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 $\Gamma_3$?

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

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