Modeling of Combustion Systems A Practical Approach 14 potx

Modeling of Combustion Systems
A Practical Approach
Joseph Colannino
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© 2006 by Taylor & Francis Group, LLC
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International Standard Book Number-10: 0-8493-3365-2 (Hardcover)
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Library of Congress Cataloging-in-Publication Data
Colannino, Joseph, 1957-
Modeling of combustion systems : a practical approach / by Joseph Colannino.
p. cm.
Includes bibliographical references and index.
ISBN 0-8493-3365-2 (alk. paper)
1. Combustion chambers Mathematical models. I. Title.
TJ254.7.C65 2006
621.402’3 dc22 2005053146
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Dedication
First, to Judy
My children and family
My father,
Frank Colannino, 12-May, 1923 to 4-August, 2004
†
†
Ed udii una voce dal cielo che diceva: “Scrivi questo: Da adesso saranno beati quelli
che moriranno nel Signore! ‘Sì’, dice lo Spirito, ‘perché possono riposare dalle loro
fatiche, e le loro opere li seguiranno in cielo!’”(Apocalisse 14:13, La Parola è Vita,
© 1997, International Bible Society)

© 2006 by Taylor & Francis Group, LLC
Contents
List of Tables
List of Figures
Concordance of Nomenclature
About the Author
Prologue
1
Introduction to Modeling 1
1.1 Model Categories 1
1.1.1 Model Validation 2
1.1.2 Fundamental Theoretical Models 2
1.1.3 Simulations 2
1.1.4 Semiempirical Models 3
1.1.5 Dimensionless Models 3
1.1.6 Empirical Models 3
1.1.7 Problems with Post Hoc Models 4
1.2 Kinds of Testing 4
1.2.1 No Physical Testing 4
1.2.2 Scale Testing 5
1.2.3 Full-Scale Testing 5
1.3 Analytical Methods 5
1.3.1 Qualitative Analysis 6
1.3.2 Dimensional Analysis 8
1.3.3 Raleigh’s Method 9
1.3.3.1 Cautions Regarding Dimensional Analysis 11
1.3.4 Function Shape Analysis 15
1.3.5 The Method of Partial Fractions 18
1.3.5.1 Limitations of Function Shape Analysis 22
1.4 Perceiving Higher Dimensionality 23

1.4.1 A View from Flatland 23
1.4.2 Contour Surfaces 24
1.4.3 Orthogonal Directions 26
1.4.4 Visualization with Cubic Regions 26
1.4.5 The Use of Color 28
1.5 Basic Data Classifications 29
1.5.1 Level of Scale 29
1.5.2 Data Quality 32
1.5.3 Planned Experiments 32
1.5.4 Unplanned Experiments 33
1.5.5 Source Classifications 33
1.5.6 Functional Classifications 33
© 2006 by Taylor & Francis Group, LLC
1.6 A Linear Algebra Primer 35
1.6.1 Matrix Addition 35
1.6.2 The Transpose Operator 36
1.6.3 Multiplication by a Constant 37
1.6.4 Matrix Multiplication 37
1.6.5 Distributive Property of Multiplication over Addition 38
1.6.6 Symmetric Matrices 39
1.6.7 The Identity Matrix 40
1.6.8 The Unity, Zero, and Constant Vectors 41
1.6.9 The Inverse 42
1.6.10 Elementary Row Operations 42
1.6.11 Solving for the Inverse 45
1.6.12 The Determinant 46
1.6.13 Orthogonality 47
1.6.14 Eigenvalues and Eigenvectors 52
1.7 Important Concepts and Notation 55
1.7.1 Summation and Matrix Notation 55

1.7.2 Converting between Summation and Matrix Notation 56
1.7.3 Averages: Mean, Mode, and Median 57
1.7.4 Various Means and the Generalized Mean 58
1.8 Least Squares 62
1.8.1 The Method of Least Squares 62
1.8.2 The Method of Least Squares: The Calculus 65
1.8.3 Least Squares for Continuous Intervals 69
1.8.4 Least Squares as a Filter 72
1.8.5 A Misconception about Least Squares 75
1.8.6 Transforming Equations for Least Squares Fitting
of the Parameters 75
1.8.7 Constrained Polynomials 77
1.8.8 Orthogonal Polynomials 80
1.8.9 General Definition of Orthogonal Polynomials 83
1.8.9.1 Discrete MOPs and Real Data 90
1.9 Addendum 93
1.9.1 Proof That M
0
Reduces to the Geometric Mean 93
1.9.2 Proof of the Monotonicity of M
p
95
1.9.3 Proof That M
p
Approaches x
max
as p → ∞ 98
1.9.4 Proof That M
p
Approaches x

min
as p → – ∞ 99
1.9.5 Proof x
min
≤ M
p
≤ x
max
for x > 0 99
1.9.6 Proof That M
p
Increases with Increasing p and the
Converse 99
References 100
2
Introduction to Combustion 101
2.1 General Overview 102
2.1.1 The Burner 102
2.1.1.1 The Fuel System 103
© 2006 by Taylor & Francis Group, LLC
2.1.1.2 About Fuels 104
2.1.1.3 Fuel Metering 105
2.1.1.4 Turndown 105
2.1.1.5 The Air System 106
2.1.1.6 The Flame Holder 108
2.1.1.7 Stabilizing and Shaping the Flame 108
2.1.1.8 Controlling Emissions 109
2.1.2 Archetypical Burners 109
2.1.2.1 Round-Flame Gas Diffusion Burners 111
2.1.2.2 Round-Flame Gas Premix Burners 111

2.1.2.3 Flat-Flame Gas Diffusion Burners 113
2.1.2.4 Flat-Flame Premix Burners 114
2.1.2.5 Flashback 115
2.1.2.6 Use of Secondary Fuel and Air 115
2.1.2.7 Round Combination Burners 116
2.1.2.8 Burner Orientations 119
2.1.2.9 Upfired 119
2.1.2.10 Downfired 120
2.1.2.11 Side-Fired 121
2.1.2.12 Balcony Fired 121
2.1.2.13 Combination Side and Floor Firing 121
2.2 Archetypical Process Units 123
2.2.1 Boilers 123
2.2.1.1 Firetube Boilers 123
2.2.1.2 Watertube Boilers 123
2.2.1.3 Fired Heaters and Reactors 123
2.2.1.4 Vertical Cylindrical 124
2.2.1.5 Cabin Style 124
2.2.1.6 Fired Reactors 126
2.2.1.7 Hydrogen Reformers 126
2.2.1.8 Ammonia Reformers 126
2.2.1.9 Ethylene Cracking Units (ECUs) 127
2.3 Important Factors and Responses 127
2.3.1 The Traditional Test Protocol 127
2.3.2 Instability, Thermoacoustic and Otherwise 128
2.3.3 Quarter-Wave Behavior 129
2.3.4 Half-Wave Behavior 131
2.3.5 Helmholtz Resonator Behavior 131
2.3.6 Mechanism for Thermoacoustic Coupling 132
2.3.7 Comments Regarding Thermoacoustic Resonance 133

2.3.7.1 Resonance in the Field 134
2.4 Mass Balance for Combustion in Air 135
2.4.1 Wet vs. Dry Measurements 137
2.4.2 Flue Gas Relations for Hydrocarbons 137
2.4.3 Accounting for Moisture 140
2.4.4 Addition of Molecular Hydrogen to the Fuel 143
© 2006 by Taylor & Francis Group, LLC
2.4.5 Addition of Flue Gas Components to Fuel 145
2.4.6 Substoichiometric Combustion 148
2.4.6.1 Lead-Lag Control 148
2.4.6.2 Substoichiometric Equations 148
2.4.7 Conservation of Mass for Flow in a Furnace 154
2.4.8 Simplifying Assumptions (SAs) 155
2.4.9 Ideal Gas Law 158
2.4.10 Dilution Correction 159
2.5 Conservation of Energy 164
2.5.1 Heat and Related Quantities 164
2.5.2 Work 165
2.5.3 Heating Value 166
2.5.4 Adiabatic Flame Temperature 167
2.5.5 Heat Capacity as a Function of Temperature 169
2.5.6 Adiabatic Flame Temperature with Preheated Air 171
2.6 Mechanical Energy Balance 173
2.6.1 Work Terms 173
2.6.2 Theoretical Mechanical Models 174
2.6.2.1 Units of Pressure 174
2.6.2.2 Natural Draft Model 175
2.6.2.3 Draft Pressure in a Furnace 175
2.6.2.4 Air Velocity Due to Natural Draft 177
2.6.2.5 Airflow through a Diffusion Burner 177

2.6.2.6 Airflow through Adjustable Dampers 182
2.6.2.7 Unknown Damper Characteristics 183
2.6.2.8 Fuel Flow as a Function of Pressure 184
2.6.2.9 Compressible Flow 185
2.6.2.10 The Fuel Capacity Curve Revisited 186
2.6.2.11 Airflow in Premix Burners 188
2.6.2.12 Gas Jets Entraining Flue Gas 189
References 189
3
Experimental Design and Analysis 191
3.1 Some Statistics 192
3.1.1 Statistics and Distributions 193
3.1.2 The Normal, Chi-Squared (χ
2
), F, and t Distributions 194
3.1.2.1 The Normal Distribution 195
3.1.2.2 Probability Distribution for Galton’s Board 196
3.1.2.3 Pascal’s Triangle 197
3.1.2.4 The Chi-Squared Distribution 200
3.1.2.5 The F Distribution 201
3.1.2.6 The t Distribution 202
3.2 The Analysis of Variance (ANOVA) 203
3.2.1 Use of the F Distribution 206
3.3 Two-Level Factorial Designs 209
3.3.1 ANOVA for Several Model Effects 211
© 2006 by Taylor & Francis Group, LLC
3.3.2 General Features of Factorial Designs 212
3.3.3 Construction Details of the Two-Level Factorial 213
3.3.4 Contrast of Factorial and Classical Experimentation 216
3.3.4.1 Statistical Properties of Classical Experimentation 219

3.3.4.2 How Factorial Designs Estimate Coefficients 221
3.3.4.3 The Sneaky Farmer 222
3.3.5 Interpretation of the Coefficients 229
3.3.6 Using Higher-Order Effects to Estimate Experimental
Error 232
3.3.6.1 Normal Probability Plots for Estimating
Residual Effects 232
3.4 Correspondence of Factor Space and Equation Form 234
3.5 Fractional Factorials 240
3.5.1 The Half Fraction 241
3.5.2 Quarter and Higher Fractions 242
3.6 ANOVA with Genuine Replicates 245
3.6.1 Bias Error 248
3.6.2 Center-Point Replicates 250
3.6.2.1 Degrees of Freedom Entries 251
3.6.2.2 Sum-of-Squares Entries 254
3.6.3 Standard Errors and the t Test 257
3.6.4 The Value of Orthogonal Designs with ANOVA 258
3.6.5 Rotatability 259
3.7 Randomization 260
3.7.1 Hysteresis 260
3.7.2 Lurking Factors 261
3.8 About Residuals 263
3.8.1 Residuals vs. Run Order 263
3.8.2 Other Residual Plots 263
3.8.3 Full and Block Randomization 264
3.8.4 Blocking 265
3.8.5 Random vs. Fixed Effects 265
3.9 Screening Designs 269
3.9.1 Simplex Designs 269

3.9.2 Highly Fractionated Factorials 272
3.9.3 Foldover 274
3.10 Second-Order Designs 275
3.10.1 Central Composites 275
3.10.1.1 Quadratic Bias Only 277
3.10.1.2 Orthogonal Components 278
3.10.1.3 Adjusting the Axial Component 280
3.10.2 Box–Behnken Designs 283
3.10.3 Multilevel Factorials 283
3.11 Sequential Experimental Design 286
3.11.1 Augmenting to Less Fractionated Factorials 287
3.11.2 Method of Steepest Ascent 287
© 2006 by Taylor & Francis Group, LLC
3.11.3 Augmenting to Second-Order Designs 289
References 291
4
Analysis of Nonideal Data 293
4.1 Plant Data 294
4.1.1 Problem 1: Events Too Close in Time 294
4.1.2 Problem 2: Lurking Factors 295
4.1.3 Problem 3: Moving Average Processes 295
4.1.4 Some Diagnostics and Remedies 297
4.1.5 Historical Data and Serial Correlation 297
4.2 Empirical Models 298
4.2.1 Model Bias from an Incorrect Model Specification 301
4.2.2 Design Bias 303
4.3 Ways to Make Designs Orthogonal 305
4.3.1 Source and Target Matrices: Morphing Factor Space 306
4.3.2 Eigenvalues and Eigenvectors 308
4.3.3 Using Eigenvectors to Make Matrices Orthogonal 316

4.3.4 Canonical Forms 318
4.3.4.1 Derivation of A Canonical Form 318
4.3.4.2 Derivation of B Canonical Form 319
4.3.4.3 Canonical Form and Function Shape 320
4.4 Regression Statistics and Data Integrity 324
4.4.1 The Coefficient of Determination, r
2
324
4.4.2 Overfit 325
4.4.3 Parsing Data into Model and Validation Sets 326
4.4.4 The Adjusted Coefficient of Determination, r
A
2
327
4.4.5 The PRESS Statistic 328
4.4.6 The Hat Matrix 329
4.4.7 The Coefficient of Determination, Predicted, r
p
2
330
4.4.8 Extrapolation 331
4.4.8.1 Failure to Detect Hidden Extrapolation 336
4.4.9 Collinearity 337
4.4.9.1 Reparameterization in Noncorrelated Factors 339
4.4.9.2 Variance Inflation Factor 342
4.4.10 Beta Coefficients 343
4.4.11 Confidence and Prediction Intervals 346
4.5 Residual Analyses 348
4.6 Categorical Factors 349
4.6.1 Multilevel Categorical Factors 349

4.6.2 Accounting for Multiple Blocks 352
4.6.3 Accounting for Hard-to-Change Factors 356
4.6.3.1 The Longest Duration Experimental Series 357
4.6.3.2 The Shortest Duration Experimental Series 358
4.6.3.3 Experimental Units 361
4.6.3.4 The Split-Plot Design 362
4.6.4 Expected Mean Squares (EMS) 367
© 2006 by Taylor & Francis Group, LLC
4.6.4.1 Methodology for Deriving EMS
for Balanced Data 367
4.6.4.2 EMS for the Factorial Design 373
4.6.4.3 EMS for a Split-Plot Design 374
4.6.4.4 Split-Plot Structure with Multiple Whole-Plot
Factors 378
4.6.4.5 Nested Factors 378
4.7 Categorical Response Values 383
4.7.1 Conversion from Qualitative to Quantitative Measures 384
4.7.2 Using the Logit and Probit Functions to Categorize
Flame Quality 385
4.8 Mixture Designs 386
4.8.1 Simplex-Centroid 388
4.8.2 Simplex-Lattice 390
4.8.3 Simplex-Axial 391
4.8.4 Generalizing to Higher Dimensions 392
4.8.5 Fuels of Many Components 395
4.8.6 Fuel Chemistry 395
4.8.6.1 Hydrogen 396
4.8.6.2 Hydrocarbon Chemistry 396
4.8.6.3 Bonding 397
4.8.6.4 Saturates 397

4.8.6.5 Olefins 399
4.8.6.6 Coke Formation 399
4.8.6.7 Mono-Olefins 400
4.8.6.8 Di-Olefins 400
4.8.6.9 Acetylenes 401
4.8.6.10 Aromatic Hydrocarbons 401
4.8.6.11 Cyclo Hydrocarbons 402
4.8.7 Representing Gaseous Fuel Blends 402
4.8.7.1 Chemical Bond Method 403
4.8.7.2 Equivalent Oxygen Method 407
4.8.7.3 Component Ranges 408
4.8.7.4 Pseudo-Components 410
4.8.8 Orthogonal Mixture Designs 410
4.8.8.1 Ratios of Mixture Fractions 411
4.8.9 Combining Mixture and Factorial Designs 413
4.8.9.1 Mixtures within Factorial 414
4.8.9.2 Mixture within Fractional Factorial 414
4.8.9.3 Fractionated Mixture within Fractional Factorial 415
References 420
5
Semiempirical Models 421
5.1 NOx and Kinetics 422
5.1.1 NOx: Some General Comments 422
5.1.2 The Thermal NOx Mechanism 422
© 2006 by Taylor & Francis Group, LLC
5.1.3 The Fuel-Bound Nitrogen Mechanism 424
5.1.4 The Prompt NOx Mechanism 426
5.1.5 Chemical Kinetic Effects for NOx in Diffusion Flames 427
5.1.5.1 NOx Response to Air in Diffusion Flames 427
5.1.5.2 Dimensional Units for NOx 432

5.1.5.3 The Relation of Referent and Objective Forms 434
5.1.5.4 NOx Response to Temperature in Diffusion
Flames 435
5.1.5.5 NOx Response to Fuel Composition 438
5.1.5.6 Chemical NOx When Prompt NOx Is Important 439
5.1.6 Chemical Kinetic Effects for NOx in Premixed Flames 440
5.1.6.1 NOx Response to Temperature in Premixed
Flames 440
5.1.6.2 NOx Response to Air in Premixed Burners 440
5.1.6.3 Solving for ζ as a Function of α
w
441
5.1.6.4 Solving for T as a Function of α
w
441
5.1.6.5 Log NOx as a Function of α
w
442
5.2 Overview of NOx Reduction Strategies 443
5.2.1 Low Excess Air (LEA) Operation 443
5.2.2 Air Staging 445
5.2.3 Overfire Air 445
5.2.4 Burners out of Service (BOOS) 446
5.2.5 Fuel Staging 446
5.2.6 Fuel Blending 447
5.2.7 Flue Gas Recirculation 447
5.2.7.1 Mass-Based Relations 447
5.2.7.2 Molar and Volumetric Definitions 450
5.2.8 Fuel Dilution, Flue Gas Inducted Recirculation (FIR) 453
5.2.9 Steam or Water Injection 456

5.2.10 Selective Noncatalytic Reduction (SNCR) 456
5.2.11 Selective Catalytic Reduction (SCR) 457
5.3 NOx Models 458
5.3.1 Categorization of Emissions Reduction Strategies 460
5.3.2 Temperature Reduction Strategies 460
5.3.2.1 Fuel Blending or Fuel Dilution 460
5.3.2.2 Flue Gas Inducted Recirculation 460
5.3.2.3 Flue Gas Recirculation 461
5.3.2.4 Steam or Water Injection 462
5.3.2.5 Air Staging 462
5.3.2.6 Fuel Staging 467
5.3.2.7 Overfire Air 467
5.3.2.8 Burners out of Service 469
5.3.3 Concentration Reduction Strategies 469
5.3.3.1 Low Excess Air (LEA) Operation 469
5.3.3.2 Air Staging with Fuel-Bound Nitrogen 470
5.3.3.3 Fuel Staging with Fuel-Bound Nitrogen 471
© 2006 by Taylor & Francis Group, LLC
5.3.4 Reagent Injection Strategies 471
5.3.4.1 Selective Noncatalytic Reduction (SNCR) 471
5.3.4.2 Selective Catalytic Reduction (SCR) 476
5.3.4.3 Limestone Injection 476
5.4 CO Models 477
5.4.1 Cold CO 478
5.4.2 Hot CO 479
5.4.3 General Behavior of Hot CO 479
5.4.4 Equilibrium Considerations 481
5.4.5 Arrested Oxidation of CO (via Ammoniacal Poisoning
of OH Catalysis) 483
5.5 Response Transformations 483

5.5.1 Empirical Considerations for Transformation of the CO
Response 483
5.5.2 Empirical Considerations for Transformation of NOx
Response 486
5.6 Heat Flux 486
5.6.1 Heat Flux Profile 487
5.6.1.1 The Normalized Heat Flux Equation 489
5.6.1.2 Data Normalization 491
5.6.1.3 Data Smoothing 492
5.6.1.4 Renormalization 497
5.6.1.5 The Heat Flux Model 500
5.6.2 Heat Flux as a Function of Furnace Temperatures 500
5.6.3 Qualitative Behavior of z
max
503
5.6.3.1 The Effect of Air Preheat 506
5.6.3.2 The Effect of Air/Fuel Ratio 507
5.6.3.3 The Effect of Fuel Pressure 507
5.6.4 Heat Flux Profile in Terms of Fractional Heat Release 509
5.6.4.1 The Effect of the Heat Sink (Process) 511
5.6.4.2 Final Heat Flux and Process Efficiency 512
5.6.4.3 Run Length and Flux Profile Curvature 512
5.6.4.4 Factors Affecting the Initial Heat Flux 513
5.6.4.5 Similarity and Scaling of Heat Flux Curves 515
5.7 Flame Shape 515
5.7.1 Flame Measurements 516
5.7.2 Flame Length 517
5.8 Visible Plumes 521
5.8.1 Bisulfite Plumes 521
5.8.2 Ammonium Chloride Plumes 522

5.8.3 Sulfur Oxides 524
5.8.3.1 Equations for Dew Point Elevation 525
References 528
Epilogue 531
References 532
© 2006 by Taylor & Francis Group, LLC
Appendices
A
Fuel and Combustion Properties 533
B
Mechanical Properties 555
C Units Conversions 573
D
Properties of the Elements 577
E
Statistical Tables 601
F
Numbers in Binary, Octal, and Hexadecimal
Representations 609
G
Kinetics Primer 613
H
Equilibrium Primer 617
© 2006 by Taylor & Francis Group, LLC
List of Tables
Table 1.1 Qualitative Analysis of NOx Formation 7
Table 1.2 Some Dimensionless Groups 15
Table 1.3 Some Potential Factors Affecting NOx Response
from a Burner 34
Table 1.4 Hypothetical Data 50

Table 1.5 Classical Orthogonal Polynomials 85
Table 1.6 Theoretical vs. Fitted Coefficients 93
Table 2.1 Burner Sampling for One Manufacturer 111
Table 2.2 Reactants and Products for Example 2.3 136
Table 3.1 The Naked ANOVA 204
Table 3.2 A Single Factor Example 208
Table 3.3 ANOVA for Example 3.1 209
Table 3.4 A Factorial Design in Three Factors 209
Table 3.5 Basic ANOVA for Table 3.4 212
Table 3.6 Partitioned ANOVA for Table 3.4 212
Table 3.7 Contrast of Classical and Factorial Designs 216
Table 3.8 A Factorial Design in Three Factors 228
Table 3.9 Partitioned ANOVA 234
Table 3.10 A Curious Experimental Design 235
Table 3.11 A Half Fraction in Five Factors 241
Table 3.12 Replicate Data 247
Table 3.13 ANOVA for Replicate Data 248
Table 3.14 ANOVA with Pooled Effects 248
Table 3.15 The Naked ANOVA with Replicates 250
Table 3.16 ANOVA with Replicates 250
Table 3.17 A 2
2
Factorial Design with Center Points 251
Table 3.18 ANOVA for Orthogonal Pooled or Unpooled Entries 252
Table 3.19 t Test 258
Table 3.20 A Nonrandomized Experiment 261
Table 3.21 Analysis of Variance for Data for a Nonrandomized
Experiment 262
Table 3.22 A Randomized Experiment 262
Table 3.23 Analysis of Variance for a Randomized Experiment 263

Table 3.24 A Randomized 2
3
Factorial Design in Two Blocks 266
Table 3.25 ANOVA for a 2
3
Factorial Design in Two Blocks 267
Table 3.26 Data for 2
3
Factorial in Two Blocks 267
Table 3.27 ANOVA for the 2
3
Factorial Design in Two Blocks 268
Table 3.28 ANOVA for the 2
3
Factorial Design without Blocks 269
Table 3.29 A 2 × 3 × 4 Factorial Design 284
© 2006 by Taylor & Francis Group, LLC
Table 3.30 Half Fraction of a 4 × 4 Factorial Derived from
the 1/2 2
4
Design 286
Table 3.31 A 12-Factor Screening Design in 16 Runs and 2 Blocks 290
Table 3.32 Additional Points for a Central Composite Design 291
Table 4.1 A Classical Design in Three Factors 303
Table 4.2 Random Data 325
Table 4.3 r
2
for All Possible 80/20 Validations 327
Table 4.4 PRESS and Other Statistics for the Data of Table 3.17 331
Table 4.5 Is This Extrapolation? 332

Table 4.6 Outlier Statistics 335
Table 4.7 A Poor Experimental Design in Three Factors 337
Table 4.8 A Weird Experimental Design in Two Factors 339
Table 4.9 A Factorial Design in Original Metrics 345
Table 4.10 Regression Statistics, Analysis for Coded Factors 346
Table 4.11 Regression Statistics, Analysis for Original Metrics 346
Table 4.12 A 2
4
Factorial Design in Four Blocks 353
Table 4.13 Annotated X
T
X Matrix for the 2
4
Design in Four Blocks 354
Table 4.14 ANOVA for 2
4
Design in Four Blocks 356
Table 4.15 Responses and Factors of Interest 357
Table 4.16 Longest Runtime Order 359
Table 4.17 Preparing to Construct Shortest Setup Time 359
Table 4.18 Run Sequence for Shortest Setup Time 360
Table 4.19 Longest and Shortest Setup Times for Furnace
Experiment 361
Table 4.20 ANOVA for Two-Factor Factorial Design 363
Table 4.21 ANOVA for Two-Factor Split-Plot Design 364
Table 4.22 Split-Plot Design 365
Table 4.23 Split-Plot Design 366
Table 4.24 Expected Mean Squares Table, Step 1 368
Table 4.25 Expected Mean Squares Table, Step 2 368
Table 4.26 Expected Mean Squares Table, Step 6 371

Table 4.27 Expected Mean Squares Table, Step 7 372
Table 4.28 ANOVA with Expected Mean Squares, Step 8 373
Table 4.29 A Factorial Design for Multilevel Factors 374
Table 4.30 Expected Mean Squares for Split Plot, Step 6 375
Table 4.31 Expected Mean Squares Table, Step 7 376
Table 4.32 ANOVA for Two-Factor Split-Plot Design 376
Table 4.33 ANOVA for Two-Factor Fully Randomized Factorial
Design 377
Table 4.34 ANOVA for Split-Plot Design with Two Factors
in the Whole Plot 380
Table 4.35 Expected Mean Squares Table for Example 4.8 382
Table 4.36 ANOVA for Example 4.8 383
Table 4.37 Formulas for Bond Types 401
Table 4.38 Sample Refinery Gas 406
© 2006 by Taylor & Francis Group, LLC
Table 4.39 Source Refinery Gas: Augmented with Numbers
of Bond Types 406
Table 4.40 Fuel Composition Ranges 408
Table 4.41 Fuel Compositions within Constraints 410
Table 4.42 A 2
2
Factorial Mixture Design in Two Orthogonal
Blocks 416
Table 4.43 A Four-Factor Central Composite Design in Three
Orthogonal Blocks 418
Table 5.1 SO
2
vs. Limestone Injection Rate 476
Table 5.2 ANOVA for SO
2

Capture 477
Table 5.3 ANOVA for SO
2
Capture, Pooled Residual 477
Table 5.4 Heat Flux vs. Normalized Elevation 493
Table 5.5 ANOVA for Fourth-Order Smoothing 493
Table 5.6 ANOVA for Example 5.14 495
Table 5.7 A Normalized Heat Flux Curve 498
Table 5.8 Heat Flux Comparison 503
Table 5.9 Heat Flux Equation Summary for y = aln(1 + z) – bz + c 514
Table 5.10 Dew Point Elevation Data 527
Table A.1 Physical Constants of Typical Gaseous Fuel Mixture
Components 534
Table A.2 Combustion Data for Hydrocarbons 535
Table A.3 Chemical, Physical, and Thermal Properties of Gases:
Gases and Vapors, Including Fuels and Refrigerants,
English and Metric Units 537
Table A.4 Physical and Combustion Properties of Fuels 543
Table A.5 Thermodynamic Properties of Selected Compounds 547
Table A.6 Heat Capacity vs. Temperature for Selected Compounds,
J/mol K 548
Table A.7 Adiabatic Flame Temperatures 552
Table A.8 Volumetric Analysis of Typical Gaseous Fuel Mixtures 553
Table A.9 Physical Constants of Typical Gaseous Fuel Mixtures 554
Table B.1 Areas and Circumferences of Circles and Drill Sizes 556
Table B.2 Physical Properties of Pipe 565
Table B.3 K Factors 570
Table C.1 Common Conversions 574
Table C.2 Unit Dimensions for Some Combustion-
Related Quantities 575

Table D.1 Periodic Table of the Elements 578
Table D.2 Standard Atomic Weights 1981 579
Table D.3 Properties of Saturated Steam and Saturated Water 584
Table D.4 Properties of Superheated Steam 591
Table E.1 Normal Probability Function 602
Table E.2 Students t Distribution 603
Table E.3 χ
2
Distribution 604
Table E.4 F-Distribution, 99%, 95%, and 90% Confidence 606
© 2006 by Taylor & Francis Group, LLC
Table F.1 Positional Number Representation in Base 10 609
Table F.2 Positional Number Representation in Base 8 609
Table F.3 Positional Number Representation in Base 2 610
Table F.4 Base Equivalents 611
© 2006 by Taylor & Francis Group, LLC
List of Figures
Figure 1.1 A graph with asymptotes 16
Figure 1.2 Basic function shapes 16
Figure 1.3 Excess air vs. oxygen for CH
4
21
Figure 1.4 Two hmh curves 23
Figure 1.5 An observer in Flatland 24
Figure 1.6 Contour surfaces for a hyperellipsoid in four dimensions 25
Figure 1.7 A four-dimensional hypercube 27
Figure 1.8 Coordinate space in three dimensions 27
Figure 1.9 Visualizing multiple dimensions with cubic regions 28
Figure 1.10 Data types 30
Figure 1.11 An ordinal scale for flame quality 31

Figure 1.12 The least squares line 63
Figure 1.13 Linear algebra with spreadsheets 66
Figure 1.14 A least squares approximation of a continuous function 71
Figure 1.15 A filter analogy for least squares 73
Figure 1.16 A bimodal distribution 88
Figure 1.17 The first four MOPs 90
Figure 1.18 The first 16 spectral coefficients 91
Figure 1.19 Discrete data case 92
Figure 1.20 The generalized mean and its derivatives 100
Figure 2.1 A typical industrial burner 103
Figure 2.2 A typical capacity curve 105
Figure 2.3 The tile ledge as flame holder 108
Figure 2.4 A gas burner in operation 109
Figure 2.5 A gas premix floor burner 112
Figure 2.6 A flat-flame gas diffusion burner 114
Figure 2.7 Floor-fired flat-flame burners 115
Figure 2.8 A typical heat flux profile 116
Figure 2.9 A flat-flame diffusion burner 117
Figure 2.10 Wall-fired diffusion burners in operation 117
Figure 2.11 A flat-flame premix burner 118
Figure 2.12 Venturi section of a premix burner 118
Figure 2.13 An oil gun 119
Figure 2.14 A combination burner 120
Figure 2.15 A downfired burner for hydrogen reforming 121
Figure 2.16 Sidewall burners in operation 122
Figure 2.17 Balcony burner 122
Figure 2.18 Some process heater types 124
Figure 2.19 Resonant modes for furnaces 129
© 2006 by Taylor & Francis Group, LLC
Figure 2.20 Flue gas relations 141

Figure 2.21 Wet–dry flue gas relations 142
Figure 2.22 A process heater 144
Figure 2.23 Substoichiometric combustion relations 152
Figure 2.24 A furnace control volume 156
Figure 2.25 Concentration vs. time for well-mixed behavior 158
Figure 2.26 Dilution correction 159
Figure 2.27 Simplified airflow analysis of burner 177
Figure 2.28 A typical airside capacity (cap) curve 181
Figure 2.29 Single-blade damper 182
Figure 2.30 Airflow vs. damper function 183
Figure 2.31 Probit and logit functions 184
Figure 2.32 Thermodynamic relations 186
Figure 3.1 The Galton board 195
Figure 3.2 The normal distribution 196
Figure 3.3 Pascal’s triangle 197
Figure 3.4 Classical and factorial experiments 217
Figure 3.5 Information fraction for a factorial and classical design 221
Figure 3.6 The sneaky farmer 223
Figure 3.7 A 2
3
factorial design 227
Figure 3.8 Representations of NOx response 231
Figure 3.9 Normal probability plot 233
Figure 3.10 Some two-factor designs 235
Figure 3.11 A curious experimental design 236
Figure 3.12 Illustration of degrees of freedom 254
Figure 3.13 Information fraction for factorial with center points 259
Figure 3.14 Residual vs. run order 264
Figure 3.15 A central composite in two factors 276
Figure 3.16 Comparison of central composite designs 282

Figure 3.17 Box–Behnken design for p = 3 283
Figure 3.18 Method of steepest ascent 288
Figure 4.1 A municipal solid waste boiler 295
Figure 4.2 A moving average with random data 296
Figure 4.3 Graphical representation of various experimental
designs 305
Figure 4.4 Various response surfaces 321
Figure 4.5 On overfitting data 326
Figure 4.6 Hidden extrapolation 333
Figure 4.7 Failure to detect extrapolation 336
Figure 4.8 A poor experimental design 338
Figure 4.9 Augmented design 341
Figure 4.10 A better experimental design 341
Figure 4.11 Actual vs. predicted with confidence interval 348
Figure 4.12 Factor space for a ternary mixture 386
Figure 4.13 Ternary coordinate systems 387
© 2006 by Taylor & Francis Group, LLC
Figure 4.14 The simplex-centroid 389
Figure 4.15 Some designs and their information fractions 392
Figure 4.16 Some simplexes in c – 1 dimensions 393
Figure 4.17 Structural features of some alkanes 397
Figure 4.18 Constrained mixtures 409
Figure 4.19 Transformation to orthogonal factors 411
Figure 4.20 Orthogonal ratios 412
Figure 4.21 Evenly spaced ratios 413
Figure 4.22 Mixed coordinates 413
Figure 4.23 Factor space: mixtures within a factorial 414
Figure 4.24 Factor space: mixtures within a fractional factorial 415
Figure 4.25 Independent mixture factors 416
Figure 4.26 Central composite in three orthogonal blocks 419

Figure 5.1 An ECU simulator 436
Figure 5.2 A CO trim strategy 444
Figure 5.3 A burner using fuel dilution and staging 447
Figure 5.4 Mass balance for a typical FGR system 448
Figure 5.5 A mass balance for a typical FIR system 453
Figure 5.6 Comparison of FGR and FIR strategies 454
Figure 5.7 CO behavior 479
Figure 5.8 Actual vs. predicted CO 482
Figure 5.9 Box–Cox response transformation 485
Figure 5.10 Box–Cox response transformation 485
Figure 5.11 Schematic of a heat flux probe 486
Figure 5.12 The y*-z* plot for heat flux 490
Figure 5.13 Graph of data for Example 5.7 496
Figure 5.14 Time-averaged flames 518
Figure A.1 Heat capacity vs. temperature for polyatomic gases 549
Figure A.2 Heat capacity vs. temperature for triatomic and other
gases 550
Figure A.3 Heat capacity vs. temperature for diatomic and other
gases 551
© 2006 by Taylor & Francis Group, LLC
Concordance of Nomenclature
lower case Roman
e error (collective or pure)
e
b
bias error
e
r
residual error
[s] concentration of species s

[ ] dimensionless
lower case Roman italic
a, b, c, … arbitrary exponents
a coefficient in normalized heat flux equation
a
0
arbitrary constant
a
1
, a
2
, … arbitrary constants
a
k
k
th
coefficient
b Arrhenius activation constant
b arbitrary offset
b coefficient in normalized heat flux equation
b
0
, b
1
, … arbitrary constants
b
0
, b
1
, … blocking factor coefficients

b
0
, b
1
, … mixture coefficients
c coefficient in normalized heat flux equation
c number of columns in matrix or vector
c speed of sound
c number of mixture components
c
0
, c
1
, … arbitrary constants
c
j
j
th
coefficient of characteristic equation
d arbitrary column or number of columns in matrix or vector
d number of dimensions
d number of points along an edge of a simplex
leverage distance
d
nn
n
th
diagonal element
d
o

orifice diameter
d
o,k
k
th
orifice diameter
e 2.71828…
e
1
number of pure components (simplex vertices) in mixture
design
e
2
number of binary blend combinations (simplex edges) in
mixture design
d
L
2
© 2006 by Taylor & Francis Group, LLC
e
3
number of ternary blend combinations (simplex faces) in
mixture design
e
4
number of quaternary blend combinations (simplex volumes)
in mixture design
e
k
number of k blend combinations in mixture design (simplex

hypervolumes when k
≥
5)
f number of factors
f
a
fraction of air staging
f
b
fraction of burners out of service
f
k
k
th
factor
f
o
fraction of overfire air
f
s
stoichiometric mixture fraction
g gravitational acceleration constant
g
c
unit dimensional constant for U.S. customary units
g(x) spectral function
h height
h horizontal asymptote (function shape analysis)
h
j,k

arbitrary element of the H matrix
h
k,k
k
th
diagonal element of the H matrix
i imaginary unit,
i(x) information fraction of x
j an index
k an index
constant of proportionality
k
f
forward rate constant
k
r
reverse rate constant
l linear asymptote (function shape analysis)
m limit (in summation operator)
m mass
m max or min (function shape analysis)
m mean square of quadratic factors
m slope
mass flowrate
mass flowrate of air to burner
mass flowrate out of windbox or plenum
mass flowrate of fuel
m
k
k

th
mean value
m
jk
arbitrary matrix element
flue gas mass flowrate recirculated to burner
n limit (in summation operator)
n number of burners
n number of independent factors
n number of factors in simplex
n number of moles
n number of observations
mass flowrate
−1

k

m

m
a

m
b

m
f

m
r


n
© 2006 by Taylor & Francis Group, LLC
molar flowrate of air to burner
n
b
number of burners out of service
molar flowrate out of windbox or plenum
n
c
number of centerpoint replicates
n
C
number of carbon atoms in fuel molecule
n
f
number of factorial points
molar flowrate of fuel
molar flowrate of flue gas out of the stack
n
H
number of hydrogen atoms in fuel molecule
n
k
number of levels of the k
th
factor
n
o
number of orifices

n
r
number of replicates
flue gas molar flowrate recirculated to burner
p generic stoichiometric product coefficient
p number of parameters in model
q generic stoichiometric product coefficient
q orthogonalization factor
instantaneous heat release at z
q
0
fractional heat release at the floor
instantaneous heat release at z = 0
q
1
fractional heat release at the roof
q
g
fractional thermal power in flue gas
q
max
maximum fractional thermal power in flue gas
q
p
fractional process heat-release
r arbitrary row or number of rows in matrix or vector
r coefficient of correlation
r generic reactant stoichiometric coefficient
r reaction rate
r

2
coefficient of determination
r
A
2
adjusted coefficient of determination
r
c
critical pressure ratio
coefficient of inlying
r
k
k
th
replicate
r
k
2
k
th
coefficient of collinearity
r
O
2
coefficient of outlying
r
P
2
coefficient of prediction
s arbitrary row or number of rows in matrix or vector

s estimated standard deviation
s generic reactant stoichiometric coefficient
s
2
estimated variance
s
C–C
number of C–C bonds in source fuel
s
C–H
number of C–H bonds in source fuel
s
H–H
number of H–H bonds in source fuel
s
k
k
th
standard deviation
t time
t
CH4
fraction of CH
4
in target fuel

n
a

n

b

n
f

n
g

n
r

q

q
0
r
I
2
© 2006 by Taylor & Francis Group, LLC
t
C3H8
fraction of C
3
H
8
in target fuel
t
H2
fraction of H
2

in target fuel
t-distribution of z
tr( ) trace operator
u unbounded curve (function shape analysis)
u number of unique points in factor space
v velocity (fluid flow)
v vertical asymptote (function shape analysis)
mean value of a vector
v
o
velocity at the jet orifice
w mass fraction of subscripted species
w
k
k
th
transformed source factor
w(x) particular weight function for an orthogonal polynomial
x arbitrary factor (independent variable)
x mass fraction
x mole fraction dissolved substance
x transformed factor
x
CO
conversion of CO
x
NO
conversion of NOx
x
S

conversion of species S
x
1
, x
2
, … arbitrary factors
x
C,k
k
th
Cartesian coordinate
x
F
fixed factor (effect)
arithmetic mean
x
k
k
th
arbitrary factor
x
R
random factor (effect)
y arbitrary response
y mole fraction
y normalized heat flux
mean response
predicted, true, or fitted response
y
0

normalized heat flux at floor
y
1
normalized heat flux at roof
y
a
mole fraction of species a
y
a,s
mole fraction of species a at the stack exit
y
b
mole fraction of species b
y* reduced form of normalized heat flux
predicted response with the k
th
residual deleted
y
CO
mole fraction CO
y
NO
mole fraction NOx
y
NO,0
initial mole fraction NOx
y
NO,r
mole fraction NOx reduced or converted by postcombustion
strategy

y
O2,b
mole fraction of oxygen in the windbox
y
O2,bt
CO breakthrough point
y
O2,v
oxygen at the venturi outlet
tnz(,)
v
x
y
ˆ
y
ˆ
*
y
k
© 2006 by Taylor & Francis Group, LLC
y
x,0
mole fraction of species x at 0% O
2

y
x,ref
mole fraction of species x at reference oxygen conditions
y
max

maximum value of normalized heat flux before
renormalization
unadulterated maximum value of normalized heat flux before
renormalization
z mixture factor
z normal ordinate
z normalized elevation
z* reduced form of normalized elevation
average burner distance
z
1
height of first combustion zone, normalized
harmonic average burner distance
z
C,k
k
th
equilateral triangular coordinate
z
k
k
th
distance or elevation
z
max
normalized elevation of maximum heat flux
UPPER CASE ROMAN
[S] concentration of species S
[ ] dimensionless
UPPER CASE ROMAN ITALIC

A area
A Arrhenius pre-exponential
A coefficient in heat flux equation
A generic numerator in partial fractions
A
k
k
th
spectral coefficient
A(x) fitted least-squares response, continuous data
B coefficient in heat flux equation
B generic numerator in partial fractions
C arbitrary constant
C coefficient in heat flux equation
C
c
coefficient for critical flow
C
o
initial centerline concentration
C
p
isobaric heat capacity
molar isobaric heat capacity
specific isobaric heat capacity
C
v
isometric heat capacity
molar isometric heat capacity
specific isometric heat capacity

C
x
centerline concentration of species x
D diameter
ˆ
max
y
z
z
−1
ˆ
C
p
C
p
ˆ
C
v
C
v
© 2006 by Taylor & Francis Group, LLC