# Mixed conductive and convective heat transfer

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!

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
• Perry and...1973 and You call that "dusty" :=) All that stuff was developed before WWII, and I doubt that there is much newer knowledge (exept maybe some computations of something uncalculable without computers) BTW Grashof, You think of some autoconvection at low Reynolds numbers? Mar 11, 2011 at 16:34
• @Georg: I know it's not that old (I went back to 1957 for my dissertation and to the forties for a paper), but it was literally dusty. What with the internet the need to consult such references in person is much reduced these days. Mar 11, 2011 at 16:38
• Thank you for your answer. Can you comment on the PDE that I gave in my question? Does it describe my problem correctly and in the full domain? And do I need a boundary condition between the fluid and the pipe? Mar 14, 2011 at 7:42
• @Till that looks correct in the bulk, though you will certainly need some kind of boundary condition (at a minimum heat out had to affect the pipes temperature, which affects the rate of heat loss). But notice that you have a velocity in there. How do you know it? And do you know if your medium is convective or not at any going point (in space and time)? There is a lot going on in this problem. Mar 14, 2011 at 14:15
• At the moment I only assume a steady flow inside the pipe, i.e. it is not affected by the temperature. I will implement this later. I know the velocity from solving the steady state Navier-Stokes equations. At the moment I solve the above PDE in both the pipe material and the fluid. On the outer boundary I allow the heat to openly flow out. My main concern is the boundary between fluid and pipe material. You say I need a boundary condition here. Which one? Neumann? I already implemented that properties like heat capacity are temperature dependent. Mar 14, 2011 at 14:35

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.

• Thank you for your answer. Do you mean a boundary condition for the flow? My current main concern is what happens between the fluid and the pipe. Do I need a BC here? Mar 14, 2011 at 7:43
• @Till: I assume you are satisfied with taking a known flow field and then solving for thermal transport. I would think you need a BC on all boundaries, for the pipe it would be heat flux from the fliud equals heat flux into the pipe. Or if you think you can simply claim the pipe is a uniform temperature than your BC is to simply impose the temperature there. It is hard for me to concoct a good argument for the form of the temp field at inflo/outflow boundaries. At the later (outflow) you could move the solution domain/BC further downflow, most likely any errors damp out upstream of it. Mar 14, 2011 at 16:00
• ok, but should "heat from fluid equals heat into solid" not be covered by the convection-diffusion equation? Currently I solve this equation over both the fluid and the solid area, where I assume zero velocity inside the solid (so it reduces to the heat equation) and I let the heat capacity and conductivity jump on the fluid-solid boundary. So "heat from fluid equals heat into solid" is just what the finite element method makes at the mesh cell facets at the boundary. Mar 14, 2011 at 16:26
• The problem depends hugely on the behavior of the fluid surrounding the pipe. That stuff---be it air, water, oil or anything else---is not static. In fact, it usually turns out to be the biggest part of the problem. If the surrounding medium is solid, the problem might reduce to something solvable in closed form. Mar 15, 2011 at 0:23

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.