Preface ...................................................... xvii
Acknowledgments ............................................... xxi
Author ...................................................... xxiii
PART I Properties and Upgrading of Heavy Oils
Chapter 1 Heavy Petroleum ...................................... 3
1.1 Definition ................................................. 3
1.2 Classification ............................................. 4
1.3 Properties ................................................. 5
1.3.1 Physical and Chemical Properties .................... 5
1.3.2 Asphaltenes ......................................... 6
1.3.3 Chemical Characterization of Asphaltenes ............ 9
1.3.3.1 Experimental.................................... 11
1.3.3.2 Results ........................................ 12
1.3.4 Tendency to Coke Formation ......................... 16
1.3.5 Viscosity .......................................... 17
1.3.5.1 Viscosity of Crude Oils ........................ 17
1.3.5.2 Viscosity of Blends of Crude Oils .............. 17
1.3.5.3 Other Properties ............................... 25
1.3.6 Stability and Compatibility ........................ 27
1.3.6.1 Definitions .................................... 27
1.3.6.2 Analytical Methods ............................. 28
1.4 Assay of Heavy Petroleum .................................. 31
1.4.1 Definition ......................................... 31
1.4.2 Applications ....................................... 32
1.4.3 Types of Assays .................................... 32
1.4.4 Examples of Assays of Heavy Crude Oils ............. 33
1.5 Problems during Upgrading and Refining of Heavy
Petroleum ................................................. 34
References ................................................ 41
Chapter 2 Technologies for Upgrading of Heavy Petroleum ....... 45
2.1 General Classification .................................... 45
2.2 Current Situation of Residue Upgrading .................... 48
2.3 Hydrogen Addition Technologies ............................ 49
2.3.1 Hydrovisbreaking ................................... 50
2.3.2 Fixed-Bed Hydroprocessing .......................... 50
2.3.3 Moving-Bed Hydroprocessing ......................... 51
2.3.4 Ebullated-Bed Hydroprocessing ...................... 52
2.3.5 Slurry-Bed Hydroprocessing ......................... 52
2.4 Carbon Rejection Technologies ............................. 54
2.4.1 Solvent Deasphalting ............................... 55
2.4.2 Gasification ....................................... 55
2.4.3 Coking ............................................. 55
2.4.4 Visbreaking ........................................ 55
2.4.5 Residue Fluid Catalytic Cracking ................... 56
2.5 Emerging Technologies ..................................... 56
2.6 Combination of Upgrading Technologies ..................... 57
2.6.1 Combination of Carbon Rejection Technologies ....... 57
2.6.2 Combination of Hydrogen Addition Technologies ...... 62
2.6.3 Combination of Both Hydrogen Addition and Carbon
Rejection Technologies ............................. 63
References ................................................ 67
PART II Modeling of Noncatalytic Processes
Chapter 3 Modeling of Visbreaking ............................. 73
3.1 Introduction .............................................. 73
3.2 Process Description ....................................... 74
3.3 Types of Visbreaking ...................................... 75
3.3.1 Coil Visbreaking ................................... 75
3.3.2 Soaker Visbreaking ................................. 76
3.3.3 Differences ........................................ 76
3.4 Process Variables ......................................... 77
3.4.1 Feed Properties .................................... 77
3.4.2 Temperature ........................................ 78
3.4.3 Pressure ........................................... 78
3.4.4 Residence Time ..................................... 79
3.4.5 Steam Injection .................................... 79
3.4.6 Main Process Variables ............................. 79
3.5 Chemistry ................................................. 80
3.5.1 C-C Bond Scission .................................. 81
3.5.2 Dehydrogenation .................................... 81
3.5.3 Isomerization ...................................... 81
3.5.4 Polymerization/Condensation ........................ 81
3.5.5 Reactions Involving Heteroatoms .................... 81
3.6 Kinetics .................................................. 81
3.7 Reactor Modeling .......................................... 85
3.7.1 Correlations ....................................... 85
3.7.2 Reactor Model ...................................... 86
3.7.2.1 General Aspects of Coil and Soaker Reactors .... 86
3.7.2.2 Modeling Coil and Soaker Reactors .............. 86
3.7.3 Simulation of the Visbreaker ....................... 92
3.7.3.1 Reactor Characteristics and Operating
Conditions ..................................... 92
3.7.3.2 Feed and Product Properties .................... 93
3.7.3.3 Results ........................................ 93
3.7.3.4 Final Remarks and Recommendations .............. 97
Nomenclature .............................................. 99
Greek Letters ............................................ 100
References ............................................... 100
Chapter 4 Modeling of Gasification ........................... 103
4.1 Introduction ............................................. 103
4.2 Types of Gasifiers ....................................... 104
4.2.1 Moving-Bed Gasifiers .............................. 104
4.2.1.1 Countercurrent Fixed-Bed ...................... 104
4.2.1.2 Co-Current Fixed-Bed .......................... 104
4.2.2 Fluidized-Bed Gasifiers ........................... 105
4.2.3 Entrained-Flow Gasifier ........................... 105
4.2.4 Others ............................................ 106
4.3 Process Variables ........................................ 106
4.3.1 Temperature ....................................... 106
4.3.2 Pressure .......................................... 107
4.3.3 Fluidization Velocity ............................. 107
4.3.4 Air/Steam Ratio ................................... 107
4.3.5 Equivalence Ratio ................................. 107
4.3.6 Particle Size ..................................... 107
4.4 Process Description ...................................... 107
4.5 Chemistry and Thermodynamics ............................. 108
4.6 Modeling of the Gasifier ................................. 110
4.6.1 Model Equations ................................... 111
4.6.1.1 Mass Balance .................................. 111
4.6.1.2 Thermodynamic Equilibrium ..................... 114
4.6.1.3 Energy Balance ................................ 119
4.6.1.4 Heating Value of Synthesis Gas and
Gasification Efficiency ....................... 121
4.6.2 Model Solution .................................... 122
4.7 Simulation of the Gasifier ............................... 123
4.7.1 Validation of the Model ........................... 123
4.7.2 Effect of Reaction Conditions ..................... 125
4.7.2.1 Effect of Pressure ............................ 125
4.7.2.2 Effect of Temperature ......................... 127
4.7.2.3 Effect of Oxygen-to-Vacuum Residue Ratio ...... 127
4.7.2.4 Effect of Water-to-Vacuum Residue Ratio ....... 129
4.7.3 Application of the Model .......................... 130
4.7.3.1 Simulation with Different Vacuum Residues as
Feedstock ..................................... 130
4.7.3.2 Simulation of the Production of Hydrogen ...... 132
References ............................................... 133
Chapter 5 Modeling of Coking ................................. 135
5.1 Introduction ............................................. 135
5.2 Coking Processes ......................................... 136
5.2.1 Delayed Coking .................................... 136
5.2.2 Fluid-Coking ...................................... 138
5.2.3 Flexi-Coking ...................................... 138
5.3 Process Description ...................................... 139
5.4 Process Variables ........................................ 141
5.4.1 Furnace Outlet Temperature/Coke Drum Inlet
Temperature ....................................... 141
5.4.2 Coke Drum Pressure ................................ 142
5.4.3 Combined Feed Ratio ............................... 142
5.4.4 Type of Feed ...................................... 143
5.5 Fundamentals of Coking ................................... 143
5.5.1 Chemistry ......................................... 143
5.5.2 Kinetics .......................................... 145
5.5.3 Thermal Decomposition of Asphaltenes .............. 145
5.6 Kinetics of Coking ....................................... 147
5.6.1 Fractionation of Atmospheric Residue .............. 147
5.6.2 Non-Isothermal Kinetics ........................... 148
5.6.3 Thermal Decomposition ............................. 149
5.6.4 Kinetic Parameters ................................ 152
5.6.5 Remarks ........................................... 154
5.7 Correlations to Predict Coking Yields .................... 156
5.7.1 Correlations ...................................... 156
5.7.1.1 Correlations of Gary and Handwerk (2001) ...... 156
5.7.1.2 Correlations of Maples (1993) ................. 158
5.7.1.3 Correlations of Schabron and Speight (1997) ... 160
5.7.1.4 Correlations of Castiglioni (1983) ............ 161
5.7.1.5 Correlations of Smith et al. (2006) ........... 164
5.7.1.6 Correlations of Volk et al. (2002) ............ 164
5.7.2 Application of the Correlations ................... 166
5.7.2.1 Effect of Feed Properties ..................... 166
5.7.2.2 Effect of Pressure ............................ 168
5.7.2.3 Effect of Temperature ......................... 168
5.7.3 Final Remarks ..................................... 169
Nomenclature ............................................. 171
References ............................................... 171
Chapter 6 Noncatalytic (Thermal) Hydrotreating ............... 175
6.1 Introduction ............................................. 175
6.2 Experimental ............................................. 177
6.2.1 Crude Oils and Residua ............................ 177
6.2.2 Experimental Setup ................................ 177
6.2.3 Reaction Conditions ............................... 179
6.2.4 Analytic Techniques ............................... 180
6.3 Results and Discussion ................................... 180
6.3.1 Two-Reactor Unit .................................. 180
6.3.1.1 Noncatalytic Hydrodesulfurization ............. 180
6.3.1.2 Selectivity toward NHDS and NHDM .............. 183
6.3.1.3 Effect on the API Gravity ..................... 184
6.3.1.4 Effect on Distillation Curves ................. 185
6.3.1.5 Effect on Liquid Product Composition .......... 185
6.3.1.6 Profiles of Axial Temperature ................. 188
6.3.2 One-Reactor Unit .................................. 190
6.3.2.1 Kinetics of NHDS and NHDM ..................... 190
6.3.2.2 Kinetics of Vacuum Residue Conversion ......... 192
6.3.2.3 Kinetics of Noncatalytic Hydrocracking ........ 194
Nomenclature ............................................. 198
Subscripts ............................................... 199
References ............................................... 199
PART III Modeling of Catalytic Processes
Chapter 7 Modeling of Catalytic Hydroprocessing .............. 203
7.1 Introduction ............................................. 203
7.1.1 Importance of Hydrotreating in Petroleum
Refining .......................................... 203
7.1.2 Current Situation ................................. 206
7.2 Process Description ...................................... 208
7.3 Types of Reactors ........................................ 210
7.3.1 Fixed-Bed Reactors ................................ 210
7.3.1.1 Quenching in FBRs ............................. 213
7.3.1.2 Reactor Internals ............................. 214
7.3.2 Moving-Bed Reactors ............................... 216
7.3.3 Ebullated-Bed Reactors ............................ 217
7.3.4 Slurry-Phase Reactors ............................. 219
7.4 Fundamentals ............................................. 219
7.4.1 Chemistry ......................................... 219
7.4.1.1 Hydrodesulfurization .......................... 220
7.4.1.2 Hydrodenitrogenation .......................... 221
7.4.1.3 Hydrodeoxygenation ............................ 223
7.4.1.4 Hydrodemetallization .......................... 223
7.4.1.5 Saturation Reactions .......................... 223
7.4.1.6 Hydrocracking ................................. 223
7.4.1.7 Hydrodeasphaltenization ....................... 224
7.4.2 Reaction Kinetics ................................. 224
7.4.3 Thermodynamics .................................... 229
7.4.4 Catalysts ......................................... 231
7.5 Process Variables ........................................ 233
7.5.1 Reaction Temperature .............................. 233
7.5.2 Hydrogen Partial Pressure ......................... 235
7.5.3 Space Velocity .................................... 236
7.5.4 Hydrogen-to-Oil Ratio and Gas Recycle ............. 236
7.5.5 Catalyst Activation ............................... 238
7.6 Modeling of Hydrotreating of Heavy-Oil-Derived Gas Oil ... 239
7.6.1 Experimental Section .............................. 240
7.6.1.1 Materials and Experimental Setup .............. 240
7.6.1.2 Experimental Tests ............................ 240
7.6.1.3 Analytical Methods ............................ 240
7.6.2 Formulation of the Reactor Model .................. 242
7.6.2.1 Model Assumptions ............................. 242
7.6.2.2 Unsteady State Mass Balances .................. 243
7.6.2.3 Unsteady State Heat Balances .................. 245
7.6.2.4 Boundary Conditions ........................... 245
7.6.2.5 Integration Method ............................ 246
7.6.3 Reaction Kinetic Models ........................... 246
7.6.3.1 Hydrodesulfurization .......................... 246
7.6.3.2 Hydrodenitrogenation .......................... 247
7.6.3.3 Hydrodearomatization .......................... 248
7.6.3.4 Olefins Hydrogenation ......................... 249
7.6.3.5 Mild Hydrocracking ............................ 249
7.6.4 Estimation of Parameters .......................... 250
7.6.4.1 Kinetic Parameters ............................ 250
7.6.4.2 Catalyst Effectiveness Factor ................. 252
7.6.4.3 Hydrodynamic Parameters ....................... 253
7.6.5 Results and Discussion ............................ 254
7.6.5.1 Dynamic Simulation of an Isothermal
HDT Bench-Scale Reactor ....................... 255
7.6.5.2 Dynamic Simulation of an Isobaric
Nonisothermal HDT Commercial Reactor .......... 256
Nomenclature ............................................. 262
Greek Letters ............................................ 264
Subscripts ............................................... 264
Superscripts ............................................. 265
References ............................................... 265
Chapter 8 Modeling and Simulation of Heavy Oil
Hydroprocessing ............................................... 271
8.1 Introduction ............................................. 271
8.2 Description of the IMP Heavy Oil Upgrading Technology .... 272
8.3 Experimental Studies ..................................... 274
8.3.1 Generation of Kinetic Data ........................ 274
8.3.2 Study of the Effect of Various Heavy Feedstocks
on Catalyst Deactivation .......................... 277
8.3.3 Long-Term Catalyst Stability Test ................. 280
8.4 Modeling Approach ........................................ 282
8.4.1 Steady-State Mass and Heat Balance Equations ...... 282
8.4.2 Dynamic Mass and Heat Balance Equations ........... 284
8.4.3 Reaction Kinetics ................................. 286
8.4.4 Scale-Up of Kinetic Data .......................... 287
8.4.5 Catalyst Deactivation ............................. 288
8.4.6 Solution Method ................................... 289
8.4.6.1 Steady-State Simulations ...................... 289
8.4.6.2 Dynamic Simulations ........................... 289
8.5 Data Fitting ............................................. 289
8.5.1 Kinetic Parameters ................................ 290
8.5.2 Deactivation Parameters ........................... 291
8.6 Simulation of the Bench-Scale Unit ....................... 292
8.6.1 Reactor Simulation under Steady Catalyst
Activity .......................................... 292
8.6.2 Reactor Simulations with Time-Varying Catalyst
Activity .......................................... 296
8.6.2.1 Effect of Feedstock Type and Reaction
Temperature on Catalyst Deactivation .......... 296
8.6.2.2 Process Performance during the Catalyst
Stability Test ................................ 297
8.7 Scale-Up of Bench-Unit Kinetic Data ...................... 300
8.8 Simulation of the Commercial Unit ........................ 302
8.8.1 Reactor Design and Simulation under Stable
Catalyst Activity ................................. 302
8.8.2 Reactor Simulation and Analysis during Time-on-
Stream ............................................ 304
8.8.3 Transient Reactor Behavior during Start-Up ........ 307
8.8.3.1 Quenching ..................................... 308
8.8.3.2 Feed Temperature .............................. 310
8.8.3.2 Start-Up Strategy ............................. 313
Nomenclature ............................................. 314
Greek Letters ............................................ 316
Subscripts ............................................... 316
References ............................................... 316
Chapter 9 Modeling of Bench-Scale Reactor for HDM and HDS
of Maya Crude Oil ............................................. 319
9.1 Introduction ............................................. 319
9.2 The Model ................................................ 320
9.2.1 Model Assumptions ................................. 320
9.2.2 Description of the Model .......................... 321
9.2.2.1 Stoichiometric Coefficients for HDS
Reaction ...................................... 322
9.2.2.2 Reaction Rate Coefficients .................... 323
9.2.2.3 Determination of Kinetic Parameters ........... 324
9.2.2.4 Estimation of Transport and Thermodynamic
Properties .................................... 326
9.2.3 Model Solution .................................... 331
9.3 Experimental ............................................. 333
9.3.1 Feedstock Characterization ........................ 333
9.3.2 Experimental Reactor .............................. 333
9.3.3 Isothermal Performance of Reactor ................. 334
9.3.4 Catalyst Properties ............................... 334
9.3.5 Catalyst Loading .................................. 334
9.3.6 Catalyst Activation ............................... 334
9.3.7 Minimizing Mass-Transfer Resistances .............. 334
9.3.8 Effect of Reaction Conditions ..................... 335
9.4 Results .................................................. 335
9.4.1 Stoichiometric Coefficient ........................ 335
9.4.2 Kinetic Parameters for HDS and HDM Reactions ...... 336
9.4.3 Simulation of the Bench-Scale Reactor ............. 339
9.4.4 Comments about Model Assumptions .................. 342
Nomenclature ............................................. 345
Subscripts ............................................... 346
Greek Letters ............................................ 346
References ............................................... 346
Chapter 10 Modeling of Ebullated-Bed and Slurry-Phase
Reactors ...................................................... 349
10.1 Introduction ............................................. 349
10.2 Characteristics of Ebullated-Bed Reactor ................. 351
10.2.1 Parts of the Ebullated-Bed Reactor ................ 352
10.2.1.1 Recycle Cup ................................... 352
10.2.1.2 Flow Distributor System ....................... 353
10.2.1.3 Distributor Grid .............................. 353
10.2.1.4 Downcomer ..................................... 353
10.2.1.5 Emulating Pumps ............................... 354
10.2.2 Advantages and Disadvantages ...................... 354
10.2.3 Catalyst Bed Inventory ............................ 355
10.2.4 Sediment Formation ................................ 356
10.2.5 Catalyst Attrition ................................ 357
10.2.6 Catalyst Deactivation ............................. 358
10.2.7 Process Economics ................................. 361
10.3 EBR Commercial Technologies .............................. 362
10.3.1 H-Oil Process ..................................... 362
10.3.2 T-Star Process .................................... 363
10.3.3 LC-Fining ......................................... 363
10.4 Modeling of Ebullated-Bed Reactor ........................ 364
10.4.1 Hydrodynamic Studies .............................. 366
10.4.2 Scaling-Down Studies .............................. 368
10.4.3 Reactor Modeling .................................. 372
10.5 Modeling of Slurry-Phase Reactors ........................ 382
10.6 Kinetic Study for Hydrocracking of Heavy Oil in CSTR ..... 384
10.6.1 Experimental ...................................... 386
10.6.1.1 Experimental Setup ............................ 386
10.6.1.2 Catalyst Loading and Activation ............... 386
10.6.1.3 Experiments and Product Analysis .............. 386
10.6.2 Results and Discussion ............................ 388
10.6.2.1 Mass-Transfer Limitations ..................... 388
10.6.2.2 Kinetic Modeling .............................. 389
10.6.3 Conclusions ....................................... 395
10.7 Final Remarks ............................................ 395
Nomenclature ............................................. 396
Greek Letters ............................................ 398
Subscripts ............................................... 399
References ............................................... 399
Chapter 11 Modeling of Hydrocracking by Continuous Kinetic
Lumping Approach .............................................. 405
11.1 Introduction ............................................. 405
11.2 Continuous Kinetic Lumping Model ......................... 409
11.2.1 Description of the Model .......................... 409
11.2.2 Model Assumptions for Fixed-Bed Reactor ........... 412
11.2.3 Solution of the Model ............................. 412
11.3 Experimental ............................................. 420
11.3.1 Hydrocracking of Maya Crude Oil ................... 420
11.3.2 Effect of Pressure on Hydrocracking of Maya
Crude Oil ......................................... 420
11.3.3 Simultaneous HDS and Hydrocracking of Heavy ОІ1 ... 420
11.4 Step-By-Step Example for Application of the Model ........ 421
11.4.1 Data Used ......................................... 421
11.4.2 Assumptions Regarding Boiling Points .............. 421
11.4.3 Numerical Solution ................................ 422
11.4.4 Results and Discussion ............................ 422
11.4.4.1 Maximum Boiling Point Temperature ............. 422
11.4.4.2 Domain Partition and Linear Approximation of
the Yield Function ............................ 423
11.4.4.3 Size of Step for Residence Time Variations .... 424
11.4.4.4 Value of Model Parameters ..................... 424
11.4.4.5 Results of the Case of Study .................. 424
11.5 Modeling Hydrocracking of Maya Crude Oil ................. 425
11.5.1 Experimental Results .............................. 425
11.5.2 Parameter Estimation .............................. 426
11.5.3 Validation of the Model ........................... 427
11.5.4 Application of the Model .......................... 429
11.6 Modeling the Effect of Pressure and Temperature on the
Hydrocracking of Maya Crude Oil .......................... 431
11.6.1 Background ........................................ 431
11.6.1.1 Literature Reports ............................ 431
11.6.1.2 Effect of Pressure ............................ 431
11.6.1.3 Importance of Pressure Effect ................. 433
11.6.2 Accounting for the Effect of Pressure ............. 433
11.6.3 Results and Discussion ............................ 434
11.6.3.1 Experiments ................................... 434
11.6.3.2 Dependence of Model Parameters on Pressure
and Temperature ............................... 436
11.6.3.3 Values of Model Parameters as Function of
Pressure ...................................... 438
11.6.3.4 Prediction of Distillation Curves ............. 439
11.7 Modeling Simultaneous HDS and HDC of Heavy Oil ........... 441
11.7.1 Description of the Model .......................... 442
11.7.1.1 Hydrocracking Model ........................... 442
11.7.1.2 Hydrodesulfurization Model .................... 444
11.7.2 Solution of Model ................................. 445
11.7.3 Results and Discussion ............................ 446
11.7.3.1 Hydrocracking Reaction ........................ 446
11.7.3.2 Hydrodesulfurization Reaction ................. 447
11.7.3.3 Final Considerations .......................... 448
11.8 Significance of Parameters of Continuous Kinetic
Lumping Model............................................. 448
11.8.1 About the Model Parameters ........................ 448
11.8.2 Other Factors Affecting the Model Parameters ...... 449
11.8.3 Unresolved Questions and Future Research .......... 449
Nomenclature ............................................. 450
Subscripts ............................................... 451
Superscripts ............................................. 451
Greek Letters ............................................ 451
References ............................................... 452
Chapter 12 Correlations and Other Aspects of Hydroprocessing .. 455
12.1 Correlations to Predict Product Properties
during Hydrotreating of Heavy Oils ....................... 455
12.1.1 Description of Correlations ....................... 457
12.1.2 Results and Discussion ............................ 461
12.1.2.1 Experimental Data ............................. 461
12.1.2.2 Predictions Using Literature Values of
Parameters .................................... 461
12.1.2.3 Prediction Using Optimized Values of
Parameters .................................... 467
12.1.2.4 Correlating Values of Parameters with
Feed Properties ............................... 468
12.2 Hydrogen Consumption during Catalytic Hydrotreating ...... 470
12.2.1 Hydrogen Consumption .............................. 473
12.2.1.1 Mass Balance of Hydrogen in Gas Stream ........ 475
12.2.1.2 Global Hydrogen Balance ....................... 475
12.2.1.3 Class of Hydrogen-Consuming Chemical
Reactions ..................................... 475
12.2.1.4 Hydrogen Consumption by Reaction Average
Contributions ................................. 478
12.2.1.5 Hydrogen Consumption by Kinetic Modeling ...... 479
12.2.2 Solubility of Hydrogen ............................ 480
12.2.3 Results and Discussion ............................ 482
12.2.3.1 Experimental Data ............................. 482
12.2.3.2 Global Hydrogen Balance ....................... 482
12.2.3.3 Hydrogen Balance in Gas Streams ............... 485
12.2.3.4 Class of Hydrogen-Consuming Chemical
Reactions ..................................... 487
12.2.3.5 Hydrogen Consumption by Reaction
Average Contributions .................................... 487
12.3 Real Conversion and Yields from Hydroprocessing of
Heavy Oils Plants ........................................ 488
12.3.1 Experimental Data ................................. 488
12.3.2 Methodology ....................................... 489
12.3.3 Results ........................................... 490
12.4 Calculation of Fresh-Basis Composition from Spent
Catalyst Analysis ........................................ 493
12.4.1 Statement of the Problem .......................... 494
12.4.2 Catalyst Samples and Characterization ............. 495
12.4.3 Results and Discussion ............................ 496
12.5 Use of Probability Distribution Functions for Fitting
Distillation Curves of Petroleum ......................... 499
12.5.1 Brief Background of Probability Distribution
Functions ......................................... 500
12.5.2 Methodology ....................................... 502
12.5.2.1 Data Source ................................... 502
12.5.2.2 Example of Parameter Estimation ............... 507
12.5.2.3 Parameter Estimation for All Distribution
Functions ..................................... 508
12.5.3 Results and Discussion ............................ 511
12.5.3.1 Ranking of Functions .......................... 511
12.5.3.2 Validation of the Best Functions .............. 516
Nomenclature ............................................. 519
Subscripts ............................................... 521
Superscripts ............................................. 521
Greek Letters ............................................ 521
References ............................................... 521
Index ......................................................... 525
| 
