Bubble Point and Dew Point

Calculation of bubble point and dew point for vapor-liquid systems apply simplifications of the Rachford-Rice mass balance, as the liquid phase fraction (0% or 100%) is already known. Default solution strategy applies first Accelerated Successive Substitutions method to update phase compositions in an inner loop, and a Quasi-Newton method to update pressure or temperature in an outer loop. If convergence is not reached in 10 iterations, or if user defines initial estimates to be good, the algorithm switches to a Phase Envelope method solving the following system of equations:

\[\begin{split}f_i &= \ln K_i + \ln \hat{\phi}_i^v(\underline{y}, T, P) -\ln \hat{\phi}_i^l(\underline{x}, T, P) \quad i = 1,...,c \\ f_{c+1} &= \sum_{i=1}^c (y_i-x_i)\end{split}\]

Bubble Point

Bubble point is a fluid state where saturated liquid of known composition and liquid fraction 100% is forming a differential size bubble. The algorithm finds vapor phase composition and either temperature or pressure of the bubble point.

The algorithm solves composition using a simplified Rachford-Rice equation:

\[FO = \sum_{i=1}^c x_i (K_i-1) = \sum_{i=1}^c y_i -1 = 0\]
>>> import numpy as np
>>> from phasepy import component, mixture, rkseos
>>> from phasepy.equilibrium import bubbleTy, bubblePy
>>> butanol = component(name='butanol', Tc=563.0, Pc=44.14, Zc=0.258, Vc=274.0, w=0.589462,
                        Ant=[10.20170373, 2939.88668723, -102.28265042])
>>> 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])
>>> Kij = np.zeros([2,2])
>>> mix = mixture(mtbe, butanol)
>>> mix.kij_cubic(Kij)
>>> eos = rkseos(mix, 'qmr')
>>> x = np.array([0.5, 0.5])
>>> P0, T0 = 1.0, 340.0
>>> y0 = np.array([0.8, 0.2])
>>> bubbleTy(y0, T0, x, 1.0, eos) # vapor fractions, temperature
(array([0.90411878, 0.09588122]), 343.5331023048577)
>>> bubblePy(y0, P0, x, 343.533, eos) # vapor fractions, pressure
(array([0.90411894, 0.09588106]), 0.9999969450754181)
bubbleTy(y_guess, T_guess, X, P, model, good_initial=False, v0=[None, None], full_output=False)[source]

Bubble point (X, P) -> (Y, T).

Solves bubble point (vapor phase composition and temperature) at given pressure and liquid phase composition.

Parameters:
  • y_guess (array) – Initial guess of vapor phase molar fractions
  • T_guess (float) – Initial guess of equilibrium temperature [K]
  • X (array) – Liquid phase molar fractions
  • P (float) – Pressure [bar]
  • model (object) – Phase equilibrium model object
  • good_initial (bool, optional) – If True uses only phase envelope method in solution
  • v0 (list, optional) – Liquid and vapor phase molar volume used as initial values to compute fugacities
  • full_output (bool, optional) – Flag to return a dictionary of all calculation info
Returns:

  • Y (array) – Vapor molar fractions
  • T (float) – Equilibrium temperature [K]
bubblePy(y_guess, P_guess, X, T, model, good_initial=False, v0=[None, None], full_output=False)[source]

Bubble point (X, T) -> (Y, P).

Solves bubble point (vapor phase composition and pressure) at given temperature and liquid composition.

Parameters:
  • y_guess (array) – Initial guess of vapor phase molar fractions
  • P_guess (float) – Initial guess for equilibrium pressure [bar]
  • X (array) – Liquid phase molar fractions
  • T (float) – Absolute temperature [K]
  • model (object) – Phase equilibrium model object
  • good_initial (bool, optional) – If True uses only phase envelope method in solution
  • v0 (list, optional) – Liquid and vapor phase molar volume used as initial values to compute fugacities
  • full_output (bool, optional) – Flag to return a dictionary of all calculation info
Returns:

  • Y (array) – Vapor molar fractions
  • P (float) – Equilibrium pressure [bar]

Dew Point

Dew point is a fluid state where saturated vapor of known composition and liquid fraction 0% is forming a differential size liquid droplet. The algorithm finds liquid phase composition and either temperature or pressure of the dew point.

The algorithm solves composition using a simplified Rachford-Rice equation:

\[FO = 1 - \sum_{i=1}^c \frac{y_i}{K_i} = 1 - \sum_{i=1}^c x_i = 0\]
>>> import numpy as np
>>> from phasepy import component, mixture, prsveos
>>> from phasepy.equilibrium import dewPx, dewTx
>>> ethanol = component(name='ethanol', Tc=514.0, Pc=61.37, Zc=0.241, Vc=168.0, w=0.643558,
...                ksv=[1.27092923, 0.0440421],
...                GC={'CH3':1, 'CH2':1, 'OH(P)':1})
>>> mtbe = component(name='mtbe', Tc=497.1, Pc=34.3, Zc=0.273, Vc=329.0, w=0.266059,
...                ksv=[0.76429651, 0.04242646],
...                GC={'CH3':3, 'CH3O':1, 'C':1})
>>> mix = mixture(mtbe, ethanol)
>>> C0 = np.array([0.02635196, -0.02855964, 0.01592515])
>>> C1 = np.array([312.575789, 50.1476555, 5.13981131])
>>> mix.rk(C0, C1)
>>> eos = prsveos(mix, mixrule = 'mhv_rk')
>>> y = np.array([0.5, 0.5])
>>> P0, T0 = 1.0, 340.0
>>> x0 = np.array([0.2, 0.8])
>>> dewPx(x0, P0, y, 340.0, eos) # liquid fractions, pressure
(array([0.20223477, 0.79776523]), 1.0478247383242278)
>>> dewTx(x0, T0, y, 1.047825, eos) # liquid fractions, temperature
(array([0.20223478, 0.79776522]), 340.0000061757033)
dewPx(x_guess, P_guess, y, T, model, good_initial=False, v0=[None, None], full_output=False)[source]

Dew point (y, T) -> (x, P)

Solves dew point (liquid phase composition and pressure) at given temperature and vapor composition.

Parameters:
  • x_guess (array) – Initial guess of liquid phase molar fractions
  • P_guess (float) – Initial guess of equilibrium pressure [bar]
  • y (array) – Vapor phase molar fractions
  • T (float) – Temperature [K]
  • model (object) – Phase equilibrium model object
  • good_initial (bool, optional) – If True uses only phase envelope method in solution
  • v0 (list, optional) – Liquid and vapor phase molar volume used as initial values to compute fugacities
  • full_output (bool, optional) – Flag to return a dictionary of all calculation info
Returns:

  • x (array) – Liquid molar fractions
  • P (float) – Equilibrium pressure [bar]
dewTx(x_guess, T_guess, y, P, model, good_initial=False, v0=[None, None], full_output=False)[source]

Dew point (y, P) -> (x, T)

Solves dew point (liquid phase composition and temperature) at given pressure and vapor phase composition.

Parameters:
  • x_guess (array) – Initial guess of liquid phase molar fractions
  • T_guess (float) – Initial guess of equilibrium temperature [K]
  • y (array) – Vapor phase molar fractions
  • P (float) – Pressure [bar]
  • model (object) – Phase equilibrium model object
  • good_initial (bool, optional) – If True uses only phase envelope method in solution
  • v0 (list, optional) – Liquid and vapor phase molar volume used as initial values to compute fugacities
  • full_output (bool, optional) – Flag to return a dictionary of all calculation info
Returns:

  • x (array) – Liquid molar fractions
  • T (float) – Equilibrium temperature [K]