theoretical chemical engineering modeling and simulation

Theoretical Chemical Engineering
Christo Boyadjiev
Theoretical Chemical
Engineering
Modeling and Simulation
123
ISBN 978-3-642-10777-1 e-ISBN 978-3-642-10778-8
DOI 10.1007/978-3-642-10778-8
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PD Dr. Christo Boyadjiev
Institute of Chemical Engineering (IChE)
Bulgarian Academy of Sciences (BAS)
Acad. G. Bonchev Str., Bl. 103
1113 Sofia, Bulgaria
email:

Motto
Experimenters are the striking force of
science. The experiment is a question which
science puts to nature. The measurement is
the registration of nature’s answer. But
before the question is put to nature, it must
be formulated. Before the measurement
result is used, it must be explained, i.e., the
answer must be understood correctly. These
two problems are obligations of the
theoreticians.
Max Planck
This book is dedicated to my parents.
Christo Boyanov Boyadjiev
Abstract
The theoretical methods of chemical engineering for modeling and simulation of
industrial processes are surveyed in this book. On this basis it is possible to
formulate correct experimental conditions and to understand correctly the exper-
imental results.
The continuous media approach is used for modeling simple processes such as
hydrodynamic processes, mass transfer processes, and heat transfer processes. The
theory of scalar, vector, and tensor fields permits one to create the basic equations
and boundary conditions. Problems of rheology, turbulence, turbulent diffusion,
and turbulent mass transfer are examined too.
The chemical processes and adsorption models and especially the stoichiom-
etry, reaction mechanism, reaction route, kinetics of simple and complex chemical
reactions, physical and chemical adsorption, and heterogeneous reactions are
discussed.

Different types of complex process models are presented depending on the
process mechanism. The relation between the mechanism and the mathematical
description is shown in the case of physical absorption. Characteristic scales,
generalized variables, and dimensionless parameters are used for analysis of the
process mechanism. Full information about this mechanism permits the creation of
theoretical models. Mass transfer in film flows is an example of such models,
where the effects of a chemical reaction and gas motion and absorption of slightly
and highly soluble gases are considered.
The very complicated hydrodynamic behavior in column apparatuses is a
reason for using diffusion-type models in the cases of mass transfer with a
chemical reaction and interphase mass transfer. An average concentration model
of an airlift reactor is presented.
Similarity theory models are demonstrated in the case of absorption in packed-
bed columns. Generalized (dimensionless) variables and generalized individual
cases are used for formulation of the similarity conditions and similarity criteria.
The dimension analysis, mathematical structure of the models, and some errors in
criteria models are discussed.
ix
Regression models are preferred when there is complete absence of information
about the process mechanism and the least-squares method is used for parameter
identification.
A theoretical analysis of models of the mass transfer theories is presented in the
cases of linear and nonlinear mass transfer. The model theories, boundary layer
theory, mass transfer in countercurrent flows, influence of the intensive mass
transfer on the hydrodynamics, boundary conditions of the nonlinear mass transfer
problem, nonlinear mass transfer in the boundary layer, and the Marangoni effect
are examined.
A qualitative theoretical analysis is presented as a generalized analysis. The use
of generalized variables permits the analysis of the models of mass transfer with a
chemical reaction, nonstationary processes, and stationary processes and the effect

of the chemical reaction rate.
The generalized analysis permits the analysis of the mechanism of gas–liquid
chemical reactions in the cases of irreversible chemical reactions, homogenous
catalytic reactions, and reversible chemical reactions and the relationships between
the chemical and physical equilibria during absorption.
A comparative qualitative analysis for process mechanism identification is
presented in the cases of different nonlinear effects, nonstationary absorption
mechanisms, and nonstationary evaporation kinetics.
A quantitative theoretical analysis is presented for solution of the scale-up
problems and statistical analysis of the models. The similarity and scale-up, scale
effect and scale effect modeling, scale-up theory and hydrodynamic modeling, and
scale effect and scale-up of column apparatuses are discussed. The statistical
analysis ranges over basic terms, statistical treatment of experimental data, testing
of hypotheses, significance of parameters, and model adequacy of different types
of models.
The stability analysis of the models examines the general theory of stability
(evolution equations, bifurcation theory), hydrodynamic stability (fundamental
equations, power theory, linear theory, stability, bifurcations, and turbulence), the
Orr–Sommerfeld equation (parallel flows, almost parallel flows, linear stability,
and nonlinear mass transfer), and self-organizing dissipative structures (interphase
heat and mass transfer between gas–liquid immovable layers, Oberbeck–Bous-
sinesq equations, gas absorption, and liquid evaporation).
The calculation problems in chemical engineering theory are related to the
solutions of differential equations and identification of the model parameters
(estimation). Different analytical methods, such as the similarity variables method,
Green’s functions, Laplace transforms, the Sturm–Liouville problem, the eigen-
value problem, and perturbation methods, are presented. Numerical methods (finite
differences method, finite elements method) are examined on the basis of com-
mercial software. Iterative solution methods are considered too.
Parameter estimation methods are discussed in the case of incorrect (ill-posed)

inverse problems. An iterative method for parameter identification is presented for
solution of correct, incorrect, and essentially incorrect problems. The optimization
methods are examined as a basis of the least squares function minimization.
x Abstract
Models of chemical plant systems are presented as a set of process models and
the relations between them. An algorithm for simulation of chemical plants is
proposed. The methods of optimal synthesis of chemical plants are considered in
the case of optimal synthesis of heat recuperation systems. The renovation of
chemical plants is formulated as a mathematical model. The main problems are the
renovation by optimal synthesis, renovation by introduction of new equipment,
and renovation by introduction of new processes.
Examples from the author’s investigations are presented at the end of all
chapters.
Christo B. Boyadjiev
Abstract xi
Preface
The role of theory in science was formulated very brilliantly by Max Planck:
Experimenters are the striking force of science. The experiment is a question which
science puts to nature. The measurement is the registration of nature’s answer. But
before the question is put to nature,itmust be formulated. Before the measurement
result is used,itmust be explained, i.e., the answer must be understood correctly.
These two problems are obligations of the theoreticians.
Chemical engineering is an experimental science, but theory permits us to
formulate correct experimental conditions and to understand correctly the exper-
imental results. The theoretical methods of chemical engineering for modeling and
simulation of industrial processes are surveyed in this book.
Theoretical chemical engineering solves the problems that spring up from the
necessity for a quantitative description of the processes in the chemical industry.
They are quite different at the different stages of the quantitative description, i.e., a

wide circle of theoretical methods are required for their solutions.
Modeling and simulation are a united approach to obtain a quantitative
description of the processes and systems in chemical engineering and chemical
technology, which is necessary to clarify the process mechanism or for optimal
process design, process control, and plant renovation.
Modeling is the creation of the mathematical model, i.e., construction of the
mathematical description (on the basis of the process mechanism), calculation of
the model parameters (using experimental data), and statistical analysis of the
model adequacy.
Simulation is a quantitative description of the processes by means of algorithms
and software for the solution of the model equations and numerical (mathematical)
experiments.
The processes in chemical engineering are composed of many simple processes,
such as hydrodynamic, diffusion, heat conduction, and chemical processes. The
models are created in the approximations of continuous media mechanics.
The complex process model is constructed on the basis of the physical mech-
anism hypothesis. In cases where full information is available, it is possible to
create a theoretical type of model. If the information is insufficient (it is not
xiii
possible to formulate the hydrodynamic influence on the heat and mass transfer),
the model is pattern theory, diffusion type or similarity criteria type. The absence
of information leads to the regression model.
The theoretical analysis of the models solves qualitative, quantitative, and
stability problems. The qualitative analysis clarifies the process mechanism or
similarity conditions. The quantitative analysis solves the problems related to the
scale-up and model adequacy. The stability analysis explains the increase of the
process efficiency as a result of self-organizing dissipative structures.
All theoretical methods are related to calculation problems. The solutions of the
model equations use analytical and numerical methods. The identification (esti-
mation) of the model parameters leads to the solutions of the inverse problems, but

very often they are incorrect (ill-posed) and need the application of regularization
methods, using a variational or an iterative approach. The solutions of many
chemical engineering problems (especially parameter identification) use minimi-
zation methods.
The book ideology briefly described above addresses the theoretical foundation
of chemical engineering modeling and simulations. It is concerned with building,
developing, and applying the mathematical models that can be applied success-
fully for the solution of chemical engineering problems. Our emphasis is on the
description and evaluation of models and simulations. The theory selected reflects
our own interests and the needs of models employed in chemical and process
engineering. We hope that the problems covered in this book will provide the
readers (Ph.D. students, researchers, and teachers) with the tools to permit the
solution of various problems in modern chemical engineering, applied science, and
other fields through modeling and simulations.
The solutions of the theoretical problems of modeling and simulations employ a
number of mathematical methods (exact, asymptotic, numerical, etc.) whose
adoption by engineers will permit the optimal process design, process control, and
plant renovation.
The modeling and the simulations of chemical systems and plants can be
achieved very often through a hierarchical modeling. This approach uses the
structural analysis of the process systems. The result of the structural analysis is a
quantitative description allowing further optimal process design, process control,
and plant renovation. The effectiveness of the optimal solutions can be enhanced if
they are combined with suitable methods of optimal synthesis. The latter is a
methodical basis and a guide for process system renovations.
The book incorporates a lot of fundamental knowledge, but it is assumed that
the readers are familiar with the mathematics at engineering level of usual uni-
versity courses.
The above comments are the main reasons determining the structure of this
book.

Part 1 concerns model construction problems. The mechanics of the continuum
approach is used for modeling hydrodynamic, diffusion, and heat conduction
processes as basic (elementary) processes in chemical engineering. The modeling
of complex processes in chemical engineering is presented on the basis of the
xiv Preface
relation between the process mechanism and the mathematical description. The
models are classified in accordance with the knowledge available concerning the
process mechanisms. This means a situation when a theoretical model is available,
i.e., sufficient knowledge of the process mechanism as well as the opposite situ-
ation of knowledge deficiency, which leads to regression models. Theoretical
diffusion, dimensionless, and regression types of models are illustrated. The linear,
nonlinear, and pattern mass transfer theories are considered too.
Part 2 focuses on theoretical analysis of chemical engineering process models.
The qualitative analysis uses generalized (dimensionless) variables and shows the
degree to which the different physical effects participate in a complex process.
On this basis, similarity criteria and physical modeling conditions are shown.
The quantitative analysis concerns the scale-up problems and statistical analysis of
the models. The stability analysis of the models permits the nonlinear mass
transfer effects to be obtained and the creation of the self-organizing dissipative
structures with very intensive mass transfer.
Part 3 addresses the calculation problems in modeling and simulation. Dif-
ferent analytical and numerical methods for the solution of differential equations
are considered. The estimation of the model parameters is related to the solutions
of the ill-posed inverse problems. An iterative method for incorrect problem
solutions is presented. Different methods for function minimization are shown for
the purposes of process optimization and model parameter identification.
Part 4 examines modeling and simulation of the chemical plant systems.
The simulation of the systems on the basis of structure system analysis is pre-
sented. The optimal synthesis of chemical plants is considered in the case of the
optimal synthesis of heat recuperation systems.

This book can be used as a basis for theoretical and experimental investigations in
the field of the chemical engineering. The methods and analyses presented permit
theoretical problems to be solved, the experimental conditions to be correctly
formulated, and the experimental results to be interpreted correctly.
The fundamental suggestion in this book is the necessity for full correspondence
(direct and inverse) between the separated physical effect in the process and the
mathematical (differential) operator in the model equation.
The main part of this book has a monographic character and the examples are
from the author’s papers. The book uses the author’s lectures ‘‘Course of modeling
and optimization’’ (subject chemical cybernetics in the Faculty of Chemistry of
Sofia University ‘‘St. Kliment Ohridski’’), ‘‘Course of modeling and simulation of
chemical plant systems’’ (Bourgas University ‘‘Prof. Asen Zlatarov’’), and
‘‘Master’s classes of theoretical chemical engineering’’ (Bourgas University ‘‘Prof.
Asen Zlatarov’’). That is why, as a whole, it is possible for it to be used as teaching
material for modeling and simulation. This book proposes an exact formulation
and the correct solution of quantitatively described problems in chemical engi-
neering. It may be useful for scientists, Ph.D. students, and undergraduate students.
Some of the results presented in the book were obtained with financial support
from the National Fund ‘‘Scientific Researches’’ of the Republic of Bulgaria
Preface xv
(contracts no. TH-154/87, TH-162/87, TH-89/91, TH-127/91, TH-508/95, TH-4/99,
TH-1001/00, TH-1506/05).
The author would like to thank Assoc. Prof. PhD Jordan Hristov, Assoc. Prof.
PhD Natasha Vaklieva-Bancheva, Assoc. Prof. PhD Boyan Ivanov, Assist. Prof.
PhD Maria Doichinova, Assist. Prof. Petya Popova, Assist. Prof. Elisaveta
Shopova and Dipl. Eng., M.Sc. Boyan Boyadjiev for their help in the preparation
of this book.
Christo Boyanov Boyadjiev
xvi Preface
Contents

Part I Model Construction Problems
Simple Process Models 3
1 Mechanics of Continuous Media . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1 Scalar and Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Stress Tensor and Tensor Field. . . . . . . . . . . . . . . . . . . . . . . 7
2 Hydrodynamic Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Laminar Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Two-Phase Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Particular Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7 Generalized Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.8 Basic Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.9 Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.10 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3 Mass and Heat Transfer Processes . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Transfer Processes Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Diffusion Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Turbulent Diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6 Turbulent Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Chemical Processes and Adsorption . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Stoichiometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Mechanism and Reaction Route . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Kinetics of Simple Chemical Reactions . . . . . . . . . . . . . . . . . 50
4.4 Kinetics of Complex Reactions. . . . . . . . . . . . . . . . . . . . . . . 52
4.5 Adsorption Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.6 Physical Adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

xvii
4.7 Chemical Adsorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.8 Heterogeneous Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1 Dissolution of a Solid Particle . . . . . . . . . . . . . . . . . . . . . . . 56
5.2 Contemporary Approach of Turbulence Modeling. . . . . . . . . . 58
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Complex Process Models 61
1 Mechanism and Mathematical Description . . . . . . . . . . . . . . . . . . . 61
1.1 Mechanism of Physical Absorption . . . . . . . . . . . . . . . . . . . . 62
1.2 Mathematical Description. . . . . . . . . . . . . . . . . . . . . . . . . . . 62
1.3 Generalized Variables and Characteristic Scales . . . . . . . . . . . 63
1.4 Dimensionless Parameters and Process Mechanism . . . . . . . . . 64
1.5 Boundary Conditions and Mechanism . . . . . . . . . . . . . . . . . . 66
1.6 Kinetics and Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2 Theoretical Models: Mass Transfer in Film Flows . . . . . . . . . . . . . . 68
2.1 Film with a Free Interface . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.2 Effect of a Chemical Reaction . . . . . . . . . . . . . . . . . . . . . . . 70
2.3 Effect of Gas Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.4 Absorption of Slightly Soluble Gas . . . . . . . . . . . . . . . . . . . . 76
2.5 Absorption of Highly Soluble Gas . . . . . . . . . . . . . . . . . . . . 78
3 Diffusion-Type Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.1 Mass Transfer with a Chemical Reaction . . . . . . . . . . . . . . . . 81
3.2 Interphase Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3 Average Concentration Models . . . . . . . . . . . . . . . . . . . . . . . 83
3.4 Airlift Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4 Similarity Theory Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1 Absorption in a Packed-Bed Column. . . . . . . . . . . . . . . . . . . 92
4.2 Generalized (Dimensionless) Variables . . . . . . . . . . . . . . . . . 92
4.3 Generalized Individual Case and Similarity . . . . . . . . . . . . . . 94

4.4 Mathematical Structure of the Models . . . . . . . . . . . . . . . . . . 95
4.5 Dimension Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.6 Some Errors in Criteria Models . . . . . . . . . . . . . . . . . . . . . . 100
5 Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.1 Regression Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2 Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3 Least-Squares Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1 Effect of Surfactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 Effect of Interface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.3 Photobioreactor Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
xviii Contents
Mass Transfer Theories 127
1 Linear Mass Transfer Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
1.1 Model Theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
1.2 Boundary Layer Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
1.3 Two-Phase Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . 132
2 Mass Transfer in Countercurrent Flows. . . . . . . . . . . . . . . . . . . . . . 134
2.1 Velocity Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
2.2 Concentration Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 137
2.3 Comparison Between Co-current and Countercurrent Flows . . . 139
3 Nonlinear Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.1 Influence of Intensive Mass Transfer on the Hydrodynamics . . . 141
3.2 Boundary Conditions of the Nonlinear Mass
Transfer Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.3 Nonlinear Mass Transfer in the Boundary Layer. . . . . . . . . . . 145
3.4 Two-Phase Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.5 Nonlinear Mass Transfer and the Marangoni Effect . . . . . . . . 157
4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

4.1 Heat Transfer in the Conditions of Nonlinear Mass Transfer . . . 163
4.2 Multicomponent Mass Transfer. . . . . . . . . . . . . . . . . . . . . . . 165
4.3 Concentration Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.4 Influence of High Concentration on the Mass Transfer Rate . . . 173
4.5 Nonlinear Mass Transfer in Countercurrent Flows . . . . . . . . . 180
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Part II Theoretical Analysis of Models
Qualitative Analysis 187
1 Generalized Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
1.1 Generalized Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
1.2 Mass Transfer with a Chemical Reaction . . . . . . . . . . . . . . . . 188
1.3 Nonstationary Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
1.4 Steady-State Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
1.5 Effect of the Chemical Reaction Rate . . . . . . . . . . . . . . . . . . 191
2 Mechanism of Gas–Liquid Chemical Reactions . . . . . . . . . . . . . . . . 192
2.1 Irreversible Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . 192
2.2 Homogenous Catalytic Reactions . . . . . . . . . . . . . . . . . . . . . 202
2.3 Reversible Chemical Reactions. . . . . . . . . . . . . . . . . . . . . . . 205
2.4 Relationships Between the Chemical Equilibrium and the
Physical Equilibrium During Absorption . . . . . . . . . . . . . . . . 208
3 Comparative Qualitative Analysis for Process Mechanism
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
3.1 Comparison of the Nonlinear Effects. . . . . . . . . . . . . . . . . . . 211
3.2 Nonstationary Absorption Mechanism . . . . . . . . . . . . . . . . . . 221
Contents xix
3.3 Nonstationary Evaporation Kinetics. . . . . . . . . . . . . . . . . . . . 228
4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
4.1 Sulfuric Acid Alkylation Process in a Film Flow Reactor . . . . 236
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
Quantitative Analysis 243

1 Scale-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
1.1 Similarity and Scale-Up. . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
1.2 Scale Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
1.3 Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
1.4 Scale-Up Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
1.5 Axial Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
1.6 Evaluation of the Scale Effect . . . . . . . . . . . . . . . . . . . . . . . 256
1.7 Hydrodynamic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
2 Average Concentration Model and Scale-Up . . . . . . . . . . . . . . . . . . 259
2.1 Diffusion-Type Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
2.2 Influence of the Radial Nonuniformity of the Velocity
Distribution on the Process Efficiency . . . . . . . . . . . . . . . . . . 260
2.3 Scale Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
2.4 Average Concentration Model . . . . . . . . . . . . . . . . . . . . . . . 264
2.5 Scale Effect Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
2.6 Scale-Up Parameter Identification . . . . . . . . . . . . . . . . . . . . . 267
3 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
3.1 Basic Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
3.2 Statistical Treatment of Experimental Data . . . . . . . . . . . . . . 281
3.3 Estimates of the Expectation and the Dispersion. . . . . . . . . . . 282
3.4 Tests of Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
3.5 Dispersion Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
3.6 Significance of Parameter Estimates and Model Adequacy . . . 289
3.7 Model Suitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
3.8 Adequacy of the Theoretical Models and Model Theories . . . . 293
4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
4.1 Statistical Analysis of Diffusion Type Models . . . . . . . . . . . . 295
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
Stability Analysis 297
1 Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

1.1 Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
1.2 Bifurcation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
1.3 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
2 Hydrodynamic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
2.1 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
2.2 Power Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
2.3 Linear Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
xx Contents
2.4 Stability, Bifurcations, and Turbulence . . . . . . . . . . . . . . . . . 311
2.5 Stability of Parallel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . 313
3 Orr–Sommerfeld Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
3.1 Parallel Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
3.2 Almost Parallel Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
3.3 Linear Stability and Nonlinear Mass Transfer. . . . . . . . . . . . . 316
4 Self-Organizing Dissipative Structures . . . . . . . . . . . . . . . . . . . . . . 328
4.1 Nonlinear Mass Transfer in the Boundary Layer. . . . . . . . . . . 330
4.2 Gas Absorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
4.3 Liquid Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
5.1 Gas–Liquid System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
5.2 Liquid–Liquid System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
5.3 Effect of Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
5.4 Effect of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
Part III Calculation Problems
Solution of Differential Equations 405
1 Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
1.1 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
1.2 Similarity Variables Method. . . . . . . . . . . . . . . . . . . . . . . . . 409
1.3 Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

1.4 Laplace Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
2 Perturbation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
2.1 Expansions with Respect to a Parameter . . . . . . . . . . . . . . . . 414
2.2 Expansions with Respect to a Coordinate. . . . . . . . . . . . . . . . 417
2.3 Nonuniform Expansions (Poincaret–Lighthill–Ho Method). . . . 418
3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
3.1 Finite Differences Method . . . . . . . . . . . . . . . . . . . . . . . . . . 422
3.2 Finite Elements Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
4.1 Application of Green’s Functions . . . . . . . . . . . . . . . . . . . . . 424
4.2 Sturm–Liouville Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
Parameter Identification (Estimation) 429
1 Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
1.1 Direct and Inverse Problems. . . . . . . . . . . . . . . . . . . . . . . . . 430
1.2 Types of Inverse Problems. . . . . . . . . . . . . . . . . . . . . . . . . . 430
1.3 Incorrectness of the Inverse Problems . . . . . . . . . . . . . . . . . . 432
Contents xxi
2 Sets and Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
2.1 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
2.2 Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
2.3 Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
2.4 Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
2.5 Functional of the Misfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
2.6 Some Properties of the Direct and Inverse Operators. . . . . . . . 439
3 Incorrectness of the Inverse Problems. . . . . . . . . . . . . . . . . . . . . . . 440
3.1 Correctness After Hadamard. . . . . . . . . . . . . . . . . . . . . . . . . 441
3.2 Correctness After Tikhonov . . . . . . . . . . . . . . . . . . . . . . . . . 442
4 Methods for Solving Incorrect (Ill-Posed) Problems . . . . . . . . . . . . . 442
4.1 Method of Selections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

4.2 Method of Quasi-Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . 444
4.3 Method of Substitution of Equations . . . . . . . . . . . . . . . . . . . 445
4.4 Method of the Quasi-Reverse . . . . . . . . . . . . . . . . . . . . . . . . 445
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
5 Methods for Solving Essentially Ill-Posed Problems . . . . . . . . . . . . . 446
5.1 Regularization Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
5.2 Variational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
5.3 Iterative Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
6 Parameter Identification in Different Types of Models . . . . . . . . . . . 456
6.1 Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
6.2 Selection Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
6.3 Variational Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . 459
6.4 Similarity Theory Models . . . . . . . . . . . . . . . . . . . . . . . . . . 461
6.5 Diffusion-Type Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
6.6 Theoretical Models and Model Theories . . . . . . . . . . . . . . . . 464
7 Minimum of the Least-Squares Function. . . . . . . . . . . . . . . . . . . . . 465
7.1 Incorrectness of the Inverse Problem . . . . . . . . . . . . . . . . . . . 465
7.2 Incorrectness of the Least Squares Function Method . . . . . . . . 466
7.3 Regularization of the Iterative Method for Parameter
Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469
7.4 Iteration Step Determination and Iteration Stop Criterion. . . . . 471
7.5 Iterative Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471
7.6 Correct Problem Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 472
7.7 Effect of the Regularization Parameter . . . . . . . . . . . . . . . . . 473
7.8 Incorrect Problem Solution. . . . . . . . . . . . . . . . . . . . . . . . . . 473
7.9 Essentially Incorrect Problem Solution . . . . . . . . . . . . . . . . . 475
7.10 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
7.11 Statistical Analysis of Model Adequacy. . . . . . . . . . . . . . . . . 478
7.12 Comparison between Correct and Incorrect Problems . . . . . . . 480
8 Multiequation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

8.1 Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484
8.2 Fermentation System Modeling. . . . . . . . . . . . . . . . . . . . . . . 486
xxii Contents
9 Experiment Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
9.1 Experimental Plans of Modeling . . . . . . . . . . . . . . . . . . . . . . 494
9.2 Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
9.3 Significance of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 498
9.4 Adequacy of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
9.5 Randomized Plans. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
9.6 Full and Fractional Factor Experiment. . . . . . . . . . . . . . . . . . 501
9.7 Compositional Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
10 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
10.1 Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
10.2 Statistical Analysis of the Parameter Significance and Model
Adequacy of the Regression Models . . . . . . . . . . . . . . . . . . . 510
10.3 Clapeyron and Antoan Models . . . . . . . . . . . . . . . . . . . . . . . 514
10.4 Incorrectness Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
10.5 Increase of the Exactness of the Identification
Problem Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
10.6 Incomplete Experimental Data Cases. . . . . . . . . . . . . . . . . . . 518
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
Optimization 531
1 Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
1.1 Unconstraints Minimization . . . . . . . . . . . . . . . . . . . . . . . . . 531
1.2 Constraints Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
1.3 Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
1.4 Solution of a Set of Nonlinear Equations . . . . . . . . . . . . . . . . 536
2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
2.1 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
2.2 Nonlinear Programming. . . . . . . . . . . . . . . . . . . . . . . . . . . . 538

3 Dynamic Programming and the Principle of the Maximum . . . . . . . . 543
3.1 Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
3.2 Principle of Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543
3.3 Principle of the Maximum . . . . . . . . . . . . . . . . . . . . . . . . . . 544
4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
4.1 Problem of Optimal Equipment Change. . . . . . . . . . . . . . . . . 546
4.2 A Calculus of Variations Problem. . . . . . . . . . . . . . . . . . . . . 548
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
Part IV Chemical Plant Systems
Systems Analysis 553
1 Simulation of Chemical Plant Systems . . . . . . . . . . . . . . . . . . . . . . 553
1.1 Model of Chemical Plant Systems. . . . . . . . . . . . . . . . . . . . . 554
1.2 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555
Contents xxiii
1.3 Sequential Module (Hierarchical) Approach . . . . . . . . . . . . . . 555
1.4 Acyclic Chemical Plant Systems. . . . . . . . . . . . . . . . . . . . . . 556
1.5 Cyclic Chemical Plant Systems. . . . . . . . . . . . . . . . . . . . . . . 558
1.6 Independent Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
1.7 Breaking Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
1.8 Optimal Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
2 Simulation for Specified Outlet Variables . . . . . . . . . . . . . . . . . . . . 563
2.1 Zone of Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564
2.2 Absolutely Independent Influence . . . . . . . . . . . . . . . . . . . . . 566
2.3 Independent Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
2.4 Combined Zones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
3 Models of Separate Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
3.1 Types of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
3.2 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570
3.3 Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
3.4 Chemical Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
Synthesis of Systems 575
1 Optimal Synthesis of Chemical Plants . . . . . . . . . . . . . . . . . . . . . . 575
1.1 Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
1.2 Optimal Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
1.3 Main Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
1.4 Methods of Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
1.5 Optimal Synthesis of a System for Recuperative
Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
2 Renovation of Chemical Plant Systems. . . . . . . . . . . . . . . . . . . . . . 581
2.1 Mathematical Description. . . . . . . . . . . . . . . . . . . . . . . . . . . 582
2.2 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
2.3 Main Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
2.4 Renovation by Optimal Synthesis of Chemical
Plant Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
2.5 Renovation by Introduction of Highly Intensive Equipment . . . 587
2.6 Renovation by Introduction of Highly Effective Processes . . . . 587
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
Conclusion 589
Index 591
xxiv Contents