Physical Consistency of Generalized Linear Driving Force Models
for Adsorption in a Particle
Jose
´
A
Ä
lvarezRamı
´
rez,* Guillermo Ferna
´
ndezAnaya,
²
Francisco J. Valde
´
sParada, and J. Alberto OchoaTapia
Departamento de Ingenierı
´
a de Procesos e Hidra
´
ulica, Universidad Auto
´
noma MetropolitanaIztapalapa,
Apartado Postal 55534, Mexico, D.F., P.C. 09340 Mexico
The socalled (firstorder) linear driving force (LDF) model for gas adsorption kinetics is frequently
and successfully used for analysis and design of adsorptive processes because it is simple,
analytical, and physically consistent. Yet, for certain operating conditions, such as cyclic
adsorption and desorption, significant differences between the LDF model and the rigorous
Fickian diffusion (FD) model can be found. In principle, increasing the order of the approximate
LDF model can yield predictions closer to the FD model. As in the classical firstorder LDF
model, generalized LDF must be consistent with the physics of the FD model. This paper provides
a minimal set of properties that generalized LDF models should meet in order to be physically
consistent. This is done by showing that the FD model describes positive real dynamics, which
are closely related to the thermodynamics of the adsorption

diffusion process. In this form, a
generalized LDF model should inherit this property in order to guarantee that the main
thermodynamic characteristics of the adsorption

diffusion dynamics will be retained to some
extent.
1. Introduction
Consider diffusion and adsorption dynamics in a
uniformly porous spherical particle. The mass balance
equation describing the concentration profile
c
(
r
p
,
t
)
inside the particle is
where
r
is the radial particle coordinate,
±
p
is the
porosity in the particle, and
D
p
is the effective pore
diffusivity based on the total area. A linear adsorption
equilibrium on the pore walls with equilibrium constant
K
was assumed. Equivalently, the above equation can
be written in dimensionless form as
where
q
)
(
±
p
+
K
)
c
/
c
* is a dimensionless concentration
per unit volume in the particle,
c
* is a characteristic
concentration, and
Œ
)
r
/
R
p
is the dimensionless radial
variable. A diffusion time constant can be defined as
By introducing the dimensionless time variable
τ
)
t
/
τ
D
,
one obtains the standard formulation referred to as the
pore diffusion model:
The boundary conditions, for all
τ
g
0, are
and the initial condition, for all
Œ
∈
[0,1], is often taken
as
Equations 1

3 will be referred to as the Fickian
diffusion (FD) model. Sometimes, for different values
of the “design parameter”
f
(
τ
), repeated solutions of eqs
1

3 have to be obtained for optimizationbased designs
or simply for simulation purposes. An important ex
ample is the cyclic operation of adsorption systems
where the solution of the diffusion model must be
repeated over many cycles of operation in order to
establish the final cyclic steadystate separation per
formance of the overall process.
1
Although the solution
of the problem (eqs 1
