The extent of fluid bypassing f does not appear to be affected by either the power input or the holding time. Some of the feed was routed directly to regions of the vessel near the exit by means of the brine distributor header; values o f f probably reflect this. As with conceptual models for most “black box” devices, one can claim no uniqueness of the model from analyzing input-output data alone. Detailed internal measurements are needed. Empirical models can probably be made to fit the data as well as the model used. The idea of the deadwater and bypassing concepts is intuitive, but obviously an oversimplification of the true facts. The variation of Vo/ V r with power input suggests, however, that there is some physical characteristic of the vessel contents which can be controlled, to some extent at least, and which affects the output in a manner analogous to a stagnant region. The value of this formulation is the assistance provided in thinking about the behavior in simple, intuitive terms of a mixed vessel containing liquid and solid phases.
Acknowledgment
The authors acknowledge the assistance of the Struthers Scientific and International Corp. during the tests on the crystallizer vessel. Nomenclature
c
concentration of dye in liquid sample reference tracer concentration tracer concentration in injected solution fraction of total feed bypassing vessel experimentally determined function of time volume liquid feed rate to vessel volumetric liquid exchange rate between regions time volume tracer concentration in primary region tracer concentration in secondary region
€o
GREEKLETTERS = volume fraction ice
dimensionless concentration, X/co dimensionless concentration, Y / c o r = holding time
( = 7 =
Conclusions
The simple series combination of the model for a tworegion vessel with fluid exchange gives an adequate representation of liquid mixing in the vessel studied, although the benefit of the two additional sections in fitting the response curves is marginal. Because of the design of this vessel, however, it cannot be recommended that the correlation of V D I V Tus. power input be used for general design of vessels containing an agitated slurry. The agitator power consumption is a gross value, the power consumption efficiency being unknown. Generalized power curves for this type of agitator are not available. The simple model can be of value in characterizing agitated slurries and the results encourage further investigation in more extensive experiments.
literature Cited
Corrigan, T. E., Beavers, W. O., unpublished paper, 1967. Levenspiel, O., “Chemical Reaction Engineering,” pp. 242 et seq., Wiley, New York, 1962. Orcutt, J. C., Mixon, F. O., Hale, F. J., Report to Office of Saline Water, Contract No. 14-01-0001-960, June 14, 1968. RECEIVED for review October 30, 1968 ACCEPTED July 24, 1969 Study suppotted by the U.S. Department of Interior, Office of Saline Water. under Contract No. 14-01-0001-960.
CONDENSATION OF IMMISCIBLE LIQUIDS ON HORIZONTAL TUBE J .
A .
S Y K E S ’
A N D
J .
M. MARCHELLO
University of Maryland, College Park, Md. 20742 Heat transfer coefficients for the condensation of eutectic mixtures of seven organic compounds with water on horizontal tubes were correlated. Existing correlations together with a laminar two-film model, a n empirical expression, and a nucleation model were tested. The proposed nucleation model provided the most uniform fit to all the data.
IN
1916, Nusselt combined his hydrodynamic analysis for falling films in laminar flow with heat conduction and obtained a simple equation for the heat transfer coefficient during the condensation of pure components on horizontal tubes.
h = 0.725 I
[ ___ Pgp2k3A D l T f 1:
Present address, Shell Development Co., Emeryville, Calif.
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 1, January 1970
(1)
Since that time, many aspects of the condensation problem, such as fog formation, presence of noncondensables, extraneous hydrodynamic effects, and vapors of miscible binary liquids, have been investigated (Sykes, 1968). One aspect that has not received sufficient attention is the condensation of vapors of immiscible liquids. Yet, the condensation of immiscible eutectics occurs frequently in industrial processes where either solvent drying or heterogeneous azeotropic extraction is used.
63
Relatively few experimental studies have been made and little or no theory exists for the condensation of immiscible liquids. The objectives of this study were to bring together the available experimental information, and explore several theoretical approaches with the goal of obtaining a better understanding of the phenomena. , A liquid-vapor eutectic for two immiscible liquids is characterized by the point on the temperature-composition phase diagram for a binary mixture where -
5
P r = P,
+ Px
in which Px,is the vapor pressure exerted by a homogeneous solution a t X,,, and PT is the total pressure. Since there are three phases in equilibrium and two components, there is only one degree of freedom-Le., the system is determined if either the temperature or the pressure is specified. For many systems, particularly organic-water systems, the mutual solubility is negligible, the activity coefficients are unity, and the eutectic satisfies Pr = P P p4 (3) Such eutectics can be called simple eutectics. When simple eutectics condense, the vapor and liquid mole fractions are equal and the weight ratio of the two components of an eutectic is given by
+
(4)
The variation of the eutectic composition with temperature or pressure is small. A twofold variation in the composition ratio requires more than a 100-fold variation in the pressure or 100°C. variation in the temperature for most systems. When a binary vapor mixture of immiscibles enters a condenser, if the mixture is not a t the eutectic composition, the component in excess condenses first. The second component acts as a noncondensable. As the vapors proceed along the condenser, the component in excess continues to condense until the eutectic is reached. From this point, the vapor mixture condenses as the eutectic mixture. Previous Investigations
The problem of condensation of vapors of immiscible liquids was first approached in a multipurpose study by Kirkbride (1933). As in all studies up to the present day, a successful theoretical attack was not made (Table I). Kirkbride's final equation for the eutectic heat transfer coefficient is simply the pure component heat transfer equation weighted by the physical property ratios for heat of vaporization, pure component vapor pressures, and molecular weights. Kirkbride tested his correlation by condensing benzene and water and cleaner's naphtha and water outside a
Table I. Studies of Condensation of lrnrniscibleson a Horizontal Tube
Authors
System Dimensions Diam., Length, in ft.
Condensing Surface
Materials Condensed with Water
Kirkbride (1933)
1.313
8.19
Iron
Benzene Cleaner's naphtha
Baker and Mueller (1937a,h)
1.313.
3.68
Copper
Benzene Toluene Trichloroethylene Mixed heptanes
Baker and Tsao (1940a,b)
1.o 0.625
3.68
Copper
Benzene Toluene Chlorobenzene Trichloroethylene Tetrachloroethylene
Hazelton and Baker (No horizontal tube studies) (1944)
(Correlated above systems)
Patton and Feagen (1941)
1.315
4.0
Copper
Turpentine
Stepanek and Standart (1958)
0.393"
3.28
Copper
Benzene Toluene Dichloroethane Chlorobenzene
Sykes (1968)
1.375
2.0
Copper
Toluene Carbon tetrachloride
Correlation Proposed
+ 801 [ 1 - -D0.0167 -] [ 1 - __ 0.85 500
he=
1 +o(
r
* Standart (1 968).
64
Ind. Eng. Chern. Process Des. Develop., Vol. 9,No. 1, January 1970
horizontal iron tube. N o attempt to maintain eutectic conditions was made. Thus, there may have been a region in the condenser where only one component was condensing. For this study only Kirkbride’s eutectic data were used. Baker and three of his students followed Kirkbride, seeking a correlation for the heat transfer coefficient during the condensation of immiscibles. I n the first study, Baker and Mueller (1937a,b) attempted to maintain waterorganic eutectic conditions using benzene, toluene, trichloroethylene, and mixed heptanes as the organics. Baker and Mueller were the first to recognize that the condensation was filmwise for the organic component and usually dropwise for water. Baker and Mueller attempted to correlate their data by noting the defects in Kirkbride’s equation, analogy to dropwise condensation, and semimechanistic arguments. The correlation presented by Baker and Mueller is given in Table I. The bracketed term is empirical and not dimensionally consistent but correlated their data and those of Kirkbride. Their equation does not hold a t the limits of’the pure components. Baker and Tsao (1940a,b) sought a correlation based on a series of mechanistic arguments. First, since the water forms droplets embedded in the surface of the organic film, they reasoned that a higher effective thermal conductivity would be found because of the higher thermal conductivity of water over organics and the roughening of the film by the droplets. The second mechanistic effect attempted to account for the fact that the underside of the tube held a thick organic film and so heat transfer through this area was probably small. Their empirical correction is presented in Table I. This correlation predicts: no temperature dependence for the heat transfer coefficient, and a universal heat transfer coefficient for all organics which condense as immiscibles with water. A second similar correlation, which reflected the -0.25 power diameter dependence obtained by Nusselt, was also proposed. Baker and Tsao’s data are too scant around the eutectic to be of use here. Baker and Hazelton (Hazelton and Baker, 1944) took the earlier data of Kirkbride, Baker and Mueller, and Baker and Tsao and proposed a correlation for horizontal tubes based on their successful vertical tube correlation. Again, this correlation shows no temperature dependence for the heat transfer coefficient, and predicts a constant pure component heat transfer coefficient for all organics which condense with water. The diameter dependence prediction agrees with Nusselt’s pure component prediction. Patton and Feagan (1941) reported data for the condensation of turpentine and water on a 1.35-inch-diameter horizontal copper tube. This eutectic forms a t about 53% water. The heat transfer coefficients of Patton and Feagan show a temperature dependence and can be correlated approximately by h, = 3000 AT/-”’ (5) Stepanek and Standart (1958) present the only recent study t o obtain heat transfer coefficients for the condensation of immiscible liquids and are the first to attempt a theoretical development. They used a thermal model and a hydrodynamic model. For the thermal model they assumed: All heat transfer resistance is by conduction in the liquid phase. Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 1, January 1970
The liquid which better wets the tube surface condenses in the form of a film; the second liquid forms droplets on this film. The droplet is modeled as a short cylinder having the maximum diameter of the droplet and an equivalent volume. The average behavior of the droplet can be described by a model with uniformly distributed, equal-sized droplets. For the hydrodynamic model, Stepanek and Standart assumed: The water droplets had the same effect as a water film of an equivalent thickness. The water film flows in plug flow with a velocity equal to the maximum velocity of the organic film. After making the hydrodynamic analysis and coupling i t with the heat transfer model, the authors then considered two special cases: The number of droplets per unit area is constant, and the area covered by the droplets is constant. Although they did not obtain an analytical solution, Stepanek and Standart’s analysis (1958) is outstanding, since it suggested that surface tension was important and that the temperature difference is more important than predicted by Nusselt’s equation (Table I ) . Stepanek and Standart’s correlation is interesting, since eight correlation constants were determined from four sets of data. Furthermore, the factor Fi is usually a very small number and a slight variance in the physical properties has a big effect on the heat transfer coefficient. I t is easy to calculate Fj as a negative number. I n addition, they did not compare their data and correlation with those of other investigators. Sykes (1968) in a recent experimental study obtained condensation heat transfer coefficients for the pure components steam, toluene, and carbon tetrachloride and for the toluene-water and carbon tetrachloride-water eutectics. The toluene-water eutectic was selected because the data of Baker and Mueller and Stepanek and Standart showed a discontinuity. KO data were available for the carbon tetrachloride-water eutectic. However, this system has a density difference between the two components which is greater than that in any system yet studied. The results from the pure component measurements indicated that Nusselt’s equation adequately predicts pure component condensation heat transfer coefficients. None of the corrections which have been proposed to Nusselt’s equation were necessary. The results for the condensation of the eutectics showed that the eutectic condensation heat transfer coefficients were distinctly a function of temperature, that the temperature function was different for each system, and that Nusselt’s equation could not adequately predict the eutectic Condensation heat transfer coefficient. The toluene-water eutectic data of this study agreed with those obtained by Baker and Mueller. A comparison of the studies and the proposed correlations indicates significant disagreement between experimenters as to the effect of different variables. There is little agreement on which physical properties are most important. Nearly all of the earlier correlations ignore the fact that the correlations can be reduced to simpler, more meaningful terms with the use of dimensionless physical and eutectic properties. The effect of diameter is apparent in some correlations and ignored in others. Most important, however, the effect of the film temperature 65
difference is virtually ignored by all experimenters up to Stepanek and Standart. All observers report that the organic phase forms a continuous film on the condensing surface, and eutectic condensation heat transfer coefficients in the range of the pure organic coefficients (300 to 100 B.t.u./hr. sq. ft. O F . ) as opposed t o the pure steam coefficients (1500 to 4000 B.t.u./hr. sq. ft. O F . ) . Thus, in this study eutectic heat transfer coefficients (Figures 1 through 7 ) are presented in the form of H us. AT,, where H is the ratio of the eutectic heat transfer coefficient to the pure organic coefficient predicted by Nusselt’s equation for the same
were used to obtain the constants for toluene, because if all of the data are used a large slope and small intercept are obtained (Table 11). However, no single set of the data reflects this condition; the use of such values does not appear merited. This anomaly reflects the fact that the toluene data of Baker and Mueller and this study
A1f.
Film Temperature Difference Dependence
The data show a general trend for H to increase as ATf increases. If the eutectic coefficient obeyed a relationship like the Nusselt prediction, all of the data would lie on a horizontal line. The results of a least squares fit for each system using
H = HIAT;
(6)
show (Table 11) that there is considerable variation in the “intercepts” (Hi) and slopes ( n ) for the various systems. The values of H i are shown for a range of In H I plus or minus one standard deviation of In Hi. Only the data of this study and Baker and Mueller
,
6
-
I
I
I
I
I
I
I 1
2
I
4
6
8
1
I
20
0
F I L M TEWPERATURE DIFFERENCE,
Figure 3. 1,2-DichIoroethane-water coefficients
6 .
,
-
I N C S S E L T PREDICTlOiV PURE H.ATER
’
I
NUSSELT PREDICTION FOR PURE WATER H = 6 . 5 - 6.0
I
I
I
,
1
40
/
,
I
80 100
O F
eutectic heat transfer
I
I
I
60
0 STEPANEK k \ 3 STANDAST
,
,
I
,
1
4-
t
T
+
\IMMISCIBLE LAMINAR F I L M S MODEL PREDICTION
+
+ H
NUSSELT PREDICTION PURE CHLOROBENZENE
I
2
4
6
8
1
I
I
0
20
F I L M TEMPERATURE D I F F E R E N C E ,
I
,
1
1
40
1
1
5
4-
I
I
I
!
Figure 4. coefficients 6
4
:I
, 2
I
DIFFESENCE,
Chlorobenzene-water
I
I
I
I
,
, 4
2.
20
6 78910
1
1
1
30
40 59 60 7 0 80 100
O F
eutectic
I
I
1
heat
I
transfer
I
1
1
’
)
t BAKER AND MUELLER
1 IMMISCIBLE LAMINAR F I L M S MODEL PREDICTION
, , , , , , 6
8
1
0
,
,
PURE
, , , , , ,] 40
20
F I L M TEMPERATURE D I F F E R E N C E ,
66
5
L N U S S E L T PREDICTION PURE B E N Z E N E
.I
Figure
4
NUSSELT PREDICTION FOR PURE WATER, H::.7-7.
0 STEPANEK AND STANDART
IMMISCIBLE LAMINAR F I L M S MODEL PREDICTION
H
3
FILM TEMPEqATURE
t BAKER ANC MUELLER
NUSSELT PREDICTION FOR PURE WATER H z 6 . 3 - 5 . 9
2
I
‘F
Figure 1 . Toluene-water eutectic heat transfer coefficents
1
I
,
80 100
60
60
80 100
OF
Benzene-water eutectic heat transfer coefficents
:;t I
I
2
,
, , 4
~
6
n - HEPTANE
,
, , ,, 8
1
F I L M TEMPERATURE
0
20
DIFFERENCE,
~
40
, , , 60
,ai
80 100
‘F
Figure 5. Heptane-water eutectic heat transfer coefficients Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 1, January 1970
NUSSELT PREDICTION FOR PURE WATER H: 7.7-7.2
@
i
THIS STUDY
c
4-
A
,
.3 -
I
I
1
I
,
,
I
l
l
Figure 6. Carbon tetrachloride-water eutectic heat transfer coefficients
compared with the data of Stepanek and Standart show a discontinuity. All studies prior to Stepanek and Standart’s have shown no temperature dependence for the eutectic heat transfer coefficient correlation. This means that n in Equation 6 should be 0.25 for all systems. The least squares fit shows that only three of the seven systems have slopes close to 0.25. For heptane and chlorobenzene, the data show a certainty that the slope is not 0.25. Furthermore, one of the slopes is greater and the other is less than 0.25. For the benzene and carbon tetrachloride data, the chance that the slope is 0.25 is considerably less than 30% for a normal distribution. The data of the remaining systems include the possibility that the eutectic heat transfer coefficient is not a function of temperature. Consideration of the fitted exponents, n, shows the unlikelihood that the earlier investigators were correct in their correlations. R Correlation
6 -
5
I
-
NUSSELT PREDICTION FOR PURE WATER, H :7.0 -6.3
r
I
I
+ BAKER AN0
MUELLER
d
43/
/ /-
+
2 r
H
HR/’
IMMISCIBLE LAMIUAR FILMS MODEL PREDlCTION
--_
t
,
2
1
I, I,2-TRICHLOROETHYLENE
-B/ I
./.
3
5 6 78910 20 F ! L M TEMPERATURE SIFFERENCE,
4
30
40
50
M)
As a result of using a simple relation to show that the eutectic heat transfer coefficient was a function of temperature, constants H I and n were generated. These constants were also found to correlate with a physical property parameter. Adamson (1967) cites the work of Langmuir, who studied the limiting thickness for a large lens supported on a horizontal liquid surface. Langmuir showed that as a droplet of an immiscible substance grew and formed a lens, the limiting thickness of the lens could be obtained from the density differences, the ratio of the densities, and the spreading coefficient. The treatment suggests (Sykes, 1968) correlation in terms of
7080 I00
(7)
‘F
Figure 7. 1,1,2-Trichl~roethylene-water eutectic heat transfer coefficients
where Ar =
[I - r ]
Table II. Results of least Squares Fit to H = HIAT;
No.
Source“
Points
Low
HI Med.
High
Stepanek and Standart Baker and Muell-er Sykes, Baker, Mueller, and Sykes (All data)
68 28 16
0.432 0.506 0.726
0.444 0.675 1.082
0.455 0.901 1.611
0.260 0.312 0.120
44 112
0.577 0.259
0.734 0.284
0.934 0.310
Baker and Mueller and Kirkbride Stepanek and Standart (All data) Stepanek and Standart
8 41 49 43
0.491 0.603 0.641 0.657
0.656 0.615 0.654 0.669
Stepanek and Standart
30
0.928
Heptanes
Baker and Mueller
25
Carbon tetrachloride
Sykes
16
1,1,2-Trichloroethylene
Baker and Mueller
8
System Toluene
Benzene
1,2-Dichloroethane Chlorobenzene
of
n
s.w
C‘
0.116 0.086 0.122
S/* 0.422 0.270 0.225
0.271 0.514
0.073 0.033
0.264 0.247
28.7 118.8
0.877 0.627 0.668 0.681
0.254 0.309 0.264 0.234
0.084 0.013 0.010 0.014
0.124 0.042 0.064 0.048
4.7 3.1 4.4 15.0
0.930
0.932
0.0825
0.0096
0.0365
3.5
0.337
0.408
0.493
0.508
0.060
0.177
18.1
0.420
0.523
0.650
0.418
0.070
0.141
12.9
0.870
1.209
1.520
0.280
0.071
0.135
12.3
S“
r,
4.0 30.0 21.4
For references see Table I .
Ind. Eng. Chern. Process Des. Develop., Vol. 9, No. 1, January 1970
67
The correlating equations to be used with Equation 6 are:
(r,
and the mass flow rates by the continuity equation = pkV2h2)are
Hi = 1 - 0.80 R (8) n = 0.67 R (9) where the physical parameter, R , was evaluated a t a "mean film temperature" of 60" C. The ability of these relations to predict the data is summarized in Table 111. The reduced heat transfer coefficient, H , predicted by these equations is shown as H R in Figures 1 through 7 . For toluene, only the data points from the correlation for the combined data of Sykes and Baker and Mueller were used. The agreement with most of the data is good, with the prominent exceptions of the trichloroethylene data of Baker and Mueller and the toluene data of Stepanek and Standart. I n general, the available data can be predicted to about +20% (Figure 8). laminar Two-Film Model
With the possible exception of Stepanek and Standart, all previous studies of heat transfer during the condensation of vapors of immiscible liquids are lacking in a theoretical approach. For Stepanek and Standart, the theoretical model proposed led to seemingly intractable mathematical relations and the data were finally correlated empirically. The correlating parameters were partially surmised by the theoretical discussion. The minimum eutectic heat transfer coefficient might be expected to result from a model which assumes that the two liquids flow as separate laminar films with one against the wall, while the other rides on top of the first. In this model, it is assumed that: The eutectic condenses at the vapor-liquid interface and instantaneously separates into two liquid layers. All of the heat given up by the condensate is deposited at the vapor-liquid interface and transferred, by conduction in series, through the liquid layers to the cold surface. The liquids are in laminar flow and have Newtonian viscosities. The hydrodynamic solution for the steady-state laminar gravity flow of two immiscible liquids over a flat plate requires the simultaneous solution of the equations of motion for each liquid. The solution to this problem is available (Sykes, 1968, 1969). The average velocities are:
I
A K E R RMUELLER - +,0 BTHIS STUDY, CARBON
TETRACHLORIDE
@ T H I S STUDY, T O L U E N E
/
4POINTS. B E N Z E N E 7
2-
'&?+'
