Transport phenomena - shell balance

[Anabelle37]

Homework Statement

Use a shell balance method to derive the equation governing the two dimensional temperature distribution,u, in a circular disk in polar coordinate system(r,theta). allow for the possibility of differing thermal conductivities in the r and theta directions.

Homework Equations

fouriers law in the theta direction of a cylinderical coordinate system is q_theta=-k_theta*(1/r)*(du/dtheta)

The Attempt at a Solution

I know I have to begin with a shell balance: heat in - heat out + generation = accumulation

I have assumed that its steady state with no generation of heat.

shell balance: (rate of heat flow in at r) - (rate of heat flow out at dr) + (rate of heat flow in at theta) - (rate of heat flow out at rdtheta) = 0

does this seem right so far??

so when i actually write equations for the rate of heat flow = area x heat flow, q
what is the area since is only a circular disk? Can i not use surface area?

 

[kurt.physics]

qin at r = kr(Tin-Tout)/drqout at dr = kr(Tout-Tdout)/drqin at theta = ktheta(Tin-Tout)/dthetaqout at dtheta = ktheta(Tout-Tdout)/dthetawhere Tin, Tout, Tdout are all temperaturessubstituting these equations into the shell balance equation:kr(Tin-Tout)/dr - kr(Tout-Tdout)/dr + ktheta(Tin-Tout)/dtheta - ktheta(Tout-Tdout)/dtheta = 0simplifying:kr(Tin-2Tout+Tdout)/dr + ktheta(Tin-2Tout+Tdout)/dtheta = 0which is the same as: kr(du/dr) + ktheta(du/dtheta) = 0so is this the equation governing the two dimensional temperature distribution?