Worksheet Grade 10 Math
Alderpoint Filter Loading Rate and Backwash Worksheet
A grade 10 worksheet on filter loading rates at a fictional treatment plant, where students find rectangular filter area, convert MGD to gpm, and size backwash flow.
Objectives
- Students will calculate the surface area of a rectangular filter and its loading rate in gallons per minute per square foot.
- Students will convert a plant flow from MGD to gpm and reason about loading limits during backwash.
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.
Also: filter loading rate, surface area, MGD
Teacher overview
The fictional Alderpoint Water Treatment Plant filters its water through two rectangular rapid sand filters before disinfection. This worksheet has students find each filter's surface area, convert the plant's flow to gallons per minute, and compute the filter loading rate that operators watch to keep filters working well. The same numbers show up again when one filter is taken offline for backwash, which raises the load on the filter left in service.
Student Worksheet
The Alderpoint Water Treatment Plant (a fictional plant used for this worksheet) runs two identical rectangular filters. Each filter box measures 20 feet long by 15 feet wide. Use these facts where needed: 1 MGD equals 694 gpm, and conventional filters typically run at a loading rate of 2 to 6 gpm per square foot.
Find the surface area of one filter box. Use area = length × width.
Both filters are in service at the same time. What is their combined surface area?
The plant is treating 2.0 MGD. Convert this flow to gallons per minute using 1 MGD = 694 gpm.
With both filters in service, the full plant flow is split across the combined area. Find the loading rate in gpm per square foot. Use loading rate = flow (gpm) / total filter area (sq ft). Round to the nearest hundredth.
Operators take one filter offline to clean it, so the full plant flow now passes through the single remaining 300-square-foot filter. Find the new loading rate in gpm per square foot. Round to the nearest hundredth.
Is the single-filter loading rate from item 5 still inside the typical range of 2 to 6 gpm per square foot? Answer yes or no and explain in one sentence.
To clean the offline filter, the crew runs a backwash at 15 gpm per square foot across the 300-square-foot filter. Find the backwash flow in gpm. Use backwash flow = backwash rate × filter area.
The backwash runs for 10 minutes. How many gallons of water does one backwash use? Use gallons = flow (gpm) × time (min).
Explain in two or three sentences why taking one filter offline raises the loading rate on the filter that stays in service.
The plant wants the loading on a single in-service filter to stay at or below 6 gpm per square foot while the other filter is offline. What is the largest plant flow, in gpm, that one 300-square-foot filter can carry at that limit? Compare it to the 2.0 MGD flow from item 3.
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: 300 sq ft. Area = 20 ft × 15 ft = 300 square feet for one filter box.
Answer: 600 sq ft. Two filters at 300 sq ft each give 300 × 2 = 600 square feet combined.
Answer: 1,388 gpm. 2.0 MGD × 694 gpm per MGD = 1,388 gpm.
Answer: 2.31 gpm per sq ft. Loading rate = 1,388 gpm / 600 sq ft = 2.313, which rounds to 2.31 gpm per square foot. This sits inside the 2 to 6 range.
Answer: 4.63 gpm per sq ft. With one filter offline, all 1,388 gpm passes through 300 sq ft: 1,388 / 300 = 4.627, which rounds to 4.63 gpm per square foot.
Answer: Yes. The value 4.63 is between 2 and 6, so the single filter still runs inside the typical conventional loading range.
Answer: 4,500 gpm. Backwash flow = 15 gpm per sq ft × 300 sq ft = 4,500 gpm.
Answer: 45,000 gal. Gallons = 4,500 gpm × 10 min = 45,000 gallons for one backwash.
Answer: The same plant flow must now pass through less filter area, because one filter is out of service. Loading rate is flow divided by area, so cutting the area in half while the flow stays the same doubles the loading rate on the remaining filter.
Answer: 1,800 gpm. At the 6 gpm per sq ft limit, maximum flow = 6 × 300 = 1,800 gpm. The current flow of 1,388 gpm (from item 3) is below 1,800 gpm, so a single filter can safely carry the whole plant flow during backwash.
Differentiation
Support
Give students the two filter areas (300 sq ft each, 600 sq ft combined) already computed so they can focus on the loading-rate division in items 4 and 5. Provide a printed strip listing "loading rate = flow / area" and "1 MGD = 694 gpm" so the conversions do not have to be recalled from memory.
Extension
Ask students to find the plant flow, in MGD, at which a single in-service filter at the fictional plant would exactly reach the 6 gpm per square foot ceiling, then explain what operating change the plant would need if demand ever pushed flow past that point.
ELL supports
Provide a labeled top-view diagram of a rectangular filter showing length, width, and the water spreading across the surface, so "surface area" and "loading rate" are visual before they are numeric. Allow item 9 to be answered with a labeled sketch and a short sentence rather than a full paragraph.
Extensions
Invite a local water utility to share the real dimensions and design flow of one of its filters, then repeat items 1 through 5 with those numbers. Students can compare their computed loading rate to the range an operator reports and discuss why plants keep a spare filter so one can be cleaned without shutting the plant down.