Question #$4(25$ marks$)$

Consider a large, well$-$insulated cylindrical tank $($which has a

radius of $20$ cm and a height of $700$ cm $)$ that is filled with air and

water $($the water depth is $10$ cm $)$ as sketched. Initially, the tank$\text{'}$s

interior absolute pressure is one atmosphere and its absolute

temperature is $300$ K $.$ The tank is fitted with a fast moving piston

that forces water upwards at a rate of $10\frac{m}{s}$ after ${10}^{-4}$ seconds

towards a converging nozzle. The nozzle exit plane is $700$ cm

above the liquid$-$vapour interface and it has a small cross$-$sectional

exit area of $100c{m}^2.$

For parts $($a$)$ through $($e$)$ you may assume dry air above the liquid$-$

vapour interface. Please state your assumptions explicitly.

$($a$)$ Calculate the distance travelled by the piston in the first ${10}^{4}sec.$ after it starts to move assuming a linear

acceleration from zero.

$($b$)$ How long does it take for the piston caused pressure pulse to reach the top of the tank?

$($c$)$ Assuming the piston is able to achieve and to maintain a constant speed of $10\frac{m}{s},$ then determine the

speed of the air flow exiting the converging nozzle at the precise moment when the piston is moving upward

at precisely $10\frac{m}{s}?$

$($d$)$ What is the maximum air mass flow rate possible through this nozzle?

$($e$)$ At the maximum mass flow rate calculated in part $($d$)$ how quickly will the piston be moving?

$($f$)$ How would your results change in part $($e$)$ if you considered the vapour phase to be an equilibrium

mixture of both air and water vapour? Hint: You may wish to consult a thermodynamics textbook to

determine the partial pressure of water vapour.