Influence mechanism of dynamic and static liquid bridge forces on particle deposition behaviors in solar photovoltaic mirrors

Abstract

Solar energy is one of the most promising forms of renewable energy for solving the energy crisis and environmental problems. Dust deposition on photovoltaic mirrors has a serious negative impact on the photoelectric conversion efficiency of solar power stations. In this paper, the influence mechanism of the dynamic and static liquid bridge forces on particle deposition behaviors on solar photovoltaic mirrors is investigated. In addition, the expression and physical meaning of the particle critical separation velocity are proposed. The research results show that the static liquid bridge force can be the primary deposition force causing dust particles to adhere to photovoltaic mirrors. However, the dynamic liquid bridge force can act as a resistance force for the particle motion process and even make dust particles roll along and finally stay on the mirror. The contact force is the primary separation force that causes dust particles to flow away from the mirror. Whether dust particles adhere to the mirror depends on the relative size of the deposition and separating forces. The particle critical separation velocity describes the relative size of the collision-rebound effect and mirror adhesion effect and is expressed in Eq. (16). These research findings can provide theoretical guidance for mirror cleaning methods in the operation process of photovoltaic mirrors.

Graphical abstract

Keywords

dust deposition / discrete element method (DEM) / photovoltaic mirrors / solar energy

1 Introduction

China’s traditional energy reserves are severely insufficient, and the energy gap is very large, substantially affecting its energy security. In addition, China’s coal-fired energy structure has led to serious pollution and environmental crises. Only the development of renewable energy can solve the energy security problem and environmental crisis of China [1]. Solar energy, as one of the most ideal renewable energy sources, has attracted much attention at home and abroad due to its advantages, such as safety, cleanliness, not threatening human beings and damaging the environment [2]. By the end of 2019, China’s cumulative photovoltaic installed capacity had reached 205 GW, accounting for 10% of the total installed capacity in China [3]. However, dust deposition on photovoltaic mirrors significantly decreases the working performance of the mirrors and thus reduces the solar energy conversion efficiency [4–6]. Thus, investigating particle deposition behaviors on solar photovoltaic mirrors is of great theoretical and engineering significance.

Most of the current research is focused on studying the influence of dust deposition on photovoltaic mirrors and solar photovoltaic power generation. Although the feasibility and reliability of solar power technology have brought considerable power generation benefits, dust deposition on solar photovoltaic mirrors has a negative impact on the efficiency of solar power generation. Because solar power plants are usually located in sunny areas, such as arid and semi-arid areas, they suffer from the loss of 1% or even more output power every day due to dust deposition on the solar mirrors [7,8]. Erdenedavaa et al. [9] investigated the effect of dust deposition on solar collectors and analyzed the best time of a year for cleaning by a transient systems (TRNSYS) simulation model. The results indicated that dust deposition had a negative effect on reflectivity and that the best cleaning time for solar collectors was between mid-January and early February. However, they did not analyze the force of dust particles in their work. The monthly fouling rate of arid and semi-arid areas in California is between 1% and 12%, while the fouling rate of California as a whole is very different. Dust deposition on solar photovoltaic mirrors can greatly reduce the conversion efficiency of a solar energy conversion system (photovoltaic panels, modules, and batteries). For example, at one site, the efficiency decreased by 17% after six days of operation and by 66% after six months of operation [10]. Therefore, studying particle motion characteristics can provide an important theoretical basis for guiding the mirror dust removal technology.

Some studies have investigated and assessed the dust deposition characteristics and current cleaning methods of solar photovoltaic mirrors. Mani and Pillai [11] analyzed the dust deposition trends of photovoltaic mirrors under different meteorological conditions. Hegazy [12] studied the microparticle deposition distribution on photovoltaic mirrors with different installation angles in solar power plants in central Egypt. Ahmed and Mohammed [13] studied the effects of dust deposition and environmental weather on the efficiency of solar cells in Iraq. They found that increasing dust accumulation and temperature, especially on warm and sunny days, can greatly reduce the efficiency of solar cells. Lu et al. [14–16] studied the effect of gravity force and particle motion velocity on dust motion behaviors by utilizing the computational fluid dynamics-discrete particle model (CFD-DPM) method. They found that gravity played an important role in the dust deposition rate and that different particle motion velocity had a different dust deposition rate. It can be inferred that there is a particle critical separation velocity at which dust particles flow away from the mirrors. Investigating the particle critical separation velocity can be helpful to remove dust particles from mirrors. However, to date, few studies have been conducted on particle critical separation velocity. At present, solar power plants adopt natural cleaning (wind, rain, snow, etc.), regular manual or automatic mechanical cleaning (soot blowers, brushes, cleaning robots, etc.), and similar methods to clean photovoltaic mirrors. Khadhim et al. [17] performed many experiments to study the effect of periodic cleaning on solar panel output power. They found that dust accumulation on solar panels decreased with increasing cleaning frequency. However, the abovementioned current cleaning methods are inefficient and expensive. Chesnutt et al. [18] studied the flow characteristics of individual spherical dust particles in solar photovoltaics (PVs) with an electrodynamic dust shield (EDS). They found that the cleaning efficiency of PV panels is closely related to the EDS design and operating parameters. However, the interactions among dust particles remain unclear. The forces acting on dust particles were analyzed and it was pointed out that the liquid bridge force should not be ignored during the particle deposition process [19]. However, few studies focused on the effect of liquid bridge forces on the dust deposition characteristics of photovoltaic mirrors. This oversight has led to a lack of theoretical guidance on mirror cleaning methods in the operation process of photovoltaic mirrors.

The discrete element method (DEM) was first proposed by Cundall in the 1970s to analyze the mechanical behavior of particle populations. The basic idea of DEM is to assume that the research object is a collection of rigid elements. Based on Newton’s Second Law, the motion process of each element is solved by the central difference method to obtain the overall flow phenomenon of the research object. Recently, the DEM method has become a popular simulation tool for investigating the particle motion process. Many researchers study the flow behavior and multiphase transport phenomena of fluids and particles in circulating fluidized beds, pneumatic beds, cyclones, impinging flow reactors, and pneumatic conveyors by using the DEM method [20–24]. The data obtained based on the above-mentioned DEM method, such as particle mixing, minimum fluidization velocity, minimum jet velocity, and bubble formation, are basically consistent with the experimental data and actual flow conditions. Therefore, it is feasible to study the particle motion process and deposition behaviors of the flow field of photovoltaic mirrors by adopting the DEM method.

To date, little information can be found which expounds the effect of liquid bridge force on particle deposition behaviors in solar photovoltaic mirrors. The liquid bridge force consists of dynamic and static force components which have a great impact on particle aggregation and adhesion behaviors [25,26]. In particular, few investigations on the effect of the dynamic liquid force on particle motion characteristics in the flow field of photovoltaic mirrors are available. Studying the effect of both dynamic and static liquid bridge forces on the particle motion process is of great importance for particle motion characteristics. The objective of this paper is to investigate the influence mechanism of dynamic and static liquid bridge forces on particle deposition behaviors in solar photovoltaic mirrors. First, it establishes the liquid force model and motion model of dust particles via the DEM method and introduces the model setting and verification, and working conditions. Then, it studies particle deposition dynamics, describes the effect of both dynamic and static liquid bridge force on particle deposition behaviors, and discusses the particle critical separation velocity. Finally, it concludes the research work conducted in this paper and proposes directions for future research.

2 Model description

This paper adopts a DEM method to study and discuss particle deposition characteristics and to discuss the effect of the liquid bridge force on particle motion behaviors in the flow field of photovoltaic mirrors.

2.1 Liquid bridge force model

In a certain humidity environment, a water film layer is formed on the surface of particles, a liquid bridge is formed between particles and other particles or walls, and then the liquid bridge force is generated. The static liquid bridge force and the dynamic liquid bridge force compose the liquid bridge force. The static liquid bridge force (F_l,s) is defined by the Makima et al. model [25], as expressed in Eqs. (1–3).

(1)F_{\mathrm{l,s}}=\mathrm{\pi }r\mathit{\gamma }({\mathrm{e}}^{\frac{AD}{\mathrm{2}r+B}}+C),

A=-\mathrm{1.1}{(V*)}^{-\mathrm{0.53}},

B=(-\mathrm{0.34}\ln V*-\mathrm{0.96}){\theta }_{\mathrm{1}}^{\mathrm{2}}-\mathrm{0.019}\ln V*+\mathrm{0.48},

(2)C=\mathrm{0.042}\ln V*+\mathrm{0.78},

A=-\mathrm{1.9}{(V*)}^{-\mathrm{0.51}},

B=(-\mathrm{0.016}\ln V*-\mathrm{0.76}){\theta }_{\mathrm{1}}^{\mathrm{2}}-\mathrm{0.12}\ln V*+\mathrm{1.2},

(3)C=\mathrm{0.013}\ln V*+\mathrm{0.18},

where γ represents the liquid surface tension, r is the particle radius, and D is the distance between two particles or a particle and mirror. Equation (2) shows the expression for parameters A, B, and C for the contact between particles. Similarly, Eq. (3) shows the expression for parameters A, B, and C for the contact between a particle and photovoltaic mirror. The dynamic liquid bridge force (F_l,d) consists of the tangential viscous component (F_l,dt) and the normal viscous component (F_l,dn) and is defined by Pitois et al. [27], as demonstrated in Eqs. (4–6).

(4)F_{\mathrm{l},\mathrm{d}}=F_{\mathrm{l},\mathrm{d}\mathrm{n}}+F_{\mathrm{l},\mathrm{d}\mathrm{t}},

(5)F_{\mathrm{l,dn}}=\mathrm{6}\mathrm{\pi }{\mu }_{\mathrm{l}}R{*}^{\mathrm{2}}{\displaystyle \frac{v_{\mathrm{r,n}}}{D}}{(\mathrm{1}-\mathrm{1}/\sqrt{\mathrm{1}+V*/(\mathrm{\pi }R*D^{\mathrm{2}})})}^{\mathrm{2}},

(6)F_{\mathrm{l,dt}}=\mathrm{6}\mathrm{\pi }{\mu }_{\mathrm{l}}R*v_{\mathrm{r,t}}({\displaystyle \frac{\mathrm{8}}{\mathrm{15}}}\ln {\displaystyle \frac{R*}{D}}+\mathrm{0.9588}),

where v_r,n is the normal relative velocity, v_r,t is the tangential relative velocity, R* is the equivalent radius, and μ_l is the dynamic viscosity. Thus, the liquid bridge force (F_l) is exhibited in Eq. (7).

(7){\overrightarrow{F}}_{\mathrm{l}}={\overrightarrow{F}}_{\mathrm{l,d}}+{\overrightarrow{F}}_{\mathrm{l,s}}.

2.2 Particle motion model

Based on Newton’s Second Law, the particle motion equation is depicted in Eq. (8), as suggested by Liu et al. [19,28].

(8)m_{\mathrm{p}}{\displaystyle \frac{\mathrm{d}{\overrightarrow{u}}_{\mathrm{p}}}{\mathrm{d}t}}={\overrightarrow{F}}_{\mathrm{g}}+{\overrightarrow{F}}_{\mathrm{b}}+{\overrightarrow{F}}_{\mathrm{e}}+{\overrightarrow{F}}_{\mathrm{v}}+{\overrightarrow{F}}_{\mathrm{l}}+{\overrightarrow{F}}_{\mathrm{c}}+{\overrightarrow{F}}_x,

where F_b and F_g are the buoyancy force and gravity force, respectively, as displayed in Eq. (9); F_e is electrostatic force, as shown in Eq. (10) [29,30]; F_v is the van der Waals force, as shown in Eq. (11) [31]; F_c is the contact force and consists of the tangential component (F_ct) and the normal component (F_cn), represented by the Hertz-Mindlin with JKR model, as shown in Eqs. (12–14) [32]; and F_x represents other forces, such as the thermal swimming force and the added mass force, etc. The abovementioned forces (F_x) can be neglected in this paper, as suggested by Zhong et al. [33].

(9)F_{\mathrm{g}}=\mathrm{\pi }{d}_{\mathrm{p}}^3{\rho }_{\mathrm{p}}g/\mathrm{6}{,}_{}{F}_{\mathrm{b}}=\mathrm{\pi }{d}_{\mathrm{p}}^3\rho g/\mathrm{6},

(10){\overrightarrow{F}}_{\mathrm{e}}={\overrightarrow{F}}_{\mathrm{ee}}+{\overrightarrow{F}}_{\mathrm{el}}+{\overrightarrow{F}}_{\mathrm{es}},

(11){\overrightarrow{F}}_{\mathrm{v}}=-{\displaystyle \frac{\varpi }{\mathrm{8}\mathrm{\pi }}}\left[{\displaystyle \frac{r}{l_{\mathrm{0}}^{\mathrm{2}}}}+{\displaystyle \frac{r}{{\left(l_{\mathrm{0}}+\mathrm{2}r\right)}^{\mathrm{2}}}}-{\displaystyle \frac{\mathrm{2}r}{l_{\mathrm{0}}\left(l_{\mathrm{0}}+\mathrm{2}r\right)}}\right],

(12){\overrightarrow{F}}_{\mathrm{c}\mathrm{n}ij}=(-k_{\mathrm{n}}{\delta }_{\mathrm{n}}^{1.5}-c_{\mathrm{n}}{\overrightarrow{v}}_{ij}·\overrightarrow{n})\overrightarrow{n},

(13){\overrightarrow{F}}_{\mathrm{c}\mathrm{t}ij}=-k_{\mathrm{t}}{\overrightarrow{\delta }}_{\mathrm{t}}-c_{\mathrm{t}}{\overrightarrow{v}}_{\mathrm{s}},\left|{\overrightarrow{F}}_{\mathrm{c}\mathrm{t}ij}\right|\le {\mu }_{\mathrm{s}}\left|{\overrightarrow{F}}_{\mathrm{c}\mathrm{n}ij}\right|,

(14){\overrightarrow{F}}_{\mathrm{c}\mathrm{t}}=-{\mu }_{\mathrm{s}}\left|{\overrightarrow{F}}_{\mathrm{c}\mathrm{n}ij}\right|{\overrightarrow{n}}_{\mathrm{t}},\left|{\overrightarrow{F}}_{\mathrm{c}\mathrm{t}ij}\right|>{\mu }_{\mathrm{s}}\left|{\overrightarrow{F}}_{\mathrm{c}\mathrm{n}ij}\right|,

where d_p and ρ_p are particle diameter and density, respectively; ρ is the air density; F_ee, F_el, and F_es are the mirror electrostatic force, the electrical double layer force, and the electric field force, respectively; l₀ denotes the average distance between molecules; \varpi is the Lifshitz constant; k_t and k_n denote the spring constant in the tangential and the normal direction, respectively; δ_n and δ_t are the tangential displacement and the normal displacement, respectively; c_t and c_n represent the damping coefficient in the tangential and the normal direction, respectively; μ_s is the friction coefficient; {\overrightarrow{v}}_{ij} denotes the relative velocity between particles i and j; \overrightarrow{n} represents the unit vector from particle i to particle j; {\overrightarrow{n}}_{\mathrm{t}} is the tangential unit vector; and {\overrightarrow{v}}_{\mathrm{s}} is the sliding velocity vector.

Figure 1 shows the schematic diagram of the forces acting on a dust particle on an inclined mirror. The electrostatic force, the van der Waals force, the liquid bridge force, gravity force, etc. can make particles deposit on mirrors. It is previously pointed out that the liquid bridge force is much greater than the electrostatic force, the van der Waals force, and gravity force [19]. However, the air buoyancy force, the collision force, etc. can make particles leave the mirror. When the liquid bridge force of a particle is greater than the combined force of contact force and buoyancy force in the vertical direction of the photovoltaic mirror, the particle will deposit on the mirror.

2.3 Model setting and verification

It is assumed in this paper that all dust particles with a uniform spherical shape do not undergo shape deformation in the particle motion process, neglecting particle-particle and particle-wall heat and mass transfer and particle rotation. Figure 2 shows the computational domain of the solar photovoltaic mirror and dust particle. The photovoltaic mirror is an inclined plane with a length and width of 0.3 mm. The research area is the area within the red frame shown in Fig. 2. The dust particles are generated on the left side of the computational domain. The incident angle is defined as the angle between the incident particle velocity and the mirror surface. The working conditions and related factors, including calculation parameters, physical parameters, and contact parameters, are summa-rized in Table 1. Meng measured the maximum value of moisture content of dust particles and found that the moisture content of dust was about 0.26%, which corresponds to a dimensionless liquid bridge volume of approximately 1.5% [34]. To investigate the influence of the particle moisture content on particle motion behaviors, eight cases are selected, with dimensionless liquid bridge volumes from 0 to 1.5%, as shown in Table 1. Similarly, to investigate the effect of the incident angle on the particle motion characteristics, five cases are selected, with particle incident angles (α) of 10°, 16°, 20°, 30°, and 40°, as presented in Table 1. In addition, the conditions used to study the effects of particle size and mirror installation angle on the particle deposition process are also given in Table 1. To study the particle deposition behaviors, the particle time step is 10⁻⁶ s, as recommended by Tsuji et al. [22].

The software EDEM 2018 with the soft-sphere model is adopted to study the influence mechanism of dynamic and static liquid bridge forces on particle deposition behaviors in solar photovoltaic mirrors. To verify the correctness of the EDEM simulation value of liquid bridge force, the theoretical and simulated values are compared in Fig. 3. Besides, to verify the accuracy of the particle motion model, EDEM simulation result of wet particle stacking experiment are shown in Fig. 4. It can be observed from Fig. 3 that the relative error of the liquid bridge force between the simulation value and theoretical value is less than 3.0% with consideration of the dynamic liquid bridge force. The liquid bridge force models built by Makimi et al. [25] and Pitois et al. [27] are reliable. It can be seen from Fig. 4 that the stacking angle obtained in this paper is 40°. With value of the stacking angle obtained by Xu [35] about 43°, the relative error of Fig. 4 between the results of Xu [35] and those of the simulation is less than 7.0%. Thus, the particle motion model established in this paper is acceptable and can be adopted.

3 Results and discussion

3.1 Particle deposition dynamics

Studying particle motion behaviors is of great importance for particle deposition dynamics in the flow field of photovoltaic mirrors. The motion behaviors of dust particles can be divided into three motion models after particle collision with a clear photovoltaic mirror [19], as shown in Figs. 5–7. Figure 5 shows the particle motion process at V* = 0.01%, α = 30°, and u₀ = 0.5 m/s. The dust particle shown in Figs. 5(a) and 5(b) will adhere to the mirror with or without consideration of the dynamic liquid bridge force. The dust particle shown in Fig. 5(c) will leave the mirror without consideration of the liquid bridge force. The liquid bridge force consists of the static and the dynamic liquid bridge forces. The liquid bridge force can cause dust particles to be deposited on the mirror. Besides, the static liquid bridge force plays a leading role in the particle deposition behaviors of Fig. 5. Moreover, a small particle inlet velocity causes a weak effect of the dynamic bridge force on the particle deposition behaviors. Thus, the influence of the liquid bridge force on particle motion dynamics should not be neglected. Furthermore, the impacts of the static liquid bridge force and dynamic liquid bridge force are different in the particle motion process.

Figure 6 shows the particle motion process at V* = 0.01%, α = 30°, and u₀ = 5.0 m/s, in which, the dust particle shown in Fig. 6(a) will stay on the mirror after rolling along the mirror with consideration of the dynamic liquid bridge force; however, the dust particle shown in Figs. 6(b) and 6(c) will leave the mirror without consideration of the dynamic liquid bridge force and liquid bridge force. Compared with Fig. 5, it can be concluded that increasing the particle inlet velocity can improve the effect of the dynamic liquid force on the particle motion characteristics. Moreover, the dynamic liquid bridge force has a great impact on the particle deposition behaviors in Fig. 6. Thus, the influence of the dynamic liquid bridge force on particle deposition dynamics should not be neglected. Furthermore, the dynamic liquid bridge force can be a leading force in the particle motion process.

Figure 7 shows the particle motion process at V* = 0.01%, α = 30°, and u₀ = 15.0 m/s. The dust particles shown in Fig. 7 will leave the mirror without or with consideration of the dynamic liquid bridge force and the liquid bridge force. A comparison of Figs. 5 and 6 indicate that increasing the particle inlet velocity can make the dust particle rebound after collision with the mirror, and the contact force can overcome the liquid bridge force working in the particle motion process. Thus, increasing the particle inlet velocity can shorten the impact of the liquid bridge force on particle motion dynamics.

The results of Figs. 5–7 show that dust particles can adhere to, roll and stay on, and flow away from a mirror after particle collision with the mirror. The same motion characteristics can be found in studying the flow field of dusty photovoltaic mirrors [19]. Under certain air humidity conditions, the liquid bridge force can cause dust particles to be deposited on the mirror. The effects of the dynamic liquid bridge force and the static liquid bridge force on particle deposition dynamics differ with increasing particle inlet velocity. Thus, the study of the influence mechanism of the dynamic and the static liquid bridge forces on particle deposition behaviors in solar photovoltaic mirrors is very helpful.

3.2 Effect of liquid bridge force on particle deposition behaviors

Based on the above analysis of Figs. 5–7, the investigation of the impact of the dynamic and the static liquid bridge forces on particle deposition behaviors can provide a theoretical basis for removing dust particles from photovoltaic mirrors. Figure 8 describes the influence of the liquid bridge force on particle motion behaviors at u₀ = 0.5 m/s. In Fig. 8(a), the static liquid bridge force is greater than the dynamic liquid bridge force after particle collision with the mirror and can act as an attractive force to make the dust particle adhere to the mirror. In Fig. 8(b), due to the large distance between the dust particle and the mirror, there is no obvious change in the dynamic liquid bridge force. When the dust particle is close enough to the mirror, the dynamic liquid bridge force changes suddenly and reaches the maximum value. After that, the particle collides with the mirror, and the particle velocity sharply decreases until reaching zero, so does the dynamic liquid bridge force. Thus, the dynamic liquid bridge force is a repulsion force that slows down the particle before contacting the mirror. However, without consideration of the dynamic liquid bridge force, the particle will attach to the mirror and then produce elastic deformation. Meanwhile, part of the kinetic energy will be stored as deformation energy. In the process of deformation recovery, part of this energy is converted into kinetic energy, and the other part is converted into heat energy. After this process repeats, the particle velocity curve exhibits damping oscillation, as shown in Fig. 8(b). The motion characteristics shown in Fig. 5 are in good agreement with the results in Fig. 8. Therefore, the static liquid bridge force is an attractive force during the contact process with the mirror, while the dynamic liquid bridge force is a repulsion force during the motion process, causing the dust particle to decelerate and finally adhere to the mirror.

Figure 9 shows the influence of the liquid bridge force on particle motion behaviors at u₀ = 5.0 m/s. Based on Eqs. (5) and (6), the maximum value of the dynamic liquid bridge force in Fig. 9(a) is greater than that in Fig. 8(a) at u₀ = 0.5 m/s. In addition, the dynamic bridge force does not rapidly decrease to zero after particle collision with the mirror, but gradually decreases for a certain time and finally reaches zero. Similarly, the static liquid bridge force is also greater than the dynamic liquid bridge force after particle collision with the mirror. In Fig. 9(b), the dust particle will decelerate and finally stay on or leave from the mirror after contacting the mirror. Without consideration of the dynamic liquid bridge force, the particle flows away from the mirror due to the coupled effect of the static liquid bridge force and contact force. However, the results of Figs. 6(a) and 6(b) indicate that the tangential viscous component of the dynamic liquid bridge force causes dust particles to roll along the mirror and finally stay on the mirror. Increasing the particle inlet velocity can result in a tangential velocity component during the particle motion process. Thus, the normal viscous component of the dynamic liquid bridge force hinders the normal motion of the dust particle, while the tangential viscous component of the dynamic liquid bridge force causes the dust particle to roll along the mirror with increasing particle inlet velocity. The static liquid bridge force is the primary force that makes the dust particle finally adhere to the mirror. Increasing the particle inlet velocity can increase the effect of the dynamic liquid force on particle deposition behaviors. The dynamic liquid force cannot be neglected in studying the motion process in the flow field of photovoltaic mirrors. Furthermore, at u₀ = 15.0 m/s, as shown in Fig. 7, the dust particle rebounds after colliding with the mirror. This rebound occurs because there is still enough kinetic energy to leave from the computation domain due to the great effect of the contact force of particles. The effect of the liquid bridge force on particle motion characteristics does not play a primary role in particle deposition behaviors when the particle inlet velocity greatly increases.

Equations (4–6) show that the dynamic liquid bridge force increases with increasing particle inlet velocity. Although the acting time and distance of the dynamic liquid bridge force are very small, the particles start to decelerate before contacting the mirror. In addition, the dust particles can still adhere to the mirror at a high particle inlet velocity. Thus, the dynamic liquid bridge force has a great impact on particle motion behaviors. Figure 10 shows the ratio of the dynamic bridge force to the static liquid bridge force at different inlet velocities. It can be seen from Fig. 10(a) that the maximum value of the dynamic liquid bridge force increases and the peak curves of Fig. 10(a) shift to the left as the particle inlet velocity increases. Additionally, for the three curves, the ratio of the dynamic liquid bridge force to the static liquid bridge force first decreases, then increases, and finally decreases to zero. This process occurs because the dynamic liquid bridge force and the static liquid bridge force increase with decreasing liquid bridge length; the increase trend is shown in Fig. 10(b). After the dust particle collides with the mirror, the particle velocity decreases and even reaches zero, resulting in the dynamic liquid bridge force reducing to zero. Therefore, increasing the particle inlet velocity can cause some particles to flow away from the mirror for a high contact force, but can decrease the particle motion velocity and cause the particles to be deposited on the mirror because the dynamic liquid bridge force plays a great resistance role in particle motions.

3.3 Particle critical separation velocity

The results of Figs. 8–10 show that when dust particles collide with photovoltaic mirrors, the dust particles can either adhere or rebound, depending on the relative size of the deposition and separating forces. In addition, different particle inlet velocities can cause dust particles to stay on the mirror or leave from the mirror. Thus, the particle critical separation velocity can be defined as the particle inlet velocity where a dust particle flows away from the mirror. The particle size, the mirror installation angle, the particle incident angle, the dimensionless liquid bridge volume, etc. play important roles in particle motion behaviors. Investigating the particle critical separation velocity is very helpful in revealing the influence mechanism of the dynamic and the static liquid bridge forces on particle deposition behaviors in solar photovoltaic mirrors.

(1) Particle size and mirror installation angle

Figure 11(a) describes the influence of the particle radius on the particle critical separation velocity. Figure 11(a) indicates that the particle critical separation velocity increases when the particle radius decreases. The result of Fig. 11(a) is consistent with the conclusion of Qasem et al. [36]. The effects of the electrostatic force, the van der Waals force, gravity force, etc. on particle sizes between 10 μm and 30 μm are much weaker than those of the liquid bridge force in a certain air humidity environment. The effect of the liquid bridge force on small particles is greater than that on large particles due to the weak inertial effect. Thus, compared with large particles, small particles have a higher particle critical separation velocity and are more likely to adhere to solar photovoltaic mirrors.

Figure 11(b) describes the influence of the mirror installation angle on the particle critical separation velocity. Figure 11(b) indicates that the particle critical separation velocity decreases when the mirror installation angle increases. The reason for this is that the contact force increases with increasing mirror installation angle and is a separation force that causes the particle to flow away from the mirror. Increasing the mirror installation angle can increase the collision velocity with the mirror and thus weaken the effect of liquid bridge forces. The resistance effect of the dynamic liquid bridge force on particle motion characteristics is weaker than the separation effect of the particle contact force. Thus, solar photovoltaic mirrors with small installation angles have large particle separation velocities, and many dust particles tend to be deposited on the mirrors.

(2) Particle incident angle and dimensionless liquid bridge volume

Figure 12(a) describes the influence of particle incident angle on particle critical separation velocity. In Fig. 12(a), the particle critical separation velocity decreases with increasing particle incident angle. The particle incident angle affects both the tangential and normal component velocities. Increasing the normal component inlet velocity can increase the contact force and the dynamic liquid bridge force, making the particles to fly away from the mirror. Besides, the normal component inlet velocity increases with increasing particle incident angle. The contact force plays a negative role in particle deposition characteristics, and the dynamic liquid bridge force has a resistance effect on particle motion characteristics. The greater effect of the contact force than dynamic liquid bridge force leads to the fact that the particle flows away from the mirror. Figure 12(b) describes the influence of dimensionless liquid bridge volume on the particle critical separation velocity. Figure 12(b) shows that the particle critical separation velocity increases when the dimensionless liquid bridge volume increases. The reason for this is that increasing the dimensionless liquid bridge volume can improve the effect of the static liquid bridge force based on Eq. (1). Thus, increasing the particle incident angle and decreasing the dimensionless liquid bridge volume can decrease the particle critical separation velocity, reducing the deposition degree of solar photovoltaic mirrors.

By analysis and fitting of the data in both Figs. 11 and 12 via MATLAB R2016a, the relationship among particle critical separation velocity (u_c), dimensionless liquid bridge volume (V*), incidence angle (α), particle size (r), and mirror installation angle (θ) can be obtained, as shown in Eqs. (15) and (16) (k is constant). Equation (16) is available in the parameter range shown in Eq. (17). The particle critical separation velocity can be determined in similar ways to that for given particle physical properties. Increasing the factors r, θ, and α or decreasing the factor V* can weaken the particle critical separation velocity. In this paper, the contact force is the primary separation force that causes dust particles to leave, while the liquid bridge force is the primary deposition force that makes particles adhere to the mirror. The static liquid bridge force causes attraction to the mirror, and the dynamic liquid force can hinder particle motion to decrease the collision-bounce effect. Whether the collision-bounce effect can overcome the effect of the liquid bridge force depends on the particle critical separation velocity. When the particle inlet velocity is smaller than the particle critical separation velocity, the dust particle will deposit on the mirror and vice versa. Thus, the particle critical separation velocity reflects the relative size of the collision-rebound effect and the mirror adhesion effect. The main adhesion force is the liquid bridge force. In addition, Eq. (16) reveals the effects of air humidity, particle size, solar photovoltaic mirror parameters, and particle inlet direction on particle deposition behaviors in the flow field of photovoltaic mirrors. Decreasing the particle critical separation velocity can shorten the mirror deposition degree and guide dust removal from the mirror.

(15)u_{\mathrm{c}}=f(r,V*,\alpha ,\theta ),

(16)u_{\mathrm{c}}=k{\mathrm{e}}^{-\mathrm{0.027}\theta -\mathrm{3185}r}(\mathrm{0.0032}{\alpha }^{\mathrm{2}}-\mathrm{0.0022}\alpha +\mathrm{4.7078})(\mathrm{3518.7}V*-\mathrm{3.22}),

(17)\begin{array}{l}\mathrm{10}\le r\le \mathrm{30}\mathrm{;}\mathrm{}\mathrm{25}°\le \theta \le \mathrm{45}°,\hfill \\ \mathrm{0.1}\mathrm{\%}\le V*\le \mathrm{0.5}\mathrm{\%}\mathrm{;}\mathrm{}\mathrm{10}°\le \alpha \le \mathrm{40}°.\hfill \end{array}

4 Conclusions

In this paper, many investigations have been conducted on particle deposition dynamics in the flow field of solar photovoltaic mirrors. In addition, the effects of the static and the dynamic liquid bridge forces on particle deposition behaviors have been analyzed. Moreover, the expression and physical meaning of the particle critical separation velocity have been described. The following conclusions are drawn:

The static liquid bridge force can be the primary deposition force causing dust particles to adhere to photovoltaic mirrors. However, the dynamic liquid bridge force can act as a resistance force for the particle motion process and even make dust particles roll along the mirror and finally stay on the mirror. The contact force is the primary separation force that causes dust particles to fly away from the mirror.

Increasing the particle inlet velocity can increase the contact force to make dust particles leave the mirror, but will decrease the particle motion velocity and cause particles to be deposited on the mirror. Whether dust particles adhere to the mirror depends on the relative size of the deposition and separating forces.

The particle critical separation velocity is proposed to describe the relative size of the collision-rebound effect and the mirror adhesion effect, as shown in Eqs. (16) and (17). Decreasing the particle critical separation velocity can decrease the mirror deposition degree and promote dust removal from the mirror.

The study of the influence mechanism of the dynamic and static liquid bridge forces on particle deposition behaviors in solar photovoltaic mirrors in this paper is expected to be of great theoretical significance. Studies on the effective removal of dust particles from solar photovoltaic mirrors will be conducted in the future.

References

[1] Mekhilef S, Saidur R, Safari A. A review on solar energy use in industries. Renewable & Sustainable Energy Reviews, 2011, 15(4): 1777–1790

[2] National Renewable Energy Laboratory. Realizing a clean energy future: highlights of NREL analysis (Brochure). NREL/BR-6A20–60894, 2013

[3] Wang S, Cui X. Photovoltaic will become the most important power source in the world. China Science Journal, 2020, 26: 003

[4] Yaghoubi M, Niknia I, Kanaan P, Experimental study of dust deposition effect on the performances of parabolic trough collectors. In: Proceedings of 17th Solar Paces Conference, Granada, Spain, 2011

[5] Şahin A D. A new formulation for solar irradiation and sunshine duration estimation. International Journal of Energy Research, 2007, 31(2): 109–118

[6] Sansoni P, Fontani D, Francini F, Optical collection efficiency and orientation of a solar trough medium-power plant installed in Italy. Renewable Energy, 2011, 36(9): 2341–2347

[7] Sarver T, Al-Qaraghuli A, Kazmerski L L. A comprehensive review of the impact of dust on the use of solar energy: history, investigations, results, literature, and mitigation approaches. Renewable & Sustainable Energy Reviews, 2013, 22: 698–733

[8] Sayyah A, Horenstein M N, Mazumder M K. Energy yield loss caused by dust deposition on photovoltaic panels. Solar Energy, 2014, 107: 576–604

[9] Erdenedavaa P, Rosato A, Adiyabat A, Model analysis of solar thermal system with the effect of dust deposition on the collectors. Energies, 2018, 11(7): 1795

[10] Caron J R, Littmann B. Direct monitoring of energy lost due to soiling on first solar modules in California. IEEE Journal of Photovoltaics, 2013, 3(1): 336–340

[11] Mani M, Pillai R. Impact of dust on solar photovoltaic (PV) performance: research status, challenges and recommendations. Renewable & Sustainable Energy Reviews, 2010, 14(9): 3124–3131

[12] Hegazy A A. Effect of dust accumulation on solar transmittance through glass covers of plate-type collectors. Renewable Energy, 2001, 22(4): 525–540

[13] Ahmed O K, Mohammed Z A. Dust effect on the performance of the hybrid PV/thermal collector. Thermal Science and Engineering Progress, 2017, 3: 114–122

[14] Lu H, Zhao W. CFD prediction of dust pollution and impact on an isolated groundmounted solar photovoltaic system. Renewable Energy, 2019, 131: 829–840

[15] Lu H, Lu L, Wang Y. Numerical investigation of dust pollution on a solar photovoltaic (PV) system mounted on an isolated building. Applied Energy, 2016, 180: 27–36

[16] Lu H, Zhang L. Numerical study of dry deposition of monodisperse and polydisperse dust on building-mounted solar photovoltaic panels with different roof inclinations. Solar Energy, 2018, 176: 535–544

[17] Khadhim I J, Mehdi I J, Muhsson I M. Periodic cleaning effect on the output power of solar panels. In: 2nd Scientific Conference, Karbala University, 2014

[18] Chesnutt J, Ashkanani H, Guo B, Simulation of microscale particle interactions for optimization of an electrodynamic dust shield to clean desert dust from solar panels. Solar Energy, 2017, 155: 1197–1207

[19] Liu X, Yue S, Lu L, Study on dust deposition mechanics on solar mirrors in a solar power plant. Energies, 2019, 12(23): 4550

[20] Chu K W, Wang B, Xu D L, CFD-DEM simulation of the gas-solid flow in a cyclone separator. Chemical Engineering Science, 2011, 66(5): 834–847

[21] Sae-Heng S, Swasdisevi T, Amornkitbamrung M. Investigation of temperature distribution and heat transfer in fluidized bed using a combined CFD-DEM model. Drying Technology, 2011, 29(6): 697–708

[22] Tsuji Y, Kawaguchi T, Tanaka T. Discrete particle simulation of two-dimensional fluidized bed. Powder Technology, 1993, 77(1): 79–87

[23] Xu B H, Yu A B. Numerical simulation of the gas-solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics. Chemical Engineering Science, 1997, 52(16): 2785–2809

[24] Zhu R R, Zhu W B, Xing L C, DEM simulation on particle mixing in dry and wet particles spouted bed. Powder Technology, 2011, 210(1): 73–81

[25] Mikami T, Kamiya H, Horio M. Numerical simulation of cohesive powder behavior in a fluidized bed. Chemical Engineering Science, 1998, 53(10): 1927–1940

[26] Hotta K, Takeda K, Iinoya K. The capillary binding force of a liquid bridge. Powder Technology, 1974, 10(4–5): 231–242

[27] Pitois O, Moucheront P, Chateau X. Liquid bridge between two moving spheres: an experimental study of viscosity effects. Journal of Colloid and Interface Science, 2000, 231(1): 26–31

[28] Liu X, Yue S, Lu L, Study on single-particle residence time of impulse, symmetric and asymmetric coaxial impinging streams. Powder Technology, 2019, 342: 118–130

[29] Behrens S H, Grier D G. The charge of glass and silica surfaces. Journal of Chemical Physics, 2001, 115(14): 6716–6721

[30] Bowling R A. An analysis of particle adhesion on semiconductor surfaces. Journal of the Electrochemical Society, 1985, 132(9): 2208–2214

[31] Dzyaloshinskii I E, Lifshitz E M, Pitaevskii L P. The general theory of van der Waals forces. Advances in Physics, 1961, 10(38): 165–209

[32] Liu X, Yue S, Lu L, Simulations of an asymmetric gas-solid two-phase impinging stream reactor. Numerical Heat Transfer Part A, 2018, 74(2): 1032–1051

[33] Zhong W Q, Xiong Y Q, Yuan Z L, DEM simulation of gas-solid flow behaviors in spout-fluid bed. Chemical Engineering Science, 2006, 61(5): 1571–1584

[34] Meng G S. Research on mechanism of dust particle adhesion and removal from solar panel surface in desert area. Dissertation for the Master’s Degree. Xining: Qinghai University, 2015 (in Chinese)

[35] Xu H B. Study on spouting and fluidization characteristics of wet particles. Dissertation for the Doctoral Degree. Nanjing: Southeast University, 2017 (in Chinese)

[36] Qasem H, Betts T R, Müllejans H, Dust-induced shading on photovoltaic modules. Progress in Photovoltaics: Research and Applications, 2014, 22(2): 218–226

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