Development and Investigation of a Separation Process Within Cross-Flow with Superimposed Electric Field

Abstract

The increasing demand for highly specific particulate products in industrial processes is a driving factor in the development of novel particle separation processes. In this work, a multidimensional separation process for wet simultaneous separation by hydrodynamic diameter and electrophoretic mobility was developed. The hydrodynamic effects and electrophoretic influences within this process were experimentally investigated on different scales with three setups for batch and continuous operation. Flow rates were varied from a few mL∙min⁻¹ to several 100 L∙h⁻¹, and electric field strengths of up to 300 V∙cm⁻¹ were employed to analyze different spherical particles in the range of 1 to 100 µm. The investigations demonstrated the limitation of the separation process due to some of the resulting effects, such as electrolysis. A scale-up approach for hydrodynamic separation was developed based on CFD simulation, which can predict the operating range of the process with the high efficiency.

1. Introduction

The advancement of particle technology and its broad industrial application is leading to increasingly sophisticated, high-quality and complex products. Particulate systems with very specific properties are used to precisely control processes and are crucial for process success. They offer the potential to develop new procedures or to increase the efficiency of existing ones. Their impact on process design is significantly determined by the specific surface area of the particles, which is strongly dependent on the size, shape and porosity of the particles. In addition to these physical properties, chemical properties, such as chemical composition or zeta potential, must also be considered, particularly in the sub-micron and micron size range, where particle size has a significant influence on the properties—and therefore the behaviour—of particulate products. These products find application in a wide range of industries, including the production of composite materials, as catalysts in reactors and in the healthcare sector. In mechanical operations, such as smoothing and planning surfaces, the size of the abrasive particles is relevant for the quality of the surface with regard to roughness [1]. As electrode materials in battery technology, the size and flow properties of particles determine whether a homogeneous and agglomerate-free electrode surface is achieved [2]. In polymer–glass composites, finer particle fractions improve ductility without affecting other properties [3]. Fine particles are typically generated through crystallization, precipitation, chemical reactions, or the comminution of coarse particles. To obtain particles in the desired sub-micron and micron size range, separation processes are required.

In the context of processes, a distinction can be made between two categories of separation. The first uses a physical particle property, such as size, shape, density or wettability, in order to obtain the particle fractions with a narrow distribution of that property [4,5,6,7]. Müller provides a comprehensive overview of solid–liquid separation processes, including limits and applicable particle size ranges [8]. This overview is illustrated in Figure 1 and extended by further separation approaches [9,10]. The second category of processes is based on the separation according to chemical particle properties, including chemical composition, dielectrophoretic and magnetic properties and zeta potential [11,12,13,14]. In the case of simultaneous separation based on more than one particle property, the process is referred to as multidimensional separation [6,15].

Separation processes have been increasingly enhanced over the last few decades and multidimensional approaches have also been implemented. In conventional hydrocyclone applications, the particles are already separated along their size and density. In [16], an additional separation property was successfully implemented in the process by applying an external magnetic field. The cut size limitation of hydrocyclones is about 5 µm, as described by the fishhook effect [17]. However, using computational fluid dynamics (CFD) and experimental data, the design of mini hydrocyclones was adapted to enable the separation of particles measuring less than 2 µm [18]. In another design approach [19], a separation cut size of about 0.5 µm was shown, albeit with a high energy consumption.

Centrifugal separation can also be achieved in terms of size and density. The acting centrifugal forces are many times greater than in the gravitational forces, thus enabling separation in the micron and sub-micron range. In [20], the blade design of a centrifugal separator was adapted using CFD simulations, resulting in enhanced particle collection efficiency and a gradual decline in pressure for particles measuring less than 0.5 µm. Ultracentrifuges enable the separation of particles smaller than 0.1 µm [21]. Flotation is a separation process that utilizes the differences in the wettability, size and density of particles. In the process of gassing a suspension, a fraction of particles with a range of 20–200 µm [22] is adhered to rising bubbles and elevated. This range can be extended by tailoring the particle or gas properties [23,24]. In our previous works [25,26], we developed a cross-flow filtration method to fractionate particles smaller than 10 µm. Under constant flow conditions, the particle size in the boundary layer of a membrane can be controlled and separated by backwashing.

In this work, a novel cross-flow separation technique with a superimposed electric field is presented. It can be used for the highly specific separation of suspensions containing particles in the micrometre range. A discontinuous operation was developed by Altmann and Ripperger [27] and was further elaborated in our current preliminary works [26,28]. This approach is related to multidimensional processes; it not only allows for particle classification according to their size but also aims to achieve a simultaneous separation along the electrophoretic mobility (EPM) of the individual particles. This paper presents the design of a batch setup investigating particle motion in the multidimensional field of forces through the combination of experimental and simulation techniques. Moreover, a setup for continuous hydrodynamic separation is developed, and the process is investigated with particular consideration for approaches to scale up using CFD simulation.

The following section outlines the theoretical basis of the developed method, elucidating the role of acting forces and the fundamental principles of separation. Based on this, the experimental setups are presented with the corresponding operating conditions and properties of the used materials. In addition to the technical implementation, the simulation and the calculation methods are explained. The obtained results are divided into two main sections: firstly, an investigation of hydrodynamic separation is described, and secondly, the particle motion in the multidimensional force field is analyzed. Both main mechanisms are analyzed using the different setups, process parameters and materials.

2. Separation Method and Mechanisms

The developed separation process (Figure 2) is based on a cross-flow filtration with a superimposed electric field. The objective is to achieve a separation of the particles, specifically in terms of their hydrodynamic diameter and charge. The main mechanisms that lead to separation are hydrodynamic and electrostatic effects, particularly the hydrodynamic forces acting on particles in a fluid flow and electrophoretic forces exerted on charged particles in a homogeneous electric field.

To guarantee a uniform and dependable separation, it is essential to maintain a laminar flow regime within the channels of the separation module. For the same reason, the electric field must be homogeneous across the entire cross-section. The separation module comprises two rectangular main channels with a suspension-carrying fluid (feed) flowing parallel to a particle-free flow (filtrate). Both streams are admitted uniformly into the module. The separation occurs at the separation medium, which is placed between the main channels and contains the separation geometry, e.g., single or multiple gaps, which are larger than the particle size. On the outside, each channel is surrounded by an electrode (anode and cathode) aligned parallel to the flow, creating a homogeneous electric field between them. To avoid the influence of the gravity on the separation, the module is adjusted vertically.

The forces acting on the particles in the near-wall boundary layer are shown in detail (Figure 2, right). The particles are transported by the flow, resulting in drag force $F_D$. The locally acting velocity gradient in the laminar profile influences the particles by creating an asymmetric flow that results a wall shear-gradient induced lift force $F_L$. The electric field exerts an electrophoretic force $F_E$ on the particles, resulting in the movement of a negatively charged particle towards the anode. By adjusting the differential pressure between the two channels, a volume of the suspension is transported through the gap into the filtrate channel, where particles from the near-wall boundary layer are collected. This flow rate is referred to as the exchange flow rate or the exchange velocity $v_{ex}$. The size of the particles in this boundary layer is directly related to the hydrodynamics and the charge, as well as the strength of the electric field.

The method described in this paper is aimed at the separation of particles in the size range of 1–100 µm. For this size range, the particle movement due to the viscous flow forces is dominant, and the effects of the Brownian movement (typically relevant for particles < 0.5 µm) can be neglected.

2.1. Hydrodynamic Forces

In the multiphase flow, the interaction between the disperse phase and the fluid is subject to a variety of effects, including surface, mass or bulk forces on particles caused by the pressure and velocity gradients of the surrounding fluid flow [29]. In the described process, the hydrodynamic effects on the particles are a consequence of two characteristic forces: drag force $F_D$ and lift force $F_L$. The drag force is caused by the relative movement between the particle and the fluid and acts in opposition to the relative velocity:

$${\overrightarrow{F}}_D={\displaystyle \frac{\pi }{8}}\rho d_P^2{C}_D\left(\overrightarrow{v}-{\overrightarrow{v}}_p\right)\left|\overrightarrow{v}-{\overrightarrow{v}}_p\right|$$

(1)

where $v$ is the fluid velocity, $v_P$ the particle velocity, ρ the fluid density and $d_P$ the particle diameter. In general, the drag coefficient $C_D$ is dependent on the shape of the particle [30,31,32]. However, the well-known correlations with the particle Reynolds number in Equation (2) exist for a spherical particle [33].

$${Re}_p={\displaystyle \frac{\rho \left|\overrightarrow{v}-{\overrightarrow{v}}_P\right|d_P}{\eta }}$$

(2)

where η is the dynamic viscosity. For low particle Reynolds numbers (${Re}_p<0.25$), the Stokes law can be applied for a spherical particle. Subsequently, the drag coefficient and the drag force are defined as

$$C_D={\displaystyle \frac{24}{{Re}_P}}$$

(3)

$${\overrightarrow{F}}_D=3\pi \eta \left(\overrightarrow{v}-{\overrightarrow{v}}_P\right)d_P$$

(4)

In this flow regime, the force is directly proportional to the particle diameter. Various correlations exist to describe the drag coefficient beyond the Stokes regime. Kaskas [34] described the drag coefficient in a wide range of ${Re}_P<2\times {10}^5$ as follows:

$$C_D={\displaystyle \frac{24}{{Re}_P}}+{\displaystyle \frac{4}{\sqrt{{Re}_P}}}+0.4$$

(5)

In addition, the Morsi and Alexander [35] model is able to describe the drag coefficient over a wide range of ${Re}_P$:

$$C_D=a_1+{\displaystyle \frac{a_2}{{Re}_P}}+{\displaystyle \frac{a_3}{{Re}_P^2}}$$

(6)

The different flow regimes surrounding the particles are considered with the constants $a_1$, $a_2$ and $a_3$ and are presented in

Table 1. Parameters of the Morsi and Alexander correlation for the determination of the drag coefficient as a function of the Reynolds number.

${\mathit{R}\mathit{e}}_{\mathit{P}}$ Ranges	${\mathit{a}}_1$	${\mathit{a}}_2$	${\mathit{a}}_3$
0.1 ≤ Re ≤ 1	0	24	0
1 < Re ≤ 10	3.69	22.73	0.0903
10 < Re ≤ 100	1.222	29.1667	−3.8889
100 < Re ≤ 1000	0.6167	46.5	−116.67
1000 < Re ≤ 5000	0.3644	98.33	−2778
5000 < Re ≤ 10,000	0.357	148.62	−47,500
Re > 10,000	0.46	578.7	−166,200

below.

For further correlation of the wide Re number, the reader is referred to the review paper by Goossens [36].

Lift force $F_L$ always acts perpendicular to the drag force, with the direction being contingent on the flow velocity around the particle, acting towards the higher relative velocity. With regard to the separation process, the wall-induced lift force is particularly relevant. At the wall, the fluid velocity is zero, resulting in a large velocity and shear gradient in the boundary layer. Conversely, at the centre of the main flow in the channel, the velocity gradient is reversed, thereby resulting in a reversed lift force [37]. The significant influence of the lift force on the particle trajectory within a Poiseuille flow (fully developed laminar flow) was demonstrated by Segré and Silberberg [38,39,40]. Within the flow, the particles converge to an equilibrium radial position depending on the channel dimensions. This finding was subsequently validated by Ho and Leal [41].

An analytical calculation model of the wall-induced lift force is based on the studies of Saffman [42]. If the particle is in contact with the wall, the following equation can be utilized:

$${\overrightarrow{F}}_{L,Saffman}=0.807{\displaystyle \frac{d_P^3·{\overrightarrow{\tau }}_w^{1.5}·{\rho }^{0.5}}{\eta }}$$

(7)

This equation includes the particle diameter scaled in the power of three and wall shear stress ${\tau }_w$, which can be calculated as

$${\overrightarrow{\tau }}_w={\displaystyle \frac{6·\eta ·\overrightarrow{v}}{D_h}}$$

(8)

The lift effect-inducing overflow velocity $v$ from the main channel and its hydraulic diameter $D_h$ include the following design parameters:

$$D_h={\displaystyle \frac{4·A}{P}}={\displaystyle \frac{2·a·b}{a+b}}$$

(9)

where $A$ is the cross-section area of the rectangular flow channel and $P$ is its wetted perimeter with width $a$ and depth $b$.

As a consequence of the size dependence to the power of three (Equation (7)), the lift force exerts an increasing effect on larger particles. Therefore, large particles are more likely to be transported into the core flow, which enables the separation of small particles from the flow closer to the wall.

In our previous work [43], it was demonstrated that the experimentally determined classification cut size in the boundary layer of a membrane in the cross-flow filtration process could be accurately predicted using lift force models according to Saffman [42], Rubin [44] and Leighton [45]. The model developed based on the lift force shows a good prediction of the cut size obtained from the measurements of the finer fraction of particles deposited on the membrane surface. Moreover, there are numerous studies on the determination of the lift force in the literature [46,47,48,49]. A comprehensive overview of the lift force calculation based on both experimental and numerical approaches can be found in the work of Shi and Rzehak [50].

2.2. Electrophoretic Forces

The superimposed homogeneous electrical field, generated by the addition of an external direct current (DC) circuit, results in the deflection of particles by an additional electrophoretic force $F_E$, which is acting orthogonally to the main flow direction (Figure 2). The direction and magnitude of this force are dependent upon the electrochemical properties of the particles, particularly the electrophoretic mobility $\mu$ (EPM) [51,52,53]. This coefficient represents a material parameter and can be measured using the Dynamic Light Scattering (DLS) method [54,55,56]. Electrophoretic mobility is calculated as the ratio of particle velocity $v_E$ in an electrical field with an electric field strength $E$.

$$\mu ={\displaystyle \frac{{\overrightarrow{v}}_E}{\overrightarrow{E}}}$$

(10)

In the case of homogeneous electric fields, the strength can be calculated by the voltage difference $U$ between the two electrodes and their distance $a$ as

$$\overrightarrow{E}={\displaystyle \frac{\overrightarrow{U}}{a}}$$

(11)

The electrophoretic force exerted on a particle with charge $q$ in electric field $E$ is described by the Coulomb law:

$${\overrightarrow{F}}_E=q\overrightarrow{E}$$

(12)

The resulting electrophoretic particle motion is countered by a drag force. In the case of spherical particles, in equilibrium and in Stokes flow, the following equation applies from Equations (4) and (12):

$$q\overrightarrow{E}=3\pi \eta d_P{\overrightarrow{v}}_E$$

(13)

After rearranging according to the velocity, Equation (10) is inserted so that the charge can be expressed as a function of the electrophoretic mobility:

$$q=3\pi \eta d_P\mu$$

(14)

Finally, this expression can be resubstituted into Equation (12) to calculate the force:

$${\overrightarrow{F}}_E=3\pi {\eta d}_P\mu \overrightarrow{E}$$

(15)

In the above equation, all variables are linearly correlated, especially the particle diameter. In terms of the separation process, this force is compared with the lift effect, which is much more dependent on the particle size ($F_L~d_P^3$). Therefore, the electrophoretic force mainly influences micron and sub-micron particles, while flow-induced effects dominate for larger particles.

2.3. Calculation of a Theoretical Separation with a Balance of Forces

In consideration of the forces described, an analytical calculation of a separation with a single gap is carried out. The equilibrium of forces on a particle in the area above the gap can be calculated to determine the maximum wall distance directly in front of the gap that a particle must reach in order to be transported into the filtrate. This value in Equation (17) is representative of the separation height $h_{sep}$, as only particles with a smaller wall distance can penetrate the gap due to the drag force of the exchange flow. The bigger particles remain in the main flow due to lift and electrostatic forces. The separation is achieved when the residence time of the vertical and horizontal particle movement is equal; this is referred to as the separation time $t_{sep}$.

$$t_{sep}={\displaystyle \frac{h_{sep}}{\left|{\overrightarrow{v}}_{p,ex}\right|}} =t_{feed}={\displaystyle \frac{L_{gap}}{\left|{\overrightarrow{v}}_{feed}\right|}}$$

(16)

$$h_{sep}=\left|{\displaystyle \frac{{\overrightarrow{v}}_{p,ex}}{{\overrightarrow{v}}_{feed}}}\right|·L_{gap}$$

(17)

$v_{p,ex}$ is the absolute velocity of the particle in the gap and $t_{feed}$ is the residence time in vertical direction. The separation length $L_{sep}$ is also determined; it corresponds to the maximum vertical distance in front of the gap that a particle can have in order to be separated. It results from the velocity ratio in the main channel and the size-dependent separation height.

$$L_{sep}=\left|{\displaystyle \frac{{\overrightarrow{v}}_{feed}}{{\overrightarrow{v}}_E+{\overrightarrow{v}}_{Lift}}}\right|·h_{sep}$$

(18)

The above equation includes the horizontal velocity components due to electrophoresis $v_E$ (Equation (10)) and the lift force $v_{Lift}$. In Figure 3, the theoretical separation is shown schematically for both parameters.

A horizontal force balance on the particle in the area around the gap in the cross-flow is used to determine this absolute particle velocity:

$${\overrightarrow{F}}_E+{\overrightarrow{F}}_{L,Saffman}={\overrightarrow{F}}_{D,filt}$$

(19)

These forces are defined by Equations (4), (7) and (15) and apply in the Stokes range

$$3\pi d_P\eta \mu \overrightarrow{E}+0.807{\displaystyle \frac{d_P^3}{\eta }}{\left({\displaystyle \frac{6·\eta ·{\overrightarrow{v}}_{feed}}{D_h}}\right)}^{1.5}·{\rho }^{0.5}=3\pi d_P\eta ({\overrightarrow{v}}_{ex}-{\overrightarrow{v}}_{p,ex})$$

(20)

Dividing Equation (19) by $3\pi d_P\eta$, the equation is simplified to

$${\overrightarrow{v}}_{p,ex}={\overrightarrow{v}}_{ex}-\mu \overrightarrow{E}-0.086 d_P^2{\left({\displaystyle \frac{6·{\overrightarrow{v}}_{feed}}{D_h}}\right)}^{1.5}·{\rho }^{0.5}{\eta }^{0.5}$$

(21)

$${\overrightarrow{v}}_{p,ex}={\overrightarrow{v}}_{ex}-{\overrightarrow{v}}_E-{\overrightarrow{v}}_{Lift}$$

(22)

Inserted into Equation (17), this gives the following analytical expression for the separation height:

$$h_{sep}=\left|{\displaystyle \frac{\left({\overrightarrow{v}}_{ex}-{\overrightarrow{v}}_E-{\overrightarrow{v}}_{Lift}\right)}{{\overrightarrow{v}}_{feed}}}\right|·L_{gap}$$

(23)

This separation height can be solved for the appropriate process parameters ($v_{ex}$, $v_{feed}$, $E$), fluid and particle properties ($\rho$, $\eta$, $\mu$, $d_P$) and module geometries ($D_h$, $L_{gap}$).

In the case of ${Re}_p>0.25$, the drag coefficient can be calculated using the Morsi and Alexander correlation in Equation (6). Subsequently, the right-hand side of Equation (20) is modified as follows:

$$\begin{array}{l}3\pi d_P\eta \mu \overrightarrow{E}+0.807{\displaystyle \frac{d_P^3}{\eta }}{\left({\displaystyle \frac{6·\eta ·{\overrightarrow{v}}_{feed}}{D_h}}\right)}^{1.5}·{\rho }^{0.5}\\ ={\displaystyle \frac{\pi }{8}}\rho d_P^2{a}_1{\left({\overrightarrow{v}}_{ex}-{\overrightarrow{v}}_{p,ex}\right)}^2+{\displaystyle \frac{\pi }{8}}{d}_P{a}_2\eta \left({\overrightarrow{v}}_{ex}-{\overrightarrow{v}}_{p,ex}\right)+{\displaystyle \frac{\pi }{8}}{\displaystyle \frac{a_3{\eta }^2}{\rho }}\end{array}$$

(24)

The solution of this quadratic equation gives

$$x={\displaystyle \frac{-B+\sqrt{B^2-4AC}}{2A}}$$

(25)

With the constant values, which include the particle and fluid properties, as well as the feed velocity and the electrophoretic mobility, this yields

$$A={\displaystyle \frac{\pi }{8}}\rho d_P^2{a}_1$$

(26)

$$B={\displaystyle \frac{\pi }{8}}{d}_P{a}_2\eta$$

(27)

$$C={\displaystyle \frac{\pi }{8}}{\displaystyle \frac{a_3{\eta }^2}{\rho }}-3\pi d_P\eta \mu \overrightarrow{E}-0.807d_P^3{\rho }^{0.5}{\eta }^{0.5}{\left({\displaystyle \frac{6·{\overrightarrow{v}}_{feed}}{D_h}}\right)}^{1.5}$$

(28)

The absolute particle velocity in the gap is then calculated as follows:

$${\overrightarrow{v}}_{p,ex}={\overrightarrow{v}}_{ex}-x$$

(29)

This equation can be used to calculate the separation height $h_{sep}$ in Equation (18) at large Reynolds numbers.

3. Experimental Setup, Simulation Parameters and Materials

3.1. Experimental Setup

In order to investigate the separation and movement of particles, three experimental setups were developed. A laboratory-scale setup was employed to separate a particle suspension under constant process conditions, with a process volume of ~100 L, flow rates of several 100 L∙h⁻¹ and field strengths of up to 300 V∙cm⁻¹. In addition, two further setups were used; these differ significantly from the first setup in terms of the size, the process parameters and the operating mode. The flow rates were reduced to mL∙min⁻¹, the process volume was limited to a few mL and discontinuous processing allowed for short test times with high repeatability. Within the discontinuous setups, the particle movement was analyzed in detail and visualized to apply the observations to the separation process.

The first experimental configuration, designated as setup no. 1, is utilized for the investigations of the continuous separation process. In accordance with the method described in the previous chapter, two main flows are used: the feed and the filtrate. To ensure the continuous operating mode, both fluids are operated in independent cycles with their individual tank, pump, bypass and process control units (Figure 4). At the start of the process (timestep $t_0$), the feed cycle contains the initial particle concentration, while the filtrate consists of deionized (DI) water only. During the process, the separation occurs in the separation module and the fine fraction of particles is transported into the filtrate. The particle size distribution (PSD) and the particle concentration undergo changes over time. Figure 4 presents both a flow sheet and an image of the actual setup no. 1, with detailed technical specifications provided below.

The main cycles, comprising the feed and filtrate, are designed almost identically to achieve a uniform flow rate. A centrifugal pump (Grundfos GmbH, Erkrath, Germany, CRE 3-5) (P1&P2) is situated below each tank (T1&T2) and is connected to a control circuit that includes one volume flow sensor (ABB Asea Brown Boveri Ltd., Zurich, Switzerland, ProcessMaster FEP 311) (F1&F2) each to ensure the process conditions. The filtrate outlet is additionally monitored by a flow rate sensor (Endress+Hauser Group Services AG, Reinach BL, Switzerland, Promag H) (F3), ensuring that the mass balance in the separation module is completely known. To each outlet of the separation module, manual membrane valves are attached to regulate the internal exchange flow rate. The total pressure of both main flows is measured before they enter into the separation module using BD sensors (BD|Sensors GmbH, Thierstein, Germany, DMP 331). In addition, the differential pressure (DP) between the outlets is recorded with BD sensors (DPT 100). Conductivity (Senseca Germany GmbH, Regenstauf, Germany, Greisinger GLMU 400MP) and temperature (PT100 sensor) are monitored inside the tanks. The process temperature is controlled via a cooling loop (Lauda Dr. R. Wobser GmbH & Co. Kg, Lauda-Königshofen, Germany, Proline RP855) in the filtrate tank. Two filter elements (FEs) with a pore size of 0.1 µm can be connected to the process for the pre- and post-cleaning of the system. The process data are recorded on a separate computer using the software Labview^® 21.0, which enables the display and configuration of the process parameters online. Suspension samples are taken directly from the tanks and the PSDs are obtained by the static light scattered method (SLS, HORIBA Europe GmbH, Oberursel, Germany, Retsch Horiba LA-950).

The separation module (SM) is based on an electro dialysis module from FUMATECH BWT GmbH. The internal geometry has been modified and consists of the two main channels and a separation medium in between. Both channels are outward-limited to an ion exchange membrane, and behind this, the electrodes are installed parallel to the flow direction and are surrounded by a rinsing solution, which supplies the electrolyte (sodium sulphate—Na₂SO₄, 3 mass% solution) to the ion exchange membrane to prevent possible electrolysis when voltage is applied. The rinse fluid is also implemented in a cycle with its own tank and pump. The electrodes are connected to a laboratory power supply (Voltcraft^®, Düsseldorf, Germany, PS405Pro), which provides a DC up to 80 V. The individual inlays that define the flow channel can be selected using a stack design, and different separation media can be installed. Both significantly influence the flow properties, as well as the internal exchange flow rate and, consequently, the separation. In this context, a single-gap profile and a three-gap profile are used, where two different channel widths and depths can be set. The geometry is shown schematically in Figure 5 below.

The single-gap profile is used in the separation medium as a model geometry. This has already been used in a preliminary work [28] and is particularly suitable for analyzing the separation process and limits. The three-gap profile is utilized to investigate a scale-up approach by numbering-up. The main requirement in this case is to balance the internal flow rates to achieve homogeneous separation. This is sufficiently sophisticated in the three-gap geometry and increases as the number of gaps increases. The separation medium is made of a conductive material (stainless steel, σ ≈ 1.5 × 10⁶ S∙m⁻¹) to guarantee a homogeneous electric field. The thickness of the separation medium and the gap length are set at 1 mm in both cases, with the gaps extending across the entire depth. The dimensions are summarized in

Table 2. Geometric dimensions of the flow channels used in setup no. 1 corresponding to the labelling in Figure 5.

		Unit	Setup No. 1
			Single-Gap Profile	Three-Gap Profile
Total length	$l_{tot}$	mm	420
Gap distance	$l_{gap}$	mm		74.5
Main length	$l$	mm	306
Main width	$a$	mm	3; 6
Main depth	$b$	mm	60
In- and Outlet length	$l_{in}$; $l_{out}$	mm	58
In- and Outlet depth	$b_{in}$; $b_{out}$	mm	22

. The inlet and outlet areas, characterized by their lengths ($l_{in}$; $l_{out}$) and depths ($b_{in}$; $b_{out}$), provide the alignment and development of the flow profile. These are not influenced by the electric field and have no influence on the separation in the main separation area. The length $l_{gap}$ (shown in Figure 5b) is the distance between the individual gaps within the multi-gap profile.

The following section describes two additional setups that are used. Setup no. 2 consists of an enclosed module, while setup no. 3 incorporates optical access. These two are operated discontinuously and are used to investigate the electro-induced effects on the flow and the particles. Moreover, to determine operating limitations for the separation process experimentally, the separation medium is replaced by an open-flow geometry. A schematic representation of the resulting process is provided in Figure 6.

In the context of the experiment, two distinct flows—one containing particles (particles and DI water) and the other devoid of particles (DI water)—are introduced into a rectangular flow module uniformly, where they combine and form a laminar flow. The region in between the walls (electrodes) is considered the combined flow area for the particle movement. The acting forces are equivalent to those previously described in the context of separation process.

Figure 7 shows a process scheme with an image of the electro flow module, designated as setup no. 2. A syringe pump (Harvard Apparatus, Holliston, MA, USA, KD Scientific, KDS200) with two identical syringes provides a fluid volume of 30 mL for each inlet flow through tubes (Ø 1 mm). The flow rate is adjustable within the range of 5 to 30 mL∙min⁻¹. The module itself consists of three main components: two identical PVC bases, each with an inlet, an outlet and an electrode; and one inlay with the geometric design and dimensions (cf.

Table 3. Geometric dimensions of the two discontinuous electro flow modules.

		Unit	Setup No. 2	Setup No. 3
			Enclosed	Optical Accessible
Total length	$l_{tot}$	mm	210	110
Main length	$l$	mm	170	90
Main width	$a$	mm	3	3
Main depth	$b$	mm	1	1
In- and Outlet length	$l_{in}/l_{out}$	mm	20	10
In- and Outlet depth	$b_{in}/b_{out}$	mm	1	1

) of the rectangular flow channels, situated between the two setup parts. At the outlets, the flows are collected and subsequently measured with a particle counter (Markus Klotz GmbH, Bad Liebenzell, Germany, Abacus^® mobil fluid touch). Sample s₁ (orange) corresponds to the outlet located on the particle inlet side, and the opposite side is referred to as sample s₂ (green). In this technical implementation, the electrodes function as the walls and are also connected to a laboratory power supply (Voltcraft^®, Düsseldorf, Germany, PS405Pro).

Due to the design of the module (setup no. 2), the particle movement can only be described passively by collecting and measuring the samples at the outlets. In order to observe the particle movement, setup no. 3 was developed. It enables online particle tracking via optical accessibility. The concept with a transparent electro flow module is shown in Figure 8 below. The dimensions are listed in Table 3 above.

To access the module for optical measurement methods, the walls of the flow channel are produced from a transparent polymer (Dongguan Godsaid Technology Co., Ltd., Tangxia, Dongguan, China, RESIONE G217) via a 3D printing process (SLA, Elegoo Inc., Shenzhen, China, Saturn). Two stainless-steel plates are inserted into the manufactured module as electrodes and side wall boundaries, and the inlets and outlets are drilled. A transparent PMMA covers the module. To determine the particle movement in the module, a camera (Basler AG, Ahrensburg, Germany, Basler ace 2 R a2A3840-45ucBAS) equipped with a telecentric lens (1×) is used, with a resolution of 0.8 MP and a pixel size of 2 µm on a region of interest (ROI) of 7.7 mm × 4.3 mm. In combination with the optics and the parallelised light, the resolution of the physical and the image pixels is identical. With this chosen ROI, the entire channel width can be captured with 1500 pixels. Consequently, the size information of particles down to approximately 10 µm. can be obtained. A LED strip is used as a light source, providing backlighting. The camera system is mounted on a linear unit to enable capturing the entire channel geometry. The peripheral suppliers (syringe pump and power supply) are the same devices as in the previously described batch setup. Likewise, it is also possible to analyze the outlets with the extinction particle counter (Abacus^® mobil fluid touch) to obtain a particle size distribution.

3.2. Modelling of the Multidimensional Separation

In addition to the experiments, a numerical study is performed using computational fluid dynamics (CFD) to determine the influence of the flow and the forces on the particle trajectories, as well as to analyze the geometry and the process parameters. Furthermore, the numerical model can be used to optimize the separation.

In CFD, Navier–Stokes equations are used for fluid flow and the Newtonian motion equation for particles [57]. The k-ω-SST turbulence model [58] is utilized to calculate and identify potential turbulent regions and to ensure that the boundary layer is calculated as precisely as possible. This also allows flows to be resolved in geometries with widenings and constrictions. The motion of each spherical particle with diameter $d_P$ and mass $m_P$ is determined by the Newtonian motion equation as

$$m_P{\displaystyle \frac{d{\overrightarrow{v}}_p}{dt}}={\overrightarrow{F}}_D+{\overrightarrow{F}}_L+{\overrightarrow{F}}_E+{\overrightarrow{F}}_G+{\overrightarrow{F}}_b$$

(30)

This depends on the acting forces: drag $F_D$, lift $F_L$, electrophoretic $F_E$, gravitational $F_G$ and buoyancy force $F_b$. Particle rotation and Magnus lift forces were not considered, as it was assumed that the particles were ideal spheres moving in a uniform flow. Moreover, the effects of temperature, diffusion and Brownian motion were not considered due to process tempering and the relatively large particle sizes (~10 µm).

The drag force is given in Equation (1). Within the CFD model, the drag coefficient is calculated using the correlation by Morsi and Alexander in Equation (6).

The implementation of the lift force in CFD corresponds to the Saffman model with a general description for multidimensional vector analysis provided by Li and Ahmadi [42,59]:

$${\overrightarrow{F}}_L=1.615d_P^2{\left(\eta \rho \right)}^{{\displaystyle \frac{1}{2}}}{\displaystyle \frac{D}{{\left|D\right|}^{\frac{1}{2}}}}\left(\overrightarrow{v}-{\overrightarrow{v}}_p\right)$$

(31)

where the lift force is proportional to the 2nd power of the particle diameter and is a function of dynamic viscosity η and a quotient of deformation tensor D, the three-dimensional description of the shear rate. As a result, in contrast to Equation (7), the lift force can be determined for any position of the particle in the flow.

The electrophoretic force acting on a particle with charge $q$ in a suspension and in electric field $E$ is described in Equation (15). To implement the electrophoretic force into the CFD simulation, a user-defined function (UDF) is developed. This UDF specifies the strength of the homogeneous electric field, the viscosity of water and the electrophoretic mobility. The particle diameter is transcribed from the pure fluid simulation. The value of the electrophoretic mobility used is taken from the experimentally determined material parameter.

The geometries of the experimental setups used are generated for the simulation and mapped with a computational mesh. The objective of selecting the mesh size is to ensure that the flow is completely resolved while minimizing the computational power and time. For this purpose, two-dimensional simulations are carried out, employing the software Fluent^® (Ansys 2024 R1). The selected simulation plane within the flow geometry is located central along the main flows to analyze the flow profile and its regime in the characteristic areas of the module. In the flow module, a uniformly square mesh with an edge length of 40 µm was utilized, while the separation module employed an even finer grid of 25 µm. The grid independence analysis performed demonstrated that a decrease in the grid size exerts no significant influence on the results. To calculate the particle trajectories, five iteration steps of the particle movement were considered within a single mesh cell.

As for boundary conditions, either velocity or flow rate were used as inlets. One outlet is also set by the velocity or the flow rate and the other is specified by pressure. In this case, the pressure is set to zero and therefore refers to the ambient pressure. The wall condition is set to no slip.

The particles’ contact behaviour with the walls of the separation module is considered in the CFD simulation using the coefficient of restitution (COR) [60]. The collision experiments [61] and simulations [62] of particles’ impact on a wall were conducted using particles and fluids that were comparable to those used in the present process. Due to high viscosity forces [63,64], the COR is close to zero in all cases, which is why this value is also assumed in the CFD. Nevertheless, should a particle be in contact with the wall, it may be detached from the surface the by fluid forces.

3.3. Particle System

In the present study, different particle systems are used in the experimental setups, including soda lime beads and polystyrene monospheres. The particles are dispersed in deionized water (Hydrotec GmbH, Selb, Germany, Hydromos^® UO50W with mixed-bed demineralizer Hydroion^®) characterized by conductivity < 10 µS∙cm⁻¹ and a pH value of ~7. The water is prefiltered with a filter cartridge (Schwegmann Filtrations-Technik GmbH, Grafschaft, Germany, pore size 0.1 µm). The particle size is determined by static light scattering (Retsch Horiba LA-950) and microscopy. To analyze the particle shape, a light microscope (Leica Microsystems GmbH, Wetzlar, Germany, Leitz Orthoplan) and a SEM (Thermo Fisher Scientific Inc., Waltham, MA, USA, Phenom XL G2) were used.

The soda lime beads (72.5% SiO₂, 13.7% Na₂O) from Potters Europe were used. The particle density was 2650 kg∙m⁻³, and the refraction index was 1.6. To ensure a broad range of particle sizes, different fractions (Potters Industries Inc., Malvern, PA, USA, Spheriglass^® A-Glass 5000|3000) were selected, the distributions and SEM images of which are shown in Figure 9.

The separation task targets a particle size of less than 100 µm, and the shown PSDs contain this size class in different proportions, as well as varying maximum particle sizes. In addition, all particles exhibit a diameter greater than 1 µm, indicating that they are considered to be free of Brownian motion [65]. The selected SEM images demonstrate that the particles exhibit slight irregularities and fine asperities, but the majority are almost homogeneously spherical. For these reasons, these particle systems are suitable as a model material. Thus, different cut sizes can be realized and investigated.

Beside the broadly distributed soda lime beads, two monomodal particle systems with a diameter of 10 µm and 80 µm are used in the present investigations. Both particle systems consist of polystyrene (Lamberti S.p.A. Microbeads^®, Gallarate, Italy, Dynoseeds^® TS10 and TS80) with a density of 1050 kg∙m⁻³. Due to this small density difference compared to water, the effect of sedimentation is minimal. The size and uniformity of the monospheres can be observed in the following microscope images (Figure 10), which demonstrate their suitability for the targeted investigation of particle movement and separation effects.

Electrophoretic mobility was measured by means of the dynamic light scattering (DLS) method, whereby the movement of particles in an electric field was tracked and the speed was correlated with the Stokes and Coulomb forces [54]. The Zetasizer Nano ZS from Malvern Panalytical GmbH (Kassel, Germany) was employed. The electrochemical property can be varied by the pH value or the general ion supply (additives) of the suspension. During the measurement, it was ensured that the suspension was adapted to the process characteristics in terms of pH value and conductivity.

As electrophoretic mobility is a material parameter, Spheriglass^® A-Glass 5000, representative of all soda lime beads, and the polystyrene spheres (Dynoseeds^® TS10) were tested. The results for the material systems are shown in Figure 11.

A negative electrophoretic mobility was determined for both material systems. The mean values were determined to be −3.2 µm∙cm∙V⁻¹∙s⁻¹ (soda lime beads) and −1.9 µm∙cm∙V⁻¹∙s⁻¹ (polystyrene spheres). Consequently, it can be deduced that the particles will migrate to the positive electrode (anode) during the separation process.

4. Results

Particle motion and multidimensional separation were analyzed by experimental studies and CFD simulations. The particle motion was investigated in the batch process, providing the basis for process development regarding continuous separation. In the continuous process, primarily, the hydrodynamic separation was investigated experimentally to determine the optimal operating parameters and minimal particle cut sizes. The effect of the multidimensionality and the scale-up approaches were described by numerical simulation.

4.1. Investigation of the Particle Motion in the Discontinuous Process

In this study, discontinuous setup no. 2 was used. First, the hydrodynamics within the flow geometry was characterized. For this purpose, different inlet flow rates were evaluated for both glass beads and polystyrene particles at varying concentrations. The flow rate of the inlets was varied from 5 to 30 mL∙min⁻¹, resulting in a maximal mean inlet velocity of 0.5 m∙s⁻¹. The investigations were performed by experiments and CFD simulations. In the experiment, particle transport in hydrodynamical and electrical fields was studied passively by analyzing the particle size and concentration at the outlets. In the simulation, the movement of the particles within the module was analyzed by calculating the trajectories, and their concentration at the two outlets was determined. The particles were generated as monomodal spheres with varied sizes of 1, 5, 10, 15 or 20 µm, which corresponds to the PSD of the experimentally studied materials. In order to facilitate evaluation, 25 particles of a single monomodal fraction were generated and distributed in a uniform manner over the cross-section of the inlet v₁ (feed). In the second step, the electric field force was implemented and its additional particle movement was analyzed.

The resulting flow profiles for the minimum and maximum flow rates are shown in Figure 12. Additionally, the particle trajectories at the module split are illustrated.

For both flow rates, a stationary, laminar flow regime can be indicated, as evidenced by the uniform distribution of the velocity profile across both inlets and the combined zone. At the end of both inlets, the flow shows a free jet behaviour, and between the two combining flows, an enclosed zone is created at the wall, where the fluid is slowed down. The flow region remains unperturbed, as no backflows or vortices are observed. A stagnation zone is formed at the centre of the channel split, resulting in equal division of the flow at this point. A large area with stationary flow is consequently available for the following investigation of particle movement in the multidimensional field of forces. With regard to particle trajectories, the influence of the hydrodynamics and the lift force acting perpendicular to the wall on the particle motion can be observed. Therefore, the trajectories were evaluated and the maximum wall distance was determined. This study was carried out for 1, 5, 10, 15 and 20 µm polystyrene particles at flow rates of 5 and 30 mL·min⁻¹. The evaluation was conducted 50 mm after the inlet, where homogeneous flow conditions were observed for both flow velocities used. These conditions remained relatively constant for 115 mm until the onset of the stagnation point effect upstream. The horizontal (lift) and vertical particle movement (main flow) were then compared over this distance, and their ratio was calculated. It should be noted that the initial particle position within the channel varied in the evaluation. The results are listed in

Table 4. Particle initial wall distance at 50 mm after the inlet and the ratio of horizontal to vertical movement over 115 mm in the flow channel depending on the particle size and flow rate in setup no. 2 obtained by the CFD simulations.

Particle Diameter		1 µm	5 µm	10 µm	15 µm	20 µm
Material	Flow Rate	Initial Wall Distances [µm] Ratio of Horizontal to Vertical Path [µm·mm−1]
polystyrene	5 mL·min−1	189.81 0.0077	189.38 0.008	189.04 0.0106	189.53 0.0174	191.29 0.0304
	30 mL·min−1	150.62 0.0864	149.54 0.0858	149.75 0.0859	149.29 0.0865	150.49 0.0896

below.

The analysis shows that the initial wall distance of the particle is greater at a flow rate of 5 mL·min⁻¹ than at 30 mL·min⁻¹; this is due to the fact that an increasing velocity leads to greater inertial forces. In addition, an increase in the relative movement, defined as the ratio between the horizontal and vertical path, has been observed to occur to be a function of the flow rate and the particle size. This correlation is also included in the Saffman model in Equation (7). Since the flow gradient around the particle decreases towards the centre, a reversal will occur at a certain point. Consequently, particles fed to the left of the symmetry line cannot migrate to the outlet s₂.

In comparison to the CFD simulations, the same set of operating parameters was investigated experimentally. Figure 13 shows the particles measured at the outlets as a proportion of the total outlet particles for the soda lime beads (Spheriglass^® A-Glass 5000) and the polystyrene particles (Dynoseeds^® TS10) with a concentration of 0.01 g·L⁻¹ at an inlet flow rate of 5 mL·min⁻¹. In order to ascertain the impact of the flow rate on particle movement, the measurements with the polystyrene were performed at different flow rates up to 30 mL·min⁻¹.

In all experiments, particles can be detected in both outlets. The proportion of particles in the outlet on the feed side is always greater than on the initially particle-free side. At the lowest flow rate, the same behaviour can be observed for both particle systems, with approximately 5 to 7% of the total particle number of particles being detected in the initially particle-free DI water side. For higher flow rates, this effect does not change significantly; only at the highest flow rate of 30 mL·min⁻¹ does the proportion of the particles in the particle-free side increase slightly to 11%. Comparison with the CFD simulation shows that this effect does not occur in the flow field, but that the particles converge towards the wall. However, the real effect of the particle interaction on the flow is not considered in the CFD due to the one-way coupling. This influence can lead to microscopic fluctuations in the flow gradient and additional particle migration. The experimental process is also affected by manufacturing inaccuracies, resulting in rounded edges or slight asymmetry in the flow channels and outlets. The overall variation in the results is small; therefore, the hydrodynamic split for the different flow rates and the given geometry is sufficient and forms the basis for the following multidimensional investigations.

In the following section, the multidimensional particle dispositioning is investigated for particles in the laminar flow field with an orthogonally acting electric field. In this context, simulations are first carried out and then compared with experiments. The effect of the electrophoresis in an electric field directs the particles either towards the cathode or the anode, depending on the particle surface charge and polarity. For the studied materials, the particles migrate to the positive electrode (anode) due to their negative electrophoretic mobility (cf. Figure 11). In the simulation, this effect is applied by the UDF; in the experiment, a voltage source is connected to the electrodes. In both cases, direction is controlled by the polarity, and the magnitude of the acting force is controlled by the voltage intensity. In the following investigations within the CFD, the particle feed will be on the cathode (left-hand side) side in order to investigate the migration towards the anode (right-hand side) in the flow field.

In order to investigate the particle movement under the influence the hydrodynamical and electrical forces, a simulation was performed at a constant flow rate of 30 mL·min⁻¹ and the particle trajectories were evaluated for different sizes and at varying field strengths. Figure 14 shows the resulting 10 µm trajectories as an example for three different operating points. The first case corresponds to pure hydrodynamics, which has also been shown and described previously. The particles were assumed to be spherical, with the physical properties of soda lime beads.

As the electric field strength increases, a clear influence on the particle deflection can be observed. The particles are increasingly tracked towards outlet s₂. This is the result of the electrophoretic force, which transports the particles towards the anode due to the negative electrophoretic mobility. From this observation, it can be concluded that the ratio between the drag force and the electrophoretic force can be used to control the movement of the particles in the flow channel and thus adjust the separation. This study was also carried out for particles with sizes of 1, 5, 15 and 20 µm, and the result of the particle movement is almost identical. This demonstrates, as predicted in Section 2.3, that the electrophoretic force only has a small size dependence compared to the flow-induced drag force. If the electric field is reversed (anode at the feed inlet), all particles are transported to outlet s₁ and no particle split occurs.

This observation is compared to experimental measurements with both particle systems, where flow rate, particle concentration and electric field strengths are adjusted. First, the tests are carried out with the soda lime beads using PSD in the range of 1–20 µm. The feed suspension is prepared with two different concentrations, a low concentration of 0.01 g·L⁻¹ and a high concentration of 5 g·L⁻¹. The experiments are carried out at a constant flow rate of 5 mL·min⁻¹ and varying electric field strengths. The particle number fractions in the outlets are measured using Klotz^® Abacus^® mobil fluid touch. To determine the mass fractions, the samples are dried and weighed. The resulting size ratios are shown in Figure 15.

For the comparison to CFD, the mass concentration is considered over the range from −130 to 260 V·cm⁻¹. In the case of negative field strength, the particle fraction in the s₂ outlet initially increases with increasing field strength due to particle migration in the electric field. This is in good agreement with the simulation (Figure 14). However, from a field strength of about −100 V/cm, the distribution of particles through the two outlets becomes approximately the same and does not change further. From theory and CFD simulation, one would expect an increase in particle migration towards the s₂ outlet.

For the positive range of the electric field strength, according to the theory, the proportion of particles at outlet s₂ should be zero. There is a tendency towards this when analyzing the number of particles is analyzed, but only at low field strengths of about +20 V·cm⁻¹ does the proportion decrease before increasing again. This indicates that the electrophoretic deflection of the particles is taking place but is disturbed by other electrochemical processes in the experiment. In order to analyze the influence of flow velocity on these effects, a further series of experimental tests was carried out with the polystyrene monospheres (Dynoseeds TS10) at varying flow rates. Figure 16 shows the resulting number ratios. Each diagram represents the distribution for one flow rate: 5 mL·min⁻¹ (a) and 30 mL·min⁻¹ (b).

In Figure 16a, the graph of the particle number ratio is almost identical to the experiment with glass beads. This is due to the similar electrophoretic mobility and particle size range of polystyrene and soda lime beads. It also shows that the lower density of the polystyrene particles does not affect this behaviour. With increasing flow velocity, the equalizing of the distribution effect is only accomplished at higher field strengths. This phenomenon is not reached in the experiments due to the limited measurement range of the velocity, but there is a tendency for the point of mixing to occur at higher field strengths. It can therefore be seen that mixing is not only dependent on the electric field strength, but also on the flow rate and thus the test duration, or the interaction of the two. On the other hand, an increased particles proportion at output s₁ can also be seen at 50 V·cm⁻¹. This strongly indicates that the interfering effects are not dominant at low electric field strengths.

For a more detailed examination of the particle movement, setup no. 3 is used subsequently. This enables the particles in the flow to be visualized and correlations between the process variables to be observed. Figure 17 below shows real images inside the flow channel near the outlets. The Dynoseeds TS10 are used, the concentration was increased to 5 g·L⁻¹ in the experiments to obtain a strong contrast between the two phases. The images shown were recorded with 16-bit grayscale information. The areas of solids or phase boundaries appear dark due to the different transmission properties of the materials used.

In all images the flow direction is upwards and the particle inlet is on the left-hand side. In Figure 17b,c, the anode is set to the left wall, according to the previously defined nomenclature, and the electric field is positive; in (d) and (e), it is reversed. For the hydrodynamic case in (a), it can be seen that the flow regime follows the laminar pattern and the particles are almost exclusively on the inlet side. In the other images, a clear difference can be identified, particularly with regard to the formation of bubbles within the channel. This process is referred to as electrolysis [66], whereby water is decomposed into its constituent elements by the electric current. The formation of oxygen (O₂) bubbles occurs at the anode, while hydrogen (H₂) bubbles form at the cathode. The formation of bubbles on the electrodes and their growth to a specified size exerts a significant impact on the flow characteristics. The cross-section of the flow and thus the flow field is changed. The narrowing of the main flow, the flow around the bubbles and the backflow regions after the bubbles show a strong influence on the particle motion. The number and maximum size of the bubbles increase as the electric field strength increases, which also increases their influence on the flow. By comparing the images with reversed field orientation, it can also be seen that the electrophoretic force has an influence on the particle position in the channel. In Figure 17b, the concentration of particles (based on grey value) near the wall is higher than in (d) and (e), the particles are driven to the anode as was shown in the simulation. This effect is especially noticeable by comparing the operating point at −100 V·cm⁻¹ (Figure 17d) and Figure 14 in the middle). The images provide a qualitative description of the current-induced effects. The time-dependent bubble formation and the growth rate are not directly included in this observation, as the time of recording varies. Nevertheless, they provide an explanation for the mixing behaviour in the experimental series of measurements shown above.

For the separation process in setup no. 1, ion exchange membranes are required to prevent the bubble growth resulting from electrolysis. These semi-permeable membranes are selective for certain ions and impermeable to particles and the continuous phase within the main channels. For reduction or oxidation, a combination with an additional rinse channel of an electrolyte solution is required between the electrodes and the ion exchange membrane (see Figure 4b). It is also possible to go below the dissociation voltage of water, but this would significantly reduce the electric field strength and therefore not be suitable for use in the separation module.

Another electro-induced effect in liquids is electro-osmosis. Due to the dipole interactions of the water molecules, a fluid movement is induced, and it always flows from the cathode to the anode. This movement can certainly help the process, on the one hand by separating the charge of the particles and on the other hand by influencing the internal flow rate. However, the impact of electro-osmosis on the process requires detailed investigation.

4.2. Investigation on Continuous Separation with Scale-Up Approaches

An essential step in process optimization is scale-up. The key scale-up factors that contribute to enhancing the overall process efficiency are the increase in the particle feed concentration and the flow rate, in addition to time- and energy-optimized operation. For this purpose, a theoretical approach based on a scale-up by numbering-up is shown to increase the separation efficiency. The basis for this is provided by investigations of the hydrodynamic separation in setup no. 1 using CFD simulations and experimental approaches [26]. As a result of systematic studies using CFD, a single-gap geometry was implemented to investigate the hydrodynamic separation. In subsequent experiments, it was shown that the separation grain size decreases with decreasing exchange flow rates between the feed and the filtrate channel, while the overflow velocity remains constant. Figure 18 shows these results for an overflow velocity of approx. 1 m·s⁻¹ with the particle collective (Spheriglass^® A-Glass 3000) and an initial concentration of 2.5 g·L⁻¹.

Each shown PSD corresponds to the final particle size distribution in the filtrate at one experimental set exchange flow rate. The upper separation limit is defined with the d₉₉ value of the filtrate PSD and can be compared to the feed PSD, which remains coarse at the end of the experiments. The results show that the hydrodynamic separation is verifiable to a separation size of approx. 25 µm. A further reduction in the exchange volume flow through improved operating parameters can lead to sharper fractionation. The increasing velocity gradient near the wall is followed by a reduced local particle size which can flow through the separation medium into the filtrate. However, in this setup interference of the process parameters occur. For the main flow rates, the coefficient of variation is >0.5%, but for the exchange flow rates this value can rise to 25–50% due to the general low flow.

In the scale-up approach, the single-gap geometry in setup no. 1 is extended to a three-gap geometry, and the distance between the gaps is equally distributed (cf. Figure 5b and Table 2). The channel width is also extended to 2.5 mm, in comparison to the single-gap profile, which has a channel width of 1 mm. As a result of the expansion, the overflow velocity and the corresponding Reynolds number decrease, enabling higher flow rates and particle concentrations to be utilized in the long term.

In order for the separation to be consistent, the flow conditions in all gaps must be identical. For this purpose, CFD simulations are used to determine the distribution of the exchange flow rate across the three gaps and thus to analyze an appropriate operating point. Firstly, the distribution of the flow rate in the individual gaps was analyzed for different total exchange flow rates. The inlet flow rates were chosen to be constant (${\dot{\mathrm{V}}}_1$ = ${\dot{\mathrm{V}}}_2$ = 216 L·h⁻¹) and the filtrate outlet (${\dot{\mathrm{V}}}_3$) was varied. The difference between the filtrate inlet (${\dot{\mathrm{V}}}_2$) and its outlet flow rate (${\dot{\mathrm{V}}}_3$) is referred to as the total exchange flow rate. This corresponds to the sum of all flow rates between the feed and the filtrate channels through the individual gaps. The feed outlet ($\mathrm{p}$) is set to a pressure of 0 bar as a reference pressure to the atmosphere and remains unchanged in all simulations. For each gap, an exchange flow rate ratio is formed, representing the proportion of the flow rate through the corresponding gap to the total exchange flow. Figure 19 shows the results as a ratio of the total exchange flow rate. The gaps are named in accordance with the direction of flow, with the first gap situated closest to the inlet.

The results show that the distribution across the gaps varies significantly. The volume flow rate ratios are almost independent of the total exchange flow rate, whereby the flow rate ratio through gap_3 is almost 0.75. This would result in the separation being dominated by this gap and a theoretically coarser separation would occur. Also, a further increase in the total exchange flow rate will not be considered as the separation grain size decreases as the exchange volume flow rate decreases. In order to equalize the gap flow rates, the boundary conditions can be adjusted. In the following steps, an exemplary total exchange flow rate of 9 L·h⁻¹ is selected and the feed inlet flow rate ($\dot{{\mathrm{V}}_1}$) is varied. This approach can be implemented well on both the CFD and the experimental sides. Optionally, the pressure at the feed outlet can also be regulated. Figure 20 shows the effect of an adjustable inlet feed flow rate on the individual gap flow rates.

The diagram illustrates the convergence of the gap flow rates as the inlet volume flow increases. The ratio in the last gap is observed to be linearly regressive, while the ratios in the previous gaps demonstrate a linear increase. The data allow for the generation of a linear equation for each gap, which demonstrates the flow rate ratio for a given exchange volume flow and varying feed inlet flow rate. It is regrettable that there is no point of intersection at which the three-gap flow rates are precisely equivalent. This indicates that the separation cannot be consistent across all gaps, thereby limiting the upper separation grain size by the maximum gap flow rate when selecting the operating point. This general procedure can also be carried out for the other exchange flow rates. The linear relationships between the controlled variables can also be found there. These correlations can then be used to predict the working range for a homogeneous separation by surface interpolation of the linear equations. This approach is visualized in Figure 21.

The stronger marked lines shown correspond to the linear equations created from the results of the individual simulations. The planes through these lines were generated using a cubic interpolation method. In this observation, also no operating point can be derived, at which all gap flow rates are equal. Instead, the intersection of the planes of gap_1 and gap_3 can be identified as an operating range in which the classification is almost homogeneous. Additionally, it can be stated that at low total exchange flow rate, high experimental control accuracy is required, as deviations in the inlet feed flow rate have a large impact on the individual gap flows.

In order to conduct a comparative experiment, an operating point was set with a feed inlet flow rate of 237 L·h⁻¹ within the processing range for the three gaps. This was then compared with the single-gap experiment at even inlet flow rates, at 216 L·h⁻¹. For both experiments, the particle concentration was set to 3.75 g·L⁻¹ and the process time was set to 120 min. At the conclusion of the test period, samples were obtained and a mass fraction was determined by drying the suspension. The analysis yielded a final mass concentration of approximately 15% of the feed concentration for the single-gap geometry. In the case of the three-gap geometry, this proportion is higher at approximately 25%. The effect of the scale-up can be achieved in this aspect; however, in further investigations, the separation must be evaluated. To analyze the separation, the PSD of the starting feed and the filtrate at the end were determined. These distributions are shown in Figure 22; for comparison,

Table 5. Comparison of resulting PSDs and filtrate mass concentration after 120 min of the different experiments with the single-gap and three-gap separation geometry with a feed particle concentration of 3.75 g·L⁻¹.

	d10/µm	d50/µm	d90/µm	d99/µm	wt.% Filtrate/g·L−1
single-gap	~5	~10	~16	~26	~0.55
three-gap	~7	~15	~30	~45	~0.93

also shows selected values from the experiments.

Both experiments show a finer distribution of the PSD in the filtrate compared to the feed distribution. Accordingly, a separation can be demonstrated in both cases. However, the proportion of larger particles is higher in the three-gap than in the single-gap geometry. The deviation that can be recognized for large particles can be explained by the non-ideal distribution of the internal flows, in which the largest exchange flow rate determines the classification quality. Further experiments with a broad parameter study are necessary to ascertain the extent to which the separation can be optimized. The CFD-based parameter study has, however, already shown that it is also valid for future process variations.

5. Conclusions

In this work, multidimensional separation was investigated in a cross-flow with a superimposed electric field. Three different setups were developed and used for this purpose. Hydrodynamic separation according to particle size in the laminar flow regime could be both described in the batch process and implemented in the continuous system. The deviations between the CFD simulations and the experiments can be explained by non-idealities and edge effects in the separation modules. The continuous setup is also subject to slight fluctuations due to the complex control system. The extension of the separation geometry with additional exchange channels shows the possibility of scaling-up with almost the same theoretical separation particle size.

The effects of the electric field on the particle movement were investigated in detail using the batch setup. The particle motion in a multidimensional force field is influenced by a number of factors. In addition to hydrodynamic and electrophoretic forces, the electrochemical effects of the continuous phase, such as electrolysis and gas bubble formation, become important at higher electric field strengths and disturb both the flow and the homogeneous electric field. To prevent and counteract such effects, an ion exchange membrane and a rinse channel with electrolyte must be implemented. In addition, the movement of the fluid due to electro-osmosis can probably not be completely prevented. Further research is underway to explore the effect of additional electrokinetic effects on the separation process and implement the ion exchange membranes.