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134 H. Hassanzadeh et al. / Computers and Chemical Engineering 33 (2009) 133143

Nomenclature

cp heat capacity (J kgl Kl)

C concentration (kg/m3)

Co Courant number

D molecular diffusion coefficient (m2/s)

E activation energy (J kmol1)

f convective flux functionH heat of reaction (J/kg)

k pre-exponential rate constant, unit depends on

reaction type

L length of reacting system (m)

Le Lewis number

N total number of grid blocks

Pe Peclet number

Q numerical solution in TVD methods

r ratio of successive gradients

R universal gas constant (8314.472J kmol1 Kl)

t time (s)

T temperature (K)

TVD total variation diminishing

u velocity (ms1)x spatial coordinate (m)

Greek letters

inverse of dimensionless activation energy inverse of dimensionless heat of reaction difference or increment

porosity dimensionless temperature weighting factor in third order method effective thermal conductivity (J m1 s1 K1)

numerical error weighting factor in third order method density (kg m3)

dimensionless time used in Section 2.1 flux limiter function square root of Thiele modulus dimensionless distance used in Section 2.1 variable in numerical error function; can be temper-

ature or concentration

weighting factor in single-point upstream method

Subscripts

D dimensionless

f front

G gas

H heat

i grid block index

inlet inlet condition

L left

M mass

mod modified

num numerical

ref reference

s system

S solid

0 initial value

* scale value

Superscripts

n time index

a cubic autocatalytic reaction is coupled to mass diffusion and

convective transport. Their calculations show that special high

resolution schemes such as ENO (essentially non-oscillatory)

are necessary to track efficiently steep moving fronts exhibited

by strongly convective problems. Alhumaizi (2004, 2007) stud-

ied the accuracy of several finite difference schemes to solve

a one-dimensional convection-diffusion-reaction problem of an

autocatalytic mutating reaction model and foundthat the Superbee

and MUSCL (Monotone Upstream-centered Schemes for Conserva-

tion Laws) flux limiters are the most appropriate for simulating

sharp fronts for convection-diffusion-reaction and convection-

diffusion systems, respectively. Ataie-Ashtiani and Hosseini (2005)

and Ataie-Ashtiani, Lockington, and Volker (1996) developed a cor-

rection for the truncation error associated with a finite difference

solution of convection and diffusion with a first order reaction.

Theycompared numericalresultswith analytical solutionsand sug-

gested that truncation errors are not negligible. It was also shown

that the CrankNicholson method is the most accurate scheme

based on truncation error analysis.

The attention that has been given in the literature to devis-

ing suitable numerical schemes for improving the accuracy of

convection-dominated problems has not been the case for prob-

lems involving convection-diffusion with Arrhenius type kinetics

reaction; such investigations are rare. There are in fact only lim-

itedstudiessuggesting suitable high-resolutionnumericalschemes

for specific applications of convection-diffusion-reaction systems.

Furthermore, there is no numerical scheme yet identified that per-

forms in a superior manner for all problems; therefore, a choice

is usually made based on experience. As far as is known to the

authors, comparative studies of higher-order and flux-limiting

methods for coupled heat and mass transfer in gassolid sys-

tems with Arrhenius type reactions have never been reported

in the literature. The objective of this study is therefore to per-

form a comparative study of flux-limiting methods for numerical

simulation of gassolid reactions with Arrhenius type reaction

kinetics, where the exothermic nature of the reactions coupled

with heat and mass transfer leads to steep temperature and con-centration gradients. Previous comparative studies of flux limiters

did not consider Arrhenius type reactions. Results of this study

will find application in numerical modeling of gassolid reactions

with Arrhenius type reaction kinetics involved in various industrial

operations.

This paper is organized as follows. First, numerical errors in a

first order convection-diffusion-reaction system is described. Next,

the governing partial differential equations for gassolid reactions

are presented. The finite difference discretization of the governing

equations and different numerical schemes for approximating con-

vective flux are then presented. Comparisons of different methods

are described in Section 5 followed by summary and conclusions

section.

2. Numerical errors in convection-diffusion-reaction

systems

2.1. First order convection-diffusion-reaction system

The governing differential equation for a linear convection-

diffusion-reaction system is given by

C

t= D

2C

x2 u

C

x kC (1)

where Cis concentration of reactant, D is molecular diffusion coef-

ficient, t is time, x is the linear spatial coordinate, u is velocity,

and k is the reaction rate constant. The forward time and single-

point upstream finite difference method of discretization lead to


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H. Hassanzadeh et al. / Computers and Chemical Engineering 33 (2009) 133143 135

the following approximation for the governing partial differential

equation:

Cn+1i

Cni

t= D

Cni+1

2Cni+ Cn

i1

(x)2 u

Cni Cn

i1

x kCni (2)

where stands for increments in time or space, and i and n denotegrid index and time step index, respectively. Neglecting third and

higher-order terms, the reactant concentration at different timesand locations can be approximated by the following expressions

(Ataie-Ashtiani et al., 1996; Chaudhari, 1971; Lantz, 1971; Moldrup,

Yamaguchi, Hansen, & Ralston, 1992; Moldrup, Kruse,Yamaguchi, &

Rolston, 1996):

Cn+1i

= Cni +tC

t+

t2

2

2C

t2+ . . . (3)

Cni+1 = C

ni +x

C

x+

x2

2

2C

x2+ . . . (4)

Cni1 = C

ni x

C

x+

x2

2

2C

x2 . . . (5)

The second time derivative of the reactant concentration in Eq.

(3) can be obtained by taking the time derivative of the governingequation as given by the following expression:

2C

t2= D

2

x2

C

t

u

x

C

t

k

C

t(6)

Substituting for C/tfrom Eq. (1) results in:

2C

t2= D2

4C

x4 Du

3C

x3+ u2

2C

x2+ 2uk

C

x 2kD

2C

x2+ k2C (7)

Neglecting the higher-order derivatives 3C/x3 and 4C/x4 gives

2C

t2= u2

2C

x2 2kD

2C

x2+ 2uk

C

x+ k2C (8)

Using Eqs. (3)(5) and (8) and substituting in Eq. (2) we obtain

C

t= D

1+

ux

2D

tu2

2D+ kt

2C

x2

u(1+ kt)C

x k

1+

kt

2

C (9)

Comparing Eqs. (9) and (1) reveals that the numerical dis-

cretization modifies the diffusion coefficient, velocity, and reaction

constant. The modified diffusion coefficient, velocity, and reaction

constant are thus given by Dmod = D(1 + ux/2Dtu2/2D + kt),

umod = u(1 + kt), and kmod = k(1 + kt/2), respectively. Thecomponents of the diffusion coefficient, velocity, and reac-

tion constant attributable to numerical discretization are

then Dnum = D(ux/2Dtu2/2D + kt), unum = ukt, and

knum = k2t/2), respectively, where the subscript num refersto the additional terms that are created because of the numerical

discretization errors.

We scale distance x by L, length of the reaction domain, and

time by diffusion time scale, L2/D. The modified form of Eq. (9) can

therefore be represented by:

C

=

1+

Pe(1 Co)

2+2

2C

2

Pes(1+ 2)

C

2

1+

2

2

C (10)

where is the dimensionless time, is the dimensionless dis-tance, Pes = uL/D is the system Peclet number, Co = ut/x and

Pe = ux/D are grid block Courant and Peclet numbers, respectively,

and 2 = kL2/D is the Thiele modulus (Fogler, 1998). The numericaldiffusion, velocity, and reaction constant can then be respectively

represented by:

DnumD

=Pe(1 Co)

2+2 (11)

unumu

= 2 (12)

knumk

=2

2 (13)

The scaling analysis shows that, for a simple linear convection-

diffusion-reaction system, numerical results are sensitive to both

temporal and spatial discretization. Therefore, the accuracy of

numerical solutions is affected by grid size.

3. Governing equations for gassolid reactions

The mathematical model used in this study to perform numer-

ical experimentations is based on a formulation presented by

Rajaiah et al. (1988). Based on this model, the exothermic

non-catalytic gassolid reaction is assumed to take place in a one-

dimensional flow system. The reaction that takes place is betweena gaseous oxidizer (oxygen or nitrogen) and a solid phase that gen-

erates a gasor a solid.This model assumes that Fourierslaw of heat

conduction in a solid is valid. Sintering effects are ignored and the

system porosity remains constant during the process. The phys-

ical properties of the materials are assumed constant. Radiation

effects are incorporated into an effective thermal conductivity and

the reaction is considered to be first order with respect to gas and

solid reactants. A pseudo-homogenous, one phase model is used.

The governing differential equations for such a systemare given by

Energy balance : (cp)T

t=

2T

x2 ucp

T

x

+k0(H)CGCS exp ERT (14)

Gas mass balance : CGt

= D2CGx2

uCGx

k0CGCS exp

E

RT

(15)

Solid mass balance : (1 )CStD

= k0CGCS exp

E

RT

(16)

where cp = cp + scps(1 ) is the effective heat capacity, Tis temperature, C is concentration of reactant, is density, cp isheat capacity, is the average thermal conductivity, k0 is the pre-exponential factor, E is activation energy, H is heat of reaction, R

is the universal gas constant, D is molecular diffusion coefficient, uis velocity, is solid material porosity, tis time, and x is the spatialcoordinate. The subscripts S and G denote solid and gas, respec-

tively. The initial and boundary conditions as suggested by Rajaiah

et al. (1988) are given by

T= T0 0 x , t= 0 (17.1)

CG = CG0 0 x , t= 0 (17.2)

CS = CS0 0 x , t= 0 (17.3)

T

x= (cp)u(Tinlet T) at x = 0 (17.4)

T

x

= 0 at x = (17.5)


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DCGx

= u(CG inlet CG) at x = 0 (17.6)

CGx

= 0 at x = (17.7)

The following dimensionless scaling groups are used to render

the equations into dimensionless form:

= E(T T)

RT2(18.1)

CDG =CG

CG inlet(18.2)

CDS =CS

CS inlet(18.3)

xD =

cpt

x (18.4)

tD =Ek0(H)CGi

cpRT2exp

E

RT

t (18.5)

t =cpRT2

Ek0(H)CG inletCSinletexp

ERT

(18.6)

x =

t

cp(18.7)

PeH =ucp

t

cp

1/2(18.8)

PeM =u

D

t

cp

1/2(18.9)

Le =

cpD(18.10)

G =cpRT2

E(H)CG inlet(18.11)

S =cpRT2

(1 )E(H)CG inlet(18.12)

=RT

E(18.13)

Using the above scaling groups, the dimensionless form of the

differential equations is given by

tD=

2

x2D PeH

xD+ CGDCSD exp

+ 1

(19)

CGDtD

= 1Le

2

CGDx2D

PeM CGDxD

GCGDCSD exp

+ 1

(20)

CSDtD

= SCGDCSD exp

+ 1

(21)

The above in fact employs a standard scaling protocol available

in the combustion literature to render the equations dimension-

less (Dandekar, Puszynski, Degreve, & Hlavacek, 1990; Dandekar,

Puszynski, & Hlavacek, 1990; Merzhanov & Borovinskaya, 1975;

Merzhanov, Filonenko, & Borovinskaya, 1973; Merzhanov and

Khaikin, 1988; Puszynski et al., 1987; Rajaiah et al., 1988). 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 (Dandekar, Puszynski et al., 1990).

4. Discretization of the governing equations

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, while the convective flux values are evaluated

based on the grid block interface values. An explicit discrete finite

difference formulation of the governing equations for temperature,

gas concentration, and solid concentration is given by

n+1i

= ni +tD

x2D(ni+1 2

ni +

ni1

) PeHtDxD

(ni+1/2

ni1/2

)

+CnGDiCnSDi

tD exp

n

i

ni+ 1

(22)

Cn+1GDi

= CnGDi+

tD

Lex2D

(CnGDi+1 2CnGDi

+ CnGDi1)

PeMtD

LexD(CnGDi+1/2

CnGDi1/2)

GCnGDi

CnSDitD exp

ni

ni+ 1

(23)

Cn+1SDi

= CnSDi SC

nGDi

CnSDitD exp

n

i

ni+ 1

(24)

where tD and xD are temporal and spatial increments, respec-tively. In the following section, different numerical schemes

appropriate for approximating convective flux are presented.

4.1. Numerical approximation of convective flux

There are several options for discretizing advection terms in the

flow and transport equations. The mostcommonly used method forapproximating block interface properties is single-point upstream

weighting. The single-point upstream weighting scheme approxi-

mates the value of a function at a grid block face with the value in

thegrid block on theupstreamside.The drawback of such a scheme

is that artificial diffusion is introduced and might be considerable,

thereby producing erroneous results unless a large number of grid

blocks (i.e., a high degree of grid block refinement) is employed. An

alternative method of reducing numerical diffusion is to employ a

higher-order method,such as two-pointupstream weighting(Todd,

ODell, & Hirasaki, 1972). Third-order methods (Leonard, 1979) as

well as TVD schemes (Harten, 1983) have also been used to con-

trol numerical diffusion. In the following, a brief review of these

numerical methods is presented.

4.1.1. Single-point upstream

This scheme is widely used in evaluating interface block prop-

erties, but as noted is prone to numerical diffusion. In this method,

net convective flux into a cell can be approximated as

x(uC) =

1

x(ui+1/2[(1)Ci +Ci+1]

ui1/2[(1)Ci1 +Ci]) (25)

where x is the grid block size and is the weighting factor and isequalto zero or onedepending on theflow direction. It is noted that

for the problem under investigation the fluid velocity u is constant

andtherefore the choice of interface values forvelocitygivenby Eq.

(25) is irrelevant.


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4.1.2. Two-point upstream

This approximation was proposed by Todd et al. (1972). In this

second-order method, the mesh interface values are obtained by

using the values at two adjacent blocks that are dependent on

the flow direction. Because the approximation is an extrapolation

process, it is important to limit the computed values to physi-

cally acceptable values. However, in such cases, the method reverts

to single-point upstream weighting which guarantees physically

acceptable results. The block interface properties for an arbitrary

grid block can be approximated as follows:

fi1/2 =fi1 +xi1

xi1 +xi2(fi1 fi2) (26)

fi+1/2 =fi +xi

xi +xi1(fi fi1) (27)

where f= uC is the convective flux function. The block interface

fluxes are calculated based on the single-point upstream value

when the following conditions are not satisfied: fi+1/2 the greater

offi or fi+1 and fi1/2 the greater offi or fi1.

4.1.3. Third order methods

For third order methods, introduced by Leonard (1979), the gridblock interface value is approximated by using three points adja-

cent to an arbitrary grid block. In this method the boundary grid

blocks are approximated by the single-point upstream weighting.

Saad, Pope, and Sepehrnoori (1990) modified Leonards method to

account for variable grid block size. For that third order method,

the block interfaceproperties are calculated based on thefollowing

approximations:

fi1/2 =fi1 +i1(fi1 fi2)+ 2i1(fi fi1) (28)

fi+1/2 =fi +i(fi fi1)+ 2i(fi+1 fi) (29)

where

i =

1

3

xi

xi +xi1 and i =

1

3

xi

xi +xi+1 (30)

The boundary points can be approximated by single-point

upstream weighting because at least two upstream points and one

down-stream point in each coordinate direction are needed for

the higher-order method. Similar to the previous case, the above

formulation involves an extrapolation process, and it is necessary

to constrain the computed values to physically admissible values.

The other condition that must be fulfilled is the monotonicity con-

straint, which requires that the interface values of the calculated

parameterbe less than orequalto thelargerconcentrationon either

side of the grid block (Saad et al., 1990).

Table 1

The literature values of various flux limiters

Flux limiter Flux limiter function (r)

van Leer (van Le er, 1974) (r+ |r|)/(r+ 1)

MC (van Leer, 1977) max[0, min(2r, 0.5(r+ 1), 2]

van Albada-1 (van Albada, van Leer, &

Roberts, 1982)

r(r+ 1)/(r2 + 1)

Superbee (Roe, 1985, 1986) max[0, min(2r, 1), min(r, 2)]

Minmod (Roe, 1986) max[0, min(r, 1)]SMART (Gaskell & Lau, 1988) max[0, min(2r, 0.75r+ 0.25, 4)]

H-QUICK (Leonard, 1987) 2(r+ |r|)/(r+ 3)

Koren (1993) max[0, min(2r, (2r+ 1)/3), 2]

OSPRE (Waterson & Deconinck, 1995) 1.5r(r+ 1)/(r2 + r+ 1)

CHARM (Zhou, 1995) r( 1 +3r)/(r+ 1)2

HCUS (Waterson & Deconinck, 1995) 1.5(r+ |r|)/(r+ 2)

van Albada-2 (Kermani et al., 2003) 2r/(r2 + 1)

4.1.4. Total variation diminishing methods

TheT VD method was first introducedby Harten (1983) and later

by Sweby (1984). The TVD property guarantees that the total vari-

ation of the solution of a function will not increase as the solution

progresses in time. A numerical method is said to be TVD if

TV(Q

n+1

) TV(Q

n

) (31)where TV is the total variation given by

TV =

j

|Qnj Qn

j+1| (32)

The scheme limits the flux between grid blocks and then lim-

its spurious growth in the grid block averages so that the above

inequality is satisfied. A general approach is to multiply the jump

in grid block averages by a limiting function. For our problem, the

interface flux can be expressed by

fi1/2 =fi1 +(r1)

2[fi fi1] (33)

fi+1/2

=fi+

(r2)

2[f

i+1f

i] (34)

where

r1 =fi1 fi2

fi fi1(35)

and

r2 =fi fi1fi+1 fi

(36)

where flux limiter function and r is the ratio of successive gra-dients and is a measure of the smoothness of the solution. Making

= 0 returns to the commonly used single-point upstreammethod

Fig. 1. (a) TVD region and (b) second order TVD region ( Sweby, 1984); = 1 and = rcorrespond to central (Lax-Wendroff) and Beam-Warming scheme, respectively.


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Table 2

Data used in numerical simulations

Parameter S G 0 inlet PeH PeM Le

Test case 1 0.2138 0.100 0.0793 4.677 0 50 500 10

Test case 2 2 0.76 0.15 0.5 0 100 200 1

Fig. 2. Dimensionless temperature (a), gas concentration (b), and solid concentration (c) for a gassolid reaction of test case 1 at tD = 2.5, and dimensionless temperature (d),

gas concentration (e), and solid concentration (f) for a gassolid reaction of test case 2 at tD = 0.6 obtained by conducting tests with 5103 , 10103 and 15103 grid blocks

using single-point upstream method.

as described previously. These types of flux limiters are called non-

linear flux limiters. There are many TVD schemes found in the

literature for solving convection dominatedproblems; Table 1 gives

some of the flux limiters reported in the literature. Fig. 1 shows

the acceptable flux limiter region for second-order TVD schemes

as suggested by Sweby (1984).

5. Results

The governing partial differential equations of gassolid reac-

tions are solved using different numerical schemes. Two test

problems that give different frontal characteristics are presented

and results of different numerical schemes are compared. To esti-

Table 3

Comparison of numerical error for total number of grid blocks N= 50

Flux limiter Error (fraction)

Test case 1 Test case 2

CGD CSD CGD CSD

Single-point upstream 0.673 0.337 0.238 0.254 0.181 0.031

van Leer (van Lee r, 1974) 0.259 0.185 0.149 0.211 0.129 0.023

MC (van Leer, 1977) 0.213 0.155 0.144 0.206 0.123 0.023

van Albada-1 (van Albada et al., 1982) 0.299 0.205 0.145 0.215 0.131 0.024

Superbee (Roe, 1985, 1986) 0.213 0.100 0.155 0.207 0.109 0.021

Minmod (Roe, 1986) 0.376 0.238 0.123 0.217 0.136 0.025

SMART (Gaskell & Lau, 1988) 0.278 0.195 0.139 0.214 0.135 0.022

H-QUICK (Leonard, 1987) 0.305 0.210 0.140 0.222 0.138 0.023

Koren (1993) 0.256 0.183 0.142 0.213 0.131 0.023

OSPRE (Waterson & Deconinck, 1995) 0.273 0.192 0.149 0.212 0.130 0.024

CHARM (Zhou, 1995) 0.296 0.204 0.143 0.227 0.139 0.024

HCUS (Waterson & Deconinck, 1995) 0.291 0.202 0.144 0.219 0.135 0.023

van Albada-2 (Kermani et al., 2003) 0.315 0.188 0.130 0.193 0.112 0.023


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gas concentration (e), and solid concentration (f) for a gassolid reaction of test case 2 at tD = 0.6 obtained by conducting tests with various flux limiters given in Table 1 and

50 grid blocks.

mate the accuracy of the various numerical solutions, we define

numerical error using the following expression as a measure of

numerical solution accuracy:

=

( ref)

2dxD

2 dxD

1/2(37)

where can be either temperature or concentration and thesubscript ref denotes reference solution. The convergence of the

numerical solutions was verified by conducting tests for 5103,

10103, and 15103 grid blocks using the single-point upstream

method. The data used in the numerical simulations are given in

Table 2. Fig. 2 shows the reference numerical solutions for the two

test cases. These reference solutions are used in the analysis that

Table 4

Comparison of numerical error for total number of grid blocks N=100



CGD CSD CGD CSD


van Leer (van Le er, 1974) 0.115 0.093 0.073 0.146 0.089 0.012

MC (van Leer, 1977) 0.114 0.078 0.063 0.139 0.084 0.011


Superbee (Roe, 1985, 1986) 0.187 0.072 0.055 0.134 0.069 0.011

Minmod (Roe, 1986) 0.190 0.133 0.091 0.153 0.097 0.013


H-QUICK (Leonard, 1987) 0.137 0.109 0.083 0.155 0.098 0.012

Koren (1993) 0.122 0.093 0.071 0.147 0.092 0.011


CHARM (Zhou, 1995) 0.138 0.114 0.086 0.158 0.099 0.012




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100 grid blocks.

follows. Results demonstrate that large numbers of grid blocks (in

excess of a thousand) are needed to find an accurate numerical

solution using the single-point upstream method. In the follow-

ing, we compare various flux-limiting schemes for the purpose

of seeking an appropriate flux limiter. Such an appropriate flux

limiter will facilitate an accurate numerical solution with fewer

grid blocks as compared to the traditional single-point upstream

method.

Numerical experiments were performed using all flux limiters

given in Table 1 and using 50, 100, 500, and 1000 grid blocks.

Fig. 3 shows temperature and concentration profiles obtained from

different flux-limiting schemes and with 50 grid blocks. The cor-

responding numerical errors for N= 50 are summarized in Table 3.

Results reveal that, for N= 50 and for both test cases, MC (van Leer,

1977), Superbee (Roe, 1985, 1986), and van Albada-2 (Kermani,

Gerber, & Stockie, 2003) flux limiters are superior as compared to

Table 5




CGD CSD CGD CSD


van Leer (van Lee r, 1974) 0.048 0.035 0.011 0.042 0.030 0.002

MC (van Leer, 1977) 0.064 0.033 0.012 0.038 0.027 0.002


Superbee (Roe, 1985, 1986) 0.089 0.036 0.014 0.029 0.016 0.002

Minmod (Roe, 1986) 0.021 0.039 0.011 0.051 0.038 0.002


H-QUICK (Leonard, 1987) 0.053 0.040 0.012 0.049 0.036 0.002

Koren (1993) 0.062 0.037 0.012 0.044 0.033 0.002


CHARM (Zhou, 1995) 0.049 0.044 0.012 0.051 0.038 0.002




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500 grid blocks.

the other schemes. Results show that the single-point upstream

method fails to model the reaction front accurately, while the flux-

limiting methods tend to correctlycapture the sharpreaction front.

Fig. 4 compares numerical solutions obtained by using different

flux limiters and choosing total number of grid blocks equal to 100.

Whilemost of theflux limiters couldaccuratelycapturealmostall of

the reaction front characteristics, the calculatednumerical errors in

Table 4 reveal thatMC (van Leer, 1977), Superbee (Roe, 1985, 1986),

and van Albada-2 (Kermani et al., 2003) are more accurate than

the other flux limiters. Similar to the previous case (N= 50), the

single-point upstream method fails to resolve the reaction front.

Figs. 5 and 6 show the calculated profiles with 500 and 1000 total

numbers of grid blocks, respectively. Results shown in Figs. 5 and 6

and Tables 5 and 6 reveal that the flux limiters accurately cap-

ture thereaction front.Again, the single-point upstreamresults are

still not accurate at N= 1000. Results therefore suggest that using

Table 6




CGD CSD CGD CSD


van Leer (van Le er, 1974) 0.043 0.023 0.006 0.020 0.015 0.001

MC (van Leer, 1977) 0.051 0.024 0.008 0.018 0.014 0.001


Superbee (Roe, 1985, 1986) 0.074 0.035 0.011 0.015 0.009 0.001

Minmod (Roe, 1986) 0.017 0.020 0.004 0.025 0.021 0.001


H-QUICK (Leonard, 1987) 0.043 0.026 0.006 0.025 0.020 0.001

Koren (1993) 0.048 0.025 0.007 0.022 0.017 0.001


CHARM (Zhou, 1995) 0.041 0.028 0.006 0.025 0.021 0.001




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1000 grid blocks.

a flux-limiting approach allows conducting numerical simulations

with significantly fewer numbers of grid blocks yet with the same

accuracy of very fine grid simulation by the single-point upstream

scheme.

6. Summary and conclusions

Physical processes involved in gassolid reactive systems

include diffusive (heat and mass) as well as reactive processes

that have different intrinsic scaling characteristics. In most cases,

the non-linearity of the processes does not allow analytical solu-

tion. Therefore, accurate numerical modeling is necessary inorder to properly interpret experimental measurements, leading

to developing a better understanding and design of industrial scale

processes. Accurate fine grid numerical simulation of sharp reac-

tion front propagation in gassolid reactions is a challenging task.

Indeed, in most practical cases, such as in situ combustion in heavy

oil reservoirs, the use of small grid blocks is not feasible. The con-

ventional single-point upstream scheme is also known to smearthe

fronts. Therefore, one needs to account for the small scale gradi-

ents that cannot be captured by coarse grid blocks when using the

single-point upstream method. One possible option for reducing

the smearing effect resulting from this method is to use flux lim-

iters. In this paper, we conducted a comparative study of various

flux limiters to find appropriate flux limiters for one-dimensional

gassolid reactive flow simulations with Arrhenius type reaction.

Relativelyfine gridnumericalsimulations of the gassolidreactions

show that most of the methods with the exception of single-point

upstream perform well. More specifically, for small number of grid

points which is of practical importance Superbee, MC, and van

Albada-2 flux limiters are superior as compared to other schemes.

These results will aid in choosing proper flux limiters in numeri-

cal modeling of gassolid reactions with Arrhenius type reaction

kinetics.

Acknowledgment

The financial support of the Alberta Ingenuity Centre for In Situ

Energy (AICISE) is acknowledged.

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