# Density solver for ideal mixture of real gases

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

1. Is the mixing rule that I used valid?
2. 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.
• Possibly related: physics.stackexchange.com/q/139524/25301 Mar 23, 2016 at 0:09
• @KyleKanos In the question that you have linked, the user wants to know how to numerically solve for density given pressure and temperature. I am asking if the equation that I want to solve is correct. Mar 23, 2016 at 0:11
• That question is also about a similar modified ideal gas EOS, that's why I suggested it was related to yours. Mar 23, 2016 at 2:06

## 2 Answers

There are numerous methods for estimating the PVT behavior of real gas mixtures from knowledge of the behavior of the individual components. These include the Mixture Equation of State (in which the parameters for the mixture are determined from the parameters for the individual species), Kay's rule (in which pseudo-critical properties are employed in conjunction with the law of corresponding states), the Additive Pressure rule (in which the pressure contribution of each species is set equal to that of the pure component at the same temperature and volume of the system), and the Additive Volume Rule (in which the volume contribution of each component is set equal to that of the pure component at the same temperature and pressure as the system). For details, see Fundamental of Engineering Thermodynamics by Moran, et al, pages 699-703.

• What about the expression for reduced density? Which one is correct? Mar 22, 2016 at 13:07
• They are all approximate, and give slightly different answers. Several cases are compared in Moran et al. Mar 22, 2016 at 13:29
• I don't see how they would give nearly the same answers. Suppose I have a mixture of carbon dioxide (CO2) and nitrogen (N2). They are in a ratio of 0.9:0.1 by mole fraction. Then, $M_{mix}$ = 42.4 If I look at the contribution of N2, then $M_{mix}/M_{N2} \approx 1.5$. The values obtained for $\delta_{N2}$ from the two expressions mentioned in the question would be very different considering similar values of $\rho$. Mar 22, 2016 at 18:20
• I don't quite understand what your problem is. Why don't you just try a couple of the methods I mentioned and see what you get? Then you can compare with the approach that you are considering. Most of these methods are based on molar volume (or molar density). You can convert of mass density in the end. Moran et al does some examples for a mixture of methane and butane. Mar 22, 2016 at 18:52
• If you look at the equation for pressure in the question, it depends on $\alpha^r_{\delta,N2}$ (considering my earlier example). This expression will yield very different values considering the two different expressions for reduced density. I want to know which one is right. Is this clearer? Mar 22, 2016 at 18:58

The case of multicomponent mixtures is explained in the paper itself, in section 2.2, p3035. You do not need to apply your own mixing rules, these are incorporated in the functions $$\rho_r$$ and $$T_r$$, introduced in equations 3 and 4, and later defined explicitly in equations 16 and 17 in terms of the pure components.