Overview

In order to conduct an Eulerian-Eulerian (EE) simulation that accounts for a particle size distribution (PSD) for the dispersed phase, a population balance model is required. ANSYS Fluent provides two fundamentally different approaches for accounting for the PSD: (1) discrete methods, which rely on a discrete representation of the PSD; and (2) moment methods, which rely on representation of the statistical moments of the PSD. This article provides a brief summary of the methods and how the Particle Bin Calculator can be used to calculate and apply the necessary boundary conditions for discrete methods.

Population Balance Equation

Before reviewing the population balance model options, it's useful to introduce the population balance equation: \[ \frac{\partial}{\partial t}[n(L,t)] + \nabla \cdot [\bar u n(L,t)] + G = A^+ - A^- + B^+ - B^- \] where the variables are defined as:

\( n(L,t) \)	Number density of particles with internal coordinate, \( L \). For a particle, the internal coordinate is generally particle diameter so this represents the number density of particles with a given diameter. Accordingly, because the density is on a volume basis, the units of \( n \) are \( \text{length}^4 \).
\( \bar u \)	Velocity vector
\( G \)	Particle growth
\( A^+ \)	Particle birth due to aggregation
\( A^- \)	Particle death due to aggregation
\( B^+ \)	Particle birth due to breakage
\( B^- \)	Particle death due to breakage

The growth, aggregation, and breakage source and sink terms are all model calculations during a CFD simulation. Our focus here is to summarize the process for calculating the boundary conditions for the number density, thus we don't need to be concerned with these additional terms.

Therefore, let's assume for simplicity that there is (1) no particle growth, (2) no particle breakage, and (3) no particle aggregation. The first of these assumptions is consistent with the physics of a dusty gas flow, as particle growth is associated with applying the population balance equation to different media such as bubbles. The validity of no particle breakage and aggregation will depend upon the material, but is irrelevant to the discussion about specifying boundary conditions. These assumptions result in the following equation:

\[ \frac{\partial}{\partial t}[n(L,t)] + \nabla \cdot [\bar u n(L,t)] = 0 \]

This (simplified) population balance transport equation is solved for both discrete methods and moment methods, but the exact version used in each case is distinct.

In discrete methods, the user specifies a set of bins defined by their diameters. During the simulation, the volume fraction of particles within bin \( i \) (represented by \(\alpha_i \)) is solved for in each grid cell according to the following version of the population balance equation:

\[ \frac{\partial}{\partial t}(\rho_s \alpha_i) + \nabla \cdot (\rho_s u_i \alpha_i) = 0 \]

It's evident from this equation that reconstruction of the PSD at any location in the CFD domain will be simple because the volume fraction data is part of the solution.

In moment methods, the population balance equation is applied to solve for the moments of the PSD within each cell of the CFD domain according to:

\[ \frac{\partial}{\partial t}(\rho_s m_k) + \nabla \cdot (\rho_s \bar{u} m_k) = 0 \]

"Moments" are statistical quantities used to characterize a distribution function, and the k^th moment can be calculated as \( m_k = \int_0^\infty n(L; \bar{x}, t) L^k dL \). The first few moments can be related to statistical quantities that many people recognize. For example, \( m_0 \) is the total particle count per unit volume, \(\mu = m_1 / m_0 \) is the mean of the distribution, and \(\sigma = \mu \sqrt{ \frac{m_0 m_2}{m_1^2} -1 } \) is the standard deviation of the distribution.

The value of a moment method approach is that moments are more "efficient" than discrete bins at capturing information about a PSD. Consider, for example, a PSD that follows a Gaussian distribution. The PSD would need only three moments to be fully characterized, resulting in the need for solving only three population balance equations. By contrast, with a discrete representation, we can easily imagine needing 10 or more bins to reasonably represent the PSD, especially if there is a significant size difference between the smallest and largest particles.

The table below summarizes the advantages and disadvantages of discrete methods in ANSYS Fluent. As discussed, discrete methods can be advantageous because the simulation directly resolves the PSD throughout the simulation domain, by tracking the fraction of material in each bin. However, to achieve accurate results, it may necessary to use a large number of bins, which will significantly increase the computational burden because it requires solution of additional equations for each bin. By contrast, moment-based methods can be more efficient, but they require subsequent determination of the PSD from the moments. Methods are available for this, but they can be difficult to use, especially for complex distributions relying on a large number of moments.

Approach for Modeling PSD	ANSYS Fluent Methods	Advantage	Disadvantage
Discrete Methods	(Homogeneous) Discrete All bins share a single momentum equation (i.e., unable to model particle segregation)	Direct Representation of the PSD Simulation directly represents the PSD with a pre-selected number of bins, making it easy to reconstruct the PSD.	Computational Expense Grows with the Number of Bins The number of transport equations to be solved scales with the number of bins. Especially challenging for the Inhomogeneous Discrete method where bins may be assigned to unique phases for which distinct momentum equations must be solved. Bins Must be Known a Priori Particles may move between phases as they break or agglomerate. The full range of these bins must be specified before the simulation.
	Inhomogeneous Discrete Bins may be assigned to unique phases, each with a distinct velocity, permitting modeling of particle segregation

Use of the Particle Bin Calculator Tool

ANSYS Fluent users may activate use of the population balance equation by selecting from the available methods shown in the following screenshot. The summary in this section shows how output from the Particle Bin Calculator can be used to generate the required model inputs for each of the methods.

(Homogeneous) Discrete Method: Upon selecting the "Discrete" method, users may input data to describe the PSD bins. Example instructions are listed below, with available GIFs that can be revealed by clicking the "Show GIF" buttons:

• Conduct Bin Calculation Show GIF
  
  Use the Particle Bin Calculator to calculate bins and corresponding bin fractions. The GIF shows selection of 10 bins after fitting PSD data with a continuous distribution.
  Download the homogeneous discrete data (both bin diameters and bin fractions) to be used as input for ANSYS Fluent.
  
• Fluent Setup Show GIF
  
  Set the number of bins in the Population Balance dialog to match the number generated in the Particle Bin Calculator.
  Load the diameter_homogeneous.txt text file which was previously downloaded from the Particle Bin Calculator.
  Print the bins to the ANSYS Fluent console to verify they were properly loaded. You will observe that Fluent numbers the bins in order of decreasing diameter with the largest diameter corresponding to "Bin-0".
  Use the data in the fraction_homogeneous.txt text file to set the boundary conditions for the fraction of the phase in each bin (not shown). The data in the text file are labeled with the corresponding bin number. These data correspond to what ANSYS Fluent documentation refers to as variable \( f_i \) .
  

Inhomogeneous Discrete Method: When using the inhomogeneous discrete method, a unique phase needs to be defined for each bin for which the user wishes to simulate segregation. Instructions include:

• Conduct Bin Calculation
  
  The process is the same as described above for the homogeneous discrete model, but instead of downloading the homogeneous data, the user should download the inhomogeneous data by clicking the Download Inhomogeneous link.
• Fluent Setup Show GIF
  
  Set the number of Eulerian phases to match the number of bins for which segregation should be modeled. In this example, we have already set the number of Eulerian phases to 11 which accounts for each of the 10 discrete bins plus one additional phase for the continuous phase. This is done on the Models tab of the Multiphase model dialog (not shown).
  Set the number of Active Secondary Phases to match the number of bins. This will generate a Phase dropdown selector below for proper assignment. These settings were established before recording of the GIF. Note that it's possible to assign more than one phase to an "active secondary phase", but this will result in a single momentum equation being modeled for the phase, and no particle segregation amongst the bins within the phase will occur.
  Load the diameter_inhomogeneous.txt text file which was previously downloaded from the Particle Bin Calculator.
  Print the bins to the ANSYS Fluent console to verify they were properly loaded.
  Use the data in the fraction_inhomogeneous.txt text file to set the boundary conditions for the fraction of the phase in each bin (not shown). The data in the text file are labeled with the corresponding bin number.