HUNGARIAN JOURNAL 
OF INDUSTRIAL CHEMISTRY 

VESZPREM . 
Vol. 30. pp. 211-214 (2002) 

MATHEMATICAL MODEL OF VARIABLE VOLUME DIAFILTRATION 

M. N. TEKIC, Z. ZAVARG6, D. KRSTIC and M. DJURIC 

(Faculty of Technology, University ofNovi Sad, Bul. CaraLazara 1, 21000 Novi Sad, YUGOSLAVIA) 

Received: April11, 2002; Revised: August 13, 2002 

The use of ultrafiltration as a separation technique is frequently accompanied by a diafiltration step to remove 
microsolutes. There are different ways to combine ultrafiltration and diafiltration to obtain the desired final concentration 
of components. Variable volume diafiltration is a continuous process in which water is continuously added at a rate Jess 
than the permeate flow rate so that the concentration of macrosolute continuously increases, and the final concentrations 
of macrosolute and microsolute can be reached simultaneously. In this paper a mathematical model of variable volume 
diafiltration is proposed. The model includes both initial volume and concentration, rejection coefficients, processing 
time as well as water volume added during diafiltration. The developed model could be served for design and 
optimization purposes. 

Keywords: modeling, ultrafiltration, diafiltration 

Introduction 

Over two decades ultrafiltration has been recognized as 
a separation technique to fractionate species according 
to their size. This technique has several advantages over 
other separation techniques such as distillation, 
evaporation solvent extraction, chromatography, etc. 

Ultrafiltration is widely used as a separation 
technique in many different fields such as chemical and 
biochemical engineering, food processing and 
pharmaceutical industry. In order to achieve 
macrosolute-microsolute separation different 
ultrafiltration processes can be used. The concentration 
of macrosolute by ultrafiltration is frequently 
accompanied by a diafiltration step to remove 
microsolute. It consists of a continuous or discontinuous 
addition of a pure solvent to retentate feed and can be 
applied to both batch and continuous ultrafiltration 
processes. 

Both, batch and continuous diafilltration can be 
performed in different ways. Batch diafiltration can be 
carried out depending on the way the pure solution is 
added as a continuous or discontinuous batch 
diafiltration. On the other hand, continuous diafiltration 
can be performed as a co-current or counter-current [1] 
process. 

An interesting process in which a pure solvent is 
continuously added at a rate less than permeate flow 
rate so that concentration of macrosolute and the 

removal of impurities occurs simultaneously. wa:c. 
proposed by Jeffrin and Charrier [2]. Thi~ way a smgle 
diafiltration process with a continuou~ly decreasing 
volume makes possible that the final concentration!'> of 
components are reached simultaneously. 

The aim of this work is to develop a process model 
of a variable volume diafiltration. In the presemed 
model the initial solution concentration, initial volum\.,. 
retention, processing time as well as the membrane area 
are included. 

This model differs from the approach given by 
Jaffrin and Charrier [2] where the rejection coefficient 
was not included. Thus, on the basis of incomplete 
macrosolute rejection and the constant flux assumption 
an analytical solution is obtained. 

Model 

Let us consider the diafiltration process schematically 
presented in Fig. I. The equation of continuity is given 
by: 

dV 
-=Qo-QF 
dt 

(1) 

Here Vis the volume of the solution in the tank and Ql> 
and Q, are diafiltration water flow rate and filtrate flow 
rate respectively. 
The mass balance of the macrosolute can be written as 



212 

d(VC) = QFC(l- R) (2) 
dt 

Table 1 Values of model parameters and coefficients 

Parameter Unit Value 

A 
? 

1 m-

J m3/m2s 2.5·10-5 

Vo m3 0.2 
R 1 0.9-1 
r 1 0-0.1 

C/Co 1 2-10 
cclc1 2-10 

Pconst variable 

Pvar 1 variable 
a 1 variable 

where c is concentration, and R is the macrosolute 
rejection coefficient. 

In order to obtain concentrated solution with the 
final volume \'J the flow rate of diafiltration water must 
be a constant smaller fraction of QF. 

QD =aQF a<1 

After rearranging Eq. (2) becomes 

0 ~ ~ 
J d(lnC) = -JQF(~-R) dt- J d(lnV) 
~ 0 ~ 

From the third term of Eq. ( 4) it follows 

V = V0 - QF(l-a)t 

(3) 

(4) 

(5) 

and the first term on. the right hand side of Eq.(4) 
becomes 

- 's' QF<I- R) dt = (1- R) In[I QF<I-a) t J (6) 
o V (l-a) Yo f 

By introducing Eq.(6) in (4). it follows 

ln Cf = (1-R)ln(l- QF(l-a)tf)-ln VI (7) 
C0 (1-a) V0 V0 

from which the relative concentration of macrosolute 
can be related with time 

cf (8) z-. R-a 
· • ( 1- (1-a)~0·A-t1 )'"" 

The processing time necessary to reach the desired final 
concentration can be calculated as 

(9) 

The processing time can be related to microsolute 
concentrations 

t - 1- -
[ ( 

C0 J~=:] Yo 
f- c.f (1-a)JA 

WATER RETENTATE 

~ 

FEED 
TANK 

T 

PERMEATE 

(10) 

Fig. I Schematic representation of the variable volume 
diafiltration process 

where r is the microsolute rejection coefficient which is 
zero or close to zero. 

From the last two Eqs.(9) and (10) the relation 
between macrosolute and microsolute concentration can 
be found 

:; =( ~; t~ (11) 
The coefficient a can now be calculated from the above 
Eq.(ll) 

rln~+Rln cf 
a 

C.r c0 (12) 

ln~+ln2 
Cf Co 

The solution volume added during the diafiltration in 
this process is 

VD,var = QDt = aJA~ 

combining with Eq.(9) one can get · 

(13) 

In the classical (constant volume) diafiltration process 
the volume ratio is 

R = VD.const =-1-lnS1_ (15) 
PcOif!St V. l 

o -r cf 

where concentration ratio cfc0 can be found from 
Eq.(ll). 

Calculation 

The calculations of the developed model were carried 
out on the basis of parameters given in Table 1. 

The influence of the relative concentration of 
macrosolute on processing time at different retention 



are given in Fig.2. The calculations were carried out for 
initial feed volume of 0.2m3, and a= 0.3. As expected, 
we have the shortest time at higher retention as it is at 
batch ultrafiltration [3]. 

180 

170 

160 

150 

140 

I130 

120 

110 

100 

90 

6 10 

Cf/CO 

213 

Conclusions 

A more realistic model which differs from Jaffin and 
Charrier's [2] and includes the rejection coefficient is 
presented. The model provides the basis for more 
complex simulations of variable volume diafiltration 
andean 

320r,:=:=:=::::.:;-----------------, 

300 r=0.1 

280 

260 

240 

c 220 
,~200 

180 

160 

140 

--u=0.3 
.......... 0.=0.5 

'"' .... 0.=0.7 

~----

Fig.2 Processing time vs. macrosolute relative concentration 120 
+-~.-~--.-~-r--,--~-~-~~ 

900 

800 

700 

600 

:s 500 
.s 
- 400 

300 

200 

100 

0.0 ~ Q2 Q3 Q4 0~ Q6 OJ Q8 0.9 

Fig.3 Processing time vs. retentate dilution 

Fig.3 shows how processing time depends on the 
ratio of the dilution flow rate to the permeate flow rate, 
a, at different retention. It can be seen that the 
processing time increases with increasing a, and that the 
value of a over 0.7leads to very high processing time. 

The processing time as the function of the relative 
dilution of the microsolute at different values of a are 
shown in Fig.4. It should be noted that at a = 0.3 the 
processing time is almost independent of the relative 
dilution. With increasing an a the processing, time rises 
significantly with the relative dilution. A simple way of 
comparing variable volume and constant volume 
diafiltration is to compare the amount of the diafiltration 
solvent used to reach desired the final concentration. 

The results of the calculation on Fig.5 represent the 
relative volume of the water added during the constant 
volume and variable volume diafiltration. It is important 
to note that the variable volume diafiltration requires 
less diluent water than the constant volume diafiltration. 

c:l. 

6 

cO/cf 
10 

Fig.4 Processing time vs. relative microsolute dilution 

2.8 

2.6 

2.4 

2.2 

2.0 

1.8 

1.6 

1.4 

1.2 

1.0 

0.8 

0.6 

0.4 

0.2 

0.0 
2 4 6 

cO/cf 

8 

I 
.---~-1 

10 

Fig.5 Relative diafiltration volume vs. microsolute dilution at 
the process with constant and variable volume diafiltration 

be useful in analysis of process paramete-rs. The 
obtained analytical solution enables simple calculations 
such as processing time, amount of diafiltration solvent, 
concentration of macrosolute and dilution of 
microsolute. The calculations also show that variable 
volume diafiltration needs less diafiltration liquid than 
the constant volume diafiltration which could be of 
practical interest. 

Acknowledgement 

This research was supported by Ministry of Science, 
Technology and Development, Republic of Serbia, 
Project No. 1362. 



214 

r 
v 

SYMBOLS 

membrane area, !if 
concentration of macrosolute in any time, wt.% 
initial concentration of macrosolute , wt.% 
final concentration of macrosolute wt.% 
initial concentration of microsolute , wt.% 
final concentration of microsolute wt.% 
permeate flux, m3 /m2s 
water (solution) flow rate added during 
diafiltration, m3/s 
filtration flow rate, m3/s 
macrosolute rejection coefficient 
microsolute rejection coefficient 
volume in the tank, m3 

Vo initial volume, m3 

Yj final volume, m3 

t time, s 
t.r final time, s 
a defined by Eq.(3) 

REFERENCES 

1. DUTRE B. and TRAGARDH G.: Dessalination, 1994, 
95,227-267 

2. JAFFRIN M. Y. and CHARRIER J. PH.: J. Membrane 
Sci., 1994, 97, 71-81 

3. TEK.IC M. N., KUJACKI J. and V ATAI GY.: Hung. J. 
Ind. Chern., 1994,22, 1-3 


	Page 213 
	Page 214 
	Page 215 
	Page 216