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Math problem set Grade 10 Careers & CTE

Kestrel Cove Desalination: Reverse-Osmosis Operator Math

A grade 10 careers math set at the fictional Kestrel Cove desal plant, where students compute permeate and concentrate flows, recovery, and energy use.

Objectives

  1. Students will calculate permeate and concentrate flows from a feed flow and a recovery rate.
  2. Students will convert flows between MGD and gpm and find the percent of demand a source meets.
  3. Students will compute and compare the daily energy use of reverse osmosis and conventional treatment.

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.

Also: permeate, concentrate, recovery rate

Teacher overview

The town of Kestrel Cove runs a seawater reverse-osmosis desalination plant, a fictional plant used for this problem set. Seawater is pushed through membranes; the clean water that passes through is called permeate, and the salty water left behind is the concentrate. The share of the feed water that becomes permeate is the recovery rate. This set walks students through the plant's flows, the percent of town demand it meets, and the energy it takes. A common misconception this set corrects is basing the recovery rate on the town's demand instead of on the feed water, and forgetting that concentrate equals feed minus permeate.

Kestrel Cove Desalination: Reverse-Osmosis Operator Math — Grade 10 Careers & CTE
Name: ____________________________ Date: ______________

Problems

At the town of Kestrel Cove's fictional desalination plant, the seawater feed to the reverse-osmosis racks is 5.0 MGD, and the plant runs at a recovery rate of 45%, meaning permeate is 45% of the feed. Use these facts: one MGD equals 694 gpm and one MGD equals 1,000,000 gallons per day. The plant uses 13 kWh of electricity per 1,000 gallons of permeate. The town's total water demand is 3.0 MGD.

  1. Find the permeate flow, in MGD. (Permeate = recovery rate x feed.)
  2. Find the concentrate flow, in MGD. (Concentrate = feed - permeate.)
  3. Convert the permeate flow from problem 1 to gpm.
  4. The plant must help meet the town's 3.0 MGD demand. How much additional treated water, in MGD, must come from the town's wells to make up the rest?
  5. What percent of the town's 3.0 MGD demand does the permeate supply?
  6. Find the plant's daily energy use for making the permeate, in kWh per day. (Permeate in problem 1 is 2,250,000 gallons per day, which is 2,250 units of 1,000 gallons.)
  7. A conventional inland plant treating the same 2.25 MGD would use only 1.5 kWh per 1,000 gallons. How many kWh per day would that conventional plant use?
  8. How many more kWh per day does the reverse-osmosis plant use than the conventional plant?
  9. The plant raises its recovery rate from 45% to 50%. What is the new permeate flow, in MGD, and does the concentrate flow rise or fall? Explain.

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: 2.25 MGD. Permeate = 0.45 x 5.0 MGD = 2.25 MGD.
  2. Answer: 2.75 MGD. Concentrate = feed - permeate = 5.0 - 2.25 = 2.75 MGD.
  3. Answer: 1,562 gpm. 2.25 MGD x 694 gpm/MGD = 1,561.5, which rounds to 1,562 gpm.
  4. Answer: 0.75 MGD. The wells must supply 3.0 MGD demand - 2.25 MGD permeate = 0.75 MGD.
  5. Answer: 75%. 2.25 MGD / 3.0 MGD = 0.75, or 75% of demand.
  6. Answer: 29,250 kWh per day. 2,250 units of 1,000 gallons x 13 kWh = 29,250 kWh per day.
  7. Answer: 3,375 kWh per day. 2,250 units x 1.5 kWh = 3,375 kWh per day.
  8. Answer: 25,875 kWh per day. 29,250 kWh - 3,375 kWh = 25,875 kWh per day more for reverse osmosis.
  9. Answer: 2.5 MGD. New permeate = 0.50 x 5.0 = 2.5 MGD. The concentrate falls, from 2.75 to 5.0 - 2.5 = 2.5 MGD, because more of the feed now leaves as permeate, so less is left behind as concentrate.

Differentiation

Support

Provide a labeled flow diagram of one membrane rack showing feed entering, permeate leaving one side, and concentrate leaving the other, with the equations permeate = recovery x feed and concentrate = feed - permeate printed beside it. Allow a calculator for every step.

Extension

Ask students to find the seawater feed flow the fictional Kestrel Cove plant would need to produce 3.0 MGD of permeate at the original 45% recovery rate, and to show the calculation that supports their answer.

ELL supports

Pre-teach "reverse osmosis," "permeate," "concentrate," and "recovery rate" with a simple picture of seawater splitting into clean water and salty water. Allow problem 9 to be answered with the number plus a short phrase rather than a full sentence.