A solid rod is manufactured from a new superconducting material, Coldmet, and is shown in Figure P5.24. The material has the property that when a current is passed through it, it actually gets colder. The heat generation rate in the rod is \(\dot{q}=-\mathcal{J}^{2} ho_{r}\) where: \(ho_{r}=2 \times 10^{-3} \Omega-\mathrm{m}\); \(\mathcal{J}\left(\mathrm{A} / \mathrm{m}^{2}\right)\). Consider a rod, \(3 \mathrm{~cm}\) in diameter and \(1 \mathrm{~m}\) long. One end \((z=0)\) is insulated and the other end \((z=L)\) is held at \(77 \mathrm{~K}\).

a. Derive the differential equation describing the axial temperature profile in the rod. Assume steady state.

b. Solve the differential equation to determine the temperature profile (no numbers needed).

c. For a current of \(1000 \mathrm{amps}\) and a thermal conductivity of \(20 \mathrm{~W} / \mathrm{mK}\) what is the temperature at the insulated end? What is the heat flow at the other end?

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b z=0 FIGURE P5.24 J k T= 77 K z = L