Compressible Flow - TME085

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1 Compressible Flow - TME085 Lecture 14 Niklas Andersson Chalmers University of Technology Department of Mechanics and Maritime Sciences Division of Fluid Mechanics Gothenburg, Sweden niklas.andersson@chalmers.se

2 Overview Compressible flow Basic Concepts Compressibility flow regimes speed of sound Thermodynamics thermally perfect gas calorically perfect gas entropy 1:st and 2:nd law High temperature effects molecular motion internal energy Boltzmann distribution equilibrium gas CFD Spatial discretization Numerical schemes Time integration Shock handling Boundary conditions PDE:s traveling waves method of characteristics finite non-linear waves acoustic waves shock reflection moving shocks governing equations Crocco s equation entropy equation substantial derivative nonconservation form conservation form Conservation laws integral form Quasi 1D Flow diffusers nozzles governing equations 2D Flow shock expansion theory expansion fans shock reflection oblique shocks 1D Flow friction heat addition normal shocks isentropic flow governing equations energy momentum continuity

3 Chapter 16 Properties of High-Temperature Gases Chapter 17 High-Temperature Flows: Basic Examples Niklas Andersson - Chalmers 3 / 83

4 Overview flow regimes momentum continuity Compressible f speed of sound thermally perfect gas calorically perfect gas Compressibility Thermodynamics Basic Concepts High temperature effects Spatia discretizatio equilibrium gas entropy 1:st and 2:nd law molecular motion internal energy Boltzmann distribution Niklas Andersson - Chalmers 4 / 83

5 Addressed Learning Outcomes 6 Define the special cases of calorically perfect gas, thermally perfect gas and real gas and explain the implication of each of these special cases A deep dive into the theory behind the definitions of calorically perfect gas, thermally perfect gas, and other models Niklas Andersson - Chalmers 5 / 83

6 Roadmap - High Temperature Effects Thermodynamic and chemical equilibrium Equilibrium gas: practical examples Gas models: Calorically perfect gas Thermally perfect gas Equilibrium gas Normal shock Nozzle flow Thermodynamic properties Boltzmann distribution Microscopic description of gases Niklas Andersson - Chalmers 6 / 83

7 Properties of High-Temperature Gases Applications: Rocket nozzle flows Reentry vehicles Shock tubes / Shock tunnels Internal combustion engines Niklas Andersson - Chalmers 7 / 83

8 Properties of High-Temperature Gases Example: Reentry vehicle Mach 32.5 Air Calorically perfect gas T = 283 Table A.2 T s /T = 206 T = 283 T s = K Niklas Andersson - Chalmers 8 / 83

9 Properties of High-Temperature Gases Example: Reentry vehicle Mach 32.5 Air Calorically perfect gas T = 283 Table A.2 T s /T = 206 T = 283 T s = K A more correct value is T s = K Something is fishy here! Niklas Andersson - Chalmers 8 / 83

10 Roadmap - High Temperature Effects Thermodynamic and chemical equilibrium Equilibrium gas: practical examples Gas models: Calorically perfect gas Thermally perfect gas Equilibrium gas Normal shock Nozzle flow Thermodynamic properties Boltzmann distribution Microscopic description of gases Niklas Andersson - Chalmers 9 / 83

11 Chapter 16.2 Microscopic Description of Gases Niklas Andersson - Chalmers 10 / 83

12 Microscopic Description of Gases Hard to make measurements Accurate, reliable theoretical models needed Available models do work quite well Niklas Andersson - Chalmers 11 / 83

13 Molecular Energy Vy y Vx x Vz z Translational kinetic energy thermal degrees of freedom: 3 Rotational kinetic energy thermal degrees of freedom: 2 for diatomic gases 2 for linear polyatomic gases 3 for non-linear polyatomic gases Vibrational energy (kinetic energy + potential energy) thermal degrees of freedom: 2 Electronic energy of electrons in orbit (kinetic energy + potential energy) O C O CO2 linear polyatomic molecule H O H2O non-linear polyatomic molecule H Translational energy Rotational energy (only for molecules - not for mono-atomic gases) Vibrational energy Electronic energy Niklas Andersson - Chalmers 12 / 83

14 Molecular Energy The energy for one molecule can be described by ε = ε trans + ε rot + ε vib + ε el Results of quantum mechanics have shown that each energy is quantized i.e. they can exist only at discrete values Not continuous! Might seem unintuitive Niklas Andersson - Chalmers 13 / 83

15 Molecular Energy The lowest quantum numbers defines the zero-point energy for each mode for rotational energy the zero-point energy is exactly zero ε o trans is very small but finite - at absolute zero, molecules still moves but not much ε jtrans = ε j trans ε o trans ε lvib = ε l vib ε o vib ε krot = ε k rot ε mel = ε m el ε o el Niklas Andersson - Chalmers 14 / 83

16 Energy States three cases with the same rotational energy different direction of angular momentum quantum mechanics different distinguishable states a finite number of possible states for each energy level Niklas Andersson - Chalmers 15 / 83

17 Macrostates and Microstates Macrostate: molecules collide and exchange energy the N j distribution (the macrostate) will change over time some macrostates are more probable than other most probable macrostates (distribution) thermodynamic equilibrium Microstate: same number of molecules in each energy level but different states the most probable macrostate is the one with the most possible microstates possible to find the most probable macrostate by counting microstates Niklas Andersson - Chalmers 16 / 83

18 Macrostates and Microstates Macrostate I Microstate I ε o : (N o = 2, g o = 5) ε 1 : (N 1 = 5, g 1 = 6) ε 2 : (N 2 = 3, g 2 = 5). ε j : (N j = 2, g j = 3) Niklas Andersson - Chalmers 17 / 83

19 Macrostates and Microstates Macrostate I Microstate II ε o : (N o = 2, g o = 5) ε 1 : (N 1 = 5, g 1 = 6) ε 2 : (N 2 = 3, g 2 = 5). ε j : (N j = 2, g j = 3) Niklas Andersson - Chalmers 17 / 83

20 Macrostates and Microstates Macrostate II Microstate I ε o : (N o = 1, g o = 5) ε 1 : (N 1 = 5, g 1 = 6) ε 2 : (N 2 = 4, g 2 = 5). ε j : (N j = 1, g j = 3) Niklas Andersson - Chalmers 17 / 83

21 Macrostates and Microstates N = j N j E = j ε j N j Niklas Andersson - Chalmers 18 / 83

22 Roadmap - High Temperature Effects Thermodynamic and chemical equilibrium Equilibrium gas: practical examples Gas models: Calorically perfect gas Thermally perfect gas Equilibrium gas Normal shock Nozzle flow Thermodynamic properties Boltzmann distribution Microscopic description of gases Niklas Andersson - Chalmers 19 / 83

23 Chapter 16.5 The Limiting Case: Boltzmann Distribution Niklas Andersson - Chalmers 20 / 83

24 Boltzmann Distribution The Boltzmann distribution: N j = N g je ε j/kt Q where Q = f(t, V) is the state sum defined as Q j g j e ε j/kt g j is the number of degenerate states, ε j is the energy above zero-level (ε j = ε j ε o ), and k is the Boltzmann constant Niklas Andersson - Chalmers 21 / 83

25 Boltzmann Distribution The Boltzmann distribution: N j = N g je ε j/kt Q For molecules or atoms of a given species, quantum mechanics says that a set of well-defined energy levels ε j exists, over which the molecules or atoms can be distributed at any given instant, and that each energy level has a certain number of energy states, g j. For a system of N molecules or atoms at a given T and V, N j are the number of molecules or atoms in each energy level ε j when the system is in thermodynamic equilibrium. Niklas Andersson - Chalmers 22 / 83

26 Boltzmann Distribution P Boltzmann distribution for a specific temperature E At temperatures above 5K, molecules are distributed over many energy levels, and therefore the states are generally sparsely populated (N j g j ) Higher energy levels become more populated as temperature increases Niklas Andersson - Chalmers 23 / 83

27 Roadmap - High Temperature Effects Thermodynamic and chemical equilibrium Equilibrium gas: practical examples Gas models: Calorically perfect gas Thermally perfect gas Equilibrium gas Normal shock Nozzle flow Thermodynamic properties Boltzmann distribution Microscopic description of gases Niklas Andersson - Chalmers 24 / 83

28 Chapter Evaluation of Gas Thermodynamic Properties Niklas Andersson - Chalmers 25 / 83

29 Internal Energy The internal energy is calculated as E = NkT 2 ( ln Q T The internal energy per unit mass is obtained as ) V e = E M = NkT 2 Nm ( ) ln Q T V = { } ( ) k ln Q m = R = RT 2 T V Niklas Andersson - Chalmers 26 / 83

30 Internal Energy - Translation ( ) ε trans = h2 n 2 1 8m a 2 + n2 2 1 a 2 + n2 3 2 a 2 3 n 1 n 3 quantum numbers (1,2,3,...) a 1 a 3 linear dimensions that describes the size of the system h Planck s constant m mass of the individual molecule ( ) 2πmkT 3/2 Q trans = V h 2 Niklas Andersson - Chalmers 27 / 83

31 Internal Energy - Translation ( ) 2πmkT 3/2 Q trans = V h 2 ln Q trans = 3 2 ln T ln 2πmk h 2 + ln V ( ) ln Qtrans T V = T ( ) ln e trans = RT 2 Qtrans = RT 2 3 T V 2T = 3 2 RT Niklas Andersson - Chalmers 28 / 83

32 Internal Energy - Rotation ε rot = h2 8π 2 J(J + 1) I J rotational quantum number (0,1,2,...) I moment of inertia (tabulated for common molecules) h Planck s constant Q rot = 8π2 IkT h 2 Niklas Andersson - Chalmers 29 / 83

33 Internal Energy - Rotation Q rot = 8π2 IkT h 2 ln Q rot = ln T + ln 8π2 Ik h 2 ( ) ln Qrot T V = 1 T ( ) ln e rot = RT 2 Qrot = RT 2 1 T V T = RT Niklas Andersson - Chalmers 30 / 83

34 Internal Energy - Vibration ( ε vib = hν n + 1 ) 2 n vibrational quantum number (0,1,2,...) ν fundamental vibrational frequency (tabulated for common molecules) h Planck s constant Q vib = 1 1 e hν/kt Niklas Andersson - Chalmers 31 / 83

35 Internal Energy - Vibration Q vib = 1 1 e hν/kt ln Q vib = ln(1 e hν/kt ) ( ) ln Qvib T V = hν/kt 2 e hν/kt 1 ( ) ln e vib = RT 2 Qvib = RT 2 hν/kt 2 T V e hν/kt 1 = hν/kt e hν/kt 1 RT lim T hν/kt e hν/kt 1 = 1 e vib RT Niklas Andersson - Chalmers 32 / 83

36 Specific Heat e = e trans + e rot + e vib + e el e = 3 hν/kt RT + RT + 2 e hν/kt 1 RT + e el C v ( ) e T V Niklas Andersson - Chalmers 33 / 83

37 Specific Heat Molecules with only translational and rotational energy e = 3 2 RT + RT = 5 2 RT C v = 5 2 R C p = C v + R = 7 2 R γ = C p C v = 7 5 = 1.4 Niklas Andersson - Chalmers 34 / 83

38 Specific Heat Mono-atomic gases with only translational and rotational energy e = 3 2 RT C v = 3 2 R C p = C v + R = 5 2 R γ = C p C v = 5 3 = Niklas Andersson - Chalmers 35 / 83

39 Roadmap - High Temperature Effects Thermodynamic and chemical equilibrium Equilibrium gas: practical examples Gas models: Calorically perfect gas Thermally perfect gas Equilibrium gas Normal shock Nozzle flow Thermodynamic properties Boltzmann distribution Microscopic description of gases Niklas Andersson - Chalmers 36 / 83

40 Calorically Perfect Gas In general, only translational and rotational modes of molecular excitation Translational and rotational energy levels are sparsely populated, according to Boltzmann distribution (the Boltzmann limit) Vibrational energy levels are practically unpopulated (except for the zero level) Characteristic values of γ for each type of molecule, e.g. mono-atomic gas, di-atomic gas, tri-atomic gas, etc He, Ar, Ne,... - mono-atomic gases (γ = 5/3) H 2, O 2, N 2,... - di-atomic gases (γ = 7/5) H2 O (gaseous), CO 2,... - tri-atomic gases (γ < 7/5) Niklas Andersson - Chalmers 37 / 83

41 Calorically Perfect Gas p = ρrt e = C v T h = C p T h = e + p/ρ C p C v = R γ = C p /C v C v = C p = R γ 1 γr γ 1 γ, R, C v, and C p are constants γp a = ρ = γrt Niklas Andersson - Chalmers 38 / 83

42 Thermally Perfect Gas In general, only translational, rotational and vibrational modes of molecular excitation Translational and rotational energy levels are sparsely populated, according to Boltzmann distribution (the Boltzmann limit) The population of the vibrational energy levels approaches the Boltzmann limit as temperature increases Temperature dependent values of γ for all types of molecules except mono-atomic (no vibrational modes possible) Niklas Andersson - Chalmers 39 / 83

43 Thermally Perfect Gas p = ρrt e = e(t) C v = de/dt h = h(t) C p = dh/dt h = e + p/ρ C p C v = R γ = C p /C v C v = C p = R γ 1 γr γ 1 R is constant γ, C v, and C p are variable (functions of T) γp a = ρ = γrt Niklas Andersson - Chalmers 40 / 83

44 High-Temperature Effects Example: properties of air 2000 K T Thermally perfect gas: e and h are non-linear functions of T the temperatur range represents standard atmospheric pressure (lower pressure gives lower temperatures) region of variable γ thermally perfect gas 600 K region of constant γ (γ=1.4) calorically perfect gas 50 K Niklas Andersson - Chalmers 41 / 83

45 High-Temperature Effects For cases where the vibrational energy is not negligible (high temperatures) lim e vib = RT C v = 7 T 2 R However, chemical reactions and ionization will take place long before that Translational and rotational energy fully excited above 5 K Vibrational energy is non-negligible above 600 K Chemical reactions begin to occur above 2000 K Niklas Andersson - Chalmers 42 / 83

46 High-Temperature Effects As temperature increase further vibrational energy becomes less important Why is that so? Niklas Andersson - Chalmers 43 / 83

47 High-Temperature Effects Example: properties of air (continued) T 9000 K O O + + e (start of ionization) 4000 K 2500 K N 2 2N (start of dissociation) O 2 2O (start of dissociation) no reactions With increasing temperature, the gas becomes more and more mono-atomic which means that vibrational modes becomes less important Niklas Andersson - Chalmers 44 / 83

48 Equilibrium Gas For temperatures T > 2500K Air may be described as being in thermodynamic and chemical equilibrium (Equilibrium Gas) reaction rates (time scales) low compared to flow time scales reactions in both directions (example: O 2 2O) Tables must be used (Equilibrium Air Data) or special functions which have been made to fit the tabular data Niklas Andersson - Chalmers 45 / 83

49 Equilibrium Gas How do we obtain a thermodynamic description? p = p(r, T) e = e(ν, T) h = h(p, T) h = e + p ρ C v = C p = ( ) e T ν ( ) h T p ( ) e a 2 p ν e = γrt ( ) h 1 ρ p T T γ = C p C v = ( ) h T ( ) e T p ν RT = p ρ Note: R is not a constant here i.e. this is not the ideal gas law Niklas Andersson - Chalmers 46 / 83

50 Roadmap - High Temperature Effects Thermodynamic and chemical equilibrium Equilibrium gas: practical examples Gas models: Calorically gas Thermally perfect gas Equilibrium gas Normal shock Nozzle flow Thermodynamic properties Boltzmann distribution Microscopic description of gases Niklas Andersson - Chalmers 47 / 83

51 Chapter 17.1 Thermodynamic and Chemical Equilibrium Niklas Andersson - Chalmers 48 / 83

52 Thermodynamic Equilibrium Molecules are distributed among their possible energy states according to the Boltzmann distribution (which is a statistical equilibrium) for the given temperature of the gas extremely fast process (time and length scales of the molecular processes) much faster than flow time scales in general (not true inside shocks) Niklas Andersson - Chalmers 49 / 83

53 Thermodynamic Equilibrium Global thermodynamic equilibrium: there are no gradients of p, T, ρ, v, species concentrations true thermodynamic equilibrium Local thermodynamic equilibrium: gradients can be neglected locally this requirement is fulfilled in most cases (hard not to get) Niklas Andersson - Chalmers 50 / 83

54 Chemical Equilibrium Composition of gas (species concentrations) is fixed in time forward and backward rates of all chemical reactions are equal zero net reaction rates chemical reactions may be either slow or fast in comparison to flow time scale depending on the case studied Niklas Andersson - Chalmers 51 / 83

55 Chemical Equilibrium Global chemical equilibrium: there are no gradients of species concentrations together with global thermodynamic equilibrium all gradients are zero Local chemical equilibrium gradients of species concentrations can be neglected locally not always true - depends on reaction rates and flow time scales Niklas Andersson - Chalmers 52 / 83

56 Thermodynamic and Chemical Equilibrium Most common cases: Thermodynamic Equilibrium Chemical Equilibrium Gas Model 1 local thermodynamic equilibrium local chemical equilibrium equilibrium gas 2 local thermodynamic equilibrium chemical non-equilibrium finite rate chemistry 3 local thermodynamic equilibrium frozen composition frozen flow 4 thermodynamic non-equilibrium frozen composition vibrationally frozen flow length and time scales of flow decreases from 1 to 4 Frozen composition no (or slow) reactions vibrationally frozen flow gives the same gas relations as calorically perfect gas! no chemical reactions and unchanged vibrational energy example: small nozzles with high-speed flow Niklas Andersson - Chalmers 53 / 83

57 Roadmap - High Temperature Effects Thermodynamic and chemical equilibrium Equilibrium gas: practical examples Gas models: Calorically gas Thermally perfect gas Equilibrium gas Normal shock Nozzle flow Thermodynamic properties Boltzmann distribution Microscopic description of gases Niklas Andersson - Chalmers 54 / 83

58 Chapter 17.2 Equilibrium Normal Shock Wave Flows Niklas Andersson - Chalmers 55 / 83

59 Equilibrium Normal Shock Wave Flows Question: Is the equilibrium gas assumption OK? Answer: for hypersonic flows with very little ionization in the shock region, it is a fair approximation not perfect, since the assumption of local thermodynamic and chemical equilibrium is not really true around the shock however, it gives a significant improvement compared to the calorically perfect gas assumption Niklas Andersson - Chalmers 56 / 83

60 Equilibrium Normal Shock Wave Flows Basic relations (for all gases), stationary normal shock: ρ 1 u 1 = ρ 2 u 2 ρ 1 u p 1 = ρ 2 u p 2 h u2 1 = h u2 2 For equilibrium gas we have: ρ = ρ(p, h) T = T(ρ, h) (we are free to choose any two states as independent variables) Niklas Andersson - Chalmers 57 / 83

61 Equilibrium Normal Shock Wave Flows Assume that ρ 1, u 1, p 1, T 1, and h 1 are known Also u 2 = ρ ( ) 2 1u 1 ρ 1 u 2 ρ1 1 + p 1 = ρ 2 u 1 + p 2 ρ 2 ρ 2 p 2 = p 1 + ρ 1 u 2 1 ( 1 ρ ) 1 ρ 2 h u2 1 = h ( h 2 = h u2 1 ( ) 2 ρ1 u 1 ρ 2 1 ( ρ1 ρ 2 ) 2 ) Niklas Andersson - Chalmers 58 / 83

62 Equilibrium Normal Shock Wave Flows initial guess ρ 1 ρ 2 when converged: calculate p 2 and h 2 ρ 2 = ρ(p 2, h 2 ) T 2 = T(ρ 2, h 2 ) update ρ 1 ρ 2 ρ 2 = ρ(p 2, h 2 ) ρ 2, u 2, p 2, T 2, h 2 known no ρ 2 ρ 2old < ε yes stop Niklas Andersson - Chalmers 59 / 83

63 Equilibrium Air - Normal Shock Tables of thermodynamic properties for different conditions are available For a very strong shock case (M 1 = 32), the table below (Table 17.1) shows some typical results for equilibrium air calorically perfect gas (γ = 1.4) equilibrium air p 2 /p ρ 2 /ρ h 2 /h T 2 /T Niklas Andersson - Chalmers 60 / 83

64 Equilibrium Air - Normal Shock Analysis: Pressure ratio is comparable Density ratio differs by factor of 2.5 Temperature ratio differs by factor of 5 Explanation: Using equilibrium gas means that vibration, dissociation and chemical reactions are accounted for The chemical reactions taking place in the shock region lead to an absorption of energy into chemical energy drastically reducing the temperature downstream of the shock this also explains the difference in density after the shock Niklas Andersson - Chalmers 61 / 83

65 Equilibrium Air - Normal Shock Additional notes: For a normal shock in an equilibrium gas, the pressure ratio, density ratio, enthalpy ratio, temperature ratio, etc all depend on three upstream variables, e.g. u 1, p 1, T 1 For a normal shock in a thermally perfect gas, the pressure ratio, density ratio, enthalpy ratio, temperature ratio, etc all depend on two upstream variables, e.g. M 1, T 1 For a normal shock in a calorically perfect gas, the pressure ratio, density ratio, enthalpy ratio, temperature ratio, etc all depend on one upstream variable, e.g. M 1 Niklas Andersson - Chalmers 62 / 83

66 Equilibrium Gas - Detached Shock calorically perfect gas equilibrium gas M = 20 M = 20 shock moves closer to body What s the reason for the difference in predicted shock position? Niklas Andersson - Chalmers 63 / 83

67 Equilibrium Gas - Detached Shock Calorically perfect gas: all energy ends up in translation and rotation increased temperature Equilibrium gas: energy is absorbed by reactions does not contribute to the increase of gas temperature Niklas Andersson - Chalmers 64 / 83

68 Roadmap - High Temperature Effects Thermodynamic and chemical equilibrium Gas models: Calorically gas Thermally perfect gas Equilibrium gas Thermodynamic properties Boltzmann distribution Microscopic description of gases Equilibrium gas: practical examples Normal shock Nozzle flow Niklas Andersson - Chalmers 65 / 83

69 Chapter 17.3 Equilibrium Quasi-One-Dimensional Nozzle Flows Niklas Andersson - Chalmers 66 / 83

70 Equilibrium Quasi-1D Nozzle Flows First question: Is chemically reacting gas also isentropic (for inviscid and adiabatic case)? entropy equation: Tds = dh νdp Quasi-1D equations in differential form (all gases): momentum equation: dp = ρudu energy equation: dh + udu = 0 Niklas Andersson - Chalmers 67 / 83

71 Equilibrium Quasi-1D Nozzle Flows udu = dp ρ = νdp Tds = udu νdp = udu + udu = 0 Isentropic flow! ds = 0 Niklas Andersson - Chalmers 68 / 83

72 Equilibrium Quasi-1D Nozzle Flows Second question: Does the area-velocity relation also hold for a chemically reacting gas? Isentropic process gives M = 1 at nozzle throat still holds da A = (M2 1) du u Niklas Andersson - Chalmers 69 / 83

73 Equilibrium Quasi-1D Nozzle Flows For general gas mixture in thermodynamic and chemical equilibrium, we may find tables or graphs describing relations between state variables. Example: Mollier diagram h p = constant T = constant for any point (h, s), we may find p, T, ρ, a, s Niklas Andersson - Chalmers 70 / 83

74 Equilibrium Quasi-1D Nozzle Flows h p o For steady-state inviscid adiabatic nozzle flow we have: T o h u2 1 = h u2 2 = h o h o assume h o is known p 1 T 1 where h o is the reservoir enthalpy 1 p 2 T 2 2 isentropic process s Niklas Andersson - Chalmers 71 / 83

75 Equilibrium Quasi-1D Nozzle Flows At point 1 in Mollier diagram we have: 1 2 u2 1 = h o h 1 u 1 = 2(h o h 1 ) Assume that u 1 = a 1 (sonic conditions) gives ρ 1 u 1 A 1 = ρ a A At any point along isentropic line, we have u = 2(h o h) and ρ, p, T, a etc are all given which means that ρu is given A A = ρ a ρu may be computed for any point along isentropic line Niklas Andersson - Chalmers 72 / 83

76 Equilibrium Quasi-1D Nozzle Flows Equilibrium gas gives higher T and more thrust During the expansion chemical energy is released due to shifts in the equilibrium composition T equilibrium gas calorically perfect gas 1 A/A Niklas Andersson - Chalmers 73 / 83

77 Equilibrium Quasi-1D Nozzle Flows Equilibrium gas gives higher T and more thrust During the expansion chemical energy is released due to shifts in the equilibrium composition T equilibrium gas calorically perfect gas 1 A/A Chemical and vibrational energy transfered to translation and rotation increased temperature Niklas Andersson - Chalmers 73 / 83

78 Equilibrium Quasi-1D Nozzle Flows - Reacting Mixture Real nozzle flow with reacting gas mixture: T equilibrium gas real case calorically perfect gas 1 A/A Niklas Andersson - Chalmers 74 / 83

79 Equilibrium Quasi-1D Nozzle Flows - Reacting Mixture Real nozzle flow with reacting gas mixture: T equilibrium gas real case calorically perfect gas 1 A/A Space nozzle applications: u e 4000 m/s Required prediction accuracy 5 m/s Niklas Andersson - Chalmers 74 / 83

80 Equilibrium Quasi-1D Nozzle Flows - Reacting Mixture Equilibrium gas: very fast chemical reactions local thermodynamic and chemical equilibrium Vibrationally frozen gas: very slow chemical reactions (no chemical reactions frozen gas) vibrational energy of molecules have no time to change calorically perfect gas! Niklas Andersson - Chalmers 75 / 83

81 Large Nozzles High T o, high p o, high reactivity Real case is close to equilibrium gas results Example: Ariane 5 launcher, main engine (Vulcain 2) H 2 + O 2 H 2 O in principle, but many different radicals and reactions involved (at least 10 species, 20 reactions) T o 3600 K, p o 120 bar Length scale a few meters Gas mixture is quite close to equilibrium conditions all the way through the expansion Niklas Andersson - Chalmers 76 / 83

82 Ariane 5 Ariane 5 space launcher extreme high temperature and high speed flow regime Niklas Andersson - Chalmers 77 / 83

83 Vulcain Engine Vulcain engine: first stage of the Ariane 5 launcher Niklas Andersson - Chalmers 78 / 83

84 Space Shuttle Launcher - SSME Niklas Andersson - Chalmers 79 / 83

85 Space Shuttle Launcher - SSME Niklas Andersson - Chalmers 80 / 83

86 Small Nozzles Low T o, low p o, lower reactivity Real case is close to frozen flow results Example: Small rockets on satellites (for maneuvering, orbital adjustments, etc) Niklas Andersson - Chalmers 81 / 83

87 Small Nozzles Niklas Andersson - Chalmers 82 / 83

88 Roadmap - High Temperature Effects Thermodynamic and chemical equilibrium Gas models: Calorically gas Thermally perfect gas Equilibrium gas Thermodynamic properties Boltzmann distribution Microscopic description of gases Equilibrium gas: practical examples Normal shock Nozzle flow Niklas Andersson - Chalmers 83 / 83

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