Mathematics in Chemical Engineering 3 Mathematics in Chemical Engineering Bruce A. Finlayson, Department of Chemical Engineering, University of Washington, Seattle, Washington, United States (Chap. 1, 2, 3, 4, 5, 6, 7, 8, 9, 11 and 12) Lorenz T. Biegler, Carnegie Mellon University, Pittsburgh, Pennsylvania, United States (Chap. 10) Ignacio E. Grossmann, Carnegie Mellon University, Pittsburgh, Pennsylvania, United States (Chap. 10) 1. Solution of Equations .......... 5 7.2. Initial Value Methods .......... 62 1.1. Matrix Properties ............ 5 7.3. Finite Difference Method ....... 63 1.2. Linear Algebraic Equations ...... 7 7.4. Orthogonal Collocation ........ 65 1.3. Nonlinear Algebraic Equations ... 10 7.5. Orthogonal Collocation on 1.4. Linear Difference Equations ..... 12 Finite Elements .............. 69 1.5. Eigenvalues ................ 12 7.6. Galerkin Finite Element Method .. 71 2. Approximation and Integration ... 13 7.7. Cubic B-Splines .............. 73 2.1. Introduction ................ 13 7.8. Adaptive Mesh Strategies ....... 73 2.2. Global Polynomial Approximation . 13 7.9. Comparison ................ 73 2.3. Piecewise Approximation ....... 15 7.10. Singular Problems and Infinite 2.4. Quadrature ................ 18 Domains .................. 74 2.5. Least Squares ............... 20 8. Partial Differential Equations .... 75 2.6. Fourier Transforms of Discrete Data 21 8.1. Classification of Equations ...... 75 2.7. Two-Dimensional Interpolation and 8.2. Hyperbolic Equations .......... 77 Quadrature ................ 22 8.3. Parabolic Equations in One 3. Complex Variables ........... 22 Dimension ................. 79 3.1. Introduction to the Complex Plane . 22 8.4. Elliptic Equations ............ 84 3.2. Elementary Functions ......... 23 8.5. Parabolic Equations in Two or Three 3.3. Analytic Functions of a Complex Dimensions ................. 86 Variable ................... 25 8.6. Special Methods for Fluid Mechanics 86 3.4. Integration in the Complex Plane .. 26 8.7. Computer Software ........... 88 3.5. Other Results ............... 28 9. Integral Equations ........... 89 4. Integral Transforms .......... 29 9.1. Classification ............... 89 4.1. Fourier Transforms ........... 29 9.2. Numerical Methods for Volterra 4.2. Laplace Transforms ........... 33 Equations of the Second Kind .... 91 4.3. Solution of Partial Differential 9.3. Numerical Methods for Fredholm, Equations by Using Transforms ... 37 Urysohn, and Hammerstein 5. Vector Analysis .............. 40 Equations of the Second Kind .... 91 6. Ordinary Differential Equations as 9.4. Numerical Methods for Eigenvalue Initial Value Problems ......... 49 Problems .................. 92 6.1. Solution by Quadrature ........ 50 9.5. Green’s Functions ............ 92 6.2. Explicit Methods ............. 50 9.6. Boundary Integral Equations and 6.3. Implicit Methods ............. 54 Boundary Element Method ...... 94 6.4. Stiffness ................... 55 10. Optimization ............... 95 6.5. Differential – Algebraic Systems ... 56 10.1. Introduction ................ 95 6.6. Computer Software ........... 57 10.2. Gradient-Based Nonlinear 6.7. Stability, Bifurcations, Limit Cycles 58 Programming ............... 96 6.8. Sensitivity Analysis ........... 60 10.3. Optimization Methods without 6.9. Molecular Dynamics .......... 61 Derivatives ................. 105 7. Ordinary Differential Equations as 10.4. Global Optimization .......... 106 Boundary Value Problems ....... 61 10.5. Mixed Integer Programming ..... 110 7.1. Solution by Quadrature ........ 62 10.6. Dynamic Optimization ......... 121 Ullmann’s Modeling and Simulation c 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-31605-2 4 Mathematics in Chemical Engineering 10.7. Development of Optimization 12. Multivariable Calculus Applied to Models .................... 124 Thermodynamics ............. 135 11. Probability and Statistics ....... 125 12.1. State Functions .............. 135 11.1. Concepts .................. 125 12.2. Applications to Thermodynamics .. 136 11.2. Sampling and Statistical Decisions . 129 11.3. Error Analysis in Experiments .... 132 12.3. Partial Derivatives of All ...... 11.4. Factorial Design of Experiments and Thermodynamic Functions 137 Analysis of Variance ........... 133 13. References ................. 138 Symbols q heat flux Variables Q volumetric flow rate; rate of heat gener- a 1, 2, or 3 for planar, cylindrical, spher- ation for heat transfer problems ical geometry r radial position in cylinder ai acceleration of i-th particle in molecu- ri position of i-th particle in molecular dy- lar dynamics namics A cross-sectional area of reactor; R radius of reactor or catalyst pellet Helmholtz free energy in thermody- Re Reynolds number namics S entropy in thermodynamics Bi Biot number Sh Sherwood number Bim Biot number for mass t time c concentration T temperature Cp heat capacity at constant pressure u velocity Cs heat capacity of solid U internal energyy in thermodynamics Cv heat capacity at constant volume vi velocity of i-th particle in molecular dy- Co Courant number namics D diffusion coefficient V volume of chemical reactor; vapor flow Da Damkohler¨ number rate in distillation column; potential en- De effective diffusivity in porous catalyst ergy in molecular dynamics; specific E efficiency of tray in distillation column volume in thermodynamics F molar flow rate into a chemical reactor x position G Gibbs free energy in thermodynamics z position from inlet of reactor hp heat transfer coefficient H enthalpy in thermodynamics Greek symbols J mass flux a thermal diffusivity k thermal conductivity; reaction rate con- δ Kronecker delta stant ∆ sampling rate ke effective thermal conductivity of ε porosity of catalyst pellet porous catalyst η viscosity in fluid flow; effectiveness kg mass transfer coefficient factor for reaction in a catalyst pellet K chemical equilibrium constant η0 zero-shear rate viscosity L thickness of slab; liquid flow rate in dis- φ Thiele modulus tillation column; length of pipe for flow ϕ void fraction of packed bed in pipe λ time constant in polymer flow mi mass of i-th particle in molecular dy- ρ density namics ρs density of solid M holdup on tray of distillation column τ shear stress n power-law exponent in viscosity for- µ viscosity mula for polymers p pressure Pe Peclet number Mathematics in Chemical Engineering 5 Special symbols for special structures. They are useful because | subject to they increase the speed of solution. If the un- : mapping. For example, h : Rn → Rm , knowns appear in a nonlinear fashion, the prob- states that functions h map real num- lem is much more difficult. Iterative techniques bers into m real numbers. There are m must be used (i.e., make a guess of the so- functions h written in terms of n vari- lution and try to improve the guess). An im- ables portant question then is whether such an iter- ∈ member of ative scheme converges. Other important types → maps into of equations are linear difference equations and eigenvalue problems, which are also discussed. 1. Solution of Equations 1.1. Matrix Properties Mathematical models of chemical engineering systems can take many forms: they can be sets A matrix is a set of real or complex numbers of algebraic equations, differential equations, arranged in a rectangular array. and/or integral equations. Mass and energy bal- a11 a12 ... a1n ances of chemical processes typically lead to a21 a22 ... a2n large sets of algebraic equations: A= . . a11x1+a12x2 = b1 am1 am2 ... amn a21x1+a22x2 = b2 The numbers aij are the elements of the matrix Mass balances of stirred tank reactors may lead T to ordinary differential equations: A,or(A)ij = aij . The transpose of A is (A )= d y aji . = f [y (t)] d t The determinant of a square matrix A is integral a11 a12 ... a1n Radiative heat transfer may lead to a21 a22 ... a2n equations: A= . . 1 . y (x)=g (x)+λ K (x, s) f (s)ds an1 an2 ... ann 0 If the i-th row and j-th column are deleted, a Even when the model is a differential equa- new determinant is formed having n−1rowsand tion or integral equation, the most basic step in columns. This determinant is called the minor of the algorithm is the solution of sets of algebraic aij denoted as Mij . The cofactor Aij of the el- equations. The solution of sets of algebraic equa- ement aij is the signed minor of aij determined tions is the focus of Chapter 1. by A single linear equation is easy to solve for i+j either x or y: Aij =(−1) Mij y = ax+b The value of |A| is given by n n If the equation is nonlinear, |A| = aij Aij or aij Aij f (x)=0 j=1 i=1 it may be more difficult to find the x satisfying where the elements aij must be taken from a this equation. These problems are compounded single row or column of A. when there are more unknowns, leading to si- If all determinants formed by striking out multaneous equations. If the unknowns appear whole rows or whole columns of order greater in a linear fashion, then an important considera- than r are zero, but there is at least one determi- tion is the structure of the matrix representing the nant of order r which is not zero, the matrix has equations; special methods are presented here rank r. 6 Mathematics in Chemical Engineering The value of a determinant is not changed if 0 if j<i−1 aij = aij otherwise the rows and columns are interchanged. If the elements of one row (or one column) of a de- 0 if j>i+1 terminant are all zero, the value of the deter- Block diagonal and pentadiagonal matrices also minant is zero. If the elements of one row or arise, especially when solving partial differential column are multiplied by the same constant, the equations in two- and three-dimensions. determinant is the previous value times that con- stant. If two adjacent rows (or columns) are in- QR Factorization of a Matrix. IfAisanm terchanged, the value of the new determinant is × n matrix with m ≥ n, there exists an m × m the negative of the value of the original determi- unitary matrix Q =[q1, q2, ...qm ] and an m × n nant. If two rows (or columns) are identical, the right-triangular matrix R such that A=QR.