Calculating Water Flow with Known Velocity at One End

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.

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.

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 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).

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.

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