Water of uniform density flows steadily at through a reducing elbow that lies wholly…

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

Fluids and Continuum Physics, Ch Conservation of mass, momentum and energy

Problem

Water of uniform density

\rho=1000\ \mathrm{kg\,m^{-3}}

flows steadily at

Q=60\ \mathrm{litres\,s^{-1}}

through a reducing elbow that lies wholly in a horizontal plane. The inlet bore is

150\ \mathrm{mm}

and the outlet bore is

100\ \mathrm{mm}

. The water enters along

+x

and leaves in a direction making

45^{\circ}

with

+x

, the turn being towards

+y

. Gauge pressures measured at the inlet and outlet cuts are

250\ \mathrm{kPa}

and

220\ \mathrm{kPa}

respectively. Take the control volume to be the water in the elbow between the two cuts, with the outward normal on each face. The flow is uniform across each cut; gauge pressure is used throughout, so the atmosphere all round the elbow contributes nothing; the elbow is horizontal, so weight has no component in the plane of the balance. Take

g=9.8\ \mathrm{m\,s^{-2}}

where it is needed. ((a)) The mean speeds at the inlet and the outlet. ((b)) The

x

and

y

components of the force the water exerts on the elbow, and the magnitude of that force. ((c)) The rate at which mechanical energy is lost in the elbow, and the equivalent head loss in metres of water.

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