Math problem set Grade 10 Math

Halden Bay Desalter: Membrane Flux and Salt Rejection

A grade 10 problem set on reverse osmosis: students find membrane flux, count elements and vessels, and compute salt rejection at a fictional brackish desalter.

Grade: Grade 10
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 through 5 carry MGD, gallons per day, and gallons per square foot per day through the flux relationship, holding units consistent as this standard requires.

All materials for HSN-Q.A.1 →

HSA-CED.A.1 Common Core Math

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

How this material aligns: Problem 4 rearranges the flux equation into area = flow divided by flux, a one-variable equation students solve to size the membrane surface.

All materials for HSA-CED.A.1 →

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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 membrane flux and use it to size a reverse osmosis system.
Students will compute salt rejection and distinguish it from water recovery.

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: flux, salt rejection, membrane element

Teacher overview

A reverse osmosis desalter pushes salty water through membranes that let water pass but hold back most of the salt. Two numbers describe how it runs: flux, the gallons that pass through each square foot of membrane per day, and salt rejection, the percent of salt kept out of the product. This set uses the fictional city of Halden and its brackish desalter. A common misconception this set corrects is treating salt rejection as if it were water recovery: a 97 percent rejection describes water quality, not the share of the feed that becomes drinking water.

Halden Bay Desalter: Membrane Flux and Salt Rejection — Grade 10 Math

Name: ____________________________ Date: ______________

Problems

The city of Halden runs a brackish reverse osmosis desalter. This scenario is fictional, and its numbers are set for this problem set. Flux is defined as permeate flow (gallons per day) divided by membrane area (square feet), reported in gallons per square foot per day.

Plant data:

Permeate (product) flow: 1.44 MGD
Feed salt content: 3,200 mg/L; permeate salt content: 96 mg/L
Each membrane element has 400 square feet of area
A pressure vessel holds 6 elements
Design flux target: 12 gallons per square foot per day

Convert the permeate flow of 1.44 MGD to gallons per day.
Convert 1.44 MGD to gpm, using 1 MGD = 694 gpm.
Find the salt rejection as a percent. Use rejection (%) = (feed - permeate) / feed x 100.
At the design flux of 12 gallons per square foot per day, find the membrane area needed, in square feet.
How many membrane elements are needed at 400 square feet each?
How many pressure vessels are needed if each holds 6 elements?
The plant actually installs 320 elements. Find the true flux at that installed area.
A classmate says a 97 percent salt rejection means 97 percent of the feed water becomes product. Explain why rejection and recovery are different measures.

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: 1,440,000 gpd.
1.44 MGD x 1,000,000 gallons per MG = 1,440,000 gallons per day.
Answer: 999 gpm.
1.44 MGD x 694 gpm per MGD = 999.36, which rounds to 999 gpm.
Answer: 97.0 percent.
(3,200 - 96) / 3,200 x 100 = 3,104 / 3,200 x 100 = 97.0 percent.
Answer: 120,000 sq ft.
Area = flow / flux = 1,440,000 gallons per day / 12 gallons per square foot per day = 120,000 square feet.
Answer: 300 elements.
120,000 square feet / 400 square feet per element = 300 elements.
Answer: 50 vessels.
300 elements / 6 elements per vessel = 50 pressure vessels.
Answer: 11.25 gfd.
Installed area is 320 x 400 = 128,000 square feet. Flux = 1,440,000 / 128,000 = 11.25 gallons per square foot per day, a little below the 12 target because more area shares the same flow.
Answer: Rejection measures salt kept out; recovery measures the share of feed turned into product.
Salt rejection is a quality number found from the salt levels. Recovery is a flow number that compares product water to feed water. A plant can reject 97 percent of the salt while recovering far less than 97 percent of the water, because some feed leaves as concentrate.

Differentiation

Support

Give students the flux definition already rearranged three ways: flux = flow / area, area = flow / flux, and flow = flux x area. Work problem 1 together so the million-to-one conversion from MGD is clear before problem 4.

Extension

Tell students the concentrate (reject) stream is a fictional 0.36 MGD. Have them find the feed flow, then compute recovery as product divided by feed, and compare that recovery percent to the 97 percent salt rejection to make the difference concrete.

ELL supports

Pre-teach "membrane," "flux," and "rejection" with a labeled sketch of water passing through a membrane while salt stays behind. Provide a sentence frame for problem 8: "Rejection and recovery differ because ___."