Numerical Simulation of the Distribution Patterns of Particle Deposition on the Vertical Wall Behind Near-Wall Heat Source

Abstract

Near-wall heat sources have a crucial part to play in the process of particle deposition. Thus, this study investigates the impact of the near-wall heat source on the distribution patterns of particle deposition on the vertical wall behind the heat source, taking into account the variability in heat source temperatures and distances from the vertical wall. A model based on the Eulerian–Lagrangian method was established for tracking the motion trajectories of 1000 particles with a density of 1400 kg/m³ and a particle size range of 0.01–10.0 μm. The temperature field, airflow field, and particle deposition distribution in six cases were analyzed. It was shown that the heat source temperature significantly affectis the temperature field, airflow field, and particle deposition distribution on the vertical wall behind the heat source. This study demonstrated that as the temperature rises, the quantity of particles deposited in the upper-right region of the vertical wall decreases more noticeably. The quantity of particles deposited onto the vertical wall is inversely related to the distance between the near-wall heat source and the vertical wall. On one hand, the deposition distribution law serves as a foundation for advancing the technology aimed at removing suspended particles via thermal plumes. On the other hand, it provides critical insights for addressing the challenges associated with harmful particle deposition linked to the attachment effects of thermal plumes.

1. Introduction

Indoor particulate matter (PM) has a range of adverse effects on human health and society. On the one hand, this suspended PM contaminates the air, which is considered one of the risk factors for preventing non-communicable diseases [1,2], especially particles with a diameter of less than 10 μm. These particles are regarded as inhalable particles and can have a negative impact on human health. Inhalation of air contaminated by these particles by humans leads to reduced lifespans [3] and exacerbates asthma [4], cancer [5], and cardiovascular disease [6]. On the other hand, particle deposition affects both production processes and daily life. For example, the deposition of particles on artifacts will damage the artifacts [7], the deposition on walls will contaminate the walls, and so on.

Indoor particle suspension and deposition are significantly affected by airflow. Numerous scholars have extensively investigated the impact of airflow on indoor particle deposition. Kong et al. [8] carried out an analysis comparing four modes of ventilation (side exhaust and supply air, top exhaust, and supply air) within hospital wards, revealing that lateral airflow effectively reduced airborne virus contamination. Cetin et al. [9] examined the impact of air change rate on particle deposition. They identified that when the particle flow rate reached a certain proportion, the pollutant content increased as the air change rate increased. Conversely, when the particle flow rate was constant, the pollutant content decreased as the air change rate increased. Conversely, when the particle flow rate was constant, the contaminant level decreased with an increase in air change rate. Notably, some larger-sized particles deviated from this trend due to agglomeration effects. Zhuang et al. [10] analyzed the movement and diffusion of particles indoors and found that the indoor particle concentration was significantly affected by particles smaller than 0.5 µm. Small particles were easily affected by airflow and rose to the upper space of the room. Chu and Yang [11] discovered that the interior PM2.5 concentrations declined with rising external wind speed, and the concentration decreased with a higher ventilation rate.

Among the multitude of airflows, thermal plumes significantly influence the distribution of indoor particles, which has attracted considerable attention from scholars. Considerable investigations have been conducted to explore the impact of thermal plumes on particle deposition distributions. Mutlu [12] conducted a three-dimensional numerical comparison study on the impact of various heating systems on particulate matter concentrations within a house with zero air exchange rate. The results pointed out that a floor heating system was more effective in removing contaminants that were uniformly distributed, while a radiator system exhibited superior efficacy in eliminating contaminants emitted near the floor. Luo et al. [13] conducted numerical simulations employing the Eulerian–Lagrangian method to study the impact of floor temperature on particle deposition distribution in capillary radiant floor heating systems. It has been shown that the dispersion height of particles with various diameters rose as the floor temperature went up. Moreover, the particle concentration reached the maximum value at a height of 0.6 m. Xiong et al. [14] found that temperature gradients and particle size played a crucial role in particle deposition efficiency, with deposition efficiency positively correlated with temperature gradients. Zhou et al. [15] researched the deposition of particles within ventilated and floor-heated rooms, considering inlet velocity and floor temperature as variables. They discovered that the particle concentration decay rate rose with increasing inlet velocity; when the inlet velocity was constant, the particle removal efficiency increased with the floor temperature. Particles tend to deposit on walls near heat sources, primarily caused by thermal convection around the heat source. The phenomenon of particles accumulating on indoor vertical or horizontal surfaces, causing the wall to turn black, is colloquially referred to as “black magic dust” [16]. Chen and Li [17] discovered that the attenuation rate loss coefficient for particles rose with rising heat source temperature and decreased with the increasing gap between the wall and the heat source. Chen [18] observed that the configuration of the heat source exerted an impact on both the indoor temperature field and the velocity field, subsequently impacting particle distribution and deposition.

In summary, although researchers have conducted extensive and in-depth research on the factors influencing the deposition of indoor particles by airflow, there is relatively limited attention paid in the field of indoor particulate matter research to the distribution of particulate matter deposited on vertical walls behind the near-wall heat source. However, the deposition of particles on the wall behind the near-wall heat source is very common. On the one hand, in daily life, when the heat dissipation equipment is in operation, the thermal plume generated by it will affect the movement of particles, causing the particles to deposit on the surrounding walls. This will not only contaminate the wall surface but also contaminate any artwork hanging on the wall, thus destroying the beauty and tidiness of the living environment. On the other hand, in the field of industrial production, the high temperature generated during the operation of machines will accelerate the deposition of particulate matter on the equipment. Such deposition may lead to equipment scaling and cause wear and tear on the equipment. When the thickness of the deposition layer continuously increases, it may even cause serious problems such as equipment blockage and corrosion. These problems will have a negative impact on the performance of the equipment, shorten service life, and increase production costs. For example, inside equipment such as combustion chambers, boilers, and heat exchangers, in such a high-temperature environment, particles gradually deposit on the wall surface under the interaction of heat flow and airflow, forming a deposition layer. The formation of this deposition layer reduces the heat exchange efficiency of the equipment and increases the heat transfer resistance, resulting in increased energy consumption, reduced thermal performance of the equipment, and ultimately shortened service life of the equipment. Therefore, it is crucial to deeply understand the particle deposition mechanism under the effect of the thermal plume attachment above the near-wall heat source. On the one hand, this deposition mechanism has important theoretical and practical significance for solving the problem of harmful particle deposition caused by the thermal plume attachment effect, especially in aspects such as controlling the “black magic dust” phenomenon, protecting cultural relic murals, and preventing the scaling of heat-producing equipment. On the other hand, mastering the deposition mechanism can provide a theoretical basis for the material selection and design of equipment and contribute to the optimization of industrial process parameters.

Consequently, this present research delves into the deposition and distribution of particulates ranging from 0.01 to 10.0 μm in diameter on the vertical wall behind the near-wall heat source (hereafter “back wall”) under the influence of the near-wall heat source. A three-dimensional airflow field in a chamber is simulated under six cases employing the RNG k-ε model. The Eulerian–Lagrangian method-based Discrete Phase Model (DPM) is used for tracking 1000 monodisperse particle trajectories. This study scrutinizes the effects of temperature fields, airflow fields, varying distances from the heat source to the back wall, and variations of the temperature of the heat source on particle deposits onto the back wall. Additionally, it analyzes the effects of these factors on the distribution of particle deposits in different regions of the back wall.

2. Method

2.1. Computational Domain and Boundary Conditions

The geometric model considered in this paper was built by imitating the experimental model of Chen et al. [19], as demonstrated by Figure 1. The chamber has the following dimensions of length (x), width (y), and height (z): 0.8 m, 0.4 m, and 0.4 m. The inlet and outlet dimensions are identical, both 0.04 m × 0.04 m. The center coordinates for the inlet are x = 0 m, y = 0.2 m, and z = 0.36 m, while those for the outlet are x = 0.8 m, y = 0.2 m, and z = 0.04 m. It can be seen from Figure 1 that the heat source has the following dimensions: length (x) of 0.14 m, width (y) of 0.03 m, and height (z) of 0.114 m. Notably, the near-wall heat source was symmetric in relation to the center plane of x = 0.4 m, and the base of the near-wall heat source was 0.02 m away from the floor of the chamber. There were two different particle release locations, as illustrated in Figure 1. Particle release location ① was located at the inlet. The particle release location ② was located in the red rectangular region directly above the heat source The dimensions of the rectangular region were established as length (x) × width (y) = 0.8 m × 0.03 m, and were positioned 0.005 m above the upper surface of the heat source. Six distinct cases were established in accordance with the condition of the heat source and the position where particles were released (see Table 1).

The inlet air velocity was 0.225 m/s, directed to the right. The initial air temperature was set to 20 °C. The outlet was selected as outflow. The wall conditions of the chamber were all set as stationary, no-slip walls with a roughness height of 0 m and a roughness constant of 0.5 under adiabatic boundary conditions. The wall turbulence was processed by means of the standard wall function method. When the particle was released from ①, the initial velocity was 0.225 m/s, directed to the right; when the particle was released from ②, the initial velocity was 0.225 m/s, directed upwards. Nine various sizes of spherical particles (0.01 μm, 0.1 μm, 0.3 μm, 0.5 μm, 1.0 μm, 3.0 μm, 5.0 μm, 7.0 μm, 10.0 μm) were selected, and the density was 1400 kg/m³. A total of 1000 monodisperse particles were released uniformly within the chamber at one time, and the particle temperature was consistent with the initial air temperature. The particle trajectories were simulated under steady-state conditions. The SIMPLE algorithm was brought into effect to address the pressure–velocity coupling situation. Furthermore, the second-order upwind scheme was applied to the convective discretization, which effectively enhances the numerical accuracy in solving multiphase flow problems [20].

2.2. Fluid Phase

The indoor airflow was considered stable, with incompressible turbulence. The Reynolds-Averaged Navier–Stokes (RANS) model was employed to conduct the turbulence simulation [21,22,23]. The RNG k-ε model demonstrated favorable attributes, including high stability, computational accuracy, and cost-effectiveness in simulating turbulent flow in heated rooms [24]. Therefore, in this numerical simulation, the RNG k-ε model was adopted to perform the simulation of the airflow field within the chamber, and the Boussinesq approximation was utilized to account for the buoyancy effect inside the cavity.

In incompressible heat flow, changes in temperature lead to changes in density, thereby producing buoyancy effects. Under the Boussinesq approximation, the temperature can be considered an active scalar, with its impact on the airflow field achieved through the buoyancy term. The governing equation is as follows [25,26].

Mass balance equation:

(1)$\nabla ·u_f=0$

Momentum balance equation:

(2)${\displaystyle \frac{\mathit{\partial }T}{\mathit{\partial }t}}+u_f·\nabla T=\alpha {\nabla }^{2}T$

Energy balance equation:

(3)${\displaystyle \frac{\mathit{\partial }{u}_f}{\mathit{\partial }t}}+u_f·\nabla u_f=-\left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{${\rho }_0$}\right.}\right)\nabla p+v_f{\nabla }^2{u}_f+g\beta \left(T-T_0\right)z$

where T, $u_f$, and p denote the fluid temperature, velocity, and pressure. ${\rho }_0$ and $T_0$ denote reference density and temperature, z represents the vertical coordinate, $\alpha$ denotes the thermal diffusivity, $v_f$ represents the dynamic viscosity. $\nabla$ denotes the gradient operator, $\beta$ represents the thermal expansion coefficient. In these equations, all transport coefficients are assumed to be constants.

2.3. Particle Phase

This study adopted a DPM based on the Eulerian–Lagrangian method [27,28]. The fluid is described as a continuous medium in Eulerian coordinates and is described within the Eulerian coordinate system, while discrete phases follow the trajectories by integrating the particle force differential equations in Lagrangian coordinates, thereby calculating the trajectories of these particles and the heat or mass transfer caused by them. Ultimately, the specific equilibrium equation of the particle force is:

(4)${\displaystyle \frac{d{u}_p}{dt}}=F_D\left(u_a-u_p\right)+{\displaystyle \frac{g\left({\rho }_p-{\rho }_a\right)}{{\rho }_p}}+F_a$

where $u_p$ represents the particle velocity, $u_a$ denotes the air velocity, $F_D$ represents the reciprocal of the relaxation time, g represents the acceleration of gravity, $F_D\left(u_a-u_p\right)$ denotes the drag force, ${\rho }_a$ and ${\rho }_p$ denote the densities of air and particles, and $F_a$ represents other forces.

According to Newton’s second law, $F=ma$, particle mass $m={\displaystyle \frac{\pi }{6}}{{\displaystyle \rho }}_p{{\displaystyle d}}_p^3$, when only considering the Stokes drag force $F=3\pi \mu d_{p}v$, the change of the particle velocity $v$ with time $t$ satisfies the equation $m{\displaystyle \frac{dv}{dt}}=-F$. Therefore, ${\displaystyle \frac{\pi }{6}}{{\displaystyle \rho }}_p{{\displaystyle d}}_p^3{\displaystyle \frac{dv}{dt}}=-3\pi \mu d_{p}v$.

By transforming both sides of the equation and then integrating, we can obtain:

${\displaystyle {\int }_{v_0}^v\frac{dv}{v}}=-{\displaystyle \frac{18\mu }{{\rho }_p{d}_p^2}}{\displaystyle {\int }_0^{t}dt}$

Consequently, it can be figured out as follows:

$v=v_0{e}^{-\frac{t}{{\tau }_p}}$

Here, ${\tau }_p$ represents the relaxation time of the particles, ${\tau }_p={\displaystyle \frac{{\rho }_p{d}_p^2}{18\mu }}$.

In the DPM, in order to deal with different flow states (different Reynolds numbers), it is necessary to correct the expression of the particle relaxation time and introduce a correction factor ${\displaystyle \frac{24}{C_D\mathrm{Re}}}$, which is introduced to correct the relaxation time of the particles.

At this time, ${\tau }_p={\displaystyle \frac{{\rho }_p{d}_p^2}{18\mu }}{\displaystyle \frac{24}{C_D\mathrm{Re}}}$.

Hence,

(5)$F_D={\displaystyle \frac{18\mu }{{\rho }_p{d}_p^2}}{\displaystyle \frac{C_{D}Re}{24}}$

where $C_D$ represents the drag coefficient, $\mu$ denotes the viscosity of the air molecules, ${\tau }_p$ denotes the relaxation time, m denotes particle mass, and $a$ denotes acceleration. Re represents Reynolds number, which is calculated by the following formula:

(6)$Re\equiv {\displaystyle \frac{\rho d_p\left|u_p-u_a\right|}{\mu }}$

In the present research, a Discrete Random Walk (DRW) model was used for calculating turbulent diffusion. The DRW model presumes that the pulse velocity parameter is a constant function of time for discrete segments and that its random value is constant for a given time period of the eddy characteristic lifetime [29].

2.4. Grid Independence Test

The model was meshed using Ansys Fluent [30]. Structured meshing was employed for grid division due to its advantages of smaller computational errors, easier convergence, faster grid generation, and better grid quality. To ensure an accurate estimation of turbulent flow and particle phase, a grid sensitivity analysis was needed. With a distance of 0.02 m from the heat source to the back wall, three different grids (see Table 2) were used to check the grid independence. The simulation data show that there is no apparent discrepancy between the average velocity calculated for the three grids; however, compared with the other two grids, the grid 1 particle deposition had significantly higher scores. The particle deposition fraction of grids 2 and 3 are relatively close, but grid 3 has more grid cells, requiring more convergence time compared to grid 2. Considering the factors of computer memory and capacity, grid 2 with 1,540,206 grid cells was selected for simulation in this study.

2.5. Simulation Validation

2.5.1. Validation of Fluid Flow

To accurately simulate the flows, the experimentally measured data of Chen et al. [19] was employed for the present research to validate the airflow field. Figure 2 displays the comparison of the velocities at three different positions x (x = 0.2 m, x = 0.4 m, and x = 0.6 m) in the y = 0.2 m plane. As can be observed from Figure 2, it can be found that the simulated data within this research are basically identical to the results of previous studies [19], which proves the accuracy and reliability of the established model.

2.5.2. Validation of Particulate Tracking Results

For validating the effectiveness of numerical computational models for particle tracking calculations, the normalized concentration distribution results from this numerical simulation were contrasted with the empirical data obtained by Chen et al. [19]. As can be noted in Figure 3, the normalized concentrations at three different positions x (x = 0.2 m, x = 0.4 m, x = 0.6 m) on a plane y = 0.2 m are compared. As shown in this figure, the results of the simulation are largely in line with the literature data [19], which shows that the model used in this simulation is feasible for predicting the particle deposition distribution.

3. Results and Discussions

3.1. Temperature Fields and Airflow Fields

Temperature and velocity are crucial factors influencing particle motion. Therefore, analyzing temperature fields and airflow fields is essential. The temperature field and airflow field of Cases III, IV, and V are analyzed at the typical locations of plane x = 0.4 m and y = 0.355 m (Figure 4 and Figure 5). The temperature contour plot in Figure 4 and Figure 5 reveals that the temperatures near a heat source surface are higher compared to other regions, and the highest temperature appears in the region of z = 0.12–0.14 m. The upper region of the chamber exhibits higher temperatures, while the bottom region shows lower temperatures. This phenomenon arises because the temperature of the air around the near-wall heat source is lower than that of the heat source. Consequently, the heat source heats this portion of the air, resulting in elevated temperatures. The heated air interacts with the cooler ambient air, creating a temperature difference and subsequent density variation. This density difference generates buoyancy that causes the air to move upward, forming a thermal plume. As the thermal plume ascends, it further heats the upper region of the chamber, which leads to the upper region of the chamber having a higher temperature than the lower region. It can be observed from Figure 4 and Figure 5 that the sequence of the maximum temperature differences for all three cases is Case IV, Case III, and Case V, indicating that the higher the temperatures of the heat source, the greater the effect on the temperature field.

Figure 4 and Figure 5 also depict the airflow fields at the x = 0.4 m and y = 0.355 m plane for Cases III, IV, and V. The velocity contour plot in Figure 4 and Figure 5 illustrates higher velocities above the heat source and near the ceiling. This phenomenon arises due to a temperature differential between the air and the surface on the upper part of the heat source, which ultimately generates a thermal plume that drives the air upwards. As air rises to the ceiling with the thermal plume, it encounters resistance from the ceiling, altering its flow direction and moving to the sides, resulting in a larger airflow field near the ceiling. From Figure 4 and Figure 5, it is observed that the velocity at the bottom of the chamber is greater than in the middle region. This discrepancy is caused by a distinct temperature difference at the bottom of the chamber, which leads to a change in density and hence airflow. In contrast, the middle region of the chamber experiences less noticeable temperature variations, resulting in lower velocities. The velocity data in Figure 4 and Figure 5 show that the regions with higher airflow velocities exhibit a slight rightward deviation along the vertical alignment with the heat source. This is likely because the direction of the incoming air velocity is rightward, and this airflow couples with an upward thermal plume generated by the heat source. Causing the original thermal plume to shift to the right. Additionally, Figure 4 and Figure 5 present that the order of maximum velocities for the three cases is Case IV, Case III, and Case V. This trend indicates that higher temperatures have a more distinct impact on the airflow field.

3.2. Effect of Near-Wall Heat Source Temperature on Particle Deposition Distribution on the Back Wall

The temperature of the heat source exercises a substantial influence on the temperature and airflow fields, which in turn affects the distribution of particle deposition. Figure 6 presents scatter plots of particle deposition onto the back wall for various particle sizes under Cases IV, III, and V. Figure 6a displays that 0.01 μm particles are most distributed on the back wall and have the widest distribution region, which is distributed throughout the entire back wall region, with the most concentrated being the upper-left region of the back wall. The cause of this is that the release location of the particles is at an upper-left corner of the chamber, and particles of 0.01 μm are very distinctly subjected to thermophoretic forces. Upon release, they are carried upward by the thermal plume created by the heat source and deposited predominantly on the upper-left corner of the back wall. Additionally, since the direction of the velocity of the incoming air is set to be rightward, some 0.01 μm particles will be carried to the right side by the rightward airflow, and that makes the 0.01 μm particles spread out over the whole back wall region. From Figure 6a, it is observed that particles ranging from 0.5 to 10.0 μm are primarily distributed between the z = 0 and 0.1 m regions, with a denser distribution at the lower-right region of the back wall. This stems from the particles being influenced by gravity, moving downward, and by the incoming airflow, moving to the right. Consequently, these particles are mainly deposited in regions below z = 0.1 m and densely distributed in the lower-right position of the back wall. Additionally, Figure 6a reveals that the particle distribution in the upper-right corner of the back wall is sparse. This is due to the upper-right corner being far away from the inlet, where smaller particles are mainly affected by the thermophoretic forces and have been basically deposited before reaching this region. Meanwhile, larger particles are influenced by gravity and predominantly deposited to the lower region of the back wall, which leads to a reduced particle distribution at the upper-right area of the back wall.

Figure 6b indicates that the 0.01 μm particles have a more clustered deposition in the upper-left portion of the back wall, and the distribution decreases in other regions. This is due to the temperature of the heat source having gone up from 50 °C to 60 °C. The heightened heat source temperature enhances thermophoretic forces, rendering the influence on 0.01 μm particles more pronounced. Consequently, upon release at the inlet, these particles are entrained upward by the thermal plume and deposited predominantly in the upper-left corner of the back wall. Additionally, Figure 6b reveals that the primary distribution of particles in the range of 0.5 to 10.0 μm has expanded to the z = 0–0.2 m region. This is because the rise in the heat source temperature strengthens the upward thermal plume, driving them upward and ultimately causing a shift in the deposition location.

Figure 6c reveals that 0.01 μm particles are more densely accumulated in the upper-left corner of the back wall compared to Figure 6b, while the primary deposition location of particles of other sizes is further shifted upward, with fewer particles deposited in the region below z = 0.05 m. This is attributed to the enhanced upward transport of particles due to elevated heat source temperatures. This enhanced thermal plume makes it easier for the particles to be carried upward and eventually deposited at a higher location. From Figure 6c, it can also be noticed that the deposition of particles in the x = 0.6–0.8 m region is minimal. This is attributed to the elevated near-wall heat source temperature, which strengthens thermophoretic effects and enhances the upward thermal plume. As a result, most of the particles released from the inlet are carried upward by the thermal plume and deposited on the back wall and ceiling before reaching the x = 0.6–0.8 m region, resulting in a reduced deposition in this region.

Comparing Figure 6a–c, it is plainly observable that following the rise in the temperature of the heat source, the primary deposition positions of the particles on the back wall shift upwards. In addition, as the heat source temperature rises, the particle deposition quantity in the upper-right corner of the back wall gradually diminishes.

3.3. Effect of the Distance Between Near-Wall Heat Source and Back Wall on Particle Deposition Distribution on Back Wall

To investigate the influence that the distance from the heat source to the back wall has on the distribution of particle deposition on the back wall, it is of great importance to examine the sedimentation of particles of various sizes on the back wall. Figure 7 illustrates the number of particles of a different size deposited onto the back wall for Cases I, II, and III. As shown in Figure 7, with regard to the particles whose size ranges from 1.0 to 10.0 μm, the amount of particle deposition diminishes as the distance from the heat source to the back wall increases, and the tendencies are similar. When the particle size is 0.01–0.5 μm, the number of particles deposited fluctuates slightly as the distance from the heat source to the back wall varies, but generally speaking, the quantity of particles deposited decreases with increasing distance from the heat source to the back wall. The reason is that as the distance from the near-wall heat source to the back wall increases, the velocity of the airflow above the heat source decays faster. A faster decay of the airflow velocity results in a reduction of the velocity gradient in the region near the back wall, which in turn reduces the turbulence of the air, ultimately causing a reduction of the quantity of particles deposited on the back wall. Figure 7 reveals that when the distance from the heat source to the back wall rises from 20 mm to 30 mm, the situation of particle sedimentation on the back wall is more obvious than when the distance rises from 10 mm to 20 mm. This may be attributed to a more significant reduction in air velocity and turbulence intensity in the region near the back wall when the distance between the heat source and the back wall is increased from 20 mm to 30 mm compared to an increase from 10 mm to 20 mm. Furthermore, Figure 7 demonstrates that particles with sizes greater than 5.0 μm in diameter have minimal deposition on the back wall. Notably, for particles of 10.0 μm, deposition is negligible, and variations in the distance from the heat source bring about a scant impact on the deposition quantity. This is because for particles larger than 5.0 μm, gravity deposition plays a dominant role and particles are more prone to deposition on the floor of the chamber, which results in a decrease in the quantity of particles deposited on the back wall.

3.4. Particle Deposition in Different Regions of the Back Wall

To facilitate a detailed analysis of particle deposition patterns on the back wall, it was divided into eleven distinct regions. This enables a detailed examination of deposition variations within these specific regions. For a clearer presentation of the regional division, in Figure 8, we added colors to the various different regions of the back wall, thereby making the regional division easier to observe. The delineation of these regions is depicted in Figure 8.

To elucidate the spatial variations in particle deposition across the back wall, the deposition patterns observed in the different regions under Case IV are chosen for analysis. Figure 9 presents the particle deposition quantities in different regions of the back wall under Case IV. As shown in Figure 9a, it is evident that both Regions 1 and 11 exhibit a comparable trend in particle deposition for varying particle sizes. When the particle size enlarges from 0.01 to 0.1 μm, the quantity of deposited particles increases, which is probably explained by the more significant influence of the thermal plume on 0.01 μm particles. Thereby resulting in more 0.01 μm particles being carried by the airflow up to the ceiling of the chamber compared to 0.1 μm ones. Thus, compared with 0.01 μm particles, the deposition amount of 0.1 μm particles on the back wall is greater. The deposition patterns of particles within the 0.1–10.0 μm size range in Regions 1 and 11 display minor fluctuations. However, the overall trend reveals that particle deposition quantities decrease with increasing particle diameter. This trend is likely caused by the fact that when the particle diameter is on the rise, the particles are subjected to a greater degree of gravitational forces, leading to a higher concentration of particulates settling in the lower part of the chamber. Consequently, the sedimentation of particles in the back wall decreases. Figure 9a further reveals that the overall particle deposition in Region 1 surpasses that in Region 11. This disparity is likely attributable to the fact that Region 1 is in closer proximity to the particle inlet, thereby giving rise to a higher level of particle deposition in this particular region. In contrast, Region 11 is situated near the outlet and is relatively distant from the inlet. Consequently, by the time particles travel from the inlet to reach this region, only a limited number of particles remain. As a result, the particle deposition within Region 11 is comparatively lower than that in Region 1.

Figure 9b,c reveals that the quantity of particles deposited onto regions 2–7 increases significantly as the particle size increases from 0.01 μm to 0.1 μm. The reason for this may be that the smaller inertia of 0.01 μm particles, in which they are more vulnerable to the airflow effects and are carried by the thermal plume to the ceiling of the chamber, results in reduced deposition on the back wall. As a result, the number of 0.01 μm particles settled onto the back wall is less than that of 0.1 μm particles. Figure 9b,c indicates that in Regions 2–7, for particles of sizes between 0.1 and 10.0 μm, the deposition of particles exhibits minor fluctuations with increasing particle diameter. However, the overarching trend indicates that the amount of particle deposition diminishes with increasing particle size. This phenomenon arises due to the escalating inertial effects associated with larger particle diameters. As particle size increases, their inertia becomes more pronounced, enhancing the likelihood of particles moving downward. Consequently, larger particles have a tendency to deposit on the floor of the chamber, ultimately resulting in fewer particles being deposited in Regions 2–7. Additionally, Figure 9b,c demonstrates significantly higher particle deposition in Region 3 compared to Regions 2 and 4. Similarly, particle deposition in Region 6 is substantially greater than in Regions 5 and 7. The elevated particle deposition observed in Regions 3 and 6 can be attributed to their direct location above the heat source, which leads to stronger thermophoretic forces on particles in these regions, leading to enhanced deposition.

The quantity of particles deposited in Regions 8–10 generally tends to increase as the particle size grows, as illustrated in Figure 9d. The reason for this is that Regions 8–10 are located in the lower region of the back wall, and the particles are more significantly affected by gravity as the particle size grows. As a result, larger particles are carried by the airflow and deposited in these regions, ultimately resulting in an increased quantity of particles deposited in these three regions. It is also evident from Figure 9d that the number of deposited particles is generally lower in Region 9 than in Regions 8 and 10. This can be explained by the fact that Region 9 is in the region obscured by the heat source, and the heat source blocks part of the particles from depositing here, resulting in the total number of particles deposited here being smaller than in Regions 8 and 10.

3.5. Effect of Particle Inlet Position on Particle Deposition in Different Regions of the Back Wall

Based on the analysis presented earlier, it is evident that the thermal plume will enhance airflow near the wall’s surface, which in turn drives the particles to move upwards. This increases the probability of particles colliding with the back wall, resulting in particles being deposited on the back wall. To deeply explore the effects of the near-wall heat source on the deposition of particles on the back wall, the particle injection position was altered to the particle injection location ② and the quantity of particles deposited within distinct regions of the back wall was counted. Figure 10 illustrates the amount of particles deposited within distinct regions of the back wall under Case VI. As is clearly manifested in Figure 10, the deposition quantities of particles in Region 1 and Region 11, Region 2 and Region 4, Region 5 and Region 7, and Region 8 and Region 10 are in a relatively proximate state. Nevertheless, certain deviations can still be discerned. Specifically, Region 11 presents a marginally higher deposition quantity in comparison to Region 1. Region 4 exhibits a deposition amount that exceeds that of Region 2, Region 7 shows a deposition level that surpasses that of Region 5, and Region 10 has a deposition quantity that outpaces that of Region 8. This particular phenomenon can be accounted for by the fact that the particle injection position ② is symmetrically arranged along the central plane x = 0.4 m, and the heat source also exhibits such symmetry. Subsequently, the particulate matter deposited onto the back wall is fundamentally symmetrically distributed along the central plane x = 0.4 m, which thereby leads to the deposition numbers of particles in these regions being essentially alike. However, the incoming airflow from the chamber inlet and directed towards the right perturbs this symmetry. As an aftermath, the particles ejected from the injection position are influenced by the rightward airflow, inducing the particles to be transported by the rightward airflow and shift to the right. This, in turn, gives rise to a slightly greater number of particles being deposited in the right region of the x = 0.4 m plane rather than in the left region. Figure 10 illustrates that the deposition of particles in Regions 1 and 11 is relatively higher compared to other regions, which may be attributed to the larger area of these regions, increasing the likelihood of particles being deposited. Ultimately, this resulted in a higher number of particles being deposited in these two regions. Additionally, Figure 10 reveals that Regions 8–10 experience comparatively lower particle deposition. This is because these three regions are located below the particle injection position, and smaller particles are greatly influenced by upward thermal plumes, which have a lower probability of being deposited in these regions. The vast majority of particles deposited in these regions are larger particles that are primarily influenced by gravity; however, since these regions are closer to the floor, there is also a higher probability of larger particles depositing on the floor, leading to a lower overall deposition number in these regions. From Figure 10, it is also noticed that the amount of particles deposited in Region 2 surpasses that in Region 5, the amount of particles deposited in Region 3 exceeds that in Region 6, and the amount of particles deposited in Region 4 is larger than that in Region 7. The result may stem from two distinct factors. Firstly, it is plausible that the particle release location ② is in close proximity to Regions 5–7. Upon release, the particles exhibit an initial upward velocity, propelling them away from these three regions. Consequently, the likelihood of particle deposition diminishes within these three regions. Secondly, the lower portions of Regions 5–7 are close to the upper surface of the heat source. The intensified airflow velocity near this surface creates a formidable barrier that impedes particle deposition. These dual factors contribute to the observed deposition patterns: greater deposition in Region 2 compared to Region 5, larger deposition in Region 3 relative to Region 6, and heightened deposition in Region 4 surpassing that in Region 7.

4. Conclusions

In this study, we used a DPM to track the particle trajectories inside a small chamber. The effect of the surface temperature of a near-wall heat source and the distance from the heat source to the back wall on the distribution of particles with diameters between 0.01 and 10.0 μm deposited on the back wall were investigated. We arrived at the following conclusions:

• (1). As the temperature of the heat source became higher, the maximum value of airflow velocity and temperature became larger. Within the chamber, the region from the upper surface of the heat source to the ceiling, as well as the region near the ceiling, had a higher air velocity compared to other regions.

• (2). The primary deposition location of particles on the back wall shifted upward as the heat source temperature rose; the number of particles deposited on the x = 0.6–0.8 m region of the back wall decreased with increasing temperature.

• (3). The number of particles deposited onto the back wall diminished as the distance from the heat source to the back wall was enlarged; the decrease in the number of particles deposited onto the back wall was more pronounced when the distance from the heat source to the back wall was increased from 20 mm to 30 mm than when the distance was increased from 10 to 20 mm. Particles with a diameter exceeding 5.0 μm exhibited a notably lower deposition quantity on the back wall.

• (4). When the particle injection position was at location ①, the overall deposition quantity of particles with diameters from 0.1 to 10.0 μm in Regions 1 and 11 exhibited a decreasing trend with increasing particle diameter. Within Region 1, particle deposition exceeded that in Region 11. The deposition number of particles in Regions 8–10 increased overall with an increase in particle diameter, while the quantity of particles deposited in Region 9 was less than that in Regions 8 and 10.

• (5). With the particle injection position at location ②, the particle deposition distribution onto the back wall exhibited overall symmetry about the center plane (x = 0.4 m). However, a slight asymmetry was observed, with a marginally higher quantity of particles deposited onto the right side than the left side at x = 0.4 m. Additionally, particle deposition in Regions 8–10, located below the injection position, was lower.

The deposition distribution law under the attachment effect of the thermal plume above the near-wall heat source obtained in this study is positive and of great significance in effectively preventing the harmful deposition of particulates. It can actually improve indoor air quality and safeguard human health. Moreover, this mechanism also has considerable guiding value for the material selection and design of industrial equipment. Meanwhile, it can also contribute to the optimization of industrial process parameters. Although we have conducted a detailed study on the particle deposition on the back wall surface caused by the heat source, it is undeniable that there are still some limitations in this research. Specifically, we only considered the situation of a single heat source during the research process. However, in real life, the situation of multiple heat sources is common. The thermal plumes generated by multiple heat sources will couple with each other, and the impact of this coupling effect on particle deposition is significantly different from that in the case of a single heat source, which requires further in-depth study. In addition, regarding the deposition wall surface, we only limited it to a fixed value in this research. In fact, different deposition wall surfaces may vary in terms of material, roughness, and other aspects, and the impacts of these differences on particle deposition should be different, which also needs to be further explored in subsequent research.

Author Contributions

Validation, L.Y.; investigation, Y.L.; resources, J.L.; data curation, J.X.; writing—original draft preparation, P.W.; writing—review and editing, X.C. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Nomenclature

a	Acceleration
CD	Drag coefficient
DPM	Discrete Phase Model
DRW	Discrete Random Walk
Fa	Other forces, kg.ms−2
FD	Inverse of the relaxation time, s−1
g	Gravitational acceleration, ms−2
m	Particle mass
p	Pressure, Pa
PM	Particulate matter
RANS	Reynolds-Averaged Navier–Stokes
Re	Reynolds number
T	Temperature, K
u	Velocity, ms−1
z	Vertical coordinate, m
Greek symbols
α	Thermal diffusivity, m2 s−1
β	Thermal expansion coefficient, K−1
μ	Dynamic viscosity, kgm−1 s−1
ρ	Density, kgm−3
τ	Relaxation time
$\nabla$	Gradient operator
Superscripts
a	Air
p	Particle
0	Reference point

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Schematic of the model chamber.

Figure 2. Comparison of simulated airflow field with experimental data [19] in x − direction velocity at three different locations on the y = 0.2 m plane. (a) x = 0.2 m; (b) x = 0.4 m; (c) x = 0.6 m.

Figure 3. Comparison of normalized particle concentration with experimental data [19] at three distinct positions on the plane y = 0.2 m. (a) x = 0.2 m; (b) x = 0.4 m; (c) x = 0.6 m.

Figure 4. Temperature fields and airflow fields in a plane at x = 0.4 m. (a) Case III; (b) Case IV; (c) Case V.

Figure 5. Temperature fields and airflow fields in the plane y = 0.355 m. (a) Case III; (b) Case IV; (c) Case V.

Figure 6. Distribution of particle deposition on the back wall at different temperatures. (a) Case IV; (b) Case III; (c) Case V.

Figure 7. Number of particles deposited on the back wall for Cases I, II, and III.

Figure 8. Back wall zoning diagram.

Figure 9. Number of particles deposited in different regions of the back wall for different particle sizes in Case IV. (a) Region 1 and Region 11; (b) Regions 2–4; (c) Regions 5–7; (d) Regions 8–10.

Figure 10. Number of particles deposited in different regions of the back wall for Case VI.

Table 1

Parameters under different cases.

Case	Surface Temperature of Heat Source (°C)	Distance Between the Heat Source and the Back Wall (mm)	Particle Injection Location
Ⅰ	60	10	①
Ⅱ	60	20	①
Ⅲ	60	30	①
Ⅳ	50	30	①
Ⅴ	70	30	①
Ⅵ	60	30	②

Table 2

Grid independent validation.

Grid Model	Number of Grid Cells	Distance of the First Grid to the Back Wall (m)	Average Velocity at the Plane y = 0.2 m	Particle Deposition Fraction
1	1,249,471	0.0008	0.0307289 m/s	81.8444%
2	1,540,206	0.0005	0.0307209 m/s	80.7556%
3	1,879,647	0.0003	0.0307141 m/s	80.7333%

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