Modeling of two-phase ﬂow - GdR...

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 •


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


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!



Model development

• Dimension reduction

• Averaging of the Navier-Stokes equations

• Source terms

• Quantities

DensityVelocityEnergyPressure

9>>=

>>;separate, mixture or equal



Model development

• Dimension reduction! Quasi-1D flow in a tube, Stewart and Wendro↵ [1]

• Averaging of the Navier-Stokes equations

• Source terms

• Quantities


9>>=




Model development


• Averaging of the Navier-Stokes equations! Introduction of void fractions ↵ Drew and Passman [2]! Baer-Nunziato type [3]

• Source terms

• Quantities


9>>=




Model development



• Source terms: Replace viscous and di↵usive terms, RELAP [4]! Use empirical laws dependent on local flow pattern

• Quantities


9>>=




Model development



• Source terms: Replace viscous and di↵usive terms, RELAP [4]! Use empirical laws dependent on local flow pattern

• Quantities


9>>=



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`↵`⇢È`

↵g

⇢g

↵g

⇢g

vg

↵g

⇢g

Eg

1

CCCCCCCCA

, f(u) =

0

BBBBBBBB@

0↵`⇢`v`

↵`(⇢`v2` + p`)

↵`(⇢È` + 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


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`↵`⇢È`

↵g

⇢g

↵g

⇢g

vg

↵g

⇢g

Eg

1

CCCCCCCCA

, f(u) =

0

BBBBBBBB@

0↵`⇢`v`

↵`(⇢`v2` + p`)

↵`(⇢È` + 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


�iFi + vi�i

vg


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


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


�iFi + vi�i

vg


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


Two-phase flow model | 2 Conservation at interface

s(u) =

0

BBBBBBBB@

�i/⇢i��i


�iFi + vi�i

vg


1

CCCCCCCCA

Heat conduction limited model

�i =1

Ei ` � Ei g

⇣Fi(vg � v`) + Qi ` + Qi g

⌘

RELAP [4].


Two-phase flow model | 3 Equation of state

f(u) =

0

BBBBBBBB@

0↵`⇢`v`

↵`(⇢`v2` + p`)

↵`(⇢È` + 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.


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.




@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] :


± wg

.


g

:vi := �v` + (1� �)v

g

with � 2 [0, 1]





@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] :


± wg

.


g

:vi := �v` + (1� �)v

g

with � 2 [0, 1]



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



t

u+ A(u) · @x

u = s(u).


@t

⌘(u) + @x

(u)! 0

with



@u (u)T != @u⌘(u)

T A(u)


@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



t

u+ A(u) · @x

u = s(u).


@t

⌘(u) + @x

(u)! 0

with



@u (u)T != @u⌘(u)

T A(u)


@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


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

)



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].



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


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




g

:⌘(u) = �(↵`⇢`s` + ↵

g

⇢g

sg

) and (u) = �(↵`⇢`s`v` + ↵g

⇢g

sg

vg

)


0


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


!=

0


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


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]



g

:⌘(u) = �(↵`⇢`s` + ↵

g

⇢g

sg

) and (u) = �(↵`⇢`s`v` + ↵g

⇢g

sg

vg

)


0


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


!=

0


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


Interphasic velocity [Hyperbolicity]vi = �v` + (1� �)v

g

with � 2 [0, 1]Interphasic pressure

pi := �p` + (1� �)pg

) � = (1��)Tg

�T`+(1��)Tg



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



g

:⌘(u) = �(↵`⇢`s` + ↵

g

⇢g

sg

) and (u) = �(↵`⇢`s`v` + ↵g

⇢g

sg

vg

)


�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



g

:⌘(u) = �(↵`⇢`s` + ↵

g

⇢g

sg

) and (u) = �(↵`⇢`s`v` + ↵g

⇢g

sg

vg

)


�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


E` = h` � p`

⇢`+ 1

2v2` , Ei ` := h` sat � pi

⇢i+ 1

2v2i

Eg

= hg

� p

g

⇢g

+ 12v

2g


⇢i+ 1

2v2i


g

with � 2 [0, 1]




g

:⌘(u) = �(↵`⇢`s` + ↵

g

⇢g

sg

) and (u) = �(↵`⇢`s`v` + ↵g

⇢g

sg

vg

)


�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


E` = h` � p`

⇢`+ 1

2v2` , Ei ` := h` sat � pi

⇢i+ 1

2v2i

Eg

= hg

� p

g

⇢g

+ 12v

2g


⇢i+ 1

2v2i


g

with � 2 [0, 1]




g

:⌘(u) = �(↵`⇢`s` + ↵

g

⇢g

sg

) and (u) = �(↵`⇢`s`v` + ↵g

⇢g

sg

vg

)


�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


E` = h` � p`

⇢`+ 1

2v2` , Ei ` := h` sat � pi

⇢i+ 1

2v2i

Eg

= hg

� p

g

⇢g

+ 12v

2g


⇢i+ 1

2v2i

Interphasic velocity

vi := �v` + (1� �)vg

) � :=p

T

g

pT`+

pT

g




g

:⌘(u) = �(↵`⇢`s` + ↵

g

⇢g

sg

) and (u) = �(↵`⇢`s`v` + ↵g

⇢g

sg

vg

)


�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


E` = h` � p`

⇢`+ 1

2v2` , Ei ` := h` sat � pi

⇢i+ 1

2v2i

Eg

= hg

� p

g

⇢g

+ 12v

2g


⇢i+ 1

2v2i


vi := �v` + (1� �)vg

) � :=p

T

g

pT`+

pT

g




g

:⌘(u) = �(↵`⇢`s` + ↵

g

⇢g

sg

) and (u) = �(↵`⇢`s`v` + ↵g

⇢g

sg

vg

)


�Qi `

T`+


⇢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


E` = h` � p`

⇢`+ 1

2v2` , Ei ` := h` sat � pi

⇢i+ 1

2v2i

Eg

= hg

� p

g

⇢g

+ 12v

2g


⇢i+ 1

2v2i


vi := �v` + (1� �)vg

) � :=p

T

g

pT`+

pT

g




g

:⌘(u) = �(↵`⇢`s` + ↵

g

⇢g

sg

) and (u) = �(↵`⇢`s`v` + ↵g

⇢g

sg

vg

)


�Qi `

T`+


⇢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


E` = h` � p`

⇢`+ 1

2v2` , Ei ` := h` sat � pi

⇢i+ 1

2v2i

Eg

= hg

� p

g

⇢g

+ 12v

2g


⇢i+ 1

2v2i


vi := �v` + (1� �)vg

) � :=p

T

g

pT`+

pT

g

Interphasic density⇢i := ⇢sat! Check entropy inequality within physical relevant region!


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

.


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

.


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.


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.

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