# Heat Transfer Boundary Conditions

luca131

## Homework Statement

A high temperature, gas cooled nuclear reactor consists of a composite cylindrical wall for which a thorium fuel element (Ka=57 W/m*K) is encased in graphite (Kb= 3 W/m*K) and gaseous helium flows through an annular coolant channel. Consider conditions for which the helium temperature is T=600k and the convection coefficient at the outer surface of the graphite is h= 2000 W/m^2*K. If thermal energy is uniformly generated in the fuel elemnt at a rate of q=10^8 W/m^3 what are the temperaturesT1 and T2 at the inner and outer surfaces, respectively, of the fuel element?

## Homework Equations

1/r d/dr (r dT/dr)+q/k=0

## The Attempt at a Solution

Not really sure how to derive this formula. The boundary conditions I have chosen was
at r=0 dT/dr=0 because the max temperature should be in the center. My second boundary condition is the T(r0)= Ts. If someone could help derive this formula that would be great been trying to derive the temperature profile for sometime now.
If someone could so me a crude derivation of the temperature profile that would be great.

If you do a heat balance on the section of one of the layers between r and r + delta r, you obtain: $$\left[-2\pi r k\frac{d T}{dr}\right]_{r+dr}-\left[-2\pi r k\frac{dT}{d r}\right]_{r}=(2\pi r dr)q$$or$$\frac{1}{r}\frac{d}{dr}\left(rk\frac{dT}{dr}\right)=-q$$Heat is generated only in the thorium fuel element, and the temperature of the surrounding helium is 600 K, although this is not the outer temperature of the graphite (because of the thermal boundary layer between the outer surface and the bulk helium).