Validation of Radiation Computations using Viewfactors andCOMSOL’s Hemicube Approaches

A. F. Emery∗1, R. J. Cochran2, H. Dillon1 and A. Mescher1

1University of Washington, Seattle, WA 98195,2Applied CHT, Seattle 98040∗ Corresponding Author: emery@u.washington.edu

Abstract: The radiation viewfactor compu-tation is frequently a major part of the compu-tational expense when solving thermal prob-lems. In some situations, the user may haveaccess either to analytic expressions or to spe-cialized viewfactor codes that are specificallytailored to the problem at hand. A method forusing user supplied radiation viewfactors withCOMSOL is described. The method providesa simple and accurate way of treating radia-tion problems. A short discussion of the gridconvergence is provided.

Keywords: Viewfactors, Radiation HeatTransfer, Hemicube

1 Introduction Although there are severaldifferent approaches to computing the radia-tion heat transfer between opaque diffuse sur-faces, one of the most common (and the oneemployed by COMSOL) is based upon radios-ity which is defined as the total radiative fluxleaving a surface and is denoted by J . Themethod is applicable only to isothermal sur-faces with constant radiative properties of emis-sivity, ε, absorbtivity, α, and reflectivity, ρ, forwhich α = ε. The condition that α = ε will besatisfied if a) either i) the irradiation of thesurface is diffuse (i.e., independent of angle)or ii) the surface is diffuse and b) either iii)the irradiation corresponds to emission froma black body at the surface temperature oriv) the surface is gray (properties are indepen-dent of wavelength). Under these conditions,the radiation from surface i with area Ai tosurface j with area Aj is given by

Qij = AiFij(Ji − Jj) (1a)

and

Qi =Aiεi(σT 4

i − Ji)

(1 − εi)(1b)

where Qi is the net thermal energy leaving Ai,Qij is the radiation from Ai to Aj and Fij isthe viewfactor, i.e., the fraction of radiationleaving Ai that is intercepted by Aj . Figure1a is a schematic of the dc electrical analogcircuit for radiation between three surfaces.

22

21

Aε

ε−

11

11

Aε

ε−

33

31

Aε

ε−

232

1

FA

131

1

FA121

1

FA

4

1Tσ

1J

2J3

J4

3Tσ 4

2Tσ

Figure 1 Radiosity Network

The problem can be solved in terms of tem-peratures and either irradiation or radiosity.Siegel and Howell [1]describe several differentapproaches. In terms of radiosity, the appro-priate equations for N surfaces are

N∑

j=1

[δkj − (1 − εk)Fkj]Jj = εkσT 4k 1 ≤ k ≤ m

(2a)N

∑

j=1

(δkj − Fkj)Jj =Qk

Ak

m + 1 ≤ k ≤ N

(2b)

where δij is the Kronecker delta and surfaces1 ≤ i ≤ m have prescribed temperatures andm + 1 ≤ i ≤ N have prescribed heat input.

The problem is clearly non linear because ofthe T 4 terms. In addition, the formulationdepends upon the assumption that the tem-perature, radiosity, and radiative propertiesare constant on a surface. Even when thetemperature and the radiative properties areconstant, the radiosity is rarely constant inreal engineering systems. As a consequenceof the requirement of constancy, we often findthat the radiative surfaces are subdivided intosmaller sub surfaces, often termed ”facets,”for which constancy is a better assumption.As a consequence, the number of radiatingsurfaces often increases to a large number.While the solution of Eqs. 2 is not computa-tionally prohibitive, the computation of theviewfactors can be extremely expensive.

Frequently the user has available either an-alytic expressions for the viewfactors [2] ora separate computer code and wishes to usethese values and avoid having COMSOL com-pute them by using either the hemicube or thedirect area integration methods. Some view-factor codes are specialized and optimized forspecific structures, this is often the case whenmany of the surfaces have obstructed views ofthe other surfaces. The question is how to in-put such viewfactors into COMSOL since inits current version (3.5a) there is no mecha-nism for doing so. The viewfactor matrix isoften stored using a sparse matrix format, aswell as a dense matrix format. The viewfactormatrix must be mapped to the finite elementfaces of the mesh that forms the enclosure.

Figure 2a: Radiative Test Problem

2 Radiative Test Problem As an exam-ple, let us look at the problem described inthe COMSOL Heat Transfer Module User’sGuide [3] on pages 131-141 and shown in Fig-ure 2a. The problem was solved using the ra-diosity formulation and dividing the surfacesinto a number of sub-surfaces, usually termed”facets.” The viewfactors were evaluated us-ing Hottel’s crossed-string method [1]. Figure2b displays the convergence of the solution asthe number of facets is increased (an equalnumber was used on each surface). Conver-gence is measured in terms of the error in theaverage value of the radiosity and tempera-ture on surfaces 2 and 3. The convergence isslightly less than quadratic which is not sur-prising because of the nonlinearity associatedwith the T 4 terms. Figure 3a depicts the spa-tial distribution of the radiosity on each sur-face in terms of the arc length. Although the

radiosity varies strongly with position it doesnot take too many facets to give excellent re-sults as shown in Figure 3b.

0 0.5 1 1.5 2−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

Log10

(Nfacets)

Log

10(%

err

or)

J(2)

J(3)

T(2)

T(3)

Figure 2b: Grid Convergence

0 1 2 3 4 53000

4000

5000

6000

7000

8000

9000

10000

11000

12000

arc length

Radio

sity

A1

A2

A3

Figure 3a: Radiosity

0 5 10 15 20 254500

5000

5500

6000

6500

7000

7500

8000

8500

9000

9500

Number of Facets on each Surface

avd J

A1

A2

A3

Figure 3b: Effect of the Number of Facetson the average Radiosity

In order to compute just the radiative problemusing COMSOL we reduced the block heightfrom 1 to 0.1 and used an orthotropic conduc-tivity of 1 in the direction parallel to the sur-face and 10000 perpendicular to the surface.Two meshes were used, a fine one (hauto=2,2912 quadratic Lagrangian triangles, 6898 dof)and a coarse one (hauto=5, 182 triangles, 637dof). Figure 4 compares the solutions. Thetemperatures, radiative flux, and irradiationmatched, as shown on Figure 4a. However,there was a disagreement for the radiosity forsurface 1, Figure 4b. The values plotted wereextracted from COMSOL running under a Mat-lab m file and it was characteristic of the COM-SOL solution that there was an anomolous be-havior at the ends of the surfaces (this will bediscussed further in section 5). Table 1 com-pares the results.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−4000

−3000

−2000

−1000

0

1000

2000

3000

arc length

Ra

dia

tive

Flu

x

A1

A2

A3

Figure 4a: Radiative Flux, Exact (heavy lines)COMSOL results (light lines)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 53000

4000

5000

6000

7000

8000

9000

10000

11000

12000

arc length

Ra

dio

sity

A1

A2

A3

Figure 4b: Radiosity, Exact (heavy lines)COMSOL results (light lines)

Table 1 COMSOL results

Coarse Fine ExactMesh Mesh Values

637 6898dof dof

∫

ndflux1 -10931 -10966 -11000∫

rflux1 -10932 -10968 -11000∫

G1 29167 29255 29337T 1 300 300 300J1 4559 4572 4584∫

ndflux2 6000 6000 6000∫

rflux2 6000 6000 6000∫

G2 21047 21081 21159

T 2 653 653 654J2 9016 9027 9053∫

ndflux3 5008 4948 5000∫

rflux3 5000 5000 5000∫

G3 31679 31767 31876T 3 603 603 604

J3 7336 7353 7375where

∫

ndflux and∫

rflux are theintegrated normal conductive and radiative

and T is the average temperature.

3 COMSOL Sample Problem

The radiative test problem is not characteris-tic of general heat transfer problems in whichthere is substantial conduction in the parts.To examine the effect of conduction we solvedthe sample problem according to the direc-tions given in the manual, i.e., thick layers,isotropic conductivity of 400 using the finemesh specified as specified, Figure 5a and acoarse mesh, Figure 5b (this is the defaultmesh available in the GUI with hauto=5)..

Figure 4a Fine Mesh forthe Sample Problem

Figure 4b Coarse Mesh for thethe Sample Problem

Using the approach described in the COMSOLmanual and computing the radiation viewfac-tors by the hemicube method, we obtain theresults given in Table 2 for the fine mesh rec-ommended in the manual, Figure 4a, and thecoarse mesh, Figure 4b. The results are listedin Table 2 and show very good agreement be-tween the two meshes. The only substantivedifference between the results were for the ir-radiation and radiative flux as shown in Figure5.

Table 2 Fine and Coarse Mesh Results

Coarse Mesh Fine Mesh270 3013

triangles triangles663 6730dof dof

∫

ndflux1 -10896 -10895∫

rflux1 -10894 -10894∫

G1 29419 29244T 1 307 307J1 4588 4588∫

ndflux2 6000 6000∫

rflux2 6000 6000∫

G2 19390 19303T 2 645 644J2 8434 8434∫

ndflux3 4993 4993∫

rflux3 5000 5000∫

G3 30709 30523T 3 601 600J3 7105 7104

0 1 2 3 4 54500

5000

5500

6000

6500

7000

7500

8000

Arc Length

Irra

dia

tion

Figure 5a Comparison of Irradiation,fine mesh (heavy lines) and coarse mesh

(light lines)

0 1 2 3 4 5−3000

−2000

−1000

0

1000

2000

3000

Arc Length

Radia

tive F

lux

Figure 5b Comparison of Radiative Flux,fine mesh (heavy lines) and coarse mesh

(light lines)

4 Using User Defined Viewfactors Theuser supplied viewfactors can be used in twodifferent ways, both starting with the sameprocedure.

1 Identify all of the radiating surfaces, A(i), 1 ≤

i ≤ m2 Define the average temperature and area of

each surface, TR(i), AR(i)3 Define the boundary condition for each

surface as one of a prescribed heat flux, qR(i).

Method 1

Use a routine that calculates the radiosity andheat flux for each radiating surface using Equa-tions 2. The computed heat fluxes are qR(i).This is most easily done if COMSOL is run un-der Matlab so that the solution to Equations2 can be programmed as a Matlab m file.

Method 2

This method is probably the easiest to imple-ment, certainly with geometries input by handor through an m file. It depends upon the useof exchange factors, usually termed script F,F . In terms of F the radiative exchange be-tween any two surfaces is given by

Qij = σAiFij(T4i − T 4

j ) (3a)

and the flux boundary condition, qR(i) is givenby

qR(i) = σ

N∑

j=1

AjFji(T4j − T 4

i )/AR(i) (3b)

Script F is computed by (note that AiFij =AjFji)

F = D−1M + e (4a)

where with ρ = 1 − ε

D =

1ε(1) −

ρ(2)ε(2)F12 . . . −

ρ(m)ε(m)F1m

......

. . ....

−ρ(1)ε(1)Fm1 −

ρ(2)ε(2)Fm2 . . . 1

ε(m)

M =

−1 F12 . . . F1m

......

. . ....

Fm1 Fm2 . . . −1

e =

ε(1) 0 0 0...

.... . .

...0 0 0 ε(m)

While∑N

j=1 Fij = 1 the exchange factors sat-

isfy∑N

j=1 Fij = εi

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�

��

���

�

��

���

�

��

�

���

�

���

�

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Figure 6 Exchange Factor Network

The results of using one to three facets on eachsurface are compared to the COMSOL resultsfor the coarse mesh in Figure 7. For one facet,the results are constant on each surface and asthe number of facets increases the distributionbegins to approach that of COMSOL using thehemicube method.

Arc Length

0 1 2 3 4 5

Flu

x

-500

0

500

1000

1500

2000

2500

3000

3500

User Viewfactors, 1 Facet

User Viewfactors, 2 Facets

User Viewfactors, 3 Facets

COMSOL Hemicube

Figure 7a Radiative Flux on A3

Arc Length

0 1 2 3 4 5

Tem

per

ature

592

594

596

598

600

602

604

606

608

User Viewfactors, 1 Facet

User Viewfactors, 2 Facets

User Viewfactors, 3 Facets

COMSOL Hemicube

Figure 7b Temperature on A3

0 1 2 3 4 5−3000

−2000

−1000

0

1000

2000

3000

arc length

Radia

tive F

lux

Comsol Coarse Mesh

A6

A9

A3

0 1 2 3 4 54500

5000

5500

6000

6500

7000

7500

8000

arc length

Irra

dia

tion

Comsol Coarse Mesh

A6

A9

A3

a) Radiative Flux b) IrradiationFigure 8 Coarse Mesh

5 Identifying Radiating Surfaces We arerelatively new users of COMSOL and althoughwe have spent significant time in plumbing thedepths of the code, we have not been able tofind an easy automatic way to identify the ra-diating surfaces. Probably the best approachis to input the geometry using a CAD programin which the radiating surfaces have a uniqueidentifier. We hope that some interested userscan develop the logical steps within the COM-SOL model to implement steps 1-3 in eithera Matlab m file or a COMSOL script file.The problem is to relate the boundary num-ber with the facet number used in the view-factor program. Depending upon the geom-etry, COMSOL assigns a boundary number.Small changes in the geometry usually resultin a change in the boundary number. Thusthe user cannot be assured that the bound-ary number is unique. See our paper ”Effectof a Correlated Surface Roughness on Con-duction through a Slab” for an example of thesteps necessary for a user to be able to define aboundary with a unique and unchanging num-ber.6 Concerns about COMSOL’s approach

An examination of Figure 5 shows that theCOMSOL results display an anomalous be-havior at the ends of the surfaces. Unfortu-nately we do not have access to the propri-etary code and were unable to determine thecause of this behavior. The coarse mesh re-sults shown on Figure 8 show this effect muchclearer.

Although the COMSOL calculations give ex-cellent results for the average and total quan-tities, this behavior at the ends of the surfacesmay lead to difficulties if the system being an-alyzed depends upon the computed fluxes asinput to other features.To gain a better understanding of this effectwe examined the simpler system shown in Fig-ure 9 which because of the boundary condi-tions should have zero radiative heat flux onall surfaces.

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��

��

Figure 9 Schematic of Isothermal Cavity

Figure 10 displays the radiative flux on thefloor of the cavity. As for the sample prob-lem, there is a discontinuity at the ends of thesurface. The temperature and the radiosityare correctly computed and the integrated

radiative flux is also correct, but the flux andthe irradiation at the corners is incorrect. With-out access to the code it is not possible for usto suggest any modifications that would elimi-nate this problem. Note that the value plottedon these figures are the nodal values extractedfrom COMSOL and are not what you wouldsee from the Domain plot, line extrusion post-processing

Arc Length

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Rad

iati

ve

Flu

x

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Figure 10 Radiative Flux on the Cavity Floor

Acknowledgment These results were obtainedas part of the research supported by theNational Science Foundation through Grant0626533.

References

1 Siegel, R. and Howell, J. R., Thermal Ra-

diation Heat Transfer, Taylor and Francis,1992

2 Howell, J. R., ”A Catlog of Radiation HeatTransfer Configuration Factors’”,http://www.me.utexas.edu/ howell/

3 COMSOL Version 3.5a, Heat Transfer

Module User’s Guide,COMSOL Multiphysics,

2008