In physics, you've probably solved conservation of energy problems that deal with a car on a hill, a mass on a spring, and a roller coaster in a loop. You may not think of the pressure of water in a pipe as a conservation of energy problem, but you should be thankful that's exactly how mathematician Daniel Bernoulli approached the problem in the 1700s. Using Bernoulli's equation, you can also calculate the flow of water through a pipe based on pressure.

## Calculating Water Flow With Known Velocity at One End

Convert all your measurements to SI units (the agreed-upon international system of measurement). Convert pressure to Pa, density to kg/m^3, height to m, and velocity to m/s. You can find conversion tables online (see Resources section).

Solve Bernoulli's equation for the desired velocity, either the initial velocity into the pipe or the final velocity out of the pipe. Bernoulli's equation is P_1 + 0.5*p*(v_1)^2 + p*g*(y_1) = P_2 + 0.5*p*(v_2)^2 + p*g*y_2 where P_1 and P_2 are initial and final pressures, respectively, p is the density of the water, v_1 and v_2 are initial and final velocities, respectively, and y_1 and y_2 are initial and final heights, respectively. Each height is measured from the center of the pipe. To find the initial water flow, solve for v_1. Subtract P_1 and p*g*y_1 from both sides, then divide by 0.5*p. Now take the square root of both sides to obtain the equation v_1 = { [P_2 + 0.5*p*(v_2)^2 + p*g*y_2 - P_1 - p*g*y_1] / (0.5*p) }^0.5 Perform an analogous calculation to find final water flow.

Substitute your measurements for each variable (the density of water is 1,000 kg/m^3), and calculate the initial or final water flow in units of m/s.

## Calculating Water Flow With Unknown Velocity at Both Ends

If both v_1 and v_2 in Bernoulli's equation are unknown, use conservation of mass to substitute v_1 = v_2*A_2 / A_1 or v_2 = v_1*A_1 / A_2 where A_1 and A_2 are initial and final cross-sectional areas, respectively (measured in m^2).

Solve for v_1 (or v_2) in Bernoulli's equation. To find initial water flow, subtract P_1, 0.5*p*(v_1*A_1 / A_2)^2, and p*g*y_1 from both sides. Divide by [0.5*p - 0.5*p*(A_1 / A_2)^2]. Now take the square root of both sides to obtain the equation v_1 = { [P_2 + p*g*y_2 - P_1 - p*g*y_1] / [0.5*p - 0.5*p*(A_1 / A_2)^2] }^0.5 Perform an analogous calculation to find final water flow.

Substitute your measurements for each variable, and calculate the initial or final water flow in units of m/s.

#### References

- "Physics"; Richard Wolfson and Jay M. Pasachoff; 1999
- NIST: Essentials of the SI: Introduction

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- Nam-Ngum-Stausee in Laos image by Digitalpress from Fotolia.com