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, 2011 6 th OpenFOAM R Workshop, PennState University, USA, 2011
Overview Motivation Physics and mathematics Validation Case setup Results Conclusion and outlook TU Bergakademie Freiberg 1
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
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
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
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
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
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
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
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
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 = 8.93 10 12 10 6 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
Validation - Single Component I Axisymmetric mesh with 160000 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
Validation - Single Component II Single component T d 0 =300 K Single component T d 0 =320 K 1 0.9998 Simulation T S =310 K analytics 1 0.9998 Simulation T S =311 K analytics 0.9996 0.9996 Diameter mm 2 /mm 0 2 0.9994 0.9992 0.999 Diameter mm 2 /mm 0 2 0.9994 0.9992 0.999 0.9988 0.9988 0.9986 0.9986 0.9984 0 5e-05 0.0001 0.00015 0.0002 0.00025 0.0003 0.9984 0 5e-05 0.0001 0.00015 0.0002 0.00025 0.0003 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
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
Validation - Binary Mixture II 1 0.9995 Simulation T S =307 K analytics 0.999 Diameter mm 2 /mm 0 2 0.9985 0.998 0.9975 0.997 0.9965 0.996 0.9955 0.995 0 5e-05 0.0001 0.00015 0.0002 0.00025 0.0003 Time in s Fig. 7: Validation binary mixture evaporation: T 1 l = 300 K TU Bergakademie Freiberg 14
Case Setup Fig. 8: Scheme of 2D-geometry Same configuration for 2D- and 3D-cases (cylindrical shape) 2D mesh resolution: 500 200 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
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 1.0 1 350 1.064 0.0206 0.05 53 2 octane 2D 4.54 1 350 1.064 0.0206 1 240 3 octane 2D 70 1 350 1.064 0.0206 250 3700 4 heptane+decane 2D 4.54 0.3 320 1.164 0.02223 1 280 5 heptane+decane 2D 4.54 0.3 350 1.064 0.02223 1 240 6 heptane+decane 2D 4.54 0.3 400 0.93 0.02223 0.9 190 7 heptane+decane 2D 4.54 0.3 600 0.6208 0.02223 0.6 95 8 hexane+dodecane 2D 4.54 0.3 350 1.064 0.0232 1 240 9 hexane+dodecane 2D 4.54 0.5 600 0.6208 0.0219 0.6 95 10 hexane+dodecane 2D 70 0.8 350 1.064 0.0197 265 3700 11 hexane+dodecane 2D 220 0.5 600 0.6208 0.0219 1370 11600 12 heptane+decane 2D 70 0.3 350 1.064 0.0223 230 3700 13 heptane+decane 2D 70 0.5 350 1.064 0.0216 240 3700 14 heptane+decane 2D 70 0.8 350 1.064 0.0205 250 3700 15 heptane+decane 2D 70 0.5 600 0.6208 0.0216 140 1470 16 heptane+decane 2D 70 0.8 600 0.6208 0.0205 150 1470 17 heptane+decane 3D 70 0.5 350 1.064 0.0216 240 3700 18 heptane+decane 3D 4.54 0.5 600 0.6208 0.0216 0.6 95 19 hexane+dodecane 3D 70 0.8 350 1.064 0.0197 265 3700 Tab. 1: Parameter list TU Bergakademie Freiberg 16
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
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 0 2 0.999 0.998 0.997 # Species We 1 octane 0.05 2 octane 1 3 octane 250 0.996 Tab. 2: Parameter list 0.995 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 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
Results III - Weber Number Weber Number Variation Diameter mm 2 /mm 0 2 1 0.998 0.996 0.994 Case 5 Case 11 Case 12 Case 15 # Species We 5 heptane+decane 1 11 hexane+dodecane 1370 12 heptane+decane 230 15 heptane+decane 140 0.992 Tab. 3: Parameter list 0.99 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 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
Results IV - Temperature Temperature Variation Diameter mm 2 /mm 0 2 1 0.998 0.996 0.994 Base Case We=1 Case 4 Case 5 Case 6 Case 7 # Species T in [K] 2 octane 350 4 heptane+decane 320 5 heptane+decane 350 6 heptane+decane 400 7 heptane+decane 600 0.992 0.99 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 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
Results V - Composition Composition Variation 1 Base Case We=70 Case 10 Case 12 Case 14 Diameter mm 2 /mm 0 2 0.998 0.996 0.994 # Species Y L1 3 octane 1 10 hexane+dodecane 0.8 12 heptane+decane 0.3 14 heptane+decane 0.8 0.992 0.99 0 0.0001 0.0002 0.0003 0.0004 0.0005 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
Results IV - 2D Case 3D Case 3D-2D Comparison 1 3D Case 17 Case 13 Diameter mm 2 /mm 0 2 0.998 0.996 0.994 0.992 0.99 0 0.0001 0.0002 0.0003 0.0004 0.0005 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
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
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. 423-456, 1999 [Turns(2000)] Turns, S.R., "An Introduction to Combustion - Concepts and Applications", McGraw-Hill Higher Education, 2000 TU Bergakademie Freiberg 24
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 040201030). Thanks to Bernhard Gschaider for his valuable comments and collaboration. Thank you for your attention! TU Bergakademie Freiberg 25