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. Water in a pipe is a conservation of energy problem too. In fact, that's exactly how mathematician Daniel Bernoulli approached the problem in the 1700s. Using Bernoulli's equation, calculate the flow of water through a pipe based on pressure.

## Calculating Water Flow with Known Velocity at One End

## Convert Measurements to SI Units

Convert all measurements to SI units (the agreed-upon international system of measurement). Find conversion tables online and convert pressure to Pa, density to kg/m^3, height to m and velocity to m/s.

## Solve Bernoulli's Equation

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. Measure each height 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. T_ake the square root of both sides to obtain the equation v_1 = { [P_2 + 0.5p(v_2)^2 + pgy_2 - P_1 - pgy_1] ÷ (0.5p) }^0.5.

Perform an analogous calculation to find final water flow.

## Substitute Measurements for Each Variable

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

## Use Conservation of Mass

If both v_1 and v_2 in Bernoulli's equation are unknown, use conservation of mass to substitute v_1 = v_2A_2 ÷ A_1 or v_2 = v_1A_1 ÷ A_2 where A_1 and A_2 are initial and final cross-sectional areas, respectively (measured in m^2).

## Solve for Velocities

Solve for v_1 (or v_2) in Bernoulli's equation. To find initial water flow, subtract P_1, 0.5_p_(v_1A_1 ÷ A_2)^2, and pgy_1 from both sides. Divide by [0.5p - 0.5p(A_1 ÷ A_2)^2]. Now take the square root of both sides to obtain the equation v_1 = { [P_2 + pgy_2 - P_1 - pgy_1] / [0.5p - 0.5p x (A_1 ÷ A_2)^2] }^0.5

Perform an analogous calculation to find final water flow.

## Substitute Measurements for Each Variable

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