Ind. En$, Chem. Process Des. Dev. 1989, 22, 687-688
Bitter are as follows: (1)The proposed equation could have been derived using “caloric concentrations” defined by Peters (1922). This was also suggested to me by Lemlich (1982) and it does result in the proposed equations. (2) The feed line is curved when the enthalpy-composition lines are straight but not parallel. However, in most cases the feed line is a short line and considering it as a straight line is an acceptable approximation. This was not emphasized in the paper (Govind, 1982) but was considered
887
in the analysis. Literature Cited Govind, R. Ind. Eng. Process Des. D e v . 1982. 21, 532-535. Lemllch, R., private communicatlon, 1982. Peters, W. A. Ind. Eng. Chem. 1922, 14, 476.
Department of Chemical Engineering University of Cincinnati Cincinnati, Ohio 45221
Rakesh Govind
Comments on “Approximate Design Equations for Reverse Osmosis Desalination by Spiral Wound Modules” Sir: In a recent article, Sirkar et al. (1982) propose a numerical approximation for the concentration polarization relationship for reverse osmosis membranes. Their method is to approximate the relationship between the solute concentration at the wall (X,) and that in the freestream (Xzl)in terms of the ratio of solute velocity to solute mass transfer coefficient, NlrVl/kl. Thus x22
= x21 e x P “ ~ l ; / ~ l l
is approximated as
in order that explicit flux expressions may be developed. The purpose of this comment is to propose an improved approximation derived from a Tchebycheff economization [cf. Dalquist and Bjorck (1974)l. For example, an approximation
over the interval [0,1.5] results in a maximum error of 2.91 90.It is not necessary to use a single approximation for the interval. Much higher levels of accuracy can be achieved using this technique if a smaller interval is chosen. The simplicity of a second-order expression is preserved even though the economization is derived from a fourthorder polynomial approximation. Thus the useful range of the approximate design equations may be considerably extended without resorting to a higher order approximation as suggested by the authors.
Literature Cited Slrkar, K. K.; Dang, P. T.; Rao, 0. H. Ind. Eng. Chem. Process D e s . D e v . 1982, 21, 517. Dalquist, G.; Bwck, A. “Numerlcal Methods”; Prentice-Hall, Engelwood Cllffs, NJ, 1974.
College of Engineering University of South Florida Tampa, Florida 33620
J. D. Jeffers 5.C. Kranc* R. P. Carnahan
Response to Comments on “Approximate Design Equations for Reverse Osmosis Desalination by Spiral Wound Modules” Sir: The improved approximation proposed by Jeffers, Kranc, and Carnahan for expressing concentration polarization in a highly rejected single solute reverse osmosis system with chopped laminar flow indeed leads to a somewhat better explicit flux expression especially in the range [1,1.5]. However, we would like to point out on the basis of further unpublished calculations that a significantly better prediction of L+ value is not necessarily achieved. This is due to the fact that a number of other assumptions and approximations were used in the deriv-
ation of the analytical result for L+. Department of Chemistry & Chemical Engineering Stevens Institute of Technology Hoboken, New Jersey 07030 Chemical Engineering Department Siddaganga Institute of Technology Tumkur-572103, Karnataka, India
K. K. Sirkar* P. T. Dang G. H. Rao
Comments on “Hydrogenation of Aromatic Hydrocarbons Catalyzed by Suifided Co0-Mo0,/y-Ai20,. Reactivities and Reaction Networks” Sir: In a recent paper, Sapre and Gates (1981) have apparently misunderstood the interpretation suggested by Patzer et al. (1979) regarding the constancy of the tetralin 0196-4305/83/1122-0687$01.50/0
(T)-to-naphthalene (N) concentration ratio observed by the latter authors in the hydroconversion of l-methylnaphthalene (M). According to Sapre and Gates, ”Patzer 0 1983 American Chemical Society
