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What is the general approach for calculating the heat transfer through a body? In this special case I want to calculate the $\Delta T$ (i.e. the temperature difference between the side and the top/bottom) for a given $P_{heat}$ through the following cylinder:enter image description here

The heat power is applied on the side of the outer cylinder (equally distributed), and is distributed both on the upper side and on the lower side of the cylinder. Without the trench in the cylinder I know the formula, but how does it change with the trench? (The trench is not going through the whole cylinder, only through half of it resulting in a common base). The formula for the whole cylinder is with $S_m$ the average way from side to the base, $A_Q$ the area covered by the heating source on the side, $A_B$ the area on the bottom of the cylinder and $\lambda_{Al}$ the heat transfer capacity: $$\dot{Q}=\frac{\lambda_{Al}\cdot(A_B-A_Q)}{S_m\cdot\ln\left(\frac{A_B}{A_Q}\right)}\cdot \Delta T$$ Btw., is that formula correct, or are there other ways to calculate that?

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    $\begingroup$ The geometry is complicated enough that I don't think you'll be able to find a closed form analytical solution to the problem, though I might be wrong. You'll probably have to accept a numerical solution. $\endgroup$ – Chris Mueller Oct 23 '15 at 11:56
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    $\begingroup$ I agree with @ChrisMueller the common base will make it difficult to have an analytical solution (like the formula you included). Any CFD code can solve this using a thermal solid model. $\endgroup$ – Algo Oct 23 '15 at 12:00
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    $\begingroup$ @arc_lupus It looks like Ansys is free for students. It is definitely worthwhile to learn to do FEM simulations; most of the problems you encounter outside of university can only be estimated with analytical calculations. $\endgroup$ – Chris Mueller Oct 23 '15 at 12:15
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    $\begingroup$ @arc_lupus There is also Autodesk Simulation CFD with a 3 years free license. $\endgroup$ – Algo Oct 23 '15 at 12:44
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    $\begingroup$ For a geometry that is not trivially simple but still not as complex as some parts, using a Finite Difference Model could be useful, and there you could code it yourself using Excel (I did this for a ice rink simulation for a class project.) I can elaborate on this in an answer later. $\endgroup$ – Trevor Archibald Oct 23 '15 at 16:05

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