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Prediction and calculation of azeotropic behaviour from an equation of state
ltlJlBPlm[ [QOIUBRiA ELSEVIER
Fluid Phase Equilibria 127 (1997) 123-127
Prediction and calculation of azeotropic behaviour from an equation of state A.V. Trotsenko The Odessa State Academy of Refrigeration, 1/3 Peter Veliky str. 270100, Odessa, Ukraine Received 3 March 1995; accepted 24 April 1996
Abstract
A new method for prediction and determination of azeotropic behaviour without performing mixture vapour-liquid equilibrium calculations is presented. It can be used for any equation of state and any mixing rule, but it is limited by the pseudocritical parameters of the mixture. The use of the method is demonstrated using cubic equations of state. Keywords: Method of calculation; Equation of state; Vapour-liquid equilibria; Mixture
The Maxwell equation P s m ( T , z ) = f v l G p ( u , T , z ) d u / ( U G - UL)
(1)
can be used not only for pure substances but also for azeotropic points of n-component mixtures. At a fixed temperature T and mixture composition z = { z/}, i = 1,2 . . . . . n - 1 the pressure Psm, and volumes v~ and v u can be determined using an equation of state and the phase equilibrium conditions for temperatures and pressures lower than the critical temperature and pressure for pure substances, and lower than the pseudocritical temperature and pressure for mixtures. Points L and G are either in the heterogeneous region or on binodals. The last situation occurs for pure substances and azeotropic points. The pressure calculated from the Maxwell rule is related to bubble point pressure Pb and dew point pressure pj by the inequalities
Po(T,z) < Psm(T,z) <-pu(r,z). For binary systems the set of points calculated by the Maxwell rule (Maxwell points) forms a continuous line (Maxwell line) Psm(Z)at constant T, or Tsm(Z) at constant p, which lies inside the heterogeneous two-phase region. The shape of the Maxwell line for different types of vapour-liquid equilibria are shown in the 0378-3812/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S0378-3812(96)03105-6
124
A.V. Trotsenko / Fluid Phase Equilibria 127 (1997) 123-127
pressure-composition diagrams in Fig. 1. The dashed curves are the Maxwell curves, and the solid lines are determined from phase equilibrium calculations, using the Redlich-Kwong-Wilson equation of state [ 1]. As noted before, the Maxwell lines exist only at Psm < Pc- This inequality determines the limitation on their use. The neighbourhood of the disappearing point for the Maxwell line is shown in Fig. 2. The function Psm(Z) at constant T has a global extremum in the azeotropic point, a maximum for positive azeotropic mixtures (Fig. 1, c and e) and a minimum for negative ones. As is shown in many of our calculations, the Maxwell curve can show an extremum only in azeotropic points. So the necessary conditions of azeotropy are given by
OPsm(T,z)/OZ i = 0, i = 1,2 . . . . . n - 1.
(2)
The determination of the derivatives in Eq. (2) must be done with conditions VG = VG(T, Z), VL = VL(T, Z), Psm( VL ,T) = Psm( VG ,T).
So one can obtain the azeotropic conditions for any equation of state p(v,T, z) as
fo°~Op(v,T,z)/OZidv=O,i = 1,2 . . . . . n - 1 .
(3)
Eqs. (1) and (3) and conditions
p(vc,T,z)-psm(T,z
) =0
and
P(VL,T,z)-psm(T,z ) =0 are the system of equations which determine the azeotropic parameters for an n-component mixture. This method and methods using the vapour-liquid calculations for the mixture are associated with the extremum search. The mathematical problem for these methods can be formulated in a similar way. But replacing the mixture phase equilibria calculations by those made with the Maxwell line brings about a great reduction of computational effort. Also, from this method some quality predictions can be made. In particular, a comparison of the signs of the derivative ~Psm(Z, Z)/~Z i at the pure component limits gives the possibility of predicting azeotropic behaviour. For example, for a binary mixture, with the exception of double azeotropy, the azeotropic condition can be written as
Opsm(T,O)/OZ" Opsm(T,1)/OZ < 0 So at fixed temperature or pressure the algorithm for azeotropic point search for binary mixtures involves the investigation of a function of one variable. A generalized method for determination of the Maxwell point parameters is known [2], but for some cubic equation of state, which can be written in the form
p = e . T / b m . f ( v / b m, am//bm)
(4)
this calculation procedure may be simplified. The van der Waals, Redlich-Kwong and Peng-Robinson equations of state and some of their modifications can be written in the form of Eq. (4). From Eqs. (1) and (2) and the other phase equilibrium conditions for pure substances one can prove that dimensionless Maxwell-point quantities 7rsm, q~I~, ~PG are universal functions of the ratio am/b m.
125
A.V. Trotsenko / Fluid Phase Equilibria 127 (1997) 123-127
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Fig. 1. Maxwell curves for different types of vapour-liquid equilibria.
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A.V. Trotsenkz) / Fluid Phase Equilibria 127 (1997) 123-127
126
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3.6
:~ 3.4
Jff/!/f//
3.2 jf / 3.0 0.85
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/
0.87
0.89
0.91
0.93
0.95
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Fig. 2. Phase diagram for Na-CH 4 mixtures near the critical state. 1, pseudocritical line; 2, Maxwellcurve; 3, binodal. The values of these functions depend only on the form of the equation of state and are independent on the nature of the components, their concentration or the mixing rules. All this information is included in the ratio am/b m. This form of representation for the Maxwell point quantities gives the possibility of creating a simple computational procedure for prediction and calculation of azeotropy and to test mixing rules. In principle, any continuous line that definitely lies in the two-phase region, can be used for the prediction and calculation of azeotropy, and the Maxwell line is one of these. The criterion of azeotropy can be formulated as the existence of a global extremum for this line. However, the Maxwell line is well grounded and can be calculated in a simple way.
1. List of symbols
a,b parameters in cubic equation of state n p T v z
number of components pressure absolute temperature specific volume mixture composition or fraction of the first component in binary system
1.1. Greek symbols 7r q~ ~-
dimensionless ratio (Psm" bm/(R" reduced specific volume (v/b m) reduced temperature (T/To)
T))
1.2. Subscripts b c
boiling point critical state for pure substances or pseudocritical state for mixture
A.V. Trotsenko / Fluid Phase Equilibria 127 (1997) 123-127
d G i L m sm
127
condensing point 'gas' phase number of components 'liquid' phase mixture Maxwell point
Acknowledgements The author thanks Mrs. E.I. Tabachnik for support in mathematical and calculational problems.
References [1] G.M. Wilson, Adv. Cryog. Eng., 9 (1964) 198-207. [2] G.K. Lavrenchenko,E.I. Tabachnik, A.V. Trotsenko, Thermal physics properties of substances and materials. GSSSD, 18 (1983) 41-46 (in Russian).
Fluid Phase Equilibria 127 (1997) 123-127
Prediction and calculation of azeotropic behaviour from an equation of state A.V. Trotsenko The Odessa State Academy of Refrigeration, 1/3 Peter Veliky str. 270100, Odessa, Ukraine Received 3 March 1995; accepted 24 April 1996
Abstract
A new method for prediction and determination of azeotropic behaviour without performing mixture vapour-liquid equilibrium calculations is presented. It can be used for any equation of state and any mixing rule, but it is limited by the pseudocritical parameters of the mixture. The use of the method is demonstrated using cubic equations of state. Keywords: Method of calculation; Equation of state; Vapour-liquid equilibria; Mixture
The Maxwell equation P s m ( T , z ) = f v l G p ( u , T , z ) d u / ( U G - UL)
(1)
can be used not only for pure substances but also for azeotropic points of n-component mixtures. At a fixed temperature T and mixture composition z = { z/}, i = 1,2 . . . . . n - 1 the pressure Psm, and volumes v~ and v u can be determined using an equation of state and the phase equilibrium conditions for temperatures and pressures lower than the critical temperature and pressure for pure substances, and lower than the pseudocritical temperature and pressure for mixtures. Points L and G are either in the heterogeneous region or on binodals. The last situation occurs for pure substances and azeotropic points. The pressure calculated from the Maxwell rule is related to bubble point pressure Pb and dew point pressure pj by the inequalities
Po(T,z) < Psm(T,z) <-pu(r,z). For binary systems the set of points calculated by the Maxwell rule (Maxwell points) forms a continuous line (Maxwell line) Psm(Z)at constant T, or Tsm(Z) at constant p, which lies inside the heterogeneous two-phase region. The shape of the Maxwell line for different types of vapour-liquid equilibria are shown in the 0378-3812/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S0378-3812(96)03105-6
124
A.V. Trotsenko / Fluid Phase Equilibria 127 (1997) 123-127
pressure-composition diagrams in Fig. 1. The dashed curves are the Maxwell curves, and the solid lines are determined from phase equilibrium calculations, using the Redlich-Kwong-Wilson equation of state [ 1]. As noted before, the Maxwell lines exist only at Psm < Pc- This inequality determines the limitation on their use. The neighbourhood of the disappearing point for the Maxwell line is shown in Fig. 2. The function Psm(Z) at constant T has a global extremum in the azeotropic point, a maximum for positive azeotropic mixtures (Fig. 1, c and e) and a minimum for negative ones. As is shown in many of our calculations, the Maxwell curve can show an extremum only in azeotropic points. So the necessary conditions of azeotropy are given by
OPsm(T,z)/OZ i = 0, i = 1,2 . . . . . n - 1.
(2)
The determination of the derivatives in Eq. (2) must be done with conditions VG = VG(T, Z), VL = VL(T, Z), Psm( VL ,T) = Psm( VG ,T).
So one can obtain the azeotropic conditions for any equation of state p(v,T, z) as
fo°~Op(v,T,z)/OZidv=O,i = 1,2 . . . . . n - 1 .
(3)
Eqs. (1) and (3) and conditions
p(vc,T,z)-psm(T,z
) =0
and
P(VL,T,z)-psm(T,z ) =0 are the system of equations which determine the azeotropic parameters for an n-component mixture. This method and methods using the vapour-liquid calculations for the mixture are associated with the extremum search. The mathematical problem for these methods can be formulated in a similar way. But replacing the mixture phase equilibria calculations by those made with the Maxwell line brings about a great reduction of computational effort. Also, from this method some quality predictions can be made. In particular, a comparison of the signs of the derivative ~Psm(Z, Z)/~Z i at the pure component limits gives the possibility of predicting azeotropic behaviour. For example, for a binary mixture, with the exception of double azeotropy, the azeotropic condition can be written as
Opsm(T,O)/OZ" Opsm(T,1)/OZ < 0 So at fixed temperature or pressure the algorithm for azeotropic point search for binary mixtures involves the investigation of a function of one variable. A generalized method for determination of the Maxwell point parameters is known [2], but for some cubic equation of state, which can be written in the form
p = e . T / b m . f ( v / b m, am//bm)
(4)
this calculation procedure may be simplified. The van der Waals, Redlich-Kwong and Peng-Robinson equations of state and some of their modifications can be written in the form of Eq. (4). From Eqs. (1) and (2) and the other phase equilibrium conditions for pure substances one can prove that dimensionless Maxwell-point quantities 7rsm, q~I~, ~PG are universal functions of the ratio am/b m.
125
A.V. Trotsenko / Fluid Phase Equilibria 127 (1997) 123-127
0.30
4.0
Cl
343,2 K / ~ ' ~ "
b
0.28 3.0
0.26 0.24
Q_
0-0,22
O-
2.0
¢-A ~
0.20
1.0
,-'.j- J
0.18 0,16
0.2
R12
0.4
z
0.8
4.5.
0.8
0.0
1.0 R22
~
!°
¸
0.0 0.2 CHI
0.30
0
.8
0.8
1.0 N2
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°
; 2.13.2 K
0.25
: Z/
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0.20
/,',
~o!
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::~/
o.o5
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(
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o
J
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1 ~
¢J
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t'~
.1 /
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0.05
o.oo
o.2
C3Ha
/
o.~ ....... I
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R23
Fig. 1. Maxwell curves for different types of vapour-liquid equilibria.
03
1.0 R14
A.V. Trotsenkz) / Fluid Phase Equilibria 127 (1997) 123-127
126
3.8 140 K
3.6
:~ 3.4
Jff/!/f//
3.2 jf / 3.0 0.85
J
/
0.87
0.89
0.91
0.93
0.95
Z
Fig. 2. Phase diagram for Na-CH 4 mixtures near the critical state. 1, pseudocritical line; 2, Maxwellcurve; 3, binodal. The values of these functions depend only on the form of the equation of state and are independent on the nature of the components, their concentration or the mixing rules. All this information is included in the ratio am/b m. This form of representation for the Maxwell point quantities gives the possibility of creating a simple computational procedure for prediction and calculation of azeotropy and to test mixing rules. In principle, any continuous line that definitely lies in the two-phase region, can be used for the prediction and calculation of azeotropy, and the Maxwell line is one of these. The criterion of azeotropy can be formulated as the existence of a global extremum for this line. However, the Maxwell line is well grounded and can be calculated in a simple way.
1. List of symbols
a,b parameters in cubic equation of state n p T v z
number of components pressure absolute temperature specific volume mixture composition or fraction of the first component in binary system
1.1. Greek symbols 7r q~ ~-
dimensionless ratio (Psm" bm/(R" reduced specific volume (v/b m) reduced temperature (T/To)
T))
1.2. Subscripts b c
boiling point critical state for pure substances or pseudocritical state for mixture
A.V. Trotsenko / Fluid Phase Equilibria 127 (1997) 123-127
d G i L m sm
127
condensing point 'gas' phase number of components 'liquid' phase mixture Maxwell point
Acknowledgements The author thanks Mrs. E.I. Tabachnik for support in mathematical and calculational problems.
References [1] G.M. Wilson, Adv. Cryog. Eng., 9 (1964) 198-207. [2] G.K. Lavrenchenko,E.I. Tabachnik, A.V. Trotsenko, Thermal physics properties of substances and materials. GSSSD, 18 (1983) 41-46 (in Russian).