CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 70, 2018 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Timothy G. Walmsley, Petar S. Varbanov, Rongxin Su, Jiří J. Klemeš 
Copyright © 2018, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-67-9; ISSN 2283-9216 

Thermocapillary Instability in the Nonstationary Process of 

Gas Absorption. Effect of Lewis and Prandtl Numbers on The 

Critical Time 

Evgeny F. Skurygin*, Taras A.  Poroyko 

Yaroslavl State Technical University, Yaroslavl, Russia 

skouryguine@rambler.ru 

Theoretical analysis of the development of surface convection in the transient process of gas absorption by an 

initially motionless layer of liquid was carried out. The main resistance to mass transfer is concentrated in the 

liquid phase. Convective instability is caused by the temperature dependence of the surface tension. Linear 

analysis showed a strong dependence of the critical time on the viscosity of the liquid. On the basis of the 

simplified nonlinear model that does not use semi-empirical assumptions, approximate estimates of the critical 

time from physical characteristics are obtained. The problem was solved in a two-dimensional formulation. The 

concentration of the absorbed substance is represented as the sum of the three terms of the Fourier series on 

the coordinate directed along the gas - liquid interface. Equations for the temperature and the fluid velocity are 

linearized. Nonlinearity was taken into account only in the equations for concentration. The calculations are 

compared with known experimental data. 

1. Introduction 

Thermocapillary instability, also known as the Marangoni effect, is of interest in its ability to exert a strong 

influence on the mass transfer rate at the liquid-gas interface (Scriven et al., 1959). The latter has practical 

application for chemical-technological processes associated with the contact of liquid and gaseous phases 

(Nepomnyashchiy, 2002). The Marangoni effect is of practical importance in medicine and ecology, for example, 

in the processes of purifying gases from impurities; absorption of carbon dioxide by water, collection of organic 

liquids from the surface (Molder et al., 2002). Modern problems of air and water purification were discussed in 

work of Phang et al. (2017). The effectiveness of new solvents for absorbing CO2 from the atmosphere was 

experimentally studied in (Ahmad et al, 2017). 

The process of gas absorption in liquid proceeds in the diffusion regime only up to a certain critical time, after 

which it is replaced by more intense convective mode (Plevan, 1966). The problem is to determine the physical 

characteristics of the system under which the convective instability arises and to calculate the critical time for 

the transition of the process to the convective regime. Theoretical estimates of the mass transfer rate under 

conditions of thermocapillary convection (Dil'man et al., 1998) are applicable in a limited range of parameters. 

The purpose of this paper is to estimate the critical time depending on physical characteristics on the basis of 

theoretical analysis and numerical simulation of the process. 

2. Statement of the problem - basic equations 

The transient process of gas absorption by an initially motionless layer of a liquid of infinite thickness is 

considered. The main resistance to mass transfer is concentrated in the liquid phase. Convective instability is 

caused by the temperature dependence of the surface tension. The process is determined by the following 

equations 

                                

 
 

 

 
   

                                                  
DOI: 10.3303/CET1870097 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Skurygin E.F., Poroyko T.A., 2018, Thermocapillary instability in the nonstationary process of gas absorption. effect 
of lewis and prandtl numbers on the critical time , Chemical Engineering Transactions, 70, 577-582  DOI:10.3303/CET1870097   

577



0,       0,

 + ( )  -  = 0,    = -

0.

z x

c T
c D c T T

t t

v v

t x z



    

 
         

 

  
  

  

 

v v

v

v  

(1) 

where c is the concentration of the absorbed substance in the liquid, T is the temperature, v = (vx, vz) is the 

liquid velocity vector, D is the diffusivity,   - the thermal diffusivity,   - the kinematic viscosity of the liquid, and 

t is the time. The problem solved in a two-dimensional formulation. The coordinate x is directed along the liquid-

gas boundary, z is directed deep into the liquid along the normal to the free surface, the value z = 0 corresponds 

to the interphase boundary. 

The boundary conditions given in the following form: 

v
,    ,    v 0,        0.

0,     ,     v 0,       v 0       .

x
B z

x z

c T T
c c HD for z

z z z x

c
T T for z

z

  



   
     

   


     



 (2) 

where cB is the equilibrium concentration at the liquid-gas interface,  ∆H is the specific heat of dissolution, 

p
c   - the thermal conductivity of the liquid,   - its density, pc  - the specific heat capacity, and the  -

temperature coefficient of surface tension. 

The initial conditions are as follows: 

( , , ) ( , ),     ( , , ) ( , ),

v ( , , ) ( , ),     v ( , , ) ( , )   0

c T

x vx z vz

c x z t c f x z T x z t T f x z

x z t f x z x z t f x z for t

 
   

  
  (3) 

The functions fc, fT, fvx, fvz represent the initial values of the perturbations of concentration, temperature, and 

fluid velocity. 

3. Linear analysis 

Let us represent the values of concentration, temperature and velocity as the sum of the unperturbed values, 

corresponding to diffusion into a stationary liquid, and a sinusoidally varying small perturbations 

(0) (1) (0) (1)

(1)

( , ) ( , ) ( , ) cos ;   ( , ) ( , ) ( , ) cos ;

( , ) v ( , ) cos .
z

c t c z t c z t kx T t T z t T z t kx

v t z t kx

   



r r

r
 (4) 

The index (1) corresponds to a small perturbation; r is the radius vector, k is the wave number. Linearized 

equations for small perturbations and the corresponding boundary conditions have the following form 

( 0) ( 0)
(1) (1) (1) (1) (1)

2
2

2

v    v    v 0,    
c T

D c T
t z t z t

k
z

 
         
                   

         


  



 (5) 

(1) (1) 2 (1)
(1) (1) 2 (1)

2

(1) (1) (1)
(1)

v
0,     ,   v ,              0

v
0     v   0    0  ,    

z

p

T c
c c HD = 0 k T for z

z z z

c T
0 for z

z z

   
  

    
  

  
    

  

 (6) 

The method of "freezing" the time of the unperturbed solution is widely known in the linear analysis of 

nonstationary processes (Lick, 1965). According to the method, small perturbations vary in proportion to exp(pt). 

Solutions with Re p> 0 correspond to increasing, with Re p <0 - decaying, and with Re p = 0 - neutral or 

oscillatory perturbations. The method gives an underestimate of the critical time (Homsy et al, 1973), but the 

values obtained by this method have a fairly clear physical meaning. Neutral perturbations correspond to a 

stationary point of the perturbation amplitudes. Equations for perturbations at the stationary point are as follows: 

578



(0) (0)
(1) (1) (1) (1) (1)0 0

(t , z) (t , z)
( ) (z) v (z) ,   ( ) (z) v (z) ,   ( - ) v ( ) 0,

Re 0.

st st st st st

c T
p D c p T p z

z z

p

 
 

          
 



 (7) 

The time t0 (k) at which the system of Eq(7) has nontrivial solutions corresponds to the termination of the 

damping and the beginning of the joint growth of the perturbations with the wave number k. We define the time 

tst as the minimum value of t0 (k) with respect to the wave numbers. The time tst and the corresponding optimal 

wave number kst can be represented in the following dimensionless form: 

2

1

* * *

* *

( ) ,  ( ) ,  ,  
( ) ( )

st st st st *

1
t Le t k K Le z t = z

D H c c H c c D

  


 



 

 
   

    
 (8) 

where  /Le D  is the Lewis number, *t  and *z  are the characteristic scales of length and time. 

For Le≥50, their numerical values with an error of not more than 5% are described by the following asymptotic 

approximation (Poroiko et al., 2013) 

1/ 3 1/ 3 2/ 3 1 2/ 3 1/ 6 1/ 6 1/ 2
(1 3 12 ),     (2 0.7 0.9 )

st st
Le Le K Le Le Le   

     
       (9) 

In the case of absorption of carbon dioxide by water under normal conditions, the time of the stationary point is 

0.15 s. The experimental time of transition of the process to the convective regime is 100 s. (Plevan, 1966). Let 

us consider the mechanism of such a long delay. The development of disturbances in the concentration of the 

absorbed substance, temperature and fluid velocity during absorption is determined by the competition between 

the forces of viscosity and surface tension. In the nonstationary process, the spatial scales of concentration, 

temperature and velocity increase, which leads to a decrease in the viscosity forces and, as is known, the 

process loses stability at a critical time. At times to a stationary point, t<tst, the viscosity forces dominate, which 

lead to a decrease in the amplitude of velocity perturbations, and thus can delay the transition of the process to 

the convective regime. The approach of Poroyko et al. (2014) to analyze the transition of the process to the 

convective regime was used. The evolution of perturbations with wave numbers kst passing through a stationary 

point is considered. Let us estimate the damping of such perturbations at times to a stationary point and their 

further development. 

The problem for unperturbed values of concentration and temperature are solved together with Eq(5) and 

boundary conditions (6) for small perturbations. As an additional condition, we require passage of the 

perturbations through the stationary point: 

(1) (1) (1) (1) (1) (1)
( , ) ( ),     ( , ) ( ),    v ( , ) v ( )    , ,  0 

st st st st st
c z t c z T z t T z z t z for t t k k p        (10) 

The solving of the equations for t < tst is an inverse problem for the evolution equations. The problem was solved 

by the method of selecting the initial conditions for t = 0. The latter are given in the form of linear combinations 

of Laguerre functions with coefficients determined from the requirement that conditions (10) be satisfied. 

Figure 1(a) shows the numerical dependences of the root-mean-square values of the concentration Crms 

temperature ϴrms, and velocity Vrms perturbations versus time *t t  for 79Le  , Pr / 7   . The data are 

presented in logarithmic coordinates. 

In the time interval st  , ( 8.57st  ), the intensity of the perturbation of concentration and temperature 

practically does not change, however, the intensity of the velocity perturbation sharply decays, its root- mean-

square value decreases approximately 100-fold. The initial intensity of the velocity is restored at times 

approximately 2,000 t*, which exceeds the time of the stationary point by about 250 times. The time of the 

stationary point corresponds to a pronounced minimum of velocity perturbations and cannot be the time of 

transition to the convective regime. The transition is possible only for many long times, when the perturbations 

reach the necessary intensity. 

Let us note that such a strong attenuation of velocity perturbations and their long-term recovery is due to the 

high viscosity of the liquid; for the carbon dioxide-water system, the kinematic viscosity exceeds the thermal 

diffusivity by 7 times, and the diffusion coefficient by more than 500 times. 

Figure 1(b) shows the time dependences of the root-mean-square velocity perturbations, raised for Le = 79 and 

different values of the Prandtl number. With an increase in the Prandtl number, the intensity of velocity 

perturbations decreases, and the time of their restoration substantially increases. The latter allows us to 

conclude that the critical time for the transition of the process to the convective regime increases with increasing 

Prandl number. 

579



  

(a) (b) 

Figure 1: Intensity of disturbances as a function of time: (a) Pr=7, 1 – 
rms
С , 2 - 

rms
 , 3 - 

rms
V , (b) normal 

velocity perturbations, 1 – Pr=2, 2-Pr=4, 3-Pr=5, 4 - Pr=7, 5 - Pr=13 

4. Nonlinear analysis 

Let us use the simplified nonlinear model (Skurygin et al, 2016) for an approximate estimate of the critical time. 

Let us represent the instantaneous values of concentration, temperature and velocity in the following form: 

0 1 2

0 1 2

1 2

1
1 2 1 2

( , , ) ( , ) ( , ) cos( ) ( , ) cos(2 )

T( , , ) ( , ) ( , ) cos( ) ( , ) cos(2 )

v ( , , ) v ( , ) cos( ) v ( , ) cos(2 )

1 v
v ( , , ) v ( , ) sin( ) v ( , ) sin(2 ),   v ,   

z

x x x x x

c z x t c z t c z t kx c z t kx

z x t T z t T z t kx T z t kx

z x t z t kx z t kx

z x t z t kx z t kx v
k z

  

  

 

  
   



2
1 v

;    
2k z





 (11) 

The wave number k was chosen equal to 0.5 kst. Such a choice of the wave number ensures the fastest growth 

of the second Fourier terms of the concentration and velocity. The equations for all terms of velocity and 

temperature are linearized. Nonlinearity is taken into account only in the concentration equations. 

For zero Fourier terms of concentration and temperature, the initial conditions were given as for unperturbed 

values: 

0 0
( , ) ,     ( , )       0c z t c T z t T for t

 
    (12) 

For nonzero Fourier terms of the concentration, the initial conditions were given as perturbations of various 
intensities and different spatial scales:

 
* *

*

     z   
( , )  0,    1, 2

0     

n n

n

n

c for z Z
c z t for t n

for z Z z

 
  


 (13) 

Where 
n

  - the intensity of the perturbations, Zn - their spatial scales in the direction normal to the surface. The 

perturbations of velocity and temperature at the initial instant were set to zero. 

The critical time is defined as the time at which perturbations developing during the absorption process can be 

observed experimentally (Trouette et al, 2012). Let us define the critical time tcr as the time at which the amount 

of absorbed substance under surface convection exceeds the corresponding value for a stationary liquid by 1 

%. 

 * (Pr, , )cr crt t Le P  (14) 

where 1 1 2 2( , , , )P Z Z   is a set of parameters characterizing the initial intensity of the perturbations. 

Figure 2(a) presents the results of calculations of the critical time as a function of the Lewis number for Pr = 7 
and various initial intensities of the concentration perturbations. The data are normalized for a critical time at 

Le=80. Figure 2(b) shows the results of calculations of cr  as a function of the Prandtl number for Le = 80 and 

the same initial perturbation intensities. The data are normalized to values at Pr = 7. The calculated values are 
aligned along the curves determined by the following formulas: 

580



2 3

Pr

80 80 Pr Pr
( ) 0.1 0.8 0.1 ,      (Pr) 0.55 0.45

7 7
Le

f Le f
Le Le

       
           

       
 (15) 

  

(a) (b) 

Figure 2: Critical time functions (a) of the Lewis number and (b) of the Prandtl number. Results of the 

calculations. Z1 = Z2 = 10/Le1/2 

Calculations of the critical time, performed in the range 2 < Pr < 12, 20 < Le < 200 at the same initial perturbation 

intensities, showed that the function 

( , Pr, )
( , Pr)

(80, 7, )

cr

cr

Le P
f Le

P




  (16) 

weakly depends on the initial intensity of the perturbations. To calculate it, we can use the following approximate 

formula: 

Pr
(Pr, ) ( ) * (Pr)[1 ( 1) ( 1) ( 1)( 1)]

80 Pr
0.05,     0.17,     0.29;        

7

Le
f Le f Le f x y x y

x y
Le

  

  

       

    
 

(17) 

5. Comparison with known experimental data 

Table 1 presents the known experimental data of the critical time and the results of calculations for the nonlinear 

model. For the case of CO2 absorption by water at a temperature of 24.5 0C, we take the experimental value of 

the critical time of 100 s. Theoretical estimates for other systems were performed using Eq(17). 

Table 1: Theoretical estimates and known experimental data 

Solvent t*, s Le Pr tcr, s, experimental data tcr, s, Eq(17) 

value reference 

water, 24.5 0C 0.0171 

 

76.6 6.19 100 Plevan, 1966, 

Dil’man,1998, 

Tan, 1992 

 

100 

water, 18 0C 0.0155 88 7.45 120 Tan, 1992 

 

115 

toluen, 25 0C 2.06×10-4 20.2 6.88 10.75 Sun, 2002 

 

6.9 

methanol, 25 0C 1.09×10-4 27.6 6.80 9.13 Sun, 2002 2.4 

 

The theoretical estimates are in agreement with the experimental data on the absorption of carbon dioxide in 

water and toluene. The underestimation of the critical time for the absorption of carbon dioxide in methanol is 

due the following factors. The solvent density changes during the absorption. Dissolution of carbon dioxide at a 

constant temperature leads to an increase in density. The released heat of absorption leads to an increase in 

581



the temperature of the solvent, and reduces the density. In this case, the density change caused by heating the 

liquid is greater than when the solute concentration is changed. Reducing the density in the upper layers of the 

solvent generates forces that impede surface convection. Due to the high equilibrium concentration of carbon 

dioxide and the high coefficient of thermal expansion of methanol, this effect becomes significant and delays 

the critical time. 

6. Conclusions 

The development of perturbations in the transient process of gas absorption is determined by two sharply 

differing time scales. The first is the time of the stationary point of the perturbation amplitudes, when the 

perturbations of the concentration cease to decay and begin to grow. The second is the critical time for the 

transition of the absorption process to the convective regime. Such a strong difference in time scales is due to 

the strong effect of the viscosity of the liquid on the critical time. Calculations using the simplified nonlinear 

model lead to approximate estimates of the critical time. For more accurate estimates, it is expected that more 

terms should be taken into account in the Fourier series of the concentration for the coordinate along the 

interface. 

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