HUNGARIAN JOURNAL · 
OF INDUSTRIAL CHEMISTRY 

VESZPREM 
Vol. 30. pp. 13- 18 (2002) 

ON THE NON-LINEAR MASS TRANSFER THEORY 

C. BOYADJIEV 

(Institute of Chemical Engineering, Bulgarian Academy of Sciences, "Acad. G.Bontchev" str.,Bl.103, 1113 Sofia, 
BULGARIA) 

Received: February 19,2001 

A theoretical analysis ofnon~linear mass transfer kinetics has been done. The comparison between Stephan flow, a flow 
induced by large concentration gradients and Marangoni effect is shown. The conditions of existing of these non-linear 
effects in mass transfer kinetics are determined. 

Keywords: non-linear effects, Stephan flow, Marangoni effect 

Introduction 

A theoretical analysis of non-linear mass transfer has 
been developed in [1]. The main idea follows from the 
non-linearity of the convection-diffusion equation: 

p(c)W(c)gradc = divfp(c)D(c)gradc]+ ken. (1) 

The velocity W is governed by the hydrodynamic 
equations. However, the principal non-linear 
phenomenon is due to the concentration effects on the 
velocity W(c), density p(c), viscosity Jl(c), diffusivity 
D( c) and on the chemical reaction rate ken (for n ¢ 1 ). 

It was shown [ 1] that there are a number of cases 
with non-linear mass transfer behavior. The well-known 
linear mass transfer theory could be successfully applied 
in these cases. However, in case of two-phase interphase 
mass transfer with a flat interface the above equation 
permits a non-linear mass transfer model to be derived 
by means of the boundary layer approximation: 

( 
OU . OU . ) {)

2 
U · dU J dV j (2) 

P; u 1 a: +v1 a:: = p,1 i)y21 +A1 • a_;--+ay-=0' 

(k. oc. i1 2c. 
u.-1 +v.-1 =D-

2
1 +B., j=1,2; 

1 i1x 'iJy <ry 1 

Contact information: E~mail: chboyadj@bas.bg 

where the index 1 is used to denote gas or liquid phase, 
while the index 2 designates liquid or solid phase. The 
terms Ai and Bi (f=l,2) are the contributions of some 
additional physical effects. 

The solution of the above set of equations permits 
the interphase mass transfer rate to oe determined as 
follows: . 

(3) 

There are a number of processes where ui, vi, Ji;, PJ·, 
Di1 Ai1 and Bi are independent on the concentration Cj 
(j=l.2). These situations are the basis of the linear mass 
transfer theory. 

The mathematical model allows the following 
principle characteristics of the linear mass transfer to be 
drawn: 

the interphase mass transfer rate J does not depend 
on the mass transfer direction; 



14' 

the interphase mass transfer coefficient Kj does not 
depend on the concentrations Cjo (j=l,2). 

The striving to decrease the size of the industrial 
devices necessitates process intensification. The systems 
with intensive mass transfer are characterized by a 
behavior that deviates considerably from the 
characteristics mentioned above. The main feature is the 
higher mass transfer rate, which differs significantly 
from the value predicted by the linear mass transfer 
theory. The non-linear effects leading to above have 
been described [ 1]. 

In systems with high concentrations and such 
exhibiting large concentration gradients, the deviations 
from the linear Pick's diffusion law are significant, too. 
Under such conditions, the higher concentrations could 
affect both the viscosity and the density of the fluid: 

Di=Di(ci), vi=J.li(ci), pi=pi(c), j=l,2.(4) 

The concentration effect introduces a non-linearity in 
the convection-diffusion equation discussed in details in 
F,3J. 

The other non-linear effect due to the non-uniform 
concentration distributions 

Ai = g(pi- Po)• Pj = Pici) (5) 

leads to a natural convection [ 4,5]. 
The next cause that may intensify the mass transfer 

process is the existence of a chemical reaction with a 

rate B j in the bulk of the phase: 

(6) 

The studies developed in [1, 6], show that in the gas-
liquid systems with a chemical reaction: B 1=0, while 
B2=kcn. Moreover, the chemical reaction rate could 
affect significantly the interphase mass transfer 
mechanism. 

The thermal effect of the chemical reactions could 
lead to the temperature non-uniformity on the interface 
and to consequent surface tension gradients. This calls 
for new boundary conditions taking into account the 
equality of the tangential components of the stress tensor 
on the interface: 

y = 0, Pi a ul = f.lz a Uz -- a (j • 
a y a y ax {7) 

The investigation of this effect (Marangoni effect) 
[1 ,8] has shown that it is negligible when there are no 
surfactants in the system. 

One of the most interesting non-linear effects arises 
from the conditions imposed by the high concentration 
gradients. The latter induce secondary flows at the 
interface. This effect has been discussed in details in { 11 
for a large number of systems taken as examples and it 
has been termed "non~linear mass transfer effect~~. 

Under the conditions imposed by high concentration 
gradients induced secondary flows occur. They provoke 
convective components of the mass transfer flux 

additional to the main mass flux. In this case the mass 
transfer rate is: 

1 = M:p* J(~c) dx, 
Po o Y y=O 

(8) 

where the secondary flow affects both the diffusion mass 

transfer J ~c ) and the convective mass transfer 
L/l y y=O 

M* 
_1!_ . In gas-liquid and liquid-liquid systems [9, 10] 

Po 
the non-linearity has been proven to be the effect of the 
induced secondary flow on the diffusion transfer. In 
liquid-solid systems the induced flow affects mainly the 
convective transfer. These effects are ·clearly 
demonstrated the electrochemical systems [1, 6] due to 
the high molecular mass (M) of the metals. 

All the non-linear effects influence the velocity 
fields, which lead to changes in the hydrodynamic 
stability of the system. The loss of stability could cause 
an increase of the amplitudes of the random disturbances 
until a new stable state or a stable periodic process is 
reached [ 1]. The latter is a self-organizing dissipative 
structure with a mass transfer rate growing sharply 
which is not the case in the conventional systems. The 
problem has been discussed in details in [ 4, 5] in the 
case of non-stationary absorption of pure gases in 
immobile liquid layer with flat interface. 

Non-linear effects in mass transfer kinetics induced 
by the secondary flows, lead to quality changes in the 
mass transfer rate, because they are related to physical 
mechanisms. 

In this sense the most interesting are the Stephan 
flow [11], (occurred by volume change at the interface), 
the flow induced by large concentration gradients [1,12] 
and the flow resulted from surface tension gradients 
[7,8]. These secondary flows require introduction of 
new boundary conditions at the interface between gas -
solid, gas - liquid, liquid - liquid and liquid - solid 
phases. A comparative analysis of the occurrence of 
these secondary flows will be presented. 

Stephan Flow 

The Stephan flow occurs in case of heterogenic 
reactions at the interface between two phases. As a 
result the substances are disappeared (reagents) or 
generated (reaction products) at this interface. This 
·~disappearance or generation" of substances at the 
boundary between two phases might be a consequence 
of chemical reactions, adsorption or desorption 
processes (the substances connect physically or 
~heroically with substances from phase boundary), 
mterphase mass transfer (the substances "disappearn at 
phase boundary as a result of the transfer into the other 
phase), l~quid - vapour phase transition (boiling, 
condensation}. Some of the above heterogenic reactions 



lead to changes in the volume of the phase at the 
interface, which leads to occurrence of hydrodynamic 
flow called Stephan flow. 

Let's consider the heterogenic reaction [11), 
presented by the stoichiometric equation: 

N 

LV;Ai=O, (9) 
i=l 

where ~ and V; (i = 1, ... , N) correspond to the symbols 
of participating substances in the reaction and their 
stoichiometric coefficients. For the initial substances 
(reagents) V; > 0, while for the reaction products 

V; < 0 . The rate of the heterogenic reaction 

j i (moll m 2 s) is defined for the separate substances 

(i = 1, ... ,N), where N is their total number. For the 
reagents ji > 0 , and for the reaction products it < 0 . 

The reagents (reaction products) are supplied to 
(took off from) the reaction surface by diffusion and 
convection: 

j = -Dgradc+ vc, (10) 

where j is the vector of the mass transfer rate , 

D(m2 / s) is the diffusivity, grad- is the vector of the 

gradient, c(mol/ m 3 ) molar concentration, v - velocity 

vector. For the separate substances the molar flux has 
the following form: 

15 

heterogenic reaction (per unit area, per unit time). They 
can be presented as: 

(15) 

where V; (m3 I m2 s) is the volume reaction rate of the 
substances in the gas (vapour) phase, and W; (m3 I mol) -
their molar volume. 

The systems gas (vapour) - liquid (solid) will be 
considered below, where the Stephan flow occurs in 
gaseous (vapour) phase, because it is practically not 
physically applicable in liquid and solid phases. 

The summation of the stoichiometric coefficients 
leads to: 

(16) 

where v > o(v < o) means the increase (decrease) of the 
mols number (the volume) of the reaction mixture as a 
result of the heterogenic reaction. 

From (14) directly follows: 

. vi . . 1 N 1i = - h' l = , ... , . (17) 
VI 

The summing up of (17) leads to: 

(18) 

(11) where the substance A1 we consider as limiting, i.e. the 

The projection of the vectors in the vector equation over 
the normal vector of the interface n (in points of the 
surface) might be noted as: 

ji = {i;.n~ ~ ~ = (gradci.n1 v = (v.n), (12) 

where ji (moll m 2 s) are molar fluxes, which have to be 

equal to the rates of the reactions of the separate 

substances, and oci - normal derivate at the interface, v 
on 

- the rate, induced as a result of heterogenic reaction at 
certain conditions and is called the velocity of the 
Stephan flow. It is positive when v is oriented to the 
phase boundary and negative in the opposite case. 

The introduction of (12) into (11) leads to: 

. oci . ]; = -Di-- + vci, z = 1, ... , N , (13) on 
where j;{j = l, ... ,N) should be satisfy the condition for 
the stoichiometry of the flows: 

jliV1=j2fV2= ... =jN/VN. (14) 

From {14) can be seen, that the stoichiometry 

coefficients V; (moll m2 s) present mols of the 
substances (i = 1, ... , N), which participate in the 

rate of its reaction limits rate of the heterogenic reaction. 
In case of gases and vapours we can express the 

concentration through the partial pressure: 

p 
c. =-1 i=l, ... ,N, 

1 RT' 
(19) 

where R is the universal gas constant, T 
temperature. In this way from (13) directly follows: 

. D; o ~ vi} 
J;== RT on+ RT' i=1, ... ,N. (20) 

The summation of (20) leads to: 

~. 1 ~Do~ vP 
L.,;h =--L.J i--+-, 
t=t RT i=I on RT 

(21) 

N 

where P = L ~ is the total pressure of the mixture. 
i=l 

The velocity of the Stephan flow is obtained directly 
from (18) and (21): 

RT . 1 ~ D a~ (Z2) v=-r h +-.£.J ;-;-· 
p p i=1 un 

In case of two component mixtures D1 = D2 = D, 
i.e.: 



16 

RT . DaP 
v=p-r h +-p;;;;· (23) 

It can be seen from (23) that the velocity of the 
Stefan flow is determined by the relative change in the 
volume of the reaction mixture y as a result from 
changes in the volume velocity vi or in case of phase 

transition (the change of the molar volume wi ). This 
velocity decreases as a result of hydraulic resistance 

( :: < 0) . At absence of the hydraulic resistance 

P = eonst and the velocity of Stephan flow takes the 
form: 

RT . 
v=-rh· p (24) 

In case of reduction of the reaction mixture volume 
as a result of heterogenic reaction the Stephan flow is 
oriented to the reaction interface ( y > O,v > 0 ). In 
opposite case ( y < O,v < 0) it is oriented from the 
reaction interface. 

In case of heterogenic chemical reaction without 
phase transition: 

i.e. v > 0 , when the total volume rate of the chemical 
reaction of the substances iri the mixture is positive and 
the volume increases. In opposite case v < 0 . 

In cases when the heterogenic reaction presents the 
liquid-vapour phase transition at the interface (boiling, 
condensation) the molar rates and (as a results) their 
volume rates are equal: 

1 1 

(26) 

i.e. in case of condensation (boiling) 
wl > W;p r < 0, (w.~ > w., r > o) and the Stephan flow 
is oriented to (from) the interface. 

In cases. when the heterogenic reaction presents 
adsorption (desorption) VI = V2 ~ f :::: 0. i.e. the 
conditions for the Stephan flow do not exist. The 
analogous is the case of absorption (desorption)~ where 
the product transfers into the second phase. i.e. 
V1 =v2 • v=O. 

The obtained result (24) shows that Stephan flow at 
interface arises when the heterogenic reaction leads to 
changes in the summary (total) volume of reaction 
mixture. Obviously this can happen only at phase 
boondary in gas or vapour phase and it is practically 
impossible at the boundary in liquid or solid phase. 

Non -linear mass transfer 

One of the significant effects of non- linear mass 
transfer arises in systems with intensive interphase mass 
transfer, when large concentration gradients induce 
secondary flows which velocity is normally oriented to 
the interface. 

For a simplicity we will consider two-component 
fluid [ 1, 6], where the component A is a substance 

dissolved into component B (solvent). The density of 

the solution p (kg I m 3 ) can be presented through the 
mass concentrations of the component A (M c) and 
solvent B (M 0 c0 ): 

p =M0 e0 +Me= Po +Me, (27) 

where M and M 0 are the molar mass (kg I mol) of 
A and B , c and c0 - their molar concentrations 

(mol 1m3 ). 
Every elementary volume of the solution has a 

velocity V , which can be expressed through velocities 
of the substances A ( v) and. B( v 0 ), i.e velocity of the 

mass flow, transferred by every elementary volume 
presents a sum of the mass flows of A and B : 

(28) 

The equation (28) can be projected over the normal 
vector n of the phase surface: 

p* (v.n)= p~(v0.n)+M c*(v.n), (29) 

where the upper index ( *) notes the values at the 
interface. 

From (29) the velocity of the secondary flow 
v(ml s) induced by the diffusion (large concentration 
gradient) can be determined: 

v =(V.n). (30) 
At the boundary between two non-mixing phases the 

mass flux is zero, i.e. 

(31) 

The molar flux of the dissolved substance (at 

interface) N (moll m2 s) can be expressed through the 
rate (moll m

2 s) of the diffusion and convective 
transfer: 

*( )* Joe)* * N=c v.n =-~lon. +vc. (32) 
The introduction of (30), (31) and (32) into (29) 

leads to: · 

MD(oc)'" * • 
v=- p~ dn ' Po =Moco~ (33) 



* where c0 is the molar concentration of B at the 
interface. For a flat phase boundary y = 0 is obtained 
directly [1, 6]: 

v=-MD(~) * a . 
Po Y y=O 

(34) 

In some approximations [12] N is expressed only 
by the diffusion flux: 

N=-j~c) ...,l y y=O (35) 

and for the velocity of the secondary flow is obtained 

MD(ac) v=--- --* a , 
Po Y y=O 

p* = p~ +Me*. (36) 

Obviously the obtained results (34) and (36) 

coincide at c * = 0 (for example at desorption of gases). 
From (34) it is seen that in the systems with an 

intensive interphase mass transfer the normal component 
of the velocity differs from zero (as in the systems with 
linear mass transfer), and depends on the concentration 
of the transferred substance, i.e. the convection-
diffusion equation is non linear. This requires the 
boundary condition at y = 0 ( v = 0) to be replaced with 
(34). 

The obtained result (34) shows that the local mass 
flux at phase boundary has diffusion and convective 
components: 

z=-M - +Mvc =-MD-* - .(37) . ~ ac) - * p * ( ac J 
ay y=O Po ay y=O 

From (37) the mass transfer rate can be directly 

determined by averaging of the mass flux i (kg I m2 s) 
over the interface. 

The comparison between Stephan flow's velocity 
(24) and the velocity of the secondary flow induced by 
large concentration gradients shows that Stephan flow 
arises in gaseous or vapour phase as a result of the 
changes in the phase volume at the interphase. Such 
changes occur in some heterogenic chemical reactions 
accompanied by changes of the reaction mixture volume 
or phase transition (boiling, condensation). 

The secondary flow at large concentration gradients 
is a result of the intensive interphase mass transfer and 
can be observed in gaseous and liquid phase (the 
Stephan flow in liquid phase is physically impossible). 

Marangoni effect 

The Marangoni effect is a result of secondary flow 
which velocity is tangentially oriented to the interface 
and is caused by the surface tension gradient, induced by 
concentration or temperature gradient at the 

17 

interface. This effect will be considered in gas - liquid 
systems. 

The influence of the secondary flows over the mass 
transfer rate is a result of the velocity component 
oriented normally to the interface. This creates an 
intensive convective transfer, which is summarized with 
the diffusion transfer. At the Marangoni effect the 
induced flow is tangential and the normal component 
appears from the flow continuity equation: 

au+~=O. 
ax ay 

(38) 

The flows in the boundary layer are characterized by 
two characteristic scales of velocity (u0 , v0 ) and two 
linear scales (8, L), which are related to the 
dimensionless variables of the flow: 

x=LX, y=oY, 8 =JJ.LL . (39) 
UoP 

The introduction of (39) into (38) leads to 
dimensionless equation 

au+ v0L av =O 
ax u08 ()y ' 

(40) 

where the continuity of the flow is retained at the 
following ratio between the characteristic scales of the 
flow: 

Let us suppose that the Marangoni effect is a result 
of temperature gradient at the interface. In this case the 
characteristic velocity of the Marangoni effect can be 
determined from the equation [ 1]: 

tt(~~ L =~: =~~ ~:. (42) 
If introducing (39) in ( 42) (and temperature scale 

At ) we reach the condition for the existence of 
Marangoni effect and its characteristic velocity: 

8 llt aa 
u =---

0 L J1. at· (43) 

The introduction of (43) in (41) allows the 
determination of the characteristic velocity of the 
secondary flow liable for the increase of the mass 
transfer rate as a result of the Marangoni effect: 

& dO' 
vo=---. 

pu0 L dt 
(44) 

In the case of absorption of C02 in H 20 and 
temperature change because of chemical reaction, the 

order of the velocity v0 can be determined: 

(45) 



18 

In cases of non-linear mass transfer, characteristic 
scales have to be introduced into (34): 

J¥. v=v0 V, c=11cC, y=ocY, oc = . (46) 0 
From (34) and (46) directly can be obtained the 

condition for an existence of the non-linear mass 
transfer effect and its characteristic velocity: 

Vo = M ~c J"•D. (47) 
Po L 

The v0 order of the velocity in (47) at analogous 
conditions with ( 45) is determined directly: 

(48) 

The obtained results (45) and (48) show, that in 
systems with an intensive interphase mass transfer the 
non-linear effects are related to the concentration 
gradients normally oriented to the interface, and are not 
related to the temperature gradients at the phase 
boundary. This proportion between the effect of non-
linear mass transfer and the Marangoni effect is based 
mainly on the following three reasons: 

the normal component of the velocity v0 is always 
smaller than the induced tangential component (see 
(41)); 

to the large concentration gradient he correspond 

small temperature gradients At because of the small 
heat effect of the absorption; · 

the absence of surface-active substances. 
The non-linear mass transfer effect and the 

Marangoni effect can determine the mass transfer rate 
not only by additional convective flows, but also by loss 
of stability. In these cases accidental disturbances lead 
to self-organized dissipative structures with very 
intensive mass transfer £1]. The stability of these 
systems depends mainly from v0 , which is the reason 
the process to be limited again from the non-linear mass 
transfer. 

Conclusion 

The presented theoretical analysis shows that the effects 
of the Stephan flow and the non-linear mass transfer at 

large concentration gradients have different physical 
nature, and as a result different mathematical models. 

The Stephan flow [12] appears at heterogenic 
chemical reactions (with changing volume of the 
reaction mixture) or liquid-vapour phase transition at 
interface (boiling, condensation). 

The non-linear mass transfer [ 1] appears at 
interphase mass transfer in gas (liquid) - solid, gas-
liquid and liquid-liquid systems as a result of large 
concentration gradients. 

The Marangoni effect does not appear in gas-liquid 
and liquid-liquid systems [7 ,8] at absence of surface 
active agents, because of the small heat effect of the 
dissolution process. 

REFERENCES 

1. BOYADJIEV C. B., BABAK V. N.: Non-Linear Mass 
Transfer and Hydrodynamic Stability, Elsevier, 
Amsterdam, pp. 500, 2000 

2. BOYADJIEV C. B.: Hung. J. Ind. Chem.,1996, 24, 
35-39 

3. BOYADJIEV C. B., HALATCHEV I.: Int. J. Heat Mass 
Transfer, 1998,41,939-944 

4. BOYADJIEV C. B.: Int. J. Heat Mass Transfer, 2000, 
43, 2749-2757 

5. BOYADJIEV C. B.: Int. J. Heat Mass Transfer, 2000, 
43, 2759-2766 

6. KR.YLOV V. S., BOYADJIEV C. B.: Non-Linear Mass 
Transfer, Institute of Thermophysics, Novosibirsk, 
pp.231, 1996, (in Russian). 

7. BOYADJIEV C. B., HALATCHEV I.: Int. J. Heat Mass 
Transfer, 1998,41, 197-202 

8. BOYADJIEV C. B., DOICHINOVA M., Hung. J. Ind. 
Chern., 1999, 27, 215-219 

9. BOYADJIEV C. B.: Hung. J. Ind. Chern., 1998, 26, 
181-187 

10. SAPUNDZHIEV T., BOYADJIEV C. B.: Russian J. Eng. 
Thermophysics, 1993, 3, 185-198 

11. FRANK-KAMENETsKll D. A.: Diffusion and Heat 
Transfer in Chemical Kinetics, Science, Moscow, 
pp.491, 1967 

12. BIRD R. B., STEWART W. E., LIGHTFOOT E. N.: 
Transport Phenomena, J. Wiley, New York, pp.687, 
1965 


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