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New equation of state based on the significant structure model
Fluid Phase Equilibria, 41 (1989) 17-38 Elsevier Science Publishers B.V., Amsterdam -
17
Printed in The Netherlands
NEW EQUATION OF STATE BASED ON THE SIGNIFICANT STRUCTURE MODEL KAZUHIKO
SUZUKI
and HARUHUSA
Central Research Laboratories,
Idemitsu
RICHARD
L. SMITH,
HIROSI
Department
of Chemical Engineering,
SUE
Kosan Co., Ltd., Sodegaura, Chiba, 299-02 (Japan)
INOMATA,
KUNIO
ARAI and SHOZABURO
SAITO
Tohoku University, Sendai, 980 (Japan)
(Received July 5, 1988; accepted in final form January 4, 1989)
ABSTRACT Suzuki, K., Sue, H., Smith, R.L., Inomata, H., Arai, K. and Saito, S., 1989. New equation of state based on the significant structure model. Fluid Phase Equilibria, 47: 17-38. A new non-cubic equation of state based on molecular thermodynamics has been developed by considering cell theory and forming the potential energy of a system from the volumetric average of the van der Waals type and the 15-6 Mie type potential energies. The proposed equation has four constants and requires knowledge of the critical temperature, the critical pressure, the critical volume, the excluded volume and an additional parameter to characterize any given fluid. The new equation yields good agreement with experimental saturation properties of 29 non-polar and 16 polar substances from reduced temperatures of 0.5 to the critical temperature. The new equation exactly fits the critical point, generally gives better predictions of liquid volumes than, and predictions of vapor pressures and heats of vaporization equally as good as commonly used cubic equations, and yet only requires two fitting parameters.
INTRODUCTION
Since the time of van der Waals, two general approaches have been taken to the development of equations of state in terms of molecular interactions. The first approach begins from the gas state. The van der Waals type of equation of state adds corrections to the ideal gas equation of state and makes it possible to predict a condensed phase. While this type of equation can predict volumetric properties in the gas region well, the prediction of volumetric properties in the liquid region is only qualitative. Various researchers have improved the van der Waals type equation by empirically modifying the form of the attractive term and its temperature dependence or by adding other parameters (Soave, 1972; Pate1 and Teja, 1982). 0378-3812/89/$03.50
0 1989 Elsevier Science Publishers B.V.
18
The second approach begins from the liquid state and predicts liquid properties using the basic properties of the pure substance such as the hard-sphere diameter and the potential energy to predict the liquid properties. These models include the cell model, the hole model and the significant liquid structure model (Eyring et al., 1958). In the present work, a new equation of state is developed from a consideration of the van der Waals equation and the cell model, and on the assumption that the potential energy of a system can be represented by the volumetric average of the van der Waals type and the 15-6 Mie type potential energies. FORMULATION
OF THE EQUATION
OF STATE
We consider the system to be described by V, T, NA, (J and assume the following. (i) The system is a group of independent cells. Each cell has a volume V-, = V/N, and no interactions exist among the cells. (ii) The free vo1ume V, is defined as the cell volume VI minus the excluded volume b/N,. (iii) The potential energy of a system is the volumetric average of the van der Waals type and the 15-6 Mie-type potential energies. On these assumptions, we derive the equation of state by the following procedures. (i) Free volume V, The free volume per particle is Q=(V-b)/N,
0)
(ii) System potential energy It is difficult to describe the potential energy of real systems over a wide
range of densities. For the dilute region, the van der Waals type potential energy can predict P- V-T properties. However, the Mie type potential is useful in the dense region, because it contains a repulsive term and is mathematically simple. To describe a liquid property X, Eyring proposed the volumetric average method based on his significant liquid structure model. That is
x=xs$+xg il-+
1
(2)
where X,, Xa are the values of the property in the solid and the gas states, respectively. Applying his definition to the partition function based on the 12-6 Lennard-Jones potential for the liquid state and on the ideal gas condition for the gas state, Eyring was able to derive his equation of state.
19
We modify his concepts and describe the potential energy of a system as the volumetric average of the van der Waals type and the Mie type potential energies. The van der Waals type potential energy is @P(V) VDW
=
-U/V
(3)
The Mie potential is +(r)Mie=C’E[(u/r)m-
=
c’c
(a/r)6]
1
(fJ/r)‘”
-
(4)
(o/r)“]
where c, = ( m/6)1’(m-6) (m-6) If we assume that liquid cells are in the close-packed state and all molecules are fixed at the center of each cell with coordination number z, the relationship between the volume and radius is a3/fi
= V,/N,
(5)
r’/fi
= V/N,
(6)
and the potential energy can be written as ~(V),,=N,C’r~
[(?i’-(+,‘I
=#!)y!i’]
(7)
where c = 3N,c’cz/2 Following Eyring, we assume that the potential energy of a system is the volumetric average of the above two potentials. @(v)vDw+
~a(v)h4ie
(iii) Cell model Following the simple cell model concept, we assume that each cell is independent of its own uniform potential and that no interactions exist among the cells. The configurational partition function Q is
(9)
20
(iv) Equation of state By using thermodynamic from eqns.-(8) and (9).
relations, the equation of state can be found
Hence, as the hard-sphere volume V, is re3NA V”=6=the relationship volume b is
b
(12)
4 between
the close-packed
volume V, and the excluded
(13) In order to improve the P- V-T corrective term d to eqn. (11)
prediction accuracy, we add a volumetric
where V, = 3b/&% Thus, there are now four independent constants a, b, c and d. The usefulness of this additional term d can be demonstrated as follows. By
1lVr
Fig. 1. Experimental saturation curve of n-octane compared with calculations from eqns. (11) (- - - - - -) and (14) ().
21
using the optimum constant a at each temperature, the predicted saturated vapor pressure can be made the same as the experimental vapor pressure. We calculated the saturated P- V-T curve of n-octane as shown in Fig. 1. The average absolute deviation of saturated liquid densities decreases from 15.7% with eqn. (11) to 1.7% with eqn. (14). EVALUATION
OF CONSTANTS
The proposed equation of state satisfies PC= p&I/,) (aP/av),c
(15) = 0
(a2P/w2),
(16)
c= 0
(17)
Therefore, the value of a, c and d at the critical point can be determined
Q41 ASSUME K,B ASSUME b
1 CALCULATE
J CALCULATE ac,c,d
Ps,i with fl'fg
J CALCULATE
with P=P_,fl=fg 1
Fig. 2. Optimization procedure for b and
K.
22
analytically from eqns. (15)-(17), if the value of b is known. For the constant a, the present work employs Heyen’s functional form (Heyen, 1980). a = a, exp[n(l
- T,‘)]
(18)
Although we determined parameter sets of 19and K for each substance at the first step, we found that 8 was approximately 0.035 for all substances and, therefore, redetermined only the constant b and parameter K under the condition B = 0.035. The parameter K is considered to be influenced by the saturated vapor pressure and the constant b by the saturated liquid volume. We determined the constant b by minimizing the average absolute deviation (AAD) in saturated liquid volumes and then used the optimum b to determine K by minimizing the deviation in saturated vapor pressures as shown in Fig. 2. The AAD is defined as AAD( X) = 100 5 i=l
where X denotes some property. When we optimized the constant b, we restricted the saturated reduced temperature range from 0.5 to 0.9, because experimental data above T, = 0.9 are scarce and the optimized b was for each case very sensitive to data above T, = 0.9. After we determined the optimum K, we checked AAD values in saturated liquid volumes again and found them not to change significantly.
OTHER EQUATIONS
OF STATE
We discuss Soave-Redlich-Kwong (SRK) and Patel-Teja (PT) equations of state, which are compared with the new equation of state, and confirm the critical point expression. Soave-Redlich-Kwong
equation
The SRK equation of state was originally proposed as an improvement over the Redlich-Kwong equation and has two constants a and b. The SRK equation is
(20)
23
where a = SI,( R’T,z/p,)a
a, = 0.42747
b = %Wc/PJ
Q2,= 0.08664
lx”.5= 1 + m (1 - T,O.5) m = 0.480 + 1.574~ - 0.1760~ It predicts a fixed critical compressibility of l/3. Patek Teja equation The PT equation of state was improved from the Peng-Robinson equation of state and has three constants a, b and c. The PT equation is p=--
1
RT V-b
V(V+
b) + c(V-
b)
(21)
where a = Q,( R2q2/PC)cx b = %(RT,/PJ c = Q,(RTJP,) (Y’.~= 1 + F(1 - T,“.5) Q, = 3g + 3(1 - 25,)52, + Cg + 1 - 35, !-J2,= 1 - 35, and 5, is the smallest positive root of the following cubic equation, 0; + (2 - 35,)Q2,2 + 3@,
- 6,’ = 0
It uses an empirical critical compressibility factor of the pure substance, t,. Values for optimum F and 4, have been presented by Georgeton et al. (1986). PURE FLUID
Saturation
PROPERTIES
curve
The proposed equation of state was used to calculate saturated molar volumes, vapor pressures and heats of vaporization of 45 pure fluids from reduced temperatures T, = 0.5 up to the critical point. Calculated values were compared with experimental values and those of the SRK equation and
24 TABLE
1
Properties
of pure fluids
No.
Component
T, (R)
P, (atm)
V, (cm3 mol-‘)
0
Non-polar 1
Hydrogen
33.23
12.99
63.86
- 0.22
2
Argon
150.86
48.34
74.57
- 0.004
3
Nitrogen
126.25
33.52
92.15
0.040
4
Oxygen
154.77
50.20
78.85
0.021
5
Methane
190.55
45.80
98.84
0.008
6
Ethane
305.50
48.49
141.71
0.098
7
Propane
370.00
42.09
195.97
0.152
8
n-Butane
425.16
37.46
254.91
0.193
9
n-Pentane
469.77
33.30
310.97
0.251
10
n-Hexane
507.85
29.91
368.26
0.296
11
n-Heptane
540.16
27.00
426.37
0.351
12
n-Octane
569.35
24.63
490.86
0.394
13
n-Nonane
595.15
22.58
543.43
0.444
14
n-Decane
619.15
20.89
602.86
0.490
1.5
n-Dodecane
659.2
17.89
718.7
0.562
16
n-Tridecane
677.2
16.98
768.1
0.623
17
n-Tetradecane
695.2
15.99
826.6
0.679
18
n-Pentadecane
710.2
15.00
885.0
0.706
19
n-Hexadecane
725.2
14.01
943.5
0.742
20
n-Heptadecane
735.2
13.03
1001.9
0.770
21
n-Octadecane
750.2
13.03
1060.4
0.790
22
n-Nonadecane
760.2
12.04
1118.8
0.827
23
n-Eicosane
775.2
10.96
1177.2
0.907
24
Ethylene
282.65
50.10
128.68
0.085
25
Propylene
365.05
45.40
180.59
0.148
26
I-Butene
419.55
39.67
240.79
0.187
27
Carbon
304.19
72.85
94.01
0.225
28
Benzene
562.6
48.60
256.98
0.212
29
Carbon
tetrachloride
556.35
45.00
275.63
0.194
Carbon
monoxide
132.92
34.52
93.06
0.049
417.15
76.08
123.74
0.073 0.048
Poiar 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Critical al. 1977.
dioxide
Chlorine Fluorine
144.0
52.55
71.16
Water
647.27
218.25
56.70
0.344
Ammonia
405.60
111.52
72.47
0.25
Sulfur dioxide
430.65
76.79
122.03
0.251
Methanol
513.15
78.5
116.36
0.559
Ethanol
516.25
62.96
166.97
0.635
I-Propanol
536.85
50.16
220.15
0.624
1-Butanol
562.93
43.55
274.53
0.59
l-Pentanol
586
38
326.48
0.58
1-Hexanol
610
40
381.26
0.56
I-Octanol
658
34
489.59
1-Decanol
700
22
599.56
0.304 -
Ethyl acetate
523.25
38.00
286.32
0.363
Ethyl ether
466.95
35.63
279.62
0.281
properties
are from the data references
in Table 3. Acentric
Factors
are from Reid et
25 TABLE 2 Optimized parameters of a new equation of state No.
Component
b~ V~
Hydrogen Argon Nitrogen Oxygen Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane Ethylene Propylene 1-Butene Carbon dioxide Benzene Carbon tetrachloride
0.3333 0.3270 0.3094 0.2945 0.3213 0.3232 0.3251 0.3075 0.3119 0.3079 0.3058 0.3011 0.2995 0.2983 0.2998 0.3051 0.3081 0.3097 0.3132 0.3120 0.3135 0.3125 0.3133 0.3214 0.3184 0.3016 0.3148 0.3106 0.3156
Carbon monoxide Chlorine Fluorine Water Ammonia Sulfur dioxide Methanol Ethanol 1-Propanol 1-Butanol 1-Pentanol 1-Hexanol 1-Octanol 1-Decanol Ethyl acetate Ethyl ether
0.3123 0.3144 0.2932 0.2804 0.2948 0.3112 0.3017 0.3068 0.3021 0.2966 0,2979 0.2989 0.3009 0.3019 0.3002 0.3122
Non -polar 1 2 3 4 5 6 7 8 9 10 11
12 13
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
-
1.609 8.465 10.85 11.00 9.185 12.32 14.44 16.42 19.63 19.82 21.55 22.99 24.22 25.41 27.82 29.09 30.09 31.22 31.85 33.26 35.10 35.77 34.50 11.50 14.08 16.31 15.75 16.62 15.65
Polar 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
11,35 11,47 13.40 18.74 16.48 16.12 25.33 30.10 30.00 31.18 30.62 33.51 33.21 28.62 21.68 18.57
26 TABLE 3 Dimensionless parameters of a new equation of state No.
Component
~a
~2b
~2c
f~d
Hydrogen Argon Nitrogen Oxygen Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane Ethylene Propylene 1-Butene Carbon dioxide Benzene Carbon tetrachloride
0.4879 0.4957 0.4786 0.4589 0.4934 0.5084 0.5119 0.4984 0.5035 0.5064 0.5088 0.5067 0.5116 0.5134 0.5215 0.5270 0.5310 0.5347 0.5411 0.5437 0.5399 0.5444 0.5523 0.5037 0.5054 0.4919 0.5024 0.5030 0.5052
0.1014 0.09523 0.09224 0.09180 0.09302 0.08859 0.08833 0.08418 0.08450 0.08139 0.07943 0.07793 0.07526 0.07395 0.07128 0.07160 0.07138 0.07056 0.06966 0.06751 0.07035 0.06749 0.06352 0.08933 0.08715 0.08369 0.08637 0.08402 0.08575
50.34 58.04 51.79 40.70 58.56 68.53 70.18 67.94 69.85 73.88 76.72 77.39 82.09 84.19 90.01 91.36 92.83 94.80 97.57 100.5 96.45 100.7 106.7 65.98 68.43 65.37 67.81 70.07 69.51
0.07186 0.09728 0.08233 0.05692 0.09980 0.1299 0.1350 0.1285 0.1340 0.1456 0.1535 0.1549 0.1677 0.1732 0.1899 0.1955 0.2007 0.2072 0.2169 0.2253 0.2136 0.2260 0.2451 0.1223 0.1298 0.1215 0.1280 0.1347 0.1330
Carbon monoxide Chlorine Fluorine Water Ammonia Sulfur dioxide Methanol Ethanol 1-Propanol 1-Butanol 1-Pentanol 1-Hexanol 1-Octanol 1-Decanol Ethyl acetate Ethyl ether
0.4833 0.5016 0.4541 0.5144 0.5151 0.5078 0.5359 0.5184 0.5136 0.5042 0.5055 0.4672 0.4651 0.5282 0.5104 0.5127
0.09199 0.08647 0.09279 0.06533 0.07159 0.08253 0.06545 0.07614 0.07573 0.07677 0.07686 0.09107 0.09277 0.06933 0.07607 0.08117
54.45 67.37 36.91 95.43 87.50 73.41 100.9 83.65 82.33 77.56 78.02 46.31 43.85 94.33 80.78 76.60
0.08928 0.1267 0.04765 0.1974 0.1815 0.1444 0.2218 0.1736 0.1689 0.1548 0.1562 0.07016 0.06284 0.2029 0.1642 0.1538
Non-polar 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
Polar 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
27 the P T e q u a t i o n . T h e critical p r o p e r t i e s a n d the acentric factors are given in T a b l e 1, the o p t i m i z e d p a r a m e t e r s b/v¢ and x in T a b l e 2 a n d d i m e n s i o n l e s s c o n s t a n t s related to a, b, c a n d d o f eqn. (14) at the critical p o i n t in T a b l e 3. T h e d i m e n s i o n l e s s c o n s t a n t s are d e f i n e d as
~,, =Pcac/ ( RT¢ )2
(22)
f~b = Pcb/( RT¢)
(23)
=
(24)
Rro)
f~a = Pcd/( RT~)
(25)
T a b l e s 4, 5 a n d 6 give A A D values in s a t u r a t e d liquid volumes, s a t u r a t e d gas v o l u m e s a n d v a p o r pressures a n d heats of v a p o r i z a t i o n , respectively. T h e new e q u a t i o n generally gives lower A A D values in liquid p h a s e s t h a n S R K a n d P T a n d generally similar or b e t t e r p r e d i c t i o n s o f v a p o r pressures a n d heats o f v a p o r i z a t i o n . A A D values in s a t u r a t e d liquid v o l u m e s are b e l o w 3% e x c e p t for h y d r o g e n . H o w e v e r , the new e q u a t i o n generally gives slightly w o r s e p r e d i c t i o n s for s a t u r a t e d gas v o l u m e s t h a n d o the S R K a n d P T models. D e v i a t i o n s b e t w e e n e x p e r i m e n t a l a n d c a l c u l a t e d s a t u r a t e d p r o p e r -
n-Octane ps 2.5~
, .............", .
| .........
/
/
"-,.....
~2
n Octane VlS -
~i
2.5
\
-5.(
0.6
02 0'8 Tr (-)
0:9
10
05
t
0.6
I
0.7 1"r
l
0.8
I
0.9
1.0
(-)
Fig. 3. AP s= 100(1- Pd~c/P~S~p),the deviation in saturated vapor pressure for n-octane as a function of reduced temperature T~: - - , this work; . . . . . , PT-EOS; . . . . . . , SRK-EOS. Fig. 4. AV~=lOO(1--V~ac/Vl~exp), the deviation in saturated liquid molar volume for n-octane as a function of reduced temperature T~: - - , this work; . . . . . , PT-EOS; . . . . . . , SRK-EOS.
28 TABLE
4
Comparison
No.
of error (AAD) Component
in saturated
liquid volumes
Number
Data
T,
of
refer-
range
points
ence
New eqn.
PT
Gpti-
Optimum
General
mum
SRK
Non-polar 1
Hydrogen
20
a
0.512-0.993
4.44
8.23
_
7.90
2
Argon
69
a
0.555-0.994
1.12
3.90
4.38
4.41
3
Nitrogen
65
a
0.500-0.998
1.30
1.32
4.65
4.80
4
Oxygen
78
a
0.504-0.995
1.44
4.86
4.33
4.40
5
Methane
27
a
0.525-0.997
1.77
3.41
5.38
6.93
6
Ethane
15
a
0.524-0.982
0.88
3.45
5.45
9.65
7
Propane
17
a
0.512-0.992
1.43
3.55
4.51
10.23
8
n-Butane
12
a
0.501-0.760
1.18
1.27
4.04
7.71
9
n-Pentane
17
a
0.645-0.986
1.84
1.85
4.52
14.38
10
n-Hexane
24
a
0.538-0.991
1.68
1.67
4.09
15.27
11
n-Heptane
27
a
0.506-0.987
1.90
1.96
3.29
17.56
12
n-Octane
27
a
0.532-0.989
1.88
2.19
3.25
19.24
13
n-Nonane
13
a
0.509-0.711
1.29
3.34
0.77
17.27
14
n-Decane
14
a
0.506-0.716
1.38
3.50
0.84
18.67
15
n-Dodecane
16
a
0.505-0.733
1.61
3.11
1.82
22.86
16
n-Tridecane
17
a
0.507-0.743
1.76
2.13
1.77
22.20
17
n-Tetradecane
14
a
0.566-0.753
1.47
1.50
22.44
18
n-Pentadecane
14
a
0.582-0.765
1.50
1.50
0.86 _
19
n-Hexadecane
10
a
0.649-0.763
1.15
1.50
-
26.04
20
n-Heptadecane
15
a
0.589-0.780
1.74
1.77
1.62
29.73
21
n-Octadecane
13
a
0.604-0.764
1.43
1.52
24.61
22
n-Nonadecane
12
a
0.609-0.754
1.35
1.45
6.15 _
23
n-Eicosane
11
0.610-0.739
1.26
1.53
3.32
36.14
24
Ethylene
25
“b
0.584-0.991
1.02
2.58
5.55
11.02
25
Propylene
12
a
0.529-0.817
1.46
2.57
1.90
6.32
26
1-Butene
9
a
0.508-0.699
0.98
2.61
1.24
7.62
27
Carbon
24.04
29.12
dioxide
43
a
0.712-0.996
1.13
1.70
6.92
17.32
28
Benzene
46
a
0.515-0.999
2.00
2.04
5.32
14.41
29
Carbon 20
a
0.617-0.994
1.33
2.00
5.41
13.31
12.91
tetrachloride Polar 30
Carbon monoxide
25
0.512-0.977
1.26
Chlorine
21
0.519-0.988
2.18
2.97 _
32
Fluorine
20
0.660-0.990
1.76
-
33
Water
69
0.502-0.999
1.03
6.62
41.48
34
Ammonia
21
0.518-0.999
2.58
3.25
30.62
35
Sulfur dioxide
19
0.750-0.994
0.84
4.70
18.75
36
Methanol
29
0.532-0.997
1.29
10.72
46.24
37
Ethanol
21
0.529-0.999
2.00
8.41
27.09
38
I-Propanol
19
0.658-0.993
1.22
6.55
23.67
39
I-Butanol
11
0.530-0.698
0.54
0.29
13.97
31
3.07 4.60
29 Table 4 (continued) No.
Component
Number of points
Polar 40 41 42 43
1-Pentanol 1-Hexanol l-Octanol 1-Decanol
44 45
Ethyl acetate Ethyl ether
Data reference
T, range
8 9 8 4
d d d d
0.551-0.671 0.513-0.645 0.506-0.613 0.547-0.590
0.37 0.78 0.72 0.46
0.93 6.05 16.46 _
13.84 3.73 2.95 -
28 22
a a
0.522-0.998 0.585-0.998
2.25 1.97
5.02 -
23.96 20.77
a, Vargaftik, 1975; b, JSME Zwolinski, 1973.
Data Book,
New eqn.
PT
Optimum
Optimum
General
1983; c, Timmermans,
1950;
SRK
d, Wilhoit and
ties are plotted against reduced temperature for n-octane in Figs. 3 to 6. The deviation at each saturated temperature is defined as A ( X) = loo@ - Xcalc/XexP)
(26)
where X denotes some property. The equation can predict the saturated liquid volume accurately up to the critical temperature.
-201 0.5
0.6
0.7 Tr
0.0 t-1
0.9
1.0
0.5
0.6
0.7 Tr
0.6
0.9
1.0
(-1
Pig. 5. APi =lOO(l-VgS,,/&&,), the deviation in saturated gas molar volume for n-octane PT-EGS; ______, this work; .---., as a function of reduced temperature T,: -, SRK-EOS.
Fig. 6. A(AfL,J =loo(l- A%p,dc/A%p.exp), the deviation in saturated heat of vaporizathis work; .-.-, PT-EOS; tion for n-octane as a function of reduced temperature T,: -, - - - - - -, SRK-EOS.
30
r~
e~
~.~
<
z
~
~ ~
==oz
~¢~=~
31
.
.
.
.
.
.
~
°
,
I
I
co
r~
~
32 TABLE
6
Comparison of error (AAD) in heats of vaporization Component
New eqn. Optimum
PT Optimum
SRK
16 17
Hydrogen Argon Nitrogen Oxygen Methane Ethane Propane n-Hexane n-Heptane n-Octane n-Nonane n-Decane Ethylene Propylene 1-Butene Carbon dioxide Benzene
3.84 3.02 3.22 3.38 3.55 1.43 1.89 1.61 1.85 1.62 2.43 2.16 3.56 2.19 0.65 7.53 9.02
_ 3.23 3.91 3.24 5.52 1.66 2.08 2.13 1.51 2.04 2.02 0.83 3.72 0.76 0.74 5.02 9.02
6.78 3.50 4.31 3.51 5.81 2.44 2.51 2.44 2.25 2.06 0.26 1.65 4.09 1.82 0.63 5.06 8.10
Polar 18 19 20 21 22 23 24 2.5 26 27
Carbon monoxide Chlorine Fluorine Water Ammonia Sulfur dioxide Methanol 1-Propanol Ethyl acetate Ethyl ether
1.83 4.28 6.87 2.82 3.45 3.76 2.61 5.03 1.73 1.10
2.25 _ 3.79 2.80 3.31 3.19 6.01 1.77
2.24 4.32 8.46 6.97 3.53 2.43 6.22 5.66 2.61 4.16
No. Non-polar
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1.5
Isotherms We checked the P- V-T properties for carbon dioxide, as shown in Fig. 7. The calculation results of our new equation of state are compared with those of SRK and PT. Our results are superior in the near critical region. GENERALIZATION
OF EQUATION
Constant b and parameter
OF STATE
K
For non-polar fluids, b and parameter K can be correlated to the critical volume V, and the acentric factor w respectively. By using data of normal
33
saturation
0
0.5
1.0 1lVr
curve
1.5
2.0
2.5
C-1
Fig. 7. Experimental isotherms of carbon dioxide compared with calculations from this work, IT-EOS and SRK-EOS: 0, T, = 0.920; A, T, =1.019; 0, T, = 1.315.
hydrocarbons from methane to n-eicosane, we determined the generalized functional form of b and K. b/K K=
= 0.31 8.821 + 41.13~ - 12.61~~
However, the generalized equations cannot be applied substances such as water, ammonia and the alcohols. generalized equations are listed in Tables 4 and 5. The 1.0% for saturated liquid volumes and 2.6% for saturated
(27) (28) to strongly polar AAD values with loss in accuracy is vapor pressures.
DISCUSSION
For polar fluids, there exist not only physical interactions but also chemical interactions, which is why a generalization of the new equation of state was not attempted in this work. However, we do outline the way in which the constants a and b differ for polar fluids. As the values of a and b are influenced by the evaluated saturated temperature range, we selected 26 fluids from Table 4 for which data exist from T, = 0.5 to 1.0. Those component numbers are 1, 2, 3, 4, 5, 6, 8, 9, 10, 11,12, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 44 and 45. Constant b Although we define the ratio b/V, to be 0.31 for non-polar fluids, b/K for polar fluids is about 0.30 for the normal alcohols as shown in Table 2.
0
30z 3E Y
0 0 0 Qlo
zo0°38
>r 0037 10-
00 Oe$
0
0
36 ’ 20
.
’ 40 V,
*
’ 60
,
’ 80
100
(cclmol)
Fig. 8. Relationship of hard-sphere volumes V, to Bondi’s van der Waals volumes V,: 0, non-polar fluid; 0, polar fluid; 33, 34, 36, 37, 38, component numbers in Table 4.
Polar molecules are more easily condensed than normal fluids because of hydrogen bonding. Bondi (1964) calculated the van der Waals volume VW using his group contribution method and showed slight decrements in volume as a result of the strong association in polar fluids. Our hard-sphere volume V,, can be related to the van der Waals volume as shown in Fig. 8. This means that there is a possibility for the constant b to be generalized even for polar fluids. Constant a For non-polar fluids, the constant a is related to the parameter K by eqn. (18) and K is related to the acentric factors as shown in Fig. 9. However, ~~~~~~ is smaller than K,__,~~~ at the same acentric factor. Hence, we consider the reason for this from the standpoint of association. As the constant a is related to the interaction in the gas phase, we can consider the second virial coefficient B. Lambert (1953) thought some polar molecules became new chemical species through the chemical association forces and divided the second virial coefficient into non-polar and polar parts. That is
non-polar is the second virial coefficient, when all polar molecules are monomer. BpOlaris determined by the degree of the chemical bond. If there is weak dimerization, BpOlarshould be
B polar = -RTK
(30)
35
Fig. 9. Relationship of the parameters K to the acentric factors w: 0, polar fluid; 33, 34, 36, 37, 38, component numbers in Table 4.
non-polar fluid; 0,
So, the second virial coefficients of polar fluids become smaller than those of non-polar fluids. The second virial coefficient B of the new equation of state at the critical temperature is (31)
B, = b - a,/RT,
Therefore, in strongly polar fluids, a,/RT,V,, should be larger than in non-polar fluids. For the previously mentioned fluids, we plotted a,/RT,V,, against critical temperatures as shown in Fig. 10. For strongly polar fluids (water, ammonia and the alcohols), (a,/RTcVh),,, is larger than
20-
15
0
0
“a”
1 200
c
t 400 Tc
*
a 6M)
.
000
(K)
Fig. 10. Relationship of a,/RT,V, to the critical temperatures. 0. non-polar fluid; 0, fluid; 33, 34, 36, 37, 38, component numbers in Table 4.
36
at the same critical temperature. So, for polar fluids, the ( at/R VII 1non-polar constant a may be divided into two parts as a=a non-polar+ ~,,I,~
(32)
The proportion of apolar is no larger than 30% as can be seen in Fig. 10. We consider it is possible for the constant a to be generalized by some polar factors, for example, the dipole moment or the parachor (Quayle, 1953). CONCLUSION
A new equation of state is proposed based on molecular thermodynamics that is capable of accurate and consistent predictions of the saturated properties of non-polar and polar fluids. It requires only the critical constants, the excluded volume and one empirical constant. Since the critical point is reproduced exactly, prediction accuracy in this region is improved. LIST OF SYMBOLS
a, b, c, d constants in eqn. (14) b
Y F k K
m N
N* P
Q R r
T V
v, v, v, vll VW X X z
excluded volume second virial coefficient fugacity constant in eqn. (21) Boltzmann’s constant equilibrium constant of association constant in eqn. (20) total data number Avogadro’s number pressure configurational partition function gas constant distance from the center of the hard sphere temperature molar volume close-packed volume cell volume free volume hard-sphere volume Bondi’s van der Waals volume property defined by Eyring in eqn. (4) property in eqns. (19) and (26) coordination number
31
Greek symbols parameter in eqns. (20) and (21) potential-energy constant parameter in eqn. (18) parameter in eqn. (18) hard-sphere diameter potential energy of interaction potential energy acentric factor dimensionless constant in eqns. (22) to (25) Subscripts C
talc exp g ; opt r s sys vdw
critical point calculated value experimental value gas state component i liquid state optimized value reduced value saturated state system van der Waals
REFERENCES Bondi, A., 1964, J. Phys. Chem., 6: 441. Eyring, H., Ree, T. and Hirai, N., 1958, Proc. Natl. Acad. Sci. (U.S.), 44: 683. Georgeton, G.K., Smith, R.L. and Teja, A.S., 1986, Equations of State, ACS Symp. Ser. 300, Chap. 21. Heyen, G., 1980, Proc. 2nd Int. Conf. on Phase Equilibria and Fluids Properties in the Chemical Industry, Vol. 1, Dechema, Frankfurt. JSME Data Book, Thermophysical Properties of Fluids, 1983, Japan Mechanical Engineering Society. Lambert, J.D., 1953, Discuss. Faraday Sot., 15: 226. Patel, N.C. and Teja, A.S., 1982, Chem. Eng. Sci., 37: 463. Quayle, O.R., 1953, Chem. Revs., 53: 439. Reid, R.C., Prausnitz, J.M. and Sherwood, T.K., 1977, The Properties of Gases and Liquids, 3rd edn., McGraw-Hill, New York. Soave, G., 1972, Chem. Eng. Sci., 27: 1197.
38 Timmermans, J., 1950. Physico-Chemical Constants of Pure Organic compounds, Elsevier, Amsterdam. Vargaftik, N.B., 1975, Tables on the Thermophysical Properties of Liquids and Gases, (2nd edn.), Wiley, New York. Wilhoit, R.C. and Zwolinski, B.J., 1973, Physical and Thermodynamic Properties of Aliphatic Alcohols, American Chemical Society, Washington, DC.
17
Printed in The Netherlands
NEW EQUATION OF STATE BASED ON THE SIGNIFICANT STRUCTURE MODEL KAZUHIKO
SUZUKI
and HARUHUSA
Central Research Laboratories,
Idemitsu
RICHARD
L. SMITH,
HIROSI
Department
of Chemical Engineering,
SUE
Kosan Co., Ltd., Sodegaura, Chiba, 299-02 (Japan)
INOMATA,
KUNIO
ARAI and SHOZABURO
SAITO
Tohoku University, Sendai, 980 (Japan)
(Received July 5, 1988; accepted in final form January 4, 1989)
ABSTRACT Suzuki, K., Sue, H., Smith, R.L., Inomata, H., Arai, K. and Saito, S., 1989. New equation of state based on the significant structure model. Fluid Phase Equilibria, 47: 17-38. A new non-cubic equation of state based on molecular thermodynamics has been developed by considering cell theory and forming the potential energy of a system from the volumetric average of the van der Waals type and the 15-6 Mie type potential energies. The proposed equation has four constants and requires knowledge of the critical temperature, the critical pressure, the critical volume, the excluded volume and an additional parameter to characterize any given fluid. The new equation yields good agreement with experimental saturation properties of 29 non-polar and 16 polar substances from reduced temperatures of 0.5 to the critical temperature. The new equation exactly fits the critical point, generally gives better predictions of liquid volumes than, and predictions of vapor pressures and heats of vaporization equally as good as commonly used cubic equations, and yet only requires two fitting parameters.
INTRODUCTION
Since the time of van der Waals, two general approaches have been taken to the development of equations of state in terms of molecular interactions. The first approach begins from the gas state. The van der Waals type of equation of state adds corrections to the ideal gas equation of state and makes it possible to predict a condensed phase. While this type of equation can predict volumetric properties in the gas region well, the prediction of volumetric properties in the liquid region is only qualitative. Various researchers have improved the van der Waals type equation by empirically modifying the form of the attractive term and its temperature dependence or by adding other parameters (Soave, 1972; Pate1 and Teja, 1982). 0378-3812/89/$03.50
0 1989 Elsevier Science Publishers B.V.
18
The second approach begins from the liquid state and predicts liquid properties using the basic properties of the pure substance such as the hard-sphere diameter and the potential energy to predict the liquid properties. These models include the cell model, the hole model and the significant liquid structure model (Eyring et al., 1958). In the present work, a new equation of state is developed from a consideration of the van der Waals equation and the cell model, and on the assumption that the potential energy of a system can be represented by the volumetric average of the van der Waals type and the 15-6 Mie type potential energies. FORMULATION
OF THE EQUATION
OF STATE
We consider the system to be described by V, T, NA, (J and assume the following. (i) The system is a group of independent cells. Each cell has a volume V-, = V/N, and no interactions exist among the cells. (ii) The free vo1ume V, is defined as the cell volume VI minus the excluded volume b/N,. (iii) The potential energy of a system is the volumetric average of the van der Waals type and the 15-6 Mie-type potential energies. On these assumptions, we derive the equation of state by the following procedures. (i) Free volume V, The free volume per particle is Q=(V-b)/N,
0)
(ii) System potential energy It is difficult to describe the potential energy of real systems over a wide
range of densities. For the dilute region, the van der Waals type potential energy can predict P- V-T properties. However, the Mie type potential is useful in the dense region, because it contains a repulsive term and is mathematically simple. To describe a liquid property X, Eyring proposed the volumetric average method based on his significant liquid structure model. That is
x=xs$+xg il-+
1
(2)
where X,, Xa are the values of the property in the solid and the gas states, respectively. Applying his definition to the partition function based on the 12-6 Lennard-Jones potential for the liquid state and on the ideal gas condition for the gas state, Eyring was able to derive his equation of state.
19
We modify his concepts and describe the potential energy of a system as the volumetric average of the van der Waals type and the Mie type potential energies. The van der Waals type potential energy is @P(V) VDW
=
-U/V
(3)
The Mie potential is +(r)Mie=C’E[(u/r)m-
=
c’c
(a/r)6]
1
(fJ/r)‘”
-
(4)
(o/r)“]
where c, = ( m/6)1’(m-6) (m-6) If we assume that liquid cells are in the close-packed state and all molecules are fixed at the center of each cell with coordination number z, the relationship between the volume and radius is a3/fi
= V,/N,
(5)
r’/fi
= V/N,
(6)
and the potential energy can be written as ~(V),,=N,C’r~
[(?i’-(+,‘I
=#!)y!i’]
(7)
where c = 3N,c’cz/2 Following Eyring, we assume that the potential energy of a system is the volumetric average of the above two potentials. @(v)vDw+
~a(v)h4ie
(iii) Cell model Following the simple cell model concept, we assume that each cell is independent of its own uniform potential and that no interactions exist among the cells. The configurational partition function Q is
(9)
20
(iv) Equation of state By using thermodynamic from eqns.-(8) and (9).
relations, the equation of state can be found
Hence, as the hard-sphere volume V, is re3NA V”=6=the relationship volume b is
b
(12)
4 between
the close-packed
volume V, and the excluded
(13) In order to improve the P- V-T corrective term d to eqn. (11)
prediction accuracy, we add a volumetric
where V, = 3b/&% Thus, there are now four independent constants a, b, c and d. The usefulness of this additional term d can be demonstrated as follows. By
1lVr
Fig. 1. Experimental saturation curve of n-octane compared with calculations from eqns. (11) (- - - - - -) and (14) ().
21
using the optimum constant a at each temperature, the predicted saturated vapor pressure can be made the same as the experimental vapor pressure. We calculated the saturated P- V-T curve of n-octane as shown in Fig. 1. The average absolute deviation of saturated liquid densities decreases from 15.7% with eqn. (11) to 1.7% with eqn. (14). EVALUATION
OF CONSTANTS
The proposed equation of state satisfies PC= p&I/,) (aP/av),c
(15) = 0
(a2P/w2),
(16)
c= 0
(17)
Therefore, the value of a, c and d at the critical point can be determined
Q41 ASSUME K,B ASSUME b
1 CALCULATE
J CALCULATE ac,c,d
Ps,i with fl'fg
J CALCULATE
with P=P_,fl=fg 1
Fig. 2. Optimization procedure for b and
K.
22
analytically from eqns. (15)-(17), if the value of b is known. For the constant a, the present work employs Heyen’s functional form (Heyen, 1980). a = a, exp[n(l
- T,‘)]
(18)
Although we determined parameter sets of 19and K for each substance at the first step, we found that 8 was approximately 0.035 for all substances and, therefore, redetermined only the constant b and parameter K under the condition B = 0.035. The parameter K is considered to be influenced by the saturated vapor pressure and the constant b by the saturated liquid volume. We determined the constant b by minimizing the average absolute deviation (AAD) in saturated liquid volumes and then used the optimum b to determine K by minimizing the deviation in saturated vapor pressures as shown in Fig. 2. The AAD is defined as AAD( X) = 100 5 i=l
where X denotes some property. When we optimized the constant b, we restricted the saturated reduced temperature range from 0.5 to 0.9, because experimental data above T, = 0.9 are scarce and the optimized b was for each case very sensitive to data above T, = 0.9. After we determined the optimum K, we checked AAD values in saturated liquid volumes again and found them not to change significantly.
OTHER EQUATIONS
OF STATE
We discuss Soave-Redlich-Kwong (SRK) and Patel-Teja (PT) equations of state, which are compared with the new equation of state, and confirm the critical point expression. Soave-Redlich-Kwong
equation
The SRK equation of state was originally proposed as an improvement over the Redlich-Kwong equation and has two constants a and b. The SRK equation is
(20)
23
where a = SI,( R’T,z/p,)a
a, = 0.42747
b = %Wc/PJ
Q2,= 0.08664
lx”.5= 1 + m (1 - T,O.5) m = 0.480 + 1.574~ - 0.1760~ It predicts a fixed critical compressibility of l/3. Patek Teja equation The PT equation of state was improved from the Peng-Robinson equation of state and has three constants a, b and c. The PT equation is p=--
1
RT V-b
V(V+
b) + c(V-
b)
(21)
where a = Q,( R2q2/PC)cx b = %(RT,/PJ c = Q,(RTJP,) (Y’.~= 1 + F(1 - T,“.5) Q, = 3g + 3(1 - 25,)52, + Cg + 1 - 35, !-J2,= 1 - 35, and 5, is the smallest positive root of the following cubic equation, 0; + (2 - 35,)Q2,2 + 3@,
- 6,’ = 0
It uses an empirical critical compressibility factor of the pure substance, t,. Values for optimum F and 4, have been presented by Georgeton et al. (1986). PURE FLUID
Saturation
PROPERTIES
curve
The proposed equation of state was used to calculate saturated molar volumes, vapor pressures and heats of vaporization of 45 pure fluids from reduced temperatures T, = 0.5 up to the critical point. Calculated values were compared with experimental values and those of the SRK equation and
24 TABLE
1
Properties
of pure fluids
No.
Component
T, (R)
P, (atm)
V, (cm3 mol-‘)
0
Non-polar 1
Hydrogen
33.23
12.99
63.86
- 0.22
2
Argon
150.86
48.34
74.57
- 0.004
3
Nitrogen
126.25
33.52
92.15
0.040
4
Oxygen
154.77
50.20
78.85
0.021
5
Methane
190.55
45.80
98.84
0.008
6
Ethane
305.50
48.49
141.71
0.098
7
Propane
370.00
42.09
195.97
0.152
8
n-Butane
425.16
37.46
254.91
0.193
9
n-Pentane
469.77
33.30
310.97
0.251
10
n-Hexane
507.85
29.91
368.26
0.296
11
n-Heptane
540.16
27.00
426.37
0.351
12
n-Octane
569.35
24.63
490.86
0.394
13
n-Nonane
595.15
22.58
543.43
0.444
14
n-Decane
619.15
20.89
602.86
0.490
1.5
n-Dodecane
659.2
17.89
718.7
0.562
16
n-Tridecane
677.2
16.98
768.1
0.623
17
n-Tetradecane
695.2
15.99
826.6
0.679
18
n-Pentadecane
710.2
15.00
885.0
0.706
19
n-Hexadecane
725.2
14.01
943.5
0.742
20
n-Heptadecane
735.2
13.03
1001.9
0.770
21
n-Octadecane
750.2
13.03
1060.4
0.790
22
n-Nonadecane
760.2
12.04
1118.8
0.827
23
n-Eicosane
775.2
10.96
1177.2
0.907
24
Ethylene
282.65
50.10
128.68
0.085
25
Propylene
365.05
45.40
180.59
0.148
26
I-Butene
419.55
39.67
240.79
0.187
27
Carbon
304.19
72.85
94.01
0.225
28
Benzene
562.6
48.60
256.98
0.212
29
Carbon
tetrachloride
556.35
45.00
275.63
0.194
Carbon
monoxide
132.92
34.52
93.06
0.049
417.15
76.08
123.74
0.073 0.048
Poiar 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Critical al. 1977.
dioxide
Chlorine Fluorine
144.0
52.55
71.16
Water
647.27
218.25
56.70
0.344
Ammonia
405.60
111.52
72.47
0.25
Sulfur dioxide
430.65
76.79
122.03
0.251
Methanol
513.15
78.5
116.36
0.559
Ethanol
516.25
62.96
166.97
0.635
I-Propanol
536.85
50.16
220.15
0.624
1-Butanol
562.93
43.55
274.53
0.59
l-Pentanol
586
38
326.48
0.58
1-Hexanol
610
40
381.26
0.56
I-Octanol
658
34
489.59
1-Decanol
700
22
599.56
0.304 -
Ethyl acetate
523.25
38.00
286.32
0.363
Ethyl ether
466.95
35.63
279.62
0.281
properties
are from the data references
in Table 3. Acentric
Factors
are from Reid et
25 TABLE 2 Optimized parameters of a new equation of state No.
Component
b~ V~
Hydrogen Argon Nitrogen Oxygen Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane Ethylene Propylene 1-Butene Carbon dioxide Benzene Carbon tetrachloride
0.3333 0.3270 0.3094 0.2945 0.3213 0.3232 0.3251 0.3075 0.3119 0.3079 0.3058 0.3011 0.2995 0.2983 0.2998 0.3051 0.3081 0.3097 0.3132 0.3120 0.3135 0.3125 0.3133 0.3214 0.3184 0.3016 0.3148 0.3106 0.3156
Carbon monoxide Chlorine Fluorine Water Ammonia Sulfur dioxide Methanol Ethanol 1-Propanol 1-Butanol 1-Pentanol 1-Hexanol 1-Octanol 1-Decanol Ethyl acetate Ethyl ether
0.3123 0.3144 0.2932 0.2804 0.2948 0.3112 0.3017 0.3068 0.3021 0.2966 0,2979 0.2989 0.3009 0.3019 0.3002 0.3122
Non -polar 1 2 3 4 5 6 7 8 9 10 11
12 13
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
-
1.609 8.465 10.85 11.00 9.185 12.32 14.44 16.42 19.63 19.82 21.55 22.99 24.22 25.41 27.82 29.09 30.09 31.22 31.85 33.26 35.10 35.77 34.50 11.50 14.08 16.31 15.75 16.62 15.65
Polar 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
11,35 11,47 13.40 18.74 16.48 16.12 25.33 30.10 30.00 31.18 30.62 33.51 33.21 28.62 21.68 18.57
26 TABLE 3 Dimensionless parameters of a new equation of state No.
Component
~a
~2b
~2c
f~d
Hydrogen Argon Nitrogen Oxygen Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane n-Dodecane n-Tridecane n-Tetradecane n-Pentadecane n-Hexadecane n-Heptadecane n-Octadecane n-Nonadecane n-Eicosane Ethylene Propylene 1-Butene Carbon dioxide Benzene Carbon tetrachloride
0.4879 0.4957 0.4786 0.4589 0.4934 0.5084 0.5119 0.4984 0.5035 0.5064 0.5088 0.5067 0.5116 0.5134 0.5215 0.5270 0.5310 0.5347 0.5411 0.5437 0.5399 0.5444 0.5523 0.5037 0.5054 0.4919 0.5024 0.5030 0.5052
0.1014 0.09523 0.09224 0.09180 0.09302 0.08859 0.08833 0.08418 0.08450 0.08139 0.07943 0.07793 0.07526 0.07395 0.07128 0.07160 0.07138 0.07056 0.06966 0.06751 0.07035 0.06749 0.06352 0.08933 0.08715 0.08369 0.08637 0.08402 0.08575
50.34 58.04 51.79 40.70 58.56 68.53 70.18 67.94 69.85 73.88 76.72 77.39 82.09 84.19 90.01 91.36 92.83 94.80 97.57 100.5 96.45 100.7 106.7 65.98 68.43 65.37 67.81 70.07 69.51
0.07186 0.09728 0.08233 0.05692 0.09980 0.1299 0.1350 0.1285 0.1340 0.1456 0.1535 0.1549 0.1677 0.1732 0.1899 0.1955 0.2007 0.2072 0.2169 0.2253 0.2136 0.2260 0.2451 0.1223 0.1298 0.1215 0.1280 0.1347 0.1330
Carbon monoxide Chlorine Fluorine Water Ammonia Sulfur dioxide Methanol Ethanol 1-Propanol 1-Butanol 1-Pentanol 1-Hexanol 1-Octanol 1-Decanol Ethyl acetate Ethyl ether
0.4833 0.5016 0.4541 0.5144 0.5151 0.5078 0.5359 0.5184 0.5136 0.5042 0.5055 0.4672 0.4651 0.5282 0.5104 0.5127
0.09199 0.08647 0.09279 0.06533 0.07159 0.08253 0.06545 0.07614 0.07573 0.07677 0.07686 0.09107 0.09277 0.06933 0.07607 0.08117
54.45 67.37 36.91 95.43 87.50 73.41 100.9 83.65 82.33 77.56 78.02 46.31 43.85 94.33 80.78 76.60
0.08928 0.1267 0.04765 0.1974 0.1815 0.1444 0.2218 0.1736 0.1689 0.1548 0.1562 0.07016 0.06284 0.2029 0.1642 0.1538
Non-polar 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
Polar 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
27 the P T e q u a t i o n . T h e critical p r o p e r t i e s a n d the acentric factors are given in T a b l e 1, the o p t i m i z e d p a r a m e t e r s b/v¢ and x in T a b l e 2 a n d d i m e n s i o n l e s s c o n s t a n t s related to a, b, c a n d d o f eqn. (14) at the critical p o i n t in T a b l e 3. T h e d i m e n s i o n l e s s c o n s t a n t s are d e f i n e d as
~,, =Pcac/ ( RT¢ )2
(22)
f~b = Pcb/( RT¢)
(23)
=
(24)
Rro)
f~a = Pcd/( RT~)
(25)
T a b l e s 4, 5 a n d 6 give A A D values in s a t u r a t e d liquid volumes, s a t u r a t e d gas v o l u m e s a n d v a p o r pressures a n d heats of v a p o r i z a t i o n , respectively. T h e new e q u a t i o n generally gives lower A A D values in liquid p h a s e s t h a n S R K a n d P T a n d generally similar or b e t t e r p r e d i c t i o n s o f v a p o r pressures a n d heats o f v a p o r i z a t i o n . A A D values in s a t u r a t e d liquid v o l u m e s are b e l o w 3% e x c e p t for h y d r o g e n . H o w e v e r , the new e q u a t i o n generally gives slightly w o r s e p r e d i c t i o n s for s a t u r a t e d gas v o l u m e s t h a n d o the S R K a n d P T models. D e v i a t i o n s b e t w e e n e x p e r i m e n t a l a n d c a l c u l a t e d s a t u r a t e d p r o p e r -
n-Octane ps 2.5~
, .............", .
| .........
/
/
"-,.....
~2
n Octane VlS -
~i
2.5
\
-5.(
0.6
02 0'8 Tr (-)
0:9
10
05
t
0.6
I
0.7 1"r
l
0.8
I
0.9
1.0
(-)
Fig. 3. AP s= 100(1- Pd~c/P~S~p),the deviation in saturated vapor pressure for n-octane as a function of reduced temperature T~: - - , this work; . . . . . , PT-EOS; . . . . . . , SRK-EOS. Fig. 4. AV~=lOO(1--V~ac/Vl~exp), the deviation in saturated liquid molar volume for n-octane as a function of reduced temperature T~: - - , this work; . . . . . , PT-EOS; . . . . . . , SRK-EOS.
28 TABLE
4
Comparison
No.
of error (AAD) Component
in saturated
liquid volumes
Number
Data
T,
of
refer-
range
points
ence
New eqn.
PT
Gpti-
Optimum
General
mum
SRK
Non-polar 1
Hydrogen
20
a
0.512-0.993
4.44
8.23
_
7.90
2
Argon
69
a
0.555-0.994
1.12
3.90
4.38
4.41
3
Nitrogen
65
a
0.500-0.998
1.30
1.32
4.65
4.80
4
Oxygen
78
a
0.504-0.995
1.44
4.86
4.33
4.40
5
Methane
27
a
0.525-0.997
1.77
3.41
5.38
6.93
6
Ethane
15
a
0.524-0.982
0.88
3.45
5.45
9.65
7
Propane
17
a
0.512-0.992
1.43
3.55
4.51
10.23
8
n-Butane
12
a
0.501-0.760
1.18
1.27
4.04
7.71
9
n-Pentane
17
a
0.645-0.986
1.84
1.85
4.52
14.38
10
n-Hexane
24
a
0.538-0.991
1.68
1.67
4.09
15.27
11
n-Heptane
27
a
0.506-0.987
1.90
1.96
3.29
17.56
12
n-Octane
27
a
0.532-0.989
1.88
2.19
3.25
19.24
13
n-Nonane
13
a
0.509-0.711
1.29
3.34
0.77
17.27
14
n-Decane
14
a
0.506-0.716
1.38
3.50
0.84
18.67
15
n-Dodecane
16
a
0.505-0.733
1.61
3.11
1.82
22.86
16
n-Tridecane
17
a
0.507-0.743
1.76
2.13
1.77
22.20
17
n-Tetradecane
14
a
0.566-0.753
1.47
1.50
22.44
18
n-Pentadecane
14
a
0.582-0.765
1.50
1.50
0.86 _
19
n-Hexadecane
10
a
0.649-0.763
1.15
1.50
-
26.04
20
n-Heptadecane
15
a
0.589-0.780
1.74
1.77
1.62
29.73
21
n-Octadecane
13
a
0.604-0.764
1.43
1.52
24.61
22
n-Nonadecane
12
a
0.609-0.754
1.35
1.45
6.15 _
23
n-Eicosane
11
0.610-0.739
1.26
1.53
3.32
36.14
24
Ethylene
25
“b
0.584-0.991
1.02
2.58
5.55
11.02
25
Propylene
12
a
0.529-0.817
1.46
2.57
1.90
6.32
26
1-Butene
9
a
0.508-0.699
0.98
2.61
1.24
7.62
27
Carbon
24.04
29.12
dioxide
43
a
0.712-0.996
1.13
1.70
6.92
17.32
28
Benzene
46
a
0.515-0.999
2.00
2.04
5.32
14.41
29
Carbon 20
a
0.617-0.994
1.33
2.00
5.41
13.31
12.91
tetrachloride Polar 30
Carbon monoxide
25
0.512-0.977
1.26
Chlorine
21
0.519-0.988
2.18
2.97 _
32
Fluorine
20
0.660-0.990
1.76
-
33
Water
69
0.502-0.999
1.03
6.62
41.48
34
Ammonia
21
0.518-0.999
2.58
3.25
30.62
35
Sulfur dioxide
19
0.750-0.994
0.84
4.70
18.75
36
Methanol
29
0.532-0.997
1.29
10.72
46.24
37
Ethanol
21
0.529-0.999
2.00
8.41
27.09
38
I-Propanol
19
0.658-0.993
1.22
6.55
23.67
39
I-Butanol
11
0.530-0.698
0.54
0.29
13.97
31
3.07 4.60
29 Table 4 (continued) No.
Component
Number of points
Polar 40 41 42 43
1-Pentanol 1-Hexanol l-Octanol 1-Decanol
44 45
Ethyl acetate Ethyl ether
Data reference
T, range
8 9 8 4
d d d d
0.551-0.671 0.513-0.645 0.506-0.613 0.547-0.590
0.37 0.78 0.72 0.46
0.93 6.05 16.46 _
13.84 3.73 2.95 -
28 22
a a
0.522-0.998 0.585-0.998
2.25 1.97
5.02 -
23.96 20.77
a, Vargaftik, 1975; b, JSME Zwolinski, 1973.
Data Book,
New eqn.
PT
Optimum
Optimum
General
1983; c, Timmermans,
1950;
SRK
d, Wilhoit and
ties are plotted against reduced temperature for n-octane in Figs. 3 to 6. The deviation at each saturated temperature is defined as A ( X) = loo@ - Xcalc/XexP)
(26)
where X denotes some property. The equation can predict the saturated liquid volume accurately up to the critical temperature.
-201 0.5
0.6
0.7 Tr
0.0 t-1
0.9
1.0
0.5
0.6
0.7 Tr
0.6
0.9
1.0
(-1
Pig. 5. APi =lOO(l-VgS,,/&&,), the deviation in saturated gas molar volume for n-octane PT-EGS; ______, this work; .---., as a function of reduced temperature T,: -, SRK-EOS.
Fig. 6. A(AfL,J =loo(l- A%p,dc/A%p.exp), the deviation in saturated heat of vaporizathis work; .-.-, PT-EOS; tion for n-octane as a function of reduced temperature T,: -, - - - - - -, SRK-EOS.
30
r~
e~
~.~
<
z
~
~ ~
==oz
~¢~=~
31
.
.
.
.
.
.
~
°
,
I
I
co
r~
~
32 TABLE
6
Comparison of error (AAD) in heats of vaporization Component
New eqn. Optimum
PT Optimum
SRK
16 17
Hydrogen Argon Nitrogen Oxygen Methane Ethane Propane n-Hexane n-Heptane n-Octane n-Nonane n-Decane Ethylene Propylene 1-Butene Carbon dioxide Benzene
3.84 3.02 3.22 3.38 3.55 1.43 1.89 1.61 1.85 1.62 2.43 2.16 3.56 2.19 0.65 7.53 9.02
_ 3.23 3.91 3.24 5.52 1.66 2.08 2.13 1.51 2.04 2.02 0.83 3.72 0.76 0.74 5.02 9.02
6.78 3.50 4.31 3.51 5.81 2.44 2.51 2.44 2.25 2.06 0.26 1.65 4.09 1.82 0.63 5.06 8.10
Polar 18 19 20 21 22 23 24 2.5 26 27
Carbon monoxide Chlorine Fluorine Water Ammonia Sulfur dioxide Methanol 1-Propanol Ethyl acetate Ethyl ether
1.83 4.28 6.87 2.82 3.45 3.76 2.61 5.03 1.73 1.10
2.25 _ 3.79 2.80 3.31 3.19 6.01 1.77
2.24 4.32 8.46 6.97 3.53 2.43 6.22 5.66 2.61 4.16
No. Non-polar
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1.5
Isotherms We checked the P- V-T properties for carbon dioxide, as shown in Fig. 7. The calculation results of our new equation of state are compared with those of SRK and PT. Our results are superior in the near critical region. GENERALIZATION
OF EQUATION
Constant b and parameter
OF STATE
K
For non-polar fluids, b and parameter K can be correlated to the critical volume V, and the acentric factor w respectively. By using data of normal
33
saturation
0
0.5
1.0 1lVr
curve
1.5
2.0
2.5
C-1
Fig. 7. Experimental isotherms of carbon dioxide compared with calculations from this work, IT-EOS and SRK-EOS: 0, T, = 0.920; A, T, =1.019; 0, T, = 1.315.
hydrocarbons from methane to n-eicosane, we determined the generalized functional form of b and K. b/K K=
= 0.31 8.821 + 41.13~ - 12.61~~
However, the generalized equations cannot be applied substances such as water, ammonia and the alcohols. generalized equations are listed in Tables 4 and 5. The 1.0% for saturated liquid volumes and 2.6% for saturated
(27) (28) to strongly polar AAD values with loss in accuracy is vapor pressures.
DISCUSSION
For polar fluids, there exist not only physical interactions but also chemical interactions, which is why a generalization of the new equation of state was not attempted in this work. However, we do outline the way in which the constants a and b differ for polar fluids. As the values of a and b are influenced by the evaluated saturated temperature range, we selected 26 fluids from Table 4 for which data exist from T, = 0.5 to 1.0. Those component numbers are 1, 2, 3, 4, 5, 6, 8, 9, 10, 11,12, 24, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 44 and 45. Constant b Although we define the ratio b/V, to be 0.31 for non-polar fluids, b/K for polar fluids is about 0.30 for the normal alcohols as shown in Table 2.
0
30z 3E Y
0 0 0 Qlo
zo0°38
>r 0037 10-
00 Oe$
0
0
36 ’ 20
.
’ 40 V,
*
’ 60
,
’ 80
100
(cclmol)
Fig. 8. Relationship of hard-sphere volumes V, to Bondi’s van der Waals volumes V,: 0, non-polar fluid; 0, polar fluid; 33, 34, 36, 37, 38, component numbers in Table 4.
Polar molecules are more easily condensed than normal fluids because of hydrogen bonding. Bondi (1964) calculated the van der Waals volume VW using his group contribution method and showed slight decrements in volume as a result of the strong association in polar fluids. Our hard-sphere volume V,, can be related to the van der Waals volume as shown in Fig. 8. This means that there is a possibility for the constant b to be generalized even for polar fluids. Constant a For non-polar fluids, the constant a is related to the parameter K by eqn. (18) and K is related to the acentric factors as shown in Fig. 9. However, ~~~~~~ is smaller than K,__,~~~ at the same acentric factor. Hence, we consider the reason for this from the standpoint of association. As the constant a is related to the interaction in the gas phase, we can consider the second virial coefficient B. Lambert (1953) thought some polar molecules became new chemical species through the chemical association forces and divided the second virial coefficient into non-polar and polar parts. That is
non-polar is the second virial coefficient, when all polar molecules are monomer. BpOlaris determined by the degree of the chemical bond. If there is weak dimerization, BpOlarshould be
B polar = -RTK
(30)
35
Fig. 9. Relationship of the parameters K to the acentric factors w: 0, polar fluid; 33, 34, 36, 37, 38, component numbers in Table 4.
non-polar fluid; 0,
So, the second virial coefficients of polar fluids become smaller than those of non-polar fluids. The second virial coefficient B of the new equation of state at the critical temperature is (31)
B, = b - a,/RT,
Therefore, in strongly polar fluids, a,/RT,V,, should be larger than in non-polar fluids. For the previously mentioned fluids, we plotted a,/RT,V,, against critical temperatures as shown in Fig. 10. For strongly polar fluids (water, ammonia and the alcohols), (a,/RTcVh),,, is larger than
20-
15
0
0
“a”
1 200
c
t 400 Tc
*
a 6M)
.
000
(K)
Fig. 10. Relationship of a,/RT,V, to the critical temperatures. 0. non-polar fluid; 0, fluid; 33, 34, 36, 37, 38, component numbers in Table 4.
36
at the same critical temperature. So, for polar fluids, the ( at/R VII 1non-polar constant a may be divided into two parts as a=a non-polar+ ~,,I,~
(32)
The proportion of apolar is no larger than 30% as can be seen in Fig. 10. We consider it is possible for the constant a to be generalized by some polar factors, for example, the dipole moment or the parachor (Quayle, 1953). CONCLUSION
A new equation of state is proposed based on molecular thermodynamics that is capable of accurate and consistent predictions of the saturated properties of non-polar and polar fluids. It requires only the critical constants, the excluded volume and one empirical constant. Since the critical point is reproduced exactly, prediction accuracy in this region is improved. LIST OF SYMBOLS
a, b, c, d constants in eqn. (14) b
Y F k K
m N
N* P
Q R r
T V
v, v, v, vll VW X X z
excluded volume second virial coefficient fugacity constant in eqn. (21) Boltzmann’s constant equilibrium constant of association constant in eqn. (20) total data number Avogadro’s number pressure configurational partition function gas constant distance from the center of the hard sphere temperature molar volume close-packed volume cell volume free volume hard-sphere volume Bondi’s van der Waals volume property defined by Eyring in eqn. (4) property in eqns. (19) and (26) coordination number
31
Greek symbols parameter in eqns. (20) and (21) potential-energy constant parameter in eqn. (18) parameter in eqn. (18) hard-sphere diameter potential energy of interaction potential energy acentric factor dimensionless constant in eqns. (22) to (25) Subscripts C
talc exp g ; opt r s sys vdw
critical point calculated value experimental value gas state component i liquid state optimized value reduced value saturated state system van der Waals
REFERENCES Bondi, A., 1964, J. Phys. Chem., 6: 441. Eyring, H., Ree, T. and Hirai, N., 1958, Proc. Natl. Acad. Sci. (U.S.), 44: 683. Georgeton, G.K., Smith, R.L. and Teja, A.S., 1986, Equations of State, ACS Symp. Ser. 300, Chap. 21. Heyen, G., 1980, Proc. 2nd Int. Conf. on Phase Equilibria and Fluids Properties in the Chemical Industry, Vol. 1, Dechema, Frankfurt. JSME Data Book, Thermophysical Properties of Fluids, 1983, Japan Mechanical Engineering Society. Lambert, J.D., 1953, Discuss. Faraday Sot., 15: 226. Patel, N.C. and Teja, A.S., 1982, Chem. Eng. Sci., 37: 463. Quayle, O.R., 1953, Chem. Revs., 53: 439. Reid, R.C., Prausnitz, J.M. and Sherwood, T.K., 1977, The Properties of Gases and Liquids, 3rd edn., McGraw-Hill, New York. Soave, G., 1972, Chem. Eng. Sci., 27: 1197.
38 Timmermans, J., 1950. Physico-Chemical Constants of Pure Organic compounds, Elsevier, Amsterdam. Vargaftik, N.B., 1975, Tables on the Thermophysical Properties of Liquids and Gases, (2nd edn.), Wiley, New York. Wilhoit, R.C. and Zwolinski, B.J., 1973, Physical and Thermodynamic Properties of Aliphatic Alcohols, American Chemical Society, Washington, DC.