# Natural convection between parallel isothermal plates at different temperatures

1. Jan 2, 2012

### Appa

I was searching this forum for a post like mine but couldn't find anything suitable. The problem is as follows:

There is an electronics cabinet with a double wall, and between the walls there is an air-flow that cools down the cabinet. The walls are isothermal and at different temperatures, both temperatures being higher than the ambient one. What I had to do first was to find the heat flow to the fluid when the temperatures were given, and I was able to do that quite easily with a few correlations that I found in a book called "Handbook of heat transfer fundamentals".

Nu=$\frac{qs}{2HW(T_{w}-T_{∞})k}$

where T$_{w}$ = $\frac{1}{2}(T_{1}+T_{2})$, s the distance between the plates, H the hight of the plates and W their width. q is the heat thransfer to the fluid from both plates.

Nu$_{fd}$=$\frac{4T*^{2}+7T*+4}{90(1+T*)^{2}}$Ra

where Ra=$\frac{gβ(T_{w}-T_{∞})s^{3}}{\nu \alpha}$$\frac{s}{H}$ and T*= $\frac{T_{2}-T_{∞}}{T_{1}-T_{∞}}$, $T_{1}≥T_{2}$

Nu= $[(Nu_{fd})^{m}+(cRa^{1/4})^{m}]^{1/m}$

I know all the constant values and can thus find q. Now to the problem itself:
After having found q I should find how much energy flows through wall no 2, if there is a constant heat flux from the environment toward wall 1, called $q_{1}$.
This I need to know in order to find the wall temperatures in another phase of the problem. I know I will have to iterate the temperatures but I am having trouble understanding the physics of this whole problem.

So, What I know is the Nusselt number and the h (from h=$\frac{k}{s}$Nu) but how can I find the temperatures? What I don't understand is whether the h I already found is only the convection heat transfer from both walls to the fluid or if I can use it in the expression q=hA($T_1-T_2$), or the heat flow from wall 1 to wall 2, as well. If this is so, I guess I should only find suitable energy balances for both points and then iterate woth some initial temperatures.

Could someone please help me with this? I am a bit stuck as I don't understand the physics of what I am doing quite clearly. Cheers.