# Calculating heated rectangle temperature rise

1. Jul 15, 2016

### Sherlock_H

Hi
I have a question about temperature rise and thermal conductivity.
If I have a small 1 watt heater (3 x 3 x 3mm) in the middle of a rectangular block (100x40x70mm) made of a material that has a thermal conductivity of 0.48W/mk, how do I work out the final temperature that the block settles at, on the surface i.e temperature rise above ambient.
I'm assuming the block is in still air, at say 20C.
Do I have to work out the thermal resistance of the block to air? I'm not sure how to do this when its in 3 dimensions.

Thanks

2. Jul 15, 2016

### MexChemE

Hi, Sherlock!

Are you familiar with the three-dimensional heat equation? Since you're only interested in the final temperature, you need to solve, either numerically or analytically, the steady state heat equation, a partial differential equation
$$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} = 0$$
This equation will model the temperature distribution in the block, but you will need to identify the boundary conditions of your system. For example, you can use convective boundary conditions at the interfaces between the block and the air. The boundary conditions at the middle of the block are what will give you most trouble, but I'm sure someone more experienced will come and help us with that.

In any case, start getting familiar with the heat equation and boundary conditions, and I also recommend consulting a heat transfer textbook. Personally, I like Fundamentals of Heat and Mass Transfer by Incropera et al.

3. Jul 15, 2016

### Staff: Mentor

Yes. You need to have some estimate of the heat transfer coefficient from the block to the air. It won't be very accurate, but you can find typical values in the literature. I would be inclined to assume a low value of say 1 BTU/hr-ft^2-F.