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
[email protected], 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)
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COMPOSITE AVERAGING THE MASS BALANCE • Space and time averaged mass balance: ∂ ∂ ARk2 < ρk >2 + ARk2 < ρk wk >2 = Γk , ∂t ∂z
Z Γk , − Ci
m ˙k
dl nk nkC
• Mean value definitions: Rk2 < ρk >2 , Rk2 ρk = αk ρk ,
Rk2 < ρk wk >2 , αk ρk vk = M k
• With these new variables, ∂ ∂ Aαk ρk + Aαk ρk vk = Γk ∂t ∂z • No assumptions, simple change of variable. α(r) 6= α and wk (r) 6= vk are non-uniform.
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MIXTURE MASS BALANCE • Add the phase mass balances, ∂ ∂ A(α1 ρ1 + α2 ρ2 ) + A(α1 ρ1 v1 + α2 ρ2 v2 ) = 0 ∂t ∂z • Mixture density (definition), ρ = α1 ρ1 + α2 ρ2 • Mixture velocity defined as to preserves the mass flow rate, ρv = α1 ρ1 v1 + α2 ρ2 v2 • Mixture mass balance, ∂ ∂ Aρ + Aρv = 0 ∂t ∂z
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MOMENTUM BALANCE • Simplified, 1 pressure (instead of 3), neglect the effect longitudinal diffusion. ∂ ∂ ∂ 2 A< Rk2 ρk wk >2 + A< Rk2 ρk wk >2 + ARk2 < pk >2 − ARk2 < ρk gz >2 ∂t ∂z ∂z Z Z dl dl =− (m ˙ k wk − nk Vk nz ) + nk Vk nz nk nkC nk nkC Ci Ck • 1D assumption: velocity space correlation C, mean pressure, pk , the so-called flat profile assumption. < Rk2 ρk wk2 >2 C, = 1, 2 αk ρk vk • Interaction terms, change of variable: Z dl − = Γk vki , (m ˙ k wk ) nk nkC Ci Z
dl nk Vk nz = −Aγτki , n n k kC Ci
Rk2
∂ ∂pk < pk >2 = αk ∂z ∂z
Z Ci
dl = A< | γ>| 2 = Aγ nk nkC
Z
dl nk Vk nz = −Pk τwk n n k kC Ck
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MOMENTUM BALANCE (CT’D) • New notations, in blue, the flat profile assumption main consequence, ∂ ∂ ∂pk 2 Aαk ρk vk + Aαk ρk vk − Aαk = Γk vki − Aγτki − Pk τwk · · · ∂t ∂z ∂z • Momentum balance for the mixture, single pressure ∂ ∂ ∂p A(α1 ρ1 v1 + α2 ρ2 v2 ) + A(α1 ρ1 v12 + α2 ρ2 v22 ) − A = −P τw ∂t ∂z ∂z • Another form of the inertia term, x , 2 G 2 2 αG ρG vG + αL ρL vL = 0, ρ
MG M ,
1 x2 (1 − x)2 = + , ρ0 αρG (1 − α)ρL
M G= . A
• Give two examples of inconsistency of the flat profile assumption.
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TOTAL ENERGY BALANCE • Energy balance, native form → enthalpy form, 1 2 ∂ 1 2 ∂ Ak < ρk uk + vk >2 + Ak < ρk wk uk + vk >2 ∂t 2 ∂z 2 ∂ + Ak < nz (qk − Tk vk ) >2 −Ak < ρk gk vk >2 ∂z Z 1 2 dl =− (m ˙ k uk + vk + nk (qk − Tk vk )) 2 nk nkC Ci ∪Ck • In the • In the
∂ ∂z ∂ ∂t
terms, uk → hk − pk /ρk , Tk → Vk , ∂ term uk → hk − pk /ρk adds − ∂t Ak < pk >2 , use the identity (2), Z ∂ ∂pk dl Ak < pk >2 = Ak < >2 + pk vi n ∂t ∂t nk nkC Ci ∪Ck
• Collect the pressure terms in the RHS,
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TOTAL ENERGY BALANCE (CT’D) • Energy balance in enthalpy form, ∂ 1 ∂pk ∂ 1 Ak < ρk hk + vk2 >2 −Ak < >2 + Ak < ρk wk hk + vk2 >2 ∂t 2 ∂t ∂z 2 ∂ + Ak < nz (qk − Vk vk ) >2 − Ak < ρk gk vk >2 ∂z Z 1 2 dl =− (m ˙ k hk + vk + nk (qk − Vk vk )) 2 nk nkC Ci ∪Ck • Neglect the diffusive term, vk2 ≈ vk2 , the mean enthalpy preserves the flux, ∂ 1 ∂pk ∂ 1 Aαk ρk hk + vk2 − Aαk + Aαk ρk vk hk + vk2 ∂t 2 ∂t ∂z 2 −Aαk ρk gk vk = Γk htki + Aγqki + Pk qkw
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MIXTURE ENERGY BALANCE • Add the two phase balances, interaction terms sum vanish at the interface, ∂ ∂ A(α1 ρ1 ht1 + α2 ρ2 ht2 ) + A(α1 ρ1 v1 ht1 + α2 ρ2 v2 ht2 ) ∂t ∂z −A
∂p − A(α1 ρ1 g1 v1 + α2 ρ2 g2 v2 ) = P qw , ∂t
• where the total enthalpy is htk , hk + 21 vk2 , • Other practical form of the enthalpy flux, AαV ρV vV htV + AαL ρL vL htL = M (xhtV + (1 − x)htL ) | {z } | {z } MV
ML
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THE HOMOGENEOUS MODEL AT THERMODYNAMIC EQUILIBRIUM (HEM) • 3 balances for the mixture + 3 assumptions – Mean velocity are equal : wV = wL ≡ α = β – Mean temperatures satisfy the equilibrium condition : TL = TV = Tsat (p) • Thermodynamique, EOS, ρL = ρLsat (p),
ρV = ρV sat (p),
hL = hLsat (p),
hV = hV sat (p)
• HEM void fraction, QG xρL = = α(x, p) α=β= QG + QL xρL + (1 − x)ρV • Balance equations are identical to that of single-phase flow, ∂ ∂ Aρ + Aρw = 0, ρ = αρV + (1 − α)ρL ∂t ∂z ∂ ∂ ∂p 2 Aρw + Aρw + A = −P τW + Aρgz ∂t ∂z ∂z ∂ 1 2 ∂p ∂ 1 2 Aρ(h + w ) − A + Aρw(h + w ) = P qW + Aρgz w ∂t 2 ∂t ∂z 2 1D-time averaged models
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HEM (CT’D) • Other practical form, combine with the mass balance, ∂ ∂ Aρ + Aρw = 0 ∂t ∂z ∂w ∂w ∂p P ρ + ρw + = − τW + ρgz ∂t ∂z ∂z A ∂ 1 2 ∂p ∂ 1 2 P ρ (h + w ) − + ρw (h + w ) = qW + ρgz w ∂t 2 ∂t ∂z 2 A • Mechanical energy balance, momentum balance ×w, ∂ 1 2 ∂ 1 2 ∂p P ρ w + ρw w +w = − wτW + ρgz w ∂t 2 ∂z 2 ∂z A dp • Entropy balance, T ds = dh − ρ ρT
∂s ∂s P + ρwT = (qW + wτW ) ∂t ∂z A
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HEM (CT’D) • Alternate form with total enthalpy and entropy, ∂ ∂ρ ∂w ρw dA ρ+w +ρ =− ∂t ∂z ∂z A dz ∂ 1 2 ∂p ∂ 1 2 P ρ (h + w ) − + ρw (h + w ) = qW + ρgz w ∂t 2 ∂t ∂z 2 A ∂s ∂s P ρT + ρwT = (qW + wτW ) ∂t ∂z A • Very important particular case: stationary flow, adiabatic, no friction nor volume forces, M = Aρw = cst 1 2 h + w = cst 2 s = xsV + (1 − x)sL = cst • Applications: flashing in long pipes, w/o heating, critical flow (no model !). 1D-time averaged models
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CLOSURES FOR THE HEM • Friction and heat flux, qW , τW , • Independent variables : x, w, et p, 1 • EOS, v = = xvV + (1 − x)vL , specific volume [m3 /kg], ρ vL = vLsat (p),
vV = vV sat (p), vV,p ,
vL,p ,
hL = hLsat (p), hV,p ,
hV = hV sat (p)
hL,p
• NB: back again to the thermodynamic consistency issue.
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DRIFT-FLUX MODEL • Different phase mean velocities, ”mechanical non-equilibrium”. • Mass balances for the liquid and vapor, momentum balance for the mixture, wV 6= wL , wm mixture velocity. • Additional closure (core modeling, FLICA) wV − wL = f (x, p, α, flow regime, · · · ) JGL = 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.
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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 ?).
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EXAMPLE: STRATIFIED FLOWS D 0
D
• Isothermal, incompressible, horizontal, µ = 0, σ = 0, m ˙ = 0, 2D. • Mass balance, ∂ h 1 ρ1 + ∂t ∂ h 2 ρ2 + ∂t
∂ h1 ρ1 < u1 >= 0 ∂z ∂ h2 ρ2 < u2 >= 0 ∂z
• Momentum balances, jump of momentum at the interface, ∂ ∂ ∂ ∂h1 h1 ρ1 < u1 > + h1 ρ1 < u21 > + h1 < p1 >= pi1 ∂t ∂z ∂z ∂z ∂ ∂ ∂ ∂h2 h2 ρ2 < u2 > + h2 ρ2 < u22 > + h2 < p2 >= pi2 ∂t ∂z ∂z ∂z pi1 = pi2 , pi 1D-time averaged models
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CLOSURES • 4 equations, 5 unknown variables h, < u1 >, < u2 >, < p1 >, < p2 >, • 3 unknown quantities: < u21 >, < u22 >, pi . • Topological relation,
F F
F
1 < p1 >= pi + ρ1 gh1 2 1 < p2 >= pi − ρ2 gh2 2 E
• Cannot be derived from momentum ⊥. • Spatial correlations, flat profile assumption, or relaxation < u2k > = 1, < uk >2
d 1 2 2 2 < uk >= < uk > − < uk >0 dt T
• Closed system. 1D-time averaged models
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STABILITY OF STRATIFIED FLOW ∂X • Solve PDE’s, A ∂X + B ∂t ∂z = 0, X = (h, u1 , u2 , p1 ) : ρ1 0 0 0 ρ1 u 1 −ρ 0 0 0 −ρ2 u2 2 A, , B , 1 0 ρ h 0 0 1 1 2 ρ1 gh1 0 0 ρ2 h2 0 (ρ1 − 12 ρ2 )gh2
ρ1 h 1
0
0
ρ2 h2
ρ1 u1 h1
h1
0
ρ2 u 2 h 2
• Use the perturbation method, Van Dyke (1975) : X = X0 + X1 + O(2 ), • Linearizes the PDE’s, separate the orders, ∂X0 ∂X0 A +B = 0, ∂t ∂z
X0 = cst
∂X1 ∂X1 + B(X0 ) =0 ∂t ∂z • X0 , base solution, X1 , first order (linear) perturbation, A(X0 )
• z ∈ [a, b], BC and IC are needed. 1D-time averaged models
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0
0 h2
STABILITY OF STRATIFIED FLOW (CT’D) e 1 exp i(ωt − kz), c = ω/k, phase velocity, • Progressive waves, X1 = X
• Temporal stability, X1 (a, t) = f (t), ω ∈ R, how (far) does perturbations propagates into the domain ? • Spatial stability: X1 (z, 0) = g(z), k ∈ R, perturbation amplification? • When the RHS of balance equations are non-zero, long wave assumption. e1 = 0 (cA(X0 ) − B(X0 ))X
e 1 ∈ ker(cA(X0 ) − B(X0 )) • One class of solution, X • Dispersion equation,
2
2
−ρ1 ρ2 h1 h2 ρ1 h2 (u1 − c) + ρ2 h1 (u2 − c) − (ρ1 − ρ2 )gh1 h2 = 0 • Stable if and only if the 2 roots are real, ρ1 h 2 + ρ2 h 1 (u1 − u2 ) 6 g(ρ1 − ρ2 ) ρ1 ρ2 2
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STABILITY OF STRATIFIED FLOW (CT’D) • Conditional stability : ∆u 6 ∆uC (u1 − u2 )2 6 g(ρ1 − ρ2 )
ρ1 h 2 + ρ2 h 1 ρ1 ρ2
– Kelvin-Helmholtz instability, – Heavy on top, light below, ρ2 > ρ1 , 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 ?
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PRESSURE DROP MODELING • Simplified flow model, – Mass balance of the mixture, – Momentum balance of the mixture, – If adiabatic: x = x0 , or solve the energy equation, – Evolution equation (wV 6= wL ), • Closures : XZ L,G
dl nk Vk nz = −P τW n n k kC Ck
=−
XZ L,G
dl nk qk = P qW n n k kC Ck
• NB: Constant pipe cross-sectional area, A = cst, nk nkC = 1
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PRESSURE DROP MODELING (CT’D) • Stationary flow, constant flow area, d ρw = 0, G = cst dz d dp P P 4 ρw2 + = − τW + ρgz , , dz dz A A Dh • Wall friction appears only in the momentum balance, d P dp dp dp dp = − ρw2 − τW + ρgz , + + dz dz A dz A dz F dz G • Experiments where
dp dz
and possibly α = RG2 are measured. • NB: evolution equation is used for dp : dz A
Use the models with the same set of assumptions.
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WALL FRICTION WITH THE HEM • Friction should not be dominant, P 4 dp = − τW = − τW dz F A Dh • NB: frictional pressure drop, f = 4Cf (quite tricky...) dp D dz τW CF = 1 2 , f = 1 2 F 2 ρw 2 ρw 1. Annular flow x ≈ 1, CF = 0, 005, flashing flows, x ≈ 0, CF = 0, 003. 2. x 1, CF = CF L , M = ML + MV , x ≈ 1, CF = CF G , M = ML + MV 3. NB: Single-phase friction factors, 0, 079 Re−0,25 , Re < 20 000 16 Poiseuille : , Blasius : 0, 046 Re−0,20 , Re > 20 000 Re 1D-time averaged models
Re ,
GD µ
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WALL FRICTION (CT’D) • Historical perspective, ”two-phase viscosity”, – Dukler (1964) : µ = βµg + (1 − β)µL , CF = 0, 0014 + 0, 125Re−0,32 – Ishii-Zuber (1978), (liquid-liquid or gas-liquid, αDM = 0, 62) µ = µC
αD 1− αDM
+0,4µC −2,5αDM µD µ +µ D
C
• Acceleration pressure drop, use the appropriate evolution (α 6= β), 2 2 dp d x (1 − x) = −G2 + dz A dz αρV (1 − α)ρL • Quality from the enthalpy balance, low velocity, thermal equilibrium, G
d 4 (xhV + (1 − x)hL ) = qW dz Dc
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TWO-COMPONENT, LOCKHART & MARTINELLI • Air-water experiments, low pressure, ∆pF et RG3 are measured (QCV). Three sets of experiments in the same horizontal pipe. Conditions
two-phase
Mass rate
M = M G + ML
dp dz
gas only
dp dz
F
MG dp dz
liquid only
G
ML dp dz
L
• Definitions on the non-dimensional friction pressure drop: (two-phase pressure drop multiplier ) Φ2L ,
dp dz
dp dz
, L
Φ2G ,
dp dz
dp dz
, G
X2 ,
dp dz
dp dz
L
G
• Blasius is used (Cf = 0, 046 Re−0,2 ), X, L. & M. parameter 0,1 0,9 0,5 µL 1−x ρG Xtt = µG x ρL 1D-time averaged models
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LOCKHART & MARTINELLI CORRELATION 1
ΦL ΦG
10
1 0.01
α
Φ
100
0.1
ΦL =
1 X
10
100
αL αG
0.1
0.01 0.01
0.1
1
10
100
X
r
1 20 1+ + 2 X X
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αL = √
X 1 + 20X + X 2
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1000
STEAM-WATER, MARTINELLI & NELSON • Steam water experiments, 34.5 ÷ 207 bar, ∆pG ≈ 0. Two experiments in the same helical tube. ∆p = ∆pF + ∆pA . Conditions Mass rate dp dz
F
two-phase
liquid only
M
ML = M
dp dz
dp dz
fo
• Two-phase multiplier,
Φ2f 0 =
dp dz
dp dz
fo
• Evolution equation (acceleration pressure drop), data and models.
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MARTINELLI & NELSON CORRELATIONS
Φ2f 0 =
dp dz
dp dz
Void fraction vs quality and pressure (bar)
fo
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BOILING FLOWS
• Boiling flows, Thom. Z xS ∆pF 1 r3 = = Φ2Lo dx ∆pF o xS 0 • Other methods, see Delhaye (2008b).
Friction multiplier vs pressure and quality (%). 1D-time averaged models
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FRIEDEL’S CORRELATION • Thousands of data reduction, various fluids, non-dimensional. Arbitrary orientation. Two-phase multiplier by Martinelli & Nelson. M and x are given, Φ2Lo
=
dp dz
dp dz
F = E +
3, 24F H Fr0,045 We0,035
Lo
−1
x 1−x G2 D G2 ρh = + , We = , Fr = ρG ρL σρh gDρ2h 0,91 0,19 0,7 µG ρL µG 1− , F = x0,78 (1 − x)0,224 H= ρG µL µL ρL CF Go CF Go = CF G (M ), CF Lo = CF L (M ), E = (1 − x)2 + x2 ρG CF Lo
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REFERENCES Delhaye, J.-M. 2008a. Thermohydraulique des r´eacteurs nucl´eaires. Collection g´enie atomique. EDP Sciences. Chap. 7-Mod´elisation des ´ecoulements diphasiques en conduite, pages 231–274. Delhaye, J.-M. 2008b. Thermohydraulique des r´eacteurs nucl´eaires. Collection g´enie atomique. EDP Sciences. Chap. 8-Pertes de pression dans les conduites, pages 275– 317. Ishii, M., & Hibiki, T. 2006. Thermo-fluid dynamics of two-phase flows. Springer. Van Dyke, M. 1975. Perturbation methods in fluid mechanics. Parabolic Press.
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