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10 FE practice problems: clarifier overflow rate and detention time, with solutions

By Dr. Mohamed Elansary, PhD Environmental EngineeringUpdated

These ten fill-in problems cover the clarifier design checks in the Water and Wastewater area of the FE Environmental specification: surface overflow rate, tank area and dimensions, and detention time. Fill-in questions have no options to lean on, so carry units through every line and round only at the end. Each solution lists the common wrong answers and the mistake behind each one.

How to use these problems

Give each problem an honest attempt before you open the solution: write the given values with units, find the equation in the FE Reference Handbook PDF, and commit to an answer. Then compare line by line. If you picked a wrong option, read the note for that option; each one names the mistake that produces it.

The problems

Problem 1 · FE Environmental, Clarifier overflow rate and detention time

A rectangular clarifier is 55 ft long and 16 ft wide, with a side water depth of 14.0 ft. The design flow is 1.26 MGD. Find the detention time (hydraulic residence time) in hours.

Enter your answer in hours (h), 3 significant figures.

Handbook: Environmental Engineering, Clarifier, FE Reference Handbook 10.6

Show the worked solution
Answer
1.76 h
Given
L = 55 ft, W = 16 ft, H = 14.0 ft, Q = 1.26 MGD, 7.481 gal/ft³ (handbook)
Find
detention time θ (h)
Handbook
Environmental Engineering, Clarifier (hydraulic residence time); Units and Conversion Factors (7.481 gal per ft³), pages 3 and 345
Equation
θ = V/Q, with V = L·W·H
Substitute
  1. V = L·W·H = (55 ft)(16 ft)(14.0 ft) = 12,320 ft³
  2. V = 12,320 ft³ × 7.481 gal/ft³ = 92,170 gal
  3. θ = V/Q = (92,170 gal)/(1.26 × 10⁶ gal/d) = 0.07315 d × 24 h/d = 1.76 h
Result
1.76 h, 3 significant figures
Check
the overflow rate is vo = Q/A = (1.26 × 10⁶ gal/d)/(55 ft × 16 ft) = 1,432 gpd/ft², and θ = H × 7.481/vo × 24 = 1.76 h, the same answer.
Common wrong answers
  • 0.235 h: divided the volume in ft³ by a flow in gallons per day without the 7.481 gal/ft³ conversion
  • 0.0731 h: stopped at V/Q in days and did not multiply by 24 h/d
  • 0.125 h: divided the surface area by Q and left out the depth

Problem 2 · FE Environmental, Clarifier overflow rate and detention time

A circular primary clarifier with a diameter of 12.5 m treats an average flow of 6,200 m³/d. Find the overflow rate (surface loading rate) in m³/(m²·d).

Enter your answer in m³/(m²·d), 3 significant figures.

Handbook: Environmental Engineering, Clarifier, FE Reference Handbook 10.6

Show the worked solution
Answer
50.5 m³/(m²·d)
Given
D = 12.5 m, Q = 6,200 m³/d
Find
overflow rate vo (m³/(m²·d))
Handbook
Environmental Engineering, Clarifier (overflow rate), page 345
Equation
vo = Q/A_surface, with A = πD²/4
Substitute
  1. A = πD²/4 = π(12.5 m)²/4 = 122.7 m²
  2. vo = Q/A = (6,200 m³/d)/(122.7 m²) = 50.5 m³/(m²·d)
Result
50.5 m³/(m²·d), 3 significant figures
Check
units (m³/d)/m² = m³/(m²·d), the same as m/d; the rate does not depend on the depth.
Common wrong answers
  • 12.6 m³/(m²·d): used πD² instead of πD²/4, so the area is 4 times too large and the rate 4 times too small
  • 158 m³/(m²·d): divided Q by the circumference πD (that is the weir loading idea, per m of weir) instead of the surface area
  • 202 m³/(m²·d): put the radius into πD²/4, so the area is 4 times too small and the rate 4 times too large

Problem 3 · FE Environmental, Clarifier overflow rate and detention time

A rectangular clarifier is 105 ft long and 25 ft wide, with a side water depth of 13.0 ft. The design flow is 2.83 MGD. Find the detention time (hydraulic residence time) in hours.

Enter your answer in hours (h), 3 significant figures.

Handbook: Environmental Engineering, Clarifier, FE Reference Handbook 10.6

Show the worked solution
Answer
2.16 h
Given
L = 105 ft, W = 25 ft, H = 13.0 ft, Q = 2.83 MGD, 7.481 gal/ft³ (handbook)
Find
detention time θ (h)
Handbook
Environmental Engineering, Clarifier (hydraulic residence time); Units and Conversion Factors (7.481 gal per ft³), pages 3 and 345
Equation
θ = V/Q, with V = L·W·H
Substitute
  1. V = L·W·H = (105 ft)(25 ft)(13.0 ft) = 34,130 ft³
  2. V = 34,130 ft³ × 7.481 gal/ft³ = 255,300 gal
  3. θ = V/Q = (255,300 gal)/(2.83 × 10⁶ gal/d) = 0.09021 d × 24 h/d = 2.16 h
Result
2.16 h, 3 significant figures
Check
the overflow rate is vo = Q/A = (2.83 × 10⁶ gal/d)/(105 ft × 25 ft) = 1,078 gpd/ft², and θ = H × 7.481/vo × 24 = 2.16 h, the same answer.
Common wrong answers
  • 0.289 h: divided the volume in ft³ by a flow in gallons per day without the 7.481 gal/ft³ conversion
  • 0.167 h: divided the surface area by Q and left out the depth
  • 0.0902 h: stopped at V/Q in days and did not multiply by 24 h/d

Problem 4 · FE Environmental, Clarifier overflow rate and detention time

A circular primary clarifier has a diameter of 36.0 m and a side water depth of 3.7 m. It treats an average flow of 24,800 m³/d. Find the hydraulic residence time (detention time) in hours.

Enter your answer in hours (h), 3 significant figures.

Handbook: Environmental Engineering, Clarifier, FE Reference Handbook 10.6

Show the worked solution
Answer
3.64 h
Given
D = 36.0 m, H = 3.7 m, Q = 24,800 m³/d
Find
hydraulic residence time θ (h)
Handbook
Environmental Engineering, Clarifier (hydraulic residence time and overflow rate), page 345
Equation
θ = V/Q, with V = A·H and A = πD²/4
Substitute
  1. A = πD²/4 = π(36.0 m)²/4 = 1,018 m²
  2. V = A·H = (1,018 m²)(3.7 m) = 3,766 m³
  3. θ = V/Q = (3,766 m³)/(24,800 m³/d) = 0.1519 d × 24 h/d = 3.64 h
Result
3.64 h, 3 significant figures
Check
θ = H/vo gives the same time: the overflow rate is vo = Q/A = 24.36 m³/(m²·d), and 3.7 m/(24.36 m/d) × 24 h/d = 3.64 h.
Common wrong answers
  • 14.6 h: used πD² instead of πD²/4 (the diameter as if it were the radius), so the area and the time are 4 times too large
  • 0.985 h: divided the surface area by Q and left out the depth
  • 0.152 h: stopped at V/Q in days and did not multiply by 24 h/d

Problem 5 · FE Environmental, Clarifier overflow rate and detention time

A rectangular clarifier is 150 ft long and 37 ft wide, with a side water depth of 12.5 ft. The design flow is 6.77 MGD. Find the detention time (hydraulic residence time) in hours.

Enter your answer in hours (h), 3 significant figures.

Handbook: Environmental Engineering, Clarifier, FE Reference Handbook 10.6

Show the worked solution
Answer
1.84 h
Given
L = 150 ft, W = 37 ft, H = 12.5 ft, Q = 6.77 MGD, 7.481 gal/ft³ (handbook)
Find
detention time θ (h)
Handbook
Environmental Engineering, Clarifier (hydraulic residence time); Units and Conversion Factors (7.481 gal per ft³), pages 3 and 345
Equation
θ = V/Q, with V = L·W·H
Substitute
  1. V = L·W·H = (150 ft)(37 ft)(12.5 ft) = 69,380 ft³
  2. V = 69,380 ft³ × 7.481 gal/ft³ = 519,000 gal
  3. θ = V/Q = (519,000 gal)/(6.77 × 10⁶ gal/d) = 0.07666 d × 24 h/d = 1.84 h
Result
1.84 h, 3 significant figures
Check
the overflow rate is vo = Q/A = (6.77 × 10⁶ gal/d)/(150 ft × 37 ft) = 1,220 gpd/ft², and θ = H × 7.481/vo × 24 = 1.84 h, the same answer.
Common wrong answers
  • 0.147 h: divided the surface area by Q and left out the depth
  • 0.246 h: divided the volume in ft³ by a flow in gallons per day without the 7.481 gal/ft³ conversion
  • 0.0767 h: stopped at V/Q in days and did not multiply by 24 h/d

Problem 6 · FE Environmental, Clarifier overflow rate and detention time

A circular primary clarifier with a diameter of 39.0 m treats an average flow of 66,100 m³/d. Find the overflow rate (surface loading rate) in m³/(m²·d).

Enter your answer in m³/(m²·d), 3 significant figures.

Handbook: Environmental Engineering, Clarifier, FE Reference Handbook 10.6

Show the worked solution
Answer
55.3 m³/(m²·d)
Given
D = 39.0 m, Q = 66,100 m³/d
Find
overflow rate vo (m³/(m²·d))
Handbook
Environmental Engineering, Clarifier (overflow rate), page 345
Equation
vo = Q/A_surface, with A = πD²/4
Substitute
  1. A = πD²/4 = π(39.0 m)²/4 = 1,195 m²
  2. vo = Q/A = (66,100 m³/d)/(1,195 m²) = 55.3 m³/(m²·d)
Result
55.3 m³/(m²·d), 3 significant figures
Check
units (m³/d)/m² = m³/(m²·d), the same as m/d; the rate does not depend on the depth.
Common wrong answers
  • 221 m³/(m²·d): put the radius into πD²/4, so the area is 4 times too small and the rate 4 times too large
  • 13.8 m³/(m²·d): used πD² instead of πD²/4, so the area is 4 times too large and the rate 4 times too small
  • 539 m³/(m²·d): divided Q by the circumference πD (that is the weir loading idea, per m of weir) instead of the surface area

Problem 7 · FE Environmental, Clarifier overflow rate and detention time

A circular primary clarifier with a diameter of 14.5 m treats an average flow of 4,300 m³/d. Find the overflow rate (surface loading rate) in m³/(m²·d).

Enter your answer in m³/(m²·d), 3 significant figures.

Handbook: Environmental Engineering, Clarifier, FE Reference Handbook 10.6

Show the worked solution
Answer
26.0 m³/(m²·d)
Given
D = 14.5 m, Q = 4,300 m³/d
Find
overflow rate vo (m³/(m²·d))
Handbook
Environmental Engineering, Clarifier (overflow rate), page 345
Equation
vo = Q/A_surface, with A = πD²/4
Substitute
  1. A = πD²/4 = π(14.5 m)²/4 = 165.1 m²
  2. vo = Q/A = (4,300 m³/d)/(165.1 m²) = 26.0 m³/(m²·d)
Result
26.0 m³/(m²·d), 3 significant figures
Check
units (m³/d)/m² = m³/(m²·d), the same as m/d; the rate does not depend on the depth.
Common wrong answers
  • 104 m³/(m²·d): put the radius into πD²/4, so the area is 4 times too small and the rate 4 times too large
  • 6.51 m³/(m²·d): used πD² instead of πD²/4, so the area is 4 times too large and the rate 4 times too small
  • 94.4 m³/(m²·d): divided Q by the circumference πD (that is the weir loading idea, per m of weir) instead of the surface area

Problem 8 · FE Environmental, Clarifier overflow rate and detention time

A circular clarifier 90 ft in diameter treats a flow of 5.30 MGD. Find the overflow rate in gpd/ft².

Enter your answer in gpd/ft², 3 significant figures.

Handbook: Environmental Engineering, Clarifier, FE Reference Handbook 10.6

Show the worked solution
Answer
833 gpd/ft²
Given
D = 90 ft, Q = 5.30 MGD
Find
overflow rate vo (gpd/ft²)
Handbook
Environmental Engineering, Clarifier (overflow rate), page 345
Equation
vo = Q/A_surface, with A = πD²/4
Substitute
  1. A = πD²/4 = π(90 ft)²/4 = 6,362 ft²
  2. Q = 5.30 MGD = 5,300,000 gal/d
  3. vo = Q/A = (5,300,000 gal/d)/(6,362 ft²) = 833 gpd/ft²
Result
833 gpd/ft², 3 significant figures
Check
units (gal/d)/ft² = gpd/ft²; 1 MGD = 1,000,000 gal/d.
Common wrong answers
  • 111 gpd/ft²: converted the flow to ft³/d before dividing, which gives ft/d, not gpd/ft²
  • 3,330 gpd/ft²: put the radius into πD²/4, so the area is 4 times too small and the rate 4 times too large
  • 208 gpd/ft²: used πD² instead of πD²/4, so the area is 4 times too large and the rate 4 times too small

Problem 9 · FE Environmental, Clarifier overflow rate and detention time

A rectangular clarifier is 175 ft long and 39 ft wide, with a side water depth of 13.0 ft. The design flow is 8.85 MGD. Find the detention time (hydraulic residence time) in hours.

Enter your answer in hours (h), 3 significant figures.

Handbook: Environmental Engineering, Clarifier, FE Reference Handbook 10.6

Show the worked solution
Answer
1.80 h
Given
L = 175 ft, W = 39 ft, H = 13.0 ft, Q = 8.85 MGD, 7.481 gal/ft³ (handbook)
Find
detention time θ (h)
Handbook
Environmental Engineering, Clarifier (hydraulic residence time); Units and Conversion Factors (7.481 gal per ft³), pages 3 and 345
Equation
θ = V/Q, with V = L·W·H
Substitute
  1. V = L·W·H = (175 ft)(39 ft)(13.0 ft) = 88,730 ft³
  2. V = 88,730 ft³ × 7.481 gal/ft³ = 663,800 gal
  3. θ = V/Q = (663,800 gal)/(8.85 × 10⁶ gal/d) = 0.07500 d × 24 h/d = 1.80 h
Result
1.80 h, 3 significant figures
Check
the overflow rate is vo = Q/A = (8.85 × 10⁶ gal/d)/(175 ft × 39 ft) = 1,297 gpd/ft², and θ = H × 7.481/vo × 24 = 1.80 h, the same answer.
Common wrong answers
  • 0.241 h: divided the volume in ft³ by a flow in gallons per day without the 7.481 gal/ft³ conversion
  • 0.138 h: divided the surface area by Q and left out the depth
  • 0.0750 h: stopped at V/Q in days and did not multiply by 24 h/d

Problem 10 · FE Environmental, Clarifier overflow rate and detention time

A circular primary clarifier with a diameter of 38.0 m treats an average flow of 28,300 m³/d. Find the overflow rate (surface loading rate) in m³/(m²·d).

Enter your answer in m³/(m²·d), 3 significant figures.

Handbook: Environmental Engineering, Clarifier, FE Reference Handbook 10.6

Show the worked solution
Answer
25.0 m³/(m²·d)
Given
D = 38.0 m, Q = 28,300 m³/d
Find
overflow rate vo (m³/(m²·d))
Handbook
Environmental Engineering, Clarifier (overflow rate), page 345
Equation
vo = Q/A_surface, with A = πD²/4
Substitute
  1. A = πD²/4 = π(38.0 m)²/4 = 1,134 m²
  2. vo = Q/A = (28,300 m³/d)/(1,134 m²) = 25.0 m³/(m²·d)
Result
25.0 m³/(m²·d), 3 significant figures
Check
units (m³/d)/m² = m³/(m²·d), the same as m/d; the rate does not depend on the depth.
Common wrong answers
  • 99.8 m³/(m²·d): put the radius into πD²/4, so the area is 4 times too small and the rate 4 times too large
  • 237 m³/(m²·d): divided Q by the circumference πD (that is the weir loading idea, per m of weir) instead of the surface area
  • 6.24 m³/(m²·d): used πD² instead of πD²/4, so the area is 4 times too large and the rate 4 times too small

A new problem posts every day inside the lab, with the full worked solution the same evening.

Frequently asked questions

What is a surface overflow rate?

It is the flow divided by the clarifier’s surface area, a velocity-like loading that sizes the tank. The handbook gives the relationships in the Environmental Engineering chapter, and each solution cites the page.

How precise should a fill-in answer be?

Enter the value in the units the question asks for. Our solutions round final answers to 3 significant figures and keep 4 in the steps.

Is this topic only on FE Environmental?

It is central to FE Environmental, and wastewater treatment also appears within the Water Resources and Environmental Engineering area of FE Civil.

Sources

  1. NCEES FE Environmental CBT exam specifications (PDF). Retrieved October 3, 2026.
  2. NCEES FE Civil CBT exam specifications (PDF). Retrieved October 3, 2026.
  3. NCEES FE Reference Handbook 10.6 (free PDF in MyNCEES). Retrieved October 3, 2026.