Modeling of two-phase flow > Direct steam generation in solar thermal power plants

Modeling of two-phase flow > Direct steam generation in solar thermal power plants Pascal Richter RWTH Aachen University EGRIN Pirirac-sur-Mer June 3, 2015 Novatec Solar GmbH

Direct steam generation Solar thermal power plant Steam turbine Condensator Deaerator Generator Cooling tower Solar collector Pump Pump Pascal Richter Modeling of two-phase flow 2/14

Direct steam generation Fresnel collector Secondary reflector Steam turbine Absorber tube (two-phase flow) Sun light Generator Cooling tower Fresnel Solar collector Condensator Deaerator Solar collector Pump Pump Pascal Richter Modeling of two-phase flow 2/14

Direct steam generation Two-phase flow Steam turbine Generator Cooling tower Condensator Deaerator Solar collector Pump 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 with Navier-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 Density Velocity Energy Pressure 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 Density Velocity Energy Pressure 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 Density Velocity Energy Pressure 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 9 >= Quantities Density Velocity Energy Pressure separate, mixture or equal >; Pascal Richter Modeling of two-phase flow 3/14

Two-phase flow model The system is in non-conservative form @ t u + @ x f(u)+b(u) @ x u = s(u), 0 1 0 1 g 0 ` ` ` `v` ` `v` `( `v 2` + p`) u = ` `E`, f(u) = `( `E` + p`)v` B g g C B g g v g C @ g g v g A @ g ( g vg 2 + p g ) A g g E g g ( g E g + p g )v g 0 1 0 1 v i 0 0 0 0 0 0 i/ i 0 0 0 0 0 0 0 i p i 0 0 0 0 0 0 F i v i i B(u) = p i v i 0 0 0 0 0 0, s(u) = v`f i + Q i ` E i ` i B 0 0 0 0 0 0 0 C B i C @ p i 0 0 0 0 0 0A @ F i + v i i A p i v i 0 0 0 0 0 0 v g F i + Q i g + E i g i Pascal Richter Modeling of two-phase flow 4/14

Two-phase flow model The system is in non-conservative form @ t u + @ x f(u)+b(u) @ x u = s(u), 0 1 0 1 g 0 Model properties ` ` ` `v` 1 Source terms ` `v` and interphase `( `v 2` + p`) quantities u = ` `E`, f(u) = `( `E` + p`)v` B g g C B g g v g 2 Conservation of mass, momentum C and energy @ at g the g v g interface A @ g ( g vg 2 + p g ) A g g E g g ( g E g + p g )v 3 Equation of state g 0 1 0 1! How vto i describe 0 0 pressure 0 0 0p 0? i/ i 0 0 0 0 0 0 0 4 Well-posedness of the model i! Hyperbolicity p i 0 0 0 0 0 0 F i v i i B(u) = p i v i 0 0 0 0 0 0, s(u) = v`f i + Q i ` E i ` i 5 Entropy inequality B 0 0 0 0 0 0 0! Consistent with 2nd C B i C @ p i 0 0 0 0 0 0A @ F i + v i i A law of thermodynamics p i v i 0 0 0 0 0 0 v g F i + Q i g + E i g i Pascal Richter Modeling of two-phase flow 4/14

Two-phase flow model 1 Source terms 0 1 0 1 v i 0 0 0 0 0 0 i/ i 0 0 0 0 0 0 0 i p i 0 0 0 0 0 0 F i v i i B(u) = p i v i 0 0 0 0 0 0, s(u) = v`f i + Q i ` E i ` i B 0 0 0 0 0 0 0 C B i C @ p i 0 0 0 0 0 0A @ F i + v i i A p i v i 0 0 0 0 0 0 v g F i + Q i g + E i g i Interphase quantities: i, v i, p i, E i `, E i g, i =??? Flow regimes for friction F i : Models for heat transfer Q i `, Q i g : Convection, Condensation, Nucleate & Film boiling Pascal Richter Modeling of two-phase flow 5/14

Two-phase flow model 2 Conservation at interface 0 1 i/ i i F i v i i s(u) = v`f i + Q i ` E i ` i B i C @ F i + v i i A v g F i + Q i g + E i g i Heat conduction limited model i = E i ` 1 E i g F i (v g v`)+q i ` + Q i g RELAP [4]. Pascal Richter Modeling of two-phase flow 6/14

Two-phase flow model 3 Equation of state 0 1 0 ` `v` `( `v 2` + p`) f(u) = `( `E` + p`)v` B g g v g C @ g ( g vg 2 + p g ) A g ( g E g + p g )v g Describe p by two state parameter p` = p( `, u`), p g = p( g, u g ) 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 T = v i, v`, v` + w`, v` w`, v g, v g + w g, v g w g with speed of sound w. Eigenvectors form a basis of R 7 as soon as the non-resonance condition is fulfilled Coquel, Hérard, Saleh, and Seguin [5] : v i 6= v` ± w` and v i 6= v g ± w g. Gallouët, Hérard, and Seguin [6] choose v i as convex combination between v` and v g : v i := 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 pair Closed quasilinear form: @ t u + A(u) @ x u = s(u). Find entropy function (u) and entropy flux (u), such that @ t (u)+@ x (u) apple! 0 Pascal Richter Modeling of two-phase flow 9/14

Two-phase flow model 5 Entropy-entropy flux pair Closed quasilinear form: @ t u + A(u) @ x u = s(u). Find entropy function (u) and entropy flux (u), such that with @ t (u)+@ x (u)! apple 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: @ 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)! apple 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 s g ) and (u) = ( ` `s`v` + g g s g v g ) 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 s g ) and (u) = ( ` `s`v` + g g s g v g ) 1 Convexity of (u): Follow proof of Coquel, Hérard, 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 s g ) and (u) = ( ` `s`v` + g g s g v g ) 2 Compatibility condition @ u (u) T! = @ u (u) T A(u): 0 p`v` p g v 1 0 g p`v` T g p` 1 +u` v` ` 2 v2` v 2` s`! v` = pg g v g +ug 1 v 2 2 g Tg v g 2 B s @ Tg g C B A @ vg Tg p g v g (p` p i )(v` v i ) T g p` +u` v` ` 1 2 v2` v 2` s` v` v g pg g +ug Tg 1 v 2 2 g v 2 g Tg vg Tg s g + (p g p i )(v g v i ) T g 1 C A 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 s g ) and (u) = ( ` `s`v` + g g s g v g ) 2 Compatibility condition @ u (u) T! = @ u (u) T A(u): 0 p`v` p g v 1 0 g p`v` T g p` 1 +u` v` ` 2 v2` v 2` s`! v` = pg g v g +ug 1 v 2 2 g Tg v g 2 B s @ Tg g C B A @ vg Tg p g v g (p` p i )(v` v i ) T g p` +u` v` ` 1 2 v2` v 2` s` v` v g pg g +ug Tg 1 v 2 2 g Interphasic velocity [Hyperbolicity] v i = v` +(1 )v g with 2 [0, 1] Interphasic pressure Gallouët, Hérard, and Seguin [6] p i := p` +(1 )p g with 2 [0, 1] v 2 g Tg vg Tg s g + (p g p i )(v g v i ) T g Pascal Richter Modeling of two-phase flow 11/14 1 C A

Two-phase flow model 5 Entropy-entropy flux pair (2) Choose mixture of physical entropy s` and s g : (u) = ( ` `s` + g g s g ) and (u) = ( ` `s`v` + g g s g v g ) 2 Compatibility condition @ u (u) T! = @ u (u) T A(u): 0 p`v` p g v 1 0 g p`v` T g p` 1 +u` v` ` 2 v2` v 2` s`! v` = pg g v g +ug 1 v 2 2 g Tg v g 2 B s @ Tg g C B A @ vg Tg p g v g (p` p i )(v` v i ) T g p` +u` v` ` 1 2 v2` v 2` s` v` v g pg g +ug Tg 1 v 2 2 g Interphasic velocity [Hyperbolicity] v i = v` +(1 )v g with 2 [0, 1] Interphasic pressure p i := p` +(1 )p g ) = (1 )T g +(1 )T g v 2 g Tg vg Tg s g + (p g p i )(v g v i ) T g Pascal Richter Modeling of two-phase flow 11/14 1 C A

Two-phase flow model 5 Entropy-entropy flux pair (3) Choose mixture of physical entropy s` and s g : (u) = ( ` `s` + g g s g ) and (u) = ( ` `s`v` + g g s g v g ) 3 Entropy inequality of entropy production: @ u (u) T s(u)! apple 0: Q i ` + E i ` E` + v 2` v`v i + p` i Q i g E i g 2 E g + vg T g p` + s` ` i v g v i + p g p g + s i g T g! g i apple 0 T g 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 s g ) and (u) = ( ` `s`v` + g g s g v g ) 3 Entropy inequality of entropy production: @ u (u) T s(u)! apple 0: Q i ` + h` sat h` + 1 2 (v` v i ) 2 + p` p i i + s` i 1 Q i g h g sat h g + 2 (v g v i ) 2 + p g p i + s g T g! i i apple 0 T g T g Spec. total energy =spec.enthalpy spec. pressure + kinetic energy (physical law) p` E` = h` + 1 ` 2 v 2`, E p i ` := i h` sat i + 1 2 v i 2 p E g = h g g g + 1 2 v g 2 p, E i g := h i g sat i + 1 2 v i 2 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 s g ) and (u) = ( ` `s`v` + g g s g v g ) 3 Entropy inequality of entropy production: @ u (u) T s(u)! apple 0: Q i ` + h` sat h` + 1 2 (v` v i ) 2 + p` p i i + s` i 1 Q i g h g sat h g + 2 (v g v i ) 2 + p g p i + s g T g! i i apple 0 T g T g Spec. total energy =spec.enthalpy spec. pressure + kinetic energy (physical law) p` E` = h` + 1 ` 2 v 2`, E p i ` := i h` sat i + 1 2 v i 2 p E g = h g g g + 1 2 v g 2 p, E i g := h i g sat i + 1 2 v i 2 Interphasic velocity v i := v` +(1 )v g with 2 [0, 1] 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 s g ) and (u) = ( ` `s`v` + g g s g v g ) 3 Entropy inequality of entropy production: @ u (u) T s(u)! apple 0: Q i ` + h` sat h` + 1 2 (v` v i ) 2 + p` p i i + s` i 1 Q i g h g sat h g + 2 (v g v i ) 2 + p g p i + s g T g! i i apple 0 T g T g Spec. total energy =spec.enthalpy spec. pressure + kinetic energy (physical law) p` E` = h` + 1 ` 2 v 2`, E p i ` := i h` sat i + 1 2 v i 2 p E g = h g g g + 1 2 v g 2 p, E i g := h i g sat i + 1 2 v i 2 Interphasic velocity v i := v` +(1 )v g ) := p T g p p + T g 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 s g ) and (u) = ( ` `s`v` + g g s g v g ) 3 Entropy inequality of entropy production: @ u (u) T s(u)! apple 0: Q i ` + h` sat h` + p` p i i p g p i i + s` i Q i g h g sat h g + + s g T g i T g T g! apple 0 Spec. total energy =spec.enthalpy spec. pressure + kinetic energy (physical law) p` E` = h` + 1 ` 2 v 2`, E p i ` := i h` sat i + 1 2 v i 2 p E g = h g g g + 1 2 v g 2 p, E i g := h i g sat i + 1 2 v i 2 Interphasic velocity v i := v` +(1 )v g ) := p T g p p + T g 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 s g ) and (u) = ( ` `s`v` + g g s g v g ) 3 Entropy inequality of entropy production: @ u (u) T s(u)! apple 0: Q i ` + h` sat h` + p` p i i p g p i i + s` i Q i g h g sat h g + + s g T g i T g T g! apple 0 Spec. total energy =spec.enthalpy spec. pressure + kinetic energy (physical law) p` E` = h` + 1 ` 2 v 2`, E p i ` := i h` sat i + 1 2 v i 2 p E g = h g g g + 1 2 v g 2 p, E i g := h i g sat i + 1 2 v i 2 Interphasic velocity v i := v` +(1 )v g ) := p T g p p + T 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 s g ) and (u) = ( ` `s`v` + g g s g v g ) 3 Entropy inequality of entropy production: @ u (u) T s(u)! apple 0: Q i ` + h` sat h` + p` p i i p g p i i + s` i Q i g h g sat h g + + s g T g i T g T g! apple 0 Spec. total energy =spec.enthalpy spec. pressure + kinetic energy (physical law) p` E` = h` + 1 ` 2 v 2`, E p i ` := i h` sat i + 1 2 v i 2 p E g = h g g g + 1 2 v g 2 p, E i g := h i g sat i + 1 2 v i 2 Interphasic velocity v i := v` +(1 )v g ) := p T g p p + T 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 pair Consistent with 2nd law of thermodynamics 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 u L and u R at its left x L and right x R 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. Gallouët, J.-M. Hérard, 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. AZein. 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. Michaël 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