A container, such as the one used for transport on ships, has dimensions of $6$ x $2\mathrm{.}4$ x $2\mathrm{.}4$ m$^3($approx$.).$ The mass is $2250$ kg and it is made of steel. It stands on the relatively well insulated floor and can transfer heat to the surroundings via the other walls. Outside, the temperatures are $4\backslash$deg C $.5$ tons $($V $=5$ m$^3,$ A $=25$ m$^2,$ other values such as water$)$ of tropical fruit are to be stored in the container for a few days. The initial temperature of the fruit is $20\backslash$deg C$,$ loading takes place at $6$ a$.$m$.$ To prevent the fruit from becoming moldy, there is a fan for a fresh air supply of $30$ m$^3/$h from various small openings taken care of$.$

The container has a heater with two heating elements, each with an output of $2$ kW$.$ The Heating elements can be switched on or off for a period of $15$ minutes at a time. The Heat transfer coefficients on the outside of the container $($as it is protected from the wind$)$ are $5$ W$/($m$^2$K$),$ inside between air be stored at temperatures above $14\backslash$deg C and the air temperature should also be kept in be in this area.

a$)$ Calculate the time course of the temperatures of fruit and air for the Unregulated operation of the heating at full and half power.

b$)$ The two heating elements can be controlled based on the air temperature. Analyze for a control temperature of $17\backslash$deg C and another temperature of your choice, such as time course of the temperature over the three days and how high it will be

associated energy savings.