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Archived Heat equation and energy transport.

1. Homework Statement
I have a rod of density [tex] \rho[/tex] and lenght [tex] l[/tex]. It's located at [tex]0\leq x\leq l [/tex]. The density of internal energy per mass is [tex]E = c(T-T_0) + E_0 [/tex] where T is the tempertature in Kelvin,[tex] E_0[/tex] is a constant and [tex]c [/tex] 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 [tex] T_0, T_l[/tex]

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

[tex] T(x) = \frac{T_l - T_0}{l}x + T_0[/tex]

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

2. Homework Equations


3. 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.
 
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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.
[tex]q'' = -k \frac{dT}{dx} = - \rho c \kappa \frac{dT}{dx}[/tex]
[tex]q'' = \rho c \kappa \left(\frac{T_0 - T_l}{l} \right)[/tex]
Now, in order to find the total thermal energy of the rod we must integrate the energy density with respect to mass
[tex]E = \int_0^m \hat{E} \ dm[/tex]
Let S be the cross-section area of the rod, then
[tex]dm = \rho S dx[/tex]
[tex]E = \rho S \int_0^l \left[c (T_l - T_0) \frac{x}{l} + E_0 \right] dx[/tex]
[tex]E = \rho S l \left[ \frac{c (T_l - T_0)}{2} + E_0 \right][/tex]
 

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