Question 12.1: Two infinite parallel black walls at T1 and T2 (Figure 11.3)...
Two infinite parallel black walls at T_1 and T_2 (Figure 11.3) are D apart, and the space between them is filled with an absorbing, emitting, and scattering gas with absorption coefficient κ_λ(x). Assuming the nearly transparent approximation is valid, derive an expression for the gas temperature distribution for the limit where radiation is dominant so that heat conduction can be neglected.
For steady state and no heat conduction, the medium is in radiative equilibrium without internal energy sources, so q_r is constant and Equation 10.37 gives (this includes scattering)
\int_{\lambda =0}^{\infty }{k_{\lambda }(x)I_{\lambda b}}[T(x)]d\lambda=\int_{\lambda =0}^{\infty }{k_{\lambda }(x)\overline{I} _{\lambda i}(x)d\lambda } (12.4)
\nabla .q_r (S)=4\pi \int_{\lambda =0}^{\infty }{k_\lambda } (S)\left[I_{\lambda b}(S)-\frac{1}{4\pi }\int_{\Omega =0}^{4\pi }{I_\lambda } (S,\Omega )d\Omega \right]d\lambda =4\pi \int_{\lambda =0}^{\infty }{k_\lambda }(S)[I_{\lambda b}(S)-\overline{I}_\lambda (S) ] d\lambda (10.37)
where the x coordinate is normal to the walls. The \overline{I} _{\lambda ,i} is obtained from the intensities reaching a volume element along paths in the positive and negative coordinate directions,
4\pi \overline{I}_{\lambda ,i}(x)=\int_{\smallfrown }^{}{I_\lambda ^+}(x,\Omega _i)d\Omega _i+\int_{\smallfrown }^{}{I_\lambda ^-}(x,\Omega _i)d\Omega _i (12.5)
Since the walls are black, the nearly transparent approximation gives, at any location between the walls, I_\lambda ^+(\Omega _i)=I_{\lambda b}(T_1) and I_\lambda ^-(\Omega _i)=I_{\lambda b}(T_2) . Then, since the blackbody intensity is independent of angle, Equation 12.5 reduces to
4\pi \overline{I}_{\lambda ,i}(x)=2\pi [I_{\lambda b}(T_1)+I_{\lambda b}(T_2)] (12.6)
Substituting Equation 12.6 into Equation 12.4 gives, at any x position between the walls,
\int_{\lambda =0}^{\infty }{k_{\lambda }(x)I_{\lambda b}}[T(x)]d\lambda=\frac{1}{2} \int_{\lambda =0}^{\infty }{k_{\lambda }(x)[I_{\lambda b}(T_1)+I_{\lambda b}(T_2)]d\lambda } (12.7)
If κ_λ(x) depends on local temperature, Equation 12.7 is solved iteratively for T(x). If κ_λ can be assumed independent of gas temperature,
\int_{\lambda =0}^{\infty }{k_{\lambda }(x)I_{\lambda b}}[T(x)]d\lambda=\frac{1}{2} \int_{\lambda =0}^{\infty }{k_{\lambda }[I_{\lambda b}(T_1)+I_{\lambda b}(T_2)]d\lambda } (12.8)
The right side is evaluated for the specified T_1 and T_2. Then the gas temperature T, which is independent of x for the specified conditions, can be found to satisfy Equation 12.8 by using a root solver. For a gray gas with temperature-independent properties, κ is constant and Equation 12.8 gives T^4=(T_1^4+T_4^2)/2.In this optically thin limit without heat conduction, the entire gray gas approaches a fourth-power temperature that is the average of the fourth powers of the black boundary temperatures.