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Math problem set Grade 9 Science

Fallowmere's River Intake: From Stream Flow to Plant Capacity

A grade 9 problem set converting river flow among cfs, MGD, and gpm at a fictional town's intake, then testing a withdrawal limit against summer low flow.

Objectives

  1. Students will convert river discharge among cubic feet per second, million gallons per day, and gallons per minute.
  2. Students will express a town's withdrawal as a percent of river flow at high flow and at summer low flow.
  3. Students will explain how a river's low flow constrains a town's water supply and its growth.

Vocabulary

Key words: discharge, cubic feet per second, million gallons per day, filter loading rate, withdrawal

Teacher overview

A river gauge reports discharge in cubic feet per second, a treatment plant reports production in million gallons per day, and a pump nameplate reports capacity in gallons per minute. An operator has to move between all three without changing the physical quantity. This set follows the intake of the fictional town of Fallowmere on the Harrow River from spring high water into summer low water, and it ends by testing whether a projected population could still be served under the town's withdrawal permit. The misconception this set is built to correct is directional: students who know that 1 MGD equals 1.547 cfs often multiply when converting cfs into MGD, which makes the answer larger than the reading they started from instead of smaller. Checking that MGD should be a smaller number than cfs for the same river catches the error immediately.

Fallowmere's River Intake: From Stream Flow to Plant Capacity — Grade 9 Science
Name: ____________________________ Date: ______________

Problems

The fictional town of Fallowmere withdraws 3.6 MGD from the Harrow River. Use these conversions: 1 MGD equals 1.547 cfs, 1 MGD equals 694 gpm, and 1 cfs equals 448.8 gpm. Round to one decimal place unless a problem says otherwise.

  1. A spring gauge reading on the Harrow River is 340 cfs. Convert that discharge to MGD.
  2. Convert Fallowmere's 3.6 MGD withdrawal to gallons per minute.
  3. Using the spring discharge from problem 1, what percent of the river is Fallowmere withdrawing? Round to the nearest tenth of a percent.
  4. In late August the gauge reads 46 cfs. Convert that discharge to MGD.
  5. Using the August discharge from problem 4, what percent of the river is the 3.6 MGD withdrawal? Round to the nearest tenth of a percent.
  6. Fallowmere's state permit allows the town to take no more than 15 percent of the river's daily flow. What is the smallest river discharge, in cfs, that still allows a 3.6 MGD withdrawal? Round to the nearest tenth.
  7. The plant splits its 3.6 MGD evenly between two filters, and each filter has 320 square feet of surface area. Find the filter loading rate for one filter in gpm per square foot, rounded to the nearest tenth. Conventional plants generally run between 2 and 6 gpm per square foot.
  8. Fallowmere's planners design for 90 gallons per person per day. If the population reaches 52,000, what daily demand would the plant have to meet, in MGD? Give your answer to the nearest hundredth.
  9. Using the August discharge from problem 4 and the demand from problem 8, what percent of the river would the town be withdrawing? Round to the nearest tenth of a percent.
  10. Using your answers from problems 4, 6, and 9, explain how the Harrow River's summer flow limits how much Fallowmere can grow, and name one step the town could take that would let it serve more people without taking a larger share of the August river.

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: 219.8 MGD. 340 cfs divided by 1.547 cfs per MGD equals 219.78, which rounds to 219.8 MGD.
  2. Answer: 2,498.4 gpm. 3.6 MGD times 694 gpm per MGD equals 2,498.4 gpm.
  3. Answer: 1.6 percent. 3.6 MGD divided by 219.8 MGD equals 0.0164, or 1.6 percent of the spring flow.
  4. Answer: 29.7 MGD. 46 cfs divided by 1.547 cfs per MGD equals 29.735, which rounds to 29.7 MGD.
  5. Answer: 12.1 percent. 3.6 MGD divided by 29.7 MGD equals 0.1212, or 12.1 percent of the August flow.
  6. Answer: 37.1 cfs. The permit requires 3.6 MGD to be no more than 15 percent of flow, so the flow must be at least 3.6 divided by 0.15, which equals 24 MGD, and 24 MGD times 1.547 cfs per MGD equals 37.128, which rounds to 37.1 cfs.
  7. Answer: 3.9 gpm per square foot. Each filter receives 1.8 MGD, and 1.8 times 694 equals 1,249.2 gpm, and 1,249.2 divided by 320 square feet equals 3.90 gpm per square foot, which sits inside the usual range.
  8. Answer: 4.68 MGD. 52,000 people times 90 gallons per person per day equals 4,680,000 gallons per day, which is 4.68 MGD.
  9. Answer: 15.8 percent. 4.68 MGD divided by 29.7 MGD equals 0.1576, or 15.8 percent, which is above the 15 percent the permit allows.
  10. Answer: Accept any explanation that uses the calculated values. In August the river carries only 29.7 MGD, so the permit caps the town near 4.46 MGD, and the projected demand of 4.68 MGD would take 15.8 percent of the river, which the permit does not allow. The binding limit is the flow of the river in its driest weeks rather than the size of the plant. Steps that would help include reducing leakage and per-person use, adding storage that is filled during spring high flow, or developing a second source such as wells or reuse so that August withdrawals stay below the permit share.

Differentiation

Support

Give students a conversion reference card listing the three relationships with an example worked in each direction, and mark on it that a discharge in MGD is always a smaller number than the same discharge in cfs. Assign problems 1 through 5 only, and provide the August gauge reading already converted for students who need to start at problem 5.

Extension

Have students find the largest population the fictional town could serve at 90 gallons per person per day while staying within the permit at the August discharge from problem 4, then calculate how much that population could grow if per-person use fell to 75 gallons per day.

ELL supports

Pre-teach discharge, withdrawal, permit, and loading rate with a labeled diagram of the fictional Fallowmere intake showing the river, the intake pipe, and the plant. Provide a sentence frame for problem 10: "In August the river carries ___, so the town may take only ___, which means ___."