Untitled - Aerobib - Universidad Politécnica de Madrid

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9.6. VORTICITY ACROSS THE FLAME 243 is continuous across the flame, by applying Stokes theorem to both faces of a surface element of the flame, the following is obtained ω n1 = ω n2 , (9.59) that is, the normal component of the vorticity is continuous across the flame. The jump of ω τ can then be obtained from (9.57) by writing this equation for each si<strong>de</strong> of the flame and forming the difference. Thus, when taking into account that v n1 = ϕ, v n2 = λϕ, T 2 = λT 1 , S 2 = S 1 + ∆S there results ω τ2 = ω τ2 λ + T [ 1 λ − 1 ∂S 1 ϕ λ ∂t + ∂(∆S) ] . (9.60) ∂t Likewise by using (9.58) we obtain for ω t2 ω t2 = ω t1 λ − T [ 1 λ − 1 ϕ λ ∂S 1 ∂τ + ∂(∆S) ] . (9.61) ∂τ On the other hand, equations (9.57) and (9.58) , when written for conditions before the flame, give ϕω τ1 = T 1 ∂S 1 ∂t , (9.62) ϕω t1 − v t ω n1 = − T 1 ∂S 1 ∂τ . (9.63) By eliminating ∂S 1 and ∂S 1 between these two equations and (9.60) and (9.61), we ∂t ∂τ finally obtain for ω τ2 and ω t2 , ω τ2 = ω τ1 + T 1 ϕ ∂(∆S) , (9.64) ∂t ω t2 = ω t1 − λ − 1 λ ω n1 cot α 1 − T 1 ϕ ∂(∆S) . (9.65) ∂τ In particular, if the motion before the flame is irrotational (ω n1 = ω t1 = ω τ1 = 0), the vorticity produced by the flame results ω n2 = 0, (9.66) ω t2 = − T 1 ϕ ω τ2 = T 1 ϕ ∂(∆S) , ∂τ (9.67) ∂(∆S) . ∂t (9.68) Furthermore, if the entropy jump can be expressed as (9.53), we have ω t2 = − c pT 1 ϕ ω τ2 = c pT 1 ϕ 1 ∂λ λ ∂τ , (9.69) 1 ∂λ λ ∂t , (9.70)
244 CHAPTER 9. FLOWS WITH COMBUSTION WAVES or else, if the motion is rotational, the equations resulting from (9.64) and (9.65) when ∆S is substituted therein for its value (9.53). In the special case of a plane motion, the only component different from zero of the vorticity is ω τ , and its value after the flame is given by (9.64) if the motion before the flame is rotational, and by (9.68) if it is potential. In Fig. 9.7.a, the rotation sense of the vorticity generated by the flame has bean indicated, in the case of a plane motion, assuming that the motion before the flame is potential. In Fig. 9.7.b, the sense corresponding to a shock wave is shown for the same case. References [1] Emmons, H. W., Ball, G. A. and Maier, A. D.: Development of a Combustion Tunnel. Army Or<strong>de</strong>nance Project Report, Harvard University, Cambridge Mass., 1954. [2] Emmons, H. W.: Fundamentals of Gas Dynamics. Sec. E, Vol. III of High Speed Aerodynamics and Jet Propulsion. Princeton University Press. 1958. [3] Gross, R. A.: Combustion Tunnel Laboratory Interim Technical Report No. 2. Harvard University, June 1952. [4] Gross, R. A. and Esch, R.: Low Speed Combustion Aerodynamics. Jet Propulsion, March-April 1954, pp. 95-101. [5] von Kármán, Th.: Aerothermodynamics and Combustion Theory. L’Aerotecnica, Vol. XXXIII, Fasc. 1st., 1953, pp. 80-86.
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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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6.10. STRUCTURE OF THE COMBUSTION W
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6.11. IGNITION TEMPERATURE 161 6.11
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6.12. GENERAL EQUATIONS FOR THE COM
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6.13. OZONE DECOMPOSITION FLAME 169
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6.14. HYDRAZINE DECOMPOSITION FLAME
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6.15. FLAME PROPAGATION IN HYDROGEN
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Page 212 and 213: 6.15. FLAME PROPAGATION IN HYDROGEN
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Page 228 and 229: 7.1. INTRODUCTION 209 300 250 TURBU
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Page 240 and 241: Chapter 8 Ignition, flammability an
Page 242 and 243: 8.2. IGNITION 223 comparison betwee
Page 244 and 245: 8.3. FLAMMABILITY LIMITS 225 The pr
Page 246 and 247: 8.4. QUENCHING 227 8.4 Quenching So
Page 248 and 249: 8.4. QUENCHING 229 [19] Simon, D. M
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Page 252 and 253: 9.2. CONDITIONS THAT MUST BE SATISF
Page 254 and 255: 9.3. NORMAL FLAME FRONT 235 9.3 Nor
Page 256 and 257: 9.4. INCLINED FLAME FRONT 237 60 50
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Page 264 and 265: Chapter 10 Aerothermodynamic field
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
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Page 288 and 289: 11.3. SCALING OF ROCKETS 269 From h
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Page 294 and 295: 11.5. FLAME STABILIZATION 275 Flame
Page 296 and 297: 11.5. FLAME STABILIZATION 277 Flame
Page 298 and 299: 11.5. FLAME STABILIZATION 279 which
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Page 302 and 303: 12.1. INTRODUCTION 283 In fact, oth
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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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Untitled - Aerobib - Universidad Politécnica de Madrid

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