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 μm 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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References

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  1. R. E. Bedford and C. K. Ma, “Emissivities of diffuse cavities: Isothermal and nonisothermal cones and cylinders,” J. Opt. Soc. Am. 64, 339–349 (1974). We have found the following errata in this paper: (i) In Eq. (7) the factor 12[∊a(zj+1)+∊a(zj)] should appear inside the summation. (ii) In the equation near the bottom of p. 341, xm should be replaced by xn. (iii) In Eq. (9) the variables xi and xi+1 should be interchanged. (iv) Delete (0.65 μ m) from the caption to Fig. 10.
    [Crossref]
  2. R. E. Bedford and C. K. Ma, “Emissivities of diffuse cavities. II: Isothermal and nonisothermal cylindro-cones,” J. Opt. Soc. Am. 65, 565–572 (1975). We have found the following errata in this paper: (i) In Eq. (5) an addition sign (+) should appear immediately before the quantity [1 − ∊(z,λ,Tz)]. (ii) The final term in the numerator of fi[following Eq. (5)] should contain cos2θ instead of cos3θ. (iii) The denominator of Eq. (7) should contain a factor 2.
    [Crossref]
  3. See, e.g., J. S. Toor, R. Viskanta, and E. R. F. Winter, “Radiant heat transfer between simply arranged surfaces with direct dependent properties,” J. Spacecr. Rockets 7, 382–384 (1970). In a series of papers these authors have compared values of local irradiances predicted by diffuse, specular, and more complex models that take account of the directional characteristics of the surfaces. For closed systems the overall heat transfer was found not very sensitive to the choice of model. The largest differences in predictions occurred for open systems having highly reflecting surfaces and large temperature differences. These are just the opposite of the conditions of interest here. For experimental measurements Viskanta et al. used polished and roughened gold-plated surfaces arranged in open configurations and even then the results frequently agreed as well with the diffuse as with the other models. We conclude therefore that the diffuse model is likely to suffice for calculating effective emissivities of blackbody simulators.
    [Crossref]
  4. M. Eppley and A. R. Karoli, “Absolute radiometry based upon a change in electrical resistance,” J. Opt. Soc. Am. 47, 748–755 (1957).
    [Crossref]
  5. J. M. Kendall and C. M. Berdahl, “Two blackbody radiometers of high accuracy,” Appl. Opt. 9, 1082–1091 (1970).
    [Crossref] [PubMed]
  6. R. P. Heinisch and R. N. Schmidt, “Development and application of an instrument for the measurement of directional emittance of blackbody cavities,” Appl. Opt. 9, 1920–1925 (1970).
    [PubMed]
  7. R. J. Chandos and R. E. Chandos, “Radiometric properties of isothermal, diffuse wall cavity sources,” Appl. Opt. 13, 2142–2152 (1974). Note that there are two typographical errors within the large square brackets in their Eq. (24).
    [Crossref] [PubMed]
  8. F. O. Bartell and W. L. Wolfe, “Cavity Radiation Theory,” Infrared Phys. 16, 13–24 (1976).
    [Crossref]

1976 (1)

F. O. Bartell and W. L. Wolfe, “Cavity Radiation Theory,” Infrared Phys. 16, 13–24 (1976).
[Crossref]

1975 (1)

1974 (2)

1970 (3)

See, e.g., J. S. Toor, R. Viskanta, and E. R. F. Winter, “Radiant heat transfer between simply arranged surfaces with direct dependent properties,” J. Spacecr. Rockets 7, 382–384 (1970). In a series of papers these authors have compared values of local irradiances predicted by diffuse, specular, and more complex models that take account of the directional characteristics of the surfaces. For closed systems the overall heat transfer was found not very sensitive to the choice of model. The largest differences in predictions occurred for open systems having highly reflecting surfaces and large temperature differences. These are just the opposite of the conditions of interest here. For experimental measurements Viskanta et al. used polished and roughened gold-plated surfaces arranged in open configurations and even then the results frequently agreed as well with the diffuse as with the other models. We conclude therefore that the diffuse model is likely to suffice for calculating effective emissivities of blackbody simulators.
[Crossref]

J. M. Kendall and C. M. Berdahl, “Two blackbody radiometers of high accuracy,” Appl. Opt. 9, 1082–1091 (1970).
[Crossref] [PubMed]

R. P. Heinisch and R. N. Schmidt, “Development and application of an instrument for the measurement of directional emittance of blackbody cavities,” Appl. Opt. 9, 1920–1925 (1970).
[PubMed]

1957 (1)

Bartell, F. O.

F. O. Bartell and W. L. Wolfe, “Cavity Radiation Theory,” Infrared Phys. 16, 13–24 (1976).
[Crossref]

Bedford, R. E.

Berdahl, C. M.

Chandos, R. E.

Chandos, R. J.

Eppley, M.

Heinisch, R. P.

Karoli, A. R.

Kendall, J. M.

Ma, C. K.

Schmidt, R. N.

Toor, J. S.

See, e.g., J. S. Toor, R. Viskanta, and E. R. F. Winter, “Radiant heat transfer between simply arranged surfaces with direct dependent properties,” J. Spacecr. Rockets 7, 382–384 (1970). In a series of papers these authors have compared values of local irradiances predicted by diffuse, specular, and more complex models that take account of the directional characteristics of the surfaces. For closed systems the overall heat transfer was found not very sensitive to the choice of model. The largest differences in predictions occurred for open systems having highly reflecting surfaces and large temperature differences. These are just the opposite of the conditions of interest here. For experimental measurements Viskanta et al. used polished and roughened gold-plated surfaces arranged in open configurations and even then the results frequently agreed as well with the diffuse as with the other models. We conclude therefore that the diffuse model is likely to suffice for calculating effective emissivities of blackbody simulators.
[Crossref]

Viskanta, R.

See, e.g., J. S. Toor, R. Viskanta, and E. R. F. Winter, “Radiant heat transfer between simply arranged surfaces with direct dependent properties,” J. Spacecr. Rockets 7, 382–384 (1970). In a series of papers these authors have compared values of local irradiances predicted by diffuse, specular, and more complex models that take account of the directional characteristics of the surfaces. For closed systems the overall heat transfer was found not very sensitive to the choice of model. The largest differences in predictions occurred for open systems having highly reflecting surfaces and large temperature differences. These are just the opposite of the conditions of interest here. For experimental measurements Viskanta et al. used polished and roughened gold-plated surfaces arranged in open configurations and even then the results frequently agreed as well with the diffuse as with the other models. We conclude therefore that the diffuse model is likely to suffice for calculating effective emissivities of blackbody simulators.
[Crossref]

Winter, E. R. F.

See, e.g., J. S. Toor, R. Viskanta, and E. R. F. Winter, “Radiant heat transfer between simply arranged surfaces with direct dependent properties,” J. Spacecr. Rockets 7, 382–384 (1970). In a series of papers these authors have compared values of local irradiances predicted by diffuse, specular, and more complex models that take account of the directional characteristics of the surfaces. For closed systems the overall heat transfer was found not very sensitive to the choice of model. The largest differences in predictions occurred for open systems having highly reflecting surfaces and large temperature differences. These are just the opposite of the conditions of interest here. For experimental measurements Viskanta et al. used polished and roughened gold-plated surfaces arranged in open configurations and even then the results frequently agreed as well with the diffuse as with the other models. We conclude therefore that the diffuse model is likely to suffice for calculating effective emissivities of blackbody simulators.
[Crossref]

Wolfe, W. L.

F. O. Bartell and W. L. Wolfe, “Cavity Radiation Theory,” Infrared Phys. 16, 13–24 (1976).
[Crossref]

Appl. Opt. (3)

Infrared Phys. (1)

F. O. Bartell and W. L. Wolfe, “Cavity Radiation Theory,” Infrared Phys. 16, 13–24 (1976).
[Crossref]

J. Opt. Soc. Am. (3)

J. Spacecr. Rockets (1)

See, e.g., J. S. Toor, R. Viskanta, and E. R. F. Winter, “Radiant heat transfer between simply arranged surfaces with direct dependent properties,” J. Spacecr. Rockets 7, 382–384 (1970). In a series of papers these authors have compared values of local irradiances predicted by diffuse, specular, and more complex models that take account of the directional characteristics of the surfaces. For closed systems the overall heat transfer was found not very sensitive to the choice of model. The largest differences in predictions occurred for open systems having highly reflecting surfaces and large temperature differences. These are just the opposite of the conditions of interest here. For experimental measurements Viskanta et al. used polished and roughened gold-plated surfaces arranged in open configurations and even then the results frequently agreed as well with the diffuse as with the other models. We conclude therefore that the diffuse model is likely to suffice for calculating effective emissivities of blackbody simulators.
[Crossref]

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Figures (10)

FIG. 1
FIG. 1

Geometry of the double cone.

FIG. 2
FIG. 2

Effective emissivities of surfaces of isothermal double cones with ∊ = 0.7, θ = 30°, L1 = L2, R1 = R2, and various ω.

FIG. 3
FIG. 3

Effective emissivities of surfaces of isothermal double cones with ∊ = 0.7, ω = 60°, L2 = (2 tan60°)−1, R1 = R2, and θ = 15°, 30°, 45°, 60°, 70°, 80°, 85° (curves 1–7, respectively).

FIG. 4
FIG. 4

Effective emissivities of surfaces of isothermal double cones with ∊ = 0.7, θ = 15°, ω = 10°, R 1 = 1 2 R 2, and L1/L2 = 1, 2, 4, 8 (curves 1–4, respectively).

FIG. 5
FIG. 5

Effective emissivities of surfaces of isothermal double cones with ∊ = 0.7, ω = 10°, L2 = 2L1 tanθ, R 1 = 1 2 R 2, and θ = 15°, 30°, 45°, 60°, 70°, 80°, 85°, (curves 1–7, respectively).

FIG. 6
FIG. 6

Integrated cavity emissivities of the isothermal double cones of Fig. 3 (left-hand axis) and Fig. 5 (right-hand axis) as functions of θ for a detector of radius R0 = 0.2R1 at positions H/L1 tanθ = 1, 5, 50, 500, and for a detector of radius R0 = R1 at position H = 0.

FIG. 7
FIG. 7

Effective emissivities of surfaces of isothermal double cones having the same conical section, aperture, and total length, but varying ω. Upper group (left-hand axis): ∊ = 0.7, θ = 45°, L1 = L2, R 1 = 1 2 L 1 tan θ, θ = 0°, 7.85°, 13.63°, 20.56°, 26.56° (curves 1–5 respectively): lower group (right-hand axis): ∊ = 0.7, θ = 20°, L = 2L2, R 1 = 1 2 L 1 tan θ, ω = 0°, 6.92°, 13.98°, 20° (curves 6–9, respectively).

FIG. 8
FIG. 8

Effective emissivities of surfaces of isothermal double cones with ∊ = 0.7, (L1 + L2) = (tan 15°)−1, R1 = R2 = 0.5, and the following pairs of values of θ, ω for curves 1–5, respectively: 15°, 90°; 19.66°, 28.19°; 28.79°, 15°; 46.98°, 10.13°; 90°, 7.63°. The black box on the right axis shows the range of ∊a(y) for curve 1, and on the left axis the range of ∊a(x) for curve 5.

FIG. 9
FIG. 9

Effective total and effective spectral emissivities (λ = 0.65 μm and 1.00 μm, T0 = 1300 K) of surfaces on nonisothermal double cones with ∊ = 0.7, θ = 30°, ω = 15°, L1 = L2, R1 = R2 (solid curves, left ordinates), and R 1 = 1 2 R 2 (dashed curves, right ordinates). Curves labeled 2 are for a linear 1% decrease in temperature from x = 0 to y = L1 + L2; curves labeled 3 are for a corresponding increase in temperature; curves labeled 1 are for isothermal cavities. Labels T and S distinguish total from spectral emissivities; labels a and b distinguish 0.65 from 1.00 μm.

FIG. 10
FIG. 10

Integrated spectral cavity emissivities of nonisothermal double cones with ∊ = 0.7, θ = 30°, ω = 15°, L1 = L2, T0 = 1300 K, R1 = R2 (labeled A), and R 1 = 1 2 R 2 (labeled B) as functions of wavelength for detectors of radius R0 = R1 at H = 0 and R0 = 0.2 R1 at H = 500 R2. Curves using the left ordinates are for a linear 1% decrease in temperature from x = 0 to y = L1 + L2, and curves using the right ordinates for a corresponding increase in temperature. Corresponding total emissivities are indicated by the bars labeled T.

Tables (1)

Tables Icon

TABLE I Normal emissivities ( N c) of two groups of cavities having intrinsic emissivity ∊ = 0.7: Within each group all of the cavities have the same aperture and overall length.

Equations (13)

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a ( x 0 , λ , T x 0 , T 0 ) = M λ ( x 0 , λ , T x 0 ) / M λ , b b ( λ , T 0 ) ,
a ( x 0 , T x 0 , T 0 ) = M ( x 0 , T x 0 ) / M b b ( T 0 ) ,
a ( x 0 , λ , T x 0 , T 0 ) = M λ , b b ( λ , T x 0 ) M λ , b b ( λ , T 0 ) + ( 1 - ) ( i = 1 n 1 2 [ a ( x i + 1 ) + a ( x i ) ] d F x 0 , x i + 1 - d F x 0 , x i + j = 1 n 1 2 [ a ( y j + 1 ) + a ( y j ) ] ( d F x 0 , y j - d F x 0 , y j + 1 ) + k = m n 1 2 [ a ( z k + 1 ) + a ( z k ) ] ( d F x 0 , z k + 1 - d F x 0 , z k ) ) ,
a ( y 0 , λ , T y 0 , T 0 ) = M λ , b b ( λ , T y 0 ) M λ , b b ( λ , T 0 ) + ( 1 - ) ( i = 1 n 1 2 [ a ( x i + 1 ) + a ( x i ) ] ( d F y 0 , x i + 1 - d F y 0 , x i ) + j = 1 n 1 2 [ a ( y j + 1 ) + a ( y j ) ] d F y 0 , y j + 1 - d F y 0 , y j + k = m n 1 2 [ a ( z k + 1 ) + a ( z k ) ] ( d F y 0 , z k + 1 - d F y 0 , z k ) ) ,
a ( z , λ , T z , T 0 ) = ) M λ , b b ( λ , T z ) M λ , b b ( λ , T 0 ) + ( 1 - ) ( i = 1 n 1 2 [ a ( x i + 1 ) + a ( x i ) ] ( d F z , x i + 1 - d F z , x i ) + j = 1 n 1 2 [ a ( y j + 1 ) + a ( y j ) ] ( d F z , y j + 1 - d F z , y j ) ) ,
d F x 0 , x i = f i ± ( x i cos 2 θ 2 x 0 sin θ - 1 2 sin θ ) ,             + ( x i < x i + 1 x 0 ) , - ( x 0 x i < x i + 1 ) , f i = { [ ( x i + x 0 ) 2 - 4 x i x 0 cos 2 θ ] 1 / 2 / 4 x 0 sin θ } { 1 + ( x 0 - x i ) ( x i + x 0 - 2 x i cos 2 θ ) / [ ( x i + x 0 ) 2 - 4 x i x 0 cos 2 θ ] } , d F x 0 , y j = 1 2 sin θ - cos 2 θ 2 x 0 sin θ ( y j + [ ( L - y j ) 2 tan 2 ω + x 0 2 tan 2 θ + ( y j - x 0 ) 2 ] [ x 0 tan 2 θ - ( y j - x 0 ) ] - 2 ( L - y j ) 2 x 0 tan 2 ω tan 2 θ { [ ( L - y j ) 2 tan 2 ω + x 0 2 tan 2 θ + ( y j - x 0 ) 2 ] 2 - 4 ( L - y j ) 2 x 0 2 tan 2 ω tan 2 θ } 1 / 2 ) , d F x 0 , z k = sin θ 2 - cos 2 θ 2 x 0 sin θ ( ( L 1 + L 2 - x 0 ) + [ z k 2 + x 0 2 tan 2 θ + ( L 1 + L 2 - x 0 ) 2 ] [ x 0 tan 2 θ - ( L 1 + L 2 - x 0 ) ] - 2 z k 2 x 0 tan 2 θ { [ z k 2 + x 0 tan 2 θ + ( L 1 + L 2 - x 0 ) 2 ] 2 - 4 z k 2 x 0 2 tan 2 θ } 1 / 2 ) .
d F y 0 , y j = g j ± ( ( L - y j ) cos 2 ω 2 ( L - y 0 ) sin ω - 1 2 sin ω ) ,             + ( y 0 y j < y j + 1 ) , - ( y j < y j + 1 y 0 ) .
d F y 0 , z k = - sin ω 2 - cos 2 ω 2 ( L - y 0 ) sin ω × { ( L 1 + L 2 - y 0 ) - [ z k 2 + ( L - y 0 ) 2 tan 2 ω + ( L 1 + L 2 - y 0 ) 2 ] [ ( L - y 0 ) tan 2 ω + ( L 1 + L 2 - y 0 ) ] - 2 z k 2 ( L - y 0 ) tan 2 ω { [ z k 2 + ( L - y 0 ) 2 tan 2 ω + ( L 1 + L 2 - y 0 ) 2 ] 2 - 4 z k 2 ( L - y 0 ) 2 tan 2 ω } 1 / 2 } d F z , x i = 1 2 ( 1 - [ z 2 + ( L 1 + L 2 - x i ) 2 - x i 2 tan 2 θ ] / { [ z 2 + ( L 1 + L 2 - x i ) 2 + x i 2 tan 2 θ ] 2 - 4 z 2 x i 2 tan 2 θ } 1 / 2 ) , d F z , y j = 1 2 ( 1 - [ z 2 + ( L 1 + L 2 - y j ) 2 - ( L - y j ) 2 tan 2 ω ] / { [ z 2 + ( L 1 + L 2 - y j ) 2 + ( L - y j ) 2 tan 2 ω ] 2 - 4 z 2 ( L - y j ) 2 tan 2 ω } 1 / 2 ) , x 1 = 0 ,             x n + 1 = L 1 ,             y 1 = L 1 ,             y n + 1 = L 1 + L 2 ,             z 1 = 0 ,             z m = R 1 ,             z n + 1 = R 2 ,             1 m n , M λ , b b ( λ , T x 0 ) / M λ , b b ( λ , T 0 ) = ( e C 2 / λ T 0 - 1 ) / ( e C 2 / λ T x 0 - 1 ) .
a ( 0 ) = [ + ( 1 - ) sin θ j = 1 n 1 2 [ a ( y j + 1 ) + a ( y j ) ] ( ( L - y j ) 2 tan 2 ω ( L - y j ) 2 tan 2 ω + y j 2 - ( L - y j + 1 ) 2 tan 2 ω ( L - y j + 1 ) 2 tan 2 ω + y j + 1 2 ) + ( 1 - ) sin θ k = m n 1 2 [ a ( z k + 1 ) + a ( z k ) ] ( z k + 1 2 z k + 1 2 + ( L 1 + L 2 ) 2 - z k 2 z k 2 + ( L 1 + L 2 ) 2 ) ] [ + ( 1 - ) sin 3 θ ] - 1 .
1 2 a ( L 1 ) ( 1 + sin θ sin ω - cos θ cos ω ) + 1 2 [ a ( L 1 ) + a ( y 2 ) ] × ( sin θ - sin θ sin ω + cos θ cos ω 2 - d F L 1 , y 2 ) ,
1 2 a ( R 2 ) ( 1 - sin ω ) .
1 2 a ( L 1 + L 2 ) ( 1 - sin ω ) + 1 2 [ a ( L 1 + L 2 ) + a ( y n ) ] [ 1 2 ( 1 + sin ω ) - d F R 2 , y n ] .
c ( λ , T , T 0 ) = 1 2 ( i = 1 n [ a ( x i + 1 ) + a ( x i ) ] ( F L 1 + L 2 + H , x i + 1 - F L 1 + L 2 + H , x i ) + j = 1 w [ a ( y i + 1 ) + a ( y i ) ] × ( F L 1 + L 2 + H , y j + 1 - F L 1 + L 2 + H , y j ) ) ( F L 1 + L 2 + H , y c ) - 1