A SHORT INTRODUCTION TO TWO-PHASE FLOWS Condensation and boiling heat transfer

Transcription

1 A SHORT INTRODUCTION TO TWO-PHASE FLOWS Condensation and boiling heat transfer 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 HEAT TRANSFER MECHANISMS Condensation heat transfer: drop condensation film condensation Boiling heat transfer: Pool boiling, natural convection, ébullition en vase Convective boiling, forced convection, Only for pure fluids. For mixtures see specific studies. Usually in a mixture, h x i h i and possibly h i. Many definitions of heat transfer coefficient, h[w/m 2 /K] = q T, Nu = hl k, k(t?) Condensation and boiling heat transfer 1/42

3 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 2/42

4 FILM CONDENSATION Thermodynamic equilibrium at the interface, T i = T sat (p ) Local heat transfer coefficient, h(z) q T i T p = q T sat T p Averaged heat transfer coefficient, h(l) 1 L L 0 h(z)dz NB: Binary mixtures T i (x α, p) and p α (x α, p). Approximate equilibrium conditions, For non condensible gases in vapor, p V = xp sat (T i ), Raoult relation For dissolved gases in water, p G = Hx G, Henry s relation Condensation and boiling heat transfer 3/42

5 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 M L P, Re F 4Γ µ L Smooth, laminar, Re F < 30, Wavy laminar, 30 < Re F < 1600 Wavy turbulent, Re F > 1600 Condensation and boiling heat transfer 4/42

6 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) = ( k 3 L ρ L g(ρ L ρ V )(h LV +0, 68C P L [T sat T P ]) 4µ L (T sat T P )z ) 1 4 Averaged heat transfer coefficient (T W = cst) : h(z) z 1 4, h(l) = 4 3 h(l) Condensate film flow rate, energy balance at the interface, Γ(L) = h(l)(t sat T P )L h LV Heat transfer coefficient-flow rate relation, h(l) k L ( µ 2 L ρ L (ρ L ρ V ) ) 1 3 = 1, 47 Re 1 3 F h LV and ρ V at saturation. k L, ρ L at the film temperature T F 1 2 (T W +T i ), µ = 1 4 (3µ L(T P ) + µ L (T i )), exact when 1/µ L linear with T. Condensation and boiling heat transfer 5/42

7 SUPERHEATED VAPOR Two heat sources: vapor (T V > T i ) and interface. Increase of heat transfer wrt to saturated conditions, empirical correction, ( h S (L) = h(l) 1 + CP V (T V T sat ) h LV Energy balance at the interface, film flow rate, Γ(L) = h S (L)(T W T sat )L h LV + C P V (T V T sat ) ) 1 4 Condensation and boiling heat transfer 6/42

8 FILM FLOW RATE-HEAT TRANSFER COEFFICIENT Laminar, h(l) k L ( µ 2 L ρ L (ρ L ρ V ) ) 1 3 = 1, 47 Re 1 3 F Wavy laminar and previous regime (Kutateladze, 1963), h(z) Re 0,22 F ), h(l) k L ( µ 2 L ρ L (ρ L ρ V ) ) 1 3 = Re F 1, 08Re 1,22 F 5, 2 Turbulent and previous regimes (Labuntsov, 1975), h(z) Re 0,25 F, h(l) k L ( µ 2 L ρ L (ρ L ρ V ) ) 1 3 = Re F Pr 0,5 F (Re 0,75 F 253) NB: Implicit relation, Re F depends on h(l) through Γ. Condensation and boiling heat transfer 7/42

9 OTHER MISCELLANEOUS EFFECTS Steam velocity, v V, when dominant effect, V v descending flow, vapor shear added to gravity, Decreases fil thickness, Delays transition to turbulence turbulence, h τ 1 2 i See for example Delhaye (2008, Ch. 9, p. 370) When 2 effects are comparable, h 1 stagnant, h 2 with dominant shear, h = (h h 2 2) 1 2 Condensation and boiling heat transfer 8/42

10 CONDENSATION ON HORIZONTAL TUBES Heat transfer coefficient definition, h = 1 π π 0 h(u)du Stagnant vapor conditions, laminar film, Nusselt (1916) h = (0.70) ( k 3 L ρ L (ρ L ρ V )gh LV µ L (T sat T p )D ) , imposed temperature, 0.70, imposed heat flux. Γ, film flow rate per unit length of tube. Condensation and boiling heat transfer 9/42

11 Film flow rate- heat transfer coefficient, energy balance, h k L ( µ 2 L ρ L (ρ L ρ V ) ) 1 3 = (1.47) Re 3 F Vapor superheat and transport proprieties, same as vertical wall Effect of steam velocity (Fujii), ( ) h u = 1.4 V (T sat T P )k L 1 < h < 1.7, h 0 gdh LV µ L h 0 Tube number effect in bundles, (Kern, 1958), h(1, N) h 1 = N 1/6 Condensation and boiling heat transfer 10/42

12 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 11/42

13 heat transfer coefficient, 1 h = h G h d h i h co G : non-condensible gas, d : drop, i : phase change, co coating thickness. Non-condensible gases effect, ω i 0, 02 h h/5 Example, steam on copper, T sat > 22 o C, h in W/cm 2 / o C, h d = min(0, 5 + 0, 2T sat, 25) Condensation and boiling heat transfer 12/42

14 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 13/42

15 BOILING CURVE G 9? Imposed heat flux, P = qπdl = UI $ Wall and wire temperature are equal, D 0 R(T ) = U I, < T > 3 T W "! #, 6 I = J + Wall super-heat: T = T W T sat Heat transfer coefficient, q h T W T sat Condensation and boiling heat transfer 14/42

16 BOILING CURVE ), -. / 0 A = JB K N + 0 * ) 9 = I K F A HD A = J Next Condensation and boiling heat transfer 15/42

17 HEAT TRANSFER REGIMES G K? A = JA > E E C. K N = N, * K H K J /, 6 G 0. E > E E C. K N E ), 6 I = J OA: Natural convection AD: Nucleate boiling DH: Transition boiling HG: Film boiling Condensation and boiling heat transfer 16/42

18 TRANSITION BOILING STABILITY G Wire energy balance, K? A = JA > E E C. K N = N, * K H K J / MC v dt dt = P qs ), 6 G 0. K N E. E > E E C, 6 I = J Linearize at T 0, q 0, T = T 0 + T 1, MC v dt 1 dt = P q 0 S S q }{{} T T 1 =0 Solution, linear ODE, T 1 = T 10 exp( αt), α = S MC v ( q T ) T 0 2 stable solutions, one unstable (DH), q T < 0 Transition boiling, imposed temperature experiments (Drew et Müller, 1937). Condensation and boiling heat transfer 17/42

19 NATURAL CONVECTION Wire diameter D, natural convection q = h(t F T sat ), Nu = hd k Pr = ν L α L, Ra = gβ(t F T sat )D 3 ν L α L Nusselt number is the non-dimensional heat transfer coefficient (h). k L, α L, ν L at the film temperature 1 2 (T F + T sat ), β à T sat. Churchill & Chu (1975), 10 5 < Ra < 10 12, Nu = 0, 60 + [ 0, 387 Ra 1/6 1 + ( 0,559 Pr ) 9/16 ] 8/27 2 Condensation and boiling heat transfer 18/42

20 NATURAL CONVECTION ON A FLAT PLATE Scales A, P, plate area and perimeter. Length scale, L = A P. Nu = hl k = ql k L (T P T ), Ra = gβ(t P T )L 3 ν L α L Two regimes, 0, 560 Ra 1/4 ( ) Nu = 1 + (0, 492Pr) 9/16 4/9 ( ) 1 + 0, 0107Pr 0, 14 Ra 1/ , 01Pr 1 < Ra < , 024 Pr 2000, Ra < Thermodynamic and transport properties Raithby & Hollands (1998). For liquids: all at T F = 1 2 (T P + T ) Condensation and boiling heat transfer 19/42

21 ONSET OF NUCLEATE BOILING H Control parameters: p L et T W = T L Super-heated wall: T L = T sat (p L ) + T Site distribution: r, R = R(r, θ) 6 Mechanical balance: p V = p L + 2σ R Thermodynamic equilibrium: p V = p sat (T Li ) T Li = T sat (p V ) T Li = T sat (p L + 2σ R ) (T L T )+ 2σ R dt dp sat T > T eq = 2σ R dt dp sat, Heat flux to interface: q > 0, Ṙ > 0 ( q = h(t L T Li ) = h T 2σ ) dt R dp sat R > R eq = 2σ T dt dp sat 1 bar, T = 3 o C, R eq = 5, 2 µm, 155 bar, T = 3 o C, R eq = 0, 08 µm Condensation and boiling heat transfer 20/42

22 NUCLEATE BOILING MECHANISMS Super-heated liquid transport, Yagumata et al. (1955) q (T P T sat ) 1.2 n 0.33 n: active sites number density, n T 5 6 sat q T 3 sat Very hight heat transfer, precision unnecessary. Rohsenow (1952), analogy with convective h. t.: Nu = CRe a Pr b, Scales : Re = ρ LV L µ L, Length: detachment diameter, capillary length: L σ g(ρ L ρ V ) Liquid velocity: energy balance, q = ṁh LV, V q ρ L h LV Ja C pl(t P T sat ) = C sf Re 0.33 Pr s L h LV C sf 0.013, s = 1 water, s = 1.7 other fluids. Condensation and boiling heat transfer 21/42

23 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, λ T = 2π σ 3 g(ρ L ρ V ), 1 2 ρ V UV 2 < π σ, λ H qa = ρ V U V A J h LV Condensation and boiling heat transfer 22/42

24 Zuber (1958), jet radius R J = 1 4 λ T, λ H = 2πR J, marginal stability, q CHF = 0.12ρ 1/2 V h LV 4 σg(ρ V ρ L ) Lienhard & Dhir (1973), jet radius R J = 1 4 λ T, λ H = λ T, q CHF = 0.15ρ 1/2 V h LV 4 σg(ρ V ρ L ) Kutateladze (1948), dimensional analysis and experiments, q CHF = 0.13ρ 1/2 V h LV 4 σg(ρ V ρ L ) Condensation and boiling heat transfer 23/42

25 FILM BOILING Analogy with condensation (Nusselt, Rohsenow), Bromley (1950), V L ( ρv g(ρ L ρ V )h LV Nu L = 0.62 D3 µ V k V (T W T sat ) ) 1 4 (, h LV = h LV C ) P V (T W T sat ) h LV Transport and thermodynamical properties: Liquid at saturation T sat, Vapor at the film temperature, T F = 1 2 (T sat + T W ). Radiation correction: T W > 300 o C, ɛ : emissivity, σ = 5, W/m 2 /K 4 h = h(t < 300 o C) + ɛσ(t 4 W T 4 sat) T W T sat Condensation and boiling heat transfer 24/42

26 TRANSITION BOILING Minimum flux, q min = Ch LV 4 σg(ρ L ρ V ) (ρ L + ρ V ) 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, T min and T max, from each neighboring regime (NB and FB), Linear evolution in between (log-log plot!). Condensation and boiling heat transfer 25/42

27 SUB-COOLING EFFECT Liquid sub-cooling, T L < T sat, T sub T sat T L Ivey & Morris (1961) q C,sub = q C,sat (1 + 0, 1 ( ρl ρ V ) 3/4 C P L T sub h LV ) Condensation and boiling heat transfer 26/42

28 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 Condensation and boiling heat transfer 27/42

29 BACK TO THE EQUILIBRIUM (STEAM) QUALITY Regime boundaries depend very much on z. Change of variable, x eq Equilibrium quality, non dimensional mixture enthalpy, x eq h h Lsat h LV Energy balance, low velocity, stationary flows, M dh dz = Mh LV dx eq dz = qp Uniform heat flux, x eq linear in z. Close to equilibrium, x eq x According to the assumptions of the HEM, 0 > x eq single-phase liquid (sub-cooled) 0 < x eq < 1 two-phase, saturated 1 < x eq single-phase vapor (super-heated) Condensation and boiling heat transfer 28/42

30 CONVECTIVE HEAT TRANSFER IN VERTICAL FLOWS Boiling flow description Constant heat flux heating, Fluid temperature evolution, (T sat ), Wall temperature measurement, Flow regime, Heat transfer controlling mechanism. Condensation and boiling heat transfer 29/42

31 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

32 Nucleate boiling suppression, Liquid film dry-out, boiling crisis (I), Single-phase vapor convection. Condensation and boiling heat transfer 31/42

33 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 Condensation and boiling heat transfer 32/42

34 BOILING SURFACE Condensation and boiling heat transfer 33/42

35 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

36 SINGLE-PHASE FORCED CONVECTION Forced convection (Dittus & Boelter, Colburn), Re > 10 4, Nu hd k L = 0, 023Re 0,8 Pr 0,4, Re = GD µ L, Pr L = µ LC P L k L Fluid temperature, T F, mixing cup temperature, that corresponding to the area-averaged mean enthalpy. Transport properties at T av Local heat transfer coefficient, q h(t W T F ), T av = 1 2 (T W + T F ) Averaged heat transfer coefficient (length L), q h( T W T F ), TF = 1 2 (T F in + T F out ), T av = 1 2 ( T W + T F ) Always check the original papers... Condensation and boiling heat transfer 35/42

37 NUCLEATE BOILING & SIGNIFICANT VOID Onset and suppression of nucleate boiling, ONB, (Frost & Dzakowic, 1967), ( ) 0,5 8σqTsat T P T sat = Pr L k L ρ V h LV Onset of signifiant void, OSV, (Saha & Zuber, 1974) St = qd Nu = k L (T sat T L ) = 455, q = 0, 0065, GC P L (T sat T L ) Pé Pé < 7 104, thermal regime > 7 104, hydrodynamic regime Condensation and boiling heat transfer 36/42

38 DEVELOPPED BOILING AND CONVECTION Weighting of two mechanisms, x eq > 0 (Chen, 1966) Nucleate boiling(forster & Zuber, 1955), S, suppression factor,same model for pool boiling, Forced convection, Dittus Boelter, F, amplification factor, h = h FZ S + h DB A 1 S = (ReF 1.25 ) /X 0.1, F = 2.35(1/X ) /X > 0.1 Condensation and boiling heat transfer 37/42

39 CHEN CORRELATION (CT D) Nucleate boiling, Forced convection h F Z = k0.79 L CpL 0.45 ρ0.49 L σµ 0.29 L h 0.24 LV ρ0.24 V h DB = k L D Re0.8 Pr 0.4 L From Clapeyron relation, slope of saturation line, Non dimensional numbers definitions, p sat = h LV (T W T sat ) T sat (v V v L ) (T W T sat ) 0.24 p 0.75 sat Re = GD(1 x eq) µ L, X = NB: implicit in (T W T sat ). ( ) 0.9 ( 1 xeq ρv x eq ρ L ) 0.5 ( µl µ V ) 0.1, Pr L = µ LC pl k L Condensation and boiling heat transfer 38/42

40 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, q CHF (p, G, L, H i,...), artificial reduction of dispersion. For tubes and uniform heating, no length effect, q CHF (p, G, x eq ) Tables by Groenveld, Bowring (1972) correlation, best for water in tubes Correlation by Katto & Ohno (1984), non dimensional, many fluids, regime identification. Condensation and boiling heat transfer 39/42

41 MAIN PARAMETERS EFFECT ON CHF After Groeneveld & Snoek (1986), tube diameter, D = 8 mm. CHF[kW/m 2 ] G=1000 kg/s/m 2 P= 10 bar P= 30 bar P= 45 bar P= 70 bar P= 100 bar P= 150 bar P= 200 bar CHF[kW/m 2 ] p=150 bar G= 0 kg/s/m 2 G=1000 kg/s/m 2 G=5000 kg/s/m 2 G=7500 kg/s/m exit quality [%] exit quality [%] Generally decreases with the increase of the exit quality. q CHF 0, x eq 1. Generally increases with the increase of the mass flux, CHF is non monotonic with pressure. Condensation and boiling heat transfer 40/42

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) Condensation and boiling heat transfer 41/42

43 REFERENCES Bejan, A. (ed) Heat transfer. John Wiley & Sons. Bird, R. B., Stewart, W. E., & Lightfoot, E. N Transport phenomena. Revised second edn. John Wiley & Sons. Collier, J. G., & Thome, J. R Convective boiling and condensation. third edn. Oxford: Clarendon Press. Delhaye, J. M Transferts de chaleur : ebullition ou condensation des corps purs. Techniques de l ingénieur. Delhaye, J.-M Thermohydraulique des réacteurs nucléaires. Collection génie atomique. EDP Sciences. Groeneveld, D. C., & Snoek, C. V 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 Raithby, G. D., & Hollands, K. G Handbook of heat transfer. 3rd edn. McGraw- Hill. W. M. Roshenow, J. P. Hartnett and Y. I Cho, Eds. Chap. 4-Natural convection, pages Roshenow, W. M., Hartnett, J. P., & Cho, Y. I Handbook of heat transfer. 3rd edn. McGraw-Hill. Condensation and boiling heat transfer 42/42