Phase Stability Analysis

Stability tests give a relative estimate of how stable a phase (given temperature, pressure and composition) is thermodynamically. A stable phase decreases the Gibbs free energy of the system compared to a reference phase (e.g. the overall composition of a mixture). To evaluate the relative stability, phasepy applies the Tangent Plane Distance (TPD) function introduced by Michelsen.

\[F_{TPD}(w) = \sum_{i=1}^c w_i \left[\ln w_i + \ln \hat{\phi}_i(w) - \ln z_i - \ln \hat{\phi}_i(z) \right]\]

First, the TPD function is minimized locally w.r.t. phase composition \(w\), starting from an initial composition. If the TPD value at the minimum is negative, it implies that the phase is less stable than the reference. A positive value means more stable than the reference.

The function tpd_min() can be applied to find a phase composition from given initial values, and the TPD value of the minimized result. Negative TPD value for the first example (trial phase is liquid) means that the resulting liquid phase composition is unstable. Positive TPD value for the second example (trial phase is vapor) means that the resulting vapor phase is more stable than the reference phase. Therefore the second solution could be used e.g. as an initial estimate for two-phase flash calculation.

>>> import numpy as np
>>> from phasepy import component, mixture, preos
>>> from phasepy.equilibrium import tpd_min
>>> water = component(name='water', Tc=647.13, Pc=220.55, Zc=0.229, Vc=55.948, w=0.344861,
                      Ant=[11.64785144, 3797.41566067, -46.77830444],
                      GC={'H2O':1})
>>> mtbe = component(name='mtbe', Tc=497.1, Pc=34.3, Zc=0.273, Vc=329.0, w=0.266059,
                     Ant=[9.16238246, 2541.97883529, -50.40534341],
                     GC={'CH3':3, 'CH3O':1, 'C':1})
>>> mix = mixture(water, mtbe)
>>> mix.unifac()
>>> eos = preos(mix, 'mhv_unifac')
>>> w = np.array([0.01, 0.99])
>>> z = np.array([0.5, 0.5])
>>> T = 320.0
>>> P = 1.01
>>> tpd_min(w, z, T, P, eos, stateW='L', stateZ='L') # molar fractions and TPD value
(array([0.30683438, 0.69316562]), -0.005923915138229763)
>>> tpd_min(w, z, T, P, eos, stateW='V', stateZ='L') # molar fractions and TPD value
(array([0.16434188, 0.83565812]), 0.24576563932356765)
tpd_min(W, Z, T, P, model, stateW, stateZ, vw=None, vz=None)[source]

Minimizes the Tangent Plane Distance (TPD) function for trial phase and calculates TPD value for the minimum.

Parameters:
  • W (array) – Initial molar fractions of the trial phase
  • Z (array) – Molar fractions of the reference phase
  • T (float) – Absolute temperature [K]
  • P (float) – Absolute pressure [bar]
  • model (object) – Phase equilibrium model object
  • stateW (string) – Trial phase type. ‘L’ for liquid phase, ‘V’ for vapor phase
  • stateZ (string) – Reference phase type. ‘L’ for liquid phase, ‘V’ for vapor phase
  • vz (vw,) – Phase molar volume used as initial value to compute fugacities
Returns:

  • W (array) – Minimized phase molar fractions
  • f (float) – Minimized TPD distance value

Phasepy function tpd_minimas() can be used to try to find several TPD minima. The function uses random initial compositions to search for minima, so it can find the same minimum multiple times.

>>> from phasepy.equilibrium import tpd_minimas
>>> nmin = 3
>>> tpd_minimas(nmin, z, T, P, eos, 'L', 'L') # minima and TPD values (two unstable minima)
(array([0.99538258, 0.00461742]), array([0.30683414, 0.69316586]), array([0.99538258, 0.00461742])),
array([-0.33722905, -0.00592392, -0.33722905])
>>> tpd_minimas(nmin, z, T, P, eos, 'V', 'L') # minima and TPD values (one stable minimum)
(array([0.1643422, 0.8356578]), array([0.1643422, 0.8356578]), array([0.1643422, 0.8356578])),
array([0.24576564, 0.24576564, 0.24576564])
tpd_minimas(nmin, Z, T, P, model, stateW, stateZ, vw=None, vz=None)[source]

Repetition of Tangent Plane Distance (TPD) function minimization with random initial values to try to find several minima.

Parameters:
  • nmin (int) – Number of randomized minimizations to carry out
  • Z (array) – Molar fractions of the reference phase
  • T (float) – Absolute temperature [K]
  • P (float) – Absolute pressure [bar]
  • model (object) – Phase equilibrium model object
  • stateW (string) – Trial phase type. ‘L’ for liquid phase, ‘V’ for vapor phase
  • stateZ (string) – Reference phase type. ‘L’ for liquid phase, ‘V’ for vapor phase
  • vz (vw,) – Phase molar volume used as initial value to compute fugacities
Returns:

  • W_minima (tuple(array)) – Minimized phase molar fractions
  • f_minima (array) – Minimized TPD distance values

Function lle_init() can be used to generate two phase compositions e.g. for subsequent liquid liquid equilibrium calculations. Note that the results are not guaranteed to be stable or even different.

>>> from phasepy.equilibrium import lle_init
>>> lle_init(z, T, P, eos)
array([0.99538258, 0.00461742]), array([0.30683414, 0.69316586])
lle_init(Z, T, P, model, vw=None, vz=None)[source]

Carry out two repetitions of Tangent Plane Distance (TPD) function minimization with random initial values to find two liquid phase compositions.

Parameters:
  • Z (array) – Molar fractions of the reference phase
  • T (float) – Absolute temperature [K]
  • P (float) – Absolute pressure [bar]
  • model (object) – Phase equilibrium model object
  • vz (vw,) – if supplied volume used as initial value to compute fugacities
Returns:

W_minima – Two minimized phase molar fractions
Return type:

tuple(array)