Pipe (1D) Element

Introduction

The Chemical Pipe consists in the discretization of one-dimensional (1D) rective advection-diffusion equation:

\begin{eqnarray*}k\phi\frac{dc}{dt} + v\frac{dc}{d\ell} - \phi D\frac{d^2c}{d\ell^2} - R = 0 \end{eqnarray*}

where :

• \( c \) is the species concentration;
• \( \ell \) is longitudinal coordinates of the conduit element;
• \( v \) is the velocity;
• \( k \) is the retardation;
• \( \phi \) is the porosity;
• \( D \) is diffusion term;
• \( R \) is reaction term;

The finite element discretization of the transport equation is made by using Galerkin-FEM method. With the application of the integration by parts followed by the Green's theorem in the dispersive and the advective terms is obtained the weak form of the transport equation:

\begin{eqnarray*} \int_{\Omega }{{{\frac{\partial \mathbf{N}}{\partial \ell }}^{\text{T}}}\left( \phi D\frac{\partial \mathbf{N}}{\partial \ell } -v\mathbf{N} \right)d\Omega {{\mathbf{c}}_{i}}}+k\phi\int_{\Omega }{{{\mathbf{N}}^{\text{T}}}\mathbf{N}d\Omega }\frac{\partial {{\mathbf{c}}_{i}}}{\partial t} +\int_{\Gamma }{\left( v{{\frac{\partial \mathbf{N}}{\partial \ell }}^{\text{T}}}\mathbf{N} -\phi D{{\mathbf{N}}^{\text{T}}}\frac{\partial \mathbf{N}}{\partial \ell } \right){{\mathbf{\eta }}_{\ell }}d\Gamma {{\mathbf{c}}_{i}}} -\int_{\Omega }{{{\mathbf{N}}^{\text{T}}}Rd\Omega }=\mathbf{0} \end{eqnarray*}

where:

• \( \mathbf{N} \) is the matrix of shape functions;
• \( \mathbf{c}_i \) is the nodal species concentration;
• \( \Omega \) is the element domain;
• \( \Gamma \) is the element boundary;
• \( \eta_\ell \) is the normal component in the longitudinal direction \( \ell \) with respect to the element boundary \( \Gamma \) ;

Properties

The chemical pipe requires the following properties:

Property	Description	Type	Def. Unit	required
Dp	Pipe diameter	Scalar	m	Yes
D	Diffusion	Scalar	m2/s	Yes
phi	Porosity	Scalar	–	Yes
v	Velocity	Vector	m/s	Yes
k	Retardation	Scalar	–	Yes
R	Reaction rate	Scalar	mol/(m3.s)	Yes

Example:

PropertySet{
        id          = 'MatProp',
        typeName    = 'GemaPropertySet',
        description = 'Material parameters',
        properties  = {
                {id = 'Dp',   description = 'Pipe diameter', unit = 'm'},
                {id = 'D',   description = 'diffusion', unit = 'm^2/s'},
                {id = 'phi', description = 'porosity',  unit = ''},
                {id = 'v',   description = 'velocity', dim=2, unit = 'm/s'},
                {id = 'k',   description = 'retardation'},
                {id = 'R',   description = 'reaction rate', unit = 'mol/(m3.s)'},

        },
        values = {
                {Dp = 0.797884560802865, D = 3.56E-6, phi = 0.4, v = {4.0e-5,0}, k = 1.0, R = 0.0 },
        }
}

In terms to use the pipe in the chemical physics, the typename in the configurations of PhysicalMethod have to be set as 'ChemicalFemPhysics.pipeFixedReaction'.

Example:

PhysicalMethod {
  id       = 'ChemistryPhysics',
  typeName = 'ChemicalFemPhysics.pipeFixedReaction',
  type     = 'fem',

  mesh = 'mesh',
  boundaryConditions = {'bc1','bc2'},
}

Material attributes

During the reactive transport analysis using the pipe elements, the following internal Gauss attributes are stored:

Attribute	Description	Type	Required	Def. Unit	History
Dp	Pipe diameter	scalar	Yes	m	Yes

Units

The Chemical Fem physics uses internally the international system of units for its calculations, as can be seen above by the default units for properties and attributes. The whole unit system can be replaced through the unitSystem physics attribute, as described at the gemaFemProcessCommonFemPhysicsOptions page.

When making the substitution, the units on the following table should be replaced by the desired units, forming a coherent unit system. The unit system of the material should be also replaced.

local newUnitSystem = {
  time     = 's',
  coord    = 'm',
  C        = 'mol/m3',
  v        = 'm/s',
  D        = 'm2/s',
  Dp       = 'm',
  C-q      = 'mol/(m2.s)',
  q        = 'mol/(m2.s)',
  R        = 'mol/(m3.s)',
}

References

1. Kovacs, A., & Sauter, M. (2007). Modelling karst hydrodynamics. In N. Goldscheider & D. Drew (Eds.), Methods in karst hydrogeology. London: Taylor & Francis Group.

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