Blackbody Radiation -- How it works.....

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Page ID: 74962

Alexander Scheeline & Thomas M. Spudich
Analytical Sciences Digital Library

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The definitive reference on blackbody radiation and radiative transfer is by Siegel and Howell. Here's a summary of some of the salient ideas.

In an enclosure any light radiated by a surface can do only two things: scatter from any object encountered or be absorbed by such an object. Any other phenomenon (taking the energy in a single photon and, perhaps, splitting that into two lower-energy photons) can be thought of as absorption of the original photon, followed by emission of new photons. Thus, eventually, every emitted photon is reabsorbed, and at equilibrium the distribution of photons assumes a steady state known as the Planck distribution. The hemispheric radiance, i.e. the total output at a particular wavelength λ for a black body at a temperature T is:

\[B_{\lambda,b}(\lambda,T) = \dfrac{2\pi C_1}{\lambda^5\left(e^{\Large\frac{C_2}{\lambda T}} - 1\right)}\]

with C₁ = 0.5944×10⁸ W µm⁴ m^-2 and C₂= 14388 µm K. These somewhat strange units indicate that wavelengths are in micrometers, and that the total radiation into a hemisphere (2π steradians) is in units of W m^-2 µm^-1. A number of practical results follow immediately from this formula:

\[B_{\lambda,b}(\lambda,\Delta\lambda,T) = \dfrac{A}{4\pi (f/\#)^2}\dfrac{2\pi {\Delta\lambda} C_1}{\lambda^5\left(e^{\Large\frac{C_2}{\lambda T}} - 1\right)}\]
Over an arbitrary wavelength range from λ₁ to λ₂, with source area A and f/number f/#, the flux is
\[B_{\lambda,b}(\lambda_1,\lambda_2,T) = \dfrac{A}{4\pi (f/\#)^2}
\int_{\lambda_1}^{\lambda_2}\dfrac{2\pi C_1}{\lambda^5\left(e^{\Large\frac{C_2}{\lambda T}} - 1\right)}d\lambda\]
For real materials at temperature T, the actual intensity is the blackbody intensity times the emissivity. Unfortunately, emissivity is usually symbolized as e, the same symbol used for dielectric constant!
The peak output is at a wavelength such that λ_maxT=2897.8 µm K
Total output P=σT⁴, where σ is the Stefan-Boltzmann constant, 5.67×10⁸ W m^-2 K^-4. Unlike previous equations this is over an entire sphere, not a hemisphere!

To play with the effect of temperature on black body radiation, load this spreadsheet. Some questions to think about:

For a graphite furnace at 2000 K, what power is radiated? What does this suggest about the size of the furnace and the likelihood that portable apparatus can be built?
At room temperature (298 K), where is the peak radiation wavelength? Considering that CO₂ has peak absorption at 9.6 and 10.6 µm. What is the connection between these absorption wavelengths and the controversies about global warming?

Citation:

R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer, 4th Edition, Taylor and Francis (New York, 2002).