Piezo Drop-on-Demand Printhead

What is modelled

A piezo wall squeezes the ink chamber. The chamber behaves as a damped acoustic resonator near 24 kHz; the pulse width chooses how well the pressure wave builds up. Pressure above the capillary threshold pushes ink out through the nozzle and part of it pinches off as a drop.

Equations

• Resonator: p'' + 2ζω0 p' + ω02 p = ω02 K V(t) with ζ = 0.06 + 0.015 μ (mPa s).
• Nozzle speed: v = cd √(2 (p - 4σ/d)/ρ).
• Printability: Z = 1/Oh = √(ρ σ d)/μ, printable roughly for 1 < Z < 14.
• Optimal pulse: tw = π/ωd (half the ring period).

Checks run in the page

• Ring-down frequency against f₀√(1-ζ²).
• Scan of pulse width finds the maximum near π/ω_d.
• A long pulse settles to K V.
• Rayleigh-Plateau drop to jet ratio 1.89, and the 20 % pinch-off bookkeeping.

Honest simplifications

A reduced-order lumped model, not CFD. A fixed 20 % of the pushed volume becomes the drop (retraction and ligament pinch-off are lumped), only the first pressure excursion after a pulse can eject ink, there is no air drag and the viscous loss in the nozzle is represented only through damping.

• Preset "Wrong pulse width" (42 us): the pressure wave cancels itself and little ink is ejected.
• Preset "Too viscous": Z falls below 1 and the head stops being printable.
• Increase the frequency until the next pulse arrives while the chamber still rings.