2. Thermodynamics of the separation
Thermodynamic properties play an important role in separation operations with respect to
energy requirements, phase equilibria, biological activity, and equipment sizing [1]. In
distillation, since liquid activity coefficient and excess properties of equilibrium state can
give predictions for K-value or property changes of mixing, analyzing phase equilibrium state
can be significant [2]. There are several models for the prediction of vapor-liquid equilibrium
(VLE). Here, we will deal with the underlying principles of VLE and the models for the
excess Gibbs energy, and show them as diagrams of ethanol-water system comparing with
experimental data.
2.1 Underlying principles
The definition of the activity coefficient is
coefficient of species i,
species i , and
, where
is the fugacity of species i in solution,
is the activity
is the fugacity of pure
is liquid mole fraction of i in solution.
By Lewis/Randall rule, this equation converts into
where
is the
fugacity of species i in ideal solution [2].
At VLE, the Gibbs free energy of the two phases should be equal, which means the fugacity
of the two phases also should be the same.
By the definition of fugacity :
. VLE measurements are very often made at
pressure low enough that the vapor phase may be assumed an ideal gas. In this case,
leads the equation to
=1,
[2].
With these principles and assumptions, the activity coefficient is re-defined as
, and this is modified Raoult’s law allowing calculation of activity coefficients
from experimental low-pressure VLE data [2].
Activity coefficient
is
also
partial
molar
property of
solution,
(for binary system). By summability relation :
which
gives
. With this equation, the models for the excess Gibbs energy suggest
different relation between liquid composition and activity coefficient, which gives the derived
liquid activity coefficient of the system. Vapor composition, pressure, and temperature also
can be derived from the activity coefficient and modified Raoult’s law [2].
Furthermore, the definition of K-value for modified Raoult’s law,
, can give
predictions for K-value from the derived activity coefficient [1].
For the analyzing of ethanol/water equilibrium state, the experimental data is from
KDB(Korea Thermophysical Properties Data Bank) of CHERIC(Chemical Engineering
Research Information Center). The information is isobaric T-x-y data at 760 mmHg.
T, deg.C
100.0
99.3
96.9
96.0
96.0
95.6
94.8
93.8
93.5
92.9
90.5
90.5
89.4
88.4
88.6
87.2
85.4
84.5
84.0
83.4
83.0
82.3
82.0
81.4
81.5
81.2
80.9
80.5
80.2
80.0
P, mmHg
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
760
x
0
0.0028
0.0118
0.0137
0.0144
0.0176
0.0222
0.0246
0.0302
0.0331
0.0519
0.0530
0.0625
0.0673
0.0715
0.0871
0.126
0.143
0.172
0.206
0.210
0.255
0.284
0.321
0.324
0.345
0.405
0.430
0.449
0.506
y
0
0.032
0.113
0.157
0.135
0.156
0.186
0.212
0.231
0.248
0.318
0.314
0.339
0.370
0.362
0.406
0.468
0.487
0.505
0.530
0.527
0.552
0.567
0.586
0.583
0.591
0.614
0.628
0.633
0.661
79.5
78.8
78.5
78.4
78.3
78.3
760
760
760
760
760
760
0.545
0.663
0.735
0.804
0.917
1.000
0.673
0.733
0.776
0.815
0.906
1.000
For the vapor pressure of the components, the equation,
[3], is used.
For ethanol, C1=73.304, C2=-7122.3, C3=-7.1424, C4=2.8853*10^(-6), C5=2.
For water C1=73.649, C2=-7258.2, C3=-7.3037, C4=4.1653*10^(-6), C5=2 [3].
2.2 Models for the excess Gibbs energy
2.2.1 Margules equation
In Margules equation, it assumes
is a linear function of
.
Activity coefficient is derived by the differentiation of the equation with respect to
[2].
.
2.2.2 Van Laar equation
This equation results when the reciprocal expression
function of
is expressed as a linear
.
New parameters are defined by the equations,
,
these definitions,
[2]
, and from
2.2.3 Local composition Models
The prediction of the molecular thermodynamics of liquid-solution behavior is often based
on the concept of local composition. In this concept, within a liquid solution, local
compositions, different from the overall mixture composition, are presumed to account for
the short-range order and nonrandom molecular orientations that result from different in
molecular size and intermolecular forces [2].
2.2.3.1 Wilson equation
The Wilson equation contains two parameters like the Margules and van Laar equations.
Here,
,
and
are the molar volumes at temperature T of pure liquid.
and
are constants independent of composition and temperature.
Activity coefficient is also derived by differentiation of
.
[2]
For ethanol/water binary system,
=339.1160(J/mol), and
=4450.75932(J/mol) [4].
2.2.3.2 NRTL(Non-Random-Two-Liquid) equation
The NRTL equation has three independent parameters for binary system.
,
, and
are constants that are dependent on particular pair of species, and
independent of composition and temperature.
[2]
For
ethanol/water
binary
=15937.56528(J/mol) [4].
system,
=0.0417,
=9868.83912(J/mol),
and
2.2.3.3 UNIFAC(Universal Quasi-Chemical Functional-group Activity Coefficients)
method
In UNIFAC method, activity coefficient is expressed as the summation of two parts. The
first part is combinatorial part which contains pure-species parameters only. The second part
is residual part to account for molecular interactions. This method considers interactions
between the functional groups(subgroups) of the molecules in solution [2].
Here,
,
,
,
,
,
,
,
,
: species
: dummy index
: subgroup
: dummy index
: the number of subgroup k that belongs to species i
,
, and
are constants that can be found in references [2].
For ethanol/water binary system, (A : Ethanol, B : Water)
Main group
Subgroup
CH3
1
0.9011
0.848
1
0
CH2
2
0.6744
0.540
1
0
5 "OH"
OH
15
1.0000
1.200
1
0
7 "H20"
H2 0
17
0.9200
1.400
0
1
1 "CH2"
=
=
=
=
=
=986.5 K,
=0K
=986.5 K,
=156.40 K,
=1318.00 K,
=156.40 K,
=353.50 K,
=1318.00 K
=300.00 K,
=300.00 K
=-229.10 K [2].
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
[1] Seader J.D., Henley E.J., and Roper D.K., “Separation Process Principles”. John Wiley &
Sons, 3rd Edition, 2011.
[2] Smith J.M., Van Ness H.C., and Abbott M.M., “Introduction to Chemical Engineering
Thermodynamics”. McGrawHill, 7th edition, 2005.
[3] Perry R.H., and Green D.W., “Perry's Chemical Engineer's Handbook”. McGrawHill, 8th
edition, 2008
[4] Gmehling J., Onken U., and Arlt W. “Vapor-Liquid Equilibrium Data Collection”.
Chemistry Data Series, Volume I, parts 1-8, 1974-1990
Appendix – Matlab Coding
- Margules equation
clear all; clc;
% Margules
t=[100 99.3 96.9 96.0 96.0 95.6 94.8 93.8 93.5 92.9 90.5 90.5 89.4 88.4
88.6 87.2 85.4 84.5 84.0 83.4 83.0 82.3 82.0 81.4 81.5 81.2 80.9 80.5 80.2
80.0 79.5 78.8 78.5 78.4 78.3 78.3]; % mmHG
T=t+273.15;
P=101.33;
% kPa
x1=[0.0001 0.0028 0.0118 0.0137 0.0144 0.0176 0.0222 0.0246 0.0302 0.033
0.0519 0.0530 0.0625 0.0673 0.0715 0.0871 0.126 0.143 0.172 0.206 0.210
0.255 0.284 0.321 0.324 0.345 0.405 0.430 0.449 0.506 0.545 0.663 0.735
0.804 0.917 0.999];
y1=[0.0001 0.032 0.113 0.157 0.135 0.156 0.186 0.212 0.231 0.248 0.318
0.314 0.339 0.370 0.362 0.406 0.468 0.487 0.505 0.530 0.527 0.552 0.567
0.586 0.583 0.591 0.614 0.628 0.633 0.661 0.673 0.733 0.776 0.815 0.906
0.999];
x2=1-x1;
y2=1-y1;
C1=73.304;
C2=-7122.3;
C3=-7.1424;
C4=2.8853*10^(-6);
C5=2;
P1_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;
C1=73.649;
C2=-7258.2;
C3=-7.3037;
C4=4.1653*10^(-6);
C5=2;
P2_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;
R=8.314;
% ---- exp ---gamma1_e=y1.*P./(x1.*P1_sat);
gamma2_e=y2.*P./(x2.*P2_sat);
GE_e=R.*T.*(x1.*log(gamma1_e)+x2.*log(gamma2_e));
% ---- derived ----
for i=1:36
if i>4 && i<36
func(i-4)=GE_e(i)./(R.*T(i).*x1(i).*x2(i));
x_a(i-4)=x1(i);
end
end
x=0:0.01:1;
c=polyfit(x_a,func,1);
y=c(2)+c(1).*x;
A12=y(1);
A21=y(101);
gamma1_d=exp((1-x1).^2.*(A12+2*(A21-A12).*x1));
gamma2_d=exp((x1.^2.*(A21+2*(A12-A21).*(1-x1))));
y_d=x1.*gamma1_d.*P1_sat./P;
K_e=y1./x1;
K_d=y_d./x1;
y=x;
figure(1)
plot(x1,y1,'r^');
hold on
plot(x1,y_d, 'b');
hold on
plot(x, y, 'k')
hold on
legend('Exp. Data', 'Margules equation', '45¢ª line', 0);
title('x-y diagram for ethanol/water VLE');
xlabel('Liquid-phase mole fraction of ethanol(x1)');
ylabel('Vapor-phase mole fraction of ethanol(y1)');
grid on
figure(2)
plot(x1, gamma1_e, 'r^');
hold on
plot(x1, gamma2_e, 'bo');
hold on
plot(x1, gamma1_d, 'r');
hold on
plot(x1, gamma2_d, 'b');
hold on
legend('¥ã1-ethanol by exp. data', '¥ã2-water by exp. data', '¥ã1-ethanol
by Margules equation', '¥ã2-water by Margules equation', 0);
title('Liquid-phase activity coefficients for ethanol/water system at 1
atm');
xlabel('Liquid phase mole fraction of ethanol(x1)');
ylabel('Liquid-phase activity coefficient');
axis([0 1 0 7]);
grid on
figure(3)
plot(x1, K_e, 'ro');
hold on
plot(x1, K_d, 'bo');
hold on
legend('K-value by exp. data', 'K-value by Margules equation', 0);
title('K-value of ethanol/water system at 1 atm');
xlabel('Liquid-phase mole fraction of ethanol(x1)');
ylabel('K-value');
axis([0 1 0 17]);
grid on
- van Laar equation
clear all; clc;
% Van Laar
t=[100 99.3 96.9 96.0 96.0 95.6 94.8 93.8 93.5 92.9 90.5 90.5 89.4 88.4
88.6 87.2 85.4 84.5 84.0 83.4 83.0 82.3 82.0 81.4 81.5 81.2 80.9 80.5 80.2
80.0 79.5 78.8 78.5 78.4 78.3 78.3]; % mmHG
T=t+273.15;
P=101.33;
% kPa
x1=[0.0001 0.0028 0.0118 0.0137 0.0144 0.0176 0.0222 0.0246 0.0302 0.033
0.0519 0.0530 0.0625 0.0673 0.0715 0.0871 0.126 0.143 0.172 0.206 0.210
0.255 0.284 0.321 0.324 0.345 0.405 0.430 0.449 0.506 0.545 0.663 0.735
0.804 0.917 0.999];
y1=[0.0001 0.032 0.113 0.157 0.135 0.156 0.186 0.212 0.231 0.248 0.318
0.314 0.339 0.370 0.362 0.406 0.468 0.487 0.505 0.530 0.527 0.552 0.567
0.586 0.583 0.591 0.614 0.628 0.633 0.661 0.673 0.733 0.776 0.815 0.906
0.999];
x2=1-x1;
y2=1-y1;
C1=73.304;
C2=-7122.3;
C3=-7.1424;
C4=2.8853*10^(-6);
C5=2;
P1_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;
C1=73.649;
C2=-7258.2;
C3=-7.3037;
C4=4.1653*10^(-6);
C5=2;
P2_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;
R=8.314;
% ---- exp ---gamma1_e=y1.*P./(x1.*P1_sat);
gamma2_e=y2.*P./(x2.*P2_sat);
GE_e=R.*T.*(x1.*log(gamma1_e)+x2.*log(gamma2_e));
% ---- derived ---x=0:0.01:1;
for i=1:36
if i>1 && i<36
func(i-1)=x1(i).*x2(i).*R.*T(i)./GE_e(i);
x_a(i-1)=x1(i);
end
end
c=polyfit(x_a,func,1);
y=c(2)+c(1).*x;
A12=1/y(1);
A21=1/y(101);
GE_d=R*T.*x1.*(1-x1).*(A12*A21./(A12.*x1+A21.*(1-x1)));
gamma1_d=exp(A12.*(1+(A12.*x1)./(A21.*(1-x1))).^(-2));
gamma2_d=exp(A21.*(1+(A21.*(1-x1))./(A12.*x1)).^(-2));
y_d=x1.*gamma1_d.*P1_sat./P;
K_e=y1./x1;
K_d=y_d./x1;
y=x;
figure(1)
plot(x1,y1,'r^');
hold on
plot(x1,y_d, 'b');
hold on
plot(x, y, 'k')
hold on
legend('Exp. Data', 'Van Laar equation', '45¢ª line', 0);
title('x-y diagram for ethanol/water VLE');
xlabel('Liquid-phase mole fraction of ethanol(x1)');
ylabel('Vapor-phase mole fraction of ethanol(y1)');
grid on
figure(2)
plot(x1, gamma1_e, 'r^');
hold on
plot(x1, gamma2_e, 'bo');
hold on
plot(x1, gamma1_d, 'r');
hold on
plot(x1, gamma2_d, 'b');
hold on
legend('¥ã1-ethanol by exp. data', '¥ã2-water by exp. data', '¥ã1-ethanol
by Van Laar equation', '¥ã2-water by Van Laar equation', 0);
title('Liquid-phase activity coefficients for ethanol/water system at 1
atm');
xlabel('Liquid phase mole fraction of ethanol(x1)');
ylabel('Liquid-phase activity coefficient');
axis([0 1 0 7]);
grid on
figure(3)
plot(x1, K_e, 'ro');
hold on
plot(x1, K_d, 'bo');
hold on
legend('K-value by exp. data', 'K-value by Van Laar equation', 0);
title('K-value of ethanol/water system at 1 atm');
xlabel('Liquid-phase mole fraction of ethanol(x1)');
ylabel('K-value');
axis([0 1 0 17]);
grid on
- Wilson equation
clear all; clc;
% Wilson
t=[100 99.3 96.9 96.0 96.0 95.6 94.8 93.8 93.5 92.9 90.5 90.5 89.4 88.4
88.6 87.2 85.4 84.5 84.0 83.4 83.0 82.3 82.0 81.4 81.5 81.2 80.9 80.5 80.2
80.0 79.5 78.8 78.5 78.4 78.3 78.3]; % mmHG
T=t+273.15;
P=101.33;
% kPa
x1=[0.0001 0.0028 0.0118 0.0137 0.0144 0.0176 0.0222 0.0246 0.0302 0.033
0.0519 0.0530 0.0625 0.0673 0.0715 0.0871 0.126 0.143 0.172 0.206 0.210
0.255 0.284 0.321 0.324 0.345 0.405 0.430 0.449 0.506 0.545 0.663 0.735
0.804 0.917 0.999];
y1=[0.0001 0.032 0.113 0.157 0.135 0.156 0.186 0.212 0.231 0.248 0.318
0.314 0.339 0.370 0.362 0.406 0.468 0.487 0.505 0.530 0.527 0.552 0.567
0.586 0.583 0.591 0.614 0.628 0.633 0.661 0.673 0.733 0.776 0.815 0.906
0.999];
x2=1-x1;
y2=1-y1;
C1=73.304;
C2=-7122.3;
C3=-7.1424;
C4=2.8853*10^(-6);
C5=2;
P1_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;
C1=73.649;
C2=-7258.2;
C3=-7.3037;
C4=4.1653*10^(-6);
C5=2;
P2_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;
R=8.314;
% ---- exp ---gamma1_e=y1.*P./(x1.*P1_sat);
gamma2_e=y2.*P./(x2.*P2_sat);
GE_e=R.*T.*(x1.*log(gamma1_e)+x2.*log(gamma2_e));
% ---- derived ---x=0:0.01:1;
a12=80.7419*4.2;
a21=1059.7046*4.2;
V1=58.68;
V2=18.0;
lambda12=V2./V1.*exp(-a12./(R.*T));
lambda21=V1./V2.*exp(-a21./(R.*T));
GE_d=R.*T.*(-x1.*log(x1+(1-x1).*lambda12)-(1-x1).*log((1-x1)+x1.*lambda21));
gamma1_d=exp(-log(x1+(1-x1).*lambda12)+(1-x1).*(lambda12./(x1+(1x1).*lambda12)-lambda21./((1-x1)+x1.*lambda21)));
gamma2_d=exp(-log((1-x1)+x1.*lambda21)-x1.*(lambda12./(x1+(1x1).*lambda12)-lambda21./((1-x1)+x1.*lambda21)));
y_d=x1.*gamma1_d.*P1_sat./P;
K_e=y1./x1;
K_d=y_d./x1;
y=x;
figure(1)
plot(x1,y1,'r^');
hold on
plot(x1,y_d, 'b');
hold on
plot(x, y, 'k')
hold on
legend('Exp. Data', 'Wilson equation', '45¢ª line', 0);
title('x-y diagram for ethanol/water VLE');
xlabel('Liquid-phase mole fraction of ethanol(x1)');
ylabel('Vapor-phase mole fraction of ethanol(y1)');
grid on
figure(2)
plot(x1, gamma1_e, 'r^');
hold on
plot(x1, gamma2_e, 'bo');
hold on
plot(x1, gamma1_d, 'r');
hold on
plot(x1, gamma2_d, 'b');
hold on
legend('¥ã1-ethanol by exp. data', '¥ã2-water by exp. data', '¥ã1-ethanol
by Wilson equation', '¥ã2-water by Wilson equation', 0);
title('Liquid-phase activity coefficients for ethanol/water system at 1
atm');
xlabel('Liquid phase mole fraction of ethanol(x1)');
ylabel('Liquid-phase activity coefficient');
axis([0 1 0 7]);
grid on
figure(3)
plot(x1, K_e, 'ro');
hold on
plot(x1, K_d, 'bo');
hold on
legend('K-value by exp. data', 'K-value by Wilson equation', 0);
title('K-value of ethanol/water system at 1 atm');
xlabel('Liquid-phase mole fraction of ethanol(x1)');
ylabel('K-value');
axis([0 1 0 17]);
grid on
- NRTL equation
clear all; clc;
% NRTL
t=[100 99.3 96.9 96.0 96.0 95.6 94.8 93.8 93.5 92.9 90.5 90.5 89.4 88.4
88.6 87.2 85.4 84.5 84.0 83.4 83.0 82.3 82.0 81.4 81.5 81.2 80.9 80.5 80.2
80.0 79.5 78.8 78.5 78.4 78.3 78.3]; % mmHG
T=t+273.15;
P=101.33;
% kPa
x1=[0.0001 0.0028 0.0118 0.0137 0.0144 0.0176 0.0222 0.0246 0.0302 0.033
0.0519 0.0530 0.0625 0.0673 0.0715 0.0871 0.126 0.143 0.172 0.206 0.210
0.255 0.284 0.321 0.324 0.345 0.405 0.430 0.449 0.506 0.545 0.663 0.735
0.804 0.917 0.999];
y1=[0.0001 0.032 0.113 0.157 0.135 0.156 0.186 0.212 0.231 0.248 0.318
0.314 0.339 0.370 0.362 0.406 0.468 0.487 0.505 0.530 0.527 0.552 0.567
0.586 0.583 0.591 0.614 0.628 0.633 0.661 0.673 0.733 0.776 0.815 0.906
0.999];
x2=1-x1;
y2=1-y1;
C1=73.304;
C2=-7122.3;
C3=-7.1424;
C4=2.8853*10^(-6);
C5=2;
P1_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;
C1=73.649;
C2=-7258.2;
C3=-7.3037;
C4=4.1653*10^(-6);
C5=2;
P2_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;
R=8.314;
% ---- exp ---gamma1_e=y1.*P./(x1.*P1_sat);
gamma2_e=y2.*P./(x2.*P2_sat);
GE_e=R.*T.*(x1.*log(gamma1_e)+x2.*log(gamma2_e));
% ---- derived ---x=0:0.01:1;
b12=-2349.7236*4.2;
b21=3794.6584*4.2;
alpha=0.0417;
tau12=b12./(R.*T);
tau21=b21./(R.*T);
G12=exp(-alpha.*tau12);
G21=exp(-alpha.*tau21);
GE_d=R.*T.*x1.*(1-x1).*((G21.*tau21)./(x1+(1-x1).*G21)+((G12.*tau12)./((1x1)+x1.*G12)));
gamma1_d=exp((1-x1).^(2).*(tau21.*(G21./(x1+(1x1).*G21)).^(2)+G12.*tau12./(((1-x1)+x1.*G12).^(2))));
gamma2_d=exp(x1.^(2).*(tau12.*(G12./((1x1)+x1.*G12)).^(2)+G21.*tau21./((x1+(1-x1).*G21).^(2))));
y_d=x1.*gamma1_d.*P1_sat./P;
K_e=y1./x1;
K_d=y_d./x1;
y=x;
figure(1)
plot(x1,y1,'r^');
hold on
plot(x1,y_d, 'b');
hold on
plot(x, y, 'k')
hold on
legend('Exp. Data', 'NRTL equation', '45¢ª line', 0);
title('x-y diagram for ethanol/water VLE');
xlabel('Liquid-phase mole fraction of ethanol(x1)');
ylabel('Vapor-phase mole fraction of ethanol(y1)');
grid on
figure(2)
plot(x1, gamma1_e, 'r^');
hold on
plot(x1, gamma2_e, 'bo');
hold on
plot(x1, gamma1_d, 'r');
hold on
plot(x1, gamma2_d, 'b');
hold on
legend('¥ã1-ethanol by exp. data', '¥ã2-water by exp. data', '¥ã1-ethanol
by NRTL equation', '¥ã2-water by NRTL equation', 0);
title('Liquid-phase activity coefficients for ethanol/water system at 1
atm');
xlabel('Liquid phase mole fraction of ethanol(x1)');
ylabel('Liquid-phase activity coefficient');
axis([0 1 0 7]);
grid on
figure(3)
plot(x1, K_e, 'ro');
hold on
plot(x1, K_d, 'bo');
hold on
legend('K-value by exp. data', 'K-value by NRTL equation', 0);
title('K-value of ethanol/water system at 1 atm');
xlabel('Liquid-phase mole fraction of ethanol(x1)');
ylabel('K-value');
axis([0 1 0 17]);
grid on
- UNIFAC method
clear all; clc;
% UNIFAC
t=[100 99.3 96.9 96.0 96.0 95.6 94.8 93.8 93.5 92.9 90.5 90.5 89.4 88.4
88.6 87.2 85.4 84.5 84.0 83.4 83.0 82.3 82.0 81.4 81.5 81.2 80.9 80.5 80.2
80.0 79.5 78.8 78.5 78.4 78.3 78.3]; % mmHG
T=t+273.15;
P=101.33;
% kPa
x1=[0.0001 0.0028 0.0118 0.0137 0.0144 0.0176 0.0222 0.0246 0.0302 0.033
0.0519 0.0530 0.0625 0.0673 0.0715 0.0871 0.126 0.143 0.172 0.206 0.210
0.255 0.284 0.321 0.324 0.345 0.405 0.430 0.449 0.506 0.545 0.663 0.735
0.804 0.917 0.999];
y1=[0.0001 0.032 0.113 0.157 0.135 0.156 0.186 0.212 0.231 0.248 0.318
0.314 0.339 0.370 0.362 0.406 0.468 0.487 0.505 0.530 0.527 0.552 0.567
0.586 0.583 0.591 0.614 0.628 0.633 0.661 0.673 0.733 0.776 0.815 0.906
0.999];
x2=1-x1;
y2=1-y1;
C1=73.304;
C2=-7122.3;
C3=-7.1424;
C4=2.8853*10^(-6);
C5=2;
P1_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;
C1=73.649;
C2=-7258.2;
C3=-7.3037;
C4=4.1653*10^(-6);
C5=2;
P2_sat=exp(C1+C2./T+C3.*log(T)+C4.*T.^(C5))./1000;
R=8.314;
% ---- exp ---gamma1_e=y1.*P./(x1.*P1_sat);
gamma2_e=y2.*P./(x2.*P2_sat);
GE_e=R.*T.*(x1.*log(gamma1_e)+x2.*log(gamma2_e));
% ---- derived ---% species A = ethanol, species B = water
% subgroup 1 = CH3, subgroup 2 = CH2, subgroup 15 = OH, subgroup 17 = H20
x=0:0.01:1;
R1=0.9011;
R2=0.6744;
R15=1.0000;
R17=0.9200;
Q1=0.848;
Q2=0.540;
Q15=1.200;
Q17=1.400;
V1A=1;
V2A=1;
V15A=1;
V17A=0;
V1B=0;
V2B=0;
V15B=0;
V17B=1;
a_1_2=0;
a_2_1=0;
a_1_1=0;
a_2_2=0;
a_15_15=0;
a_17_17=0;
a_1_15=986.5;
a_2_15=986.5;
a_1_17=1318.00;
a_2_17=1318.00;
a_15_1=156.40;
a_15_2=156.40;
a_17_1=300.00;
a_17_2=300.00;
a_15_17=353.50;
a_17_15=-229.10;
t_1_2=1;
t_2_1=1;
t_1_1=1;
t_2_2=1;
t_15_15=1;
t_17_17=1;
t_1_15=exp(-a_1_15./T);
t_2_15=exp(-a_2_15./T);
t_1_17=exp(-a_1_17./T);
t_2_17=exp(-a_2_17./T);
t_15_1=exp(-a_15_1./T);
t_15_2=exp(-a_15_2./T);
t_17_1=exp(-a_17_1./T);
t_17_2=exp(-a_17_2./T);
t_15_17=exp(-a_15_17./T);
t_17_15=exp(-a_17_15./T);
rA=V1A*R1+V2A*R2+V15A*R15+V17A*R17;
rB=V1B*R1+V2B*R2+V15B*R15+V17B*R17;
qA=V1A*Q1+V2A*Q2+V15A*Q15+V17A*Q17;
qB=V1B*Q1+V2B*Q2+V15B*Q15+V17B*Q17;
e1A=V1A*Q1/qA;
e2A=V2A*Q2/qA;
e15A=V15A*Q15/qA;
e17A=V17A*Q17/qA;
e1B=V1B*Q1/qB;
e2B=V2B*Q2/qB;
e15B=V15B*Q15/qB;
e17B=V17B*Q17/qB;
bA1=e1A.*t_1_1 +
bA2=e1A.*t_1_2 +
bA15=e1A.*t_1_15
bA17=e1A.*t_1_17
bB1=e1B.*t_1_1 +
bB2=e1B.*t_1_2 +
bB15=e1B.*t_1_15
bB17=e1B.*t_1_17
c1=(x1.*qA.*e1A +
c2=(x1.*qA.*e2A +
c15=(x1.*qA.*e15A
c17=(x1.*qA.*e17A
s1=c1.*t_1_1 +
s2=c1.*t_1_2 +
s15=c1.*t_1_15
s17=c1.*t_1_17
e2A.*t_2_1 + e15A.*t_15_1 + e17A.*t_17_1;
e2A.*t_2_2 + e15A.*t_15_2 + e17A.*t_17_2;
+ e2A.*t_2_15 + e15A.*t_15_15 + e17A.*t_17_15;
+ e2A.*t_2_17 + e15A.*t_15_17 + e17A.*t_17_17;
e2B.*t_2_1 + e15B.*t_15_1 + e17B.*t_17_1;
e2B.*t_2_2 + e15B.*t_15_2 + e17B.*t_17_2;
+ e2B.*t_2_15 + e15B.*t_15_15 + e17B.*t_17_15;
+ e2B.*t_2_17 + e15B.*t_15_17 + e17B.*t_17_17;
(1-x1).*qB.*e1B)./(x1.*qA + (1-x1).*qB);
(1-x1).*qB.*e2B)./(x1.*qA + (1-x1).*qB);
+ (1-x1).*qB.*e15B)./(x1.*qA + (1-x1).*qB);
+ (1-x1).*qB.*e17B)./(x1.*qA + (1-x1).*qB);
c2.*t_2_1 + c15.*t_15_1 + c17.*t_17_1;
c2.*t_2_2 + c15.*t_15_2 + c17.*t_17_2;
+ c2.*t_2_15 + c15.*t_15_15 + c17.*t_17_15;
+ c2.*t_2_17 + c15.*t_15_17 + c17.*t_17_17;
JA=rA./(rA.*x1+rB.*(1-x1));
JB=rB./(rA.*x1+rB.*(1-x1));
LA=qA./(qA.*x1+qB.*(1-x1));
LB=qB./(qA.*x1+qB.*(1-x1));
gammaA_C=exp(1-JA+log(JA)-5.*qA.*(1-JA./LA+log(JA./LA)));
gammaB_C=exp(1-JB+log(JB)-5.*qB.*(1-JB./LB+log(JB./LB)));
gammaA_R=exp(qA.*(1-(c1.*bA1./s1 + c2.*bA2./s2 + c15.*bA15./s15 +
c17.*bA17./s17 - e1A.*log(bA1./s1) - e2A.*log(bA2./s2) e15A.*log(bA15./s15) - e17A.*log(bA17./s17))));
gammaB_R=exp(qB.*(1-(c1.*bB1./s1 + c2.*bB2./s2 + c15.*bB15./s15 +
c17.*bB17./s17 - e1B.*log(bB1./s1) - e2B.*log(bB2./s2) e15B.*log(bB15./s15) - e17B.*log(bB17./s17))));
gamma1_d=gammaA_C.*gammaA_R;
gamma2_d=gammaB_C.*gammaB_R;
GE_d=R.*T.*(x1.*log(gamma1_d)+(1-x1).*log(gamma2_d));
y_d=x1.*gamma1_d.*P1_sat./P;
K_e=y1./x1;
K_d=y_d./x1;
y=x;
figure(1)
plot(x1,y1,'r^');
hold on
plot(x1,y_d, 'b');
hold on
plot(x, y, 'k')
hold on
legend('Exp. Data', 'UNIFAC method equation', '45¢ª line', 0);
title('x-y diagram for ethanol/water VLE');
xlabel('Liquid-phase mole fraction of ethanol(x1)');
ylabel('Vapor-phase mole fraction of ethanol(y1)');
grid on
figure(2)
plot(x1, gamma1_e, 'r^');
hold on
plot(x1, gamma2_e, 'bo');
hold on
plot(x1, gamma1_d, 'r');
hold on
plot(x1, gamma2_d, 'b');
hold on
legend('¥ã1-ethanol by exp. data', '¥ã2-water by exp. data', '¥ã1-ethanol
by UNIFAC method', '¥ã2-water by UNIFAC method', 0);
title('Liquid-phase activity coefficients for ethanol/water system at 1
atm');
xlabel('Liquid phase mole fraction of ethanol(x1)');
ylabel('Liquid-phase activity coefficient');
axis([0 1 0 7]);
grid on
figure(3)
plot(x1, K_e, 'ro');
hold on
plot(x1, K_d, 'bo');
hold on
legend('K-value by exp. data', 'K-value by UNIFAC method', 0);
title('K-value of ethanol/water system at 1 atm');
xlabel('Liquid-phase mole fraction of ethanol(x1)');
ylabel('K-value');
axis([0 1 0 17]);
grid on