Problem $6(25$ points$)$

In a dairy plant, milk at $4C$ is pasteurized continuously at $72C$ at a rate of $1,200$ liters per minute for $24$ hours per day and $365$ days a year. The milk is heated to the pasteurizing temperature by hot process water generated by a natural gas fired boiler. The pasteurized milk is then cooled from $72C$ back to $4C$ by chilled water.

To save energy, operating costs, and CO$2$ emissions, the plant installs a regenerator $($i$.$e$.,$ a heat exchanger that cools the pasteurized milk while simultaneously pre$-$heating the incoming cold milk$).$ The regenerator has an effectiveness of $82$ percent $($i$.$e$.,Q_{\text{a}\text{c}\text{t}}^{}=Q_{\text{m}\text{a}\text{x}}^{}$~$=m^{}{c}_p{T}_{\text{m}\text{a}\text{x}}).$ If the cost of natural gas is $),$ determine how much energy $(k\frac{J}{y}r)$ and money $($$$/$year$)$ the regenerator will save the company per year by reducing the natural gas usage in the boiler.

For milk $$~~$1030k\frac{g}{m^3},c_p$~~$3\mathrm{.}79k\frac{J}{k}g-C\{TA-3\}$

a$)$ The maximum rate of heat transfer in the regenerator is $(kW)$ :

b$)$ The actual rate of heat transfer in the regenerator is $(kW)$ :

c$)$ The outlet temperature of the milk that is going to the pasteurizer is $(T_{2}C)$ :

d$)$ The outlet temperature of the cold milk that is going to the bottle line is $(T_{4}C)$:

e$)$ The annual energy savings is $(k\frac{J}{y}r)$:

f$)$ The annual operating cost savings is: $(\frac{\text{\$}}{y}r)$ :