Peristaltic Pump Flow

What is modelled

A rotor carries rollers that pinch a flexible tube against a curved housing. Fluid trapped between two pinch points is pushed ahead at the roller speed. As each roller leaves the tube the tube springs back and briefly draws fluid backward, so the outlet flow shows a dip once per roller passage. A small gap left under a roller lets some fluid slip back when the outlet pressure is high.

Equations

• Ideal flow Qi = A ω R, with A the bore area and R the roller path radius, so each revolution moves A · 2π R.
• Slip through the residual gap (Poiseuille slit) Qleak = w h³ Δp / (12 μ L), shared between the n_eng rollers engaged in series.
• Net flow Q = Qi (1 − dip) − Qleak; efficiency ηv = Q / Qi; pulse frequency f = n N / 60.
• Pressure at which the flow stops Δp0 = Qmean / G, with G the leak conductance.

Checks run in the page

• Swept volume per revolution against the integrated waveform.
• Slit leakage by numerical integration of the parabolic velocity profile.
• The peak of a discrete Fourier transform at rollers x revolutions per second.
• Zero-flow pressure by bisection against the closed form.

Honest simplifications

The dip is a smooth empirical shape with an adjustable depth, not a computed tube-collapse model. Tube wall elasticity, hysteresis, fatigue, fluid compressibility, pulsation damping in the pipework and wear are ignored, and the residual gap is a simple function of occlusion. Real pumps need calibration against measured flow.

• Raise the back-pressure until the net flow in the third plot drops to zero.
• Choose Two rollers, fast: the pulse rate follows the roller count and the ripple is bigger.
• Reduce the occlusion: slip leakage rises quickly because it grows with the cube of the gap.