Comparative data from the present and other specific volume methods are listed in Table I, for commercial samples of linear polyethylene and two polystyrenes. All samples contained 0.3 weight % antioxidant, to avoid degradation. The antioxidant was incorporated by hot milling or prolonged tumbling with polymer granules. The figures quoted under “literature values” refer to data published as characteristic of comparable polymers. The specific volumes listed under “extrusion rheometer” refer to values measured b y an extrusion method described recently (Rudin et al., 1968). This procedure has been shown to give results which agree well with those from dilatometry, for a number of polyolefin plastics. Replicate samples of the same materials were used for the present and extrusion rheometer experiments. The present technique is reasonably accurate. I t s relative speed and convenience recommend it for applications, such as copolymer studies, which may involve a large number of samples. Coincidence of polystyrene data from the present procedure and dilatometry at both 170” and 186°C indicates the absence of trapped gas. If air had been present in the rheometer barrel, its expansion would have caused a temperaturedependent deviation between the two sets of results. The present method is applicable to lower molecular weight species than the extrusion rheometer technique, in which leakage between the piston and reservoir wall becomes troublesome when the viscosity of the polymer melt is low. For polystyrene, which wets steel well, this occurs a t about 220°C with a polymer of molecular weight 20,000 (viscosity about 1000 poises). The new method is applicable to such relatively fluid polymers, but it is not suitable for study of nonpolymeric species. Excessive leakage past the two plugs was found, for example, with hexatriacontane at 150°C (viscosity about 2 centipoises).
The procedure described is evidently suitable only for polymer melts. It could conceivably be extended to solids, at the cost of additional complexity, b y dispersing these in a viscous liquid which did not interact significantly with t h e solid polymer. Polyisobutenes, which are readily available in a range of viscosities, might be suitable carriers for polar polymers and polyoxyethylenes might perform the same function for polyolefins. We suggest that the simple technique described for elimination of trapped air could be applied to compressibility measurements. For axial stability under heavy loads the driving piston would be a steel core with polytetrafluoroethylene cladding. Acknowledgment
The authors thank the National Research Council of Canada for financial support. literature Cited Bagley, E. B., J . A p p l . Phys. 28,624 (1957). Bekkedahl, N., J . Res. Natl. Bur. Std. 43, 145 (1945). Bianchi, U., Magnasco, V., J. Polymer Sci. 41,177 (1959). Chiang, R., J . Polymer Sci. 36,91 (1959). Foster, G. N., Waldman, N., Griskey, R. G., Polymer Eng. Sci. 6, 131 (1966). Fox, T. G., Flory, P. J., J . A p p l . Phys. 21,581 (19.50). Gubler, 11.G., Kovacs, A. J., J . Polymer Sci. 34,551 (1959). Hellwegge, K.-H., Knappe, W., Lehman, P., Kolloid-Z. 183, 110 ,*nI.n\ \lYOL).
Heydemann, P., Guicking, H. D., Kolloid-2. 193, 16 (1963). Parks, W., Richards, R. B., Trans. Fnraday SOC.45,20 (1949). Rehage, G., Breuer, H., J . Polymer Sci. C16,2299 (1967). Richardson, M. J., Flory, P. J., Jackson, J. B., Polymer 4, 221 ( 1963 ).
Rudin, A., Chee, K. K., Shaw, J. H., IUPAC Macromolecular Symposium, Toronto, 1968, J . Polymer Sci. C , in press. Terry, B. W., Yang, K., SPE JournalZO, 540 (1964). RECEIVED for review March 5, 1969 ACCEPTED SEPTEMBER 26, 1969
COMMUNICATIONS
Analytic Driving-Force Relation for Pore-Diffusion Kinetics in Fixed-Bed Adsorption An equation describes the course of pore-diffusion breakthrough in fixed-bed adsorption in terms of bulk concentrations a t each point in the bed.
THE
EFFECT of pore-diffusion kinetics in fixed-bed adsorption or ion exchange on the breakthrough for the column effluent was studied by Hall and coworkers (1966), who performed the needed numerical integration of the secondorder partial differential equations describing the system. I n this study a search has been made for a n equation t h a t would describe the course of pore-diffusion breakthrough in terms of bulk concentrations at each point in the bed. Where no axial dispersion occurs and there is no interference from other resistances, such as equation should yield the same
breakthrough curve as Hall’s exact solution. An explicit differential equation can be integrated analytically in constant-pattern cases, where the equilibrium parameter r [ = x(1 - y*)/y*(l - x)] is appreciably less than unity. Here x is the dimensionless fluid-phase concentration, c/co, y is the dimensionless solid-phase concentration, q / Q , and * denotes the equilibrium value. Acrivos (Vermeulen, 1958) found that a square-root driving force best fits pore diffusion at r = 0. By analogy with other mechanisms, it is inferred that a linear driving VOL. 9 NO. 1 FEBRUARY 1970
I&EC FUNDAMENTALS
179
Table 1.
Effluent Concentration-History Points Determined by Numerical and Empirical Methods r = 0.2
r = 0.4
N(T
-
Exact
X
0.044 0.241 0.519 0.789 0.917
N(T
1) Approx.
-2.00 -1.00 0.00 1.oo 2.00
Exact
X
-2.06 -1.00 -0.01 1.03 2.26
0.017 0.182 0.534 0.851 0.987
force should apply when r = 1. Among several expressions investigated which conformed to these limits, the best fit was obtained by the function:
Here N,,,, is the number of mass-transfer units and T is the throughput parameter. Under constant-pattern conditions, where y = x, this equation can be integrated to yield
-4,35 -1.85 0.15 2.15 4.65
r = 0.6
- 1) Approx.
-4.21 -1.78 0.16 2.37 5.99
0.028 0.227 0.498 0.771 0.974
-
+ 6-
I+-
d 1
1-r
Npore(T- 1)
- (1 - r)x + d F
dr
adjusts the results from the assumed The factor driving force to match the midpoint slope with that of the three-dimensional solution. A new evaluation of Hall’s data including the ranger = 0 to r = 0.7, with a new fit for r = 0 as described below, yields the following correlation for $pore: $pore
=
1.o rZ.Oo 1.83 (1
+
(3)
- r)0,92
T h e “exact” values of +pore as determined by computer matching of midpoint slopes are:
r r
= = =
+pore
=
0.0 0.548 0.5 0.825
0.1 0.609 0.6 0.871
0.2 0.655 0.7 0.901
0.3 0.4 0.711 0 ,769 0.8 0.921
The constant of integration, I,, appearing in Equation 2 may be evaluated by determining the center of gravity of the x us. NporeTcurve, by integration. An alternate method used in this case involved comparison of the results calculated using Equation 2 with those given by Hall. The region used was that in the range near T = 1. It was found that the integration constant is a function of r. When Equation 3 is used for jlpare,the constant is given by:
I , = 2.44
- 2.15r
(4)
Table I shows the agreement obtained between the approximate results of Equations 2 to 4 and the exact values from numerical analysis. The average uncertainty in N(T - 1) at each of several x levels is: at 2 0.02, about 5%; a t x 0.20,
-
180
l&EC FUNDAMENTALS
-7.00 -2.50 0.00 2.50 7.00
-6.62 -2.44 -0.04 2.55 7.85
-
= 0)
For this case, Hall’s fit to the exact data can be put in the form:
l/r
d 1 - (1 - r)x -
N(T - 1 ) Approx.
about 3%; a t x 0.80, about 8%; at x 0.95, about 15%. Otherwise stated, at given N and NT, x is around 15% low for x < 0.05, and less than 3% low for x > 0.95. Thus the analytic relation represents the concentration history relatively well. For example, the fit is much better than that given by the linear driving force, y* - y, frequently used to describe the particle-phase (“solid”) diffusion rate. I n the present relation (Equation l ) , the denominator term relates the effective pore-phase concentration to the coexisting solidphase concentration. The Irreversible Case (r
1 (1 - r)x l-dl-(l-r)x
Exact
X
-
VOL. 9 NO. 1 FEBRUARY 1970
=
Const
Hall et al. determined the constant to be 3.59, whereas a restudy of the problem suggests that 3.66 will give a better over-all agreement. The constant in Equation 4 is two thirds, and the constant in Equation 3 is half, of 3.66. Nomenclature
I, = constant of integration N,,,, = number of mass-transfer units (Equation 17 of Hall et al., 1966) r = equilibrium parameter T = throughput parameter (Equation 16 of Hall et aZ., 1966) x = dimensionless fluid-phase concentration y = dimensionless solid-phase concentration $pore = correction factor for matching midpoint slope in analytical approximation * = equilibrium value literature Cited
Hall, K. R., Eagleton, L. C.. Acrivos, A., Vermeulen, T., IND. END.CHEM.FUNDAMENTALS 5, 212 (1966). Vermeulen, T., Advan. Chem. Eng. 2 , 177-8 (1958). THEODORE VERMEULEN ROBERT E. QUILICI’ University of California Berkeley, Calif. 94720 1 Present address, Aerojet-General Nucleonics, San Ramon, Calif. 94583. RECEIVED for review February 26, 1969 ACCEPTED July 17, 1969
Work supported in part by a grant from the Office of Saline Water, U. S. Department of the Interior.
