- Email: [email protected]
Data driven temperature dependence of EOS parameters
ELSEVIER
Fluid Phase Equilibria 140 (1997) 97- 105
Data driven temperature dependence of EOS parameters A. Deiik * , S. Kern&y. Trchnicul
Uniwrsity
of Budapest, Department
of Chemicul
’
I. Farkas Engirwering.
H- 1521 B~~dup~~.sr. Hutqq
Received 7 October 1996: accepted 6 August I997
Abstract Using three kinds of experimental data (p,, V, and T, at the critical state or p,. V,“’ and 11,‘. at different temperatures for saturation data) three parameters of the EOS may be directly determined and the temperature dependence of the parameters may be established from thermodynamic conditions of vapour-liquid critical point or vapour-liquid phase equilibria respectively. The principle was demonstrated on the BACK EOS. Examples of argon and n-alkanes were used to demonstrate the idea. It was found, that there are two parameter sets of the BACK equation that satisfy the critical or saturation conditions for certain pure compounds. The BACK equation is able to reproduce experimental Z, values for compounds above Zc = 0.2764 (that is, for argon, methane, ethane), but it is improper for higher alkanes. In case of n-alkanes we found that there is no simple function for T dependence of BACK parameters. Parameter values obtained in the way demonstrated may be useful to give initial values for parameter estimation from experimental (e.g.. VLE) data. ((3 1997 Elsevier Science B.V. Key,t,o&c:
Equation of state: Vapour-liquid
equilibria;
Critical state
1. Introduction Soave [ 11 in his famous equation of state determined the two parameters (a and h) partly from pure component critical data CT,,, p,) and partly from pure component vapour pressure data (T,. p,), allowing the data to tell the proper temperature dependence of the a parameter. At the vapour-liquid critical point the following two conditions hold for a pure substance:
(‘1
* Corresponding author. E-mail: [email protected]. ’Preliminary version of this work was presented as a poster (P3.157) on the CHISA ‘96 Conference. 037X-38I2/97/$17.00 6 1997 Elsevier Science B.V. All rights reserved PI/ SO378-38 I2(97)00200-8
Prague. 1996
A. Decik et al./ Fluid Phase Equilibria 140 (1997) 97-105
98
i-1 d2P
dV2
T=j”,
The pressure temperature: p, =
(2)
0 =
explicit
P(T,,V,,@) -
equation
of state yields
a value of the critical
pressure
at the critical
(3)
These three equations allow the determination of three unknowns. These three unknowns may be three EOS parameters (if the EOS has at least three adjustable parameters), but in that case we must have experimental data for the critical pressure, temperature and volume in order to solve the set of equations. Alternatively, the three unknowns may consist of the missing experimental data as well, thus the number of EOS parameters to be determined and the number of missing experimental critical coordinates add up to three. Soave used the experimental critical pressure and temperature in his two-parameter EOS model, and chose the critical volume as the third unknown; thus he did not use experimental information on the critical volume. The resulting parameters were a, and b,. For vapour-liquid equilibrium also, three equations hold:
(4) (5) (6) This means that not more than three EOS parameters can be calculated directly from experimental saturation PVT data. If density data (V,” and v,“> are not used (they are treated as unknowns), only one EOS parameter may be determined. This route was followed exactly by Soave, who calculated the a parameter for each experimental temperature, using pure component vapour pressure data, fixing the value of b parameter at that determined for critical conditions (b = b,). The two remaining unknowns for the set of equations were the densities of the two equilibrium phases, which he used unavoidably for the EOS calculations. He thus obtained the a parameter as a function of the temperature, which allowed to fit a function. The three equations (both at critical or saturation state) allow the determination of three EOS parameters from experimental data. If saturation data are utilised, this would require the use of all three kinds of experimental data: pressure and density of both phases at different temperatures. Obviously the cubic EOS models are not expected to reflect properly the volumetric properties, therefore this approach (that is, the determination of three parameters from experimental data at each temperature) is hopeless for cubic EOS models. If, however, a more sophisticated EOS model is used, capable of representing volumetric behaviour as well, three of its parameters may directly be obtained Fig. 1. (a) The (Y parameter of BACK EOS obtained from experimental data vs. temperature for argon and methane [3]. Mach& and Boublik [4]; 0 argon, 1st solution; 0 argon, 2nd solution; ??methane, 1st solution; 0 methane, 2nd solution; large circle: critical point. (b) The V * parameter of BACK EOS obtained from experimental data vs. temperature for argon and methane [3]. -Machat and Boublik [4]; 0 argon, 1st solution; 0 argon, 2nd solution; ?? methane, 1st solution; 0 methane, 2nd solution; large circle: critical point. (c) The u/k parameter of BACK EOS obtained from experimental data vs. temperature for argon and methane [3]. ~ Machat and Boublik [4]; 0 argon, 1st solution: 0 argon, 2nd solution; W methane, 1st solution; 0 methane, 2nd solution; large circle: critical point.
A. D&k
80
90
et al./
Fluid Phase Equilibria
100
110
120
130
140 (1997) 97-105
140
IS0
160
170
180
190
200
T, K
1
19’t(b) Oo 17.00
IS.00
!
C”4 II
II;d
??
13.00 .* 11.00
.
.
.
.
.
_o
/ i1
5.00
.
?? *+ Ar
coo?/
u--. 80
90
100
110
120
130
140 T,
150
160
170
180
I90
200
K
170
@
150 .
.
.
@
. .
130
“ml”‘t’l””
110 80
90
I’ 100
110
120
“““‘,“’ 130
140 T,
K
150
160
I70
IX0
190
200
A. Decik et ul./ Fluid Phase Equilibria I40 (1997) 97-105
100
from experimental data, thus a sensible BACK EOS for calculations.
temperature
dependence
may be deduced.
We chose the
2. Results To reiterate, the procedure is as follows: Using three kinds of experimental data ( p,, V, and T, at the critical state or p,, V,” and VsL at different temperatures for saturation data) three parameters of the EOS are determined from Eqs. (l)-(3) or Eqs. (4)-(6), respectively, hopefully showing sensible temperature dependence of the parameters. The main formulae for the BACK model are summarised below:
z =zrep +zattr 1 -p Zrep- (1 -y)
3ay + (1 -Y)’
‘y2Y2(3 -Y) +
(1 -Y)3
where y=V*/V=y$
n=l
m
The constants Dnm were determined by Chen and Kreglewski [2], based on argon data. Their set of constants was used in this work. The values for the three parameters ((Y, V * and u/k) are determined from experimental data. Upon solving the set of non-linear equations (Eqs. (l)-(3) or Eqs. (4)-(6)), multiple roots were obtained, from which only the physically meaningful ones were retained. The first condition of selection is that of mechanical stability: (g), I 0 for the whole physically sound range should be satisfied. Further the solutions from the physically unacceptable region (e.g., too high core volume) were excluded. Fig. la-c show the parameters obtained for argon and methane data, together with the values reported by Mach& and Boublik [4]. We found two sets of parameters at each temperature, and also for the critical data. The values adjusted by Machat and Boublik [4] are consistently close to one set. A similar picture was obtained for ethane. The existence of two sets of solutions needs an explanation, especially since they do not appear as results of parameter estimation. This is facilitated more properly by Fig. 2, which gives the mutual position of parameters obtained for ethane data. In both Figs. 1 and 2 the parameter values belonging to the critical point are circled. Note that the points in the (Y, V * , u/k space belong to different temperature values. The two sets behave in a different manner. The first one spans a narrow range both in cy and V *, the second one is situated in a wide cx and V * range. When parameters are adjusted to experimental data (not done here), usually rather weak temperature dependence is assumed and reflected by the function fitted; this way only the first set of parameters remains feasible. It is also seen in Fig. 2 that the parameters are strongly correlated.
IO I
420 ~-----_T---‘-
Fig. 2. cy. L’’ and u/k obtained from experimental large circle: critical point.
ethane data [5] in the parameter space. ?? I st solution:
.__~._~
0.350
III 2nd solution;
~
0.340 CK 0.330 r/T 0.320
0.310
is
/
0.260
I/
I
I!
i-
1
r
0.260 t
0.270
/'i'
LL
w
W”
C,%
0.7
0.6
09
, 1.0
1.1
1.2
1.3
14
1.5
1.6
alpha Fig. 3. The compressibility factor at the critical point vs. LYfor BACK EOS. CK: D,,,,, from Chen and Kreglewski D,,,,, from Saager et al., [7]: F: BACKONE EOS, developed by Miiller at al. [8].
[2]: S:
102
A. De&k et al. / Fluid Phase Equilibria 140 (1997) 97-105
1.400
1.200
1.000
0.800
t. 150
150
200
200
250
250
300
300
350
350
400
450
500
550
T, K 800.00
700.00
600.00
500.00
400.00
300.00
200.00 " 150
'
'
200
250
' / / 300
Ii 350 T, K
400
L 450
' 500
1
550
A. Decik et al./ Fluid Phase Equilibria 140 (1997) 97-105
I03
It is well known experimentally [6] that the change in Z, may well be represented as a function of a parameter reflecting material properties (such as o, Pitzer’s acentric factor). Instead of o we use here the explicit shape parameter cy. Fig. 3 is a plot of Z, with a, obtained by solving Eqs. ( 1) and (2) at given (Y values. The shape of the Z, vs. (Y plot is characteristic to the BACK equation. At Z, > 0.2764 two LY values belong to the same Z,, while below that there is no solution. Thus, in principle, the BACK equation is able to reproduce experimental Z, values for compounds above Z, = 0.2764 (that is for argon, methane, ethane), but it is improper for higher alkanes. If the D,,,, constants are different from those used by Chen and Kreglewski (1977) [2], the qualitative picture remains the same. The second curve (S) in Fig. 3 was obtained by using D,,,,, constants of Saager et al. [7], who fitted them to ethane data, arbitrarily fixing the (Y value at 1.2 126. It is interesting to note that the minimum Z value is obtained almost at that (Y value. The third curve (F) shows the results obtained using BACKONE EOS [8], while the BACKONE critical Z is not as low as it should be in comparison with real fluids; it remains at least lower than 0.30 from a I .O up to (Y 1.3. This seems to be already some progress in comparison with the Chen-Kreglewski and Saager correlations. Fig. 4a-c suggest that the temperature dependence of EOS parameters changes gradually with the number of carbon atoms in the homologous series for the lower alkanes, but the behaviour of rz-hexane is anomalous, similar to the findings of Mach& and Boublik [4]. The energy parameter u/k is decreasing with increasing temperature, conforming with the usual assumption. The form of temperature dependence of the parameters is unexpected: however: - V ’ is too much T dependent with extremum in many cases, * a is T dependent in spite of the often applied constancy assumption, but its scattering suggests larger uncertainty of this parameter, involving its minor role in phase equilibrium calculations. One should not forget that the T dependence obtained obviously reflects model errors as well; actually, the improper volume dependence appears as temperature dependence. We found that not all experimental points offer a physically sound mathematical solution of non-linear equations (Eqs. (l)-(3) and Eqs. (4)-(6)) for BACK EOS and not all successful EOS models belonging to the BACK ‘family’ (SAFT, PHCT) give a solution to the system of non-linear equations (Eqs. (l)-(3)) and Eqs. (4)-(6)).
3. Conclusion There are two (feasible) parameter sets conditions for certain pure compounds, estimation. The other set does not appear limited assumed temperature dependence
of the BACK equation that satisfy the critical or saturation one of them is close to the set obtained by parameter in the course of the usual parameter estimation due to the of the parameters.
Fig. 3. (a) The LYparameter of BACK EOS obtained from experimental data vs. temperature for n-alkanes [5]. ~ Machat and Boublik [3]; W ethane, 0 propane, + n-butane, ?? n-pentane, LII n-hexane, A n-heptane. (b) The V * Mach&t and Boublik parameter of BACK EOS obtained from experimental data vs. temperature for n-alkanes [s]. ~ [3]; ??ethane, 0 propane, + n-butane, 0 n-pentane, 0 n-hexane, A n-heptane. (c) The u/k parameter of BACK EOS Machat and Boublfk [3]; ?? ethane. 10 obtained from experimental data vs. temperature for n-alkanes [5]. ~ propane. + rr-butane, 0 n-pentane, 0 n-hexane, A n-heptane.
A. D&k et al. / Fluid Phase Equilibria 140 (1997) 97-105
104
In case of n-alkanes we found that there is no simple function for T dependence of BACK parameters. Critical points may not be obtained below Z, = 0.2764 with the BACK equation. Parameter values obtained in the way demonstrated may be useful to give initial values for parameter estimation from experimental (e.g., VLE) data.
4. List of symbols
D nm k P T u V V0 V* Z Greek letters CY
cp 0
-0 Subscripts attr
universal constants in Eq. (9) Boltzmann’s constant pressure temperature interaction energy molar volume molar close-packed volume of the hard core molar core volume compressibility
non-sphericity parameter, its value is equal to unity for spheres, but greater than one forother convex bodies fugacity coefficient Pitzer’s acentric factor vector of EOS parameters
S
attractive critical repulsive saturated
Superscripts L V
liquid phase vapour phase
C
rep
Acknowledgements This work has been supported by the Hungarian National Science Foundation No. T 016880. Suggestions by Prof. Johann Fischer (Wien) are acknowledged.
(OTKA) under grant
A. De&k et al. /Fluid
Phase Equilibriu
140 (1’297) 97- 105
I 0s
References [I] G. Soave. Equilibrium constants from a modified Redlich-Kwong equation of state, Chem. Eng. Sci. 27 (1972) 1197-1’03. [2] S.S. Chen, A. Kreglewski, Applications of the augmented van der Waals theory for tluids: I. Pure fluids. Bcr. Bunsenges. Phys. Chem. X1 (1977) 1049-1052. [3] Vargaftik. N.B., Spravotchnik po teplofizicseskim svoistvam gazov i zsidkostei, Nauka. Moscow, 1972 (in Russian). [4] V. Mach&, T. Boublik. Vapour-liquid equilibria at elevated pressures from the BACK equation of state: I. One-component systems, Fluid Phase Equilib. 2 I (1985) 1-9. [s] D.B. Smith, R. Srivastava, Thermodynamic data for pure compounds: Part A. Hydrocarbons and ketones. Elsevier, 19x6. [6] R.C. Reid, J.M. Prausnitz. B.E. Poling, The properties of gases and liquids, 4th edn., McGraw-Hill. New York. 1988. p. 23. 171 B. Saager, R. Hennenberg. J. Fischer, Construction and application of physically based equations of state. Fluid Phase Equilib. 72 (1992) 41-66. (81 A. Miller, J. Winkelmann, J. Fischer, Backonefamily of equations of state: I. Nonpolar and pure fluids. AICHE J. -12 (19%) I 116-l 126.
Fluid Phase Equilibria 140 (1997) 97- 105
Data driven temperature dependence of EOS parameters A. Deiik * , S. Kern&y. Trchnicul
Uniwrsity
of Budapest, Department
of Chemicul
’
I. Farkas Engirwering.
H- 1521 B~~dup~~.sr. Hutqq
Received 7 October 1996: accepted 6 August I997
Abstract Using three kinds of experimental data (p,, V, and T, at the critical state or p,. V,“’ and 11,‘. at different temperatures for saturation data) three parameters of the EOS may be directly determined and the temperature dependence of the parameters may be established from thermodynamic conditions of vapour-liquid critical point or vapour-liquid phase equilibria respectively. The principle was demonstrated on the BACK EOS. Examples of argon and n-alkanes were used to demonstrate the idea. It was found, that there are two parameter sets of the BACK equation that satisfy the critical or saturation conditions for certain pure compounds. The BACK equation is able to reproduce experimental Z, values for compounds above Zc = 0.2764 (that is, for argon, methane, ethane), but it is improper for higher alkanes. In case of n-alkanes we found that there is no simple function for T dependence of BACK parameters. Parameter values obtained in the way demonstrated may be useful to give initial values for parameter estimation from experimental (e.g.. VLE) data. ((3 1997 Elsevier Science B.V. Key,t,o&c:
Equation of state: Vapour-liquid
equilibria;
Critical state
1. Introduction Soave [ 11 in his famous equation of state determined the two parameters (a and h) partly from pure component critical data CT,,, p,) and partly from pure component vapour pressure data (T,. p,), allowing the data to tell the proper temperature dependence of the a parameter. At the vapour-liquid critical point the following two conditions hold for a pure substance:
(‘1
* Corresponding author. E-mail: [email protected]. ’Preliminary version of this work was presented as a poster (P3.157) on the CHISA ‘96 Conference. 037X-38I2/97/$17.00 6 1997 Elsevier Science B.V. All rights reserved PI/ SO378-38 I2(97)00200-8
Prague. 1996
A. Decik et al./ Fluid Phase Equilibria 140 (1997) 97-105
98
i-1 d2P
dV2
T=j”,
The pressure temperature: p, =
(2)
0 =
explicit
P(T,,V,,@) -
equation
of state yields
a value of the critical
pressure
at the critical
(3)
These three equations allow the determination of three unknowns. These three unknowns may be three EOS parameters (if the EOS has at least three adjustable parameters), but in that case we must have experimental data for the critical pressure, temperature and volume in order to solve the set of equations. Alternatively, the three unknowns may consist of the missing experimental data as well, thus the number of EOS parameters to be determined and the number of missing experimental critical coordinates add up to three. Soave used the experimental critical pressure and temperature in his two-parameter EOS model, and chose the critical volume as the third unknown; thus he did not use experimental information on the critical volume. The resulting parameters were a, and b,. For vapour-liquid equilibrium also, three equations hold:
(4) (5) (6) This means that not more than three EOS parameters can be calculated directly from experimental saturation PVT data. If density data (V,” and v,“> are not used (they are treated as unknowns), only one EOS parameter may be determined. This route was followed exactly by Soave, who calculated the a parameter for each experimental temperature, using pure component vapour pressure data, fixing the value of b parameter at that determined for critical conditions (b = b,). The two remaining unknowns for the set of equations were the densities of the two equilibrium phases, which he used unavoidably for the EOS calculations. He thus obtained the a parameter as a function of the temperature, which allowed to fit a function. The three equations (both at critical or saturation state) allow the determination of three EOS parameters from experimental data. If saturation data are utilised, this would require the use of all three kinds of experimental data: pressure and density of both phases at different temperatures. Obviously the cubic EOS models are not expected to reflect properly the volumetric properties, therefore this approach (that is, the determination of three parameters from experimental data at each temperature) is hopeless for cubic EOS models. If, however, a more sophisticated EOS model is used, capable of representing volumetric behaviour as well, three of its parameters may directly be obtained Fig. 1. (a) The (Y parameter of BACK EOS obtained from experimental data vs. temperature for argon and methane [3]. Mach& and Boublik [4]; 0 argon, 1st solution; 0 argon, 2nd solution; ??methane, 1st solution; 0 methane, 2nd solution; large circle: critical point. (b) The V * parameter of BACK EOS obtained from experimental data vs. temperature for argon and methane [3]. -Machat and Boublik [4]; 0 argon, 1st solution; 0 argon, 2nd solution; ?? methane, 1st solution; 0 methane, 2nd solution; large circle: critical point. (c) The u/k parameter of BACK EOS obtained from experimental data vs. temperature for argon and methane [3]. ~ Machat and Boublik [4]; 0 argon, 1st solution: 0 argon, 2nd solution; W methane, 1st solution; 0 methane, 2nd solution; large circle: critical point.
A. D&k
80
90
et al./
Fluid Phase Equilibria
100
110
120
130
140 (1997) 97-105
140
IS0
160
170
180
190
200
T, K
1
19’t(b) Oo 17.00
IS.00
!
C”4 II
II;d
??
13.00 .* 11.00
.
.
.
.
.
_o
/ i1
5.00
.
?? *+ Ar
coo?/
u--. 80
90
100
110
120
130
140 T,
150
160
170
180
I90
200
K
170
@
150 .
.
.
@
. .
130
“ml”‘t’l””
110 80
90
I’ 100
110
120
“““‘,“’ 130
140 T,
K
150
160
I70
IX0
190
200
A. Decik et ul./ Fluid Phase Equilibria I40 (1997) 97-105
100
from experimental data, thus a sensible BACK EOS for calculations.
temperature
dependence
may be deduced.
We chose the
2. Results To reiterate, the procedure is as follows: Using three kinds of experimental data ( p,, V, and T, at the critical state or p,, V,” and VsL at different temperatures for saturation data) three parameters of the EOS are determined from Eqs. (l)-(3) or Eqs. (4)-(6), respectively, hopefully showing sensible temperature dependence of the parameters. The main formulae for the BACK model are summarised below:
z =zrep +zattr 1 -p Zrep- (1 -y)
3ay + (1 -Y)’
‘y2Y2(3 -Y) +
(1 -Y)3
where y=V*/V=y$
n=l
m
The constants Dnm were determined by Chen and Kreglewski [2], based on argon data. Their set of constants was used in this work. The values for the three parameters ((Y, V * and u/k) are determined from experimental data. Upon solving the set of non-linear equations (Eqs. (l)-(3) or Eqs. (4)-(6)), multiple roots were obtained, from which only the physically meaningful ones were retained. The first condition of selection is that of mechanical stability: (g), I 0 for the whole physically sound range should be satisfied. Further the solutions from the physically unacceptable region (e.g., too high core volume) were excluded. Fig. la-c show the parameters obtained for argon and methane data, together with the values reported by Mach& and Boublik [4]. We found two sets of parameters at each temperature, and also for the critical data. The values adjusted by Machat and Boublik [4] are consistently close to one set. A similar picture was obtained for ethane. The existence of two sets of solutions needs an explanation, especially since they do not appear as results of parameter estimation. This is facilitated more properly by Fig. 2, which gives the mutual position of parameters obtained for ethane data. In both Figs. 1 and 2 the parameter values belonging to the critical point are circled. Note that the points in the (Y, V * , u/k space belong to different temperature values. The two sets behave in a different manner. The first one spans a narrow range both in cy and V *, the second one is situated in a wide cx and V * range. When parameters are adjusted to experimental data (not done here), usually rather weak temperature dependence is assumed and reflected by the function fitted; this way only the first set of parameters remains feasible. It is also seen in Fig. 2 that the parameters are strongly correlated.
IO I
420 ~-----_T---‘-
Fig. 2. cy. L’’ and u/k obtained from experimental large circle: critical point.
ethane data [5] in the parameter space. ?? I st solution:
.__~._~
0.350
III 2nd solution;
~
0.340 CK 0.330 r/T 0.320
0.310
is
/
0.260
I/
I
I!
i-
1
r
0.260 t
0.270
/'i'
LL
w
W”
C,%
0.7
0.6
09
, 1.0
1.1
1.2
1.3
14
1.5
1.6
alpha Fig. 3. The compressibility factor at the critical point vs. LYfor BACK EOS. CK: D,,,,, from Chen and Kreglewski D,,,,, from Saager et al., [7]: F: BACKONE EOS, developed by Miiller at al. [8].
[2]: S:
102
A. De&k et al. / Fluid Phase Equilibria 140 (1997) 97-105
1.400
1.200
1.000
0.800
t. 150
150
200
200
250
250
300
300
350
350
400
450
500
550
T, K 800.00
700.00
600.00
500.00
400.00
300.00
200.00 " 150
'
'
200
250
' / / 300
Ii 350 T, K
400
L 450
' 500
1
550
A. Decik et al./ Fluid Phase Equilibria 140 (1997) 97-105
I03
It is well known experimentally [6] that the change in Z, may well be represented as a function of a parameter reflecting material properties (such as o, Pitzer’s acentric factor). Instead of o we use here the explicit shape parameter cy. Fig. 3 is a plot of Z, with a, obtained by solving Eqs. ( 1) and (2) at given (Y values. The shape of the Z, vs. (Y plot is characteristic to the BACK equation. At Z, > 0.2764 two LY values belong to the same Z,, while below that there is no solution. Thus, in principle, the BACK equation is able to reproduce experimental Z, values for compounds above Z, = 0.2764 (that is for argon, methane, ethane), but it is improper for higher alkanes. If the D,,,, constants are different from those used by Chen and Kreglewski (1977) [2], the qualitative picture remains the same. The second curve (S) in Fig. 3 was obtained by using D,,,,, constants of Saager et al. [7], who fitted them to ethane data, arbitrarily fixing the (Y value at 1.2 126. It is interesting to note that the minimum Z value is obtained almost at that (Y value. The third curve (F) shows the results obtained using BACKONE EOS [8], while the BACKONE critical Z is not as low as it should be in comparison with real fluids; it remains at least lower than 0.30 from a I .O up to (Y 1.3. This seems to be already some progress in comparison with the Chen-Kreglewski and Saager correlations. Fig. 4a-c suggest that the temperature dependence of EOS parameters changes gradually with the number of carbon atoms in the homologous series for the lower alkanes, but the behaviour of rz-hexane is anomalous, similar to the findings of Mach& and Boublik [4]. The energy parameter u/k is decreasing with increasing temperature, conforming with the usual assumption. The form of temperature dependence of the parameters is unexpected: however: - V ’ is too much T dependent with extremum in many cases, * a is T dependent in spite of the often applied constancy assumption, but its scattering suggests larger uncertainty of this parameter, involving its minor role in phase equilibrium calculations. One should not forget that the T dependence obtained obviously reflects model errors as well; actually, the improper volume dependence appears as temperature dependence. We found that not all experimental points offer a physically sound mathematical solution of non-linear equations (Eqs. (l)-(3) and Eqs. (4)-(6)) for BACK EOS and not all successful EOS models belonging to the BACK ‘family’ (SAFT, PHCT) give a solution to the system of non-linear equations (Eqs. (l)-(3)) and Eqs. (4)-(6)).
3. Conclusion There are two (feasible) parameter sets conditions for certain pure compounds, estimation. The other set does not appear limited assumed temperature dependence
of the BACK equation that satisfy the critical or saturation one of them is close to the set obtained by parameter in the course of the usual parameter estimation due to the of the parameters.
Fig. 3. (a) The LYparameter of BACK EOS obtained from experimental data vs. temperature for n-alkanes [5]. ~ Machat and Boublik [3]; W ethane, 0 propane, + n-butane, ?? n-pentane, LII n-hexane, A n-heptane. (b) The V * Mach&t and Boublik parameter of BACK EOS obtained from experimental data vs. temperature for n-alkanes [s]. ~ [3]; ??ethane, 0 propane, + n-butane, 0 n-pentane, 0 n-hexane, A n-heptane. (c) The u/k parameter of BACK EOS Machat and Boublfk [3]; ?? ethane. 10 obtained from experimental data vs. temperature for n-alkanes [5]. ~ propane. + rr-butane, 0 n-pentane, 0 n-hexane, A n-heptane.
A. D&k et al. / Fluid Phase Equilibria 140 (1997) 97-105
104
In case of n-alkanes we found that there is no simple function for T dependence of BACK parameters. Critical points may not be obtained below Z, = 0.2764 with the BACK equation. Parameter values obtained in the way demonstrated may be useful to give initial values for parameter estimation from experimental (e.g., VLE) data.
4. List of symbols
D nm k P T u V V0 V* Z Greek letters CY
cp 0
-0 Subscripts attr
universal constants in Eq. (9) Boltzmann’s constant pressure temperature interaction energy molar volume molar close-packed volume of the hard core molar core volume compressibility
non-sphericity parameter, its value is equal to unity for spheres, but greater than one forother convex bodies fugacity coefficient Pitzer’s acentric factor vector of EOS parameters
S
attractive critical repulsive saturated
Superscripts L V
liquid phase vapour phase
C
rep
Acknowledgements This work has been supported by the Hungarian National Science Foundation No. T 016880. Suggestions by Prof. Johann Fischer (Wien) are acknowledged.
(OTKA) under grant
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