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Novatec Solar GmbH
Modeling of two-phase flow> Direct steam generation in solar thermal power plants
Pascal RichterRWTH Aachen University
EGRIN | Pirirac-sur-Mer | June 3, 2015
Direct steam generation | Solar thermal power plant
Solar collector
Steamturbine
Generator
Coolingtower
Conden-sator
Pump
Deaerator
Pump
Pascal Richter | Modeling of two-phase flow | 2/14
Direct steam generation | Fresnel collector
Solar collector
Steamturbine
Generator
Coolingtower
Conden-sator
Pump
Deaerator
Pump
Sun
light
FresnelSolar collector
•
Absorber tube(two-phase flow)
•
Secondary reflector •
Pascal Richter | Modeling of two-phase flow | 2/14
Direct steam generation | Two-phase flow
Solar collector
Steamturbine
Generator
Coolingtower
Conden-sator
Pump
Deaerator
Pump
• Liquid and steam phase in absorber tubes
• Exchange of mass, momentum and energy across the phases
• Interaction of the phases at the wall
• Network coupling
Pascal Richter | Modeling of two-phase flow | 2/14
How to model two-phase flow?
Fluid k, that occupies the observed domain, is described withNavier-Stokes equations: Continuity, momentum and total energy
Plenty of models in the literature!
Pascal Richter | Modeling of two-phase flow | 3/14
How to model two-phase flow?
Model development
• Dimension reduction
• Averaging of the Navier-Stokes equations
• Source terms
• Quantities
DensityVelocityEnergyPressure
9>>=
>>;separate, mixture or equal
Pascal Richter | Modeling of two-phase flow | 3/14
How to model two-phase flow?
Model development
• Dimension reduction! Quasi-1D flow in a tube, Stewart and Wendro↵ [1]
• Averaging of the Navier-Stokes equations
• Source terms
• Quantities
DensityVelocityEnergyPressure
9>>=
>>;separate, mixture or equal
Pascal Richter | Modeling of two-phase flow | 3/14
How to model two-phase flow?
Model development
• Dimension reduction! Quasi-1D flow in a tube, Stewart and Wendro↵ [1]
• Averaging of the Navier-Stokes equations! Introduction of void fractions ↵ Drew and Passman [2]! Baer-Nunziato type [3]
• Source terms
• Quantities
DensityVelocityEnergyPressure
9>>=
>>;separate, mixture or equal
Pascal Richter | Modeling of two-phase flow | 3/14
How to model two-phase flow?
Model development
• Dimension reduction! Quasi-1D flow in a tube, Stewart and Wendro↵ [1]
• Averaging of the Navier-Stokes equations! Introduction of void fractions ↵ Drew and Passman [2]! Baer-Nunziato type [3]
• Source terms: Replace viscous and di↵usive terms, RELAP [4]! Use empirical laws dependent on local flow pattern
• Quantities
DensityVelocityEnergyPressure
9>>=
>>;separate, mixture or equal
Pascal Richter | Modeling of two-phase flow | 3/14
How to model two-phase flow?
Model development
• Dimension reduction! Quasi-1D flow in a tube, Stewart and Wendro↵ [1]
• Averaging of the Navier-Stokes equations! Introduction of void fractions ↵ Drew and Passman [2]! Baer-Nunziato type [3]
• Source terms: Replace viscous and di↵usive terms, RELAP [4]! Use empirical laws dependent on local flow pattern
• Quantities
DensityVelocityEnergyPressure
9>>=
>>;separate, mixture or equal
Pascal Richter | Modeling of two-phase flow | 3/14
Two-phase flow modelThe system is in non-conservative form
@t
u+ @x
f(u) + B(u) @x
u = s(u),
u =
0
BBBBBBBB@
↵g
↵`⇢`↵`⇢`v`↵`⇢`E`
↵g
⇢g
↵g
⇢g
vg
↵g
⇢g
Eg
1
CCCCCCCCA
, f(u) =
0
BBBBBBBB@
0↵`⇢`v`
↵`(⇢`v2` + p`)
↵`(⇢`E` + p`)v`↵g
⇢g
vg
↵g
(⇢g
v2g
+ pg
)↵g
(⇢g
Eg
+ pg
)vg
1
CCCCCCCCA
B(u) =
0
BBBBBBBB@
vi 0 0 0 0 0 00 0 0 0 0 0 0pi 0 0 0 0 0 0pivi 0 0 0 0 0 00 0 0 0 0 0 0
�pi 0 0 0 0 0 0�pivi 0 0 0 0 0 0
1
CCCCCCCCA
, s(u) =
0
BBBBBBBB@
�i/⇢i��i
�Fi � vi�i�v`Fi + Qi ` � Ei `�i
�iFi + vi�i
vg
Fi + Qi g + Ei g�i
1
CCCCCCCCA
Pascal Richter | Modeling of two-phase flow | 4/14
Two-phase flow modelThe system is in non-conservative form
@t
u+ @x
f(u) + B(u) @x
u = s(u),
u =
0
BBBBBBBB@
↵g
↵`⇢`↵`⇢`v`↵`⇢`E`
↵g
⇢g
↵g
⇢g
vg
↵g
⇢g
Eg
1
CCCCCCCCA
, f(u) =
0
BBBBBBBB@
0↵`⇢`v`
↵`(⇢`v2` + p`)
↵`(⇢`E` + p`)v`↵g
⇢g
vg
↵g
(⇢g
v2g
+ pg
)↵g
(⇢g
Eg
+ pg
)vg
1
CCCCCCCCA
B(u) =
0
BBBBBBBB@
vi 0 0 0 0 0 00 0 0 0 0 0 0pi 0 0 0 0 0 0pivi 0 0 0 0 0 00 0 0 0 0 0 0
�pi 0 0 0 0 0 0�pivi 0 0 0 0 0 0
1
CCCCCCCCA
, s(u) =
0
BBBBBBBB@
�i/⇢i��i
�Fi � vi�i�v`Fi + Qi ` � Ei `�i
�iFi + vi�i
vg
Fi + Qi g + Ei g�i
1
CCCCCCCCA
Model properties
1 Source terms and interphasequantities
2 Conservation of mass, momentumand energy at the interface
3 Equation of state! How to describe pressure p ?
4 Well-posedness of the model! Hyperbolicity
5 Entropy inequality! Consistent with 2nd
law of thermodynamics
Pascal Richter | Modeling of two-phase flow | 4/14
Two-phase flow model | 1 Source terms
B(u) =
0
BBBBBBBB@
vi 0 0 0 0 0 00 0 0 0 0 0 0pi 0 0 0 0 0 0pivi 0 0 0 0 0 00 0 0 0 0 0 0
�pi 0 0 0 0 0 0�pivi 0 0 0 0 0 0
1
CCCCCCCCA
, s(u) =
0
BBBBBBBB@
�i/⇢i��i
�Fi � vi�i�v`Fi + Qi ` � Ei `�i
�iFi + vi�i
vg
Fi + Qi g + Ei g�i
1
CCCCCCCCA
• Interphase quantities: �i, vi, pi, Ei `, Ei g , ⇢i = ???
• Flow regimes for friction Fi:
• Models for heat transfer Qi `, Qi g :
Convection, Condensation, Nucleate & Film boiling
Pascal Richter | Modeling of two-phase flow | 5/14
Two-phase flow model | 2 Conservation at interface
s(u) =
0
BBBBBBBB@
�i/⇢i��i
�Fi � vi�i�v`Fi + Qi ` � Ei `�i
�iFi + vi�i
vg
Fi + Qi g + Ei g�i
1
CCCCCCCCA
Heat conduction limited model
�i =1
Ei ` � Ei g
⇣Fi(vg � v`) + Qi ` + Qi g
⌘
RELAP [4].
Pascal Richter | Modeling of two-phase flow | 6/14
Two-phase flow model | 3 Equation of state
f(u) =
0
BBBBBBBB@
0↵`⇢`v`
↵`(⇢`v2` + p`)
↵`(⇢`E` + p`)v`↵g
⇢g
vg
↵g
(⇢g
v2g
+ pg
)↵g
(⇢g
Eg
+ pg
)vg
1
CCCCCCCCA
Describe p by two state parameter
p` = p(⇢`, u`), pg
= p(⇢g
, ug
)
with density ⇢ and specific inner energy u.
Pascal Richter | Modeling of two-phase flow | 7/14
Two-phase flow model | 4 Hyperbolicity
Rewrite system in terms of primitive quantities
@t
w +M(w)@x
w = s(w)
Eigenvalues
� =�vi, v`, v` + w`, v` � w`, v
g
, vg
+ wg
, vg
� wg
�T
with speed of sound w .
Eigenvectors form a basis of R7 as soon asthe non-resonance condition is fulfilled Coquel, Herard, Saleh,and Seguin [5] :
vi 6= v` ± w` and vi 6= vg
± wg
.
Gallouet, Herard, and Seguin [6] choose vi as convex combinationbetween v` and v
g
:vi := �v` + (1� �)v
g
with � 2 [0, 1]
Non-resonance condition will always be fulfilled! M is diagonalisable ! quasilinear system is hyperbolic.
Pascal Richter | Modeling of two-phase flow | 8/14
Two-phase flow model | 4 Hyperbolicity
Rewrite system in terms of primitive quantities
@t
w +M(w)@x
w = s(w)
Eigenvalues
� =�vi, v`, v` + w`, v` � w`, v
g
, vg
+ wg
, vg
� wg
�T
with speed of sound w . Eigenvectors form a basis of R7 as soon asthe non-resonance condition is fulfilled Coquel, Herard, Saleh,and Seguin [5] :
vi 6= v` ± w` and vi 6= vg
± wg
.
Gallouet, Herard, and Seguin [6] choose vi as convex combinationbetween v` and v
g
:vi := �v` + (1� �)v
g
with � 2 [0, 1]
Non-resonance condition will always be fulfilled! M is diagonalisable ! quasilinear system is hyperbolic.
Pascal Richter | Modeling of two-phase flow | 8/14
Two-phase flow model | 4 Hyperbolicity
Rewrite system in terms of primitive quantities
@t
w +M(w)@x
w = s(w)
Eigenvalues
� =�vi, v`, v` + w`, v` � w`, v
g
, vg
+ wg
, vg
� wg
�T
with speed of sound w . Eigenvectors form a basis of R7 as soon asthe non-resonance condition is fulfilled Coquel, Herard, Saleh,and Seguin [5] :
vi 6= v` ± w` and vi 6= vg
± wg
.
Gallouet, Herard, and Seguin [6] choose vi as convex combinationbetween v` and v
g
:vi := �v` + (1� �)v
g
with � 2 [0, 1]
Non-resonance condition will always be fulfilled! M is diagonalisable ! quasilinear system is hyperbolic.
Pascal Richter | Modeling of two-phase flow | 8/14
Two-phase flow model | 5 Entropy-entropy flux pairClosed quasilinear form: @
t
u+ A(u) · @x
u = s(u).
Find entropy function ⌘(u) and entropy flux (u), such that
@t
⌘(u) + @x
(u)! 0
with
1 Convex entropy (decreasing behaviour):⌘00(u) > 0
2 Compatibility condition of Tadmor [8]:
@u (u)T != @u⌘(u)
T A(u)
3 Entropy production condition:
@t
u + A(u) · @x
u = s(u)
, @u⌘(u)T @
t
u + @u⌘(u)T A(u) · @
x
u = @u⌘(u)T s(u)
, @t
⌘(u) + @x
(u) = @u⌘(u)T s(u)
! 0
Pascal Richter | Modeling of two-phase flow | 9/14
Two-phase flow model | 5 Entropy-entropy flux pairClosed quasilinear form: @
t
u+ A(u) · @x
u = s(u).
Find entropy function ⌘(u) and entropy flux (u), such that
@t
⌘(u) + @x
(u)! 0
with
1 Convex entropy (decreasing behaviour):⌘00(u) > 0
2 Compatibility condition of Tadmor [8]:
@u (u)T != @u⌘(u)
T A(u)
3 Entropy production condition:
@t
u + A(u) · @x
u = s(u)
, @u⌘(u)T @
t
u + @u⌘(u)T A(u) · @
x
u = @u⌘(u)T s(u)
, @t
⌘(u) + @x
(u) = @u⌘(u)T s(u)
! 0
Pascal Richter | Modeling of two-phase flow | 9/14
Two-phase flow model | 5 Entropy-entropy flux pairClosed quasilinear form: @
t
u+ A(u) · @x
u = s(u).
Find entropy function ⌘(u) and entropy flux (u), such that
@t
⌘(u) + @x
(u)! 0
with
1 Convex entropy (decreasing behaviour):⌘00(u) > 0
2 Compatibility condition of Tadmor [8]:
@u (u)T != @u⌘(u)
T A(u)
3 Entropy production condition:
@t
u + A(u) · @x
u = s(u)
, @u⌘(u)T @
t
u + @u⌘(u)T A(u) · @
x
u = @u⌘(u)T s(u)
, @t
⌘(u) + @x
(u) = @u⌘(u)T s(u)
! 0
Pascal Richter | Modeling of two-phase flow | 9/14
Two-phase flow model | 5 Entropy-entropy flux pair (1)Choose mixture of physical entropy s` and s
g
:⌘(u) = �(↵`⇢`s` + ↵
g
⇢g
sg
) and (u) = �(↵`⇢`s`v` + ↵g
⇢g
sg
vg
)
Pascal Richter | Modeling of two-phase flow | 10/14
Two-phase flow model | 5 Entropy-entropy flux pair (1)Choose mixture of physical entropy s` and s
g
:⌘(u) = �(↵`⇢`s` + ↵
g
⇢g
sg
) and (u) = �(↵`⇢`s`v` + ↵g
⇢g
sg
vg
)
1 Convexity of ⌘(u):
Follow proof of Coquel, Herard, Saleh, and Seguin [5],using results of Godlewski and Raviart [7].
Pascal Richter | Modeling of two-phase flow | 10/14
Two-phase flow model | 5 Entropy-entropy flux pair (2)Choose mixture of physical entropy s` and s
g
:⌘(u) = �(↵`⇢`s` + ↵
g
⇢g
sg
) and (u) = �(↵`⇢`s`v` + ↵g
⇢g
sg
vg
)
2 Compatibility condition @u (u)T!= @u⌘(u)T · A(u):
0
BBBBBBBBBBBBBBBBBBBBB@
p`v`
T`� p
g
v
g
T
g
v` ·p`⇢`
+u`�12v
2`
T`
v
2`
T`� s`
� v`T`
v
g
·p
g
⇢g
+u
g
� 12v
2g
T
g
v
2g
T
g
� s
g
� v
g
T
g
1
CCCCCCCCCCCCCCCCCCCCCA
!=
0
BBBBBBBBBBBBBBBBBBBBB@
p`v`
T`� p
g
v
g
T
g
� (p`�pi)(v`�vi)T`
+(p
g
�pi)(vg�vi)T
g
v` ·p`⇢`
+u`�12v
2`
T`
v
2`
T`� s`
� v`T`
v
g
·p
g
⇢g
+u
g
� 12v
2g
T
g
v
2g
T
g
� s
g
� v
g
T
g
1
CCCCCCCCCCCCCCCCCCCCCA
Pascal Richter | Modeling of two-phase flow | 11/14
Two-phase flow model | 5 Entropy-entropy flux pair (2)Choose mixture of physical entropy s` and s
g
:⌘(u) = �(↵`⇢`s` + ↵
g
⇢g
sg
) and (u) = �(↵`⇢`s`v` + ↵g
⇢g
sg
vg
)
2 Compatibility condition @u (u)T!= @u⌘(u)T · A(u):
0
BBBBBBBBBBBBBBBBBBBBB@
p`v`
T`� p
g
v
g
T
g
v` ·p`⇢`
+u`�12v
2`
T`
v
2`
T`� s`
� v`T`
v
g
·p
g
⇢g
+u
g
� 12v
2g
T
g
v
2g
T
g
� s
g
� v
g
T
g
1
CCCCCCCCCCCCCCCCCCCCCA
!=
0
BBBBBBBBBBBBBBBBBBBBB@
p`v`
T`� p
g
v
g
T
g
� (p`�pi)(v`�vi)T`
+(p
g
�pi)(vg�vi)T
g
v` ·p`⇢`
+u`�12v
2`
T`
v
2`
T`� s`
� v`T`
v
g
·p
g
⇢g
+u
g
� 12v
2g
T
g
v
2g
T
g
� s
g
� v
g
T
g
1
CCCCCCCCCCCCCCCCCCCCCA
Interphasic velocity [Hyperbolicity]vi = �v` + (1� �)v
g
with � 2 [0, 1]Interphasic pressure Gallouet, Herard, and Seguin [6]
pi := �p` + (1� �)pg
with � 2 [0, 1]
Pascal Richter | Modeling of two-phase flow | 11/14
Two-phase flow model | 5 Entropy-entropy flux pair (2)Choose mixture of physical entropy s` and s
g
:⌘(u) = �(↵`⇢`s` + ↵
g
⇢g
sg
) and (u) = �(↵`⇢`s`v` + ↵g
⇢g
sg
vg
)
2 Compatibility condition @u (u)T!= @u⌘(u)T · A(u):
0
BBBBBBBBBBBBBBBBBBBBB@
p`v`
T`� p
g
v
g
T
g
v` ·p`⇢`
+u`�12v
2`
T`
v
2`
T`� s`
� v`T`
v
g
·p
g
⇢g
+u
g
� 12v
2g
T
g
v
2g
T
g
� s
g
� v
g
T
g
1
CCCCCCCCCCCCCCCCCCCCCA
!=
0
BBBBBBBBBBBBBBBBBBBBB@
p`v`
T`� p
g
v
g
T
g
� (p`�pi)(v`�vi)T`
+(p
g
�pi)(vg�vi)T
g
v` ·p`⇢`
+u`�12v
2`
T`
v
2`
T`� s`
� v`T`
v
g
·p
g
⇢g
+u
g
� 12v
2g
T
g
v
2g
T
g
� s
g
� v
g
T
g
1
CCCCCCCCCCCCCCCCCCCCCA
Interphasic velocity [Hyperbolicity]vi = �v` + (1� �)v
g
with � 2 [0, 1]Interphasic pressure
pi := �p` + (1� �)pg
) � = (1��)Tg
�T`+(1��)Tg
Pascal Richter | Modeling of two-phase flow | 11/14
Two-phase flow model | 5 Entropy-entropy flux pair (3)Choose mixture of physical entropy s` and s
g
:⌘(u) = �(↵`⇢`s` + ↵
g
⇢g
sg
) and (u) = �(↵`⇢`s`v` + ↵g
⇢g
sg
vg
)
3 Entropy inequality of entropy production: @u⌘(u)T s(u)! 0:
�Qi `
T`+
Ei ` � E` + v
2` � v`vi +
p`
⇢i� p`
⇢`+ s`T`
T`· �i
�Qi g
T
g
�Ei g � E
g
+ v
2g
� v
g
vi +p
g
⇢i� p
g
⇢g
+ s
g
T
g
T
g
· �i
! 0
Pascal Richter | Modeling of two-phase flow | 12/14
Two-phase flow model | 5 Entropy-entropy flux pair (3)Choose mixture of physical entropy s` and s
g
:⌘(u) = �(↵`⇢`s` + ↵
g
⇢g
sg
) and (u) = �(↵`⇢`s`v` + ↵g
⇢g
sg
vg
)
3 Entropy inequality of entropy production: @u⌘(u)T s(u)! 0:
�Qi `
T`+
Ei ` � E` + v
2` � v`vi +
p`
⇢i� p`
⇢`+ s`T`
T`· �i
�Qi g
T
g
�Ei g � E
g
+ v
2g
� v
g
vi +p
g
⇢i� p
g
⇢g
+ s
g
T
g
T
g
· �i
! 0
Spec. total energy = spec. enthalpy � spec. pressure + kinetic energy (physical law)
E` = h` � p`
⇢`+ 1
2v2` , Ei ` := h` sat � pi
⇢i+ 1
2v2i
Eg
= hg
� p
g
⇢g
+ 12v
2g
, Ei g := hg sat � pi
⇢i+ 1
2v2i
Interphasic velocityvi := �v` + (1� �)v
g
with � 2 [0, 1]
Interphasic density⇢i := ⇢sat
Pascal Richter | Modeling of two-phase flow | 12/14
Two-phase flow model | 5 Entropy-entropy flux pair (3)Choose mixture of physical entropy s` and s
g
:⌘(u) = �(↵`⇢`s` + ↵
g
⇢g
sg
) and (u) = �(↵`⇢`s`v` + ↵g
⇢g
sg
vg
)
3 Entropy inequality of entropy production: @u⌘(u)T s(u)! 0:
�Qi `
T`+
h` sat � h` + 12(v` � vi)2 +
p`�pi
⇢i+ s`T`
T`· �i
�Qi g
T
g
�h
g sat � h
g
+ 12(v
g
� vi)2 +p
g
�pi
⇢i+ s
g
T
g
T
g
· �i
! 0
Spec. total energy = spec. enthalpy � spec. pressure + kinetic energy (physical law)
E` = h` � p`
⇢`+ 1
2v2` , Ei ` := h` sat � pi
⇢i+ 1
2v2i
Eg
= hg
� p
g
⇢g
+ 12v
2g
, Ei g := hg sat � pi
⇢i+ 1
2v2i
Interphasic velocityvi := �v` + (1� �)v
g
with � 2 [0, 1]
Interphasic density⇢i := ⇢sat
Pascal Richter | Modeling of two-phase flow | 12/14
Two-phase flow model | 5 Entropy-entropy flux pair (3)Choose mixture of physical entropy s` and s
g
:⌘(u) = �(↵`⇢`s` + ↵
g
⇢g
sg
) and (u) = �(↵`⇢`s`v` + ↵g
⇢g
sg
vg
)
3 Entropy inequality of entropy production: @u⌘(u)T s(u)! 0:
�Qi `
T`+
h` sat � h` + 12(v` � vi)2 +
p`�pi
⇢i+ s`T`
T`· �i
�Qi g
T
g
�h
g sat � h
g
+ 12(v
g
� vi)2 +p
g
�pi
⇢i+ s
g
T
g
T
g
· �i
! 0
Spec. total energy = spec. enthalpy � spec. pressure + kinetic energy (physical law)
E` = h` � p`
⇢`+ 1
2v2` , Ei ` := h` sat � pi
⇢i+ 1
2v2i
Eg
= hg
� p
g
⇢g
+ 12v
2g
, Ei g := hg sat � pi
⇢i+ 1
2v2i
Interphasic velocityvi := �v` + (1� �)v
g
with � 2 [0, 1]
Interphasic density⇢i := ⇢sat
Pascal Richter | Modeling of two-phase flow | 12/14
Two-phase flow model | 5 Entropy-entropy flux pair (3)Choose mixture of physical entropy s` and s
g
:⌘(u) = �(↵`⇢`s` + ↵
g
⇢g
sg
) and (u) = �(↵`⇢`s`v` + ↵g
⇢g
sg
vg
)
3 Entropy inequality of entropy production: @u⌘(u)T s(u)! 0:
�Qi `
T`+
h` sat � h` + 12(v` � vi)2 +
p`�pi
⇢i+ s`T`
T`· �i
�Qi g
T
g
�h
g sat � h
g
+ 12(v
g
� vi)2 +p
g
�pi
⇢i+ s
g
T
g
T
g
· �i
! 0
Spec. total energy = spec. enthalpy � spec. pressure + kinetic energy (physical law)
E` = h` � p`
⇢`+ 1
2v2` , Ei ` := h` sat � pi
⇢i+ 1
2v2i
Eg
= hg
� p
g
⇢g
+ 12v
2g
, Ei g := hg sat � pi
⇢i+ 1
2v2i
Interphasic velocity
vi := �v` + (1� �)vg
) � :=p
T
g
pT`+
pT
g
Interphasic density⇢i := ⇢sat
Pascal Richter | Modeling of two-phase flow | 12/14
Two-phase flow model | 5 Entropy-entropy flux pair (3)Choose mixture of physical entropy s` and s
g
:⌘(u) = �(↵`⇢`s` + ↵
g
⇢g
sg
) and (u) = �(↵`⇢`s`v` + ↵g
⇢g
sg
vg
)
3 Entropy inequality of entropy production: @u⌘(u)T s(u)! 0:
�Qi `
T`+
h` sat � h` + p`�pi
⇢i+ s`T`
T`· �i
�Qi g
T
g
�h
g sat � h
g
+p
g
�pi
⇢i+ s
g
T
g
T
g
· �i
! 0
Spec. total energy = spec. enthalpy � spec. pressure + kinetic energy (physical law)
E` = h` � p`
⇢`+ 1
2v2` , Ei ` := h` sat � pi
⇢i+ 1
2v2i
Eg
= hg
� p
g
⇢g
+ 12v
2g
, Ei g := hg sat � pi
⇢i+ 1
2v2i
Interphasic velocity
vi := �v` + (1� �)vg
) � :=p
T
g
pT`+
pT
g
Interphasic density⇢i := ⇢sat
Pascal Richter | Modeling of two-phase flow | 12/14
Two-phase flow model | 5 Entropy-entropy flux pair (3)Choose mixture of physical entropy s` and s
g
:⌘(u) = �(↵`⇢`s` + ↵
g
⇢g
sg
) and (u) = �(↵`⇢`s`v` + ↵g
⇢g
sg
vg
)
3 Entropy inequality of entropy production: @u⌘(u)T s(u)! 0:
�Qi `
T`+
h` sat � h` + p`�pi
⇢i+ s`T`
T`· �i
�Qi g
T
g
�h
g sat � h
g
+p
g
�pi
⇢i+ s
g
T
g
T
g
· �i
! 0
Spec. total energy = spec. enthalpy � spec. pressure + kinetic energy (physical law)
E` = h` � p`
⇢`+ 1
2v2` , Ei ` := h` sat � pi
⇢i+ 1
2v2i
Eg
= hg
� p
g
⇢g
+ 12v
2g
, Ei g := hg sat � pi
⇢i+ 1
2v2i
Interphasic velocity
vi := �v` + (1� �)vg
) � :=p
T
g
pT`+
pT
g
Interphasic density⇢i := ⇢sat
Pascal Richter | Modeling of two-phase flow | 12/14
Two-phase flow model | 5 Entropy-entropy flux pair (3)Choose mixture of physical entropy s` and s
g
:⌘(u) = �(↵`⇢`s` + ↵
g
⇢g
sg
) and (u) = �(↵`⇢`s`v` + ↵g
⇢g
sg
vg
)
3 Entropy inequality of entropy production: @u⌘(u)T s(u)! 0:
�Qi `
T`+
h` sat � h` + p`�pi
⇢i+ s`T`
T`· �i
�Qi g
T
g
�h
g sat � h
g
+p
g
�pi
⇢i+ s
g
T
g
T
g
· �i
! 0
Spec. total energy = spec. enthalpy � spec. pressure + kinetic energy (physical law)
E` = h` � p`
⇢`+ 1
2v2` , Ei ` := h` sat � pi
⇢i+ 1
2v2i
Eg
= hg
� p
g
⇢g
+ 12v
2g
, Ei g := hg sat � pi
⇢i+ 1
2v2i
Interphasic velocity
vi := �v` + (1� �)vg
) � :=p
T
g
pT`+
pT
g
Interphasic density⇢i := ⇢sat! Check entropy inequality within physical relevant region!
Pascal Richter | Modeling of two-phase flow | 12/14
Two-phase flow model | Summary
Model properties
X Source terms and interphase quantities
X Conservation of mass, momentum and energy at the interface
X Equation of state
X Well-posed hyperbolic model
X Entropy-Entropy flux pairConsistent with 2nd law of thermodynamics
� No linear degenerated field for first eigenvalue �1 = vi,only i↵ vi := v`, or vi := v
g
, or vi :=↵`⇢`v`+↵
g
⇢g
v
g
↵`⇢`+↵g
⇢g
.
Pascal Richter | Modeling of two-phase flow | 13/14
Two-phase flow model | Summary
Model properties
X Source terms and interphase quantities
X Conservation of mass, momentum and energy at the interface
X Equation of state
X Well-posed hyperbolic model
X Entropy-Entropy flux pairConsistent with 2nd law of thermodynamics
� No linear degenerated field for first eigenvalue �1 = vi,only i↵ vi := v`, or vi := v
g
, or vi :=↵`⇢`v`+↵
g
⇢g
v
g
↵`⇢`+↵g
⇢g
.
Pascal Richter | Modeling of two-phase flow | 13/14
Two-phase flow model | Next steps
Numerical schemes for quasilinear system
1 Path-conservative scheme, Castro et al. [10]• transform into homogeneous system in non-conservative form
• path connects two states uL and uR at its left xL and right xR limits
across a discontinuity
2 Relaxation, Baudin, Berthon, Coquel, Masson, and Tran [11]• transform system, such that it is linearly degenerated (?)
• extend system with relaxed pressure and temperature equations
• system linearly degenerate ! easy to find Riemann solution.
Pascal Richter | Modeling of two-phase flow | 14/14
Bibliography I
H.B. Stewart and B. Wendro↵. “Two-phase flow: models and methods”. In: Journal of
Computational Physics 56.3 (1984), pp. 363–409.
D.A. Drew and S.L. Passman. Theory of Multicomponent Fluids. Applied mathematical
sciences. Springer, 1998.
M.R. Baer and J.W. Nunziato. “A two-phase mixture theory for the
deflagration-to-detonation transition (DDT) in reactive granular materials”. In:
International Journal of Multiphase Flow 12.6 (1986), pp. 861–889.
Idaho National Laboratory. RELAP5-3D c�Code Manual Volume I: Code Structure,
System Models and Solution Methods. Tech. rep. INL Idaho National Laboratory (INL),
INEEL Report EXT-98-00834, Revision 4.0, 2012.
F. Coquel et al. “Two properties of two-velocity two-pressure models for two-phase flows”.
In: (2013).
T. Gallouet, J.-M. Herard, and N. Seguin. “Numerical modeling of two-phase flows using
the two-fluid two-pressure approach”. In: Mathematical Models and Methods in Applied
Sciences 14.05 (2004), pp. 663–700.
Pascal Richter | Modeling of two-phase flow | 13/14
Bibliography II
E. Godlewski and P.-A. Raviart. Numerical approximation of hyperbolic systems of
conservation laws. Vol. 118. Springer, 1996.
E. Tadmor. “The numerical viscosity of entropy stable schemes for systems of conservation
laws. I”. In: Mathematics of Computation 49.179 (1987), pp. 91–103.
A Zein. “Numerical methods for multiphase mixture conservation laws with phase
transition”. PhD thesis. University of Magdeburg, 2010.
M.J. Castro et al. “Entropy Conservative and Entropy Stable Schemes for Nonconservative
Hyperbolic Systems”. In: SIAM Journal on Numerical Analysis 51.3 (2013), pp. 1371–1391.
Michael Baudin et al. “A relaxation method for two-phase flow models with hydrodynamic
closure law”. In: Numerische Mathematik 99.3 (2005), pp. 411–440.
Pascal Richter | Modeling of two-phase flow | 14/14
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