International Journal of Chemical Engineering

Volume 2016, Article ID 8743105, 6 pages

http://dx.doi.org/10.1155/2016/8743105

## Comparison of the Accuracy and Performance of Different Numbers of Classes in Discretised Solution Method for Population Balance Model

^{1}College of Urban Construction and Environmental Engineering, Chongqing University, Chongqing 400044, China^{2}Chongqing Research Academy of Environmental Sciences, Chongqing 401147, China^{3}State Environmental Protection Key Laboratory of Environmental Planning and Policy Simulation, Beijing 100011, China

Received 7 June 2016; Revised 5 September 2016; Accepted 15 September 2016

Academic Editor: Doraiswami Ramkrishna

Copyright © 2016 Zhenliang Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

One way of solving population balance model (PBM) in a time efficient way is by means of discretisation of the population property of interest. A computational grid, for example, ( is the volume of particle in class ), could be used to classify the particles in discretisation techniques. However, there are still disagreements in the appropriate number of classes divided by the grids. In this study, the different numbers of classes for solving PBM were compared in terms of accuracy and performance to describe the particle size distribution (PSD) from the flocculation of activated sludge. It is found that the simulated PSDs are similar to the experimental data for all the geometric grids , and there is no obvious difference among the values of calibrated parameter, ratio of breakage rate coefficient and collision efficiency, for each velocity gradient. However, the simulation results with less error could be obtained with larger number of classes, and more computational times, which show exponential relationship with the number of classes, are needed. Considering numerical accuracy and efficiency, the classes 35 or a geometric grid with factor 1.6, aligning with the Fibonacci sequence , is recommended for the particles in the size range of 5.5~1086 *μ*m.

#### 1. Introduction

Population balance model (PBM) has been widely used to describe the processes involving dynamical behaviour of population properties. Applications can be found in a variety of scientific areas such as flocculation, crystallisation, polymerisation, granulation, and aerosol and cell culture dynamics [1]. The numerical techniques to solve this type of equations are summarised by Ramkrishna [2]. One of these techniques resulting in an acceptable computational time and accuracy is the discretisation of the particle size.

Discretisation techniques divide the property range of interest into a finite number of classes () according to a given computational grid, transforming the PBM into a set of ordinary differential equations that can be solved simultaneously [3]. A computational grid, for example, ( is the volume of particle in class ), could be used to classify the particles for solving PBMs. However, there are still disagreements in the appropriate number of classes. McAnally and Mehta [4] concluded that an available geometric grid used to classify the particles should be in order to maintain the continuity of particles distribution after aggregation. Calabrese et al. [5] demonstrated that the geometric grid with factor 2 is not as suitable as another grid which kept in accordance with the Fibonacci sequence, . Xu et al. [6] suggested a computational grid to achieve a reasonable numerical accuracy. Verney et al. [7] believed that the number of classes could obtain the satisfying simulation results when the particle size is in the range of 4~800 *μ*m, whereas Mietta et al. [8] recommended the number of classes for nearly the same range of particles. Absolutely, the computational cost increases with the increasing number of classes. Acceptable computational speeds are important when using the model for parameter estimation or in combination with CFD [9]. Therefore, the problem needs to be firstly solved in the application of PBM: what is the appropriate number of classes in discretisation techniques? In addition, it is still not clear whether the number of classes has influence on the results of parameters calibration.

In this study, different numbers of classes in the discretised solution for PBM were compared, in terms of accuracy and performance to describe the particle size distribution (PSD) from the flocculation of activated sludge, and the appropriate number of classes was recommended according to a reasonable compromise between accuracy and efficiency of PBM.

#### 2. Theory and Experiments

##### 2.1. Theoretical Section

The discretised PBM developed by Kumar and Ramkrishna [10] is used to describe the variation rate in the number of particles with a given size:where is the number concentration of particles of class , is Dirac delta function, is the collision efficiency, is the collision frequency of particles in classes and , is the breakage rate of particles in class , is the breakage distribution function defining the fraction of daughter-particle of class size breaking from a particle of class size , and is a proportional coefficient assigning the fraction of the particle from the aggregate ():The collision efficiency is generally considered as a constant which needs to be calibrated with the experiments [11]. The collision frequency due to shear rate has usually been considered as function of particles size , as well as the breakage rate [12, 13]:where is the average velocity gradient:where represents the homogeneous turbulent energy dissipation rate of the mixed tank and is the kinematic viscosity of the suspending fluid.where is the breakage rate coefficient.

For a geometric grid adopted to solve the population balance model, the binomial breakage function can be used for the daughter-particle distribution [14]:where is the parameter which is defined to determine the location at which the probability mass function reaches its peak and can be calculated from

##### 2.2. Modelling Method

Integrating (1) involved computing the discrete of values. The geometric grids with a variable factor of were used to classify the particle according to the size. The solution equations were derived using the Euler method, which involved establishing 0.1~1 s as the iterative calculation of to maintain the calculation stability [15]. The final state of flocculation is that particle aggregation and breakage counterbalance each other; that is, = 0 in (1). Connecting (5) and (1) yieldsEquation (8) shows the correlation between the breakage rate coefficient and the collision efficiency . Given the ratio of breakage rate coefficient and collision efficiency , the number concentration of particles in each class could be obtained. The volume percentage distribution was chosen as the fitting variables of the minimum error for parameters calibration [16]:where and were the measured and simulated volume percentage of particles in class , respectively.

The optimization results of parameter, the ratio of breakage rate coefficient and collision efficiency, were estimated by the exhaust algorithm in the range of possible values. The computational speed was recorded during the simulation using Matlab 2012 (MathWorks Inc., USA) on the PC with Intel i5 processor systems (3.0 GHz).

##### 2.3. Experimental Section

Flocculation experiments were conducted in a mixing tank (ZR4-6, China) using activated sludge collected from a municipal wastewater treatment plant. The volume of mixed liquid in the tank was 1 L and the concentration of sludge was 0.1 kg·m^{−3}. The PSD (or volume percentage distribution) of activated sludge during flocculation was measured using a laser particle size analyzer (S3500, Microtrac, in the range of 0.02~2800 *μ*m) at the velocity gradients of 28.2 s^{−1}, 64.7 s^{−1}, and 101.7 s^{−1}, which are the typical values of velocity gradient for flocculation occurrence. In order to reduce the possible effect on the particle size distribution caused by the difference of flowing shear between the sampling pipe and the mixing tank, the velocity gradients in the sampling pipe were set to be similar to those in the mixing tank by controlling suitable sample flow rates in the sampling pipe [17].

The raw output of the laser particle size analyzer is volume percentage distribution for a given grid. However, PBM is formulated on a number concentration basis and might use a different grid. The cumulative volume percentage distribution was first calculated from the raw volume percentage distribution and then interpolated at the pivots of the new grid, allowing the recalculation of the volume percentage distribution [18]. Finally, the initial number concentrations of the models were generated by transforming the volume distribution data to a number distribution.

#### 3. Results and Discussion

Figure 1 shows the variation rate in the number of particles for three numbers of classes from different computational grids (take, e.g., the simulation of flocculation at the velocity gradient of 28.2 s^{−1}). For s, of some particles, especially whose sizes are smaller than 10 *μ*m, is so far from zero (~−2.0 10^{6}). For s, of all particles is nearly zero, indicating that the PSD at steady state of flocculation would be achieved soon. When s, is far smaller than zero and unchanged with time; so 5400 s can be used for the total simulation time at which the steady-state PSD is obtained. Moreover, it seems that three different geometric grids make negligible difference to the variation rate in number of particles.