Ind. Eng. Chem. Fundam. 1983, 22, 177-182
177
Application of the Shrinking, Hard-Sphere Theory to the UNIQUAC Equation for Calculating Phase-Equilibrium Properties in Liquid Metal Systems KI-pung Yo0 and Thomas F. Anderson* Department of Chemical Engineering, University of Connecticut, Storrs, Connecticut 06268
The structural parameters and coordination number in the UNIQUAC model were made functions of temperature by introducing the concept of the shrinking hard-sphere. The simultaneous correlatlon of activity coefficients and excess enthalpy for liquid metal mixtures by the modified UNIQUAC model was remarkably successful. Both for positive and for negative deviations from ideal behavior, the proposed equation gives a good representation for binary alloy systems even at high temperatures. The approach is also applicable to multicomponent, multiphase metal alloys and allows reasonable extrapolation of limited data to elevated temperatures.
During the past several years, a growing amount of research has been concerned with processing liquid metal mixtures at high temperatures. For an economic and rational design of such processes, a knowledge of the thermodynamic properties of liquid-metal mixtures is required. Unfortunately, suitable experimental data are scarce and frequently of poor quality, because of the difficulties associated with high-temperature technology. Experimentation is further hindered by phenomena such as valency, electronegativity, structural factors, eutectic formation, and intermetallic compound formation, which are common in liquid-metal systems. Because of these experimental difficulties, increased emphasis is being placed on theoretical models for such systems. Early attempts to represent metallic solutions (Hardy, 1953; Mott, 1957) were based on the regular or quasiregular theories of Hildebrand (1929). Later, quasi-chemical theory (Guggenheim, 1952) formed the basis for the surrounded-atom models of Mathieu et al. (1965) and Hicter et al. (1967). A similar basis was used by Lupis and Elliott (1966,1967) in their central-atom theory. Recently, Paulaitis and Eckert (1981) proposed a perturbed, hardsphere model based on corresponding-states principles; they applied their equation to numerous liquid-metal mixtures. Eckert et al. (1982) also extended the chemical solution theory to strongly solvating liquid metal mixtures. In this work the universal quasi-chemical (UNIQUAC) equation of Abrams and Prausnitz (1975) is applied to metal mixtures. However, to achieve a good representation of mixture properties and to provide for good extrapolation with respect to temperature, the structural parameters and coordination number in the UNIQUAC model are correlated as functions of temperature. In developing our correlation we recognize that, although repulsive forces represented by a hard-sphere potential dominate the structure of most dense fluids, the metal atoms are not completely hard spheres. The atomic radius shrinks with increasing temperature because kinetic energies and colliding frequencies increase with temperature. The correct temperature dependence is achieved by incorporating the pertinent concepts of the shrinking, hard-sphere theory (Chandler and Weeks, 1970; Weeks et ai., 1971). The result is an analytical expression for the van der Waals radius as a function of temperature and a semiempirical temperature-dependent function for the coordination number. These new functions are incorporated into the UNIQUAC equation. 0196-4313/83/1022-0177$01.50/0
Temperature Dependence of van der Waals Radius Hard-sphere potential theory was first employed by Enskog and Svensk (1921). In its simplest form, this potential is characterized by an infinitely strong repulsion when an attempt is made to bring the spheres closer together than a distance a; the distance u may be taken as the hard-sphere diameter of the molecules. At separations greater than u, there are no attractive forces whatsoever. Using the hard-sphere potential with the single parameter u, reasonable values for density and transport properties can be calculated for liquids and liquid mixtures. Dymond and Alder (1966) were the first to suggest using a temperature dependent hard-sphere diameter with the hard-sphere potential theory. The temperature dependence they proposed was obtained by assuming that for the liquid the repulsive and attractive pressure components in the van der Waals equation of state counterbalanceeach other. Ascarelli and Paskin (1967) and others successfully applied this theory to the calculation of self-diffusion coefficients of liquid metals. An alternative approach was developed by Protopapas et al. (1973,1975) for the temperature dependence of the hard-sphere diameter. Since their equation tends to show consistently better agreement with experimental results than that of Dymond and Alder, we have adopted the concepts of their theory to arrive at a temperature-dependent van der Waal radius. This equation was also used by Paulaitis and Eckert (1981) to provide a temperature dependence of hard-sphere diameter in a correspondingstates model for representing solutions behavior. Briefly, Protopapas chose a parabolic curve to describe the repulsive potential field. The minimum of this potential coincided with a separation, uo, which is the hardsphere diameter at the melting point. At this separation, the potential energy is zero. Protopapas then assumed that “perfect” collisions occur in the liquid such that the kinetic energy is totally converted into potential energy. Since an expression for the potential energy has been established as a function of the separation distance, the temperature dependence of u can then be determined by simply equating the kinetic and potential energies. The expression obtained by Protopapas is u = 1.288(WA/p,)0.334[1- 0.112(T/T,)0.5] X lo-@ (1) In eq 1, WAis the atomic weight, pm is the density (g/cm3) at the melting point, and T, is the melting point tem0 1983 American Chemical Society
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Ind. Eng. Chern. Fundam., Vol. 22, No. 2, 1983
perature. The hard-sphere diameter, u, has units of centimeters. The hard-sphere diameter calculated with eq 1 actually decreases with increasing temperature (for T > Tm); hence, the equation is called the shrinking, hard-sphere theory. For further details on the derivation of eq 1 readers are referred to the works of Protopapas et al. (1973, 1975). In the UNIQUAC equation, the van der Waals radius is required. The atomic radius is just half of the hardsphere diameter calculated with eq 1, but this is not the same as the van der Waals radius. However, it is logical to assume that the van der Waals radius is proportional to the atomic radius. Based on several metals for which van der Waal radii and atomic radii are reported, we found the constant of proportionality to be 2.5. When this constant is used with the equation for the hard-sphere diameter, the following expression for the van der Waals radius, R,, is obtained
R, = 1 . 6 1 ( W A / / ~ , ) ~ . ~-~0.112(T/T,)o~5] ~[l X lo-*
(2)
Equation 2 can be used directly in the UNIQUAC equation for liquid metals.
Modification of the UNIQUAC Model The UNIQUAC equation (Abrams and Prausnitz, 1975) is principally an extension of the quasi-chemical model of Guggenheim (1952). The essential feature is that the effects of molecular size and shape are introduced into the model in the form of the van der Waals area and volume parameters, qi and ri, respectively. For a binary system, there are two adjustable parameters, AU12and AUzl, which must be determined from phase equilibrium data. The activity coefficient y, is given as the sum of two contributions, a combinatorial and a residual In
71
= In
Ylmmb+
In
(3)
Ylrmid
The combinatorial contribution is given by
Table I. Pure-Comnonent Parameters for Liauid Metals component AWi p m i , g/cm3 Tmi,K A1 26.98 2.41 933 Bi 544 208.98 10.07 112.41 594 Cd 8.02 7.67 1768 co 58.93 cu 1536 7.96 63.55 Au 1337 196.97 17.28 Fe 1808 55.85 7.13 Pb 601 207.20 10.71 1244 Mn 54.94 6.43 24.31 922 1.59 Mg Ni 58.70 1726 7.91 2.52 Si 1683 28.09 1235 9.32 107.87 Ag 505 Sn 6.98 118.69 Zn 693 6.57 65.38
T ~K ~ , 2740 1833 1038 3143 2840 3080 3032 2013 2235 1363 3005 2628 2485 2543 1180
number 2,which appears in eq 4 and 6, was assigned a value of 10. When 2 is taken as a constant, it is conveniently absorbed into the interaction parameter by letting Au, = (2/2)AUij. In this form the UNIQUAC equation has been shown to give a good representation of both vapor-liquid and liquid-liquid equilibria for a wide variety of polar and nonpolar liquids (Anderson and Prausnitz, 1978; Prausnitz et al., 1980). With liquid metal systems extremely high temperatures and very wide temperature ranges are encountered. At these conditions the structural parameters and the coordination number can no longer by considered temperature independent. A suitable temperature dependence can be obtained by making use of the concept of the shrinking hard sphere discussed in the previous section. The structural parameters are related to the van der Waals volume and area. The volume and area can be calculated from the van der Waals radius, which is now given as a function of temperature by eq 2. Omitting the mathematical details, we find that the UNIQUAC structural parameters for metals as functions of temperature are given by qr = 1.063(WA,/p,,)0.667(1- 0.112Tr:.5)2
(10)
r, = 0 . 9 7 2 ( w ~ , / ~ , , ) (-i 0.112~,:.5)3
(11)
and
where
ll = (2/2)(r1 - qd - (rl - 1); l2 = (2/2)(r2 - q 2 ) - (
~ 2
1) (5)
The residual contribution is given by
where
In eq 4,4 is the average segment fraction
x2r2 (8) Xlrl + x272 Similarly, in eq 4 and 6, 0 is the average area fraction Xlrl
$1
= xlrl
+ x2r2' *
42 =
XlQl x2q2 ; $2 = (9) xlql + x2q2 x1q1 + x2q2 For component 2, y2 can be found by interchanging subscripts 1 and 2. In the original UNIQUAC equation the values for ri and qi were taken as constants. In addition, the coordination
%1
=
where T,, = TITmz.The atomic weights, densities, and melting points for a number of metals are given in Table I. These are used directly in eq 10 and 11. The coordination number, 2, is a measure of the average number of nearest neighbors to a randomly selected atom in the fluid. Qualitatively, it indicates how well-packed the fluid is. The maximum theoretical limit is 12, which occurs in a perfectly ordered crystalline lattice. For liquid metals the value of 2 lies between 12 at the melting point and 6 (or 4) at the boiling point. To adequately describe mixtures of liquid metals this temperature dependence of 2 must also be taken into account. Unfortunately, no exact analytic method is available to describe how 2 varies with temperature. It is known, however, that this number varies directly with the van der Waals radius (Protopapas et al., 1973,1975). This suggests that the temperature dependence of 2 should have the same general form as that given for the van der Waals radius, eq 2. In addition, we require that the value of 2 lie between 12 and 6 over the range of temperature from the melting point to the boiling point. Based on a large number of metals, we find that the boiling point temperature is about 2.5 times the melting point temperature. It is reasonable, therefore, to require 2 to vary from a value of 12 at T , to
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Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
T 1
Trn 1.0
1.6
b'
t
2.5
2.0
I.o Reduced Temperature,
1,
= TIT,,,
Hultgren et a N I S 7 3 )
Q
-a
E c
in
C a l c d w i t h UNIQUAC equatlon
A-
I* -E-
-
1.2
1 .o
a E
n n
=T/Tm
Figure 3. Coordination number for metal solution.
Q
2 m 3
2.5
2.0
Reduced Temperature. T ,
Figure 1. UNIQUAC pure-component area parameter for liquid metal.
-
1.5
0.0
o -
mm
?
r
0.8
m' 1 .o
Tb 1.5
Reduced Temperature,
2.6
2.0
Tr=TITm
I
Figure 2. UNIQUAC pure-component volume parameter for liquid metal.
6 at 2.5Tm. The resulting expression for Z as a function of T is Z = 22.43[1 - 0.4635(T/T',)0.5]
6.0
4.0
Ysn
(12)
where prnis the average melting point for the mixture given by
2.0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
M o l e F r a c t i o n o f Tin
The constants in eq 12 were determined by the constraints Z = 12 at T = T, and Z = 6 at T = 2.5T,. Equation 12 is not exact, but it provides a physically reasonable approximation for the variation of Z with temperature. The proposed structural parameters and coordination number as functions of temperature are shown in Figures 1-3. The dimensionless form of these figures allows one to read directly the appropriate parameters for liquidmetal mixtures. In practice, eq 10, 11, and 12 as well as the UNIQUAC equation are programmed directly and are calculated automatically by computer. For temperatures outside the range T, = 1to T, = 2.5, some precautions are necessary since the equations would give unreasonable values for ri, qi,and Z. We recommend that for reduced temperatures outside the normal range the limiting values be used, i.e., 12 for T, < 1 and 6 for T, > 2.5. Applications to Binary and Multicomponent Systems To test the model developed here, data were reduced
Figure 4. Comparison between calculated and experimental molar excess energies and activity coefficienta for the aluminum-tin system at 973 K.
for 14 binary metal solutions. Systems showing both positive and negative deviations from Raoult's law were examined. The data were those reported by Hultgren et al. (1973). The estimated maximum errors in these data are fO.O1 to 0.06 for activity coefficients, f0.1 to 1.75 kcal/mol for excess Gibbs energies, and f0.10 to 2.00 kcal/mol for excess enthalpies. The optimum values of the adjustable parameters AU12 and AU,, were obtained by fitting the experimental activity-coefficient data. The parameter estimation routine was based on a suitable statistical criterion, in this case the maximum likelihood principle (Anderson et al., 1978). The parameters estimated with this method are those that minimize the sum of the squared difference between the calculated and experimental activity coefficients. Table I1 gives the estimated parameters for each of the binary systems examined here.
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Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
1 .o
1
HE GE
1 i
":I
Data reported by Hultsren et al(1973) 1.2
A Y J
T
. 0
-m E
I
Data reported b y Hultgren e1 al(1973)
A T
-
-
m
0
I
0.8
C a l c d w i t h UNIQUAC
-E
0.8
equation
a
x0
>a
0.4
xu
W
0.8
w 0 0
0.0
W
0.4
c1
w
-0.4
0.2
-0.8 0.0 4.0
'co 'C"
2 .o
0.0
0.0
0.2
0.4
0.8
0.8
1.0
Mole Fraction of Iron M o l e F r a c t i o n of Copper
Figure 5. Comparison between calculated and experimental molar excess energies and activity coefficients for the silver-copper system at 1423 K.
Figure 7. Comparison between calculated and experimental molar excess energies and activity coefficients for the cobalt-iron system at 1863 K.
Data reported b y Hultgren et a l ( 1 9 7 3 )
Data reported b y
0.0
-
1 .o
m
-
0
E I: a
u
e 0
Calcd with UNIQUAC equatlon
E
>
-2.0
m0 r
0.8
% .
-4.0
w
Y
X
0.8 L.
0
0
w
-6.0
W
0.4
F -8.0
0.2
-10.0 0.8
i\
t.0
'C d
'F e 0.4
Y 2.0
Pd
0.0
0.0 0.0
0.2
0.4
0.6
0.8
0.0
0.2
0.4
0.8
0.8
1.0
1.0 M o l e F r a c t l o n o f SlllcOn
Mole Fraction o f Lead
Figure 6. Comparison between calculated and experimental molar excess energies and activity coefficients for the cadmium-lead system at 773 K.
The calculated activity coefficients are compared with experimental data in Figures 4-11. Agreement between calculated and experimental values is very good for most systems regardless of any characteristic physical effects.
Figure 8. Comparison between calculated and experimental molar excess energies and activity coefficients for the iron-silicon system at 1873 K.
Systems are shown which exhibit both positive and negative deviations from ideal behavior. In general, systems which show positive deviations from ideal behavior are represented slightly better than those which show negative deviations. The systems aluminum-
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
I
Table 11. Interaction Parameters for Binary Metal Metal Solutions components (112) temp, K Al/Sn Co/Fe AWg Cu/Mg Al/cu Al/Si Al/Fe AdCu Cd/Pb Fe/Si &/AI Bi/Pb Mn/Pb Mn/Bi
AUlZ,
973 1863 1073 1100 1373 1100 1873 1423 773 1873 1243 1253 1253 1253
AU21,
kcal/mol
kcal/mol
18.525 80.667 13.810 -0.685 54.285 5.785 -38.437 22.422 2.140 -28.550 49.615 3.617 94.600 35.220
-8.507 -53.790 -21.715 -16.372 13.357 -17.285 -59.1 1 2 -9.930 0.540 -59.275 -49.682 -15.105 -30.447 --8.125
I
I
I
I
OHE, D a t a reported by Hultgran et a i ( l 9 7 3 )
2.0
-
O A GT E \
-
Caiod with UNIQUAC
-I
-8.0} I
I
I
181
8
Data reported by Hultgren et a i ( 1 9 7 3 )
0.0
'
-0.5
AI Fe
C a l c d n l t h UNIQUAC
-m 0
-1.0
E
0.0
2
0.2
0.4
0.8
0.8
1.0
Ip
.r
.
-1.5
Mole F r a c t i o n o f I r o n
-2.0
Figure 10. Comparison between calculated and experimental molar excess energies and activity coefficients for the aluminum-iron system at 1873 K.
w L
0
we,
D a t a reported by
-2.6
Huiteren et a i ( 1 9 7 3 )
Fitted
-3.0 0 .o
'cu
-0.5
I
I
0.0
0.2
0.4
0.6
0.6
'
0.0
-1.0
1.0
Mole F r a c t i o n o f M a g n e s i u m
Figure 9. Comparison between calculated and experimental molar excess energies and activity coefficients for the copper-magnesium system at 1000 K.
tin (Figure 4), silver-copper (Figure 5), and cadmium-lead (Figure 6) are all systems of this type. Iron-silicon (Figure 8), copper-magnesium (Figure 9), aluminum-iron (Figure lo), and aluminum4licon (Figure 11)are all mixtures which show negative deviations from ideal behavior. This phenomenon is principally due to the formation of intermetallic compounds, i.e., weak chemical bonds between unlike atoms. In spite of this rather complex behavior, the model proposed here represents these mixtures remarkably well. The cobalt-iron mixture (Figure 7) is especially interesting. This system shows slight positive deviations from Raoult's law, but negative values for the excess enthalpy. Again, the model is effective in representing this phenomenon. While the calculated activity coefficients compare favorably for nearly all systems, the calculate excess enthalpies tend to show greater deviations from the corresponding experimental values. This is not unexpected,
-1.5
-2.0
\ 0.0
0.2
0.4
0.8
0.8
0.0
1.0
Mole Fraction of Slllcon
Figure 11. Comparison between calculated and experimental molar excess energies and activity coefficients for the aluminum-silicon system at 1100 K.
however, since none of the excess enthalpy data were used in estimating the parameters. With this fact in mind, the agreement appears to be quite good. In all cases the correct qualitative behavior is predicted; in several systems the calculated values are predicted to within the experimental uncertainty reported for the data. Such agreement
r7T
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
182
6.0
5.0
port from The Ministry of Education, Republic of Korea.
Nomenclature
D a t a of Weinstein and E l l l o t t ( l P 6 3 )
Calcd with UNIQUAC equation
C
I
1
0.0
li = pure-component parameter related to z , ri, and qi n = number of component q; = molecular geometric-area parameter for pure component i r = atomic radius ri = molecular volume parameter for pure component i R = gas constant RWi= van der Waals atomic radius, 8, T = absolute temperature Tb, = boiling point for component i Tm,= melting point for camponent i T, = average melting point for multicomponent system T,, = reduced temperature for component i Au;.= UNIQUAC binary interaction parameter related to T~~ = UNIQUAC interaction parameter from auij x i = liquid phase mole fraction of component i WA8= atomic weight 2 = lattice coordination number
~d~ 1 0.0
0.1
0.2
0.3
0.4
0.5
0.6
Mcle Fraction o f Manganese
F i g u r e 12. Comparison between calculated a n d experimental act i v i t y coefficients o f manganese for t h e manganese-lead-bismuth system a t 1253 K.
is essential if the model is to be used to predict activity coefficients at temperatures either higher or lower than where the data are measured. The last example, Figure 12, compares predicted and experimental activity coefficients for manganese in the ternary system manganese-lead-bismuth. Two curves are shown which represent different mole ratios of lead to bismuth. In general, the predicted activity coefficients agree with the experimental values to within 10%. The important feature to note is that no additional adjustable parameters are required when the UNIQUAC equation is extended to multicomponent systems. This appears to indicate that reliable estimates can be made by phase equilibria in multicomponent metal systems using only two adjustable parameters per binary. Conclusions A shrinking hard-sphere theory is incorporated into the UNIQUAC model to provide temperature dependence for the structural parameters and the coordination number. The modified UNIQUAC model gives a good representation of metal-solution properties using only two adjustable parameters per binary. Its extension to multicomponent systems is straightforward and requires no higher-order parameters. The usefulness of the new model follows from its temperature dependence which is semitheoretically based and allows extrapolation of mixture data to higher temperatures. Acknowledgment The authors are grateful to the Computer Center, University of Connecticut, Storrs, for the use of its facilities. K.Y. acknowledges with thanks the fellowship sup-
Greek Letters B; = area fraction of component i in combinatorial contribution to the activity coefficient u = hard-sphere diameter, 8,
hard-sphere diameter at melting point density at melting point y, = activity coefficient ~ i = , UNIQUAC binary parameter & = segment fraction of component i uo =
pmi =
Literature Cited Abrams, D. S.;Prausnitz, J. M. AIChE J. 1975, 21, 116. Anderson, T. F.; Abrams, D. S.; Grens, E. A. AIChE J. 1978, 24, 20. Anderson, T. F.; Prausnitz, J. M. I n d . Eng. Chem. Process Des. Dev. 1978. 77,552. Ascarelli, C. P.; Paskln, A. Phys. Rev. 1967, 165, 222. Chandler, D.; Weeks, J. D. Phys. Rev. Left. 1970, 25, 149. Chandler, D. Acc. Chem. Res. 1974, 7 ,243. Dymond, J. H.; Alder, B. J. J. Chem. Phys. 1966, 4 5 , 2061. Eckert, C. A.; Smith, J. S.;Irwin, R. B.; Cox, K. R. AIChE J . 1982, 28 325. Enskog, D.; Svensk, K. Vet. Akad. Hand. 1921, 63, 4. Guggenheim, E. A. "Mixtures", Clarendon Press: London, 1952; Chapter 3. Hardy, H. K. Acta Metall. 1953, 7, 203. Hicter, P.; Mathleu, J. C.; Durand, F.; Bonnier, E. J. Chim. Phys. 1967, 6 4 , 523. Hildebrand, J. H. J. Am. Chem. Soc. 1929, 51,66. Hultgren, R.; Desai, P. D.; Hawkins, D. T.; Gleiser, M.; Kelly, K. K. "Selected Values of Thermodynarnlc Properties of Binary Alloys", American Society for Metals, 1973; Chapter 2. Lupis, C. H. P.; Elllott, J. F. Acta Metall. 1966, 74, 1019. Lupis, C. H. P.; Elliott, J. F. Acta Metall. 1967, 15, 265. Mathieu, J. C.; Durand, F.; Bonnier, E. J. Chim. Phys. 1965, 62, 1289. Mott, 0.W. Phil. Mag. 1957, 2 , 259. Paulaitis, M. E.; Eckert, C. A. AIChE J. 1982, 27, 418. Prausnitz, J. M.; Anderson, T. F.; Grens, E. A.; Eckert, C. A,; Hsieh, R.; 0'Connell, J. P. "Computer Calculations for Multlcomponent Vapor-Liquid and Liquid-Liquid Equilibria"; Prentice-Hail, Inc.: Englewocd Cllffs, NJ, 1980; Chapter 4. Protopapas, P.; Anderson, ti. C.; Parlee, A. D. J. Chem. Phys. 1973, 5 9 , 15. Protopapas, P.; Parlee, A. D. High Temp. Sci. 1975, 7 , 259. Weeks, J. D.; Chandler, D.: Anderson, H. C. J. Chem. Phys. 1971, 55, 5239. Weinstein, M.: Elliott, J. F. J. Nectrochem. Soc. 1963, 110, 792.
Receiued for review Accepted
D e c e m b e r 31, 1981 D e c e m b e r 16, 1982
