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Math problem set Grade 8 Science

Quarry Bend's Elevated Tank: Stored Energy and Water Pressure

A grade 8 science problem set where students find the weight and stored potential energy of water in a fictional elevated tank and explain why height sets pressure.

Objectives

  1. Students will calculate the volume, weight, and gravitational potential energy of the water held in a spherical elevated tank.
  2. Students will compare the potential energy stored in the Earth-water system when the same water sits at different heights.
  3. Students will explain why water pressure at the base of a tank depends on the height of the water surface rather than on the tank's volume or shape.

Vocabulary

Standpipe
A standpipe is a tall, cylindrical water storage tank that sits directly on the ground instead of standing on legs like a water tower. It stores water and helps maintain steady pressure in the pipes nearby.

Also: gravitational potential energy, foot-pound, weight, pressure, elevated tank

Teacher overview

The fictional Quarry Bend water utility keeps treated water in a buried clearwell at the plant, and its high-service pump lifts that water into a spherical tank on a tower, mostly at night. This set uses those tanks to teach the physics of elevated storage: lifting water stores gravitational potential energy in the Earth-water system, and that stored energy is what pushes water through the mains when the pump is off. Students find volumes only as a first step toward weight, stored energy, and pressure, then compare the Quarry Bend sphere with a standpipe in a neighboring town. A common misconception is that a bigger or wider tank produces more water pressure; pressure at the base depends only on how high the water surface stands, while stored energy depends on both the weight of the water and its height, and problems 8 and 9 are built to separate the two ideas. Watch also for students who put the diameter into a volume formula where the radius belongs.

Quarry Bend's Elevated Tank: Stored Energy and Water Pressure — Grade 8 Science
Name: ____________________________ Date: ______________

Problems

Use the following information about the fictional Quarry Bend utility for every problem.

  • The clearwell is a buried cylinder with a radius of 20 feet and a depth of 12 feet. When it is full, its water surface is level with the ground.
  • The ground at the plant and at the base of the tower is at the same elevation. Call that elevation height 0 feet.
  • The elevated tank is a sphere with a radius of 15 feet. The bottom of the sphere is 105 feet above the ground, its center is 120 feet above the ground, and its top is 135 feet above the ground. A riser pipe connects the tank to the water main at ground level. Ignore the water inside the riser pipe when you find volumes.
  • Use 3.14 for pi. One cubic foot holds 7.48 gallons, one gallon of water weighs 8.34 pounds, and each foot of water height above a point adds 0.433 psi (pounds per square inch) of pressure at that point.

Energy model. When the Earth pulls on an object, raising the object higher stores gravitational potential energy in the Earth-object system. In this set, measure that energy in foot-pounds:

potential energy (foot-pounds) = weight (pounds) × height above the ground (feet)

The water in a full sphere is spread from the bottom of the sphere to the top. Because a sphere is the same shape above and below its center, use the height of the center as the height of all the water in a full sphere.

  1. Find the volume of the clearwell in cubic feet, using Volume = pi × radius² × depth. Then convert it to gallons, rounded to the nearest gallon.
  2. Find the volume of the elevated tank in cubic feet, using Volume = four-thirds × pi × radius³. Then convert it to gallons, rounded to the nearest gallon.
  3. Using your rounded gallon answer from problem 2, find the weight of the water in the full elevated tank. Round to the nearest thousand pounds.
  4. Using your weight from problem 3, find the gravitational potential energy stored in the Earth-water system by the water in the full tank, measured from the ground.
  5. Before the present tower was built, Quarry Bend's tank sat on a shorter tower with its center only 60 feet above the ground. Find the potential energy the same full tank of water stored at that height. Then describe how the energy changed when the water was moved twice as far from the ground, and explain why.
  6. Find the water pressure at the base of the tower, rounded to the nearest tenth of a psi: (a) when the tank is full and the water surface is 135 feet above the ground, and (b) when the water surface has fallen to the bottom of the sphere, 105 feet above the ground. Distribution systems typically run at about 40 to 80 psi. Are both pressures in that range?
  7. On a busy morning the pump is off, and the tank drains from full to the bottom of the sphere as people use water. By how many psi does the pressure at the base of the tower fall? The water that left the tank carried stored potential energy with it. Describe where that energy went.
  8. The neighboring town of Hollin Gap (also fictional) stores its water in a standpipe, a tall cylinder that stands on the ground. It has a radius of 5 feet and is filled with water to a height of 135 feet. Find its volume in cubic feet and convert it to gallons, rounded to the nearest gallon.
  9. Find the water pressure at the base of the full Hollin Gap standpipe, to the nearest tenth of a psi, and compare it with the pressure at the base of Quarry Bend's full tower from problem 6. Then, without calculating, decide which full tank stores more potential energy in its Earth-water system and give two reasons.
  10. The clearwell in problem 1 holds more water than the elevated tank, yet Quarry Bend's water pressure comes from the elevated tank. Draw a labeled model of the Earth-water system that shows the ground at height 0, the clearwell, and the elevated tank, with an arrow for the Earth's pull on the water. Then use your model and your answer to problem 4 to explain (a) why the utility spends electricity pumping water up the tower, and (b) why the tank can keep water flowing to homes for a while during a power failure while the clearwell cannot.

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: 112,739 gallons.
    Volume = 3.14 × 20² × 12 = 3.14 × 400 × 12 = 3.14 × 4,800 = 15,072 cubic feet. Then 15,072 cu ft × 7.48 gal/cu ft = 112,738.56, which rounds to 112,739 gallons.
  2. Answer: 105,692 gallons.
    Volume = (4/3) × 3.14 × 15³ = (4/3) × 3.14 × 3,375 = (4/3) × 10,597.5 = 14,130 cubic feet. Then 14,130 cu ft × 7.48 gal/cu ft = 105,692.4, which rounds to 105,692 gallons.
  3. Answer: 881,000 pounds.
    Weight = 105,692 gal × 8.34 lb/gal = 881,471.28 pounds, which rounds to 881,000 pounds to the nearest thousand.
  4. Answer: 105,720,000 foot-pounds.
    The center of the full sphere is 120 feet above the ground, so potential energy = 881,000 lb × 120 ft = 105,720,000 foot-pounds.
  5. Answer: 52,860,000 foot-pounds.
    Potential energy = 881,000 lb × 60 ft = 52,860,000 foot-pounds. At 120 feet the system stored 105,720,000 foot-pounds, and 105,720,000 ÷ 52,860,000 = 2, so moving the same water twice as far from the ground doubled the stored energy. The weight of the water did not change; only the arrangement of the Earth-water system changed. Because the stored energy equals weight × height, for the same weight the energy grows in proportion to the height.
  6. Answer: 58.5 psi when full and 45.5 psi at the bottom of the sphere; both are in the 40 to 80 psi range.
    (a) 135 ft × 0.433 psi/ft = 58.455 psi, which rounds to 58.5 psi. (b) 105 ft × 0.433 psi/ft = 45.465 psi, which rounds to 45.5 psi. Both values fall between 40 and 80 psi.
  7. Answer: 13.0 psi.
    58.5 psi − 45.5 psi = 13.0 psi. (Check: the water surface fell 135 − 105 = 30 feet, and 30 ft × 0.433 psi/ft = 12.99 psi, which rounds to 13.0 psi.) The stored potential energy changed into other forms. As the water fell through the riser and flowed into the mains, it gained kinetic energy (the energy of moving water). Friction between the moving water and the pipe walls, valves, and meters turned much of that energy into thermal energy, warming the water and pipes very slightly. Some was used to push water up into the upper floors of buildings and out through faucets.
  8. Answer: 79,269 gallons.
    Volume = 3.14 × 5² × 135 = 3.14 × 25 × 135 = 3.14 × 3,375 = 10,597.5 cubic feet. Then 10,597.5 cu ft × 7.48 gal/cu ft = 79,269.3, which rounds to 79,269 gallons.
  9. Answer: 58.5 psi, the same as at the base of Quarry Bend's full tower.
    135 ft × 0.433 psi/ft = 58.455 psi, which rounds to 58.5 psi. The standpipe holds 79,269 gallons and the sphere holds 105,692 gallons, yet the base pressures are equal, because pressure at a point depends only on how high the water surface stands above that point, not on the amount of water or the tank's shape. Quarry Bend's sphere stores more potential energy for two reasons. First, it holds more water, so the water weighs more (105,692 gallons compared with 79,269). Second, its water sits higher on average: the sphere's water is centered 120 feet above the ground, while the standpipe's water fills it from the ground up, so its middle is only 135 ÷ 2 = 67.5 feet above the ground.
  10. Answer: The clearwell's water sits at or below the ground, so measured from the ground it stores no gravitational potential energy that could push water to homes, while the elevated tank's water stores about 105,720,000 foot-pounds.
    A complete model shows the ground line marked height 0, the clearwell below that line, the sphere with its bottom, center, and top marked at 105, 120, and 135 feet, and a downward arrow on the water labeled as the Earth's pull. (a) The clearwell's water sits at or below the ground, so it has no stored energy that could push water up and out to homes at ground level or above. The pump uses electrical energy to do work against the Earth's pull, raising the water and storing about 105,720,000 foot-pounds in the Earth-water system (problem 4). The pump actually supplies more than that, because the water starts below ground in the clearwell and because some of the motor's energy becomes heat. (b) During a power failure the pump stops, yet the water in the tank is still high above the town. The Earth's pull on that water keeps the mains pressurized, and the stored potential energy is released as water flows out, until the tank runs low. The clearwell has no stored energy above the ground, so without a pump its water cannot reach the mains.

Differentiation

Support (below grade level)

Give students a reference card with the cylinder and sphere volume formulas, the three constants (7.48 gallons per cubic foot, 8.34 pounds per gallon, and 0.433 psi per foot), and the energy model written as weight × height. Provide a side-view sketch of the tower with the 105, 120, and 135 foot marks already labeled, and work problem 1 together as a class before students attempt problem 2. For problem 9, let students list "amount of water" and "height of water" as two columns and fill them in for each tank before deciding.

Extension (above grade level)

Have students check their answer to problem 9 by calculating. The Hollin Gap standpipe's 79,269 gallons weigh 79,269 × 8.34 = 661,103.46 pounds, or about 661,000 pounds, and its water is centered 67.5 feet above the ground, so it stores about 661,000 × 67.5 = 44,617,500 foot-pounds. Students then find how many times as much energy the Quarry Bend sphere stores (105,720,000 ÷ 44,617,500 is about 2.4) and explain why the water in the lower part of a standpipe adds weight but stores little energy, so that a utility cannot let a standpipe drain far before pressure in town falls below the typical 40 to 80 psi range.

ELL supports

Provide a labeled side-view drawing of the clearwell, the tower, and the standpipe with height, radius, weight, and pressure named in English and, where possible, in students' home languages. Post the sentence frames "The pressure is the same because..." and "The tank stores more energy because it holds more water and..." for problems 9 and 10, and allow students to answer problem 10 with a labeled drawing and short phrases.

Extensions

Ask students to find out, from a local utility's website or a short interview, how high their own town's elevated tanks or standpipes stand and when the utility runs its pumps to fill them. Students can then explain, using the Quarry Bend energy model, why many utilities fill their tanks at night and let them drain during the day.