General Rate Model¶

Group /input/model/unit_XXX - UNIT_TYPE - GENERAL_RATE_MODEL¶

For information on model equations, refer to General rate model (GRM).

UNIT_TYPE

Specifies the type of unit operation model

Type: string

Range: $GENERAL_RATE_MODEL$

Length: 1

NCOMP

Number of chemical components in the chromatographic medium

Type: int

Range: $\geq 1$

Length: 1

ADSORPTION_MODEL

Specifies the type of binding model of each particle type (or of all particle types if length is $1$ )

Type: string

Range: Binding models

Length: $1 / NPARTYPE$

ADSORPTION_MODEL_MULTIPLEX

Multiplexing mode of $ADSORPTION_MODEL$ . If set to $0$ , each particle type has a different binding model and the length of $ADSORPTION_MODEL$ is $NPARTYPE$ . If set to $1$ , all particle types share the same binding model and the length of $ADSORPTION_MODEL$ is $1$ . This field is optional and inferred from the length of $ADSORPTION_MODEL$ if left out.

Type: int

Range: ${0, 1}$

Length: 1

NBOUND

Number of bound states for each component in each particle type in type-major ordering

Type: int

Range: $\geq 0$

Length: $NCOMP$ / $NPARTYPE \cdot NCOMP$

REACTION_MODEL_BULK

Specifies the type of reaction model of the bulk volume. The model is configured in the subgroup $reaction_bulk$ .

Type: string

Range: See Section Reaction models

Length: 1

REACTION_MODEL_PARTICLES

Specifies the type of reaction model of each particle type (or of all particle types if length is $1$ ). The model is configured in the subgroup $reaction_particle$ , or $reaction_particle_XXX$ in case of disabled multiplexing.

Type: string

Range: See Section Reaction models

Length: $1 / NPARTYPE$

REACTION_MODEL_PARTICLES_MULTIPLEX

Multiplexing mode of $REACTION_MODEL_PARTICLES$ . If set to $0$ , each particle type has a different reaction model and the length of $REACTION_MODEL_PARTICLES$ is $NPARTYPE$ . If set to $1$ , all particle types share the same reaction model and the length of $REACTION_MODEL_PARTICLES$ is $1$ . This field is optional and inferred from the length of $REACTION_MODEL_PARTICLES$ if left out.

Type: int

Range: ${0, 1}$

Length: 1

INIT_C

Initial concentrations for each component in the bulk mobile phase

Unit: $mol m_{IV}^{- 3}$

Type: double

Range: $\geq 0$

Length: $NCOMP$

INIT_CP

Initial concentrations for each component in the bead liquid phase (optional, $INIT_C$ is used if left out). The length of this field can be $NCOMP$ (same values for each particle type) or $NPARTYPE \cdot NCOMP$ Values for each particle type can only be given when $ADSORPTION_MODEL_MULTIPLEX$ is $0$ . The ordering is type-major.

Unit: $mol m_{MP}^{- 3}$

Type: double

Range: $\geq 0$

Length: $NCOMP / NPARTYPE \cdot NCOMP$

INIT_Q

Initial concentrations for each bound state of each component in the bead solid phase. If $ADSORPTION_MODEL_MULTIPLEX$ is $0$ , values for each particle type are required in type-component-major ordering (length is $NTOTALBND$ ). If $ADSORPTION_MODEL_MULTIPLEX$ is $1$ , values for one particle type are required in component-major ordering (length is $\sum_{i = 0}^{NCOMP - 1} {NBND}_{i}$ ).

Unit: $mol m_{SP}^{- 3}$

Type: double

Range: $\geq 0$

INIT_STATE

Full state vector for initialization (optional, $INIT_C$ , $INIT_CP$ , and $INIT_Q$ will be ignored; if length is $2 NDOF$ , then the second half is used for time derivatives)

Unit: $v a r i o u s$

Type: double

Range: $R$

Length: $NDOF / 2 NDOF$

COL_DISPERSION

Axial dispersion coefficient

Unit: $m_{IV}^{2} s^{- 1}$

Type: double

Range: $\geq 0$

Length: see $COL_DISPERSION_MULTIPLEX$

COL_DISPERSION_MULTIPLEX

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

Component-independent, section-independent; length of $COL_DISPERSION$ is $1$

Component-dependent, section-independent; length of $COL_DISPERSION$ is $NCOMP$

Component-independent, section-dependent; length of $COL_DISPERSION$ is $NSEC$

Component-dependent, section-dependent; length of $COL_DISPERSION$ is $NCOMP \cdot NSEC$ ; ordering is section-major

Type: int

Range: ${0, \dots, 3}$

Length: 1

COL_LENGTH

Column length

Unit: $m$

Type: double

Range: $> 0$

Length: 1

COL_POROSITY

Column porosity

Type: double

Range: $(0, 1]$

Length: 1

FILM_DIFFUSION

Film diffusion coefficients for each component of each particle type

Unit: $m s^{- 1}$

Type: double

Range: $\geq 0$

Length: see $FILM_DIFFUSION_MULTIPLEX$

FILM_DIFFUSION_MULTIPLEX

Multiplexing mode of $FILM_DIFFUSION$ . Determines whether $FILM_DIFFUSION$ is treated as component-, type-, and/or section-independent. This field is optional. When left out, multiplexing behavior is inferred from the length of $FILM_DIFFUSION$ . Valid modes are:

Component-dependent, type-independent, section-independent; length of $FILM_DIFFUSION$ is $NCOMP$

Component-dependent, type-independent, section-dependent; length of $FILM_DIFFUSION$ is $NCOMP \cdot NSEC$ ; ordering is section-major

Component-dependent, type-dependent, section-independent; length of $FILM_DIFFUSION$ is $NCOMP \cdot NPARTYPE$ ; ordering is type-major

Component-dependent, type-dependent, section-dependent; length of $FILM_DIFFUSION$ is $NCOMP \cdot NPARTYPE \cdot NSEC$ ; ordering is section-type-major

Type: int

Range: ${0, \dots, 3}$

Length: 1

PAR_POROSITY

Particle porosity of all particle types or for each particle type

Type: double

Range: $(0, 1]$

Length: $1 / NPARTYPE$

PAR_RADIUS

Particle radius of all particle types or for each particle type

Unit: $m$

Type: double

Range: $> 0$

Length: $1 / NPARTYPE$

PAR_CORERADIUS

Particle core radius of all particle types or for each particle type (optional, defaults to $m$ )

Unit: $m$

Type: double

Range: $[0, PAR_RADIUS)$

Length: $1 / NPARTYPE$

PORE_ACCESSIBILITY

Pore accessibility factor of each component in each particle type (optional, defaults to $1$ ). Note: Should not be used in combination with any binding model!

Type: double

Range: $(0, 1]$

Length: see $PORE_ACCESSIBILITY_MULTIPLEX$

PORE_ACCESSIBILITY_MULTIPLEX

Multiplexing mode of $PORE_ACCESSIBILITY$ . Determines whether $PORE_ACCESSIBILITY$ is treated as component-, type-, and/or section-independent. This field is optional. When left out, multiplexing behavior is inferred from the length of $PORE_ACCESSIBILITY$ . Valid modes are: 0. Component-dependent, type-independent, section-independent; length of $PORE_ACCESSIBILITY$ is $NCOMP$ 1. Component-dependent, type-independent, section-dependent; length of $PORE_ACCESSIBILITY$ is $NCOMP \cdot NSEC$ ; ordering is section-major 2. Component-dependent, type-dependent, section-independent; length of $PORE_ACCESSIBILITY$ is $NCOMP \cdot NPARTYPE$ ; ordering is type-major 3. Component-dependent, type-dependent, section-dependent; length of $PORE_ACCESSIBILITY$ is $NCOMP \cdot NPARTYPE \cdot NSEC$ ; ordering is section-type-major

Type: int

Range: ${0, \dots, 3}$

Length: 1

PAR_DIFFUSION

Effective particle diffusion coefficients of each component in each particle type

Unit: $m_{MP}^{2} s^{- 1}$

Type: double

Range: $> 0$

Length: see $PAR_DIFFUSION_MULTIPLEX$

PAR_DIFFUSION_MULTIPLEX

Multiplexing mode of $PAR_DIFFUSION$ . Determines whether $PAR_DIFFUSION$ is treated as component-, type-, and/or section-independent. This field is optional. When left out, multiplexing behavior is inferred from the length of $PAR_DIFFUSION$ . Valid modes are:

Component-dependent, type-independent, section-independent; length of $PAR_DIFFUSION$ is $NCOMP$

Component-dependent, type-independent, section-dependent; length of $PAR_DIFFUSION$ is $NCOMP \cdot NSEC$ ; ordering is section-major

Component-dependent, type-dependent, section-independent; length of $PAR_DIFFUSION$ is $NCOMP \cdot NPARTYPE$ ; ordering is type-major

Component-dependent, type-dependent, section-dependent; length of $PAR_DIFFUSION$ is $NCOMP \cdot NPARTYPE \cdot NSEC$ ; ordering is section-type-major

Type: int

Range: ${0, \dots, 3}$

Length: 1

PAR_SURFDIFFUSION

Particle surface diffusion coefficients of each bound state of each component in each particle type (optional, defaults to all 0 $m_{SP}^{2} s^{- 1}$ )

Unit: $m_{SP}^{2} s^{- 1}$

Type: double

Range: $\geq 0$

Length: see $PAR_SURFDIFFUSION_MULTIPLEX$

PAR_SURFDIFFUSION_MULTIPLEX

Multiplexing mode of $PAR_SURFDIFFUSION$ . Determines whether $PAR_SURFDIFFUSION$ is treated as component-, type-, and/or section-independent. This field is optional. When left out, multiplexing behavior is inferred from the length of $PAR_SURFDIFFUSION$ . Valid modes are:

Component-dependent, type-independent, section-independent; length of $PAR_SURFDIFFUSION$ is $NBND$ ; ordering is component-major
Component-dependent, type-independent, section-dependent; length of $PAR_SURFDIFFUSION$ is $NBND \cdot NSEC$ ; ordering is section-component-major
Component-dependent, type-dependent, section-independent; length of $PAR_SURFDIFFUSION$ is $NTOTALBND$ ; ordering is type-component-major
Component-dependent, type-dependent, section-dependent; length of $PAR_SURFDIFFUSION$ is $NTOTALBND \cdot NSEC$ ; ordering is section-type-component-major

PAR_SURFDIFFUSION_MULTIPLEX

Type: int

Range: ${0, \dots, 3}$

Length: 1

PAR_SURFDIFFUSION_DEP

Parameter dependence of $PAR_SURFDIFFUSION$ , please refer to Parameter Dependencies, section parameter-state dependencies, for more information.

VELOCITY

Interstitial velocity of the mobile phase (optional if $CROSS_SECTION_AREA$ is present, see Section Specification of flow rate / velocity and direction) Unit: $m s^{- 1}$

Type: double

Range: $R$

Length: $1 / NSEC$

CROSS_SECTION_AREA

Cross section area of the column (optional if $VELOCITY$ is present, see Section Specification of flow rate / velocity and direction) Unit: $m^{2}$

Type: double

Range: $> 0$

Length: 1

NPARTYPE

Number of particle types. Optional, inferred from the length of $NPAR$ or $NBOUND$ if left out.

Type: int

Range: $\geq 1$

Length: 1

PAR_GEOM

Specifies the particle geometry for all or each particle type. Valid values are $SPHERE$ , $CYLINDER$ , $SLAB$ . Optional, defaults to $SPHERE$ .

Type: string

Length: $1$ / $NPARTYPE$

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 $SENS_SECTION$ 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: $NPARTYPE / NCOL \cdot NPARTYPE$

Group /input/model/unit_XXX/discretization - UNIT_TYPE - GENERAL_RATE_MODEL¶

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

PAR_DISC_TYPE

Specifies the discretization scheme inside the particles for all or each particle type. Valid values are $EQUIDISTANT_PAR$ , $EQUIVOLUME_PAR$ , and $USER_DEFINED_PAR$ .

Type: string

Length: $1$ / $NPARTYPE$

PAR_DISC_VECTOR

Node coordinates for the cell boundaries (ignored if $PAR_DISC_TYPE \neq USER_DEFINED_PAR$ ). The coordinates are relative and have to include the endpoints $0$ and $1$ . They are later linearly mapped to the true radial range $[r_{c, j}, r_{p, j}]$ . The coordinates for each particle type are appended to one long vector in type-major ordering.

Type: double

Range: $[0, 1]$

Length: $\sum_{i} ({NPAR}_{i} + 1)$

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 $FV$

Type: string

Range: ${FV, DG}$

Length: 1

Finite Volumes (Default)¶

NCOL

Number of axial column discretization points

Type: int

Range: $\geq 1$

Length: 1

NPAR

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

Type: int

Range: $\geq 1$

Length: $1$ / $NPARTYPE$

PAR_BOUNDARY_ORDER

Order of accuracy of outer particle boundary condition. Optional, defaults to $2$ .

Type: int

Range: ${1, 2}$

Length: 1

RECONSTRUCTION

Type of reconstruction method for fluxes

Type: string

Range: $WENO$

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: $MAX_KRYLOV = NCOL \cdot NCOMP \cdot NPARTYPE$ )

Type: int

Range: ${0, \dots, NCOL \cdot NCOMP \cdot NPARTYPE}$

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

FIX_ZERO_SURFACE_DIFFUSION

Determines whether the surface diffusion parameters $PAR_SURFDIFFUSION$ are fixed if the parameters are zero. If the parameters are fixed to zero ( $FIX_ZERO_SURFACE_DIFFUSION = 1$ , $PAR_SURFDIFFUSION = 0$ ), 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 discretization parameters, see also Flux reconstruction methods (FV specific)), and Nonlinear solver for consistent initialization.

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 cellselements. 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 = $⌊$ NCOL / (POLYDEG + 1 ) $⌋$

Type: int

Range: $\geq 1$

Length: 1

EXACT_INTEGRATION

Specifies the DG integration variant. Optional, defaults to 0

Type: int

Range: ${0, 1}$

Length: 1

PAR_POLYDEG

DG particle (radial) polynomial degree. Optional, defaults to 3. The total number of particle (radial) discrete points is given by (PARPOLYDEG + 1 ) * PAR_NELEM.

Type: int

Range: $\geq 1$

Length: 1

PAR_NELEM

Number of particle (radial) discretization DG elements for each particle type. For the particle discretization, it is usually most performant to fix PAR_NELEM = 1 and to increase the polynomial degree for more accuracy.

Type: int

Range: $\geq 1$

Length: $1$ / $NPARTYPE$

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

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