Mathematical Modeling of Physical Systems

Description

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