FiPyexamples.elphf.diffusion.mesh1Ddimensional¶
In this example, we present the same three-component diffusion problem
introduced in examples/elphf/diffusion/mesh1D.py
but we demonstrate FiPy’s facility to use dimensional quantities.
>>> import warnings
>>> warnings.warn("\n\n\tSupport for physical dimensions is incomplete.\n\tIt is not possible to solve dimensional equations.\n")
>>> from fipy import CellVariable, FaceVariable, PhysicalField, Grid1D, TransientTerm, DiffusionTerm, PowerLawConvectionTerm, LinearLUSolver, Viewer
>>> from fipy.tools import numerix
We solve the problem on a 40 mm long 1D mesh
>>> nx = 40
>>> dx = PhysicalField(1., "mm")
>>> L = nx * dx
>>> mesh = Grid1D(dx = dx, nx = nx)
Again, one component in this ternary system will be designated the “solvent”
>>> class ComponentVariable(CellVariable):
... def __init__(self, mesh, value = 0., name = '',
... standardPotential = 0., barrier = 0.,
... diffusivity = None, valence = 0, equation = None):
... CellVariable.__init__(self, mesh = mesh, value = value,
... name = name)
... self.standardPotential = Variable(standardPotential)
... self.barrier = Variable(barrier)
... self.diffusivity = Variable(diffusivity)
... self.valence = valence
... self.equation = equation
...
... def copy(self):
... return self.__class__(mesh = self.mesh,
... value = self.value,
... name = self.name,
... standardPotential =
... self.standardPotential,
... barrier = self.barrier,
... diffusivity = self.diffusivity,
... valence = self.valence,
... equation = self.equation)
>>> solvent = ComponentVariable(mesh = mesh, name = 'Cn', value = "1 mol/m**3")
We can create an arbitrary number of components, simply by providing a Tuple or list of components
>>> substitutionals = [
... ComponentVariable(mesh = mesh, name = 'C1', diffusivity = "1e-9 m**2/s",
... standardPotential = 1., barrier = 1., value = "0.3 mol/m**3"),
... ComponentVariable(mesh = mesh, name = 'C2', diffusivity = "1e-9 m**2/s",
... standardPotential = 1., barrier = 1., value = "0.6 mol/m**3"),
... ]
>>> interstitials = []
>>> for component in substitutionals:
... solvent -= component
We separate the solution domain into two different concentration regimes
>>> x = mesh.cellCenters[0]
>>> substitutionals[0].setValue("0.3 mol/m**3")
>>> substitutionals[0].setValue("0.6 mol/m**3", where=x > L / 2)
>>> substitutionals[1].setValue("0.6 mol/m**3")
>>> substitutionals[1].setValue("0.3 mol/m**3", where=x > L / 2)
We create one diffusion equation for each substitutional component
>>> for Cj in substitutionals:
... CkSum = ComponentVariable(mesh = mesh, value = 0.)
... CkFaceSum = FaceVariable(mesh = mesh, value = 0.)
... for Ck in [Ck for Ck in substitutionals if Ck is not Cj]:
... CkSum += Ck
... CkFaceSum += Ck.harmonicFaceValue
...
... convectionCoeff = CkSum.faceGrad \
... * (Cj.diffusivity / (1. - CkFaceSum))
...
... Cj.equation = (TransientTerm()
... == DiffusionTerm(coeff=Cj.diffusivity)
... + PowerLawConvectionTerm(coeff = convectionCoeff))
If we are running interactively, we create a viewer to see the results
>>> if __name__ == '__main__':
... viewer = Viewer(vars=[solvent] + substitutionals,
... datamin=0, datamax=1)
... viewer.plot()
Now, we iterate the problem to equilibrium, plotting as we go
>>> solver = LinearLUSolver()
>>> from builtins import range
>>> for i in range(40):
... for Cj in substitutionals:
... Cj.updateOld()
... for Cj in substitutionals:
... Cj.equation.solve(var = Cj,
... dt = "1000 s",
... solver = solver)
... if __name__ == '__main__':
... viewer.plot()
Since there is nothing to maintain the concentration separation in this problem, we verify that the concentrations have become uniform
>>> print(substitutionals[0].scaled.allclose("0.45 mol/m**3",
... atol = "1e-7 mol/m**3", rtol = 1e-7))
1
>>> print(substitutionals[1].scaled.allclose("0.45 mol/m**3",
... atol = "1e-7 mol/m**3", rtol = 1e-7))
1
Note
The absolute tolerance atol must be in units compatible with the value to be checked, but the relative tolerance rtol is dimensionless.