Modeling of two-phase ﬂow - GdR...

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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

Deaerator

Pascal Richter | Modeling of two-phase flow | 2/14

Direct steam generation | Fresnel collector

Solar collector

Steamturbine

Generator

Coolingtower

Conden-sator

Deaerator

FresnelSolar collector

Absorber tube(two-phase flow)

Secondary reflector •

Direct steam generation | Two-phase flow

Solar collector

Steamturbine

Generator

Coolingtower

Conden-sator

Deaerator

• 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

>>;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

Model development

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

• Source terms

• Quantities

Model development

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

• Quantities

Model development

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

• Quantities

Two-phase flow modelThe system is in non-conservative form

f(u) + B(u) @x

u = s(u),

BBBBBBBB@

↵`⇢`↵`⇢`v`↵`⇢`E`

CCCCCCCCA

, f(u) =

BBBBBBBB@

0↵`⇢`v`

↵`(⇢`v2` + p`)

↵`(⇢`E` + p`)v`↵g

CCCCCCCCA

B(u) =

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

CCCCCCCCA

, s(u) =

BBBBBBBB@

�i/⇢i��i

�Fi � vi�i�v`Fi + Qi ` � Ei `�i

�iFi + vi�i

Fi + Qi g + Ei g�i

CCCCCCCCA

Two-phase flow modelThe system is in non-conservative form

f(u) + B(u) @x

u = s(u),

BBBBBBBB@

↵`⇢`↵`⇢`v`↵`⇢`E`

CCCCCCCCA

, f(u) =

BBBBBBBB@

0↵`⇢`v`

↵`(⇢`v2` + p`)

↵`(⇢`E` + p`)v`↵g

CCCCCCCCA

B(u) =

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

CCCCCCCCA

, s(u) =

BBBBBBBB@

�i/⇢i��i

�iFi + vi�i

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) =

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

CCCCCCCCA

, s(u) =

BBBBBBBB@

�i/⇢i��i

�iFi + vi�i

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) =

BBBBBBBB@

�i/⇢i��i

�iFi + vi�i

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) =

BBBBBBBB@

0↵`⇢`v`

↵`(⇢`v2` + p`)

↵`(⇢`E` + p`)v`↵g

CCCCCCCCA

Describe p by two state parameter

p` = p(⇢`, u`), pg

= p(⇢g

with density ⇢ and specific inner energy u.

Two-phase flow model | 4 Hyperbolicity

Rewrite system in terms of primitive quantities

w +M(w)@x

w = s(w)

Eigenvalues

� =�vi, v`, v` + w`, v` � w`, v

� wg

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

Gallouet, Herard, and Seguin [6] choose vi as convex combinationbetween v` and v

:vi := �v` + (1� �)v

with � 2 [0, 1]

Non-resonance condition will always be fulfilled! M is diagonalisable ! quasilinear system is hyperbolic.

w +M(w)@x

w = s(w)

Eigenvalues

� =�vi, v`, v` + w`, v` � w`, v

� wg

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 := �v` + (1� �)v

with � 2 [0, 1]

w +M(w)@x

w = s(w)

Eigenvalues

� =�vi, v`, v` + w`, v` � w`, v

� wg

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 := �v` + (1� �)v

with � 2 [0, 1]

Two-phase flow model | 5 Entropy-entropy flux pairClosed quasilinear form: @

u+ A(u) · @x

u = s(u).

Find entropy function ⌘(u) and entropy flux (u), such that

⌘(u) + @x

(u)! 0

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:

u + A(u) · @x

u = s(u)

, @u⌘(u)T @

u + @u⌘(u)T A(u) · @

u = @u⌘(u)T s(u)

⌘(u) + @x

(u) = @u⌘(u)T s(u)

u+ A(u) · @x

u = s(u).

⌘(u) + @x

(u)! 0

@u (u)T != @u⌘(u)

T A(u)

u + A(u) · @x

u = s(u)

, @u⌘(u)T @

u + @u⌘(u)T A(u) · @

u = @u⌘(u)T s(u)

⌘(u) + @x

(u) = @u⌘(u)T s(u)

u+ A(u) · @x

u = s(u).

⌘(u) + @x

(u)! 0

@u (u)T != @u⌘(u)

T A(u)

u + A(u) · @x

u = s(u)

, @u⌘(u)T @

u + @u⌘(u)T A(u) · @

u = @u⌘(u)T s(u)

⌘(u) + @x

(u) = @u⌘(u)T s(u)

Two-phase flow model | 5 Entropy-entropy flux pair (1)Choose mixture of physical entropy s` and s

:⌘(u) = �(↵`⇢`s` + ↵

) and (u) = �(↵`⇢`s`v` + ↵g

:⌘(u) = �(↵`⇢`s` + ↵

) and (u) = �(↵`⇢`s`v` + ↵g

1 Convexity of ⌘(u):

Follow proof of Coquel, Herard, Saleh, and Seguin [5],using results of Godlewski and Raviart [7].

:⌘(u) = �(↵`⇢`s` + ↵

) and (u) = �(↵`⇢`s`v` + ↵g

2 Compatibility condition @u (u)T!= @u⌘(u)T · A(u):

BBBBBBBBBBBBBBBBBBBBB@

T`� p

v` ·p`⇢`

+u`�12v

T`� s`

� v`T`

� 12v

CCCCCCCCCCCCCCCCCCCCCA

T`� p

� (p`�pi)(v`�vi)T`

�pi)(vg�vi)T

v` ·p`⇢`

+u`�12v

T`� s`

� v`T`

� 12v

:⌘(u) = �(↵`⇢`s` + ↵

) and (u) = �(↵`⇢`s`v` + ↵g

T`� p

v` ·p`⇢`

+u`�12v

T`� s`

� v`T`

� 12v

T`� p

� (p`�pi)(v`�vi)T`

�pi)(vg�vi)T

v` ·p`⇢`

+u`�12v

T`� s`

� v`T`

� 12v

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

with � 2 [0, 1]Interphasic pressure Gallouet, Herard, and Seguin [6]

pi := �p` + (1� �)pg

with � 2 [0, 1]

:⌘(u) = �(↵`⇢`s` + ↵

) and (u) = �(↵`⇢`s`v` + ↵g

T`� p

v` ·p`⇢`

+u`�12v

T`� s`

� v`T`

� 12v

T`� p

� (p`�pi)(v`�vi)T`

�pi)(vg�vi)T

v` ·p`⇢`

+u`�12v

T`� s`

� v`T`

� 12v

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

with � 2 [0, 1]Interphasic pressure

pi := �p` + (1� �)pg

) � = (1��)Tg

�T`+(1��)Tg

:⌘(u) = �(↵`⇢`s` + ↵

) and (u) = �(↵`⇢`s`v` + ↵g

3 Entropy inequality of entropy production: @u⌘(u)T s(u)! 0:

�Qi `

Ei ` � E` + v

2` � v`vi +

⇢i� p`

⇢`+ s`T`

T`· �i

�Qi g

�Ei g � E

⇢i� p

· �i

:⌘(u) = �(↵`⇢`s` + ↵

) and (u) = �(↵`⇢`s`v` + ↵g

�Qi `

Ei ` � E` + v

2` � v`vi +

⇢i� p`

⇢`+ s`T`

T`· �i

�Qi g

�Ei g � E

⇢i� p

· �i

Spec. total energy = spec. enthalpy � spec. pressure + kinetic energy (physical law)

E` = h` � p`

⇢`+ 1

2v2` , Ei ` := h` sat � pi

⇢i+ 1

, Ei g := hg sat � pi

⇢i+ 1

Interphasic velocityvi := �v` + (1� �)v

with � 2 [0, 1]

Interphasic density⇢i := ⇢sat

:⌘(u) = �(↵`⇢`s` + ↵

) and (u) = �(↵`⇢`s`v` + ↵g

�Qi `

h` sat � h` + 12(v` � vi)2 +

p`�pi

⇢i+ s`T`

T`· �i

�Qi g

g sat � h

+ 12(v

� vi)2 +p

⇢i+ s

· �i

E` = h` � p`

⇢`+ 1

2v2` , Ei ` := h` sat � pi

⇢i+ 1

with � 2 [0, 1]

:⌘(u) = �(↵`⇢`s` + ↵

) and (u) = �(↵`⇢`s`v` + ↵g

�Qi `

h` sat � h` + 12(v` � vi)2 +

p`�pi

⇢i+ s`T`

T`· �i

�Qi g

g sat � h

+ 12(v

� vi)2 +p

⇢i+ s

· �i

E` = h` � p`

⇢`+ 1

2v2` , Ei ` := h` sat � pi

⇢i+ 1

with � 2 [0, 1]

:⌘(u) = �(↵`⇢`s` + ↵

) and (u) = �(↵`⇢`s`v` + ↵g

�Qi `

h` sat � h` + 12(v` � vi)2 +

p`�pi

⇢i+ s`T`

T`· �i

�Qi g

g sat � h

+ 12(v

� vi)2 +p

⇢i+ s

· �i

E` = h` � p`

⇢`+ 1

2v2` , Ei ` := h` sat � pi

⇢i+ 1

Interphasic velocity

vi := �v` + (1� �)vg

) � :=p

:⌘(u) = �(↵`⇢`s` + ↵

) and (u) = �(↵`⇢`s`v` + ↵g

�Qi `

h` sat � h` + p`�pi

⇢i+ s`T`

T`· �i

�Qi g

g sat � h

⇢i+ s

· �i

E` = h` � p`

⇢`+ 1

2v2` , Ei ` := h` sat � pi

⇢i+ 1

vi := �v` + (1� �)vg

) � :=p

:⌘(u) = �(↵`⇢`s` + ↵

) and (u) = �(↵`⇢`s`v` + ↵g

�Qi `

⇢i+ s`T`

T`· �i

�Qi g

g sat � h

⇢i+ s

· �i

E` = h` � p`

⇢`+ 1

2v2` , Ei ` := h` sat � pi

⇢i+ 1

vi := �v` + (1� �)vg

) � :=p

:⌘(u) = �(↵`⇢`s` + ↵

) and (u) = �(↵`⇢`s`v` + ↵g

�Qi `

⇢i+ s`T`

T`· �i

�Qi g

g sat � h

⇢i+ s

· �i

E` = h` � p`

⇢`+ 1

2v2` , Ei ` := h` sat � pi

⇢i+ 1

vi := �v` + (1� �)vg

) � :=p

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

, or vi :=↵`⇢`v`+↵

↵`⇢`+↵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

, or vi :=↵`⇢`v`+↵

↵`⇢`+↵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.

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.

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.

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