Duncannon Elevated Tank: A Storage and Pressure Design Project

Driving question: How can we design an elevated water tank for the town of Duncannon that stores at least 500,000 gallons, keeps pressure at the highest homes between 45 psi and 80 psi as the tank drains and refills, and keeps the materials cost at or below 1,500,000 dollars?

Scenario: The Duncannon Water Board (a fictional board created for this project) has asked student engineering teams to propose an elevated tank for a new part of town. The highest homes sit at the same ground level as the base of the tank's tower. Water pressure at those homes comes from the height of the water surface above the ground. That surface rises and falls every day as the tank refills overnight and drains during the day, so the pressure at the homes rises and falls with it. When the tank is nearly empty, the water surface is close to the tank floor, which sits on top of the tower. When the tank is full, the water surface is at the tower height plus the tank's water depth.

Constraints (all set for this fictional project):

• Storage: the tank must hold at least 500,000 gallons. Since one cubic foot holds 7.48 gallons, that is 500,000 ÷ 7.48 = 66,845 cubic feet.
• Pressure: the pressure at the highest homes must stay between 45 psi and 80 psi at every water level. For the low end, the tank floor must be at least 104 feet above the ground, because 45 psi × 2.31 feet per psi = 104 feet of head, so the tower must be at least 104 feet tall. For the high end, the full water surface may be no more than 184 feet above the ground, because 80 psi × 2.31 = 184.8 feet, which is rounded down to 184 so the pressure does not go over 80 psi. In symbols: tower height ≥ 104 and tower height + water depth ≤ 184.
• Budget: the materials cost must be at or below 1,500,000 dollars. Tank shell steel costs 45 dollars per square foot of tank surface area, and the supporting tower costs 9,000 dollars per foot of tower height.

Reference formulas: cylinder volume (cubic feet) = 0.785 × diameter² × depth; cylinder surface area (square feet) = 2 × 0.785 × diameter² (top and bottom) + 3.14159 × diameter × depth (side); pressure (psi) = height of water above the point (feet) × 0.433.

Worked example of one candidate: A cylinder 50 feet across and 35 feet deep holds 0.785 × 50² × 35 = 68,688 cubic feet, or about 513,800 gallons, which clears the storage minimum. The tower is set at the 104-foot minimum, so when the tank is nearly empty the pressure at the homes is 104 × 0.433 = 45.0 psi. When the tank is full, the water surface is 104 + 35 = 139 feet above the ground, which is under the 184-foot limit and gives 139 × 0.433 = 60.2 psi. Tank surface area is 2 × 0.785 × 50² + 3.14159 × 50 × 35 = 3,925 + 5,498 = 9,423 square feet, so shell steel costs about 424,000 dollars. The 104-foot tower costs 104 × 9,000 = 936,000 dollars, for a total near 1,360,000 dollars, which is under budget.

Final deliverable: A labeled scale drawing of the tank and tower on grid paper, plus a one-page written proposal to the Duncannon Water Board that reports the design's storage, its tower height and water depth, the pressure at the highest homes when the tank is nearly empty and when it is full, and an itemized materials cost proving it meets all three constraints.

1. Session 1 (60 min) — Set the targets: Teams convert the 500,000-gallon storage minimum to cubic feet, compute the 104-foot tank-floor height needed for 45 psi, and compute the 184-foot ceiling on the full water surface that keeps pressure at or below 80 psi. They write each of the three constraints (storage, pressure, budget) as an inequality and explain in their own words why the pressure constraint has both a lower and an upper limit. Output: a completed constraints sheet stating each target with its units and its inequality.

2. Session 2 (60 min) — Draft two candidate tanks: Each team sketches two cylinder designs of clearly different shape, one narrower and deeper and one wider and shallower, choosing a diameter and depth for each so that both hold at least 500,000 gallons. Teams compute each candidate's volume in gallons and check that its depth fits under the pressure ceiling (with a 104-foot tower, the water depth may be at most 184 − 104 = 80 feet). Output: two labeled sketches on grid paper, each with its volume calculation and depth check shown.

3. Session 3 (60 min) — Cost the candidates and choose one: Teams choose a tower height for each candidate, compute the pressure when the tank is nearly empty and when it is full, compute the tank surface area and total materials cost, and mark each design as viable or not, naming the constraint it fails if it is not. They choose the viable design they judge best. Output: a cost comparison table and a one-sentence justification for the chosen design.

4. Session 4 (60 min) — Write and present the proposal: Teams turn their chosen design into a final scale drawing and a one-page written proposal, then present both to the class acting as the Duncannon Water Board. Output: the finished drawing, the written proposal, and a short team presentation.