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6.9. SOLUTION OF THE FLAME EQUATIONS 149 second iteration of the Boys-Corner method but it is very arduous to perform and, furthermore, like the first one it does not converge towards the correct solution for increasing activation temperatures. 1.5 1.4 1.3 ZELDOVICH 1.2 Λ −1/2 / Λ −1/2 exact 1.1 1.0 0.9 0.8 WILDE KARMAN 3 SENDAGORTA BOYS−CORNER 2nd IT. KARMAN 2 KARMAN 1 0.7 BOYS−CORNER 0.6 2 4 6 8 10 12 14 16 θ a Figure 6.5: Comparison between the flame propagation problem eigenvalue obtained by different approximated methods and the exact value for θ 0 = 0.125. 1.3 Λ −1/2 / Λ −1/2 exact 1.2 1.1 1.0 0.9 0.8 0.7 ZELDOVICH WILDE SENDAGORTA KARMAN 3 BOYS−CORNER 2nd IT. KARMAN 2 KARMAN 1 BOYS−CORNER 0.6 2 4 6 8 10 12 14 16 θ a Figure 6.6: Comparison between the flame propagation problem eigenvalue obtained by different approximated methods and the exact value for θ 0 = 0.250.
150 CHAPTER 6. LAMINAR FLAMES The other methods are considerably better mainly when applied to normal activation temperatures. This is specially true about Wilde’s method, the second and third alternatives of von Kármán’s one and that proposed by Sendagorta [23], which is the best and does not require more work than the others. 1.0 0.8 BOYS−CORNER 0.6 EXACT θ 0.4 0.2 KARMAN 3 0.0 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0 ε Figure 6.7: Comparison of the distributions of ɛ vs θ obtained by two approximate methods with the exact result. Figure 6.7 shows, for one of the case calculated, the law of the variation of ε with θ of the Boys-Corner approximation, Kármán’s third alternative and the exact solutions. It is evident that the approximation reached with Kármán’s method is very satisfactory. The following studies the nature of this method after referring the reader to the above references for a more detailed analysis of the other methods mentioned. If in Eq. (6.67) we bring the denominator of the right hand side into the left one, and this equation is integrated between the cold boundary (6.71) and the hot boundary (6.70), we obtain 1 − θ 0 2 − ∫ 1 0 ∫ 1 ( ) n λ 1 − Y (1 − θ) dε = Λ(1 − θ 0 ) θ δ−n e −θ 1 − θ a θ dθ. θ i λ f 1 + aY (6.73) Consequently, in order to determine Λ, the problem reduces now to obtaining an approximation of θ vs ε on the integral of the left side of Eq. (6.73) and an approximation of Y vs θ on the integral of the right side of the same equation. These integrals
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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
Page 118 and 119: 4.4. PROPAGATION VELOCITY 99 p H’
Page 120 and 121: 4.5. APPLICATIONS 101 By substituti
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Page 124 and 125: Chapter 5 Structure of the combusti
Page 126 and 127: 5.2. WAVE EQUATIONS 107 of diffusio
Page 128 and 129: 5.2. WAVE EQUATIONS 109 2) Burnt ga
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Page 134 and 135: 5.5. DETONATIONS 115 Of these two v
Page 136 and 137: 5.5. DETONATIONS 117 waves. Such th
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Page 140 and 141: 5.5. DETONATIONS 121 curves, repres
Page 142 and 143: 5.6. DEFLAGRATIONS 123 mum temperat
Page 144 and 145: 5.6. DEFLAGRATIONS 125 8 7 v/v 1 =T
Page 146 and 147: 5.7. TRANSITION FROM DEFLAGRATION T
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Page 150 and 151: Chapter 6 Laminar flames 6.1 Introd
Page 152 and 153: 6.1. INTRODUCTION 133 papers presen
Page 154 and 155: 6.3. BOUNDARY CONDITIONS 135 c) Dif
Page 156 and 157: 6.4. MODIFICATION OF THE CONDITIONS
Page 158 and 159: 6.4. MODIFICATION OF THE CONDITIONS
Page 160 and 161: 6.6. EXAMPLE 141 generally impossib
Page 162 and 163: 6.6. EXAMPLE 143 Two more relations
Page 164 and 165: 6.7. REACTION VELOCITY 145 1.0 0.8
Page 166 and 167: 6.8. FLAME EQUATIONS 147 6.8 Flame
Page 170 and 171: 6.9. SOLUTION OF THE FLAME EQUATION
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
Page 188 and 189: 6.13. OZONE DECOMPOSITION FLAME 169
Page 196 and 197: 6.14. HYDRAZINE DECOMPOSITION FLAME
Page 206 and 207: 6.15. FLAME PROPAGATION IN HYDROGEN
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6.15. FLAME PROPAGATION IN HYDROGEN
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Chapter 7 Turbulent flames 7.1 Intr
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7.1. INTRODUCTION 209 300 250 TURBU
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7.2. TURBULENT COMBUSTION THEORIES
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7.4. COMPARISON WITH EXPERIMENTAL R
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Chapter 8 Ignition, flammability an
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8.2. IGNITION 223 comparison betwee
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8.3. FLAMMABILITY LIMITS 225 The pr
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8.4. QUENCHING 227 8.4 Quenching So
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8.4. QUENCHING 229 [19] Simon, D. M
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Chapter 9 Flows with combustion wav
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9.2. CONDITIONS THAT MUST BE SATISF
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9.3. NORMAL FLAME FRONT 235 9.3 Nor
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9.4. INCLINED FLAME FRONT 237 60 50
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9.5. ENTROPY JUMP ACROSS THE FLAME
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9.5. ENTROPY JUMP ACROSS THE FLAME
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9.6. VORTICITY ACROSS THE FLAME 243
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Chapter 10 Aerothermodynamic field
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10.1. INTRODUCTION 247 form numeric
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10.1. INTRODUCTION 249 ∆ P /∆ P
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10.2. TSIEN METHOD 251 As aforesaid
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10.2. TSIEN METHOD 253 1.0 λ=6.0 M
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10.3. METHOD OF FABRI-SIESTRUNCK-FO
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10.3. METHOD OF FABRI-SIESTRUNCK-FO
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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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