Conduction through cylinder wall

[RTW69]

Homework Statement

I am trying to find the temperature profile through a cylinder wall. It is a one dimensional, steady state conduction problem without internal heat generation

Homework Equations

d/dr(r*dT/dr)=0 where r is radius and T is temperature

The Attempt at a Solution

I can find the solution easy enough if I assume internal heat generation
d/dr(r*dT/dr)=-r*q_gen/k I can separate variables and let the q_gen term equal 0 since there is no q_gen to get T(r)=c_1*ln(r)+c_2 I know the boundary conditions so the solution is pretty straight forward. What I would like to do is solve d/dr(r*dT/dr)=0 directly. I can take the derivative of r*dT/dr with respect to r and get d^2T/dr^2+1/r*dT/dr=0 I know I must integrate twice but my calculus is pretty much limited to separation of variables. I am nor sure how to integrate this equation. Any suggestions?

 

[LCKurtz]

Homework Statement

I am trying to find the temperature profile through a cylinder wall. It is a one dimensional, steady state conduction problem without internal heat generation

Homework Equations

d/dr(r*dT/dr)=0 where r is radius and T is temperature

The Attempt at a Solution

I can find the solution easy enough if I assume internal heat generation
d/dr(r*dT/dr)=-r*q_gen/k I can separate variables and let the q_gen term equal 0 since there is no q_gen to get T(r)=c_1*ln(r)+c_2 I know the boundary conditions so the solution is pretty straight forward. What I would like to do is solve d/dr(r*dT/dr)=0 directly. I can take the derivative of r*dT/dr with respect to r and get d^2T/dr^2+1/r*dT/dr=0 I know I must integrate twice but my calculus is pretty much limited to separation of variables. I am nor sure how to integrate this equation. Any suggestions?

If d/dr(r*dT/dr)= 0 then r*dT/dr = C (a constant)

So dT/dr = C/r ...