Question

Assume frictionless flow in a long horizontal conical pipe. The diameter is 12 inches at one end and 6 inches at the other, The pressure head at the smaller end is 16 feet of water, if water flows through this cone from the larger to the smaller end at a rate of 150 gallons per minute (gpm) calculate the fluid velocities at the two ends and the pressure head at the larger end. Draw a picture of the problem and show all work

Answer 1

Finding the velocities at both the ends

Water going through the bigger end = water coming through the smaller end.
So 150 gallons per minute is same for both the ends.
Mass rate = 150 gpm = 6.31 x 10^-5 m³/s.
A₁ = 0.073 m².
A₁ v₁ = mass rate = 6.31 x 10^-5 m³/s.
v₁ = (6.31 x 10^-5 m³/s) / 0.073 m².
= 8.64 x 10^-4 m/s.

Using the equation of continuity,
A₁ v₁ = A₂ v₂
A₂ = 0.018 m².
v₂ = (6.31 x 10^-5 m³/s) / 0.018 m².
= 35.1 x 10^-4 m/s.

Finding the pressure head at the larger head.

Using Bernoulli's theorem,
P₁ + 0.5 v₁² = P₂ + 0.5 v₂².
16 feet = 4.8768 m
P₂ = gh = 1000 x 10 x 4.8768 m = 48768 N/m².
P₁ = 48768 + 0.00557 = 48768.00557 N/m².