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Worksheet Grade 9 Math

Underhill Plant Filters: Loading Rate and Turbidity Worksheet

A grade 9 worksheet treating filter loading rate as gallons per minute per square foot, sizing Underhill's filters and checking finished-water turbidity limits.

Objectives

  1. Students will calculate filter loading rate as a flow-per-area density.
  2. Students will size filter surface area to keep loading within the operating range.
  3. Students will compare finished-water turbidity readings to regulatory limits.

Vocabulary

Turbidity
Turbidity is a measure of how cloudy or murky water looks because of tiny particles floating in it. Clear water has low turbidity, and muddy water has high turbidity.

Also: loading rate, surface area, continuity, cubic feet per second

Teacher overview

The fictional Underhill treatment plant sends water through sand filters. Operators track the filter loading rate, which is the flow in gallons per minute divided by the filter surface area in square feet. It is a kind of density: how much flow each square foot of filter must handle. This worksheet has students compute loading rates, size a filter for a higher flow, and check finished water against turbidity limits.

Underhill Plant Filters: Loading Rate and Turbidity Worksheet — Grade 9 Math
Name: ____________________________ Date: ______________

Student Worksheet

Underhill (a fictional plant) runs sand filters. Loading rate equals flow (gpm) divided by filter area (sq ft). Conventional plants run 2 to 6 gpm per square foot. Finished water must stay at or below 0.3 NTU in 95 percent of samples, and never rise above 1 NTU.

  1. One filter takes 800 gpm and has 200 sq ft of surface area. Find its loading rate in gpm per square foot.
  2. Is that loading rate inside the usual 2 to 6 gpm per square foot range?
  3. Flow rises to 1,200 gpm on the same 200 sq ft filter. Find the new loading rate.
  4. The plant caps loading at 6 gpm per square foot. What is the largest flow the 200 sq ft filter can take?
  5. Underhill must treat 1,800 gpm and keep loading at or below 6 gpm per square foot. How many square feet of filter are needed?
  6. Underhill already has 200 sq ft of filter. Using item 5, how much more filter area must it add?
  7. A finished-water sample reads 0.25 NTU. Does it meet the 0.3 NTU standard?
  8. Another sample reads 0.8 NTU. It is below the never-exceed level of 1 NTU, but is it within the 0.3 NTU standard? Explain what this tells the operator.
  9. The filter's outlet pipe has a cross-section area of 0.5 sq ft, and water moves through it at 3.6 ft per second. Flow Q equals area times velocity. Find Q in cubic feet per second.
  10. Convert the 1.8 cfs from item 9 to gpm, using 1 cfs equal to 448.8 gpm. Does it roughly match the 800 gpm from item 1?

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: 4 gpm/sq ft. 800 gpm / 200 sq ft = 4 gpm per square foot.
  2. Answer: Yes. 4 gpm per square foot sits inside the 2 to 6 gpm per square foot range.
  3. Answer: 6 gpm/sq ft. 1,200 gpm / 200 sq ft = 6 gpm per square foot.
  4. Answer: 1,200 gpm. 6 gpm/sq ft x 200 sq ft = 1,200 gpm.
  5. Answer: 300 sq ft. 1,800 gpm / 6 gpm/sq ft = 300 sq ft of filter area.
  6. Answer: 100 sq ft. 300 sq ft needed minus 200 sq ft on hand equals 100 sq ft to add.
  7. Answer: Yes. 0.25 NTU is at or below the 0.3 NTU standard.
  8. Answer: No. 0.8 NTU is under the 1 NTU never-exceed level but above the 0.3 NTU standard, which warns the operator to check the filter before it worsens.
  9. Answer: 1.8 cfs. Q = area x velocity = 0.5 sq ft x 3.6 ft/s = 1.8 cfs.
  10. Answer: 808 gpm. 1.8 cfs x 448.8 gpm/cfs = 807.84 gpm, about 808 gpm, close to the 800 gpm in item 1.

Differentiation

Support

Give students the loading-rate formula on a reference card and one fully worked example that uses numbers not on this worksheet. Print the conversion "1 cubic foot per second equals 448.8 gallons per minute" for item 10, and allow a calculator throughout.

Extension

Ask students to find the loading rate if two 200-square-foot filters share a 2,000-gallon-per-minute flow equally, then decide whether each filter stays within the usual operating range and explain their reasoning.

ELL supports

Pre-teach "loading rate," "surface area," and "turbidity" with a picture of water spreading across a flat filter bed, and show turbidity as cloudy versus clear water in two jars. Provide a sentence frame for item 8.