High-Moisture Extrusion: 2D Radial Fiber-Alignment Field

🌱 2D Radial Alignment Field (HMEC)

Process Controls

Screw speed 300 rpm

Cook-zone temperature 140 °C

Moisture content 60 %

Cooling-die temperature 45 °C

Live Readouts

Wall shear rate γ̇

0 s⁻¹

Spec. mech. energy

0 kJ/kg

Wall alignment (exit)

0.00

Core alignment (exit)

0.00

How this 2D model differs from the 3D version

The 3D version assigns every protein parcel the same single shear rate and evaluates one closed-form alignment curve along the barrel axis, regardless of where a parcel sits in the cross-section. This 2D version is a genuinely different, independently-computed model: it solves an advection-reaction-diffusion PDE on a length × radius grid, so shear, temperature and alignment all vary across the channel's radius as material flows downstream:

Local shear:     γ̇(r) = γ̇_wall · (r/R)              (drag-flow profile: 0 at core, max at wall)
Alignment PDE:   ∂S/∂t = −u·∂S/∂x + α·γ̇(r)·(1−S) − S/τ(T) + D_S·∂²S/∂r²
Heat PDE:        ∂T/∂t = −u·∂T/∂x + D_T·∂²T/∂r²,   T(x, wall) = T_zone(x)  (forced)
Relaxation:      τ(T) = τ0·exp[−β(T−T0)]

The barrel/die wall temperature is a directly-controlled boundary condition, but the channel's core has to catch up to it by radial heat diffusion — exactly like a real extrudate, whose center lags the wall thermally. Because shear vanishes at the centerline (γ̇(0)=0), the reaction term there is pure relaxation (dS/dt = −S/τ), so core material can only align by advecting in from upstream, never by local shear — a real, physically distinct failure mode the 3D single-value model cannot show at all: a well-quenched skin with a mushier, less-aligned core.

Screw speed (N) — sets the wall shear γ̇_wall linearly; the radial profile then spreads that shear from 0 at the core to the full value at the wall.
Cook temperature (T) — shortens τ(T) everywhere, but the wall reaches it first — the core only catches up as heat diffuses inward.
Moisture — lowers melt consistency, trading alignment for a softer, less energy-intensive melt (same Arrhenius-type proxy as the 3D version, for readout consistency).
Cooling-die temperature — the die-exit wall boundary condition; a colder die quenches the wall's alignment fast, but a mushy, still-hot core can persist several centimeters in if radial diffusion hasn't caught up — visible here as a lighter core streak surviving into the die.

Same real-world subject as the 3D version — this is the cook-then-quench mechanism plant-based "meat" extruders use to turn isotropic soy/pea protein into muscle-like fiber — but reached here through a resolved radial field instead of one global value broadcast to every parcel.

FeedTransitionCook (shear)Cooling die

Raw dough (isotropic) Cooked melt (shearing) Locked fiber (exiting)

↔ length (feed→die)
↕ radius (core↔wall)