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In summary, some measure of control over the isomer distribution resulting from mixed-acid nitration of toluene is possible but only under severe constraints and far from the conditions normally employed for efficient mononitration. The effects that are observable appear to be best explained by mass-transfer influences. Although alternative methods of nitration do offer greater latitude in the determination of the isomeric content, none can be accomplished with the convenience, speed, and low cost of a mixed-acid process. Acknowledgment
The capable experimental work of D. Miller, W. Guyer, and F. Leh, analyses by the Air Products analytical staff, statistical evaluation by E. Ascher, and helpful discussions with J. E. Sawicki and B. A. Toseland are gratefully acknowledged as well as permission to publish these results from Air Products & Chemicals, Inc. Literature Cited Amos, D. W.; Baines, D. A.; Flewett, G. W. Tetrahedron Lett. 1973, 3191. Bakke, J. M.; Liaskar, J. Ger. Offen. 2 826 433, 1979 (to Aktiebolag Bofors; CA:90: 121181b). Berman, L. V.; Crouse, R. H. U S . Patent 3 149 169, 1964 (to Union Carbide Corp.; CA:61:14585). Chapman, J. W.; Cox, P. R.: Strachan, A. N. Chem. Eng. Sci. 1974, 29, 1247. Coombes, R. G.;Russell, L. W. J . Chem. Soc., Perkin Trans. 2 1974a. 830. Coombes, R. G.; Russell, L. W. J . Chem. Soc.. Perkin Trans. 7 1974b, 1751. Coon, C. L.; Blucher, W. G.; Hill, M. E. J . Org. Chem. 1973a, 3 8 , 4243. Coon, C. L.: Hill, M. E. U.S. Patent 3 714 272, 1973b (to Stanford Research Institute; CA:78:97303x). Coon, C. L.; McDonald, G. J.; Hill, M. E. U S . Patent 3708546, 1973c (to Stanford Research Institute; CA:78:58008k). Cox, P. R.; Strachan, A. N. Chem. Eng. Sci. 1972a, 27,457. Cox, P. R.; Strachan, A. N. Chem. Eng. J . 1972b, 4 , 253. Cox, P. R.; Strachan, A. N. Chem. Eng. Sci. 1974, 29, 1247. Davies, B.; Thomas, C. B. J . Chem. SOC.1975, 65. Effenberger, F.; Bantel, F. H.; Eilingsfeld, H. Br. Patent 1436 954, 1976 (to BASF A G CA:81:13324f).
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Hanson, C.; Ismail, H. A. M. J . Appi. Chem. Biotechnoi. 1975, 25, 319. Hanson, C.; Ismail, H. A. M. J . Appi. Chem. Biotechnol. 1976, 2 6 , 111. Hoggett, J. G.; Moodie, R. B.; Penton, J. R.; Schofield, K. Nitration and Aromatic Reactivity; Cambridge University Press: Cambridge, MA, 197 1: pp 13-25. Holleman, M. A. F. R e d . Trav. Chim. Pays-6as 1914, 3 3 , 1. Lawrence, F. R . U.S. Patent 4234470, 1980 (to E. I.DuPont de Nemours & Co.; CA:94:83754p). Manabe, 0.; Kaneo, T. U S . Patent 3965200, 1976 (to Osaka City; CA:78:97302w). Manabe, 0.; Kaneo, T.; Ikeda, T.; Yamamoto, T.; Nishimura, S.Jpn. Kokai 50059336-W42, 1975 (to Sumitomo Chem. Ind. KK; CA:83:96705w). Milligan, B.; Miller, D. G. US. Patent 3957889, 1976 (to Air Products & Chemicals Inc.; CA:85:77857x). Narang, S. C.; Thompson, M. J. Aust. J . Chem. 1978, 3 7 , 1839. Olah, G. A.; Lin, H. C.; Olah, J. A.; Narang, S. C. Proc. Nati. Acad. Sci. U . S . A . 1970a, 7 5 , 1045. Olah, G. A.; Malhotra, R.; Narang, S. C. J . Org. Chem. 1978b, 4 3 , 4628. Olah, G. A.; Narang, S. C.; Malhotra, R.; Olah, J. A. J . Am. Chem. SOC. 1979, 107. 1805. Owsley, D. C.; Bloomfield, J. J. U.S. Patent 4 107 220, 1978 (to Monsanto Co.; CA:90: 137459j). Perrin, C. L. J . Am. Chem. SOC. 1977, 99,5516. Ridd, J. H. Acc. Chem. Res. 1971, 4 , 248. Roberts, R. M.; Browder, H. D.; Kobe, K. A. J . Am. Chem. SOC.1959a, 8 7 , 1165. Roberts, R. M.; Heiberger, P.; Watkins, J. D.; Browder, H. P., Jr.; Kobe, K. A. J . Am. Chem. SOC. 1956, 8 0 , 4205. Roberts, R. M.; Watkins, J. D.; Kobe, K. A. J . Am. Chem. SOC.1959b, 8 1 , 1167. Schubert, H.: Wunder, F. Br. Patent 1560349, 1980 (to Hoechst AG; CA:88: 104874~). Sogn, A. W.; Natoli, J. G. U S . Patent 3 196 186, 1964 (to Allied Chemical Corp.; CA:63:11427h). Stock, L. M.; Wright, T. L. J . Org. Chem. 1979, 4 4 , 3467. Tsang, S. M. US. Patent 3 126417, 1964a (to American Cyanamide Co.; CA:61:612c). Tsang, S. M.;Paul, A. P.; Digiaimo, M. P. J . Org. Chem. 1964, 29, 3387. Vaughan, R. J. U S . Patent 3976704, 1976 (to Varen Technology, CA:86:72 177177). Wright, 0.L. U.S. Patent 2 948 759, 1960 (to Pittsburgh Coke and Chemical Co.; CA:55:3544D). Wright, 0. L.; Teipel, J.; Thoennes, D. J . Org. Chem. 1965, 3 0 , 1301. Yamamoto, R.; Yamamoto, K.; Takagi, A. Jpn. Patent Sho 55-33406, 1980 (to Mitsui Toatsu; CA:93:46167h).
Received for reuiew July 15, 1985 Accepted February 10, 1986
A Novel Approach to Transfer Limiting Phenomena during Ultrafiltration of Macromolecules Pierre Aimar' and Victor Sanchez Laboratoire de Chimie-Physique et Nectrochimie (Laboratoire associs au C.N.R.S. No. 192), Universits Paul Sabatier, 3 1062 Toulouse Cedex, France
Solvent flux, V, during macromolecular ultrafiltration reaches a limiting value at high applied pressures, A P . From a theoretical point of view, we demonstrate that the existence of this limiting flux can be due only to a decrease of the average mass-transfer coefficient when the concentration increases in the boundary layer. The membrane permeability to the pure solvent has no effect except on the value of the pressure difference necessary to reach the limiting flux. From a practical point of view, we propose an expression for the limiting flux as a function of module geometry, hydrodynamic conditions, and bulk concentration and a system of equations to compute the characteristic curves, V = f ( A P ) ,the parameters of which are to be deduced from experimental results. A good agreement of these models with experimental data published elsewhere or obtained in our laboratory confirms the validity of our assumptions. Thus, we can deduce qualitative information concerning control of ultrafiltration units.
I.I n t r o d u c t i o n Membrane processes occupy a special place in separation science since they do not involve any change of state. This reduces energy requirements and makes them suitable for
processing biological or food products. Expected performances are strongly limited, however, by an accumulation of retained solutes at the membrane-solution interface. During macromolecular ultraf-
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iltration this so-called concentration polarization phenomenon has four major effects: (i) changes in the physicochemical properties of the interfacial solution, (ii) changes in membrane properties due to interactions with solutes, (iii) rise in osmotic pressure that partially balances the applied pressure, and (iv) the gelation of the interfacial solution in the case of high polarization. Theories involving one or another of these phenomena give qualitative, and sometimes quantitative, explanations of many experimental results. In order to describe the performances of an ultrafiltration unit throughout the whole range of operating conditions, we are obliged to use, one after another, several of these theories or, numerically, to solve the mass-transfer equations in the boundary layer along the membrane (Clifton et al., 1984; Rautenbach and Holtz, 1980). The aim of this work is to evaluate the relative importance of each limiting phenomenon according to the membrane and solution properties and the hydrodynamic conditions. Thus, we show, from a theoretical point of view, that the variations in the physicochemical properties of the interfacial solution are sufficient in themselves to explain the existence of a limiting flux. The present model makes it possible to calculate the mass transfer for different operating conditions, provided a few experiments are done as will be shown in the experimental part of this paper. Furthermore, the solute concentration and the applied pressure corresponding to the limiting flux may be calculated. When the gelation assumption is added to the basic assumptions, the equations deduced from our model are in good agreement with those obtained from the gel theory.
TI. Theory A. Osmotic Pressure and Gel Theory. As long as no solute is retained, the flux, V, through the membrane is given by Darcy's law as a function of the applied pressure, SP V = AP/R (1) where R is the hydraulic resistance of the membrane. When a solute of bulk concentration, Co, is fully rejected by the membrane, the interfacial concentration, C,, increases with the flux, V , as predicted by the following equation deduced from the film theory V = k In (C,/Co) (2) where k is the mass-transfer coefficient. The concentration difference between the two sides of the membrane creates an osmotic pressure difference, AT, whose effect is described by the following equation, first derived by Kedem and Katchalsky (1958) and then applied by Goldsmith (1971) for macromolecular ultrafiltration. V = (AP- AT)/R (3) The osmotic pressure of concentrated macromolecular solutions varies with the concentration, C, according to the relationship AT = a,C + a2C2+ a3C3 (4) where a , is the coefficient in van't Hoff s law for infinitely dilute solutions; a2 and a3 represent the nonideality of the solution. Combining eq 2-4 leads to an expression for the pressure, LIP,that is necessary to reach a given flux, V. 3
AP = RV + Ca,CoLexp(iV/k) [=I
(5)
Using an analogy with electrical circuits, we define a quantity that we call "mass-transfer impedance", given by the slope of the curve. A I', vs. V at each point.
3
daP/dV = R
+ (CiaLC
