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VALIDATING YOUR BINARY VLE DATA
The method described here measures the thermodynamic consistency of your vapor-liquid equilibrium data. This method does not ensure that
The following vapor-liquid equilibrium data was measured for a 50/50 by volume mixture of methanol-benzene at atmospheric pressure (101.325 kPa):
The vapor pressure shown in the two right hand columns were provided from a process simulator, but they could be estimated using Roult's Law or another relation. The vapor pressures are for the pure components with 1 designating methanol and 2 designating benzene. This naming convention will be used for the remainder of the analysis. You will also need the following values for this analysis: critical pressure and temperatures for the pure components and molar volumes for the pure components. For this system, these values are:
Pc1=7974.3 kPa Pc2=4833.2 kPa Tc1=2400C Tc2=288.50C
Vm1=4.05 x 10-5 m3/mol and Vm2=8.89 x 10-5 m3/mol
These equations will be explained in more detail when they are used for calculations later.
Note for Isobaric Systems
When deciding whether or not the data from an isobaric system is thermodynamically consistent, or "good", the net area under the curve produced by plotting the suggested curve should be within the range from -0.10 to the result of the right side of Equation 5 (the Gibbs-Duhem Equation). This range is specified because liquids with similar boiling points will fall into the lower half of this range while liquids with normal boiling points are more than 300C apart may be in the upper half of this range. Unfortunately, this includes the enthalpy of mixing for the system. This is usually not available. Therefore, a good estimate to use to approximate the right side of Equation 5 is the following:
For the first point of our data:
Fugacity Coefficient Calculation:
Recall that the sample calculations shown here are for the first data point in our example:
Activity Coefficient Calculation:
For our first data point:
Determining the point on the graph:
Now, the first point to be plotted on the graph can be found:
This procedure is repeated for all points in the equilibrium data collection. As you can see here, you would not want to do this by hand calculation. Below is a good spreadsheet setup for doing this calculation repetitively.
The graph that is produced is shown below, notice that a trendline has been fit to this curve. The R2 value and the equation for the trendline are shown on the graph. The closer the R2 value is to one, the better the fit to the data. These features are both readily available on Microsoft Excel. I recommend that you have an R2 of at least 0.96 for accurate results.
Remember, if you don't have time to get you're own spreadsheet ready, check out the Spreadsheets Solutions portion of The Chemical Engineers' Resource Page. You'll find one there ready for download!
So...Is My Data Any Good or Not?
Now that you've constructed a great looking graph, it's time to find out if the data passed this thermodynamic test for validity. As you recall from earlier, the key is the net area under the curve that you've constructed. For an isobaric system, the net area should be:
For the system shown here, we have an upper limit of 0.357. Therefore, in order for the data to be considered 'thermodynamically sound', the net area under the curve must be greater than -0.10 but less than 0.357. This range should specify that the net area is less than 8% of the total area. This leaves room for estimation error with the critical values and other constants. If your data is not inside this range, check to see if the net area exceeds 8% of the total area. If this is also true, you should recheck you data.
The net area is evaluated by integrating the equation for the trendline from 0 to 1, for the trendline shown on the above graph, this corresponds to a value of -0.070. This falls within the suggested range and it is about 7.4% of the total area. Both checks for consistency are good. This really isn't surprising since the data is from a well regarded source. But why is the result in the lower part of the range? This can be expected for binary systems with normal boiling points that are less than 300C apart. Methanol's normal boiling point is 64.7 0C while benzene boils at 80.1 0C.
If you have any questions about this analysis, just drop me a line at the email link below. I hope you find it useful, I certainly found it interesting!