# What are the heat equation boundary conditions with heating?

I'm trying to figure out how to do a simplified model of heat flow through a wall when the wall is also sunlit. The simplified model is a 1-dimensional heat equation, $$\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}.$$ So, the boundary condition inside is a heat bath at some temperature ($$T_{\mathrm{in}}$$). How are things handled on the outside, though? The outside is in contact with the air, heat bath at temperature $$T_{\mathrm{out}}$$. Stopping there, the answer would be straightforward. Say we add the sun shining, which provides additional heating intensity to the outside surface, $$I=I_0\cos^2\theta (1-\epsilon)$$ ($$\epsilon$$ the albedo).

What, then, is the boundary condition on $$T$$ and/or its derivative at the outside surface? In such a situation do we need to consider the outside to be not a heat bath and consider heat transport in the outside air?

On the outside, you have a combination of radiative and convective heat transfer in parallel. So the heat flux at the wall is given by $$q=-k\left(\frac{\partial T}{\partial x}\right)_{x=L}=-solar+\epsilon \sigma(T_{x=L}^4-T_{\infty}^4)+h(T_{x=L}-T_{\infty})$$where the first term is the unreflected solar flux at the wall, the second term is the Stefan-Boltzman radiative heat flux, the third term is the convective heat flux, with $$T_{\infty}$$ representing the outside air temperature and h representing the convective heat transfer coefficient (determined by the outside wind speed).