A SHORT INTRODUCTION TO TWO-PHASE FLOWS Condensation and boiling heat transfer 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
HEAT TRANSFER MECHANISMS • Condensation heat transfer: – drop condensation – film condensation • Boiling heat transfer: – Pool boiling, natural convection, ´ebullition en vase – Convective boiling, forced convection, • Only for pure fluids. For mixtures see specific studies. Usually in a P mixture, h 6 xi hi and possibly hi . • Many definitions of heat transfer coefficient, q h[W/m /K] = , ∆T 2
hL Nu = , k
Condensation and boiling heat transfer
k(T ?)
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CONDENSATION OF PURE VAPOR • Flow patterns – Liquid film flowing. – Drops, static, hydrophobic wall (θ ≈ π). Clean wall, better htc. • Fluid mixture non-condensible gases: – Incondensible accumulation at cold places. – Diffusion resistance. – Heat transfer deteriorates. – Traces may alter significantly h
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FILM CONDENSATION • Thermodynamic equilibrium at the interface, Ti = Tsat (p∞ ) • Local heat transfer coefficient, h(z) ,
q q = T i − Tp Tsat − Tp
• Averaged heat transfer coefficient, Z 1 L h(L) , h(z)dz L 0 • NB: Binary mixtures Ti (xα , p) and pα (xα , p). Approximate equilibrium conditions, – For non condensible gases in vapor, pV = xPsat (Ti ), Raoult relation – For dissolved gases in water, pG = HxG , Henry’s relation Condensation and boiling heat transfer
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CONTROLLING MECHANISMS • Slow film, little convective effect, conduction through the film (main thermal resistance) • Heat transfer controlled by film characteristics, thickness, waves, turbulence. • Heat transfer regimes, ML , Γ, P
4Γ ReF , µL
– Smooth, laminar, ReF < 30, – Wavy laminar, 30 < ReF < 1600 – Wavy turbulent, ReF > 1600
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CONDENSATION OF SATURATED STEAM • Simplest situation, only a single heat source: interface, stagnant vapor, • Laminar film (Nusselt, 1916, Rohsenow, 1956), correction 10 to 15%, h(z) =
3 kL ρL g(ρL
− ρV )(hLV +0, 68CP L [Tsat − TP ]) 4µL (Tsat − TP )z
14
1
• Averaged heat transfer coefficient (TW = cst) : h(z) ∝ z − 4 , h(L) = 43 h(L) • Condensate film flow rate, energy balance at the interface, Γ(L) =
h(L)(Tsat − TP )L hLV
• Heat transfer coefficient-flow rate relation, 1
3 2 ¯ h(L) µL − 13 = 1, 47 ReF kL ρL (ρL − ρV ) • hLV and ρV at saturation. kL , ρL at the film temperature TF , 12 (TW +Ti ), • µ = 41 (3µL (TP ) + µL (Ti )), exact when 1/µL linear with T . Condensation and boiling heat transfer
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SUPERHEATED VAPOR • Two heat sources: vapor (TV > Ti ) and interface. • Increase of heat transfer wrt to saturated conditions, empirical correction, 14 1 + CP V (TV − Tsat ) ¯ ¯ hS (L) = h(L) hLV • Energy balance at the interface, film flow rate, ¯ S (L)(TW − Tsat )L h Γ(L) = hLV + CP V (TV − Tsat )
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FILM FLOW RATE-HEAT TRANSFER COEFFICIENT • Laminar, 1
3 2 ¯ µL h(L) − 13 = 1, 47 ReF kL ρL (ρL − ρV ) −0,22 • Wavy laminar and previous regime (Kutateladze, 1963), h(z) ∝ ReF ), 1
3 2 ¯ h(L) µL ReF = kL ρL (ρL − ρV ) 1, 08Re1,22 − 5, 2 F • Turbulent and previous regimes (Labuntsov, 1975), h(z) ∝ Re0,25 F , 1
3 2 ¯ h(L) µL ReF = 0,75 kL ρL (ρL − ρV ) 8750 + 58Pr−0,5 (Re − 253) F F • NB: Implicit relation, ReF depends on h(L) through Γ.
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OTHER MISCELLANEOUS EFFECTS • Steam velocity, vV , when dominant effect, • Vv descending flow, vapor shear added to gravity, • Decreases fil thickness, • Delays transition to turbulence turbulence, 1 2
h ∝ τi
• See for example Delhaye (2008, Ch. 9, p. 370) • When 2 effects are comparable, h1 stagnant, h2 with dominant shear , h=
(h21
+
2 12 h2 )
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CONDENSATION ON HORIZONTAL TUBES • Heat transfer coefficient definition, Z π 1 ¯= h h(u)du π 0 • Stagnant vapor conditions, laminar film, Nusselt (1916) ¯= h
0.728 (0.70)
3 kL ρL (ρL
− ρV )ghLV µL (Tsat − Tp )D
14
• 0.728, imposed temperature, 0.70, imposed heat flux. • Γ, film flow rate per unit length of tube.
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• Film flow rate- heat transfer coefficient, energy balance, 13 2 ¯ 1.51 µL h − 13 = ReF kL ρL (ρL − ρV ) (1.47) • Vapor superheat and transport proprieties, same as vertical wall • Effect of steam velocity (Fujii), 2 0.05 ¯ h uV (Tsat − TP )kL = 1.4 h0 gDhLV µL
¯ h 1< < 1.7, h0
• Tube number effect in bundles, (Kern, 1958), h(1, N ) = N −1/6 h1
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DROP CONDENSATION • Mechanisms, – Nucleation at the wall, – Drop growth, – Coalescence, – Dripping down (non wetting wall) • Technological perspective, – Wall doping or coating – Clean walls required, fragile – Surface energy gradient walls. Selfdraining
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• heat transfer coefficient, 1 1 1 1 1 = + + + h hG hd hi hco • G : non-condensible gas, d : drop, i : phase change, co coating thickness. • Non-condensible gases effect, ωi ≈ 0, 02 ⇒ h → h/5 • Example, steam on copper, Tsat > 22o C, h in W/cm2 /o C, hd = min(0, 5 + 0, 2Tsat , 25)
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POOL BOILING 1 7
• Nukiyama (1934) • Only one heat sink, stagnant saturated water, • Wire NiCr and Pt, – Diameter: ≈ 50µm, – Length: l
6
– Imposed power heating: P
I= J
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BOILING CURVE • Imposed heat flux, q (W /c m 2
)
P = qπDl = U I • Wall and wire temperature are equal, D→0
1 1 6
U R(T ) = , I
| 3 ≈ TW
• Wall super-heat: ∆T = TW − Tsat 2 4
• Heat transfer coefficient, 3 5
2 0 0
, T
sa t
(°C )
Condensation and boiling heat transfer
h,
TW
q − Tsat
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BOILING CURVE G F E
H e a t flu x
AD C H
B A
W a ll s u p e rh e a t
http://www-heat.uta.edu, Next
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HEAT TRANSFER REGIMES q B u rn -o u t
N u c le a te b o ilin g
F lu x m a x .
G D , T 0, q 0
H
F ilm
b o ilin g
F lu x m in .
A
, T
• OA: Natural convection
• DH: Transition boiling
• AD: Nucleate boiling
• HG: Film boiling
Condensation and boiling heat transfer
sa t
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TRANSITION BOILING STABILITY • Wire energy balance, q B u rn -o u t
F lu x m a x .
N u c le a te b o ilin g
G D , T 0, q
M Cv
0
H
dT = P − qS dt
• Linearize at ∆T0 , q0 , T = T0 + T1 , F ilm
b o ilin g
F lu x m in .
A
, T
M Cv sa t
dT1 ∂q = P − q0 S −S T1 | {z } dt ∂∆T =0
• Solution, linear ODE, T1 = T10 exp(−αt),
α=
S M Cv
∂q ∂∆T
T0
• 2 stable solutions, one unstable (DH), ∂q 0, R˙ > 0 2σ dT q = h(TL∞ − TLi ) = h ∆T − R dp sat ∆T > ∆Teq =
2σ dT R dp sat ,
R > Req =
2σ dT ∆T dp sat
1 bar, ∆T = 3o C, Req = 5, 2 µm, 155 bar, ∆T = 3o C, Req = 0, 08 µm Condensation and boiling heat transfer
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NUCLEATE BOILING MECHANISMS • Super-heated liquid transport, Yagumata et al. (1955) q ∝ (TP − Tsat )1.2 n0.33 • n: active sites number density, 5÷6 3 n ∝ ∆Tsat ⇒ q ∝ ∆Tsat
• Very hight heat transfer, precision unnecessary. • Rohsenow (1952), analogy with convective h. t.: Nu = CRea Prb , ρL V L • Scales : Re = , µL q – Length: detachment diameter, capillary length: L ≈ g(ρLσ−ρV ) – Liquid velocity: energy balance, q = mh ˙ LV , V ≈
q ρL hLV
CpL (TP − Tsat ) = Csf Re0.33 PrsL hLV ≈ 0.013, s = 1 water, s = 1.7 other fluids. Ja ,
• Csf
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BOILING CRISIS, CRITICAL HEAT FLUX
• Flow pattern close to CHF: critical heat flux ), Rayleigh-Taylor instability, • Stability of the vapor column: Kelvin-Helmholtz, • Energy balance over A, √ r σ , λT = 2π 3 g(ρL − ρV )
1 σ ρV UV2 < π , 2 λH
Condensation and boiling heat transfer
qA = ρV UV AJ hLV
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• Zuber (1958), jet radius RJ = 14 λT , λH = 2πRJ , marginal stability, p 1/2 qCHF = 0.12ρV hLV 4 σg(ρV − ρL ) • Lienhard & Dhir (1973), jet radius RJ = 14 λT , λH = λT , p 1/2 qCHF = 0.15ρV hLV 4 σg(ρV − ρL ) • Kutateladze (1948), dimensional analysis and experiments, p 1/2 qCHF = 0.13ρV hLV 4 σg(ρV − ρL ) Condensation and boiling heat transfer
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FILM BOILING
• Analogy with condensation (Nusselt, Rohsenow), Bromley (1950), V L NuL = 0.62
ρV )h0LV
ρV g(ρL − D µV kV (TW − Tsat )
3
41
,
h0LV
= hLV
CP V (TW − Tsat ) 1 + 0.34 hLV
• Transport and thermodynamical properties: – Liquid at saturation Tsat , – Vapor at the film temperature, TF = 21 (Tsat + TW ). • Radiation correction: TW > 300o C, : emissivity, σ = 5, 67 10−8 W/m2 /K4 4 4 σ(TW − Tsat ) h = h(T < 300 C) + TW − Tsat o
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TRANSITION BOILING
• Minimum flux, qmin = ChLV
s 4
σg(ρL − ρV ) (ρL + ρV )2
– Zuber (1959), C = 0.13, stability of film boiling, – Berenson (1960), C = 0, 09, rewetting, Liendenfrost temperature. • Scarce data in transition boiling, • Quick fix, ∆Tmin and ∆Tmax , from each neighboring regime (NB and FB), • Linear evolution in between (log-log plot!). Condensation and boiling heat transfer
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SUB-COOLING EFFECT
• Liquid sub-cooling, TL < Tsat , ∆Tsub , Tsat − TL • Ivey & Morris (1961) qC,sub = qC,sat
1 + 0, 1
ρL ρV
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CP L ∆Tsub hLV
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!
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CONVECTIVE BOILING REGIMES
→ Increasing heat flux, constant flow rate → 1. Onset of nucleate boiling
3. Liquid film dry-out
2. Nucleate boiling suppression
4. Super-heated vapor
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BACK TO THE EQUILIBRIUM (STEAM) QUALITY • Regime boundaries depend very much on z. Change of variable, xeq • Equilibrium quality, non dimensional mixture enthalpy, xeq ,
h − hLsat hLV
• Energy balance, low velocity, stationary flows, dh dxeq M = M hLV = qP dz dz • Uniform heat flux, xeq linear in z. Close to equilibrium, xeq ≈ x • According to the assumptions of the HEM, 0 > xeq 0 < xeq < 1 1 < xeq
single-phase liquid (sub-cooled) two-phase, saturated single-phase vapor (super-heated)
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CONVECTIVE HEAT TRANSFER IN VERTICAL FLOWS
Boiling flow description • Constant heat flux heating, • Fluid temperature evolution, (Tsat ), • Wall temperature measurement, • Flow regime, • Heat transfer controlling mechanism.
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From the inlet, flow and heat transfer regimes, • Single-phase convection • Onset of nucleate boiling, ONB • Onset of signifiant void, OSV • Important points for pressure drop calculations, flow oscillations. Condensation and boiling heat transfer
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• Nucleate boiling suppression, • Liquid film dry-out, boiling crisis (I), • Single-phase vapor convection. Condensation and boiling heat transfer
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HEAT TRANSFER COEFFICIENT
DO: dry-out, DNB: departure from nucleate boiling (saturated, sub-cooled), PDO: post dry-out, sat FB: saturated film boiling, Sc Film B: sub-cooled film boiling
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BOILING SURFACE
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S-Phase conv: single-phase convection, PB: partial boiling, NB: nucleate boiling (S, saturated, Sc, subcooled), FB: film boiling, PDO: post dry-out, DO: dry-out, DNB: departure from nucleate boiling. Condensation and boiling heat transfer
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SINGLE-PHASE FORCED CONVECTION • Forced convection (Dittus & Boelter, Colburn), Re > 104 , Nu ,
hD = 0, 023Re0,8 Pr0,4 , kL
Re =
GD , µL
PrL =
µL CP L kL
• Fluid temperature, TF , mixing cup temperature, that corresponding to the area-averaged mean enthalpy. • Transport properties at Tav – Local heat transfer coefficient, q , h(TW − TF ),
Tav =
1 (TW + TF ) 2
– Averaged heat transfer coefficient (length L), ¯ T¯W − T¯F ), q¯ , h(
1 T¯F = (TF in + TF out ), 2
Tav =
1 ¯ (TW + T¯F ) 2
• Always check the original papers... Condensation and boiling heat transfer
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NUCLEATE BOILING & SIGNIFICANT VOID
• Onset and suppression of nucleate boiling, ONB, (Frost & Dzakowic, 1967), 0,5 8σqTsat TP − Tsat = PrL kL ρV hLV • Onset of signifiant void, OSV, (Saha & Zuber, 1974) qD Nu = = 455, kL (Tsat − TL ) q St = = 0, 0065, GCP L (Tsat − TL )
P´e < 7 104 , thermal regime P´e > 7 104 , hydrodynamic regime
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DEVELOPPED BOILING AND CONVECTION
• Weighting of two mechanisms, xeq > 0 (Chen, 1966) – Nucleate boiling(Forster & Zuber, 1955), S, suppression factor,same model for pool boiling, – Forced convection, Dittus Boelter, F , amplification factor, h = hFZ S + hDB A 1 1 = 1 + 2.53 10−6 (ReF 1.25 )1.17 , F = 2.35(1/X + 0.213)0.736 S Condensation and boiling heat transfer
1/X 6 0.1 1/X > 0.1 37/42
CHEN CORRELATION (CT’D) • Nucleate boiling, hF Z
0.79 0.45 0.49 kL CpL ρL = 0.00122 0.29 0.24 0.24 (TW − Tsat )0.24 ∆p0.75 sat σµL hLV ρV
• Forced convection
kL 0.8 0.4 hDB = 0.023 Re PrL D • From Clapeyron relation, slope of saturation line, ∆psat
hLV (TW − Tsat ) = Tsat (vV − vL )
• Non dimensional numbers definitions, 0.9 0.5 0.1 GD(1 − xeq ) 1 − xeq ρV µL , X= , Re = µL xeq ρL µV
PrL =
µL CpL kL
• NB: implicit in (TW − Tsat ). Condensation and boiling heat transfer
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CRITICAL HEAT FLUX • No general model. – Dry-out, multi-field modeling – DNB, correlations or experiment in real bundles • Very sensitive to geometry, mixing grids, • Recourse to experiment is compulsory, • In general, qCHF (p, G, L, ∆Hi , ...), artificial reduction of dispersion. • For tubes and uniform heating, no length effect, qCHF (p, G, xeq ) – Tables by Groenveld, – Bowring (1972) correlation, best for water in tubes – Correlation by Katto & Ohno (1984), non dimensional, many fluids, regime identification.
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MAIN PARAMETERS EFFECT ON CHF After Groeneveld & Snoek (1986), tube diameter, D = 8 mm. 10000 9000 8000
p=150 bar G= 0 kg/s/m22 G=1000 kg/s/m G=5000 kg/s/m22 G=7500 kg/s/m
5000
4000 CHF[kW/m2]
7000 CHF[kW/m2]
6000
G=1000 kg/s/m2 P= 10 bar P= 30 bar P= 45 bar P= 70 bar P= 100 bar P= 150 bar P= 200 bar
6000 5000 4000
3000
2000
3000 2000
1000
1000 0 −20
0
20
40 exit quality [%]
60
80
100
0 −20
0
20
40 exit quality [%]
60
80
• Generally decreases with the increase of the exit quality. qCHF → 0, xeq → 1. • Generally increases with the increase of the mass flux, • CHF is non monotonic with pressure. Condensation and boiling heat transfer
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100
MORE ON HEAT TRANSFER • Boiling and condensation, – Delhaye (1990) – Delhaye (2008) – Roshenow et al. (1998) – Collier & Thome (1994) – Groeneveld & Snoek (1986) • Single-phase, – Bird et al. (2007) – Bejan (1993)
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REFERENCES Bejan, A. (ed). 1993. Heat transfer. John Wiley & Sons. Bird, R. B., Stewart, W. E., & Lightfoot, E. N. 2007. Transport phenomena. Revised second edn. John Wiley & Sons. Collier, J. G., & Thome, J. R. 1994. Convective boiling and condensation. third edn. Oxford: Clarendon Press. Delhaye, J. M. 1990. Transferts de chaleur : ebullition ou condensation des corps purs. Techniques de l’ing´enieur. Delhaye, J.-M. 2008. Thermohydraulique des r´eacteurs nucl´eaires. Collection g´enie atomique. EDP Sciences. Groeneveld, D. C., & Snoek, C. V. 1986. Multiphase Science and Technology. Vol. 2. Hemisphere. G. F. Hewitt, J.-M. Delhaye, N. Zuber, Eds. Chap. 3: a comprehensive examination of heat transfer correlations suitable for reactor safety analysis, pages 181–274. Raithby, G. D., & Hollands, K. G. 1998. Handbook of heat transfer. 3rd edn. McGrawHill. W. M. Roshenow, J. P. Hartnett and Y. I Cho, Eds. Chap. 4-Natural convection, pages 4.1–4.99. Roshenow, W. M., Hartnett, J. P., & Cho, Y. I. 1998. Handbook of heat transfer. 3rd edn. McGraw-Hill.
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