Mathematical modeling solution process

From the end of sixties, the principal studies in the theory of chemical technology were based on mathematical and physical modelling of the total set of superimposed processes. Vigorous development of computers and numerical methods of analysis promoted a fast development of investigations and continuous complication of models, which enable us to percive new details of the processes. At present, the fundamental physical principles and phenomena are understood in principle, mathematical models of processes have been developed in the main types of reactors, the fields of their application have been determined and computational methods for solution and analysis have been defined. Since the mid 1970s, the main attention of the researchers has been attracted to the study of the peculiarities of processes. 

The solution adopted by us is the use of computer simulations of mathematical models of the process and the mock-up situations. Eventually, simulation techniques will become so accurate, that the mock-up step can be discarded. For the time being it is reasonable to use such models to generate corrections for smaller differences between mock-up and process. 

Transport Models. Many mechanistic and mathematical models have been proposed to describe reverse osmosis membranes. Some of these descriptions rely on relatively simple concepts others are far more complex and require sophisticated solution techniques. Models that adequately describe the performance of RO membranes are important to the design of RO processes. Models that predict separation characteristics also minimize the number of experiments that must be performed to describe a particular system. Excellent reviews of membrane transport models and mechanisms are available (9,14,25-29). 

No single method or algorithm of optimization exists that can be apphed efficiently to all problems. The method chosen for any particular case will depend primarily on (I) the character of the objective function, (2) the nature of the constraints, and (3) the number of independent and dependent variables. Table 8-6 summarizes the six general steps for the analysis and solution of optimization problems (Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-HiU, New York, 1988). You do not have to follow the cited order exac tly, but vou should cover all of the steps eventually. Shortcuts in the procedure are allowable, and the easy steps can be performed first. Steps I, 2, and 3 deal with the mathematical definition of the problem ideutificatiou of variables and specification of the objective function and statement of the constraints. If the process to be optimized is very complex, it may be necessaiy to reformulate the problem so that it can be solved with reasonable effort. Later in this section, we discuss the development of mathematical models for the process and the objec tive function (the economic model). 

MATHEMATICAL MODELING OF THE PROCESS TAKEN PLACE IN THE SOLID SUPPORT - SOLUTION , TYPE INDICATOR PIPES... 

Any numerical experiment is not a one-time calculation by standard formulas. First and foremost, it is the computation of a number of possibilities for various mathematical models. For instance, it is required to find the optimal conditions for a chemical process, that is, the conditions under which the reaction is completed most rapidly. A solution of this problem depends on a number of parameters (for instance, temperature, pressure, composition of the reacting mixture, etc.). In order to find the optimal workable conditions, it is necessary to carry out computations for different values of those parameters, thereby exhausting all possibilities. Of course, some situations exist in which an algorithm is to be used only several times or even once. 

The mathematical model can only be an approximation of real-life processes, which are often extremely complex and often only partially understood. Thus models are themselves neither good nor bad but, as pointed out by Kapur, will either give a good fit or a bad fit to actual process behaviour. Similarly, it is possible to develop several different models for the same process, and these will all differ in some respect in the nature of their predictions. Indeed it is often desirable to try to approach the solution of a given problem from as many different directions as possible, in order to obtain an overall improved description. The purpose of the model also needs to be very clearly defined, since different models of a process, each of which has been developed with a particular purpose in mind, may not satisfy a different aim for which the model was not specifically constructed. 

In the study of a process or a phenomenon to solve specific problems, mathematical modeling is the process of representing mathematically the essential elements of a process or a phenomenon of the system and the interactions of the elements with one another. Computer simulation is the process of experimenting with the model by using the computer as a tool, l.e. a computer is used to obtain solutions to the mathematical relationships of the model. The model usually is not a complete representation of the system, which often Involves Inclusion of so many details that one can be overwhelmed by its complexity. Computer is not a required tool to carry out simulation as there are mathematical models which have analytical solutions. 

Simplified mathematical models These models typically begin with the basic conservation equations of the first principle models but make simplifying assumptions (typically related to similarity theory) to reduce the problem to the solution of (simultaneous) ordinary differential equations. In the verification process, such models must also address the relevant physical phenomenon as well as be validated for the application being considered. Such models are typically easily solved on a computer with typically less user interaction than required for the solution of PDEs. Simplified mathematical models may also be used as screening tools to identify the most important release scenarios however, other modeling approaches should be considered only if they address and have been validated for the important aspects of the scenario under consideration. 

The fundamental basis for virtually all a prion mathematical models of air pollution is the statement of conservation of mass for each pollutant species. The formulation of a mathematical model of air poUution involves a number of basic steps, the first of which is a detaUed examination of the basis of the description of the diffusion of material released into the atmosphere. The second step requires that the form of interaction among the various physical and chemical processes be specified and tested against independent experiments. Once the appropriate mathematical descriptions have been formulated, it is necessary to implement suitable solution procedures. The final step is to assess the ability of the model to predict actual ambient concentration distributions. 

The eontrol of pH is a very important problem in maity processes, particularly in effluent wastewater treatment. The development and solution of mathematical models of these systems is, therefore, a vital part of chemical engineering dynamic modeling. 

The mathematical models used to infer rates of water motion from the conservative properties and biogeochemical rates from nonconservative ones were flrst developed in the 1960s. Although they require acceptance of several assumptions, these models represent an elegant approach to obtaining rate information from easily measured constituents in seawater, such as salinity and the concentrations of the nonconservative chemical of interest. These models use an Eulerian approach. That is, they look at how a conservative property, such as the concentration of a conservative solute C, varies over time in an infinitesimally small volume of the ocean. Since C is conservative, its concentrations can only be altered by water transport, either via advection and/or turbulent mixing. Both processes can move water through any or all of the three dimensions... 

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