Physics Asked on May 26, 2021
I have this problem involving conduction of heat:
Suppose that $z = 0$ represents the ground level on a street where an
electric cable is buried at $x = 0$ and $z = −D$. The ground is kept at a
fixed temperature $u = 0$ and the cable is releasing heat (Joule effect)
at $Q$ units of energy per unit time and unit length of the cable. The
cable lies inside a protecting pipe of radius $d$ ($0 < d < D$). Calculate
the steady-state temperature $u=u_0$ of the protecting pipe. We suppose
$u_0$ = constant, the conductivity of the pipe is supposed to be high.
The result will depend on the thermal conductivity $k$ of the ground.
I’ve studied some 1-D case with a uniform exterior temperature, but in this case being a 3-D problem with a boundary (at $z=0$) where the temperature is fixed I do not think that it is the same way of procedure. Am I right? Does the steady-state temperature of the protecting pipe is constant all over their area?
I have an approximate method of solving this, based on the method of images. Treat the cable as a line source of heat flow Q (per unit length) at z = -D, and place an image line sink of heat flow -Q at z = +D. (Of course, the medium would be extended upward to infinite z). These boundary conditions will guarantee a temperature u = 0 at z = 0. Then solve for the average temperature around the circle of radius d surrounding the line source. It won't be exact (because the temperature at the pipe won't be exactly constant), but you can see how much the temperature varies circumferentially, and judge whether it is not too excessive. Certainly, this method will be very accurate at large values of D/d.
Based on this analysis, I get $$u_0=frac{Q}{2pi k}ln{(2D/d)}$$
Answered by Chet Miller on May 26, 2021
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