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6.14. HYDRAZINE DECOMPOSITION FLAME 185 The parabolic approximation given by Eq. (6.87) is adopted for Y vs θ ( ) L θ − θ0 Y = , (6.214) 1 − θ 0 which satisfies Eq. (6.203) for ε = 0, and for ε = 1 is tangent to the straight line 1 − Y = L 1 − θ 1 − θ 0 , (6.215) which, in turn, is also tangent to the solution of Eq. (6.216) as it can easily be verified when considering that for θ ≃ 1 is and 1 − ε ≫ 1 − Y, (6.216) 1 − ε ≫ 1 − θ . (6.217) 1 − θ 0 When approximation (6.214) is taken into Eq. (6.213), it results where 1 − θ 0 2 − ∫ 1 ∫ ⎡ ( ) 1 θ − L ⎤3/2 θ0 1 − ⎢ 1 − θ I K = ⎣ 0 ⎥ ( ) θ − L ⎦ θ0 θ 0 1 + 1 − θ 0 The approximation of θ vs ε for the computation of 0 (1 − θ) dε = ΛI K , (6.213.a) e −θ 1 − θ a θ dθ. (6.218) ∫ 1 θ (1 − θ) dε can be obtained from the solution of Eq. (6.206) for values of ε close to 1. Through the procedure applied in Ref. [30] , it results 11 Therefore ∫ 1 0 [ ( ) ] 3/2 −2/5 1 − θ = (1 − ε) 4/5 4 L 1 − θ 0 5 Λ . (6.219) 2 [ (1 − θ) dε ≃ 5 ( ) ] 3/2 −2/5 9 (1 − θ 4 L 0) Λ −2/5 . (6.220) 5 2 If Eqs. (6.218) and (6.220) are taken into Eq. (6.213) the following equation will be obtained for the eigenvalue Λ ⎛ Λ = 1 − θ 0 2I K ⎝1 − 10 9 which is solved through iteration or else graphically. [ ( ) ] ⎞ 3/2 −2/5 4 L Λ −2/5 ⎠ , (6.221) 5 2 11 Essentially, it reduces to neglecting (1 − θ) respect to (1 − ε) in the denominator of Eq. (6.206) and to apply the linear approximation given by Eq. (6.215) in order to eliminate Y in the numerator of the same.
186 CHAPTER 6. LAMINAR FLAMES In order to perform the numerical calculation of the flame velocity the following values given by Hirschfelder were applied. These same values were also used by Gilbert and Altman in his work: ρ 0 = 9.17 × 10 −4 gr/cm 3 , T 0 = 423 K, T f = 1933 K, λ = 0.00067 cal cm s K , c p = 0.6623 cal gr K , L = 1.33. (6.222) With them, the reduced activation temperature results to be θ a = The numerical integration of Eq. (6.218) gives When the precedent values are taken into (6.221) which, when taken into (6.212), finally results 37 000 = 9 632. (6.223) 1.933 × 1.9872 I K 1 − θ 0 = 270.9 × 10 −5 . (6.224) Λ −1/2 = 8.2962 × 10 −2 , (6.225) u 0 = Λ−1/2 3.97 ≃ 209 cm/s. (6.226) The agreement between this value and the 200 cm/s given by experimental results is better than it could be expected considering the dubiousness of many of the numerical values used. Influence of pressure As shown in Eq. (6.212) the flame velocity varies with pressure in accordance with the following law u 0 ∼ p1/2 ρ 0 ∼ p −1/2 , (6.227) in agreement with the predictions of theory for first-order reaction flames Eq. (6.75) and as verified by the experiments carried-out by Adams and Stocks [36].
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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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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 168 and 169: 6.9. SOLUTION OF THE FLAME EQUATION
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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
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Page 240 and 241: Chapter 8 Ignition, flammability an
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
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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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