Original Research Paper
The numerical analysis of particle-size distribution of clusters in shear flow at one-dimensional closed system and three-dimensional open system

https://doi.org/10.1016/j.apt.2019.01.006Get rights and content

Highlights

  • A method for analyzing the three-dimensional particle size distribution was proposed.

  • The accuracy was proved in a numerical simulation of backward-step flow.

  • The calculated local particle size distribution agree well with the experimental value.

  • In the low velocity region, the numerical and experiment result disagree slightly.

Abstract

Aimed at optimizing the resin-molding process, a method for numerically analyzing aggregation and dispersion behavior of the filler in resin composite was proposed. The flow of a resin composite during molding was calculated by using computational fluid dynamics (CFD), and particle-size distribution (PSD) of the cluster in each computational domain of the CFD model was estimated by solving the population balance equation (PBE). The proposed numerical-analysis method is based on the thixotropy model of Usui et al. In the thixotropy model, PSD is calculated by taking into account the aggregation and dispersion rate of the cluster by Brownian coagulation, shear coagulation, and shear breakage. Shear-breakage rate of the cluster is evaluated by solving the energy balance of bonding energy of the primary particles cut at the breakage and the drag applied to the cluster by the flow of the fluid. The composite viscosity was calculated using Krieger and Dougherty’s model based on apparent-solid-volume fraction estimated from the calculated PSD. To solve the PBE at low calculation cost, it was discretized using the fixed-pivot technique of Kumar and Ramkrishna. The proposed method was incorporated into the general-purpose CFD software FLOW-3D®, and its accuracy was proved.

Introduction

To improve the performance of a resin-based composite, such as a semiconductor-encapsulation material, electric-wire coating, and insulating composite for high-voltage equipment, the filler needs to become finer. As the filler becomes finer, the surface-area-to volume ratio becomes larger, so it easily aggregates due to the surface effect. This aggregation can greatly affect the performance of the resin composite material in terms of heat resistance, insulation, and thermal conductivity. Therefore, to maximize the performance, it is necessary to optimize the kneading process and the molding process at the design stage in a short time. To achieve that optimization, it is effective to numerically analyze the particle dispersion during the kneading process and the molding process.

Methods for numerical analysis of solid-liquid multiphase flow with the particle aggregation and dispersion can be roughly categorized in terms of the model they use, namely, a solid-liquid two-phase flow model or a mixture model. In the solid-liquid two-phase flow model, solid-phase particle motion and liquid-phase flow are calculated, and the interaction between the solid and liquid phases is evaluated. Various numerical methods, which differ in terms of the means used to calculate motion and interaction of each phase, have been proposed [1], [2], [3], [4], [5], [6], [7], [8], [9]. In the mixture model, which assumes a virtual fluid in which solid-phase particles are dispersed in the liquid phase, solid-phase particle motion is calculated from the stress applied by the flow of the virtual fluid [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]. However, the relative velocity between the solid and liquid phases is not calculated, so the interaction between the solid and liquid phases due to the relative velocity cannot be taken into account. In addition, it is necessary to accurately model the aggregation and dispersion kinetics of the solid-phase particles by using the flow of the virtual fluid. Although under these restrictions, the calculation cost of the mixture model is lower than that of the two-phase flow model, it is effective in estimating the particle dispersion in solid-liquid multiphase flow during molding process.

The two-dimensional numerical-simulation method proposed Masuda et al. [10], [11], [12] calculates the aggregation and dispersion behaviors of clusters in the microchannel flow on the basis of the thixotropy model established by Usui [13], [14]. In that thixotropy model, particle aggregation and dispersion kinetics are modeled by Brownian coagulation, shear coagulation, and shear breakage. In the model of Masuda et al., a non-uniform shear flow is calculated by computational fluid dynamics (CFD), and the particle aggregation and dispersion rate in that shear flow is evaluated by the thixotropy model. Then, the particle-size distribution (PSD) of the cluster at each element of the CFD model is calculated by solving the population-balance equation (PBE) for the clusters. The apparent-solid-volume fraction including the voids in the cluster is estimated from the obtained PSD, and the change in suspension viscosity is calculated. When applying the model of Masuda et al. to optimization of the molding process of resin composites, it is necessary to extend it to large-size clusters and three-dimensional flows with free surfaces.

In this study, which aimed to develop a numerical-analysis method for particle dispersion Masuda et al.'s method was extended to three-dimensional space. In three-dimensionalization, the PBE was discretized using the fixed-pivot technique of Kumar and Ramkrishna [27], [28] to solve the PBE at low calculation cost. The proposed method was incorporated into general-purpose CFD software FLOW-3D®, and its accuracy in a numerical simulation of backward-step flow was proved.

Section snippets

Numerical methods

The following PBE for calculating the PSD profile in a shear flow was developed by Masuda et al. [10], [11], [12]:nit+Unkx+Vnky=12αbj=1,j+k=ii-12kbT3η0rj+rk1rj+1rknjnk-αbj=12kbT3η0ri+rj1ri+1rjninj+12αsj=1,j+k=ii-14γ̇3rj+rk3njnk-αsj=14γ̇3ri+rj3ninj-34π·γ̇2F0Nb,iηsdp3i-1i1-ε-1ni+32π·γ̇2F0Nb,iηsdp32i-12i1-ε-1n2i+34π·γ̇2F0Nb,iηsdp32i-22i-11-ε-1n2i-1+34π·γ̇2F0Nb,iηsdp32i2i+11-ε-1n2i+1where t [s] is time, i [-], j [-], and k [-] are numbers of primary particles in the cluster, ni [m−3] is

One-dimensional closed system (Case 1)

To evaluate whether the PBE results depend on the initial condion of the PSD, time-dependent changes of the average number of particles in cluster at various initial condition in Case 1-1 are shown in Fig. 4. The initial condition of PSD were set to ϕc,1 = 1, ϕc,5 = 1, ϕc,10 = 1, ϕc,15 = 1, and ϕc,20 = 1, respectively. In this case, the PBE results at steady state do not change when the PBE calcuration starts from well-dispersed smaller particles (ϕc,1 = 1) or larger aggregates (ϕc,20 = 1).

Conclusions

A method for analyzing the three-dimensional PSD in shear flow was proposed. The PSD is calculated by taking into account the aggregation and dispersion rate of the cluster by Brownian coagulation, shear coagulation, and shear breakage. The proposed method was incorporated into the general-purpose CFD software FLOW-3D®, and its accuracy was proved in a numerical simulation of backward-step flow. In the downstream region, the numerical and experiment result of distributions of average number of

References (31)

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    Guichard et al. [18] modelled the dynamics of nanoparticles under Brownian motion and turbulent effects using a differential-algebraic framework. Shimada et al. [19] analysed the three dimensional particle size distribution in a shear flow considering Brownian coagulation, shear coagulation and shear breakage. Besides these numerical studies, Swift and Friedlander [20] provided a theoretical analysis of the change in the particle number concentration for simultaneous Brownian coagulation and laminar shear in the continuum regime.

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