Stochastic simulation of agglomerate formation in fluidized bed spray drying: A micro-scale approach
Introduction
Agglomeration is a size enlargement process used to modify physical properties of powders such as size, shape, density or porosity. Fluidized bed spray drying is one of the most successful agglomeration operations. Its main advantage over other processes is the easy combination of distribution and drying in a single apparatus, avoiding the need of an additional dryer. The process principle is quite simple: a liquid feed is dispersed in form of droplets onto a bed of solid particles. Particle collisions within the bed promote the formation of liquid bridges among particles or agglomerates. The liquid evaporates in the fluidization gas forming solid bridges and producing “blueberry-like” agglomerates. Once the agglomerates reach a desired size, they are discharged from the fluidized bed.
Many attempts to describe the processes involved in particle enlargement have been made during the last years. Theoretical (Peglow et al., 2007) as well as experimental work (Hemati et al., 2003) is, however, highly focused on the influence of process conditions and equipment design from a macroscopic point of view. Very few studies regarding the particle–particle interactions within the fluidized bed have been reported. To address these interactions we should be able to describe the dynamics of dispersed systems. Traditionally, this is done based on population balances. Population balance equations can in principle be solved once initial conditions and all rate laws are specified. In practice this is a complex task because of to the integro-differential nature of the equations which require discretization in both, space and time. Furthermore, since every fraction has to be described by an individual equation, solution becomes very difficult in case of multi-parametric systems. Analytical solution of PBE is possible in only few cases, generally solutions have to be obtained numerically (Kumar et al., 2006, Kumar et al., 2008).
The difficulties of solving complicated multivariate PBE can be overcome by the Monte Carlo (MC) method. First proposed by Metropolis and Ulam (1949), MC methods use probabilistic tools to simulate the evolution of a finite sample of the particle population. In MC-based methods, discretization is not required because their inherent discrete nature adapts naturally to processes involving discrete events. This simplifies the programming effort compared to deterministic methods and allows the inclusion of multiple mechanisms (agglomeration, breakage, drying, etc.) in a straightforward manner. With the spread of fast computational machines, MC simulations have become an important alternative for the solution of complex problems. Several different areas of engineering involving coagulation or fragmentation have been treated with MC approaches: chemical reaction (Lísal et al., 2005), polymers (Tan, 1999), colloids and suspensions (Tandon and Rosner, 1999) crystallization (Falope et al., 2001).
MC approaches applied to modeling of agglomeration processes represent the stochastic numerical solution of the population balance equations that describe mathematically the evolution of particle size distribution during size enlargement (Ramkrishna, 1980). MC methods are divided into two main groups according to the treatment of the time step: time driven and event driven methods. In the time driven approach, a time step is first specified and then all the possible events within this time step are implemented (Van Peborgh Gooch and Hounslow, 1996). In event driven MC, an event happens first and then time runs by a previously designated amount (Ramkrishna and Borwanker, 1977). An immediate disadvantage of time driven methods in comparison to event driven techniques arises when the time of the event exceeds the computational time step. Under such circumstances, the computational effort increases because of the non-event simulation cycles. On the other hand, since under event driven methods an event must happen in every time step, (tevent is always equal to tstep), the simulation speed is significantly increased even if the size of time step changes during simulation. Event driven methods are hence more appropriate when the duration of the event is not previously known but determined in the course of simulation, as is the case in the present study.
An inherent issue of MC methods for solving PBE is the size of the “simulation box” or the total number of particles Np being simulated. The simulation box should be understood as a representative sample of the particle population. The number of particles within the simulation box changes according to the process that dominates during the simulation. In coalescence dominated processes and prolonged simulations, the number of particles will decrease and tend to only one. On the other hand, under breakage or fragmentation conditions, the total number of particles will increase overfilling of the simulation box and making the simulation extremely slow. These problems can be overcome by regulating the number of entities continuously or periodically. Constant number MC and constant volume MC (CVMC) methods make continuous and periodical particle number regulation, respectively (Smith and Matsoukas, 1997). The choice between these two methods should be done according to their features and to the mechanism governing the process. Zaho et al. (2007) found that the CVMC method shows a better performance for processes governed by coalescence such as size enlargement processes. Therefore, the CVMC method is used in the present study to simulate the transient behavior of a particle population undergoing size enlargement. The connection between CVMC and the particulate system is provided by the event driven nature of the method. An “event” is defined as a pairwise collision between two randomly chosen groups of particles in the fluidized bed. These particle collisions promote interactions inside the bed that result in the formation of agglomerates.
Agglomerate formation has been recognized to be a complex network of discrete micro-scale interactions among particles, agglomerates and binder droplets which occur in series and in parallel (Tan et al., 2006). Initially, the primary particles capture the sprayed droplets, wetting their particle surface. Deposited droplets that do not produce any bridge dry out and vanish. When wetted particles come in contact with each other two scenarios are possible: the liquid layer cannot dissipate the initial kinetic energy of collision causing the particles to rebound or the particles stick together due to viscous forces forming a liquid bridge. The liquid bridge may again break if it is not strong enough to withstand the collision forces otherwise it may endure and become a solid bridge, which can subsequently break because of bed turbulence. As it can be inferred, there are many parameters affecting the path of a single entity as it moves through the network to finally produce an agglomerate.
A first attempt to describe agglomerate formation taking into account some of these interactions has been communicated by Thielmann et al. (2008). The main objective of this study was to investigate the effect of particle surface energy on agglomeration rate, distinguishing between hydrophobic and hydrophilic particles. However, no importance was given to the capture rate of sprayed droplets. In the simulation, the total amount of binder was supplied at the beginning of the process and maintained constant during agglomeration. Drying of the deposited binder droplets, which reduces the droplet volume and increases the viscosity, was neglected.
In the present study, a model which includes continuous binder addition and simultaneous drying is presented. Agglomerate structure is considered in order to correlate the number of primary particles in the agglomerate with a characteristic diameter. Breakage of the already formed agglomerates is introduced to the model to accomplish the equilibrium between agglomeration and rupture which is typical for fluidized bed processes.
Section snippets
Continuous binder addition
Fluidized bed spray drying combines continuous binder addition and drying in a single process. At the beginning of the process the particles in the bed do not carry any droplet. In the present model, just as in the real processes and in contrast to the studies of Thielmann et al. (2008), binder is continuously added to the particles in form of discrete droplets. The equivalence between real process and model is established by the number of droplets per unit time and particle in the simulation
Experimental methods
Agglomeration experiments were carried out in a lab-scale fluidized bed plant. The plant is equipped with gas dosing units providing the fluidization and atomization air. The cylindrical agglomeration chamber has a diameter of 15 cm and a height of 45 cm. The throughput of binder solution was controlled by a piston pump. Additionally, a balance was used for the online measurement of the sprayed amount. The binder was added to the particles in a top spray configuration. Experiments were performed
Experimental results
The objective of the experiments carried out in this study was to provide data that enable to discuss the influence of variation of process parameters on the micro-scale interactions occurring in the agglomerator during fluidized bed spray drying. In Table 2, the agglomeration rates obtained from representative experiments are presented. For each couple of experiments, all the conditions were maintained constant except for the tested parameter. Fig. 3 depicts the particle size distributions at
Conclusions
A novel stochastic modeling approach capable of simulating the formation of agglomerates during fluidized bed spray drying has been presented in this study. The solid phase is represented by a population of non-deformable, non-porous, monosized and spherical particles. The model is based on the single micro-level interactions among the main entities within the fluidized bed, namely primary particles and droplets, which generate discrete events that result in a blueberry-like particle
Notation
a base radius of spherical cap, m B binder concentration, wt% d diameter, m Df fractal dimension, dimensionless e restitution coefficient, dimensionless fc collision frequency, 1/s h binder layer thickness, m ha particle surface asperities, m Kmax maximum coordination number, dimensionless l characteristic length, m M mass, kg mass flow rate, kg/s molecular weight, kg/kmol np number of particles in the agglomerate, dimensionless droplet flow rate, 1/s Np number of particles in simulation box, dimensionless p number
Acknowledgments
The first author would like to acknowledge financial support by CONACyT (Mexico) and DAAD (Germany).
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