Heat us generated in an electric wire of radius r0 at a volu

Heat us generated in an electric wire of radius r_0 at a volumetric rate q\"_0. The wire moves with velocity U through a large chamber where it exchanges heat by convection. The heat transfer coefficient is h and chamber temperature is T_infinity. The wire enters the chamber at temperature T_i. Determine the steady state two-dimensional temperature distribution.

Solution

Heat balance with convection:

Q= pi r₀² L qo =h (Ts-T inf)2pi*r₀L

DT = r₀ q_0/(2h) , hence surface temp is found.

Now heat flux is also kdT/dr , total heat flux over cylinder is 2pir L k ( dT/dr)

We also have dQ/dr = 2pi r L q₀

from the above eqns, d/dr(rdT/dr) = -q₀/kr

Integrating and using BC, T =-q₀/4k r² +Const

at the surface, T =Ts

const can be found, so soln is T = Ts + qo/4k (r₀²-r²), where Ts is known from the relation DT=r₀q₀/2h