Maple documents for ChE 7110

This page contains links to Maple documents that complement or illustrate the material in Hjortsø and Wolenski, "Linear mathematical models in chemical engineering".

Contents

1. Model formulation
1.1 Classical Models
1.1.1 Macroscopic balances
1.1.1.1 Mass and Energy Balances
example_1.1_Steady_State_Balances_on_a_Recycle_Reactor.mw
Solves the algebraic equations in the model of the recycle reactor. Also checks Maple results against the rfesults in the book.
1.1.1.2 Balances Involving Chemical Kinetics
1.1.2 The quasi steady state assumption
example_1.7_Density_gradient_solution.mw
Numerical solution of the balance equations, liquid volume and mass of solute, of the density gradient apparatus. Comparison to the analytical solutions obtain after applying the quasi steady state assumption.
1.1.3 Differential Balances
1.1.3.1 Coordinate Systems
1.1.3.2 Constitutive Equations
1.1.3.3 Operator Notation
1.1.3.4 Mass and Energy Balances
example_1.8_Effectiveness_factor_for_catalytic_pellets.mw
This document solves for the concentration profile for a first order reaction in a catalytic pellet for rectangular, cylindrical and spherical geometries and finds the effectiveness factors versus the Thiele modulus.
example_1.12_zero_order_reaction_in_catalyst.mw
This document solves for the concentration profile for a zero order reaction in a catalytic pellet.
1.1.3.5 Problems in Fluid Mechanics
1.1.3.6 Summary of Common Boundary Conditions
1.1.3.7 Symmetry
1.2 Abstract Control Volumes

2. Some Ordinary Differential Equations
2.1 First Order Equations
2.1.1 Separable equations
2.1.2 Linear first order equations
2.1.3 Exact equations
2.1.4 Homogeneous equations
2.1.5 Bernoulli equation
2.1.6 Clairaut’s equation
2.1.7 Riccati equation
2.2 Second Order Equations
2.2.1 Dependent Variable does not occur explicitely
2.2.2 Free Variable does not occur Explicitly
2.2.3 Homogeneous equations
2.3 Higher Order Equations
2.4 Variable Transformations
2.5 The importance of being Lopschitz

3. Finite dimensional vector spaces.
3.1 Basic concepts
3.2 Examples
3.3 Span, linear independence, and basis
3.3.1 Coordinates
3.4 Isomorphisms
3.4.1 Isomorphisms of vector spaces
3.4.2 Subspaces
3.4.3 Sums
3.4.4 Representation of subspaces
3.5 Matrices
3.5.1 Matrix algebra
3.5.2 Gauss elimination
3.5.3 Determinants
3.5.3.1 Basic properties of determinants
3.5.3.2 Calculation of determinants
3.5.3.3 The derivative of a determinant
3.5.4 The classical adjoint matrix
3.6 Systems of linear algebraic equations
3.6.1 Rank
3.6.2 Applications of rank
Solution_of_the_linear_combinations_needed_to_specify_chemical_reactions.mw
This document finds three linearly independent reactions that determines the equilibrium composition in a mixture of mixture are: CO, H2, CH3OH, C2H6, CO2 and H2O. Similar but not identical to example 3.9.
3.6.3 Solution structure
3.6.4 The null and range space of a matrix
3.6.5 Overdetermined systems
3.7 The algebraic eigenvalue problem
3.7.1 Finding eigenvalues and eigenvectors
3.7.2 Multiplicity
3.7.3 Similar matrices
3.7.3.1 Equivalence relations
3.7.4 Eigenspaces and eigenbases
3.7.4.1 Diagonalization of simple and semi-simple matrices
3.7.5 Generalized eigenspaces
3.7.5.1 Generalized eigenbases
3.7.6 Jordan canonical form
Protocol_for_similarity_transformation_to_Jordan_form.mw
This document does not use the full power of Maple but goes through the step by step procedure of constructing a similarity transformation to a Jordan matrix.
3.7.7 Jordan form of real matrices with complex eigenvalues.
3.7.8 Powers and exponentials of matrices
3.7.9 Location of eigenvalues
3.8 Geometry of vector spaces
3.8.1 Vector products
3.8.1.1 Inner product
3.8.1.2 Cross product
3.8.1.3 Triple Scalar Product
3.8.1.4 Dyad or outer product
3.8.2 Gram-Schmidt orthogonalization
3.8.3 Eigenrows
3.8.4 Real, symmetric matrices

4 Tensors
4.1 Definitions and basic concepts
4.2 Examples
4.2.1 Matrices as operators
4.2.2 Equivalence transformations
4.3 The adjoint operator
4.4 Tensors
4.4.1 Transformation rules
4.4.2 Invariants of tensors
4.5 Some tensors from physics and engineering
4.5.1 Fourier's law
4.5.2 The stress tensor
4.6 Vectors and Tensors in curvilinear coordinates
4.6.1 Proper transformations
4.6.2 Vectors and transformations at a point
4.6.3 Covariance and contravariance
4.6.4 The physical components

5. Linear difference equations.
5.1 Linear equations with constant coefficients.
5.1.1 Homogeneous solutions
5.1.2 Particular solutions
5.2 Single, first order equations.
5.3 Single, higher order equations.
5.3.1 Solution by variable transformation
5.3.1.1 Euler's equation
5.3.2 Reduction of order
5.3.3 Particular solution by variation of parameters
5.4 Systems of linear difference equations
5.4.1 Basic theorems
5.4.2 Particular solution by variation of parameters
5.4.3 Equations with constant coefficients
5.4.3.1 Homogeneous solutions
5.4.3.2 Particular solutions for constant inhomogeneous term
5.5 Non Linear Equations
5.5.1 Riccati's equation

6. Linear differential equations.
6.1 Linear equations with constant coefficients
6.1.1 Homogeneous solutions
6.1.2 Particular solutions
6.2 Single, higher order equations.
6.2.1 Solution by variable transformation
6.2.1.1 Euler's equation
6.2.2 Reduction of order
6.2.3 Particular solution by variation of parameters
6.3 Systems of linear differential equations
6.3.1 Basic theorems
6.3.2 Particular solution by variation of parameters
6.3.3 Equations with constant coefficients
6.3.3.1 Homogeneous solutions
6.3.3.2 Particular solutions for constant inhomogeneous term
6.3.3.3 Dealing with complex eigenvalues
6.3.3.4 Classification of steady states
6.3.3.5 Stability of nonlinear ODEs.
6.4 Series solutions.
6.5 Some common functions defined by ODEs.
6.5.1 Exponential and trigonometric functions
6.5.2 Bessel functions
6.5.3 Legendre functions

7. Hilbert Spaces.
7.1 Infinite Dimensional Vector Spaces.
7.1.1 Countable and uncountable inifinties.
7.1.2 Normed Spaces.
7.1.3 Bases in infinite dimensional spaces
7.1.4 The function spaces Lp[0,1]
7.2 Hilbert Spaces.
7.2.1 Inner products
7.2.2 Examples
7.2.3 Orthogonality
7.2.4 Orthogonal projections
7.2.5 Orthogonal complements
7.3 Linear Operators in Hilbert Spaces.
7.3.1 Adjoint operators.
7.3.2 The Sturm-Liouville operator
7.4 Eigenvalue Problems.
7.4.1 Sturm-Liouville Problems.
7.4.2 SLP with periodic boundary conditions.
7.5 Fourier series.
7.5.1 Fourier sine series.
7.5.2 Fourier cosine series.
7.5.3 Complete Fourier series.
Example7.7_Fourier_series.mw
Assembles and plots the sine, cosine and complete Fourier series for t*(1-t). The Fourier series for any other functions can be obtained by simply changing the line in which the function is defined.
7.5.4 Gibb's phenomena
7.5.5 Generalized Fourier series.

8. Partial differential equations
8.1 Fourier series methods.
8.1.1 Classification of second order PDEs
8.1.2 Inner product method
8.1.3 PDEs with Sturm-Liouville operators
8.1.3.1 Homogeneous problem
8.1.3.2 Homogeneous problem with transcendental equation for eigenvalues
8.1.3.3 Inhomogeneous PDE
8.1.3.4 Inhomogeneous, time varying boundary conditions
8.1.4 Other self-adjoint PDEs
Example_8.8_Leaching_of_hazardous_waste.mws
Assembles the Fourier series for this problem and animates the solution.
8.2 Finite Fourier transform.
8.3 First order PDEs.
8.4 First Oder PDE and Cauchy's method.
8.4.1 Cauchy's method for linear equations
8.5 Similarity Transformation.

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