Id.Eng. Chem. Fundam. 1981, 20, 300-302
300
Nomenclature do = jet nozzle diameter d = particle size of size fraction i d: = average particle size determined by dp = l/CXi/dpi g = gravitational acceleration Ga = Galileo number defined in eq 6 Rd = ratio of complete fluidization velocity at pressure P over that at atmospheric pressure, (Ud)p/(U&,b (Re)d = Reynolds number at complete fluidization Ud = complete fluidization velocity ( Ud)atm= complete fluidization velocity at atmospheric pressure (Ud)p= complete fluidization velocity at pressure P (U& = minimum fluidization velocity of particle size fraction I
Uo= average jet nozzle velocity
Xi = weight fraction of particle size fraction i
= fluid viscosity
Literature Cited Ergun, S. Chem. Eng. Progr. 1952, 48, 89. King. D. F. Ph.D. Mssertatbn, Unlverdty of C a m , July 1979. Knowlton, T. M. AIChESymp. Ser. 1977, 73, No. 761, 22. Knowbn, T. M.; Hfrsen, I. “The Effect of PrbMHlre on Jet penetratbn In SemKyIlKklcal Qas-Fluldlred Beds”; paper presented at the Intemetbnel Conference on Fluidization, Hennker, NH, Aug 3-8, 1980. Rlchardson, J. F.; Zakl, W. N. Trans. Inst. Chem. Eng. 1964, 32, 35. Wen, C.Y.; Yu, Y. H. AIChE J. 1986, 72, 610. Yang, W. C.; Kealms, D. L. “Design and OperaParamstars for a Fb#lzsd Bed Aggkmwating Combustcf/Qasitler”; Proceedings of the Intematknal Conference on Fluldlzetlon, J. F. Davidson and D. L. Keakns, Ed.; Cambrkige University Press, 1978; p 208. Yang. W. C.; Keairns, D. L. I d . Eng. Chem. Fundem. 1079, 78, 317.
Research and Development Center Westinghouse Electric Corporation Pittsburgh, Pennsylvania 15235
pf = fluid density pp = particle density ecf = voidage at complete fluidization
Wen-chingYang
Received for review July 31,1980 Accepted March 16,1981
Simple Model for Predicting Fluid Phase Equilibria A simple model for nonideal mixtures was proposed, where the equilibrium of interaction energies acting simul-
taneously between several molecules instead of the interactbn energy dmerences actlng between pairs of mdecuies are considered as being effective for the excess properties. The differential equation expressingthe model relates the parameters in the equation given by Wiison (1964) for the Gibbs excess energy 8’ to the numbers of mdecuies ngand n, whose interactions are effective for 8’ and the excess enthalpy he, respectively. The correletkns derhred predict accurately the limit for computing he and g’ by unique sets of Wilson parameters. By use of the Wilson parameters given in extensive iiqukl-vapor equHlbrkrm data reductions, an empirical correlation was obtained which reduces the Wilson equation knpricstfv to a one-parameter form. The correlations derived suggest unitary treatments of various problems in the fleM of predicting phase equilibria.
A unitary treatment of several problems of the actual research in the field of predicting fluid phase equilibria becomes possible, if one starts from a simple model of the interaction energy equilibrium in systems of neighbor molecules, whose interactions should be considered as being effective for exhibiting a given excess property of the mixture, rather than the differences between the interaction energies acting between pairs of molecules, as considered in the known models based on the local concentration concept. Assuming that the equilibrium of the interaction energiea acting simultaneouslybetween all molecules in systems restrained to nearest neighbors is performed by a process analogous to a chemical one, governed by the mass action law, the contribution of the interaction energies acting around ni molecules of the component i, and effective for the excess property considered, to the sum I of the interaction energies working in the whole restrained molecular system, as “reaction product”, is proportional to the concentration of this component xi to the power ni, thus
In order to obtain correlations relating ni to the deviations from ideal behavior, first a correlation expressing ni in terms of the parameters of the equation given by Wilson (1964) for the Gibbs excess energy was derived. Starting from this equation in the binary form
p / R T = -xi In (xi + Aij*xj)- x j In ( x i
k
E1 Xin’.dIi = 0
(2)
where Ii is the interaction energy acting around ni molecules of the component i. 0196-4313/81/1020-0300$01.25/0
(3)
where the interaction energy differences Xij - Xii from the parameters Aij = ” i exp( vi
X.. - X.. -%) I
(4)
are considered as being effective for all excess properties; in eq 2, Ii = Xij - Xii, I j = Xji - Xjj, and eq 2 becomes xi”i*d(Xij- hii) + Xjnj*d(Xji- A j j ) = 0 (5)
By differentiating eq 3 at constant composition, g“ and T, and assuming in a first approximation ni = nj = ng,eq 5 leads to -
Because at equilibrium of the interaction energies dI = 0, this equilibrium may be expreased in terms of the model proposed by
+ Aji*xi)
d(Xij - Xii) - Aji(xi + Aijxj) = d(Aji - Xjj) Aij(xj + Aji-xi)
(:)
(6)
If in eq 6 the Wilson parameters A, and Aii obtained by reduction of experimental liquid-vapor equilibrium data are used, and the concentrations xi* and xj* for the extreme g“*/RT of the dimensionless Gibbs excess energy are introduced, one obtains for ng values which show a dependence on g * / R T . For obtaining an empirical correlation expressing this dependence, the results of the data reductions published 0 1981 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 20, No.
3, 1981 301
Table I. Data Points in Figure 1 from the Wilson Parameters of Other Authors
acetone-benzene acetone-carbon tetrachloride acetone-chloroform acetone-ethanol acetone-methanol acetone-methyl acetate acetone-2-propanol acetone-water benzene-cyclohexane benzene-2-propanol carbon tetrachloridebenzene carbon tetrachloridecyclohexane carbon tetrachloride2-propanol chloroform-benzene cyclohexane-2propanol ethanol-benzene ethanol-cyclohexane ethanol-heptane ethanol-water methanol-2-propanol methanol-water methyl acetate-benzene methyl acetate-chloroform methyl acetate-cyclohexane l-propanol-water 2-propanol-water
r T - - ' I
-04
-0:1
c
Figure 1. The dependence of the exponent ns on $*/RT.
by Holmes and Van Winkle (1970), Nagata (19731, and Hdla (1974) were considered. The 67 values of n represented in Figure 1vs. $*/RT computed from 70kilson parameters pairs (Table I), selected in this order by Gothard (1980a) to have g"*/RT from -0.18 up to +0.56 from various binary systems, may be correlated by the equation
ns = 2 e~p[4.64($*/RT)~]
(7)
with root mean square deviation of 9%. About 4% of the Wilson parameter pairs given in the extensive vapor-liquid equilibrium data reductions considered lead to large deviations from eq 7. The results obtained show that in nearly ideal solutions ($*/RT 0), n, 2, and according to the significance assigned to n,, small deviations from ideal behavior, exhibited in the Gibbs excess energy, may be attributed to interactions between pairs of molecules of the components, considered also in all known models of solutions, based on the local concentration concept. Near the upper limit of $*/RT in homogeneous liquid systems (0.55 0.6) n, 8. This number may be considered as being the largest number of molecules effectively interacting in stable liquid systems, and it suggests that in explanations concerning the limit of stability of nonideal liquid mixtures symmetry contributions should also be considered. Starting from comparisons of the deviations from eq 7 and the deviations of liquid-vapor equilibrium data predicted by use of the Wilson parameters considered in computing n, and g*/ RT, Gothard (1980b,c) suggested the use of eq 6 and 7 as selecting criteria of the parameters respectively of the experimental data. For the excew enthalpy, the Wilson equation (eq 3) leads to
- -
- -
In a similar way as for the Gibbs excess energy, and assuming that ni = nj = nh is the number of molecules
Holmes and Van Winkle Nagata HGla (1970) (1973) (1974) 1 2 3 4 5 6 8 10 13 19 20 24 28 29
7
12 14 16,17
9 11 15
18 21 23 27 30
22 25 26 31
32
33
Qs
35
34
39 41
37
38 44
43 45 47 50 53 56 59
48 51 54 57 61
62
63
65
64
66
69
68 70
67
44 46 49 52 55 58 60
whose interactions are effective for exhibiting the excess enthalpy, one obtains from eq 2 and 8 d(Aij - Aii) -
-
d(Aji - A j j ) [ ( x i + AjixJAji - xjAji(Aji - Ajj)/RT](xi
+ hijxj)' [ ( x i + Aijxj)Aij - xiAij(Aij - X,)/RT](xj + A j i ~ i ) . ~
(;)
(9)
According to the new model, predicting of various excess properties of a mixture by use of a unique interaction energy difference pair may be expected only when at least the numbers of molecules whose interactions are exhibited by the excess properties considered are identical. By comparison of eq 6 and 9 it becomes clear that n, = nhonly if the terms xjAji(Aji - Ajj)/RT and xiAij(Aij - AG)/RTin eq 9 may be neglected as being much smaller than ( x . + Ajixi)Ajiand ( x i + Aijx.)Aij,respectively. This is possible up to he of about 500 jmol-', in very good agreement with the upper limit given recently by Wilkinson (1979) for predicting accurately liquid-vapor equilibria from heat of mixing data, by use of a unique pair of Wilson parameters for a given system. It should be noted, finally, that the model proposed may be applied also to other equations based on the local concentration concept, and for other excess properties, with similar results, to be presented in further papers, and that eq 6 leads to a one-parameter Wilson equation in implicit form, without the model statements used in this order by
Ind. Eng. Chem. Fundam. 1981, 20,302-303
302
Tassios (1971)and also by H d a (1972) for obtaining the well-known boundary conditions of the Wilson parameters in multicomponent systems. The predicting performances of the new one-parameter form of the Wilson equation derived from this model are discussed in by Gothard (1980a) in a paper to be published elsewhere. Acknowledgment The authors wish to express his gratitude to Mrs. Valeria Suarasanu for the computations performed. Literature Cited
ternatlonal hference on Thermodynamics, 1980c, Merseburg, East Germany, Aug 26-29, Poster Session Paper 61. Wla, E. Collect. Czech. Chem. Commun. 1972, 37. 2817. ala,E. Chem. Tech& 1974, 26(8), 482, Supplement. Holmes, M. J.; Van Wlnkle, M. Ind. fng. Chem. 1970, 62(1), 21-31A. Nagata, 1. J. Chem. €ng. Jpn. 1973, 6(1). 18-30. Tasslos, D. AlChE J. 1971, 76, 1387. Witkinson, S. Trans. Inst. Chem. Eng. Symp. Ser. 1979, 56(1), 1-15. Wilsm, G. M. J . Am. Chem. Soc.1984, 86, 127.
Romanian Institute for Research Deuelopment and Design for Oil Refineries Ploiesti, Romania
Gothard, F. A. F M phese fqu#lb. 1980a, in press. Gothard, F. A. Rev. Chh. (eUCwest0 IBOOb, 37(3), 304. Gothard, F. A. “Conelatkns Derived From a Model of the Equilibrlwn of Interactlon Energles in Molecular Systems of Nearest Neighburs”, 6th In-
Francisc A. Gothard
Received for review February 27, 1980 Accepted November 24, 1980
CORRESPONDENCE Crltlcal Remarks on Using Moments Method Sir: In a paper in this journal (I),the moments methods has been recommended for curve evaluation in a rather uncritical manner, not taking into account a number of shortcomingspreviously discussed in the literature (2-7). Below, the statistical properties of the moments method, e.g., confidence, efficiency, and sufficiency (B), are considered in a qualitative sense. The principal problem of the moments method is the increasing error of experimental pfn,exwith increasing moment order n (see Table I), mainly caused by the unfavorable noise level in the tail due to the term tn in the equation
Table I. Typical Standard Deviations of Exuerimental Moments u,
56
0-1
1-2
2-4
4->30
6-10
10->lo0 Wm
I
This holds even more for the central moments
9.5
where the relative weighting of the curve parts (see Figure 1)
0
causes a strong underemphasis of the most accurately measured central curve portion (Wr,n = 0 for t = dl), whereas the more scattered tailsegments are overweighted. Therefore, central moments p,, scatter more than the pfn (see Table I). This has the following consequences. 1. Since a small number of experimental moments (usually n = 2) can be evaluated within a reasonable error, but a function f(t) is uniquely defined only by all existing moments (12),the moments method is not critical relative to model errors. This property of the method is illustrated by Figure 2 (9). Four different time functions are represented, all of which coincide in their zeroth, first, and second moments, but differ in the higher moments. Of course, the model describing the physical process may be still another one. Because only a small number of moments can be evaluated, not all information involved in the experiment is utilized-the method is not sufficient. 0198-4313/81/1020-0302$01.25/0
-0.5
Figure 1. Relative weighting W,,,, of curve portions in the experimental central moments pn,*=.
2. The larger the number of parameters to be estimated, the more moments of higher order must be evaluated. Some of the parameters may then be strongly erroneous, especially those exerting only a small influence in the model. The method is in such cases of low efficiency. The lack of efficiency may be circumvented by variation of parameters such as flow rate or particle size, leading to evaluation of series of experiments instead of individual response curves (14). 3. The moments method provides unbiased (confident) values, if the scattering in the tailing edge of the measured curve is purely stochastic. Discrepancies between model 0 1981 American Chemical Society
