Heat equation and energy transport.

[center o bass]

Homework Statement

I have a rod of density $$\rho$$ and length $$l$$. It's located at $$0\leq x\leq l$$. The density of internal energy per mass is $$E = c(T-T_0) + E_0$$ where T is the tempertature in Kelvin,$$E_0$$ is a constant and $$c$$ is the specific heat capacity. We assume that the temperature is not varying across the rodd. The temperature at the two ends of the rod is $$T_0, T_l$$

a) this question was to find the time independet solution to the heatequation
$$\frac{\partial T}{\partial t} = \kappa \nabla^2 T$$ and I found this one by using the conditions to be

$$T(x) = \frac{T_l - T_0}{l}x + T_0$$

b) Find the transport of energy, per unit time, out of a cross-section of the rod at $$x = l$$.Also find the total thermal energy in the rod.

Homework Equations

The Attempt at a Solution

Im thinking that the solution to this probably is a flux integral, but I don't know how to proceed and what to integrate.

 

[MexChemE]

Transport of energy, per unit time, out of a cross-section of the rod. That sounds like heat flux. We use Fourier's Law in order to obtain the expression for heat flux.
$$q'' = -k \frac{dT}{dx} = - \rho c \kappa \frac{dT}{dx}$$
$$q'' = \rho c \kappa \left(\frac{T_0 - T_l}{l} \right)$$
Now, in order to find the total thermal energy of the rod we must integrate the energy density with respect to mass
$$E = \int_0^m \hat{E} \ dm$$
Let S be the cross-section area of the rod, then
$$dm = \rho S dx$$
$$E = \rho S \int_0^l \left[c (T_l - T_0) \frac{x}{l} + E_0 \right] dx$$
$$E = \rho S l \left[ \frac{c (T_l - T_0)}{2} + E_0 \right]$$