Question

Water is flowing into a factory in a horizontal pipe with a radius of 0.0183 m at ground level. This pipe is then connected to another horizontal pipe with a radius of 0.0300 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.

Answer 1

Use continuity and Bernoulli

Continuity

V1*A1 = v2*A2

V1*pi*r1^2 = v2*pi*r2^2

V1*r1^2 = v2*r2^2

Bernoulli

P1 + 1/2*rho*v1^2 + rho*g*h1 = P2 + 1/2*rho*v2^2 + rho*g*h2

From the problem statement, P1 = P2, so they cancel.

Assume h1 = 0, so h2 = 11.6 m

1/2*rho*v1^2 = 1/2*rho*v2^2 + rho*g*h2

All the densities cancel

V1^2 = v2^2 + 2*g*h2

Use continuity and solve for v1 or v2 and sub in.

(R2/r1)^4*v2^2 = v2^2 + 2*g*h2

V2^2*((r2/r1)^4 - 1) = 2*g*h2

V2 = sqrt ((2*g*h2)/((r2/r1)^4 - 1))

Plug in number

V2 = 6.04 m/s

Then use that for the volume flow

V_dot = v2*A2 = v2*pi*r2^2

Plug in numbers

V_dot = 0.0171 m^3/s