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Simulation of 3D Radiative Heat Transfer Using a Hybrid Numerical Method.

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Dev. Chem. Eng. Mineral Process., 8(3/4),pp.219-232, 2000.
Simulation of 3D Radiative Heat Transfer
Using a Hybrid Numerical Method
X. Huawei, Z. Chuguan", L. Zhaohui and R. Wei
National Laboratory of Coal Combustion, Huazhong University of
Science and Technology, Wuhan, Hubei 430074, P.R. CHINA
A hybrid-combinationof the Monte Carlo and zone methods [I] is based on the zone
method and uses the Monte Carlo method to calculate direct radiative exchange
This approach is developed to analyze three-dimensional radiative heat
transfer with anisotropically scatrering media. Special technique of the Monte Carlo
method for slop walls is presented. The code is applied to two test cases. In the first
test case the program is validated by simulating a rectangular encbsure, in which the
computed temperatures and wall heat fluxes are compared with Hottel's zone method.
In the second case the code is applied to a 100 MW pulverized coal-fired utility boilel:
The geometry is discretised on a Cartesian grid.
Radiation is the dominant mode of heat transfer in many high temperature processes.
In such situations an accurate representation of the radiative heat transfer
phenomenon is important in making accurate predictions of the overall heat transfer.
There are several different computational methods by which radiative heat transfer
problems may be analyzed. The most common methods are zone and Monte Carlo.
The zone method [2] is a well-established method for determining radiative heat
transfer calculations. This method has a long and well-established pedigree and has
been used successfully in a range of applications, notably for furnaces and other high
* Authorfor correspondence.
X. Huawei, Z chuguan, L Zhaohui and R. Wei
temperature heating devices [3].
However, the zone method has two major
disadvantages: (1) it is unable to model heat transfer of variable absorption
coefficients and scatter coefficients, or for an intricate geometry; (2) it is time
consuming when calculating direct exchange areas which include multi-integration.
The Monte Carlo method [4] is a statistical method in which the history of large
numbers of photon bundles is traced. It has the advantage that almost any probiem of
arbitrary complexity can be addressed with relative ease, nevertheless it has the
disadvantages of statistical scatter in the results and of its long computer time.
In this paper, a hybrid method that combines the advantages of zone method and
Monte Carlo method is presented. It uses the Monte Carlo method to calculate direct
exchange area between zones, which avoids the intricate calculation of multiintegration and can treat a complex geometry of zones and non-uniform medium. It
also can reduce statistical errors to a small extent and less computer time required
with fewer bundles for tracing. Therefore, this method fits the criteria for radiation
modeling. A 3-D program is coded to calculate temperature distribution in the
furnace, which considers the scattering influence of radiative media. This paper also
gives a thorough description of the Monte Carlo method in complicated wall
enclosures,and calculationsof two test cases using this hybrid method are presented.
Formulation of Direct Exchange Area
In the zone method, the enclosure surfaces and gas volumes are divided into a finite
number of surface and volume zones. Each gas zone is to be isothermal and have
uniform radiative properties. The surfaces are considered to be diffuse grey emitters.
An energy balance is written for each zone and the radiation exchange between all the
surface volume zones is then calculated.
Central to the zone method are exchange areas. Direct Exchange Areas (DEAs)
give a measure of the amount of radiation emitted by one zone, which is directly
intercepted by another zone. Total Exchange Areas (TEAS) are a measure of the
amount of the radiation emitted by one zone, which is eventually absorbed by another
zone. DEAs and TEASare calculated from the geometric orientation of the zones, gas
attenuation coefficient, and surface emissivities. Formulation of DEAs is given here.
Simulation of 3 0 Radiative Heat Transfer using Hybrid Numerical Method
I Gas-to-SurfaceExchange
The amount of radiant energy emitted by volume element dVi that is directly incident
on a surface element dAj after travelling a distance r through the grey medium is
given by:
The transmittance 2(r) for a medium of varying absorption coefficient (k) is given by:
The radiative energy emitted by an isothermal volume Vi and is transmitted directly to
a surface area A, is then given by:
Similarly, the amount of energy leaving the isothermal area A, that is directly
intercepted by the volume Vi is given by:
Hence the net direct radiative heat exchange between the gas and surface zone is:
The double integral in Equation ( 5 ) is called the gas-to-surface direct area and is
denoted by
X. Huawei, Z Chuguan, L Ulaohui and R. Wei
II Gas-to-Gas Exchange
The amount of radiant energy emitted from volume element dVi , which is absorbed
by volume element dVj, oriented so that four of its edges of length dr and its face of
area dA, are respectively parallel and perpendicular to r, the distance connecting the
volume centers, is found to be given by:
Thus, by the same argument as in the definition of gas-surface exchange, the net
direct radiative heat exchange between two isothermal volume zones Vi and Vj is:
And the gas-to-gas direct exchange area is:
III Surjiwe-to-Su~uceExchange
The amount of radiant energy by a black surface element dAi which is directly
intercepted by another surface element dAj after being partially attenuated by gray
medium is found to be :
Emission in direction
per unit solid angle
Solid angle subtended by
dAj at dAi
The net direct radiative heat exchange between two black, isothermal surface zones Ai
and Aj is given by:
The surface-to-surface direct exchange area is defined as:
s,s, =-'
Simulation of 30 Radiative Heat Transfer using Hybrid Numerical Method
Thus the direct exchange areas are complicated functions of the zone size, shape,
disposition and the distribution of absorption coefficients among the volume zones.
The evaluation of these quantities by analytical methods is normally very difficult
even for some simple zone shapes and dispositions. However, they can be obtained
relatively easily by the Monte Car10 method as follows.
Monte Car10 Calculations of Direct Exchange Areas
From the above derivations the following can be implied: "The direct exchange area
of a zone pair is simply the energy amount absorbed by one zone due to the emission
of the other zone when the blackbody emissive power of the emitting zone f14 is I
and all walls of the system are black so that there is no radiative energy reflection
from the wall. ''
Therefore, we can use the Monte Car10 method to calculate DEAs. Let the initial
emissive power of an energy bundle of the emitting zone in this case be:
(for a
volume zone), P," = 2 (for a surface zone.)
Let I+ be the true path length of an energy coefficient kj (not considering scatter), after
travelling Kis distance the emissive power of the energy bundle lost to zone j is:
This is the contribution of a single energy bundle to the rate of absorption of zone j.
All such contributions are accumulated in a memory
Pa,which on completion of
tracking all energy bundles of zone j, will contain an estimate of the total radiative
flux to zone j. Thus the exchange area between zone i and zone j will be:
(15 )
After crossing volume zone j the energy bundle enters zone k, and if zone k is a
volume zone the above tracking procedure is repeated with the new value of P,"
which is now updated to that of
p,'" . If zone k is a surface zone, then the remaining
energy of the bundle is all absorbed by zone k, and the exchange area between i and j
will be obtained after tracing all bundles lost in j, accumulated in a memory PW':
g i s p ( o r z ) = PWk
X. Huawei, 2 Chuguan,L Zhaohui and R. Wei
If the volume zone medium has a scattering coefficient &d, then the pathlength of the
emitting bundle in zone j will be changed. Here a "probable pathlength to scattering"
is determined by:
1, =-- 1 In R,
(17 )
is a random number in (0,l).
the energy bundle is assumed to pass through zone j without being
scattered. The power loss to the volume zone is calculated with pathlength Lj. The
tracking is repeated in the next zone.
Lj >Is, the energy bundle is assumed to be scattered after travelling a
pathlength I, in zone j. In this case the power loss to zone j is calculated with I , . The
scattering angle a is again determined by R, =
)sins da
(& is a random number in (O,l), f(a) is the scattering angular distribution function).
The tracking is then continued with the energy bundle still in zone j but travelling in a
new direction.
Monte Car10 Techniques The transformation of direction cosines
between Merent system of co-ordinates
Let MI,RYl and RZ1 be the direction cosines of surface energy particles with
respect to a co-ordinatesystem on this surface (see Figure 1):
Is =RX.T+RY .;+RZ .k'
The tracking procedure demands an equivalent expression for direction cosines with
respect to a fixed co-ordinate system OXYZ.
This can be done by a transformation of co-ordinates as follows.
Let i, j, k be the unit vector of the OXYZ co-ordinate system. Let i',j*, k' be the unit
vector of the OX'Y'Z' co-ordinate system. A vector expressed in OX'Y'Z' system as:
Simulation of 3 0 Radiative Heat Transferusing Hybrid Numerical Method
f' =
can be expressed in OXYZ system as:
+ v 2 j -tv3L
unit vectors i',j', k' in the OXYZ co-ordinate system can be written as:
+:527 + I &
I'= I,: + 12j + 13k 5' = .I,
k' = KIT -+ K 2 j + K3k
Thus the first step of the transformation procedure is to find the vector equations for
the unit vectors of the co-ordinate system on the surface in question.
The orthogonal vector of a surface is normally chosen as one axis of the coordinate system on this surface. If the surface is not a plane (a cylindrical or conical
surface) this co-ordinate system is established on the tangent plane of the surface. The
normal vector of this plane is found from the scalar function of the plane:
f (x, y, 2,) = constant
X. Huawei, Z Chuguan,L Zhaohui and R. Wei
The n o d vector is found to be:
where grad( jJis a vector:
4 f z + -4fj+-k
and lgrudfll is the magnitude of grad(f). This vector serves as unit vector l’ in our
The second vector, j’ is on the plane and is freely chosen in such a way that its
components are readily found. The third vector, T‘ is found as the cross product of i’
3’: *; = ?* xg*.
Thus direction cosines of surface particles from whatever source can be expressed:
RX =a,,RX1+021RYl+a,,RZ1 RY = ~ , , R X ~ + C L ~ R Y ~RZ
+ ~=al,RX1+a,RYl+~,RZ1
as the general form.
For the determination of RXl, RYl,RZ1,see reference (51.
Cornparision of DEAs Calculated from Monte Carlo Method and
Analytical Calculation
Here an example of DEAs calculation using Monte Carlo method is given, which
compares the analytical results from reference [a]. For an enclosed cube filled with
Greymedium, we subdivide it into 2~2x2
cells. Given different optical pathlength KB
and radiation bundle number 8000, we obtained different DEAs shown in Figure 2.
For comparison reasons, the dimensionless DEAs are given the following
gs(i,j ) =z(i.
Gb = 4kB3-&I).
gg(i, j ) =z(i,
Simulation of 3 0 Radiative Heat Transfer using Hybrid Numerical Method
Figure 2. Comparision of DEAs bemeen Monte Carlo method and analytical results.
(a)DEAs between two volume zones; (b)DEAs between volume zone and sulface zone;
(c)DEAs between two parallel surface zones; (d)DEAs between two vertical surface zones.
22 7
X. Huawei, Z Chuguan,L zhaohui and R. Wei
As shown in Figure 2.. the results of the Monte Carlo method agree well with the
analytical results. Inevitably, the Monte Carlo approach will bring errors into the
method as it is a statistical method. However, as the number of bundles increases, the
error will be reduced. It has been proved that the accuracy of the Monte Carlo method
is in direct ratio of
(where Ni is the number of bundles).
Radiation Heat lkansfer in a Rectangular Combustion Chamber
The geometry of combustion and medium properties for the base problem examined
by Menguc and Viskanta [7]are Iisted here.
Enclosure: X=2m, Y=2m, M m , boundaries at z = 0, T = 1200K, E = 0.85;
boundaries at z = Z, T = 400K, E = 0.70; other boundaries T = 900K,E = 0.70.
Properties of the medium: B 4 S m , q = 5.0 kW/m3.
Note when other values are employed in the calculations, they are indicated either
directly on the figures or in the captions.
With a rectangular combustion geometry, it can be subdivided into 5x5~10,and
radiation bundles number taking 5000. Temperature distribution of the enclosure and
wall heat flux are calculated as shown in Figures 3 to 6, compared with results of
reference [2] which are calculated using Hottel’s zone method.
Figure 3 shows the temperature distributions in the medium at three axial
locations. The hybrid method and zone method are in very good agreement with
reference [2]. The maximum error does not exceed 6%. Figure 4 compares the heat
flux profiles at the hot and cold surfaces. The error between hybrid method and zone
method is lower than 10%.
Figure 5 shows the effect of phase function on temperature. For isotropic scatter
(g = 0), the temperatures in the medium generally increase because of scattering.
While for anisotropic scattering, the temperatures also increase, but at the same time
the distributions become uniform because forward scattering increases heat transfer
from a hot surface to cold surface. The effect of scattering on the radiative flux
profiles at the hot and cold walls is illustrated in Figure 6.
It is clear from the figures that the hybrid method used for calculating radiative
heat transfer shows a high accuracy compared with the Hottel’s zone method.
Simulation of 3 0 Radiative Heat Transfer using Hybrid Numerical Method
I b.25m'
; &.om"
Figure 3. Temperature distributions in the
medium at three axial locations.
zone method
Ahybrid method
Figure 4. Heat flux profiles at hot and
cold sulface.
zone method
Ahybrid method
z=O.m *
. . . -.
Figure 5. Temperature distributions in the
medium at three axial locations.
zone method
Ahybrid method
Figure 6. Heatflux profiles at hot and
cold surfaces.
Ahybrid method
Zone method
X. Huawei, Z Chuguan, L Zhaohui and R. Wei
Computation of a Coal-Fired Utility Boiler
In the previous section the validity of the hybrid method was identified. In the
following section, computation of radiation of a real boiler is performed to show
whether this method is effective. The structure parameters of the furnace under
investigation and the heat source profile are taken from reference [8].
Figure 7. Temperature distribution in furnace (K).
Simulation of 3 0 Radiative Heat Transfer using Hybrid Numerical Method
Rear wall
Right wall
Front wall
LejI wall
Figure 8. Incident wall radiative heatflux ( x 1.163xld kW)
In order to use this hybrid method, the furnace is subdivided into 20x13~10,the
absorption coefficient at the flame zone is 0.97 and at other zone is 0.3 rn-', scattering
coefficient is 0.3m-', heat resistance is 0.0012 mz.h.K/J and the emissivity of the walls
is 0.8, The temperature distribution in the furnace is calculated and is shown in
Figure 7, the exit average temperature is 1104OC, about 10°C higher than the
measured results. Figure 8 shows that the wall incident radiative heat flux profiles are
in good agreement with the experimental values.
(1) The hybrid method (combined Monte Carlo method and zone method) is
developed to treat three-dimensional radiative heat transfer in absorbingemitting and anisotropically scattering media. A special technique is introduced
to treat slop walls in Monte Carlo method.
X.Huawei, Z Chuguan, L Zhaohui and R. Wei
(2) Direct radiative exchange areas are calculated using the Monte Carlo method
with fewer bundles for good precision. The enclosure test case shows that this
hybrid method is almost as exact as the zone method.
(3) According to the real boiler calculation, this method is valid in modeling
temperature distribution and wall heat flux profiles in the furnace.
This Project was supported by The National Science Foundation of China.
1. Vercammcn, A.J., and Frornent, G.F. An improved zone method using Monte Carlo techniques for
simulation of radiation in industrial furnaces. Int. J. Heat Mass Transfer. 1980,23, pp.329-336.
2. Hottel, H.C., and Sarofun, AF. Radiative Transfer. McGraw-Hill. New Yo& 1967.
3. Steward, F.R.,and Gunrz, H.R. Mathematical simulation of an indushial boiler by the zone method
anaiysis. Heat Transfer in Flames,(N.H.Afgan and J.M. Beer, &). Scrip% Washington, 1974, pp.4771.
4. Howell, J.R., and Perlmutter, M. Monte Carlo solution of thermal transfer through radiant media
betwten gray walls, J. Heat Transfa, 1964,86(1), pp.116-122.
5. Wang, Y.S., Fang, W.C.. Zhou, L.X.,and Xu, X.C. Numerical calculation in combustion processes,
Science Publishing House,China, 1986.
6. Tucker, R.J. Direct exchange azcas for calculating radiation transfer in rectangular fumaccs, J. Heat
Transfer, August 1986, ~01.108.p.707.
7. Menguc, M.P., and Vtskanta, R. Radiative transfer in duee-dimensional mtangular enclosures
containing inhomogeneous, anisotmpically scattuing media, J. Quant. Spectroscopy Rad. Transfer,
I985,33(6), pp.533-549.
8. Xu, X.C. Mathematical modeling of three-dimensional heat transfer from the flame in combustion
chambns. 18th Intemational Symposium on Combustion, pp.1919-1926.
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