Modeling of two-phase flow > Direct steam generation in solar - PowerPoint PPT Presentation

Novatec Solar GmbH 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

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

Direct steam generation | Fresnel collector Steam Secondary reflector turbine • • t h Generator g i Absorber tube l n u (two-phase flow) S Cooling tower • Conden- sator Fresnel Deaerator Solar collector Pump Pump Solar collector Pascal Richter | Modeling of two-phase flow | 2/14

Direct steam generation | Two-phase flow Steam turbine Generator Cooling tower Conden- sator Deaerator Pump Pump Solar collector • 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 9 Density > > Velocity = • Quantities separate , mixture or equal Energy > > Pressure ; 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 ff [1] • Averaging of the Navier-Stokes equations • Source terms 9 Density > > Velocity = • Quantities separate , mixture or equal Energy > > Pressure ; 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 ff [1] • Averaging of the Navier-Stokes equations ! Introduction of void fractions α Drew and Passman [2] ! Baer-Nunziato type [3] • Source terms 9 Density > > Velocity = • Quantities separate , mixture or equal Energy > > Pressure ; 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 ff [1] • Averaging of the Navier-Stokes equations ! Introduction of void fractions α Drew and Passman [2] ! Baer-Nunziato type [3] • Source terms: Replace viscous and di ff usive terms, RELAP [4] ! Use empirical laws dependent on local flow pattern 9 Density > > Velocity = • Quantities separate , mixture or equal Energy > > Pressure ; 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 ff [1] • Averaging of the Navier-Stokes equations ! Introduction of void fractions α Drew and Passman [2] ! Baer-Nunziato type [3] • Source terms: Replace viscous and di ff usive terms, RELAP [4] ! Use empirical laws dependent on local flow pattern 9 Density > > Velocity = • Quantities separate , mixture or equal Energy > > Pressure ; 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 α g 0 α ` ρ ` α ` ρ ` v ` B C B C B C B α ` ( ρ ` v 2 C α ` ρ ` v ` ` + p ` ) B C B C B C B C u = α ` ρ ` E ` , f ( u ) = α ` ( ρ ` E ` + p ` ) v ` B C B C B C B C α g ρ g α g ρ g v g B C B C α g ( ρ g v 2 B C B C α g ρ g v g g + p g ) @ A @ A α g ρ g E g α g ( ρ g E g + p g ) v g 0 1 0 1 v i 0 0 0 0 0 0 Γ i / ρ i 0 0 0 0 0 0 0 � Γ i B C B C B C B C p i 0 0 0 0 0 0 � F i � v i Γ i B C B C B C B C B ( u ) = p i v i 0 0 0 0 0 0 , s ( u ) = � v ` F i + Q i ` � E i ` Γ i B C B C B C B C 0 0 0 0 0 0 0 Γ i B C B C B C B C � p i 0 0 0 0 0 0 F i + v i Γ i @ A @ A � p i v i 0 0 0 0 0 0 v g F i + Q i g + E i g Γ 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 α g 0 Model properties α ` ρ ` α ` ρ ` v ` B C B C B C B α ` ( ρ ` v 2 C 1 Source terms and interphase α ` ρ ` v ` ` + p ` ) B C B C B C B C quantities u = α ` ρ ` E ` , f ( u ) = α ` ( ρ ` E ` + p ` ) v ` B C B C B C B C α g ρ g α g ρ g v g 2 Conservation of mass, momentum B C B C α g ( ρ g v 2 B C B C α g ρ g v g g + p g ) and energy at the interface @ A @ A α g ρ g E g α g ( ρ g E g + p g ) v g 3 Equation of state ! How to describe pressure p ? 0 1 0 1 v i 0 0 0 0 0 0 Γ i / ρ i 0 0 0 0 0 0 0 4 Well-posedness of the model � Γ i B C B C B C B C p i 0 0 0 0 0 0 � F i � v i Γ i ! Hyperbolicity B C B C B C B C B ( u ) = p i v i 0 0 0 0 0 0 , s ( u ) = � v ` F i + Q i ` � E i ` Γ i 5 Entropy inequality B C B C B C B C 0 0 0 0 0 0 0 Γ i ! Consistent with 2nd B C B C B C B C � p i 0 0 0 0 0 0 F i + v i Γ i law of thermodynamics @ A @ A � p i v i 0 0 0 0 0 0 v g F i + Q i g + E i g Γ 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 Γ i / ρ i 0 0 0 0 0 0 0 � Γ i B C B C B C B C p i 0 0 0 0 0 0 � F i � v i Γ i B C B C B C B C B ( u ) = p i v i 0 0 0 0 0 0 , s ( u ) = � v ` F i + Q i ` � E i ` Γ i B C B C B C B C 0 0 0 0 0 0 0 Γ i B C B C B C B C � p i 0 0 0 0 0 0 F i + v i Γ i @ A @ A � p i v i 0 0 0 0 0 0 v g F i + Q i g + E i g Γ i • Interphase quantities: Γ i , v i , p i , E i ` , E i g , ρ 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 Γ i / ρ i � Γ i B C B C � F i � v i Γ i B C B C s ( u ) = � v ` F i + Q i ` � E i ` Γ i B C B C Γ i B C B C F i + v i Γ i @ A v g F i + Q i g + E i g Γ i Heat conduction limited model 1 ⇣ ⌘ Γ i = F i ( v g � v ` ) + Q i ` + Q i g E i ` � E i g RELAP [4]. Pascal Richter | Modeling of two-phase flow | 6/14

Two-phase flow model | 3 Equation of state 0 1 0 α ` ρ ` v ` B C B α ` ( ρ ` v 2 C ` + p ` ) B C B C f ( u ) = α ` ( ρ ` E ` + p ` ) v ` B C B C α g ρ g v g B C α g ( ρ g v 2 B g + p g ) C @ A α g ( ρ g E g + p g ) v g Describe p by two state parameter p ` = p ( ρ ` , u ` ) , p g = p ( ρ g , u g ) with density ρ 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 , λ = v ` , v ` + w ` , v ` � w ` , v g , v g + w g , v g � w g with speed of sound w . 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 � T � v i , λ = 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, H´ erard, Saleh, and Seguin [5] : v i 6 = v ` ± w ` and v i 6 = v g ± w g . 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 � T � v i , λ = 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, H´ erard, Saleh, and Seguin [5] : v i 6 = v ` ± w ` and v i 6 = v g ± w g . Gallou¨ et, H´ erard, and Seguin [6] choose v i as convex combination between v ` and v g : v i := β v ` + (1 � β ) v g with β 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 η ( u ) and entropy flux ψ ( u ), such that ! ∂ t η ( u ) + ∂ x ψ ( u )  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 η ( u ) and entropy flux ψ ( u ), such that ! ∂ t η ( u ) + ∂ x ψ ( u )  0 with 1 Convex entropy (decreasing behaviour): η 00 ( u ) > 0 2 Compatibility condition of Tadmor [8]: = ∂ u η ( u ) T A ( u ) ! ∂ u ψ ( u ) T Pascal Richter | Modeling of two-phase flow | 9/14