simulation of binary mixture droplet evaporation … departmentofenergyprocessengineering...

Peter Keller and Christian HasseDepartment of Energy Process Engineeringand Chemical EngineeringTU Bergakademie Freiberg

Simulation of Binary MixtureDroplet Evaporation using VOF MethodsJune 14, 2011

6th OpenFOAMr Workshop,PennState University, USA, 2011

Overview

MotivationPhysics and mathematicsValidationCase setupResultsConclusion and outlook

2011-06-146th OFW

Keller/HasseDroplet Evaporation

NTFDTU Bergakademie Freiberg 1

Motivation I

Fig. 1: Fuel injectionand ignition, source:BMW

Main aim: simulation of multicomponent fuelcombustionWhole process too complex to validateAnalysis of single steps from fuel injection untilflame expansion

Fig. 2: Simulation ofn-heptane combustion

Validation of atomization of open jetscomputationally very expensiveAlmost no experimental data for multicomponentfuel evaporationMathematical validation just with simplificationspossible

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Motivation II

First checks of atomization behaviour depending on nozzle design →turbulent inflow necessaryStudies on secondary breakup of n-heptane dropletsValidation of single component fuel droplet evaporation in dependence ontemperature below and above boiling temperatureCombination of atomization, evaporation and chemical reaction (describedin [Keller et al.])Current work: binary mixture evaporation implemented in OpenFOAMr

using Volume of Fluid (VOF) approachFurther implementations due to multicomponent mixtures andCantera-/flamelet-coupling in preparation

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Physics and Mathematics I

Basic solver: interMixingFoam - interface capture of3 incompressible fluids (miscible liquids) using VOFapproachSource code extended due to gas mixture, sourceterms for evaporation, enthalpy equation, mixinglaws for ideal gases and liquidsVOF special: scalar transport equation for liquidvolume fraction

Volume-of-fluid equation (liquid tracking):

∂α

∂t+∇ · (αU) = 0, with α

= 0 , if gas∈ (0, 1) , if interface= 1 , if liquid

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Physics and Mathematics II - Modified Conservation Equations

Momentum equation (original)

∂(ρu)∂t

+∇ · (ρuu) = ∇ · µ[∇u+ (∇u)T

]+ρg −∇p− Fs

with surface tension force Fs = σmκn.

VOF-equations (in original version for gas phase α1 and first liquid phase α2)

∂α2

∂t+∇ · (φα2) = ∇ ·D23∇α2 − Sα2

∂α3

∂t+∇ · (φα3) = ∇ ·D32∇α3 − Sα3

α1 = 1− α2 − α3

with volumetric source terms Sα2 and Sα3 and special ”OpenFOAM-fluxes” φα2

and φα3

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Physics and Mathematics III - Modified Conservation Equations

Species transport equations

∂YG1

∂t+∇ · (YG1u) = ∇ ·DG1∇YG1 + SYG1

∂YG2

∂t+∇ · (YG2u) = ∇ ·DG2∇YG2 + SYG2

YG3 = 1− YG1 − YG2

with mass related evaporation source terms SYG1and SYG2

Enthalpy equation

∂hs∂t

+∇ · (hsu) = ∇ · λρcp∇hs + SH

with evaporation source SH

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Physics and Mathematics IV - Modified Conservation Equations

Mass conservation (until now still incompressible)

∇ · u = Sp

with volume balance term Sp

Source terms (exemplary):

SYG1= δ1 ·

[DG1

1− YG1

ρGρG1

MC1

MG∇YG1 · (−κ)

]+(1− δ1) ·

[λ

ρG1∆Hv,1

MC1

MG∇T · (−κ)

]SYG2

= δ2 ·[

DG2

1− YG2

ρGρG2

MC2

MG∇YG2 · (−κ)

]+(1− δ2) ·

[λ

ρG2∆Hv,1

MC2

MG∇T · (−κ)

]

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Physics and Mathematics VI - Gradients

Fig. 3: Gradientcalculation

Discretization of gradients according

∇YGi = YGi,sat − YGiδx

and ∇T = T − Tboil,iδx

Calculation of saturation mass fractions usingRaoult’s law and Wagner equation according:

YGi,sat = MC1

MG

pi,satp

XLi

ln pi,satpc

= Tc,iT

(Ai

(1− T

Tc,i

)+Bi

(1− T

Tc,i

)1.5

+Ci(

1− T

Tc,i

)3+Di

(1− T

Tc,i

)6)

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Physics and Mathematics VII - PLIC

Fig. 4: 3D PLIC,source:[Gueyffier et al.]

Distance δx of mass center of gas and surfacecalculated using piecewise-linear interface calculationPLICAccording Gueyffier et al. volume of liquid in cell

V = 16nxnynz

d3 −3∑j=1

H(d− njdj)(d− njdj)3

+3∑j=1

H(d− dmax + njdj)(d− dmax + njdj)3

with d = nxx+ nyy + nzz, surface normal ~n =

nxnynz

, cell lengths ~l =

dxdydz

With mass center of gas phase (xs, ys, zs) determination of distance:

δx =

∣∣∣∣∣∣xxxs + nyys + nzzs − d√n2x + n2

y + n2z

∣∣∣∣∣∣2011-06-146th OFW



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, . . . ) andNASA-polynomials (heat capacities, enthalpies)

Example: Tyn/Calus

D∞ijm2/s

= 8.93 · 10−12(

106MCj

ρLj

)−1/3(106MCi

ρLi

)1/6(PjPi

)0.6· T · (103ηLj )−1

and henceDAB,L = (D∞AB)XB · (D∞BA)XA

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Validation - Single Component I

Axisymmetric mesh with 160000 cellsInitial diameter D = 100µmDifferent initial liquid temperatures T 1

l = 300K and T 2l = 320K

Inflow temperature T = 350K, Reynolds number Re < 1

Validation done using D2-law (see [Turns(2000)])

dD2

dt= −K

with K = 8ρDAB

ρlln(

1− YA,∞1− YA,sat

)

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Validation - Single Component II

0.9984

0.9986

0.9988

0.999

0.9992

0.9994

0.9996

0.9998

1

0 5e-05 0.0001 0.00015 0.0002 0.00025 0.0003

Dia

mete

r m

m2/m

m0

2

Time in s

Single component Td0=300 K

SimulationTS=310 K analytics

Fig. 5: Validation single componentevaporation: T 1

l = 300K

0.9984

0.9986

0.9988

0.999

0.9992

0.9994

0.9996

0.9998

1

0 5e-05 0.0001 0.00015 0.0002 0.00025 0.0003

Dia

mete

r m

m2/m

m0

2

Time in s

Single component Td0=320 K


Fig. 6: Validation single componentevaporation: T 2

l = 320K

Good agreement between analytics and simulation resultsDroplet heating/cooling from different initial state to almost equal surfacetemperature

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Validation - Binary Mixture I

Same mesh and diameter as beforeInitial liquid temperature Tl = 300KInflow temperature T = 350K, Reynolds number Re < 1Droplet composition: α2 = 0.8, α3 = 0.2

Expanding to multicomponent mixtures, D2-law reads:

dD2

dt= −K

K = 8ρρl

J∑j=1

DjM ln1−

∑Jj=1 Yvap,j,∞

1−∑Jj=1 Yvap,j,sat

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Validation - Binary Mixture II

0.995

0.9955

0.996

0.9965

0.997

0.9975

0.998

0.9985

0.999

0.9995

1

0 5e-05 0.0001 0.00015 0.0002 0.00025 0.0003

Dia

me

ter

mm

2/m

m0

2

Time in s


Fig. 7: Validation binary mixture evaporation: T 1l = 300K

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Case Setup

Fig. 8: Scheme of 2D-geometry

Same configuration for 2D- and3D-cases (cylindrical shape)2D mesh resolution: 500 × 200cells3D mesh resolution: ≈ 2Mio. cellsDroplet: 2D ≈ 300, 3D ≈ 3000cellsCFL=0.2

Parameter variations due to temperature and composition influence as wellas impact of Weber number

We = ρgu2reld

σ

3 3D- and 16 2D-simulations (see table next slide)

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Case Setup - Parameter Variation List, D=1mm

# Species Dim Uin [m/s] YL1 T [K] ρG [kg/m3 ] σ [ m/s2 ] We Re

1 octane 2D 1.0 1 350 1.064 0.0206 0.05 532 octane 2D 4.54 1 350 1.064 0.0206 1 2403 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 2805 heptane+decane 2D 4.54 0.3 350 1.064 0.02223 1 2406 heptane+decane 2D 4.54 0.3 400 0.93 0.02223 0.9 1907 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 2409 hexane+dodecane 2D 4.54 0.5 600 0.6208 0.0219 0.6 9510 hexane+dodecane 2D 70 0.8 350 1.064 0.0197 265 370011 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 370013 heptane+decane 2D 70 0.5 350 1.064 0.0216 240 370014 heptane+decane 2D 70 0.8 350 1.064 0.0205 250 370015 heptane+decane 2D 70 0.5 600 0.6208 0.0216 140 147016 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 370018 heptane+decane 3D 4.54 0.5 600 0.6208 0.0216 0.6 9519 hexane+dodecane 3D 70 0.8 350 1.064 0.0197 265 3700

Tab. 1: Parameter list

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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

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Results II - Weber Number

0.995

0.996

0.997

0.998

0.999

1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Dia

me

ter

mm

2/m

m0

2

Time in s

Base Cases n-Octane

Base Case We=0.1

Base Case We=1

Base Case We=70

Fig. 13: Base cases, Weber number variation

# Species We

1 octane 0.052 octane 13 octane 250


Reference cases with n-octane (same solver - similar properties)Higher Weber number → evaporation faster due to surface enlargementand transport of vapor

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Results III - Weber Number

0.99

0.992

0.994

0.996

0.998

1

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Dia

me

ter

mm

2/m

m0

2

Time in s

Weber Number Variation

Case 5

Case 11

Case 12

Case 15

Fig. 14: Weber number variation

# Species We

5 heptane+decane 111 hexane+dodecane 137012 heptane+decane 23015 heptane+decane 140


Same as before for higher Weber numbersTemperature difference (case 12 and 15) → higher evaporation rate atbeginning and earlier achievement of saturation concentration at surface

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Results IV - Temperature

0.99

0.992

0.994

0.996

0.998

1

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

Dia

me

ter

mm

2/m

m0

2

Time in s

Temperature Variation

Base Case We=1Case 4Case 5Case 6Case 7

Fig. 15: Inflow temperature variation

# Species T in [K]

2 octane 3504 heptane+decane 3205 heptane+decane 3506 heptane+decane 4007 heptane+decane 600


Before breakup single component case (Tbase = T5) slower than binary onesWith breakup and surface enlargement acceleration of evaporation of singlecomponent case higher

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Results V - Composition

0.99

0.992

0.994

0.996

0.998

1

0 0.0001 0.0002 0.0003 0.0004 0.0005

Dia

me

ter

mm

2/m

m0

2

Time in s

Composition Variation

Base Case We=70Case 10Case 12Case 14

Fig. 16: Composition variation

# Species YL1

3 octane 110 hexane+dodecane 0.812 heptane+decane 0.314 heptane+decane 0.8


Evaporation rate strongly dependent on compositionHigher liquid concentration of high volatile components (case 10 n-hexane,case 14 n-heptane) → higher evaporation rate

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Results IV - 2D Case ↔ 3D Case

Fig. 17: 3D Case (17)

0.99

0.992

0.994

0.996

0.998

1

0 0.0001 0.0002 0.0003 0.0004 0.0005

Dia

mete

r m

m2/m

m0

2

Time in s

3D-2D Comparison

3D Case 17Case 13

Fig. 18: Comparison 2D-3D (13-17)

Similar results for 2D- and 3D-caseTransient behaviour and temperature drop observable

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Conclusion

ConclusionsNew VOF-solver implemented to solve for binary mixture evaporation andbreakupValidation done for single component and binary mixture dropletevaporationDifferences shown between single component and binary mixture dropletevaporation caused by temperature differences, composition and inflowvelocity

OutlookGeneralization of solver due to multicomponent mixturesCoupling with flamlet library and hence chemical reactionsCoupling with Cantera to compute species properties of gas phase

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References

[Keller et al.] Keller, P.; Nikrityuk, P.A.; Meyer, B.; Müller-Hagedorn, M., "NumericalSimulation of Evaporating Droplets with Chemical Reactions using a Volume ofFluid Method", 7th 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 forThree-Dimensional Flows", Journal of Computational Physics 152, p. 423-456,1999

[Turns(2000)] Turns, S.R., "An Introduction to Combustion - Concepts andApplications", McGraw-Hill Higher Education, 2000

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Acknowledgement

The research has been funded by the Bavarian Science Foundationin the project WiDiKO - Wirkkette Direkteingespritzter Kraftstoffeim Ottomotor (project number NP:275) and by the Federal Ministryof Education and Research of Germany in framework of Virtuhcon(project number 040201030).Thanks to Bernhard Gschaider for his valuable comments andcollaboration.

Thank you for your attention!

2011-06-146th OFW



simulation of binary mixture droplet evaporation … departmentofenergyprocessengineering...

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