the point on onset of fluidization. The correlations of the other investigators (Ramamurthy and Subbaraju, 1973), while adequate over certain ranges, are not as accurate as the equations presented in this study. Acknowledgment This work is sponsored by the National Science Foundation Grant No. GK-10977.
.1;
,
,
,,,,,
,
, , , , , ,,,
, ,
, C:E;R P ;
Nomenclature a = correlation constant, dimensionless b = correlation constant, dimensionless d , = particle diameter, L g = gravitational acceleration, LO-2 NG, = Galileo number, dp3(pp- p l ) p g / p 2 , dimensionless NRe = Reynolds number, d p u p l / p ,dimensionless (NRe)mf = Reynolds number a t incipient fluidization point, dpUmfPI/p, dimensionless u = superficial velocity of fluid through beds, LO-1 umf = minimum fluidization velocity, LO-l
1
10 10
IO6
IO’
100
NGP
Figure 10. Relationship between NGa,N R ~and , fluidized beds for NGa> 105.
c
in liquid-solid
Under the conditions lo5 > Nc, >> 19, the denominator of this equation can be simplified such that the equation becomes ( A’,,)mi = 0.00138N(;,0s9“ (8) This equation is practically identical with eq 6. In addition, for Nca > 105,the Bena, et al., correlation is ( ICrKe)n,f =
0.03865N(;c,U’”OL
(9)
Examining eq 7 and 9 more closely, one notes that only a small difference exists between the coefficients while the exponents are equal. Once again, the resemblance between the two equations is very close. Reviewing the conclusions drawn previously, it is evident that the present correlation can be applicable from
Greek Letters t = bed voidage, dimensionless e m f = bed voidage at incipient fluidization point, dimensionless de = [l - 1.21(1 - t ) 2 ’ 3 ] - 1 , dimensionless p = viscosity of the fluid, M L - I T - l p1 = density of the fluid, M L - 3 p , = density of the particle, ML-3 Literature Cited Bena, J., Havalda, I., Matas, J., Collect. Czech. Chem. Comrnun., 36, 3563 (1971). Lewis, W. K., Gilliland, E. R., Bauer, W. C., lnd. Eng. Chem., 41, 1104 (1949). Ramamurthy, K., Subbaraju. K., lnd. Eng. Chem., Process Des. Develop., 12, 184 (1973). Richardson, J. F., Zaki, W. N.. Trans. lnst. Chem. Eng., 32,35 (1954). Wen, C. Y., Yu, Y. H., Chem. Eng. Prog. Symp. Ser., 62, 100 (1966). Wilhelm, R. H., Kwauk, M., Chem. €ng..Progr., 44, 201 (1948). D e p a r t m e n t of C h e m i c a l Engineering West Virginia University Morgantown, West Virginia 26506
C.
Y. Wen* L. S. Fan
Receioed for reuieu! July 12, 1973 Accepted December 5 , 1973
CORRESPONDENCE Some Remarks to Marina’s Modification of the NRTL Equation
Sir: Recently Marina and Tassios (1973) have proposed a modification of the NRTL equation (Renon and Prausnitz, 1968) which consists of substituting for the adjustable parameter CY the value -1. The mentioned authors deduced, on the basis of analysis of the measured and calculated values of activity coefficients, that this value could substitute rather ambiguous rules for the estimation of this parameter (Renon and Prausnitz, 1968) or its relatively complicated optimalization from the measured equilibrium data. It is very probable that the proposed value will be sufficient for a number of the homogeneous and heterogeneous systems, but, on the other hand, the use of any particular value of cy restricts, in our opinion, the applicability of the NRTL equation. A part of our former calculations extended into the region of negative cy values is illustrated in
Figure 1. By using the values in this figure, one can judge the limits of the applicability of the NRTL equation with a certain value of cy dependent on ( and (G11)xo. The parameter l denotes - 0.51, where X O corresponds to the compositions at which the quality G11 = 0.43429 x a2(g/RT)/dX12 reaches its minimum value (G1l)xo. The symbols are defined as follows: g is the molar Gibbs energy, T the absolute temperature, XIthe mole fraction, and R the gas constant. The critical isotherm is therefore characterized by (G1l)xo = 0 and X o = X,,where X, is the composition corresponding to the critical point. The analogous values for an ideal symmetrical system are (G11)xo = 1.73’72andXo = 0.5, i e . , t; = 0. The limits of the applicability of the NRTL equation are also determined by the following facts. Let us consider, for example, CY = 0.5; in this case the NRTL equation
1x0
Ind. Eng. C h e m . , Process Des. Develop., Vol. 13, No. 2,1974
197
higher positive values of the parameter a restrict the applicability of the NRTL equation in the heterogeneous region. On the contrary, the modification proposed by Marina and Tassios with the value a = -1 makes it impossible to describe in a quantitative correct way the behavior of some homogeneous systems, e.g., all systems with .$ > 0.35. With the systems water-polar substance the value of ,$ lies very often in the range 0.2-0.4 and if the value of ,$ is greater than 0.35, it is necessary to use the parameter a < -1 to preserve the position of this minimum. The relations for the determination of the G11 were summarized by Su‘ska (1972). A more detailed discussion of the NRTL equation and calculation procedures have been prepared by Novik (1973). Literature Cited Marina, J. M . , Tassios, D. P., lnd. Eng. Chern., Process Des. Develop.,
-1
I 0,5
I
06
I
03
4
12,67 (1973).
1
0,2
0;1
NovBk, J. P., SuSka, J., MatouS, J., Collect. Czech. Chern. Cornmun., in press, 1973. Renon, H.,Prausnitz, J. M., AlChEJ., 14,135 (1968). SuSka, J . , NovBk. J. P., Matoub, J., Pick, J., Collect. Czech. Chem. Cornrnun., 37, 2663 (1972).
0
Figure 1 .
-
is not able to describe the behavior of systems which have XO 0.3 and (Gll)xo < 0.2, without a shift of this minimum. Considering the data in Figure 1 it follows, too, that
Institute of Chemical Technology Department of Physical Chemistry &aha 6, 1905, Czechoslovakia
Josef P. Novlk* Josef SuBka Jaroslav MatouS
Received for review April 16, 1973 Accepted October 17,1973
Some Remarks to Marina’s Modification of the NRTL Equation
Sir: Dr. Novak and his coworkers with their letter concerning the LEMF equation (Le., the NRTL equation with a = -l), their presentation in the 4th Congress CHISA 72 concerning the NRTL and Redlich-Kister equations (Novak, et al., 1972), and their recent publication (Suska, et al., 1972) demonstrate the limitation of the aforementioned empirical equations in correlating VLE data. In the case of the LEMF equation the limitation applies to a very limited number of systems for which 0.35 < ( < 0.4 (Le., 0.1 < XO < 0.15) since they did not find any systems with .$ > 0.4. According to Suska, et al. (1972), however, the values of XO cannot be determined uniquely because the calculation of G11 involves differentiation of the experimental data. For example, for the system ethanol (1)-n-heptane (2), use of the P-X data yields a minimum value for G11 at X I = 0.55 while use of the y-n data yields G11 min at X I = 0.35. Because of this uncertainty it appears to the writer that it would be safer that the quantity (Gll)%o, evaluated from the constants obtained by regressing the VLE data, be checked. If it is found negative a value of -1 > a > -2 should be used.
Such a value should cover systems with 0.4 > 6 > 0.35, as seen from Novak’s graph, and, as shown by Marina and Tassios (1973) and Larson and Tassios (1972), should yield standard deviations not too far from the minimum one. Since, however, the magnitude of the 710 and 7z0values iml pose some restrictions in the NRTL equation (CHISA paper) and no such reference is made for the LEMF equation, the whole subject of the limitations of the LEMF equation needs further study. Literature Cited Larson, D., Tassios,
D.,lnd.
Eng. Chern., Process Des. Develop., 11, 37
(1972). Marina, J., Tassios, D., lnd. Eng. Chem., Process Des. Develop., 12,67
(1973). Novak. J. P., et a/., paper presented at the 4th Congress, CHISA 72,1972. Suska, J., Novak, J. P., Matous, J , Pick, J., Collect. Czech. Chern. Commun., 37,2664 (1972).
Department of Chemical Engineering Dimitrios Tassios and Chemistry Newark College of Engineering Newark, New Jersey 07102 Received for reuieu, September 19, 1973 Accepted October 17, 1973
CORRECTION In the article, “Calculation of High-pressure Phase Equilibria and Molecular Weight Distribution in Partial Decompression of Polyethylene-Ethylene Mixtures,” by D. C. Bonner, D. P. Maloney, and J. M. Prausnitz [Ind. Eng. Chem., Process Des. Develop., 13, 91 (1974)], the last line in the Appendix on p 95 should read: The value of X21 at 260°C is -49.9 atm. ~
198
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 2, 1974
~-
