Blackbody fraction for radiation

From Thermal-FluidsPedia

Jump to: navigation, search
Plot of the blackbody fraction vs. λT product
Figure 1: Plot of the blackbody fraction vs. λT 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

   {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 F0-λT of at least 0.99812 (at λT = 100,000), and four terms is accurate to five significant figures up to λT = 6,000, where F0-λT = 0.73778. Figure 1 is a plot of eq. (1) or (2).

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

 {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 F0-λT values on the right-hand-side of eq. (3).

Blackbody energy in the ranges 0 <λT < (λT) 1  and 0 <λT < (λT) 2


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.

Further Reading

External Links