Heat transfer problem - conduction in a cylinder

ana111790

1. The problem statement, all variables and given/known data
The following cylinder has a temperature inside Ti and temperature outside To. Using the general equation for heat conduction in a cylinder, write the temperature distribution equation as a function of the radius T(r). What is the temperature midway at r=a? (Take the heat conductivity = k, and length of cylinder is L).
Assume no convection and constant temperature across the length of the cylinder.



2. Relevant equations

Fourier's Law in cylindrical coordinates: q''= -k (dT/dr)

3. The attempt at a solution
Boundary conditions:
r=r_i, T=T_i
r=r_o, T=T_o

So integrating Fourier's equation with these boundary points I get:
T_o-T_i= -r_oq'' ln(r_o/r_i)

I think this gives the temperature difference though, not the distribution and I also have the q'' (flux term) still in the equation as an unknown. How would I find the temperature distribution and T(r=a)?

RTW69

Steady-state, no heat generation for cylinder: d/dr(r*dT/dr)=0

integrate twice with respect to r: T(r)=C_1*ln(r)+C_2