$4.$

A cylindrical vessel supported on the circular base has a diameter of $10$ m and a

height of $15$ m$,$ is used for storing process water. The discharge pipe from the

tank in the base of the vessel is quite long and hence offers resistance to the

flow of water through it$.$ The vessel is supplied by a pumped water supply. In a

series of tests in which the tank is initially empty, and the inlet feed flow, Qo$,$ is

fixed to a pre$-$determined value, it is found that the water level in the tank initially

increases, but, after some time reaches a steady height, h$,$ above the tank base;

at which point the inlet flow rate, Qo and the exit flow rate, Q$,$ are equal. The

following Table of test results were recorded,

Q$/($m$3/$min$)1$

$2$

$5$

h$/$m

$1\mathrm{.}27$

$2\mathrm{.}55$

$7$

$9$

$6\mathrm{.}37$

$($a$)$

$8\mathrm{.}91$

$11\mathrm{.}46$

From the above data find the relationship between the exit volumetric flow

rate and the volume of water in the tank.

$($b$)$

$[20\%]$

For a particular inlet flow rate the steady state level in the tank is found to

be $9\mathrm{.}0$ m$.$ If the inlet flow is now set to $5\mathrm{.}0$ m$3/$min$,$ describe in words what

you think will happen to the water level in the tank.

$[20\%]$

$($c$)$

$($d$)$

Set up the unsteady state mass balance for the water in the tank, for the

conditions specified in $($b$)$ and obtain an expression which describes how

the water volume in the tank varies with time.

$[40\%]$

What is the final steady state water level in the tank for the conditions

specified in part $($b$),$ and how long will it take the water level in the tank to

reach $95\%$ of this final steady state value?

$[20\%]$

Note:

$1$

$()$

$=$ln$()+$

C is an integration constant, which is not required if the integral is solved with

limits$.$