21. Additive-Volume Real-Fluid EOS Closure#

The additive-volume closure constructs mixture properties from pure-species equations of state at a common pressure and temperature. It combines a rule for mixture volume with a separate rule for mixture internal energy. Together, these determine density and enthalpy for a specified composition.

21.1. Assumptions and Additive-Volume Closure#

Consider a mixture of \(N_s\) species with masses \(m_k\) and total mass \(M=\sum_{k=1}^{N_s}m_k\). The mass fractions are \(Y_k=m_k/M\), with \(Y_k\geq 0\) and \(\sum_{k=1}^{N_s}Y_k=1\). All species share the same pressure \(p\) and temperature \(T\).

Let \(\rho_k(p,T)\) be the density of pure species \(k\) at this pressure and temperature, obtained from its equation of state:

(21.1.1)#\[p=p_k(\rho_k,T) \qquad k=1,\ldots,N_s\]

The pure-species density \(\rho_k\) differs from the partial density \(\rho Y_k\), which is the mass of species \(k\) per unit mixture volume. Here \(\rho\) is the mixture density.

The additive-volume assumption states that the mixture volume \(V\) is the sum of the volumes that the species masses would occupy as pure fluids at the common \(p\) and \(T\):

(21.1.2)#\[V=\sum_{k=1}^{N_s}\frac{m_k}{\rho_k(p,T)}\]

Dividing by \(M\) gives the mixture specific volume \(v=V/M=1/\rho\):

(21.1.3)#\[v(p,T,Y) =\frac{1}{\rho(p,T,Y)} =\sum_{k=1}^{N_s}Y_k v_k(p,T) =\sum_{k=1}^{N_s}\frac{Y_k}{\rho_k(p,T)}\]

Here \(v_k=1/\rho_k\) is the pure-species specific volume, and \(Y\) denotes the collection of mass fractions. This is often called an Amagat mixture rule. It includes real-fluid behavior through the pure-species equations of state, but adds no excess volume of mixing.

21.2. Mixture Energy and Enthalpy#

Let \(e_k(p,T)\) be the specific internal energy of pure species \(k\) at the same state. The energy closure assumes a mass-weighted sum:

(21.2.1)#\[e(p,T,Y)=\sum_{k=1}^{N_s}Y_k e_k(p,T)\]

This is an additional modeling assumption; it does not follow from volume additivity. Each \(e_k\) includes the real-fluid contribution from its caloric model, but the mixture rule adds no excess internal energy of mixing.

The mixture static specific enthalpy is \(h=e+pv\). Substituting the volume and energy closures gives

(21.2.2)#\[h =\sum_{k=1}^{N_s}Y_k e_k +p\sum_{k=1}^{N_s}Y_k v_k =\sum_{k=1}^{N_s}Y_k(e_k+p v_k)\]

Thus enthalpy is also a mass-weighted sum of pure-species values:

(21.2.3)#\[h(p,T,Y)=\sum_{k=1}^{N_s}Y_k h_k(p,T) \qquad h_k=e_k+p v_k\]

All energies in this closure are specific thermodynamic quantities; they exclude the kinetic energy of the flow.

21.3. Recovering the Thermodynamic State#

Given \(p\), \(T\), and \(Y\), evaluate the pure-species properties and apply Eqs. (21.1.3), (21.2.1), and (21.2.3). When density or enthalpy is specified instead of temperature, the same closure provides a scalar equation for \(T\).

For prescribed pressure, density \(\rho_{\mathrm{given}}\), and composition, hold \(p\) and \(Y\) fixed and solve

(21.3.1)#\[R_\rho(T) =\rho_{\mathrm{given}} -\left(\sum_{k=1}^{N_s}\frac{Y_k}{\rho_k(p,T)}\right)^{-1} =0\]

For prescribed pressure, static specific enthalpy \(h_{\mathrm{given}}\), and composition, hold \(p\) and \(Y\) fixed and solve

(21.3.2)#\[R_h(T) =h_{\mathrm{given}}-\sum_{k=1}^{N_s}Y_k h_k(p,T) =0\]

The pure-species properties must be reevaluated at each trial temperature. Once the temperature is found, the volume and energy closures supply the remaining mixture properties.

Density and composition may also be specified through partial densities \(q_k=\rho Y_k\). For \(\rho>0\), the inverse relations are

(21.3.3)#\[\rho=\sum_{k=1}^{N_s}q_k \qquad Y_k=\frac{q_k}{\rho}\]

This representation contains the same information as density and normalized mass fractions; it does not add a thermodynamic constraint.

21.4. Consistency and Limitations#

At a cell or reconstructed face state, density, energy, enthalpy, and pure-species properties must satisfy the closure at the same \(p\), \(T\), and \(Y\). Reconstructing extra thermodynamic quantities independently can produce an inconsistent state. For example, a specified \(p\), \(\rho\), and \(Y\) determine enthalpy through the recovered temperature; enthalpy cannot then be prescribed independently.

The closure has the following limitations:

  • Mixing effects. Pure-fluid nonideality is retained, but excess volume and excess internal energy caused by mixing are omitted.

  • Equilibrium. Common pressure and temperature impose mechanical and thermal equilibrium. The closure takes composition as given and does not determine chemical equilibrium or phase partitioning.

  • State recovery. The chosen pure-species EOS branches must admit a common state within their ranges of validity. A prescribed density or enthalpy need not have a temperature solution, and a solution need not be unique across branches. If the relevant mixture property changes only weakly with temperature, its inversion is poorly conditioned.