A cylindrical vessel with circular base has a diameter of $10$ 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, Q$0,$ 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, Q$0$ and the exit flow rate, Q$,$ are equal. The following

test results were recorded.

Q $[$L$/$min$](1,2,5,7,9)$

h $[$m$](1\mathrm{.}13,2\mathrm{.}58,6\mathrm{.}72,9\mathrm{.}01,12\mathrm{.}36)$

$1.$ Plot flow rate as a function of volume of water in the tank.

$2.$ Using Excel or Matlab, find the relationship between the exit volumetric flow rate and the

volume of water in the tank.

$3.$ 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 $5\mathrm{.}0$ L$/$min$,$ describe in words what you think will happen to the water level in

the tank.

$4.$ Set up the unsteady state mass balance for the water in the tank, for the conditions specified

in $3$ and obtain an expression which describes how the water volume in the tank varies with

time.

$5.$ Plot the water volume in the tank and the flow rate as a function of time for times between $0$

and $300$ minutes. Describe the plots and discuss briefly. Compare with your assumption

stated in $3.$