Water at 20°C flows by gravity through a smooth pipe from one reservoir to a lower one. The elevation difference is 60 m. The pipe is 360 m long, with a diameter of 12 cm. For the given system a pump is used at night to drive water back to the upper reservoir. If the pump delivers 15,000 W to the water, estimate the flow rate. For water at 20°C, take rho = 998 kg/m3 and μ = 0.001 kg/m-s. Round the answer to the nearest whole number.

Flow Rate = 80 m^3 / hours (Rounded to the nearest whole number)

Explanation:

Given

Hf = head loss f = friction factor L = Length of the pipe = 360 m V = Flow velocity, m/s D = Pipe diameter = 0.12 m g = Gravitational acceleration, m/s^2 Re = Reynolds's Number rho = Density = 998 kg/m^3 μ = Viscosity = 0.001 kg/m-s Z = Elevation Difference = 60 m

Calculations

Moody friction loss in the pipe = Hf = (f*L*V^2) / (2*D*g)

The energy equation for this system will be,

Hp = Z + Hf

The other three equations to solve the above equations are:

Re = (rho*V*D) / μ

Flow Rate, Q = V * (pi/4) * D^2

Power = 15000 W = rho*g*Q*Hp

1/f^0.5 = 2*log ((Re*f^0.5) / 2.51)

We can iterate the 5 equations to find f and solve them to find the values of:

Re = 235000

f = 0.015

V = 1.97 m/s

And use them to find the flow rate,

Q = V * (pi/4) * D^2

Q = (1.97) * (pi/4) * (0.12) ^2 = 0.022 m^3/s = 80 m^3 / hours