Scalar

The Scalar statistics provide time-dependent, reduced-order data describing the evolution of scalar fields within the simulation domain. These outputs capture global measures of scalar transport, including overall concentration, reaction rates, addition or removal, spatial uniformity, etc. A tab-separated ASCII .txt file is created for each scalar field. The name of this file is Scalar_{DynamicName}.txt, where the dynamic name corresponds to the name of the scalar field in the Model Tree. The output files are updated at the Statistics Output Write Interval.

These outputs complement 3D spatial Plane (Slice) and Volume datasets by providing time-resolved summaries suited for convergence analysis, system-level characterization, and comparison across operating conditions.

The scalar statistics can be grouped into four primary categories: inventory metrics, mixing metrics, rate-based (flux) metrics, and particle addition/removal metrics. These categories provide a structured view of scalar behavior, separating how much scalar exists, how it is distributed, and the mechanisms by which it evolves over time.

• Inventory Metrics (e.g., Total Amount, Total Volume, Conc Mean, Vol Frac Mean) describe the total scalar content in the system. The total quantities define the absolute scalar inventory, while the mean values provide a normalized (per-volume) representation of that quantity. Together, these are used to track accumulation, depletion, and overall mass balance.

• Mixing Metrics (e.g., Conc Std Dev, Conc RSD, Vol Frac Std Dev, Vol Frac RSD) describe how the scalar is distributed spatially. These metrics quantify spatial variability and degree of homogeneity, with the relative standard deviation (RSD) serving as a primary indicator of mixing quality.

• Rate-based (Flux) Metrics (e.g., Injection Rate, Reaction Rate, Surface Reaction Rate, Particle Reaction Rate, Particle-Scalar Coupling Rate, Free Surface Flux) describe the mechanisms by which scalar is added, removed, transferred, or transformed. These terms define the pathways governing scalar evolution and enable decomposition of changes in scalar inventory over time.

• Particle Addition/Removal Metrics (e.g., Particle Set Injection Rate, Particle Set Removal Rate) describe scalar transport associated specifically with particle inflow and outflow. These terms track the scalar carried into and out of the system by particle sets and provide a direct link between particle dynamics and scalar mass balance.

Statistics Table

The index table below shows the statistics that can appear in the Scalar Output file. Within this table, each statistic corresponds to a column in the output table that evolves with the time column.

Statistics	Units	Details	When Appears
Time	s	simulation time
[dynamic] Transfer Rate	[dynamic]	total transfer rate for
[dynamic] Transfer Rate	[dynamic]	total transfer rate for
Conc Mean	[dynamic]	spatial mean of scalar field concentration
Conc RSD	%	spatial relative standard deviation of scalar field concentration
Conc Std Dev	[dynamic]	spatial standard deviation of scalar field concentration
Free Surface Flux	[dynamic]	total scalar field flux through free surface
Free Surface Flux	m^3/s	total miscible scalar field flux through free surface
Injection Rate	[dynamic]	total injection rate
Inlet/Outlet Link Amount in Recirculation	[dynamic]	total amount of scalar in recirculation through inlet/outlet link
Inlet/Outlet Link Amount in Recirculation	m^3	total amount of miscible scalar in recirculation through inlet/outlet link
Particle Reaction Rate	[dynamic]	total rate for scalar in particle-based reaction
Particle Reaction Rate	m^3/s	total rate for miscible scalar in particle-based reaction
Particle Set Injection Rate	[dynamic]	total rate of scalar addition from particle injection
Particle Set Injection Rate	m^3/s	total rate of miscible scalar addition from particle injection
Particle Set Removal Rate	[dynamic]	total rate of scalar decrease from particle removal
Particle Set Removal Rate	m^3/s	total rate of miscible scalar decrease from particle removal
Particle-Scalar Coupling Rate	[dynamic]	total rate for scalar transfer between fluid and particles, positive indicates scalar going into particles
Particle-Scalar Coupling Rate	m^3/s	total rate for miscible scalar transfer between fluid and particles, positive indicates miscible scalar going into particles
Reaction Rate	[dynamic]	total rate for scalar field in reaction
Reaction Rate	m^3/s	total rate for miscible scalar field in reaction
Surface Reaction Rate	[dynamic]	total rate for scalar in surface reaction
Surface Reaction Rate	m^3/s	total rate for miscible scalar in surface reaction
Total Amount	[dynamic]	total amount of scalar in system
Total Volume	m^3	total volume of miscible scalar in system
Vol Frac Mean	vf	spatial mean of miscible scalar field volume fraction
Vol Frac RSD	%	spatial relative standard deviation of miscible scalar field volume fraction
Vol Frac Std Dev	vf	spatial standard deviation of miscible scalar field volume fraction

Usage and Interpretation

Inventory Metrics

The concentration and volume fraction metrics are used to quantify both mixing performance and overall scalar inventory. The mean values (Conc Mean, Vol Frac Mean) represent the system-wide average scalar level and are calculated as

\[\mu = \frac{1}{N} \sum_{i=1}^{N} x_i,\]

where \(x_i\) is the local species concentration or volume fraction within voxel \(i\), and \(N\) is the total number of voxels.

The total quantity metrics (Total Amount, Total Volume) represent the absolute scalar content in the system, independent of spatial distribution.

\[\text{Total} = \sum_{i=1}^{N} x_i\]

These total values define how much scalar exists in the system, while the mean provides a normalized (per-volume) representation of that quantity.

The Inlet/Outlet Link Amount in Recirculation provides an additional inventory measure associated with linked boundary conditions. This term represents the amount of scalar currently contained within the recirculation pathway, accounting for any transport delay between outlet and inlet connections.

Mixing Metrics

The variability metrics (Conc Std Dev, Conc RSD, Vol Frac Std Dev) describe how uniformly the scalar is distributed. The standard deviation, \(σ\) , is calculated as

\[\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}.\]

The corresponding relative standard deviation (RSD) is

\[\text{RSD} = \frac{\sigma}{\mu} \times 100.\]

The RSD is a key indicator of homogeneity—values approaching zero indicate a well-mixed state. As a rule of thumb, RSD values below ~5% are typically considered well mixed in many engineering applications.

Rate-Based (Flux) Metrics

The rate-based quantities (e.g., Injection Rate, Reaction Rate, Surface Reaction Rate, Particle Reaction Rate, Particle-Scalar Coupling Rate, Free Surface Flux) define the pathways governing scalar evolution and enable decomposition of changes in scalar inventory over time.

The Injection Rate describes scalar addition through inlets and outlets, with corresponding inlet or outlet concentrations reported alongside these values.

The Free Surface Flux captures scalar transfer across the free surface and represents an additional pathway for scalar entering or leaving the system, particularly in cases involving interfacial transport (e.g., gas absorption or volatilization).

The Reaction Rate and Surface Reaction Rate distinguish between volumetric and boundary-driven transformations within the fluid and along static geometries, respectively.

The Particle-Scalar Coupling Rate describes interphase transfer between the scalar field and particle sets (e.g., dissolution or absorption). In gas–liquid systems, this term corresponds to the total interphase mass transfer rate and is directly related to quantities such as \(kLa\).

The Particle Reaction Rate captures intra-particle reactions for scalar fields coupled to particles, distinct from interphase transfer.

Particle Addition/Removal Metrics

The particle-related scalar terms (e.g., Particle Set Injection Rate, Particle Set Removal Rate) quantify the scalar entering and leaving the system via particle addition and removal.

These terms represent the scalar carried into and out of the domain with particle sets and provide a direct measure of particle-phase scalar transport.

The difference between the Particle Set Injection Rate and Particle Set Removal Rate reflects the net transfer of scalar between particles and the fluid, in the absence of particle-phase reactions.

For example, in a gas–liquid system, bubbles may enter carrying oxygen that is coupled to a scalar field. As the bubbles rise, oxygen transfers into the fluid. The reduction in scalar carried by the particles between injection and removal corresponds to the amount of oxygen transferred to the fluid.

Similarly, in CO₂ stripping, bubbles entering with little or no CO₂ absorb dissolved CO₂ from the fluid. When the bubbles exit, they carry this CO₂ out of the system. Under steady-state conditions, the scalar leaving with the particles balances the interphase mass transfer from the fluid to the bubbles.

These metrics are critical for ensuring consistency between particle and fluid scalar budgets and for quantifying interphase transport in multiphase systems.