Problem 1352 A golf course sprinkler system discharges water

Problem 13.52

A golf course sprinkler system discharges water from a horizontal pipe at the rate of 7150 cm3/s . At one point in the pipe, where the radius is 3.85 cm , the water\'s absolute pressure is 2.45×105 Pa . At a second point in the pipe, the water passes through a constriction where the radius is 1.80 cm .

Part A

What is the water\'s absolute pressure as it flows through this constriction?

Solution

using an bernoulli\'s equation, we have

p₁ /  + g h₁ + (1/2) v₁² = p₂ /  + g h₂ + (1/2) v₂²   

where, h₁ = h₂ = h

then, we have

p₁ /  + (1/2) v₁² = p₂ /  + (1/2) v₂²

we know that,   volumetric flow rate is defined by -

Q = A v

or v = Q / r²

Now, p₁ /  + (1/2) (Q² / ² r₁⁴) = p₂ /  + (1/2) (Q² / ² r₂⁴)

p₂ = p₁ +  Q² / 2 ² [1 / r₁⁴ - 1 / r₂⁴]

where, p₁ = water\'s absolute pressure at one point = 2.45 x 10⁵ Pa

= density of water = 1000 kg/m³

Q = volumetric flow rate = 7150 cm³/s = 7.15 x 10^-3 m³/s

r₁ = radius of the pipe at one point = 0.0385 m

r₂ = radius of the pipe at second point = 0.0180 m

then, we have

p₂ = (2.45 x 10⁵ Pa) + (1000 kg/m³) (7.15 x 10^-3 m³/s) / 2 (3.14)² [1 / (0.0385 m)⁴ - 1 / (0.0180 m)⁴]

p₂ = 233241.82 =   2.33 x 10^5 Pa