Numerical Study on Particulate Fouling Characteristics of Flue with a Particulate Fouling Model Considering Deposition and Removal Mechanisms

Abstract

Due to a large amount of particulate matter in industrial flue gas, the formation of particulate deposits on the flue wall will increase the instability of equipment operation, which needs to be solved urgently. In this paper, a numerical investigation on the characteristics of particulate deposition and removal in the furnace flue was carried out for waste heat and energy recovery. This research adopted a comprehensive fouling model combined with the discrete phase model (DPM) which was performed by the CFD framework and extended by user-defined functions (UDFs). Firstly, the particulate deposition and removal algorithms were proposed to develop the judgment criterion of particle fouling based on the Grant and Tabakoff particle–wall rebound model and the Johnson–Kendall–Roberts (JKR) theory. This model not only considered the particles transport, sticking, rebound, and removal behaviors, but also analyzed the deposition occurring through the multiple impactions of particles with the flue wall. Then, the influence of furnace gas velocity, particle concentration, and inflection angle α of the tee section on the particulate fouling were predicted. The results show that the furnace gas velocity, particle concentration, and flue structure have significant effects on particle fouling and distribution, and the particle fouling mainly occurs in the blind elbow section and the tee sections of the flue. In addition, the fouling mass of particles decreases with an increasing furnace gas velocity and the decrease in particle concentration. Lastly, the fouling mass of particles decreases with the increase in the inflection angle α of the tee section, and the location of particle fouling gradually transfers from the blind elbow section to the tee section.

1. Introduction

The rotary kiln-electric furnace (RKEF) process is the one of the most widely adopted technological processes for ferronickel smelting production due to the advantages of yielding high grade nickel/iron from crude ferronickel products, less harmful elements, high production efficiency, and maturing process [1]. Nonetheless, the RKEF process also suffers from the high energy consumption and a large amount of by-product (e.g., slag and off-gas), which restricts its development; thus, reducing the energy consumption while achieving the same quality product has become a pressing need for the current ferronickel smelting industry. For this reason, the furnace gas which contains a vast amount of CO is recycled into rotary kilns as auxiliary fuel and combusted with bituminous coal, with a goal to improve the utilization rate of residual heat and energy. In the smelting process of electric furnaces, the ores will undergo the reduction reaction and slagging reaction. Meanwhile, the furnace gas is produced, and then escapes from the ores with many pores. Moreover, the furnace gas will carry a large amount of ore particles into the flue. In the flue, the ore particles are likely to impact the flue wall and deposit on the wall, which can cause severe hazards to the flue structure, such as fouling, erosion, corrosion, and capacity-limiting plugging. In serious cases, the ore particle deposition even leads to significant deterioration of the flue service life and the furnace gas delivery rate. In view of the fact that the flue is the transportation equipment for utilizing residual heat and energy of furnace gas in the RKEF process, a profound study on the mechanism of particle transportation, deposition, and removal in the flue is the key to ensure the stability and efficient recycling of residual furnace gas energy. Therefore, the research on the characteristics of particulate deposition and removal in flues is needed.

In the flue, the transportation process of ore particles with high-temperature furnace gas is a complex process of multiphase flow and multi-physical field coupling, which includes the hydrodynamics, heat and mass transfer of gas-solid two-phase, as well as the collision, rebound, sticking, and removal of particles. Thus, in order to accurately predict the motion and fouling behaviors of the ore particles, the method of combining experimental investigation and numerical simulation is applied to research the particulate transport and deposition mechanisms. Hosseini et al. [2] studied particle deposition in compact heat exchangers by using experimental and numerical simulation methods, and analyzed the influence of flow velocity on the particle deposition characteristics and mechanism for different particle sizes. Waclawiak et al. [3] proposed a model of powdery, medium-temperature particulate deposition on superheater tubes, which is not only using the DEFINE_DPM_EROSION procedure to calculate the deposition rate, but also using the DEFINE_GRID_MOTION procedure to consider the change in morphologies of the fouled tube wall caused by the growth of the deposits. Moreover, the effectiveness of the particle deposition model is verified by the experimental data. With the rapid development of computational fluid dynamics (CFDs), the researchers carried out a more in-depth study of developing the mathematical models for describing the formation mechanism of particulate fouling. Tang et al. [4] proposed particle deposition and removal algorithms to establish the judgment criterion of particle fouling based on the energy conservation model and the critical moment theory, and simulated the particle fouling process for different particle sizes and flue gas velocities. Lee et al. [5] established the sticking probability model of a deposit surface based on the energy conservation theory. The model not only considers the deposition through the multiple impactions of particles with the deposit surface, but also analyzes the influence of different parameters on particle sticking such as particle viscosity, surface tension, impact velocity, impact angle, and the thickness of the sticky layer on a particle. Xu et al. [6] developed a predictive particle fouling model based on the Johnson–Kendall–Roberts (JKR) theory and the critical deposition velocity criterion, and applied the DEFINE_DPM_BC macro to modify the wall boundary conditions, with the goal to determine whether the particles deposited. Moreover, the effects of gas drag, gravity, buoyancy, Brownian, Saffman lift, and thermophoretic forces on the particle deposition process were investigated. In recent years, researchers found that the fouling rate on the deposit surface is dependent upon the particle transport mechanism and the particle impact behavior [7]. The mechanisms of transporting particles toward the deposit surface are expressed in terms of inertial impact, thermophoresis, condensation, and turbulent diffusion [5,8,9,10]. Inertial impact is considered to be the dominant transport mechanism for particles with its size larger than 10 μm, while the particle with its size smaller than 10 μm is mainly transported by eddy transport and thermophoresis toward the deposit surface [11].

In addition, particle fouling depends not only on the transport mechanism, but also on the physical properties of the particles (e.g., the particle size, viscosity, and chemical properties etc.), the physical properties of the collision surface (e.g., wall temperature and roughness etc.), and the particle impact behavior (e.g., particle impact velocity and angle etc.) [7,8,11,12,13]. In addition, other factors such as the equipment geometry, fluid dynamics, and the chemical reactions in the transportation process also have a significant effect on the particle fouling process [7,11]. Lee et al. [5] proposed the adhesion probability of the particle is determined by particle viscosity, surface tension, impact speed and angle, and the sticky layer thickness on a particle. Zhou et al. [14] pointed out that the critical deposition velocity u_cr is a criterion for comparing the normal impact velocity of particles with the critical deposition velocity based on the JKR theory. It is found that when the impact velocity of particles is less than the critical sticking velocity, the particles may adhere to the wall, otherwise particles may rebound or remove other deposited particles [15,16]. Han et al. [7] established a numerical model to numerically predict the deposition rate of particles by comprehensively considering particle transport, impact velocity, and impact angle. Lee et al. [17] used CFD software to simulate the deposition of particles and evaluated whether the particles may deposit on the surface by considering the gravity, rebound, and adhesion forces during the particle–wall impact.

These things considered, Knudsen et al. [18] proposed that chemical reaction fouling is determined by two opposite mechanisms: deposition and removal. Meanwhile, it is found in Ref. [4] that a comprehensive particle fouling model should consist of both particle deposition and the removal sub-models based on Kern–Seaton theory. Thus, the study of particle fouling needs to consider not only the mechanism of particle deposition and rebound, but also discuss the particle removal mechanism. El-Batsh et al. [19] studied the deposition and removal mechanism of particles on turbine blades based on the analysis of the critical capture velocity and critical shear velocity as particle–wall collision. Pan et al. [20] proposed a comprehensive particle fouling model to study the process of particle deposition and fouling removal in inertial impaction process. Abd-Elhady et al. [16] numerically predicted the effect of critical sticking velocity and the removal velocity on the growth rate of particulate fouling layers during particle–wall collision.

Although the research on the mechanism of particle deposition, rebound, and removal has achieved significant progress in boilers, there are still relatively few research in published literature that focus on particle deposition and removal in electric furnace flues. This working condition has the characteristics of large range of particle size (e.g., 5~140 μm), large proportion of big particles (e.g., about 90% particles larger than 10 μm), high furnace gas temperature (e.g., 800~1000°C), and high flow velocity. This study comprehensively considers the influence of Saffman’s lift, gravity, and drag forces on the particle deposition process [21], combined with the physical properties of particles actually measured by Dalian Huarui Heavy Industry Group Co., Ltd. (Dalian, China). In order to accurately predict the particle fouling mass, the particle deposition mass rate and removal mass rate are evaluated. Meanwhile, the time amplification factor is applied to simulate the actual fouling process, and to convert the simulation results and simulation time into the actual time scale. Furthermore, the effects of flow velocity, particle concentration, and inflection angle α of the tee section on the particle fouling mass and location were investigated, and the optimization scheme of flue structure was proposed.

2. Physical Model

2.1. Technology Process

As shown in Figure 1, in the RKEF process, the semi-dry laterite ores are firstly mixed with the anthracite and the limestone in the raw proportioning station, and then fed into the rotary kiln for roasting and pre-reduction processes. Finally, the calcined laterite ores are transported into the enclosed electric furnace for reduction smelting. The electric furnace uses arc heat and resistance heat to smelt the calcined laterite ores. Undergoing the reduction and slagging reaction, the ferronickel and slag are then generated and the furnace gas (containing large amounts of carbon monoxide) is also produced. In order to improve the utilization rate of residual heat and energy of RKEF process, the flue is used to recover the furnace gas as auxiliary fuel, which is also help to reduce the unit coal consumption of ferronickel alloy. In this process, the high-temperature furnace gas may carry a large amount of ores particles into the flue. During the transportation process, a series of particle behaviors will take place, such as particle transport, impact, sticking, rebound, and removal.

2.2. Physical Model of the Flue

In this paper, the structure of the flue is shown in Figure 2. The electric furnace flue is composed of the furnace mouth section, flue section I, flue section II, flue section III, tee section, blind elbow section and outlet section. The flue model is characterized with the angle of tee section α = 80°. The flue is composed of shell, cement layer,. and fluid domain. Q235b steel is selected for the shell according to the actual survey data. The specific dimensions are shown in

Table 1. The parameters used by the model.

Flue	Length (mm)
Furnace mouth section	1600
Section I	5290
Section II	3770
Section III	8344
Tee section	2300 × 1740
Blind elbow section	1300
Outlet section	7000

.

3. Mathematical Model

In order to study the gas–solid two-phase flow characteristics between furnace gas and solid particles in the flue, the Eulerian–Lagrangian method is applied to solve the particle-laden furnace gas flow problem. The Eulerian model is employed to describe the gas phase flow, while the stochastic Lagrangian model is used to predict the particle’s motion. In this study, the furnace gas is a three-dimensional turbulent flow with characteristics of particle-laden, heat, and mass transfers. In addition, the fluid is considered as unsteady during the simulation of furnace gas flow [6], and the furnace gas parameters are shown in

Table 2. Physical parameters of furnace gas.

Properties	Value
Temperature, K	300	500	800	1000	1200	1400	1600	1800
Density, kg/m3	1.1262	0.6751	0.4219	0.3376	0.2813	0.2411	0.211	0.1876
Specific heat capacity, J/(kg·K)	1123.3	1187.6	1283.1	1338.4	1376.4	1399.2	1426.8	1501.0
Thermal conductivity, W/(m·K)	0.0279	0.0426	0.0625	0.0751	0.0866	0.0954	0.0883	0.0944
10−5 Viscosity, kg/(m·s)	1.862	2.626	3.594	4.156	4.675	5.162	5.625	6.067

. The RNG k-ε turbulence model with enhanced wall treatment is employed to forecast the gas flow field and heat transfer process [22]. The governing equations of mass, momentum, and energy of the continuous phase, as well as the equations of the RNG k-ε turbulence model exist in a large portion of the literature and will not be repeated here for simplicity [11,17,23,24].

According to the field monitoring data, the furnace gas is composed of carbon monoxide (i.e., 55.8~75%), hydrogen (i.e., 1.8~2.6%), methane (i.e., 3.5~4.8%), nitrogen (i.e., 15.8~35.6%), and carbon dioxide (i.e., 1.9~2.6%). Thus, the density of mixed gas was solved according to the ideal gas state equation, and given as:

$${\rho }_m=\sum \frac{M_{A}p}{RT}$$

(1)

where M_A is the molar mass of component A; R is the universal gas constant (molar gas constant), R = 8.314, J/mol·K; T is the gas temperature, K; p is the gas pressure, Pa.

The viscosity of mixed gas is expressed as:

$${\mu }_m=\frac{\sum y_A{\mu }_A{M}_A^{\frac{1}{2}}}{\sum y_A{M}_A^{\frac{1}{2}}}$$

(2)

where y_A is the mole fraction of component A in the mixed gas; μ_A is the viscosity of component A at the same temperature, Pa·s.

3.1. Particle Equations of Motion

For particle phase modeling, the Lagrange method is applied to track the trajectory of particle combined with considering the force balance on the particle. The particle dispersion due to turbulence is analyzed numerically by using a stochastic tracking model associated with discrete random walk (DRW) model [4,7]. Meanwhile, the DRW model is also employed to evaluate the influence of instantaneous turbulent velocity fluctuations of the gas on the particle trajectories [22]. We assume that the ore particles are spherical and sufficiently dense. In addition, the particle are transported in the gas flow without rotating. The force balance equation of particles is expressed in terms of the drag force, gravity force, and additional forces per unit mass:

$$\frac{d{u}_p}{dt}=F_D(u-u_p)+\frac{g({\rho }_p-\rho )}{{\rho }_p}+F_{ai}$$

(3)

where F_D(u − u_p) is the drag force per unit mass acting on the particle (m/s²), and $F_D=\frac{3\mu C_D{\mathrm{Re}}_P}{4{\rho }_P{d}_P^2}$, u and u_p are the velocity of air and particle, respectively (m/s), ρ and ρ_p are furnace gas and particle density, respectively (kg/m³), μ is the molecular viscosity of furnace gas (kg/(m·s)), d_p is the diameter of the particle (m), g is the gravitational acceleration (m/s²), C_D is the non-linear drag coefficient [25], Re_p is the relative particle Reynolds number, and ${\mathrm{Re}}_p=\frac{\rho \left|u-u_p\right|d_p}{u}$, $\frac{g({\rho }_p-\rho )}{{\rho }_p}$ is the gravity force per unit mass (m/s²), F_ai is the additional forces per unit mass (m/s²).

Within the range of particle sizes (i.e., 5~140 μm) analyzed in this paper, the drag force plays an important role in the particle motion and deposition process, which may be one order of magnitude larger than other forces [16]. Furthermore, the gravity force is the body force acting on the particle, and needs to be considered due to some of large particles (i.e., d_p > 20 μm) used in this study. In this paper, the additional forces contain the Basset force, Brownian, thermophoretic, pressure gradient, virtual mass, and Saffman’s lift forces, and the magnitude of these forces are dependent on the particle flow and property (such as particle size and density). Thus some necessary discussions are implemented to evaluate whether some of the forces are small enough to be ignored. The Basset force caused by the unsteady flow is negligible due to the particles with density being significantly greater than gas density. It is reasonable to ignore the thermophoretic and Brownian forces since both the thermophoretic and Brownian forces are valid for submicron particles (i.e., d_p < 1 μm).

According to Ref. [26], the pressure gradient and virtual mass forces are several orders of magnitude smaller than other external forces, since the ratio of gas density to particulate density is quite small (i.e., approximately 0.0002178), and thus leading to neglect the pressure gradient and virtual mass forces. Saffman’s lift force may influence some small particles (i.e., 5~10 μm) motion near the flue wall, because the Saffman’s lift force may play a dominant role in the near-wall area with the high-velocity gradient to consider Saffman’s lift force. Its general form is provided by Saffman: $F_{saff}=\frac{2K_C{v}^{0.5}\rho d_{ij}}{{\rho }_P{d}_P{(d_{lk}{d}_{kl})}^{0.25}}(u_i-u_{pi})$, where K_C = 2.594, d_ij the deformation tensor [27]. Therefore, only the drag, gravity, and Saffman’s lift forces for particulate motion are considered in this study.

3.2. Particle-Surface Interactions

3.2.1. Deposition Model

As the particle–wall collision process occurs, the particle may have a chance to deposit on the wall if it meets deposition criterion. In this work, the evaluation criterion is implemented based on the JKR theory [28]. In this model, the critical deposition velocity is applied as a criterion to determine whether the particle deposits on the wall or rebounds to the flow. When the impact velocity of particles is less than the critical sticking velocity, the particles may stick to the wall; otherwise particles may rebound or remove other deposited particles. In addition, the deposition model also analyzes the particle sticking caused by the multiple impacts of particles. Figure 3 is a schematic diagram of particle transportation and fouling formation in the flue. When particle deposition occurs on the flue wall, the mass of deposited particles will be recorded by user-defined memory (UDM). According to Ref [28], the critical deposition velocity can be written as:

$$u_{cr}={(\frac{2K}{d_P{R}_r^2})}^{(10/7)}$$

(4)

where K is the effective stiffness parameter, and can be defined as:

$$K=0.51{[\frac{5\pi (k_s+k_P)}{4{\rho }_p^{3/2}}]}^{2/5}$$

(5)

$$k_s=\frac{1-{\nu }_s^2}{\pi E_s}$$

(6)

$$k_P=\frac{1-{\nu }_P^2}{\pi E_P}$$

(7)

where E_s and E_p represent the young’s modulus of the wall and particle, respectively, v_s and v_p represent the Poisson’s ratios of the wall and particle, respectively, d_p is particle diameter, ρ_p is the particle density, R_r is the particle restitution coefficient. Equation (4) shows that the smaller R_r, the greater the critical deposition velocity, which will lead to an increase in the number of deposited particles. The particle restitution coefficient is specified as 0.9 considering that is commonly used in similar researches [29].

3.2.2. Rebound Model

The particle–wall rebound model is very important to obtain the rebound angle and velocity of particle impacting the wall [30]. It is widely known that when the particles collide with the wall, particles will lose a fraction of their kinetic energy due to plastic deformation of the particle. In addition, the energy loss is commonly much smaller than the initial kinetic energy of incident particles, and then the particles will rebound into the flow, resulting in the change in rebound velocity and angle for the particles. In this paper, the particle rebound model proposed by Grant and Tabakoff [31] is used to calculate the velocity change during the particle–wall collision. In addition, the model is based on the coefficient of restitution to predict the energy loss caused by particle impaction, and is written as:

$$\left\{\begin{array}{l}{e}_n=\frac{u_{2n}}{u_{1n}}=0.993-1.76\theta +1.56{\theta }^2-0.49{\theta }^3\\ e_t=\frac{u_{2t}}{u_{1t}}=0.998-1.66\theta +2.11{\theta }^2-0.67{\theta }^3\end{array}\right.$$

(8)

where e_n and e_t are the normal and tangential velocity restitution coefficients, respectively, u_1n and u_2n are the normal velocities before and after particle collision with the wall respectively, u_1t and u_2t are the tangential velocities before and after particle collision with the wall, respectively, θ is the particle impact angle.

3.2.3. Remove Model

Most research shows that fouling formation is the result of the simultaneous action of two opposing mechanisms, such as deposition and removal [32]. For the moment, the theoretical analysis and experiments are implemented to show that the removal rate depends upon the fouling thickness, shear action and strength of the deposit layer. According to Ref. [33], the removal rate of fouling is considered to be directly proportional to the local wall shear stress and the fouling thickness, and inversely proportional to the strength bond factor, as shown in Equation (9) [4,6]:

$${\dot{m}}_r=k(\frac{{\tau }_w}{\psi })x_f$$

(9)

where ${\dot{m}}_r$ is the removal rate, k is the removal constant, ψ is the strength bond factor of the deposit layer, ε is deposit porosity, τ_w is the local wall shear stress on the deposit surface, and is expressed as ${\tau }_{\omega }=\mu {\left(\frac{du}{dy}\right)}_{y_w}$, x_f is the fouling layer thickness, and $x_f=\frac{m_f}{\epsilon ·{\rho }_p}$, m_f is fouling mass per unit area.

As mentioned above, fouling mass is the difference between the deposition and removal mass. The fouling mass can be expressed as:

$$m_f=m_d-m_r$$

(10)

where m_d is the deposition mass, m_r is the removal mass, and defined as:

$$m_r=m_{r,t}+{\dot{m}}_r\Delta t$$

(11)

where m_r,t is the removal mass starting at time t, ${\dot{m}}_r\Delta t$ represents the additional removal mass within time step Δt.

4. Time Magnification Factor

In the actual fouling formation process, it usually takes a few hours or even some months to stabilize fouling accumulation in a flue for a dynamic balance of deposition and removal. In the numerical simulation process, the calculation time step must be small enough to accurately describe the particle fouling process. However, it is unrealistic to calculate the actual fouling accumulation process (in the order of several hours or even some months) with such a small time step. Thus, it is necessary to expand the simulation time scale of fouling accumulation, which can enlarge the simulation results and time of fouling accumulation to match the actual time scale [4].

In the simulation, if the deposition particle number of the fluid node is bigger than critical value (i.e., critical deposition number n_cr), the fluid node will become the solid node causing the fluid node shift the critical thickness δ_cr in a certain orientation [4,34]. In addition, the fouling growth rate is dependent upon n_cr. Thus, in order to scale up the simulation results of fouling to the actual scale, it is assumed that when the number of deposited particles per square millimeter area is equal to n_cr, the thickness of the deposition layer will correspondingly increase by δ_cr. Furthermore, for scaling up the simulation time to the actual time scale, it is necessary to multiply the simulation time by time magnification factor N. The N is as given in Equation (12):

$$N=N_1{N}_2=\frac{l{n}_{cr}\varphi {\rho }_p{\delta }_{cr}}{C_F{n}_{cr}{u}_{in}{S}_{in}{t}_{step}}$$

(12)

where the magnification factor N₁ and N₂ are shown as:

$$N_1=\frac{l\varphi {\delta }_{cr}}{n_{cr}\frac{4}{3}\pi {(\frac{d_p}{2})}^3}$$

(13)

$$N_2=\frac{C_P}{C_F}=\frac{n_{in}{\rho }_p\pi d_p^3}{6C_F{u}_{in}{S}_{in}{t}_{step}}$$

(14)

where ϕ = 1–ε, ε is the fouling porosity, and l is the length of the wall grid. Where C_p, C_F are the particle concentration in the simulation and the actual process, n_in is the particle number injected into the computational domain, u_in is the flow velocity, S_in is the cross-sectional area of the computational domain, and t_step is the time step.

It is necessary to determine the values of n_cr and δ_cr before the simulation. In this study, we set n_cr = 40~60, u_in = 3 m/s, d_p = 28.13 μm, and δ_cr = 1.0 mm. So the simulated time can be transformed into the actual time by Equation (12), and then the relationship between fouling mass and time for different n_cr may be obtained, as shown in Figure 4. When n_cr is larger than 55, the increase in fouling resistance is independent of the value of n_cr. Therefore, n_cr = 60 and δ_cr = 1 mm are chosen in the following research.

5. Boundary Condition

In this paper, the velocity inlet and pressure outlet are used as the inlet and outlet boundary conditions. Moreover, both the inlet and outlet conditions are escape, and the wall is set as reflect. At the flue surfaces, the no-slip boundary condition is specified [35,36]. The inlet temperature of the flue is 1073 K, the inlet speed of furnace gas is 3 m/s to 7 m/s, and the particle concentration is 5 g/m³ to 10 g/m³. The physical properties of furnace gas given by the enterprise are shown in Table 2. Moreover, by assuming that the initial particle velocity equals to inlet velocity of furnace gas and 9000 particles are injected at flue inlet each time step with a time step size of 0.001 s. The specific physical properties are shown in

Table 3. Physical properties of the deposition model.

Properties	Symbol	Value
Density of gas, kg/m3	ρp	1550
Poisson’s ratios of the particle	vp	0.18
Poisson’s ratios of the wall	vs	0.13
Young’s modulus of the particle, GPa	Ep	192
Young’s modulus of the wall, Gpa	Es	215
Particle concentration, g/m3	Cp	5~10
Furnace gas temperature, K	T	1073
Furnace gas velocity, m/s	u	3~7

.

6. Size Distribution of Particles

For predicting the deposition characteristics in the flue, the particle size provided by Dalian Huarui Heavy Industry Group Co. Ltd. was selected for following simulation, and the particle size considered in this paper follows Rosin–Rammler distribution between 5~140 μm for matching the actual size distribution of incident particles, as shown in Figure 5. As can be seen in the figure, particle size is mainly distributed in the range of less than 100 μm, with an average particle size of 28.13 μm.

7. Numerical Method and Validation

7.1. Numerical Method

In present study, the commercial software FLUENT is used to numerically simulate particle fouling process in a flue with a scheme of double precision. The second-order upwind scheme is applied to discretize the convective and diffusive terms. The SIMPLE algorithm is adopted to carry out velocity–pressure coupling. The convergence criterion is that the residual of each governing equation is less than 1.0 × 10⁻⁴ except that the energy equation is 1.0 × 10⁻⁶. It is assumed that the interaction or impact between particles and the influence of particle phase on fluid phase are ignored, which allows for an adoption of a one-way coupling for the particle phase [37]. A stochastic tracking model associated with discrete random walk is used to predict the turbulent dispersion of particles in the continuous phase, and the particles are monitored till they exit the computational domain. The fouling model shown in Figure 6 is carried out using fluent user-defined files (UDFs), which modifies the simulation procedure. In addition, the present work adopts the DEFINE_DPM_EROSION macro to modify the boundary conditions of deposit surface to evaluate whether particles were deposited. The user-defined memory (UDM) is applied to store the mass of deposited particles. In the procedure, the fouling mass depends on the difference between the deposition and removal masses.

7.2. Grid Generation and Independence Validation

The commercial software ICEM is used to establish a structural grid for the discretization of the computational domain, and the computational domain is divided into three parts: carbon steel layer, cement layer, and fluid domain layer, as shown in Figure 7. It is well known that the mesh number may influence the simulation result of particle fouling. Therefore, the grid independence verification is necessary to perform for ensuring the accuracy of the simulation. The numerical simulation was carried out for different number of meshes, and the values of Re and Nu at the flue outlet were plotted against the mesh number, as shown in Figure 8. It is clear from this figure that the values of Re and Nu no longer change significantly with increasing grid number when the grid number is approximately 1.95 million under the same conditions. In order to meet the requirements of calculation accuracy and efficiency, the overall mesh number of the physical model is taken to be 1.95 million.

7.3. Numerical Validation

In order to validate the accuracy of the mathematical models and numerical methods in terms of particle fouling, a numerical simulation on the deposition mass is performed and is compared with the experimental results presented in Ref. [4] and simulation results obtained in Xu et al. [6] and Tang et al. [4]. Figure 9 shows the comparison of experimental and numerical results on the fouling mass. It can be found that our simulation result is in good agreement with the experimental result presented in Ref. [4], and the relative difference among the both is 12.2%. The comparative results of the fouling mass for 5 μm particle diameter are shown in Figure 9. As can be seen from the Figure 10, the deposit mass increases with increasing time and the relative errors between our simulation result and the other two results are 8.52% and 4.56%, respectively. Thus our simulation result basically coincided with the results of two references, which can verify the feasibility of the particle fouling model developed in this paper.

8. Results and Discussion

In order to solve the particle fouling problem in the flue of electric furnace, in the following research, the particle fouling behavior in the flue was first analyzed, and then the effects of furnace gas velocity, particle concentration, and inflection angle α of the tee section on particle fouling were discussed, and the gas–solid two-phase heat and mass transfer mechanism and particle fouling mechanism were investigated.

Figure 11 shows a steady-state particle deposition mass rate graph. It can be found from Figure 11 that particle fouling generally occurs where the furnace structure changes, such as cross-sectional area alterations, flue inflections, blind elbows, and tee sections. As the particles flow into the flue with the furnace gas, particle fouling occurs first at the furnace mouth section due to the sudden reduction in cross-sectional area of the flue and the inertia impaction of the particles. Then, the particles will flow through the section Ⅰ, section Ⅱ and section Ⅲ of flue in turn, and the fouling occurs at the corner of the flue. There are two reasons for this result: firstly, as the furnace gas carrying particles flows to the inside corner of the flue, due to the influence of the corner structure, turbulence and the boundary layer separation of the fluid occurs and vortexes are formed, and then the fouling is caused by turbulent eddies and Brownian diffusion [21]; secondly, the particle flow with the furnace gas to the outside corner of the flue, and particle fouling will occur due to Brownian diffusion and particle inertia [21]. Afterwards fouling also occurs when the particles flow with the furnace gas through the tee section, blind elbow section, and outlet section. In addition, the blind elbow section and the tee section are the main fouling areas in the flue, and the reasons of fouling will be introduced in detail in Section 8.3.

8.1. Furnace Gas Velocity

In this study, to investigate the effects of flow velocity on particle deposition, removal, and fouling masses, the furnace gas velocity was varied from 5 m/s to 10 m/s during the simulation, as shown in Figure 12. It can be found that the deposition and fouling masses decrease with the increase in furnace gas velocity. At the same time, the deposition and fouling masses show an asymptotic tendency [20]. At lower velocities (5 m/s), the inertial and diffusion forces are smaller than the drag force; the smaller the Stokes number, the poorer the particle following, and so the deposition and fouling masses grow rapidly. As the furnace gas velocity increases, the impact velocity of the particles increases and produces a lower deposition rate; leading to a decrease in deposition and fouling masses. Wang et al. [4] also pointed out that the kinetic energy of particles increase as the velocity of flow increasing and the adhesion energy is not sufficient to cause deposition. As a result, the number of particles that can adhere to the wall will decrease, leading to a decrease in deposition and fouling masses.

In addition, it is also seen from the Figure 12 that the removal mass decreases with increasing the furnace gas velocity. This is because increasing the velocity of gas leads to an enhancement of the fluid shear stress. From Equation (9), it is obvious that the higher the wall shear stress of the fluid, the higher the particle removal force, and the increase in the removal force is also able to increase the fouling removal mass rate, leading to an increase in the removal mass with the increase in the furnace gas velocity.

Based on the above two reasons, this paper concludes that the increase in the velocity of the gas leads to a decrease in the deposition and fouling masses, conversely, an increase in the removal mass. Thus, to better compare and predict the asymptotic behavior of fouling mass, a curve was fitted to the simulated data. The extension line in the figure indicates the extension of the fitted curve beyond the simulation time.

8.2. Particle Concentration

It is found that particle concentration is also one of the main factors affecting particle deposition. Therefore, to examine the effect of particle concentration on fouling characteristics, the particle concentration was varied from 5 g/m³ to 10 g/m³ during the simulation. Figure 13 plots the particle deposition, removal, and fouling masses versus time at various concentrations. As shown in Figure 13a,c, the deposition and fouling masses increase continuously with the increase in particle concentration, and the removal mass also increases gradually. The reason for the above results is that when the particle concentration increases, the amount of particles per unit volume of furnace gas increases, which will increase the number of particles collision with the wall, and subsequently increase the particle deposition mass. As a result, the thickness of the deposition layer will also increase. From Equation (9), it can be seen that the increase in deposition layer thickness leads to increasing removal rate, which, however, has a relatively small order of magnitude effect on the removal rate. To sum up, the fouling mass increases with increasing particle concentration.

8.3. Flue Structure Optimization

The present study finds that the particles are mainly deposited in the blind elbow section and the tee section, and so the effect of the inflection angle α of the tee section on the deposition characteristics of the flue blind elbow section and the tee section was investigated based on the actual working conditions of the flue equipment in the field, and the range of angle α in the simulation is 80°~120°. Figure 14 plots the variations of the steady-state deposition mass rate versus angle α for the tee section, blind elbow section, and outlet section. It can be seen from Figure 14 that the deposition mass rate decreases with increasing the angle α for the blind elbow section and outlet section, but the deposition mass rate increases with increasing the angle α for the tee section.

Figure 15 shows the particle deposition characteristic in the blind elbow, outlet, and tee sections for different inflection angle α. From the figure, it can be seen that the deposition mass rate of the blind elbow section and outlet section decreases with the increase in the inflection angle α, and the range of the deposition also decreases; however, the deposition mass rate in the deposition zone A of the tee section increases with the increase in the angle α, and the particle deposition location is transferred from the blind elbow section to the deposition zone A. At the same time, the deposition mass rate in the deposition zone B of the tee section decreases with the increase in the angle α, but the range of deposition is more dispersed. The reasons for this phenomenon are as follows.

Firstly, with the increase in the inflection angle α, the amount of furnace gas flowing into the blind elbow section gradually decreases, which leads to the decrease in the pressure of the blind elbow section from 106 Pa to 34 Pa. This conclusion can be reflected by the pressure distribution diagram of the blind elbow section as shown in Figure 16. Therefore, the number of particles entering the blind elbow section also decreases, and a large number of particles flow directly into the flue outlet section with the mainstream furnace gas, as shown in Figure 17. Figure 17a shows that the collision zone of particles with the flue wall can be predicted based on the trajectory of the particles. As the angle α increases, the collision zone between the particles and the wall in the blind elbow section is shrinking, causing the range of the deposition zone to decrease correspondingly. In addition, the number of particles flowing into the blind elbow section decreases with the angle α increases, leading to a decrease in the number of particles deposited, which in turn causes a decrease in the deposition mass rate in the blind elbow section.

Secondly, it can be seen from Figure 17 that the particles enter the blind elbow section with the furnace gas and first flow to the lower part of the blind elbow section, then flow along the wall to the upper part of the blind elbow section, and finally flow to the outlet section. In this process, the particles are transported to the wall by inertial collision and the turbulent diffusion mechanism. Thus the fouling behaviors of the blind elbow section and deposition zone A are mainly caused by the particle inertial and Brownian diffusion. As shown in Figure 18, when the furnace gas carrying particles flows to the outlet section, due to the influence of the tee structure, turbulence and boundary layer separation of the fluid occurs and vortexes are formed. So the deposition behavior in the deposition zone B is mainly caused by the turbulent eddies and Brownian diffusion [21].

Thirdly, Figure 18 shows the variation of vortex in the tee section for different inflection angle α. It can be found from the Figure 18 that with increasing angle α, the turbulence intensity and boundary-layer separation phenomenon of the fluid will gradually become weaker, which leads to decrease the size of vortex core in the deposition zone B, and then makes the deposition behavior in the deposition zone B change from deposition caused by turbulent eddies to deposition caused by Brownian diffusion. Furthermore, the velocity distribution of the cross-section C also reflects the decrease in the size of vortex core with increasing angle α.

Figure 19 depicts the particle deposition, removal, and fouling masses versus time for different inflection angle α. It can be seen from Figure 15 that the deposition, removal, and fouling masses decrease with the angle α increases. This is because the deposition of particles in the flue mainly occurs in the blind elbow section, where the deposition mass is much larger than the deposition mass in other sections of the flue. Meanwhile, the deposition mass of the blind elbow section decreases with the increasing angle α, which causes the deposition mass of the whole flue to decrease with the increasing angle α.

9. Conclusions

The numerical simulation is used to investigate the particle fouling characteristics of the flue in the rotary kiln-electric furnace smelting process. In this study, the particle fouling model considering the particle deposition and removal mechanism is developed. The model adopts Grant and Tabakoff particle–wall rebound model and Johnson–Kendall–Roberts (JKR) theory with UDFs to develop the judgment criterion of particle fouling. The time amplification factor is applied to simulate the actual fouling process, and to convert the simulation results and simulation time into the actual time scale. The effects of furnace gas velocity, particle concentration, and inflection angles α of the tee section on the fouling mass and location of particles were investigated comparatively. We obtained the following conclusions:

(1)

Particle fouling generally occurs where the furnace structure changes, such as cross-sectional area alteration, flue inflection, blind elbows, and tee sections where fouling behaviors are mainly caused by turbulent eddies, Brownian diffusion, and particle inertia. Meanwhile, the blind elbow section and the tee section are the main locations of particle fouling.

(2)

The fouling mass is strongly dependent on the furnace gas velocity. The deposition mass and fouling mass of particles decrease with the increase in furnace gas velocity, and show an asymptotic behavior. Meanwhile, the removal mass of particles increases with the increase in furnace gas velocity. Therefore, appropriately increasing the furnace gas velocity is an effective measure to reduce particle fouling in the flue.

(3)

Particle concentration is another major factor affecting particle fouling. The deposition, removal, and fouling masses of particles increase with increasing particle concentration.

(4)

The fouling mass rate in the blind elbow and outlet sections of the flue decreases with increasing the inflection angle α of the tee section. However, the fouling mass rate of the tee section increases with the increase in the inflection angle α of the tee section. Meanwhile, the particle fouling position is transferred from the blind elbow section to the deposition zone A of the tee section. In addition, the fouling mass rate of the tee section deposition zone B decreases with the increase in the tee section inflection angle α, and the deposition area is more dispersed.

(5)

The deposition mass, removal mass, and fouling mass decrease with increasing the inflection angle α of the tee section. Therefore, appropriately increasing the tee section inflection angle α is an effective measure to reduce particle fouling in the flue.