Abstract

A method to calculate the directional local effective emissivity for a cavity having non-Lambertian isothermal surfaces is presented. Numerical calculation is carried out for a cylindrocone. The directional local effective emissivity directed to the cavity bottom is smaller than that directed to the cavity opening, and the local effective emissivity for a Lambertian takes a value close to the middle. The integrated effective emissivities are computed for a nonimaging system and an imaging optical system.

© 1999 Optical Society of America

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References

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  1. E. M. Sparrow, R. D. Cess, Radiation Heat Transfer (Hemisphere, Washington, D.C., 1978).
  2. R. E. Bedford, C. K. Ma, “Emissivities of diffuse cavities. II. Isothermal and non-isothermal cylindro-cones,” J. Opt. Soc. Am. 65, 565–572 (1975).
    [CrossRef]
  3. C. Shouren, C. Zaixiang, C. Hongpan, “Precise calculation of the integrated emissivity of baffled blackbody cavities,” Metrologia 16, 69–72 (1980).
    [CrossRef]
  4. Y. Ohwada, “Numerical calculation of multiple reflections in diffuse cavities,” J. Opt. Soc. Am. 71, 106–111 (1981).
    [CrossRef]
  5. A. Ono, “Calculation of the directional emissivities of cavities by the Monte Carlo method,” J. Opt. Soc. Am. 70, 547–554 (1980).
    [CrossRef]
  6. V. I. Sapritsky, A. V. Prokhorov, “Calculation of the effective emissivities of specular-diffuse cavities by the Monte Carlo method,” Metrologia 29, 9–14 (1992).
    [CrossRef]
  7. M. J. Ballico, “Modeling of the effective emissivity of a graphite tube black body,” Metrologia 32, 259–265 (1996).
    [CrossRef]
  8. J. Ishii, M. Kobayashi, F. Sakuma, “Effective emissivities of black-body cavities with grooved cylinders,” Metrologia 35, 175–180 (1998).
    [CrossRef]
  9. Y. S. Torokian, D. P. Dewitt, Thermophysical Properties of Matter (IFI/Plenum, New York, 1972), Vol. 8. (Data compiled by the Purdue Research Foundation.)
  10. T. S. Trowbridge, K. P. Reitz, “Average irregularity representation of a rough surface for ray reflection,” J. Opt. Soc. Am. 65, 531–536 (1975).
    [CrossRef]
  11. T. Iuchi, T. Tsurukawaya, “New radiation thermometry using directional emissivity,” in Proceedings of TempMECO ’96, 6th International Symposium on Temperature and Thermal Measurements in Industry and Science (Levrotto & Bella, Torino, Italy, 1996), pp. 359–364.
  12. Y. Ohwada, “Influence of deviation from Lambertian reflectance on the effective emissivity of a cavity,” Metrologia 32, 713–716 (1996).
    [CrossRef]
  13. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972).
  14. Y. Ohwada, “Mathematical proof of an extended Kirchhoff law for a cavity having direction-dependent characteristics,” J. Opt. Soc. Am. A 5, 141–145 (1988).
    [CrossRef]
  15. F. E. Nicodemus, “Reflectance nomenclature and directional reflectance and emissivity,” Appl. Opt. 9, 1474–1475 (1970).
    [CrossRef] [PubMed]
  16. Y. Ohwada, “A method for calculating the temperature variation along a cavity wall,” Meas. Sci. Technol. 2, 907–911 (1991).
    [CrossRef]

1998

J. Ishii, M. Kobayashi, F. Sakuma, “Effective emissivities of black-body cavities with grooved cylinders,” Metrologia 35, 175–180 (1998).
[CrossRef]

1996

Y. Ohwada, “Influence of deviation from Lambertian reflectance on the effective emissivity of a cavity,” Metrologia 32, 713–716 (1996).
[CrossRef]

M. J. Ballico, “Modeling of the effective emissivity of a graphite tube black body,” Metrologia 32, 259–265 (1996).
[CrossRef]

1992

V. I. Sapritsky, A. V. Prokhorov, “Calculation of the effective emissivities of specular-diffuse cavities by the Monte Carlo method,” Metrologia 29, 9–14 (1992).
[CrossRef]

1991

Y. Ohwada, “A method for calculating the temperature variation along a cavity wall,” Meas. Sci. Technol. 2, 907–911 (1991).
[CrossRef]

1988

1981

1980

C. Shouren, C. Zaixiang, C. Hongpan, “Precise calculation of the integrated emissivity of baffled blackbody cavities,” Metrologia 16, 69–72 (1980).
[CrossRef]

A. Ono, “Calculation of the directional emissivities of cavities by the Monte Carlo method,” J. Opt. Soc. Am. 70, 547–554 (1980).
[CrossRef]

1975

1970

Ballico, M. J.

M. J. Ballico, “Modeling of the effective emissivity of a graphite tube black body,” Metrologia 32, 259–265 (1996).
[CrossRef]

Bedford, R. E.

Cess, R. D.

E. M. Sparrow, R. D. Cess, Radiation Heat Transfer (Hemisphere, Washington, D.C., 1978).

Dewitt, D. P.

Y. S. Torokian, D. P. Dewitt, Thermophysical Properties of Matter (IFI/Plenum, New York, 1972), Vol. 8. (Data compiled by the Purdue Research Foundation.)

Hongpan, C.

C. Shouren, C. Zaixiang, C. Hongpan, “Precise calculation of the integrated emissivity of baffled blackbody cavities,” Metrologia 16, 69–72 (1980).
[CrossRef]

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972).

Ishii, J.

J. Ishii, M. Kobayashi, F. Sakuma, “Effective emissivities of black-body cavities with grooved cylinders,” Metrologia 35, 175–180 (1998).
[CrossRef]

Iuchi, T.

T. Iuchi, T. Tsurukawaya, “New radiation thermometry using directional emissivity,” in Proceedings of TempMECO ’96, 6th International Symposium on Temperature and Thermal Measurements in Industry and Science (Levrotto & Bella, Torino, Italy, 1996), pp. 359–364.

Kobayashi, M.

J. Ishii, M. Kobayashi, F. Sakuma, “Effective emissivities of black-body cavities with grooved cylinders,” Metrologia 35, 175–180 (1998).
[CrossRef]

Ma, C. K.

Nicodemus, F. E.

Ohwada, Y.

Y. Ohwada, “Influence of deviation from Lambertian reflectance on the effective emissivity of a cavity,” Metrologia 32, 713–716 (1996).
[CrossRef]

Y. Ohwada, “A method for calculating the temperature variation along a cavity wall,” Meas. Sci. Technol. 2, 907–911 (1991).
[CrossRef]

Y. Ohwada, “Mathematical proof of an extended Kirchhoff law for a cavity having direction-dependent characteristics,” J. Opt. Soc. Am. A 5, 141–145 (1988).
[CrossRef]

Y. Ohwada, “Numerical calculation of multiple reflections in diffuse cavities,” J. Opt. Soc. Am. 71, 106–111 (1981).
[CrossRef]

Ono, A.

Prokhorov, A. V.

V. I. Sapritsky, A. V. Prokhorov, “Calculation of the effective emissivities of specular-diffuse cavities by the Monte Carlo method,” Metrologia 29, 9–14 (1992).
[CrossRef]

Reitz, K. P.

Sakuma, F.

J. Ishii, M. Kobayashi, F. Sakuma, “Effective emissivities of black-body cavities with grooved cylinders,” Metrologia 35, 175–180 (1998).
[CrossRef]

Sapritsky, V. I.

V. I. Sapritsky, A. V. Prokhorov, “Calculation of the effective emissivities of specular-diffuse cavities by the Monte Carlo method,” Metrologia 29, 9–14 (1992).
[CrossRef]

Shouren, C.

C. Shouren, C. Zaixiang, C. Hongpan, “Precise calculation of the integrated emissivity of baffled blackbody cavities,” Metrologia 16, 69–72 (1980).
[CrossRef]

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972).

Sparrow, E. M.

E. M. Sparrow, R. D. Cess, Radiation Heat Transfer (Hemisphere, Washington, D.C., 1978).

Torokian, Y. S.

Y. S. Torokian, D. P. Dewitt, Thermophysical Properties of Matter (IFI/Plenum, New York, 1972), Vol. 8. (Data compiled by the Purdue Research Foundation.)

Trowbridge, T. S.

Tsurukawaya, T.

T. Iuchi, T. Tsurukawaya, “New radiation thermometry using directional emissivity,” in Proceedings of TempMECO ’96, 6th International Symposium on Temperature and Thermal Measurements in Industry and Science (Levrotto & Bella, Torino, Italy, 1996), pp. 359–364.

Zaixiang, C.

C. Shouren, C. Zaixiang, C. Hongpan, “Precise calculation of the integrated emissivity of baffled blackbody cavities,” Metrologia 16, 69–72 (1980).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Meas. Sci. Technol.

Y. Ohwada, “A method for calculating the temperature variation along a cavity wall,” Meas. Sci. Technol. 2, 907–911 (1991).
[CrossRef]

Metrologia

Y. Ohwada, “Influence of deviation from Lambertian reflectance on the effective emissivity of a cavity,” Metrologia 32, 713–716 (1996).
[CrossRef]

C. Shouren, C. Zaixiang, C. Hongpan, “Precise calculation of the integrated emissivity of baffled blackbody cavities,” Metrologia 16, 69–72 (1980).
[CrossRef]

V. I. Sapritsky, A. V. Prokhorov, “Calculation of the effective emissivities of specular-diffuse cavities by the Monte Carlo method,” Metrologia 29, 9–14 (1992).
[CrossRef]

M. J. Ballico, “Modeling of the effective emissivity of a graphite tube black body,” Metrologia 32, 259–265 (1996).
[CrossRef]

J. Ishii, M. Kobayashi, F. Sakuma, “Effective emissivities of black-body cavities with grooved cylinders,” Metrologia 35, 175–180 (1998).
[CrossRef]

Other

Y. S. Torokian, D. P. Dewitt, Thermophysical Properties of Matter (IFI/Plenum, New York, 1972), Vol. 8. (Data compiled by the Purdue Research Foundation.)

T. Iuchi, T. Tsurukawaya, “New radiation thermometry using directional emissivity,” in Proceedings of TempMECO ’96, 6th International Symposium on Temperature and Thermal Measurements in Industry and Science (Levrotto & Bella, Torino, Italy, 1996), pp. 359–364.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (McGraw-Hill, New York, 1972).

E. M. Sparrow, R. D. Cess, Radiation Heat Transfer (Hemisphere, Washington, D.C., 1978).

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Figures (11)

Fig. 1
Fig. 1

The cavity surface is subdivided into a mesh for numerical calculation.

Fig. 2
Fig. 2

The left-hand side is the facet distribution function, where g1 and g2 are for g(γ)=exp(-γ2) and g(γ)=exp(-2γ2), respectively; g3 and g4 are for η=π/30 and η=π/18, respectively, with g(γ)=exp(-100γ2) for γη and g(γ)=exp[-(γ2+99η2)] for γη; and g5 and g6 are for α0=0.7 and α0=0.4, respectively, with g(γ)=α04/[α02 cos2(γ)+sin2(γ)]2. The right-hand side is the bidirectional reflectance distribution function, where ϕ is the angle of incidence, with ϕ=-30° for curves having a smaller maximum at φ=30° and ϕ=-80° for those having a maximum normalized to unity at φ=80°.

Fig. 3
Fig. 3

Schematic illustration of the geometry, where θ is the apex angle, l is the cavity length, and the radius of the cylinder is normalized to unity.

Fig. 4
Fig. 4

Deviation of the directional local effective emissivity from unity at xQ=0.5 for l=8, where the group of curves labeled A is for (ϕ)=0.8 and the one labeled B is for (ϕ)=0.9. Lbt and g0 are for a Lambertian, where the former is for the band method and the latter is for g(γ)=1, and g3, g5, and g6 are the same as those in Fig. 2.

Fig. 5
Fig. 5

Deviation of the directional local effective emissivity from unity for  (ϕ)=0.8, l=16, and θ=120°, where Lbt and g0 are the same as those in Fig. 4 and gi with i=1,, 6 are the same as those in Fig. 2. The left-hand side is for yQ=0.3, and the right-hand side is for yQ=3.

Fig. 6
Fig. 6

Deviation of the directional local effective emissivity from unity at yQ=0.3 for 0=0.8, where θ=120°, l=16, and g3, g5, and g6 are the same as those in Fig. 2. For ϕ>70° the left-hand and right-hand sides are for  (ϕ)=0.008ϕ+0.24 and  (ϕ)=-0.02ϕ+2.2, respectively.

Fig. 7
Fig. 7

Deviation of the directional local effective emissivity from unity versus position on the side wall with emission angle as a parameter for 0=0.8, g(γ)=g3, where g3 is the same as that in Fig. 2, θ=120°, and l=16. Lbt and g0 are the same as those in Fig. 4. For ϕ>70° the left-hand side is for  (ϕ)=0.8, and the right-hand side is for  (ϕ)=-0.02ϕ+2.2.

Fig. 8
Fig. 8

Integrated effective emissivity for the nonimaging system with l=16 and 0=0.8, where 1=0.8 is for  (ϕ)=0, 1=0.96 is for  (ϕ)=0.008ϕ+0.24, and 1=0.4 is for  (ϕ)=-0.02ϕ+2.2 for ϕ>70°, Lbt and g0 are the same as those in Fig. 4, and g1 and g3 are the same as those in Fig. 2.

Fig. 9
Fig. 9

Schematic illustration for calculation of the penumbra region.

Fig. 10
Fig. 10

Deviation of the integrated effective emissivity from unity for the imaging optical system with f=70, where Lbt and g0 are the same as those in Fig. 4, gi with i=1,, 6 are the same as those in Fig. 2, 0=0.8, and θ=120°. The group of curves labeled A is for l=8, and the one labeled B is for l=16. For ϕ>70° the left-hand side is for  (ϕ)=0.8, and the right-hand side is for  (ϕ)=-0.02ϕ+2.2.

Fig. 11
Fig. 11

Same as Fig. 10 but with f=40.

Tables (1)

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Table 1 Hemispherical Local Effective Emissivities Computed with and without Use of a Reciprocity Rule for a Non-Lambertian Surface a

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

en(ΨQ,R)=en-1(ΨQ,R)+i=1wj=1mf (ΨQ, Pij ;ΨQ,R)×[en-1(ΨPij,Q)-en-2(ΨPij,Q)]FdQ-ΔPij,
e0(ΨQ,R)=0,e1(ΨQ,R)=(ΨQ,R)forn2,
Q=1, 2,, m×w,R=1, 2,, m×(w+u),
FdQ-ΔPij=ΔPij cos(ϕQ)dωQ,Ph cos(ϕQ)dωQ,P
fori=1, 2,, w,j=1, 2,, m,
f (ΨQ,Pi j ; ΨQ,R)=ρ(ΨQ,Pi j)g(γPij, R)/G(sr-1),
Gk=1w+un=1mg(γPi j, Rkn)ΔωQ,Rkn,
ρ(ΨQ,Pij)=1-(ΨQ,Pi j),
ΔωQ,Rkn=ΔRkn cos(ϕQ)dωQ,Rkn(sr),
i=1w+uj=1mf(ΨQ,Pi j ; ΨQ,R)ΔωQ,Pij=ρ(ΨQ,R)
f (ΨQ,Pi j ; ΨQ, R)=f (ΨQ,R: ΨQ,Pij).
en(ΨQ,R)=en-1(ΨQ,R)+i=1wj=1mf(ΨQ, R ; ΨQ, Pi j)×[en-1(ΨPi j,Q)-en-2(ΨPi j,Q)]FdQ-ΔPi jforn2, Q=1, 2,, m×w,
R=1, 2,, m×(w+u),
i=1w+uj=1mf(ΨQ, R ; ΨQ,Pi j)ΔωQ,Pij=ρ(ΨQ, R).
f(ΨQ,R; ΨQ,Pi j)=ρ(ΨQ,R)g(γR,Pi j)/G,
Gk=1w+un=1mg(γR,Pkn)ΔωQ,Pkn.
g(γ)=α04/[α02 cos2(γ)+sin2(γ)]2.
g(γ)=exp(-100α1γ2)forγηwithπ/2η>0exp[-α1(γ2+99η2)]forγη,
(ϕ)=αϕ+β,
α=(1-0)/20°,β=(90-71)/2,
eh(Q)=i=1w+uj=1me(ΨQ,Rij)FdQ-ΔRiji=1w+uj=1mFdQ-ΔRij,
E(d)=Q=1m×wA=1ke(ΨQ,A)ΔQFdQ-ΔAQ=1m×wA=1kΔQFdQ-ΔA,
E(ymax)=i=1wmaxj=1mA=1kQe(ΨQij,A)ΔQijFdQij-ΔAi=1wmaxj=1mA=1kQΔQijFdQij-ΔA,
r1=r1(f+s+l-yQ)/(s+l-yQ),
x1=xQ f /(s+l-yQ),
z1=0.
ci=(i-0.5)δc+c0fori=1, 2,, u,
c0=|x1-r1|,δc=(r2-c0)/u.
χi=arccos({[(x1+ci)2-(r1)2] / (4x1ci)}1/2)
fori=1, 2,, u.
FdQ-ΔAi=1πΔAi cos(ϕQ)dωQ,A
fori=1, 2,, u, χA<2χi,
FdQ-ΔAij=1πΔAi j cos(ϕQ)dωQ, A
fori=1, 2,, u,j=1, 2,, v,

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