Although they foundthat the qualitative aspects of the performance matched well with that equation, their results were approximately 30% higher for the overall heat transfer coefficient..
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Heat transfer results for liquid–liquidspray columns were correlatedby Plass et
al (1979) For holdups greater than 5% (φ > 0.05),
U V (Btu/hr-ft2-°F) = 4.5 × 104(φ − 0.05)e −0.75( ˙m d / ˙m c )+ 600 (19.44) For columns where vaporization is taking place, the heat transfer can be calculated
as follows (Jacobs andBoehm, 1980):
St= ˙m U v V
L C P,L =
Q
LMTD· ˙m L C P,L = 2(Ja V · PrV )0.6 R m −0.15 (19.45)
To determine this result, data from a variety of systems were analyzed, including some where boiling took place on a liquidsurface
An even simpler result was given by Walter (1981) For all configurations with light hydrocarbons or refrigerants vaporizing in water, the following was recom-mended:
The only complication in applying this relationship is that theφ is the value coming
into the boiling section Hence, if preheating andboiling are taking place in the same column, this wouldbe the holdup leaving the preheater portion
More recently, Siquerios andBonilla (1999) have evaluatedthe correlation above against experimental results they determined from vaporizing normal pentane in water Although they foundthat the qualitative aspects of the performance matched well with that equation, their results were approximately 30% higher for the overall heat transfer coefficient
Direct contact boiling phenomena have been proposedfor a variety of applications, including nuclear reactors A proposal has been presented(Kinoshita andNishi, 1994;
Kinoshita et al., 1995) for an innovative steam generator system for fast breeder reactors that utilizes water anda molten metal In experiments using Wood’s alloy (Bi–Pb–Sn–Cd) as the continuous phase fluidandwater, volumetric heat transfer coefficients in the range 4 to 34 kW/m3· K were demonstrated
Spray columns have been appliedfor waste heat recovery, andseveral of the papers cited in this chapter address this application One clever approach is to use
a falling cloudof solidparticles in a stream of exhaust gas (Sagoo, 1981, 1982) No specific performance correlations were given, but the approach was deemed to be quite successful
The simplicity of the spray columns can leadto some undesirable aspects in overall performance In many situations, the spray pattern may deteriorate at some distance from the nozzle Spray columns are also prone to backmixing Even in relatively organized flows, the wake following a droplet can impede heat transfer (Letan, 1988)
A methodwas reportedby Letan (1988) for designing liquid–liquidspray columns
Crucial aspects of this are the column diameter and the column length Of course, the appropriate diameter of the column is that required to pass the necessary flows As
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the relative flow rate increases, the drag on the droplets increases At some point,
too much holdup of the dispersedfluidcouldoccur Flooding is the point where
the flow rate of the dispersedfluidandits imposeddrag on the droplets do not al-low the dispersedfluidto flow completely through the column Considering these types of interactions, Letan presented a result to determine the proper diameter of the column:
Dcolumn = 2 ˙V C[(V d /V c )(1 − ¯φ) + ¯φ]
πV T ¯φ(1 − ¯φ)ζ
1/2
(19.47)
In this equation, the exponentζ is given by the following empirical relationships:
Re ≤ 0.2 ζ = 4.65
0.2 < Re < 1 ζ = 4.35Re −0.03
1≤ Re ≤ 500 ζ = 4.45Re −0.01
Here the Reynolds number is based on the terminal velocity of the droplets The length of the column is determined to accomplish the necessary heat transfer Letan (1988) has given a method that includes the effect of the wake of the droplets This results in a fairly lengthy computational method, andthe reference notedshouldbe consultedto use this approach
An approximate approach has been recommended by Jacobs (1988a) This uses
a formulation basedon concepts usedin closedheat exchangers to determine their length:
When the flow of the two fluids is in the same direction, several column operational characteristics are quite different No longer is flooding a limitation to operation,
as both fluids are being forced through the column co-currently On the other hand, the residence time is generally less than in a counterflow tower, and the temperature differentials are less favorable A recent comprehensive study of such a system was reportedby Shiina (1997) In this work, R-113 was usedfor the dispersedphase injectedthrough a jet, andwater was usedfor the continuous phase Work was performedin a 50-mm-inner-diameter column with a maximum length of 1.5 m The injector level couldbe variedto show the effects of contact height on the results A complicatedset of correlations was presentedthat fit the general form
St= ˙m U V V
d C P,d = c1+ c2(R m ) c3(Ja · Pr d,V ) c4
Z
Dcolumn
c5D
nozzle
Dcolumn
c6 (19.49) The constants for this equation are shown in Table 19.1
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TABLE 19.1 Empirical Correlation Constants for Eq (19.49)
Source: Shiina (1997).
aRange of applicability for table is 6.3 < Rm < 380, 4.8 < Ja < 37.
Differential Treatment If predictive information on the detailed behavior of dis-crete zones inside a spray column is desired, a differential approach can be used In this formulation, the behavior of a typical bubble is predicted when it is interacting with various elevations of the continuous phase Models of this type have been cast
in terms of a one-dimensional equation set, with the height in the column being the independent parameter There is no reason why two- and three-dimensional formu-lations cannot be used, but one-dimensional models generally yield good predictions for the overall performance of a spray column The added complexity of higher-order models is normally not considered to be worthwhile
Several formulations have appeared Among those found in the literature are for-mulations for liquid–liquidspray columns given by Jacobs andGolafshani (1989), Hutchins andMarschall (1989), andSummers andCrowe (1991), andthree-phase exchanges in spray columns by Çoban andBoehm (1988), Tadrist et al (1987, 1991),
Ay et al (1994), andSong et al (1998) Brickman andBoehm (1994a,b) useda for-mulation similar to that of Çoban and Boehm (1988) to study design optimization of various configurations
A basic approach to the analysis of a three-phase spray column will now be out-lined This draws on the work of Çoban andBoehm (1988) andBrickman andBoehm (1994a) First, consider the one-dimensional equations representing continuity, mo-mentum, andenergy These are written for the region influencedby each stream of bubbles as follows:
dP
dz = −
˙m d
A
dv d
dz −
˙m c
A
dv c
dz − [ρc (1 − φ) + ρ d φ]g (19.50)
dh d
dz =
A
˙m d
Q d
dh c
dz =
A
˙m c
ηQ c
Here the termη is used to denote the amount of heat that leaves (enters) the dispersed
phase andis transferredto (from) the continuous phase Heat loss to the surroundings from the column will render this parameter less than unity
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A relationship for the droplet velocity can be written in terms the drag coefficient andthe other parameters is modifiedslightly from the result given by Raina et al
(1984)
v = 1.1547{[1 − (ρ d /ρ c )(R0/R)3](2Rg/C D )}1/2 R[(5/6)−(R/Tc)]
[(T2
c + T2
d )/2T d T c R (C Pµc /k c ) R0/1.6R (19.53)
In this equation, the subscript 0 denotes beginning values,C Dis the drag coefficient
on a single droplet defined below,R is the droplet radius, and R is the average droplet
radius from insertion to the point of interest in the column This average droplet radius
is given by the relationship
R ≡ R20+ R2
The drag coefficient on a single droplet can now be found (White, 1974):
C D =
24
1+ Re1/2 + 0.4
1+ (2µ c /3µ d )
1+ (µ c /µ d ) (19.55)
In general, heat transfer coefficient calculations must be made for both the interior andthe exterior surfaces of the droplet A variety of correlations can be usedfor this,
as discussed earlier in the chapter One distinction is if the droplet is all liquid, all vapor, or if there is evaporation or condensation taking place inside For purposes of discussion here, it is assumed that the droplet is initially all liquid, and then at some point in the column vaporization begins
On the outside of the droplet, a variety of correlations are available, as noted previously For example, eqs (19.1), (19.2), or (19.3) couldbe applied, but a variety
of other correlations appear in the literature [see, e.g., Sideman and Shabtai (1964)]
Any of these can be usedto findthe external heat transfer coefficienth e Then a correlation for the interior-to-the-bubble heat transfer is determined Again,
a wide variety of correlations are available, depending on the presumed flow (or stationary) situation inside the bubble Remember that the shape and size of the bubble have an influence on these aspects, as discussed by Grace (1983) One possible approach to this if the droplets are small is to assume that the internal heat transfer is solely by conduction when the droplet is all liquid If and when evaporation begins,
eq (19.22) can be appliedto findtheh i With both the external andinternal heat transfer coefficients found, the overall heat transfer coefficient can be calculated:
U = h e h i
With this heat transfer coefficient determined, the heat transfer can be calculated This
is given by
Q d = U(4πR2)N(T c − T d ) (19.57)
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Here the termN denotes the number of bubble trains in a given spray column Note
that all of these factors just discussedare foundat a given elevationz, andthen
the effect of height in the column is foundfrom the mass, momentum, andenergy balances shown in eqs (19.50)–(19.52)
If the droplet number density does not change throughout the column, the droplet radius follows the dispersed phase density in the following manner
R = R0
ρd0
ρd
1/3
(19.58) This may not always be a good assumption, depending on possible breakup of drop-lets, agglomeration, andother factors
Droplet agglomeration or breakup couldbe includedin the model This couldalso include a variety of droplet sizes as may be the actual situation in a column Both of these general aspects have been discussedby Song et al (1998) They usedprobability functions to describe each of these aspects
A general design problem is one where the two fluids are flowing countercurrently andtheir corresponding mass flow rates andincoming temperatures are known From
a design standpoint, it is desired to estimate the outgoing temperatures of the two fluids as they might be inflenced by various physical parameters of the column (e.g., column diameter, column length, droplet diameter, operational pressure)
Analysis next considers the differential zone An example of this is shown in Fig
19.8 HereN injection areas are considered for the dispersed fluid It is assumed that
these injectors are essentially equidistant from one another so that the volume of the active zone is made up ofN more or less equal subvolumes that run the full length
of the active portion of the column
Note that a particular arrangement (dispersedfluidin at the bottom that is vapor-izedby the continuous fluidas it travels up the column) has been assumed This is done here only to give specific discussion points The method can be applied to any general spray column combination of fluids, or particles flowing through a fluid
To proceed, thermodynamic and thermal–physical properties are needed Many such routines have been developedover the years andare now foundalmost rou-tinely using thermodynamics and heat transfer texts or commercially available math-ematical solver software A variety of numerical approaches can be usedto solve this system of equations Brickman andBoehm (1994a) useda fourth-order Runge–Kutta solution algorithm At the bottom of the column, the dispersed phase temperature is known This is the location where the solution begins Unknown from the standpoint
of the solution methodare the velocity gradients andthe temperature of continuous fluid What is known is the incoming temperature of the continuous fluid at the top of the column To handle this problem, a shooting type of method is used The unknown temperature at the bottom is estimated, the solution carriedout, andthe continuous phase temperature calculatedat the top is comparedto the one given If these temper-atures do not correspond to within a predetermined limit, a new value for the exiting continuous-fluidtemperature is estimatedandthe calculation repeated
Using an approach similar to the one just described, Çoban and Boehm (1988) made comparisons to experimental results determined in large-scale direct contact
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Figure 19.8 One-dimensional differential analysis volume that can be used to study perfor-mance of spray columns
spray towers usedin energy conversion systems Very goodcomparisons were found
Later Brickman andBoehm (1994a,b, 1995) appliedthis type of solution technique
to optimize designs of direct contact heat exchangers of the spray column type They showedthe influence of a variety of parameters on the overall performance of the direct contact heat exchanger It was noted that a wide range in overall performance couldbe effectedby simple adjustments to the operational parameters
Melting and Solidification Applications One application of spray columns that has drawn interest over the years is that of solidification of a medium, typically for thermal storage or chemical separation This can be accomplishedin a batch mode, where the solidifying medium does not circulate outside the column Alternatively, a means of moving a slurry can be usedto circulate the partially solidifiedmedium for
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steady operation The number of studies is sufficiently limited, and the working sub-stances andphysical systems sufficiently numerous, that no broadly encompassing predictive technique has been developed However, individual summaries of some the specific work are given here
Kim andMersmann (1998) studiedmelt crystallization ofdodecanol from a
n-decanol/n-dodecanol mixture for separation purposes In this set of experiments they
used various coolants, including a gas, a liquid, and a vapor Several semiempirical relationships were given to describe the data they found Agitated columns were usedfor some of these studies, andthe impeller diameter was usedas a correlation
variable in those cases For a gas–liquid system, the following relationship fit the data
very well:
U V D2
kmelt = 1.79φWe −1.2 (19.59)
Here the Weber number (We) is basedon the droplet superficial velocity andthe
column diameter For the situation when a liquid–liquid system is used, the authors
separatedthe correlation into three ranges of holdup ratio:
0< φ < 0.2: U V D2im
k m = 1.06φWe −1.2 im (19.60a)
0.2 < φ < 0.8: U V D2im
k m = 0.18We −1.2 im (19.60b)
0.8 < φ < 1: U V D2im
k m = 0.34φWe −1.2 im (19.60c)
In these correlations, the impeller characteristics are used
Moderately high temperature thermal storage (around 48°C) using direct contact processes was examined by Kiatsiriroat et al (2000) In these studies, sodium thio-sulfate pentahydrate exchanged heat with a heat transfer oil A direct contact storage unit made of acrylic and having a diameter of 0.12 m and a length of 0.7 m was used
The phase-change material remainedin the unit They foundan empirical equation that fit the solidification data very well:
St= 67.48 Ste −1.4033· Pr−0.3508 (19.61)
This also fit data they foundfor water–oil andwater/R-12 systems
A direct contact technique applicable to coldstorage investigatedby Utaka et al
(1998) was relatedto earlier work they performed(Utaka et al., 1987) Their approach useda closedvessel partially filledwith water where the hydration of HCFC142b took place For this system, the critical decomposition temperature of the gas hydrate
is 13.1°C, and the formation temperature of the gas hydrate depends on the system pressure For their system, the pressure was controlled Although no heat transfer correlation was recommended in this study of transient phenomena, they did show
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how the overall heat transfer coefficient variedwith time (from about 4 kW/m3· K
down to an order of magnitude smaller) and showed how this was dominated by the heat transfer coefficient on the liquidside
A study of direct-contact freezing of tetradecane by a water–ethylene glycol so-lution in a spray column was reportedby Inaba andSato (1996) Both solidification rate versus temperature andsome visual results were given No correlation of heat transfer rate was noted
In a study related to the one just noted, latent heat transfer for cold storage using direct contact means has also been investigatedby Inaba andSato (1997) In this work, tetradecane (paraffin) oil was usedas the phase-change material, andcoldwater was usedto accomplish the solidification This phase change takes place at 5.8°C and liberates 229 kJ/kg of heat Heat transfer characteristics were not given explicitly, but they did correlate volume fraction solidified as a function of the Stefan number based
on the solidphase, the Reynolds number, anda dimensionless temperature ratio
A combineddirect contact freezer andice slurry district cooling system has been reported(Knodel, 1989) In this approach, the evaporator section of the refrigeration system was a direct contact heat transfer device The ice slurry that resulted was then movedthrough a distribution system No specific heat transfer performance was given
Thermal storage for solar heating was investigatedby Fouda et al (1980, 1984)
A mixture of 68.2% Na2SO4and31.8% water was usedthroughout the work as the latent heat storage material andthe paraffinic solvent Varsol was usedas the immisci-ble heat transfer fluid For most runs a spray column was used, but a limited number were performedusing a screen packing (see the discussion on packedcolumns be-low) Volumetric heat transfer data, basedon the salt volume, were correlatedwith the relationship
In this equation, ˜V is the superficial velocity of the heat transfer fluidbasedon the
total column cross-sectional area The correlation constants variedwith the size of the column, the fluidcombination used, andwhether or not the column hadpacking
in it These are listedin Table 19.2
TABLE 19.2 Correlation Constants of Volumetric Heat Transfer Coefficients
in Eq (19.62)
Adapted from data presented by Fouda et al (1984).
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FaridandYacoub (1989) andFaridandKhalaf (1994) have reportedstudies on latent heat storage at temperatures in the approximate range 20 to 50°C In the first
of these studies a single column was used, and three hydrated salts were the focus
of the work: Na2CO2· 10H2O, Na2SO4· 10H2O, andNa2HPO4· 12H2O Volumetric heat transfer coefficients in the range 2 to 12 kW/m3· K were demonstrated The
authors fit these data with a correlation like that shown eq (19.62) for a variety of cases with values for the first constant in the range 23.1 ≤ c1≤ 46.5 andthe second
constant in the range 0.70 ≤ c2≤ 1.28 In the secondstudy, two columns were used
with Na2CO2· 10H2O andNa2S2O3· 5H2O The volumetric heat transfer coefficients were not correlated, but they were in the same general range as noted in the earlier paper In this secondwork, a thermal efficiency was definedas the ratio of the actual energy removed(or stored) from both columns to the energy theoretically available
in both columns Values for this parameter of 45.3 to 91% were demonstrated
Baffledcolumns are usedas means of improving on the performance of spray columns
These devices rearrange flow that can be more favorable to heat transfer in both the dispersed and continuous phases However, as internals are added to the col-umn, the capital cost increases: hence the improvements in performance needto outweigh the additional expense of installation
The flow andheat transfer situation in baffled-tray columns is more complicated
to describe than that in spray columns Figure 19.9 is a schematic showing some elements of this type of column Important variables are the diameter of the column, the curtain area (shown in the figure), andthe window area (column area minus the tray area) For this situation, Fair (1972, 1988) has given a heat transfer correlation for typical designs of baffles spacedabout 0.6 m apart andwindow areas of 40 to 50% of the column area:
U v (W/m3· K) = 585G0.7
G G0.4
This correlation fits data approximately for a range of systems, including air–water
as well as hydrogen–light hydrocarbon/oil units
Baffledcolumns have also been the basis of increasing heat transfer to immersed tubes The use of a slurry in a bubble column has been reportedby Saxena andChen (1994) They gave the following correlation for the heat transfer in this situation:
St= 0.0389(Re · Fr · Pr2) −0.25[µL (Pa · s)] −0.15 (19.64)
In this equation, the liquidviscosity has the units shown
Sieve tray or perforatedtray columns are a form of the baffledtype These can
be usedfor applications of liquid–liquidandthree-phase systems In this arrange-ment, each of the baffles serves to catch the dispersedfluidandre-form the droplets
A typical tray interaction is shown in Fig 19.10 At least two positive characteris-tics can result from this One is that the droplet can be resized, which could be very
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Figure 19.9 Conceptual schematic of a baffledtray column with the liquidindicated
Figure 19.10 Fluidinteractions at one unit in a perforatedtray, baffledcolumn direct-contact heat exchanger