Worksheet Grade 12 Science

Filter Loading and Backwash Rates at the Marrowbrook Plant

A grade 12 science worksheet using operator math to find filter loading and backwash rates at a fictional plant and check them against the turbidity standard.

Grade: Grade 12
Subject: Science
Material: Worksheet
Time: 40 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: Students carry units through the flow, area, and loading-rate conversions in items 1 through 10, the multi-step dimensional analysis this standard requires of operator-style math.

All materials for HSN-Q.A.1 →

HSG-MG.A.2 Common Core Math

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

How this material aligns: Items 3, 5, 7, and 8 treat filter loading rate as a density, gallons per minute spread over square feet of filter surface, the density-modeling skill this standard names.

All materials for HSG-MG.A.2 →

HS-ESS3-4 NGSS

Evaluate or refine a technological solution that reduces impacts of human activities on natural systems.

How this material aligns: Items 4, 8, and 12 evaluate the plant's filters against the 2 to 6 gpm per square foot design range and tie loading rate to holding finished water at or below 0.3 NTU.

All materials for HS-ESS3-4 →

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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 filter loading rates in gallons per minute per square foot from plant flow and filter area.
Students will determine how taking a filter offline for backwash changes the loading on the remaining filters.
Students will relate filter loading rate to the turbidity standard that finished water must meet.

Vocabulary

Backwash: Backwash is when a treatment plant reverses the flow of water through a filter to flush out the dirt and particles it has trapped. It is like rinsing out a dirty sponge so the filter can keep working well.
Turbidity: Turbidity is a measure of how cloudy or murky water looks because of tiny particles floating in it. Clear water has low turbidity, and muddy water has high turbidity.

Also: filter loading rate, surface area, rapid sand filter

Teacher overview

The fictional town of Marrowbrook runs a conventional surface-water plant with four identical rapid sand filters. This worksheet has students calculate the filter loading rate, in gallons per minute per square foot of filter surface, at normal and peak flows, and see what happens to that rate when one filter drops offline for backwashing. A common mistake in this math is dividing the whole plant flow by a single filter's area. The flow splits across every filter that is in service, so the correct area is the combined area of the filters actually running, not one filter and not always all four.

Loading rate matters because finished water must stay at or below 0.3 NTU in 95 percent of monthly samples. Pushing filters past their design loading tends to let more turbidity slip through, so items 11 and 12 ask students to connect the arithmetic to that standard.

Filter Loading and Backwash Rates at the Marrowbrook Plant — Grade 12 Science

Name: ____________________________ Date: ______________

Student Worksheet

Marrowbrook (a fictional town used for this worksheet) treats a normal flow of 3.6 MGD through four identical rapid sand filters. Each filter measures 20 feet by 15 feet. Conventional filters are typically designed to run at 2 to 6 gpm/sq ft, and the plant's finished water must stay at or below 0.3 NTU in 95 percent of monthly samples. Use these formulas and conversions for every item.

Filter loading rate = flow through the filters (gpm) / filter surface area (sq ft)
Filter surface area = length x width
Conversion: 1 MGD = 694 gpm

Convert the plant's normal flow of 3.6 MGD to gallons per minute. Round to the nearest whole gpm.
Calculate the total filter surface area, in square feet, with all four filters in service.
With all four filters in service, calculate the filter loading rate in gpm per square foot. Round to the nearest hundredth.
Is the loading rate from item 3 within the typical design range of 2 to 6 gpm/sq ft? Answer yes or no.
One filter is taken offline for backwashing, leaving three in service to carry the full 3.6 MGD flow. Calculate the new loading rate on the three remaining filters. Round to the nearest hundredth.
During a summer peak, flow rises to 5.0 MGD. Convert this peak flow to gallons per minute.
At the 5.0 MGD peak with all four filters in service, calculate the loading rate. Round to the nearest hundredth.
At the 5.0 MGD peak with one filter offline for backwashing (three in service), calculate the loading rate. Round to the nearest hundredth.
Each filter is backwashed at a rate of 15 gpm/sq ft. Calculate the backwash flow, in gpm, needed for one 20-by-15-foot filter.
The backwash in item 9 runs for 10 minutes. Calculate the total volume of water used in one backwash, in gallons.
Explain why taking one filter offline for backwashing raises the loading rate on the filters that stay in service.
Using your peak-flow answers, explain why higher loading rates, and especially rates above the design range, can make it harder to keep finished water at or below 0.3 NTU.

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: 2,498 gpm.
3.6 MGD x 694 gpm per MGD = 2,498.4 gpm, which rounds to 2,498 gpm.
Answer: 1,200 sq ft.
Each filter is 20 ft x 15 ft = 300 sq ft. Four filters: 4 x 300 = 1,200 sq ft.
Answer: 2.08 gpm/sq ft.
Loading = flow / area = 2,498.4 gpm / 1,200 sq ft = 2.082, which rounds to 2.08 gpm/sq ft.
Answer: Yes.
2.08 gpm/sq ft falls inside the 2 to 6 gpm/sq ft design range, near the low end.
Answer: 2.78 gpm/sq ft.
Three filters give 3 x 300 = 900 sq ft. Loading = 2,498.4 gpm / 900 sq ft = 2.776, which rounds to 2.78 gpm/sq ft.
Answer: 3,470 gpm.
5.0 MGD x 694 gpm per MGD = 3,470 gpm.
Answer: 2.89 gpm/sq ft.
Loading = 3,470 gpm / 1,200 sq ft = 2.892, which rounds to 2.89 gpm/sq ft.
Answer: 3.86 gpm/sq ft.
Loading = 3,470 gpm / 900 sq ft = 3.856, which rounds to 3.86 gpm/sq ft. This is still inside the 2 to 6 gpm/sq ft range but noticeably higher than normal.
Answer: 4,500 gpm.
Backwash flow = 15 gpm/sq ft x 300 sq ft = 4,500 gpm for one filter.
Answer: 45,000 gal.
4,500 gpm x 10 minutes = 45,000 gallons used in one backwash.
Answer: The same flow is carried by less filter area.
When a filter is pulled for backwashing, the plant's flow does not stop. It redistributes over the remaining filters, so each square foot of the filters still running handles more gallons per minute, which raises the loading rate.
Answer: Higher loading pushes water through faster, carrying more turbidity.
At the peak-flow loading rates in items 7 and 8, water moves through the sand faster and spends less time being filtered, so more fine particles can pass through. That makes it harder to hold finished water at or below 0.3 NTU, which is why operators watch loading closely during peaks and backwashes.

Differentiation

Support (below grade level)

Provide the loading-rate formula and the 1 MGD = 694 gpm conversion on a printed card, and work item 1 and item 2 together as a class before students continue. Accept unrounded decimal answers alongside the rounded values.

Extension (above grade level)

Ask students to imagine a peak at which three in-service filters (900 sq ft) would run at 6 gpm/sq ft, the top of the design range, and find the plant flow in MGD that produces it. They should set up and solve the loading equation for the unknown flow rather than guess and check.

ELL supports

Pre-teach "loading rate," "backwash," and "surface area" with a labeled diagram of a rectangular filter and an arrow showing water spreading across its top. Allow students to answer items 11 and 12 with a labeled sketch or a short bulleted list instead of full sentences.

Extensions

Invite students to look up how many filters their local plant operates, or to write a short question for the utility asking how the plant keeps loading rates in range during a backwash. Comparing a real plant's filter count with Marrowbrook's four filters turns the arithmetic into a question about their own water system.