Engineering mathematical description

Let us think about how resistance of a wire gives way to information production. Rational inferences will be considered and at the final stage we will see that the linguistic words after their representation by symbols will lead to a formulation with a parameter of which the numerical value can be determined by experimentation only. An engineer should be aware of the fact that not only rational inferences but also their support by experimentation can lead to useful formulations. Let the resistance of a wire denoted by R, and then ask what the major variables are that affect the resistance This question can be explored further by considering the geometrical dimensions of the wire, which are its cross sectional (circular, the simplest form) area. A, and length, L as in Fig. 4.7. 

After rational thinking one can write implicitly the dependence of R onto L and A as R = f(A, L), which means that the resistance is a function of (i.e. depends on) the cross-sectional area and the length, which represent the geometric dimensions of the wire. If one reasons rationally about how these geometric quantities affect the resistance then s/he can realize that the relationship between R and A is inversely proportional, whereas R is directly proportional with L. This linguistic and logical relationship (in a way correlation) statements enables one to write the following combined proportionality. 

Herein a implies proportionality. Rational thinking does not indicate equality, but if one wants to write equality, then a constant must be imported on one of the sides, for instance, here the constant c is added on the right hand side and finally. 

This expression is derived with linguistic, philosophical thinking and rational reasoning without any numerical thoughts. Now the question is what are the meaning, interpretation and relevance of the constant hi order to answer this question, the constant can be left as subject as follows. 

This expression tells us much verbal information that may not be thought at the first glance. Let us list some of these interpretations as follows. 

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Many models have been proposed for adsorption and ion exchange equilibria. The most important factor in selecting a model from an engineering standpoint is to have an accurate mathematical description over the entire range of process conditions. It is usually fairly easy to obtain correcl capacities at selected points, but isotherm shape over the entire range is often a critical concern for a regenerable process. 

Optimization in the design and operation of a reactor focuses on formulating a suitable objective function plus a mathematical description of the reactor the latter forms a set of constraints. Reactors in chemical engineering are usually, but not always, represented by one or a combination of... 

Part I comprises three chapters that motivate the study of optimization by giving examples of different types of problems that may be encountered in chemical engineering. After discussing the three components in the previous list, we describe six steps that must be used in solving an optimization problem. A potential user of optimization must be able to translate a verbal description of the problem into the appropriate mathematical description. He or she should also understand how the problem formulation influences its solvability. We show how problem simplification, sensitivity analysis, and estimating the unknown parameters in models are important steps in model building. Chapter 3 discusses how the objective function should be developed. We focus on economic factors in this chapter and present several alternative methods of evaluating profitability. 

Models in general are a mathematical representation of a conceptual picture. Rate equations and mass balances for the oxidants and their reactants are the basic tools for the mathematical description. As Levenspiel (1972, p.359) pointed out the requirement for a good engineering model is that it be the closest representation of reality which can be treated without too many mathematical complexities. It is of little use to select a model which closely mirrors reality but is so complicated that we cannot do anything with it. In cases where the complete theoretical description of the system is not desirable or achievable, experiments are used to calculate coefficients to adjust the theory to the observations this procedure is called semi-empirical modeling. 

Within the constraints imposed, particularly by the requirements of sterility, for example, when mono-cultures are used, a major role of biochemical engineering is to provide appropriate mathematical descriptions of biochemical reaction processes and from these to devise suitable design criteria for economically viable processes. 

It is my opinion that recent developments in the mathematical description of nonlinear dynamical systems have the potential for an enormous impact in the fields of fluid mechanics and transport phenomena. However, an attempt to assess this potential, based upon research accomplishments to date, is premature in any case, there are others better qualified than myself to undertake the task. Instead, I will offer a few general observations concerning the nature of the changes that may occur as the mathematical concepts of nonlinear dynamics become better known, better understood, more highly developed, and, lastly, applied to transport problems of interest to chemical engineers. 

In general, the methods of modeling, analysis, and optimization in engineering begin with deciding on the system geometry, architecture, and components, and the manner in which the components are connected. Engineering analysis involves the mathematical description of the conceptualized system and its performance. Finally, optimization leads to the most favorable conditions for maximum performance (e.g., minimum entropy production or minimum cost). 

To determine physical and heat engineering SRU parameters, a calculated study of potential emergency situations related to water penetration into the core was performed. Mathematical description of the processes was based on dot description of both the neutron kinetics and the equations for heat transfer in the storage facility container (under such container SRU arranged within steel cup with frozen alloy was understood). 

Several models of the droplet easily penetrable roughness were suggested to demonstrate that the Eulerian mathematical description of a canopy flow can be generalized to represent the more complex structures met in practice. Linking the numerical methods with the analytical solutions of simplified models, one examines the correctness of the models and obtains the analytical estimations useful for engineering purposes. 

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