A blending tank is used to mix two different components, A and B$.$ Due to enthalpy of mixing, the temperature within the vessel would increase if no cooling method is supplied. The vessel has an internal diameter of $5m$ and straight side height of $15m.$ The liquid level in the tank is maintained at $11m.$ The process fluids have a liquid density of $1200k\frac{g}{m^3},$ a viscosity of $25cP,$ a thermal conductivity of $350m\frac{W}{m}-K,$ and a heat capacity of $5\mathrm{.}3k\frac{J}{k}g-K.$

To remove the process heat, treated cooling water is supplied through a $180,7cm$ outer diameter half$-$piped jacket. The inside diameter of the pipe is $6\mathrm{.}2cm.$ The cooling water is supplied at $25C$ with a flow rate of $0\mathrm{.}01\frac{m^3}{sec},$ but leaves at an unknown temperature. It is assumed that the Dean number may be ignored and the viscosity at the wall is $1\mathrm{.}5$ times greater than bulk viscosity. The other properties can be taken for water at $25C.$

The agitator in the tank consists of three, $1m$ diameter concave radial turbines that operate at $84rpm$ or $\frac{1\mathrm{.}4}{sec}.$ The goal is to determine the outlet temperature of the cooling water and verify if the cooling water flow rate is sufficient. $($i$.$e$.$ Cooling water can not exceed $60C$ when returned to the cooling towers.$)$

a$)$ Calculate the Nu number and $h_i$ for the cooling media side. $($Half Jacket$)$ Assume heat is only transferred through the straight wall side of the tank $($i$.$e$.$ heat transfer through the top and bottom of the tank can be ignored.$)$

b$)$ Calculate the Nu number and $h_0$ for the agitated side. $($Assume the bulk viscosity is equal to the viscosity at the wall$)$

c$)$ Assuming a fouling factor for treated cooling tower water, provided in the paper, calculate the overall heat transfer coefficient. $($Assume the tank and jacket is constructed of stainless steel $k=16\mathrm{.}3\frac{W}{m^{**}}K)$

d$)$ Solve for the outlet temperature of the cooling water and explain if the cooling capacity for the system is adequate. Why is there a limit to the return temperature on cooling water?