Thermal Conductivity

[AirForceOne]

1. The problem statement, all variables and given/known data

Rod of cross sectional area A and length l has its left end held at constant temperature t1 and its right end held at t2<t1. If the conductivity varies with distance from the left end, x, according to the relationship k= x/R + k0 (R and k0 are positive), what is the steady state heat flow, H, through the rod.

2. Relevant equations

heat flow = kA*((t1-t2)/l)

3. The attempt at a solution

I've always had a hard time understanding what to integrate and what to take the derivative of. As far as I know, I need to integrate k from some x to some final x...

[hikaru1221]

Consider a small element dx on the rod, corresponding to a change dT in temperature. Temperature T is a function of position x. We have:

H=\frac{dQ}{dt}=kA\frac{dT}{dx}=A(k_o+\frac{x}{R}) \frac{dT}{dx}

Therefore: \int^{T_2}_{T_1}AdT = \int^{L}_{0}H\frac{dx}{k_o +\frac{x}{R}}

Now as H is constant, the above integrals can be solved, right? Then you can deduce H from that.

[AirForceOne]

Thanks a ton!