Engineering problems statistics

Optimization pervades the fields of science, engineering, and business. In physics, many different optimal principles have been enunciated, describing natural phenomena in the fields of optics and classical mechanics. The field of statistics treats various principles termed maximum likelihood, minimum loss, and least squares, and business makes use of maximum profit, minimum cost, maximum use of resources, minimum effort, in its efforts to increase profits. A typical engineering problem can be posed as follows A process can be represented by some equations or perhaps solely by experimental data. You have a single performance criterion in mind such as minimum cost. The goal of optimization is to find the values of the variables in the process that yield the best value of the performance criterion. A trade-off usually exists between capital and operating costs. The described factors—process or model and the performance criterion—constitute the optimization problem. ... 

In many engineering problems, the physical mechanism ofthe system is too complex and not suffidently understood to permit the formulation of even an approximately accurate model, such as the perfect gas law. However, when such complex systems are in question, it is recommended to use statistical models that to a greater or lesser, but always well-known accuracy, describe the behavior of a system. 

Fig. 5.18. A contour plot of fitted yield when x3 = 1.75. (From Statistics for Engineering Problem Solving (1st Ed.) by S. B. Vardeman 1994. Reprinted with permission of Brooks/Cole, a Division ofThomson Learning www.thomsonlearning.com. FAX 800-730-2215.)...

Vardeman, S. B., Statistics for Engineering Problem Solving, PWS-Kent, Boston, MA, 1994. 

In order to describe the three-phase (Hydrate - Liquid Water - Vapor) equihbria (H-Lw-V) the theory developed by van der Waals-Platteeuw [9, 10] is traditionally used. The theory is based on Statistical Thermodynamics and according to Sloan and Koh [1] it is probably one of the best examples of using Statistical Thermodynamics to solve successfully a real engineering problem. An excellent description of the theory for the three-phase equilibrium calculations is provided in a number of publications [1,9, 10] and will not be repeated here. In addition, extensive details on the methodology for the calculation of two-phase equilibrium (H-L ) conditions can be foimd in the review papers by Holder et al. [12], and Tsimpanogiannis et al., [11]. 

In the estimation of diffusivities for applications to chemical engineering problems, various approaches, kinetic theory, absolute theory, hydrodynamic theory, statistical-mechanical theory and both empirical and semi-empirical correlation had been employed for the calculation of binary diffusion coefficient. It is essential to appreciate that different theories are necessary for non-electrolytes and electrolytes solutions, therefore, different estimation methods are required for each case. A fact that all methods had been overlooked and limitations of each one were recognized too. 

Chapters 6, 7, 8 and 9 provide important applications of nonlinear models to example engineering problems. Topics covered include data fitting to nonlinear models using least squares techniques, various statistical methods used in the analysis of data and parameter estimation for nonlinear engineering models. These important topics build upon the basic nonlinear analysis techniques of the preceding chapters. These topics are not extensively covered by most introductory books on numerical methods. 

Because this problem is complex several avenues of attack have been devised in the last fifteen years. A combination of experimental developments (protein engineering, advances in x-ray and nuclear magnetic resonance (NMR), various time-resolved spectroscopies, single molecule manipulation methods) and theoretical approaches (use of statistical mechanics, different computational strategies, use of simple models) [5, 6 and 7] has led to a greater understanding of how polypeptide chains reach the native confonnation. 

Molecular modeling has evolved as a synthesis of techniques from a number of disciplines—organic chemistry, medicinal chemistry, physical chemistry, chemical physics, computer science, mathematics, and statistics. With the development of quantum mechanics (1,2) ia the early 1900s, the laws of physics necessary to relate molecular electronic stmcture to observable properties were defined. In a confluence of related developments, engineering and the national defense both played roles ia the development of computing machinery itself ia the United States (3). This evolution had a direct impact on computing ia chemistry, as the newly developed devices could be appHed to problems ia chemistry, permitting solutions to problems previously considered intractable. 

For certain types of stochastic or random-variable problems, the sequence of events may be of particular importance. Statistical information about expected values or moments obtained from plant experimental data alone may not be sufficient to describe the process completely. In these cases, computet simulations with known statistical iaputs may be the only satisfactory way of providing the necessary information. These problems ate more likely to arise with discrete manufactuting systems or solids-handling systems rather than the continuous fluid-flow systems usually encountered ia chemical engineering studies. However, there ate numerous situations for such stochastic events or data ia process iadustries (7—10). 

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