# Confusion in fourier's law

My book starts section 2.2 like this:

Equation 1.1 is this: $$q_x=-k\frac {\partial T}{\partial x}$$

As you can see in the picture I posted, my book says that one can go from equation 1.1 to equation 2.1 by integration. I don't see how this is done by integration at all.

Thank you

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I wouldn't call it integration, but knowing that the thermal conductivity is a constant, as well as the boundary temperature, you can calculate

$$\frac{dT}{dx}=\frac{T_2-T_1}{\Delta x}.$$

And then there is the difference between heat flow and heat flux, the latter being the heat flow per unit area, hence the factor $A$.

Formally, you would not know that the temperature profile is linear, but you can derive it from Fourier's law. Basically, you assume that the heat flux on point 1 and point 2 are equal, which givees you the steady state form of the heat equation

$$\frac{d^2T}{dx^2}=0,$$ which is integrated twice to obtain the linear profile.

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+1 Convincing. It's awkward that the book said its integration. I guessed that this was obtained by assuming a linear profile, but I did not know how to conclde that it is linear. Now when you wrote $\frac{\partial^2 T}{\partial x^2}=0$, you are assuming that there is not heat generation, steady state and that $\frac{\partial^2 T}{\partial y^2}=0$..... –  Amr Aug 11 '13 at 11:48
If this is the case, then there are other cases in which i think these assumptions don't work. Thank you –  Amr Aug 11 '13 at 11:50