Simulation of Binary Mixture Droplet Evaporation using VOF Methods

Transcription

1 Peter Keller and Christian Hasse Department of Energy Process Engineering and Chemical Engineering TU Bergakademie Freiberg Simulation of Binary Mixture using VOF Methods June 14, th OpenFOAM R Workshop, PennState University, USA, 2011

2 Overview Motivation Physics and mathematics Validation Case setup Results Conclusion and outlook TU Bergakademie Freiberg 1

3 Motivation I Fig. 1: Fuel injection and ignition, source: BMW Main aim: simulation of multicomponent fuel combustion Whole process too complex to validate Analysis of single steps from fuel injection until flame expansion Fig. 2: Simulation of n-heptane combustion Validation of atomization of open jets computationally very expensive Almost no experimental data for multicomponent fuel evaporation Mathematical validation just with simplifications possible TU Bergakademie Freiberg 2

4 Motivation II First checks of atomization behaviour depending on nozzle design turbulent inflow necessary Studies on secondary breakup of n-heptane droplets Validation of single component fuel droplet evaporation in dependence on temperature below and above boiling temperature Combination of atomization, evaporation and chemical reaction (described in [Keller et al.]) Current work: binary mixture evaporation implemented in OpenFOAM R using Volume of Fluid (VOF) approach Further implementations due to multicomponent mixtures and Cantera-/flamelet-coupling in preparation TU Bergakademie Freiberg 3

5 Physics and Mathematics I Basic solver: intermixingfoam - interface capture of 3 incompressible fluids (miscible liquids) using VOF approach Source code extended due to gas mixture, source terms for evaporation, enthalpy equation, mixing laws for ideal gases and liquids VOF special: scalar transport equation for liquid volume fraction Volume-of-fluid equation (liquid tracking): α = 0, if gas + (αu) = 0, t with α (0, 1), if interface = 1, if liquid TU Bergakademie Freiberg 4

6 Physics and Mathematics II - Modified Conservation Equations Momentum equation (original) (ρu) t with surface tension force F s = σ m κˆn. + (ρuu) = µ [ u + ( u) T ] +ρg p F s VOF-equations (in original version for gas phase α 1 and first liquid phase α 2 ) α 2 t + (φ α 2 ) = D 23 α 2 Ṡα 2 α 3 t + (φ α 3 ) = D 32 α 3 Ṡα 3 α 1 = 1 α 2 α 3 with volumetric source terms Ṡα 2 and Ṡα 3 and special OpenFOAM-fluxes φ α2 and φ α3 TU Bergakademie Freiberg 5

7 Physics and Mathematics III - Modified Conservation Equations Species transport equations Y G1 t Y G2 t + (Y G1 u) = D G1 Y G1 + ṠY G1 + (Y G2 u) = D G2 Y G2 + ṠY G2 Y G3 = 1 Y G1 Y G2 with mass related evaporation source terms ṠY G1 and ṠY G2 Enthalpy equation with evaporation source ṠH h s t + (h λ su) = h s + ρc ṠH p TU Bergakademie Freiberg 6

8 Physics and Mathematics IV - Modified Conservation Equations Mass conservation (until now still incompressible) with volume balance term Ṡp u = Ṡp Source terms (exemplary): [ ] DG1 ρ G M C1 Ṡ YG1 = δ 1 Y G1 ( κ) 1 Y G1 ρ G1 M G [ ] λ M C1 +(1 δ 1 ) T ( κ) ρ G1 H v,1 M G [ ] DG2 ρ G M C2 Ṡ YG2 = δ 2 Y G2 ( κ) 1 Y G2 ρ G2 M G [ ] λ M C2 +(1 δ 2 ) T ( κ) ρ G2 H v,1 M G TU Bergakademie Freiberg 7

9 Physics and Mathematics VI - Gradients Discretization of gradients according Y Gi = Y G i,sat Y Gi δx and T = T T boil,i δx Fig. 3: Gradient calculation Calculation of saturation mass fractions using Raoult s law and Wagner equation according: Y Gi,sat = M C 1 p i,sat M G p ln p i,sat = T c,i p c T X L i (A i ( 1 T T c,i +C i ( 1 T T c,i ) + B i ( 1 T T c,i ) 3 + D i ( 1 T T c,i ) 1.5 ) 6 ) TU Bergakademie Freiberg 8

10 Physics and Mathematics VII - PLIC Fig. 4: 3D PLIC, source: [Gueyffier et al.] Distance δx of mass center of gas and surface calculated using piecewise-linear interface calculation PLIC According Gueyffier et al. volume of liquid in cell 1 3 V = d 3 H(d n j dj)(d n j dj) 3 6n x n y n z j=1 3 + H(d d max + n j dj)(d d max + n j dj) 3 j=1 with d = n x x + n y y + n z z, surface normal n = n x n y n z, cell lengths dx l = dy dz With mass center of gas phase (x s, y s, z s ) determination of distance: δx = x x x s + n y y s + n z z s d n 2 x + n 2 y + n 2 z TU Bergakademie Freiberg 9

11 Physics and Mathematics VIII - Species and Mixture Properties Determination of substance-specific properties according: Watson equation (evaporation enthalpies), Fuller relation (gas mixture diffusion coefficients), Tyn/Calus method (liquid mixture diffusion coefficient), Hugill/Welsenes equation (surface tension),... different polynomials (thermal conductivity, viscosities,... ) and NASA-polynomials (heat capacities, enthalpies) Example: Tyn/Calus Dij ( m 2 = M ) 1/3 ( C j 10 6 M C i /s ρ Lj ρ Li ) 1/6 ( ) 0.6 Pj T (10 3 η Lj ) 1 P i and hence D AB,L = (D AB) XB (D BA) X A TU Bergakademie Freiberg 10

12 Validation - Single Component I Axisymmetric mesh with cells Initial diameter D = 100 µm Different initial liquid temperatures Tl 1 = 300 K and Tl 2 = 320 K Inflow temperature T = 350 K, Reynolds number Re < 1 Validation done using D 2 -law (see [Turns(2000)]) with dd 2 dt K = 8ρD AB ρ l = K ( ) 1 YA, ln 1 Y A,sat TU Bergakademie Freiberg 11

13 Validation - Single Component II Single component T d 0 =300 K Single component T d 0 =320 K Simulation T S =310 K analytics Simulation T S =311 K analytics Diameter mm 2 /mm Diameter mm 2 /mm e e Time in s Time in s Fig. 5: Validation single component evaporation: Tl 1 = 300 K Fig. 6: Validation single component evaporation: Tl 2 = 320 K Good agreement between analytics and simulation results Droplet heating/cooling from different initial state to almost equal surface temperature TU Bergakademie Freiberg 12

14 Validation - Binary Mixture I Same mesh and diameter as before Initial liquid temperature T l = 300 K Inflow temperature T = 350 K, Reynolds number Re < 1 Droplet composition: α 2 = 0.8, α 3 = 0.2 Expanding to multicomponent mixtures, D 2 -law reads: dd 2 dt = K K = 8ρ ρ l J j=1 D jm ln 1 J j=1 Y vap,j, 1 J j=1 Y vap,j,sat TU Bergakademie Freiberg 13

15 Validation - Binary Mixture II Simulation T S =307 K analytics Diameter mm 2 /mm e Time in s Fig. 7: Validation binary mixture evaporation: T 1 l = 300 K TU Bergakademie Freiberg 14

16 Case Setup Fig. 8: Scheme of 2D-geometry Same configuration for 2D- and 3D-cases (cylindrical shape) 2D mesh resolution: cells 3D mesh resolution: 2Mio. cells Droplet: 2D 300, 3D 3000 cells CFL=0.2 Parameter variations due to temperature and composition influence as well as impact of Weber number W e = ρ gu 2 rel d σ 3 3D- and 16 2D-simulations (see table next slide) TU Bergakademie Freiberg 15

17 Case Setup - Parameter Variation List, D=1 mm # Species Dim U in [m/s] Y L1 T [K] ρ G [kg/m 3 ] σ [ m/s 2 ] We Re 1 octane 2D octane 2D octane 2D heptane+decane 2D heptane+decane 2D heptane+decane 2D heptane+decane 2D hexane+dodecane 2D hexane+dodecane 2D hexane+dodecane 2D hexane+dodecane 2D heptane+decane 2D heptane+decane 2D heptane+decane 2D heptane+decane 2D heptane+decane 2D heptane+decane 3D heptane+decane 3D hexane+dodecane 3D Tab. 1: Parameter list TU Bergakademie Freiberg 16

18 Results I - Weber Number Fig. 9: Case 7, We=0.5 Fig. 10: Case 5, We=1 Fig. 11: Case 15, We=140 Fig. 12: Case 11, We=1370 TU Bergakademie Freiberg 17

19 Results II - Weber Number Base Cases n-octane 1 Base Case We=0.1 Base Case We=1 Base Case We=70 Diameter mm 2 /mm # Species We 1 octane octane 1 3 octane Tab. 2: Parameter list Time in s Fig. 13: Base cases, Weber number variation Reference cases with n-octane (same solver - similar properties) Higher Weber number evaporation faster due to surface enlargement and transport of vapor TU Bergakademie Freiberg 18

20 Results III - Weber Number Weber Number Variation Diameter mm 2 /mm Case 5 Case 11 Case 12 Case 15 # Species We 5 heptane+decane 1 11 hexane+dodecane heptane+decane heptane+decane Tab. 3: Parameter list Time in s Fig. 14: Weber number variation Same as before for higher Weber numbers Temperature difference (case 12 and 15) higher evaporation rate at beginning and earlier achievement of saturation concentration at surface TU Bergakademie Freiberg 19

21 Results IV - Temperature Temperature Variation Diameter mm 2 /mm Base Case We=1 Case 4 Case 5 Case 6 Case 7 # Species T in [K] 2 octane heptane+decane heptane+decane heptane+decane heptane+decane Time in s Tab. 4: Parameter list Fig. 15: Inflow temperature variation Before breakup single component case (T base = T 5 ) slower than binary ones With breakup and surface enlargement acceleration of evaporation of single component case higher TU Bergakademie Freiberg 20

22 Results V - Composition Composition Variation 1 Base Case We=70 Case 10 Case 12 Case 14 Diameter mm 2 /mm # Species Y L1 3 octane 1 10 hexane+dodecane heptane+decane heptane+decane Time in s Tab. 5: Parameter list Fig. 16: Composition variation Evaporation rate strongly dependent on composition Higher liquid concentration of high volatile components (case 10 n-hexane, case 14 n-heptane) higher evaporation rate TU Bergakademie Freiberg 21

23 Results IV - 2D Case 3D Case 3D-2D Comparison 1 3D Case 17 Case 13 Diameter mm 2 /mm Time in s Fig. 17: 3D Case (17) Fig. 18: Comparison 2D-3D (13-17) Similar results for 2D- and 3D-case Transient behaviour and temperature drop observable TU Bergakademie Freiberg 22

24 Conclusion Conclusions New VOF-solver implemented to solve for binary mixture evaporation and breakup Validation done for single component and binary mixture droplet evaporation Differences shown between single component and binary mixture droplet evaporation caused by temperature differences, composition and inflow velocity Outlook Generalization of solver due to multicomponent mixtures Coupling with flamlet library and hence chemical reactions Coupling with Cantera to compute species properties of gas phase TU Bergakademie Freiberg 23

25 References [Keller et al.] Keller, P.; Nikrityuk, P.A.; Meyer, B.; Müller-Hagedorn, M., "Numerical Simulation of Evaporating Droplets with Chemical Reactions using a Volume of Fluid Method", 7 th International Conference on Multiphase Flows, 2010 [Gueyffier et al.] Gueyffier, D.; Li, J.; Nadim, A.; Scardovelli, R.; Zaleski, S., "Volume-of-Fluid Interface Tracking with Smoothed Surface Stress Methods for Three-Dimensional Flows", Journal of Computational Physics 152, p , 1999 [Turns(2000)] Turns, S.R., "An Introduction to Combustion - Concepts and Applications", McGraw-Hill Higher Education, 2000 TU Bergakademie Freiberg 24

26 Acknowledgement The research has been funded by the Bavarian Science Foundation in the project WiDiKO - Wirkkette Direkteingespritzter Kraftstoffe im Ottomotor (project number NP:275) and by the Federal Ministry of Education and Research of Germany in framework of Virtuhcon (project number ). Thanks to Bernhard Gschaider for his valuable comments and collaboration. Thank you for your attention! TU Bergakademie Freiberg 25