Radiative Heat Transfer in Acute Forward Anisotropic ... · Radiative Heat Transfer in Acute Forward Anisotropic Scattering Media Using the Discrete Ordinates Method Pedro M. Granate

Radiative Heat Transfer in Acute Forward Anisotropic Scattering Media Usingthe Discrete Ordinates Method

Pedro M. Granate

Instituto Superior Tecnico, Universidade de Lisboa, Lisboa, [email protected]

Abstract

The discrete ordinates method (DOM) is a widely known computational method for solving the radiative transferequation (RTE). However, when in presence of acute forward anisotropic scattering media, this method fails toverify the conservation conditions regarding the scattering energy and phase function asymmetry factor. Becauseof this, the phase function must undergo a normalization procedure so that the conservation of the scatteringenergy and/or asymmetry factor are ensured.

Various authors have proposed different normalization techniques, from which the following stand out: Liu etal. [1], Hunter and Guo 2011 [2] and 2014 [3], Boulet et al. [4] with a finite volume method (FVM) approach andEvans [5] with a hybrid method entitled spherical harmonics discrete ordinates method (SHDOM).

In this study, all these techniques are applied to four benchmark test cases. The first being a two-dimensional en-closure, the second an axi-symmetric geometry, the third a three-dimensional cubic geometry with pure scatteringmedia and the fourth a three-dimensional transient problem with collimated radiation.

It was concluded that the asymmetry factor together with the phase function shape, are the most important factorswhen deciding whether or not to implement a normalization technique and which technique to implement. All testsshow that both Hunter and Guo normalizations, and SHDOM are able to produce accurate results. FVM producesresults which are on agreement with the previous three except for the second test. Liu et al.’s technique revealsto be the most inaccurate for g > 0.90 over-predicting the results when compared to the previous techniques.

Keywords: Radiative transfer, Anisotropic Scattering, DOM, SHDOM, Renormalization

1. Introduction

The discrete ordinates method (DOM) was initially developed in 1960, by Chandrasekhar, in his work about inter-stellar radiation [6]. A few years later, Carlson and Lathrop [7] and Lee [8] made the method popular by applyingit to the neutron transport theory. There were some attempts to reapply the method to thermal radiation, fromwhich the works of Love et al. [9], Hottel et al. [10] and Roux and Smith [11] stand out. However, it was only in the80’s, with the outstanding work published by Fiveland [12,13] and Truelove [14,15], that the method truly becameoptimized and received great attention by the radiative heat transfer scientific community. The DOM consistsbasically in calculating the radiative transfer equation (RTE) integrals by numerical approximation and solving thisequation for a certain number of directions which span over the total solid angle of 4π [16]. As such, the DOM issimply a finite differencing method of the directional dependence of the radiation. Over the years, many angularand spatial discretization schemes have been developed such as the step, clam (spatial) and the SN quadratures(angular).

Associated with this method there are several documented problems, being the most well-known the false scatter-ing and ray effect, both studied by Chai et al. [17] and Coelho [18], among others. Other limitations arise when thismethod is applied to acute forward anisotropic scattering media, specially, those in which their phase functionsexhibit very strong peaks. This problem was studied by many authors, such as Kim and Lee [19] and Boulet et

1

Nomenclature

β Extinction coefficientχl Coeficient of the Legendre polinomial of or-

der lη Diretor cosin for direction yκ Absorption coefficientΛ Normalized associated Legendre Functionsµ Diretor cosin for direction zΩ Solid angleω AlbedoΦ Phase functionρ Reflectivityσs scattering coefficientε Emissivityξ Diretor cosin for direction xA Area

g Asymmetry factorI IntensityJ Source termM Number of quadrature directionsn Normal to the wallrw Position on the walls Direction of propagationV Volumew Quadrature weightb Blackbody.i Diferent direction.P In node Pj In one direction, jl In one direction, lm In one direction, m

al. [4], and it is intrinsically connected to the angular dicretization, which, for these particular phase functions,neither conserves the scattering energy nor the asymmetry factor. In the attempt to solve this problem, severalauthors have proposed normalization techniques for the phase function. These remarkable attempts include thework of Liu et al. [1] to conserve the scattering energy and the works of Hunter and Guo, 2011 [2, 20, 21] and2014 [3], which conserve both quantities. Boulet et al. claim that the FVM approach applied to the DOM alsoconserves the scattering energy and the asymmetry factor [4]. Leaving the branch of thermal radiation, Evans,in 1996, proposed a new hybrid method, which combines the DOM with the spherical harmonics (SH). Accordingto the author, this method manages to calculate the integral part of the RTE, which relates do the in-scatteringphenomenon, accurately, without using the angular discretization of DOM, using only the Legendre coefficientsregarding the phase function in study [5].

The main goal of this study is to compare all the phase function normalization methods previously mentioned.That will be achieved firstly by studying the phase function itself, and what changes the several normalization tech-niques produce in its shape and energy conservation. Secondly, by applying the several normalization techniquesto benchmark tests where their accuracy in calculating heat fluxes will be determined and compared. Thesebenchmark tests include several types of geometry, two-dimensional, axi-symmetric and three-dimensional; sev-eral types of media either absorbingor non absorbing, optically thick and optically thin; applied to stationaryproblems with diffuse emission and time-dependent problems with collimated radiation.

Although the behaviour of several normalization techniques has already been documented, it is the first time, tothe author’s best knowledge, that so many techniques are compared in so many different case studies. Also,the application of the SHDOM to thermal radiation problems with acute forward anisotropic scattering media is anovelty, to the author’s knowledge.

2. Main Concepts

2.1. Mathematical formulation - RTE and DOM

The steady-state RTE for a gray, absorbing-emiting and scattering medium can be written as [16]:

∂I∂s

= s ·∇I = κ Ib − βI +σs

4π

∫4π

I(si)Φ(si ,s) dΩi . (1)

where β is the extinction coefficient β = κ + σs . The therm κ Ib is the emission term, βI is the attenuation due toabsorption and out-scattering term and the

∫4πI(si)Φ(si ,s) dΩi is the in-scattering term from direction si to direction

s.

2

The enclosure walls are taken to be grey, diffuse emitters (if Tw > 0). The intensity at any given point of theenclosure wall is given by:

I(rw , s) = ε(rw )Ib (rw ) +ρ(rw )π

∫n·s<0

I(rw , s′)|n · s|dΩ′. (2)

where n is the surface outward normal. The first term accounts for emission, while the second term accounts forreflection.

When using the DOM these equations are discretized and become [16]:

ImP =

|ξm |Ax Imx,in + |ηm |Ay Imy,in + |µm |Az Imz,in + κ IbP V + σs4πV

M∑l=1

(l,m)

wl IlP Φ(s li ,s

m)

|ξm |Ax + |ηm |Ay + |µm |Az + βV − σs4πwmΦ(sm

i ,sm)V

. (3)

for equation 1 and

Imw = εw Ibw +ρw

π

M∑j=1

(nf ·s j )<0

Ijwwj |n · s j | (4)

for equation 2.

Another thing which is important to define are the heat fluxes at the walls which are given by:

(q)in =∫

cos θi<0I(si) cos θi dΩi ⇒

M∑j=1

(n·s j )<0

wj Ij |n · s j |. (5a)

(q)out =∫

cos θo>0I(so) cos θo dΩo ⇒

M∑j=1

(n·s j )>0

wj Ij |n · s j |. (5b)

(q)net =∫

4πI(s) n ·s dΩ ⇒

M∑j=1

wj Ij |n · s j |. (5c)

being the left side mathematical definition and the right side for its discretization.

In order to satisfy the energy conservation constrain, the phase function must satisfy the following equation

14π

∫4πΦ(si ,s) dΩ⇒

14π

M∑l=1

Φ(li , l) wl = 1 having li = 1, 2, 3...M. (6)

Furthermore the asymmetry factor g, which is a measure of the degree of anisotropy of scattering is related to thephase function in the following manner:

g =14π

∫4πΦ(si ,s) si · s dΩ⇒

14π

M∑l=1

Φ(li , l) wl cos(θli ,l

)having li = 1, 2, 3...M. (7)

3. Henyey- Greenstein Phase Function and Normalization Techniques

3.1. Henyey-Greenstein Phase function

The Henyey-Greenstein phase function is defined as [22] :

ΦHG (θ) =1 − g2[

1 + g2 − 2g cos (θ)]3/2 . (8)

3

where θ is the angle between the two propagation directions (before and after scattering) and g is the asymmetryfactor.

This function can be approximated by a Legendre Polynomial expansion, being the coefficients for such definedas:

χn = (2n + 1) gn. (9)

For axi-symmetric problems, the DOM only needs half as many directions in order to calculate the radiativeintensity field. This is introduced in order to save computacional time, however, the phase function must bechanged, in order to produce physically realistic results, to the following [2]:

Φ(li , l) =12

[ΦHG (θ1) +ΦHG (θ2)] . (10)

where:

cos θ1 = ξliξl + ηliηl + µliµl (11a)

cos θ2 = ξliξl + ηliηl − µliµl (11b)

3.2. Liu et al. normalization

The technique proposed by Liu et al. [1] is very simple. It consists in multiplying the phase function by a normal-ization coefficient, so that the new phase function Φ is given by:

Φ(li , l) = Φ(li , l) ×

14π

M∑l=1

Φ(li , l)wl

−1

having li = 1, 2, 3...M (12)

This technique, however, does not conserve the asymmetry factor g, which distorts the phase function and pro-duces inaccurate results.

3.3. Hunter and Guo Normalizations

In 2011, Hunter and Guo [2,20,21] proposed a new normalization technique, which ensures not only the conser-vation of the scattering energy but also the asymmetry factor. The new normalized phase function Φ must thensatisfy the following constrains:

14π

M∑l=1

Φ(li , l) wl = 1 having li = 1, 2, 3...M. (13a)

14π

M∑l=1

Φ(li , l) wl cos(θli ,l

)= g havng li = 1, 2, 3...M. (13b)

3.3.1. 2011 version

Hunter and Guo stated that the new normalized function Φ was the result of the old non normalized phase functionΦ multiplied by a normalization coefficient in the following way:

Φ(l, li) = [1 + A (l, li)]Φ(l, li) para li e l = 1, 2, 3...M. (14)

where A (l, li) is a two-dimensional vector with the same dimensions as Φ(l, li), which contains the normalizingfactors. The new normalized phase function Φ must necessarily satisfy equations 13a and 13b. The new phasefunction has also a symmetry condition imposed such as:

Φ(li , l) = Φ(l, li) (15)

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These three equations (13a, 13b e 15) create an undetermined system with 2M equations and(M2 + M

)/2 un-

knowns. The desired solution can be achieved by calculating the minimum norm of the system through a QRdecomposition, where the system Fx = b, in which F is a m × n matrix, is decomposed as:

F = QR (16)

where Q is an orthogonal matrix m × n and R, an upper triangular matrix n × n. The solution can be calculated inthe following way, assuming R is non singular:

x = QT bR−1 (17)

This method can also be used to normalize the ballistic phase function Φ(l, lB ) where lB is the direction of thecollimated beam. Equations 13a and 13b are still valid for li = lB if the ballistic direction doesn’t happen to be withinthe quadrature set of directions, M. The system obtained is also undetermined with 2 equations and M unknownsand can be solved in the same way, by calculating a minimum norm solution through a QR decomposition [20,23].

3.3.2. 2014 Version

Consider the directions li , li , for which cos(θli ,li

)= 1, and li , l−i for which cos

(θli ,l−i

)= −1, directions for forward and

backward scattering, respectively. Hunter and Guo state that only the terms of the phase function correspondingto these two directions should be normalized, and so:

Φ(li , li) = [1 + A (li)]Φ(li , li) having li , li = 1, 2, 3...M. (18a)

Φ(li , l−i) = [1 + B(l−i)]Φ(li , l−i) having li , l−i = 1, 2, 3...M. (18b)

where A (li) and B(li) are both vectors of dimension M which contain the normalization factor for forward andbackward scattering respectively. Equations 13a and 13b can be rewritten by applying equations 18a and 18b,which, after some manipulation, originate the following explicit solution:

A (li) =1

2Φ(li , li)wli

4π (1 + g) −M∑l=1

Φ(li , l)wl

(1 + cos

(θli ,l

)) . (19a)

B(l−i) =1

2Φ(li , l−i)wl−i

4π (1 − g) +M∑l=1

Φ(li , l)wl

(cos

(θli ,l

)− 1

) . (19b)

having li e l−i = 1, 2, 3...M.

3.4. FVM Normalization

This normalization procedure consists in applying the same treatment given to the phase function in the FVM tothe DOM. The new normalized function Φ(li , l) is given by [4]:

Φ(li , l) =

∫∆Ωl

∫∆ΩliΦ(si , s)dΩli dΩl

∆Ωl ∆Ωli. (20)

In order to be able to apply this normalization technique, a polar/azimuthal quadrature scheme, or equivalent,where the boundaries of the solid angles ∆Ωl and ∆Ωli are known, must be used used. In this study this integralis calculated numerically using the Gauss-Legendre integration method with six points for each integral (64 for thetotal integral of equation 20).

This normalization, in theory, conserves both the scattering energy and the asymmetry factor. However, its preci-sion is strongly influenced by the numerical integration method, which may actually lead to the non conservation ofboth these quantities for very high values of g. Hunter and Guo in 2012 [24], proposed a normalization techniquefor the FVM, which presupposes that presented FVM technique is not sufficient to ensure the conservation of thescattering energy and the asymmety factor.

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3.5. SHDOM

This method’s approach is very different from the previous ones, since it does not act on the phase functiondirectly. In 1996, Evans [5], suggested that the in-scattering integration over the solid angle of 4π should be madeusing spherical harmonics (SH). This introduction was suggested with the purpose of joining the advantages ofthe SH, which accurately calculates this integral, with the advantages of the DOM, which excels in calculatingthe propagation of radiation within a domain. However, this approach requires that conversions between SH andDOM, and vice-versa are made.

The hybrid method proposed by Evans presupposes the use of a Gaussian quadrature and works in the followingway. In the first iteration, the radiative intensity field is calculated with DOM. Before the first iteration ends this fieldis converted from the DOM domain to the SH domain using the following equation:

IDOM → ISH Il,m =Nθ∑j=1

wgauss,j Λl,m(θj)Nφ∑i=1

wi,ju(m φi)Ii,j (21)

where Λlm(θj) are the normalized associated Legendre functions, wgauss,j the Gauss weights, wi,j the quadratureweights normalized by the corresponding Gauss weight wgauss,j , which means that, wi,j = wi,j/wgauss,j , u(mφk ) =cos(mφ) for m ≥ 0 and u(mφk ) = sin(mφ) for m < 0.

After that, the second iteration begins and the SH source term Jl,m is calculated from the intensity field throughthe following expression:

SourceSH Jl,m =ωχl

2l + 1Il,m (22)

where ω is the scattering albedo and χl the Legendre coefficient l given by equation 9.

The source term is then converted back from the SH domain into the DOM domain with the following equation:

SourceSH → SourceDOM Ji,j =M∑

m=−M

u(m φi)L∑

l=|m|

Λl,m(θj)Jl,m (23)

The subscripts i and j correspond to the directions subscripts of the Gauss quadrature. These indicate onedirection and, for each pair, a generalized subscript m may be generated. For example for m = 1,i = 1 and j = 1,for m = 2, i = 1 and j = 2, and so on. The in-scattering goes from Ji,j to Jm which is equivalent to:

Jm ⇔

M∑l=1

wl IlP Φ(s li ,s

m) (24)

where the right side represents the calculation currently made by the DOM of the in-scattering term. This term,presented in equation 3 is replaced by Jm and a new radiative intensity field is generated using DOM. Theseconversions are made for each control volume of the domain and the procedure continues in a loop until the con-vergence criteria is achieved. The results obtained with this method are very accurate but it has the disadvantageof not being able to be studied independently, it can only be tested when applied to a certain problem.

4. Results and discussion

4.1. Test Case 1

The first test case is a two-dimensional enclosure, 1mx1m subdivided in 27x27 control volumes, with black wallsin which three are cold and the south wall has unity emissive power. The medium has ω = 1.0 and it is cold.The phase function used is the F1 phase function with g = 0.85. The solution presented was calculated withquadratures of 288 directions. In this test, the net heat flux in y direction is evaluated for x = 0.5 with the stepscheme, figure 1 and with the clam scheme 1. More information regarding the test is available in [19]. As it canbe seen from both figures, both spacial discretization schemes produce solutions very close to each other andvery close to a no normalized solution. It can be seen however, that by using the clam scheme, in FVM and

6

SHDOM, which use different quadrature schemes than the SN , ray effect is evident. This effect was describedby Coelho [18] as the compensation effect, due to the implementation of a spacial scheme that reduces falsescattering. Despite having a high g this phase function’s peak value is four time lower than the one of the HGphase function with the same g. This leads to the results being very close to each other and very close to asolution obtained without the use of any normalization technique. Which indicates that not only the asymmetryfactor g is important in the decision weather or not to implement a normalization technique, but also the peakvalue that the function exhibits. In order to produce a solution with evident differences between the normalizationmethods, the optical thickness should be raised and HG phase function should be used.

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Qy n

et

y

Central Line - STEP

No Normalization Liu et al. Hunter Guo 2011

Hunter Guo 2014 FVM SHDOM

Ref. Kim & Lee

Figure 1: Test case 1 - Net heat flux in y direction for x = 0.5

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Qy n

et

y

Central Line - CLAM

No Normalization Liu et al. Hunter Guo 2011

Hunter Guo 2014 FVM SHDOM

Ref. Kim & Lee

Figure 2: Test case 1 - Net heat flux in y direction for x = 0.5

4.2. Test Case 2

The second test case is an axi-symmetric enclosure with height=2m and radius=1m where all walls are black. Bothaxial walls are cold and the radial wall has unity emissive power. The domain is subdivided in NR × Nz = 25 × 25.The HG phase function is used. In this test, several g were tested for the pure-scattering medium with τ = 25.0 andfor the absorbing medium several τ were tested with g = 0.95. The medium is always cold. Further informationabout this test can be read from [2]. In all the cases the net heat flux was calculated in a line parallel to therevolution axis along the radius wall. Figures 3 and 4 show the results for the non-absorbing medium while figures5 and 6 for the clam scheme, show the results for the absorbing medium. Quadrature schemes with 288 directionswere used. It can be seen that, for g = 0.80 (ω = 1.0), the differences between the several techniques are almostnegligible for both schemes. By increasing g, it is seen that Liu et al. normalization previews a net heat fluxmuch higher than the rest of the normalization techniques. This is caused by the distortion this normalizationtechnique produces to the shape of the phase function. For the FVM, the results obtained show that the heat fluxis considerably lower when compared to the other normalization techniques, and, for g = 0.95 it even producesphysical non-realistic results, being the heat flux negative (not shown in the picture). The remaining normalizationtechniques, SHDOM, Hunter Guo 2011 and 2014, show good agreement with each other. The introduction ofabsorption in the medium attenuates the differences between the normalization methods. It can be seen that byincreasing the optical thickness the differences between the methods are more evident and the results, for theclam scheme, show no signs of ray effect.

4.3. Test Case 3

The third test case is a three-dimensional cubic enclosure with unity side length. All the walls are black andcold except the bottom wall which has unity emissive power. The domain is subdivided into 25x25x25 controlvolumes. The medium is cold, w = 1.0 and τ = 10.0. The phase function used was the HG phase functionwith several asymmetry factors. In this problem the incident heat flux is evaluated on the top wall, along the liney = 0.5. Figures 7 and 8, show the results obtained for the step scheme and for the clam scheme. The reference

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0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Qr n

et

z

STEP τ=25.0 ω=1

Liu et al. SHDOM Hunter Guo 2011

Hunter Guo 2014 FVM Ref. Hunter Guo 2011

g=0.95

g=0.90

g=0.80

Figure 3: Test case 2 - Net heat flux across the wall for r = 1

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Qr n

et

z

CLAM τ=25.0 ω=1

Liu et al. SHDOM Hunter Guo 2011

Hunter Guo 2014 FVM 0.9 Ref. Hunter Guo 2011

g=0.95

g=0.90

g=0.80

Figure 4: Test case 2 - Net heat flux across the wall for r = 1

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0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Qr n

et

z

STEP g=0.95 ω=0.90

Liu et al. SHDOM Hunter Guo 2011

Hunter Guo 2014 FVM Ref. Hunter Guo 2001

τ=1.0

τ=5.0

τ=25.0

Figure 5: Test case 2 - Net heat flux across the wall for r = 1

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Qr n

et

z

CLAM g=0.95 ω=0.90

Liu et al. SHDOM Hunter Guo 2011

Hunter Guo 2014 FVM Ref. Hunter guo 2011

τ=1.0

τ=5.0

τ=25.0

Figure 6: Test case 2 - Net heat flux across the wall for r = 1

solution presented was obtained with the Monte Carlo method. For g = 0.80, it is seen once again that all thenormalization techniques preview the same results and that they are very close to the reference solution. Withthe clam scheme the results are a little closer than with the step scheme. By increasing g one can observe thatLiu et al.’s normalization technique continues to produce results significantly different than the other remainingtechniques, suggesting that this normalization is very limited. On this test case, the results for FVM show goodagreement with the other methods and with the reference solution. This suggests that if this technique is appliedwith cautious, and if the results are compared with a reference solution it can produce very good results. Theremaining three techniques, SHDOM and Hunter Guo 2011 and 2014, produce results in good agreement witheach other and very close to the reference solution. The results with the clam scheme are slightly closer to theMonte Carlo solution which shows the importante of having an accurate spacial discretization scheme.

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0.00

0.05

0.10

0.15

0.20

0.25

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Qz i

n

x

STEP

Liu et al. FVM Hunter Guo 2011 Hunter Guo 2014 SHDOM Monte Carlo

g=0.20

g=0.80

g=0.93

Figure 7: Test case 3 - Incident heat flux on the wall for z = 1, y = 0.5

0.00

0.05

0.10

0.15

0.20

0.25

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Qz i

n

x

CLAM

Liu et al. FVM Hunter Guo 2011 Hunter Guo 2014 SHDOM Monte Carlo

g=0.20

g=0.80

g=0.93

Figure 8: Test case 3 - Incident heat flux on the wall for z = 1, y = 0.5

4.4. Test case 4

The forth test case adresses a transient problem with collimated radiation. This test has significant errors incomparison with the reference Monte Carlo solution as discussed by Pereira in [25]. The details of this test arenot shown here for convenience, one can see them in [25]. Basically, it consists of a cubical enclosure with acold absorbing medium surrounded by cold black walls. On the face x = 0 there is collimated irradiation, whoseintensity follows a Gaussian curve over time. The HG function was used with g = 0.90. The normalizations usedwere Hunter Guo 2011, Liu et al. and FVM. Figures 9 and 10 show the results obtained for the transmittance andreflectance, respectively. Generally it can be seen that the results obtained with a normalization technique aremuch closer to the reference Monte Carlo solution than the results without normalization applied. Hunter e Guo’s2011 normalization seems to be the one to produce better results of reflectance. However, when analysing theresults for transmittance, the Liu et al.’s normalization is able to produce results closer to the peak of Monte Carlosolution. From these results, a general conclusion about which normalization produces the most accurate resultsis imprudent.

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

3.00E-09 3.20E-09 3.40E-09 3.60E-09 3.80E-09 4.00E-09 4.20E-09 4.40E-09 4.60E-09 4.80E-09

T

t

Transmittance

Monte Carlo Hunter Guo 2011 FVM Liu et al. Sem Normalização

Figure 9: Test case 4 - Transmittance over time

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

0.00E+00 5.00E-10 1.00E-09 1.50E-09 2.00E-09 2.50E-09 3.00E-09 3.50E-09

R

t

Reflectance

Monte Carlo Hunter Guo 2011 FVM Liu et al. No Normalization

Figure 10: Test case 4 - Reflectance over time

5. Conclusions

Several normalization techniques were applied to benchmark problems and their results compared. It was shownthat, for the stationary tests, the Liu et al.’s normalization should not be used for high g. FVM’s results were

9

contradictory since this technique performs well on 2D and 3D problems but not in axi-symmetric. The other nor-malization techniques (SHDOM, Hunter and Guo 2011 and 2014) produced very good results, in good agreementwith each other and with the reference solution when provided. For future work, the adaptation of the SHDOMmethod to work with collimated emission whould be investigated.

References

[1] Liu, L., Luan, L., and Tan, H., International Journal of Heat and Mass Transfer 45 (2001) 3259.

[2] Hunter, B. and Guo, Z., International Journal of Heat and Mass Transfer 55 (2012) 1544.

[3] Hunter, B. and Guo, Z., Numerical Heat Transfer, Part B 65 (2014) 195.

[4] Boulet, P., Collin, A., and Consalvi, J., Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007)460.

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