Mathematical models and scale

Ultrasound can thus be used to enhance kinetics, flow, and mass and heat transfer. The overall results are that organic synthetic reactions show increased rate (sometimes even from hours to minutes, up to 25 times faster), and/or increased yield (tens of percentages, sometimes even starting from 0% yield in nonsonicated conditions). In multiphase systems, gas-liquid and solid-liquid mass transfer has been observed to increase by 5- and 20-fold, respectively [35]. Membrane fluxes have been enhanced by up to a factor of 8 [56]. Despite these results, use of acoustics, and ultrasound in particular, in chemical industry is mainly limited to the fields of cleaning and decontamination [55]. One of the main barriers to industrial application of sonochemical processes is control and scale-up of ultrasound concepts into operable processes. Therefore, a better understanding is required of the relation between a cavitation coUapse and chemical reactivity, as weU as a better understanding and reproducibility of the influence of various design and operational parameters on the cavitation process. Also, rehable mathematical models and scale-up procedures need to be developed [35, 54, 55]. 

Gu, T. Mathematical Modeling and Scale-up of Liquid Chromatography, Springer Verlag, New York, 1995. 

Mezzomo, N Martinez, J Ferreira, SRS. Supercritical fluid extraction of peach (Prunus persica) almond oil Kinetics, mathematical modeling and scale-up. Journal of Supercritical Fluids, 2009 51, 10-16. 

The introduction of computers to many companies allows proprietary software to be used for layout design. Spreadsheet, mathematical modeling and computer-aided design (CAD) techniques are available and greatly assist the design process, and have added to the resources available to planners. However, the traditional scale models described above will still be useful to present the result to management and shop floor personnel. 

The work of Crank [38] provides a review of the mathematical analysis of well defined component transport in homogeneous systems. These mathematical models and measured concentration profile data may be used to estimate diffu-sivities in homogenized samples. The use of MRI measurements in this way will generate diffusivities applicable to models of large-scale transport processes and will thereby be of value in engineering analysis of these processes and equipment. 

Cold flow studies have several advantages. Operation at ambient temperature allows construction of the experimental units with transparent plastic material that provides full visibility of the unit during operation. In addition, the experimental unit is much easier to instrument because of operating conditions less severe than those of a hot model. The cold model can also be constructed at a lower cost in a shorter time and requires less manpower to operate. Larger experimental units, closer to commercial size, can thus be constructed at a reasonable cost and within an affordable time frame. If the simulation criteria are known, the results of cold flow model studies can then be combined with the kinetic models and the intrinsic rate equations generated from the bench-scale hot models to construct a realistic mathematical model for scale-up. 

The principles and methods of scale-up can be applied to chemical reactors. In the absence of significant thermal effects, i.e., when the ratio <2r/ Vr may be considered negligible, ideal batch reactors do not show any problem of scale-up, because the volume Vr does not appear in the mathematical model (2.17), so that their performance is only determined by chemical kinetics (see Sect. 2.3). On the contrary, a very complex behavior is expected for real reactors in fact, this behavior cannot be analyzed in terms of mathematical models, and the design procedures must be largely based on semi-empirical rules of scale-up. 

The following reviews scale-up of chemical reactors, considers the dimensionless parameters, mathematical modeling in scale-up, and scale-up of a batch system. 

The photoresist technologies involve many steps each can be characterized by many parameters. It is very difficult to optimize the processes. Previously used pilot experiments now proved to be very costly and difficult. It is not possible to optimize all parameters by performing large series of the lithography experiments, at all possible sets of the parameters. The problem may be simplified and sometimes completely solved by means of mathematical modeling and simula-tion.f ° Photoresist technology simulation provides the opportunity to perform experiments in a virtual environment that can be much faster and cheaper than the full-scale wafer experiments. 

In contrast to fluid process technologies such as distillation, absorption, and extraction, mechanical processes are only accessible to a limited extent to mathematical modeling, and process design is dependent on experiment. However, only a few experimental techniques that allow reliable scale-up to be performed are available. 

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