Let's say that there is a circular conical section that has diameter $D=.25x$ without any heat generation and I need to find the temperature distribution.

Originially I thought I could use the heat diffusivity equation at steady state to find the temperature distribution. The differential equation would be:


I am looking at the solution to the example in the book and they use Fourier's Law $$q_{x}=-kA\frac{dT}{dx}$$ and their result is $T(x)=T_{1}-\frac{4q}{\pi a^{2}k}(\frac{1}{x_{1}}-\frac{1}{x_{2}})$

Why do they use one as opposed to the other? Will the two methods produce the same result?

The reason I ask is because they also provide a derivation for the temperature distribution of a plane wall with no heat generation and they use the heat diffusivity equation


Both methods should thus give you the same result. The diffusion equation for heat can be derived from Fourier's law.

For a one-dimensional problem, you can do that easily yourself, by taking a inifinitesimal part of the rod of size $dx$ and realizing that $q_x$ at either side of this part should be the same (since there is no heat source or sink in this piece). Taking the limit for $dx \rightarrow 0$, will give you the diffusion equation. If your problem becomes more complex, the diffusion equation will be easier to solve.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.