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Prediction of liquid density of LNG, N2, H2S and CO2 multicomponent systems by the CORGC equation of state
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um
llll ELSEVIER
Fluid Phase Equilibria 112 (1995) 89-99
Prediction of liquid density of LNG, N2, HzS and C O 2 multicomponent systems by the CORGC equation of state Khashayar Nasrifar, Mahmood Moshfeghian * Department of Chemical Engineering, Shiraz University, Shiraz, Iran
Received 29 April 1994; accepted 11 May 1995
Abstract The capability of the CORGC equation of state for prediction of liquid density of multicomponent mixtures has been tested. In order to improve its accuracy, a new set of mixing rules has been developed which gives better predictions than the best available correlations. The average of absolute deviation for 25 multicomponent systems of nonpolar was only 0.67% and 2.7% for 5 slightly polar systems.
Keywords: Theory; Equation of state; Group contributions; Liquid density
1. Introduction Thermodynamic properties and their availability for design of chemical engineering processes are essential part of any computer simulation program. In the absence of experimental data, it is customary to rely on the predicted values by a powerful tool that is named equation of state. Unfortunately, the capability of equations of state such as Soave-Redlich-Kwong (Soave, 1972) and Peng and Robinson (1976) to predict liquid density has been very poor and engineers have been forced to use correlations that are more reliable. Recently, an equation of state has been introduced by (Pults et al., 1989), that was based on chain of rotator (COR) equation of state. This equation of state is named the chain of rotator group contribution (CORGC) equation of state. The CORGC equation of state has lower errors in prediction of thermodynamic properties in comparison with other equations of state. These results can be found in the work of (Pults et al., 1989), (Shariat et al., 1993), and (Zia-Razzaz and Moshfeghian, 1993).
* C o r r e s p o n d i n g author. 0378-3812/95/$09.50 © 1995 Elsevier Science B.V. All fights reserved SSDI 0 3 7 8 - 3 8 1 2 ( 9 5 ) 0 2 7 8 9 - 0
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90
Surprisingly, the error for prediction of pure component liquid density was comparable with the best correlations, (Zia-Razzaz and Moshfeghian, 1993). In this paper, the CORGC equation of state has been applied to predict liquid density of multicomponent systems such as hydrocarbons and mixtures containing non hydrocarbon components. In order to improve its accuracy, a new set of mixing rule has been proposed. Its capabilities have been compared with the best correlations such as (Hankinson and Thomson, 1979) correlation (COSTALD) and the modified Rackett correlation (RSD) which was developed by (Spencer and Danner, 1973).
2. The C O R G C equation of state
The CORGC equation of state as developed by (Pults et al., 1989), was accorded to Van der waals and perturbation theory. They considered the pressure of a fluid as the sum of repulsive and attractive contribution. In the CORGC equation of state the repulsive pressure is that from translational motion and group rotation of molecules in COR equation of state. The attractive pressure has been represented by a form of the Redlich-Kwong equation. They have introduced the CORGC equation of state as:
Z
PV RT
l+y+y2-y3
=--=
(l-y) 3
c +
(a-l) "2"
3y+3ay
2-(a+l)y 3
(l-y) 3
-
a(T) RT[V+b(T)]
(1)
In this equation, y = b(T)/4V is the reduced density, b(T) is the molecular covolume, V is the molar volume, c is the number of external degree of freedom, a is a constant equal to 1.078, a(T) is the attractive interaction parameter and Z is the compressibility factor. The equation of state parameters a, b and c are made up of group contributions according to the following equations: NG NG aij=
E EVimVjnqmqnamn(T) m n
(2)
NC NC
Ex xja j
a(T)= E i
(3)
j
NG
bi(T) =
~.~vimbm(T)
(4)
m NC
b(T) = EXibi(T) i NC
NG
C = EXiEUimCm i
(5)
m
(6)
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89-99
91
Where Uim is the number of groups m in molecule i, NG is the number of group species in molecule i, NC represents the total number of components in the fluid mixture and qm is the normalized surface area of group m based on a value of 10 for methane. Other parameters have been given by:
amn -----amn
[_:]
exp
b,,(T) = bmexp[- -~T- ]
[ -
T~n
-;-7-
(7) (8)
Where T ~ and amn(T) are the interactions' parameters between group m and n, and b m and T~ are the molecular covolume and temperature constant parameters for each group, respectively. * * + + Each group takes six parameters. They are b m, T.~ , Cm, qm, Tmm, atom. In addition to these parameters, the group interactions between unlike groups in one molecule or two different molecules must be considered. Parameters amn+ and Tm+~ are the required ones. The numbers of these two parameters for satisfying the unlike interaction between groups depend on the number and types of molecules present in a multicomponent fluid. These parameters can be found in the work of (Pults et al., 1989), or in the revised form that have been reported by (Shariat et al., 1993).
3. Calculation procedure The calculation procedure is fairly simple. At a given pressure, temperature, and mixture composition, based on the group parameters, the equation of state parameters are calculated. Then by assuming a value for the molar volume and performing a simple direct substitution, the molar volume is found by Eq. (1) in less than ten iterations. It should be noted that no curve fitting of group parameters nor binary group or molecular interaction optimization was performed in this study. Only the parameters reported by previous investigators were used.
4. Model evaluation In order to test the ability and accuracy of the CORGC EOS in predicting liquid density of multicomponent system, experimental data for 30 multicomponent system at bubble point condition were obtained from literature. Table 1 presents the constituents, reference, composition (or composition range), temperature range and a designated code name for each system. Several theoretical mixing rules for cubic equation of state, (Mansoori, 1986), in addition to empirical ones have been tested. In addition the original mixing rules of the CORGC EOS were considered using both the Pults's et al. (PGC) and Shariat's et al. (SDM) revised parameters. For the sake of completeness, theoretical mixing rules have been evaluated and compared with each other and the mixing rules with the best results, the VDW mixing rules (Redlich-Kwong equation of state), was selected for further comparison and evaluation. Next the original mixing rules of the CORGC EOS were evaluated. Table 2 indicates the original mixing rules of the CORGC EOS does not give accurate enough results no matter which set of group
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89-99
92
Table 1 Multicomponent systems considered in this study System
Ref.
T range (K)
Code name
(0.0075 - 0.4924)N 2 + (0.5075 - 0.9525)C l 0.0593N 2 + 0.9407C 2 (0.0201 - 0.0674)N 2 + (0.9326 - 0.9798)123 (0.3546 - 0.6800)C l + (0.3199 - 0.6454)C 2 (0.2954 - 0.8579)C 2(0.1420 - 0.7006)C 3 (0.7833 - 0.9204)C ~ + (0.0795 - 0.2167)iC 4 (0.7798 - 0.9278)C l + (0.0721 - 0.7798)nC 4 (0.0593 N 2 + 0.8907C ~ + 0.04998nC 4 (0.8600C ~ + 0.0460C 2 + 0.0470C 3 + 0.0457iC 4 (0.0505 - 0.338)N 2 + (0.3414 - 0.8409C ~ + (0.1086 - 0.3206)C2 (0.497 - 0.0995)N z + (0.7977 - 0.9055)C l + (0.0097 - 0.1028)C 3 0.0084N 2 +0.8526C l +0.0083C 2 +0.0507C 3 (0.7644 - 0.9737)C ~ + (0.0080 - 0.00730)C 2 + (0.0034 - 0.02612)(73 0.8513C~ + 0.0576C 2 +0.0481C 3 + 0.0430nC 4 0.0025N 2 +0.8130C 3 + 0.0475C 2 + 0.0087C 3 0.0554N 2 + 0.7909C ~ + 0.0560C 2 + 0.0500C3 + 0.0477nC 4 0.009N 2 + 0.8060C ~ + 0.0468C 2 + 0.0082C3 + 0.05iC4 0.8534C t + 0.0789C 2 + 0.0073C 3 + 0-0085iC4 + 0.0099nC 4 + 0.0097iC 5 + 0.0008nC 5 0.888544C ~ + 0.0500C 2 + 0.0404C 3 + 0.0258iC 4 + 0.2901nC 4 0.0059N 2 +0.7427C~ +0.1650C 2 + 0.0655C 3 + 0.0084iC 4 + 0.0089NC 4 + 0.0007iC 5 + 0.0007nC 5 (0.1464 - 0.8637)C6 H 6 +(0.1363 - 0.8536)nC 4 (0.6304 - 0.8546)cyc - C6(0.2352 - 0.9567)nC 3 (0.4326 - 0.7648)C6 H 6 + (0.2352 - 0.9567)nC 3 ( 0 . 1 7 0 5 - 0.8174)1C4H 8 + ( 0 . 1 8 2 6 - 0.8295)C3 H 6 ( 0 . 2 1 4 3 - 0.8640)nC 3 + ( 0 . 1 3 6 - 0.7857)nC 8 (0.1103 - 0.2221)C 2 + (0.7779 - ).8897)H 2S ( 0 . 1 6 1 4 - 0.9009)CO z + ( 0 . 0 9 9 1 - 0 . 8 3 8 6 ) H 2 S (0.0750 - 0.9192)H 2 S + 0.0808 - 0.925)nC 1o (0.2501 - 0.7988)H 2 S + (0.2012 - 0.7499)nC 5 (0.000-0.126)N 2 + ( 0 . 8 7 4 - 1.000)H2S
Hiza et al., 1977 Hiza et al., 1977 Hiza et al., 1977 Hiza et al., 1977 Hiza et al., 1977 Hiza et al., 1977 Hiza et al., 1977 Haynes, 1982 Haynes, 1982 Hiza and Haynes, 1980
95-140 105-120 100-115 105-140 105-130 110-125 120-130 110-125 115-135 105-120
MLD1 MLD2 MLD3 MLD4 MLD5 MLD6 MLD7 MLD8 MLD9 MLD 10
Hiza and Haynes, 1980
105-120
MLD 11
Hiza and Haynes, 1980 Hiza and Haynes, 1980
105-120 105-125
MLD12 MLD13
Haynes, 1982 Hiza and Haynes, 1980 Hiza and Haynes, 1980
115-135 105-120 105-110
MLD 14 MLD 15 MLD16
Haynes, 1982
115-130
MLDI7
Haynes, 1982
110-125
MLD18
Hiza and Haynes, 1980
110-120
MLD19
Haynes, 1982
110-125
MLD20
Chen et al., 1981 Chen et al., 1981 Glanville et al., 1950 Goff et al., 1950 Kay et al., 1974 Kay and Rambosek, 1953 Bierlein and Kay, 1953 Reamer et al., 1953a Reamer et al., 1953b Bsserer and Robinson, 1975
225.0-270.0 195.0-245.0 200.0-265.0 277.1-311.1 342.2-447.3 278-350 277-320 277.7-444.4 277.7-377.6 256.6322.0
MLD21 MLD22 MLD23 MLD24 MLD25 MLD26 MLD27 MLD28 MLD29 MLD30
p a r a m e t e r s a r e u s e d . H o w e v e r , t h e P u l t s ' s et al. ( P G C ) g r o u p p a r a m e t e r s g i v e m u c h b e t t e r r e s u l t s t h a n t h e S h a r i a t ' s et al. ( S D M ) g r o u p p a r a m e t e r s . T a b l e 2 a l s o i n d i c a t e s t h e o r i g i n a l m i x i n g r u l e s o f t h e CORGC EOS are better than the VDW mixing rules. In view of the above
results, the effect of mixing
investigation. In order to improve liquid density of multicomponent
the accuracy
r u l e is p r o n o u n c e d
of the CORGC
a n d it w a r r a n t s
further
equation of state for prediction of
mixtures, several theoretical mixing rules for cubic equation of state
K. Nasrifar, M. Moshfeghian/ Fluid Phase Equilibria 112 (1995) 89- 99
93
in addition to empirical ones have been evaluated. Based on our evaluation, the following set of one-fluid-mixing rules are proposed for the a and b terms; everything else kept the same: NC NC
a ( T ) t / 3 = ~_, ~ , X i X j a l j / 3 i j
(9)
NC
b(T)'/2= ~.,X~b~(T)~/2
(1o)
i
In other words, Eqs. (3) and (5) will be replaced by Eqs. (9) and (10), respectively.
5. Results In order to demonstrate the accuracy of the proposed mixing rules, Eqs. (9) and (10), for the CORGC equation of state in prediction of liquid density of mixtures, 30 different multicomponent systems have been considered. The calculated results have been compared with experimental data. Table 2 also presents the comparison between the original and proposed mixing rules. Table 2
Table 2 Accuracy of the CORGC equation of state in terms of AAD% a for liquid density prediction System
MLD 1 MLD2 MLD3 MLD4 MLD5 MLD6 MLD7 MLD8 MLD9 MLD 10 MLD I 1 MLD 12 MLD 13 MLD 14 MLD15 MLD 16 MLD17 MLD 18 MLD 19 MLD20 Average
NPT
21 4 6 20 20 4 4 4 5 7 4 4 10 5 4 2 4 5 4 4
Original mixing rules
VDW mixing rules
Proposed mixing rules
PGC
SDM
PGC
SDM
PGC
SDM
1.44 0.42 0.95 0.33 0.49 2.34 2.58 2.07 1.60 2.12 0.44 0.45 0.46 0.27 0.37 0.50 1.9 l 0.99 0.23 0.52 0.94
1.44 1.86 4.79 1.63 5.52 6.84 8.26 0.24 1.51 0.68 2.54 1.55 4.34 4.01 1.65 3.78 1.18 2.06 3.29 2.65 2.95
3.55 0.22 0.28 0.45 1.32 1.62 3.55 1.62 1.49 2.29 0.51 2.57 0.92 1.60 1.56 1.09 2.35 4.12 2.35 1.94 1.59
1.72 0.92 19.72 4.06 6.10 22.27 4.58 13.82 19.41 8.21 8.86 11.11 7.09 6.21 7.64 5.29 17.77 6.96 7.08 5.22 8.00
0.48 0.54 0.95 0.21 0.17 1.52 0.37 1.59 1.01 0.51 0.14 0.12 0.19 0.23 0.13 0.31 1.11 0.77 0.23 0.11 0.45
0.37 1.91 5.10 1.16 3.18 4.55 4.34 0.75 0.90 0.45 1.59 0.97 2.55 2.03 1.55 1.95 0.82 1.40 1.76 1.62 1.84
PGC: Pults-Greenkorn-Chao group parameters are used. SDM: Shariat-Dehghani-Moshfeghian used. ~ AAD% = 1 / N P T "~iNr'IT-((1 pexpi -- pcali 1)/pexpiX 100).
group parameters are
94
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89-99
Table 3 Ability of the CORGC EOS for prediction of liquid density of mixtures in comparison with two well known correlations System MLDI MLD2 MLD3 MLD4 MLD5 MLD6 MLD7 MLD8 MLD9 MLD10 MLD 11 MLDI2 MLD 13 MLD 14 MLD 15 MLD 16 MLD17 MLD 18 MLDI9 MLD20 MLD21 MLD22 MLD23 MLD24 MLD25 Average
NPT 21 4 6 20 20 4 4 4 5 7 4 4 10 5 4 2 4 5 4 4 10 11 14 13 10
CORGC
COSTALD
RSD
AAD%
Bias%a
AAD%
Bias%
AAD%
0.48 0.54 0.96 0.21 0.17 1.52 0.37 1.59 1.01 0.51 0.14 0.12 0.19 0.23 0.13 0.31 1.11 0.77 0.23 0.11 0.66 0.24 1.71 1.52 1.70 0.67
0.44 - 0.54 - 0.96 - 0.11 0.07 1.52 - 0.37 1.56 1.01 0.51 - 0.14 - 0.005 0.09 - 0.23 0.06 - 0.3 l 1.11 0.75 0.23 0.11 0.52 0.09 1.71 1.52 0.16 0.37
1.05 0.21 0.21 0.07 0.48 1.61 1.48 0.75 0.45 0.82 0.73 0.46 0.37 0.35 0.44 0.82 0.73 0.93 0.69 0.7 l 1.44 0.33 0.55 0.45 2.12 0.69
1.05 0.21 - 0.21 0.06 0.48 1.61 1.48 0.75 0.45 0.82 0.73 0.46 0.37 0.35 0.44 0.82 0.73 0.93 0.69 0.71 1.44 0.33 0.55 - 0.44 - 1.98 0.42
2.08 1.32 0.58 1.27 1.00 0.45 0.17 2.65 2.22 2.51 2.27 1.88 0.80 2.08 1,64 2,54 2.43 2.01 2.19 1,93 1.79 0.15 1.04 1.40 2.34 1.43
Bias%
-
-
-
2.08 1.32 0.58 1.27 0.79 0.45 0.13 2.65 2.22 2.51 2.27 1.88 0.80 2.08 1.64 2.54 2.43 2.01 2.19 1.93 1.79 0.15 1.04 0.05 0.16 1.16
Pults-Greenkorn-Chao group parameters are used. RSD, modified Rackett equation by Spence and Danner. aBias% = 1/NPT ~i N~ = I ((I pexpi - pcali I)/pexpi)(100). indicates the superiority o f proposed mixing rules with respect to the original and theoretical ones. As can be seen, average of A A D % (percent o f absolute average deviation) has been improved. It is also noted that for the systems considered, the group parameters of P u l t s - G r e e n k o r n - C h a o gives much better results for either set o f mixing rules. The ability o f the C O R G C equation o f state to predict liquid density o f multicomponent systems is also compared with the most accurate and widely used correlations. Table 3 presents the A A D % and bias% (percent o f relative deviation) for the C O R G C equation o f state and two o f the best correlations of liquid density, Hankinson and T h o m s o n (1979), and Spencer and Danner (1973). As can be seen the C O R G C equation of state predicts liquid density of mixtures better than the two correlations. The average of A A D % for 25 mixtures of hydrocarbon and n i t r o g e n / h y d r o c a r b o n was only 0.67% for the C O R G C equation of state, while for H a n k i n s o n T h o m s o n was 0.69% and for S p e n c e r - D a n n e r was 1.43%. Also Table 3 introduces the least bias% for the C O R G C equation o f state with respect to the two correlations. In order to demonstrate the ability o f the proposed mixing rules further, the accuracy o f the C O R G C equation o f state has been tested for liquid density of slightly polar components. Table 4
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89- 99
95
Table 4 Ability of the CORGC EOS for prediction of liquid density of slightly polar mixtures in comparison with two well known correlations System
MLD26 MLD27 MLD28 MLD29 MLD30 Average
NPT
CORGC
14 11 24 12 20
COSTALD
RSD
AAD%
Bias%
AAD%
Bias%
AAD%
3,08 4,38 3,41 1,74 1,16 2.70
- 3.08 - 4.38 1.86 0.07 - 0.04 - 0.57
3.48 2.80 2.69 4.58 3.05 3.21
2.18 1.28 2.69 1.37 - 3.05 0.78
3.17 2.57 3.08 3.67 2.99 3.09
Bias%
-
1.06 0.99 2.40 1.85 2.99 1.40
P u l t s - G r e e n k o r n - C h a o group parameters are used. RSD, modified Rackett equation by Spencer and Danner.
presents the summary of comparison results for the three methods. It is obvious that the prediction of three methods are not as accurate as the predictions of hydrocarbon and hydrocarbon/nitrogen systems, but the results are in the same range for the three methods. Fig. I shows graphically the percent of liquid density deviation (bias%) as a function of temperature for MLD1 system. The pressure range and number of points used for plotting are 0.14-0.92 MPa and 8, respectively. Figs. 2 and 3 are for MLD13 and MLD29 systems. The pressure ranges and number of plotted points are 0.04-0.13 MPa and 6, and 0.45-2.46 MPa and 4, respectively. It should be emphasized that Fig. 1 represents only a portion of data points for system of
2.00 RSD
!
1.00
0.00 r~
Z
-1.00 O"
-2.00
J 100
I 110
~
I , I 120 130 TEMPERATURE(K)
J
I 140
Fig. 1. Comparison of the ability of the CORGC equation of state with 2 correlations for 0.0475 N 2 + 0.9525 CH 4 (MLD1) system.
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89-99
96
0.50 COST~[J L ,
-
-
L
_ .
-
0.30 RSD r'- "" ..~t--" ~ S C
0.10
O
R
G
~
ca
m
-0.10
-
Z ca
ca - 0 . 3 0 Q.
-0.50
I
i
I
115
105
125
TEMPERATURE(K) Fig. 2. Comparison of the ability of the CORGC equation of state with 2 correlations for 0.3424 CH4+0.3137 C2H 6 +0.3439 C3H 6 (MLD29) system.
MLD 1. The points selected have the same composition. However not all of the points of MLD 1 are at the same composition. This statement is also true for Figs. 2 and 3. Figs. 1-3 each presents the constant composition systems while the composition for each system in
5.00 l
STALD
3.00
i 1.00 m
Z
-I.00
~
~_.
CORGC
-
ca - 3 . 0 0 I--I
<:Y
-5.00
*
270
I
290
i
I
*
I
t
I
310 330 350 TEMPERATURE(K)
i
I
370
i
390
Fig. 3. Comparison of the ability of the CORGC equation of state with 2 correlations ofr 0.7499nCsH~2 +0.2501H2S (MLD29) system.
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89- 99
97
Table 1 is not necessarily constant. For example, Fig. 1 presents the comparison results for those points of MLD1 whose composition range is 4.75 mol.% N 2 and 95.25 mol.% C H 4 (only 8 points) while the composition range for this system in Table 3 is 4.75 to 49.24 mol.% N 2 and 50.75 to 95.25 mol.% C H 4 (altogether 21 points).
6. Conclusions In addition to the original mixing rules of the CORGC EOS, several other reported mixing rules have been evaluated for prediction of liquid densities of multicomponent systems. Unfortunately, none of them gave accurate enough results. Therefore, a new set of simple mixing rule has been proposed along with the CORGC equation of state for accurate prediction of liquid density of multicomponent systems. The accuracy of the proposed method is compared with the best available correlations. Based on the evaluation performed for 30 multicomponent systems, except for polar fluids, excellent accuracy has been obtained. It is probably the first time that a relatively simple equation of state with a limited number of parameters predicts liquid density better than the known correlations. Contrary to the COSTALD method by Hankinson-Thomson and the Spencer-Danner modification of Rackett equation which require additional parameters such as ZRA, OJSRK and characteristic volume, V *, the proposed method does not require any specific molecular parameters. In other words, with the same group parameters that pure component thermodynamic properties or vapor liquid equilibrium behaviors are calculated, liquid densities are also calculated with the desired accuracy. Specially, no curve fitting of interaction parameters were required. Based on our evaluations, the recommended group values for liquid density calculations are the original ones as reported by (Pults et al., 1989) and these values were used in our evaluations. The superiority of the Pult" s e t al. group parameters in comparison to (Shariat et al., 1993) could be due to the fact that the former researchers used only vapor pressure and liquid density as the basis for group parameter determination. However, Shariat et al. used vapor volume and heat of vaporization in addition to vapor pressure and liquid density as the basis for group parameter determination. Therefore, it seems possible that the CORGC equation of state can be used not only for prediction of vapor phase properties which is common in other equations of state, but also may be used for prediction of liquid density with the desired accuracy.
7. List of symbols AAD AAAD a
b COR c
NC NG NPT
average absolute deviation average of average absolute deviation molecular attraction, c m 6 bar mol-2 molecular covolume, cm 3 mol- 1 chain of rotator no. of external degree of freedom, based on 10 for methane no. of components in the system no. of groups in the system no. of points
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89-99
98
P PGC q R SDM SRK T V V* Y Z ZRA
absolute pressure (atm) Pults-Greenkorn-Chao group parameters normalized surface area of a group gas constant, 82.053 cm 3 atm m o l - l K-1 Shariat-Dehghani-Moshfeghian group parameters Soave-Redlich-Kwong equation of state absolute temperature (K) molar volume, cm 3 m o l curve fitted characteristic molar volume for COSTALD method, cm 3 m o l reduced density compressibility factor. curve fitted critical compressibility factor for Rackett equation
Greek letters a v O.)SRK
constant equal to 1.078, in CORGC EOS. no. of group in each molecule curve fitted acentric factor based the Soave-Redlich-Kwong equation of state
Superscript + •
characteristic group energy parameter characteristic group covolume parameter in CORGC EOS
Subscripts c I j m n
critical property molecule i molecule j group m group n
Acknowledgements The authors wish to express their appreciation to the School of Engineering for use of computer facilities and the financial support of the Vice-Chancellor for Research of Shiraz University.
References Bierlein, J.A. and Kay, W.B., 1953. Phase equilibrium properties of system carbon dioxide-hydrogen sulfide, Ind. Eng. Chem., 45(3): 618-624. Bsserer, G.J. and Robinson, D.B., 1975. Equilibrium-phaseproperties of nitrogen-hydrogensulfide system, J. Chem. Eng. Data, 20 (2): 157-161. Chen, W.L., Luks, K.D. and Khon, J.P., 1981. Three phase solid-liquid-vapor Equilibria of binary hydrocarbonsystems propane-benzene, propane cyclohexane,n-butane-benzene,n-butane-cyclohexane,n-butane-n-decane, and n-butane-ndodecane, J. Chem. Eng. Data, 26(3): 310-312.
K. Nasrifar, M. Moshfeghian / Fluid Plu~se Equilibrm 112 (1995) 89- 99
9t,~
Goff, G.H., Farrington, P.S. and Sage, B.H., 1950. Volumetric and phase behavior of Propene-I-butene system, Ind. Eng. Chem., 42(4): 735-743. Glanville, J.W., Sage, B.H. and Lacey, W.W., 1950. Volumetric and phase behavior of propane-benzene system, Ind. Eng. Chem., 42(3): 508-513. Hiza, M.J. and Haynes, W.M., 1980. Orthorombic Liquid densities and excess volumes for multicomponent mixtures of low molar-mass alkanes and nitrogen between 105 and 125 K, J. Chem. Thermodynamics, 1(12): 1-10. Hiza, M.J., Haynes, W.M. and Parrish, W.R., 1977. Orthorombic Liquid densities and excess volumes tbr binary mixtures of low molar mass alkanes and nitrogen between 105 and 140 K, J. Chem. Thermodynamics, 9(9): 873-896. Hankinson, R.W. and Thomson, G.H., 1979. A new correlation for saturated densities of liquids and their mixtures, AIChE J., 25(4): 653-663. Haynes, W.M., 1982. Measurements of orthorhombic-liquid densities of multicomponent mixtures of LNG components (N 2. CH 4. C 2 H 6, C3H s, CH3CH(CH3)CH3, C 4 Hto, CH3CH(CH3)C2Hs, and CsHI2) between 110 and 130 K. J. Chem. Thermodynamics, 14(7): 603-612. Kay, W.B. and Rambosek, G.M., 1953. Liquid-vapor equilibrium relations in binary systems, Ind Eng Chem, 45 (1): 221 -226. Kay, WB., Genco, J. and Fichtner, D.A., 1974. Vapor-liquid equilibrium relationships of binary systems propane-n-octane and n-butane-n-octane, J. Chem. Eng. Data, 19(3): 275-280. Mansoori, G.A., 1986. Mixing rules for cubic equations of state, American Chemical Society, 15:314-330. Peng. D.Y. and Robinson, D., 1976. A new two-constant equation of state, lnd Eng Chem Fund., 15 (I): 59-64. Pults, J.D., Greenkorn, R.A. and Chao, K.C., 1989. Chain of rotator group contribution equation of state, Chem Eng Sci., 11 (44): 2553-2564. Reamer, H.H., Sage, B.H. and Lacey, W.N., 1953a. Phase equilibrium in hydrocarbon system, Ind. Eng. Chem., 45(8): 1805-1809. Reamer, H.H., Selleck, F.T., Sage, B.H. and Lacey, W.N., 1953b. Volumetric and phase behavior of decane-hydrogen sulfide system, Ind. Eng. Chem., 45(8): 1810-1812, Spencer. C.F. and Danner, R.P., 1973. Prediction of bubble-point density of mixtures, J. Chem. and Eng. Data, 18(2): 230-234. Shariat, M.H., Dehghany, F. and Moshfeghian, M., 1993. Extension and evaluation of the chain of rotor group contribution equation of state for prediction of thermodynamic properties of polar and nonpolar compounds, Fluid Phase Equilibria, 85: 19-40. Soave, G. 1972. Equilibrium constants from a modified Redlich-Kwong equation of state, Chem. Eng. Sci., 27(7): 1197-1203. Zia-Razzaz, M. and Moshfeghian, M., 1993. Application of the CORGC equation of state with a new mixing rule to polar and nonpolar fluids, Fluid Phase Equilibria, 91: 239-263.
,.
um
llll ELSEVIER
Fluid Phase Equilibria 112 (1995) 89-99
Prediction of liquid density of LNG, N2, HzS and C O 2 multicomponent systems by the CORGC equation of state Khashayar Nasrifar, Mahmood Moshfeghian * Department of Chemical Engineering, Shiraz University, Shiraz, Iran
Received 29 April 1994; accepted 11 May 1995
Abstract The capability of the CORGC equation of state for prediction of liquid density of multicomponent mixtures has been tested. In order to improve its accuracy, a new set of mixing rules has been developed which gives better predictions than the best available correlations. The average of absolute deviation for 25 multicomponent systems of nonpolar was only 0.67% and 2.7% for 5 slightly polar systems.
Keywords: Theory; Equation of state; Group contributions; Liquid density
1. Introduction Thermodynamic properties and their availability for design of chemical engineering processes are essential part of any computer simulation program. In the absence of experimental data, it is customary to rely on the predicted values by a powerful tool that is named equation of state. Unfortunately, the capability of equations of state such as Soave-Redlich-Kwong (Soave, 1972) and Peng and Robinson (1976) to predict liquid density has been very poor and engineers have been forced to use correlations that are more reliable. Recently, an equation of state has been introduced by (Pults et al., 1989), that was based on chain of rotator (COR) equation of state. This equation of state is named the chain of rotator group contribution (CORGC) equation of state. The CORGC equation of state has lower errors in prediction of thermodynamic properties in comparison with other equations of state. These results can be found in the work of (Pults et al., 1989), (Shariat et al., 1993), and (Zia-Razzaz and Moshfeghian, 1993).
* C o r r e s p o n d i n g author. 0378-3812/95/$09.50 © 1995 Elsevier Science B.V. All fights reserved SSDI 0 3 7 8 - 3 8 1 2 ( 9 5 ) 0 2 7 8 9 - 0
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89-99
90
Surprisingly, the error for prediction of pure component liquid density was comparable with the best correlations, (Zia-Razzaz and Moshfeghian, 1993). In this paper, the CORGC equation of state has been applied to predict liquid density of multicomponent systems such as hydrocarbons and mixtures containing non hydrocarbon components. In order to improve its accuracy, a new set of mixing rule has been proposed. Its capabilities have been compared with the best correlations such as (Hankinson and Thomson, 1979) correlation (COSTALD) and the modified Rackett correlation (RSD) which was developed by (Spencer and Danner, 1973).
2. The C O R G C equation of state
The CORGC equation of state as developed by (Pults et al., 1989), was accorded to Van der waals and perturbation theory. They considered the pressure of a fluid as the sum of repulsive and attractive contribution. In the CORGC equation of state the repulsive pressure is that from translational motion and group rotation of molecules in COR equation of state. The attractive pressure has been represented by a form of the Redlich-Kwong equation. They have introduced the CORGC equation of state as:
Z
PV RT
l+y+y2-y3
=--=
(l-y) 3
c +
(a-l) "2"
3y+3ay
2-(a+l)y 3
(l-y) 3
-
a(T) RT[V+b(T)]
(1)
In this equation, y = b(T)/4V is the reduced density, b(T) is the molecular covolume, V is the molar volume, c is the number of external degree of freedom, a is a constant equal to 1.078, a(T) is the attractive interaction parameter and Z is the compressibility factor. The equation of state parameters a, b and c are made up of group contributions according to the following equations: NG NG aij=
E EVimVjnqmqnamn(T) m n
(2)
NC NC
Ex xja j
a(T)= E i
(3)
j
NG
bi(T) =
~.~vimbm(T)
(4)
m NC
b(T) = EXibi(T) i NC
NG
C = EXiEUimCm i
(5)
m
(6)
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89-99
91
Where Uim is the number of groups m in molecule i, NG is the number of group species in molecule i, NC represents the total number of components in the fluid mixture and qm is the normalized surface area of group m based on a value of 10 for methane. Other parameters have been given by:
amn -----amn
[_:]
exp
b,,(T) = bmexp[- -~T- ]
[ -
T~n
-;-7-
(7) (8)
Where T ~ and amn(T) are the interactions' parameters between group m and n, and b m and T~ are the molecular covolume and temperature constant parameters for each group, respectively. * * + + Each group takes six parameters. They are b m, T.~ , Cm, qm, Tmm, atom. In addition to these parameters, the group interactions between unlike groups in one molecule or two different molecules must be considered. Parameters amn+ and Tm+~ are the required ones. The numbers of these two parameters for satisfying the unlike interaction between groups depend on the number and types of molecules present in a multicomponent fluid. These parameters can be found in the work of (Pults et al., 1989), or in the revised form that have been reported by (Shariat et al., 1993).
3. Calculation procedure The calculation procedure is fairly simple. At a given pressure, temperature, and mixture composition, based on the group parameters, the equation of state parameters are calculated. Then by assuming a value for the molar volume and performing a simple direct substitution, the molar volume is found by Eq. (1) in less than ten iterations. It should be noted that no curve fitting of group parameters nor binary group or molecular interaction optimization was performed in this study. Only the parameters reported by previous investigators were used.
4. Model evaluation In order to test the ability and accuracy of the CORGC EOS in predicting liquid density of multicomponent system, experimental data for 30 multicomponent system at bubble point condition were obtained from literature. Table 1 presents the constituents, reference, composition (or composition range), temperature range and a designated code name for each system. Several theoretical mixing rules for cubic equation of state, (Mansoori, 1986), in addition to empirical ones have been tested. In addition the original mixing rules of the CORGC EOS were considered using both the Pults's et al. (PGC) and Shariat's et al. (SDM) revised parameters. For the sake of completeness, theoretical mixing rules have been evaluated and compared with each other and the mixing rules with the best results, the VDW mixing rules (Redlich-Kwong equation of state), was selected for further comparison and evaluation. Next the original mixing rules of the CORGC EOS were evaluated. Table 2 indicates the original mixing rules of the CORGC EOS does not give accurate enough results no matter which set of group
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89-99
92
Table 1 Multicomponent systems considered in this study System
Ref.
T range (K)
Code name
(0.0075 - 0.4924)N 2 + (0.5075 - 0.9525)C l 0.0593N 2 + 0.9407C 2 (0.0201 - 0.0674)N 2 + (0.9326 - 0.9798)123 (0.3546 - 0.6800)C l + (0.3199 - 0.6454)C 2 (0.2954 - 0.8579)C 2(0.1420 - 0.7006)C 3 (0.7833 - 0.9204)C ~ + (0.0795 - 0.2167)iC 4 (0.7798 - 0.9278)C l + (0.0721 - 0.7798)nC 4 (0.0593 N 2 + 0.8907C ~ + 0.04998nC 4 (0.8600C ~ + 0.0460C 2 + 0.0470C 3 + 0.0457iC 4 (0.0505 - 0.338)N 2 + (0.3414 - 0.8409C ~ + (0.1086 - 0.3206)C2 (0.497 - 0.0995)N z + (0.7977 - 0.9055)C l + (0.0097 - 0.1028)C 3 0.0084N 2 +0.8526C l +0.0083C 2 +0.0507C 3 (0.7644 - 0.9737)C ~ + (0.0080 - 0.00730)C 2 + (0.0034 - 0.02612)(73 0.8513C~ + 0.0576C 2 +0.0481C 3 + 0.0430nC 4 0.0025N 2 +0.8130C 3 + 0.0475C 2 + 0.0087C 3 0.0554N 2 + 0.7909C ~ + 0.0560C 2 + 0.0500C3 + 0.0477nC 4 0.009N 2 + 0.8060C ~ + 0.0468C 2 + 0.0082C3 + 0.05iC4 0.8534C t + 0.0789C 2 + 0.0073C 3 + 0-0085iC4 + 0.0099nC 4 + 0.0097iC 5 + 0.0008nC 5 0.888544C ~ + 0.0500C 2 + 0.0404C 3 + 0.0258iC 4 + 0.2901nC 4 0.0059N 2 +0.7427C~ +0.1650C 2 + 0.0655C 3 + 0.0084iC 4 + 0.0089NC 4 + 0.0007iC 5 + 0.0007nC 5 (0.1464 - 0.8637)C6 H 6 +(0.1363 - 0.8536)nC 4 (0.6304 - 0.8546)cyc - C6(0.2352 - 0.9567)nC 3 (0.4326 - 0.7648)C6 H 6 + (0.2352 - 0.9567)nC 3 ( 0 . 1 7 0 5 - 0.8174)1C4H 8 + ( 0 . 1 8 2 6 - 0.8295)C3 H 6 ( 0 . 2 1 4 3 - 0.8640)nC 3 + ( 0 . 1 3 6 - 0.7857)nC 8 (0.1103 - 0.2221)C 2 + (0.7779 - ).8897)H 2S ( 0 . 1 6 1 4 - 0.9009)CO z + ( 0 . 0 9 9 1 - 0 . 8 3 8 6 ) H 2 S (0.0750 - 0.9192)H 2 S + 0.0808 - 0.925)nC 1o (0.2501 - 0.7988)H 2 S + (0.2012 - 0.7499)nC 5 (0.000-0.126)N 2 + ( 0 . 8 7 4 - 1.000)H2S
Hiza et al., 1977 Hiza et al., 1977 Hiza et al., 1977 Hiza et al., 1977 Hiza et al., 1977 Hiza et al., 1977 Hiza et al., 1977 Haynes, 1982 Haynes, 1982 Hiza and Haynes, 1980
95-140 105-120 100-115 105-140 105-130 110-125 120-130 110-125 115-135 105-120
MLD1 MLD2 MLD3 MLD4 MLD5 MLD6 MLD7 MLD8 MLD9 MLD 10
Hiza and Haynes, 1980
105-120
MLD 11
Hiza and Haynes, 1980 Hiza and Haynes, 1980
105-120 105-125
MLD12 MLD13
Haynes, 1982 Hiza and Haynes, 1980 Hiza and Haynes, 1980
115-135 105-120 105-110
MLD 14 MLD 15 MLD16
Haynes, 1982
115-130
MLDI7
Haynes, 1982
110-125
MLD18
Hiza and Haynes, 1980
110-120
MLD19
Haynes, 1982
110-125
MLD20
Chen et al., 1981 Chen et al., 1981 Glanville et al., 1950 Goff et al., 1950 Kay et al., 1974 Kay and Rambosek, 1953 Bierlein and Kay, 1953 Reamer et al., 1953a Reamer et al., 1953b Bsserer and Robinson, 1975
225.0-270.0 195.0-245.0 200.0-265.0 277.1-311.1 342.2-447.3 278-350 277-320 277.7-444.4 277.7-377.6 256.6322.0
MLD21 MLD22 MLD23 MLD24 MLD25 MLD26 MLD27 MLD28 MLD29 MLD30
p a r a m e t e r s a r e u s e d . H o w e v e r , t h e P u l t s ' s et al. ( P G C ) g r o u p p a r a m e t e r s g i v e m u c h b e t t e r r e s u l t s t h a n t h e S h a r i a t ' s et al. ( S D M ) g r o u p p a r a m e t e r s . T a b l e 2 a l s o i n d i c a t e s t h e o r i g i n a l m i x i n g r u l e s o f t h e CORGC EOS are better than the VDW mixing rules. In view of the above
results, the effect of mixing
investigation. In order to improve liquid density of multicomponent
the accuracy
r u l e is p r o n o u n c e d
of the CORGC
a n d it w a r r a n t s
further
equation of state for prediction of
mixtures, several theoretical mixing rules for cubic equation of state
K. Nasrifar, M. Moshfeghian/ Fluid Phase Equilibria 112 (1995) 89- 99
93
in addition to empirical ones have been evaluated. Based on our evaluation, the following set of one-fluid-mixing rules are proposed for the a and b terms; everything else kept the same: NC NC
a ( T ) t / 3 = ~_, ~ , X i X j a l j / 3 i j
(9)
NC
b(T)'/2= ~.,X~b~(T)~/2
(1o)
i
In other words, Eqs. (3) and (5) will be replaced by Eqs. (9) and (10), respectively.
5. Results In order to demonstrate the accuracy of the proposed mixing rules, Eqs. (9) and (10), for the CORGC equation of state in prediction of liquid density of mixtures, 30 different multicomponent systems have been considered. The calculated results have been compared with experimental data. Table 2 also presents the comparison between the original and proposed mixing rules. Table 2
Table 2 Accuracy of the CORGC equation of state in terms of AAD% a for liquid density prediction System
MLD 1 MLD2 MLD3 MLD4 MLD5 MLD6 MLD7 MLD8 MLD9 MLD 10 MLD I 1 MLD 12 MLD 13 MLD 14 MLD15 MLD 16 MLD17 MLD 18 MLD 19 MLD20 Average
NPT
21 4 6 20 20 4 4 4 5 7 4 4 10 5 4 2 4 5 4 4
Original mixing rules
VDW mixing rules
Proposed mixing rules
PGC
SDM
PGC
SDM
PGC
SDM
1.44 0.42 0.95 0.33 0.49 2.34 2.58 2.07 1.60 2.12 0.44 0.45 0.46 0.27 0.37 0.50 1.9 l 0.99 0.23 0.52 0.94
1.44 1.86 4.79 1.63 5.52 6.84 8.26 0.24 1.51 0.68 2.54 1.55 4.34 4.01 1.65 3.78 1.18 2.06 3.29 2.65 2.95
3.55 0.22 0.28 0.45 1.32 1.62 3.55 1.62 1.49 2.29 0.51 2.57 0.92 1.60 1.56 1.09 2.35 4.12 2.35 1.94 1.59
1.72 0.92 19.72 4.06 6.10 22.27 4.58 13.82 19.41 8.21 8.86 11.11 7.09 6.21 7.64 5.29 17.77 6.96 7.08 5.22 8.00
0.48 0.54 0.95 0.21 0.17 1.52 0.37 1.59 1.01 0.51 0.14 0.12 0.19 0.23 0.13 0.31 1.11 0.77 0.23 0.11 0.45
0.37 1.91 5.10 1.16 3.18 4.55 4.34 0.75 0.90 0.45 1.59 0.97 2.55 2.03 1.55 1.95 0.82 1.40 1.76 1.62 1.84
PGC: Pults-Greenkorn-Chao group parameters are used. SDM: Shariat-Dehghani-Moshfeghian used. ~ AAD% = 1 / N P T "~iNr'IT-((1 pexpi -- pcali 1)/pexpiX 100).
group parameters are
94
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89-99
Table 3 Ability of the CORGC EOS for prediction of liquid density of mixtures in comparison with two well known correlations System MLDI MLD2 MLD3 MLD4 MLD5 MLD6 MLD7 MLD8 MLD9 MLD10 MLD 11 MLDI2 MLD 13 MLD 14 MLD 15 MLD 16 MLD17 MLD 18 MLDI9 MLD20 MLD21 MLD22 MLD23 MLD24 MLD25 Average
NPT 21 4 6 20 20 4 4 4 5 7 4 4 10 5 4 2 4 5 4 4 10 11 14 13 10
CORGC
COSTALD
RSD
AAD%
Bias%a
AAD%
Bias%
AAD%
0.48 0.54 0.96 0.21 0.17 1.52 0.37 1.59 1.01 0.51 0.14 0.12 0.19 0.23 0.13 0.31 1.11 0.77 0.23 0.11 0.66 0.24 1.71 1.52 1.70 0.67
0.44 - 0.54 - 0.96 - 0.11 0.07 1.52 - 0.37 1.56 1.01 0.51 - 0.14 - 0.005 0.09 - 0.23 0.06 - 0.3 l 1.11 0.75 0.23 0.11 0.52 0.09 1.71 1.52 0.16 0.37
1.05 0.21 0.21 0.07 0.48 1.61 1.48 0.75 0.45 0.82 0.73 0.46 0.37 0.35 0.44 0.82 0.73 0.93 0.69 0.7 l 1.44 0.33 0.55 0.45 2.12 0.69
1.05 0.21 - 0.21 0.06 0.48 1.61 1.48 0.75 0.45 0.82 0.73 0.46 0.37 0.35 0.44 0.82 0.73 0.93 0.69 0.71 1.44 0.33 0.55 - 0.44 - 1.98 0.42
2.08 1.32 0.58 1.27 1.00 0.45 0.17 2.65 2.22 2.51 2.27 1.88 0.80 2.08 1,64 2,54 2.43 2.01 2.19 1,93 1.79 0.15 1.04 1.40 2.34 1.43
Bias%
-
-
-
2.08 1.32 0.58 1.27 0.79 0.45 0.13 2.65 2.22 2.51 2.27 1.88 0.80 2.08 1.64 2.54 2.43 2.01 2.19 1.93 1.79 0.15 1.04 0.05 0.16 1.16
Pults-Greenkorn-Chao group parameters are used. RSD, modified Rackett equation by Spence and Danner. aBias% = 1/NPT ~i N~ = I ((I pexpi - pcali I)/pexpi)(100). indicates the superiority o f proposed mixing rules with respect to the original and theoretical ones. As can be seen, average of A A D % (percent o f absolute average deviation) has been improved. It is also noted that for the systems considered, the group parameters of P u l t s - G r e e n k o r n - C h a o gives much better results for either set o f mixing rules. The ability o f the C O R G C equation o f state to predict liquid density o f multicomponent systems is also compared with the most accurate and widely used correlations. Table 3 presents the A A D % and bias% (percent o f relative deviation) for the C O R G C equation o f state and two o f the best correlations of liquid density, Hankinson and T h o m s o n (1979), and Spencer and Danner (1973). As can be seen the C O R G C equation of state predicts liquid density of mixtures better than the two correlations. The average of A A D % for 25 mixtures of hydrocarbon and n i t r o g e n / h y d r o c a r b o n was only 0.67% for the C O R G C equation of state, while for H a n k i n s o n T h o m s o n was 0.69% and for S p e n c e r - D a n n e r was 1.43%. Also Table 3 introduces the least bias% for the C O R G C equation o f state with respect to the two correlations. In order to demonstrate the ability o f the proposed mixing rules further, the accuracy o f the C O R G C equation o f state has been tested for liquid density of slightly polar components. Table 4
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89- 99
95
Table 4 Ability of the CORGC EOS for prediction of liquid density of slightly polar mixtures in comparison with two well known correlations System
MLD26 MLD27 MLD28 MLD29 MLD30 Average
NPT
CORGC
14 11 24 12 20
COSTALD
RSD
AAD%
Bias%
AAD%
Bias%
AAD%
3,08 4,38 3,41 1,74 1,16 2.70
- 3.08 - 4.38 1.86 0.07 - 0.04 - 0.57
3.48 2.80 2.69 4.58 3.05 3.21
2.18 1.28 2.69 1.37 - 3.05 0.78
3.17 2.57 3.08 3.67 2.99 3.09
Bias%
-
1.06 0.99 2.40 1.85 2.99 1.40
P u l t s - G r e e n k o r n - C h a o group parameters are used. RSD, modified Rackett equation by Spencer and Danner.
presents the summary of comparison results for the three methods. It is obvious that the prediction of three methods are not as accurate as the predictions of hydrocarbon and hydrocarbon/nitrogen systems, but the results are in the same range for the three methods. Fig. I shows graphically the percent of liquid density deviation (bias%) as a function of temperature for MLD1 system. The pressure range and number of points used for plotting are 0.14-0.92 MPa and 8, respectively. Figs. 2 and 3 are for MLD13 and MLD29 systems. The pressure ranges and number of plotted points are 0.04-0.13 MPa and 6, and 0.45-2.46 MPa and 4, respectively. It should be emphasized that Fig. 1 represents only a portion of data points for system of
2.00 RSD
!
1.00
0.00 r~
Z
-1.00 O"
-2.00
J 100
I 110
~
I , I 120 130 TEMPERATURE(K)
J
I 140
Fig. 1. Comparison of the ability of the CORGC equation of state with 2 correlations for 0.0475 N 2 + 0.9525 CH 4 (MLD1) system.
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89-99
96
0.50 COST~[J L ,
-
-
L
_ .
-
0.30 RSD r'- "" ..~t--" ~ S C
0.10
O
R
G
~
ca
m
-0.10
-
Z ca
ca - 0 . 3 0 Q.
-0.50
I
i
I
115
105
125
TEMPERATURE(K) Fig. 2. Comparison of the ability of the CORGC equation of state with 2 correlations for 0.3424 CH4+0.3137 C2H 6 +0.3439 C3H 6 (MLD29) system.
MLD 1. The points selected have the same composition. However not all of the points of MLD 1 are at the same composition. This statement is also true for Figs. 2 and 3. Figs. 1-3 each presents the constant composition systems while the composition for each system in
5.00 l
STALD
3.00
i 1.00 m
Z
-I.00
~
~_.
CORGC
-
ca - 3 . 0 0 I--I
<:Y
-5.00
*
270
I
290
i
I
*
I
t
I
310 330 350 TEMPERATURE(K)
i
I
370
i
390
Fig. 3. Comparison of the ability of the CORGC equation of state with 2 correlations ofr 0.7499nCsH~2 +0.2501H2S (MLD29) system.
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89- 99
97
Table 1 is not necessarily constant. For example, Fig. 1 presents the comparison results for those points of MLD1 whose composition range is 4.75 mol.% N 2 and 95.25 mol.% C H 4 (only 8 points) while the composition range for this system in Table 3 is 4.75 to 49.24 mol.% N 2 and 50.75 to 95.25 mol.% C H 4 (altogether 21 points).
6. Conclusions In addition to the original mixing rules of the CORGC EOS, several other reported mixing rules have been evaluated for prediction of liquid densities of multicomponent systems. Unfortunately, none of them gave accurate enough results. Therefore, a new set of simple mixing rule has been proposed along with the CORGC equation of state for accurate prediction of liquid density of multicomponent systems. The accuracy of the proposed method is compared with the best available correlations. Based on the evaluation performed for 30 multicomponent systems, except for polar fluids, excellent accuracy has been obtained. It is probably the first time that a relatively simple equation of state with a limited number of parameters predicts liquid density better than the known correlations. Contrary to the COSTALD method by Hankinson-Thomson and the Spencer-Danner modification of Rackett equation which require additional parameters such as ZRA, OJSRK and characteristic volume, V *, the proposed method does not require any specific molecular parameters. In other words, with the same group parameters that pure component thermodynamic properties or vapor liquid equilibrium behaviors are calculated, liquid densities are also calculated with the desired accuracy. Specially, no curve fitting of interaction parameters were required. Based on our evaluations, the recommended group values for liquid density calculations are the original ones as reported by (Pults et al., 1989) and these values were used in our evaluations. The superiority of the Pult" s e t al. group parameters in comparison to (Shariat et al., 1993) could be due to the fact that the former researchers used only vapor pressure and liquid density as the basis for group parameter determination. However, Shariat et al. used vapor volume and heat of vaporization in addition to vapor pressure and liquid density as the basis for group parameter determination. Therefore, it seems possible that the CORGC equation of state can be used not only for prediction of vapor phase properties which is common in other equations of state, but also may be used for prediction of liquid density with the desired accuracy.
7. List of symbols AAD AAAD a
b COR c
NC NG NPT
average absolute deviation average of average absolute deviation molecular attraction, c m 6 bar mol-2 molecular covolume, cm 3 mol- 1 chain of rotator no. of external degree of freedom, based on 10 for methane no. of components in the system no. of groups in the system no. of points
K. Nasrifar, M. Moshfeghian / Fluid Phase Equilibria 112 (1995) 89-99
98
P PGC q R SDM SRK T V V* Y Z ZRA
absolute pressure (atm) Pults-Greenkorn-Chao group parameters normalized surface area of a group gas constant, 82.053 cm 3 atm m o l - l K-1 Shariat-Dehghani-Moshfeghian group parameters Soave-Redlich-Kwong equation of state absolute temperature (K) molar volume, cm 3 m o l curve fitted characteristic molar volume for COSTALD method, cm 3 m o l reduced density compressibility factor. curve fitted critical compressibility factor for Rackett equation
Greek letters a v O.)SRK
constant equal to 1.078, in CORGC EOS. no. of group in each molecule curve fitted acentric factor based the Soave-Redlich-Kwong equation of state
Superscript + •
characteristic group energy parameter characteristic group covolume parameter in CORGC EOS
Subscripts c I j m n
critical property molecule i molecule j group m group n
Acknowledgements The authors wish to express their appreciation to the School of Engineering for use of computer facilities and the financial support of the Vice-Chancellor for Research of Shiraz University.
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