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

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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.

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