How it works — slider-crank kinematics
A crank and connecting rod drive each plunger. With crank radius \(r\), rod length \(L\), ratio \(C=L/R\) and crank angle \(\theta\), the plunger velocity is
The \(\sin 2\theta\) term is the finite-rod distortion: it sharpens the velocity peak and shifts it ahead of 90° — smaller C, sharper peak, bigger ripple. One cylinder's instantaneous delivery is \(q(\theta)=A\,v(\theta)\) on its discharge stroke; the simulator sums the contributions of \(N\) cylinders — phased at \(2\pi/N\) single-acting, and at \(\pi/N\) double-acting (a duplex DA carries its two cranks 90° apart):
$$q_{SA}(\theta)=\sum_{k=0}^{N-1}\max\!\bigl(0,\;v(\theta-2\pi k/N)\bigr)\qquad q_{DA}(\theta)=\sum_{k=0}^{N-1}\bigl|\,v(\theta-\pi k/N)\,\bigr|$$The pulsation amplitude is measured against the mean flow \(\bar q\):
$$\text{above}=\frac{q_{max}-\bar q}{\bar q}\qquad \text{below}=\frac{\bar q-q_{min}}{\bar q}\qquad \text{total}=\frac{q_{max}-q_{min}}{\bar q}$$| Symbol | Meaning | Units |
|---|---|---|
| \(r,\ L,\ C\) | crank radius, rod length, ratio \(C=L/R\) | m, m, — |
| \(\omega,\ \theta\) | crank angular speed and angle | rad/s, ° |
| \(v,\ X\) | plunger velocity and displacement | m/s, m |
| \(q,\ \bar q\) | instantaneous combined flow and its mean | m³/s |
| \(N\) | number of cylinders (throws), phased \(2\pi/N\) | — |
| \(A\) | plunger area (cancels in the normalised curve) | m² |
1 · Cylinder count rules the ripple (C ≈ 6:1)
The classic pulsation table, reproduced by the simulator within a percent — note how the odd counts beat both even neighbours (triplex 23% sits between duplex 46% and quadruplex 33%):
| Pump type | Plungers | Above mean | Below mean | Total | Phase angle |
|---|---|---|---|---|---|
| Duplex (double-acting) | 2 | 24% | 22% | 46% | 180° |
| Triplex | 3 | 6% | 17% | 23% | 120° |
| Quadruplex | 4 | 11% | 22% | 33% | 90° |
| Quintuplex | 5 | 2% | 5% | 7% | 72° |
| Sextuplex | 6 | 5% | 9% | 14% | 60° |
| Septuplex | 7 | 1% | 3% | 4% | 51.5° |
| Nonuplex | 9 | 1% | 2% | 3% | 40° |
That is why power-pump crankshafts always take an odd number of throws — the duplex double-acting, whose two cranks stand 90° apart and whose rod-side stroke is counted without rod-area correction in the classic table, is the standing exception.
2 · The L/R ratio is the fine adjustment (triplex)
| C = L/R | Above mean | Below mean | Total |
|---|---|---|---|
| 4 : 1 | 8.2% | 20.0% | 28.2% |
| 5 : 1 | 7.6% | 17.6% | 25.2% |
| 6 : 1 | 6.9% | 16.1% | 23.0% |
| 7 : 1 | 6.4% | 15.2% | 21.6% |
A shorter rod (small C) steepens the velocity peak and worsens the ripple; a longer rod smooths the flow but stretches the frame. The usual industrial compromise is 4:1–6:1 — C also enters volumetric efficiency, dampener sizing, the liquid-separation speed and the acceleration head, so it is never chosen on pulsation alone.
3 · When the ripple is too big — dampeners and acoustic filters
Above 25% total pulsation, fit a discharge dampener (🔴); 10–25% is workable with sturdy piping (🟡); ≤ 10% is smooth (🟢). A bladder dampener precharged to 50–66% of system pressure can attenuate pulsations below ~50 Hz by up to 90%, and a well-designed suction stabiliser makes the suction line behave like only 5–15 pipe diameters of straight pipe. For stubborn cases an acoustic low-pass filter — a volume–choke–volume Helmholtz device — works when its natural frequency sits at or below half the plunger frequency:
a = speed of sound in the liquid, d = choke diameter, D = chamber diameter, L = chamber and choke length. Pipe runs themselves have organ-pipe modes, \(f_N=Na/2L\) (open–open or closed–closed) and \(f_N=Na/4L\) (closed–open): a pulse frequency landing on one of them can multiply vibration twenty-fold — check before blaming the pump.
4 · The suction side pays twice
The kinematics that ripple the delivery also swing the suction pressure. Acceleration head — the pressure spent accelerating the liquid column in the suction line at every stroke — can be the largest single NPSHA component, in bad layouts up to ten times the sum of all other suction losses. Keep the effective NPSHA comfortably above NPSHR with the pulsation margin included, and remember the absolute pressure at the plunger must stay above vapour pressure through the whole cycle, not just on average.
Worked example (anchor)
Triplex single-acting, C = 6:1, 300 rpm. The simulator reproduces the table row:
| Quantity | Simulator | Classic table |
|---|---|---|
| Above mean | ≈ +6% | 6% |
| Below mean | ≈ −17% | 17% |
| Total pulsation | ≈ 23% | 23% |
| Discharge pulses / rev | 3 | 3 (fundamental) |
| Pulse frequency | 3 × 300 / 60 = 15 Hz | — |
| Verdict | 🟡 10–25% band — suction stabiliser pays on long suction lines | |
Slide to quintuplex and the same readouts fall to ≈ 2% / 5% / 7% (🟢); slide to duplex double-acting and they climb to ≈ 24% / 22% / 46% (🔴 — dampener required).
Engineering criteria applied
- Pulsation traffic light — total > 25% 🔴 dampener required; 10–25% 🟡 moderate; ≤ 10% 🟢 smooth.
- Throw-count rule — odd counts smoother than the neighbouring even ones; crankshafts take odd throws except the classic duplex.
- L/R band — industrial 4:1–6:1; the slider spans 3–8 to show the full sensitivity.
- Dampener design cues — bladder precharge 50–66% of system pressure; acoustic-filter cutoff ≤ ½ plunger frequency; relief valves set +10% (triplex and up) or +25% (duplex DA) above operating pressure so normal ripple never lifts them.
- NPSH — acceleration head may reach 10× all other suction losses; effective NPSHA must clear NPSHR with the pulsation margin included.