Worksheet Grade 10 Math
Morrowfen Lime Softening: Dose, Cost, and Sludge Worksheet
A grade 10 worksheet on a fictional softening plant: hardness removal, quicklime dose in pounds per day, purity correction, chemical cost, and sludge production.
Objectives
- Students will calculate the hardness a softening plant must remove and express it as a daily mass in pounds.
- Students will convert a chemical dose into an as-delivered quantity by correcting for purity, then price it per day and per year.
- Students will estimate the dry sludge a softening plant produces from the hardness it removes.
Vocabulary
Key words: hardness, quicklime, softening, purity, sludge, pounds formula
Teacher overview
Groundwater that has moved through limestone carries dissolved calcium and magnesium, and a plant that removes them has to buy lime by the ton and haul the resulting solids away. This worksheet walks through one day at a softening plant in the fictional town of Morrowfen, from the hardness the plant must remove to the quicklime it feeds, the money that lime costs, and the sludge the process leaves behind. A misconception worth naming before students begin is that a plant should soften water as far as it can; operators leave a deliberate residual hardness, because water with almost no hardness is aggressive toward pipe walls and can worsen corrosion in the distribution system.
Student Worksheet
Morrowfen is a fictional town created for this worksheet. Its softening plant treats groundwater from a limestone aquifer. Use the plant data below. Round money to the nearest cent.
| Plant data | Value |
|---|---|
| Plant flow | 2.4 MGD |
| Raw total hardness | 285 mg/L |
| Calcium hardness | 195 mg/L |
| Magnesium hardness | 90 mg/L |
| Target finished hardness | 85 mg/L |
| Jar test result | 0.9 mg/L of quicklime per 1.0 mg/L of hardness removed |
| Quicklime purity as delivered | 92 percent |
| Quicklime price | $0.14 per pound |
| Sludge produced | 2.0 pounds of dry solids per pound of hardness removed |
Use these two relationships. Pounds per day equals concentration in mg/L times flow in MGD times 8.34. One MGD equals 694 gpm.
How much hardness must the plant remove? Subtract the target from the raw hardness. Give your answer in mg/L.
Find the pounds of hardness removed each day. Use the removal from item 1, the 2.4 MGD flow, and the factor 8.34.
What percent of the raw hardness does the plant remove? Divide the removal from item 1 by 285 and multiply by 100.
What percent of the raw total hardness is magnesium hardness? Divide 90 by 285 and multiply by 100.
Find the quicklime dose in mg/L. Multiply the removal from item 1 by the jar test factor of 0.9.
Find the pounds of pure quicklime the plant feeds each day. Use the dose from item 5, the 2.4 MGD flow, and the factor 8.34.
The delivered lime is only 92 percent pure. Divide your answer from item 6 by 0.92 to find the pounds of delivered lime needed each day.
Lime costs $0.14 per pound. Find the daily lime cost. Use the delivered pounds from item 7.
Find the yearly lime cost. Multiply your daily cost from item 8 by 365 days.
Convert the plant flow of 2.4 MGD to gallons per minute. Use 694 gpm per MGD.
The plant makes 2.0 pounds of dry sludge for every pound of hardness removed. Find the dry sludge in pounds per day. Use your answer from item 2.
Explain why the plant leaves 85 mg/L of hardness in the finished water instead of removing all of it.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
Answer: 200 mg/L.
The plant must remove the difference between the raw and target values: 285 minus 85 equals 200 mg/L of hardness.Answer: 4,003.2 lbs/day.
Using the pounds formula: 200 mg/L times 2.4 MGD times 8.34 equals 4,003.2 pounds of hardness removed each day.Answer: about 70.2 percent.
Dividing 200 by 285 gives 0.7018, and multiplying by 100 gives about 70.2 percent removal.Answer: about 31.6 percent.
Dividing 90 by 285 gives 0.3158, and multiplying by 100 gives about 31.6 percent. Magnesium is the smaller share, which matters because magnesium hardness is the more expensive part to remove.Answer: 180 mg/L.
Multiplying the 200 mg/L removal by the jar test factor of 0.9 gives a quicklime dose of 180 mg/L.Answer: about 3,602.9 lbs/day.
Using the pounds formula: 180 mg/L times 2.4 MGD times 8.34 equals 3,602.88, which rounds to about 3,602.9 pounds of pure quicklime per day.Answer: about 3,916.2 lbs/day.
Dividing 3,602.88 by 0.92 gives 3,916.17, which rounds to about 3,916.2 pounds of delivered lime per day. Dividing by purity increases the quantity, because part of every delivered pound is not quicklime.Answer: $548.26 per day.
Multiplying 3,916.17 pounds by 0.14 dollars per pound gives 548.264, which rounds to 548.26 dollars per day.Answer: about $200,116 per year.
Multiplying 548.264 dollars per day by 365 days gives 200,116.5, which rounds to about 200,116 dollars per year. Accept answers within a few dollars, since rounding at item 8 shifts the total.Answer: 1,665.6 gpm.
Multiplying 2.4 MGD by 694 gpm per MGD gives 1,665.6 gallons per minute.Answer: 8,006.4 lbs/day.
Multiplying the 4,003.2 pounds of hardness removed by 2.0 pounds of dry solids per pound gives 8,006.4 pounds of dry sludge per day. That mass has to be thickened, dewatered, and hauled, which is a large share of a softening plant's operating budget.Answer: Very soft water is corrosive to pipes.
Accept any answer explaining that water with almost no hardness and low alkalinity tends to dissolve metal and cement pipe surfaces, so operators leave a residual on purpose. Strong answers also note that removing the last portion of hardness costs more lime and makes more sludge for each milligram removed, so the plant would pay twice for a result customers would not notice.
Differentiation
Support
Give these students items 1, 2, and 5 worked as examples on a reference card, with the pounds formula written out and each factor labeled. Let them skip item 4 and work the rest in order, and provide a filled-in unit map showing that mg/L times MGD times 8.34 yields pounds per day so the units stop being a guessing game.
Extension
Ask these students to recalculate items 5 through 9 for the fictional Morrowfen plant if a new jar test lowers the lime factor to 0.8, and to state how much the plant would save in a year. Have them also explain why the plant's magnesium share would change the lime dose if the aquifer chemistry shifted, even at the same total hardness.
ELL supports
Provide a labeled diagram of the softening process showing raw water entering, lime being fed, solids settling out, and finished water leaving, with each stage named in short phrases. Post a strip pairing hardness with dissolved minerals, dose with how much is added, purity with how much of the bag is the real chemical, and sludge with the solids left over.
Extensions
Have students look up whether their own utility softens its water and what finished hardness it reports in its annual consumer confidence report, then compare that value with the fictional Morrowfen target. Students can also research how a plant reuses softening sludge, since some utilities send it to farms as an agricultural liming material rather than to a landfill, and estimate how many truckloads a year the plant in this worksheet would ship at a stated load size.