# Modeling technique for Joule heating

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1. May 14, 2015

### jatin1990

Good Day, I am looking for mathematical modelling of Joule heating of a simple cantilever beam . Can anybody provide me good source of relevant material.

2. May 14, 2015

### Staff: Mentor

What is the relevance of the beam being cantilevered? Is it that the beam is extended out into the air? What is the thermal boundary condition at the cantilever end of the beam. Do you need to include the heat transfer from the beam to the air, or is it just that the cantilever end is a heat sink at fixed temperature?

Chet

3. May 14, 2015

### jatin1990

Thanks Chestermiller for reply, "What is the relevance of the beam being cantilevered?" good point there no practical relevance the only thing is V is connected to the fixed end and V0 is connected to the free end. "What is the thermal boundary condition at the cantilever end of the beam." Actually there is another cantilever on the other end , both are connected at the center so we can assume free end surface is loosing heat (equivalent to the actual conduction into the other cantilever) "Do you need to include the heat transfer from the beam to the air" No we can neglect the convection lose.

Thanks.

4. May 14, 2015

### jatin1990

Thanks Chestermiller for reply, "What is the relevance of the beam being cantilevered?" good point there no practical relevance the only thing is V is connected to the fixed end and V0 is connected to the free end. "What is the thermal boundary condition at the cantilever end of the beam." Actually there is another cantilever on the other end , both are connected at the center so we can assume free end surface is loosing heat (equivalent to the actual conduction into the other cantilever) "Do you need to include the heat transfer from the beam to the air" No we can neglect the convection lose.
Thanks.

5. May 14, 2015

### Staff: Mentor

The differential equation that describes this system is given by:
$$0=k\frac{d^2T}{dx^2}+Q$$
where k is the thermal conductivity, T is the temperature, x is the distance along the beam, and Q is the rate of heat generation per unit volume. Do you know how to solve this for the temperature distribution along the beam?

Chet