Literature Cited
and
Aris, R., IND.ENG.CHEM.FUNDAMENTALS 8, 603 (1969). Piaford, R. L., Baker B. 111, Blum, D. E., IND.ENG.CHEM. FUNDAMENTALS 8,144 (1969). Rhee, H., Ph.D. thesis, University of Minnesota, 1968. Wilhelm, R. H., Rice, A. W., Rolke, R. W., Sweed, N. H., IND. ENG.CHEM.FUNDAMENTALS 7, 337 (1968).
respectively, where the bracket [ ] denotes the jump of the quantity enclosed across the sharp boundary. This acceleration evidently inhibits the effect of the parametric pumping and promotes the approach toward a limiting separation. The separation factor here depends upon both the initial concentration and the temperature pair chosen, but is independent of the cycle time. The variety of situations that might arise by variation of these factors requires individual treatment and bears further investigation.
Hyun-Ku Rhee Neal R. Amundson University of Minnesota Minneapolis, Minn. 55455
is not applicable here, even though it may be true in the gas phase if y has partial pressure units. Since the data for the present case indicate that
SIR: Since the recently published equilibrium data for the toluene-n-heptane-silica gel system (Sweed and Wilhelm, 1969) used in the parametric pumping experiment which we analyzed are best described by nonlinear Langmuir-type isotherms, the comments of the above authors are well taken. The curvature of the isotherms a t high concentration mill affect the plot of separation factor vs. number of cycles (Figure 6). Both the shift in concentration a t temperature shift points and the concentration-wave velocities during the flow periods will reflect this curvature. A cycleby-cycle graphical method capable of handling these effects was suggested by Sweed and Wilhelm. An analytical solution for n cycles analogous to our solution for the linear case is not yet available. The ultimate separation factor (a,, n + m ) is not necessarily limited, because the isotherm is nonlinear as the above authors suggest. Since the fluid concentration units we used were moles of solute per mole of fluid (Sweed and Wilhelm used milliliters of solute per milliliter of fluid), the expression
there would not seem to be any restriction preventing all of the solute being ultimately pumped to one end under the equilibrium theory; in other words, a, --L a ,n + m . I n the general case (Kipling, 1965), it may be true that f T .-,0 , 0 5 y _< 1; in such cases, one would expect to find a limiting separation factor, since the concentrations for which f T + 0 are, in effect, azeotropes for parametric pumping. literature Cited
Kipling, J. J., “Adsorption from Solutions of Non-Electrolytes,” Chap. 4, Academic Press, New York, 1965. Sweed, N. H., Wilhelm, R. H., IND.ENG.CHEM.FUNDAMENTALS 8, 221 (1969).
Burke Baker III Shell Oil Co. Deer Park, Tex. 77538
Correction CONCENTRATION POLARIZATION IN REVERSE OSMOSIS UNDER NATURAL CONVECTION I n this communication by A. R. Johnson and Andreas Acrivos [IsD. EXG.CHEM.FUA-DAM. 8, 359 (1969)l Table I should have been given as follows: Table I. Coefficients n
0
1
2
0
1.58316 1.05175 0.30697 -0.02291 -0.05138 -0.01670 - 0.00077
- 1.91070 -3.72886 -3.03079 - 1.00795 0.26412 0.43237
1,91257 7.77960 12.26927 9.60402 2.59026
1
2 3 4 5 6
304 Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
(pm.n
(0) in Equation 18
m 3
- 1.62652
- 12.11562 -32.14680 -43.44920
4
5
1.17600 15.32022 63.62862
- 16.34317
-0.70774
6
0.33251
