$$

$[$Total $20$ points$]$ Consider the following shell$-$and$-$tube exchanger. A hot fluid $(c_h=4180\frac{J}{k}g*K)$ enters the circular tube with a mass flow rate of $0\mathrm{.}10k\frac{g}{s}$ and a cold fluid $(c_c=2500\frac{J}{k}g*K)$ enters the shell with a mass flow rate of $0\mathrm{.}25k\frac{g}{s}.$ The tube wall is very thin. $($a$,10$ points$)$ If the inlet temperatures of both fluids are $95$ and $20C,$ respectively and the outlet temperature of the hot fluid is $70C,$ determine the total heat transfer rate and the outlet temperature of the cold fluid. $($b$,10$ points$)$ How long must the heat exchanger be to achieve the desired outlet temperature? $($additional information: convection coefficients for both fluids are $700($hot$)$ and $1300($cold$)\frac{W}{m^2}*K,$ respectively$)$