with the process liquor. The main challenges in
employing a data-driven model are the minimization of
necessary a-priori experiments and the conversion of
raw data into useful information. In this study, we
consider the following empirical relations which were
reported earlier in [14]:

1,62,5
2

2,4 )(
32,2

2
2,1 )(

fff cscscs
ff escscsq

++
++= (7)

)()( 42,31,22,11 zczczczR fff +++= (8)

1,62,5
2

2,4 )(
32,2

2
2,12 )(

fff cwcwcw
ff ewcwcwR

++
++= (9)

where s1, ..., s6, z1, ..., z4, w1, ..., w6 are suitable
coefficients that were previously determined from
laboratory experiments with the test solution as described
later.

Special cases and analytical solutions

The complexity of the modelling problem originates
from the fact that in most of the membrane filtration
processes the solute rejections are concentration-
dependent quantities. Since the concentrations are due
to change while processing the feed, the rejections of
both microsolute and macrosolute are affected by the
extent to which the microsolute concentration is reduced
and also to which the macrosolute is concentrated.
Analogously, the permeate flux also depends on the
actual feed concentration of both components. In general,
the model equations require numerical techniques to
solve them, since no closed form solutions exist.
However, when the effect of the feed concentrations on
the rejections is neglected, then a constant rejection
coefficient σ can be introduced such that Ri(t) = σi =
constant for i = 1, 2. When introducing this simplifying
assumption on the rejections, the differential equations

can be reduced to simple algebraic equations, The
resulting exact solutions are reviewed below:
1. Concentration mode: Since no diluant is applied,

u(t) = 0 and cd,i = 0. The concentration of component
i at the end of the operation is given by

(10)

where the expression
)(
)0(

tV
V

is by definition the

concentration factor n.
2. Constant-volume dilution mode: The solute free-

diluant is continuously added to the feed tank in a
rate equal to the permeate flow. Thus, cd,i = 0 and
u(t) = q(t). The component concentration is related to
the total volume of wash-water Vw can be written as

)(
)1(

,, )0()(
ff

tV
V

iffif ectc
−

i = 1, 2, ... (11)

where the expression
)( ff

tV
V

is by definition the

dilution factor D.
3. Variable-volume dilution mode: Solute-free wash-

water is added at a rate αq(t) , where α is a parameter
with value 0 ≤ α ≤ 1. Assuming that the permeate
flux remains unchanged during the process, Krstić et
al. [6] gave the expression for the component balance:

α
ασ

α −
−

⎟
⎟
⎠

⎞
⎜
⎜
⎝

⎛ −
−

=
1

,
,

)0(
)()1(

)0(
)(

if
fif

V
tV

c
tc i =1, 2, ... (12)

Note, that the main pitfall of the commonly used
modelling approaches is often the assumption of constant
rejection coefficients. These simplifying assumptions
can easily be misused when their appropriateness is not
carefully checked for the given separation process. For
instance, a typical rejection profile of an inorganic salt
nanofiltration is illustrated in Fig. 2.

Figure 2: Rejection of the membrane Desal-DK5 for

NaCl as a function of feed concentration (30 bar, 25 ºC,
0.55 m2 spiral-wound element, 1.0 m3h-1 recirculation

flow-rate). Solid line is for eye guidance

162

The complexity of the problem further increases in
the presence of more than one solute, due to their
interdependent permeation.

Optimization problem formulation

We define the optimization problem as follows:

minimize (J = cf,2(tf)) (13)

such that

tf ≤ 6 (14)

n = 3. (15)

Thus, the objective of the separation is to reduce the
concentration of component 2 in the final product as
much as possible with the restriction that the total
operation time should not exceed 6 hours and a total
concentration factor 3 is achieved.

In the case of TD, the objective is to find the optimal
set of variables of pre-concentration factor n1, dilution
factor D, and post-concentration factor n2. In the case of
PVVD, the optimal set of variables n1 and α is to be
determined.

Note that the numerical values of the constraints in
Eqs. (14) and (15) are chosen according to the processing
conditions and the specifications of our laboratory
system. However, the concept itself can find a general
interest. Industrial problems can be handled in an
analogous way, when the optimal operational parameters
of an existing membrane plant with a defined membrane
area are to be found.

Experimental

In this study we use the filtration data from our earlier
work [13]. These data serve as input for the mathematical
analysis. The laboratory apparatus, applied chemicals,
and sample analysis have been described in details
earlier. In brief, nanofiltration experiments were carried
out with the membrane Desal-DK5 separating a binary
aqueous solution at constant temperature and pressure.
The process liqueur was a test system consisting of
sucrose (hereafter called component 1) and sodium
chloride (component 2). A limited number of a-priori
experiments were used to determine the dependence of
R and q on concentration. The resulting functions are
reported in section “Rejection and permeate flow”.

Results

The dynamics of a diafiltration process can be evaluated
by simultaneous solving of Eqs. (5) and (6). Considering
a TD process with a fixed pre-concentration factor n1,
the post-concentration factor n2 is readily given with the

use of the constraint on the total concentration factor as
n2=n/n1. It is evident that longer dilution results in lower
final microsolute concentration. Thus, for each pre-
concentration factor, a maximal dilution factor can be
found so that the given constraint on the total operation
time is still satisfied. For instance, when the initial
solution is pre-concentrated with a factor 2, then a
maximal operational time for CVD can be calculated so
that the total operation time including the post-
concentration step does not exceed the given 6 hours.
This example is illustrated in Figs. 3a and 3b.

Figure 3: The estimated 6-hour time-course of the

concentrations and the volumes of feed and permeate
for a traditional diafiltration process with a

preconcentration-factor of 2

The optimization problem of PVVD is analogous to
TD. Here, an optimal α has to be found for each fixed n1
so that the objective function is minimized while
satisfying the constraints. Fig. 4 shows the calculated
values of α for fixed n1 values. Obviously, when n1=n, α
must be 1 in order to satisfy the constraint on n.

In both cases of TD and PVVD, the respective
operation parameters of D and α for a fixed n1 were found
by applying iterative methods similar to as reported in
[13]. The optimization results obtained by varying n1
stepwise form 1 to n are illustrated in Figs. 5 and 6.

163

Figure 4: Optimized α values as a function of pre-

concentration factor for the PVVD process

Figure 5: Optimization diagram for traditional

diafiltration. Final microsolute concentration (dashed
line) and required wash-water volume (continuous line)

are plotted versus pre-concentration factor

Figure 6: Optimization diagram for diafiltration

involving pre-concentration combined variable-volume
dilution. Final microsolute concentration (dashed line)
and required wash-water volume (continuous line) are

plotted versus pre-concentration factor

When comparing the TD and the PVVD processes,
from the optimization diagrams shown in Figs. 5 and 6,

we can conclude that the best diafiltration strategy is a
specific case when n1 = n and α = 1. In other words, the
optimal strategy is to pre-concentrate the process
liqueur to its minimum volume and then to apply a
constant-volume dilution without a post-concentration
step. We would like to draw attention to the fact that a
great care is needed when interpreting and generalizing
such finding. The here presented methodology for
choosing an appropriate diafiltration technique is general
in the sense that it can be readily adopted for different
solute/membrane systems without the need of major
changes in the provided procedure. However, the output
of the optimization is unique for each application. The
choice of TD versus PVVD depends primary on
1. the response of the particular membrane to the

specific solution that is expressed in terms of rejection
Ri and permeate flow q,

2. the terms involved in the objective function (i.e. the
definition of the separation goal),

3. the involved constraints (technological demands)
and their numerical values that need to be satisfied.

Any changes in these above listed specifications
may modify the output of the optimization, and lead to a
different optimal strategy of diafiltration.

Further optimization aspects

It should be pointed out that the main difference
between the various types of operational modes is due
to the quantity and the duration of the diluant stream
introduced in the feed tank during the entire operation.
In this context, diafiltration techniques differ in their
strategies for controlling the introduction of the diluant
stream u(t). In the widely applied conventional
diafiltration processes, such as TD or PVVD, the
trajectory of the control variable u(t) is arbitrarily pre-
defined for the entire operational time. However, it may
happen that the optimal time-dependent profile of the
diluant flow is not among these arbitrarily constructed
scenarios. The optimal control trajectory can be
determined by formulating an optimization problem
subject to process model described by differential
equations. Using a dynamic optimization solver called
Dynopt developed by Čižniar et. al [15], we are
currently developing a unified technology for water
utilization control that addresses generality versus special
cases. This approach is currently under investigation
and will be published soon.

Conclusions

We provide a methodology that is useful for the design
of batch diafiltration processes. A general mathematical
model in a compact form is presented. It unifies the
existing models for constant-volume dilution mode,
variable-volume dilution mode, and concentration mode
operations. A rich representation of the separation
process is given due to the employment of concentration-

164

dependent solute rejections in the design equations.
Thus, a formal tool is provided for describing the
engineering design that supports the disciplined use of
data-driven and mechanism-driven permeation models.
The use of such a mathematical framework allows the
optimization of the overall diafiltration process. The
provided methodology is particularly applicable for
decision makers to choose an appropriate diafiltration
technique for a given separation design problem. Further
research effort is directed at the dynamic optimization
of diafiltration processes.

ACKNOWLEDGMENT

This research is a cooperative effort. The first author
would like to thank the Hessen State Ministry of Higher
Education, Research and the Arts for the financial
support within the Hessen initiative for scientific and
economic excellence (LOEWE-Program). The second
author acknowledges the support of the Slovak
Research and Development Agency under the contract
No. VV-0029-07.

LIST OF SYMBOLS

A − membrane area (m2)
c – concentration (mol m-3)
D − dilution factor
J − permeate flux (m h-1)
n – concentration factor
q – permeate flow-rate
R – rejection
t – operation time (h)
u – diluant flow-rate (m3 h-1)
x – state variables (mol m-3)
V – volume

GREEK SYMBOLS

α – proportionality factor of diluant flow to permeate
flow

SUBSCRIPTS

d – diluant
f – feed
i – component (i = 1 macro-solute, and i = 2 micro-

solute)
p – permeate
w − wash-water

ABBREVIATIONS

C − concentration mode
CVD – constant-volume dilution mode
VVD – variable-volume dilution mode
PVVD − diafiltration involving pre-concentration and

variable-volume dilution mode
TD – traditional diafiltration

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