Post-dryout heat transfer is characterized by the following modes: • Wall-to-vapor heat transfer: turbulent or laminar convection to the continuous vaporphase • Wall-to-droplet heat tran
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the wall is too hot to be rewetted by the liquid, and a continuous stable, if chaotic, vapor film is formed between the wall and the continuous liquid core For horizontal and inclined tubes, dryout typically initiates on the upper perimeter of the tube while the lowerperimeterremains wet, and may also occuronly on one side of a vertical tube heated nonuniformly by, say, a radiant heat source Post-dryout heat transfer is characterized by the following modes:
• Wall-to-vapor heat transfer: turbulent (or laminar) convection to the continuous
vaporphase
• Wall-to-droplet heat transfer: evaporation of droplets that impinge on the hot
wall
• Vapor-to-droplet heat transfer: convection from the bulk superheated vapor to
the saturated liquid in the droplets, including any droplets passing through the thermal boundary layer on the wall that do not actually contact the wall
• Radiation heat transfer from wall-to-droplets/vapor/upstream wall: net radiation
flux dependent on the view factor, emissive properties, transparency of the vapor, and respective temperatures
9.11.4 Inverted Annular Flow Heat Transfer
This regime is also referred to as forced-convective film boiling From observations
of film boiling inside a vertical tube, Dougall and Rohsenow (1963) observed that the flow consisted of a central liquid core surrounded by a thin annular film of vapor on the heated wall when occurring at low vapor quality and low flow rates The interface was not smooth but wavy Because of the density difference between the two phases, the vaporwas assumed to be traveling at a much highervelocity than the liquid core
Depending on the conditions imposed on vertical upward flow, the liquid core was observed to flow upward, remain more or less stationary, or even flow downward
Entrained vapor bubbles were also observed in the liquid core
The simplest inverted annular flow to analyze is heat transfer through a laminar vaporfilm Fora vertical flat surface, this is similarto the Nusselt solution forfalling film condensation on a vertical flat plate The local heat transfer coefficientα(z) at a
distance z from the point of onset of film boiling is
α(z) = C
λ3
GρG (ρ L− ρG )gh LG
zµ G ∆T
1/4
(9.149)
where the wall superheat is∆T = T w − Tsat and the value ofC depends on the
boundary conditions:C = 0.707 for zero interfacial stress and C = 0.5 forzero
interfacial velocity, for example
Forturbulent flow in the vaporfilm, the heat transfercoefficient is inversely dependent on the distancez according to Wallis and Collier (1968):
α(z)z
λG = 0.056Re0G .2 (Pr G· GrG )1/3 (9.150)
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where the vaporReynolds numberReGis that of the vaporfraction flowing alone in the tube, PrGis the vaporPrandtl number, and GrGis the vapor Grashof number:
GrG= z3gρ G (ρ L− ρG )
µ2
G
(9.151)
Other important effects on inverted annular flow heat transfer are the flow structure
at the interface (waves, periodic disturbances, instabilities), subcooling of the liquid core, thermal radiation, and so on
9.11.5 Mist Flow Heat Transfer
A heat transfer model for the mist flow regime should ideally include the heat transfer mechanisms mentioned earlier and nonequilibrium effects on the local temperature and vaporquality However, most methods include only one orseveral of these
For single-phase turbulent convection from the wall to the continuous vapor phase, Dougall and Rohsenow (1963) have used the Dittus–Boelter correlation:
NuG =αdi
λG = 0.023Re0GH .8 · Pr0G .4 (9.152)
where the velocity was assumed to be the homogeneous velocityu H:
u H = ˙m
ρH = ˙m
χ
ρG +
1− χ
ρL
(9.153)
so that the Reynolds numberof the homogeneous vaporis
ReGH = ˙mdi
µG
χ +ρG
ρL (1 − χ)
(9.154)
They used the equilibrium vapor qualityχe in this expression with all properties evaluated at the saturation temperature Hence, nonequilibrium effects and other heat transfer modes are neglected, so this method should only be used as a first approximation
Groeneveld (1973) added another multiplying factorY to this approach, where
Y = 1 − 0.1
ρL
ρG − 1
(1 − χ)
0.4
(9.155)
NuG = 0.00327
˙md
i
µG
χ +ρρG
L (1 − χ)
0.901
· Pr1.32
The equilibrium vapor qualityχe and saturation properties are used His database covers flows in vertical and horizontal tubes and in vertical annuli for the following conditions:
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• 2.5 mm < d i < 25 mm
• 34 bar < p < 215 bar
• 700 kg/m2· s < ˙m < 5300 kg/m2· s
• 0.1 < χ < 0.9
• 120 kW/m2 < q < 2100 kW/m2
Use of his correlation beyond the range above is not recommended Groeneveld and Delorme (1976) subsequently proposed a new correlation that accounts for non-equilibrium effects, whose simplified version is as follows First the parameterψ is
obtained from
ψ = 0.13864Pr0.203
G · Re0.2
GH,e
qd
i c pG,e
λL h LG
1.307 − 1.083χ e + 0.846χ2
e
(9.157)
which is valid for0≤ ψ ≤ π/2 (Note: When ψ < 0, its value is set to 0.0; when
ψ > π/2, it is set to π/2.) Theirhomogeneous Reynolds numberRe GH,e based on the equilibrium vapor quality is
ReGH,e = ˙mdi
µG
χe+ρG
ρL
1− χe (9.158)
Forvalues ofχegreater than unity (i.e., when the enthalpy added to the fluid places its equilibrium state in the superheated vapor region),χeis set equal to 1.0 in the expres-sions above To determine the values ofT G,aandχa, an energy balance is used where
h G,a is the actual vaporenthalpy andh L,sat is the enthalpy of the saturated liquid, whileχeis the equilibrium vapor quality andh LGis the latent heat of evaporation
The actual vaporquality is obtained from
χa= h LGχ e
and the change in enthalpy is
h G,a − h L,sat = h LG+
T G,a
Tsat
c pG dT G (9.160)
The difference between the actual vapor enthalpy h G,a and the equilibrium vapor enthalpyh G,eis
h G,a − h G,e
For0 ≤ χe ≤ 1, the equilibrium vapor enthalpy h G,e is that of the saturated vapor, that is,
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Forχe > 1, the equilibrium vapor enthalpy h G,eis calculated as
Finally, the heat transfer coefficient, defined asα = q/(T w − T G,a ), is obtained from
αd i
λG,f =
qd i (T w − T G,a )λ G,f = 0.008348
˙md
i
µG,f
χa+ρρG
L (1 − χ a )
0.8774
· Pr0.6112 G,f
(9.164) where the subscript “G,f ” in these expressions indicates that the vapor properties
should be evaluated at the film temperature
T G,f = T w + TG,a
This iterative method predicts the values ofα, TG,a, andχawhen given those of ˙m, χ e, andq The method includes the effect of departure from equilibrium but still ignores
the contributions of wall-to-droplet, vapor-to-droplet, and radiation heat transfer This method is more accurate than that of Groeneveld (1973) and has a similar application range
A more complete model has been proposed by Ganic and Rohsenow (1977) In theirmodel the total heat flux in mist flow was assumed to be the sum of wall-to-vaporconvectionq G, wall-to-droplet evaporationq L, and radiationqrad, so that
The wall-to-vapor convection contribution was obtained by introducing the actual vapor velocity into the McAdams turbulent flow correlation as
q G = 0.0023
λ
L
d i
˙mχd
i
εµG
0.8
· Pr0.4
G (T w − Tsat) (9.167)
using the void fractionε (e.g., see Section 9.9 foran expression to calculate the void
fraction), while the physical properties are evaluated at the saturation temperature
The total radiant heat flux from the wall to droplets and from the wall to vapor is
qrad = FwLσSB
T4
w − T4 sat
+ FwGσSB
T4
w − T4 sat
(9.168)
whereF wLandF wGare the respective view factors,σSBis the Stephan–Boltzmann constant (σSB= 5.67×10−8W/m2·K4) and blackbody radiation is assumed.F wG= 0
for a transparent vapor The radiant heat flux is in fact negligible except at very large wall temperatures
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Although ignoring nonequilibrium effects, they predicted the heat flux due to impinging droplets on the wall as
q L = u d (1 − ε)ρ L h LG f cdexp 1−
T w
Tsat
2
(9.169)
where the droplet deposition velocityu d is
u d = 0.15 ˙mχ
ρLε
f G
f cd is the cumulative deposition factor, and f G is the single-phase friction factor calculated at the effective vaporReynolds number(i.e., ˙mχ e d i /εµ G) The value of
f cd is a complicated function of droplet size The importance ofq L relative to the total heat flux is significant at low to medium vapor qualities, where a large fraction
of liquid is present in the flow
For heat transfer from the superheated vapor to a single, isolated entrained liquid droplet, Ganic and Rohsenow (1977) predicted the convective heat transfer coefficient from the vapor to the droplet,αD, in terms of a droplet Nusselt number as
NuD = 2 + 0.6Re1/2
D · Pr1/3
where
NuD= αλD D
ReD= ρG D(u G − uD )
D is the droplet diameter, u G the vaporvelocity,u D the droplet velocity, and the Prandtl number PrGis based on vapor properties:
PrG= µG c pG
The factor of 2 on the right-hand side of eq (9.171) is that resulting for pure conduc-tion to the droplet, and the second term accounts for convecconduc-tion The heat transfer rate to the droplet is
Q = πD2αD (T G,a − T D ) (9.175)
whereT G,ais the superheated vapor temperature andT D is the droplet temperature
T Dis assumed to be the saturation temperature, but a method for estimating the value
ofT G,ais required
Methods to model mist flow heat transfer for internal flows numerically have pro-gressed rapidly in recent years For example, Andreani and Yadigaroglu (1997) have
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developed a three-dimensional Eulerian–Lagrangian model of dispersed flow boiling that includes a mechanistic description of the formation of the droplet spectrum
Evaporation of pure fluids is typical of systems that operate on a closed cycle, such
as waterin a powerplant ora refrigerant in an air-conditioning system Instead, evaporation of mixtures is an important heat transfer process in the petrochemical processing industries, in the production of polymers, occurs in refrigeration systems when a miscible oil enters the refrigerant charge in the compressor, and so on Thus, in this section, the basic principles of mixture boiling are presented For a more complete discussion of this topic, refer to Thome and Shock (1984)
9.12.1 Vapor–Liquid Equilibria and Properties
Some knowledge of the principles of vapor–liquid equilibria is required to understand the basics of mixture boiling Phase equilibrium of a binary mixture sytem is typically presented on a phase diagram such as the one depicted in Fig 9.20 for a mixture that does not exhibit an azeotrope This diagram shows the bubble point and dew point temperature curves at constant pressure, where the vertical axis is temperature and the horizontal axis gives the mole fraction of the two components in both the liquid and vaporphases The component with the lowerboiling point temperature at the particular pressure (i.e., in this case the one to the right at a mole fraction of 1.0) is
referred to as the more volatile or lighter component; the other fluid is referred to as the less volatile or heavier component.
For a mixture with the mole fractionx in the liquid phase, its equilibrium mole
fraction in the vapor phase isy The bubble point temperatures Tbub are given by
Figure 9.20 Phase equilibrium diagram of a binary mixture
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the bottom curve as a function ofx, and the dew point temperatures Tdeware given
by the uppercurve as a function ofy The bubble point temperature represents the
temperature at which a subcooled mixture of the liquid will first form vapor when heating the fluid Similarly, the dew point temperature is the temperature at which
a superheated vapor will first form liquid upon being cooled The difference in the liquid and vapor mole fractions is caused by the different partial pressures exerted
by each fluid, and (y − x) is the driving force for mass transfer to the bubble during
boiling The temperature difference between the two curves at any vertical line at a liquid mole fractionx is referred to as the boiling range or temperature glide of the
mixture,∆T bp The physical properties of mixtures often do not follow a linear interpolation between the pure component properties, whether using a mixing law based on mass fraction or on mole fraction For instance, the liquid viscosity of a binary mixture may
be higherorlowerthan that of its two components at the same temperature Thus, it
is important to use appropriate methods for the prediction of mixture properties for use in thermal design A comprehensive treatment is available in Reid et al (1987)
9.12.2 Nucleate Boiling of Mixtures
The basic relationship between the mixture boiling heat transfer coefficientαmixtwith respect to the ideal pure fluid boiling heat transfer coefficientαid is
αmixt
αid = F c=
∆Tid
The ideal boiling coefficient is that which would be obtained by inserting the mixture physical properties into a nucleate pool boiling correlation in Section 9.5, so this can
also be referred to as the boiling coefficient of the equivalent pure fluid Fora heat flux
ofq, the wall superheat for boiling of the equivalent pure fluid is ∆T id Fora zeotropic mixture, mass transfer of the more volatile component to the bubble interface and evaporation into the bubble to provide its larger equilibrium mass fraction in the vapor phase has the effect of forming a mass diffusion layer around the bubble Hence, the local mass fraction of the more volatile component is lower at the bubble interface, and thus the bubble point temperature of the mixture at the bubble interface is higher than in the bulk liquid This rise in bubble point temperaturedT bp diminishes the
superheat available for evaporation and hence slows down the bubble growth rate The value ofdT bpranges from a minimum value of zero for a pure fluid or an azeotrope up
to the maximum possible value, which is the boiling range (or temperature glide) of
a zeotropic mixture The value ofdT bpis controlled by the combined heat and mass
transfer diffusion process A mixture in which the liquid and vapor mole fractions are
the same is referred to as an azeotrope.
Schl¨under (1983) proposed a simple solution predictingdT bpfora bubble growing
in a uniformly superheated fluid That method was extended by Thome (1989) to multicomponent mixtures in terms of the boiling range∆Tbp, and his analytically derived mass transfer factorF cfornucleate pool boiling of mixtures is
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F c=
1+αid
q ∆T bp
1− exp
−q
ρL h LGβ L
−1
(9.177)
For zeotropic mixtures,F c < 1.0 since ∆T bp > 0, but F cequals 1.0 forpure fluids and azeotropes since for these fluids∆T bp = 0 The nucleate boiling heat transfer
coefficient for zeotropic mixtures is thus obtained by includingF cin, forexample, the Cooper correlation to give
αnb= 55p0.12
r (−log10p r ) −0.55 M −0.5 q0.67 F c (9.178) whereq is the heat flux and p r andM are those of the liquid mixture The ideal
heat transfer coefficientαid is first determined with eq (9.178) by setting∆Tbp to 0.0 such thatF c = 1.0 This method is valid forboiling ranges up to 30 K and
hence covers many of the zeotropic refrigerant blends and hydrocarbon mixtures of industrial interest In these expressions, the heat fluxq is in W/m2, the liquid density
ρL is in kg/m3, the latent heath LGis in J/kg, and the mass transfer coefficientβL
equals a fixed value of 0.0003 m/s
9.12.3 Flow Boiling of Mixtures
Evaporation of mixtures inside vertical and horizontal tubes is typical of numerous industrial processes More recently, with the retirement of the older refrigerants and their replacements in some cases by three-component zeotropic mixtures, for example R-407C, evaporation of mixtures has now become common to the design of air-conditioning systems Mixtures have three important effects on thermal design First,
as a mixture evaporates along a tube, its local bubble point temperature rises as the more volatile component preferentially evaporates into the vapor phase Hence the change in enthalpy also includes sensible heating of the liquid and vaporup this temperature gradient in addition to the latent heat of vaporization The vapor produced has a mole fraction ofy, while that of the liquid has a mole fraction of x.
Hence, it is the enthalpy change between these two states that is used to calculate the latent heat change during flow boiling Consequently, an enthalpy curve is required
to do the energy balance between the evaporating fluid and the heating fluid in a heat exchanger
The second effect is that of the mass transfer on nucleate pool boiling contribution
to the flow boiling heat transfer coefficient This can be introduced into heat transfer models that have explicit nucleate boiling and convective boiling contributions, such
as most of those discussed in Sections 9.8 and 9.9 This can be accomplished by multiplying the nucleate boiling contribution in those methods by the mixture boiling correction factorF c, discussed in Section 9.12.2
The third effect is the gas-phase heat transfer resistance The most widely used approach for predicting in-tube condensation of miscible mixtures where all
com-ponents are condensable is commonly referred to as the Silver–Bell–Ghaly method.
This method accounts forthe need to cool the vaporphase when condensing a mix-ture along its dew point temperamix-ture curve in addition to removal of the latent heat
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Similarly, the vapor phase must be heated up when a mixture proceeds up the bubble point curve during evaporation The effective flow boiling heat transfer coefficient
αeff for evaporation of a mixture is calculated by proration as
1
αeff
αtp(χ)+
Z G
The flow boiling heat transfer coefficientαtp (χ) is obtained with one of the in-tube
correlations cited previously for pure fluids but using the local physical properties
of the mixture and including the effect of F c on the nucleate boiling coefficient
The otherheat transfercoefficient is that of the vaporαG, which is calculated with the Dittus–Boelter turbulent flow correlation using the vapor fraction of the flow in calculating the vaporReynolds number The parameterZ Gis the ratio of the sensible
heating of the vapor to the total heating rate, which can be written as
In this expressionχ is the local vaporquality, c pGis the specific heat of the vapor, anddT bp /dh is the slope of the bubble point curve with respect to the enthalpy of the
mixture as it evaporates (i.e., the slope of the enthalpy curve) This approach is based
on two important assumptions in determining the value ofαG: (1) Mass transfer has
no effect; and (2) the value ofαGis determined assuming that the vapor occupies the entire cross section of the tube
Since the error in ignoring the first assumption becomes significant for mixtures with large boiling ranges, the method above is reliable only for mixtures with small to medium-sized boiling ranges (perhaps up to about 15 K) The second assumption, on the other hand, tends to be conservative since the interfacial waves in annular flows tend to enhance the vapor-phase heat transfer coefficient above that obtained with the Dittus–Boelter correlation
9.12.4 Evaporation of Refrigerant–Oil Mixtures
Introduction of a miscible lubricating oil in a refrigerant often has a detrimental effect
on nucleate pool boiling heat transfer, similar to that described for zeotropic mixtures discussed in Section 9.12.2 A refrigerant–oil mixture is, in fact, a very wide-boiling-range mixture with values of∆T bp of up to 300 K or more Other effects are also important here, such as the three-order-of-magnitude difference between the oil’s dynamic viscosity and that of the refrigerant liquid For flow boiling of refrigerant–oil mixtures, similar to zeotropic mixtures, it is necessary to calculate the bubble point temperature of the mixture as a function of the local oil mass fraction and to apply the enthalpy profile approach to determine the change of enthalpy of the mixture Simple algebraic methods were proposed by Thome (1995b) for both these purposes
For flow boiling of refrigerant–oil mixtures, experimental data sometimes dem-onstrate an increase in the local heat transfer coefficient at low to medium vapor qualities, but typically, oil has only a detrimental effect, causing substantially reduced heat transfer coefficients at high vapor qualities, where the local oil mass fraction
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rises rapidly For plain tubes, Z¨urcher et al (1998) conducted refrigerant–oil evapo-ration tests with local oil mass fractions up to 50 wt % oil and liquid viscosities up to 0.035 N· s/m2(35 cP in common engineering units) with R-134a and R-407C They showed that the Kattan–Thome–Favrat method described in Section 9.9 gave reason-able accuracy for these increasingly viscous mixtures using only the liquid viscosity
of the mixture in the appropriate equations They utilized the well-known Arrhenius logarithmic mixing law for viscosities of liquid mixtures, rearranged in the form
µref−oil
µref =
µ
oil
µref
w
(9.181)
The local viscosity of the refrigerant–oil mixture, µref −oil, thus calculated is used
directly for the liquid viscosityµLin the boiling model and flow pattern map calcula-tions described in Section 9.9, while refrigerant properties are used for all others The refrigerant–oil viscosity, which changes by several orders of magnitude with respect
to that of the pure refrigerant, completely dominates the much smaller changes expe-rienced by other physical properties The oil and refrigerant viscosities,µrefandµoil, are those at the respective local bubble point temperature of the mixture The local oil mass fraction,w, in terms of the nominal inlet oil mass fraction winletcirculating
in the system, is
w = winlet
whereχ is the local vaporquality, including the mass of the oil in the liquid phase
Note, however, that the expression above is undefined whenχ = 1 and that a local
vaporquality greaterthan (1− winlet) cannot be achieved since the oil is nonvolatile and remains totally in the liquid phase Figure 9.21 illustrates the effect of oil on the
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Plain tube G300
Vapor quality (%)
K)
Figure 9.21 Simulation of Kattan–Thome–Favrat model for R-134a/oil mixtures