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Engineering Approaches
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Thermal Conductivity of Two-Phase Materials
I
GEORGE TSU-NING TSAO’ The University of Michigan, Ann Arbor, Mich.
6
This equation is proposed to correlate experimental data on
4 porous materials 4 packed beds
4two-phase liquid-liquid systems
A N is derived to relate the effective thermal conductivity of a twoEQUATION
*
phase heterogeneous material to the conductivities of the two constituents and to two parameters which describe the spatial distribution of the two phases. Porous materials, packed beds, two-phase liquid-liquid emulsions, and others are systems to which this equation can be applied. The effective thermal conductivity of heterogeneous materials has been of absorbing interest to many researchers for years. Many equations have been derived and many models have been postulated. However, it appears that there is no general solution to this problem as yet. The major difficulty is caused by the irregularity of the spatial distribution of the constituent materials, and hence the mathematics involved. I n 1934, Schumann and Voss (8) reported an approximate formula which gives the thermal conductivity of a heterogeneous system. Wilhelm, Johnson, Wynkoop, and Collier (77) analyzed the problem of heat conduction through granular media by using an electrical network. The effective conductivity of packed beds was studied by Smith and coworkers (7, 2). Heat transfer through packed beds was also studied by Yagi and Kunii (72, 73). Loeb (7) presented a theory relating the effective conductivity of porous materials to the conductivities of the solid constituent. Kingery, Francl, Coble, and Vasilos (5) experimentally measured the effective conductivities of several Present address, Stonewall Process Development, Merck and Go., Inc., Elkton, Va.
oxide ceramics. Recently, the problem of transfer of heat through porous rocks was investigated by Kunii and Smith (6). The same problem involved in several kinds of oil-water emulsions was studied by Wang and Knudsen (70). Heat transfer through two-phase heterogeneous materials can be considered as an over-all result of three mechanisms -namely, conduction through continuous and discontinuous phases ; convection; and radiation. For granular materials, if the sizes of the grains are small, the contribution of convection to the over-all transfer of heat is negligible (9). When the temperature is low, radiation effect will also be small ( 3 ) . Therefore, there are physical systems in which only thermal conduction is important. The equation derived in this work is for the combined thermal conduction through continuous and discontinuous phases without considering convection and radiation. A cube (1 X 1 X 1) of a two-phase heterogeneous material as shown in Figure A is assumed to have the surfaces parallel to xy and xz planes perfectly insulated-i.e., the direction of the overall propagation of heat is along x axis. In this article, the shaded space is referred to as phase d, the discontinuous phase, and the rest within the cube is referred to as phase c, the continuous phase. Phases c and d in a cube of heterogeneous material may be arranged in series or in parallel with respect to the direction of the over-all heat propagation. They are probably the two simplest models of distribution of the two phases. With bulk porosity defined as
the volume of phase d per unit volume of the heterogeneous material, the effective thermal conductivities of the two models may be expressed as follows: Parallel
k, = (1
Series k ,
=
- P) k ,
+ Pkd
(1)
P
(2)
1
(1
- P) k,
+&
Equations 1 and 2 are based upon the additivities of conductances in parallel and resistances in series. The k , ’ ~ obtained from the two equations are not equal. Therefore, bulk porosity alone is not enough to describe the characteristics of a heterogeneous material with regard to its effective thermal conductivity. I n this article, other parameters are postulated and used to formulate a general equation.
Modified Model A modified model, shown in Figure B, is in no way a general one. However, by using this model as a n illustration, three parameters can be postulated which will be used in the later derivation of a general equation. The first parameter is called “two dimensional porosity.” For a plane parallel to the yz-plane (Figure B ) , the two dimensional porosity, PB,is defined as area occupied b y phase d P* = unit area I n Figure B, the two dimensional porosity of the planes between x o and X I is constant and it is zero between XI and XZ. A seaond parameter is designated as “one dimensional porosity.” For a line drawn in the cube parallel VOL. 53, NO. 5
MAY 1961
395
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I
h
II Using the equations for resistances in parallel and resistances in series, a model can be devised which has the same thermal conductivity as the irregular two-phase material
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INDUSTRIAL AND ENGINEERING CHEMISTRY
THERMAL CONDUCTIVITY to the x axis, one dimensional porosity is defined as length occupied by phase d PI = unit length In a similar manner, the third parameter, “three dimensional porosity,” is defined as volume occupied by phase d Pa = unit volume I t is clear that the three dimensional porosity is the same as the conventionally defined bulk porosity. For the modified model in Figure B, we have P1 Pz
=
P$
(3)
P
and
”J
(4) From Equation 4, P3 alone is not enough to describe the characteristics of a heterogeneous material.
Development of a General Equation A two-phase heterogeneous material as shown in Figure A is sliced into many thin layers which are parallel to yt-plane. Each of the layers is so thin that we may assume Pz to be constant within each given layer. Based upon the additivity of conductances in parallel, phase d of each layer can be rearranged into a rectangle (in Figure C) without changing the effective thermal conductance of each individual layer. In doing so, it is assumed that the driving potential for the thermal conduction in x direction is uniform throughout each layer. The sequence of the layers is then rearranged into the one shown in Figure D in which P Z decreases in the x direction. Based upon the additivity of resistances in series, this rearrangement has no effect on the over-all thermal resistance of the whole cube. After these rearrangements, a model is obtained which is geometrically invariant along z axis. A unit area parallel to the xy plane is shown in Figure E. I n developing the model shown in Figure E from the one in Figure C,it is assumed that the thickness of each layer in Figure C is infinitesimal that a continuous curve is obtained. I t is possible that the curve in Figure D can have discontinuities. I n the following derivation, the curve is assumed to be, in general, continuous. However, it may easily be shown that the final derived equation can also apply to discontinuous cases. Within the unit area (Figure E ) , PI and P Z can be used as coordinates. We have, therefore, PI
=
J1
PZ dP1 =
h1
P I dP2
(5)
For a unit length along x axis, we have,
k c -- - 1= _ _1 =
sdR
L1
k,
1 dPi
+ (kd - k,)
p1 (6)
T o solve Equation 6, a function between PI and Pz is needed. I t is not difficult to derive an equation relating PI to Pz in the cases of regular distribution of phase d such as that of regular packing of spheres. However, in the general case where irregular distribution of phase d exists, an equation between P1 and PZ cannot be analytically derived. A probabilistic approach is used in the following derivation. I n Figure E, the horizontal line ABC intersects the curve a t B. This curve is such that P Z decreases in the x direction. If a second horizontal line D E F is drawn, the probability of the event that the D E F line has a one dimensional porosity larger than PI is equal to P z . This is true because this event only happens when the second horizontal line is drawn in the rectangle which has a width of P2 below line A B C and the whole model has a width of one. Therefore, we can write Pa
=
W(>Pl)
(7)
where PI/ is the notation for probability. Equation 7 represents the curve in Figure E. An infinite number of horizontal lines such as lines ABC and D E F can be drawn in a unit area. Supposedly, each line is divided into n segments. The value of n can be made so large that each segment is “occupied” by only one phase, either phase d or phase c. Each segment will have a probability of P3 to be occupied by phase d and (1 P3) to be occupied by phase c. The probability of having Pln out of n segments occupied by phase d follows a binominal distribution. When n is very large, a binominal distribution approaches a normal distribution ( 4 ) . Therefore, we can write Pz = W(>P1) =
where PI is the fraction of n segments occupied by phase d in the above experiment and is also the one dimensional porosity as previously defined. I n the equation, I.L is the mean of PI. From Equation 5, it is not difficult to see that p equals Pa. The standard deviation, U , of P I for any particular two-phase heterogeneous material has to be determined by actual experiment. The
value of Pz for any particular value of P1 can therefore be determined by Equation 8. Combination of Equations 6 and 8 yields,
Discussion Contrary to most of the equations reported in the literature which use only one parameter, usually bulk porosity, to describe the characteristics of a heterogeneous material, Equation 7 uses two parameters, /.l and U , to express the effective thermal conductivity. Both p and u can be experimentally determined for a particular two-phase material.
Nomenclature = continuous phase d = discontinuous phase k, = thermal conductivity of the continuous phase k d = thermal conductivity of the discontinuous phase k , = effective thermal conductivity of the two-phase heterogeneous material P = porosity PI,Pa,P3 = three newly postulated parameters R = resistance to the thermal conduction W = notation for probability p = mean of P I u = standard deviation of PI
c
literature Cited (1) Argo, W. E., Smith, J. M., Chem. Eng. Prom. 49. 443 (1953). (2) BGnneli, D. G . , Irvin, H. B., Olson, R. W., Smith, 3 . M., IND.ENG. CHEM. 41, 1977 (1949). (3) Hill, F. B., Wilhelm, R. H., A.Z.Ch.E. Journal 5 , 486 (1959). (4) Hoel, P. G., “Introduction to Mathe-
matical Statistics,” 2nd ed., Wilev. New York, 1954. (5) Kingery, W. D., Francl, J., Coble, R. L., Vasilos, T., J . Am. Ceram. SOC.37, 107 (1954). (6) Kinii, D., Smith, J. M., A.I.Ch.E. Journal 6, 71 (1960). (7) Loeb, A. L., J . A m . Ceram. SOC.37, 96 (1954). ( 8 ) Schumann, T. E. W., Voss, V., Fuel 13,249 (1934). (9) Waddams, A. L., J . SOC. Chem. Ind. (London) 63, 337 (1944). (10) Wanp. R. H.. Knudsen, J. G.. IND. ‘ ENG.cHEM. 50, 1667 ( i m j . (11) Wilhelm. R. H.. Johnson. W. C.. ‘ Wynkoop, ’R., Coliier, D. W., Chem: Eng. Progr. 44, 105 (1948). (12) Yagi, S., Kunii, D., A.Z.Ch.E. Journal 3, 373 (1957). (13) Yagi, S., Kunii, D., Ibid., 6, 97 (1960). RECEIVED for review August 30, 1960 ACCEPTED March 1, 1961 VOL. 53, NO. 5
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