# How can i convert a boundary flux condition into an internal source term?

Imagine a rectangular box that is thermally insulated on all sides except for one, where a heat flux is applied.

Now imagine the same box with the same conditions on all sides except that the side with the heat flux is now exposed to open air at a certain temperature and flowing with a certain velocity. The heat flux is still applied only to the surface of the box, but this surface is no longer a boundary of the domain of the overall problem.

As i see it there are two strategies to handle this:

1. Treat the problem as two separate domains with information passing through the interface between the air and the box.

2. Convert the surface heat flux (in some way) into an equivalent volumetric source term which applies only to the elements at the interface between the box and the air.

I want to know if there is a canonical strategy for approach #2. How can i convert surface fluxes into internal source terms such that convergence is assured in the limit as the mesh is refined?

• Your question seems incomplete. What problem are you solving? – Vikram Mar 8 '16 at 9:56
• Can you please include information about the equations you want to solve? It seems that you're interested in solving the energy equation, but are you only interested in steady state? Do you need to know details of the air velocity? Are you interested in implementing finite difference, finite volume or finite element method? Or do you want an analytic solution? This information will better help answer what "canonical strategy" (or strategies) exist for your problem. – Charles May 8 '16 at 16:10

If i understand correctly you currently have the boundary conditions as follows: $$-\lambda\frac{dT}{dx}=Q$$ where $\lambda$ is the conductivity, $\frac{dT}{dx}$ is the temperature gradient and $Q$ is the value of the heat flux.
If you now have a boundary subjected to forced convection you need to augment this boundary condition by: $$-\lambda\frac{dT}{dx}=Q+h\left(T-T_\infty\right)$$ where $h$ is the heat transfer coefficient from the surface to the air and $T_\infty$ is the temperature of the air in the bulk (far away from the boundary). The heat transfer coefficient depends on the flow velocity by an appropriate Nusselt-Reynolds relation: $$\mathrm{Nu}=\begin{cases} 0.664\mathrm{Re}{}^{\frac{1}{2}}\mathrm{Pr}{}^{\frac{1}{3}} & \mathrm{Re}<5\cdot10^{5},~\mathrm{Pr}>0.6\\ 0.037\mathrm{Re}{}^{\frac{4}{5}}\mathrm{Pr}{}^{\frac{1}{3}} & 5\cdot10^{5}<\mathrm{Re}<10^{7},~0.6<\mathrm{Pr}<60 \end{cases}$$ where: $$\mathrm{Nu}=\frac{hl}{\kappa}\quad\mathrm{Re}=\frac{ul}{\nu}\quad\mathrm{Pr}=\frac{\nu}{\alpha}$$ where $l$ is the length of the boundary, $u$ is the flow speed, $\nu$ is the kinematic viscosity and $\alpha=\frac{\kappa}{\rho c_p}$ is the thermal diffusivity.
• @Paul - but you don't want to do it by treating the system as two separate domains? If you only want to consider the box domain but need to take the air domain into account we generally lump the effects of heat transfer in the air domain into the heat transfer coefficient $h$ as i have detailed in my answer. I am not aware of any other (productive) method to do this besides modeling the whole system. – nluigi Aug 2 '17 at 7:26