Moment Properties#

Properties related to Population Balance Methods using Quadrature Method of Moments (QMOM)

Moment Weight Function#

Moment Weight Function = {GALERKIN | SUPG } <float>

Description / Usage#

This card specifies the weight function to be used on the weighted residual of the Moment equations. Definitions of the input parameters are as follows:

GALERKIN

Name of the model for the weight functions for a full Galerkin formulation. This is the default when this card is absent. * <float> - set but unused for Galerkin

SUPG

Name of the model for the weight functions for a streamwise upwinded Petrov-Galerkin formulation.

<float> - the value of the weight function, a number between 0. and 1.; a value of 1. corresponds to a full SUPG

Examples#

The following is a sample input card:

Moment Weight Function = GALERKIN 0.0

Moment Weight Function = SUPG 1.0

Technical Discussion#

References#

No References.

Moment Time Integration#

Moment Time Integration = {STANDARD | TAYLOR_GALERKIN }

Description / Usage#

This card specifies the time integration method to be used for time-stepping of the Moment equations. Definitions of the input parameters are as follows:

STANDARD: Name of the method for time-stepping in the standard formulation. This is the default when this card is absent.
TAYLOR_GALERKIN: Name of the method for time-stepping in using the Implicit Taylor-Galerkin formulation.

Examples#

The following is a sample input card:

Moment Time Integration = STANDARD

Moment Time Integration = TAYLOR_GALERKIN

Technical Discussion#

References#

No References.

Moment Shock Function#

Moment Shock Function = {NONE | YZBETA } <float>

Description / Usage#

This card specifies the shock capturing function to be used on the Moment equations. Definitions of the input parameters are as follows:

NONE

No shock capturing is applied. This is the default when this card is absent

YZBETA

A YZBETA shock capturing term is applied.

<float> - the value of the weight function, a number between 0. and 1.; a value of 1. corresponds to a full YZBETA shock capturing

Examples#

The following is a sample input card:

Moment Weight Function = NONE (default)

Moment Weight Function = YZBETA 1.0

Technical Discussion#

References#

No References.

Moment Growth Kernel#

Moment Growth Kernel = {CONSTANT | VISCOSITY_SCALED | VISCOSITY_PRESSURE_SCALED} <float1> [float2]

Description / Usage#

This card specifies the growth kernel to be used for the Moment equations. Currently only used for PMDI_10 QMOM models. Definitions of the input parameters are as follows:

CONSTANT

Constant growth rate * <float1> the prefactor

VISCOSITY_SCALED

Growth rate scaled by viscosity \(G = G_0 \eta_0 / \mu\), for use with PMDI_10 models * <float1> - \(G_0\)

VISCOSITY_PRESSURE_SCALED

Growth rate scaled by viscosity and pressure \(G = G_0 (\eta_0 / \mu) / (p - p_ref)^2\), for use with PMDI_10 models

<float1> \(G_0\)
<float2> \(p_ref\)

Examples#

The following is a sample input card:

Moment Growth Kernel = CONSTANT 1e-6

Moment Growth Kernel = VISCOSITY_SCALE 1e-3

Moment Growth Kernel = VISCOSITY_PRESSURE_SCALED 1e-8

Technical Discussion#

Note this is multiplicative with the growth rate in the FOAM_PMDI_10 moment source model

References#

Ortiz, Weston, et al. “Population balance modeling of polyurethane foam formation with pressure‐dependent growth kernel.” AIChE Journal (2021): e17529.

Moment Coalescence Kernel#

Moment Coalescence Kernel = {ADDITION | VISCOSITY_SCALED_ADDITION | VISCOSITY_ADDITION_BUBBLE_RATIO} <float1>

Description / Usage#

This card specifies the coalescence kernel to be used for the Moment equations. Currently only used for PMDI_10 QMOM models. Definitions of the input parameters are as follows:

ADDITION: \(\beta(v,v') = \beta_0 (v + v')\) * <float1> \(\beta_0\)
VISCOSITY_SCALED_ADDITION: \(\beta(v,v') = \beta_0 (\eta_0 / \mu) (v + v')\) * <float1> \(\beta_0\)
VISCOSITY_ADDITION_BUBBLE_RATIO: \(\beta(v,v') = \beta_0 (\eta_0 / \mu) (v + v') (v / (v' + \epsilon))\) * <float1> \(\beta_0\)

Examples#

The following is a sample input card:

Moment Coalescence Kernel = ADDITION 1e-12

Moment Coalescence Kernel = VISCOSITY_SCALED_ADDITION 1e-3

Moment Coalescence Kernel = VISCOSITY_ADDITION_BUBBLE_RATIO 1e-8

Technical Discussion#

Currently only for FOAM_PMDI_10 moment source model

References#

Ortiz, Weston, et al. “Population balance modeling of polyurethane foam formation with pressure‐dependent growth kernel.” AIChE Journal (2021): e17529.