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Let us say we have a metal rod. Along that rod there is a rate of heat generation of H per unit length. If we assume we are in the steady state then I would expect us (from the thermal diffusion equation) to have the following expression: $$\kappa \frac{\partial^2 T}{\partial x^2}=-\frac{H}{A}$$ Where $A$ is the cross sectional area of the rod. I have however seen this written* instead as: $$\kappa \frac{\partial^2 T}{\partial x^2}=-H$$

Which of these is right and why?

*Blundell, S. and Blundell, K., Concepts in thermal physics, 2nd ed, page 101

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  • $\begingroup$ I normally see it as $\dot q/\kappa$ (obviously $H=\dot q$). You might want to check your assumed units of H with the actual units of H as claimed by others (e.g., Blundell$^2$) $\endgroup$
    – Kyle Kanos
    Jan 11, 2016 at 15:11

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H in the second equation must be W/m3 i.e. per unit volume to be correct(Check homogeneity of dimensions). You should also check if the rod is having unit cross-section area,in that case the 2nd equation is also right.

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