Question

Water is flowing into a factory in a horizontal pipe with a radius of 0.0163 m at ground level. This pipe is then connected to another horizontal pipe with a radius of 0.0380 m on a floor of the factory that is 11.6 m higher. The connection is made with a vertical section of pipe and an expansion joint. Determine the volume flow rate that will keep the pressure in the two horizontal pipes the same.
_________ m³/s

Answer 1

Density of water, p = 1000 kg/m^3
h = 11.6 m
r1 = 0.0163 m
r2 = 0.0380 m

Using continuity eq,
v1 * a1 = v2 * a2
v1 * π * r1^2 = v2 * π * r2^2
v1 = v2 * (r2/r1)^2

Using Benoulli's theorem
p1 + 1/2 * p*v1^2 = p2 + p*g*h + 1/2*p*v2^2
As pressure is same, p1 = p2, we can cancel them out !!
1/2 * 1000 * ( v2 * (r2/r1)^2)^2 = 1000 * 9.8 * 11.6 + 1/2 * 1000 *  v2^2
1/2 * 1000 * ( v2 * (0.0380/0.0163)^2)^2 = 1000 * 9.8 * 11.6 + 1/2 * 1000 *  v2^2
v2 = 2.82 m/s

Volume flow rate, = 2.82 * π * 0.0380^2 m^3/s
Volume flow rate, = 0.0128 m^3/s