In addition to this explanation, some examples may be found in the pyExamples directory.

Initializing a model

The available models are HardSphere - The RET for Hard Spheres, MieKinGas - The RET-Mie. They are initialized by passing the appropriate component identifiers to the class constructors.

from pykingas.HardSphere import HardSphere
from pykingas.MieKinGas import MieKinGas

mie = MieKinGas('CO2,C1') # RET-Mie for CO2/CH4 mixture
hs = HardSphere('AR,KR,XE') # RET-HS for Ar/Kr/He mixture

The component identifiers are equivalent to the file names in the pykingas/fluids directory, and are consistent with the identifiers used by ThermoPack. A list of all available fluids and their identifiers can be found in the Fluid identifiers section.

Note on pure components

When doing computations for a single component, two mole fractions must be supplied.

Internally pure components are treated as binary mixtures of equivalent species, such that a model initialized with e.g. MieKinGas('H2') will treat pure hydrogen as a mixture of “Hydrogen with hydrogen”. This allows computation of the self-diffusion coefficient through the normal interdiffusion method, but carries the caveat mentioned above.

Properties are not dependent on the supplied mole fractions, but it has been found that for numerical stability, the choice x = [0.5, 0.5] is best.

This may be changed in future versions, such that no mole fraction needs to be supplied when working with pure fluids.

Specifying parameters

If we wish to pass specific parameters to the models, this is done through various keyword arguments, as

# Continued 
mie = MieKinGas('LJF,LJF', mole_weights=[5, 10], sigma=[2.5e-10, 3e-10], eps_div_k=[150, 200], la=[6, 7], lr=[12, 13])

the mole_weights argument sets the molar masses of the components, the sigma argument sets the mie-potential $\sigma$-parameters (in m), the eps_div_k argument sets the mie-potential $\epsilon$-parameters, the la argument sets the attractive exponents ($\lambda_a$), and the lr argument sets the repulsive exponents ($\lambda_r$).

Classes will only accept keyword arguments that are relevant to them, i.e.

hs = HardSphere('LJF,LJF', eps_div_k=[100, 200]) # Throws an error

will throw an error.

To specify the parameters for only one component, and use default parameters for another, set the parameter for the components that are to use default values to None, as

# Continued 
mie = MieKinGas('AR,KR', la=[None, 7], lr=[None, 14]) # Uses the default values for Ar, and specified values for Kr
mie = MieKinGas('AR,KR', la=[6, None], lr=[None, 14]) # Uses default la for Kr, and default lr for Ar.

For isotopic mixtures, one can specify masses in the same way

from pykingas.MieKinGas import MieKinGas
mie = MieKinGas('CH4,CH4,CH4,CH4', mole_weights=[16, 17, 18, 19]) # Isotopic mixture of 1-, 2-, 3-, and 4 times deuterised methane

The Equation of State

KineticGas uses an Equation of State (EoS) internally to compute the derivatives of chemical potential with respect to molar density. Additionally, the tp-inteface methods for predicting transport coefficients use the EoS to compute molar volume at a given T, p, x. This each models stores its own equation of state in the self.eos attribute. By default, this is a ThermoPack equation of state object, which can be specified using the use_eos kwarg upon initialization, as

from pykingas.MieKinGas import MieKinGas
from thermopack.cubic import cubic

comps = 'AR,H2O' # The components we wish to model
eos = cubic(comps, 'SRK') # Soave-Redlich-Kwong EoS for Argon-water mixture
mie = MieKinGas(comps, use_eos=eos)

This can be useful if the components to be modeled do not have parameters for the default eos (thermopack.saftvrmie for MieKinGas), or if one wishes to use some other eos.

In the latter case, the only requirement is that the EoS object implements a method with signature equivalent to thermopack’s chemical_potential_tv. If the tp-interface is to be used, the object must also implement a method with signature equivalent to thermopack’s specific_volume.

Properties at infinite dilution

Properties at infinite dilution can be of interest. Note that at infinite dilution, viscosity, thermal conductivity, and the thermal diffusion factor are independent of density, while the diffusion coefficient and thermal diffusion coefficient are inversely proportional to the density. To initialize a model where the species have negligible covolume (i.e. the radial distribution function is uniformly equal to one), set the kwarg is_idealgas=True, as

from pykingas.MieKinGas import MieKinGas
mie = MieKinGas('H2', is_idealgas=True) # Properties of hydrogen at infinite dilution

Making predictions

In addition to the methods here, a Tp-interface exists for the same methods, consisting of the methods thermal_conductivity_tp, viscosity_tp, interdiffusion_tp, theramal_diffusion_coeff_tp and thermal_diffusion_factor_tp. These methods are only wrappers for ease of use, that use the internal equation of state of the object (self.eos) to compute the molar volume at given (T, p, x) (assuming vapour phase), and passes the call to the methods documented here. Those methods have signatures equivalent to these, but with molar volume swapped out for pressure.

Please note that the Enskog solutions are explicit in density (not pressure), such that when making predictions as a function of pressure, an accurate equation of state is required to translate from a (T, V, n) state to a (T, p, n) state.

Thermal conductivity

Thermal conductivities are predicted with the method thermal_conductivity(self, T, Vm, x, N=None), where T is the temperature, Vm is the molar volume, x is the molar composition and N is the Enskog approximation order.

Example:

from pykingas.MieKinGas import MieKinGas

kin = MieKinGas('O2,N2,CO2,C1') # Mixture of air with carbon dioxide and methane, modeled with RET-Mie
T = 800 # Kelvin
Vm = 0.0665 # cubic meter per mole, approximately equivalent to a pressure of 1 bar
x = [0.05, 0.25, 0.5, 0.2] # Molar composition

cond = kin.thermal_conductivity(T, Vm, x, N=2) # Thermal conductivity [W / m K]

Shear viscosity

Shear viscosities are predicted with the method viscosity(self, T, Vm, x, N=None), where T is the temperature, Vm is the molar volume, x is the molar composition and N is the Enskog approximation order.

Example:

from pykingas.MieKinGas import MieKinGas

kin = MieKinGas('O2,N2,CO2,C1') # Mixture of air with carbon dioxide and methane, modeled with RET-Mie
T = 800 # Kelvin
Vm = 0.0665 # cubic meter per mole, approximately equivalent to a pressure of 1 bar
x = [0.05, 0.25, 0.5, 0.2] # Molar composition

visc = kin.viscosity(T, Vm, x, N=2) # Shear viscosity [Pa s]

Diffusion coefficients

Diffusion coefficients may be defined in many different ways, and depend upon the frame of reference (FoR). For a more in-depth discussion on this see the supporting information of Revised Enskog Theory for Mie fluids: Prediction of diffusion coefficients, thermal diffusion coefficients, viscosities and thermal conductivities. For more details on the definitions available in the KineticGas package, see the memo on definitions of the diffusion coefficient.

The interface to all diffusion coefficients is the method interdiffusion(self, T, Vm, x, N), where T is the temperature, Vm is the molar volume, x is the molar composition and N is the Enskog approximation order.

The default definition of the diffusion coefficient is

\[J_i^{(n)} = - \sum_{i \neq l} D_{ij} \nabla n_j\]

where $J_i$ is the molar flux of species $i$ in the centre of moles (CoN) FoR, and $i \neq l$ are the independent molar density gradients. $l$ is by default the last component in the mixture, such that for a binary system, this reduces to

\[J_1 = - D \nabla n_1\]

The common Fickean diffusion coefficient. The diffusion coefficients are then computed as

from pykingas.MieKinGas import MieKinGas

kin = MieKinGas('AR,KR') # RET-Mie for a mixture of argon and krypton
T = 300 # Kelvin
Vm = 0.025 # cubic meter per mole, approximately equivalent to a pressure of 1 bar
x = [0.3, 0.7] # Molar composition

D = kin.interdiffusion(T, Vm, x, N=2) # Binary diffusion coefficient [m^2 / s]

Note: For binary mixtures, if the kwarg use_binary=True and use_independent=True (default behaviour), only a single diffusion coefficient is returned (not an array).

Variations of the diffusion coefficient

To compute diffusion coefficients in other frames of reference, use the frame_of_reference kwarg, the valid options are 'CoN' (centre of moles, default), 'CoM' (centre of mass / barycentric), 'CoV' (centre of volume), and 'solvent', in combination with the solvent_idx kwarg.

Example:

from pykingas.MieKinGas import MieKinGas

kin = MieKinGas('AR,KR') # RET-Mie for a mixture of argon and krypton
T = 300 # Kelvin
Vm = 0.025 # cubic meter per mole, approximately equivalent to a pressure of 1 bar
x = [0.3, 0.7] # Molar composition

D_CoN = kin.interdiffusion(T, Vm, x, N=2, frame_of_reference='CoN') # Diffusion coefficient in the CoN FoR
D_CoM = kin.interdiffusion(T, Vm, x, N=2, frame_of_reference='CoM') # Diffusion coefficient in the CoM FoR (barycentric)
D_CoV = kin.interdiffusion(T, Vm, x, N=2, frame_of_reference='CoV') # Diffusion coefficient in the CoV FoR
D_solv_Ar = kin.interdiffusion(T, Vm, x, N=2, frame_of_reference='solvent', solvent_idx=0) # Diffusion coefficient in the solvent FoR, with Argon as the solvent
D_solv_Kr = kin.interdiffusion(T, Vm, x, N=2, frame_of_reference='solvent', solvent_idx=1) # Diffusion coefficient in the solvent FoR, with Krypton as the solvent

When using the solvent FoR, the dependent molar density gradient is by default set to be the solvent.

To explicitly set the dependent molar density gradient (default is the last component), use the dependent_idx kwarg, as

# Continued

D_1 = kin.interdiffusion(T, Vm, x, N=2, dependent_idx=0) # Diffusion coefficeint in the CoN FoR, with \nabla n_{Ar} as the dependent gradient
D_2 = kin.interdiffusion(T, Vm, x, N=2, dependent_idx=1) # Diffusion coefficeint in the CoN FoR, with \nabla n_{Kr} as the dependent gradient

The dependent_idx, the specifies the value of $l$ in the equation

\[J_i^{(n)} = - \sum_{i \neq l} D_{ij} \nabla n_j\]

defining the diffusion coefficient. The two diffusion coefficients computed above would thus correspond to the diffusion coefficients

\(J_1^{(n)} = D_1 \nabla n_2\) \(J_2^{(n)} = D_1 \nabla n_2\)

and

\(J_1^{(n)} = D_2 \nabla n_1\) \(J_2^{(n)} = D_2 \nabla n_1\)

where the superscript $^(n)$ denotes that the fluxes are in the centre of moles frame of reference.

To compute diffusion coefficients corresponding to a dependent set of fluxes and forces, defined by

\[J_i^{(FoR)} = - \sum_j D_{ij} \nabla n_j,\]

set the kwarg use_independent=False, as

# Continued

D = kin.interdiffusion(T, Vm, x, N=2, use_independent=False) # Dependent diffusion coefficients in the CoN FoR

For the current system this corresponds to the coefficients of the equation

\[J_1^{(n)} = - D[0, 0] \nabla n_1 - D[0, 1] \nabla n_2\]

and

\[J_2^{(n)} = - D[1, 0] \nabla n_1 - D[1, 1] \nabla n_2.\]

where D[i, j] are the elements of the matrix returned by kin.interdiffusion(T, Vm, x, N=2, use_independent=False).

The frame_of_reference kwarg works as normal when use_independet=False.

Thermal diffusion

Thermal diffusion is characterised by several common coefficients, the thermal diffusion coefficients $D_{T,i}^{(FoR)}$, the thermal diffusion factor $\alpha_{ij}$, the thermal diffusion ratios $k_{T, i}$ and the Soret coefficients $S_{T,i}$.

Of these, the thermal diffusion coefficients, $D_{T,i}^{(FoR)}$, carry the same ambiguity as the diffusion coefficients in their dependency on the frame of reference (FoR) and choice of dependent gradient. For more details on the definitions available in the KineticGas package, see the memo on definitions of the diffusion coefficient.

The Thermal diffusion factors

The thermal diffusion factor gives the ratio

\[\nabla \ln (x_i / x_j) = - \alpha_{ij} \nabla \ln T\]

in the absence of mass fluxes, and can be directly related to the Onsager phenomenological coefficients. They are computed as

from pykingas.MieKinGas import MieKinGas

kin = MieKinGas('C1,C3,CO2') # RET-Mie for a mixture of methane, propane and CO2
T = 300 # Kelvin
Vm = 0.025 # cubic meter per mole, approximately equivalent to a pressure of 1 bar
x = [0.3, 0.6, 0.1] # Molar composition

alpha = kin.thermal_diffusion_factor(T, Vm, x, N=2) # Thermal diffusion factors [dimensionless]

The thermal diffusion ratios

The thermal diffusion ratios satisfy the relation

\[\nabla n_i = - k_{T,i} \nabla \ln T\]

in the absence of mass fluxes, and can be directly related to the Onsager phenomenological coefficients. They are computed as

# Continued 
kT = kin.thermal_diffusion_ratio(T, Vm, x, N=2) # Thermal diffusion ratios [dimensionless]

The thermal diffusion coefficients

The thermal diffusion coefficients are by default defined by

\[J_i^{(n)} = D_{T, i} \nabla \ln T - \sum_{j \neq l} D_{ij} \nabla n_j,\]

where $J_i^{(n)}$ is the molar flux of species $i$ in the centre of moles (CoN) FoR, $\nabla n_j$ is the molar density gradient of component $j$, and $l$ is the index of the dependent gradient. This is computed by

from pykingas.MieKinGas import MieKinGas

kin = MieKinGas('C1,O2,CO2') # RET-Mie for a mixture of methane, oxygen and CO2
T = 300 # Kelvin
Vm = 0.025 # cubic meter per mole, approximately equivalent to a pressure of 1 bar
x = [0.3, 0.6, 0.1] # Molar composition

DT = kin.thermal_diffusion_coeff(T, Vm, x, N=2) # Thermal diffusion coefficients in the CoN FoR [mol / m s]

Variations of the thermal diffusion coefficients

For other frames of reference, use the frame_of_reference kwarg, with options equivalent to those for interdiffusion, that is: 'CoN' (centre of moles, default), 'CoM' (centre of mass / barycentric), 'CoV' (centre of volume), and 'solvent', in combination with the solvent_idx kwarg.

Example:

# Continued
DT_CoN = kin.thermal_diffusion_coeff(T, Vm, x, N=2, frame_of_reference='CoN') # Thermal diffusion coefficient in the CoN FoR
DT_CoM = kin.thermal_diffusion_coeff(T, Vm, x, N=2, frame_of_reference='CoM') # Thermal diffusion coefficient in the CoM FoR (barycentric)
DT_CoV = kin.thermal_diffusion_coeff(T, Vm, x, N=2, frame_of_reference='CoV') # Thermal diffusion coefficient in the CoV FoR
DT_solv_C1 = kin.thermal_diffusion_coeff(T, Vm, x, N=2, frame_of_reference='solvent', solvent_idx=0) # Thermal diffusion coefficient in the solvent FoR, with methane as the solvent
DT_solv_C3 = kin.thermal_diffusion_coeff(T, Vm, x, N=2, frame_of_reference='solvent', solvent_idx=1) # Thermal diffusion coefficient in the solvent FoR, with propane as the solvent
DT_solv_CO2 = kin.thermal_diffusion_coeff(T, Vm, x, N=2, frame_of_reference='solvent', solvent_idx=2) # Thermal diffusion coefficient in the solvent FoR, with CO2 as the solvent

To explicitly select the dependent molar gradient (default is the last component), use the dependent_idx kwarg, equivalently to interdiffusion.

Example:

# Continued
DT = kin.thermal_diffusion_coeff(T, Vm, x, N=2, dependent_idx=0) # Thermal diffusion coefficient in the CoN FoR, with \nabla n_{C1} as the dependent gradient
D = kin.interdiffusion(T, Vm, x, N=2, dependent_idx=0) # Diffusion coefficient in the CoN FoR with \nabla n_{C1} as the dependent gradient

This gives the coefficients corresponding to the flux equations

\[J_{C1} = D_{T}[0] \nabla \ln T - D[0, 1] \nabla n_{O2} - D[0, 2] \nabla n_{CO2},\] \[J_{O2} = D_{T}[1] \nabla \ln T - D[1, 1] \nabla n_{O2} - D[1, 2] \nabla n_{CO2},\] \[J_{CO2} = D_{T}[2] \nabla \ln T - D[2, 1] \nabla n_{O2} - D[2, 2] \nabla n_{CO2}.\]

To compute coefficients corresponding to flux equation with all forces and fluxes (not an independent set), set the kwarg use_independent=False, as

# Continued
DT = kin.thermal_diffusion_coeff(T, Vm, x, N=2, use_independent=False) # Thermal diffusion coefficient in the CoN FoR, with all gradients
D = kin.interdiffusion(T, Vm, x, N=2, use_independent=False) # Diffusion coefficient in the CoN FoR with all gradients

This gives the coefficients corresponding to the flux equations

\[J_{C1} = D_{T}[0] \nabla \ln T - D[0, 0] \nabla n_{C1} - D[0, 1] \nabla n_{O2} - D[0, 2] \nabla n_{CO2},\] \[J_{O2} = D_{T}[1] \nabla \ln T - D[1, 0] \nabla n_{C1} - D[1, 1] \nabla n_{O2} - D[1, 2] \nabla n_{CO2},\] \[J_{CO2} = D_{T}[2] \nabla \ln T - D[2, 0] \nabla n_{C1} - D[2, 1] \nabla n_{O2} - D[2, 2] \nabla n_{CO2}.\]

The frame_of_reference kwarg works as normal when setting use_independent=False.