Ind. Eng. Chem. Res. 1989,28, 165-173 McHugh, M. A.; Subramaniam, B. Reactions in Supercritical Fluids-A Review. Ind. Eng. Chem. Process. Des. Dev. 1986,25, 1-12. Moore, J. W.; Pearson, R. G. Kinetics and Mechanism; New York, 1981. Paulaitis, M. E.; Penninger, J. M. L.; Gray, R. D., Jr.; Davidson, P. Chemical Engineering a t Supercritical Fluid Conditions; Ann Arbor Science: Ann Arbor, MI, 1983. Quint, J. R.; Wood, R. H. Thermodynamics of a Charged HardSphere Ion in a Compressible Dielectric Fluid. 2. Calculation of the Ion-Solvent Pair Correlation Function, the Excess Solvation, the Dielectric Constant near the Ion, and the Partial Molar Volume of the Ion in a Water-like Fluid above the Critical Point. J . Phys. Chem. 1985,89,380-384.
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Shaposhnikov, Yu. K.; Kosyukova, L. V. Pereabotka Drev., Ref. Inform. 1965,3,6. Sourirajan, S.; Kennedy, G. C. The System H20-NaCl a t Elevated Temperature and Pressures. Am. J . Sci. 1962,266,115-141. Townsend, S. H.; Klein, M. T. Dibenzyl ether as a probe into the supercritical fluid solvent extraction of volatiles from coal with water. Fuel 1985,64, 635-638. Townsend, S. H.; Abraham, M. A.; Huppert, G. L.; Klein, M. T.; Paspek, S. C. Solvent Effects during Reactions in and with Supercritical Fluids. Ind. Eng. Chem. Res. 1988,27,143-149. Vuori, A.; Bredenberg, J. B-son. Holzforschung 1984,3,133. Received for review February 18, 1988 Accepted September 26, 1988
Heterogeneous Model of a Moving Bed Reactor. 2. Parametric Analysis of the Steady-State Structure Pedro E. Arce,t Orlando M. Alfano,t Irma M. B. Trigatti,$and Albert0 E. Cassano*l§ INTEC,l Casilla de Correo No. 91, 3000 S a n t a Fe, Argentina
This work describes a parametric study performed with a heterogeneous model of a countercurrent moving bed reactor which, in the general case, includes heat transfer with the surroundings. An adequate dimensionless form of the mass and energy balances of the reactor model allows us to identify the characteristic numbers related to the behavior of the reactor or the solid pellets. The analysis of the structure of steady states is performed through the representation of the multiplicity surfaces in a three-dimensional space of dimensionless characteristic numbers. In general, a structure of 1-3-1 steady states is obtained for the adiabatic and nonadiabatic cases. However, for the latter it is possible to detect a pathology of 1-3-5-3-1 steady states in a narrow range of the involved parameters. Through this analysis, it is possible to diagnose the probable operating points of the reactor and draw useful criteria for its analysis and design.
I. Introduction A moving bed operation is convenient for a number of processes in chemical and metallurgical industries. Coal combustion and gasification and the direct reduction of iron ores are two important examples at present. Even though counter- and cocurrent operation may be employed, for the case of liquid-liquid (Luss and Amundson, 1967) or gas-solid (Szekely et al., 1976) heterogeneous systems, the countercurrent system seems to be more widely used. For the mathematical modeling of this type of reactor, we need to take into consideration the two phases in countercurrent movement exchanging heat and mass between each other and, eventually, with the environment. The presence of a chemical reaction in one or both phases is also a possibility to be taken into account. These features, indeed, define a physicochemical system whose modeling is highly complex, and therefore, suitable assumptions must be invoked to reduce it to a level of mathematical tractability. The earliest systematic analysis of this type of reactor goes back to a series of papers published by Amundson et al. (Munro and Amundson, 1950; Amundson, 1956a,b; Siegmund et al., 1956). These authors carefully delineated *To whom correspondence concerning this paper should be addressed. Research Assistant from CONICET and U.N.L. t Supporting Research Staff member of CONICET. SMember of CONICET’s Research Staff and Professor a t U.N.L. Instituto de Desmollo Tecnoldgico para la Industria Qdmica. Universidad Nacional del Litoral (U.N.L.) and Consejo Nacional de Investigaciones Cientificas y TBcnicas (CONICET), 3000 Santa Fe, Argentina. 0888-5885/89/2628-0165$01.50/0
the problem under study and presented their solutions in terms of the Fourier transform. In their analysis, both lumped and intragradient solid particles were considered. However, they limited the study only to the linear range of the reaction rate function with respect to the solid temperature. In this way, they obtained analytical solutions which are valid only for certain particular systems and within a restricted temperature range. Afterwards, Schaefer et al. (1974) studied the behavior of a moving bed reactor at steady state, but employing an exclusively thermal model. The authors showed the possible existence of multiple steady states, using a step function to represent the heat generation curve. In the area of coal gasification and/or combustion, we may refer to the paper of Rudolph (1976), who dealt with the problem of a Lurgi gasifier qualitatively. Yoon et al. (1978) simulated the same type of reactor by using a single temperature for both phases and a shrinking core model for the carbon pellet, with and without an ash layer. Later on, Amundson and Arri (1978) and Arri and Amundson (1978) presented the description of a Lurgi gasifier using a flame-front model for the pellet. They considered different temperatures for the gas and the solid phases and a temperature gradient in the ash layer. More recently, Cho and Joseph (1981) extended Yoon et al.’s model (1978) by including different temperatures for each phase and Caram and Fuentes (1982) presented a simplified model of a countercurrent gasifier which allows the temperature profiles in both phases to be obtained analytically. Regarding the literature in the metallurgical area, there are models of moving bed reactors applied to the direct reduction of iron ores. Tsay et al. (1976a,b), using a three-interface concept for the pellet, proposed a reactor model for the direct reduction of hematite with a mixture 0 1989 American Chemical Society
166 Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989 S'
equations for the gas and solid phases:
0
df
S
Go
=-D~WY,E,~,)~€,,~,)
(1)
The following dimensionless energy balances are also required for the description of the solid and fluid temperature profiles: L ________._._......... 2
101 I bl Figure 1. Schematic representation of (a) the moving bed reactor and (b) the shrinking core pellet model.
of hydrogen and carbon monoxide. Later, Yagi and Szekely (1979) studied the effect of nonuniform gas and solid flows on the performance of a moving bed reactor for the case of reduction of iron ores with hydrogen. Kam and Hughes (1981) proposed a simple representation for the reduction of hematite using a one-interface model for the pellet, and finally Hughes and Kam (1982) analyzed the influence of the water gas shift reaction on the reactor behavior. Although the existence of multiple steady states has been clearly demonstrated, the object of most of these papers has been oriented toward the discussion of specific features of a given system. Hence, it is important to consider aspects related to the existence of steady-state multiplicity zones, as Thoma and Vortmeyer (1978) did from the theoretical and experimental points of view. In this way a more general characterization and a deeper understanding of the model behavior may be achieved. By employing a model previously presented (Arce et al., 1982), this paper attempts to characterize the reactor behavior through the analysis of the effects that changes in an appropriate set of dimensionless numbers could cause on its performance. Based on this representation of the reactor, a parametric study of the model is performed and the steady-state multiplicity zones are delimited. In addition, the different types of controlling resistances in the solid pellet are analyzed as well as the modification of the relative contributions of these resistances during the pellet movement inside the reactor. Finally, some useful criteria for an analysis and design of this type of reactor are pointed out, emphasizing the usefulness of the previously defined characteristic dimensionless numbers. 11. Description of t h e Model 11.1. Reactor Model. In this paper, we consider a moving bed reactor of the type illustrated in Figure la. The solid pellets are introduced a t the top of the reactor and fall by gravity to a gas stream injected from the bottom. The main assumptions used in the model formulation are the following (Arce et al., 1982): (1) steady-state operation; (2) the gas and solid phases move in plug flow with negligible axial dispersion; (3) radial concentration and temperature gradients are neglected; (4) the bed is uniform, formed by spherical particles of constant radius; (5) ideal gas behavior for the fluid; (6) constant physical properties; ( 7 ) the radiation transfer between particles is neglected; and (8) the exchange of energy between the reactor and the outside can be modeled assuming that the solid particles are essentially always surrounded by the gas which interacts through the reactor boundaries with the environment. From these assumptions, the mathematical model is represented by the following dimensionless mass balance
do, df
-=
HI@,- 8,)
- H&
- 8).
(3)
The boundary conditions for the model are the following: a t the bottom (f = 0 ) Y(0)= 1
(5)
8,(0) = 0;
(6)
w(1) = 1
(7)
a t the top (f = 1) 8J1) = 8:
(8)
According to the formulation written above, the model can be analyzed in terms of the following dependent variables: the composition and temperature of the gas phase b,S,), and the composition and temperature of the solid phase (w,O,). In order to complete the description of the problem, it is necessary to supply the explicit functional relations for [,(w), Ny,&,8.), and ~ ( [ c , S , ) , which will be presented in the following section. In this way, from the mathematical point of view, the model is represented by a system of four nonlinear, coupled, first-order ordinary differential equations with split boundary conditions. This model becomes, essentially, a two-point boundary value problem in the axial coordinate. 11.2. Pellet Model. A noncatalytic gas-solid reaction of the following type takes place in the heterogeneous moving bed reactor: At) + B$) products (9)
-
To describe the pellet behavior inside the reactor, a model with inaccessible gas to the interior of the reacting solid (Alfano and Arce, 19821, such as the heterogeneous or shrinking core model (Yagi and Kunii, 1955) illustrated in Figure lb, is used. The limiting situation in which the porous layer is not removed from the reacting core has been used. This situation gives rise to a resistance to mass transfer. An additional restriction to mass and energy transfer in the gas film surrounding the particle is also assumed. The description of the pellet model is completed with the following assumptions: (1) irreversible chemical reaction (Wen, 1968), with intrinsic kinetics of first order with respect to the gas reactant; the specific reaction constant is represented by an Arrhenius-type law; (2) pseudo steady state for the mass balance of the gas reactant (Bischoff, 1963); and (3) negligible temperature gradients inside the pellet (Srinivas and Amundson, 1980; Kam and Hughes, 1981). Based on the previous assumptions and expressing the reaction rate according to Cunningham and Calvelo (1970) and Lemcoff et al. (1971), the following dimensionless equation can be obtained:
Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989 167
nb,t,,e,) = tc2 Y exp[Ar(l - 1/0Jl
(10)
From pure geometric considerations and using integral mass balances for the pellet, the following relationship is readily written: 1 - wwo
The solution of the mass balance for the gas reactant in the porous layer, making use of the assumptions stated for the pellet model, yields (Wen, 1968; Ishida and Wen, 1968)
(12)
11.3. Numerical Method. Due to the mathematical complexity, especially because of its nonlinear characteristics, the model must be solved numerically. Generally, the model can be written in the following compact notation:
where
The strategy to obtain the solution consists in assuming a certain value for the solid-phase variables, temperature and concentration, at the bottom of the reactor; i.e., V,(O) = v,"
(15)
In this way, the problem is transformed into an initial value problem, which is numerically integrated from t = 0 to = 1 using a fourth-order, variable-step RungeKutta-Gill method. The value of the solid-phase variables thus obtained at the top of the reactor, V,C(l),is compared to the corresponding boundary condition, Vs(l). If the difference IV,(l) - V,"I
