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Math problem set Grade 9 Math

Perrin Bay Desalination: RO Recovery Rate and Energy Set

A grade 9 problem set on Perrin Bay's seawater reverse osmosis plant: recovery rate, concentrate flow, and the energy each thousand gallons of drinking water costs.

Objectives

  1. Students will calculate recovery rate and concentrate flow for a reverse osmosis plant.
  2. Students will find energy use per thousand gallons and compare it to conventional treatment.
  3. Students will choose appropriate quantities to describe how a desalination plant performs.

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: recovery rate, permeate, concentrate, kilowatt-hour

Teacher overview

The fictional Perrin Bay desalination plant pushes seawater through reverse osmosis membranes to make drinking water. Only part of the feed becomes clean product; the rest leaves as salty concentrate. Desalination also uses more energy than ordinary treatment, so operators watch the energy each thousand gallons costs. This set has students choose the right quantities, recovery rate, concentrate flow, and energy per thousand gallons, to describe how the plant performs. A common misconception is confusing recovery rate (product divided by feed) with the concentrate left over, or forgetting to divide gallons by 1,000 before multiplying by an energy rate.

Perrin Bay Desalination: RO Recovery Rate and Energy Set — Grade 9 Math
Name: ____________________________ Date: ______________

Problems

Perrin Bay (a fictional plant) takes in 10 MGD of seawater. Its recovery rate is 45 percent, so 45 percent of the feed becomes clean product, called permeate.

  1. Find the permeate flow in MGD. Multiply the 10 MGD feed by the 45 percent recovery rate.
  2. Find the concentrate flow. Subtract the permeate from the feed.
  3. Convert the 4.5 MGD of permeate to gallons per minute, using 1 MGD equal to 694 gpm.
  4. Seawater reverse osmosis uses about 13 kWh for every 1,000 gallons. The plant makes 4.5 million gallons of permeate per day. Find the daily energy use in kWh.
  5. Ordinary treatment uses about 2 kWh per 1,000 gallons. Find the daily energy to treat the same 4.5 million gallons, then say how many times more energy reverse osmosis uses.
  6. Energy per 1,000 gallons is a useful quantity for comparing plants. If Perrin Bay used 54,000 kWh to make 4.5 million gallons, find its energy rate in kWh per 1,000 gallons.
  7. To raise the permeate to 5 MGD while holding recovery at 45 percent, what feed flow is needed? Divide the target permeate by the recovery rate.
  8. Carlsbad, California runs the nation's largest seawater plant at about 50 MGD. About how many Perrin Bay plants, each making 4.5 MGD of permeate, would equal Carlsbad's output?

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: 4.5 MGD. 0.45 x 10 MGD = 4.5 MGD of permeate.
  2. Answer: 5.5 MGD. 10 MGD feed - 4.5 MGD permeate = 5.5 MGD of concentrate.
  3. Answer: 3,123 gpm. 4.5 MGD x 694 gpm per MGD = 3,123 gpm.
  4. Answer: 58,500 kWh. 4.5 million gallons is 4,500 units of 1,000 gallons; 4,500 x 13 kWh = 58,500 kWh per day.
  5. Answer: 6.5 times. 4,500 x 2 kWh = 9,000 kWh per day; 58,500 / 9,000 = 6.5 times as much energy.
  6. Answer: 12 kWh. 54,000 kWh / 4,500 units of 1,000 gallons = 12 kWh per 1,000 gallons.
  7. Answer: 11.1 MGD. 5 MGD / 0.45 = 11.11 MGD of feed, about 11.1 MGD.
  8. Answer: 11 plants. 50 MGD / 4.5 MGD = 11.11, so about 11 Perrin Bay plants would match Carlsbad's output. Also accept 12, since 11 plants make 49.5 MGD and 12 are needed to fully reach 50 MGD.

Differentiation

Support

Provide the recovery-rate definition (permeate divided by feed) and the conversion "1 MGD equals 694 gallons per minute" on a card. Work one energy example with numbers not in this set, showing the step of dividing gallons by 1,000 first, and allow a calculator.

Extension

Ask students to find Perrin Bay's concentrate flow if the recovery rate improves to 50 percent, and to explain what happens to the amount of salty brine as recovery goes up.

ELL supports

Pre-teach "feed," "permeate," "concentrate," and "recovery" with a diagram showing a seawater arrow splitting into a clean stream and a salty stream. Provide a sentence frame for problem 5.