Solution templates#

Class Templates#

class multipy.templates.Templates#

Contains solution templates and building blocks for common multicomponent mass trasfer problems.

Templates.stefan_diffusion#

multipy.templates.Templates.stefan_diffusion(self, alpha, beta)#

Computes a matrix \(\pmb{\Phi}\) and a vector \(\pmb{\phi}\) such that:

\[ \begin{align}\begin{aligned}\Phi_{i, i} = \frac{\alpha_i}{\beta_{i, n}} + \sum_{j \neq i}^{n} \frac{\alpha_j}{\beta_{i,j}}\\\Phi_{i, j} = - \alpha_i \Big( \frac{1}{\beta_{i, j}} - \frac{1}{\beta_{i, n}} \Big)\end{aligned}\end{align} \]

and

\[\phi_i = - \frac{\alpha_i}{\beta_{i, n}}\]

where \(n\) is the number of species and \(\alpha_i\) and \(\beta_{i,j}\) are free user-specified coefficients.

This template can be used in Stefan diffusion -type problems.

Parameters

  • alpha – scalar numpy.ndarray specifying the cofficients \(\alpha_{i}\). It should be of size (n_species,1).

  • beta – scalar numpy.ndarray specifying the coefficients \(\beta_{i,j}\). It should be of size (n_species,n_species).

Returns

  • phi - scalar numpy.ndarray \(\pmb{\phi}\). It has size (n_species-1,1).

  • Phi - scalar numpy.ndarray \(\pmb{\Phi}\). It has size (n_species-1,n_species-1).