simulation of fixed bed processes
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Abstract
The numerical simulation of fixed bed processes using the Method of Lines is analyzed from
an efficiency point of view and a discretization scheme is proposed that is easy to implement
and more efficient than conventional schemes. Biasing of the intervals used to discretize the
partial differential equations that describe bulk transport in a fixed bed process is studied.
The amount of biasing that results in the least error is derived. The effect of using unequally
biased intervals is also discussed. The improvement in accuracy that results from using
biased intervals can be used to reduce the number of equations required to get the desired
accuracy . Discretization of the diffusion and adsorption equation using a geometric grid is
studied to determine the improvement in accuracy that can be achieved using a non-linear
grid. The behavior of different adsorption isotherms on the optimal geometric grid is also
discussed.
Two test problems are simulated to validate the discretization scheme developed. The
adsorption of Cadmium on novel organo-silicates developed by Gomez-Salazar et. al.[7] is
used to demonstrate the improvement in efficiency obtained when a non-linear isotherm is
used. Using a geometric grid for discretizing the pellet equations does not result in a sig-
nificant improvement in accuracy over a linear grid because the non-linear isotherm results
in a concentration front moving through the pellet. Biased intervals in the discretization of
the bed equation are used successfully to improve the accuracy of the breakthrough curve
obtained. The adsorption of Toluene on an activated Carbon packed bed present in a room-
air cleaner is simulated assuming a linear isotherm. Discretization of the bed equation using
biased intervals does not result in significant gains in the accuracy of the solution because
the solution profiles in the bed are flat. A geometric grid is used for discretizing the pellet
equation and the solution obtained is more efficient than that obtained with conventional
methods.
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An Efficient Numerical Method for the Simulation of Fixed-bed
Processes
by
Manuj Swaroop
BTech, Indian Institute of Technology Kanpur, 2002
Masters thesis
Submitted in partial fulfillment of the requirements for the degree
of Master of Science in Chemical Engineering in the GraduateSchool of Syracuse University
December 2004
Approved: _____________________Prof. John C. Heydweiller
Date: _____________________
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Copyright 2004 Manuj Swaroop
All rights reserved.
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Contents
List of Figures vii
Acknowledgements ix
1 Introduction 1
1.1 Mathematical model for a fixed bed . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Numerical methods for simulation of fixed bed . . . . . . . . . . . . . . . . 6
1.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 The fixed-bed equation 10
2.1 Biased differencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Integration over biased intervals . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Temporal truncation error . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 Explicit scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.2 Error in trapezoidal integration . . . . . . . . . . . . . . . . . . . . . 20
2.4.3 Unequal biasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Convection-Diffusion-Reaction equation . . . . . . . . . . . . . . . . . . . . 23
2.5.1 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.8 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
v
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CONTENTS vi
3 The pellet equation 35
3.1 Geometric grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Equation at r=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.2 Equation at r=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.1 Linear isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.2 Non Linear isotherm . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Simulation of fixed beds 49
4.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 Cadmium adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Room air cleaner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Bibliography 66
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LIST OF FIGURES viii
3.8 Average absolute error for a non linear isotherm based on amount contained
in the pores, with a linear grid and a geometric grid . . . . . . . . . . . . . 47
3.9 Average absolute error for a non linear isotherm based on surface concentra-
tion of pellet, with a linear grid and a geometric grid . . . . . . . . . . . . . 48
4.1 Breakthrough curve obtained using simple upwind differencing compared
with the benchmark and optimal biasing curves . . . . . . . . . . . . . . . . 57
4.2 Breakthrough curve obtained using central differencing compared with the
benchmark and optimal biasing curves . . . . . . . . . . . . . . . . . . . . . 57
4.3 Breakthrough curve obtained using partial upwind biasing in the spatial part
compared with the benchmark and optimal biasing curves . . . . . . . . . . 58
4.4 Model of the room air cleaner used for the adsorption of toluene on activated
carbon. For the numerical simulation, the bed is modeled as a thick, short
bed as shown on the right side, obtained by unfolding the cylindrical bed. . 59
4.5 Simulation results for Toluene adsorption on activated carbon in a room air
cleaner, at flow rate Q = 7.44 104cm3/s for (a), (b), and (c) and Q =2.36104cm3/s for (d) usingnb = 21 (no. of grid points in the bed), np = 10(no. of grid points in pellets) and m = 1.4 (geometric factor) . . . . . . . . 63
4.6 Concentration profiles in the first pellet in the bed at high flow rates for time
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Acknowledgements
I would like to express my sincere gratitude and respect to my advisor, Dr. John C.
Heydweiller for his valuable guidance, motivation and patience throughout this research
work. It has been a privilege and a very good learning experience for me to work under his
supervision. I would also like to thank Dr. Tavlarides for allowing me to carry out experi-
ments in his lab and Dr. Jianshun Zhang for introducing me to some practical applications
of this research work and providing me with experimental data.
Dr. Thong Q. Dang gave me valuable insights into various numerical methods that are
applicable to this research work. I am grateful to Dr. Ashok S. Sangani for his support,
encouragement and interest in my research and for the fruitful discussions I had with him.
All the Chemical Engineering staffincluding Ms. Dawn Long and Mickey Hunter were
very helpful and supportive during my stay at Syracuse University and I would like to take
this opportunity to thank them.
I would also like to thank my friends and colleagues Nitin Agarwal, Bhushan Hole,
Francisco Nam, Shailesh Ozarkar, Wenhao Chen and Gautam Bisht for their encouragement
and assistance with this thesis.
I am very thankful to my friends, parents and brother for their friendship and love.
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Chapter 1
Introduction
Adsorption phenomena play an important role in many natural, biological and chemicalsystems. Adsorption operations are used extensively in the chemical and petrochemical
industries and are being increasingly used for biotechnology and environmental control
applications. Adsorption maybe used as a separation process or as a step in multiphase
reaction systems. The process of adsorption involves separation of a component of a mixture
from one phase and its accumulation on the surface of another phase, usually solid. The
adsorbing phase is called the adsorbent and the material adsorbed at the surface of that
phase is called the adsorbate.The most common configuration used for adsorption processes is a fixed bed. In a fixed
bed, adsorbent pellets are held together in place to form a bed, through which the incoming
fluid is made to flow. The fluid flows through interstitial spaces present in the bed and
diffuses into the pores of the adsorbent pellets. Accumulation of the adsorbate takes place
mostly on the surface inside the pores of the pellets, which provide a very large surface
area. This configuration is quite easy to accommodate in a variety of designs for different
applications. The particular application of interest in the present work is indoor room aircleaners. They consist of a fixed bed usually made of activated carbon cloth or pellets. A
fan is used to blow air from the outside into the bed. The fixed bed is usually small in length
and has a large cross-sectional area. This kind of structure results in faster adsorption of
gases.
Indoor air cleaners are used to remove volatile organic compounds (VOCs) from the air.
1
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CHAPTER 1. INTRODUCTION 2
These gases are typically emitted from adhesives, upholstery, manufactured wood products,
cleaning supplies and some household appliances. The concentration of such gases in indoor
air is usually extremely low (measured in parts per billion). As a result, accurate isotherms
for their adsorption are not available. The low concentration of the gases also means that
the adsorbent will take a very long time to become saturated with the adsorbate. Thus,
indoor room air cleaners are designed to last for several years.
The lifespan and efficiency of an adsorber is characterized by its breakthrough curve.
This curve is a plot of the concentration of the adsorbate at the outlet of the fixed bed
versus time. The breakthrough curve is used to determine the time for which the adsorber
operated within acceptable limits. A breakthrough curve with a sharp slope implies that the
adsorber was working efficiently and has a longer lifetime. On the other hand, if the pellets
are unable to adsorb efficiently, the breakthrough occurs earlier and the slope is less steep,
so that the bed does not saturate optimally. The long life span of these adsorbers poses a
problem in proper testing and design of the fixed beds used in the air cleaner. Experiments
with the same concentrations that are present in indoor air would take too long to complete
to be of any practical use. On the other hand, if higher concentrations of gases are used, the
physics of adsorption is expected to be different and so the results may not be extrapolated
to low concentrations.
Simulation of fixed bed adsorbers for indoor air quality can be used to calculate the
values of the adsorption parameters for VOCs at very low concentrations, and that infor-
mation can be used to predict the breakthrough curve of an adsorber. However, the nature
of the adsorption isotherm at such low concentrations, combined with the variations in the
kind of flow that takes place in different regions of the bed makes it a difficult task to make
an accurate prediction of the breakthrough curve.
1.1 Mathematical model for a fixed bed
The simplified mathematical model used to simulate a fixed bed can be described as follows.
The flow through the bed is mostly convective. Hence, a one dimensional plug flow approx-
imation in the direction of the flow can be used to represent the flow in the bed. A mass
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CHAPTER 1. INTRODUCTION 3
transfer coefficient coupled with the concentration difference between the concentration in
the bulk fluid and the surface of an average pellet in the same region is used to describe
mass transfer to the pellets. The equation for the bed flow can be written in the following
form[21]:
uscbz
+cbt
+b3kfRp
(cb c|r=R) = 0 (1.1)
cb =
0 z 0 t 0cbin z= 0 t >0
where
z = axis along the direction of flow
cb = concentration in bed along z
us = superficial bed velocity
= bed packing fraction
b = density of the bed
p = density of pellets
R = radius of pellets
kf = bulk mass transfer coefficient
This partial differential equation contains a convection term, along with a source term to
represent mass transfer from the bulk to the surface of a pellet. The source term couples
the bed equation with the pellet equations.
To solve the fixed bed equation, the bed is discretized into equal intervals. The pellets
contained in each interval are represented by a single averaged pellet in that interval (see
Figure 1.1). The total amount transferred to the pellets in a given region can be determined
using a mass balance between the bulk and the pellets. This balance is incorporated in the
source term in the bed equation.
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CHAPTER 1. INTRODUCTION 4
Figure 1.1: Model of a fixed bed. The axial coordinate is denoted by x. The bed isdiscretized into equal intervals of size x. The concentration profile inside the pellets ineach interval is represented by a single averaged pellet.
The adsorbate that is transferred to the surface of a pellet diffuses into its pores and
then adsorbs on the surface inside the pores. The pellets are modeled as spherical particles
with identical cylindrical pores. The flow inside the pores is mostly due to diffusion. The
flow and adsorption equation for the pellets can be written as follows[21]:
p+p
q
c
c
t = De
1
r2
r
r2
c
r
+ Dsp
1
r2
r
r2
q
r
(1.2)
c= 0 for t 0; 0 r Rc
r = 0 at r= 0
Dpc
r =kf(cb c) at r= R
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CHAPTER 1. INTRODUCTION 5
where
c = concentration inside pores of pellet
r = radial axis of pellet
p = porosity of pellet
p = density of pellet
De = effective pore diffusion coefficient
Ds = surface diffusion coefficient
q = adsorption isotherm (function of c)
The above partial differential equation is a parabolic equation in spherical coordinates and
it represents diffusion inside the pores and on the surface along with the adsorption. It is
linked to the bed equation through the boundary condition at r = R.
The averaged pellets in each interval of the bed are expected to have different concen-
tration profiles because the bulk concentration in the bed is a function of the distance along
the bed axis. The solution of the pellet equation also depends on the rate of change of
the bulk concentration in its section. Hence the pellet equation has to be solved for each
section of the bed. Thus, there is 1 PDE for the bed andnb PDEs for the pellets, where nb
is the number of intervals used in the bed, that have to be solved for a complete fixed-bed
simulation. This approach results in a nested structure of equations.
It should be noted that there are significant differences in the physics and numerical
characteristics of the bed equation and the pellet equation. They can be summarized as
follows:
The bed equation is a convective equation with a linear source term whereas the pelletequation is a diffusion equation. Typically, the spatial discretization scheme and the
time integration schemes used for these two types of equations are different.
If the isotherm to be used is non-linear, then the pellet equation becomes non-linear
and has to be solved numerically for each section with appropriate approximations.
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CHAPTER 1. INTRODUCTION 6
The two equations are in different spatial domains. As a result, different spatial grid
spacing would be needed for each kind of equation and that would affect the size of
the time step that can be used to solve them simultaneously.
If the two equations are non-dimensionalized, it is noted that the time scale that
emerges for each is quite different. So, the solution would have to proceed at the
smaller of the two time scales and might slow down the solution procedure significantly.
A unified approach to the simulation procedure for such problems, which can solve the
equations resulting from different models simultaneously is desirable and is presented in
this study.
1.2 Numerical methods for simulation of fixed bed
Numerous studies have been published which have dealt with the problem of solving an
advective equation with high accuracy and stability[6]. Similarly, there are many good
techniques for the solution of the diffusion equation. However, not all of them meet the
requirements of an efficient numerical method suitable for simulation of fixed bed prob-
lems. High order discretization schemes are usually computationally intensive. Second
order upwind schemes lead to problems in evaluating the boundary condition. The numeri-
cal approach must also be capable of solving both the advective equation and the nonlinear
diffusion equation with similar accuracy. Hence specialized methods for either kind of equa-
tion may not be suitable and may also be difficult to implement. A suitable numerical
method would have to be able to handle different grid discretizations, be able to solve
non-linear equations, be computationally efficient and easy to implement.
A concise review of the methods that can be used to solve equations arising in fixed
beds is given by Le Lann et. al.[11]. They presented the following classification of numerical
methods used to solve partial differential equations:
Method of Lines (MOL)
Finite difference Methods
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CHAPTER 1. INTRODUCTION 7
Weighted Residuals Methods
Finite Element Methods
Finite Volume Methods
Adaptive Grid Methods
Moving grid Methods
A more specific review of numerical methods suitable for adsorption models was presented
by Costa and Rodrigues[5]. Sun and Meunier[19] developed an improved finite difference
method for fixed bed sorption problems using a higher order implicit scheme. Solution meth-
ods based on the method of characteristics were presented by Loureiro and Rodrigues[12].Weighted residual methods may be used to solve the complete set of equations but they
are generally not suitable for problems that have sharp moving fronts. Sharp fronts or
steep gradients can arise in the bed equation as well as the pellet equations with a non
linear isotherm. Oscillations and negative values of concentrations are often observed in
such cases when weighted residual methods are used[9]. Adaptive grid and moving grid
techniques[16, 17] can be used to reduce the error and enhance stability at the cost of
computational effi
ciency but they are diffi
cult to implement when both particle and bedequations have to be solved simultaneously.
The Method of Lines (MOL) converts each Partial Differential Equation (PDE) into a
set of Ordinary Differential Equations (ODEs) in time by discretizing the spatial variable.
The collection of ODEs resulting from each PDE can be combined to form a set of ODEs
that represent the complete problem. The resulting of equations can be solved simulta-
neously using an ODE solver. The ODE solver integrates the equations in time using a
specified numerical scheme. The advantage with this method is that well established in-tegration routines may be used for solving large sets of ODEs with good accuracy. These
include algorithms that can solve stiffas well as non stiffsystems of equation and feature
automatic step size adjustment and integration-order selection to maintain a user-specified
error tolerance and to solve the equations with high efficiency. The main drawback is that
non-stiffODE solvers may have problems with estimating and controlling the impact of the
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CHAPTER 1. INTRODUCTION 8
space discretization scheme on the overall numerical scheme. Finite element, finite volume
or finite difference methods may also be incorporated in the MOL for the spatial discretiza-
tion. Only the spatial discretization needs to be done by the user when using the MOL and
as a result the MOL requires less effort on the part of the user as compared to a full finite
element simulation.
The nature of the adsorption problem is such that as the solution proceeds, the time
derivative decreases in magnitude. So a numerical method in which the size of the time
increments can be increased as the solution proceeds is desirable[22]. The MOL can be used
with sophisticated algorithms for controlling the time step, and so it is suitable for such
applications. The method of lines can also be used to simultaneously solve sets of PDEs
which require different kinds of spatial differencing schemes as all of them will result in
ODEs in time, which can be solved together.
The MOL when used with the proper spatial discretization and a reasonably fast and
accurate (depending on the problem) numerical integration scheme turns out to be a very
suitable and convenient method for fixed bed applications. The present work attempts to
formulate a simple and easy to use numerical method based on the MOL, which can be used
to simulate fixed bed adsorbers. The numerical scheme formulation should be more efficient
than conventional methods used in fixed bed simulations. This can be done by reducing
the number of equations required to get the same accuracy as compared to conventional
methods. The numerical scheme should allow solution of advection and diffusion equations
simultaneously. It should also be independent of grid spacing and time step so that it can
be implemented easily in different problems.
1.3 Objective
The goal of the present work is to develop an easy to implement and reasonably fast and
accurate method based on the MOL for the simulation of fixed bed adsorber problems.
This study builds on the ideas presented by Heydweiller and Patel[10] about upwind biased
discretization schemes and attempts to present an in-depth analysis of the problem, leading
to an improved numerical scheme based on biased intervals and the MOL. Other techniques
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CHAPTER 1. INTRODUCTION 9
to make the solution more efficient, like using a non-linear grid for discretization of the
pellet equations have also been investigated. The resulting techniques have been applied to
predict the breakthrough curve of a room air cleaner and also to investigate the adsorption
isotherms of the VOCs adsorbed on it.
The following chapter describes a general scheme for deriving the biasing matrices based
on certain parameters. The non-dimensionalized advection equation is used to demonstrate
the technique throughout the derivation. An error analysis of the resulting set of equations
is done and is used to find the best values for the biasing parameters. This analysis also
shows how different biasing for the temporal and the spatial part can be used to improve
the accuracy further. The amount of biasing would also depend on the particular numerical
scheme used for time integration. The derivation is then generalized to include equations
with diffusion and source terms.
The pellet equations are also solved with the MOL using a non-linear grid in the next
chapter. The effect of different non-linear grids on the accuracy of the solution is investigated
and the grid that results in the least error for the same number of grid points is used. The
optimal geometric grid was determined for various grid sizes for a linear and a non-linear
isotherm[7].
The last chapter deals with applying the numerical schemes developed in this study to
the simulation of two fixed-bed problems. The bed equation and the pellet equations are
solved simultaneously and the solution profiles and the breakthrough curve are presented.
The first problem is the simulation of adsorption of Cadmium on novel organo-ceramic
adsorbents developed by Gomez-Salazar et. al.[7]. The results for this setup were already
available and were used to check the derived numerical scheme for efficiency. Simulation of
adsorption of toluene on activated carbon in a room-air cleaner at very low concentrations
is also performed. The results are then compared with experimental data to examine theimprovements obtained in efficiency.
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Chapter 2
The fixed-bed equation
Diff
erent spatial discretization schemes can be used with the method of lines for solvingdifferent kinds of PDEs. Second-order centered finite differences yield stable and efficient
solutions for parabolic equations. However, using central differencing for hyperbolic equa-
tions results in an unstable solution procedure. Backward differencing can be used with
hyperbolic equations to make the solution procedure stable but the relative inaccuracy of
first order backward differencing results in an inefficient solution as a large number of grid
points are needed to reduce the numerical dissipation that is introduced by this scheme.
Second order backward diff
erencing can also be used but it is more diffi
cult to implementat boundaries. There are several other schemes available for the solution of hyperbolic
equations[6] including explicit, implicit and several semi implicit schemes. However, only a
few of them are suitable for use with the MOL.
A discretization scheme to be used with the MOL should be based on spatial discretiza-
tion only, as the time discretization is handled by the ODE solver. If the same numerical
scheme is used discretize the time domain as well as the spatial domain for different kinds
of equations, then the temporal discretization should yield an accurate and stable solutionfor all the ODEs being solved simultaneously. This renders many conventional numerical
schemes unusable with the MOL if more than one kind of PDE is being solved.
Discretization schemes that have been proposed for solving the hyperbolic equation using
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CHAPTER 2. THE FIXED-BED EQUATION 11
the method of lines will be discussed using the advection equation:
ut = c0ux (2.1)
Carver and Hinds[3] proposed the following formula, which incorporates biased differencing
of the spatial domain while also using a spatially biased time derivative:
1
6
1 +
3
2
(ut)i1+ 4(ut)i+
1 3
2
(ut)i+1
= c02 x[(1 )ui+1+ 2ui (1 +)ui1] (2.2)
The parameter was determined empirically using numerical experiments such that the
total error was minimized. Although the solution of the advective equation with c0 =1was stable and accurate with the parameter set to 0.3, the computation time required was
50% greater than the time required using either backward or upstream biased differences.
Using eq. (2.2) necessitates the solving of a matrix problem at each time step even though
a non-stiff integrator is employed.
Heydweiller and Patel[10] formulated an upstream biased differencing scheme for the
solution of hyperbolic PDEs. This scheme does not involve a mass matrix, i.e., the time
derivative at only one grid point is used in any given ODE that results from the spatial
discretization. Hence, it is computationally more efficient than the scheme proposed by
Carver and Hinds. The biased differencing results in a stable solution for hyperbolic equa-
tions which have smooth solutions. It also permits sets of coupled hyperbolic and parabolic
equations to be solved simultaneously by the method of lines. This kind of coupling is very
common in mathematical models used by chemical engineers. The parameters used in the
difference formula are functions of the grid spacing and can give accuracy between first and
second order. The biased upwind differencing can be written in the form
(ux)i=aui+1+ cui bui1
2x + C
(x)p
P (uxx)i+
(x)2
6 (uxxx) (2.3)
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CHAPTER 2. THE FIXED-BED EQUATION 12
or
(ux)i=ui+1 ui1
2 x + Dcui+1 2ui+ ui1
(x)2 (2.4)
where Dc =C(x)p/P , C = (c0/ |c0|) and p and P are parameters independent of
the grid spacing x. They also showed that single values of the parameters (p = 1.5 and
P= 1.0) could be used for most problems on a normalized spatial domain.
The idea of discretizing the equations over biased intervals on the spatial grid has been
investigated in detail in the present study. The analysis has been used to formulate an
efficient and easy to use method for solving fixed bed problems, based on the improvements
in accuracy that result from using biased intervals.
An integral formulation for discretizing PDEs in a particular domain has been used in the
present work. This method is essentially a simplified form of the finite volume technique.
The solution domain is divided into intervals, usually of equal size. The equation to be
discretized is integrated over the discretization variable in each interval. The resulting set
of equations are independent of the discretization variable. Numerical integration formulas
can be used to approximate terms that cannot be integrated directly. The accuracy of the
resulting equations depends on the method used for numerical integration. This method
can be used to discretize a variety of equations along with their boundary conditions in a
consistent manner. Several finite difference methods run into problems at a boundary if
one or more imaginary point is needed to incorporate the boundary condition. The integral
method does not have this problem as it can be integrated over a partial interval at the
boundary, within the solution domain.
The integral method can be used very effectively for discretizing equations over a biased
interval. In this chapter, the idea of discretizing PDEs over a biased interval to improve the
accuracy has been explored further using the integral method.
2.1 Biased differencing
The upstream-biased difference scheme presented by Heydweiller and Patel (eq. 2.3) can be
derived by integrating the partial differential equation for each grid point over an interval
xshifted upstream (see Figure 2.1) by an amount defined by biasing parameters, a and b.
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CHAPTER 2. THE FIXED-BED EQUATION 13
Figure 2.1: An upwind-biased interval about a grid point, (c0< 0)
The parameter a represents the part of the interval to the right of the point xi that forms
part of the integration domain while the parameter b represents the part of the interval to
the left ofxi that is used. The advection equation (eq. 2.1) has been used to illustrate the
integration method.
The advection equation can be integrated over a biased interval as follows:
xi+(ax/2)xi(bx/2)
u
tdx= c0
xi+(ax/2)xi(bx/2)
u
xdx (2.5)
The parameters a and b are chosen such that a+b = 2, in order to have non-overlapping
intervals of length x each. Using the trapezoidal rule for the integral on the left hand
side (l.h.s.) of the equation and making suitable approximations, the following difference
equation was obtained:
duidt
=c0
aui+1+ (b a)ui bui1
(a + b)x
(2.6)
The right hand side (r.h.s.) of this equations is the same as eq. (2.3). At the boundaries,
integration is done over a partial interval and the same approximations are used as above
so that the order of accuracy is maintained throughout and the boundaries can be handled
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CHAPTER 2. THE FIXED-BED EQUATION 14
conveniently. The integration formula at the left boundary is given by:
x0+(ax/2)x0
u
tdx= c0
x0+(ax/2)x0
u
xdx (2.7)
wherex0 denotes the leftmost grid point in the bed. The formula for the right boundary is
given by: xIxI(bx/2)
u
tdx= c0
xIxI(bx/2)
u
xdx (2.8)
wherexIdenotes the rightmost grid point in the bed.
The numerical integration used to obtain the r.h.s. of eq. (2.6) is of order (x)2 as the
value ofuiat the two ends of the biased interval was obtained using the lever rule. However,
the integration scheme used for evaluating the l.h.s. results in a larger truncation error. Ingeneral, higher order integration techniques can be used to evaluate both sides of eq. (2.5)
to improve the accuracy of the resulting equations. The number of grid points involved in
the resulting equation for each interval depends on the order of accuracy of the integration
scheme used. If a higher order scheme is to be used for integration, then the integration
domain should encompass more grid points, resulting in a larger interval. However, this
approach can run into problems at the boundary. The number of points available for the
last interval would be less than the number of points available for the adjacent interval.This would result in a difference between the order of accuracy of the integration used in
the two intervals and can lead to numerical errors.
The resulting set of equations can be expressed with the help of a l.h.s. and a r.h.s.
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CHAPTER 2. THE FIXED-BED EQUATION 15
matrix as follows:
Ll1 Ll2
Lc1 Lc2 Lc3
Lc1 Lc2 Lc3
Lr1 Lr2
u1/t
u2/t
un1/t
un/t
=c0
Rl1 Rl2
Rc1 Rc2 Rc3
Rc1 Rc2 Rc3
Rr1 Rr2
u1
u2
un1
un
(2.9)
2.2 Integration over biased intervals
The advection equation (eq. 2.1) has been used to demonstrate the integration over biased
interval formulation used in this study. The trapezoidal rule is used to integrate terms
that cannot be integrated algebraically. The lever rule is used to calculate the value of any
expression in between grid points. Both these methods are second order accurate. Hence
the order of accuracy of the resulting equation is second order.
Consider a pointxi in the space domain. The equation is integrated over an interval of
length x around xi. The interval is upwind -biased as shown in Figure 2.1. The biasing
parametersa and b are used to define the amount of biasing. Since the size of the interval
is kept constant over the entire domain (a+ b = 2), only one of them is an independent
parameter (a in the present study). The integration technique used to derive the set of
equations also plays an important role in determining the accuracy of the solution, along
with the amount of biasing.
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CHAPTER 2. THE FIXED-BED EQUATION 16
Integration of the spatial derivative results in the following expression:
c0
xi+(ax/2)xi(bx/2)
u
xdx = c0
u|xi+(ax/2) u|xi(bx/2)
= c0bui1+ (b a)ui+ aui+1a + b (2.10)
The discretization of the spatial part obtained with this method is the same as that presented
by Heydweiller and Patel (eq. 2.6).
The integration of the time derivative is evaluated as the area under the two trapezoids
ABEF and BCDE in Figure 2.1 (assuming that ui/t is used instead ofui).
xi+(ax/2)
xi(bx/2)
u
t
dx = EF
u
t
dx + DE
u
t
dx
=
b2
ui1t
+ (2ab + 4)uit
+ a2ui+1t
x
8 (2.11)
The complete discretized equation can be written as:
b2
ui1t
+ (2ab + 4)uit
+ a2ui+1t
1
8 = c0
bui1+ (b a)ui+ aui+1(a + b)x
(2.12)
The equations obtained using this method of integration are more accurate than eq.
(2.6) but the solution requires the solution of an additional matrix problem at each time
step. In this respect, the set of equations is similar to eq. (2.2). The difference is that
there is no empirical correlation used to derive the biasing parameters. The equations have
been obtained by simply improving the accuracy of the integration. The only independent
parameter is the amount of biasing and an optimum value for this parameter will be de-
termined using an error analysis. The accuracy of the solution can be improved by using
a more accurate scheme of integration, although it will reduce the efficiency of solution if
more than three adjacent grid points are used per solution point.
This method also makes handling of the boundary condition a simple task. The null
boundary condition can be handled by integrating over the partial interval left at the end:
b2
uI1t
+ (ab + 2b)uIt
1
8 = c0
buI1+ (2 a)uI(a + b)x
(2.13)
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CHAPTER 2. THE FIXED-BED EQUATION 17
wherexI is the last grid point in the spatial domain.
2.3 Error analysis
A complete error analysis was done for the spatially discretized equations. The coefficients
of each term were treated as independent parameters so that the results can be used to find
the truncation error in equations derived using different integration schemes by replacing
the coefficients with their corresponding expressions. The error analysis does not take into
account the truncation error arising from discretization of the time derivative. This is
intentional because the numerical integration in time is done by the ODE solver. Further,
a simple explicit or implicit scheme might not be appropriate for solving all the equations
being solved simultaneously. Hence the time discretization is best left to the ODE solver,
which can use sophisticated algorithms to do the integration in time efficiently. However,
an explicit scheme was analyzed in order to provide a comparison.
After integrating the advection equation over a biased interval for use with the method
of lines as shown in the previous section, it can be expressed in the following manner:
k(ut)i1+ (1 (k+ l))(ut)i+ l(ut)i+1= c0x
(pui1 (p + q)ui+ qui+1) (2.14)
The actual value of the biasing parameters is a function of the amount of biasing and the
scheme used for integration. Only schemes using three grid points for a single equation
have been considered in this case. The results from this analysis can therefore be used to
evaluate the truncation error using all the integration schemes used in this study.
Using the Taylor series expansion foru, the r.h.s. can be written as
RHS=
c0
(qp)(ux)i+ (p + q) (x)
2 (uxx)i+ (qp) (x)
2
6 (uxxx)i+ (p + q)
(x)324
(uxxxx)i+
(2.15)
Comparing with eq. (2.1), qp= 1. Substituting in the above expression and replacingui
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CHAPTER 2. THE FIXED-BED EQUATION 18
with u gives
RHS=c0ux+ c0(p + q)(x)
2 uxx+ c0
(x)26
uxxx+ c0(p + q)(x)3
24 uxxxx+ (2.16)
Similarly, using the Taylor series for ut, the l.h.s. can be written as:
LHS = ut+ (l k)(x)utx+ (l+ k) (x)2
2 utxx+ (l k) (x)
3
6 utxxx+ (2.17)
LHS = ut (l k)(x)uxx (l+ k) (x)2
2 uxxx (l k) (x)
3
6 uxxxx+ (2.18)
using relations from eq. (2.59).
The complete equation can now be written as:
ut = c0ux+ (x)
c0uxx
(p + q)
2 (l k)
+(x)2
c0uxxx
1
6 (l+ k)
2
+(x)3
c0uxxxx
(p + q)
24 (l k)
6
+ (2.19)
Substituting the values of the coefficients ofut and u from eq. (2.12), the error terms can
be written as:
ET{O[(x)]} (p + q)2
(l k) = a 12
a 12
= 0
ET{O[(x)2]} 1
6 (l+ k)
2 =
a2 2a +2
3
(2.20)
The first order error term is identically zero if either both sides of the equation are integrated
over the same interval or if both sides are integrated over a centered interval ( l = k and
p + q= 0). The second order term can be reduced to zero by using a root of the quadratic
expression in that term as the value ofa. The resulting value of the biasing parameter a is:
a= 1 13
0.423 (2.21)
This value of the biasing parameter would yield a third order accurate discretization in
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CHAPTER 2. THE FIXED-BED EQUATION 19
space. However, the truncation error derived here does not include the error due to time
discretization. In the actual solution, there would be some error added to the first and
second order error terms depending on the numerical scheme used for time integration. In
principle, if a third order accurate scheme is used for integration in time, the solution would
also be third order accurate.
Caveat: The derivative relations (eq. 2.59) used to obtain this result are applicable only
if the profile ofuis continuous and differentiable at all points. If there is any discontinuity
in the first derivative ofu, none of the derivatives are defined at that point and hence those
relations are not valid there. The solution is expected to be less accurate at and adjacent
to the points where u is non-differentiable but the accuracy should be unaffected at the
remaining grid points.
2.4 Temporal truncation error
The left hand side can be discretized using a first order time discretization scheme as follows:
LHS= k
un+1i1 uni1
t
+ [1 (k+ l)]
un+1i uni
t
+ l
un+1i+1 uni+1
t
(2.22)
where the superscript ndenotes the current time step.
2.4.1 Explicit scheme
The r.h.s for an explicit scheme can be written as:
RHS=c0unx+ c0(p + q)
(x)
2 unxx+ c0
(x)26
unxxx+ c0(p + q)(x)3
24 unxxxx+ (2.23)
Let t = r(x). This substitution enables us to consolidate the truncation error in
the temporal and spatial domains and hence, to obtain the overall truncation error. The
variable r can be considered to be the velocity of the numerical solution. Replacinguni with
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CHAPTER 2. THE FIXED-BED EQUATION 20
u and using the Taylor series expansion of the terms on the left hand side gives:
ut = c0ux+ (x)
c0
(p + q)
2 uxx r
2utt (l k)utx
+(x)2
c01
6 uxxx r2
6uttt (k+ l)
2 utxx (l
k)r
2 uttx
+(x)3
c0(p + q)
24 uxxxx (l k)
6 utxxx r
3
24utttt (k+ l)r
4 uttxx (l k)r
2
6 utttx
+ (2.24)
Substituting the relations given in the appendix for derivatives ofu in eq. (2.24), the
following expression is obtained:
ut = c0ux+ (x)uxx
c0 (p + q)2 c20 r2 c0(l k)
+(x)2uxxx
c0
1
6 c30
r2
6 c0 (k+ l)
2 c20
(l k)r2
+(x)3uxxxx
c0
(p + q)
24 c0 (l k)
6 c40
r3
24 c20
(k+ l)r
4 c30
(l k)r26
+ (2.25)
2.4.2 Error in trapezoidal integration
The expressions for the biasing variables k , l, pand qfor integration of equations using the
trapezoidal rule can be obtained by comparing eq. (2.14) to eq. (2.12).
k = b2
8
l = a2
8
p = b2
q = a
2 (2.26)
Substituting these expressions in eq. (2.25) and simplifying, we get the total error using
the trapezoidal integration on biased intervals.
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CHAPTER 2. THE FIXED-BED EQUATION 21
Explicit scheme:
ET{O[(x)]} 12
c20r (2.27)
ET{O[(x)2]}
1
12c0+
1
4c0a
1
8c0a
2 +1
4c20r
1
6c30r
2 (2.28)
ET{O[(x)3]} c0
24c0a
24 +
c20ra
8 c
20r
8 c
20ra
2
16 c
30r
2a
12 +
c30r2
12 c
40r
3
24 (2.29)
These results clearly show that the spatial discretization is second order accurate. The
time discretization is first order accurate, so the first order error is non-zero and is directly
proportional to the size of the time step relative to the grid size. Hence smaller time steps
will result in better accuracy. However, for a given value ofr, a suitable value of the biasing
parametera can be chosen, such that the second order error is identically zero. This results
in improved accuracy of the overall solution. The accuracy would be better if a higher order
time discretization was used.
2.4.3 Unequal biasing
It should be noted in the preceding error analysis that the first order error term could not
be eliminated because the terms resulting from the l.h.s. and r.h.s. which contained the
biasing parameter, canceled out. If the biasing were diff
erent on the two sides, it wouldprovide an additional degree of freedom that can be used to eliminate the first order error
term. This is the reason for using different biased intervals for the temporal and spatial
parts of the equation.
Using unequal biasing for the l.h.s. and the r.h.s., the biasing variables can be written
as:
k = b2
8
l = a2
8
p = b
2
q = a
2 (2.30)
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CHAPTER 2. THE FIXED-BED EQUATION 22
where a is the biasing factor for the l.h.s. of the equation with b = 2 a and a is thebiasing factor for the r.h.s. of the equation with b = 2 a. Substituting these expressionsin eq. (2.25) and simplifying, the following truncation error are obtained:
ET{O[(x)]} c0a + c0a c20r
2 (2.31)
ET{O[(x)2]} 1
12c0+
1
4c0a 1
8c0a
2 +1
4c20r
1
6c30r
2 (2.32)
ET{O[(x)3]} (c0a
2c0a + c0)24
+c20ra
8 c
20r
8 c
20ra
2
16 c
30r
2a
12 +
c30r2
12 c
40r
3
24
(2.33)
The above expressions have three degrees of freedom, including the time step. This
implies that, in principle, the two biasing parameters and a suitable time step can be
chosen to eliminate the error up to the third order. However, if a limit is imposed on the
time step, for example due to stability considerations, the error can be eliminated up to
the first order and the second order error can be minimized. In the case of a fixed bed
simulation, the time step may also be limited by the time scale of the pellet equations.
The results for unequal intervals are applicable only if the time step is fixed. A similar
analysis can be done for an implicit scheme or any other time discretization scheme and
equations can be formulated for minimizing the truncation error. In most problems involving
fixed beds, a fixed time step is not suitable for solving all the equations simultaneously. As
a result, this approach has not been investigated further in this study. The above analysis
has been presented to illustrate how using different biasing for integration of the two sides
of an equation can be used to improve the accuracy. Even when an ODE solver is used,
there may be first or second order error terms resulting from the time discretization used
by the ODE solver, and slightly different biasing for the integration intervals can be used
to reduce that error. The actual difference in biasing needed depends on the problem and
the ODE solver used and may need to be determined by experimentation.
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CHAPTER 2. THE FIXED-BED EQUATION 23
2.5 Convection-Diffusion-Reaction equation
The technique used above to calculate the biasing parameters can be applied to a typical
convection-diffusion-reaction equation. In an adsorption problem, the reaction term would
be replaced with a mass transfer term. In general, this term can be called the source term.
The non-dimensional equation can be written as:
u
t =c0
u
x+
1
P e
2u
x2+ f(u) (2.34)
The parameters in this equation could have been reduced further by including c0 in the
non-dimensional time but in some practical applications, the time scale used for non-
dimensionalization may belong to another set of equations (like the ones for particles inthe bed) and in that case, the c0 term will remain as it is. The source term actually
used for computation is usually the linearized form of the actual source term, so that
f(u) =constant and f(u) = 0.
Integrating this equation over a biased spatial interval, we get:
xi+(ax/2)xi(bx/2)
u
tdx
=c0
xi
+(a
x/2)
xi(bx/2)
ux
dx + 1P e
x
i
+(a
x/2)
xi(bx/2)
2ux2
dx +
xi
+(a
x/2)
xi(bx/2)f(u)dx (2.35)
The temporal and spatial parts have been integrated over differently biased intervals. The
biasing parameter for the temporal part is a and for the spatial part it is a. Using the
trapezoidal rule for integration, we get:
xi+(ax/2)xi(bx/2)
u
tdx =
b2
ui1t
+ (2ab + 4)uit
+ a2ui+1t
x
8
c0
xi+(a
x/2)
xi(bx/2)
ux
dx = c0
bui1+ (b a)ui+ aui+1a + b
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CHAPTER 2. THE FIXED-BED EQUATION 24
1
P e
xi+(ax/2)xi(bx/2)
2u
x2dx =
1
P e
u
x
xi+(ax/2)
ux
xi(bx/2)
= 1
P e
ui1 2ui+ ui+1
x
xi+(ax/2)xi(bx/2)
f(u)dx =
b
2f(ui1) + (2a
b
+ 4)f(ui) + a
2f(ui+1) x
8 (2.36)
It should be noted that integrating over a biased interval results in the same formula for
the parabolic term as a regular finite difference approximation over a centered interval.
The regular approximation for the diffusion term is second order accurate and so trying to
increase the order of accuracy of the integration using biased intervals does not affect its
discretization. Also note that the coefficients of the discretized source term have the same
form as the coefficients of the temporal part. The only difference is that they are based on
different biasing parameters.
2.5.1 Error analysis
In general, the discretized equation can be written as:
k(ut)i1+ (1 (k+ l))(ut)i+ l(ut)i+1 = c0x
[pui1 (p + q)ui+ qui+1]
+
1
P e(x)2[ui1 2ui+ ui+1]+
kf(ui1) + [1 (k + l)]f(ui) + lf(ui+1)
(2.37)
The error terms resulting from the convective term have already been derived in Section
2.3. The truncation error of the diffusion term is well known. It can be written as:
ETdiff=
1
12P e (
x)2
uxxxx+ O[(
x)4
] (2.38)
The source term can be written using a Taylor series expansion as:
f(ui1) = f(ui) + (u)f(u)
u
i
+(u)2
2
2f(u)
u2
i
+
where u= ui1 ui (2.39)
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CHAPTER 2. THE FIXED-BED EQUATION 25
The source term f(u) has been linearized, hence only the first order error term in u is
considered without any loss of accuracy. A Taylor series expansion (2.55) can be used to
write:
f(ui1) = f(ui) +x(ux)i+(x)
2
2 (uxx)i (x)
3
6 (uxxx)i+
f(ui)
f(ui+1) = f(ui) +
x(ux)i+
(x)22
(uxx)i+(x)3
6 (uxxx)i+
f(ui) (2.40)
An error analysis similar to that done in the previous sections can be done to write:
ut = c0ux+ 1
P euxx+ f(u)
+(
x)c0(p + q)
2 uxx (l k)utx + ux(l k)f(ui)+(x)2
c06
uxxx (k+ l)2
utxx
+ uxxf
(ui)(k + l)
2 +
uxxxx12P e
+(x)3
c0(p + q)
24 uxxxx (l k)
6 utxxx
+ uxxxf
(ui)(l k)
6
+ (2.41)
Using the relations derived from the generalized equation (eq. 2.34) ( given in eq. 2.60),
the above equation can be written as:
ut = c0ux+ 1P e
uxx+ f(u)
+(x)
uxxc0
(p + q)
2 (l k)
+ uxf
(l k) (l k) (l k) uxxxP e
+(x)2
uxxxc0
1
6 (k+ l)
2
+ fuxx
k + l
2 k+ l
2
+
uxxxxP e
1
12k + l
2
+ (2.42)
The error terms in the above equation contain different derivatives ofu. There is no obvious
way to compare the coeffi
cients of these terms, hence each should be minimized separately.
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CHAPTER 2. THE FIXED-BED EQUATION 26
This results in the following equations for improving the accuracy of the solution:
(p + q)
2 (l k) = 0 (2.43)
(l
k)
(l
k) = 0 (2.44)
(l k) = 0 (2.45)1
6 (k+ l)
2 = 0 (2.46)
k + l
2 k+ l
2 = 0 (2.47)
1
12k+ l
2 = 0 (2.48)
Equation (2.43) implies that the biased interval should be the same for the time derivative
term and the convective term. Equation (2.44) implies that the biased interval should
be the same for the time derivative term and the source term. Equation (2.45) means
that the interval of integration should be centered. These three equations are sufficient
to determine the biasing parameters (a = a = 1) and they yield a second order accurate
spatial discretization.
However, if the diffusion (uxx) term is not present or is very small relative to the con-
vective term in the original equation, as is the case in many fixed bed problems, a more
accurate discretization can be achieved. Equation (2.46) can be used to determine the value
of the biasing parameter instead of equation (2.45). This equation gives the same value of
a as eq. (2.21) (a = 0.423). Equation (2.47) would be satisfied automatically as equation
(2.44) has already been satisfied if the interval of integration for the source term is the
same as that for the time derivative term. The last equation would be negligible since the
diffusion term is negligible. This formulation would result in a third order accurate spatial
discretization.
2.6 Conclusion
The results obtained in Section 2.4 and Section 2.5 are subject to the same caveat as the
results in Section 2.3. These results are applicable only if the solution is differentiable at
all points. The biasing parameters derived here may still be used for solutions which have
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CHAPTER 2. THE FIXED-BED EQUATION 27
a discontinuity in the derivative but the solution may show oscillations in the region of
discontinuity. However, this restriction does not preclude the simulation of fixed bed ad-
sorbers using the above method because the concentration profile along the bed is observed
to be smooth for the most part. The profile has a discontinuity at very short times but it
smoothes out quickly.
The analysis done above resulted in a very straightforward approach to solving a fixed
bed equation using the method of lines. If an ODE solver that controls the time step is to be
used then the values of the biasing parameters are predetermined. They do not depend on
the physical parameters or the spatial or temporal grid. The only factor that they depend
on is the nature of the equation being solved. If the equation is convection dominated
then the appropriate biasing parameter is given by eq. (2.21). If the equation is diffusion
dominated then no biasing is necessary and the interval should be centered about a grid
point. The actual discretized equations can be obtained by integrating over the biased or
non-biased interval. On the other hand, if a fixed time step is to be used, which can be
controlled by the user, then the method for deriving the appropriate biasing factors has
been presented. The biasing factor has been determined for an advective equation to be
solved using an explicit scheme as an example.
2.7 Results and discussion
The formula obtained in the previous section was tested on the following advection equation:
u
t = u
x
for 0 x 1
Boundary condition u(0, t) =u0(0); t
0
Initial condition u(x, 0) =u0(x) (2.49)
The exact solution, given byu(x, t) =u0(x t) is shown with a dashed line in all the figuresat t = 0.5. A stiffODE solver was used for the simulation in all cases. The simulation
was done in MATLAB r[13] using the ode15sODE solver. The results obtained using the
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CHAPTER 2. THE FIXED-BED EQUATION 28
biased interval formulation were compared to the results using a simple backward difference.
A smooth sinusoidal initial condition was used for the comparison to avoid any errors that
could be introduced by discontinuities in the derivative. The following initial condition was
used:
u0(x) =
1+cos(+ x0.2
2)
2 0 x 0.20 0.2 x 1
(2.50)
It is evident from Figure 2.2 that a drastic improvement in accuracy can be achieved by
using the proper biasing. A biasing parameter different from the one obtained in Section
2.3 results in larger oscillations either leading or trailing the wave depending on the value of
the parameter. This behavior shows that numerical dissipation can be eliminated to a great
extent by using equally biased intervals . However, improper biasing can lead to instability
similar to that observed with using central differencing. The correct biasing can mitigate
both dissipation and dispersion error to a large extent.
The simulation was also carried out for the following initial conditions:
Smooth front:
u0(x) =
1+cos( x0.2
)
2 0 x 0.20 0.2 x 1
(2.51)
Triangular wave:
u0(x) =
x0.1 0 x 0.1
1 (x0.1)0.1 0.1 x 0.20 0.2 x 1
(2.52)
Step input:
u0(x) =
1 x= 0
0 0< x 1(2.53)
The results of these simulations are shown in Figure 2.3. The smooth front represents
the kind of solution profile that is encountered typically in fixed beds. It is evident from
the figure that there is very little dissipation and dispersion in the solution to the smooth
front. The triangular wave and the step input have at least one discontinuity in their spatial
derivative and as such the solution is not expected to be very accurate. This can be verified
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CHAPTER 2. THE FIXED-BED EQUATION 29
in Figure 2.3. There is some numerical dissipation in the overall solution and some dispersion
near the points of discontinuity. However, the solution obtained using biased intervals is
more accurate than that obtained by using a simple backward or central difference with the
same value ofx.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x
u
deltax=0.01 time=0.5 no mass matrix aRHS=0
(a) Simple backward difference
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x
u
deltax=0.01 time=0.5 aLHS=0.423 aRHS=0.423
(b) Equally biased intervals with a=0.423
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x
u
deltax=0.01 time=0.5 aLHS=0.2 aRHS=0.2
(c) Equally biased intervals with a=0.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x
u
deltax=0.01 time=0.5 aLHS=0.8 aRHS=0.8
(d) Equally biased intervals with a=0.8
Figure 2.2: Advection equation for a smooth function using biased intervals
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CHAPTER 2. THE FIXED-BED EQUATION 30
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x
u
deltax=0.01 time=0.5 aLHS=0.423 aRHS=0.423
(a) Sinusoidal wave
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x
u
deltax=0.01 time=0.9 aLHS=0.423 aRHS=0.423
(b) smooth front
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x
u
deltax=0.01 time=0.5 aLHS=0.423 aRHS=0.423
(c) Triangular wave
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x
u
deltax=0.01 time=0.5 aLHS=0.423 aRHS=0.423
(d) Step input
Figure 2.3: Simulation results with different initial conditions using equally biased intervals
To show that the above formulation gives reasonable solutions for a range of values of
x, (eq. 2.49) was solved for x = 0.04, 0.02, 0.01 and 0.005 using the initial condition
given in (eq. 2.50). The results are shown in Figure 2.4. It can be seen that there is greater
dissipation and dispersion error with larger values ofx, although the solution remains
stable.
The above results show that the biased intervals formulation can be used very effectively
for typical fixed bed equations. It can reduce the error to a great extent in smooth solution
profiles as compared to backward differencing or central differencing. Even for profiles with
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CHAPTER 2. THE FIXED-BED EQUATION 31
discontinuous derivatives, the solution obtained is better. It is be appropriate to mention
here that there are other numerical schemes available for solving the advective equation[6].
Some of them can also capture regions with discontinuous derivatives with little or no
oscillations. However, these schemes are specific to the advection equation and most of
them also use a fixed time step.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x
u
deltax=0.04 time=0.5 aLHS=0.423 aRHS=0.423
(a) dx=0.04
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x
u
deltax=0.02 time=0.5 aLHS=0.423 aRHS=0.423
(b) dx=0.02
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x
u
deltax=0.01 time=0.5 aLHS=0.423 aRHS=0.423
(c) dx=0.01
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
x
u
deltax=0.005 time=0.5 aLHS=0.423 aRHS=0.423
(d) dx=0.005
Figure 2.4: Simulation results with different grid sizes using equally biased intervals
The emphasis in the present study was to develop a consistent and easy to implement
numerical scheme that can be used for solving the advection equation along with a source
term and a diffusion term. It should also be independent of the time step used. The biased
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CHAPTER 2. THE FIXED-BED EQUATION 32
interval scheme developed above provides a straightforward approach to solving this kind
of problems. This scheme is also easy to implement if the advection equation is coupled
with a parabolic equation. The parabolic equation can be discretized over intervals different
from those used for the advection equation without affecting the solution of the advection
equation. There is no optimum time stepping required to get the desired accuracy for a
given discretization, hence the ODE solver can solve both equations simultaneously without
losing any more accuracy.
2.8 APPENDIX
Evaluating the terms in the integration of eq. (2.5).
Terms occurring in the spatial part:
u|xi(bx/2) = bui1+ aui
a + b
u|xi+(ax/2) = bui+ aui+1
a + b
u|xi+(ax/2) u|xi(bx/2) = bui1+ (b a)ui+ aui+1
a + b (2.54)
Terms occurring in the temporal part:
u
t
xi(bx/2)
= b
ui1t + a
uit
a + b
u
t
xi+(ax/2)
= bui
t + aui+1t
a + bF E
u
tdx =
ut
xi(bx/2)
+ uit
2
bx
2
=
b2 ui1
t + (2a + b)buit
4(a + b)
x
ED
u
tdx =
ut
xi+(ax/2)
+ uit
2
ax
2
=
a2
ui+1t + (2b + a)a
uit
4(a + b)
x
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CHAPTER 2. THE FIXED-BED EQUATION 33
Taylor series expansion of variables:
ui1 = ui x(ux)i+(x)2
2 (uxx)i (x)
3
6 (uxxx)i+
(x)424
(uxxxx)i+
ui+1 = ui+
x(ux)i+
(
x)2
2 (uxx)i+
(
x)3
6 (uxxx)i+
(
x)4
24 (uxxxx)i+ (2.55)
un+1i =uni + (t)(ut)
ni +
(t)2
2 (utt)
ni +
(t)3
6 (uttt)
ni +
(t)4
24 (utttt)
ni + (2.56)
un+1i1 = uni1+ (t)(ut)
ni1+
(t)2
2 (utt)
ni1+
(t)3
6 (uttt)
ni1+
(t)4
24 (utttt)
ni1+
= uni (x)(ux)ni +(x)2
2 (uxx)
ni
(x)3
6 (uxxx)
ni +
(x)4
24 (uxxxx)
ni +
+(t) (ut)n
i(x)(u
tx)n
i +
(x)2
2 (u
txx)n
i(x)3
6 (u
txxx)n
i +
+(t)2
2
(utt)
ni (x)(uttx)ni +
(x)2
2 (uttxx)
ni +
+(t)3
6 [(uttt)
ni (x)(utttx)ni + ] +
(t)4
24 [(utttt)
ni + ] + (2.57)
un+1i+1 = uni+1+ (t)(ut)
ni+1+
(t)2
2 (utt)
ni+1+
(t)3
6 (uttt)
ni+1+
(t)4
24 (utttt)
ni+1+
= uni + (x)(ux)ni +
(x)2
2
(uxx)ni +
(x)3
6
(uxxx)ni +
(x)4
24
(uxxxx)ni +
+(t)
(ut)
ni + (x)(utx)
ni +
(x)2
2 (utxx)
ni +
(x)3
6 (utxxx)
ni +
+(t)2
2
(utt)
ni + (x)(uttx)
ni +
(x)2
2 (uttxx)
ni +
+(t)3
6 [(uttt)
ni + (x)(utttx)
ni + ] +
(t)4
24 [(utttt)
ni + ] + (2.58)
The following relations can be derived from the hyperbolic equation (eq. 2.1). They are
used to simplify the error expression for the equation:
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CHAPTER 2. THE FIXED-BED EQUATION 34
ut = c0ux
utt = (c0ux)t = (c0ut)x= c20uxx
utx = (c0ux)x = c0uxx
uttt = (utt)t= (c20uxx)t = c
20(utx)x= c
30uxxx
uxxt = (uxx)t= (utx)x= c0uxxx
uttx = (utt)x= c20uxxx
utxxx = (ut)xxx = (c0ux)xxx = c0uxxxx
utttt = (uttt)t = (c30uxxx)t= (c
30uxxt)x= c
40uxxxx
uttxx = (uttx)x= c20uxxxx
utttx = (uttt)x= c30uxxxx (2.59)
The following set of relations are used to simplify the terms in the convection diffusion
reaction equation (eq. 2.34):
ut = c0ux+ 1
P euxx+ f(u)
utx = c0uxx+ 1
P euxxx+ f
(u)ux
uxxt = c0uxxx+ 1
P euxxxx+ f
(u)uxx
utt = c0uxt+ 1
P euxxt+ f
(u)ut
= f f + 2c0fux+
c0+
2f
P e
uxx+
2c0P e
uxxx+ 1
P e2uxxxx (2.60)
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Chapter 3
The pellet equation
The mass transfer inside pellets in a fixed bed is usually dominated by diff
usion. The solutediffuses from the pellet surface into the pores and gets adsorbed on the surface inside the
pores. The pore structure has been approximated with cylindrical pores arranged radially
in the pellet for the purpose of simulation. The pore diffusion constant was calculated using
the average pore diameter that was determined from experiments. This results in a simple
diffusion equation in spherical coordinates. The pellet is assumed to be symmetric, hence
the equation has spatial dependence only in r. The resulting equation can be written in
non dimensional form as follows:
c
= (c)kp
2c
r2+
2
rc
r
(3.1)
c = 0 for 0; 0 r 1c
r
r=0
= 0
c
r
r=1
=Bi(cbc|r=1)
where (c) represents the adsorption term, which may be a constant (in the case of a
linear isotherm) or a function of c (in the case of a non linear isotherm). A non-linear
term inc/rresulting from the surface diffusion term was not used in this analysis for the
sake of simplicity. The non-linear term disappears in both the fixed-bed problems that are
simulated in the last chapter and hence, the results are not affected by this omission.
35
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CHAPTER 3. THE PELLET EQUATION 36
The non dimensional parameters used in this equations are:
c = c/cb in
r = r/R
= t/
= time scale used for non dimensionalization
Bi = Rkf
Dp
kp = Dp
R2
where
cb in = inlet concentration
R = radius of pellet
kf = bulk mass transfer coefficient
Dp = pore diffusion coefficient
As the adsorbate starts diffusing from the surface of a pellet into an empty pore, the
concentration gradient near the boundary is very steep. In order to capture this part of
the process using the method of lines, a very fine grid is needed near the external bound-
ary. However, towards the center of the pellet, the concentration profile becomes flatter in
accordance with the boundary condition of zero slope at the center. This characteristic of
mass transfer in pellets suggests that a non-linear grid with more points near the external
boundary and fewer towards the center would be able to capture the concentration profile
near the surface with greater accuracy than a linear grid using the same number of grid
points. The non linear grid would be more computationally efficient than a linear grid as the
number of equations required for obtaining the same accuracy as a linear grid is reduced.
Hence, a geometric grid was used for simulating the pellet equations.
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CHAPTER 3. THE PELLET EQUATION 37
Figure 3.1: Two adjacent intervals in a geometric grid
3.1 Geometric grid
A geometric grid is made of grid elements whose size is defined by a geometric progression.
For a pellet, the smallest grid element is near the external boundary (r = 1) while the
largest one is near the center (r = 0). The length of any two adjacent intervals has a
predefined ratiom such that the length of an interval is m times the length of its adjacent
outer interval. Two adjacent grid elements are shown in Figure 3.1. The geometric factor
m that gives the most efficient and accurate solution depends on several factors including
the number of grid points used and the adsorption isotherm. The optimal value ofm to be
used in a simulation can not be derived but rather has to be determined for each problem
by numerical experimentation. However, using a geometric grid does improve the efficiency
of the simulation by reducing the number of equations required to be solved in order to
obtain the same degree of accuracy as that obtained with a linear grid.
The difference approximations for the first and second derivatives at rj are obtained
from the Taylor series expansions ofcj1 and cj+1 aboutcj :
cj = q2cj1 (p2 q2)cj+ p2cj+1
p2q+pq2
cj = 2qcj1 2(p + q)cj+ 2pcj+1
p2q+pq2 (3.2)
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CHAPTER 3. THE PELLET EQUATION 38
where p = mn+1r and q=mnr (see Figure 3.1) and n = 0 for the outermost interval.
The base interval r is defined as:
1
r
=1 m(np1)
1 m (3.3)
where np is the number of grid points in the pellet. Substituting the above expressions in
eq. (3.1), the following discretized pellet equation is obtained:
cjt
=(cj )kp
2q
1 qr
cj1 2(p + q)
1 + pqr
cj+ 2p
1 + pr
cj+1
p2q+pq2
(3.4)
3.1.1 Equation at r=0
The zero-slope boundary condition at the center of the pellet leads to an indeterminate form
for the last term on the r.h.s. of eq. (3.1). This term can be converted to a determinate
form using the LHospital rule as follows:
limr0
2
r
c
r = 2
2c
r2
r=0
(3.5)
Substituting this expression into eq. (3.1), the following equation is obtained at the center
of the pellet:c
=(c)kp
32c
r2
(3.6)
This equation can be discretized using eq. (3.2) to get the spatial derivative as:
cj01 = (p2 q2)cj0+ p2cj0+1
q2
cj0 = 6
q2(cj0+ cj0+1) (3.7)
wherej0 denotes the grid point at the center of the pellet. Substituting in eq. (3.6)
cj0t
=(cj0)kp
6
q2(cj0+ cj0+1)
(3.8)
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CHAPTER 3. THE PELLET EQUATION 39
3.1.2 Equation at r=1
The derivative boundary condition at the surface of the pellet is discretized using an imag-
inary point on the geometric grid outside the pellet. Using eq. (3.2):
cJ+1 = Bi(p2q+pq2)
p2 c +
1 q
2
p2 Bi(p
2q+pq2)
p2
cJ+
q2
p2cJ1
cJ = 2
p2cJ1+
2
p2 2Bi
p
cJ+
2Bi
p c (3.9)
whereJdenotes the grid point at the surface of the pellet. Substituting in eq. (3.1):
cJt
=(cJ)kp
2
p2cJ1
2
p2+
2Bi
p + 2Bi
cJ+ 2Bi
1 +
1
p
c
(3.10)
wherec is the bulk concentration near the surface of the pellet.
3.2 Simulation procedure
The discretized pellet equations do not use biased intervals, hence there is no mass matrix
involved. The spatial discretization is expressed in the form of a tridiagonal matrix. How-
ever, the adsorption term (c) may be non-linear and thus needs to be evaluated at each
time step. A stiff ODE solver is used to integrate the equations in time. The simulation
was done in MATLAB r[13] using the ode15s ODE solver.Initially the pellet is empty, hence the initial condition is:
c(r, 0) = 0; t= 0 (3.11)
The non-dimensional bulk concentration near the surface of the pellet is set to 1. This
leads to a very steep slope of the concentration profile near the boundary. Theoretically, a
very large number of grid points would be needed to capture the slope. A geometric grid
puts more grid points near the boundary and as a result, the accuracy of the solution is
improved. A criterion is needed to decide the number of grid points and the geometric
ratio that is needed to get a reasonably accurate solution. The criterion used here is either
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CHAPTER 3. THE PELLET EQUATION 40
the total amount contained in the pores of the pellet or the concentration on the surface
of the pellet. These quantities are plotted against time. The profile obtained with a large
number of equally spaced points is used as the benchmark to compare solutions from other
discretizations against.
The optimal value of the geometric factor m for a range of grid sizes was calculated
by minimizing the error from the benchmark solution. The results depend on the kind of
isotherm used and are given in the next section. The error based on the amount in the
pores is defined as follows:
Enp(pores) =
timet=0 |(qp)bench (qp)np|
Nt(3.12)
where
time = total simulation time
np = number of grid points
Nt = number of samples in time
qp = amount in pores
The error based on the surface concentration is defined as follows:
Enp(surf) =
timet=0 |(Cs)bench (Cs)np|
Nt(3.13)
The number of samples used to calculate the error was based on the time steps resulting
from the integration of the benchmark solution, for consistency across all grid sizes. Linear
interpolation was used to calculate the values of the error at points for which the actual
solution for a given grid size was not available.
3.3 Results and discussion
The simulation was performed with a linear isotherm and a non linear isotherm. The opti-
mum values of the geometric parameter m and the corresponding error have been reported.
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CHAPTER 3. THE PELLET EQUATION 41
The solution profile inside the pellet and the amount contained in the pores and the surface
concentration versus time is also shown. The results demonstrate that the improvement in
accuracy obtained by using a geometric grid depends heavily on the form of the isotherm.
3.3.1 Linear isotherm
Linear isotherms are particularly relevant to room air cleaners. Air cleaners operate at very
low concentrations and the adsorption isotherms are typically linear at low concentrations.
The following results have been obtained with a linear isotherm, (c) = 1. The resulting
ODEs are also linear. The solution is then controlled by diffusion. The solution profiles
obtained using the optimal geometric factor for a grid size of 10 nodes are shown in Figure
3.2. The surface concentration rises very rapidly at short times and then rises slowly to thebulk concentration. The slope of the concentration profile inside the pellet is very steep at
short times but it becomes flatter as time increases.
The optimal geometric factors for a linear isotherm are shown in Figure 3.3. Both the
criteria, surface concentration and amount in the pores lead to the same optimal geometric
factor. The optimal factor m 1 as n as expected. It can be seen in Figure 3.4and Figure 3.5 that there is more than an order of magnitude improvement in accuracy if a
geometric grid with the optimal geometric factor is used, as compared to a linear grid withthe same number of grid points. The results given in these figures have been fitted with
smooth curves to show the trend.
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CHAPTER 3. THE PELLET EQUATION 42
0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1
0
0.2
0.4
0.6
0.8
1
r
c
npoints=10 geom ratio =1.4 time=0.0005
(a) profile at very short time, t
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CHAPTER 3. THE PELLET EQUATION 43
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
0 5 10 15 20 25 30 35Number of grid points (n)
geometricfactor(m)
based on amount inpellets
based on surfaceconcentration
Figure 3.3: Geometric factors for a linear isotherm based on amount contained in the poresand surface concentration
1.0E-04
1.0E-03
1.0E-02
1.0E-01
0 5 10 15 20 25 30 35
Number of grid points (n)
Average
absolute
error
using optimalgeometric grid
using a linear grid
Figure 3.4: Average absolute error for a linear isotherm based on amount contained in thepores, with a linear grid and a geometric grid
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CHAPTER 3. THE PELLET EQUATION 44
1.0E-04
1.0E-03
1.0E-02
1.0E-01
0 5 10 15 20 25 30 35Number of grid points (n)
Average
absolute
error
using optimalgeometric grid
using a linear grid
Figure 3.5: Average absolute error for a linear isotherm based on surface concentration ofpellet, with a linear grid and a geometric grid
3.3.2 Non Linear isotherm
The non linear isotherm used has been taken from research done by Gomez-Salazar[8] on
the adsorption of Cadmium on a novel organo-ceramic adsorbent. The isotherm is given in
non-dimensional form by:
q = AcBc + h2 + h
2 (3.14)The slope of this isotherm is given by:
q
c =
ABc + h2 + h
2 ABcBc + h2 + h
3 Bc + h2
(3.15)
and
(c) = 1p+1
q
c
(3.16)where A, B and h are derived from experiments. The values given by Gomez-Salazar
et. al.[7] were used in the above equation. The slope of this isotherm is very large at
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CHAPTER 3. THE PELLET EQUATION 45
very low concentrations and falls very rapidly to small values at higher concentrations.
As a result, the resulting equations are highly non-linear. The solution profiles have the
shape of a concentration front moving radially inwards into the pellet. The concentration
profile remains steep as the pellet fills up. The simulation results are shown in Figure
3.6. The profiles shown have been made using a grid of 101 points. The geometric factor
used is m = 1.01. Although m is very close to 1, it results in more even distribution of
irregularities in the solution profile as compared to m = 1. As the number of points is
reduced, the reduction in accuracy can be seen in the form of irregularities in the solution
profile as shown in Figure 3.6(d). These irregularities are a result of the non-linearity of the
isotherm.
The optimal geometric parameters are shown in Figure 3.7. The average error is shown
in Figure 3.8 and Figure 3.9. It can be seen that the geometric grid does not help much
with this isotherm, which is a result of a front moving through the bed. As a result, using a
finer grid near the surface and a coarse grid near the center will not improve the accuracy of
solution for the entire simulation period. However, the moving front is observed to become
less steep towards the center, hence a geometric grid with a small geometric factor will
provide some improvement in accuracy. These irregularities shown in Figure 3.6(d) are
small at short times, when the concentration profile in the pellet is close to the surface and
larger at longer times, when the front in the concentration profile has moved towards the
center. This is a result of using a geometric grid with a larger than optimal geometric factor.
The number of grid points close to the
top related