Real-Time Modeling of Volume and Form Dependent Nanoparticle Fractionation in Tubular Centrifuges

Abstract

A dynamic process model for the simulation of nanoparticle fractionation in tubular centrifuges is presented. Established state-of-the-art methods are further developed to incorporate multi-dimensional particle properties (traits). The separation outcome is quantified based on a discrete distribution of particle volume, elongation and flatness. The simulation algorithm solves a mass balance between interconnected compartments which represent the separation zone. Grade efficiencies are calculated by a short-cut model involving material functions and higher dimensional particle trait distributions. For the one dimensional classification of fumed silica nanoparticles, the numerical solution is validated experimentally. A creation and characterization of a virtual particle system provides an additional three dimensional input dataset. Following a three dimensional fractionation case study, the tubular centrifuge model underlines the fact that a precise fractionation according to particle form is extremely difficult. In light of this, the paper discusses particle elongation and flatness as impacting traits during fractionation in tubular centrifuges. Furthermore, communications on separation performance and outcome are possible and facilitated by the three dimensional visualization of grade efficiency data. Future research in nanoparticle characterization will further enhance the models use in real-time separation process simulation.

Keywords: dynamic modeling; fractionation; multi-dimensional particle properties; real-time simulation; solid–liquid separation; tubular centrifuges.

Figure A1

(a) Global grade efficiency for the RPS classification with boundary conditions listed in Table 1; markers show the ADC reference measurement and solid lines show the 1D-TCM output for different spatial discretization settings; (b) RSS calculated with Equation (A1) for different spatial discretization settings; the dashed line indicates the setpoint used for all simulations shown in the main manuscript.

Figure A2

Simulation time factor in dependence of the total distribution class count for different time step sizes. Unity indicates a real-time measurement. Values greater than one factorize the simulation speed increase.

Figure 1

Image of three different three dimensional (3D) aggregates and their corresponding inertia ellipsoid illustrated as a transparent hull. Elongation e and flatness f values for each ellipsoid are displayed below each body.

Figure 2

Schematic cross section of a tubular centrifuge rotor highlighting the geometry and spatial discretization with $j=\{1,2,\cdots ,J\}$ cylindrical compartments of equal volume.

Figure 3

Spatial discretization of the separation zone in $j=\{1,2,\cdots ,J\}$ compartments with illustration of incoming and outgoing flows (a); Flow chart of the simulation algorithm (b).

Figure 4

Schematic overview of the pipeline used to compute three dimensional (3D) particle trait distributions (PTDs) based on virtual silicon dioxide (SiO${}_2$) aggregates.

Figure 5

Two dimensional (2D) image of eleven unique nanoparticles (NPs) of the virtual particle system (VPS) created by the custom diffusion-limited-aggregation (DLA) algorithm (M1). Two different projections are shown to visualize differences is nanocluster form.

Figure 6

Cumulative sum distribution $G^{*}\left(s_{\mathrm{ext}}\right)$ derived by least-squares boundary modeling (LSBM) code based on analytical centrifuge (AC) extinction profiles (a). Reciprocal ext. weighted sedimentation coefficient $s_{25,\mathrm{ext}}^{-1}$, $s_{50,\mathrm{ext}}^{-1}$ and $s_{75,\mathrm{ext}}^{-1}$ plotted against $\varphi$ of the analyzed silicon dioxide (SiO${}_2$) suspension; dotted, dashed and dash-dotted lines indicate a linear least squares fit (Equation (31)) with an associated coefficient of determination $R^2$ (b). Note that the lines indicate the connection between the data extracted from (a) and plotted in (b).

Figure 7

(a) Volume weighted density distribution of feed and centrate suspension (left ordinate) and global grade efficiency (right ordinate) at $t_N=360s$. Markers indicate experimental data acquired by analytical disk centrifuge (ADC) measurements, solid lines mark the tubular centrifuge model (TCM) case output. (b) Global separation efficiency versus time calculated by the TCM model. Experimentally acquired reference with two independent methods drawn with horizontal dashed and dotted lines, gray areas highlight the standard deviation of both measurements.

Figure 8

(a) Three dimensional (3D) particle trait distribution (PTD) ${\tilde{q}}_{3,\mathrm{in}}(f,e,d)$ of virtual silicon dioxide (SiO${}_2$) particle system: discretization over the three distinct traits d, e and f; (b–d) Marginalized distributions ${\tilde{q}}_{3,\mathrm{in}}$ visualized in a two dimensional (2D) grid.

Figure 8

(a) Three dimensional (3D) particle trait distribution (PTD) ${\tilde{q}}_{3,\mathrm{in}}(f,e,d)$ of virtual silicon dioxide (SiO${}_2$) particle system: discretization over the three distinct traits d, e and f; (b–d) Marginalized distributions ${\tilde{q}}_{3,\mathrm{in}}$ visualized in a two dimensional (2D) grid.

Figure 9

(a) Three dimensional (3D) grade efficiency ${\tilde{T}}_{\mathrm{g}}(f,e,d)$ of virtual silicon dioxide (SiO${}_2$) particle system after separation: discretization over the three distinct traits d, e and f; (b) Marginalized data over particle traits e and d.

Figure 10

Heatmap of drag correction factor $k_{\mathrm{S},\mathrm{B}}$ for ellipsoids versus their elongation and flatness based on Equation (9) found in [53].

Figure 11

Volume weighted density distribution of virtual particle system (VPS) feed- and centrate suspension (left ordinate) and global grade efficiency (right ordinate). Distribution data are acquired by a marginalization of three dimensional (3D) tubular centrifuge model (TCM) output. Dimensional reduction applied for comparison between $T_{\mathrm{g}}$ and ${\tilde{T}}_{\mathrm{g}}$.