Answer to Question #183120 in Mechanical Engineering for luke

Question

A 2mm diameter electrical wire is insulated by a 2mm thick rubberized sheath (k=0.13W/m-K). The convection heat transfer coefficient at the outer surface of the sheath is 10W/m2-K and the temperature of the ambient air is 20C. What is the critical diameter of the insulation? (meters)

Answer 1

Diameter of wire is 2 mm= $2\times 10^{-3}$

Thermal heat conduction = k=0.13W/m-K,

thickness of sheath= t= $2\times 10^{-3}$ m, Ambient temperature= $T_a$ = $20^o C$

insulation= 50$^o C$

Now net thermal resistance,

$R=\frac{ln \frac{r+t}{r}}{2 \times \pi k} + \frac{1}{2 \pi rh}$

$R=\frac{ln \frac{1+2}{1}}{2 \times \pi k} + \frac{1}{2 \pi 1\times 10}$

R= 6.650 m²K/W

Maximum allowable power per unit length

Q'=$\frac{T_{in}- T_{\infty}}{6.650}=\frac{50-20}{6.65}= 4.511 \frac{W}{m}$