How to Apply Bernoulli's Principle

Finds the pressure at another point along a pipe. Pressure, speed and height together stay constant along the flow, so the pressure drops where the liquid runs faster or climbs higher.

In a liquid flowing through a pipe, pressure, speed and height taken together stay constant along the flow. Writing 1 and 2 for two sections along it gives the following.

p_1 + \dfrac{1}{2}\rho v_1^2 + \rho gh_1 = p_2 + \dfrac{1}{2}\rho v_2^2 + \rho gh_2

$p$ is the pressure, $\rho$ the density, $v$ the velocity and $h$ the height. The second term is the dynamic pressure and the third the pressure due to height. This is conservation of energy written for a fluid, the three terms standing for pressure, motion and position.

Fast means low pressure

Since the sum is fixed, a rise in speed must be paid for by a fall in pressure. A narrowing pipe has to move the same volume through a smaller opening, so the liquid speeds up and the pressure there drops. It is why a wing lifts and why a spray bottle draws liquid up its tube.

Example

At section 1 the water is at 200 kPa, moving at 2 m/s, at a height of 0 m. At section 2 it moves at 8 m/s at a height of 3 m. Speeding up costs $\frac{1}{2} \times 1000 \times (2^2 - 8^2) = -30$ kPa and climbing costs $1000 \times 9.8 \times (0 - 3) = -29.4$ kPa, leaving $200 - 30 - 29.4 = 140.6$ kPa at section 2.

Notes

The formula assumes an inviscid fluid with no losses to friction. Real pipework loses further pressure against the pipe wall, and a long run needs that allowance made separately.

It compares two points along the same flow. Two points on separate streams cannot be set against each other this way.