MVKraposhin OFW8 Presentation

8th OpenFOAM Workshop, Jeju, Korea, 1114 June 2013

NATIONAL RESEARCH CENTER«KURCHATOV INSTITUTE»

BAUMAN MOSCOW STATE UNIVERSITY

Development of finite volume model of thermo and hydrodynamic of two phase flow with

condensation, evaporation, temperature dilation and compressibility of media using OpenFOAM:

current state and perspectives

M.V. KraposhinA.M. TagirovA.V. YuskinS.V. Strizhak


CURRENT STATE OF THE MULTIPHASE MODELING IN OPENFOAM

● Mixture models: twoLiquidMixingFoam, settlingFoam

● VoF with the interface capturing: interFoam, compressibleInterFoam, interPhaseChangeFoam

● Euler-Euler models: twoPhaseEulerFoam, bubbleFoam

● Thermodynamic cavitation model: cavitatingFoam

It would be nice to have a model that:

1) Takes into account the effects of temperature and pressure on the density and phase transformations due to changes in pressure and temperature

2) Which could be used to test the empirical relations


MODEL DESCRIPTION

New model is based on the interPhaseChangeFoam equations:● Single system of equations for all

phases● Transport equation for the liquid

volume fraction● Properties of the mixture are functions of phase

properties and volume fraction● Mass balance formulated in volumetric fluxes

α1 =V 1

V= 1−α2

α2 =V 2

V= 1−α1

Σiαi = 1

ρ = α1ρ1 + α2ρ2

∂αiρi∂ t

=−∇⋅(αiρiU ) + mi


GOVERNING EQUATIONS

● The volume balance for the liquid phase● The momentum balance for mixture in mass fluxes● The energy balance for mixture in volumetric

fluxes● Equations of state for each phase● Saturation curve● Mass source semi-empirical models

Thermodynamic equilibrium model


VOLUME FRACTION BALANCE EQUATION

● Mass balance is converted to the volume balance

∂αiρi∂ t

=−∇⋅(αiρiU ) + mi

∂αi

∂ t+ ∇⋅(αiU ) =−

αiρi

dρid t

+miρi

∂∂ t (αiρi ζ ) + ∇⋅(αiρiU ζ ) =

ρiαi (∂ζ∂ t

+ ∇⋅(U ζ )) + ζαi

dρidt

+ ζρidαi

dt

m1 = m+ + m−

m+ =ρ2ρ1ρ (1−α1)α1

3ℜB √ 2

3∣p−ps∣

ρ1pos(p− ps)

m− =ρ2ρ1ρ α1(1−α1+αnuc)

3ℜB √ 2

3∣ps− p∣

ρ1pos(ps−p)

Schnerr-Sauer phase change model(based on Rayleigh-Taylor equation)


EQUATIONS OF STATE

● For the liquid phase:

● For gas (vapor) phase

● Two equations can be combined in the one

ρ = ρ0 +∂ρ

∂TΔT +

∂ρ

∂ pΔ p = ρ0 − ρβΔT +

∂ρ

∂ pΔ p

ρ =1

C p

C v

R /MT

p

ρ = ρ +∂ρ

∂ pp , or ρ = ρ −

∂ρ

∂ pp


CONTINUITY (PRESSURE) EQUATION

● We will use PISO method to couple velocity and pressure

● Momentum equation formulated for mixture in mass fluxes

● Inhomogeneous continuity equation obtained by summing volume transport equation for all phases:

α1ρ1

dρ1

d t+

α2ρ2

dρ2

d t+ ∇⋅(U ) = ( 1

ρ1−

1ρ2 )m1


MATERIAL DERIVATIVE OF DENSITY

● If the density of the phase varies linearly with pressure, then this derivative is also the pressure derivative:

● Or in correction form:

● In OpenFOAM: tAcorr = A - (A & A.psi());

αiρi

dρid t

=d pd t (

αiψiρi ) + p(αi

ρi

dψi

d t ) + (αiρi

d ρid t )

αiρi

dρid t

=αiρi ([ dρi

d t− ψi

d pd t ]

Explicit

+[ψid pd t ]

Implicit

)


FINAL VERSION OF THE PRESSURE EQUATION

● Substituting equations of state and momentum equation in the inhomogeneous continuity equation we can get pressure equation

(α1ρ1

ψ1 +α2ρ2

ψ2) d pd t + (α1ρ1

d ρ1

d t+

α2ρ2

d ρ2

d t ) +

∇⋅( HA ) − ∇⋅1A

∇ p= ( 1ρ1

−1ρ2 ) m1

m1 = m1(α1, p)


PHASE TEMPERATURE BALANCE

● Phase temperature balance is derived from from the phase enthalpy balance:∂αiρihi

∂ t+ ∇⋅(αiρihiU ) + ∇⋅qi = mihi − mi

pρi

+dαi p

d thi = hi + C p , iT

αi(∂T∂ t + ∇⋅(U T )) + Tαiψi

ρid pdt

+ Tdαi

dt−

1ρiC p ,i

∇⋅κiEff

∇ T =

T1ρimi −

1ρiC p ,i

pρimi +

1ρiC p ,i

( dαi p

d t ) −

( ∂αiρi hi∂ t

+ ∇⋅(αiρiU hi )) 1ρiC p ,i

+hi

ρiC p ,i

mi

When is constant, last term vanisheshi


MIXTURE TEMPERATURE BALANCE

● By summing temperature balances for each phase, we can get mixture temperature balance (thermodynamically equilibrium approximation

∂T∂ t

+ ∇⋅(U T ) − T ∇⋅U −α1

ρ1C p ,1

∇⋅κ1Eff

∇ T −α2

ρ2C p ,2

∇⋅κ2Eff

∇ T =

( −1ρ1C p ,1

1ρ1

+1

ρ2C p ,2

1ρ2 ) p m1 +

1ρ1C p ,1

( dα1 p

d t ) +1

ρ2C p ,2( dα2 p

d t )

T ∇⋅(U ) = T (( 1ρ1

−1ρ2 )m1 − (α1

ρ1

dρ1

d t+

α2ρ2

dρ2

d t ))From continuity equation


SEGREGATED APPROACH: HOW TO MAKE ALL BALANCES NUMERICALLY CONSERVATIVE

● The key idea — use terms from pressure equation everywhere, when material derivatives of density or pressure, or volume fraction met

● Try to combine balance terms to obtain ● For any constant value (L, f.e.):

∇⋅(U )

L(∂α2ρ2

∂ t+ ∇⋅(α2ρ2U )−m2) = 0


CONSERVATION OF THE LIQUID VOLUME FRACTION

● Subtract and add to right hand side, replace with expression from pressure equation to link it with the volume fraction equation

● And to obtain expression for volume fraction material derivative

∂α1

∂ t+ ∇⋅(α1U ) = α1α2( 1

ρ2

dρ2

d t−

1ρ1

dρ1

d t ) +

m1( 1ρ1

− α1( 1ρ1

−1ρ2 )) + α1 ∇⋅U

dα1

d t= α1α2( 1

ρ2

dρ2

d t−

1ρ1

dρ1

d t ) + m1( 1ρ1

− α1( 1ρ1

−1ρ2 ))

dα2

d t=−α1α2( 1

ρ2

dρ2

d t−

1ρ1

dρ1

d t ) + m1(− 1ρ2

− α2( 1ρ1

−1ρ2 ))

α1 ∇⋅(U )α1 ∇⋅(U )α1 ∇⋅(U )


CONSERVATION OF THE MIXTURE TEMPERATURE

● By combining terms in the right hand side we can get:

( −1ρ1C p ,1

1ρ1

+1

ρ2C p ,2

1ρ2 ) p m1 +

1ρ1C p ,1

( dα1 p

d t ) +1

ρ2C p ,2(dα2 p

d t ) =

d pd t (

α1

ρ1C p ,1

+α2

ρ2C p ,2 ) + α1α2( 1ρ2

dρ2

d t−

1ρ1

dρ1

d t )( 1ρ1C p ,1

−1

ρ2C p ,2) +

pm1( 1ρ1

−1ρ2 )(−

α1

ρ1C p ,1

−α2

ρ2C p ,2 )


SATURATED PRESSURE

● Roche Magnus

● Boltzman law

ps = 610.94 e17.625(T−273.15

T−30.11 )

ln ps =−L

(R /M )T+ lnT + C


SOLUTION PROCEDURE

● Solve for the liquid volume fraction phase (using explicit/implicit MULES of FVM method)

● Update mixture density● Predict velocity (momentum equation)● Formulate and solve pressure equation● Update pressure-dependent parts of densities● Solve temperature equation● Update saturated pressure and temperature-

dependent parts of densities● Update turbulent properties


TEST CASES: HEMISPERICAL BODY

U∞ = 5.35m / s

p∞ = 7.8 kPa

T = 300 K , ps = 3.5kPa

σ = 0.3


HEMISPHERICAL BODY: TEST CASES

1) Incompressible solver, Schnerr-Sauer evaporation model

2) Compressible solver, densities do not depend on pressure

3) Compressible solver, psiCorrection, without source terms in the energy equation

4) Compressible solver, matrix correction form of pressure equation, w/o source terms in energy equation

5) Compressible solver, phase change and and dp/dt are accounted

0 0,5 1 1,5 2 2,5 3 3,5 4

-1

-0,5

0

0,5

1

1,5Merkle

non-dimensional distance

Cp

Without cavitation, incompressible

0 0,5 1 1,5 2 2,5 3 3,5 4

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

1,2


Cp

With cavitation, incompressible, Merkle model


RESULTS

0 0,5 1 1,5 2 2,5 3 3,5 4

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

Rouse&McNown Cp1

Cp2 Cp3

Cp4


Cp

0 0,5 1 1,5 2 2,5 3 3,5 4

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

Rouse&McNown Cp5

non-dimensional distanceC

p

Case 2

Case 1

Case 3

Case 4

Case 5


BUBBLE GROWTH AND RISE

p∞ = 98.1kPa

p∞ = 98.1kPa

T = 328K

T = 329KT = 349K

T = 345K


RESULTS

0 2 4 6 8 10 12 14 16 18 2040

50

60

70

80

90

100

110

120

130

140

Time, ms

R,

mkm

Growth of the bubble

Bubbledepartureand formationof the new one

Here R — effectiveradius, calculatedfrom volume fractionof the void phase


FC72 PROPERTIES● Site

http://detector-cooling.web.cern.ch/detector-cooling/data/C6F14Prop.htm

● Saturated pressure — 100kPa

● Saturation temperature — 56 C

● Liquid density — 1674 kg/m3

● Kinematic viscosity — 3.61x10-7 м2/с

● Constant pressure heat capacity — 1060 Дж/кг/К

● Heat conduction coeff — 0.056Вт/м/К

● Prandtl number — 11.23

● Volumetric expansion coeff — 0.0015 1/C

● Surface tension — 10.04Н/м!!!




FC72 PROPERTIES OF VAPOUR

● Site - http://chemeo.com/cid/51-127-4 ● Const. pressure heat capacity — 305.26 J/mole/K● Molecular mass — 338 kg/mole

http://chemeo.com/cid/51-127-4


STEAM JET SUBMERGED IN THE SUBCOOLED

p=1MPaT=450K

p=0.1MPa

p=0.1MPaT=293K

steamwater


RESULTS

CFD ANALYSIS OF A TURBULENT JET BEHAVIOR INDUCED BY A STEAM JETDISCHARGE THROUGH A SINGLE HOLE IN A SUBCOOLED WATER POOL,Korea Atomic Energy Research Institute

MEASURED TEMPERATURE 10cm FROM THE OUTLET IS ABOUT 320K


MODEL'S ASSUMPTIONS SUMMARY (1)

● Thermodynamic equilibrium mixture --> energy and pressure are cell averaged. Do we need a non-equilibrium model?

● Constant physical properties of phases (Cp, L, e.t.c). It's not a common case. Do we need to take into account this dependencies?

● The energy equation was formulated in enthalpy. Usage of internal energy removes complexity in the source terms of energy equation.

● Constant contact angle strongly impacts on solution


MODEL'S ASSUMPTIONS SUMMARY (2)● Mass source term depends on the pressure difference

(simplified Rayleigh-Taylor equation). Possibly, we need to consider explicitly the impact of the difference in temperature on the evaporation/condensation

● The mass source term is defined in the volume. But the phase transition process occurs through surface.

● Anything else, that i forgot to mention Water

Steam


QUESTIONS, PLEASE

MVKraposhin OFW8 Presentation

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