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

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

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

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