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Bernoulli Equation, Gage Pressure Water Pipe

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Prequisite Knowledge
Bernoulli's Equation
Absolute, Gage, Vacuum, and Atmospheric Pressures

Problem
A pipe at ground level ruptures, and the water shoots up 29 m. Estimate the gage pressure of the water in the pipe.

Solution

All necessary assumptions are made in order for the use of the Bernoulli equation.
Assume that the water pressure in the pipe at the burst section is equal to the water main pressure.
Assume that the friction between the water and air is negligible.
Assume that the irreversibilities that may occur at the burst section of the pipe due to abrupt expansion are negligible.

Now, let's look at the Bernoulli equation:

p1 ρg

+

v12 2g

+ z1 =

p2 ρg

+

v22 2g

+ z2

Assume that the velocity of water inside the hose is 0, v₁ = 0.
z₁ is the height of the pipe at ground level, thus z₁ = 0.
At the top of the water trajectory, v₂ = 0.
The pressure at the top of the water trajectory is atmospheric pressure, p₂ = p_atm

The equation can then simplify to:

p1 ρg

=

patm ρg

+ z2

Further simplifying:

p1 − patm ρg

= z2

Then:

pgage ρg

= z2

And finally:

p_gage = pg z₂

Substituting:

p_gage = (1000 kg/m³)*(9.81 m/s²)*(29 m)*[(1 kPa)/(1 kN/m²)]*[(1 kN)/(1000 kgm/s²)]

p_gage = 284.5 kPa