saturation_pressure(model::EoSModel, T)
saturation_pressure(model::EoSModel,T,method::SaturationMethod)
saturation_pressure(model,T,x0::Union{Tuple,Vector})

Performs a single component saturation equilibrium calculation, at the specified temperature T, of one mol of pure sustance specified by model. Returns (p₀, Vₗ, Vᵥ) where p₀ is the saturation pressure (in [Pa]), Vₗ is the liquid saturation volume (in [m³]) and Vᵥ is the vapour saturation volume (in [m³]).

If the calculation fails, returns (NaN, NaN, NaN)

By default, it uses ChemPotVSaturation

Examples:

julia> pr = PR(["water"])
PR{BasicIdeal, PRAlpha, NoTranslation, vdW1fRule} with 1 component:
  "water"
Contains parameters: a, b, Tc, Pc, Mw

julia> p,vl,vv = saturation_pressure(pr,373.15) #default, uses Clapeyron.ChemPotVSaturation
(96099.38979351855, 2.2674781912892906e-5, 0.03201681565699426)

julia> p,vl,vv = saturation_pressure(pr,373.15,IsoFugacitySaturation()) #iso fugacity
(96099.38979351871, 2.2674781912892933e-5, 0.03201681565699359)

julia> p,vl,vv = saturation_pressure(pr,373.15,IsoFugacitySaturation(p0 = 1.0e5)) #iso fugacity, with starting point
(96099.38979351871, 2.2674781912892933e-5, 0.03201681565699547)

saturation_temperature(model::EoSModel, p, kwargs...)
saturation_temperature(model::EoSModel, p, method::SaturationMethod)
saturation_temperature(model, p, T0::Number)

Performs a single component saturation temperature equilibrium calculation, at the specified temperature T, of one mol of pure sustance specified by model. Returns (T₀, Vₗ, Vᵥ) where T₀ is the saturation Temperature (in [K]), Vₗ is the liquid saturation volume (in [m³]) and Vᵥ is the vapour saturation volume (in [m³]).

If the calculation fails, returns (NaN, NaN, NaN)

Available methods

• AntoineSaturation: (default) direct VT solver (fast and robust).
• ClapeyronSaturation: Clapeyron-equation descent that repeatedly calls a saturation solver (reliable but slower; supports T0/crit).
• SuperAncSaturation: superancillary correlations for supported EoS (cubic, PC-SAFT via extension); requires enabling superancillaries and is only available on supported models.

Examples

julia-repl

julia> pr = PR(["water"])
PR{BasicIdeal, PRAlpha, NoTranslation, vdW1fRule} with 1 component:
  "water"
Contains parameters: a, b, Tc, Pc, Mw

julia> Ts,vl,vv = saturation_temperature(pr,1e5) # AntoineSaturation by default
(374.24014010712983, 2.269760164801948e-5, 0.030849387955737825)

julia> saturation_pressure(pr,Ts)
(100000.00004314569, 2.269760164804427e-5, 0.03084938795785433)

enthalpy_vap(model::EoSModel, T,method = ChemPotVSaturation(x0_sat_pure(model,T)))

Calculates ΔH, the difference between saturated vapour and liquid enthalpies at temperature T [K], in [J]

crit_pure(model::EoSModel,x0=nothing)

Calculates the critical point of a single component modelled by model. Returns a tuple, containing:

• Critical temperature [K]
• Critical pressure [Pa]
• Critical molar volume [m³·mol⁻¹]

acentric_factor(model::EoSModel;crit = crit_pure(model), satmethod = ChemPotVSaturation())

Calculates the acentric factor using its definition:

ω = -log10(psatᵣ) -1, at Tᵣ = 0.7

To do so, it calculates the critical temperature (using crit_pure) and performs a saturation calculation (with saturation_pressure(model,0.7Tc,satmethod))

pm,vs,vl = melting_pressure(model::CompositeModel,T;v0=x0_melting_pressure(model,T))

Calculates the melting pressure of a CompositeModel containing a solid and fluid phase EoS, at a specified temperature T. You can pass a tuple of initial values for the volumes (vs0,vl0).

returns:

• pm is melting Pressure [Pa]
• vs is melting solid volume at specified temperature [m³]
• vl is melting liquid volume at specified temperature [m³]

pm,vs,vl = melting_temperature(model::CompositeModel,p;v0=x0_melting_pressure(model,T))

Calculates the melting temperature of a CompositeModel containing a solid and fluid phase EoS, at a specified pressure p. You can pass a tuple of initial values for the volumes (vs0,vl0).

returns:

• Melting Temperature [K]
• melting solid volume at specified pressure [m³]
• melting liquid volume at specified pressure [m³]

psub,vs,vv = sublimation_pressure(model::CompositeModel,T;v0=x0_sublimation_pressure(model,T))

Calculates the sublimation pressure psub of a CompositeModel containing a solid and fluid phase EoS, at a specified temperature T. You can pass a tuple of initial values for the volumes (vs0,vv0).

returns:

• Sublimation Pressure [Pa]
• Sublimation solid volume at specified temperature [m³]
• Sublimation vapour volume at specified temperature [m³]

pm,vs,vv = sublimation_temperature(model::CompositeModel,p;v0=x0_sublimation_pressure(model,T))

Calculates the sublimation temperature of a CompositeModel containing a solid and fluid phase EoS, at a specified pressure p. You can pass a tuple of initial values for the volumes (vs0,vv0).

returns:

• Sublimation Temperature [K]
• Sublimation solid volume at specified pressure [m³]
• Sublimation vapour volume at specified pressure [m³]

Tt,pt,vs,vl,vv = triple_point(model::CompositeModel;v0 = x0_triple_point(model))

Calculates the triple point of a CompositeModel containing solid and fluid phase EoS.

returns:

• Triple point Temperature [K]
• Triple point Pressure [Pa]
• Solid volume at Triple Point [m³]
• Liquid volume at Triple Point [m³]
• Vapour volume at Triple Point [m³]

ChemPotVSaturation <: SaturationMethod
ChemPotVSaturation(V0)
ChemPotVSaturation(;vl = nothing,
                    vv = nothing,
                    crit = nothing,
                    crit_retry = true
                    f_limit = 0.0,
                    atol = 1e-8,
                    rtol = 1e-12,
                    max_iters = 10^4)

Default saturation_pressure Saturation method used by Clapeyron.jl. It uses equality of Chemical Potentials with a volume basis. If no volumes are provided, it will use x0_sat_pure. When crit_retry is true, if the initial solve fail, it will try to obtain a better estimate by calculating the critical point. f_limit, atol, rtol, max_iters are passed to the non linear system solver.

ChemPotDensitySaturation <: SaturationMethod
ChemPotDensitySaturation(;vl = nothing,
                        vv = nothing,
                        crit = nothing,
                        f_limit = 0.0,
                        atol = 1e-8,
                        rtol = 1e-12,
                        max_iters = 10^4)

Saturation method for saturation_pressure. It uses equality of Chemical Potentials with a density basis. If no volumes are provided, it will use x0_sat_pure. vl and vl are initial guesses for the liquid and vapour volumes. f_limit, atol, rtol, max_iters are passed to the non linear system solver.

IsoFugacitySaturation <: SaturationMethod
IsoFugacitySaturation(;p0 = nothing,
    vl = nothing,
    vv = nothing,
    crit = nothing,
    max_iters = 20,
    p_tol = sqrt(eps(Float64)))

Saturation method for saturation_pressure. Uses the isofugacity criteria. Ideal for Cubics or other EoS where the volume calculations are cheap. If p0 is not provided, it will be calculated via x0_psat.

ClapeyronSaturation <: SaturationMethod
ClapeyronSaturation(T0 = nothing, crit = nothing, satmethod = ChemPotVSaturation())

Saturation method for saturation_temperature. It solves iteratively saturation_temperature(model,Ti,satmethod) until convergence, by using the Clapeyron equation:

dp/dT = ΔS/ΔV

It descends from the critical point (or T0, if provided). Reliable, but slow.

It is recommended that T0 > Tsat, as the temperature decrease iteration series is more stable. Default method for saturation_temperature until Clapeyron 0.3.7

AntoineSaturation <: SaturationMethod
AntoineSaturation(;T0 = nothing,
                    vl = nothing,
                    vv = nothing,
                    f_limit = 0.0,
                    atol = 1e-8,
                    rtol = 1e-12,
                    max_iters = 10^4,
                    crit = nothing,
                    crit_retry = false)

Saturation method for saturation_temperature. Default method for saturation temperature from Clapeyron 0.3.7. It solves the Volume-Temperature system of equations for the saturation condition.

If only T0 is provided, vl and vv are obtained via x0_sat_pure. If T0 is not provided, it will be obtained via x0_saturation_temperature. It is recommended to overload x0_saturation_temperature, as the default starting point calls crit_pure, resulting in slower than ideal times. f_limit, atol, rtol, max_iters are passed to the non linear system solver.

CritExtrapolationSaturation <: SaturationMethod
CritExtrapolationSaturation(;crit = nothing)

Saturation method that extrapolates the liquid and vapor volumes from the critical point. the extrapolation is defined as:

ρₓ = ρc ± sqrt(6* Tc * (Tc - T)/Tc * ∂²p∂ρ∂T / ∂³p∂ρ³)

This method is specially useful at calculating the liquid and vapor volumes when T > 0.999Tc.

References

1. Bell, I. H., & Deiters, U. K. (2023). Superancillary equations for nonpolar pure fluids modeled with the PC-SAFT equation of state. Industrial & Engineering Chemistry Research. doi:10.1021/acs.iecr.2c02916

SuperAncSaturation <: SaturationMethod
SuperAncSaturation()

Saturation method for saturation_pressure. It uses Chebyshev expansions constructed with extended precision arithmetic to obtain the saturation curves of supported EoS within Float64 precision. Models supported are:

• vdW models and variants
• RK models and variants
• PR models and variants
• PCSAFT models and some variants (via EoSSuperancillaries.jl package extension)

References

1. Bell, I. H., & Deiters, U. K. (2021). Superancillary equations for cubic equations of state. Industrial & Engineering Chemistry Research, 60(27), 9983–9991. doi:10.1021/acs.iecr.1c00847

use_superancillaries!(val::Bool = true)

Enable the use of cubic and PC-SAFT superancillaries as initial points for saturation_pressure. For PCSAFT it also enables the use of superancillaries for critical point calculations. This function requires EoSSuperancillaries.jl to be loaded in the current session (using EoSSuperancillaries).

pw, vw = widom_pressure(model, T; p0 = nothing, v0 = nothing)

Calculate the Widom pressure and corresponding volume at a given temperature for a thermodynamic model.

The Widom point represents the temperature T where maxima of the isobaric heat capacity occurs along an isobar in the supercritical region. In particular, widom_pressure calculates the widom point at a specified temperature.

Arguments

• model: Thermodynamic model used for calculations.
• T: Temperature at which to calculate the Widom pressure [K].
• p0: Optional initial pressure guess [Pa]. If not provided, a default value will be used.
• v0: Optional initial volume guess [m³]. If not provided, a default value will be used.

Returns

• pw: Widom pressure at the specified temperature [Pa].
• vw: Volume at the Widom pressure and specified temperature [m³].

Example

model = PCSAFT(["methane"])
T = 200.0  # 200 K
pw, vw = widom_pressure(model, T)

Tw, vw = widom_temperature(model, p; T0 = nothing, v0 = nothing)

Calculate the Widom temperature and corresponding volume at a given pressure for a thermodynamic model.

The Widom point represents the temperature T where maxima of the isobaric heat capacity occurs along an isobar in the supercritical region. In particular, widom_temperature calculates the widom point at a specified temperature.

Arguments

• model: Thermodynamic model used for calculations
• p: Pressure at which to calculate the Widom temperature [Pa]
• T0: Optional initial temperature guess [K]. If not provided, a default value will be used.
• v0: Optional initial volume guess [m³]. If not provided, a default value will be used.

Returns

• Tw: Widom temperature at the specified pressure [K].
• vw: Volume at the Widom temperature and specified pressure [m³].

Example

model = PCSAFT(["methane"])
p = 150e5  #150 bar, over critical pressure
Tw, vw = widom_temperature(model, p)

p_ciic, v_ciic = ciic_pressure(model, T; p0 = nothing, v0 = nothing)

Calculate the Characteristic Isobaric Inflection Curve (CIIC) point at a given temperature for a thermodynamic model.

The CIIC represents points of maximum isobaric heat capacity (Cp) along isotherms, in contrast to Widom lines which represent maxima of Cp along isobars.

Arguments

• model: Thermodynamic model used for calculations.
• T: Temperature at which to calculate the CIIC pressure [K].
• p0: Optional initial pressure guess [Pa]. If not provided, a default value will be used.
• v0: Optional initial volume guess [m³]. If not provided, a default value will be used.

Returns

• p_ciic: Pressure at the CIIC point for the specified temperature [Pa].
• v_ciic: Volume at the CIIC point [m³].

Example

model = PCSAFT(["methane"])
T = 200.0  # 200 K
p_ciic, v_ciic = ciic_pressure(model, T)

T_ciic, v_ciic = ciic_temperature(model, p; T0 = nothing, v0 = nothing)

Calculate the Characteristic Isobaric Inflection Curve (CIIC) point at a given pressure for a thermodynamic model.

The CIIC represents points of maximum isobaric heat capacity (Cp) along isotherms, in contrast to Widom lines which represent maxima of Cp along isobars.

Arguments

• model: Thermodynamic model used for calculations.
• p: Pressure at which to calculate the CIIC temperature [Pa].
• T0: Optional initial temperature guess [K]. If not provided, a default value will be used.
• v0: Optional initial volume guess [m³]. If not provided, a default value will be used.

Returns

• T_ciic: Temperature at the CIIC point for the specified pressure [K].
• v_ciic: Volume at the CIIC point [m³].

Example

model = PCSAFT(["methane"])
p = 150e5  # 150 bar
T_ciic, v_ciic = ciic_temperature(model, p)