12 Thermochemistry and Mixture Properties

12.1 Thermally perfect species

For a thermally perfect ideal-gas species \(k\), the specific heat depends on temperature while the ideal-gas relation is retained. With species gas constant \[R_k=\frac{R_u}{W_k},\] where \(R_u\) is the universal gas constant and \(W_k\) is molecular weight, \[c_{v,k}=c_{p,k}-R_k.\]

The species enthalpy relative to a reference temperature \(T_0\) is \[h_k(T) =h_k(T_0)+\int_{T_0}^{T}c_{p,k}(\theta)\,d\theta, \](12.1) and the internal energy is \[e_k(T)=h_k(T)-R_kT. \](12.2) If \(h_k(T_0)\) contains the standard formation enthalpy, Eq. (12.1) naturally includes both formation and sensible contributions.

12.2 NASA seven-coefficient form

CMPS supports the common NASA polynomial representation used by equilibrium and thermochemistry databases (Gordon and McBride 1971). Within one temperature interval, \[\frac{c_{p,k}}{R_k} =a_{1,k}+a_{2,k}T+a_{3,k}T^2+a_{4,k}T^3+a_{5,k}T^4, \](12.3) \[\frac{h_k}{R_kT} =a_{1,k}+\frac{a_{2,k}}{2}T +\frac{a_{3,k}}{3}T^2 +\frac{a_{4,k}}{4}T^3 +\frac{a_{5,k}}{5}T^4 +\frac{a_{6,k}}{T}, \](12.4) and \[\frac{s_k^{\circ}}{R_k} =a_{1,k}\ln T+a_{2,k}T +\frac{a_{3,k}}{2}T^2 +\frac{a_{4,k}}{3}T^3 +\frac{a_{5,k}}{4}T^4+a_{7,k}. \](12.5) The appropriate low- or high-temperature coefficient set is selected for the current temperature range.

A convenient standard reference is \(T_0=298.15\,\mathrm{K}\). The sensible enthalpy relative to that state is \[h_{s,k}(T)=h_k(T)-h_k(T_0),\] which is identically zero at \(T=T_0\).

12.3 Mixture thermodynamics

For mass fractions \(Y_k\) satisfying \(\sum_kY_k=1\), mass-specific mixture properties are \[\begin{aligned} h(T,\mathbf{Y})&=\sum_{k=1}^{N}Y_kh_k(T),\\ c_p(T,\mathbf{Y})&=\sum_{k=1}^{N}Y_kc_{p,k}(T),\\ R_m(\mathbf{Y})&=\sum_{k=1}^{N}Y_kR_k,\\ \frac{1}{W_m}&=\sum_{k=1}^{N}\frac{Y_k}{W_k}. \end{aligned}\](12.6) For an ideal-gas mixture, \[\rho=\frac{p}{R_mT}.\] For the constant-density incompressible regime, that equation of state is not used to update carrier density; the material must provide the validated constant physical density required by Chapter 4.

12.4 Derivatives in the coupled Jacobian

Because temperature and independent species mass fractions are primitive columns, thermodynamic functions evaluated with AD contribute derivatives such as \[\frac{\partial h}{\partial T}=c_p, \qquad \frac{\partial h}{\partial Y_s}, \qquad \frac{\partial\rho}{\partial p}, \qquad \frac{\partial\rho}{\partial T}\] to the active residual where the equation of state requires them. In the constant-density incompressible carrier path, \(\partial\rho/\partial p\) and \(\partial\rho/\partial T\) are not used as physical density couplings.

12.5 Transport-property qualification

Thermodynamic mixture properties such as \(h\) and \(c_p\) can be composition/temperature dependent, but this must not be generalized automatically to all transport properties. The current implementation review found that the active viscosity, conductivity, and molecular-diffusivity accessors do not yet provide a fully general \(T\)–\(\mathbf{Y}\) mixing law in every mixture configuration. A case that requires such transport-property dependence should therefore verify the selected material model explicitly rather than assuming it from the thermochemistry polynomial support.

12.6 Current fluid EOS families

The current carrier material implementation supports ideal-gas, stiffened-gas and constant-density incompressible closures. The pressure-temperature density laws are

\[\rho_{ig}=\frac{p}{RT},\qquad \rho_{sg}=\frac{p+p_\infty}{R_sT},\qquad \rho_{inc}=\rho_0,\]

where the stiffened-gas effective gas constant is

\[R_s=C_p\frac{\gamma-1}{\gamma},\qquad C_v=\frac{C_p}{\gamma}.\]

The stiffened-gas internal-energy pressure relation and sound speed are

\[p=(\gamma-1)\rho e-\gamma p_\infty,\qquad a^2=\gamma\frac{p+p_\infty}{\rho}.\]

The corresponding ideal-gas expressions are recovered with \(p_\infty=0\) when the caloric model is compatible. The constant-density closure has \(\rho_p=\rho_T=0\) and is coupled through artificial compressibility rather than a physical acoustic EOS.

A complete source-aligned derivation, including thermodynamic derivatives, total-state relations, caloric formulas and homogeneous-VOF embedding, is given in Equations of State and Caloric Models.

12.7 Transport-property closures

The active source contains temperature-dependent viscosity and conductivity closures in addition to the thermodynamic functions above. Constant and piecewise-polynomial forms are supplemented by Sutherland and Gupta-type viscosity laws and kinetic/Gupta-type conductivity laws. Their exact implementation forms and range behavior are documented in Transport Properties and Material Closures.