Scale-Up Rules for Extruders

Reference ID: MET-395F | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

Scale-up of polymer extruders is a critical process engineering task that involves transitioning from laboratory or pilot-scale equipment to production-scale machinery. Because extrusion involves complex non-Newtonian fluid dynamics, heat transfer, and mechanical energy dissipation, simple geometric scaling is insufficient. Engineers must apply specific scaling rules to maintain product quality, ensure consistent melt temperature, and prevent material degradation.

These calculations are typically used during the design phase of manufacturing lines to predict the required screw speed and throughput capacity of a larger extruder based on the performance of a smaller, validated reference unit. Maintaining similarity in shear history and residence time distribution is essential for ensuring that the physical properties of the extruded product remain constant across different machine sizes.

Methodology & Formulas

The scaling methodology relies on the geometric similarity of the screw and barrel assembly, where the screw diameter D is the primary characteristic dimension. The scale ratio is defined as S = D₂ / D₁. Depending on the process requirements, one of three primary scaling regimes is selected to determine the new screw speed N₂ and throughput ṁ₂.

The power requirement P is estimated based on the Newtonian approximation, assuming constant viscosity, which scales according to the following relationship:

\[ P_{\mathrm{ratio}} = S^{3} \cdot \left( \frac{N_{2}}{N_{1}} \right)^{2} \]

The specific formulas for each scaling regime are as follows:

Scaling Regime	Screw Speed Formula	Throughput Formula
Constant Shear Rate	\( N_{2} = N_{1} \)	\( \dot{m}_{2} = \dot{m}_{1} \cdot S^{3} \)
Constant Melt Temperature	\( N_{2} = N_{1} \cdot \sqrt{\frac{D_{1}}{D_{2}}} \)	\( \dot{m}_{2} = \dot{m}_{1} \cdot S^{2.5} \)
Constant Residence Time	\( N_{2} = N_{1} \)	\( \dot{m}_{2} = \dot{m}_{1} \cdot S^{3} \)

Engineering Considerations:

Condition	Threshold/Criteria
Geometric Similarity	Assumes constant L/D ratio and channel depth proportional to D.
Scale Factor Validity	\( 0.1 \leq \frac{D_{2}}{D_{1}} \leq 10.0 \)
Operational Limits	All parameters (D, N, ṁ) must be strictly positive.

To maintain geometric similarity during scale-up, process engineers must ensure that the ratio of the screw diameter to the barrel length remains constant. Key considerations include:

Maintaining a constant ratio of screw channel depth to screw diameter.
Ensuring the flight pitch remains proportional to the screw diameter.
Keeping the clearance between the screw flight and the barrel wall consistent relative to the diameter.

The approach to screw speed depends on the specific process objective. The three primary scaling regimes and their corresponding speed adjustments are:

Constant Shear Rate: Keep screw speed constant (N₂ = N₁). Under geometric similarity, the shear rate in the screw channel is directly proportional to the screw speed, so maintaining the same speed preserves the same shear intensity.
Constant Melt Temperature: Reduce screw speed according to N₂ = N₁ · √(D₁/D₂). This balances the increased viscous dissipation in a larger machine against the reduced surface-area-to-volume ratio for heat transfer, maintaining a similar melt temperature profile.
Constant Residence Time: Keep screw speed constant (N₂ = N₁). The mean residence time in the extruder is inversely proportional to screw speed, so a constant speed ensures the material spends the same amount of time in the process section.

Note: The constant shear rate and constant residence time regimes both yield N₂ = N₁ under geometric similarity, although they represent distinct physical constraints.

As the extruder size increases, the volume of the material grows by the cube of the diameter, while the surface area grows only by the square. This leads to significant thermal challenges:

Reduced heat transfer efficiency per unit volume.
Increased risk of localized overheating in the melt.
Greater reliance on internal viscous dissipation rather than external barrel heating.

Worked Example: Scale-Up of a Single-Screw Extruder Using the Constant Melt Temperature Rule

Scenario: A process engineer needs to scale up a single-screw extruder from a laboratory-scale machine to a production-scale machine. The objective is to maintain a similar melt temperature profile between the two scales. The base extruder has a screw diameter D₁ = 50.0 mm, runs at a speed N₁ = 100.0 rpm, and achieves a throughput ṁ₁ = 20.0 kg/h. The target screw diameter is D₂ = 100.0 mm. Geometric similarity is assumed (i.e., the length-to-diameter ratio and the channel depth-to-diameter ratio remain constant). The chosen scaling rule is constant melt temperature, which is the most common industrial rule for maintaining comparable viscous dissipation and melt quality.

Knowns:

Base screw diameter, D₁ = 50.0 mm
Base screw speed, N₁ = 100.0 rpm
Base throughput, ṁ₁ = 20.0 kg/h
New screw diameter, D₂ = 100.0 mm
Scaling rule: constant melt temperature

Step-by-Step Calculation:

Calculate the geometric scale ratio.
\[ S = \frac{D_{2}}{D_{1}} = \frac{100.0\ \text{mm}}{50.0\ \text{mm}} = 2.0 \]

The scale ratio is 2.0, meaning the new extruder is twice the diameter of the base extruder.
Apply the constant melt temperature rule for screw speed.
Under this rule, the screw speed scales inversely with the square root of the diameter ratio:

\[ N_{2} = N_{1} \cdot \sqrt{\frac{D_{1}}{D_{2}}} = 100.0\ \text{rpm} \cdot \sqrt{\frac{50.0\ \text{mm}}{100.0\ \text{mm}}} \]

\[ N_{2} = 100.0\ \text{rpm} \cdot \sqrt{0.5} = 100.0\ \text{rpm} \cdot 0.7071067811865476 = 70.71067811865476\ \text{rpm} \]

Rounded to two decimal places: \(N_{2} = 70.71\ \text{rpm}\).
Apply the constant melt temperature rule for throughput.
Throughput scales with the diameter ratio raised to the power of 2.5:

\[ \dot{m}_{2} = \dot{m}_{1} \cdot (S)^{2.5} = 20.0\ \text{kg/h} \cdot (2.0)^{2.5} \]

\[ \dot{m}_{2} = 20.0\ \text{kg/h} \cdot 5.656854249492381 = 113.13708498984761\ \text{kg/h} \]

Rounded to two decimal places: \(\dot{m}_{2} = 113.14\ \text{kg/h}\).
Compute the relative power scaling factor (Newtonian approximation).
Under the assumption of constant viscosity, the power requirement scales with D³N². The relative power ratio, taking the base power as 1.0, is:

\[ P_{\mathrm{ratio}} = (S)^{3} \cdot \left(\frac{N_{2}}{N_{1}}\right)^{2} = (2.0)^{3} \cdot \left(\frac{70.71067811865476\ \text{rpm}}{100.0\ \text{rpm}}\right)^{2} \]

\[ P_{\mathrm{ratio}} = 8.0 \cdot (0.7071067811865476)^{2} = 8.0 \cdot 0.5 = 4.0 \]

The relative power ratio is 4.0.

Final Answer:

New screw speed: N₂ = 70.71 rpm
New throughput: ṁ₂ = 113.14 kg/h
Relative power requirement (base = 1.0): P_ratio = 4.0

Note: The speed reduction from 100.0 rpm to 70.71 rpm and the throughput increase from 20.0 kg/h to 113.14 kg/h reflect the constant melt temperature scaling rule, which balances viscous heating and heat transfer to maintain a similar melt temperature profile. The power ratio of 4.0 indicates that the production-scale extruder will require four times the power of the lab-scale unit under Newtonian assumptions.

"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