The frequency of this 10 20 30 40 50 60 70 80 90 100 Steam Mass Flow Flux kg/s m 2 Bubbling region Chugging region Condensation oscillation region Transition region Figure 19.3 Map of co
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Phenomena Related to LOCA Interest in the condensation of a vapor in a liquid has been motivatedby concern about loss-of-coolant accidents (LOCA) in light-water nuclear reactors In this situation, unwantedsteam generatedin the core is forced through a pool of water These phenomena can be quite complicatedbecause of the various forms of unsteady steam flow that can exist The heat transfer coefficients and the liquidtemperatures can experience wide-ranging time fluctuations here
Aya andNariai (1991) have summarizedthe various modes of condensation that can occur in this situation They have indicated the importance of pool water subcool-ing Note that these types of phenomena occur normally near standard atmospheric pressure They gave an approximate map that shows the differentiation between the various types of behavior This is shown in Fig 19.3 andis discussedbelow
If a steam jet is directedinto a liquidwater pool at lower mass flux rates (i.e., the
lower-left-handregion of Fig 19.3), a phenomenon denotedas chugging can occur.
For this situation, the results of Young et al (1974) can be used:
h = 6.5ρ L C P,L v0.6
S d L
0.4
(19.19) Aya andNariai (1991) notedthat this result didnot include the effect of subcooling, which they felt was very important
At higher mass flux rates (say, greater than about 25 kg/s· m2) than those asso-ciatedwith eq (19.19), another type of oscillation can occur The frequency of this
10 20 30 40 50 60 70 80 90 100
Steam Mass Flow Flux (kg/s m )2
Bubbling region
Chugging region
Condensation oscillation region
Transition region
Figure 19.3 Map of condensation phenomena that can develop in loss of coolant accidents
(After Aya andNariai, 1991.)
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one is closely relatedto that of the periodic growth andreduction of a steam bubble attachedto the jet exit This phenomenon is referredto in the literature simply as
condensation oscillation The region where this phenomenon takes place is shown
on the right-hand side of Fig 19.3 A correlation that represents the data quite well has been given by Fukuda (1982):
¯h = 43.78k L
d
dG S
ρLνL
0.9
C P,L ∆T
Aya andNariai (1991) also outlinedthe magnitudes of the various types of con-densation phenomena when steam is condensed in pool water These are shown in Fig 19.4 Although the values are affectedby numerous variables, including time, and are difficult to illustrate exactly, general trends can be shown
The highest of these modes is the film coefficient on the vapor side (not the overall value) This is denoted as “steam-side interfacial” in the figure Moving down in Fig
19.4, the next variation is shown for the “chugging” region Along the line indicated,
Steam side interfacial Chugging
Condensation oscillation Jet in steam flow Jet in vessel Liquid drop Stratified
Laminar jet
107
106
105
104
103
102
Subcooling (K)
K)
Figure 19.4 Various regimes of condensation of steam in water according to Aya and Nariai (1991)
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the higher values are for the individual transient processes, while the lower values denote the average values “Condensation oscillation” is affected by both the pool subcooling as well as the steam mass flow flux Higher values of the latter are found
in the upper portion of the shaded region Heat transfer coefficients for this regime can be predicted using eq (19.20) Near the middle levels, the “jet in steam flow” and the “jet in vessel” typical ranges are shown Values in the former category are affected
by the steam mass flux at higher levels of subcooling The water pool water can be stratifiedin certain situations When this is the case, the heat transfer coefficients are then decreased This variation is shown by the range denoted as “stratified” in Fig
19.4 Here the heat transfer is influencedby many variables, including the mass flux andthe degree of stratification
Ranges of the heat transfer coefficients for laminar jets and droplets (Sideman and Moalem-Maron, 1982) are also shown for comparison purposes The jet values are influenced by the diameter of the jet as well as mass flux rate The droplet variations shown assume a small contact time Longer contact times wouldgreatly reduce the magnitudes ofh.
More recently, Ju et al (2000) reported a study on details of the condensation process They usedholographic interferometry to determine the heat transfer to con-densing bubbles associated with the application of core makeup tanks
19.3.3 Evaporation of a Liquid by a Surrounding Vapor, Gas or Liquid
Droplet Evaporation in a Vapor or Gas A problem of great importance in applications is the spray cooling of a hot gas In this situation, liquiddroplets are evaporatedby the warmer gas, cooling the latter Unlike condensing systems, the presence of the gas does not impede the process Several workers have focused on this problem, motivatedby a variety of applications Using a stochastic modeling approach, Carey andHawks (1995) analyzedsmall droplets evaporating in their own superheatedvapor For larger microdroplets, they foundthe following relationship:
hd
k = 2
ln(Ja v − 1)
where the Jakob number is given by Ja= C pv (T∞− T R )/h fg A more complicated result is given for microdrops
Tong andSirignano (1984) analyzedthe evaporation of multicomponent droplets
in a hot gas They useda simplifiedmodel for this problem that has application to evaporation of fuel in combustion systems No correlation was presentedin this work, but time-varying results were shown for some specific cases
Droplet Evaporation in a Liquid Çoban andBoehm (1989) modifiedthe results given by several others for the evaporation of immiscible droplets in a continuous liquid Their particular application was for organic liquids evaporating in water In the model, the extremely high heat transfer coefficients associated with the evaporating
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portion of the droplet were combined in appropriate ways with the poorer heat transfer through the vaporizedportion of the droplet This result is
hd
k L = (Re c· Prc )1/3 +5k c
k L (1 − ) (19.22) where
≡ 0.466(π − β + 0.5 sin 2β)2/3
andβ is vapor half-opening angle of the droplet, assuming that the vapor portion of
the evaporating droplet accumulates in the top part of the droplet When the flow of the two fluids is in the same direction, operational aspects change considerably This
is discussed further in Section 19.4.1
Work has also been performedfor a three-phase exchanger where the dispersed phase is injectedinto a stagnant continuous phase (Smith et al., 1982) Specific results were foundfor cyclopentane injectedinto a vessel of water, andthese results were comparedto a numerical approach that usedthe drift flux model Only short transient runs were possible for this situation Both a preagglomeration state anda postagglomeration stage were considered In both cases, the thermal resistance of the dispersedphase was assumedto be negligible, andsingle-droplet velocity was assumedto be of the form
In this equation andthe two below, the 0 subscript denotes the initial value of the droplet Further, the single droplet heat transfer was calculated as
hd
k c = c2· Rec3
C · Pr1/3
The value ofc3is known to be in the range 0.7 to 1.0 For the preagglomeration stages, the heat transfer of the bubble as a function of the travel distance in the stagnant water was expressedby
h(z) = 2ψφ0 h d0 d0 (1 + ψz) ψBz3/ψ− 1 (19.25)
where
B ≡ 2h0 ∆T
ν0d0hfg
ρd,L− ρd,V
ρd,Lρd,V and ψ ≡ (1 − c1)(1 + c3) + 1
andwithc1,c2, andc3as definedby comparison to eqs (19.23) and(19.24) For the postagglomeration correlation, Smith et al (1982) foundthe following result:
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h(z)d0Bz
2h d0 = φ0[(1 + ψ0Bz a )3/ψ0− 1] + φmax
1− φmax
×
1+ ψa Bz a + 6(1 − φmax)1−c3ψa B(z − z a )1/2ψ a
− (1 + ψ a Bz a )1/2ψ a
(19.26)
In this equation the subscript a is usedto denote the onset of agglomeration The
same order of magnitude of predicted heat transfer coefficients for evaporation were notedas those for results reportedin the literature
Studies of direct-contact evaporation of R114 injected into stagnant water near saturation were reportedby Celata et al (1995) Photographic means were usedto quantify the rate at which the refrigerant evaporatedandthe amount of the liquid remaining at after the droplet More recently they investigateda relatedproblem, but with set amounts of subcooling (Celata et al., 1999)
19.4.1 Spray Columns
Columns, including spray columns andcolumns with internals, can be usedfor a variety of direct contact heat transfer processes Columns with internals can include tray columns andcolumns with packing Packings can range from spheres to more
complicatedgeometries calledstructured packings The latter are preferredfrom
the perspective of minimizing cost andmaximizing performance Columns allow intimate contact between a dispersedphase anda continuous phase In condensation situations, the liquid(sink) phase is usually continuous A review of a variety of process heating devices has been given by Jacobs (1988b)
Three analysis approaches have been used One draws on previous design practice for conventional, indirect heat exchangers, another uses concepts from mass transfer device analysis, and the third uses more detailed analyses of the behavior of a typical droplet in the device and accumulates the effects for all, to find overall performance
Both the first andthirdapproaches are discussedhere in some detail
First consider the design approach analogous to that used for closed heat exchang-ers No matter which of the many types of heat transfer phenomena are present (e.g., desuperheating, condensation, and subcooling could all be taking place in a given device), the overall heat transfer is calculated using the LMTD approach If the area between the two separate fluids is known, the calculation can be made:
Q = U A A i · LMTD = U A A i ∆T1 − ∆T2
ln(∆T1/∆T2) (19.27)
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In this equation,U Adenotes the overall heat transfer coefficient based upon the fluid interface areaA i The subscripts 1 and 2 denote the temperature differences between
the two fluids at each of the ends of the contacting device Several problems exist with this approach For one, the area-basedheat transfer coefficient on the fundamental droplet level may be difficult to estimate Second, the area of that interface can also
be difficult to determine Third, even if the area is known, it is not unusual for it to vary considerably throughout the contactor A more easily evaluated approach to this that has been usedin many studies reportedin the literature is to replace the area-based heat transfer characteristics with volume-basedvalues This is given as
Q = U V V · LMTD = U V V ∆T1 − ∆T2
ln(∆T1/∆T2) (19.28)
In this equation,V denotes the total contactor internal volume and U V is the volumet-ric heat transfer coefficient basedon that volume Although the latter can suffer from imprecision about details of the heat transfer variations as indicated in the area-based approach, the assumption made here is that the result is a composite for the overall situation
Another way of dealing with the elusive concept of the basis for the overall heat transfer coefficient is to use a term for the surface area of the droplets per unit volume
of the column In this manner, the heat transfer coefficient can be written asU A a Here
theU Ais the traditional surface heat transfer coefficient, but it is now based on the area of the droplets, whilea is the area of the droplets per unit volume of the heat
exchanger Hence
This does not simplify the problems of determining the area-to-volume information explainedabove; it is simply another notation usedin the literature
Reminiscent of mass transfer operations, where columns have traditionally been
usedmost frequently, the transfer unit technique is often applied In this approach, a
volumetric heat transfer coefficient, usually determined from empirical data, is used
The concept of a stage comes into play This is illustratedin Fig 19.5.
One situation that occurs frequently is that of partial condensation of a superheated vapor by a cool liquid In this case, the volumetric heat transfer coefficient can be written as (Fair, 1972; Sideman and Moalem-Maron, 1982)
1/h V,L + (1/βh V,G )(Q G /Q T ) (19.30)
In this equation,h V,L andh V,Grepresent the heat transfer coefficients on the liquid side and the gas (vapor) side in direct contact, respectively; andQ T andQ G,
respec-tively, are the total heat transfer andthe amount of heat transfer involvedin removing the superheat The subscriptG is usedto differentiate it from locations where actual
condensation is occurring Finally, the termβ is usedfor the following:
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Figure 19.5 Temperature versus distance plot of heat exchanger with stages outlined The first stage is at the left, the secondis indicatedexplicitly, anda portion of a thirdexists at the right
β ≡ NCˆ P,G /h V,G V
1− e− ˆNC P,G /h V,G V (19.31)
Here the term ˆN is usedto denote the mass transfer rate Note that when the contactor
only includes condensation (no desuperheating) and the liquid (sink) is the same compoundas the vapor, the vapor–liquidportion is not present, andthe vapor-side heat transfer coefficient in the overall computation can be neglected For this case, the following results:
The overall height of a heat transfer unit(Z i,T ) can be written as
Z i,T = Z i,G + Z i,L G G C P,G
G L C P,L
Q L
This is a characteristic of the specific heat transfer equipment andprocess Finally, the number of transfer units (NTUs) can be found:
NTUg =
T g2
T g1
dT g
T g − T ≈ ln
T − T g1
T − T g2 (19.34)
From this the total column height can be found:
Z T =V A T = NTUg · Z i,T (19.35)
In this equation,A denotes the cross-sectional area of the empty column.
Global Treatments Spray columns offer the simplest contactor configuration
In this arrangement, a column is outfittedwith inlets for each of the continuous
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anddispersedfluids andtheir corresponding outlets The lighter fluidis admittedat the bottom of the column and flows upward in a down-coming heavier fluid, due
to the influence of gravity The lighter fluidcan serve as either the dispersed or the continuous component, although most applications findit as the former Special arrangements (often called disengagement zones) are incorporatedin both the top and
bottom of the column to allow the two fluids to be separated A simple schematic of one of these types of columns is shown in Fig 19.6 Although the details of the actual introduction of the two fluids is not shown in this figure, the general arrangement is quite simple
One of the more important variables that affect heat transfer in a spray column
is the holdup (denoted here by the symbolφ) This is defined as the amount of the
Figure 19.6 Spray column when the dispersed phase enters from the bottom Other than as
a means of introducing the two fluids, the column does not contain any internals
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dispersedfluidin the continuous fluidat any time In a general heat transfer device, the holdup can vary with distance up the column The holdup is often used to correlate both heat transfer andmass transfer
Other variables of interest in discussions of spray columns are the superficial
velocities ( ˜V ) of each of the phases andthe slip velocity The superficial velocity is
the actual velocity that the dispersed or continuous phase is moving, and this depends
on the holdup For the dispersed phase, this is
while the continuous phase velocity is given as
˜V c= v c
Finally, the slip velocity ˜ Vslipis given as the difference of the two component velocities shown above As shown by Letan (1988), the operational relation of the system, in terms of the terminal velocityv T, is
v T (1 − φ)γ−1= ˜V d
φ +
˜V c
The flooding holdup can then be found The flooding situation is found at the following condition (Letan, 1988):
∂ ˜V c
∂φ
˜V d
With this condition, the flooding limit on holdup can be found:
φf = (γ + 1)2+ 4γ(1/R − 1)
1/2
− (γ + 1)
whereR ≡ ˜V d / ˜V c andγ is as defined in eq (19.38) At the limit where the
con-tinuous phase velocity is zero (R becomes unlimited), the maximum holdup can be
determined:
Finally, the pressure drop in the continuous phase in a vertical column is found as follows:
∆P
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As will be notedagain in the discussion of columns that incorporate internals (baffles, packings, etc.), the spray column offers some benefits andshortcomings
Of the benefits, by far the most important is the relative simplicity andresulting low cost of these devices Also, the heat transfer performance can be very high One of the shortcomings is that there couldbe a fair amount of backmixing, which would hinder overall performance Examples of possible temperature versus distance traces that might be encounteredare given in Fig 19.7
Means of estimating the heat transfer in these devices have been addressed in the literature For example, for gas–liquidsystems, Fair (1988) has given the following correlation:
U V,G (W/m3· K) = 867G0.82G G0.47L Z −0.38 T (19.43)
In this equation,Z T denotes the height of a single spray zone Fair indicates that this equation is only for the gas side of the heat transfer process, but because of circulation inside the droplets, there is relatively little resistance to heat transfer in the liquid phase
Figure 19.7 Various types of performance of a spray column that can result from mixing phenomena (After Letan, 1988.)