$(35$ pts $)$ Consider a double pipe heat exchanger in which water is heated by oil as shown in the figure. The inner pipe is of thin wall construction with diameter $D=32mm.$ An engineer wants to know the mass flow rate of oil $m_h^{}$ that leads to the effectiveness $=0\mathrm{.}72.$ It is known that

$($a$)$ Determine the mass flow rate of oil, $m_h^{}$ that leads to $=0\mathrm{.}72.($If you don't have an answer to $($a$),$ assume that $m_h^{}=1k\frac{g}{s}$ for the next sub$-$problems.$)$

$($b$)$ Determine the hot oil outlet temperature, $T_{h,\text{o}\text{u}\text{t}}.$

$($c$)$ Determine the LMTD $T_{lm}$ and the heat exchanger surface area $A_s.$

$($d$)$ Determine the overall heat transfer coefficient, $U.$ with the knowledge of $T_{lm},A_s,$ and $Q^{}.$

$($e$)$ Determine the heat transfer coefficient $h_i$ for the cold water from the $$ number of the forced convection.

$($f$)$ Determine that fouling factor $R_f$ if $h_o=2000\frac{W}{m^2}-K$ with the knowledge of $U$ and $h_i.$