When to use Heat Diffusivity eqn and when to use Fourier's law to find temperature distribution?

Question

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:

$$\frac{d}{dx}(k\frac{dT}{dx})=0$$

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

Answer 1

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.