$($i$)$ Why is it sometimes not possible use a constant U$-$value in heat exchanger

design where large temperature gradients exist?

$105k\frac{g}{s}$ of high$-$pressure saturated hydrocarbon vapour is to be condensed in a single$-$

pass shell$-$and$-$tube exchanger containing $120$ stainless steel tubes with $17mm$ in

outer diameter $($treat the heat exchanger as being thin walled$).30k\frac{g}{s}$ of cooling water

at $10C$ enters the $Hx$ through the tubes while the vapour condenses on the shell side.

The heat transfer coefficient at any point on the condensing vapour$-$side of the heat

exchanger is estimated using the correlation:

$h=112(T_{\text{v}\text{a}\text{p}\text{o}\text{u}\text{r}}-T_{\text{w}\text{a}\text{l}\text{l}}{)}^{2\mathrm{.}1}$

while that on the tube$-$side it may be calculated using the Dittus Boelter equation:

$=0\mathrm{.}023R^{0\mathrm{.}8}P{r}^{0\mathrm{.}3}$

$($ii$)$ Accounting for the variation of shell side heat transfer coefficient with

temperature, estimate the required heat exchanger tube area required

$($iii$)$ Approximate the error which would have been encountered had the

temperature variation not been correctly accounted for

$($iv$)$Briefly list what recommendations would you provide in relation to this heat

exchanger design in order to improve it$?$

Data

Water Properties:

Heat capacity

Viscosity

Thermal conductivity

Hydrocarbon Properties

Heat of vaporisation

$-,1\mathrm{.}4{10}^2\frac{J}{k}g$

Boiling temperature

Numerical Integrations

Numerical integrations should be carried out using the trapezoidal rule:

${\displaystyle \underset{a}{\overset{b}{}}}f(x)dx$~~$(b-a)\frac{f(b)+f(a)}{2}$

In the interest of time, you should approximate an integral using only

two points.

For example, if you are to integrate a function between $x$ coordinates $50$ and $90,$ then

let $a=50,b=90.$ Do not worry about including any intermediate points in this exam.