Ind. Eng. Chem. Res. 1989, 28, 341-355 Ely, J. F. Prediction of the Viscosity and Thermal Conductivity in Hydrocarbon Mixtures-Computer Program TRAPP. Presented at the 60th Annual Gas Processors Association Convention, San Antonio, TX, March 21-23, 1981. Ely, J. F.; Hanley, H. J. M. Prediction of Transport Properties. 1. Viscosity of Fluids and Mixtures. Znd. Eng. Chem. Fundam. 1981, 20, 323-332. Garbow, B. S., et al. MINPACK Project. Argonne National Laboratory, March 1980. Goodwin, R. D. The Thermophysical Properties of Methane, from 90 to 500 K at Pressures to 700 Bar. National Bureau of Standards Technical Note 653, 1974. Hall, K. R.; Yarborough, L. New, Simple Correlation for Predicting Critical Volume. Chem. Eng. 1971, Nov 1, 76-77. Jhaveri, B. D.; Youngren, G. K. Three-Parameter Modification of the Peng-Robinson Equation of State to Improve Volumetric Predictions. SOC.Pet. Eng. Res. Eng. 1988, 3, 1033-1040. Leach, J. W.; Chappelear, P. S.; Leland, T. W. AZChE J . 1968, 14, 568-576. Leland, T. W.; Rowlinson, J. S.; Sather, G. A. Trans. Faraday SOC. 1968, 64, 1447-1460.
347
Martin, J. J. Cubic Equations of State-Which?, Znd. Eng. Chem. Fundam. 1979, 18, 81-97. Ortega, J. M.; Rheinboldt, W. C. Zterative Solution of Nonlinear Equations in Several Variables: Academic Press: New York, 1970. Peng, D.-Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59-64. Riazzi, M. R.; Daubert, T. E. Simplify Property Predictions. Hydrocarbon Process. 1980, March, 115-116. Robinson, D. B.; Peng, D.-Y. The Characterization of the Heptanes and Heavier Fractions for the GPA Peng-Robinson Programs. Gas Processors Association Research Report RR-28, March 1978. Rowlinson, J. S.; Watson, I. D. Chem. Eng. Sci. 1969,24,1565-1573. Turek, E. A.; Metcalfe, R. S. Phase Equilibria in Carbon DioxideMulticomponent Hydrocarbon Systems: Experimental Data and an Improved Prediction Technique. Presented at the 55th Annual Fall Technical Conference of the SPE, Sept 21-24, 1980; SPE 9231.
Received for review July 13, 1988 Accepted October 17, 1988
A Radially Perfused Cell for Measuring Diffusion in Compacted, Highly Sorbing Media Frank M. Jahnket and Clayton J. Radke* Chemical Engineering Department, University of California, Berkeley, California 94720
A unique radially perfused cell designed specifically t o measure the transient diffusion of species in compacted, highly adsorbing and swelling porous media is described in this paper. The perfused cell exhibits extremely high mass-transfer coefficients while resisting large swelling pressures. It also allows a simultaneous determination of the effective diffusion and sorption retardation coefficients of species from a single experiment. The concentration boundary layer resulting from the radial flow configuration is sufficiently uniform that one-dimensional, axial diffusion is studied in the sample medium. Over the range of Peclet numbers studied, from lo3 to 3 X lo5,overall Nusselt numbers range from 70 to 550, in excellent agreement with a theoretical convective dispersion prediction containing no adjustable parameters. Use of the cell is illustrated by an experiment in which cesium chloride diffuses into a moderately compacted sodium montmorillonite clay gel. We find that surface diffusion controls the migration of cesium in these clays. 1. Introduction
The transient diffusion of solute species through compacted, highly adsorbing porous media is of interest in such diverse areas of technology as enhanced oil recovery, ionexchange chromatography, plant nutrition, and toxic waste disposal. We are particularly interested in the suitability of the clay mineral montmorillonite as a component of the scheme proposed to dispose of high-level nuclear wastes in underground repositories (Waste Isolation Systems Panel, 1983). Many diffusion cells have been proposed for quantifying species diffusion rates (for reviews, see Crank (1975) or Crank and Park (1968); the best allow for a simultaneous determination of the diffusion and sorption coefficients of species in the sample medium from a single experiment. None, however, have considered the unique requirements imposed by particulate and compacted, sorbing media. Specifically, a diffusion cell designed for these materials must satisfy three general but conflicting requirements. First, the cell must be rigid enough to resist the swelling pressures the medium exerts. Common is the swelling of
* Author to whom correspondence should be addressed. +Presentaddress: Manufacturing Research and Engineering Organization, Research Laboratories, Eastman Kodak Company, Rochester, NY 14650.
0888-588518912628-0341$0 1,5010
materials exhibiting high species exchange capacities, such as ion-exchange resins in wetting solvents (Helfferich, 1962). Compacted clays demand particular attention to this requirement: swelling pressures of up to 10 MPa have been measured for highly compacted montmorillonite clays (Barclay and Ottewill, 1970). Next, the cell must retain the individual particles comprising the medium. Again clays prove difficult; the clay particles we study have diameters of but 50 nm. Finally, external mass-transfer resistances must not obfuscate a proper interpretation of the diffusion experiment. As demonstrated below, experiments designed to measure diffusion rates through highly sorbing media are very sensitive to the external mass-transfer resistances a diffusion cell imposes. The new radially perfused cell we describe in this paper is designed to meet these seemingly irreconcilable requirements. The perfused cell introduces solute species a t one end of a sorbing, linear medium from a well-stirred tank of finite volume. This method has multiple advantages. The progress of the experiment is followed easily by monitoring the concentration history of the tank. If in addition the concentration profile in the medium is determined at the conclusion of an experiment, we can readily ascertain the internal consistency of an experiment by overall solute mass conservation. Additionally, this concentration-history-with-profiling technique allows both sorption and diffusion coefficients 0 1989 American Chemical Society
348 Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989
of the solute in the medium to be calculated from a single experiment. As solute diffuses through the medium, two phenomena generally modify transport rates relative to those expected in the bulk solution. Solutes adsorb onto the solid, which in the Henry's law region is accounted for by the sorption retardation coefficient, a , = 1 + (1 em)K,/em, where ,6 is the porosity of the medium and K , is Henry's adsorption constant. The structure of the medium also influences transport rates as a result of the presence of the particles. The tortuosity, T,~, reduces the molecular diffusion coefficient, D,, found in the bulk solution, yielding an apparent diffusion coefficient, D, = Dm/rm2. Both concentration histories and profiles are parameterized by the quantities D,/a, and amDm,in the absence of external mass-transfer resistances. (The parameter D,/a, is known commonly as the effective diffusion coefficient.) Hence, D, and a, can be determined uniquely if both the concentration history and profile are measured. Finally, use of a finite-volume tank permits an analysis quantifying the effects of external mass-transfer resistances. It can be shown (Jahnke, 1987) that the masstransfer properties of a semiinfinite sorbing medium dominate external mass-transfer resistances of the cell as long as
when the duration of an experiment is identical with the time constant of the tank,
Here, k,, and VIA denote the overall mass-transfer coefficient of the cell and the tank volume to medium area ratio, respectively. External mass-transfer resistances may be neglected entirely when k,, is an order of magnitude greater than that required by eq 1,that is, when /3 < 0.1. Typically, mass-transfer coefficients must be large for diffusion experiments to be unencumbered by resistances of the cell. For example, consider the diffusion of a sorbate with D, = cm2/s from a finite-volume tank with a volume to area ratio of 50 cm through a moderately sorbing, semiinfinite medium with em2a, = 200. From eq 1 and 2, we find that k,, must be greater than 8 X cm/s for external mass-transfer effects to be ignored over a 1-week experiment. This value is nearly the same as the steady-state mass-transfer coefficient of typical low-resistance membranes, such as those manufactured by the Nuclepore Corporation (Pleasanton, CA) (1984). A properly designed diffusion cell requires a porous membrane to prevent the fine particles of the medium from sloughing into the tank. It must also provide support for the fragile membrane to resist swelling pressures which would otherwise cause a catastrophic rupture during the experiment. Any mass-transfer resistances the cell adds in excess of the membrane must be near zero or the volume to area ratio, VIA, must be altered so that lower values of k,, still satisfy eq 1. This course rarely proves fruitful, however, as times of experiments increase with the square of VIA, whereas decreases only linearly with VIA. Hence, we reduce the mass-transfer resistances of the cell to values as low as are possible and thereby minimize the duration of an experiment. To meet these requirements, we propose a radially perfused diffusion cell, as shown schematically in Figure 1. The sorbing medium under study is located in a cylindrical chamber between a rigid but movable piston and
I I
-
A
1zy I
Gasket) Teflon
r i
I
Sintered 'Metol Disk
1
Well. stirred Tonk of finitevolume
bution tch
Figure 1. Schematic of the radially perfused diffusion cell illustrating its primary components. Not shown is an exit line within the cell used to eliminate trapped air.
a Nuclepore membrane with pores sufficiently small to retain the particulate medium. A porous sintered metal disk and the head of the cell provide structural strength for the fragile membrane. The key feature of this design is that sorbing species are introduced by a flow configuration wherein the sorbate circulates from an external tank, through the cell, and back, hence the appellation of a perfused cell. Once inside the cell, the fluid distributes circumferentially in a notch near the outside edge of the metal disk, assuring that a uniform angular concentration bathes the sintered disk. As fluid enters and traverses the sintered disk radially toward the center of the cell, sorbate transfers through the membrane and into the medium. Fluid channeling around the sintered disk is prevented by Viton O-rings (two on the head of the cell and one on either side of the distribution notch) and by a Teflon gasket (between the head of the cell and the sintered disk). At the conclusion of an experiment, the head of the cell, the sintered disk, and the membrane are removed. The piston, whose excursion is monitored, is incrementally advanced to extrude the medium, which is then sectioned and analyzed to determine the final concentration profile. Quenching to immobilize species within the medium during profiling is usually not necessary because the time required to section is small compared to the time of the experiment. A more detailed description of the cell may be found in Jahnke (1987). External mass-transfer resistances of the perfused cell, which we wish to minimize so that eq 1is satisfied and the time of an experiment is minimized, are confined to the membrane and the concentration boundary layer which develops inside the porous disk as a result of species transfer into the medium. Because the mass-transfer resistance of the membrane is fixed, we first develop in section 2 a quantitative theory describing the formation of the concentration boundary layer within the porous disk. Our objective is to investigate the uniformity of the boundary layer or, equivalently, to determine whether the concentration of diffusant along the inlet face of the sorbing medium is sufficiently uniform that a one-dimensional diffusion description is valid. Section 3 describes the experimental determination of the dependence of the overall mass-transfer coefficient of the perfused cell with flow rate. In the proposed theory arise a number of physical parameters; these are measured from independent experiments whose description follows in section 4. Then, in section 5, the mass-transfer coefficient data are compared with the theoretical prediction which now contains no adjustable parameters. We conclude in section 6 with an illustration of cell use: an experiment where aqueous cesium chloride diffuses through a moderately compacted
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 349 i:= 1-r/R
I
Sintered Disk
z' = z/R
gives
t
where the Peclet number is defined as Pe I(u)lR/Dand E Om/?. As an adequate description of the variation of fi (i = 1, t) over the entire fluid velocity range found in the perfused cell, we follow Perkins and Johnston (1963) and write
2.0
D
6
fi(r) = 1
r=O
Figure 2. Enlarged view of the mass-transfer region of the radially perfused cell.
sodium montmorillonite clay gel.
2. The Concentration Boundary Layer of the Perfused Cell Figure 2 depicts the mass-transfer region of the perfused cell. Fluid of interstitial velocity u carries a dilute solute initially of uniform concentration co radially toward r = 0 through a nonsorbing sintered disk, semiinfinite axially and infinite radially, of porosity e and tortuosity r2. Once the fluid reaches r = R, solute diffuses axially first through a porous membrane, which exhibits a steady-state masstransfer coefficient k,, and whose face is located at z = 0. Diffusion then commences into the sorbing medium. The concentration boundary layer resulting from solute transfer grows from r = R to the center of the cell a t r = 0, where the fluid exits. Also shown on the figure is the Brinkman (Brinkman, 1947) or hydrodynamic boundary layer, which as drawn is much thinner than the concentration boundary layer. It can be shown that mass-transfer resistances of typical membranes dwarf those of the Brinkman layer (Jahnke, 1987); thus, this layer is neglected. Also neglected are exit effects on the flow profile. We assume uniform radial plug flow. In normal cell use, the inlet conceniration, co, diminishes with time due to the depletion of solute from the external tank. Because the residence time of fluid in the sintered disk is much smaller than t* in eq 2, the pseudo-steadystate approximation applies. Species concentrations in the pore fluid of the sorbing medium, cm,also vary with time because of mass transfer into the medium. To isolate the mass-transfer properties of the cell, we consider a shorttime development where c, = 0. With the pseudo-steady-state assumption, the convective dispersion equation for species migration through the sintered metal disk becomes
where f i and ft correct the molecular diffusion coefficient for longitudinal and transverse dispersion, as discussed below. Because the interstitial velocity is a function of radius alone, fluid continuity demands that
+ yiPe,
(6)
where Peg is the local Peclet number based on the grain diameter d,, Peg = tlu(r)ld,/D. yLis the proportionality constant, universal for geometrically similar porous media. Along with continuity in eq 4,eq 6 becomes QiPe fi@) = 1 (7) 1-i: where the dispersivity, Qi, is defined as Ri = yicd,/2R. For our cell, dispersivities are typically on the order of lob4. For large values of Pe, we note that the coefficient
+
(1 - i:)f,(i:) = -1 - i: + 9,
