Mathematical Modeling of Physical Systems: An Introduction

Mathematical Modeling of Physical Systems provides a concise and lucid introduction to mathematical modeling for students and professionals approaching the topic for the first time. It is based on the premise that modeling is as much an art as it is a science--an art that can be mastered only by sustained practice. To provide that practice, the text contains approximately 100 worked examples and numerous practice problems drawn from civil and biomedical engineering, as well as from economics, physics, and chemistry. Problems range from classical examples, such as Euler's treatment of the buckling of the strut, to contemporary topics such as silicon chip manufacturing and the dynamics of the human immunodeficiency virus (HIV). The required mathematics are confined to simple treatments of vector algebra, matrix operations, and ordinary differential equations. Both analytical and numerical methods are explained in enough detail to function as learning tools for the beginner or as refreshers for the more informed reader. Ideal for third-year engineering, mathematics, physics, and chemistry students, Mathematical Modeling of Physical Systems will also be a welcome addition to the libraries of practicing professionals.

Table of Contents:
Preface Notation 1. Getting Started and Beyond 1.1: When Not to Model Example 1.1: The Challenger Space Shuttle Disaster Example 1.2: Loss of Blood Vessel Patency 1.2: Some Initial Tools and Steps 1.3: Closure Example 1.3: Discharge of Plant Effluent into a River Example 1.4: Electrical Field Due to a Dipole Example 1.5: Design of a Thermocouple Example 1.6: Newton's Law for Systems of Variable Mass: A False Start and the Remedy Example 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 m 2.1.5: The Scalar or Dot Product 2.1.6: The Vector or Cross Product Example 2.1: Distance of a Point from a Plane Example 2.2: Shortest Distance Between Two Lines Example 2.3: Work as an Application of the Scalar Product Example 2.4: Extension of the Scalar Product to n Dimensions: A Sale of Stocks Example 2.5: A Simple Model Economy 2.2: Matrices 2.2.1: Types of Matrix 2.2.2: The Echelon Form, Rank r 2.2.3: Matrix Equality 2.2.4: Matrix Addition Example 2.6: Acquisition Costs 2.2.5: Multiplication by a Scalar 2.2.6: Matrix Multiplication Example 2.7: The Product of Two Matrices Example 2.8: Matrix-Vector Representation of Linear Algebraic Equations 2.2.7: Elementary Row Operations Example 2.9: Application of Elementary Row Operations: Algebraic Equivalence 2.2.8: Solution of Sets of Linear Algebraic Equations: Gaussian Elimination Example 2.10: An Overspecified System of Equations with a Unique Solution Example 2.11: A Normal System of Equations with no Solutions 2.3: Ordinary Differential Equations (ODEs) Example 2.12: A Population Model Example 2.13: Newton's Law of Cooling 2.3.1: Order of an ODE 2.3.2: Linear and Nonlinear ODEs 2.3.3: Boundary and Initial Conditions Example 2.14: Classification of ODEs and Boundary Conditions 2.3.4: Equivalent Systems Example 2.15: Equivalence of Vibrating Mechanical Systems and an Electrical RLC Circuit 2.3.5: Analytical Solution Methods Example 2.16: Solution of NonLinear ODEs by Separation of Variables Example 2.17: Mass on a Spring Subjected to a Sinusoidal Forcing Function Example 2.18: Application of Inversion Procedures Example 2.19: The Mass-Spring System Revisited: Resonance Practice Problems 3. Geometrical Concepts Example 3.1: A Simple Geometry Problem: Crossing of a River Example 3.2: The Formation of Quasi Crystals and Tilings from Two Quadrilateral Polygons Example 3.3: Charting of Market Price Dynamics: The Japanese Candlestick Method Example 3.4: Surveying: The Join Calculation and the Triangulation Intersection Example 3.5: The Global Positioning System (GPS) Example 3.6: The Orthocenter of a Triangle Example 3.7: Relative Velocity and the Wind Triangle Example 3.8: Interception of an Airplane Example 3.9: Path of Pursuit Example 3.10: Trilinear Coordinates: The Three-Jug Problem Example 3.11: Inflecting Production Rates and Multiple Steady States: The van Heerden Diagram Example 3.12: Linear Programming: A Geometrical Construction Example 3.13: Stagewise Adsorption Purification of Liquids: The Operating Diagram Example 3.14: Supercoiled DNA Practice Problems 4. The Effect of Forces 4.1: Introduction Example 4.1: The Stress-Strain Relation: Stored Strain Energy and Stress Due to the Impact of a Falling Mass Example 4.2: Bending of Beams: Euler's Formula for the Buckling of a Strut Example 4.3: Electrical and Magnetic Forces: Thomson's Determination of e/m Example 4.4: Pressure of a Gas in Terms of Its Molecular Properties: Boyle's Law and the Ideal Gas Law, Velocity of Gas Molecules Example 4.5: Path of a Projectile Example 4.6: The Law of Universal Gravitation: Escape Velocity and Geosynchronous Satellites Example 4.7: Fluid Forces: Bernoulli's Equation and the Continuity Equation Example 4.8: Lift Capacity of a Hot Air Balloon Example 4.9: Work and Energy: Compression of a Gas and Power Output of a Bumblebee Practice Problems 5. Compartmental Models Example 5.1: Measurement of Plasma Volume and Cardiac Output by the Dye Dilution Method Example 5.2: The Continuous Stirred Tank Reactor (CSTR): Model and Optimum Size Example 5.3: Modeling a Bioreactor: Monod Kinetics and the Optimum Dilution Rate Example 5.4: Nonidealities in a Stirred Tank. Residence Time Distributions from Tracer Experiments Example 5.5: A Moving Boundary Problem: The Shrinking Core Model and the Quasi-Steady State Example 5.6: More on Moving Boundaries: The Crystallization Process Example 5.7: Moving Boundaries in Medicine: Controlled-Release Drug Delivery Example 5.8: Evaporation of a Pollutant into the Atmosphere Example 5.9: Ground Penetration from an Oil Spill Example 5.10: Concentration Variations in Stratified Layers Example 5.11: One-Compartment Pharmacokinetics Example 5.12: Deposition of Platelets from Flowing Blood Example 5.13: Dynamics of the Human Immunodeficiency Virus (HIV) Practice Problems 6. One-Dimensional Distributed Systems Example 6.1: The Hypsometric Formulae Example 6.2: Poiseuille's Equation for Laminar Flow in a Pipe Example 6.3: Compressible Laminar Flow in a Horizontal Pipe Example 6.4: Conduction of Heat Through Various Geometries Example 6.5: Conduction in Systems with Heat Sources Example 6.6: The Countercurrent Heat Exchanger Example 6.7: Diffusion and Reaction in a Catalyst Pellet: The Effectiveness Factor Example 6.8: The Heat Exchanger Fin Example 6.9: Polymer Sheet Extrusion: The Uniformity Index Example 6.10: The Streeter-Phelps River Pollution Model: The Oxygen Sag Curve Example 6.11: Conduction in a Thin Wire Carrying an Electrical Current Example 6.12: Electrical Potential Due to a Charged Disk Example 6.13: Production of Silicon Crystals: Getting Lost and Staging a Recovery Practice Problems 7. Some Simple Networks Example 7.1: A Thermal Network: External Heating of a Stirred Tank and the Analogy to the Artifical Kidney (Dialysis) Example 7.2: A Chemical Reaction Network: The Radioactive Decay Series Example 7.3: Hydraulic Networks Example 7.4: An Electrical Network: Hitting a Brick Wall and Going Around It Example 7.5: A Mechanical Network: Resonance of Two Vibrating Masses Example 7.6: Application of Matrix Methods to Stoichiometric Calculations Example 7.7: Diagnosis of a Plant Flow Sheet Example 7.8: Manufacturing Costs: Use of Matrix-Vector Products Example 7.9: More About Electrical Circuits: The Electrical Ladder Networks Example 7.10: Photosynthesis and Respiration of a Plant: An Electrial Analogue for the CO2 Pathway Practice Problems 8. More Mathematical Tools: Dimensional Analysis and Numerical Methods 8.1: Dimensional Analysis 8.1.1: Introduction Example 8.1: Time of Swing of a Simple Pendulum Example 8.2: Vibration of a One-Dimensional Structure 8.1.2: Systems with More Variables than Dimensions: The Buckingham p Theorem Example 8.3: Heat Transfer to a Fluid in Turbulent Flow Example 8.4: Drag on Submerged Bodies, Horsepower of a Car Example 8.5: Design of a Depth Charge 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 Example 8.6: The Global Positioning System Revisited: Using the MATHEMATICA Package for Gaussian Elimination 8.2.4: Numerical Solution of Single Nonlinear Equations: Newton's Method Example 8.7: Chemical Equilibrium: The Synthesis of Ammonia by the Haber Process 8.2.5: Numerical Simulation of Simultaneous Nonlinear Equations: The Newton-Raphson Method Example 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 Example 8.9: The Effect of Drag on the Trajectory of an Artillery Piece Practice Problems Index

About the Author :
Diran Basmadjian is Professor (Emeritus) of Chemical Engineering and Applied Chemistry at the University of Toronto. He is the author of two books and over forty journal papers in the areas of adsorption, biochemical engineering, and mathematical modeling.