To predict local heat transfer coefficients in the subcooled boiling regime, Gungor and Winterton 1986 have adapted their corre-lation by using separate temperature differences for drivin
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Vapor quality
Vapor quality
s)
K)
n-butane sat = 60°CT D= 19.89mm = 15kW/mq 2
n-butane sat = 60°CT D= 19.89mm = 15kW/mq 2
SW
I
A
MF
S
G = 20kg/m s2
G = 60kg/m s2
G = 200kg/m s2
Figure 9.16 Simulation of Kattan–Thome–Favrat model for puren-butane at 60°C, showing
flow pattern map and heat transfer coefficients
• Fully stratified flow for ˙m = 20 kg/m2 · s at all χ values, with a monotonic
decrease inαtpwith increasingχ
• Stratified–wavy flow for ˙m = 60 kg/m2 · s at all χ values shown that give a
moderate peak inαtpversusχ
• Intermittent flow for ˙m = 200 kg/m2· s with χ ≤ 0.4 that shows a moderate rise
inαtpversusχ
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• Annular flow for ˙m = 200 kg/m2 · s with 0.4 < χ < 0.93 that results in an
ever-steeper rise inαtpversusχ as the annularfilm thins out before the onset of
dryout occurs atχ = 0.93
• Annular flow with partial dryout (modeled as stratified–wavy flow) for ˙m =
200 kg/m2· s with χ ≥ 0.93, where a sharp decline in α tp versusχ afterthe
peak occurs
Although not illustrated,αtp goes to its natural limit ofαvaporatχ = 0 On the
otherhand, forall liquid flow, the convective boiling heat transfercoefficientαcbfor
liquid film flow does not go to the tubularvalue Hence,αcbshould be obtained with the Dittus–Boelter or Gnielinski correlation whenχ = 0 Including more recent data
forevaporation of ammonia formass velocities as low as 16.3 kg/m2·s and results for
evaporation of refrigerant–oil mixtures, their flow boiling model is applicable over the following parameter ranges:
• 1.12 ≤ psat≤ 8.9 bar
• 0.0085 ≤ p r ≤ 0.225
• 16.3 ≤ ˙m ≤ 500 kg/m2· s
• 0.01 ≤ χ ≤ 1.0
• 440 ≤ q ≤ 71,600 W/m2
• 17.03 ≤ M ≤ 152.9 (but up to about 300 for refrigerant–oil mixtures)
• 74 ≤ ReL≤ 20,399 and 1300 ≤ ReG≤ 376,804
• 1.85 ≤ Pr L ≤ 5.47 (but up to 134 for refrigerant–oil mixtures)
• 0.00016 ≤ µ L ≤ 0.035N · s/m2(i.e., 0.16 to 35 cP)
• Tube metals (copper, carbon steel, and stainless steel)
Forannularflows, the accuracy of this new method is similarto those of the Shah (1982), Jung et al (1989), and Gungor-Winterton (1986, 1987) correlations; however, the latter methods do not provide a method to determine when annular flow conditions exist For stratified–wavy flows, the Kattan–Thome–Favrat model has been shown
to be twice as accurate as the best of these other methods, even though these other correlations have stratified flow threshold criteria and corresponding heat transfer correction factors Atχ > 0.85, typical of direct-expansion evaporator applications,
the Kattan–Thome–Favrat model is three times more accurate than the best of these othermethods, which have standard deviations of over±80%
The Kattan–Thome–Favrat model is implemented as follows for a given tube internal diameter, specific design conditions, and fluid physical properties:
1 Determine the local flow pattern corresponding to the local design condition using the Kattan–Thome–Favrat flow pattern map (Section 9.7) together with the local heat flux, vaporquality, and mass velocity
2 Calculate the local vaporvoid fractionε with eq (9.130)
3 Calculate the local liquid cross-sectional areaA Lwith eq (9.131)
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4 If the flow is annular or intermittent (intermittent flow is thermally modeled
as if it were annular), determine the annular liquid film thicknessδ from eq
(9.136) withθdryset to 0
5 If the flow is stratified–wavy (note that the flow pattern map classifies annular flow with partial dryout at the top of the tube as being stratified–wavy), eq
(9.132) is first utilized to calculateθstrat, then the values of ˙mhigh and ˙mlow
are determined with the flow pattern map at vapor qualityχ Next, θdry is calculated with eq (9.134) ifχ ≤ χmaxorwith eq (9.135) ifχ > χmax, and then the annularliquid film thicknessδ is determined from eq (9.136) with
this value ofθdry
6 If the flow is fully stratified, use eq (9.132) to calculateθstratand then deter-mine the annularliquid film thicknessδ from eq (9.136) using the value of
θstratforθdry
7 Determine the convective boiling heat transfer coefficientαcbwith eq (9.128).
8 Calculate the vapor-phase heat transfer coefficientαvapor with eq (9.129) if part of the wall is dry
9 For a pure, single-component liquid or an azeotropic mixture, the nucleate pool boiling heat transfer coefficientαnbis determined with eq (9.127) using the total local heat fluxq.
10 Calculate the heat transfer coefficient on the wetted perimeter of the tubeαwet
with eq (9.126) using the values ofαnbandαcb
11 Determine the local flow boiling coefficientαtp with eq (9.125)
For evaporation of zeotropic mixtures and refrigerant–oil mixtures, refer to Sec-tion 9.12
Fully developed subcooled boiling is characterized by vapor formation at the heated wall in the form of single bubbles oras a bubbly layerparallel to the wall These bubbles are swept into the subcooled area of the liquid flow by the variable shear stress on their boundary imposed by the turbulent flow velocity profile The bubbles then condense in the subcooled core To predict local heat transfer coefficients in the subcooled boiling regime, Gungor and Winterton (1986) have adapted their corre-lation by using separate temperature differences for driving the respective nucleate boiling and convective boiling processes so that the heat flux is calculated as a sum
of theircontributions as
q = α L[Tw − T L (z)] + Sα nb (Twall− Tsat) (9.137)
This formula predicted their database with a mean error of±25% The methods
pre-sented earlier for saturated forced-convective evaporation may be adapted to sub-cooled flow boiling in an analogous manner
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In the foregoing sections we addressed flow boiling when it occurs inside tubes
Boiling on the outside of horizontal tube bundles is another important process, typical
of kettle and thermosyphon reboilers, waste heat boilers, fire tube steam generators, and flooded evaporators
Figure 9.17 depicts a simplified tube bundle layout Subcooled liquid enters the bun-dle from below and flows upward past the tubes Until the wall temperature surpasses the saturation temperature of the liquid, single-phase convective heat transfer occurs
Once the wall temperature is aboveTsat, subcooled boiling may occurand the bubbly flow regime begins Farther up the bundle, the bulk fluid temperature reaches the sat-uration temperature and saturated boiling begins The rapid departure of sequential bubbles tends to form bubble jets from the top of tubes With coalesce of these bub-bles into a larger size, sliding bubbub-bles are formed as they pass between adjacent tubes, characterized by a thin evaporating film of liquid between the bubble and the wall The flow becomes ever more chaotic and locally unstable and the chugging flow regime is
Figure 9.17 Boiling on a horizontal tube bundle (From Collier and Thome, 1994.)
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encountered At highervaporqualities with increasing vaporshearon the liquid films
on the tubes, the liquid becomes entrained in the vapor, which becomes the contin-uous phase This is the spray flow regime and the tubes are wetted by the impact of droplets that may maintain a continuous liquid film on the tubes Thus, the active heat transfer modes in bundle boiling are nucleate boiling, convective boiling, and thin-film evaporation At some critical conditionx crillustrated in the diagram, dryout of the tubes may occur with a substantial decrease in the local heat transfer coefficient
Typically, forsaturated inlet conditions the local flow boiling coefficient at the bottom of the tube is similarin value to that fornucleate pool boiling on a single tube (methods described in Section 9.5) As the local vapor quality rises from bottom to top in the bundle, the influence of convection becomes more and more important At the top of the bundle, the heat transfer coefficient may become as high as three to four times that at the bottom
Bundle boiling coefficients can be analyzed by normalizing the bundle coefficientαb
with the single tube nucleate pool boiling coefficientαst, whereαstis eithermeasured
or calculated using a nucleate pool boiling correlation andαb may refereitherto a local value within the bundle orto the mean value forthe entire tube bundle This
ratio, known as the bundle boiling factor F b, indicates the relative enhancing effect of
two-phase convection in the bundle compared to the pool boiling coefficient, such that
F b = αb
The value of F b tends toward 1.0 at high heat fluxes and high reduced pressures because the nucleate boiling coefficient becomes dominant Forplain tubes and low finned tubes, local values ofF b tend to range from about 1.0 at the bottom of the bundle, when the inlet flow is all liquid and thus the convective effect is minimal, up
to as high as 3 or 4 near the top tube rows, where the convective contribution is very pronounced Mean bundle values ofF b, on the other hand, are normally in the range 1.5 to 2.0
The simplest thermal design method is to assume a value of the bundle boiling factor, such as F b = 1.5 as a conservative value Then, after calculating the single-tube
boiling heat transfer coefficient using one of the methods in Section 9.5, the mean bundle boiling coefficient is obtainable from eq (9.138) Another simple approach has been proposed by Palen (1983), where the mean bundle boiling heat transfer coefficientαb is assumed to be a superposition of the contributions of boiling and natural convection as
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Hereαst is the single-tube nucleate pool boiling coefficient,F b the bundle boiling factor,F cthe mixture boiling correction factor (see Section 9.12), andαncthe natural convection heat transfercoefficient forthe tube bundle Palen recommends using
αnc = 250 W/m2 · K and F b = 1.5 For pure fluids and azeotropic mixtures, F c
is equal to 1.0; for zeotropic mixtures, its value may vary from 0.1 to 1.0
Various Chen-type in-tube boiling correlations have been proposed for evaporation
on the outside of plain tube bundles In this approach, the liquid-only heat transfer
co-efficient to the liquid phase in eq (9.89) is calculated using a correlation for turbulent crossflow over a tube bundle rather than the Dittus–Boelter in-tube correlation, and the nucleate boiling coefficient is predicted with one of the methods in Section 9.5
New expressions for the boiling suppression factorS and two-phase multiplier F are
then formulated, sometimes with the boiling suppression factor set to unity So far, these methods have had only limited success in predicting local bundle boiling heat transfer coefficients since they are typically based on small databases composed of only one combination of tube diameterand tube pitch and one ortwo fluids, and hence are not applicable for general use
Post-dryout heat transfer occurs during forced-flow evaporation when the heated surface becomes dry before complete evaporation It refers to the heat transfer process downstream from the point at which the surface became dry and may occur at any vapor quality or even during subcooled flow boiling Post-dryout heat transfer is also
referred to as the liquid-deficient regime oras mist flow heat transfer; however, these
terms do not describe the process when it occurs at low vapor quality In general, the post-dryout heat transfer regime is entered from the wet wall regime by passing through one of the following transitions in the evaporation process:
• Dryout of the liquid film A liquid film (such as in an annularflow) may
com-pletely evaporate, leaving only the entrained liquid droplets in the vapor to be evaporated
• Entrainment of the liquid film Fora high vaporshearstress on the liquid film,
the liquid may be pulled from the surface and become entirely entrained in the vaporphase
• Critical heat flux Imposing a large heat flux or wall superheat at the wall may
create a continuous layer of vapor on the wall, starting from dryout under a single small bubble, a large Taylorbubble, ora dense packing of bubbles
For the transition from annular to mist flow, refer to Section 9.7 on two-phase flow maps For further discussion on the mechanisms and prediction of the critical heat flux, refer to the reviews by Weisman (1992), Katto (1994, 1996), and Celata (1997)
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The heated wall is not completely dry in the post-dryout heat transfer regime En-trained droplets impinge on the surface and wet it locally before either evaporating orrebounding back into the vaporphase Second, in horizontal channels, the upper portion of the heated periphery of the flow channel becomes dry while the bottom remains wetted by the flowing liquid, such that there is simultaneously flow boiling heat transfer on the bottom and post-dryout heat transfer around the top This partial dryout of the perimeter of the heated channel may also occur when there is a signifi-cant variation in the peripheral heat flux, such as a boiler tube exposed to radiant heat from only one direction The post-dryout regime may be reached during saturated boiling in a channel but may also be encountered during subcooled boiling at high heat fluxes Only saturated boiling is discussed here
The post-dryout regime may be encountered in fossil-fuel boilers and fired heaters,
on nuclearpowerplant fuel rod assemblies during a hypothetical loss-of-coolant accident, in direct-expansion evaporators and air-conditioning coils, and in cooling of various high-heat-flux devices Heat transfer coefficients in the post-dryout regime are significantly lower than those for wet wall evaporation In this chapter, first thermal nonequilibrium effects and heat transfer phenomena particular to post-dryout flow are described, and then methods for predicting heat transfer under post-dryout conditions inside channels with uniform boundary conditions are presented
9.11.2 Thermal Nonequilibrium
The wall superheat during wet wall evaporation remains relatively small, typically low 15 to 30 K In the post-dryout regime, instead, the local wall temperature may be-come significantly higher than the saturation temperature, such that a departure from equilibrium occurs The two limiting cases are illustrated in Fig 9.18 For complete departure from equilibrium, heat is transferred only to the continuous vapor phase If heat absorbed by the entrained droplets is insignificant, the vapor temperatureT G (z)
downstream from the point of dryout rises with the sensible heating of the vapor
Similarly, the wall temperatureT w (z) rises like that of a single-phase convective flow, giving the temperature profile illustrated in Fig 9.18a Post-dryout evaporation tends
toward the case of complete thermal nonequilibrium at low pressures and low mass flow rates at high vapor qualities
For complete thermodynamic equilibrium, illustrated in Fig 9.18b, the rate of heat
transfer to the entrained droplets is assumed to be so effective that the vapor tem-peratureT G (z) remains at the saturation temperature as the droplets evaporate The
wall temperatureT w (z) varies depending on the intensity of the droplet evaporation
process Evaporation tends toward thermal equilibrium at high reduced pressures and very high mass flow rates
A typical process path is illustrated in Fig 9.19, where the local vapor temperature
is lower than that occurring for complete nonequilibrium, but is still significantly above the local saturation temperature of the complete equilibrium case Hence, for post-dryout heat transfer the temperature of the vapor is not known a priori but is part
of the solution Thermodynamic equilibrium means that all the heat absorbed by the fluid is utilized to evaporate the liquid, and hence the local equilibrium vapor quality
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Figure 9.18 Thermodynamic states in the post-dryout regime (From Collier and Thome, 1994.)
isχe (z) If, instead, all the heat does into superheating the vaporafterthe onset of
dryout, the vapor quality remains that at the dryout pointχDO(z) In between, the
actual local vaporqualityχa (z) is somewhere between these two limits such that
χDO(z) < χ a (z) < χ e (z).
Consider Fig 9.19, which depicts post dryout in a vertical tube of internal diameter
d i heated uniformly with a heat flux ofq Dryout occurs at a length zDO from the inlet, and it is assumed that thermodynamic equilibrium exists at the dryout point
If complete equilibrium is maintained after dryout, all the liquid will be evaporated when pointz eis reached However, in the actual situation, only a fraction (κ) of the
surface heat flux is used to evaporate the remaining liquid in the post-dryout region while the remainder is used to superheat the bulk vapor The liquid is thus evaporated completely only when a downstream distance ofz ais reached Assuming that the total
heat fluxq(z) from the tube wall to the fluid is comprised of the heat flux associated
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Dryout Post-dryoutregion Superheatingregion
Uniform heat flux q
z e
zDO
z a
Wall temp
Bulk vapor thermodynamicBased on
equilibrium Saturation temp
Based on thermodynamic equilibrium
T z w( )
T z G( )
Thermodynamic qualitye( )z
a( )z
DO ( )z Actual
variation of vapor quality 1
Length ( )z
Figure 9.19 Departure from thermodynamic equilibrium in the post-dryout regime (From Collierand Thome, 1994.)
with droplet evaporationq L (z) and the heat flux associated with vaporsuperheating
q G (z), then
q(z) = q L (z) + q G (z) (9.140) Furthermore, let
κ = q L (z)
whereκ is considered independent of tube length, so that the profiles of the actual bulk
vapor temperature and actual vapor quality are linear The vapor quality forz < z eis given by an energy balance
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χ(z) − χDO= 4
d i ˙mh LG (z − zDO) (9.142) whereh LGis the latent heat of vaporization, ˙m is the total mass velocity, and point
z eis given by
z e=
d
i ˙mh LG
4q
1− χDO
The variation in the actual vapor qualityχa (z) with length for z < z ais
χa (z) − χDO= 4κq
d i ˙mh LG (z − zDO) (9.144) wherez ais
z a =
d
i ˙mh LG
4κq (1 − χDO)
Combining eq (9.142) with (9.145) yields
κ = χa (z) − χDO
χ(z) − χDO
=z a − zDO
z e − zDO
(9.146) The actual bulk vapor temperatureT G,a (z) is thus
T G,a (z) = Tsat+4(1 − κ)q(z − zDO)
forz < z a, while forz a > z it is
T G,a (z) = Tsat+4q(z − z e )
The two limiting cases in Fig 9.18 are obtained by settingκ = 0 and κ = 1,
respectively, in the expressions above In reality,κ is not independent of tube length
and must be predicted from the actual process conditions As illustrated in Fig 9.19, small droplets may remain entrained in the vapor wall beyond the location ofχe (z),
where one is tempted to believe that all the flow is superheated vapor
Post-dryout heat transfer may occur in the dispersed flow regime, in which the vapor
phase becomes the continuous phase and all the liquid is entrained as dispersed
droplets or as inverted annular flow, in which the vaporforms an annularfilm on the
tube wall and the liquid is in the central core The first typically occurs after dryout
or entrainment of an annular film flow, while the second occurs when the critical heat flux is exceeded at low vaporquality orin a subcooled liquid In inverted annularflow,