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Math problem set Grade 12 Careers & CTE

Kestrel Bay Desalination: Recovery and Energy Math

A grade 12 careers math set on a fictional reverse osmosis desalination plant, chaining recovery, flow, and energy math to find the daily cost of fresh water.

Objectives

  1. Students will calculate the permeate, concentrate, and recovery of a reverse osmosis plant.
  2. Students will chain unit conversions to find a desalination plant's daily energy use and cost.

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, recovery, concentrate

Teacher overview

A seawater desalination plant only turns part of the water it takes in into fresh drinking water. The share it recovers, called recovery, drives every later number, from how much energy the plant burns to what the water costs. This set walks students through one fictional plant so each answer feeds the next. A common error is using the full feed flow where only the recovered permeate belongs, for example computing energy on the seawater taken in rather than on the fresh water actually produced.

Kestrel Bay Desalination: Recovery and Energy Math — Grade 12 Careers & CTE
Name: ____________________________ Date: ______________

Problems

Kestrel Bay is a fictional seawater reverse osmosis desalination plant used for this problem set. All figures below are stated for this exercise. The plant takes in 12 MGD of seawater and recovers 45 percent of it as fresh permeate. Use these facts: recovery means the fresh permeate is 45 percent of the feed; one MGD equals 694 gallons per minute; the plant uses 13 kWh of energy for every 1,000 gallons of permeate; and electricity costs $0.14 per kWh.

  1. The plant takes in 12 MGD of seawater and recovers 45 percent as fresh permeate. How many MGD of permeate does it produce?
  2. The rest of the feed leaves as concentrate. How many MGD of concentrate does the plant discharge?
  3. Convert the permeate flow of 5.4 MGD to gallons per minute, using 1 MGD = 694 gpm. Round to the nearest whole gpm.
  4. The permeate flow is 5.4 MGD. How many gallons of fresh water is that per day?
  5. The plant uses 13 kWh for every 1,000 gallons of permeate. Using the 5,400,000 gallons per day from problem 4, how many kWh does the plant use per day?
  6. Electricity costs $0.14 per kWh. Using the 70,200 kWh per day from problem 5, what is the daily energy cost?
  7. Using the daily energy cost of $9,828 and the daily permeate of 5,400,000 gallons, find the energy cost per 1,000 gallons of fresh water, to the nearest cent.
  8. A nearby conventional plant makes fresh water using about 1.5 kWh per 1,000 gallons. How many times more energy per 1,000 gallons does Kestrel Bay use than the conventional plant? Round to the nearest tenth.

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: 5.4 MGD. Permeate = 12 MGD times 0.45 = 5.4 MGD.
  2. Answer: 6.6 MGD. Concentrate = 12 MGD minus 5.4 MGD = 6.6 MGD.
  3. Answer: 3,748 gpm. 5.4 MGD times 694 gpm per MGD = 3,747.6, which rounds to 3,748 gallons per minute.
  4. Answer: 5,400,000 gal. 5.4 MGD means 5.4 million gallons per day, which is 5,400,000 gallons per day.
  5. Answer: 70,200 kWh. 5,400,000 gallons divided by 1,000 is 5,400 units of 1,000 gallons; 5,400 times 13 kWh = 70,200 kWh per day.
  6. Answer: $9,828. 70,200 kWh times $0.14 per kWh = $9,828 per day.
  7. Answer: $1.82. $9,828 divided by 5,400 (the number of 1,000-gallon units) = $1.82 per 1,000 gallons, to the nearest cent.
  8. Answer: 8.7 times. 13 kWh divided by 1.5 kWh = 8.67, which rounds to about 8.7 times as much energy per 1,000 gallons.

Differentiation

Support

Provide a filled-in flow diagram showing feed, permeate, and concentrate, with the 45 percent recovery marked, and give students the value 5,400,000 gallons already converted so they can focus on the energy chain in problems 5 through 7.

Extension

Ask students to find what recovery percent the plant would need to reach in order to produce 6.0 MGD of permeate from the same fictional 12 MGD feed, and to explain what would happen to the concentrate flow at that higher recovery.

ELL supports

Pre-teach "permeate," "concentrate," and "recovery" with a labeled sketch of water splitting into a fresh stream and a salty stream. Allow students to show their work as a labeled chain of numbers and units rather than in full sentences.

Extensions

Have students look up the energy figure a real desalination plant reports per 1,000 gallons and compare it with Kestrel Bay's, then discuss why desalinated water usually costs more to produce than water from a river or a well. Public water authorities often publish these figures in their planning documents.