Ever since the concept of effectiveness factor was first introduced by Sir James Jeans (15) and further developed by Thiele (23), Wheeler (30), and Aris (I), this subject has been studied by a number of investigators, both theoretically and experimentally ( 2 , 3, 6 to 13, 16 to 22, 24 to 29).With the advent of the electronic digital computer, virtually no problem remains unsolved. However, due to the ease and the increasing use of numerical methods, we have often neglected or abandoned our attempt to obtain analytical solutions which generally offer better insight into the problem and solution. This paper describes the analytical solutions of cases heretofore thought t o be impossible. By the proper use of transformation of variables, the effectiveness factors for the zerourder reaction A -+bRoccurring in the pores of cylindrical and spherical catalyst particles, as well as thin disks, can be obtained. The reactant exhaustion phenomenon associated with this zeroorder reaction will also be discussed. It should be noted that in an important piece of previous work (26), the case of a spherical particle has been solved both numerically and asymptotically. In the latter method an approximation was made such that the starting equation became identical to that for thin-disk geometry. In that case, solutions for effectiveness factors for zero-, fist-, and second-order reactions were obtained. Even so, the mathematical procedure employed was such that the difficult task of obtaining concentration distribution itself was bypassed. In our present work the difficulty has been overcome and a complete set of expressions for pertinent quantities i s tabulated and similarities compared. PROBLEM FORMU LATlONIn solving any problem in the physical sciences, two types of equations are usually required. One, the conservation equation, or the equation of change, describes the system, while the other, the constitutive equation, or the equation of state, describes the properties of the material involved. state equation of continuity ( 4 )In our present case, the equation of change i s the steadywhere NA is the total molar flux (diffusional plus bulk flow) of species A, and R A the molar rate of production of A. The negative sign exists since 4 i s being consumed rather than produced.

On p. 2411 the statement 'An important conclusion to be drawn from theory (Miller et al. 1989), which was largely verified, was that the intense lines in the clumps arose from transitions with small kinetic energy releases' contains an incorrect reference. In fact, the prediction that the intense lines in the clumps arose from transitions with small kinetic energy releases was predicted using semi-classical mechanics by E. Pollack and co-workers (Gomez Llorente & Pollack 1988; Berblinger et al. 1989).

Asymptotic analytic expressions for the concentration and temperature profiles as well as effectiveness factor inside porous, thin-disk shaped catalyst particles have been developed for zeroth-order exothermic reaction with reactant-to-product volume change. The individual and combinative effects of volume-change, Thiele and thermal Thiele moduli, and of the dimensionless activation energy are discussed. As expected, the expressions revert to those for the corresponding isothermal and constant-volume cases. Errors arising from the approximation were calculated and found to be quite acceptable. MARSHALL SCOPEThe general objective of the authors' recent works has been to study the influence of volume or mole change on the mass transfer efficiency inside catalyst particles. In this paper, the heat transfer effect is and the system studied is a zeroth-order exothermic reaction which frequently occurs in industrial practice such as the hydrogenation and dehydrogenation of cyc!ic hydrocarbons. characteristic of zeroth-order reactions in exhibiting the reactant-exhaustion phenomenon which makes the interaction of heat and mass transfer highly interesting. HOWever, because of mathematical complications, the simplifying approximation of moderate temperature rise has been made. However, analysis shows that the general applicability of the result is not seriously affected. ~h~ present study is also confined to thin-disk geometry.This reaction was chosen also because of the unique The heat and transfer the film surrounding the catalyst particle is treated as a separate problem and thus not included here. CONCLUSIONS AND SIGNIFICANCEWe have developed analytic solutions for the concentration and temperature profiles as well as for the effectiveness factor for zeroth-order reaction occurring in porous, thin-disk type catalyst particles with reactant-toproduct volume change. Although an approximation was made, the error introduced is small enough to render the results useful in a large number of reaction systems, including such catalytic reactions as hydrogenation and dehydrogenation of cyclic hydrocarbons wherein the strong hydrocarbon adsorption allows the rate to be approximated by zeroth-order behavior.The results show that the effectiveness factor decreases it increases with increasing combination of the thermal Thiele modulus and dimensionless activation energy. However, the temperature profile inside the pellet does depend on the latter two factors separately. The study also confirms that when a reactant is exhausted before reaching the center of the pellet a flat temperature profile is maintained in this no-reaction central layer and the level of this temperature is a function of the thermal Thiele modulus (dimensionless heat generation-to-mass diffusion ratio) but not of the dimensionless activation energy. These dependencies and behaviors can be qualitatively applied to cylindrical and spherical particles as r re1. --with increasing volume-change modulus (expansion) while well.In a recent series of papers (Lih and Lin...