MathematicsinChemicalEngineering - Wiley-VCH
Mathematics in Chemical Engineering3Mathematics in Chemical EngineeringBruce 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. .2.6.3.6.4.6.5.6.6.6.7.6.8.6.9.7.7.1.Solution of Equations . . . . . . . . . .Matrix Properties . . . . . . . . . . . .Linear Algebraic Equations . . . . . .Nonlinear Algebraic Equations . . .Linear Difference Equations . . . . .Eigenvalues . . . . . . . . . . . . . . . .Approximation and Integration . . .Introduction . . . . . . . . . . . . . . . .Global Polynomial Approximation .Piecewise Approximation . . . . . . .Quadrature . . . . . . . . . . . . . . . .Least Squares . . . . . . . . . . . . . . .Fourier Transforms of Discrete DataTwo-Dimensional Interpolation andQuadrature . . . . . . . . . . . . . . . .Complex Variables . . . . . . . . . . .Introduction to the Complex Plane .Elementary Functions . . . . . . . . .Analytic Functions of a ComplexVariable . . . . . . . . . . . . . . . . . . .Integration in the Complex Plane . .Other Results . . . . . . . . . . . . . . .Integral Transforms . . . . . . . . . .Fourier Transforms . . . . . . . . . . .Laplace Transforms . . . . . . . . . . .Solution of Partial DifferentialEquations by Using Transforms . . .Vector Analysis . . . . . . . . . . . . . .Ordinary Differential Equations asInitial Value Problems . . . . . . . . .Solution by Quadrature . . . . . . . .Explicit Methods . . . . . . . . . . . . .Implicit Methods . . . . . . . . . . . . .Stiffness . . . . . . . . . . . . . . . . . . .Differential – Algebraic Systems . . .Computer Software . . . . . . . . . . .Stability, Bifurcations, Limit CyclesSensitivity Analysis . . . . . . . . . . .Molecular Dynamics . . . . . . . . . .Ordinary Differential Equations asBoundary Value Problems . . . . . . .Solution by Quadrature . . . . . . . 0616162Ullmann’s Modeling and Simulationc 2007 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: .5.10.6.Initial Value Methods . . . . . . . . . .Finite Difference Method . . . . . . .Orthogonal Collocation . . . . . . . .Orthogonal Collocation onFinite Elements . . . . . . . . . . . . . .Galerkin Finite Element Method . .Cubic B-Splines . . . . . . . . . . . . . .Adaptive Mesh Strategies . . . . . . .Comparison . . . . . . . . . . . . . . . .Singular Problems and InfiniteDomains . . . . . . . . . . . . . . . . . .Partial Differential Equations . . . .Classification of Equations . . . . . .Hyperbolic Equations . . . . . . . . . .Parabolic Equations in OneDimension . . . . . . . . . . . . . . . . .Elliptic Equations . . . . . . . . . . . .Parabolic Equations in Two or ThreeDimensions . . . . . . . . . . . . . . . . .Special Methods for Fluid MechanicsComputer Software . . . . . . . . . . .Integral Equations . . . . . . . . . . .Classification . . . . . . . . . . . . . . .Numerical Methods for VolterraEquations of the Second Kind . . . .Numerical Methods for Fredholm,Urysohn, and HammersteinEquations of the Second Kind . . . .Numerical Methods for EigenvalueProblems . . . . . . . . . . . . . . . . . .Green’s Functions . . . . . . . . . . . .Boundary Integral Equations andBoundary Element Method . . . . . .Optimization . . . . . . . . . . . . . . .Introduction . . . . . . . . . . . . . . . .Gradient-Based NonlinearProgramming . . . . . . . . . . . . . . .Optimization Methods withoutDerivatives . . . . . . . . . . . . . . . . .Global Optimization . . . . . . . . . .Mixed Integer Programming . . . . .Dynamic Optimization . . . . . . . . 59596105106110121
4Mathematics in Chemical Engineering10.7. Development of OptimizationModels . . . . . . . . . . . . . . . . . . . .11.Probability and Statistics . . . . . . .11.1. Concepts . . . . . . . . . . . . . . . . . .11.2. Sampling and Statistical Decisions .11.3. Error Analysis in Experiments . . . .11.4. Factorial Design of Experiments andAnalysis of Variance . . . . . . . . . . 4125125129132133SymbolsVariables1, 2, or 3 for planar, cylindrical, spherical geometryacceleration of i-th particle in molecular dynamicscross-sectional area of reactor;Helmholtz free energy in thermodynamicsBiot numberBiot number for massconcentrationheat capacity at constant pressureheat capacity of solidheat capacity at constant volumeCourant numberdiffusion coefficientDamköhler numbereffective diffusivity in porous catalystefficiency of tray in distillation columnmolar flow rate into a chemical reactorGibbs free energy in thermodynamicsheat transfer coefficiententhalpy in thermodynamicsmass fluxthermal conductivity; reaction rate constanteffective thermal conductivity ofporous catalystmass transfer coefficientchemical equilibrium constantthickness of slab; liquid flow rate in distillation column; length of pipe for flowin pipemass of i-th particle in molecular dynamicsholdup on tray of distillation columnpower-law exponent in viscosity formula for polymerspressurePeclet numberMultivariable Calculus Applied toThermodynamics . . . . . . . . . . . .12.1. State Functions . . . . . . . . . . . . .12.2. Applications to Thermodynamics .12.3. Partial Derivatives of AllThermodynamic Functions . . . . .13.References . . . . . . . . . . . . . . . .qQrriRReSShtTuUviVxzaδ εηη0φϕλρρsτµ. 135. 135. 136. 137. 138heat fluxvolumetric flow rate; rate of heat generation for heat transfer problemsradial position in cylinderposition of i-th particle in molecular dynamicsradius of reactor or catalyst pelletReynolds numberentropy in thermodynamicsSherwood numbertimetemperaturevelocityinternal energyy in thermodynamicsvelocity of i-th particle in molecular dynamicsvolume of chemical reactor; vapor flowrate in distillation column; potential energy in molecular dynamics; specificvolume in thermodynamicspositionposition from inlet of reactorGreek symbolsthermal diffusivityKronecker deltasampling rateporosity of catalyst pelletviscosity in fluid flow; effectivenessfactor for reaction in a catalyst pelletzero-shear rate viscosityThiele modulusvoid fraction of packed bedtime constant in polymer flowdensitydensity of solidshear stressviscosity
Mathematics in Chemical Engineering : Special symbolssubject tomapping. For example, h : Rn Rm ,states that functions h map real numbers into m real numbers. There are mfunctions h written in terms of n variablesmember ofmaps into5for special structures. They are useful becausethey increase the speed of solution. If the unknowns appear in a nonlinear fashion, the problem is much more difficult. Iterative techniquesmust be used (i.e., make a guess of the solution and try to improve the guess). An important question then is whether such an iterative scheme converges. Other important typesof equations are linear difference equations andeigenvalue problems, which are also discussed.1. Solution of Equations1.1. Matrix PropertiesMathematical models of chemical engineeringsystems can take many forms: they can be setsof algebraic equations, differential equations,and/or integral equations. Mass and energy balances of chemical processes typically lead tolarge sets of algebraic equations:a11 x1 a12 x2 b1a21 x1 a22 x2 b2Mass balances of stirred tank reactors may leadto ordinary differential equations:dy f [y (t)]dtRadiative heat transfer may lead to integralequations: 1y (x) g (x) λ K (x, s) f (s) d s0Even when the model is a differential equation or integral equation, the most basic step inthe algorithm is the solution of sets of algebraicequations. The solution of sets of algebraic equations is the focus of Chapter 1.A single linear equation is easy to solve foreither x or y:y ax bIf the equation is nonlinear,f (x) 0it may be more difficult to find the x satisfyingthis equation. These problems are compoundedwhen there are more unknowns, leading to simultaneous equations. If the unknowns appearin a linear fashion, then an important consideration is the structure of the matrix representing theequations; special methods are presented hereA matrix is a set of real or complex numbersarranged in a rectangular array. a11 a21 A . .am1a12a22.am2 a1na2n . . amn.The numbers aij are the elements of the matrixA, or (A)ij aij . The transpose of A is (AT ) aji .The determinant of a square matrix A is a11 a21 A . . an1a12a22.an2. a1n a2n . . ann If the i-th row and j-th column are deleted, anew determinant is formed having n 1 rows andcolumns. This determinant is called the minor ofaij denoted as M ij . The cofactor A ij of the element aij is the signed minor of aij determinedbyA ij ( 1)i j MijThe value of A is given byn A j 1aij A ij orni 1aij A ijwhere the elements aij must be taken from asingle row or column of A.If all determinants formed by striking outwhole rows or whole columns of order greaterthan r are zero, but there is at least one determinant of order r which is not zero, the matrix hasrank r.
6Mathematics in Chemical EngineeringThe value of a determinant is not changed ifthe rows and columns are interchanged. If theelements of one row (or one column) of a determinant are all zero, the value of the determinant is zero. If the elements of one row orcolumn are multiplied by the same constant, thedeterminant is the previous value times that constant. If two adjacent rows (or columns) are interchanged, the value of the new determinant isthe negative of the value of the original determinant. If two rows (or columns) are identical, thedeterminant is zero. The value of a determinantis not changed if one row (or column) is multiplied by a constant and added to another row (orcolumn).A matrix is symmetric ifaij ajiand it is positive definite ifTnnx Ax aij xi xj 0i 1j 1for all x and the equality holds only if x 0.If the elements of A are complex numbers, A*denotes the complex conjugate in which (A*)ij a*ij . If A A* the matrix is Hermitian.The inverse of a matrix can also be used tosolve sets of linear equations. The inverse is amatrix such that when A is multiplied by its inverse the result is the identity matrix, a matrixwith 1.0 along the main diagonal and zero elsewhere.AA 1 IIf AT A 1 the matrix is orthogonal.Matrices are added and subtracted elementby element.A B is aij bijTwo matrices A and B are equal if aij bij .Special relations are(AB) 1 B 1 A 1 , (AB)T B T AT 1T, (ABC) 1 C 1 B 1 A 1A 1 ATA diagonal matrix is zero except for elementsalong the diagonal.aij aii , i j0, i jA tridiagonal matrix is zero except for elementsalong the diagonal and one element to the rightand left of the diagonal. 0 if j i 1aij aij otherwise 0 if j i 1Block diagonal and pentadiagonal matrices alsoarise, especially when solving partial differentialequations in two- and three-dimensions.QR Factorization of a Matrix. If A is an m n matrix with m n, there exists an m munitary matrix Q [q1 , q2 , . . .qm ] and an m nright-triangular matrix R such that A QR. TheQR factorization is frequently used in the actual computations when the other transformations are unstable.Singular Value Decomposition. If A is anm n matrix with m n and rank k n, consider the two following matrices.AA and A AAn m m unitary matrix U is formed from theeigenvectors ui of the first matrix.U [u1 ,u2 ,. . .um ]An n n unitary matrix V is formed from theeigenvectors vi of the second matrix.V [v1 ,v2 ,. . .,vn ]Then the matrix A can be decomposed intoA U ΣV where Σ is a k k diagonal matrix with diagonalelements dii σi 0 for 1 i k. The eigenvalues of Σ Σ are σi2 . The vectors ui for k 1 i m and vi for k 1 i n are eigenvectors associated with the eigenvalue zero; the eigenvaluesfor 1 i k are σi2 . The values of σi are calledthe singular values of the matrix A. If A is real,then U and V are real, and hence orthogonal matrices. The value of the singular value decomposition comes when a process is represented by alinear transformation and the elements of A, aij ,are the contribution to an output i for a particular variable as input variable j. The input maybe the size of a disturbance, and the output isthe gain [1]. If the rank is less than n, not all thevariables are independent and they cannot all becontrolled. Furthermore, if the singular valuesare widely separated, the process is sensitive tosmall changes in the elements of the matrix andthe process will be difficult to control.
Mathematics in Chemical Engineering71.2. Linear Algebraic EquationsConsider the n n linear systema11 x1 a12 x2 . . . a1n xn f1a21 x1 a22 x2 . . . a2n xn f2.an1 x1 an2 x2 . . . ann xn fnIn this equation a11 , . . . , ann are known parameters, f 1 , . . . , f n are known, and the unknowns are x 1 , . . . , x n . The values of all unknowns that satisfy every equation must befound. This set of equations can be representedas follows:naij xj fj or Ax fj 1The most efficient method for solving a set oflinear algebraic equations is to perform a lower– upper (LU) decomposition of the corresponding matrix A. This decomposition is essentiallya Gaussian elimination, arranged for maximumefficiency [2, 3].The LU decomposition writes the matrix asA LUThe U is upper triangular; it has zero elementsbelow the main diagonal and possibly nonzerovalues along the main diagonal and above it (seeFig. 1). The L is lower triangular. It has the value1 in each element of the main diagonal, nonzerovalues below the diagonal, and zero values abovethe diagonal (see Fig. 1). The original problemcan be solved in two steps:Ly f , U x y solves Ax LU x fEach of these steps is straightforward becausethe matrices are upper triangular or lower triangular.When f is changed, the last steps can bedone without recomputing the LU decomposition. Thus, multiple right-hand sides can be computed efficiently. The number of multiplicationsand divisions necessary to solve for m right-handsides is:Operation count 1 3 1n n m n233The determinant is given by the product ofthe diagonal elements of U. This should be calculated as the LU decomposition is performed.Figure 1. Structure of L and U matricesIf the value of the determinant is a very large orvery small number, it can be divided or multiplied by 10 to retain accuracy in the computer;the scale factor is then accumulated separately.The condition number κ can be defined in termsof the singular value decomposition as the ratioof the largest d ii to the smallest d ii (see above).It can also be expressed in terms of the norm ofthe matrix:κ (A) A A 1 where the norm is defined asn A supx 0 Ax ajk maxk x j 1If this number is infinite, the set of equationsis singular. If the number is too large, the matrix is said to be ill-conditioned. Calculation of
8Mathematics in Chemical Engineeringthe condition number can be lengthy so anothercriterion is also useful. Compute the ratio of thelargest to the smallest pivot and make judgmentson the ill-conditioning based on that.When a matrix is ill-conditioned the LUdecomposition must be performed by using pivoting (or the singular value decomposition described above). With pivoting, the order of theelimination is rearranged. At each stage, onelooks for the largest element (in magnitude);the next stages if the elimination are on the rowand column containing that largest element. Thelargest element can be obtained from only thediagonal entries (partial pivoting) or from allthe remaining entries. If the matrix is nonsingular, Gaussian elimination (or LU decomposition)could fail if a zero value were to occur along thediagonal and were to be a pivot. With full pivoting, however, the Gaussian elimination (or LUdecomposition) cannot fail because the matrixis nonsingular.The Cholesky decomposition can be used forreal, symmetric, positive definite matrices. Thisalgorithm saves on storage (divide by about 2)and reduces the number of multiplications (divide by 2), but adds n square roots.The linear equations are solved byThe LU decomposition algorithm for solvingthis set isb 1 b1for k 2, n doa k b ak , b k bk k 1akb k 1ck 1enddod 1 d1for k 2, n dod k dk a k d k 1enddoxn d n /b nfor k n 1, 1doxk d k ck xk 1b kenddoThe number of multiplications and divisions fora problem with n unknowns and m right-handsides isOperation count 2 (n 1) m (3 n 2)If bi ai ci no pivoting is necessary. For solving two-pointboundary value problems and partial differentialequations this is often the case.x A 1 fGenerally, the inverse is not used in this waybecause it requires three times more operationsthan solving with an LU decomposition. However, if an inverse is desired, it is calculated mostefficiently by using the LU decomposition andthen solvingAx(i) b(i)(i)bj 0j i1 j iThen set A 1 x(1) x(2) x(3) ··· x(n)Solutions of Special Matrices. Special matrices can be handled even more efficiently. Atridiagonal matrix is one with nonzero entriesalong the main diagonal, and one diagonal aboveand below the main one (see Fig. 2). The corresponding set of equations can then be writtenasai xi 1 bi xi ci xi 1 diFigure 2. Structure of tridiagonal matricesSparse matrices are ones in which the majority of elements are zero. If the zero entries occur in special patterns, efficient techniques canbe used to exploit the structure, as was doneabove for tridiagonal matrices, block tridiagonal
Mathematics in Chemical Engineeringmatrices, arrow matrices, etc. These structurestypically arise from numerical analysis appliedto solve differential equations. Other problems,such as modeling chemical processes, lead tosparse matrices but without such a neatly definedstructure — just a lot of zeros in the matrix. Formatrices such as these, special techniques mustbe employed: efficient codes are available [4].These codes usually employ a symbolic factorization, which must be repeated only once foreach structure of the matrix. Then an LU factorization is performed, followed by a solution stepusing the triangular matrices. The symbolic factorization step has significant overhead, but thisis rendered small and insignificant if matriceswith exactly the same structure are to be usedover and over [5].The efficiency of a technique for solving setsof linear equations obviously depends greatly onthe arrangement of the equations and unknownsbecause an efficient arrangement can reduce thebandwidth, for example. Techniques for renumbering the equations and unknowns arising fromelliptic partial differential equations are available for finite difference methods [6] and for finite element methods [7].Solutions with Iterative Methods. Sets oflinear equations can also be solved by using iterative methods; these methods have a rich historical background. Some of them are discussedin Chapter 8 and include Jacobi, Gauss – Seidel, and overrelaxation methods. As the speedof computers increases, direct methods becomepreferable for the general case, but for largethree-dimensional problems iterative methodsare often used.The conjugate gradient method is an iterative method that can solve a set of n linear equations in n iterations. The method primarily requires multiplying a matrix by a vector, whichcan be done very efficiently on parallel computers: for sparse matrices this is a viable method.The original method was devised by Hestenesand Stiefel [8]; however, more recent implementations use a preconditioned conjugate gradient method because it converges faster, provided a good “
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L. Dai et al. 1152 ¹ 2003 Wiley-VCH Verlag GmbH&Co. KGaA, Weinheim www.chemphyschem.org CHEMPHYSCHEM 2003, 4, 1150 1169 example, BN, B xC yN z, GaP, GaAs), oxide nanotubes (for example, SiO 2,Al 2O 3,V 2O 5,TiO 2), and polymer nanotubes (for example, polyacrylonitrile, polyaniline,
Organic and organometallic chemistry of the Master 1's level - Basic knowledge of kinetic and thermodynamic chemistry of the L3's level. REFERENCES Catalysis, From principles to Applications, M. Beller, A. Renken, R. A. van Santen, Wiley-VCH, 2012. Concepts of Nanochemistry, L. Cademartiri, G.A. Ozin, Wiley-VCH, 2009.
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behavior is best done with an understanding of behavior change theories and an ability to use them in practice (1990, p. 19). the goal of this Gravitas, therefore, is to introduce three major theories of behav-ior change, describe the key variables of behavior change models, and to explore the link between behav-ior change and attitude.
