- Email: [email protected]
Comparison between a noncubic equation of state particularly suitable for polar fluids and the Soave equation of state
Shorter Communications
3096 Table 2. Explicit expressions for d,, Figure
and the required conditions for their use (C = 8 and r = 0.747 for the Rushton impeller) Required condition
Expression for d,.. d-/D { 1 + 0.209[
= (1 /C Re.)J’4 x
= (l/C
5’ We/(rC3’4Rez’4)
N R R R; Rer t t .%+.I, T We
11
number of cycIes in a sequence of deformation and relaxation ~s~~~ds~d&t$p.s.) value of R corresponding to d,, drop Reynolds number (= p$N’/& tank Reynolds number (= p,DNZ/pc) time, s eddy life time, s tank diameter, m Weber number (- p,NzD3/o)
Greek letters ratio of impeller diameter to tank diameter c (= BIT) dimensionless time ;I dimensionless deformation of drop deformation state at the beginning of the defor6*(x) mation cycle for the nth cycle deformation state at the end of the deformation @,(a) zone or at the beginning of the relaxation zone for the nth cycle dispersed-phase viscosity, Pas Fd density of continuous phase, kg/m3 P, a interfacial tension, N/m * defined as the product of volume fraction of relaxation zone and circulation group number for the vessel. For the Rushton impeller its value is 0.747.
Chemical Engimrfng Science. Printed in Great Britain.
Comparison
+C
Re,+)3/4 x 2.735
i
Q 0.1351 X
We/c’ > 605 +
-0.1485
Subscripts
f
We/l5
(r2CRe,/~4)‘.‘542
We,(CL”Re~~~)~1’os27~‘s’33
d-/D
n
.5’2c
2.686(js’CRe,/c4
-
1.341)‘.*’
final state initial state
REFERENCES Arai, K., Konno, M., Matunga, Y. and Saito, S., 1977, Effect of dispersed phase viscosity on the maximum stable drop size for break-up in turbulent flow. J. &em. Engng Japan 10, 325-330. Calabrcse, R. V., Chang, T. P. K. and Dan& P. T., 1986, Drop breakup in turbulent stirred-tank contactors; Part I: effect of dispersed phase viscosity. A.Z.Ch.E. J. 32, 657-666. Davies, J. T.. 1985, Drop sizes of emulsions related to turbulent energy dissipation rates. Chem. Engng Sci. 40, 839-842. Hinxe, O., 1955, Fundamentals of hydrodynamic mechanisms of splitting in dispersion processes. A.1.Ch.E. J. 1, 289-295. Holmes, D. B., Voncken, R. M. and Dekker, J. A., 1964, Fluid flow in turbine-stirred, baffled tanks--I. Circulation time. Chem. Engng Sci. 19, 201-208. Kumar, S., Kumar. R. and Gandhi, K. S., 1992, A multi-stage model for drop breakage in stirred vessels. Cheat. Engng Sci. 47, 97 l-980. Lagisetty, J. S., Das, P. K., Kumar, R. and Gandhi, K. S., 1986, Breakage of viscous and non-Newtonian drops in stirred dispersions. Chem. Engng Sci. 41, 65-72.
Vol. 48, No. 1% pp. 3O963100.1993.
OLW-2509f93 S6.00 + 0.00 Rrgamon Press Ltd
between a noncubic equation of state particularly suitable for polar fluids and the Soave equation of state (Received
19 January
1993; accepted
INTRODWCTiON Recently, Brandani et al. (1992) have presented a new equation of state based on theoretical results of a system of hard spheres with dipoles and on the computer simulation data
for publication
11 March
1993)
for n-butane. This new equation of state was applied to the correlation of thermodynamic properties for 10 pure polar and nonpolar fluids. In this communication, the new equation of state has been testcd using a data base constituted by the vapor pressures of
Shorter Communications 59 pure polar and nonpolar fluids. The aim of this work was
= 3.10026 + O.O24838fi, - 0.027299/1,2 RT,
to find a correlation for the two adjustable parameters of the
equation as a function of the acentric factor. The results were compared with those obtained using the equation of state of Soave (1972), which is widely used by chemical engineers for VLE calculations, even for polar substances (Baztia, 1983; Joffe et ol., 1983). THE EQUATION OF STATE
The compressibility factor I is the sum of two contributions: one from the reference system (t”‘) and one which takes into account the effect of dispersion forces (z”‘): r = -&
= r=f + Z.“.
+/t**+
-f%$
(1 - v)’
P-m
(5)
I’]’ = 1 - 1.5334/Z* + O.l2993j?
(6)
f’2’ = 1 + 4.1339fl* + 0.10764@
(7)
f13’ = 1 + 1.6024fi& + 0.73728fi6.
(8)
For the term in eq. (1) which takes into account the effect of dispersion forces, we choose the expression proposed by Dohrn and Prausnitz (1990): (9)
The universal parameters k, and kt were determined by fitting the computer simulation data of n-butane taken from and Donohue (1989) and their best Fig. 4 of Vim&hand values were found to be k, = - 1.43279 and k2 = 3.97055. For the dependence of the parameters E and b on temperature the following expressi\ons were chosen: : = z[l
+ a(1 - co.‘)]*
b = b,[l + /l(l - Z-$‘)]’
c
= 0.16552 - 0.0015668&
(15)
from which pm, _ pL.pY(AY - A’-)
(16)
rc
mz- 1
I
-dq orl
= CI In(1 - q) + CI~ l--9
+ C r112- tl) - $+ 3(1-
+ tklrt + +k&)
(17)
and c, = - (1 --fIl’) C* =
f -p-
cs = $(l i”fi”
+
(18) 2fI3J
+_f’2’ -f”‘).
(19) (20)
The two adjustable parameters of the equation of state were determined minimizing the objective function:
where Pz\=;.is calculated from eq. (16) using the equation state and eq. (17).
of
RESULTS AND DISCUSSION The experimental vapor pressures were taken from Boublik et al. (1984) for the 59 compounds which constitute the data base. Using OT and B as adjustable parameters the fit gives a mean absolute percent deviation of 0.92 which is a good result if compared with those which can be obtained from a simple linear interpolation of the function In Pm’vs l/T. Moreover, while for dothere is a correlation trend with w, fi does not show a systematic pattern. Therefore, we assumed for /l the average vaiue of 1.4986 and we proceeded to a new fit using e as the only adjustable parameter. Figure 1 shows this parameter as a function of OL with o, the acentric factor taken from Reid et al. (1987). The straight line is described by the following equation: Q = - 0.23549 + 0.882260.
(11)
Water and 1-hexanol deviate from this parameter and for these substancn and higher alcohols we suggest the following equation for a:
+ O.O048321ji,z
- O.OCNM4518fi: - 0.0035999/i:
A= + Py’vL = AY + P”‘vy
(10)
where OL and pare the two adjustable parameters of equation of state, T, is the reduced temperature and s, and b, are the parameters at the critical point, obtained from the equation of state by using the critical point constraints: RT
(I41
the Helmholtz free energy we obtain
A-A -= RT
The coefficients (li and b, were determined by fitting fluid volumetric properties obtained from an equation of state for hard spheres with point dipoles at their contrast (Rushbrooke et al., 1973). The results were:
+kltl+k2$).
p‘ = p”. Introducing
(3)
where @ is in Debyes, b, in cm’/mol and T in Kelvin. With the goal of extending the equation of state to fluid mixtures, we assumed the following expression for the coefficients f Ii’:
z.” = -$+(I
EVALUATION OF PARAMETERS
For a pure substance at a given temperature, a single value of P exists which satisfies the saturation condition:
where A’ and AL are the Helmholtz free energy of saturated vapor and saturated liquid, respectively, and their expression is
(4)
I”’ = 1 + a&+ + b&b.
(13)
(2)
As proposed by Bryan and Prausnitz (1987). the coefficients f”l, f”’ andf”’ are a function of temperature through the reduced dipole moment ji given by 95.56fl
- 0.2 1575ji:.
(PY - P?
where the reduced density is q = b&4.
- 0.16059ji:
(1)
For the compressibility factor of the reference system we use a general&d Carnahan-Starling expression: Z,C’_ 1 +f[‘$
3097
(12)
do= - 0.23657 + 1.30974w - 0.563460’.
(22)
(23)
Table 1 gives the P-range and average absolute percent deviation (AADP) calculated for the 59 fluids of the data base wing our equation of state with the proposed comlation for CLand the corresponding AADP calculated with the
3098
Shorter Communications Soave equation of state for comparison. Our results give greater accuracy. Table 2 gives the P-range and the AADP obtained with the Iwo equation of state for the 10 compounds not present in the data base. The results of our equation of state are better than those of the Soave equation, especially in the case of l-octanol.
CONCLUSIONS The results that the new equation
of state gives are much better than those of the Soave equation of state when two adjustable parameters are used. It has been found that parameter iz can be linearly correlated with the acentric factor. When this correlation is used in the equation of state to predict vapor pressures, we find that the AAPD is greater than when we use the two parameter fit; however, it is apparent that the prediction shows a substantial improvement over the Soave equation of state. Fig. 1. The parameter a as a function of the acentric factor when p = 1.4986.
Acknowledgement-The authors are grateful to the Italian Minister0 dell’Universita’ e della Ricerca Scientifica e Tecnologica for providing financial support for this work.
Table 1. Comparison between the AAPD on vapor pressures using our equation of state with a predicted with the acentric factor and that using the Soave equation of state for the 59 substances of the data base P-range (bar) Methane Elhane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane Iso-butane 2,2-Dimethyl propane 2-Methyl butane 2,2-Dimethyl butane 2,3-Dimethyl butane 2-Methyl pentane 3-Methyl pentane 2,2-Dimethyl pentane 2,3-Dimethyl pentane 2,4-Dimethyl pentane 3,3-Dimethyl pentane 3-Ethyl pentane 2,2.3-Trimethyl butane
Ethylene
Propylene 1 -Butene 1-Pentene 2-Methyl 2-butene 3-Methyl I-butene 1-Hexene 2,3-Dimethyl 2-butene 1-Heptene Methyl formate Ethyl formate Propyl formate Methyl acetate Ethyl acetate Propyl acetate Methyl propionate Butyraldehyde Acetone
0.242.41 10-3-48.5 0.016-I .03 O.OlG16.51 0.32-29.00 0.11-1.07 0.06&l .04 0.08-l .04 10-3-31.89 0.12-1.01 0.02-l .04 O-29- 1.04 0.19-1.04 0.17-1.04 0.17-1.04 0.0% 1.04 0.06-I .a4 0.08-t .04 0.06-l .04 0.06-I .04 0.08-L .04 10-347.80 0.02- 1.04 0.03-1.11 0.54-l .04 0.25-2.70 0.47-2.70 0.17-1:04 0.2&1.01 0.06 1.04 0.70-1.02
0.12-1.00 0.12-1.06 0.0%0.98 0.084.97 0.09- 1.Oo 0.0% 1.oo 0.19-0.98 0.05-0.97
AAPD This work
AAPD Soave
1.1 5.8 1.5 1.4 0.7 2.3 4.0 2.7 2.3 7.4 2.3 0.6 1.2 2.1 2.7 2.4 3.7 3.4 3.1 13.6
3.2 5.1 2.1 1.7 1.1 I.2 1.8 0.4 3.4 7.4 1.5 0.3 0.7 1.3 1.9 1.3 2.3 2.0 2.2 12.1 1.2 0.8 4.7 3.0 9.9 1.4 5.4 1.5 10.4 10.3 2.2 1.o 1.4 1.7 1.1 1.3 2.7 4.4 5.7
1.8
2.3 4.5 3.0 8.6 2.0 4.1 3.0 1.6 13.1 1.7 2.6 1.5 1.1 32:: 3.8 0.4 to.4
Shorter Communications Table 1. (Confd) Methyl ethyl ketone Methyl isobutyl ketone Methyl propyl ketone Methanol Ethanol I-Propanol I-Butanol 2-Butanol I-Pentanol I-Hexanol Water Carbon dioxide Acetonitrile Ammonia Ethylene oxide Carbon disulfide Phenol Iso-butyl benzene o-Xylene Pyridine
0.27-1.33 0.02-1.01 0.20-1.33 0.1-2.06 0.06-1.79 0.2s1.33 o-33-1.33 0.33-1.34 0.07-1.77 0.01-1.01 0.06-22 1.0 34.8&73.50 0.04-48.3 0.1&l 10.0 0.68-2.21 0.02-l .06 0.08-1.01 0.06-1.04 0.01-1.04 0.20-2.70
Mean Max
4.4 11.9 2.4
1.9 3.2 1.2 8.9 0.2 5.0 4.4 2.2 3.7 8.5 4.1 5.8 1.4
4.: 1:2 2.5 2.8 9.0 55.5 10.4 0.8 4.3 2.2 0.8 2.0 2.8 2.3 3.2 0.3
3.6 13.6
5:::
Table 2. Comparison between the AAPD on vapor pressures using our equation of state with a predicted with the acentric factor and that using the Soave equation of state for the 10 substances not included in the data base P-range (bar)
AAPD This work
AAPD Soave
0.44-l .68 0.09-l .Ol 0. I 5-2.06 1.28-2.58 0.013-1.01 10-3-1.35 0.05-1.01 0.01-1.01 0.25-l .02 0.06-l -04
7.97 4.11 0.95 0.94 7.51 4.70 3.33 3.16 0.69 0.31
5.10 0.83 0.77 1.37 2.92 70.40 1.83 2.85 5.09 1.51
3.37 7.97
9.27 70.40
Acetaldehyde Butyl acetate Benzene Acetylene n&red n-Octyl alcohol Dimethyl ether Styrene Propionic aldehyde Toluene Mean Max
PIETRO BRANDANI
KTI Co., San Diego CA 91173. U.S.A.
VINCENZO BRANDANI’ GIOVANNI DEL RE LORIS MASCIULLI Dipartimento di Chimica Ingegneria Chimica e Materiali Universira’ de L’Aquila I-67040 Monteluco di Roio L’Aquila, Italy NOTATION coefficients in eq. (5) Helmholtz free energy, J/mol covolume, cms/mol ‘Author
to whom correspondence
should be addressed.
coefficients in eq. (5) defined by eq. (18) defined by eq. (19) defined by eq. (20) defined by eq. (6) defined by eq. (7) defined by eq. (8) universal constant universal constant pressure, bar gas constant, J/mol K temperature, K molar volume, cmg/mol compressibility factor Greek letters adjustable parameter adjustable parameter ; E energy parameter, J/mol reduced density ?I
Shorter Communications
3100 p” ii P 0
chemical potential, J/m01 dipole moment, D reduced dipole moment molar density, cn?/mol acentric factor
Subscripts
c CalC cxp r
critical calculated experimental reduced
Superscripts att IG L ref sat V
attractive ideal gas liquid reference saturation vapor REFERENCES
Baztia, E. R., 1983, Cubic equation of state For mixtures containing polar compounds, in Chemical Engineering Thermodynamics (Edited by S. A. Newman), pp. 195210. Ann Arbor Science Publishers, Ann Arbor, MI. Boublik, T., Fried, V. and Hala, E., 1984, The Vapour Pressures of Pure Substances. Elsevier, Amsterdam.
Brandani, V., Del Re, G., Di Giacomo, G. and Brandani, P., 1992, A new equation of state for polar and nonpolar pure fluids. Fluid Phase Equilibria 75, 81-87. Bryan, P. F. and Prausnitz. J. M., 1987, Thermodynamic properties of polar fluids from a perturbcd-dipolar-hardsphere equation of state. Fluid Phase Equilibria 38, 201-216. Dohm, R. and Prausnitz, J. M., 1990, A simple perturbation term for the Carnahan-Starling equation of state. Fluid Phase Equilibria 61, 53-69. Joffe, J., Joseph, H. and Tassios, D., 1983, Vapor-liquid equilibria with a modified Martin equation of state, in Chemical Engineering Thermodynamics (Edited by S. A. Newman), pp. 211-219. Ann Arbor Science Publishers, Ann Arbor, Mt. Reid, R. C., Prausnitz, J. M. and Polina B. E., 1987, The Properties of Gases and Liquids, pp. 656732. McGrawHill, New York. Rushbrooke, G. S., Stell, G. and Hoje, J. S., 1973, Theory of polar fluids. I. Dipolar hard spheres, Mol. Phys. 26, 1199-1215. Soave, G., 1972, Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Engng Sci. 27, 1197-1203. Vimalchand, P. and Donohue, D., 1989, Comparison of equation of state for chain molecules. J. Phys. Chem. 93, 43554360.
OK@-25W93 56.00 + 0.00 @ 1993 Per~amon Press Ltd
Estimation
of solid molar volumes using the neighboring factor with application to isumeric compounds (First received 16 October
1992; accepted
INTRODUCTION
Polycyclic aromatic compounds and heterocyclic compounds in coal liquids provide important raw materials for speciality chemicals, and the solid molar volumes of such compounds are among the fundamental data required in treatment of coal liquids. A general method to obtain the density of organic compounds is the group additivity approach. Exner (1967) estimated the densities of 870 organic liquids using this approach, while Fedors (1974) calculated the molar volumes of polymers using group parameters of liquids. Some group contribution methods and group molar volumes for rubber and glassy amorphous polymers are available in a handbook (Barton, 1988). However, these group additivity approaches give the same molar volumes for isomers, and volume parameters of crystalline compounds have not been reported. This work examines a new approach to estimate the solid molar volumes of aromatic compounds using a neighboring factor, which relates to the difference in size between two neighboring groups. THEORY Assignment of groups A monocyclic compound contains a-f groups bonded directly to a benzene ring. Each group is usually assigned
in revised form 10 March
1993)
according to one of the two concepts. Tarver (1979) assigns them to six groups, C.-a, C.-b, C.-c, C,---d, C,-e and C.-f, where C, designates an aromatic carbon atom. The other method assigns them to six substituents and benzene carbon backbone, i.e. V, - V, and F’-, where V, is the molar volume of group i and V,, is that of benzene carbon backbone. In this study, the latter method is selected. Isomers of monocyclic compound Solid molar volume usually increases with the increase of the difference in size between two neighboring groups, perhaps duo to the existence of empty space caused by the volume difference between the two substituents. This work attempts to distinguish the volume difference between isomers by introducing a new coefficient, termed the neighboring factor: Tij = I/&&. where & is the ratio of molar volume of substltuent group i to the average molar volume of two neighboring groups i and j. This factor is similar to the molecular connectivity (Kier and Hall, 1986). Based on this concept, 4 and F are given for i and j groups as follows:
vj
V, 9r = (v, + V,)/2’
&‘=(s+
VJ)/2
(I)
3096 Table 2. Explicit expressions for d,, Figure
and the required conditions for their use (C = 8 and r = 0.747 for the Rushton impeller) Required condition
Expression for d,.. d-/D { 1 + 0.209[
= (1 /C Re.)J’4 x
= (l/C
5’ We/(rC3’4Rez’4)
N R R R; Rer t t .%+.I, T We
11
number of cycIes in a sequence of deformation and relaxation ~s~~~ds~d&t$p.s.) value of R corresponding to d,, drop Reynolds number (= p$N’/& tank Reynolds number (= p,DNZ/pc) time, s eddy life time, s tank diameter, m Weber number (- p,NzD3/o)
Greek letters ratio of impeller diameter to tank diameter c (= BIT) dimensionless time ;I dimensionless deformation of drop deformation state at the beginning of the defor6*(x) mation cycle for the nth cycle deformation state at the end of the deformation @,(a) zone or at the beginning of the relaxation zone for the nth cycle dispersed-phase viscosity, Pas Fd density of continuous phase, kg/m3 P, a interfacial tension, N/m * defined as the product of volume fraction of relaxation zone and circulation group number for the vessel. For the Rushton impeller its value is 0.747.
Chemical Engimrfng Science. Printed in Great Britain.
Comparison
+C
Re,+)3/4 x 2.735
i
Q 0.1351 X
We/c’ > 605 +
-0.1485
Subscripts
f
We/l5
(r2CRe,/~4)‘.‘542
We,(CL”Re~~~)~1’os27~‘s’33
d-/D
n
.5’2c
2.686(js’CRe,/c4
-
1.341)‘.*’
final state initial state
REFERENCES Arai, K., Konno, M., Matunga, Y. and Saito, S., 1977, Effect of dispersed phase viscosity on the maximum stable drop size for break-up in turbulent flow. J. &em. Engng Japan 10, 325-330. Calabrcse, R. V., Chang, T. P. K. and Dan& P. T., 1986, Drop breakup in turbulent stirred-tank contactors; Part I: effect of dispersed phase viscosity. A.Z.Ch.E. J. 32, 657-666. Davies, J. T.. 1985, Drop sizes of emulsions related to turbulent energy dissipation rates. Chem. Engng Sci. 40, 839-842. Hinxe, O., 1955, Fundamentals of hydrodynamic mechanisms of splitting in dispersion processes. A.1.Ch.E. J. 1, 289-295. Holmes, D. B., Voncken, R. M. and Dekker, J. A., 1964, Fluid flow in turbine-stirred, baffled tanks--I. Circulation time. Chem. Engng Sci. 19, 201-208. Kumar, S., Kumar. R. and Gandhi, K. S., 1992, A multi-stage model for drop breakage in stirred vessels. Cheat. Engng Sci. 47, 97 l-980. Lagisetty, J. S., Das, P. K., Kumar, R. and Gandhi, K. S., 1986, Breakage of viscous and non-Newtonian drops in stirred dispersions. Chem. Engng Sci. 41, 65-72.
Vol. 48, No. 1% pp. 3O963100.1993.
OLW-2509f93 S6.00 + 0.00 Rrgamon Press Ltd
between a noncubic equation of state particularly suitable for polar fluids and the Soave equation of state (Received
19 January
1993; accepted
INTRODWCTiON Recently, Brandani et al. (1992) have presented a new equation of state based on theoretical results of a system of hard spheres with dipoles and on the computer simulation data
for publication
11 March
1993)
for n-butane. This new equation of state was applied to the correlation of thermodynamic properties for 10 pure polar and nonpolar fluids. In this communication, the new equation of state has been testcd using a data base constituted by the vapor pressures of
Shorter Communications 59 pure polar and nonpolar fluids. The aim of this work was
= 3.10026 + O.O24838fi, - 0.027299/1,2 RT,
to find a correlation for the two adjustable parameters of the
equation as a function of the acentric factor. The results were compared with those obtained using the equation of state of Soave (1972), which is widely used by chemical engineers for VLE calculations, even for polar substances (Baztia, 1983; Joffe et ol., 1983). THE EQUATION OF STATE
The compressibility factor I is the sum of two contributions: one from the reference system (t”‘) and one which takes into account the effect of dispersion forces (z”‘): r = -&
= r=f + Z.“.
+/t**+
-f%$
(1 - v)’
P-m
(5)
I’]’ = 1 - 1.5334/Z* + O.l2993j?
(6)
f’2’ = 1 + 4.1339fl* + 0.10764@
(7)
f13’ = 1 + 1.6024fi& + 0.73728fi6.
(8)
For the term in eq. (1) which takes into account the effect of dispersion forces, we choose the expression proposed by Dohrn and Prausnitz (1990): (9)
The universal parameters k, and kt were determined by fitting the computer simulation data of n-butane taken from and Donohue (1989) and their best Fig. 4 of Vim&hand values were found to be k, = - 1.43279 and k2 = 3.97055. For the dependence of the parameters E and b on temperature the following expressi\ons were chosen: : = z[l
+ a(1 - co.‘)]*
b = b,[l + /l(l - Z-$‘)]’
c
= 0.16552 - 0.0015668&
(15)
from which pm, _ pL.pY(AY - A’-)
(16)
rc
mz- 1
I
-dq orl
= CI In(1 - q) + CI~ l--9
+ C r112- tl) - $+ 3(1-
+ tklrt + +k&)
(17)
and c, = - (1 --fIl’) C* =
f -p-
cs = $(l i”fi”
+
(18) 2fI3J
+_f’2’ -f”‘).
(19) (20)
The two adjustable parameters of the equation of state were determined minimizing the objective function:
where Pz\=;.is calculated from eq. (16) using the equation state and eq. (17).
of
RESULTS AND DISCUSSION The experimental vapor pressures were taken from Boublik et al. (1984) for the 59 compounds which constitute the data base. Using OT and B as adjustable parameters the fit gives a mean absolute percent deviation of 0.92 which is a good result if compared with those which can be obtained from a simple linear interpolation of the function In Pm’vs l/T. Moreover, while for dothere is a correlation trend with w, fi does not show a systematic pattern. Therefore, we assumed for /l the average vaiue of 1.4986 and we proceeded to a new fit using e as the only adjustable parameter. Figure 1 shows this parameter as a function of OL with o, the acentric factor taken from Reid et al. (1987). The straight line is described by the following equation: Q = - 0.23549 + 0.882260.
(11)
Water and 1-hexanol deviate from this parameter and for these substancn and higher alcohols we suggest the following equation for a:
+ O.O048321ji,z
- O.OCNM4518fi: - 0.0035999/i:
A= + Py’vL = AY + P”‘vy
(10)
where OL and pare the two adjustable parameters of equation of state, T, is the reduced temperature and s, and b, are the parameters at the critical point, obtained from the equation of state by using the critical point constraints: RT
(I41
the Helmholtz free energy we obtain
A-A -= RT
The coefficients (li and b, were determined by fitting fluid volumetric properties obtained from an equation of state for hard spheres with point dipoles at their contrast (Rushbrooke et al., 1973). The results were:
+kltl+k2$).
p‘ = p”. Introducing
(3)
where @ is in Debyes, b, in cm’/mol and T in Kelvin. With the goal of extending the equation of state to fluid mixtures, we assumed the following expression for the coefficients f Ii’:
z.” = -$+(I
EVALUATION OF PARAMETERS
For a pure substance at a given temperature, a single value of P exists which satisfies the saturation condition:
where A’ and AL are the Helmholtz free energy of saturated vapor and saturated liquid, respectively, and their expression is
(4)
I”’ = 1 + a&+ + b&b.
(13)
(2)
As proposed by Bryan and Prausnitz (1987). the coefficients f”l, f”’ andf”’ are a function of temperature through the reduced dipole moment ji given by 95.56fl
- 0.2 1575ji:.
(PY - P?
where the reduced density is q = b&4.
- 0.16059ji:
(1)
For the compressibility factor of the reference system we use a general&d Carnahan-Starling expression: Z,C’_ 1 +f[‘$
3097
(12)
do= - 0.23657 + 1.30974w - 0.563460’.
(22)
(23)
Table 1 gives the P-range and average absolute percent deviation (AADP) calculated for the 59 fluids of the data base wing our equation of state with the proposed comlation for CLand the corresponding AADP calculated with the
3098
Shorter Communications Soave equation of state for comparison. Our results give greater accuracy. Table 2 gives the P-range and the AADP obtained with the Iwo equation of state for the 10 compounds not present in the data base. The results of our equation of state are better than those of the Soave equation, especially in the case of l-octanol.
CONCLUSIONS The results that the new equation
of state gives are much better than those of the Soave equation of state when two adjustable parameters are used. It has been found that parameter iz can be linearly correlated with the acentric factor. When this correlation is used in the equation of state to predict vapor pressures, we find that the AAPD is greater than when we use the two parameter fit; however, it is apparent that the prediction shows a substantial improvement over the Soave equation of state. Fig. 1. The parameter a as a function of the acentric factor when p = 1.4986.
Acknowledgement-The authors are grateful to the Italian Minister0 dell’Universita’ e della Ricerca Scientifica e Tecnologica for providing financial support for this work.
Table 1. Comparison between the AAPD on vapor pressures using our equation of state with a predicted with the acentric factor and that using the Soave equation of state for the 59 substances of the data base P-range (bar) Methane Elhane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane Iso-butane 2,2-Dimethyl propane 2-Methyl butane 2,2-Dimethyl butane 2,3-Dimethyl butane 2-Methyl pentane 3-Methyl pentane 2,2-Dimethyl pentane 2,3-Dimethyl pentane 2,4-Dimethyl pentane 3,3-Dimethyl pentane 3-Ethyl pentane 2,2.3-Trimethyl butane
Ethylene
Propylene 1 -Butene 1-Pentene 2-Methyl 2-butene 3-Methyl I-butene 1-Hexene 2,3-Dimethyl 2-butene 1-Heptene Methyl formate Ethyl formate Propyl formate Methyl acetate Ethyl acetate Propyl acetate Methyl propionate Butyraldehyde Acetone
0.242.41 10-3-48.5 0.016-I .03 O.OlG16.51 0.32-29.00 0.11-1.07 0.06&l .04 0.08-l .04 10-3-31.89 0.12-1.01 0.02-l .04 O-29- 1.04 0.19-1.04 0.17-1.04 0.17-1.04 0.0% 1.04 0.06-I .a4 0.08-t .04 0.06-l .04 0.06-I .04 0.08-L .04 10-347.80 0.02- 1.04 0.03-1.11 0.54-l .04 0.25-2.70 0.47-2.70 0.17-1:04 0.2&1.01 0.06 1.04 0.70-1.02
0.12-1.00 0.12-1.06 0.0%0.98 0.084.97 0.09- 1.Oo 0.0% 1.oo 0.19-0.98 0.05-0.97
AAPD This work
AAPD Soave
1.1 5.8 1.5 1.4 0.7 2.3 4.0 2.7 2.3 7.4 2.3 0.6 1.2 2.1 2.7 2.4 3.7 3.4 3.1 13.6
3.2 5.1 2.1 1.7 1.1 I.2 1.8 0.4 3.4 7.4 1.5 0.3 0.7 1.3 1.9 1.3 2.3 2.0 2.2 12.1 1.2 0.8 4.7 3.0 9.9 1.4 5.4 1.5 10.4 10.3 2.2 1.o 1.4 1.7 1.1 1.3 2.7 4.4 5.7
1.8
2.3 4.5 3.0 8.6 2.0 4.1 3.0 1.6 13.1 1.7 2.6 1.5 1.1 32:: 3.8 0.4 to.4
Shorter Communications Table 1. (Confd) Methyl ethyl ketone Methyl isobutyl ketone Methyl propyl ketone Methanol Ethanol I-Propanol I-Butanol 2-Butanol I-Pentanol I-Hexanol Water Carbon dioxide Acetonitrile Ammonia Ethylene oxide Carbon disulfide Phenol Iso-butyl benzene o-Xylene Pyridine
0.27-1.33 0.02-1.01 0.20-1.33 0.1-2.06 0.06-1.79 0.2s1.33 o-33-1.33 0.33-1.34 0.07-1.77 0.01-1.01 0.06-22 1.0 34.8&73.50 0.04-48.3 0.1&l 10.0 0.68-2.21 0.02-l .06 0.08-1.01 0.06-1.04 0.01-1.04 0.20-2.70
Mean Max
4.4 11.9 2.4
1.9 3.2 1.2 8.9 0.2 5.0 4.4 2.2 3.7 8.5 4.1 5.8 1.4
4.: 1:2 2.5 2.8 9.0 55.5 10.4 0.8 4.3 2.2 0.8 2.0 2.8 2.3 3.2 0.3
3.6 13.6
5:::
Table 2. Comparison between the AAPD on vapor pressures using our equation of state with a predicted with the acentric factor and that using the Soave equation of state for the 10 substances not included in the data base P-range (bar)
AAPD This work
AAPD Soave
0.44-l .68 0.09-l .Ol 0. I 5-2.06 1.28-2.58 0.013-1.01 10-3-1.35 0.05-1.01 0.01-1.01 0.25-l .02 0.06-l -04
7.97 4.11 0.95 0.94 7.51 4.70 3.33 3.16 0.69 0.31
5.10 0.83 0.77 1.37 2.92 70.40 1.83 2.85 5.09 1.51
3.37 7.97
9.27 70.40
Acetaldehyde Butyl acetate Benzene Acetylene n&red n-Octyl alcohol Dimethyl ether Styrene Propionic aldehyde Toluene Mean Max
PIETRO BRANDANI
KTI Co., San Diego CA 91173. U.S.A.
VINCENZO BRANDANI’ GIOVANNI DEL RE LORIS MASCIULLI Dipartimento di Chimica Ingegneria Chimica e Materiali Universira’ de L’Aquila I-67040 Monteluco di Roio L’Aquila, Italy NOTATION coefficients in eq. (5) Helmholtz free energy, J/mol covolume, cms/mol ‘Author
to whom correspondence
should be addressed.
coefficients in eq. (5) defined by eq. (18) defined by eq. (19) defined by eq. (20) defined by eq. (6) defined by eq. (7) defined by eq. (8) universal constant universal constant pressure, bar gas constant, J/mol K temperature, K molar volume, cmg/mol compressibility factor Greek letters adjustable parameter adjustable parameter ; E energy parameter, J/mol reduced density ?I
Shorter Communications
3100 p” ii P 0
chemical potential, J/m01 dipole moment, D reduced dipole moment molar density, cn?/mol acentric factor
Subscripts
c CalC cxp r
critical calculated experimental reduced
Superscripts att IG L ref sat V
attractive ideal gas liquid reference saturation vapor REFERENCES
Baztia, E. R., 1983, Cubic equation of state For mixtures containing polar compounds, in Chemical Engineering Thermodynamics (Edited by S. A. Newman), pp. 195210. Ann Arbor Science Publishers, Ann Arbor, MI. Boublik, T., Fried, V. and Hala, E., 1984, The Vapour Pressures of Pure Substances. Elsevier, Amsterdam.
Brandani, V., Del Re, G., Di Giacomo, G. and Brandani, P., 1992, A new equation of state for polar and nonpolar pure fluids. Fluid Phase Equilibria 75, 81-87. Bryan, P. F. and Prausnitz. J. M., 1987, Thermodynamic properties of polar fluids from a perturbcd-dipolar-hardsphere equation of state. Fluid Phase Equilibria 38, 201-216. Dohm, R. and Prausnitz, J. M., 1990, A simple perturbation term for the Carnahan-Starling equation of state. Fluid Phase Equilibria 61, 53-69. Joffe, J., Joseph, H. and Tassios, D., 1983, Vapor-liquid equilibria with a modified Martin equation of state, in Chemical Engineering Thermodynamics (Edited by S. A. Newman), pp. 211-219. Ann Arbor Science Publishers, Ann Arbor, Mt. Reid, R. C., Prausnitz, J. M. and Polina B. E., 1987, The Properties of Gases and Liquids, pp. 656732. McGrawHill, New York. Rushbrooke, G. S., Stell, G. and Hoje, J. S., 1973, Theory of polar fluids. I. Dipolar hard spheres, Mol. Phys. 26, 1199-1215. Soave, G., 1972, Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Engng Sci. 27, 1197-1203. Vimalchand, P. and Donohue, D., 1989, Comparison of equation of state for chain molecules. J. Phys. Chem. 93, 43554360.
OK@-25W93 56.00 + 0.00 @ 1993 Per~amon Press Ltd
Estimation
of solid molar volumes using the neighboring factor with application to isumeric compounds (First received 16 October
1992; accepted
INTRODUCTION
Polycyclic aromatic compounds and heterocyclic compounds in coal liquids provide important raw materials for speciality chemicals, and the solid molar volumes of such compounds are among the fundamental data required in treatment of coal liquids. A general method to obtain the density of organic compounds is the group additivity approach. Exner (1967) estimated the densities of 870 organic liquids using this approach, while Fedors (1974) calculated the molar volumes of polymers using group parameters of liquids. Some group contribution methods and group molar volumes for rubber and glassy amorphous polymers are available in a handbook (Barton, 1988). However, these group additivity approaches give the same molar volumes for isomers, and volume parameters of crystalline compounds have not been reported. This work examines a new approach to estimate the solid molar volumes of aromatic compounds using a neighboring factor, which relates to the difference in size between two neighboring groups. THEORY Assignment of groups A monocyclic compound contains a-f groups bonded directly to a benzene ring. Each group is usually assigned
in revised form 10 March
1993)
according to one of the two concepts. Tarver (1979) assigns them to six groups, C.-a, C.-b, C.-c, C,---d, C,-e and C.-f, where C, designates an aromatic carbon atom. The other method assigns them to six substituents and benzene carbon backbone, i.e. V, - V, and F’-, where V, is the molar volume of group i and V,, is that of benzene carbon backbone. In this study, the latter method is selected. Isomers of monocyclic compound Solid molar volume usually increases with the increase of the difference in size between two neighboring groups, perhaps duo to the existence of empty space caused by the volume difference between the two substituents. This work attempts to distinguish the volume difference between isomers by introducing a new coefficient, termed the neighboring factor: Tij = I/&&. where & is the ratio of molar volume of substltuent group i to the average molar volume of two neighboring groups i and j. This factor is similar to the molecular connectivity (Kier and Hall, 1986). Based on this concept, 4 and F are given for i and j groups as follows:
vj
V, 9r = (v, + V,)/2’
&‘=(s+
VJ)/2
(I)