Equation that describes the axial flow of heat from a solid cylinder I'd like your help regarding this thermodynamics problem:
When trying to solve it I found that there is a problem with the leaks in the cylinder caps, and it occurred to me that to avoid the problem with caps it would be useful to consider it to be infinite in length and
than the amount of heat per unit
length would be given by an equality as follows (I don't know if this helps anything yet):
Thanks in advanced!
 A: Define a smaller concentric cylinder of radius $r$ and height $\ell$ within the solid cylinder in question. At steady state, the power sourced from inside this cylinder is equal to the power leaving across its boundary. Due to axial and azimuthal symmetry, there is no power leaving through the top and bottom of this smaller cylinder, and the heat flux $q$ leaving through the sides is the same everywhere. Conservation of power yields
$$2\pi r \ell \dot q = \dot q \ell$$
$$q = \frac{\dot q}{2\pi r}. $$
From Fourier's law of heat conduction,
$$\vec q = -k\nabla T = - k\frac{\partial T}{\partial r} \hat r,$$
where $k$ is the thermal conductivity of the cylinder and the second equation uses the axial and azimuthal symmetries. We have
$$ \frac{\partial T}{\partial r} = -\frac{\dot q}{2\pi k r}.$$
The solution to this differential equation is
$$ T(r) = -\frac{\dot q}{2\pi k} \log\left(\frac r {r_0}\right)$$
where $r_0$ is a constant determined by the outer boundary condition. You will notice that the temperature blows up for $r = 0$. This arises from the fact that an infinitely narrow heat source is an idealization and not very realistic.
If you have a time-varying source or boundary conditions, you will need to solve the time-dependent heat equation
$$\nabla^2 T - \frac{C_V}{k}\frac{\partial T}{\partial t} = -\frac{\dot q}{k}\delta^2(r)$$
with the relevant boundary conditions. With the symmetries mentioned above,
$$\frac{\partial^2 T}{\partial r^2} - \frac{C_V}{k}\frac{\partial T}{\partial t} = -\frac{\dot q}{k}\delta^2(r).$$
