Simulation of Condensing Compressible Flows

Transcription

1 Simulation of Condensing Compressible Flows Maximilian Wendenburg

2 Outline Physical Aspects Transonic Flows and Experiments Condensation Fundamentals Practical Effects Modeling and Simulation Equations, Solver, Boundary Conditions Results Validation Effects in Laval Nozzles, Turbines and Compressors 2

3 Schlieren Photography: Visualizing y ρ x subsonic supersonic x lamp mirror sensitive! wind tunnel mirror camera 3

4 Supersonic Wind Tunnel valve vacuum tank 10 mbar 34 m³ window with Laval nozzle atmosphere A* 4

5 Variation of Static Properties in Isentropic Flows T/T 0 p/p 0 ρ/ρ 0 M = r u a 5

6 Condensation of Dissolved Water Vapor in Air low cooling rate dt 3 10 dt K s dt 5 10 dt K s dt 6 10 dt high cooling rate K s 6

7 Partial Pressure of Vapor p v in Isentropic Expansion p v [Pa] 2000 Φ 0 = 65 % S = pv p, ( T ) s i s e n t r o p i c e x p a n s i o n Φ 0 = pv p, ( T s ) T 0 = 295 K p 0 = 10 5 Pa 100% 500 p s, (T) (( )) T [K] 7

8 Types of Non-Equilibrium Condensation Homogeneous Nucleation cluster nucleus droplet Droplet growth Dominates at high cooling rates Heterogeneous Particles serve as seeds for droplets Dominates at low cooling rates 8

9 Practical Effects Aircraft Influence on lift and drag Loss of thrust Steam Turbines Erosion Oscillation 9

10 Physical Aspects Isentropic expansion: T, ρ, p Transonic flows dt 6 K High cooling rates ( 10 ) Supersaturation non-equilibrium dt Homogeneous/heterogeneous condensation Influence of condensation on flow s 10

11 FLM 11 Euler Equations 0 ) ( ) ( 0 ) ( ) ( 0 ) ( = + + = + + = + pu Eu t E pi u u t u u t r r r r r r r ρ ρ ρ ρ ρ ρ RT a T p p e T T u E e κ ρ = = = = ), ( ) ( r Dt D u t φ ρ φ ρ ρφ = + ) ( ) ( Thermodynamics, EOS:

12 Equations for Homogeneous Condensation ( ρn t ( ρg t hom hom ) ) + ( ρn + ( ρg hom hom r u) r u) = J hom 4π *3 = ρl 3 r J hom nucleation n kg -1 number density of droplets g - condensate mass fraction J m -3 s -1 nucleation rate + 2 drhom ρl 4πrhomρnhom dt 3 droplet growth J r hom hom σ ( T ) 3 m l g n l dr dt *2 r m l v σ R 2 = v exp hom = 3 2 π 3 4π ρ v hom ρ ρ 4π 3 hom = α ρ v p ( T ) T v p 2π R s, r v T g = ml ma + mv + ml total masses Hertz-Knudsen Law 12

13 Multiphase-Multicomponent Thermodynamics e T p κ a = = = E = T ( e, g p( ρ, T, g = κ( g κ u r 1 2 Air as carrier gas with 2 vapor: max, g) both ideal gases max p ρ max, g, T), g) Pure water vapor: real gas behavior 13

14 Solver CATUM Condensation density-based FVM solver Cell-averaged values Considering fluxes over cell boundaries: conservative 1-D, 2-D, 3-D on structured multiblock grids approach: solve local Riemann problems 14

15 Finite Volume Method L FLUX R 15

16 Reconstruction at Cell Faces Average values stored in cell center Reconstruct the required value at the cell face 4 adjacent cells for 2 nd order accuracy Limiter functions: high order smoothness where continuous, low order sharpness at shocks 16

17 1-D Riemann Problem L R Q t F ( Q) = 0 + x Q ρ = ρu ρe F ( Q) = 1 0 ρu u + p E up expansion wave contact wave shock p L > p R 17

18 Boundary Conditions: Ghost Cells General: 2 ghost cells for 2 nd order accuracy Inlet Pressure extrapolated from outermost cell, other values calculated from stagnation conditions (p 0, T 0, ) by isentropic relationships, e.g. with p p = 1+ κ 1 2 κ 1 2 M M κ κ 1 Wall Velocity normal to wall is zero Outlet Subsonic: set outlet pressure, extrapolate other values Supersonic: extrapolate all 18

19 Modeling and Simulation Euler equations Additional Eqns. for Condensation Influence on EOS Reconstruction Riemann problem BCs via ghost cells 19

20 Validation of the Implementation A 1 * M<1 M>1 20

21 Subcritical Heat Addition in Nozzle S2 M<1 M> x [m] log 10 J hom p/p 01 log 10 J hom g/g max diabactic diabatic adiabatic p/p 01 g/g max x [m] 0 21

22 Supercritical Heat Addition in Nozzle S2 Nozzle S2 T 0 = 295 K M<1 M>1 M>1 shock: increase of T, ρ, p p 0 = 10 5 Pa Φ 0 = 65 % x [m] M<1 log 10 J hom p/p 01 onset earlier log 10 J hom shock g/g max diabactic diabatic adiabatic p/p 01 g/g max x [m] 0 22

23 Condensing Flows in Nozzle S2 p v [Pa] Φ Φ 0 = 65 % Φ 0 = 50 % 1000 M=1 500 p s, (T) Nozzle S2 T 0 = 295 K p 0 = 10 5 Pa 0 (( )) T [K] 23

24 Unsteady Effects at High Relative Humidity t 24

25 Validation of the Implementation T 0 = K p 0 = x 10 5 Pa Φ 0 = 91.1 % Mode 1 Result f = 948 Hz ½ grid: 241x21 cells 25

26 Symmetric Oscillation in Nozzle S2 T 0 = K p 0 = x 10 5 Pa Φ 0 = 91.1 % Mode 1 f = 948 Hz ½ grid: 241x21 cells 26

27 Hysteresis in Nozzle A1 28

28 Flow in Nozzle A1 T 0 = 305 K p 0 = 10 5 Pa Φ 0 = 77 % f = 1082 Hz full grid: 220 x 41 29

29 Asymmetric Oscillation in Nozzle S2 T 0 = 305 K p 0 = 10 5 Pa Φ 0 = 95 % enforced disturbance f = 3073 Hz full grid: 241x41 cells 30

30 Turbine Stage VKI with viscous effects 31

31 Rotor Stator Interaction Homogeneous Heterogeneously dominated g r p 01 = bar T 01 = K β 1,1 = 120 Re rotor = M 2,2,is = 1.13 u = 175 m/s n het,0 = m -3 r het = 10-8 m f rs = 2.46 khz f vs,stator = 10.8 khz f vs,rotor = 18.5 khz 33

32 Supersonic Axial Compressor NORD-1500 Griffon II (1957) M max =2.2, H max =16400m, kn Alpha Jet (1973) M max =0.85, H max =14630m, kn 34

33 First Stage of a Supersonic Axial Compressor Developed view of a cylindrical cut 35

34 Section of Axial Compressor Stage 36

35 Cascade Element 37

36 Experimental Setup 38

37 Cascade Element of an Axial Transonic Compressor π=p 2 /p 1 =1.764 adiabatic flow M 1 =1.3 π π diabatic = adiabatic reduction of ca. 18% π=p 2 /p 1 =1.440 T 01 =291 K p 01 =1.008 bar φ 0 =80 % x=10 g/kg diabatic flow 39

38 Cascade Element of an Axial Transonic Compressor π=p 2 /p 1 =1.630 adiabatic flow M 1 =1.3 π π diabatic = adiabatic reduction of ca. 17% π=p 2 /p 1 =1.358 T 01 =292 K p 01 =1.006 bar φ 0 =69.8 % x=9.6 g/kg diabatic flow 40

39 Cascade Element of an Axial Transonic Compressor π=p 2 /p 1 =1.055 adiabatic flow M 1 =1.3 π π diabatic = adiabatic reduction of ca. 15% π=p 2 /p 1 =0.890 T 01 =294 K p 01 =1.005 bar φ 0 =80.3 % x=9.6 g/kg diabatic flow 41

40 Condensation Effects in Transonic Compressors Reduction of inlet Mach number stage compression ratio Loss of thrust efficiency 42

41 Results Steady flows Subcritical Supercritical Unsteady flows Symmetric/asymmetric oscillations Different modes Effects in Turbomachinery 43

42 Cпасибо Thank you for your attention!

43 Discussion

44 2 σ ( T ) * Critical Radius r = hom ρ ( T ) R T ln( S) l v p v [Pa] Φ 0 = 90 % 200 surface creation i s e n t r o p i c e x p a n s i o n Φ 0 = 65 % Φ 0 = 50 % G / k B / T expansion work r* m M = 1.0 (( p 0 = 10 5 Pa 0 )) T [K] E-10 1E E-09-9 r [m] p s, (T) T 0 = 295 K p 0 = 10 5 Pa Φ 0 = 65 % Φ 0 = 35 % Nozzle S2 T 0 = 295 K 46

45 Critical Radius 3E-19 r* m M = 0.8 2E-19 G [J] 1E-19 r* m M = r* m M = E E E E-09-9 r [m] T 0 = 295 K p 0 = 10 5 Pa Φ 0 = 65 % 47

46 Transition in Nozzle A1 p( t) 0-1 = p o ( t) p 01 p u ( t) log 10 ( p +ε) t [ms] Nozzle A1 T 0 = 305 K p 0 = 10 5 Pa Φ 0 = 77 % 48

47 Transition in Nozzle S2 49

48 Appendix 1 50

49 Validation with Subcritical Flow in Nozzle S2 51

50 Heat Addition by Condensation latent heat L [kj/kg] of gases in air: H 2 O: 2260 Increase of static specific heat capacity c p [kj/kg/k]: air: