Flow is geometry, power is the load
A positive displacement pump moves a fixed volume every turn. One cubic inch per revolution at 1,800 rpm sweeps 1,800 cubic inches a minute, and a gallon is 231 cubic inches, so that is 7.79 GPM before any losses. At 92 percent volumetric it delivers 7.17 GPM. Nothing about the pressure changes that number, which is the first thing people get wrong about hydraulics — a pump makes flow, and pressure is whatever the load puts in the way of it.
Power is where the load appears. GPM times psi divided by 1,714 is hydraulic horsepower: 7.17 GPM at 2,000 psi is 8.36 hp of useful output, and at 85 percent overall the shaft is taking 9.84 hp to produce it. Drop the load to 500 psi and the same pump at the same speed is taking 2.46 hp. That four-to-one swing happens within one second of a bucket touching the pile.
Torque does not care how fast the shaft turns
Work per revolution is displacement multiplied by pressure — 1 cubic inch at 2,000 psi is 2,000 inch-pounds of work per turn, and dividing by 2π gives 318 lbf-in, or 26.5 lbf-ft, of shaft torque. Change the speed and the torque stays put; only the horsepower moves, because horsepower is torque multiplied by speed. That is why a pump on a small engine can stall it at low idle and pull the same pressure fine at 1,800 rpm.
The default form with its 85 percent overall and 92 percent volumetric lands at 28.7 lbf-ft rather than 26.5, because the mechanical share of the losses — overall divided by volumetric — is the part that shows up as extra torque on the shaft rather than as missing oil.
What the pump throws away by itself
Eight percent slip on the default is 0.62 GPM going nowhere useful, and the 15 percent overall loss is 1.48 horsepower. That 1.48 hp is 3,755 BTU an hour of heat put into the oil by the pump alone, before a single valve or hose has done anything. It is the floor of the cooling problem and it exists whenever the pump is turning.
Put the flow over a relief valve and the number changes character entirely. Deadheaded at 2,000 psi, the whole 8.36 hp of hydraulic output goes to heat: 21,281 BTU an hour, well over five times the pump loss. A machine idling with a closed-centre valve and a fixed pump is doing exactly this, which is the usual explanation for oil that runs hot with nothing on the machine moving.
Two efficiencies, and why one caps the other
Volumetric efficiency is about oil: how much of the swept volume actually leaves the port instead of slipping back past clearances. Overall efficiency is about energy: how much of the shaft power comes out as hydraulic power. Overall is always the lower of the two, because it carries the leakage plus the friction, and the page holds overall down to volumetric if you enter it higher — hydraulic output cannot be more than the delivered flow carries.
Both numbers come off a curve for your model at your pressure and your speed, and both fall with wear. A worn pump betrays itself in the volumetric figure first: same shaft speed, same noise, less oil, and a cycle time that has crept up by a couple of seconds over a season.
Questions people ask
How do I convert pump displacement to GPM?
Multiply cubic inches per revolution by rpm and divide by 231. One cubic inch per revolution at 1,800 rpm is 7.79 GPM swept. Then multiply by the volumetric efficiency from the pump curve at your pressure — at 92 percent that is 7.17 GPM actually delivered. If your displacement is in cc/rev, divide by 16.387 to get cubic inches first.
What is the formula for hydraulic horsepower?
GPM multiplied by psi, divided by 1,714. That is the power leaving the pump port. To get shaft power you divide by the overall efficiency: 7.17 GPM at 2,000 psi is 8.36 hydraulic hp, and at 85 percent overall the shaft is absorbing 9.84 hp. The 1,714 is not arbitrary — it falls out of converting gallons and psi into foot-pounds per minute against the 33,000 in a horsepower.
Why does pump torque stay the same at different speeds?
Because torque is set by the work per revolution, which is displacement times pressure, and neither of those changes with speed. One cubic inch per revolution at 2,000 psi is 318 lbf-in of shaft torque at 600 rpm and at 3,000 rpm alike. What changes with speed is horsepower, which is torque times rpm, and that is why a pump that runs fine at full throttle can stall an engine at idle.
How much heat does a hydraulic pump make?
Whatever the overall efficiency loses, and one horsepower is 2,544 BTU an hour. A pump taking 9.84 hp at the shaft to deliver 8.36 hydraulic hp is putting 1.48 hp, or 3,755 BTU an hour, into the oil. That is the floor. Flow crossing a relief valve does far more, because none of that power does work — the full hydraulic output turns into temperature.
What is the difference between volumetric and overall efficiency?
Volumetric compares oil out with oil swept, so it measures leakage. Overall compares energy out with energy in, so it carries leakage plus friction and is always the lower figure. Both come from the pump curve at your working pressure and speed rather than being properties of the pump, and both fall as clearances open up with hours on the machine.
Does a bigger pump mean higher pressure?
No. Displacement sets flow, and flow sets speed at a cylinder or a motor. Pressure is set by the load resisting that flow and capped by a relief valve setting. A larger pump on the same load makes everything move faster at the same pressure and takes proportionally more power to do it, which is a drive sizing question rather than a pressure one.