Question $3$

A Newtonian liquid in uniform film flow on the internal surface of a long vertical tube

is contacted with a gas in upward counterflow.

$($i$)$ Demonstrate by clearly outlining any assumptions made in your analysis.

that the volumetric flowrate of the liquid can be described by:

$\frac{Q}{d_i}=(\frac{p}{L}-g)\frac{{}^3}{3}+\frac{{}_{\text{s}\text{u}\text{r}f}{}^2}{2}$

where $\frac{p}{L}$ is the liquid film pressure gradient, ${}_{\text{s}\text{u}\text{r}\text{f}}$ the magnitude of the interfacial

frictional shear stress between the liquid and the gas which is assumed to be acting

vertically downwards $($i$.$e$.$: parallel with the direction of flow$),$ the film thickness, and

$d_i$ is the inner pipe diameter.

A liquid is admitted at a fixed rate of $12k\frac{g}{h}r$ to the top of a vertical tube, $25mm$ in

internal diameter and $3m$ in length. The liquid flows in a uniform thin film down the

internal surface of the tube. In normal operation the liquid film is in contact with a gas

in upward countercurrent flow.

$($ii$)$ Calculate the residence time of the liquid in the tube in the absence of any

gas flow.

$($iii$)$ Estimate the maximum allowable gas throughput if the residence time of

the liquid is not to exceed $40$ seconds.

You may assume that a constant friction factor of $0\mathrm{.}006$ applies to the gas$-$liquid

interface.

DATA:

Liquid properties:

Density$=990$ kg$/$m$3$

Viscosity $=0\mathrm{.}005$Ns$/$m$2$

Gas properties:

Density $=1\mathrm{.}2$ kg$/$m$3$