Mathematical Modeling of Physical Systems: An Introduction

The aim of this text is to provide the beginning student or professional with introduction to the topic in an easily understood student friendly manner. It is based on the premise that modeling is as much art as it is science and that mastering the art can only be achieved by sustained practice. To provide that practice, the text contains some 100 odd worked examples as well as numerous practice problems drawn from variety of disciplines. They range from classical examples such as Euler's treatment of the buckling of the strut, to the contemporary topics of silicon chip manufacturing and the dynamics of the human immunodeficiency virus (HIV). Other examples are drawn from mechanical, electrical, chemical and environmental engineering disciplines as well as from the fields of economics, physics and chemistry. The mathematics required are confined to simple treatment of vector algebra, matrix operations and the solutions of ordinary differential equations. Both analytical and numerical methods are taken up and are described in sufficient detail to serve as a learning tool for the beginner or as a refresher for the informed reader. The text is designed for 3rd and 4th yr students in all engineering disciplines as well as mathematics, physics and chemistry. It should also serve as a welcome addition to libraries of practicing professionals.

Table of Contents:
PREFACE ; NOTATION ; 1. GETTING STARTED AND BEYOND ; 1.1 When Not to Model ; Illustration 1.1 The Challenger Space Shuttle Disaster ; Illustration 1.2 Loss of Blood Vessel Patency ; 1.2 Some Initial Tools and Steps ; 1.3 Closure ; Illustration 1.3 Discharge of Plant Effluent into a River ; Illustration 1.4 Electrical Field Due to a Dipole ; Illustration 1.5 Design of a Thermocouple ; Illustration 1.6 Newton's Law for Systems of Variable Mass: A False Start and the Remedy ; Illustration 1.7 Release of a Substance into a Flowing Fluid. Determination of a Mass Transfer Coefficient ; Practice Problems ; 2. SOME MATHEMATICAL TOOLS ; 2.1 Vector Algebra ; 2.1.1 Definition of a Vector ; 2.1.2 Vector Equality ; 2.1.3 Vector Addition and Subtraction ; 2.1.4 Multiplication by a Scalar ; 2.1.5 The Scalar or Dot Product ; 2.1.6 The Vector or Cross Product ; Illustration 2.1 Distance of a Point from a Plane ; Illustration 2.2 Shortest Distance Between Two Lines ; Illustration 2.3 Work as an Application of the Scalar Product ; Illustration 2.4 Extension of the Scalar Product to n Dimensions. A Sale of Stocks ; Illustration 2.5 A Model Economy ; 2.2 Matrices ; 2.2.1 Types of Matrices ; 2.2.2 The Echelon Form. Rank r ; 2.2.3 Matrix Equality ; 2.2.4 Matrix Addition ; Illustration 2.6 Acquisition Costs ; 2.2.5 Multiplication by a Scalar ; 2.2.6 Matrix Multiplication ; Illustration 2.7 The Product of Two Matrices ; Illustration 2.8 Matrix-Vector Representation of Linear Algebraic Equation ; 2.2.7 Elementary Row Operations ; Illustration 2.9 Application of Elementary Row Operation. Algebraic Equivalence ; 2.2.8 Solution of Sets of Linear Algebraic Equations. Gaussian Elimination ; Illustration 2.10 An Overspecified System of Equations with a Unique Solution ; Illustration 2.11 A Normal System of Equations with no Solutions ; 2.3 Ordinary Differential Equations ; Illustration 2.12 A Population Model ; Illustration 2.13 Newton's Law of Cooling ; 2.3.1 Order of an ODE ; 2.3.2 Linear and Non-linear ODE's ; 2.3.3 Boundary and Initial Conditions ; Illustration 2.14 Classification of ODE's and Boundary Conditions ; 2.3.4 Equivalent Systems ; Illustration 2.15 Equivalence of Vibrating Mechanical Systems and an Electrical RLC Circuit ; 2.3.5 Analytical Methods ; A. Solution by Separation of Variables ; Illustration 2.16 Solution of Non-Linear ODE's by Separation of Variables ; B. The D-Operator and Eigenvalue Methods. Particular Integrals ; Illustration 2.17 Mass on a Spring Subjected to a Sinusoidal Forcing Function ; C. The Laplace Transformation ; Illustration 2.18 Application of Inversion Procedures ; Illustration 2.19 The Mass-Spring System Revisited. Resonance ; Practice Problems ; 3. GEOMETRICAL CONCEPTS ; 3.1 Introduction ; Illustration 3.1 A Simple Geometry Problem: Crossing of a River ; Illustration 3.2 The Formation of Quasi Crystals and Tilings from Two Quadrilateral Polygons ; Illustration 3.3 Charting of Market Price Dynamics: The Japanese Candlestick Method ; Illustration 3.4 Surveying: The Join Calculation. The Triangulation Intersection ; Illustration 3.5 The Global Positioning System (GPS) ; Illustration 3.6 The Orthocenter of a Triangle ; Illustration 3.7 Relative Velocity and the Wind Triangle ; Illustration 3.8 Interception of an Airplane ; Illustration 3.9 Path of Pursuit ; Illustration 3.10 Trilinear Coordinates. The Three Jug Problem ; Illustration 3.11 Inflecting Production Rates and Multiple Steady States. The Van Heerden Diagram ; Illustration 3.12 Linear Programming: A Geometrical Construction ; Illustration 3.13 Stagewise Adsorption Purification of Liquids. The Operating Diagram ; Illustration 3.14 Supercoiled DNA ; Practice Problems ; 4. THE EFFECT OF FORCES ; 4.1 Introduction ; Illustration 4.1 The Stress-Strain Relation. Stored Strain Energy. Stress Due to the Impact of a Falling Mass ; Illustration 4.2 Bending of Beams. Euler's Formula for the Buckling of a Strut ; Illustration 4.3 Electrical and Magnetic Forces. Thomson's Determination of e/m ; Illustration 4.4 Pressure of a Gas in Terms of its Molecular Properties. Boyle's Law and the Ideal Gas Law. Velocity of Gas Molecules ; Illustration 4.5 Path of a Projectile ; Illustration 4.6 The Law of Universal Gravitation. Escape Velocity. The Synchronous Satellite ; Illustration 4.7 Fluid Forces. Bernoulli's Equation and Its Applications. The Continuity Equation ; Illustration 4.8 Lift Capacity of a Hot Air Balloon ; Illustration 4.9 Work and Energy. Compression of a Gas. Power Output of a Bumblebee Practice Problems ; 5. COMPARTMENTAL MODELS ; 5.1 Introduction ; Illustration 5.1 Measurement of Plasma Volume and Cardiac Output by the Dye Dilution Method ; Illustration 5.2 The Continuous Stirred Tank Reactor (CSTR). Model and Optimum Size ; Illustration 5.3 Modeling of a Bioreactor. Monod Kinetics. The Optimum Dilution Rate ; Illustration 5.4 Non-Idealities in a Stirred Tank. Residence-Time Distributions from Tracer Experiments ; Illustration 5.5 A Moving Boundary Problem: The Shrinking Core Model and the Quasi-Steady State ; Illustration 5.6 More on Moving Boundaries. The Crystallization Process ; Illustration 5.7 Moving Boundaries in Medicine: Controlled Release Drug Delivery ; Illustration 5.8 Evaporation of a Pollutant into the Atmosphere ; Illustration 5.9 Ground Penetration from an Oil Spill ; Illustration 5.10 Concentration Variations in Stratified Layers ; Illustration 5.11 One-Compartment Pharmocokinetics ; Illustration 5.12 Deposition of Platelets from Flowing Blood ; Illustration 5.13 Dynamics of the Human Immunodeficiency Virus (HIV) ; Practice Problems ; 6. ONE-DIMENSIONAL DISTRIBUTED SYSTEMS ; 6.1 Introduction ; Illustration 6.1 The Hypsometric Formula ; Illustration 6.2 Poiseuille's Equation for Laminar Flow in a Pipe ; Illustration 6.3 Compressible Laminar Flow in a Horizontal Pipe ; Illustration 6.4 Conduction of Heat Through Various Geometries ; Illustration 6.5 Conduction in Systems with Heat Sources ; Illustration 6.6 The Countercurrent Heat Exchanger ; Illustration 6.7 Diffusion and Reaction in a Catalyst Pellet. The Effectiveness Factor ; Illustration 6.8 The Heat Exchanger Fin ; Illustration 6.9 Polymer Sheet Extrusion. The Uniformity Index ; Illustration 6.10 The Streeter-Phelps River Pollution Model. The Oxygen Sag Curve ; Illustration 6.11 Conduction in a Thin Wire Carrying an Electrical Current ; Illustration 6.12 Electrical Potential Due to a Charged Disk ; Illustration 6.13 Production of Silicon Crystals: Getting Lost and Staging a Recovery ; Practice Problems ; 7. SOME SIMPLE NETWORKS ; 7.1 Introduction ; Illustration 7.1 A Thermal Network: External Heating of a Stirred Tank ; Illustration 7.2 A Chemical Reaction Network. The Radioactive Decay Series ; Illustration 7.3 Hydraulic Networks ; Illustration 7.4 An Electrical Network: Hitting a Brick Wall and Going Around It ; Illustration 7.5 A Mechanical Network. Resonance of Two Vibrating Masses ; Illustration 7.6 Application of Matrix Methods to Stoichiometric Calculations ; Illustration 7.7 Diagnosis of a Plant Flow Sheet ; Illustration 7.8 Manufacturing Costs. Use of Matrix-Vector Products ; Illustration 7.9 More About Electrical Circuits. The Electrical Ladder Networks ; Illustration 7.10 Networks in Plant Physiology: Photosynthesis and Respiration ; Practice Problems ; 8. MORE MATHEMATICAL TOOLS: DIMENSIONAL ANALYSIS AND NUMERICAL METHODS ; 8.1 Dimensional Analysis ; 8.1.1 Introduction ; Illustration 8.1 Time of Swing of a Simple Pendulum ; Illustration 8.2 Vibration of a One-Dimensional Structure ; 8.1.2 Systems with More Variables than Dimensions. The Buckingham pi Theorem ; Illustration 8.3 Heat Transfer to a Fluid in Turbulent Flow ; Illustration 8.4 Drag on Submerged Bodies. Horsepower of a Car ; Illustration 8.5 Design of a Depth Charge ; Practice Problems ; 8.2 Numerical Methods ; 8.2.1 Introduction ; 8.2.2 Numerical Software Packages ; 8.2.3 Numerical Solution of Simultaneous Linear Algebraic Equations. Gaussian Elimination ; Illustration 8.6 The Global Positioning System Revisited: Gaussian Elimination Using the MATHEMATICA Package ; 8.2.4 Numerical Solution of Single Nonlinear Equations. Newton's Method ; Illustration 8.7 Chemical Equilibrium: The Synthesis of Ammonia by the Haber Process ; 8.2.5 Numerical Solution of Simultaneous, Non-Linear Equations. The Newton-Raphson Method ; Illustration 8.8 More Chemical Equilibria: Producing Silicon Films by Chemical Vapor Deposition (CVD) ; 8.2.6 Numerical Solution of Ordinary Differential Equations. The Euler and Runge-Kutta Methods ; Illustration 8.9 The Effect of Drag on the Trajectory of an Artillery Piece ; Practice Problems ; Index