## Abstract

Local hemispherical effective emissivities and integrated cavity emissivities are computed for isothermal and nonisothermal diffuse double cones (a conical section joined to a conical frustum) with and without lids, and are compared with corresponding values for cylindrocones. The local emissivities increase and become more uniform with increasing taper of the frustum. They are also considerably higher when there is a lid. For cavities with the same conical section, length, and aperture, a lidded double cone is blacker than a lidded cylindrocone when the front half of the frustum is invisible, but less black otherwise. For double cones of the same length, diameter, and aperture, the best choice of cone and frustum angles depends upon the particular viewing conditions. The integrated cavity emissivities vary only slightly with the angles of cone and frustum when the frustum is relatively long, and the normal emissivity (for a small on-axis detector a large distance away) is higher than the hemispherical emissivity (for a detector that fills the cavity aperture). When the frustum is relatively short, all of these vary substantially with angle, and the hemispherical emissivity can be higher than the normal emissivity. There is a marked variation of both local and integrated emissivities with wavelength in nonisothermal double cones; e.g., for the particular cases illustrated, the normal spectral emissivities change by from 4% to 6% and the hemispherical spectral emissivities by from 17% to 20% between 0.3 and 1 *μ*m for a 1% temperature variation at 1300 K. The amount of the change in these spectral emissivities also depends upon the geometry of the cavity.

© 1976 Optical Society of America

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### Equations (13)

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(1)
$${\mathrm{\u220a}}_{a}({x}_{0},\mathrm{\lambda},{T}_{{x}_{0}},{T}_{0})={M}_{\mathrm{\lambda}}({x}_{0},\mathrm{\lambda},{T}_{{x}_{0}})/{M}_{\mathrm{\lambda},bb}(\mathrm{\lambda},{T}_{0}),$$
(2)
$${\mathrm{\u220a}}_{a}({x}_{0},{T}_{{x}_{0}},{T}_{0})=M({x}_{0},{T}_{{x}_{0}})/{M}_{bb}({T}_{0}),$$
(3)
$$\begin{array}{l}{\mathrm{\u220a}}_{a}({x}_{0},\mathrm{\lambda},{T}_{{x}_{0}},{T}_{0})=\frac{\mathrm{\u220a}{M}_{\mathrm{\lambda},bb}(\mathrm{\lambda},{T}_{{x}_{0}})}{{M}_{\mathrm{\lambda},bb}(\mathrm{\lambda},{T}_{0})}+(1-\mathrm{\u220a})\hspace{0.17em}(\sum _{i=1}^{n}{\scriptstyle \frac{1}{2}}[{\mathrm{\u220a}}_{a}({x}_{i+1})+{\mathrm{\u220a}}_{a}({x}_{i})]\mid d{F}_{{x}_{0},{x}_{i+1}}-d{F}_{{x}_{0},{x}_{i}}\mid \\ +\hspace{0.17em}\sum _{j=1}^{n}{\scriptstyle \frac{1}{2}}[{\mathrm{\u220a}}_{a}({y}_{j+1})+{\mathrm{\u220a}}_{a}({y}_{j})](d{F}_{{x}_{0},{y}_{j}}-d{F}_{{x}_{0},{y}_{j+1}})+\sum _{k=m}^{n}{\scriptstyle \frac{1}{2}}[{\mathrm{\u220a}}_{a}({z}_{k+1})+{\mathrm{\u220a}}_{a}({z}_{k})](d{F}_{{x}_{0},{z}_{k+1}}-d{F}_{{x}_{0},{z}_{k}})),\end{array}$$
(4)
$$\begin{array}{l}{\mathrm{\u220a}}_{a}({y}_{0},\mathrm{\lambda},{T}_{{y}_{0}},{T}_{0})=\frac{\mathrm{\u220a}{M}_{\mathrm{\lambda},bb}(\mathrm{\lambda},{T}_{{y}_{0}})}{{M}_{\mathrm{\lambda},bb}(\mathrm{\lambda},{T}_{0})}+(1-\mathrm{\u220a})\hspace{0.17em}(\sum _{i=1}^{n}{\scriptstyle \frac{1}{2}}[{\mathrm{\u220a}}_{a}({x}_{i+1})+{\mathrm{\u220a}}_{a}({x}_{i})](d{F}_{{y}_{0},{x}_{i}+1}-d{F}_{{y}_{0},{x}_{i}})\\ +\hspace{0.17em}\sum _{j=1}^{n}{\scriptstyle \frac{1}{2}}[{\mathrm{\u220a}}_{a}({y}_{j+1})+{\mathrm{\u220a}}_{a}({y}_{j})]\mid d{F}_{{y}_{0},{y}_{j+1}}-d{F}_{{y}_{0},{y}_{j}}\mid +\sum _{k=m}^{n}{\scriptstyle \frac{1}{2}}[{\mathrm{\u220a}}_{a}({z}_{k+1})+{\mathrm{\u220a}}_{a}({z}_{k})](d{F}_{{y}_{0},{z}_{k+1}}-d{F}_{{y}_{0},{z}_{k}})),\end{array}$$
(5)
$$\begin{array}{l}{\mathrm{\u220a}}_{a}(z,\mathrm{\lambda},{T}_{z},{T}_{0})=)\frac{\mathrm{\u220a}{M}_{\mathrm{\lambda},bb}(\mathrm{\lambda},{T}_{z})}{{M}_{\mathrm{\lambda},bb}(\mathrm{\lambda},{T}_{0})}+(1-\mathrm{\u220a})\hspace{0.17em}(\sum _{i=1}^{n}{\scriptstyle \frac{1}{2}}[{\mathrm{\u220a}}_{a}({x}_{i+1})+{\mathrm{\u220a}}_{a}({x}_{i})](d{F}_{z,{x}_{i+1}}-d{F}_{z,{x}_{i}})\\ +\hspace{0.17em}\sum _{j=1}^{n}{\scriptstyle \frac{1}{2}}[{\mathrm{\u220a}}_{a}({y}_{j+1})+{\mathrm{\u220a}}_{a}({y}_{j})](d{F}_{z,{y}_{j+1}}-d{F}_{z,{y}_{j}})),\end{array}$$
(6)
$$\begin{array}{l}d{F}_{{x}_{0},{x}_{i}}={f}_{i}\pm \left(\frac{{x}_{i}\hspace{0.17em}{\text{cos}}^{2}\theta}{2{x}_{0}\hspace{0.17em}\text{sin}\theta}-\frac{1}{2\hspace{0.17em}\text{sin}\theta}\right),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\begin{array}{c}+\hspace{0.17em}({x}_{i}<{x}_{i+1}\le {x}_{0}),\\ -({x}_{0}\le {x}_{i}<{x}_{i+1}),\end{array}\\ {f}_{i}=\{{[{({x}_{i}+{x}_{0})}^{2}-4{x}_{i}{x}_{0}\hspace{0.17em}{\text{cos}}^{2}\theta ]}^{1/2}/4{x}_{0}\hspace{0.17em}\text{sin}\theta \}\{1+({x}_{0}-{x}_{i})({x}_{i}+{x}_{0}-2{x}_{i}\hspace{0.17em}{\text{cos}}^{2}\theta )/[{({x}_{i}+{x}_{0})}^{2}-4{x}_{i}{x}_{0}\hspace{0.17em}{\text{cos}}^{2}\theta ]\},\\ d{F}_{{x}_{0},{y}_{j}}=\frac{1}{2\hspace{0.17em}\text{sin}\theta}-\frac{{\text{cos}}^{2}\theta}{2{x}_{0}\text{sin}\theta}\hspace{0.17em}\left({y}_{j}+\frac{[{(L-{y}_{j})}^{2}{\text{tan}}^{2}\omega +{x}_{0}^{2}\hspace{0.17em}{\text{tan}}^{2}\theta +{({y}_{j}-{x}_{0})}^{2}][{x}_{0}\hspace{0.17em}{\text{tan}}^{2}\theta -({y}_{j}-{x}_{0})]-2{(L-{y}_{j})}^{2}{x}_{0}\hspace{0.17em}{\text{tan}}^{2}\omega \hspace{0.17em}{\text{tan}}^{2}\theta}{{\{{[{(L-{y}_{j})}^{2}\hspace{0.17em}{\text{tan}}^{2}\omega +{x}_{0}^{2}\hspace{0.17em}{\text{tan}}^{2}\theta +{({y}_{j}-{x}_{0})}^{2}]}^{2}-4{(L-{y}_{j})}^{2}{x}_{0}^{2}\hspace{0.17em}{\text{tan}}^{2}\omega \hspace{0.17em}{\text{tan}}^{2}\theta \}}^{1/2}}\right),\\ d{F}_{{x}_{0},{z}_{k}}=\frac{\text{sin}\theta}{2}-\frac{{\text{cos}}^{2}\theta}{2{x}_{0}\hspace{0.17em}\text{sin}\theta}\left(({L}_{1}+{L}_{2}-{x}_{0})+\frac{[{z}_{k}^{2}+{x}_{0}^{2}\hspace{0.17em}{\text{tan}}^{2}\theta +{({L}_{1}+{L}_{2}-{x}_{0})}^{2}][{x}_{0}\hspace{0.17em}{\text{tan}}^{2}\theta -({L}_{1}+{L}_{2}-{x}_{0})]-2{z}_{k}^{2}{x}_{0}\hspace{0.17em}{\text{tan}}^{2}\theta}{{\{{[{z}_{k}^{2}+{x}_{0}\hspace{0.17em}{\text{tan}}^{2}\theta +{({L}_{1}+{L}_{2}-{x}_{0})}^{2}]}^{2}-4{z}_{k}^{2}{x}_{0}^{2}\hspace{0.17em}{\text{tan}}^{2}\theta \}}^{1/2}}\right).\end{array}$$
(7)
$$d{F}_{{y}_{0},{y}_{j}}={g}_{j}\pm \left(\frac{(L-{y}_{j})\hspace{0.17em}{\text{cos}}^{2}\omega}{2(L-{y}_{0})\hspace{0.17em}\text{sin}\omega}-\frac{1}{2\hspace{0.17em}\text{sin}\omega}\right),\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\begin{array}{c}+\hspace{0.17em}({y}_{0}\le {y}_{j}<{y}_{j+1}),\\ -({y}_{j}<{y}_{j+1}\le {y}_{0}).\end{array}$$
(8)
$$\begin{array}{l}d{F}_{{y}_{0},{z}_{k}}=-\frac{\text{sin}\omega}{2}-\frac{{\text{cos}}^{2}\omega}{2(L-{y}_{0})\hspace{0.17em}\text{sin}\omega}\\ \times \hspace{0.17em}\left\{({L}_{1}+{L}_{2}-{y}_{0})-\frac{[{z}_{k}^{2}+{(L-{y}_{0})}^{2}\hspace{0.17em}{\text{tan}}^{2}\omega +{({L}_{1}+{L}_{2}-{y}_{0})}^{2}][(L-{y}_{0}){\text{tan}}^{2}\omega +({L}_{1}+{L}_{2}-{y}_{0})]-2{z}_{k}^{2}(L-{y}_{0}){\text{tan}}^{2}\omega}{{\{{[{z}_{k}^{2}+{(L-{y}_{0})}^{2}\hspace{0.17em}{\text{tan}}^{2}\omega +{({L}_{1}+{L}_{2}-{y}_{0})}^{2}]}^{2}-4{z}_{k}^{2}{(L-{y}_{0})}^{2}\hspace{0.17em}{\text{tan}}^{2}\omega \}}^{1/2}}\right\}\\ d{F}_{z,{x}_{i}}={\scriptstyle \frac{1}{2}}(1-[{z}^{2}+{({L}_{1}+{L}_{2}-{x}_{i})}^{2}-{x}_{i}^{2}\hspace{0.17em}{\text{tan}}^{2}\theta ]\hspace{0.17em}/{\{{[{z}^{2}+{({L}_{1}+{L}_{2}-{x}_{i})}^{2}+{x}_{i}^{2}\hspace{0.17em}{\text{tan}}^{2}\theta ]}^{2}-4{z}^{2}{x}_{i}^{2}\hspace{0.17em}{\text{tan}}^{2}\theta \}}^{1/2}),\\ d{F}_{z,{y}_{j}}={\scriptstyle \frac{1}{2}}(1-[{z}^{2}+{({L}_{1}+{L}_{2}-{y}_{j})}^{2}-{(L-{y}_{j})}^{2}{\text{tan}}^{2}\omega ]/{\{{[{z}^{2}+{({L}_{1}+{L}_{2}-{y}_{j})}^{2}+{(L-{y}_{j})}^{2}\hspace{0.17em}{\text{tan}}^{2}\omega ]}^{2}-4{z}^{2}{(L-{y}_{j})}^{2}\hspace{0.17em}{\text{tan}}^{2}\omega \}}^{1/2}),\\ {x}_{1}=0,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{x}_{n+1}={L}_{1},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{y}_{1}={L}_{1},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{y}_{n+1}={L}_{1}+{L}_{2},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{z}_{1}=0,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{z}_{m}={R}_{1},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{z}_{n+1}={R}_{2},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}1\le m\le n,\\ {M}_{\mathrm{\lambda},bb}(\mathrm{\lambda},{T}_{{x}_{0}})/{M}_{\mathrm{\lambda},bb}(\mathrm{\lambda},{T}_{0})=({e}^{{C}_{2}/\mathrm{\lambda}{T}_{0}}-1)/({e}^{{C}_{2}/\mathrm{\lambda}{T}_{{x}_{0}}}-1).\end{array}$$
(9)
$$\begin{array}{l}{\mathrm{\u220a}}_{a}(0)=[\mathrm{\u220a}+(1-\mathrm{\u220a})\hspace{0.17em}\text{sin}\theta \sum _{j=1}^{n}{\scriptstyle \frac{1}{2}}[{\mathrm{\u220a}}_{a}({y}_{j+1})+{\mathrm{\u220a}}_{a}({y}_{j})]\hspace{0.17em}\left(\frac{{(L-{y}_{j})}^{2}{\text{tan}}^{2}\omega}{{(L-{y}_{j})}^{2}\hspace{0.17em}{\text{tan}}^{2}\omega +{y}_{j}^{2}}-\frac{{(L-{y}_{j+1})}^{2}\hspace{0.17em}{\text{tan}}^{2}\omega}{{(L-{y}_{j+1})}^{2}\hspace{0.17em}{\text{tan}}^{2}\omega +{y}_{j+1}^{2}}\right)\\ +\hspace{0.17em}(1-\mathrm{\u220a})\hspace{0.17em}\text{sin}\theta \sum _{k=m}^{n}{\scriptstyle \frac{1}{2}}[{\mathrm{\u220a}}_{a}({z}_{k+1})+{\mathrm{\u220a}}_{a}({z}_{k})]\left(\frac{{z}_{k+1}^{2}}{{z}_{k+1}^{2}+{({L}_{1}+{L}_{2})}^{2}}-\frac{{z}_{k}^{2}}{{z}_{k}^{2}+{({L}_{1}+{L}_{2})}^{2}}\right)]\hspace{0.17em}{[\mathrm{\u220a}+(1-\mathrm{\u220a})\hspace{0.17em}{\text{sin}}^{3}\theta ]}^{-1}.\end{array}$$
(10)
$${\scriptstyle \frac{1}{2}}{\mathrm{\u220a}}_{a}({L}_{1})(1+\text{sin}\theta \hspace{0.17em}\text{sin}\omega -\text{cos}\theta \hspace{0.17em}\text{cos}\omega )+{\scriptstyle \frac{1}{2}}[{\mathrm{\u220a}}_{a}({L}_{1})+{\mathrm{\u220a}}_{a}({y}_{2})]\times \hspace{0.17em}\left(\frac{\text{sin}\theta -\text{sin}\theta \hspace{0.17em}\text{sin}\omega +\text{cos}\theta \hspace{0.17em}\text{cos}\omega}{2}-d{F}_{{L}_{1},{y}_{2}}\right),$$
(11)
$${\scriptstyle \frac{1}{2}}{\mathrm{\u220a}}_{a}({R}_{2})(1-\text{sin}\omega ).$$
(12)
$${\scriptstyle \frac{1}{2}}{\mathrm{\u220a}}_{a}({L}_{1}+{L}_{2})(1-\text{sin}\omega )+{\scriptstyle \frac{1}{2}}\hspace{0.17em}[{\mathrm{\u220a}}_{a}({L}_{1}+{L}_{2})+{\mathrm{\u220a}}_{a}({y}_{n})][{\scriptstyle \frac{1}{2}}(1+\text{sin}\omega )-d{F}_{{R}_{2},{y}_{n}}].$$
(13)
$$\begin{array}{l}{\mathrm{\u220a}}^{c}(\mathrm{\lambda},T,{T}_{0})=\frac{1}{2}\hspace{0.17em}(\sum _{i=1}^{n}[{\mathrm{\u220a}}_{a}({x}_{i+1})+{\mathrm{\u220a}}_{a}({x}_{i})]\hspace{0.17em}({F}_{{L}_{1}+{L}_{2}+H,{x}_{i+1}}\\ -\hspace{0.17em}{F}_{{L}_{1}+{L}_{2}+H,{x}_{i}})+\sum _{j=1}^{w}[{\mathrm{\u220a}}_{a}({y}_{i+1})+{\mathrm{\u220a}}_{a}({y}_{i})]\\ \times \hspace{0.17em}({F}_{{L}_{1}+{L}_{2}+H,{y}_{j+1}}-{F}_{{L}_{1}+{L}_{2}+H,{y}_{j}}))\hspace{0.17em}{({F}_{{L}_{1}+{L}_{2}+H,{y}_{c}})}^{-1}\end{array}$$