Equation of state and thermodynamic properties for mixtures of H2O, O2, N2, and CO2 from ambient up to 1000 K and 280 MPa

Graphical abstract

S. Supporting Information S1 VTBMSR-III 2 S1.1 Parameters ci, pi and k In the subsequent sections, all regressed parameter sets found by regression (see Section 2.3) for the volume translation parameters c i , the polar parameters p i and the binary interaction parameters k (z) b,ij are listed. For oxygen, the parameter set numbering is given by a two-digit number due to two different parameter sets for the volume translation parameters. The first digit indicates the parameter set for the volume translation, the second digit the one for the polar parameters (indicated with placeholder x). As an example parameter set 21 for oxygen: The volume translation parameter c i are taken from parameter set 2 (regarding volume translation) obtained by regression on density data of the liquid phase of Sychev and IUPAC [B, C]. The polar parameters p i are taken from parameter set 1 (regarding polarity) obtained by regression on heat capacity data of the vapor and supercritical phase of Sychev [B]. The parameter sets of water, nitrogen and carbon dioxide are given by one digit since only one reference data set was applied for determination of the volume translation parameters c i . Therefore, the digit has only to indicate the parameter set for the polar parameters p i . S1.1.1. Volume Translation Parameters c i Table S1.1: Volume translation parameters for water, oxygen, nitrogen, and carbon dioxide. To avoid rounding errors the values are listed with full digits despite limited significance. In the column "Regressed on" the considered phase is specified along with the property: l (liquid), v (vapor), sc (supercritical). References see Section S4.
The relation for the ideal enthalpy is independently of the EOS and given in Pilz [11]: the one for the residual enthalpy in Lieball [17]: The ideal specific heat capacity at constant pressure (c 0 p ) is determined based on a polynomial approach [35].
The parameters α, β, γ, δ and ε for the particular substances are listed in Table S1.7 for completeness. α β · 10 3 γ · 10 6 δ · 10 9 ε · 10 12 Heat Capacity at Constant Pressure. The specific heat capacity at constant pressure (c p ) is defined as [35]: Since the equation of state is pressure-explicit the definition of the heat capacity at constant pressure cannot be used directly. Instead, it can be related to the heat capacity at constant volume by the subsequent relation [36]. Fugacity.
The fugacity (f ) can be described as the product of the pressure and the fugacity coefficient: where the fugacity coefficient is calculated by [36]: (S1.9)

S1.2.2. Volume Translation Corrected Equation for Fugacity by Kutney
Kutney [18] proposed corrected equations for derived thermodynamic properties when a volume translation is used.

S1.3.3. Discussion of Derivatives of α and c
In the result section, it is mentioned that the second derivative of parameter α causes discontinuities in the calculation of the heat capacities c v and c p . In Figure S1.1, the temperature dependencies of α, c and their derivatives are illustrated within the range of an SCWO process (200 -800 K). The limit values of the α-function and its derivatives at T r = 1 are determined in Section S3.5.5.
Figures S1.1a and S1.1b show that the parameters α and c are continuous in the considered temperature range for all of the four substances water, oxygen, nitrogen, and carbon dioxide. The same applies to the first derivatives of both parameters, see Figure S1.1c and S1.1d. But the second derivatives are no longer continuous for all of the four substances, see Figure S1.1e and S1.1f. The second derivatives of water and carbon dioxide are disrupt at their critical temperatures, T c,H2O = 647.14 K and T c,CO2 = 304.12 K, respectively. Since the critical temperatures of oxygen and nitrogen, T c,O2 = 154.58 K and T c,N2 = 126.20 K, are lower than the considered temperature range, the second derivatives of α for oxygen and nitrogen are continuous. Compared to the second derivative of α, only the second derivative of c of water is discontinuous. Since the parameters c 1 and c 2 are 0 for oxygen, nitrogen, and carbon dioxide, also c of carbon dioxide is continuous. Therefore, it is identifiable that the second derivative of parameter α causes the discontinuities in the heat capacities.
This discontinuity of the second derivative of the parameter α is a known issue. Le Guennec et al.
[37] have developed requirements for the α function so that vapor-liquid equilibrium and derived thermodynamic properties can be properly predicted at all temperatures. No common α-function fulfills the proposed constraints without any restrictions on the adjustable parameters.
As our proposed combining rule does not take the therm l ij term into account, Equation (S1.49) becomes For a two compound system For a three compound system Pseudo three compound system using Eq. (S1.59): with a 3 = a 2 , k a,33 = k a,22 , k a,13 = k a,12 Therefore, when a 3 = a 2 , k a,33 = k a,22 and k a,13 = k a,12 , Equation (S1.65) corresponds to the equation for a two compound system (Eq. (S1.54)) and the Michelsen-Kistenmacher syndrome is not relevant for our calculations. The mentioned criteria are fulfilled when splitting a substance since the same parameter a and interaction parameter coefficients k (z) a,ij are applied at the same temperature.
Mixing and combining rule for For a two compound system For a three compound system (S1.79) Pseudo three compound system using Eq. (S1.79): Therefore, when b 3 = b 2 , k b,33 = k b,22 and k b,13 = k b,12 , Equation (S1.85) corresponds to the equation for a two compound system (Eq. (S1.73)) and the Michelsen-Kistenmacher syndrome is not relevant for our calculations. The mentioned criteria are fulfilled when splitting a substance since the same parameter b and interaction parameter coefficients k (z) b,ij are applied at the same temperature.

S1.4.2. Dilution Effect
Michelsen and Kistenmacher [28] point to the dilution effect caused by the l ij term. The l ij term is calculated in the combining rule in a double summation but is involved in a product of three molar fractions. Since the l ij term is omitted in our combining rule, the calculations are not affected by the dilution effect.

S1.4.3. Pseudo Quaternary System of Water
To investigate a potential Michelsen-Kistenmacher syndrome, the one compound system of water vapor is recalculated with the EOS for a quaternary system varying the molar fractions and compared to the results of the EOS for a single compound system.
In all variations of calculations the same results for molar volume, enthalpy, and the heat capacities are calculated. The results are shown in a summarized form in Table S1.8 where the averaged relatives errors are listed. The equal values for the particular properties represent that the underlying results are equal otherwise the averaged absolute relative errors would vary. Therefore, no Michelsen-Kistenmacher syndrom is observable. Further, a potential Michelsen-Kistenmacher syndrome is tested with the binary mixture H 2 O/N 2 . The binary system is extended to a ternary and quaternary system the molar fractions of water and nitrogen.
In all variations of calculations the same results for molar volume, enthalpy, and the heat capacities are calculated. The results are shown in a summarized form in Table S1.9 where the averaged relatives errors are listed. The equal values for the particular properties represent that the underlying results are equal otherwise the averaged absolute relative errors would vary. Therefore, no Michelsen-Kistenmacher syndrom is observable. 1 0 1 0 6.6 ± 0.6 0.5 0.5 1 0 6.6 ± 0.6 1 0 0.5 0.5 6.6 ± 0.6 0.5 0.5 0.5 0.5 6.6 ± 0.6 Remark: The pseudo molar fractionx i indicates the fraction of the particular molar fraction x i in the mixture.

S2.1. Redlich-Kwong
Ideal enthalpy Heat capacity at constant volume Heat capacity at constant pressure Residual enthalpy Ideal enthalpy Heat capacity at constant volume Heat capacity at constant pressure

S2.3. Peng-Robinson
Residual enthalpy Heat capacity at constant volume Heat capacity at constant pressure

S2.4. VTBMSR-I Equation of state
Ideal enthalpy Heat capacity at constant volume Heat capacity at constant pressure

S2.5. VTBMSR-II Equation of state
Ideal enthalpy Heat capacity at constant volume Heat capacity at constant pressure Fugacity coefficient S2.5.1. Parameter α and its Derivatives α

S3.5. Derivatives of Different α-functions
The first and second derivatives of parameter α with respect to temperature are not generally continuous when α is given by temperature-dependent determination. This causes discontinuities in derived thermodynamic properties (e.g. c v and c p ) that includes these derivatives, see also S1.3.3. In the following sections parameter α and its derivatives are evaluated at the limit temperature T r = 1 for both cases (T r ≤ 1 and T r > 1) for the different α functions given by Boston and Mathias [13], Mathias [14], VTBMSR-I [26], -II [11] and -III, see Table 1. Further, the profiles of the parameters are presented graphically (Supporting Information. S3.5.7).
Equation (S3.206) and Equation (S3.207) yield a priori not the same result indicating that alpha(T) is not obliged to be continuous.
Equation (S3.228) and Equation (S3.229) yield a priori not the same result indicating that alpha(T) is not obliged to be continuous.

S3.5.5. VTBMSR-III
• Left-hand limit for T r = 1 lim Tr 1 α = 1 (S3.233) • Right-hand limit for T r = 1 lim • Right-hand limit for T r = 1 lim Equation (S3.239) and Equation (S3.240) yield a priori not the same result indicating that alpha(T) is not obliged to be continuous.   Figure S3.5: Parameter α(Tr) and its first and second derivatives with respect to the reduced temperature Tr for water in liquid and vapor phase.