In the zonal method, considerable computational resources are needed to calculate the direct exchange areas (DEA) among the isothermal zones due to integrals with up to six dimensions, while strong singularities occur in the integrands when two zones are adjacent or overlaping (self-irradiation). A special transformation of variables to reduce a double integral into several single integrals is discussed in this paper. This technique was originally presented by Erkku (1959) for calculation of DEA using a uniform zone system in a cylindrical enclosure. However, nonuniform zones are needed for applications with large thermal gradients. Thus we extended this technique to calculate the DEA for non-uniform zones in an axisymmetrical cylinder system. A six-fold reduction in computational time was observed in calculating DEA compared with cases without a variable transformation. It is shown that accuracy and efficiency of estimation of radiation heat flux is improved when using a nonuniform zone system. Reasonable accuracy of all DEA are calculated without resorting to the conservative equations. Results compared well with analytical solutions and numerical results of previous researchers. This technique can be readily extended to rectangular enclosures with similar reduction in computation time expected.

Issue Section:
Radiative Heat Transfer
Keywords:
heat radiation, Heat Transfer, Laser, Numerical Methods, Participating Media, Radiation
Topics:
Computation, Cylinders, Dimensions, Heat flux, Heat transfer, Irradiation (Radiation exposure), Lasers, Radiation (Physics), Temperature gradient, Numerical analysis, Thermal radiation
1.
Hottel
,
H. C.
, and
Cohen
,
E. S.
,
1958
, “
Radiant Heat Exchange in a Gas-Filled Enclosure: Allowance for Nonuniformity of Gas Temperature
,”
AIChE J.
,
4
, pp.
3
–
14
.
2.
Modest, M. F., 2003, Radiative Heat Transfer, 2nd ed., Academic Press, San Diego, CA.
3.
Hottel, H. C., and Sarofim, A. F., 1967, Radiative Transfer, McGraw-Hill Inc., New York.
4.
Goyhe´ne´che
,
J. M.
, and
Sacadura
,
J. F.
,
2002
, “
The Zonal Method: A New Explicit Matrix Relation to Calculate the Total Exchange Areas in Anisotropically Scattering Medium Bounded by Anisotropically Reflecting Walls
,”
ASME J. Heat Transfer
,
124
, pp.
696
–
703
.
5.
Stroud, A. H., 1971, Approximate Calculation of Multiple Integrals, Prentice-Hall, Englewood Cliffs, NJ.
6.
Siddall, R. G., 1986, “Accurate Evaluation of Radiative Direct-Exchange Areas for Rectangular Geometries,” in Proceedings of the Eighth International Heat Transfer Conference, Hemisphere Publishing Corp., Washington, DC, pp. 751–756.
7.
Tucker
,
R. J.
,
1986
, “
Direct Exchange Areas for Calculating Radiation Transfer in Rectangular Furnaces
,”
ASME J. Heat Transfer
,
108
, pp.
707
–
710
.
8.
Erkku, H., 1959, “Radiant Heat Exchange in Gas-Filled Slabs and Cylinders,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
9.
Larsen
,
M. E.
, and
Howell
,
J. R.
,
1986
, “
Least-Squares Smoothing of Direct Exchange Areas in Zonal Analysis
,”
ASME J. Heat Transfer
,
108
, pp.
239
–
242
.
10.
Sika
,
J.
,
1991
, “
Evaluation of Direct-Exchange Areas for a Cylindrical Enclosure
,”
ASME J. Heat Transfer
,
113
, pp.
1040
–
1044
.
11.
Murty
,
C. V. S.
,
1993
, “
Evaluation of Radiation Reception Factors in a Rotory Kiln Using a Modified Monte-Carlo Scheme
,”
Int. J. Heat Mass Transfer
,
36
, pp.
119
–
133
.
12.
Tian, W., and Chiu, W. K. S., 2002, “Numerical Modeling of CO2 Laser-Heated Moving Glass Rods,” in Proceedings of ASME International Mechanical Engineering Congress and Exposition, CD-ROM, 1, ASME, New York, Paper No. IMECE-33082.
13.
Walton, G., 2002, “Calculation of Obstructed View Factors by Adaptive Integration,” Tech. Rep. NISTIR 6925, NIST.
14.
Siegel, R., and Howell, J., 2002, Thermal Radiation Heat Transfer, 4th ed., Taylor & Francis, London.
15.
Azad
,
F. H.
, and
Modest
,
M. F.
,
1981
, “
Evaluation of the Radiative Heat Flux in Absorbing, Emitting and Linear-Anisotropically Scattering Cylindrical Media
,”
ASME J. Heat Transfer
,
103
, pp.
350
–
356
.
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