Upload
diego-franco
View
216
Download
0
Embed Size (px)
Citation preview
8/10/2019 2011 Aplicado a Radiacion
1/6
Numerical simulations of a coupled radiativeconductive heat transfer model
using a modified Monte Carlo method
Andrey E. Kovtanyuk a,b, Nikolai D. Botkin c,, Karl-Heinz Hoffmann c
a Institute of Appl. Math. FEB RAS, Radio St. 7, 690041 Vladivostok, Russiab Far Eastern Federal University, Sukhanova St. 8, 690950 Vladivostok, Russiac Technische Universitt Mnchen, Zentrum Mathematik, Boltzmannstr. 3, D-85747 Garching b. Mnchen, Germany
a r t i c l e i n f o
Article history:
Received 1 March 2011
Received in revised form 18 October 2011
Accepted 24 October 2011
Available online 22 November 2011
Keywords:
Radiative heat transfer
Conductive heat transfer
Monte Carlo method
Diffusion approximation
a b s t r a c t
Radiativeconductive heat transfer in a medium bounded by two reflecting and radiating plane surfaces
is considered. This process is described by a nonlinear system of two differential equations: an equation
of the radiative heat transfer and an equation of the conductive heat exchange. The problem is character-
ized by anisotropic scattering of the medium and by specularly and diffusely reflecting boundaries. For
the computation of solutions of this problem, two approaches based on iterative techniques are consid-
ered. First, a recursivealgorithm based on some modification of theMonteCarlomethod is proposed. Sec-
ond, the diffusion approximation of the radiative transfer equation is utilized. Numerical comparisons of
the approaches proposed are given in the case of isotropic scattering.
2011 Elsevier Ltd. All rights reserved.
1. Introduction
The study of the coupled heat transfer [13] where the radiative
and conductive contributions are simultaneously taken into ac-
count is important for many engineering applications. So, Andre
and Degiovanni[4,5], Banoczi and Kelley[6], and Klar and Siedow
[7]have studied the thermal properties of some semi-transparent
and insulating materials in the context of a coupled radiativecon-
ductive model. The mathematical treatment of this nonlinear mod-
el is studied in[811]. In[8], Siewert and Thomas use the simple
iteration method and a computationally stable version of the PNapproximation. In work[9],Siewert has applied the Newton itera-
tion method instead of the simple iteration procedure. This allows
the author to calculate some numerical examples which are not
feasible using the simple iteration method (compare [8]). Kelley
has provided existence and uniqueness theorems for the consid-ered problem in the case of isotropic scattering and non-reflecting
boundaries [10]. An analytical version of the discrete-ordinates
method along with Hermites cubic splines and Newtons method
to solve a class of coupled nonlinear radiationconduction heat
transfer problems in a solid cylinder is proposed in [11]. The algo-
rithm is implemented to establish high-quality results for various
data sets which include some difficult cases.
In our paper, some iterative algorithm for solving this problem
is considered. For the calculation of solutions of the radiative trans-
fer equation, two ways are used. The first approach proposed by
the authors utilizes a recursive algorithm based on some modifica-
tion of the Monte Carlo method. This algorithm suits for the appli-
cation of parallel calculations, and hence it can provide a good
accuracy within a reasonable computing time. The second ap-
proach uses the diffusion approximation of the radiative transfer
equation. It is shown that using this approximation gives a good
description of the solution behavior. A numerical comparison of
the approaches proposed is done in the case of isotropic scattering
and reflecting boundaries. The calculations are implemented on a
computer cluster of the Technical University of Munich using the
technology of parallel computing supported by the application pro-
gramming interface OpenMP.
2. Problem formulation
Let us consider the coupled radiativeconductive heat transfer
problem which is formulated as in[8,9]. The equation of the radi-
ation transfer for a homogenous layer is written in the normalized
form as
lIss;l Is;l x2
Z 11pl;l0Is;l0dl0 1xH4s; 1
whereI(s,l) is the normalized density of the radiation flux at thepoint s e[0,s0] in the direction which angle cosine with the positive
0017-9310/$ - see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2011.10.045
Corresponding author.
E-mail addresses: [email protected] (A.E. Kovtanyuk), [email protected]
(N.D. Botkin),[email protected](K.-H. Hoffmann).
International Journal of Heat and Mass Transfer 55 (2012) 649654
Contents lists available atSciVerse ScienceDirect
International Journal of Heat and Mass Transfer
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j h m t
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2011.10.045mailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2011.10.045http://www.sciencedirect.com/science/journal/00179310http://www.elsevier.com/locate/ijhmthttp://www.elsevier.com/locate/ijhmthttp://www.sciencedirect.com/science/journal/00179310http://dx.doi.org/10.1016/j.ijheatmasstransfer.2011.10.045mailto:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2011.10.0458/10/2019 2011 Aplicado a Radiacion
2/6
direction of the axiss isl e[1,1];x < 1 the albedo of single scat-tering;p(l,l0) the phase function;H(s) the normalize temperature.Note that the case of non absorbing media (x = 1) is excluded fromthe consideration as unrealistic. Introduce the following sets for the
definition of boundary conditions:
C f0g 0;1 [ fs0g 1; 0;C f0g 1;0 [ fs0g 0;1:We supply Eq.(1)with the boundary conditions
In;l hn BIn;l; n;l 2C; 2
where the functionh and the operator B are defined by
h0 :e1H41; Bf0;l :
qs1I0;l 2qd1Z 10
I0;l0l0dl0; l >0;
hs0 :e2H42; Bfs0;l :
qs2Is0;l 2qd2Z 10
Is0;l0l0dl0; l< 0:
Here, H1 and H2 are the normalized temperatures on the bound-
aries;qsi andqdi the coefficients of specular and diffuse reflections,
respectively; ei 1 qsi qdi the emissivity coefficients for the
boundary surfaces. It is assumed that e1, e2> 0, which provides the
estimate kBk < 1 (see Section 3). Note that the first summand onthe right-hand side of the definition of the operator B describes
the contribution of the specular reflection, whereas the second
one describes the contribution of the diffuse reflection.
The equation of the conductive heat transfer is written as
H00s 12Nc
Z 11Is;lldl
0; 3
andNcis the conduction-to-radiation parameter[8]. For Eq.(3), we
set the following boundary conditions:
H0 H1; Hs0 H2: 4For finding the solution of system(1)(4), we will use a simple
iteration method with parameter. According to that, choose an
initial approximation of the temperature H(s) (for example, thelinear approximation which corresponds to zero value of the
right-hand side of (3)) and denote it as Hh0is. Then, substituteHh0is into(1) instead of the function H(s), find the solution ofthe problem(1) and (2), and denote it as Ih1is;l. Then, find thesolution of the problem (3) and (4) under the given function
Ih1is;l and denote it as ~Hh1is. Choose a small positive real aand set Hh1is a ~Hh1is 1 aHh0is to be the next approxi-mation ofH(s). Then, putHh1isinstead of the function H(s) intoEq. (1), find the next approximation Ih2is;l, and so on. Thus, inthejth step, we use the functions Hhj1isand ~Hhjisto determinethe next approximation of the function H(s) by the followingformula:
Hhj
is a~
Hhj
is 1 aHhj
1
is: 5The main complexity in the numerical realization of this itera-
tive method is finding the solution of the radiative transfer Eq.
(1). For its treatment, we will mainly use a recursive algorithm
based on the Monte Carlo method. As alternative, we will construct
a diffusion approximation of Eq. (1)(P1 approximation). We will
compare the results of these approaches with the numerical data
from[8,9].
3. Solvability of the radiative transfer equation
Let us consider the problem (1) and (2). We assume that the
function H(s) is nonnegative, and H(s) eCb(0, s0), where Cb(X) isthe Banach space of functions bounded and continuous on Xwith
the normkukCbX supx2X
juxj. Also, letpl;l0 2CbX X, where
X 1; 0 [ 0; 1, and
1
2
Z 11pl;l0dl01:
Note that the operatorB : CbC ! CbC
is linear, bounded, non-
negative, and
kBk maxi
qsi qdi< 1:
DenoteX 0; s0 1; 0 [ 0; 1. We define a class D(X) wheresolutions Iof the problem(1) and (2)are sought.
A function I(s, l) belongs to D(X), if the following properties hold:
(1) I(s, l) is absolutely continuous in se
(0, s0] for alll > 0, andabsolutely continuous ins e[0, s0) for alll < 0;
Nomenclature
A an integral operatorB operator of reflectionCb Banach space of bounded and continuous functionsD a functional classI normalized density of the radiation flux
Ihji radiation flux in the jth step of the iterative procedureIn radiation flux in thenth step of the recursive procedureh input radiation fluxL a linear operatorM number of recursive trajectoriesN number of summands of the truncated Neumann seriesNc conduction-to-radiation parameterp phase functionS an integral operatorT operator of the Neumann seriesX the set of optical and angular variables
Greek symbolsa iteration parameterC a set used in the definition of boundary conditions
C+ a set used in the definition of boundary conditionsH1 normalized temperature on the left boundaryH2 normalized temperature on the right boundaryH normalized temperatureHhji temperature in the jth step of the iterative procedure
e1 emissivity coefficient of the left boundarye2 emissivity coefficient of the right boundaryx albedo of single scatteringl angular variables optical depth (point of the layer)s0 optical thickness of the layern boundary pointqd1 coefficient of diffuse reflection of the left boundary
qd2 coefficient of diffuse reflection of the right boundary
qs1 coefficient of specular reflection of the left boundary
qs2 coefficient of specular reflection of the right boundary
/0 diffuse approximation of the average flux
650 A.E. Kovtanyuk et al. / International Journal of Heat and Mass Transfer 55 (2012) 649654
8/10/2019 2011 Aplicado a Radiacion
3/6
(2)lIs0(s, l) + I(s, l) eCb(X);(3) I(s, l) eCb(C
) .
For further purposes, we introduce the following function
nl 0; l 2 0;1;s0; l 2 1;0:
The differential expression LIs;l lIss;l Is;l defines alinear operator L: D(X)? Cb(X). In the space D(X), we introducethe norm
kukDmaxfkukCbC; kLukCbXgand notice that the inclusion D(X) Cb(X) holds.
The expressions
Aus;l 1l
Z snlexp s s
0
l
us0;lds0;
Sus;l x2
Z 11pl;l0us;l0dl0;
TIs;l BInl;lexp s
n
ll ASIs;l 6
define linear operators A: Cb(X)?D(X), S: Cb(X)? Cb(X), and
T: D(X)?D(X).
According to[12], the following statements hold:
Theorem 1. A function I is a solution of the problem (1) and (2),iff it
is a solution of the operator equation
Is;l I0s;l TIs;l;
I0s;l exp snll
hnl 1xAH4s;l; 7
in the class DX.
Theorem 2. Assuming that the inequalities ||B|| < 1 and x < 1 hold,there exists a unique solution of the problem (1) and (2) (or of the
integralEq.(7))that can be found in the form of the Neumann series
Is;l X1k0
TkI0s;l 8
converging in the norm of CbX.
Remember that kBk < 1 in our case, and therefore the condi-
tions ofTheorem 2are satisfied.
4. Recursive algorithm based on the Monte Carlo method
Let us consider the iterative algorithm described in Section 2.
For computing a solution to the problem (1) and (2)corresponding
to a given function H(s), we propose a recursive algorithm basedon the Monte Carlo method. According toTheorem 2, there exists
a unique solution of the problems (1) and (2) that can be found
in the form of the Neumann series (8). The Monte Carlo method
is appropriate for computing the finite sums
INs;l XNn0
TnI0s;l: 9
To implement the computation, rewrite(9)as the following recur-
rence relation:
Ins;l TIn1s;l I0s;l; n1;2; . . . ;N:
Let us consider a structure of the operatorT(see Eq.(6)). It containstwo summands: the first one describes reflection effects, the second
one describes the contribution of scattering effects. Consider the
second summand in more detail. Applying simple transformations,
we rewrite it in the form
Js;l : ASIs;l x2 1exp sn
l
Z s
n Z 1
1
exps s0=ll1
exp
s
n
=l
pl;l0Is0;l0dl0ds0;
10where n= n(l). According to the Monte Carlo method, we canapproximate the integral in this expression as the mean value of a
random sequence defined by the random variables s0 andl0 distrib-uted over the intervals (n, s) and (1, 1) with the densities
exps s0=ll1expsn=l ; and
1
2pl;l0; 11
respectively. Therefore, the integral (10) is being approximated
with the following finite sum:
Js;l xM
1 exp s nll
XM
k
1
Izk;lk:
Here,zk,lk,k = 1, 2, . . . , Mare independent realizations of the ran-dom variables s0 andl0 distributed over the intervals (n, s) and(1, 1) with the densities (11). Hence, we can approximate the
functions In(s, l),n = 1, 2, . . . , N, as follows:
Ins;l Ins;l 1M
XMk1sn;ks;l; I0s;l I0s;l; 12
sn;ks;l BIn1nl;lexp snll
x 1 exp snll
In1zk;lk I0s;l: 13
Thus, the finite sum (9)can be calculated using the recurrence rela-
tions(12) and (13).For computing the second summand of the right-hand side of
recursive relation (13), we have to generate points zk and lk,k= 1, . . . , M, distributed on the intervals (n(l), s) and (1, 1),respectively, with the densities given by (11), respectively. The
points zk are defined as follows:
zks lln1 ak1exps n=l;whereak are independent realizations of a random variable uni-formly distributed on the interval (0, 1). The generation of the angu-
lar values,lk, is governed by the phase function p(l, l0). Thus,
lk2ak1in the case of isotropic scattering, and
lkl11ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 l2 4lakq
in the case where p(l, l0) = 1 + ll0 (binomial scattering law, see[13]).
For computing the term BIn1 appearing in the first summand of
the right-hand side of(13), the angular variable involving in the
definition of the specular part of the operatorB (see the definition
ofB and the remarks to its structure) remains deterministic and
equals tol. The diffusive part ofB is being computed using ran-dom values generated as
l0k sgnlffiffiffiffiffiak
p ;
whereak are independent realizations of a random variable uni-formly distributed on the interval (0, 1) .
It should be noted that the above recursive algorithm based onthe Monte Carlo method is suitable for the utilization of parallel
A.E. Kovtanyuk et al. / International Journal of Heat and Mass Transfer 55 (2012) 649654 651
8/10/2019 2011 Aplicado a Radiacion
4/6
computing technologies. There are two basic ways for the parallel-
ization of the computing process. First, the calculation of the func-
tionIat each point of the layer is performed by a separate thread.
Second, the generation of each recursive trajectory of the Monte
Carlo method is performed by a separate thread. Thus, the algo-
rithm proposed is aimed to multiprocessor systems and moderne
grid computing.
Note, that the method of discrete ordinates, which is rather
popular in the one-dimensional case, shows a restricted paralleliz-
ability so that it is hardly applicable in three dimensions.
5. Implementations of the iterative method
In this section, different approaches to the numerical solution of
the coupled problem(1)(4)are considered. Section5.1proposes
two kinds of recursive relations based of the Monte Carlo method.
The first one is valid in the general case of anisotropic scattering,
and the second one is applicable to the case of isotropic scattering
only. In Section5.2, a model based on the diffusion approximation
of the radiation transfer equation is derived in the case of isotropic
scattering.
5.1. Recursive relations based on the Monte Carlo method for the
coupled heat transfer problem
For more stable numerical implementation of the iterative pro-
cedure described in Section2, we express the solution of the prob-
lem(3) and (4)for a given function I. After integrating Eq.(3), we
obtain
Hs 12Nc
Z s0
Z 11If;lldldfC1sC2; 14
Here the constantsC1 andC2are defined from the boundary condi-
tions(4):
C1s10 H2H1 1
2Nc
Z s0
0
Z 1
1If;lldldf
; C2H1:
Assume that the approximation Hhj1is of the temperature is al-ready obtained in the (j1)th step of the iterative procedure of
Section 2. On the basis of(5) and (14), using the Monte Carlo meth-
od, we obtain the following approximation of the temperature in
thejth step:
Hhjis 1 aHhj1is a sMNc
XMk1
lkIhjiNzk;lk C1s C2
!;
15
where the random variables zk andlk are uniformly distributedover the intervals (0, s) and (1, 1), respectively. The computationof the I
hjiNs;l is implemented on the base of formulas(12) and
(13)assuming that the temperature is equal to Hhj1is. The con-stant C1 is also computed on the base of the Monte Carlo method.
Thus, the approximationHhjis of the function Hhjis is computedby formulas(12), (13), and (15).
The next approach, which is considered under assumptions of
isotropic scattering, is based on different analytical representations
of solutions of the coupled heat transfer problem(1)(4). Accord-
ing to[10], we obtain from Eq. (1):
Z 11Is;lldl
021x H4s 1
2
Z 11Is;ldl
:
After substitution of this expression into (3)and the integration, weobtain
Hs 1 xNc
Z s0
Z f0
H4x 12
Z 11Ix;ldl
dxdfC1s C2:
16The constantsC1 andC2 are determined from the boundary condi-
tions(4)as follows:
C1s
10 H
2H
11
x
NcZ s00
Z f0
H4
x
1
2Z 1
1Ix;ldl
dxdf ;C2H1:On the basis of(5) and (16), using the Monte Carlo method, we ob-
tain the following approximation of the temperature in thejth step
of the iterative procedure:
Hhjis 1 aHhj1is
a s1 xMNc
XMk1xk H
hj1izk 4 IhjiNzk;lk C1sC2
!;
17
wherexk,zk,lkare numerical realizations of random variables uni-
formly distributed on the intervals (0, s), (0,xk) and (1, 1), respec-tively. The computation ofI
hjiNs;l is implemented on the base of
formulas(12) and (13) assuming that the temperature is equal to
Hhj1is. The constantC1is also computed on the base of the MonteCarlo method. Thus, the approximation Hhjis of the functionHhjis is computed using formulas(12), (13), and (17).
Thus, two kinds of recursive relations based on the Monte Carlo
method are proposed in this subsection. The proposed approaches
allow us to avoid the instability that occurs due to differentiating
in the right-hand side of Eq. (3).
5.2. Diffusion approximation for the coupled heat transfer problem
Now we consider an approach based on the diffusion approxi-
mation (also named P1 approximation, see [14]). Note that thismethod is mostly applicable in the case of isotropic scattering.
We represent the function I(s, l) by the two first summands inthe Fourier expansion in Legendre polynomials:
Is;l /0s l/1s: 18It gives us the following approximation of Eq.(1):
/000s 31x/0s 31xH4s; 19There are different approaches to the derivation of the boundary
conditions for the diffusion approximation(19). Exemplary discus-
sions of this issue can be found in book [14]. In the present work, we
choose the following two ways. In the first way, we substitute the
expansion (18) into the boundary conditions (2) instead of the func-
tion I(s, l) and integrate(2) over all incoming directions l of thelayer. This yields
e1/00 1
2 1 qs1
4
3qd1
/000 e1H41; 20
e2/0s0 1
2 1 qs2
4
3qd2
/00s0 e2H42: 21
In the second way, we use the Marshak boundary conditions[15]. It
gives
e1/00 2
31 qs1qd1/000 e1H41; 22
e2/0s0 2
3 1 qs2qd2/00s0 e2H
42: 23
652 A.E. Kovtanyuk et al. / International Journal of Heat and Mass Transfer 55 (2012) 649654
8/10/2019 2011 Aplicado a Radiacion
5/6
With new notations, Eq. (3)is rewritten as follows:
H00s r/000s; r 1
3Nc: 24
Note that the function /0s is interpreted here as the functionI(s, l) averaged over all directionsl .
From Eq.(24),we obtain
Hs r/0s C1sC2: 25The constantsC1 andC2 are defined from the boundary conditions
(4)as follows:
C1s10H2H1r/0s0 /00; C2H1r/00:Thus, the coupled problem (1)(4) is reduced to the coupled system
of Eqs.(19), (20), (21), and (25) (or(19), (22), (23), and (25)).
In the next section, we present the results of numerical experi-
ments based on the above proposed approaches.
6. Numerical experiments
Numerical experiments are carried out for two problems con-
sidered by Siewert and Thomas[8]and Siewert[9]where the sim-ple iteration procedure [8] and Newtons iteration method [9]
combined with a computationally stable version of thePNapprox-
imation have been used. In both cases, the following values of
parameters are taken:x 0:9, s0= 3, H1= 1, qs1 0:1, qd1 0:2,
e1 0:7, H2= 0.5,qs2 0:3, qd2 0:1, and e2 = 0.6. The difference
between the two considered problems consists in the value of
the conduction-to-radiation parameter Nc. The calculations are
implemented forNcequals 0.05 and 0.00001. The last value ofNccorresponds to the case of high temperatures.
Fig. 1 presents the following approximations of the temperature
H(s) (Nc= 0.05): first, computed on the basis of the Monte Carlorecursive algorithm(12), (13), and (15); second, computed on the
basis of the diffusion approximation (19), (20), (21), and (25);
and third, obtained by Siewert and Thomas[8]. For the implemen-tation of the Monte Carlo method, the following values are taken:
the number of the summands of the Neumann series, N= 14; the
number of the generated trajectories, M= 10000. The diffusion
approximation is implemented with the Maple 9.5. For both ap-
proaches, 20 steps of the iterative algorithm are used. The param-
eter a of the iteration method described in Section 2 is chosen to beequal 0.5. It is seen all approximations are close enough to each
other. We do not demonstrate results for the diffusion approxima-
tion with the Marshak boundary conditions, because the corre-
sponding plots are visually indistinguishable from those for the
boundary conditions(20) and (21).
Fig. 2 presents the following approximations of the temperature
H(s) for Nc = 0.00001 (this corresponds to higher temperaturescompared with the previous case): first, computed on the basis
of the Monte Carlo recursive algorithm (12), (13), and (15); second,
computed on the basis of the diffusion approximations (19), (20),
(21), and (25); third, computed on the basis of the diffusion
approximation corresponding to (19), (22), (23), and (25); and
fourth, obtained by Siewert [9]. In the implementation of the
numerical method, 500 steps of the iterative procedure are used.
The parameter a of the iteration method is chosen to be equal0.0001. It is seen that the deviation of the temperature curves is
more essential than in Fig. 1. Nevertheless, the diffusion approxi-
mation describes the behavior of the temperature properly. Thus,
it can be successfully applied to various heat transfer problems
which are not require obtaining very high accuracy.
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3
Normalizedtemperature
Optical thickness
Fig. 1. Results of numerical simulation for Nc= 0.05: the iterative algorithm based
on the Monte Carlo method after 20 iteration steps (solid curve); the diffusion
approximation based on the boundary conditions (20) and (21) (dashed curve); anddata from[8](squares).
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3
Normalizedtemperature
Optical thickness
Fig. 2. Results of numerical simulation for Nc= 0.00001: the iterative algorithm
based on the Monte Carlo method after 500 iteration steps (solid curve); the
diffusion approximation based on the boundary conditions(20) and (21) (dashed
curve); the diffusion approximation based on the Marshak boundary conditions
(dotted curve); and data from[9](squares).
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3
Normalizedtempera
ture
Optical thickness
Fig. 3. Numerical experiments, Nc= 0.00001, demonstrating a convergence of the
iterative procedure based on the Monte Carlo method. The plots correspond to 50
steps (dotted curve), 150 steps (dashed curve), and 500 steps (solid curve) of theiterative procedure.
A.E. Kovtanyuk et al. / International Journal of Heat and Mass Transfer 55 (2012) 649654 653
8/10/2019 2011 Aplicado a Radiacion
6/6
Fig. 3shows numerical experiments that demonstrate the con-
vergence of the iterative procedure based on the Monte Carlo
method whenNc= 0.00001. The plots correspond to 50 steps, 150
steps and 500 steps of the iterative procedure.
Fig. 4 shows numerical experiments (Nc= 0.00001) that demon-
strate an instability of the iterative procedure based on the Monte
Carlo method. This instability occurs in the case of insufficient
number of the trajectories, M= 2000. The plots correspond to 300
and 900 steps of the iterative procedure. A similar effect is ob-
served in the case of utilizing the diffusion approximation when
few decimal places were used in the computation.
The presented calculations are implemented on a computer
cluster of the Technical University of Munich using the technology
of parallel computing supported by the application programminginterface OpenMP.
7. Conclusion
This paper proposes a modified Monte Carlo algorithm for the
numerical treatment of nonlinear coupled radiativeconductive
heat transfer problems. Compared with PN approximations, the
algorithm proposed allows us to obtain more precise results, be-
cause it deals with the exact model, whereas PN approximations
utilize simplified equations. Compared with the method of discrete
ordinates, the modified Monte Carlo algorithm is well appropriate
for parallelization, because trajectories can be randomly generated
independently on each other, and additionally parallelization over
points of the layer in which the normalized temperature is calcu-
lated can easily be implemented. The potential of parallelization
can be recognized from the second test example given in this pa-
per. Here, the computation of the temperature in each of 20 points
is based on 104 randomly generated trajectories. Therefore, there
are 2 105 independently computable blocks. Thus, the develop-
ment of multiprocessor systems will provide the permanently
growing speedup of the modified Monte Carlo algorithm so that
it expects to show a good performance in complicated cases, in
particular, for thee dimensional problems.
Acknowledgements
This publication was supported in part by the German Aca-
demic Exchange Service (DAAD); German Research Society (DFG),
SPP 1253; Award No. KSA-C0069/UK-C0020, made by King Abdul-
lah University of Science and Technology (KAUST); and Ministry of
Education and Science of Russian Federation (state contracts
14.740.11.0289, 14.740.11.1000, 16.740.11.0456, 07.514.11.4013).
References
[1] M.N. Ozisik, Radiative Transfer and Interaction with Conduction and
Convection, John Wiley, New York, 1973.[2] M.F. Modest, RadiativeHeat Transfer, 2nded., Academic Press, NewYork,2003.
[3] R. Viskanta, Heat transfer by conduction and radiation in absorbing and
scattering materials, J. Heat Transfer 87 (1965) 143150.
[4] S. Andre, A. Degiovanni, A theoretical study of the transient coupled
conduction and radiation heat transfer in glass: phonic diffusivity
measurements by the Nash technique, Int. J. Heat Mass Transfer 38 (18)
(1995) 34013412.
[5] S. Andre, A. Degiovanni, A new way of solving transient radiativeconductive
heat transfer problems, J. Heat Transfer 120 (4) (1998) 943955.
[6] J.M. Banoczi, C.T. Kelley, A fast multilevel algorithm for the solution of
nonlinear systems of conductiveradiative heat transfer equations, SIAM J. Sci.
Comp. 19 (1) (1998) 266279.
[7] A. Klar, N. Siedow, Boundary layers and domain decomposition for radiative
heat transfer and diffusion equations: applications to glass manufacturing
process, Eur. J. Appl. Math. 9 (4) (1998) 351372.
[8] C.E. Siewert, J.R. Thomas, A computational method for solving a class of
coupled conductiveradiative heat-transfer problems, J. Quant. Spectrosc.
Radiat. Transfer 45 (5) (1991) 273281.
[9] C.E. Siewert, An improved iterative method for solving a class of coupledconductiveradiative heat-transfer problems, J. Quant. Spectrosc. Radiat.
Transfer 54 (4) (1995) 599605.
[10] C.T. Kelley, Existence and uniqueness of solutions of nonlinear systems of
conductiveradiative heat transfer equations, Transport Theory Statist. Phys.
25 (2) (1996) 249260.
[11] L.B. Barichello, P. Rodrigues, C.E. Siewert, An analytical discrete-ordinates
solution for dual-mode heat transfer in a cylinder, J. Quant. Spectrosc. Radiat.
Transfer 73 (2002) 583602.
[12] I.V. Prokhorov, I.P. Yarovenko, T.V. Krasnikova, An extremum problem for the
radiation transfer equation, J. Inverse Ill-Posed Problems 13 (2005) 365382.
[13] H.G. Kaper, J.K. Shultis, J.G. Veninga, Numerical evaluation of the slab albedo
problem solution in one-speed anisotropic transport theory, J. Comp. Phys. 6
(1970) 288313.
[14] D.S. Anikonov, A.E. Kovtanyuk, and I.V. Prokhorov, Transport Equation and
Tomography, VSP, Utrecht, 2002.
[15] R. Marshak, Note on the spherical harmonic method as applied to the Milne
problem for sphere, Phys. Rev. 71 (7) (1947) 443446.
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3
Normalize
dtemperature
Optical thickness
Fig. 4. Numerical experiments, Nc = 0.00001, demonstrating instability of the
iterative procedure based on Monte Carlo method. This instability occurs in the
case of insufficient number of trajectories. The plots correspond to 300 steps (solid
curve) and 900 steps (dashed curve) of the iterative procedure.
654 A.E. Kovtanyuk et al. / International Journal of Heat and Mass Transfer 55 (2012) 649654