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7.1. INTRODUCTION 209 300 250 TURBULENT FLAME SPEED, cm/s 200 150 100 50 0 0 10 20 30 40x10 3 REYNOLDS NUMBER OF PIPE FLOW Figure 7.2: Variation of flame speed with Reynolds number of flow. locity is half of that for a Reynolds number 3 × 10 4 . Other measurements have given turbulent velocities considerably higher than these observed here. From the theoretical stand-point several attempts have been made to explain the activation of combustion due to turbulence. The first attempt was made by Damköhler [2] who pointed-out two causes for the action of turbulence. One is the increase in the transport coefficients (conductivity and diffusion) due to turbulent diffusivity. The second cause is the distortion of the laminar flame front due to the turbulent oscillations of the velocity which would increase, considerably, the effective surface of the combustion front. The importance of either one of these two factors would depend in each case on the relation between the scale of turbulence and the thickness of the laminar
210 CHAPTER 7. TURBULENT FLAMES flame. In technical applications this ratio is generally very large, hence the main cause for the activation of combustion would be in this case the distortion of the flame front. Figure 7.1b seems to confirm this point of view. The major part of the later works on the subject have attempted to estimate the importance of either one of these causes, preserving Damköhler’s model. In 1952 von Kármán and Marble [12] proposed a different model which considered the turbulent flame as a zone whose structure should be defined by mean values of the characteristic variables (temperature, reaction rate, etc.) to which were superimposed static oscillations of turbulent nature. Later, this model was adopted and developed by Sommerfield and his collaborators [13], who obtained some experimental evidence (not sufficient) that the approximation is correct. The comparison between theoretical and experimental results is not conclusive enough in any case to establish either one of the proposed theories, which are actually in a preliminary phase. Consequently the present study will only be a brief exposure of the basic principles and fundamental results of these theories. 7.2 Turbulent combustion theories So far, the only type of turbulence whose influence on combustion has been studied is the isotropic. This turbulence is characterized by two magnitudes: its scale l and its intensity v ′ . 1 Damköhler studies the influence of turbulence on the flame by comparing l and v ′ with the corresponding magnitudes of the laminar combustion wave of the mixture; these being thickness d l of the flame and the laminar propagation velocity u l . When performing this comparison the following cases arise. flame 1) The turbulence scale is small when compared to the thickness of the laminar l d l ≪ l (7.1) In this case the action of turbulence is reduced to an activation of the transport coefficients conductivity and diffusion at the flame). Let x = λ ρc p (7.2) and ε ∼ lv ′ , (7.3) 1 For an exposure of the principles of turbulence, see, i.e., H.L. Dryden: A review of the Statistical Theory of Turbulence. Quart. Applied Math., Vol. 1, 1943, pp. 7-42.
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Aerothermochemistry Gregorio Millá
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PRESENTACIÓN Amable Liñán Es un
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que no cambiaron los resultados, y
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INSTITUTO NACIONAL DE TÉCNICA AERO
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aim to become acquainted with Aerot
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Tarifa, Manuel de Sendagorta and Ig
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3.4 Energy equation . . . . . . . .
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8.4 Quenching . . . . . . . . . . .
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Chapter 1 Thermochemistry 1.1 Intro
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1.1. INTRODUCTION 3 GAS ε/k (K) σ
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1.1. INTRODUCTION 5 14 12 10 N 2 CH
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1.2. THERMODYNAMIC FUNCTIONS OF AN
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1.2. THERMODYNAMIC FUNCTIONS OF AN
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1.3. MIXTURE OF GASES 11 Helmholtz
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1.3. MIXTURE OF GASES 13 is the par
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1.3. MIXTURE OF GASES 15 Entropy Si
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1.4. CALCULATION OF THE THERMODYNAM
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1.4. CALCULATION OF THE THERMODYNAM
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1.5. CHEMICAL REACTIONS IN A MIXTUR
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1.6. CHEMICAL EQUILIBRIUM 23 Since
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∆G 0 j = ∑ i 1.7. CASE OF A MIX
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1.7. CASE OF A MIXTURE OF IDEAL GAS
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1.8. CHEMICAL KINETICS 29 1.8 Chemi
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1.8. CHEMICAL KINETICS 31 Mechanics
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1.8. CHEMICAL KINETICS 33 This stru
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1.8. CHEMICAL KINETICS 35 [3] Hirsc
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Chapter 2 Transport phenomena in ga
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2.2. DIFFUSION 39 Hereinafter, nota
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2.2. DIFFUSION 41 where r is the di
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2.2. DIFFUSION 43 T ∗ Ω (1,1)∗
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2.2. DIFFUSION 45 Gas Pair σ 12 (
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2.2. DIFFUSION 47 coefficients. 14
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2.3. VISCOSITIES 49 2.3 Viscosities
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2.3. VISCOSITIES 51 Here µ and µ
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2.3. VISCOSITIES 53 Its value depen
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2.4. THERMAL CONDUCTIVITY 55 2000 1
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2.4. THERMAL CONDUCTIVITY 57 4000 3
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Chapter 3 General equations 3.1 Int
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3.2. EQUATION OF CONTINUITY 61 The
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3.2. EQUATION OF CONTINUITY 63 A i
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3.4. ENERGY EQUATION 65 where the s
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3.4. ENERGY EQUATION 67 The energy
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3.5. GENERAL EQUATIONS 69 obtained
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3.6. ENTROPY VARIATION 71 Taking in
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3.7. ONE-DIMENSIONAL MOTIONS 73 red
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3.8. STATIONARY, ONE-DIMENSIONAL MO
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3.9. THE CASE OF ONLY TWO CHEMICAL
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3.10. STATIONARY, ONE-DIMENSIONAL M
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3.11. APPENDIX: NOTATION AND REPERT
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3.11. APPENDIX: NOTATION AND REPERT
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Chapter 4 Combustion Waves 4.1 Deto
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4.2. KINDS OF DETONATIONS AND DEFLA
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4.2. KINDS OF DETONATIONS AND DEFLA
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4.3. VELOCITY OF THE BURNT GASES 95
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4.3. VELOCITY OF THE BURNT GASES 97
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4.4. PROPAGATION VELOCITY 99 p H’
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4.5. APPLICATIONS 101 By substituti
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4.7. INDETERMINACY OF THE SOLUTION
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Chapter 5 Structure of the combusti
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5.2. WAVE EQUATIONS 107 of diffusio
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5.2. WAVE EQUATIONS 109 2) Burnt ga
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5.4. LIMITING FORM OF THE WAVE EQUA
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5.4. LIMITING FORM OF THE WAVE EQUA
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5.5. DETONATIONS 115 Of these two v
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5.5. DETONATIONS 117 waves. Such th
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5.5. DETONATIONS 119 This relation
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5.5. DETONATIONS 121 curves, repres
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5.6. DEFLAGRATIONS 123 mum temperat
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5.6. DEFLAGRATIONS 125 8 7 v/v 1 =T
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5.7. TRANSITION FROM DEFLAGRATION T
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5.7. TRANSITION FROM DEFLAGRATION T
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Chapter 6 Laminar flames 6.1 Introd
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6.1. INTRODUCTION 133 papers presen
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6.3. BOUNDARY CONDITIONS 135 c) Dif
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6.4. MODIFICATION OF THE CONDITIONS
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6.4. MODIFICATION OF THE CONDITIONS
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6.6. EXAMPLE 141 generally impossib
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6.6. EXAMPLE 143 Two more relations
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6.7. REACTION VELOCITY 145 1.0 0.8
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6.8. FLAME EQUATIONS 147 6.8 Flame
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6.9. SOLUTION OF THE FLAME EQUATION
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Page 178 and 179: 6.10. STRUCTURE OF THE COMBUSTION W
Page 180 and 181: 6.11. IGNITION TEMPERATURE 161 6.11
Page 182 and 183: 6.12. GENERAL EQUATIONS FOR THE COM
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Page 242 and 243: 8.2. IGNITION 223 comparison betwee
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Page 248 and 249: 8.4. QUENCHING 229 [19] Simon, D. M
Page 250 and 251: Chapter 9 Flows with combustion wav
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Page 254 and 255: 9.3. NORMAL FLAME FRONT 235 9.3 Nor
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Page 266 and 267: 10.1. INTRODUCTION 247 form numeric
Page 268 and 269: 10.1. INTRODUCTION 249 ∆ P /∆ P
Page 270 and 271: 10.2. TSIEN METHOD 251 As aforesaid
Page 272 and 273: 10.2. TSIEN METHOD 253 1.0 λ=6.0 M
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10.5. CHAMBER WITH SLOWLY VARYING C
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Chapter 11 Similarity in combustion
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11.2. DIMENSIONLESS PARAMETERS OF A
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11.2. DIMENSIONLESS PARAMETERS OF A
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11.3. SCALING OF ROCKETS 267 Furthe
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11.3. SCALING OF ROCKETS 269 From h
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11.4. SCALING OF ROCKETS FOR NON-ST
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11.4. SCALING OF ROCKETS FOR NON-ST
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11.5. FLAME STABILIZATION 275 Flame
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11.5. FLAME STABILIZATION 277 Flame
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11.5. FLAME STABILIZATION 279 which
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Chapter 12 Diffusion flames 12.1 In
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12.1. INTRODUCTION 283 In fact, oth
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12.1. INTRODUCTION 285 which preced
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12.1. INTRODUCTION 287 remains appr
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12.2. GENERAL EQUATIONS FOR LAMINAR
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12.3. BOUNDARY CONDITIONS ON THE FL
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12.4. SIMPLIFIED EQUATIONS 293 As f
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12.4. SIMPLIFIED EQUATIONS 295 whil
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12.5. SOLUTIONS OF THE SIMPLIFIED S
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12.5. SOLUTIONS OF THE SIMPLIFIED S
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Chapter 13 Combustion of liquid fue
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13.2. ATOMIZATION 303 lem has been
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13.4. COMBUSTION 305 solution corre
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13.5. NOTATION 307 2) The phenomeno
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13.6. CONTINUITY EQUATIONS 309 Chem
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13.7. ENERGY EQUATION 311 The value
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13.8. DIFFUSION EQUATIONS 313 13.8
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13.9. COMBUSTION VELOCITY OF THE DR
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13.10. INTEGRATION OF THE EQUATIONS
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13.11. NUMERICAL APPLICATION 319 wh
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13.11. NUMERICAL APPLICATION 321 10
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13.12. COMPARISON WITH EXPERIMENTAL
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13.14. COMBUSTION OF FUEL SPRAYS 32
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13.15. DROPLET EVAPORATION 327 4 0.
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13.15. DROPLET EVAPORATION 329 1 0.
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13.16. APPENDIX: APPLICATION OF PRO
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