Diffusion#

Class Diffusion#

class multipy.diffusion.Diffusion(binary_diffusion_coefficients, species_names=None)#

Supports computing and storing diffusion-related quantities.

Parameters

  • binary_diffusion_coefficients – scalar numpy.ndarray specifying the binary diffusion coefficients, \(\pmb{\mathcal{D}}\), in \([m^2/s]\) for all species. It should be a symmetric matrix of size (n_species,n_species).

  • species_names – (optional) list of str specifying the names for all species. It should match the number and ordering of species as per the binary_diffusion_coefficients parameter. If not specified, species will be tagged with consecutive integers, i.e. '1', '2' and so on.

Getters:

  • get_binary_diffusion_coefficients (is set at class init)

  • get_n_species read only - (is set to 0 at class init)

  • get_fickian_diffusion_coefficients_molar_molar (is set to None at class init)

  • get_fickian_diffusion_coefficients_mass_mass (is set to None at class init)

  • get_fickian_diffusion_coefficients_molar_volume (is set to None at class init)

Setters:

  • set_binary_diffusion_coefficients setter for get_binary_diffusion_coefficients

  • set_fickian_diffusion_coefficients_molar_molar setter for get_fickian_diffusion_coefficients_molar_molar

  • set_fickian_diffusion_coefficients_mass_mass setter for get_fickian_diffusion_coefficients_mass_mass

  • set_fickian_diffusion_coefficients_molar_volume setter for get_fickian_diffusion_coefficients_molar_volume

Diffusion.print_binary_diffusion_coefficients#

multipy.diffusion.Diffusion.print_binary_diffusion_coefficients(self, table_format='pandas', float_format='.8f')#

Prints the binary diffusion coefficients matrix, \(\pmb{\mathcal{D}}\).

If table_format is set to 'pandas', a table will be printed in the pandas.DataFrame format:

../_images/print_binary_diffusion_coefficients-pandas.png

If table_format is set to 'raw', a table will be printed in the raw text format:

|            | Acetone    | Methanol   | Air        |
| Acetone    | 0.0        | 8.48e-06   | 1.372e-05  |
| Methanol   | 8.48e-06   | 0.0        | 1.991e-05  |
| Air        | 1.372e-05  | 1.991e-05  | 0.0        |
Parameters

  • table_format – (optional) str specifying the printing format. It can be 'pandas' or 'raw'.

  • float_format – (optional) str specifying the float formatting when printing the table.

Diffusion.fickian_diffusion_coefficients_molar_molar#

multipy.diffusion.Diffusion.fickian_diffusion_coefficients_molar_molar(self, species_mole_fractions)#

Computes the molar Fickian diffusion coefficients expressed in a molar-averaged velocity reference frame, \(\mathbf{D}\), from:

\[\mathbf{D} = \mathbf{B}^{-1}\]

where the elements of \(\mathbf{B}\) are given by:

\[ \begin{align}\begin{aligned}B_{i, i} = \frac{X_i}{\mathcal{D}_{i, n}} + \sum_{j \neq i}^{n} \frac{X_j}{\mathcal{D}_{i,j}}\\B_{i, j} = - X_i \Big( \frac{1}{\mathcal{D}_{i, j}} - \frac{1}{\mathcal{D}_{i, n}} \Big)\end{aligned}\end{align} \]

and \(n\) is the number of species.

Parameters

species_mole_fractions – scalar numpy.ndarray specifying the species mole fractions, \(\mathbf{X}_i\), in \([-]\). It should be of size (n_species,n_observations).

Returns

  • fickian_diffusion_coefficients - scalar numpy.ndarray of molar Fickian diffusion coefficients expressed in a molar-averaged velocity reference frame, \(\mathbf{D}\), in \([m^2/s]\). It has size (n_species-1,n_species-1,n_observations).

Diffusion.fickian_diffusion_coefficients_mass_mass#

multipy.diffusion.Diffusion.fickian_diffusion_coefficients_mass_mass(self, species_mole_fractions, species_mass_fractions)#

Computes the mass Fickian diffusion coefficients expressed in a mass-averaged velocity reference frame, \(\mathbf{D}^o\), from:

\[\mathbf{D}^o =\]
Parameters

  • species_mole_fractions – scalar numpy.ndarray specifying the species mole fractions, \(\mathbf{X}_i\), in \([-]\). It should be of size (n_species,n_observations).

  • species_mass_fractions – scalar numpy.ndarray specifying the species mass fractions, \(\mathbf{Y}_i\), in \([-]\). It should be of size (n_species,n_observations).

Returns

  • fickian_diffusion_coefficients - scalar numpy.ndarray of mass Fickian diffusion expressed in a mass-averaged velocity reference frame, \(\mathbf{D}^o\), in \([m^2/s]\). It has size (n_species-1,n_species-1,n_observations).

Diffusion.fickian_diffusion_coefficients_molar_volume#

multipy.diffusion.Diffusion.fickian_diffusion_coefficients_molar_volume(self, species_volume_fractions)#

Computes the molar Fickian diffusion coefficients expressed in a volume-averaged velocity reference frame, \(\mathbf{D}^V\), from:

\[\mathbf{D}^V =\]
Parameters

species_volume_fractions – scalar numpy.ndarray specifying the species volume fractions, \(\mathbf{V}_i\), in \([-]\). It should be of size (n_species,n_observations).

Returns

  • fickian_diffusion_coefficients - scalar numpy.ndarray of molar Fickian diffusion expressed in a volume-averaged velocity reference frame, \(\mathbf{D}^V\), in \([m^2/s]\). It has size (n_species-1,n_species-1,n_observations).

Diffusion.effective_diffusivity#

multipy.diffusion.Diffusion.effective_diffusivity(self, case=None)#

Computes the effective diffusivity using one of the selected assumptions:

Diffusion.diffusive_flux_molar_molar#

multipy.diffusion.Diffusion.diffusive_flux_molar_molar(self, grad_species_mole_fractions, mixture_molar_density)#

Computes the molar diffusive flux relative to a molar-averaged velocity, \(\mathbf{J}_i\), from the generalized Fick’s law:

\[\mathbf{J}_i = - c \mathbf{D} \nabla \mathbf{X}_i\]

where \(\mathbf{D}\) are the molar Fickian diffusion coefficients expressed in a molar-averaged velocity reference frame.

Parameters

  • grad_species_mole_fractions – vector numpy.ndarray specifying the species mole fractions gradients, \(\nabla \mathbf{X}_i\), in \([-]\). It should be of size (n_species,n_observations).

  • mixture_molar_density – mixture molar density, \(c\), in \([mole/m^3]\).

Returns

  • diffusive_flux - vector numpy.ndarray of molar diffusive fluxes relative to a molar-averaged velocity, \(\mathbf{J}_i\), in \([mole/(m^2s)]\). It has size (n_species,n_observations).

Diffusion.diffusive_flux_mass_mass#

multipy.diffusion.Diffusion.diffusive_flux_mass_mass(self, grad_species_mass_fractions, mixture_mass_density)#

Computes the mass diffusive flux relative to a mass-averaged velocity, \(\mathbf{j}_i\), from the generalized Fick’s law:

\[\mathbf{j}_i = - \rho \mathbf{D}^o \nabla \mathbf{Y}_i\]

where \(\mathbf{D}^o\) are the mass Fickian diffusion coefficients expressed in a molar-averaged velocity reference frame.

Parameters

  • grad_species_mass_fractions – vector numpy.ndarray specifying the species mass fractions gradients, \(\nabla \mathbf{Y}_i\), in \([-]\). It should be of size (n_species,n_observations).

  • mixture_mass_density – scalar numpy.ndarray specifying the mixture mass density, \(\pmb{\rho}\), in \([kg/m^3]\). It should be of size (1,n_observations).

Returns

  • diffusive_flux - vector numpy.ndarray of mass diffusive fluxes relative to a mass-averaged velocity, \(\mathbf{j}_i\), in \([kg/(m^2s)]\). It has size (n_species,n_observations).

Diffusion.diffusive_flux_molar_volume#

multipy.diffusion.Diffusion.diffusive_flux_molar_volume(self, grad_species_molar_densities)#

Computes the molar diffusive flux relative to a volume-averaged velocity, \(\mathbf{J}_i^V\), from the generalized Fick’s law:

\[\mathbf{J}_i^V = - \mathbf{D}^V \nabla \mathbf{c}_i\]

where \(\mathbf{D}^V\) are the molar Fickian diffusion coefficients expressed in a volume-averaged velocity reference frame.

Parameters

grad_species_molar_densities – vector numpy.ndarray specifying the species molar densities gradients, \(\nabla \mathbf{c}_i\), in \([-]\). It should be of size (n_species,n_observations).

Returns

  • diffusive_flux - vector numpy.ndarray of molar diffusive fluxes relative to a volume-averaged velocity, \(\mathbf{J}_i^V\), in \([mole/(m^2s)]\). It has size (n_species,n_observations).