Thermodynamics and Propulsion

19.1 Ideal Radiators

An ideal thermal radiator is called a ``black body.'' It has several properties:

1. It has $\alpha = 1$ , and absorbs all radiation incident on it.
2. The energy radiated per unit area is $E_b = \sigma T^4$ where $\sigma$ is the Stefan-Boltzmann constant,
   $$\sigma = 5.67 \times 10^{-8}\textrm{ W/m\textsuperscript{2}K\textsuperscript{4}}.$$
   The units of $E_b$ are therefore W/m²

The energy of a black body, $E_b$ , is distributed over a range of wavelengths of radiation. We can define $e_\lambda = dE_b/d\lambda \approx \Delta E_b/\Delta \lambda$ , the energy radiated per unit area for a range of wavelengths of width $\Delta \lambda$ . The behavior of $e_\lambda$ is given in Figure 19.2.

Figure 19.2: Emissive power of a black body at several temperatures, predicted and observed; $(\lambda T)_{e_{\lambda_\textrm{max}}}= 0.2898 \textrm{ cm K}$ [from: A Heat Transfer Textbook by Lienhard, J.]

The distribution of $e_\lambda$ varies with temperature. The quantity $\lambda T$ at the condition where $e_\lambda$ is a maximum is given by $(\lambda T)_{e_{\lambda_\textrm{max}}}= 0.2898 \textrm{ cm K}$ . As $T$ increases, the wavelength for maximum energy emission shifts to shorter values. The frequency of the radiation, $f$ , is given by $f = c/\lambda$ so high energy means short wavelengths and high frequency.

Figure 19.3: A cavity with a small hole (approximates a black body)

Figure 19.4: A small black body inside a cavity

A physical realization of a black body is a cavity with a small hole (Figure 19.3). There are many reflections and absorptions. Very few entering photons (light rays) will get out. The inside of the cavity has radiation which is homogeneous and isotropic (the same in any direction, uniform everywhere).

Suppose we put a small black body inside the cavity as seen in Figure 19.4. The cavity and the black body are both at the same temperature.

The radiant energy absorbed by the black body per second and per m² is $\alpha_B H$ , where $H$ is the irradiance, the radiant energy falling on any surface inside the cavity. The radiant energy emitted by the black body is $E_B$ . Since $\alpha_B = 1$ for a black body, $H = E_B$ . The irradiance within a cavity whose walls are at temperature $T$ is therefore equal to the radiant emittance of a black body at the same temperature and irradiance is a function of temperature only.