Toward High Level Modeling of Fluids: Application of Laws of Corresponding States to SAFT Equations.
The paper I'll discuss in this post is this one: Application of the Corresponding-State Law to the Parametrization of Statistical Associating Fluid Theory (SAFT)-Type Models: Generation and Use of “Generalized Charts” (Romain Privat,*,† Edouard Moine,† Baptiste Sirjean,† Rafiqul Gani,‡ and Jean-Noël Jaubert, Ind. Eng. Chem. Res. 2019, 58, 9127?9139)
After Chernobyl blew up in 1996 - Chernobyl was and is an example of the worst case in the use of nuclear power, although there is, these days, a lot of popular bad thinking about Fukushima and focus on it - I changed my mind about nuclear energy, and began to consider, in light of the fact that the worst case was now a known, and not a speculative unknown, that nuclear energy is the only acceptable form of energy there is, given the magnitude of the environmental crisis we are experiencing, whether we really care or not.
For people traveling the route I traveled, evolving from dumb-assed, ignorant, anti-nuke to nuclear energy advocate, in my case, a fierce advocate, it seems to me that there is a phase in which one becomes enamored of liquid fluoride molten salt reactors, often referred to by the abbreviation LFTR, for liquid fluoride thorium reactor. This fascination sometimes involves the belief that thorium is an acceptable nuclear fuel whereas the plutonium/uranium system of fuels is not, something with which I disagree, although certainly went through this phase myself.
Around the time I was going through my LFTR phase, I decided to raise my level of understanding of nuclear technology to a higher level, applying my general educational background to expanding my knowledge, and consider in detail the chemistry of nuclear fuels, in the process, learning all about some very interesting things and discovering significant gaps in my knowledge.
I am not really a molten salt kind of guy these days, although I wish all the molten salt people well, but am now more of a "breed and burn" kind of guy, exploiting a very different type of molten fuel. I am still interested in molten salts, not really in power production settings, but as a useful tool for recovering valuable resources from used nuclear fuel, often described by people - excuse the oxymoronic statement - who simultaneously lack imagination and have overactive imaginations, as "nuclear waste."
The chemistry of used nuclear fuels, and nuclear fuels undergoing use, is decidedly complex and represents a scientific challenge inasmuch as nuclear fuel is a continuously evolving substance and by definition, will contain a broad array of elements, some elements from the fourth period of the periodic table, all of the elements in the fifth period, some of the elements in the sixth period, all of the light lanthanides, as well as many, depending on design, of the light actinide elements, in addition to additives, which, again depending on design, might include lithium, beryllium, fluorine, structural elements such as iron, cobalt and nickel and moderators like carbon and hydrogen.
Recently I was looking into the failure of a molten salt nuclear startup which was funded, in part, by a gay Trumper - speaking of oxymorons - run by two MIT graduates, a startup which seems to have advertised a notion that the founders cheerfully admitted to be mistaken after further analysis before disbanding the company and releasing its IP. I decided to leaf through the graduate thesis of one of the founders, and her thesis involved modeling of molten salts got me thinking about equations of state for liquids, which brings me to a discussion of the paper referenced at the opening of this post.
From the introduction to the paper:
As a well-known result, the classical van der Waals, Peng–Robinson (PR), and Soave–Redlich–Kwong (SRK) models obey the corresponding-state law. For a pure substance, these equations of state (EoS) can be written as a universal relationship between the state variables Tr (the reduced temperature), Pr (the reduced pressure), and vr (the reduced molar volume):
where Tc is the critical temperature, Pc the critical pressure, and vc the critical molar volume. For the SRK and PR EoS, an additional dimensionless size parameter (the acentric factor ? ) is added in the universal relationship, which can be written in the general form Pr = f(Tr,vr,? ). It is said that the SRK and PR EoS obey the three-parameter corresponding state law.
To apply such models to a given pure species, one must switch from a space of dimensionless reduced state-variables (Tr,Pr,vr) to a space involving dimensional state variables (T,P,v ). To do so, these EoS require knowledge of three input parameters: the critical temperature (Tc,exp), critical pressure (Pc,exp), and acentric factor (?exp ).
What about SAFT EoS? Derived from molecular-potential models, these EoS consider, as input variables, the ones used in the underlying molecular-potential models. For most of them (PC-SAFT, CK-SAFT, SOFT-SAFT, ...), these are segment number (m), segment diameter (? ), and energy parameter (? ), described below. Therefore, most SAFT-type EoS also obey the three-parameter corresponding-state law. When dealing with this class of models, the reduced state variables involved in the corresponding-state law are normalized using combinations of the input model parameters (m, ?, ? ). In this molecular approach, the reduced temperature, pressure, and density are denoted as T*, P*, and ?, respectively. In their universal form, SAFT-type EoS for nonassociating pure species obey the three-parameter corresponding-state law and, therefore, they can be written as a universal relationship between the three aforementioned reduced state variables and the size parameter m. As a consequence, it can be shown that
the reduced critical coordinates of a critical point are only dependent on the m parameter,
the reduced coordinates of a liquid–liquid–vapor triple point are only dependent on m (note that SAFT-type EoS are known to predict triple points, although this behavior is physically meaningless(1?3)), and
the acentric factor of a pure component is only dependent on m.
These observations can be of special interest when thinking about the parametrization of SAFT-type EoS. The present article aims at discussing two potential applications of such results in terms of EoS parametrization.
(SAFT refers to "Statistical Association Fluid Theory" and PC-SAFT refers to "Perturbed Chain Statistical Association Fluid Theory" )
The paper makes reference to the more familiar gas laws in a nice review, focusing on the general correspondence principles with which they are associated. To wit:
(1)
s an example, the well-known van der Waals EoS obeys the corresponding-state law:
(2)
s highlighted by eq 2, two parameters among the three reduced variables Tr, Pr, and vr must be specified to apply the corresponding-state law. Equivalently, it can be said that two input parameters, to be chosen among the critical temperature, pressure, or molar volume, must be preliminarily known to apply eq 2 to a specific compound. For illustration, the pressure expression provided by the van der Waals EoS is
(3)
Consequently, experimental values of the critical temperature Tc,exp and critical pressure Pc,exp must be preliminary known to apply the van der Waals EoS to a given pure compound.
As a limitation, it is generally observed that two size-asymmetric pure fluids having the same Tr and Pr deviate significantly from the two-parameter corresponding-state law. To overcome this issue, an additional parameter must be introduced. The three-parameter corresponding-state law frequently introduces the acentric factor or the critical compressibility factor as an additional parameter. The universal relationship between reduced variables then becomes
As an example, the well-known Soave–Redlich–Kwong (SRK) EoS(10) obeys the three-parameter corresponding-state law:
(5)
with
where the acentric factor ?exp is considered to be an experimental property, since it is a straightforward function of experimental quantities (i.e., the vapor-pressure at Tr = 0.7 and the critical pressure). Consequently, knowing the experimental values of three input parameters (Tc,exp, Pc,exp, ?exp) is a prerequisite to apply the three-parameter corresponding-state law, as illustrated with the SRK EoS.
(6)
The three-parameter corresponding-state law entails that reduced vapor–liquid equilibrium (VLE) properties depend solely on the reduced temperature and the acentric factor:
(7)
where Prsat is the reduced vapor pressure and ?vapH is the enthalpy of vaporization of pure species.
...and so on. (A well known descendent of the SRK equation of state is the Peng-Robinson equation of state, to which this type of thinking applies.)
What the authors seem to be after is building a database for physical chemists like those that exist, for example, for protein chemists, which has the experimental data readily available for use in complex fluid systems, for example, fluids containing the valuable components of nuclear fuels.
Some pictures from the text:
The caption:
The caption:
The caption:
...and so on...
Here's a nice figure showing the power of systematization:
The caption:
The authors note in their conclusion:
Parameterization methods for associating compounds will be proposed in the future.
That is something to love, scientific and engineering ambition.
(There is so much to love in life, despite the tragedy of our times.)
It's a lovely paper for inducing reflection. It's a shame that I won't live long enough to really know it intimately. Well, my sons at least are smarter than I am.
I trust you're enjoying this weekend. For me, it's a little noisy in this library, as it's reunion weekend and people are walking through it, talking loudly about their old days, but the remaining days are too beautiful to look back all that much.
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