Math problem set Grade 10 Math
Larkhollow Lift Station Pump Cycle Problem Set
A grade 10 problem set on a fictional wastewater wet well: students find its working volume, fill and pump-down times, cycle length, and daily pump starts.
Objectives
- Students will calculate a rectangular wet well's working volume in cubic feet and gallons.
- Students will compute fill time, pump-down time, cycle length, and daily pump starts from stated flow rates.
Vocabulary
- Lift station
- A lift station is a pumping setup that raises wastewater from a low spot so it can keep flowing toward the treatment plant. Sewers mostly flow downhill, so low areas need a lift.
Also: wet well, pump cycle, net drawdown rate
Teacher overview
The fictional Larkhollow lift station pumps wastewater uphill out of a rectangular wet well. Sewage flows in steadily; when the water reaches the pump-on level, the pump runs until the water drops to the pump-off level, then shuts off and the cycle repeats. This set has students find the working volume, then the fill time, pump-down time, and how many times per day the pump starts. A common misconception here is treating the pump's full capacity as the rate the water level drops; students who forget that inflow keeps arriving during pumping will get the pump-down time wrong.
Problems
The Larkhollow lift station (a fictional wastewater pumping station used for this set) has a rectangular wet well. Between the pump-on and pump-off levels, the well measures 10 feet long, 8 feet wide, and 5 feet of depth. Wastewater flows in at an average of 250 gpm. The pump moves 700 gpm when it runs. Use 1 cubic foot = 7.48 gallons and 1 MGD = 694 gpm where needed.
Find the working volume of the wet well between the pump-on and pump-off levels, in cubic feet. Use volume = length × width × depth.
Convert the working volume to gallons using 1 cubic foot = 7.48 gallons.
With the pump off, inflow of 250 gpm fills the working volume. How many minutes does it take to fill? Round to the nearest tenth.
When the pump runs, it removes 700 gpm while inflow of 250 gpm keeps arriving. Find the net rate at which the water level drops, in gpm.
Using the net rate from problem 4, how many minutes does the pump take to lower the water from pump-on to pump-off? Round to the nearest tenth.
One full cycle is the fill time plus the pump-down time. Find the cycle length in minutes. Round to the nearest tenth.
On average, how many cycles happen in one hour? Round to the nearest tenth.
How many times does the pump start in a full 24-hour day? Round to the nearest whole start.
During one pump-down, how many gallons does the pump actually lift out of the well? Use the pump's full 700 gpm and the pump-down time from problem 5. Explain why this is more than the working volume from problem 2.
Convert the pump's 700 gpm capacity to MGD using 1 MGD = 694 gpm. Round to the nearest hundredth.
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: 400 cu ft. Volume = 10 ft × 8 ft × 5 ft = 400 cubic feet.
Answer: 2,992 gal. 400 cubic feet × 7.48 gallons per cubic foot = 2,992 gallons.
Answer: 12.0 min. Fill time = 2,992 gallons / 250 gpm = 11.968, which rounds to 12.0 minutes.
Answer: 450 gpm. Net drawdown rate = 700 gpm pumped − 250 gpm inflow = 450 gpm.
Answer: 6.6 min. Pump-down time = 2,992 gallons / 450 gpm = 6.649, which rounds to 6.6 minutes.
Answer: 18.6 min. Cycle length = 11.968 + 6.649 = 18.617, which rounds to 18.6 minutes.
Answer: 3.2 cycles. 60 minutes / 18.617 minutes per cycle = 3.223, which rounds to 3.2 cycles per hour.
Answer: 77 starts. 3.223 cycles per hour × 24 hours = 77.3, which rounds to 77 pump starts per day.
Answer: 4,654 gal. The pump runs 6.649 minutes at 700 gpm: 6.649 × 700 = 4,654 gallons. This is more than the 2,992-gallon working volume because inflow of 250 gpm keeps adding water during the 6.649 minutes the pump runs, and the pump must remove that added water too.
Answer: 1.01 MGD. 700 gpm / 694 gpm per MGD = 1.009, which rounds to 1.01 MGD.
Differentiation
Support
Provide the working volume in gallons (2,992) already computed so students can move straight to the timing problems. Give a labeled diagram showing the pump-on level, the pump-off level, and arrows for inflow and pumping, so the idea of a net drawdown rate in problem 4 is visible before it is calculated.
Extension
Ask students to find how many minutes a full cycle would take if the inflow to the fictional station doubled to 500 gpm during a storm, and to explain what happens to the number of daily pump starts. Have them comment on why frequent starts can shorten a pump motor's life.
ELL supports
Pre-teach "wet well," "cycle," and "net rate" with a simple animation or hand-drawn sequence showing the water rising, the pump switching on, and the water falling. Provide sentence frames for problem 9: "The pump lifts more than the working volume because ___."
Extensions
Have students graph the water level in the wet well over two full cycles, with time on the horizontal axis and depth on the vertical axis. The sawtooth shape, rising slowly during fill and dropping quickly during pumping, gives a visual model of the cycle and connects the timing answers to a picture an operator would recognize.