Worksheet Grade 12 Math

Ospreyfield Digester Gas: Volatile Solids to Kilowatt-Hours

A grade 12 worksheet tracing a fictional plant's sludge from volatile solids destroyed to biogas, energy density in BTU per cubic foot, and generated electricity.

Grade: Grade 12
Subject: Math
Material: Worksheet
Time: 50 min
Prep: No prep
Materials: Calculator

Standards

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 4 and 5 apply an energy density in BTU per cubic foot of biogas, the exact volume-based density modeling this standard names.

All materials for HSG-MG.A.2 →

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: Items 1 through 6 chain pounds per day, cubic feet per day, BTU per day, and kilowatt-hours per day, with units guiding every conversion.

All materials for HSN-Q.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 volatile solids destroyed from a stated feed rate and destruction percentage.
Students will apply an energy density in BTU per cubic foot to a daily biogas volume.
Students will convert fuel energy into generated electricity and compare it with plant demand.

Vocabulary

Detention time: Detention time is how long water stays inside a tank or basin before flowing out. Water must stay long enough for each treatment step, like settling or disinfecting, to finish its job.

Also: volatile solids, anaerobic digestion, biogas, energy density, cogeneration

Teacher overview

An anaerobic digester is a heated tank where bacteria convert part of a plant's sludge into methane, and a plant with an engine on that gas line can run a share of its own load on it. This worksheet follows one day of sludge through the fictional Ospreyfield Water Reclamation Facility, a 3.2 MGD plant, from pounds of volatile solids fed to kilowatt-hours generated. The common error at this grade is multiplying the full solids load by the gas yield, when only the volatile fraction that is actually destroyed produces gas. A second error is treating biogas as pure methane, which overstates its energy density by more than half.

Ospreyfield Digester Gas: Volatile Solids to Kilowatt-Hours — Grade 12 Math

Name: ____________________________ Date: ______________

Student Worksheet

All values below describe the fictional Ospreyfield Water Reclamation Facility.

Digester data for one day:

Sludge fed to the digester: 8,400 pounds of total solids per day, at 26,000 gallons per day.
Volatile solids: 72 percent of the total solids fed.
Volatile solids destruction: 58 percent of the volatile solids fed.
Gas yield: 15 cubic feet of biogas per pound of volatile solids destroyed.
Biogas composition: 63 percent methane, with the balance carbon dioxide, which carries no fuel value.
Methane energy content: 1,012 BTU per cubic foot.
Engine set: converts 32 percent of the fuel energy it burns into electricity, at 3,412 BTU per kilowatt-hour.
Plant electrical demand: 14,800 kilowatt-hours per day.
Purchased electricity price: $0.11 per kilowatt-hour.
Digester tank: a cylinder 60 feet in diameter with 25 feet of side water depth. Use volume (cu ft) = 0.785 x diameter squared x depth, and 1 cubic foot = 7.48 gallons.

How many pounds of volatile solids does the digester receive per day?
How many pounds of volatile solids are destroyed per day?
How many cubic feet of biogas does the digester produce per day? Round to the nearest 10 cubic feet.
Carbon dioxide carries no fuel value, so only the methane fraction supplies energy. Find the energy density of this biogas in BTU per cubic foot, rounded to the nearest whole BTU.
Using your item 3 and item 4 answers, find the total fuel energy in the day's biogas. Report it in million BTU per day, rounded to the nearest tenth.
How many kilowatt-hours of electricity does the engine set generate per day? Round to the nearest kilowatt-hour.
What percent of the plant's 14,800 kilowatt-hour daily demand does the item 6 answer cover? Round to the nearest tenth of a percent.
At $0.11 per kilowatt-hour of purchased electricity avoided, what is one day of generation worth?
What is a full 365-day year of that generation worth, to the nearest dollar?
How many pounds of total solids leave the digester each day, assuming the destroyed volatile solids are the only mass lost?
What percent of the total solids mass fed to the digester is destroyed? Round to the nearest tenth of a percent.
Find the digester's volume in gallons.
Find the digester's detention time in days, where detention time = volume divided by the daily flow into the tank. Round to the nearest tenth of a day.
A new operator suggests raising gas production by thickening the sludge, so each gallon fed to the digester carries more total solids. Explain in two or three sentences why total solids alone does not determine gas production, and name the two quantities that do.

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: 6,048 lb/day.
8,400 pounds of total solids x 0.72 = 6,048 pounds of volatile solids per day.
Answer: 3,508 lb/day.
6,048 x 0.58 = 3,507.84, which rounds to 3,508 pounds of volatile solids destroyed per day.
Answer: 52,620 cu ft/day.
3,508 pounds destroyed x 15 cubic feet per pound = 52,620 cubic feet of biogas per day. Carrying the unrounded 3,507.84 pounds gives 52,617.6, which rounds to the same 52,620 cubic feet at this precision.
Answer: 638 BTU/cu ft.
Only 63 percent of each cubic foot is methane: 0.63 x 1,012 BTU = 637.56, which rounds to 638 BTU per cubic foot. Treating the gas as pure methane would claim 1,012 BTU per cubic foot, overstating the fuel by about 59 percent.
Answer: 33.6 million BTU/day.
52,620 cubic feet x 638 BTU per cubic foot = 33,571,560 BTU per day, or 33.6 million BTU per day.
Answer: 3,149 kWh/day.
Energy to electricity: 33,571,560 x 0.32 = 10,742,899 BTU per day. Convert: 10,742,899 / 3,412 BTU per kilowatt-hour = 3,148.6, which rounds to 3,149 kilowatt-hours per day.
Answer: 21.3 percent.
3,149 / 14,800 = 0.2128, or 21.3 percent of the plant's daily electrical demand.
Answer: $346.39.
3,149 kilowatt-hours x $0.11 = $346.39 of purchased power avoided in one day.
Answer: $126,432.
$346.39 x 365 days = $126,432.35, which rounds to $126,432 per year. That figure is what the utility can weigh against the cost of maintaining the engine set.
Answer: 4,892 lb/day.
8,400 pounds fed - 3,508 pounds destroyed = 4,892 pounds of total solids leaving the digester each day, which is the mass that goes on to dewatering.
Answer: 41.8 percent.
3,508 / 8,400 = 0.4176, or 41.8 percent of the total solids mass. Note that this is well below the 58 percent destruction figure, because only the volatile fraction was ever available to destroy.
Answer: 528,462 gal.
Volume = 0.785 x 60 x 60 x 25 = 70,650 cubic feet. In gallons: 70,650 x 7.48 = 528,462 gallons.
Answer: 20.3 days.
Detention time = 528,462 gallons / 26,000 gallons per day = 20.32, which rounds to 20.3 days. That sits in the normal range for a heated digester, so the tank is not the limiting factor here.
Answer: volatile solids fed and percent destroyed.
Gas comes only from the volatile portion of the solids, and only from the share of that portion the bacteria actually break down. Thicker sludge with the same volatile content and the same destruction rate delivers the same pounds destroyed per day at a lower flow, so it raises detention time rather than gas production. Raising output requires either more volatile solids fed or a higher destruction percentage.

Differentiation

Support

Provide a printed flow diagram of the fictional Ospreyfield digester with five labeled boxes reading total solids, volatile solids, volatile solids destroyed, cubic feet of gas, and kilowatt-hours, and have students write each computed value into its box as they work items 1 through 6. Give the item 4 setup as 0.63 x 1,012 already written out.

Extension

Have students find the break-even purchased-power price at which one year of generation at the fictional plant would cover a $95,000 annual engine maintenance contract, then state what would happen to that break-even price if volatile solids destruction fell from 58 percent to 50 percent.

ELL supports

Pre-teach "volatile," "destruction," "yield," and "detention time" with a labeled cross-section of a digester showing sludge in, gas out, and solids out. Provide a units-tracking column beside items 1 through 6 with the unit for each step already printed, so students confirm that pounds become cubic feet and cubic feet become BTU. Offer sentence frames for item 14.

Extensions

Have students redo items 1 through 6 for the fictional plant under a co-digestion scenario in which a food-waste hauler delivers an additional 1,900 pounds of volatile solids per day at the same 58 percent destruction and the same gas yield, then report the new percent of plant demand covered. Ask students to identify which item in the original chain that added feed does not change, and to explain why the detention time answer from item 13 falls when the added material arrives at 3,000 gallons per day.