Question 15.12: A double acting reciprocating pump has plunger diameter 17.5......
A double acting reciprocating pump has plunger diameter 17.5 cm and a stroke length 35 cm. The suction pipe has a diameter 15 cm, and is fitted to an air vessel.
Determine the crank angle θ at which there is no flow of water to or from the air vessel. Take crank speed to be 150 rpm, and the plunger has simple harmonic.
Given: D = 17.5 cm = 0.175 m, L = 35 cm = 0.35 m, d_{s} = 15 cm = 0.15 m, N= 150 rpm, and r = 0.35/2 = 0.175 m
Area of the plunger A={\frac{\pi}{4}}(0.175)^{2}=0.02405\,\mathrm{m^{2}\quad}
Angular velocity of crank for pump {{\omega}}={\frac{2\pi N}{60}}={\frac{2\pi\times150}{60}}=5\pi\,\operatorname{rad}/{\mathbf{s}}
Discharge from the double acting pump Q=\,{\frac{2A L N}{60}}={\frac{2A\times2r\times60\,\omega}{60\times2\pi}}={\frac{2r A\omega}{\pi}}
or Q={\frac{2\times0.175\times0.02405\times5\pi}{\pi}}=0.0420875\,{\mathrm{m}}^{3}/{\mathrm{s}} (i)
Discharge beyond the air vessel = Arω sin θ
= 0.02405 × 0.175 × 5π × sinθ
or = 0.06611 sinθ (ii)
For no flow to or from the air vessel, discharges given by (i) and (ii) must be equal.
Hence, 0.06611 sin θ = 0.0420875
or sin θ = 0.636628
Therefore, θ = 39°32′