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 $=0\mathrm{.}5$ W$/$m$.$K$.$ The drum contains $180$ kg of solid paraffin at its melting temperature of $50$ oC$.$ 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 Th$=80$ oC 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 $=0\mathrm{.}1$ W$/$m$.$K The convective heat transfer coefficient over the outside surface of the insulation is h $=5$ W$/$m$2$K$.$ The inside surface temperature of the drum is at the melting point of paraffin, i$.$e$.,$ Ts$,$i $=50$ oC$.$

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

b$.$ Find the power Pelec $($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 $(50$oC$).$ If the heat of fusion for

paraffin is $200$ kJ$/$kg $($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.