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New method to determine the BWR coefficients in saturated regions
New method to determine the BWR coefficients in saturated regions T. A s a m i Air Separation Plant Department, Engineering Division, Kobe Steel Limited, 4-chome, Iwayanakamachi Nada-ku, Kobe 657, Japan Received 27 August 1987 This Paper describes a new method of obtaining an optimum set of coefficients in the original Benedict-Webb-Rubin (BWR) equation using the vapour and liquid densities in the saturated region. A step to further improve its applicability utilizing the high accuracy equations of state for pure substance is also detailed. The eight BWR coefficients are determined using appropriate object functions containing selected deviations of variables such as saturated pressure, liquid density, vapour density and fugacity. BWR coefficients thus obtained are found to broadly reproduce satisfactorily the experimental data of PVT relation in the saturated and superheated region. The accuracy of vapour-liquid equilibria for binary systems have been also verified for more than 500 experimental points. Furthermore, the second virial coefficient, B, is calculated and compared with the values in literature. These results indicate that the original BWR equation can be satisfactorily used in designing industrial plants.
Keywords: mathematical models; phase equilibria; argon; nitrogen; oxygen Knowledge of the equation of state is essential for the design of various industrial processes, especially air separation processes. For many years various attempts have been made to determine accurately the P VFrelationship. However, in the region of saturated vapour and liquid, for gases such as nitrogen, argon and oxygen, the simple equations of state are seldom satisfactory and high accuracy equations are too complex for industrial purposes. The inadequacy of the original BWR equation I in the saturated region comes from the fact that the coefficients have been determined using the P V T data obtained in the region of supercritical temperatures 2 and it is found to be almost impossible to estimate the P V T relation in the saturated region. Therefore, some modifications of the original equation have been attempted in the past to fit the P V T data in the saturation, by means of selected coefficients expressed as functions of temperature 3, or by using the law of corresponding states 4'5. However, these methods are too cumbersome to calculate the P V T relations. In this Paper, a new method to determine the BWR coefficients by a non-linear optimization and by selecting an appropriate object function with the utilization of the law of corresponding states is proposed from the viewpoint that the original BWR equation is simple in comparison with other equations of state. Using the eight coefficients thus determined, the original BWR equation is verified to be useful for accurate calculations of both the P V T relations in saturated and superheated regions over a wider range and the vapour-liquid equilibria of the binary mixtures.
Selection of c o n s t r a i n t f a c t o r s The original BWR equation and the derived fugacity
0011-2275/88/080521 06 $03.00 '~' 1988 Butterworth & Co (Publishers) Ltd
equation are expressed as follows
• ['cp3"~, 1 -J¢ + \1-/
+a p ° + l
3
+~T
p2)exp(-rp 2)
(1)
6a ( R T b - a ) o 2 + 5 ~ ps
3p2c[ 1-exp(-yp2) + ~ T~ 7p2
1 exp(_ yp2)] 2
2cp2RT 3 I 1 - exp(-YPZ)Tp 2 - exp(_ypZ)
,
]
2 ypZexp(- ypZ)
(2)
where P, f, T and p denote pressure, fugacity, temperature and density, respectively. R is a gas constant and a, b, c, Ao, Bo, Co, a and y are BWR coefficients. To determine the eight coefficients simultaneously, the saturated pressure, P, liquid density, PL, vapour density, Pv, and fugacity, f, are selected as constraint factors. Using the preliminary approximate values of BWR coefficients as initial values, the densities of liquid, PL, and vapour, Pv, and P and f at fixed temperature Tk(k = 1, 2 . . . . n), are calculated from the following simultaneous equations.
P(p, ~)Bw. P %) for values o f k = l , 2 . . . . n f(P, Tk)BWR-- f = 0
(3a)
Cryogenics 1988 Vol 28 August
521
New method to determine BWR coefficients: 7". Asami where subscript BWR denotes the BWR equation or the equation deduced from it. From the above separate equations for liquid density (p = PL) and vapour density (P = Pv) can be written. Applying Taylor's expansion to the above four equations and neglecting terms of second order and above, we get P(Pk, Tk)BWR -F
IOP(P' L s 0 T) j[
T = T~
(P -- Pk) -- P = 0
P P~
(3b)
=
r-,.,lOj~p, T)]~
f(Pk' Tk)BWR"1- L
ap
~
r
later for three different cases). The optimum BWR coefficients are determined by minimizing this object function. Powell's method combined with techniques of linear search is adopted 7 to minimize the object function. For simplicity, six kinds of deviations of constraint factors are defined as follows
(p -- pk) - f = 0
= P=P,
Af
wherein Pk is substituted by PLk and P,k in each equation. The above equation can further be simplified as follows
[_%?w,]
Af
,,_,,, p = p,
Ap
= --[e(pk, T~)Bw. Pk] -
(3c)
r
_T
L
r
lOf(pT_,o=~)aTWk l. (P
A)
_1 P = Pk =
-
~).w.
~(p~,
-
A]
The deviations of constraint factors can then be written as
Pk
The characteristics of the object function, Y, for different combinations of pressure and density are explained below.
k
Case I
A =T, Pk
The object function is conditioned by minimizing the square of deviations of saturated pressure at saturated vapour and liquid densities k
Y = £ (WlkE~k + W2kE2~)
Substituting in Equation (3c) above, we get
(4)
k=l r=
fl = Pk
p
k
-
where l'Vjk(j= 1, 2. . . . 6; k = 1, 2. . . . n) are the weighting factors of each deviation value.
\ P A
= -- [P(P~, Tk)BWR-- f , ]
(AP) _ fk(ff )
FOf(p, T)sw,]
(3d)
P =Pk
-- - [_f(p,. ~ ) . w . - A ] Using the law of corresponding states 6, one set of fixed values for all of the BWR coefficients is determined as the initial value and the following deviations are calculated
PL ,/k and
P~/~ The squares of these deviations are then substituted into an appropriate equation of object function Y (described
522
Cryogenics 1988 Vol 28 August
Case II The differential of saturated pressure with respect to the densities of saturated vapour and saturated liquid varies widely in the region away from the critical point. The following relation exists as shown in Figure 1
(5) where Ps denotes saturated vapour pressure. If the object function is based on the deviation of saturated pressure, the deviation of liquid density alone can converge to the same degree. However, from the relation in Equation (5) it is observed that the deviation of saturated vapour density does not converge to the same degree as the saturated liquid density even at small differential pressure values. The following object function is adopted to obtain the BWR coefficients which satisfy the PVT relation in the saturated region
Y= £ (WlkE2k -I- Wsk E2k q'- W6kE2k) k=l
(6)
New method to determine BWR coefficients: T. Asami Source data used In the process of calculation, appropriate empirical equations of pressure, liquid and vapour densities in saturated state are used at temperatures between the critical point and triple point to avoid scattering of raw data. 1 Wagner's equation is used to fit the saturated pressure 11"12 for nitrogen, argon and oxygen as follows Inn = T c T - l ( n l r + n2 rl'5 q- n3 z3 +tlar 6 + nsz 7
Q.
0=
+ n6 z9)
o_
2
Ap
where n = P/Pc and r = 1 - T/T~. Pc and Tc denote the critical pressure and temperature, respectively, and n~ n 6 are coefficients; an empirical equation of saturated liquid density, PL, in accordance with Guggenheim-Wagner is used as follows 13 PL/Pc = NO + ~ Ni Ti/3 for i = 1, 2 . . . . n
/XPv
Density p , (rnol/rn 3)
ln(pv/pc) = ~ Nit i/3 for i = 1, 2 . . . . n
To further improve the object function of case II and to satisfy the relation of vapour-liquid equilibria, we add fugacity as a constraint factor to the previous object function and determine BWR coefficients by minimizing the object function using the least squares method. Thus, the object function is expressed as follows
Expression of deviations The deviations of the calculated values from the original data are calculated using the following formula
~"=
Procedure The following steps were used to determine the BWR coefficients:
3
(calculated v a l u e ) - (value from data)l (val~ue f r o m d ~ a ) I × 100
(11)
j=l
Determination of the BWR coefficients
2
(10)
where Pc denotes the critical density and N i is a numerical coefficient.
Case III
l
(9)
where Pc denotes the critical density and N i is a numerical coefficient; and an equation of saturated vapour density, p,, in accordance with simplified Guggenheim-Wagner is also used as follows la
~PL
Figure 1 P p diagram illustrating the general characteristics of isotherms in the saturated vapour and liquid phase
k=l
(8)
the values of saturated vapour pressure, saturated liquid and vapour densities and fugacity, were selected from data published by NBS (for fixed temperatures8- lO); one set of fixed values of BWR coefficients were determined and deviations Elk -- E6k were calculated using Equation (3d); and the square of deviations was minimized and the BWR coefficients were obtained from the minimum value of the object function. Substituting the above BWR coefficients into Equation (1), each factor constrained in the object function can be compared with other published data, e.g. NBS reports, etc.
Results and discussion The BWR coefficients were determined for nitrogen, argon and oxygen according to the above procedure. The weighting factor, W, for each term of the object function was ignored since the scattering of data is negligible and has no effect on the calculation. Three alternative cases of object functions in Equations (4), (6) and (7) have been verified. The first case uses saturated vapour pressure as the constraint factor, the second uses saturated pressure, saturated liquid and vapour densities as constraint factors, and the third uses the previously mentioned constraint factors with the addition of fugacity. The results are tabulated in Table 1. It can be seen that in case I the deviations of vapour density are about 3~ , % , notwithstanding the deviations of liquid density which are less than about 2 %. However, both the deviations in cases II and III are approximately of the same degree except for the deviation of fugacity. Since the deviations of all constraint factors in case III converge to the same degree and are the minimum in all three cases, the evaluation
Cryogenics 1988 Vol 28 August
523
New method to determine BWR coefficients. T. Asami Table 1
Comparison of the deviations, ~(%), of constraint factors #L, Pv, P and f, in cases I, II and III Nitrogen (63.15-126.20 K)
/3(,OL) e(Pv) ~(P) ~(f)
I
II
0.54 3.04 0.78 0.85
0.47 0.40 0.37 0.48
Argon (83.8-150,69 K)
III 0.52 0.36 0.36 0.27
I 0.60 2.42 0.50 1.24
of the BWR equation is best executed using the set of BWR coefficients in case III. The BWR equation and the Starling and Han (BWRS) equation have been verified for their accuracy in the saturated region using a set of BWR coefficients published by other authors ~5't6. The results are given in Table 5. These BWR coefficients were obtained based on data in the region of supercritical temperature in one case and vapour pressure and liquid density in the other. Hence, the BWR coefficients determined by this method cannot be used to accurately predict the P VT relation in the saturated region or the vapour liquid equilibria. The prediction performances of saturated densities by cubic equations of state are usually poor, especially in the case of saturated liquid densities. For example, the Peng-Robinson equation 17 has been proposed to improve the prediction performance of the Soave-RedlichKwong equation ~a for saturated liquid densities. However, the deviation in liquid densities cannot be ignored ~7. On the other hand, a virial expansion type equation of state, such as the BWR equation, can be used accurately to calculate the P V T relation in the high density region. Nevertheless, eight constants of the BWR equation have sometimes been determined by using P V T data from the gaseous phase. In this case, they should not be used in
Table 2
II 0.41 0.17 0.22 1.30
Oxygen (70-154,58 K) III
0.34 0.76 0.43 0.51
I
II
2.21 3.98 5,05 6.00
0.90 0.70 0.50 0.80
saturated regions because extreme deviations occur. The BWR constants of nitrogen3 and argon2 were determined using P V T data at vapour pressure in the supercritical gaseous region. For comparison, the BWR constants for oxygen were evaluated using coefficients derived by Seshadri et al.lS. Those for nitrogen were obtained from the correlation of Starling and Han (BWRS) 19. The deviations between the data and the calculated results obtained using the above BWR constants are shown in Table 5. It can be seen that those deviations are larger than in the present work. The deviations of vapour density, Pv, between the predicted value and data in the superheated region are indicated in Table 2. The deviations of equilibrium temperature between the calculated value and data from the literature, for oxygen + argon, argon + nitrogen and nitrogen + oxygen in binary mixtures, are shown in Table 3 2°. In particular, in case III by adding fugacity as a constraint factor to the object function, BWR coefficients are obtained which satisfy both the P V T and vapourliquid equilibria relations. The optimum coefficients for nitrogen, argon and oxygen are given in Table 4. To verify the BWR equation using the calculated BWR coefficients, the second virial coefficient, B, is calculated and compared with the values published by other authors.
Deviations, ~, of vapour density, Pv, in the superheated region
Pressure (MPa)
Temperature (K)
Number of
Absolute mean deviation,
data points
8(%)
300~88.88 300-100 300-1 oo 300-125 300-125
10 9 9 8 8
0,03734 0.05560 0.1216 0,2335 0,3790
300-100 300-125 300-125 300-125 300-150 300-150
9 8
0.1286 0.1919 0.3784 0.6665
311.1 - I 00 311.1 -130.6 311.1 -130.6 311.1 -130.6 311.1-161.1 311.1 -161.1
8 7 7 7 6 6
Nitrogen 0.2069 0.3448 0.6895 1.379 2.758
Argon 0.2027 0.4053 0.8106 1.520 3.040 4.053
8
8 7 7
1.446 1.849
Oxygen 0.2069 0.3448 0.6895 1.379 2.758 3.792
524
0,85 1.59 0.50 1.26
III
Cryogenics 1 9 8 8 Vol 2 8 A u g u s t
0.1282 0.1316 0.2661 0.5790 1.235 1.762
New method to determine BWR coefficients: T. Asami Table 3
Deviations, e, between the calculated and experimental vapour-liquid equilibrium temperatures in three binary mixtures
System
Pressure a (MPa)
Number of data points
Absolute mean deviation, ~(%)
Ar-O 2
0.1013-0.4053 0.6079-1.01 31 1.21 61-2.6342
45 45 110
1.6 1.6 2.0
N 2 Ar
0.1013-0.4053 0.6079-1.01 31 1.21 61 2.6342
46 44 96
1.8 1.4 1.8
N2 02
0.1013-0.4053 0.6079-1.01 31 1.2161 2.6342
24 27 86
0.4 0.3 0.3
aSource of experimental data: G.M. Wilson et al. 2°
T a b l e 4 BWR coefficients determined in case II1: A o ( M P a m 6 k m o l 2 ) , B0(m3kmol (m 6 kmol 2), c (MPa K2 m 9 kmol 3), ~ (m 9 kmol- 3), 7 ( m6 kmol 2) BWR coefficients
Nitrogen
Ao Bo CO a b c
0.15019143 0.55931 023 x 10 0.30769511 x 103 0.26460522×10 0.368481 98 x 1 0 0.46882840 × 102 0.79014894 x 1 0 0.56821964 x 10
7
Argon
2 2 4 2
B = Be--RT
Co
(12)
RT 3
The calculated values and data from the literature are presented in Table 621 . The values in the low temperature region show good consistency. However, in the region of supercritical temperature, as in the case of the P V T relation, the deviation increases slightly. Also, the P V T relation in the superheated region, between atmospheric and critical pressure, is accurately satisfied. It can be verified from Table 2 that the deviations of vapour density in the superheated region are very small.
T a b l e 5 Deviations of f)L, Pv and P calculated using the coefficients of equations of state between BWR and BWRS of other authors and NBS data Oxygen (BWR)
Nitrogen (BWRS)
17.0 82.0 66.0
1.58 2.24
13 70 1 54.58 Reference 15
63 77.4-116.5 Reference 1 (
Absolute mean deviation (%):
~;(PL) ~:(Pv) ~:(P) Number of data points Temperature range (K) Source
3), b
Oxygen
0.85989511 × 10 0.941 21657 x 10 0.8465301 5 x 103 0.82459105×10 0.56534201 × 1 0 0.10571768 × 103 0.221 22512 x 1 0 0.33078947 x 10
1
B is deduced from the BWR equation by the following formula Ao
1), Co(MPaK2m6kmol-2), a (MPam9kmol
1 2 2 2 4 2
0.14632602 0.35604172 x 10 1 0.36064256 x 103 0.21228725x10 2 0.29022418 x 10 2 0.52748418 x 102 0.41 330800 x 10 4 0.46113980 x 10 -2
Conclusions A new method to obtain an optimum set of BWR coefficients which widely satisfies both the P V T and vapour-liquid equilibrium relations was presented. The liquid density, vapour density, vapour pressure and fugacity were selected as constraint factors, and all the BWR coefficients were determined simultaneously by minimizing the appropriate object functions wherein each factor has a high differential value for saturated vapour pressure, and liquid and vapour densities in the region of saturation. Thus by minimizing the object function based on the deviation of saturated pressure and the deviations of densities of saturated vapour and saturated liquid, the results were far more accurate than those obtained by minimizing the object function based on deviation of saturated pressure in saturated vapour and liquid regions. We can say that if the BWR coefficients are determined using liquid density, vapour density and vapour pressure in saturation, the object function described in case II is also suited to determine the optimum BWR coefficients. The results thus obtained indicate that the reproducibility in the saturated region is satisfactory for designing industrial plants, such as air separation plants. One of the advantages of determining the BWR coefficients is that the mixture combination rule can also be applied to the original BWR equation of state. If there are vapour-liquid equilibrium data or any other high accuracy equations of state for pure substances wherein the mixture combination rule cannot be applied, the
Cryogenics 1988 Vol 28 August
525
New method to determine BWR coefficients: T. Asami Table 6
Comparison of calculated second virial coefficients (in cm 3 mo1-1) with those in the literature 21 Nitrogen
Temperature (K)
Bca I
Bli t
75 80 85 90 95 100 110 125 150 175 200 250 300
-273 -242
- (275 + 8) - (243 + 7)
- 196
- (197 _+ 5)
-162 -136 -108 -75.5 -39.O -18.7 -5.7
Argon
Bca I
-(160+3) -(132_+2) -(104+2) - (71.5 + 2)
- 278 -245 -218 -196 -161 -125 -89.7
- (251 _+ 3) - (225 + 3) - (202.5 + 2) - ( 1 8 3 . 5 + 1) -(154.5+1) -(123.0+1) - (86.2 _+ 1 )
- (35.2 + 1 ) -(16.2+1) -(4.2_+0.5)
-55.0 -38.5 -28.8
- (47.4 _+ 1 ) - ( 2 7 . 9 + 1) -(15.5_+0.5)
equilibrium constant obtained from the vapour-liquid equilibrium data or the fugacity obtained from the high accuracy equations of state of pure substances, can be used to accurately determine the BWR equation of state of pure substances. This can further be extended to mixed component systems and is simpler than other high accuracy equations of state.
References
1 Benedict, M., Webb, G.B. and Rubin, LC. J. Chem Phys. (1940) 8 334-345 2 Zndlkeviteh, D. and Kaufmann, T.G. A1ChE J (1966)12 577-580 3 Lin, MS. and Naphtali, L.M. AIChEJ(1963) 9 580-584 4 Cooper, H.W. and Goldfrank, J.C. Hydrocarbon Processing (1967) 46 141-146 5 Opfell, J.B., Sage, B.H. and Pitzer, K.S. Ind Eng Chem (1956) 48 2069-2076 6 Joffe, J. Chem Eng Prog(1949) 45 160-166 7 Powell, MJ.D. The Computer Journal (1965) 7 303-307 8 Jacobson,R.T, Stewart, R . L McCarty, R.D. and Hanley, H.J.M. Thermophysicai properties of nitrogen from the fusion line to 3500R (1944K) for pressures to 150000 PSIA, NBS Technical Note No. 648, US Department of Commerce, Washington DC,
526
Cryogenics
1988
Bli t
Vol 28 August
Oxygen
Bca I
-219 -183 -157 -127 -94.6 -73.1 - 57.8 -37.8 -24.7
Bli t
- (241 + 10) -(194+7) - ( 1 6 1 _+7) -(126+5) - (89 + 3) -(65+3) - (49 + 2) -(28+2) --(16+1)
USA (1973) 30-33 9 Gosman, A.L., McCarty, R.D. and Hust, J.G. Thermodynamic properties of argon from the triple point to 300 K at pressures to 1000 atmospheres, Report NSRDS-NBS 27, US Department of Commerce, Washington DC, USA (1969) 32 10 MeCarty, R.D. and Weber, L.A. Thermophysical properties of oxygen from the freezing line to 600 R for pressures to 5000 PSIA, NBS Technical Note No. 384, US Department of Commerce, Washington DC, USA (1971) 22-27 11 Wagner, W. Cryogenics (1973) 13 470-482 12 Wagner, W., Ewers, J. and Pentermann, W. J Chem Thermodynamics (1976) 8 1049-1060 13 Guggenheim, E.A. J Chem Phys (1945) 13 253-261 14 Pentermann, W. and Wagner, W. J. Chem Thermodynamics (1978) 10 1161 1172 15 Seshadri, D.N., Viswanath, D.S. and Kuloor, N.R. J Chem Eng Data (1967) 12 70-71 16 Shiki, N., Fukuzato, R., Asada, K. and Tanigaki, Y. Kobe Steel Engineering Reports (1980) 30 116-117 17 Peng, D.Y. and Robinson, D.B. lnd Eng Chem Fundam (1976) 15 59~4 18 Soave, G. Chemical Engineering Science (1972) 27 1197-1203 19 Starling, K.E. and Han M.S. Hydrocarbon Processing (1972) 51 129-132 20 Wilson, G.M., Silverberg, P.M. and Zellner, M.G. Adv Cryog Eng (1965) 10 192-208 21 Dymond, J.H. and Smith, E.B. The Virial Coefficients of Pure Gases and Mixtures Clarendon Press, Oxford, USA (1980) 1-246
Keywords: mathematical models; phase equilibria; argon; nitrogen; oxygen Knowledge of the equation of state is essential for the design of various industrial processes, especially air separation processes. For many years various attempts have been made to determine accurately the P VFrelationship. However, in the region of saturated vapour and liquid, for gases such as nitrogen, argon and oxygen, the simple equations of state are seldom satisfactory and high accuracy equations are too complex for industrial purposes. The inadequacy of the original BWR equation I in the saturated region comes from the fact that the coefficients have been determined using the P V T data obtained in the region of supercritical temperatures 2 and it is found to be almost impossible to estimate the P V T relation in the saturated region. Therefore, some modifications of the original equation have been attempted in the past to fit the P V T data in the saturation, by means of selected coefficients expressed as functions of temperature 3, or by using the law of corresponding states 4'5. However, these methods are too cumbersome to calculate the P V T relations. In this Paper, a new method to determine the BWR coefficients by a non-linear optimization and by selecting an appropriate object function with the utilization of the law of corresponding states is proposed from the viewpoint that the original BWR equation is simple in comparison with other equations of state. Using the eight coefficients thus determined, the original BWR equation is verified to be useful for accurate calculations of both the P V T relations in saturated and superheated regions over a wider range and the vapour-liquid equilibria of the binary mixtures.
Selection of c o n s t r a i n t f a c t o r s The original BWR equation and the derived fugacity
0011-2275/88/080521 06 $03.00 '~' 1988 Butterworth & Co (Publishers) Ltd
equation are expressed as follows
• ['cp3"~, 1 -J¢ + \1-/
+a p ° + l
3
+~T
p2)exp(-rp 2)
(1)
6a ( R T b - a ) o 2 + 5 ~ ps
3p2c[ 1-exp(-yp2) + ~ T~ 7p2
1 exp(_ yp2)] 2
2cp2RT 3 I 1 - exp(-YPZ)Tp 2 - exp(_ypZ)
,
]
2 ypZexp(- ypZ)
(2)
where P, f, T and p denote pressure, fugacity, temperature and density, respectively. R is a gas constant and a, b, c, Ao, Bo, Co, a and y are BWR coefficients. To determine the eight coefficients simultaneously, the saturated pressure, P, liquid density, PL, vapour density, Pv, and fugacity, f, are selected as constraint factors. Using the preliminary approximate values of BWR coefficients as initial values, the densities of liquid, PL, and vapour, Pv, and P and f at fixed temperature Tk(k = 1, 2 . . . . n), are calculated from the following simultaneous equations.
P(p, ~)Bw. P %) for values o f k = l , 2 . . . . n f(P, Tk)BWR-- f = 0
(3a)
Cryogenics 1988 Vol 28 August
521
New method to determine BWR coefficients: 7". Asami where subscript BWR denotes the BWR equation or the equation deduced from it. From the above separate equations for liquid density (p = PL) and vapour density (P = Pv) can be written. Applying Taylor's expansion to the above four equations and neglecting terms of second order and above, we get P(Pk, Tk)BWR -F
IOP(P' L s 0 T) j[
T = T~
(P -- Pk) -- P = 0
P P~
(3b)
=
r-,.,lOj~p, T)]~
f(Pk' Tk)BWR"1- L
ap
~
r
later for three different cases). The optimum BWR coefficients are determined by minimizing this object function. Powell's method combined with techniques of linear search is adopted 7 to minimize the object function. For simplicity, six kinds of deviations of constraint factors are defined as follows
(p -- pk) - f = 0
= P=P,
Af
wherein Pk is substituted by PLk and P,k in each equation. The above equation can further be simplified as follows
[_%?w,]
Af
,,_,,, p = p,
Ap
= --[e(pk, T~)Bw. Pk] -
(3c)
r
_T
L
r
lOf(pT_,o=~)aTWk l. (P
A)
_1 P = Pk =
-
~).w.
~(p~,
-
A]
The deviations of constraint factors can then be written as
Pk
The characteristics of the object function, Y, for different combinations of pressure and density are explained below.
k
Case I
A =T, Pk
The object function is conditioned by minimizing the square of deviations of saturated pressure at saturated vapour and liquid densities k
Y = £ (WlkE~k + W2kE2~)
Substituting in Equation (3c) above, we get
(4)
k=l r=
fl = Pk
p
k
-
where l'Vjk(j= 1, 2. . . . 6; k = 1, 2. . . . n) are the weighting factors of each deviation value.
\ P A
= -- [P(P~, Tk)BWR-- f , ]
(AP) _ fk(ff )
FOf(p, T)sw,]
(3d)
P =Pk
-- - [_f(p,. ~ ) . w . - A ] Using the law of corresponding states 6, one set of fixed values for all of the BWR coefficients is determined as the initial value and the following deviations are calculated
PL ,/k and
P~/~ The squares of these deviations are then substituted into an appropriate equation of object function Y (described
522
Cryogenics 1988 Vol 28 August
Case II The differential of saturated pressure with respect to the densities of saturated vapour and saturated liquid varies widely in the region away from the critical point. The following relation exists as shown in Figure 1
(5) where Ps denotes saturated vapour pressure. If the object function is based on the deviation of saturated pressure, the deviation of liquid density alone can converge to the same degree. However, from the relation in Equation (5) it is observed that the deviation of saturated vapour density does not converge to the same degree as the saturated liquid density even at small differential pressure values. The following object function is adopted to obtain the BWR coefficients which satisfy the PVT relation in the saturated region
Y= £ (WlkE2k -I- Wsk E2k q'- W6kE2k) k=l
(6)
New method to determine BWR coefficients: T. Asami Source data used In the process of calculation, appropriate empirical equations of pressure, liquid and vapour densities in saturated state are used at temperatures between the critical point and triple point to avoid scattering of raw data. 1 Wagner's equation is used to fit the saturated pressure 11"12 for nitrogen, argon and oxygen as follows Inn = T c T - l ( n l r + n2 rl'5 q- n3 z3 +tlar 6 + nsz 7
Q.
0=
+ n6 z9)
o_
2
Ap
where n = P/Pc and r = 1 - T/T~. Pc and Tc denote the critical pressure and temperature, respectively, and n~ n 6 are coefficients; an empirical equation of saturated liquid density, PL, in accordance with Guggenheim-Wagner is used as follows 13 PL/Pc = NO + ~ Ni Ti/3 for i = 1, 2 . . . . n
/XPv
Density p , (rnol/rn 3)
ln(pv/pc) = ~ Nit i/3 for i = 1, 2 . . . . n
To further improve the object function of case II and to satisfy the relation of vapour-liquid equilibria, we add fugacity as a constraint factor to the previous object function and determine BWR coefficients by minimizing the object function using the least squares method. Thus, the object function is expressed as follows
Expression of deviations The deviations of the calculated values from the original data are calculated using the following formula
~"=
Procedure The following steps were used to determine the BWR coefficients:
3
(calculated v a l u e ) - (value from data)l (val~ue f r o m d ~ a ) I × 100
(11)
j=l
Determination of the BWR coefficients
2
(10)
where Pc denotes the critical density and N i is a numerical coefficient.
Case III
l
(9)
where Pc denotes the critical density and N i is a numerical coefficient; and an equation of saturated vapour density, p,, in accordance with simplified Guggenheim-Wagner is also used as follows la
~PL
Figure 1 P p diagram illustrating the general characteristics of isotherms in the saturated vapour and liquid phase
k=l
(8)
the values of saturated vapour pressure, saturated liquid and vapour densities and fugacity, were selected from data published by NBS (for fixed temperatures8- lO); one set of fixed values of BWR coefficients were determined and deviations Elk -- E6k were calculated using Equation (3d); and the square of deviations was minimized and the BWR coefficients were obtained from the minimum value of the object function. Substituting the above BWR coefficients into Equation (1), each factor constrained in the object function can be compared with other published data, e.g. NBS reports, etc.
Results and discussion The BWR coefficients were determined for nitrogen, argon and oxygen according to the above procedure. The weighting factor, W, for each term of the object function was ignored since the scattering of data is negligible and has no effect on the calculation. Three alternative cases of object functions in Equations (4), (6) and (7) have been verified. The first case uses saturated vapour pressure as the constraint factor, the second uses saturated pressure, saturated liquid and vapour densities as constraint factors, and the third uses the previously mentioned constraint factors with the addition of fugacity. The results are tabulated in Table 1. It can be seen that in case I the deviations of vapour density are about 3~ , % , notwithstanding the deviations of liquid density which are less than about 2 %. However, both the deviations in cases II and III are approximately of the same degree except for the deviation of fugacity. Since the deviations of all constraint factors in case III converge to the same degree and are the minimum in all three cases, the evaluation
Cryogenics 1988 Vol 28 August
523
New method to determine BWR coefficients. T. Asami Table 1
Comparison of the deviations, ~(%), of constraint factors #L, Pv, P and f, in cases I, II and III Nitrogen (63.15-126.20 K)
/3(,OL) e(Pv) ~(P) ~(f)
I
II
0.54 3.04 0.78 0.85
0.47 0.40 0.37 0.48
Argon (83.8-150,69 K)
III 0.52 0.36 0.36 0.27
I 0.60 2.42 0.50 1.24
of the BWR equation is best executed using the set of BWR coefficients in case III. The BWR equation and the Starling and Han (BWRS) equation have been verified for their accuracy in the saturated region using a set of BWR coefficients published by other authors ~5't6. The results are given in Table 5. These BWR coefficients were obtained based on data in the region of supercritical temperature in one case and vapour pressure and liquid density in the other. Hence, the BWR coefficients determined by this method cannot be used to accurately predict the P VT relation in the saturated region or the vapour liquid equilibria. The prediction performances of saturated densities by cubic equations of state are usually poor, especially in the case of saturated liquid densities. For example, the Peng-Robinson equation 17 has been proposed to improve the prediction performance of the Soave-RedlichKwong equation ~a for saturated liquid densities. However, the deviation in liquid densities cannot be ignored ~7. On the other hand, a virial expansion type equation of state, such as the BWR equation, can be used accurately to calculate the P V T relation in the high density region. Nevertheless, eight constants of the BWR equation have sometimes been determined by using P V T data from the gaseous phase. In this case, they should not be used in
Table 2
II 0.41 0.17 0.22 1.30
Oxygen (70-154,58 K) III
0.34 0.76 0.43 0.51
I
II
2.21 3.98 5,05 6.00
0.90 0.70 0.50 0.80
saturated regions because extreme deviations occur. The BWR constants of nitrogen3 and argon2 were determined using P V T data at vapour pressure in the supercritical gaseous region. For comparison, the BWR constants for oxygen were evaluated using coefficients derived by Seshadri et al.lS. Those for nitrogen were obtained from the correlation of Starling and Han (BWRS) 19. The deviations between the data and the calculated results obtained using the above BWR constants are shown in Table 5. It can be seen that those deviations are larger than in the present work. The deviations of vapour density, Pv, between the predicted value and data in the superheated region are indicated in Table 2. The deviations of equilibrium temperature between the calculated value and data from the literature, for oxygen + argon, argon + nitrogen and nitrogen + oxygen in binary mixtures, are shown in Table 3 2°. In particular, in case III by adding fugacity as a constraint factor to the object function, BWR coefficients are obtained which satisfy both the P V T and vapourliquid equilibria relations. The optimum coefficients for nitrogen, argon and oxygen are given in Table 4. To verify the BWR equation using the calculated BWR coefficients, the second virial coefficient, B, is calculated and compared with the values published by other authors.
Deviations, ~, of vapour density, Pv, in the superheated region
Pressure (MPa)
Temperature (K)
Number of
Absolute mean deviation,
data points
8(%)
300~88.88 300-100 300-1 oo 300-125 300-125
10 9 9 8 8
0,03734 0.05560 0.1216 0,2335 0,3790
300-100 300-125 300-125 300-125 300-150 300-150
9 8
0.1286 0.1919 0.3784 0.6665
311.1 - I 00 311.1 -130.6 311.1 -130.6 311.1 -130.6 311.1-161.1 311.1 -161.1
8 7 7 7 6 6
Nitrogen 0.2069 0.3448 0.6895 1.379 2.758
Argon 0.2027 0.4053 0.8106 1.520 3.040 4.053
8
8 7 7
1.446 1.849
Oxygen 0.2069 0.3448 0.6895 1.379 2.758 3.792
524
0,85 1.59 0.50 1.26
III
Cryogenics 1 9 8 8 Vol 2 8 A u g u s t
0.1282 0.1316 0.2661 0.5790 1.235 1.762
New method to determine BWR coefficients: T. Asami Table 3
Deviations, e, between the calculated and experimental vapour-liquid equilibrium temperatures in three binary mixtures
System
Pressure a (MPa)
Number of data points
Absolute mean deviation, ~(%)
Ar-O 2
0.1013-0.4053 0.6079-1.01 31 1.21 61-2.6342
45 45 110
1.6 1.6 2.0
N 2 Ar
0.1013-0.4053 0.6079-1.01 31 1.21 61 2.6342
46 44 96
1.8 1.4 1.8
N2 02
0.1013-0.4053 0.6079-1.01 31 1.2161 2.6342
24 27 86
0.4 0.3 0.3
aSource of experimental data: G.M. Wilson et al. 2°
T a b l e 4 BWR coefficients determined in case II1: A o ( M P a m 6 k m o l 2 ) , B0(m3kmol (m 6 kmol 2), c (MPa K2 m 9 kmol 3), ~ (m 9 kmol- 3), 7 ( m6 kmol 2) BWR coefficients
Nitrogen
Ao Bo CO a b c
0.15019143 0.55931 023 x 10 0.30769511 x 103 0.26460522×10 0.368481 98 x 1 0 0.46882840 × 102 0.79014894 x 1 0 0.56821964 x 10
7
Argon
2 2 4 2
B = Be--RT
Co
(12)
RT 3
The calculated values and data from the literature are presented in Table 621 . The values in the low temperature region show good consistency. However, in the region of supercritical temperature, as in the case of the P V T relation, the deviation increases slightly. Also, the P V T relation in the superheated region, between atmospheric and critical pressure, is accurately satisfied. It can be verified from Table 2 that the deviations of vapour density in the superheated region are very small.
T a b l e 5 Deviations of f)L, Pv and P calculated using the coefficients of equations of state between BWR and BWRS of other authors and NBS data Oxygen (BWR)
Nitrogen (BWRS)
17.0 82.0 66.0
1.58 2.24
13 70 1 54.58 Reference 15
63 77.4-116.5 Reference 1 (
Absolute mean deviation (%):
~;(PL) ~:(Pv) ~:(P) Number of data points Temperature range (K) Source
3), b
Oxygen
0.85989511 × 10 0.941 21657 x 10 0.8465301 5 x 103 0.82459105×10 0.56534201 × 1 0 0.10571768 × 103 0.221 22512 x 1 0 0.33078947 x 10
1
B is deduced from the BWR equation by the following formula Ao
1), Co(MPaK2m6kmol-2), a (MPam9kmol
1 2 2 2 4 2
0.14632602 0.35604172 x 10 1 0.36064256 x 103 0.21228725x10 2 0.29022418 x 10 2 0.52748418 x 102 0.41 330800 x 10 4 0.46113980 x 10 -2
Conclusions A new method to obtain an optimum set of BWR coefficients which widely satisfies both the P V T and vapour-liquid equilibrium relations was presented. The liquid density, vapour density, vapour pressure and fugacity were selected as constraint factors, and all the BWR coefficients were determined simultaneously by minimizing the appropriate object functions wherein each factor has a high differential value for saturated vapour pressure, and liquid and vapour densities in the region of saturation. Thus by minimizing the object function based on the deviation of saturated pressure and the deviations of densities of saturated vapour and saturated liquid, the results were far more accurate than those obtained by minimizing the object function based on deviation of saturated pressure in saturated vapour and liquid regions. We can say that if the BWR coefficients are determined using liquid density, vapour density and vapour pressure in saturation, the object function described in case II is also suited to determine the optimum BWR coefficients. The results thus obtained indicate that the reproducibility in the saturated region is satisfactory for designing industrial plants, such as air separation plants. One of the advantages of determining the BWR coefficients is that the mixture combination rule can also be applied to the original BWR equation of state. If there are vapour-liquid equilibrium data or any other high accuracy equations of state for pure substances wherein the mixture combination rule cannot be applied, the
Cryogenics 1988 Vol 28 August
525
New method to determine BWR coefficients: T. Asami Table 6
Comparison of calculated second virial coefficients (in cm 3 mo1-1) with those in the literature 21 Nitrogen
Temperature (K)
Bca I
Bli t
75 80 85 90 95 100 110 125 150 175 200 250 300
-273 -242
- (275 + 8) - (243 + 7)
- 196
- (197 _+ 5)
-162 -136 -108 -75.5 -39.O -18.7 -5.7
Argon
Bca I
-(160+3) -(132_+2) -(104+2) - (71.5 + 2)
- 278 -245 -218 -196 -161 -125 -89.7
- (251 _+ 3) - (225 + 3) - (202.5 + 2) - ( 1 8 3 . 5 + 1) -(154.5+1) -(123.0+1) - (86.2 _+ 1 )
- (35.2 + 1 ) -(16.2+1) -(4.2_+0.5)
-55.0 -38.5 -28.8
- (47.4 _+ 1 ) - ( 2 7 . 9 + 1) -(15.5_+0.5)
equilibrium constant obtained from the vapour-liquid equilibrium data or the fugacity obtained from the high accuracy equations of state of pure substances, can be used to accurately determine the BWR equation of state of pure substances. This can further be extended to mixed component systems and is simpler than other high accuracy equations of state.
References
1 Benedict, M., Webb, G.B. and Rubin, LC. J. Chem Phys. (1940) 8 334-345 2 Zndlkeviteh, D. and Kaufmann, T.G. A1ChE J (1966)12 577-580 3 Lin, MS. and Naphtali, L.M. AIChEJ(1963) 9 580-584 4 Cooper, H.W. and Goldfrank, J.C. Hydrocarbon Processing (1967) 46 141-146 5 Opfell, J.B., Sage, B.H. and Pitzer, K.S. Ind Eng Chem (1956) 48 2069-2076 6 Joffe, J. Chem Eng Prog(1949) 45 160-166 7 Powell, MJ.D. The Computer Journal (1965) 7 303-307 8 Jacobson,R.T, Stewart, R . L McCarty, R.D. and Hanley, H.J.M. Thermophysicai properties of nitrogen from the fusion line to 3500R (1944K) for pressures to 150000 PSIA, NBS Technical Note No. 648, US Department of Commerce, Washington DC,
526
Cryogenics
1988
Bli t
Vol 28 August
Oxygen
Bca I
-219 -183 -157 -127 -94.6 -73.1 - 57.8 -37.8 -24.7
Bli t
- (241 + 10) -(194+7) - ( 1 6 1 _+7) -(126+5) - (89 + 3) -(65+3) - (49 + 2) -(28+2) --(16+1)
USA (1973) 30-33 9 Gosman, A.L., McCarty, R.D. and Hust, J.G. Thermodynamic properties of argon from the triple point to 300 K at pressures to 1000 atmospheres, Report NSRDS-NBS 27, US Department of Commerce, Washington DC, USA (1969) 32 10 MeCarty, R.D. and Weber, L.A. Thermophysical properties of oxygen from the freezing line to 600 R for pressures to 5000 PSIA, NBS Technical Note No. 384, US Department of Commerce, Washington DC, USA (1971) 22-27 11 Wagner, W. Cryogenics (1973) 13 470-482 12 Wagner, W., Ewers, J. and Pentermann, W. J Chem Thermodynamics (1976) 8 1049-1060 13 Guggenheim, E.A. J Chem Phys (1945) 13 253-261 14 Pentermann, W. and Wagner, W. J. Chem Thermodynamics (1978) 10 1161 1172 15 Seshadri, D.N., Viswanath, D.S. and Kuloor, N.R. J Chem Eng Data (1967) 12 70-71 16 Shiki, N., Fukuzato, R., Asada, K. and Tanigaki, Y. Kobe Steel Engineering Reports (1980) 30 116-117 17 Peng, D.Y. and Robinson, D.B. lnd Eng Chem Fundam (1976) 15 59~4 18 Soave, G. Chemical Engineering Science (1972) 27 1197-1203 19 Starling, K.E. and Han M.S. Hydrocarbon Processing (1972) 51 129-132 20 Wilson, G.M., Silverberg, P.M. and Zellner, M.G. Adv Cryog Eng (1965) 10 192-208 21 Dymond, J.H. and Smith, E.B. The Virial Coefficients of Pure Gases and Mixtures Clarendon Press, Oxford, USA (1980) 1-246