A Critical Analysis of Cereal Extrusion

Background

A number of popular cereal products are made using a manufacturing process known as extrusion. There is a stage during this process where some of the liquid component of the cereal mixture changes phase to a gas (see diagram below). The subsequent evolution of gas within the mixture will affect the thermodynamic and rheological properties of the mixture. The mechanisms at play include the transfer of: mass, momentum, and heat, as well as the change in phase of the components and the changing volume of the product. The interactions between these mechanisms will determine the final shape and texture of the product. Previous attempts to model the system during this stage of extrusion, often referred to as the flash, have only succeeded for specific, reduced cases. For example, the system becomes much more tractable if you only consider axisymmetric cases. 

Figure 1: Schematic of a cereal extruder where, in this illustration, the cereal is flowing from left to right. A) the die, a contracting nozzle through which the extrudate is extruded. B) the extrudate pre-bubble nucleation. C) the flash, which starts at the nucleation of bubbles and ends once the vaporisation of liquid slows down. D) downstream the growth of bubbles slows down and the cereal is post-processed. Not shown is the screw system to the left of point B.

The difficulty in simulating extrusion comes from the high degree of coupling between the physical processes. The evolution of every component in the system, from the size of the vapour bubbles that form to the temperature of the product, is dependent on the evolution of just about every other component. One way to approach this problem is to first write down a model of the fully-coupled system, and then systematically examine the impact of weakening the coupling between between the components using ideas from asymptotic analysis.

A tool which can perform simulations of extrusion can be used to aid experimentation and product development by providing a relatively cheap way of testing ideas. Some of the experiments that are currently performed in labs, requiring physical equipment, can be performed theoretically. For example, when trying to design a product with a different shape, the proposed outlet shape must be built and tested in a lab. If this shape is not ideal, a new one must be built until the desired cereal shape is achieved. With the aid of simulations, the effect of different outlet shapes can be tested theoretically, informing the experimentalist of a good starting design. Such a tool may also be useful in identifying potential issues, such as the formation of slugs in the extruder. 

The goal of this project is to construct a model of the flash, and then use computational and theoretical advancements made since this problem was last attempted to construct a solver capable of simulating the flash.

Progress

We have constructed a model for extrusion which approximates the gas-liquid mixture with a single compressible fluid. The compressible flow equations need to be closed by assuming an appropriate constitutive law. Conventional constitutive laws used to close the compressible flow equations assume some functional relationship between the pressure, density, and sometimes temperature. In order to account for the complexity of the evolving gas within the cereal mixture, we will close the flow equations using model derived from an understanding of the ongoing physics on the length scale of a bubble. We have investigated the dynamics of bubble evolution in a cereal mixture and constructed a model relating bubble size, and therefore the volume fraction of gas in our mixture, to the other state variables in our system. We are currently investigating the evolution of a compressible fluid undergoing extrusion by using a range of simpler constitutive laws. 

Future Work

After investigating the dynamics of general, simplified, compressible fluids in the situations we are interested in, we will combine our physics based model of bubble growth to the compressible flow equations and work on solving the system for the physically relevant case.