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Mathematical Modeling of Physical Systems

An Introduction

Diran Basmadjian

Publication Date - 19 December 2002

ISBN: 9780195153149

368 pages
7-1/2 x 9-1/4 inches

This volume provides a concise and lucid introduction to mathematical modeling for students and professionals approaching the topic for the first time.


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.

About the Author(s)

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.

Table of Contents

    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

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