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Math problem set Grade 12 Engineering & Design

Vantage Mesa Raw-Water Pump Main: Flow, Velocity, and Sizing Problems

A grade 12 engineering problem set on a fictional raw-water pump main: students convert MGD to gpm and cfs, then use continuity to size the discharge pipe.

Objectives

  1. Students will convert a pump flow rate among MGD, gpm, and cfs using standard operator conversions.
  2. Students will apply the continuity equation to find pipe velocity and compare it against a design range.

Vocabulary

Flow rate
Flow rate measures how much water moves past a point in a certain amount of time, like gallons per minute through a garden hose. Bigger pipes and stronger pushes give bigger flow rates.

Also: continuity, velocity, cross-sectional area

Teacher overview

The fictional Vantage Mesa Water System is installing a new raw-water pump that draws from a river intake and pushes water to the treatment plant. This set has students size the pump's discharge main by first converting the flow among the units operators use, then applying the continuity equation to check the water velocity against a design range. A common misconception in this math is using the pipe diameter directly in the continuity equation instead of first finding the cross-sectional area, so watch for that error in problems 3 and 4.

Vantage Mesa Raw-Water Pump Main: Flow, Velocity, and Sizing Problems — Grade 12 Engineering & Design
Name: ____________________________ Date: ______________

Problems

Use these facts about the fictional Vantage Mesa system for every problem:

  • The new pump moves 2.0 million gallons per day (MGD) from the intake to the plant.
  • One option for the discharge main is a 12-inch pipe with a 1.0-foot inside diameter.
  • A second option is a 10-inch pipe with a 0.833-foot inside diameter.
  • The system's design velocity range for the main is 2 to 5 feet per second.
  • Conversions: 1 MGD = 694 gpm; 1 MGD = 1.547 cfs; a circle's area = 0.785 x Diameter squared.
  • Continuity: Flow (cfs) = Area (square feet) x Velocity (feet per second).
  1. Convert the pump's flow of 2.0 MGD to gallons per minute (gpm).
  2. Convert the pump's flow of 2.0 MGD to cubic feet per second (cfs). Round to the nearest hundredth.
  3. Find the inside cross-sectional area of the 12-inch main, whose inside diameter is 1.0 foot. Use Area = 0.785 x Diameter squared.
  4. Using continuity, find the water velocity in the 12-inch main in feet per second. Use the flow from problem 2 and the area from problem 3. Round to the nearest hundredth.
  5. Does the velocity in the 12-inch main fall within the 2 to 5 feet per second design range?
  6. The team also priced the 10-inch main, whose inside diameter is 0.833 foot. Find its cross-sectional area, then its velocity at the same flow of 3.094 cfs. Round the area to the nearest thousandth and the velocity to the nearest hundredth.
  7. Does the 10-inch main meet the design velocity limit? State which pipe the team should choose and why.
  8. The plant sits 88 feet above the pump. Find the pressure the pump must add to overcome that elevation, using the rule that 1 foot of head equals 0.433 psi. Round to the nearest tenth.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: 1,388 gpm.
    2.0 MGD x 694 gpm/MGD = 1,388 gpm.
  2. Answer: 3.09 cfs.
    2.0 MGD x 1.547 cfs/MGD = 3.094 cfs, which rounds to 3.09 cfs. The unrounded value 3.094 cfs is used in later problems.
  3. Answer: 0.785 square feet.
    Area = 0.785 x Diameter squared = 0.785 x (1.0 ft) squared = 0.785 x 1.0 = 0.785 square feet.
  4. Answer: 3.94 feet per second.
    Rearrange continuity to Velocity = Flow / Area = 3.094 cfs / 0.785 square feet = 3.941, which rounds to 3.94 feet per second.
  5. Answer: Yes.
    The velocity of 3.94 feet per second falls between 2 and 5 feet per second, so the 12-inch main meets the design range.
  6. Answer: 0.545 square feet and 5.68 feet per second.
    Area = 0.785 x (0.833 ft) squared = 0.785 x 0.694 = 0.545 square feet. Velocity = 3.094 cfs / 0.545 square feet = 5.677, which rounds to 5.68 feet per second.
  7. Answer: No; choose the 12-inch main.
    The 10-inch main's velocity of 5.68 feet per second is above the 5 feet per second limit, so it does not meet the design range. The team should choose the 12-inch main, whose velocity of 3.94 feet per second is within range.
  8. Answer: 38.1 psi.
    88 ft x 0.433 psi/ft = 38.104, which rounds to 38.1 psi. This is the pressure needed just to lift the water the 88-foot elevation difference, before any friction or delivery pressure is added.

Differentiation

Support

Provide the two pipe areas (0.785 and 0.545 square feet) already calculated, so students focus on the conversions and the continuity step. Give a printed reference strip listing the three conversions and the continuity equation.

Extension

Suppose the fictional system upsizes the pump to move more water through the 12-inch main. Ask students to find the flow, in cfs, at which the velocity in that main would reach the 5 feet per second limit, working backward from the continuity equation and the fixed area.

ELL supports

Provide a labeled diagram of a pipe cross-section showing diameter, radius, and area as separate ideas before problem 3. Allow students to show the continuity work as a labeled equation with numbers substituted in, rather than a written explanation.

Extensions

Ask a local utility's engineering staff whether they will share the design velocity range they use for transmission mains, then compare it with the 2 to 5 feet per second range in this set and discuss why very high velocities raise both friction losses and the risk of water hammer.