# Population Balance Model for Nucleation, PBM Tutorial

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• The problem numerically simulates Population Balance Model (PBM) for Nucleation using ANSYS Fluent software.
• We design the 3-D model by the Design Modeler software.
• We Mesh the model by ANSYS Meshing software, and the element number equals 112077.
• We use the Species Transport model to define a mixture of water, calcium, and oxalate as the initial phase.
• We use the Eulerian Multiphase model to define a two-phase flow, in which the secondary phase is calcium-oxalate, resulting from a chemical reaction.
• We use The population balance model (PBM) to study produced particles’ behavior and predict the density or population of particles in specific sizes.
• We use a UDF function to define the nucleation and growth rates in PBM analysis.

## Population Balance Model (PBM) for Nucleation, ANSYS Fluent CFD Simulation Training

The present problem simulates the production process of Calcium-Oxalate based on the Population Balance Model (PBM) using ANSYS Fluent software. We perform this CFD project and investigate it by CFD analysis.

We model the present model n three dimensions using Design Modeler software. The study model consists of a small rectangular cubic chamber connecting two elbow tubes.

The chamber has a length, width, and thickness of 12 mm, 6 mm, and 2 mm, and the two inlet pipes have a circular cross-section with a diameter of 1.5 mm. The model consists of a rotating zone and a stationary zone.

The meshing of this present model has been generated by ANSYS Meshing software. The total cell number is 112077.

## CFD Methodology

In this project, the calcium-oxalate production process is first modeled, and then the PBM is analyzed using the production, growth, and displacement of production particles.

The mechanism of the studied system is such that the flow of water carrying calcium and oxalate enters a chamber through two pipes; From one tube, only a mixture of water with calcium is injected, and from the other tube, only a stream of water is injected with oxalate.

The two inlet streams are then mixed, and a chemical reaction occurs in which the combination of calcium and oxalate produces calcium-oxalate. Multiphase and Species Transport models must be used to define this simulation process in software.

Also, PBM, chemical reaction, and mass transfer should be activated.

First, the Eulerian multiphase model defines a two-phase flow; The primary phase is a mixture of water, calcium, and oxalate. The secondary phase is calcium-oxalate, resulting from a chemical reaction.

A chemical reaction within a multiphase flow is then defined. Calcium and oxalate react with each other from the initial phase as a reactant and lead to the production of calcium-oxalate as a product.

The rate of this reaction is defined based on Arrhenius, and the activation energy of the reaction is equal to 1e + 8 j.kg-1.mol-1. In addition to defining the chemical reaction, the process of mass transfer between materials must also be defined.

Therefore, three stages of mass transfer are defined so that in these three stages of mass transfer, three substances, including water, calcium, and oxalate, can be converted to calcium-oxalate.

Also, the species transport model must define a mixture of water, calcium, and oxalate as the initial phase.

After defining the process of conversion of primary phase materials to secondary phase and the occurrence of chemical reaction, it is possible to study the behavioral pattern of particles produced in the secondary phase.

The population balance model (PBM) can study produced particles’ behavior and predict the density or population of particles in specific sizes.

When several particles are formed in an environment, various stages occur, including forming the initial particle nucleation, the growth of formed particles, the aggregation of several small particles, and the breakage of a large particle into smaller particles.

There are different models for defining the PBM in the software, and in this simulation, the discrete method is used. Several bins must be defined for different particle sizes to estimate the particle population or density in each size using this method.

It should be noted that recognizing the number of categories and the definition range for categories is based on inductive reasoning and several steps of simulation solution in the form of trial and error.

In this simulation, 48 bins are used; So that the smallest size with a diameter of 5e-7 m is defined and the largest possible size for each particle up to a diameter of about 2.9345e-5 m.

Also, in this simulation, it is assumed that aggregation and breakage do not occur, and only nucleation and growth processes occur. A UDF function is used to define the nucleation rate and growth rate. Moreover, the laminar model is used to solve the fluid flow equations.

## PBM Conclusion

After the simulation, two-dimensional results related to the mass fraction of the present chemical species for different bin numbers at different times are obtained.

For instance, a two-dimensional contour is obtained related to the volume fraction of the primary phase (including water, calcium, and oxalate) and the second phase (calcium-oxalate).

These contours show that calcium-oxalate is formed by a chemical reaction and mass transfer from the primary to the secondary phase. The formation of the desired material occurs in the initial area of the chamber, which is the junction of two inlet pipes. Then the table related to the sizes of each definition category (bin) is presented. There are 48 categories for particle size classification, starting at a minimum of 5e-7m at bin-47 and continuing to a maximum at bin-0. In addition, we represent the bar graph of number density in terms of the bin number.

Number density represents the ratio of particles per unit volume; That is, how many particles with the desired volume (according to the size of each bin) are within the desired area. Finally, we obtain two-dimensional contours of the volume fraction of calcium-oxalate produced in different bins. The results show that from about bin-30 onwards, the amount of material produced becomes significant.

This means that the volume of particles produced in the categories between bin-0 to bin-30 is imperceptible. Therefore, we can conclude that the produced particles are formed and grow in smaller volumetric sizes (between bin-47 to bin-30) and are not noticeable in larger volumetric sizes.

This video is the 4th episode of the Population Balanced Model Training Course.

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