Cubic Equations of State (thermo.eos)

This module contains implementations of most cubic equations of state for pure components. This includes Peng-Robinson, SRK, Van der Waals, PRSV, TWU and many other variants.

For reporting bugs, adding feature requests, or submitting pull requests, please use the GitHub issue tracker.

Base Class

class thermo.eos.GCEOS

Bases: object

Class for solving a generic Pressure-explicit three-parameter cubic equation of state. Does not implement any parameters itself; must be subclassed by an equation of state class which uses it. Works for mixtures or pure species for all properties except fugacity. All properties are derived with the CAS SymPy, not relying on any derivations previously published.

$$P=\frac{RT}{V-b}-\frac{a\alpha(T)}{V^2 + \delta V + \epsilon}$$

The main methods (in order they are called) are GCEOS.solve, GCEOS.set_from_PT, GCEOS.volume_solutions, and GCEOS.set_properties_from_solution.

GCEOS.solve calls GCEOS.check_sufficient_inputs, which checks if two of T, P, and V were set. It then solves for the remaining variable. If T is missing, method GCEOS.solve_T is used; it is parameter specific, and so must be implemented in each specific EOS. If P is missing, it is directly calculated. If V is missing, it is calculated with the method GCEOS.volume_solutions. At this point, either three possible volumes or one user specified volume are known. The value of a_alpha, and its first and second temperature derivative are calculated with the EOS-specific method GCEOS.a_alpha_and_derivatives.

If V is not provided, GCEOS.volume_solutions calculates the three possible molar volumes which are solutions to the EOS; in the single-phase region, only one solution is real and correct. In the two-phase region, all volumes are real, but only the largest and smallest solution are physically meaningful, with the largest being that of the gas and the smallest that of the liquid.

GCEOS.set_from_PT is called to sort out the possible molar volumes. For the case of a user-specified V, the possibility of there existing another solution is ignored for speed. If there is only one real volume, the method GCEOS.set_properties_from_solution is called with it. If there are two real volumes, GCEOS.set_properties_from_solution is called once with each volume. The phase is returned by GCEOS.set_properties_from_solution, and the volumes is set to either GCEOS.V_l or GCEOS.V_g as appropriate.

GCEOS.set_properties_from_solution is a large function which calculates all relevant partial derivatives and properties of the EOS. 17 derivatives and excess enthalpy and entropy are calculated first. Finally, it sets all these properties as attibutes for either the liquid or gas phase with the convention of adding on _l or _g to the variable names, respectively.

Attributes

Tfloat

Temperature of cubic EOS state, [K]

Pfloat

Pressure of cubic EOS state, [Pa]

afloat

a parameter of cubic EOS; formulas vary with the EOS, [Pa*m^6/mol^2]

bfloat

b parameter of cubic EOS; formulas vary with the EOS, [m^3/mol]

deltafloat

Coefficient calculated by EOS-specific method, [m^3/mol]

epsilonfloat

Coefficient calculated by EOS-specific method, [m^6/mol^2]

a_alphafloat

Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]

da_alpha_dTfloat

Temperature derivative of $a \alpha$ calculated by EOS-specific method, [J^2/mol^2/Pa/K]

d2a_alpha_dT2float

Second temperature derivative of $a \alpha$ calculated by EOS-specific method, [J^2/mol^2/Pa/K**2]

Zcfloat

Critical compressibility of cubic EOS state, [-]

phasestr

One of ‘l’, ‘g’, or ‘l/g’ to represent whether or not there is a liquid-like solution, vapor-like solution, or both available, [-]

raw_volumeslist[(float, complex), 3]

Calculated molar volumes from the volume solver; depending on the state and selected volume solver, imaginary volumes may be represented by 0 or -1j to save the time of actually calculating them, [m^3/mol]

V_lfloat

Liquid phase molar volume, [m^3/mol]

V_gfloat

Vapor phase molar volume, [m^3/mol]

Vfloat or None

Molar volume specified as input; otherwise None, [m^3/mol]

Z_lfloat

Liquid phase compressibility, [-]

Z_gfloat

Vapor phase compressibility, [-]

PIP_lfloat

Liquid phase phase identification parameter, [-]

PIP_gfloat

Vapor phase phase identification parameter, [-]

dP_dT_lfloat

Liquid phase temperature derivative of pressure at constant volume, [Pa/K].

$$\left(\frac{\partial P}{\partial T}\right)_V = \frac{R}{V - b} - \frac{a \frac{d \alpha{\left (T \right )}}{d T}}{V^{2} + V \delta + \epsilon}$$

dP_dT_gfloat

Vapor phase temperature derivative of pressure at constant volume, [Pa/K].

$$\left(\frac{\partial P}{\partial T}\right)_V = \frac{R}{V - b} - \frac{a \frac{d \alpha{\left (T \right )}}{d T}}{V^{2} + V \delta + \epsilon}$$

dP_dV_lfloat

Liquid phase volume derivative of pressure at constant temperature, [Pa*mol/m^3].

$$\left(\frac{\partial P}{\partial V}\right)_T = - \frac{R T}{\left( V - b\right)^{2}} - \frac{a \left(- 2 V - \delta\right) \alpha{ \left (T \right )}}{\left(V^{2} + V \delta + \epsilon\right)^{2}}$$

dP_dV_gfloat

Gas phase volume derivative of pressure at constant temperature, [Pa*mol/m^3].

$$\left(\frac{\partial P}{\partial V}\right)_T = - \frac{R T}{\left( V - b\right)^{2}} - \frac{a \left(- 2 V - \delta\right) \alpha{ \left (T \right )}}{\left(V^{2} + V \delta + \epsilon\right)^{2}}$$

dV_dT_lfloat

Liquid phase temperature derivative of volume at constant pressure, [m^3/(mol*K)].

$$\left(\frac{\partial V}{\partial T}\right)_P =-\frac{ \left(\frac{\partial P}{\partial T}\right)_V}{ \left(\frac{\partial P}{\partial V}\right)_T}$$

dV_dT_gfloat

Gas phase temperature derivative of volume at constant pressure, [m^3/(mol*K)].

$$\left(\frac{\partial V}{\partial T}\right)_P =-\frac{ \left(\frac{\partial P}{\partial T}\right)_V}{ \left(\frac{\partial P}{\partial V}\right)_T}$$

dV_dP_lfloat

Liquid phase pressure derivative of volume at constant temperature, [m^3/(mol*Pa)].

$$\left(\frac{\partial V}{\partial P}\right)_T =-\frac{ \left(\frac{\partial V}{\partial T}\right)_P}{ \left(\frac{\partial P}{\partial T}\right)_V}$$

dV_dP_gfloat

Gas phase pressure derivative of volume at constant temperature, [m^3/(mol*Pa)].

$$\left(\frac{\partial V}{\partial P}\right)_T =-\frac{ \left(\frac{\partial V}{\partial T}\right)_P}{ \left(\frac{\partial P}{\partial T}\right)_V}$$

dT_dV_lfloat

Liquid phase volume derivative of temperature at constant pressure, [K*mol/m^3].

$$\left(\frac{\partial T}{\partial V}\right)_P = \frac{1} {\left(\frac{\partial V}{\partial T}\right)_P}$$

dT_dV_gfloat

Gas phase volume derivative of temperature at constant pressure, [K*mol/m^3]. See GCEOS.set_properties_from_solution for the formula.

dT_dP_lfloat

Liquid phase pressure derivative of temperature at constant volume, [K/Pa].

$$\left(\frac{\partial T}{\partial P}\right)_V = \frac{1} {\left(\frac{\partial P}{\partial T}\right)_V}$$

dT_dP_gfloat

Gas phase pressure derivative of temperature at constant volume, [K/Pa].

$$\left(\frac{\partial T}{\partial P}\right)_V = \frac{1} {\left(\frac{\partial P}{\partial T}\right)_V}$$

d2P_dT2_lfloat

Liquid phase second derivative of pressure with respect to temperature at constant volume, [Pa/K^2].

$$\left(\frac{\partial^2 P}{\partial T^2}\right)_V = - \frac{a \frac{d^{2} \alpha{\left (T \right )}}{d T^{2}}}{V^{2} + V \delta + \epsilon}$$

d2P_dT2_gfloat

Gas phase second derivative of pressure with respect to temperature at constant volume, [Pa/K^2].

$$\left(\frac{\partial^2 P}{\partial T^2}\right)_V = - \frac{a \frac{d^{2} \alpha{\left (T \right )}}{d T^{2}}}{V^{2} + V \delta + \epsilon}$$

d2P_dV2_lfloat

Liquid phase second derivative of pressure with respect to volume at constant temperature, [Pa*mol^2/m^6].

$$\left(\frac{\partial^2 P}{\partial V^2}\right)_T = 2 \left(\frac{ R T}{\left(V - b\right)^{3}} - \frac{a \left(2 V + \delta\right)^{ 2} \alpha{\left (T \right )}}{\left(V^{2} + V \delta + \epsilon \right)^{3}} + \frac{a \alpha{\left (T \right )}}{\left(V^{2} + V \delta + \epsilon\right)^{2}}\right)$$

d2P_dTdV_lfloat

Liquid phase second derivative of pressure with respect to volume and then temperature, [Pa*mol/(K*m^3)].

$$\left(\frac{\partial^2 P}{\partial T \partial V}\right) = - \frac{ R}{\left(V - b\right)^{2}} + \frac{a \left(2 V + \delta\right) \frac{d \alpha{\left (T \right )}}{d T}}{\left(V^{2} + V \delta + \epsilon\right)^{2}}$$

d2P_dTdV_gfloat

Gas phase second derivative of pressure with respect to volume and then temperature, [Pa*mol/(K*m^3)].

$$\left(\frac{\partial^2 P}{\partial T \partial V}\right) = - \frac{ R}{\left(V - b\right)^{2}} + \frac{a \left(2 V + \delta\right) \frac{d \alpha{\left (T \right )}}{d T}}{\left(V^{2} + V \delta + \epsilon\right)^{2}}$$

H_dep_lfloat

Liquid phase departure enthalpy, [J/mol]. See GCEOS.set_properties_from_solution for the formula.

H_dep_gfloat

Gas phase departure enthalpy, [J/mol]. See GCEOS.set_properties_from_solution for the formula.

S_dep_lfloat

Liquid phase departure entropy, [J/(mol*K)]. See GCEOS.set_properties_from_solution for the formula.

S_dep_gfloat

Gas phase departure entropy, [J/(mol*K)]. See GCEOS.set_properties_from_solution for the formula.

G_dep_lfloat

Liquid phase departure Gibbs energy, [J/mol].

$$G_{dep} = H_{dep} - T S_{dep}$$

G_dep_gfloat

Gas phase departure Gibbs energy, [J/mol].

$$G_{dep} = H_{dep} - T S_{dep}$$

Cp_dep_lfloat

Liquid phase departure heat capacity, [J/(mol*K)]

$$C_{p, dep} = (C_p-C_v)_{\text{from EOS}} + C_{v, dep} - R$$

Cp_dep_gfloat

Gas phase departure heat capacity, [J/(mol*K)]

$$C_{p, dep} = (C_p-C_v)_{\text{from EOS}} + C_{v, dep} - R$$

Cv_dep_lfloat

Liquid phase departure constant volume heat capacity, [J/(mol*K)]. See GCEOS.set_properties_from_solution for the formula.

Cv_dep_gfloat

Gas phase departure constant volume heat capacity, [J/(mol*K)]. See GCEOS.set_properties_from_solution for the formula.

c1float

Full value of the constant in the a parameter, set in some EOSs, [-]

c2float

Full value of the constant in the b parameter, set in some EOSs, [-]

A_dep_g

Departure molar Helmholtz energy from ideal gas behavior for the gas phase, [J/mol].

A_dep_l

Departure molar Helmholtz energy from ideal gas behavior for the liquid phase, [J/mol].

beta_g

Isobaric (constant-pressure) expansion coefficient for the gas phase, [1/K].

beta_l

Isobaric (constant-pressure) expansion coefficient for the liquid phase, [1/K].

Cp_minus_Cv_g

Cp - Cv for the gas phase, [J/mol/K].

Cp_minus_Cv_l

Cp - Cv for the liquid phase, [J/mol/K].

d2a_alpha_dTdP_g_V

Derivative of the temperature derivative of a_alpha with respect to pressure at constant volume (varying T) for the gas phase, [J^2/mol^2/Pa^2/K].

d2a_alpha_dTdP_l_V

Derivative of the temperature derivative of a_alpha with respect to pressure at constant volume (varying T) for the liquid phase, [J^2/mol^2/Pa^2/K].

d2H_dep_dT2_g

Second temperature derivative of departure enthalpy with respect to temperature for the gas phase, [(J/mol)/K^2].

d2H_dep_dT2_g_P

Second temperature derivative of departure enthalpy with respect to temperature for the gas phase, [(J/mol)/K^2].

d2H_dep_dT2_g_V

Second temperature derivative of departure enthalpy with respect to temperature at constant volume for the gas phase, [(J/mol)/K^2].

d2H_dep_dT2_l

Second temperature derivative of departure enthalpy with respect to temperature for the liquid phase, [(J/mol)/K^2].

d2H_dep_dT2_l_P

Second temperature derivative of departure enthalpy with respect to temperature for the liquid phase, [(J/mol)/K^2].

d2H_dep_dT2_l_V

Second temperature derivative of departure enthalpy with respect to temperature at constant volume for the liquid phase, [(J/mol)/K^2].

d2H_dep_dTdP_g

Temperature and pressure derivative of departure enthalpy at constant pressure then temperature for the gas phase, [(J/mol)/K/Pa].

d2H_dep_dTdP_l

Temperature and pressure derivative of departure enthalpy at constant pressure then temperature for the liquid phase, [(J/mol)/K/Pa].

d2P_drho2_g

Second derivative of pressure with respect to molar density for the gas phase, [Pa/(mol/m^3)^2].

d2P_drho2_l

Second derivative of pressure with respect to molar density for the liquid phase, [Pa/(mol/m^3)^2].

d2P_dT2_PV_g

Second derivative of pressure with respect to temperature twice, but with pressure held constant the first time and volume held constant the second time for the gas phase, [Pa/K^2].

d2P_dT2_PV_l

Second derivative of pressure with respect to temperature twice, but with pressure held constant the first time and volume held constant the second time for the liquid phase, [Pa/K^2].

d2P_dTdP_g

Second derivative of pressure with respect to temperature and, then pressure; and with volume held constant at first, then temperature, for the gas phase, [1/K].

d2P_dTdP_l

Second derivative of pressure with respect to temperature and, then pressure; and with volume held constant at first, then temperature, for the liquid phase, [1/K].

d2P_dTdrho_g

Derivative of pressure with respect to molar density, and temperature for the gas phase, [Pa/(K*mol/m^3)].

d2P_dTdrho_l

Derivative of pressure with respect to molar density, and temperature for the liquid phase, [Pa/(K*mol/m^3)].

d2P_dVdP_g

Second derivative of pressure with respect to molar volume and then pressure for the gas phase, [mol/m^3].

d2P_dVdP_l

Second derivative of pressure with respect to molar volume and then pressure for the liquid phase, [mol/m^3].

d2P_dVdT_g

Alias of GCEOS.d2P_dTdV_g

d2P_dVdT_l

Alias of GCEOS.d2P_dTdV_l

d2P_dVdT_TP_g

Second derivative of pressure with respect to molar volume and then temperature at constant temperature then pressure for the gas phase, [Pa*mol/m^3/K].

d2P_dVdT_TP_l

Second derivative of pressure with respect to molar volume and then temperature at constant temperature then pressure for the liquid phase, [Pa*mol/m^3/K].

d2rho_dP2_g

Second derivative of molar density with respect to pressure for the gas phase, [(mol/m^3)/Pa^2].

d2rho_dP2_l

Second derivative of molar density with respect to pressure for the liquid phase, [(mol/m^3)/Pa^2].

d2rho_dPdT_g

Second derivative of molar density with respect to pressure and temperature for the gas phase, [(mol/m^3)/(K*Pa)].

d2rho_dPdT_l

Second derivative of molar density with respect to pressure and temperature for the liquid phase, [(mol/m^3)/(K*Pa)].

d2rho_dT2_g

Second derivative of molar density with respect to temperature for the gas phase, [(mol/m^3)/K^2].

d2rho_dT2_l

Second derivative of molar density with respect to temperature for the liquid phase, [(mol/m^3)/K^2].

d2S_dep_dT2_g

Second temperature derivative of departure entropy with respect to temperature for the gas phase, [(J/mol)/K^3].

d2S_dep_dT2_g_V

Second temperature derivative of departure entropy with respect to temperature at constant volume for the gas phase, [(J/mol)/K^3].

d2S_dep_dT2_l

Second temperature derivative of departure entropy with respect to temperature for the liquid phase, [(J/mol)/K^3].

d2S_dep_dT2_l_V

Second temperature derivative of departure entropy with respect to temperature at constant volume for the liquid phase, [(J/mol)/K^3].

d2S_dep_dTdP_g

Temperature and pressure derivative of departure entropy at constant pressure then temperature for the gas phase, [(J/mol)/K^2/Pa].

d2S_dep_dTdP_l

Temperature and pressure derivative of departure entropy at constant pressure then temperature for the liquid phase, [(J/mol)/K^2/Pa].

d2T_dP2_g

Second partial derivative of temperature with respect to pressure (constant volume) for the gas phase, [K/Pa^2].

d2T_dP2_l

Second partial derivative of temperature with respect to pressure (constant temperature) for the liquid phase, [K/Pa^2].

d2T_dPdrho_g

Derivative of temperature with respect to molar density, and pressure for the gas phase, [K/(Pa*mol/m^3)].

d2T_dPdrho_l

Derivative of temperature with respect to molar density, and pressure for the liquid phase, [K/(Pa*mol/m^3)].

d2T_dPdV_g

Second partial derivative of temperature with respect to pressure (constant volume) and then volume (constant pressure) for the gas phase, [K*mol/(Pa*m^3)].

d2T_dPdV_l

Second partial derivative of temperature with respect to pressure (constant volume) and then volume (constant pressure) for the liquid phase, [K*mol/(Pa*m^3)].

d2T_drho2_g

Second derivative of temperature with respect to molar density for the gas phase, [K/(mol/m^3)^2].

d2T_drho2_l

Second derivative of temperature with respect to molar density for the liquid phase, [K/(mol/m^3)^2].

d2T_dV2_g

Second partial derivative of temperature with respect to volume (constant pressure) for the gas phase, [K*mol^2/m^6].

d2T_dV2_l

Second partial derivative of temperature with respect to volume (constant pressure) for the liquid phase, [K*mol^2/m^6].

d2T_dVdP_g

Second partial derivative of temperature with respect to pressure (constant volume) and then volume (constant pressure) for the gas phase, [K*mol/(Pa*m^3)].

d2T_dVdP_l

Second partial derivative of temperature with respect to pressure (constant volume) and then volume (constant pressure) for the liquid phase, [K*mol/(Pa*m^3)].

d2V_dP2_g

Second partial derivative of volume with respect to pressure (constant temperature) for the gas phase, [m^3/(Pa^2*mol)].

d2V_dP2_l

Second partial derivative of volume with respect to pressure (constant temperature) for the liquid phase, [m^3/(Pa^2*mol)].

d2V_dPdT_g

Second partial derivative of volume with respect to pressure (constant temperature) and then presssure (constant temperature) for the gas phase, [m^3/(K*Pa*mol)].

d2V_dPdT_l

Second partial derivative of volume with respect to pressure (constant temperature) and then presssure (constant temperature) for the liquid phase, [m^3/(K*Pa*mol)].

d2V_dT2_g

Second partial derivative of volume with respect to temperature (constant pressure) for the gas phase, [m^3/(mol*K^2)].

d2V_dT2_l

Second partial derivative of volume with respect to temperature (constant pressure) for the liquid phase, [m^3/(mol*K^2)].

d2V_dTdP_g

Second partial derivative of volume with respect to pressure (constant temperature) and then presssure (constant temperature) for the gas phase, [m^3/(K*Pa*mol)].

d2V_dTdP_l

Second partial derivative of volume with respect to pressure (constant temperature) and then presssure (constant temperature) for the liquid phase, [m^3/(K*Pa*mol)].

d3a_alpha_dT3

Method to calculate the third temperature derivative of $a \alpha$, [J^2/mol^2/Pa/K^3].

da_alpha_dP_g_V

Derivative of the a_alpha with respect to pressure at constant volume (varying T) for the gas phase, [J^2/mol^2/Pa^2].

da_alpha_dP_l_V

Derivative of the a_alpha with respect to pressure at constant volume (varying T) for the liquid phase, [J^2/mol^2/Pa^2].

dbeta_dP_g

Derivative of isobaric expansion coefficient with respect to pressure for the gas phase, [1/(Pa*K)].

dbeta_dP_l

Derivative of isobaric expansion coefficient with respect to pressure for the liquid phase, [1/(Pa*K)].

dbeta_dT_g

Derivative of isobaric expansion coefficient with respect to temperature for the gas phase, [1/K^2].

dbeta_dT_l

Derivative of isobaric expansion coefficient with respect to temperature for the liquid phase, [1/K^2].

dfugacity_dP_g

Derivative of fugacity with respect to pressure for the gas phase, [-].

dfugacity_dP_l

Derivative of fugacity with respect to pressure for the liquid phase, [-].

dfugacity_dT_g

Derivative of fugacity with respect to temperature for the gas phase, [Pa/K].

dfugacity_dT_l

Derivative of fugacity with respect to temperature for the liquid phase, [Pa/K].

dH_dep_dP_g

Derivative of departure enthalpy with respect to pressure for the gas phase, [(J/mol)/Pa].

dH_dep_dP_g_V

Derivative of departure enthalpy with respect to pressure at constant volume for the liquid phase, [(J/mol)/Pa].

dH_dep_dP_l

Derivative of departure enthalpy with respect to pressure for the liquid phase, [(J/mol)/Pa].

dH_dep_dP_l_V

Derivative of departure enthalpy with respect to pressure at constant volume for the gas phase, [(J/mol)/Pa].

dH_dep_dT_g

Derivative of departure enthalpy with respect to temperature for the gas phase, [(J/mol)/K].

dH_dep_dT_g_V

Derivative of departure enthalpy with respect to temperature at constant volume for the gas phase, [(J/mol)/K].

dH_dep_dT_l

Derivative of departure enthalpy with respect to temperature for the liquid phase, [(J/mol)/K].

dH_dep_dT_l_V

Derivative of departure enthalpy with respect to temperature at constant volume for the liquid phase, [(J/mol)/K].

dH_dep_dV_g_P

Derivative of departure enthalpy with respect to volume at constant pressure for the gas phase, [J/m^3].

dH_dep_dV_g_T

Derivative of departure enthalpy with respect to volume at constant temperature for the gas phase, [J/m^3].

dH_dep_dV_l_P

Derivative of departure enthalpy with respect to volume at constant pressure for the liquid phase, [J/m^3].

dH_dep_dV_l_T

Derivative of departure enthalpy with respect to volume at constant temperature for the gas phase, [J/m^3].

dP_drho_g

Derivative of pressure with respect to molar density for the gas phase, [Pa/(mol/m^3)].

dP_drho_l

Derivative of pressure with respect to molar density for the liquid phase, [Pa/(mol/m^3)].

dphi_dP_g

Derivative of fugacity coefficient with respect to pressure for the gas phase, [1/Pa].

dphi_dP_l

Derivative of fugacity coefficient with respect to pressure for the liquid phase, [1/Pa].

dphi_dT_g

Derivative of fugacity coefficient with respect to temperature for the gas phase, [1/K].

dphi_dT_l

Derivative of fugacity coefficient with respect to temperature for the liquid phase, [1/K].

drho_dP_g

Derivative of molar density with respect to pressure for the gas phase, [(mol/m^3)/Pa].

drho_dP_l

Derivative of molar density with respect to pressure for the liquid phase, [(mol/m^3)/Pa].

drho_dT_g

Derivative of molar density with respect to temperature for the gas phase, [(mol/m^3)/K].

drho_dT_l

Derivative of molar density with respect to temperature for the liquid phase, [(mol/m^3)/K].

dS_dep_dP_g

Derivative of departure entropy with respect to pressure for the gas phase, [(J/mol)/K/Pa].

dS_dep_dP_g_V

Derivative of departure entropy with respect to pressure at constant volume for the gas phase, [(J/mol)/K/Pa].

dS_dep_dP_l

Derivative of departure entropy with respect to pressure for the liquid phase, [(J/mol)/K/Pa].

dS_dep_dP_l_V

Derivative of departure entropy with respect to pressure at constant volume for the liquid phase, [(J/mol)/K/Pa].

dS_dep_dT_g

Derivative of departure entropy with respect to temperature for the gas phase, [(J/mol)/K^2].

dS_dep_dT_g_V

Derivative of departure entropy with respect to temperature at constant volume for the gas phase, [(J/mol)/K^2].

dS_dep_dT_l

Derivative of departure entropy with respect to temperature for the liquid phase, [(J/mol)/K^2].

dS_dep_dT_l_V

Derivative of departure entropy with respect to temperature at constant volume for the liquid phase, [(J/mol)/K^2].

dS_dep_dV_g_P

Derivative of departure entropy with respect to volume at constant pressure for the gas phase, [J/K/m^3].

dS_dep_dV_g_T

Derivative of departure entropy with respect to volume at constant temperature for the gas phase, [J/K/m^3].

dS_dep_dV_l_P

Derivative of departure entropy with respect to volume at constant pressure for the liquid phase, [J/K/m^3].

dS_dep_dV_l_T

Derivative of departure entropy with respect to volume at constant temperature for the gas phase, [J/K/m^3].

dT_drho_g

Derivative of temperature with respect to molar density for the gas phase, [K/(mol/m^3)].

dT_drho_l

Derivative of temperature with respect to molar density for the liquid phase, [K/(mol/m^3)].

dZ_dP_g

Derivative of compressibility factor with respect to pressure for the gas phase, [1/Pa].

dZ_dP_l

Derivative of compressibility factor with respect to pressure for the liquid phase, [1/Pa].

dZ_dT_g

Derivative of compressibility factor with respect to temperature for the gas phase, [1/K].

dZ_dT_l

Derivative of compressibility factor with respect to temperature for the liquid phase, [1/K].

fugacity_g

Fugacity for the gas phase, [Pa].

fugacity_l

Fugacity for the liquid phase, [Pa].

kappa_g

Isothermal (constant-temperature) expansion coefficient for the gas phase, [1/Pa].

kappa_l

Isothermal (constant-temperature) expansion coefficient for the liquid phase, [1/Pa].

lnphi_g

The natural logarithm of the fugacity coefficient for the gas phase, [-].

lnphi_l

The natural logarithm of the fugacity coefficient for the liquid phase, [-].

more_stable_phase

Checks the Gibbs energy of each possible phase, and returns ‘l’ if the liquid-like phase is more stable, and ‘g’ if the vapor-like phase is more stable.

mpmath_volume_ratios

Method to compare, as ratios, the volumes of the implemented cubic solver versus those calculated using mpmath.

mpmath_volumes

Method to calculate to a high precision the exact roots to the cubic equation, using mpmath.

mpmath_volumes_float

Method to calculate real roots of a cubic equation, using mpmath, but returned as floats.

phi_g

Fugacity coefficient for the gas phase, [Pa].

phi_l

Fugacity coefficient for the liquid phase, [Pa].

rho_g

Gas molar density, [mol/m^3].

rho_l

Liquid molar density, [mol/m^3].

sorted_volumes

List of lexicographically-sorted molar volumes available from the root finding algorithm used to solve the PT point.

state_specs

Convenience method to return the two specified state specs (T, P, or V) as a dictionary.

U_dep_g

Departure molar internal energy from ideal gas behavior for the gas phase, [J/mol].

U_dep_l

Departure molar internal energy from ideal gas behavior for the liquid phase, [J/mol].

Vc

Critical volume, [m^3/mol].

V_dep_g

Departure molar volume from ideal gas behavior for the gas phase, [m^3/mol].

V_dep_l

Departure molar volume from ideal gas behavior for the liquid phase, [m^3/mol].

V_g_mpmath

The molar volume of the gas phase calculated with mpmath to a higher precision, [m^3/mol].

V_l_mpmath

The molar volume of the liquid phase calculated with mpmath to a higher precision, [m^3/mol].

Methods

Hvap(T)	Method to calculate enthalpy of vaporization for a pure fluid from an equation of state, without iteration.
PT_surface_special([Tmin, Tmax, Pmin, Pmax, ...])	Method to create a plot of the special curves of a pure fluid - vapor pressure, determinant zeros, pseudo critical point, and mechanical critical point.
P_PIP_transition(T[, low_P_limit])	Method to calculate the pressure which makes the phase identification parameter exactly 1.
P_discriminant_zero_g()	Method to calculate the pressure which zero the discriminant function of the general cubic eos, and is likely to sit on a boundary between not having a vapor-like volume; and having a vapor-like volume.
P_discriminant_zero_l()	Method to calculate the pressure which zero the discriminant function of the general cubic eos, and is likely to sit on a boundary between not having a liquid-like volume; and having a liquid-like volume.
P_discriminant_zeros()	Method to calculate the pressures which zero the discriminant function of the general cubic eos, at the current temperature.
P_discriminant_zeros_analytical(T, b, delta, ...)	Method to calculate the pressures which zero the discriminant function of the general cubic eos.
P_max_at_V(V)	Dummy method.
Psat(T[, polish, guess])	Generic method to calculate vapor pressure for a specified T.
Psat_errors([Tmin, Tmax, pts, plot, show, ...])	Method to create a plot of vapor pressure and the relative error of its calculation vs.
T_discriminant_zero_g([T_guess])	Method to calculate the temperature which zeros the discriminant function of the general cubic eos, and is likely to sit on a boundary between not having a vapor-like volume; and having a vapor-like volume.
T_discriminant_zero_l([T_guess])	Method to calculate the temperature which zeros the discriminant function of the general cubic eos, and is likely to sit on a boundary between not having a liquid-like volume; and having a liquid-like volume.
T_max_at_V(V[, Pmax])	Method to calculate the maximum temperature the EOS can create at a constant volume, if one exists; returns None otherwise.
T_min_at_V(V[, Pmin])	Returns the minimum temperature for the EOS to have the volume as specified.
Tsat(P[, polish])	Generic method to calculate the temperature for a specified vapor pressure of the pure fluid.
V_g_sat(T)	Method to calculate molar volume of the vapor phase along the saturation line.
V_l_sat(T)	Method to calculate molar volume of the liquid phase along the saturation line.
Vs_mpmath()	Method to calculate real roots of a cubic equation, using mpmath.
a_alpha_and_derivatives(T[, full, quick, ...])	Method to calculate $a \alpha$ and its first and second derivatives.
a_alpha_and_derivatives_pure(T)	Dummy method to calculate $a \alpha$ and its first and second derivatives.
a_alpha_for_Psat(T, Psat[, a_alpha_guess])	Method to calculate which value of $a \alpha$ is required for a given T, Psat pair.
a_alpha_for_V(T, P, V)	Method to calculate which value of $a \alpha$ is required for a given T, P pair to match a specified V.
a_alpha_plot([Tmin, Tmax, pts, plot, show])	Method to create a plot of the $a \alpha$ parameter and its first two derivatives.
as_json([cache, option])	Method to create a JSON-friendly serialization of the eos which can be stored, and reloaded later.
check_sufficient_inputs()	Method to an exception if none of the pairs (T, P), (T, V), or (P, V) are given.
d2phi_sat_dT2(T[, polish])	Method to calculate the second temperature derivative of saturation fugacity coefficient of the compound.
dH_dep_dT_sat_g(T[, polish])	Method to calculate and return the temperature derivative of saturation vapor excess enthalpy.
dH_dep_dT_sat_l(T[, polish])	Method to calculate and return the temperature derivative of saturation liquid excess enthalpy.
dPsat_dT(T[, polish, also_Psat])	Generic method to calculate the temperature derivative of vapor pressure for a specified T.
dS_dep_dT_sat_g(T[, polish])	Method to calculate and return the temperature derivative of saturation vapor excess entropy.
dS_dep_dT_sat_l(T[, polish])	Method to calculate and return the temperature derivative of saturation liquid excess entropy.
discriminant([T, P])	Method to compute the discriminant of the cubic volume solution with the current EOS parameters, optionally at the same (assumed) T, and P or at different ones, if values are specified.
dphi_sat_dT(T[, polish])	Method to calculate the temperature derivative of saturation fugacity coefficient of the compound.
exact_hash()	Method to calculate and return a hash representing the exact state of the object.
from_json(json_repr[, cache])	Method to create a eos from a JSON serialization of another eos.
model_hash()	Basic method to calculate a hash of the non-state parts of the model This is useful for comparing to models to determine if they are the same, i.e. in a VLL flash it is important to know if both liquids have the same model.
phi_sat(T[, polish])	Method to calculate the saturation fugacity coefficient of the compound.
resolve_full_alphas()	Generic method to resolve the eos with fully calculated alpha derviatives.
saturation_prop_plot(prop[, Tmin, Tmax, ...])	Method to create a plot of a specified property of the EOS along the (pure component) saturation line.
set_from_PT(Vs[, only_l, only_g])	Counts the number of real volumes in Vs, and determines what to do.
set_properties_from_solution(T, P, V, b, ...)	Sets all interesting properties which can be calculated from an EOS alone.
solve([pure_a_alphas, only_l, only_g, ...])	First EOS-generic method; should be called by all specific EOSs.
solve_T(P, V[, solution])	Generic method to calculate T from a specified P and V.
solve_missing_volumes()	Generic method to ensure both volumes, if solutions are physical, have calculated properties.
state_hash()	Basic method to calculate a hash of the state of the model and its model parameters.
to([T, P, V])	Method to construct a new EOS object at two of T, P or V.
to_PV(P, V)	Method to construct a new EOS object at the spcified P and V.
to_TP(T, P)	Method to construct a new EOS object at the spcified T and P.
to_TV(T, V)	Method to construct a new EOS object at the spcified T and V.
volume_error()	Method to calculate the relative absolute error in the calculated molar volumes.
volume_errors([Tmin, Tmax, Pmin, Pmax, pts, ...])	Method to create a plot of the relative absolute error in the cubic volume solution as compared to a higher-precision calculation.
volume_solutions(T, P, b, delta, epsilon, ...)	Halley's method based solver for cubic EOS volumes based on the idea of initializing from a single liquid-like guess which is solved precisely, deflating the cubic analytically, solving the quadratic equation for the next two volumes, and then performing two halley steps on each of them to obtain the final solutions.
volume_solutions_full(T, P, b, delta, ...[, ...])	Newton-Raphson based solver for cubic EOS volumes based on the idea of initializing from an analytical solver.
volume_solutions_mp(T, P, b, delta, epsilon, ...)	Solution of this form of the cubic EOS in terms of volumes, using the mpmath arbitrary precision library.

property A_dep_g

Departure molar Helmholtz energy from ideal gas behavior for the gas phase, [J/mol].

$$A_{dep} = U_{dep} - T S_{dep}$$

property A_dep_l

Departure molar Helmholtz energy from ideal gas behavior for the liquid phase, [J/mol].

$$A_{dep} = U_{dep} - T S_{dep}$$

property Cp_minus_Cv_g

Cp - Cv for the gas phase, [J/mol/K].

$$C_p - C_v = -T\left(\frac{\partial P}{\partial T}\right)_V^2/ \left(\frac{\partial P}{\partial V}\right)_T$$

property Cp_minus_Cv_l

Cp - Cv for the liquid phase, [J/mol/K].

$$C_p - C_v = -T\left(\frac{\partial P}{\partial T}\right)_V^2/ \left(\frac{\partial P}{\partial V}\right)_T$$

Hvap(T)

Method to calculate enthalpy of vaporization for a pure fluid from an equation of state, without iteration.

$$\frac{dP^{sat}}{dT}=\frac{\Delta H_{vap}}{T(V_g - V_l)}$$

Results above the critical temperature are meaningless. A first-order polynomial is used to extrapolate under 0.32 Tc; however, there is normally not a volume solution to the EOS which can produce that low of a pressure.

Parameters

Tfloat

Temperature, [K]

Returns

Hvapfloat

Increase in enthalpy needed for vaporization of liquid phase along the saturation line, [J/mol]

Notes

Calculates vapor pressure and its derivative with Psat and dPsat_dT as well as molar volumes of the saturation liquid and vapor phase in the process.

Very near the critical point this provides unrealistic results due to Psat’s polynomials being insufficiently accurate.

References

1

Walas, Stanley M. Phase Equilibria in Chemical Engineering. Butterworth-Heinemann, 1985.

N = 1

The number of components in the EOS

PT_surface_special(Tmin=0.0001, Tmax=10000.0, Pmin=0.01, Pmax=1000000000.0, pts=50, show=False, color_map=None, mechanical=True, pseudo_critical=True, Psat=True, determinant_zeros=True, phase_ID_transition=True, base_property='V', base_min=None, base_max=None, base_selection='Gmin')

Method to create a plot of the special curves of a pure fluid - vapor pressure, determinant zeros, pseudo critical point, and mechanical critical point.

The color background is a plot of the molar volume (by default) which has the minimum Gibbs energy (by default). If shown with a sufficient number of points, the curve between vapor and liquid should be shown smoothly.

When called on a mixture, this method does not have physical significance for the Psat term.

Parameters

Tminfloat, optional

Minimum temperature of calculation, [K]

Tmaxfloat, optional

Maximum temperature of calculation, [K]

Pminfloat, optional

Minimum pressure of calculation, [Pa]

Pmaxfloat, optional

Maximum pressure of calculation, [Pa]

ptsint, optional

The number of points to include in both the x and y axis [-]

showbool, optional

Whether or not the plot should be rendered and shown; a handle to it is returned if plot is True for other purposes such as saving the plot to a file, [-]

color_mapmatplotlib.cm.ListedColormap, optional

Matplotlib colormap object, [-]

mechanicalbool, optional

Whether or not to include the mechanical critical point; this is the same as the critical point for a pure compound but not for a mixture, [-]

pseudo_criticalbool, optional

Whether or not to include the pseudo critical point; this is the same as the critical point for a pure compound but not for a mixture, [-]

Psatbool, optional

Whether or not to include the vapor pressure curve; for mixtures this is neither the bubble nor dew curve, but rather a hypothetical one which uses the same equation as the pure components, [-]

determinant_zerosbool, optional

Whether or not to include a curve showing when the EOS’s determinant hits zero, [-]

phase_ID_transitionbool, optional

Whether or not to show a curve of where the PIP hits 1 exactly, [-]

base_propertystr, optional

The property which should be plotted; ‘_l’ and ‘_g’ are added automatically according to the selected phase, [-]

base_minfloat, optional

If specified, the base property will values will be limited to this value at the minimum, [-]

base_maxfloat, optional

If specified, the base property will values will be limited to this value at the maximum, [-]

base_selectionstr, optional

For the base property, there are often two possible phases and but only one value can be plotted; use ‘l’ to pefer liquid-like values, ‘g’ to prefer gas-like values, and ‘Gmin’ to prefer values of the phase with the lowest Gibbs energy, [-]

Returns

figmatplotlib.figure.Figure

Plotted figure, only returned if plot is True, [-]

P_PIP_transition(T, low_P_limit=0.0)

Method to calculate the pressure which makes the phase identification parameter exactly 1. There are three regions for this calculation:

• subcritical - PIP = 1 for the gas-like phase at P = 0

• initially supercritical - PIP = 1 on a curve starting at the critical point, increasing for a while, decreasing for a while, and then curving sharply back to a zero pressure.

• later supercritical - PIP = 1 for the liquid-like phase at P = 0

Parameters

Tfloat

Temperature for the calculation, [K]

low_P_limitfloat

What value to return for the subcritical and later region, [Pa]

Returns

Pfloat

Pressure which makes the PIP = 1, [Pa]

Notes

The transition between the region where this function returns values and the high temperature region that doesn’t is the Joule-Thomson inversion point at a pressure of zero and can be directly solved for.

Examples

>>> eos = PRTranslatedConsistent(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> eos.P_PIP_transition(100)
0.0
>>> low_T = eos.to(T=100.0, P=eos.P_PIP_transition(100, low_P_limit=1e-5))
>>> low_T.PIP_l, low_T.PIP_g
(45.778088191, 0.9999999997903)
>>> initial_super = eos.to(T=600.0, P=eos.P_PIP_transition(600))
>>> initial_super.P, initial_super.PIP_g
(6456282.17132, 0.999999999999)
>>> high_T = eos.to(T=900.0, P=eos.P_PIP_transition(900, low_P_limit=1e-5))
>>> high_T.P, high_T.PIP_g
(12536704.763, 0.9999999999)

P_discriminant_zero_g()

Method to calculate the pressure which zero the discriminant function of the general cubic eos, and is likely to sit on a boundary between not having a vapor-like volume; and having a vapor-like volume.

Returns

P_discriminant_zero_gfloat

Pressure which make the discriminants zero at the right condition, [Pa]

Examples

>>> eos = PRTranslatedConsistent(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> P_trans = eos.P_discriminant_zero_g()
>>> P_trans
149960391.7

In this case, the discriminant transition does not reveal a transition to two roots being available, only negative roots becoming negative and imaginary.

>>> eos.to(T=eos.T, P=P_trans*.99999999).mpmath_volumes_float
((-0.0001037013146195082-1.5043987866732543e-08j), (-0.0001037013146195082+1.5043987866732543e-08j), (0.00011799201928619508+0j))
>>> eos.to(T=eos.T, P=P_trans*1.0000001).mpmath_volumes_float
((-0.00010374888853182635+0j), (-0.00010365374200380354+0j), (0.00011799201875924273+0j))

P_discriminant_zero_l()

Method to calculate the pressure which zero the discriminant function of the general cubic eos, and is likely to sit on a boundary between not having a liquid-like volume; and having a liquid-like volume.

Returns

P_discriminant_zero_lfloat

Pressure which make the discriminants zero at the right condition, [Pa]

Examples

>>> eos = PRTranslatedConsistent(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> P_trans = eos.P_discriminant_zero_l()
>>> P_trans
478346.37289

In this case, the discriminant transition shows the change in roots:

>>> eos.to(T=eos.T, P=P_trans*.99999999).mpmath_volumes_float
((0.00013117994140177062+0j), (0.002479717165903531+0j), (0.002480236178570793+0j))
>>> eos.to(T=eos.T, P=P_trans*1.0000001).mpmath_volumes_float
((0.0001311799413872173+0j), (0.002479976386402769-8.206310112063695e-07j), (0.002479976386402769+8.206310112063695e-07j))

P_discriminant_zeros()

Method to calculate the pressures which zero the discriminant function of the general cubic eos, at the current temperature.

Returns

P_discriminant_zeroslist[float]

Pressures which make the discriminants zero, [Pa]

Examples

>>> eos = PRTranslatedConsistent(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> eos.P_discriminant_zeros()
[478346.3, 149960391.7]

static P_discriminant_zeros_analytical(T, b, delta, epsilon, a_alpha, valid=False)

Method to calculate the pressures which zero the discriminant function of the general cubic eos. This is a quartic function solved analytically.

Parameters

Tfloat

Temperature, [K]

bfloat

Coefficient calculated by EOS-specific method, [m^3/mol]

deltafloat

Coefficient calculated by EOS-specific method, [m^3/mol]

epsilonfloat

Coefficient calculated by EOS-specific method, [m^6/mol^2]

a_alphafloat

Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]

validbool

Whether to filter the calculated pressures so that they are all real, and positive only, [-]

Returns

P_discriminant_zerosfloat

Pressures which make the discriminants zero, [Pa]

Notes

Calculated analytically. Derived as follows.

>>> from sympy import *
>>> P, T, V, R, b, a, delta, epsilon = symbols('P, T, V, R, b, a, delta, epsilon')
>>> eta = b
>>> B = b*P/(R*T)
>>> deltas = delta*P/(R*T)
>>> thetas = a*P/(R*T)**2
>>> epsilons = epsilon*(P/(R*T))**2
>>> etas = eta*P/(R*T)
>>> a_coeff = 1
>>> b_coeff = (deltas - B - 1)
>>> c = (thetas + epsilons - deltas*(B+1))
>>> d = -(epsilons*(B+1) + thetas*etas)
>>> disc = b_coeff*b_coeff*c*c - 4*a_coeff*c*c*c - 4*b_coeff*b_coeff*b_coeff*d - 27*a_coeff*a_coeff*d*d + 18*a_coeff*b_coeff*c*d
>>> base = -(expand(disc/P**2*R**3*T**3))
>>> sln = collect(base, P)

P_max_at_V(V)

Dummy method. The idea behind this method, which is implemented by some subclasses, is to calculate the maximum pressure the EOS can create at a constant volume, if one exists; returns None otherwise. This method, as a dummy method, always returns None.

Parameters

Vfloat

Constant molar volume, [m^3/mol]

Returns

Pfloat

Maximum possible isochoric pressure, [Pa]

P_zero_g_cheb_limits = (0.0, 0.0)

P_zero_l_cheb_limits = (0.0, 0.0)

Psat(T, polish=False, guess=None)

Generic method to calculate vapor pressure for a specified T.

From Tc to 0.32Tc, uses a 10th order polynomial of the following form:

$$\ln\frac{P_r}{T_r} = \sum_{k=0}^{10} C_k\left(\frac{\alpha}{T_r} -1\right)^{k}$$

If polish is True, SciPy’s newton solver is launched with the calculated vapor pressure as an initial guess in an attempt to get more accuracy. This may not converge however.

Results above the critical temperature are meaningless. A first-order polynomial is used to extrapolate under 0.32 Tc; however, there is normally not a volume solution to the EOS which can produce that low of a pressure.

Parameters

Tfloat

Temperature, [K]

polishbool, optional

Whether to attempt to use a numerical solver to make the solution more precise or not

Returns

Psatfloat

Vapor pressure, [Pa]

Notes

EOSs sharing the same b, delta, and epsilon have the same coefficient sets.

Form for the regression is inspired from [1].

No volume solution is needed when polish=False; the only external call is for the value of a_alpha.

References

1

Soave, G. “Direct Calculation of Pure-Compound Vapour Pressures through Cubic Equations of State.” Fluid Phase Equilibria 31, no. 2 (January 1, 1986): 203-7. doi:10.1016/0378-3812(86)90013-0.

Psat_cheb_range = (0.0, 0.0)

Psat_errors(Tmin=None, Tmax=None, pts=50, plot=False, show=False, trunc_err_low=1e-18, trunc_err_high=1.0, Pmin=1e-100)

Method to create a plot of vapor pressure and the relative error of its calculation vs. the iterative polish approach.

Parameters

Tminfloat

Minimum temperature of calculation; if this is too low the saturation routines will stop converging, [K]

Tmaxfloat

Maximum temperature of calculation; cannot be above the critical temperature, [K]

ptsint, optional

The number of temperature points to include [-]

plotbool

If False, the solution is returned without plotting the data, [-]

showbool

Whether or not the plot should be rendered and shown; a handle to it is returned if plot is True for other purposes such as saving the plot to a file, [-]

trunc_err_lowfloat

Minimum plotted error; values under this are rounded to 0, [-]

trunc_err_highfloat

Maximum plotted error; values above this are rounded to 1, [-]

Pminfloat

Minimum pressure for the solution to work on, [Pa]

Returns

errorslist[float]

Absolute relative errors, [-]

Psats_numlist[float]

Vapor pressures calculated to full precision, [Pa]

Psats_fitlist[float]

Vapor pressures calculated with the fast solution, [Pa]

figmatplotlib.figure.Figure

Plotted figure, only returned if plot is True, [-]

T_discriminant_zero_g(T_guess=None)

Method to calculate the temperature which zeros the discriminant function of the general cubic eos, and is likely to sit on a boundary between not having a vapor-like volume; and having a vapor-like volume.

Parameters

T_guessfloat, optional

Temperature guess, [K]

Returns

T_discriminant_zero_gfloat

Temperature which make the discriminants zero at the right condition, [K]

Notes

Significant numerical issues remain in improving this method.

Examples

>>> eos = PRTranslatedConsistent(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> T_trans = eos.T_discriminant_zero_g()
>>> T_trans
644.3023307

In this case, the discriminant transition does not reveal a transition to two roots being available, only to there being a double (imaginary) root.

>>> eos.to(P=eos.P, T=T_trans).mpmath_volumes_float
((9.309597822372529e-05-0.00015876248805149625j), (9.309597822372529e-05+0.00015876248805149625j), (0.005064847204219234+0j))

T_discriminant_zero_l(T_guess=None)

Method to calculate the temperature which zeros the discriminant function of the general cubic eos, and is likely to sit on a boundary between not having a liquid-like volume; and having a liquid-like volume.

Parameters

T_guessfloat, optional

Temperature guess, [K]

Returns

T_discriminant_zero_lfloat

Temperature which make the discriminants zero at the right condition, [K]

Notes

Significant numerical issues remain in improving this method.

Examples

>>> eos = PRTranslatedConsistent(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> T_trans = eos.T_discriminant_zero_l()
>>> T_trans
644.3023307

In this case, the discriminant transition does not reveal a transition to two roots being available, only to there being a double (imaginary) root.

>>> eos.to(P=eos.P, T=T_trans).mpmath_volumes_float
((9.309597822372529e-05-0.00015876248805149625j), (9.309597822372529e-05+0.00015876248805149625j), (0.005064847204219234+0j))

T_max_at_V(V, Pmax=None)

Method to calculate the maximum temperature the EOS can create at a constant volume, if one exists; returns None otherwise.

Parameters

Vfloat

Constant molar volume, [m^3/mol]

Pmaxfloat

Maximum possible isochoric pressure, if already known [Pa]

Returns

Tfloat

Maximum possible temperature, [K]

Examples

>>> e = PR(P=1e5, V=0.0001437, Tc=512.5, Pc=8084000.0, omega=0.559)
>>> e.T_max_at_V(e.V)
431155.5

T_min_at_V(V, Pmin=1e-15)

Returns the minimum temperature for the EOS to have the volume as specified. Under this temperature, the pressure will go negative (and the EOS will not solve).

Tsat(P, polish=False)

Generic method to calculate the temperature for a specified vapor pressure of the pure fluid. This is simply a bounded solver running between 0.2Tc and Tc on the Psat method.

Parameters

Pfloat

Vapor pressure, [Pa]

polishbool, optional

Whether to attempt to use a numerical solver to make the solution more precise or not

Returns

Tsatfloat

Temperature of saturation, [K]

Notes

It is recommended not to run with polish=True, as that will make the calculation much slower.

property U_dep_g

Departure molar internal energy from ideal gas behavior for the gas phase, [J/mol].

$$U_{dep} = H_{dep} - P V_{dep}$$

property U_dep_l

Departure molar internal energy from ideal gas behavior for the liquid phase, [J/mol].

$$U_{dep} = H_{dep} - P V_{dep}$$

property V_dep_g

Departure molar volume from ideal gas behavior for the gas phase, [m^3/mol].

$$V_{dep} = V - \frac{RT}{P}$$

property V_dep_l

Departure molar volume from ideal gas behavior for the liquid phase, [m^3/mol].

$$V_{dep} = V - \frac{RT}{P}$$

property V_g_mpmath

The molar volume of the gas phase calculated with mpmath to a higher precision, [m^3/mol]. This is useful for validating the cubic root solver(s). It is not quite a true arbitrary solution to the EOS, because the constants b,`epsilon`, delta and a_alpha as well as the input arguments T and P are not calculated with arbitrary precision. This is a feature when comparing the volume solution algorithms however as they work with the same finite-precision variables.

V_g_sat(T)

Method to calculate molar volume of the vapor phase along the saturation line.

Parameters

Tfloat

Temperature, [K]

Returns

V_g_satfloat

Gas molar volume along the saturation line, [m^3/mol]

Notes

Computes Psat, and then uses volume_solutions to obtain the three possible molar volumes. The highest value is returned.

property V_l_mpmath

The molar volume of the liquid phase calculated with mpmath to a higher precision, [m^3/mol]. This is useful for validating the cubic root solver(s). It is not quite a true arbitrary solution to the EOS, because the constants b,`epsilon`, delta and a_alpha as well as the input arguments T and P are not calculated with arbitrary precision. This is a feature when comparing the volume solution algorithms however as they work with the same finite-precision variables.

V_l_sat(T)

Method to calculate molar volume of the liquid phase along the saturation line.

Parameters

Tfloat

Temperature, [K]

Returns

V_l_satfloat

Liquid molar volume along the saturation line, [m^3/mol]

Notes

Computes Psat, and then uses volume_solutions to obtain the three possible molar volumes. The lowest value is returned.

property Vc

Critical volume, [m^3/mol].

$$V_c = \frac{Z_c R T_c}{P_c}$$

Vs_mpmath()

Method to calculate real roots of a cubic equation, using mpmath.

Returns

Vslist[mpf]

Either 1 or 3 real volumes as calculated by mpmath, [m^3/mol]

Examples

>>> eos = PRTranslatedTwu(T=300, P=1e5, Tc=512.5, Pc=8084000.0, omega=0.559, alpha_coeffs=(0.694911, 0.9199, 1.7), c=-1e-6)
>>> eos.Vs_mpmath()
[mpf('0.0000489261705320261435106226558966745'), mpf('0.000541508154451321441068958547812526'), mpf('0.0243149463942697410611501615357228')]

__repr__()

Create a string representation of the EOS - by default, include all parameters so as to make it easy to construct new instances from states. Includes the two specified state variables, Tc, Pc, omega and any kwargs.

Returns

recreationstr

String which is valid Python and recreates the current state of the object if ran, [-]

Examples

>>> eos = PR(Tc=507.6, Pc=3025000.0, omega=0.2975, T=400.0, P=1e6)
>>> eos
PR(Tc=507.6, Pc=3025000.0, omega=0.2975, T=400.0, P=1000000.0)

a_alpha_and_derivatives(T, full=True, quick=True, pure_a_alphas=True)

Method to calculate $a \alpha$ and its first and second derivatives.

Parameters

Tfloat

Temperature, [K]

fullbool, optional

If False, calculates and returns only a_alpha, [-]

quickbool, optional

Legary parameter being phased out [-]

pure_a_alphasbool, optional

Whether or not to recalculate the a_alpha terms of pure components (for the case of mixtures only) which stay the same as the composition changes (i.e in a PT flash); does nothing in the case of pure EOSs [-]

Returns

a_alphafloat

Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]

da_alpha_dTfloat

Temperature derivative of coefficient calculated by EOS-specific method, [J^2/mol^2/Pa/K]

d2a_alpha_dT2float

Second temperature derivative of coefficient calculated by EOS-specific method, [J^2/mol^2/Pa/K^2]

a_alpha_and_derivatives_pure(T)

Dummy method to calculate $a \alpha$ and its first and second derivatives. Should be implemented with the same function signature in each EOS variant; this only raises a NotImplemented Exception. Should return ‘a_alpha’, ‘da_alpha_dT’, and ‘d2a_alpha_dT2’.

Parameters

Tfloat

Temperature, [K]

Returns

a_alphafloat

Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]

da_alpha_dTfloat

Temperature derivative of coefficient calculated by EOS-specific method, [J^2/mol^2/Pa/K]

d2a_alpha_dT2float

Second temperature derivative of coefficient calculated by EOS-specific method, [J^2/mol^2/Pa/K^2]

a_alpha_for_Psat(T, Psat, a_alpha_guess=None)

Method to calculate which value of $a \alpha$ is required for a given T, Psat pair. This is a numerical solution, but not a very complicated one.

Parameters

Tfloat

Temperature, [K]

Psatfloat

Vapor pressure specified, [Pa]

a_alpha_guessfloat

Optionally, an initial guess for the solver [J^2/mol^2/Pa]

Returns

a_alphafloat

Value calculated to match specified volume for the current EOS, [J^2/mol^2/Pa]

Notes

The implementation of this function is a direct calculation of departure gibbs energy, which is equal in both phases at saturation.

Examples

>>> eos = PR(Tc=507.6, Pc=3025000, omega=0.2975, T=299., P=1E6)
>>> eos.a_alpha_for_Psat(T=400, Psat=5e5)
3.1565798926

a_alpha_for_V(T, P, V)

Method to calculate which value of $a \alpha$ is required for a given T, P pair to match a specified V. This is a straightforward analytical equation.

Parameters

Tfloat

Temperature, [K]

Pfloat

Pressure, [Pa]

Vfloat

Molar volume, [m^3/mol]

Returns

a_alphafloat

Value calculated to match specified volume for the current EOS, [J^2/mol^2/Pa]

Notes

The derivation of the solution is as follows:

>>> from sympy import * 
>>> P, T, V, R, b, a, delta, epsilon = symbols('P, T, V, R, b, a, delta, epsilon') 
>>> a_alpha = symbols('a_alpha') 
>>> CUBIC = R*T/(V-b) - a_alpha/(V*V + delta*V + epsilon) 
>>> solve(Eq(CUBIC, P), a_alpha)
[(-P*V**3 + P*V**2*b - P*V**2*delta + P*V*b*delta - P*V*epsilon + P*b*epsilon + R*T*V**2 + R*T*V*delta + R*T*epsilon)/(V - b)]

a_alpha_plot(Tmin=0.0001, Tmax=None, pts=1000, plot=True, show=True)

Method to create a plot of the $a \alpha$ parameter and its first two derivatives. This easily allows identification of EOSs which are displaying inconsistent behavior.

Parameters

Tminfloat

Minimum temperature of calculation, [K]

Tmaxfloat

Maximum temperature of calculation, [K]

ptsint, optional

The number of temperature points to include [-]

plotbool

If False, the calculated values and temperatures are returned without plotting the data, [-]

showbool

Whether or not the plot should be rendered and shown; a handle to it is returned if plot is True for other purposes such as saving the plot to a file, [-]

Returns

Tslist[float]

Logarithmically spaced temperatures in specified range, [K]

a_alphalist[float]

Coefficient calculated by EOS-specific method, [J^2/mol^2/Pa]

da_alpha_dTlist[float]

Temperature derivative of coefficient calculated by EOS-specific method, [J^2/mol^2/Pa/K]

d2a_alpha_dT2list[float]

Second temperature derivative of coefficient calculated by EOS-specific method, [J^2/mol^2/Pa/K^2]

figmatplotlib.figure.Figure

Plotted figure, only returned if plot is True, [-]

as_json(cache=None, option=0)

Method to create a JSON-friendly serialization of the eos which can be stored, and reloaded later.

Returns

json_reprdict

JSON-friendly representation, [-]

Examples

>>> import json
>>> eos = MSRKTranslated(Tc=507.6, Pc=3025000, omega=0.2975, c=22.0561E-6, M=0.7446, N=0.2476, T=250., P=1E6)
>>> assert eos == MSRKTranslated.from_json(json.loads(json.dumps(eos.as_json())))

property beta_g

Isobaric (constant-pressure) expansion coefficient for the gas phase, [1/K].

$$\beta = \frac{1}{V}\frac{\partial V}{\partial T}$$

property beta_l

Isobaric (constant-pressure) expansion coefficient for the liquid phase, [1/K].

$$\beta = \frac{1}{V}\frac{\partial V}{\partial T}$$

c1 = None

Parameter used by some equations of state in the a calculation

c2 = None

Parameter used by some equations of state in the b calculation

check_sufficient_inputs()

Method to an exception if none of the pairs (T, P), (T, V), or (P, V) are given.

property d2H_dep_dT2_g

Second temperature derivative of departure enthalpy with respect to temperature for the gas phase, [(J/mol)/K^2].

$$\frac{\partial^2 H_{dep, g}}{\partial T^2} = P \frac{d^{2}}{d T^{2}} V{\left(T \right)} - \frac{8 T \frac{d}{d T} V{\left(T \right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha} {\left(T \right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{ \left(\delta + 2 V{\left(T \right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{2 T \operatorname{atanh}{\left( \frac{\delta + 2 V{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{3}}{d T^{3}} \operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}} + \frac{16 \left(\delta + 2 V{\left(T \right)} \right) \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)} - \operatorname{a\alpha}{\left(T \right)}\right) \left( \frac{d}{d T} V{\left(T \right)}\right)^{2}}{\left(\delta^{2} - 4 \epsilon\right)^{2} \left(\frac{\left(\delta + 2 V{\left(T \right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)^{2}} - \frac{4 \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)} - \operatorname{a\alpha}{\left(T \right)}\right) \frac{d^{2}}{d T^{2}} V{\left(T \right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T \right)} \right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{2 \operatorname{atanh}{\left(\frac{\delta + 2 V{\left(T \right)}} {\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}}$$

property d2H_dep_dT2_g_P

Second temperature derivative of departure enthalpy with respect to temperature for the gas phase, [(J/mol)/K^2].

$$\frac{\partial^2 H_{dep, g}}{\partial T^2} = P \frac{d^{2}}{d T^{2}} V{\left(T \right)} - \frac{8 T \frac{d}{d T} V{\left(T \right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha} {\left(T \right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{ \left(\delta + 2 V{\left(T \right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{2 T \operatorname{atanh}{\left( \frac{\delta + 2 V{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{3}}{d T^{3}} \operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}} + \frac{16 \left(\delta + 2 V{\left(T \right)} \right) \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)} - \operatorname{a\alpha}{\left(T \right)}\right) \left( \frac{d}{d T} V{\left(T \right)}\right)^{2}}{\left(\delta^{2} - 4 \epsilon\right)^{2} \left(\frac{\left(\delta + 2 V{\left(T \right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)^{2}} - \frac{4 \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)} - \operatorname{a\alpha}{\left(T \right)}\right) \frac{d^{2}}{d T^{2}} V{\left(T \right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T \right)} \right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{2 \operatorname{atanh}{\left(\frac{\delta + 2 V{\left(T \right)}} {\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}}$$

property d2H_dep_dT2_g_V

Second temperature derivative of departure enthalpy with respect to temperature at constant volume for the gas phase, [(J/mol)/K^2].

$$\left(\frac{\partial^2 H_{dep, g}}{\partial T^2}\right)_V = \frac{2 T \operatorname{atanh}{\left(\frac{2 V + \delta}{\sqrt{ \delta^{2} - 4 \epsilon}} \right)} \frac{d^{3}}{d T^{3}} \operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}} + V \frac{\partial^{2}}{\partial T^{2}} P{\left(V,T \right)} + \frac{2 \operatorname{atanh}{\left(\frac{ 2 V + \delta}{\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{2}} {d T^{2}} \operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}}$$

property d2H_dep_dT2_l

Second temperature derivative of departure enthalpy with respect to temperature for the liquid phase, [(J/mol)/K^2].

$$\frac{\partial^2 H_{dep, l}}{\partial T^2} = P \frac{d^{2}}{d T^{2}} V{\left(T \right)} - \frac{8 T \frac{d}{d T} V{\left(T \right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha} {\left(T \right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{ \left(\delta + 2 V{\left(T \right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{2 T \operatorname{atanh}{\left( \frac{\delta + 2 V{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{3}}{d T^{3}} \operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}} + \frac{16 \left(\delta + 2 V{\left(T \right)} \right) \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)} - \operatorname{a\alpha}{\left(T \right)}\right) \left( \frac{d}{d T} V{\left(T \right)}\right)^{2}}{\left(\delta^{2} - 4 \epsilon\right)^{2} \left(\frac{\left(\delta + 2 V{\left(T \right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)^{2}} - \frac{4 \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)} - \operatorname{a\alpha}{\left(T \right)}\right) \frac{d^{2}}{d T^{2}} V{\left(T \right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T \right)} \right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{2 \operatorname{atanh}{\left(\frac{\delta + 2 V{\left(T \right)}} {\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}}$$

property d2H_dep_dT2_l_P

Second temperature derivative of departure enthalpy with respect to temperature for the liquid phase, [(J/mol)/K^2].

$$\frac{\partial^2 H_{dep, l}}{\partial T^2} = P \frac{d^{2}}{d T^{2}} V{\left(T \right)} - \frac{8 T \frac{d}{d T} V{\left(T \right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha} {\left(T \right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{ \left(\delta + 2 V{\left(T \right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{2 T \operatorname{atanh}{\left( \frac{\delta + 2 V{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{3}}{d T^{3}} \operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}} + \frac{16 \left(\delta + 2 V{\left(T \right)} \right) \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)} - \operatorname{a\alpha}{\left(T \right)}\right) \left( \frac{d}{d T} V{\left(T \right)}\right)^{2}}{\left(\delta^{2} - 4 \epsilon\right)^{2} \left(\frac{\left(\delta + 2 V{\left(T \right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)^{2}} - \frac{4 \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)} - \operatorname{a\alpha}{\left(T \right)}\right) \frac{d^{2}}{d T^{2}} V{\left(T \right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T \right)} \right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{2 \operatorname{atanh}{\left(\frac{\delta + 2 V{\left(T \right)}} {\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}}$$

property d2H_dep_dT2_l_V

Second temperature derivative of departure enthalpy with respect to temperature at constant volume for the liquid phase, [(J/mol)/K^2].

$$\left(\frac{\partial^2 H_{dep, l}}{\partial T^2}\right)_V = \frac{2 T \operatorname{atanh}{\left(\frac{2 V + \delta}{\sqrt{ \delta^{2} - 4 \epsilon}} \right)} \frac{d^{3}}{d T^{3}} \operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}} + V \frac{\partial^{2}}{\partial T^{2}} P{\left(V,T \right)} + \frac{2 \operatorname{atanh}{\left(\frac{ 2 V + \delta}{\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{2}} {d T^{2}} \operatorname{a\alpha}{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}}$$

property d2H_dep_dTdP_g

Temperature and pressure derivative of departure enthalpy at constant pressure then temperature for the gas phase, [(J/mol)/K/Pa].

$$\left(\frac{\partial^2 H_{dep, g}}{\partial T \partial P}\right)_{T, P} = P \frac{\partial^{2}}{\partial T\partial P} V{\left(T,P \right)} - \frac{4 T \frac{\partial}{\partial P} V{\left(T,P \right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha}{\left(T \right)}} {\left(\delta^{2} - 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T,P \right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{16 \left(\delta + 2 V{\left(T,P \right)}\right) \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)} - \operatorname{a\alpha}{\left(T \right)}\right) \frac{\partial} {\partial P} V{\left(T,P \right)} \frac{\partial}{\partial T} V{\left(T,P \right)}}{\left(\delta^{2} - 4 \epsilon\right)^{2} \left(\frac{\left(\delta + 2 V{\left(T,P \right)}\right)^{2}} {\delta^{2} - 4 \epsilon} - 1\right)^{2}} + \frac{\partial} {\partial T} V{\left(T,P \right)} - \frac{4 \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)} - \operatorname{a\alpha} {\left(T \right)}\right) \frac{\partial^{2}}{\partial T\partial P} V{\left(T,P \right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T,P \right)}\right)^{2}} {\delta^{2} - 4 \epsilon} - 1\right)}$$

property d2H_dep_dTdP_l

Temperature and pressure derivative of departure enthalpy at constant pressure then temperature for the liquid phase, [(J/mol)/K/Pa].

$$\left(\frac{\partial^2 H_{dep, l}}{\partial T \partial P}\right)_V = P \frac{\partial^{2}}{\partial T\partial P} V{\left(T,P \right)} - \frac{4 T \frac{\partial}{\partial P} V{\left(T,P \right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha}{\left(T \right)}} {\left(\delta^{2} - 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T,P \right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{16 \left(\delta + 2 V{\left(T,P \right)}\right) \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)} - \operatorname{a\alpha}{\left(T \right)}\right) \frac{\partial} {\partial P} V{\left(T,P \right)} \frac{\partial}{\partial T} V{\left(T,P \right)}}{\left(\delta^{2} - 4 \epsilon\right)^{2} \left(\frac{\left(\delta + 2 V{\left(T,P \right)}\right)^{2}} {\delta^{2} - 4 \epsilon} - 1\right)^{2}} + \frac{\partial} {\partial T} V{\left(T,P \right)} - \frac{4 \left(T \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)} - \operatorname{a\alpha} {\left(T \right)}\right) \frac{\partial^{2}}{\partial T\partial P} V{\left(T,P \right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T,P \right)}\right)^{2}} {\delta^{2} - 4 \epsilon} - 1\right)}$$

property d2P_dT2_PV_g

Second derivative of pressure with respect to temperature twice, but with pressure held constant the first time and volume held constant the second time for the gas phase, [Pa/K^2].

$$\left(\frac{\partial^2 P}{\partial T \partial T}\right)_{P,V} = - \frac{R \frac{d}{d T} V{\left(T \right)}}{\left(- b + V{\left(T \right)}\right)^{2}} - \frac{\left(- \delta \frac{d}{d T} V{\left(T \right)} - 2 V{\left(T \right)} \frac{d}{d T} V{\left(T \right)} \right) \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)}} {\left(\delta V{\left(T \right)} + \epsilon + V^{2}{\left(T \right)}\right)^{2}} - \frac{\frac{d^{2}}{d T^{2}} \operatorname{a\alpha}{\left(T \right)}}{\delta V{\left(T \right)} + \epsilon + V^{2}{\left(T \right)}}$$

property d2P_dT2_PV_l

Second derivative of pressure with respect to temperature twice, but with pressure held constant the first time and volume held constant the second time for the liquid phase, [Pa/K^2].

$$\left(\frac{\partial^2 P}{\partial T \partial T}\right)_{P,V} = - \frac{R \frac{d}{d T} V{\left(T \right)}}{\left(- b + V{\left(T \right)}\right)^{2}} - \frac{\left(- \delta \frac{d}{d T} V{\left(T \right)} - 2 V{\left(T \right)} \frac{d}{d T} V{\left(T \right)} \right) \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)}} {\left(\delta V{\left(T \right)} + \epsilon + V^{2}{\left(T \right)}\right)^{2}} - \frac{\frac{d^{2}}{d T^{2}} \operatorname{a\alpha}{\left(T \right)}}{\delta V{\left(T \right)} + \epsilon + V^{2}{\left(T \right)}}$$

property d2P_dTdP_g

Second derivative of pressure with respect to temperature and, then pressure; and with volume held constant at first, then temperature, for the gas phase, [1/K].

$$\left(\frac{\partial^2 P}{\partial T \partial P}\right)_{V, T} = - \frac{R \frac{d}{d P} V{\left(P \right)}}{\left(- b + V{\left(P \right)}\right)^{2}} - \frac{\left(- \delta \frac{d}{d P} V{\left(P \right)} - 2 V{\left(P \right)} \frac{d}{d P} V{\left(P \right)} \right) \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)}} {\left(\delta V{\left(P \right)} + \epsilon + V^{2}{\left(P \right)}\right)^{2}}$$

property d2P_dTdP_l

Second derivative of pressure with respect to temperature and, then pressure; and with volume held constant at first, then temperature, for the liquid phase, [1/K].

$$\left(\frac{\partial^2 P}{\partial T \partial P}\right)_{V, T} = - \frac{R \frac{d}{d P} V{\left(P \right)}}{\left(- b + V{\left(P \right)}\right)^{2}} - \frac{\left(- \delta \frac{d}{d P} V{\left(P \right)} - 2 V{\left(P \right)} \frac{d}{d P} V{\left(P \right)} \right) \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)}} {\left(\delta V{\left(P \right)} + \epsilon + V^{2}{\left(P \right)}\right)^{2}}$$

property d2P_dTdrho_g

Derivative of pressure with respect to molar density, and temperature for the gas phase, [Pa/(K*mol/m^3)].

$$\frac{\partial^2 P}{\partial \rho\partial T} = -V^2 \frac{\partial^2 P}{\partial T \partial V}$$

property d2P_dTdrho_l

Derivative of pressure with respect to molar density, and temperature for the liquid phase, [Pa/(K*mol/m^3)].

$$\frac{\partial^2 P}{\partial \rho\partial T} = -V^2 \frac{\partial^2 P}{\partial T \partial V}$$

property d2P_dVdP_g

Second derivative of pressure with respect to molar volume and then pressure for the gas phase, [mol/m^3].

$$\frac{\partial^2 P}{\partial V \partial P} = \frac{2 R T \frac{d}{d P} V{\left(P \right)}}{\left(- b + V{\left(P \right)}\right)^{3}} - \frac{\left(- \delta - 2 V{\left(P \right)} \right) \left(- 2 \delta \frac{d}{d P} V{\left(P \right)} - 4 V{\left(P \right)} \frac{d}{d P} V{\left(P \right)}\right) \operatorname{a\alpha}{\left(T \right)}}{\left(\delta V{\left(P \right)} + \epsilon + V^{2}{\left(P \right)}\right)^{3}} + \frac{2 \operatorname{a\alpha}{\left(T \right)} \frac{d}{d P} V{\left(P \right)}}{\left(\delta V{\left(P \right)} + \epsilon + V^{2} {\left(P \right)}\right)^{2}}$$

property d2P_dVdP_l

Second derivative of pressure with respect to molar volume and then pressure for the liquid phase, [mol/m^3].

$$\frac{\partial^2 P}{\partial V \partial P} = \frac{2 R T \frac{d}{d P} V{\left(P \right)}}{\left(- b + V{\left(P \right)}\right)^{3}} - \frac{\left(- \delta - 2 V{\left(P \right)} \right) \left(- 2 \delta \frac{d}{d P} V{\left(P \right)} - 4 V{\left(P \right)} \frac{d}{d P} V{\left(P \right)}\right) \operatorname{a\alpha}{\left(T \right)}}{\left(\delta V{\left(P \right)} + \epsilon + V^{2}{\left(P \right)}\right)^{3}} + \frac{2 \operatorname{a\alpha}{\left(T \right)} \frac{d}{d P} V{\left(P \right)}}{\left(\delta V{\left(P \right)} + \epsilon + V^{2} {\left(P \right)}\right)^{2}}$$

property d2P_dVdT_TP_g

Second derivative of pressure with respect to molar volume and then temperature at constant temperature then pressure for the gas phase, [Pa*mol/m^3/K].

$$\left(\frac{\partial^2 P}{\partial V \partial T}\right)_{T,P} = \frac{2 R T \frac{d}{d T} V{\left(T \right)}}{\left(- b + V{\left(T \right)}\right)^{3}} - \frac{R}{\left(- b + V{\left(T \right)} \right)^{2}} - \frac{\left(- \delta - 2 V{\left(T \right)}\right) \left(- 2 \delta \frac{d}{d T} V{\left(T \right)} - 4 V{\left(T \right)} \frac{d}{d T} V{\left(T \right)}\right) \operatorname{ a\alpha}{\left(T \right)}}{\left(\delta V{\left(T \right)} + \epsilon + V^{2}{\left(T \right)}\right)^{3}} - \frac{\left( - \delta - 2 V{\left(T \right)}\right) \frac{d}{d T} \operatorname{ a\alpha}{\left(T \right)}}{\left(\delta V{\left(T \right)} + \epsilon + V^{2}{\left(T \right)}\right)^{2}} + \frac{2 \operatorname{a\alpha}{\left(T \right)} \frac{d}{d T} V{\left(T \right)}}{\left(\delta V{\left(T \right)} + \epsilon + V^{2}{\left( T \right)}\right)^{2}}$$

property d2P_dVdT_TP_l

Second derivative of pressure with respect to molar volume and then temperature at constant temperature then pressure for the liquid phase, [Pa*mol/m^3/K].

$$\left(\frac{\partial^2 P}{\partial V \partial T}\right)_{T,P} = \frac{2 R T \frac{d}{d T} V{\left(T \right)}}{\left(- b + V{\left(T \right)}\right)^{3}} - \frac{R}{\left(- b + V{\left(T \right)} \right)^{2}} - \frac{\left(- \delta - 2 V{\left(T \right)}\right) \left(- 2 \delta \frac{d}{d T} V{\left(T \right)} - 4 V{\left(T \right)} \frac{d}{d T} V{\left(T \right)}\right) \operatorname{ a\alpha}{\left(T \right)}}{\left(\delta V{\left(T \right)} + \epsilon + V^{2}{\left(T \right)}\right)^{3}} - \frac{\left( - \delta - 2 V{\left(T \right)}\right) \frac{d}{d T} \operatorname{ a\alpha}{\left(T \right)}}{\left(\delta V{\left(T \right)} + \epsilon + V^{2}{\left(T \right)}\right)^{2}} + \frac{2 \operatorname{a\alpha}{\left(T \right)} \frac{d}{d T} V{\left(T \right)}}{\left(\delta V{\left(T \right)} + \epsilon + V^{2}{\left( T \right)}\right)^{2}}$$

property d2P_dVdT_g

Alias of GCEOS.d2P_dTdV_g

property d2P_dVdT_l

Alias of GCEOS.d2P_dTdV_l

property d2P_drho2_g

Second derivative of pressure with respect to molar density for the gas phase, [Pa/(mol/m^3)^2].

$$\frac{\partial^2 P}{\partial \rho^2} = -V^2\left( -V^2\frac{\partial^2 P}{\partial V^2} - 2V \frac{\partial P}{\partial V} \right)$$

property d2P_drho2_l

Second derivative of pressure with respect to molar density for the liquid phase, [Pa/(mol/m^3)^2].

$$\frac{\partial^2 P}{\partial \rho^2} = -V^2\left( -V^2\frac{\partial^2 P}{\partial V^2} - 2V \frac{\partial P}{\partial V} \right)$$

property d2S_dep_dT2_g

Second temperature derivative of departure entropy with respect to temperature for the gas phase, [(J/mol)/K^3].

$$\frac{\partial^2 S_{dep, g}}{\partial T^2} = - \frac{R \left( \frac{d}{d T} V{\left(T \right)} - \frac{V{\left(T \right)}}{T} \right) \frac{d}{d T} V{\left(T \right)}}{V^{2}{\left(T \right)}} + \frac{R \left(\frac{d^{2}}{d T^{2}} V{\left(T \right)} - \frac{2 \frac{d}{d T} V{\left(T \right)}}{T} + \frac{2 V{\left(T \right)}}{T^{2}}\right)}{V{\left(T \right)}} - \frac{R \frac{d^{2}}{d T^{2}} V{\left(T \right)}}{V{\left(T \right)}} + \frac{R \left(\frac{d}{d T} V{\left(T \right)} \right)^{2}}{V^{2}{\left(T \right)}} - \frac{R \frac{d^{2}}{dT^{2}} V{\left(T \right)}}{b - V{\left(T \right)}} - \frac{R \left( \frac{d}{d T} V{\left(T \right)}\right)^{2}}{\left(b - V{\left(T \right)}\right)^{2}} + \frac{R \left(\frac{d}{d T} V{\left(T \right)} - \frac{V{\left(T \right)}}{T}\right)}{T V{\left(T \right)}} + \frac{16 \left(\delta + 2 V{\left(T \right)}\right) \left(\frac{d}{d T} V{\left(T \right)}\right)^{2} \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)}}{\left(\delta^{2} - 4 \epsilon\right)^{2} \left(\frac{\left(\delta + 2 V{\left(T \right)}\right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)^{2}} - \frac{8 \frac{d}{d T} V{\left(T \right)} \frac{d^{2}}{d T^{2}} \operatorname{a\alpha}{\left(T \right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T \right)} \right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} - \frac{4 \frac{d^{2}}{d T^{2}} V{\left(T \right)} \frac{d}{d T} \operatorname{a\alpha}{\left(T \right)}}{\left(\delta^{2} - 4 \epsilon\right) \left(\frac{\left(\delta + 2 V{\left(T \right)} \right)^{2}}{\delta^{2} - 4 \epsilon} - 1\right)} + \frac{2 \operatorname{atanh}{\left(\frac{\delta + 2 V{\left(T \right)}}{\sqrt{\delta^{2} - 4 \epsilon}} \right)} \frac{d^{3}} {d T^{3}} \operatorname{a\alpha}{\left(T \right)}} {\sqrt{\delta^{2} - 4 \epsilon}}$$