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
where and .
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 |
---|
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 |
---|
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).
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
where .
Results and discussion
F |
---|
Abaqus | Analytical |
---|
0.2356 |
0.2361 |
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
where , , , and . Reference solution: F = 0.4974.
Results and discussion
Abaqus results for F: 0.4974 (xrvd38n4.inp); 0.4974 (xrvds3n4.inp).
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
where and .
Results and discussion
F |
---|
Abaqus | Analytical |
---|
0.4335 |
0.4337 |
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
where and . Reference solution: F = 0.1746.
Results and discussion
Abaqus results for F: 0.1746 (xrvd38n6.inp); 0.1746 (xrvds6n6.inp).
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
where .
Results and discussion
F |
---|
Abaqus | Analytical |
---|
0.2221 |
0.2229 |
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: ; .
C | F |
---|
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 |
---|
Abaqus | Analytical |
---|
0.6195 |
0.6190 |
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 | F |
---|
30° | 45° | 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 |
---|
Abaqus | Analytical |
---|
0.5732 |
0.5740 |
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
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, 60°:
NR | F |
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:
| F |
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, 60°:
# of elements on bottom plate | F |
---|
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 |
References
- Siegel, R., and J. R. Howell, Thermal Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, 3rd, 1992.
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