copyright 2006, vikram shabde
TRANSCRIPT
OPTIMAL DESIGN AND CONTROL OF A SPRAY
DRYING PROCESS THAT MANUFACTURES
HOLLOW MICRO-PARTICLES
by
VIKRAM SHABDE, B.E.
A DISSERTATION
IN
CHEMICAL ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
Approved
Karlene A. Hoo Chairperson of the Committee
Uzi Mann
Gary Gladysz
Nazmul Karim
Accepted
John Borrelli Dean of the Graduate School
December, 2006
Copyright 2006, Vikram Shabde
ACKNOWLEDGEMENTS
I am deeply grateful to my research advisor, Dr. Karlene A. Hoo, for her guid-
ance and support during the four years that I was her graduate student. She has
shown extreme patience in helping me understand the concepts of process control
and optimization. I am also thankful to her for due diligence in proofreading all
of my manuscripts and this dissertation. Her financial support also is appreciated
for my studies and travel to the national meeting of the American Institute of
Chemical Engineering conference and technical workshops.
This work would not have been possible without my parents and their constant
love and support. This work is dedicated to them.
I would like to thank Dr. Uzi Mann for being on my committee and providing
me with the project that ultimately became central to my dissertation. I take
away the need to always define the problem.
Thanks also go to Dr. Gary Gladysz, for providing me with a rewarding in-
ternship at Los Alamos National Laboratory and for his service on my committee;
and to Dr. Naz Karim for agreeing to serve on my committee.
My academic career here has been enriched by many things including the
industrial sponsors of the TTU Process Control and Optimization Consortium;
and the faculty, helpful staff, and graduate students within our research group
and those in the Chemical Engineering Department.
ii
CONTENTS
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . ii
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Spray drying . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Design of spray dryers . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Modeling the spray drying process . . . . . . . . . . . . . . . 4
1.4 Regulation of spray dryers . . . . . . . . . . . . . . . . . . . 6
1.4.1 Brief literature review on process control . . . . . . . . . 6
1.4.2 Literature review on spray dryer regulation . . . . . . . . 9
1.5 Brief literature review of simultaneous design and control . . 11
1.6 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
II DESIGN OF A SPRAY DRYER . . . . . . . . . . . . . . . . . . . 17
2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Experimental system . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Modeling a single droplet . . . . . . . . . . . . . . . . . . . . 24
2.3.1 Physical phenomena . . . . . . . . . . . . . . . . . . . . 25
2.3.2 Model development . . . . . . . . . . . . . . . . . . . . . 27
2.3.3 Initial and boundary conditions . . . . . . . . . . . . . . 29
iii
2.3.3.1 Heating step . . . . . . . . . . . . . . . . . . . . . 30
2.3.3.2 Evaporation step . . . . . . . . . . . . . . . . . . . 31
2.3.3.3 Surface motion . . . . . . . . . . . . . . . . . . . . 32
2.3.4 Dimensionless model . . . . . . . . . . . . . . . . . . . . 32
2.3.5 Model parameters . . . . . . . . . . . . . . . . . . . . . . 36
2.4 Spray dryer modeling . . . . . . . . . . . . . . . . . . . . . . 38
2.4.1 Spray dryer design . . . . . . . . . . . . . . . . . . . . . 39
2.4.2 Single-droplet model . . . . . . . . . . . . . . . . . . . . 45
2.4.3 Particle size distribution model . . . . . . . . . . . . . . 48
2.4.4 Design steps . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . 52
2.5.1 Numerical solution of the single droplet model . . . . . . 52
2.5.1.1 Mathematical formulation . . . . . . . . . . . . . . 54
2.5.1.2 Residual minimization . . . . . . . . . . . . . . . . 55
2.5.2 Numerical solution of the spray dryer model . . . . . . . 56
2.6 Results and discussion . . . . . . . . . . . . . . . . . . . . . 57
2.6.1 Single droplet modeling results . . . . . . . . . . . . . . . 57
2.6.2 Spray dryer model results . . . . . . . . . . . . . . . . . 60
2.6.2.1 Hollow particles . . . . . . . . . . . . . . . . . . . . 62
2.6.2.2 Variation in particle size . . . . . . . . . . . . . . . 63
2.6.2.3 Thermal efficiency . . . . . . . . . . . . . . . . . . 64
2.6.2.4 Heat loss . . . . . . . . . . . . . . . . . . . . . . . 64
iv
2.6.2.5 Effect of feed conditions . . . . . . . . . . . . . . . 65
2.6.2.6 Effects of interparticle collision . . . . . . . . . . . 65
2.6.2.7 Effect of droplet distribution . . . . . . . . . . . . 66
2.6.2.8 Model validation . . . . . . . . . . . . . . . . . . . 67
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
III OPTIMIZATION OVERVIEW . . . . . . . . . . . . . . . . . . . . 87
3.1 Optimization in process design . . . . . . . . . . . . . . . . . 87
3.1.1 Formulation of an optimization problem . . . . . . . . . 88
3.1.2 Classification of optimization problems . . . . . . . . . . 90
3.2 Non-linear programming . . . . . . . . . . . . . . . . . . . . 91
3.2.1 Unconstrained non-linear optimization . . . . . . . . . . 93
3.2.2 Constrained non-linear optimization . . . . . . . . . . . . 95
3.2.3 Solution Techniques for NLP . . . . . . . . . . . . . . . . 97
3.3 Mixed-integer programming (MIP) . . . . . . . . . . . . . . . 102
3.3.1 Mixed-integer linear programming (MILP) . . . . . . . . 103
3.3.2 Mixed-integer non-linear programming (MINLP) . . . . . 104
3.3.2.1 Duality of optimization problems . . . . . . . . . . 105
3.4 Dynamic optimization . . . . . . . . . . . . . . . . . . . . . . 111
3.4.1 Variational methods . . . . . . . . . . . . . . . . . . . . 112
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
v
IV SIMULTANEOUS DESIGN AND CONTROL . . . . . . . . . . . 118
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.3 Design for flexibility . . . . . . . . . . . . . . . . . . . . . . . 125
4.3.1 Design under uncertainty . . . . . . . . . . . . . . . . . . 125
4.4 Modified analytical hierarchical procedure . . . . . . . . . . . 130
4.5 Bi-level optimization . . . . . . . . . . . . . . . . . . . . . . 131
4.6 Example: reactor and flash system . . . . . . . . . . . . . . . 135
4.6.1 Modeling and simulation . . . . . . . . . . . . . . . . . . 136
4.6.2 Sequential design and control . . . . . . . . . . . . . . . 137
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
V SPRAY DRYER CONTROL . . . . . . . . . . . . . . . . . . . . . 145
5.1 Inferential modeling . . . . . . . . . . . . . . . . . . . . . . . 147
5.2 Principal component estimator . . . . . . . . . . . . . . . . . 149
5.2.1 Principal component analysis . . . . . . . . . . . . . . . 149
5.2.2 Principal Component Regression . . . . . . . . . . . . . . 150
5.3 Inferential control . . . . . . . . . . . . . . . . . . . . . . . . 153
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
VI IMPLEMENTATION OF THE BI-LEVEL OPTIMIZATION STRAT-
EGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.1 Sequential optimization strategy . . . . . . . . . . . . . . . . 163
vi
6.2 Bi-level optimization strategy . . . . . . . . . . . . . . . . . 165
6.3 Design and control of the spray drying process . . . . . . . . 166
6.3.1 Optimal control of spray dryer . . . . . . . . . . . . . . . 168
6.3.2 Bi-level optimization of spray dryer . . . . . . . . . . . . 170
6.3.2.1 Comparison between bi-level and sequential opti-
mization strategy . . . . . . . . . . . . . . . . . . . 171
6.3.3 Effect of tighter operational constraints . . . . . . . . . . 172
6.3.4 Effect of the weights wp and wc . . . . . . . . . . . . . . 173
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
VII SUMMARY AND FUTURE WORK . . . . . . . . . . . . . . . . . 181
7.1 Modeling and design . . . . . . . . . . . . . . . . . . . . . . 181
7.2 Control synthesis . . . . . . . . . . . . . . . . . . . . . . . . 182
7.3 Integration of design and control . . . . . . . . . . . . . . . . 183
7.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 184
I Lumped model of the spray dryer . . . . . . . . . . . . . . . . . . 186
II Computer code for Matlab c© . . . . . . . . . . . . . . . . . . . . . 187
vii
ABSTRACT
Spray drying processes have a wide variety of applications in industries from
the manufacture of food to pharmaceuticals and more recently hollow polymeric
microparticles. In spite of the wide uses of the spray drying process the approach
to the design of the spray dryer has remained experiential. One reason for this is
that the spray drying process itself exhibits certain characteristics, such as rapid
heat and mass transfer and a distributed nature, that are challenging from a first-
principles modeling perspective. Additionally, the control of these processes is
difficult since direct on-line realtime measurements of the product quality (e.g.,
particle size) are not available.
In the spray drying process, a liquid spray is brought into contact with a heating
gas to evaporate the solvent present in the solution. The product is in particle
form. The contact between the heating gas and liquid spray may be achieved in
either a co-current or a counter-current manner. In the design of the spray dryer,
the residence time of individual particles inside the spray dryer is important. A
particle should remain in the spray dryer long enough for complete evaporation of
the solvent but not too long to cause product degradation, e.g., by charring.
In this research, a rigorous first-principles based model of the spray dryer is
developed and presented. The model is in two parts: the first, studies the changes
occurring to a single droplet as it is exposed to the heating gas; the second, tracks
individual particles and calculates the total heat and mass transfer between the
droplet phase and the heating gas. The first part of the model consists of a coupled
set of parabolic partial differential equations with a moving boundary condition.
This model is solved using the Gradient Weighted Moving Finite Element Method.
To solve the second part of the model, a Lagrangian framework is used to track the
individual droplets. The flow rate of the heating gas is calculated from empirical
equations available in the literature. The model’s results are validated against an
existing experimental laboratory scale spray drying unit. The validated model is
used to develop an optimal design of the spray drying system.
A control structure to regulate the spray dryer’s product properties is devel-
oped. The two important property variables to be controlled are the mean particle
size (diameter of the particles) and the residual solvent concentration in the prod-
uct. Reliable and efficient on-line, realtime measurements of these variables are
difficult to obtain. Therefore, these variables must be estimated from available
viii
measurements such as the temperature of the heating gas measured at points
along the length of the spray dryer. Applying the method of Principal Compo-
nent Regression, inferential models are developed to estimate the particle size and
residual solvent concentration from the heating gas temperature measurements.
Using these estimates an appropriate control strategy is developed and tested in
the presence of expected disturbances.
Finally, the coupling of design and control that exists naturally due to the
interconnection of unit operations and the recycle of material and energy is ad-
dressed. Traditionally, process design and controller synthesis have been treated
as distinct tasks with the controller synthesis following process design flowsheet
generation. This serial approach ignores the effects the process design decisions
has on the control structure synthesis. Thus, the achievable control performance
if a function of the process design. In this work, a novel bi-level optimization
strategy is used to optimize both the process design and controller synthesis. The
bi-level strategy operates at two levels. At the first level, the combined design and
control problem is solved subject only to the constraints of the design problem.
At the second level, the control problem is solved parameterized by the design so-
lution. The solutions found from the bi-level and serial approaches are compared
and analyzed.
ix
LIST OF FIGURES
1.1 Left panel: Single-input single-output system. Right panel: Multiple-
input multiple-output system. . . . . . . . . . . . . . . . . . . . . . 8
2.1 Schematic diagram of the experimental system. . . . . . . . . . . . 24
2.2 Hollow micro-particles made using the experimental system. Left
panel: thin skin. Right panel: thick skin (Courtesy: S.V. Emets). . 25
2.3 Temperature and concentration profiles during droplet evaporation
[3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Shell balances inside the droplet. . . . . . . . . . . . . . . . . . . . 29
2.5 Left: Simple schematic of a spray dryer with co-current liquid feed
and drying gas. Center: Schematic of the liquid spray inside the
dryer. Right: Zones in a spray shower [22]. . . . . . . . . . . . . . . 38
2.6 Model predictions at design conditions. The polymer and temper-
ature profiles are those at the surface. ◦ : cP (polymer), 3 : θ
(temperature), and 2 : X (radius) [3]. . . . . . . . . . . . . . . . . . 70
2.7 Spatial polymer (top) and water (bottom) profiles during the evap-
oration step. ◦ : τ = 0.00, 2 : τ= 0.005, . : τ= 0.0057, 3 : τ=
0.0063, / : τ= 0.0080, 4 : τ= 0.0087, + : τ =0.0102, × : τ= 0.0139
[3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.8 Polymer concentration profile at the surface for changes in the inlet
feed concentration. ◦ : +10%, 2 : +5%, 3 : nominal, 4 : -5%, and
/ : -10% [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.9 Polymer concentration profile at the surface for changes in the air
temperature. ◦ : +50oC, 2 : +25oC, 3 : nominal, 4 : -25oC, and
/ : -50oC [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.10 Polymer concentration profile at the surface to changes in the heat
transfer coefficient. ◦ : +25%, 2 : +15%, 3 : nominal, 4 : -15%,
and / : -25% [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.11 Polymer concentration profile at the surface for changes in the dif-
fusivity coefficient. ◦ : +25%, 2 : +15%, 3 : nominal, 4 : -15%,
and . : -25 [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.12 Axial and radial trajectories of a 60 µ droplet starting from the
point of injection [22]. . . . . . . . . . . . . . . . . . . . . . . . . . 74
x
2.13 Axial and tangential velocities of the droplet as it travels the length
of the spray-drying chamber [22]. . . . . . . . . . . . . . . . . . . . 74
2.14 Polymer concentration at the surface of the droplet for the assumed
droplet distribution [22]. . . . . . . . . . . . . . . . . . . . . . . . . 75
2.15 The size distribution at the inlet (top) and the outlet (bottom) of
the spray-drying chamber [22]. . . . . . . . . . . . . . . . . . . . . . 75
2.16 Temperatures of a representative droplet and the air stream as they
pass through the spray-drying chamber [22]. . . . . . . . . . . . . . 76
2.17 Effect of heat loss on air temperature. ?: 50% increase from the
nominal, ◦: 50% decrease from the nominal, 4: nominal (20% heat
loss) [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.18 Effect of feed flowrate on the evaporation rate. Solid line: nominal;
dashed line: 15% increase; dotted line: 15% decrease [22]. . . . . . . 77
2.19 Effect of air temperature on the evaporation rate. Solid line: nom-
inal; dashed line: 15% increase; dotted line: 15% decrease [22]. . . . 77
2.20 Effect of collisions on the final particle size distribution [22] . . . . . 78
2.21 Effect of change in mean droplet size for a uniform inlet distribu-
tion, Uniform droplet mean size of 60 microns (solid line), Uniform
droplet mean size of 80 microns (dashed line) . . . . . . . . . . . . 78
2.22 Effect of changes in inlet droplet size distribution, ◦: Uniform
droplet size of 60 microns, ?: Log-normal (std. dev. = 30 microns)
[22] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.23 Comparison of the air temperature profile inside the chamber be-
tween the experimental data: ?, /, . and 4 and the model predic-
tions (◦)[22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.1 Classification of different optimization problems. . . . . . . . . . . . 92
4.1 Sequential design and control strategy. . . . . . . . . . . . . . . . . 119
4.2 Steps in the simultaneous design and control approach of [52]. . . . 128
4.3 Nested Optimization. . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.4 Schematic of a jacketed continuous stirred-tank reactor and a flash
that is used to produce product B from reactant A. . . . . . . . . . 135
xi
5.1 Comparison of the reconstructed temperature data using 2 principal
components with temperature measurements at the inlet (T1), the
outlet(T7) and the center (T4) of the spray dryer. . . . . . . . . . . 153
5.2 Predictions of residual moisture content. Comparison of the PCR
model predictions to data. Solid line: Full model. Dashed line:
PCR predictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.3 Predictions of final particle size (diameter). Comparison of the
PCR model predictions to data. Solid line: Full model. Dashed
line: PCR predictions. . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.4 A schematic of the inferential model in a feedback control strategy. 155
5.5 Closed-loop performance of the residual moisture content to a 10%
decrease in polymer feed concentration. Dashed line: desired setpoint.158
5.6 Closed-loop performance of the mean particle diameter to a 10%
decrease in polymer feed concentration. Dashed line: desired setpoint.158
5.7 Air flowrate response to a 10% decrease in polymer feed concentration.159
5.8 Inlet droplet mean diameter to a 10% decrease in polymer feed
concentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.9 Closed-loop performance of the residual moisture content to a 15%
increase in feed flowrate. Dashed line: desired setpoint. . . . . . . . 161
5.10 Closed-loop performance of the residual moisture content to a 15%
increase in feed flowrate: Dashed line: desired setpoint. . . . . . . . 161
5.11 Air flowrate response to a 15% increase in feed flowrate. . . . . . . 162
5.12 Inlet droplet mean diameter response to a 15% increase in feed
flowrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.1 Comparison between lumped model (dashed line) and full model
(<) for final mean particle size. . . . . . . . . . . . . . . . . . . . . 175
6.2 Comparison between lumped model (dashed line) and full model
(<) for residual moisture content. . . . . . . . . . . . . . . . . . . . 175
6.3 Optimal trajectory for residual moisture content (top) along with
trajectory for air flowrate (bottom). . . . . . . . . . . . . . . . . . . 176
6.4 Optimal trajectory for residual moisture content (top) along with
trajectory for air flowrate (bottom) for the bi-level strategy. . . . . 177
6.5 Effect of changes in the ratio wp/wc . . . . . . . . . . . . . . . . . . 178
xii
LIST OF TABLES
2.1 Dimensionless variables. . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2 Model parameter values. . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 Inlet droplet distribution . . . . . . . . . . . . . . . . . . . . . . . . 61
2.4 Nominal operating conditions . . . . . . . . . . . . . . . . . . . . . 61
2.5 Spray dryer design values . . . . . . . . . . . . . . . . . . . . . . . . 61
2.6 Percentage breakage predicted by the population-balance model . . 66
4.1 Reactor and flash variable definitions . . . . . . . . . . . . . . . . . 138
4.2 Design optimization results: sequential . . . . . . . . . . . . . . . . 141
4.3 Dynamic optimization results: sequential . . . . . . . . . . . . . . . 141
4.4 Design optimization results: bi-level, leader . . . . . . . . . . . . . . 142
4.5 Dynamic optimization results: bi-linear, follower . . . . . . . . . . . 142
4.6 Comparison of sequential and bi-level optimization . . . . . . . . . 142
5.1 Nominal Operating Conditions . . . . . . . . . . . . . . . . . . . . . 151
5.2 Eigenvalues of the covariance matrix of the X. . . . . . . . . . . . . 152
5.3 The PI controller parameters. . . . . . . . . . . . . . . . . . . . . . 157
6.1 Sequential design optimization of the spray dryer . . . . . . . . . . 168
6.2 Sequential optimal control of the spray dryer . . . . . . . . . . . . . 169
6.3 Bi-level design optimization of the spray dryer: Leader . . . . . . . 171
6.4 Bi-level optimal control of the spray dryer: Follower . . . . . . . . . 171
6.5 Comparison of bi-level and sequential optimization strategy (design) 172
6.6 Comparison of bi-level and sequential optimization strategy (opti-
mal control) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
6.7 Comparison of bi-level and sequential optimization strategy (opti-
mal control) for different constraints . . . . . . . . . . . . . . . . . 173
xiii
CHAPTER 1
INTRODUCTION
1.1 Spray drying
Spray drying is the process of contacting an atomized stream to be dried with
a gas stream that is at a higher temperature than the liquid stream. The higher
temperature of the gas stream causes evaporation of the liquid from the droplets,
forming particles. Spray drying has been used extensively in the food industry for
example the manufacture of milk powder; the pharmaceutical industry to form
powders for pelletization [1, 2]; and the agricultural industry to produce different
granular materials, to name a few examples. Recently, the use of this process to
manufacture hollow, micron and sub-micron particles has also been demonstrated
[3, 4]. In spite of such a variety of applications, the mechanisms of spray drying
to form particles is not completely understood. One of the reasons for this is
that the spray drying process, characterized by rapid and simultaneous heat and
mass transfer between the droplets of the feed solution and the heating gas, is
complex to describe in a mathematical model as many of the model parameters
are not readily measurable [5, 6, 7]. Such difficulties also limit the regulation of
the spray dryer operation to tailor the end point properties (mechanical, thermal,
size, density, etc. ) of the particles [8].
Spray drying chambers typically are vertical vessels with a cylindrical cross-
section and a conical bottom [8]. The size of the cylindrical and conical sections
1
depends on the application needs. Different spray dryer designs can be obtained by
passing the heating gas either co-currently or counter-currently to the feed solution
[9], by a different choice of atomizer. Mujumdar et al. [10] have investigated
horizontal spray dryers using computational fluid dynamic (CFD) simulations.
1.2 Design of spray dryers
Chemical processes are concerned with using known chemistry, kinetics, and
thermodynamics to manufacture different products based on specified require-
ments. Process design for chemical processes involves such activities as selecting
different unit operations, generating the process flowsheet, and optimizing the
flowsheet to improve the economics.
In this work, the emphasis will be on optimization of the process design rather
than flowsheet synthesis. The flowsheet may be optimized for a particular objec-
tive function, such as profit (to be maximized) or loss (to be minimized). The
process optimization problem may consist of continuous variables such as equip-
ment sizes, flowrates, and compositions, to name a few and also binary variables
that denote the presence or absence of a particular unit operation. The constraints
are the physical constraints on the process variables (inequality constraints) and
the process models (equality constraints). It should be noted that the process de-
sign task is carried out at steady state to achieve the nominal designed conditions.
The concept of dynamics associated with startups and operability is addressed by
the process control specialist but the selection and placement of actuators and
2
sensors, two devices used to develop an effective control strategy, are selected not
by the process control specialist but rather by the process designer.
A number of techniques are available for flowsheet synthesis. The hierarchical
design proposed by Douglas is widely followed though alternative methods also
are used [11, 12]. Mann and Hoo [13] present a variant of the Douglas approach.
In their work, they place an emphasis on the design of the reactor system and
classify the material recycle and energy integration as economic drivers rather
than requirements of the design hierarchy. Other noteworthy flowsheet synthesis
techniques are the penalty function approach of Diwekar et al. [14] and the two-
level stochastic programming approach proposed by Pistikopoulos and Acevedo
[15]. Daichendt and Grossmann [16] propose a decomposition approach within a
mathematical mixed integer non-linear programming (MINLP) framework.
The design of spray dryers usually is based on knowledge obtained through pilot
plant tests or prior experience [8]. The choice of spray dryer configuration – co-
current, counter-current or mixed-flow, is decided based on the desired qualities of
the final product [8]. The residence time and the amount of evaporation required
are the two most important factors to be considered in the spray dryer design.
These two factors are used to determine equipment size and the heat duty. The
operating conditions can be determined by knowing the throughput required, the
residence time, and the heat duty.
The use of theoretical methods for spray dryer design has been limited [17].
One of the reasons for this limitation has been the difficulty in predicting the
3
flow patterns of the heating gas. Another difficulty has been the estimation of
proper drying rates. The advent of commercially available CFD tools has re-
duced the difficulty in determining the flow patterns; and drying rates can be
calculated using suitable rate-based models. Sometimes, an experimentally deter-
mined drying curve or heating gas flow patterns is used to design the spray dryer
[1, 5, 17, 18, 19, 20, 21].
In this work, a rate-based model approach will be used in conjunction with
empirical relationships for the heating gas flow pattern to determine the design
parameters for the spray dryer when the final product is the manufacture of hollow
particles [22].
1.3 Modeling the spray drying process
Developing a mathematical model of a spray drying process is a difficult task,
for instance, modeling the rapid heat and mass transfer that occurs between the
droplet phase and the liquid phase; the moving boundary of the droplet, and the
presence of multiphase flows. Other aspects of the spray drying process include
accounting for the angle of entrance of the heating gas into the spray dryer, the
spray dryer geometry, and the flow pattern.
In general, four approaches can be used to design a spray drying process. The
following discussion is taken from the work of Oakley [5].
• Mass and energy balance models. These models are constructed based on
steady state energy and mass balances of the spray drying chamber and are
4
useful to investigate the feasibility of a particular separation [23].
• Equilibrium models. These models incorporate equilibrium relationships
on the amount of solvent in the particle into the energy and mass balance
models. Thus, the exit concentration of the solvent can be predicted. The
advantage of equilibrium models is that they can be used to predict the exit
concentration of the solvent and the effect of different operating conditions.
The main limitation of these models is that the equilibrium relationships
(e.g., desorption isotherm) have to be determined experimentally [5].
• Rate-based models. This type of model is dynamic, thus, it describes the
rate at which the solvent is removed from the droplets as the particles travel
through the spray drying chamber. These models also track the trajectory of
the droplets in the chamber and can be used to predict residence times [22].
The final particle size for a given initial droplet size also can be determined
using these models. A limitation of these models is knowledge of the heating
gas flow pattern and the injection pattern. Katta and Gauvin [18] and
Palencia [24] based their studies on rate-based models.
• Particle-source-in-cell models. Crowe et al. [25] pioneered the numerical ap-
proach of CFD to solve first-principles models that accounted for both the
droplet and heating gas flow patterns. Crowe et al. introduced a particle-
source-in-cell (PSI-Cell) model. The dispersed phase (the droplets) is as-
sumed to be a source of mass, energy, and momentum in a grid of the
5
heating gas phase. The main advantage of the PSI-Cell model is that they
are not limited by the spray chamber geometry. Computational fluid dy-
namic (CFD) techniques are used to solve PSI models. The disadvantages
of the CFD numerical approach include: long calculation times, numerous
model parameter values that may have no physical counterparts [6], and the
generation of a very detailed grid to arrive at a solution.
Rate-based and CFD approaches represent the particle motion using a La-
grangian technique. Recently, population balance methods also have been used to
track the droplet distribution as the droplets travel through the chamber [1, 22].
Christofides et al. have modeled a spray pyrolysis process using the population
balance approach [26]. Bertling et al. have used population balance models to
describe agglomerated particles in a spray drying process [27].
1.4 Regulation of spray dryers
Before discussing the regulation of spray dryers, a brief review of some of the
basic concepts of process control is presented.
1.4.1 Brief literature review on process control
Even though a process is designed to operate at the nominal operating con-
ditions, it will deviate from the designed steady state due to disturbances or to
pre-planned changes to the product quality and quantity. Thus, the dynamic as-
pects of the process operation have to be considered. It is the task of the control
6
system to ensure that the process is operating at the desired conditions in spite of
the transient nature of the process. Regulation is achieved by manipulating a set
of independent variables to maintain the controlled variables at the desired values.
The control action can be found from closed-loop information, that is, informa-
tion from the process is applied to determine the magnitude and direction of the
control action. The control action also may be found from open-loop information
where the size and direction of the manipulated variables are determined based
only on a model of the process or said another way disregarding any current pro-
cess information. Open-loop optimal control trajectory is generally found using
dynamic optimization [28].
Closed-loop control strategies can be further classified based on the control law
that is used. A control law is the rule that determines how the control action is
obtained. The most widely used control law is the proportional-integral-derivative
(PID) control law. Mathematically, the PID control law is given by [29],
p(t) = p+Kc
[ε(t) +
1
τI
∫ε(t)dt+ τD
dε
dt
](1.1)
where p(t) is the controller output at time t, Kc is the controller gain, ε(t) is
the error between the set-point (desired value of the controlled variable) and the
current value of the controlled variable, τI is the integral time or reset time, τD is
the derivative time and p is the controller bias.
In the case of digital control, the PID law can be expressed in two forms, the
7
position form and the velocity form [29],
Position: pk = p+Kc
[εk + ∆t
τI
k∑1
εj +τD∆t
(εk − εk−1)
]Velocity: ∆pk = Kc
[(εk − εk−1) +
∆t
τIεk +
τD∆t
(εk − 2εk−1 + εk−2)
] (1.2)
where ∆t is the sampling time, pk is the controller output at the kth sampling
instant, εk is the error at the kth sampling instant, ∆pk = (pk−pk−1) is the change
in controller output from the (k − 1)th to the kth sampling instant.
The PID control law is based on a single-input single-output (SISO) system.
Figure 1.1 illustrates both SISO (left figure) and MIMO (multiple-input multiple-
output) (right figure) systems. The PID control law has been applied to MIMO
systems with considerable success. Tools such as the relative gain array (RGA)
or the Neiderlinski index are used to select the control-manipulated variable pairs
under the assumption of steady state or small perturbations from steady state
[30, 31].
Figure 1.1: Left panel: Single-input single-output system. Right panel: Multiple-input multiple-output system.
In the last twenty years, modern control laws have been synthesized based on a
8
strategy called model predictive control (MPC). This controller has its roots in the
classical linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG)
theories [32, 28]. The LQG calculates an optimal feedback control law based on
minimizing a quadratic function of the error in not driving the controlled variables
to their desired values and of the cost of using large control actions to minimize this
error. The LQG’s formulation differs from the LQR in that the former is based on
statistical assumptions of the measurement and process noise. Both the LQR and
LQG are solved assuming an infinite solution horizon and using only estimates of
the outputs as feedback information. In contrast, the MPC formulation requires
that the optimal control actions be found at each sampling instant using the
current measurement of the process, and finite control and prediction horizons. A
recent review of MPC can be found in [33] and excellent textbooks on the subject
include Brosilow and Joseph [34] and Camacho and Bordens [35].
1.4.2 Literature review on spray dryer regulation
The most important factor in the regulation of the quality of the product of
the spray drying process producing non-hollow particles is the amount of solvent
in the product. For hollow particles, both the amount of solvent and particle size
are important. On-line measurements of the solvent concentration in the product
and the mean particle size are very difficult to obtain, this makes the regulation
of the product properties especially difficult.
Allen et al. [36] use a cascaded PI controller strategy to regulate the average
9
particle size for a spray dryer with a pneumatic nozzle atomizer, using on-line
measurements of the particle size. The manipulated variable chosen was the nozzle
air flowrate. Masters [8] discusses three different control schemes and has provided
some guidelines for the design of a control system for a general spray dryer. The
heating gas flowrate or heating gas temperature is chosen to regulate the amount of
solvent that remains in the particle. The application of model predictive controllers
to spray dryers are not very common. Christofides et al. [26] demonstrated one of
the earliest applications of model-based control on a spray pyrolysis process using
a population balance model of the droplets as the model. The final particle size
distribution is controlled using the wall temperature.
For producing hollow particles, the spray dryer configuration, the amount of
residual solvent in the product, and the final particle size all are important factors
to obtain the desired property known as the tap density1. However, instrumen-
tation to provide accurate and on-line measurements of particle size and residual
solvent concentration are either not available or expensive to purchase and main-
tain. In lieu of actual measurements, these variables (particle size and residual
solvent concentration) are estimated from measurable quantities using inferential
models.
There are several techniques available for estimation, including the well known
Kalman filter [28, 32]. However, the distributed nature of the spray dryer model
precludes the use of state equation-based estimators. Statistical tools such as
1A packing density used to characterize powders. It is calculated as specified by ASTM-B527
10
Principal Component Analysis (PCA), Principal Component Regression (PCR),
and Projection to Latent Squares (aka partial Least Squares) (PLS), multiway-
PLS [37] and others [38] may be used. In [39], it was shown how to use these
methods to develop subspace controllers for systems such as a distillation column.
In the current process, the available measurements are the input polymer solution
conditions,the heating gas conditions, and the temperatures along the length of
the spray chamber. The temperature data are highly collinear (not independent)
thus, the use of a common inferential model technique such as multiple linear
regression (MLR) should not be used. In this work, a PCR approach to develop a
model to estimate the product quality properties as a function of the heating gas
temperature is employed [40].
1.5 Brief literature review of simultaneous design and control
Zeigler and Nichols stated that control applied to any plant cannot be inde-
pendent of the plant design [41]. In spite of this valuable insight, there has been
neglect in recognizing the coupling between control design and the process de-
sign. Typically, control engineers develop a control structure after the process
flowsheet (selection of the unit operations and their connectivity) is completed.
The drawback of such a serial approach is that the control synthesis choices are
limited by the design decisions. Rising costs and more stringent environmental
and operational constraints have motivated more complicated designs that invari-
ably are more challenging from a control point of view. In addition, for industries
11
such as fine chemicals, drugs, etc., a greater flexibility is demanded to address a
wide slate of different products. As a consequence, processes are now required to
produce multiple product types whereas before, (up to mid 1980s) processes were
operated to produce a single product type. Recently, the coupling between design
and control has gained renewed importance in part due to drivers such as energy
efficiency, environmental impact and maintenance costs (less mechanical parts).
Since the decisions made to achieve the design affects the synthesis and ef-
fectiveness of the control structure, incorporating control requirements into the
decisions made during process design exercise will result in a system (process and
control structure) that is operable at the designed conditions in comparison to
a system that is economically superior but cannot be regulated satisfactorily at
the designed conditions. A comprehensive study of this coupling between design
and control has been minimally addressed in the chemical engineering literature.
Generally, some open-loop controllability measures such as the relative gain array,
minimum singular value, etc. have been used to include control analysis in the
design exercise [42], but this practice is limited. Floudas has applied open-loop
controllability, singular value analysis (SVD) to quantify coupling [43]. Ulsoy and
coworkers [44, 45, 46] derive a coupling term from optimality conditions.
Vasbinder et al. provide a decision based approach to synthesizing a control
structure for a given design flowsheet based on a modified Analytical Hierarchical
Process (mAHP) [47]. Grossmann and coworkers have applied decomposition ap-
proaches to address design and operations simultaneously [48, 49]. Pistikopoulos
12
and coworkers have extended the approach of Grossmann and coworkers to include
design under uncertainty for simultaneous design and control [50, 51, 52, 53].
The objective of the research by Grossmann and coworkers is to design flexible
processes [48, 49]. They use a decomposition strategy to solve a simultaneous
design and operation optimization problem. The decomposition is based on sepa-
rating the design stage from the operating stage. That is, during the operation of
the process, changes in the process conditions may result in changes in the ther-
modynamic or kinetic parameters. Thus, a process that is optimal at the original
design conditions may no longer remain optimal at the new conditions. It is as-
sumed that the range over which the parameters may vary is known and a cost
function is minimized over that range to find the optimal design. The contribution
of this method is that it introduces a technique of incorporating operational con-
siderations into the design optimization. It however, does not explicitly consider
the control aspects of the process. Also, the range over which the parameters
vary is fixed and the solution to the optimization may not be valid beyond this
range. Pistikopoulos provides a stochastic optimization approach to incorporate
parameter variations in the process design [54].
Pistikopoulos and coworkers extend the decomposition concept of Grossmann
and coworkers to include control aspects. In their approach they explicitly in-
clude the cost of incorporating the control structure in the objective function of
the optimization problem. Pistikopoulos and coworkers also introduce a feasibility
analysis step in the design process. The feasibility step is introduced to ensure
13
that the inequality constraints on the optimization problem are satisfied. Another
modification of the work of Grossmann is in the consideration of parameter un-
certainty. Rather than evaluate the parameters over the entire range, only those
changes that are more likely to occur, are considered. This reduces the compu-
tational burden. An important limitation of the approach of Pistikopoulos and
coworkers is that the choice of the control structure has to be made a priori.
Floudas and coworkers [42, 43] use open-loop controllability measures such as
the minimum condition number or relative gain array, within the decomposition
framework proposed by Grossmann [48, 49], to account for control effects.
Ulsoy and coworkers [44, 45] use a decomposition approach to the simultaneous
problem of design and control by finding the numerical solution of both problems as
two nested dependent problems. A quantitative measure of the degree of coupling
between design and control based on optimality conditions also was provided.
Alyaquot et al. [55] extended this work to multi-disciplinary optimization (MDO)
problems. In an MDO problem, complex processes are decomposed into smaller,
more tractable problems that are optimized individually. These optimized units
are then combined through a coordination of sub-systems. Coordination is used
to imply that the design variables in one sub-system are known to other sub-
systems. Not surprisingly some sub-systems naturally will be more coupled than
others. The work by Alyaquot et al. [55] determines the extent of the coupling
by studying the sensitivity of different sub-system output variables to each other.
Output variables are the variables that transfer information from one sub-system
14
to another. Based on these sensitivity considerations, a decision can be made to
remove some sub-systems from the optimization formulation. The optimization
problem is solved without the removed sub-systems and the solution is compared
to the solution where all sub-systems are present. It is shown that the system
objective function is not affected by removing these weakly coupled systems while
making the computational task easier.
The objective of this work is the process design and control of a spray dryer
that produces hollow micron and sub-micron polymeric particles. The objective
is achieved in two ways. The first is the serial approach of developing the design
followed by the control structure to regulate the product quality properties at the
desired set-points. The second is to employ the bi-level optimization strategy of
Ulsoy and coworkers to the design and control of the spray dryer. A comparison
between the two approaches is provided to delineate the advantages and disadvan-
tages of both approaches.
1.6 Organization
The dissertation is organized as follows. Chapter 2 introduces the particu-
lar problem – that of producing hollow micron and sub-micron particles. First-
principles models that describe the mechanisms that produces a single particle
and a distribution of particles are developed, numerical solved, and validated us-
ing data from a laboratory scale experimental system. Then, the design of the
spray dryer itself is developed using the rate-based approach of Katta and Gau-
15
vin [18]. Chapter 3 reviews optimization theory by presenting the different types
of optimization problems that can be formulated and the techniques used to solve
these problems. Chapter 4, introduces the difficult problem of simultaneous de-
sign and control with the notion that the optimal design must admit an optimal
controller design from the outset. Chapter 5 design a control strategy to regulate
the spray dryer following the serial approach to design and control. The control
strategy employs an inferential control scheme to compensate for the lack of on-
line realtime measurements of the product quality variables. Chapter 6 presents
the bi-level approach to design and control applied to the spray dryer. A compar-
ison between the serial and bi-level approach is presented and analyzed. Lastly,
Chapter 7 summarizes the contributions of this work and provides directions for
future work.
16
CHAPTER 2
DESIGN OF A SPRAY DRYER
Spray-drying technology is used in a wide variety of processes ranging from
manufacture of food products to pharmaceuticals. Most recently, spray-drying
technology has been investigated to produce hollow micro-particles [3]. This chap-
ter presents an approach to design a spray-drying chamber using a rate-based de-
scription of the drying process combined with a droplet size distribution model.
The primary spray-drying chamber design criterion is the moisture content of the
final particle. The prediction of the final particle properties are compared to ex-
perimental data obtained from a laboratory spray-drying unit. The results show
that the final spray-drying chamber design is sensitive to the liquid feed flowrate,
the inlet drying gas temperature, and heat loss. Most of the material presented in
this chapter can be found in published articles by Shabde et. al [3, 22].
2.1 Background
Spray-drying technology is used extensively in many industries such as the
food industry; for example, to dry a feed solution in order to generate particular
products from the solution [5, 56]. Spray-drying technology also has been used to
manufacture hollow or solid micro-particles for different applications (e.g., light-
weight composites) [3, 57]. Hollow spherical particles have a number of potential
applications, but one of the most important application is their use as fillers in
17
syntactic foams. Hollow particles provide a means to produce light composite
materials (foams) with desirable mechanical, thermal, and electrical properties
that can be easily molded and machined due to the small size of the particles.
The properties of the hollow particles affect the properties of the syntactic
foams – most notably, the density of the particles and their mechanical properties.
Both of these properties depend mainly on three factors [3]:
• The type of raw material used.
• The diameter of the particles.
• The thickness of the skin.
For a given particle diameter, the thinner the particle skin, the smaller the
particle density. Mechanical properties depend on the rigidity and thickness of the
skin. It is essential to produce reliably and economically micro-hollow polymeric
particles with desirable properties. To date, there are three main methods to
produce hollow micro-particles [57],
1. Spray-drying based process
2. Sacrificial core method
3. Emulsion and phase separation techniques
The most widely used process is based on spray-drying technology. In this
process, a polymer solution (the material of the particle) is atomized in a spray-
drying chamber. The solution also contains a latent gas (blowing agent), which,
18
as described below, leads to the formation of the hollow core. As the droplets are
exposed to hot air, the solvent evaporates forming a layer of higher concentration
of polymer at the outer boundary of the droplet. As the droplet continues to
shrink an impermeable layer is formed.
Upon completion of the solvent evaporation, the temperature of the droplet
(now a particle) rises, resulting in the decomposition of the blowing agent trapped
in the center of the particle; thus forming the hollow core. The sacrificial core
method uses cores that are coated with the material of interest. These coated
cores are exposed to substances that cause the cores to dissolve leaving behind
intact hollow micro-spheres. Whilst such methods provide a greater control over
the geometry of the hollow micro-particles, care needs to be taken to ensure proper
coating. Also, these methods are not easily adapted for mass production.
The various phase separation techniques utilize surface tension forces to form
emulsions that consist of spherical globules of the material of interest [57]. A
good review of the various aspects related to hollow micro-spheres can be found
in Bertling [27]. A review of different manufacturing methods along with some
discussion on characterization of the particles and modeling of the manufacturing
process also are provided.
Here, the development of a physics-based model for the phenomena associated
with an individual particle is discussed. This model can be used to predict the
onset of skin formation on the surface of a particle.
The literature does not provide a fundamental model that describes the pro-
19
duction of hollow micro-particles. A large volume of work has been carried out
on the modeling of conventional spray drying operations (formation of non-hollow
particles) [1, 8], mainly as they apply in the food industry. However, these models
do not apply to the production of hollow particles because they do not account
for the formation of an impermeable skin and generation of gas in the core. When
considering the production of hollow particles, it is important that the tempera-
tures and concentrations of the individual droplet be determined because spatial
variations in the temperatures and concentrations of the droplets are necessary
to allow the formation of the outer skin. Thus, the time to skin formation is an
important factor in deciding the residence time of the droplets inside the drying
chamber, which in turn affects the chamber dimensions.
Farid provides a fairly rigorous model of a single droplet, however concentration
gradients within the droplet are omitted [58]. A similar model for a slurry droplet
can be found in [59].
In the field of combustion, Sirignano and coworkers investigate different situ-
ations during the burning of individual fuel droplets [60]. They apply a number
of modeling and numerical analysis techniques to describe the burning of a single
fuel droplet and arrays of droplets.
In a spray-drying operation, there are three main phenomena:
1. Atomization of the liquid feed
2. Drying of the droplets once they are formed
20
3. Motion of the droplet in the spray-drying unit.
The design of spray dryers has remained mainly empirical for several reasons,
but primary among these is a fundamental understanding of the atomization pro-
cess. In general, experimental correlations are relied upon to describe the atomized
spray. Masters provides a procedure for the design of spray-drying chambers based
mainly on experiential knowledge of the process [8]. Oakley [5] and Kieviet [61]
have attempted to design spray dryers using fundamental principles. The mod-
els used to describe the spray drying process may contain material and energy
balances between the two phases – the droplet phase and the bulk gas phase; or
material and energy balances between these two phases and a description of the
equilibrium between the dispersed and continuous phases; or rate-based descrip-
tions, which do not assume the existence of an equilibrium.
Mass and energy balances are used during a preliminary design because their
solutions give the feed rates and temperatures to achieve the desired quality of
drying. Rate-based models are typically more complex and are further classified
depending on whether the flow of the gas phase is modeled mechanistically or with
empirical correlations. The mechanistic rate-based models to model the heating
gas flow patterns are rigorous but usually require sophisticated computational fluid
dynamics (CFD) techniques to solve these models [6, 25]. The empirical rate-based
models of the gas phase are easier to solve in general, but are dependent on the
geometry of the chamber.
Katta and Gauvin [18] showed that rate-based models that contain empirical
21
correlations to model the flow of the gas phase can represent the particle phase ac-
curately . However, their work neither investigated hollow particles nor combined
the entire spray (macroscale) with the droplet (microscale) to obtain the design
parameters. Palencia it et al. [24] present a mechanistic approach to spray-dryer
modeling by representing the spray dryer to be a cascade of mixing tanks. In the
present work, a droplet size distribution is assumed and the motion of a repre-
sentative droplet is followed. Tracking the droplet trajectory is useful since this
information affects the chamber dimensions.
Modeling the heat- and mass-transfer rates between the gas and liquid phases,
and the particle motion are critical to the design of the spray dryer. In the case of
solid particles, it is reasonable to assume uniform temperatures and concentrations
[5, 18]. For example, Farid [58] provides a constant temperature model for crust
formation on the surface of a single droplet. Similar models have been proposed
by Cheong et al. [59] for drying slurry droplets.
When considering the production of hollow particles, it is important that the
temperatures and concentrations of the individual droplet be determined because
spatial variations in the temperatures and concentrations of the droplets are nec-
essary to allow the formation of the outer skin. Thus, the time to skin formation is
an important factor in deciding the residence time of the droplets inside the drying
chamber, which in turn affects the chamber dimensions. In this work, the single
droplet model of the phenomena occurring on a droplet, developed by Shabde et
al. [3], is used to determine the temperature and concentration gradients within
22
the droplet.
The chapter is organized as follows: Section (2.2) provides a description of
the experimental system. This is followed by a discussion of modeling of a single
droplet in Section(2.3). Section (2.4) describes the phenomena occurring within
the spray dryer and develops a mathematical model for it. The numerical tech-
niques used to solve the models are explained in Section (2.5). The results ob-
tained and a discussion of the results are presented in Section (2.6). Finally, the
contributions of this chapter are summarized in Section(2.7).
2.2 Experimental system
The spray drying system consists of the following:
• A spray drying chamber has a cylinder-on-cone geometry. The atomizer
feed to the top of the vessel and the heating gas feeds to the side wall near
the top. Thermocouples are set along the wall of the vessel to measure the
temperature of the heating gas. The conical section has an opening at the
bottom to remove the spent gas and particles. A cyclone is used to separate
the gas from the particles.
• The atomizer is used to introduce the liquid feed to the spray drying cham-
ber. The atomizer used in this work can be set to produces 10-100 micron
sized droplets.
• Heaters are provided to heat the heating gas before it enters the spray drying
chamber. The heaters are used to generate temperatures as high as 1000oF .
23
• A peristaltic pump is used to meter the feed solution to the atomizer.
• A cyclone is used which separates the product from the spent heating gas.
A schematic of the spray drying system is shown in Figure 2.1.
Figure 2.1: Schematic diagram of the experimental system.
Figure 2.2 shows samples of the hollow micro-particles produced by the exper-
imental system. These hollow micro-particles are spherical in shape with imper-
meable skins. The mean particle diameters are between 10 to 80 microns with a
shell thickness of 2 to 10 microns.
2.3 Modeling a single droplet
As the droplets from the atomizer travel down the spray chamber, they come in
contact with the heating gas. Applying heat and mass balances to a control volume
around an individual droplet leads to first-principles models of the phenomena
affecting the droplets. Due to the evaporation of the solvent, the droplet surface
24
Figure 2.2: Hollow micro-particles made using the experimental system. Leftpanel: thin skin. Right panel: thick skin (Courtesy: S.V. Emets).
regresses. This change in the droplet size has to be accounted for in the description
of the model of the droplet.
2.3.1 Physical phenomena
A number of different phenomena occur as the droplet travels the length of
the spray chamber. Since the heating gas is at a higher temperature than the
droplet, heat is transferred to the droplet. This heat is conductively transferred
from the surface of the droplet toward the interior thus increasing the temperature
inside the droplet. When the surface temperature reaches the boiling temperature
of the solvent, evaporation of solvent begins to occur at the surface. Thus, the
droplet passes through two steps: a heating step where sensible heat is supplied
to the droplet to raise its temperature to the evaporation temperature and an
evaporation step during which evaporation of the solvent occurs. In the latter
step, the heat transferred is used (a) to provide the latent heat to evaporate the
solvent and (b) to provide the sensible heat which is conducted into the core of
25
the droplet.
Solvent vapors are released into the heating gas. There are two driving forces
for the loss of solvent to the gas phase. The major force is due to the heat that
is provided to the droplet, causing evaporation while the smaller force is the mass
transfer due to the difference in the solvent concentration in the droplet or liquid
phase and the gas phase. The entering heating gas is assumed to be carrying a
negligible amount of solvent.
Similarly, the movement of the solvent from the interior of the droplet to the
surface occurs due to diffusion. Thus, the evaporation of the solvent provides a
driving force for the diffusion of the solvent.
As the solvent evaporates, the droplet shrinks. Because of a lower polymer
mobility, the polymer concentration near the surface increases. This phenomenon
is known as skinning [62]. Thus, the polymer concentration gradient at the surface
is steeper as compared to the profile away from the surface (interior to the particle).
A sufficiently high heating rate can evaporate almost all the water leaving only
polymer at the surface that forms the impermeable shell. Thus, for the formation
of a hollow particle, it is necessary that the temperature gradient between the
heating gas and the droplet surface be sufficiently large, otherwise, the end product
will not be a hollow particle[18, 63].
As mentioned earlier, the formation of a hollow particle is divided into two
steps: (a) evaporation of the solvent (shrinking) and (b) decomposition of the
trapped blowing agent. Figure 2.3 represents an illustration of the hypothesized
26
mechanisms and the temperature and concentration profiles during droplet evap-
oration.
The following assumptions are made in the development of the model:
1. Because of the relatively large amount of heating gras, the conditions of the
hot gas (e.g., temperature) do not vary with time. Hence, variations in the
heating gas conditions are omitted from the model.
2. A spherically symmetric field exists in and around the droplet. Thus, internal
circulation is absent. This is reasonable since the droplet size is small.
3. There is negligible relative velocity between the droplet and the air. Hence,
the external heat and mass transfers are approximated by conduction and
diffusion, respectively.
4. There is a negligible amount of solvent in the incoming heating gas.
2.3.2 Model development
In order to develop the mathematical model that describes the physical phe-
nomena, heat and mass balances are taken over a differential shell element, as
shown in the figure 2.4.
The energy balance equation is given by,
∂(ρCpT )
∂t=
1
r2
∂
∂r
(kr2∂T
∂r
)0 ≤ r ≤ R(t), t > 0
27
Temperature
PolymerConcentration
Water Concentration
Vapor
Heat R(t)
Figure 2.3: Temperature and concentration profiles during droplet evaporation [3].
where T is the temperature of the heating gas, k is the thermal conductivity of
the solution, ρ is the density of the solution, Cp is the specific heat capacity of
the solution, and the independent variables are time, t, and radial distance, r. It
is assumed that k is independent of radial position and time. It then follows that
the above equation can be written as,
∂(T )
∂t= α
1
r2
∂
∂r
(r2∂T
∂r
)0 ≤ r ≤ R(t), t > 0 (2.1)
where α = k/ρCp is the thermal diffusivity of the solution.
Assuming that the binary diffusion coefficient of the solvent-polymer solution
does not vary with time and position, the following equation represents the con-
28
Figure 2.4: Shell balances inside the droplet.
centration of the polymer inside the droplet,
∂(cP )
∂t= D 1
r2
∂
∂r
(r2∂cP∂r
)0 ≤ r ≤ R(t) (2.2)
A species balance of the solvent, which in this case is water, gives,
∂(cW )
∂t= D 1
r2
∂
∂r
(r2∂cW
∂r
)0 ≤ r ≤ R(t) (2.3)
The notation can be found in the nomenclature section at the end of the chapter.
2.3.3 Initial and boundary conditions
It is assumed that the solution is well mixed and therefore, the concentration
inside the droplet is uniform. Similarly it is assumed that the droplet is initially
at a uniform temperature, equal to the temperature of the liquid feed. Thus, the
29
initial conditions for Equations (2.1) to (2.3) are,
T (r, 0) = T0 0 ≤ r ≤ R
cP (r, 0) = cP,0 0 ≤ r ≤ R
cW (r, 0) = cW,0 0 ≤ r ≤ R
(2.4)
The boundary conditions at the center of the droplet arise due to the symmetry
of the droplet and they are unchanged during both heating and evaporations steps,
∂T
∂r(0, t) = 0 t ≥ 0
∂cP∂r
(0, t) = 0 t ≥ 0
∂cW∂r
(0, t) = 0 t ≥ 0
(2.5)
Since the droplet goes through two steps of heating and evaporation, two sets
of boundary conditions are required at the surface.
2.3.3.1 Heating step
An energy balance at the surface provides the boundary condition during the
heating step,
−k∂T∂z
(R, t) = h(Tair − T |R) T |R < Tsat (2.6)
where h is the external heat transfer coefficient, Tsat is the evaporation temperature
of water, and Tair is the temperature of the heating gas, which in this case is an
30
air stream.
During the heating step there is no flux of either water or polymer at the
surface of the droplet. Thus,
D∂cP∂r
= 0 0 ≤ r ≤ R(0)
D∂cW∂r
= 0 0 ≤ r ≤ R(0)
(2.7)
2.3.3.2 Evaporation step
In the evaporation step, the temperature at the surface remains constant at
the boiling point of the solvent. In the case of the polymer, a mass conservation
on the polymer is used to obtain the boundary condition. Since the polymer is not
transferred out of the droplet, the total amount of polymer in the droplet (MP )
remains constant,
MP =
R(t)∫0
cP r2dr (2.8)
Taking the time derivative of the above equation gives,
∂MP
∂t=
R(t)∫0
∂cP∂t
r2dr + cP (R(t), t)R2(t)dR(t)
dt= 0 (2.9)
where R(t) is the radius of the droplet at time t and dR(t)/dt is the rate at which
the droplet surface changes. Combining Equations (2.8) and (2.9) gives,
D∂cP∂r
∣∣∣∣∣r = R(t)+ cP
dR(t)
dt= 0.
31
The boundary condition for water at the surface is obtained by recognizing that
water losses are proportional to the motion of the surface. Thus, the boundary
conditions at the droplet surface are given by,
T |R(t) = Tsat
D∂cP∂r
∣∣∣∣∣r = R(t)+ cP
dR(t)
dt= 0
D∂cW∂r
∣∣∣∣∣r = R(t)=ρW
λ
(h(T |R(t) − Tair
)− k
∂T
∂r
(2.10)
where λ is the heat of vaporization water and ρW is the density of water.
2.3.3.3 Surface motion
During the heating step, the radius of the droplet does not change. However,
in the evaporation step, the surface and hence the radius changes with time as
water is evaporated. The radius can be determined from the following equations,
Heating step: R(t) = R(0) t ≤ tH
Evaporation step:dR
dt=ρW
λ(h(T |R(t) − Tair)− k
∂T
∂rt > tH
(2.11)
where tH is the time to reach the evaporation temperature of water.
2.3.4 Dimensionless model
To simplify the analysis, it is convenient to reduce the equations and boundary
conditions to a dimensionless set of equations. Table 2.1 lists the dimensionless
32
variables.
The characteristic time constant, t0, is defined by,
t0 ≡R(0)2
D.
The initial radius R(0) is the characteristic length of the droplet; thus, the char-
acteristic time gives an estimate of how much time will be required for diffusion
from the center of the droplet to the surface. Concentrations and temperature are
made dimensionless by using the initial conditions. The non-dimensionalization
leads to different non-dimensional groups such as the Biot number (BiH : ratio
of convective heat transfer to conductive heat transfer [64]) for heat transfer, the
Lewis number (Le: ratio of the thermal diffusivity to the diffusion co-efficient
[65]), and others.
Using the dimensionless variables in Table 2.1, the differential equations in
(2.1) to (2.3) and initial conditions in(2.4) become,
∂(cW )
∂τ=
1
x2
∂
∂x(x2∂cW
∂x) cW (x, 0) = cW,0
∂(cP )
∂τ=
1
x2
∂
∂x(x2∂cP
∂x) cP (x, 0) = cP,0
∂θ
∂τ= Le
1
x2
∂
∂x(x2∂θ
∂r) θ(x, 0) = θ0
(2.12)
The dimensionless boundary conditions are:
33
Table 2.1: Dimensionless variables.
Variable Symbol Definition
Dimensionless time τt
t0
Characteristic time t0R(0)2
D
Dimensionless radial position x(τ)r(t)
R(0)
Dimensionless droplet radius X(τ)R(t)
R(0)
Dimensionless water concentration cW (x, τ)cW (x, t)
ρ0
Dimensionless polymer concentration cP (x, τ)cP (x, t)
ρ0
Dimensionless temperature θ(x, τ)T (x, t)− T (x, 0)
Tair(0)− T (x, 0)
Dimensionless Heat of vaporization Hevapλ
Cp(Tair(0)− T (x, 0))
Lewis number Le α/D
Nusselt number Nuhdp
kair
Sherwood number Shkx dp
cWDair
Biot number BiHh dp
k
Reynolds number Redp v ρair
µair
Schmidt number Scµair
ρair Dair
Prandtl number PrCp µ
kair
34
(i) Center of the droplet,
x = 0
∂cW∂x
(0, τ) = 0, τ ≥ 0
∂cP∂x
(0, τ) = 0, τ ≥ 0
∂θ
∂x(0, τ) = 0, τ ≥ 0
(2.13)
(ii) Surface of the droplet during the heating step,
x = X(τ) =R(t)
R0
∂θ
∂x= BiH(θair − θ(R(τ)))
∂cP∂x
= 0
∂cW∂x
= 0
(2.14)
(iii) Surface of the droplet during the evaporation step,
x = X(τ) =R(t)
R(0)
θ|X(τ) = θsat
∂cP∂x
= −cPdX
dτ
∂cW∂x
= (1− cW )dX
dτ
(2.15)
(iv) Surface motion during the heating step,
X(τ) = 1.0 (2.16)
35
(v) Surface motion during the evaporation step,
dX
dτ= LeHevap
(BiH (θair − θ (X(τ)))− ∂θ
∂x
)(2.17)
2.3.5 Model parameters
The parameters that are necessary to solve the above system of equations
include the mass diffusivity (binary diffusion coefficient) of the water-polymer
mixture, the thermal diffusivity of the mixture, and the external heat (h) and
mass (kx) transfer coefficients. The external heat and mass transfer coefficients
are obtained from the following expressions [64],
Nu = 2 + 0.6Re1/3Pr2/3
Sh = 2 + 0.6Re1/3Sc2/3
(2.18)
The Nusselt number (ratio of convective to conductive heat transfer), which is a
measure of heat transfer occurring at the droplet surface, is defined by [64],
Nu ≡ hdp
kair
The Sherwood number (ratio of total and molecular mass transfer) is defined by
Sh ≡ kxdp
cWDair
36
Since the diameter of the droplet is small, the relative velocity between the droplet
and the gas (air) is small. It then follows that the Re is small (∼ 10−2). Thus,
the values of h and kx can be found by,
h ∼ 2kair
dp
kx ∼ 2cWDair
dp
(2.19)
The thermal conductivity of water is estimated from the following expression
[66],
k = −3.8538× 10−1 + 5.25× 10−3(T )− 6.369× 10−6(T )2
where, T is the temperature. and k is the thermal conductivity in Wm−1K−1.
The binary diffusivity coefficient of water and polymer is estimated using Wilke
and Chang correlation [65]. The parameters, for the experimental conditions used
here, are given in Table 2.2.
Table 2.2: Model parameter values.
Parameter Symbol Value
Binary diffusion coefficient D 1.25× 10−9 m2s−1
Mass transfer coefficient kx 1.265 m s−1
Heat transfer coefficient h 1640 Wm−2K−1
Thermal diffusivity of water α 1.824×10−7 m2s−1
Thermal conductivity of air kair 0.0328 Wm−1K−1
Diffusion coefficient of air-water Dair 0.375 ×10−4 m2s−1
Reynolds number Re 1.65 ×10−2
Schmidt number Sc 2.5×10−2
Prandtl number Pr 1.7
37
2.4 Spray dryer modeling
In a typical spray-drying operation, the feed solution is sprayed into a ver-
tical chamber using a suitable atomizer configuration. The common atomizer
configurations are centrifugal pressure nozzle, pneumatic nozzle, rotating disc and
ultrasonic atomizers [8, 18].
Heat is supplied to the liquid mixture by passing a hot gas (e.g., air) either co-
current or counter-current to the liquid. The left panel in Figure 2.5 is a simplified
schematic of a spray dryer. The ensuing flow pattern is a function of both the
chamber geometry and the introduction of the gas. The contact between the
droplets and the hot gas provides direct heating, as a result the solvent in the
solution evaporates and the droplets are converted into solid particles.
Figure 2.5: Left: Simple schematic of a spray dryer with co-current liquid feedand drying gas. Center: Schematic of the liquid spray inside the dryer. Right:Zones in a spray shower [22].
In the manufacture of hollow particles, an additive, originally a part of the
liquid solution, decomposes due to the temperature difference and imparts the
property of hollowness to the final particle.
38
2.4.1 Spray dryer design
The design of the spray-drying chamber is dependent on the desired final prod-
uct properties. The most common product specifications relate to the size of the
final particle and the amount of solvent in the final product. The amount of gas
used to dry the droplet affects the rate of solvent evaporation, whereas the choice
of the atomizer influences, to a large degree, the particle size distribution.
The following design approach is adapted from the work of Gauvin and Katta
[18]. Given the feed and product specifications, the size of the spray-drying cham-
ber, the heating requirements, and the flow rate of the drying gas can be deter-
mined iteratively. The design calculations are performed in three steps:
• Step one – involves calculation of the liquid droplet trajectories;
• Step two – determines the drying gas flow pattern; and
• Step three uses the results of the previous steps to obtain the heat- and
mass-transfer rates between the liquid droplet and the gas, the heating re-
quirements, and the size of the spray-drying chamber.
The size of the spray chamber is determined by the trajectory of the largest
droplet. The design criterion is that the largest droplet must contain less than
10% moisture before the droplet contacts the spray chamber wall. Knowing the
distance (both radial and axial) travelled by the droplet before the 10% moisture
limit is reached provides the radius and length of the spray-drying chamber.
The droplet’s trajectories are determined using a Lagrangian formulation [61].
39
It has been shown that the Eulerian formulation results in numerical errors when
tracking particulates [60]. Additionally, the computational burden is reduced in
the Lagrangian formulation because the trajectory of an individual droplet rather
than the trajectories of the entire ensemble of droplets are followed.
Since the gas is introduced tangentially, the geometry of the flow pattern within
the spray-drying chamber has a spiral shape as shown schematically in the right
panel of Figure 2.5. Thus, it is reasonable to assume that the droplets are traveling
in a centrifugal field [8].
Following Masters [8], tangential, radial, and axial components of the droplet
velocity are considered. Using the notation in [18], the momentum balances for a
representative droplet are given by,
dVt
dt= −VtVr
r− 3CDρgVf (Vt − Vgt)
4diρ
dVr
dt= −V
2t
r− 3CDρgVf (Vr − Vgr)
4diρ+FL
m
dVv
dt= g − 3CDρgVf (Vv − Vgv)
4diρ
(2.20)
where
Vf =√
(Vv − Vgv)2 + (Vt − Vgt)
2 + (Vr − Vgr)2
is magnitude of the relative velocity between the particle and the gas phase; Vt, Vr,
and Vv are the tangential, radial, and axial velocities of the droplet, respectively;
and Vg represents the drying gas velocity. The reader is referred to the nomencla-
ture section for the definition of the variables and parameters. The initial velocities
40
of the droplets are usually determined by the velocity at which the droplets are
ejected from the atomizer and the air velocity.
The variable FL represents the shear lift force on the droplets and is a function
of the gas density and the radius of the droplet,
FL = 20.25ρgd2i (Vgv/K)0.5KVf
K = 1.4Vgv(r/(x tan(θ)))2 = 1.4Vgv(r/r5)2.
(2.21)
The shear lift force is transverse to the direction of flow and thus acts in the radial
direction [67].
In the above equation x is the axial distance from the entrance of the liquid
feed, θ is the half angle of the cone formed by the liquid spray with the drying
chamber’s centerline, and r5 is the width of the spray a distance x from the entrance
of the liquid feed (center panel of Figure 2.5). The half angle is assumed to be 15o
[68]. The initial conditions for Equation (2.20) are the droplet velocities at the
exit of the atomizer. Since the atomizer used in this work releases droplets with
very low energy [68], the initial velocities of the droplets are taken to be zero in
the axial and radial direction while the initial velocity in the tangential direction
is taken to be equal to the inlet air velocity. The air velocity is estimated from
the air flowrate.
The liquid spray, as it travels down the spray-drying chamber can be classified
into two zones, the nozzle zone and the entrainment zone (right panel in Figure
2.5). In the nozzle zone, the spray’s velocity remains influenced by the atomizer
41
while in the entrainment zone, the spray’s velocity is influenced by the drying
gas. In this work, it is assumed that the atomizer creates sprays with very low
velocities. Thus, only changes to the spray’s velocity in the entrainment zone are
assumed non-negligible.
At any vertical distance from the liquid spray’s entrance, for a tangentially
introduced gas in a chamber with a circular cross-section [18], the radial variation
in the tangential velocity of the gas stream is given by,
Vgt = C1(r/Rx)0.5 0 ≤ r ≤ Rx (2.22)
where the radius of the chamber (Rx) is a function of the axial distance, x, as
measured from the entry point of the liquid feed.
The radial variation in the axial velocity of the gas stream is given by,
Vgv = C2(r/Rx)2.5 0 ≤ r ≤ Rx. (2.23)
The coefficients, C1 and C2, in Equations (2.22) and (2.23) are found experimen-
tally. The above relations were obtained in a system very similar in geometry and
gas flow pattern to the one under consideration [18].
To model the heat– and mass– transfer between the droplet phase and the air
phase, the following assumptions are made:
1. The solution is well-mixed and all droplets contain the same amount of
42
solvent and solute.
2. The droplets are exposed to the same amount of heat.
3. The heat lost from the unit to the surroundings is 20% [18].
The heat (q) supplied to the droplets and the amount of solvent lost by the
droplets due only to the mass-transfer between the droplets and the gas are given
by,
dq
dt= kg Nuπ di ni (Tg − Ts)
dm
dt= πDv ρg Sh di ni (cS − c)
(2.24)
where ni is the number of droplets, kg is the thermal conductivity of the gas, Dv is
the solvent-gas binary diffusion coefficient, cS is the solvent concentration at the
surface of the droplet, and c is solvent concentration in the gas surrounding the
droplet.
The Nusselt number, Nu, represents the ratio of heat transferred due to con-
vection and conduction [64]. The Sherwood number, Sh, represents the ratio of
mass transferred by convection and diffusion [64]. Both the Nusselt and Sherwood
numbers can be determined from the Reynolds number, Re, which is a ratio be-
tween the inertial and viscous forces. The following equations are used to calculate
the Nu and Sh numbers [64],
Nu = 2.0 + 0.6(Re)0.6(Pr)0.33
Sh = 2.0 + 0.6(Re)0.6(Sc)0.33.
(2.25)
43
The Schmidt number, Sc, is a ratio between kinematic and diffusive viscosities
and the Prandtl number, Pr, represents a ratio between momentum diffusivity
and thermal diffusivity. It is found that the contribution to the Nusselt and
Sherwood numbers from the Reynolds and Prandtl numbers is negligible, hence
both Nu and Sh are approximately equal to two. This finding implies that the
main mechanisms of mass and heat transport in the reactor are diffusion and
conduction, respectively.
The amount of gas, Wg, required to dry the droplet is found by taking an
overall energy balance over the spray dryer,
WgCs(Tgi−Tgo) = Wwλ+WwCf (Tgo−Tw)+WfCf (Tw−Tf )+WpCps(Tp−Tw)+q`
(2.26)
where q` is the heat loss, which is assumed to be 20% of the total heat supplied.
The evaporation rate, Ex up to an axial distance x, is estimated from Equation
(2.24) as
Ex =1
λ
dq
dt+dm
dt.
In the entrainment zone, the gas temperature at any distance x from where
the liquid feed entered the chamber can be determined analogously to the nozzle
zone,
Tx = Tgi− (Exλ+ (Ex −Wp)Cf (Tgo − Tw) +WpCps(Tp − Tw) + q`x)
WgCs
(2.27)
44
In the case where the solvent is water, the average humidity in the gas sur-
rounding the droplet is obtained from,
c =
nozzle zone
Ex
Me
+ cSi
entrainiment zoneEx
Wg
+ cSi
(2.28)
where cSiis the humidity of the gas (in this case air) at the inlet of the chamber,
Me is the entrainment rate, and the ratios, Ex/Me and Ex/Wg, represent the
fraction of water that is added to the air due to evaporation.
2.4.2 Single-droplet model
Equations (2.20) to (2.28) do not account for changes in the solvent concen-
tration inside the droplet. In the case of solid particles, a uniform solvent concen-
tration and a uniform droplet temperature are reasonable assumptions. However,
when the particles are not solid (hollow), these assumptions may not be valid.
The single droplet model was developed in Section 2.3. The model equations are
repeated here for the reader’s convenience.
∂(Td)
∂t= α
1
r2d
∂
∂rd
(r2d
∂Td
∂rd
)0 ≤ rd ≤ R(t), t > 0 (2.29)
∂(cA)
∂t= D 1
r2d
∂
∂rd
(r2∂cP∂rd
)0 ≤ rd ≤ R(t), t > 0 (2.30)
∂(cB)
∂t= D 1
r2d
∂
∂rd
(r2d
∂cB∂rd
)0 ≤ rd ≤ R(t), t > 0 (2.31)
45
where Td is the droplet temperature, cA is the polymer concentration, cB is the
water concentration, α = k/ρCp is the thermal diffusivity, and D is the binary
diffusivity coefficient.
Initial conditions for the above equations are,
T (rd, 0) = T0 0 ≤ rd ≤ R
cA(rd, 0) = cA,0 0 ≤ rd ≤ R
cB(rd, 0) = cB,0 0 ≤ rd ≤ R
(2.32)
The boundary conditions at the center arise due to the symmetry of the droplet
and are unchanged during the heating and evaporation steps,
∂T
∂r(0, t) = 0 t ≥ 0
∂cA∂r
(0, t) = 0 t ≥ 0
∂cB∂r
(0, t) = 0 t ≥ 0
(2.33)
The boundary conditions at the surface of the droplet are different in the
heating and evaporation steps. In the heating step, there is no mass flux of either
the solvent or the polymer but there is convective heat transfer from the bulk gas
46
to the droplet. Thus, for the heating step,
−k ∂T∂rd
(R, t) = h(Tg − Ts) T |R < Tsat
D∂cA∂rd
= 0 0 ≤ rd ≤ R(0)
D∂cA∂rd
= 0 0 ≤ rd ≤ R(0)
(2.34)
The boundary conditions for the evaporation step must account for the decrease
in the droplet size. Since evaporation occurs on the surface, the area around the
droplet remains cool and prevents super-heating of the droplet surface. Thus, the
droplet surface temperature must be equal to the saturation temperature. The
boundary conditions on the components are obtained by a species balance on the
concentration of A around the droplet,
D∂cA∂rd
∣∣∣∣∣rd = R(t)+ cA
dR(t)dt
= 0
D∂cB∂rd
∣∣∣∣∣rd = R(t)=
ρB
λ(h(T |R(t) − Tg)− kg
∂T
∂rd
T |R(t) = Tsat
(2.35)
The boundary condition for the rate at which the surface regresses is given by,
Heating step: R(t) = R(0) t ≤ tH
Evaporation step:dR
dt=ρs
λ(h(T |R(t) − Tg)− k
∂T
∂rt > tH
(2.36)
where tH is the time to reach the evaporation temperature of component B (sol-
47
vent).
2.4.3 Particle size distribution model
The single-particle model provides the properties of the particles that are being
tracked but neglects any collisions between different droplets sizes or between the
droplet and the chamber wall. A population balance approach is applied, which
can account for collisions between droplets or particles.
Fundamental to the understanding of a population-balance model is the con-
cept of a particle phase space [69]. In the discussion that follows the use of the terms
particle and droplet are interchangeable. In the particle phase space, two sets of
coordinates are defined – internal coordinates, which are the physical properties
of the population (e.g., particle size in a distribution) and external coordinates,
which give the physical location of the distribution. To develop a population-
balance model of the droplets, the internal coordinate chosen is the size of the
droplets and the external coordinates are the axial and radial directions that the
droplets travel as they pass through the chamber.
An initial droplet distribution is assumed. Thus, instead of tracking a single
droplet, a droplet distribution is analyzed. Let N(X, t) represent the number of
particles at any time t, where the particle state space vector X=[x r dp], with
x and r the axial and radial distances, respectively; and dp the particle diameter
48
(internal coordinate). The population-balance equation is given by,
∂N(X, t)
∂t+ Vav
∂N(X, t)
∂x+ Var
∂N(X, t)
∂r+∂(ExN(X, t))
∂dp
= B(X) (2.37)
where B(X) represents the birth and death of the particles due to collisions. Since
design is the purpose of this work, the first term on the left hand side of Equation
(2.37) is zero.
Since experimental data accounting for collisions is not available, certain as-
sumptions have to be made. The first assumption is that only binary collisions
are considered. Secondly, since the heating phase is very small, collisions between
droplets and particles are not considered. It is assumed that collisions between
droplets do not lead to breakage of the droplets. However, the collisions between
particles is assumed to cause breakage. This assumption is justified since particles
have a solid skin, coalescence is unlikely to occur. Thus, two types of collisions
are assumed to occur.
• During the heating stage, some of the droplets collide with each other. It is
assumed that collisions between two droplets lead to coalescence and that
the droplet formed due to the collision has a volume equal to the sum of
the colliding droplets. While such an assumption may not be completely
accurate, this simplification assists in maintaining a tractable problem.
• It is assumed that collision between the hollow particles results in breakage
of the particles and thus a loss in the total number of particles.
49
The following equation describes the birth and death of the particles due to
collisions [69],
B(X) =1
2
X∫0
a(X−X′)N(X−X′, t)dX−
N(X, t)
∞∫0
a(X,X′)N(X′, t)dX
(2.38)
where a(X,X′) is the collision frequency, which can be determined by a stochastic
collision model [70],
a(X,X′) =π
4(dp + d′p)2 | Vfdp
−Vf ′dp| N(X′, t) (2.39)
The boundary conditions associated with Equation (2.37) result from the zero flux
assumption at the boundary,
N(X, t)X = 0 (2.40)
This condition corresponds to no flux of the particles across the boundary of the
particle phase space [69].
2.4.4 Design steps
In the design of the spray dryer, the important design parameters are the cham-
ber dimensions, the heating gas flowrate and the residence time of the particles in
the spray dryer.
The design equations for the spray-drying chamber are given by Equations
50
(2.20) to (2.27). The equations for the droplet are given by (2.1) to (2.36). The
approach to obtain the design parameters for the spray dryer can be summarized
as follows:
1. Choose a design criterion.
The criterion selected is that the largest particle contains ≤ 10% water. To
compare to existing laboratory data, a binary liquid mixture of polymer and
water with air as the drying gas is selected.
2. Choose the atomizer type and spray-drying chamber geometry.
An atomizer and a chamber geometry of a cylindrical top and conical bot-
tom are selected to provide a comparison to the results obtained from a
similar laboratory scale unit. The choice of the atomizer fixes the droplet
size distribution.
3. Track the largest droplet as it passes through the chamber by solving the
resulting spray-dryer design and droplet equations simultaneously. The po-
sition of the largest particle reaches less than 10% moisture gives the di-
mensions of the spray chamber. The axial distance of the particle from the
atomizer at this point gives the length while the radial distance gives the
radius of the spray drying chamber.
4. The residence time of the particles is also obtained from the velocity equa-
tions.
The numerical solver must be able to account for the moving boundary of the
51
particle to obtain accurate estimates of the chamber size, the gas flowrate and
the heating requirements. The process of obtaining the design parameters is an
iterative process.
2.5 Numerical methods
2.5.1 Numerical solution of the single droplet model
To solve the system of equations given in Equation (2.12) with boundary con-
ditions given by Equations (2.13) to (2.15), the method of Gradient Weighted
Moving Finite Elements (GWMFE) is used [71]. This method is a moving node
finite element method that provides a natural framework for solving the evapora-
tion phase of the droplet because the change in the particle size with time and has
to be followed accurately.
Different approaches have been used to solve moving boundary problems. Farid
[58] used a front fixing finite difference method [72] to solve a system of partial
differential equations. Seydel [73] employed a change of variables to a normalized
coordinate system where the surface position is fixed. Such front fixing algorithms
give rise to a pseudo-convection term, which increases the complexity of the system
of equations. In addition, the domain of the problem has to be discretized at
each step. The discretization step may be particularly difficult in two and three
dimensional problems. Other techniques to solve moving boundary problems are
discussed in [74].
The different finite difference methods used to solve moving boundary problems
52
may be classified as fixed-grid methods and fixed-time methods. In the fixed
grid methods, the entire domain of the problem is discretized. As the boundary
moves, the time step is varied iteratively such that every new position of the
boundary coincides with a node point. In the fixed-time methods, the time step
for integration is fixed and the domain has to be re-discretized after every time
step. Finlayson discusses different methods for solving moving boundary problems
[75].
In this work, the temperature difference between the inlet air and the liquid
feed is very large, (≈100 K, degree of magnitude). Thus, the convective effect
dominates the diffusive effect, which results in the formation of steep fronts at
the droplet surface. To capture these steep fronts accurately fixed node finite
difference methods will require a larger number of nodes, at least in the region
near the front [76, 77].
The notion of moving finite elements (MFE) is that if the nodes are free to
move then as a steep front develops, more nodes can be moved into that area to
improve the accuracy of the solution while in the region away from the front, less
number of nodes are necessary to obtain an accurate solution. The other advantage
of MFE methods is that because the solution remains smooth in spite of shocks,
larger time steps can be taken, thus reducing the number of computational steps.
The conventional MFE methods may suffer due to overlap between nodes and
a resultant fold-over of the solution. To address this issue, the nodal positions
are weighted to prevent overlapping. The gradient weighted moving finite ele-
53
ment method is one such variation of the MFE where the weighting is the normal
component of the velocity of the nodes [71, 78].
2.5.1.1 Mathematical formulation
The following explanation of the GWMFE method is adapted from the work of
Miller and Carlson [78]. Consider a partial differential equation of the form given
by,
ut = Lu
where L is a differential operator and u represents a position. Then, the above
equation represents the vertical motion ofu(t). Consider, normal motion n, where
n ≡ ut/√
1 + u2x and ux means the partial derivative with respect to position.
The above equation can be converted into the normal motion form by dividing by√1 + u2
x,
n = K(u). (2.41)
Let U(x, t) given by,
U(x, t) =∑
j
βjUj(t) (2.42)
be an approximate solution to the above equation. It then follows that the motion
of U is given by
U(x, t) =∑
j
βjUj(t)
where βj are the basis functions. In this work, the set βj is chosen to be piecewise
54
linear hat functions,
βj =
x− xj−1
xj − xj−1
xj−1 ≤ x ≤ xj
xj+1 − x
xj+1 − xj
xj ≤ x ≤ xj+1
(2.43)
Other choices of basis functions can be used. However, in this work, higher order
basis functions do not increase the accuracy of the solution.
The normal motion with this assumed function is then given by,
n = u · n =∑
j
xj(βjn1) + uj(β
jn2)
where
n ≡ (n1, n2) ≡(
1/
√1 + U2
x , Ux/
√1 + U2
x
)
is a unit normal vector, xj are the nodal motions, and Uj are the nodal amplitudes
(value of approximation (U(x, t)) at every node).
2.5.1.2 Residual minimization
Define a residual of Equation (2.41) as,
ψ ≡∫
(n−K(u))2dS
The solution for Uj is obtained by minimizing the residual, ψ, with respect
to the nodal velocities and the nodal amplitudes. The residual minimization is
55
carried out by setting the derivatives of the residual with respect to the nodal
velocities and amplitudes to zero.
1
2
∂ψ
∂xj
=
∫(n−K(u))βjn1dS = 0
1
2
∂ψ
∂Uj
=
∫(n−K(u))βjn2dS = 0
The result is a system of ordinary differential equations (ODEs) in the independent
variable time that can be solved by any initial value problem solver. A Newton
method is used for time integration along with a non-linear Krylov subspace ac-
celerator [78].
2.5.2 Numerical solution of the spray dryer model
The spray dryer model is a set of ordinary differential equations (ODE). The
initial conditions are the initial velocities of the individual droplets. Using the size
of the atomizer on the experimental unit, the initial positions of the droplets are
determined. The initial velocities in the axial and radial direction are taken to be
zero. The initial velocity in the tangential direction is taken to be equal to the
inlet air velocity. The ODEs are solved using a variable time-step fourth-order
Range-Kutta solver.
56
2.6 Results and discussion
2.6.1 Single droplet modeling results
The model equations are solved using the GWMFE method. The results are
in two parts: firstly, the dimensionless model is solved at nominal conditions and
non-nominal conditions; then the sensitivity of the model to changes in a selected
set of parameter values is investigated.
The dimensionless equations are solved at the designed (nominal) conditions
(θin=0, cP (0)=0.192, θair=1, BiH=0.0938, Le= 145.92, Hevap=1.97, and R(0)=
60 microns). The non-nominal conditions tested are:
• The inlet polymer concentration of the droplet is varied by ± 10%.
• The inlet droplet temperature by ± 15%.
• The air temperature by ± 8%.
To test the sensitivity, the heat transfer coefficient within the droplet and the
binary diffusion coefficient between water and polymer are varied ± 25%.
Figure (2.6) shows how the droplet radius, the polymer concentration and the
temperature at the surface vary as functions of dimensionless time at the design
conditions.
From the figure, it is observed that initially when only the sensible heat is
being transferred to the droplet, only the temperature of the droplet increases.
57
There is no change in the concentration of the polymer at the surface. The evap-
oration temperature is reached at τ=0.005 dimensionless time units (t=0.0144 s
for t0=2.88 s). Simultaneously, the surface (2) of the droplet begins to recede due
to the evaporation of the water. A polymer concentration profile (◦) is developed
as the water evaporates; the temperature at the surface remains constant at the
evaporation temperature.
Figure (2.7) shows the corresponding spatial profiles of the polymer and the
water concentrations, respectively. As the evaporation starts, the droplet surface
regresses as shown in the figure. Also, it can be seen that a sharp front is being
developed at the surface due to the extremely large difference in the temperatures
of the droplet surface and the air. Thus, more and more nodes are pushed in
to the region of the shock and the polymer and water concentration surface can
be described accurately. The time at which the surface concentration of polymer
reaches 1.00 is called the time to skin formation. The model predicts the con-
centration and temperature variations that take place during the operation (cf.
Figure 2.3).
The effects of different operating conditions on the droplet are also studied.
The inlet concentration of the droplet is varied to investigate its effect on skin
formation. For changes of ±10 % in the inlet concentration of the polymer there
appears to be negligible changes in the thermal and mass transfer characteristics
of the droplet. However, it is observed that a decrease in the initial polymer
concentration increases the time required for skin formation. This is due to the
58
fact that a lower polymer concentration means a greater amount of water in the
droplet. At a given constant heat transfer rate, the time required to evaporate
the additional water is more than the nominal, which in turn results in a longer
time for skin formation. It can be seen that if the polymer concentration were to
decrease further, more time would be required until, in the limit, hollow particle
wouldn’t be formed at all unless an even higher inlet temperature of air was used
[63]. The effect of changes in the polymer concentration on the time to skin
formation is shown in Figure( 2.8).
The inlet droplet temperature and the gas temperature are varied to study
their effect on skin formation. A ±10oC change in the inlet temperature of the
droplet did not produce a significant change in the time to skin formation. This
indicates that the air supplied is at a sufficiently high temperature to compensate
for reasonable fluctuations in the inlet droplet temperature. However, variations
in the air temperature have significant effect on the time for skin formation. This
is observed when changes of 8% (±50oC) to the air inlet temperature of air is
made. The results are shown in Figure (2.9) The response shows that the time for
skin formation varies inversely with increases in the inlet air temperature. The
inlet air temperature determines the heat transfer rate to the droplet. Thus, an
increase in the inlet air temperature increases the amount of heat transferred to
the droplet. This leads to faster evaporation rates and therefore shorter times for
skin formation.
The effect of uncertainty in the heat transfer parameters inside the droplet is
59
studied by changing the heat transfer coefficient by ± 15% and ± 25% from its
nominal value. The resulting polymer concentration profile is shown in Figure
(2.10). It can be seen that the time to skin formation is sensitive to the rate of
heat transferred. The dependence on the amount of heat transferred is non-linear.
That is, when the heat transfer coefficient is increased by 15 and 25%, the time
for skin formation does not change by the same amount when decreased by the
same amount. It is not surprising that beyond a certain point, increasing the
heat transfer rate does not affect appreciably the time for skin formation and that
decreasing the heat transferred will retard the rate of skin formation.
Similar simulations are carried out for changes in the diffusivity coefficient.
The polymer concentration profiles are shown in Figure (2.11) From the graph, it
is observed that as the diffusion coefficient increases, the time required for skin
formation decreases. The relationship between the binary mass diffusivity and
the time for skin formation is non-linear with shorter lengths of time when the
diffusivity is increased than when the diffusivity is decreased by the same amount.
2.6.2 Spray dryer model results
The mass-based droplet distribution is known to be log-normal [68]. Using
this distribution, six droplet sizes, listed in Table 2.3, are selected. The assumed
nominal operating conditions are given in Table 2.4. A numerical solution of the
droplet and spray-dryer design models gives the design parameters listed in Table
2.5 for the nominal operating conditions.
60
Table 2.3: Inlet droplet distribution
Bin no. Droplet size, Fraction of particles
microns in the bin1 20 0.102 30 0.103 45 0.304 60 0.155 80 0.156 100 0.20
Table 2.4: Nominal operating conditions
Operating parameters Value
Liquid feed flowrate 25 mLmin−1
Feed temperature 298 KDrying gas (air) temperature 644 KAmbient temperature 298 KFinal moisture content ≤10%
The main chamber design criterion is that the largest size (100 microns) particle
should contain ≤10% moisture in the final product. The spray-dryer and droplet
Table 2.5: Spray dryer design values
Design parameter Value
Length of the chamber 0.68 mRadius of the chamber 0.32 mLength of cylindrical section 0.30 mLength of conical section 0.38 mHeat required 9 KWDrying gas flowrate 40.77 g/sThermal efficiency 57%
61
models are solved iteratively until this requirement is achieved. The chamber size
(axial and radial parameters) is obtained by calculating the distance a represen-
tative droplet travels from the point at which it is injected into the spray-drying
chamber to the point where it is completely dry (≤ 10% moisture). The length
of the chamber is calculated based on the final particle moisture content and the
final air temperature. The radius of the chamber is determined by the criterion
that none of the droplets hit the chamber wall.
The results show that the cylindrical length of the chamber is ∼50% smaller
than the conical length. The axial and radial trajectories of a 60 µ droplet are
shown in Figure 2.12. The distance covered by the droplet at 2.4 s gives the
dimensions of the chamber. The axial distance covered provides the length of the
chamber and the radial distance, the radius of the chamber.
Figure 2.13 shows the tangential and axial velocities of a representative droplet
as it travels the length of the chamber. It is observed that the axial velocity
becomes extremely large while the tangential velocity decreases in the conical
section because the conical shape acts as a converging nozzle. The hump in the
tangential velocity profile is when the geometry of the spray dryer changes from
cylindrical to conical.
2.6.2.1 Hollow particles
The polymer concentration on the surface of the droplet is shown in Figure
2.14. Once the solvent concentration at the surface the droplet is 10% a skin
62
forms [3]. However, the inside of the droplet has to meet the criterion of ≤10%
final moisture. Drying is either convective mass-transfer controlled or diffusion
controlled. Until the boiling temperature of the solvent is reached, the removal of
water from the droplets occurs due to convective mass transfer only. Thus, the
evaporation is mass-transfer controlled, which causes the polymer concentration
to change linearly. Once the evaporation temperature is reached, the removal of
water is controlled by the rate of diffusion of water to the surface of the droplet.
The rate at which water can be brought to the surface by diffusion is less than the
rate at which water can be removed when it is at the surface. This effect along
with the effect of the curvature of the particle causes the polymer concentration
change to be nonlinear. The nonlinear change in the polymer concentration is
present at all droplet sizes but is not as pronounced in the smaller droplet sizes
since their water content is less as compared to the larger droplets.
2.6.2.2 Variation in particle size
The design model, (Equations (2.20) to (2.36)), can used to predict the final
particle distribution for a given feed droplet distribution. The droplets shrink due
to solvent evaporation, hence the particles that are formed are smaller in size than
the droplets that entered at the inlet of the spray dryer. The top graph in Figure
2.15 represents the inlet droplet size distribution (see Table 2.3). The bottom
graph shows the particle size distribution at the outlet of the spray dryer. Here, it
is assumed that no droplets are destroyed due to collisions with the chamber wall
63
or with each other. Since all the droplets are assumed to be exposed to identical
temperature conditions, the reduction in size of the droplets is almost the same
except at the larger sizes where the amount of water evaporated is slightly greater.
2.6.2.3 Thermal efficiency
An energy balance provides the amount of heat that must be supplied to meet
the product moisture criterion. Thermal efficiency (η) is defined as the ratio of
the heat used for vaporization to the total heat supplied by the drying gas [18],
η =Ww λ
Wg (Tgi− Tw)Cs +Wf (Tf − Tw)Cf
. (2.44)
The results show that for the present chamber design (cf. Table 2.5), the thermal
efficiency is ∼57%. The efficiency may be improved by lowering the exit temper-
ature of the product. However, by so doing, the chamber length will increase.
Figure 2.16 contrasts the temperatures of a representative droplet and the air
stream as both pass through the length of the spray-drying chamber.
2.6.2.4 Heat loss
The assumed heat loss is 20% of the nominal design [18]. The sensitivity of
the predicted air temperature to an uncertainty in this assumption is investigated.
Figure 2.17 shows the result of a ± 50% change from the nominal assumed heat
loss. It is found that the predicted air temperature is inversely proportional to the
heat loss. The predicted air temperature throughout the chamber is lower than
64
the nominal design if greater heat loss is assumed. The opposite is true for smaller
heat losses.
2.6.2.5 Effect of feed conditions
The effect of different operating conditions on the evaporation rate is investi-
gated because the evaporation rate is an accepted measure of chamber performance
[8]. The results of a ±15% change from the nominal feed flowrate and inlet air
temperature are shown in Figures 2.18 and 2.19. It is observed that the evapora-
tion rate is directly proportional to the feed flowrate. The Integral Time Squared
Error (ITSE) [29] for a ±15% change in the feed flowrate is 313.28 and 556.45,
respectively, which shows some degree of nonlinear behavior.
An increase in the inlet air temperature increases the evaporation rate as ex-
pected since a higher air temperature provides a greater driving force to dry the
droplet. The ITSE for the increase is 2381.54 and for the decrease is 2047.58. The
sensitivity of the evaporation rate to changes in inlet air temperature is not as
pronounced when compared to similar changes in the feed flowrate.
2.6.2.6 Effects of interparticle collision
The design of the spray dryer does not provide for any droplet or particle
collisions. Collisions result in a smaller number of particles at the outlet and
changes in the size distribution. Using the assumed log-normal distribution of the
droplets at the inlet along with the population-balance model given by Equations
(2.37) to (2.40), the effect of collisions is illustrated in Figure 2.20.
65
Table 2.6: Percentage breakage predicted by the population-balance model
Bin no. % Breakage
1 12.542 12.673 5.114 8.195 4.026 5.04
The collisions between the droplets can lead to either coalescence or breakage.
The reduction in the number of droplets indicates that most collisions occur when
the skin has already been formed, i.e. in the evaporation zone. This evidence
also is supported by the fact that the droplets of a larger size, greater than 100
microns (6th bin) are not formed, which may have been expected if there had been
coalescence.
The percentage of droplets in each bin that are destroyed due to collisions is
listed in Table 2.6. From the values in this table the number of droplets that are
lost as debris because of interparticle collision is significant. This has important
implications in the control of the final product size distribution, which is crucial
in some manufacturing processes.
2.6.2.7 Effect of droplet distribution
The effect of changes in the droplet distribution on the operation of the spray
dryer is examined. Figure 2.22 shows the changes in the evaporation rate if the
inlet droplet distribution is changed from log-normal (?) to uniform (◦) distribution
66
of 60 µ. It was found that for a uniform droplet distribution an increase in the
droplet size causes a decrease in the evaporation rate (cf. Figure (2.21)). Since the
log-normal distribution has a higher fraction of droplets with smaller diameters,
this explains why the evaporation rate for the log-normal distribution is greater
when compared to a uniform distribution (cf. Table 2.3).
2.6.2.8 Model validation
The model’s prediction can be compared to experimental data collected from
a laboratory scale spray-drying unit (located at Texas Tech University) that has
a cylindrical/conical spray-drying chamber.
1. Air temperature data at different points along the length of the laboratory
spray-drying chamber are available. However, since the length of the labora-
tory and design chambers are different, the comparison is made by normal-
izing the two chamber lengths between 0 and 1. Four different temperature
data profiles are compared between the experiment and the simulated results
(see Figure 2.23). The maximum error between the two sources of data is
11%.
2. An important criterion for product quality, especially when the final particle
is hollow is the tap density (ρtap). The tap density is the packing density
(ASTM standard test method B527) and may be related to the individual
67
particle density by the following equation [79],
ρtap = (1− ε) ρpart
where ε is the porosity of the packed bed and ρpart is the density of the
particle. For a packed bed of spheres, ε = 0.45 [79]. The experimental tap
density varied between 0.05 to 0.5, whereas the tap density predicted by the
model for similar experimental conditions was found to be between 0.17 to
0.23. The predicted values are well within the range of the experimental tap
densities.
2.7 Summary
The modeling and design of a spray-drying chamber that manufactures hol-
low micro-particles was accomplished using a rate-based approach coupled with
a fundamental model of the heat- and mass-transfer processes that describe the
formation of hollow micro-particles. The different physical phenomena occurring
inside a representative were described. A first-principles model of these different
phenomena was developed. Due to the evaporation of the solvent, such a model
leads to a moving boundary problem with steep fronts. Therefore the model was
solved using a Gradient Weighted Finite Element technique. The time to skin for-
mation was determined by solving the model at nominal conditions. The model
was further exercised to test the effects of different operating conditions (inlet
polymer concentration, inlet air temperature) and parameter uncertainty ( heat
68
transfer co-efficient, binary diffusivity). The spray-drying chamber design was
compared and validated against an existing laboratory scale unit operation. A
number of factors influencing the design and operation of the spray-drying cham-
ber were identified; namely, feed flowrate, inlet air temperature, and heat loss. A
degree of nonlinearity was observed in the dependence of the chamber performance
(e.g., evaporation rate) on these factors.
A population-balance model was used to model a provide a more realistic
droplet distribution for design. It was found that a significant percentage of the
particles was predicted to be broken due to collisions. These results have to be
validated experimentally.
69
Figure 2.6: Model predictions at design conditions. The polymer and temperatureprofiles are those at the surface. ◦ : cP (polymer), 3 : θ (temperature), and 2 : X(radius) [3].
70
0.5 0.6 0.7 0.8 0.9 1Dimensionless radial distance
0.2
0.4
0.6
0.8D
imen
sion
less
pol
ymer
con
cent
ratio
n
Increasing time
0.5 0.6 0.7 0.8 0.9 1Dimensionless radial distance
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss w
ater
ceo
ncen
trat
ion
Increasing time
Figure 2.7: Spatial polymer (top) and water (bottom) profiles during the evapo-ration step. ◦ : τ = 0.00, 2 : τ= 0.005, . : τ= 0.0057, 3 : τ= 0.0063, / : τ=0.0080, 4 : τ= 0.0087, + : τ =0.0102, × : τ= 0.0139 [3].
71
0 0.01 0.02Dimensionless time
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss p
olym
er c
once
ntra
tion
0.025
Figure 2.8: Polymer concentration profile at the surface for changes in the inletfeed concentration. ◦ : +10%, 2 : +5%, 3 : nominal, 4 : -5%, and / : -10% [3].
0 0.005 0.01 0.015 0.02 0.025Dimensionless time
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss p
olym
er c
once
ntra
tion
Figure 2.9: Polymer concentration profile at the surface for changes in the airtemperature. ◦ : +50oC, 2 : +25oC, 3 : nominal, 4 : -25oC, and / : -50oC [3].
72
0 0.005 0.01 0.015 0.02 0.025Dimensionless time
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss p
olym
er c
once
ntra
tion
Figure 2.10: Polymer concentration profile at the surface to changes in the heattransfer coefficient. ◦ : +25%, 2 : +15%, 3 : nominal, 4 : -15%, and / : -25% [3].
0 0.01 0.02Dimensionless time
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss p
oly
mer
con
cent
ratio
n
0.025
Figure 2.11: Polymer concentration profile at the surface for changes in the dif-fusivity coefficient. ◦ : +25%, 2 : +15%, 3 : nominal, 4 : -15%, and . : -25[3].
73
Figure 2.12: Axial and radial trajectories of a 60 µ droplet starting from the pointof injection [22].
Figure 2.13: Axial and tangential velocities of the droplet as it travels the lengthof the spray-drying chamber [22].
74
Figure 2.14: Polymer concentration at the surface of the droplet for the assumeddroplet distribution [22].
Figure 2.15: The size distribution at the inlet (top) and the outlet (bottom) ofthe spray-drying chamber [22].
75
Figure 2.16: Temperatures of a representative droplet and the air stream as theypass through the spray-drying chamber [22].
Figure 2.17: Effect of heat loss on air temperature. ?: 50% increase from thenominal, ◦: 50% decrease from the nominal, 4: nominal (20% heat loss) [22].
76
Figure 2.18: Effect of feed flowrate on the evaporation rate. Solid line: nominal;dashed line: 15% increase; dotted line: 15% decrease [22].
Figure 2.19: Effect of air temperature on the evaporation rate. Solid line: nominal;dashed line: 15% increase; dotted line: 15% decrease [22].
77
Figure 2.20: Effect of collisions on the final particle size distribution [22]
Figure 2.21: Effect of change in mean droplet size for a uniform inlet distribution,Uniform droplet mean size of 60 microns (solid line), Uniform droplet mean sizeof 80 microns (dashed line)
.
78
Figure 2.22: Effect of changes in inlet droplet size distribution, ◦: Uniform dropletsize of 60 microns, ?: Log-normal (std. dev. = 30 microns) [22]
Figure 2.23: Comparison of the air temperature profile inside the chamber betweenthe experimental data: ?, /, . and 4 and the model predictions (◦)[22].
79
NOMENCLATURE
BiH Biot number for heat transfer, dimensionless
CD Drag coefficient, dimensionless
Cf Specific heat of the feed liquid, 327 J kg−1K−1
Cs Specific heat of the gas, 1055 J kg−1K−1
Cps Specific heat of the product, J kg−1K−1
C1 Experimental constant depending on the axial distance from atomizer, cm s−1
C3 Experimental constant, cm s−1
Cp Specific heat capacity of the solution, J kg−1 K−1
D Binary diffusion coefficient of the water and polymer, m2s−1
Dv Diffusion coefficient between gas and solvent, 0.375 ×10−4 m2 s−1
FL Shear lift force on the representative droplet, ergs
K Curl of the fluid velocity, s−1
Me Amount of gas entrained, g s−1
MP Mass of polymer inside the droplet, kg
N(X,t) Particle density at any time, t s, m−3
80
Nu Nusselt number, dimensionless
Pr Prandtl number, dimensionless
R(0) Initial droplet radius, m
R(t) Radius of the droplet at t s, m
R(t) Radius of the droplet at time t, m
Rx Radius of the chamber at a distance x cm from the atomizer, cm
Re Reynolds number, dimensionless
Sc Schmidt number, dimensionless
Sh Sherwood number, dimensionless
T Temperature of the droplet, K
T0 The temperature of the droplet at t = 0s
Td Temperature of the droplet, K
Tg Temperature of gas, K
Ts Temperature of droplet surface, K
Tf Temperature of feed, K
Tp Temperature of product, K
Tw Wet bulb temperature of drying gas, K
81
Tx Temperature of air at a distance x from the atomizer, K
TgiTemperature of gas at inlet, K
Tgo Temperature of gas at outlet, K
Tl(r, 0) Initial local temperature, K
Tsat Evaporation temperature of water, K
Vgt Tangential velocity of the representative droplet, cm s−1
Vgv Axial velocity of gas, cm s−1
Vrv Axial velocity of the representative droplet, cm s−1
Vr Radial velocity of the representative droplet, cm s−1
Vt Tangential velocity of the representative droplet, cm s−1
Wf Feed flowrate, g s−1
Wg Amount of gas fed to the spray dryer, g s−1
Ww Amount of solvent evaporated, g s−1
cP (x, τ) Dimensionless polymer concentration
cW (x, τ) Dimensionless water concentration
cSiConcentration of solvent in the air at the inlet, dimensionless
q` Heat losses, W
82
r5 Width of the spray, cm
x Axial distance of the representative droplet from the atomizer, cm
x′ Entrainment length, cm
a(X,X′) Collision frequency, s−1
c Concentration of solvent in gas, dimensionless
cS Concentration of solvent on surface of droplet, dimensionless
cA,0 The initial concentration of species A in the droplet, kgm−3
cB,0 The initial concentration of species B in the droplet, kgm−3
cW,0 The initial water concentration in the droplet, kgm−3
cW The concentration of water in the droplet at any time, kgm−3
di Diameter of droplets, cm
dp Droplet radius, m
h External heat transfer coefficient, 1640 Wm−2K
Hevap Dimensionless heat of vaporization
k Thermal Conductivity of the solution, W m −1K
kg Thermal conductivity of the gas, 0.0328 Wm−1 K−1
kair Thermal conductivity of air, Wm−1K−1
83
kx Mass transfer coefficient, ms−1
ni Number of droplets
q Heat transferred from the gas to the droplets W
r Radial distance of the representative droplet from the center of the spray cham-
ber, cm
rd Radial distance from the center of the droplet, m
t Time, s
t0 Characteristic time, s
Tair The inlet temperature of air, K
tH Time for the droplet surface to reach the evaporation temperature, s
tH Time required for surface to reach evaporation temperature,s
X(τ) Dimensionless radius
x(τ) Dimensionless radial distance
B(X) Particle birth and death function, s−1
cP,0 The initial polymer concentration in the droplet, kgm−3
cP The concentration of polymer in the droplet at any time, kgm−3
Le Lewis number, dimensionless
84
di Diameter of droplet, cm
m Mass of droplet, g
Vv Axial velocity of the representative droplet, cm s−1
λ Latent heat of vaporization of solvent, 2260 J g−1
D Diffusion coefficient between solvent and solute in the droplet, 1.25× 10−9
m2 s−1
ρ Density of droplet, g cm−3
ρ Density of solution, kgm−3
ρg Density of gas, 1.265 kgm−3
ρs Density of solvent, 995 kgm−3
ρtap Tap density of hollow particle, g cm−3
θ Half angle of the spray with the vertical axis, degrees
ε Porosity of the packed bed
r Radial distance, m
X Particle state space vector
α The thermal diffusivity of the solution, m2/s
α The thermal diffusivity of the solution
85
λ Latent heat of vaporization, J kg −1
ρ0 Initial solution density, kgm−3
ρW Density of water, kgm−3
τ Dimensionless time
θ(x, τ) Dimensionless temperature of the droplet
86
CHAPTER 3
OPTIMIZATION OVERVIEW
Design for a chemical process involves choosing the right equipment, sizing the
equipment to obtain an optimal size, flowsheet synthesis, and identifying operating
conditions under which the process will generate maximum profit.
Once a flowsheet has been selected, the optimum design parameters and operat-
ing conditions for the process to operate at maximum profit usually are determined
by solving an appropriately formulated optimization problem.
The chapter is organized as follows. In section 3.1, the basics of optimization
are introduced and the different types of optimization are classified. Section 3.2
deals with non-linear optimization problems. Section 3.3 provides a discussion on
mixed integer programming (MIP) and section 3.4 follows with a similar discourse
on dynamic optimization. A summary of the chapter follows in section 3.5.
3.1 Optimization in process design
Process design optimization can be either single-unit optimization or plantwide
optimization [80]. In either case, the problem is a steady-state optimization prob-
lem whose solution gives (i) the equipment sizes and the (ii) the operating con-
ditions for maximum profit. In this section, the basics of optimization will be
discussed. Then, the different types of optimization problems that arise will be
presented followed by relevant solution techniques.
87
3.1.1 Formulation of an optimization problem
Any optimization problem has to have the following basic elements, [42, 80, 81].
1. Objective function.
The objective function, also called a functional is a scalar function to be
maximized or minimized. This function can represent the cost of operating
the process, the cost of not meeting a desired quality specification, or the
profit after accounting for operating costs. Often, there may be multiple
goals that must be satisfied by the optimization. Such problems are known
as multi-objective optimization problems. Sometimes, these objectives can
be combined, using a transformation or an appropriate scalarization, into
a single objective function. For example the net profit function is usually
the combination of minimizing capital and operating costs and maximizing
product throughput.
2. Equality constraints.
Equality constraints generally arise from the process itself, i.e. conservation
of mass, momentum, and energy.
3. Inequality constraints.
Inequality constraints are usually physical constraints that arise due to
the physics of the problem, i.e. non-negative volumes, compositions, and
flowrates; or constraints imposed by devices, i.e. a minimum or maximum
pressure on a steam line.
88
4. Design variables.
Design or decision variables are variables whose values are changed to ar-
rive at the optimal solution [81]. The design variables may be continuous
quantities like mass, volume, etc., or discrete quantities such as the number
of trays in a distillation column. The discrete variables may be integers or
non-integers. Examples of integer design variables include, the number of
trays in a distillation column, number of catalyst beds, etc. Design variables
may also be logical or boolean (0 or 1), generally referring to the presence
or absence of a particular unit [82].
A typical steady-state optimization problem may be formulated as follows,
mind
e(d)
subject to: h(d) = 0, g(d) ≤ 0
(3.1)
where d is the vector of design variables, e(d) is the objective function, h(d) and
g(d) are the equality constraints and inequality constraints, respectively. The
domain defined by the design variables is known as the design space. The region
within the design space which satisfies the set of constraints is called the feasible
region. The point within the feasible region where the objective function attains
its optimum value is called the optimum point. If the optimum lies outside the
feasible region then the solution is said to be infeasible.
89
3.1.2 Classification of optimization problems
Optimization problems can be classified as wither unconstrained or constrained
optimization problems depending on whether the constraints given in Equation
(3.1) are enforced. Real problems are in general constrained, thus, this chapter
will focus only on constrained optimization problems.
Optimization problems also can be further classified on the basis of whether
or not the objective function or constraints are linear in the design variables. The
classification is as follows [81].
• Linear programming (LP) problems are problems where both the constraints
and the objective function are linear in the design variables. These problems
have a unique solution which usually lie at one or more of the constraints.
• The optimization is called a non-linear programming (NLP) problem if either
the constraints or the objective function is not linear in the design variables.
In this case a unique optimum cannot be guaranteed and the NLP solution
is in general not found at the constraints. Also, NLP problems may lead to
local optima different from the global optimum if the problem is non-convex.
A problem is a convex programming problem if the objective function and
the inequalities are convex functions of the design variables [80]1
The quadratic programming (QP) problem is a special subset of NLPs [80].
In a QP, the objective function is quadratic2 in the design variables while
1A function f(x) is convex if f [γx1 + 1− γ)x2] ≤ γf(x1) + (1− γ)f(x2) for 0 ≤ γ ≤ 1.2f(x) is said to be quadratic if f(x) = 1/2xTAx + bx + c, where A is a symmetric matrix,
b is a vector of coefficients of xi and c is a scalar [81].
90
the constrains are linear in the design variables. In many cases, the QP is
solved successfully using a least-squares minimization method and a unique
optimum is guaranteed.
• Optimization problems can be further classified depending on the type of
design variables. For example, optimizations involving integer variables are
known as integer programming (IP) problems. When the design variables
consists of integer and continuous variables, the optimization is called a
mixed integer problem (MIP) which is further classified according to whether
the continuous problem is linear (MILP – mixed integer linear programming)
or non-linear (MINLP – mixed integer non-linear programming) in the design
variables.
• If the process models or the objective function is dependent on time, the
optimization is said to be dynamic optimization (DP) [83].
Of the various types of optimization problems, the NLP, MINLP and DP problems
are the most common in chemical engineering. A classification of the different
optimization problems is shown in Figure 3.1.
3.2 Non-linear programming
Non-linear programming problems arise frequently in chemical engineering pro-
cesses. For example in a unit-operation optimization, the process models (equality
constraints) are generally non-linear, algebraic quantities [80]. The objective func-
91
Optimization
Steady state Dynamic
DiscreteContinuous
NLPLP
MILP MINLP
Multi-period Optimal Control
Figure 3.1: Classification of different optimization problems.
tion may be a minimization of some least-squared error in the decision variables
and is therefore non-linear [84].
A typical NLP problem, involving only continuous variables, may be repre-
sented as follows,
mind
e(d)
subject to: h(d) = 0, g(d) ≤ 0
(3.2)
If both e(d) and g(d) are convex functions and h(d) is linear in d, the problem
is a convex NLP. The convex problem has the useful property that the solution is
unique and further, the local solution also is the global solution. If however, the
equality constraints are non-linear, the solution is not unique and multiple local
(minima) optima may exist. In this case, special techniques are used to solve this
class of global optimization problems, however these techniques do not guarantee
finding the global solution.
Another sub-class of the NLP problem is the LP problem. These are problems
92
which have linear objective functions and inequality constraints,
mind
e(d)
subject to: A d = 0, d` ≤ d ≤ du.
(3.3)
where, A is the matrix of coefficients of the design variables and d` and du denote
the lower and upper bounds on the design variables respectively. Such problems
are usually solved using for example the simplex algorithm [80].
A number of methods have been developed to handle LP problems. The most
common is the simplex algorithm in its various forms, barrier methods [80] and
interior point methods [83]. The barrier or interior point methods are discussed
in subsection 3.2.3.
3.2.1 Unconstrained non-linear optimization
Before focusing on constrained NLP problems, a few concepts about the un-
constrained case are important. The formulation of an unconstrained NLP is given
by,
mind
e(d). (3.4)
The necessary optimality conditions for unconstrained NLP problems are [84],
First-order necessary conditions: ∇e(d) = 0
Second-order necessary conditions:
∇e(d) = 0
zT∇2e(d)z ≥ 0, y ∈ <n
93
where z is any non-zero vector.
Most of the unconstrained optimization solution methods are based on choosing
a search direction and minimizing along that direction. The two most popular
search methods are the conjugate gradient3 methods and the steepest descent
methods [80]. Fletcher [80] provides a conjugate gradient formula to calculate the
new search direction as follows,
sk+1 = −∇e(dk+1) + sk∇Te(dk+1)∇e(dk+1)
∇Te(dk)∇e(dk)
where sk is the kth search direction.
Newton methods have also been developed for solving unconstrained opti-
mization problem. These methods use second-derivative information (Hessian)
to calculate the new search direction [80]. If the matrix consisting of the second-
order derivatives of the objective function is positive-semidefinite, the method con-
verges. However, if the matrix of second-order derivative is not always positive-
semidefinite, then this matrix is replaced by another, which is positive-definite.
This is accomplished using Marquardt’s method by making the diagonal terms
positive and large enough to make the matrix positive definite. Finally, line search
methods use the gradient of the objective function to search in the direction of the
gradient. These are known as line search methods. While methods using second-
3Two directions si and sj are said to be conjugate if, given a positive-definite matrix Q,
(si)T Qsj = 0.
94
order derivative information converge quicker, the second-order derivatives are
not as accurately known as the first-order derivatives. Therefore, methods using
first-order information in a Newton’s method framework are quite popular. These
methods are called quasi-Newton methods and they use various updating formulae
to update the first-order information. The most popular updating formula is the
BFGS (Broyden-Fletcher-Goldfarb-Shanno) update,
Hk+1 = Hk +yk(yk)T
(dk)Tyk− (Hkdk)(Hkdk)T
dkHkdk
3.2.2 Constrained non-linear optimization
Constrained non-linear optimization problems may be stated as follows,
mind
e(d)
subject to: h(d) = 0 g(d) ≤ 0
(3.5)
Constrained non-linear optimization problems are characterized by a non-linear
objective function while the constraints may or may not be non-linear. One
method to solve constrained non-linear optimization problems is the use of La-
grange multipliers. The conversion of a constrained optimization problem to that
of an unconstrained one usually involves a transformation of some combination of
the objective function and the constraints. Given the original NLP formulated in
95
Equation (3.2), a Lagrange function may be defined as,
L(d, λ) = e(d) + λT h(d) + µT g(d) µ ≥ 0, (3.6)
where λT = (λ1, . . . , λm) and µT = (µ1, . . . , µp) are the Lagrangian multipliers
associated with the constraints. Using the framework of the Lagrange function,
the original constrained problem is transformed into an unconstrained one with
the Lagrange function as the objective function, that is,
mind,λ,µ
L(d, λ) = e(d) + λT h(d) + µT g(d) (3.7)
The Lagrange function provides the necessary optimal conditions for constrained
optimization problems (Karush-Kuhn-Tucker (KKT) optimality conditions). If
the vector of points d∗ is the optimum then the KKT conditions are,
∇e(d∗) + λT∇h(d∗) + µT g(d∗) = 0 (3.8)
h(d∗) = 0 (3.9)
g(d∗) ≤ 0 (3.10)
µTj g(d∗) = 0, j = 1, 2, . . . , p (3.11)
µj ≥ 0, j = 1, 2, . . . , p (3.12)
The KKT conditions are sometimes referred to as first-order KKT conditions.
Second-order KKT conditions are employed if the curvature of the functions are
96
important in generating complete information about the optimum [84].
Second-order optimality conditions are obtained from the optimality conditions
for the unconstrained case. If the first-order conditions are satisfied, the second-
order optimality condition are [84],
zT∇2L(d, λ, µ)z ≥ 0
where z is any non-zero vector.
3.2.3 Solution Techniques for NLP
The solution techniques for NLPs can be classified as follows [81, 84].
Exterior penalty function methods. In these methods, the constrained
problem is converted into an unconstrained problem by the use of a penalty
parameter, r. The product of the penalty parameter and the constraint ex-
pression gives the penalty function, which is added to the original objective
function. The resultant problem is an unconstrained problem which is solved
for increasing r-values. As r increases, there is a greater penalty on violat-
ing the constraints. Thus, the solution obtained almost always satisfy the
constraints. As a result, the value of the unconstrained objective function
approaches the value of the constrained objective function; the latter being
the optimum point.
97
Consider the following NLP,
mind
e(d)
subject to: h(d) = 0
(3.13)
The unconstrained objective function is given by,
P(d) = e(d) + r (h(d))2 (3.14)
where r ≥ 0 is a penalty parameter and P(d) is the unconstrained objective
function. If both the objective function and the constraints are convex, an
optimum is guaranteed.
Interior point (barrier) methods. Barrier methods are similar to penalty
function methods except that they are designed only for inequality con-
straints. Recall that equality constraints can be transformed into inequality
constraints [81]. As with the penalty function method, a parameter called
the barrier parameter is used to scale the expression in the inequality. The
product of the barrier parameter and the inequality is called the barrier
function. The barrier function is added to the original objective function
to produce an unconstrained problem. The unconstrained problem is solved
for decreasing values of the barrier parameter. The choice of the barrier
function should be such that violating the constraint incurs a large penalty.
As a result, when the barrier parameter approaches zero, the solution of the
98
unconstrained problem approaches the solution of the constrained problem.
Suppose that the following problem is to be optimized,
mind
e(d)
subject to: g(d) ≤ 0
(3.15)
A suitable barrier function may be formulated as,
B(d) = e(d)− 1
rln(−g(d)) (3.16)
As the value of the inequality g(d) approaches zero (approaches the bound-
ary) the barrier function becomes larger and thus pushes the solution away
from the boundary.
Gradient projection methods. The previous methods relied upon trans-
forming the constrained optimization problem into an unconstrained prob-
lem or a series of unconstrained problems. The gradient projection methods
function by generating the negative gradient of the objective function. The
gradient provides a search direction for the optimum solution. The gradient
is projected onto the null space of the gradients of the equalities and in-
equalities [84]. The gradient projection methods may not result in a feasible
search direction if the constraints are non-linear.
Generalized reduced gradient (GRG) methods. The GRG methods are
99
based on the algorithm proposed by Wolfe [81]. The method finds the search
directions using the gradient of the objective function. By moving along the
gradient with an appropriate stepping procedure the optimal point is reached
[80]. The equality constraints are handled by converting one variable in terms
of another and using the objective function thus obtained to solve an un-
constrained optimization problem. Inequality constraints are converted into
equality constraints through the use of slack variables [81].
Consider the non-linear optimization problem shown in Equation (3.1). The
inequality constraints can be converted into equality constraints by using
slack variables [81].
minw
e(w)
subject to: h(w) = 0
w` ≤ w ≤ wu
(3.17)
where w is the new set of variables including the slack variables and w` and
wu signify lower and upper bounds on w. The variables w are partitioned as
basic and non-basic variables, w = [wnb wb]. The reduced search direction
is obtained as follows,
s =∂e
∂wnb
−[∂e
∂wb
]T
Jwb(3.18)
where s is the search direction, wnb are non-basic variables, wb are basic
variables, and J denotes the Jacobian.
100
Successive linear programming methods The successive linear program-
ming methods use a first-order Taylor’s series approximation around an ini-
tial point to convert the non-linear functions into approximate linear func-
tions. The original variables are replaced by the deviations around the initial
point [80]. The resulting LP problem is solved until there is an improvement
in the value of the objective function. The values of the design variables thus
obtained now become the initial point for further linearization and resolving
of the resulting LP problem. This process is repeated until an optimum is
reached. The optimum for the successive LPs will be the optimum of the
original NLP if the original NLP is not highly non-linear. If it is highly
non-linear, the linear approximations may lead to totally incorrect search
directions.
Successive quadratic programming methods. Analogous to successive
linear programming methods, successive quadratic programming (SQP) meth-
ods exist that generate a quadratic approximation to the objective function.
The SQP approach also generates linear approximations to the constraints.
The approximate quadratic problem then can be solved in a number of ways.
The inequality constraints can be converted to equality constraints by the
use of slack variables [81].
101
3.3 Mixed-integer programming (MIP)
The decision variables in an optimization may be non-continuous [80]. For
example, a decision may have be whether a unit should be on (true or the use
of a ’0’) or off (false or the use of a ’1’). In some cases, the solution may be
rounded to the nearest integer without a serious loss of optimality. However, in
a number of cases, that is not possible. When a problem has both continuous
and integer decision variables, the problem is called a mixed integer programming
(MIP) problem and can be represented as follows,
mind,y
e(d,y) d ∈ <n, y ∈ {0, 1}q
subject to: h(d,y) = 0
g(d,y) ≤ 0
(3.19)
where q is the number of binary variables.
In this discussion only binary integer variables are considered. However, a sim-
ilar problem might be set up for other integer variables as well. If the optimization
problem is linear in the objective function and the constraints, it is a mixed-integer
linear programming (MILP) problem. However, if either the objective function or
the constraints is non-linear then the problem is said to be a mixed integer non-
linear programming (MINLP) problem. A special case when only binary integer
variables (e.g., {0,1}) are present is referred to as a binary integer programming
(BIP) problem.
Generally, integer programming problems, whether linear or non-linear, are
102
more difficult to solve than their counterparts and usually require a large compu-
tational effort due to the combinatorial nature of the discrete variables [80]. Any
value of the integer variable leads to an LP problem. Thus, as the number of
integer variables increase so does the complexity and solution time.
3.3.1 Mixed-integer linear programming (MILP)
MILP problems arise in the chemical industry in a number of processes. For
instance, gasoline blending [85], scheduling [86], batch process design [54, 87], and
flexibility of chemical processes [88], to name a few.
An MILP problem with binary variables can be represented by,
mind,y
cT d + fT y d ∈ <n, y ∈ {0, 1}q
subject to: Ad + by ≤ 0
(3.20)
where c and f are vectors of parameters; d and y are continuous and binary
variables, respectively; and A and b are coefficients associated with the inequality
constraints.
A brute force approach to examine all LPs formed due to the combinatorial
nature of the discrete variables is almost always an intractable problem. Thus,
different methods have been developed to solve MILP problems. The two most
popular approaches are the branch-and-bound algorithms and the cutting plane
algorithms.
Branch-and-bound methods. In this approach, the integer variables are
103
allowed to take continuous values. This is known as LP relaxation [89]. The
initial LP problem is solved. If all the discrete variables have discrete values
at the solution, then this is the solution to the MILP problem. If only some
of the discrete variables are integer valued then for all variables which have
non-integer values, two subsequent LP problems are formulated and solved,
one at the upper bounds and another at the lower bound of these variables.
This process is repeated until discrete values for all discrete variables are
obtained [80].
The variables at which branching is done are called the branching variables.
The choice of these variables is crucial and may be based on user-specified
priorities [89].
Cutting plane methods. After the relaxed LP problem is solved, artificial
linear constraints, called cuts, are generated, which force the non-integer
variables to take integral values. By so doing, the feasible region is made
successively smaller until the optimum is reached.
3.3.2 Mixed-integer non-linear programming (MINLP)
Duality theory is used in the numerical techniques to solve MINLP problems.
Therefore, some concepts about duality theory are first presented.
104
3.3.2.1 Duality of optimization problems
Primal and dual problems are the two representations of non-linear problems
[84]. Consider an optimization problem defined as,
mind
e(d)
subject to: h(d) = 0 g(d) ≤ 0
(3.21)
This problem is said to be in the primal form. The dual objective function of this
problem is obtained from the Lagrange function of the primal objective function.
That is,
maxλ,µ≥0
infd
L(d, λ, µ)
subject to: L(d, λ, µ) = e(d) + λTh(d) + µTg(d)
(3.22)
Thus, the dual is a maximization problem where as the primal is a minimization.
In addition, the objective function of the maximization in the dual problem is a
function of the Lagrange multipliers. The definitions and properties of the primal
and dual problems give rise to the concepts of weak and strong duality.
Theorem 3.3.1. Weak duality: The objective function of the primal problem
provides the upper bound for the objective function for the dual problem for any
feasible solution of the primal problem.
For any feasible solution d∗ and let λ∗ and µ∗ be the corresponding feasible
solution for the dual problem. Then,
e(d∗) ≥ infd
L(d, λ∗, µ∗)
105
The class of MIPs where either the objective function or the constraints is non-
linear is called a mixed-integer non-linear programming (MINLP) problem. Non-
linearity can be exclusively in the integer domain (quadratic assignment model
[84]), exclusively in the continuous domain (input-output relationships between
flows, temperatures, etc.), or in a joint integer-continuous domain.
MINLP problems arise in chemical engineering processes. For example, the
integration of design and control [52, 15, 42, 50], batch process design [84], and
the design of multi-product plants [48, 49].
These problems combine the complexity arising due to the combinatorial ten-
dency of the discrete variables with the non-linearity in the continuous variables
[84] and therefore are quite difficult to solve. However, a few algorithms have had
success with solving this problem. These algorithms are based on decomposition
approaches that make use of the duality of the optimization problem. Two of the
more common approaches are outlined.
Generalized Benders Decomposition (GBD). The GBD formulation is
based on the concept of complicating variables [90]. Complicating variables
are those variables that when held constant promote finding a solution to
the optimization because,
1. The optimization problem can be decomposed into a number of inde-
pendent problems.
2. The optimization problem becomes convex even though the original
106
problem is not.
3. A well known structure of the problem results, for which efficient algo-
rithms already are available to find a solution.
In the GBD, at each step, the original problem is split into the dual problem
and the primal problem using the duality of the original problem. The primal
problem generates upper bounds on the solution while the dual problem
generates the lower bounds.
Consider the original MINLP problem given by,
mind,y
e(d,y) d ∈ <n, y ∈ {0, 1}q
subject to: h(d,y) = 0
g(d,y) ≤ 0
(3.23)
The primal problem is obtained by fixing the y-variables and solving the re-
sultant problem only in the d-space. The primal problem at the kth iteration
can be stated as,
mind
e(d,yk) d ∈ <n
subject to: h(d,yk) = 0
g(d,yk) ≤ 0
(3.24)
107
The dual problem makes use of non-linear duality theory [84],
miny
α α ∈ <
subject to:
e(dk,yk) +∇ye(dk,yk) (y − yk) +
µk[g(dk,yk) +∇yg(dk,yk) (y − yk)
]≤ α
λk[g(dk,yk) +∇yg(dk,yk) (y − yk)
]≤ 0
d ∈ <n
(3.25)
where α is based on a Lagrange function given as,
L(d,y, λ, µ) = e(dk,yk) +∇ye(dk,yk) (y − yk)+
µk[g(dk,yk) +∇yg(dk,yk) (y − yk)
]and λ and µ are the Lagrangian multipliers.
Once the primal and master (or dual) problems are formulated, the solution
procedure is as follows:
1. Choose a feasible initial values for the set of discrete variables, y = y1.
Using these values, solve the primal problem to obtain the optimal d
and the Lagrange multipliers λ1 and µ1.
2. Using the optimal values found in the previous step, solve the master
problem.
3. Solve the primal problem using the solution found from the master
108
problem. Update the upper bound. If the difference between the upper
and lower bounds are smaller than some user-defined error criterion,
terminate the program, the solution found at this iterate is the optimal
solution. If the criterion does not hold, the solution of the primal
problem is used in the next iteration of the master problem. If the
solution of the master problem is found to be infeasible, solve instead a
feasible problem i.e., the primal problem is re-solved at the point where
it became infeasible using certain minimization procedures [90]. Once
a feasible solution to the primal problem is obtained, the Lagrange
multipliers are found and the master problem is resolved.
Other variations on the procedure associated with applying the GBD are
available based on the calculation of the Lagrangian in the master problem
(3.25) and on whether the master problem objective function can actually
be calculated. The interested reader is referred to Floudas [84] for further
information.
Outer Approximation (OA). The outer approximation method and its
variants are based on decomposing the initial problem into primal and master
problems. The OA methods exploit the convexity of the objective function
(if present or the objective function is transformed to a convex problem)
and the constraints to generate an outer approximation (linearization) to
the original problem. The convex problem is an MILP and provides lower
109
bounds on the original problem. The MILP forms the master problem while
the primal problem is obtained by choosing particular values of the discrete
variables. The steps in the OA algorithm are as follows:
• The primal problem (Equation (3.24)) is solved for different constant
values of the discrete variables.
• Using the solution of the primal problem, the master problem is for-
mulated by linearizing the objective function and constraints and using
the duality of the original problem.
miny
α α ∈ <
subject to: e(dk,yk) +∇e(dk,yk)T
d− dk
y − yk
≤ α
g(dk,yk) +∇g(dk,yk)
d− dk
y − yk
≤ 0
d ∈ <n
(3.26)
The solution of these problem gives the corresponding values of both
discrete and continuous variables. The master problem provides the
lower bound on the solution while the primal problem provides the
upper bound and both the dual objective function and the primal ob-
jective function are non-decreasing and non-increasing (monotonic), re-
spectively. Therefore, after a finite number of iterations, the upper and
110
lower bounds converge to within a small ε of each other. Thus, as the
iterations increase, the bounds move towards each other till they come
within some error of each other. The optimum solution is the solution
at the bounds.
3.4 Dynamic optimization
The operation of a chemical process is a dynamic proposition [83]. Appli-
cations of dynamic optimization are in the areas such as process control [28],
scheduling[85], fault detection, and monitoring [37]. A typical dynamic optimiza-
tion problem may be formulated as follows,
minu
Φ(x(tf ),d(tf ),u(tf ),p) +
∫ tf
0
F(x,d,u,p)dt
subject to: x = f(x,u), x0 = 0
g(x,d,u,p) = 0
xL ≤ x ≤ xU
dL ≤ d ≤ dU
uL ≤ u ≤ uU
tf` ≤ tf ≤ tf
u
(3.27)
where tf is the final time, F is the objective function, Φ is the objective function
at the final time, x are the state variables, u are the control variables, and p are
parameters of the system, such as the heat transfer coefficient, etc. There are
usually two approaches to solve this class of problems. The first is the variational
111
approach based on the variational principle [28, 91] and the other is the full or
partial discretization approach [28, 92].
3.4.1 Variational methods
Analogous to the Lagrangian for NLP problems, the following Lagrangian equa-
tion can be defined for DP problems,
H ≡ F (x,u) + λT(t) f(x,u) (3.28)
where λ(t) are time-dependent multipliers [28]. The above equation is often re-
ferred to as the Hamiltonian. The necessary conditions for optimality of the solu-
tion to Equation (3.28) are given by the weak maximum principle [28],
Theorem 3.4.1. Weak Maximum Principle
In order for a control u to be optimal, the Hamiltonian H must be maximum
along the unconstrained portions of the control trajectory,∂H
∂u= 0, with the time-
dependent multipliers, λ, chosen such that
dλ
dt= −∂H
∂x
λ(tf ) =∂Φ
∂x
Φ is the objective function defined in Equation (3.27). Using the maximum
principle, a boundary value problem is developed, which is solved along with
the constraints. The variational method is computationally expensive and not
112
feasible for larger problems [83]. Therefore, discretization techniques have been
formulated based on discretizing the control variables or both the control and the
state variables in the time domain. Partial discretization methods are classified
as either dynamic programming or direct sequential methods.
Dynamic Programming. In the dynamic programming approach, the
time horizon is divided into a number of stages, each with the same fixed
length [92, 93]. Over each stage, the control variables are represented by
linear or constant functions of time. The best control variable value at each
stage is determined and the bounds on the control variables are reduced at
each successive iteration until the optimal profile for the objective function
is obtained. Then, the collection of control variable values at each stage give
the control trajectory. These techniques are not useful for large problems as
their convergence rate is very slow as compared to other methods.
Direct Sequential Methods. The direct sequential methods also are known
as control parametrization methods [28, 94]. The control variables are dis-
cretized in time and can be represented by a set of trial functions,
ui(t) =s∑
j=1
aij ψij(t)
Using an NLP solver, the values of aij are determined to optimize the ob-
jective function. The choice of trial functions is important because the trial
functions also determine the functional form of the control variables. This
113
is one primary drawback of the control vector parametrization method since
the trial functions have to be chosen a priori [28, 91, 92].
Full discretization methods discretize the control variables and the state vari-
ables in the time domain and use NLP solvers to solve the optimization at each
step. Since the problem is made very large due to the discretization, special de-
composition techniques have to be developed. In spite of these difficulties, full
discretization methods are preferred for systems where instabilities are likely to
occur or where path constraints are important [83].
Multiple Shooting Methods. In these methods, both the control vari-
ables and the state variables are discretized in time. The control variables
are expressed as in the direct sequential methods,
ui(t) =s∑
j=1
aij ψij(t)
The variables aij form one set of unknowns. The other set of unknowns is
the state variables. Thus, at every stage, the differential-algebraic system
is solved for the values of the coefficients and the state variables. While at
every stage, the constraints have to be satisfied, between different stages, the
solution may not remain feasible. This infeasibility is handled by specifying
using interior point penalty (barrier) functions (Equation(3.16)).
Collocation Methods. In this method, the state and control variables are
discretized in time using appropriate trial functions [94]. Biegler et al. have
114
used Lagrange polynomials to approximate the state variables [95]; Mahade-
van et al have used wavelets to approximate the state and control variable
variables [96]. The advantages of the collocation methods are that the meth-
ods (i) allow the control variables to be discretized to the same accuracy as
differential and algebraic state variables, (ii) can address unstable modes and
(iii) can be used in conjunction with moving finite element methods [76] for
accurate DAE solutions [83]. Applications of these methods have been in the
area of model order reduction and model-based control [97] and parameter
estimation [98].
3.5 Summary
This chapter provided a review of different types of optimization problems
and the techniques that may be used to solve these problems. This sets the
background for a later chapter where both steady-state optimization and dynamic
optimization are used to obtain the optimal design and the optimal controller of
a spray dryer. The study of the methods used for solving MINLP problems is
useful to understand the different decomposition techniques used for handling the
problem of simultaneous design and control.
115
NOMENCLATURE
A Matrix of coefficients of the design variables
B(d) Barrier function
H Positive definite matrix used for determining search direction.
L Lagrange function
P(d) Penalty function
d Vector of design variables
d(` Lower bound on the design variables
du Upper bound on the design variables
e(d) Design objective function
g(d) Inequality constraints for the design optimization problem
h(d) Equality constraints for the design optimization problem
q Number of integer variables
r Penalty parameter
s Search direction in GRG method
wb Basic variables
116
wnb Non-basic variables
y Integer variables
z Some non-zero vector
λ Lagrange multiplier for the equality constraints
µ Lagrange multiplier for the equality constraints
117
CHAPTER 4
SIMULTANEOUS DESIGN AND CONTROL
4.1 Introduction
The design of the process is a steady state concept that results in a process
flowsheet of connected unit operations. On the other hand, the operation of the
process at the designed conditions is not a steady state proposition. Rather, the
process is subjected constantly to disturbances in the feed operating conditions,
utility conditions, and infrequent phenomena such as fouling, catalyst poisoning
or inactivity, and product requirement changes. An approach to account for these
disturbances during the design phase is to employ safety factors, which are typi-
cally based on experiential knowledge. Often, these safety factors are overdesigns
of – a release valve or capacity of a vessel; or redundancy in the number and type
of unit operation. These add-ons, in many instances, drive the process economics
to less than the true optimum, which may lead to discarding the process flow-
sheet for the wrong reasons than for the right ones. Another means to address
disturbances is during the controller synthesis phase that serially follows process
design.
Traditionally, design and control have been developed in a sequential manner
with the control structure synthesis following the design. A simplified list of the
decision steps involved in generating a sequential design and control strategy once
the process flowsheet is known, is shown in Figure 4.1. In a design project, differ-
118
Figure 4.1: Sequential design and control strategy.
ent alternatives to manufacture the product are proposed. Once an alternative is
accepted, the next step is to generate a flowsheet. This involves equipment sizing,
determining all stream flowrates, and the operating conditions. These activities
are carried out using the concepts of optimization [80, 86]. The optimization is
solved to satisfy an economic objective function (profit or loss). Design parame-
ters include: vessel volumes, feed flowrates, stream compositions, pressures, and
temperatures, to name a few. Once the process design and operating conditions
are determined, the control structure is developed assuming that the designer also
has provided the sensors and actuators at locations critical to the regulation of
the plant. The role of the control structure is to produce a satisfactory dynamic
response to expected disturbances, to regulate the controlled variables at their
desired operating setpoints, to maintain safe operations, and to minimize environ-
mental impact.
119
Control structure synthesis involves [99],
• Determining the controlled variables. This is the first step in control struc-
ture synthesis. The controlled variables may be the conversion, product
purity, reaction temperature, etc.
• Determine the control degrees of freedom. The control degrees of freedom
represent the number of variables that can be manipulated to achieve the
control objectives.
The next steps deal with utilizing the control degrees of freedom based on priori-
tizing among different tasks.
1. Energy management. Ensure efficient energy dissipation to thermal utility
systems.
2. Product rate. Determine the variables that most affect the product and use
them to control the product rate.
3. Product quality. The product quality should be controlled.
4. Recycle flows. Recycle streams are flow-controlled to prevent propagation of
disturbances.
5. Individual units. The regulation of individual units may require sophisti-
cated control strategies.
120
6. Plant performance. Overall performance can be achieved by improving the
economics of the process or by improving the dynamic controllability of the
process.
In addition to being subjected to unexpected disturbances, a continuous pro-
cess may be expected to produce different grades of the same product (process
flexibility) [100]. Factors such as controllability, stability, dynamic operability,
switchability, and flexibility should be designed into the process.
The chapter is organized as follows. Section (4.2) introduces simultaneous de-
sign and control concepts. Section (4.3) discusses the issues of design for flexibility.
The integration of design and control is addressed in sections (4.4) and (4.5. The
former introduces a decision-based methodology that relies on an already com-
pleted flowsheet while the latter uses optimization theory to define a the design
and control solutions. An example problem to illustrate these concepts is provided
in section (4.6). The chapter is summarized in section (4.7).
4.2 Introduction
Consider the following design problem,
Design: mind
e(d)%subject to: h(d) = 0, g(d) ≤ 0 (4.1)
Here, d are design variables, e(d) is the objective function, and h(d) and g(d)
represent the equality and inequality constraints, respectively [80, 81].
121
The corresponding optimal control problem is given as follows [91, 28],
Control: minu(t),x(t),t0
Φ(x(T),T) +
T∫t0
L(x(t),u(t)) dt
subject to: x(t) = f(x(t),u(t), t)
η(u(t), t) ≤ 0, Ψ(x(T),T) = 0, x(t0) = x0
(4.2)
where u(t) are the manipulated variables; x(t) are design variables; and η(u(t), t)
and Ψ(x(T ), T ) are constraints; Φ(x(T ), T ) is the terminal cost function; L(x(t),u(t))
is the running cost, x0 is the initial state of the process; and T represents time
horizon for the process to meet the endpoint constraints. For continuous pro-
cesses, T is selected to be a multiple of the largest open-loop time constant, while
for batch processes, T is usually the final batch time [42].
The design and control problems in Equations (6.1) and (6.2) are solved se-
quentially in a traditional framework. Thus, the optimal values of the design
variables that affect the process flowsheet are found first. The design variables are
fixed at these optimal values and a control structure is developed in response to
these design variables. Since the design variables are fixed, any dynamic coupling
between the design and control variables only can be addressed on the control
side. It is possible that a process design that may be optimal in the steady state
sense may not admit a feasible control structure. If however, control issues are
considered during the design phase, infeasibility may be avoided.
Simultaneous optimization of design and control is a multi-objective optimiza-
122
tion problem which may be stated by combining Equations (6.1) and (6.2). Thus,
the simultaneous design and optimization problem can be formulated as follows
[28, 45, 91],
minu(t),x(t),t0,d
e(d) + Φ(x(T),T) +
T∫t0
L(x(t),u(t)) dt
subject to: x(t) = f(x(t),u(t), t,d)
η(u(t), t) ≤ 0, Ψ(x(T),T) = 0, x(t0) = x0
h(d) = 0, g(d) ≤ 0
(4.3)
The simultaneous design and control optimization problem (Equation (6.3))
with its corresponding constraints can be numerically intractable or computation-
ally exhaustive. Different approaches to address these issues are developed on
the premise of a decomposition of the design and control problem to simplify the
solution of the combined problem while at the same time retaining optimality.
Grossmann and coworkers [48, 49] developed a decomposition strategy to in-
corporate flexibility into the design optimization problem. The plant is designed
to operate at certain nominal conditions but it may be required to operate at
non-nominal conditions. By not operating at the nominal, the optimum may no
longer be valid. To address this, Grossmann and coworkers investigated the ef-
fect of parameter uncertainty on the design optimum. They choose to distinguish
between the design of the process for nominal conditions (design stage) and op-
eration under uncertainty (operating stage). In the operating stage, the process
123
parameters are assumed to vary within a known range. This distinction between
the design and operating stages leads to a decomposition. In the operating stage,
the optimization problem is solved by fixing the design variables over a range of
uncertainty. The control variables found for an optimum operating stage are then
used as parameters in the optimization of the design stage. The decomposition
approach of Grossmann and coworkers has been extended by others to incorporate
dynamic aspects, for example, the work by Pistikopoulos and coworkers [52, 50, 51]
to incorporate a dynamic analysis of the process design and controller synthesis.
Floudas et al. considered open-loop controllability measures such as the mini-
mum singular value, relative gain array or the Integral Squared Error (ISE) within
the framework of a multi-objective design optimization problem to account for the
effect of design and control [42]. Floudas et al. make use of several controllabil-
ity criteria in the design phase. Thus, the objective function in the formulation
of the optimization problem reflects the controllability measures chosen. The ε-
constraint method [84], where the control objective is treated as a constraint, is
used to convert the multi-objective optimization formulation to a single objec-
tive optimization. The resultant optimization problem is solved using generalized
Bender’s decomposition and outer approximation methods [90].
Narraway and Perkins propose a back-off method that involves generating a
set of optimal plants. It is expected that the optimum can be found at one of the
constraints. Then, the set of plants are backed away from the active constraints;
the candidate process that requires the least costs in the new sub-optimal location
124
is chosen as the best design.
More recently, Ulsoy and coworkers [44, 45] use a nested optimization strategy
when addressing the problem of design and control. The simultaneous design
and control problem is decomposed into two sub-problems. Rigorous proofs are
provided that show that the proposed objective function guarantees the optimal
process and controller designs.
4.3 Design for flexibility
The simultaneous design and control problem may not be numerically tractable.
Grossmann and coworkers [48, 49] propose a decomposition approach to incorpo-
rate process flexibility in the optimal design. This decomposition approach has
resulted in additional research by Pistikopoulos and coworkers [52, 15, 50, 54] and
Floudas and coworkers [42, 43]. A brief overview of the technique is given below.
4.3.1 Design under uncertainty
In general, process design is carried out by assuming reasonable values for
different parameters (e.g., equilibrium constants, kinetic rate constants, heat and
mass transfer coefficients) associated with the process. A process design optimized
at the assumed parameter values may become sub-optimal at other non-nominal
conditions. The strategy proposes to account for parameter uncertainty by par-
titioning the optimization into two different optimization problems, the design
stage and the operating stage. In the operating stage, the optimal operation of
the process is obtained by minimizing a cost function, while in the design stage
125
the solution obtained from the operating stage is minimized over the range of
uncertain parameters.
Let θ represent a vector of parameters (θL ≤ θ ≤ θU) with bounded
uncertainty intervals. The optimization problem during the process operating
stage is formulated as,
minu(t)
C(d,u(t), θ)
subject to: h(d,u(t),x(t), θ) = 0, g(d,u(t),x(t), θ) ≤ 0
(4.4)
where it is assumed that a feasible design can be found. The optimal solution,
C∗(d, θ), represents the optimal operation of the process over a bounded range of
uncertainty [49].
In the design stage, C∗(d, θ) is minimized over the range of parameters. The
design stage optimization problem is formulated as,
mind
C∗(d, θ)
subject to: ∀θ ∈ T (∃ (j ∈ J [gj(d,u(t),x(t), θ) ≤ 0])
(4.5)
where J is the total number of inequalities.
The inequality condition on gj is re-written as a max-min-max problem as
follows,
χ(d) = maxθ∈T
mind
maxj∈J
[gj(d,u(t),x(t), θ)].
The condition χ(d) ≤ 0, implies that at critical values of θ, points at which the
126
constraints are most likely violated, there exists a feasible design. To reduce the
computational burden, the cost function is discretized over the parameter space.
The important contributions of the approach of Grossmann and coworkers are
the decomposition technique and the incorporation of process flexibility [48, 49].
However, their approach (i) does not include control effects during the design and
(ii) is restricted to multi-period but steady-state problems, i.e. the assumption
that the transients between different operating stages are negligible [48]. Pis-
tikopoulos et al. [50] and Floudas et al. [43] extended the decomposition technique
to simultaneous design and control problems.
Mohideen et al. [52] addresses control effects by using a similar decomposition
to that proposed by Grossmann [48, 49] wherein a particular type of control struc-
ture is assumed and related costs of implementing the control scheme are used
in the objective function of the multi-objective optimization. Another important
feature of the work by Mohideen is the addition of a check on dynamic feasibility
of the process design. The simultaneous design and control optimization strategy
developed by Mohideen et al. [52] are shown in Figure 4.2.
The multi-period optimization problem is formulated in a manner similar to
Grossmann [49]. However, Mohideen et al. choose discrete values of the param-
eters to generate a finite number of scenarios as opposed to Grossmann [48, 49],
who considered all possible values of the parameters. Choosing discrete scenar-
ios reduces the computational burden of solving the optimization problem. The
cost function in the current case (Equation (4.6)) , explicitly considers the costs
127
Initialization
Determine Optimal Design and Control Scheme
Feasibility-Suitability Analysis
Optimal Design and Control Scheme
Update Critical Scenarios
Assume critical scenarios
Fix Design and Control Scheme
Feasible and Stable
Figure 4.2: Steps in the simultaneous design and control approach of [52].
incurred due to a control structure along with the cost function proposed by Gross-
mann [49]. The simultaneous optimization problem is then given as,
mind,y,vv
E [φ(x(tf ),x(tf ),xa(tf ),u(tf ),vv(tf ), θ(tf ),d,y, tf )]
subject to: h (x(t),xa(t),u(t),vv(t), θ(t),d,y) = 0
g (x(t),xa(t),u(t),vv(t), θ(t),d,y) ≤ 0
x(t, ) = f (x(t),xa(t),u(t),vv(t), θ(t),d,y)
x(t0) = x0
(4.6)
where xa(t) are algebraic variables, y are integer variables, vv(t) are the setpoints
and θ are parameters.
128
The dynamic feasibility condition is tested over the time of operation. If χ ≤ 0,
the process and control designs are feasible [51]. The dynamic feasibility condition
is given by,
χ = maxθ
minvv
maxj∈J,t∈[0,tf ]
[gj(d,u(t),x(t), θ)]
The uncertain parameters are assumed to be of two types: (i) high-frequency
and time-dependent, which are treated as disturbances and (ii) low-frequency and
time-invariant. The solution methodology for this optimization problem is based
on using a finite element collocation technique that discretizes both the state and
the manipulated variables. This discretization reduces the differential equations to
an algebraic form. Then, the uncertain parameters and the disturbance variables
are approximated at a finite number of values (scenarios, previously identified).
By considering only a finite number of scenarios the complexity of the problem is
reduced.
Once a solution is obtained for the optimization problem, the next step is
a feasibility analysis of the optimized process and control structure. Since the
optimization is carried out only for discrete values of the parameters (scenarios),
feasibility tests are done to show that the solution obtained is indeed feasible over
the entire range of scenarios. If the solution is found to be infeasible, the values
of the parameters for which the solution is rendered infeasible are used in the
optimization to ensure feasibility (cf. Figure 4.2).
129
4.4 Modified analytical hierarchical procedure
Another methodology, the modified Analytical Hierarchical Procedure (mAHP)
proposes the use of a decision-based methodology for prioritizing among compet-
ing objectives [11, 12]. The mAHP is based on a singular value assessment of the
objectives. Such a technique can be used when a flowsheet already exists and a
plantwide control strategy is to be synthesized. The first step in this process is to
determine the objectives of the process design (conversion, production rate, etc.).
The objectives are associated with the major unit operations in the process flow-
sheet. For example, conversion may be associated with the reactor, purity with
the separation system, etc. The process flowsheet is then decomposed into groups
of unit operations based around the primary unit operations associated with the
process objectives.
Steady-state sensitivity tests are conducted on the different sets of unit oper-
ations and the effect of disturbances on the objectives are tallied. These effects
are used to develop a series of matrices from which a quantitative ranking among
the objectives is obtained for each group of unit operations. A steady state op-
timization of each group of unit operations is carried out to identify any control
constraints. Finally, control structures are synthesized and tested for each group
of unit operations. The groupings with their control structures are combined and
the new plantwide control structure is tested over an expected set of disturbances.
130
4.5 Bi-level optimization
Ulsoy et al. [44, 45] have proposed and implemented bi-level and simultaneous
optimization strategies to account for the effect of coupling between process design
and control. Bi-level optimization is a subset of hierarchical optimization [101].
Hierarchical optimization is a concept in decision-making theory wherein a number
of critical decisions made affects the other decisions [102]. A special case of bi-level
optimization is where a particular set of decisions (leaders) influences other sets
(followers) [103]. When there is only one follower, the entire system is said to be
a bi-level system.
The bi-level optimization concept is natural for design and control problems
because the design decisions can influence the control decisions. The following is
taken from [44, 45, 46].
• A weighted sum of the objective functions in the design (Equation (6.1))
and the control (Equation (6.2)) problems constitutes the objective function
of the leader problem.
• The traditional control objective function (Equation (6.2)) is chosen to be
the objective function of the follower problem.
• The decision variables for the design problem are chosen as the decision
variables for the leader problem, while the decision variables of the control
problem are chosen to be the decision variables of the follower problem.
• The leader optimization problem is solved first. The optimal decision vari-
131
ables found for the leader optimization problem parameterize the follower
problem. The follower problem is then optimized to find a set of optimal
decision variables for both the design and control problems.
• To solve the leader problem, initial estimates of the control decision variables
are required. Care has to be taken that the initial control decision variables
are feasible.
In the nested optimization framework, the problem is formulated as follows,
Leader: mind
wpe(d) + wc
Φ(x(T ), T ) +
T∫t0
L(x(t),u(t)) dt
subject to: h(d) = 0, g(d) ≤ 0
Follower: minu(t),x(t),t0
Φ(x(T ), T ) +
T∫t0
L(x(t),u(t)) dt
subject to: x(t) = f(x(t),u(t), t, ,d)
η(u(t), t), Ψ(x(T ), T ) = 0 x(t0) = x0
(4.7)
where wp and wc are vectors of weights used to specify the relative importance of
the process design to the controller design. The strategy is illustrated schemati-
cally in Figure 4.3.
Ulsoy and coworkers have shown that there is some degree of coupling between
the design and the control. Their approach is based on solving the proposed
bi-level optimization problem, thus, they quantify the coupling term based on
the Karush-Kuhn-Tucker conditions for optimality and Pontryagin’s maximum
132
Figure 4.3: Nested Optimization.
principle [28, 91].
Since the sequential optimization of the process and controller problems ne-
glects this coupling, the solution found in this serial manner can be shown to be
sub-optimal [45, 103]. In the special case when there is no coupling between de-
sign and control, the results of a sequential strategy should be the same as that
of a bi-level optimization strategy. In such a situation, the sequential strategy is
preferred since the computational requirements are less [44].
For the simultaneous design and controller optimization problem, it can be
shown that the combined problem may not be convex even if the underlying sub-
problems are convex [45, 104, 105].
The bi-level optimization strategy proposed reduces the complexity of the prob-
lem by relaxing the controller constraints on the combined problem to arrive at
a set of process design variables in the first step (leader problem). In the second
step, the control formulation is optimized (follower problem).
Different techniques can be used to solve the bi-level optimization problem.
The leader problem (also called the outer loop problem), is a multi-objective opti-
133
mization similar to the optimization formulations considered by Pistikopoulos and
coworkers [52, 15, 50]. The follower problem is a dynamic optimization problem
and may be solved using control-vector parametrization techniques [28, 43]. The
leader optimization results in a Pareto set, i.e. either of the objective functions can
be reduced further only at the expense of the other. Once the leader problem is
solved, the process design variables obtained appear as parameters in the follower
problem.
The bi-level and simultaneous optimization formulations both account for cou-
pling between the process and controller designs. However, the full-blown si-
multaneous optimization problem is computationally exhaustive. The traditional
sequential optimization approach to process and controller designs neglects any
coupling. Thus, the bi-level approach provides a compromise between the two ex-
tremes of simultaneous and sequential optimization approaches. In fact, wp/wc →
∞ corresponds to the sequential optimization problem. On the other hand, the
simultaneous design and control problem is similar to the leader problem in the
bi-level optimization, with the difference being that the leader problem is opti-
mized over only the process design variables while the simultaneous problem is
optimized over all the variables (both process and control). The bi-level approach
will be used to determine the optimal process and controller designs for the spray
dryer introduced in chapter 2.
134
4.6 Example: reactor and flash system
A hypothetical system consisting of a jacketed continuous stirred-tank reactor
(CSTR) followed by a flash drum is used to demonstrate the coupling between
process and controller designs. A liquid mixture of two generic components, A
and B, is the feed to the reactor. A first-order reaction occurs converting A to
product B.
A→ B (4.8)
The reactor effluent stream (product B and unreacted A) is the feed to the flash
drum where A and B are separated at a constant pressure. Unreacted A is returned
to the reactor while the product B is removed from the system. A schematic of
the flowsheet is shown in Figure (4.4).
Figure 4.4: Schematic of a jacketed continuous stirred-tank reactor and a flashthat is used to produce product B from reactant A.
The reaction inside the reactor occurs in the liquid phase with a dependence on
135
temperature given by an Arrhenius expression. The requirements on the process
are: 70% conversion of reactant A per pass of the reactor and a 98% purity of
product B from the flash drum.
4.6.1 Modeling and simulation
First-principles models of the reactor and flash system are developed. The
values of the reaction rate constant and other parameters are taken from [106].
The process models are solved using the gPROMS c© (PSE, UK) modeling and
simulation software. The flash drum is treated as an equilibrium stage; as such
an equilibrium calculation is carried out to simulate the dynamic behavior of the
flash.
• CSTR
dV
dt= F0 − F
d(V CA)
dt= F0CA0 − FCA − V kCA
d(V T )
dt= F0T0 − FT − λV kCA
ρCP
− UAH(T − TJ)
ρCP
TJ
dt=
FJ(TJ0 − TJ)
VJ
+UA(T − TJ)
ρJVJCJ
(4.9)
136
• Flash drum2∑
j=1
xj = 1
2∑j=1
yj = 1
Fv = f Fin
Fl = (1− f)Fin
Fin x0j = (Fv) yj + (Fl)xj
v = 0.064
√ρl − ρv
ρv
Mv =2∑
j=1
yj Mwj
Ml =2∑
j=1
xj Mwj
(4.10)
where subscripts J and f stand for the jacketed CSTR and the flash drum, re-
spectively; and subscripts l and v stand for liquid and vapor phases, respectively.
All of unreacted A recovered from the flash is recycled back to the reactor,
F0 = Ffresh + Fv
Definition of the variables can be found in Table 4.1.
4.6.2 Sequential design and control
The design and control objectives for this process can be stated as follows:
• Reactor design: Find the smallest volume that will satisfy at least 70%
conversion of A.
• Flash design: Find the smallest volume (drum diameter) that will satisfy a
137
Tab
le4.
1:R
eact
oran
dflas
hva
riab
ledefi
nit
ions
Var
iable
Defi
nitio
nU
nit
s
AH
eat
tran
sfer
area
ft2
CA
Fee
dm
ole
frac
tion
ofA
Dim
ensi
onle
ssC
PH
eat
capac
ity
ofco
olan
tB
tu-(
lbm
-oR
)−1
CP
Hea
tca
pac
ity
ofpro
cess
liquid
sB
tu-(
lbm
-oR
)−1
CA
0Fee
dco
mpos
itio
nofA
lbm
ol-ft−
3
Df
Fla
shdru
mdia
met
erft
FR
eact
orex
itflow
rate
ft3h−
1
Fin
Fee
dflow
rate
ft3h−
1
F0
Rea
ctor
inle
tflow
rate
ft3h−
1
FJ
Ste
amflow
rate
ft3h−
1
Fv
Fla
shdru
mva
por
flow
rate
ft3h−
1
Fl
Fla
shdru
mliquid
flow
rate
ft3h−
1
Mv
Vap
orpro
duct
mol
ecula
rw
eigh
tlb
-(lb
mol
)−1
Mw
jM
olec
ula
rw
eigh
tof
com
pon
entj
lb-(
lbm
ol)−
1
TR
eact
orte
mper
ature
oR
T0
Init
ialfe
edte
mper
ature
oR
TJ
Jac
ket
tem
per
ature
oR
TJ0
Inle
tja
cket
tem
per
ature
oR
UO
vera
llhea
ttr
ansf
erco
effici
ent
Btu
-(s-
ft2-o
R)−
1
VC
ST
Rvo
lum
eft
3
fFra
ctio
nof
feed
that
isva
por
ized
Dim
ensi
onle
ssk
Rea
ctio
nra
tes−
1
vFla
shdru
mfluid
fth−
1
xj
Liq
uid
mol
efr
acti
ons
ofco
mpon
entj
Dim
ensi
onle
ssx
oIn
itia
lliquid
mol
efr
acti
onof
com
pon
entA
Dim
ensi
onle
ssy j
Vap
orm
ole
frac
tion
sof
com
pon
entj
Dim
ensi
onle
ssλ
Hea
tof
vapor
izat
ion
Btu
/lbm
olρ
Den
sity
ofth
epro
cess
liquid
slb
m-ft−
3
ρl,ρ
vLiq
uid
and
vapor
den
siti
eslb
m-ft−
3
ρj
Cool
ant
den
sity
lb-ft−
3
138
bottoms purity of 80% B.
The objective functions in the design problems are cost functions associated with
the capital costs of both units. The operating costs are not considered in this
example. The constraints on the conversion and purity are inequality constraints.
The design variables are chosen to be the reactor volume (V ) and the fraction of
feed to the flash drum that is vaporized (f). The design problems for the reactor
and flash system is defined as follows,
minV,f
0.5 (V 2 +D6f )
subject to: fR(CA, T, TJ , V, FJ) = 0 fF (Fl, Fv, x, y,Df ) = 0
500 ≤ T ≤ 725 500 ≤ TJ ≤ 725
XA ≥ 0.7 Conversion of A
Df ≥√
1.27Mv Fv
45 ρv v0.8 ≤ yB ≤ 1.0
0 ≤ Fl ≤ 100 0 ≤ FJ ≤ 200
(4.11)
The optimal control problem may be stated as follows:
• Reactor control: Maintain the conversion of A at the optimum determined
from the process design solution by manipulating the steam flowrate (FJ).
• Flash drum control: Maintain the fraction of feed vaporized, by manipulating
the flash drum bottoms product flowrate (Fl).
In other words, the optimal control problem is to maintain the specified reactor
conversion and flash product purity. The manipulated variables are the steam
139
flowrate and the flash drum liquid effluent.
minFl,FJ
tf∫t0
((XA −X∗A)2 + (f − f ∗)2) dt
subject to: z = fR(CA, T, TJ , V, FJ) z ≡ [CA, T, TJ ]
fF (Fl, Fv, x, y,Df ) = 0
500 ≤ T ≤ 725 500 ≤ TJ ≤ 725
XA ≥ 0.7 0.8 ≤ xB ≤ 1.0
0 ≤ Fl ≤ 100 0 ≤ FJ ≤ 200
(4.12)
The optimization results for the sequential procedure are given in Tables 4.2
and 4.3) while the results for the nested optimization procedure are given in Tables
4.4 and 4.5. Since the optimum value of the control objective functions are much
smaller than that of the design objective functions, a normalization is appropri-
ate. Let each of the sequential objective functions have a value of 1. The bi-level
objective functions are taken to be whatever fraction they are of their correspond-
ing sequential objective function i.e. the leader objective function corresponds to
the sequential design objective function since both optimizations are carried out
over the design variables; while the follower objective function of the bi-level op-
timization is taken to be correspond to the sequential dynamic optimization. The
normalized bi-level objective function is about 15% of the sequential objective
function. This shows that the two solutions are very different. The bi-level opti-
mization strategy can provide better results especially when the coupling between
140
the process design and the controller design is not weak.
Table 4.2: Design optimization results: sequential
Definition Variable Optimum
Design variable V 39.0 ft3
Design variable f 0.2905Objective function 0.5 (V 2 +D6
f ) 3.707e5
Table 4.3: Dynamic optimization results: sequential
Definition Variable Optimum
Control variable FJ 51.25 ft3/hrControl variable Fl 19.66 ft3/hr
Objective function
tf∫t0
((XA −X∗A)2 + (f − f ∗)2) dt 5.45e-4
4.7 Summary
In this chapter, the integration between process design and controller design
was introduced. The consideration of the controller design during the process
design stage itself was motivated. Different techniques for integration of design
and control were discussed.
The bi-level strategy proposed by Ulsoy et al. [44, 45] was compared to the
traditional serial strategy on a simple example that of a combined continuous
stirred-tank reactor and flash drum. The results indicate the potential of the bi-
level strategy when there is strong coupling between the process design and the
controller design. The bi-level strategy is preferred over the simultaneous opti-
mization of the design and control problems because of the reduced computational
141
Table 4.4: Design optimization results: bi-level, leader
Definition Variable Optimum
Design variable V 48.23 ft3
Design variable f 0.2425Objective function 0.5 (V 2 +D6
f )+ 1.035e5tf∫
t0
((XA −X∗A)2 + (f − f ∗)2)dt
Table 4.5: Dynamic optimization results: bi-linear, follower
Definition Variable Optimum
Control variable FJ 53.27 ft3/hrControl variable Fl 21.69 ft3/hr
Objective function
tf∫t0
((XA −X∗A)2 + (f − f ∗)2) dt 3.27e-7
Table 4.6: Comparison of sequential and bi-level optimization
Design obj Dynamic obj Norm sum Product purity
Sequential 3.707e5 5.45e-4 2 0.91Bi-level 1.104e5 (leader) 3.27e-7 (follower) 0.3 0.91
burden.
142
NOMENCLATURE
C(d, u(t), θ) Overall cost function for the plant
C∗(d, θ) Optimal operation of a given plant over a fixed range of uncertainty
L(x, t) Running cost
T Time horizon
d Vector of design variables
e(d) Design objective function
g(d) Inequality constraints for the design optimization problem
h(d) Equality constraints for the design optimization problem
t Time
t0 Initial time
u(t) Vector of manipulated variables
vv(t) Set-points
x(t0) Initial value of state variables
x(t) State variables
xa(t) Algebraic variables
143
y Integer variables
Φ(x, t) Terminal cost function
Ψ(x(T),T) End point constraints on the state variables
η(u(t), t) Constraints on the manipulated variables
θ Vector of uncertain parameters
144
CHAPTER 5
SPRAY DRYER CONTROL
In this chapter, it is assumed that the spray dryer design is complete and a
control strategy is to be designed to regulate the product properties.
In spray drying, the control objective is to regulate the moisture content in
the final product and the final particle size. If on-line particle size measurements
are available, these measurements can be used directly for controller synthesis.
One such control scheme has been demonstrated by Allen et al. [36] where they
used on-line particle size measurements to develop a cascaded proportional-integral
(PI) control strategy to regulate the average particle size for a spray dryer. The
reliability of the on-line sensor was not discussed. The manipulated variable chosen
was the nozzle air flowrate. Other, control schemes are discussed by Masters [8].
On-line measurements of the solvent concentration in the product and product
particle size are difficult to obtain [107]. Since the simplest of control strategy,
feedback regulation, relies on measurements of the controlled variables, then esti-
mates of these variables must be obtained from other measurements.
This work discusses three different control schemes in brief and provides some
guidelines for the synthesis of a control strategy for the spray dryer introduced in
chapter 2. A way to regulate the product quality is to maintain the outlet temper-
ature of the product. The product quality is inferred from the outlet temperature.
The regulation of the outlet temperature can be achieved either by varying the feed
145
flowrate or by varying the heating gas temperature. Perez-Correa et al. employ
a first-principles model of the spray drying process to synthesize a control scheme
for a spray drying process where milk is the feed and air is the heating gas [108].
They propose a cascaded control scheme to control product quality. The outlet
air moisture is regulated by the feed flowrate using a PI control structure. Mea-
surements of the outlet product moisture is obtained from off-line measurements
of the product moisture. A first-principles model is used to provide the setpoint
for the product moisture. The error between this setpoint value and the value
obtained from off-line measurements drives the setpoint of the outlet air moisture.
In this work, a control strategy to regulate the spray dryer described in Chapter
2 is to be designed. This chapter restricts its attention to simple PI controllers
and feedback strategies. The controlled variables chosen are the product particle
size and the final product moisture content and the manipulated variables are the
air feed rate and the inlet droplet size. Reliable on-line measurements of either
controlled variables are not assumed to be available mirroring the corresponding
laboratory scale system.
The on-line, realtime output measurements available are the air temperatures
throughout the spray drying chamber. Thus, a soft sensor approach is taken in
which, the temperature measurements are used to infer the final product moisture
and the final product particle size. The model presented in Chapter 2 is used to
generate data to develop the inferential models. Once the inferential models are
validated, sensitivity analysis is carried out to choose the pairing of the input to
146
output. The control structure is implemented on the spray dryer system and the
closed-loop performance in the face of disturbances are presented and analyzed.
The remainder of the chapter is organized as follows: The next section 5.1
explains the concept of inferential modeling. Section 5.2.1 develops two principal
component regression (PCR) models to estimate the final particle size (particle
diameter) and residual moisture content (fraction of solvent in particles). The
control structure used for the regulation of the spray dryer is discussed in section
5.3 along with results for disturbance rejection. A summary of the work done is
given in section 5.4.
5.1 Inferential modeling
Inferential modeling has been applied to a number of chemical process appli-
cations such as control of a distillation column [109, 110, 111], control of batch
processes [40, 37], and process monitoring [112, 113]. Inferential models, also
known as soft sensors, are useful when measurements of the variables of interest
are not available or if the measurements are not timely to be useful in any on-line
realtime application.
Inferential models rely on other on-line measured variables to estimate the de-
sired unmeasured variables. Multivariate statistical methods are quite popular for
the generation of inferential models due to their ease of use. Methods within this
category include principal component analysis (PCA), projection to latent struc-
tures also known as partial least squares, (PLS), principal component regression
147
(PCR), and multiple linear regression (MLR) [39, 114, 115]. Related methods
include the class of ARMA (auto-regressive moving average) models [116] and
NARMA (non-linear auto-regressive moving average) models of which artificial
neural networks is a special class [117, 118].
If first-principles models of the process are available, then state estimation
techniques that can be used to develop a state estimator include the Kalman filter
and its variants [32], and the Luenberger observer [119]. The Kalman filter is a
stochastic state estimator which seeks to reduce the variances in the unmeasured
output variables [28]. The primary limitations on the Kalman filter are the as-
sumed distributions on the process and measurement noise and the convergence
of the covariance of the state errors.
In some cases however, using a state estimator is not possible or feasible. For
instance, if the measurement of the primary variable has a large delay or if the
amount of computational effort in synthesizing the estimator is prohibitively high,
as is the case for distributed systems, then statistical process control methods such
as PCR and PLS may be used to provide estimates of desired variables from other
available measurements. For example, Rokhlenko et al. [120] developed a robust
estimator for a continuous-stirred tank system.
148
5.2 Principal component estimator
5.2.1 Principal component analysis
Fitting a set of observations to highly correlated data using the usual regression
methods may lead to errors due to the collinearity of the data set [121]. Principal
component analysis is a technique for fitting data to a set of observations that may
be correlated. It is based on projecting the data into a subspace of the observations
that is obtained by generating principal components of the observations. The
principal components are independent, linear combinations of the observations
arranged in the order of decreasing variability [115, 122]. Principal component
analysis is similar to singular value decomposition [114].
Consider a data set of primary variables Y ∈ Rm×r and secondary variables
X ∈ Rm×n, where n and p are the number of variables in each data set and x and
y are vectors of observations at sample time m. Usually, n << m and r << n.
The objective is to estimate Y from the secondary variable measurements. To
obtain the principal components of X, the data are scaled to unit variance and
zero mean. Then, principal components (PCs) of the normalized data set, X, are
then given by
tk = p′kx, k = 1, . . . , n (5.1)
where pk is the eigenvector corresponding to the k-th largest eigenvalue of the
covariance matrix of X [122]. The total number of PCs is equal to the number of
variables in the data set. However, only the first few PCs are needed to reconstruct
149
the data set quite accurately, especially if the data are collinear. The PCs are found
in an ordered manner, with the first PC accounting for the greatest direction of
variability, the second PC for the second greatest direction of variability and so
on. By selecting only the first few this reduces the dimensionality of the data.
The neglected PCs are usually process noise.
The projection of the set X is given by
X = TPT + E (5.2)
where P ∈ Rn×l is a matrix whose columns are the eigenvectors of the covariance
matrix of X. The matrix P also is called the loadings matrix. The matrix T ∈
Rm×l is called the scores matrix and consists of coefficients that also are the
coordinates of the X variables in the principal component space. The error vector,
E, is due to considering less than n number of PCs [38].
5.2.2 Principal Component Regression
To estimate Y from X, the Y-set is regressed against the scores, T of the
X-set. In the current work, a linear function of the set of scores is chosen as the
functional form,
Y = TB (5.3)
150
where the B is the matrix of coefficients found from,
B = (TTT)−1TTY. (5.4)
In the spray dryer studied, the measurable output variables are the air tem-
peratures. The nominal operating conditions are the design conditions listed in
Chapter 2 and tabulated in Table 5.1 for the reader’s convenience.
Table 5.1: Nominal Operating Conditions
Operating parameters Value
Liquid feed flowrate 25 mLmin−1
Feed temperature 298 KDrying gas (air) temperature 644 KAmbient temperature 298 KFraction of polymer in feed 0.2Final moisture content ≤10%
To generate the calibration data set, the model developed in Chapter 2 is
exercised about the nominal operating point using ±10% changes in the feed
concentration and ±5% changes in the feed flowrate. The X set is composed
of the inlet and exit temperatures and five different temperature measurements
located inside and along the length of the spray drying chamber. The number
five is selected to match the experimental spray dryer. The eigenvalues of the
covariance matrix of the X set show that the first two eigenvalues are very large
compared to the others. Thus, the first two principal component vectors represent
better than 99% of variability in the measurements. The first four eigenvalues are
151
listed in Table 5.2.
The first two PCs are used to reconstruct the calibration data set. The R-
squared value for the fit of the reconstructed data to the actual data is found to
be 93%.
Table 5.2: Eigenvalues of the covariance matrix of the X.
# Value
1 451.952 100.543 1.14 0.06
Figure 5.1 compares the temperatures reconstructed from the first two PCs to
the data.
Using the scores obtained from the projection of the temperature data onto
the first two PCs, PCR models of the residual moisture content and final particle
size are developed. The PCA scores are regressed against the residual moisture
content and final particle size data, respectively. The models obtained are,
RMC = 0.1103 + 0.7430∆T1 + 0.724∆T2 + 0.7293∆T3 + 0.7385∆T4
+0.7423∆T5 + 0.7297∆T6 + 0.6941∆T7
Dp = 3.67× 10−3 + 0.1234∆T1 + 0.1261∆T2 + 0.1282∆T3 + 0.1286∆T4
+0.1258 ∗∆T5 + 0.1191∆T6 + 0.1200∆T7
(5.5)
where RMC is the residual moisture content, Dp is the final droplet size and Ti
152
Figure 5.1: Comparison of the reconstructed temperature data using 2 principalcomponents with temperature measurements at the inlet (T1), the outlet(T7) andthe center (T4) of the spray dryer.
(i=1,. . .,7) are the air temperatures. The performance of the PCR models to
predict data (validation data) not used to generate the PCs are shown in Figures
5.2 and 5.3. The validation data set is generated by making step changes of ±15%
in the feed concentration and ±10% in the total feed flowrate.
5.3 Inferential control
The PCR models are used to estimate the product residual moisture and mean
particle size in a multiple loop PI structure. The schematic of the control structure
is shown in Figure 5.4.
153
Figure 5.2: Predictions of residual moisture content. Comparison of the PCRmodel predictions to data. Solid line: Full model. Dashed line: PCR predictions.
Figure 5.3: Predictions of final particle size (diameter). Comparison of the PCRmodel predictions to data. Solid line: Full model. Dashed line: PCR predictions.
154
Controller Plant
Inferentialmodel
y* yp
yest ys
u+-
Figure 5.4: A schematic of the inferential model in a feedback control strategy.
The manipulated variables (u) are the inlet droplet size and the inlet air
flowrate. Pairing the controlled and manipulated variables is based both on pro-
cess knowledge and the use of the Relative Gain Array (RGA) [30]. Using the
steady-state gains of the controlled variables (y) to unit changes in u, the RGA
is found to be
RGA =
0.6914 0.3086
0.3086 0.6914
(5.6)
where
u =
inlet air flowrate
inlet droplet size
y =
residual moisture content
final particle size
Thus, the mean particle size will be regulated by the inlet droplet size and the
residual moisture content by the inlet air flowrate. The RGA provides only steady-
state information ignoring any dynamics. However, since the time constant of the
spray drying process is about 2.5 s, the use of the RGA is justified.
155
The selected pairing is checked for structural stability using the Niederlinski
index [30]. If the Niederlinski index (N) is less than zero then the pairing is struc-
turally unstable. This result is both necessary and sufficient for a 2×2 system. The
Niederlinski index for this system is 1.45. Thus the proposed control-manipulated
pairing is structurally stable.
The velocity form of the PI control law is implemented. The form was presented
in Chapter 1 but is repeated here for the reader’s convenience,
∆pk = Kc
[(εk − εk−1) +
∆t
τIεk +
τD∆t
(εk − 2εk−1 + εk−2)
]
The two PI loop parameters, the gain (Kc) and the reset time (τI) are selected
to reject disturbances in the feed composition and the feed flowrate. The values
of the tuning parameters for both controllers are listed in Table 5.3. The particle-
size controller is a reverse acting controller, i.e. the output increases as the input
increases while the residual moisture content controller is a direct acting controller.
The performance of the controllers is shown in Figures 5.5 and 5.6. Figure
5.5 shows the closed-loop performance for a 10% decrease in the polymer feed
concentration. The residual moisture content initially decreases before returning
to its nominal value. The controller responds by decreasing the air flowrate. Due to
the decrease, less water is evaporated and therefore, the residual moisture content
remains constant. The process takes about 26 seconds to return to the nominal
value from the time the disturbance occurs. The maximum overshoot is 0.023. A
156
decrease in the feed concentration of the polymer means there is more water in
the droplets. To keep residual moisture content constant, the air flowrate should
increase. This is shown in Figure 5.7.
Figure 5.6 shows the closed-loop performance of the particle size loop. As can
be seen from the RGA, there is some cross-interaction between these two loops.
Hence, the controllers are de-tuned to diminish some of the cross-interaction. The
settling time is about 40 seconds for the mean particle diameter. The maximum
overshoot is 0.76×10−3 cm. The inlet particle diameter is shown in Figure 5.8. As
the feed concentration decreases, the inlet droplet size decreases to adjust for the
final particle size becoming larger.
Table 5.3: The PI controller parameters.
Loop Kc τI
Particle size 0.92 (dimensionless) 0.3 (s)Residual moisture -6 (g s−1) 0.4 (s)
The closed-loop performance for an increase of 15% in the feed flowrate is
investigated. The results are shown in Figures 5.9 and 5.10. Initially, the closed-
response of both controlled variables exhibit some oscillatory behavior. This be-
havior is due to the significant difference in the properties of the streams entering
the spray dryer and the properties of the air-particulate system that already exists
inside the spray dryer. However, once the transient has past, the PI controllers
are able to regulate the system. The settling time is about 45 seconds for both
the residual moisture content and the mean particle diameter. The maximum
157
Figure 5.5: Closed-loop performance of the residual moisture content to a 10%decrease in polymer feed concentration. Dashed line: desired setpoint.
Figure 5.6: Closed-loop performance of the mean particle diameter to a 10% de-crease in polymer feed concentration. Dashed line: desired setpoint.
158
Figure 5.7: Air flowrate response to a 10% decrease in polymer feed concentration.
Figure 5.8: Inlet droplet mean diameter to a 10% decrease in polymer feed con-centration.
159
overshoot is 0.0525 for the residual moisture content and 1.15×10−3 for the mean
particle diameter. As the feed flowrate increases, the air flowrate also increases as
more water also is fed to the spray dryer. At the same time, an increase in the
amount of water leads to an increase in the feed droplet size. Figure 5.11 shows
the response of the air flowrate while Figure 5.12 shows the response of the feed
droplet size.
5.4 Summary
In this chapter, a simple feedback control strategy that uses a proportional-
integral (PI) control law was used to regulate product residual moisture content
and the mean particle size. The manipulated variables used were the input droplet
size and the inlet air flowrate. It was noted that the controlled variables cannot be
directly measured on-line in realtime. To address this problem, linear inferential
sensors using principal component regression (PCR) were developed. The PCR
model’s predictions appear to capture the relationship between the air tempera-
ture measurements and the controlled variables satisfactorily. The PCR models
were used in an inferential control strategy. Proportional-integral controllers were
synthesized and tuned to reject disturbances in the feed composition and feed
flowrate. Due to multivariable interactions between the input and output vari-
ables, the PI controllers were de-tuned, providing sluggish closed-loop responses.
160
Figure 5.9: Closed-loop performance of the residual moisture content to a 15%increase in feed flowrate. Dashed line: desired setpoint.
Figure 5.10: Closed-loop performance of the residual moisture content to a 15%increase in feed flowrate: Dashed line: desired setpoint.
161
Figure 5.11: Air flowrate response to a 15% increase in feed flowrate.
Figure 5.12: Inlet droplet mean diameter response to a 15% increase in feedflowrate.
162
CHAPTER 6
IMPLEMENTATION OF THE BI-LEVEL OPTIMIZATION STRATEGY
In this chapter, the bi-level optimization strategy is applied to the spray dryer
system and compared to the solution found using the sequential strategy.
The chapter is organized as follows. Section 6.1 re-iterates the concepts of se-
quential optimization. The bi-level optimization concepts are discussed in section
6.2. The spray dryer problem as a case study is presented in section 6.3 and the
results obtained by using the sequential and bi-level optimization strategies are
compared. The results are summarized in section 6.4.
6.1 Sequential optimization strategy
Consider the following design problem [80, 81],
Design problem: mind
e(d)
subject to: h(d) = 0 g(d) ≤ 0
(6.1)
where d are design variables, e(d) is the objective function, and h(d) and g(d)
represent the equality and inequality constraints, respectively.
163
The corresponding optimal control problem is given as follows [28, 91],
Control problem: minu(t),x(t),t0
Φ(x(T),T) +
T∫t0
L(x(t),u(t)) dt
subject to: x(t) = f(x(t),u(t), t)
η(u(t), t) ≤ 0, Ψ(x(T),T) = 0, x(t0) = x0
(6.2)
where u(t) are the manipulated variables; x(t) are design variables; and η(u(t), t)
and Ψ(x(T),T) are constraints; Φ(x(T ), T ) is the terminal cost function; L(x(t),u(t))
is the running cost, x0 is the initial state of the process; and T represents time
horizon for the process to meet the endpoint constraints.
The design and control problems in Equations (6.1) and (6.2) are solved se-
quentially in a traditional framework. The design problem is solved first subject to
the steady state model of the process and other equality and inequality constraints.
The optimal control problem pertains to choosing the operating conditions to min-
imize some objective function. Since any coupling between design and control is
neglected, it is possible that a process design that is optimal in the steady state
sense may not admit a feasible control structure [45, 46]. One technique of trying
to improve the process and controller optima is to iterate between the design and
control optimizations [44]. In the iterative technique an initial plant and controller
design is available. The design objective is then optimized without compromising
the control objective. Thus, the control objective appears as a constraint in the
design problem. Next, the control objective, Equation (6.2), is optimized with the
164
optimum design variables parameterizing the control problem. The two optimiza-
tion problems are solved repeatedly until there is further improvement in either
objective functions.
The effect of design on control can be accounted for explicitly by solving the
design and control problem simultaneously. Simultaneous optimization of design
and control is a multi-objective optimization problem which may be stated by com-
bining Equations (6.1) and (6.2). Thus, the simultaneous design and optimization
problem can be formulated as follows [28, 45, 91],
minu(t),x(t),t0,d
e(d) + Φ(x(T),T) +
T∫t0
L(x(t),u(t)) dt
subject to: x(t) = f(x(t),u(t), t,d)
η(u(t), t) ≤ 0, Ψ(x(T),T) = 0, x(t0) = x0
h(d) = 0, g(d) ≤ 0
(6.3)
The simultaneous design and control optimization problem (Equation (6.3))
with its corresponding constraints may be numerically intractable or at least com-
putationally exhaustive [44]. Different approaches to address these problems have
been developed [44, 49, 50].
6.2 Bi-level optimization strategy
An alternative to the sequential, sequential-iterative and simultaneous opti-
mization strategies is a bi-level optimization strategy [44, 103]. The mathematical
formulation of this strategy has been addressed in Chapter 4. The equations
165
pertaining to this strategy are,
Leader: mind
wpe(d) + wc
Φ(x(T ), T ) +
T∫t0
L(x(t),u(t),d) dt
subject to: h(d) = 0, g(d) ≤ 0
Follower: minu(t),x(t),t0
Φ(x(T ), T ) +
T∫t0
L(x(t),u(t)) dt
subject to: x(t) = f(x(t),u(t), t,d)
η(u(t), t), Ψ(x(T ), T ) = 0 x(t0) = x0
(6.4)
where wp and wc are vectors of weights used to specify the relative importance of
the plant design to the controller design.
6.3 Design and control of the spray drying process
In this section, the spray drying process is used to demonstrate the a bi-level
optimization strategy. The reader is referred to Chapter 2 for details on the spray
dryer design.
The requirements on the product quality for the spray dryer are a residual
moisture content that is less than 10% while the desired throughput is 2 kg prod-
uct/hr. Given a particular mean inlet droplet size, a mean outlet particle size is
desired. The final particle size becomes another product quality requirement.
The product quality requirements and throughput act as inequality and equal-
ity constraints, respectively. The objective function chosen is a profit function. It
is the difference between the profit obtained by selling the product and the cost
166
of operating the equipment, i.e. the cost of utilities and the cost of the feedstock.
The utility costs are obtained from [80]. A reliable source for the product pric-
ing wasn’t available. Hence, the product price has been estimated from various
supplier catalogs. The objective function for the design problem is given by,
e(Lc, Dc, T1, τr) = (ProdRate)(D)−[(FRateW)(Water) + (FRateA)(Air) + (FRateP)(Polymer)]
The dimensions of the chamber, the residence time and the hot gas inlet tem-
perature are chosen as the design variables. The solvent is water while the heating
gas is air. Thus, the steady state design optimization problem for the spray dryer
is formulated as follows,
Design problem: maxLc,Dc,T1,τr
e(Lc, Dc, T1, τr)
subject to: Design Model = 0
RMC ≤ 0.1 0.002cm ≤ Dp ≤ 0.004cm
(6.5)
where FRateW is the amount of water in the spray, Water is the price per unit of
water, FRateA is the air flowrate, Air is the cost per unit of air, FRateP is the
amount of polymer used and Polymer is the cost per unit of polymer. Produc-
tion Rate, ProdRate, is the amount of product obtained while D is the cost of
a unit mass of product (micro-particles). The optimization is carried out using
gPROMS c© (PSE, Inc., UK) software environment. The results of the optimiza-
tion are tabulated in Table 6.1. The residence time obtained from the sequential
167
design optimization is 2.55 s.
Table 6.1: Sequential design optimization of the spray dryer
Definition Symbol Optimal value
Air temperature T1 640 KLength of the chamber Lc 0.61 m
Diameter of the chamber Dc 0.55 mResidence time τr 2.55 s
Objective Function Ξ 0.0056 $/s
6.3.1 Optimal control of spray dryer
The spray dryer is a distributed parameter system and optimization techniques
for such systems are inherently more difficult [28]. Recognizing this, and also
the fact that there are no actuators along the length of the reactor, a lumped
parameter model for the spray dryer is developed. This model is a system of
ordinary differential equations (ODEs) and known techniques are available for
optimal control of such systems [83]. The lumped parameter model is validated
against residual moisture content and the final mean particle size obtained from
the distributed model by inserting a step change of 5% in the feed concentration.
The validation results are shown below.
168
The optimal control problem is stated as follows:
mindin,Wg
T∫t0
((RMC −RMC∗)2 + ((Dp −Dp∗)/Dp
∗)2 + (Wg −Wg∗)2)dt
subject to: Lumped Model
0.005 cm ≤ din ≤ 0.012 cm
20 g/s ≤ Wg ≤ 90 g/s
360 K ≤ Tout ≤ 450 K
(6.6)
where RMC is the residual moisture content, Dp is the final particle size, and Wg
is the air flowrate. The manipulated variables are the inlet mean droplet size and
the inlet air flowrate. The optimization is carried out over a horizon of 50 seconds
and the manipulated variables are discretized into five regions of 10 seconds each.
The manipulated variables are assumed to be piece-wise constant in each region.
The optimal trajectory for the residual moisture content is shown in the Figure
6.3. The remaining results are tabulated in Table 6.2.
Table 6.2: Sequential optimal control of the spray dryer
Name Symbol Optimal value at time = 50 s
Air flowrate Wg 39.7 g/sInlet droplet size din 0.006 cm
Objective Function - 0.1158
169
6.3.2 Bi-level optimization of spray dryer
The bi-level optimization strategy converges if it is initialized with a feasible
design and controller [44]. It follows that the design and controller generated in
the sequential stage can be used as the starting point for the bi-level optimization.
In the leader problem, the combined design and controller problem is solved
with the constraints of the design problem. The design variables are the decision
variables while the manipulated variables are obtained from the solution of an
optimal control problem and act as parameters in the leader problem. Equal
weights are chosen i.e.
wp = wc = 0.5.
The choice of 1/2 means that both the design and control problems are of equal
importance.
The leader problem is:
maxLc,Dc,T1,τr
0.5e(Lc, Dc, T1, τr)− 0.5
T∫t0
((RMC −RMC∗)2 + ((Dp −Dp∗)/Dp
∗)2 + (Wg −Wg∗)2) dt
subject to: Lumped Model
0.005 cm ≤ din ≤ 0.012 cm
20 g/s ≤ Wg ≤ 90 /s
360 K ≤ Tout ≤ 450 K
(6.7)
The results from the leader problem parameterize the follower problem which
is a pure optimal trajectory problem. The results from the leader problem are
170
Table 6.3: Bi-level design optimization of the spray dryer: Leader
Name Symbol Optimal value
Air temperature T1 800 KLength of the chamber Lc 0.52 m
Diameter of the chamber Dc 0.48 mObjective Function Ξ 0.054
tabulated in Table 6.3. The residence time obtained from the nested optimization
is 1.88 s. The optimal trajectory is shown in the Figure 6.4. It can be seen
that the manipulated variable in the nested strategy does not make large moves
as compared to that in the sequential strategy. This may be due to the higher
temperature the bi-level design is operating at as seen in the Table 6.3.
Table 6.4: Bi-level optimal control of the spray dryer: Follower
Name Symbol Optimal value at time = 50 s
Air flowrate Wg 32.01 g/sInlet droplet size din 60e-6 m
Objective Function - 0.1023
6.3.2.1 Comparison between bi-level and sequential optimization strategy
Comparing the optimal values of the objective function obtained from both, the
sequential and the bi-level, optimization strategy is not straightforward. One way
is to do a normalization of the objective functions using the sequential objective
function as unity. This was done in the CSTR and flash example system (see
Chapter 4).
171
An alternative is to subtract the objective function value of the follower prob-
lem from the leader problem. The difference roughly represents the weighted
design objective function. In this problem, the choice of the weights makes this
easier. These design objective function values are presented in Table 6.5.
Table 6.5: Comparison of bi-level and sequential optimization strategy (design)
Name Optimal Value
Leader objective function 0.0058Design (sequential) objective function 0.0056
The optimal control objective functions of the two strategies can be compared
directly (see Table 6.6).
Table 6.6: Comparison of bi-level and sequential optimization strategy (optimalcontrol)
Name Optimal Value
Follower objective function 0.1023Control (sequential) objective function 0.1158
6.3.3 Effect of tighter operational constraints
The effect of changing the operating constraints on both the sequential and bi-
level strategies is tested by varying the constraints. For example, when the outlet
temperature constraint is varied in the sequential strategy, the control objective
function increased to 0.2452, an increase of 112% while the objective function
for the follower in the bi-level optimization increased marginally to 0.1205 (20%
172
increase). These results indicate that the controller in the bi-level optimization
works less than the controller in the sequential strategy. While the bi-level design
is a little less profitable, it is more controllable and more flexible. Table 6.7 lists
the controller objective functions for different constraints.
Table 6.7: Comparison of bi-level and sequential optimization strategy (optimalcontrol) for different constraints
Constraint Follower objective function Sequential objective function
360K ≤ Tout ≤ 380K 0.1205 0.24520.1 ≤ RMC ≤ 0.2 0.1932 0.3252
0.05 ≤ RMC ≤ 0.08 0.3954 0.4401
6.3.4 Effect of the weights wp and wc
Figure 6.5 shows the effect of different values of the ratio wp/wc on the optimal
leader value. As this ratio gets very large the bi-level optimum is the sequential
optimum meaning that the design problem is more important than the controller
design problem.
6.4 Summary
In this chapter, a bi-level optimization strategy was applied to the spray dryer
system. The bi-level strategy was compared to the traditional sequential strategy
and was found to be superior to the sequential strategy. The bi-level was also found
173
to be more flexible than the sequential design and control strategy. In addition,
the bi-level strategy is guaranteed to obtain the optimum if it is initialized from
a feasible point [44]. This is not the case with a sequential optimization strategy.
Indeed, it is possible in a sequential optimization to arrive at a design which admits
no feasible controller but has great return on investment. In the bi-level strategy,
since the control is considered along with the design itself, the design will always
have a feasible controller associated with it.
174
Figure 6.1: Comparison between lumped model (dashed line) and full model (<)for final mean particle size.
Figure 6.2: Comparison between lumped model (dashed line) and full model (<)for residual moisture content.
175
Figure 6.3: Optimal trajectory for residual moisture content (top) along withtrajectory for air flowrate (bottom).
176
Figure 6.4: Optimal trajectory for residual moisture content (top) along withtrajectory for air flowrate (bottom) for the bi-level strategy.
177
Figure 6.5: Effect of changes in the ratio wp/wc
178
NOMENCLATURE
L(x, t) Running cost
RMC Residual moisture content
T Time horizon
Wg Air flowrate
d Vector of design variables
dp Final particle size
e(d) Design objective function
g(d) Inequality constraints for the design optimization problem
h(d) Equality constraints for the design optimization problem
t Time
t0 Initial time
u(t) Vector of manipulated variables
x(t0) Initial value of state variables
x(t) State variables
Φ(x, t) Terminal cost function
179
Ψ(x(T),T) End point constraints on the state variables
η(u(t), t) Constraints on the manipulated variables
180
CHAPTER 7
SUMMARY AND FUTURE WORK
This work has contributions in the area of modeling and design of spray dryers,
spray dryer control and integration of design and control. These will be summa-
rized in the following sections.
7.1 Modeling and design
A first-principles model for a spray dryer was developed. This fundamental
model consists of two sub-models. The lower level model describes the phenomena
occurring on the surface of a single droplet as it is exposed to a hot gas. The
model is a system of three coupled, parabolic, partial differential equations. The
boundary conditions of the model give rise to a moving boundary problem due to
the evaporation of solvent from the droplet. This set of equations was solved using
the Gradient Weighted Moving Finite Element Method (GWMFE). One validation
of the model was that the model predicted the formation of a skin on the surface
of the droplet. The time to skin formation was defined as the time required for the
polymer concentration at the surface to reach 100%. The sensitivity of the time to
skin formation to different parameters such as heat and mass transfer coefficients
and to operating conditions was studied.
The upper level model used a Lagrangian framework to track the particles
as they travelled the length of the spray drying chamber. The air flow patterns
181
were modelled using empirical expressions. This model was validated against
experimental data of the air temperatures taken along the length of the spray
dryer. The heat and mass transfer between the droplet phase and air also was
modelled. This model, known as the design model, consisted of a system of five
partial differential equations. These equations were solved in a Lagrangian frame
of reference to actually track individual particles. his model was then used to
develop a design of a spray dryer to manufacture hollow micro-particles.
7.2 Control synthesis
A regulatory control structure was developed for the spray dryer. The prod-
uct moisture and mean droplet size were recognized to be the two most impor-
tant quantities as far as the product was concerned. Since on-line measurements
of these quantities are usually neither available nor reliable, inferential models
based on air temperature measurements along the length of the spray dryer were
developed. The full model was exercised by injecting disturbances in the feed
concentration and feed flowrate to collect data from which to construct a Princi-
pal Component Regression (PCR) estimator. The manipulated variables available
were the feed air flowrate and the inlet droplet size. Using a Relative Gain Array
and the Niederlinksi index, the air flowrate was paired to product moisture content
and particle size to inlet droplet size. Simple, feedback that uses a proportional-
integral control law was synthesized and tested against disturbances in the feed
concentration and the feed flowrate.
182
7.3 Integration of design and control
A novel optimization strategy to account for the coupling between design and
control was studied. It was shown by comparing with the results from the bi-level
optimization strategy that the solution obtained using a sequential (i.e. design
first followed by control synthesis) strategy was sub-optimal. The bi-level strategy
accounts for the coupling between design and control by explicitly considering the
control objective function in the design problem. This is called the Leader problem.
In the follower problem, only the control objective function is optimized. This
strategy was tested on the spray dryer system and compared with the sequential
strategy. The spray dryer design and controller obtained from both strategies were
compared by testing against different operating constraints than what they were
designed for. It was found that the bi-level solution provided greater flexibility as
compared to the sequential solution.
7.4 Future work
During the course of this work, a number of areas were identified for future
research. Some of the recommendations are listed below.
Spray dryer modeling : The modeling work for spray dryers hampered due to
lack of experimental data to validate the models. For example, velocity data
for flow inside the spray dryer was not available to test against the models.
Thus better on-line or in-line data collection instrumentation is necessary.
Spray dryer design : In this work, design for spray dryers was developed using
183
first-principles models. However, far more research can be conducted in the
field of developing spray dryer design using more sophisticated modeling
techniques. The use of Computational Fluid Dynamics (CFD) models is one
particular area. While CFD models are not of much utility from a control
perspective, they can provide rigor to the model which may lead to new
insights in spray dryer design.
Integration of design and control : In this work, a bi-level optimization strat-
egy was proposed to account for the coupling between design and control.
This work however, did not address the issues of the extent of coupling, the
nature of coupling, nor any theoretical proof about the extent of the effect of
coupling. Further, other approaches to the integration between design and
control need to be considered.
7.5 Contributions
• A first-principles model for phenomena on a single droplet was developed.
This model was solved using a Gradient Weighted Moving Finite Element
(GWMFE) method. This work resulted in a refereed publication.
V.S. Shabde, S.V. Emets, U. Mann, K. A. Hoo, N.N. Carlson and G.M.
Gladysz (2005) “Modeling a hollow particle production process”, Computers
and Chemical Engineering, 29: 2420-2428, 2005.
• A theoretic approach to design of spray dryers was developed based on first
principles modeling of the spray drying process. This work resulted in a
184
refereed publication.
V.S. Shabde and K.A. Hoo(2006) “Design and Operation of a Spray Dryer
for the Manufacture of Hollow Micro-particles”, Industrial and Engineering
Chemistry Research, 2006.
• An inferential control strategy for regulating the product moisture and mean
particle size of a spray drying system was developed. This work will be
submitted for refereed publication.
• A bi-level optimization strategy was presented for the integration of design
and control. This strategy was applied to the spray dryer system to demon-
strate its efficacy. This work will be submitted for refereed publication.
185
APPENDIX A
Lumped model of the spray dryer
The lumped model of the spray dryer is obtained by taking energy and mass
balances across the liquid (droplet) and gas (air) phases and neglecting the dis-
tributed nature of the events that occur between the entrance and exit of the
chamber.
An energy balance around the air phase leads to an equation for the air tem-
perature at the exit of the spray dryer,
d Ta
dt= (Tain
− Ta)/τ − π kaNudi ni(Ta − Tf )
WgCs(1.1)
Species balance for the solvent (water) around the liquid phase:
dCw
dt=
(Cwin− Cw)
τ− π kaNudi ni
(Ta − Tf )
λWp ρw
(1.2)
The equation for changes in the droplet size:
d
dtdi =
(main
−ma
τ
)4Cp(Ta − Tf )
πdi2 − π kaNu
(Ta − Tf )
λ ρw di
(1.3)
186
APPENDIX B
Computer code for Matlab c©
Matlab c© (v 7.0 R14) has been used extensively in this work to simulate the
spray dryer model. This spray dryer model code is also the basis for generating
the principal components. Some of this code has been shown in this appendix.
Spray dryer modeling
This section contains the computer code for the full model used in the design
of the spray dryer.
Spray dryer modeling
This model contains the computer code for some of the data generation for
usedd to calculate the Principal Components of the spray drying process. The
Principal Components themselves are obtained using an inbuilt Matlab function
“princomp”.
187
188
189
190
191
BIBLIOGRAPHY
[1] D. Fletcher, B. Guo, D. Harvie, T. Langrish, J. Nijdam, and J. Williams.
What is important in the simulation of spray dryer performance and how do
current CFD models perform ? In 3rd International conference on CFD in
minerals and process industries, 2003.
[2] Y. Maa and S.J. Prestrelski. Biopharmaceutical powders: Particle forma-
tion and formulation considerations. Current Pharmaceutical Biotechnology,
1(3):283–302, 2000.
[3] V.S. Shabde, S. V. Emets, U. Mann, K. A. Hoo, N.N. Carlson, and G. M.
Gladysz. Modeling a hollow micro-particle production process. Computers
and Chemical Engineering, 29(11-12):2420–2428, 2005.
[4] P. Narayan, D. Marchant, and M.A. Wheatley. Optimization of spray dry-
ing by factorial design for production of hollow microspheres for ultrasound
imaging. Journal of Biomedical Materials Research, 56(3):333–341, 2001.
[5] D. Oakley. Spray dryer modeling in theory and practice. Drying Technology,
22(6):1371–1402, 2004.
[6] D. Fletcher and T. Langrish. Prospects for modelling and design of spray
driers in the 21st century. Drying Technology, 21:197–215, 2003.
192
[7] N.V. Menshutina and T. Kudra. Computer aided drying technologies. Dry-
ing Technology, 19(8):1825–1849, 2001.
[8] K. Masters. Spray Drying Handbook. John Wiley and Sons., New York, NY,
5th edition, 1991.
[9] I. Zbicinski and R. Zietara. CFD model of counter-current spray drying
process. In 14th International Drying Symposium, volume A, pages 169–176,
2004.
[10] L. Huang, K. Kumar, and A.S. Muzumdar. Use of computational fluid
dynamics to evaluate alternative spray dryer chamber configurations. Drying
Technology, 21(3):385–412, 2003.
[11] E.M. Vasbinder and K.A. Hoo. A decision based approach to plantwide
control structure synthesis. Ind. & Eng. Chem. Res., 42:4586–4598, 2003.
[12] E.M. Vasbinder. Design based approach to integration of chemical process
design and control structure synthesis. Doctor of philosophy, Texas Tech
University, Lubbock,TX, 2004.
[13] S. Emets. An examination of modified heirarchical design structure for chem-
ical process. Master of science, Texas Tech University, Lubbock,TX, 2002.
[14] U.M. Diwekar and P.D. Chaudhari. Process synthesis under uncertainty: A
penalty function approach. AIChE Journal, (3):742–752, 1996.
193
[15] E. N. Pistikopoulos and J. Acevedo. Stochastic optimization based algo-
rithms for process synthesis under uncertainty. Computers and Chemical
Engineering, 22(4):647–671, 1998.
[16] M.M Daichendt and I.E. Grossmann. Integration of hierarchical decompo-
sition and mathematical programming for synthesis of process flowsheets.
Computers and Chemical Engineering, (1-2):147–175, 1998.
[17] R.B. Keey and Q.T. Pham. Behavior of spray dryers with nozzle atomizers.
Chemical Engineer, 311:516–521, 1976.
[18] S. Katta and W. Gauvin. Basic concepts of spray dryer design. AICHE
Journal, 22(4):713–724, 1976.
[19] J. Nijdam, B. Guo, D. Fletcher, and T. Langrish. Lagrangian and eulerian
models for simulating turbulent dispersion and agglomeration of droplets
within a spray. In 3rd International conference on CFD in minerals and
process industries, 2003.
[20] L. Ozmen and T.A.G. Langrish. A study of the limitations to spray dryer
outlet performance. Drying Technology, 21(5):895–917, 2003.
[21] D.E. Oakley, R.E. Bahu, and D. Reay. The aerodynamics of cocurrent spray
dryers. In Sixth International Drying Symposium, pages 373–378, 1988.
194
[22] V.S. Shabde and K. A. Hoo. Design and operation of a spray dryer for the
manufacture of hollow microparticles. Industrial and Engineering Chemistry
Research, 2006.
[23] R.A. Goffredi and E.J. Crosby. Limiting analytical relationships for predic-
tion of spray dryer performance. Industrial and Engineering Process Design
and Development, 22:665–672, 1983.
[24] C. Palencia, J. Nava, E. Herman, G. Rodrıguez-Jimenes, and M. Garcıa-
Alvarado. Spray dryer dynamic modeling with a mechanistic model. Drying
Technology, 20(3):569–586, 2002.
[25] C.T Crowe. Modelling spray-air contact in spray drying systems, volume 1
of Advances in Drying, chapter 3. Hemisphere Publishing, New York, NY,
1980.
[26] P.D. Christofides. Model-based control of particulate processes. Kluwer Aca-
demic Publishers, Norvell, MA, 2002.
[27] J. Bertling, J. Blomer, and R. Kummel. Hollow microspheres. Chemical
Engineering Technology, 27(8):829–837, 2004.
[28] W.H. Ray. Advanced Process Control. Butterworths, Boston, MA, 1989.
[29] D. Seborg, T. F. Edgar, and D. A. Mellichamp. Process Dynamics and
Control. John Wiley, New York, NY, 2nd edition, 2004.
195
[30] B.A. Ogunnaike and W.H. Ray. Process Dynamics, Modeling and Control.
Oxford University Press, New York, NY, 1994.
[31] P. Grosdidier and M. Morari. Closed loop properties from steady-state gain
information. Industrial Engineering Chemistry Fundamentals, 24:22–235,
1985.
[32] R.E. Kalman. A new approach to linear filtering and prediction problems.
Transactions of the ASME–Journal of Basic Engineering, 82(Series D):35–
45, 1960.
[33] D. Zheng. System Identification and Model-based Control for a Class of
Distributed Parameter Systems. Doctor of philosophy, Texas Tech University,
Lubbock, TX, 2002.
[34] C. Brosilow and B. Joseph. Techniques of Model-based Control. Prentice
Hall, Upper Saddle River, NJ, 2002.
[35] e. Camacho and C. Bordons. Model Predictive Control. Springe-Verlag,
London, UK, 2nd edition, 2004.
[36] R.M. Allen and H.H.C. Bakker. Spray dryer control based on online particle
size analysis. Chemical Engineering Research and Design, 72(A2):251–254,
1994.
196
[37] K. A. (Hoo) Kosanovich, K. S. Dahl, and M.J. Piovoso. Improved under-
standing using multiway principal component analysis. Ind. & Eng. Chem.
Res., 35(1):138–146, 1996.
[38] B. M. Wise, N. B. Gallagher, S. W. Butler, D. White, and G. G. Barna.
Development and benchmarking of multivariate statistical process control
tools for a semiconductor etch process: impact of measurement selection
and data treatment on sensitivity. In IFAC SAFEPROCESS’97, pages 35–
42.
[39] K. A. (Hoo) Kosanovich and M.J. Piovoso. Applications of multivariable
statistical methods to process monitoring and controller design. Int. Journal
of Control, 59(3):743–765, 1994.
[40] P. Nomikos and J. McGregor. Batch control using multiway principal com-
ponent analysis. AIChE Journal, 40:1361–1375, 1994.
[41] J. G. Zeigler and N. B. Nichols. Process lags in automatic-control circuits.
In Transactions of the A.S.M.E., volume 65, pages 433–444, 1943.
[42] M. L. Luyben and C. A. Floudas. Analyzing the interaction of design and
control- I & II. Computers and Chemical Engineering, 18:933–993, 1994.
[43] C.A. Schweiger and C.A. Floudas. Interaction of design and control: opti-
mization with dynamic models. In W. Hager and P. M. Pardalos (Editors),
197
editors, Optimal control: Theory, Algorithms and Applications, New York,
NY, 1997. Kluwer Academic Publishers.
[44] H.K. Fathy, J.A. Reyer, P.Y. Papalambros, and A.G. Ulsoy. On the coupling
between plant and controller optimization problems. In American Control
Conference, pages 1864–1869, 2001.
[45] G.A. Brusher, P.T. Kabamba, and A.G. Ulsoy. Coupling between model-
ing and controller-design problems-part ii: Design. In American Control
Conference, pages 2613–2617, 1994.
[46] G.A. Brusher, P.T. Kabamba, and A.G. Ulsoy. Coupling between model-
ing and controller-design problems-part i: Analysis. In Transactions of the
A.S.M.E., volume 119, pages 498–502, 1997.
[47] E.M. Vasbinder, K.A. Hoo, and U. Mann. Synthesis of plantwide control
structures using a decision-based methodology. In P. Seferlis and M. Geor-
giadis, editors, The Integration of Design and Control. Elsevier Science, Am-
sterdam, 2004.
[48] I.E. Grossmann and R.W.H. Sargent. Optimum design of multipurpose
chemical plants. Industrial and Engineering Chemistry Proces Design and
Development, 18(2):343–349, 1978.
[49] I.E. Grossmann and K.P. Halemane. Optimal process design under uncer-
tainty. AIChE Journal, 29(3):425–433, 1982.
198
[50] R. Ross, J. D. Perkins, E. N. Pistikopoulos, G. L.M Koot, and J. M. G. van
Schijndel. Optimal design and control of a high-purity industrial distillation
system. Computers and Chemical Engineering, 25(1):141–150, 2001.
[51] P. Seferlis and M. C. Georgiadis (Editors). The integration of process design
and control. Elsevier, San Diego, CA, 2004.
[52] M. J. Mohideen, J. D. Perkins, and E. N. Pistikopoulos. Towards an effi-
cient numerical procedure for mixed integer optimal control. Computers and
Chemical Engineering, 21:S457–S462, 1997.
[53] V. Visweswaran, C. Floudas, M. Ierapatritou, and E.N. Pistikopoulos. A
decomposition-based global optimization approach for solving bilevel linear
and quadratic programs. In C.A. Floudas and P.M. Pardalos, editors, State
of the Art in Global Optimization. Kluwer Academic Publishers, 1996.
[54] E.N. Pistikopoulos and M.G. Ierapetritou. Novel approach for optimal
process design under uncertainty. Computers and Chemical Engineering,
19(10):1089–1110, 1995.
[55] S.F. Alyoquot, P.Y. Papalambros, and A.G. Ulsoy. Quantification and use
of system coupling in decomposed design optimization problems. In ASME
International Mechanical Engineering Congress and Exposition, 2005.
[56] R. Verdumren, H. Straatsma, M. Verschuren, j Haren, E. Smit, G. Barge-
man, and P. Jong. Modelling spray drying processes for dairy products. In
199
Communication at the 1st International Symposium on Spray Drying of Milk
Products, pages 453–463, 2001.
[57] D. Wilcox and M. Berg. Microsphere fabrication and applications: an
overview. Material Research Society, 1994.
[58] M. Farid. A new approach to modelling of single droplet drying. Chemical
Engineering Science, 58(13):2985–2993, 2003.
[59] H. Cheong, G. Jefferys, and C. Mumford. A receding interface model for the
drying of slurry droplets. International Journal of Heat and Mass Transfer,
13:537–546, 1969.
[60] W.A. Sirignano. Fluid dynamics and transport of droplets and sprays. Cam-
bridge University Press, Cambridge, UK, 1st edition, 1999.
[61] F. Kieviet. Modeling Quality in Spray Drying. Doctor of philosophy, Eind-
hoven University of Technology, The Netherlands, 1997.
[62] M. Vinjamur and R. Cairncross. Non-Fickian non-isothermal model for dry-
ing of polymer coatings. AICHE J., 48(11):2444–2458, 2002.
[63] W Lenggoro and K Okuyama. Preparation of nanoparticles via spray route.
Chemical Engineering Science, 58:537–547, 2003.
[64] B. Bird, W. Stewart, and E. Lightfoot. Transport Phenomena. John Wiley
and Sons., New York, NY, 6th edition, 2002.
200
[65] R. Treybal. Mass transfer operations. McGraw-Hill, 3rd edition, 1980.
[66] J. Prausnitz, B. Poling, and R. Reid. Properties of gases and liquids.
McGraw-Hill, New York, NY, 4th edition, 1987.
[67] D. Legendre and J. Magnaudet. The lift force on a spherical bubble in a
viscous linear shear flow. Journal of Fluid Mechanics, 368:81–126, 1998.
[68] G.G. Nasr, A. J. Yule, and L. Bendig. Industrial Sprays and Atomization:
Design, Analysis and Applications. Springer, New York, NY, 1st edition,
2002.
[69] D. Ramakrishna. Population Balances: Theory and Applications to Partic-
ulate Systems in Engineering. Academic Press, New York, NY, 2000.
[70] M. Sommerfeld and S. Blei. Lagrangian modeling of agglomeration during
spray drying processes. In Proceedings of the 9th International Conference
on Liquid Atomization and Spray Systems, 2001.
[71] K. Miller. A geometrical-mechanical application of gradient weighted moving
elements. SIAM journal on numerical analysis, 35(1):67–90, 1997.
[72] J. Crank. Free and moving boundary problems. Oxford University Press,
Oxford, UK, 1987.
[73] P. Seydel, A. Sengespick, J. Blomer, , and J. Bertling. Experimental and
mathematical modeling of solid formation at spray drying. Chem. Eng.
Tech., 27(5):505–510.
201
[74] M. Zerroukat and C. Chatwin. Computational moving boundary problems.
John Wiley and Sons, New York,NY, 1994.
[75] B. Finlayson. Numerical methods for problems with moving fronts. Ravenna
Park Publishers, New York, NY, 1992.
[76] Miller K. and Miller R. Moving finite elements, i. Society for Industrial and
Applied Mathematics journal on numerical analysis, 18(7):1019–1032, 1981.
[77] A.N. Hryamk and A.W. Westerberg. Computational considerations for mov-
ing finite element methods. Chemical Engineering Science, 41(6):1673–1680,
1985.
[78] K. Miller and N. Carlson. Design and application of the gradient weighted
moving finite element method in one dimension. SIAM journal on numerical
analysis, 19(3):766–798, 1998.
[79] M. McCabe, J. Smith, and P. Harriott. Unit Operations of Chemical Engi-
neering. McGraw-Hill, New York, NY, 6th edition, 2000.
[80] T. F. Edgar and D. M. Himmelblau. Optimization of chemical processes.
Mc-Graw Hill, New York, NY, 1988.
[81] C. Onwubiko. Introduction to engineering optimization. Prentice Hall Inc.,
Upper Saddle River, NJ, 2000.
[82] L.T. Beigler, I.E. Grossmann, and A.W. Westerberg. Systematic Methods of
Chemical Process Design. Prentice Hall, New Jersey, 1997.
202
[83] L.T. Biegler and I.E. Grossmann. Retrospective on optimization. Computers
and Chemical Engineering, 28:1169–1192, 2004.
[84] C.A. Floudas. Non-linear and mixed-integer optimization. Oxford University
Press, New York, NY, 1995.
[85] J. Zhenya and M. Irerapetritou. Mixed-integer linear programming model
for gasoline blending and distribution scheduling. Industrial and Engineering
Chemistry Research, 42(4):825–835, 2003.
[86] J. Cerda, G. Henning, and I. Grossmann. A mixed-integer linear program-
ming model for short-term scheduling of single-stage multiproduct batch
plants with parallel lines. Industrial and Engineering Chemistry Research,
36(5):1695–1707, 1997.
[87] E.N. Pistikopoulos and M.G. Ierapetritou. Batch plant design and oper-
ations under uncertainty. Industrial and Engineering Chemistry Research,
35(3):772–787, 1996.
[88] I. E. Grossmann and C. A. Floudas. Active constraint strategy for flexi-
bilty analysis in chemical processes. Computers and Chemical Engineering,
11(6):675–693, 1987.
[89] J. Kallrath. Mixed integer optimization in the chmical process industry:
Experience, potential and future perspectives. Transactions of the Institute
of Chemical Engineers, 78(A):809–822, 2000.
203
[90] A. M. Geffrion. Generalized benders decomposition. Journal of Optimization
Theory and Applications, 10(4):237–260, 1972.
[91] Y.C. Ho and A.E. Bryson. Applied Optimal Control: Optimization, Estima-
tion and Control. Blaisdell Pub. Co., Waltham, MA, 1st edition, 1969.
[92] D.P. Bertsekas. Dynamic programming and optimal control. Athena Scien-
tific, Belmont, MA.
[93] S.A. Dadebo and K.B. McAuley. Iterative dynamic programming for mini-
mum energy control problems with time delay. Optimal Control Application
Methodology, 16:217–227, 1995.
[94] L. Cervantes and L. T. Beigler. Optimization strategies for dynamic systems,
1999.
[95] J. E. Cuthrell and L.T. Biegler. On the optimization of differential-algebraic
process systems. AIChE Journal, 33:1257–1270, 1987.
[96] N. Mahadevan and K. A. Hoo. Model reduction and controller development
for a class of distributed parameter systems. Chem. Eng. Sci., 55:4271–4290,
2000.
[97] D. Zheng and K. A. Hoo. System identification and model-based control for
distributed parameter systems. Compt. & Chem. Engng., 28(8):1361–1375,
2004.
204
[98] G. E. Suares and K. A. (Hoo) Kosanovich. Parameter & state estimation of
a proton-exchange-membrane fuel cell using a sequential quadratic program-
ming approach. Ind. & Eng. Chem. Res., 36(10):4264–4272, year=1997,.
[99] Luyben M and Tyreus B.D.and Luyben W.L. Plantwide Process control.
McGraw-Hill, New York, NY, 1st edition, 1999.
[100] L. Narraway and J. D. Perkins. Selection of process control structure based
on economics. Computers and Chemical Engineering, 18:S511–S515, 1994.
[101] V. Oduguwa and R. Roy. Bi-level optimisation using genetic algorithm. In
2002 IEEE Conference on Artificial Intelligence Systems, pages 322–327,
2002.
[102] P.W. Ue and H. Shuh-Tzy. Linear bi-level programming problems, a review.
The Journal of Operational Research Soviety, 42(2):125–133, 1991.
[103] G. Anandalingam and T.L. Friesz. Hierarchical optimization: An introduc-
tion. Annals of operation research, 34:1–11, 1992.
[104] L. N. Vicente. Bilevel Programming: Introduction, History, and Overview,
volume 1, pages 178–180. Kluwer Academic Publishers.
[105] O. Ben-Ayed and C.E. Blair. Operations Research, 38(3):556–560, 1990.
[106] W.L. Luyben. Process Modeling, Simulation and Control for Chemical En-
gineers. McGraw-Hill, New York, NY, 1990.
205
[107] P. Cunningham and D. O’Callaghan. Modern process control techniques in
the production of dried milk processes - a review. Le Lait, pages 335–342,
2005.
[108] J. R. Perez-Correa and F. Farıas. Modeling and control of a spray dryer: a
simulation study. Food Control, (4):219–227, 1995.
[109] J. Zhang. Inferential feedback control of distillation composition based on
pcr and pls models. In American Conrol Conference, pages 1196–1201, 2001.
[110] T. Mejdell and S. Skogestad. Estimation of distillation compositions from
multiple temperature measurements using partial-least-squares regression.
Industrial & Engineering Chemistry Research, 30(12):2543–2555, 1991.
[111] M. Miyazaki Kano, S. K. Hasebe, and I. Hashimoto. Inferential control
system of distillation compositions using dynamic partial least squares re-
gression. Journal of process control, 10(2-3):157–166, 2000.
[112] J. Kresta, T. Marlin, and J. McGregor. Development of inferential process
models using pls. Computers & Chemical Engineering, 18(7):597–611, 1994.
[113] S.J. Qin, Yuem H., and R. Dunia. Self-validating inferential sensors with
application to air emission monitoring. Industrial & Engineering Chemistry
Research, 36:1675–1685, 1997.
[114] H. Martens and T. Naes. Multivariate Calibration. John Wiley & Sons, New
York, NY, 1989.
206
[115] J.S. Jackson. A User’s Guide to Principal Components. John Wiley & Sons,
New York, NY, 1991.
[116] G. Box, G. Jenkins, and G. Reinsel. Time Series Analysis: Forecasting and
Control. Prentice Hall, Upper Saddle River, NJ, 3rd edition, 1994.
[117] J. Hertz, A. Krogh, and R. Palmer. Introduction to The Theory of Neural
Computation. Addison Wesley, Redwood City, CA, 1991.
[118] K. A. Hoo, E. D. Sinzinger, and M.J. Piovoso. Improving the predictions of
neural networks. J. Process Control, 12(1):193–202, 2001.
[119] D. Luenberger. Observers for multivariable systems. In IEEE Transactions
on Automatic Control, volume AC-11, pages 190–197, 1066.
[120] V. Rokhlenko, A. Guez, K. A. (Hoo) Kosanovich, and M.J. Piovoso. Nonlin-
ear adaptive control with parameter estimation of a cstr. J. Process Control,
5(3):137–148, 1995.
[121] J. Hwang and D. Nettleton. Principal components regression with data-
chosen components and related methods. Technometrics, 45(1):70–79, 2003.
[122] I. Jolliffe. Principal Component Analysis. Springer-Verlag, New York, NY,
1986.
207