A SHORT INTRODUCTION TO TWO-PHASE FLOWS Critical flow phenomenon

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1 A SHORT INTRODUCTION TO TWO-PHASE FLOWS Critical flow phenomenon Hervé Lemonnier DM2S/STMF/LIEFT, CEA/Grenoble, Grenoble Cedex 9 Ph. +33(0) , herve.lemonnier@cea.fr herve.lemonnier.sci.free.fr/tpf/tpf.htm ECP,

2 INDUSTRIAL OCCURRENCE Depressurization of a nuclear reactor, LOCA (small or large break) Industrial accidents prevention Safety valves sizing, SG, chemical reactor. liquid helium storage in case of vacuum loss. LPG storage in case of fire. Two typical situations, A pressurized liquid becomes super-heated due to the break, flashing occurs. A gas is created in a vessel, exothermal chemical reaction, pressurizes the vessel, thermal quenching to recover control. Critical flow: for given reservoir conditions (pressure), and varying outlet conditions, there exists a limit to the flow rate that can leave the system. Critical flow phenomenon 1/42

3 SUMMARY Two-component flows Experimental characterization Geometry and inlet effects Steam water flows, saturation and subcooling Theory and modeling, 2 particular simple cases, Single-phase flow of a perfect gas Two-phase flow at thermodynamic equilibrium General theory, if time permits... Critical flow phenomenon 2/42

4 SINGLE-PHASE GAS FLOW, LONG NOZZLE Non dimensional pressure P/P 0 60A10E00.PRE 60A10M00.PRE 60B10M00.PRE 60A16M00.PRE 60A20M00.PRE 60A30M00.PRE 60A41M00.PRE 60A50M00.PRE 60A57M00.PRE Abscissa (mm) File M G P back kg/h bar 60A10E00.PRE A10M00.PRE B10M00.PRE A16M00.PRE A20M00.PRE A30M00.PRE A41M00.PRE A50M00.PRE A57M00.PRE air, T G o C P 0 6 bar, D = 10 mm Choking occurs when p t /p Critical flow phenomenon 3/42

5 SINGLE-PHASE GAS FLOW, SHORT NOZZLE Non dimensional pressure P/P 0 60A10A00.PRE 60A13B00.PRE 60A19B00.PRE 60A33B00.PRE 60A41B00.PRE 60A47B00.PRE 60A56B00.PRE Abscissa (mm) File M G P back kg/h bar 60A10A00.PRE A13B00.PRE A19B00.PRE A33B00.PRE A41B00.PRE A47B00.PRE A56B00.PRE air, T G 19 o C P 0 6 bar, D = 5 mm Choking occurs when p t /p Critical flow phenomenon 4/42

6 TWO-PHASE AIR-WATER FLOW Non dimensional pressure P/P 0 60A10M36.PRE 60A15M36.PRE 60A21M36.PRE 60A28M36.PRE 60A37M36.PRE 60A49M36.PRE 60A56M36.PRE Abscissa (mm) File M G P back kg/h bar 60A10M36.PRE A15M36.PRE A21M36.PRE A28M36.PRE A37M36.PRE A49M36.PRE A56M36.PRE T L T G 19 o C P 0 6 bar, D = 10 mm, M L 358 kg/h. Choking occurs when p t /p 0 < 0.5 Critical flow phenomenon 5/42

7 TWO-PHASE AIR-WATER FLOWS Non dimensional pressure P/P 0 60B10A50.PRE 60A14B50.PRE 60A19B50.PRE 60A24B50.PRE 60A36B50.PRE 60A45B50.PRE 60A55B50.PRE Abscissa (mm) File M G P back kg/h bar 60B10A50.PRE A14B50.PRE A19B50.PRE A24B50.PRE A36B50.PRE A45B50.PRE A55B50.PRE T L T G 19 o C P 0 6 bar, D = 5 mm, M L 500 kg/h. Choking simple criterion lost. Critical flow phenomenon 6/42

8 SAFETY VALVE CAPACITY REDUCTION Gas mass flow rate, kg/h Long throat EFGH, D = 10 mm P 0 = 2 bar -Annular injection (E)- P 0 = 4 bar P 0 = 6 bar P 0 = 2 bar -Central injection (G)- P 0 = 4 bar P 0 = 6 bar Liquid mass flow rate, kg/h Critical flow phenomenon 7/42

9 QUALITY EFFECT ON GAS CAPACITY 1.0 Critical mass flow rate / Single-phase mass flow rate Long throat EFGH, D = 10 mm P 0 = 2 bar -Annular injection (E)- P 0 = 4 bar P 0 = 6 bar P 0 = 2 bar -Central injection (G)- P 0 = 4 bar P 0 = 6 bar Gas quality Critical flow phenomenon 8/42

10 SAFETY VALVE CAPACITY REDUCTION, SHORT NOZZLE 1.0 Critical gas mass flow rate / Single-phase gas flow rate Truncated short nozzle P 0 = 2 bar - Annular (S) P 0 = 4 bar P 0 = 6 bar P 0 = 2 bar- Central (R) P 0 = 4 bar P 0 = 6 bar Gas quality Critical flow phenomenon 9/42

11 STEAM-WATER FLOWS p sat (T L0 ) = bar Critical flow phenomenon 10/42

12 SUCOOLING EFFET ON CRITICAL FLOW Critial mass flux [kg/m 2 /s] Data 60 bar HEM Steam quality [%] Super Moby Dick data, 60 bar, saturated and subcooled In HEM here, friction is neglected. Critical flow phenomenon 11/42

13 MAIN FEATURES Gas flow rate reaches a limit when the back pressure drops. In single-phase flow, this limit depends on Mainly on pressure M G SP 0 Geometry, throat length, effect is second order. In two-phase flow, The gas flow rate depends on quality. The maximum flow rate of gas and the back pressure for choking depend on geometry, and on inlet effects, mechanical non-equilibrium, w G w L, history effects. In steam water flows, thermodynamic non-equilibrium plays the same role. In flashing flows mechanical non-equilibrium may be secondary. Critical flow phenomenon 12/42

14 MODELING OF CHOKED FLOWS Single-phase gas or steam and water at thermal equilibrium. Theory of choked flows: Time dependent 1D-model, analysis of propagation Stationary flows, critical points of ODE s Selected results in two-phase flows, Non equilibrium effects on critical flow. Some numerical results. Critical flow is a mathematical property of the 1D flow model. Critical flow phenomenon 13/42

15 PRIMARY BALANCE EQUATIONS (1D) Mixture mass balance Mixture momentum balance ρ t + w ρ z + w ρ z = ρw A w t + w w z + 1 ρ Mixture total energy balance (u + 12 ) t w2 + w z Volume forces have been neglected. da dz p z = P A τ W (h + 12 w2 ) = P A q W τ W : wall sher stress, q W : heat flux to the flow. P: Common wetted and heated perimeter Closures must be provided and remain algebraic (no differential terms). Critical flow phenomenon 14/42

16 SECONDARY BALANCE EQUATIONS (1D) mixture enthalpy balance, h t 1 ρ Mixture entropy balance, p t + w h z = P Aρ (τ W w + q W ) s t + w s z = P AρT (τ W w + q W ) NB: secondary equations were derived from primary ones. Mixture equations remain valid if mechanical or thermodynamic nonequilibrium are accounted for. Critical flow phenomenon 15/42

17 PROPAGATION ANAYSIS Mass, momentum, entropy balances for the mixture, A X t + B X z = C Equation of state, p(ρ, s), the pressure p should be eliminated. ρ w ρ 0 X = w, A = 0 1 0, B = p ρ/ρ 1 p s/ρ s w, Waves are small perturbations, perturbation method,, X = X 0 +ɛx 1 +, X 0 : Steady state solution. Taylor expansion, polynomials in ɛ... Critical flow phenomenon 16/42

18 SOLUTIONS Steady flow, Linear perturbation, B X 0 z = C RHS are evaluated at X 0, A(X 0 ) X 1 t + B(X 0 ) X 1 z = DX 1 DX 1 = C X X 1 A X X X 0 1 t B X X X 0 1 z. Perturbation as waves, X 1 = X 1 e i(ωt kz) c, phase velocity of small perturbations, c ω ( k, ca B D ) ik X 1 = 0 Dispersion equation. Critical flow phenomenon 17/42

19 DISPERSION EQUATION, SOUND VELOCITY Large wave number assumption, k (X 0 : is quasi uniform), Non zero solutions if and only if, det (ca B) = (w c)(w 2 p ρ) = 0 3 propagation modes: w, w ± a, a: so called isentropic speed of sound, ( ) p a 2 = p ρ ρ Examples, Perfect gas, R = R M, R p. g. cst., γ = C P C V, a 2 = γrt Steam and water at thermal equilibrium, a 2 h x = ρ ph x + ρ x(1/ρ h p), h(x, p) = xh V sat + (1 x)h Lsat, v(x, p) = xv V sat + (1 x)v Lsat = 1 ρ s Critical flow phenomenon 18/42

20 WAVE PROPAGATIONS AND CHOKING t w-a w w+a w-a w w+a t w-a w w+a w-a w w+a z z (a) subsonic flow (b) supersonic flow When w a < 0 every where, subsonic flow, flow rate depends on back pressure If somewhere, w a > 0, supersonic flow, Waves can no longer propagate from downstream the point where w = a is the critical (sonic) section. Waves are stationary. Critical flow phenomenon 19/42

21 THE CRITICAL VELOCITY OF THE HEM Air at 20 o C, HEM sound velocity (m/s) steam quality 1 bar 50 bar 100 bar 150 bar 200 bar a 343 m/s Steam and water at 5 bar, Thermal equilibrium mixture 1 < a < 439 m/s Saturated Water only at 5 bar a 1642 m/s Saturated steam only at 5 bar : a 494 m/s The mixture compressibility results from the change in composition at thermodynamic equilibrium. Critical flow phenomenon 20/42

22 PRACTICAL IMPLEMENTATION Transient analysis shortcomings: Physical consistency of the two-fluid one-pressure models Conditionally hyperbolic, Terrible numerical analysis (non-conservative schemes) Time and space requirements are large to resolve waves. Critical flow can be analyzed with the stationary model Steady equations are much simpler, EDO s instead of EDP s, no physical consistency problems, Initial value problem, Simple and accurate schemes (Runge-Kutta, adaptative step) Price to pay: critical points, where solutions is not unique (there s no free lunch...) Critical flow phenomenon 21/42

23 STATIONARY FLOW OF A PERFECT GAS Mass balance, d dz Aρw = 0 Momentum balance, circular pipe, C F is the friction coefficient, ρw dw dz + dp dz = 4 D 1 2 C F ρw 2 = 0 Energy balance, adiabatic flow, d (h + 12 ) dz w2 = 0 For a perfect gas and a variable section, D(z), Only one ODE, Ma = w a, dm 2 dz = 4M 2 (1 + γ 1 2 M 2 )(γc F M 2 D ) D(1 M 2 ) Critical flow phenomenon 22/42

24 SOLUTIONS ANALYSIS Variable section nozzle, D = F (x), y = Ma 2, dy dx = Y (x, y) X(x, y) γ 1 4y(1 + 2 = y)(γc F y F ), F (1 y) Autonomous form, dx du = X(x, y) dy du = Y (x, y) u dummy, advancement parameter, the system is autonomous when u is not explicit in the RHS. u > 0 selected, signe of X. Solve the initial value problem: Draw the current lines of vector (X, Y ). The analysis of the solution topology does not require to calculate them! Critical flow phenomenon 23/42

25 ADIABATIC FLOW WITH CONSTANT SECTION, Constant section, F =cst, C F =cst. X = F (1 y) and Y = 4γC F y 2 ( 1 + γ 1 2 ) y. Evolution equation, Ma M Initial conditions, z = 0, M = M 0 Solution analysis, signs of X and Y. back to EDO s, integration is possible, 1 M 2 M 4 (1 + γ 1 2 M 2 ) dm 2 = 4γC F D dz. 4C F z D = G(M) G(M 0), G(M) = γ + 1 2γ for given M = M 0, z cannot exceed z, limiting length. γ ln M 2 M 2 1 γm 2. 4C F z D = G(1) G(M) = 1 M 2 γm 2 + γ + 1 2γ ln (γ + 1)M 2 2(1 + γ 1 2 M 2 ) Critical flow phenomenon 24/42

26 ADIABATIC FLOW WITH FRICTION: FANNO FLOW Because of friction the flow is not isentropic, M evolution parameter, Velocity Temperature, Density Pressure, v v = M γ + 1 2(1 + γ 1 2 M 2 ) T T = γ + 1 2(1 + γ 1 2 M 2 ) ρ ρ = 1 M p p = 1 M 2(1 + γ 1 2 M 2 ) γ + 1 γ + 1 2(1 + γ 1 2 M 2 ) Critical flow phenomenon 25/42

27 ADIABATIC FLOW WITH VARIABLE SECTION Variable section, adiabatic flow, X = F (1 y) Y = 4y(1 + γ 1 y)(γc F y F ) 2 No analytic solution, Signs of X and Y, 4 quadrants, X = 0 y = 1, Y = 0 F (x) = γc F Critical point: X and Y are both zero. (x, y ) downstream the throat. Linear analysis around the critical point. Critical flow phenomenon 26/42

28 CRITICAL POINTS FEATURES Linearize around critical point, change of variable, dy dx = Y 1 X 1 = Y x(x, y )x + Y y (x, y )y X x (x, y )x + X y (x, y )y, Slope of solutions at the critical point, λ = y /x = Y 1 /X 1, 2 real roots since F (x*), x = x x y = y y F (x )λ 2 + 2C F γ(γ + 1)λ 2(γ + 1)F (x ) = 0 = 4γ 2 (γ + 1) 2 C 2 F + 8(γ + 1)F (x )F (x ), λ 1 λ 2 = 2(γ + 1)F (x ) F (x ) At the critical point, 2 branches one is subsonic the other is supersonic, other critical points may occur (Kestin & Zaremba, 1953). For a variable and decreasing back pressure, subsonic flow, critical flow and supersonic flow. Some range of back pressure can not be reached. In agreement with experiments provided that 1D assumption is correct. Critical flow phenomenon 27/42

29 ISENTROPIC FLOW WITH VARIABLE CROSS SECTION Very important particular case: isentropic flow. (y 1)dy 2y(1 + γ 1 2 y) = 2F dx F = da A No history effect (C F = 0), A is the main variable, no longer z. Critical points can only be at the throat or at the end. Evolution, Pressure, Mass flux, p 0 p = ( A A = 1 M G = ρw = p 0 γ RT γ 1 ) γ M 2 2 [( ) ( γ 1 )] γ+1 M 2 γ γ 1, M ( γ γ 1 2 M 2) 2(γ 1) p p 0 = 2(γ 1) ( ) γ 2 γ 1 0, 5283 γ + 1, G = p 0 RT0 γ ( 2 ) γ+1 γ 1 γ + 1 Critical flow phenomenon 28/42

30 STEAM WATER FLOW WITH THE HEM Important and simple particular case: isentropic flow with saturated reservoir conditions, p 0, x 0 h 0, s 0 Mass, energy and entropy for the mixture, closed form solution, m = Sρw, h 0 = h w2, s 0 = s. Mixture variables, thermodynamical variables at saturation 1 (x, p) = x ρ ρ V (p) + 1 x ρ L (p) = v(x, p) = xv V (p) + (1 x)v L (p) h(x, p) = xh V (p) + (1 x)h L (p), s(x, p) = xs V (p) + (1 x)s L (p), Critical flow phenomenon 29/42

31 A SIMPLE ALGORITHM Look for the back pressure, p, that makes G = ρw maximum, Get the quality from entropy, x = s 0 s L (p) s V (p) s L (p) get the velocity from energy w = 2(h 0 h) Calculates the mass flux, G = ρw = w v P x h w ρ G c bar - kj/kg m/s kg/m 3 kg/m 2 /s m/s Critical flow phenomenon 30/42

32 SATURATED WATER 5 BAR G (kg/m 2 /s) w et c (m/s) P (bar) 4.4 G G (tableau 4) w c Critical flow phenomenon 31/42

33 CRITICAL FLOW WITH THE HEM (SATURATED INLET) x 0 = 0 7 x 0 = 0 G (kg/m 2 /s) P c (bar) x 0 = x 0 = 1 G c, x 0 = P c, x 0 = P (bar) P (bar) Critical flow phenomenon 32/42

34 CRITICAL FLOW WITH THE HEM (CT D) x 0 = x 0 = G (kg/m 2 /s) P c (bar) 60 x 0 = x 0 = G c, x 0 = P c, x 0 = P (bar) P (bar) Critical flow phenomenon 33/42

35 TWO-PHASE FLOW WITH THE TWO-FLUID MODEL Two-fluid model, 6 equations or more. Wealth of behavior, non-equilibriums, numerical integration is required Critical conditions are mathematical properties of the system. The consistency of the velocity propagation depend on the closure consistency. Wave propagation may differ from sound velocity, However the solution topology is identical to that of single-phase flow. The critical section may differ in position (greatly) Many choked flow models assume the critical section position, though it results from the calculation The critical nature of the flow is assumed, though it derives from the calculation. Critical flow phenomenon 34/42

36 CRITICAL POINTS ANALYSIS ODE s n equations solved wrt derivatives, Autonomous form, B dx dz = C, Critical point condition, (Bilicki et al., 1987) showed, dz du =, dx i dz = i, dx i du = i i = 1, n = 0, i = 0, i = 1, n = 0 and i0 = 0 i = 0, i i 0 [1, n] Only two independent critical conditions: = 0, critical condition (i), same as w = a in single phase flow, i = 0, sets the critical section location, same as F (x) = γf. Critical flow phenomenon 35/42

37 SOLUTION TOPOLOGY After Bilicki et al. (1987). Critical flow phenomenon 36/42

38 NUMERICAL SOLUTION OF EQUATIONS n equations, dimension of phase space is n + 1 = 0 ou i =0 defines a manifold of dimension n. All critical points S manifold of dimension n 1. Topology at the critical point, identical to single-phase flow The linearized system has only tow non-zero eigenvalues. The corresponding eigenvectors, solution near the critical point. One step there and resume numerical integration. Shooting problem: Find the point in S connected to X 0 by a solution. Critical flow phenomenon 37/42

39 PRACTICAL IMPLEMENTATION Boundary problem with a free boundary, z < L. n-1 upstream conditions are given, shooting on the last one (ex. gas flow rate). PIF algorithm by Yan Fei (Giot, 1994, 2008) Assume the critical point is a saddle. Dichotomic search. If calculation goes up to the nozzle end, flow is subcritical. increase the gas flow rate. If solution turns back, changes sign z < L. Decrease the flow rate. This method cannot cross the critical point. Cannot reach the supercritical branch. Critical flow phenomenon 38/42

40 Direct method (Lemonnier & Bilicki, 1994, Lemaire, 1999). Assume there is a saddle. Check later. Find an estimate of critical point by PIF. Set two its coordinates to satisfy exactly. ( = p = 0). Backward integration (linearization, eigen-values, chek here for the saddle, eigen-vectors...) Solve (Newton) for the remaining (n 2) coordinates to reach X 0 Allows the full determination of the critical point topology. Get the two downstream branches afterwards. NB: with non equilibriums, backward integration may become unstable is the system is stiff: change the model... Critical flow phenomenon 39/42

41 NON-EQUILIBRIUM EFFECTS Thermal non-equilibrium in steam water flows, Homogeneous relaxation model, HRM 3 mixture equations, x x eq, h L < h Lsat (p), h V = h V sat (p) dx dz = x x eq, ρ = ρ(p, h, x) wθ θ is a closure, from Super Moby Dick experiment (Downar-Zapolski et al., 1996). A A = PC F w 2 2A ρ p + x x eq θρ The critical section shifts downstream due to non-equilibrium. Mechanical non-equilibrium air-water flows, Two-component isothermal flow Mechanical on equilibrium: liquid inertia and interfacial friction, τ i A A = P τ W A ρ G wg 2 3(1 α)τ ( i 4R d ρ G wg 2 1 ρ GwG 2 ) ρ L wl 2 The critical section shifts downstream. May leave the nozzle... ρ x Critical flow phenomenon 40/42

42 MORE ON CRITICAL FLOW Two-phase choked flows, Introduction, Giot (1994) Text, Giot (2008), in French Non-equilibrium effects, Lemonnier & Bilicki (1994) Voir aussi, Single-phase flow, very tutorial Kestin & Zaremba (1953) Math aspects and critical points, Bilicki et al. (1987) Modeling, HRM, Downar-Zapolski et al. (1996) Two-component flows, Lemonnier & Selmer-Olsen (1992) Critical flow phenomenon 41/42

43 REFERENCES Bilicki, Z., Dafermos, C., Kestin, J., Majda, J., & Zeng, D. L Trajectories and singular points in steady-state models of two-phase flows. Int. J. Multiphase Flow, 13, Downar-Zapolski, P., Bilicki, Z., Bolle, L., & Franco, J The non-equilibrium relaxation model for one-dimensional flashing liquid flow. Int. J. Multiphase Flow, 22(3), Giot, M Two-phase releases. J. Loss Prev. Process Ind., 7(2), Giot, M Thermohydraulique des réacteurs nucléaires. Collection génie atomique. EDP Sciences. Chap. 11-Blocage des écoulements diphasiques, pages Kestin, J., & Zaremba, S. K One-dimensional high-speed flows. Aircraft Engineering, June, 1 5. Lemaire, C Caractérisation et modélisation du blocage de débit en écoulement dispersé à deux constituants en géométrie tridimensionnelle. Ph.D. thesis, Institut National Polytechnique de Grenoble, Grenoble, France. Lemonnier, H., & Bilicki, Z Multiphase Science and Technology. Vol. 8. Begell House. G.F. Hewitt, J.M. Delhaye and N. Zuber, Eds. Chap. 6, Steady two-phase choked flow in channels of variable cross sectional area. Lemonnier, H., & Selmer-Olsen, S Experimental investigation and physical modelling of two-phase two-component flow in a converging-diverging nozzle. International Journal of Multiphase Flow, 18(1), Critical flow phenomenon 42/42

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

A SHORT INTRODUCTION TO TWO-PHASE FLOWS 1D-time averaged models Hervé Lemonnier DM2S/STMF/LIEFT, CEA/Grenoble, 38054 Grenoble Cedex 9 Ph. +3304 38 78 45 40 herve.lemonnier@cea.fr, herve.lemonnier.sci.free.fr/tpf/tpf.htm