How to Calculate Water Flow Through a Pipe Based on Pressure

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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

  1. Convert Measurements to SI Units

  2. 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.

  3. Solve Bernoulli's Equation

  4. 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.

  5. Substitute Measurements for Each Variable

  6. 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

  1. Use Conservation of Mass

  2. 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).

  3. Solve for Velocities

  4. 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.

  5. Substitute Measurements for Each Variable

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

References

About the Author

Allison Boley writes both fiction and nonfiction, having placed as a semifinalist in the international Scriptapalooza Semi-Annual Television Writing Competition. Boley graduated summa cum laude from the Barrett Honors College at Arizona State University, where she is concurrently pursuing her doctorate in physics.

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