Population balance modeling of formation and breakage of nanoparticle agglomerates in a spouted bed

Abstract

A population balance framework for fluidization of Al2O3 nano-agglomerates in a ProCell-type spouted bed considering agglomeration and breakage phenomena is developed. An equipartition of kinetic energy (EKE kernel) is used to model the agglomeration phenomena. For breakage, two different fragment distribution functions are considered, the Dirac delta function and a function with confined uniform binary breakage. The fractal dimension property is incorporated in both the agglomeration and breakage models. The model parameters are extracted from batch fluidization experiments conducted by varying the air pressure of the top nozzle. The developed model shows a good agreement with experimental observation in regard of the temporal change in the area and volume-weighted average size of nano-agglomerates and of their final size distribution. The developed population balance model is simulated to study the influence of model parameters on the evolution of average size and coefficient of variation of the nano-agglomerates.

Introduction

Nanoparticles are particulate materials with particle sizes ranging between 1 and 100 nm. Due to their large surface area per unit mass, nanoparticles possess unique optical, electrical, mechanical, electrochemical, and thermal properties depending on their size [1], [2], [3]. The processing of nanoparticles in a fluidized bed can have several purposes such as to coat, granulate, separate or dry, and offers a wide range of commercial applications in drug delivery, semiconductors, electronics, sensors, etc. [4], [5], [6].

Nanoparticles tend to agglomerate due to their strong interparticle forces, mainly van der Waals forces, but frequently also electrostatic and capillary forces. Hence, nanoparticles are fluidized in the form of agglomerates rather than primary particles. The fluidization behavior of the nanoparticle agglomerates can be classified into two types, agglomerate particulate fluidization (APF) and agglomerate bubbling fluidization (ABF). The APF is characterized by uniform and bubbleless fluidization with high bed expansion, whereas the ABF is characterized by limited bed expansion with the occurrence of bubbles and channels [7]. In order to enhance or improve the fluidization behavior of the nanoparticle agglomerates, various fluidization assistance techniques have been proposed, such as ultrasonic vibrations [8], mechanical vibrations [9], magnetic field disturbance [10], secondary gas flow using microjets [11], use of flow conditioners [12], etc. The main purpose of these various types of assistance is to break large agglomerates. The agglomeration of nanoparticles during fluidization is governed by the dynamic equilibrium between the formation and breakage of agglomerates.

Several studies have been performed related to the estimation of the size of nanoparticle agglomerates during fluidization. Quevedo et al. [13] used the Richardson-Zaki criterion coupled with fractal analysis to estimate the approximate size of nano-agglomerates in a rotating fluidized bed. Valverde and Castellanos [14] proposed a simple equation to estimate the size of agglomerates in the gas-fluidized bed of nanoparticles derived from the balance between the interparticle adhesive force and the local shear force on the particles present at the outer layer of the agglomerates. Similarly, Tahmasebpoor et al., Tamadondar et al. [15], [16] also derived a comprehensive model based on the balance between van der Waals and hydrogen bond as adhesive forces and collision, gravity, and drag as separation forces to estimate the equilibrium size of agglomerates formed during the nanoparticle fluidization. Matsuda et al. [17] developed a comprehensive model for agglomeration during the fluidization of nanoparticles based on an energy balance with respect to energy consumption for disintegration of agglomerates.

The evolution of the size distribution of nanoparticle agglomerates due to agglomeration and breakage phenomena encountered during fluidization can be described by population balance equations (PBE). With just one distributed property, the PBE is an integro-partial differential equation that requires closures in terms of expressions for, e.g., the agglomeration kernel, the breakage function, etc. in order to be solved. Those can be of theoretical, semi-empirical or empirical origin. The use of PBE to describe the evolution of particle size distribution during fluidized bed agglomeration has been reported extensively [18], [19], [20], [21], [22], though not for nanoparticles and without consideration of breakage.

Alternatively, a whole family of discrete models, so-called Monte Carlo models has been proposed for the simulation of spray fluidized bed agglomeration, which typically take breakage into account, through in simplified form [23], [24], [25]. Dosta et al. [26] developed a multiscale simulation approach considering the PBE in the macroscale and a hybrid computational fluid dynamics (CFD) and discrete element method (DEM) scheme in the microscale to model agglomerate breakage in a fluidized bed, again with larger primary particles. The DEM has also been used for simulations with agglomerating nanoparticles in a fluidized bed, either alone [27] or combined with CFD [8], [28], [29], [30]. However, PBE simulations that would consider simultaneous agglomeration and breakage of fluidized nanoparticles are still missing. Such simulations rather exist for the agglomeration and breakage of nanoparticle agglomerates in other operations and in different kinds of equipment, such as in stirred media mill [31], [32], in rotor-stator mixer [33], and during chemical synthesis [34].

In this present study, we develop a mathematical model based on the population balance framework to model agglomeration and breakage of nanoparticle agglomerates during fluidization. In order to estimate the model parameters, batch fluidization experiments were conducted in a ProCell-type spouted bed considering aluminum oxide ( ${Al}_{2} O_{3}$ ) as the material.

The rest of the article is organized as follows: Section 2 describes the experimental procedure of fluidization experiments of aluminum oxide nano-agglomerates. Section 3 presents the development of the population balance model considering agglomeration and breakage phenomena for the fluidization process. The solution scheme of the PBE is presented in Section 4. The simulation results are presented and discussed in Section 5, followed by conclusions and an outlook in Section 6.