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[Doc] Revise species thermo documentation

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Expand the description to cover both the calculation of standard-state
cp/h/s and the models that provide parameters affecting the equation
of state.

Also try to provide a description of the difference between the typical
definition of standard state and the distinction that Cantera sometimes
tries to make between "reference state" and "standard state" properties.
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doc/sphinx/reference/thermo/species-thermo.md
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@ -3,48 +3,89 @@
The phase models discussed in the [](phase-thermo) section implement specific models for
the thermodynamic properties appropriate for the type of phase or interface they
represent. Although each one may use different expressions to compute the properties,
they all require thermodynamic property information for the individual species. For the
phase types implemented at present, the properties needed are:
they all require thermodynamic property information for the individual species.
1. the molar heat capacity at constant pressure $\hat{c}^\circ_p(T)$ for a range of
temperatures and a reference pressure $p^\circ$;
2. the molar enthalpy $\hat{h}(T^\circ, p^\circ)$ at $p^\circ$ and a reference
temperature $T^\circ$;
3. the absolute molar entropy $\hat{s}(T^\circ, p^\circ)$ at $(T^\circ, p^\circ)$.
Generally, the phase models require a parameterization of the standard state heat
capacity, enthalpy, and entropy for each species at a fixed pressure $p^\circ$ as a
function of $T$. In addition, phase models may require information describing how each
species affects the equation of state, either in terms of the species standard molar
volume, or through parameters that are specific to the equation of state.
The superscript $^\circ$ here represents the *reference state*--a specified state (that
is, a set of conditions $T^\circ$ and $p^\circ$ and fixed chemical composition) at which
thermodynamic properties are known.
(sec-standard-state-species-thermo)=
## Models for Species Standard State Enthalpy, Entropy, and Heat Capacity
The models described in this section can be used to provide thermodynamic data for each
species in a phase. Each model implements a different *parameterization* (functional
form) for the heat capacity. Note that there is no requirement that all species in a
phase use the same parameterization; each species can use the one most appropriate to
represent how the heat capacity depends on temperature.
Many of Cantera's phase thermodynamic models are formulated to make use of the
[standard state](https://goldbook.iupac.org/terms/view/S05925) thermodynamic properties
for individual species, defined at a standard pressure $p^\circ$ and for the composition
specified by the phase model (for example, a pure gas in the case of the ideal gas
model, or an ion at infinite dilution in water in the case of aqueous solutions).
Currently, several types are implemented that provide species properties appropriate for
models of ideal gas mixtures, ideal solutions, and pure compounds.
```{caution}
In some parts of the Cantera documentation, properties calculated at the standard
pressure $p^\circ$ are referred to as "reference-state" thermodynamic properties, while
properties calculated using the composition defining the standard state but at any
pressure are referred to as "standard state" properties. This nomenclature is fairly
unique to Cantera, based on the desire to distinguish these different steps in the
calculation of the full thermodynamic properties, and is not often seen in other
descriptions of solution thermodynamics.
```
The necessary properties are:
1. $\hat{c}^\circ_p(T)$: the standard-state molar heat capacity at constant pressure as
a function of temperature;
2. $\hat{h}^\circ(T)$, the standard-state molar enthalpy as a function of temperature;
3. $\hat{s}^\circ(T)$: the standard-state molar entropy as a function of temperature.
While each of these functions is implemented explicitly in Cantera for computational
efficiency, $\hat{h}^\circ(T)$ and $\hat{s}^\circ(T)$ can be expressed in terms of
$\hat{c}^\circ_p(T)$ using the relations
$$ \hat{h}^\circ(T) = \hat{h}^\circ(T_\mathrm{ref}) +
\int_{T_\mathrm{ref}}^T \hat{c}^\circ_p(T) \; dT $$
and
$$ \hat{s}^\circ(T) = \hat{s}^\circ(T_\mathrm{ref}) +
\int_{T_\mathrm{ref}}^T \frac{\hat{c}^\circ_p(T)}{T} \; dT $$
respectively. This means that a parameterization (functional form) of
$\hat{c}_p^\circ(T)$ plus the constants $\hat{h}^\circ(T_\mathrm{ref})$ and
$\hat{s}^\circ(T_\mathrm{ref})$ at a reference temperature $T_\mathrm{ref}$ is
sufficient to define the standard state properties for a species.
The models described in this section can be used to provide standard state thermodynamic
data for each species in a phase. They are implemented by classes deriving from
{ct}`SpeciesThermoInterpType`.
```{note}
There is no requirement that all species in a phase use the same parameterization; each
species can use the one most appropriate to represent how the heat capacity depends on
temperature.
```
(sec-thermo-nasa7)=
## The NASA 7-Coefficient Polynomial Parameterization
### The NASA 7-Coefficient Polynomial Parameterization
The NASA 7-coefficient polynomial parameterization is used to compute the species
reference-state thermodynamic properties $\hat{c}^\circ_p(T)$, $\hat{h}^\circ(T)$, and
standard-state thermodynamic properties $\hat{c}^\circ_p(T)$, $\hat{h}^\circ(T)$, and
$\hat{s}^\circ(T)$.
The NASA parameterization represents $\hat{c}^\circ_p(T)$ with a fourth-order
polynomial:
$$
\frac{\hat{c}_p^\circ(T)}{\overline{R}} &= a_0 + a_1 T + a_2 T^2 + a_3 T^3 + a_4 T^4
\frac{\hat{c}_p^\circ(T)}{R} &= a_0 + a_1 T + a_2 T^2 + a_3 T^3 + a_4 T^4
\frac{\hat{h}^\circ (T)}{\overline{R} T} &= a_0 + \frac{a_1}{2} T + \frac{a_2}{3} T^2 +
\frac{\hat{h}^\circ (T)}{R T} &= a_0 + \frac{a_1}{2} T + \frac{a_2}{3} T^2 +
\frac{a_3}{4} T^3 + \frac{a_4}{5} T^4 + \frac{a_5}{T}
\frac{\hat{s}^\circ(T)}{\overline{R}} &= a_0 \ln T + a_1 T + \frac{a_2}{2} T^2 +
\frac{\hat{s}^\circ(T)}{R} &= a_0 \ln T + a_1 T + \frac{a_2}{2} T^2 +
\frac{a_3}{3} T^3 + \frac{a_4}{4} T^4 + a_6
$$
This model is implemented by the C++ classes {ct}`NasaPoly1` and {ct}`NasaPoly2`.
:::{note}
This is the "old" NASA polynomial form, used in the original NASA equilibrium program
and in Chemkin, which uses 7 coefficients in each of two temperature regions. It is not compatible with the form used in more recent versions of the NASA equilibrium program,
@ -58,7 +99,7 @@ A NASA-7 parameterization can be defined in the YAML format by specifying
:::
(sec-thermo-nasa9)=
## The NASA 9-Coefficient Polynomial Parameterization
### The NASA 9-Coefficient Polynomial Parameterization
The NASA 9-coefficient polynomial parameterization {cite:p}`mcbride2002` ("NASA9" for
short) is an extension of the NASA 7-coefficient polynomial parameterization which
@ -69,19 +110,20 @@ The NASA9 parameterization represents the species thermodynamic properties with
following equations:
$$
\frac{\hat{c}_p^\circ(T)}{\overline{R}} &= a_0 T^{-2} + a_1 T^{-1} + a_2 + a_3 T
\frac{\hat{c}_p^\circ(T)}{R} &= a_0 T^{-2} + a_1 T^{-1} + a_2 + a_3 T
+ a_4 T^2 + a_5 T^3 + a_6 T^4
\frac{\hat{h}^\circ(T)}{\overline{R} T} &= - a_0 T^{-2} + a_1 \frac{\ln T}{T} + a_2
\frac{\hat{h}^\circ(T)}{R T} &= - a_0 T^{-2} + a_1 \frac{\ln T}{T} + a_2
+ \frac{a_3}{2} T + \frac{a_4}{3} T^2 + \frac{a_5}{4} T^3 +
\frac{a_6}{5} T^4 + \frac{a_7}{T}
\frac{\hat{s}^\circ(T)}{\overline{R}} &= - \frac{a_0}{2} T^{-2} - a_1 T^{-1} + a_2 \ln T
\frac{\hat{s}^\circ(T)}{R} &= - \frac{a_0}{2} T^{-2} - a_1 T^{-1} + a_2 \ln T
+ a_3 T + \frac{a_4}{2} T^2 + \frac{a_5}{3} T^3 + \frac{a_6}{4} T^4 + a_8
$$
A common source for species data in the NASA9 format is the
[NASA ThermoBuild](/userguide/thermobuild) tool.
[NASA ThermoBuild](/userguide/thermobuild) tool. This model is implemented by the C++
classes {ct}`Nasa9Poly1` and {ct}`Nasa9PolyMultiTempRegion`.
:::{admonition} YAML Usage
:class: tip
@ -90,7 +132,7 @@ A NASA-9 parameterization can be defined in the YAML format by specifying
:::
(sec-thermo-shomate)=
## The Shomate Parameterization
### The Shomate Parameterization
The Shomate parameterization is:
@ -104,10 +146,11 @@ $$
$$
where $t = T / 1000\textrm{ K}$. It requires 7 coefficients $A$, $B$, $C$, $D$, $E$,
$F$, and $G$. This parameterization is used to represent reference-state properties in
$F$, and $G$. This parameterization is used to represent standard-state properties in
the [NIST Chemistry WebBook](http://webbook.nist.gov/chemistry). The values of the
coefficients $A$ through $G$ should be entered precisely as shown there, with no units
attached. Unit conversions to SI is handled internally.
attached. Unit conversions to SI is handled internally. This model is implemented by the
C++ classes {ct}`ShomatePoly` and {ct}`ShomatePoly2`.
:::{admonition} YAML Usage
:class: tip
@ -116,23 +159,25 @@ A Shomate parameterization can be defined in the YAML format by specifying
:::
(sec-thermo-const-cp)=
## Constant Heat Capacity
### Constant Heat Capacity
In some cases, species properties may only be required at a single temperature or over a
narrow temperature range. In such cases, the heat capacity can be approximated as
constant, and simple expressions can be used for the thermodynamic properties:
$$
\hat{c}_p^\circ(T) &= \hat{c}_p^\circ(T^\circ)
\hat{c}_p^\circ(T) &= \hat{c}_p^\circ(T_\mathrm{ref})
\hat{h}^\circ(T) &= \hat{h}^\circ\left(T^\circ\right) + \hat{c}_p^\circ \left(T-T^\circ\right)
\hat{h}^\circ(T) &= \hat{h}^\circ\left(T_\mathrm{ref}\right) + \hat{c}_p^\circ \left(T-T_\mathrm{ref}\right)
\hat{s}^\circ(T) &= \hat{s}^\circ(T^\circ) + \hat{c}_p^\circ \ln{\left(\frac{T}{T^\circ}\right)}
\hat{s}^\circ(T) &= \hat{s}^\circ(T_\mathrm{ref}) + \hat{c}_p^\circ \ln{\left(\frac{T}{T_\mathrm{ref}}\right)}
$$
The parameterization uses four constants: $T^\circ, \hat{c}_p^\circ(T^\circ),
\hat{h}^\circ(T^\circ)$, and $\hat{s}^\circ(T)$. The default value of $T^\circ$ is
298.15 K; the default value for the other parameters is 0.0.
The parameterization uses four constants: $T_\mathrm{ref}$,
$\hat{c}_p^\circ(T_\mathrm{ref})$, $\hat{h}^\circ(T_\mathrm{ref})$, and
$\hat{s}^\circ(T)$. The default value of $T_\mathrm{ref}$ is 298.15 K; the default value
for the other parameters is 0.0. This model is implemented by the C++ class
{ct}`ConstCpPoly`.
:::{admonition} YAML Usage
:class: tip
@ -140,3 +185,149 @@ A constant heat capacity parameterization can be defined in the YAML format by
specifying [`constant-cp`](sec-yaml-constcp) as the `model` in the species `thermo`
field.
:::
(sec-thermo-piecewise-Gibbs)=
### Piecewise Gibbs Free Energy
This parameterization uses a piecewise interpolation of the Gibbs free energy between specified values, with a constant heat capacity between each pair of
interpolation points. This model is implemented by the C++ class {ct}`Mu0Poly`.
:::{admonition} YAML Usage
:class: tip
A piecewise Gibbs parameterization can be defined in the YAML format by specifying
[`piecewise-Gibbs`](sec-yaml-constcp) as the `model` in the species `thermo` field.
:::
## Models for Species Contributions to the Equation of State
Besides enthalpy, entropy, and heat capacity data, some phase models require additional
parameters that describe how each species affects the equation of state. These models
are used in combination with one of the above parameterizations for the standard state
enthalpy, entropy, and heat capacity. This applies to most models that represent
condensed phases.
### Constant Volume Model
This species equation of state model can be used for phase models that require standard
state density information for each species, and simply specifies a constant specific
volume, density or molar density for the species. Implemented by the C++ class
{ct}`PDSS_ConstVol`.
:::{admonition} YAML Usage
:class: tip
A constant volume parameterization can be defined in the YAML format by specifying
[`constant-volume`](sec-yaml-eos-constant-volume) as the `model` in the species
`equation-of-state` field.
:::
### Temperature-dependent Density
This species equation of state model can be used for phase models that require the
standard state molar volume $V_{m,k}^\circ$ for each species $k$. The standard state
density $\rho^\circ_k$ is described using a third-order polynomial:
$$ \rho^\circ_k(T) = \frac{W_k}{V_{m,k}^\circ(T)} = a_0 + a_1 T + a_2 T^2 + a_3 T^3 $$
where $W_k$ is the molecular weight of the species. This model is implemented by class
{ct}`PDSS_SSVol`.
:::{admonition} YAML Usage
:class: tip
A temperature-dependent density parameterization can be defined in the YAML format by
specifying [`density-temperature-polynomial`](sec-yaml-eos-density-temperature-polynomial)
as the `model` in the species `equation-of-state` field.
:::
### Temperature-dependent Molar Volume
This species equation of state model can be used for phase models that require the
standard state molar volume $V_{m,k}^\circ$ for each species $k$. The standard state
molar volume is described using a third-order polynomial:
$$ V_{m,k}^\circ(T) = a_0 + a_1 T + a_2 T^2 + a_3 T^3 $$
This model is implemented by class {ct}`PDSS_SSVol`.
:::{admonition} YAML Usage
:class: tip
A temperature-dependent molar volume parameterization can be defined in the YAML format
by specifying [`molar-volume-temperature-polynomial`](sec-yaml-eos-molar-volume-temperature-polynomial)
as the `model` in the species `equation-of-state` field.
:::
### Peng-Robinson Equation of State
This species equation of state model is parameterized using the coefficients $a$, $b$,
and $\omega$ for a pure species that follows the Peng-Robinson equation of state:
$$ P = \frac{RT}{V_m - b} - \frac{a\alpha}{V_m^2 + 2bV_m - b^2} $$
where $V_m$ is the molar volume,
$$ \alpha = \left[ 1 + \kappa \left(1 - \sqrt{T_r}\right) \right]^2 $$
$$ \kappa =
\begin{cases}
0.37464 + 1.54226\omega - 0.26992\omega^2, & \omega \le 0.491 \\
0.379642 + 1.487503\omega - 0.164423\omega^2 + 0.016667\omega^3 , & \omega > 0.491
\end{cases}
$$
$T_r = T / T_c$ is the reduced temperature, and $T_c$ is the critical temperature.
These pure-species properties are combined in the multi-species Peng-Robinson phase
model, implemented by class {ct}`PengRobinson`.
:::{admonition} YAML Usage
:class: tip
A Peng-Robinson parameterization for a species can be defined in the YAML format by
specifying [`Peng-Robinson`](sec-yaml-eos-peng-robinson) as the `model` in the species
`equation-of-state` field.
:::
### Redlich-Kwong Equation of State
This species equation of state model is parameterized using the coefficients $a$ and $b$
for a pure species that follows the Redlich-Kwong equation of state:
$$ P = \frac{RT}{V_m-b} - \frac{a}{\sqrt{T} V_m (V_m + b) } $$
where $V_m$ is the molar volume.
These pure-species properties are combined in the multi-species Redlich-Kwong phase
model, implemented by class {ct}`RedlichKwongMFTP`.
:::{admonition} YAML Usage
:class: tip
A Redlich-Kwong parameterization for a species can be defined in the YAML format by
specifying [`Redlich-Kwong`](sec-yaml-eos-redlich-kwong) as the `model` in the species
`equation-of-state` field.
:::
## Other Species Equation of State Models
The following models provide complete standard state parameterizations for a species,
including molar volume, enthalpy, entropy, and heat capacity.
### Helgeson-Kirkham-Flowers-Tanger Model
The Helgeson-Kirkham-Flowers-Tanger model for aqueous species. Implemented by the C++
class {ct}`PDSS_HKFT`.
:::{admonition} YAML Usage
:class: tip
An HKFT parameterization can be defined in the YAML format by specifying
[`HKFT`](sec-yaml-eos-hkft) as the `model` in the species `equation-of-state` field.
:::
### Liquid Water
A model for liquid water that uses the IAPWS95 formulation {cite:p}`wagner2002`.
Implemented by the C++ class {ct}`PDSS_Water`.
:::{admonition} YAML Usage
:class: tip
A liquid water parameterization can be defined in the YAML format by specifying
[`liquid-water-IAPWS95`](sec-yaml-eos-liquid-water-iapws95) as the `model` in the
species `equation-of-state` field.
:::