Example P05.10 Regression of Isobaric VLE Data for the System Methanol -

Toluene

problem:

Calculate the activity coefficients in the system methanol (1) – toluene (2) from the data measured by Ocon, et al. [71] at atmospheric pressure assuming ideal vapor phase behavior. Try to fit the untypical behavior of the activity coefficients of methanol as function of composition using temperature independent gE-model parameters (Wilson, NRTL, UNIQUAC).

Explain why the activity coefficients of methanol show a maximum at high toluene concentration.

The vapor pressure constants are given in Appendix A. Experimental data as well as molar volumes, r and q values can be downloaded from the textbook page on www.ddbst.com. For the calculation, ideal vapor phase behavior should be assumed.

[71] Ocon J., Tojo G., Espada L., Anal.Quim., 65, 641-648, 1969

general

constants

and

definitions:

input data

The following area contains the experimental data points (T, x, y):

Solution

In the first step, the required molar volume for the Wilson model is calculated:

The activity coefficient functions for the three models are:

Wilson

UNIQUAC

Please note that in all three cases, the binary interaction parameters are used in the unit Kelvin. All functions include the interaction parameter matrix as a variable. This is important in case of regression.


In order to calculate the VLE behavior, a function for the pure component vapor pressure is required. The Wagner equation with parameters from Appendix A is used here:

Vapor pressure constants for the Wagner equation (Appendix A):

Verification:

In the next step the activity coefficients can be calculated from the experimental data and the pure component vapor pressures:

In order to regress the binary interaction parameters to these activity coefficient values, a suitable objective function must be defined. The following procedure is identical for all three models and will only be commented in case of the Wilson model.

Regression Using the Wilson Model

Definition of the objective Function:

The following function transforms two dimensionless parameters into the interaction parameter matrix required for UNIQUAC.

Now parameter regression can be performed using a Given - Minerr block (for details see the Mathcad documentation):

starting values:

The resulting parameters are now stored in the interaction parameter matrix:

A graphical representation of the activity coefficients derived from experiment and the regressed curve is shown on the next page.

Wilson

As can be seen from the next diagram, there is a strong decrease in equilibrium temperature between pure toluene and the mixture with 20 mol% of methanol. Temerature independent parameters are not able to describe this behavior and thus the activity coefficients of both components are not well reproduced in this concentration range. The same is observed for the other two models as shown below:

Regression Using the NRTL Model

Setting a fixed value of the nonrandomness factor a:

Definition of the objective Function:

The following function transforms two dimensionless parameters into the interaction parameter matrix required for UNIQUAC.

Now parameter regression can be performed in the same way as above:

starting values:

Regression Using the UNIQUAC Model

Definition of the objective Function:

This function transforms two dimensionless parameters into the interaction parameter matrix required for UNIQUAC.

Now parameter regression can be performed in the same way as above:

starting values:

Regression Using the UNIQUAC Model

Temperature dependent parameter function:

Definition of the objective Function:

This function transforms two dimensionless parameters into the interaction parameter matrix required for UNIQUAC.

Now parameter regression can be performed in the same way as above:

starting values:

Temperature dependent parameters give a nearly perfect description.