In this example, we will learn how to perform an advection-diffusion simulation of a given chemical species through a Cubic network. The algorithm can be applied to more complex networks in the same manner as described in this example. For the sake of simplicity, a one layer 3D cubic network is used here. On OpenPNM, 4 different space discretization schemes for the advection-diffusion problem are available and consist of:

1. Upwind

2. Hybrid

3. Powerlaw

4. Exponential

Depending on the Peclet number characterizing the transport (ratio of advective to diffusive fluxes), the solutions obtained using these schemes may differ. In order to achive a high numerical accuracy, the user should use either the powerlaw or the exponential schemes.

## Generating network#

First, we need to generate a Cubic network. For now, we stick to a one layer 3d network, but you might as well try more complex networks!

[1]:

import numpy as np
import openpnm as op
%config InlineBackend.figure_formats = ['svg']
np.random.seed(10)
%matplotlib inline
ws = op.Workspace()
ws.settings["loglevel"] = 40
np.set_printoptions(precision=5)
shape = [1, 20, 30]
net = op.network.Cubic(shape=shape, spacing=1e-4)


Next, we need to add a geometry to the generated network. A geometry contains information about size of the pores/throats in a network. OpenPNM has tons of prebuilt geometries that represent the microstructure of different materials such as Toray090 carbon papers, sand stone, electrospun fibers, etc. For now, we stick to a sample geometry called SpheresAndCylinders that assigns random values to pore/throat diameters.

[2]:

geom = op.geometry.SpheresAndCylinders(network=net, pores=net.Ps, throats=net.Ts)


Next, we need to add a phase to our simulation. A phase object(s) contain(s) thermophysical information about the working fluid(s) in the simulation. OpenPNM has tons of prebuilt phases as well! For this simulation, we use air as our working fluid.

[3]:

air = op.phases.Air(network=net)

---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
Input In [3], in <cell line: 1>()
----> 1 air = op.phases.Air(network=net)

AttributeError: module 'openpnm' has no attribute 'phases'


Finally, we need to add a physics. A physics object contains information about the working fluid in the simulation that depend on the geometry of the network. A good example is diffusive conductance, which not only depends on the thermophysical properties of the working fluid, but also depends on the geometry of pores/throats.

[4]:

phys_air = op.physics.Standard(network=net, phase=air, geometry=geom)

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Input In [4], in <cell line: 1>()
----> 1 phys_air = op.physics.Standard(network=net, phase=air, geometry=geom)

NameError: name 'air' is not defined


# Performing Stokes flow#

Note that the advection diffusion algorithm assumes that velocity field is given. Naturally, we solve Stokes flow inside a pore network model to obtain the pressure field, and eventually the velocity field. Therefore, we need to run the StokesFlow algorithm prior to running our advection diffusion. There’s a separate tutorial on how to run StokesFlow in OpenPNM, but here’s a simple code snippet that does the job for us.

[5]:

sf = op.algorithms.StokesFlow(network=net, phase=air)
sf.set_value_BC(pores=net.pores('front'), values=200.0)
sf.set_value_BC(pores=net.pores('back'), values=0.0)
sf.run();

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Input In [5], in <cell line: 1>()
----> 1 sf = op.algorithms.StokesFlow(network=net, phase=air)
2 sf.set_value_BC(pores=net.pores('front'), values=200.0)
3 sf.set_value_BC(pores=net.pores('back'), values=0.0)

NameError: name 'air' is not defined


It is essential that you attach the results from StokesFlow (i.e. pressure field) to the corresponding phase, since the results from any algorithm in OpenPNM are by default only attached to the algorithm object (in this case to sf). Here’s how you can update your phase:

[6]:

air.update(sf.results())

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Input In [6], in <cell line: 1>()
----> 1 air.update(sf.results())

NameError: name 'air' is not defined


Now that everything is set up, it’s time to perform our advection-diffusion simulation. For this purpose, we need to add corresponding algorithm to our simulation. As mentioned above, OpenPNM supports 4 different discretizations that may be used with the AdvectionDiffusion and Dispersion algorithms. Setting the discretization scheme can be performed when defining the physics model as follows:

[7]:

mod = op.models.physics.ad_dif_conductance.ad_dif

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Input In [7], in <cell line: 2>()

NameError: name 'phys_air' is not defined


Then, the advection-diffusion algorithm is defined by:

[8]:

ad = op.algorithms.AdvectionDiffusion(network=net, phase=air)

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Input In [8], in <cell line: 1>()

NameError: name 'air' is not defined


Note that network and phase are required parameters for pretty much every algorithm we add, since we need to specify on which network and for which phase do we want to run the algorithm.

Note that you can also specify the discretization scheme by modifying the settings of our AdvectionDiffusion algorithm. You can choose between upwind, hybrid, powerlaw, and exponential. It is important to note that the scheme specified within the algorithm’s settings is only used when calling the rate method for post processing.

Next, we need to add some boundary conditions to the simulation. By default, OpenPNM assumes zero flux for the boundary pores.

[9]:

inlet  = net.pores('front')
outlet = net.pores(['back', 'top', 'bottom'])

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Input In [9], in <cell line: 3>()
1 inlet  = net.pores('front')
2 outlet = net.pores(['back', 'top', 'bottom'])

NameError: name 'ad' is not defined


set_value_BC applies the so-called “Dirichlet” boundary condition to the specified pores. Note that unless you want to apply a single value to all of the specified pores (like we just did), you must pass a list (or ndarray) as the values parameter.

## Running the algorithm#

Now, it’s time to run the algorithm. This is done by calling the run method attached to the algorithm object.

[10]:

ad.run();

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Input In [10], in <cell line: 1>()

NameError: name 'ad' is not defined


# Post processing#

When an algorithm is successfully run, the results are attached to the same object. To access the results, you need to know the quantity for which the algorithm was solving. For instance, AdvectionDiffusion solves for the quantity pore.concentration, which is somewhat intuitive. However, if you ever forget it, or wanted to manually check the quantity, you can take a look at the algorithm settings:

[11]:

print(ad.settings)

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Input In [11], in <cell line: 1>()

NameError: name 'ad' is not defined


Now that we know the quantity for which AdvectionDiffusion was solved, let’s take a look at the results:

[12]:

c = ad['pore.concentration']

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Input In [12], in <cell line: 1>()

NameError: name 'ad' is not defined


## Heatmap#

Since the network is 2d, we can simply reshape the results in form of a 2d array similar to the shape of the network and plot the heatmap of it using matplotlib.

[13]:

print('Network shape:', shape)
c2d = c.reshape(shape)

Network shape: [1, 20, 30]

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Input In [13], in <cell line: 2>()
1 print('Network shape:', shape)
----> 2 c2d = c.reshape(shape)

NameError: name 'c' is not defined

[14]:

import matplotlib.pyplot as plt
plt.imshow(c2d[0,:,:]);
plt.title('Concentration (mol/m$^3$)');
plt.colorbar();

---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
Input In [14], in <cell line: 2>()
1 import matplotlib.pyplot as plt
----> 2 plt.imshow(c2d[0,:,:]);
3 plt.title('Concentration (mol/m$^3$)');
4 plt.colorbar()

NameError: name 'c2d' is not defined