# 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

## 2 Answers

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.

• Your approach assumes that i only want to model the heat transfer in the box. I want to model both the heat transfer in the box and the fluid flow around the box. The problem I'm asking is how to convert the external boundary condition to an internal interface condition. – Paul Mar 4 '17 at 23:55
• @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
• the interface boundary may not be fixed in time: it can be dynamic. Think of the box not as a solid, but a fluid whose interface is also affected by the the airflow. The heat flux condition must follow the interface without mesh motion. – Paul Aug 3 '17 at 4:14
• @Paul that is a different situation than I understood from your question; it seemed you were describing a box with 6 closed sides but with one side which was not insulated. If the box is fluid you will get mixing because of shear at the fluid-gas interface, sort of like a cavity flow problem. Perhaps that is useful to look into. – nluigi Aug 3 '17 at 5:51
• Initially, they are the same. The problem is imposing the a surface heat flux along an infinitesimally thin internal region (not on a boundary). – Paul Aug 3 '17 at 22:10

If you know the flow conditions upstream of the volumetric energy source term, you could extend the computational domain in that direction. The energy term then becomes an internal energy source. You can, then apply a new set of boundary conditions to the entrance of the extended domain upstream.