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Heat transfer problem  conduction in a cylinder 
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#1
Feb1511, 03:34 PM

P: 43

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_{o}q'' 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)? 


#2
Feb1611, 01:59 PM

P: 374

Steadystate, 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 


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