A copper rod and an aluminum rod of equal diameter are joined end to end in good

Question

A copper rod and an aluminum rod of equal diameter are joined end to end in good thermal contact. The temperature of the free end of the copper rod is held constant at 100°C, and that of the far end of the aluminum rod is held at 0°C. If the copper rod is 0.42 m long, what must be the length of the aluminum rod so that the temperature at the junction is 50°C? Solve in meters.


*Note
Consider how the rate at which copper transfers energy to the junction must be related to the rate that the aluminum transfers energy away. Then express each rate in terms of the length of the rod, the temperature at the ends of the rod, and the thermal conductivity of the rod in order to solve for the unknown length.

Explanation / Answer

The key is that the heat flow is the same through both. Writethe heat flow for each: copper: H = ( kc A / Lc ) ?T aluminum: H = (ka A / La ) ?T and set them equal: ( kc A/ Lc ) ?T = ( kaA / La ) ?T You are told that the temp in the middle is 50, so the ?Twill be equal... Also, the cross section areas are the same, so: ( kc / Lc) = ( ka / La) and finally: So: La = Lc ( ka / kc ) = 0.42 m * ( 238 / 397 ) = 0.251 m

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