Transcript Solution Thermodynamics: Applications

Solution Thermodynamics:
Applications
Chapter 12-Part II
Example of data reduction
• The following is a set of VLE data for the system
methanol(1)/water(2) at 333.15K
P/kPa x1
19.953 0
y1
0
P/kPa x1
y1
60.614 0.5282 0.8085
39.223 0.1686 0.5714 63.998 0.6044 0.8383
42.984 0.2167 0.6268 67.924 0.6804 0.8733
48.852 0.3039 0.6943 70.229 0.7255 0.8922
52.784 0.3681 0.7345 72.832 0.7776 0.9141
56.652 0.4461 0.7742 84.562 1
1
Find parameter values for the Margules equation that give the best fit of GE/RT to
the data, and prepare a P x y diagram that compares the experimental points with
curves determined from the correlation
1) Calculate EXPERIMENTAL values of activity coefficients g1 and g2 and GE
y1 P
y2 P
g1 
;g 2 
sat
sat
x1 P1
x2 P2
G / RT  x1 ln g 1  x2 ln g 2
E
We have shown that:
E
G
 A21 x1  A12 x2
x1 x2 RT
ln g 1  x [ A12  2( A21  A12 ) x1 ]
2
2
ln g 2  x [ A21  2( A12  A21 ) x2 ]
2
1
Now we have our analytical model
E
G
 0.475x1  0.683x2
x1 x2 RT
Lets calculate ln g1, ln g2, GE/x1x2RT, and:
P
calc
 xg
calc
1 1
P
x1g 1 P1

calc
P
calc
y
calc
1
sat
1
sat
 x2g
calc
2
sat
2
P
90
80
70
P, P calc
60
50
x1
40
y1
Pcalc
30
y1calc
20
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x1, y1
RMS= SQRT (S (Pi-Picalc)2/n = 0.398 kPa
0.8
0.9
1
Thermodynamic consistency
• We need to check that the experimentally
obtained activity coefficients satisfy the
Gibbs-Duhem equation.
• If the experimental data are inconsistent
with the G-D equation, they are not correct
Consistency test
*
G 
*
*

  x1 ln g 1  x2 ln g 2
 RT 
E
*
d (G / RT )

dx1
E
d (G E / RT )
g1
 ln
dx1
g2
d (G E / RT )
g 1  d ln g 1*
d ln g 2* 

  ln   x1
 x2
dx1
g2 
dx1
dx1 
Consistency test
*
*

d (G / RT )
g1
d ln g 1
d ln g 2 

  ln   x1
 x2
dx1
g2 
dx1
dx1 
*
*

g1
d ln g 1
d ln g 2 

0   ln   x1
 x2
g2 
dx1
dx1 
E
g1
 ln  0
g2
Solid lines are the result of data reduction adjusting GE/RT
Avg values within +0.1 and -0.1
are acceptable
The experimental data is not thermodynamically consistent
An alternative objective function:
Barker’s method
• Fit the model GE/RT to make the
calculated pressures the closest possible
to the experimental data.
• For example, obtain A12 and A21 for the
Margules equation to minimize the
calculated pressures with respect to the
experimental values. (see dashed lines in
Figure 12.7)
example
• Using Barker’s method, find parameters
for the Margules eqn that provide the best
fit of GE/RT to the data, and prepare a Pxy
diagram that compares the experimental
points with curves determined form the
correlation.
solution
ln g 1  x [ A12  2( A21  A12 ) x1 ]
2
2
ln g 2  x [ A21  2( A12  A21 ) x2 ]
2
1
Minimize the sum of squares of the following function:
[ P
i
exp
 x1g 1 ( A12 , A21, x1 , x2 ) P1
sat

i
x2g 2 ( A12 , A21, x1 , x2 ) P2 ]
sat 2
Starting with A12=0.5, A21=1, get A12= 0.758, A21=0.435
Calculate the RMS for P
RMS= SQRT (S (Pi-Picalc)2/n = 0.167 kPa
0.4
P-Pcalc
0.3
y - ycalc
P-Pcalc, y-ycalc
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
-0.1
-0.2
-0.3
-0.4
x1
0.6
0.7
0.8
0.9
1