Dynamic Effectiveness Factor for Catalyst Particles
Jose
´
A
Ä
lvarezRamı
´
rez,* J. Alberto OchoaTapia, and Francisco J. Valde
´
sParada
Di
V
isio
´
n de Ciencias Ba
´
sicas e Ingenierı
´
a, Uni
V
ersidad Auto
´
noma Metropolitana

Iztapalapa,
Apartado Postal 55534, Mexico D.F. 09340, Mexico
Recei
V
ed: January 17, 2005; In Final Form: March 31, 2005
The effectiveness factor (EF) is a nondynamic concept that has been demonstrated to be useful for the analysis
and design of reaction

diffusion systems (e.g., catalyst particles). The aim of this paper is to introduce a
dynamic EF factor (DEF) concept that extends the existing nondynamic one. In the first step, the standard EF
is interpreted as a scaling factor that transforms total reaction rates from surface/bulk to catalyst particle
conditions. Through the use of Fourier transform (i.e., frequency domain) to deal with time variations, the
above interpretation is extended to dynamic conditions by defining the DEF as a linear operator transforming
total reaction rate signals from surface/bulk to catalyst particle conditions. It is shown that the classical
nondynamic EF concept is recovered in the steadystate limit of the DEF definition. Interestingly, the DEF
can be easily computed from the nondynamic EF expressions by introducing a complex Thiele modulus.
Results show that for reaction

diffusion processes where the diffusion mechanism is governed by Fick’s
law the magnitude of the DEF decreases with the frequency. This means that the best reaction rate performance
is obtained when the process operates at steadystate (i.e., nondynamic) conditions. However, when a diffusion
model with relaxation time is assumed to hold, resonant peaks at nontrivial frequencies can be displayed.
Physically, this behavior implies that dynamic (e.g., periodic) operation of the reaction

diffusion process
can yield better performance when compared with its nondynamic counterpart.
1. Introduction
In most practical heterogeneous reaction systems involving
catalyzed chemical reactions, catalyst particles are made large
enough to minimize the pressure drop in the packed bed such
that intraparticle resistance to mass transfer becomes significant.
By departure from mass and energy balances, the description
of reaction

diffusion phenomena in a heterogeneous reaction
system commonly involves distributed parameter models (e.g.,
partial differential equations (PDEs)) for both a fluid and catalyst
particles. Despite its quite good accuracy, this type of model is
rarely used for design and/or simulation purposes because of
the difficulty to find a computationally nonexpensive solution.
For instance, excessive computational burden can occur within
an optimization framework where the distributed parameter
model of the catalyst particle should be solved at each
optimization trial. A commonly used approach to avoid the
drawback of solving the catalyst particle model is the use of an
effectiveness factor (EF) concept.
1