Phase equilibrium¶

Vapour-liquid equilibrium by both routes: an equation of state for both phases (phi-phi) and an activity-coefficient liquid with an EOS or ideal vapour (gamma-phi). The unified EOSModel / GammaPhiModel interface lets the rest of the stack switch thermodynamic models without changing call sites.

See the non-ideal phase equilibrium guide for worked examples.

EOS (phi-phi) equilibrium¶

equilibrium ¶

Phase equilibrium: K-values, Rachford-Rice, flash, saturation, and stability.

This module turns the cubic equation of state (fugacio.thermo.eos) into the equilibrium calculations a process simulator actually calls:

wilson_k: the classic K-value initial guess;
rachford_rice: the material-balance root for the vapour fraction;
flash_pt: an isothermal-isobaric two-phase flash;
psat_eos: pure-component saturation pressure by equifugacity;
bubble_pressure_eos / dew_pressure_eos: phase envelopes;
stability_analysis: Michelsen's tangent-plane-distance test.

Every iterative result is differentiable end-to-end: the scalar solves carry hand-written implicit-function-theorem rules (jax.custom_jvp) and the flash/saturation loops reuse fugacio.thermo.implicit.fixed_point. You can therefore take a gradient of any equilibrium output with respect to T, P, composition, or model parameters, the property that makes Fugacio a differentiable core rather than just another flash package.

Classes:

Name	Description
`FlashResult`	Result of an isothermal-isobaric flash.
`StabilityResult`	Result of a tangent-plane stability analysis.

Functions:

Name	Description
`wilson_k`	Wilson correlation for initial K-values `K_i = y_i / x_i`.
`rachford_rice`	Solve the Rachford-Rice equation for the vapour fraction `beta`.
`flash_pt`	Isothermal-isobaric two-phase flash by accelerated successive substitution.
`psat_eos`	Pure-component saturation pressure (Pa) from the EOS by equifugacity.
`bubble_pressure_eos`	Bubble-point pressure and incipient vapour composition at fixed `T`, `x`.
`dew_pressure_eos`	Dew-point pressure and incipient liquid composition at fixed `T`, `y`.
`stability_analysis`	Michelsen tangent-plane stability test for a feed `z` at `(T, P)`.

FlashResult ¶

Bases: NamedTuple

Result of an isothermal-isobaric flash.

Attributes:

Name	Type	Description
`beta`	`Array`	Vapour molar fraction (mol vapour / mol feed).
`x`	`Array`	Liquid-phase mole fractions.
`y`	`Array`	Vapour-phase mole fractions.
`k`	`Array`	Equilibrium ratios `K_i = y_i / x_i` at the solution.

StabilityResult ¶

Bases: NamedTuple

Result of a tangent-plane stability analysis.

Attributes:

Name	Type	Description
`stable`	`Array`	`True` if the feed is single-phase stable.
`tpd`	`Array`	The smallest (most negative) modified tangent-plane distance found.

wilson_k ¶

wilson_k(
    t: ArrayLike,
    p: ArrayLike,
    tc: Array,
    pc: Array,
    omega: Array,
) -> Array

Wilson correlation for initial K-values K_i = y_i / x_i.

K_i = (Pc_i / P) * exp[5.373 (1 + omega_i)(1 - Tc_i / T)].

rachford_rice ¶

rachford_rice(z: Array, k: Array) -> Array

Solve the Rachford-Rice equation for the vapour fraction beta.

Returns beta clamped to [0, 1]: 0 for a subcooled liquid, 1 for a superheated vapour, and the interior root otherwise. The residual sum_i z_i (K_i - 1) / (1 + beta (K_i - 1)) is monotonically decreasing in beta on (0, 1), so a bisection is used for the interior root.

flash_pt ¶

flash_pt(
    eos: CubicEOS,
    t: ArrayLike,
    p: ArrayLike,
    z: Array,
    tc: Array,
    pc: Array,
    omega: Array,
    *,
    kij: Array | None = None,
    tol: float = 1e-12,
    max_iter: int = 300,
) -> FlashResult

Isothermal-isobaric two-phase flash by accelerated successive substitution.

Solves the equal-fugacity conditions phi_i^L x_i = phi_i^V y_i together with the Rachford-Rice material balance, starting from Wilson K-values. The converged solution (and therefore beta, x, y) is differentiable with respect to (T, P, z, ...) via implicit differentiation of the fixed point.

psat_eos ¶

psat_eos(
    eos: CubicEOS,
    t: ArrayLike,
    tc: ArrayLike,
    pc: ArrayLike,
    omega: ArrayLike,
    tol: float = 1e-10,
    max_iter: int = 100,
) -> Array

Pure-component saturation pressure (Pa) from the EOS by equifugacity.

Solves ln phi^L(T, P) = ln phi^V(T, P) for P with a Newton iteration in ln P (which keeps the pressure positive), initialised from the Wilson vapour-pressure estimate. Differentiable in T (and the critical constants) via the Clapeyron-like implicit derivative dP/dT.

bubble_pressure_eos ¶

bubble_pressure_eos(
    eos: CubicEOS,
    t: ArrayLike,
    x: Array,
    tc: Array,
    pc: Array,
    omega: Array,
    *,
    kij: Array | None = None,
    tol: float = 1e-12,
    max_iter: int = 300,
) -> tuple[Array, Array]

Bubble-point pressure and incipient vapour composition at fixed T, x.

Returns (P, y). Solved as a coupled fixed point in (ln P, y) so the result is differentiable in temperature and composition.

dew_pressure_eos ¶

dew_pressure_eos(
    eos: CubicEOS,
    t: ArrayLike,
    y: Array,
    tc: Array,
    pc: Array,
    omega: Array,
    *,
    kij: Array | None = None,
    tol: float = 1e-12,
    max_iter: int = 300,
) -> tuple[Array, Array]

Dew-point pressure and incipient liquid composition at fixed T, y.

Returns (P, x), differentiable in temperature and composition.

stability_analysis ¶

stability_analysis(
    eos: CubicEOS,
    t: ArrayLike,
    p: ArrayLike,
    z: Array,
    tc: Array,
    pc: Array,
    omega: Array,
    *,
    kij: Array | None = None,
    iters: int = 40,
) -> StabilityResult

Michelsen tangent-plane stability test for a feed z at (T, P).

Performs two trial-phase searches (vapour-like and liquid-like). If the modified tangent-plane distance tm dips below zero for either trial, a second phase can lower the Gibbs energy and the feed is unstable (it will split). Returns the worst (smallest) tm found and a stability flag.

Gamma-phi equilibrium¶

gammaphi ¶

Gamma-phi vapour-liquid equilibrium (activity-coefficient liquid model).

The cubic-EOS flash in fugacio.thermo.equilibrium describes both phases with one equation of state (the phi-phi approach). For the low-pressure, polar, strongly non-ideal mixtures that dominate real separations (ethanol/water and other azeotropes, alcohol/ketone/water systems), a cubic EOS with zero binary interaction parameters is simply the wrong tool. The standard answer is the gamma-phi approach: model the liquid with an activity-coefficient model and the vapour with an equation of state (or as an ideal gas at low pressure).

Equilibrium equates the component fugacities

x_i gamma_i(x, T) f_i^{0,L}(T, P) = y_i phi_i^V(y, T, P) P

so the K-values are

K_i = y_i / x_i = gamma_i f_i^{0,L} / (phi_i^V P).

With an ideal vapour (phi^V = 1) and the plain saturation reference (f^{0,L} = Psat), this collapses to modified Raoult's law K_i = gamma_i Psat_i / P, enough to reproduce azeotropes that the zero-kij cubic cannot. The richer reference (saturation fugacity coefficient + Poynting, see fugacio.thermo.reference) and an EOS vapour are available via keyword flags.

Every routine here, K-values, the four saturation calculations (bubble/dew at fixed T or P), and the isothermal flash, is a fixed point or a bracketed root solved by the implicit-diff primitives in fugacio.thermo.implicit, so each output is differentiable end-to-end with respect to T, P, composition, and the activity-model parameters. That last point is what turns parameter regression (fugacio.thermo.regression) into plain gradient descent.

Functions:

Name	Description
`gamma_phi_k_values`	Gamma-phi K-values `K_i = gamma_i f_i^{0,L} / (phi_i^V P)`.
`bubble_pressure_gamma`	Bubble-point pressure and incipient vapour composition at fixed `T`, `x`.
`dew_pressure_gamma`	Dew-point pressure and incipient liquid composition at fixed `T`, `y`.
`bubble_temperature_gamma`	Bubble-point temperature and incipient vapour at fixed `P`, `x`.
`dew_temperature_gamma`	Dew-point temperature and incipient liquid at fixed `P`, `y`.
`flash_pt_gamma`	Isothermal-isobaric gamma-phi flash by accelerated successive substitution.

gamma_phi_k_values ¶

gamma_phi_k_values(
    model: ActivityModel,
    t: ArrayLike,
    p: ArrayLike,
    x: Array,
    y: Array,
    tc: Array,
    pc: Array,
    omega: Array,
    *,
    eos: CubicEOS = PR,
    kij: Array | None = None,
    vapor: str = "ideal",
    poynting: bool = False,
    phi_saturation: bool = False,
) -> Array

Gamma-phi K-values K_i = gamma_i f_i^{0,L} / (phi_i^V P).

Parameters:

Name	Type	Description	Default
`model`	`ActivityModel`	Liquid activity-coefficient model.	required
`t`	`ArrayLike`	Temperature (K).	required
`p`	`ArrayLike`	Pressure (Pa).	required
`x`	`Array`	Liquid mole fractions.	required
`y`	`Array`	Vapour mole fractions (only matters for an EOS vapour).	required
`tc`	`Array`	Component critical temperatures (K).	required
`pc`	`Array`	Component critical pressures (Pa).	required
`omega`	`Array`	Component acentric factors.	required
`eos`	`CubicEOS`	Cubic EOS for the saturation reference and (if selected) the vapour.	`PR`
`kij`	`Array \| None`	Optional binary interaction matrix for the vapour EOS.	`None`
`vapor`	`str`	`"ideal"` (phi^V = 1) or `"eos"`.	`'ideal'`
`poynting`	`bool`	Include the Poynting pressure correction in the reference.	`False`
`phi_saturation`	`bool`	Include the saturation fugacity coefficient in the reference.	`False`

Returns:

Type	Description
`Array`	K-values aligned with `x`.

bubble_pressure_gamma ¶

bubble_pressure_gamma(
    model: ActivityModel,
    t: ArrayLike,
    x: Array,
    tc: Array,
    pc: Array,
    omega: Array,
    *,
    eos: CubicEOS = PR,
    kij: Array | None = None,
    vapor: str = "ideal",
    poynting: bool = False,
    phi_saturation: bool = False,
    tol: float = 1e-12,
    max_iter: int = 200,
) -> tuple[Array, Array]

Bubble-point pressure and incipient vapour composition at fixed T, x.

Solved as a coupled fixed point in (ln P, y): K-values give an unnormalised vapour y* = K x whose sum scales the pressure until it is one. Returns (P, y), differentiable in T, x and the model parameters.

dew_pressure_gamma ¶

dew_pressure_gamma(
    model: ActivityModel,
    t: ArrayLike,
    y: Array,
    tc: Array,
    pc: Array,
    omega: Array,
    *,
    eos: CubicEOS = PR,
    kij: Array | None = None,
    vapor: str = "ideal",
    poynting: bool = False,
    phi_saturation: bool = False,
    tol: float = 1e-12,
    max_iter: int = 200,
) -> tuple[Array, Array]

Dew-point pressure and incipient liquid composition at fixed T, y.

Coupled fixed point in (ln P, x): with x_i = y_i / K_i (and the activity coefficients re-evaluated at the updated x), the pressure is scaled until the liquid sums to one. Returns (P, x).

bubble_temperature_gamma ¶

bubble_temperature_gamma(
    model: ActivityModel,
    p: ArrayLike,
    x: Array,
    tc: Array,
    pc: Array,
    omega: Array,
    *,
    eos: CubicEOS = PR,
    kij: Array | None = None,
    vapor: str = "ideal",
    poynting: bool = False,
    phi_saturation: bool = False,
    t_min: float = 150.0,
    t_max: float = 700.0,
    tol: float = 1e-09,
    max_iter: int = 200,
    inner_iter: int = 0,
) -> tuple[Array, Array]

Bubble-point temperature and incipient vapour at fixed P, x.

The bubble temperature is the root of sum_i K_i(T) x_i = 1 (the saturation sum is monotone in T), found with the bracketed solver and differentiated by the implicit function theorem. Returns (T, y). For an ideal vapour the K-values are independent of y and inner_iter can stay zero; raise it for an EOS vapour so the incipient y settles before the bracket step.

dew_temperature_gamma ¶

dew_temperature_gamma(
    model: ActivityModel,
    p: ArrayLike,
    y: Array,
    tc: Array,
    pc: Array,
    omega: Array,
    *,
    eos: CubicEOS = PR,
    kij: Array | None = None,
    vapor: str = "ideal",
    poynting: bool = False,
    phi_saturation: bool = False,
    t_min: float = 150.0,
    t_max: float = 700.0,
    tol: float = 1e-09,
    max_iter: int = 200,
    inner_iter: int = 30,
) -> tuple[Array, Array]

Dew-point temperature and incipient liquid at fixed P, y.

Root of sum_i (y_i / K_i(T)) = 1 in T with an inner sweep that settles the incipient liquid x (on which the activity coefficients depend) at each trial temperature. Returns (T, x).

flash_pt_gamma ¶

flash_pt_gamma(
    model: ActivityModel,
    t: ArrayLike,
    p: ArrayLike,
    z: Array,
    tc: Array,
    pc: Array,
    omega: Array,
    *,
    eos: CubicEOS = PR,
    kij: Array | None = None,
    vapor: str = "ideal",
    poynting: bool = False,
    phi_saturation: bool = False,
    tol: float = 1e-12,
    max_iter: int = 300,
) -> FlashResult

Isothermal-isobaric gamma-phi flash by accelerated successive substitution.

Iterates the gamma-phi K-values to a fixed point in ln K with the Rachford-Rice material balance closing the phase split at each step. The converged beta, x, y are differentiable with respect to (T, P, z) and the activity-model parameters.

Unified model interface¶

phase ¶

A uniform phase-equilibrium model: one interface over EOS and gamma-phi.

The two routes to vapour-liquid equilibrium, a cubic equation of state for both phases (fugacio.thermo.equilibrium) and an activity model for the liquid with an EOS/ideal vapour (fugacio.thermo.gammaphi), have, until now, different call signatures. That made the rest of the stack (flashes inside unit operations, column K-values, the copilot tools) hard-wire the EOS. This module unifies them behind a single EquilibriumModel interface so a flowsheet can be switched from Peng-Robinson to NRTL by swapping one object.

Both concrete models (EOSModel and GammaPhiModel) bundle the component constants they need and expose the same five calls: flash_pt, bubble_pressure/bubble_temperature and dew_pressure/dew_temperature. They are registered JAX pytrees whose parameters (critical constants, kij, and, for the gamma-phi model, the activity-model parameters) are differentiable leaves, while structural choices (which cubic, ideal-vs-EOS vapour) are static metadata. A flowsheet built on a model therefore stays end-to-end differentiable, including with respect to the thermodynamic model's own parameters.

Classes:

Name	Description
`EquilibriumModel`	Structural type for a vapour-liquid equilibrium model.
`EOSModel`	Equation-of-state (phi-phi) equilibrium model.
`GammaPhiModel`	Gamma-phi equilibrium model: activity-coefficient liquid, EOS/ideal vapour.

Functions:

Name	Description
`eos_model`	Construct an `EOSModel` from component constants.
`gamma_phi_model`	Construct a `GammaPhiModel` from an activity model and constants.

EquilibriumModel ¶

Bases: Protocol

Structural type for a vapour-liquid equilibrium model.

Every model maps operating conditions to a phase split (flash_pt) and to the four saturation calculations. Implementations bundle their own component constants, so callers pass only state (T, P, composition).

Methods:

Name	Description
`flash_pt`	Isothermal-isobaric two-phase flash.
`bubble_pressure`	Bubble pressure and incipient vapour `(P, y)` at fixed `T`, `x`.
`dew_pressure`	Dew pressure and incipient liquid `(P, x)` at fixed `T`, `y`.
`bubble_temperature`	Bubble temperature and incipient vapour `(T, y)` at fixed `P`, `x`.
`dew_temperature`	Dew temperature and incipient liquid `(T, x)` at fixed `P`, `y`.

flash_pt ¶

flash_pt(
    t: ArrayLike, p: ArrayLike, z: Array
) -> FlashResult

Isothermal-isobaric two-phase flash.

bubble_pressure ¶

bubble_pressure(
    t: ArrayLike, x: Array
) -> tuple[Array, Array]

Bubble pressure and incipient vapour (P, y) at fixed T, x.

dew_pressure ¶

dew_pressure(t: ArrayLike, y: Array) -> tuple[Array, Array]

Dew pressure and incipient liquid (P, x) at fixed T, y.

bubble_temperature ¶

bubble_temperature(
    p: ArrayLike, x: Array
) -> tuple[Array, Array]

Bubble temperature and incipient vapour (T, y) at fixed P, x.

dew_temperature ¶

dew_temperature(
    p: ArrayLike, y: Array
) -> tuple[Array, Array]

Dew temperature and incipient liquid (T, x) at fixed P, y.

EOSModel `dataclass` ¶

EOSModel(
    tc: Array,
    pc: Array,
    omega: Array,
    kij: Array | None,
    eos: CubicEOS,
)

Equation-of-state (phi-phi) equilibrium model.

Attributes:

Name	Type	Description
`tc,`	`(pc, omega)`	Component critical constants and acentric factors.
`kij`	`Array \| None`	Binary interaction matrix (`None` => zeros).
`eos`	`CubicEOS`	Cubic equation of state (static; default Peng-Robinson).

Methods:

Name	Description
`flash_pt`	Isothermal-isobaric two-phase flash via the cubic EOS.
`bubble_pressure`	Bubble pressure and incipient vapour at fixed `T`, `x`.
`dew_pressure`	Dew pressure and incipient liquid at fixed `T`, `y`.
`bubble_temperature`	Bubble temperature and incipient vapour at fixed `P`, `x`.
`dew_temperature`	Dew temperature and incipient liquid at fixed `P`, `y`.
`stability`	Michelsen tangent-plane stability of feed `z` at `(T, P)`.

flash_pt ¶

flash_pt(
    t: ArrayLike, p: ArrayLike, z: Array
) -> FlashResult

Isothermal-isobaric two-phase flash via the cubic EOS.

bubble_pressure ¶

bubble_pressure(
    t: ArrayLike, x: Array
) -> tuple[Array, Array]

Bubble pressure and incipient vapour at fixed T, x.

dew_pressure ¶

dew_pressure(t: ArrayLike, y: Array) -> tuple[Array, Array]

Dew pressure and incipient liquid at fixed T, y.

bubble_temperature ¶

bubble_temperature(
    p: ArrayLike,
    x: Array,
    *,
    t_min: float = 150.0,
    t_max: float = 700.0,
) -> tuple[Array, Array]

Bubble temperature and incipient vapour at fixed P, x.

Found by inverting bubble_pressure for the temperature whose bubble pressure equals P (the saturation pressure rises monotonically with temperature), then returning the incipient vapour there.

dew_temperature ¶

dew_temperature(
    p: ArrayLike,
    y: Array,
    *,
    t_min: float = 150.0,
    t_max: float = 700.0,
) -> tuple[Array, Array]

Dew temperature and incipient liquid at fixed P, y.

stability ¶

stability(
    t: ArrayLike, p: ArrayLike, z: Array
) -> StabilityResult

Michelsen tangent-plane stability of feed z at (T, P).

GammaPhiModel `dataclass` ¶

GammaPhiModel(
    activity: ActivityModel,
    tc: Array,
    pc: Array,
    omega: Array,
    kij: Array | None,
    eos: CubicEOS,
    vapor: str,
    poynting: bool,
    phi_saturation: bool,
)

Gamma-phi equilibrium model: activity-coefficient liquid, EOS/ideal vapour.

Attributes:

Name	Type	Description
`activity`	`ActivityModel`	Liquid activity-coefficient model (a differentiable pytree).
`tc,`	`(pc, omega)`	Component critical constants and acentric factors.
`kij`	`Array \| None`	Binary interaction matrix for an EOS vapour (`None` => zeros).
`eos`	`CubicEOS`	Cubic EOS used for the saturation reference and (if selected) vapour.
`vapor`	`str`	`"ideal"` (phi^V = 1) or `"eos"` (static).
`poynting`	`bool`	Include the Poynting correction in the reference (static).
`phi_saturation`	`bool`	Include the saturation fugacity coefficient (static).

Methods:

Name	Description
`flash_pt`	Isothermal-isobaric gamma-phi flash.
`bubble_pressure`	Bubble pressure and incipient vapour at fixed `T`, `x`.
`dew_pressure`	Dew pressure and incipient liquid at fixed `T`, `y`.
`bubble_temperature`	Bubble temperature and incipient vapour at fixed `P`, `x`.
`dew_temperature`	Dew temperature and incipient liquid at fixed `P`, `y`.

flash_pt ¶

flash_pt(
    t: ArrayLike, p: ArrayLike, z: Array
) -> FlashResult

Isothermal-isobaric gamma-phi flash.

bubble_pressure ¶

bubble_pressure(
    t: ArrayLike, x: Array
) -> tuple[Array, Array]

Bubble pressure and incipient vapour at fixed T, x.

dew_pressure ¶

dew_pressure(t: ArrayLike, y: Array) -> tuple[Array, Array]

Dew pressure and incipient liquid at fixed T, y.

bubble_temperature ¶

bubble_temperature(
    p: ArrayLike,
    x: Array,
    *,
    t_min: float = 150.0,
    t_max: float = 700.0,
) -> tuple[Array, Array]

Bubble temperature and incipient vapour at fixed P, x.

dew_temperature ¶

dew_temperature(
    p: ArrayLike,
    y: Array,
    *,
    t_min: float = 150.0,
    t_max: float = 700.0,
) -> tuple[Array, Array]

Dew temperature and incipient liquid at fixed P, y.

eos_model ¶

eos_model(
    tc: Array,
    pc: Array,
    omega: Array,
    *,
    kij: Array | None = None,
    eos: CubicEOS = PR,
) -> EOSModel

Construct an EOSModel from component constants.

gamma_phi_model ¶

gamma_phi_model(
    activity: ActivityModel,
    tc: Array,
    pc: Array,
    omega: Array,
    *,
    kij: Array | None = None,
    eos: CubicEOS = PR,
    vapor: str = "ideal",
    poynting: bool = False,
    phi_saturation: bool = False,
) -> GammaPhiModel

Construct a GammaPhiModel from an activity model and constants.

Tangent-plane stability¶

stability ¶

Tangent-plane stability analysis for an arbitrary fugacity model.

A mixture of overall composition z is stable as a single phase only if no trial composition w lowers the Gibbs energy, i.e. if the (modified) tangent-plane distance

tm(w) = 1 + sum_i W_i (ln W_i + ln coeff_i(w) - d_i - 1),   d_i = ln z_i + ln coeff_i(z)

stays non-negative at every stationary point (w = W / sum W). Here coeff_i is whatever turns a composition into a fugacity: the activity coefficient gamma_i for a liquid activity model, or the fugacity coefficient phi_i for an equation of state. Casting the test in terms of a generic ln_coeff_fn makes one implementation serve both worlds.

The companion of equilibrium: while fugacio.thermo.equilibrium answers "given that it splits, into what?", this module answers "does it split at all?", the test that decides whether a feed is one phase, needs a VLE flash, or (for a liquid activity model with a miscibility gap) needs the liquid-liquid solver in fugacio.thermo.lle. The most negative trial result also gives an excellent initial guess for those splits, which those modules consume directly.

Classes:

Name	Description
`TangentPlaneResult`	Outcome of a tangent-plane stability search.

Functions:

Name	Description
`tangent_plane_distance`	Gibbs tangent-plane distance of trial phase `w` relative to feed `z`.
`stability_analysis_general`	Michelsen stability test for a generic `ln_coeff_fn` and trial set.
`liquid_stability`	Test a liquid of composition `z` for splitting into two liquids at `T`.

TangentPlaneResult ¶

Bases: NamedTuple

Outcome of a tangent-plane stability search.

Attributes:

Name	Type	Description
`stable`	`Array`	`True` if the feed is single-phase stable (no negative `tm`).
`tpd`	`Array`	The smallest modified tangent-plane distance found across all trials.
`split`	`Array`	The (normalised) trial composition that achieved `tpd`, a ready initial guess for the incipient phase when the feed is unstable.

tangent_plane_distance ¶

tangent_plane_distance(
    ln_coeff_fn: LnCoeffFn, z: Array, w: Array
) -> Array

Gibbs tangent-plane distance of trial phase w relative to feed z.

tpd(w) = sum_i w_i (ln w_i + ln coeff_i(w) - ln z_i - ln coeff_i(z)); a negative value means the trial phase w lies below the tangent plane at z and the feed can lower its Gibbs energy by forming it.

stability_analysis_general ¶

stability_analysis_general(
    ln_coeff_fn: LnCoeffFn,
    z: Array,
    trials: Array,
    *,
    iters: int = 50,
    tol: float = 1e-08,
) -> TangentPlaneResult

Michelsen stability test for a generic ln_coeff_fn and trial set.

Parameters:

Name	Type	Description	Default
`ln_coeff_fn`	`LnCoeffFn`	Maps a composition to `ln(coeff_i)` (`ln gamma` or `ln phi`) at the fixed temperature/pressure of interest.	required
`z`	`Array`	Feed composition.	required
`trials`	`Array`	Stack of initial trial-phase compositions, shape `(n_trials, n)`.	required
`iters`	`int`	Successive-substitution iterations per trial.	`50`
`tol`	`float`	Negative-`tm` threshold below which the feed is declared unstable.	`1e-08`

Returns:

Type	Description
`TangentPlaneResult`	A `TangentPlaneResult`.

liquid_stability ¶

liquid_stability(
    model: ActivityModel,
    t: ArrayLike,
    z: Array,
    *,
    iters: int = 60,
    tol: float = 1e-08,
    strength: float = 0.9,
) -> TangentPlaneResult

Test a liquid of composition z for splitting into two liquids at T.

Uses the activity coefficients as the ln_coeff_fn and trial phases enriched toward each pure component. A negative tpd flags a miscibility gap; the returned split seeds fugacio.thermo.lle.flash_lle.

Phase equilibrium¶

EOS (phi-phi) equilibrium¶

equilibrium ¶

FlashResult ¶

StabilityResult ¶

wilson_k ¶

rachford_rice ¶

flash_pt ¶

psat_eos ¶

bubble_pressure_eos ¶

dew_pressure_eos ¶

stability_analysis ¶

Gamma-phi equilibrium¶

gammaphi ¶

gamma_phi_k_values ¶

bubble_pressure_gamma ¶

dew_pressure_gamma ¶

bubble_temperature_gamma ¶

dew_temperature_gamma ¶

flash_pt_gamma ¶

Unified model interface¶

phase ¶

EquilibriumModel ¶

flash_pt ¶

bubble_pressure ¶

dew_pressure ¶

bubble_temperature ¶

dew_temperature ¶

EOSModel dataclass ¶

flash_pt ¶

bubble_pressure ¶

dew_pressure ¶

bubble_temperature ¶

dew_temperature ¶

stability ¶

GammaPhiModel dataclass ¶

flash_pt ¶

bubble_pressure ¶

dew_pressure ¶

bubble_temperature ¶

dew_temperature ¶

eos_model ¶

gamma_phi_model ¶

Tangent-plane stability¶

stability ¶

TangentPlaneResult ¶

tangent_plane_distance ¶

stability_analysis_general ¶

liquid_stability ¶

EOSModel `dataclass` ¶

GammaPhiModel `dataclass` ¶