Solution of Unsteady-State Shrinking-Core Models by Means of Spectral/Fixed-Point Methods: Nonuniform Reactant Distribution and Nonlinear Kinetics Alessandra Adrover* ,†,‡ and Massimiliano Giona †,§ Centro Interuniversitario sui Sistemi Disordinati e sui Frattali nell’Ingegneria Chimica, Universita ´ di Roma “La Sapienza”, Via Eudossiana 18, 00184 Roma, Italy We present a method for solving moving-boundary problems in the presence of nonuniformities and nonlinearities. The method is based on the spectral decomposition of the concentration field of reacting/diffusing species in a suitable system of eigenfunctions and on the solution of the corresponding system of ordinary differential equations for the Fourier coefficients by means of fixed-point methods. The method proposed by Selim and Seagrave is improved upon and extended to solve moving-boundary problems in the presence of position-dependent diffusion coefficients and nonlinearities in kinetic rates. We apply this method to solve isothermal and nonisothermal shrinking-core models for fluid-solid noncatalytic reactions by focusing on the influence of spatial nonuniformities in the solid reactant distribution. 1. Introduction The analysis of noncatalytic fluid-solid reactions can be grouped into three main classes of models in which (1) the solid is initially nonporous and reaction occurs at a moving boundary, (2) the solid is porous and reaction occurs in a distributed way throughout the entire pore space, and (3) the porous pellet is regarded as consisting of elementary initially nonporous particles (grains). For a review of these approaches see Szekely et al. (1976) and Ramachadran and Doraiswamy (1982). In cases 1 and 3, the resulting balance equations involve diffusion with chemical reaction at a moving interface located either at the length scale of the entire solid pellet or at the smaller length scale corresponding to the grains. The first class of models is usually referred to as shrinking-core models and has been extensively considered in many different cases: linear kinetics (Wen, 1968), nonlinear Langmuir-Hinshelwood kinetics (Sohn and Szekely, 1973), and in a modified form by Erk and Dudukovic (1984), nonuniformly distributed solid reactants (Dudukovic, 1984; Sohn and Xia, 1986; 1987; Krishnan and Sotirchos, 1993), noniso- thermal conditions (Ishida and Wen, 1968; Wen, 1970; Ishida et al., 1971). Structural models belonging to class 3, referred to for the sake of simplicity as grain models, have been studied under a variety of different conditions and assumptions aimed at improving the effects of structural changes (sintering, pore-plugging, etc.) during the reac- tion evolution (Ishida and Wen, 1971; Ramachandran and Smith, 1977a,b; Ranade and Harrison, 1979; Bhatia and Perlmutter, 1980; Sotirchos and Yu, 1985; Prasanna et al., 1985; Batarseh et al., 1989). A comparison of different structural models (cylindrical pores with ran- dom intersections, network models, partially sintered spherical models) is carried out by Lindner and Simo- nsson (1981). To some extent, grain models can be regarded as a hierarchical refinement of shrinking-core models in which the shrinking units are subunits of the whole pellet. In the overwhelming majority of these papers, the unifying mathematical simplification is a pseudo-steady- state (PSS) approximation in the evolution of concentra- tion and temperature profiles within the porous product matrix. Bischoff (1963, 1965) has analyzed the accuracy of the PSS approximation by means of perturbation analysis in the case of first-order shrinking-core kinetics with vanishing concentration at the reaction surfaces, reach- ing the conclusion that for those systems where the ratio of the fluid to solid density is rather small (as in the case of gas-solid reactions) the PSS approximation is a reasonable and acceptable assumption, while if this ratio is of the order of magnitude of unity (as in many liquid-solid systems), PSS solutions deviate from the solution of the unsteady-state balance equations. Simi- lar results have been recently obtained by Carey and Murray (1989) by applying a more refined perturbation scheme and more general boundary conditions. The validity of the PSS solution is still more question- able in the case of nonuniform reactant distributions and particularly for nonisothermal reactions. In the nonisothermal case, PSS analysis furnishes a time- independent picture of the reaction evolution, expressed, e.g., by means of effectiveness-conversion plots (Bev- eridge and Goldie, 1968; Ishida and Wen, 1968). These plots furnish a quantitative description of the transi- tions between reaction- and diffusion-controlled regimes and between ignition and extinction phenomena. This description is not, however, sufficient for prediction of the temporal evolution of the reaction in those (exo- thermal) cases for which steady-state multiplicity may occur (Cannon and Denbigh, 1957a,b). Moreover, as noted by Beveridge and Goldie (1968), PSS approxima- tion is not acceptable for high solid heat capacities, and there are many examples (as discussed by Wen and Wang, 1970) in which PSS predictions clash with the corresponding dynamic solutions. Several numerical methods have been proposed for the solution of moving-boundary problems: simple * Corresponding author. Telephone: +39-6-44585892. Fax: +39-6-44585339. E-mail: alex@giona.ing.uniroma1.it. † Centro Interuniversitario sui Sistemi Disordinati e sui Frattali nell’Ingegneria Chimica. ‡ Dipartimento di Ingegneria Chimica, Universite ´ di Roma “La Sapienza”. § Present address: Dipartimento di Ingegneria Chimica, Universita ` di Cagliari, Piazza d’Armi, 09123 Cagliari, Italy. 2452 Ind. Eng. Chem. Res. 1997, 36, 2452-2465 S0888-5885(96)00653-7 CCC: $14.00 © 1997 American Chemical Society