Modeling Thermodynamic Properties of Mixtures of CO₂ + O₂ in the Allam Cycle by Equations of State

Abstract

Different equations of state (EOS) were applied for describing thermodynamic properties of the system $\text {CO}_{2} + \text {O}_{2},$ which is important for the Allam cycle: cubic EOS (Soave–Redlich–Kwong, Peng–Robinson), molecular-based EOS (PC-SAFT, PCP-SAFT, SAFT-VR Mie, polar soft-SAFT, BACKONE, sCPA), and multiparameter EOS (GERG-2008, EOS-CG). The pure component models were taken from the literature. The results for the mixture were compared to experimental data from the literature for two cases: (i) the thermodynamic properties of the mixture were modeled using predictive mixing and combination rules; (ii) additional binary interaction parameters were fitted to experimental data for improving the performance. In the predictive mode (i), the best results were obtained with molecular-based EOS. In the adjusted mode (ii) the best results were obtained with the multi-parameter GERG-2008 EOS and EOS-CG.

1 Introduction

Power plants based on the Allam cycle generate electric energy as well as $\text {CO}_{2}$ of high purity at high pressure from burning hydrocarbon fuels with pure oxygen [1]. The process is attractive, as it combines a high thermal efficiency with efficient $\text {CO}_{2}$ capture [2, 3]. In the simplest picture, the Allam cycle can be described as a Joule-Brayton cycle with internal heat recuperation in which the main working fluid is $\text {CO}_{2}$. Additionally, oxygen and water are present in the working fluid. Water is removed before the compression and most of the $\text {CO}_{2}$ is removed after the compression. Hence, for modeling the compression, the knowledge of the properties of mixtures of $\text {CO}_{2}$ and $\text {O}_{2}$ is essential. To some extent, also other components can be present such as argon, nitrogen, and trace elements from the fuel. In this work, we focus on the main components $\text {CO}_{2}$ and $\text {O}_{2}$ of the working fluid.

For the design of power plants based on the Allam cycle, information on the thermal and caloric properties as well as phase equilibria of the working fluid is needed. The interest in the Allam cycle has led to a number of experimental studies of thermophysical properties of mixtures relevant for the Allam cycle in recent years [4,5,6,7,8,9,10] that can be used as basis for the thermodynamic modeling by equations of state (EOS). In principle, EOS can be used to predict mixture properties from pure component properties using predictive mixing and combination rules. Nevertheless, for many applications, such predictions are not accurate enough, so that adjustable binary parameters are introduced that are fitted to selected experimental data of the mixture. In this work, the mixture $\text {CO}_{2} + \text {O}_{2}$ was modeled using different EOS based on pure component models that were taken from the literature. Both, predictive mixing and combination rules as well as mixing and combination rules with adjustable parameters were tested.

Different types of EOS have been used for modeling mixtures of $\text {CO}_{2}$ with other components, see, e.g. Refs. [4, 11,12,13]. In 2011, Diamantonis et al. [11] have assessed predictions of cubic EOS and EOS based on statistical association fluid theory (SAFT) for phase equilibria of binary mixtures of $\text {CO}_{2}$ with ($\text {CH}_4,\,\text {N}_{2},\,\text {O}_{2},\,\text{SO}_{2},\,\text {Ar},\,\text {and}\,\text {H}_{2}\text {S}$). The studied EOS included the Redlich–Kwong [14], Soave–Redlich–Kwong (SRK) [15], Peng-Robinson (PR) [16], SAFT [17], and perturbed chain-SAFT (PC-SAFT) [18] EOS. Diamantonis et al. [11] concluded that the PC-SAFT EOS is overall the most accurate EOS for predicting $\text {CO}_{2}$ + gas mixture properties. In 2014, Mazzoccoli et al. [12] studied different EOS regarding their predictions of phase equilibria and homogeneous state density data of binary mixtures of $\text {CO}_{2}$ with ($\text {N}_{2},\,\text {O}_{2},\,\text {and}\,\text {Ar}$). The studied EOS included the Benedict-Webb-Rubin (BWR) [19], PC-SAFT, and GERG-2008 [20] EOS. The GERG-2008 and PC-SAFT EOS were found to perform well in most cases. In 2016, Lasala et al. [4] compared different cubic EOS with different combination rules for predicting phase equilibria of binary mixtures of $\text {CO}_{2}$ with ($\text {N}_{2},\, \text {Ar},\, \text {and}\, \text {O}_{2}$) and found that using a temperature-dependent parameter in the combination rule improved the accuracy significantly. In 2017, Perez et al. [13] studied the modeling of vapor–liquid equilibria and homogeneous state densities of 108 binary mixtures related to carbon capture and storage using the SRK, PR, PC-SAFT, and SAFT-VR Mie [21] EOS using a temperature-dependent combination rule parameter. Perez et al. [13] concluded that the SAFT-VR Mie EOS is the most accurate of the studied EOS.

Overall, the findings reported in the literature give no clear picture, which EOS to use for modeling mixture properties of $\text {CO}_{2}+\text {O}_{2}$. Furthermore, a significant amount of new data has become available only recently and was not included in most studies discussed above [4,5,6,7,8,9,10]. Therefore, in this work, we have studied the modeling of mixture properties of $\text {CO}_{2}+\text {O}_{2}$ using different EOS. First, the available literature data are reviewed and compiled in a database. This includes vapor–liquid equilibrium data $Tpx^{\prime}x^{\prime \prime}$ as well as homogeneous state property data (density $\rho$, speed of sound w, and isothermal compressibility $\beta$) for a given composition, pressure, and temperature. Ten EOS were used for modeling the mixture properties, namely the SRK [15], PR [16], PC-SAFT [18], PCP-SAFT [18, 22], SAFT-VR Mie [21], polar soft-SAFT [23, 24], BACKONE [25, 26], sCPA [27, 28], GERG-2008 [20], and EOS-CG [29] EOS. The EOS-CG as well as pure component EOS models for $\text {CO}_{2}$ and $\text {O}_{2}$ [30, 31] were developed by Roland Span and co-workers.

Two cases were considered: (i) the thermodynamic properties of the mixture were modeled using fully predictive mixing and combination rules (“predictive mode”); (ii) temperature-dependent binary interaction parameters were fitted to a training dataset of experimental data for improving the performance of the models (“adjusted mode”). The binary interaction parameters were fitted to the same training dataset for all studied EOS; except the EOS-CG [29]. The binary interaction parameters for the mixture $\text {CO}_{2} + \text {O}_{2}$ were already fitted in the original publication by Gernert and Span [29] and they are adopted here for the EOS-CG in the “adjusted mode”. The results are compared for both cases (i) and (ii) for the complete literature data, a training, and a test dataset. This allows a comparison of the different EOS on an equal basis.

This paper is structured as follows: first, the experimental data basis for mixtures of $\text {CO}_{2}+\text {O}_{2}$ that was established from literature data is presented; then, the studied EOS are described, focusing in the modeling of mixture properties. Finally, the results of the application of the EOS in the predictive and adjusted mode are presented and critically compared.

2 Methods

2.1 Experimental Data Basis

Tables 1 and 2 give an overview of the available experimental data on the vapor–liquid equilibrium and homogeneous state property data, respectively, in the system $\text {CO}_{2}+\text {O}_{2}$. In total, 255 phase equilibrium data points and 1866 homogeneous state data points have been reported. The phase equilibrium datasets report pressure p and the compositions of both phases x', x″ for a given temperature T. For the homogeneous states, data on the density $\rho,$ the speed of sound w, and the isothermal compressibility $\beta$ at a given pressure and temperature are available. The database compiled within this work is provided in the electronic Supplementary Information.

Table 1 Overview of studies reporting experimental data on vapor–liquid equilibrium properties of the system $\text {CO}_{2}+\text {O}_{2}$

Table 2 Overview of studies reporting experimental data on homogeneous state point properties of the system $\text {CO}_{2}+\text {O}_{2}$

Phase equilibrium data (cf. Table 1) have been reported in the literature in the range $218\,\text {K}\le T \le 298\,\text {K}$, $0.9\,\text {MPa}\le p \le 14.4$ MPa, and $0.0\,\text {mol}{\cdot}\text {mol}^{-1} < x_{\text {O}_{2}} \le 0.55$ mol·mol⁻¹. The phase equilibrium data covers almost the complete range between the temperature of the triple point and the critical temperature of pure $\text {CO}_{2}$ ($T_{\text {tr}, \text {CO}_{2}} = 216.58\,\text {K}$ [32, 33], $T_{\text {c}, \text {CO}_{2}} = 304.15\,\text {K}$ [34]). The data on homogeneous state points lie in the region $250\, \text {K}\le T \le 423\,\text {K},\,0.5\,\text {MPa}\le p \le 47.8\, \text {MPa}$, and $0.0\,\text{mol}{\cdot}\text{mol}^{-1}< x_{\text {O}_{2}} \le 0.75$ mol·mol⁻¹. The critical temperature of oxygen is $T_{\text {c,}{\text {O}_{2}}}=154.77\,\text {K}$ [34]. Hence, oxygen is supercritical in the entire studied temperature range. The mixture $\text {CO}_{2}+\text {O}_{2}$ is a type I mixture according to the scheme of van Konynenburg and Scott [35].

The literature data were checked for clear outliers by comparison of each data point to the rest of the data along isotherms and isobars at equal compositions. Overall, seven data points for VLE and 12 data points for homogeneous state points were removed for the EOS evaluation; for details see Supplementary Information. The remaining experimental data were split into a training dataset and a test dataset. The training dataset comprises of data between $273\, \text {K}\le T \le 333\, \text {K},\, 5\, \text {MPa}\le p \le 30\, \text {MPa}$, and $0.0\,\text {mol}{\cdot}\text {mol}^{-1}< x_{\text {O}_{2}} \le 0.5\,\text {mol}{\cdot}\text {mol}^{-1}$. The test dataset includes the remaining data. An overview of the number of experimental data points in each dataset is given in Table 3. Additionally, Fig. 1 gives an overview of the temperatures and pressures for which data are available.

Table 3 Number of experimental data for homogeneous state point properties and vapor–liquid equilibrium properties of the system $\text {CO}_{2}+\text {O}_{2}$ used in the comparison of the EOS

Fig. 1

Pressure–temperature diagram with the homogeneous state data and phase equilibrium data compiled in the database, cf. Tables 1 and 2. Additionally, the vapor pressure curves of $\text {CO}_{2}$ and $\text {O}_{2}$ are shown (computed from the reference EOS models from Refs. [30, 43]). The star indicates the critical points. The area of the training data are marked by the red box (Color figure online)

2.2 Equations of State

In this work, ten EOS were used for the modeling of mixture properties for the system $\text {CO}_{2}+\text {O}_{2}$. Table 4 gives an overview of the EOS used in this work. For all calculations carried out in this work, an in-house code was used that was validated by comparing the results with data from the EOS developers.

All EOS were implemented using the Helmholtz energy formulation. The Helmholtz energy per particle $a = A/N$ can be written as a sum of the ideal gas contribution and a configurational contribution as

$$\begin{aligned} \frac{a}{k_\text {B}T}={\tilde{a}}={\tilde{a}}_{\text {id}} + {\tilde{a}}_{\text {config}}. \end{aligned}$$

(1)

The configurational contribution ${\tilde{a}}_{\text {config}}$ is calculated by the respective EOS. For the ideal gas contribution ${\tilde{a}}_{\text {id}}$, the pure component models from Kunz and Wagner [20] were used in all cases. The ideal gas contribution of the total Helmholtz energy of a mixture is calculated as

$$\begin{aligned} {\tilde{a}}_{\text {id}}=\sum\limits _{i=1}^{N} x_i[{\tilde{a}}_{\text {id},i}(\rho ,T) + \text {ln}\, x_i ], \end{aligned}$$

(2)

where ${\tilde{a}}_{\text {id},i}$ is the ideal gas contribution of the pure components $i = \text {CO}_{2}$ and $\,\text {O}_{2}$ and $x_i$ indicates the mole fraction of component i. The pure component EOS models for $\text {CO}_{2}$ and $\text {O}_{2}$ were taken from the literature in all cases. In all cases, VLE data were used for their parametrization [20, 22, 25, 34, 36,37,38,39,40,41]—in some cases, additionally, homogeneous state property data were used [20, 22]. The pure component model parameters are summarized in the Supplementary Material.

Table 4 Overview of the equation of states (EOS) used in this work including information on the basic publication of the EOS and its year, and the publications from which the pure component models were taken together with the number of parameters (indicated by #)

2.2.1 Multi-parameter EOS

The GERG-2008 [20] EOS and the EOS-CG [29] are empirical multi-parameter EOS. The pure component parameters have no direct physical meaning. They are substance-specific and were correlated in the original works [30, 31, 42, 43] to a large number of experimental data for the respective substance. This leads to very accurate models for the pure substances regarding the data that was used for the fitting. Yet, using multi-parameter EOS for extrapolation, i.e. outside the state range that was considered for the model parametrization, has to be done with caution [44,45,46,47,48]. The two multi-parameter EOS models used here for describing the mixture $\text {CO}_{2}+\text {O}_{2}$ use different pure component models, cf. Table 4. The GERG-2008 EOS uses the pure component EOS models from Refs. [31, 42], whereas the EOS-CG uses those from Refs. [30, 43].

For multi-parameter EOS, the extension to mixtures is usually carried out by defining a reducing function for the mixture density $\rho _\text {r}({x})$ and mixture temperature $T_\text {r}({x})$. Thereby, the configurational contribution to the Helmholtz energy of the mixture can be written as

$$\begin{aligned} {\tilde{a}}^\text {multi-parameter}_\text {config} = {\tilde{a}}_\text {config}\left( \frac{\rho }{\rho _\text {r}}, \frac{T_\text {r}}{T}, {x}\right) . \end{aligned}$$

(3)

The reducing functions are calculated according to Ref. [20] as

$$\begin{aligned} \frac{1}{\rho _r} = \sum\limits^{N}_{i=1}\sum ^{N}_{j=1}x_i x_j \beta _{v,ij} \gamma _{v,ij} \cdot \frac{x_i + x_j}{\beta ^2_{v,ij} x_i + x_j} \cdot \frac{1}{8}\left( \frac{1}{\rho _{c,i}^{1/3}} + \frac{1}{\rho _{c,j}^{1/3}}\right) ^3, \end{aligned}$$

(4)

and

$$\begin{aligned} T_r = \sum ^{N}_{i=1}\sum ^{N}_{j=1}x_i x_j \beta _{T,ij} \gamma _{T,ij}\cdot \frac{x_i + x_j}{\beta ^2_{T,ij} x_i + x_j}\cdot (T_{\text {c},i} T_{\text {c},j})^{0.5}, \end{aligned}$$

(5)

with the critical temperature $T_{\text {c},i},\,T_{\text {c},j}$ and critical density $\rho _{\text {c},i},\,\rho _{\text {c},j}$ of component i and j. Both, the GERG-2008 EOS and the EOS-CG use the mixture ansatz outlined in Eqs. 3–5. The parameters ($\beta _{v,ij},\,\beta _{T,ij},\,\gamma _{v,ij},\,\gamma _{T,ij})$ are adjustable mixture parameters in the GERG-2008 and EOS-CG model. They obey the relations [20]

$$\begin{aligned} \beta _{v,ij} = 1/\beta _{v,ji},\quad \quad \beta _{T,ij} = 1/\beta _{T,ji}, \end{aligned}$$

(6)

and

$$\begin{aligned} \gamma _{v,ij} = \gamma _{v,ji},\quad \quad \gamma _{T,ij} = \gamma _{T,ji}. \end{aligned}$$

(7)

An additional departure function can be regressed to mixture properties for multi-parameter EOS to improve their accuracy [20, 49] which was, however, not used here.

2.2.2 Cubic EOS

The SRK [15] and PR [16] EOS were implemented according to Ref. [50] in the Helmholtz energy form. The pure component models are specified by the numbers for the critical temperature $T_\text {c}$, the critical pressure $p_\text {c}$, and the acentric factor $\omega$. The corresponding numbers are given in the Supplementary Information. The standard one-fluid mixing rules were applied for the calculation of the mixture attraction energy parameter $a_\text {m}$ and the mixture co-volume parameter $b_\text {m}$:

$$\begin{aligned} a_\text {m} = \sum ^{N}_{i=1}\sum ^{N}_{j=1}x_i x_j a_{ij}(T), \end{aligned}$$

(8)

$$\begin{aligned} b_\text {m} = \sum ^{N}_{i=1}\sum ^{N}_{j=1}x_i x_j b_{ij}. \end{aligned}$$

(9)

For the calculation of the cross-interaction parameters between the components i and j, the modified Lorentz-Berthelot combination rules were used, i.e.

$$\begin{aligned} a_{ij}(T) = \xi _{ij} \sqrt{a_{ii}(T) a_{jj}(T)}, \end{aligned}$$

(10)

and

$$\begin{aligned} b_{ij} = \frac{1}{2}({b_{ii}+b_{jj})}, \end{aligned}$$

(11)

where $\xi _{ij}$ is a binary interaction parameter.

2.2.3 Molecular-based EOS

Six molecular-based EOS were used in this work: PC-SAFT [18], PCP-SAFT [18, 22], SAFT-VR Mie [21], polar soft-SAFT [23, 24], BACKONE [25, 26], and sCPA [27, 28] EOS. In these EOS the different molecular features are represented by independent terms in the Helmholtz energy. The configurational Helmholtz energy is formulated as

$$\begin{aligned} {\tilde{a}}_\text {config} = {\tilde{a}}_\text {rep} + {\tilde{a}}_\text {disp} + {\tilde{a}}_\text {chain} + {\tilde{a}}_\text {polar} + {\tilde{a}}_\text {assoc}, \end{aligned}$$

(12)

with the repulsion ${\tilde{a}}_\text {rep}$, dispersion ${\tilde{a}}_\text {disp}$, chain ${\tilde{a}}_\text {chain}$, polar ${\tilde{a}}_\text {polar}$, and association ${\tilde{a}}_\text {assoc}$ contribution. Table 5 gives an overview of the terms used in the different molecular-based EOS. For $\text {O}_{2}$, the basic modeling approach was the same for all EOS, i.e. dispersive and repulsive interactions as well as the molecule elongation are modeled. For $\text {CO}_{2}$, in addition to these terms, in some cases, the polarity of the molecule was explicitly described by an additional term. Different approaches are used for this in the EOS shown in Table 5: a polar term is used for the PCP-SAFT, polar soft-SAFT and BACKONE and an association term in the sCPA.

Table 5 Overview of the configurational Helmholtz energy terms used in the modeling of $\text {CO}_{2}$ and $\text {O}_{2}$ for each of the studied molecular-based EOS models

The pure component models used for the PC-SAFT [18], PCP-SAFT [18, 22], SAFT-VR Mie [21], and polar soft-SAFT [23, 24] have (at least) three parameters: the size parameter $\sigma$, dispersion energy parameter $\varepsilon$, and the chain length parameter m. The SAFT-VR Mie [21] models have an additional adjustable parameter $\lambda _r$, which indicates the repulsion exponent for the Mie potential [51]. Moreover, the quadrupole terms of the PCP-SAFT and polar soft-SAFT have the quadrupole moment as an additional parameter. The quadrupole term used in the polar soft-SAFT EOS has an additional parameter xp, which describes the number of quadrupole elements in the molecule (modeling the delocalization of the quadrupole). The BACKONE [25, 26] EOS incorporates the parameter $\alpha$ (instead of the chain length parameter m) modeling the elongation of the molecule. The sCPA [27, 28] EOS combines the Helmholtz energy term from the SRK EOS ${\tilde{a}}^\text {SRK}$ for modeling dispersive and repulsive interactions with an association term ${\tilde{a}}_\text {assoc}$. Since $\text {CO}_{2}$ is modeled with the association term, it has two additional parameters: the association energy parameter ${{\hat{\varepsilon }}}$ and the association volume parameter ${{\hat{\beta }}}$. Details and the numeric values of the parameters of the pure component models, that were taken from the literature [22, 24, 25, 36, 38,39,40,41], are given in the Supplementary Information.

The mixing rules used in the molecular-based EOS were directly adopted from the original works [18, 21,22,23,24,25,26,27,28]. In the sCPA EOS, the cross-interaction parameters between the components were computed according to Eqs. 8–11 (see above). For the PC-SAFT, the PCP-SAFT, polar soft-SAFT, and the BACKONE EOS, the modified Lorentz-Berthelot combination rules were used, i.e.

$$\begin{aligned} \varepsilon _{ij} = \xi _{ij} \sqrt{\varepsilon _{ii} \varepsilon _{jj}}, \end{aligned}$$

(13)

$$\begin{aligned} \sigma _{ij} = \frac{1}{2} ({\sigma _{ii} + \sigma _{jj}}). \end{aligned}$$

(14)

In the SAFT-VR Mie EOS, the energy cross-interaction parameter was computed with the combination rule

$$\begin{aligned} \varepsilon _{ij} = \xi _{ij} \frac{\sqrt{\sigma _{ii}^3 \sigma _{jj}^3}}{\sigma _{ij}^3} \sqrt{\varepsilon _{ii} \varepsilon _{jj}}, \end{aligned}$$

(15)

while $\sigma _{ij}$ was calculated according to Eq. 14. Therein, the binary interaction parameter $\xi _{ij}$ is treated the same as the binary interaction parameter of the cubic EOS, cf. Eq. 10. For the chain term, polar term, and association term, no combination rule applies.

Table 6 Binary interaction parameters for the mixture $\text {CO}_{2}+\text {O}_{2}$ obtained from the regression to experimental data, cf. Eq. 16

Table 7 Binary interaction parameters for the mixture $\text {CO}_{2}+\text {O}_{2}$ obtained from the regression to experimental data for the GERG-2008 EOS, cf. Eqs. 4 and 5

2.2.4 Regression of Binary Interaction Parameters

First, all EOS were used in a purely predictive mode, i.e. no mixture parameters were adjusted. In the predictive mode, we have used the multi-parameter EOS with $\beta _{v,\text {CO}_{2},\text {O}_{2}}=\beta _{T, \text {CO}_{2},\text {O}_{2}}= \gamma _{v,\text {CO}_{2},\text {O}_{2}}=\gamma _{T,\text {CO}_{2},\text {O}_{2}}=1$, cf. Eqs. 4 and 5. The cubic and molecular-based EOS were used in the predictive mode with $\xi _{\text {CO}_{2},\text {O}_{2}}=1$, cf. Eqs. 10, 13, and 15. For the multi-parameter EOS, in the predictive mode, the Lorentz–Berthelot combination rules were applied. For the GERG-2008 EOS, the predictive mode corresponds to the EOS model as originally proposed by Kunz and Wagner [20], which was not adjusted to $\text {CO}_{2}+\text {O}_{2}$ mixture data. For the EOS-CG, the predictive mode reflects a simplified version of the original EOS-CG model that was fitted to $\text {CO}_{2}+\text {O}_2$ mixture data by Gernert and Span [29].

Additionally, the binary interaction parameters were regressed to the training set of the mixture data in an adjusted mode. The obtained models were compared to the predictive mode models. For the cubic and molecular-based EOS, the cross-interaction energy parameter $\xi _{\text {CO}_{2},\text {O}_{2}}$ was adjusted. Therefore, a linear temperature-dependent function was used

$$\begin{aligned} \xi _{\text {CO}_{2},\text {O}_{2}} = o_1 + {o_2}\cdot (T/\text {K}), \end{aligned}$$

(16)

where $o_1$ and $o_2$ are the adjustable parameters. For the GERG-2008 EOS, the four binary interaction parameters ${\overline{o}}=\beta _{v,\text {CO}_{2},\text {O}_{2}},\,\beta _{T,\text {CO}_{2},\text {O}_{2}},\,\gamma _{v,\text {CO}_{2},\text {O}_{2}},\,\gamma _{T,\text {CO}_{2},\text {O}_{2}}$ were adjusted. For the EOS-CG, the binary interaction parameters $\gamma _{v,\text {CO}_{2},\text {O}_{2}},\,\gamma _{T,\text {CO}_{2},\text {O}_{2}}$ were adapted for the adjusted mode from the original work by Gernert and Span [29] with $\beta _{v,\text {CO}_{2},\text {O}_{2}},\,\beta _{T,\text {CO}_{2},\text {O}_{2}}=1$. Hence, for the cubic, molecular-based EOS, and the EOS-CG two adjustable parameters were used, whereas four adjustable parameters were used for the GERG-2008 EOS. This follows the usual way these model classes are used for mixture modeling [20, 29, 49, 52,53,54,55].

The binary interaction parameters ${\overline{o}}^k$ for the EOS k were obtained by minimizing a least-squares objective function F given by

$$\begin{aligned} \min _{{\bar{o}^{k} }}F (\bar{o}^{k} ) & = \frac{1}{{N_{\rho } }}\sum\limits_{{t = 1}}^{{N_{\rho } }} {\left[ {\frac{{\rho _{t}^{{{\text{EOS}}}} - \rho _{t}^{{{\text{Ref}}}} }}{{\rho _{t}^{{{\text{Ref}}}} }}} \right]} ^{2} + \frac{1}{{100 \cdot N_{\beta } }}\sum\limits_{{t = 1}}^{{N_{\beta } }} {\left[ {\frac{{\beta _{t}^{{{\text{EOS}}}} - \beta _{t}^{{{\text{Ref}}}} }}{{\beta _{t}^{{{\text{Ref}}}} }}} \right]} ^{2} \\ & \quad + \frac{1}{{10 \cdot N_{w} }}\sum\limits_{{t = 1}}^{{N_{w} }} {\left[ {\frac{{w_{t}^{{{\text{EOS}}}} - w_{t}^{{{\text{Ref}}}} }}{{w_{t}^{{{\text{Ref}}}} }}} \right]} ^{2} + \frac{{0.5}}{{N_{{x_{{{\text{O}}_{{\text{2}}} }}^{\prime } }} }}\sum\limits_{{t = 1}}^{{N_{{x_{{{\text{CO}}_{{\text{2}}} }}^{\prime } }} }} {\left[ {x_{{{\text{O}}_{{\text{2}}} ,t}}^{{\prime {\text{,EOS}}}} - x_{{{\text{O}}_{{\text{2}}} ,t}}^{{\prime {\text{,Ref}}}} } \right]} ^{2} \\ & \quad + \frac{{0.5}}{{N_{{x_{{{\text{O}}_{{\text{2}}} }}^{{\prime \prime}} }} }}\sum\limits_{{t = 1}}^{{N_{{x_{{{\text{O}}_{{\text{2}}} }}^{{\prime \prime} } }} }} {\left[ {x_{{{\text{O}}_{{\text{2}}} ,t}}^{{{\prime \prime}{\text{,EOS}}}} - x_{{{\text{O}}_{{\text{2}}} ,t}}^{{{\prime \prime} {\text{,Ref}}}} } \right]} ^{2} \\ \end{aligned}$$

(17)

where the superscript Ref indicates the experimental data points and the superscript EOS indicates the value computed by a given EOS and $N_l$ indicates the number of data points for the properties $l = \rho ,\,\beta ,\,w,\,x^{\prime}_{\text {O}{_{2}}},\,\text{and}\,x^{\prime \prime}_{\text {O}{_{2}}}$. Only experimental data from the training set were considered in the fitting procedure, cf. Fig. 1. The weights used in the fit, cf. Eq. 17, were determined in an iterative approach such that a good compromise between the individual objectives was obtained. The isothermal compressibility data and speed of sound data were weighted considerably less (1/100 and 1/10 times less, respectively) compared to the homogeneous state density data since they were only available for a single composition of the mixture ($x_{\text {O}{_{2}}} = 0.0652$ mol$\cdot$mol⁻¹).

The adjusted parameters $o_1$ and $o_2$ for the cubic and molecular-based EOS are given in Table 6. The adjusted binary interaction parameters for the GERG-2008 [20] EOS $\beta _{v,\text {CO}_{2},\text {O}_{2}},\,\beta _{T,\text {CO}_{2},\text {O}_{2}},\,\gamma _{v,\text {CO}_{2},\text {O}_{2}},\,\gamma _{T,\text {CO}_{2},\text {O}_{2}}$ are given in Table 7. Note, that the mixture parameters for the EOS-CG as proposed in Ref. [29] were obtained by regression to a wider temperature range compared to the training dataset used in this work.

Table 8 Overview of the quality of the description of experimental VLE data for the system $\text {CO}_2+\text {O}_2$ by different EOS. Results from predictions based on pure component models are compared to results that were obtained after adjusting binary parameters (given in brackets) to the training data. Results are reported for each EOS from top to bottom for all data, the training data, and the test data. For details, see text

Fig. 2

Phase envelope of the system $\text {CO}_2+\text {O}_2$ calculated by different EOS (lines) compared to literature data (symbols) [4, 6, 61, 63,64,65,66]. Results from predictions based on pure component models (top) are compared to results that were obtained after adjusting binary parameters (bottom) to the training data. The GERG-2008 and EOS-CG results collapse to one line in the top figures (predictive mode). For details, see text (Color figure online)

2.2.5 Assessment of the EOS

The performance of the different EOS used in both the predictive and adjusted mode was evaluated in a systematic way. For all EOS, the performance for the homogeneous state properties and the phase equilibrium properties was evaluated. Therefore, mean deviations from a given model and the experimental data were computed. For the homogeneous state properties, the mean absolute relative deviation ${\delta _{lm}^{\text {rel},k}}$ was calculated for each of the studied EOS k for the properties $l=\rho ,\,\beta ,\, \text {and}\,w$ for the fluid region $m=\text {vapor},\,\text {liquid},\,\text {and supercritical}$ by

$$\begin{aligned}{\delta _{lm}^{\text{rel},k}} = \frac{1}{N_{lm}} \mathop{\sum}\limits_{t=1}^{N_{lm}}\left|100\,\% \frac{{Y_{lmt}^\text {EOS}} - {Y_{lmt}^\text {Ref}}}{{Y_{lmt}^\text {Ref}}}\right| , \end{aligned}$$

(18)

where $Y^\text {Ref}$ indicates the value of the experimental data point taken from the literature. For the homogeneous state point properties, the assignment of the data points from the literature to the fluid regions was carried out by estimating the critical temperatures depending on the composition from the experimental phase boundary data (see electronic Supplementary Information). The mean absolute relative deviation ${\delta _{lm}^{\text {rel},k}}$ is therefore a measure for the performance of the EOS k for the property l in the fluid region m.

The performance of the EOS models for describing the vapor–liquid equilibrium phase boundary was evaluated by the mean absolute deviation ${\delta _{n}^{\text {abs},k}}$ for phase composition for a given temperature and pressure defined as

$${\delta_{n}^{{\rm abs},k}} = \frac{1}{N_n} \sum\limits_{t=1}^{N_n}|{x^{\text{EOS}}_{\text{O}_{2},nt}} -{{x^{\rm Ref}_{\text{O}_{2},nt}}}|,$$

(19)

where n represents the liquid $(n=\,\text {liq})$ and vapor $(n=\,\text {vap})$ side of the phase boundary.

Both mean deviations ${\delta _{lm}^{\text {rel},k}}$ and ${\delta _{n}^{\text {abs},k}}$ were calculated for three cases: (i) for all data points available for the mixture $\text {CO}_2+\text {O}_2$, (ii) for the data points considered for the mixture parametrization, i.e. the training data (cf. Fig. 1), and (iii) the test data.

3 Results and Discussion

3.1 Vapor–Liquid Equilibrium

The results for the mean absolute deviation in the liquid and vapor phase mole fraction ${\delta _{\text {liq}}^{\text {abs}}}$ and ${\delta _{\text {vap}}^{\text {abs}}}$, respectively, are given in Table 8. Additionally, Fig. 2 shows the phase envelopes obtained from the EOS in comparison with experimental data for three different temperatures. Both, Table 8 and Fig. 2 show the results for the predictive mode and the adjusted mode.

The critical pressure obtained from the predictive calculations by most studied EOS significantly overestimates the highest pressures reported by the experimental data for the studied temperatures, cf. Fig. 2. The only exceptions are the PR and SRK EOS, which in return yield high deviations for the liquid phase composition, cf. Table 8. A good prediction of the studied phase envelopes (top row in Fig. 2) is obtained for the SAFT-VR Mie and the PCP-SAFT EOS, which is also reflected in the mean deviations given in Table 8. The inclusion of the quadrupole in the model of $\text {CO}_2$ of the PCP-SAFT EOS compared to the PC-SAFT model reduces the deviations in the liquid phase composition and overall improves the description of the phase boundary for the temperatures $T\,/\,\text {K}= 288.15,\, 298.15$, cf. Fig. 2. The multi-parameter GERG-2008 EOS yields high deviations for the liquid side composition ${\delta _{\text {liq}}^{\text {abs}}}$ and also significantly overestimates the critical point. The EOS-CG yields almost identical results for the predictive calculations as the GERG-2008 EOS. This is interesting, since different pure component models are used in the GERG-2008 EOS and EOS-CG.

For the adjusted mode, the performance of all models significantly improves for the training dataset—as expected. In particular, the results obtained from the adjusted GERG-2008 EOS are in very good agreement with the experimental data. One reason for this may be that four parameters were used for the adjustment. The PR EOS also yields significantly lower deviations compared to the predictive mode and is on a par with the adjusted GERG-2008 EOS with only two parameters. For the liquid phase concentration deviations, the EOS-CG and the adjusted GERG-2008 EOS yield practically identical results (especially for the training dataset). On the contrary, for the vapor phase, the adjusted GERG-2008 EOS yields significantly lower concentration deviations compared to the EOS-CG, cf. Table 8. For the molecular-based EOS, the SAFT-VR Mie and the PCP-SAFT EOS yield a good description of the phase envelope in the adjusted mode (only two adjusted parameters). Both EOS overestimate the critical point, but the SAFT-VR Mie EOS overestimates the critical point stronger compared to the PCP-SAFT EOS, cf. Fig. 2. A more detailed discussion on the overestimation of the critical point by the molecular-based EOS is given in the Supplementary Information. Interestingly, the description of the phase envelope by the polar soft-SAFT, BACKONE, and sCPA show almost no improvement by the adjustment of the binary interaction parameters. For the sCPA, this could be the result of modeling $\text {CO}_2$ as an associating fluid. The association contribution is not influenced by the binary interaction parameter since only $\text {CO}_2$ exhibits associating sides in the mixture.

The models (both predictive mode and adjusted mode) were also tested for describing the phase behavior at low temperatures, i.e. for the test dataset. The parameter adjustment also improves the performance of the PR, SRK, and PC-SAFT for the test data liquid phase composition significantly. On the other hand, the mean deviations for the vapor phase composition ${\delta _{\text {vap}}^{\text {abs}}}$ increase only slightly. The temperature-dependent binary interaction parameter also results in an improvement in the description of the phase envelope at lower temperatures for the more complex models PCP-SAFT and SAFT-VR Mie. For the adjusted GERG-2008 EOS, some deviations occur for the phase equilibria at $T \lessapprox 240\, \text {K}$. This is not surprising considering the empirical character of the GERG-2008 model type. The EOS-CG on the other hand, yields low deviations even for $T \lessapprox 240\, \text {K}$, which is probably due to the fact that the mixture model parameters were adjusted to a wider temperature range [29]. Adjusting the mixture parameters of the GERG-2008 model to a wider temperature range most likely would improve the model performance.

3.2 Homogeneous State Properties

The mean average relative deviation ${\delta ^{\text {rel}}}$ results for homogeneous state properties are reported in Table 9. Additionally, deviation plots for the homogeneous state densities are given in the Supplementary Information. In the predictive mode, i.e. not using adjusted mixture parameters, the GERG-2008 EOS and (simplified) EOS-CG yield overall the best results, especially for the speed of sound and the liquid density. The molecular-based PCP-SAFT and SAFT-VR Mie EOS also yield low deviations for the density. For most molecular-based and both cubic EOS, relatively large deviations are obtained for the 2nd-order derivative properties (isothermal compressibility $\beta$ and speed of sound w). This is in line with results reported in the literature [56,57,58,59] and probably due to the fact that the pure component models were in most cases not adjusted to homogeneous state property data. The isothermal compressibility $\beta$ is predicted with mean deviations of approximately $5\%$ by the PC-SAFT EOS. Interestingly, the PCP-SAFT EOS, which uses additionally a quadrupole contribution for $\text {CO}_2$, yields lower deviations for the density, but almost equivalent mean deviations for the 2nd-order derivative properties (isothermal compressibility $\beta$ and speed of sound w) compared to the PC-SAFT EOS. This is likely a result of different fitting strategies used for the development of the pure component models [22, 60]. A detailed discussion is given in the Supplementary Information. The speed of sound w is the only studied quantity, where the rotational and vibrational degrees of freedom, included in the ideal gas term ${\tilde{a}}_{\text {id}}$, cf. Eq. 1, influence the results. The GERG-2008 EOS yields the lowest deviations $\le\, 0.5\%$ followed by the (simplified) EOS-CG with deviations $\le\, 0.7\%$. From the remaining studied EOS, the PC-SAFT EOS yields the lowest deviations for the speed of sound $\le\, 5\%.$

Table 9 Overview of the quality of the description of experimental homogeneous state data for the system $\text {CO}_2+\text {O}_2$ by different EOS

The results obtained from the adjusted models provide further interesting insights. While in most cases, significant improvements were observed for the modeling of the VLE phase behavior by adjusting binary interaction parameters (cf. Table 8 and Fig. 2), a more ambivalent picture evolves for the homogeneous state properties (cf. Table 9). In some cases, significant improvements are observed, e.g. for the adjusted GERG-2008 EOS for modeling the isothermal compressibility β as well as for the PC-SAFT EOS for most properties. Yet, in several cases, the model accuracy slightly decreases using an adjusted binary mixture parameter. This is likely due to the fact, that the description of the phase equilibrium and homogeneous state properties are conflicting objectives. Accordingly, improvements in the description of the phase equilibrium are bought at the expense of the decreasing accuracy for the modeling of the homogeneous state properties. This is particularly prominent for liquid state densities and speed of sound described by the adjusted GERG-2008 EOS and for the 2nd-order derivatives β and w described by the PCP-SAFT EOS. The EOS-CG yields slightly higher deviations for the liquid state densities and isothermal compressibility but lower deviations for the vapor state densities compared to the corresponding predictive mode results. For other properties and EOS, this conflicting objectives effect is also present, but less prominent. Overall, the EOS-CG and adjusted GERG-2008 yield the lowest deviations in the adjusted mode for homogeneous state properties. Also, the adjusted PCP-SAFT and PC-SAFT EOS yield a reasonably accurate description of the homogeneous state properties—in many cases well competitive to the multi-parameter EOS.

The deviations for the test data are in most cases lower compared to the deviations for the training data, cf. Table 9. This is likely due to the fact, that the data in the vicinity of the critical point of $\text{CO}_2$ is only included in the training data. Modeling this data is a challenging task for EOS and therefore yields larger deviations compared to the remaining data.

In most cases, the adjusted mode results for the test data show the same trend compared to the adjusted mode results for the training data. A lower deviation ${\delta^{\text {rel}}}$ for a given property in the training dataset leads to a lower deviation for this property in the test dataset, when compared to the predictive mode results. The adjusted parameters can, therefore, also be used to extrapolate to lower (or higher) temperatures and pressures for all studied EOS. Yet, these extrapolations should be done with caution.

4 Conclusions

Accurate models for the thermodynamic properties of the mixture $\text {CO}_2+\text {O}_2$ are crucial for designing Allam cycle processes. In this work, different EOS were used for modeling the mixture $\text {CO}_2+\text {O}_2$. First, the available literature data were critically reviewed and compiled in a database, which is provided in the electronic Supplementary Information. Two modeling approaches were used: (i) the EOS were used in a purely predictive mode and (ii) in an adjusted mode by parametrizing binary interaction parameters to experimental mixture data. In total, ten EOS were studied: two cubic [15, 16], six molecular-based [18, 21,22,23,24,25,26,27,28], and two multi-parameter EOS [20, 29].

Using adjusted mixture parameters does not automatically result in an improvement in the accuracy of all considered properties due to the conflicting objects during the fit, i.e. correct phase envelope and homogeneous state property description. The phase equilibrium predictions of the GERG-2008 EOS (i.e. no mixture parameters adjusted) show significant deviations to the experimental phase equilibrium data. The regression of the binary interaction parameter reduces the deviations of the compositions of the coexisting phases obtained from the GERG-2008 EOS drastically. Yet, this goes in hand with a decrease of the accuracy for some homogeneous state properties. Also, using the adjusted GERG-2008 model for extrapolation to lower temperatures results in significant deviations for the phase envelope.

The classical cubic PR EOS yields low deviations for the phase equilibrium predictions for the adjusted mode, but comparably large deviations for homogeneous state properties. The molecular-based EOS models, on the other hand, are an attractive alternative to the multi-parameter EOS. They provide robust extrapolation capabilities, but yield slightly less accurate description of the thermodynamic properties. In particular, the PCP-SAFT and SAFT-VR Mie models considered in this work yield a good performance. In both cases (cubic and molecular-based EOS), the extrapolation by the temperature-dependent binary interaction parameter results in lower deviations for the phase envelope at low temperatures ($T \lessapprox 240\, \text {K}$) compared to the GERG-2008 EOS. The EOS-CG (where the binary interaction parameters were regressed using data from a wider temperature range [29]) yields overall the lowest deviations for the phase envelopes at $T \lessapprox 240\, \text {K}$. In this work, reliable models for describing the mixture behavior $\text {CO}_2+\text {O}_2$ were developed and compared to the EOS-CG model that was adjusted to mixture data in the literature. In particular, the adjusted GERG-2008, the PCP-SAFT, and the SAFT-VR Mie EOS as well as the EOS-CG from the literature provide accurate models for the thermodynamic properties of the mixture $\text {CO}_2+\text {O}_2$. For future work, it would be interesting to apply these thermodynamic models for describing the actual Allam cycle in an exemplaric use case. Also, for a future work, testing the effects of more advanced mixing rules for the different EOS would be interesting.

The working fluid in the Allam cycle primarily consists of $\text {CO}_2+\text {O}_2$. Yet, to some extent also water, nitrogen, argon, and other residue components are present in the process [3]. Extending the models to include these components in the parametrization and studying the influence of the obtained models on the description of the Allam cycle would be interesting.