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Math problem set Grade 8 Careers & CTE

Otterbourne's Digester Tanks: Volume Math for a Biosolids Operator

A grade 8 problem set where students use cylinder, cone, and sphere volume formulas to check the capacity of a fictional digester, grit hopper, and biogas holder.

Objectives

  1. Students will calculate the volume of a cylindrical digester in cubic feet and convert it to gallons.
  2. Students will apply the cone and sphere volume formulas to a grit hopper and a gas holder.
  3. Students will compare two tank volumes and explain how changing a diameter changes a volume.

Vocabulary

Biosolids
Biosolids are the nutrient-rich solid material left over after sewage sludge has been treated and made safe. Many farms use treated biosolids as fertilizer for their fields.
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: digester, biogas, cylinder, cone, sphere

Teacher overview

An anaerobic digester is a heated, sealed tank where bacteria break down the solids removed from wastewater, producing biogas the plant can burn for heat and power. Biosolids operators run those digesters, and a routine part of the job is confirming how much a tank actually holds before deciding how much sludge to feed it. This set follows one operator at a fictional treatment plant through the three tank shapes that show up on a single walk across the site. The misconception it targets is substituting the diameter where a formula calls for the radius, which makes a cone volume four times too large and a sphere volume eight times too large; the second common slip is forgetting the one-third factor in the cone formula.

Otterbourne's Digester Tanks: Volume Math for a Biosolids Operator — Grade 8 Careers & CTE
Name: ____________________________ Date: ______________

Problems

Imagine the Otterbourne Water Resource Recovery Facility, a fictional plant serving a small city. An operator is checking three tanks. Use 3.14 for pi, and use the conversion that one cubic foot holds 7.48 gallons.

  • The digester is a cylinder 60 feet in diameter with 25 feet of side water depth. Cylindrical tank volume in cubic feet equals 0.785 x diameter squared x depth.
  • The grit hopper is a cone 8 feet across the top and 6 feet deep. Cone volume equals one-third x pi x radius squared x height.
  • The gas holder is a sphere 20 feet in diameter. Sphere volume equals four-thirds x pi x radius cubed.
  1. Find the volume of the digester in cubic feet.
  2. Convert the digester volume from problem 1 to gallons.
  3. The plant feeds 26,400 gallons of thickened sludge into the digester each day. Using detention time equals volume divided by flow, how many days does an average gallon stay in the digester? Round to the nearest whole day.
  4. Find the volume of the grit hopper in cubic feet. Round to the nearest whole cubic foot.
  5. Convert the grit hopper volume from problem 4 to gallons, using the unrounded volume from before you rounded. Round to the nearest whole gallon.
  6. Find the volume of the gas holder in cubic feet. Round to the nearest whole cubic foot.
  7. The digester produces 12,000 cubic feet of biogas per day. How many hours does the plant need to fill the empty gas holder at that rate? Round to the nearest tenth of an hour.
  8. How many times as large as the gas holder is the digester? Round to the nearest tenth.
  9. The plant's engineer proposes a second digester with the same 25-foot depth but a 120-foot diameter. Without computing the new volume, explain whether it holds twice as much as the first digester, more than twice, or less than twice, and why.

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.
    Volume = 0.785 x 60 x 60 x 25. First, 60 x 60 = 3,600. Then 0.785 x 3,600 = 2,826. Finally 2,826 x 25 = 70,650 cubic feet.
  2. Answer: 528,462 gallons.
    70,650 cubic feet x 7.48 gallons per cubic foot = 528,462 gallons.
  3. Answer: 20 days.
    Detention time = 528,462 gallons divided by 26,400 gallons per day = 20.02 days, which rounds to 20 days.
  4. Answer: 100 cubic feet.
    The radius is half of 8 feet, or 4 feet. Volume = (1/3) x 3.14 x 4 x 4 x 6. First, 4 x 4 = 16. Then 3.14 x 16 = 50.24, and 50.24 x 6 = 301.44. Finally 301.44 divided by 3 = 100.48, which rounds to 100 cubic feet.
  5. Answer: 752 gallons.
    Using the unrounded volume, 100.48 cubic feet x 7.48 gallons per cubic foot = 751.59 gallons, which rounds to 752 gallons.
  6. Answer: 4,187 cubic feet.
    The radius is half of 20 feet, or 10 feet. Volume = (4/3) x 3.14 x 10 x 10 x 10. First, 10 x 10 x 10 = 1,000. Then 3.14 x 1,000 = 3,140, and 3,140 x 4 = 12,560. Finally 12,560 divided by 3 = 4,186.67, which rounds to 4,187 cubic feet.
  7. Answer: 8.4 hours.
    4,186.67 cubic feet divided by 12,000 cubic feet per day = 0.3489 day. Then 0.3489 x 24 hours = 8.37 hours, which rounds to 8.4 hours.
  8. Answer: 16.9 times.
    70,650 divided by 4,186.67 = 16.88, which rounds to 16.9 times as large.
  9. Answer: More than twice as much, because volume grows with the square of the diameter.
    Doubling the diameter from 60 feet to 120 feet doubles the radius, and the 0.785 x diameter squared term multiplies the volume by 2 x 2 = 4 when the depth stays the same. The second digester would hold about four times the first one, not twice.

Differentiation

Support

Give students a labeled sketch of each tank in this fictional scenario with the radius already marked next to the diameter, so the halving step is visible rather than remembered. Provide problems 1 through 3 as a guided example worked line by line, then have students start independently at problem 4.

Extension

Ask students to find the side water depth that would give the invented digester exactly 600,000 gallons of capacity at the same 60-foot diameter, then check the answer by recomputing the volume from that depth.

ELL supports

Post the three volume formulas with the words "diameter," "radius," "depth," and "height" labeled on matching diagrams, and let students point to the matching part of the diagram before substituting. Allow problem 9 to be answered with a labeled sketch and the phrase "four times" rather than a written paragraph.

Extensions

Have students calculate how many days of biogas production the gas holder in this invented scenario can store if the plant burns gas at a steady rate you supply, then discuss why a plant sizes storage for a night's production rather than a week's. Students interested in the career can compare the license classes their own state requires for operating a plant with anaerobic digestion.