Math problem set Grade 11 Math

Torvale Coast RO Blending: Recovery and a TDS Mass Balance

A grade 11 problem set on a fictional brackish RO plant: find recovery, solve a blended-TDS mass balance for the bypass fraction, and estimate daily energy.

Grade: Grade 11
Subject: Math
Material: Math problem set
Time: 45 min
Prep: No prep
Materials: Calculator

Standards

HSN-Q.A.1 Common Core Math

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

How this material aligns: Problems 1, 4, and 6 track MGD, mg/L, gallons, and kWh through the recovery, blending, and energy calculations, using unit reasoning to guide each multi-step conversion this standard targets.

All materials for HSN-Q.A.1 →

HSA-CED.A.3 Common Core Math

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.

How this material aligns: Problems 3 and 7 set up the blended-TDS equation as a constraint at the 500 mg/L target and solve for the viable bypass fraction, the constraint modeling this standard requires.

All materials for HSA-CED.A.3 →

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This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-23.

Objectives

Students will calculate reverse-osmosis recovery and product flows from a stated feed flow.
Students will solve a flow-weighted mass balance for the bypass fraction that meets a TDS target.
Students will estimate daily treatment energy from a per-thousand-gallon rate.

Vocabulary

Reverse osmosis: Reverse osmosis cleans water by pushing it through a membrane with holes so tiny that salt and most other dissolved things cannot fit through. It can even turn seawater into fresh water.
Total dissolved solids (TDS): Total dissolved solids is a measure of everything dissolved in water that you cannot see or filter out, like minerals and salts. Water with very high TDS can taste salty or bitter, even though it might look perfectly clear.

Also: permeate, recovery, mass balance

Teacher overview

Brackish-water reverse-osmosis plants rarely treat every gallon through the membranes. Instead they blend clean, low-salt permeate with a small stream of untreated water, which saves energy while keeping the delivered water below a total-dissolved-solids (TDS) target. This set has grade 11 students work through recovery, a flow-weighted blending mass balance, and an energy estimate for the Torvale Coast Water District, a fictional district. A common student error is to average the two TDS values directly instead of weighting each stream by its flow; the blend depends on how much of each stream is used, not on a plain average of their concentrations.

Torvale Coast RO Blending: Recovery and a TDS Mass Balance — Grade 11 Math

Name: ____________________________ Date: ______________

Problems

The Torvale Coast Water District is a fictional district. Its brackish RO plant uses these values:

RO feed water TDS: 2,100 mg/L. RO permeate TDS: 100 mg/L.
District target for delivered water: at most 500 mg/L TDS.
RO unit feed flow: 2.0 MGD. RO recovery: 75%.
Energy: this brackish RO plant uses about 3 to 4 kWh per 1,000 gallons.

The RO unit takes 2.0 MGD of feed water at 75% recovery. How much permeate (MGD) does it make, and how much concentrate?
Recovery is permeate flow divided by feed flow, written as a percent. Confirm that 1.5 MGD of permeate from 2.0 MGD of feed gives 75% recovery.
The district blends permeate (100 mg/L) with untreated feed water (2,100 mg/L). For a blend that is fraction f untreated water and (1 minus f) permeate, write the blended-TDS expression, then solve for the fraction f that gives exactly 500 mg/L.
Using f = 0.20, the plant delivers all 1.5 MGD of permeate as the 80% permeate portion. What total blended flow (MGD) results, and how much untreated bypass water (MGD) is added?
Verify the blend from problem 4 by a mass balance: add the TDS load from each stream and divide by the total flow. Does it equal 500 mg/L?
The plant's RO units use about 3 to 4 kWh per 1,000 gallons. Using 3.5 kWh per 1,000 gallons, estimate the daily RO energy for treating the 2.0 MGD feed.
After a dry spell the feed water TDS rises to 2,600 mg/L while permeate stays at 100 mg/L. Recompute the bypass fraction f that still meets the 500 mg/L target.
Using your problem 3 and problem 7 answers, explain why saltier feed water forces the plant to bypass less untreated water and lean harder on the RO units.

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: 1.5 MGD permeate and 0.5 MGD concentrate.
Permeate = 0.75 x 2.0 = 1.5 MGD. Concentrate = 2.0 minus 1.5 = 0.5 MGD.
Answer: 75%.
1.5 MGD divided by 2.0 MGD = 0.75, or 75% recovery, which matches the stated value.
Answer: 0.20.
The blended TDS is 100(1 minus f) + 2,100 f. Setting it equal to 500 gives 100 + 2,000 f = 500, so 2,000 f = 400 and f = 0.20, meaning 20% untreated water.
Answer: 1.875 MGD total, with 0.375 MGD bypass.
The 1.5 MGD of permeate is the 80% portion, so total = 1.5 divided by 0.80 = 1.875 MGD. Bypass = 1.875 minus 1.5 = 0.375 MGD.
Answer: 500 mg/L.
Permeate load = 100 x 1.5 = 150; bypass load = 2,100 x 0.375 = 787.5; total load = 937.5. Divide by 1.875 MGD to get 500 mg/L, which matches the target.
Answer: 7,000 kWh/day.
2.0 MGD = 2,000,000 gallons per day; divided by 1,000 gives 2,000 units; 2,000 x 3.5 kWh = 7,000 kWh per day.
Answer: 0.16.
With feed at 2,600 mg/L: 100(1 minus f) + 2,600 f = 500 gives 100 + 2,500 f = 500, so 2,500 f = 400 and f = 0.16, meaning only 16% bypass.
Answer: Saltier feed allows less bypass.
As the untreated water gets saltier, each unit of bypass adds more salt to the blend, so a smaller bypass fraction is needed to stay at 500 mg/L. The plant must then route more water through the membranes, which raises energy use.

Differentiation

Support

Provide the blended-TDS equation already set up for problem 3, with only the final algebra left to complete. Give students a two-stream diagram showing permeate and bypass merging into the delivered flow so the mass balance is concrete.

Extension

Ask students to find the bypass fraction and total delivered flow if the fictional district lowered its target to 350 mg/L at the original feed strength, then explain in one sentence how a stricter target changes the plant's energy use.

ELL supports

Pre-teach "permeate," "bypass," "blend," and "recovery" with a labeled sketch of two pipes joining into one. Provide a sentence frame for problem 8: "When the feed water is saltier, the plant can bypass ___ water because ___."