Applied Numerical Methods for Chemical Engineers

Applied Numerical Methods for Chemical Engineers emphasizes the derivation of a variety of numerical methods and their application to the solution of engineering problems, with special attention to problems in the chemical engineering field. These algorithms encompass linear and nonlinear algebraic equations, eigenvalue problems, finite difference methods, interpolation, differentiation and integration, ordinary differential equations, boundary value problems, partial differential equations, and linear and nonlinear regression analysis. MATLAB is adopted as the calculation environment throughout the book because of its ability to perform all the calculations in matrix form, its large library of built-in functions, its strong structural language, and its rich graphical visualization tools. Through this book, students and other users will learn about the basic features, advantages and disadvantages of various numerical methods, learn and practice many useful m-files developed for different numerical methods in addition to the MATLAB built-in solvers, develop and set up mathematical models for problems commonly encountered in chemical engineering, and solve chemical engineering related problems through examples and after-chapter problems with MATLAB by creating application m-files. Applied Numerical Methods for Chemical Engineers Copyright Dedication Preface Contents 1 Nonlinear equations Motivation 1.1 Introduction 1.2 Types of roots and their approximation 1.3 The method of successive substitution 1.4 The Wegstein method 1.5 The bisection method 1.6 The method of linear interpolation 1.7 The Newton-Raphson method 1.8 Synthetic division algorithm 1.9 The eigenvalue method 1.10 Newton’s method for solving system of nonlinear equations 1.11 Homotopy method 1.12 Using the built-in MATLAB and Excel functions 1.13 Summary Problems References 2 Simultaneous linear algebraic equations Motivation 2.1 Introduction 2.2 Review of selected matrix and vector operations 2.2.1 Matrices and determinants 2.2.2 Matrix transformations 2.2.3 Matrix polynomials and power series 2.2.4 Vector operations 2.3 Consistency of equations and existence of solutions 2.4 Cramer’s rule 2.5 Gauss elimination method 2.5.1 Gauss elimination in formula form 2.5.2 Gauss elimination in matrix form Calculation of determinants by the Gauss method 2.6 Gauss-Jordan Reduction Method 2.6.1 Gauss-Jordan reduction in formula form 2.6.2 Gauss-Jordan reduction in matrix form 2.6.3 Gauss-Jordan reduction with matrix inversion 2.7 Gauss-Seidel substitution method 2.8 Jacobi method 2.9 Homogeneous algebraic equations and the characteristic-value problem 2.9.1 The Faddeev-Leverrier method 2.9.2 Elementary similarity transformations 2.9.3 The QR algorithm of successive factorization 2.10 Using built-in MATLAB® and Excel functions 2.11 Summary Problems References 3 Finite difference methods and interpolation Motivation 3.1 Introduction 3.2 Symbolic operators 3.3 Backward finite differences 3.4 Forward finite differences 3.5 Central finite differences 3.6 Interpolating polynomials 3.7 Interpolation of equally spaced points 3.7.1 Gregory-Newton interpolation 3.7.2 Stirling’s interpolation 3.8 Interpolation of unequally spaced points 3.8.1 Lagrange polynomials 3.8.2 Spline interpolation 3.9 Using built-in MATLAB® functions 3.10 Summary Problems References 4 Differentiation and integration Motivation 4.1 Introduction 4.2 Differentiation by backward finite differences 4.2.1 First derivative 4.2.2 Second derivative 4.3 Differentiation by forward finite differences 4.3.1 First derivative 4.3.2 Second derivative 4.4 Differentiation by central finite differences 4.4.1 First derivative 4.4.2 Second derivative 4.5 Spline differentiation 4.6 Integration formulas 4.7 Newton-Cotes formulas of integration 4.7.1 The trapezoidal rule 4.7.2 Simpson’s 1/3 rule 4.7.3 Simpson’s 3/8 rule 4.7.4 Summary of Newton-Cotes integration 4.8 Gauss quadrature 4.8.1 Two-point Gauss-Legendre quadrature 4.8.2 Higher-point Gauss-Legendre formulas 4.9 Spline integration 4.10 Multiple integrals 4.11 Using built-in MATLAB® functions 4.12 Summary Problems References 5 Ordinary differential equations: initial value problems Motivation 5.1 Introduction 5.2 Classifications of ordinary differential equations 5.3 Transformation to canonical form 5.4 Linear ordinary differential equations 5.5 Nonlinear ordinary differential equations 5.5.1 The Euler and modified Euler methods 5.5.2 The Runge-Kutta methods 5.5.3 The Adams and Adams-Moulton methods 5.5.4 Simultaneous Differential Equations 5.6 Using built-in MATLAB® functions 5.7 Difference equations and their solutions 5.8 Propagation, stability, and convergence 5.8.1 Stability and Error Propagation of Euler Methods 5.8.2 Stability and error propagation of Runge-Kutta methods 5.8.3 Stability and error propagation of multistep methods 5.9 Step size control 5.10 Stiff differential equations 5.11 Summary Problems References 6 Ordinary differential equations: boundary value problems Motivation 6.1 Introduction 6.2 The shooting method 6.3 The finite-difference method 6.4 Collocation methods 6.5 Using built-in MATLAB® functions 6.6 Summary Problems References 7 Partial differential equations Motivation 7.1 Introduction 7.2 Classification of partial differential equations 7.3 Initial and boundary conditions 7.4 Solution of partial differential equations using finite differences 7.4.1 Elliptic partial differential equations 7.4.2 Parabolic partial differential equations Implicit methods Method of lines 7.4.3 Hyperbolic partial differential equations 7.4.4 Irregular boundaries and polar coordinate systems 7.5 Using built-in MATLAB® functions 7.6 Stability analysis 7.7 Summary Problems References 8 Linear and nonlinear regression analysis Motivation 8.1 Process analysis, mathematical modeling, and regression analysis 8.2 Review of statistical terminology used in regression analysis 8.2.1 Population and sample statistics 8.2.2 Probability density functions and probability distributions 8.2.3 Confidence intervals and hypothesis testing 8.3 Linear regression analysis 8.3.1 The least squares method 8.3.2 Properties of the estimated vector of parameters 8.4 Nonlinear regression analysis 8.4.1 The method of steepest descent 8.4.2 The Gauss-Newton method 8.4.3 Newton’s method 8.4.4 The Marquardt method1 8.4.5 Multiple nonlinear regression 8.5 Analysis of variance and other statistical tests of the regression results 8.6 Using built-in MATLAB® and Excel functions 8.7 Summary Problems References Appendix Orthogonal polynomials Index