Species Thermodynamic Models#

The phase models discussed in the Phase Thermodynamic Models 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.

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.

Models for Species Standard State Enthalpy, Entropy, and Heat Capacity#

Many of Cantera’s phase thermodynamic models are formulated to make use of the standard state thermodynamic properties for individual species, defined at a standard pressure \(p^\circ\) and for the composition specified by the phase model. For example, this could include 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. The value of \(p^\circ\) is not fixed by Cantera, and may vary among different sources of thermodynamic data.

Caution

In some parts of the Cantera documentation, properties calculated at the user-specified standard pressure \(p^\circ\) are referred to as reference-state thermodynamic properties, as they represent a well-known reference state and properties for all other states are calculated according to their departure from this known reference condition. In these same parts of the documentation, the term standard-state properties refers to properties calculated using the composition defining the standard state but at the mixture’s current pressure.

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:

\(\hat{c}^\circ_p(T)\): the standard-state molar heat capacity at constant pressure as a function of temperature;
\(\hat{h}^\circ(T)\), the standard-state molar enthalpy as a function of temperature;
\(\hat{s}^\circ(T)\): 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

\[ \hat{h}^\circ(T) = \hat{h}^\circ(T_\t{ref}) + \int_{T_\t{ref}}^T \hat{c}^\circ_p(T) \; dT \]

and

\[ \hat{s}^\circ(T) = \hat{s}^\circ(T_\t{ref}) + \int_{T_\t{ref}}^T \frac{\hat{c}^\circ_p(T)}{T} \; dT \]

respectively. This means that a parameterization of \(\hat{c}_p^\circ(T)\) plus the constants \(\hat{h}^\circ(T_\t{ref})\) and \(\hat{s}^\circ(T_\t{ref})\) at a reference temperature \(T_\t{ref}\) 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 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.

The NASA 7-Coefficient Polynomial Parameterization#

The NASA 7-coefficient polynomial parameterization is used to compute the species 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:

\[ \begin{align}\begin{aligned} \frac{\hat{c}_p^\circ(T)}{R} &= a_0 + a_1 T + a_2 T^2 + a_3 T^3 + a_4 T^4\\\frac{\hat{h}^\circ (T)}{R T} &= a_0 + \frac{a_1}{2} T + \frac{a_2}{3} T^2 + \frac{a_3}{4} T^3 + \frac{a_4}{5} T^4 + \frac{a_5}{T}\\\frac{\hat{s}^\circ(T)}{R} &= a_0 \ln T + a_1 T + \frac{a_2}{2} T^2 + \frac{a_3}{3} T^3 + \frac{a_4}{4} T^4 + a_6 \end{aligned}\end{align} \]

This model is implemented by the C++ classes NasaPoly1 and NasaPoly2.

Note

This is sometimes referred to as the NASA7 polynomial within Cantera. It was 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, which uses 9 coefficients for each temperature region.

YAML Usage

A NASA7 parameterization can be defined in the YAML format by specifying NASA7 as the model in the species thermo field.

The NASA 9-Coefficient Polynomial Parameterization#

The NASA 9-coefficient polynomial parameterization [McBride et al., 2002], often called NASA9 within Cantera for short, is an extension of the NASA 7-coefficient polynomial parameterization that includes two additional terms in each temperature region and supports an arbitrary number of temperature regions.

The NASA9 parameterization represents the species thermodynamic properties with the following equations:

\[ \begin{align}\begin{aligned} \frac{\hat{c}_p^\circ(T)}{R} &= a_0 T^{-2} + a_1 T^{-1} + a_2 + a_3 T + a_4 T^2 + a_5 T^3 + a_6 T^4\\\frac{\hat{h}^\circ(T)}{R T} &= - a_0 T^{-2} + a_1 \frac{\ln T}{T} + a_2 + \frac{a_3}{2} T + \frac{a_4}{3} T^2 + \frac{a_5}{4} T^3 + \frac{a_6}{5} T^4 + \frac{a_7}{T}\\\frac{\hat{s}^\circ(T)}{R} &= - \frac{a_0}{2} T^{-2} - a_1 T^{-1} + a_2 \ln T + a_3 T + \frac{a_4}{2} T^2 + \frac{a_5}{3} T^3 + \frac{a_6}{4} T^4 + a_8 \end{aligned}\end{align} \]

A common source for species data in the NASA9 format is the NASA ThermoBuild tool. This model is implemented by the C++ classes Nasa9Poly1 and Nasa9PolyMultiTempRegion.

YAML Usage

A NASA-9 parameterization can be defined in the YAML format by specifying NASA9 as the model in the species thermo field.

The Shomate Parameterization#

The Shomate parameterization is:

\[ \begin{align}\begin{aligned} \hat{c}_p^\circ(T) &= A + Bt + Ct^2 + Dt^3 + \frac{E}{t^2}\\\hat{h}^\circ(T) &= At + \frac{Bt^2}{2} + \frac{Ct^3}{3} + \frac{Dt^4}{4} - \frac{E}{t} + F\\\hat{s}^\circ(T) &= A \ln t + B t + \frac{Ct^2}{2} + \frac{Dt^3}{3} - \frac{E}{2t^2} + G \end{aligned}\end{align} \]

where \(t = T / 1000\textrm{ K}\). It requires 7 coefficients \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), and \(G\). This parameterization is used to represent standard-state properties in the NIST Chemistry WebBook. 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. This model is implemented by the C++ classes ShomatePoly and ShomatePoly2.

YAML Usage

A Shomate parameterization can be defined in the YAML format by specifying Shomate as the model in the species thermo field.

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:

\[ \begin{align}\begin{aligned} \hat{c}_p^\circ(T) &= \hat{c}_p^\circ(T_\t{ref})\\\hat{h}^\circ(T) &= \hat{h}^\circ\left(T_\t{ref}\right) + \hat{c}_p^\circ \left(T-T_\t{ref}\right)\\\hat{s}^\circ(T) &= \hat{s}^\circ(T_\t{ref}) + \hat{c}_p^\circ \ln{\left(\frac{T}{T_\t{ref}}\right)} \end{aligned}\end{align} \]

The parameterization uses four constants: \(T_\t{ref}\), \(\hat{c}_p^\circ(T_\t{ref})\), \(\hat{h}^\circ(T_\t{ref})\), and \(\hat{s}^\circ(T)\). The default value of \(T_\t{ref}\) is 298.15 K; the default value for the other parameters is 0.0. This model is implemented by the C++ class ConstCpPoly.

YAML Usage

A constant heat capacity parameterization can be defined in the YAML format by specifying constant-cp as the model in the species thermo field.

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 Mu0Poly.

YAML Usage

A piecewise Gibbs parameterization can be defined in the YAML format by specifying piecewise-Gibbs 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 mechanical 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 PDSS_ConstVol.

YAML Usage

A constant volume parameterization can be defined in the YAML format by specifying 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 PDSS_SSVol.

YAML Usage

A temperature-dependent density parameterization can be defined in the YAML format by specifying 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 PDSS_SSVol.

YAML Usage

A temperature-dependent molar volume parameterization can be defined in the YAML format by specifying 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,

\[ \alpha = \left[ 1 + \kappa \left(1 - \sqrt{T_r}\right) \right]^2 \]

\[\begin{split} \kappa = \begin{cases} 0.37464 + 1.54226\omega - 0.26992\omega^2, & \omega \le 0.491 \\ 0.379642 + 1.487503\omega - 0.164423\omega^2 + 0.016667\omega^3 , & \omega > 0.491 \end{cases} \end{split}\]

\(T_r = T / T_c\) 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 PengRobinson.

YAML Usage

A Peng-Robinson parameterization for a species can be defined in the YAML format by specifying 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 RedlichKwongMFTP.

YAML Usage

A Redlich-Kwong parameterization for a species can be defined in the YAML format by specifying 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 PDSS_HKFT.

YAML Usage

An HKFT parameterization can be defined in the YAML format by specifying HKFT as the model in the species equation-of-state field.

Liquid Water#

A model for liquid water that uses the IAPWS95 formulation [Wagner and Pruß, 2002]. Implemented by the C++ class PDSS_Water.

YAML Usage

A liquid water parameterization can be defined in the YAML format by specifying liquid-water-IAPWS95 as the model in the species equation-of-state field.