Lessons · Engineering · Bernoulli between two points
Bernoulli: pressure, speed and height trade with each other
Between two points along a flow with no losses, p + ½ ρ v² + ρ g z is the same at both. Faster means lower pressure; higher means lower pressure; and a tank drains at v = sqrt(2 g h).
Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.
What it is for
A pipe narrows at a valve and the pressure gauge after it reads lower than the one before, and the operator thinks the valve is leaking. Nothing is leaking; the water sped up and the pressure paid for it. Bernoulli is why.
How to think about it
Write all six terms first, three on each side. Then cross out what is the same on both sides or is zero: a free surface open to the air is at atmospheric pressure and nearly zero speed. Solve for the one term left.
Worked example
p_1 + ½ρv_1² + ρgz_1 = p_2 + ½ρv_2² + ρgz_2All six terms. Write them every time.
A tank open to the air with a hole 3 m below the surface: p_1 = p_2 = atmospheric, v_1 ≈ 0The surface is still and both points see the air, so pressure cancels.
ρ g × 3 = ½ ρ v_2² → v_2 = sqrt(2 × 9.81 × 3) = 7.67 m/sHeight on one side, speed on the other. Density cancels too.
Horizontal pipe, same height: p_1 + ½ρv_1² = p_2 + ½ρv_2². Faster means lower pressureThe other common case. The height terms cancel and pressure trades against speed.
Your turn
A hole 6 m below the surface of an open tank. Write the outlet speed.
v = sqrt(2 × 9.81 × ) = 10.85 m/s
Solve one, graded on the server
The trap
Crossing out a term that is not the same on both sides. Write all six first, then cross out with a reason for each. A term that is zero is not the same as a term that was forgotten.