Wilson Activity Coefficient Model - how to go from y --> x ? 1

[Lams001]

Hi all,

I've been trying to figure out how to "invert" the Wilson model to get liquid fractions from supplied vapour fractions, and I'm confusing myself thoroughly in the process. HYSYS is spitting out a result that I'm not happy with so I want to check the composition profiles with a stage-by-stage, and for that I want to be able to do dew point calcs.

For the system I'm modeling I can assume an ideal gas phase as my column is atmospheric, but the liquid behaviour is non-ideal. No LLE need be considered because the activity coefficients are still relatively low, so I'm using modified Raoult's Law.

So - is there an incredibly obvious way of doing this that I'm missing somehow? All I need is a way of getting from y --> x using Wilson, but it's been a LONG time since I had thermo.

 

[PaoloPemi]

for each component in the mixture gas mole fraction

y(i) = x(i) * Kvalue(i)

so you need access to compositions and fugacities, if you intend to do data regression the derivatives vs. composition would be useful...
for these calc's I would suggest a different tool as a thermodynamic library which give access to all these properties (compositions, fugacities, derivatives etc.), see for example

http://www.prode.com/en/properties.htm

by the way there is also a data regression utility for calculating BIP for Wilson ot other models, once you have calculated the required parameters you can easily export to your simulator.

 

[Latexman]

I think you are asking if there is a way to form an explicit equation for x. There is not. Given y, the liquid activity coefficients must be available to calculate x, and x must be available to calculate the liquid activity coefficients in the Wilson Equation. It is iterative.

Good luck,
Latexman

 

[Lams001]

Thanks to both of you for the responses.

So my unpleasant conclusion is confirmed - the only way to do this using Wilson is to either write a solver in MATLAB or C+ and crunch my ternary system through an optimisation algorithm, or to build a (very large) database of VLE data. Since I have a 3-component system, this second option probably isn't the greatest of ideas.

Given y1, y2 and y3 I can build three Wilson expressions and substitute in a modified Raoult's law term to eliminate either the x or gamma dependence, then solve three equations in parallel by iteratively selecting combinations of x1, x2 and x3. Anyone see any obvious problems with this approach?

Also - what about this alternative approach. What I really want to do is confirm my feed location using simple Lewis-Matherson (or similar). To do this, ideally I would do bottom-up operating line and bubble-point calculations and match them to top-down dew-point calculations. If I carried out bubble-point calculations right up to my mass-balance specified condenser fractions, I could avoid having to carry out any dew point calculations until the very last one. I could then solve this single one by hand, taking guesses around the ideal predictions (and avoid writing any code). Any thoughts on whether this would be a reasonable approach?

Thanks again for the responses so far.

 

[PaoloPemi]

if you have the compostions (on stages) you shouldn't have problems to calculate dew point and bubble points, for selecting feed location a common approach is to calculate iteratively the column with feed(s) at different stages and select the best solution.
If (instead) you wish to calculate an approximate solution you can define a simplified model based on Kvalues (see inside-out) in the simplified model you can even include a composition dependence (based on derivatives of fugacity vs. composition) see my first post, but this approach would require certainly some work...

 

[Lams001]

This would be good if I knew stage-by-stage compositions, but that is what I want to do a hand calculation to determine. All I know is the specified feed, bottoms and distillate fractions and flows. HYSYS has calculated these composition profiles, but I want to confirm them stage-by-stage.

I've selected a reflux ratio and number of stages using Fenske-Underwood-Gilliland correlations, and confirmed that HYSYS gives a sensible result roughly the same as these hand calculations. I'm planning to do feed location optimisation in Aspen HYSYS, moving the feed around and using the economic evaluation features until I'm happy that I've found roughly the best arrangement.

What I want to do then is a stage-by-stage to confirm the temperature and composition profiles that HYSYS has generated at this "optimum" configuration. Simultaneously, using Lewis-Matherson I should be able to get my temperature and composition profiles AND confirm feed tray location. I should also then be able to estimate my condenser and reboiler duties independently of HYSYS.

Thank you for responding so quickly once again!

 

[PaoloPemi]

classic Lewis-Matherson model (where kvalues are calculated from relative volatility to a reference component) perhaps is not the best to describe a mixture with highly non-ideal behaviour (but you may adopt some variant), nowadays simulators (even my little Prode Properties) include full-Newton and Russel's solvers for the column, specifically full-Newton is in my opinion quite reliable for these cases.

 

[Lams001]

Provided modified Raoult's law is sufficient (i.e. significant liquid-phase non-ideality, but no LLE and reasonable vapour phase ideality), would Lewis-Matherson not be perfectly adequate? Algorithmically, once you have a means of computing relative volatilities stage-by-stage, I see no difference in the way Lewis-Matherson should work out.

Correct me if I'm wrong!

Thanks

 

[PaoloPemi]

the problem is that for mixtures with high non-ideal behaviour you cannot assume constant relative volatility as that means that individual fugacities are almost independent from molar fractions (not true for Wilson as noted by Latexman)...
you can define locally a composition dependence or solve iteratively a simplified model as in Russel's method, for a local composition dependence you need a tool (see Prode Properties or equivalent) which exports composition derivatives, see Holland book for additional information.