I
T. B. JEFFERSON,
0.W.
WITZELL, and W. L. SIBBITT'
Purdue University, Lafayette, Ind.
Thermal Conductivity of Graphite-Silicone Oil and Graphite-Water Suspensions Reliable data can be calculated to replace experimental determinations of manyb types of mixtures
OVER
the past 50-0dd years, numerous methods for the calculation of thermal conductivities of two-component twophase mixtures have been proposed. Many of these (7, 4, 5, 7, 70, 77) are analytical, with conductivity expressed in equation form. Some (3, 9 ) are graphical methods for predicting conductivity value. All are based on simplified models consisting of uniform particles of one phase distributed throughout a second phase. Rather simple geometries have been assumed for the dispersed particles, to keep the mathematics tolerable. The analytical approaches work rather well for many mixtures of two phases having individual conductivities of the same order of magnitude. Analytical and experimental values often differ widely, however, where the two conductivities are in a ratio of, say, 1000 to 1. Thus an explicit equation is needed, applicable to such mixtures.
Figure 1. Analytical conductivity is determined by considering the fluid-solid suspension cubes as divided into
e
A
Analytical Conductivity Determination Consider the fluid-solid suspension under analysis to be divided into a number of cubes (Figure l , u ) , each having a sphere of solid a t its center. If a representative cube is taken out and cut in half on a plane perpendicular to heat flow, the result is a half-cube of fluid with a half-sphere of solid inside. This may be further subdivided into components A, B, and C (Figure l , b and c), which will comprise the composite model used in the following development. In the theoretical calculation of thermal conductivity, this composite is considered a parallel-series thermal circuit such as the one illustrated in Figure 1,d.
Assumptions Made in Derivation. The solid particles are spherical and uniform in size. The particles are distributed in an orderly fashion. Conductivities and densities of both solid and fluid are constant. The entire solid-fluid suspension sample is confined between two constanttemperature planes perpendicular to the Present address, Los Alamos Scientific Laboratory, Los Alamos, N. M.
tt)
direction of heat flow. Thus it follows, by symmetry, that any other normal plane which separates two adjacent layers of cubes will be isothermal. Further, any normal plane passing through the centers of a layer of spheres will be isothermal. Heat flow is in the x-direction only (Figure 1, e ) . Components A, B, and C are to be
treated as resistors in the thermal circuit, the individual resistances of which do not change with changes in solid concentration. ,Once a value is obtained for the thermal resistance of A , this will be added to the resistance of C. Such an operation requires that the plane separating A and C be isothermal. This would be strictly true only where adjacent spheres are in contact.
~ S P A O E R ~ H E A T E R j - POTENTIAL LEADS
-
-
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--!t
FLUID L A Y E R
THdRMlSTOR
FlLLlNQ TUBE WELLS
SECTION-AA
Figure 2. An instrument of the concentric cylinder type was used for measuring conductivity VOL. 50, NO. 10
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GALVANOMETER LAMP
TRANSFORMER ,PHOTOCELL AMPLIFIER
I I
3".
THERMISTOR
7
1
SELECTOR SWITCH POWER STATSCOOLIN COILS
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-T E S T INSTRUMENT
THERMISTORS -
CONTROL
THERMOSWITCH
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bUXILIARY 1.E L ATE
Figure 3.
= qA
+
Since F = _ .a - .
(1)
qB
If the temperature difference across the model is t , and a spherical model of unit diameter is used as shown in Figure l,f, q A and q B can be written as:
(3)
From the relation for qA, the combined conductivity of the series system made up of A and C may be calculated as:
and combination of Equations 1, 2, and 3 yields
The dimensions of components of the model were chosen for convenience in calculation, and will not affect the final expression for conductivity of the suspension. Before the suspension conductivity, ks, can be calculated using Equation 6, the conductivity of A must be evaluated. Using the dimensions shown in Figure l,e, the heat flow through A may be written :
4
.4B = (1
1590
A,- = (1
+ 2n)'
+ 2n)2 - 4 7r
-
INDUSTRIAL AND ENGINEERING CHEMISTRY
I+
- 4(1 +iT 2 n ) 2
ks = k~ [1
::.dO'[ 2n)Z
4(1
2 fitL]
04
1
, "10
'
, 100
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1
120
140
TEMPERATURE,
(lo)
The conductivity of a two-phase mixture can be obtained from Equations 7 , 9, and 10. The analytical relations just derived have been applied to two two-phase systems--graphite-gelled water and graphite-oil suspensions-and calculated values for conductivities us. per cent graphite by weight plotted. When kp/kL is very large (about 300 and 1000 for the two suspensions in this investigation), Equations 7, 9, and 10 can be combined to give :
L
Here, A*, AB, As, and n depend on the volume fraction of solid material in the suspension. From the geometry of the model, ?
Equation 6 may be written as
where -4 = T , dA = 2 ~ r d r ,and At' is the temperature difference across A. Equation 8 yields the following expression for kA:
from which the following expression for the conductivity of the whole model is obtained:
Aa
OIL
The instrument was immersed in a constant-temperature oil bath
The heat flow through the composite model is: qS
110 V. A.C.
1&
DEQ. F.
1
1
180
'
l 200
l
~
220
Figure 4. The conductivity of graphite in DC-550 oil was determined as a function of the weight composition
l
THERMAL C 9 NDUC T I V I TY
L
\
Experimental Conductivity Determination The conductivities of these same suspensions were experimentally determined. Finely divided graphite powder was suspended in Dow-Corning silicone oil D.C.-550 and in a 5% tragacanth-water gel (Table I).
Table I.
Properties of Components of Suspensions Thermal Conductivity, B.t.u./ Spec. Hr. Ft. O F . Component Gravity loOD F. Graphite 2.15" 93b 5 % tragacanth-water gel l.OOc 0.323c D.C.450 silicone oil 1.07d 0.0879 Information supplied by United States Graphite Co., Saginaw, Mich., and checked by authors in approximate measurements on samples of solid material from which powdered product is ground. * Measured by Powell (8) for solid samples of graphite. Measured by authors. Information supplied by Dow-Coming Gorp., Midland, Mich.
The suspensions were made by slowly mixing No. 205 lubricating graphite powder (United States Graphite Co.) into the liquid until the desired solid concentration was obtained. The tragacanth-water gel was prepared with the aid of the Purdue School of Pharmacy. Tragacanth powder (5% by weight of the final mixture) was slowly stirred into hot water until it was all dissolved. The resulting gel was capable of holding the
graphite particles in suspension. As predicted by Boggs and Sibbitt ( 2 ) , conductivity (and density) properties remained essentially the same as for water. The density values in Table I were used to convert volume fractions of the suspended solid material (F,Equation 11) to weight fractions. Graphite weight concentrations of from 0 to 48.7% in tragacanth-water gel and from 0 to 59.4% in D.C.-550 were tested. Higher concentrations would have' resulted in voids in the sample. A primary test instrument of the concentric cylinder type was used for actual measurement of conductivity values. The instrument (Figure 2) was designed and built by Boggs (2). The annular space, which was approximately 0.11 inch thick, contained the suspension sample to be tested. Heat was supplied electrically by a direct current heating coil in the inner cylinder, and was transferred radially through the test sample. The center 1-inch length was used as the test section; axial heat flow to and from the test section was held to a minimum by means of guard heating a t the ends of the cylinder. The conductivity of the sample was calculated from the equation for two-dimensional radial conduction in a cylinder. The power dissipated in the test section, q, was measured electrically. The voltage drops across the test section heater, and across a standard 1-ohm resistor placed in series with the heater, were measured by means of a Leeds & Northrup K-2 potentiometer. Readings represented voltage and current, respectively ,
0 GRAPHITE FRACTION BY WEIGHT
Figure 5. Calculated conductivities of gelled water-graphite suspensions compare favorably with experiment
0.1
The temperature difference across the sample, of the order of 1' F., was determined through use of a thermistor probe. I n this application, the 14-A Western Electric thermistor served as a very sensitive resistance thermometer. The thermistor was moved from one well to the other and the temperature difference obtained from readings from the same thermistor a t each location. A five-dial Leeds & Northrup wheatstone bridge was used to measure the thermistor resistance. The entire test instrument was immersed in a constant-temperature oil bath during runs (Figure 3), so that the temperature distribution throughout the instrument and test sample was dependent only upon the power input to the instrument and the conductivity of the sdmple. The temperature-sensitive element was a Western Electric Thermistor, connected in one leg of a Wheatstone bridge. A Type R mirror galvanometer, connected across the bridge, deflected when a slight change in bath temperature caused a corresponding change in the thermistor resistance. Through a reflected beam of light, this activated a photocell, which controlled the bath heaters through a system of relays. This control system was capable of de. tecting temperature changes of a few thousandths of 1O F. The maximum expected error in measured conductivity values was f 4 . 2 % . A detailed error analysis and a more thorough description of the test apparatus are available (2, 6). Results Conductivities of six concentrations of graphite-silicone oil and four concentrations of graphite-gelled water suspensions were experimentally determined. Concentrations ranged from 0 to 59.4% solid by weight in the case of graphite-
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44
46
OS
0.7
GRAPHITE FRACTION BY WEIGHT
Figure 6. A comparison was made of the calculated and experimental conductivities of oil-graphite suspensions VOL. 50, NO. 10
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0.3
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OJ
GRAPHITE FRACTION BY WEIGHT
Figure 7. Considerable deviation exists between various analytical determinations and the experimental conductivities for gelled water-graphite suspensions
silicone oil, and 0 to 48.7% solid by weight for graphite-gelled water. A typical plot of conductivities as a’function of temperature and solids concentration is shown in Figure 4. Using the 100OF. data from these graphs, conductivity was plotted as a function of solid concentration (Figures 5 and 6). Comparison of the theoretical and experimental curves indicates that the analytical method is capable of predicting conductivities of the right order of magnitude. However, discrepancies between the analytical and experimental results ranged from 0 to 30%, with a n average deviation of about 25%. The conductivities of the liquid components, which made the major contribution to the suspension conductivity values, were experimentally determined as a part of this study. Erroneous k L values were thus eliminated as possible causes for disagreement between the experimental and analytical results. The conductivity of the suspension, ks, is insensitive to the conductivity of the graphite particles. As a matter of fact, the effect of using kp = 120 B.t.u./hr.ft.-” F. instead of 93 in Equations 7, 9, and 10 was negligible. The suspension conductivity is, however, rather sensitive to the component densities when conductivity is considered as a function of weight concentration of the dispersed solid material. Experimental error is expected to account for only about 4% of the discrepancy. Much of the disagreement is probably due to the simplifying assumptions regarding the configuration of the model and heat transfer through the model. Theoretical conductivities, calculated by several methods (7, 3-5, 77) are plotted for comparison in Figures 7 and 8. The good agreement among the several theories for low solid concentrations is to be expected; the shape of the model used in the development of the analytical method has less influence
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02 0.3 0.4 0.5 GRAPHITE FRACTION B Y WEIGHT
0.6
0.7
Figure 8. Several analytical determinations of the conductivity of oil-graphite suspensions agree with experimental information = parameter, defined
n
by Figure 1, f related to volume fraction of solid material in two-phase mixture = heat transfer rate = heat transfer rate through component A of model = heat transfer rate through component B of model = heat transfer rate through component C of model = heat transfer rate through composite model = radial distance from center line of model, defined by Figure 1,e = temperature difference, measured in direction of heat flow, across entire model = temperature difference across component A of model = temperature difference across sample in experimental apparatus = distance through which heat is conducted in liquid = distance through which heat is conducted in solid particle
on the result as particle separation increases, and the conductivity of the mixture approaches the limiting value of kL. The theoretical curves most nearly approximating the shape of the experimental curves are those of Deissler (3) and the writers. Of these, absolute values obtained by the Deissler method are somewhat nearer to the measured ones. The Deissler values were determined by a graphical interpolation procedure; an explicit equation relating conductivity and solid concentration was not given. The analytical methods developed in this paper is believed to be an improvement over the earlier theories, in that, of those explicitly relating thermal conductivity to the component conductivities and solid concentration, it is in closest agreement with experiment over the range of concentrations reported when both magnitude and trend are considered.
r
Nomenclature
literature Cited
A,
(1) Austin, J. B., Symposium on Thermal
area, measured perpendicular to heat flow, of component A of model AB = area, measured perpendicular to heat flow, of component B of model A s = area, measured perpendicular to heat flow, of composite model F = volume fraction of solid (dispersed) material in two-phase mixture k A thermal conductivity of component A of model k B thermal conductivity of component B of model kc thermal conductivity of component C of model RAC = thermal conductivity of combination of components A and C k s = thermal conductivity of composite model kL = thermal conductivity of liquid (continuous) phase in mixture k, = thermal conductivity of solid (dispersed) phase in mixture In = natural logarithm
INDUSTRIAL AND ENGINEERING CHEMISTRY
=
q qA
qB qc qs
At At’
AT xL xp
Insulating Materials, Columbus Regional Meeting, A.S.T.M., Philadelphia, 1939. (2) Boggs, J. H., Sibbitt, W. L., IKD.ENG. CHEM.47, 289 (1955). (3) Deissler, R. C., Eian, C. S., Natl. Advisory Comm. Aeronaut. NACA RME 52 C 0 5 (June 1952). (4) Eucken, A., Forsch. Gebiel e Ingeniewzu. 11, 6 (1940). (5) Jakob, M., “Heat Transfer,” vol. I, p. 87, New York, 1949. (6) Jefferson, T. B., Ph.D. thesis, Purdue University, June 1955. (7) Lees, C. H., Phil. Trans. Roy. Soc. London A 191, 428-38 (1898). (8) Powell, R. W., Proc. Phys. Soc. (London) 49, 419-26 (1937). (9) Schumann, T. E. W., Voss, V., Fuel in Science and Practice 13. D. 249 (August 1934). (10) Strickler, H. S., J . Chem. Phys. 20, 1333-4 (1952). (11) Tareev, B. M., Kolloid. Zhur. 6, 545-50 (1940). RECEIVED for review October 16, 1957 ACCEPTED June 12, 1958
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