Three-dimensional elemental cavity radiation view factor calculations

These examples verify the use of three-dimensional elemental cavity radiation view factor calculations in Abaqus.

This page discusses:

Identical, directly opposed parallel rectangles
Two infinitely long, directly opposed parallel plates of the same finite width
Coaxial parallel squares of different sizes
Two infinitely long parallel plates of different widths; the centerlines of each plate are connected by the perpendicular between the plates
Two finite rectangles of the same length, having one common edge and at an angle of 90° to each other
Two infinitely long plates of unequal widths h and w, having one common edge and at an angle of 90° to each other
Two rectangles with one common edge and an included angle of Φ
Rectangles having a common edge and forming an arbitrary angle; one rectangle is infinitely long
Two infinitely long plates of equal finite width w, having one common edge and having an included angle of α to each other

ProductsAbaqus/Standard

Relatively simple configurations were selected for these verification problems to ensure that analytical solutions or tabulated results could be found. In some cases certain parameters such as the distance between two surfaces or the number of elements on a surface were varied to illustrate the effects of these parameters on view factor calculations within Abaqus. To duplicate the tabulated results for the cases where parameters were varied, the user can modify the input files provided with the Abaqus release.

Identical, directly opposed parallel rectangles

Problem description

Analytical solution

\begin{matrix} F ​_{1 - 2} & = \frac{2}{π X Y} (\ln {[\frac{(1 + X^{2}) (1 + Y^{2})}{1 + X^{2} + Y^{2}}]}^{\frac{1}{2}} + X \sqrt{1 + Y^{2}} \arctan \frac{X}{\sqrt{1 + Y^{2}}} \\ + Y \sqrt{1 + X^{2}} \arctan \frac{Y}{\sqrt{1 + X^{2}}} - X \arctan X - Y \arctan Y), \end{matrix}

where $X = a / c$ and $Y = b / c$ .

Results and discussion

One element per area (xrvd38n1.inp, xrvd38m1.inp, xrvds4n1.inp and xrvds8n1.inp); c can be varied to obtain the following results:

c F $_{1 - 2}$
Abaqus Analytical
1 0.7370 0.7374
3 0.4236 0.4237
6 0.2090 0.2090
10 0.1001 0.1001
15 0.0502 0.0502
25 0.0195 0.0195
35 0.0102 0.0102
40 0.0078 0.0078
Two elements per area (xrvd38n2.inp); c can be varied to obtain the following results:

c F $_{1 - 2}$
Abaqus Analytical
1 0.7370 0.7374
3 0.4236 0.4237
6 0.2090 0.2090
10 0.1011 0.1001
15 0.0502 0.0502
25 0.0197 0.0195
35 0.0102 0.0102
40 0.0078 0.0078
The Abaqus results for c = 15 are 0.0502 (xrvds3n1.inp and xrvds6n1.inp).

Input files

xrvd38n1.inp: One DC3D8 element is used to discretize each surface of the cavity; c = 15.
xrvd38m1.inp: One DC3D8 element is used to discretize each surface of the cavity; the MOTION option is used to vary the distance between the rectangles.
xrvd38m1.f: User subroutine UMOTION used in xrvd38m1.inp.
xrvd38n2.inp: Two DC3D8 elements are used to discretize each surface of the cavity; c = 6.
xrvds3n1.inp: Two DS3 elements are used to discretize each surface of the cavity; c = 15.
xrvds4n1.inp: One DS4 element is used to discretize each surface of the cavity; c = 15.
xrvds6n1.inp: Two DS6 elements are used to discretize each surface of the cavity; c = 15.
xrvds8n1.inp: One DS8 element is used to discretize each surface of the cavity; c = 15.

References

Howell, J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill Book Company, New York, 1982.

Two infinitely long, directly opposed parallel plates of the same finite width

Problem description

Analytical solution

F {}_{1 - 2}= F ​_{2 - 1} = \sqrt{1 + H^{2}} - H,

where $H = h / w$ .

Results and discussion


F $_{1 - 2}$
Abaqus	Analytical
0.2356	0.2361

Input files

xrvd38p3.inp: One DC3D8 element is used to discretize each surface of the cavity. The infinite extent of the cavity is modeled with three-dimensional periodic symmetry (NR = 15).

References

Howell, J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill Book Company, New York, 1982.

Coaxial parallel squares of different sizes

Problem description

Analytical solution

\begin{matrix} F ​_{1 - 2} & = \frac{{(A B)}^{2}}{π} for A < 0.2 \\ and \\ F ​_{1 - 2} & = \frac{1}{π A^{2}} (\ln \frac{{[A^{2} (1 + B^{2}) + 2]}^{2}}{(Y^{2} + 2) (X^{2} + 2)} + \sqrt{Y^{2} + 4} [Y \arctan \frac{Y}{\sqrt{Y^{2} + 4}} - X \arctan \frac{X}{\sqrt{Y^{2} + 4}}] \\ + \sqrt{X^{2} + 4} [X \arctan \frac{X}{\sqrt{X^{2} + 4}} - Y \arctan \frac{Y}{\sqrt{X^{2} + 4}}]) for A \geq 0.2, \end{matrix}

where $A = a / c$ , $B = b / c$ , $X = A (1 + B)$ , and $Y = A (1 - B)$ . Reference solution: F $_{2 - 1}$ = 0.4974.

Results and discussion

Abaqus results for F $_{2 - 1}$ : 0.4974 (xrvd38n4.inp); 0.4974 (xrvds3n4.inp).

Input files

xrvd38n4.inp: One DC3D8 element is used to discretize each surface of the cavity.
xrvds3n4.inp: Two DS3 elements are used to discretize each surface of the cavity.

References

Howell, J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill Book Company, New York, 1982.

Two infinitely long parallel plates of different widths; the centerlines of each plate are connected by the perpendicular between the plates

Problem description

Analytical solution

F ​_{1 - 2} = \frac{1}{2 B} [\sqrt{{(B + C)}^{2} + 4} - \sqrt{{(C - B)}^{2} + 4}],

where $B = b / a$ and $C = c / a$ .

Results and discussion


F $_{1 - 2}$
Abaqus	Analytical
0.4335	0.4337

Input files

xrvd38p5.inp: One DC3D8 element is used to discretize each surface of the cavity. The infinite extent of the cavity is modeled with three-dimensional periodic symmetry (NR = 15).

References

Howell, J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill Book Company, New York, 1982.

Two finite rectangles of the same length, having one common edge and at an angle of 90° to each other

Problem description

Analytical solution

\begin{matrix} F ​_{1 - 2} & = \frac{1}{π W} (W \arctan \frac{1}{W} + H \arctan \frac{1}{H} - \sqrt{H^{2} + W^{2}} \arctan \frac{1}{\sqrt{H^{2} + W^{2}}} \\ + \frac{1}{4} \ln [\frac{(1 + W^{2}) (1 + H^{2})}{1 + W^{2} + H^{2}} {\frac{W^{2} (1 + W^{2} + H^{2})}{(1 + W^{2}) (W^{2} + H^{2})}}^{W^{2}} {\frac{H^{2} (1 + H^{2} + W^{2})}{(1 + H^{2}) (H^{2} + W^{2})}}^{H^{2}}]), \end{matrix}

where $H = h / l$ and $W = w / l$ . Reference solution: F $_{1 - 2}$ = 0.1746.

Results and discussion

Abaqus results for F $_{1 - 2}$ : 0.1746 (xrvd38n6.inp); 0.1746 (xrvds6n6.inp).

Input files

xrvd38n6.inp: One DC3D8 element is used to discretize each surface of the cavity.
xrvds6n6.inp: Two DS6 elements are used to discretize each surface of the cavity.

References

Siegel, R., and J. R. Howell, Thermal Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, 3rd, 1992.

Two infinitely long plates of unequal widths h and w, having one common edge and at an angle of 90° to each other

Problem description

Analytical solution

F ​_{1 - 2} = \frac{1}{2} (1 + H - \sqrt{1 + H^{2}}),

where $H = h / w$ .

Results and discussion


F $_{1 - 2}$
Abaqus	Analytical
0.2221	0.2229

Input files

xrvd38p7.inp: One DC3D8 element is used to discretize each surface of the cavity. The infinite extent of the cavity is modeled with three-dimensional periodic symmetry (NR = 15).

References

Siegel, R., and J. R. Howell, Thermal Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, 3rd, 1992.

Two rectangles with one common edge and an included angle of Φ

Problem description

Analytical solution

Definitions: $A = a / b$ ; $C = c / b$ .


C	F $_{1 - 2}$
C	A = 0.6	A = 1.0	A = 2.0
0.1	0.894753	0.898003	0.899505
0.2	0.859340	0.868201	0.871800
0.4	0.777610	0.812110	0.822722
0.6	0.665734	0.754703	0.778772
1.0	0.452822	0.619028	0.700100
2.0	0.233632	0.350050	0.521308
4.0	0.117384	0.177461	0.286713
6.0	0.078311	0.118499	0.192535
10.0	0.047002	0.071148	0.115803
20.0	0.023504	0.035583	0.057945

Results and discussion


F $_{1 - 2}$
Abaqus	Analytical
0.6195	0.6190

Input files

xrvd38n8.inp: One DC3D8 element is used to discretize each surface of the cavity; $a = b = c = $ 8.0.
xrvds4n8.inp: One DS4 element is used to discretize each surface of the cavity; $a = b = c = $ 8.0.

References

Howell, J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill Book Company, New York, 1982.

Rectangles having a common edge and forming an arbitrary angle; one rectangle is infinitely long

Problem description

Analytical solution

Definition: $A = a / b$ .


A	F $_{1 - 2}$
A	$Φ = $ 30°	$Φ = $ 45°	$Φ = $ 60°
0.1	0.900022	0.804838	0.690483
0.2	0.872918	0.767740	0.648105
0.4	0.825360	0.706295	0.581494
0.6	0.783499	0.655351	0.529168
1.0	0.711717	0.573951	0.450407
2.0	0.579571	0.441004	0.332686
4.0	0.426592	0.307875	0.225049
6.0	0.341612	0.240643	0.173501
10.0	0.249219	0.171450	0.121970
20.0	0.154976	0.104189	0.073151

Results and discussion


F $_{1 - 2}$
Abaqus	Analytical
0.5732	0.5740

Input files

xrvd38n9.inp: DC3D8 elements are used to discretize the surfaces of the cavity; one element for the finite surface and nine elements with an edge length of eight units for the infinite surface; $a = b = $ 10.0; $Φ = $ 45°.

References

Howell, J. R., A Catalog of Radiation Configuration Factors, McGraw-Hill Book Company, New York, 1982.

Two infinitely long plates of equal finite width w, having one common edge and having an included angle of α to each other

Problem description

Analytical solution

F {}_{1 - 2}= F ​_{2 - 1} = 1 - \sin \frac{α}{2} .

Results and discussion

For this test three parameters can be varied: the angle, the number of reflections, and the number of elements used to model the bottom plate. All of the variations shown in the following tables can be verified by modifying input file xrvd38p0.inp.

One element per plate, $α = $ 60°:

NR F $_{1 - 2}$
Abaqus Analytical
2 0.4969 0.5000
4 0.4900 0.5000
8 0.4993 0.5000
12 0.4994 0.5000
16 0.4994 0.5000
20 0.4994 0.5000

One element per plate, NR $=$ 12:


$α$	F $_{1 - 2}$
$α$	Abaqus	Analytical
10°	0.9123	0.9128
20°	0.8258	0.8264
30°	0.7406	0.7412
40°	0.6574	0.6580
50°	0.5768	0.5774
60°	0.4994	0.5000
70°	0.4258	0.4264
80°	0.3566	0.3572
90°	0.2923	0.2929

NR $=$ 2, $α = $ 60°:

# of elements on bottom plate F $_{1 - 2}$
Abaqus Analytical
1 0.4969 0.5000
3 0.4969 0.5000
6 0.4969 0.5000
9 0.4969 0.5000
12 0.4969 0.5000
15 0.4969 0.5000

Input files

xrvd38p0.inp: One DC3D8 element is used to discretize each surface of the cavity. The infinite extent of the cavity is modeled with three-dimensional periodic symmetry (NR = 12); $α = $ 60°.

References

Siegel, R., and J. R. Howell, Thermal Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, 3rd, 1992.