Heat Transfer: Calculating heat transfer rate?

Homework Statement

Given the two domains pictured below, calculate the heat transfer rate (Q [W]) for each case.
http://img6.imagebanana.com/img/83fe4j54/Selection_001.png

The domain on the left is insulated on the left, right, and right half of the bottom side, while the domain on the right is only insulated on the left and right sides. The domain on the left has the left half of the bottom side held at constant Tc and the domain on the right has the entire bottom side held at Tc. Both domains have the top surfaces held at Th.

Homework Equations

$Q=-\frac{kA}{L}\Delta{T}$

The Attempt at a Solution

I've solved for the temperature distributions numerically; however, I don't know how to calculate the heat transfer rate for the left scenario.

I believe the domain on the right would simply be:
$Q=-\frac{kA}{L}(Th-Tc)$

But I am not sure how to treat the domain on the left because of the half-insulated base. Any help/hints would be appreciated. Thanks!

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Chestermiller
Mentor
If you have the numerical solution, use the first two rows of grid points at the top to estimate the temperature gradient variation at the top. Multiply by the thermal conductivity k to get the heat flux variation along the top. Add up all the heat fluxes time the incremental widths along the top the get the overall heat flow rate.

Chestermiller,

Thank you for your reply. The answer is so obvious one you explained it. I've calculated the two heat transfer rates and they are definitely different (good). I also checked their dependencies on grid resolution and they always converge to the same numbers (good).

Thanks very much for your help.

Hi

I am trying to solve one problem:

A wall is required to be insulated by embedding wood within concrete. the total thickness of the concrete and wood are 10cm and 1 cm respectively and are fixed due to structural constraints. determine the minimum heat leak into the room if the location of the wood can be changed within the concrete. Take K for wood as 0.1w/mk and for concrete 1.0w/mk

I tried solving using thermal resistance concept by equating derivative of the total thermal resistance to 0, but not able to get the answer.

Hi

I am trying to solve one problem:

A wall is required to be insulated by embedding wood within concrete. the total thickness of the concrete and wood are 10cm and 1 cm respectively and are fixed due to structural constraints. determine the minimum heat leak into the room if the location of the wood can be changed within the concrete. Take K for wood as 0.1w/mk and for concrete 1.0w/mk

I tried solving using thermal resistance concept by equating derivative of the total thermal resistance to 0, but not able to get the answer.