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Math problem set Grade 9 Math

Caldbeck Treatment Plant: Alum Coagulant Dosing Problem Set

A grade 9 problem set using the pounds formula to size alum coagulant feed, plan bag supply, and stay within a fictional plant's pump and cost limits.

Objectives

  1. Students will use the pounds formula to convert a chemical dose and flow into a daily feed weight.
  2. Students will check whether a feed rate fits within stated pump, supply, and cost limits.

Vocabulary

Alum
Alum is the most common chemical that water plants add to make tiny dirt particles clump together. A pinch of alum can help a whole jar of muddy water turn clear.

Also: coagulant, dose, pounds formula

Teacher overview

The Caldbeck Water Treatment Plant (a fictional plant used for this set) adds alum, a coagulant, so that fine particles clump together and settle out before filtration. Operators size the alum feed with the pounds formula, which turns a dose in milligrams per liter and a flow in million gallons per day into a feed weight in pounds per day. This set has students find feed rates, plan bag supply, and check each result against real operating limits. A common misconception is that a larger plant always needs a larger dose; the dose depends on the raw water, while the flow decides how many pounds that dose adds up to.

Caldbeck Treatment Plant: Alum Coagulant Dosing Problem Set — Grade 9 Math
Name: ____________________________ Date: ______________

Problems

Use these facts about Caldbeck for every problem. Round to one decimal place unless told otherwise.

  • Normal flow: 6.0 MGD (million gallons per day).
  • Normal alum dose from the jar test: 10 mg/L.
  • Pounds formula: feed (lbs/day) = dose (mg/L) x flow (MGD) x 8.34.
  • Alum is delivered in 50-pound bags.
  1. Find the alum feed in pounds per day at the normal dose and flow.
  2. How many pounds of alum does the plant use over a 30-day month at the normal feed rate?
  3. Over a 7-day week at the normal feed rate, how many full 50-pound bags must the plant open? (You cannot open part of a bag.)
  4. A high-turbidity day raises the dose to 14 mg/L at the same 6.0 MGD flow. Find the new feed rate in pounds per day.
  5. Write an equation for alum feed F (in lbs/day) as a function of dose D (in mg/L) at the fixed 6.0 MGD flow. Then find the feed at a dose of 8 mg/L.
  6. The alum storage silo holds 20,000 pounds. At the normal feed rate from item 1, how many whole days of supply is that?
  7. The chemical-feed pump works accurately only between 300 and 900 lbs/day. Is the high-turbidity feed from item 4 inside that range?
  8. In summer the flow rises to 9.0 MGD at a dose of 12 mg/L. Find the feed rate, and state whether it stays within the pump's 900 lbs/day maximum.
  9. Alum costs $0.45 per pound (a fictional local price). Find the daily alum cost at the normal feed rate from item 1.
  10. Explain why the plant runs a jar test to choose the dose each day instead of always using the same number, and what a dose set too low could do to the finished-water turbidity limit of 0.3 NTU.

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: 500.4 lbs/day. Feed = 10 mg/L x 6.0 MGD x 8.34 = 60 x 8.34 = 500.4 pounds per day.
  2. Answer: 15,012 lbs. 500.4 lbs/day x 30 days = 15,012 pounds in the month.
  3. Answer: 71 bags. A week uses 500.4 x 7 = 3,502.8 pounds. Dividing by 50 gives 70.06 bags, and since a partial bag must be opened, the plant opens 71 bags.
  4. Answer: 700.6 lbs/day. Feed = 14 mg/L x 6.0 MGD x 8.34 = 84 x 8.34 = 700.56, which rounds to 700.6 pounds per day.
  5. Answer: 400.3 lbs/day. The equation is F = D x 6.0 x 8.34 = 50.04D. At D = 8 mg/L, F = 50.04 x 8 = 400.32, about 400.3 pounds per day.
  6. Answer: 39 days. 20,000 pounds / 500.4 lbs/day = 39.97 days, which is 39 whole days of supply.
  7. Answer: Yes. The high-turbidity feed is about 700.6 lbs/day, which is between the pump's 300 and 900 lbs/day limits.
  8. Answer: 900.7 lbs/day. Feed = 12 mg/L x 9.0 MGD x 8.34 = 108 x 8.34 = 900.72 pounds per day, which is just above the pump's 900 lbs/day maximum, so the pump cannot keep up at that setting.
  9. Answer: $225.18. Cost = 500.4 pounds x $0.45 per pound = $225.18 for the day.
  10. Answer: Raw water changes with weather and runoff, so the dose that makes particles clump well changes too; the jar test finds the right dose for that day. A dose set too low makes weak floc that does not settle, letting more particles through and pushing finished-water turbidity toward or past the 0.3 NTU limit.

Differentiation

Support

Provide the pounds formula on a card with each unit labeled, and pre-fill the first multiplication for item 1 (10 x 6.0 = 60). Remind students that bag and day counts round in the direction that keeps the plant supplied, and allow calculators.

Extension

Ask students to find the highest dose, in mg/L, the plant could feed at the summer flow of 9.0 MGD while staying at or below the pump's 900 lbs/day maximum, working backward through the pounds formula.

ELL supports

Provide a picture sequence showing alum added, particles clumping into floc, and floc settling. Pre-teach "coagulant," "dose," and "feed rate," and post the pounds formula with its three units written underneath in students' home languages where possible.