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Mass transfer in gas–liquid contactors: a new technique for numerical solution of the film equations
Chemical Engineering and Processing 43 (2004) 1339–1346
Mass transfer in gas–liquid contactors: a new technique for numerical solution of the film equations Benoˆıt Haut∗ , Véronique Halloin Chemical Engineering Department, Université Libre de Bruxelles, C.P. 165/67, Avenue F.D. Roosevelt, 50, 1050 Bruxelles, Belgium Received 15 November 2002; received in revised form 7 July 2003; accepted 7 July 2003 Available online 26 June 2004
Abstract The two film theory is often used to evaluate the mass transfer rate in gas–liquid contactors. The analytical solving of the film mass balance equations is often not possible because of the non-linearity of the reaction terms. Therefore, a numerical solution of the equations of the contactor model has to be developed. However, this becomes a heavy task when the bulk phases are not perfectly mixed. An original numerical solving technique is presented here. The technique is based on a Taylor series development of all the unknowns of the contactor model. The technique is presented on three applications. Two of these applications are simple enough to allow the comparison of our results with the results obtained with a classical shooting technique. © 2004 Elsevier B.V. All rights reserved. Keywords: Mass transfer; Two film theory; Gas–liquid contactor; Asymptotic technique
1. Introduction Many chemical and biochemical processes involve a reaction between a gas and some materials present in a liquid. The gas has to be dissolved in the liquid phase in order that the chemical reactions can take place. The mass transfer rates in gas–liquid contactors (the usual name given to devices used for such processes) may vary considerably according to specific design aspects such that the rate of energy dissipation. The prediction of a gas–liquid contactor performance requires a model development in which an essential step is the description of mass transfer between phases. Among the numerous models existing to describe this transfer, the two film theory [1,2] is probably the simplest, and therefore the most used one. This model assumes that on both sides of the gas–liquid interface there is a thin stagnant layer, or film, through which the different components diffuse. The mass flux across this layer is proportional to the difference between the interfacial and the bulk concentrations. The coefficient of proportionality is the mass transfer coefficient. Although not entirely accurate, the film theory provides ∗
Corresponding author. Tel.: +32 2 650 40 96; fax: +32 2 650 29 10. E-mail address: [email protected] (B. Haut).
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a helpful conceptual picture of interfacial mass transfer. Other frequently encountered models are Higbie penetration theory and the Danckwerts surface renewal model [3]. Within each stagnant film, the concentration profile of a component is linear as far as the reaction of this component is negligible in the film, i.e. the reaction is slow with respect to mass transfer. If this is not the case, the reaction changes the concentration profile and modifies the mass transfer rate. Correction to mass transfer rate due to chemical reaction is calculated through the resolution of the mass balance equation in the films, for each component: Di
d 2 Ci = Ri dx2
(1)
This set of equations is subject to boundary conditions: at the interfaces between the stagnant films and the bulks of the phases, continuity of the concentrations is assumed. At the gas–liquid interface, thermodynamic equilibrium and flux continuity are assumed for the transferred component in the isothermal case [4]. A zero flux is assumed at the gas–liquid interface for the non-transferred components. Modeling of a gas–liquid contactor is realized by coupling these mass balance equations in the films with the balance equations written on the bulk of the phases. The ideal procedure to solve the resulting system is the following one:
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1. An analytical solution of the film balance Eqs. (1) is derived. This solution depends on the concentrations in the bulk of the gaseous and liquid phases. These concentrations are indeed involved in the boundary conditions of Eqs. (1). 2. The mass fluxes between phases are derived from the analytical expressions of the concentration profiles in the films. 3. These mass fluxes are introduced in the bulk mass balance equations. Depending on the contacting pattern of the considered gas–liquid contactor, the bulk phases are described using plug flow models, perfectly mixed tank models, dispersion models or compartment models. Space discretization must be operated in case of plug flow or dispersion models. The resulting model consists in a set of N algebraic linear equations with N unknowns that can be easily solved by an iterative process. Unfortunately, because of the non-linear terms in the production rate, it is often impossible to find an analytical solution to the film Eqs. (1), except for first-order reactions. A numerical solution of these equations has therefore to be set up. It is important to observe that this numerical resolution has to be coupled with the resolution of the mass balance equations on the bulk of the phases, as boundary conditions of the film equations involve bulk concentrations. Various numerical techniques are available, among which discretization of mass balance equations in the liquid film, shooting technique or linearization of the film equations [5,6]. The first two of these three approaches proved to be efficient if a perfectly mixed zone can model each of the bulk phases. But for other situations, these two approaches can in practice become very heavy to implement. One of the major problems of numerical simulations is indeed the estimation of a consistent set of initial guesses for all variables. Besides, simultaneously solving such large-scale systems may cause significant computational difficulties. In this paper, a new technique is proposed to solve the equations of a gas–liquid contactor model. It is based on an asymptotic approach, in which all the unknowns are developed in Taylor series. As it will be shown, the main advantage of this technique lies in the reduction of the whole problem into a number n of simpler sub-problems that can be solved in turn. Besides, as the film equations can be solved analytically for each of the n sub-problems, the so-called ideal method described here before can be applied for each of the n sub-problems. This technique cannot be applied systematically, but it is shown in this paper that it offers significant advantages in several cases.
2. General presentation of the technique As the mathematical formalism of the technique is fastidious to expose in detail, the main ideas of the technique are
briefly summarized in this section. In the following section, the technique is presented in detail on three cases. N components possibly involved in R reactions in the stagnant liquid film are considered. The mass transport of these components in the film is described by a system of N second order differential equations. This system is first reduced to a system of R second order equations. This resulting system is written down in dimensionless form, revealing dimensionless numbers that can be divided into two groups: a first group of dimensionless numbers characterizing the competition between reaction and diffusion in the liquid film, and a second group of dimensionless numbers involving characteristic times of mass supply and mass withdrawal. It can be observed that within this second group of numbers, it is sometimes possible to identify some of them whose set to zero will transform the system into a new one that can be solved analytically. Therefore, all the unknowns of the gas–liquid contactor model are developed in Taylor series of these numbers. The developments are of degree n. Introduction of these developments in each equation of the model leads to n sub-problems that can be solved in turn. Moreover, the film mass balance equations can be solved analytically for each of the n sub-problems. Therefore, the ideal three-steps solving procedure described in the introduction can be applied for each of the n sub-problems. Symbolic calculation software such as Mathematica® can be used to optimize the identification and solving of the different sub-problems.
3. Applications of the technique The technique is first presented and validated against two test cases for which the solution can also be derived by a classical shooting technique. A third application where a discretization of the film equations or a shooting technique is by far cumbersome to implement than our technique is then presented. 3.1. Application no. 1 (test case) Let us consider the case of a reaction A + B → C of order 1–1, in which component A is transferred from gas to liquid phase under isothermal conditions. The resistance to mass transfer is located within the stagnant liquid film. Gas and liquid phases are perfectly mixed and only steady state is considered here. The only inlets for A and B are respectively the gas and the liquid phase. The transfer between phases and the reaction does not significantly modify the volumetric flow rate of the phases. This first application is chosen because the corresponding equations can be solved easily using a shooting technique. Mass balance equations in the liquid film are given by: DA
d 2 CA = k1 CA CB dx2
(2)
B. Haut, V. Halloin / Chemical Engineering and Processing 43 (2004) 1339–1346
d 2 CB DB = k1 CA CB dx2
(3)
These equations are supplemented by the corresponding boundary conditions: dCB (g) (l) CA (0) = HCA , CA (xl ) = CA , = 0, dx 0 CB (xl ) =
(l) CB
Returning with Eq. (5) into Eq. (2), yields: A dx2
(l) k1 CB
x , xl
XA = (l)
(l)
XA =
CA
(g)
HCA,0
CA (ξxl ) (g) HCA,0 (l)
,
XB =
,
XB = (l)
CB
(6)
(l)
CB,0
CB (ξxl ) (l)
CB,0 (g)
,
XA =
,
(g)
CA
(g)
CA,0
Eq. (6) can then be written in dimensionless form: d 2 XA (l) (l) 2 2 = Ha1 XA XB + β XA − X A XA dξ 2 dXA (ξ − 1)XA − dξ 0
(7)
=
DA
T =
,
β=
DA HCA,0 (l) DB CB,0
εV DA a, Q g xl
R1 =
V(1 − ε) (l) k1 CB,0 , Ql
(g)
Qg S= , Ql
F=
Qg CA,0
(13)
(l)
Ql CB,0
At the limit β → 0, Eq. (8) can be solved analytically. This corresponds to the classical first-order approximation, only valid in practice for large excess of B or when the diffusivity of component B is significantly larger than the diffusivity of component A. Therefore, all the unknowns of the model are developed in Taylor series in function of β. Any unknown y is then written as: n
yi β i
(14)
where the new unknowns yi are functions of ξ if y refers to a concentration in the liquid film, or are real numbers if y refers to a bulk concentration. The order of development n is chosen accordingly to the desired accuracy. The equations of the n sub-problems are generated by introducing developments Eq. (14) into Eqs. (8) and (10)–(12) and by equalizing coefficients of equal powers of β in the resulting expressions. The following system, generated by equalizing βi coefficients, is called the system of order i: dXA,i (g) = δ1,i+1 (15) XA,i − HT dξ 0
(16)
j=0
(g)
(8)
(l)
F(δ1,i+1 − XA,i ) = XA,i
HF (l) + (δ1,i+1 − XB,i ) S
(17)
i i−1 d2 XA,i (l) 2 = Ha X X + XA,i−1−j A,j 1 B,j−i dξ 2
(g)
(l)
k1 CB,0 xl2
(l)
+ (1 − XB ) (12) S where T, S, R1 and F are dimensionless numbers defined as:
i dXA,i (l) (l) (l) + R XA,i−j XB,j = 0 XA,i + ST dξ 1
where the Hatta number Ha1 and β are defined by: Ha21
(11)
i=0
In order to write the model equations in dimensionless form, the following dimensionless variables are defined: ξ=
(l) HF
(g)
y=
k1 (l) 2 CA = CA + − C A CA DA DB dCA − (x − xl )CA dx 0
dXA (l) (l) + R 1 XA XB = 0 dξ 1
F(1 − XA ) = XA
(4)
This system of two second order equations can be reduced into one second order equation. Indeed, the subtraction of Eq. (3) from Eq. (2) eliminates the non-linear term, and the resulting equation can be integrated twice: DA DA dCA (l) (l) CB (x) = CB + (CA (x) − CA ) − (x − xl ) DB DB dx 0 (5)
d2 C
(l)
XA + ST
1341
(9)
Dimensionless number Ha1 compares the characteristic time of diffusion of A in the stagnant liquid film to the characteristic time of reaction A + B → C. Dimensionless number β compares mass supply rates of A and of B in the liquid film. Eq. (8) is supplemented by mass balance equations written in the bulk of each phase: dXA (g) XA − HT =1 (10) dξ 0
j=0
j=0
dXA,j (l) × XA,j − XA,j − (ξ − 1) dξ 0
(18)
Note that: 1. the n systems can be solved in turn, as no unknown of index higher than i appears in the system of order i. Therefore, the first term of the Taylor development of the unknowns is determined by solving the system of order 0, the second term is determined by solving the system of order 1, . . .
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Fig. 1. S = 10, T = 1.5, R1 = 30, H = 0.4, F = 1, Ha1 = 3, β = 1.
2. whatever the i value, Eq. (18) can be solved analytically. Indeed, the only term containing XA,i in the left side of Eq. (18) is proportional to XA,i . The different systems can be solved in turn following the ideal procedure described in the introduction, the number of system to be solved depending on the chosen accuracy. In Fig. 1, several approximation orders are compared to the solution obtained by a shooting technique using a fourth-order Runge–Kutta integration method. β is chosen equal to 1, in order to be in a situation where a numerical resolution is necessary. It can be observed that the deviation between the solution obtained by the shooting technique and the solution obtained with our technique decreases when the order of the approximation (n) increases. Let us define the average deviations between two profiles 1 y1 (x) and y2 (x) as 0 |y1 (ξ) − y2 (ξ)| dξ. At the order 0, the average deviation between a concentration profile in the liquid film computed with our technique and the corresponding profile computed with the shooting technique is found to be 0.1 for component A and 0.2 for component B. These deviations fall to 0.02 and 0.1 for the approximation of order 1, and to 0.008 and 0.04 for the approximation of order 2. Results obtained with an approximation of order 3 are not
represented in Fig. 1. They are visually undistinguishable from the results calculated using the shooting technique. Several simulations have been performed for values of β less than 1 and values of Ha1 less than 3. The average deviation between a profile calculated with our technique and the corresponding profile obtained by the shooting technique is always found to be less than 0.1, as long as an approximation of order 2 is used. The maximum observed deviation falls to 0.025 when β is inferior to 0.5, even for values of Ha1 up to 4. In Fig. 2, several profiles of concentration in the liquid film are presented for various values of parameters Ha1 and R1 . They are computed with our technique using an order 2 approximation. As expected, the curvature of the profiles increases with the kinetic constant of the reaction. The average deviations between these profiles and the profiles calculated by a shooting technique are found to be less than 0.025. In conclusion, our technique proved to be adequate to solve the equations of this first model, as long as at least a second order approximation is chosen. Indeed, under this condition, the profiles of concentration in the liquid film are found to be close to the profiles calculated by a shooting technique, within a large range of values of the dimensionless numbers of the model. However, for such a simple case, our technique does not bring significant advantages in term
Fig. 2. S = 10, T = 1.5, H = 0.4, F = 1, β = 0.5.
B. Haut, V. Halloin / Chemical Engineering and Processing 43 (2004) 1339–1346
1343
of computer time with comparison to the classical shooting technique. 3.2. Application no. 2 (test case) As most of the industrial problems involve multiple reactions, the second application chosen to test our technique considers a secondary reaction between the reactant B and the product C of the main reaction: A+B → C; B+C → D. It is supposed here that component C is not introduced at the contactor inlet. All the hypotheses assumed for test case 1 prevail: both reactions are of order 1–1, the stagnant film model is used to describe the transfer of component A from gas to liquid under isothermal conditions, and the gas and liquid phases are considered to be perfectly mixed. The three mass balance equations in the stagnant liquid film can be reduced into two equations that are written in dimensionless form as: d 2 XA (l) (l) 2 2 = Ha1 XA XB + 2β XA − X A XA dξ 2 dXA (l) − (ξ − 1)XA + ηβ[XC XA − XC XA ] dξ 0 (19)
d2 XC −Ha21 (l) (l) 2 = X X + 2β XA − X A XA A B η dξ 2 dXA (l) − (ξ − 1)XA + ηβ[XC XA − XC XA ] dξ 0 (l) (l) 2 + Ha2 XC XB + 2β XC XA − XA XC dXA (l) 2 − (ξ − 1)XC + ηβ[XC − XC XC ] dξ 0 (20) The following new definitions are introduced: (l)
Ha22 =
k2 CB,0 xl2 DC
,
η=
DC , DA
Fig. 3. S = 10, T = 2/3, R1 = 30, 2k1 = 3k2 , H = 0.4, F = 1, Ha1 = 2, Ha2 = 1, β = 0.5.
deviations are 0.006, 0.036 and 0.010 in the conditions of Fig. 4. Several simulations have been performed for values of β inferior to 0.5 and values of Ha1 , Ha2 and η inferior to 3. The average deviation between a profile calculated with our technique and the corresponding profile obtained by the shooting technique is always found to be less than 0.1, as long as an approximation of order 2 is used. In conclusion, our technique proved to be adequate to solve the equations of this second model, as long as at least a second order approximation is chosen. Indeed, under this condition, the profiles of concentration in the liquid film are found to be close to the profiles calculated by a shooting technique, within a large range of values of the dimensionless numbers of the model. 3.3. Application no. 3 The two previous applications are helpful to validate our technique against numerical results obtained with the proven
(l)
XC (ξ) =
CC (xl ξ) (g)
HCA,0 (21)
As in application no. 1, these equations have an analytical solution only at the limit β → 0. Out of this limit, the unknowns of the problem are developed in Taylor series in function of β and the same approach as presented in application no. 1 can be followed. Concentration profiles in the stagnant liquid film computed with our technique are compared in Figs. 3 and 4 to profiles computed with a classical shooting technique. For the conditions of Fig. 3, average deviations of 0.002, 0.012 and 0.003 are respectively observed between the profiles of concentration in A, B and C computed with our technique and the profiles computed with a shooting technique. These
Fig. 4. S = 10, T = 2, R1 = 300, 2k1 = 3k2 , H = 0.4, F = 1, Ha1 = 2, Ha2 = 3, β = 0.5.
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B. Haut, V. Halloin / Chemical Engineering and Processing 43 (2004) 1339–1346
classical shooting technique. However, the complexity of these cases is not high enough to fully display the potential advantages of our technique. The third application is chosen for this purpose. In this problem, the gas phase is modeled as a plug flow while the liquid phase is assumed to be perfectly mixed. This can be considered as a realistic hydrodynamic model of a bubble column [4]. Let us consider the case of a reaction A+B → C of order 1–1, in which component A is transferred from gas to liquid phase under isothermal conditions. The resistance to mass transfer is located within the stagnant liquid film. Only steady state is considered here. The only inlets for A and B are respectively the gas and the liquid phase. It is assumed that the transfer between phases and the reaction does not significantly modify the volumetric flow rate of the phases. A dimensionless coordinate z is introduced in the plug flow model of the gas phase: z = 0 at the inlet and z = 1 at the outlet. Balance equations on the bulk of the phases are given by: (g) ∂XA ∂XA = HT ∂z ∂ξ ξ=0,z (l) XA
1
+ ST 0
(22)
∂XA (l) (l) dz + R1 XA XB = 0 ∂ξ ξ=1,z (l) HF
(g)
F(1 − XA (z = 1)) = XA
S
(l)
+ (1 − XB )
(23)
(24)
Dimensionless number F, S, T and R1 are still defined by Eqs. (13). The following equation for the mass balance in the liquid film is obtained: ∂ 2 XA (l) (l) 2 2 = Ha1 XA XB + β XA − X A XA ∂ξ 2 ∂XA (25) − (ξ − 1)XA ∂ξ ξ=0,z
(l)
XA,i + ST
i ∂XA,i (l) (l) dz + R XA,i−j XB,j = 0 ∂ξ ξ=1,z
1 z=0
j=0
(28) (g)
(l)
F(δ1,i+1 − XA,i (z = 1)) = XA,i
HF (l) + (δ1,i+1 − XB,i ) S (29)
i i−1 ∂2 XA,i (l) 2 = Ha X X + XA,i−1−j A,j B,j−i 1 ∂ξ 2 ×
j=0
(l) XA,j − XA,j
j=0
∂XA,j − (ξ − 1) ∂ξ ξ=0,z (30)
Each of these systems is completed by adequate boundary conditions. As in the two test cases presented in the previous section, these systems can be solved in turn and following the so-called ideal procedure presented in the introduction. However, a suitable discretization of Eq. (27) has to be done. A side effect of the Taylor development of all the unknowns of the model is that for i > 0, Eq. (28) is linear, thus enabling an optimization of the solution of the ith sub-problem. Several concentrations profiles obtained with our technique are presented in Figs. 5 and 6. As 40 points are used for the discretization of Eq. (27), our technique leads to the transformation of the problem into n successive sub-problems composed of 42 algebraic equations. As the approximate solutions for n = 2 and 3 are visually indistinguishable in Figs. 5 and 6, it can be considered that a satisfying level of convergence is reached for n = 2. This is coherent with the observations made on the first test case. It is interesting to observe that a shooting technique would have necessitate to find the zero of a function of 80 variables whose evaluation necessitates the resolution of 160 first
XA is here a function of z and ξ. This equation is supplemented by the following boundary conditions: (g)
XA (ξ = 0, z) = XA (z),
(l)
XA (ξ = 1, z) = XA
(26)
As an analytical solution exists for Eq. (8) in the limit β → 0, all the unknowns of the model are developed in Taylor series in function of β. Inserting these developments into Eqs. (22)–(25) and equalizing coefficients of equal powers of β in the resulting expressions, yields the following equations for the n sub-problems: (g)
∂XA,i ∂z
∂XA,i = HT ∂ξ ξ=0,z
(27) Fig. 5. S = 10, T = 1.5, R1 = 30, H = 0.4, Ha1 = 3, β = 0.5.
B. Haut, V. Halloin / Chemical Engineering and Processing 43 (2004) 1339–1346
1345
the mass balance equations in the bulk of the phases are linearized. This facilitates the determination of the correction. The technique can be applied for multiple or reversible reactions, but it is restricted to reactions of integer order.
Acknowledgements F.N.R.S. (Belgian national fund for scientific research) fellow researcher Benoˆıt Haut acknowledges the fund for its financial support.
Appendix A. Nomenclature Fig. 6. S = 10, T = 1.5, R1 = 30, H = 0.4, F = 1/10, Ha1 = 3, β = 0.5, n = 2.
a Ci (x)
order ordinary differential equations. A 10 points discretization of the film equations leads to the resolution of more than 400 non-linear algebraic equations. This heavy numerical handling can be avoided with the help of our technique. As appears in Fig. 5, the outlet concentration of A in the gaseous phase is significantly modified when n goes from zero (n = 0 corresponds to the assumption usually made in industrial practice) to two. Pertinent corrections to the usual industrial approach can be computed quickly with our technique.
4. Conclusions In this paper, a new technique is proposed to solve the equations of a gas–liquid contactor model. It is based on an asymptotic approach, in which all the unknowns are developed in Taylor series. The solution of the system of equations is composed of a first approximation (order 0) to which a series of corrections is added. The ith correction is defined as the correction of order i. When computing a correction, an analytical solution of the mass balance equations in the liquid film is derived. This solution is then inserted in the mass balance equations written on the bulk of the phases. Solving of the resulting system of equations yields the correction. This one is evaluated once for all but is needed to start the computation of the correction of higher order. The technique has been first presented and validated against two test cases for which the solution can be derived more or less easily by a classical shooting technique. A third application where a discretization of the film equations or a shooting technique is by far less effective than our technique is finally presented. The existence of an analytical solution of the mass balance equations in the film permits to consider any degree of mixing in the bulk of the phases. A side effect of the development into Taylor series is that, except for order 0,
(l)
Ci
(g)
Ci
(l)
Ci,0 (g)
Ci,0 Di F H
Ha1 Ha2 k1 k2 n Ql Qg S T R1 Ri (x) V Xi (x) (l)
Xi
interfacial area in the contactor expressed in m2 of interface per m3 of gas phase (m−1 ) concentration of component i at the position x within the liquid stagnant layer (mol/m3 ) concentration of component i in the bulk zone of the liquid phase (mol/m3 ) concentration of component i in the bulk zone of the gas phase (mol/m3 ) concentration of component i in the liquid feed stream (mol/m3 ) concentration of component i in the gas feed stream (mol/m3 ) diffusion coefficient of component i (m2 /s) dimensionless number, see Eq. (13) dimensionless number related to the Henry’s coefficient of component A Hatta number associated with reaction A + B → C, see Eq. (9) Hatta number associated with reaction B + C → D, see Eq. (21) kinetic coefficient of reaction A + B → C (m3 /mol s) kinetic coefficient of reaction B + C → D (m3 /mol s) order of the Taylor series development, see Eq. (14) volumetric flow rate of the liquid phase (m3 /s) volumetric flow rate of the gaseous phase (m3 /s) dimensionless number, see Eq. (13) dimensionless number, see Eq. (13) dimensionless number, see Eq. (13) consumption of rate of component i at the position x within the liquid layer (mol/m3 s) volume of the contactor (m3 ) reduced concentration of component i at the position ξ within the liquid stagnant layer, see Eqs. (7) and (21) reduced concentration of component i in the bulk zone of the liquid phase, see Eq. (7)
1346 (g)
Xi x
xl z
B. Haut, V. Halloin / Chemical Engineering and Processing 43 (2004) 1339–1346
reduced concentration of component i in the bulk zone of the gas phase, see Eq. (7) position within the liquid film. x = 0 at the gas–liquid interface and x = xl at the interface between the liquid film and the bulk of the liquid phase (m) thickness of the liquid stagnant film dimensionless coordinate
Greek letters β dimensionless number, see Eq. (9) δi,j Kronecker delta. δi,j = 1 if i = j and δi,j = 0 if i = j ε gas hold-up in the contactor η dimensionless number, see Eq. (21) ξ dimensionless position within the liquid stagnant layer, see Eq. (7)
References [1] J. Villermaux, Génie de la Réaction Chimique, Lavoisier Tec & Doc, Paris, 1995. [2] E.S. Gaddis, Mass transfer in gas–liquid contactors, Chem. Eng. Process. 38 (1999) 503–510. [3] J.A. Hogendoorn, R.D. Vas Bhat, J.A.M. Kuipers, W.P.M. van Swaaij, G.F. Versteeg, Approximation for the enhancement factor applicable to reversible reactions of finite rate in chemically loaded solutions, Chem. Eng. Sci. 52 (1997) 4547–4559. [4] W.D. Deckwer, Bubble Column Reactors, Wiley, Chichester (GB), 1992. [5] H. Benbelkacem, H. Cano, J.C. Huard, M. Roustan, H. Debellefontaine, Modélisation du transfert de matière gaz/liquide couplé avec une réaction chimique. Application au régime cinétique intermédiaire, Récents Progrès en Génie des Procédés (80), Lavoisier Tec & Doc, Paris, 2001. [6] E.Y. Kenig, R. Scheinder, A. Gorak, Multicomponent unsteady-state film model: a general analytical solution to the linearized diffusionreaction problem, Chem. Eng. J. 83 (2001) 85–94.
Mass transfer in gas–liquid contactors: a new technique for numerical solution of the film equations Benoˆıt Haut∗ , Véronique Halloin Chemical Engineering Department, Université Libre de Bruxelles, C.P. 165/67, Avenue F.D. Roosevelt, 50, 1050 Bruxelles, Belgium Received 15 November 2002; received in revised form 7 July 2003; accepted 7 July 2003 Available online 26 June 2004
Abstract The two film theory is often used to evaluate the mass transfer rate in gas–liquid contactors. The analytical solving of the film mass balance equations is often not possible because of the non-linearity of the reaction terms. Therefore, a numerical solution of the equations of the contactor model has to be developed. However, this becomes a heavy task when the bulk phases are not perfectly mixed. An original numerical solving technique is presented here. The technique is based on a Taylor series development of all the unknowns of the contactor model. The technique is presented on three applications. Two of these applications are simple enough to allow the comparison of our results with the results obtained with a classical shooting technique. © 2004 Elsevier B.V. All rights reserved. Keywords: Mass transfer; Two film theory; Gas–liquid contactor; Asymptotic technique
1. Introduction Many chemical and biochemical processes involve a reaction between a gas and some materials present in a liquid. The gas has to be dissolved in the liquid phase in order that the chemical reactions can take place. The mass transfer rates in gas–liquid contactors (the usual name given to devices used for such processes) may vary considerably according to specific design aspects such that the rate of energy dissipation. The prediction of a gas–liquid contactor performance requires a model development in which an essential step is the description of mass transfer between phases. Among the numerous models existing to describe this transfer, the two film theory [1,2] is probably the simplest, and therefore the most used one. This model assumes that on both sides of the gas–liquid interface there is a thin stagnant layer, or film, through which the different components diffuse. The mass flux across this layer is proportional to the difference between the interfacial and the bulk concentrations. The coefficient of proportionality is the mass transfer coefficient. Although not entirely accurate, the film theory provides ∗
Corresponding author. Tel.: +32 2 650 40 96; fax: +32 2 650 29 10. E-mail address: [email protected] (B. Haut).
0255-2701/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2003.07.002
a helpful conceptual picture of interfacial mass transfer. Other frequently encountered models are Higbie penetration theory and the Danckwerts surface renewal model [3]. Within each stagnant film, the concentration profile of a component is linear as far as the reaction of this component is negligible in the film, i.e. the reaction is slow with respect to mass transfer. If this is not the case, the reaction changes the concentration profile and modifies the mass transfer rate. Correction to mass transfer rate due to chemical reaction is calculated through the resolution of the mass balance equation in the films, for each component: Di
d 2 Ci = Ri dx2
(1)
This set of equations is subject to boundary conditions: at the interfaces between the stagnant films and the bulks of the phases, continuity of the concentrations is assumed. At the gas–liquid interface, thermodynamic equilibrium and flux continuity are assumed for the transferred component in the isothermal case [4]. A zero flux is assumed at the gas–liquid interface for the non-transferred components. Modeling of a gas–liquid contactor is realized by coupling these mass balance equations in the films with the balance equations written on the bulk of the phases. The ideal procedure to solve the resulting system is the following one:
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1. An analytical solution of the film balance Eqs. (1) is derived. This solution depends on the concentrations in the bulk of the gaseous and liquid phases. These concentrations are indeed involved in the boundary conditions of Eqs. (1). 2. The mass fluxes between phases are derived from the analytical expressions of the concentration profiles in the films. 3. These mass fluxes are introduced in the bulk mass balance equations. Depending on the contacting pattern of the considered gas–liquid contactor, the bulk phases are described using plug flow models, perfectly mixed tank models, dispersion models or compartment models. Space discretization must be operated in case of plug flow or dispersion models. The resulting model consists in a set of N algebraic linear equations with N unknowns that can be easily solved by an iterative process. Unfortunately, because of the non-linear terms in the production rate, it is often impossible to find an analytical solution to the film Eqs. (1), except for first-order reactions. A numerical solution of these equations has therefore to be set up. It is important to observe that this numerical resolution has to be coupled with the resolution of the mass balance equations on the bulk of the phases, as boundary conditions of the film equations involve bulk concentrations. Various numerical techniques are available, among which discretization of mass balance equations in the liquid film, shooting technique or linearization of the film equations [5,6]. The first two of these three approaches proved to be efficient if a perfectly mixed zone can model each of the bulk phases. But for other situations, these two approaches can in practice become very heavy to implement. One of the major problems of numerical simulations is indeed the estimation of a consistent set of initial guesses for all variables. Besides, simultaneously solving such large-scale systems may cause significant computational difficulties. In this paper, a new technique is proposed to solve the equations of a gas–liquid contactor model. It is based on an asymptotic approach, in which all the unknowns are developed in Taylor series. As it will be shown, the main advantage of this technique lies in the reduction of the whole problem into a number n of simpler sub-problems that can be solved in turn. Besides, as the film equations can be solved analytically for each of the n sub-problems, the so-called ideal method described here before can be applied for each of the n sub-problems. This technique cannot be applied systematically, but it is shown in this paper that it offers significant advantages in several cases.
2. General presentation of the technique As the mathematical formalism of the technique is fastidious to expose in detail, the main ideas of the technique are
briefly summarized in this section. In the following section, the technique is presented in detail on three cases. N components possibly involved in R reactions in the stagnant liquid film are considered. The mass transport of these components in the film is described by a system of N second order differential equations. This system is first reduced to a system of R second order equations. This resulting system is written down in dimensionless form, revealing dimensionless numbers that can be divided into two groups: a first group of dimensionless numbers characterizing the competition between reaction and diffusion in the liquid film, and a second group of dimensionless numbers involving characteristic times of mass supply and mass withdrawal. It can be observed that within this second group of numbers, it is sometimes possible to identify some of them whose set to zero will transform the system into a new one that can be solved analytically. Therefore, all the unknowns of the gas–liquid contactor model are developed in Taylor series of these numbers. The developments are of degree n. Introduction of these developments in each equation of the model leads to n sub-problems that can be solved in turn. Moreover, the film mass balance equations can be solved analytically for each of the n sub-problems. Therefore, the ideal three-steps solving procedure described in the introduction can be applied for each of the n sub-problems. Symbolic calculation software such as Mathematica® can be used to optimize the identification and solving of the different sub-problems.
3. Applications of the technique The technique is first presented and validated against two test cases for which the solution can also be derived by a classical shooting technique. A third application where a discretization of the film equations or a shooting technique is by far cumbersome to implement than our technique is then presented. 3.1. Application no. 1 (test case) Let us consider the case of a reaction A + B → C of order 1–1, in which component A is transferred from gas to liquid phase under isothermal conditions. The resistance to mass transfer is located within the stagnant liquid film. Gas and liquid phases are perfectly mixed and only steady state is considered here. The only inlets for A and B are respectively the gas and the liquid phase. The transfer between phases and the reaction does not significantly modify the volumetric flow rate of the phases. This first application is chosen because the corresponding equations can be solved easily using a shooting technique. Mass balance equations in the liquid film are given by: DA
d 2 CA = k1 CA CB dx2
(2)
B. Haut, V. Halloin / Chemical Engineering and Processing 43 (2004) 1339–1346
d 2 CB DB = k1 CA CB dx2
(3)
These equations are supplemented by the corresponding boundary conditions: dCB (g) (l) CA (0) = HCA , CA (xl ) = CA , = 0, dx 0 CB (xl ) =
(l) CB
Returning with Eq. (5) into Eq. (2), yields: A dx2
(l) k1 CB
x , xl
XA = (l)
(l)
XA =
CA
(g)
HCA,0
CA (ξxl ) (g) HCA,0 (l)
,
XB =
,
XB = (l)
CB
(6)
(l)
CB,0
CB (ξxl ) (l)
CB,0 (g)
,
XA =
,
(g)
CA
(g)
CA,0
Eq. (6) can then be written in dimensionless form: d 2 XA (l) (l) 2 2 = Ha1 XA XB + β XA − X A XA dξ 2 dXA (ξ − 1)XA − dξ 0
(7)
=
DA
T =
,
β=
DA HCA,0 (l) DB CB,0
εV DA a, Q g xl
R1 =
V(1 − ε) (l) k1 CB,0 , Ql
(g)
Qg S= , Ql
F=
Qg CA,0
(13)
(l)
Ql CB,0
At the limit β → 0, Eq. (8) can be solved analytically. This corresponds to the classical first-order approximation, only valid in practice for large excess of B or when the diffusivity of component B is significantly larger than the diffusivity of component A. Therefore, all the unknowns of the model are developed in Taylor series in function of β. Any unknown y is then written as: n
yi β i
(14)
where the new unknowns yi are functions of ξ if y refers to a concentration in the liquid film, or are real numbers if y refers to a bulk concentration. The order of development n is chosen accordingly to the desired accuracy. The equations of the n sub-problems are generated by introducing developments Eq. (14) into Eqs. (8) and (10)–(12) and by equalizing coefficients of equal powers of β in the resulting expressions. The following system, generated by equalizing βi coefficients, is called the system of order i: dXA,i (g) = δ1,i+1 (15) XA,i − HT dξ 0
(16)
j=0
(g)
(8)
(l)
F(δ1,i+1 − XA,i ) = XA,i
HF (l) + (δ1,i+1 − XB,i ) S
(17)
i i−1 d2 XA,i (l) 2 = Ha X X + XA,i−1−j A,j 1 B,j−i dξ 2
(g)
(l)
k1 CB,0 xl2
(l)
+ (1 − XB ) (12) S where T, S, R1 and F are dimensionless numbers defined as:
i dXA,i (l) (l) (l) + R XA,i−j XB,j = 0 XA,i + ST dξ 1
where the Hatta number Ha1 and β are defined by: Ha21
(11)
i=0
In order to write the model equations in dimensionless form, the following dimensionless variables are defined: ξ=
(l) HF
(g)
y=
k1 (l) 2 CA = CA + − C A CA DA DB dCA − (x − xl )CA dx 0
dXA (l) (l) + R 1 XA XB = 0 dξ 1
F(1 − XA ) = XA
(4)
This system of two second order equations can be reduced into one second order equation. Indeed, the subtraction of Eq. (3) from Eq. (2) eliminates the non-linear term, and the resulting equation can be integrated twice: DA DA dCA (l) (l) CB (x) = CB + (CA (x) − CA ) − (x − xl ) DB DB dx 0 (5)
d2 C
(l)
XA + ST
1341
(9)
Dimensionless number Ha1 compares the characteristic time of diffusion of A in the stagnant liquid film to the characteristic time of reaction A + B → C. Dimensionless number β compares mass supply rates of A and of B in the liquid film. Eq. (8) is supplemented by mass balance equations written in the bulk of each phase: dXA (g) XA − HT =1 (10) dξ 0
j=0
j=0
dXA,j (l) × XA,j − XA,j − (ξ − 1) dξ 0
(18)
Note that: 1. the n systems can be solved in turn, as no unknown of index higher than i appears in the system of order i. Therefore, the first term of the Taylor development of the unknowns is determined by solving the system of order 0, the second term is determined by solving the system of order 1, . . .
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Fig. 1. S = 10, T = 1.5, R1 = 30, H = 0.4, F = 1, Ha1 = 3, β = 1.
2. whatever the i value, Eq. (18) can be solved analytically. Indeed, the only term containing XA,i in the left side of Eq. (18) is proportional to XA,i . The different systems can be solved in turn following the ideal procedure described in the introduction, the number of system to be solved depending on the chosen accuracy. In Fig. 1, several approximation orders are compared to the solution obtained by a shooting technique using a fourth-order Runge–Kutta integration method. β is chosen equal to 1, in order to be in a situation where a numerical resolution is necessary. It can be observed that the deviation between the solution obtained by the shooting technique and the solution obtained with our technique decreases when the order of the approximation (n) increases. Let us define the average deviations between two profiles 1 y1 (x) and y2 (x) as 0 |y1 (ξ) − y2 (ξ)| dξ. At the order 0, the average deviation between a concentration profile in the liquid film computed with our technique and the corresponding profile computed with the shooting technique is found to be 0.1 for component A and 0.2 for component B. These deviations fall to 0.02 and 0.1 for the approximation of order 1, and to 0.008 and 0.04 for the approximation of order 2. Results obtained with an approximation of order 3 are not
represented in Fig. 1. They are visually undistinguishable from the results calculated using the shooting technique. Several simulations have been performed for values of β less than 1 and values of Ha1 less than 3. The average deviation between a profile calculated with our technique and the corresponding profile obtained by the shooting technique is always found to be less than 0.1, as long as an approximation of order 2 is used. The maximum observed deviation falls to 0.025 when β is inferior to 0.5, even for values of Ha1 up to 4. In Fig. 2, several profiles of concentration in the liquid film are presented for various values of parameters Ha1 and R1 . They are computed with our technique using an order 2 approximation. As expected, the curvature of the profiles increases with the kinetic constant of the reaction. The average deviations between these profiles and the profiles calculated by a shooting technique are found to be less than 0.025. In conclusion, our technique proved to be adequate to solve the equations of this first model, as long as at least a second order approximation is chosen. Indeed, under this condition, the profiles of concentration in the liquid film are found to be close to the profiles calculated by a shooting technique, within a large range of values of the dimensionless numbers of the model. However, for such a simple case, our technique does not bring significant advantages in term
Fig. 2. S = 10, T = 1.5, H = 0.4, F = 1, β = 0.5.
B. Haut, V. Halloin / Chemical Engineering and Processing 43 (2004) 1339–1346
1343
of computer time with comparison to the classical shooting technique. 3.2. Application no. 2 (test case) As most of the industrial problems involve multiple reactions, the second application chosen to test our technique considers a secondary reaction between the reactant B and the product C of the main reaction: A+B → C; B+C → D. It is supposed here that component C is not introduced at the contactor inlet. All the hypotheses assumed for test case 1 prevail: both reactions are of order 1–1, the stagnant film model is used to describe the transfer of component A from gas to liquid under isothermal conditions, and the gas and liquid phases are considered to be perfectly mixed. The three mass balance equations in the stagnant liquid film can be reduced into two equations that are written in dimensionless form as: d 2 XA (l) (l) 2 2 = Ha1 XA XB + 2β XA − X A XA dξ 2 dXA (l) − (ξ − 1)XA + ηβ[XC XA − XC XA ] dξ 0 (19)
d2 XC −Ha21 (l) (l) 2 = X X + 2β XA − X A XA A B η dξ 2 dXA (l) − (ξ − 1)XA + ηβ[XC XA − XC XA ] dξ 0 (l) (l) 2 + Ha2 XC XB + 2β XC XA − XA XC dXA (l) 2 − (ξ − 1)XC + ηβ[XC − XC XC ] dξ 0 (20) The following new definitions are introduced: (l)
Ha22 =
k2 CB,0 xl2 DC
,
η=
DC , DA
Fig. 3. S = 10, T = 2/3, R1 = 30, 2k1 = 3k2 , H = 0.4, F = 1, Ha1 = 2, Ha2 = 1, β = 0.5.
deviations are 0.006, 0.036 and 0.010 in the conditions of Fig. 4. Several simulations have been performed for values of β inferior to 0.5 and values of Ha1 , Ha2 and η inferior to 3. The average deviation between a profile calculated with our technique and the corresponding profile obtained by the shooting technique is always found to be less than 0.1, as long as an approximation of order 2 is used. In conclusion, our technique proved to be adequate to solve the equations of this second model, as long as at least a second order approximation is chosen. Indeed, under this condition, the profiles of concentration in the liquid film are found to be close to the profiles calculated by a shooting technique, within a large range of values of the dimensionless numbers of the model. 3.3. Application no. 3 The two previous applications are helpful to validate our technique against numerical results obtained with the proven
(l)
XC (ξ) =
CC (xl ξ) (g)
HCA,0 (21)
As in application no. 1, these equations have an analytical solution only at the limit β → 0. Out of this limit, the unknowns of the problem are developed in Taylor series in function of β and the same approach as presented in application no. 1 can be followed. Concentration profiles in the stagnant liquid film computed with our technique are compared in Figs. 3 and 4 to profiles computed with a classical shooting technique. For the conditions of Fig. 3, average deviations of 0.002, 0.012 and 0.003 are respectively observed between the profiles of concentration in A, B and C computed with our technique and the profiles computed with a shooting technique. These
Fig. 4. S = 10, T = 2, R1 = 300, 2k1 = 3k2 , H = 0.4, F = 1, Ha1 = 2, Ha2 = 3, β = 0.5.
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classical shooting technique. However, the complexity of these cases is not high enough to fully display the potential advantages of our technique. The third application is chosen for this purpose. In this problem, the gas phase is modeled as a plug flow while the liquid phase is assumed to be perfectly mixed. This can be considered as a realistic hydrodynamic model of a bubble column [4]. Let us consider the case of a reaction A+B → C of order 1–1, in which component A is transferred from gas to liquid phase under isothermal conditions. The resistance to mass transfer is located within the stagnant liquid film. Only steady state is considered here. The only inlets for A and B are respectively the gas and the liquid phase. It is assumed that the transfer between phases and the reaction does not significantly modify the volumetric flow rate of the phases. A dimensionless coordinate z is introduced in the plug flow model of the gas phase: z = 0 at the inlet and z = 1 at the outlet. Balance equations on the bulk of the phases are given by: (g) ∂XA ∂XA = HT ∂z ∂ξ ξ=0,z (l) XA
1
+ ST 0
(22)
∂XA (l) (l) dz + R1 XA XB = 0 ∂ξ ξ=1,z (l) HF
(g)
F(1 − XA (z = 1)) = XA
S
(l)
+ (1 − XB )
(23)
(24)
Dimensionless number F, S, T and R1 are still defined by Eqs. (13). The following equation for the mass balance in the liquid film is obtained: ∂ 2 XA (l) (l) 2 2 = Ha1 XA XB + β XA − X A XA ∂ξ 2 ∂XA (25) − (ξ − 1)XA ∂ξ ξ=0,z
(l)
XA,i + ST
i ∂XA,i (l) (l) dz + R XA,i−j XB,j = 0 ∂ξ ξ=1,z
1 z=0
j=0
(28) (g)
(l)
F(δ1,i+1 − XA,i (z = 1)) = XA,i
HF (l) + (δ1,i+1 − XB,i ) S (29)
i i−1 ∂2 XA,i (l) 2 = Ha X X + XA,i−1−j A,j B,j−i 1 ∂ξ 2 ×
j=0
(l) XA,j − XA,j
j=0
∂XA,j − (ξ − 1) ∂ξ ξ=0,z (30)
Each of these systems is completed by adequate boundary conditions. As in the two test cases presented in the previous section, these systems can be solved in turn and following the so-called ideal procedure presented in the introduction. However, a suitable discretization of Eq. (27) has to be done. A side effect of the Taylor development of all the unknowns of the model is that for i > 0, Eq. (28) is linear, thus enabling an optimization of the solution of the ith sub-problem. Several concentrations profiles obtained with our technique are presented in Figs. 5 and 6. As 40 points are used for the discretization of Eq. (27), our technique leads to the transformation of the problem into n successive sub-problems composed of 42 algebraic equations. As the approximate solutions for n = 2 and 3 are visually indistinguishable in Figs. 5 and 6, it can be considered that a satisfying level of convergence is reached for n = 2. This is coherent with the observations made on the first test case. It is interesting to observe that a shooting technique would have necessitate to find the zero of a function of 80 variables whose evaluation necessitates the resolution of 160 first
XA is here a function of z and ξ. This equation is supplemented by the following boundary conditions: (g)
XA (ξ = 0, z) = XA (z),
(l)
XA (ξ = 1, z) = XA
(26)
As an analytical solution exists for Eq. (8) in the limit β → 0, all the unknowns of the model are developed in Taylor series in function of β. Inserting these developments into Eqs. (22)–(25) and equalizing coefficients of equal powers of β in the resulting expressions, yields the following equations for the n sub-problems: (g)
∂XA,i ∂z
∂XA,i = HT ∂ξ ξ=0,z
(27) Fig. 5. S = 10, T = 1.5, R1 = 30, H = 0.4, Ha1 = 3, β = 0.5.
B. Haut, V. Halloin / Chemical Engineering and Processing 43 (2004) 1339–1346
1345
the mass balance equations in the bulk of the phases are linearized. This facilitates the determination of the correction. The technique can be applied for multiple or reversible reactions, but it is restricted to reactions of integer order.
Acknowledgements F.N.R.S. (Belgian national fund for scientific research) fellow researcher Benoˆıt Haut acknowledges the fund for its financial support.
Appendix A. Nomenclature Fig. 6. S = 10, T = 1.5, R1 = 30, H = 0.4, F = 1/10, Ha1 = 3, β = 0.5, n = 2.
a Ci (x)
order ordinary differential equations. A 10 points discretization of the film equations leads to the resolution of more than 400 non-linear algebraic equations. This heavy numerical handling can be avoided with the help of our technique. As appears in Fig. 5, the outlet concentration of A in the gaseous phase is significantly modified when n goes from zero (n = 0 corresponds to the assumption usually made in industrial practice) to two. Pertinent corrections to the usual industrial approach can be computed quickly with our technique.
4. Conclusions In this paper, a new technique is proposed to solve the equations of a gas–liquid contactor model. It is based on an asymptotic approach, in which all the unknowns are developed in Taylor series. The solution of the system of equations is composed of a first approximation (order 0) to which a series of corrections is added. The ith correction is defined as the correction of order i. When computing a correction, an analytical solution of the mass balance equations in the liquid film is derived. This solution is then inserted in the mass balance equations written on the bulk of the phases. Solving of the resulting system of equations yields the correction. This one is evaluated once for all but is needed to start the computation of the correction of higher order. The technique has been first presented and validated against two test cases for which the solution can be derived more or less easily by a classical shooting technique. A third application where a discretization of the film equations or a shooting technique is by far less effective than our technique is finally presented. The existence of an analytical solution of the mass balance equations in the film permits to consider any degree of mixing in the bulk of the phases. A side effect of the development into Taylor series is that, except for order 0,
(l)
Ci
(g)
Ci
(l)
Ci,0 (g)
Ci,0 Di F H
Ha1 Ha2 k1 k2 n Ql Qg S T R1 Ri (x) V Xi (x) (l)
Xi
interfacial area in the contactor expressed in m2 of interface per m3 of gas phase (m−1 ) concentration of component i at the position x within the liquid stagnant layer (mol/m3 ) concentration of component i in the bulk zone of the liquid phase (mol/m3 ) concentration of component i in the bulk zone of the gas phase (mol/m3 ) concentration of component i in the liquid feed stream (mol/m3 ) concentration of component i in the gas feed stream (mol/m3 ) diffusion coefficient of component i (m2 /s) dimensionless number, see Eq. (13) dimensionless number related to the Henry’s coefficient of component A Hatta number associated with reaction A + B → C, see Eq. (9) Hatta number associated with reaction B + C → D, see Eq. (21) kinetic coefficient of reaction A + B → C (m3 /mol s) kinetic coefficient of reaction B + C → D (m3 /mol s) order of the Taylor series development, see Eq. (14) volumetric flow rate of the liquid phase (m3 /s) volumetric flow rate of the gaseous phase (m3 /s) dimensionless number, see Eq. (13) dimensionless number, see Eq. (13) dimensionless number, see Eq. (13) consumption of rate of component i at the position x within the liquid layer (mol/m3 s) volume of the contactor (m3 ) reduced concentration of component i at the position ξ within the liquid stagnant layer, see Eqs. (7) and (21) reduced concentration of component i in the bulk zone of the liquid phase, see Eq. (7)
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Xi x
xl z
B. Haut, V. Halloin / Chemical Engineering and Processing 43 (2004) 1339–1346
reduced concentration of component i in the bulk zone of the gas phase, see Eq. (7) position within the liquid film. x = 0 at the gas–liquid interface and x = xl at the interface between the liquid film and the bulk of the liquid phase (m) thickness of the liquid stagnant film dimensionless coordinate
Greek letters β dimensionless number, see Eq. (9) δi,j Kronecker delta. δi,j = 1 if i = j and δi,j = 0 if i = j ε gas hold-up in the contactor η dimensionless number, see Eq. (21) ξ dimensionless position within the liquid stagnant layer, see Eq. (7)
References [1] J. Villermaux, Génie de la Réaction Chimique, Lavoisier Tec & Doc, Paris, 1995. [2] E.S. Gaddis, Mass transfer in gas–liquid contactors, Chem. Eng. Process. 38 (1999) 503–510. [3] J.A. Hogendoorn, R.D. Vas Bhat, J.A.M. Kuipers, W.P.M. van Swaaij, G.F. Versteeg, Approximation for the enhancement factor applicable to reversible reactions of finite rate in chemically loaded solutions, Chem. Eng. Sci. 52 (1997) 4547–4559. [4] W.D. Deckwer, Bubble Column Reactors, Wiley, Chichester (GB), 1992. [5] H. Benbelkacem, H. Cano, J.C. Huard, M. Roustan, H. Debellefontaine, Modélisation du transfert de matière gaz/liquide couplé avec une réaction chimique. Application au régime cinétique intermédiaire, Récents Progrès en Génie des Procédés (80), Lavoisier Tec & Doc, Paris, 2001. [6] E.Y. Kenig, R. Scheinder, A. Gorak, Multicomponent unsteady-state film model: a general analytical solution to the linearized diffusionreaction problem, Chem. Eng. J. 83 (2001) 85–94.