Formulation of multiple reflections within a diffuse cavity radiator: comments

Yoshiko Ohwada National Research Laboratory of Metrology, 1.-1-4, Ume-zono, Sakura-mura, Niihaгi-gun, Ibaraki 305, Japan. Received 16 July 1984. 0003-6935/84/234181-02$02.00/0. © 1984 Optical Society of America. An equation used for the iterative calculation of multiple

reflections within a diffuse cavity radiator, which is given by Eq. (1) in Ref. 1, is mathematically derived. The consider­ation is based on the multiple reflection analysis or series methods given by previous workeгs.2-5

The nth-ordeг approximation Mn(xi) to the radiant exi-tance M(xi) of a point dxi on the cavity wall is given by

or

with M1(xi) = eMb(xi), where ε is the intrinsic emissivity, Mb(xi) is the radiant exitance of a blackbody at the same temperature as that of dxi, and Zk(xi) is the radiation re­flected from dxi per unit area that was emitted and underwent (k — I) reflections elsewhere in the cavity. The quantity Zn(xi) is related to Zn-1(x) by

with Z1(x) = εMb(x), where p is the surface reflectance, ∫x indicates the integration over the whole cavity surface, and d∫xi-x is the angle factor from dxi to dx.

The expression in Eq. {1) is similar to that given by Sydnor5

and is equivalent to the first « terms of the series of nested integrals given by Refs, 2-4. To compute an arbitrary order approximation Mn(xi) iteгatively with an electronic computer, it is convenient to express Mn (xi) in terms of the successive loweг-oгder approximations and known quantities with initial values given.

The quantity Zn(xi) in Eq. (2) is eliminated in favor of Mn-1(x) and Mn-2(x) as follows: using Eq. (2), Eq. (3) is rewritten as

with M0(x) = 0. Inserting Eq. (4) into Eq. (2) gives

with M0(x) = 0, and M1(x) = eMb(x). This is the equation given in Eef. 1, and Mn(xi) is solved simultaneously for all points dxi on the cavity wall.

Inserting Eq. (4) into Eq. (1) with n replaced by k, one ob­tains

This may be written as

The negative terms Mi (x) with i = 1,... ,n -2in brackets of Eq. (7) are canceled by the corresponding positive terms, and since M0(x) = 0, Eq. (7J reduces to

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Equation (8) is essentially the same as the integral equation given by Sparrow et al.6

In the actual numerical calculation of Eq. (5) which was carried out in Refs. 1, 7, and 8, the zonal approximation method described in Refs. 9-11 was used, and the above in­tegrals were replaced by sums.2,9-11

The author is indebted to R. E. Bedford of NRC and to the staff of the thermal measurement section for enlightening discussions.

References 1. Y. Ohwada, "Numerical Calculations of Multiple Reflections in

Diffuse Cavities," J. Opt. Soc. Am. 71, 106 (1981). 2. R E . Bedford, "Theory and Practice of Radiation Theníiometty,"

National Bureau of Standards, in press, (1983), Chap. 9. 3. S. Takata, "A Formulation of the Theory of Interreflection," J.

Ilium. Eng. Inst. Jpn. 51, 702 (1967). 4. F. 0. Bartell and W. L. Wolfe, "Cavity Radiators; an Ecumenical

Theory," Appl. Opt, 15, 84 (1976). 5. C. L. Sydnor, "Series Representation of the Solution of the In­

tegral Equation for Emissivity of Cavities," J. Opt. Soc. Am. 59, 1288 (1969).

6. E. M, Sparrow, L. U. Albers, and E. R Eckert, "Thermal Radia­tion Characteristics of Cylindrical Enclosures," J. Heat. Transfer C84, 73 (1962).

7. Y. Ohwada, "Numerical Calculation of Effective Emissivities of Diffuse Cones with a Series Technique," Appl. Opt. 20, 3332 (1981).

8. Y. Ohwada, "Evaluation of Effective Emissivities of Noniso­thermal Cavities," Appl. Opt. 22, 2322 (1983).

9. R. E. Bedford, and C. K, Ma, "Emissivities of Diffuse Cavities: Isothermal and Nonisotheгmal Cones and Cylinders," J. Opt Soc. Am. 64, 339 (1974).

10. R. E. Bedford and C.K. Ma, "Emissivities of Diffuse Cavities. II: Isothermal and Nonisotherma! Cylindro-cones," i. Opt. Soc. Am. 65, 565(1975).

11, RE . Bedford and C.K. Ma, "Emissivities of Diffuse Cavities. III. Isothermal and Nonigotheгmai Double Cones," J. Opt. Soc. Am, 66, 724 (1976).

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