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Introduction to Chemical Engineering Mathematics

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Introduction to Chemical Home Page Title Page Engineering Mathematics Contents JJ II J I Vitaly Alexandrov Department of Chemical and Biomolecular Engineering Page1 of 128 University of Nebraska-Lincoln Go Back Full Screen July 22, 2020 Close Quit Home Page Title Page Contents JJ II J I Page1 of 128 Go Back "Simplicity is the ultimate sophistication." Full Screen − Leonardo da Vinci Close Quit Contents Preface8 Recommended Literature8 Home Page 1 Mathematical Modeling in Chemical Engineering in a Nutshell9 Title Page 1.1 Governing Equations.......................... 10 Contents 1.2 Building a Realistic Model....................... 12 JJ II 1.3 Initial and Boundary Conditions................... 14 J I Page2 of 128 1.4 Simplification.............................. 15 Go Back Linearization.............................. 16 Full Screen Use of Symmetry and Geometry Simplification........... 19 Close Decoupling Equations......................... 20 Quit Lumping................................. 22 1.5 Solving the Governing Equations................... 22 1.6 Model Testing.............................. 25 2 Linear Algebra 27 2.1 Vector Spaces.............................. 27 Home Page 2.2 Scalar Product............................. 31 Title Page 2.3 Vector Product............................. 35 Contents 2.4 Linear Maps, Matrices and Determinants.............. 41 JJ II J I 2.5 Eigenvalue Problem and Matrix Diagonalization.......... 50 Page3 of 128 2.6 Hermiticity and Unitarity....................... 54 Go Back 2.7 Numerical Aspects of Linear Algebra................. 58 Full Screen 2.8 Case Study: Flow Graphs in Multicomponent Flash Separation.. 59 Close 2.9 Case Study: Protein Phosphorylation................ 61 Quit 3 Fourier Calculus 64 3.1 Fourier Series.............................. 65 3.2 Fourier Transform, Convolution and Correlation.......... 70 3.3 Laplace Transform and Transfer Function.............. 77 Home Page 4 Differential Equations 83 Title Page 4.1 Classification of Differential Equations................ 83 Contents 4.2 Separable ODEs............................ 84 JJ II J I 4.3 Logistic Equation, Population Growth and Tragedy of the Commons 85 Page4 of 128 4.4 Systems of Linear First-Order ODEs................. 88 Go Back 4.5 System Stability............................ 92 Full Screen 4.6 Harmonic Oscillator and Green’s Function Method......... 94 Close 4.7 Numerical Methods for Solving ODEs................ 100 Quit 4.8 Partial Differential Equations (PDEs)................ 103 4.9 Case Study: Unsteady Heat Conduction and the Method of Lines. 103 5 Nonlinear Dynamical Systems: From Fireflies to Reactors 109 5.1 Prelude................................. 109 Home Page 5.2 Nonlinear Algebraic Equations.................... 111 Title Page 5.3 Case Study: Antoine Equation for Nonlinear Flash Separation... 112 Contents 5.4 Nonlinear Dynamics, Steady States and Multistability....... 113 JJ II J I 5.5 Lotka-Volterra Model: Biological and Chemical Oscillators..... 114 Page5 of 128 5.6 Case Study: Glycolytic Metabolic Oscillator............. 119 Go Back 5.7 Case Study: Michaelis-Menten Enzyme Kinetics........... 122 Full Screen Close Quit Preface Albert Einstein said that "The most incomprehensible thing about the world is that it is comprehensible." It is comprehensible quantitatively through mathemat- ics. What is even more remarkable is that many areas of mathematics started off as purely abstract theories (e.g., geometry, number theory, group theory) turned out to be able to explain the natural phenomena. Take the number π (the first letter of the Greek word "perimetros"), why does this irrational number help ratio- nalize so many real phenomena with no (direct) relation to circle’s circumference? Home Page Joseph Fourier developed a mathematical technique (Fourier analysis) in the 19th century to describe heat transfer. The human inner ear (cochlea), however, has Title Page performed Fourier transforms to process sounds for ages. We know that bees are smart enough to build the honeycomb in the form of regular hexagons. However, Contents humans (mathematician Thomas Hales to be precise) only in 1999 proved the Hon- eycomb Conjecture stating that a regular hexagonal grid is the best way to divide JJ II a surface into regions of equal area with the least total perimeter. J I I strongly believe that it is essential to develop skills to understand mathemati- Page6 of 128 cal ideas using abstract formalism. This empowers you with the ability to ana- lyze diverse phenomena through the prism of the same fundamental mathematical Go Back concepts. Abstraction makes things simpler and reveals counter-intuitive. On the other hand, specific examples enable immediate connections with practice helping Full Screen build physical intuition. So, in these lecture notes I have tried to strike a balance between theory and applications. For generality purposes, I have not confined the Close pool of examples solely to chemical engineering, despite the title. Moreover, it should be stressed that there is no mathematics for Chemical Engineering. The Quit thing is that Nature does not know what we decided to call Chemical Engineering, Physics or Biology. In fact, the modern chemical engineering is so broad that the mathematical repertoire of a good chemical engineer should be quite extensive. Therefore, it is wise to develop a more universal understanding of mathematics as fundamental concepts resurface in many areas. These lecture notes should be considered as a companion to the lectures that I teach at the University of Nebraska-Lincoln. To adhere to the lecture notes style, formal mathematical proofs are left out as they can be easily found elsewhere. The Home Page notes are not supposed to substitute exercises, but rather complement them. By solving a lot of practice problems by yourself, you will be able to translate your Title Page passive learning into active understanding of the subject. Remember that nobody made it to the NHL by watching ice hockey on Youtube. Since a lot of problems Contents of practical relevance cannot be solved analytically, homework assignments pro- vide ample opportunities to practice numerical modeling. But bear in mind what JJ II Eugene Wigner had to say: "It is nice to know that the computer understands the problem. But I would like to understand it too." In computing, the GIGO J I (Garbage In, Garbage Out) principle works particularly well. Page7 of 128 The covered topics include linear algebra, Fourier calculus and Laplace transforms, Go Back ordinary and partial differential equations, nonlinear dynamical systems. Before diving into specifics of mathematical methods, I outline a general framework for Full Screen mathematical methods applied in chemical engineering. To get the most out of the class it is recommended to read the appropriate section before coming to a class. Close Also, understand all the examples, solve all practice problems by yourself and subsequently analyze the provided solutions, complete all homework assignments. Quit Recommended Literature General mathematics: 1. K. F. Riley, M. P. Hobson, S. J. Bence, "Mathematical Methods for Physics and Engineering" Cambridge University Press 2006 (all-in-one: a complete treatise on mathematical methods). 2. C. Henry Edwards and David Penney, "Elementary Differential Equations Home Page with Boundary Value Problems" Pearson 2007 (very good treatment of ODEs with lots of solved problems). Title Page 3. Alexander Altland and Jan von Delft, "Mathematics for Physicists" Cam- Contents bridge University Press 2019 (an ultimate textbook on undergraduate mathematics, except probability). JJ II J I Chemical engineering mathematics and numerical modeling: Page8 of 128 4. Kevin Dorfman and Prodromos Daoutidis, "Numerical Methods with Chem- ical Engineering Applications" Cambridge University Press 2017 Go Back (geared to practical applications with lots of MATLAB problems). Full Screen 5. Kenneth Beers, "Numerical Methods for Chemical Engineering: Applications in MATLAB" Cambridge University Press 2019. Close (more theoretical treatment of mathematical modeling) Quit 1. Mathematical Modeling in Chemical Engineer- ing in a Nutshell The goal of this chapter is to outline a general framework for mathematical mod- eling in chemical engineering. It may be worth coming back to this chapter while studying the following topics to better connect the general principles described here with specific mathematical approaches and examples. To construct a math- ematical model of a physical phenomena two types of equations need to be com- Home Page bined - conservation laws (material-independent) and constitutive relations (material-dependent) - to obtain the governing equations. Title Page REMARK: Regarding scientific models in general a famous statement by George Contents Box is in order: "All models are wrong, but some are useful." What it really means is that all models and even fundamental laws of nature as we know them now are JJ II inaccurate. For example, the Newton’s second law (F = ma) with mass m being constant was long believed to be applicable to any objects. Einstein demonstrated J I that this is wrong (or better inaccurate) − mass is not constant and increases when Page9 of 128 the object speeds up. The Newton’s law becomes more and more inaccurate as the speed of an object approaches the speed of light. Einstein figured out the science Go Back behind this phenomenon in his theory of special relativity. The Newton’s law of motion, however, is accurate enough when dealing with slow-moving
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