I n d . Eng. Chem. Res. 1987,26,633-634
633
Mijares, G.; Cole, J. D.; Naugle, N. W.; Preisig, H. A,; Holland, C. D.AZChE J. 1986, 32, 1439.
and the closed-loop system is unstable.
Gerard0 Mijares, Charles D. Holland*
Literature Cited
Chemical Engineering Department Texas A&M University College Station, Texas 77843
Garcia, C.; Morari, M. Znd. Eng. Chem. Process. Des. Dev. 1985,24, 472.
Response to Comments on “Internal Model Control. 2. Design Procedure for Multivariable Systems” Sir: We would like to thank Mijares and Holland for pointing out that Theorem 2 in the paper by Garcia and Morari (1985) was stated incorrectly. The following two statements are correct. Theorem 2A. Assume that the filter F(z) is of the exponential type 1-a F(z) = I OIa 0 where Aj(A) denotes the jth eigenvalue of A, G is the model of the system, and G, is an IMC-structured controller. Theorem 2B. Let the plant have two inputs an$ two outputs, and assume the diagonal elements of G(l)G(l)-l to be positive. Assume also that the filter F(z) is diagonal and of the exponential type
(
F(z) = diag l::;-l) O I a i < l
more restrictive assumptions. In particular, they assume the plant (and model) dynamics to be described by Gz-l, where G is a constant matrix. Garcia and Morari (1985) allow arbitrary (stable) plant dynamics and prove that a first-order filter is sufficient for robust stabilization as long as the eigenvalue condition mentioned above is satisfied. Also, the value of a* cited in the correspondence by Mijares and Holland is only meaningful in the context of the very restrictive plant dynamics assumption and has no general significance. Finally, Mijares and Holland conjecture in their correspondence that there might be a lower bound a* for which different ai(a* 5 ai< 1)may be allowed in different loops. This is not true in general. It is easy to show that for their specific example the eigenvalue condition is violated for all values of ai= 1- y(1 - &J where s1= 0.41, hz = 0.619, kYQ = 0.969, and 0 < y I 1. Thus, there does not exist any lower bound a* < 1. Grosdidier and Morari (J986) have derived the sufficient condition p ( E ) = p((G - G)G-l(l)) < 1 which guarantees that the ais can be chosen independently ( p is computed with respect to the structure of the filter F-a diagonal matrix). For the Mijares and Holland example, p ( E ) = 1.64.
i=land2
and that G, = G--l. There exists an a* (0 Ia* < 1)such that the system is closed-loop stable for all aiin the opeg interval a* Iai< 1 (i = 1 and 2) if and only if G and G satisfy j = 1 and 2 Re [Aj[G(l)6(1)-1]] > 0
Literature Cited Garcia, C. E.; Morari, M. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 472-484. Grosdidier, P.; Morari, M. Automatica 1986,22(3),309-319. Mijares, G.; Cole, J. D.; Naugle, N. W.; Preisig, H. A.; Holland, C. D. AZChE J . 1986,32(9), 1439-1449.
where Aj[A] denotes the jth eigenvalue of A, 6 is the model of the system, and G, is an IMC-structured controller. The comparison with the results by Mijares et al. (1986) is misleading, however. Though Mijares et al. derive an equivalent eigenvalue condition, they do so under much
Manfred Morari Chemical Engineering, 206-41 California Institute of Technology Pasadena, California 91125
Comments on “Improved Equation for the Calculation of Minimum Fluidization Velocity” Sir: The various expressions proposed for the prediction of minimum fluidization velocities are often attemr>ts to approximate the factors given as eq 2 and 3 which appear in eq 1relating Remfto Ar which was derived by applying Ergun’s equation to incipient fluidization. Remf =
[(
4%3fj72)2
”-I1”
+ 1.75C1
CZ
- 42*857-
C1
(l)
C1 = l / $ e m P
(2)
Cz = (1 - cmf)/@cmP
(3)
On the basis of a very limited amount of data, Narsimhan (1965) related the voidage at minimum fluidization 0888-5885/87/2626-0633$01.50/0
for particle sizes greater than 0.5 mm to the shape factor as Emf = 0.768 - 0.424 (4) Wen and Yu (1966) suggested that the value for emf given by the correlation of Narsimhan (1965) for spherical particles a t 0.35 seems to be too small, and they showed that Narsimhan’s correlation was only valid for a limited range - of conditions. As a result, Wen and Yu (1966) gave C1 = 1/@,f3 14 (2b)
-
= (1 - Emf)/&*emf3
(3b) For spherical particles, i.e., 4 = 1.0, eq 2b and 3b give emf = 0.414 and 0.383, respectively. These values were, c2
0 1987 American Chemical Society
E
11
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- emf)/@emf3
= 5.9 f 0.6
(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 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
