A gate valve is partly closed in a horizontal pipe of cross-sectional area carrying…
PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow · Kinematics, conservation laws and ideal flow
Fluid Dynamics, Ch Euler's equation in Lamb's form and the four Bernoulli theorems
Problem
A gate valve is partly closed in a horizontal pipe of cross-sectional area A1=100cm2 carrying water (density 1000kgm−3) at U1=2.0ms−1, measured well upstream (section 1) where the flow is smooth and steady. The gap left by the plate forces the flow into a vena contracta of area Av=50cm2 just downstream of the plate, where the jet has separated from the plate edges and has not yet felt the pipe walls; behind this jet a churning, recirculating wake fills the rest of the pipe cross-section. Farther downstream (section 2) the jet has mixed with this wake and the flow again fills the pipe uniformly, at the same area A1. Take g=9.8ms−2 and treat the fluid as incompressible. Someone argues: "Bernoulli's theorem holds along a streamline from section 1 to section 2: the flow is steady, gravity is conservative, and an incompressible fluid is certainly barotropic. Since the pipe has the same area A1 at both sections, mass conservation gives U2=U1, so p1+21ρU12=p2+21ρU22 gives p2=p1: a partly closed valve produces no permanent pressure drop, only a temporary dip and recovery at the throat."
((a)) Say precisely which condition of the steady-streamline Bernoulli theorem fails, and where along the path from section 1 to section 2 it fails.
((b)) Using Bernoulli's theorem where it is valid and a momentum balance where it is not, find the actual pressure drop p1−p2 and express it as a head loss in metres of water.