Zf= height of liquid in bubble column when gas is flowing, cm ZL = height of liquid in bubble column when gas is not flowing, cm Greek Letters a = frequency factor i n Arrhenius equation, eq 8 CYD = frequency factor defined by eq 9 fl = slope of the least-squares line in Figures 2-5 6 = ratio of diffusivities B = dimensionless time defined by tr/CAi p = viscosity, CP v = number of moles of absorbent reacting with one mole of solute p = density, g/cm3 T = dimensionless concentration 4 = enhancement factor Subscripts 0 = initial conditions A = oxygen B = propionaldehyde c = continuous phase d = dispersed phase g = gas phase i = gas-liquid interface M = catalyst
Literature Cited Allen, G. C., Augilio, A., Adwan. Chem. Ser. No. 76, 362 (1968). Bawn, C. E. H., Discuss. FaradaySoc. 14, 181 (1953). Bawn, C. E. H., Hobin, T. P., Raphael, L., Proc. Roy. SOC., Ser. A 327, 313 (1956). Bawn, C. E. H., Jolley, J. B., f l o c . Roy. SOC.,Ser. A 327, 296 (1956). Bawn, C. E. H., Williamson, J. B., Trans. FaradaySoc. 47, 721 (1951).
Box, G. E. P., Hunter, W. G., Technometrics 4(3), 301 (1962) Brian,P.L.T..A.l.Ch.E.J.l0,5(1964). Calderbank. P. H., Moo-Young, M. B., Chem. Eng. Sci. 16, 39 (1961). Carpenter, B. H., Ind. Eng. Chem., Process Des. Develop. 4, 105 (1965). De Waai, K . J. A,, Okeson, J. C., Chem. Eng. Sci. 21,559 (1966). Emanuel, N. M., Knorre, D. G., Zh. Fiz. Khim. 26,425 (1952) Gal-Or, B.. Hoelscher, H. E., A./.Ch.E. J. 12,499 (1966). Gal-Or. B.. Resnick, W., Adwan. Chem. Eng. 7, 296 (1968). Gal-Or, 6.. Waslo, S.,Chem. Eng. Sci. 26,829 (1971). Govindarao, V. M. H., Gururnurthy, C. V., J. Appl. Chem. Biotech. (London) 23,465 (1973). Gururnurthy, C. V., Ph.D. Thesis, Indian Institute of Science, Bangalore, India, 1971. Gururnurthy, C. V.. Govindarao, V. M. H., lndian J. Chem. 6, 327 (1968). Gururnurthy, C. V., Govindarao, V. M. H., lndian J. Tech. 10, 183 (1972). Havel, S., Vys. Sk. Chem. Tech. Pardubice 13(1). 31 (1966). Hunter, W.G., Mezaki, R.,A.l.Ch.E. J. 10,315 (1964). Hurst, P.. Skirrov, G., Tipper, C. F. H., J. Phys. Chem. 66, 191 (1962). Ingold, K. U.. "Lipids and Their Oxidation," Chapter 5, Avi Publishing Co.. Inc., Westport, Conn., 1962. Langdon, W. K., Schwoegler, E. J.. lnd. Eng. Chem. 43, 1010 (1951). Marta, F., Boga, E., Matok, M., Discuss. FaradaySoc. 46, 173 (1968). Melle, U. D., British Patent, 603,175 (June 10, 1948); [Chem. Abstr. 43, 669' (1949)]. Prausnitz, J. M., Shair, F. H.,A.l.Ch.E. J. 7, 683 (1961). Reid, R. C., Sherwood, T. K . , "The Properties of Gases and Liquids," 2nd ed, McGraw-Hill, New York. N. Y . , 1966. Rudkovskii, D. M., lnst. Neft. Prot. 2, 98 (1960); [Chem. Abstr. 57, 16376 (1962)]. Russel, T. W. F , Schaftlein, R. W., lnd. Eng. Chem. 60(5),12 (1968) Steacie, E. W. R , Hater, N. H., Rosenberg, S , J. Phys. Chem. 38, 1189 (1934). Tuiney, T. A., "Oxidation Mechanisms," Butterwor.ths, London, 1965. Whitefield, G. H.. U. S. Patent. 2.959.613 (Jan 20. 1958): IChem. Abstr. 56,58388 (1962)] Whitefield, G H , Kemp, E , British Patent, 886,324 (Jan 3 1962), [Chem Abstr 57,58041 (1962)]
Received f o r review June 16, 1972 Accepted October 31, 1973
The Thermal and Electrical Conductivity of Two-Phase Systems Lawrence E. Nielsen' Chemical Engineering Department, Washington University, St. Louis, Mo., and Monsanto Company, St. Louis, Mo. 63766
Equations from the theory of the elastic moduli of composite materials are used to calculate transport properties, such as electrical and thermal conductivities, of two-phase systems. The nature of the packing of the dispersed particles and their shape are important factors. The effect of particle shape can be incorporated into the theory a s a generalized Einstein coefficient, which has been published for many shapes. The effect of particle packing can be taken care of by the maximum packing fraction, which can be experimentally measured in many cases and calculated in other cases. The theory appears to agree well with experimental results.
Introduction The estimation of the electrical and thermal conductivity of many kinds of two-phase systems is a problem often encountered in chemical engineering and other engineering sciences. The types of systems vary from the electrical conductivity of metal-filled plastics to the thermal conductivity of packed beads, porous materials, composites, and foams. Many equations have been proposed for the transport properties, such as electrical and thermal conductivity, of two-phase systems (Ashton, et al., 1969; Behrens, 1968; Bruggeman, 1935; Cheng and Vachon, 1969; Hamilton and Crosser, 1962; Kerner, 1956; Springer and Tsai, 1967;
' Permanent address, Monsanto Co., St. Louis, Mo. 63166
Sundstrom and Chen, 1970; Tsao, 1961; and Zinsmeister and Purohit, 1970). Nearly all of these theories neglect several important factors, however. Generally, only the conductivities of the two components and their concentration are considered. The effects due to the shape of the particles, how the particles pack, and possible anisotropy are neglected. Certainly, the conductivity of a system containing two components is not the same when the dispersed phase is in the form of spheres as it is when the dispersed phase is aligned rods. Thus, shape and anisotropy of the system should be considered. Likewise, the conductivity should change rapidly in the concentration range near the maximum packing fraction of the dispersed phase since in this concentration range the large number of particle-particle contacts provides a through-going path Ind. Eng. Chem., Fundam., Vol. 13, No. 1 , 1974
17
for the flow of heat or electricity. Most of the theoretical equations, however, assume uniform changes up to the point where the dispersed phase makes u p the complete system. During the past few years the theory of the elastic moduli of composite materials has developed to the extent that one can predict these moduli with a high degree of confidence (Ashton, et al., 1969; Halpin, 1969; Hashin and Rosen, 1964; Hashin and Shtrikman, 1963; Kerner, 1956; Lewis and Nielsen, 1970; Nielsen, 1969, 1970; Rosen, 1970; Tsai, 1968; and Whitney and Riley, 1966). These equations also may be used to calculate electrical and thermal conductivities. The justification for using the same equations for conductivity as for elastic moduli has been discussed by Ashton, et al., 1969, and by Springer and Tsai, 1967. The conductivity becomes the analog of the stiffness or elastic shear modulus, and the disturbance of the flux field becomes analogous to the disturbance of the stress field by the dispersed particles. Theory The theories of the moduli of composite materials are well covered in the literature and need not be reviewed here. Two classes of theories are the self-consistent models (Hill, 1965; Kerner, 1956) and the variational methods based on energy theorems of classical elasticity (Hashin, 1964). The point of this paper is to bring to the attention of engineers that some of the equations can be used to estimate conductivity of two-phase systems more accurately than what bas been possible in the past. Of the numerous equations proposed for calculating the elastic moduli of all kinds of composite materials, the best and most versatile are the so-called Halpin-Tsai equations (Ashton, et al., 1969; Halpin, 1969; Tsai, 1968) as modified by Nielsen (1969, 1970) and Lewis (1970). It is important to emphasize that, in principle, there are no adjustable parameters in these equations if enough information is available on a given system to characterize the shape of the dispersed particles and the type of packing of the particles. In most cases, however, estimates have to be made of these quantities as they were not reported in the original work. It turns out that these theoretical equations are very general types of mixing equations similar to those which are constantly being used by engineers to calculate many kinds of properties when two materials are mixed. The original Halpin-Tsai equations are identical with the theoretical equation of Kerner when the dispersed particles are spheres. The modified equations for two-phase systems are (Lewis and Nielsen, 1970; Nielsen, 1970)
(3) (4) The property of interest, such as the thermal or electrical conductivity of the two-phase material, is K . Subscripts 1 and 2 refer to the continuous and dispersed phases, respectively. The constant A depends primarily upon the shape of the dispersed particles and how they are oriented with respect to the direction of flow of thermal or electrical currents; the constant A is related to the generalized Einstein coefficient k E (Einstein, 1905, 1906). The factor B is a constant which takes into account the relative con18
Ind. Eng. Chem.. Fundam., Vol. 13, No. 1, 1974
Table I. Value of A for Various Two-Phase Systems
Type of dispersed phase
Direction of heat flow
Spheres Aggregates of spheres
Any Any
Randomly oriented rods Aspect ratio = 2 b Randomly oriented rods Aspect ratio = 4 Randomly oriented rods Aspect ratio = 6 Randomly oriented rods Aspect ratio = 10 Randomly oriented rods Aspect-ratio = 15 Uniaxially oriented fibers Uniaxiallv oriented fibers
Any
A 1.50 2.50
4a
-
la
1.58
Any
2.08
Any
2.8
Any
4.93
Any
8.38
Parallel to fibers 2L/D Peruendicular 0.5 to fibers
a $a = maximum packing fraction of the spheres in the aggregates, i.e., 4mof the particles which make u p the aggregates. b Aspect ratio = length, L/diameter, D.Taken from various sources including Ashton, et al., 1969; Burgers, 1938; and Lewis and Nielsen, 1968.
ductivity of the two components. The factor $ is determined by the maximum packing fraction $m of the dispersed particles. The volume fraction of the dispersed phase is 42. The maximum packing fraction $m is defined as the true volume of the particles divided by the volume they appear to occupy when packed to their maximum extent. Thus, $ 4 2 can be considered as a reduced concentration which approaches 1.0 when & = &, rather than a t $2 = 1. Table I lists the value of A for various kinds of composites. Some of these values have been calculated for the viscosity of suspensions. For instance, the Einstein coefficient for a suspension of rigid spheres is 2.5, so A = 1.5 (Einstein, 1905, 1906). Table I1 lists typical maximum packing fractions for spheres and rods packed in various ways. Equation 4 is not exact, but for composite systems it has been found to be a very good approximation in most cases. Equations 1-4 predict values of conductivity which are somewhat too high when 4 2 = @m if the discontinuous dispersed phase is the more conducting of the two phases. However, a t lower concentrations, theory and experiment generally agree very well. Although this paper will be limited to cases where one phase is dispersed in another, Nielsen (1969) has shown how these same equations can be applied to foams or modified to apply to systems in which both phases are continuous. The use of eq 1-4 to calculate the thermal conductivity will be illustrated by several experimental examples from the literature. I t will be shown that there is good agreement between the theory and the experiments if the shape of the dispersed particles and the nature of their packing is taken into account. The same equations probably are applicable to electrical conductivity as well. An example of the agreement between theory and experiment is shown in Figure 1 for the thermal conductivity of aluminum spheres and rods embedded in rubber (Hamilton and Crosser, 1962). Random close packing ($m = 0.64) was assumed for the spheres, and for the randomly oriented rods it was assumed that $m = 0.52. From Table I, the values of A are 1.5 for spheres and 4.93 for rods with an aspect ratio of 10 to 1 (Burgers, 1938). Since aluminum has much greater conductivity than rubber, B = 1. The experimental results are fitted much more closely by the modified equations using a reduced concentra-
-
45
6
i
I1
.-
Y Y
3
A' X : EXPERIMENTAL
1. t
VOLUME FRACTION
Figure
Thermal conductivity of composites consisting of aluminum spheres and aluminum cylinders (aspect ratio 1 O : l ) in rubber. Solid lines are theoretical predictions using eq 1-4. Dotted lines are predictions of the original Halpin-Tsai equations: 0, experimental values for spheres; X , experimental value for cylinders 1.
Table 11. Maximum Packing Fractions
.+m
Shape of particle
Type of packing
4m
Spheres Spheres Spheres Spheres Spheres Spheres Rods or fibers Rods or fibers Rods or fibers Rods or fibers
Hexagonal close Face centered cubic Body centered cubic Simple cubic Random close Random loose Uniaxial hexagonal close Uniaxial simple cubic Uniaxial random Three dimensional random
0.7405 0.7405 0.60 0.524 0.637 0.601 0.907 0.785 0.82 0.52
Figure 2. Thermal conductivity of uniaxially oriented graphite fibers in epoxy resin measured perpendicular to the direction of fiber alignment. Curve A, theoretical prediction using A = 1, &, = 0.82; curve B, theoretical curve using A = 0.5, &, = 0.82; curve C, original Halpin-Tsai equations using A = 0.5; X , experimental values
1 0
50 40
tion rC.42 than by the original Kerner or Halpin-Tsai equations, which are shown by the dotted lines. A second example is the thermal conductivity of a graphite-epoxy composite in the direction perpendicular to the graphite fiber axis (Ashton, et al., 1969). Table I gives A = 0.5 for dispersed aligned fibers. However, the fibers used in composites are generally yarns made u p of many, fibers in a bundle, so the yarns are analogous to the case of aggregated spheres in which the generalized Einstein coefficient is (Lewis and Nielsen, 1968)
-
hE=-- A + l - 1.5 = 1.84 @a 0.82
so the proper value of A for aggregated fibers is approximately 0.84, or roughly 1. The value of da (the packing fraction in the agglomerate) is 0.82 for random packing of aligned fibers according to Table II. From the original literature source, it is not clear whether the matrix wetted and penetrated the graphite fibers to give a behavior analogous to monofilaments or whether the yarn should be considered as an aggregate. However, if the fibers are perfectly aligned, the experimental data should be bounded by values of A between 0.5 and 1. Figure 2 shows that this assumption is reasonably accurate, as most of the experimental points are within or near these limits of A = 0.5 (curve B) and A = 1.0 (curve A). The agreement is much poorer if 4mis not considered.
V O L U M E FRACTION
$2
VOLUME FRACTION
$2
Figure 3. A. Thermal conductivity (cal/sec cm "C) of polyethyltheory using A = 1.5 and &, ene containing glass spheres; -, = 0.64; - - - , unmodified Halpin-Tsai and Kerner equations; X, experimental points. B. Thermal conductivity of aluminum oxide powder in polystyrene: , eq 1-4 with A = 3.0 and @, = 0.64; - - - , unmodified Kerner equation; X , experimental points Figure 3 shows more extensive data on two other systems (Sundstrom and Lee, 1972). Equations 1-4 fit the data very well. For glass spheres in polyethylene, there is little doubt that the theoretical values of 2.5 for the Einstein coefficient and of 0.64 for random close packing of spheres are valid. For the case of aluminum oxide powder in polystyrene, the powder was not characterized in the original article, but it can be assumed the particles were of irregular shapes approximating roughened spheres with reasonable values of A = 3 and 4m = 0.64. The broken lines in Figure 3 are the original Kerner equation; the fit of this equation is poor even for the case of spheres since the packing is not considered. Sundstrom and Lee (1972) fitted their data to several other theories. An inspection of their graphs show that eq 1-4 generally fit the data better than any of the other theoretical equations they tried. Equations 1-4 include an infinite number of mixture m and c $ ~ = 1, are in the original Hallaws. When A
-
Ind. Eng. Chern., Fundam., Vol. 13, No. l , 1974
19
pin-Tsai equations, the equations become the ordinary "rule of mixtures," that is When A mixtures
-
K
=
K1$1
+
K2$2
(6)
0, the equations become the inverse rule of (7)
Intermediate values of A give mixture rules which are between these upper and.lower bounds. Acknowledgment The work described in this paper is a part of the research conducted by the Monsanto/Washington University Association sponsored by the Advanced Research Projects Agency, Department of Defense, under Office of Naval Research Contract N00014-67-C-0218, formerly N00014-66-C-0045.
Behrens, E., J. Compos. Mater. 2,2 (1968). Eruggeman, D. A. G., Ann. Phys. 24,636 (1935). Burgers, J. M., "Second Report on Viscosity and Plasticity," p 113, Nordemann, New York, N. Y., 1938. Cheng, S. C.. Vachon, R. I., lnt. J. Heat Mass Transfer 12,249 (1969). Einstein, A,, Ann. Phys. 17, 549 (1905). Einstein, A,, Ann. Phys. 19, 289 (1906). Halpin, J. C., J. Compos. Mater. 3, 732 (1969). Hamilton, R. L., Crosser, 0. K., lnd. Eng. Chem., Fundam. 1, 187 (1962). Hashin, Z.,Appl. Mech. Rev. 17, 1 (1964). Hashin, Z.,Rosen. E. W., J. Appl. Mech. E31,223 (1964). Hashin, Z. Shtrikman, S..J. Mech. Phys. Solids 11, 127 (1963). Hill, R., J. Mech. Phys. Solids 13, 189 (1965). Kerner, E. H., Proc. Phys. SOC.B69, 802, 808 (1956). Lewis, T. B., Nielsen, L. E., Trans. SOC.Rheol. 12,421 (1968). Lewis, T. B., Nielsen, L. E., J. Appl. Polym. Scl. 14, 1449 (1970). Nielsen, L. E., Appl. Polym. Sympos. No. 12, 249 (1969). Nielsen, L. E., J. Appl. Phys. 41,4626 (1970). Rosen, B. W., Proc. Roy. SOC.,Ser. A 319, 79 (1970). Springer, G. S., Tsai, S. W., J. Compos. Mater. 1, 166 (1967). Sundstrom, D., Chen, S., J. Compos. Mater. 4, 113 (1970). Sundstrom, D., Lee, Y-D.. J. Appl..Po/ym. Sci. 16, 3159 (1972). Tsai, S.W., U. S. Dept. of Commerce Rept. AD834-851 (1968). Tsao, G. T.. lnd. Eng. Chem. 53,395 (1961). Whitney, J. M., Riley, M. E., A l A A J. 4, 1537 (1966). Zinsmeister, G. E., Purohit, K. S., J. Compos. Mater. 4, 278 (1970).
Literature Cited Receiuedfor review September 29, 1972 Accepted July 30,1973
Ashton, J. E., Halpin, J. C., Petit, P. H., "Primer on Composite Analysis," Chapter 5, p 72, Technomic, Stamford, Conn., 1969.
Kinetics of a Moving Boundary Ion-Exchange Process Paul R. Dana' and Thomas D. Wheelock* Department of Chemical Engineering, lo wa State University, Ames, Iowa 50010
The acid elution of the dark blue cupric ammine complex from a cation-exchange resin produced a sharply defined moving boundary within each bead, which was photographed and measured to provide rate data. 'Analysis of the data by means of a theoretical model indicated that the overall elution process was controlled by a combination of internal and external mass transfer when acid concentrations less than 1.0 N were employed. At these acid concentvations the interdiffusion coefficient for the resin phase was found to be 3 X cm2/sec, independent of acid concentration but somewhat dependent on particle size. Data collected at. higher acid concentrations were not amenable to analysis by the method employed. The model should be useful for predicting the time required to elute the cupric ammine complex or for designing equipment to carry out this process.
Introduction The theoretical possibility of moving boundary processes in ion-exchange beads was suggested by Helfferich (1965). He anticipated that such processes would result when conversion of the resin involves consumption (or fixation) of the entering ion by a chemical reaction which is fast compared to diffusion. Generally such processes cannot be observed directly. However, while studying the ion-exchange behavior of the deep blue copper ammine complex, it was discovered that the acid elution of this complex from a strong-acid resin bead proceeds by a shrinking core process which can be followed readily because of a distinct color change. The color change is due to the destruction of the copper ammine complex by the hydrogen ions which penetrate the resin, thereby producing the following reaction within the resin phase
' Present address, Dupont Co., Wilmington, Del. 20
Ind. Eng. Chem., Fundam., Vol. 13!No. 1 , 1974
(The overbars used in this and subsequent expressions designate the resin phase.) The process described here differs from ordinary ion exchange where there is no accompanying chemical reaction except ion exchange and where all of the counterions in any given resin bead are thought to diffuse simultaneously so that no moving boundary develops. It has been found that the rates of ordinary ion-exchange processes are controlled either by film diffusion, intraparticle diffusion, or a combination of both (Boyd, et al., 1947). Film diffusion tends to be the rate-controlling mechanism when high capacity resins of small diameter and low degree of crosslinking are contacted with slowly moving, dilute solutions, and conversely intraparticle diffusion tends to control when highly cross-linked, low-capacity resins are contacted with rapidly moving, concentrated solutions (Helfferich, 1962). Furthermore, the initial stage of an ion-ex-
