GROUP $-$ C

$11.($a$)$ The velocity distribution in a circular pipe is given by

$\frac{u}{u_{\text{m}\text{a}\text{x}}}=(1-\frac{r}{R}{)}^n$

where u is the velocity at a distance r from the centre, ${?}^{\text{m}\text{a}\text{x}}$ is the maximum velocity at the centre of the pipe and R is the pipe radius. Find the ratio of average velocity of flow in the pipe to the maximum velocity.

$[($CO$3)($Evaluate$/$HOCQ$)]$

$($b$)$ Define momentum correction factor. Also obtain its expression for a Newtonian fluid flowing through a smooth, round tube.

$[($CO$3)($Apply$/$IOCQ$)$

$6+6=12$

$($a$)$

Water at $25C$ is pumped at a constant rate of $11\frac{m^3}{h}$ from a large reservoir resting on the floor to the open top of an experimental absorption tower. The point of discharge is $5\mathrm{.}6$ m above the floor, friction losses in the $52$ mm pipe from the reservoir to the tower is $2\mathrm{.}6\frac{J}{K}g.$ At what height in the reservoir must the water level be kept if the pump can deliver only $0\mathrm{.}12$ kW $.$

Given data: Density of water $=998K\frac{g}{m^3},$ viscosity of water $=0\mathrm{.}9cP.$

$[($CO$3)($Evaluate$/$HOCQ$)]$

$($b$)$ Write down the x $-$component of Navier$-$Stokes equation. State its significance.

$[($CO$3)($Analyze$/$IOCQ$)]$

$($C$)$ Mention the difference between hydraulically smooth pipe and rough pipe.

$[($CO$3)($Remember$/$LOCQ$)]$

$8+2+2=12$

$($a$)$ A pump $($efficiency $55\%)$ draws a solution $($sp$.$ Gravity $1\mathrm{.}84)$ from a storage tank through a pipe of $75$ mm inside diameter. The velocity in the suction line is $0\mathrm{.}9\frac{m}{s}.$ The pump discharges through a pipe having $50$ mm inside diameter. The end of discharge pipe is $14\mathrm{.}5$ m above the level of solution in the storage tank. Friction losses in the entire system are $29\frac{J}{K}g.$

i$)$ What is the power of the pump?

ii$)$ What pressure must the pump develop?

$[($CO$3)($Evaluate$/$HOCQ$)]$

$($b$)$ Derive an expression of the velocity profile in case of Coutte flow without pressure gradient.

$[($CO$3)($Analyze$/$IOCQ$)]$

$9+3=12$

$($a$)$ Define 'kinetic energy correction factor'. Obtain its value for laminar flow of a Newtonian fluid flowing through a smooth, circular pipe.

$[($CO$3)($Analyse$/$IOCQ$)]$

$($b$)$ Water $($density $998K\frac{g}{m^3},$ viscosity $($:$-0\mathrm{.}9c.p.$