$(40$ pts$)$ Consider a $1\mathrm{.}0-$meter tall plastic drum with $2\mathrm{.}0$ cm of wall thickness and $50$ cm of inner

diameter, i$.$e$.,$ with an outer diameter of $54$ cm $.$ The thermal conductivity of the plastic is $k=0\mathrm{.}5$

$\frac{W}{m}.$K$.$ The drum contains $180$ kg of solid paraffin at its melting

temperature of $50C.$ The outside air temperature is $20$ C $.$

Paraffin is to be melted using a belt$/$strip heater, as depicted in a figure,

which wraps over$/$around the 'entire cylindrical surface' of the drum. The

tank$\text{'}$s circular top and bottom are insulated; thus, there is no heat loss. The

heater is set to operate at $T_h=80C$ for safety. To reduce heat loss to the

outside, we wrap the exposed heater surface with a $5\mathrm{.}0$ cm thick insulation

with a thermal conductivity of $k=0\mathrm{.}1\frac{W}{m}.$K The convective heat transfer

coefficient over the outside surface of the insulation is $h=5\frac{W}{m^2}K.$ The

inside surface temperature of the drum is at the melting point of paraffin,

i$.$e$.,T_{s,i}=50C.$

a$.$ Draw the resistance diagram representing the heat transfer process.

b$.$ Find the power $P_{\text{e}\text{l}\text{e}\text{c}}($Watts$)$ necessary to operate the

heater. If the heater is plugged into a $116$ V outlet, what

would be the current draw in Amperes?

c$.$ How much heat is lost to the surroundings?

d$.$ The tank has $180$ kg of paraffin initially all solid at the

melting temperature $(50C).$ If the heat of fusion for

paraffin is $200k\frac{J}{k}g($i$.$e$.,$ heat necessary to melt the

paraffin$),$ how long would it take to melt the entire paraffin? Assume that the process is

slow enough that the temperature of the paraffin remains uniform until all paraffin melts.