Question $3$

A Newtonian liquid in uniform laminar film flow on the internal surface of a

long vertical tube is contacted with a gas in upward counterflow.

$($i$)$ Show that the volumetric flowrate in the film can be described by:

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

where $\frac{p_{\text{s}\text{u}\text{r}\text{f}}}{L}$ is the pressure gradient across the surface of the liquid film,

${}_{\text{s}\text{u}\text{r}\text{f}}$ the magnitude of the interfacial shear stress and $$ the film thickness.

In a process contactor, liquid is fed to the top of a vertical tube $25mm$ in

intemal diameter and $3m$ in length. The liquid flows dowrward in a uniform

thin film on the internal surface of the tube. In normal operation the liquid film

is in contact with a gas in upward countercurrent flow. The process under

consideration requires that the average residence time of the liquid in the

contactor be equal to $30$ seconds.

Calculate the liquid flowrate which satisfies this condition for

$($ii$)$ Liquid film flow in the absence of any gas flow.

$($iii$)$ Film flow with a countercurrent gas flow of $0\mathrm{.}01\frac{m^3}{s}.$

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

liquid interface.

DATA:

Liquid Properties:

Density $({}_L)$

$1000k\frac{g}{m^3}$

Viscosity $({}_L)$

$0\mathrm{.}005N\frac{s}{m^2}$

Gas Properties:

Density $({}_g)$

$1\mathrm{.}2k\frac{g}{m^3}$