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 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 M0 Reduces to the Geometric Mean .................... 93 1.9.2 Proof of the Monotonicity of Mp ................................ 95 1.9.3 Proof That Mp Approaches xmax as p → ∞ .......................... 98 1.9.4 Proof That Mp Approaches xmin as p → -∞ ......................... 99 1.9.5 Proof xmin≤Mp≤xmax for x>0 ....................................... 99 1.9.6 Proof That Mp 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 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 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 (χ2), 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 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 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, r2 ......................... 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, rA2 ................ 327 4.4.5 The PRESS Statistic .......................................... 328 4.4.6 The Hat Matrix ............................................... 329 4.4.7 The Coefficient of Determination, Predicted, rp2 .............. 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 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 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 far 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 αw ..................... 441 5.1.6.4 Solving for T as a Function of αw ..................... 441 5.1.6.5 Log NOx as a Function of αw ........................... 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 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 zmax ................................... 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 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 Index ................................................................... 619