A. Heat transfer at a black wall

Often for design purposes a combustion dumber or chemical reactor can be modeled as containing a well-mixed gas, one with uniform temperature and composition, excluding the wall boundary layer. Suppose, for now, that the boundary layer has a negligible effect. The net spectral radiant heat flux across any plane can be written

$$\label{eq1} q=\int^{2\pi}_{0}\int^{\pi/2}_{0}(I^{+}-I^{-})\cos{\theta}\sin{\theta}\,d\theta\, d\phi\tag{1}$$

At a black wall,

$$\label{eq2} I^{\ast}=I_{b}(T_{w})=\frac{B_{w}}{\pi} \tag{2}$$