Scalar Coupling and Thermal Coupling¶

Scalar Coupling: Species can be exchanged between particles and a scalar field.
Thermal Coupling: Energy can be exchanged between particles and a fluid field.

Scalar Coupling¶

Overview¶

With the scalar coupling option, users can model species exchange between particles and fluid, as well as associated changes in particle size and composition. Scalar coupling offers a variety of possible applications. When applied to solid particles, scalar coupling can include particle growth, as during crystallization, or particle dissolution, as part of a buffer preparation. When applied to gas bubbles, scalar coupling can include oxygen transfer in bioreactors or \(CO_2\) stripping during a pH control process.

Transport is handled on a particle-by-particle basis using the local particle and fluid properties. The process is two-way coupled such that particle transport acts as a source/sink term to the scalar field while also causing particle volume and composition to change over time.

Model CO2-DO Scalar Coupling file

Model DO Scalar Coupling file

Framework¶

The coupling method determines if particle-fluid scalar coupling is active in the simulation and whether it is modeled using built-in or user-defined functions. The following section details the governing concepts and parameters for scalar coupling.

Convection¶

When the interfacial mass transfer is modeled using a Convection Framework, the species mass transfer rate between a particle and the surrounding fluid is informed by the mass transfer coefficient of the particle and the local species concentration difference,

(1)¶\[\dot{q}_p = k_{l,p} A_p (C_p^* - C_p)\]

where \(\dot{q}_p\) is the mass transfer rate between particle \(p\) and the surrounding fluid; \(k_{l,p}\) is the convective transfer coefficient of the fluid surrounding the particle; \(A_p\) is the area of particle; \(C_p^*\) is the saturation concentration of the species participating in the transport process; and \(C_p\) is the concentration of the species in the fluid surrounding the particle.

To solve this convective transport equation (1) for each particle, the code combines the kL UDF with the instantaneous particle surface area and a user-defined saturation concentration. For gas bubbles, the saturation concentration is calculated automatically from the gas-phase concentration using a user-specified dimensionless solubility (e.g., dimensionless Henry’s law constant). The effects of any changes in gas-phase concentration due to pressure and/or gas composition on saturation concentration are handled internally.

For inertial particles, the units on saturation concentration are automatically adjusted to maintain dimensional homogeneity with the coupled scalar field. If the coupled scalar field has units of moles, for example, the saturation concentration will be characterized by a molar volume (mol/L). Alternatively, if the coupled scalar field has units of grams, the saturation concentration will be characterized by its mass density (g/L). These saturation concentrations are assumed to remain constant over time.

In addition to this saturation concentration, the code also prompts for the particle phase density and the initial volume fraction.

The particle phase density is the density of the scalar field when contained in the particle. This density is used to link the species transport rate to changes in particle volume. The units characterizing the particle phase density are linked to the units of the coupled scalar field.

The initial volume fraction describes the volume fraction of the scalar field within each particle when it enters the system. The initial volume fraction, combined with the particle phase density, determines the initial quantity of each species within each particle. The initial volume fraction is a dimensionless quantity that is unaffected by the units of the coupled scalar field.

Dissolution¶

When the interfacial mass transfer is modeled using a Dissolution Framework, the species mass transfer rate between a particle and the surrounding fluid is modeled as

(2)¶\[ \dot{q}_p = \phi_p A_p\]

where \(\dot{q}_p\) is the mass transfer rate from particle \(p\) to the surrounding fluid; \(\phi_p\) is the solid particle dissolution rate; and \(A_p\) is the area of the particle. Note that this equation appeals to a sink boundary condition, which assumes that the dissolved concentration remains very low compared to the saturation concentration.

To solve this dissolution transport equation (2) for each particle, the code multiplies the dissolution rate UDF by the instantaneous particle surface area. Since the dissolution rate UDF is provided in kg/m2-s, the nominal output from this expression is a mass transfer rate, measured in kg/s. If the coupled scalar field has units of moles, the code will prompt the user for the species molecular weight. This molecular weight is used to convert the mass transfer rate into a molar transfer rate (mol/s) to maintain dimensional homogeneity.

As with the convection model, the code prompts for the particle phase density and the initial volume fraction. The particle phase density is the density of the scalar field when contained in the particle. This density is used to link the species transport rate to changes in particle volume. The units characterizing the particle phase density are linked to the units of the coupled scalar field.

The initial volume fraction is a dimensionless quantity that is unaffected by the units of the coupled scalar field. It describes the volume fraction of the scalar field within each particle when it enters the system. When combined with the particle phase density, it determines the initial quantity of each species within each particle.

More sophisticated models which handle processes such as crystal growth and surface wetting phenomena are discussed at length in the literature.

The settings and selections presented in the Property Grid will depend on the particle type, the interfacial mass transfer framework, and the units of the coupled scalar field.

Coupling Method¶

Disabled

This option does not consider particle-fluid scalar coupling. Although a convective transfer coefficient or dissolution rate may be defined, no species will be transferred between the particles and fluid.

Automatic

With this setting, scalar coupling automatically follows the interfacial mass transfer framework, whether convection or dissolution.

Scalar: {Scalar}

This is the name of the coupled scalar field. This selection has a dynamic name that is linked to the name of the component in the model tree.

Track Scalar

Options for Track Scalar.

Off

Track Scalar is off.

On

Track Scalar is on.

Initial Volume Fraction UDF

dimensionless | This dimensionless value defines the fraction of the particle initially occupied by each scalar. If multiple coupled scalar fields are present, the sum of all volume fractions must not exceed 1. If the total volume fraction of all coupled scalar fields is less than 1, the complement represents media in the particle not participating in any coupled transport process. This is a System UDF.

Download Sample File: Automatic Convection

Download Sample File: Automatic Dissolution

If your scalar Base Units are Moles, this selection will appear:

Molar Volume: m ³ /mol | The volume of space occupied by each mole of solid in the particle for each species. This value links the change in particle volume due to mass exchange with the surrounding fluid.

If your scalar Base Units is Mass, this selection will appear:

Particle Phase Density: kg/m ³ | The density of the coupled species when it is incorporated as part of the particle. For bubbles, this value represents the gas-phase density of any coupled species. For inertial particles, this value represents the solid-phase density of the coupled species.

Saturated Concentration: mol/L or g/L | The (maximum) saturation concentration of the coupled species in the surrounding fluid. For inertial particles, the units on saturation concentration are automatically adjusted to maintain dimensional homogeneity with the coupled scalar field. If the coupled scalar field has units of moles, the saturation concentration will be characterized by a molar volume (mol/L). Alternatively, if the coupled scalar field has units of grams, the saturation concentration will be characterized by its mass density (g/L). These saturation concentrations are assumed to remain constant over time.
Dimensionless Solubility: dimensionless | The dimensionless ratio between the aqueous-phase concentration of a species and its gas-phase concentration. The dimensionless solubility of oxygen and carbon dioxide at 298.15 K are 0.032 and 0.83, respectively.
Molecular Weight: g/mol | The molecular weight of a molecule is the sum of the atomic weights of all the atoms in the molecule.

Custom

Scalar coupling is a user-defined function. Unlike the automatic option, which applies the same mass transfer coefficient and scalar coupling representation to each species participating in mass transfer, the scalar coupling expression here can be tailored for each species.

Scalar Coupling UDF

base units /s | This UDF defines the mass transfer rate between particles and the surrounding fluid. One output must be defined within the UDF: a floating-point variable named rate_{scalar}, where {scalar} is the dynamic name of the scalar field. This output variable defines the rate at which species is transferred across the particle/fluid interface.

The transfer rate is typically a function of the convective mass transfer coefficient, the bubble surface area, and the local concentration difference. Positive values imply mass transfer into the particle. Negative values imply mass transfer away from the particle. A unique transport model can be specified for each species. The user must ensure that the transport parameters are dimensionally homogeneous. This is a local UDF, calculated on a particle-by-particle basis using the local particle/fluid properties.

Download Sample File: Scalar Coupling

Scalar: {Scalar}

This selection has a dynamic name that is linked to the name of the component in the model tree.

Track Scalar

Options for Track Scalar.

Off

Track Scalar is off.

On

Track Scalar is on.

Initial Volume Fraction UDF

Download Sample File: Custom Convection

Download Sample File: Custom Dissolution

If your scalar Base Units are Moles, this selection will appear:

Molar Volume: m ³ /mol | The volume of space occupied by each mole of solid in the particle for each species. This value links the change in particle volume due to mass exchange with the surrounding fluid.

If your scalar Base Unit is Mass, this selection will appear:

Particle Phase Density: kg/m ³ | The density of the coupled species when it is incorporated as part of the particle. For bubbles, this value represents the gas-phase density of any coupled species. For inertial particles, this value represents the solid-phase density of the coupled species.

Thermal Coupling¶

Overview¶

With the thermal coupling option, users can model thermal exchange between particles and fluid, as well as associated changes in local fluid and particle temperature. Transport is handled on a particle-by-particle basis using the local particle and fluid properties. The process is two-way coupled such that thermal transport acts as a source/sink term to the fluid thermal field while also causing the particle temperature to change over time. Total energy is conserved in this transport process.

Model Particle-Fluid Thermal Coupling file

Fluid-Particle Heat Transfer Coupling Method¶

Convection¶

When the interfacial mass transfer is modeled using a Convection Framework, the species mass transfer rate between a particle and the surrounding fluid is informed by the heat transfer coefficient, the particle surface area, and the local temperature difference between the particle and the fluid:

\[\dot{q}_{T_p} = h_p A_p (T_p - T_f)\]

where \(\dot{q}_{T}\) is the heat transfer rate between particle \(p\) and the surrounding fluid, \(h_p\) is the convective transfer coefficient of the fluid surrounding the particle, \(A_p\) is the area of particle, \(T_p\) is the temperature of the particle, and \(T_f\) is the temperature of the fluid surrounding the particle.

Within the automatic framework, the transfer heat transfer coefficient is calculated using the model of Deckwer [Chemical Engineering Journal, 43 (1990) 819–841], which combines Higbie’s penetration theory with Kolmogoroff’s theory of isotropic turbulence:

\[h_p = 0.1 C_v (\varepsilon \nu)^{1/4} \left( \frac{\alpha}{\nu} \right)^{1/2}\]

where \(C_v\), \(\nu\), and \(\alpha\) are the specific heat (at constant volume), kinematic viscosity, and thermal diffusivity of the fluid. Alternative convection models can be specified using the Custom approach.

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