$11\mathrm{.}22$ A shell$-$and$-$tube heat exchanger must be designed to heat $2\mathrm{.}5k\frac{g}{s}$ of water from $15$ to $85C.$ The heating is to be accomplished by passing hot engine oil, which is available at $160C,$ through the shell side of the exchanger. The oil is known to provide an average convection coefficient of $h_o=400\frac{W}{m^2}*K$ on the outside of the tubes. Ten tubes pass the water through the shell. Each tube is thin walled, of diameter $D=25mm,$ and makes eight passes through the shell. If the oil leaves the exchanger at $100C,$ what is its flow rate? How long must the tubes be to accomplish the desired heating?

Hint: use the effectiveness$/$NTU method. ie$..$

STEP $1$: Firstly find effectiveness using:

$=\frac{T_{c,o}-T_{c,i}}{T_{h,i}-T_{c,i}}$

Shell$-$and$-$tube

One shell pass

$(2,4,$dots tube passes$)$

$(NTU{)}_1=-(1+C_r^2{)}^{-\frac{1}{2}}ln(\frac{E-1}{E+1})$

$E=\frac{\frac{2}{{}_1}-(1+C_r)}{(1+C_r^2{)}^{\frac{1}{2}}}$

STEP$2$: Noting that $UA=$NTU$C_{\text{m}\text{i}\text{n}}$

We need to find $U$ based noting that the heat transfer coefficient in the tube is $3060\frac{W}{m^2}.K$ and with oil in outside of the tube is $400\frac{W}{m^2}.K.$

Calculate $U$ using

$U=[\frac{1}{h_c}+\frac{1}{h_l}{]}^{-1}$

STEP $3$: Find the Length required