A SHORT INTRODUCTION TO TWO-PHASE FLOWS 1D-time averaged models - PowerPoint PPT Presentation

A SHORT INTRODUCTION TO TWO-PHASE FLOWS 1D-time averaged models Herv´ e Lemonnier DM2S/STMF/LIEFT, CEA/Grenoble, 38054 Grenoble Cedex 9 Ph. +33(0)4 38 78 45 40 herve.lemonnier@cea.fr , herve.lemonnier.sci.free.fr/TPF/TPF.htm ECP, 2011-2012

TIME- AND AREA-AVERAGED (1D-) MODELS • Homogeneous model, • Drift-flux model, • Two-fluid model, – Closure issue, – Some unexpected consequences of some modeling assumptions. 1. Physical consistency of the two-fluid model, 2. Mathematical nature of the PDE’s • Back to composite averaged equations (common assumptions) 1D-time averaged models 1/30

COMPOSITE AVERAGING THE MASS BALANCE • Space and time averaged mass balance: � ∂tAR k 2 < ρ k > 2 + ∂ ∂ d l Γ k � − ∂z AR k 2 < ρ k w k > 2 = Γ k , m k ˙ n k � n kC C i • Mean value definitions: R k 2 < ρ k > 2 � R k 2 ρ k = α k ρ k , R k 2 < ρ k w k > 2 � α k ρ k v k = M k • With these new variables, ∂tAα k ρ k + ∂ ∂ ∂z Aα k ρ k v k = Γ k • No assumptions, simple change of variable. α ( r ) � = α and w k ( r ) � = v k are non-uniform. 1D-time averaged models 2/30

MIXTURE MASS BALANCE • Add the phase mass balances, ∂tA ( α 1 ρ 1 + α 2 ρ 2 ) + ∂ ∂ ∂z A ( α 1 ρ 1 v 1 + α 2 ρ 2 v 2 ) = 0 • Mixture density (definition), ρ = α 1 ρ 1 + α 2 ρ 2 • Mixture velocity defined as to preserves the mass flow rate, ρv = α 1 ρ 1 v 1 + α 2 ρ 2 v 2 • Mixture mass balance, ∂tAρ + ∂ ∂ ∂z Aρv = 0 1D-time averaged models 3/30

MOMENTUM BALANCE • Simplified, 1 pressure (instead of 3), neglect the effect longitudinal diffusion. ∂tA< R k 2 ρ k w k > 2 + ∂ ∂ ∂ ∂z A< R k 2 ρ k w 2 k > 2 + AR k 2 ∂z < p k > 2 − AR k 2 < ρ k g z > 2 � � d l d l m k w k − n k � V k � n z ) n k � V k � n z = − ( ˙ + n k � n kC n k � n kC C i C k • 1D assumption: velocity space correlation C , mean pressure, p k , the so-called flat profile assumption. C � < R k 2 ρ k w 2 k > 2 ∂ ∂p k = 1 , R k 2 ∂z < p k > 2 = α k α k ρ k v 2 ∂z k • Interaction terms, change of variable: � � d l d l − ( ˙ m k w k ) = Γ k v ki , = A< | γ> | 2 = Aγ n k � n kC n k � n kC C i C i � � d l d l n k � V k � n z n k � V k � n z = − Aγτ ki , = − P k τ wk n k � n kC n k � n kC C i C k 1D-time averaged models 4/30

MOMENTUM BALANCE (CT’D) • New notations, in blue, the flat profile assumption main consequence, ∂tAα k ρ k v k + ∂ ∂ ∂p k ∂z Aα k ρ k v 2 k − Aα k ∂z = Γ k v ki − Aγτ ki − P k τ wk · · · • Momentum balance for the mixture, single pressure ∂tA ( α 1 ρ 1 v 1 + α 2 ρ 2 v 2 ) + ∂ ∂ 2 ) − A∂p ∂z A ( α 1 ρ 1 v 2 1 + α 2 ρ 2 v 2 ∂z = − Pτ w • Another form of the inertia term, x � M G M , L = G 2 x 2 + (1 − x ) 2 1 G = M α G ρ G v 2 G + α L ρ L v 2 ρ ′ , ρ ′ = , A . αρ G (1 − α ) ρ L • Give two examples of inconsistency of the flat profile assumption. 1D-time averaged models 5/30

TOTAL ENERGY BALANCE • Energy balance, native form → enthalpy form, � � � � ∂ u k + 1 > 2 + ∂ u k + 1 2v 2 2v 2 ∂tA k < ρ k ∂z A k < ρ k w k > 2 k k + ∂ ∂z A k < n z � ( q k − T k � v k ) > 2 − A k < ρ k g k � v k > 2 � � � u k + 1 d l 2v 2 + n k � ( q k − T k � v k )) = − ( ˙ m k k n k � n kC C i ∪ C k ∂ • In the ∂z terms, u k → h k − p k /ρ k , T k → V k , ∂t term u k → h k − p k /ρ k adds − ∂ ∂ • In the ∂t A k < p k > 2 , use the identity (2), � ∂tA k < p k > 2 = A k < ∂p k ∂ d l p k v i � n ∂t > 2 + n k � n kC C i ∪ C k • Collect the pressure terms in the RHS, 1D-time averaged models 6/30

TOTAL ENERGY BALANCE (CT’D) • Energy balance in enthalpy form, � � � � ∂ h k + 1 > 2 − A k < ∂p k ∂t > 2 + ∂ h k + 1 2v 2 2v 2 ∂tA k < ρ k ∂z A k < ρ k w k > 2 k k + ∂ ∂z A k < n z � ( q k − V k � v k ) > 2 − A k < ρ k g k � v k > 2 � � � h k + 1 d l 2v 2 + n k � ( q k − V k � v k )) = − ( ˙ m k k n k � n kC C i ∪ C k • Neglect the diffusive term, v 2 k ≈ v 2 k , the mean enthalpy preserves the flux, � � � � ∂ h k + 1 ∂p k ∂t + ∂ h k + 1 2 v 2 2 v 2 ∂tAα k ρ k − Aα k ∂z Aα k ρ k v k k k − Aα k ρ k g k v k = Γ k h t ki + Aγq ki + P k q kw 1D-time averaged models 7/30

MIXTURE ENERGY BALANCE • Add the two phase balances, interaction terms sum vanish at the interface, ∂ 2 ) + ∂ ∂tA ( α 1 ρ 1 h t 1 + α 2 ρ 2 h t ∂z A ( α 1 ρ 1 v 1 h t 1 + α 2 ρ 2 v 2 h t 2 ) − A∂p ∂t − A ( α 1 ρ 1 g 1 v 1 + α 2 ρ 2 g 2 v 2 ) = Pq w , k � h k + 1 • where the total enthalpy is h t 2 v 2 k , • Other practical form of the enthalpy flux, h t h t L = M ( xh t V + (1 − x ) h t Aα V ρ V v V V + Aα L ρ L v L L ) � �� � � �� � M V M L 1D-time averaged models 8/30

THE HOMOGENEOUS MODEL AT THERMODYNAMIC EQUILIBRIUM (HEM) • 3 balances for the mixture + 3 assumptions – Mean velocity are equal : w V = w L ≡ α = β – Mean temperatures satisfy the equilibrium condition : T L = T V = T sat ( p ) • Thermodynamique, EOS, ρ L = ρ L sat ( p ) , ρ V = ρ V sat ( p ) , h L = h L sat ( p ) , h V = h V sat ( p ) • HEM void fraction, Q G xρ L α = β = = = α ( x, p ) Q G + Q L xρ L + (1 − x ) ρ V • Balance equations are identical to that of single-phase flow, ∂tAρ + ∂ ∂ ∂z Aρw = 0 , ρ = αρ V + (1 − α ) ρ L ∂tAρw + ∂ ∂ ∂z Aρw 2 + A∂p ∂z = − Pτ W + Aρg z ∂tAρ ( h + 1 ∂ 2 w 2 ) − A∂p ∂t + ∂ ∂z Aρw ( h + 1 2 w 2 ) = Pq W + Aρg z w 1D-time averaged models 9/30

HEM (CT’D) • Other practical form, combine with the mass balance, ∂tAρ + ∂ ∂ ∂z Aρw = 0 ρ∂w ∂t + ρw∂w ∂z + ∂p ∂z = − P Aτ W + ρg z ρ ∂ ∂t ( h + 1 2 w 2 ) − ∂p ∂t + ρw ∂ ∂z ( h + 1 2 w 2 ) = P Aq W + ρg z w • Mechanical energy balance, momentum balance × w , ρ ∂ 1 2 w 2 + ρw ∂ 2 w 2 + w∂p 1 ∂z = − P Awτ W + ρg z w ∂t ∂z • Entropy balance, T d s = d h − d p ρ ρT ∂s ∂t + ρwT ∂s ∂z = P A ( q W + wτ W ) 1D-time averaged models 10/30

HEM (CT’D) • Alternate form with total enthalpy and entropy, ∂tρ + w∂ρ ∂ ∂z + ρ∂w ∂z = − ρw d A A d z ρ ∂ ∂t ( h + 1 2 w 2 ) − ∂p ∂t + ρw ∂ ∂z ( h + 1 2 w 2 ) = P Aq W + ρg z w ρT ∂s ∂t + ρwT ∂s ∂z = P A ( q W + wτ W ) • Very important particular case: stationary flow, adiabatic, no friction nor volume forces, M = Aρw = cst h + 1 2 w 2 = cst s = xs V + (1 − x ) s L = cst • Applications: flashing in long pipes, w/o heating, critical flow (no model !). 1D-time averaged models 11/30

CLOSURES FOR THE HEM • Friction and heat flux, q W , τ W , • Independent variables : x , w , et p , • EOS, v = 1 ρ = xv V + (1 − x ) v L , specific volume [m 3 /kg], v L = v L sat ( p ) , v V = v V sat ( p ) , h L = h L sat ( p ) , h V = h V sat ( p ) v V,p , v L,p , h V,p , h L,p • NB: back again to the thermodynamic consistency issue. 1D-time averaged models 12/30

DRIFT-FLUX MODEL • Different phase mean velocities, ”mechanical non-equilibrium”. • Mass balances for the liquid and vapor, momentum balance for the mixture, w V � = w L , w m mixture velocity. • Additional closure (core modeling, FLICA) w V − w L = f ( x, p, α, flow regime , · · · ) J GL = f ( α, flow regime , · · · ) • Slow transients NOT inertia controlled. • Can also be used in 3D, see for example Delhaye (2008a), Ishii & Hibiki (2006). • Main advantage: only one momentum balance. 1D-time averaged models 13/30

THE TWO-FLUID MODEL • Mechanical and thermal non-equilibriums, – 3 balance equations per phase (6), or – 3 balance equations for the mixture and 3 equations for the dispersed phase. • Closures – Topological relations, < pq > , < p >< q > , the pressure issue, – Interactions at the interface, – Interactions of each phase at the wall. • Consequences of the closure assumptions, – Propagation characteristics, – Critical flow, – Mathematical nature of the PDE’s (hyperbolicity ?). 1D-time averaged models 14/30

� � � � � EXAMPLE: STRATIFIED FLOWS • Isothermal, incompressible, horizontal, µ = 0, σ = 0, ˙ m = 0, 2D. • Mass balance, ∂th 1 ρ 1 + ∂ ∂ ∂z h 1 ρ 1 < u 1 > = 0 ∂th 2 ρ 2 + ∂ ∂ ∂z h 2 ρ 2 < u 2 > = 0 • Momentum balances, jump of momentum at the interface, ∂th 1 ρ 1 < u 1 > + ∂ ∂ 1 > + ∂ ∂h 1 ∂z h 1 ρ 1 < u 2 ∂z h 1 < p 1 > = p i 1 ∂z ∂th 2 ρ 2 < u 2 > + ∂ ∂ 2 > + ∂ ∂h 2 ∂z h 2 ρ 2 < u 2 ∂z h 2 < p 2 > = p i 2 ∂z p i 1 = p i 2 � p i 1D-time averaged models 15/30

� � � � � � CLOSURES • 4 equations, 5 unknown variables h , < u 1 > , < u 2 > , < p 1 > , < p 2 > , • 3 unknown quantities: < u 2 1 > , < u 2 2 > , p i . • Topological relation, < p 1 > = p i + 1 2 ρ 1 gh 1 < p 2 > = p i − 1 2 ρ 2 gh 2 • Cannot be derived from momentum ⊥ . • Spatial correlations, flat profile assumption, or relaxation < u 2 � � k > d k > = 1 d t < u 2 < u 2 k > − < u 2 < u k > 2 = 1 , k > 0 T • Closed system. 1D-time averaged models 16/30