Introduction: The Unavoidable Challenge of Scale Up

In material development using twin screw kneading extruders, once a certain level of research results has been obtained on lab scale or pilot scale machines, engineers inevitably face the difficult task of moving to production equipment, namely scale up. The objective of increasing throughput is simple and clear in itself, yet the approach to achieving it is far from simple.

In particular, the corotating fully intermeshing twin screw kneading extruder carries out melting, dispersive and distributive mixing, and chemical reactions simultaneously within a single machine. For this reason, even slight changes in conditions such as screw diameter, rotational speed, or fill ratio can substantially alter the shear history experienced by the material and its residence behavior.

The Basics of Scale Up Theory for Twin Screw Kneading Extruders

Scale up of twin screw kneading extruders is generally discussed under the assumption of geometric similarity, with the barrel inner diameter D taken as the characteristic length. The most important question in this discussion is the design philosophy itself: what should serve as the basis for similarity? In practice, one is tempted to match all conditions at the same time, but in real twin screw kneading processes this is virtually impossible. The work is closer to deciding what to prioritize and what to give up. To organize this thinking, scale up theory presents two representative limiting models.

The Cubic Law: Scale Up Based on Three Dimensional Throughput

The first approach bases scale up on throughput. The internal flow volume of the extruder increases in proportion to the cube of the inner diameter D. Throughput Q follows the relation

Q ∝ D3

and increases accordingly.
This view treats the interior of the extruder as a three dimensional space and uses volumetric processing capacity as the basis. It is commonly known as the cubic law. The underlying assumption is that scale up proceeds at identical rotational speed and identical shear rate.

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 is 4. Under the cubic law, the throughput is calculated to increase by a factor of 64. In other words, an extremely large gain in production capacity is theoretically possible simply by enlarging the equipment.

Scale up based on the cubic law is highly attractive because it raises production capacity efficiently. On the other hand, the screw peripheral speed also rises, and the shear history that the material experiences per unit mass changes more easily. From the standpoint of mixing quality and reproducibility of physical properties, this calls for careful attention. The increase in shear heating and the resulting rise in resin temperature become particular concerns.

The Square Law: Scale Up Based on Two Dimensional Heat Transfer and Energy

The second approach bases scale up on energy input and heat transfer. In a twin screw kneading extruder, electrical heating and the heat exchange that takes place through the barrel occur mainly via the barrel surface. The capacity of this exchange is proportional to the surface area, that is, to the square of the inner diameter D. Throughput Q then approaches the relation

Q ∝ D2

.
This square law views the extruder as a two dimensional heat transfer and energy supply system. Its premise is that the thermal energy exchanged between the barrel and the resin, together with the peripheral speed at the screw outer diameter, remain the same.

Applying the same case of scale up from KZW15 to KZW60, the screw diameter ratio is 4, so the throughput increase under the square law is a factor of 16. The rate of increase is smaller than under the cubic law.

Scale up based on the square law is effective when reproducibility of the temperature history and mixing state of the material is the priority. However, the gain in throughput from enlarging the equipment is limited compared with the cubic law.

Designing Between the Square Law and the Cubic Law

Real operating conditions rarely match either the square law or the cubic law in full. In practice, scale up conditions most often sit between the three dimensional throughput basis (the cubic law) and the two dimensional heat transfer and energy basis (the square law).

Moving toward the cubic law raises productivity in terms of discharge rate, but the increase in shear heating and the rise in resin temperature tend to reduce reproducibility of quality. Moving toward the square law makes the material history easier to preserve, yet the gain in production capacity is limited. Designing a scale up is close to searching for the point where you can hold quality while pushing volume as far as possible.

One topic that frequently comes up during scale up is the reproducibility of the discharge resin temperature. Resin temperature is determined by a combination of factors: shear heating from screw rotation, and heating or cooling through the barrel. It is not a quantity that can be controlled easily through a single operating variable. As the screw diameter grows, the internal flow volume rises with the cube of the inner diameter, while the barrel surface area rises only with the square. Because of this geometric difference, the relative effectiveness of heat transfer to the bulk of the material declines. As a result, applying the same temperature settings and rotational conditions used on a smaller machine will not reproduce the same temperature history. This is one of the reasons scale up is said to be difficult.

The Choice Between Larger Machines and Multiple Units

From a theoretical standpoint, when the reproducibility of material quality is the top priority, one option is to install several twin screw kneading extruders of the same class as the existing machine and operate them in parallel. This makes it possible to keep the shear history, residence time, and temperature history nearly identical, which helps to limit quality risk.

From the standpoint of floor space, operational efficiency, and capital investment, however, there are many situations where consolidated production on a single larger machine is required. In that case, simple application of the scaling laws is not enough. Optimization of the screw segment configuration, adjustment of L/D, control of rotational speed, and redesign of side feed positions and vent conditions all need to be considered together with the equipment design and operating conditions as one integrated picture.

Summary: Designing the Optimal Solution Between the Square and the Cube

The scale up of a twin screw kneading extruder is not completed by a simple enlargement of dimensions or by the direct application of a single formula. As the equipment grows, many factors change at the same time, so matching every metric exactly is difficult both in theory and in practice. The cubic law presented by scale up theory is a three dimensional throughput based view, while the square law is a two dimensional heat transfer and energy based view. The actual twin screw kneading process does not follow either one completely; it sits somewhere between the two.

If productivity comes first, the operating point shifts toward the cubic law; if reproducibility of material history matters more, it shifts toward the square law. This trade off cannot be avoided. That is exactly why scale up is difficult, and at the same time why it is such an interesting subject. Scale up of twin screw kneading extruder is a design process of selecting the single point best suited to the objective from the countless options that exist between the square and the cube. It is a field where the philosophy and experience of the engineer operating the extruder come through clearly.

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Technovel Corporation — Extrusion Machinery Specialists

Osaka based Technovel specializes in extrusion machinery. We built the world’s first horizontally multi screw extruder, and our Quad and Octa screw extruders now serve diverse industries. Our twin screw range runs from the world’s smallest 6 mm lab unit, through our best-selling 15 mm model, to large production machines. This column shares the know how behind them.

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