Combustion Theory

Forman A. Williams9780201407778, 0201407779

Table of contents :
Preface to the Second Edition……Page 5
Preface to the First Edition……Page 11
Contents……Page 13
1. Summary of Relevant Aspects of Fluid Dynamics and Chemical Kinetics……Page 24
1.1. THE CONSERVATION EQUATIONS FOR MULTICOMPONENT, REACTING, IDEAL-GAS MIXTURES……Page 25
1.2.1. Unsteady flow……Page 27
1.2.2. Steady flow……Page 30
1.3. COUPLING FUNCTIONS……Page 32
1.4. CONSERVATION CONDITIONS AT AN INTERFACE……Page 36
1.5. DISCUSSION OF THE APPROACH ADOPTED IN THE FOLLOWING DEVELOPMENT OF COMBUSTION THEORY……Page 40
REFERENCES……Page 41
2. Rankine-Hugoniot Relations……Page 42
2.1.1. Derivation of the equations……Page 43
2.1.2. The cold-boundary difficulty……Page 45
2.1.3. Use of the Rankine-Hugoniot equations……Page 46
2.2.1. Simplification of the Rankine-Hugoniot equations……Page 47
2.2.3. Properties of the Hugoniot curve……Page 49
2.2.4. Analysis of the detonation branch……Page 51
2.2.6. Properties of Chapman-Jouguet waves……Page 53
2.3.2. Frozen versus equilibrium sound speeds……Page 56
2.3.3. Proof that v_infty= a_e,infty at the Chapman-Jouguet points……Page 57
2.3.4. Summary of the properties of Hugoniot curves……Page 58
REFERENCES……Page 60
3. Diffusion Flames and Droplet Burning……Page 61
3.1.1. Definition of the problem……Page 62
3.1.2. Assumptions……Page 63
3.1.3. Solution of the species conservation equation for the coupling
function p……Page 64
3.1.4. The flame shape and the flame height……Page 65
3.1.5. The flame-sheet approximation……Page 67
3.1.6. The validity of the other approximations……Page 68
3.1.7. Comments on the formulation and the analysis……Page 69
3.2.2. The nature of carbon combustion……Page 71
3.2.3. Analysis……Page 73
3.3.1. Background and definition of the problem……Page 75
3.3.2 Assumptions……Page 77
3.3.3. Analysis predicting the burning rate……Page 79
3.3.4. Discussion of the burning-rate formula……Page 82
3.3.5. Predictions of other characteristics of burning droplets……Page 84
3.3.6. Further realities of droplet burning……Page 85
3.4.1. Approaches to structure questions……Page 92
3.4.2. The mixture-fraction variable……Page 96
3.4.3. Activation-energy asymptotics……Page 99
3.4.4. Ignition and extinction……Page 103
3.5. MONOPROPELLANT DROPLET BURNING……Page 107
REFERENCES……Page 109
4.1. IGNITION DELAY AND THE WELL-STIRRED REACTOR……Page 115
4.2.1. Steady-state, quasi-onedimensional conservation equations……Page 119
4.2.2. Specific impulse of rockets……Page 122
4.2.3. Near-equilibrium and near-frozen flows……Page 123
4.2.4 Application to the reaction A $ B with species A and B present in only trace amounts……Page 126
4.2.5. Freezing of reactions……Page 128
4.2.6. Two-phase nozzle flow……Page 129
4.3.1. Conservation equations; characteristic surfaces……Page 131
4.3.2. The method of characteristics for steady, two-dimensional
(axially symmetrical and plane) flows……Page 136
4.3.3. The method of characteristics for one-dimensional, unsteady flows……Page 141
4.3.4.1. Preliminary relationships……Page 142
4.3.4.3. Reduction to a single partial differential equation……Page 143
4.3.4.4. Dispersion relations [62]……Page 146
4.3.4.5. An initial-value problem [59]……Page 147
4.3.4.6. Related problems……Page 149
REFERENCES……Page 150
5. Theory of Laminar Flames……Page 153
5.1.1. Experiments……Page 154
5.1.2. Phenomenological analysis of a deflagration wave……Page 158
5.2.1. Introductory remarks……Page 159
5.2.2. Preliminary assumptions and equations……Page 160
5.2.3. Approximations that further simplify the energy equation……Page 161
5.2.4. Simplifications in the energy and diffusion equations for unimolecular reactions in binary mixtures……Page 162
5.2.6. Dimensionless forms for the momentum equation and the species-conservation equation……Page 164
5.2.7. Summary of the simplified mathematical problem……Page 165
5.3.1. Governing equatiow……Page 166
5.3.2. The cold-boundary difficulty [16], [17], [18]……Page 168
5.3.3. Bounds on the burning-rate eigenvalue……Page 172
5.3.4. Iterative procedures and variational methods……Page 174
5.3.5. The approximations of Zeldovich, Frank-Karnenetskii,
and von Karman……Page 176
5.3.6. Asymptotic analysis for strongly temperature-dependent rates……Page 177
5.3.7. Generalizations to other flames……Page 183
5.4.1.1. Formulation……Page 188
5.4.1.3. Literature……Page 190
5.4.1.4. Burning-velocity calculations……Page 193
5.4.2. The extended steady-state approximation……Page 195
5.4.3. Conservation equations for reaction intermediaries……Page 196
5.4.4. A criterion for the applicability of the extended steady-state
approximation……Page 197
5.4.5. Methods of analysis for testing steady-state approximations……Page 198
5.4.6. Observations on theories of flame structure……Page 200
REFERENCES……Page 202
6. Detonation Phenomena……Page 205
6.1.2. Properties of the governing equations……Page 206
6.1.2.1. Location of Singularities……Page 207
6.1.2.2. Solutions in the neighborhoods of singular points……Page 210
6.1.2.3. General properties of the integral curves……Page 211
6.1.3. Remarks on deflagrations……Page 213
6.1.4. Approximate solution for the structure of a detonation……Page 214
6.1.5. Discussion of detonation structure……Page 215
6.1.6. The structure of ZND detonations……Page 217
6.2.1. Basic considerations for plane waves……Page 220
6.2.3. Effects of tube walls……Page 222
6.2.4. Ambiguities associated with frozen and equilibrium sound speeds……Page 224
6.2.5. Effects of three-dimensional structures……Page 226
6.3.1. Spinning detonations and stability considerations……Page 227
6.3.2. Theories of transverse structures……Page 231
6.3.4. Detonability limits and quenching thickness……Page 235
6.3.5. The transition from deflagration to detonation……Page 240
6.4. DETONATIONS IN SOLIDS, LIQUIDS, AND SPRAYS……Page 242
REFERENCES……Page 244
7. Combustion of Solid Propellants……Page 252
7.1. DESCRIPTION OF STEADY DEFLAGRATION OF A HOMOGENEOUS SOLID……Page 253
7.2. APPLICATIONS OF TRANSITION-STATE THEORY……Page 256
7.3 APPROACH TO INTERFACIAL EQUILIBRIUM……Page 258
7.4. DEFLAGRATION CONTROLLED BY CONDENSED-PHASE REACTION RATES……Page 261
7.5. DEFLAGRATION CONTROLLED BY GAS-PHASE REACTION RATES……Page 266
7.6. DISPERSION PHENOMENA AND OTHER INFLUENCES……Page 272
7.7. COMBUSTION OF HETEROGENEOUS PROPELLANTS……Page 274
7.8. EROSIVE BURNING……Page 281
REFERENCES……Page 284
8. Ignition, Extinction, and Flammability Limits……Page 288
8.1. MINIMUM IGNITION ENERGIES AND QUENCHING DISTANCES……Page 291
8.2.1. Methods of analysis……Page 294
8.2.3. Concentration limits of flammability……Page 300
8.2.5. Estimates of heat loss……Page 302
8.3. ACTIVATION-ENERGY ASYMPTOTICS IN IGNITION THEORY……Page 307
REFERENCES……Page 314
9. Combustion Instabilities……Page 317
9.1.1. Oscillation modes……Page 318
9.1.2. Conservation of acoustic energy……Page 321
9.1.3. The acoustic admittance……Page 324
9.1.4.1. Relative importance……Page 327
9.1.4.2. Nozzle damping…….Page 328
9.1.4.3. Wall damping……Page 331
9.1.4.5. Solid vibrations……Page 332
9.1.4.6. Relaxation damping……Page 334
9.1.4.7. Particle damping……Page 335
9.1.5.2. Amplification criteria……Page 338
9.1.5.3. Time-lag theories……Page 341
9.1.5.4. Combustion response……Page 342
9.1.5.5. Heterogeneity effects……Page 346
9.1.6. Nonlinear effects……Page 347
9.2. INHERENT OSCILLATIONS OF BURNING SOLIDS……Page 351
9.3. OSCILLATORY BURNING IN LIQUID-PROPELLANT
ROCKET MOTORS……Page 359
9.4. SYSTEM INSTABILITIES IN COMBUSTION EQUIPMENT……Page 362
9.5.1. Formulation through asymptotic methods……Page 364
9.5.2. Cellular flames……Page 372
9.5.2.1. Body-force instabilities……Page 373
9.5.2.2. Hydrodynamic instabilities……Page 375
9.5.2.3. Diffusive-thermal instabilities……Page 380
REFERENCES……Page 388
10. Theory of Turbulent Flames……Page 396
10.1.1. Probability density functionals……Page 398
10.1.3. Properties of probability-density functions……Page 403
10.1.4. Fourier decompositions……Page 407
10.1.5. Scales of turbulence……Page 410
10.2.1. Objectives of Analyses……Page 414
10.2.2. Use of coupling functions……Page 416
10.2.3. Production of trace species……Page 424
10.2.4. Average rates of heat release……Page 427
10.2.5. Effects of strain on flame sheets……Page 430
10.3.1. Objectives of analysis……Page 433
10.3.2. Effects of strain on laminar flames……Page 437
10.3.3. Theory of wrinkled laminar flames……Page 445
10.3.4 Turbulent flame speeds……Page 451
10.3.5. Flames in turbulence of high intensity or small scale……Page 459
REFERENCES……Page 462
11. Spray Combustion……Page 468
11.1.1. Particle size and shape……Page 470
11.1.3. The spray equation……Page 471
11.2.1. The model……Page 472
11.2.2. Simplified spray equation……Page 473
11.2.3 Solution of the spray equation……Page 474
11.2.4. Droplet size distributions……Page 475
11.2.5. The combustion efficiency and other spray properties……Page 477
11.3.1. Motivation……Page 480
11.3.2. Overall continuity……Page 481
11.3.4. Momentum conservation……Page 482
11.3.6. Comments on formulations……Page 484
11.4.2. Overall continuity……Page 485
11.4.4. Momentum conservation……Page 486
11.4.5. Energy conservation……Page 487
11.5.1. The model……Page 488
11.5.2. The spray equation……Page 489
11.5.4. Droplet drag……Page 490
11.5.6. Solution to the problem……Page 491
11.5.7. The chamber length for complete combustion……Page 493
11.6.1. Description……Page 494
11.6.2. Overall continuity and the spray equation……Page 496
11.6.3. Species conservation……Page 497
11.6.4. Momentum and energy conservation……Page 498
11.6.5. The mathematical problem and boundary conditions……Page 500
11.6.6. Solution to the problem……Page 501
11.7. SPRAY PENETRATION AND CLOUD COMBUSTION……Page 502
REFERENCES……Page 503
12. Flame Attachment and Flame Spread……Page 507
12.1.1. Derivation of simplified governing equations……Page 508
12.1.2. Generalizations……Page 511
12.2.1. Definition of the problem……Page 517
12.2.2. Boundary conditions……Page 518
12.2.3. Solution……Page 520
12.2.4. The burning rate……Page 522
12.2.5. The force on the plate……Page 523
12.2.6. Related studies……Page 524
12.3. MECHANISMS OF FLAME STABILIZATION……Page 525
12.4. PROCESSES OF FLAME SPREAD……Page 531
REFERENCES……Page 538
A. Summary of Applicable Results of Thermodynamics and Statistical Mechanics……Page 542
A.1.1. The laws of thermodynamics……Page 543
A.1.2. Thermodynamic functions……Page 544
A.2.1. Background……Page 545
A.2.2. Summary of results……Page 547
A.2.3. Evaluation of partition functions……Page 548
A.3.1. General equilibrium condition……Page 550
A.3.2. Phase equilibria……Page 552
A.3.3. Ideal-gas reactions……Page 553
A.3.5. Reactions in condensed phases……Page 554
A.3.7. Calculation of equilibrium compositions……Page 556
A.4.1. Definition of heat of reaction……Page 559
A.4.2. Differential heat of reaction……Page 560
A.4.3. Heat of formation and other properties……Page 561
A.4.4 The equations of Kirchhoff and vant Hoff……Page 562
A.4.5. The adiabatic flame temperature……Page 564
A.5.1. The phase rule……Page 565
A.5.2. Vapor pressures of binary mixtures……Page 566
A.5.3. Boiling points of binary mixtures……Page 568
A.5.4. Temperature dependence of vapor pressures of binary mixtures……Page 569
A.5.5. Colligative properties of solutions……Page 571
REFERENCES……Page 573
B.1.1. Statement of the law……Page 575
B.1.2. Multiple reactions; equilibrium constant……Page 576
B.1.3. Reaction order and molecularity……Page 578
B.2.1. General methods……Page 579
B.2.2. First-order reactions and unimolecular reactions……Page 580
B.2.4. Opposing reactions……Page 582
B.2.5.1. Initiation, propagation, and termination steps.*……Page 584
B.2.5.2. The steady-state and partial-equilibrium approximations……Page 586
B.2.5.3. Branched-chain explosions……Page 591
B.2.5.4. Thermal explosions……Page 597
B.2.5.5. Kinetics of hydrocarbon combustion……Page 602
B.2.6. Catalysis……Page 605
B.3.2. The activation energy……Page 606
B.3.3. Collision reaction-rate theory [2], [3], [11]……Page 608
B.3.4. Transition-state theory [2], [3], [11], [51], [52]……Page 610
B.3.5. Comparison between transition-state theory and collisional theory……Page 612
B.3.7. Modern developments in reaction-rate theory……Page 614
B.4.3. Adsorption of desorption rate-controlling……Page 616
B.4.4. Surface reaction rate-controlling……Page 619
REFERENCES……Page 622
C. Continuum Derivation of the Conservation Equations……Page 625
C.l. DEFINITIONS AND BASIC MATHEMATICAL RELATIONS……Page 626
C.2. CONTINUITY EQUATIONS……Page 628
C.3. MOMENTUM EQUATION……Page 629
C.4. ENERGY EQUATION……Page 631
C.5. COMPARISON BETWEEN THE CONSERVATION LAWS DERIVED FOR INDEPENDENT COEXISTENT CONTINUA AND THE KINETIC-THEORY RESULTS FO……Page 633
C.5.1. Definitions of kinetic theory……Page 634
C.5.2. Comparison of conservation equations……Page 635
C.6. PROOF OF EQUATION (6)……Page 636
REFERENCES……Page 638
D.l. THE VELOCITY DISTRIBUTION FUNCTION AND THE BOLTZMANN EQUATION……Page 639
D.2. DEFINITIONS OF FLUID-DYNAMICAL VARIABLES……Page 641
D.4. SUMMATIONAL INVARIANTS……Page 645
D.5.2. Momentum conservation……Page 646
D.5.4. Species conservation……Page 647
REFERENCES……Page 648
E. Transport Properties……Page 649
E.1. COLLISION INTEGRALS……Page 650
E.2.1. Physical derivation of the multicomponent diffusion equation 1101……Page 652
E.2.2. Simplified diffusion equations……Page 655
E.2.3. Binary diffusion coefficients……Page 656
E.2.4. Multicomponent diffusion coefficients……Page 657
E.2.5. Thermal diffusion coefficients……Page 658
E.3. UNIFIED ELEMENTARY TREATMENT OF TRANSPORT PROCESSES……Page 659
E.4.1. Coefficient of viscosity……Page 661
E.4.2. The pressure tensor……Page 662
E.5.1. Thermal conductivity……Page 663
E.5.2. The heat-flux vector……Page 664
E.6. DIMENSIONLESS RATIOS OF TRANSPORT COEFFICIENTS……Page 667
REFERENCES……Page 668
Subject Index……Page 685
Author Index……Page 671