Energy equation with convective heat transfer.
 

Hi,
In books, while driving conservative equations we use control-volume approach and in the energy equation, we consider only conduction equation (Fourier’s law). Please tell me how can we incorporate convection into it.

http://www.bakker.org/dartmouth06/engs150/03-cnsrv.pdf

I think we can't directly put q_{convection}=h*A_surface*(T1-T2) into it because convection is surface phenomena, not a volumetric phenomenon. For example, let us assume we have one quad element at the interface between the fluid and solid, where the cross-sectional area ( direction of grad(T)) of the face of the element is not equal to surface area so assuming

q=q_{convective} +q_{conductive} could introduce some error. Am I right?

Please provide some link where they incorporate all heat transfer mode in the energy equation.

If you are considering convection as that appearing in conjugate heat transfer that is a BC.s applied to the surface that ensures that q is the same at the interface. The equation for the total energy has only a general expression for the surface integral of the normal compononent of the q vector.

In my understanding direction of conductive heat transfer is along gradient(T), since we are not having complete Temperature field we can't calculate grad(T) but at least we can derive that locally.

http://i.imgur.com/9ZPnz01.png

In the figure "s" stands for solid and "f" stands for fluid.

In figure (a) we have split the solid block into two elements(mesh) using slant line, heat transfer between those elements should be same as the figure (b), so the area to calculate heat transfer rate is the cross-sectional area.

Figure (c) and (d) having interface between fluid and solid, where conduction area should be same as figure(a) and (b) but convection area is wetted area that will differ from figure (c) and (d) so convection heat transfer will be higher in figure (c) then figure (d)

In figure(c) and (d), I think conduction heat transfer area is lower than convection but we know B.C we use never take care of that for example

This is one of the examples I could think of. My intention is: Is there any pre-ready Energy equation available that take care of all heat transfer mode in it.

You may consider a fluid element, that can transfer heat to another element by all mode of heat transfer (convection and radiation could be dominant). How can I account for all heat transfer mode in an element?

Generally the continuity of the flux is prescribed at the interface solid/fluid.
If you want to have an idea of general heat transfer models that can be practically applied in CFD I suggest a reading of
http://www.afs.enea.it/fluent/Public.../PDF/chp11.pdf

Then, you can focus on theoretical aspects on texbooks of heat transfer.

What I have understood so for is, there is no need to put explicit convection heat transfer in energy equation because that would be taken care by convective part of the energy equation (but I didn't understand the explicit relation between convection part and h*(T1-T2)) and radiation is treated as a source term.

Thank you.

Thank you prof. Filippo for that nice reference for H.T. Do you know any reference relates heat transfer to phase change phenomena in connection with OpenFoam capabilities?
Thank you in advance!


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The link is working on my PC ...:confused:

No, sorry...

Thanks for replying!
Actually, I am asking because we have very nice experimental setup for two phase flow in our lab. Therefore, it would be great if we integrate that with OpenFoam numerical capabilities.


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Try to ask in the OF forum ;)

Probably my internet might be disconnected at that time :).

As already pointed out by Filippo, the convective part in the energy equation just contains all the convection. Expressing it as h(T-T_inf) is just a mean to handle overall convection effects at the system level or, for numerical simulations, when you don't actually want to simulate convection (i.e., your domain consists of the solid part only and you approximate convection in the fluid as a bc). Note however that, for actual simulations with such fluid-solid interfaces, you typically need an actual grid interface to separate them. Then the only difference among the different cases ends up being in the different numerical accuracies (typically lower for distorted interfaces).

Simply put, convection goes into energy equation as a flux term qh= hdT, where dT is simply temperature variation. Now, if you're implying natural convection and Boussinesq assumption for buoyancy, then this is a whole new matter; still, the heat part is as I described. Frank Incropera has a good chapter in convection, and O. C. Zienkiewicz The FE set vol3 states the full energy equation and how to discretise it.

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Well, I was trying to post the picture, but it's not happening... So here it is:

Convection is a heat flux, just like radiation (in the surface sense); thus, it crosses the volume boundary as a flux qh, and is defined as is, i.e., don't need to change the classical definition. Now, of course the area is dA, meaning that this is simply a boundary term. When a weak form is obtained for the energy equation, this term adds to other Neumann BCs. Then, the difference lies in that one temperature is unknown; this is a mixed, or Cauchy condition, in which both a Dirichlet and a Neumann term are collected into the surface. So, when a discrete form Ku=f is achieved, the known term hTref is a known flux and moves to the rhs. The unknown term modifies part of the element matrix in which it occurs. Now, the catch is that h depends on the flow: so a coupling relation with the momentum equation is introduced to interact h, T and flow properties (velocity and buoyancy). In a thermal FE software, you assume h known, and the coupling disappears, as the flux is well defined. But it is always driven by temperature difference, obeying zeroth law of thermodynamics. Hope it helps!

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