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Axial Flow Column 1D

Group /input/model/unit_XXX - UNIT_TYPE - COLUMN_MODEL_1D

UNIT_TYPE

Specifies the type of unit operation model

Type: string

Range: \(\texttt{COLUMN_MODEL_1D}\)

Length: 1

GEOMETRY

Column geometry

Type: string

Range: \(\texttt{AXIAL_FLOW_CYLINDER}\)

Length: 1

NCOMP

Number of chemical components in the chromatographic medium

Type: int

Range: \(\geq 1\)

Length: 1

CROSS_SECTION_AREA

Cross section area of the column, see Section Specification of flow rate / velocity and direction

Unit: \(\mathrm{m}^{2}\)

Type: double

Range: \(>0\)

Length: 1

BED_LENGTH

Column length

Unit: \(\mathrm{m}\)

Type: double

Range: \(> 0\)

Length: 1

FORWARD_FLOW

Flow direction of the column.

Type: bool

Range: \(\{0,1\}\)

Length: 1 or NSEC

COL_POROSITY

Column porosity of the interstitial volume, either as a single value for the whole column or as one value per DG node. Used only if model is not a Lumped Rate Model without Pores, in which case TOTAL_POROSITY is used instead.

The second form describes a packing density that varies along the flow path and is available for every geometry, see Axially varying bulk porosity. The values are given in the ordering of the bulk solution, i.e. element by element from the column inlet at \(x = 0\) to \(x = \texttt{BED_LENGTH}\), with the nodes of each element in ascending order, exactly as CROSS_SECTIONAL_AREA_AT_NODES; the two entries that share an element interface node must be equal. CADET-Core multiplies the profile with the geometric cross section area of the chosen geometry itself, so no further field has to be adjusted. It requires SPATIAL_METHOD \(= \texttt{DG}\) and USE_COLLOCATION_DG \(= 0\), since the collocation operator bakes a constant coefficient into its operators.

Type: double

Range: \((0,1]\)

Length: \(1\) or \((\texttt{POLYDEG} + 1) \cdot \texttt{NELEM}\)

TOTAL_POROSITY

Total porosity including the porosity of the particles, either as a single value for the whole column or as one value per DG node. Only applicable for the Lumped Rate Model without pores, i.e. when NPARTYPE = 1 AND particle_type_000\HAS_FILM_DIFFUSION = 0

The second form describes a packing density that varies along the flow path and is available for every geometry under the same conditions as a nodal COL_POROSITY above, see Axially varying bulk porosity. The solid phase source is evaluated at the discrete points, i.e. by collocation of the element integral that weights it with the solid cross section area, which is exact whenever the porosity is constant on an element.

Type: double

Range: \((0,1]\)

Length: \(1\) or \((\texttt{POLYDEG} + 1) \cdot \texttt{NELEM}\)

FLOW_AREA_QUADRATURE_POLYDEG

Polynomial degree the flow through area \(w(x) = \varepsilon_c(x) A(x)\) is integrated exactly with when the element integrals are evaluated by Gauss quadrature, i.e. for GEOMETRY \(= \texttt{SMOOTHLY_VARYING}\) and for an axially varying porosity on any geometry, see Axially varying bulk porosity. Optional; defaults to POLYDEG, which integrates the nodal interpolant of \(w\) exactly and is all the transport operator can profit from. The bulk side operator of the film transfer term is integrated with the same rule but carries an additional factor \(1 / \varepsilon_c(x)\), which is not a polynomial, so the rule is not exact there. For a smooth porosity profile the default already reaches that integral to round-off, so raising the degree is an escape hatch for a profile that is hard to integrate rather than a knob that routinely pays off. Ignored by the analytic geometries with a constant porosity, which integrate their weight in closed form.

Type: int

Range: \(\geq \texttt{POLYDEG}\)

Length: 1

NPARTYPE

Number of particle types. A value of \(0\) means that no particles are present.

Type: int

Range: \(\geq 0\)

Length: 1

PAR_TYPE_VOLFRAC

Volume fractions of the particle types. The volume fractions can be set for all axial cells together or for each individual axial cell. For each cell, the volume fractions have to sum to \(1\). In case of a spatially inhomogeneous setting, the data is expected in cell-major ordering and the \(\texttt{SENS_SECTION}\) field is used for indexing the axial cell when specifying parameter sensitivities. This field is optional in case of only one particle type.

Type: double

Range: \([0,1]\)

Length: \(\texttt{NPARTYPE} / \texttt{NCOL} \cdot \texttt{NPARTYPE}\)

COL_DISPERSION

Axial dispersion coefficient

Unit: \(\mathrm{m}_{\mathrm{IV}}^{2}\,\mathrm{s}^{-1}\)

Type: double

Range: \(\geq 0\)

Length: see \(\texttt{COL_DISPERSION_MULTIPLEX}\)

COL_DISPERSION_MULTIPLEX

Multiplexing mode of \(\texttt{COL_DISPERSION}\). Determines whether \(\texttt{COL_DISPERSION}\) is treated as component- and/or section-independent. This field is optional. When left out, multiplexing behavior is inferred from the length of \(\texttt{COL_DISPERSION}\). Valid modes are:

  1. Component-independent, section-independent; length of \(\texttt{COL_DISPERSION}\) is \(1\)

  2. Component-dependent, section-independent; length of \(\texttt{COL_DISPERSION}\) is \(\texttt{NCOMP}\)

  3. Component-independent, section-dependent; length of \(\texttt{COL_DISPERSION}\) is \(\texttt{NSEC}\)

  4. Component-dependent, section-dependent; length of \(\texttt{COL_DISPERSION}\) is \(\texttt{NCOMP} \cdot \texttt{NSEC}\); ordering is section-major

Type: int

Range: \(\{0, \dots, 3 \}\)

Length: 1

INIT_C

Initial concentrations for each component in the bulk mobile phase

Unit: \(\mathrm{mol}\,\mathrm{m}_{\mathrm{IV}}^{-3}\)

Type: double

Range: \(\geq 0\)

Length: \(\texttt{NCOMP}\)

INIT_STATE

Full state vector for initialization (optional, \(\texttt{INIT_C}\), \(\texttt{INIT_CP}\), and \(\texttt{INIT_CS}\) will be ignored; if length is \(2\texttt{NDOF}\), then the second half is used for time derivatives). The ordering of the state vector is defined in Unit Operation state vector ordering.

Unit: \(various\)

Type: double

Range: \(\mathbb{R}\)

Length: \(\texttt{NDOF} / 2\texttt{NDOF}\)

NREAC_LIQUID

Number of liquid phase reaction models (optional, only if liquid reactions are present).

Type: int

Range: \(\geq 0\)

Length: 1

Group /input/model/unit_XXX/liquid_reaction_YYY

Each reaction is specified in another subgroup liquid_reaction_YYY, see Reaction models.

Group /input/model/unit_XXX/particle_type_XXX

Each particle type is specified in another subgroup particle_type_XXX, see Particle Model.

Group /input/model/unit_XXX/discretization - UNIT_TYPE - COLUMN_MODEL_1D

USE_ANALYTIC_JACOBIAN

Determines whether analytically computed Jacobian matrix (faster) is used (value is 1) instead of Jacobians generated by algorithmic differentiation (slower, value is 0)

Type: int

Range: \(\{0, 1\}\)

Length: 1

Spatial discretization - Numerical Methods

CADET offers two spatial discretization methods: Finite Volumes (FV) and Discontinuous Galerkin (DG). Each method has its own set of input fields. While both methods approximate the same solution to the same underlying model, they may differ in terms of computational performance. With our currently implemented variants of FV and DG, FV perform better for solutions with steep gradients or discontinuities, while DG can be much faster for rather smooth solutions. For the same number of discrete points, DG will generally be slower but often more accurate.

For further information on the choice of discretization methods and their parameters, see Spatial discretization methods.

SPATIAL_METHOD

Spatial discretization method

Type: string

Range: \(\{\texttt{FV}, \texttt{DG}\}\)

Length: 1

Discontinuous Galerkin

POLYDEG

DG polynomial degree. Optional, defaults to 4 and \(N_d \in \{3, 4, 5\}\) is recommended. The total number of axial discrete points is given by (POLYDEG + 1 ) * NELEM

Type: int

Range: \(\geq 1\)

Length: 1

NELEM

Number of axial column discretization DG elements. The total number of axial discrete points is given by (POLYDEG + 1 ) * NELEM

Type: int

Range: \(\geq 1\)

Length: 1

NCOL

Number of axial discrete points. Optional and ignored if NELEM is defined. Otherwise, used to calculate NELEM = \(\lfloor\) NCOL / (POLYDEG + 1 ) \(\rfloor\)

Type: int

Range: \(\geq 1\)

Length: 1

USE_COLLOCATION_DG

Specifies whether the collocation DG variant should be used. Optional, only available for axial cylinder columns, defaults to 1.

Type: int

Range: \(\{0, 1\}\)

Length: 1

LINEAR_SOLVER

Specifies the linear solver variant used to factorize the semidiscretized system. Optional, defaults to SparseLU. For more information on these solvers, we refer to the Eigen documentation

Type: string

Range: \(\{\texttt{SparseLU}, \texttt{SparseQR}, ..., \texttt{BiCGSTAB}\}\)

Length: 1

When using the DG method, we generally recommend specifying USE_MODIFIED_NEWTON = 1 in Group /input/solver/time_integrator, i.e. to use the modified Newton method to solve the linear system within the time integrator. For further information on discretization parameters, see also Nonlinear solver for consistent initialization.

Finite Volumes

NCOL

Number of axial column discretization points, i.e. FV cells

Type: int

Range: \(\geq 1\)

Length: 1

RECONSTRUCTION

Type of reconstruction method for fluxes

Type: string

Range: \(\texttt{WENO, KOREN}\)

Length: 1

For further information on discretization parameters for reconstruction methods, see also Flux reconstruction methods (FV specific).

The following FV discretization parameters are only required if particles are present:

GS_TYPE

Type of Gram-Schmidt orthogonalization, see IDAS guide Section 4.5.7.3, p. 41f. A value of \(0\) enables classical Gram-Schmidt, a value of 1 uses modified Gram-Schmidt.

Type: int

Range: \(\{0, 1\}\)

Length: 1

MAX_KRYLOV

Defines the size of the Krylov subspace in the iterative linear GMRES solver (0: \(\texttt{MAX_KRYLOV} = \texttt{NCOL} \cdot \texttt{NCOMP} \cdot \texttt{NPARTYPE}\))

Type: int

Range: \(\{0, \dots, \texttt{NCOL} \cdot \texttt{NCOMP} \cdot \texttt{NPARTYPE} \}\)

Length: 1

MAX_RESTARTS

Maximum number of restarts in the GMRES algorithm. If lack of memory is not an issue, better use a larger Krylov space than restarts.

Type: int

Range: \(\geq 0\)

Length: 1

SCHUR_SAFETY

Schur safety factor; Influences the tradeoff between linear iterations and nonlinear error control; see IDAS guide Section 2.1 and 5.

Type: double

Range: \(\geq 0\)

Length: 1

FV_ARROW_HEAD_OPTIMIZATION

Optional, specifies whether an optimized Jacobian block structure is exploited by a customized linear solver module for improved computational efficiency. Prohibits mix of particle types and mix of FV, DG discretizations for bulk and particles.

Type: int

Range: \(\{0,1\}\)

Length: 1

FIX_ZERO_SURFACE_DIFFUSION

Determines whether the surface diffusion parameters \(\texttt{SURFACE_DIFFUSION}\) are fixed if the parameters are zero. If the parameters are fixed to zero (\(\texttt{FIX_ZERO_SURFACE_DIFFUSION} = 1\), \(\texttt{SURFACE_DIFFUSION} = 0\)), the parameters must not become non-zero during this or subsequent simulation runs. The internal data structures are optimized for a more efficient simulation. This field is optional and defaults to \(0\) (optimization disabled in favor of flexibility).

Type: int

Range: \(\{0, 1\}\)

Length: 1

For further information on discretization parameters, see also Flux reconstruction methods (FV specific), and Nonlinear solver for consistent initialization.