Numerical calculation of effective emissivities of diffuse cones with a series technique

Yoshiko Ohwada

doi:10.1364/AO.20.003332

Applied Optics
Vol. 20,
Issue 19,
pp. 3332-3335
(1981)
•https://doi.org/10.1364/AO.20.003332

Numerical calculation of effective emissivities of diffuse cones with a series technique

Yoshiko Ohwada

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Yoshiko Ohwada, "Numerical calculation of effective emissivities of diffuse cones with a series technique," Appl. Opt. 20, 3332-3335 (1981)

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Abstract

Effective emissivities of conical cavities having diffuse surfaces are computed by an iterative procedure and effective reflectances from series that involve powers of the reflectance of the wall material. If reflections after the nth are neglected in the computation, an upper bound for the truncation error in the effective reflectance is 1.2ρⁿ⁺¹I_n₊₁/(1 − ρ) for cone half-angles between 3.75 and 60° and for 5 ≤ n, where ρ is the surface reflectance and I_n₊₁ is the (n + 1)th-order coefficient of the effective reflectance.

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		E(X) ± σa
		R = 1.0		R = 0.5

θ°	X\∊	0.50	0.75	0.50	0.75
60.00	0.10	0.5868 ± 1	0.81083 ± 1	0.7374 ± 1	0.88249 ± 1
	0.25	0.5677 ± 1	0.79814 ± 1	0.7135 ± 1	0.86772 ± 1
	0.50	0.5450 ± 1	0.78235 ± 1	0.7465 ± 1	0.88292 ± 3
30.00	0.10	0.8470 ± 2	0.94746 ± 2	0.9447 ± 2	0.98241 ± 2
	0.25	0.7994 ± 1	0.92775 ± 2	0.9229 ± 1	0.97369 ± 2
	0.50	0.7228 ± 1	0.88888 ± 2	0.8839 ± 1	0.95422 ± 1
15.00	0.10	0.9733 ± 1	0.99190 ± 1	0.9921 ± 1	0.99778 ± 1
	0.25	0.9524 ± 1	0.98619 ± 2	0.9853 ± 1	0.99604 ± 1
	0.50	0.8937 ± 2	0.96597 ± 3	0.9648 ± 1	0.98921 ± 2
7.50	0.10	0.9968 ± 1	0.99897 ± 1	0.9991 ± 1	0.99973 ± 1
	0.25	0.9937 ± 1	0.99817 ± 1	0.9983 ± 1	0.99952 ± 1
	0.50	0.9773 ± 1	0.99386 ± 1	0.9933 ± 1	0.99831 ± 1
3.75	0.10	0.99961 ± 1	0.99987 ± 1	0.99990 ± 1	0.99997 ± 1
	0.25	0.99929 ± 1	0.99978 ± 1	0.99981 ± 1	0.99994 ± 1
	0.50	0.99719 ± 1	0.99921 ± 1	0.99923 ± 1	0.99980 ± 1

		I_k(X)

θ°	X/k	1	2	3	4	5	6	7	8	9	10
60.00	0.10	0.69576	0.22802	0.05912	0.01346	0.00288	0.0006	0.0001
	0.25	0.75593	0.19579	0.03918	0.00738	0.00139	0.0003	0.0001
	0.50	0.83422	0.13992	0.02159	0.00353	0.00061	0.0001
30.00	0.10	0.15661	0.16413	0.15720	0.13588	0.10839	0.0818	0.0595	0.042	0.029	0.020
	0.25	0.21880	0.22572	0.18168	0.12866	0.08594	0.0560	0.0362	0.023	0.015	0.010
	0.50	0.36603	0.26875	0.15484	0.08671	0.04940	0.0288	0.0172	0.010	0.006	0.004
15.00	0.10	0.02319	0.02635	0.03019	0.03444	0.03868	0.0424	0.0454	0.047	0.048	0.048
	0.25	0.03749	0.04990	0.06137	0.06922	0.07257	0.0720	0.0687	0.064	0.058	0.052
	0.50	0.09462	0.12580	0.12596	0.11087	0.09248	0.0756	0.0615	0.050	0.041	0.034
7.50	0.10	0.00303	0.00321	0.00340	0.00363	0.00388	0.0042	0.0045	0.005	0.005	0.006
	0.25	0.00514	0.00611	0.00724	0.00857	0.01008	0.0117	0.0134	0.015	0.017	0.018
	0.50	0.01590	0.02355	0.03124	0.03752	0.04166	0.0437	0.0441	0.043	0.042	0.040
3.75	0.10	0.00038	0.00039	0.00040	0.00041	0.00041	0.0004	0.0004
	0.25	0.00066	0.00071	0.00075	0.00080	0.00086	0.0009	0.0010	0.001	0.001	0.001
	0.50	0.00217	0.00272	0.00335	0.00410	0.00495	0.0059	0.0069	0.008	0.009	0.010

		I_k(X)

θ°	X/k	1	2	3	4	5	6	7	8	9	10
60.00	0.10	0.43181	0.12973	0.08144	0.03445	0.03887	0.0220	0.0266	0.017	0.020	0.014
	0.25	0.50000	0.09647	0.06966	0.03065	0.03615	0.0209	0.0249	0.016	0.019	0.013
	0.50	0.44880	0.05119	0.09620	0.03125	0.04962	0.0234	0.0329	0.019	0.024	0.016
30.00	0.10	0.05100	0.05572	0.06581	0.06459	0.06265	0.0579	0.0536	0.049	0.045	0.041
	0.25	0.07880	0.08068	0.07835	0.06765	0.06108	0.0542	0.0492	0.045	0.041	0.037
	0.50	0.15465	0.09046	0.07434	0.06018	0.05421	0.0479	0.0438	0.040	0.036	0.033
15.00	0.10	0.00621	0.00717	0.00967	0.01196	0.01450	0.0169	0.0192	0.021	0.023	0.024
	0.25	0.01043	0.01441	0.02016	0.02450	0.02794	0.0302	0.0316	0.032	0.032	0.032
	0.50	0.02973	0.03936	0.04319	0.04267	0.04149	0.0398	0.0381	0.036	0.035	0.033
7.50	0.10	0.00077	0.00082	0.00101	0.00117	0.00134	0.0015	0.0017	0.002	0.002	0.002
	0.25	0.00132	0.00159	0.00214	0.00271	0.00337	0.0041	0.0049	0.006	0.006	0.007
	0.50	0.00427	0.00654	0.00955	0.01215	0.01433	0.0160	0.0172	0.018	0.019	0.019
3.75	0.10	0.00010	0.00010	0.00012	0.00013	0.00014	0.0002	0.0002
	0.25	0.00017	0.00018	0.00022	0.00025	0.00028	0.0003	0.0004	0.0004
	0.50	0.00055	0.00070	0.00096	0.00125	0.00158	0.0020	0.0024	0.0028	0.003	0.004

		E(X) ± σa
		R = 1.0		R = 0.5

θ°	X\∊	0.50	0.75	0.50	0.75
60.00	0.10	0.5868 ± 1	0.81083 ± 1	0.7374 ± 1	0.88249 ± 1
	0.25	0.5677 ± 1	0.79814 ± 1	0.7135 ± 1	0.86772 ± 1
	0.50	0.5450 ± 1	0.78235 ± 1	0.7465 ± 1	0.88292 ± 3
30.00	0.10	0.8470 ± 2	0.94746 ± 2	0.9447 ± 2	0.98241 ± 2
	0.25	0.7994 ± 1	0.92775 ± 2	0.9229 ± 1	0.97369 ± 2
	0.50	0.7228 ± 1	0.88888 ± 2	0.8839 ± 1	0.95422 ± 1
15.00	0.10	0.9733 ± 1	0.99190 ± 1	0.9921 ± 1	0.99778 ± 1
	0.25	0.9524 ± 1	0.98619 ± 2	0.9853 ± 1	0.99604 ± 1
	0.50	0.8937 ± 2	0.96597 ± 3	0.9648 ± 1	0.98921 ± 2
7.50	0.10	0.9968 ± 1	0.99897 ± 1	0.9991 ± 1	0.99973 ± 1
	0.25	0.9937 ± 1	0.99817 ± 1	0.9983 ± 1	0.99952 ± 1
	0.50	0.9773 ± 1	0.99386 ± 1	0.9933 ± 1	0.99831 ± 1
3.75	0.10	0.99961 ± 1	0.99987 ± 1	0.99990 ± 1	0.99997 ± 1
	0.25	0.99929 ± 1	0.99978 ± 1	0.99981 ± 1	0.99994 ± 1
	0.50	0.99719 ± 1	0.99921 ± 1	0.99923 ± 1	0.99980 ± 1

		I_k(X)

θ°	X/k	1	2	3	4	5	6	7	8	9	10
60.00	0.10	0.69576	0.22802	0.05912	0.01346	0.00288	0.0006	0.0001
	0.25	0.75593	0.19579	0.03918	0.00738	0.00139	0.0003	0.0001
	0.50	0.83422	0.13992	0.02159	0.00353	0.00061	0.0001
30.00	0.10	0.15661	0.16413	0.15720	0.13588	0.10839	0.0818	0.0595	0.042	0.029	0.020
	0.25	0.21880	0.22572	0.18168	0.12866	0.08594	0.0560	0.0362	0.023	0.015	0.010
	0.50	0.36603	0.26875	0.15484	0.08671	0.04940	0.0288	0.0172	0.010	0.006	0.004
15.00	0.10	0.02319	0.02635	0.03019	0.03444	0.03868	0.0424	0.0454	0.047	0.048	0.048
	0.25	0.03749	0.04990	0.06137	0.06922	0.07257	0.0720	0.0687	0.064	0.058	0.052
	0.50	0.09462	0.12580	0.12596	0.11087	0.09248	0.0756	0.0615	0.050	0.041	0.034
7.50	0.10	0.00303	0.00321	0.00340	0.00363	0.00388	0.0042	0.0045	0.005	0.005	0.006
	0.25	0.00514	0.00611	0.00724	0.00857	0.01008	0.0117	0.0134	0.015	0.017	0.018
	0.50	0.01590	0.02355	0.03124	0.03752	0.04166	0.0437	0.0441	0.043	0.042	0.040
3.75	0.10	0.00038	0.00039	0.00040	0.00041	0.00041	0.0004	0.0004
	0.25	0.00066	0.00071	0.00075	0.00080	0.00086	0.0009	0.0010	0.001	0.001	0.001
	0.50	0.00217	0.00272	0.00335	0.00410	0.00495	0.0059	0.0069	0.008	0.009	0.010

Abstract

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Equations (10)

Applied Optics