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Wärme - und Stoffübertragung

, Volume 7, Issue 1, pp 31–44 | Cite as

Algebraic methods for the calculation of radiation exchange in an enclosure

  • John A. Clark
  • Michael E. Korybalski
Article

Abstract

The algebraic methods of Hottel and Sarofim, Gebhart and one recently published by the authors for the calculation of radiant interchange in an enclosure are examined. Each of these formulations, while written in a different form, is shown to be mathematically equivalent to the others. This equivalence is established by deriving the formulations of Hottel and Sarofim and Gebhart from that of the authors. Because of the analytical complexity of these forms their mathematical equivalence cannot be demonstrated by explicit methods. For this reason proof of equivalence is established by numerical programming using a digital computer. Three different numerical examples are used for this purpose.-In addition, the mathematical structure underlying the absorption factor Bij of Gebhart and the total exchange factor\(\overline {S_i S_j }\) and total view factor ℱij of Hottel and Sarofim is developed.

Keywords

Absorption Factor Mathematical Structure Digital Computer Algebraic Method Explicit Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Abbreviations

Nomenclature

A

area, [ft2]

Aij

cofactor of matrix [aij], Eq. (15), [1]

[aij]

geometric view factor-reflectance matrix, Eq. (8)

Bij

Gebhart absorption factor, Eq. (1), [1]

D

determinant of [aij] matrix, [1]

DERGEBHOT

name of authors' computer program to calculate Gebhart's Bij and Hottel's\(\overline {S_i S_j }\) = A ℱij

Fij

geometric view factor, [1]

[Fij]

geometric view factor matrix, Eq. (9), [1]

ij

Hottel's script-F or total view factor, Eq. (32), [1]

H

incident radiation flux, Eq. (3), [Btu/hr-ft2]

[I]

identity matrix, Eq. (10), [1]

J

radiosity flux, Eqs. (14), (16), (17), [Btu/hr-ft2]

K

(ε/ρ) σT4, Eq. (13), [Btu/hr-ft2]

n

number of surfaces in enclosure, [1]

qij*

net radiation heat transfer rate between surfaces Ai and Aj, Eq. (31) [Btu/hr]

qij

energy absorbed at surface Aj originating as emission at surface Ai, [Btu/hr]

qj*

net radiation heat transfer at surface Aj, [Btu/hr]

RADTQO

name of authors' computer program to use with formulation in [12]

T

temperature, [‡R]

x

coordinate

Greek

α

absorptance or absorptivity, [1]

Β

angle, [1]

[Βij]

matrix, Eq. (25), [1]

γij

coefficients of the inverted matrix [aij]−1, Eq. (15), [1]

ε

emittance or emissivity, [1]

ρ

reflectance or reflectivity, [1]

σ

Stefan-Boltzmann constant

Subscripts

i

designation of row number in matrix; also ith surface

j

designation of column number in matrix; also jth surface

Zusammenfassung

Die algebraischen Methoden zur Berechnung des Strahlungsaustausches in HohlrÄumen von Hottel und Sarofim, Gebhart und von den Autoren dieser Arbeit werden untersucht. Trotz verschiedener Schreibweisen sind die einzelnen Formulierungen mathematisch gleichwertig. Das zeigt sich, wenn man die Formulierungen von Hottel und Sarofim sowie von Gebhart aus der unsrigen ableitet. Wegen der komplizierten mathematischen Form kann diese Äquivalenz nicht explizit gezeigt werden. Sie wurde daher numerisch mit Hilfe eines Digitalrechners an drei Beispielen nachgewiesen. Au\erdem wurde die mathematische Struktur des Absorptionsfaktors By nach Gebhart, des totalen Austauschfaktors\(\overline {S_i S_j }\) und des totalen FlÄchenverhÄltnisses ℱij nach Hottel und Sarofim abgeleitet.

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References

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    Hottel, H. C: Radiant-Heat Transmission. Chap. 4 in Heat Transmission by W. H. McAdams, McGraw-Hill Book Co., Third Edition, 1954.Google Scholar
  2. 2.
    Hottel, H. C, Sarofim, A. F.: Radiative Transfer, McGraw Hill Book Co., 1967.Google Scholar
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    Gebhart, B.: Heat Transfer, McGraw Hill Book Co., Chap. 5, 1971.Google Scholar
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    Gebhart, B.: Surface Temperature Calculations in Radiant Surroundings of Arbitrary Complexity for Gray, Diffuse Radiation. Intern. J. Heat Mass Transfer 3 (1961) 341/346.Google Scholar
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    Sparrow, E. M.: On the Calculation of Radiant Interchange Between Surfaces. In Modern Developments in Heat Transfer, edited by W. Ibele, Academic Press (1963) 181/212.Google Scholar
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    Eckert, E. R. G., Drake, R. M., Jr.: Heat and Mass Transfer, McGraw Hill Book Co., 1959.Google Scholar
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    Love, T. J.: Radiative Heat Transfer, Charles E. Merrill Publishing Co., 1968.Google Scholar
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    Wiebelt, J. A.: Engineering Radiation Heat Transfer, Holt, Rinehart and Winston, 1966.Google Scholar
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    Siegel, R., Howell, J. R.: Thermal Radiation Heat Transfer, McGraw Hill Book Co., 1972.Google Scholar
  12. 12.
    Clark, J. A., Korybalski, M. E.: Radiation Heat Transfer in an Enclosure Having Surfaces Which Are Adiabatic or of Known Temperature. Proceedings, First National Heat and Mass Transfer Conference, Madras, India, Dec. 6–8, 1971.Google Scholar
  13. 13.
    Gebhart, B.: Personal Communication, J. A. Clark, July 23, 1971.Google Scholar

Copyright information

© Springer Verlag 1974

Authors and Affiliations

  • John A. Clark
    • 1
  • Michael E. Korybalski
    • 2
  1. 1.Department of Mechanical EngineeringUniversity of MichiganAnn ArborUSA
  2. 2.Department of Mechanical EngineeringUniversity of MichiganUSA

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