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

Transcription

1 A SHORT INTRODUCTION TO TWO-PHASE FLOWS 1D-time averaged models Hervé Lemonnier DM2S/STMF/LIEFT, CEA/Grenoble, Grenoble Cedex 9 Ph herve.lemonnier@cea.fr, herve.lemonnier.sci.free.fr/tpf/tpf.htm ECP,

2 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

3 COMPOSITE AVERAGING THE MASS BALANCE Space and time averaged mass balance: t AR k2 < ρ k > 2 + z AR k2 < ρ k w k > 2 = Γ k, Mean value definitions: dl Γ k ṁ k C i n k n kc R k2 < ρ k > 2 R k2 ρ k = α k ρ k, R k2 < ρ k w k > 2 α k ρ k v k = M k With these new variables, t Aα kρ k + z Aα kρ k v k = Γ k No assumptions, simple change of variable. αr α and w k r v k are non-uniform. 1D-time averaged models 2/30

4 MIXTURE MASS BALANCE Add the phase mass balances, t Aα 1ρ 1 + α 2 ρ 2 + z Aα 1ρ 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, t Aρ + z Aρv = 0 1D-time averaged models 3/30

5 MOMENTUM BALANCE Simplified, 1 pressure instead of 3, neglect the effect longitudinal diffusion. t A< R k2ρ k w k > 2 + z A< R k2ρ k wk 2 > 2 + AR k2 z < p k > 2 AR k2 < ρ k g z > 2 dl dl = ṁ k w k n k V k n z + n k V k n z C i n k n kc C k n k n kc 1D assumption: velocity space correlation C, mean pressure, p k, the so-called flat profile assumption. C < R k2ρ k w 2 k > 2 α k ρ k v 2 k = 1, R k2 z < p p k k > 2 = α k z Interaction terms, change of variable: dl ṁ k w k = Γ k v ki, C i n k n kc C i n k V k n z dl n k n kc = Aγτ ki, C i dl n k n kc = A< γ> 2 = Aγ C k n k V k n z dl n k n kc = P k τ wk 1D-time averaged models 4/30

6 MOMENTUM BALANCE CT D New notations, in blue, the flat profile assumption main consequence, t Aα kρ k v k + z Aα kρ k v 2 k Aα k p k z = Γ kv ki Aγτ ki P k τ wk Momentum balance for the mixture, single pressure t Aα 1ρ 1 v 1 + α 2 ρ 2 v 2 + z Aα 1ρ 1 v α 2 ρ 2 v 2 2 A p z = P τ w Another form of the inertia term, x M G M, α G ρ G v 2 G + α L ρ L v 2 L = G2 ρ, 1 ρ = x2 αρ G + 1 x2 1 αρ L, G = M A. Give two examples of inconsistency of the flat profile assumption. 1D-time averaged models 5/30

7 TOTAL ENERGY BALANCE Energy balance, native form enthalpy form, u k v2 k t A k < ρ k > 2 + z A k < ρ k w k u k v2 k > 2 + z A k < n z q k T k v k > 2 A k < ρ k g k v k > 2 = ṁ k u k + 1 dl C i C k 2 v2 k + n k q k T k v k n k n kc In the z terms, u k h k p k /ρ k, T k V k, In the t term u k h k p k /ρ k adds t A k < p k > 2, use the identity 2, t A k < p k > 2 = A k < p k t > dl 2 + p k v i n C i C k n k n kc Collect the pressure terms in the RHS, 1D-time averaged models 6/30

8 TOTAL ENERGY BALANCE CT D Energy balance in enthalpy form, t A k < ρ k h k v2 k > 2 A k < p k t > 2 + z A k < ρ k w k h k v2 k > 2 + z A k < n z q k V k v k > 2 A k < ρ k g k v k > 2 = ṁ k h k + 1 dl C i C k 2 v2 k + n k q k V k v k n k n kc Neglect the diffusive term, vk 2 v2 k, the mean enthalpy preserves the flux, t Aα kρ k h k + 1 p k 2 v2 k Aα k t + z Aα kρ k v k h k v2 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

9 MIXTURE ENERGY BALANCE Add the two phase balances, interaction terms sum vanish at the interface, t Aα 1ρ 1 h t 1 + α 2 ρ 2 h t 2 + z Aα 1ρ 1 v 1 h t 1 + α 2 ρ 2 v 2 h t 2 A p t Aα 1ρ 1 g 1 v 1 + α 2 ρ 2 g 2 v 2 = P q w, where the total enthalpy is h t k h k v2 k, Other practical form of the enthalpy flux, Aα V ρ V v }{{ V h t V + Aα } L ρ L v L h t L = Mxh t V + 1 xh t }{{} L M V M L 1D-time averaged models 8/30

10 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 Lsat p, ρ V = ρ V sat p, h L = h Lsat p, h V = h V sat p HEM void fraction, α = β = Q G Q G + Q L = xρ L xρ L + 1 xρ V = αx, p Balance equations are identical to that of single-phase flow, t Aρ + Aρw = 0, z ρ = αρ V + 1 αρ L t Aρw + z Aρw2 + A p z = P τ W + Aρg z t Aρh w2 A p t + z Aρwh w2 = P q W + Aρg z w 1D-time averaged models 9/30

11 HEM CT D Other practical form, combine with the mass balance, ρ w t t Aρ + z Aρw = 0 + ρw w z + p z = P A τ W + ρg z ρ t h w2 p t + ρw z h w2 = P A q W + ρg z w Mechanical energy balance, momentum balance w, ρ t 1 2 w2 + ρw z Entropy balance, T ds = dh ρ ρt s t 1 2 w2 + w p z = P A wτ W + ρg z w + ρwt s z = P A q W + wτ W 1D-time averaged models 10/30

12 HEM CT D Alternate form with total enthalpy and entropy, t ρ + w ρ z + ρw z = ρw A da ρ t h w2 p t + ρw z h w2 = P A q W + ρg z w ρt s t Very important particular case: volume forces, + ρwt s z = P A q W + wτ W stationary flow, adiabatic, no friction nor M = Aρw = cst h w2 = cst s = xs V + 1 xs L = cst Applications: flashing in long pipes, w/o heating, critical flow no model!. 1D-time averaged models 11/30

13 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 xv L, specific volume [m 3 /kg], v L = v Lsat p, v V = v V sat p, h L = h Lsat 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

14 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 = fx, p, α, flow regime, J GL = fα, flow regime, Slow transients NOT inertia controlled. Can also be used in 3D, see for example Delhaye 2008a, Ishii & Hibiki Main advantage: only one momentum balance. 1D-time averaged models 13/30

15 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

16 EXAMPLE: STRATIFIED FLOWS 0 D D Isothermal, incompressible, horizontal, µ = 0, σ = 0, ṁ = 0, 2D. Mass balance, t h 1ρ 1 + z h 1ρ 1 < u 1 >= 0 t h 2ρ 2 + z h 2ρ 2 < u 2 >= 0 Momentum balances, jump of momentum at the interface, t h 1ρ 1 < u 1 > + z h 1ρ 1 < u 2 1 > + z h h 1 1 < p 1 >= p i1 z t h 2ρ 2 < u 2 > + z h 2ρ 2 < u 2 2 > + z h h 2 2 < p 2 >= p i2 z p i1 = p i2 p i 1D-time averaged models 15/30

17 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. F F E Topological relation, < p 1 >= p i ρ 1gh 1 F < p 2 >= p i 1 2 ρ 2gh 2 Cannot be derived from momentum. Spatial correlations, flat profile assumption, or relaxation < u 2 k > < u k > 2 = 1, d dt < u2 k >= 1 T [ < u 2 k > < u 2 k > 0 ] Closed system. 1D-time averaged models 16/30

18 STABILITY OF STRATIFIED FLOW Solve PDE s, A X t + B X z = 0, X = h, u 1, u 2, p 1 : ρ ρ 1 u 1 ρ 1 h ρ ρ 2 u 2 0 ρ 2 h 2 0 A, B 0 ρ 1 h ρ 1gh 1 ρ 1 u 1 h 1 h ρ 2 h 2 0 ρ ρ 2gh 2 0 ρ 2 u 2 h 2 h 2 Use the perturbation method, Van Dyke 1975 : X = X 0 + ɛx 1 + Oɛ 2, Linearizes the PDE s, separate the orders, A X 0 t + B X 0 z = 0, X 0 = cst AX 0 X 1 + BX 0 X 1 t z = 0 X 0, base solution, X 1, first order linear perturbation, z [a, b], BC and IC are needed. 1D-time averaged models 17/30

19 STABILITY OF STRATIFIED FLOW CT D Progressive waves, X 1 = X 1 exp iωt kz, c = ω/k, phase velocity, Temporal stability, X 1 a, t = ft, ω R, how far does perturbations propagates into the domain? Spatial stability: X 1 z, 0 = gz, k R, perturbation amplification? When the RHS of balance equations are non-zero, long wave assumption. cax 0 BX 0 X 1 = 0 One class of solution, X 1 kercax 0 BX 0 Dispersion equation, ρ 1 ρ 2 h 1 h 2 ρ1 h 2 u 1 c 2 + ρ 2 h 1 u 2 c 2 ρ 1 ρ 2 gh 1 h 2 = 0 Stable if and only if the 2 roots are real, u 1 u 2 2 gρ 1 ρ 2 ρ 1h 2 + ρ 2 h 1 ρ 1 ρ 2 1D-time averaged models 18/30

20 STABILITY OF STRATIFIED FLOW CT D Conditional stability : u u C Kelvin-Helmholtz instability, u 1 u 2 2 gρ 1 ρ 2 ρ 1h 2 + ρ 2 h 1 ρ 1 ρ 2 Heavy on top, light below, ρ 2 > ρ 1, always unstable hopefully. Flat pressure profiles, g = 0, always unstable. Nature of PDE s: conditionally hyperbolic, g = 0, the IC problem is ill-posed Hadamard No possibility to get a stationary state from a transient calculation. With no differential terms in the closures, the two-fluid model with one pressure leads to ill-posed problems. Why codes produce a solution? 1D-time averaged models 19/30

21 PRESSURE DROP MODELING Simplified flow model, Mass balance of the mixture, Momentum balance of the mixture, If adiabatic: x = x 0, or solve the energy equation, Evolution equation w V w L, Closures : L,G = L,G C k n k V k n z C k n k q k dl n k n kc = P τ W dl n k n kc = P q W NB: Constant pipe cross-sectional area, A = cst, n k n kc = 1 1D-time averaged models 20/30

22 PRESSURE DROP MODELING CT D Stationary flow, constant flow area, d ρw = 0, G = cst d ρw2 + = P A τ W + ρg z, P A 4 D h Wall friction appears only in the momentum balance, = d ρw2 P A τ W + ρg z + Experiments where and possibly α = R G2 are measured. NB: evolution equation is used for : Use the models with the same set of assumptions. A A F + G 1D-time averaged models 21/30

23 WALL FRICTION WITH THE HEM Friction should not be dominant, = P A τ W = 4 τ W D h NB: frictional pressure drop, f = 4C f quite tricky... C F = τ D W 1, f = F 2 ρw2 1 2 ρw2 1. Annular flow x 1, C F = 0, 005, flashing flows, x 0, C F = 0, x 1, C F = C F L, M = M L + M V, x 1, C F = C F G, M = M L + M V 3. NB: Single-phase friction factors, 16 Poiseuille : Re, Blasius : 0, 079 Re 0,25, Re < , 046 Re 0,20, Re > F Re GD µ 1D-time averaged models 22/30

24 WALL FRICTION CT D Historical perspective, two-phase viscosity, Dukler 1964 : µ = βµ g + 1 βµ L, C F = 0, , 125Re 0,32 Ishii-Zuber 1978, liquid-liquid or gas-liquid, α DM = 0, 62 µ µ C = 1 α D α DM 2,5αDM µ D +0,4µ C µ D +µ C Acceleration pressure drop, use the appropriate evolution α β, = G 2 d [ ] x 2 1 x2 + A αρ V 1 αρ L Quality from the enthalpy balance, low velocity, thermal equilibrium, G d xh V + 1 xh L = 4 D c q W 1D-time averaged models 23/30

25 TWO-COMPONENT, LOCKHART & MARTINELLI Air-water experiments, low pressure, p F et R G3 are measured QCV. Three sets of experiments in the same horizontal pipe. Conditions two-phase gas only liquid only Mass rate M = M G + M L M G M L F Definitions on the non-dimensional friction pressure drop: two-phase pressure drop multiplier G L Φ 2 L L, Φ 2 G G, X 2 L G Blasius is used C f = 0, 046 Re 0,2, X, L. & M. parameter X tt = µl µ G 0,1 0,9 1 x ρg x ρ L 0,5 1D-time averaged models 24/30

26 LOCKHART & MARTINELLI CORRELATION 100 Φ L Φ G 1 α L α G Φ 10 α Φ L = X X + 1 X α L = X X X + X 2 1D-time averaged models 25/30

27 STEAM-WATER, MARTINELLI & NELSON Steam water experiments, bar, p G 0. Two experiments in the same helical tube. p = p F + p A. Two-phase multiplier, Conditions two-phase liquid only Mass rate M M L = M F Φ 2 f0 = fo Evolution equation acceleration pressure drop, data and models. fo 1D-time averaged models 26/30

28 MARTINELLI & NELSON CORRELATIONS Void fraction vs quality and pressure bar Φ 2 f0 = fo 1D-time averaged models 27/30

29 BOILING FLOWS Boiling flows, Thom. r 3 = p F p F o = 1 x S xs Other methods, see Delhaye 2008b. 0 Φ 2 Lodx Friction multiplier vs pressure and quality %. 1D-time averaged models 28/30

30 FRIEDEL S CORRELATION Thousands of data reduction, various fluids, non-dimensional. Arbitrary orientation. Two-phase multiplier by Martinelli & Nelson. M and x are given, H = Φ 2 Lo = F Lo = E + 3, 24F H Fr 0,045 We 0,035 x ρ h = + 1 x 1, We = G2 D, Fr = G2 ρ G ρ L σρ h gdρ 2 h 0,91 0,19 µg 1 µ 0,7 G, F = x 0,78 1 x 0,224 ρ G µ L µ L ρl C F Go = C F G M, C F Lo = C F L M, E = 1 x 2 + x 2 ρ LC F Go ρ G C F Lo 1D-time averaged models 29/30

31 REFERENCES Delhaye, J.-M. 2008a. Thermohydraulique des réacteurs nucléaires. Collection génie atomique. EDP Sciences. Chap. 7-Modélisation des écoulements diphasiques en conduite, pages Delhaye, J.-M. 2008b. Thermohydraulique des réacteurs nucléaires. Collection génie atomique. EDP Sciences. Chap. 8-Pertes de pression dans les conduites, pages Ishii, M., & Hibiki, T Thermo-fluid dynamics of two-phase flows. Springer. Van Dyke, M Perturbation methods in fluid mechanics. Parabolic Press. 1D-time averaged models 30/30