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Worksheet Grade 9 Math

Farnhollow Digester Biogas: Volume, Detention Time, and Energy

A grade 9 worksheet on a wastewater plant's anaerobic digester: cylinder volume, detention time, solids loading, and the electricity its biogas engine makes.

Objectives

  1. Students will calculate a cylindrical digester's volume in cubic feet and in gallons.
  2. Students will find detention time, solids loading, and removal efficiency from given plant data.
  3. Students will convert a daily biogas volume into a yearly electricity total and a dollar value.

Vocabulary

Anaerobic digester
An anaerobic digester is a sealed tank where bacteria break down sewage sludge without any oxygen, shrinking its volume and producing methane gas as a byproduct. Some treatment plants capture that gas and burn it to generate their own electricity or heat.
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: biogas, solids loading, removal efficiency

Teacher overview

An anaerobic digester is a heated, sealed tank where bacteria break down the solids removed from wastewater. The bacteria give off biogas, and many plants burn that gas in an engine to make part of their own electricity. This worksheet works through one fictional digester from three angles: its geometry, the time sludge spends inside it, and the energy its gas produces. The calculations are the same ones a plant operator runs, and every figure in the worksheet describes the fictional Farnhollow facility rather than any real plant. Two errors are worth watching for while students work. The first is inverting the detention-time division, dividing the daily flow by the volume rather than the volume by the flow, which yields a small decimal that students rarely question. The second is entering the 4.2 percent solids as 4.2 rather than 0.042 in question 4, which inflates the loading by a factor of one hundred.

Farnhollow Digester Biogas: Volume, Detention Time, and Energy — Grade 9 Math
Name: ____________________________ Date: ______________

Student Worksheet

All figures below describe the fictional Farnhollow Water Reclamation Facility. Suppose its anaerobic digester is a cylinder 60 feet across, holding sludge to a depth of 25 feet. The plant feeds the digester 26,000 gallons of sludge per day, and that sludge is 4.2 percent solids. The digester makes 12,000 cubic feet of biogas per day. An engine turns every 100 cubic feet of biogas into 1.8 kWh of electricity. The whole plant uses 1,200 kWh per day, and electricity costs $0.12 per kWh in this scenario. The wastewater entering the plant has a BOD of 240 mg/L, and the treated water leaving has a BOD of 25 mg/L.

  1. Find the volume of the digester in cubic feet. Use volume = 0.785 x diameter squared x depth.
  2. Convert your answer from question 1 to gallons. One cubic foot holds 7.48 gallons.
  3. Find the detention time in days. Detention time is the volume in gallons divided by the daily flow in gallons.
  4. Find the pounds of solids fed to the digester each day. Multiply 26,000 gallons by 8.34 pounds per gallon, then by 0.042.
  5. Find the kilowatt-hours the engine makes each day from 12,000 cubic feet of biogas.
  6. Find the kilowatt-hours the engine makes in a 365-day year.
  7. Find the value of one year of that electricity at $0.12 per kWh.
  8. Find what percent of the plant's daily electricity use of 1,200 kWh the engine covers.
  9. Find the BOD removal efficiency as a percent. Use (influent minus effluent) divided by influent, times 100.
  10. Secondary treatment rules set a monthly average effluent BOD of 30 mg/L or less. Does the plant's 25 mg/L effluent meet that limit? Answer yes or no.
  11. Explain in two or three sentences why raising the sludge depth in the same tank would increase the detention time.

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: 70,650 cubic feet.
    Substituting gives 0.785 x 60 x 60 x 25. The diameter squared is 3,600, so 0.785 x 3,600 = 2,826 square feet of surface, and 2,826 x 25 = 70,650 cubic feet.

  2. Answer: 528,462 gallons.
    Multiplying gives 70,650 cubic feet x 7.48 gallons per cubic foot = 528,462 gallons.

  3. Answer: 20.3 days.
    Dividing gives 528,462 gallons divided by 26,000 gallons per day = 20.33 days, which rounds to 20.3 days. The gallons cancel and leave days, which is the check that the division went the right way.

  4. Answer: 9,107 pounds per day.
    Multiplying gives 26,000 x 8.34 = 216,840 pounds of sludge per day, and 216,840 x 0.042 = 9,107.28 pounds of solids per day, which rounds to 9,107.

  5. Answer: 216 kWh per day.
    The gas amounts to 12,000 divided by 100 = 120 units of 100 cubic feet, and 120 x 1.8 = 216 kWh per day.

  6. Answer: 78,840 kWh per year.
    Multiplying gives 216 kWh per day x 365 days = 78,840 kWh in a year.

  7. Answer: $9,460.80.
    Multiplying gives 78,840 kWh x $0.12 per kWh = $9,460.80 for the year.

  8. Answer: 18 percent.
    Dividing gives 216 divided by 1,200 = 0.18, and 0.18 x 100 = 18 percent of the plant's daily electricity use.

  9. Answer: 89.6 percent.
    Substituting gives (240 - 25) divided by 240 = 215 divided by 240 = 0.8958, and 0.8958 x 100 = 89.58 percent, which rounds to 89.6 percent.

  10. Answer: Yes.
    The effluent BOD of 25 mg/L is below the 30 mg/L monthly average limit, so this month's result meets the standard.

  11. Answer: Detention time rises because the tank holds more sludge while the daily feed stays the same.
    Depth is a factor in the volume formula, so a greater depth gives a larger volume in cubic feet and therefore more gallons. Detention time is that larger volume divided by the same 26,000 gallons per day, and a larger numerator over the same denominator gives a longer time.

Differentiation

Support

Give these students the digester volume from question 1 already computed, so they begin at the conversion in question 2, and provide a printed conversion strip listing 7.48 gallons per cubic foot and 8.34 pounds per gallon. Working questions 5 through 7 as a guided sequence with the teacher before independent work keeps the energy chain from stacking up all at once.

Extension

Ask students to find the diameter a second fictional digester would need to hold twice the volume of the Farnhollow tank at the same 25-foot depth, then explain why doubling the diameter does not double the volume. A follow-up question is how many days of the plant's electricity the engine's yearly output would cover if it ran only during the six warmest months.

ELL supports

Pre-teach "digester," "detention time," "influent," and "effluent" with a simple labeled cross-section showing sludge going in on one side and gas leaving the top. Provide a worked example of the volume formula with different numbers than the ones on this worksheet, and give a sentence frame for question 11.

Extensions

Students can compare the fictional Farnhollow engine's share of plant electricity with the share reported by a real utility in its published annual report or sustainability summary. A second task is to calculate how many years of the engine's electricity savings it would take to pay back an installation cost the class chooses and states as an assumption.