A method of representing the nonisothermal effectiveness factor for fixed bed calculations

Abstract

A parametric representation of the effectiveness factor for o first-order, nonisathermal irreversible reoction in a catalyst pellet has geen devised that amply covers the realistic range of parometers for the exothermic case. Given the Thiele modulus, Prater temperature, and dimensionless activation energy, the effectiveness factor can be recovered in about 1/1000 sec. (CDC 6600) to an accuracy of better than 5% over most of the range. When heat and mass transfer resistance a t the surfoce is added the time required is about 0.03 sec. This makes fixed bed calculations more than 50 times faster than would otherwise be possible.When a chemical reaction takes place in a catalyst pellet the interaction of diffusion and reaction is commonly expressed in terms of an effectiveness factor, the ratio of the actual reaction rate to that which would obtain were there no diffusive limitations. This characteristically chemical engineering device appears in the independent, but almost simultaneous, work of Damkohler ( l o ) , Thiele ( 1 8 ) , and Zeldovich (21) in the late 1930's and early 1940's and is one of the few aspects of the early work that the physicist Juttner did not anticipate in 1909 (11). This first work treated the isothermal case for several orders of reaction, and it was not until the early 1960's that nonisothermal effectiveness factors were published again, independently and almost simultaneously, by Weiss and Hicks (19), Amundson and Schilson ( 2 ) , Carberry (6, 7 , 9 ) , Metzner and Tinkler (15). From the nonisothermal studies it became evident that for sufficiently exothermic reactions there could be three steady states over a certain range of the Thiele modulus, a discovery that has produced much interesting theoretical work in the last decade.The effectiveness factor is a functional of the solution of a second-order differential equation with two-point boundary conditions and although with the proper precautions it is a straightforward enough matter to compute, it is sufficiently time-consuming to be out of the question normally to compute it for each particle in a fixed bed calculation. Realizing this, some workers have pointed out that in many cases the particle can be regarded as isothermal with heat transfer resistance concentrated at the surface. This allows the comparatively simple formulae for the isothermal effectiveness factor to be exploited. Liu (12) however has taken a rather different approach and sought to fit the computed results by a simpler empirical expression which can be called up on the computer in :I fraction of the time it would take to solve the differential J. G. Jouven is with Universal Oil Products, Des Plaines, Illinoia. Page 402March, 1972equation. On the basis of this he was able to do fixed bed calculations taking into account the diffusive effects in thc particle. The present paper reports work in which a bestfitting interpolation formula has been used to provide ;I rapid way of calculating the effectiveness over a much \vider range of parameters than Liu was able to tak...