Population Balance Model (PBM)
Population Balance Model (PBM): Theoretical Foundations
Overview
Professor, what is the Population Balance Method (PBM)?
The PBM (Population Balance Model) is a model that tracks the spatiotemporal changes in the size distribution of bubbles, droplets, and particles. It describes the process by which the size distribution changes due to coalescence, breakup, nucleation, and growth.
In what situations is it used?
It is used in a wide range of problems where the size distribution of a dispersed phase is important, such as bubble size distribution in bubble columns, droplet size distribution in emulsification processes, crystal size distribution in crystallization, and particle size changes in aerosols. It is commonly used in combination with the Euler-Euler method.
Governing Equations
Please tell me the PBM equation.
The transport equation for the number density $n(V, \mathbf{x}, t)$ of particles (bubbles/droplets) with volume $V$ is as follows.
Each term on the right-hand side represents the following.
- $B_{coal}$: Generation by coalescence (two smaller particles coalesce to form size V)
- $D_{coal}$: Disappearance by coalescence (a particle of size V coalesces with another to become larger)
- $B_{break}$: Generation by breakup (a large particle breaks up to produce size V)
- $D_{break}$: Disappearance by breakup (a particle of size V breaks up into smaller ones)
How are the coalescence and breakup rates modeled?
Let me show you some representative closure models.
| Process | Model Example | Driving Mechanism |
|---|---|---|
| Coalescence Frequency | Prince & Blanch (1990) | Turbulent collision + film drainage |
| Coalescence Frequency | Luo (1993) | Turbulent energy |
| Breakup Frequency | Luo & Svendsen (1996) | Breakup by turbulent eddies |
| Breakup Frequency | Martínez-Bazán (2010) | Inertia-surface tension balance |
Sauter Mean Diameter
How do you obtain a representative diameter from the size distribution?
The Sauter mean diameter $d_{32}$ is most commonly used.
$m_k = \int_0^\infty V^{k/3} n(V) dV$ is the $k$-th moment of the distribution. $d_{32}$ corresponds to the volume-to-surface area ratio and is suitable for evaluating mass transfer and reaction rates.
Population Balance—Multiphase Flow Theory as Engineering's "Evolution Theory"
The Population Balance Equation (PBE) is a versatile framework for describing the spatiotemporal changes of populations possessing "internal coordinates" such as size, age, and composition. It is mathematically isomorphic to biological population dynamics (Lotka-Volterra equations) and can uniformly handle everything from bubble distributions and crystal size distributions to cell concentration distributions. The application of PBE in chemical engineering was pioneered by Randolph & Larson's 1971 paper on crystallization, and 50 years later, it has evolved into a design tool for pharmaceutical manufacturing, bubble columns, and liquid-liquid extraction as C-PBE (Coupled PBE) coupled with CFD.
Computational Methods for Population Balance Model (PBM)
Details of Numerical Methods
Please tell me how to solve the PBM numerically.
The PBE is an integro-differential equation with volume (size) as an additional independent variable, making it difficult to solve directly. The following discretization methods are used.
| Method | Overview | Computational Cost | Accuracy |
|---|---|---|---|
| MUSIG (Multi-Size Group) | Divides size space into bins | High (proportional to number of bins) | Depends on bin count |
| QMOM (Quadrature MOM) | Method of Moments + quadrature formula | Low (6-8 moments) | Good |
| DQMOM | Direct Quadrature Method of Moments | Low | Good |
| S-Gamma | Two-parameter distribution assumption | Lowest | Constrained by distribution shape |
Please explain the difference between MUSIG and QMOM.
MUSIG divides the size space into 10-30 bins (size groups) and solves a transport equation for the number density in each bin. It is accurate, but since it solves additional transport equations equal to the number of bins, the computational cost is high.
QMOM (McGraw, 1997) does not directly solve for the size distribution, but solves transport equations for the first few moments ($m_0, m_1, ..., m_5$, etc.). It determines the nodes and weights of the quadrature formula using the Product-Difference Algorithm (PDA) or Wheeler's method to calculate the source terms for coalescence and breakup.
Implementation in Fluent
Implementation in STAR-CCM+
In STAR-CCM+, the S-Gamma model (Lo, 2000) is standard, tracking the distribution with two parameters (mean diameter and variance). It has the lowest computational cost, but the distribution shape is constrained to a gamma distribution. MUSIG is also available.
Implementation in OpenFOAM
In OpenFOAM, the populationBalanceModel class has been available since v2006. MUSIG (including iMUSIG) and QMOM are implemented. Configuration is done in populationBalanceCoeffs within constant/phaseProperties.
QMOM vs DQMOM—Trade-offs in the Method of Moments
The Method of Moments is used to realistically control the computational cost of CFD-PBE. QMOM (Quadrature Method of Moments) approximates the distribution function at quadrature points and solves only the moment equations to track the temporal evolution of the entire distribution. However, in systems with complex breakup/coalescence, the "moment inversion problem" occurs where quadrature points intersect. DQMOM (Direct QMOM) avoids this problem by directly solving transport equations for the position and weight of each quadrature point, but the nonlinearity of the source terms makes convergence difficult. In CFD benchmarks for bubble columns, there are many cases where QMOM and DQMOM predictions for bubble size distribution differ by 10-20%, indicating that model selection has a non-negligible impact on results.
Population Balance Model (PBM) in Practice
Practical Guide
Please tell me the procedure for PBM analysis.
Let me explain using bubble size distribution analysis in a bubble column as an example.