Ind. Eng. Chem. Fundam. 1902,21, 379-385
379
Mass Transfer Enhanced by Equilibrium Reactions Chung-Shlh Chang' and Gary 1.Rochelle Depaflment of Chemical Engineering, The University of Texas at Austin, Austin. Texas 78712
Enhancement factors for mass transfer with single and multiple instantaneous reversible reactions are modeled by closed-form solutions of film theory and by numerical solutions of surface renewal theory. Surface renewal theory is approximated within 10% by using the film theory solutions with diffusivtty ratios replaced by their square roots.
Introduction The most general case of mass transfer enhanced by chemical reaction would consider a system of multiple, reversible reactions with finite rates. In this paper we consider the important subset of mass transfer enhancement by single and multiple, reversible, instantaneous reactions. Previous work has primarily dealt with irreversible reactions or with single reversible reactions. The effect of multiple, reversible, instantaneous reactions is important in mass transfer problems such as SO2absorption in buffer solutions. Dankwerts (1968) gave a simple model for the effect of equilibrium (reversible, instantaneous) reactions which is valid for film theory or surface renewal theory when the diffusivities of all species are equal. Reactions involving H+or OH- in aqueous solution can give diffusivity ratios as large as 10. In this paper we utilize both film theory and surface renewal theory to quantify mass transfer enhancement when diffusivities are unequal. Previous work (Olander, 1960; Danckwerts, 1970; Astarita and Savage, 1980) used film theory to model the effect of single reactions and systems of multiple reactions. In this paper surface renewal theory, which is more representative of commercial gas/liquid contactors, is solved rigorously for several single reactions and one simple system of multiple reactions. We show that the rigorous solutions of surface renewal theory can be accurately approximated by replacing diffusivity ratios by their square roots in the film theory solution, as previously suggested but not demonstrated (Brian et al., 1961; Rochelle and King, 1977; Astarita and Savage, 1980). This approximation is especially useful for systems of multiple equilibrium reactions. The general problem considered in this paper is the effect of instantaneous, reversible reactions on the diffusion rate of.solute A from a phase boundary into a bulk solvent. The analyses presented here are limited by the following assumptions. (1) Only species A can diffuse through the phase boundary. Other species participating in equilibrium reactions with species A do not cross the boundary. (2) The concentration of species A is constant and specified a t the phase boundary and in the bulk fluid. Concentrations of other species are constant and specified in the bulk fluid as constrained by the equilibrium reactions. (3) The chemical reactions are instantaneous and reversible. Hence chemical equilibrium is valid at every point in the boundary layer. This assumption gives the
approximation that species other than A do not diffuse across the phase boundary even though their concentration gradients at that point are nonzero. Even though no other species cross the phase boundary, part of species A which crosses the boundary is instantaneously converted into other equilibrium species. (4) The diffusion coefficient of each species is constant and can be represented by the binary diffusion coefficient of that species in the bulk solvent. There is no allowance for multicomponent diffusion coefficients. Therefore the models are limited to systems where species A and other equilibrium species are present at relatively low concentrations in the solvent. Both film theory and surface renewal theory can be useful models for diffusion across a phase boundary. For example, film theory can represent diffusion from one boundary to another through a stagnant fluid. Surface renewal theory is most useful for representing liquid-phase diffusion in gas/liquid contacting. This paper quantifies the effect of chemical reaction as the enhancement factor, 4, the ratio of the actual diffusion rate of A to its diffusion rate in the absence of chemical reaction. Predictions of the enhancement fador by surface renewal theory are equivalent to predictions by penetration theory. If the diffusivities of all equilibrium species are equal, predictions of surface renewal theory are equivalent to those of film theory. Results are presented in this paper for both film theory and surface renewal theory with species of unequal diffusivities. This theory was developed as part of an effort to model SO2absorption into CaO/CaC03 slurries. Additional details on this work and its application can be found in the dissertation by Chang (1979) and in papers by Chang and Rochelle (1980, 1981, 1982). Film Theory for Single Reactions Film theory assumes the steady-state diffusion of component A through a stagnant fluid from the phase boundary (subscript i) to a secondary boundary (subscript 0) where the concentrations of A and other equilibrium species are specified. The mass transfer rate varies directly with the species diffusivity. Olander (1960) and Danckwerts (1970) derived analytical solutions of film theory for several types of single reactions, as given in Table I. The general method of the solution of film theory is illustrated by the following analysis for one of the reactions not modeled by Olander and Danckwerts A=B+C
(1)
The equilibrium is given by
* Acurex Corporation, Route 1,Box 423, Morriaville,NC
27560.
C*K, = C&c
0196-4313/82/1021-0379$01.25/00 1982 American Chemical Society
(2)
380
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982
Table I. Film Tileory Enhancement Factors for Single Reactions (Olander, 1960; Danckwerts, 1970) reaction type
@ ~-1
A ~ B + C (CAK = C B C C )
(this work)
This reaction is important in the absorption of SOz with dissociation to H+ and HS03- (Chang and Rochelle, 1981). Material balance for the steady-state problem gives two homogeneous ordinary differential equations (3)
(4)
The boundary conditions are
a t x = 6, CA =
CAo,
CB =
CBo
The flux of species A across the phase boundary is given by
The mass transfer enhancement factor is given by
rb=
N6 DA(CAi - CAo)
For reaction 1, solution of eq 3, 4, 5 , and 6 gives the result in Table I. If the diffusivities of all species are equal, Danckwerts (1968) showed that the enhancement factor for both film
theory and surface renewal theory is given by an expression equivalent to ACAt (7) cAi - CAo where ACA, represents the moles per liter of A which must be added to an equilibrium system to raise the concentration of A in its unreacted form from CAoto CAiwhile also increasing the concentration of A in reacted forms as required by equilibrium. Equation 7 corresponds to the limit of the expressions for C#J in Table I as the diffusivity ratios go to 1. The Danckwerts concept is also applicable to systems of multiple reactions when the diffusivities of all species are equal. Film Theory with Noninteracting Multiple Reactions Systems of reactions with finite rates must be represented as a combination of two reaction types, competing or parallel reactions and consecutive or series reactions. The relative positions of the reactants, intermediate products, and fiial products are not changeable. However, systems of instantaneous, reversible reactions are more versatile and interconvertible and can usually be expressed by several different sets of specific reactions. In particular, a system of multiple equilibrium reactions participating in the mass transfer enhancement of species A can always be expressed as a series of single parallel reactions, each containing species A as a reactant. A system of reactions is noninteracting if the component A is the only common component in each reaction. For this system, only the distribution of reacted A between the products has to be considered. The stoichiometry relations
rb=
Ind. Eng. Chem. Fundam., Vol. 21,
between all of the species other than A are fixed. The generalized noninteracting system can be expressed as
A
J,
J,'
j=l
j=l
+ CrmjRmJ* C P m j P m j
The total steady-state material balance for component A in terms of the first product of each single reaction gives
No. 4, 1982 381
Film Theory with Interacting Multiple Reactions For a system of interacting multiple reactions, there is more than one common component in all the constituting single reactions. Therefore, the distribution of those common components must be considered in the material balances. Furthermore, the constituting single reactions no longer contribute independently to the mass transfer enhancement factor. All the reactants and products affect each other both in concentration distribution and mass transfer rate. For example, consider the system combining reaction 13 with the reaction A + B e D; CACBK~ = CD (15) This is an interacting multiple reaction system because those two reactions have two common components, A and B. The total A component material balance is
The general solution of eq 8 is (9)
The total B component material balance is
where bl and b2 are integration constants. The flux of species A across the interface is The boundary conditions are Combination of eq 9 and 10 gives N in terms of the boundary conditions CAI, C A ~Ce,li, , and Ce,lo
The definition of the mass transfer enhancement factor is given by eq 6. According to this definition and eq 11, the mass transfer enhancement factor for a system with only one single reaction e can be expressed as &=1+-
De,1 (Ce,li - Ce,lo)
DAPe,l
- cAo)
Substituting the boundary conditions and the equilibrium relations into eq 16 and 17, we can obtain the interface concentration of species B
Equation 18 shows CBiis a function of bulk concentrations, diffusivities, and equilibrium constants of all the components and reactions of this system. The mass transfer enhancement factor is given by
Comparison of eq 6 and 11 shows that the overall mass transfer enhancement factor for a system of m reactions can be expressed as m
$ = I + C(@e-l) e=l
(12)
Therefore, for a system of noninteracting multiple reactions, each constituting single reaction contributes one additive term to the overall mass transfer enhancement factor. This term is derived from the mass transfer enhancement factor of the individual single reaction by itself. That is, there is no interaction among the constituting reactions and those single reactions affect the mass transfer rate independently. Therefore, mass transfer enhancement factors for a system of noninteracting equilibrium reactions can be determined as a linear combination of factors given for single reactions as in Table I. For example, by the use of eq 12, the mass transfer enhancement factor of the system A + B & C; cAc$(2 = CC (13)
D; c ~ K = 3 CD can be obtained from Table I as A
F?
(14)
The enhancement factor is quite different from those of noninteracting systems in the previous section. There is no obvious relation between the mass transfer enhancement factor of the system of interacting multiple reactions and of the constituting single reactions. Not all multiple reaction systems have analytical solutions for mass transfer rates. When there are more than three products, the algebraic equation which relates the interface concentration and other known properties becomes a complicated form with order higher than 3. Numerical techniques, such as Newton-Rapson's method, must be used to solve the algebraic equation. Once the interface concentration is obtained, the mass transfer rate can also be calculated. Additional examples of such systems are given by Chang and Rochelle (1980, 1982).
Surface Renewal Theory for Single Reactions Surface renewal theory assumes unsteady-state diffusion of component A from the phase boundary into the bulk fluid. Concentrations of A and other equilibrium species
382 Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982
are specified in the bulk fluid at time zero. The average flux of A in the absence of chemical reaction is given by
-dB* - - B’
(20)
-dC* - - C’
Nphy
= (DAS)’”(CAi - CAJ
where s is the fractional rate of surface renewal used to characterize the physical system. There are few rigorous solutions of surface renewal theory with equilibriumreactions reported in the literature. Secor and Beutler (1967) extended their numerical solutions of finite rate reversible reactions to several specific cases of instantaneous, reversible reaction. However, their method is too cumbersome to provide general results for instantaneous reversible reactions. This paper presents a detailed procedure and results for a rigorous numerical solution of surface renewal theory for reaction 1. Detailed development of numerical solutions for reaction 13 and the reaction
A
+ B s C + D; CACBK5 = CcCD
(21)
are given by Chang (1979). For reaction 1 the unsteady-state component material balances for species A and B in all forms, reacted and unreacted, are given by
species B:
acB acc
-- - - D at
at
a2cB B
S -D
a2cc C
z
(23)
du
du
where
The boundary conditions become: at u = 0, B* = 0, C* = 0, and at u = 1, B* = 1, C* = 1. The numerical method, the Kutta-Simpson rule, was used to solve the simultaneous nonlinear differential equations 24, 25, 26, and 27. Due to the nature of the boundary value problem, two initial conditions need to be guessed to start the numerical procedure. They are either CBi,B’, or Cci, C’. Iterative adjustments were made to continue the calculation until the final computed values of B’ and C’ matched the known boundary conditions. The instantaneous flux of species A across the phase boundary can be expressed as
The equilibrium relation is given by eq 2. The initial and boundary conditions for this case are: at t = 0, CA = CAo, CB = CBo; at t > 0 and x = 0, CA = CAi; at t I 0 and x --, In accordance with surface renewal theory, the average m, C A = CAo, CB = CBo. The boundary concentration of steady-state absorption rate is given by species C can be obtained from the equilibrium relation. The final boundary condition which reflects the inability of B or C to diffuse across the interface is at t
acB- Dc-acc = 0
> 0 and x = 0, DB- ax
ax
Equations 22 and 23 can be transformed by ordinary differential equations with a finite domain by making the substitution
Substitution of eq 2 with appropriate transformations and integration gives
N,, = (DA~~)1/2ACcgClu=o where
CBi g=-+K1 The dimensionless forms of the transformed equations are given by
(30)
DcCci DBKl
+ -Dc DA
The mass transfer enhancement factor is
where Clu=oand Cci are constants obtained from the numerical solution in eq 24, 25, 26, and 27. Figure 1 compares values of the enhancement factor calculated by this numerical solution of surface renewal theory and by film theory. For values of Kl/CK less than 0.1, the effect of chemical reaction is negligible. The enhancement factor is larger at higher values of K,/CAi and varies with the diffusivity ratios. The predictions of film theory and surface renewal theory are equal at diffusivity ratios of 1.0 but differ as much as the square root of the diffusivity ratio at values different from 1.0.
Surface Renewal Theory for Multiple Reactions Because of increased complexity, numerical solutions of surface renewal theory for multiple reactions are not easy to obtain. If the diffusivities of all species are equal,
Ind. Eng. Chem. Fundam., Vol. 21, No. 4, 1982
I -I
383
50
0
150
100
IO
IO
I
:
01
I
,
0.01
01
K l
CA,
Figure 1. Comparison of enhancement factors predicted by film
*
theory and surface renewal theory for A B + C (C, = Cco = 0). Film theory is represented by the dashed lines.
surface renewal theory is equivalent to film theory. The following simplest case of multiple reactions can be solved analytically
A A
A
Bl
CAK6,1
B2 cAK6,2
...
* Brn
= CB,l = CB,2
9..
cAK6,rn
=
(32)
cB,rn
-15 - 0
The total unsteady-state material balance for component A gives
0305
50
IO
100
DE DA
Figure 3. Deviation of the approximate enhancement factor from the rigorous enhancement factor based on surface renewal theory for A + B F? C. (DB = Dc,Cco = 0, CBo/CAo = 10.)
The initial and boundary conditions are: at t = 0, CA = CAo;at t > 0 and x = 0, CA = C,; at t > 0 and x m, CA = CAo. The boundary conditions of species Be can be obtained from the equilibrium relations. Equation 33 can be solved for the instantaneous flux of A across the phase boundary defined by +
where
to yield the time-average flux
N,, = JmNse“t d t =
(DAs)’/’(CAi
rately approximated by the film theory solution with diffusivity ratios replaced by their square roots. For example, the approximate solution for reaction 1 is given by modifying the film theory solution in Table I to get
- CAo)$
(35)
where the enhancement factor, 4, is given by
For comparison, the film theory solution obtained from Table I and eq 12 gives
Surface Renewal Theory Approximation The rigorous solution of surface renewal theory for single or multiple equilibrium reactions frequently involves numerical integration and tedious trial and error procedures. On the other hand, the exact solution of film theory is easily obtained in algebraic form, even for somewhat complicated systems of multiple reactions. We have found that the solution for surface renewal theory can be accu-
Figure 2 shows a comparison of the rigorous and approximated enhancement factors for this reaction based on surface renewal theory. The approximation is good for diffusivity ratios near 1 and for high values of K l / C A i , where mass transfer is controlled primarily by diffusion of the reaction products. This approximation has been used to model data for absorption of dilute SO2 in water (Chang and Rochelle, 1980). Similar results were obtained for reactions 13 and 21, as shown in Figures 3 and 4. Calculations with all three reaction types over a wide range of K values and boundary conditions and with diffusivity ratios from 0.3 to 10 generally approximated the rigorous solution of surface renewal theory within 10% (Chang, 1979). It can be shown from Dankwerts’ theory (1968) that f i i theory and surface renewal theory me equivalent when all
384
Ind. Eng. Chern. Fundarn., Vol. 21, No. 4, 1982
0
0
- -15 0 0305
50 I O 0
IO DE -
DA
Figure 4. Deviation of the approximate enhancement factor from the rigorous enhancement factor based on surface renewal theory for A + B s C D. (DB = Dc, DD = Dc'f', CA, = CG = CD,= 0, C k / C f i = 10.)
+
components have equal diffusivities. Therefore the approximation should be exact a t diffusivity ratios of 1. Astarita and Savage (1980) showed theoretically that the square root approximation should be accurate with values of the diffusivity ratio near 1.0. With very large equilibrium constants, systems with reactions 13 or 21 approach the problem of instantaneous, irreversible reactions where the enhancement factor is given by
This corresponds to replacing the diffusivity ratio by its square root in Hatta's equation (1928) of this type. Brian et al. (1961) proposed eq 38 as an approximation of surface renewal theory and suggested that it should be accurate at high values of 4 and diffusivity ratios near 1.0. Rochelle and King (1977) utilized this approximation in a model of SOz absorption by lime/limestone slurries. The approximation to surface renewal theory is also applicable to systems of multiple reversible reactions. Reaction system 32 is approximated by replacing the diffusivity ratios in eq 37 with their square roots to give
Enhancement factors calculated rigorously from eq 36 were within 10% of those from eq 40 with DBJDA = 0.3 to 10, = 0.1 to 10, and m = 2 to 10 (Chang, 1979). Material balance for the surface renewal theory of the interacting system with reactions 13 and 15 gives two second-order nonlinear partial differential equations. Solutions of those equations involves complicated numerical methods and tedious trial and error procedures. However, surface renewal theory can be easily approximated by replacing diffusivities by their square roots in the corresponding film theory solution as shown in eq 19 to give
In other papers we use the approximation method to solve systems of reactions involving SOz absorption in NaOH/Na2S03solutions (Chang and Rochelle, 1980) and SOz absorption in organic acid buffer solutions (Chang and Rochelle, 1982). Conclusions and Significance Film theory is easily solved for the effects of single equilibrium reactions. The solution of a system of reactions containing only one common component is given by a linear combination of enhancement factors calculated for the single reactions. The solution for systems of interacting reactions requires numerical solution of algebraic equations. Surface renewal theory has been solved by numerical integration for single equilibrium reactions as complicated as A B i- C D. Film theory and surface renewal theory give identical enhancement factors with equal diffusivities, but differ as much as a factor of 3 with a diffusivity ratio of 10. Only the simplest system of reactions, A s Be, has been solved for surface renewal theory. Surface renewal theory is effectively approximated by using the solution for film theory with diffusivity ratios replaced by their square roots. The approximation is most accurate at diffusivity ratios near 1.0 and at extreme values of the equilibrium constants. Even with diffusivity ratios as large as 10, the enhancement factor can be estimated with an error less than 10%. The approximation method should be especially useful with systems of multiple equilibrium reactions.
+
+
Nomenclature bl, b, = constants in eq 9 B*, C* = dimensionless concentrations for species B and C B', C' = dimensionless concentration gradients for species B and C C = concentration, g-mol/L D = diffusivity, cm2/s d , g = dimensionless groups in eq 24, 25, 30, and 31 J , J' = number of reactants and products K = equilibrium constant K , = equilibrium constant for reaction 1, g-mol/L K,, K , = equilibrium constants for reactions 13 and 15, L/ g-mol K3,K,, fKg,e = equilibrium constants for reactions 14, 21, and 32, dimensionless m = number of reactions N = instantaneous flux of A across the phase boundary, gmol/s-cm2 N,, = average flux of A across the phase boundary, g-mol/ s-cm2 Nphy = average flux of A across the interface without chemical reaction, g-mol/s-cm2 p , r = stoichiometric coefficients s = fractional rate of surface renewal, s-l t = time, s u = dimensionless distance x = distance, cm AC = Ci - C,, g-mol/L ACA, = total concentration difference for species A, g-mol/L 6 = film thickness, cm 4 = mass transfer enhancement factor da = approximated mass transfer enhancement factor Subscripts A, B, C , D = components in equilibrium realtions
e = reaction number i = phase boundary or interface j = component number 0 = bulk fluid
Literature Cited Astarita, G.; Savage, D. W. Chern. Eng. Sci. 1980, 35, 1755. Brian, P. L. T.; Hurley, J. F., Hasseltine, E. H. AIChE J . 1961, 7 ,226.
385
Ind. Eng. Chem. Fundam. 1982, 21, 385-390 Halta, S. Toh&u Imp Unlv. Tech. Repts. 1928, 8 , 1. Olander, D. R. AIChE J. 1960, 6 , 223. Rochelle, 0. T.; King, C. J. Ind. Eng. Chem. Fundarn. 1977, 76, 67. Secor, R. M.; Beutler, J. A. AIChE J. 1987, 73, 365.
Chang. C. S., Ph.D. Dissertation, The University of Texas at Austin, 1979. Chang, C. S.; Rochelle, 0. T. “SO2 Absorption into NaOH and Na,SO, Aqueous Solutlons”; presented at the AIChE 88th Natlonai Meetlng, Philadelphia, June 8-12, 1980. Chang, C. S.; Rochelle, 0. T. AIChE J. 1981, 27, 292. Chang, C. S.; Rochelle,’G. T. AICh€ J. 1982, 28, 261. Danckwerts, P. E. Chem. Eng. Sci. 1968, 23, 1045. Danckwerts, P. V. “Gas-Liquid Reactions”; McGraw-Hill: New York, 1970.
Received for review June 1, 1981 Accepted June 11, 1982
Methanation over Transition-Metal Catalysts. 4. Co/AI,O,. Behavior and Kinetic Modeling
Rate
Pradeep K. Agrawal,” James R. Katzer,’ and Wllllam H. Manogue’ Center for Catalytic Science and Technology. Department of Chemical Engjneering, University of Deb ware, Newark. Delaware 79777
in three pseudosteady-state operating regions-clean Co, carbondeactivated Co, and sulfur-poisoned Co. The intrinsic rate data in all three regions are correlated well by the “carbide” model in which the rate-limiting step is the reaction between a surface carbon atom and a surface hydrogen atom. Kinetic modeling suggests the existence of two types of CO hydrogenation sites-one with the higher heat of CO adsorption as compared to the other. As cobalt undergoes carbon deactivation, the active sites with higher heat of CO adsorption are reduced considerably and are poisoned completely with sulfur. However, no change in either the reaction mechanism or the rate-controllingsteps is caused by catalyst deactivation. The kinetic behavior of CO hydrogenation over alumina-supported cobalt has been presented
Introduction Many catalysts undergo changes from their initial activity to a more industrially important pseudo-steady state characterized by a slow continuous activity loss. Kinetic studies have sometimes been carried out on the initial activity behavior of catalysts and on the steady-state activity, but the authors know of no kinetic studies carried out in all activity regions. Furthermore, it is possible for catalysts to exhibit more than one pseudo-steady state activity region; yet examples of such situations have not been clearly documented, and there are no reported examples of kinetic studies. Careful kinetic studies of the initial catalytic activity and of each of the pseudosteady-state activity regions, if more than one exist, can help clarify the nature of the deactivation process. If deactivation is due only to a reduction in the number of active sites, then the kinetic behavior should remain unchanged, and the reduction in the rate constant could be attributed to the fractional reduction in the number of active sites, properly calculated to account for the kinetic order. If the deactivation process is due to other factors, such as electronic changes, then the form of the kinetic rate expression could change, the value of the rate constant could change, and the values of the kinetic rate parameters would be expected to change, leading to new insights into the deactivation mechanism.
Previous papers in this series (Agrawal et al., 1981a,b; Fitzharris et al., 1982) report CO hydrogenation studies over Ni and Co catalysts supported on AlZ0,. Reliable experimental techniques have been developed for measuring intrinsic kinetics of CO hydrogenation at specified CO concentrations, at 1atm total pressure, at temperatures to 673 K, and with controlled amounts of HzS to concentrations as low as 13 ppb. This reactor system allows reaction rates and rates of deactivation to be measured directly rather than being calculated with the aid of an assumed transport, kinetic, and deactivation model. In this research the kinetic behavior of Co/AlZ0, in CO hydrogenation has been determined in three pseudosteady-state regimes having activities that differ by more than four orders of magnitude. The emphasis of this work has been on quantifying the kinetic behavior adequately to determine the best kinetic model which describes CO hydrogenation over Co/A120, and to determine the form of the rate expression in each pseudo-steady-state regime. The information gained provides strong support for the rate-determining step in CO hydrogenation. The kinetic behavior observed in the three pseudo steady-state regions indicates the nature and cause for the deactivation.
Theory In spite of an extensive literature devoted to the adsorption of H2 and CO on transition metals, comparatively few studies have been made of the adsorption of these molecules and their mixtures under conditions close to those of catalytic processes. Insufficient characterization of the surface intermediates found under reaction conditions and the great disparity of conditions between surface science studies and reaction studies have kept researchers divided on the identification of intermediate surface complex of methanation. All suggestions in the literature can
*Author to whom all correspondence should be addressed School of Chemical Engineering,Georgia Institute of Technology, Atlanta, GA 30332. ‘Central Research Department, Mobil R & D Co., Princeton, NJ 08540. Experimental Station, E. I. DuPont de Nemours and Company, Inc., Wilmington, DE 19898. 0196-4313/82/1021-0385$01.25/0
0
1982 American Chemical Society
