Study on Dynamic Characteristics of Magnetic Coagulation of Fe-Based Fine Particles in Iron and Steel Industry

Abstract

Fine dust, represented by Fe-based fine particles and emitted from the production process of the iron and steel industry, is the primary factor causing many diseases represented by industrial pneumoconiosis, and ultra-low dust emission has always been a thorny problem to be solved urgently. To explore the magnetic coagulation effect of Fe-based fine particles in the magnetic field when removing them from industrial flue gas by the magnetic field effect in the iron and steel industry, using FLUENT software, magnetic dipole force was added between particles through user defined function (UDF) based on the computational fluid dynamics-discrete phase model (CFD-DPM) method so that the collision process of particles was then equivalent to their mutual trapping process. Next, the effects of particle size, particle volume fraction, external magnetic field strength, and particle magnetic susceptibility on the magnetic coagulation process were comprehensively studied. Meanwhile, the proton balance equation (PBE) was solved using the partition method on the basis of the computational fluid dynamics-population balance model (CFD-PBM) to compare the coagulation removal effect under random and aligned orientations of magnetic dipoles, respectively. The results showed that the magnetic coagulation strength under the random orientation of magnetic dipoles was greater than that under the aligned orientation. When the particle size of Fe-based fine particles increased from 0.5 μm to 1.5 μm, the magnetic coagulation coefficient decreased from 0.5414 to 0.2882, and the difference in the removal efficiency under the two different orientations of magnetic dipoles became smaller. When the particle volume fraction increased from 0.01 to 0.03, the magnetic coagulation coefficient increased from 0.2353 to 0.5061, and the difference in the removal efficiency under two orientations was enlarged. When the applied external magnetic field strength increased from 0.5 T to 1.0 T, the magnetic coagulation coefficient increased from 0.3940 to 0.5288, and the magnetic susceptibility increased from 0.0250 to 0.0500, the coagulation coefficient increased from 0.3940 to 0.5288, and the difference under two orientations basically stayed unchanged.

1. Introduction

China is in a rapid stage of industrial development, accompanied by deep-seated conflicts and problems such as production overcapacity, serious repeated construction, and increased pressure on industrial environmental protection. As a big steel producer, China has ranked first in the world steel production since 1996 [1]. As a resource-intensive and energy-intensive industry, the iron and steel industry shows a large amount of energy consumption and pollutant emission in the production process. Compared with other industries, it is a typical heavily polluting industry [2,3], becoming one of the key targets of air pollution control in China. Especially, the emitted fine particles with a diameter less than or equal to 2.5 μm (PM₂.₅ for short) can be carriers of other pollutants because of their large quantity, slow settling speed, and large specific surface area, having a serious impact on the quality of the atmospheric environment [4]. It is worth noting that PM₂.₅ is also an important inducement for the development of various respiratory diseases [5]. Therefore, it is urgent to solve the problem of particulate pollutant emission in heavily polluting industries such as the iron and steel industry.

The filter material filtration method has been widely used in the field of industrial flue gas purification by virtue of its high particle trapping efficiency and a wide applicable range of particle sizes [6]. To further improve particle trapping performance, the fiber cross-sectional shape (profiled fiber) [7,8,9] and fiber diameter [10] can be changed from the fiber level. From the level of filtering media, the internal fiber distribution density [11], fiber axial curvature [12], fiber bending degree [13], fiber arrangement structure [14], and the geometric structure of filtering media can be changed, in which the filtering media mainly include flat plate type [15], fold type [16], and topological structure type [17]. From the level of filter materials, the surface of the traditional filter material can be loaded with nanofibers [18] or the planar filter material can be designed as folded filter material [19]. When the above methods are used for improvement, however, the particles in the range of 0.1–0.5 μm [20,21,22] show an unobvious Brownian diffusion effect and inertial collision effect, and thus fine particles escape, a problem that cannot be radically solved yet.

By analyzing the physical properties of the dust produced in the production process, it was found that after the raw Fe ore is crushed and steam-condensed at high temperature, there are many metal components in the particles, and the content of Fe element is much higher than that of other elements [23,24], and Fe magnetic substances such as Fe₃O₄ and Fe₂O₃ are the main components of the dust [25]. Therefore, it can be seen that many easily magnetized Fe-based fine particles are produced in the production process of the iron and steel industry, which can be magnetically coagulated by the magnetic field effect and then removed. In the traditional flow field, the motion state of particles after collision depends on the relative magnitude of drag force, gravity, viscous force (Van der Waals force), and collision elastic force [26]. When d_p < 30 μm, the viscous force, represented by Van der Waals force, is dominant in this region. When particles collide or contact with each other, the collision coagulation effect occurs [27]. When a uniform magnetic field is added to the flow field, the particles in the flow field are not only subjected to drag force and gravity but also to magnetic dipole force before collision. When d_p < 10 μm, the ratio of magnetic dipole force to drag force is less than 1, and the ratio to gravity exceeds the order of magnitude of 10² [28]. Although the magnetic dipole force is smaller than the drag force, they are in the same order of magnitude. Before particles collide with each other, the existence of magnetic dipole force accelerates the collision coagulation effect. When particles collide with each other, the colliding particles are subjected to the combined action of five forces: drag force, gravity force, viscous force, collision elastic force, and magnetic dipole force, and the existence of magnetic dipole force enhances the collision coagulation effect. Chinese and foreign scholars have done a lot of research on the magnetic coagulation process of ferromagnetic particles in the magnetic field.

Based on the magnetic dipole theory, Ku [29] analyzed the magnetic potential energy characteristics of magnetic particles in the magnetic field, derived the expression of the interaction force between magnetic particles, and simulated their dynamic characteristics. The results showed that magnetic particles are coagulated along the direction of external magnetic field and arranged in a chain structure. Ke [30] numerically simulated the dynamic characteristics of magnetic particles in vertical and horizontal uniform magnetic fields. The results revealed that the external magnetic field has a significant influence on the dynamic characteristics of magnetic particles, and the particles form some fragmented chains along the flow direction, which are continuously elongated and aligned, and finally approach a stable state. Senkawa [31] studied the magnetic interaction between two magnetic particles and obtained different expressions of the magnetic interaction between two magnetic particles. Karvelas [32] explored the magnetic coagulation behavior of Fe₃O₄ nanoparticles in a uniform magnetic field, and the results showed that Fe₃O₄ nanoparticles will form a chain structure under the action of the magnetic field. Hleis [25] investigated the particle deposition process in a high-gradient magnetic field. The results showed that under the conditions of low external magnetic field strength and low flow rate, the particles will form chain structures or chain beams on the way to the magnetic filter media. It was found that the ferromagnetic particles in the external magnetic field, no matter whether it is a uniform magnetic field or a gradient magnetic field, will form a chain structure under the action of the magnetic dipole force. In this study, therefore, the dynamic process of ferromagnetic particles coagulated into chains is not discussed, and the magnetic coagulation effect and strength of Fe-based fine particles in a uniform magnetic field were mainly studied.

When the magnetic coagulation dynamic process of Fe-based fine particles was calculated using the CFD-DPM method, the force on the two particles colliding with each other was analyzed, and the particle motion equilibrium equation in the composite field composed of flow field and magnetic field was established. The magnetic coagulation process could be analogized to the trapping process of one particle to another. The magnetic dipole force between particles was added by UDF, and a binary collision model was established for calculation, aiming to explore the effects of particle size, particle volume fraction, external magnetic field strength, and particle magnetic susceptibility on the magnetic coagulation process. To study the effect of the magnetic dipole orientation on magnetic coagulation and removal, moreover, the CFD-PBM method was used to calculate the dynamic process of magnetic coagulation and removal of Fe-based fine particles, and the magnetic coagulation kernel function was added to the PBM module through UDF, and the magnetic coagulation removal effects under random orientation and aligned orientation of magnetic dipoles were compared. This study has certain guiding significance for the subsequent magnetic coagulation removal of Fe-based fine particles.

2. Physical Model and Boundary Conditions

Figure 1 shows the Fe-based fine particle collision model. The collision process of Fe-based fine particles in a uniform magnetic field can be regarded as a particle-to-particle trapping process, as shown in Figure 1a. It can be seen from Figure 1b that the direction of the uniform magnetic field was consistent with the movement speed of the inlet particles, and the gravity was in the -Z axis direction, the model inlet was a velocity inlet with uniform velocity distribution, the model outlet was a free outflow, the air density was 1.225 kg/m³, the periphery boundary was set as a wall, and a no-slip boundary condition was adopted for the core particle surface [33]. When the length and height of the calculation domain were greater than 5 times and 1.5 times of the core particle diameter, a stable flow field was formed in the calculation area without affecting the particle trapping efficiency [34,35]. Therefore, the distance from the inlet and outlet of the model to the core particle center was 3.5D, the width and height were 3D, and the core particle size was the same as that of the inlet particle.

In addition, taking converter ash and refined ash as examples, Figure 2 shows the magnetic characteristics of Fe-based fine particles emitted during steelmaking in the iron and steel industry. As seen from Figure 2a, Fe-based fine particle XRD testing showed that Fe-based fine particles contained Fe₂O₃, Fe₃O₄ and other ferromagnetic substances, which had the characteristics of easy magnetization in the magnetic field. Figure 2b shows the VSM curve of converter ash and refined ash. The VSM hysteresis loop shows that when the external magnetic field intensity reached 0.5 T, the specific saturation magnetization of converter ash and refined ash was 22.5 emu/g and 3.32 emu/g, respectively. The calculated magnetic susceptibility of the two kinds of dust was 0.1363 and 0.01989, respectively, indicating that the magnetic properties of converter ash are greater than those of refined ash, and it is more easily magnetized in the magnetic field.

3. Numerical Calculation Equations

3.1. Governing Equations for the Fluid Phase

The Euler–Euler double-fluid model was used as the gaseous field model. The Reynolds number was calculated as $Re=ud/\nu <1$, categorized as a laminar flow ($u=0.1 \mathrm{m}/\mathrm{s}$, $0.5 \mathsf{\mu }\mathrm{m}\le \mathrm{d}\le 2.5 \mathsf{\mu }\mathrm{m}$, $\nu =15.70\times {10}^{-6} {\mathrm{m}}^2/\mathrm{s}$), with the continuity equation and momentum equation as follows [15]:

$$\frac{\partial \left(\alpha \rho u\right)}{\partial t}+\nabla ·\left(\alpha \rho u\right)=0$$

(1)

$$\frac{\partial \left(\alpha \rho u\right)}{\partial t}+\nabla ·\left(\alpha \rho uu\right)=-\alpha \nabla P+\alpha \rho g+\nabla ·\tau -F$$

(2)

where $\alpha$ is the volume fraction; $\rho$ is the density of the fluid, kg/m³; $u$ is the velocity of the fluid, m/s; $\nabla$ is the Hamilton operator; $P$ is the pressure of the grid cell, Pa; $\tau$ is the fluid adhesion stress tensor; $g$ is the gravitational acceleration, m/s²; and $F$ is the resultant force on the grid cells, Pa.

3.2. Governing Equations for the Particles

In a gas-solid two-phase flow, the motion of discrete phase particles is described by Newton’s second law, and the motion equilibrium equation of particles is expressed as follows [14]:

$$m_p\frac{d{\overrightarrow{u}}_p}{d_t}={\overrightarrow{F}}_D+{\overrightarrow{F}}_B+{\overrightarrow{F}}_g+{\overrightarrow{F}}_{other}$$

(3)

$${\overrightarrow{F}}_D{=m}_p\left(\frac{18\mu }{{\rho }_p{d_p}^2{C}_c}\left(\overrightarrow{u}-{\overrightarrow{u}}_p\right)\right)$$

(4)

$${\overrightarrow{F}}_B{=m}_p\left(G_i\sqrt{\frac{\pi S_0}{∆t}}\right)$$

(5)

$${\overrightarrow{F}}_g{=m}_p\left(\frac{\overrightarrow{g}\left({\rho }_p-\rho \right)}{{\rho }_p}\right)$$

(6)

$$Kn=2\lambda /d_p$$

(7)

$$C_c=1+Kn\left(1.257+0.4e^{-1.1/Kn}\right)$$

(8)

$$S_0=216\nu k_{b}T/{\pi }^2{{\rho }_p}^2{d_p}^5{C}_c$$

(9)

where $F_D$, $F_B$, and $F_g$ are respectively the drag force, Brownian force, and gravity, N; $m_p$ is the mass of particles, kg; $u_p$ is the velocity of the particle, m/s; $\nu$ is the dynamic viscosity of the fluid, Pa·s; $G_i$ is a Gaussian random function with a mean of 0 and a variance of 1; $S_0$ is the spectral intensity of noise, m²/s³; $Kn$ is the particle Knudsen number; $C_c$ is the Cunningham correction factor; $d_p$ is the particle diameter, μm; $k_b$ is the Boltzmann constant; $\lambda$ is the average free path of air molecules, nm; ${\rho }_p$ is the density of particle, kg/m³; $F_{\mathit{\text{other}}}$ includes negligible forces, such as pressure gradient force, Bassett force, virtual mass force, and Magnus force, N, and $F_{\mathit{\text{other}}}$ = 0 by default.

The force of Fe-based fine particles under the action of magnetic dipole force was analyzed, and the force diagram is shown in Figure 3. The motion equilibrium equation of particles is expressed as follows:

$$m_p\frac{d{\overrightarrow{u}}_p}{d_t}={\overrightarrow{F}}_D+{\overrightarrow{F}}_B+{\overrightarrow{F}}_g+{\overrightarrow{F}}_{\mathit{\text{Magnetic dipole force}}}+{\overrightarrow{F}}_{\mathit{\text{other}}}$$

(10)

(1) Magnetic dipole force in three-dimensional space

In the compound field composed of flow field, magnetic field, and gravity field, Fe-based fine particles will form magnetic dipoles after being magnetized under the action of the uniform magnetic field. The schematic diagram magnetic dipole forces between Fe-based fine particles is shown in Figure 4.

The magnetic dipole force between the two particles is decomposed along the radial direction and tangential direction, and the calculation formulas for the radial force ${F_r}^{\prime }$ and tangential force ${F_{\theta }}^{\prime }$ of particle $j$ on particle $i$ are as follows [36]:

$$\left\{\begin{array}{c}{F_r}^{\prime }=\frac{3{\mu }_0{m}_i{m}_j\left[3\cos {{\theta }^{\prime }}_i\cos {{\theta }^{\prime }}_j-\cos \left({\phi }_i-{\phi }_j\right)\right]}{4\pi r^4}\hfill \\ {F_{\theta }}^{\prime }=-\frac{3{\mu }_0{m}_i{m}_j\sin \left({{\theta }^{\prime }}_i+{{\theta }^{\prime }}_j\right)}{4\pi r^4}\hfill \end{array}\right.$$

(11)

where the permeability of vacuum is ${\mu }_0=1.256\times {10}^{-6} \mathrm{N}/{\mathrm{A}}^2$; $m_i$ and $m_j$ are the magnetic dipole moment of particle $i$ and $j$, respectively, A·m²; and r is the linear distance between two particles, m. Since the magnetic field is consistent with magnetic dipole moment in direction, ${\phi }_i={\phi }_j=\phi$, ${{\theta }^{\prime }}_i={{\theta }^{\prime }}_j={\theta }^{\prime }$, and $m_i=4/3\pi {\left(d_i/2\right)}^3{M}_i$, $m_j=4/3\pi {\left(d_j/2\right)}^3{M}_j$, so the calculation formulas for the radial force ${F_r}^{\prime }$ and tangential force ${F_{\theta }}^{\prime }$ of particle $j$ on particle $i$ can be simplified as follows:

$$\left\{\begin{array}{c}{F_r}^{\prime }=\frac{\pi {\mu }_0{m}_i{{m_{j}d}_i}^3{d_j}^3\left(3{cos}^2{\theta }^{\prime }-1\right)}{48r^4}\\ {F_{\theta }}^{\prime }=-\frac{\pi {\mu }_0{m}_i{{m_{j}d}_i}^3{d_j}^3\mathit{sin}2{\theta }^{\prime }}{48r^4} \end{array}\right.$$

(12)

where $d_i$ and $d_j$ are the particle sizes of particle $i$ and $j$, respectively.

In addition, the saturation magnetization of the two colliding particles is the same; that is, $M_i=M_j=M_p$, and the final calculation formulas for the radial force ${F_r}^{\prime }$ and tangential force ${F_{\theta }}^{\prime }$ of particle $j$ on particle $i$ can be expressed as follows:

$$\left\{\begin{array}{c}{F_r}^{\prime }=\frac{\pi {\mu }_0{M_p}^2{d_i}^3{d_j}^3\left(3{cos}^2{\theta }^{\prime }-1\right)}{48r^4}\\ {F_{\theta }}^{\prime }=-\frac{\pi {\mu }_0{M_p}^2{d_i}^3{d_j}^3\mathit{sin}2{\theta }^{\prime }}{48r^4} \end{array}\right.$$

(13)

According to Newton’s third law, the force action is mutual, and the calculation formulas for the radial force ${F_r}^{\prime }$ and tangential force ${F_{\theta }}^{\prime }$ of particle $i$ on particle $j$ can be expressed as follows:

$$\left\{\begin{array}{c}{F_r}^{\prime }=-\frac{\pi {\mu }_0{M_p}^2{d_i}^3{d_j}^3\left(3{cos}^2{\theta }^{\prime }-1\right)}{48r^4}\\ {F_{\theta }}^{\prime }=\frac{\pi {\mu }_0{M_p}^2{d_i}^3{d_j}^3\mathit{sin}2{\theta }^{\prime }}{48r^4} \end{array}\right.$$

(14)

When the direction of the uniform magnetic field and inlet particle are parallel along the Z axis, particles attract each other due to the magnetic attraction at $3{\mathrm{c}\mathrm{o}\mathrm{s}}^2{\theta }^{\prime }-1>0$, and particles repel each other due to the magnetic repulsion at $3{\mathrm{c}\mathrm{o}\mathrm{s}}^2{\theta }^{\prime }-1<0$. The radial and tangential components of the magnetic dipole force under the Cartesian coordinate system are expressed as follows:

$$\left\{\begin{array}{c}{F}_{M\left(x\right)}=F_r\sin \theta \sin \phi -F_{\theta }\cos \theta \sin \phi \hfill \\ F_{M\left(y\right)}=F_r\sin \theta \cos \phi -F_{\theta }\cos \theta \cos \phi \hfill \\ F_{M\left(z\right)}=F_r\cos \theta +F_{\theta }\sin \theta \hfill \end{array}\right.$$

(15)

According to $\sin \theta =\sqrt{x^2+y^2}/r$, $\cos \theta =z/r$, $\sin \phi =x/\sqrt{x^2+y^2}$, and $\cos \phi =y/\sqrt{x^2+y^2}$, the magnetic dipole force equation under the Cartesian coordinate system is expressed as follows:

$$\left\{\begin{array}{c}{F}_{M\left(x\right)}=\frac{\pi {\mu }_0{M_p}^2{d_i}^3{d_j}^3\left(x^2+y^2-4z^2\right)·x}{48{\left(x^2+y^2+z^2\right)}^{7/2}}\hfill \\ F_{M\left(y\right)}=\frac{\pi {\mu }_0{M_p}^2{d_i}^3{d_j}^3\left(x^2+y^2-4z^2\right)·y}{48{\left(x^2+y^2+z^2\right)}^{7/2}}\hfill \\ F_{M\left(z\right)}=\frac{\pi {\mu }_0{M_p}^2{d_i}^3{d_j}^3\left({3x}^2+3y^2-2z^2\right)·z}{48{\left(x^2+y^2+z^2\right)}^{7/2}}\hfill \end{array}\right.$$

(16)

(2) Motion equilibrium equation of particles

The forces acting on Fe-based fine particles in the composite field composed of flow field, magnetic field, and gravity field mainly include gas drag force, gravity force, Brownian force, and magnetic dipole force, and the motion equation of the particles in the three-dimensional space is expressed as below [36]:

$$\left\{\begin{array}{c}{m}_p\frac{d^{2}x}{d{t}^2}=F_{Dx}+F_{Mx}+F_{Bx}\hfill \\ m_p\frac{d^{2}y}{d{t}^2}=F_{Dy}+F_{My}+F_{By}\hfill \\ m_p\frac{d^{2}z}{d{t}^2}=F_{Dz}+F_{Mz}+F_{Bz}-m_{p}g\hfill \end{array}\right.$$

(17)

where $F_{Dx}$, $F_{Mx}$, and $F_{Bx}$ are the drag force on particles, the magnetic dipole between particles, and the Brownian force they bear along the X axis, N; $F_{Dy}$, $F_{My}$, and $F_{By}$ are the drag force on particles, the magnetic dipole between particles, and the Brownian force along the Y axis, N; $F_{Dz}$, $F_{Mz}$, and $F_{Bz}$ are the drag force, the magnetic dipole between particles, and the Brownian force on particles along the Z axis, N.

(3) Calculation method of coagulation coefficient

Figure 5 shows the motion trajectory of particle $j$ relative to core particle $i$ in three-dimensional space from different initial positions in space, and all relative initial positions where particle $j$ can collide with core particle $i$ in unit time are determined. In this case, such initial position points ((x_j0, y_j0, z_j0), (x_j1, y_j1, z_j1)…(x_jn, y_jn, z_jn)) constitute an effective collision scope in the three-dimensional space, and its volume is the coagulation coefficient between particles $i$ and $j$.

3.3. Calculation of the Magnetic Coagulation of Fe-Based Fine Particles Based on CFD-PBM

When calculating the magnetic coagulation process of particles in the composite field composed of flow field, magnetic field, and gravity field based on CFD-PBM, the coagulation cores of all coagulation types of particles in the flow field can be introduced into the PBM model in the form of UDF for calculation, and then the particle equilibrium equation can be solved by the partition method. For the magnetic coagulation core of Fe-based fine particles in a magnetic field, Kumar [37] has proposed the calculation formula for the magnetic coagulation core of particles in the free molecular zone and the continuous zone, as follows:

(1) Anisotropic magnetic coagulation core

$${\beta }_{mag,co}^{sat}=\frac{2^{4/3}}{3^{7/6}}{\left[\frac{m_i^{sat}{m}_j^{sat}{\mu }_0}{\pi }\right]}^{1/3}{\left(k_{b}T\right)}^{2/3}\frac{1}{\mu }\frac{\left(d_i+d_j\right)}{d_i{d}_j}$$

(18)

$${\beta }_{mag,fm}^{sat}=3^{1/6}{2}^{1/3}{\left[\frac{m_i^{sat}{m}_j^{sat}{\mu }_0}{\pi }\right]}^{1/3}{\left(k_{b}T\right)}^{1/6}\frac{1}{{{\rho }_p}^{1/2}}{\left[\frac{1}{{d_i}^3}+\frac{1}{{d_j}^3}\right]}^{1/2}\left(d_i+d_j\right)$$

(19)

(2) Isotropic magnetic coagulation core

$${\beta }_{mag,co}^{sat}=\frac{2^{5/3}}{9}{\left[\frac{m_i^{sat}{m}_j^{sat}{\mu }_0}{\pi }\right]}^{1/3}{\left(k_{b}T\right)}^{2/3}\frac{1}{\mu }\frac{\left(d_i+d_j\right)}{d_i{d}_j}$$

(20)

$${\beta }_{mag,fm}^{sat}=\frac{2^{2/3}}{3^{2/3}}{\left[\frac{m_i^{sat}{m}_j^{sat}{\mu }_0}{\pi }\right]}^{1/3}{\left(k_{b}T\right)}^{1/6}\frac{1}{{{\rho }_p}^{1/2}}{\left[\frac{1}{{d_i}^3}+\frac{1}{{d_j}^3}\right]}^{1/2}\left(d_i+d_j\right)$$

(21)

where T is the flow field temperature, K; $m_i^{sat}$ and $m_j^{sat}$ are the saturation magnetic moments of particle $i$ and $j$, respectively, A × m².

Whether particles are located in the free molecular zone or continuous zone can be judged according to $Kn=2\lambda /d_p$ [14]; according to Allen and Raabe [38], the average free path $\lambda$ of air is calculated as follows:

$$\lambda ={\lambda }_0\frac{p_0}{p}{\left(\frac{T}{T_0}\right)}^2\frac{T_0+T_s}{T+T_s}$$

(22)

where T₀ = 296.15 K; p₀ = 1.013 × 10⁵ Pa; ${\lambda }_0$ is the average free path of air molecules at normal temperature and pressure, generally 67.3 nm; T_s = 110.4 K [38]; $p$ is the absolute pressure of ambient gas, Pa; T is the absolute temperature of ambient gas, K. Through calculation, the size of particle $Kn$ in the range of 0.5–2.5 μm is shown in

Table 1. Kn value of particles of different diameters (300 K).

dp (μm)	λ (nm)	Kn	Cc
0.5	68.41	0.2736	2.010
1.0	68.41	0.1368	1.933
1.5	68.41	0.09121	1.597
2.0	68.41	0.06841	1.438
2.5	68.41	0.05473	1.346

.

According to the size of Kn and the range in

Table 2. Kn value and region of particles of different diameters (300~2000 K).

dp (μm)	0.01	0.1	1	10
Kn	10~100	1~10	0.1~1	0.01~0.1
Zone	Free molecular zone	Transition zone	Near-continuous zone/Slip zone	Continuous zone

, the particles in the range of 0.5–1.0 μm belong in the near continuous zone/slip zone (0.1 ≤ Kn ≤ 1), and the particles in the range of 1.5–2.5 μm belong in the continuous zone (0.01 ≤ Kn ≤ 0.1). In order to simplify the study, the particles in the range of 0.5–1.0 μm also belong in the continuous zone.

After being determined, the magnetic coagulation kernel is added to the PBM model through UDF, the proton balance equation (PBE) or general dynamics equation (GDE) is solved to describe the evolution process of coagulation dynamics in the particle swarm equilibrium model, and its coagulation dynamics equation is as follows [39]:

$$\frac{dn\left(w,t\right)}{dt}=\frac{1}{2}{\int }_0^w\beta \left(w-v,v\right)n\left(w-v,t\right)n\left(v,t\right)dv-n\left(w,t\right){\int }_0^{\infty }\beta \left(w,v\right)n\left(v,t\right)dv$$

(23)

where $n\left(w,t\right)$ is the particle number concentration distribution function of particles with a volume of $w$ at time t, ((N/m³)/m³, N is the particle number); $\beta \left(w,v\right)$ is the coagulation kernel of two particles with volumes $w$ and $v$, respectively, (m³/N)/s; 1/2 means that two particles simultaneously participate in one event of particle coagulation; the first term on the right side of the equation represents that the particle volume $w$ is generated by the coagulation of particle volume $\left(w,v\right)$ and particle volume $v$; the second term on the right side of the equation indicates the particle volume $w$ is lost by coagulation with other particles.

The partition method has the advantages of less calculated quantity, high calculation accuracy, flexible partition of subintervals, and simple coupling solving of the particle swarm equilibrium equation and two-fluid governing equation [40]. Therefore, the partition method was used to solve PBE, and then the whole coagulation process was numerically simulated, thus obtaining the time-dependent evolution process of the particle size distribution function. When the size of particles introduced from the inlet was 1.0 μm, for example, the particle swarm was divided into eight intervals (Bin-7–Bin-0) during analog computation, where the particle coagulation direction in the flow field was Bin-7→Bin-6→Bin-5→Bin-4→Bin-3→Bin-2→Bin-1→Bin-0. When the ratio exponent was taken as 1.0 (the particle volume of the next interval and the particle volume of the previous interval satisfy V_k+1 = f_sV_k, and 1.08 ≤ f_s ≤ 3.0), the average particle size corresponding to each interval within Bin-7–Bin-0 was calculated as seen in

Table 3. Average particle diameter in each interval from Bin-7 to Bin-0.

Interval Number	Average Particle Diameter (m)	Proportion (%)	Particle Volume Fraction (VF)
Bin-0	5.040 × 10−6	0	0
Bin-1	4.000 × 10−6	0	0
Bin-2	3.175 × 10−6	0	0
Bin-3	2.520 × 10−6	0	0
Bin-4	2.000 × 10−6	0	0
Bin-5	1.587 × 10−6	0	0
Bin-6	1.260 × 10−6	0	0
Bin-7	1.000 × 10−6	100	0.01420

. In addition, the average particle size from different inlets within Bin-7–Bin-0 is listed in

Table 4. Average diameter in Bin-7 to Bin-0 subranges with different particle diameters of inlet.

	Average Particle Diameter (μm)	Bin-7	Bin-6	Bin-5	Bin-4	Bin-3	Bin-2	Bin-1	Bin-0
Particle Diameter (μm)		d7	d6	d5	d4	d3	d2	d1	d0
0.5000		0.500	0.630	0.793	1.000	1.260	1.587	2.000	2.520
1.000		1.000	1.260	1.587	2.000	2.520	3.175	4.000	5.040
1.500		1.500	1.890	2.381	3.000	3.780	4.762	6.000	7.560
2.000		2.000	2.520	3.175	4.000	5.040	6.350	8.000	10.08
2.500		2.500	3.150	3.969	5.000	6.300	7.937	10.00	12.60

.

4. Simulation Correctness Verification

4.1. Grid Independence Test

In order to eliminate the influence of the number of grids on the numerical simulation, the grid independence test was carried out on the calculation model, and the optimal grid number was determined by calculating the collision probability between particles under different grid numbers. Due to the symmetry rule of the calculation area, the grid used in the calculation was a hexahedral structured grid, and the spherical wall of the core particle was refined. The calculation results are shown in Figure 6, with the increase in the number of grids, the magnetic coagulation removal efficiency first increased and then stabilized. When the number of grids increased from 270,000 to 650,000, the efficiency of magnetic coagulation removal changed to 0.5%. When the number of grids increased from 820,000 to 1,290,000, the efficiency of magnetic coagulation removal changed to 0.44%, and when the number of grids increased from 1,290,000 to 2,200,000, the efficiency of magnetic coagulation removal changed to 0.34%; that is, when the number of multi-fiber trapping structure grids reached about 650,000, the efficiency of magnetic coagulation removal remained basically unchanged. Therefore, 820,000 grids were selected for numerical simulation calculation.

4.2. Experiment Verification of the Simulation Method

In order to verify the correctness of the numerical simulation method, the simulation was carried out under the same conditions as the particle collision experiment. The difference was that the collision in the experiment was between particles and droplets, but the collision mechanism was the same. Figure 7 shows the comparison between the numerical simulation value and the experiment value of particle collision probability. As shown in figure, Ranz and Wong [41] reported experiment values were higher than the simulation values, while those of Walton and Woolcock [42] and Schmidt and Loeffler [43] were close to the numerical simulation results, because the fluid Reynolds number in the Ranz and Wong experiment was much higher than the one in the numerical calculation model. In addition, the Langmuir and Blodgett theory value [44] and the Slinn [45] semi-empirical formula calculation values were in good agreement with the changing trend of the numerical simulation results. The above results show that the numerical calculation model and method in this paper are suitable and can be used to calculate the particle collision probability.

5. Numerical Simulation Calculation Results

5.1. Effect of Particle Size on Magnetic Coagulation Removal Efficiency of Fe-Based Fine Particles

Figure 8 shows the influence of the particle size on the magnetic coagulation of Fe-based fine particles in the composite field. As shown in the figure, the magnetic coagulation removal efficiency of Fe-based fine particles gradually increased with the change of residence time, and the larger the particle size, the worse the magnetic coagulation effect of Fe-based fine particles. When the particle size of Fe-based fine particles increased from 0.5 μm to 1.5 μm, the magnetic coagulation coefficient of Fe-based fine particles decreased from 0.5414 to 0.2882. This is because the larger the particle size is, the more difficult it is to change the motion state of particles, and the smaller the relative motion intensity between particles under the same magnetic dipole force. The magnetic coagulation coefficient of Fe-based fine particles presented a quadratic functional relationship with the particle size, and its formula is expressed as follows: $K=0.7304-0.4196d_p+0.0832{d_p}^2$. In addition, with the increase of the dust particle size, the magnetic coagulation strength of Fe-based fine particles under a random magnetic dipole orientation was greater than that under the aligned magnetic dipole orientation, and the difference between the magnetic coagulation removal efficiency under the random magnetic dipole orientation and that under the aligned magnetic dipole orientation was smaller.

5.2. Effect of Particle Volume Fraction on Magnetic Coagulation Removal Efficiency of Fe-Based Fine Particles

Figure 9 shows the influence of the particle volume fraction on the magnetic coagulation of Fe-based fine particles in the composite field. As shown in the figure, the magnetic coagulation removal efficiency of Fe-based fine particles gradually increased with the change of residence time. The greater the particle volume fraction, the better the magnetic coagulation effect of Fe-based fine particles. When the particle volume fraction grew from 0.01 to 0.03, the magnetic coagulation coefficient of Fe-based fine particles increased from 0.2353 to 0.5061. The reason is that the increase of the particle volume fraction increases the number of particles in the composite field. Although the relative motion intensity of particles with the same particle size was unchanged under the same magnetic dipole force, the change in the quantity increased the probability of the collision between particles. The magnetic coagulation coefficient of Fe-based fine particles presented a quadratic functional relationship with the volume fraction, expressed as follows: $K=0.0216+23.56VF-247{VF}^2$. In addition, with the increase of the particle volume fraction, the magnetic coagulation strength of Fe-based fine particles under the random magnetic dipole orientation was greater than that under the aligned magnetic dipole orientation, and the difference between the magnetic coagulation removal efficiency under the random magnetic dipole orientation and that under the aligned magnetic dipole orientation was enlarged.

5.3. Effect of External Magnetic Field Strength on Magnetic Coagulation Removal Efficiency of Fe-Based Fine Particles

Figure 10 shows the influence of external magnetic field strength on the magnetic coagulation of Fe-based fine particles in the composite field. As shown in the figure, the magnetic coagulation removal efficiency of Fe-based fine particles gradually increased with the change of residence time. The greater the applied external magnetic field strength, the better the magnetic coagulation effect of Fe-based fine particles. When the applied external magnetic field strength increased from 0.5 T to 1.0 T, the magnetic coagulation coefficient of Fe-based fine particles increased from 0.3940 to 0.5288. This is because the greater the applied external magnetic field strength, the greater the magnetic dipole force between two particles after magnetization, and the greater the relative motion intensity of particles under the magnetic dipole force. The magnetic coagulation coefficient of Fe-based fine particles showed a quadratic functional relationship with the applied external magnetic field strength, expressed as follows: $K=0.1808+10.096H-62.72H^2$. In addition, as the applied external magnetic field strength increased, the magnetic coagulation strength of Fe-based fine particles under the random magnetic dipole orientation was greater than that under the aligned magnetic dipole orientation, and the difference between the magnetic coagulation removal efficiency under the random magnetic dipole orientation and that under the aligned magnetic dipole orientation remained basically unchanged.

5.4. Effect of Particle Magnetic Susceptibility on Magnetic Coagulation Removal Efficiency of Fe-Based Fine Particles

Figure 11 exhibits the influence of particle magnetic susceptibility on the magnetic coagulation of Fe-based fine particles in the composite field. As shown in the figure, the magnetic coagulation removal efficiency of Fe-based fine particles gradually increased with the change of residence time. The greater the particle magnetic susceptibility, the better the magnetic coagulation effect of Fe-based fine particles. When the magnetic susceptibility of Fe-based fine particles increased from 0.0250 to 0.0500, the coagulation coefficient of Fe-based fine particles increased from 0.3940 to 0.5288. This is because the greater the saturation magnetization of particles, the greater the magnetic dipole force between two particles after magnetization, and the greater the relative motion intensity of particles under the action of magnetic dipole force. The magnetic coagulation coefficient of Fe-based fine particles displayed a quadratic functional relationship with the magnetic susceptibility of particles, expressed as follows: $K=0.1808+10.096x_p-62.72{x_p}^2$. In addition, with the increase of particle magnetic susceptibility, the magnetic coagulation strength of Fe-based fine particles under the random magnetic dipole orientation was greater than that under the aligned magnetic dipole orientation, and the difference between the magnetic coagulation removal efficiency under the random magnetic dipole orientation and that under the aligned magnetic dipole orientation remained basically unchanged.

6. Conclusions

(1) The larger the particle size, the worse the coagulation effect of Fe-based fine particles. When the particle size of Fe-based fine particles increased from 0.5 μm to 1.5 μm, the magnetic coagulation coefficient of Fe-based fine particles decreased from 0.5414 to 0.2882. The multi-field coupling coagulation coefficient and the particle size of Fe-based fine particles satisfied a one-variable quadratic functional relationship as expressed by $K=0.7304-0.4196d_p+0.0832{d_p}^2$.

(2) The larger the particle volume fraction, the better the coagulation effect of Fe-based fine particles. As the particle volume fraction grew from 0.01 to 0.03, the magnetic coagulation coefficient of Fe-based fine particles increased from 0.2353 to 0.5061. The relationship between the magnetic coagulation coefficient of Fe-based fine particles and the particle volume fraction was a quadratic function, expressed as follows: $K=0.0216+23.56VF-247{VF}^2$.

(3) The greater the applied external magnetic field strength and particle magnetic susceptibility, the better the coagulation effect of Fe-based fine particles. When the applied external magnetic field strength increased from 0.5 T to 1.0 T or the magnetic susceptibility of Fe-based fine particles increased from 0.0250 to 0.0500, the magnetic coagulation coefficient of Fe-based fine particles increased from 0.3940 to 0.5288. The magnetic coagulation coefficient of Fe-based fine particles and their magnetic susceptibility conformed to a one-variable quadratic functional relationship, expressed as follows: $K=0.1808+10.096H-62.72H^2$, $K=0.1808+10.096x_p-62.72{x_p}^2$.

(4) The magnetic coagulation strength of Fe-based particles under a random magnetic dipole orientation was greater than that under the aligned orientation. As the dust particle size increased, the difference between the magnetic coagulation removal efficiency under the random magnetic dipole orientation and that under the aligned orientation became smaller, but the difference value was enlarged with the increase in the particle volume fraction. When the applied external magnetic field strength and particle magnetic susceptibility grew, the difference between the magnetic coagulation removal efficiency under the random magnetic dipole orientation and that under the aligned orientation basically unchanged.