up using modelling and simulation

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up using modelling and simulation LEON LEVINE

An underused resource in the food industry, mathematical modelling can be extremely usefal if applied appropriately

Mathematical modelling will often reveal hidden details about processes

hen discussing a concept, it is often best to begin with a definition. Morton Denn 1 defines a mathematical model of a process as 'a system of equations whose solution, given specified input data, is representative of the response of the process to a corresponding set of inputs.' Stanley Middleman2 eloquently points out that there is a significant amount of creativity associated with mathematical modelling. He quotes J R R Tolkien, 3 suggesting that the engineer, in designing a model, creates an alternative reality. 'Every writer [engineer] making a secondary world ... wishes to be a real maker, hopes that he [the engineer] is drawing on Reality, or that the peculiar quality of his secondary world [mathematical model], if not the details, are derived from Reality, or are flowing into it. If he [the engineer] indeed achieves a quality that can fairly be described by the dictionary definition: "inner consistency of reality", it is difficult to conceive how this can be, if the work does not partake of Reality.' The bracketed additions are Middleman's suggestions on how to read Tolkien's thesis.

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Types of models Two major types of models are used in process development and engineering: empirical and fundamental. Empirical models need 346

Chemistry & Industry 5 May 1997

a lot less knowledge and creativity, and are far more commonly used in both industry and academia. There are two types of empirical model: response surface methodology (RSM) and dimensional analysis. The RSM models are the ones most often seen in the food literature. They simply require the laying out of an orderly set of experiments over a defined experimental region. No real physical understanding of the problem is required. Dimensional analysis, meanwhile, is underused in the food industry, though the principles behind i~ have been known and taught for almost a century. Part of the the problem is a poor understanding of its power and applicability. Dimensional analysis is a technique that combines physical parameters that describe the problem in such a way to produce new, dimensionless, variables. In doing so, 'hidden' interactions between physical parameters and process scale are revealed. On the other hand, fundamental models are theoretically based. In food engineering problems, this usually means defining the underlying transport (momentum, heat and mass transfer) phenomena. Using this type of modelling needs considerably more skill and creativity. I will discuss both types of models, but my preference is bes: summed up by paraphrasing the mediaeval philosophe::Maimonides, 4 who was trying to reconcile Aristotelian logic tc religious thought: 'the theoretical is the highest form of thought·

Why model?

work for consumer responses. This, in most cases, prevents it from being applied to such things as textures and flavours. However, for engineering, dimensional analysis is superior to the nonnal RSM techniques. The variables are directly tied to the underlying physics of the problem. The scale of the equipment being tested is directly included in the variables, making them innately suitable for answering questions about scale up. In addition, since the relationships obtained are tied to the underlying physics, the results can, to a limited degree, be extrapolated. The constants obtained are also related to physical reality. For example, the size of certain exponents clearly points to the presence of laminar or turbulent flow, or gravitational versus inertial flows, or to the size of activation energies. Finally, better statistics are obtained than from simple minded RSM techniques, and because the variables used are actually combinations of 'real' variables, the number of experiments needed is always lower compared with the number required by RSM techniques. The application of dimensional analysis does not mean that RSM should be abandoned. Rather, RSM techniques should be Empirical models used to do experiments that are going to be analysed by dimenWhen the understanding of a problem, or the time available to sional analysis. This means the experimental plan of the RSM is reach an answer, is limited. empirical modelling is the appropribased on dimensionless variables rather than 'real' variables. ate approach. Both RS:\1 and dimensional analysis have been widely and successfuJly used in the food industry. This may entail some re-education for corporate statisticians. RSM is, simply, graphical representation of the statistical relaOne area that lends itself to dimensional analysis is the flow tionship between process output and independent variables. It is of solids, which is extremely important in food processing. the most efficient ,vay to optimise product fonnulae, or the perSome examples are mixing powders; coating solids as in breakformance of existing processes. 7-9 Although there are several fast, snack and confectionery products; and agglomerating cautions for the use of this technique. The results of an RSM materials such as in cereal cooking drums and in 'instantising' study are completely non-general. The response surfaces generpowders. The movement and behaviour of powder flows is ated apply only to the experimental region studied and no other. extremely complex and poorly understood. 10 Fundamental · As a result, the conclusions cannot be extrapolated. This of study of the flows requires the solution of partial differential course is not a difficulty when equations containing discontinuities. Dimensional analysis trying to optimise a fonnula, Paddle mixer power consumption but is a considerable problem 10-a of such problems reduces in process development. It is in them to a manageable, underd/R= 1.76 this area where the limitations standable, size. of RSM have the potential to be For example, one variable o Pilot (2001b/hr) + Plant (20001b/hr) most widely abused. that occurs repeatedly in these 10-4 An RSM on processing varianalyses is the Froude number d/R:0,63 ables at the laboratory or pilot (the ratio of inertial to gravitational flow). Kinematic simiplant scale may have no relationship to the performance of larity means that for gravity another pilot plant, or more controlled flows, such as in importantly another process most powder mixers and coatscale (the plant). As a result the ing systems, the Froude numconsiderable effort that is often ber should be fixed. This proundertaken to run RSM in the vides the simple scale up rules pilot plant may be wasted. At for these operations. As the d = depth of poWder, R = mixer radius, N = mixer rotational rate, P = mixer power consumption best it can only be assumed that tumbling operations - whose the qualitative nature of the performance is determined by response predicts the qualitative response of the plant. This mixing quality - are scaled up, the rotational speed of the means the model has limitations for resolving scale up issues. equipment must be reduced with the square root of the diameter, The constants determined by RSM normally have no physical and the time needed to finish the mixing increases with the basis. They are simply statistical values that describe the data. square root of the diameter. For continuous flow processes, this As a result the data obtained teach little or nothing about the implies that the equipment's capacity rises with the 2.5 power of underlying physics. In fact, the ignorant use of RSM can cloud scale, not the cube of scale, as is commonly assumed. understanding of the physical reality. Consider what results are The power consumption of powder processes can also be obtained by using a simple RSM on the temperature/time behavanalysed by dimensional analysis. Figure 1 shows a correlation between the dimensionless power number (in reality a term proiour of a simple first-order chemical reaction. Modelling based on dimensional analysis also has its limitations, portional to the power number) and a term proportional to the but is far more useful than RSM for engineering problems. 7-9 Froude number in a twin shaft paddle mixer. The parameter is There are many good reasons to undertake the sometimes time consuming and difficult task of mathematical modelling. The two best are that modelling codifies and expresses knowledge and provides the user with hidden details. Mathematical expressions are very efficient methods for expressing physical behaviour. They can often express a world of knowledge in a simple statement. Perhaps more importantly. especially for those in industry, mathematical modelling reduces the scope (cost) of experimentation. Invariably. the use of models has a number of advantages. The amount of experiments needed to analyse a particular problem are significantly reduced. Physically based models can be extrapolated to unexplored, or unexplorable regions - the essence to solving the scale-up problem, which implies extrapolation. Modelling allows alternatives to be considered which may be difficult, or expensive to test. It also allows the sensitivity (stability) of a process to variables (upsets), and the design of optimal control strategies to be studied.

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