A liquid food with a specific heat of $4300\frac{J}{k}gC$ flows through the inner tube of a

double$-$pipe heat exchanger. The flow rate of liquid food is $0\mathrm{.}62k\frac{g}{s}$ and it enters

the heat exchanger at $12C$ and exits at $54C.$ Hot water at $110C$ enters the

annular gap between the two pipes of the heat exchanger and flows

countercurrently with a flow rate of $4320k\frac{g}{h}.$ Since the average specific heat of

water is $5\mathrm{.}4k\frac{J}{k}gC,$ accepting the steady state conditions:

a$)$ Calculate the outlet temperature of the water.

b$)$ Calculate the logarithmic temperature difference.

c$)$ If the total heat transfer coefficient according to the internal surface area is

$2400\frac{W}{\left[\begin{array}{c}{m}^{2}C\end{array}\right]}$ and the internal diameter of the inner pipe is $7cm,$ calculate

the length of the heat exchanger.