6140
Ind. Eng. Chem. Res. 2007, 46, 6140-6146
Analysis of Mass Transfer with Reversible Chemical Reaction† Jerry H. Meldon,* Olanrewajo O. Olawoyin,‡ and Diego Bonanno§ Department of Chemical and Biological Engineering, Tufts UniVersity, Medford, Massachusetts 02155
An approximate analytical technique is presented for solving nonlinear differential equations that govern steady-state mass transfer with reversible reaction. Its essential feature, which K. A. Smith devised to analyze membrane transport with rapid reaction, is linearization of the kinetics on the assumption of small local departures from equilibrium. It is shown here to yield precise estimates of enhancement factors for absorption and effectiveness factors for catalysis over complete ranges of operating regimes, from exceedingly slow reaction to instantaneous reaction. I. Introduction Simultaneous mass transfer and reaction is an abiding subject of analysis with applications to gas scrubbing, catalysis, and numerous other chemical processes.1-11 When the reaction kinetics are nonlinear, exact solution of the governing differential equations generally requires numerical methods of analysis.12 The literature also abounds with handier but approximate closedform solutions that are useful for scoping, if not final design purposes. The errors they introduce vary based on how linearization is achieved. The great majority of analyses, numerical and closed-form alike, target cases with irreVersible reaction. Although this stipulation restricts applicability, process economics have a tendency to favor operating conditions that suppress back reactions. Van Krevelen and Hoftijzer13 (abbreviated hereafter as VKH) developed their classic linearization technique for the case in which nonvolatile B reacts irreversibly with dissolved gas A at the rate r ) k1[A][B]. Its simplifying approximation (suggested by the non-flux condition at the interface, (d[B]/dx)x)0 ) 0) is r ≈ k1[A][B]x)0. Absorption rates calculated on this basis are remarkably accurate. Peaceman14 achieved the same success by extending the VKH technique to cases involving a single reVersible reaction. Two decades later, Smith15 examined membrane transport that was governed by analogous differential equations but different boundary conditions. He developed an alternative method, which was suggested by the approach to reaction equilibrium as the reaction rate increases. Unlike the VKH approach, Smith’s methodology is readily applicable to absorption with coupled reactions of more than one gas. The following sections illustrate its utility in modeling both absorption with reaction and heterogeneous catalysis.
Figure 1. Concentration profile in the steady-state Film Theory model.
it requires the solution of ordinary differential equations (ODEs). More-realistic non-steady-state models (such as the Penetration Theory18 and Surface Renewal Theory19) entail partial differential equations. Notably, the steady and non-steady-state models predict identical absorption rates when the diffusivities of reacting species are all equal. Moreover, when diffusivity ratios that arise in the steady-state analyses are replaced by their square roots, the calculated absorption rates approach those yielded by the transient models;12,20-22 this correction was applied to the steady-state analyses in preparing Figures 2-4 (given later in this work). The goal here is to calculate the local rate of absorption of gas A, given its partial pressure in bulk gas (pG), its concentration in bulk liquid ([A]L), and the following liquid-phase reaction:
|νA|A + | νB|B T νCC + νDD
(1)
where B, C, and D are nonvolatile. For simplicity, heat, electrostatic, and thermodynamic nonideality effects will be neglected. The reaction rate is assumed to be
r ) k1[A]|νA|[B]|νB| - k2[C]νC[D]νD
(2)
and its effective equilibrium constant is II. Absorption and Reaction The analysis that follows is based on steady-state film theory (see Figure 1), which Whitman16 formulated and Hatta17 first applied to reactive systems. It assumes well-mixed bulk fluids and molecular diffusion in interfacial gas and liquid films, and * To whom correspondence should be addressed. Tel.: 617 6273570, Fax: 617 627-3991, E-mail address:
[email protected]. † In addition to Ken Smith, J.H.M. dedicates this paper to Thibaut Brian and the memories of Gianni Astarita, Peter Danckwerts, and Tom Sherwood. ‡ Currently with Aspentech, Inc., Cambridge, MA. § Currently with Cubist Pharmaceuticals, Inc., Lexington, MA.
Keq )
(
k1 [C]νC[D]νD ) k2 [A]|νA|[B]|νB|
)
(3)
eq
(A) Transport Formulation. Absorption analysis is based on species balances that account for liquid-film diffusion and reaction. Assuming dilute fluids, constant material parameters, and the applicability of Fick’s Law, the balances assume the following form:
d2[i] Di 2 ) -νir dx
10.1021/ie0705397 CCC: $37.00 © 2007 American Chemical Society Published on Web 08/22/2007
(4)
Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007 6141
The boundary conditions, assuming phase equilibrium at the gas/liquid interface, are
{
(
)
[A]0 d[A] ) kG pG dx R d[B] d[C] d[D] ) ) )0 dx dx dx
At x ) 0
-DA
D≡
(5)
(6a)
(i ) C, D)
1 d2C θBAθCB dy2
)-
1 d2D θBAθDB dy2
F ≡ A|νA|B|νB| Ha ≡ L
CνCDνD K
x
k1(RpG)|νA|-1[B]S|νB| DA
θBA t θiB t
(9)
νADB[B]S νBDARpG
|νB|Di νiDB
(10a)
(10b)
(11a)
(i ) C, D)
(11b)
where Ha is known as the “Hatta number”. The boundary conditions become
(6b) At y ) 0:
νC+νD
)-
where
It follows that eq 3, applied to bulk liquid, can be expressed as
(νC )(νD )(Λ
(8e)
d2A 1 d2B ) |νA|Ha2F ) 2 θBA dy2 dy
[B]L ) (1 - Λ)[B]S
νD
x L
Insertion of eqs 2 and 8 into eq 4, and rearrangement, yields
The first equation equates the fluxes of volatile/soluble species A in the liquid and gas films, at the gas/liquid interface. The set of three equations that follows expresses the vanishing of the fluxes of nonvolatile species at the interface. The last equation sets the liquid film concentrations at x ) L equal to those in well-stirred bulk liquid (which are treated here as known quantities). Bulk liquid concentrations, [i]L, are governed by the specification of [A]L and [B]S (the value of [B] in fully stripped liquid), plus the conventional assumption of reaction equilibrium (i.e., the applicability of eq 3) in bulk liquid. The latter assumption is reasonable under most conditions of practical significance2 but must break down when reaction is sufficiently slow. We invoke it here, both for simplicity and to explore the range of mathematical accuracy of the linearized analyses. We also assume that B is the only solute in fully stripped solvent. Accordingly, [B]L, [C]L, and [D]L are fixed by the “loading” (Λ), which is defined as the fractional conversion of B in bulk liquid, i.e.,
νC
(8d)
y≡
At x ) L: [i] ) [i]L
νiΛ[B]S [i]L ) |ν|B
[D] [B]S
dB dC dD dA ) -µ(1 - A), ) ) ) 0 (12a) dy dy dy dy At y ) 1: i ) iL
(12b)
νB+νC+νD
)[B]S
A| |ν|νBC+νD(1 - Λ)|νB| [A]|ν L
) Keq
(7)
Next, we transform the equations in terms of dimensionless variables and exploit the stoichiometry to replace all but one differential equation with algebra. We then identify behavior in the limiting cases of negligibly slow reaction, instantaneous reaction, and irreversible pseudo-first-order reaction. Finally, we linearize and solve the equations, apply the solution over the complete range of reaction times, and compare calculated absorption rates with those obtained via exact numerical analysis and via application of the VKH approach within the frameworks of both Film Theory and Surface-Renewal Theory. (1) Transformation to Dimensionless Variables. To facilitate elucidation of asymptotic behavior, we normalize the variables as follows:
A≡
[A] RpG
(8a)
B≡
[B] [B]S
(8b)
C≡
[C] [B]S
(8c)
where
µ≡
k GL DAR
(12c)
Equations 6 become
BL ) 1 - Λ iL )
νiΛ |ν|B
(13a)
(i ) C, D)
(13b)
Equation 7 transforms to
ννCCννDDΛνC+νD A| |ν|νBC+νD(1 - Λ)|νB|A|ν L
)K
(14a)
where
K≡
Keq(RpG)|νA| [B]νSC+νD-|νB|
(14b)
(B) Linkage Relations. Linear combinations of eqs 9 to eliminate F give
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d2A d2B d2B d2C - θBA 2 ) 2 + θCB 2 2 dy dy dy dy )
d2A ) |νA|Ha2A|νA| dy2
d2D d2B + θ DB dy2 dy2
With νA ) -1 and negligible gas-phase mass-transfer resistance (µ ) ∞),
)0
(15)
Integration and application of the y ) 0 boundary conditions yields
dB dA θBA ) φ ) µ(1 - A0) dy dy
(16a,b)
dC dB dD dB + θCB ) + θDB ) 0 dy dy dy dy
(16c)
φf
A - θBAB ) φ(1 - y) + AL - θBABL
(17a)
B + θCBC ) BL + θCBCL
(17b)
B + θDBD ) BL + θDBDL
(17c)
Ha tanh Ha
()
D
F≈
∑ δi i)A
A|νA|B|νB| -
(1) Limiting Behavior. When the reaction is extremely slow (i.e., Ha f 0), A varies linearly with y and
(18)
(
limHaf∞ A|νA|B|νB| -
C D K
)
νD
)0
[(BL + θCBCL - B0)/θCB] [(BL + θDBDL - B0)/θDB] νC
B| A0|νA|B|ν 0
CνCDνD )0 K
(26) (27a)
B + θCBC ) BL + θCBCL
(27b)
B + θDBD ) BL + θDBD
(27c)
It follows from eqs 17, 25, and 27 that
δA θBA
(28a)
δC ) -
δA θBAθCB
(28b)
δD ) -
δA θBAθDB
(28c)
δB )
(19)
i.e., near-instantaneous reaction implies effective local reaction equilibrium.20 Combining eqs 17b, 17c, and19 and applying the result at y ) 0 gives
(25)
A - θBAB ) φ(1 - y) + AL - θBABL
At the opposite extreme (when Ha f ∞), note that, because d2A/dy2 must remain finite, eqs 9a and 10a, when combined, establish that νC
(24)
j*i
and position-dependent equilibrium concentrations (i) are defined by
(9a)
)
∂i
δi ≡ i - i
2
µ (1 - AL) φf 1+µ
∂F
where departures from reaction equilibrium are defined by
The above “linkage relations” reduce the number of independent differential equations to one; we select the following:
dA ) |νA|Ha2F 2 dy
(23)
(2) Linearization. Nonlinearity of the reaction term F in eq 9a generally precludes analytical solution. Therefore, Peaceman14 adapted the VKH linearization technique. An alternative strategy is suggested in eq 19. For cases in which the Hatta number is large (Ha . 1), we follow Smith15 and retain only first-order terms in the Taylor series expansion of F about local equilibrium composition A, B, C, D; i.e.,
Note that φ is the dimensionless absorption rate, because µ(1 - A0) is the dimensionless flux of A in the gas film. Second integrations and insertion of the y ) 1 boundary conditions give
(
(22)
Equation 24 expands, in turn, to
F ≈ γ ‚δA
(29a)
νD
)K
(20)
Setting y ) 0 in eq 17a and combining the result with eq 16a gives
where
(
γ t A|νA|-1B|νB|-1 |νA|B +
)
|νB|A + θBA
(
)
φ ) A0 - AL - θAB(B0 - BL) ) µ(1 - A0) (21a,b)
DνD-1 νCD νDC + (29b) KθBA θCB θDB
Equations 20, 21a, and 21b determine A0, B0, and φ∞ (≡ limHaf∞ φ). A third limiting case is irreversible reaction (i.e., K f ∞). When, in addition, there is negligible depletion of B within the liquid film, eq 9a becomes
To complete the linearization of eq 9a, we neglect d2A/dy2 and fix γ at γ0 (these are simplifications that are consistent with the level of approximation in eq 24, i.e., they can be shown to imply neglect of higher-order departures from equilibrium). It follows that
C
νC-1
Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007 6143
d2δA 2
dy
≈
δA
(30a)
λ2
where
λ≡
1
(30b)
Hax|νA|γ0
The solution to eq 30a, subject to bulk liquid equilibrium (δAL ) 0), is
δA )
δA0 sinh[(1 - y)/λ]
(31)
sinh(1/λ)
Figure 2. Enhancement factor (E) vs Hatta number (Ha) and dimensionless gas-phase mass-transfer resistance (µ) for reaction A + B ) C + D. The horizontal broken line at top denotes Emax defined by eq 37. Solid lines represent data from Smith’s linearized analysis. Top solid line is for µ ) ∞. Legend: (]) Surface-Renewal VKH analysis.22 Numerical analysis data: (0) µ ) 10, (4) µ ) 2.5, (×) µ ) 1, and (+) µ ) 0.25.
It follows that
lim (E) ) The y ) 0 boundary condition for A (the first of the equations given as eqs 12) requires that
Haf0
lim (E) ≡ E∞ )
Haf∞
φ ) µ(1 - Ao - δAo)
(32)
A second expression for φ follows from eq 27a, evaluated at y ) 0; i.e.,
φ ) A0 - AL - θBA(B0 - BL)
µ 1+µ φ∞ 1 - AL
(33) µf∞
Finally,
φ)-
[( ) ( ) ] ( dδA dA + dy 0 dy
) 1-
0
)
B] B[ν 0
φ+ γ0 δA0 coth(1/λ) (34) λ
The dimensionless parameters (νi, K, Λ, θBA, θCB, θDB, µ, Ha), plus eqs 32-34 (three equations) and the auxiliary relations (e.g., definitions such as those given in eq 29b and 30b), determine δAo, Ao, and φ (three unknowns). Note that kG and L (the latter of which is equal to DA/kL) are typically calculated in advance from empirical correlations for specific packing types and vary with the superficial gas and liquid velocities, as well as fluid flow properties. The solution is easily implemented as a search for Bo: (i) a guess of Bo is made; then (ii) Co and Do follow from eqs 27b and 27c; and (iii) γo follows from eq 29b, λo follows from eq 30b, φ follows from eq 33, and δAo follows from eq 32. Equation 34 tests the guess of Bo. For design purposes, the dimensionless flux (φ) is the key calculated quantity. From φ, one may calculate the “enhancement factor” (E), by which the reaction (plus gas-phase mass transfer resistance) multiplies the absorption rate, i.e.,
E≡
φ 1 - AL
(35)
(36b)
Equation 36a expresses the factor by which gas-phase mass transfer resistance reduces the absorption rate in the absence of reaction effects. Equation 36b expresses the limiting factor by which the reaction (plus gas-phase mass-transfer resistance) multiplies the absorption rate, when the reaction is effectively instantaneous. Furthermore,
lim (E∞) ≡ Emax )
A]-1 [νA]A[ν 0
(36a)
φmax 1 - AL
(37)
Equation 37 expresses the factor by which instantaneous reaction multiplies the absorption rate, when there is negligible gas-phase mass-transfer resistance. Finally,
lim (E) ) K,µf∞
Ha tanh Ha (where νA ) -1, νB is effectively zero) (38)
Equation 38 expresses the factor by which the reaction multiplies the absorption rate, when there is negligible depletion of B in the liquid film, an effectively irreversible reaction (K f ∞), and, again, negligible gas-phase mass-transfer resistance. III. Results Figures 2-4 show the enhancement factor (E) versus the Hatta number (Ha), and, respectively, dimensionless gas-phase mass-transfer resistance (µ) (with reaction A + B ) C + D) (Figure 2), loading (Λ) (with reaction A + 2B ) 2C + D) (Figure 3), and K (with reaction A + B ) C) (Figure 4). Except when indicated otherwise, µ ) ∞, Λ ) 0, K ) 10, DB ) 2.25DC ) 0.25DD, and [B]S/(RpG) ) 100. Comparison is made of results based on Smith’s linearization technique, numerical analysis, and published formulas derived via VKH linearization within the frameworks of both steady-state Film Theory and transient Surface-Renewal Theory. These three figures depict the characteristic behaviors of E at small and large Ha (see eqs 36a and 36b). Figure 2 also portrays the negative effect of gas-phase masstransfer resistance on absorption. The Smith linearization and VKH approaches yield essentially the same E values, which closely match those obtained via exact numerical analysis.
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Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007
IV. Heterogeneous Catalysis The objective here is to predict a porous catalyst’s “effectiveness factor” (η, i.e., the ratio of reactant consumption to that in the absence of diffusion limitations) for a reversible reaction with nonlinear kinetics. We illustrate our approach with the following reaction:
2A T B
(39)
The assumed catalytic kinetics are Figure 3. Enhancement factor (E) versus Hatta number (Ha) and loading (Λ) for reaction A + 2B ) 2C + D. Solid lines represent data from Smith’s linearized analysis; symbols represent data from numerical analysis. Legend: (0) Λ ) 0, (]) Λ ) 0.3, (4) Λ ) 0.6, (×) Λ ) 0.75, and (+) Λ ) 90 (desorption occurs).
(
r ) kf [A]2 -
)
[B] Keq
(40)
Equations 4 are presumed to be applicable to a representative, straight, dead-ended pore of length L. Neglecting heat and external mass-transfer effects, the boundary conditions are
At x ) 0: [A] ) [A]0, [B] ) [B]0 d[A] d[B] ) )0 dx dx
At x ) L:
(41a) (41b)
We seek the effectiveness factor η, which may be calculated as follows:
η) Figure 4. Enhancment factor (E) versus Hatta number (Ha) and dimensionless equilibrium constant (K) for the reaction A + B ) C. The broken line represents data from eq 38, the solid lines represent data from Smith’s linearized analysis, and the symbols represent data from Film Theory VKH analysis.14 Legend: (]) K ) 10, (0) K ) 1, (4) K ) 0.3, (×) K ) 0.1, and (+) K ) 0.03.
-DA 2(kF[A]20
0
- kr[B]0)L
(42)
Proceeding as in the absorption case, we normalize the variables as follows:
A≡ Figure 3, besides further validating Smith’s linearization method, illustrates the negative effect of increased bulk loading Λ on E, which reflects the downward concavity of the [C] vs [A] isotherms. Note that, when Λ ) 90%, AL > 1 and, therefore, desorption occurs. Finally, Figure 4 illustrates the positiVe effect of increasing K (vanishing of the reverse reaction) on absorption; its negatiVe effect on desorption is not shown. Values of E based on the Smith and VKH linearization methods overlap, even as irreversibility is approached. Note that eq 38, which was derived assuming irreversible reaction and negligible depletion of B within the liquid film, delimits E. Clearly, linearization of reaction rate expressions introduces very little error in absorption rate predictions, when undertaken either via the VKH approach of fixing [B] at [B]0 (which is strictly justified only as Ha f 0) or by including only firstorder departures from equilibrium (strictly justified only as Ha f ∞). If there is a significant difference between the utilities of the linearization methods, it seems to be in the ease with which they may be extended to cases in which multiple gases undergo coupled liquid-phase reactions. We have successfully applied perturbation methods to analyze simultaneous absorption of CO2 and H2S in alkaline solutions;24 we are unaware of similar extension of the VKH approach. In the following section, we illustrate the application of a simple linearization method to the derivation of catalyst effectiveness factors.
(d[A] dx )
B≡
[A] CT [B]
KeqCT2
y≡
x L
(43a)
(43b)
(43c)
where
CT ≡ [A]0 + [B]0
(43d)
Equations 4 become
d 2B d2A 2 2 ) Φ (A B) ) -ω dy2 dy2
(44a,b)
where
ω≡
2DBKeqCT DA
(44c)
and the “Thiele modulus” (Φ), which is analogous to the Hatta number Ha, is defined by
Φ≡L Equation 42 becomes
x
2kfCT DA
(45)
Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007 6145
η≡
(dAdy )
0
(46)
Φ (A0 - B0) 2
2
The following linkage equation may be derived following the steps that were taken in treating absorption:
A + ωB ) A0 + ωB0 ≡ ψ
(47a,b)
Note that eq 47 differs from eq 17a, because of the difference in boundary conditions. After eliminating B based on eqs 47, eq 44a becomes
ψ-A d2A ) Φ2 A 2 2 ω dy
(
)
(48)
Next, we linearize eq 48 in terms of the departure variable δA. Although we defined δA in the absorption analysis relative to position-dependent A, here (because of the difference between eqs 17a and 47), we define it as follows:
δA ≡ A - A 0
(49)
Inserting eq 49 into eq 48 and neglecting the δA2 term yields
d2δA dy2
≈ τ 2δ A + χ
(50a)
1 ω
(50b)
where
(
τ2 ≡ Φ2 2A0 +
)
and
(
χ ≡ Φ2 A02 +
)
A0 - ψ ω
(50c)
The boundary conditions become
At y ) 0: δA ) 0
(51a)
dδA )0 At y ) 1: dy
(51b)
It follows from the solution, δA(y), that
η)
χ Φτ tanh τ(A02 - B0)
Figure 5. Effectiveness factor (E) versus the Thiele modulus (Φ) and ω for the reaction 2A f B. The broken line represents the exact solution when ω ) ∞, the solid lines represent the linearized analysis, and the symbols represent the numerical solution. Legend: (]) ω ) 1, (0) ω ) 0.1, and (4) ω ) 0.01.
(52)
Effectiveness factors based on eq 52, and on numerical solution to the nonlinear ODEs, are plotted in Figure 5, versus Φ and ω; A0 is fixed at 1.0 (bulk pure A). Linearization again retains the accuracy. Note also the small differences between the η values calculated with ω ) 1 and ω ) ∞; i.e., the reaction is already effectively irreversible when ω ) 1. V. Conclusions Linearization techniques developed by K. A. Smith sacrifice little accuracy when deployed to solve the differential equations that govern mass transfer with reversible chemical reaction in scrubbers and heterogeneous catalytic reactors. This suggests that the same techniques may be successfully applied to the
analysis of reactive extraction, phase-transfer catalysis, and other phenomena in which transport and reaction are coupled. Notation A, B, C, D ) reactive species A, B, C, D ) dimensionless concentrations defined by eqs 8 and 43 A, B, C, D ) dimensionless equilibrium concentrations defined by eqs 26 and 27 CT ) defined by eq 43d Di ) diffusion coefficient of species i (m2/s) E ) enhancement factor, defined by eq 35 Ha ) Hatta number, defined by eq 10b i ) species i; dimensionless concentration of species I j ) species j [i] ) concentration of species i (mol/m3) K ) defined by eq 14b Keq ) reaction equilibrium constant kG ) gas-phase mass transfer coefficient (mol m-2 s-1 bar-1) k1,k2,kf ) reaction rate constants L ) liquid film thickness (m) p ) partial pressure (bar) r ) reaction rate (mol m-3 s-1) x ) distance from the gas/liquid interface (m) y ) x/L Greek Symbols R ) solubility coefficient (mol m-3 bar-1) γ ) defined by eq 29b δi ) defined by eq 28 or 49 η ) defined by eq 42 θBA ) defined by eq 11a θCB, θDB ) defined by eq 11b Λ ) defined by eq 6a λ ) defined by eq 30b µ ) defined by eqs 12 ν ) stoichiometric coefficient χ ) defined by eq 50c F ) defined by eq 10a τ ) defined by eq 50b Φ ) defined by eq 45 φ ) defined by eq 16a φ∞ ) defined by eq 36v φmax ) defined by eq 37 ψ ) defined by eq 47b ω ) defined by eq 44c Subscripts A, B, C, D ) species A, B, C, D
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Ind. Eng. Chem. Res., Vol. 46, No. 19, 2007
eq ) reaction equilibrium G ) bulk gas i ) species i L ) x ) L; bulk liquid S ) stripped liquid 0)x)0 Literature Cited (1) Astarita, G. Mass Transfer with Chemical Reaction; Elsevier: Amsterdam, 1967. (2) Danckwerts, P. V. Gas-Liquid Reactions; McGraw-Hill: New York, 1970. (3) Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975. (4) Astarita, G.; Savage, D.; Bisio, A. Gas Treating with Chemical SolVents; Wiley: New York, 1983. (5) Doraiswamy, L. K.; Sharma, M. M. Heterogeneous Reactions; Wiley: New York, 1984. (6) Carra, S.; Morbidelli, M. Gas-Liquid Reactors. In Chemical Reactors and Reaction Engineering; Carberry, J. J., Varma, A., Eds.; Marcel Dekker: New York, 1987. (7) Van Swaaij, W. P. M.; Versteeg, G. F. Mass Transfer Accompanied with Complex Reversible Chemical Reactions in Gas-Liquid Systems: An Overview. Chem. Eng. Sci. 1992, 47, 3181. (8) Zarzycki, R.; Chacuk, A. Absorption; Pergamon: Oxford, U.K., 1993. (9) Satterfield, C. N. Mass Transfer in Heterogeneous Catalysis; MIT Press: Cambridge, MA, 1970. (10) Aris, R. The Mathematical Theory of Diffusion and Reaction in Porous Catalysts; Clarendon Press: Oxford, U.K., 1975. (11) Froment, G. F.; Bischoff, K. B. Chemical Reactor Analysis and Design; Wiley: New York, 1979. (12) Brian, P. L. T.; Hurley, J. F.; Hasseltine, E. H. Penetration Theory for Absorption Accompanied by a Second Order Chemical Reaction. AIChE J. 1961, 7, 226.
(13) Van Krevelen, D. W.; Hoftijzer, P. J. Kinetics of Gas-Liquid Reactions. I. General Theory. Recl. TraV. Chim. 1948, 67, 563. (14) Peaceman, D. W. Liquid-Side Resistance in Absorption with and without Chemical Reaction, Sc.D. Thesis, MAssachusetts Institute of Technology (MIT), Cambridge, MA, 1951. (15) Smith, K. A.; Meldon, J. H.; Colton, C. K. An Analysis of CarrierFacilitated Transport. AIChE J. 1973, 19, 102. (16) Whitman, W. Preliminary Experimental Confirmation of the TwoFilm Theory of Gas Absorption. Chem. Metal. Eng. 1923, 29, 146. (17) Hatta, S. On the Absorption Velocity of Gases by Liquids. Tech. Rep. Tohoku Imp. UniV. 1932, 10, 119. (18) Higbie, R. The Rate of Absorption of a Pure Gas into a Still Liquid during Short Periods of Exposure. Trans. Am. Inst. Chem. Eng. 1935, 31, 365. (19) Danckwerts, P. V. Significance of Liquid-Film Coefficient in Gas Absorption. Ind. Eng. Chem. 1951, 43, 1460. (20) Olander, D. R. Simultaneous Mass Transfer and Equilibrium Chemical Reaction. AIChE J. 1960, 6, 233. (21) Chang C. S.; Rochelle, G. T. Mass Transfer Enhanced by Equilibrium Reactions. Ind. Eng. Chem. Fundam. 1982, 21, 379. (22) Glasscock, D. A.; Rochelle, G. T. Numerical Simulation of Theories for Gas Absorption with Chemical Reaction. AIChE J. 1989, 35, 1271. (23) DeCoursey, W. J. and Thring, R. W. Effect of Unequal Diffusivities on Enhancement Factors for Reversible and Irreversible Reaction. Chem. Eng. Sci. 1989, 44, 1715. (24) Meldon, J. H. Absorption with Reaction. In ReactiVe Separation Processes; Kulprathipanja, S., Ed.; Taylor and Francis: New York, 2002.
ReceiVed for reView April 17, 2007 ReVised manuscript receiVed July 19, 2007 Accepted July 20, 2007 IE0705397
