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Axial Flow Column 2D

Group /input/model/unit_XXX - UNIT_TYPE - COLUMN_MODEL_2D

UNIT_TYPE

Specifies the type of unit operation model

Type: string

Range: \(\texttt{COLUMN_MODEL_2D}\)

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. This parameter is optional and will be ignored if COL_RADIUS is provided. Unit: \(\mathrm{m}^{2}\)

Type: double

Range: \(>0\)

Length: 1

COL_LENGTH

Column length / height

Unit: \(\mathrm{m}\)

Type: double

Range: \(> 0\)

Length: 1

COL_RADIUS

Column radius. This parameter is optional if CROSS_SECTION_AREA is provided.

Unit: \(\mathrm{m}\)

Type: double

Range: \(> 0\)

Length: 1

COL_POROSITY

Column porosity, either constant (length is 1) or for each radial zone (length is \(\texttt{NRAD}\)). In case of a spatially inhomogeneous setting, the \(\texttt{SENS_PARTYPE}\) field is used for indexing the radial zone when specifying parameter sensitivities.

Type: double

Range: \((0,1]\)

Length: \(1 / \texttt{NRAD}\)

PAR_TYPE_VOLFRAC

Volume fractions of the particle types. The volume fractions can be set homogeneous or individually along both axes. For each cell, the volume fractions have to sum to 1. In case of a spatially inhomogeneous setting, the \(\texttt{SENS_SECTION}\) field is used for indexing the axial cell and the \(\texttt{SENS_REACTION}\) field is used for indexing the radial cell when specifying parameter sensitivities. This field is optional in case of only one particle type.

Type: double

Range: \([0,1]\)

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

PAR_TYPE_VOLFRAC_MULTIPLEX

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

0. Radial-independent, axial-independent; length of \(\texttt{PAR_TYPE_VOLFRAC}\) is \(\texttt{NPARTYPE}\)

1. Radial-dependent, axial-independent; length of \(\texttt{PAR_TYPE_VOLFRAC}\) is \(\texttt{NRAD} \cdot \texttt{NPARTYPE}\); ordering is radial-major

2. Axial-dependent; length of \(\texttt{PAR_TYPE_VOLFRAC}\) is \(\texttt{NCOL} \cdot \texttt{NPARTYPE}\); ordering is axial-major

3. Radial-dependent, axial-dependent; length of \(\texttt{PAR_TYPE_VOLFRAC}\) is \(\texttt{NCOL} \cdot \texttt{NRAD} \cdot \texttt{NPARTYPE}\); ordering is axial-radial-major

VELOCITY

Indicates flow direction in each radial zone (forward if value is positive, backward if value is negative), see Section Specification of flow rate / velocity and direction). In case of a spatially inhomogeneous setting, the \(\texttt{SENS_PARTYPE}\) field is used for indexing the radial cell when specifying parameter sensitivities.

Type: double

Range: \(\mathbb{R}\)

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

VELOCITY_MULTIPLEX

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

0. Radial-independent, section-independent; length of \(\texttt{VELOCITY}\) is 1

1. Radial-dependent, section-independent; length of \(\texttt{VELOCITY}\) is \(\texttt{NRAD}\)

2. Section-dependent; length of \(\texttt{VELOCITY}\) is \(\texttt{NSEC}\)

3. Radial-dependent, section-dependent; length of \(\texttt{VELOCITY}\) is \(\texttt{NRAD} \cdot \texttt{NSEC}\); ordering is section-major

COL_DISPERSION_AXIAL

Axial dispersion coefficient

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

Type: double

Range: \(\geq 0\)

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

COL_DISPERSION_AXIAL_MULTIPLEX

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

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

1. Component-independent, radial-dependent, section-independent; length of \(\texttt{COL_DISPERSION_AXIAL}\) is \(\texttt{NRAD}\)

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

3. Component-dependent, radial-dependent, section-independent; length of \(\texttt{COL_DISPERSION_AXIAL}\) is \(\texttt{NCOMP} \cdot \texttt{NRAD}\); ordering is radial-major

4. Component-independent, radial-independent, section-dependent; length of \(\texttt{COL_DISPERSION_AXIAL}\) is \(\texttt{NSEC}\)

5. Component-independent, radial-dependent, section-dependent; length of \(\texttt{COL_DISPERSION_AXIAL}\) is \(\texttt{NRAD} \cdot \texttt{NSEC}\); ordering is section-major

6. Component-dependent, radial-independent, section-independent; length of \(\texttt{COL_DISPERSION_AXIAL}\) is \(\texttt{NCOMP} \cdot \texttt{NSEC}\); ordering is section-major

7. Component-dependent, radial-dependent, section-dependent; length of \(\texttt{COL_DISPERSION_AXIAL}\) is \(\texttt{NCOMP} \cdot \texttt{NRAD} \cdot \texttt{NSEC}\); ordering is section-radial-major

Type: int

Range: \(\{0, \dots, 7 \}\)

Length: 1

COL_DISPERSION_RADIAL

Radial dispersion coefficient. In case of a spatially inhomogeneous setting, the \(\texttt{SENS_PARTYPE}\) field is used for indexing the radial zone when specifying parameter sensitivities.

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

Type: double

Range: \(\geq 0\)

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

COL_DISPERSION_RADIAL_MULTIPLEX

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

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

1. Component-independent, radial-dependent, section-independent; length of \(\texttt{COL_DISPERSION_RADIAL}\) is \(\texttt{NRAD}\)

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

3. Component-dependent, radial-dependent, section-independent; length of \(\texttt{COL_DISPERSION_RADIAL}\) is \(\texttt{NCOMP} \cdot \texttt{NRAD}\); ordering is radial-major

4. Component-independent, radial-independent, section-dependent; length of \(\texttt{COL_DISPERSION_RADIAL}\) is \(\texttt{NSEC}\)

5. Component-independent, radial-dependent, section-dependent; length of \(\texttt{COL_DISPERSION_RADIAL}\) is \(\texttt{NRAD} \cdot \texttt{NSEC}\); ordering is section-major

6. Component-dependent, radial-independent, section-independent; length of \(\texttt{COL_DISPERSION_RADIAL}\) is \(\texttt{NCOMP} \cdot \texttt{NSEC}\); ordering is section-major

7. Component-dependent, radial-dependent, section-dependent; length of \(\texttt{COL_DISPERSION_RADIAL}\) is \(\texttt{NCOMP} \cdot \texttt{NRAD} \cdot \texttt{NSEC}\); ordering is section-radial-major

Type: int

Range: \(\{0, \dots, 7 \}\)

Length: 1

REACTION_MODEL_BULK

Specifies the type of reaction model of the bulk volume. The model is configured in the subgroup \(\texttt{reaction_bulk}\).

Type: string

Range: See Section Reaction models

Length: 1

INIT_C

Initial concentrations for each component in all radial zones the bulk mobile phase (length \(\texttt{NCOMP}\)), or for each component in each radial zone (length \(\texttt{NCOMP} \cdot \texttt{NRAD}\), ordering radial-major)

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

Type: double

Range: \(\geq 0\)

Length: \(\texttt{NCOMP} / \texttt{NCOMP} \cdot \texttt{NRAD}\)

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)

Unit: \(various\)

Type: double

Range: \(\mathbb{R}\)

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

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_2D

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 it’s 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. Optional, defaults to \(\texttt{FV}\)

Type: string

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

Length: 1

RADIAL_DISC_TYPE

Specifies the radial discretization scheme. Valid values are \(\texttt{EQUIDISTANT}\), \(\texttt{EQUIVOLUME}\), and \(\texttt{USER_DEFINED}\).

Type: string

Length: 1

RADIAL_COMPARTMENTS

Coordinates for the radial compartment boundaries (ignored if \(\texttt{RADIAL_DISC_TYPE} \neq \texttt{USER_DEFINED}\)). The coordinates are absolute and have to include the endpoints 0 and \(\texttt{COLUMN_RADIUS}\). The values are expected in ascending order (i.e., 0 is the first and \(\texttt{COLUMN_RADIUS}\) the last value in the array).

Unit: \(\mathrm{m}\)

Type: double

Range: \([0,\texttt{COLUMN_RADIUS}]\)

Length: \(\texttt{NRAD} + 1\)

Discontinuous Galerkin

AX_POLYDEG

DG polynomial degree for axial discretization. Optional, defaults to 4 and \(N^z_d \in \{3, 4, 5\}\) is recommended. The total number of axial discrete points is given by (AX_POLYDEG + 1 ) * AX_NELEM

Type: int

Range: \(\geq 1\)

Length: 1

AX_NELEM

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

Type: int

Range: \(\geq 1\)

Length: 1

RAD_POLYDEG

DG polynomial degree for radial discretization. Optional, defaults to 4 and \(N^r_d \in \{3, 4, 5\}\) is recommended, and should generally be the same as the axial degree. The total number of radial discrete points is given by (RAD_POLYDEG + 1 ) * RAD_NELEM

Type: int

Range: \(\geq 1\)

Length: 1

RAD_NELEM

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

Type: int

Range: \(\geq 1\)

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: int

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

Length: 1

For further information on discretization parameters, see also Nonlinear solver for consistent initialization.

Finite Volumes

NCOL

Number of axial column discretization cells

Type: int

Range: \(\geq 1\)

Length: 1

NRAD

Number of radial column discretization cells

Type: int

Range: \(\geq 1\)

Length: 1

NPAR

Number of particle (radial) discretization cells for each particle type

Type: int

Range: \(\geq 1\)

Length: \(1 / \texttt{NPARTYPE}\)

LINEAR_SOLVER_BULK

Linear solver used for the sparse column bulk block. This field is optional, the best available method is selected (i.e., sparse direct solver if possible). Valid values are:

• \(\texttt{DENSE}\) Converts the sparse matrix into a banded matrix and uses regular LAPACK. Slow and memory intensive, but always available.

• \(\texttt{UMFPACK}\) Uses the UMFPACK sparse direct solver (LU decomposition) from SuiteSparse. Fast, but has to be enabled when compiling and requires UMFPACK library.

• \(\texttt{SUPERLU}\) Uses the SuperLU sparse direct solver (LU decomposition). Fast, but has to be enabled when compiling and requires SuperLU library.

Type: string

Range: \(\{\texttt{DENSE},\texttt{UMFPACK},\texttt{SUPERLU}\}\)

Length: 1

RECONSTRUCTION

Type of reconstruction method for fluxes

Type: string

Range: \(\texttt{WENO}\)

Length: 1

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: \(\texttt{MAX_KRYLOV} = \texttt{NCOL} \cdot \texttt{NRAD} \cdot \texttt{NCOMP} \cdot \texttt{NPARTYPE}\))

Type: int

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

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

For further discretization parameters, see also Flux reconstruction methods, and Nonlinear solver for consistent initialization.