Blackbody fraction for radiation

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$Plot of the blackbody fraction vs. λT product$

Figure 1: Plot of the blackbody fraction vs. $λ$ T product.

It is often necessary or useful to find the fraction of blackbody spectral emissive power that lies within a particular range of wavelength. This is most conveniently done through use of the following equation:

$\frac{{{E_{\lambda b}}}}{{{T^5}}} = \frac{{2\pi {C_1}}}{{{{\left( {\lambda T} \right)}^5}\left( {{e^{{C_2}/\lambda T}} - 1} \right)}}$

from Planck distribution. Integration of that equation over the range $0 \le \left( {\lambda T} \right)^{*} \le \lambda T$ results in the fraction of blackbody emission in that range as

$\begin{array}{l} {F_{0 - \lambda T}} = \int_{\left( {\lambda T} \right)^{*} = 0}^{\lambda T} {\frac{{{E_{\lambda b}}}}{{{T^5}}}} d(\lambda T)^{*}/\int_{\left( {\lambda T} \right)^{*} = 0}^\infty {\frac{{{E_{\lambda b}}}}{{{T^5}}}} d(\lambda T)^{*} \\ = \frac{1}{\sigma }\int_{\left( {\lambda T} \right)^{*} = 0}^{\lambda T} {\frac{{2\pi {C_1}}}{{\left( {\lambda T} \right)^{*}{^5}\left( {{e^{{C_2}/\left( {\lambda T} \right)^{*}}} - 1} \right)}}d\left( {\lambda T} \right)^{*}} \\ \end{array} \qquad \qquad(1)$

This can be integrated by parts using the same change of variables as for

${E_b} = \int_{\lambda = 0}^\infty {\frac{{2\pi {C_1}}}{{{\lambda ^5}\left( {{e^{{C_2}/\lambda T}} - 1} \right)}}d\lambda } = \frac{{2\pi {C_1}{T^4}}}{{C_2^4}}\int_{\xi = 0}^\infty {\frac{{{\xi ^3}}}{{{e^\xi } - 1}}} d\xi = \sigma {T^4}$

from Planck distribution and its consequences (Chang and Rhee, 1984)) to give the analytical relation

${F_{0 - \lambda T}} = \frac{{15}}{{{\pi ^4}}}\sum\limits_{n = 1}^\infty {\left[ {\frac{{{e^{ - n\xi }}}}{n}\left( {{\xi ^3} + \frac{{3{\xi ^2}}}{n} + \frac{{6\xi }}{{{n^2}}} + \frac{6}{{{n^3}}}} \right)} \right]} \qquad \qquad(2)$

where $\xi \equiv {C_2}/\lambda T$ . Equation (2) converges rapidly, generally requiring only a few terms for accurate evaluation. Keeping only five terms in the summation gives results accurate to five significant figures up to $F$ _{0- $λ$ T} of at least 0.99812 (at $λ$ $T$ = 100,000), and four terms is accurate to five significant figures up to $λ$ T = 6,000, where $F$ _{0- $λ$ T}= 0.73778. Figure 1 is a plot of eq. (1) or (2).

Given eq. (1), the blackbody emissive power in any interval ( $λ$ $T$ )₁ to ( $λ$ $T$ )₂ can be found from

$\begin{array}{l} {F_{{{\left( {\lambda T} \right)}_2} - {{\left( {\lambda T} \right)}_1}}} = \frac{1}{\sigma }\int_{\left( {\lambda T} \right)^{*} = {{\left( {\lambda T} \right)}_1}}^{{{\left( {\lambda T} \right)}_2}} {\frac{{2\pi {C_1}}}{{\left( {\lambda T} \right){*^5}\left( {{e^{{C_2}/\left( {\lambda T} \right)^{*}}} - 1} \right)}}d\left( {\lambda T} \right)^{*}} \\ = \frac{1}{\sigma }\left[ {\int_{(\lambda T)^{*} = 0}^{{{(\lambda T)}_2}} {\frac{{2\pi {C_1}}}{{(\lambda T)^{*}{^5}[{e^{{C_2}/(\lambda T)^{*}}} - 1]}}d(\lambda T)^{*}} - \int_{(\lambda T)^{*} = 0}^{{{(\lambda T)}_1}} {\frac{{2\pi {C_1}}}{{(\lambda T)^{*}{^5}[{e^{{C_2}/(\lambda T)^{*}}} - 1]}}d(\lambda T)^{*}} } \right] \\ = {F_{0 - {{(\lambda T)}_2}}} - {F_{0 - {{(\lambda T)}_1}}} \\ \end{array} \qquad \qquad(3)$

Interpreting eq. (3) graphically, Fig. 2 shows the difference in the blackbody normalized emissive power (the cross-hatched area) in the range

${{\lambda T}_1} \le \lambda T \le {{\lambda T}_2}$

If this is divided by the area under the entire curve (which has a value of $σ$ ), then the areas are the $F$ _{0- $λ$ T} values on the right-hand-side of eq. (3).

References

Chang, S.L. and Rhee, K.T., 1984, ‘’Blackbody Radiation Functions,” Int. Comm. Heat and Mass Transfer, Vol. 11, pp. 451-455.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

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Blackbody fraction for radiation

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