8.2.5. Liquid mixtures
Properties of liquid mixtures are defined similarly to gas mixtures. Again, we define the required components for this particular mixture and can select one species as passive field, i.e. the composition field which is not explicitly solved for. When we define a pure liquid name "glycerol" analogous to the pure liquid "water", we can define the mixture properties e.g. as follows:
@MaterialProperties.register()
class MixtureLiquidGlycerolWater(MixtureLiquidProperties):
components={"water","glycerol"}
passive_field="water"
def __init__(self,pure_properties):
super().__init__(pure_properties)
self.set_by_weighted_average("mass_density") # realistic assumption here: rho=rho_water*w_water+rho_glyc*w_glyc
self.set_by_weighted_average("thermal_conductivity")
self.set_by_weighted_average("specific_heat_capacity")
yG=self.get_mass_fraction_field("glycerol") # will just expand to var("massfrac_glycerol")
# Model for the dynamic viscosity
TCelsius = subexpression(var("temperature") / kelvin-273.15)
a=0.705 - 0.0017 * TCelsius
b = (4.9 + 0.036 * TCelsius) * a ** 2.5
muG=12100 * exp((-1233 + TCelsius) * TCelsius / (9900 + 70 * TCelsius))
muW =1.790 * exp((-1230 - TCelsius) * TCelsius / (36100 + 360 * TCelsius))
alpha = subexpression(1 - yG + a * b * yG * (1 - yG) / (a * yG + b * (1 - yG)))
self.dynamic_viscosity= subexpression(muW* (muG/muW) ** (1-alpha)* 0.001*pascal * second)
# Surface tension function
self.default_surface_tension["gas"]=subexpression(72.45e-3 * ((1.0 - 0.1214690683 * yG + 0.4874796412 * yG ** 2 - 2.208295376 * yG ** 3 + 3.412242927 * yG ** 4 - 1.698619738 * yG ** 5) - (0.0001455 * (1 - yG) + 0.00008845 * yG) * (TCelsius))* newton / meter)
# Diffusion coefficient fit
D=1.024e-11 * (-0.721 * yG + 0.7368) / (0.49311e-2 * yG + 0.7368e-2)*meter ** 2 / second
self.set_diffusion_coefficient(D)
# Set activity coefficients by AIOMFAC
self.set_activity_coefficients_by_unifac("AIOMFAC")
Again, as in the case of gas mixtures, the components and passive_field must be set. The constructor takes again a dict of the pure properties.
If one does not know details on the particular change of the liquid properties with the composition, one always can use set_by_weighted_average() to calculate the average of the pure properties weighted by the local mass fractions. This makes at least sure that the properties are correct when taking the pure limits. One can also modify the optional argument fraction_type to "mole_fraction" to blend between the pure properties weighted by the mole fractions instead the mass fractions. The local mass and mole fractions of each component can be obtained by get_mass_fraction_field() and get_mole_fraction_field(), respectively. Alternatively, one can directly use e.g. var("massfrac_water") or var("molefrac_glycerol") to bind these fields to form arbitrary expressions.
As shown in the above example for the dynamic_viscosity(), one can assemble functions of the composition and temperature easily. Here, we have used a viscosity model developed by Cheng [6], while the surface tension was obtained by a fit of experimental data [41]. The same holds true for the diffusion coefficient based on the data of D'Errico et al. [9].
We can set the activity coefficients either directly by setting the dict values of the member activity_coefficients, e.g. activity_coefficients["water"]=.... If the vapor pressure shall be calculated by Raoult’s law (cf. (8.4) later on), one has to call set_vapor_pressure_by_raoults_law() afterwards. Alternatively, the vapor pressure of each component can be set directly by the dict vapor_pressure_for, e.g. vapor_pressure_for["water"]=.... One can also invoke the various UNIFAC models to calculate the activity coefficients and set the vapor pressure according to Raoult’s law with these activity coefficients. To that end, a simple call of set_activity_coefficients_by_unifac() will do the trick. One has to select a particular UNIFAC model ("Original", "Dortmund" or "AIOMFAC"). Of course, to use these models, one has to set the group contributions in the pure liquids (cf. Section 8.8 later on).