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

1963

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

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.

Int. J. Heat Mass Transfer

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.

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

J. Heat Transfer

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.

Other

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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