A very thin, lead ((mathrm{Pb})) wire of diameter (left(4 times 10^{-5} mathrm{~m} ight)) and length ((1 mathrm{~m}))

A very thin, lead \((\mathrm{Pb})\) wire of diameter \(\left(4 \times 10^{-5} \mathrm{~m}\right)\) and length \((1 \mathrm{~m})\) is placed in an air stream \(\left(1 \mathrm{~atm}, T_{\infty}=300 \mathrm{~K}\right)\) having a flow velocity, \(v_{\infty}\), of \(50 \mathrm{~m} / \mathrm{s}\). The air is in crossflow over the wire.

melting point of \(\mathrm{Pb}-600 \mathrm{~K} \quad k_{P b}-35.3 \mathrm{~W} / \mathrm{m} \mathrm{K}\)

resistance of \(\mathrm{Pb}\) wire \(-164 \Omega\)

a. How much heat can be dissipated by the wire? Ignore any temperature gradients in the wire, i.e., assume the wire is isothermal.

b. Calculate the maximum electrical current that can be passed through the wire.

c. Show, numerically, whether the assumption of no radial temperature gradient in the wire is valid (part (a)).