A siphon of uniform bore

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow

Fluids and Continuum Physics, Ch Euler's equation and Bernoulli's theorem

Problem

A siphon of uniform bore, already primed and running, carries water out of a wide open tank. The free surface of the tank is the datum z=0z = 0 and the atmospheric pressure above it is 1.013×105 Pa1.013\times10^{5}\ \mathrm{Pa}. The crest of the tube is 1.00 m1.00\ \mathrm{m} above the free surface and the open outlet is 1.50 m1.50\ \mathrm{m} below it. Assume steady, inviscid, incompressible flow of water of uniform density ρ=1000 kg m−3\rho = 1000\ \mathrm{kg\,m^{-3}} along the streamline that runs from the free surface, over the crest, to the outlet; the tank is so wide that the speed at its free surface is negligible, and because the bore is uniform the speed is the same at every point inside the tube. Take g=9.8 m s−2g = 9.8\ \mathrm{m\,s^{-2}}. ((a)) Find the speed of the water at the outlet. ((b)) Find the absolute pressure at the crest. ((c)) The water is at 20 ∘C20\ ^{\circ}\mathrm{C}, where its vapour pressure is 2.3 kPa2.3\ \mathrm{kPa}; the siphon fails if the pressure at the crest falls to that value. Predict the greatest height of the crest above the free surface for which the siphon can still run, the outlet staying 1.50 m1.50\ \mathrm{m} below the free surface.

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