# Modelling multiple heat sources?

I am working on a project to model heat transfer in a data center. I have basic models for a one dimensional heat transfer using conduction and convection. Such as, taking a server and modelling the temperature decrease over time and models taking distance into account based Fourier's equations. However I am now looking to step the model into 2 dimensions and this involves considering a stack of servers with multiple heat sources stacked on each other (such as a server rack).

I am struggling to find any useful journals that detail the interactions of multiple heat sources, would people mind pointing me in the direction of some useful sources or explain the basics to me so I can research and expand on them?

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A "stack of heat sources" still sounds to me like it may be a one dimensional heat transfer problem. Can you describe the geometry you're interested in a bit more? Are you interested in the air between racks perhaps? –  pentane Nov 13 '14 at 1:44

Heat transfer by conduction or convection can only take place if there is a temperature difference between two bodies/air etc. If you have a stack of servers in a rack, each at the same temperature, no heat transfer will occur between them so you could consider them a single thermal mass, adding together their individual heat outputs. http://formulas.tutorvista.com/physics/heat-transfer-formula.html

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The energy balance is still the same in higher dimensions. You have that for a given control volume (rate of energy in) - (rate of energy out) + (rate of energy generation from heat source) = (rate of accumulation).

The Fourier equation in 3 dimensions is

$$\frac{q}{A}=-k \nabla T$$

where the derivative has been replaced with a gradient. Your convection term remains the same if there's no spatial variation in $h$. And your heat generation term should be the same.

The equation you get for the temperature inside the medium is a second order PDE. Solving this will give you the temperature profile. You can use the convection expression as one of the boundary conditions.

$$\frac{\partial T}{\partial t} = \nabla^2 T + q(\mathbf{r})$$

Here I have left out density, heat capacity, etc. just to show the general form.

There's a lot of literature on this stuff. Try looking at this: http://www.ewp.rpi.edu/hartford/~wallj2/CHT/Notes/ch05.pdf

Edit: now that I'm rereading your question, I'm not sure I answered it. If you have multiple heat sources, they will all contribute to the $q$ term in the equation. So if you have two sources, your $q$ equation will just be the sum of the two terms. If you were modeling each as a point source, you would have the sum of two Dirac deltas centered at two different points $r_{1}$ and $r_{2}$.

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