Math problem set Grade 9 Engineering & Design

Birch Hollow Pump Station: Head, Pressure, and Flow Problems

A grade 9 engineering problem set on a booster pump station: students find total dynamic head, discharge pressure, flow conversions, and tank fill time.

Grade: Grade 9
Subject: Engineering & Design
Material: Math problem set
Time: 45 min
Prep: No prep
Materials: Calculator

Standards

HSN-Q.A.1 Common Core Math

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

How this material aligns: Students carry units through the multi-step head, pressure, and flow work in problems 1 through 5, converting feet to psi and MGD to gpm and cfs consistently.

All materials for HSN-Q.A.1 →

HSA-CED.A.1 Common Core Math

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

How this material aligns: Problem 6 asks students to write and solve a one-variable equation for the head in feet that produces the pump's 95 psi relief setting, then interpret the result.

All materials for HSA-CED.A.1 →

HS-ETS1-3 NGSS

Evaluate a solution to a complex real-world problem based on prioritized criteria and trade-offs that account for a range of constraints, including cost, safety, reliability, and aesthetics as well as possible social, cultural, and environmental impacts.

How this material aligns: Problem 6 converts the 95 psi relief setting into a head limit, and problem 8 evaluates the proposed higher-ridge tank site against that limit, finding the 100.5 psi result exceeds it so the pump or the valve setting has to be revisited.

All materials for HS-ETS1-3 →

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This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-17.

Objectives

Students will calculate total dynamic head and convert it to discharge pressure using the 0.433 psi per foot rule.
Students will convert a design flow among MGD, gpm, and cfs and use it to find a tank fill time.

Vocabulary

Total dynamic head (TDH): Total dynamic head is the total amount of lift a pump has to provide to move water where it needs to go, adding up the height it must lift the water, the friction it must overcome in the pipes, and the pressure needed at the far end. Engineers calculate it to pick a pump that is powerful enough for the job.
Friction loss: Friction loss is the pressure water loses as it rubs against the inside of a pipe on its way to you. Longer, narrower, and rougher pipes cause more of this pressure loss.
Flow rate: Flow rate measures how much water moves past a point in a certain amount of time, like gallons per minute through a garden hose. Bigger pipes and stronger pushes give bigger flow rates.

Also: static lift, discharge pressure

Teacher overview

A booster pump station lifts finished water from a low storage tank up to an elevated tank on higher ground. To size the pump, an engineer adds the vertical lift and the friction loss in the pipe to get the total dynamic head, then converts that head to a pressure the pump must produce. This set walks through those steps for one fictional station, then converts the design flow between units and finds a fill time.

A common error in this math is multiplying feet of head by 2.31 instead of 0.433 when finding pressure, which makes the pressure come out several times too large. A quick check helps: pressure in psi should always be a smaller number than the head in feet. Utilities aim for 40 to 80 psi at the tap, and most states require at least 20 psi at every connection, so answers can be compared against those service numbers.

Birch Hollow Pump Station: Head, Pressure, and Flow Problems — Grade 9 Engineering & Design

Name: ____________________________ Date: ______________

Problems

Birch Hollow's booster pump station (a fictional station used for this set) pumps finished water from an underground clearwell up to an elevated storage tank on a ridge. Use these values for all problems:

Static lift (vertical rise from the clearwell surface to the tank inlet): 185 feet.
Friction loss in the pipe at the design flow: 22 feet.
Design flow: 1.5 MGD.
Conversions: 1 foot of head = 0.433 psi; 1 psi = 2.31 feet; 1 MGD = 694 gpm; 1 MGD = 1.547 cfs.

Find the total dynamic head (TDH) the pump must overcome. TDH = static lift + friction loss.
Convert the total dynamic head to the discharge pressure the pump must produce, using 0.433 psi per foot. Round to the nearest tenth of a psi.
Convert the design flow of 1.5 MGD to gallons per minute (gpm).
Convert the design flow of 1.5 MGD to cubic feet per second (cfs). Round to the nearest hundredth.
After a heavy-use afternoon, the operator must add 500,000 gallons to refill the tank. At the design flow in gpm from problem 3, how many minutes does the refill take? Give the answer in minutes and in hours, rounded to the nearest tenth of an hour.
The pump's pressure-relief valve is set to open at 95 psi. Write and solve an equation for the head, in feet, that produces 95 psi. Then state whether the station's operating head of 207 feet stays below that limit.
During a peak day the tank is drawn down and refilled twice, adding 500,000 gallons each time. How many total gallons does the pump move, and how many hours does it run at the design flow?
The utility studies moving the tank to a higher ridge, which would raise the static lift from 185 feet to 210 feet while friction loss stays at 22 feet. Find the new total dynamic head and the new discharge pressure, and explain whether the new pressure stays within the 95 psi relief setting.

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: 207 ft.
Total dynamic head = static lift + friction loss = 185 ft + 22 ft = 207 ft.
Answer: 89.6 psi.
Pressure = head x 0.433 psi/ft = 207 ft x 0.433 = 89.631, which rounds to 89.6 psi. This is below the 95 psi relief setting.
Answer: 1,041 gpm.
1.5 MGD x 694 gpm/MGD = 1,041 gpm.
Answer: 2.32 cfs.
1.5 MGD x 1.547 cfs/MGD = 2.3205, which rounds to 2.32 cfs.
Answer: 480 minutes.
500,000 gallons / 1,041 gpm = 480.3 minutes. Dividing by 60 gives 480.3 / 60 = 8.0 hours.
Answer: 219 ft.
The equation is 0.433 x h = 95. Solving, h = 95 / 0.433 = 219.4 feet, about 219 ft. The station's operating head of 207 ft is below 219 ft, so at 207 ft the pressure (about 89.6 psi) stays below the 95 psi relief setting.
Answer: 1,000,000 gallons.
Two refills of 500,000 gallons = 1,000,000 gallons. Run time = 2 x 8.0 hours = 16.0 hours at the design flow.
Answer: 100.5 psi (TDH 232 ft).
New TDH = 210 ft + 22 ft = 232 ft. New pressure = 232 x 0.433 = 100.456, about 100.5 psi. Because 100.5 psi is above the 95 psi relief setting, the higher ridge would push the pump past its current relief limit, so the pump selection or the valve setting would have to be revisited.

Differentiation

Support

Provide a printed strip with the two conversions labeled as arrow rules, "feet to psi: multiply by 0.433" and "psi to feet: multiply by 2.31," and give the total dynamic head (207 ft) already computed so students can start from problem 2. Allow unrounded calculator answers.

Extension

Ask students to find the pump run time in hours if the design flow were raised to 2.0 MGD, and to explain how a higher flow shortens the fill time but raises the friction loss, which in turn raises the total dynamic head the pump must overcome.

ELL supports

Pre-teach "head," "static lift," "friction loss," and "discharge pressure" with a labeled side-view diagram of the clearwell, pump, pipe, and elevated tank. Allow students to show work as a labeled diagram plus numbers rather than full sentences.

Extensions

If your district's public works department is willing, ask what elevation their main tank or standpipe sits at and what pressure a school or neighborhood receives. Students can compare the real reading with the pressure predicted by the 0.433 psi per foot rule and discuss why friction loss makes the delivered pressure slightly lower than the simple height calculation.