Boundary conditions for heat transfer in the pipe

[Sobak]

Consider the heat equation
dT/dt - aΔT + v⋅∇T = S
where S is a source term dependent of the radiation intensity I and the temperature T. The fluid velocity v is prescribed.

We also consider the radiative transfer equation describing the radiative intensity I(x,ω,t) where ω is the ray direction.

The boundary Γ consists of three parts: solid walls Γ1, inflow part Γ2 and outflow part Γ3. The domain Ω may be, for example, a rectangular channel.

What boundary conditions for T should we specify on Γ2 and Γ3? What BCs are the most natural? Please give a reference if possible.

What boundary conditions can be specified for I on these parts?

 

[Geofleur]

If you assume that you have an infinite reservoir of fluid at a certain temperature $T_L$ entering from the left, you can use a constant temperature boundary condition on that side. The boundary condition for temperature on the right would normally be a "flow-through" boundary condition. If you are solving the problem numerically, and I assume you are, the flow-through condition amounts to setting the temperature at the rightmost nodes equal to those at nodes just to the left. All this is assuming that the flow is always from left to right. In another words, use "upwind weighting" to get the temperatures on the right hand boundary. I have less experience concerning the radiative part, but you might check the book Heat Conduction, by Carslaw and Jaegar, on radiative boundary conditions. The other conditions I have mentioned are common, at least in my field, for this type of problem. But in my field, the pipe would ordinarily be filled with fluid-saturated, fractured rock. :-) Anyway, I don't see why that would change the relevant boundary conditions!

Oh, and one more thing. If you can assume that the pipe is insulated on the sides parallel to the flow, you can set the normal derivative of the temperature to zero on them.