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 y1 P/kPa x1 y1

19.953 0 0 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 withcurves determined from the correlation

1) Calculate EXPERIMENTAL values of activity coefficients 1 and 2 and GE

2211

22

22

11

11

lnln/

;

xxRTG

Px

Py

Px

Py

E

satsat

We have shown that:

21212121

xAxARTxx

GE

])(2[ln

])(2[ln

2211221212

1122112221

xAAAx

xAAAx

Now we have our analytical model

2121

683.0475.0 xxRTxx

GE

satcalcsatcalccalc PxPxP 222111

Lets calculate ln 1, ln 2, GE/x1x2RT, and:

calc

satcalccalc

P

Pxy 111

1

0

10

20

30

40

50

60

70

80

90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x1, y1

P,

P c

alc

x1

y1

Pcalc

y1calc

RMS= SQRT ( (Pi-Picalc)2/n = 0.398 kPa

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

1

*2

21

*1

12

1

1

2

1

1

1

*

*22

*11

*

lnlnln

)/(

ln)/(

)/(

lnln

dx

dx

dx

dx

dx

RTGd

dx

RTGd

dx

RTGd

xxRT

G

E

E

E

E

Consistency test

0ln

lnlnln0

lnlnln

)/(

2

1

1

*2

21

*1

12

1

1

*2

21

*1

12

1

1

dx

dx

dx

dx

dx

dx

dx

dx

dx

RTGd E

Solid lines are the result of data reduction adjusting GE/RT

The experimental data is not thermodynamically consistent

Avg values within +0.1 and -0.1are acceptable

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

])(2[ln

])(2[ln

2211221212

1122112221

xAAAx

xAAAx

Minimize the sum of squares of the following function:

2221211222

121211211

exp

]),,,(

),,,([

sat

sat

ii

PxxAAx

PxxAAxP

Starting with A12=0.5, A21=1, get A12= 0.758, A21=0.435

Calculate the RMS for PRMS= SQRT ( (Pi-Pi

calc)2/n = 0.167 kPa

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x1

P-P

calc

, y-

ycal

c

P-Pcalc

y - ycalc