Introduction to Population Balance Modeling (PBM) in CFD

Kiarash Kord
Reading time: 12 min

What is Population Balance Modeling?

In the world of fluid dynamics, we often need to simulate systems with particles, droplets, or bubbles moving inside a fluid. Simple models often assume all these particles are the same size. But in the real world, this is rarely true. Think of the bubbles in a soda bottle or the water droplets in a spray – they all have different sizes. This is called a polydispersed system.

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Population Balance Modeling (PBM) is the key tool used to track these different sizes. The need for the population balance model arises from the fact that particles in nature are not uniform in size but rather poly dispersed. PBM helps us understand and predict how the Particle Size Distribution (PSD) changes over time. The PSD is simply a way to describe how many particles of each size exist in our system.

Figure 1: A typical polydispersed system. PBM tracks the different sizes of the dispersed phase particles (like bubbles or droplets) within the continuous phase (like a liquid or gas).

Why is PBM So Important?

Tracking the particle size is crucial for many reasons. The size of a particle affects how it behaves and how it interacts with the fluid around it. Understanding the PSD is important for:

Product Quality: In industries like pharmaceuticals, the size of a crystal can determine how well a medicine works.
Heat and Mass Transfer: The total surface area of all particles changes with their size. This directly impacts heating, cooling, and chemical reaction rates.
Downstream Processing: The efficiency of filters and separators depends on the size of the particles they need to handle.

Without PBM, our simulations could give inaccurate results because they miss the important effects of having many different particle sizes.

Figure 2: PBM as a unifying scientific principle applicable to diverse systems, including solid particles, liquid droplets, gas bubbles, polymers, and biological cells..

The “Birth” and “Death” of Particles

The core idea of PBM is to keep a record, or a “balance,” of the number of particles. This balance changes because of “birth” and “death” processes. These are not living particles; these terms simply describe how the number of particles of a certain size changes.

Birth: This is any process that creates new particles of a specific size.
Death: This is any process that removes particles of a specific size.

These processes happen because of four main physical phenomena:

Nucleation: New particles are formed (born) from the fluid itself, like tiny crystals appearing in a solution.
Growth: Existing particles get bigger. This is a “death” for small particles and a “birth” for larger particles.
Aggregation: Two or more smaller particles collide and stick together, creating one larger particle. This is a “death” for the small particles and a “birth” for the new, larger one.
Breakage: A large particle breaks apart into several smaller particles. This is a “death” for the large particle and a “birth” for the new, smaller ones

Figure 3: The four key phenomena that drive population balance: Nucleation (birth of new particles), Growth (increase in size), Aggregation (merging of particles), and Breakage (fragmentation of particles).

Where is PBM Used?

Because so many natural and industrial processes involve particles of different sizes, PBM is used everywhere. It is a unifying principle for many applications. Some key applications include:

Bubble Columns & Reactors: Tracking bubble sizes to optimize chemical reactions.
Crystallization: Controlling the formation of solid particles in liquids to achieve a desired product.
Sprays: Modeling liquid fuel injection or paint sprays to understand droplet size and distribution.
Granulation: Simulating the process of forming larger granules from fine powders, common in the food and drug industries.

Figure 4: Key industrial applications of PBM, highlighting its role in optimizing chemical bubble columns, controlling pharmaceutical crystallization, analyzing fuel sprays, and managing powder granulation.

In the next sections, we will explore the core mathematics behind PBM, including the famous Population Balance Equation, that allows us to model these complex systems.

Mathematical Foundation and Core Equations

To understand how Population Balance Modeling (PBM) works, we first need to look at its mathematical foundation. This section explains the core ideas and the main equation that makes PBM so powerful. We will break it down step-by-step.

Internal and External Coordinates: More Than Just Location

When we think about a particle, we usually think about its location in space (its x, y, and z coordinates). These are called external coordinates.

But particles have other important properties too. For example:

What is its size?
What is its temperature?
How old is the particle?

These properties are called internal coordinates. They describe the state of the particle itself, not just where it is. As stated, internal coordinates can be properties like size, surface area, or composition.

Figure 5: The core concept of a particle state, which is defined by both its external coordinates (position x, y, z) and its internal coordinates (physical properties like size v, temperature, etc.).

The Particle State Vector

To keep track of all these properties, we use something called a particle state vector. This is simply a list of all the variables we need to describe a particle completely.

These are some great examples:

For a group of polymers, the “degree of polymerization” can be the particle state.
For a population of cells, the “cell age” can be the particle state.
For cylindrical crystals, both “length” and “diameter” can be the particle states.

The particle state vector gives us a complete description of a particle’s properties at any moment.

The Number Density Function: n(v)

This is the most important concept in the Population Balance Equation. The number density function, written as n(v), is a mathematical tool that tells us how many particles of a certain size exist. Number density function is a mathematical representation that describes the distribution of particles within a system based on their size… It provides information about the number of particles per unit volume or per unit size interval. In simple terms, n(v) answers the question: “In a small volume of fluid, how many particles have a size of ‘v’?”

Figure 6: The Number Density Function, n(v), graphically plotted to show the distribution of particles across a range of sizes, forming the Particle Size Distribution (PSD).

The Population Balance Equation (PBE)

The main goal of PBM is to solve the Population Balance Equation (PBE). This equation looks complicated, but its core idea is very simple. It is based on a conservation principle, just like conserving mass or energy. The idea is:

Accumulation = Input – Output + Net Generation

This means the change in the number of particles of a certain size is equal to the number of particles that move in, minus the particles that move out, plus the particles that are created or destroyed inside the volume.

This idea gives us the general PBE. The following is a common form of the equation

 $\frac{\partial n(in)}{\partial t} + \nabla \cdot (u n(in)) + \nabla_{in} \cdot (G n(in)) = B(in) - D(in)$

This single equation balances all the ways a particle population can change. By solving it, we can predict the complete particle size distribution (PSD) everywhere in our CFD simulation.

Term	Name	What it Means
$\frac{\partial n(v)}{\partial t}$	Transient Term	How the number of particles of size v changes over time.
$\nabla \cdot (u n(v))$	Convection Term	How particles of size v move around with the fluid flow.
$\nabla_v \cdot (G n(v))$	Growth Term	How particles change size (grow or shrink) and move into or out of the size v category.
$B(v)$	Birth Term	The rate at which new particles of size v are created (born).
$D(v)$	Death Term	The rate at which particles of size v are removed (die).

The “Birth” and “Death” terms, B(v) and D(v), are where the real physics happens. Particles are “born” into a certain size category, or they “die” from it. This happens because of physical events:

Birth B(v): New particles of size v can be formed from the breakage of larger particles or the aggregation (sticking together) of smaller particles.
Death D(v): Particles of size v are removed when they break into smaller pieces or aggregate with other particles to form a larger one.

We will explore these physical phenomena—breakage, aggregation, nucleation, and growth—in detail in the next section.

Figure 7: Diagram illustrating the complex lifecycle of a particle population, from the formation of stable nuclei to their eventual transformation through growth, aggregation, and breakage. (Copyright 2017, Gebauer, D. et al.)

The Four Key Phenomena That Change Particle Size

In the last section, we learned that Population Balance Modeling tracks the “birth” and “death” of particles of different sizes. But what causes these changes? Four main physical processes are responsible for changing the particle size distribution. Let’s look at each one.

Nucleation: The Birth of New Particles

Nucleation is the process where completely new particles are formed from the continuous phase. Imagine steam (gas) cooling down and forming tiny water droplets (liquid). Those first tiny droplets are created by nucleation.

What it is: The formation of a new phase (solid, liquid, or gas) from a primary phase.
Simple Example: Sugar crystals appearing in a concentrated sugar syrup.
Effect: Nucleation is a “birth” process. It always increases the total number of particles in the system.

Common examples include crystallization and boiling. In simulations, the nucleation rate is often set as a specific value or defined by the user.

Figure 8: A schematic of the nucleation process, showing the ‘birth’ of new, stable particles (nuclei) from a precursor phase, which is the starting point for population growth.

Growth: Existing Particles Get Bigger

Growth is when existing particles increase in size. This can happen through processes like condensation or crystallization, where more material from the surrounding fluid attaches to the particle’s surface.

What it is: The increase in size of existing particles.
Simple Example: A small snowball rolling down a hill gets bigger as more snow sticks to it.

** Growth does not change the total number of particles. It just moves particles from smaller size categories to larger ones.

Mathematically, the growth rate (G) is simply the rate of change of a particle’s volume or size over time.

Aggregation: Particles Sticking Together

Aggregation happens when two or more smaller particles collide and merge to form a single, larger particle. This is also sometimes called coalescence, especially for liquid droplets.

What it is: The sticking together of particles to form larger ones.
Simple Example: Two small raindrops on a window running into each other and forming one bigger drop.
Effect: Aggregation is a “death” process for small particles and a “birth” process for large particles. It always decreases the total number of particles.

The rate at which aggregation happens depends on how often particles collide and how efficiently they stick together. This is modeled using an aggregation kernel.

Figure 9: A visual comparison of aggregation, where two smaller particles collide and merge, and breakage, where one larger particle shatters into multiple smaller ‘daughter’ particles.

Breakage: Particles Breaking Apart

Breakage is the opposite of aggregation. It occurs when a larger particle fragments into two or more smaller particles. This can be caused by forces within the fluid or collisions with other objects.

What it is: The fragmentation of large particles into smaller ones.
Simple Example: A large cookie crumbling into smaller pieces.
Effect: Breakage is a “death” process for large particles and a “birth” process for small particles. It always increases the total number of particles.

Two important factors control breakage are:

Breakage Frequency:How often a particle of a certain size breaks.

Daughter Distribution Function: Describes the sizes of the smaller pieces that are formed after a break.

This table provides a simple summary of the four key phenomena.

Phenomenon	Description	Effect on Total Particle Number
Nucleation	New particles are formed.	Increases
Growth	Existing particles get bigger.	Stays the same
Aggregation	Particles stick together.	Decreases
Breakage	Particles break apart.	Increases

How ANSYS Fluent Solves the Population Balance Equation: An Overview

The Population Balance Equation (PBE) is a powerful mathematical tool, but it is too complex to solve by hand. To find a solution, we need to use numerical methods that a computer can understand. These methods help us calculate how the particle sizes change throughout our simulation.

There are two main families of methods used to solve the PBE:

The Discrete Method
The Method of Moments (MOM)

Figure 10: An overview of the Population Balance Models available in ANSYS Fluent, divided into the two main solution families: the Discrete Method and the Method of Moments (MOM), including its advanced forms QMOM and DQMOM.

The Discrete Method

The Discrete Method works by sorting the particles into different size groups. Think of it like having several boxes, where each box is for a specific size range: a box for small particles, a box for medium particles, and a box for large particles. These boxes are called “bins” or “classes”.

The simulation then counts how many particles are in each bin. This approach gives us a direct picture of the Particle Size Distribution. There are different types of the Discrete method, including the Inhomogeneous Discrete method.

Method of Moments (MOM)

The Method of Moments is a clever and often faster approach. Instead of tracking every single size bin, this method tracks the overall statistical properties of the particle population. These properties are called “moments”. Moments describe the shape of the size distribution. For example, they can tell us:

The total number of particles (the zeroth moment).
The average particle size (the first moment).
How wide the distribution is (the second moment).

There are several advanced types of the Method of Moments available in ANSYS Fluent:

Standard Method of Moments (SMOM)
Quadrature Method of Moments (QMOM)
Direct Quadrature Method of Moments (DQMOM)

Each of these solution methods—from the Discrete vs moment method PBM—has specific advantages and is chosen based on the problem you need to solve.

We will provide a full guide on how to setup PBM in ANSYS Fluent, including the Discrete method, QMOM, and DQMOM, in our next blog which focuses on practical implementation.

Conclusion

This guide has walked you through the fundamental theory behind the Population Balance Model (PBM). This powerful technique is essential for any CFD engineer dealing with systems where particle sizes change. By understanding these core concepts, you are now ready to dive into the practical side of setting up and running a PBM simulation.

For those ready to apply this theory, our next blog will be a detailed, step-by-step guide on How to Setup PBM in ANSYS Fluent. We will cover everything from choosing the right model to defining boundary conditions and post-processing your results.

For more specialized CFD simulation projects, including those involving complex multiphase flows and Population Balance Modeling, don’t hesitate to explore the services offered at the CFDLand Multiphase CFD Simulations. Our team of experts is ready to help you tackle your most challenging simulation problems.