# Improving Accuracy of Coarse Grid Numerical Solution of Solid-Solid Reactions by Taylor Series Expansion of the Reaction Term

Authors: Hassan Hassanzadeh; Mehran Pooladi-Darvish; Jalal Abedi
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

Exothermic solid-solid reactions lead to sharp reaction fronts that cannot be captured by coarse spatial mesh size numerical simulations that are often required for large-scale simulations. We present a coarse-scale formulation with high accuracy by using a Taylor series expansion of the reaction term. Results show that such expansion could adequately maintain the accuracy of fine-scale behavior of a constant pattern reaction front while using a smaller number of numerical grid cells. Results for a one-dimensional solid-solid reacting system reveal reasonable computational time saving. The presented formulation improves our capabilities for conducting fast and accurate numerical simulations of industrial-scale solid-solid reactions.


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## Body

## 1. Introduction

Modeling of reactive flow has diverse applications in engineering
and science. Applications include heavy oil recovery processes, combustion in
porous media, ground water flow, and transport and reaction processes in
biofilms. An important class of reactions is that of solid-solid reactions.
Modeling of solid-solid reactions is of significant interest in various
industrial operations, including oxidation of metallic and nonmetallic mixed
powders, cement industry [1], ferrites manufacturing, solid-state
polymerization [2, 3], ceramic manufacturing [4], catalyst preparation [5], and drug storage (for more details, see a comprehensive review by
Tamhankar and Doraiswamy [6]). Numerous investigations on modeling reactive
flow have been reported over the years that greatly improved the understanding
of such systems, where the emphasis has primarily been on approximate
analytical solutions for special cases, fine grid direct numerical simulation,
and upscaling of reaction-transport from pore scale to continuum scale [7–18].Accurate numerical simulation of solid-solid reactions is a
challenging task due to the multiscale nature of the physical phenomena.
Physical processes involved in solid-solid reactive systems include diffusive
(heat and mass) and reactive processes. Reactions in porous media intrinsically
often take place at the small scale, causing development of subdiffusive-scale
concentration and temperature gradients, while heat and mass diffusive
processes have scales orders of magnitude larger than the reactions.
Large-scale simulation of such coupled processes is computationally expensive
due to limitations in computational resources. Therefore, a formulation that
captures the subdiffusive scale improves our capabilities for conducting
industrial-scale simulation of the involved processes with high accuracy. In
this paper, we provide a coarse-scale formulation to capture the subgrid-scale
phenomena appropriate for large-scale numerical simulations. The paper is
organized as follows. First, the fine-scale mathematical model used in this
study is presented. Next, the coarse-scale model is described. Then,
application of the coarse-scale model is given for a constant pattern reaction
front, followed by summary and conclusions.


## 2. Fine-Scale Conservation Equations

The exothermic solid-solid reaction is assumed to take place in a
one-dimensional semi-infinite domain. The system, which consists of a mixture
of two solid materials, is initially at temperatureT0. The
concentration of one of the solids is in excess such that the reaction can be
considered of first order with respect to the second solid. The initial
concentration of the second solid is C0. At time t=0, the initial
temperature T0 at x=0 is suddenly raised to ignition temperature
high enough to initiate the reaction by local heating. The temperature
dependence of the reaction rate is assumed to follow Arrhenius type behavior,
and all physical properties are assumed to be constant. Mass diffusion is
considered to be negligible, and the reaction front is planar and nonoscillatory.
It is further assumed that the solid reacting mixture acts as an isotropic
homogeneous system and that radiation effects are negligible [11]. The
dimensionless energy and mass balances can be presented by parabolic coupled
partial differential equations given by (2.1)∂θ∂tD=∂2θ∂xD2+CDexp(θβθ+1),∂CD∂tD=−γCDexp(θβθ+1), where the
following dimensionless groups are used [11, 13]: (2)θ=E(T−T*)RT*2,CD=CC0,tD=tt*,t*=ρcpRT*2Ek(−ΔH)C0exp(ERT*),xD=xx*,x*=λt*ρcp¯,γ=ρcp¯RT*2E(−ΔH)C0,β=RT*E, where T is temperature, C is concentration of
reactant, ρ is density, cp is heat
capacity, λ is the average thermal conductivity, k is the pre-exponential factor, E is activation energy, ΔH is the heat of reaction, R is the
universal gas constant, t is time, T* is the scale
temperature, and x is the spatial coordinate. We used a standard scaling
available in the combustion literature to render the equations dimensionless
[13, 19–21]. The scaling variable x* corresponds to an approximate measure of the
heating zone length, and x*/t* is a measure of the reaction front velocity
[13].The initial and boundary conditions are then given by(2.3)θ=θ0,0≤xD≤∞,tD=0,CD=1,0≤xD≤∞,tD=0,θ=0,atxD=0,tD>0,∂θ∂xD=0,x→∞,tD>0,∂CD∂xD=0,x→∞,tD>0.   The behavior of such a reacting convection diffusion system is
primarily determined by the inverse of dimensionless activation energy and
inverse of dimensionless heat of reaction, namely, β and γ,
respectively. Puszynski et al. [11] presented a detailed analysis of the
frontal behavior of such a system. Small β or γ values correspond to a reacting system with
high activation energy or heat of reaction and vice versa. Large values of γ (small values of heat of reaction and/or small
values of activation energy) lead to a degenerated combustion regime, whereas
low values of γ (large values of heat of reaction and/or large
values of activation energy) result in an oscillatory reaction front. Low β values (large activation energy) and
intermediate γ values result in a constant pattern profile
regime. Choosing large grid cells results in significant smearing of the
reaction front, as we will see in the following sections. In the subsequent
part of the paper, we present the model formulation that is able to maintain
the accuracy of fine grid behavior while using coarse grid cell sizes.


## 3. Coarse-Scale Conservation
Equations

Numerical simulation of reactive front propagation with a large
number of grid cells is computationally expensive and therefore we are interested in using a coarse model that
preserves the accuracy of fine grid behavior. The
coarse-scale variable can be defined by(3.1)ψ¯=1v∫vψdv, where v represents the coarse grid.We intend to represent the differential equations (2.1) with
their corresponding coarse grid equivalents (3.2). In these equations, ℜ¯ represents the “equivalent” reaction rate that
would allow close agreement between the (2.1) set and the (3.2) set, correspondingly. The coarse-scale numerical model can be expressed by [18] (3.2)∂θ¯∂tD=∂2θ¯∂xD2+ℜ¯,∂C¯D∂tD=−γℜ¯,  where (3.3)ℜ=CDexp(θβθ+1),ℜ¯=1v∫vCDexp(θβθ+1)dv=CDexp(θβθ+1)¯ are the fine-scale reaction rate and coarse-scale average reaction
rate, respectively. The reaction rate ℜ can be approximated by a Taylor series
expansion around the coarse-scale temperature and concentration. The above
expansion can be represented in terms of deviation from the coarse-scale
variables as given by (3.4)ℜ≈ℜ(C¯D,θ¯)+∂ℜ∂CD|C¯D,θ¯CD′+∂ℜ∂θ|C¯D,θ¯θ′+12!(∂2ℜ∂CD2|C¯D,θ¯(CD′)2+2∂2ℜ∂CD∂θ|C¯D,θ¯CD′θ′+∂2ℜ∂θ2|C¯D,θ¯(θ′)2)+⋯,  where CD′=CD−C¯D and θ′=θ−θ¯ are the concentration and temperature
deviations from the fine scale solution, respectively. The terms involving
derivatives account for the information that is normally lost in a coarse grid
numerical solution; those will be evaluated and accounted for herein. By using (3.1),
the average coarse-scale reaction rate can be expressed by (3.5)ℜ¯≈ℜ(C¯D,θ¯)+12!(∂2ℜ∂CD2|C¯D,θ¯(CD′)2¯+2∂2ℜ∂CD∂θ|C¯D,θ¯CD′θ′¯+∂2ℜ∂θ2|C¯D,θ¯(θ′)2¯)+⋯,(3.6)∂2ℜ∂θ2|C¯D,θ¯=1−2β(1+βθ¯)(βθ¯+1)4C¯Dexp(θ¯βθ¯+1),(3.7)∂2ℜ∂CD2|C¯D,θ¯=0,(3.8)∂2ℜ∂CD∂θ|C¯D,θ¯=exp(θ¯/(βθ¯+1))(βθ¯+1)2, where by definition the terms with first derivatives are dropped in
the averaging process [18]. Equation (3.5) is a coarse-scale representation of
the reaction rate that includes both coarse-scale concentration and temperature
and their deviations from fine scale, namely, CD′ and θ′,
respectively. We do not have an explicit definition of the deviation terms CD′ and θ′ as functions of the coarse-scale concentration
and temperature. Instead, we intend to express the quantities (θ′)2¯ and CD′θ′¯ in a form proportional to ∂θ¯/∂xD and (∂θ¯/∂xD)(∂C¯D/∂xD) such that the final coarse-scale model can be
presented as a function of coarse-scale temperature and concentration only.
Similar approach has been used by Meile and Tuncay [18] for linear
convection-diffusion-reaction problem.
The common practice in a volume averaging method [14–17, 22] is to
solve the closure problem, which is obtained by subtraction of the fine-scale
and the coarse-scale equations. Here, due to the complexity arising from the
nonlinearity of the problem, subtraction does not lead to (an) equation(s) in
terms of the perturbed quantities. However, consistent with the volume
averaging method [14–16], an appropriate approximation for the deviation terms
is linear proportionality of temperature and concentration deviations with
their corresponding gradient. Using this approximation, one may write (3.9)(θ′)2¯=α(∂θ¯∂xD)2,CD′θ′¯=α(∂θ¯∂xD)(∂C¯D∂xD). By substituting (3.9) in (3.5), the coarse-scale equations
(3.2) can be written as (3.10)∂θ¯∂tD=∂2θ¯∂xD2+ℜ(C¯D,θ¯)+α2!(2∂2ℜ∂CD∂θ|C¯D,θ¯(∂θ¯∂xD)(∂C¯D∂xD)+∂2ℜ∂θ2|C¯D,θ¯(∂θ¯∂xD)2),∂C¯D∂tD=−γℜ(C¯D,θ¯)−γ{α2!(2∂2ℜ∂CD∂θ|C¯D,θ¯(∂θ¯∂xD)(∂C¯D∂xD)+∂2ℜ∂θ2|C¯D,θ¯(∂θ¯∂xD)2)}, where α is the proportionality constant or coarse-scale
parameter and is a function of coarse grid size. The coarse-scale formulation
is not closed so far, and the proportionality constant α as a function of coarse grid size needs to be
determined. Numerical experiments reveal that the term containing cross
derivatives in the internal bracket in the right-hand side in (3.10) is small as compared to the second term. Given that the cross
derivatives are small as compared to the second term, it may be ignored.
Therefore, the coarse-scale formulation can be rendered as (3.11)∂θ¯∂tD=∂2θ¯∂xD2+ℜ(C¯D,θ¯)+α2!∂2ℜ∂θ2|C¯D,θ¯(∂θ¯∂xD)2,∂C¯D∂tD=−γℜ(C¯D,θ¯)−γ{+α2!∂2ℜ∂θ2|C¯D,θ¯(∂θ¯∂xD)2}. The coarse grid model represented by (3.11) is of the same
form as (2.1) with the same dimensionless groups, γ and β;
however, the reaction term is replaced by ℜ(C¯D,θ¯)+α(∂2ℜ/∂θ2)C¯D,θ¯(∂θ¯/∂xD)2/2.In order to find the proportionality constantα as a function of coarse spatial mesh size, a
series of numerical experiments needs to be conducted for different spatial
mesh sizes for each specific reacting system (i.e., using fixed γ and β). The functionality (α versus spatial mesh size) for a specific reacting system can be determined by
matching the coarse spatial mesh size solution with the fine scale or reference
solution. In matching process, to estimate the deviations of the coarse-scale
model predictions from the fine-scale reference solution, we define a numerical
error using the following expression as a measure of accuracy: (3.12)ε={[∫(ψ−ψ¯)2dxD][∫ψ2dxD]}1/2, where ψ can be either temperature or concentration.The proportionality constant is obtained by the minimization of the
numerical error given by (3.12). By repeating the matching process for different
number of spatial mesh sizes, the proportionality constant for each spatial
mesh size can be obtained. Once the functionality of the proportionality
constant with respect to coarse spatial mesh size is obtained, the coarse-scale
formulation (3.11) is complete and can be used for subsequent
large-scale simulation of a solid-solid reactive system. In the following, we
use the above procedure for determining the parameter α as a function of coarse spatial mesh size for
some solid-solid reacting systems.


## 4. Application of the Coarse-Scale Formulation

Based on the scaling groups used in nondimensionalizing the
governing equations, the frontal behavior of a solid-solid reaction depends on
two parameters, namely,β (inverse of dimensionless activation energy)
and γ (inverse of dimensionless heat of reaction). The coarse-scale formulation
described previously is applied for two solid-solid reactions. The governing
differential equations are discretized using an explicit-in-time finite difference
approximation. A block-centered scheme is used, where the diffusive flux is
calculated based on grid block center values. Numerical simulations are
conducted to determine the parameter α
appropriate for large-scale numerical
simulation of such reacting systems. The dimensionless constants for these
reacting systems are given in Table 1. The data are taken from examples of
solid-solid reactions given by Puszynski et al. [11]. For each reacting
system, temperature at one end is rapidly increased to an ignition temperature.
Since the temperature is scaled with the adiabatic temperature of the reaction
(Ta=T*), the minimum dimensionless temperature in the
system is equal to the negative of the dimensionless heat of reaction (1/γ).
Figure 1 shows α as a function of numerical spatial mesh size for different
solid-solid reaction systems given in Table 1. In all cases, the conditions are
selected such that a constant pattern profile exists. The condition for
existence of a constant pattern reaction front can be predicted by the
following expression [11]: (4.1)γc=12(θc−θ0), where (4.2)θc=θ0+21−2β. For γ<γc, a reacting system demonstrates constant
pattern reaction front propagation. The parameter γc for reactions
studied is given in Table 1.Table 1
Dimensionless parameters for
solid-solid reacting systems used in this study.
Reactionβγθ0θiγc10.10.2−500.220.06450.14−7.14300.218Figure 1
Coarse-scale
parameterα as a function of grid size for two reactions in Table 1.Figure1 shows the coarse-scale parameter αobtained
for the two reactions given in Table 1. Using this methodology, the coarse-scale
parameter αas a function spatial mesh size can be
obtained for a specific solid-solid reaction system. This parameter then can be
used for large-scale numerical simulation of a reactive system. Figure 1 shows
that for both reactions the coarse parameter α  starts
increasing about a dimensionless grid size of 5 suggesting that x*=0.2Δx.
Figure 2 shows the reaction rate as a function of dimensionless distance.  The dimensionless size of the reaction zone ΔxDf=Δxf/x* is approximated by the region, where the
reaction rate is larger than 0.01 times of the peak reaction rate. Results show
that for both reactions the approximate dimensionless size of the reaction zone
is about 13 suggesting that using fine-scale model one needs to use grid size
of 5/13 of the size of the reaction zone to roughly capture the reaction front.
The presented coarse-scale formulation allows us to use larger grid size while
maintaining accuracy of the solution. Figures 3 and 4 show comparisons of
concentration and temperature profiles with and without use of the presented
coarse-scale formulation. Results show that the coarse-scale formulation could
represent the location of the reaction front accurately with a small number of
grid cells. Figures 3 and 4 also show that by using coarse-scale formulation
one might choose a coarse spatial mesh size two times of the reaction zone
while maintaining the numerical solution accuracy.Figure 2
Reaction rate for the two reactions as a function of dimensionless distance.Concentration and temperature
distributions attD=400
for Reaction 1 given in Table 1 with and without coarse-scale
formulation for three grid sizes of 25, 16.7, and 9.1. (—)
reference solution, () without using
coarse-scale parameter, and () with
using coarse-scale parameter.Concentration and temperature
distributions attD=650
for Reaction 2 given in Table 1 with and without coarse-scale
formulation for three grid sizes of 25, 16.7, and 9.1. (—)
reference solution, () without using
coarse-scale parameter, and () with
using coarse-scale parameter.The calculated numerical errors for the reactions given in Table1
are presented in Figure 5 for concentration and temperature for different grid
sizes. Results show that the presented coarse-scale formulation could
significantly reduce the numerical error. In addition, the ratio of CPU time
for coarse-scale formulation and fine grid reference solutions is presented in
Figure 6. Results show that the CPU time ratio scales with the inverse of grid
size, suggesting a ten-fold reduction in CPU time with a ten-fold increase in
grid size. Currently, we are working toward finding scaling or proportionality
constant for multidimensional problems. CPU time savings is expected to be much
more significant for multidimensional problems.Numerical error in temperature (left) and
concentration (right) distributions for reactions given in Table1 with and
without coarse-scale formulation for different grid sizes.
(a)(b)(c)(d)Figure 6
Ratio of coarse to fine grid (reference)
CPU times versus dimensionless grid size for the two reactions given in Table1.


## 5. Concluding Remarks

Propagation of solid-solid reaction fronts often results
in a thin reaction zone that is difficult to resolve numerically unless a large
number of numerical grid cells are used. Such numerical simulations are
computationally expensive to perform. In this study, a coarse-scale formulation
for numerical modeling of a one-dimensional solid-solid reacting system is
presented. The presented formulation is based on a Taylor
series expansion of the reaction term
and presents the modification of the reaction term in the coarse model that
would allow an accurate solution. A key parameter in this formulation isα or so-called coarse-scale parameter, which is
a function of the coarse-scale spatial mesh size. Using the presented
formulation, this parameter as a function of spatial mesh size can be obtained.
This parameter then can be used for numerically solving a large scale and
computationally intensive reacting system. It is shown that this formulation
could reasonably obtain the accuracy of a fine grid numerical solution. It is
shown that the coarse-scale formulation considerably reduces the numerical
error. In addition, it is revealed that the ratio of CPU times of a coarse-scale
model to that of a fine grid solution scales with the inverse of grid size.
Such inverse proportionality implies a significant reduction in CPU time of a
coarse-scale model compared to that of a fine grid-scale model for large-scale
simulations. Results obtained in this study for a one-dimensional reacting
system are promising in terms of reducing computational time. However, the
presented methodology has a number of limitations that are the subject of our
current research.  First, the coarse-scale
parameter needs to be characterized as a function of the governing
dimensionless numbers. In addition, the applicability of this method needs to be
tested for multidimensional problems, where we expect a major reduction in CPU
time. Presently, we are working toward implementation of the presented
methodology for multidimensional problems and determination of the coarse-scale
parameter as a function of dimensionless numbers for solid-solid reactions. We
anticipate that the computational time saving for multidimensional problems is
more promising. The presented formulation improves our capabilities for
conducting more accurate and faster numerical simulation of industrial-scale
solid-solid reactions.



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*Source: 696253-2009-04-28.xml*