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Wed Sep 26, 2018, 08:47 PM

A Very Cool Rumination on Vapor-Liquid Cubic Equations of State.

The paper to which I'll refer here, written by scientists at Institute of Process System Engineering, Qingdao University of Science & Technology, People’s Republic of China is this one: Research into the Polynomial Alpha Function for the Cubic Equation of State (Wenying Zhao , Xiaoyan Sun, Li Xia, and Shuguang Xiang, Ind. Eng. Chem. Res., 2018, 57 (38), pp 12602–12623)

When I was a kid, I was doing a bunch of hydrogenations and because this sort of thing can be a little boring, I decided to program my HP41C calculator to solve cubic equations for the Van der Waals. The traditional way to monitor these things was to wait until the reaction stopped taking up hydrogen and work it up, but I was a dorky kid and so I programmed the calculator to do it. If I recall correctly, it came in within 2 or 3 percent of the theoretical value, so it kind of worked, but was more or less useless, because the reaction worked just as well if you didn't calculate the number of moles of hydrogen uptake from the P,V,T data.

Before my son went off to college, I challenged him to do the same thing (using Mathematica) for the Peng Robinson cubic equation, and he sort of made a decent stab at it, but largely blew it off to spend the last free days with his high school friends.

More and more, as I contemplate the awful state of our environment with respect to carbon dioxide (and other gases), I fantasize about reactors with mixtures of gases in them at high temperature - often involving supercritical fluids, especially the strange supercritical state of water - and I find myself thinking how I should find the time to think more about equations of states and their refinements.

It was thus with some joy that I came across the interesting paper linked herein. Here's a graphic from the paper:



The intro:

Equations of state (EoS) play an essential role in chemical simulation and design. An appropriate EoS can accurately calculate vapor−liquid equilibria (VLE) and the thermodynamic properties of pure compounds and mixtures in a wide range of temperatures and pressures. Since 1873, when van der Waals1 first proposed a simple cubic equation of state for real gases, thermodynamic models have developed rapidly and represent a milestone in the development of thermodynamics.2−5 The cubic EoS contains an attractive term and a repulsive term. The attractive term is a significant factor that influenced the prediction of the saturated vapor pressure.6,7 In 1949, Redlich and Kwong2 first modified the van der Waals (VDW) EoS by introducing a reduced temperature function to the attractive term. Wilson in 19648 proposed an alpha function with physical properties and reduced temperature as variables, but the prediction of the vapor pressures of pure fluids was poor, and this concept was ignored for a long time,9,10 until Soave proposed a generalized alpha function in 1972.3 Soave’s 1972 alpha function was a qualitative leap in the modification of the EoS and gave direction for the modification and development of the EoS. Since then, the alpha function has been vigorously developed.

Formally, alpha functions can be divided into polynomial and exponential functions. In this work, we review and analyze the functional forms, parameter correlations, and scope of the polynomial alpha functions developed over the past 40 years. The research methods and developing trend for this kind of function are also reported. A review of exponential alpha functions will be made in a separate publication.


The "evolutionary tree" of cubic equations of state:



I suppose only a dork could be filled with joy at this sort of thing, but I'm a dork, and damned proud of it too.

I have to find some time to play with this paper.

I'd send it to my son, but he, apparently, has a life.

I'd just thought I'd throw this out there, because I think it beautiful, and more than ever, we need beautiful things.

I just realized the paper is open sourced if you'd like to try it.



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Reply A Very Cool Rumination on Vapor-Liquid Cubic Equations of State. (Original post)
NNadir Sep 2018 OP
Glorfindel Sep 2018 #1
NNadir Sep 2018 #2
byronius Sep 2018 #3
TomSlick Sep 2018 #4
NNadir Sep 2018 #6
TomSlick Sep 2018 #7
erronis Sep 2018 #5

Response to NNadir (Original post)

Wed Sep 26, 2018, 08:53 PM

1. Thank you, my dear NNadir. I only wish I understood it, but if it makes you happy,

I'm damn well pleased. And you're right: we do need beautiful things in our lives.

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Response to Glorfindel (Reply #1)

Wed Sep 26, 2018, 08:59 PM

2. We do, especially with these awful men in our government doing so many terrible things.

Thanks for your kind words.

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Response to NNadir (Original post)

Wed Sep 26, 2018, 09:48 PM

3. Fascinating.

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Response to NNadir (Original post)

Wed Sep 26, 2018, 10:11 PM

4. I get it.

I spent a couple days tracing the history of an Arkansas state statute from the just after the Civil War to date. I was so excited to find Gnatt's Digest of Arkansas Statutes (1874) on line - in a searchable format. I was able to trace all amendments from there to date. I was pulling dusty old volumes off the firm library shelves that hadn't been touched in decades.

I geeked out! I can't even find another lawyer who thinks it's interesting. (I'm sure opposing counsel will not enjoy the history detailed in my brief.) Nevertheless, I had a ball - and got paid for it!

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Response to TomSlick (Reply #4)

Thu Sep 27, 2018, 01:20 PM

6. You do indeed get it, venturing into the obscure...

Punctilios of why you love your work.

I don't know you, but I'll bet you're an excellent attorney, because you believe that what you do matters and your field got to where it is also matters.

Thanks for your comment. It made my day.

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Response to NNadir (Reply #6)

Thu Sep 27, 2018, 01:32 PM

7. Cool word.

I've got to find a way to work "punctilios" into a brief.

It will drive my secretary nuts.

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Response to NNadir (Original post)

Thu Sep 27, 2018, 09:14 AM

5. This is what I really love/like: science is reality, politics is ephemeral

I think politics and the local/national/world stage has a huge impact on how we are able to live our lives now.

But science tells us what is possible and gives us some hard evidence about how to achieve the possibilities. All hard sciences should have an ability to explain how they work and be subject to reproducibility.

I'm a firm believer (if that computes) in the progression of scientific knowledge:
- Mathematics - the underpinning of anything that is scientific
- Physics - the observation and hypotheses of how the universe works - include astonomy
- Chemistry - how the elemental particles described within physics interact and form compounds and molecules
- Biology - how various organic molecules work together to form what we know of as life

Beyond these hard sciences are a realm of real ones, but harder to quantify as to evidence-based and reproducibility.
- Medicine
- Psychology

And then a host of pseudo sciences such as:
- Astrology
- Naturopathy
- Chiropracty
- Tea leaf reading

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