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:

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

F1-2=2πXY(ln[(1+X2)(1+Y2)1+X2+Y2]12+X 1+Y2arctanX 1+Y2+Y 1+X2arctanY 1+X2-XarctanX-YarctanY),

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:

    cF1-2
    AbaqusAnalytical
    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:

    cF1-2
    AbaqusAnalytical
    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

  1. 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 F2-1= 1+H2-H,

where H=h/w.

Results and discussion

F1-2
AbaqusAnalytical
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

  1. 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

F1-2=(AB)2π    for    A<0.2andF1-2=1πA2(ln[A2(1+B2)+2]2(Y2+2)(X2+2)+Y2+4[YarctanYY2+4-XarctanXY2+4]+X2+4[XarctanXX2+4-YarctanYX2+4])    for    A0.2,

where A=a/c, B=b/c, X=A(1+B), and Y=A(1-B). Reference solution: F2-1 = 0.4974.

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

  1. 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

F1-2=12B[(B+C)2+4-(C-B)2+4],

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

Results and discussion

F1-2
AbaqusAnalytical
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

  1. 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

F1-2=1πW(Warctan1W+Harctan1H-H2+W2arctan1H2+W2+14ln[(1+W2)(1+H2)1+W2+H2{W2(1+W2+H2)(1+W2)(W2+H2)}W2{H2(1+H2+W2)(1+H2)(H2+W2)}H2]),

where H=h/l and W=w/l. Reference solution: F1-2 = 0.1746.

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

  1. Siegel R. and JRHowell, 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

F1-2=12( 1+H- 1+H2),

where H=h/w.

Results and discussion

F1-2
AbaqusAnalytical
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

  1. Siegel R. and JRHowell, 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.

CF1-2
A = 0.6A = 1.0A = 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

F1-2
AbaqusAnalytical
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

  1. 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.

AF1-2
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

F1-2
AbaqusAnalytical
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

  1. 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 F2-1=1-sinα2.

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°:

    NRF1-2
    AbaqusAnalytical
    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:

    αF1-2
    AbaqusAnalytical
    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 plateF1-2
    AbaqusAnalytical
    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

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