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Unformatted text preview: ENU 4134 – Boiling Heat Transfer – Part 1/3
D. Schubring August 23, 2010 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 (1/2) 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 Learning Objectives (2/2) 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-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 :
2 (pb − pl ) π r = 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 Use Equation 2: Tb − Tsat = 2σ RTb Tsat ln 1 + hfg pl r (7) = RTb Tsat ln hfg pb pl (6) Nucleation Superheat (3)
By the perfect gas law, RTb /pb = volb ≈ volfg : Tb − Tsat When 2σ << pl r : Tb − Tsat But pb ≈ pl , so: Tb − Tsat ≈ 2σ Tsat volfg hfg r (10) = 2σ Tsat volfg pb hfg pl r (9) = Tsat volfg pb 2σ ln 1 + hfg pl r (8) 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 When ﬂuid is cooled: Nu = 0.023Re 0.8 Pr 0.3 (14) (13) World’s Fastest Single-Phase Convection Review (3/3) Application to Subchannels Nusubchannel = ψ Nuc .t . Examples on pages 444-448, such as: ψ = 1.826 Nuc .t . P − 1.0430 D = 0.023Re 0.8 Pr 0.333 (16) (17) (15) (Square array, water, 1.1 ≤ P /D ≤ 1.3) 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 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) (22) 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 ...
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This note was uploaded on 02/07/2011 for the course ENU 4134 taught by Professor Schubring during the Spring '11 term at University of Florida.
- Spring '11