Single-Screw Extruder Throughput Calculation

Introduction & Context

The single-screw extruder is a fundamental piece of equipment in polymer processing, food extrusion, and chemical engineering. Throughput calculation determines the mass flow rate of material being extruded, which is critical for:

Process Design: Sizing screws, dies, and downstream equipment.
Operational Control: Ensuring consistent product quality and output.
Energy Efficiency: Optimizing screw speed and pressure drop to minimize power consumption.

This calculation balances two competing effects: Drag flow (material pushed forward by the rotating screw) and Pressure flow (backflow due to resistance in the die). The net throughput is the difference between these two terms.

Methodology & Formulas

1. Drag-Flow Term (\(Q_{\text{drag}}\))

Drag flow represents the forward transport of material due to the screw's rotational motion. It depends on screw geometry (diameter \(D\), channel width \(W\), flight height \(H\), and flight angle \(\theta\)) and rotational speed (\(N\)):

\[ Q_{\text{drag}} = \frac{\pi D N W H \cos(\theta)}{2} \]

where:
\(N\) is in radians per second (convert from RPM using \(N_{\text{rad/s}} = N_{\text{RPM}} \cdot \frac{2\pi}{60}\)).

2. Pressure-Flow Term (\(Q_{\text{press}}\))

Pressure flow accounts for the backward leakage of material caused by the pressure gradient (\(\Delta P\)) along the screw channel. It is derived from the lubrication approximation for viscous flow in a shallow channel:

\[ Q_{\text{press}} = \frac{W H^3 \Delta P}{12 \mu L} \]

where:
\(\mu\) = dynamic viscosity (Pa·s),
\(L\) = axial length of the process section (m).

3. Net Volumetric Flow (\(Q_{\text{net}}\))

The net throughput is the difference between drag and pressure flow:

\[ Q_{\text{net}} = Q_{\text{drag}} - Q_{\text{press}} \]

4. Mass Flow Rate (\(\dot{m}\))

Convert volumetric flow to mass flow using the melt density (\(\rho\)):

\[ \dot{m} = Q_{\text{net}} \cdot \rho \]

Common units:
\(\dot{m}\) in kg/s or kg/h (multiply by 3600 to convert from kg/s to kg/h).

5. Validity Checks

The following empirical criteria ensure the calculation remains valid for Newtonian, laminar flow regimes:

Parameter	Formula	Validity Criterion	Warning Threshold
Reynolds Number (\(Re\))	\(Re = \frac{\rho V_{\text{char}} H}{\mu}\)	Laminar flow (\(Re < 2000\))	\(Re > 2000\)
Shear Rate (\(\dot{\gamma}\))	\(\dot{\gamma} = \frac{V_{\text{char}}}{H}\)	Newtonian behavior (\(\dot{\gamma} < 10^4\) s\(^{-1}\))	\(\dot{\gamma} > 10^4\) s\(^{-1}\)
Die Pressure Drop (\(\Delta P\))	—	Mechanical limits (\(\Delta P < 5\) bar)	\(\Delta P > 5\) bar

Note: \(V_{\text{char}}\) is the characteristic velocity, approximated as \(V_{\text{char}} = \frac{Q_{\text{drag}}}{A_{\text{die}}}\), where \(A_{\text{die}} = \frac{\pi D^2}{4}\).

6. Assumptions & Limitations

Newtonian Fluid: Assumes viscosity (\(\mu\)) is constant (independent of shear rate).
Isothermal Flow: Neglects viscous heating and temperature gradients.
Fully Developed Flow: Assumes no entrance/exit effects in the screw channel.
No Slip: Material velocity at the screw surface equals screw speed (no wall slip).
Shallow Channel: Lubrication approximation requires \(H \ll W\).

For non-Newtonian fluids (e.g., polymer melts), replace \(\mu\) with an effective viscosity or use the Power-Law model.

The theoretical throughput (Q) can be estimated with the following steps:

Determine the screw channel geometry: measure the screw diameter (D) and the effective channel depth (L).
Obtain the screw rotational speed (N) in revolutions per minute.
Calculate the volumetric displacement per revolution: V = (π × D² ÷ 4) × L.
Convert the displacement to mass flow by multiplying with the melt density (ρ) and the overall screw efficiency (η, typically 0.8‑0.9): Q = V × N × ρ × η.
Express Q in the desired units (e.g., kg/h) by applying the appropriate time conversion.

Several practical influences can reduce or increase the real throughput:

Material melt viscosity variations due to temperature gradients or shear‑thinning behavior.
Screw wear, wear‑induced clearance changes, or damaged flight geometry.
Back‑pressure from downstream equipment (die, cooling, or cutting).
Inadequate barrel heating or uneven temperature profile along the barrel.
Presence of fillers, reinforcements, or moisture that alter flow characteristics.

To account for viscosity effects, follow these steps:

Obtain the material’s viscosity model (e.g., Carreau‑Yasuda) from the supplier data sheet.
Measure or estimate the average melt temperature (T) and shear rate (γ̇) in the screw channel.
Calculate the apparent viscosity (η_app) using the chosen model and the measured T and γ̇.
Adjust the screw efficiency (η) in the theoretical formula by the ratio η_app / η_ref, where η_ref is the reference viscosity used for the baseline efficiency.
Re‑compute the throughput with the updated efficiency value.

An on‑line validation routine typically includes:

Installing a calibrated mass flow meter or a gravimetric collection system at the extruder exit.
Recording the screw speed (N) and barrel temperature profile simultaneously.
Comparing the measured mass flow rate with the calculated theoretical value.
Adjusting process parameters (e.g., temperature zones, screw speed) to bring the measured flow within an acceptable tolerance (usually ±5 %).
Documenting the deviation and any corrective actions for future reference.

Worked Example: Estimating the Throughput of a 50 mm Laboratory Extruder

A process engineer is commissioning a single-screw extruder for a new bio-compound that has a bulk density of 1.5 t m^-3. The 50 mm diameter screw has a 20 mm channel width, a 10 mm metering depth, and a 30° helix angle. The metering section is 0.5 m long, and the screw is scheduled to run at 200 rpm. The resin behaves as a Newtonian fluid with a viscosity of 0.5 Pa·s, and the head pressure is expected to be 1 bar. Determine whether the extruder will deliver a positive net flow under these conditions.

Knowns

Screw diameter, \(D = 0.050\) m
Channel width, \(W = 0.020\) m
Channel depth, \(H = 0.010\) m
Helix angle, \(\theta = 30°\) (0.524 rad)
Metering length, \(L = 0.500\) m
Screw speed, \(N = 200\) rpm
Pressure rise, \(\Delta P = 1\) bar = \(1 \times 10^5\) Pa
Melt viscosity, \(\mu = 0.5\) Pa·s
Bulk density, \(\rho = 1500\) kg m^-3

Step-by-step calculation

Convert screw speed to rad s^-1:
\(N = 200 \times 0.1047 = 20.944\) rad s^-1
Compute drag-induced volumetric flow:
\(Q_{\text{drag}} = \frac{1}{2} W H V_{\text{char}}\)
where \(V_{\text{char}} = \pi D N \sin\theta = \pi \times 0.05 \times 20.944 \times \sin 30° = 1.646\) m s^-1
\(Q_{\text{drag}} = 0.5 \times 0.02 \times 0.01 \times 1.646 = 1.646 \times 10^{-4}\) m³ s^-1
Compute pressure-induced back-flow:
\(Q_{\text{press}} = \frac{W H^3 \Delta P}{12 \mu L} = \frac{0.02 \times 0.01^3 \times 1 \times 10^5}{12 \times 0.5 \times 0.5} = 6.667 \times 10^{-4}\) m³ s^-1
Net volumetric throughput:
\(Q_{\text{net}} = Q_{\text{drag}} - Q_{\text{press}} = 1.646 \times 10^{-4} - 6.667 \times 10^{-4} = -5.021 \times 10^{-4}\) m³ s^-1
Convert to mass flow:
\(\dot{m} = Q_{\text{net}} \rho = -5.021 \times 10^{-4} \times 1500 = -0.753\) kg s^-1
\(\dot{m} = -0.753 \times 3600 = -2710.8\) kg h^-1

Final Answer

The negative sign indicates back-flow; under the stated 1 bar head pressure, the extruder cannot deliver a positive throughput. The machine would lose approximately 2711 kg h^-1 of melt back through the screw channel, so either the screw speed must be raised or the head pressure reduced to achieve a viable production rate.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle