Cubic Equation of State Model

This phase equilibrium model class applies equation of state (EoS) model for both vapor and liquid phases. EoS formulation is explicit:

\[P = \frac{RT}{v-b} - \frac{a}{(v+c_1b)(v+c_2b)}\]

Phasepy includes following cubic EoS:

  • Van der Waals (VdW)
  • Peng Robinson (PR)
  • Redlich Kwong (RK)
  • Redlich Kwong Soave (RKS), a.k.a Soave Redlich Kwong (SRK)
  • Peng Robinson Stryjek Vera (PRSV)

Both pure component EoS and mixture EoS are supported.

Pure Component EoS

Pure component example using Peng-Robinson EoS:

>>> from phasepy import preos, component
>>> ethanol = component(name='ethanol', Tc=514.0, Pc=61.37, Zc=0.241, Vc=168.0, w=0.643558,
                        Ant=[11.61809279, 3423.0259436, -56.48094263],
                        GC={'CH3':1, 'CH2':1, 'OH(P)':1})
>>> eos = preos(ethanol)
>>> eos.psat(T=350.0) # saturation pressure, liquid volume and vapor volume
(array([0.98800647]), array([66.75754804]), array([28799.31921623]))

Density can be computed given the aggregation state (L for liquid, V for vapor):

>>> eos.density(T=350.0, P=1.0, state='L')
0.01497960198094922
>>> eos.density(T=350.0, P=1.0, state='V')
3.515440899573752e-05

Volume Translation

Volume translated (VT) versions of EoS are available for PR, RK, RKS and PRSV models. These models include an additional component specific volume translation parameter c, which can be used to improve liquid density predictions without changing phase equilibrium. EoS property volume_translation must be True to enable VT.

>>> ethanol = component(name='ethanol', Tc=514.0, Pc=61.37, Zc=0.241, Vc=168.0, w=0.643558,
                        Ant=[11.61809279, 3423.0259436, -56.48094263],
                        GC={'CH3':1, 'CH2':1, 'OH(P)':1},
                        c=5.35490936)
>>> eos = preos(ethanol, volume_translation=True)
>>> eos.psat(T=350.0) # saturation pressure, liquid volume and vapor volume
(array([0.98800647]), array([61.40263868]), array([28793.96430687]))
>>> eos.density(T=350.0, P=1.0, state='L')
0.01628597159790686
>>> eos.density(T=350.0, P=1.0, state='V')
3.5161028012629526e-05

Mixture EoS

Mixture EoS utilize one-fluid mixing rules, using parameters for hypothetical pure fluids, to predict the mixture behavior. The mixing rules require interaction parameter values as input (zero values are assumed if no values are specified).

Classic Quadratic Mixing Rule (QMR)

\[a_m = \sum_{i=1}^c \sum_{j=1}^c x_ix_ja_{ij} \quad a_{ij} = \sqrt{a_ia_j}(1-k_{ij}) \quad b_m = \sum_{i=1}^c x_ib_i\]

Example of Peng-Robinson with QMR:

>>> from phasepy import preos
>>> mix = mixture(ethanol, water)
>>> Kij = np.array([[0, -0.11], [-0.11, 0]])
>>> mix.kij_cubic(Kij)
>>> eos = preos(mix, mixrule='qmr')

Modified Huron Vidal (MHV) Mixing Rule

MHV mixing rule is specified in combination with an activity coefficient model to solve EoS. In MHV model, the repulsive (covolume) parameter is calcuated same way as in QMR

\[b_m = \sum_{i=1}^c x_ib_i\]

while attractive term is an implicit function

\[g^e_{EOS} = g^e_{model}\]

Example of Peng-Robinson with MHV and NRTL:

>>> alpha = np.array([[0.0, 0.5597628],
                      [0.5597628, 0.0]])
>>> g = np.array([[0.0, -57.6880881],
                  [668.682368, 0.0]])
>>> g1 = np.array([[0.0, 0.46909821],
                   [-0.37982045, 0.0]])
>>> mix.NRTL(alpha, g, g1)
>>> eos = preos(mix, mixrule='mhv_nrtl')

Example of Peng-Robinson with MHV and Modified-UNIFAC:

>>> mix.unifac()
>>> eos = preos(mix, mixrule='mhv_unifac')

Wong Sandler (WS) Mixing Rule

WS mixing rule is specified in combination with an activity coefficient model to solve EoS.

Example of Peng-Robinson with WS and Redlich Kister:

>>> C0 = np.array([1.20945699, -0.62209997,  3.18919339])
>>> C1 = np.array([-13.271128, 101.837857, -1100.29221])
>>> mix.rk(C0, C1)
>>> eos = preos(mix, mixrule='ws_rk')
cubiceos(mix_or_component, c1=-0.41421356237309515, c2=2.414213562373095, alpha_eos=<function alpha_soave>, mixrule='qmr', volume_translation=False)[source]

Returns cubic EoS object. with custom c1, c2, alpha function and mixing rule.

Parameters:
  • mix_or_component (object) – phasepy.mixture or phasepy.component object
  • c2 (c1,) – Constants of the cubic EoS
  • alpha_eos (function) – Function that returns the alpha parameter of the cubic EoS. alpha_eos(T, alpha_params, Tc)
  • mixrule (str) – Mixing rule specification. Available opitions include ‘qmr’, ‘mhv_nrtl’, ‘mhv_unifac’, ‘mhv_rk’, ‘mhv_wilson’, ‘mhv_uniquac’, ‘mhv_original_unifac’, ‘ws_nrtl’, ‘ws_wilson’, ‘ws_rk’, ‘ws_unifac’, ‘ws_uniquac’, ‘ws_original_unifac, mhv1_nrtl’, ‘mhv1_unifac’, ‘mhv1_rk’, ‘mhv1_wilson’, ‘mhv1_uniquac, ‘mhv1_original_unifac’
  • volume_translation (bool) – If True, the volume translated version of this EoS will be used.
preos(mix_or_component, mixrule='qmr', volume_translation=False)[source]

Returns Peng Robinson EoS object.

Parameters:
  • mix_or_component (object) – phasepy.mixture or phasepy.component object
  • mixrule (str) – Mixing rule specification. Available opitions include ‘qmr’, ‘mhv_nrtl’, ‘mhv_unifac’, ‘mhv_rk’, ‘mhv_wilson’, ‘mhv_uniquac’, ‘mhv_original_unifac’, ‘ws_nrtl’, ‘ws_wilson’, ‘ws_rk’, ‘ws_unifac’, ‘ws_uniquac’, ‘ws_original_unifac, mhv1_nrtl’, ‘mhv1_unifac’, ‘mhv1_rk’, ‘mhv1_wilson’, ‘mhv1_uniquac, ‘mhv1_original_unifac’
  • volume_translation (bool) – If True, the volume translated version of this EoS will be used.
prsveos(mix_or_component, mixrule='qmr', volume_translation=False)[source]

Returns Peng Robinson Stryjek Vera EoS object.

Parameters:
  • mix_or_component (object) – phasepy.mixture or phasepy.component object
  • mixrule (str) – Mixing rule specification. Available opitions include ‘qmr’, ‘mhv_nrtl’, ‘mhv_unifac’, ‘mhv_rk’, ‘mhv_wilson’, ‘mhv_uniquac’, ‘mhv_original_unifac’, ‘ws_nrtl’, ‘ws_wilson’, ‘ws_rk’, ‘ws_unifac’, ‘ws_uniquac’, ‘ws_original_unifac, mhv1_nrtl’, ‘mhv1_unifac’, ‘mhv1_rk’, ‘mhv1_wilson’, ‘mhv1_uniquac, ‘mhv1_original_unifac’
  • volume_translation (bool) – If True, the volume translated version of this EoS will be used.
rkeos(mix_or_component, mixrule='qmr', volume_translation=False)[source]

Returns Redlich Kwong EoS object.

Parameters:
  • mix_or_component (object) – phasepy.mixture or phasepy.component object
  • mixrule (str) – Mixing rule specification. Available opitions include ‘qmr’, ‘mhv_nrtl’, ‘mhv_unifac’, ‘mhv_rk’, ‘mhv_wilson’, ‘mhv_uniquac’, ‘mhv_original_unifac’, ‘ws_nrtl’, ‘ws_wilson’, ‘ws_rk’, ‘ws_unifac’, ‘ws_uniquac’, ‘ws_original_unifac, mhv1_nrtl’, ‘mhv1_unifac’, ‘mhv1_rk’, ‘mhv1_wilson’, ‘mhv1_uniquac, ‘mhv1_original_unifac’
  • volume_translation (bool) – If True, the volume translated version of this EoS will be used.
rkseos(mix_or_component, mixrule='qmr', volume_translation=False)[source]

Returns Redlich Kwong Soave EoS object.

Parameters:
  • mix_or_component (object) – phasepy.mixture or phasepy.component object
  • mixrule (str) – Mixing rule specification. Available opitions include ‘qmr’, ‘mhv_nrtl’, ‘mhv_unifac’, ‘mhv_rk’, ‘mhv_wilson’, ‘mhv_uniquac’, ‘mhv_original_unifac’, ‘ws_nrtl’, ‘ws_wilson’, ‘ws_rk’, ‘ws_unifac’, ‘ws_uniquac’, ‘ws_original_unifac, mhv1_nrtl’, ‘mhv1_unifac’, ‘mhv1_rk’, ‘mhv1_wilson’, ‘mhv1_uniquac, ‘mhv1_original_unifac’
  • volume_translation (bool) – If True, the volume translated version of this EoS will be used.
vdweos(mix_or_component)[source]

Returns Van der Waals EoS object.

Parameters:mix_or_component (object) – phasepy.mixture or phasepy.component object