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International Journal of Mechanical Engineering and Technology (IJMET)

Volume 9, Issue 9, September 2018, pp. 230–247, Article ID: IJMET_09_09_028

Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=9&IType=9

ISSN Print: 0976-6340 and ISSN Online: 0976-6359

© IAEME Publication Scopus Indexed

EFFECTS OF THE COMBUSTION GASES ON

THE RADIATION HEAT TRANSFER

Dr. Abbas Alwi Sakhir Abed

University of Al-Qadisiyah, College of Engineering, Iraq

ABSTRACT

A Monte Carlo program that simulates radiative heat transfer in participating

media for two-dimensional has been developed. This work has added the ability to

simulate the absorption and emission of the combustion gases: 2CO ,CO , OH 2 , NO ,

2SO , and 4CH . The theory of gas radiation and the formulation and implementation

of the thermal model are presented. Several aspects of the simulation are verified by

numerical solutions, comparison to published results and experimental data. The

accuracy and utility of the simulation are demonstrated by comparison to other one-

dimensional solutions for radiative heat transfer in combustion gases and by

comparison to experimental data.

Key words: radiative, combustion gases, absorption, emission.

Cite this Article: Dr. Abbas Alwi Sakhir Abed, Effects of the combustion Gases on

the Radiation Heat Transfer, International Journal of Mechanical Engineering and

Technology 9(8), 2018, pp. 230–247.

http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=8

NOMENCLATURE

A : Area, 2m .

b : Self-broadening to foreign-gas broadening ratio.

EC : Constant defined in eq. (30).

d : Spacing between the centers of spectral lines (1cm ).

E : Energy of a photon packet (J), or energy state (J).

e : Component of the emission vector.

F : Exchange factor.

f : User-defined cutoff fractions.

k : Absorption coefficient (1/m).

L : Length of a surface (m), path length for absorption (m), or distance between two slabs (m).

CL : Length over which absorption is calculated(m).

n : Index of refraction, or empirical factor.



P : Pressure (atm), or probability.

eP : Effective broadening pressure parameter.

Q : Energy per unit time (W ).

q : Radiative heat flux ( 2/ mW ).

R : Uniformly distributed random number between 0 and 1.

S : Line intensity ( gmcm /. 21 ).

s : Path length (m).

T : Temperature (K).

V: Volume(3m )

v : Vibrational quantum number.

x : X coordinate (m).

y : Y coordinate (m).

: Gas absorptivity.

: Mean line width to spacing parameter.

: Spectral half width of a line(1cm )

: Displacement between nodes (m), change in a quantity, or interval size.

int : Step size for piecewise integration

v : Wavenumber interval

: Emissivity.

: Line width to spacing parameter

: Dimensionless temperature.

: Cone angle(Rad).

: Mass absorption coefficient ( gm /2 ).

: Reflectivity.

: Gas transmissivity.

L : Optical depth.

: Dimensionless heat flux.

: Exponential decay width (1cm ).

SUBSCRIPTS

0 : Original value, or reference state.

1 : First node of a surface.

2 : Second node of a surface.

D2 : Two-dimensional.

abs : Absorbed.

b : Black body, gas absorptoion band, or bottom plate.

E : Emission point.

EL : Elsasser model.

Effects of the combustion Gases on the Radiation Heat Transfer


el : Element.

em : Emitted.

gp : Gauss points.

I : Incident.

i : ith of a set.

int : Integration, or interception.

j : jth of a set.

NR : Not reflected.

s : Specular, or surface.

sm : Smoothed quantity.

surf : Surface.

t : Top plate.

u : Upper limit for a band.

w : Wall.

: Absorptance.

1. INTRODUCTION

Most early work applying Monte Carlo to gases dealt only with grey gases [Howell and

Perlmutter, 1964a; Perlmutter and Howell, 1964]. Howell and Perlmutter [1964b] were the

first to consider non grey media with Monte Carlo when they modeled hydrogen gas between

infinite black plates. The spectral absorption coefficient for hydrogen at high temperatures

and pressures was obtained from experimental data. Steward and Cannon [1971] used Monte

Carlo and the weighed-sum-of-grey-gases method to model a cylindrical furnace.

Several good descriptions of the use of the Monte Carlo method in radiative heat transfer,

including participating media, exist [Howell, 1968; Haji-Sheikh, 1988; Modest, 1993]. More

recently, a review of the Monte Carlo method was done by Walters and Buckius [1994] and

concentrates on the reverse Monte Carlo method and the modeling of scattering.

More recently there has been much interest in the use of Monte Carlo in modeling the

spectral behavior of real gases. Besides the work done by Farmer and Howell mentioned

above, Modest [1992] has formulated a Monte Carlo method for gases with spectral line

structure. One-dimensional Monte Carlo simulations have been done for water vapor [Liu and

Tiwari, 1994] and for a mixture of water vapor and carbon dioxide [Taniguchi et al., 1992].

Farmer and Howell [1994b] have also investigated reducing the computational time required

to model spectral effects by using the diffusion approximation [Modest, 1993; Siegel and

Howell, 1992] in optically thick cases.

Although much work has been done on modeling gas radiation by Monte Carlo, very little

work has been done using real geometries except by Farmer and Howell. To find the effects

of gas radiation for most furnaces, efforts must be made to model the complex geometries and

surfaces properties as well.

The present work describes Monte Carlo techniques to model real gas properties and

surface properties which vary with incident angle in complex, two-dimensional enclosures.

The implementation is based on the programs MONT2D and MONT3D [Maltby et al, 1994;

Burns and Pryor, 1987; Burns et al., 1992; Burns et al., 1990] which simulate radiative heat

transfer in geometries with nonparticipating media. The current work relies on the same

photon tracing algorithm and geometry and surface property models as MONT2D.



2. FORMALIZATION AND SOLUTION

The geometry is divided into surfN surfaces of uniform temperature and emittance and gridN

grid cells containing gas of uniform composition, temperature, and pressure. The sum of the

surfN surfaces and the gridN grid cells gives elN elements which emit and absorb photons.

Now, the total rate of energy, surfemQ , , emitted from a surface of area A, is [Modest, 1993;

Siegel and Howell, 1992]:

ATnQ gridem

42

, (1)

While the total rate of energy, gridemQ ,

, emitted from a gas volume of volume V not

including self-absorption is [Modest, 1993]:

VTnQ Pgridem

42

, )(4 (2)

whereP)( is the Planck mean density mass absorption coefficient product, n is

essentially one for most gas mixtures including combustion gases. For this reason, from this

point on, n is assumed to be one and is not explicitly shown in any of the equations. The

amount of energy emitted by element j that is absorbed by element i, jiabQ , is:

jemjijiabs QFQ ,, (3)

jiF is the exchange factor from element j to element i which is defined as:

j

ji

jiE

EF (4)

where iE is the total energy of photon packets released by element j, and jiE is the total

energy of photon packets released by element j which are absorbed by element i, no matter

what the path. Both jiE and jE result directly from the Monte Carlo photon tracing. Enough

photon bundles must be emitted in the Monte Carlo to reduce the statistical scatter in jiF to

an acceptable level. The total energy emission rate for the element i, iQ , is:

elN

j

jemjiiemi QFQQ1

,, (5)

If heat transfer is only by radiation and the element is at equilibrium, iQ is 0. If the

element is not at equilibrium and/or there is significant heat transfer by other means such as

conduction, convection, or chemical reaction, iQ is generally not 0. Particularly noteworthy

is that the summation in eq. (5) includes the element i. This is why eq. (2) does not include

self-absorption. Self-absorption is accounted for in the iiF term. The expanded version of eq.

(5) for surfaces is:

el

surf

surf N

Nj

jijjjP

N

j

jijjjiiii FVTFATATQ1

4

1

44 ])[(4 (6)

surfNi ..........3,2,1

and for gases it is:

ji

N

Nj

jjjPjij

N

j

jjiiiPi FVTFATVTQel

surf

surf

1

4

1

44 ])[(4])[(4 (7)



elsurf NNi ,1

For a two-dimensional case, since the geometry is infinite in the z direction, if iQ is not 0,

then it is infinite. To circumvent this, eqs. (6) and (7) are divided by a unit length in the z

direction and become:

el

surf

surf N

Nj

jijjjP

N

j

jijjjiiii FATFLTLTQ1

4

1

44 ])[(4 (8)

surfNi ..........3,2,1

ji

N

Nj

jjjPjij

N

j

jjiiiPi FATFLTATQel

surf

surf

1

4

1

44 ])[(4])[(4 (9)

elsurf NNi ,1

where L is the two-dimensional length of a surface in the x-y plane, and A is the area of

gas inside each grid in the x-y plane. The primes signify quantities per unit length in the z

direction. The gas absorptance model used in this work is the Goody narrow band model with

parameters from the Edwards exponential-tailed wide band model [Edwards, 1976; Edwards

and Balakrishnan, 1973]. By assuming that the spectrum decays exponentially in the tails of

the absorption band, Edwards has formulated four parameters that have been shown to give

accurate results for the major combustion gases when used with the Goody narrow band

model. The reasons for choosing this model are the following. Line by line integration gives

more accurate results [Tiwari and Gupta, 1978], but it requires huge databases of information

on individual spectral lines and involves lengthy calculations. Several other wide band models

exist [Edwards, 1976; Modest, 1993], such as Penner’s box model. The advantage of the

Edwards/Goody model over the other models, is that it has parameters defined for all major

combustion gases: 2CO , OH 2 , CO , 4CH , NO , and 2SO and has been found to correlate to

within 15% of experimental data. More recently, Li et al. [1995] have developed wide band

parameters for 22HC and all of the above gases except 2SO which when used with the

Elsasser narrow band model obtain results as accurate as the Edwards/Goody model.

The Goody model theorizes that the line spectral distribution is a uniform random

distribution of mean line spacing d. The line intensities of each line, iS , follow an exponential

distribution. The probability of a particular value of S , )(SP , is equal to: )/exp()/1( SSS ,

where S is the mean line intensity. For this model, )(v is not defined, except statistically,

but:

smgsmgXdS

XdS,2/1, 1

])/)/(1[

)/(exp1

(10)

Where is the line width to spacing parameter, defined by:

d

(11)

For a symmetrical band with a band center, cv , i.e. for most bands:

)/2exp()/(/ cvvdS (12)

For an asymmetric band with an upper limit, uv , such as the m3.4 band of 2CO :

)/2exp()/(/ vvdS u (13)



For an asymmetric band with an lower limit, lv , such as the rotational band of water:

)/exp()/(/ lu vvdS (14)

T

To

o [Edwards,1976] (15)

T

To

o (16)

}1

( BA

o

o

o

Pb

PPT

T

(17)

The values of ooouc vv ,,,, and lv for various gases are given Table 1.

From eqs. (11) and (17), , the line width to spacing parameter is defined as:

ee

o

o PPT

T

dd

(18)

eP , the effective broadening pressure, is derived from eq. (17). Using the mole ratio,

PPx A / , eP is:

n

oe xbPPP ]})1(1][/{[ (19)

The exponent, n , is an “empirical fudge factor,” used to obtain better agreement with

experimental data [Edwards, 1976]. n is probably a correction for anharmonic effects and

varies from 0.6 to 0.8. Values for n and b for various gases are given in Table 1. is a

measure of the degree to which individual spectral lines overlap. Large values of represent

large overlap. The two parameters and eP represent respectively the effects of temperature

and pressure on line overlap. For water vapor, this division between pressure and temperature

does not hold true because the self-broadening to foreign-gas broadening ratio, b , use to

calculate eP is found experimentally to be a function of temperature.

5.0100

6.82

T

Kb OH

This may be due to the fact that water vapor is much more polar than the other combustion

gases. The polar ends of the water molecules are attracted to each other, and this will promote

collisions. For overlapping bands, the transmittance is essentially the products of the

transmittance in each band [Edwards and Balakrishnan, 1973]. For a mixture of gN gases,

each with )(iNb overlapping bands:

)1(11 ,,,

)(

11,,,

)(

11,, jivg

iN

j

N

ijivg

iN

j

N

imixvg

bgbg

(21)

A photon is emitted from a surface in a direction given by the emission vector, e , from

the emission point, Ex . The Cartesian equation that defines the emission line is:

z

E

y

E

x

E

e

zz

e

yy

e

xx

(22)

The equation that described each cartesian surface i , is:



i

i

i

i

i

i

z

zz

y

yy

x

xx

,1,1,1 (23)

where ),,( ,1,1,1 iii zyx are the coordinates of the first node of the surface, and, ix iy , and

iz are the coordinate displacements to node 1. Using these equations, the surface which

intercepts the photon in the shortest distance can be found. The particulars of the tracing

algorithm for prismatic geometries are given by Burns and Pryor [1987]. To determine

whether a set is an object or an enclosure, the surface vector ix for surface i is defined from

node 1 to 2 as:

iii xxx ,1,2

The cumulative probability that an event will occur is determined by integrating the

PDF over the independent variable(s) and is termed the cumulative distribution function, or

CDF. For a gas [Modest, 1993]:

0

0

)(

)(

)(

dvE

dvE

vCFD

bvv

v

bvv

(25)

For a mixture of gN gases, each with )(iNb bands:

)(

1 ,,1

)(iN

j jiv

N

i

iv

bg

d

S (26)

As the photon packet travels through the geometry, the attenuation of the photon packet in

each grid cell is calculated, and a cumulative record of the energy absorbed by each cell is

kept. The photon packet is traced until its energy drops below a specified fraction, ef , of its

original energy. ef is generally set to 7101 .

Figure 1 shows an example of the photon path. When the photon packet enters a new grid

cell, first a check is done to determine if it is intercepted by any surface. If it is intercepted by

a surface, then is , the distance travelled through the grid cell, is computed to the point of

interception. If it is not intercepted, then is is calculated using the point on the grid where the

photon exits. Due to the effects of wavenumber averaging, the calculation of the amount of

energy absorbed by a grid is complicated when the photon is reflected from a surface. For this

reason, the absorption of energy by a surface will be discussed before the absorption in a grid.

For a surface:

incIsurf EE )( (27)

where

)(sin 1 R , R Randomnumbersbetween1and0 (28)

The amount of energy absorbed by the grid cell i is:

emmixgmixgEgrid EisisCE )]}1([)]([{ ,, (29)

where emE is the energy of the photon packet at emission, and s(i) is the distance travelled

from the point of emission to the point the photon packet exits the grid i. If the photon packet

is reflected, then the calculations are done to find the energy absorbed before and after hitting



the wall. EC is a correction factor that indicates what fraction of the energy of the energy is

absorbed once reflections are taken into account. If there are no reflections, the amount of

energy available to be absorbed by the gas past a certain point is the energy transmitted up to

that point, NRtrE ,

. When reflections are considered, this amount must be reduced by the

energy absorbed in all reflections with surfaces, surfabsE , . As a first approximation, EC might

be considered just NRtrsurfabsNRtr EEE ,,, /)( . The error in this formulation is that the difference

between the energy absorbed by the gas if there were no reflections, NRabsE , , and the energy

absorbed if reflections are taken into account, NRabsE , is subtracted twice. It is considered part

of energy absorbed if there were no reflections and also as part of the energy absorbed by the

surfaces. Counting this energy only once, the correction factor is:

NRtr

NRabNRabssurfabsNRtr

EE

EEEEC

,

,,,, (30)

The Monte Carlo require the calculation of several lengths, areas, and volumes. For a

two-dimensional surface i with nodes 1 and 2, since the surfaces are linear, the length of the

surface is given by:

2

,1,2

2

,1,2,2 )()( iiiiiD yyxxL (31)

To calculate the area of set i in a two-dimensional Cartesian geometry, Green’s theorem in

the plane is used [Kreyszig, 1993]

A

iD dxdyA ,2 (31)

Again, the segN surface segments that make up the set are linear. Since the contour around

the set might be traversed in either direction, the absolute value of the result is taken:

segseg i

k

N

k

kkkk

N

k

y

y

iD xxyyxdyA1

,1,2,1,2

1

,2 ))((5.0,2

,1

(32)

To establish spectral windows, the following procedure is used. First, the gas emission

spectrum, 0 to 8,0001cm , is divided into intervals of usually 5

1cm . Next, an arbitrary

characteristic length, CL , is chosen. For the work in this research, CL is taken as 1 m. For

each interval, the total absorptivity over the distance CL is calculated for the conditions in

each grid. The maximum of all calculated absorptivities is found. If, for all grid cells, the

absorptivity at the beginning of the interval is less than a fractional threshold, 1f , of the

maximum absorptivity, the interval is marked as a spectral window. No absorption

calculations are done in a spectral window, and the gas wavelength inverse CDF does not

include this region.

To further reduce calculations, the absorptivity of individual absorption bands also are

compared to the maximum total absorptivity. For each interval, only bands for which the

absorptivity for at least one grid cell in the geometry is above a second fractional threshold,

2f , are used in calculations. Typical values for the cutoff fractions are 4

1 101 f and 4

2 101 f . The numerical parameters, , 1f , and 2f must be chosen as a compromise

between accuracy and efficiency. Too small a value of wastes computer memory. Too

large a value of causes the wrong absorption bands to be used in calculations. If 1f and



2f are too small, too many absorption bands will be used in calculations, and spectral

windows will not be used to advantage. Choosing values too large for 1f and 2f will cause

the long range absorptivity, which occurs in the wings where the absorptivity is low, to be

modeled improperly.

3. RESULTS AND DISCUSSION

3.1. Gas-Gas Transmittance verification

The concept of a correction factor for gas to gas absorptance is another idea requiring

verification. The gas to gas absorptance formulas are not given by Modest [1992] but are

derived from random number relations he gives for a probabilistic Monte Carlo formulation.

To verify that the idea is being applied correctly, the two Monte Carlo test cases given by

Modest in his paper [1992] are done. Both tests use a one-dimensional geometry of

participating media between two infinite plates 1 meter apart, as shown in Fig. 2. Between the

infinite plates is a Goody model gas with one spectral absorption band. This band is

symmetric with the band center at 3,000 1cm . For this gas, ,

do not vary with temperature. 200 1cm and and

below. The first case considered is an isothermal slab of gas at 1,000 K between 0 K

nonemitting plates. For this problem, the quantity of interest is the nondimensional heat flux,

which is defined in this case as:

4

mT

q

(33)

mT is the temperature of the medium (1,000 K).

from the center line. In this test problem, is measured at the wall. The second case is that

of the gas in radiative equilibrium bound by two isothermal black plates. The temperature of

the top plate, tT , is 1,200 K while the temperature of the lower plate, bT , is 800 K. Again, the

quantity of interest is which is defined for

this case as:

)( 44

bt TT

q

(34)

Since the problem is at radiative equilibrium, is constant. The geometry for which

GAS2D is used to model a slab of gas between two infinite plates is shown in Fig. 2. The two

plates are modeled by a rectangle with totally specularly reflecting sides. The left and right

boundaries exhibit symmetry (i.e. they are perfect specular reflectors). The space between the

slabs is divided into a number of gas regions, each of which is isothermal. For these cases, the

values of the parameters used are: 7101 ef , 4101

1

f , 5

2 101 f , CL =1.0m,

15 cmv , 001,5, gIN 1

int, 01.0 cmg , 1, ggpN , 001,1, sIN , 1

int, 5.0 cms ,

1, sgpN , and 4101 vf . Variation of these parameters, indicates that these values give

results accurate to within 2%. The results are shown in Table 2. The results match nearly

exactly to two significant figures except in the last set of values for case 2 where there is a

1.3% difference. Therefore, the gas to gas absorptance model appears correct.

3.2. Gray gas in a rectangular enclosure

Although an emphasis of this work is two-dimensional geometries, no simple two-

dimensional cases with which to compare could be found. To verify that GAS2D converges to



the proper exchange fractions for two-dimensional surfaces, the case of a two-dimensional

rectangular enclosure filled with grey gas is solved numerically. From the basic definition of

exchange fractions in Siegel and Howell [1992], the exchange fraction between surfaces 1 and

2 within an intervening grey gas is formulated as a three-dimensional integral:

122

21

1

12

1 2

coscos)exp(1dAdA

s

ks

AF

A A

(35)

where s is the path length between 1dA and

2dA . A similar expression can be used to

obtain the exchange fraction between a volume of gas and a surface. To test the validity of the

code for two-dimensional geometries, four cases are solved for a square with sides of length

one.

Case 1: all surfaces black, no gas, k = 0

Case 2: all surfaces black, k=0.5

Case 3: surface 2 perfectly specularly reflecting, all other surfaces black, no gas

Case 4: surface 2 perfectly specularly reflecting, all other surfaces black, k = 0.5

Figure 3 shows the details of the geometry. For these trials, only side 1 and the gas are

emitting, and the gas exists entirely in one grid. In the GAS2D runs, one million photons each

are emitted from side 1 and the gas. Gaussian quadrature is used to perform the integrations

with 160 points per integral. The Gaussian quadrature points are generated using a subroutine

from Numerical Recipes in FORTRAN, 2nd Edition [Press et al., 1992]. The solution of the

integrals in two dimensions requires integrating from to in z direction, but

Gaussian quadrature can only be done over finite intervals. To overcome this problem, the

substitutions are made,

)tan(tz (36)

dttdz ))tan(1( 2 (37)

For cases in which there is no gas, the exact solution is obtained using Hottel’s crossed

strings method [Modest, 1993]. For cases with a specularly reflecting side, the solution by

numerical integration and the crossed strings method are obtained by replacing side 2 with a

mirror image of the other sides. Details are shown in Fig. 4. For this geometry, the exchange

fractions from emitting surface or volume i to side j are related to the exchange fractions to

side i in the original geometry by:

iajbiajaji FFF (38)

The results are shown in Tables 3 and 4. In general, the numerical solution converges

asymtotically from below to the correct answer as the number of Gaussian integration points

is increased. For the calculation of 12F , the integration converges to the solution with many

fewer Gauss points. The results are extremely good. The GAS2D solutions are within 0.21%

of the exact solutions and 2.1% of the numerical solutions in all cases. Considering the error

of up to 1.2% between the numerical and exact solutions, the GAS2D solutions are very close

to the numerical solutions.

3.3. Gray gas between infinite parallel plates

A common benchmark problem in gas radiation is the one-dimensional non-isothermal grey

gas between two parallel infinite plates in radiative equilibrium. A well known numerical

solution for this problem is one derived by Heaslet and Warming [Heaslet and Warming,

1965; Modest, 1993]. For this problem the di , is defined in eq. (34).



Since the problem is at radiative equilibrium, is constant. For black slabs, the

nondimensional heat flux, b , is a function of optical depth, L , only.

kLL (39)

Where k is the absorption coefficient (a constant for a grey gas), and L is the distance

between the two slabs. Table 5 lists b as a function of optical depth. If the plates are

diffusely reflecting and emitting with total hemispherical emissivities, 1 and

2 , then:

211

121

b

b (40)

Besides acting as a general test of GAS2D, this problem also tests the accuracy of the

energy balance code. The first test determines the number of grid cells needed to model the

problem as L . For this test, a geometry of Fig. 2 is used. One million photons

are emitted similar to that described in gas-gas transmittance test is used. One million photons

are emitted per element. The results are shown in Fig. 5. b is overestimated in every case

except for L = 0.1. The number of grids required to model the problem properly increases as

L increases. Errors less than 0.1% are due to statistical scatter in the Monte Carlo results. As

seen in the graph, this level is reached using only one grid when L = 0.1, but nine grids are

required foL = 0.5, and even forty-five grids are not sufficient for

L = 5.0.

3.4. Water Vapor between Infinite Plates

This case assumed the distance between the plates is 1 m, the number of grids is 20, and the

number of photons emitted per grid is 50,000. From past experience with MONT2D, the

number of photons seems too low to yield accurate results. For this reason, another case

where one million photons are emitted per grid is done. The results are shown below in Fig. 6

and in Table 6. As the graph shows, the GAS2D results for 50,000 photons seems as accurate

as Liu’s results for 50,000 photons, but both exhibit fluctuations. When the number of

photons is increased to one million, a smooth curve is obtained. Table 6 compares the

calculated value of wq , the heat flux at the wall, for all three cases. For each GAS2D case,

three runs are done. The range shown for wq results from the statistical scatter for these three

runs. The difference in wq between Liu’s results and GAS2D’s ranges from 8.6% to 10.5%.

Since the original Edwards’ gas model only correlates to about 15% with experimental data,

the results are as close as can be expected. The effect on wq of the larger number of photons

is not as dramatic as that for yq / . By increasing the number of photons by factor of 20,

the statistical scatter in wq dropped from 0.533/ mkW to 0.13

3/ mkW . Increasing the number

of photons has a roughly proportional effect on run time.

3.5. 2CO and OH 2 between infinite parallel plates

Another Monte Carlo problem involving molecular gases is one by Taniguchi et al. [1992].

This problem involves a mixture of 2CO : OH 2 : 2N in a mole percent ratio of 9.5:19.0:71.5 at

one atmosphere pressure. The gas mixture is typical of products of combustion of methane.

Here, the gas mixture is in radiative equilibrium between two plates one meter apart. The

bottom plate is at 1,500 K, and the top plate is at 1,000 K. To match the original problem, the

geometry is divided into five grids, and since the number of photons is not specified, one



million are emitted from each element. To determine the effects of increasing the number of

grids, this case is repeated using nine and twenty-seven grids. Since the problem is in

radiative equilibrium, it must be solved iteratively. In addition, the surface wavenumber

parameters are set to: 001,1, sIN , 1

int, 5.0 cms , 1, sgpN , and 4101 vf . The results

are shown in Fig. 7 and Table 7. The graph clearly shows that the GAS2D results for

much higher than Taniguchi’s, while the table shows that wq differs only by 6.1% to 6.5%.

Where,

44

44

)(tb

t

TT

TTy

(41)

The reason the temperature profiles differ by so much may be that Taniguchi’s gas model

is incorrect. The gas model Taniguchi uses is the Elsasser model with Edwards’ wide band

parameters. Edwards’ wide band parameters have been formulated to work with the Goody

model and have never before been used with the Elsasser model. When presenting this model,

Taniguchi proves that Edwards’ wide band parameters can be used to calculate the emissivity

of gases; he does not prove that his formulation of the Elsasser model is correct. As evident

from the graph and the table, increasing the number of grids has very little effect; all the ’s

are about the same, and wq drops by 1.1 2/ mkW or 0.55% at most. Five grids are therefore

sufficient. When the number of grid cells is increased, the number of elements emitting

photons is increased, and the number of grids cells a photon must traverse also increases. For

the one-dimensional geometries of the test cases, there are only three or four surfaces and

many photons must be traced through all grid cells. In general, increasing the number of grid

cells in a one-dimensional geometry by afactor of N increases the run time by a factor of 2N

4. CONCLUSIONS

1-Comparison of the calculations of the Edwards/Goody model to experimental results

obtained by Hines [1964], proves that the model can accurately calculate the absorption

spectra of mixtures of carbon dioxide and water.

2-The comparison to Liu and Tiwari’s [1994] results demonstrates the accuracy of the

Edwards/Goody gas model in GAS2D. The results of Taniguchi et al. [1992] do not match as

well, but this is believed to be caused by an error in their gas model.

3-The real gas model does give much better answers than using the grey gas approximation or

totally ignoring gas effects. The results of using the real gas model give answers that are two

to three times more accurate then those obtained ignoring the effects of the gas completely.

The grey gas approximation is found always to overestimate the wall heat flux by from 24%

to 154%.

4-For the Goody model, using the Curtis-Godson approximation, the spectrally smoothed

surface to gas absorptance is very different than the spectrally smoothed gas to gas

absorptance. This result suggests that care must be taken when using spectrally averaged

quantities or large errors may occur. Since most solution methods for radiative heat transfer in

participating media use some form of spectral averaging, this result has far ranging

implications.



Table 1 Values of nbvv ooouc ,,,,,, , and lv for Combustion Gases (Edwards [1976]).

Parameter 2CO OH 2

4CH CO NO

2SO

cv 1cm 960 1600 1526 2143 1876 1151

uv 1cm 2410 3756 3020 2143 1876 1361

o 1cm 11.2 69.3 56.0 25.5 20 24.83

o 0.24723 0.14311 0.06973 0.07506 0.18050 0.05952

o

21 ./ mgcm

110.0 44205.0 46.0 20.9 9.0 3.674

lv 1cm 667 1595 1306 2143 1876 519

b 1.3 5.0

1006.8

2

T

Kb OH

1.3 1.1 1.0 1.28

n 0.8 1 0.8 0.8 0.65 0.7

Table 2 Comparison to Modest’s Results for

C

A

S

E

1

Vaues Modest GAS2D

01.0,000,2 0.04 0.04

1.0,000,10 0.15 0.15

100,000,20 0.27 0.27

C

A

S

E

2

01.0,000,2 0.98 0.98

1.0,000,10 0.9 0.9

100,000,20 0.78 0.79

Table 3 Comparison of Exchange Fractions from Side 1



Table 4 Comparison of Exchange Fractions from the Gas

Table 5 Nondimensional Radiative Heat Transfer

L b L b

0.0 1 0.5 0.704

0.1 0.9157 1.0 0.5532

0.2 0.8491 2.0 0.39

0.4 0.7458 5.0 0.2077

Tiwari’s Results for wq Table 6 Comparison to Liu and

Case )/( 2mkWqw

Liu 50,000 photons -27.6

GAS2D 50,000 photons -25.23,-24.7

GAS2D 1 million photons -25.07,-24.94

Table 7 Comparison to Liu and Taniguch et al. Results for wq

Case )/( 2mkWqw

Taniguchi 5 grids 213.2

GAS2D 5 grids 199.9,200.3

GAS2D 9 grids 199.4,199.7

GAS2D 27 grids 199.2,199.3

Figure 1 Photon Path

L

bL

42089.1

3/4,1



Figure 2 Parallel Infinite Slabs

Figure 3 Square Geometry

Figure 4 Model of a Specularly Reflecting Square

Figure 5 Percent Error as a Function of L and Number of Grid Cells



Figure 6 Comparison to Liu and Tiwari’s Results for yq /

Figure 7 Comparison to Taniguchi et al. Results for

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