A jacketed vessel is used to cool the process stream as shown below. Some information is provided for this process:

(i) The volume of liquid inside the tank, V, and the volume of coolant inside the jacket, VJ, are constant. The volumetric flow rate, qF, is constant, but qJ changes over time.

(ii) Heat transfer coefficients can be neglected for the jacketed vessel.

(iii) The contents inside the tank and the jacket are well-mixed and have distinct heat capacities.

(iv) The thermal capacitances of the tank wall and jacket wall can be neglected.

(v) The overall heat transfer coefficient used for heat conduction between the tank liquid and the coolant varies with the coolant flow rate:

U = K.qJ0.8

Here, U (Btu/hour.ft2.F), qJ (ft3/hour), and K (constant).

Derive the dynamic model for this system. Perform a degree of freedom analysis. Please include additional assumptions!