Numerical calculation of multiple reflections in diffuse cavities

Yoshiko Ohwada

doi:10.1364/JOSA.71.000106

Journal of the Optical Society of America
Vol. 71,
Issue 1,
pp. 106-111
(1981)
•https://doi.org/10.1364/JOSA.71.000106

Numerical calculation of multiple reflections in diffuse cavities

Yoshiko Ohwada

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Yoshiko Ohwada, "Numerical calculation of multiple reflections in diffuse cavities," J. Opt. Soc. Am. 71, 106-111 (1981)

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Abstract

The equivalence of the effective emissivity and effective absorptivity of an isothermal diffuse cavity is proved by mathematical induction. The effective emissivity, effective absorptivity, and difference between the effective reflectivity and unity, which are numerically computed for cylindrical cavities, agree with each other within computer round-off error. Error in the above three quantities arises mostly from the assumption of constant effective emissivities across subdivisions of the cavity surfaces. The precise determination of effective emissivities shows that the minimum emissivity on the base of a long cylinder does not necessarily occur at the center.

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		∊_a(X) ± σ

L	X\∊₀	0.25	0.50	0.75
0.5	0.0	0.31384 ± 2	0.56935 ± 1	0.793713 ± 2
1.0	0.0	0.43835 ± 4	0.68777 ± 2	0.862578 ± 2
2.0	0.0	0.64016 ± 5	0.83943 ± 2	0.938870 ± 4
4.0	0.0	0.84408 ± 6	0.94603 ± 2	0.981491 ± 2
8.0	0.0	0.96187 ± 5	0.98801 ± 1	0.995573 ± 1
	0.0	0.98691 ± 3	0.99533 ± 1	0.998111 ± 1
12.0	0.5	0.98695 ± 3	0.99534 ± 1	0.998118 ± 1
	0.9	0.98697 ± 3	0.99534 ± 1	0.998130 ± 1
	0.0	0.99142 ± 3	0.99671 ± 1	0.998629 ± 1
14.0	0.5	0.99142 ± 3	0.99671 ± 1	0.998632 ± 1
	0.9	0.99136 ± 3	0.99669 ± 1	0.998636 ± 1
	0.0	0.99402 ± 2	0.99756 ± 1	0.998959 ± 1
16.0	0.5	0.99400 ± 2	0.99755 ± 1	0.998961 ± 1
	0.9	0.99392 ± 2	0.99753 ± 1	0.998962 ± 1
	0.0	0.99665 ± 2	0.99850 ± 1	0.999342 ± 1
20.0	0.5	0.99663 ± 2	0.99849 ± 1	0.999342 ± 1
	0.9	0.99653 ± 2	0.99847 ± 1	0.999341 ± 1
r = 0.2	0.0	0.99983	0.99994	0.999973
20.0	0.5	0.99983	0.99993	0.999973
	0.9	0.99982	0.99993	0.999973

		Δf(X,Y)

X	Δf(X,A)\Y	0.05	0.15	0.25	0.35	0.45
0.0	0.00249	0.50000	0.30000	0.10000	0.04118	0.02036
0.5	0.00249	0.56202	0.25328	0.08907	0.03841	0.01947
0.9	0.00248	0.68769	0.15716	0.06841	0.03287	0.01761

	G_k(0)

L\k	1	2	3	4	5	6	7	8	9	10
0.5	0.20000	0.11642	0.05560	0.03102	0.01635	0.0090	0.005	0.003	0.002	0.001
4.0	0.94118	0.89921	0.84279	0.78517	0.72763	0.6727	0.621	0.572	0.528	0.486
8.0	0.98462	0.97874	0.96898	0.95793	0.94479	0.9299	0.913	0.896	0.877	0.857
14.0	0.99492	0.99394	0.99200	0.99014	0.98802	0.9857	0.983	0.980	0.977	0.973
20.0	0.99751	0.99719	0.99644	0.99580	0.99511	0.9944	0.994	0.993	0.992	0.991

		G_k₋₁(X) − G_k(X)

L	X\k	1	2	3	4	5	6	7	8	9	10
	0.0	0.80000	0.08358	0.06032	0.02458	0.01467	0.0073	0.004	0.002
0.5	0.5	0.72361	0.11361	0.08466	0.03443	0.02066	0.0103	0.006	0.003
	0.9	0.47005	0.19349	0.16646	0.07258	0.04587	0.0228	0.013	0.007	0.004
	0.0	0.50000	0.16464	0.11532	0.06992	0.04843	0.0323	0.022	0.015	0.010	0.007
1.0	0.5	0.43798	0.17535	0.13209	0.08046	0.05616	0.0374	0.026	0.017	0.012	0.008
	0.9	0.31231	0.17246	0.16907	0.10556	0.07733	0.0514	0.036	0.024	0.017	0.011
	0.0	0.20000	0.13951	0.12399	0.09634	0.07908	0.0643	0.053	0.043	0.036	0.029
2.0	0.5	0.18470	0.13035	0.12523	0.09907	0.08240	0.0672	0.055	0.045	0.037	0.031
	0.9	0.15515	0.10422	0.12581	0.10298	0.09012	0.0741	0.062	0.056	0.042	0.034
	0.0	0.05882	0.04197	0.05642	0.05762	0.05754	0.0549	0.052	0.048	0.045	0.042
4.0	0.5	0.05723	0.03926	0.05472	0.05614	0.05690	0.0548	0.052	0.049	0.045	0.042
	0.9	0.05386	0.03298	0.05136	0.05198	0.05508	0.0540	0.052	0.049	0.046	0.043
	0.0	0.02703	0.01407	0.02195	0.02498	0.02834	0.0303	0.032	0.032	0.032	0.032
6.0	0.5	0.02668	0.01346	0.02159	0.02426	0.02771	0.0298	0.031	0.032	0.032	0.031
	0.9	0.02591	0.01201	0.02127	0.02244	0.02624	0.0283	0.030	0.031	0.031	0.031
	0.0	0.01538	0.00588	0.00976	0.01105	0.01314	0.0149	0.0165	0.018	0.019	0.020
8.0	0.5	0.01527	0.00570	0.00978	0.01082	0.01290	0.0146	0.016	0.018	0.019	0.019
	0.9	0.01501	0.00526	0.01017	0.01024	0.01239	0.0139	0.016	0.017	0.018	0.019
10.0	0.5	0.00985	0.00284	0.00516	0.00540	0.00642	0.0073	0.0083	0.0093	0.010	0.011
12.0	0.5	0.00687	0.00159	0.00308	0.00302	0.00353	0.0039	0.004	0.005	0.006	0.006
14.0	0.5	0.00506	0.00097	0.00202	0.00186	0.00213	0.0023	0.003	0.003	0.003
16.0	0.5	0.00388	0.00063	0.00141	0.00124	0.00139	0.0015	0.002	0.002
18.0	0.5	0.00307	0.00043	0.00104	0.00088	0.00097	0.0010	0.001
20.0	0.5	0.00249	0.00031	0.00080	0.00065	0.00071	0.0007	0.001
r = 0.4
20.0	0.5	0.00040	0.00005	0.00019	0.00015	0.00018
r = 0.2
20.0	0.5	0.00010	0.00001	0.00005

		∊_a(X) ± σ

L	X\∊₀	0.25	0.50	0.75
0.5	0.0	0.31384 ± 2	0.56935 ± 1	0.793713 ± 2
1.0	0.0	0.43835 ± 4	0.68777 ± 2	0.862578 ± 2
2.0	0.0	0.64016 ± 5	0.83943 ± 2	0.938870 ± 4
4.0	0.0	0.84408 ± 6	0.94603 ± 2	0.981491 ± 2
8.0	0.0	0.96187 ± 5	0.98801 ± 1	0.995573 ± 1
	0.0	0.98691 ± 3	0.99533 ± 1	0.998111 ± 1
12.0	0.5	0.98695 ± 3	0.99534 ± 1	0.998118 ± 1
	0.9	0.98697 ± 3	0.99534 ± 1	0.998130 ± 1
	0.0	0.99142 ± 3	0.99671 ± 1	0.998629 ± 1
14.0	0.5	0.99142 ± 3	0.99671 ± 1	0.998632 ± 1
	0.9	0.99136 ± 3	0.99669 ± 1	0.998636 ± 1
	0.0	0.99402 ± 2	0.99756 ± 1	0.998959 ± 1
16.0	0.5	0.99400 ± 2	0.99755 ± 1	0.998961 ± 1
	0.9	0.99392 ± 2	0.99753 ± 1	0.998962 ± 1
	0.0	0.99665 ± 2	0.99850 ± 1	0.999342 ± 1
20.0	0.5	0.99663 ± 2	0.99849 ± 1	0.999342 ± 1
	0.9	0.99653 ± 2	0.99847 ± 1	0.999341 ± 1
r = 0.2	0.0	0.99983	0.99994	0.999973
20.0	0.5	0.99983	0.99993	0.999973
	0.9	0.99982	0.99993	0.999973

Abstract

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Tables (4)

Equations (32)

Journal of the Optical Society of America