# Scalar Fields¶

## Basic Concepts¶

Scalar fields represent a continuous scalar value defined at all points across the fluid domain. They are typically used to describe species concentrations across the domain. Scalar fields can be coupled to the fluid to inform the local fluid density and viscosity. They can also be coupled to bubbles or particles to describe mass transfer processes and reaction/dissolution processes. Multiple scalar fields can participate in reactions, with local reaction rates informed by the local fluid properties and species concentrations.

Scalars can enter the fluid domain via:
1. initial background concentration,

2. injection within a child geometry,

3. non-zero concentration at an inlet,

4. transport across a free surface interface,

5. transfer from gas bubbles,

6. transfer from solid particles,

7. a reaction rate that drives scalar production.

Scalars can be removed from a system by specifying
1. an outlet,

2. a reaction rate that drives scalar consumption,

3. transport across a free surface interface,

4. transfer to gas bubbles,

5. transfer to solid particles.

Each scalar field is advanced using a high-resolution van Leer scheme. This approach links the time-varying fluid flow to the advection term in the advection-diffusion equation. Chemical reactions, we present, are advanced at each time-step using a fourth order Runge-Kutta solver.

Scalar field output files are created for each species, and record the time-evolution of the mean species concentration, the standard deviation of the species concentration across the tank volume and, when necessary, the species reaction and transfer rates. The concentrations of each species at each inlet and outlet are also recorded in the boundary condition output files.

Scalar fields are characterized by a parent-child relationship. Within this relationship, the parent defines the properties of the scalar field. The children define the shape/geometry/topology of any initial injection conditions.

## Parent Properties¶

Scalar Name

Variable name used in reaction expressions. Name has the following restrictions:

• Some names are reserved for other quantities: s, t, and T are reserved for strain, time, and temperature.

• First character must be a letter

• Must not contain spaces or special characters

• Must only contain letters, numbers, and under score

Unit of Measure

Indicates how the initial background concentration is defined.

• Molarity: Define the initial condition as a molarity

• Number Of Moles: Define the initial condition as a total number of moles

Initial Background Concentration

Initial background concentration

This is the uniform background concentration of the species at the start of the simulation. Units defined by the Unit of Measure (defined above).

Diffusion Coefficient

Diffusion coefficient [m^2/s]

Diffusion coefficient of the scalar through the base fluid.

Scalar Density Option

Indicates if the scalar field density is to be modeled explicitly.

• Off: Scalar field has no affect on local fluid density

• On: Scalar field informs the local fluid density via a user-provided density and molar mass

Transport Across Free Surface (Free Surface Simulations Only)

Indicates if the scalar field is transported across the free surface interface. If enabled, user can define a mass transfer rate as a function of the fluid properties and the local fluid species concentration.

## Child Geometry Injection Conditions¶

Injection Geometry
• Inject In Volume: the scalar boundary condition is imposed on all points contained inside the geometry.

• Inject At Surface: the scalar boundary condition is imposed on the surface of the child geometry.

These conditions are identical for children geometries that are thin relative to the lattice spacing.

Unit of Measure
• Molarity: Input from Initial Concentration Expr is used with units molarity

• Number Of Moles: Input from Initial Moles Expr is used with units moles

• Molar Rate: Input from Initial Moles Rate Expr is used with units moles/s

Start Time

Injection start time [s].

Time at which to begin adding scalar into child geometry

Stop Time

Scalar box stop time [s].

Time at which to stop adding scalar into the defined box. Between the Start and Stop times, the scalar field inside the box will be maintained at the user-defined value. To mimic the effects of an instantaneous dump, set the Start Time equal to the Stop Time.

## Boundary Conditions¶

Boundary Condition Type

For each boundary condition present in the simulation, the following inputs are available

Specified Value

The specified value defines the scalar field value at boundary condition, as measured in mol/L. The specified value can be a function of time and position across the inlet face.

A zero gradient is a Neumann boundary condition, which implies no variation in the scalar field concentration at the boundary condition in the direction of flow. Most outlets are zero gradient boundary condition.

Zero Flux

A zero flux implies no species transport though the boundary.

Recirculation

For recirculation inlets, the molar flow rate is a function of the scalar field exiting through the suction side of the coupled boundary condition.

• If this value is set to 0.5c, the molar flow rate of the scalar returning to the vessel through the recirculation loop will be 50% smaller than the molar flow rate leaving the vessel through the suction side of the recirculation line. Under these settings, the concentration in the tank will decrease. This behavior represents a system were some of the scalar field is removed from the recirculation line prior to returning to the vessel.

• If this value is set to c and Recirculation Link Flow Ratio is 1, the molar flow rate of the scalar returning to the vessel through the recirculation loop will be equal to the molar flow rate leaving the vessel. This behavior represents a system were all of the scalar field leaving through suction side of the recirculation loop returns to the inlet.

• If this value is set to 0, the molar flow rate of the scalar returning to the vessel through the recirculation loop will zero. This behavior represents a system were all of the scalar field is removed from the recirculation line prior to returning to the vessel.

• If this value is set to 2c and Recirculation Link Flow Ratio is 1, the molar flow rate of the scalar returning to the vessel through the recirculation loop will 2x larger than the molar flow rate leaving the vessel through the suction side of the recirculation line. This behavior represents a system were scalar field is added to the recirculation line prior to returning to the vessel.

• If this value were set to 1 , the molar flow rate of the miscible fluid returning to the vessel through the recirculation loop will be equal in magnitude to the volume flow rate of the return inlet.

• Et cetera

Note that, consistent with most outlet boundary conditions, the suction side of the recirculation inlet is typically set to zero gradient. If multiple recirculation inlets are coupled to a single suction-side boundary condition, the ratio of species entering through each coupled inlet will be proportional to the surface area of each inlet to the total surface area across all coupled inlets.

Stencil

Stencil used to define advection [Lattice, Finite Volume] The finite volume stencil considers fluxes in the +/- x,y, and z-directions. The lattice stencil considers fluxes in the directions of the fluid lattice

Limiter

Flux limiter used to contain convection [VanLeer, Minmod, MUSCL, Superbee, DonorCell, LaxWendroff, DonorCell]

The Donor Cell is an upwind difference technique valid for systems with a Peclet number greater than 2 (a condition typically satisfied in turbulent systems). The other schemes are total variation diminishing, and necessary in laminar systems.

## Output Data¶

Statistics Output Frequency

For each scalar field, the instantaneous total number of moles, mean concentration, concentration standard deviation, and concentration percent relative standard deviation is printed to a unique output text file. If scalar reactions, a free-surface flux, bubble-scalar coupling and/or particle-scalar couplings are present, the corresponding mass transfer rates to/from the scalar field are reported (for each coupling). Note that, when reporting the flux, positive values indicating transfer into the aqueous phase. The instantaneous concentration is also reported at each boundary conditions, as well as the species variation across the boundary condition area.

Slices Output Frequency

The spatial variation in the species concentration is reported on each slice.

Volume Output Frequency

The spatial variation in the species concentration is reported on the fluid volume output.