I
R. H. WANG'
and JAMES G. KNUDSEN
Department of Chemical Engineering, Oregon State College, Corvallis, Ore.
Thermal Conductivity of Liquid-Liquid Emulsions These three equations for predicting thermal conductivity of two-phase fluid mixtures will be of special interest to the petroleum industry where mixtures are commonplace and liquid-liquid dispersions a bound
T
INPREDICTING heat
transfer rates for flow in pipelines, thermal conductivity is one of the physical properties of the fluid which must be known. The thermal conductivity of pure, single-phase fluids is usually accurately known. However, the effective thermal conductivity of two-phase fluid mixtures is not easily determined, and little experimental data are available. In the present work the thermal conductivity of liquid-liquid dispersions was measured to show the effect af phase concentration. Emulsions were made of water and petroleum solvent, water and mineral oil, and water and carbon tetrachloride. These data should be useful in correlating heat transfer data on liquid-liquid dispersions. Two-phase fluids are encountered in the petroleum industry where mixtures of gasoline and water (or water solutions) are frequently handled; liquid-phase reactions often involve liquid-liquid dispersions. From thermal conductivities of constituent materials in a two-phase system, the thermal conductivity of the system may be predicted by an equation derived by Tareef ( 5 ) . He reasoned that the thermal field in a two-phase system was analogous to the electrical field in a similar system and obtained the relationship,
where k, is the thermal conductivity of continuous phase and kd is the thermal conductivity of the discontinuous phase. From thermal conductivity values of each pure phase and the volume fracPresent address, Department of Chemical Engineering, University of Texas, Austin, Tex.
tion of the discontinuous phase in the mixture, conductivity of the emulsion may be estimated. Two other equations for predicting thermal conductivities of emulsions may be derived by assuming a highly idealized state of the emulsion. Consider the emulsion to be made up of uniform filaments of oil and water arranged parallel to the direction of heat flow. This idealized picture of the emulsion is shown in Figure 1,a. The rate of heat transfer through the emulsion is given by
of liquid present. Thus, from Equation 3 the thermal conductivity of the emulsion may be calculated from the volume fractions and the conductivities of the pure components. The third equation for predicting thermal conductivity is derived assuming the emulsion to be made up of uniform layers of oil and water arranged perpendicular to the direction of heat flow (Figure 1,b) The rate of heat conduction is given by
from which where subscripts e, tu, and o refer to emulsion, water, and oil, respectively. Simplifying Equation 2 and solving for k,, k, = k,(+)
+k,
(2)
Because
=
kwxw
+
A T, koxo
-I- AT, = AT
(3)
where x represents the volume fraction
;IE
1' t i
I L
I
Arya A
WATER OIL
'
It
WATER OIL
WATER mL WATER OIL
WATER OIL
AT
-I
Figure 1. Idealized diagrams of the emulsions show uniform filaments of oil and water arranged parallel to direction of heat fiow (Equation 2) and uniform layers of oil and water perpendicular to direction of heat flow (Equation 4) VOL. 50, NO. 11
NOVEMBER 1958
1667
and solving for k,,
(A) WATER
0.38
Equation 7 may be used to predict thermal conductivities of emulsions from the volume fraction and conductivities of the pure components. Experimental conductivities obtained in this investigation are compared with values given by Equations 1, 3, and 7. By means of the Fourier differential equation for one-dimensional heat conduction, the steady state rate of heat flow across a thickness AL of material under a temperature difference AT is:
t : LL
,0 . 3 4
LL' 0
- 0 WATER
01
F*
u: %
A 0.30-m
1
!
WITH 2 % A G A R WATER WITH 3 % SANTOMERSE
WATER W I T H 3 % T W E E N - 2 0
\
J t 2
I-
-l
2 0
1
-
I
MCADAMS
l
-
L) l
-L-
-I
where k,, is the value of k a t the arithmetic mean temperature of the material. Most methods for measuring thermal conductivities of materials employ Equation 8. The quantities q, A , A T , and AL are measured under steady conditions of heat conduction, and k,, is calculated. Experimental
(A) 2 0 % SOLVENT
IN 8 0 % WATER
0.4
4
- ---- --
0.3
c: LL 2
(y"
c'
LL
I
--. 2
0.2' ( 6 ) 40% SOLVENT
0.351
IN 6 0 % WATER
I
I
- - --- .-
1668
-- -
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INDUSTRIAL AND ENGINEERING CHEMISTRY
-E3 _o .
-
Equipment. The construction and arrangement of the apparatus are similar to that used by Orr and DallaValle (3) to investigate thermal conductivities of slurries. The apparatus consisted of four chambers separated by horizontal brass plates with the upper most chamber maintained at a high constant temperature and the lowest one a t low constant temperature. The middle chambers were the conductivity chambers. Of these, the upper contained fluid of unknown thermal conductivity; material of known conductivity was placed in the lower conductivity chamber. The brass plates on top of the conductivity chambers were provided with vents to remove air bubbles from the chambers. The vertical walls of the two conductivity chambers were Lucite plastic sealed to the brass plate with sealing compound (Sealit, Fisher Scientific Co.). An iron-Constantan thermocouple was installed in the center of each plate bounding rhe conductivity chambers. These calibrated thermocouples were located just under the surface of the plate. Thermocouple voltages were measured by a Leeds & Northrup Type K-2 portable precision potentiometer. The upper constant temperature chamber contained a thermostat, a stirrer, and a thermometer. Cold tap water could be circulated through the lower chamber which also contained an agitator and a thermometer. During a test, the whole apparatus was wrapped with glass fiber insulation to prevent heat flow in the horizontal direction. Emulsion Preparation. The petroleum solvent and mineral oil used were
EMULSION THERMAL CONDUCTIVITY manufactured by Standard Oil Co. of California and E. R. Squibb, respectively. Emulsifying agents were used to produce the emulsions, and the concentration of any emulsifying agent was kept
Table
I.
Experimentally Determined Thermal Conductivities of Emulsions
Emulsion Comp. (Free of -Emulsifying 'Agent), % Dise. Cont. phasea p hasesa 100 w 100 w 100 w
w 100 w 100 w 100 w 100 w 100 w 100
2OPS 20 PS 20 PS 40 PS 40 PS 40 P S 60 W 60 W 60 W 60 PS 60 PS 60 PS 40 W 40 W 40 W 20w 20w
!
as low as possible-2 to 3% could be tolerated. Usually 2% (weight) agar in the water phase was sufficient to produce a stable emulsion, but in some instances i t was necessary to use an
100 PS 100 PS 100 PS 80W 80 w 80W 60 W 60 W 60 W 40 PS 40 PS 40 PS 40 W 40 W 40 W 60 PS 60 P S 60 PS 80 PS 80 P S 100 MO 100 MO
Emulsifying Agent, %"
A
T
8
0.8 2 3 3 3 3
3
3 3 2 2 2 1
1 1 1 1 1 1 1 1 0.5 0.5 0.5
1.5 1.5 1.5 2 2 2 2 2 2 2.5 2.5 2.5 0.5 0.5
2 2 2 2 2
100 MO
20 MO 20 MO 20 MO 40 MO 40 MO 40 MO 60 W 60 W 60 W 60 MO 60 MO 60 MO 40 W 40 W 40 W 20w 20 w
80W 80 w 80 w 60 W 60W , 60 W 40 MO 40 Y O 40 MO 40 W 40 W 40 W 60 MO 60 MO 60 MO 80 MO 80 MO 100 cc14 100 Cc14 100 CClp 80W 80 w 80 w 60 W 60 W 60 W 40 CClr 40 CCl4 40 cc1a 40w 40 W 60 CClr 60 cc14 80 cc14 80 cc14
SO
1 1
1 1
1 1 1 1 1 1 1 1 1
1 1 1
1 1
1
3
3 3 1 1
1 1 1
0.5 0.5 0.5 0.5 0.5 0.5
1
1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25
0.5 0.5 0.5 0.5 1
A ~Temp., . 0
F.
65.7 72.8 83.3 84.0 85.2 74.4 85.9 77.2 85.1 64.6 73.0 81.6 63.3 78.6 86.0 63.5 72.3 80.0 62.4 74.6 90.0 62.8 76.8 90.4 64.0 80.2 86.8 75.3 86.3 60.9 68.7 81.6 65.1 81.8 92.6 61.9 73.6 82.2 63.0 72.7 81.3 60.5 711.4 80.7 62.9 , 71.1 81.4 60.5 80.1 61.4 69.8 76.8 62.4 71.6 82.9 61.2 71.0 75.0 60.8 68.0 77.3 61.3 74.1 64.3 75.2 61.5 71.5
, B.t.u./(Hr.) (s4,kFt,) F.)/(Ft.) (0
0.378 0.370 0.372 0.352 0.364 0.376 0.373 0.357 0.341 0.117 0.108 0.098 0.331 0.285 0.263 0.283 0.245 0.212 0.269 0.234 0.198 0.235 0.187 0.160 0.225 0.180 0.147 0.154 0.127 0.095 0.091 0.086 0.322 0.282 0.255 0.272 0.222 0.193 0.258 0.227 0.200 0.225 0.199 0.172 0.197 0.184 0.169 0.166 0.135 0.113 0.109 0.104 0.261 0.234 0.196 0.237 0.209 0.194 0.209 0.180 0.141 0.196 0.151 0.168 0.139 0.141 0.136
* I
'
1 W = pure water: PS = petroleum solvent; MO = mineral oil;A = agar; T = Tween 20; S = Santomerse; SO = sodium oleate. a
emulsifying agent in the oil phase. Emulsifying agents used were Santomerse (Monsanto Chemical Co.), Tween 20 (Atlas Powder Co.), commercial grade sodium oleate. The emulsion to be tested was prepared by dissolving 1 to 2% (weight) agar in hot water and then adding-the desired quantity of oil phase with less than 3% (weight) emulsifying agent. The mixture was cooled slowly. While cooling, the mixture was stirred sufficiently to maintain a uniform suspension. As soon as the emulsion became stable, it was carefully poured into the upper conductivity chamber of the atmaratus. No measurements were made on the size of the dispersed phase droplets. Emulsions were milky white in appearance and droplets were not discernible with the naked eye. Procedure. Before every determination, the brass surfaces of the chambers were carefully polished with fine emery cloth. Distilled water was placed in the lower conductivity chamber. Care was taken that no air was present in the water. The emulsion to be tested was placed in the upper conductivity chamber, and again precaution was taken to exclude all air. During a test, temperatures of the brass plates were determined periodically. When these became constant, it was assumed that equilibrium had been attained. This usually required 6 to 8 hours. Temperatures in the upper bath were maintained between room temperature and 110" F. The emulsions separated above this higher temperature. The lower constant temperature bath was maintained a t about 45' F. Below 39" F. excessive convection currents occurred in the water conductivity chamber. When equilibrium was attained in the apparatus, Equation 8 could be applied to each fluid in the system:
q1
= 42, and A I = A2
From the space ratio dl/ds, the ratio of the temperature difference, A Tw/AT,, and the known thermal conductivity of water, k, (at the mean temperature of the water in the lower chamber), the thermal conductivity of the emulsion could be determined. .. . . . .. This method for measuring thermal conductivity of emulsions depends upon knowing the thermal conductivity of water* 'lawecki and (4) measured the thermal conductivity of water and, using their data and those VOL. 50, NO. 11
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NOVEMBER 1958
1669
of other workers, obtained the following equation for the thermal conductivity of water:
+
k , = 141.2 (1 0.00232T 0.0000072 T 2 ) X 10-6 cal./(sec.) (sq. cm.) ( O C.)/(cm.) (12)
Equation 12 ( T is in ’ C.), which gives thermal conductivities 1 to 2% above those obtained experimentally by Timrot and Vargaftif ( 6 ) , is used to calculate the value of k, in Equation 11. Results and Discussion
The conductivity of pure water was first measured to determine if the apparatus was operating properly and to evaluate the magnitude of convection effects in the relatively thick layers of fluid being studied. Convection effects were minimized by placing the high temperature bath on top and the low temperature bath on the bottom. Because water has a maximum density a t 39.2’ F. it was necessary to keep the cold chamber slightly above this temperature. The measured conductivities of pure water, water with 0.8 and 2% agar, and water with emulsifying agents are shown compared with Equation 12 in Figure 2,A. Because each chamber was colder at its lower boundary there was little tendency for convection; however, the temperature gradient across the layer of fluid undoubtedly caused some circulation. At the lower temperatures studied for pure water, there was about 0.035% density change across the lower chamber and 0.1% density change across the upper chamber. At the higher temperatures, density changes were 0.2 and 0.44%, respectively. Thus more circulation would occur in the upper chamber, and the high results for pure water bear this out particularly a t the lower temperatures. The results on water containing agar lie quite close to Equation 12 but below it indicating that convection in the
upper chamber is reduced because of the presence of agar. Hence, when convection is eliminated from the upper chamber, the data show that at lower temperatures convection in the lower chamber is negligible and true conductivities are measured; at higher temperatures convection in the lower chamber becomes more significant and measured conductivities are slightly lower than the correct value. The experimental accuracy of measured temperatures was 1 0 . 2 ’ F., so the conductivities are accurate to about 1 2 to 47& Most of the data for pure water and water containing agar or emulsifying agent fall within this range. Thus up to 2% agar or 3% emulsifying agent has small effect on the thermal conductivity of water. Thermal conductivities of carbon tetrachloride with 3% sodium oleate, petroleum solvent with 3% sodium oleate, and mineral oil with 2% sodium oleate are shown in Figure 2,B and C. The data for carbon tetrachloride are compared with those of McAdams ( I ) . The present results are somewhat higher and have a greater negative temperature coefficient. This is in accordance with the conclusions regarding convection in the chambers. Experimental conductivities for petroleum solvent and mineral oil agree with published values for similar hydrocarbons (2). All conductivity data on emulsions are shown in Table I. Figure 3 shows data for three emulsions compared with values predicted by Equations 1, 3, and 7 . In most cases, measured conductivities have a greater negative temperature coefficient than is given by predicted values probably because convection currents in the lower chamber are more significant a t higher temperatures. To compare the results, measured and calculated thermal conductivities were considered at two temperatures, 65’ and 80’ F., and the percentage deviation of the predicted value from the experimental value was determined (Table 11).
For emulsions made up of petroIeum solvent and water, Equation 3 shows the least deviation a t 65’ F., and Equation 1 shows the least deviation of 80’ F. Over the whole temperature range, Equation 3 appears to be most satisfactory for emulsions of either the oilin-water or water-in-oil type. I t will predict thermal conductivities within 10% of the measured value in most cases. Equation 3 shows the least deviation from experimental values over the temperature range for oil-in-water or water-in-oil emulsions made of mineral oil and water. For these emulsions, the thermal conductivity predicted by Equation 3 is also within 10% in most cases. Thermal conductivities of emulsions containing water and carbon tetrachloride are best predicted by Equation 7. In the temperature range studied, this equation shows an average deviation of about 15% from experimental values. Acknowledgment
Appreciation is expressed to the Engineering Experiment Station, Oregon State College, for a research assistantship to support this investigation. Nomenclature
A = area through which heat flows, sq. ft. d = thickness of conductivity chamber, in. k = thermal conductivity at temperature T, B.t.u./(hr.) (sq. ft.) (” F.)/jft.) L = length ,’ft, 4 = steady rate of heat flow, B.t.u./hr. T = OF. x = volume fraction of liquids ~I
Subscripts c = continuousphase
d = discontinuous phase e = emulsion o = oil w = water Literature Cited
Table II.
Deviation of Calculated and Experimental Thermal Conductivity Values
Equation 3 is the most satisfactory for water-petroleum solvent and water-mineral oil, while water-carbon tetrachloride emulsions are best predicted by Equation 7
Av. Deviation, % 65’ F.
Components
Eq. 1
Eq. 3
Eq. 7
Petroleumsolvent and water
Water in oil Oilinwater Either
-14.5 -15.6 -15.1
-3.3 -10.6 -7.0
-24.6 -27.5 -26.1
-5.3 0.0 -2.7
12.4 7.4 9.9
-19.1 -27.9 -23.5
Mineral oil and water
Water in oil Oil in water Either
-22.1 -14.8 -18.5
-5.6 -8.5 -7.1
-34.4 -36.9 -35.7
-12.4 2.6 -4.9
9.1 11.1 10.1
Carbontetrachloride and water
Water in oil Oil in water Either
-0.7 -5.1 -2.9
57.8 57.6 57.7
85.2 66.5 75.9
-27.5 -27.1 -27.3 32.2 23.6 27.9
17.4 22.8 20.1
36.9 25.5 31.2
Eq. 1
SOo F. Eq. 3
Type
Eq.
7
(1) Mchdams, W. H., “Heat Transmission,” 3rd ed., McGraw-Hill, New York, 1954. (2) Nelson, W. L., “Petroleum Refinery Engineering,” 3rd ed., McGraw-Hill, New York, 1949. (3) Orr, Clyde, Jr., DallaValle, J. M., Chem. Eng. Progr. Symp. Ser. No. 9, 50, -29 -
11054’1. ,-_-
(4) Slawecki, T. K., Molstad, M. C., IND. ENG.CHEM.48, 1100-3 (1956).
(5) Tareef. B. M., J . Colloid (U.S.S.R.) 6, 545 ‘(1940). . (6) Timrot, D. L., Vargaftif, N. B., J . Tech. Phys. (U.S.S.R.) 10, 1063 (1940). ’
RECEIVED for review February 11, 1958 ACCEPTED August 7,1958
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INDUSTRIAL AND ENGINEERING CHEMISTRY
