Sizing a
hydraulic tank
A hydraulic power pack needs a reservoir big enough to let the oil cool and settle, small enough to fit the bay, and cheap enough to build. You will use volume, surface area and density to find one that does all three.
The job
Read the spec like an engineerThe power pack driving Ashfield's press needs a fabricated steel oil reservoir. It is an open-topped box with a bolted lid, 0.45 m wide, with the length and height for you to choose.
- The pump moves 45 litres per minute, and the tank must hold at least three times that
- Add 10% on top for an air gap above the oil — so 150 litres minimum
- The bay allows a maximum tank length of 0.90 m
- Mild steel plate in 3, 4 or 5 mm: density 7850 kg/m³, priced at £22, £28 and £34 per m²
- Budget for plate: £45. Hydraulic oil has a density of 870 kg/m³
Your question: what length and height give you 150 litres or more without running out of floor space or money?
Explore the drawing
Drag the corner grip to resizeStretch the tank by its corner and watch both gauges. Capacity grows with three dimensions but plate area with only two, so a bigger tank is always better value per litre — it is the bay and the budget that stop you, never the geometry.
Worked solution
Open one step at a timeTake the 0.80 m long, 0.45 m high tank and check it properly.
The rule of thumb is that a reservoir holds three minutes' worth of pump flow, so the oil has time to shed heat and let air bubbles rise out.
The 10% is the air gap above the oil. Fill a hydraulic tank to the brim and it vents oil across the workshop floor the moment it warms up.
162 ≥ 150, so the capacity rule is satisfied. Passes.
1 m³ = 1000 litres. It comes up constantly in engineering questions and it is worth knowing cold rather than deriving it under exam pressure.
A closed box has three pairs of identical faces.
Write the three products down separately before doubling. Almost every lost mark on surface area comes from missing a pair of faces, and a written list makes that impossible.
The plate is 3 mm thick, so its volume is area × thickness — but the thickness must be in metres.
Worth knowing before you plan how to lift it onto the machine.
Full tank, oil and steel together: about 170 kg. That is a floor-loading question as well as a maths one.
Try 0.35 m high: capacity falls to 126 litres and the tank fails the cooling rule. Stretch the length to 1.0 m to recover it and you break the 0.90 m bay limit instead.
Because volume grows faster than surface area, a bigger tank is always better value per litre. The limits that stop you are the bay and the budget, never the geometry.
Your turn
Answers checked instantlyA tank measures 0.75 m by 0.40 m by 0.50 m. Work out its capacity in litres.
That tank is filled to 90% with oil of density 870 kg/m³. Work out the mass of the oil, to 1 decimal place.
Work out the total surface area of that closed tank, to 2 decimal places.
Exam-style question
Full mark schemeAn alternative reservoir is a closed cylinder of diameter 0.60 m and height 0.80 m, rolled from the same 3 mm plate (density 7850 kg/m³).
Work out its capacity in litres, to the nearest litre. The mark scheme also asks for the mass of the plate, so find that too.
Mark scheme — 5 marks
- Radius = 0.60 ÷ 2 = 0.30 m B1
- V = πr²h = π × 0.30² × 0.80 = 0.2262 m³ M1 A1
- Capacity = 0.2262 × 1000 = 226 litres A1
- SA = 2πr² + 2πrh = 0.5655 + 1.5080 = 2.0735 m²; mass = 2.0735 × 0.003 × 7850 = 48.8 kg A1
Accepted answer: 226 litres. Using the diameter in place of the radius is the classic error and it makes the answer four times too big — if your capacity came out near 900 litres, that is what happened. Halve the diameter before anything else touches the calculator.
Produced for EAL engineering learners.
Maps to GCSE Mathematics (9–1) — volume and surface area of prisms and cylinders, and density.