Introduction — Scale up as an Unavoidable Challenge

In material development using twin screw extruders, once satisfactory results are obtained on laboratory scale or pilot scale machines, the next inevitable step is scale up to production equipment. While the objective of increasing throughput is straightforward, the effects of machine enlargement on mixing behavior, thermal history, and ultimately final material properties are far from simple.

In particular, co rotating, fully intermeshing twin screw extruders simultaneously perform multiple functions within a single machine, including melting, dispersive and distributive mixing, and chemical reactions. As a result, even small changes in parameters such as screw diameter, screw speed, or degree of fill can lead to significant differences in shear history and residence behavior experienced by the material.

Fundamentals of Scale up Theory for Twin Screw Extruders

Scale up of twin screw extruders is generally discussed based on geometric similarity, using the barrel inner diameter D as the characteristic length. A key issue in this context is determining which parameter should be used as the basis for similarity.

In practice, it is tempting to attempt to preserve all operating conditions simultaneously. However, in real twin screw compounding processes, this is fundamentally impossible. Therefore, scale up theory introduces two representative limiting models to clarify the discussion.

Cubic Law — Throughput Based, Three Dimensional Scale up

The first approach to scale up is based on throughput. The internal flow volume of an extruder increases in proportion to the cube of the barrel inner diameter D. Accordingly, the throughput Q increases according to the relationship

Q ∝ D³

This concept treats the interior of the extruder as a three dimensional space and uses the volumetric processing capacity as the scaling criterion. It is commonly referred to as the cubic law. The underlying assumption is that scale up is carried out under identical screw speed and shear rate conditions.

For example, when scaling up from the KZW15 with a screw diameter of 15 mm to the KZW60 with a screw diameter of 60 mm, the screw diameter ratio becomes four. Based on the cubic law, the throughput is therefore expected to increase by a factor of 64. In other words, a substantial increase in production capacity can theoretically be achieved simply by enlarging the machine size.

Scale up based on the cubic law is highly attractive because it enables efficient improvement in productivity. On the other hand, as screw tip speed increases, the shear history per unit mass and the residence time distribution experienced by the material tend to change significantly. From the standpoint of mixing quality and property reproducibility, this requires careful consideration. In many cases, increased shear heating and excessive melt temperature rise become critical issues.

Square Law — Energy and Heat Transfer Based, Two Dimensional Scale up

The second approach to scale up is based on energy input and heat transfer. In twin screw extruders, electrical heating and heat transfer through the barrel are primarily achieved via the barrel surface. The heat transfer capacity therefore scales with surface area, which is proportional to the square of the barrel inner diameter D. As a result, the throughput Q approaches the relationship

Q ∝ D²

This concept, known as the square law, treats the extruder as a two dimensional heat and energy transfer system. The underlying assumptions are that the amount of thermal energy exchanged between the barrel and the polymer, as well as the circumferential speed at the screw outer diameter, are maintained at equivalent levels.

Similarly, when considering scale up from the KZW15 to the KZW60, the screw diameter ratio is four. Based on the square law, the corresponding increase in throughput is therefore sixteen fold, which is significantly smaller than that predicted by the cubic law.

Scale up based on the square law is effective when reproducibility of thermal history and mixing state is prioritized. However, the improvement in throughput achieved through machine enlargement is inherently limited compared with scale up based on the cubic law.

Designing Between the Square Law and the Cubic Law

In practice, operating conditions rarely correspond exactly to either the square law or the cubic law. Most real scale up conditions are established somewhere between the three dimensional throughput based criterion represented by the cubic law and the two dimensional energy based criterion represented by the square law.

Moving closer to the cubic law improves productivity but tends to reduce quality reproducibility due to increased shear heating and melt temperature rise. Conversely, approaching the square law helps preserve material history but limits gains in production capacity. Scale up design is therefore the process of determining an appropriate balance within this unavoidable trade off.

Why Reproducing the Same Melt Temperature Is Difficult

One of the most frequently discussed issues during scale up is the reproducibility of melt temperature at the die. Melt temperature is determined by a complex combination of shear heating generated by screw rotation and heating or cooling through the barrel. It cannot be controlled by a single operating variable.

As screw diameter increases, internal flow volume grows with the cube of D, while barrel surface area increases only with the square of D. Because of this geometric disparity, the effectiveness of heat transfer relative to the total material volume decreases. Consequently, even if identical temperature settings and screw speeds are applied, reproducing the same thermal history observed on a smaller machine becomes increasingly difficult. This is a fundamental reason why scale up is often regarded as challenging.

The Choice Between Machine Enlargement and Parallelization

From a theoretical standpoint, if reproducibility of material quality is the highest priority, installing multiple twin screw extruders of the same size in parallel can be a rational option. This approach allows shear history, residence time, and thermal history to be maintained almost identically, thereby reducing quality related risks.

On the other hand, considerations such as installation space, operational efficiency, and capital investment often favor consolidated production using a single large scale machine. In such cases, scale up cannot be achieved by applying scaling laws alone. Instead, it becomes essential to redesign the process as a whole, including optimization of screw segment configuration, adjustment of the L over D ratio, screw speed control, and reconfiguration of side feeding, venting positions, and operating conditions.

Conclusion — Designing the Optimal Solution Between Square and Cubic Scaling

Scale up of twin screw extruders cannot be accomplished by simple geometric enlargement or straightforward application of scaling equations. As machine size increases, throughput, shear history, residence behavior, and heat transfer conditions all change simultaneously, making it both theoretically and practically impossible to match all performance indicators perfectly.

The cubic law represents a three dimensional, throughput based scaling concept, while the square law represents a two dimensional, energy based concept. Actual twin screw compounding processes do not strictly follow either model but instead operate somewhere between them.

Prioritizing productivity pushes operation toward the cubic law, whereas emphasizing reproducibility of material history shifts it toward the square law. This trade off is unavoidable, which is precisely why scale up is both difficult and intellectually engaging.

Ultimately, scale up of twin screw extruders is a design process in which the engineer selects the most appropriate operating point from countless possibilities between the square and cubic laws. It is here that the true expertise of the extrusion engineer becomes evident.

Twin Screw Extruders by Technovel
– A Dedicated Japanese Extruder Manufacturer

　

As a world-first innovation, we have developed a new type of horizontal multi screw extruder. Our Quad screw and Octa screw extruders are already being utilized across various industries. Even in the field of conventional twin screw extruders, we offer a wide lineup ranging from ultra-compact models—such as the world’s smallest 6 mm diameter screw and our best-selling 15 mm model—to large-scale production machines.

Japanese Extrusion Experts, Always Advancing