Ind. Eng. Chem. Fundam. 1984, 23, 132-134
132
Literature Cited
\ P t ?
P t P
? t 9
pj
f - - - l
P t P
- _dm
t
j e t gas
?
lluidizing gas
1 ,
Figure 1. Development o f a j e t bubble, and definitions o f L,,, and
L,,
according t o K n o w l t o n a n d H i r s a n (1980).
minimum, and complete fluidization velocities. According
to Hirsan et al. (1980), U, is not the appropriate reference variable for this purpose and should be replaced by Ucf, which is larger than Umfin the case of wide size range of solids (Knowlton, 1977). For given solids, excess of fluidizing velocity, and jet momentum flux at the orifice, the jet penetration may be further affected by the bed to orifice size ratio, as at low values of this ratio the discharge is influenced by the hindering of the solids circulation promoted by the jet. The combined effects of the properties of the fluidized dense phase and of the geometry of the nozzle and bed containing walls are serious drawbacks toward the development of a sound correlation for 0 in the present state of knowledge.
Basov, V. A.: Markhevka, V. I.; Melik-Akhnazarov, T. Kh.; Orochko, D. I. Int. Chem. Eng. 1989, 9 , 263. Fiiia, M.; Massimilla, L.; Vaccaro, S. I n t . J. MuItIiphese Flow 1983, 9 , 259. Hirsan, I.; Sishtla, C.; Knowlton, T. M. 73rd AIChE Annual Meetlng, Chlcago, Nov 1980. Knowlton, T. M. AIChE Symp. Ser. 1977, 73(161), 22. Knowlton, T. M.; Hirsan, I.I n "Fluidization"; Grace, J. R.: Matsen, J. M., Ed.; Plenum: New Yqrk, 1980; p 315. Massimllla, L.; Donsi, G.; Migliaccio, N. AIChE Symp. Ser. 1981, 77(205), 17. Merry, J. M. AIChE J. 1975, 27, 507. Rowe, P. N.; MacGilllvray, H. J.; Cheesman, D. J. Trans. Inst. Chem. Eng. 1979, 57, 194. Shakhova, N. A. Inzh. Flz. Zh. 1988, 14, 61. Vakhrushev, I. A. Teor. Osn. Khim. Tekhn. 1972, 6 , 89. Wallis, G. B. "One-Dlmensional Two-Phase Flow"; McGraw-Hill: New York, 1969. Wen, C. Y.; Horlo, M.; Krishnan, R.; Khosravi, R.; Rengarajan, P. "Proceedings, 2nd Pacific Chem. Eng. Congress"; Denver: CO, 1977; p 1182. Wen, C. Y.: Deoie, N. R.; Chen. L. H. Powder Techno/. 1982, 3 1 , 175. Yang, W. C.; Keairns, D. L. I n "Fluidization"; Davidson, J. F.; Keairns, D. L., Ed.; Cambridge University Press: Cambridge, 1976; p 208. Yang, W. C.; Keairns, D. L. I d . €ng. Chem. Fundam. 1979, 76, 317. Yang, W. C. Ind. Eng. Chem. Fundam. 1981, 2 0 , 297. Yang, W. C.; Kim, S. S.;Rylatt, J. A. Coal Proc. Techno/. 1981, 7 , 98. Yang, W. C.; Revay, D.; Anderson, R. G.; Chelen, E. J.; Keairns. D. L.; Cicero, D. C. "Proceedings, 4th International Conference on Fluidization"; Kashikojima, Japan, 1983; p 1.10.1. Zenz, F. A. Inst. Chem. Eng. Symp. Ser. 1988. 30, 136.
Istituto di Chimica Industriale Marco Filla e Impianti Chimici Leopoldo Massimilla* Istituto di Ricerche sulla Combustione, C.N.R. Universitii di Napoli 80125 Napoli, Italy Received for review F e b r u a r y 28, 1983 Accepted S e p t e m b e r 19, 1983
Asymptotic Analysis of Pore Closure Reactions The asymptotic behavior of reactions in porous particles accompanied by structural changes that lead to pore closure is examined. Exact solutions are developed for semiinfinite pores, or equivalently, for a finite pore with large Thiele modulus. The results are applicable to noncatalytic gas-solid reactions and to catalyst deactivation problems.
Chemical reactions in porous particles leading to the formation of a solid phase are relevant to a broad range of chemical processes. Typical examples include sulfation of limestone (Hartman and Coughlin, 1974; Christman and Edgar, 1980), chlorination of magnesia (Ulrichson and Mahoney, 1980), and many metal oxide reduction reactions (Szekely et al., 1976) as well as catalyst deactivation reactions accompanied by fouling (Androutsopoulos and Mann, 1978). The mathematical description of such reactions typically consists of formulating single pore models for diffusion, reaction, and solid product deposition, and solving the resulting set of nonlinear coupled equations by numerical algorithms. In some applications involving large values of the Thiele modulus the concentration profiles are rather steep so that elaborate semianalytical numerical schemes have to be used (Chrostowski and Georgakis, 1978). In this communication, the asymptotic behavior of reactions accompanied by pore-structural changes leading to pore closure is examined. Closed form solutions are obtained for semiinfinite pores, or equivalently, for a finite pore with a corresponding large Thiele modulus. These results provide considerable insight into the behavior of pore closure reactions in the limit of large diffusional resistances.
Mathematical Formulation and Results We consider a single pore model representing gaseous or liquid phase diffusion of a reacting species A toward a receding or fixed reaction surface where solid product is formed according to the irreversible reaction A(g or 1) + ... C(s) + ... (1) For a reaction following first-order kinetics the model equations under quasi-steady-state conditions can be cast in the following dimensionless form
-
ax
au - = -~G(v)u (3) at subject to initial and boundary conditions u = l ; t=O (4a) u=l; x = o (4b) au/ax = 0; x = 1 (44 u = u&); t = 0 (4d) where uo(x)is the solution of the quasi-steady-state equation at t = 0
0196-4313/84/1023-0132$01.50/00 1984 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 23, No. 1, 1984
133
ax
Here, u and u represent normalized reactant concentration and solid product extent (e.g., pore cross-sectional area), respectively. The functions D(u), G(u) are generally positive, monotonically increasing functions of u reflecting structurally dependent diffusion and reaction rates, and they are subject to the constraints D(1) = 1, G(1) = 1,D(0) = 0
(5)
The parameters (6, k in (2) and (3) represent generalized Thiele and Biot moduli, respectively. For reactions leading to pore closure, u ( x , t ) is a decreasing function of t; thus k > 0.
Closed Form Solutions for Large Thiele Modulus Solutions to eq 2-4 are typically sought by standard numerical techniques. Of particular efficiency are schemes that convert the boundary-value problem to an initial-value problem (DudukoviE, 1976), resulting in considerable savings in numerical computations. For long, narrow porea, fast reactions, or large diffusional resistances, the zero-flux condition (4c) can be replaced by the equivalent condition (6)
u=o; x-m
&(t)f(uo(t)) = -kf(u)G(u)u (13) where the dot indicates a time derivative. The variable u can be eliminated from (12) and (13) to yield an ordinary differential equation for f ( u )
Then, we will show that the system of eq 2 and 3 admits closed form solutions which also provide the asymptotic behavior of a finite length pore at large values of the Thiele modulus. The exact results are obtained via a hodograph transformation on (2) and (3) by interchanging x and u as independent and dependent variables. We obtain
Introducing the variable W = l/f(u)G(u), eq 14 can be integrated. We obtain
which, on further integration, yields
Letting for convenience l(u) represent the integral
we finally get
The corresponding expressions for the concentration distribution u(x,t) are determined implicitly from (lo), (12), and (13)
and subject to B.C. u = 1; u = uo(t)
(74
u=o; u = l
(7b)
where uo(t) is the solution to the solid product growth equation at the pore mouth ( x = 0, u = 1)
and (9)
Direct substitution of the above expressions into eq 7 and 9 verifies that expressions 19 and 20 satisfy the original differential equations and the boundary conditions. Furthermore, since the original eq 2 is linear with respect to the variable u, it is conjectured that expressions 19 and 20 provide the unique solution to the asymptotic behavior a t large values of (6. Thus, in the limit of large (6 the concentration of species A is only a function of the solid product extent. As expected, at small times the concentration profile reduces to the usual expression u = exp[-(6x]
subject to B.C. u = uo(t); x = 0
u-1;
(gal (9b)
x-m
The solution of eq 7-9 is subsequently postulated to be in the form
where the function f(u), as yet unspecified, satisfies the boundary condition f(u)
- m;
u
1
Substitution of (10) in (7) and (9) gives
(21) corresponding to quasi-steady diffusion and reaction for constant diffusivity and reaction constants (see also Chrostowski and Georgakis, 1978). From (19) and (20) the concentration gradient at the pore mouth can be explicitly calculated
Note that at pore closure time t, determined from (8) 1
du
(11)
expression 22 indicates that the concentration gradient at
Ind. Eng. Chem. Fundam. 1984, 23, 134-135
134
-
the pore mouth becomes infinitely large a t pore closure (D(u) un,n > 0) implying that, within the limits of the quasi-steady-state approximation, reaction practically ceases within the pore. The conversion q(t) for a finite-length pore can be appropriately defined as
where X denotes the normalized amount of reactant originally available (A = 1 for catalytic reactions, X = [(lto)(a- l)]/2kto for noncatalytic gas-solid reactions with structural changes). Substitution of (22) into (24) eventually leads to the conversion profile
of validity of the asymptotic results is expected to be independent of k. Nomenclature D = diffusion coefficient function, dimensionless De = effective diffusivity through solid layer, cmz/s D, = molecular diffusivity within pore, cmz/s G = reaction rate function, dimensionless k = K r o / D , = Biot modulus, dimensionles K = reaction rate constant, cm/s L = length of pore, cm ro = initial pore radius, cm t = time, dimensionless u = reactant concentration, dimensionless u = solid product extent, dimensionless x = axial distance, dimensionless Greek Letters a = chemical parameter to account for structural changes,
and the final conversion a t pore closure time
dimensionless = initial porosity, dimensionless 7 = conversion, dimensionless 4 = (KL2/r,Jlp)1/z = Thiele modulus, dimensionless to
Literature Cited The above asymptotic results simulate accurately the conversion performance of a reaction scheme leading to pore closure in a finite pore provided that condition (412) is not violated. Due to the continuous shrinking of the pore radius and the resultant steepening of the concentration profiles, the accuracy of the asymptotic analysis improves with time. Thus, it normally suffices to test the zero-flux condition during the initial stages of the process. A sensitivity analysis carried out by Shankar and Yortsos (1983) for gas-solid reactions has shown that good agreement is observed for values of 4 as low as 3 and k values not exceeding -O(lO). It should be noted that G(u) is also a function of k for gas-solid reactions; thus large values of k lead to significant consumption of the solid reactant beyond x = 1initially, implying a violation of the zero-flux condition. On the other hand, G(u) does not depend on k for most catalyst-deactivation reactions; thus the domain
Androutsopouios, G. P. ; Mann, R. Chem. Eng. Sci. 1878, 33, 673. Christman, P. G.; Edgar, T. F. 73rd AIChE Annual Meeting, Chicago, IL, 1980; Paper No. 10f. Chrostowski, J. W.; Georgakis, C. Proceedings of the Fifth International Symposium on Chemical Reaction Engineering, Houston, TX, 1978. DudukoviE, M. P. AIChE J. 1878, 22(5).945. Hartman, M.; Coughlin, R. W. Ind. Eng. Chem. Process Des. Dev. 1874, 13, 24%. Shankar, K. ; Yortsos, Y. C. Chem. Eng. S d . 1883, 38(8).1159. Szekely, J.; Evans, J. W.; Sohn, H. Y. ”Gas-Solid Reactions”; Academic Press: New York, 1976. Uirichson, D. L.; Mahoney, D. J. Chem. Eng. Sci. 1980, 35, 567.
Departments of Chemical and Yanis C. Yortsos* Petroleum Engineering Krishnamurthy Shankar University of Southern California Los Angeles, California 90089 Received for review November 2, 1982 Revised manuscript received September 12, 1983 Accepted October 12, 1983
CORRESPONDENCE Comments on “ Isothermal Creeping Flow In Rectangular Channels” Sir: Sen (1982) claims to have found a solution in terms of a single Fourier series for steady isothermal creeping
flow in rectangular channels. His solution is said to converge much more rapidly than a double Fourier series obtained earlier by Srinivasan et al. (1979). The analytical solution by Srinivasan is said to be more convenient than a numerical solution of Johnston (1973). This problem has been one of particular interest for screw extrusion of plastics, and a rapid-converging solution equivalent to Sen’s has been used for many years as discussed in Bernhardt (1959) and applied in Edwards et al. (1964). A shape factor F, as given by Squires (1958) clearly demonstrates the effect of the drag of the sidewalls upon the one-dimensional pressure-induced flow between two
Table I. Typical Convergence Patterns for @ e q 1(Squires), e q 7a (Sen), b 3a F~ i 12 dF(a) n a = 3 a = ‘13 a=3 CY = ’ I 3 1
2 3 4 5
0.21519 0.19920 0.19786 0.19761 0.19754
0.19772 0.19751 0.19749 0.19749 0.19749
0.19772 0.19751 0.19749 0.19749 0.19749
0.21519 0.19920 0.19786 0.19761 0.19754
horizontal parallel plates of infinite width or zero aspect ratio, a = b / a . Equations 1 and 2 from Squires are rewritten as Q = (b3a/12p)(-~/dx)~p (1) 0 1984 American Chemical Society
