Analytical criteria for validity of pseudohomogeneous models of packed-bed catalytic reactors

Pseudohomogeneous models are used extensively to simulate the behavior of packed-bed catalytic reactors. The use of highly active catalysts increases the importance of the transport resistances between the solid and the fluid, and hence, may invalidate the applicability of the pseudohomogeneous models. There exist criteria in the literature that predict when pseudohomogeneous models can be used to approximate the behavior of catalytic reactors. These criteria are derived by assuming that the predictions of pseudohomogeneous and the two-phase models differ only by a small amount and hence Taylor expansions can be used to quantify the difference between the two models. In this work, we show that the existing literature criteria may lead to erroneous conclusions and present new criteria that may be used to determine the regions of validity of pseudohomogeneous models. For example, when Le(p) = h/(rho f)C(pf)k(c) greater than or equal to 1, we show that, for a single exothermic first-order reaction, the pseudohomogeneous model has the same qualitative features as the two-phase model, if K(T-0)(k(c)a(v)) < exp{4Le(p) - 2 - [E/(RT0)][(-Delta H)c(0)]/[rho fCpfT0)}, where Le(p) is the particle Lewis number, rho(f) is the density of the fluid phase, C-pf is the specific heat at constant pressure of the fluid phase, k(T-0) is the reaction rate constant at the feed temperature T-0, R is the universal gas constant, E is the activation energy, a(v) is the external particle surface area per unit volume of the reactor, lit, is the mass-transfer coefficient, h is the fluid-solid heat-transfer coefficient, (Delta H) is the heat of the reaction, and co is the feed concentration. A similar criterion is presented for 0 < Le(p) < 1. These criteria are very different from those in the literature and are derived from a rigorous analysis of the single-phase and two-phase models and comparison of their qualitative features (bifurcation diagrams). Application of the criterion to typical industrial situations show that they could differ from that of Mears (Mears, D. E. J. Catal. 1971, 20, 127),(1) by a factor of 10-10(4). It is also shown that the two-phase plug flow model has 10 qualitatively different types of bifurcation diagrams. Some of these contain high-temperature branches on which the solid temperature exceeds the adiabatic temperature rise.