Abstract

We have calculated, to first order, the apparent emissivity of the bounding diffuse surfaces of a high-emissivity cylindrical–spherical cavity enclosure. Our calculations indicate that to achieve emissivities close to a perfectly absorbing blackbody cavity along the bounding surfaces of the spherical enclosure, the radius of the sphere must be equal to or greater than a factor of 4 times the cylinder radius Rs4Rc. Furthermore, to achieve emissivities approaching a blackbody cavity along the lower bounding surfaces of the cylindrical enclosure, the length of the cylinder must be a factor of 4 times greater than the radius of the cylinder L4Rc. In addition, we present the mathematical framework necessary to calculate radiant transfer within a cavity enclosure that contains obscuration. These results can be applied to the design of high-emissivity blackbody calibration cavities and to the reduction of stray light in terrestrial and spaceborne optical systems.

© 2004 Optical Society of America

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References

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  1. E. M. Sparrow, L. U. Albers, E. R. G. Eckert, “Thermal radiation characteristics of cylindrical enclosures,” J. Heat Transfer C84, 73–81 (1962).
    [CrossRef]
  2. E. M. Sparrow, V. K. Jonsson, “Absorption and emission characteristics of diffuse spherical enclosures,” J. Heat Transfer C84, 188–189 (1962).
    [CrossRef]
  3. E. M. Sparrow, V. K. Jonsson, “Radiant emission characteristics of diffuse conical cavities,” J. Opt. Soc. Am. 53, 816–821 (1963).
    [CrossRef]
  4. E. M. Sparrow, V. K. Jonsson, “Thermal radiation absorption in rectangular-groove cavities,” J. Appl. Mech. E30, 237–244 (1963).
    [CrossRef]
  5. E. M. Sparrow, S. H. Lin, “Absorption of thermal radiation in V-groove cavities,” Int. J. Heat Mass Transfer 5, 1111–1115 (1962).
    [CrossRef]
  6. S. R. Meier, “Characterization of highly absorbing black appliqués in the infrared,” Appl. Opt. 40, 2788–2795 (2001).
    [CrossRef]
  7. S. R. Meier, “Reflectance and scattering properties of highly absorbing black appliqués over a broadband spectral region,” Appl. Opt. 40, 6260–6262 (2001).
    [CrossRef]
  8. R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer (Hemisphere, Washington, D.C., 1992).

2001 (2)

1963 (2)

E. M. Sparrow, V. K. Jonsson, “Radiant emission characteristics of diffuse conical cavities,” J. Opt. Soc. Am. 53, 816–821 (1963).
[CrossRef]

E. M. Sparrow, V. K. Jonsson, “Thermal radiation absorption in rectangular-groove cavities,” J. Appl. Mech. E30, 237–244 (1963).
[CrossRef]

1962 (3)

E. M. Sparrow, S. H. Lin, “Absorption of thermal radiation in V-groove cavities,” Int. J. Heat Mass Transfer 5, 1111–1115 (1962).
[CrossRef]

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, “Thermal radiation characteristics of cylindrical enclosures,” J. Heat Transfer C84, 73–81 (1962).
[CrossRef]

E. M. Sparrow, V. K. Jonsson, “Absorption and emission characteristics of diffuse spherical enclosures,” J. Heat Transfer C84, 188–189 (1962).
[CrossRef]

Albers, L. U.

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, “Thermal radiation characteristics of cylindrical enclosures,” J. Heat Transfer C84, 73–81 (1962).
[CrossRef]

Eckert, E. R. G.

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, “Thermal radiation characteristics of cylindrical enclosures,” J. Heat Transfer C84, 73–81 (1962).
[CrossRef]

Howell, J. R.

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

Jonsson, V. K.

E. M. Sparrow, V. K. Jonsson, “Thermal radiation absorption in rectangular-groove cavities,” J. Appl. Mech. E30, 237–244 (1963).
[CrossRef]

E. M. Sparrow, V. K. Jonsson, “Radiant emission characteristics of diffuse conical cavities,” J. Opt. Soc. Am. 53, 816–821 (1963).
[CrossRef]

E. M. Sparrow, V. K. Jonsson, “Absorption and emission characteristics of diffuse spherical enclosures,” J. Heat Transfer C84, 188–189 (1962).
[CrossRef]

Lin, S. H.

E. M. Sparrow, S. H. Lin, “Absorption of thermal radiation in V-groove cavities,” Int. J. Heat Mass Transfer 5, 1111–1115 (1962).
[CrossRef]

Meier, S. R.

Siegel, R.

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

Sparrow, E. M.

E. M. Sparrow, V. K. Jonsson, “Thermal radiation absorption in rectangular-groove cavities,” J. Appl. Mech. E30, 237–244 (1963).
[CrossRef]

E. M. Sparrow, V. K. Jonsson, “Radiant emission characteristics of diffuse conical cavities,” J. Opt. Soc. Am. 53, 816–821 (1963).
[CrossRef]

E. M. Sparrow, S. H. Lin, “Absorption of thermal radiation in V-groove cavities,” Int. J. Heat Mass Transfer 5, 1111–1115 (1962).
[CrossRef]

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, “Thermal radiation characteristics of cylindrical enclosures,” J. Heat Transfer C84, 73–81 (1962).
[CrossRef]

E. M. Sparrow, V. K. Jonsson, “Absorption and emission characteristics of diffuse spherical enclosures,” J. Heat Transfer C84, 188–189 (1962).
[CrossRef]

Appl. Opt. (2)

Int. J. Heat Mass Transfer (1)

E. M. Sparrow, S. H. Lin, “Absorption of thermal radiation in V-groove cavities,” Int. J. Heat Mass Transfer 5, 1111–1115 (1962).
[CrossRef]

J. Appl. Mech. (1)

E. M. Sparrow, V. K. Jonsson, “Thermal radiation absorption in rectangular-groove cavities,” J. Appl. Mech. E30, 237–244 (1963).
[CrossRef]

J. Heat Transfer (2)

E. M. Sparrow, L. U. Albers, E. R. G. Eckert, “Thermal radiation characteristics of cylindrical enclosures,” J. Heat Transfer C84, 73–81 (1962).
[CrossRef]

E. M. Sparrow, V. K. Jonsson, “Absorption and emission characteristics of diffuse spherical enclosures,” J. Heat Transfer C84, 188–189 (1962).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (1)

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

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Figures (7)

Fig. 1
Fig. 1

Cavity geometry.

Fig. 2
Fig. 2

Total angle factor F(zo) for a band in the sphere of radius Rs=2.

Fig. 3
Fig. 3

Total angle factor F(zo) for a band in the sphere of radius Rs=4.

Fig. 4
Fig. 4

Total angle factor F(zo) for a band in the sphere of radius Rs=6.

Fig. 5
Fig. 5

Total angle factor F(χo) for a band in the cylinder of length L=6 for an arbitrary sphere radius.

Fig. 6
Fig. 6

Apparent emissivity along the bounding surfaces of the sphere for Rs=4 and o=0.95 for cylinder lengths L=15.

Fig. 7
Fig. 7

Apparent emissivity surfaces along the bounding surfaces of the cylinder for L=4 and o=0.95.

Equations (55)

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a(χo)=o+(1-o)0L/da(χ)×1-|χo-χ| 2(χ-χo)2+32[(χ-χo)2+1]3/2dχ+4(1-o)(L/d-χo)01a(r)×[4(L/d-χo)2+1-r2]r{[4(L/d-χo)2+1+r2]2-4r2}3/2dr,
a(r)=o+8(1-o)0L/da(χ)(L/d-χ)×4(L/d-χ)2+1-r2{[4(L/d-χ)2+1+r2]2-4r2}3/2 dχ,
a(χo)=o+(1-o)o(1+χo)-12 2χo2+1χo2+1,
a(r)=o+12(1-o)o×1+4(L/d)2+4r2-1[4(L/d)2+4r2+1]2-16r2.
a(χo, λ, T)=(χo, λ, T)+[1-(χo, λ, T)]a(χ, λ, T)dFχoχ,
a(χo)=(χo)+[1-(χo)]a(χ, λ, T)dFχoχ.
c(χo)=o+(1-o)0Lc(χ)dFband(χo)band(χ)+-RsRs2-Rc2s(z)dFband(χo)band(z),
s(zo)=o+(1-o)×-RsRs2-Rc2s(z)dFband(zo)band(z)+0Lc(χ)dFband(zo)band(χ),
c(χo)=o+(1-o)o0LdFband(χo)band(χ)+-RsRs2-Rc2dFband(χo)band(z),
s(zo)=o+(1-o)o-RsRs2-Rc2dFband(zo)band(z)+0LdFband(zo)band(χ).
F12=12 1+1+R22R12-1+(1+R22)R122-4R2R12,
dFband(χo)band(χ)=dFband(χ)disk(χ+dχ)-dFband(χo)disk(χ)forχ<χo
=dFband(χo)disk(χ)-dFband(χo)disk(χ+dχ)forχ>χo.
dFband(χo)band(χ)=χ(dFband(χo)disk(χ))dχforχ<χo
=-χ(dFband(χo)disk(χ))dχforχ>χo.
dFband(χo)disk(χ)=Adisk(χ)Aband(χo) dFdisk(χ)band(χo)=πRc22πRcdχo dFdisk(χ)band(χo).
dFdisk(χ)band(χo)=Fdisk(χ)disk(χo)-Fdisk(χ)disk(χo+dχo)forχ<χo
=Fdisk(χ)disk(χo+dχo)-Fdisk(χ)disk(χo)forχ>χo.
dFdisk(χ)band(χo)=-χo(Fdisk(χ)disk(χo))dχoforχ<χo
=χo(Fdisk(χ)disk(χo))dχoforχ>χo.
dFband(χo)band(χ)=-Rc2 χ χoFdisk(χ)disk(χo)dχ.
0LdFband(χo)band(χ)=0L-Rc2 χ χoFdisk(χ)disk(χo)dχ.
=-Rc2 0L χ χo 2Rc2+(χ-χo)2-[2Rc2+(χ-χo)2]2-4Rc42Rc2dχ.
=-14Rc 0L χ χo(χ-χo)2dχ+14Rc 0L χ χo|χ-χo|(χ-χo)2+4Rc2dχ.
0LdFband(χo)band(χ)=1+L2Rc-2χo2+4Rc24Rcχo2+4Rc2-2(L-χo)2+4Rc24Rc(L-χo)2+4Rc2.
dFband(zo)band(z)=dFband(zo)disk(z+dz)-dFband(zo)disk(z)forzo>z
=dFband(zo)disk(z)-dFband(zo)disk(z+dz)forzo<z.
dFband(zo)band(z)=z(dFband(zo)disk(z))dzforzo>z
=-z(dFband(zo)disk(z))dzforzo<z.
dFband(zo)disk(z)=Adisk(z)Aband(zo) dFdisk(z)band(zo)=π(Rs2-z2)2πRsdz dFdisk(z)band(zo).
dFdisk(z)band(zo)=Fdisk(z)disk(zo)-Fdisk(z)disk(zo+dzo)forz<zo
=Fdisk(z)disk(zo+dzo)-Fdisk(z)disk(zo)forz>zo.
dFdisk(z)band(zo)=-zo(Fdisk(z)disk(zo))dzoforzo>z
=zo(Fdisk(z)disk(zo))dzoforzo<z.
dFband(zo)band(z)=-(Rs2-z2)2Rs z zoFdisk(z)disk(zo)dz.
-RsRs2-Rc2dFband(zo)band(z)=-RsRs2-Rc2-(Rs2-z2)2Rs z zoFdisk(z)disk(zo)dz.
Fdisk(z)disk(zo)=2Rs2-2zzo2(Rs2-z2)-[(z-zo)2(z+zo)2+2(z-zo)2(2Rs2-z2-z02)+(z-zo)4]1/22(Rs2-z2),
Fdisk(z)disk(zo)=Rs2-zzo(Rs2-z2)-|z-zo|Rs(Rs2-z2).
-(Rs2-z2)2Rs -RsRs2-Rc2 z zoFdisk(z)disk(zo)dz=-(Rs2-z2)2Rs×-RsRs2-Rc2 z zo Rs2-zzo-|z-zo|Rs(Rs2-z2)dz,
-RsRs2-Rc2dFband(zo)band(z)=12+Rs2-Rc22Rs.
dFdA1dA2=cos θ1 cos θ2πS2 dA2,
dFdA1dA2=(nˆ1·S)(nˆ2·S)πS4 dA2.
S=(Rc cos ϕ-Rs2-z12)iˆ+(Rc sin ϕ)jˆ+(z2-z1)kˆ,
nˆsph=-Rs2-z12Rsiˆ-z1Rskˆ,
nˆcyl=(-cos ϕ)iˆ-(sin ϕ)jˆ.
dFdA1dA2=(nˆcyl·S)(nˆsph·S)πS4 dA2
dFdA1dA2=(Rs2-z1z2-RcRs2-z12 cos ϕ)(Rc2-RcRs2-z12 cos ϕ)πRsRc(Rc2+Rs2+z22-2z1z2-2RcRs2-z12 cos ϕ)2 dA2.
ρ2=1(z2-z1)2{Rc2(z-z1)2+(Rs2-z12)(z-z2)2+2RcRs2-z12[z(z1+z2-z)-z1z2]cos ϕ}.
Rc2(z2-z1)2=Rc2(zH-z1)2+(Rs2-z12)(z2-zH)2+2RcRs2-z12[zH(z1+z2-zH)-z1z2]cos ϕc.
dFband(zo)band(χ)=2ϕcπdFdA1dA2Rcdϕdχ
=2ϕcπ (Rs2-z1z2-RcRs2-z12 cos ϕ)(Rc2-RcRs2-z12 cos ϕ)πRs(Rc2+Rs2+z22-2z1z2-2RcRs2-z12 cos ϕ)2 dϕdχ.
a(zo)=o+(1-o)oF(zo),
F(zo)=-RsRs2-Rc2dFband(zo)band(z)+0LdFband(zo)band(χ).
a(χo)=o+(1-o)oF(χo),
F(χo)=0LdFband(χo)band(χ)+-RsRs2-Rc2dFband(χo)band(z).

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