# Blackbody fraction for radiation

### From Thermal-FluidsPedia

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:

from Planck distribution. Integration of that equation over the range results in the fraction of blackbody emission in that range as

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

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

where. 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-λT } of at least 0.99812 (at λ*T* = 100,000), and four terms is accurate to five significant figures up to λT = 6,000, where *F*_{0-λT }= 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

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

If this is divided by the area under the entire curve (which has a value of σ), then the areas are the *F*_{0-λT } 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.