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I get a number of questions about which equation-of-state (EOS) models PYroMat uses. It's a good question, so it deserves a good answer.
As of version 2.2.X, PYroMat uses three primary EOS models: (1) ideal gases with the Shomate equation, (2) ideal gases with NASA polynomials, and (3) the Span and Wagner model. If you really want to dive into the details, they are documented in chapters 6 and 7 of the PYroMat User and Developer Handbook. Here's the short version:
Ideal Gases
All ideal gases use the ideal gas equation of state that is taught in every chemistry class ever conceived, $p=\rho RT$. However, the ideal gas equation only relates density, temperature, and pressure -- it has absolutely nothing to say about how the gas stores energy. For that, we need to know the specific heat.
Specific heats of ideal gases are presumed to change as a function of temperature only -- neither pressure nor density. The Shomate and NASA polynomials are just different styles of empirical models to form an equation for specific heat. For example, the older NASA polynomials take the form $c_p(T) = c_0 + c_1 T + c_2 T^2 + \ldots$
Once specific heat is known, all other energy-derived properties can be calculated (including entropy).
Multi-phase Models
When a substance is no longer ideal, calculations get very messy, and there are lots of different ways of dealing with it. The simplest if you're working with pen-and-paper, is to slice up the phase diagram into phases and use an ideal liquid, gas, solid, etc. based on where you happen to be. That's fine for back-of-the envelope estimates, but a computer can do much better.
The most broadly accepted method in the most competitive codes today is to use a single EOS model, which expresses a property (usually Helmholtz Free Energy) as a function of temperature and density, $f(T,\rho)$. Many people are surprised to learn that this is a single (very large) formula that applies everywhere - no piecewise nonsense. Because it uses density (and not pressure) there are no discontinuities at phase changes, and the model was intentionally derived to more closely resemble the energies at play when molecules are tightly packed. A discussion of the Span and Wagner EOS occupies all of Chapter 7 of the the Handbook, so it won't fit here.
This implies a number of challenging problems: (1) numerical iteration is required to evaluate properties in terms of pressure, (2) phase changes are implicit in the EOS, and have to be found numerically, (3) there are many terms, so there is some computational cost, and coding errors are more likely, (4) it is difficult to write a generic class that will accommodate many substances. Problems (2) and (4) are addressed by an impressive and growing body of works that use a (somewhat) standardized form called Span and Wagner models with very helpful approximations for the saturation conditions. Problems (1) and (3) are addressed by PYroMat and the time and effort we have put into testing the outputs against published validation sets.
More Information
If this still doesn't quite answer your question, post your question here! I'll be happy to dive in deeper.
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I get a number of questions about which equation-of-state (EOS) models PYroMat uses. It's a good question, so it deserves a good answer.
As of version 2.2.X, PYroMat uses three primary EOS models: (1) ideal gases with the Shomate equation, (2) ideal gases with NASA polynomials, and (3) the Span and Wagner model. If you really want to dive into the details, they are documented in chapters 6 and 7 of the PYroMat User and Developer Handbook. Here's the short version:
Ideal Gases
All ideal gases use the ideal gas equation of state that is taught in every chemistry class ever conceived,$p=\rho RT$ . However, the ideal gas equation only relates density, temperature, and pressure -- it has absolutely nothing to say about how the gas stores energy. For that, we need to know the specific heat.
Specific heats of ideal gases are presumed to change as a function of temperature only -- neither pressure nor density. The Shomate and NASA polynomials are just different styles of empirical models to form an equation for specific heat. For example, the older NASA polynomials take the form
$c_p(T) = c_0 + c_1 T + c_2 T^2 + \ldots$
Once specific heat is known, all other energy-derived properties can be calculated (including entropy).
Multi-phase Models
When a substance is no longer ideal, calculations get very messy, and there are lots of different ways of dealing with it. The simplest if you're working with pen-and-paper, is to slice up the phase diagram into phases and use an ideal liquid, gas, solid, etc. based on where you happen to be. That's fine for back-of-the envelope estimates, but a computer can do much better.
The most broadly accepted method in the most competitive codes today is to use a single EOS model, which expresses a property (usually Helmholtz Free Energy) as a function of temperature and density,$f(T,\rho)$ . Many people are surprised to learn that this is a single (very large) formula that applies everywhere - no piecewise nonsense. Because it uses density (and not pressure) there are no discontinuities at phase changes, and the model was intentionally derived to more closely resemble the energies at play when molecules are tightly packed. A discussion of the Span and Wagner EOS occupies all of Chapter 7 of the the Handbook, so it won't fit here.
This implies a number of challenging problems: (1) numerical iteration is required to evaluate properties in terms of pressure, (2) phase changes are implicit in the EOS, and have to be found numerically, (3) there are many terms, so there is some computational cost, and coding errors are more likely, (4) it is difficult to write a generic class that will accommodate many substances. Problems (2) and (4) are addressed by an impressive and growing body of works that use a (somewhat) standardized form called Span and Wagner models with very helpful approximations for the saturation conditions. Problems (1) and (3) are addressed by PYroMat and the time and effort we have put into testing the outputs against published validation sets.
More Information
If this still doesn't quite answer your question, post your question here! I'll be happy to dive in deeper.
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