CHEMICAL ENGINEERING TRANSACTIONS VOL. 52, 2016 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Guest Editors: Petar Sabev Varbanov, Peng-Yen Liew, Jun-Yow Yong, Jiří Jaromír Klemeš, Hon Loong Lam Copyright © 2016, AIDIC Servizi S.r.l., ISBN 978-88-95608-42-6; ISSN 2283-9216 On Modelling the Convective Mass Transfer over Moving Films Interface for the Reaction-Diffusion Systems of the Second Order Leila M. Musabekova*a, Nurjamal N. Dausheyevaa, Nurlybek S. Zhumataeva, Madina A. Jamankarayevab aState University of South Kazakhstan, Tauke Khan, 5, Shymkent, 160012, Kazakhstan bSouth Kazakhstan State Pedagogical Institute, Baytursynov str., 13, Shymkent, 160000, Kazakhstan mleyla@bk.ru The paper deals with the methods for approximately analytical and numerical solving the non-self-similar problems of film absorption, accompanied by the partly reversible reaction of a second order. The Peclet number in this case is taken as sufficiently high, and the diffusion boundary layer is very thin. The methods to obtain approximate analytical solutions of the stationary convective diffusion equation for the coupled problem of isothermal film chemisorption at high Peclet numbers have been worked out. The approximate analytical method for solving the following problems: physical absorption of poorly soluble gas and chemisorption with partly reversible reaction of the second order with specially allowance for the kinetics of surface reaction has been submitted. The known Crank-Nicholson method expanded by the iteration procedures has been modified applying to mentioned problems, and preliminary results of computer simulations have been discussed. The obtained results are likely to be of importance to mass transfer modeling and engineering practice in industrial processes and apparatuses design. 1. Introduction The film absorption processes occur in many industrial chemical and power enterprises (Chen, 1998). It is used for purifying gases from toxic admixtures or for extracting the valuable components (Grossman, 1986). However, the very limited resources to increase the interfacial mass transfer in the film processes still have been found (Cussler, 2009). Among the various causes of those failures (Ouyang et al., 1989), a certain cause, which may be not main but important in our opinion, is the difficulty to solve and to analyze the coupled equations of film motion and mass transfer (Fujita, 1993), especially it is true for the specific peculiarities of the processes at films surfaces (Taitelbaum and Koza, 2000). The main problem lies in the fact that while solving the coupled problem the surface velocity and the concentrations are initially unknown and must be determined in the process of the solution (Huang and Liu, 2015). Therefore some approximations should be used for these functions, which would be clarified during the numerical process (Bo et al., 2010). However, solving the problem of convective diffusion with boundary conditions at moving films surface, we have to expose the transformation of beforehand unknown function in the boundary conditions (Wassenaar and Segal, 1999). The basic premise which was used to simplify the approach to creating the model consists in that the chemisorption purification is usually applied in those cases when gases are sparingly soluble, and the process of physical absorption turns out little effective. Therefore, the Peclet number in this case is sufficiently high, and the diffusion boundary layer is very thin. This circumstance gives the opportunity to select such small control parameter as the gradient of surface velocity along the film motion direction. Besides that in the presence of big DOI: 10.3303/CET1652037 Please cite this article as: Musabekova L. M., Dausheyeva N. N., Zhumataev N. S., Jamankarayeva M. A., 2016, On modelling the convective mass transfer over moving films interface for the reaction-diffusion systems of the second order, Chemical Engineering Transactions, 52, 217-222 DOI:10.3303/CET1652037 217 mailto:mleyla@bk.ru http://www.aidic.it/cet/15/46/indice.html#A66A http://www.aidic.it/cet/15/46/indice.html#A67A or small control parameters there appears the possibility to use effective methods of the asymptotic analysis (Acampora et al, 2015). The main goal of our paper is to submit both the methods for transforming mathematical model for obtaining both the approximate analytical solutions of the coupled problem of isothermal film chemisorption at high Peclet numbers and appropriate numerical tools for the computer simulation. The submitted method allows also obtaining the approximate analytical solution of the convective diffusion equation taking into account the phenomenon of changing velocity of the film surface induced by the concentration-dependent surface tension. 2. Theoretical details This section presents the method for solving the problems of the non-self-similar (Schlichting and Gersten, 2003) film absorption, accompanied by a partially reversible second order chemical reaction at the film surface. 2.1 Master equations and boundary conditions The simplest scheme of a reversible second-order reaction reads (Gàlfi and Ràcz, 1988) EBA k k 1 2    , (1) where A (absorbed component) and B (absorbent) are the reagents, E is the reaction product, 1k and 2k are the reaction rate constants for direct reverse stages. The basic system of hydrodynamics and convective diffusion equations in the presence of a chemical source of mass is as follows (Yuste, Acedo, Lindenberg, 2004):   EBAAAAA CkCCkCDCV t C 21,     ,   BBABBBB CkCCkCDCV t C 21,     ,   EBAEEEE CkCCkCDCV t C 21,     (2) VPg t V lll        , (3) 0div V  (4) Here AC , BC , EC are the concentration of substances; V  is the liquid velocity; P is the pressure; AD , BD , ED are the diffusion coefficients into the liquid layer; g is the gravitational acceleration; l is the liquid density; l is the liquid dynamic viscosity . The momentum equation for a gas boundary layer is assumed to obey (Cebeci and Bradshaw, 2012) ydx dU U y U V x U U g e e g g g g       1  , (5) where  gg VU , are the longitudinal and normal components of the gas velocity, eU is the velocity at the outer boundary of gas layer; x and y are the longitudinal (along the film motion direction) and normal coordinates accordingly; g is the gas velocity;  is the tangential stress in a gas phase (Hihara and Saito, 1993). The wall conditions are 00  ll VUy , 0 y C   . (6) 218 The interfacial boundary conditions at the liquid film surface are the equality of normal mass fluxes of reagents through the interface, the no-slip condition and the balance condition for shear stresses with accounting of the concentrate-induced surface tension gradient along the interface (Brener, 1999):  hy y C D y C D g g l l      , dx dC dC d y U y V sg g l l         ,  S l l mCC Dy C      . (7) Here  is the surface tension;  is the mass transfer coefficient into gas phase; h is the liquid film thickness; gPHm  , where H is the Henry constant; indices “ S ” and “  ” denote film surface and gas phase far from the interface. In addition to these conditions, for the gas phase we take quite physically reasonable conditions, consisting in the fact that at the outer edge of the boundary layer the longitudinal component of the velocity is equal to the velocity in the flow core (Chopard et al., 1993), and the tangential stresses are absent (Vijayendran, Ligler, Leckband, 1999). In this case, you can integrate the equation of gas motion over the boundary layer thickness  following the known method (Cebeci and Bradshaw, 2012), which will result in the expression     S g eSe h h S gS SS VUV dx d dy U U dx dU UU dx d      12     (8) Here dy U U U U S g h h S g             1   , y Ug gg     (9) Expressions (8) and (9) are needed to calculate mass transfer coefficient into gas phase. 2.2 Variables and parameters transformation For solutions the obtained systems it is necessary to determine the concentration profiles over the film thickness (Kang et al., 2000). In its most general form, the problem is too complex, and effective viable approaches are not offered up today (Bo et al., 2010). At the same time, applying to the most important cases when the Peclet number is large enough, the problem can be simplified (Brener, 1999). In this limit, the thickness of the diffusion boundary layer in the liquid phase in the vicinity of the free surface is much smaller than the film thickness (Richardson and Kafri, 1999). In this regard, one can replace the value of the fluid velocity in the convective diffusion equation directly by its value at the interface. Then, using the continuity equation, we get after the change of the variable yhz  : 2 2 z C D z C dx dU z x C U l S S        (10) The appropriate boundary conditions read z C Dz l    0   CmC ; 0 z C z   . (11) In contrast to the problem which was discussed in some works (Pérez, Picioreanu, van Loosdrecht, 2005), in this case, as the liquid film moves it will be saturated by the absorbed reagent and by the reaction product, and their concentrations will change both over the film thickness and along the film surface (Tomé and de Oliveira, 2015). Therefore, any transformations should take into account the longitudinal coordinate, in addition to the self- transverse coordinate (Wassenaar and Segal, 1999). The suitable transformations are (Brener, 1999) 219  x SlS dtUDUzz 0 2  , dtdqUUXx x t SS            0 (12) Now, after the change of variables the convective diffusion equation reduces to the form with constant coefficients. EBA AAA CkCCk C n C X C 212 2 1 2 1            , EBA BBB CkCCk CC X C 212 2 2 1            , EBA EEE CkCCkn CC X C 2122 2 2 1            (13) Here BA DDn 1 , BE DDn 2 . The appropriate boundary conditions are 0    C , )1(0 0 2 S l X S l mC D dt U U D C                 (14) In many cases, the chemisorption process is carried out under conditions of supersaturation of the solution by the active absorbent, as by this the most effective absorption of the target component can be achieved. Thus in the considered case the most quickly changing concentration is AC , but the gradient of concentration BC is small, and its concentration is close to the one at the surface (Bo et al., 2010). Therefore, taking the concentration of the absorbent close to the surface concentration, we can write the approximate equality.                        A BS B BSBBSABA C C C CCCCCC 1 (15) Now there appears a small parameter BS B C C  , and it opens the opportunity to linearize the master equations (Giga et al., 2010). For this small parameter it can be given a simple sense of the relationship between the characteristic time of the conversion and the residence time of the reaction mixture in the absorber. From this it follows the main limitation for the mentioned approximation. Namely, the described asymptotic approach is admissible only under the weak reversibility of the reaction, i.e., as an engineering evaluation of the parameter, we suggest using the ratio between rate constants of inverse and direct reactions 12 kk . (16) In the future, we propose on the base of these ideas to develop a method for analysing the described model with the help of the Laplace transform and asymptotic tools. It may be relevant to many industrial processes. In the next section the numerical scheme for investigating non-linear problem Eqs(13), (14) and some preliminary results of computer simulation are submitted. 3. Numerical scheme and computer experiment As coefficients of the model system remain to be constant the known Crank - Nicholson method (Musabekova et al., 2012) added by the iteration procedures can be used with sufficient efficiency. The numerical algorithm has been realized by the software of PTC Mathcad 15.0. The used code is shown in Figure 1. Figure 2 depicts certain preliminary results of the computer simulation based on model – see Eqs.(13) and (14). 220 Figure 1: Mathcad code for the computer simulation Figure 2: Concentration of absorbed reagent over the film thickness for 1.012 kk : X= 0.02 (1); X=0.1 (2); X=0.2 (3); X=0.4 (4); X=0.5 (5); X=0.6 (6) 4. Conclusions It can be concluded that using special coordinate transformation (12), we can offer a method for solving the problem of non-self-similar film absorption, accompanied by a second-order reversible chemical reaction. Namely, the complex system of convective diffusion equations can be reduced to the form with constant coefficients (13) and boundary conditions (14). Moreover, at low reversibility of the chemical reaction and slow change of surface features along the longitudinal coordinate the model system can be reduced to the linear form. It opens up the possibility of an asymptotic analysis under the boundary conditions of sufficiently general n 10 A dx dy m n( ) Uaj i A0 Ubj i B0 Uej i E0 Fj i dy a a b m as( ) k1s as[ ] i 0 nfor j 0 mfor E1 1 y y0 i dy A dx Bi dy 2 0.5 dy dx y 2dx Ci dx 0.5 dx dy y Dj k2 dx dy2 Uej 1 i 1 0.5 y dy dx Uaj 1 i Uaj 1 i 1  dy 2 Uaj 1 i Dj 1 i dx Uaj 1 i 1 2 Uaj 1 i Uaj 1 i 1 i  dx dy 2  k1 Uaj 1 i 1 Ubj 1 i 1 Dj Ei 1 A Bi Ci Ei  Fj 1 i 1 Dj 1 i Ci Fj 1 i Bi Ci Ei  i 1 n 1for Uaj n Fj 1 n 1 1 En 1  Uaj i 1 Ei 1 Uaj i Fj 1 i 1 i n 1for j 1 mfor v 1 1for Ua  Calculating initial concentrations Calculating coefficients with tridiagonal matrix algorithm Calculating concentrations to the right of the interface border n 10 A dx dy m n( ) Uaj i A0 Ubj i B0 Uej i E0 Fj i dy a a b m as( ) k1s as[ ] i 0 nfor j 0 mfor E1 1 y y0 i dy A dx Bi dy 2 0.5 dy dx y 2dx Ci dx 0.5 dx dy y Dj k2 dx dy2 Uej 1 i 1 0.5 y dy dx Uaj 1 i Uaj 1 i 1  dy 2 Uaj 1 i Dj 1 i dx Uaj 1 i 1 2 Uaj 1 i Uaj 1 i 1 i  dx dy 2  k1 Uaj 1 i 1 Ubj 1 i 1 Dj Ei 1 A Bi Ci Ei  Fj 1 i 1 Dj 1 i Ci Fj 1 i Bi Ci Ei  i 1 n 1for Uaj n Fj 1 n 1 1 En 1  Uaj i 1 Ei 1 Uaj i Fj 1 i 1 i n 1for j 1 mfor v 1 1for Ua  Calculating initial concentrations Calculating coefficients with tridiagonal matrix algorithm Calculating concentrations to the right of the interface border 0 0.2 0.4 0.6 0.8 0.33 0.67 Relative normal coordinate, y/h R e la ti v e c o n c e n tr a ti o n r e a g e n t A 1 2 3 4 5 6 1 1 0 0.2 0.4 0.6 0.8 0.33 0.67 Relative normal coordinate, y/h R e la ti v e c o n c e n tr a ti o n r e a g e n t A 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 0.33 0.67 Relative normal coordinate, y/h R e la ti v e c o n c e n tr a ti o n r e a g e n t A 000 0.2 0.4 0.6 0.8 0.33 0.67 Relative normal coordinate, y/h R e la ti v e c o n c e n tr a ti o n r e a g e n t A 1 2 3 4 5 6 1 1 221 form implemented in practical terms. In general case system (13), (14) can be solved numerically using the modified Crank-Nicholson method with subsequent calculating the surface parameters in iteration process. The results of preliminary calculations show that the noticeable effect of the concentration-dependent capillary stresses at the film surface occurs only on very short, initial areas near the starting point of absorption (Figure 2). This effect manifests itself in the growth rate of the surface of the film along the longitudinal coordinates and the more rapid growth of the surface concentration. References Acampora L., Mancusi E., Marra F.S., 2015, Bifurcation Analysis of Perfectly Stirred Reactors with Large Reaction Mechanisms, Chemical Engineering Transactions, 43, 877-882. Bo S., Ma X., Lan Z., Chen J., Chen H, 2010, Numerical simulation on the falling film absorption process in a counter-flow absorber, Chemical Engineering Journal, 156(3), 607-612. 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