I am referring the paper on GERG-2008 equation of state. I am interested in calculating the properties of ideal mixtures of real gases. The GERG-2008 EOS is in Helmholtz free energy and hence the independent variables are mixture molar density ($\rho$) and mixture temperature ($T$). But usually the total pressure and temperature of the mixture are available to me. Consequently, I want to write a density solver.
If I suppose that the mixing rules derived for ideal gases are valid for real gases as well, then I can write the total pressure of the mixture as
$$p(T,\rho,\bar{x}) = \sum_{k=1}^{N}x_k p_k$$
where $\bar{x}$ is the mixture composition, $N$ is the number of pure components in the mixture, $x_k$ is the mole fraction of the $k$-th component and $p_k$ is the partial pressure of the $k$-th component. Following the formula for pressure in the paper mentioned above, I can write
$$p = \sum_{k=1}^{N} x_k \rho R T \left( 1 + \frac{\rho}{\rho_{c,k}} \alpha^r_{\delta,k}\ \left(\delta_k, \tau_k \right) \right) $$
where the reduced density of the $k$-th component $\delta_k$ and inverse reduced temperature of the $k$-th component $\tau_k$ are given by
$$\delta_k = \frac{\rho}{\rho_{c,k}}$$
$$\tau_k = \frac{T_{c,k}}{T}$$
and $\alpha^r_{\delta,k} = \frac{\partial \alpha_{r,k}}{\partial \delta}$ is the residual Helmholtz energy, $\rho_{c,k}$ is the critical molar density and $T_{c,k}$ is the critical temperature of the $k$-th component. Now, I need to solve the above equation with an iterative solver to obtain the molar density of the mixture.
I want to confirm if the above formulation is correct. My main questions are
- Is the mixing rule that I used valid?
- Instead of using molar densities for evaluating $\delta_k$, should I be using mass densities i.e. $\delta_k = \frac{\rho_m}{\rho_{m,c,k}}$ ($m$ indicates mass density)? Note that this is different from the above expression. In fact, here $\delta_k = \frac{M_{mix}}{M_k} \frac{\rho}{\rho_{c,k}}$ where $M$ is molar mass.
