Herv´ Lemonnier e 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

HEAT TRANSFER MECHANISMS

• Condensation heat transfer: – drop condensation – ﬁlm condensation • Boiling heat transfer: – Pool boiling, natural convection, ´bullition en vase e – Convective boiling, forced convection, • Only for pure ﬂuids. For mixtures see speciﬁc studies. Usually in a mixture, h xi hi and possibly hi . • Many deﬁnitions of heat transfer coeﬃcient, q , h[W/m /K] = ∆T

2

hL Nu = , k

k(T ?)

Condensation and boiling heat transfer

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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

Condensation and boiling heat transfer

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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, h(L) 1 L

L

h(z)dz

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 3/42

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, Γ ML , P ReF 4Γ µL

– Smooth, laminar, ReF < 30, – Wavy laminar, 30 < ReF < 1600 – Wavy turbulent, ReF > 1600

Condensation and boiling heat transfer

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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

1

1 4

• Averaged heat transfer coeﬃcient (TW = cst) : h(z) ∝ z − 4 , h(L) = 4 h(L) 3 • Condensate ﬁlm ﬂow rate, energy balance at the interface, Γ(L) = h(L)(Tsat − TP )L hLV

• Heat transfer coeﬃcient-ﬂow rate relation, ¯ h(L) kL µ2 L ρL (ρL − ρV )

1 3

=

−1 1, 47 ReF 3 1 2 (TW

• hLV and ρV at saturation. kL , ρL at the ﬁlm temperature TF

1 • µ = 4 (3µL (TP ) + µL (Ti )), exact when 1/µL linear with T .

+Ti ),

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SUPERHEATED VAPOR

• Two heat sources: vapor (TV > Ti ) and interface. • Increase of heat transfer wrt to saturated conditions, empirical correction, ¯ ¯ hS (L) = h(L) 1 + CP V (TV − Tsat ) hLV

1 4

• Energy balance at the interface, ﬁlm ﬂow rate, ¯ hS (L)(TW − Tsat )L Γ(L) = hLV + CP V (TV − Tsat )

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FILM FLOW RATE-HEAT TRANSFER COEFFICIENT

• Laminar, ¯ h(L) kL µ2 L ρL (ρL − ρV )

1 3

=

−1 1, 47 ReF 3

−0,22 • Wavy laminar and previous regime (Kutateladze, 1963), h(z) ∝ ReF ),

¯ h(L) kL

µ2 L ρL (ρL − ρV )

1 3

ReF = 1, 08Re1,22 − 5, 2 F

• Turbulent and previous regimes (Labuntsov, 1975), h(z) ∝ Re0,25 , F ¯ h(L) kL µ2 L ρL (ρL − ρV )

1 3

ReF = 8750 + 58Pr−0,5 (Re0,75 − 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, h ∝ τi

1 2

• See for example Delhaye (2008, Ch. 9, p. 370) • When 2 eﬀects are comparable, h1 stagnant, h2 with dominant shear , h= (h2 1 +

2 1 h2 ) 2

Condensation and boiling heat transfer

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CONDENSATION ON HORIZONTAL TUBES

• Heat transfer coeﬃcient deﬁnition, ¯= 1 h π π h(u)du

0

• Stagnant vapor conditions, laminar ﬁlm, Nusselt (1916) ¯ h= 0.728 (0.70)

3 kL ρL (ρL

− ρV )ghLV µL (Tsat − Tp )D

1 4

• 0.728, imposed temperature, 0.70, imposed heat ﬂux. • Γ, ﬁlm ﬂow rate per unit length of tube.

Condensation and boiling heat transfer

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• Film ﬂow rate- heat transfer coeﬃcient, energy balance, ¯ h kL µ2 L ρL (ρL − ρV )

1 3

=

1.51 (1.47)

−1 ReF 3

• Vapor superheat and transport proprieties, same as vertical wall • Eﬀect of steam velocity (Fujii), ¯ h = 1.4 h0 u2 (Tsat − TP )kL V gDhLV µL

0.05

¯ h 1< < 1.7, h0

• Tube number eﬀect in bundles, (Kern, 1958), h(1, N ) = N −1/6 h1

Condensation and boiling heat transfer

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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

Condensation and boiling heat transfer

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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)

Condensation and boiling heat transfer

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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 – Imposed power heating: P

6

I= J

Condensation and boiling heat transfer

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BOILING CURVE

• Imposed heat ﬂux, q (W /c m

2

)

P = qπDl = U I

1 1 6

• Wall and wire temperature are equal, D→0 U R(T ) = , I < T > 3 ≈ TW | |

• Wall super-heat: ∆T = TW − Tsat

2 4 3 5 2 0 0 , T sa t

• Heat transfer coeﬃcient,

(°C )

h

TW

q − Tsat

Condensation and boiling heat transfer

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BOILING CURVE

G F AD H e a t flu x C H B A E

W a ll s u p e rh e a t

http://www-heat.uta.edu, Next

Condensation and boiling heat transfer

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HEAT TRANSFER REGIMES q N u c le a te b o ilin g F lu x m a x . B u rn -o u t

D , T 0, q

0

G

H

F lu x m in .

F ilm

b o ilin g

A

, T

sa t

• OA: Natural convection • AD: Nucleate boiling

• DH: Transition boiling • HG: Film boiling

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TRANSITION BOILING STABILITY q N u c le a te b o ilin g F lu x m a x . B u rn -o u t

• Wire energy balance,

D , T 0, q

0

G

M Cv

dT = P − qS dt

H

F lu x m in .

• Linearize at ∆T0 , q0 , T = T0 + T1 ,

F ilm b o ilin g

A

, T

M Cv sa t

dT1 ∂q = P − q0 S −S T1 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 q = h(TL∞ − TLi ) = h ∆T − ∆T > ∆Teq =

2σ dT R dp sat ,

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 20/42

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 – Length: detachment diameter, capillary length: L ≈ – Liquid velocity: energy balance, q = mhLV , V ≈ ˙ Ja q ρL hLV σ g(ρL −ρV )

• Csf

CpL (TP − Tsat ) = Csf Re0.33 Prs L hLV ≈ 0.013, s = 1 water, s = 1.7 other ﬂuids.

Condensation and boiling heat transfer 21/42

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, λT = 2π 3 √ σ , g(ρL − ρV ) 1 σ 2 ρV UV < π , 2 λH qA = ρV UV AJ hLV

Condensation and boiling heat transfer

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• Zuber (1958), jet radius RJ = 1 λT , λH = 2πRJ , marginal stability, 4 qCHF = 0.12ρV hLV

1/2

4

σg(ρV − ρL )

• Lienhard & Dhir (1973), jet radius RJ = 1 λT , λH = λT , 4 qCHF = 0.15ρV hLV

1/2

4

σg(ρV − ρL )

• Kutateladze (1948), dimensional analysis and experiments, qCHF = 0.13ρV hLV

1/2

4

σg(ρV − ρL )

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Condensation and boiling heat transfer

FILM BOILING

• Analogy with condensation (Nusselt, Rohsenow), Bromley (1950), V NuL = 0.62 ρV g(ρL − ρV )hLV D µV kV (TW − Tsat )

3

1 4

L

,

hLV = hLV

CP V (TW − Tsat ) 1 + 0.34 hLV

• Transport and thermodynamical properties: – Liquid at saturation Tsat ,

1 – Vapor at the ﬁlm temperature, TF = 2 (Tsat + TW ).

• Radiation correction: TW > 300o C,

: emissivity, σ = 5, 67 10−8 W/m2 /K4 o 4 4 σ(TW − Tsat ) h = h(T < 300 C) + TW − Tsat

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TRANSITION BOILING

• Minimum ﬂux, qmin = ChLV

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 25/42

SUB-COOLING EFFECT

• Liquid sub-cooling, TL < Tsat , ∆Tsub • Ivey & Morris (1961) qC,sub = qC,sat 1 + 0, 1

Tsat − TL

ρL ρV

3/4

CP L ∆Tsub hLV

Condensation and boiling heat transfer

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CONVECTIVE BOILING REGIMES

→ Increasing heat ﬂux, constant ﬂow rate → 1. Onset of nucleate boiling 2. Nucleate boiling suppression 3. Liquid ﬁlm dry-out 4. Super-heated vapor

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Condensation and boiling heat transfer

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 hLV = qP M 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)

Condensation and boiling heat transfer

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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 30/42

• Nucleate boiling suppression, • Liquid ﬁlm dry-out, boiling crisis (I), • Single-phase vapor convection.

Condensation and boiling heat transfer 31/42

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 34/42

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), q ¯ ¯ ¯ ¯ h(TW − TF ), 1 ¯ TF = (TF in + TF out ), 2 Tav = 1 ¯ ¯ (TW + TF ) 2

• Always check the original papers...

Condensation and boiling heat transfer 35/42

NUCLEATE BOILING & SIGNIFICANT VOID

• Onset and suppression of nucleate boiling, ONB, (Frost & Dzakowic, 1967), TP − Tsat = 8σqTsat kL ρV hLV

0,5

PrL

• Onset of signiﬁant void, OSV, (Saha & Zuber, 1974) qD Nu = = 455, kL (Tsat − TL ) q St = = 0, 0065, GCP L (Tsat − TL ) P´ < 7 104 , thermal regime e P´ > 7 104 , hydrodynamic regime e

36/42

Condensation and boiling heat transfer

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

0.1

1/X > 0.1

37/42

CHEN CORRELATION (CT’D)

• Nucleate boiling, hF Z • Forced convection

0.79 0.45 kL CpL ρ0.49 L = 0.00122 0.29 0.24 0.24 (TW − Tsat )0.24 ∆p0.75 sat σµL hLV ρV

kL 0.8 0.4 Re PrL hDB = 0.023 D • From Clapeyron relation, slope of saturation line, ∆psat hLV (TW − Tsat ) = Tsat (vV − vL )

• Non dimensional numbers deﬁnitions, GD(1 − xeq ) , Re = µL X= 1 − xeq xeq

0.9

ρV ρL

0.5

µL µV

0.1

,

PrL =

µL CpL kL

• NB: implicit in (TW − Tsat ).

Condensation and boiling heat transfer 38/42

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 7000 CHF[kW/m ] 6000 5000 4000 3000 2000 1000 0 −20 0 20 40 exit quality [%] 60 80 100 0 −20 0 20 40 exit quality [%] 60 80 100

2

G=1000 kg/s/m P= 10 bar P= 30 bar P= 45 bar P= 70 bar P= 100 bar P= 150 bar P= 200 bar

2

6000

5000

p=150 bar G= 0 kg/s/m2 G=1000 kg/s/m2 G=5000 kg/s/m2 G=7500 kg/s/m2

4000 CHF[kW/m2]

3000

2000

1000

• 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 40/42

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´nieur. e Delhaye, J.-M. 2008. Thermohydraulique des r´acteurs nucl´aires. Collection g´nie atome e e ique. 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.

Condensation and boiling heat transfer

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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´nieur. e Delhaye, J.-M. 2008. Thermohydraulique des r´acteurs nucl´aires. Collection g´nie atome e e ique. 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. Condensation and boiling heat transfer 42/42