# Robin boundary conditions for the heat equation

Consider a 3D channel with fluid or gas with walls $\Gamma_1$, inflow part $\Gamma_2$ and outflow part $\Gamma_3$.

The temperature is described by the heat equation: $$\frac{\partial T}{\partial t} - a\Delta T + \mathbf v \cdot \nabla T = 0,$$ where $a = \dfrac{k}{\rho c_p}$.

Is it correct to specify Robin boundary conditions $$a\frac{\partial T}{\partial n} + \beta(T - T_b) = 0$$ on $\Gamma_2$ and $\Gamma_3$? Here $\beta = \dfrac{h}{\rho c_p}$ where $h$ is the convective heat transfer coefficient.

What does $h$ equal to on $\Gamma_2$ and $\Gamma_3$?

• The heat equation alone isn't sufficient to describe a flowing system. You'll need the advection-diffusion equation (at least). – user3823992 Oct 29 '14 at 21:15
• @user3823992 Yes, a term with the velocity was missed. The question is edited. – jokersobak Oct 30 '14 at 2:17