A tube bank consisting of $200$ tubes in ten rows high is used to heat air $.$ In a Rankine power system, $1\mathrm{.}5k\frac{g}{s}$ of steam leaves the turbine as saturated vapor at $0\mathrm{.}51$ bar and $2304k\frac{J}{k}g).$ The steam is condensed to a saturated liquid by passing it over the tubes of a shell$-$and$-$tube heat exchanger, while liquid water, having an inlet temperature of $T_{c,i}=280K,$ is passed through the tubes. The condenser contains $100$ thin$-$walled tubes, each $10mm$ in diameter, and the total water flow rate through the tubes is $15k\frac{g}{s}.$ The average convection coefficient associated with condensation on the outer surface and inside of the tubes may be approximated as $h_0=5000\frac{W}{m^2}K$ and $h_i=9408\frac{W}{m^2}K,$ respectively. The appropriate property values for the liquid water are $c_p=4178\frac{J}{k}g.K,=700{10}^{-6}k\frac{g}{s}.m,k=0\mathrm{.}628\frac{W}{m}.K,$ and $Pr=4\mathrm{.}6.$ Using the $-$ NTU method, determine: $($Note: minimum heat capacity ratio will be on the cooled side$).$

a$)$ Determine the water outlet temperature?

$C_{\text{m}\text{a}\text{x}}^{\text{m}\text{a}\text{n}}=0$

$c_{\text{m}\text{i}\text{n}}=c_c$

b$)$ Calculate the required tube length $($per tube$)?$

c$)$ After extended use, fouling accumulating on the inner and outer tube surfaces provide a cumulative fouling factor of $0\mathrm{.}0003m^2\frac{K}{?}.$ For the prescribed inlet conditions and the computed tube length, what mass fraction of the vapor is condensed?

$hgg=2304k\frac{J}{c}g$

$15k\frac{g}{s}$