Size a hydraulic pump with three formulas: flow in gpm = displacement (cu in/rev) × rpm ÷ 231, input horsepower = gpm × psi ÷ 1714 ÷ overall efficiency, and input torque in lb-in = displacement × psi ÷ (2π × mechanical efficiency). In metric, flow in l/min = displacement (cc/rev) × rpm ÷ 1000, power in kW = l/min × bar ÷ 600 ÷ efficiency, and torque in N·m = cc × bar ÷ (20π × efficiency).
Start from the job, work out the flow and pressure it needs, pick a displacement that delivers that flow at the speed you can drive the pump, then check that the engine or electric motor can supply the horsepower and torque at maximum pressure. The formulas below are the ones the pump makers print in their own catalogs, and every worked example uses a real pump from a factory catalog so you can check the arithmetic against the book.
Want the answer without the math? Our hydraulic calculator does flow, horsepower, torque, motor speed and cylinder speed in US or metric units. This page explains what goes into it, and where people go wrong.
What is pump displacement?
Displacement is the volume of oil a pump moves in one revolution of its shaft, given in cubic inches per revolution (cu in/rev or in³/rev) or cubic centimeters per revolution (cc/rev or cm³/rev). One cubic inch is 16.387 cc. Commercial Intertech describes its gear pumps exactly this way: for every complete turn of the gear shaft the pump displaces a fixed amount of oil, so output flow is proportional to drive speed.
Displacement is a geometric number. It does not change with pressure. What changes with pressure, speed, oil temperature and wear is how much of that geometric volume actually reaches the outlet, which is volumetric efficiency.
Hydraulic pump formulas: US and metric
| To find | US units | Metric units |
|---|---|---|
| Output flow | gpm = in³/rev × rpm × ηv ÷ 231 | l/min = cm³/rev × rpm × ηv ÷ 1000 |
| Displacement needed | in³/rev = gpm × 231 ÷ (rpm × ηv) | cm³/rev = l/min × 1000 ÷ (rpm × ηv) |
| Speed needed | rpm = gpm × 231 ÷ (in³/rev × ηv) | rpm = l/min × 1000 ÷ (cm³/rev × ηv) |
| Input torque | lb-in = in³/rev × psi ÷ (2π × ηm) | N·m = cm³/rev × bar ÷ (20π × ηm) |
| Input power | hp = gpm × psi ÷ (1714 × ηt) | kW = l/min × bar ÷ (600 × ηt) |
| Power from torque and speed | hp = lb-in × rpm ÷ 63,025, or lb-ft × rpm ÷ 5,252 | kW = N·m × rpm ÷ 9,549 (Rexroth: 2π × T × n ÷ 60,000) |
| Overall efficiency | ηt = ηv × ηm (volumetric times mechanical) | |
These match the formula pages in the Danfoss H1 pump basic information manual (11062168), which prints the US and metric versions side by side, the Bosch Rexroth A10VO data sheet (RE 92703), the Vickers fluid power formula sheet and Permco's pump specification sheet. Permco also gives a lb-ft version of torque: lb-ft = gpm × psi × 3.06 ÷ rpm ÷ efficiency. The 231 is the number of cubic inches in a US gallon, and 1714 converts psi × gpm into horsepower.
How do I calculate pump gpm from displacement and rpm?
Multiply displacement by speed and divide by 231 for theoretical gpm, then multiply by volumetric efficiency for real flow.
US example: a Parker PGP050 gear pump
Parker's catalog HY09-030/US lists the PGP050 with a 2" gear at 5.10 cu in/rev (83.6 cc). At 1,800 rpm: 5.10 × 1,800 ÷ 231 = 39.7 gpm theoretical. Parker's own average output table shows 37.5 gpm for that pump at 1,800 rpm and 2,500 psi, so the tested pump delivers about 94% of its theoretical flow at that condition.
Metric example: a Rexroth A10VO size 45
Rexroth's data sheet lists the A10VO series 52/53 size 45 at 45 cm³/rev maximum and a maximum speed of 2,600 rpm at full displacement. Flow is 45 × 2,600 ÷ 1000 = 117 l/min, and the data sheet prints 117 l/min, because Rexroth's table values are theoretical, without efficiency.
Working backward: what displacement do I need?
Need 20 gpm at 1,800 rpm? 20 × 231 ÷ 1,800 = 2.57 cu in/rev theoretical. Allow for volumetric efficiency and you need a little more: at 96% it is 2.67 cu in/rev. Then pick the nearest catalog size and check its tested flow table at your speed and pressure, which is better than any assumed efficiency.
How much horsepower does a hydraulic pump need?
Horsepower = gpm × psi ÷ 1714, divided by overall efficiency. Eaton's load sensing training manual uses this example: 20 gpm at 2,000 psi needs 23.3 hp, and Eaton points out that the plain formula ignores mechanical friction losses. Divide by an overall efficiency and the number goes up. Using the efficiencies Eaton's heavy-duty sizing guide tells you to assume for its piston pumps (0.96 volumetric, 0.95 torque, so 0.912 overall), 23.3 hp becomes about 25.6 hp at the shaft.
In metric, the Rexroth A10VO size 45 at 2,600 rpm and 250 bar: 117 l/min × 250 ÷ 600 = 48.75 kW, which Rexroth's table rounds to 49 kW (about 66 hp). Again, that is theoretical; the real input power is higher by the efficiency.
Size the drive for maximum pressure, not typical pressure. Horsepower rises in a straight line with pressure, so a pump that is happy at 1,500 psi can stall an engine or trip a motor overload at the relief setting. If the drive cannot carry full flow at full pressure, you need a smaller pump, a lower relief setting or a pump control that limits horsepower.
How do I calculate pump input torque?
Torque in lb-in = displacement × pressure ÷ (2π × torque efficiency). Eaton's heavy-duty sizing guide uses exactly that form and tells you to use 0.95 for torque efficiency on its pumps. The Vickers formula sheet gives the theoretical version (pressure × displacement ÷ 2π) and the power version, lb-in = hp × 63,025 ÷ rpm.
For the 2" gear at 2,500 psi (the PGP051 rating; the PGP050 2" gear is rated 2,000 psi): 5.10 × 2,500 ÷ 6.283 = 2,029 lb-in theoretical, or 169 lb-ft. Metric, from Rexroth: size 45 at 250 bar is 45 × 250 ÷ (20π) = 179 N·m, the figure Rexroth prints. Torque is what breaks shafts and couplings. Muncie uses a simpler shaft check for PTO pumps, displacement × pressure, and publishes a limit for each spline; our PTO dump pump guide works through it.
What are volumetric and mechanical efficiency?
- Volumetric efficiency (ηv) is actual flow divided by theoretical flow. Vickers defines it on its formula sheet as flow at the given pressure divided by flow at zero pressure, times 100. It is the internal leakage figure: worn pumps lose it first.
- Mechanical (torque) efficiency (ηm) is theoretical torque divided by the torque you actually have to put into the shaft. Friction in bearings, seals and sliding faces takes the difference.
- Overall efficiency (ηt) is the two multiplied together, and it is what goes in the horsepower formula.
Efficiency is not one number. It moves with speed, pressure and oil temperature. Parker's tested flow table for the PGP050 at 2,500 psi shows it plainly:
| PGP050 gear | Speed | Theoretical flow | Parker tested flow at 2,500 psi | Volumetric efficiency (our arithmetic) |
|---|---|---|---|---|
| 1" (2.55 in³) | 900 rpm | 9.9 gpm | 8.5 gpm | about 86% |
| 1" (2.55 in³) | 1,800 rpm | 19.9 gpm | 18 gpm | about 91% |
| 2" (5.10 in³) | 900 rpm | 19.9 gpm | 17.5 gpm | about 88% |
| 2" (5.10 in³) | 1,800 rpm | 39.7 gpm | 37.5 gpm | about 94% |
| 2" (5.10 in³) | 2,400 rpm | 53.0 gpm | 51 gpm | about 96% |
Look at the 2" gear: the shortfall stays between about 2.0 and 2.4 gpm at every speed in Parker's table, so the same leakage eats a bigger share of the flow at low rpm. That is why a gear pump that is fine at full throttle can feel weak at idle. For planning numbers where you have no curve, Eaton's heavy-duty hydrostatic sizing guide uses 0.96 pump volumetric efficiency, 0.95 pump torque efficiency, 0.97 volumetric efficiency for a fixed motor and 0.94 for a variable motor. Those are Eaton's figures for Eaton piston units. Use the maker's own curve for anything else.
On a pump already in service, falling volumetric efficiency shows up as slow functions and extra heat. On a piston pump, the case drain flow test measures that leakage directly.
Fixed or variable displacement: which do I need?
A fixed displacement pump (most gear pumps and many vane pumps) moves the same volume every turn, so flow only changes with drive speed. A variable displacement pump (usually a piston pump) changes its stroke, so it can deliver anything from zero to full flow at the same shaft speed. Eaton's load sensing manual lays out the three classic circuits:
| Circuit | Pump | What happens at idle or stall | Trade-off |
|---|---|---|---|
| Open center | Fixed, usually a gear pump | Flow goes back to tank through the valve; at maximum pressure full flow goes over the relief valve | Simple and cheap; wastes horsepower and makes heat at relief |
| Closed center, pressure compensated | Variable piston pump with a compensator | Valve blocks flow; the compensator destrokes the pump and it stands by at maximum pressure | No relief losses; but the pump holds high pressure all the time |
| Load sensing | Variable piston pump with a pressure-flow compensator and a load sensing valve | Pump stands by at low pressure (200 psi on Eaton's example) and delivers only the flow and pressure the work needs | Most efficient; more parts and a signal line to look after |
More detail on the third type is in load sensing hydraulic pumps explained, and how to set a compensator is in how to adjust hydraulic pump pressure.
Can you deadhead a hydraulic pump?
A fixed displacement pump cannot be deadheaded safely without a relief valve: it has to put its full flow somewhere. With a relief valve, deadheading means every gallon goes over the relief, and all of that power turns into heat. Eaton's example: a 20 gpm pump blowing over a 3,000 psi relief wastes 3,000 × 20 ÷ 1714 = 35 hp, and since one horsepower-minute is 42.4 BTU, that is 1,484 BTU of heat every minute. Without a working relief, Commercial's failure guide lists what follows: cracked housings, broken drive shafts, failed bearings and coined or cracked thrust plates.
A pressure compensated variable pump is built to deadhead. At the compensator setting it destrokes to near zero flow and only makes up internal leakage. In Eaton's example that is a maximum of 2.7 gpm at 3,000 psi, or 4.7 hp, against the 35 hp of the open center system.
Can a variable displacement pump be used in a closed center circuit?
Yes. That is what the closed center circuit is for: Eaton describes it as offering variable flow from zero to full and removing the need for a system relief, because the compensator caps pressure. Put a fixed pump on a closed center valve without an unloading arrangement and you are running against the relief every time the valve is centered.
How do I read displacement off a pump nameplate?
Most model codes carry the displacement, but each maker codes it differently. These are the ones we have factory data for:
| Pump | Where displacement is in the code | Example | Decoder |
|---|---|---|---|
| Rexroth A10VSO, A10VO | Size number = maximum cm³/rev | Size 45 = 45 cm³/rev | A10VSO / A10VO decoder |
| Danfoss / Sauer Series 90 | Frame size = maximum cc/rev | 100 = 100 cc (6.10 in³) | Series 90 decoder |
| Vickers 20V, 25V, 35V, 45V vane | Ring code = rated gpm at 1,200 rpm and 100 psi | 25V21 = 21 gpm, 4.13 in³ (67 cc) | Vickers model number guide |
| Commercial / Parker P30-P76, PGP | Two digits on each gear housing code = gear width; width × displacement per inch | P50 "20" = 2" gear = 5.10 in³ | Commercial / Parker gear pump ID |
| Commercial C101/102 and G101/102 dump pumps | Chelsea code suffix = gear width | C101/102-25 = 2.5" gear = 6.38 in³ | PTO dump pump guide |
Watch the Vickers vane code: it is a rated flow, not a displacement. The 25V21 has 4.13 cu in, so theoretical flow at 1,200 rpm is 21.5 gpm and the rated 21 gpm is that less leakage at 100 psi. The ring code is also stated in the Vickers 35V overhaul manual (I-3197-S, PDF). For other brands, start at identifying a Vickers pump, the Denison decoder or identifying a pump with no tag.
How do I size a hydraulic pump, step by step?
- Define the work. For a motor drive, the torque and speed the load needs. Danfoss's advice for hydrostatic drives is to size the motor first for the maximum torque, then pick the pump as the flow source for the maximum motor speed. For a cylinder, the force (pressure × area) and the speed you want.
- Turn it into flow. Motor flow in gpm = motor in³/rev × motor rpm ÷ 231 (Vickers gives motor speed as rpm = 231 × gpm ÷ in³/rev). Cylinder flow in gpm = 720 × speed in ft/sec × piston area in in² ÷ 231, from the same Vickers sheet.
- Find the pump speed. Electric motor speed, or engine speed times the drive ratio (Eaton writes pump speed = engine speed × input drive ratio; on a truck PTO it is engine rpm × PTO %).
- Calculate displacement. in³/rev = gpm × 231 ÷ (rpm × volumetric efficiency). Pick the nearest catalog size and check its tested flow at your speed and pressure.
- Check the speed limits. Eaton's guide checks pump speed at the highest engine speed the machine can reach, including when an overrunning load drives the engine up, against the pump's maximum rated speed. Overspeed is a cavitation cause on Commercial's list.
- Check horsepower and torque at maximum pressure. hp = gpm × psi ÷ (1714 × overall efficiency); torque = in³ × psi ÷ (2π × torque efficiency). Make sure the engine, motor, coupling and shaft can carry it.
- Plan for heat. Every horsepower that does not do work becomes heat, at 42.4 BTU per horsepower-minute. Relief losses, throttling and leakage all count.
Frequently asked questions
What is the formula for hydraulic pump gpm?
Gpm equals displacement in cubic inches per revolution times rpm, divided by 231, times volumetric efficiency. In metric, liters per minute equals displacement in cc per revolution times rpm, divided by 1000, times volumetric efficiency. A Parker PGP050 with a 2 inch gear (5.10 cubic inches) at 1,800 rpm is 39.7 gpm theoretical, and Parker tests it at 37.5 gpm at 2,500 psi.
How much horsepower does it take to drive a hydraulic pump?
Horsepower equals gpm times psi divided by 1714, divided by overall efficiency. Eaton's example is 20 gpm at 2,000 psi, which needs 23.3 hp before friction losses, and about 25.6 hp at an overall efficiency of 0.912. Always size the drive at the maximum pressure the relief or compensator allows.
What does pump displacement mean?
Displacement is the volume of oil the pump moves per shaft revolution, in cubic inches or cc per revolution. It is fixed by the pump's geometry. Flow is displacement times speed, less internal leakage, so displacement tells you the size of the pump and flow tells you what it delivers at a given speed.
How do I convert cc per revolution to gpm?
Multiply cc per revolution by rpm and divide by 1000 to get liters per minute, then divide by 3.785 to get US gpm. Or divide cc by 16.387 to get cubic inches and use cubic inches times rpm divided by 231. A Rexroth size 45 pump at 2,600 rpm is 117 liters per minute, about 30.9 gpm, theoretical.
What is the difference between fixed and variable displacement pumps?
A fixed displacement pump moves the same volume every revolution, so flow only changes with speed and excess flow goes over a relief or back to tank. A variable displacement pump changes its stroke, so a compensator can cut flow to what the circuit needs and stand by at pressure with almost no flow. Variable pumps run cooler in closed center and load sensing systems but cost more and have more to adjust.
Can you deadhead a positive displacement pump?
Only against a relief valve, and then all the flow turns into heat: Eaton's example of 20 gpm over a 3,000 psi relief wastes 35 hp, or 1,484 BTU per minute. Without a relief, pressure climbs until something breaks. A pressure compensated variable pump can deadhead, because it destrokes and only makes up internal leakage.
How do I calculate hydraulic pump torque?
Torque in lb-in equals displacement in cubic inches times pressure in psi, divided by 2 pi times torque efficiency. Eaton uses 0.95 for torque efficiency on its heavy-duty pumps. In metric, torque in N·m equals cc times bar divided by 20 pi times efficiency; a Rexroth size 45 at 250 bar is 179 N·m theoretical.
About the technical information on this page. Figures, procedures and part numbers here come from manufacturer service literature and are provided for reference only. They are only as accurate as the factory data, and the revision we have on file may not be the latest for your unit. Always confirm against the current manufacturer documentation for your exact model and follow safe work practices. RestoPower is not responsible for damage, injury or loss resulting from the use or misuse of this information, or from incorrect assembly, adjustment, installation or operation of any unit.
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