Question

A long homogeneous resistance wire of radius ro = 5 mm is being used to heat the air in a room by the passage of electric current. Heat is generated in the wire uniformly at a rate of g=5'107 W/m as a result of resistance heating. If the temperature of the outer surface of the wire remains at 180°C, determine the temperature at r = 2 mm after steady operation conditions are reached. Take the thermal conductivity of the wire to be k = 8 W/m x °C.

Answer 1

Answer:

T = 212.8125°C

Explanation:

Given

radius of the wire, $r_{0}$ = 5 mm 0.005 m

heat generated, g = 5 x $10^{7}$ W/$m^{3}$

outer surface temperature, $T_{S}$ = 180°C

Thermal conductivity, k = 8 W / m-k

Now maximum temperature occurs at the center of the wire

that is at r=0,

Therefore, $T_{o}=T_{S}+\frac{g\times r_{o}^{2}}{4\times k}$

                  $T_{o}=180+\frac{5\times 10^{7}\times 0.005^{2}}{4\times 8}$

                 $T_{o}=219.0625$°C

Therefore, temperature at r = 2 mm

$\frac{T-T_{S}}{T_{O}-T_{S}}= 1-\left (\frac{r}{r_{O}}  \right )^{2}$

$\frac{T-180}{219.0625-180}= 1-\left (\frac{2}{5}  \right )^{2}$

Therefore, T = 212.8125°C