A SHORT INTRODUCTION TO TWO-PHASE FLOWS Critical flow phenomenon - PowerPoint PPT Presentation

A SHORT INTRODUCTION TO TWO-PHASE FLOWS Critical flow phenomenon 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

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

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

SINGLE-PHASE GAS FLOW, LONG NOZZLE File M G P back kg/h bar 1 2 3 4 5 67 8 9 10 11 12 13 1415 16 17 18 19 20 21 22 23 60A10E00.PRE 363.9 0.973 Non dimensional pressure P/P 0 60A10M00.PRE 364.3 1.127 1.0 60B10M00.PRE 362.9 1.135 60A16M00.PRE 364.6 1.650 60A20M00.PRE 364.5 1.986 0.75 60A30M00.PRE 364.1 3.023 60A41M00.PRE 364.4 4.088 60A50M00.PRE 361.3 5.022 60A10E00.PRE 60A10E00.PRE 0.5 60A10M00.PRE 60A10M00.PRE 60A57M00.PRE 246.6 5.749 60B10M00.PRE 60B10M00.PRE 60A16M00.PRE 60A16M00.PRE air, T G ≈ 18 ÷ 22 o C 60A20M00.PRE 60A20M00.PRE 60A30M00.PRE 60A30M00.PRE 0.25 P 0 ≈ 6 bar, D = 10 mm 60A41M00.PRE 60A41M00.PRE 60A50M00.PRE 60A50M00.PRE Choking occurs when 60A57M00.PRE 60A57M00.PRE p t /p 0 ≈ 0 . 5 0.0 0.0 100.0 200.0 300.0 400.0 Abscissa (mm) Critical flow phenomenon 3/42

SINGLE-PHASE GAS FLOW, SHORT NOZZLE File M G P back 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 kg/h bar Non dimensional pressure P/P 0 60A10A00.PRE 94.8 0.891 1.0 60A13B00.PRE 94.9 1.281 60A19B00.PRE 94.9 1.929 60A33B00.PRE 94.9 3.288 0.75 60A41B00.PRE 95.0 4.058 60A47B00.PRE 94.9 4.695 60A56B00.PRE 88.4 5.619 0.5 60A10A00.PRE 60A10A00.PRE air, T G ≈ 19 o C 60A13B00.PRE 60A13B00.PRE 60A19B00.PRE 60A19B00.PRE P 0 ≈ 6 bar, D = 5 mm 60A33B00.PRE 60A33B00.PRE 0.25 60A41B00.PRE 60A41B00.PRE Choking occurs when 60A47B00.PRE 60A47B00.PRE 60A56B00.PRE 60A56B00.PRE p t /p 0 ≈ 0 . 5 0.0 0.0 100.0 Abscissa (mm) Critical flow phenomenon 4/42

TWO-PHASE AIR-WATER FLOW File M G P back kg/h bar 1 2 3 4 5 67 8 9 10 11 12 13 1415 16 17 18 19 20 21 22 23 Non dimensional pressure P/P 0 60A10M36.PRE 215.2 0.912 1.0 60A15M36.PRE 217.4 1.489 60A21M36.PRE 216.9 2.050 60A28M36.PRE 216.1 2.798 0.75 60A37M36.PRE 204.1 3.731 60A49M36.PRE 155.3 4.897 60A56M36.PRE 94.3 5.593 0.5 T L ≈ T G ≈ 19 o C 60A10M36.PRE 60A10M36.PRE 60A15M36.PRE 60A15M36.PRE P 0 ≈ 6 bar, D = 10 mm, 60A21M36.PRE 60A21M36.PRE 60A28M36.PRE 60A28M36.PRE M L ≈ 358 kg/h. 0.25 60A37M36.PRE 60A37M36.PRE 60A49M36.PRE 60A49M36.PRE Choking occurs when 60A56M36.PRE 60A56M36.PRE p t /p 0 < 0 . 5 0.0 0.0 100.0 200.0 300.0 400.0 Abscissa (mm) Critical flow phenomenon 5/42

TWO-PHASE AIR-WATER FLOWS File M G P back kg/h bar 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Non dimensional pressure P/P 0 60B10A50.PRE 18.50 0.942 1.0 60A14B50.PRE 18.50 1.385 60A19B50.PRE 19.10 1.925 60A24B50.PRE 18.20 2.444 0.75 60A36B50.PRE 15.40 3.626 60A45B50.PRE 10.00 4.490 60A55B50.PRE 3.20 5.540 0.5 T L ≈ T G ≈ 19 o C 60B10A50.PRE 60B10A50.PRE 60A14B50.PRE 60A14B50.PRE P 0 ≈ 6 bar, D = 5 mm, 60A19B50.PRE 60A19B50.PRE 60A24B50.PRE 60A24B50.PRE 0.25 M L ≈ 500 kg/h. 60A36B50.PRE 60A36B50.PRE 60A45B50.PRE 60A45B50.PRE Choking simple criterion 60A55B50.PRE 60A55B50.PRE lost. 0.0 0.0 100.0 Abscissa (mm) Critical flow phenomenon 6/42

SAFETY VALVE CAPACITY REDUCTION Gas mass flow rate, kg/h 400.0 Long throat EFGH , D = 10 mm Long throat EFGH , D = 10 mm P 0 = 2 bar -Annular injection (E)- P 0 = 2 bar -Annular injection (E)- P 0 = 4 bar P 0 = 4 bar P 0 = 6 bar P 0 = 6 bar 300.0 P 0 = 2 bar -Central injection (G)- P 0 = 2 bar -Central injection (G)- P 0 = 4 bar P 0 = 4 bar P 0 = 6 bar P 0 = 6 bar 200.0 100.0 0.0 0.0 200.0 400.0 600.0 800.0 1000.0 Liquid mass flow rate, kg/h Critical flow phenomenon 7/42

QUALITY EFFECT ON GAS CAPACITY Critical mass flow rate / Single-phase mass flow rate 1.0 0.8 0.6 Long throat EFGH , D = 10 mm Long throat EFGH , D = 10 mm P 0 = 2 bar -Annular injection (E)- P 0 = 2 bar -Annular injection (E)- P 0 = 4 bar P 0 = 4 bar P 0 = 6 bar P 0 = 6 bar 0.4 P 0 = 2 bar -Central injection (G)- P 0 = 2 bar -Central injection (G)- P 0 = 4 bar P 0 = 4 bar P 0 = 6 bar P 0 = 6 bar 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Gas quality Critical flow phenomenon 8/42

SAFETY VALVE CAPACITY REDUCTION, SHORT NOZZLE Critical gas mass flow rate / Single-phase gas flow rate 1.0 0.8 0.6 Truncated short nozzle Truncated short nozzle P 0 = 2 bar - Annular (S) P 0 = 2 bar - Annular (S) P 0 = 4 bar P 0 = 4 bar P 0 = 6 bar P 0 = 6 bar 0.4 P 0 = 2 bar- Central (R) P 0 = 2 bar- Central (R) P 0 = 4 bar P 0 = 4 bar P 0 = 6 bar P 0 = 6 bar 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Gas quality Critical flow phenomenon 9/42

STEAM-WATER FLOWS p sat ( T L 0 ) = 2 . 09 ÷ 2 . 11 bar Critical flow phenomenon 10/42

SUCOOLING EFFET ON CRITICAL FLOW 60000 50000 Critial mass flux [kg/m 2 /s] 40000 30000 20000 10000 Data 60 bar HEM 0 −10 −8 −6 −4 −2 0 2 4 6 Steam quality [%] Super Moby Dick data, 60 bar, saturated and subcooled In HEM here, friction is neglected. Critical flow phenomenon 11/42

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

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

PRIMARY BALANCE EQUATIONS (1D) • Mixture mass balance ∂ρ ∂t + w∂ρ ∂z + w∂ρ ∂z = − ρw d A A d z • Mixture momentum balance ∂w ∂t + w∂w ∂z + 1 ∂p ∂z = −P Aτ W ρ • Mixture total energy balance � � � � ∂ u + 1 + w ∂ h + 1 = P 2 w 2 2 w 2 Aq W ∂t ∂z • Volume forces have been neglected. • 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

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

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 1 0 0 w ρ 0             X =  , A =  , B =  , w 0 1 0 p ′ ρ /ρ 1 p ′ s /ρ    s 0 0 1 0 0 w • Waves are small perturbations, perturbation method,, X = X 0 + ǫ X 1 + · · · , • X 0 : Steady state solution. • Taylor expansion, polynomials in ǫ ... Critical flow phenomenon 16/42

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