Mixed conductive and convective heat transfer

Question

I want to make a finite element analysis of a cold airflow through warmer pipes. In particular I want to see how the pipes cool down and the air heats up, as it travels through the pipes. Wich are the equations and boundary conditions that I have to consider?

I assume that the velocity field is already given, call it $v$. Currently I solve the following equation $$ \rho c_p(\frac{\partial T}{\partial t} + v\cdot \nabla T) - k \Delta T = 0 $$ ($T\ $ temperature, $\rho\ $ density, $c_p\ $ heat capacity at constant pressure, $k\ $ heat conductivity) over the whole domain, where inside the pipes I use the given velocity field and in the pipe material I assume $v = 0$. The constants like $\rho\ $ jump on the boundary. I have an initial condition of the same temperature everywhere and then apply a Dirichlet BC at the pipe inlet of a colder temperature.

I know that this describes convective heat transfer and for $v=0$ the equation reduces to conduction heat transfer. But does it describe my problem correctly? I do not have a boundary condition on the pipe/air boundary. Do I need one? Which one?

The heatflow inside the metal pipe seems to be very slow, I expected it to be somewhat faster. The thermal diffusivity, i.e. the term $\frac{k}{\rho c_p}$ is of the magnitude $10^{-5}$, I took this value from the literature. However this makes the diffusion part of the equation very slow. Is this correct?

Thank you!

Answer 1

As Georg says, there are a set of dimensionless numbers that control which physics are important under which conditions. To the set he suggested I'll add the Grashof number

There are existing treatment of pipes embedded in a bulk medium (because this problem comes up over and over again...), and they are very complete if the pipe is either horizontal or vertical. Using these will insure that you are using the same definition of the dimensionless numbers as the people who determined (experimentally) where important changes occur (because you've made the same choices of ambiguous length scales (i.e. diameter or radius)).

Last time I worked thee kind of problems I used a couple of textbooks borrowed from the better half:

• Çengel, Y. Heat Transfer: A practical Approach 1998
• Levenspiel, O. Engineering Flow and Heat Exchange 1998

and a dusty reference from the bowels of the our university library:

• Perry and Chilton Chemical Engineers’ Handbook 1973

Answer 2

Given the stated problem: he says he knows the flow, and with low Reynolds number thats probably a pretty good approximation, it reduces to a 2D PDE (cyllindrical coordinates assuming your flow has the same symmetry) plus time. He almost certainly needs some sort of outflow end boundary condition, although I doubt the details of it will have much effect upstream. So as a start, pick anything reasonable at the outflow boundary.

Answer 3

I think your set-up is correct. Unlike what other commentators have suggested, I do not think it is necessary for you to consider a boundary condition at the interface. It is only necessary to have a jump in the properties, as you mention. Making your discretization nodes coincide with the interface you should have no major problems. Some researchers have studied the (small) temperature jump that can appear at the interface, which at moderate temperature gradients, you should be able to neglect.

I do now know what thermal conductivity you used for the pipe, but I calculated that the factor $k / \rho c_p$ should be of the order of about $0.12 m^2/K$ for copper. I suggest that you check these numbers making sure the order of magnitudes are correct.

Also, take into account that, due to the local fluid deformation, there is an increased rate of diffusion that exceeds the molecular one. The increase of the effective diffusion due to shear strain in a pipe is known as Taylor dispersion.

So, if your discretization of the fluid domain near the interface is not fine enough you could be getting artificially slow cooling.

Furthermore, note that if the flow is turbulent the mixing will become very efficient, thus increasing the cooling rate even more, so that you probably need to run a very detailed simulation or else search the literature for equivalent turbulent diffusivities or empirical models specifically developed for pipe systems.