How To Calculate Water Flow Through A Pipe Based On Pressure

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.

1. 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/m³, height to m and velocity to m/s.

2. 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:

\(P_1+0.5pv_1^2+pgy_1=P_2+0.5pv_2^2+pgy_2\)

where P₁ and P₂ are initial and final pressures, respectively, p is the density of the water, v₁ and v₂ are initial and final velocities, respectively, and y₁ and y₂ are initial and final heights, respectively. Measure each height from the center of the pipe.

To find the initial water flow, solve for v₁. Subtract P₁ and p​g​y₁ from both sides, then divide by 0.5​p. T​ake the square root of both sides to obtain the equation:

\(v_1=\sqrt{\frac{P_2+0.5pv_2^2+pgy_2-P_1-pgy_1}{0.5p}}\)

Perform an analogous calculation to find final water flow.

3. 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.

1. Use Conservation of Mass

If both v₁ and v₂ in Bernoulli's equation are unknown, use conservation of mass to substitute:

\(v_1=\frac{v_2A_2}{A_1}\text{ or }v_2=\frac{v_1A_1}{A_2}\)

where A₁ and A₂ are initial and final cross-sectional areas, respectively (measured in m2).

2. Solve for Velocities

Solve for v₁ (or v₂) in Bernoulli's equation. Then find the initial water flow,

\(v_1=\sqrt{\frac{P_2+pgy_2-P_1-pgy_1}{0.5p-0.5p(\frac{A_1}{A_2})^2}}\)

Perform an analogous calculation to find final water flow.

3. Substitute Measurements for Each Variable

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