former definitions and thus provide a common basis for comparison with the predictions made for the same system crystallized by our method. The comparison of the two processes at a specified yield shows that our process provides higher thermal economy, smaller temperature gradients, higher temperatures in the refrigerator, and lower flow rates of coolant. However, three or four thermal units are operated in our process as compared t o only two thermal units in the other process. A general optimization analysis would provide a more definite estimation of the economy of each process. Conclusions
(1) The process of crystallization by direct contact cooling has been analyzed with regard t o the product yield and the thermal economy. ( 2 ) With regard t o conventional crystallizers, the described method has the advantage of continuous operation. (3) With regard to another direct contact cooled crystallizer, the described method shows the advantage of a higher economy and the disadvantage of operating additional thermal units. (4) The process is, in general, economically attractive for its energy economy, its simple structural design, and its continuous operation. Nomenclature
B C cB CK
E,
mass flow rate of feed solution (brine) mass flow rate of crystals = heat capacity of solution (brine) = heat capacity of coolant = yield of crystals, defined by eq 1
= =
E = Et = H, = K =
process economy, defined by eq 22 thermal economy, defined by eq 5 heat of crystallization mass flow rate of coolant s = mass flow rate of depleted solution tl = temperature of coolant a t outlet in crystallizer t2 = temperature of coolant at inlet in cooler 63 = temperature of coolant at inlet in crystallizer tBl = temperature of solution (brine) at inlet in crystallizer tB2 = temperature of suspension at outlet in crvstallizer tB3 = temperature of solution disposed from liquid-liquid heat exchanger At = temperature gradient between solution and coolant A ~ B = ( ~ B I- t ~ n )= total temperature difference of solution in crystallizer AtK = ( t 2 - t 3 ) = total temperature difference of coolant in cooler Y = yields of crystals per degree of temperature difference of the crystallizing solution literature Cited
Bamforth, A. W., “Industrial Crystallization,” Macmillan, New York, N. Y., 1966, p 57. Barak, A., Dagan, G., AZChE J., 16, 9 (1970). Cerny, J., British Patent 932,215 (1963). French, K. H. W., Znd. Chem., 39, 9 (1963). Gas Council, British Patent 931,154 (1963). Kehat, E., Letan, R., Znd. Eng. Chem., Process Des. Develop., 7, 385 (1968). Kehat, E., Letan, R., Brit. Chem. Eng., 14, 803 (1969). Letan, R., Kehat, E., AZChE J., 14, 398 (1968). Negev University, Dead Sea Works Ltd., Letan, R., Israeli Patent Application 39099 (1972). Zmora, Y., DSW Internal Report No. 573 (1970). Zmora, Y., M. S.Thesis, Negev University, in preparation, 1972. Zmora, Y., Letan R., Proceedings 42nd Annual Meeting of Israel Chemical Society, Rehovot, 1972. RECEIVED for review August 3, 1972 ACCEPTED March 26, 1973
Thermodynamic Properties of Quadrupolar Mixtures in the Gaseous and Liquid Regions V. Ramaiah and Leonard 1. Stiel*’ Cniversity of Jlissouri, Columbia,Missouri 65201
Procedures recently have been developed for the calculation of the thermodynamic properties of nonpolar mixtures from the critical constants and acentric factors of the components (Eisenman and Stiel, 1971; Ramaiah and Stiel, 1972). It was found that these methods are not completely applicable for mixtures containing quadrupolar components such as carbon dioxide, nitrogen, and ethylene. Since components of this type are present in mixtures of practical interest, the previous approach has been extended in this study to quadrupolar systems. I n addition, this study provides insight into the treatment of more complex molecular interactions. The Second Virial Coefficient of Quadrupolar Mixtures
second virial coefficient can be expressed as follows Bij ~
011 ..a
-[
- 2nN0 - 3 Fa(Zij)
+ 6 (pa::)
F,(Zij)
+
+
+
where Zij = dEiCj/KT,aij = ( a i aj)/2, poij = (poi p0j)/2, and e , a, and po are the parameters of the pure components (i = j). Tee, et al. (1966), obtained spherical core parameters for a number of nonpolar substances and developed the following dimensionless relationships
The Kihara spherical core relationship for the interaction Present address, Allied Chemical Corporation, Specialty Chemicals Division, Box 1069, Buffalo, N. Y. 14240.
_ e - 1.0042 + 3.0454~
(2)
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
305,
xTe
G)
2.5412 = 1.1501
a
- = 0.0669 PO
- 0.9638~
+ 2.3724~
+ 1.0567~
(3) (4)
Eisenman and Stiel (1971) showed that eq 1-4 enabled the calculation of interaction virial coefficients from the critical constants and acentric factors of the components for nonpolar mixtures, including systems containing dissimilar components. However, for some mixtures containing carbon dioxide, larger errors resulted from the use of these relationships. For example, for ethane-carbon dioxide (Zaalishvili, 1956) a standard error of estimate of 21.8 cc/mol for seven points resulted from this procedure. Most pure quadrupolar substances such as carbon dioxide follow normal fluid behavior, so that the conclusion was reached that the behavior of a quadrupolar mixture is more sensitive to the exact type of molecular interaction than the properties of the pure components. The intermolecular potential function for a quadrupolar fluid can be expressed as a superposition of the spherical core potential with terms for induced quadrupole and quadrupolequadrupole interactions, as follows 3aQ2 p8
7 Q‘ (5) 40 K T ~ ”
where (Y is the polarizability and Q the quadrupole moment. Alternatively, eq 5 can be approximated by a potential of the spherical core form, by assuming p8 N p6p02 and p l 0 N p6pO4, as follows $(p) =
€‘
[
-2
($)12
($7
where
and
The use of the potential of eq 6 is more accurate a t high temperatures. Tee, et al. (1966), found that second virial coefficients for pure carbon dioxide, ethylene, and nitrogen could be calculated to within the accuracy of the experimental data by the use of the Kihara spherical core relationship and temperature-independent parameters e ’ / ~ , po’, and alpo‘ calculated from eq 2-4 and the critical constants and acentric factors of the components. Larger errors were obtained for the second virial coefficient of benzene by this procedure, which were attributed to inaccuracies in the experimental data. Nonpolar-Quadrupolar Mixtures
For a nonpolar-quadrupolar spherical core potential is ddP) =
where
€12*
[(--)
POl2*
mixture, the approximate l2
-2
components, and are presently not available for quadrupolar fluids for the model of eq 6. Equations 10 and 11 can be expressed alternately as
($71
(9)
-
- POI
+
PO2*
2
where e ~ *and p0z* are the parameters of the quadrupolar component which characterize shape and induction effects. Initially, experimental interaction virial coefficient data for mixtures of carbon dioxide with argon (Menon, 1965), methane and ethane (Zaalishvili, 1956), and propane and nbutane (Huff and Reed, 1963) were utilized to determine by a non-linear least-squares procedure the parameters e12*, polz*, and a12/~012* which result in the minimum deviations between the experimental values and those calculated from the spherical core relationship (eq 1 written in terms of the starred variables). The search procedure also was utilized to determine optimum values of the parameters of the more accurate potential function in place of eq 9 in which the induction term is treated as a separate term varying as pa. Although slight improvement in the deviations resulted from this procedure, best results were obtained when the quadrupolar moment was treated as an adjustable parameter. The optimum values were quite similar to those obtained for the simpler potential which was therefore subsequently employed. For most of the mixtures, it was found that the optimum values of po12* and a12/p012*, were close to those obtained from arithmetic averages of the parameters a l , UZ, pol, and p02’ calculated from eq 3 and 4 and the critical constants and acentric factors of the components. However, the optimum values of e12* were in general considerably lower than those obtained as a geometric mean of the parameters calculated from eq 2 and the critical temperatures and acentric factors of the components. It can be seen from eq 8 and 13 that po2* N po2‘, and therefore po12* N (pol pO2’)/2. Similarly, it can be seen from eq 7 and 12 that since quadrupole-quadrupole interactions are present in the pure fluid that are absent in the mixtures, €2’ is larger than the parameter e2* needed to calculate e12*. Since e2* accounts for only dispersion and induced-quadrupole interactions, it should be possible to calculate this parameter from eq 2 by the use of an appropriate acentric factor which is lower than the actual w of the substance. I n order to estimate these modified acentric factors, the results of the study of Danon and Pitzer (1962) were utilized. For quadrupolar fluids, Danon and Pitzer separated shape and quadrupolar contributions to the acentric factor by comparing the macroscopic equation for the reduced second virial coefficient (Pitzer and Curl, 1957) with the equation for the LennardJones potential combined with linear corrections for shape and quadrupolar effects. They obtained the relationship
+
,
7~
3.2Q2 + 0.24 = 1.5-P1O + -
(14)
€Po6
or POlZ*
dG,
=
POlZ[l
+
51211-1’6
+
(11)
and €12 = p012 = (pol p02)/2. The parameters e and PO are those characterizing dispersion effects for the pure 306
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
W
=
Wh
+
WQ
(15)
where W h = 0.2l(l/po) - 0.034, and 1 is the length of a thin rod core. By this procedure W h includes the effects of dispersion
Procedures have been developed for the calculation of thermodynamic properties of quadrupolar mixtures in the gaseous and liquid regions. Relationships were initially obtained for the interaction virial coefficients, BIZ, of binary mixtures containing carbon dioxide, nitrogen, ethylene, inert gases, and normal alkanes through , WM. For n-butane. The approach was then extended to the calculation of effective parameters T c ~ P, c ~ and quadrupolar-quadrupolar mixtures the procedure i s the same as that utilized previously for nonpolar mixtures. For nonpolalcquadrupolar mixtures, the interaction parameters have to b e modified to exclude the contributions of quadrupole-quadrupole interactions. Relationships are presented for the calculation of the modified parameters from the critical constants and vapor pressure data for the quadrupolar components. Experimental compressibility factors in the gaseous and liquid regions were compared with values calculated with the effective parameters for mixtures of nonpolar substances with carbon dioxide, nitrogen, ethylene, and benzene. The results compare favorably with those for other methods, and indicate that the approach of this study can be utilized to calculate interaction virial coefficients and thermodynamc properties of quadrupolar mixtures from only data for the pure components.
and induced quadrupole interactions. From vapor-pressure data for nitrogen and the parameters for this substance presented by Danon and Pitzer (Wh = 0.013, W Q = 0.027), the following additional equation was deduced for quadrupolar fluids log
= -1.552
- 1 . 7 0 ~ h- 1 . 3 0 ~ ~ (16)
Values of e2* were calculated from eq 2 and T,and W h for nitrogen, carbon dioxide (Wh = 0.074, W Q = 0.151), and ethylene (Wh = 0.059, W Q = 0.036). From the value of a*, pot* can be calculated from the relationship PO* = po'[l
+
5g]l/6
= po'(e'/c*)'/'2
(17)
For mixtures of carbon dioxide, nitrogen, and ethylene with inert gases, methane, ethane, propane, and n-butane, interaction virial coefficients were calculated from the spherical core expression with e1z* = pOl2* = (pol p 0 2 * ) / 2 , and a12 = (a1 az)/Z. For the nonpolar component, the parameters were calculated from eq 2 4 . The parameter a2 was calculated from eq 4 from p02' and the actual acentric factor of the quadrupolar component. The references for the data, temperature ranges, number of points, and standard errors between calculated and experimental values are included in
+
42,
+
Table I for carbon dioxide mixtures and in Table I1 for nitrogen and ethylene mixtures. The sign of the average deviation (experimental - calculated) is also included for each reference. It can be seen t h a t in general good results were obtained for all the mixtures considered. For carbon dioxide and nitrogen mixtures, the optimum values of EZ* were also determined for the same values of the other parameters. For each substance, it was found t h a t the average of the corresponding optimum values of W h was close t o the value utilized. The optimum value of W h was slightly higher for COZ-argon (0.0924) and lower for COz-n-butane. As shown in Table I, the use of the optimum W h for COz-argon results in substantial improvement for the low-temperature data (Brewer, 1967) for this system. The negative deviations for inert gas mixtures a t low temperatures would also be corrected by the use of the more complete potential function in which the induction term is treated separately. Since €121 in eq 12 increases with increasing w of the nonpolar component, e2* should be larger for quadrupolar mixtures with n-butane than for those containing argon. Therefore, the positive deviations obtained for some of the data for propane and n-butane mixtures are possibly due to errors in the experimental virial coefficients.
Table 1. Comparison of Interaction Second Virial Coeff icients for Carbon Dioxide Mixtures No. of Standard Mixture
Reference
Temperature range,
O K
points
COrAr
Menon (1965)
303.2-363.2
5
COrAr
Brewer (1967)
223.2-273.2
3
COrKr COrXe COrCH4
Brewer (1967) Brewer (1967) Mason and Eakin (1961)
223.2-273.2 223.2-273.2 288.9
2 2 1
COzCHa
Zaalishvili (1956)
310.9-510.8
7
COzCH4
Brewer (1967)
273.2
1
COzCzHs Zaalishvili (1956) CO~C~HB Mason and Eakin (1961) C02-C3Hs Mason and Eakin (1961) Huff and Reed (1963) COrCi" C02-C4Hio Mason and Eakin (1961) C02-C4Hio Huff and Reed (1963) Standard error of estimate = [ X I N (Bexptl- BO,1,d)*/N]l'*. for methane.
310.9-510.8 7 288.9 1 288.9 1 6 310.9-510.9 288.9 1 377.6-477.6 4 * Calculated with optimum parameter, W h
estimate:
=
error of cc/moi
2 . 1 2 (-) 1.07b (*) 7 . 3 4 (-) 5.0gb(-) 12.86 (-) 10.85 (-) 5 . 8 7 (-) 1.26b (+) 9 . 6 4 (-) 1.77b (-) 12.54 (-) 4.756 (-) 6 . 6 1 (-) 3 . 8 9 (-) 6 . 9 6 (-) 4.21 (i) 5 . 5 7 (+) 4.70 (+I 0.092 for argon and 0.122
Ind. Eng. Chern. Process Des. Develop., Vol. 12,
No. 3, 1973 307
Table 11. Comparison of Interaction Second Virial Coefficients for Nitrogen and Ethylene Mixtures Mixture
Reference
Knobler, et al. (1959) Brewer (1967) Magasanik and Ellington (1963) Brewer (1967) Brewer (1967) Mason and Eakin (1961) Brewer (1967) Mason and Eakin (1961) Huff and Reed (1963) Mason and Eakin (1961) Brewer (1967) Mason and Eakin (1961) Huff and Reed (1963) Mason and Eakin (1961) Lee and Edmister (1969)
Nz-A K'z-A NTA KrKr N2-Xe NrCH4 x2-CH4 ?;2-CZH6 Nz-C& Nz-C~HS N r CsHs ;\;Z-C4Hlo NT CiHio C2H4-CH4 C2H4-CH4
No. of points
Standard error, cc/rnol
123.2-323.2 323.4-398.2
1 9 3
4 . 7 3 (+) 4 . 0 5 (-) 1 . 2 0 (-)
148.2-323.2 173.2-323.2 288.9 273.2 288.9 277.6-510.9 288.9 248.2-298.2 288.9 427.6-477.6 288.9 268.8-348.2
6 4 1 1 1 5 1 3 1 4 1 6
2 . 5 2 (-) 7 . 7 2 (+I 3 . 1 6 (+) 0 . 1 4 (-) 6 . 5 1 (-) 4 . 7 3 (-) 5 . 9 8 (-) 17.71 (+) 14.37 (+) 14.14 (+) 4 . 5 7 (+I 4 . 8 3 (+)
Temperature range,
OK
90
Table 111. Comparison of Interaction Second Virial Coefficients for Quadrupolar-Quadrupolar Mixtures Mixture
Reference
COz-Nz COzN2 C0z-Nz
Mason and Eakin (1961) Markham and Kobe (1941) Edwards and Roseveare (1942) Pfefferle, et al. (1955) Cottrell, et al. (1956) Brewer (1967) Edwards and Roseveare (1942) Ku and Dodge (1967) Sass, et al. (1967) Edwards and Roseveare (1942)
COz-Nz COz-N2 C0z-Nz COrCzH4
COz-CzH4 COzCzH4 NTCZH~
Quadrupolar-Quadrupolar Mixtures
For quadrupolar-quadrupolar mixtures, the intermolecular potential function is of the form of eq 6. The relationships for the parameters are
and Po12
, - Pol'
+
POZ'
2
where E12Q = 7&1z4/80~1z~Tpo1210 and &I2 = dQ1&2. Thus for these mixtures s12', polz', and a12can be calculated from the parameters obtained from eq 2-4 and the critical constants and actual acentric factors of the substances. Interaction virial coefficients were calculated in this manner for mixtures of carbon dioxide with nitrogen and ethylene and the nitrogenethylene system. The results are presented in Table 111. I n general, good results were obtained for most of the data. Somewhat larger errors were obtained for carbon dioxideethylene, but the results for two references (Ku and Dodge, 1967; Sass, et al., 1967) deviate in opposite directions from the calculated values. For carbon dioxide-methane, negative deviations resulted between the experimental and calculated values for all the data, and the optimum value of W h for this system was found to be 0.122. Methane is not quadrupolar but does have a substantial octupole moment (Datta and Singh, 1971). It can 308
Ind. Eng. Chern. Process Des. Dovelop., Vol. 12,
No. 3, 1973
No. of points
Standard error, cc/rnol
288.9 298.2 298.2
1 1 1
2 . 4 4 (+I 1 . 8 8 (+) 5.80 (-)
303.2 303,2-363.2 223.2-273.2 298.2
1 3 3 1
0 . 7 2 (-) 4 . 7 0 (-) 4 . 6 9 (+I 6 . 5 0 (+)
373,2 348.2-398.2 298.2
1 3 1
13.02 (-) 11.20 (+) 10.61 (-)
Temperature range,
O K
be seen from eq 12 and 18 that the optimum c2* for this system should be higher than that calculated from W h = 0.074 and less than e2'. As shown in Table I, the use of W h = 0.122 resulted in substantial improvement for all the data for this system. For the data a t higher temperatures for this system, the use of eq 18 and 19 and the parameters from the actual acentric factors of the components also resulted in substantial improvement. For mixtures of methane with nitrogen and ethylene, no change in the procedure utilized for nonpolar-quadrupolar systems was found to be necessary. Compressibility Factor of Quadrupolar Mixtures
For the calculation of effective parameters of nonpolar mixtures, Ramaiah and Stiel (1972) approximated eq 1 as -Bij =
ff0
..a
(2) + f f l (s) Zij1.4 + Po i j
0 11
Poij
where cyo
=
[
0.4915
+ 4.3643 (") + 7.6981 Po i j
21.4812 =
[
("")
(2)' f
(Z)']
-2.7516 - 6.4574 -2 - 10.8186 Po i j
(")'I
Po ij
loz4 (21)
1024
Table IV. Comparisons of Experimental and Calculated Compressibility Factors
System
Temperature range, O K
Reference
Pressure range, atm
No. of paints
50-600
64
311.1-511.1
13.6-476.2
72
Reamer, et al. (1945)
311.1-511.1
13.6-476.2
92
CO?-C3Hs
Reamer, el al. (1951)
277.7-511.1
13.6-408.2
89
COrCaHio
Olds, et al. (1949)
311.1-511.1
13.6-476.2
232
NzCH4 (gas) N r C H 4 (liq) NrCH4 Nz-C2He
Bloomer and Parent (1953) Bloomer and Parent (1953) Keyes and Burks (1928) Reamer, et al. (1952)
125.7-177.9 114.0-135.6 273.2-473.2 277.7-511.1
16.5-43.8 15.0-38.2 28.9-327.4 13.6-340.1
31 33 110 132
399.5-422.0 N2-CsHs Watson, et al. (1954) 427.4-477.4 NrC4Hio Evans and Watson (1956) 298,2-398.2 Nz-COz Haney and Bliss (1944) NzC02 Kritschewsky and Markov (1940) 273,2-473.2 CsHd-Ar Masson and Dolley (1923) 298,l CzHrCH4 Lee and Edmister (1969) 298.2-348.2 271 .9-526.4 C~H~-TL-C,HI~ Kay (1948) 323.2 C2H4-N2 Hagenbach and Comings (1953) 313.2-398.2 C2H4-CO2 Ku and Dodge (1967), Sass, et al. (1.967) 278.9-474,6 CA-CzHe Kay and Xevens (1952) 444.4-511.1 CJfe-CaHs Glanville, et al. (1950) a Calculated with optimum Wh. Calculated by method of Pitaer and Hultgren.
17.0-340.1 13.6-272.1 30-450 50-500 30-125 14.1-405.3 10.2-102.0 9.7-374.2 12.4-507.0 20.4-91.8 13.6-408.2
57 172 130 40 100 61 102 103 221 99 70
COTAr
Abraham and Bennett (1960)
COrCH4
Reamer, et al. (1944)
COrCzHe
a2
=
[
-0.0669 - 0.5157
ff3
= -0.0012
323.2
(23)
-
(24)
1024
(Pad:j)"
Equations 20-24 reproduce the virial coefficients calculated from eq 1 with an average deviation of less than 2 cc/mol for practical ranges of Z i j (approximately 250-600°K for most mixtures). If eq 20 is substituted into both sides of the theoretical expression for the composition dependence of the second virial coefficient, and like powers of Z are equated, four equations in the three unknowns ( a / p o ) h f , (€/K)>f, and p 0 h 1 result (Ramaiah and Stiel, 1972). The last two of these equations can be added to obtain the following relationships
(%)
P~M~CYO
PoM
poMa
[
a2
ZiZj poij3ao
i
(%)
(25)
zM7.s] =
zM4.0
PoM
j
j
(s) + a3(s) ZiZj~oij'
i
=
PoM
[ (%) CY^
+ a3 (5)
Zij4.O
Poij
Zij7.81
(27)
Poij
where ZM = ( ~ / K ) M /Equations T. 20-27 are also applicable for quadrupolar mixtures, with the interaction 1)arameters s i j and p O l j determined by the use of eq 12 and 13 for nonpolar-quadrupolar mixtures and eq 18 and 19 for quadrupolar-quadrupolar mixtures. The parameters of the components are calculated from eq 2-4 and the critical constants and acentric factors (or Wh for ez*) of the substances. Equations 25-27 are solved for U M , p o ~ and , ( c / K ) ~ I , and W M is then cal-
Average Average % % error error (this (Barner and study) Quinlan)
4.35 (3.40)a 3.00 ( 0 .59)b (1.84). 1.94 0.79 (0.61)* 0.97 1.79 ( 1 . 14)b 1.20 1.53 (1.75)b 1.82 1.77 3.61 3.60 0.21 0.23 1.51 0.53 (0.54)b 3.44 1.90 3.00 3.96 6.38 2.24 1.05 1.14 5.36 5.05 0.56 0.87 2.03 4.68 3.95 0.89 0.79
culated from eq 4 and UM/P,,M. The value of Tohl is then calculated from eq 2 and ( ~ / K ) Mand OM, and Pox is then obtained from eq 2 and T c and ~ WM. The calculated effective parameters are somewhat dependent on temperature (Ramaiah and Stiel, 1972). Values of the effective parameters to^, Po51, and W h f were calculated in this manner for a number of nonpolar-quadrupolar and quadrupolar-quadrupolar mixtures. Compressibility factors were calculated for these mixtures by the use of the generalized correlation of Pitzer, et al. (1955), and the effective parameters, and were compared with the corresponding experimental values. The mixtures are listed in Table I V along with the references for the data, numbers of points, temperature and pressure ranges, and average deviations between the calculated and experimental values. It can be seen that good results are obtained by this procedure for most of the mixtures considered. For COz-argon, some improvement in the error was obtained by the use of the optimum Wh, as shown in Table IV. For COz-methane, considerable improvement was obtained with Wh = 0.122 in place of 0.074, but even better results (0.98% for the 72 points) were obtained for this system by the use of eq 18 and 19 and w = 0.225 for carbon dioxide. For nitrogen-propane and nitrogenbutane, higher errors were obtained with the lower values of Wh obtained from the data for Biz for these systems, indicating that these virial coefficients are somewhat in error. The larger error for Con-nitrogen mixtures may be partly due to the fact that a t high pressures the linear correlations for the cornpressibility factor of Pitzer, et al. (1955), are not completely applicable (Ramaiah and Stiel, 1972). I n Figure 1, esperimental compressibility factors for C02-propane (Reamer, et al., 1951) in the gaseous region a t temperatures of 377.8 and 511.1°K and a mole fraction of COz of 0.5884 are compared with the calculated values. Ind. Eng. Chem. Procerr Der. Develop., Vol. 12, No. 3, 1973
309
Table V. Comparison of Effective Macroscopic Parameters for Equimolal Mixtures at 400'K Thir study
TOM
Syrtem
COrCH4
Barner and Quinlan
POM
W bf
COrC2He
231.8 (237.6)" 285.7
53.8 (55.4)" 54.3
0.20 (0.19)Q 0.16
COrCaHa
321.6
50.2
0.18
COrCiHio
355.2
46.7
0.23
N2-CH4 NrCzHe
156.3 219.4
39.4 42.7
0.03 0.16
N~-C~HIO 299.7 38.8 0.32 C2HcCH4 234.2 47.6 0.09 C~H~-~-CTHI~ 433.7 34.2 0.39 43.6 0.14 C2HrNz 208.2 0 Calculated with O h = 0.122. * Determined by Pitzer and Hultgren. cop
-c+, Cop
- 0.5884 MOLE METHOD
FRACTION
OF THIS STUDY
..
0
300
2 00
100
PRESSURE
,
Pobl
o r . !
(242)
(57. 6)b
(0.15)b
292.6 (292)b 325.3 (322)b 351.9 (352)O 156.1 218.0 (213)b 293.6 238.1 438.5 205.7
55.6 (55.3)O 51.2 (49.5)b 47.1 (46. l ) b 39.0 43.3 (41 4)b 41 .O
0.17 (0.10)b 0.19 ( 0 .14)b 0.21 (0.20)b 0.03 0.07 (0.05)b 0.12 0.04 0.21
50.4 37.9 45.4
0.06
culated from constants obtained from interaction virial coefficient data, P,Mis obtained as z C ~ R T c ~ and / V Zc ~~ Mand T I c are ~ assumed to be linearly related in the mole fractions to the corresponding parameters of the components. For both methods, a linear relationship in composition is assumed for WM. It can be seen from Table IV that very good results for the compressibility factors are obtained by these procedures for systems for which the necessary binary interaction data are available. For nonpolar mixtures, Ramaiah and Stiel (1972) found that the major difference between the effective parameters calculated by the indicated procedure and those obtained by the other methods was that WM was larger than that calculated from the linear relationship in composition. For systems containing dissimilar components, W M was found to have a maximum a t an intermediate composition. I n Table V, values of T c ~P ,c ~and , WM a t 400°K and an equimolar composition are compared with the effective parameters for several systems calculated by the relationships of Pitzer and Hultgren (1958) and Barner and Quinlan (1969). It can be seen that in general the values of W M calculated by the approach of this study are larger than those for the other methods, and in some cases, WM is larger than the acentric factors of the components.
- EXPERIMENTAL 0
TOM
400
4TM
Discussion
Figure 1 . Comparison between experimental and calculated compressibility factors for carbon dioxide-propane
Compressibility factors were also calculated for mixtures of benzene (Wh = 0.209, uQ = 0.006) with propane and n-butane, and compared with the corresponding experimental values. The results for these systems included in Table IV indicate that the approach of this study can also be utilized for mixtures containing benzene. Calculations of the compressibility factor were also performed with effective parameters obtained by the methods of Barner and Quinlan (1969) and Pitzer and Hultgren (1958). In both of these methods, a quadratic relationship is assumed for T c ~
T ~= M
i
Cj X i Z j T c i j
(28)
I n the method of Pitzer and Hultgren (1958), a quadratic relationship is also assumed for P c ~and , Tcij and Pcij are obtained from experimental compressibility factor data. For the method of Barner and Quinlan (1969), Tcij is cal310 Ind.
Eng. Chem. Procear Der. Develop., Vol. 12, No. 3, 1973
The results of this study indicate that the methods of Eisenman and Stiel (1971) for the calculation of interaction virial coefficients and of Ramaiah and Stiel (1972) for the effective parameters of nonpolar mixtures can be extended to systems containing quadrupolar components. For quadrupolar-quadrupolar mixtures the procedure is the same as that for nonpolar mixtures, and the interaction parameters are calculated from the critical constants and acentric factors of the components by the use of the Tee, Gotoh, and Stewart relationships for molecular parameters and the combining rules for the spherical core potential. For nonpolar-quadrupolar mixtures, the major difference in the procedure is that the interaction parameter elz* is calculated by the use of a parameter e2* for the quadrupolar component which is less than the actual parameter €2' of the substance. The value of e2* is calculated from eq 2 and a modified acentric factor which can be obtained from eq 15 and 16 and experimental vapor-pressure data, This procedure effectively requires one vapor-pressure point in addition to the input data required for nonpolar mixtures.
The approach for nonpolar mixtures could also be extended for quadrupolar systems by the use of the more complete potential function, eq 5 . This procedure would require one additional macroscopic parameter for quadrupolar substances, such as wQI which can also be estimated from a n additional vapor-pressure point. Although slightly improved results could be obtained by this procedure, the results of this study indicate that the properties of quadrupolar mixtures can be represented by relationships based on the spherical core potential, with pure component and interaction parameters which are independent of temperature. More attention t o the exact details of the interactions present are required for quadrupolar mixtures. The sensitivity of the behavior of mixtures t o the exact types of interaction is illustrated by the Con-methane system. Methane is generally considered t o be nonpolar but does have a n octupole moment, the effect of which could be detected on the properties of this mixture. The methods of Pitzer and Hultgren (1958) and Barner and Quinlan (1969), which are also based on a three-parameter model analogous t o the spherical core potential, also enable the accurate calculation of compressibility factors for quadrupolar mixtures. However, these procedures cannot be utilized in general for the calculation of interaction virial coefficients B12, and these approaches require experimental interaction data for each system considered. For the approach of this study only the critical constants and vapor-pressure data for the components are required. Quadrupolar substances probably represent the limit of the complexity of the components for which temperatureindependent pure component and interaction parameters can be utilized t o obtain the effective parameters, and for which the thermodynamic properties can be calculated from generalized correlations based on the acentric factor. For polar mixtures the relationships for the effective parameters and thermodynamic properties can be extended by the inclusion of a fourth macroscopic parameter. Acknowledgment
The authors are grateful to Edith James for help in punching the data cards. Nomenclature
radius of a spherical core B second virial coefficient, cm*/mol F,(Z) = (-s/12)2,,0” Z(3’2fs’12) r [ ( 6 j - s)/12], foreq 1 1 = length of thin rod core N o = Avogadro’s number P, = critical pressure, a t m P R ~= reduced vapor pressure Q = quadrupole moment, esu em2 R = gas constant, 1.987 cal/mol O K T = temperature, OK T, = critical temperature, OK T R = reduced temperature, T / T , V , = critical volume, cm3/mol z = molefraction Z = E/KT zc = critical compressibility factor, P,V,/RT, a
= =
GREEKLETTEFS a = polarizability E = negative of minimum energy of attraction, erg E‘ = energy parameter for quadrupolar substance, calculated from eq 2, including shape, induction, and quadrupole interactions e* = energy parameter for quadrupolar substance, calculated from eq 2 and W h , including shape and induced quadrupole interactions
Boltzmann constant, 1.3805 X
=
K
erg/OK
= 3/4[(d&’ -I- ~ Z & I ’ ) / E I Z P O I Z ~ ] = 7/80 (&12&Z)/E12KTP01210)
5121 flZQ
shortest distance between molecular cores shortest distance between molecular cores when the potential energy is minimum p0’ = distance parameter calculated from eq 3 po* = distance parameter calculated from eq 17 w = acent,ric factor, -log P R ~ ~ ~-~1.0 = o . ~ Wh = shape group, 0.21 l / p o - 0.034 WQ = quadrupole group, 0.46 ( Q 2 / e p O 5 ) p = po =
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Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
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