An electrical cable consists of a $2$ mm$-$diameter copper wire encased in a $4$ mmthick insulator of thermal conductivity $0\mathrm{.}2$ W$/$m$$K$.$ The cable is placed in still air

at T$=298$ K$.$ The natural convection heat transfer coefficient can be

approximated as hnc $=1\mathrm{.}09[($Ts $-$ T$)/$ro$]$

$1/4$ W$/$m$2$

$$K where Ts is the temperature of

the insulator surface in kelvins and ro is the outer radius of the cable in metres, i$.$e$.,$

wire plus insulator.

If the maximum allowable temperature in the insulator is $150$C and the copper

wire can be considered as isothermal, what is the maximum rate of ohmic $($I

$2$R$)$

losses per unit length of the wire that can be allowed in the wire if

$($i$)$ thermal radiation is ignored, and

$($ii$)$ thermal radiation is considered for emissivity of the insulator surface $=$

$0\mathrm{.}8?$ Assume that the temperature of the surroundings is the same as the

temperature of the air.

Draw the thermal circuit for the heat transfer process in which both convection and

thermal radiation are present. State all assumptions made.

Compare and comment on the results obtained in Parts $($i$)$ and $($ii$).$