Liquid-liquid & vapour-liquid-liquid equilibrium¶

Rigorous two-liquid (LLE) and three-phase (VLLE) flashes, heterogeneous azeotropes, and the tangent-plane machinery that decides when a second liquid appears.

Liquid-liquid equilibrium¶

lle ¶

Liquid-liquid equilibrium (LLE): two-liquid flash, tie-lines, and binodals.

Partially miscible liquids split into two phases I and II whose component fugacities are equal. With an activity-coefficient model that equality is the isoactivity condition

x_i^I gamma_i(x^I, T) = x_i^II gamma_i(x^II, T),

i.e. K_i = x_i^II / x_i^I = gamma_i^I / gamma_i^II, closed by the same Rachford-Rice material balance as a vapour-liquid flash, with the phase fraction psi now the mole fraction in phase II:

sum_i z_i (K_i - 1) / (1 + psi (K_i - 1)) = 0,
x_i^I = z_i / (1 + psi (K_i - 1)),   x_i^II = K_i x_i^I.

The flash is stability-first. The tangent-plane search of fugacio.thermo.stability.liquid_stability decides whether a split exists: an established stable liquid is returned as one phase (psi = 0), and an unstable one seeds the isoactivity iteration from its most negative trial phase, which avoids the ever-present trivial solution x^I = x^II = z. An iteration that still collapses onto it is reported as TRIVIAL. The converged split is differentiable in temperature, feed, and the model parameters via implicit differentiation of the fixed point, so tie-lines move smoothly under a gradient, which matters for solvent-selection optimisation and parameter fitting to mutual-solubility data.

Only an activity model can describe an LLE; Wilson's model is excluded by construction (it has no miscibility gap), so use NRTL, UNIQUAC, or UNIFAC.

Classes:

Name	Description
`LLEResult`	Result of a two-liquid (LLE) flash.
`LLESolveResult`	Two-liquid flash and its report.

Functions:

Name	Description
`flash_lle_with_info`	Split feed `z` into two liquids at temperature `t` (isoactivity flash).
`flash_lle`	Two-liquid flash value; NaN when `flash_lle_with_info` reports failure.
`binary_binodal`	Mutual-solubility (binodal) compositions of a binary at temperature `t`.
`tie_line`	One ternary tie-line through feed `z`: `(x_raffinate, x_extract, psi)`.
`binodal_curve`	Binary binodal branches over a temperature range.

LLEResult ¶

Bases: NamedTuple

Result of a two-liquid (LLE) flash.

Attributes:

Name	Type	Description
`psi`	`Array`	Mole fraction of the feed in liquid phase `II`.
`x_i`	`Array`	Phase `I` (raffinate-like) mole fractions.
`x_ii`	`Array`	Phase `II` (extract-like) mole fractions.
`k`	`Array`	Distribution ratios `K_i = x_i^II / x_i^I` at the solution.

LLESolveResult ¶

Bases: NamedTuple

Two-liquid flash and its report.

flash_lle_with_info ¶

flash_lle_with_info(
    model: ActivityModel,
    t: ArrayLike,
    z: Array,
    *,
    k_guess: Array | None = None,
    tol: float = 1e-12,
    max_iter: int = 400,
) -> LLESolveResult

Split feed z into two liquids at temperature t (isoactivity flash).

Parameters:

Name	Type	Description	Default
`model`	`ActivityModel`	Liquid activity-coefficient model (must admit a miscibility gap).	required
`t`	`ArrayLike`	Temperature (K).	required
`z`	`Array`	Overall (feed) mole fractions; absent components stay absent.	required
`k_guess`	`Array \| None`	Optional initial distribution ratios `K_i`. If omitted, the tangent-plane stability test decides whether a split exists and seeds it.	`None`
`tol`	`float`	Fixed-point convergence tolerance.	`1e-12`
`max_iter`	`int`	Maximum number of fixed-point iterations.	`400`

Returns:

Type	Description
`LLESolveResult`	The split and its report. An established stable liquid converges to one
`LLESolveResult`	phase (`psi = 0`, both compositions equal to `z`); a collapse onto the
`LLESolveResult`	trivial solution from an unstable feed is `TRIVIAL`.

flash_lle ¶

flash_lle(
    model: ActivityModel,
    t: ArrayLike,
    z: Array,
    *,
    k_guess: Array | None = None,
    tol: float = 1e-12,
    max_iter: int = 400,
) -> LLEResult

Two-liquid flash value; NaN when flash_lle_with_info reports failure.

binary_binodal ¶

binary_binodal(
    model: ActivityModel,
    t: ArrayLike,
    *,
    feed: ArrayLike = 0.5,
    tol: float = 1e-12,
    max_iter: int = 400,
) -> tuple[Array, Array]

Mutual-solubility (binodal) compositions of a binary at temperature t.

Returns (x1_phase_I, x1_phase_II), the mole fraction of component 1 in each conjugate liquid (the tie-line ends). Any feed inside the gap gives the same pair; the default 50/50 feed sits squarely in a symmetric gap. Outside the gap (no split) both are NaN.

tie_line ¶

tie_line(
    model: ActivityModel,
    t: ArrayLike,
    z: Array,
    *,
    tol: float = 1e-12,
    max_iter: int = 400,
) -> tuple[Array, Array, Array]

One ternary tie-line through feed z: (x_raffinate, x_extract, psi).

A thin wrapper over flash_lle returning the two conjugate-phase compositions and the phase fraction, the unit of a ternary LLE diagram.

binodal_curve ¶

binodal_curve(
    model: ActivityModel,
    temperatures: Array,
    *,
    feed: ArrayLike = 0.5,
    tol: float = 1e-12,
    max_iter: int = 400,
) -> tuple[Array, Array]

Binary binodal branches over a temperature range.

Maps binary_binodal across temperatures and returns (x1_phase_I, x1_phase_II) arrays aligned with the input, the two branches of the solubility envelope that meet at the upper (or lower) critical solution temperature (NaN beyond it).

Vapour-liquid-liquid equilibrium¶

vlle ¶

Three-phase vapour-liquid-liquid equilibrium (VLLE).

When a vapour coexists with two partially miscible liquids (the heterogeneous azeotropes that make water/organic distillation and decantation work), neither a two-phase VLE flash nor an LLE flash alone suffices. VLLE couples them: a vapour V and two liquids I and II all at equal temperature, pressure, and component fugacity.

Following Michelsen, both non-reference phases are referred to liquid I through two sets of K-values,

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

so the material balance gives x_i^I = z_i / D_i with D_i = 1 + beta_V (K_i^V - 1) + beta_II (K_i^L - 1). The two phase fractions (beta_V, beta_II) solve the pair of Rachford-Rice equations sum_i z_i (K_i^V - 1)/D_i = 0 and sum_i z_i (K_i^L - 1)/D_i = 0 (a small 2x2 Newton with an analytic Jacobian), and the K-values are updated from the fugacity equalities until consistent.

The flash is seeded from a VLE flash followed by a stability-driven split of its liquid. When no three-phase solution exists (the Rachford-Rice root leaves the phase-fraction simplex) the flash reports INFEASIBLE rather than clipping phase fractions away from their compositions: use the two-phase VLE or LLE flash there. A converged three-phase state is differentiable through the fixed point.

Classes:

Name	Description
`HeterogeneousAzeotrope`	A binary heterogeneous (two-liquid) azeotrope at fixed pressure.
`VLLEResult`	Result of a three-phase vapour-liquid-liquid flash.
`VLLESolveResult`	Three-phase flash and its report.

Functions:

Name	Description
`flash_vlle_with_info`	Isothermal-isobaric three-phase (V-L-L) flash of feed `z` at `(T, P)`.
`flash_vlle`	Three-phase flash value; NaN when `flash_vlle_with_info` reports failure.
`heterogeneous_azeotrope`	Binary heterogeneous azeotrope: the boiling temperature of two conjugate liquids.

HeterogeneousAzeotrope ¶

Bases: NamedTuple

A binary heterogeneous (two-liquid) azeotrope at fixed pressure.

Attributes:

Name	Type	Description
`t`	`Array`	Azeotrope temperature (K).
`x_i`	`Array`	Composition of liquid `I`.
`x_ii`	`Array`	Composition of liquid `II`.
`y`	`Array`	Common vapour composition in equilibrium with both liquids.

VLLEResult ¶

Bases: NamedTuple

Result of a three-phase vapour-liquid-liquid flash.

Attributes:

Name	Type	Description
`beta_v`	`Array`	Vapour mole fraction of the feed.
`beta_l1`	`Array`	Mole fraction in liquid `I` (the reference liquid).
`beta_l2`	`Array`	Mole fraction in liquid `II`.
`y`	`Array`	Vapour composition.
`x_i`	`Array`	Liquid `I` composition.
`x_ii`	`Array`	Liquid `II` composition.
`three_phase`	`Array`	`True` when all three phase fractions are strictly positive.

VLLESolveResult ¶

Bases: NamedTuple

Three-phase flash and its report.

flash_vlle_with_info ¶

flash_vlle_with_info(
    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-11,
    max_iter: int = 300,
) -> VLLESolveResult

Isothermal-isobaric three-phase (V-L-L) flash of feed z at (T, P).

Seeds a vapour/liquid split from the gamma-phi VLE flash and a liquid/liquid split of its liquid from fugacio.thermo.lle.flash_lle_with_info, then drives both K-value sets to consistency with a 2x2 Rachford-Rice inner solve. Phase fractions and compositions come from the same root, so the returned state always closes the material balance.

Returns:

Type	Description
`VLLESolveResult`	The state and its report. The report is `INFEASIBLE` (and the value
`VLLESolveResult`	must not be used) when the converged root has a phase fraction outside
`VLLESolveResult`	`[0, 1]`, i.e. the feed isn't three-phase at `(T, P)`.

flash_vlle ¶

flash_vlle(
    model: ActivityModel,
    t: ArrayLike,
    p: ArrayLike,
    z: Array,
    tc: Array,
    pc: Array,
    omega: Array,
    **options: Any,
) -> VLLEResult

Three-phase flash value; NaN when flash_vlle_with_info reports failure.

heterogeneous_azeotrope ¶

heterogeneous_azeotrope(
    model: ActivityModel,
    p: ArrayLike,
    tc: Array,
    pc: Array,
    omega: Array,
    *,
    eos: CubicEOS = PR,
    feed: ArrayLike = 0.5,
    t_min: float = 250.0,
    t_max: float = 500.0,
    tol: float = 1e-09,
    max_iter: int = 200,
) -> HeterogeneousAzeotrope

Binary heterogeneous azeotrope: the boiling temperature of two conjugate liquids.

Inside a miscibility gap the two conjugate liquids have equal component activities (a_i = x_i gamma_i matches across the LLE tie-line), so they necessarily boil to the same vapour y_i = a_i Psat_i / P. The heterogeneous azeotrope is the temperature at which that shared vapour's total pressure sum_i a_i Psat_i reaches the system pressure P, a single, well-posed scalar root, located on a scanned bracket and differentiable in P and the model parameters.

Returns:

Type	Description
`HeterogeneousAzeotrope`	A `HeterogeneousAzeotrope` (the temperature, both liquid
`HeterogeneousAzeotrope`	compositions, and the common vapour), or NaN when no two-liquid boiling
`HeterogeneousAzeotrope`	point exists on `[t_min, t_max]`. Use a `feed` mole fraction (of
`HeterogeneousAzeotrope`	component 1) that lies inside the miscibility gap.