Question

Water flows at a rate of 30 gpm through a vertical pipe that changes size and contains a partially-closed valve that causes headloss. Point X is in the 2-inch diameter part of the pipe at elevation 127.15 ft and has a pressure of 65 psi and point Y is in the I inch diameter part of the pipe at elevation 135.23 ft and has a pressure of 62.7 psi. Which direction is water flowing-from X to Y or from Y to X? Why? Is this upward or downward?

Answer 1

given,

volume flow rate, r = 30 gallons per minute = 0.5 gallons per s = 0.00189271 m^3/s = 115.5 in^3/s

Point X

height hx = 127.15 ft

radius, rx = 1 inch

Pressure, Px = 65 psi

Point Y

height, hy = 135.23 ft

radius, ry = 0.5 in

Pressure, Py = 62.7 psi

from bernoulli's theorem

Px + 0.5*rho*vx + rho*g*hx = Py +0.5*rho*vy + rho*g*hy

rho = 0.000036127 lb/in^3

g = 386.4 in/s^2

2.3 + 0.5*0.000036127(vx^2 - vy^2) + 0.000036127*386.4 (127.15 - 135.23) = 0

vx^2 - vy^2 = -121084.3668

also, from continuity equation

A1Vx = A2Vy

rx^2*vx = ry^2*vy

4vx = vy

hence vx = 8072.291 in/s

vy = 32289.16448 in/s

but from the volume flow rate

pi*rx^2*vx = 115.5

vx = 36.764 in / s

hence vy = 147.05916 in /s

form unsing these values of vx and vy

Px + 0.5*rho*vx + rho*g*hx = Py +0.5*rho*vy + rho*g*hy

(Px - Py) = 0.479 psi

as Px - Py is more than 0.479 psi

that means that the fluid is flowing from X to Y as pressure of Y is lesser than it should be

hence

the flow flows upwards