Modeling of Combustion Systems A Practical Approach Joseph Colannino A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc. Boca Raton London New York © 2006 by Taylor & Francis Group, LLC Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10987654321 International Standard Book Number-10: 0-8493-3365-2 (Hardcover) International Standard Book Number-13: 978-0-8493-3365-1 (Hardcover) Library of Congress Card Number 2005053146 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. 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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 Visit the Taylor & Francis Web site at and the CRC Press Web site at Taylor & Francis Group is the Academic Division of Informa plc. © 2006 by Taylor & Francis Group, LLC (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, http://www.crcpress.com http://www.taylorandfrancis.com For permission to photocopy or use material electronically from this work, please access www.copyright.com 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 [...]... constant of proportionality forward rate constant reverse rate constant linear asymptote (function shape analysis) limit (in summation operator) mass max or min (function shape analysis) mean square of quadratic factors slope mass flowrate mass flowrate of air to burner mass flowrate out of windbox or plenum mass flowrate of fuel kth mean value arbitrary matrix element flue gas mass flowrate recirculated... parameters in model generic stoichiometric product coefficient orthogonalization factor instantaneous heat release at z fractional heat release at the floor instantaneous heat release at z = 0 fractional heat release at the roof fractional thermal power in flue gas maximum fractional thermal power in flue gas fractional process heat-release arbitrary row or number of rows in matrix or vector coefficient of. .. generalized mean flame quality according to Figure 1.11 heat release heat loss from first combustion zone resident thermal power of the flue gas at a given elevation thermal power transferred to the process resident thermal power of the flue gas at a given elevation universal gas constant internal mass ratio of flue gas recirculation external mass ratio of flue gas recirculation internal molar ratio of flue gas... 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 Table 1.2 Table 1.3 Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table... power transformation wavelength nth eigenvalue (latent root) kth quadratic coefficient in A or B canonical form viscosity true mean value frequency kth normalization constant arbitrary untransformed factor arbitrary untransformed factors arithmetic mean of untransformed factor minimum untransformed factor maximum untranformed factor kth arbitrary factor 3 .141 59… probability of success probability operator... of air staging fraction of burners out of service kth factor fraction of overfire air stoichiometric mixture fraction gravitational acceleration constant unit dimensional constant for U.S customary units spectral function height horizontal asymptote (function shape analysis) arbitrary element of the H matrix kth diagonal element of the H matrix imaginary unit, −1 information fraction of x an index an... elevation average burner distance height of first combustion zone, normalized harmonic average burner distance kth equilateral triangular coordinate kth distance or elevation normalized elevation of maximum heat flux UPPER CASE ROMAN [S] [] concentration of species S dimensionless UPPER CASE ROMAN ITALIC A A A A Ak A( x) B B C C Cc Co Cp ˆ C area Arrhenius pre-exponential coefficient in heat flux equation generic... © 2006 by Taylor & Francis Group, LLC Prologue A process engineer at a refinery may ask, “How can I model flame quality, or NOx and CO emissions, or unit performance as a function of various fuel and air preheat scenarios?” A controls engineer may wonder, “How can I construct a model for a feedforward algorithm from available plant data?” A control room operator at an electrical utility may ask, “How... straightforward Throughout the text, we reinforce the material with fully worked examples We also work nearly all of the examples using spreadsheet software — not because this is the ideal platform (often it is not), but because access to such a computer tool is widespread Dedicated statistical or mathematical software has major advantages and occasionally we use some The overwhelming emphasis is on practical. .. recently, he served as director of engineering for the burner group at John Zink, LLC, and is currently the manager of knowledge systems there He is a current or past member of the following institutions and societies: Air and Waste Management Association (AWMA), American Chemical Society (ACS), American Institute of Chemical Engineers (AICHE), American Statistical Association (ASA), Combustion Institute, . Modeling of Combustion Systems A Practical Approach Joseph Colannino A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of. 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. 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