Flow Dynamics Calibration

Physics of Viscous Polymer Flow and Extruder Pressure

During FDM printing, the extruder drives solid filament into the hotend melt zone, where it liquefies and is forced through a small nozzle orifice. Molten polymers are not ideal fluids; they are non-Newtonian viscoelastic fluids that exhibit shear-thinning behavior and elasticity. This elasticity acts like a mechanical spring inside the hotend.

1.1 The Pressure Lag Phenomenon

When the print head accelerates to print a line, the extruder motor pushes filament into the hotend. However, because of the elasticity of the solid filament, the compressibility of the melt zone, and the high flow resistance of the nozzle orifice, the pressure does not build up instantaneously. Conversely, when the toolhead decelerates at the end of a line, the residual pressure in the melt zone continues to force plastic out of the nozzle, even if the extruder motor stops spinning.

Without compensation, this pressure lag leads to specific print defects:

• During Acceleration: Under-extrusion occurs, resulting in thin, starved line starts.
• During Deceleration / Corners: Over-extrusion occurs, resulting in thick, bulging corners and blobbing.

Without Pressure Advance:
Start (Slow)       Middle (Fast)       End (Slow)
  [ Bulge ] --------- [ Thin ] --------- [ Bulge ]

With Pressure Advance:
  ================================================  (Uniform Width)

1.2 Mathematical Model of Pressure Advance (K-Factor)

To resolve this, modern firmwares (such as Klipper and Bambu OS) implement Pressure Advance (also known as Linear Advance). This algorithm models the relationship between nozzle pressure ($P$) and extrusion speed ($v$) as a linear function.