Scale mathematical modeling

Remark 1 Note that the borderlines between the three main approaches are not necessarily distinct. For instance, the targets in (ii) can be viewed as heuristics or rales that simplify the combinatorial problem and allow for its decomposition into smaller, more tractable problems (see chapter on heat exchanger network synthesis via decomposition approaches). The optimization approach (iii) can formulate thermodynamic targets, or targets on the attainable region of reaction mechanisms as optimization models, and can either utilize them so as to decompose the large-scale problem or follow a simultaneous approach that treats the full-scale mathematical model. The first... 

Many methods have been used to size relief systems area/volume scaling, mathematical modeling using reaction parameters and flow theory, and empirical methods by the Factory Insurance Association (FIA). The Design Institute for Emergency Relief Systems (DIERS) of the AIChE has performed studies of sizing reactors undergoing runaway reactions. Intricate laboratory instruments as described earlier have resulted in better vent sizes. 

Future trends in electrochemical engineering will be influenced by the need to develop molecular-based discoveries into new and improved products and processes. What is needed is to develop a multiscale systems approach that builds upon the traditional base of continuum-scale mathematical models. 

The purpose of the series is to provide high quality advanced reviews of topics of both fundamental and practical importance for the experienced reader. This volume focuses on spectacular recent developments in electrochemical surface modification for the purpose of forming thin films and surface regions that have unique functional properties and that can be well-characterized by multi-scale mathematical models. 

In the present day a different scale-up procedure is usually followed. Various aspects of the chemical reaction are studied extensively in very small but often continuously operating bench scale reactors and in "mini-plants , equipped with very accurate measuring devices. Specific physical effects are studi in the absence of chemical reaction on larger scales. Mathematical models are developed that will describe the process accurately as a function of scale, so that they can be extrapolated with confidence to predict the behaviour of larger equipment. 

Langrish, T. A. G. (2009). Multi-scale mathematical modelling of spray dryers. Journal of... 

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. 

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. 

The first step is to define the objectives of the flow model, and to identify those flow aspects that are relevant for the performance of the reactor. Then, the engineer must identify and quantify the various times and space scales involved, as well as the geometry of the system. These actions allow the problem to be represented by a mathematical model. Creating this model accurately is the most crucial task in the flow modeling project. 

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. 

In process design, it is frequently discovered that many of the basic data needed to rmderstand a process are lacking. Because most crrrrent mathematical models are not sufficiently accrrrate to permit direct scale-up of the process from laboratory data to full plant size, a pilot plant must be constracted. As models are improved, it may become possible to evaluate design decisions with more confidence, and bypass the pilot plant stage. 

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. 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

The scale-up challenges for microchannel reactors are addressed through integrated mathematical models to describe all elements of the physics. Integrated... 

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. 

In this paper we attempt a preliminary investigation on the feasibility of catalytic combustion of CO/ H2 mixtures over mixed oxide catalysts and a comparison in this respect of perovskite and hexaaluminate type catalysts The catalysts have been characterized and tested in the combustion of CO, H2 and CH4 (as reference fuel). The catalytic tests have been carried out on powder materials and the results have been scaled up by means of a mathematical model of the catalyst section of the Hybrid Combustor. 

---