Ind. Eng. Chem. Res. 1987,26, 972-976
972
p = static pressure, Pa A p = bed pressure drop, Pa r = regression coefficient ReL = Reynolds number, d & / M L u = superficial gas velocity, m/s u’ = interstitial gas velocity, m/s uF = flooding velocity, m/s umF= minimum fluidizing velocity, m/s U,Fd = minimum velocity for developed fluidization, m/s V = volume of the bed, m3 VG= volume of gas in the bed, m3 VL = volume of liquid in the bed, m3
Vp = volume of particles in the bed, m3 z = axial position in the bed, m Greek Symbols t = bed voidage, t = tG q,, m3/m3 to = initial bed voidage, m3/m3
+
EG
= gas holdup related to the operating bed volume, m3/m3
q- = liquid holdup related to the operating bed volume, m3/m3
= holdup of particles related to the operating bed volume liquid viscosity, k /(m s) pG = gas density, kg/m p~ = liquid density, kg/m3 pp = particle density, kg/m3 4 = free cross-sectional area of the supporting grid, m2/m2 tp
pL =
5
Literature Cited Akselrod, L. S.; Yakovenko, M. M. Theor. Found. Chem. Eng. (Engl. Transl.) 1969,3,124. Balabekov. 0. S.: Romankov. P. G.: Tarat. E. Ya.: Mikhalev.’ M. F. J . Appl.‘ Chem. USSR (Engl. Transl.) 1969a,42, 1454. Balabekov, 0. S.; Tarat, E. Ya.; Romankov, P. G.; Mikhalev, M. F. J . Appl. Chem. USSR (Engl. Transl. 1969b,42,2128. Balabekov, 0. S.; Tarat, E. Ya.; Romankov, P. G.; Mikhalev, M. F. J. Appl. Chem. USSR (Engl. Transl. 1971,44,1068. Blyakher, I. G.; Zhivaikin, L. Ya.; Yurovskaya, N. A. Khim. Neft. Mashinostr. 1967,2, 18. Burkat, V. S.; Tarat, E. Ya.; Baevskii, V. A. Tsuetn. Met. (Moscow) 1977,9,30. Chen, B. H.; Douglas, W. J. M. Can. J . Chem. Eng. 1968,46,245. Epstein, M., First Progress Report, Sept 1975; EPA, Washington, DC. Gelperin, N. I.; Kruglykov, B. S. Zh. Vses. Khim. 0-ua. im. D. I. Mendeleeoa 1979,24(1),59.
Gelperin, N. I.; Liferenko, V. A,; Grishko, V. Z.; Sokolov, V. I. Prom. Sanit. Ochistka Gazov. 1976,3,14. Handl, R. Ph.D. Dissertation, Technical University Clausthal, F. R.G., 1976. Ivaniukov, D. V.; Kupriyanov, V. N.; Kan, S. V.; Yatskov, A. D.; Konovalov, V. I., private communication, 1968. Kito, M.; Kayama, Y.; Sakai, T.; Sugiyama, S. Int. Chem. Eng. 1976, 16(4), 710. Koch, R.;Kubisa, R. Monogr. Wroclaw Techn. Uniu. 1973,19, 1. Kupriyanov, V. N.; Kan, S. V.; Yatskov, A. D., private communication, 1969. Kuroda. M.: Tabei. K. Int. Chem. EnP. 1981.21(2)219. Levsh, I: P.;Krainev, N. I.; Niyazov, M.-I. Int. &em: Eng. 1968,8(2), 311. Levsh,I. P.; Niyazov, M. I.; Krainev, N. I. Tr. Tashk. Politekh. Inst. 1970,64, 136. Muroyama, K.; Fan, L. S. AIChE J . 1985,31(1), 1. O’Neill, B. K.; Nicklin, D. J.; Morgan, N. J.; Leung, L. S. Can. J . Chem. Eng. 1972,50,595. O’Neill, B. K.; Nicklin, D. J.; Leung, L. S. Presented at the International Conference on Fluidization and Its Application, Toulouse, 1973. Strumilo, C.; Adamiec, J.; Kudra, T. Presented at the 2nd International Conference on Control of Gaseous Sulfur-Nitrogen Compounds, University of Salford, U.K., 1976. Tarat, E. Ya.; Burkat, V. S.; Dudorova, V. S. J. Appl. Chem. USSR (Engl. Transl.) 1974,47,105. Tichy, J.; Wong, A.; Douglas, W. J. M. Can. J . Chem. Eng. 1972,50, 215. Tichy, J.; Douglas, W. J. M. Can. J . Chem. Eng. 1972,50, 702. Ushida, S.; Chang, C. S.; Wen, C. Y. Can. J . Chem. Eng. 1977,55, 392. Volak, Z.; Palaty, Z. Presented a t the 6th International Congress CHISA ’78, Prague, Czechoslovakia, 1978. Vunjak-NovakoviE, G. V. Ph.D. Dissertation, Belgrade University, Yugoslavia, 1980. Vunjak-NovakoviE, G . V.; VukoviE, D. V.; Vogelpohl, A.; Obermayer, A. In Fluidization Proceedings o f the Third Engineering Foundation Conference on FZuidization; Henniker, N. H., Grace, J. R., Matsen, J. M., Eds.; Plenum: New York, 1980. Vunjak-NovakoviE, G. V.; VukoviE, D. V.; Littman, H. Ind. Eng. Chem. Res. 1987,preceding paper in this issue. Zhivaikin, L. Ya.; Blyakher, I. G.; Yurovskaya, N. A. Khim. (Leningrad) 1967,14, 21.
Received for review October 8, 1985 Accepted February 11, 1987
Boundary Layer Analysis for the Modeling of Noncatalytic Gas-Solid Reactions Arup
K.C h a k r a b o r t y and Gianni Astarita*
Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
Several distributed models are available in the literature for estimating the rates of noncatalytic gas-solid reactions. These models are all based on the consideration of the same basic phenomena, but different simplifying assumptions are made in each; the most common simplification is the quasi-steady-state approximation. The general model where no simplifying assumptions are made is discussed in this paper. In its dimensionless formulation, it contains two parameters. It is shown that as the values of these parameters approach zero or infinity, some of the simplifying assumptions are justified in a well-defined asymptotic sense. Only one of the models available in the literature is an exact asymptotic solution; all the others either include some unjustified simplification or retain terms of the same order of magnitude as terms which are neglected. An excellent discussion of both structured and distributed models for noncatalytic gas-solid reactions is given by Doraiswamy and Sharma (1984). Specific reactions are discussed by Mantri et al. (1976) and Ishida and Wen (1968). A recent review has been given by Ramachandran and Doraiswamy (1982). In this paper, attention is restricted to isothermal distributed models. Four essential
types can be identified: (1) the shrinking core or sharp interface model (SI); (2) the homogeneous or volume reaction model (VR); (3) the two-stage model (TS); and (4) the finite reaction zone model (FR). The VR model is the only one in which no discontinuities are assumed to exist within the system. The SI and TS models assume the existence of one discontinuity, while
0888-5885/87/2626-0972$01.50/0 0 1987 American Chemical Society
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 973 the FR model assumes the existence of two. However, it should be borne in mind that the problem considered is one of diffusion and reaction, and in the absence of any approximations, no discontinuities could exist except in some asymptotic sense. In this paper, a general model is formulated by considering all the elementary phenomena considered in earlier models but without making any simplifying assumptions. The purpose of this work is to provide rigorous criteria for the validity of possible simplifying assumptions, rather than to obtain a solution to the general model equations. The problem under consideration is that of a gas, A, diffusing and reacting in a porous solid, B. The reaction is considered to be irreversible and can be represented as A p B = products (1)
+
Let a and b be the concentrations of A and B at any point within the solid. Let a, be the value of a at the external surface and bo be the initial value of b. Dimensionless conversions are a = (a0 - a ) / a , (2)
B = (bo - b)/bo
(3)
For the flat-plate geometry, let x be the distance from the exposed surface, and let x = L be the position of a no-flux surface (which could be the midplane if both faces of the slab are exposed). The dimensionless distance from the exposed surface, y, is y =x/L (4) The diffusivity of A through the solid will in general depend on both a and b; in particular, the diffusivity through the “ash” may be significantly different from that through the unreacted solid. However, the main issue can be addressed by regarding the diffusivity as constanG extension to the variable diffusivity case is straightforward. While A may diffuse through the solid, B cannot, and hence p may change in time only as a consequence of the reaction. Let r be the local rate of disappearance of A due to the reaction (so that the rate of disappearance of B is pr). An important parameter of the problem is the concentration ratio, R R = bo/Pao (5) In most practical cases, R >> 1, because bo is the concentration of a solid phase and a. is the concentration of a gas. The SI model is based on the assumption that the reaction is instantaneous and thus that at any position and time either a or @ must be unity. Hence, a sharp retreating front, located at y = p ( t ) , is assumed to exist. At y < p the solid is ash (p = l), and the only phenomenon taking place is diffusion of A
L2(aa/at) = D(aza/ey2)
(6)
At y > p, the solid is unreacted (p = 0, a = 1). A mass balance at the moving front and the assumed continuity of a yield y =p D(da/ay) = RL2(dp/dt) (Y = 1 (7) The problem is closed by imposing a(0, t ) = 1 p(0) = 0 (8) The problem as formulated above is exactly equivalent to the mathematical description of the freezing of water discussed by Stefan (1891) almost 100 years ago. The solution is analytical. Equation 7 implies that the moving front reaches the midplane at a time of the order of RL2/D. Notice that this is significantly larger than the diffusion
time L 2 / D , in spite of the reaction being instantaneous. The physical explanation for this is simple. In the absence of the reaction, L 2 / D is the time required to saturate the solid, i.e., to absorb a mass of A which is a, times the volume of the solid; the driving force is a,,. In the presence of an instantaneous reaction, the time required is that needed for enough A to enter the solid to chemically consume all the solid reactant. The mass required is pb, times the volume of the solid, but the driving force is still ao.
The VR model is concerned with the case when the chemical reaction is very slow compared to the diffusion time. Thus, a quasi-steady-state approximation is introduced to justify dropping the da f at term from the differential equation describing the mass balance on A. The problem is closed by assigning a constitutive equation for the rate of reaction a0f at. The TS model was conceptualized by Ausman and Watson (1962) and later developed by Ishida and Wen (1968). The process is regarded as occurring in two stages. In the first stage, there is no ash layer and the quasisteady-state assumption is made. In the second stage, an ash layer of variable thickness p is formed near the exposed surface where p = 1,and hence no reaction occurs. The mass balance for A is the ordinary diffusion equation, subject to a matching boundary condition at y = p. In the core where p < 1, the VR model equation applies. The problem is closed by a mass balance at y = p , which for the case of constant diffusivity reduces to the requirement that daldy be continuous across the front. The resulting discontinuity is thus a mild one. The FR model was formulated by Mantri et al. (1976). At sufficiently short times, it is assumed that an unreacted core exists. A reaction-diffusion zone is assumed to initially grow adjacent to the exposed surface. This is analogous to the first stage of the TS model and is described by the same differential equations, but a is taken to be unity at y = p. At longer times, a layer of ash is assumed to exist near y = 0. This is followed by a reaction zone of finite thickness sandwiched between the ash layer and an inner unreacted core. The reaction zone travels inward, until the unreacted core disappears and there are only two zones left: an ash layer and a diffusion-reaction zone. This last condition is equivalent to the second stage of the TS model. In the intermediate time range when three zones are present, the ash layer is described as in the TS model. For the unreacted core, @ = 0 and a = 1,and in the central zone, the quasi-steady-state approximation is made again. The problem is closed by appropriate matching conditions and by writing a constitutive equation for the rate of reaction. Except for the SI model, which regards the reaction as instantaneous, all other models require the formulation of a constitutive equation for the rate of reaction. This can, in general, be written in the form atpt = P) (9)
w,
The function f has nonnegative values and a zero value only if either a or is unity. k is a pseudo-first-order kinetic constant. The value of k can be chosen so that f ( 0 , 0) = 1; with this, 1f k is recognized as the intrinsic time scale of the chemical reaction. The phenomenon considered reaches final equilibrium when the reaction is completed and gas A has entirely saturated the resulting ash. The reaction cannot be completed in a time less than its own intrinsic time scale, 1f k . However, even if the reaction is instantaneous, completion of the process requires the finite time, RL21D. Diffusion of A through the solid cannot be completed in less than
974 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987
the diffusion time, L 2 / D . Thus, the final equilibration time, t*, is the largest of the three time scales involved
t* = max ( l / k , R L 2 / D , L 2 / D )
(10)
In any practical case, R will be at least of the order of unity, and thus the two time scales of interest are in fact R L 2 / D and l/k; their ratio will be seen to be the key parameter. Based only on the assumption that the system considered is isothermal, a general model for gas-solid noncatalytic reactions can now be formulated. The dimensionless time is T
= kt
(11)
so that ap/a7 = 1unless either a or /3 is very close to unity = f ( a ,P) (12)
w/a7
and the mass balance on A becomes
aa/a7
+~ ( a p / a ~ )
=
(13)
where 4 = k L 2 / D is recognized as being analogous to the Thiele modulus. The boundary conditions are
ab, 0)= Pb,0) = 0
(14)
(15) No quasi-steady-state assumption is made. Should an ash layer develop, this should arise directly from the equations as a region where the solution yields p = 1. Analogously, an unreacted core would manifest itself as a region where the solution yields a = 1. The fact that in such regions practically no reaction occurs would be guaranteed by the fact that the function f takes a value of zero when either a or p is unity. For any assigned form of the kinetic function, f(a, p), the numerical solution of the general model equations should present no difficulty-except in those cases where stiffness may occur because the solution approaches a discontinuous limit. An ordering analysis is now possible. First consider the case where 4 1/R, the left side of eq 13 can be set to zero; i.e., the quasi-steady-state approximation is justified. Let t again represent the thickness of the zone over which a is significantly different from unity ( t may be the entire thickness of the solid, t = 1). Since the order of magnitude of the two terms on the right side of eq 13 must be the same 6a = R+t2
(18) Notice, however, that neither &anor t can be larger than
unity, yet one of them must be = 1. Two possibilities thus arise. If R 4 > 1; i.e., the reaction is fast compared to diffusion. This limit is clearly seen to be a singular one, since the coefficient of the highest order derivative in eq 13 becomes vanishingly small. Singular problems of this type can be analyzed within the framework of boundary layer theory. As the parameter 4 becomes very large, the first term on the right side of eq 13 can be dropped. Since the second term is of the order R , unless either a or is very close to unity, dropping the first term is justified only if R4 >> 1. When the first term is dropped, the following “reduced equations” are obtained aa/aT = ~ ( a p / a =~ Rf(a, ) p) (20)
These are first order, and not all the boundary conditions can be imposed on them. This is the classical situation encountered in boundary layer (BL) theory. An “outer” solution to the reduced equations is first sought, which satisfies as many boundary conditions as possible. A BL is then assumed to exist near the boundary where a condition has been left unimposed, and the behavior within the BL is analyzed. As will be seen, for the present problem the BL eventually detaches itself from the boundary but remains thin afterwards. For the case at hand, the outer solution is trivial
p = l
(21)
a=o
This solution to eq 20 satisfies all the boundary conditions except a(0, 7 ) = 1. The outer solution describes the unreacted core, the existence of which is thus seen to emerge directly from the classical BL procedure. Following the classical procedure, a BL is assumed to exist near y = 0; this BL has thickness t, which may depend on time. By definition, the term which has been dropped in order to obtain the reduced equations is not negligible within the boundary layer. Since = 1, unless either a or p is almost unity, its value is of the order of unity near y = 0 up to T = 1. Since the final equilibration time cannot be less than 7 = 4, unity is only a small fraction of the final equilibration time. Hence, only over such a small fraction is p significantly different from unity near y = 0 or, in other words, no ash layer is formed. Over this short time range, diffusion of A is not yet complete, and hence 6a = 1. The three terms in eq 13 have the following orders of magnitude 1/r
This short time
(T
1/4t2
R
(22)
< 1) region is subdivided in two su-
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 975 bregions when R >> 1. As long as T
