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Population balance model

The PBM in CADET is implemented as part of the reaction module and can thus be used in any unit operation that includes reactions. Typical applications consider crystallization in a Continuous stirred tank reactor model or, to model continuous processes, in a Dispersive Plug-Flow Reactor (DPFR), which is modelled by a Lumped Rate Model Without Pores.

The particle size domain (internal coordinate) is discretized by the FV method, giving us a finite set of particle sizes under consideration \(\{x_1, \dots, x_{N_x}\}\). Every particle size considered is treated as an individual component of the unit operation and the field NCOMP of that unit operation in which the crystallization happens, must be specified accordingly as \(N_x + 2\). The two additional components account for the solute \(c\) and solubility \(c_\text{eq}\). That is, by setting the ``NCOMP`` field, you specify the number of FV cells for the internal coordinate.

Note that the first component must be solute \(c\) and the last component must be the solubility \(c_\text{eq}\).

Example code to configure the PBM can be found in CADET-Verification .

Group /input/model/unit_XXX

NCOMP

Number of components, which is given by two plus the number discrete particle sizes, which is given by the number of FV cells discretizing the internal coordinate

Type: int

Range: \(\geq 3\)

Length: 1

REACTION_MODEL

The crystallization code is implemented as a reaction module, which is why crystallization needs to be specified here

Type: String

Range: \(\{ CRYSTALLIZATION \}\)

Length: 1

Group /input/model/unit_XXX/reaction_bulk - REACTION_MODEL = CRYSTALLIZATION - UNIT_TYPE = CSTR

The following parameters need to be specified under Group /input/model/unit_XXX/reaction/reaction_bulk/ for CSTR units, and Group /input/model/unit_XXX/reaction/reaction/ for transport units like the LRM.

CRY_BINS

Coordinates of the cell faces, e.g. equidistant or logarithmic discretization of the internal coordinate \(x \in [x_c, x_\text{end}]\), including the end points.

Type: double

Range: \(\geq 1\)

Length: \(\mathrm{N_x} + 1\)

CRY_NUCLEI_MASS_DENSITY

Nulcei mass density

Type: double

Range: \(\geq 0\)

Length: 1

CRY_VOL_SHAPE_FACTOR

Volumetric shape factor of the particles

Type: double

Range: \(\geq 0\)

Length: 1

CRY_PRIMARY_NUCLEATION_RATE

Primary nucleation rate constant \(k_p\)

Type: double

Range: \(\geq 0\)

Length: 1

CRY_SECONDARY_NUCLEATION_RATE

Secondary nucleation rate \(k_b\)

Type: double

Range: \(\geq 0\)

Length: 1

CRY_GROWTH_RATE_CONSTANT

Growth rate constant \(k_g\)

Type: double

Range: \(\geq 0\)

Length: 1

CRY_GROWTH_CONSTANT

Growth constant \(\gamma\)

Type: double

Range: \(\geq 0\)

Length: 1

CRY_A

Defines constant \(a\) used to determine the growth rate

Type: double

Range: \(\geq 0\)

Length: 1

CRY_G

Defines constant \(g\) used to determine the growth rate

Type: double

Range: \(\geq 0\)

Length: 1

CRY_P

Defines constant \(p\) used to determine the growth rate

Type: double

Range: \(\geq 0\)

Length: 1

CRY_GROWTH_DISPERSION_RATE

Growth dispersion rate \(D_g\)

Type: double

Range: \(\geq 0\)

Length: 1

CRY_U

Defines constant \(u\) used to determine the primary nucleation

Type: double

Range: \(\geq 0\)

Length: 1

CRY_B

Defines constant \(b\) used to determine the secondary nucleation

Type: double

Range: \(\geq 0\)

Length: 1

CRY_K

Defines constant \(k\) used to determine the secondary nucleation

Type: double

Range: \(\geq 0\)

Length: 1

CRY_GROWTH_SCHEME_ORDER

Defines the growth flux FV reconstruction scheme. It can only be \(1\): upwind scheme; \(2\): HR Koren scheme; \(3\): WENO23 scheme; \(4\): WENO35 scheme. We recommend using the HR Koren scheme, which showed to be the most performant in our benchmarks.

Type: int

Range: \([1, \dots, 4]\)

Length: 1