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Worksheet Grade 10 Science

Chlorine Loss in Sarnwick's Pipes: Testing Two Decay Models

Grade 10 students test a constant-loss model against a halving model of chlorine residual, using sample readings from a fictional town's pressure zone.

Objectives

  1. Students will compare a constant-loss model and a halving model of chlorine residual against measured samples from a distribution system.
  2. Students will explain why chlorine disappears faster where more chlorine is present.
  3. Students will use the better-fitting model to infer an unknown water age and to judge a proposed change in the residual leaving the plant.

Vocabulary

Chlorine residual
Chlorine residual is the small amount of chlorine still left in the water after it leaves the treatment plant. It keeps protecting the water from germs all the way to your faucet.
Dead-end main
A dead-end main is a section of water pipe that stops instead of connecting to another pipe on the far end, so water can only flow in and out from one direction. Water can sit still longer in a dead-end main, so utilities flush them regularly to keep the water fresh.
Distribution system
A distribution system is the network of pipes, pumps, valves, and storage tanks that carries treated water from a treatment plant to homes, schools, and businesses. Most of it is buried underground beneath streets.

Also: water age, decay model, proportional loss

Teacher overview

Chlorine does not stay put once treated water leaves a plant. It reacts with pipe walls, with biofilm, and with material dissolved in the water, so the residual an operator measures at a hydrant depends on how long that water has been in the system. This 45-minute worksheet gives students five sample readings from the fictional town of Sarnwick and two competing models of how the residual falls: one that subtracts a fixed amount of chlorine each hour, and one that removes half of whatever chlorine is present in each 12-hour stretch. Both models are fitted to the same first reading pair, so they agree at 12 hours and disagree everywhere after that, which is what makes the comparison decidable.

Students run both models forward, measure each against the sampling data, and find that the constant-loss model predicts a negative concentration by 36 hours. The point to draw out in discussion is the mechanism behind the better fit: the rate at which chlorine disappears depends on how much chlorine is there to react, so equal time intervals remove equal fractions rather than equal amounts. The last three items put that model to work on a dead-end main that reads lower than the map says it should, and on a proposal to raise the residual leaving the plant. Students produce two model predictions, an inferred water age, and a written judgment of the proposal.

Chlorine Loss in Sarnwick's Pipes: Testing Two Decay Models — Grade 10 Science
Name: ____________________________ Date: ______________

Student Worksheet

Operators in the town of Sarnwick, a fictional town used for this worksheet, sampled free chlorine at the clearwell outlet and at four points along one pressure zone on the same day. Water age is the estimated time the sampled water has spent in the pipes since it left the plant.

Sample point Water age (hours) Free chlorine residual (mg/L)
Clearwell outlet at the plant 0 2.4
Ridgeway pump station 12 1.2
Marlow Road sample station 24 0.6
Fenn Street sample station 36 0.3
Quill Court dead-end main 36 (from the system map) 0.15

Two models are fitted to the first 12 hours, where the residual fell from 2.4 mg/L to 1.2 mg/L. Model A treats the loss as a fixed 0.10 mg/L for every hour in the pipes. Model B treats the loss as half of whatever chlorine is present in each 12 hours.

  1. Model A removes 0.10 mg/L of free chlorine every hour, whatever the residual happens to be. Starting from 2.4 mg/L at the clearwell outlet, what residual does Model A predict at a water age of 24 hours?
  2. Using the same fixed rate of 0.10 mg/L per hour and the same 2.4 mg/L starting point, what does Model A predict at 36 hours? Say what that result tells you about the model.
  3. Model B removes half of whatever free chlorine is present during each 12 hours in the pipes. Starting from 2.4 mg/L, what residual does Model B predict at 24 hours?
  4. Using Model B and the same 2.4 mg/L starting point, what does it predict at a water age of 36 hours?
  5. The Fenn Street station measured 0.3 mg/L at 36 hours of water age, while Model A predicts -1.2 mg/L at that age. How far apart are the prediction and the measurement?
  6. Between the clearwell outlet and the Ridgeway pump station the residual fell from 2.4 mg/L to 1.2 mg/L in 12 hours. Between Marlow Road and Fenn Street it fell from 0.6 mg/L to 0.3 mg/L, also in 12 hours. How many times larger was the loss in the first stretch than in the second?
  7. The dead-end main on Quill Court is mapped at a water age of 36 hours, where Model B predicts 0.3 mg/L, but the sample there read 0.15 mg/L. How far below the prediction is that reading?
  8. Suppose the map is wrong and the Quill Court water follows Model B from the same 2.4 mg/L clearwell reading. What water age would Model B need to bring 2.4 mg/L down to 0.15 mg/L?
  9. Operators propose raising the residual leaving the clearwell from 2.4 mg/L to 3.0 mg/L. Using Model B and a water age of 48 hours, what residual does that proposal predict at Quill Court? The field colorimeter reads to the nearest hundredth of a milligram per liter, so report the value that way.

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: 0.0 mg/L.
    Model A subtracts the same quantity every hour: 0.10 mg/L x 24 hours = 2.4 mg/L of loss, and 2.4 - 2.4 = 0.0 mg/L. The model puts the residual at exactly zero after one day in the pipes.
  2. Answer: -1.2 mg/L.
    0.10 mg/L x 36 hours = 3.6 mg/L of loss, and 2.4 - 3.6 = -1.2 mg/L. A sample cannot hold less than no chlorine, so Model A has already failed before 36 hours; a model that removes equal amounts over equal intervals cannot describe this system.
  3. Answer: 0.6 mg/L.
    Halving twice gives 2.4 mg/L at 0 hours, 1.2 mg/L at 12 hours, and 0.6 mg/L at 24 hours, which is 2.4 x 0.5 x 0.5. Equal intervals remove equal fractions here, not equal amounts.
  4. Answer: 0.3 mg/L.
    A third halving takes 0.6 mg/L to 0.3 mg/L, that is 2.4 x 0.5 x 0.5 x 0.5. Model B reproduces both the 24-hour and the 36-hour readings exactly.
  5. Answer: 1.5 mg/L.
    0.3 - (-1.2) = 1.5 mg/L. Model B misses the same reading by nothing at all, so the sampling data support the halving model and rule out the constant-loss model.
  6. Answer: 4 times.
    The first stretch lost 2.4 - 1.2 = 1.2 mg/L and the later stretch lost 0.6 - 0.3 = 0.3 mg/L, so 1.2 / 0.3 = 4. Both stretches cover 12 hours, so time alone does not explain the difference. Chlorine is consumed by reactions with pipe walls, biofilm, and dissolved material, and the more chlorine present, the more of it reacts per hour. That concentration-dependent rate is what Model B builds in and Model A leaves out.
  7. Answer: 0.15 mg/L.
    0.3 - 0.15 = 0.15 mg/L below the prediction, which is half of what the model expects at that age. Two explanations fit the gap: the water sitting in a dead-end main may be older than the map assumes, or that stretch of pipe may consume chlorine faster than the rest of the zone because of corrosion or biofilm on the pipe wall.
  8. Answer: 48 hours.
    Keep halving: 2.4 mg/L at 0 hours, 1.2 at 12, 0.6 at 24, 0.3 at 36, and 0.15 mg/L at 48 hours, which is four halvings of 12 hours each. The age explanation therefore carries a testable prediction: a tracer study or a timed hydrant flush at Quill Court should show roughly 48 hours of travel time rather than the 36 hours on the map. If it shows 36, the extra chlorine consumption in that pipe is the better explanation.
  9. Answer: 0.19 mg/L.
    Four halvings of 3.0 mg/L give 1.5, 0.75, 0.375, and 0.1875 mg/L, which the colorimeter's hundredth-place reading reports as 0.19 mg/L. Raising the plant residual by 25 percent lifts the 48-hour residual only from 0.15 mg/L to 0.19 mg/L, still under the 0.2 mg/L bottom of the range utilities typically maintain, because a model that multiplies scales the whole curve instead of lifting its tail. Shortening the water age, through a scheduled flushing program or by looping the dead-end main, does more for that street than a higher dose, which would also give disinfection byproducts more chlorine to form from.

Differentiation

Support (below grade level)

Give students a blank halving ladder for the fictional Sarnwick zone, with 2.4 mg/L printed in the first box and the 12-hour steps already labeled, so items 3, 4, and 8 become a matter of filling boxes rather than setting up powers. Work item 1 aloud as a modeled example. Students who lose the thread of the comparison can be asked one question at a time: what does each model say at 36 hours, and which one matches the reading.

Extension (above grade level)

Ask students to write Model B for the fictional Sarnwick data as a function of water age in hours, then find the water age at which the residual first falls below the 0.2 mg/L bottom of the maintained range, starting from 2.4 mg/L. Students who have met logarithms can solve it exactly; others can bracket the answer between the 36-hour and 48-hour steps and refine by halving the interval.

ELL supports

Pre-teach "water age," "residual," and "dead-end main" with a simple map of the fictional Sarnwick zone showing one path from the plant to a looped street and one path to a street that ends. Provide sentence frames for the written parts: "Model A predicts ___ mg/L at 36 hours, which cannot happen because ___," and "The reading at Quill Court is lower than the model because ___."