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multiphase flows Application to evaporation phenomena Olivier Le - - PowerPoint PPT Presentation
multiphase flows Application to evaporation phenomena Olivier Le - - PowerPoint PPT Presentation
Dynamic relaxation processes in compressible multiphase flows Application to evaporation phenomena Olivier Le Mtayer Aix-Marseille University, Polytech Marseille UMR CNRS 7343 Some industrial applications where evaporation is crucial Fuel
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Typical physical processes we are interested in
Low pressure chamber
- Liquid/vapor mixture (drops, bubbles,
pockets, ,…) Liquid flow from a high pressure chamber
- Drops evaporation and jet explosion
- « Flashing » (rapid depressurization)
generated by the initial pressure ratio Valve/diaphragm
- Cavitation created by geometric
singularities (strong rarefaction waves) Interface problems Two-velocities flow (inter-penetration) How can be solved the multiphase flow when the topology strongly varies ? A way is to consider relaxation effects
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Plan
- Presentation of the relaxation effects
- Some multidimensional numerical results
- Description of the non-equilibrium multiphase flow model
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Non-equilibrium multiphase compressible flow model
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Multiphase compressible flow model
) u ( . t ) (
k k
k I k k
α P I P u u ρ . t ) u (
k I I k k
α . u P u P ρE . t ) E (
α . u t α
k I k
- Each phase has its own set of equations and associated variables
(velocity, pressure temperature, entropy,…) : + Equation of state :
) , P ( e
k k k
- Solved by the Discrete Equations Method (Abgrall & Saurel, JCP, 2003 ; Saurel & al.,
JFM, 2003 ; Chinnayya & al., JCP, 2004 ; Le Métayer & al., JCP, 2005 ; Saurel & al., IJNMF, 2007 ; Le Métayer & al., JCP, 2011)
- Initially proposed by (Baer & Nunziato,IJMF,1986) for two phases
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Fixed control volume = sum of the sub-volumes filled with the fluids Each phase has a time-dependent sub-volume (volume fraction)
Discrete Equations Method : Godunov’s concept applied to multiphase cells
- DEM=averaging procedure by solving Riemann problems between pure fluids at each cells interface
Cell i Cell (i-1) Cell (i+1)
- Godunov’s method=averaging procedure by solving a single Riemann problem at each cells interface
Cell i Cell (i-1) Cell (i+1) Fixed control volume = volume filled with the fluid
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Major contributions of DEM (1)
Discrete equations traducing the time evolution of phases and mixture variables are
- btained = Numerical scheme
When dealing with a two-phases bubbly flow, a continuous model has been obtained : All interface variables are determined !
x α P x P ρu t ) u (
1 I 1 2 1
1 2
u u λ
x α u t α
1 I 1
2 1
P P μ
x α u P x u P ρE t ) E (
1 I I 1 1
2 1 I
P P μP
1 2 I
u u λu
2 1 I
Z Z A
2 1 2 1 I
Z Z Z Z A
2 1 2 2 1 1 I
Z Z u Z u Z u
2 1 2 2 1 1 I
Z 1 Z 1 Z P Z P P
Interfacial area between phases
µ and λ express the rates at which pressure and velocity equilibrium are reached respectively
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Major contributions of DEM (2)
- This non-equilibrium model is able to treat simultaneously :
- mixtures with several velocities, temperatures,…
- material interfaces (contact)
- permeable interfaces (interfaces separating a cloud of drops and a gas for example,…)
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Evaporation effects
- This model is completed by heat and mass transfer source terms (evaporation,
condensation) at infinite rates (relaxations)
- They are particularly relevant when :
- the interfacial area and the flow topology are unknown,
- the Direct Numerical Simulation of interface problems is considered.
- Relaxation processes (at infinite rate) allow the obtention of solutions in limit cases
corresponding to reduced models.
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Two-phase compressible flow model + relaxation terms
m
I
H ~ m
I
u ~ m
Pressure relaxation Velocity relaxation Mass transfer Heat transfer Hyperbolic
1 1
) u .( t ) (
1 2
u u λ
1 I I 1 1
α . u P u P ρE . t ) E (
2 1 I
P P P μ
1 2 I
u u . u λ
1 2 T
T T H
2 1 I
P P P μ
I
H ~ m
1 2 T
T T H ) g g ( ~ m
2 1 I
1 2
u u λ
1 2 I
u u . u λ
2 2
) u .( t ) (
1 I 2 2
α P I P u u ρ . t ) u (
1 I I 2 2
α . u P u P ρE . t ) E (
1 I 1 1
α P I P u u ρ . t ) u (
m
2 1
P P μ
I
u ~ m
I
~ m
1 I 1
α . u t
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Relaxation coefficients
1 2
u u λ
1 2 T
T T H ) g g ( ~ m
2 1 I
2 1
P P μ
- µ and λ express the rate at which pressure and velocity equilibria are
reached respectively :
- HT expresses the rate at which temperature equilibrium is reached :
- ν expresses the rate at which Gibbs free energy equilibrium is reached :
Heat exchange between phases Mass transfer (evaporation/condensation process) Mechanical interactions (drags, compressibility ratio) between phases
μ , HT , ν → ∞
Instantaneous exchanges : relaxation effects
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Relaxation effects at infinite rates
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Hierarchy of compressible flow models : reduced models
Multi-velocity flow model Non-equilibrium flows (u1,u2,p1,p2,T1,T2,g1,g2) Single-velocity flow model Interface problems (u,p,T1,T2,g1,g2) Homogeneous Euler model Thermodynamic equilibrium flows (u,p,T,g)
Thermal relaxation (temperature) Mechanical relaxation (velocity and pressure) Chemical relaxation (Gibbs free energy)
Multi-species Euler mixture model Thermal equilibrium flows (u,p,T, g1,g2)
,
T
H
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Velocity relaxation procedure : λ → ∞
t ) (
1
t ) u (
1
1 2
u u λ t α1
1 2 I
u u . u λ
t ) (
2
cst ) ( Y
1 1
cst ) ( Y
2 2
2 2 1 1 *
u Y u Y u ) u u ).( u u ( 2 1 e e
1 * 1 * 1 * 1
EOS of both phases are not used explicitly
) u u ).( u u ( 2 1 e e
2 * 2 * 2 * 2
cst
1
1 2
u u λ
-
t ) u (
2
1 2 I
u u . u λ
-
t ) E (
1
t ) E (
2
Momentum and energy conservation
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Pressure relaxation procedure : μ → ∞
t ) (
1
t ) u (
1
t α1 cst ) ( Y
2 2
2 1 I
P P P μ
cst u1 cst u2
1 * 1 * 1 * 1
1 1 p e e
Use of EOS
) , p ( e
* 1 * * 1
2 1 I
P P P μ
t ) (
2
2 1
P P μ
t ) E (
1
t ) u (
2
t ) E (
2
cst ) ( Y
1 1
2 * 2 * 2 * 2
1 1 p e e
) , p ( e
* 2 * * 2
) p (
* * 1
) p (
* * 2
) p ( Y ) p ( Y 1
* * 2 2 * * 1 1
Function of the final relaxed pressure only When considering ideal gas or ‘Stiffened Gas’ EOS an analytical relation is available for the pressure Mass and energy conservation
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Single-velocity flow model : Interface problems
Each phase has its own temperature and Gibbs free energy (Kapila & al., 2001; Saurel & al., 2008) Mechanical equilibrium Mixture variables : Asymptotic analysis of the full non-equilibrium model Wood sound speed :
10 100 1000 10000 0.2 0.4 0.6 0.8 1 Fraction volumique eau Vitesse du son de wood dans un melange eau/air (m/s)
2 1 2 1
) E ( ) E ( E ) ( ) ( p p p u u u
2 1 2 1
u ) p E ( . t ) E ( I p u u . t ) u ( u ) ( . t ) ( u ) ( . t ) ( u . c c ) c c ( . u t
2 2 1 1 2 2 2 2 1 2 1 1 2 1 1 2 2 2 1 1
2 2 2 2 2 1 1 1 2 wood
c c c 1
,
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Pressure and temperature relaxation procedure : μ , HT → ∞
t ) (
1
t ) u (
1
t ) E (
1
t α1
t ) (
2
t ) u (
2
t ) E (
2
cst ) ( Y
1 1
cst ) ( Y
2 2
2 1 I
P P P μ
2 1
P P μ
2 1 I
P P P μ
cst u1 cst u2 cte Y Y 1
* 2 2 * 1 1
Mass and energy conservation
1 2 T
T T H
1 2 T
T T H
cte e Y e Y e
* 2 2 * 1 1
Use of EOS
) T , p (
* * * k
) T , p ( e
* * * k
) T , p ( Y ) T , p ( Y 1
* * * 2 2 * * * 1 1
When considering ideal gas or ‘Stiffened Gas’ EOS analytical relations are available for pressure and temperature
) T , p ( e Y ) T , p ( e Y e
* * * 2 2 * * * 1 1
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Two-species Euler model
u ) p E ( . t E I p u u . t u u ) ( . t ) ( u ) ( . t ) (
2 2 1 1
Mixture variables :
u . u 2 / 1 e E
u ) p E ( . t E I p u u . t u ) u Y .( t Y ) u Y .( t Y
2 2 1 1
Mechanical and thermal equilibrium Asymptotic analysis of the full non-equilibrium model
T T T p p p u u u
2 1 2 1 2 1
T
H , ,
Each phase has its own Gibbs free energy
2 2 1 1
Y Y 1
2 2 1 1
e Y e Y e
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Pressure, temperature and Gibbs free energy relaxation procedure : μ , HT , ν→ ∞
I
~ m
m
I
H ~ m
I
u ~ m
t ) (
1
t ) u (
1
t ) E (
1
t α1
t ) (
2
t ) u (
2
t ) E (
2
2 1 I
P P P μ
2 1
P P μ
2 1 I
P P P μ
Mass and energy conservation
1 2 T
T T H
1 2 T
T T H
Use of EOS
) T , p (
* * * k
) T , p ( e
* * * k
m
I
u ~ m
I
H ~ m
) g g ( ~ m
2 1 I
* 2 * 2 * 1 * 1
e Y e Y e
+ Gibbs free energy equality
) p ( T T
* sat * * 2 * 2 * 1 * 1
Y Y 1 ) p ( h ) p ( h p e ) p ( h Y
* * 1 * * 2 * * * 2 * 1
) p ( 1 ) p ( 1 1 ) p ( 1 Y
* * 1 * * 2 * * 2 * 1
Latent heat of vaporization
) p ( L
* V
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Homogeneous Euler model
u ) p E ( . t E I p u u . t u ) u .( t
g g g T T T p p p u u u
2 1 2 1 2 1 2 1
Mechanical and thermodynamic equilibrium Closure relations :
u . u 2 / 1 e E
Equilibrium sound speed :
2 2 2 , p 2 2 1 1 , p 1 2 wood 2 eq
dp ds C Y dp ds C Y T c 1 c 1
wood eq
c c
Asymptotic analysis of the full non-equilibrium model
, H , ,
T
) p ( T T
sat
1 2 2 1
1 1 1 1 Y
2 2 1 1
e Y e Y e
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For an arbitrary number of phases…
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Multiphase compressible flow model
Pressure relaxation Velocity relaxation Temperature relaxation Hyperbolic
k k
) u ( . t ) (
k I k k
α P I P u u ρ . t ) u (
k I I k k
α . u P u P ρE . t ) E (
k I k
α . u t α
N , 1 l k l kl
) u u (
N , 1 l l k kl
) P P (
N , 1 l k l kl I
) u u ( . u
N , 1 l l k kl I
) P P ( P
N , 1 l k l kl , T
) T T ( H
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Pressure, temperature relaxation and Gibbs free energy relaxation (liquid/vapor) : μkl → ∞, HT,kl → ∞, ν → ∞
m t ) (
1
I 1
u ~ m t ) u (
I N , 1 l l 1 l 1
~ m ) P P (
t α1
N , 1 l l 1 l 1 I
) P P ( P t ) E (
1 I N , 1 l 1 l l 1 , T
H ~ m ) T T ( H
m t ) (
2
I 2
u ~ m t ) u (
I N , 1 l l 2 l 2
~ m ) P P (
t α2
N , 1 l l 2 l 2 I
) P P ( P t ) E (
2 I N , 1 l 2 l l 2 , T
H ~ m ) T T ( H
t ) (
k
t ) u (
k
N , 1 l l k kl
) P P (
t αk
N , 1 l l k kl I
) P P ( P t ) E (
k
N , 1 l k l kl , T
) T T ( H
Phase k (inert) : Phase 1 (liquid) : Phase 2 (vapor) :
- pressure
- temperature
Relaxation : Relaxation :
- Gibbs free energy
- pressure
- temperature
) g g ( ~ m
2 1 I
- pressure
- temperature
Relaxation :
3 k
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Pressure, temperature and Gibbs free energy (liquid/vapor) relaxation for an arbitrary number of phases
Mass and energy conservation Use of EOS
) T , p (
* * * k
) T , p ( e
* * * k
N , 3 k * k k * 2 * 2 * 1 * 1
e Y e Y e Y e
+ Gibbs free energy equality
) p ( T T
* sat *
N , 3 k * k k * 2 * 2 * 1 * 1
Y Y Y 1
) p ( h ) p ( h ) p ( h Y p e ) p ( h Y Y Y
* * 1 * * 2 N , 3 k * * k k * * * 2 2 1 * 1
) p ( 1 ) p ( 1 ) p ( Y 1 ) p ( Y Y Y
* * 1 * * 2 N , 3 k * * k k * * 2 2 1 * 1
1 Y Y Y
N , 3 k k * 2 * 1
) N , 3 k ( cst Y Y
k * k
Modification of the final thermodynamic state between the liquid and its vapor
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One-dimensional examples
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Shock tube (water, steam, air)
Bar P 1 Bar P 10
K 467 ) P ( T T
sat
4 Liq
10 . 5
Cloud of droplets in a gaz (air/steam) mixture
K 373 ) P ( T T
sat
2 .
Air
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Mechanical equilibrium :
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Mass fraction Liq Vap Air 1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Pressure(Bar) Liq Vap Air 50 100 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Velocity(m/s) Liq Vap Air 300 350 400 450 500 550 600 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Temperature(K) Liq Vap Air
,
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Mass fraction Liq Vap Air 1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Pressure(Bar) Liq Vap Air 50 100 150 200 250 300 350 400 450 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Velocity(m/s) Liq Vap Air
No interaction between phases
300 350 400 450 500 550 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Temperature(K) Liq Vap Air
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Mechanical equilibrium
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Mass fraction Liq Vap Air 1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Pressure(Bar) Liq Vap Air 50 100 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Velocity(m/s) Liq Vap Air 300 350 400 450 500 550 600 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Temperature(K) Liq Vap Air
,
Mechanical/thermal equilibrium
T
H , ,
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Mass fraction Liq Vap Air 1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Pressure(Bar) Liq Vap Air 50 100 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Velocity(m/s) Liq Vap Air 360 380 400 420 440 460 480 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Temperature(K) Liq Vap Air
Mechanical/thermodynamical equilibrium , H , ,
T
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Mass fraction Liq Vap Air 1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Pressure(Bar) Liq Vap Air 50 100 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Velocity(m/s) Liq Vap Air 380 390 400 410 420 430 440 450 460 470 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Temperature(K) Liq Vap Air
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Another possibility : heat exchanges between inert phases and liquid/vapor are absent
m t ) (
1
I 1
u ~ m t ) u (
I N , 1 l l 1 l 1
~ m ) P P (
t α1
N , 1 l l 1 l 1 I
) P P ( P t ) E (
1 I N , 1 l 1 l l 1 , T
H ~ m ) T T ( H
m t ) (
2
I 2
u ~ m t ) u (
I N , 1 l l 2 l 2
~ m ) P P (
t α2
N , 1 l l 2 l 2 I
) P P ( P t ) E (
2 I N , 1 l 2 l l 2 , T
H ~ m ) T T ( H
t ) (
k
t ) u (
k
N , 1 l l k kl
) P P (
t αk
N , 1 l l k kl I
) P P ( P t ) E (
k
N , 1 l k l kl , T
) T T ( H
Phase k (inert) : Phase 1 (liquid) : Phase 2 (vapor) :
- pressure
- temperature
Relaxation : Relaxation :
- Gibbs free energy
- pressure
- temperature
) g g ( ~ m
2 1 I
- pressure
- temperature
Relaxation :
3 k
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Mechanical/thermodynamical equilibrium , H , ,
T
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Mass fraction Liq Vap Air 1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Pressure(Bar) Liq Vap Air 50 100 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Velocity(m/s) Liq Vap Air 380 390 400 410 420 430 440 450 460 470 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Temperature(K) Liq Vap Air
Thermodynamical equilibrium between liquid and vapor only
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Mass fraction Liq Vap Air 1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Pressure(Bar) Liq Vap Air 50 100 150 200 250 300 350 400 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Velocity(m/s) Liq Vap Air 300 350 400 450 500 550 600 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X(m) Temperature(K) Liq Vap Air
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Multi-dimensional examples
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Cryogenic liquid and inert gas injections
Inert gas injection Tank
Bar 1 P
Tank filled with cryogenic liquid ~50000 tetrahedrons Inert gas
Bar 1 P
bar 20 p
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Pressure and temperature equilibrium flow
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Pressure, temperature and Gibbs free energy equilibrium flow
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Vapor volume fraction evolution
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Filling of cavity under gravity effect
Walls Connection to atmosphere Hot steam
Bar 1 P
Air
Bar 1 P
Cold water Gravity
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Liquid volume fraction evolution : pressure relaxation
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Pressure and temperature relaxation
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Pressure, temperature and Gibbs free energy relaxation
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Perspectives
- Exact steady solutions through converging-diverging nozzles should be
investigated under mechanical, thermal and Gibbs free energy equilibria for an arbitrary number of phases Reference solutions
- The knowledge of the interfacial area at each point of the flow must be strongly
improved and particularly the change of the multiphase flow topology. Additional equations are thus required (number of particles per unit volume, geometrical equations,…) Big challenge !!!
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