Ind. Eng. Chem. Res. 1987, 26, 634-635
634
however, not borne out in the graphical comparison of eq 2b and 3b and the data gathered and presented by Wen and Yu (1966). Equations 2b and 3b were substituted into eq 1 to result in Remf = (33.72 0.0408Ar)l” - 33.7 (5)
Analysis a n d Comparison By intuition, it was found that by choosing
Various modifications to eq 5 have been given in the literature by later workers in an attempt to better correlate their data; e.g. Saxena and Vogel(1977) revised eq 2b and 3b to C1 = 1/d,tmf3 = 10.0 f 0.4 (2c)
the data given by Wen and Yu (1965) may be quite well represented, much better than either eq 2b and 3b or 2c and 3c. Substitution of eq 2d and 3d into eq 1 results in
+
Cz = (1- ~ ~ f ) / d , =~ 5.9 ~ ~ ff 0.6 3
(3C)
Recently, Lucas et al. (1986) pointed out that eq 5 of Wen and Yu was accurate only for particles with 4 1.0 and that the inaccuracy of applying equations derived for a particular system to some other systems was associated with the variation of the factors C1 and C2 with particle shape. These writers therefore classified particles into three categories: round (0.8 < 4 < l.O), sharp (0.5 < d, < O B ) , and others (0.1 < d, < 0.5). The optimum values of the constants C, and C2 were determined for each category of particles. When we made use of the emf vs. 4 data gathered by Wen and Yu (1965), various lines for eq 2 and 3 were shown corresponding to different numerical values assigned to C, and Cz. Many errors are obvious; e.g., in the plot of emf vs. 4 for varying C, values, it is obvious that the lines must pass through emf = 1.0 and 4 = 0. Nevertheless, the essence of the work of Lucas et al. (1986) was that optimum values of C, and C2 were chosen for the three categories of particles, and the appropriate equations are then applied to the conditions under consideration. They gave, for the round, sharp, and other particles, the values of C1 and C2, respectively, as 16.0 and 11.0, 10.0 and 7.5. and 8.5 and 5.0. Inspection of their Figures 2 and 3 indicates that this involved the correct piecewise choice of C, and C2 for the various ranges of IC# values. The equations by Lucas et al. (1986) were round particles
Remf= (29.52 + 0.0357.4r)’/’ sharp particles
-
29.5
(6)
-
Remf = (32.1’
+ 0.0571.4r)’!’
32.1
(7)
Remf = (25.2,
+ 0.0672Ar)’12- 25.2
(8)
others
C, = l/d,t,f3
= 14d,0.45
C? = (1 - e m f ) / $ 2 t m f 3
=
11f#l0.”~
(2d) (3d)
Remf = [(33.67+0.1)2+ (Ar/24.5d,0.45)]1’2 - 33.674°,1 (9) Thus, a single equation is obtained instead of the three equations given by Lucus et al. (1986). For the case of spherical particles when 4 = 1.0, eq 9 simply reduces to eq 5, the equation given by Wen and Yu (1966), where Lucas et al. (1986) had already shown it to give similar predictions to eq 6 for round particles. For the case of sharp particles (0.5 < d, < 0.8),it is interesting to note that the predictions of eq 9 with 4 = 0.5 match almost exactly the predictions of eq 7 of Lucas et al. (1986) in the range 1 < Ar < lo9. For the category listed as other by Lucas et al. (1986), it may be readily shown that the predictions of eq 9 with 4 = 0.25 again match almost exactly the predictions of eq 8 in the range 1 < Ar < lo9. Conclusions a n d Significance It is suggested that, being one single equation, eq 9 is easier to use than the three separate equations proposed by Lucas et al. (1986). If the shape factor is known, it is then simply a matter of substituting the value of d, into eq 9. However, if 4 is not known, one could apply the rough guide as given by Lucas et al. (1986) by using 4 = 1.0 for near-spherical particles, 4 = 0.5 for sharp particles, and 6 = 0.25 for others. Literature Cited Lucas, A.; Arnaldos, J.; Casal, J.; Puigjaner, L. fnd. Eng. Chem. Process Des. Deu. 1986, 25, 426-429. Narsimhan, G. AfChE J . 1965, 11, 550-554. Saxena, S. C.; Vogel, G. J. Trans. Inst. Chem. Eng. 1977,55, 184-189. Wen. C. Y.: Yu, Y. H. AfChEJ. 1966, 12, 610-612.
J. J. J. Chen Chemical and Materials Engineering Department The University of Auckland A u c k h d . Neu: Zealand
Response to Comments on “Improved Equation for the Calculation of Minimum Fluidization Velocity” Sir: Chen correctly points out the existence of obvious errors in the paper by Lucas et al. (1986). Such are the errata in eq 1 and 2 which should be corrected as follows: Remf=
[
(42.8572)
+ L]1’2 1.75C1
-
42.857-cz C,
(1)
Fortunately, these corrections have no effect on the validity of the results presented nor on the conclusions and significance of this study. Surprisingly enough, the only obvious error cited by Chen refers to the plot of emf vs. 4 0888-588518712626-0634$01.50/0
(Figure 2 of Lucas et al. (1986)), where, according to him the lines for varying Cz values must pass through emf = 1.0 and d, = 0. It is worth note that the value of C2 as given by eq 2 was first introduced by Wen and Yu (1966), with C, = 11. It should be obvious by inspection of this expression that emf approaches assimptotically to 1 for insignificant values of d,. In other words, when C$< 0.05, the slope of the curve starts decreasing to zero to meet the theoretical limit of emf = 1. This transition has been omitted in Figure 2 of Lucas et al. (1986) for the sake of clarity in the drawing and, overall, for the main reason that in practice, when the value of the shape factor falls below 0.1, it no longer has physical meaning. We discovered long ago that the exQ 1987 American
Chemical Society
Ind. Eng. Chem. Res. 1987, 26, 635-637 Table I. A and B Values for Three Categories“ of Particles oarticle A B round sharp other
29.5 32.1 25.2
0.0357 0.0571 0.0672
“Lucas et al., 1986.
The work by Lucas et al. (1986) produces the following equation to calculate Remf
Remf = (A2 + BAr)l12 - A
1.00 0.50 0.25
Chen, 1986. *Lucas et al., 1986.
cellent paper of Wen and Yu (1966) commits a more serious drawing error for the same kind of representation by joining with a straight line passing through tmf = 1 from 4 = 0.2 to 0. Nevertheless, the main contribution of the paper by Lucas et al. (1986) is to show that constants A and B are in fact dependent on the shape factor 4. This is a relevant result which had not been reported by the different authors cited in Table I of Lucas et al. (1986) nor by more recent works (e.g., Thonglimp et al., 1984). In this sense, the calculation of umf becomes highly simplified, since the determination of 4 is no longer required-following the work by Wen and Yu (1966)-thus avoiding the cumbersome experimental work involved in the evaluation of the shape factor. While the use of appropriate methods (Ergun, 1952; Casal et al., 1985) gives both 4 and emf to calculate umfthrough various equations existing in the literature, eq 1 provides a practical and simplified approach to its calculation by classifying the particles into three categories, as “round”, “sharp”, and “other”; this also takes into account the variation of C, and C2 with 4. Clearly enough, Chen recognizes the interest of our work and incorporates explicitly the value of 4 into a “single” equation that uses three values of 4 according to the three categories indicated by Lucas et al. (1986). Summing it up, the situation is as follows.
(3)
where A and B are constants given in Table I, while Chen, picking up the ideas proposed by Lucas et al. (1986), presents the equation
Table 11. Shape Factor Values for Three Categories of Particles particleb d” round sharp other
635
1’’
+ 24.5~pO.~~ Ar
(33.674°.1)2
~
- 33.674O.I (4)
where the value of 4 has been categorized as in Table 11. It should be obvious that both formulations of the problem are identical as far as they present a single equation, but while the second requires us explicitly to know the value of the shape factor to obtain Remf,the first one presented by Lucas et al. (1986) offers a practical and accurate way to calculate umfin the most general case even if 4 is unknown, by categorizing the particles in three kinds as they are commonly found in practice. Apparently, Chen, while benefiting from the relationship between C,, C2, and 4 established by Lucas et al. (1986), misses the main issue of their work: a simplified way with improved accuracy to calculate the minimum fluidization velocity while avoiding the difficulty associated in finding the value of 4. We hope that our remarks as well as Chen’s comments will help to give a further insight in this topic essential to fluidization practitioners.
Literature Cited Casal, J.; Lucas, A,; Arnaldos, J. Chem. Eng. J . 1985, 30, 155. Chen, J. J. J. Ind. Eng. Chem. Res. 1987, preceding paper in this issue. Ergun, S. Anal. Chem. 1952, 24(2), 388. Lucas, A.; Arnaldos, J.; Casal, J.; Puigjaner, L. Ind. Eng. Chem. Process Des. Deu. 1986, 25, 426. Thonglimp, V.; Hiquily, N.; Laguerie, C. Powder Technol. 1984, 38, 233.
Wen, C. Y.; Yu, Y. H. Chem. Eng. Prog., Symp. Ser. 1966,62, 100.
A. Lucas, J. Arnaldos J. Casal,* L. Puigjaner Chemical Engineering Department, U.P.C. Barcelona 08028, Spain
Comments on “Kinetic Model for Methanol Conversion to Olefins” with Respect to Methane Formation at Low Conversion Sir: Methanol conversion to gasoline-range hydrocarbons is known to be catalyzed by a range of catalysts including the pentasil zeolite H-ZSM-5 (Chang and Silvestri, 1977) and bifunctional acid-base catalysts (Olah et al., 1984). Although considerable research effort has been expounded to elucidate the mechanism of carbon-carbon bond formation, an agreement has yet to be reached. To date, both a trimethyloxonium ylide (van den Berg et al., 1980; Olah, 1981) and a surface-bound carbene species (Chang and Silvestri, 1977; Lee and Wu, 1985) have emerged as being the most likely reaction intermediates. Until now, most mechanistic research effort has been concerned solely with carbon-carbon bond formation, whilst methane formation has received comparatively scant attention. By use of the pentasil zeolite H-ZSM-5 at high methanol conversions, methane is only formed in small quantities, and most workers consider that it originates mainly from thermal cracking of higher hydrocarbons catalyzed by the highly acidic sites of H-ZSM-5. It has
been proposed (Olah et al., 1984) that some methane could originate via a radical pathway involving dimethyl ether, but the initial study (Benson and Jain, 1959) involved gas-phase reactions at a temperature in excess of 500 “C, and it is unlikely that such a reaction would contribute significantly to methane formation at temperatures normally encountered with catalysts for the methanol conversion reaction. However, at low methanol conversion, i.e.,
