- #1

jackkk_gatz

- 45

- 1

- Homework Statement
- Two parallel conducting plates have a high potential difference ##\Delta \phi## and are held by a bundle of rods of circular cross section with length L and radius R where L>>R. Each rod is not a perfect electrical insulator, and has an electrical resistivity (not to be confused with electrical resistance) as a linear function of temperature ##\rho_e=\rho_{e,0}+\beta*(T-T_0)##, where ##\rho_(e,0), T_0, \beta## are constant values. Since the clamping rods are not perfect electrical insulators, heat generation by electrical conduction is induced along the rod, which is prevented by the continuous circulation of an oil flow which has a bulk temperature ##T_\infty## and a convective heat transfer coefficient h over the rods. On the other hand, it has been established that the thermophysical properties of the rods are not strongly affected by the temperature variation along it: density, specific heat capacity, thermal conductivity. To facilitate the solution of the problem, consider that the plates are in thermal equilibrium with the oil flow. Establish the mathematical model that describes the equilibrium the energy balance along the rod as a function of temperature and the physical and geometrical properties indicated.

- Relevant Equations
- ?

I know that ##\dot e_{gen} = \frac {R_{e}i^2} {\pi r^2 L}## the thing is I don't know the value of i. I didn't write it, but another thing that the problem asks is to determine the flow of electric current. The model I came up with using heat generation due to electrical current is $$ \frac {d(A \frac {dT} {dx})} {dx}+\frac {(\Delta \phi)^2} {(\rho_{e,0}+\beta*(T-T_0))*Akr^2}-\frac {hP} {kA}*(T-T_\infty)=0$$ which i think is overly complicated. So even though the problem states there is heat generation, should i ignore it when obtaining my model? Another thing that I know for sure is i have to find the electric current this way ## i=\frac {(\Delta \phi)} {(\rho_{e,0}+\beta*(T-T_0))}## , i'd expect to find the temperature distribution for T and then substitute it on here ## i=\frac {(\Delta \phi)} {(\rho_{e,0}+\beta*(T-T_0))}## but in order to obtain the temperature distribution i have to establish my mathematical model. So i'm starting to think i should work with $$ \frac {d(A \frac {dT} {dx})} {dx}-\frac {hP} {kA}*(T-T_\infty)=0$$ the thing is that here, on this model, it appears that there is no heat generation. But still im not sure of what to do, I don't know for sure if i'm not ignoring something. Another thing, why is it mentioned in the problem that the rods are not perfect electrical insulators? Why is that important? shouldn't there be heat generation anyway?