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The role of models in chemical engineering
Published in Edwin Zondervan, A Numerical Primer for the Chemical Engineer, 2019
Ultimately, this book is about solving the developed models in a numerical fashion. We could consider Ptolemy’s Amalgest (150 BC) as one of the first recorded studies on modeling and numerical analysis in which numerical approximations to describe the motions of the heavenly bodies with accuracy matching reality sufficiently were developed (Figure 1.1). This is basically the essence of numerical analysis. Numerical analysis is concerned with obtaining approximate solutions to problems while maintaining reasonable bounds of error, because it is often impossible to obtain exact answers. Numerical analysis makes use of algorithms to approximate solutions. Model development and solving the models is important to the world, for example in astronomy, construction, agriculture, architecture, and, of course, in engineering! In chemical engineering we use models and their (numerical) solutions to describe reactors and separators (dynamic and steady state), to perform computational fluid dynamics, to solve thermodynamic equations of state, to optimize process performance, to design and synthesize processes, and to regress experimental data, e.g., isotherms, kinetics, and so forth.
Errors in Numerical Computations
Published in Santanu Saha Ray, Numerical Analysis with Algorithms and Programming, 2018
In numerical analysis, mathematical problems may have solutions that are quite sensitive to small computational errors. In numerical computation, one usually starts with initial data, then one computes in turn all intermediate values, finally arriving at the results. If the initial data involves errors, they will generally affect the final result. Even each arithmetic operation performed by the computer introduces round-off error. There are also errors arising out because of conversion of numbers during machine representation. Sometimes these errors grow and the cumulative effect of the round-off errors becomes unbounded. In such cases, computation is said to be unstable. To reckon with this phenomenon, we now present the concept of condition number and stability.
Interpolation, Differentiation, and Integration
Published in Julio Sanchez, Maria P. Canton, Software Solutions for Engineers and Scientists, 2018
Julio Sanchez, Maria P. Canton
Numerical analysis is the field of mathematics that studies the methods of approximation used in the solution of mathematical problems. Numerical methods, on the other hand, refers to the computerized solution of the problems of numerical analysis. In this general sense, this book is almost entirely devoted to numerical methods. However, a more restrictive, and perhaps more common definition of numerical methods, focuses on the computerized solution of problems encountered in the calculus. In this chapter we select three common topics of numerical methods: interpolation, differentiation, and integration.
Proposal of quantification method of dynamic system reliability model of digital RPS using Markov state-transition model
Published in Journal of Nuclear Science and Technology, 2023
Masanobu Haruhara, Hitoshi Muta, Yasuki Ohtori, Shohei Yamagishi, Shota Terayama
Basically, the first choice in analysis is to obtain an algebraic solution. Algebraic solutions are the solutions expressed in mathematical formulas that clearly express how the interaction among variables affects the results, and since there is no need to integrate each time, the calculation is efficient and error-free. However, algebraic solutions are considered to only be applied to relatively simple models, and the state-transition simultaneous differential equations defined in this paper are considered to be difficult to solve algebraically. Numerical analysis is used in place of algebraic solutions to approximate the solution of analytical problems that are impossible to solve by algebraic methods, usually using finite-precision numerical values.
Modeling and computing parameters of three-dimensional Voronoi models in nonlinear finite element simulation of closed-cell metallic foams
Published in Mechanics of Advanced Materials and Structures, 2018
Xiaoyang Zhang, Yidong Wu, Liqun Tang, Zejia Liu, Zhenyu Jiang, Yiping Liu, Huifeng Xi
Computing parameters significantly affect simulation accuracy even when using commercial software, such as ABAQUS, yet comprehensive investigations into the seriousness of the effects of computing parameters on simulation results has been lacking. For large and complex numerical analysis, attaining high efficiency by choosing appropriate computing parameters is challenging.
Numerical analysis of an air-to-air heat pipe heat exchanger and comparison with experimental data
Published in International Journal of Ambient Energy, 2022
Three basic methods are commonly used in solving engineering problems. These are referred to as experimental, numerical and theoretical methods. Numerical analysis is one of the most frequently used methods for solving various problems due to its cheapness, ease of application and clearer understanding of the results of the study (Çiftçi and Sözen 2020).