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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
This book provides a rigorous treatment of the fundamental concepts and techniques involved in process modeling and simulation. The book allows the reader to:
(i) Get a solid grasp of “under-the-hood” mathematical results
(ii) Develop models of sophisticated processes
(iii) Transform models to different geometries and domains as appropriate
(iv) Utilize various model simplification techniques
(v) Learn simple and effective computational methods for model simulation
(vi) Intensify the effectiveness of their research
Modeling and Simulation for Chemical Engineers: Theory and Practice begins with an introduction to the terminology of process modeling and simulation. Chapters 2 and 3 cover fundamental and constitutive relations, while Chapter 4 on model formulation builds on these relations. Chapters 5 and 6 introduce the advanced techniques of model transformation and simplification. Chapter 7 deals with model simulation, and the final chapter reviews important mathematical concepts.
Presented in a methodical, systematic way, this book is suitable as a self-study guide or as a graduate reference, and includes examples, schematics and diagrams to enrich understanding. End of chapter problems with solutions and computer software available online at www.wiley.com/go/upreti/pms_for_chemical_engineers are designed to further stimulate readers to apply the newly learned concepts.Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 406
Veröffentlichungsjahr: 2017
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Table of Contents
Cover
Title Page
Copyright
Dedication
Preface
Notation
Chapter 1: Introduction
1.1 System
1.2 Process
1.3 Process Modeling
1.4 Process Simulation
1.5 Development of Process Model
1.6 Learning about Process
1.7 System Specification
Bibliography
Exercises
Chapter 2: Fundamental Relations
2.1 Basic Form
2.2 Mass Balance
2.3 Mole Balance
2.4 Momentum Balance
2.5 Energy Balance
2.6 Equation of Change for Kinetic and Potential Energy
2.7 Equation of Change for Temperature
2.A Enthalpy Change from Thermodynamics
2.B Divergence Theorem
2.C General Transport Theorem
2.D Equations in Cartesian, Cylindrical and Spherical Coordinate Systems
Bibliography
Exercises
Chapter 3: Constitutive Relations
3.1 Diffusion
3.2 Viscous Motion
3.3 Thermal Conduction
3.4 Chemical Reaction
3.5 Rate of Reaction
3.6 Interphase Transfer
3.7 Thermodynamic Relations
3.A Equations in Cartesian, Cylindrical and Spherical Coordinate Systems
References
Bibliography
Exercises
Chapter 4: Model Formulation
4.1 Lumped-Parameter Systems
4.2 Distributed-Parameter Systems
4.3 Fluxes along Non-Linear Directions
4.A Initial and Boundary Conditions
4.B Zero Derivative at the Point of Symmetry
4.C Equation of Motion along the Radial Direction in Cylindrical Coordinates
References
Exercises
Chapter 5: Model Transformation
5.1 Transformation between Orthogonal Coordinate Systems
5.2 Transformation between Arbitrary Coordinate Systems
5.3 Laplace Transformation
5.4 Miscellaneous Transformations
5.A Differential Operators in an Orthogonal Coordinate System
References
Bibliography
Exercises
Chapter 6: Model Simplification and Approximation
6.1 Model Simplification
6.2 Model Approximation
6.A Linear Function
6.B Proof of Buckingham Pi Theorem
6.C Newton’s Optimization Method
References
Bibliography
Exercises
Chapter 7: Process Simulation
7.1 Algebraic Equations
7.2 Differential Equations
7.3 Partial Differential Equations
7.4 Differential Equations with Split Boundaries
7.5 Periodic Differential Equations
7.6 Programming of Derivatives
7.7 Miscellanea
7.A Partial Pivoting for Matrix Inverse
7.B Derivation of Newton–Raphson Method
7.C General Derivation Of Finite Difference Formulas
References
Bibliography
Chapter 8: Mathematical Review
8.1 Order of Magnitude
8.2 Big-O Notation
8.3 Analytical Function
8.4 Vectors
8.5 Matrices
8.6 Tensors
8.7 Differential
8.8 Taylor Series
8.9 L'Hôpital's Rule
8.10 Leibniz's Rule
8.11 Integration by Parts
8.12 Euler's Formulas
8.13 Solution of Linear Ordinary Differential Equations
Bibliography
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
Process Modeling and Simulation for Chemical Engineers
Theory and Practice
Simant Ranjan Upreti
Department of Chemical Engineering, Ryerson University, Toronto, Canada
This edition first published 2017
© 2017 John Wiley & Sons Ltd
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permision to reuse material from this title is available at http://www.wiley.com/go/permissions.
The right of Simant Ranjan Upreti to be identified as the author of this work has been asserted in accordance with law.
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Library of Congress Cataloging-in-Publication Data
Names: Upreti, Simant Ranjan, author.
Title: Process modeling and simulation for chemical engineers : theory and practice / Simant Ranjan Upreti.
Description: Chichester, UK ; Hoboken, NJ : John Wiley & Sons, 2017. | Includes bibliographical references and index.
Identifiers: LCCN 2016053339| ISBN 9781118914687 (cloth) | ISBN 9781118914663 (epub)
Subjects: LCSH: Chemical processes-Mathematical models. | Chemical processes-Data processing.
Classification: LCC TP155.7 .U67 2017 | DDC 660/.284401-dc23
LC record available at https://lccn.loc.gov/2016053339
Cover image: shulz/Gettyimages
Cover design by Wiley
to my wife and kids
Preface
I am delighted to present this book on process modeling and simulation for chemical engineers. It is a humble attempt to assimilate the amazing contributions of researchers and academicians in this area.
The goal of this book is to provide a rigorous treatment of fundamental concepts and techniques of this subject. To that end, the book includes all requisite mathematical analyses and derivations, which could be sometimes hard to find. Target readers are those at the graduate level. This book endeavors to equip them to model sophisticated processes, develop requisite computational algorithms and programs, improvise existing software, and solve research problems with confidence.
Chapter 1 provides the groundwork by introducing the terminology of process modeling and simulation. Chapter 2 presents the fundamental relations for this subject. Chapter 3 incorporates important constitutive relations for common systems. Chapter 4 presents model formulation with the help of several examples. Transformation techniques are introduced in Chapter 5. Model simplification and approximation methods are discussed in Chapter 6. The numerical solution of process models is the theme of Chapter 7. Review of important mathematical concepts is provided in Chapter 8.
This book can be used as a primary text for a one-semester course. Alternatively, it could serve as a supplementary text in graduate courses related to modeling and simulation. Readers could also study the book on their own. During an initial reading, one could very well skim quickly through a derivation, accept the result for the time being, and learn more from applications. Computer programs for the solutions of book examples can be obtained from the publisher's website, www.wiley.com/go/upreti/pms_for chemical_engineers/.
I am grateful to the editorial team at John Wiley & Sons for providing excellent support from start to finish.
Finally, I am deeply indebted to my wife Deepa, and children Jahnavi and Pranav. I could not have completed this book without their unsparing support and understanding.
Simant R. Upreti
Ryerson University, Toronto
Notation
Symbol
Description
Units
a
surface area per unit volume
m
−1
A
m
area of moving surfaces
m
2
A
p
area of a port of flow
m
2
A
area
m
2
A
area vector
m
2
c
average concentration of a mixture
kmol m
−3
c
i
concentration of the
i
th
species
kmol m
−3
specific heat capacity of mixture
J kg
−1
K
−1
of the
i
th
species in pure form
J kg
−1
K
−1
molar specific heat capacity of the
i
th
species in a mixture
J kmol
−1
K
−1
partial specific heat capacity of the
i
th
species in a mixture
J kmol
−1
K
−1
d
x
differential change in
x
of
x
D
diffusivity of species
m
2
s
−1
D
AB
binary diffusivity of A in a mixture of A and B
m
2
s
−1
D
matrix of multicomponent diffusivities
m
2
s
−1
e
i
the
i
th
component of energy flux
J m
−2
s
−1
E
sum of the squared errors in
y
in a population
of
y
2
E
total energy of a system
J
E
activation energy of reaction
J kmol
−1
energy per unit mass
J kg
−1
f
i
fugacity of the
i
th
species
Pa
F
volumetric flow rate
m
3
s
−1
f
i
mass flux of the
i
th
species
kg m
−2
s
−1
f
overall mass flux of a mixture
kg m
−2
s
−1
F
i
molar flux of the
i
th
species
kmol m
−2
s
−1
F
overall molar flux of a mixture
kmol m
−2
s
−1
F
force vector
N
G
Gibbs free energy
J
Gibbs free energy per unit mass of the
i
th
species
J kg
−1
g
gravity, 9.806 65
m s
−2
h
heat transfer coefficient
W m
−2
K
−1
H
enthalpy
J
enthalpy per unit mass
J kg
−1
H
i
Henry's law constant for the
i
th
species
Pa
enthalpy per unit mass of the
i
th
species
J kg
−1
partial specific enthalpy of the
i
th
species in a mixture
J kg
−1
partial molar enthalpy of the
i
th
species in a mixture
J kmol
−1
molar enthalpy of the
i
th
species in pure form
J kmol
−1
standard
J kmol
−1
H
Hessian matrix
I
identity matrix
j
i
diffusive mass flux of the
i
th
species
kg m
−2
s
−1
vector of
j
i
kg m
−2
s
−1
J
Jacobian matrix
J
i
diffusive molar flux of the
i
th
species
kmol m
−2
s
−1
vector of
J
i
kmol m
−2
s
−1
k
reaction rate coefficient
as per reaction
k
thermal conductivity
W m
−1
K
−1
k
0
frequency factor
as per reaction
k
c
mass transfer coefficient
m s
−1
K
equilibrium constant of a chemical reaction
L
lower triangular matrix
m
mass
kg
m
i
mass of the
i
th
species
kg
M
i
molecular weight of the
i
th
species
kg kmol
−1
N
c
number of components or species
N
i
number of moles of the
i
th
species
N
r
number of chemical reactions
unit vectors
p
i
partial pressure of the
i
th
species
Pa
P
pressure
Pa
P
c
critical pressure
Pa
p
momentum
kg m s
−1
q
rate of heat transfer
J s
−1
q
i
the
i
th
component of conductive heat flux
J m
−2
s
−1
Q
heat, i.e., energy in transit
J
q
conductive heat flux
J m
−2
s
−1
r
correlation coefficient
r
rate of reaction
kg(kmol) m
−3
s
−1
r
radial direction in cylindrical and spherical coordinates
m
r
2
coefficient of determination
R
universal gas constant, 8.314 46 × 10
3
J kmol
−1
K
−1
r
gen,
i
mass rate of
i
th
species generated per unit volume
kg m
−3
s
−1
R
gen,
i
molar rate of
i
th
species generated per unit volume
kmol m
−3
s
−1
s
y
standard deviation in values of
y
of
y
S
entropy
J K
−1
entropy per unit mass of the
i
th
species
J K
−1
kg
−1
S
sum of the squared errors from the average of
y
in a population
of
y
2
t
time
s
T
temperature
K, °C
T
c
critical temperature
°C
U
internal energy
J
internal energy per unit mass
J kg
−1
internal energy per unit mass of the
i
th
species
J kg
−1
U
upper triangular matrix
v
magnitude of velocity
m s
−1
v
i
the
i
th
component of
v
, or average velocity along the
x
i
-direction
m s
−1
V
volume
m
3
molar volume
m
3
kmol
−1
specific volume
m
3
kg
−1
specific volume of the
i
th
species
m
3
kg
−1
v
mass average velocity
m s
−1
molar average velocity
m s
−1
W
s
shaft work
J
change of
x
per unit time
x
-units s
−1
x
ext
amount of
x
from external source
of
x
x
gen
generated amount of
x
of
x
X
extent of chemical reaction
extent of chemical reaction per unit volume
m
−3
x
⊤
transpose of
x
(vector or matrix)
unit vector along the
x
i
-direction
∥
x
∥
norm or magnitude of
x
(vector or matrix)
z
axial direction in cylindrical coordinates
m
Greek Symbols
rate-of-strain tensor
s
−1
δ
ij
Kronecker delta
δQ
net
Q
involved in differential changes, d
U
and d
S
J
δ
unit dyadic (unit tensor)
standard heat of formation of the
i
th
species
J kmol
−1
Δ
H
r
heat of reaction
J kmol
−1
standard heat of reaction
J kmol
−1
Δ
x
small amount of, or change in
x
of
x
Δ[
x
]
loss of
x
from a system through its ports
of
x
∇
gradient operator (a vector)
η
Non-Newtonian viscosity
Pa s
θ
θ
-direction in cylindrical and spherical coordinates
rad, °
κ
dilatational viscosity
Pa s
μ
viscosity
Pa s
ν
i
stoichiometric coefficient of
i
th
species in a chemical reaction
π
molecular stress tensor
Pa
Π
dimensionless number
ρ
density, mass concentration
kg m
−3
ρ
i
density, mass concentration of the
i
th
species
kg m
−3
τ
ij
,
π
ij
,
φ
ij
the
j
th
component of stress acting on the
i
th
area component
Pa
τ
time period
s
τ
viscous stress tensor
Pa
φ
φ
-direction in spherical coordinates
rad, °
φ
overall stress tensor
Pa
potential energy per unit mass
J kg
−1
ω
mass fraction
ω
acentric factor
ω
vorticity tensor
s
−1
Chapter 1Introduction
Process modeling and simulation is our intellectual endeavor to explain real-world processes, foresee their effects, and improve them to our satisfaction. Using foundational rules and the language of mathematics, we describe a process, i.e., develop its model. Depending on what needs to be known, we pose the model as a problem. Its solution provides the needed information, thereby simulating the process as it would unfold in the real world.
This chapter lays the groundwork for process modeling and simulation. We explain the basic concepts, and introduce the involved terminology in a methodical manner. Our starting point is the definition of a system.
1.1 System
A system is defined as a set of one or more units relevant to the knowledge that is sought. Eventually, that knowledge is obtained as system characteristics, and their behavior in time and space.
We specify a system based on what we want to know about it. Consider for example a well-mixed reactor shown in Figure 1.1 below. The reactor is fed certain amounts of non-volatile species A and B in a liquid phase. Inside the reactor, the species react to form a non-volatile liquid product C. Given that we wish to know the concentration of C in the liquid phase, the system is precisely the reaction mixture as shown in the figure. Anything not relevant – such as the reactor wall, and the vapor phase over the mixture – is not included in the system.
Figure 1.1 A system of reaction mixture in a reactor
Everything external to a system constitute its surroundings. A region of zero thickness in the system separating it from the surroundings is called the boundary. Any interaction between a system and its surroundings requiring physical contact takes place across the boundary. For instance, this interaction could be transfer of mass.
For the above system of reaction mixture, the surroundings comprise the reactor wall, and the vapor phase over the reaction mixture. The system boundary is made of the surface of mixture in contact with (i) the reactor wall, and (ii) air. An example of interaction between this system and its surroundings is the evaporation of the species from the mixture through its top surface (i.e., across the boundary) to air.
1.1.1 Uniform System
A system is said to be uniform or homogenous if it stays the same, regardless of any recombination of its parts. As an illustration, consider a system in the shape of a cube. We split it into a set of arbitrary number of small cubes of identical size. Next, we recombine them in all possible ways to form the initial cube. The system would be uniform if each recombination (for each set of small cubes) resulted in the original system. If even one recombination produced a different system, the system would be non-uniform or heterogenous.
1.1.2 Properties of System
We associate a system with the properties it possesses. By property we mean any measurable characteristic that is related to matter, energy, space, or time. Some common examples of property are mass, concentration, temperature, enthalpy, pressure, volume, diffusivity, etc. With the help of the properties of a system, we can keep track of it, and compare it to other systems of interest.
System properties can be classified into intensive and extensive properties. Given a uniform system, an extensive property is proportional to the size or extent of the system. Examples of extensive properties are mass and volume. Thus, the mass of a fraction (say, 1/10th) of a system is the same fraction (1/10th) of the total mass of the system. On the other hand, an intensive property of a uniform system does not depend on its size or extent, and is the same, i.e., has the same value, for each part of the system. Examples of intensive property are concentration, temperature and pressure.
Thus, if a uniform system is at a certain pressure then any part of the system is at the same pressure. Equivalently, if all intensive properties of a system do not vary then the system is uniform. An example is the reaction mixture of Figure 1.1 on the previous page. The mixture has
the same value of concentration of the species A throughout the system (or uniform concentration of A),
uniform concentration of each of the remaining species B and C, and
a similar uniformity of any other intensive property, e.g., temperature.
For a non-uniform system, one or more intensive properties vary within the system. More precisely, the properties vary with space inside the system. For example, if the reaction mixture of Figure 1.1 on p. 1 were not well-mixed then the species concentrations, and temperature would not be the same throughout the mixture. In that situation, the mixture would be a non-uniform system.
1.1.3 Classification of System
Based on whether or not the intensive properties of a system vary with space, we can call a system uniform, or non-uniform. A non-uniform system is also known as a distributed-parameter system. If the intensive properties have variations that are small enough to be insignificant then the system may be considered as a uniform system by taking into account the space-averaged values of intensive properties. This system is then called a lumped-parameter system. Thus, the reaction mixture of Figure 1.1 would be a lumped-parameter system if the temperature varied slightly within the mixture but the latter was considered to be at some average temperature throughout.
Depending on the degree of separation from the surroundings, systems are also classified into open, closed and isolated systems. An open system allows exchanges of mass and energy with the surroundings. On the other hand, a closed system allows only the exchange of energy. An isolated system does not allow any exchange of mass, or energy.
Thus, the reaction mixture shown in Figure 1.1 would be an open system if it were heated, or cooled, and the species volatilized to the air. The mixture would be a closed system if it were heated, or cooled, but the top surface were covered to prevent any escape of the species. With the top surface covered and perfectly insulated along with the reactor vessel, the mixture would become an isolated system.
1.1.4 Model
The reason we conceive a system is that we want to learn about it. This learning is synonymous with figuring out relations between system properties. These relations give rise to a model. In mathematical terms, a model is a set of equations that involve system properties. A simple example of a model is the ideal gas law,
where , and are, respectively, the pressure, molar volume, and temperature (properties) of a system at sufficiently low pressure, and is the universal gas constant. The properties in the model do not depend on time, and the associated system is unchanging or at equilibrium to be exact.
When a system undergoes a change, it appears as an effect on one or more of the system properties. This is where the notion of process emerges.
1.2 Process
A process is defined as a set of activities taking place in a system, and resulting in certain effects on its properties. A process is either natural, or man-made. Natural processes – such as blood circulation in a human body, photosynthesis in plants, or planetary motion in the solar system – happen without human volition, and are responsible for certain effects on the associated systems. For instance, the process of blood circulation in a human body primarily involves pulmonary circulation between the heart and lungs, and systemic circulation between the heart and the rest of the body excluding lungs. This process results in, among other things, specific levels of oxygen and carbon dioxide concentrations in different parts of the body, i.e., the system.
Man-made processes on the other hand are contrived by human beings to produce results of utility. Common examples include processes to produce various synthetic chemicals and materials, to extract and refine natural resources, to treat gaseous emissions and wastewaters, and to control climate in living spaces. The reactor shown in Figure 1.1 on p. 1 enables a man-made process. It involves a chemical reaction between the reactants A and B, which results in the product C.
1.2.1 Classification of Processes
Based on how system properties change with time, processes are classified into unsteady state and steady state processes. A process that changes any property of a system with time is called an unsteady state process, and the system is called unsteady. Thus, the process of chemical reaction in the reactor of Figure 1.1 is an unsteady state process. It causes the reactant and product concentrations to, respectively, decrease and increase with time.
In contrast, a steady state process does not result in any change in system properties with time. The reason is that any decrease in a property gets instantaneously offset by an equal increase in the same.
A simple example of a steady state process is the filling of a tank with a non-volatile liquid, and its simultaneous drainage from the tank at the same flow rate as that of filling. Here, the system and the property of interest are, respectively, the liquid and its volume inside the tank. Due to equal inflow and outflow rates, the volume does not change with time.
Another example of a steady state process is a chemical reaction that continues after a transient period in a constant volume stirred tank reactor with constant flow rates of incoming reactants, and the outgoing mixture of residual reactants and products. In this process, the species concentrations, and the volume of the reaction mixture inside the tank do not change with time.
Steady State versus Equilibrium
It may be noted that the time derivatives of all properties are zero in a system under the influence of a steady state process. That is also true for a system at equilibrium. But there is a subtle difference. A system at equilibrium does not sustain any process since the gradients (i.e., spatial derivatives) of all potentials in the system have decayed to zero. For this reason, the properties in such a system do not have any propensity to change. However, the properties in a system under a steady state process undergo simultaneous increments and decrements in such a way that the net change in each property is zero. In other words, the properties end up being time-invariant. If a steady state process is stopped then the properties of the system would begin to change with time until the system eventually arrived at equilibrium.
1.2.2 Process Model
A process model is a set of equations or relations involving properties of the system under the influence of a process. The properties represent observable occurrences or phenomena classifiable into the categories of (i) initiating events, (ii) specifications, and (iii) effects. Thus, a process model is a scheme according to which a process with given specifications and initiating events would generate effects in the system.
Figure 1.2 below shows the concept of the model as a triangle, each side of which represents the relations between the two ends or vertices denoting the categories. If we know these relations, i.e., the process model, we can use it to unravel one or more unknown phenomena when the remaining ones are known. And we can do this without having to execute the process in the real world. Of course, the better the process model the better is our ability to explain the involved phenomena.
Figure 1.2 Concept of a process model
In particular, we can predict the effects of a process based on its model, and the knowledge of initiating events, and specifications. As a matter of fact, we can predict the effects for different initiating events, and specifications. Doing that enables us to isolate desirable effects, and the related initiating events, and specifications. We can then apply the latter two in the real world to achieve the desirable effects from the process. This exercise basically is process optimization and control.
Referring to Figure 1.1 on p. 1, if we know the model of the process taking place in the reaction mixture (system) then we can predict the species concentrations (effects) corresponding to different initial concentrations of A and B (initiating events), and reaction temperatures (specifications). From the predictions over a given time of reactor operation, we can pick out a desirable effect, say, the maximum concentration of C, and the related (optimal) set of initial concentrations, and reaction temperature. Based on this exercise, we can then expect to achieve the maximum concentration of C in the real world by feeding the reactor with A and B such that the mixture has the optimal initial concentrations, and is maintained at the optimal temperature for the given operation time.
In general, using a process model we can predict initiating events, specifications, or effects in the system. The prediction requires the following courses of action:
Process modeling – the development of a process model, and
Process simulation – the solution or simulation of the model, which mimics the process as it would unfold in the real world.
Types of Process Models
Process models can be categorized based on the process being represented. Thus, a steady state model represents a steady state process, and has no time derivatives. An unsteady state model represents a unsteady process, and involves one or more time derivatives.
Based on the nature of system properties, a process model may be a lumped-parameter model, or distributed-parameter model. The former involves uniform properties while the latter has at least one property that varies with space.
Based on the type of equations involved, process models are also classified into algebraic, differential, or differential-algebraic models. Moreover, if all involved equations are linear then the process model is called linear. Otherwise, the model is called non-linear.
1.3 Process Modeling
Process modeling is essentially an exercise that involves relating together the properties of a system influenced by a process. Represented as mathematical symbols, the properties are associated with each other using relevant relations under one or more assumptions. The outcome is a set of mathematical equations, which is a process model. The system properties – and through them, the initiating events, specifications and effects of the process – are expected to abide by the model thus developed. The model is therefore said to represent or describe the process.
As an example, consider the reaction process in the reactor shown in Figure 1.1 on p. 1. The involved properties are the concentrations of A, B and C, which can be represented as , and , respectively, with initial values , and . If we assume that during the process the reaction mixture
is well-mixed,
has constant volume, and
is at constant temperature
then based on certain relations, the concentrations can be associated with each other at any time through the following equations:
where (i) , , and are, respectively, the rate, frequency factor, activation energy, and temperature of the reaction, (ii) and are reaction-specific parameters, and (iii) is the universal gas constant.
The above set of equations is the model of the reaction process taking place in the reactor. This model relates the initiating events, specifications and effects of the process with each other [see Figure 1.2, p. 5]. The initiating events are represented by the initial concentrations, , and . The specifications are the values of and . Finally, the process effects at any time , are represented by the concentrations and . Changes in these properties, or, equivalently, the phenomena they represent, are expected to be in accordance with the model. Thus, for a given set of initiating events, and specifications, we expect to find the process effects from the model.
1.3.1 Relations
Relations are the ground rules that are used in process modeling to interlink the properties of a system under the influence of a process. These rules comprise fundamental relations based on scientific laws, and constitutive relations. The rules manifest as equations that constitute process models.
A scientific law is a statement that is generally accepted to be true about one or more phenomena on the basis of repeated experiments and observations. A common example is the law of conservation of mass. This law states that if a system does not exchange mass, or energy (a form of mass) with the surroundings then the mass of the system stays the same, or is conserved. By applying this law to an individual species, and accounting for its generation, or consumption in a system, we derive a fundamental relation called the mass balance of species.
A constitutive relation on the other hand is a rule that is true for systems with a specific makeup or constitution. An example is Newton's law of viscosity, which relates shear stress to strain for a system that is a Newtonian fluid.
In the model given by Equations (1.1)–(1.3) on the previous page, the last equation is a constitutive relation. It is valid for the specific reaction mixture of species A, B and C. The two differential equations are the mass balances of the species.
1.3.2 Assumptions
Assumptions of a process model are the necessary conditions that must be satisfied during the execution of the process in order for it to be represented by the model. In equivalent terms, if any assumption of a model is not satisfied during the execution of the process then the latter is not represented by the model.
For example, the assumptions of the model given by Equations (1.1)–(1.3) are perfect mixing, and constant volume as well as temperature. If any of these assumptions is not satisfied during the reaction process then it would become different from the process ‘assumed’ by the model, and would not be represented by the model. For instance, if mixing is not sufficiently close to perfect then there would be spatial changes in the concentrations of species, and temperature. Consequently, the reaction process would get dominated by diffusion, convection of species, and heat transfer, which are not accounted for by the model.
The above example shows that the assumption of perfect mixing excludes from the model the sub-processes of diffusion, convection and heat transfer in the reaction mixture. In general, an assumption implies the exclusion of one or more sub-processes from the model. Hence, an assumption restricts the model to a specific type of process, and in doing so simplifies the model. Note that the assumption of perfect mixing of the reaction mixture markedly simplifies the model by obviating the need for the diffusive fluxes of the species, and partial differential equations of momentum and heat transfer. Because of this assumption, the concentrations of species, and temperature can be considered uniform throughout the reaction mixture.
