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Unformatted text preview: ENU 4134 – Boiling Heat Transfer D. Schubring Fall 2011 Outline Introduction to pool and flow boiling Nucleation superheat Boiling incipience and bubble departure Correlations for nucleate boiling heat transfer coefficient Critical heat flux (CHF) mechanisms CHF correlations Film boiling correlations (post-CHF heat transfer) Learning Objectives (1/3) 2-a-i Identify and characterize the four regimes of pool boiling 2-a-ii Reproduce the “pool boiling diagram” (heat flux or heat transfer coefficient vs. temperature difference) for water at atmospheric pressure 2-b-i Identify physical mechanism for nucleation superheat 2-b-ii Compute nucleation superheat and boiling incipience point 2-c-i Identify and characterize the regimes of flow boiling and their relationships to flow regimes 2-c-ii Select and implement appropriate boiling heat transfer correlations, with special emphasis given to complexity of correlation vs. time/resources available for analysis Learning Objectives (2/3) 2-d-i Explain the physical mechanism of critical heat flux (CHF) in pool boiling. 2-d-ii Explain the physical mechanisms of CHF in flow boiling and identify the conditions (including type of reactor) in which they are seen. 2-d-iii Select and implement general CHF correlations 2-d-iv Develop familiarity with nuclear-specific terms/acronyms relating to CHF and apply (regulatory) limits to a nuclear system Learning Objectives (3/3) 5-a Develop familiarity with terminology and mathematical symbols common to nuclear TH, including those symbols which do not map one-to-one to terminology 5-b Use correlations and/or models to analyze problems in nuclear thermal hydraulics 5-d Identify assumptions used in development of models and critically evaluate the applicability of these assumptions for the TH conditions being modeled 5-f Identify TH-related safety limits for light water reactor operation 5-g Consider conservatism (or lack thereof) present in a model and evaluate implications of this for reactor safety analysis Pool Boiling Experiment 1: Enclose saturated liquid water in a reservoir open to pressure P at the top. Assume the reservoir is much large than bubble sizes, but not so large that the bottom of the reservoir is at an elevated pressure (e.g., large stovetop pot). Increase the heat flux at the bottom of the reservoir and observe the (increase) in temperature of the bottom surface of the reservoir. Experiment 2: Starting at the high heat flux, reduce the heat flux and measure the temperature. Experiment 3: Starting a low superheat temperature (Twall − Tsat ), increase the temperature and observe the changes in the heat flux. Pool Boiling Curve – Figure 12-2 in T&K Experiment 1 is curve A-B-C-C -E. Experiment 2 is E-D-B-A. Experiment 3 is curve A-B-C-D-E. (Provided dq /dt is small.) The line C-C is actually closer to 1.5 MW m−2 for real pool boiling a 1 atmosphere. This heat flux is termed the critical heat flux, as it represents the edge of a sudden transition in behavior. Flow Boiling (Fig. 12-4) Bubbly flow and subcooled boiling have exactly the same quality at onset; drop flow and convection to vapor have exactly the same equilibrium quality (xe ) at onset. The others depend on geometry, G , fluid properties, and q . Note that xe = x when the wall temperature is above Tsat and Tbulk (fluid far from wall) is below. [x : fraction of mass flow traveling as vapor. xe : (he − hf ) /hfg .] Flow Boiling Behaviors with q and xe (Fig. 12-7) Types of CHF: DNB: departure from nucleate boiling (PWR failure mode). Dryout (BWR failure mode) The line going up and to the left from xe = 1, q = 0 is (approximately) the line for CHF. Region G is predominantly post-CHF. Flow Boiling Behaviors with q and xe (Fig. 12-6) Constant fluids, P , G . Note: DNB in a PWR is not numerically much above dryout in a BWR – CHF in water peaks at about 5 MPa. Nucleation Superheat How much “extra” pressure and temperature are required to move from liquid to a gas bubble? Liquid pressure pl at constant temperature Tl requires a vapor (bubble) pressure, pb large enough to account for surface tension in a bubble of radius r : (pb − pl ) π r 2 = 2π r σ 2σ (pb − pl ) = r (1) (2) Assuming saturated conditions in the bubble and using the Clausius-Clapeyron relation between saturation pressure & temperature dp dT = hfg hfg ≈ Tsat (volg − volf ) Tsat volg (3) Nucleation Superheat (2) Divide C-C relation by perfect gas law (doubtful assumption): pg volg dpg pg = RTg hfg = dTg 2 RTsat (4) (5) Integrate LHS between pb and pl and reorganize: Tb − Tsat = RTb Tsat ln hfg pb pl (6) Use Equation 2: Tb − Tsat = 2σ RTb Tsat ln 1 + hfg pl r (7) Nucleation Superheat (3) By the perfect gas law, RTb /pb = volb ≈ volfg : Tb − Tsat = Tsat volfg pb 2σ ln 1 + hfg pl r (8) When 2σ << pl r : Tb − Tsat = 2σ Tsat volfg pb hfg pl r (9) 2σ Tsat volfg hfg r (10) But pb ≈ pl , so: Tb − Tsat ≈ Limitations: Equation 10 is not valid at high pressure, where the C-C relation doesn’t work well. For these conditions, Equation 7 is better. Boiling example #1. (Subcooled) Boiling Onset zNB : onset of nucleate boiling (boiling incipience). zD : onset of bubble departure before collapse (need to consider two-phase behavior). zB : onset of saturated bulk liquid. zE : onset of temperature equilibrium. World’s Fastest Single-Phase Convection Review (1/3) Combination of conduction in the fluid and mass transfer Usually phrased as a heat transfer coefficient, htc (usually h, but that looks too much like enthalpy; occasionally α, which is void fraction): q (11) htc = ∆T Often correlated as the dimensionless Nusselt number, Nu Nu = htcL k (12) L is a length scale (e.g., D for round tubes, Dh other geometries), and k , the thermal conductivity, is evaluated in the fluid (not for the wall). World’s Fastest Single-Phase Convection Review (2/3) In laminar flow, Nu can sometimes be determined analytically (with considerable effort) – see Table 10-6. Typically, order of a few. It is generally not possible to analytically determine Nu in turbulent flow. Correlations, e.g., Dittus-Boelter. Assumptions: Re > 10, 000, L/D > 60, 0.7 < Pr < 100, µbulk ≈ µwall When fluid is heated: Nu = 0.023Re 0.8 Pr 0.4 (13) Nu = 0.023Re 0.8 Pr 0.3 (14) When fluid is cooled: World’s Fastest Single-Phase Convection Review (3/3) Application to Subchannels Nusubchannel = ψ Nuc .t . (15) Examples on pages 444-448, such as: P − 1.0430 D = 0.023Re 0.8 Pr 0.333 ψ = 1.826 Nuc .t . (Square array, water, 1.1 ≤ P /D ≤ 1.3) (16) (17) Boiling Incipience Boiling begins to occur when the temperature of the wall is sufficient to make a bubble (liquid sufficiently superheated): After some algebra (subscript i : at incipience): qi = kl hfg (Tw − Tsat )2 i 8σ Tsat volfg (18) Applying Equation 11 with ∆T = Tw − Tbulk : (Tw − Tbulk )i (Tw − Tsat )2 i = kl hfg 8σ Tsat volfg htcc (19) Rohsenow’s (implicit) formula for htcc : −1 htcc = kl rmax 1+ kl hfg 1+ (Tsat − Tbulk ) (20) 2σ Tsat volfg htcc rmax : largest cavity radius available (≈ 1 µm in water). Boiling example #2. Bubble Departure (Net Vapor Generation) Thermally controlled departure: based on Tsat − Tbulk , such as the model of Dix: Tsat − Tbulk = 0.00135 q 1/2 Re htclo l (21) At higher flow rates, departure is hypothesized to be controlled by both thermal and hydrodynamic means: Pe = Re × Pr (22) High Pe : high flow rate; low Pe : low flow rate Saha and Zuber correlation (not including PWR conditions): Tsat − Tbulk = 0.0022 q kDe l q 154 Gcpl if Pe < 7 × 104 if Pe > 7 × 104 (23) Subcooled Boiling – Flow vs. Equilibrium Quality In subcooled boiling, the vapor fraction is greater than zero, but the equilibrium quality is still negative. The flow quality is for use in two-phase hydrodynamic correlations, but thermodynamics requires an equilibrium quality. Levy proposed the following correlation for a reactor application: x (z ) = xe (z ) − xe (zD ) exp xe (z ) −1 xe (ZD ) (24) ... where ZD is the axial location of first bubble departure. For many applications, the difference in axial distance between boiling incipience and bubble departure is small – for example, a single-channel analysis could assume departure occurs at the point of boiling incipience. (Consider: is this a conservative assumption vis-´-vis safety?) a Correlations for Saturated Nucleate Boiling Once Tbulk = Tsat , essentially all heat transfer occurs in the form of phase change. In this regime (regions C and D in T&K’s Figure 12-4), heat transfer is usually understood through a correlation that (at best) is mostly empirical. The correlations presented in T&K and considered in this course take one of three forms: Direct, purely empirical curve-fits for q in terms of Tw − Tsat (simple) Correlations of the two-phase heat transfer coefficient as htc2φ = f (...) htclo (moderate) Correlations of the two-phase heat transfer coefficient as htc2φ = htcNB + htcc (complex) Boiling Correlations (Simple) The correlations of Thom et al. (first) and Jens and Lottes (second): 2p 8.7 22.72 (Tw − Tsat )2 (25) 4p 6.2 254 (Tw − Tsat )4 (26) exp q = exp q = ... p in MPa, T in either K or ◦ C, q in MW m−2 . Thom et al. for lower ∆T ; both applicable to 3-6 MPa systems (extension to BWRs’ 7 MPa produces acceptable results). Boiling example #3. Boiling Correlations (Moderate) – 1 Correlations of moderate difficulty usually take the form of htc2φ = f (...) htclo (the term “boiling two-phase multiplier” or similar is not in common use, but perhaps it should be). Several such correlations take the following more specific form: q − + a2 Xtt b Ghfg kl 0 = 0.023Relo.8 Prl0.4 De 0.9 ρg 0.5 µf 1−x = x ρf µg htc2φ = htclo a1 (27) htclo (28) Xtt 0.1 (29) Note: Details of hlo can vary – Dittus-Boelter heating as example. Note that Xtt always takes the exponents above (assumes McAdams-like friction factors), but the x used is ambiguous between hydrodynamic and thermal equilibrium; typically for saturated boiling, they are about the same. Boiling Correlations (Moderate) – 2 Physics implicit in correlation: First term: relates to bubble nucleation, departure. Second: convective heat transfer, enhanced (or potentially suppressed) by presence of second phase (turbulence issue). In annular flow, a1 = 0 (no bubbles; convection through film). Boiling Correlations (Moderate) – 3 Schrock and Grossman (multi-regime): htc2φ = htclo 7400 q − + 1.11Xtt 0.66 Ghfg (30) Collier and Pulling (multi-regime): htc2φ = htclo 6700 q − + 2.34Xtt 0.66 Ghfg (31) Dengler and Addoms (annular): − htc2φ = htclo 3.5Xtt 0.5 (32) Bennett et al. (annular): − htc2φ = htclo 2.9Xtt 0.66 Boiling example #4, in-class problem set #1. (33) Boiling Correlations (Complex) htc2φ = htcNB + htcc (34) This family of correlations explicitly separates boiling heat transfer into two components. htcNB : term to due nucleate boiling and htcc : term due to (turbulent) convective heat transfer, including turbulence enhancement due to the presence of the second phase. We will consider only one example, the Chen correlation. The correlations of Collier and Bjorge et al. (pages 540-542) are explicitly excluded from course coverage. Chen Correlation – 1/2 Convective term: htcc = 0.023 G (1 − x )De µf 0.8 Prf0.4 kf ×F De (35) F ≥ 1: enhancement to turbulence and convection in liquid due to presence of vapor. Originally, F was a graphical correlation as a function of Xtt , which was difficult to use. A good approximation in explicit form is: F= − 2.35 0.213 + Xtt 1 1 0.736 if Xtt < 10 if Xtt > 10 (36) Chen Correlation – 2/2 Nucleate boiling part: hNB ∆Tsat 0.24 = 0.00122S ∆Tsat ∆p 0.75 0 0 kf0.79 cp .45 ρf .49 0. 0 σ 0.5 µ0.29 hfg24 ρg.24 f = Tw − Tsat (38) ∆p = psat (Tw ) − psat (Tsat ) S = Re = (37) −6 1.17 −1 1 + 2.53 × 10 Re G (1 − x )De F 1.25 µf (39) (40) (41) WARNING: the formula for hNB is dimensionally-consistent (figuring this out is non-trivial, however), so use “pure” SI when calculating this (J not kJ , etc.) Comments on Correlations Simple correlations only for a factor-of-two estimate. For a wide range of flow conditions and types of analysis, the “boiling multiplier” correlations work better for little marginal effort (although those that allow bubbly flow are iterative). Of the options in these notes, the Chen correlation is best within its range of flow conditions: 0.17MPa < 0.06 m s −1 < P G ρl < 6.9MPa < 4.5 m s 0< q < 2.4 MW m 0< < 0.7 x (42) −1 (43) −2 (44) (45) Industry practice: real vendors and utilities have even more complex, geometry-specific correlations, most often built into codes. There is great economic incentive in keeping good htc2φ correlations corporate secrets. CHF Mechanism – Pool Boiling Fluidization of the pool (Kutateladze) – liquid is suspended by vapor streams at the wall. Vapor velocity: jg = C1 σ (ρf − ρg )g ρ2 g 1/4 (46) C1 ≈ 0.13 − 0.18. qcr = ρg jg hfg (47) Remark: critical heat flux does not usually have anything to do with critical flow. The repeated use of the “cr” subscript can be confusing. CHF Mechanism – Pool Boiling (2) Once CHF occurs, film boiling is present. A lower jg (q ) is necessary to sustain it: jg = C2 σ (ρf − ρg )g (ρf + ρg )2 1/4 (48) C2 ≈ 0.09 (49) q (50) = jg ρg hfg Mechanism for film boiling limit: stability analysis – heavy liquid on top of lighter vapor. (Enrichment: Rayleigh-Taylor instability, which has a visually appealing Wikipedia page as of 8-23-10). Boiling example #5. CHF Mechanisms – Flow Boiling (1) – Fig. 12-21 in T&K CHF Mechanisms – Flow Boiling (2) Left of 12-21: departure from nucleate boiling – DNB, dominant at low quality and high heat flux, dominant failure mode for PWR. CHF by DNB is a local phenomenon; it is not strongly dependent on the upstream conditions of the flow and can be caused by a single, local spike in q . Right of 12-21: dryout, dominant at higher quality and lower heat flux for same G and P , dominant failure mode for BWR. In annular flow, the liquid film is continually vaporized or atomized (made into entrained droplets) due to q . When it is large enough, the entire film can leave the wall and expose the wall to vapor. Dryout is usually correlated as a function of channel power – it is highly dependent on the history of the flow. CHF Correlations (General) Three useful general CHF correlations are provided in your text: CISE-4 – 5 to 7 MPa, BWR-type mass fluxes Biasi – 0.27 to 14 MPa, intermediate and lower mass fluxes Bowring – 0.2 to 18 MPa, wide range of mass fluxes CISE-4 Correlation Designed for BWR conditions, including mass fluxes, pressures, and hydraulic diameters. It is based on computation of a critical quality, xcr , and includes a critical length, Lcr : xcr = Dh De a Lcr Lcr + b Dh : equivalent heated diameter; De equivalent hydrodynamic diameter. (51) CISE-4 Correlation (2) Computation of parameter a: G = 3375 kg m−2 s −1 Pc : thermodynamic critical pressure. 1−P /Pc /3 (G /1000)1 a= 1 + 1.481 × 10−4 (1 − P /Pc )−3 G 1− P Pc (52) if G > G (53) −1 if G < G Computation of parameter b : b = 0.199 (Pc /P − 1)0.4 GD 1.4 G in kg m−2 s−1 and Lcr and D in m. (54) CISE-4 Correlation (3) In example 12-3, the Lcr is computed as a function of L and the enthalpy associated with the subcooling, ∆hsub for the case of constant heating in a round tube: Lcr = L− GD ∆hsub 4qcr (55) One equation from an energy balance: qcr = mxhfg + m∆hsub ˙ ˙ ˙ ˙ qcr Lcr π D = mxhfg + m∆hsub qcr Lcr π D ∆hsub x= − mhfg ˙ hfg ∆hsub 4qcr Lcr − x= GDhfg hfg (56) (57) (58) (59) The details of the computation are in Example 12-3; another approach is to use Equations 51 and 59 directly in equation-solving software. Biasi/Bowring Correlations The solution for the Biasi and Bowring correlations passes through a 2-by-2 system for q and xcr . One equation from an energy balance (same as CISE-4): x = 4qcr Lcr ∆hsub − GDhfg hfg (60) For constant heating, CHF occurs first at the end of the tube, so that L = Lcr . The other equation comes directly from the correlation. Biasi Correlation(s) P in MPa, D in m, G in kg m−2 s−1 , q in W m−2 qBiasi ,a = qBiasi ,b = 2.764 × 107 1.468FG −1/6 − x (100D )n G 1/6 15.048 × 107 H (1 − x ) (100D )n G 0.6 (61) (62) For G > 300 kg m−2 s−1 , use the larger value. For G < 300 kg m−2 s−1 , use b . pbar F = 10P (63) = 0.7249 + 0.099pbar exp (−0.032pbar ) (64) 9pbar (65) H = −1.159 + 0.149pbar exp (−0.019pbar ) + 2 10 + pbar n= 0.4 if D ≥ 0.01 m 0.6 if D < 0.01 m (66) Bowring Correlation Again, P in MPa, D in m, G in kg m−2 s−1 , q in W m−2 q = A= B= C pR = A − Bhfg x C h DG 2.317 fg4 F1 √ 1 + 0.0143F2 G D GD 4 0.077F3 DG 1 + 0.347F4 = 0.145P n = 2 − 0.5pR n G 1356 (67) (68) (69) (70) (71) (72) The factors F1 , F2 , F3 , and F4 are then computed as functions of pR Bowring Correlation – F factors When pR > 1: − F1 = pR 0.368 exp [0.648 (1 − pr )] − F2 = F1 pR 0.448 exp [0.245 (1 − pR )] F3 = F4 = (73) −1 0 pR.219 1 F3 pR.649 (74) (75) (76) When pR < 1 1 p 18.942 exp [20.89 (1 − pR )] + 0.917 1.917 R −1 1 = 1.309F1 pR.316 exp [2.444 (1 − pR )] + 0.309 1 17 × pR .023 exp [16.658 (1 − pR )] + 0.667 = 1.667 1 = F3 pR.649 F1 = (77) F2 (78) F3 F4 When pR = (P ≈ 6.9 MPa [BWR]), F1 = F2 = F3 = F4 = 1. (Example #6, in-class problem set #2). (79) (80) Nuclear-Specific CHF Correlations Within the nuclear industry (and its canonical codes), there is little-to-no interest in predicting the physics of CHF, but rather just getting the right value for specific conditions. Example correlation for DNB: qcr = {(2.022 − 0.06238p ) + (0.1722 − 0.01427p ) exp[(18.177 −0.5987p )xe ]}[(0.1484 − 1.596xe + 0.1729xe |xe |)2.326G +3271][1.157 − 0.869xe ][0.2664 + 0.8357 exp(−124.1Dh )][0.8258 + 0.0003413(hf − hin ) (81) q , p , G , h, Dh in kW/m2 , MPa, kg/m2 s, kJ/kg, and m, respectively. hf and hin : saturated liquid and inlet enthalpies. Reactor vendors have long since developed better ones that are highly proprietary information – how we’ll apply CHF correlations to model reactors will be discussed in the single-channel analysis unit. Nuclear-Specific CHF Correlations (2) Some terms: For PWRs, the departure from nuclear boiling ratio (DNBR): DNBR (z ) = qcr (z ) q (z ) (82) The minimum DNBR (the MDNBR) is a design constraint on reactors – typically set to at least 1.3. Correlations for dryout tend to be a bit simpler (Equation 12-66), but are still purely empirical curve fits. In a BWR, two limits are applied: MCHFR during transient – 1.9 or higher CPR (critical power ratio) Finally, all subchannels (center, side, corner) must be studied for MDNBR, MCHFR, and/or CPR. Even though the CHF correlations are complex, they are usually explicit, as they must be evaluated many, many times in a code. Correlations for Post-CHF Heat Transfer In the nuclear industry, it is usually more important to avoid CHF rather than understand what happens in detail on recovery, but post-CHF correlations do exist: In stable film boiling, correlations often take the following form: Nug = a Reg x + ρg (1 − x ) ρf b c Prg Y (83) Most frequently, Y = 1 − 0.1 ρf − ρg ρg d 0.4 (1 − x ) 0.4 (84) At low pressure (e.g., Slaughterbeck et al.): Y =q e kg kcr f kcr : conductivity at critical point. q : must be in kW m−2 . (Boiling example #7.) (85) ...
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