Abstract

Monte Carlo methods are often applied to the calculation of the apparent emissivities of blackbody cavities. However, for cavities with complex as well as some commonly encountered geometries, the emission Monte Carlo method experiences problems of convergence. The emission and absorption Monte Carlo methods are compared on the basis of ease of implementation and convergence speed when applied to blackbody sources. A new method to determine solution convergence compatible with both methods is developed, and the convergence speeds of the two methods are compared through the application of both methods to a right-circular cylinder cavity. It is shown that the absorption method converges faster and is easier to implement than the emission method when applied to most blackbody and lower emissivity cavities.

© 2002 Optical Society of America

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References

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  1. E. M. Sparrow, R. P. Heinisch, N. Shamsundar, “Apparent hemispherical emittance of baffled cylindrical cavities,” J. Heat Transfer 96, 112–114 (1974).
    [Crossref]
  2. R. P. Heinisch, E. M. Sparrow, N. Shamsundar, “Radiant emission from baffled conical cavities,” J. Opt. Soc. Am. 63, 152–158 (1973).
    [Crossref]
  3. N. Shamsundar, E. M. Sparrow, R. P. Heinisch, “Monte Carlo radiation solutions—effect of energy partitioning and number of rays,” Int. J. Heat Mass Transfer 16, 690–694 (1973).
    [Crossref]
  4. R. C. Corlett, “Direct Monte Carlo calculation of radiative heat transfer in vacuum,” J. Heat Transfer 88, 376–382 (1966).
    [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. J. Ishii, M. Kobayashi, F. Sakuma, “Effective emissivities of black-body cavities with grooved cylinders,” Metrologia 35, 175–180 (1998).
    [Crossref]
  7. A. V. Prokhorov, “Monte Carlo simulation of the radiative heat transfer from a blackbody to a cryogenic radiometer,” in Optical Radiation Measurements III, J. M. Palmer, ed., Proc. SPIE2815, 160–168 (1996).
    [Crossref]
  8. 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]
  9. V. Sapritsky, A. Prokhorov, “Spectral effective emissivities of nonisothermal cavities calculated by the Monte Carlo method,” Appl. Opt. 34, 5645–5652 (1995).
    [Crossref] [PubMed]
  10. M. J. Ballico, “Modelling of the effective emissivity of a graphite tube black body,” Metrologia 32, 259–265 (1996).
    [Crossref]
  11. Y. Ohwada, “Numerical calculation of multiple reflections in diffuse cavities,” J. Opt. Soc. Am. 71, 106–111 (1981).
    [Crossref]
  12. W. J. Minkowycz, Handbook of Numerical Heat Transfer (Wiley-Interscience, New York, 1988).
  13. P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–774 (1988).
    [Crossref]
  14. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Hemisphere, Washington, D.C., 1992).

1998 (1)

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

1996 (1)

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

1995 (1)

1992 (1)

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]

1988 (1)

P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–774 (1988).
[Crossref]

1981 (1)

1980 (1)

1974 (1)

E. M. Sparrow, R. P. Heinisch, N. Shamsundar, “Apparent hemispherical emittance of baffled cylindrical cavities,” J. Heat Transfer 96, 112–114 (1974).
[Crossref]

1973 (2)

R. P. Heinisch, E. M. Sparrow, N. Shamsundar, “Radiant emission from baffled conical cavities,” J. Opt. Soc. Am. 63, 152–158 (1973).
[Crossref]

N. Shamsundar, E. M. Sparrow, R. P. Heinisch, “Monte Carlo radiation solutions—effect of energy partitioning and number of rays,” Int. J. Heat Mass Transfer 16, 690–694 (1973).
[Crossref]

1966 (1)

R. C. Corlett, “Direct Monte Carlo calculation of radiative heat transfer in vacuum,” J. Heat Transfer 88, 376–382 (1966).
[Crossref]

Ballico, M. J.

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

Corlett, R. C.

R. C. Corlett, “Direct Monte Carlo calculation of radiative heat transfer in vacuum,” J. Heat Transfer 88, 376–382 (1966).
[Crossref]

Heinisch, R. P.

E. M. Sparrow, R. P. Heinisch, N. Shamsundar, “Apparent hemispherical emittance of baffled cylindrical cavities,” J. Heat Transfer 96, 112–114 (1974).
[Crossref]

N. Shamsundar, E. M. Sparrow, R. P. Heinisch, “Monte Carlo radiation solutions—effect of energy partitioning and number of rays,” Int. J. Heat Mass Transfer 16, 690–694 (1973).
[Crossref]

R. P. Heinisch, E. M. Sparrow, N. Shamsundar, “Radiant emission from baffled conical cavities,” J. Opt. Soc. Am. 63, 152–158 (1973).
[Crossref]

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Hemisphere, Washington, D.C., 1992).

Ishii, J.

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

Kobayashi, M.

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

L’Ecuyer, P.

P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–774 (1988).
[Crossref]

Minkowycz, W. J.

W. J. Minkowycz, Handbook of Numerical Heat Transfer (Wiley-Interscience, New York, 1988).

Ohwada, Y.

Ono, A.

Prokhorov, 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]

A. V. Prokhorov, “Monte Carlo simulation of the radiative heat transfer from a blackbody to a cryogenic radiometer,” in Optical Radiation Measurements III, J. M. Palmer, ed., Proc. SPIE2815, 160–168 (1996).
[Crossref]

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.

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]

Shamsundar, N.

E. M. Sparrow, R. P. Heinisch, N. Shamsundar, “Apparent hemispherical emittance of baffled cylindrical cavities,” J. Heat Transfer 96, 112–114 (1974).
[Crossref]

N. Shamsundar, E. M. Sparrow, R. P. Heinisch, “Monte Carlo radiation solutions—effect of energy partitioning and number of rays,” Int. J. Heat Mass Transfer 16, 690–694 (1973).
[Crossref]

R. P. Heinisch, E. M. Sparrow, N. Shamsundar, “Radiant emission from baffled conical cavities,” J. Opt. Soc. Am. 63, 152–158 (1973).
[Crossref]

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Hemisphere, Washington, D.C., 1992).

Sparrow, E. M.

E. M. Sparrow, R. P. Heinisch, N. Shamsundar, “Apparent hemispherical emittance of baffled cylindrical cavities,” J. Heat Transfer 96, 112–114 (1974).
[Crossref]

N. Shamsundar, E. M. Sparrow, R. P. Heinisch, “Monte Carlo radiation solutions—effect of energy partitioning and number of rays,” Int. J. Heat Mass Transfer 16, 690–694 (1973).
[Crossref]

R. P. Heinisch, E. M. Sparrow, N. Shamsundar, “Radiant emission from baffled conical cavities,” J. Opt. Soc. Am. 63, 152–158 (1973).
[Crossref]

Appl. Opt. (1)

Commun. ACM (1)

P. L’Ecuyer, “Efficient and portable combined random number generators,” Commun. ACM 31, 742–774 (1988).
[Crossref]

Int. J. Heat Mass Transfer (1)

N. Shamsundar, E. M. Sparrow, R. P. Heinisch, “Monte Carlo radiation solutions—effect of energy partitioning and number of rays,” Int. J. Heat Mass Transfer 16, 690–694 (1973).
[Crossref]

J. Heat Transfer (2)

R. C. Corlett, “Direct Monte Carlo calculation of radiative heat transfer in vacuum,” J. Heat Transfer 88, 376–382 (1966).
[Crossref]

E. M. Sparrow, R. P. Heinisch, N. Shamsundar, “Apparent hemispherical emittance of baffled cylindrical cavities,” J. Heat Transfer 96, 112–114 (1974).
[Crossref]

J. Opt. Soc. Am. (3)

Metrologia (3)

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]

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

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

Other (3)

A. V. Prokhorov, “Monte Carlo simulation of the radiative heat transfer from a blackbody to a cryogenic radiometer,” in Optical Radiation Measurements III, J. M. Palmer, ed., Proc. SPIE2815, 160–168 (1996).
[Crossref]

W. J. Minkowycz, Handbook of Numerical Heat Transfer (Wiley-Interscience, New York, 1988).

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 3rd ed. (Hemisphere, Washington, D.C., 1992).

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Figures (7)

Fig. 1
Fig. 1

Conceptual drawings of (a) the emission method and (b) the absorption method that show the principal differences between the two methods. In the emission method, the bundles are emitted from the cavity surfaces and are traced until absorbed by the cavity wall or exit the cavity. In the absorption method, the bundles are emitted from the aperture and are traced until they exit the aperture or until the reflection truncation limit m* is reached.

Fig. 2
Fig. 2

Plot of apparent emissivity versus N along with the calculated least-squares fit (L.S.F.) lines showing how the estimated slope of the least-squares lines approaches zero as N increases.

Fig. 3
Fig. 3

Plot of the apparent emissivity versus L/R for a right cylinder for both emission and absorption methods showing agreement with previous results (see Sparrow et al.1) at multiple surface emissivities and a range of values of L/R.

Fig. 4
Fig. 4

Average number of calls to the ray-tracing subroutine is plotted versus L/R for both the emission and the absorption methods for various values of cavity wall emissivity ∊w. The average number of ray traces per bundle for the absorption method is much larger than for the emission method at all surface emissivities and values of L/R. The emission method approaches the theoretical 1/(1 - ρw) = 1/∊w limit at large L/R.

Fig. 5
Fig. 5

Apparent emissivity is plotted versus the truncation limit m* for the absorption method showing that the limit used in this paper is much larger than what is needed for accurate solution convergence.

Fig. 6
Fig. 6

Total number of calls to the ray-tracing subroutine is plotted versus L/R for the emission and absorption methods. For complex geometry (i.e., large values of L/R), the absorption method converges more rapidly than the emission method.

Fig. 7
Fig. 7

Plots of the number of least-squares sets required for convergence versus L/R for the emission and absorption methods showing that the advantage in speed of convergence of the absorption method is caused by the reduced number of least-squares sets required for a convergent solution.

Equations (23)

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

Ebun=wσTw4AwN,
app=EapσTw4Aap=NoutEbunσTw4Aap=Nout/AapN/Aww,
ρappX=f1ρw+f2ρw2+f3ρw3+,
f1=appdFX, A,f2=appdFX, AX1dFX, X1,f3=appdFX, AX1X2dFX, X2dFX2, X1, fi=appdFX, AX1X2  XidFX, XidF×X3, X2dFX2, X1, ,
ρapp=i=1m*NiNρwi.
m*<Intlog Erlog ρw.
ρapp=1Nj=1Nρwmj,
app=wNj=1Nk=1mjρwk-1.
Pixjdxj=1AdA,
Rij=0xjPixjdxj,
ρB=NAw.
tNr¯τ,
r¯emissl=0ρwl=11-ρw.
s2=1n-1k=1nk-0-a*Nk-N02.
a*=k=1nNk-N0k-0k=1nNk-N02,
t=¯*-¯s/n,
¯*=1nk=1na*Nk-N0+0,
¯=1nk=1naNk-N0+0.
t=a*-aN¯s/n,
Prob|t|<δ=Proba*-aN¯s/n<δ=erfδ2,
Prob|a*-aN¯|<δsn=erfδ2.
|a*-aN¯|<β.
δsnβ.

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