Radial-Piston Pump

What is modelled

A shaft with an eccentric cam turns inside a ring of radial cylinders. Each piston is pushed in while the cam moves toward it, delivering fluid, and drawn out again on the other half turn. One piston delivers a half-sine pulse, so several pistons spaced evenly around the ring add up to a flow that is nearly, but not exactly, constant.

Equations

• Piston speed vi = e ω sin(θ − 2πi/z), delivering only while it is positive.
• Mean flow Q = z A (2e) n with piston area A = π d²/4.
• Ripple δ = (Qmax − Qmin) / Qmean: (π/2z) tan(π/4z) for odd z, (π/z) tan(π/2z) for even z.
• Shaft torque T = Σ p A e sin(θi) over the discharging pistons; hydraulic power p Q.
• Leakage taken as laminar, Qleak = kl p.

Checks run in the page

• The mean of the summed piston flows against the displacement formula.
• The numerical ripple against the closed form for odd and even z.
• The integral of one piston velocity against the stroke 2e.
• The mean of torque times speed against the hydraulic power.

Honest simplifications

The cam is treated as a pure sine motion (piston speed eω sinθ, a small-eccentricity approximation), the fluid is incompressible, there are no valve timing losses, pressure ripple and pulsation of the load, and leakage is a single linear coefficient. Use it for the trends in ripple and efficiency, not for sizing a pump.

• Press Four pistons (even) and Seven pistons (smoother) and watch the ripple jump between about 33 % and 1.6 %: odd counts overlap the half-sine pulses better.
• Press High pressure, worn to see the volumetric efficiency fall as leakage grows with pressure.
• Raise the eccentricity and watch the flow and the torque scale with the stroke.