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CHAPMAN & HALL/CRC
Diran Basmadjian
The Art of
MODELING
in
SCIENCE
and
ENGINEERING
Boca Raton London New York Washington, D.C.
© 1999 By CRC Press LLC

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model,

as used in this text, is understood to refer to the ensemble of
equations which describe and interrelate the variables and parameters of a physical
system or process. The term

modeling

in turn refers to the derivation of appropriate
equations that are solved for a set of system or process variables and parameters.
These solutions are often referred to as simulations, i.e., they simulate or reproduce
the behavior of physical systems and processes.
Modeling is practiced with uncommon frequency in the engineering disciplines
and indeed in all physical sciences where it is often known as “Applied Mathemat-
ics.” It has made its appearance in other disciplines as well which do not involve
physical processes per se, such as economics, finance, and banking. The reader will
note a chemical engineering slant to the contents of the book, but that discipline
now reaches out, some would say with tentacles, far beyond its immediate narrow
confines to encompass topics of interest to both scientists and engineers. We address
the book in particular to those in the disciplines of chemical, mechanical, civil, and
environmental engineering, to applied chemists and physicists in general, and to
students of applied mathematics.
The text covers a wide range of physical processes and phenomena which
generally call for the use of mass, energy, and momentum or force balances, together
with auxiliary relations drawn from such subdisciplines as thermodynamics and
chemical kinetics. Both static and dynamic systems are covered as well as processes
which are at a steady state. Thus, transport phenomena play an important but not
exclusive role in the subject matter covered. This amalgam of topics is held together
by the common thread of applied mathematics.

definitive monographs by Carslaw and Jaeger (1959) and by Crank (1978).
Even here, however, one often encounters solutions which reduce to PDEs
of lower dimensionality, to ordinary differential equations (ODEs) or even
algebraic equations (AEs). The motto must therefore be “PDEs if neces-
sary, but not necessarily PDEs.”
• The second difficulty lies in the absence of precise solutions, even with
the use of the most sophisticated models and computational tools. Some
systems are simply too complex to yield exact answers. One must resort
here to what we term

bracketing the solution,

i.e., establishing upper or
lower bounds to the answer being sought. This is a perfectly respectable
exercise, much practiced by mathematicians and theoretical scientists and
engineers.
• The third difficulty lies in making suitable simplifying assumptions and
approximations. This requires considerable physical insight and engineer-
ing skill. Not infrequently, a certain boldness and leap in imagination is
called for. These are not easy attributes to satisfy.
Overcoming these three difficulties constitute the core of

The
Art of Modeling.

Although we will not make this aspect the exclusive domain of our effort, a large
number of examples and illustrations will be presented to provide the reader with


from the stirred tank and the 1-d pipe models. Although deceptively simple in
retrospect, the application of these models to real problems will lead to a first
encounter with the art of modeling. A first glimpse will also be had of the skills
needed in setting upper and lower bounds to the solutions. We do this even though
more accurate and elaborate solutions may be available. The advantage is that the
bounds can be established quickly and it is surprising how often this is all an engineer
or scientist needs to do. The examples here and throughout the book are drawn from
a variety of disciplines which share a common interest in transport phenomena and
the application of mass, energy, and momentum or force balances. From classical
chemical engineering we have drawn examples dealing with heat and mass transfer,
fluid statics and dynamics, reactor engineering, and the basic unit operations (dis-
tillation, gas absorption, adsorption, filtration, drying, and membrane processes,
among others). These are also of general interest to other engineering disciplines.
Woven into these are illustrations which combine several processes or do not fall
into any rigid category.
These early segments are followed, in Chapter 3, by a more detailed exposition
of mass, energy and momentum transport, illustrated with classical and modern
examples. The reader will find here, as in all other chapters, a rich choice of solved
illustrative examples as well as a large number of practice problems. The latter are
worth the scrutiny of the reader even if no solution is attempted. The mathematics
up to this point is simple, all ODE solutions being obtained by separation of variables.
An intermezzo now occurs in which underlying mathematical topics are taken
up. In Chapter 4, an exposition is given of important analytical and numerical
solutions of ordinary differential equations in which we consider methods applicable
to first and second order ODEs in some detail. Considerable emphasis is given to
deducing the qualitative nature of the solutions from the underlying model equations
and to linking the mathematics to the physical processes involved. Both linear and
nonlinear analysis is applied. Linear systems are examined in more detail in a follow-
up chapter on Laplace transformation.

and Professor C. Yip. I am also grateful to my colleagues, Professor M.V. Sefton,
Professor D.E. Cormack, and Professor Emeritus S. Sandler for providing me with
problems from their consulting and teaching practices.
Many former students were instrumental in persuading the author to convert
classroom notes into a text, among them Dr. K. Gregory, Dr. G.M. Martinez, Dr. M.
May, Dr. D. Rosen, and Dr. S. Seyfaie. I owe a special debt of gratitude to S. (VJ)
Vijayakumar who never wavered in his support of this project and from whom I
drew a good measure of inspiration. A strong prod was also provided by Professor
S.A. Baldwin, Professor V.G. Papangelakis, and by Professor Emeritus J. Toguri.
The text is designed for undergraduate and graduate students, as well as prac-
ticing professionals in the sciences and in engineering, with an interest in modeling
based on mass, energy and momentum or force balances. The first six chapters
contain no partial differential equations and are suitable as a basis for a fourth-year
course in Modeling or Applied Mathematics, or, with some boldness and omissions,
at the third-year level. The book in its entirety, with some of the preliminaries and
other extraneous material omitted, can serve as a text in Modeling and Applied
Mathematics at the first-year graduate level. Students in the Engineering Sciences
in particular, will benefit from it.
It remains for me to express my thanks to Arlene Fillatre who undertook the
arduous task of transcribing the hand-written text to readable print, to Linda Staats,
University of Toronto Press, who miraculously converted rough sketches into pro-
fessional drawings, and to Bruce Herrington for his unfailing wit. My wife, Janet,
bore the proceedings, sometimes with dismay, but mostly with pride.

REFERENCES

R.B. Bird, W.R. Stewart, and E.N. Lightfoot.

Transport Phenomena,


Process Modeling, Simulation and Control, 2nd ed.,

McGraw-Hill, New York,
1990.

248/fm/frame Page 8 Tuesday, June 19, 2001 11:45 AM
© 1999 By CRC Press LLC

B. Ogunnaike and W.H. Ray.

Process Dynamics, Modeling and Control,

Oxford University
Press, Oxford, U.K., 1996.
G.V. Reklaitis.

Introduction to Material and Energy Balances,

John Wiley & Sons, New York,
1983.
G. Stephanopoulos.

Chemical Process Control,

Prentice-Hall, Upper Saddle River, NJ, 1984.
V.L. Streeter, E.B. Wylie, and K.W. Bedford.

Fluid Mechanics, 9th ed.,

McGraw-Hill, New

The quantities listed are expressed in SI Units. Note the equivalence 1 N = 1 kg
m/s

2

, 1 Pa = 1 kg/ms

2

, 1 J = 1 kg m

2

/s

2

.
A Area, m

2

A

C

Cross-sectional area, m

2



C Speed of light, m/s
c Speed of sound, m/s
C{ } Cosine transform operation
C

D

Drag coefficient, dimensionless
C

p

Heat capacity at constant pressure, J/kg K or J/mole K
C

v

Heat capacity at constant volume, J/kg K or J/mole K
D D-operator = d/dx
D Diffusivity, m

2

/s
D Dilution Rate, 1/s
D Distillation or evaporation rate, mole/s
D Oxygen deficit =
D


Isothermal catalyst effectiveness factor, dimensionless
E

ni

Non-isothermal catalyst effectiveness factor, dimensionless
ETC Effective therapeutic concentration, kg/m

3

erf(x) Error function =
F Force, N
F Mass flow rate, kg/s
Fo Fourier number =

α

t/L

2

, dimensionless
f Friction factor, dimensionless
C C kg m
OO
22
3
*
,/−
(/ )2

s
G

s

Carrier or solvent mass velocity, kg/m

2

s
G(P,Q) Green’s function
Gr Grashof number =

ρ

2

β

gL

3

∆

T/

µ

2

∆

H

v

Enthalpy of vaporization, J/kg or J/mole
H Height, m
H Henry’s constant, m

3

/m

3

HTU Height of a transfer unit, m
H{ } Hankel transform operator
h Heat transfer coefficient, J/m

2

sK
h

f

Friction head, J/kg
I


D

Dissociation constant = k

r

/k

f

, dimensionless
K

k

Modified Bessel function of second kind and order k
K

m

Michaelis-Menten constant, mole/m

3

K

0

Overall mass transfer coefficient, various units, see Table 3.6
K


e

Elimination rate constant, 1/s
k

r

Reaction rate constant, 1/s (first order)
k

eff

Effective thermal conductivity in porous medium, J/msK
L Length or characteristic length, m
L Ligand concentration, mol/m

3

L Liquid flow rate, kg/s or mole/s
L Pollutant concentration, kg/m

3

L{ } Laplace transform operator
LMCD Log-mean concentration difference, kg/m

3

, mole/m


∞

, kg
Ma Mach number, dimensionless
m Mass, kg
N Dimensionless distance = K

0

a z/v
N Mass transfer rate, kg/s or mole/s
N

′

Mass flux, kg/m

2

s or mole/m

2

s
N Pipe number = fL/d, dimensionless
NMa

2



µ

/k, dimensionless
P

T

Total pressure
p Pressure
p dy/dx (p-substitution)
Q Strength of heat source, m

3

K
Q Volumetric flow rate, m

3

/s
q Amount adsorbed, kg/kg
q Rate of heat transfer, J/s
q′ Heat flux, J/m
2
s
q′ Vehicle flux, vehicles/m
2
s
q

3
s
S Shape factor, m
(S) Substrate concentration, mole/m
3
S Steam consumption, kg/s
S{ } Sine transform operator
Sc Schmidt number = µ/ρD, dimensionless
Sh Sherwood number = kL/D, dimensionless
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© 1999 By CRC Press LLC
St Stanton number = Nu/RePr or Sh/ReSc, dimensionless
s Arc length, m
s Laplace transform parameter, dimensionless
s Specific gravity, dimensionless
T Dimensionless time = K
0
a(ρ
f
/ρ
b
)(t/H)
T Temperature, K or °C
t Time, s
U Internal energy, J/kg or J/mole
U Outer field velocity, m/s
U Overall heat transfer coefficient, J/m
2
sK
V Electrostatic potential, V

/Q
D
, dimensionless
GREEK SYMBOLS
α Separate factor, dimensionless
α Thermal diffusivity, m
2
/s
α Filter cake resistance = [K(1 – ε)ρ
s
)
–1
β Compressibility =
β Thermal parameter = (–∆H
r
D
eff
/k
eff
T
s
)
β Volumetric coefficient of expansion =
Γ(n) Gamma function =
γ Activity coefficient, dimensionless
γ Ratio of heat capacities = C
p
/C
v
, dimensionless

δ(x – x
0
) Dirac delta function
ε Emissivity, dimensionless
ε Heat exchanger efficiency, dimensionless
ε Void fraction, dimensionless
η Similarity variable
θ Angle in cylindrical or spherical coordinate, radians
θ Dimensionless temperature
λ Characteristic value, or eigenvalue (linear systems)
λ Damping coefficient, dimensionless
µ Characteristic value or eigenvalue (nonlinear systems)
µ Chemical potential
µ Viscosity, Pas
µ
Max
Maximum monod growth rate, 1/s
ν Frequency, 1/s
ν Kinematic viscosity = µ/ρ, m
2
/s
ρ Density, kg/m
3
or C/m
3
(charge)
ρ Reflectivity, dimensionless
σ Stefan-Boltzmann constant = 5.767 × 10
–8
J/m

B
Blood
BM
Log-mean partial pressure or concentration of inert component
b
Bulk fluid
b
Bed
c
Cold
D
Dialysate
db
Dry-bulb
e
External
F
Feed
F
Fish
FW
Fish-water
f
Fluid
f
Forward
g
Gas
h
Hot

s
Shell
s
Solid
t
Terminal
t
Tube
W
Water
wb
Wet-bulb
x,y,z
Component in x, y, z direction
x,y,z
Differentiation with respect to x, y, z
SUPERSCRIPTS
0
Pure component
o
Reference state
* Equilibrium
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© 1999 By CRC Press LLC
Table of Contents
Chapter 1 Introduction
1.1 Conservation Laws and Auxiliary Relations
1.1.1 Conservation Laws
1.1.2 Auxiliary Relations
1.2 Properties and Categories of Balances

Illustration 2.3 Design of a Gas Scrubber Revisited
Illustration 2.4 An Example from Industry: Decontamination
of a Nuclear Reactor Coolant
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© 1999 By CRC Press LLC
Illustration 2.5 Thermal Treatment of Steel Strapping
Illustration 2.6 Batch Filtration: The Ruth Equations
Illustration 2.7 Drying of a Nonporous Plastic Sheet
Practice Problems
References
Chapter 3 More About Mass, Energy, and Momentum Balances
3.1 The Terms in the Various Balances
3.2 Mass Balances
3.2.1 Molar Mass Flow in Binary Mixtures
3.2.2 Transport Coefficients
Illustration 3.2.1 Drying of a Plastic Sheet Revisited:
Estimation of the Mass Transfer Coefficient k
y
Illustration 3.2.2 Measurement of Diffusivities by the
Two-Bulb Method: The Quasi-Steady State
3.2.3 Chemical Reaction Mass Balance
Illustration 3.2.3 CSTR with Second Order Homogeneous
Reaction A + B → P
Illustration 3.2.4 Isothermal Tubular Reactor with First
Order Homogeneous Reaction
Illustration 3.2.5 Isothermal Diffusion and First Order
Reaction in a Spherical, Porous Catalyst Pellet:
The Effectiveness Factor E
3.2.4 Tank Mass Balance
Illustration 3.2.6 Waste-Disposal Holding Tank

Illustration 3.3.11 The Steady-State Energy Balance for
Flowing (Open) Systems
Illustration 3.3.12 A Moving Boundary Problem:
Freeze-Drying of Food
Practice Problems
3.4 Force and Momentum Balances
3.4.1 Momentum Flux and Equivalent Forces
3.4.2 Transport Coefficients
Illustration 3.4.1 Forces on Submerged Surfaces:
Archimides’ Law
Illustration 3.4.2 Forces Acting on a Pressurized Container:
The Hoop-Stress Formula
Illustration 3.4.3 The Effects of Surface Tension: Laplace’s
Equation; Capillary Rise
Illustration 3.4.4 The Hypsometric Formulae
Illustration 3.4.5 Momentum Changes in a Flowing Fluid:
Forces on a Stationary Vane
Illustration 3.4.6 Particle Movement in a Fluid
Illustration 3.4.7 The Bernoulli Equation: Some Simple
Applications
Illustration 3.4.8 The Mechanical Energy Balance
Illustration 3.4.9 Viscous Flow in a Parallel Plate Channel:
Velocity Distribution and Flow Rate — Pressure Drop
Relation
Illustration 3.4.10 Non-Newtonian Fluids
Practice Problems
3.5 Combined Mass and Energy Balances
Illustration 3.5.1 Nonisothermal CSTR with Second Order
Homogeneous Reaction A + B → P
Illustration 3.5.2 Nonisothermal Tubular Reactors: The

4.1 Definitions and Classifications
4.1.1 Order of an ODE
4.1.2 Linear and Nonlinear ODEs
4.1.3 ODEs with Variable Coefficients
4.1.4 Homogeneous and Nonhomogeneous ODEs
4.1.5 Autonomous ODEs
Illustration 4.1.1 Classification of Model ODEs
4.2 Boundary and Initial Conditions
4.2.1 Some Useful Hints on Boundary Conditions
Illustration 4.2.1 Boundary Conditions in a Conduction
Problem: Heat Losses from a Metallic Furnace Insert
4.3 Analytical Solutions of ODEs
4.3.1 Separation of Variables
Illustration 4.3.1 Solution of Complex ODEs by Separation of
Variables
Illustration 4.3.2 Repeated Separation of Variables: The
Burning Fuel Droplet as a Moving Boundary Problem
4.3.2 The D-Operator Method: Solution of Linear nth Order ODEs with
Constant Coefficients
Illustration 4.3.3 The Longitudinal Heat Exchanger Fin
Revisited
Illustration 4.3.4 Polymer Sheet Extrusion: The Uniformity
Index
4.3.3 Nonhomogeneous Linear Second Order ODEs with Constant
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© 1999 By CRC Press LLC
Coefficients
Illustration 4.3.5 Vibrating Spring with a Forcing Function
4.3.4 Series Solutions of Linear ODEs with Variable Coefficients
Illustration 4.3.6 Solution of a Linear ODE with Constant

References
Chapter 5 The Laplace Transformation
5.1 General Properties of the Laplace Transform
Illustration 5.1.1 Inversion of Various Transforms
5.2 Application to Differential Equations
Illustration 5.2.1 The Mass Spring System Revisited:
Resonance
Illustration 5.2.2 Equivalence of Mechanical Systems and
Electrical Circuits
Illustration 5.2.3 Response of First Order Systems
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© 1999 By CRC Press LLC
Illustration 5.2.4 Response of Second Order Systems
Illustration 5.2.5 The Horizontal Beam Revisited
5.3 Block Diagrams: A Simple Control System
5.3.1 Water Heater
5.3.2 Measuring Element
5.3.3 Controller and Control Element
5.4 Overall Transfer Function; Stability Criterion; Laplace Domain
Analysis
Illustration 5.4.1 Laplace Domain Stability Analysis
Practice Problems
References
Chapter 6 Special Topics
6.1 Biomedical Engineering, Biology, and Biotechnology
Illustration 6.1.1 One-Compartment Pharmacokinetics
Illustration 6.1.2 Blood–Tissue Interaction as a Pseudo
One-Compartment Model
Illustration 6.1.3 A Distributed Model: Transport Between
Flowing Blood and Muscle Tissue

6.3 Welcome to the Real World
Illustration 6.3.1 Production of Heavy Water by Methane
Distillation
Illustration 6.3.2 Clumping of Coal Transported in Freight
Cars
Illustration 6.3.3 Pop Goes the Vessel
Illustration 6.3.4 Debugging of a Vinyl Chloride Recovery
Unit
Illustration 6.3.5 Pop Goes the Vessel (Again)
Illustration 6.3.6 Potential Freezing of a Water Pipeline
Illustration 6.3.7 Failure of Heat Pipes
Illustration 6.3.8 Coating of a Pipe
Illustration 6.3.9 Release of Potentially Harmful Chemicals
to the Atmosphere
Illustration 6.3.10 Design of a Marker Particle (Revisited)
Practice Problems
References
Chapter 7 Partial Differential Equations: Classification, Types, and
Properties; Some Simple Transformations and Solutions
7.1 Properties and Classes of PDEs
7.1.1 Order of a PDE
7.1.1.1 First Order PDEs
7.1.1.2 Second Order PDEs
7.1.1.3 Higher Order PDEs
7.1.2 Homogeneous PDEs and BCs
7.1.3 PDEs with Variable Coefficients
7.1.4 Linear and Nonlinear PDEs: A New Category — Quasilinear
PDEs
7.1.5 Another New Category: Elliptic, Parabolic, and Hyperbolic
PDEs

of Variables
Illustration 7.3.1 Heat Transfer in Boundary Layer Flow over
a Flat Plate: Similarity Transformation
7.3.2 Elimination of Dependent Variables: Reduction of Number of
Equations
Illustration 7.3.2 Use of the Stream Function in Boundary
Layer Theory: Velocity Profiles Along a Flat Plate
7.3.3 Elimination of Nonhomogeneous Terms
Illustration 7.3.3 Conversion of a PDE to Homogeneous
Form
7.3.4 Change in Independent Variables: Reduction to Canonical Form
Illustration 7.3.4 Reduction of ODEs to Canonical Form
7.3.5 Simplification of Geometry
7.3.5.1 Reduction of a Radial Spherical Configuration into a
Planar One
7.3.5.2 Reduction of a Radial Circular or Cylindrical Configuration
into a Planar One
7.3.5.3 Reduction of a Radial Circular or Cylindrical Configuration
to a Semi-Infinite One
7.3.5.4 Reduction of a Planar Configuration to a Semi-Infinite
One
7.3.6 Nondimensionalization
Illustration 7.3.5 Nondimensionalization of Fourier’s
Equation
7.4 PDEs PDQ: Locating Solutions in Related Disciplines; Solution by
Simple Superposition Methods
7.4.1 In Search of a Literature Solution
Illustration 7.4.1 Pressure Transients in a Semi-Infinite Porous
Medium
Illustration 7.4.2 Use of Electrostatic Potentials in the Solution of

2
Illustration 8.1.2 Derivation of the Divergence
Illustration 8.1.3 Derivation of Some Relations Involving
∇, ∇ ·, and ∇ ×
8.1.3 Integral Theorems of Vector Calculus
Illustration 8.1.4 Derivation of the Continuity Equation
Illustration 8.1.5 Derivation of Fick’s Equation
Illustration 8.1.6 Superposition Revisited: Green’s Functions
and the Solution of PDEs by Green’s Functions
Illustration 8.1.7 The Use of Green’s Functions in Solving
Fourier’s Equation
Practice Problems
8.2 Transport of Mass
Illustration 8.2.1 Catalytic Conversion in a Coated Tubular
Reactor: Locating Equivalent Solutions in the Literature
Illustration 8.2.2 Diffusion and Reaction in a Semi-Infinite
Medium: Another Literature Solution
Illustration 8.2.3 The Graetz–Lévêque Problem in Mass
Transfer: Transport Coefficients in the Entry Region
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© 1999 By CRC Press LLC
Illustration 8.2.4 Unsteady Diffusion in a Sphere: Sorption
and Desorption Curves
Illustration 8.2.5 The Sphere in a Well-Stirred Solution:
Leaching of a Slurry
Illustration 8.2.6 Steady-State Diffusion in Several
Dimensions
Practice Problems
8.3 Transport of Energy
Illustration 8.3.1 The Graetz-Lévêque Problem (Yet Again!)

Resistance: Arbitrary Initial Distribution
Illustration 9.1.5 Steady-State Conduction in a Hollow
Cylinder
Practice Problems
9.2 Laplace Transformation and Other Integral Transforms
9.2.1 General Properties
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© 1999 By CRC Press LLC