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 (postCHF heat transfer) Learning Objectives (1/3)
2ai Identify and characterize the four regimes of pool boiling
2aii Reproduce the “pool boiling diagram” (heat ﬂux or
heat transfer coeﬃcient vs. temperature diﬀerence) for water
at atmospheric pressure
2bi Identify physical mechanism for nucleation superheat
2bii Compute nucleation superheat and boiling incipience
point
2ci Identify and characterize the regimes of ﬂow boiling and
their relationships to ﬂow regimes
2cii 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) 2di Explain the physical mechanism of critical heat ﬂux
(CHF) in pool boiling.
2dii Explain the physical mechanisms of CHF in ﬂow boiling
and identify the conditions (including type of reactor) in
which they are seen.
2diii Select and implement general CHF correlations
2div Develop familiarity with nuclearspeciﬁc
terms/acronyms relating to CHF and apply (regulatory) limits
to a nuclear system Learning Objectives (3/3)
5a Develop familiarity with terminology and mathematical
symbols common to nuclear TH, including those symbols
which do not map onetoone to terminology
5b Use correlations and/or models to analyze problems in
nuclear thermal hydraulics
5d Identify assumptions used in development of models and
critically evaluate the applicability of these assumptions for
the TH conditions being modeled
5f Identify THrelated safety limits for light water reactor
operation
5g 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 122 in T&K
Experiment 1 is curve ABCC E. Experiment 2 is EDBA.
Experiment 3 is curve ABCDE. (Provided dq /dt is small.) The line CC 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. 124) 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. 127) 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
postCHF. Flow Boiling Behaviors with q and xe (Fig. 126) 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
ClausiusClapeyron relation between saturation pressure &
temperature
dp
dT = hfg
hfg
≈
Tsat (volg − volf )
Tsat volg (3) Nucleation Superheat (2)
Divide CC 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
CC 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 twophase behavior).
zB : onset of saturated bulk liquid.
zE : onset of temperature equilibrium. World’s Fastest SinglePhase 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 SinglePhase Convection Review (2/3)
In laminar ﬂow, Nu can sometimes be determined analytically (with
considerable eﬀort) – see Table 106. Typically, order of a few.
It is generally not possible to analytically determine Nu in
turbulent ﬂow.
Correlations, e.g., DittusBoelter. 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 SinglePhase Convection Review (3/3) Application to Subchannels
Nusubchannel = ψ Nuc .t . (15) Examples on pages 444448, 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 twophase 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
singlechannel 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 124), 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 curveﬁts for q in terms of Tw − Tsat
(simple)
Correlations of the twophase heat transfer coeﬃcient as
htc2φ = f (...) htclo (moderate)
Correlations of the twophase 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 36 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 twophase 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 – DittusBoelter heating as example.
Note that Xtt always takes the exponents above (assumes
McAdamslike 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 (multiregime):
htc2φ = htclo 7400 q
−
+ 1.11Xtt 0.66
Ghfg (30) Collier and Pulling (multiregime):
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, inclass 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 540542) 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 dimensionallyconsistent
(ﬁguring this out is nontrivial, however), so use “pure” SI when
calculating this (J not kJ , etc.) Comments on Correlations
Simple correlations only for a factoroftwo 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, geometryspeciﬁc 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: RayleighTaylor instability,
which has a visually appealing Wikipedia page as of 82310).
Boiling example #5. CHF Mechanisms – Flow Boiling (1) – Fig. 1221 in T&K CHF Mechanisms – Flow Boiling (2)
Left of 1221: 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 1221: 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:
CISE4 – 5 to 7 MPa, BWRtype mass ﬂuxes
Biasi – 0.27 to 14 MPa, intermediate and lower mass ﬂuxes
Bowring – 0.2 to 18 MPa, wide range of mass ﬂuxes CISE4 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) CISE4 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) CISE4 Correlation (3)
In example 123, 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 123; another
approach is to use Equations 51 and 59 directly in equationsolving
software. Biasi/Bowring Correlations The solution for the Biasi and Bowring correlations passes through
a 2by2 system for q and xcr .
One equation from an energy balance (same as CISE4):
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, inclass problem set #2). (79)
(80) NuclearSpeciﬁc CHF Correlations
Within the nuclear industry (and its canonical codes), there is
littletono 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 singlechannel analysis
unit. NuclearSpeciﬁc 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 1266),
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 PostCHF Heat Transfer
In the nuclear industry, it is usually more important to avoid CHF
rather than understand what happens in detail on recovery, but
postCHF 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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 Fall '11
 Schubring
 Heat, Heat Transfer

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