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Worksheet Grade 12 Math

Flow and Pipe Velocity at the Cormorant Bay Pump Station

A grade 12 worksheet where students convert a booster pump station's flow among MGD, gpm, and cfs, then use the continuity equation to find pipe velocity.

Objectives

  1. Students will convert a stated flow rate among MGD, gallons per minute, and cubic feet per second.
  2. Students will apply the continuity formula to calculate pipe velocity from flow and cross-sectional area.
  3. Students will explain how pipe diameter affects velocity at a constant flow rate.

Vocabulary

Key words: continuity equation, cross-sectional area, velocity, cfs, gpm

Teacher overview

Cormorant Bay, a fictional town used for this worksheet, moves water through its distribution system with a booster pump station that reports flow in different units depending on who is reading the gauge. This worksheet has students convert the same flow rate among million gallons per day (MGD), gallons per minute (gpm), and cubic feet per second (cfs), then use the continuity formula, Flow = Area x Velocity, to find how fast water actually moves through the station's discharge pipe.

A common misconception in this math is assuming a larger pipe always means faster-moving water. Because velocity equals flow divided by area, a larger pipe carrying the same flow actually moves water more slowly, which items 6 and 7 ask students to work out directly rather than assume.

Flow and Pipe Velocity at the Cormorant Bay Pump Station — Grade 12 Math
Name: ____________________________ Date: ______________

Student Worksheet

Cormorant Bay (a fictional town used for this worksheet) runs a booster pump station that moves a flow of 1.5 MGD through its discharge pipe. Use these conversions and the continuity formula for all items:

  • 1 MGD = 694 gallons per minute (gpm)
  • 1 MGD = 1.547 cubic feet per second (cfs)
  • 1 cfs = 448.8 gallons per minute (gpm)
  • Pipe cross-sectional area (sq ft) = 0.785 x Diameter² (ft)
  • Continuity formula: Flow (Q) = Area (A) x Velocity (V), so Velocity = Flow / Area

  1. Convert the station's flow of 1.5 MGD to gallons per minute. Round to the nearest whole gpm.

  2. Convert the station's flow of 1.5 MGD to cubic feet per second. Round to the nearest hundredth.

  3. Convert your cfs answer from item 2 back to gallons per minute using the 448.8 gpm-per-cfs conversion, and confirm it is close to your answer to item 1.

  4. The discharge pipe has an inside diameter of 1 foot. Using Area = 0.785 x Diameter², calculate the pipe's cross-sectional area in square feet.

  5. Using the continuity formula and your cfs flow rate from item 2, calculate the water's velocity in feet per second inside this 1-foot pipe.

  6. The station is upgraded to a larger discharge pipe with an inside diameter of 1.5 feet, still carrying the same 2.32 cfs flow rate. Calculate the new cross-sectional area and the new velocity.

  7. Compare your velocity answers from items 5 and 6. Explain why velocity dropped when the pipe got larger even though the flow rate in cfs stayed the same.

  8. A plant operator checking the station's flow meter would most naturally read the flow in gpm, while an engineer sizing a new pipeline would more likely work in cfs or MGD. Explain in your own words why the same flow rate is described in different units depending on who is using the number.

  9. For this project, station engineers set a design guideline of keeping pipe velocity at or below 3 feet per second in the original 1-foot pipe. Does the velocity you calculated in item 5 meet that guideline? By how much margin?

  10. Explain in your own words why engineers care about keeping pipe velocity within a limited range, considering what could go wrong if velocity runs too high or too low.

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: 1,041 gpm.
    1.5 MGD x 694 gpm per MGD = 1,041 gpm.

  2. Answer: 2.32 cfs.
    1.5 MGD x 1.547 cfs per MGD = 2.3205 cfs, which rounds to 2.32 cfs.

  3. Answer: about 1,041 gpm.
    2.32 cfs x 448.8 gpm per cfs = 1,041.2 gpm, which matches item 1's answer of 1,041 gpm within rounding, confirming the two conversion paths agree.

  4. Answer: 0.785 sq ft.
    Area = 0.785 x 1² = 0.785 x 1 = 0.785 sq ft.

  5. Answer: about 2.96 ft/s.
    Velocity = Flow / Area = 2.32 cfs / 0.785 sq ft = 2.955 ft/s, which rounds to about 2.96 ft/s.

  6. Answer: area 1.77 sq ft, velocity about 1.31 ft/s.
    New area = 0.785 x 1.5² = 0.785 x 2.25 = 1.766 sq ft, which rounds to about 1.77 sq ft. New velocity = 2.32 cfs / 1.766 sq ft = 1.313 ft/s, which rounds to about 1.31 ft/s.

  7. Answer: Velocity dropped because the same flow spread across a larger area.
    Velocity equals flow divided by area, so when the pipe's cross-sectional area grows while the flow rate stays fixed, each square foot of the pipe carries less water per second, which shows up as a lower velocity.

  8. Answer: Different units suit different purposes.
    An operator reading a flow meter at the pump station typically sees gpm because that unit matches the scale of a single station's output, while an engineer designing pipe sizes across the whole distribution system more often works in cfs or MGD because those units scale better to larger system-wide calculations.

  9. Answer: yes, by about 0.04 ft/s.
    The design guideline is 3 ft/s; the calculated velocity in item 5 is about 2.96 ft/s, which is under the guideline by about 3 - 2.96 = 0.04 ft/s.

  10. Answer: Velocity limits protect the pipe and the water quality inside it.
    Water moving too fast through a pipe can erode the pipe's interior and create pressure surges when valves open or close, while water moving too slowly can allow sediment to settle out and accumulate inside the pipe, so engineers aim for a velocity range that avoids both problems.

Differentiation

Support (below grade level)

Provide the three conversion factors and the continuity formula on a labeled reference card, and complete item 1 together as a class before releasing students to work independently on the rest.

Extension (above grade level)

Ask students to find the pipe diameter, to the nearest tenth of a foot, that would bring the velocity in item 6's flow rate down to exactly 2.0 ft/s. They should set up and solve the continuity equation for the unknown diameter rather than guess and check.

ELL supports

Pre-teach "velocity," "cross-sectional area," and "continuity" with a simple diagram of a pipe in cross-section, labeled with its diameter and a shaded circle representing area. Allow items 7, 8, and 10 to be answered as short bulleted explanations rather than full paragraphs.