A dilute solution at $20\backslash$deg C is added to a well$-$stirred tank at the rate of $80$kg$/$h$.$ A heating coil having an area of $1\mathrm{.}0$ m$2$ and overall heat transfer coefficient of $400$ W$/$m$2.$K is located in the tank and contains steam condensing at $150\backslash$deg C$.$ The heated liquid leaves at $55$ kg$/$h at the temperature of the solution in the tank, maintained uniform by effective agitation. There is $225$ kg of solution at $40\backslash$deg C in the tank at the start of the operation. The heat capacity of the liquid streams can be assumed constant $(4\mathrm{.}18$ kJ$/$kg$.$K$).$

a$)$ Write the assumptions for the conservation of energy regarding this system.

b$)$ Calculate the outlet temperature after $1$ h$.($Hint: Answer: at t$=1$ h$,$ T$=107\backslash$deg C$)$

c$)$ Estimate the minimum volume of the tank if the desired temperature of the outlet stream is $120\backslash$deg C at unsteady state. $($Hint: Answer: V $=0\mathrm{.}0425$ m$3)$