Ind. Eng. C h e m . Res. 1993,32, 42-48
42
Catalytic Storage of Hydrogen: Hydrogenation of Toluene over a Nickel/Silica Aerogel Catalyst in Integral Flow Conditions. Appl. Catal. 1988,42, 121. Lindfors, L. P.; Salmi, T. Kinetics of toluene hydrogenation on a Ni/A1,03 catalyst. Submitted for publication in Chem. Eng. Sci. 1992.
Marangozis, J. K.; Mantzouranis, B. G.; Sophos, A. N. Intrinsic Kinetics of Hydrogenation of Benzene on Nickel Catalysts Supported on Kieselguhr. Ind. Eng. Chem. Prod. Res. Deu. 1979,18, 61.
Marecot, P.; Paraiso, E.; Dumas, J. M.; Barbier, J. Benzene hydrogenation on nickel catlaysts. Role of weakly bound hydrogen. Appl. Catal. 1991,74,261. Minot, C.; Gallezot, P. Competitive Hydrogenation of Benzene and Toluene: Theoretical Study of Their Adsorption on Ruthenium, Rhodium, and Palladium. J . Catal. 1990,123, 341. Mirodatos, C.; Dalmon, J. A,; Martin, G. A. Steady-State and Isotopic Transient Kinetics of Benzene Hydrogenation on Nickel Catalysts. J . Catal. 1987,105, 405. Motard, R. L.; Burke, R. F.; Canjar, L. N.; Beckmann, R. B. The kinetics of the catalytic hydrogenation of benzene on supported nickel and nickel oxide catalysts. J . Appl. Chem. 1957, 7, 1. Nicolai, J.; Martin, R.; Jungers, J. C. La cinetique de l'hydrogenation du benzene. Bull. SOC.Chim. Belg. 1948,57,555. Orozco, J. M.; Webb, G. The adsorption and hydrogenation of benzene and toluene on alumina- and silica-supported palladium and platinum catalysts. Appl. Catal. 1983,6 , 67. Rahaman, M. V.; Vannice, M. A. The hydrogenation of Toluene and 0-, m-, and p-Xylene over Palladium. I. Kinetic Behavior and o-Xylene Isomerization. J . Catal. 1991a,127, 251. Rahaman, M. V.; Vannice, M. A. The hydrogenation of Toluene and 0 - , m-, and p-Xylene over Palladium. 11. Reaction Model. J . Catal. 1991b,127, 267. Roethe, K. P.; Roethe, A.; Rosahl, B.; Gelbin, D. Vergleichende Untersuchung einer Modellreaktion in Verschiedenen Versuchsreaktoren. Chem. Ing. Tech. 1970,42, 805. Rooney, J. J. The Exchange with Deuterium of Two Cycloalkanes on Palladium Films. *-Bonded Intermediates in Heterogeneous Catalysis. J . Catal. 1963,2, 53. Ross, R. A.; Martin, G. D.; Cook, W. G. Hydrogenation of Olefins over Nickel/Silica Catalysts. Ind. Eng. Chem. Prod. Res. Deu. 1975,14, 151. Selwood, P. W. The Mechanism of Chemisorption: Benzene and
Cyclohexane on Nickel, and the Catalytic Hydrogenation of Benzene. J . Am. Chem. SOC. 1957, 79,4637. Selwood, P. W. Adsorption and Paramagnetism; Academic Press: New York, 1962. Shopov, D.; Palazov, A.; Andreev, A. Cyclohexane, benzene and hydrogen interaction with nickel catalyst. Proc. Int. Congr. Catal. 1968,4th,388. Snagovskii, Y. S.; Lyubarskii, G. D.; Ostrovskii, G. M. Kinetika Gidrirovaniya benzola na nikele, I. Reaktsii v kineticheskoj oblasti. Kinet. Katal. 1966,2, 258. Tetenyi, P.; Barbernics, L. Investigations on the Chemisorption of Benzene on Nickel and Platinum. J . Catal. 1967,8, 215. Tetenyi, P.; Babernics, L.; Schachter, K. Kinetic study of hydrocarbon adsorption on nickel catalyst. Acta Chem. Acad. Sci. Hung. 1969,61 (4), 367. Vajda, S.; Valko, P. Reproche, Regression Program for Chemical Engineers, Eurecha: Budapest, 1985. van Meerten, R. Z. C.; Coenen, J. W. E. Gas Phase Benzene Hydrogenation on a Nickel-Silica Catalyst I. Experimental Data and Phenomenological Description. J . Catal. 1975,37,37. van Meerten, R. Z. C.; Coenen, J . W. E. Gas Phase Benzene Hydrogenation on a Nickel-Silica Catalyst IV. Rate Equations and Curve Fitting. J . Catal. 1977,46, 13. van Meerten, R. Z. C.; Verhaak, A. C. M.; Coenen, J. W. E. Gas Phase Benzene Hydrogenation on a Nickel-Silica Catalyst 11. Gravimetric Experiments of Benzene, Cyclohexene, and Cyclohexane. Adsorption and Benzene Hydrogenation. J. Catal. 1976, 44,217. van Meerten, R. Z. C.; de Graaf, T. F. M.; Coenen, J. W. E. Gas Phase Benzene Hydrogenation on a Nickel-Silica Catalyst 111. Low-Field Magnetization Measurements on Hydrogen, Benzene, Cyclohexene and Cyclohexane Adsorption, and Benzene Hydrogenation. J . Catal. 1977,46,1. Volter, J. A r-Complex Mechanism for Catalytic Hydrogenation of the Benzene Ring. J . Catal. 1964,3,297. Zlotina, N. E.; Kiperman, S. L. 0 kinetike reaktsii gidrirovania benzola na nikelevyh katalizatorah v bezgradientnoi sisteme, 11. Kineticheskie isotopnye effekty reaktsii. Kinet. Katal. 1967,6 , 1335.
Received for review April 28,1992 Revised manuscript received September 10, 1992 Accepted October 5,1992
Simplified Treatment of the Rates of Gas-Solid Reactions Involving Multicomponent Diffusion Eric G.Eddingst Department of Chemical Engineering, University of U t a h , Salt L a k e City, U t a h 84112
Hong Yong Sohn* Departments of Metallurgical Engineering and Chemical Engineering, University of U t a h , Salt L a k e City, U t a h 84112
An approximate analytical solution is presented which describes the rate of a gas-solid reaction involving multicomponent diffusion in the gas phase. The model obviates the numerical solution of the nonlinear Stefan-Maxwell equations by defining pseudobinary diffusivities a t the diffusion boundaries which take into consideration the Stefan-Maxwell dependence of the binary diffusivities. A linear dependence on mole fraction is subsequently assumed for the pseudobinary diffusivities across the control volume. The model has been generalized to account for reversible reactions and bulk flow effects due to volume change of the gas. The approximate solution is compared with the rigorous numerical solution over a variety of conditions, and the resulting comparisons show that, even for very large differences in binary diffusivity values, good agreement is obtained between the two solutions. Introduction The theoretical analysis of gas-solid reaction systems is important in a variety of engineering applications such *To whom all correspondence should be addressed. Current address: Reaction Engineering International, Salt Lake City, Utah. +
as the removal of sulfur oxides from flue gases by injection of a solid sorbent or the reduction of metal oxides by gaseous compounds. In general, systems that involve more than two gaseous species will require solution of the nonlinear system of Stefan-Maxwell equations coupled with the applicable form of the equation of continuity (Ramachandran and Doraiswamy, 1982). The exact numerical
Q888-5885/93/2632-QQ42$Q4.QO/Q 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 1, 1993 43
= constant); conditions are isothermal; total pressure is constant; assumptions are made that pertain to the Stefan-Maxwell equations (ideal gas, etc.); a pseudo-steadystate approximation is used (concentration profile is established quickly relative to changes in radius of unreaded core); and binary molecular diffusivities are independent of concentration. Governing Equations. Using the assumption$ listed above, the equations of continuity for the gaseous species A and C within the reacted porous layer reduce to SolidP&D
Figure 1. Idealization of porous particle of material B reacting with gaseous species A with the overall rate under diffusion control.
solution of this system can be quite cumbersome, in particular, when generalized to multiparticle systems for reactor design. As a simplification, such systems are often analyzed as pseudobinary mixtures using a constant average effective diffusivity such that existing analytical solutions for gasaolid reactions or simpler numerical solutions can be utilized. The use of pseudobinary diffusivities has been shown to give reasonable approximations of the more rigorous solution (Sohn and Braun, 1980; Szekely et al., 1977);however, this only appeared to be true if there were no relatively large differences between the individual binary diffusivities. The results of such an approach can be less than satisfactory due to neglect of the interactions between the fluxes of the various diffusing species as well as any concentration dependence (Szekely et al., 1977; Shain, 1961). It is desirable, therefore, to obtain an approximate analytical solution for the case of multicomponent diffusion within a reacting, porous particle which incorporates the flux relationships and the concentration dependence of such a problem. In the following, an approximate solution is presented for the conversion-wtime relationship of a diffusion-controlled reaction within an initially porous particle. The development follows that of the generalized gas-solid reaction model of Sohn and Szekely (1972) and the final form is similar to other variations of the generalized model as described in Szekely et al. (1976). An expression for the effective pseudobinary diffusivity is defmed which reflects the Stefan-Maxwell dependency of the binary diffusivities and a linear dependence on mole fraction within the control volume. This expression is combined with Fick's first law (defined in terms of the molar flux relative to stationary axes, NA)to obtain an integrable expression for Nh It will be shown that a relatively simple expression for the conversion-vs-time relationship can be obtained which gives a g o d approximation of the more rigorous, direct numerical solution of the Stefan-Maxwell equations. Although the analytical expression requires the solution of a single, nonlinear expression for the equilibrium condition at the reaction interface, this need only be performed once for any given experimental system. Model Formulation The system under evaluation, shown in Figure 1,is the diffusion-controlled reaction of an initially porous solid in the presence of inerta according to the following generalized reaction: A(g) + bB(s) uC(g) + dD(s) (1) In addition, a number of conditions were asaumed in order to adequately define the problem: the reaction is under pore diffusion control (diffusion within the solid product layer); the overall size of the particle does not change (rp Q
where NAand Nc are the molar fluxes of species A and C, respectively. The relationship between the flux, NA,and concentration for a multicomponent system is given by the Stefan-Maxwell relation (Bird et al., 1960) (4)
where xiis the mole fraction of species i, c is the total molar concentration of the gas, and Dij is the effectiue binary molecular diffusivity for the ij pair (Le., modified to account for porosity of the product layer). Given the pseudo-steady-state approximation,
-vNA = Nc (5) by stoichiometry, and NI = 0. Using these simplifications, in addition to the relationship xI = 1- xA - xc, the Stefan-Maxwell equations for species A and C can be simplified to the following:
d ~ c 1 NA(UXA + I C ) - VNAU- XA - X C ) -
-[
d r c DAC DCI If the continuity equation for species A, eq 2, is included, the result is three equations in three unknowns: xA, xc, N A (Nc is related to N A through stoichiometry). The boundary conditions for these three equations are at r = rp
xA
= XA,b,
XC
=XCb
(8)
at r = r, Equation 9 represents the equilibrium condition at the reaction boundary for a diffusion-controlled system. The expressions thus far describe the problem for a fixed reaction boundary a t r,; therefore, the movement of this boundary with time must considered as well. The rate of consumption of solid reactant B at r = r. can be related to the movement of the reaction boundary as follows:
where pB and aB are respectively the molar density of the solid reactant and the volume fraction of the pellet occupied by B and b is the stoichiometric coefficient in eq 1. The initial condition for eq 10 is given by att=0
r, = rp
(11)
In order to solve the initial-value problem defined by eqs 10 and 11,the nonlinear boundary-value problem defined
44
Ind. Eng. Chem. Res., Vol. 32, No. 1, 1993
by eqs 2-9 must be solved at each time step. Rigorous Solution. A comparison between exact solution of the governing equations and the solution obtained by approximate means was desired; therefore, a numerical solution of the nonlinear problem was pursued. The solution of the pseudo-steady-state problem, eqs 2-9, was accomplished by using a finite difference approximation of the derivatives in r to obtain nonlinear algebraic equations which were solved by a Gauss-Siedel iterative technique. Due to the fact that only the boundary values at rp were known explicitly, the solution required a backward finite differencing of the differential equations and solution of the nonlinear algebraic equations proceeded inward from the outer radius. The solution algorithm proceeded as follows: (1) Guess an initial value for NA,+at r = rp. (2) Calculate new values for xA,xc, and NA at decreasing radii using the known values from the previous iteration. (3) Compare the value of xAcalculated at the boundary ro with the value of xA calculated from the equilibrium condition, eq 9, using the current xc. (4) Iterate using new values of NA,*at r = rp to drive the error, axA,to zero using nonlinear root-finding techniques. The root-finding approach taken involved driving the function bA = f(NA,+) to zero using the secant method, where each step of the secant method required solving for the concentration and flux profiles within the entire product layer. Higher-order solution methods such as the Newton-Raphson (N-R) method would not have been beneficial here since the functionality of f(NA,+) was not known explicitly. The derivatives necessary for the N-R method would have to be approximated numerically, and the additional function evaluations would reduce the order of convergence; therefore, the secant method is considered a better method in this situation (Press et al., 1989). Although a fairly reasonable initial guess was required in order for the algorithm to converge, solution usually occurred within four to seven iterations of the secant method. A reliable ODE solver (Hindmarsh, 1980) w a coupled with the algorithm described above to perform the initial-value problem integration, and the solution required repetition of the entire algorithm at each time step of the ODE solver. Approximate Analytical Method. Application of the numerical solution described in the previous section to engineering problems becomes quite difficult when single particle results are scaled-up to analyze multiparticle reactors; therefore, it would help to have a simpler, approximate closed-form solution. An analytical solution which utilizes several simplifications is presented in this section for the problem of diffusion-controlled reactions. The diffusion of a species i in a mixture can be represented by an effective binary diffusivity, D,,, according to the following equation (Bird et al., 1960): n
N , = -cD,,VX,
+X,CN~ ]=I
(12)
for species A at rp and rc, respectively, and similarly for species C we obtain
Xc,eq(u - 1) - v Dcrn,eq = 'Aw(
& &) -
+
'cw(
& &) & -
(17)
It should be noted that these diffusivities are constant for a given problem. To obtain an analytical solution, a suitable, simple expression for D,, is necessary. The assumption of linear variation with composition, as given in eq 18, has proven useful (Bird et al., 1960): D,, = m,x, + b, (18) The slope, m,,and intercept, b,, can then be defined in terms of the constants expressed in eqs 14-17 as follows:
- Dim.eq - x,,eq
(19)
b~ = Dirn,b - X,,bmr
(20)
!')irn,b
m ,= -xi,b
where i represents either species A or species C. Other assumptions were attempted in this work with much less success, as discussed in the Discussion. Although not given in this paper, the values of D,, calculated by eq 12 indicate approximate linearity with composition in most cases of practical interest. When the binary diffusivitieshave large differences (by more than an order of magnitude), the variation of D,, becomes less linear. It is the purpose of this work, however, to find a mathematical expression for D,, which will yield an analytical solution which, in turn, is a reasonable approximation of the exact solution over as wide a range of variables as possible. The actual accuracy of eq 18 is not an issue here. The flux expression, eq 12, can now be written using eq 18 as follows:
If eq 12 is combined with eq 4, an expression for D,, can be obtained which has the form
-
(21)
for species A, and
~ ~ / C D J ( ~ , N , X,N,)
Nc =
-c(mcxc + bc)
[ + .(; 1
This effective mixture diffusivity can be evaluated for species A and C at each boundary of the control volume using the assumptions of the pseudo-steady-state approximation, and the results simplify to
-
- l)]
dxc
(22)
dr
for species C. Equations 21 and 22 can be combined with the corresponding equations of continuity, eqs 2 and 3, and integrated using the boundary conditions, eqs 8 and 9, to ob-
tain the following expressions for the flux profiles within the control volume:
+
+
C
)]
0
(24)
If these fluxes are related through stoichiometry according to eq 5, then the following relationship is obtained between the bulk and equilibrium mole fractions: mA(XA,b
-v[
- XA,eq)
v-1
(
+
~ A ( V-
1 + XA,b(v - 1) 1 + xA,eq(v - 1)
1) - mA
(v - 1)2
)] [ =
500
1wO
1500
Zoo0
2500
3wO
Time (sec)
Figure 2. Comparison between approximate and exact solutions for the case of equimolar counterdiffusion.
For the special case of v = 1, application of l'H6pital's rule to eq 28 gives
X
- k,eq) (l/v) - 1
mC(XC,b
+
This equation can be combined with the equilibrium expression at the reaction interface
Equation 27, evaluated at r = rc, can then be combined with eqs 10 and 11 and integrated to obtain a conversion-vs-time relationship in terms of the diffusion-controlled conversion function for a spherical particle, as defined by Sohn and Szekely (1972). The resulting expression can be generalized for other geometries using a shape factor, Fp,to obtain (30)
to yield two equations in two unknowns for the equilibrium mole fractions. Explicit solution of eqs 25 and 26 is not possible due to the implicit nature of eq 25; therefore, nonlinear root-finding techniques are required to obtain the equilibrium compositions. It should be noted,however, that this need only be performed once for a given problem. In addition, since the equation is nonlinear, special care should be exercised in obtaining the correct root. The authors have obtained multiple roots between 0 and 1 for x ~ ,however, ~ ~ ; only one root will satisfy eqs 25 and 26 as well as the condition x x L = 1. Results The Case of Reversible Reaction with Gas Volume Change. After evaluation of the constants in eq 23, analytical expressions for the conversion-vs-time relationship can be readily obtained. For the general case of reversible reactions, eq 23 can be written in the simplified form (27)
where the constant, PA, is defined by
where the generalized diffusion-controlled conversion function is defined as (Sohn and Sohn, 1980)
Fp(l- X)'IFp - 2(1 - X) PF,(X) = 1 Fp- 2
'
(31)
and the fractional conversion, X,is defined by the expression (rc/rp)FP= 1 - X
(32)
The shape factor, Fp,represents the geometries of infinite slabs, long cylinders, and spheres for Fp = 1, 2, and 3, respectively. Comparisons between results obtained using eq 30 for spherical geometry and results from the exact numerical solution of the rigorous expressions (eqs 2-11) are shown for different values of v in Figures 2 and 3. The comparisons utilized an equilibrium constant (Kp)equal to 1 and a factor of 10 between the lowest and highest effective binary gas diffusivities. Figure 2 displays the results for the case of equimolar counterdiffusion (v = 1) for differing bulk concentrations of A. As is shown, the approximate solution gives a reasonable estimation of the exact solution over the entire concentration range. The difference in the two solutions are in the range of typical reproducibility in the experimental measurements of gmaolid reaction rates. The curves become flat at high conversions, indicating that a small increase in conversion requires a long time. Thus the difference between the exact and the approximate solution in this region should be measured by the difference in the conversion at a given reaction time. For the case of net generation of gaseous volume ( u =: 4), Figure
46 Ind. Eng. Chem. Res., Vol. 32, No. 1, 1993 .oo
1
r
8
0.60
'$
3
0.40
-
0.80
0.80
i,,
8
0.M)
B
0.40
'g>
0.20
0.20
0.00
omy,,
- - - -. Approarmate Solution 0
Time (sec)
Figure 3. Comparison between approximate and exact solutions for the case of net generation of gas volume. 1.00
0.80
0.60
0.40 0.20
0.00 1.00
0.80
0.60
0.40
O.m 0.00
0
50
100
150
200
250
300
350
400
450
500
Time (sec)
----. Approximate Solution .._._....__ Binary Solufion Figure 4. Comparison between approximate and exact solutions for large differences in effective binary diffusivity. Also included for comparison are the solutions for the binary case at two different binary diffusivity values.
3 displays even better agreement between the exact and the approximate solution. Calculations also showed that the approximate solution provides a reasonable engineering estimate of the exact solution when v < 1. An alternative to using either the exact numerical solution or the approximate solution presented in this work is to u8e the binary equation (eq 12) with either DAC or DN, for Dim.This yields the familiar analytical solution given by eq 30 with PA replaced by cD(xA,~ - x ~ , ~The ~ ) utility . of this approach is examined in Figure 4,where the binary solution is compared with both the exact numerical solution and the approximate solution of this work. The case where the value for DAC was used (diffusivity of 1.0) is
, ,
I
500
,
,
,
,
I
,
loo0
I
.
I
-j 1503
2wO
Time (sec)
Figure 5. Comparison between approximate and exact solutions as equilibrium constants appreach the irreversible case (KP = a).
shown in the upper portion of Figure 4,and the case where DAIwas used (diffusivity of 10) is shown in the lower portion of the figure. As might be expected, the agreement of the exact binary solution using DAC becomes fairly good at low inert concentrations, as shown in the upper portion of Figure 4. The use of DN with the exact binary solution, however, does not appear to provide useful results even at high inert concentrations (bottom of Figure 4). Figure 4 also demonstrates the agreement of the approximate solution with the numerical solution for the case of vastly different binary diffusivities. The comparisons shown in Figures 2 and 3 utilized a difference in effective diffusivity of a single order of magnitude which would be representative of the extremes in diffwivity values for most diffusion problems. The effective binary diffusivities used in the approximate and numerical solutions shown in Figure 4 ranged over 2 orders of magnitude, certainly greater than would normally be expected, but the agreement between the two solutions is still very satisfactory. For the case where the various effective binary diffusivity values approach a single value, a comparison was possible between the approximate solution and the exact analytical binary solution. For this asymptotic case, the approximate solution yielded exactly the same prediction as the analytical solution. The Case of Irreversible Reaction. For the case of irreversible reactions, the boundary condition at r, changes to XA,, = 0, and the expression obtained for the conversion relationship, eq 30, can again be used with the following expression for PA:
For the special case of v = 1,eq 29 is again the appropriate expression with the added substitution, X,, = 0. Without a separate computer program being written for this case, there was considerable difficulty in obtaining numerical solutions for high values of the equilibrium constant, K,; therefore, direct comparisons with the approximate solution were not made for the irreversible case. Figure 5, however, shows a comparison between the approximate and numerical solutions for increasing values of Kp, and includes the approximate irreversible solution for comparison. As is shown, there is increasing agreement between the approximate and exact solutions with increasing Kp, with an apparent approach to the irreversible solution. Application of the Approximate Model. A comparison was made between the approximate solution and the
Ind. Eng. Chem. Res., Vol. 32, No. 1, 1993 47 pellet diffusion control. If the surface chemical reaction is given by
I .00
0.80
(36) 0.M
then the expression for conversion in the absence of intrapellet diffusion resistance, or under chemical reaction control, can be expressed as
0.40 ***-
-Exact Solution - - - -. AppmximaleSolution
0.20
0.00
,
0
,
.
,
I
,
loo0
,
I
, I ,
2000
I
,
I
I
3000
I
DAc = 1.78
I
,
,
I
n
m
.
.
I
5000
Time (sec)
Figure 6. Application of the approximate model to the hydrogen reduction of FeO a t 700 "C.
exact solution for an actual engineering application; in particular, the reduction reaction Hz(g) + FeO(s)
Q
HzO(g) + Fe(s)
(34)
in the presence of inert nitrogen at 700 OC. The three binary diffusivities for the gaseous H2-H20-N2 system were computed using standard techniques and were modified to account for particle porosity by the methods prescribed by Szekely et al. (1976). The values obtained were (utilizing the generalized nomenclature of eq 1)DAC = 1.778 cm2/s, DAI = 0.601 cm2/s, and DcI = 0.367 cm2/s. For a reactant gas taken to be initially free of moisture, calculations were made with varying bulk concentrations of Hz and the resulting solutions were compared with the corresponding rigorous numerical predictions in Figure 6. As is shown, the approximate model gave predictions of the more rigorous numerical solution that would be very good for most engineering applications over the entire range of reactant concentrations. It should be noted here that the general behavior of the approximate solution is essentially the same as the numerical solution; therefore, if the model was being utilized with adjustable parameters fit to experimental data, then the subsequent predictions made using the approximate solution would undoubtedly yield results comparable to the predictions made with a parameter fit to the numerical solution.
Discussion The Case of Mixed Rate Control. Initially Nonporous Solids. For the case of the reaction rate under both kinetic and pore diffusion control within the porous product layer, the approximate solution can be easily modified by utilizing the law of additive reaction times which has been found to be exact under certain conditions and to be a useful approximation under others (Sohn, 1978). The law can be stated as follows: in the absence of external mass-transfer resistance, the time required to attain a certain conversion is equal to the time required to attain the same conversion in the absence of resistance due to intrapellet diffusion of gaseous reactant added to the time required to attain the same conversion under the control of intrapellet diffusion of gaseous reactant. This can be stated mathematically as t(x)
t(X)labsence of diffusion resistance
+ t(X)ldiffusion
control
(35) Equation 30 is the appropriate expression for the time required to achieve the desired conversion under intra-
where the chemical-reaction controlled conversion function for an initially nonporous pellet is defined as (Szekely et al., 1976) gj$X) = 1 - (1- X)l/FP
(38)
The Case of Mixed Rate Control. Initially Porous Solids. For the case of initially porous solids, the conceptualization of the unreacted pellet as being made up of individual grains of reactant has been shown to be useful (Sohn and Szekely, 1972). Following the grain model approach, the only modification required to the procedure for initially nonporous solids involves the utilization of the chemical-reaction controlled conversion function for grains as opposed to pellets. The expression for the time required to achieve a particular conversion function for the intrapellet grains is given by
where A,, Vg,and Fg are the surface area, volume, and shape factor for the grain, respectively, and the reactioncontrolled conversion function for grains is defined by gFg(X= ) 1- (1 - X)"
(40)
where the fractional conversion is defined by eq 32. Other appropriate forms of g ( X ) functions may also be used in eq 35 (Sohn, 1991; Sohn and Chaubal, 1986; Sohn, 1978). Alternative Approaches. Several other assumptions were attempted in the formulation of the approximate solution which resulted in less than satisfactory results. The assumption of linear variation of diffusivity with particle radius was attempted as an alternative to the concentration dependence of eq 18; however, the resulting analytical expression would not reduce to a manageable form comparable to eq 30. The assumption of a cofistant arithmetic mean diffusivity between the values at the external surface and the reaction interface was also attempted utilizing a generalized form of the diffusivity expression, eq 13. The resulting comparison with the rigorous numerical solution varied depending on the conditions. Particularly when systems involving high equilibrium constants or a net change in gas volume are analyzed, diffusivity values can be significantly different at the two boundaries of the particle leading to increased discrepancies between the two solutions. Also, in order to avoid having to solve the nonlinear expression given by eq 25, attempts were made to solve the problem assuming that the mole fraction of inerts remained constant throughout the control volume. The resulting expression, xA + xc = constant, could then be combined with eq 26 to obtain directly the equilibrium composition at the reaction interface. Comparisons with the numerical solution showed this assumption to be fairly
48 Ind. Eng. Chem. Res., Vol. 32, No. 1, 1993
good at low concentrations of inerta (e.g., Xl,b = 0.1); however, there were serious discrepancies at high inert concentrations (e.g., x I , b = 0.9).
Conclusions An approximate analytical solution for the conversionvs-time relationship for gas-solid reactions involving multicomponent diffusion was developed in terms of a diffusion parameter, BA. The approximate solution was found to compare well with the exact numerical solution of the rigorous description of the problem. The expression for the approximate solution is of a form consistent with other generalized analytical solutions to gas-solid reaction problems and can be easily integrated over a multiparticle system for full-scale engineering applications. The complexity of the exact numerical solution for a single-particle system would make its application to multiparticle systems difficult, particularly since the authors found difficulties in obtaining convergence for certain conditions even a t the single-particle level. The approximate solution appears to yield better agreement with the exact solution at higher values of the equilibrium constant and higher values of v. The method also handles very large ranges in binary diffusivity values very well. A difference of 2 orders in magnitude between the highest and lowest binary diffusivity values still resulted in good agreement with the numerical solution. In addition, the use of the approximate solution with laboratory data for parameter fitting and subsequent equipment scale-up should also yield very good results as the general behavior of the approximate solution is very similar to the numerical solution. Nomenclature a, b, d = stoichiometriccoefficients of the generalized gas-solid reaction defined by eq 1 A , = surface area of internal grain, cm b, = intercept of diffusivity-concentration relation defined by eq 20, cm2/s c = total gas concentration, mol/cm3 D,, = effective binary gas diffusivity for the i-j pair, cm2/s D,,b = effective pseudobinary gas diffusivity for species i at the bulk gas interface, r = rp, defined by eqs 14 and 16, cmz/s D,,eq= effective pseudobinary gas diffusivity for species i at the reaction interface, r = rc, defined by eqs 15 and 17, cm2/s Fg= shape factor for internal grains defined as 1 for infinite slabs, 2 for long cylinders, and 3 for spheres Fp = shape factor for pellets defined as 1 for infinite slabs, 2 for long cylinders, and 3 for spheres &Ax) = reaction-controlledconversion function for intrapellet grains defined by eq 40 gF,(X) = reaction-controlled conversion function for pellets defined by eq 38 k = rate constant for surface reaction defined by eq 36 Kp = equilibrium constant for generalized reaction (eq 1) defined by eq 26 mi = slope of diffusivity-concentration relation for species i defined by eq 19, cm2/s N , = molar flux relative to stationary axes for species i, mol/(cm2.s) P = total gas pressure, atm PFJx) = generalized diffusion-controlledconversion function defined by eq 31
r = distance from the center of symmetry, cm rc = distance from the center of symmetry to the reaction interface, cm rp = distance from the center of symmetry to the outer surface of the particle, cm xi = gaseous mole fraction of species i x,,b = gaseous mole fraction of species i in the bulk gas at the particle surface x , , , ~= equilibrium gaseous mole fraction of species i at the reaction interface X = fractional conversion of solid reactant B defined by eq 32
Greek Symbols aB = void fraction of the solid reactant B (including inerts)
PA = generalized diffusion parameter defined by eq 28,
mol/ (cms) = diffusion parameter for irreversible reactions defiied by eq 33, mol/(cmd p i = diffusion parameter for equimolar counterdiffusion defined by eq 29, mol/(cm.s) pB = solid density of the reactant B, mol/cm3 v = stoichiometric coefficient of gaseous product C defined by eq 1 Subscripts
A = refers to gaseous reactant A defined by eq 1 B = refers to solid reactant B defined by eq 1 C = refers to gaseous product C defined by eq 1 D = refers to solid product D defined by eq 1 I = refers to inert gaseous species
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