Scale using mathematical modeling

In general, the desorptive behavior of contaminated soils and soHds is so variable that the requited thermal treatment conditions are difficult to specify without experimental measurements. Experiments are most easily performed in bench- and pilot-scale faciUties. Full-scale behavior can then be predicted using mathematical models of heat transfer, mass transfer, and chemical kinetics. 

Based on the pilot scale measurements mathematical modeling was started in order to describe the processes. Discrete element method (DEM) was used to model the mixing. This was originated from a simple mechanical model to formulate... 

The use of models and model simulations are extremely useful in all design and scale-up considerations. Mathematical methods to solve model equations of any degree of complexity are available now, and fast numerical techniques have been developed. In addition, almost everywhere abundant computer facilities are at hand. Therefore, a reliable design and scale-up should use mathematical models formulated on the basis of first principles, even if these models are very sophisticated. Such models and simulations based on them present the most efficient and probably the cheapest way in today s design works. 

Membrane science and technology is interdisciplinary, involving polymer chemists to develop new membrane structures physical chemists and mathematicians to describe the transport properties of different membranes using mathematical models to predict their separation characteristics and chemical engineers to design separation processes for large scale industrial utilization. The most important element in a membrane process, however, is the membrane itself. To... 

Chemical engineering is involved in every step in bringing a process from the laboratory to full-scale production. This involves determining methods to make the process continuous, safe, environmentally compatible, and economically sound. During these steps, chemical engineers determine the methods and procedures needed for a full-scale plant. Mathematical modeling is used to test various steps in the process, controls, waste treatment, environmental concerns, and economic feasibility. 

The test runs were used to verify certain critical parts of the design made for a world scale isooctane plant. The design itself was based mainly on laboratory scale experiments, mathematical models and computer simulation. However, although the modern models are very useful it was still necessary to verify that all reactions and separations will take place as designed when all components of the industrial feed are present. For this purpose the scale of the devices is not critical, provided that the essential unit operations work properly and that correct concentrations, temperatures and pressures are realized in the unit. The HUT miniplant was used for this verification step. 

Another very useful mathematical model has been proposed to specifically reveal the fractal characteristics of signals. A detailed description of this technique, also called rescaled range analysis or the R/S technique [where R or R(, s) stands for the sequential range of the data point increments for a given lag s and time t, and S or S(/,s) stands for the square root of the sample sequential variance], can be found in Fan et al. " Hurst and later Mandelbrot and Wallis have proposed that the ratio R(/,s)/S(t,s) is itself a random function with a scaling property described by relation (7.8), where the scaling... 

Scale- Up of Electrochemical Reactors. The intermediate scale of the pilot plant is frequendy used in the scale-up of an electrochemical reactor or process to full scale. Dimensional analysis (qv) has been used in chemical engineering scale-up to simplify and generalize a multivariant system, and may be appHed to electrochemical systems, but has shown limitations. It is best used in conjunction with mathematical models. Scale-up often involves seeking a few critical parameters. Eor electrochemical cells, these parameters are generally current distribution and cell resistance. The characteristics of electrolytic process scale-up have been described (63—65). 

Catalytic crackings operations have been simulated by mathematical models, with the aid of computers. The computer programs are the end result of a very extensive research effort in pilot and bench scale units. Many sets of calculations are carried out to optimize design of new units, operation of existing plants, choice of feedstocks, and other variables subject to control. A background knowledge of the correlations used in the "black box" helps to make such studies more effective. 

A combination of dimensional similitude and the mathematical modeling technique can be useful when the reactor system and the processes make the mathematical description of the system impossible. This combined method enables some of the critical parameters for scale-up to be specified, and it may be possible to characterize the underlying rate of processes quantitatively. 

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. 

Mathematical models of the reaction system were developed which enabled prediction of the molecular weight distribution (MWD). Direct and indirect methods were used, but only distributions obtained from moments are described here. Due to the stiffness of the model equations an improved numerical integrator was developed, in order to solve the equations in a reasonable time scale. 

Since electrochemical processes involve coupled complex phenomena, their behavior is complex. Mathematical modeling of such processes improves our scientific understanding of them and provides a basis for design scale-up and optimization. The validity and utility of such large-scale models is expected to improve as physically correct descriptions of elementary processes are used. 

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]. 

The several industrial applications reported in the hterature prove that the energy of supersonic flow can be successfully used as a tool to enhance the interfacial contacting and intensify mass transfer processes in multiphase reactor systems. However, more interest from academia and more generic research activities are needed in this fleld, in order to gain a deeper understanding of the interface creation under the supersonic wave conditions, to create rehable mathematical models of this phenomenon and to develop scale-up methodology for industrial devices. 

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