Theoretical study on water film area and dust removal efficiency of string grille

Breaking through the bottleneck of low resistance and high efficiency in mine exhaust dust control has been a hot issue in the industry. Aiming at the key problems of the relationship between water film area and the dust removal efficiency in wet string grille, based on the multiphase flow theory and capillary mechanics, the dimensionless bond number expression of the influence of string grille wire spacing on droplet spreading is derived. Furthermore, a liquid film formed by spreading droplets based on Kelvin correlation, young Laplace formula and Hagen Poiseuille theory, a formula for calculating the thickness and height of liquid film is established with temperature, relative humidity and molar volume of liquid phase as independent variables. According to the theory of string grille filtration and dust removal, the dust removal efficiency calculation model of string grille wire group is established between the liquid film thickness, height, wire diameter, water film area and vortex shedding frequency. Finally, the theoretical analysis of the influence of water film area on the efficiency of wet string grille dust removal is carried out based on the spray pressure and the ratio of string grille wire distance to wire diameter. It is found that the effect of spray pressure on water film area and dust removal efficiency is more significant than string grille wire distance diameter ratio. Moreover, the optimized combination of wet string grille wire distance diameter ratio 0.84, wind speed 3m/s and spray pressure 0.8 MPa is found, which provides important reference for engineering application.


Introduction
In the process of mining, a large amount of dust is produced, which comes to the airshaft with the underground air flow and then discharged, which pollutes the atmospheric environment. About mine dust removal, china and foreign scholars have developed a variety of dust removal methods, mainly divided into dry and wet dust removal. Suitable for large air volume, high pollution limited space, string grille dust removal technology is the first choice. The string grille dust removal technology is a kind of filtering dust removal technology. The droplet group ejected from the nozzle collides with the surface of the string grille, and the droplets are capillary wetted on the surface of the string grille due to the capillary effect. Under the pneumatic disturbance of the dust laden air flow, a downward flowing water film is formed in the longitudinal fine string grille gap. When the dust laden air passes through the water film, it is captured by the water film and flows downward with the water flow under the action of the flow, the fracture is generated and new water film is generated continuously under the action of continuous spray droplet group. During this process, the wet string grille dynamic water film dust removal mainly depends on the water film area, which directly affects the dedusting efficiency of string grille.
In terms of wet dust removal, Zhou [1,2] and Zhou [3] were studied on the basis of spray pressure. The spray dedusting technology was optimized, not only the engineering application was carried out, but the dust removal efficiency was high. Further, Wang [4,5] analyzed the relationship between the atomization characteristics of the atomizing nozzle and the dust removal efficiency. However, in the aspect of theory and experiment, having research of string grille dust removal, for example, Fairs [6] discovered the existence of a liquid film by studying the formation of a sulfuric acid vinegar fog on four kinds of fibers with different properties. Using a continuous spray of water mist on the fiber, Agranovski [7] designed a method to form a water film after a few minutes. Theoretical and experimental studies show that the water film thickness increases with fiber equivalent diameter and differs along different heights of the fiber grille. In the case of equilibrium saturation, Jakub [8,9] and Ryan [10] observed uneven liquid coverage on the fiber and studied the efficiency of filter-trapping aerosol. Gong [11] studied the dominant mechanism of water film formation on the fiber grille and pointed out the capillary effect of the fiber spacing. Jiang [12] and Jin [13] theoretically analyzed the dedusting mechanism of water films and assumed that dust particles that are difficult to moisten tend to penetrate into the water film. Li [14] and Wu [15] indirectly explained dust removal efficiency from the aspect of fiber grille gap, material, and grille, as well as nozzle spacing according to water films observed in experiments. However, most of these studies analyzed the existence of water films from a macro and indirect perspective and there remains a lack of systematic analysis of the physical mechanism in water film formation and fragmentation processes, which makes the relationship between water film state and dust removal efficiency poorly understood.
In order to study the relationship between water film and dust removal efficiency, it is important to understand the interaction between string grille structure and droplets. In this paper, theoretical analysis is carried out for the process of removing water film from the string grille. The mathematical expressions of the water film area and the dust removal efficiency are established by analyzing the influence rule of the water film area on the spray pressure, the wind speed and the structural parameters of the string grille. By limiting the wind speed, dust particle size and water irreducible saturation, changing the spray pressure and the string grille wire distance diameter ratio to the water film area and the efficiency of dust removal are compared and analyzed. It is concluded that the spray pressure and the string grille wire distance diameter ratio have a significant effect on the water film area and the efficiency of 3 dust removal, which provides a reference for theoretical and engineering applications of string grille dust removal.

Water film formation mechanism
The water mist contacts the two adjacent wire and forms a water film owing to surface adhesion forces and capillary action. The Bond number is used to describe the capillary phenomenon in surface mechanics [16] and is the relative magnitude of static water mist gravity and surface tension: where  c is capillary length, which is considered here to be half of the string grille wire spacing d0 (mm),  is the water mist density (kg/m 3 ), g is gravitational acceleration (N/kg), and  lv is the liquid gas surface tension (N/m). When the Bond number is large, surface tension produced by static water mist under capillary pressure can be ignored. A small Bond number (<0.1) shows that the surface tension strong affects the hydrostatic gravity. In the film formation test, the water is 20 °C ,   -3 =7.3 10 lv N/m,  9.8 g N/kg,   994 g/m 3 , combined with reasonable reference values [17] and  0.0036 Bo , From Eq. (1), it can be calculated that: the distance between the string grille wire is calculated from the equations above to be less than 0.34 mm. The water mist generates a capillary phenomenon between the string grille wire.

Basic physical parameters of the string grille
The diameter of the string grille wire is a particularly important physical parameter. The string grille wire is cylindrical in shape and uniformly distributed in the wire grille with equal spacing. In order to facilitate the formation of capillary water film on the string grille and improve the dust removal efficiency of the string grille, the distance of string grille wire must be less than 0.34mm, the diameter of string grille wire is represented by df, and the ratio of string grille wire distance to diameter is the string grille wire distance diameter ratio, expressed in d  . According to analysis of the filtering mechanism of the string grille, the string grille wire is uniformly distributed on the rectangular string grille and parallel to the airflow direction. The percentage of the string grille wire area to the whole string grille area prior to wetting is expressed by  s .
The wire-filling rate is determined during production. String grille voidage  s is obtained according to   1 s . When the airflow carries water mist through the string grille, the filling rate of the wet string grille wire includes the water mist attached to the string grille wire and the residual water film after the film ruptures. The filling rate of the string grille is therefore composed of the filling rates of the unwetted wire and water: where  ws and  w are the filling rates of the wet wire and water, respectively (dimensionless). 4 The liquid holding capacity of the string grille wire is measured and calculated experimentally and  represents the wet string grille voidage (dimensionless).

Calculation of water film thickness and height on a string grille
When the water mist and string grille wire meet conditions of film formation, a liquid film force is produced, also known as a liquid bridge force [18,19]. Capillary action theory can be used to analyze the liquid film force. The left and right sides of the water film between the string grille wire are connected owing to adhesion. On the other hand, there is surface tension on the upper and lower film surfaces, which forms a concave water film surface in the arc ( Fig.  1) and leads to a pressure difference between the gas and body side of the water film (i.e. pressure outside the water film is greater than that inside). This negative pressure on the film itself has a pulling effect, which is essentially the liquid film force. (1) Derivation and calculation of water film thickness During the tests, water mist is sprayed continuously on the string grille plate with a small gap between the wire, and a water film forms under capillary action. The outer outline of the water film between the wire is a circular arc concave surface owing to the influence of the water film force (Fig. 2). Let d0 represent the distance between the two adjacent string grille wire. The average curvature radius of the water film side surface is r, r1 and r2 are the curvature radius of the curved surface perpendicular and parallel to the string grille wire, respectively, and the surface tension of the water film is δ. According to the Young-Laplace [20] equation, the negative pressure difference of the water film between the string grille wire is: In a humid environment, when the water mist and parallel body of the wettable string grille wire are close to one another, the string grille surface absorbs the water mist molecules to form a water film. According to the physical and mechanical knowledge of surfaces and interfaces, the separation pressure P0 in the water film is expressed as: where H is the Hamaker constant (J) and L is the water film thickness (mm). Under mechanical equilibrium conditions, the negative pressure of the water film between the 5 string grille wire is equal to the separation pressure in the water film. Combined with the relationship between the curvature radius of the water film surface and relative humidity in the continuous spray environment, the Kelvin equation [21] shows: where k r is the kelvin radius (m), m V is the molar volume of the water mist (m 3 /mol),  is the angle, taken here as 0°, 0 R is the universal gas constant (J· K −1 · mol −1 ), T is the absolute temperature (K), g p is the saturated vapor pressure of the water mist (Pa), l p is the saturated vapor pressure of the water (Pa), and / g l p p is the saturated vapor pressure ratio of water mist to water (dimensionless), fitting the reference [22] data, the relationship between the function of saturated vapor pressure ratio and the spray pressure p (MPa) can be obtained as follows: The negative pressure difference of the water membrane and the separation pressure are a pair of equilibrium forces, combined with Eqs. (4)- (7): The thickness of the available water film is: Under continuous spraying at 20 º C, the molar volume of water fog is Vm = 1.8 × 10 5 m 3 /mol, the universal gas constant R0 = 8.31 J· K −1 · mol −1 , the absolute temperature T = 293 K, and the Hamack constant of water mist H = 3.72 × 10 −20 J.
(2) Derivation and calculation of water film height Water mist between the wire forms a water film under capillary action and presents a circular concave liquid surface under magnification. The height of the concave water film is affected by the total capillary force (negative pressure difference of the total water film) Fm, adjacent string grille wire viscosity force Fv, and water mist gravity Fg.
According to the Young-Laplace equation [20], the capillary force (negative pressure difference of the water film) can be written as follows: As can be seen in Fig. 2, If the distance between the string grille wire is d0 = 2r, the total capillary force is as follows: The capillary motion of the water mist between string grille wire is unsteady and satisfies the Navier-Stokes equation. To simplify the calculation, the flow of water mist between the string grille wire is regarded as steady flow, the radial and circumferential direction components of the flow velocity is set to 0, and the flow is symmetrically distributed. The Navier-Stokes 6 equation [23] can therefore be expressed by the special Hagen-Poiseuille equation [24]. The viscous force between string grille wire can be obtained by the internal friction law of a where p is the spray pressure (MPa) and z is the capillary tube length direction coordinate (m). Taking into account the uniform change of the water film force between the string grille wire,    The internal friction force on the surface of string grille wire can be obtained by replacing Eq. (16) with the Newtonian viscous fluid constitutive equation (13): The viscous force of the adjacent string grille wire is: The gravity of the water mist between string grille wire can be expressed as: where h is the rising water film height in the string grille wire clearance (mm) and  is the water mist density (kg/m 3 ). Capillary action of the water film between the string grille wire is the result of the above three forces, according to the action direction of the force, namely:  (22), the rising height of water film capillarity depends on the distance between adjacent string grille wire, contact angle between the water film and wire, surface tension of water film, and average flow velocity.

Results and Discussion
Relationship between water film area on string grille wire and dust removal efficiency The dust filtration efficiency of a string grille depends on the uniform filling rate of the string grille wire (Fig. 3). The filtered area of the string grille plate used is 590 mm× 550 mm, the cylindrical wire diameter is df, the wire distance is d0, the filling rate is βs, and the total length of the double string grille wire diameter is: where Ls is the total diameter and length of the double string grille wire (mm), a is the height of single layer string grille wire (mm), and b is the width of the single layer string grille wire (mm).
where R is the low interception settlement, dimensionless, as shown in Eq. (27), Stk is the Stokes number, dimensionless, as shown in Eq. (28).
where p d is the dust particle size (mm), p  is the true density of dust particles (kg/m 3 ), u C is the Cunningham correction coefficient, often taken as 1, and  g is the viscosity of air(Pa·S). For a string grille wire dust catcher that is uniformly distributed in the string grille, a unit filter dx is assumed on the string grille wire surface, and the dust concentration entering the unit is C (%). In a unit time, through the string grille with a filter area of  ab (A, m 2 ) , the layer thickness can be regarded as water film thickness L (mm), the total number of dust particles in the filter unit is   2 4 s f s f s d L dx d L . Therefore, the amount of dust particles captured by the filter unit per unit time is: The dust capture efficiency of simplified string grille wire is: For a string grille wire under spray wetting condition, the trapping of dust particles by the string grille wire and water film occurring between the strings is included in the process of dust particle capture. According to the previous analysis, the water film initially forms under the action of capillary condensation and adsorption condensation, and spreads out by liquid film force. The water film area is due to the formation of water mist between the wires, which is related to the vortex shedding frequency, the filling rate of wet string grille wire and the effective saturation of water mist. Assuming that the water film between the string grille wire uniformly fills the wire gap, the water film area function formula can be expressed as follows: where  is the correction coefficient, considering that water film also exists on the string wire, 1.5 is taken here, r S is the dimensionless vortex shedding frequency, and we S is the effective saturation of the water on the plate (dimensionless). 9 The vortex shedding frequency can be obtained by the relationship between vortex shedding dimensionless shedding frequency and string grille wire Reynolds number studied by Webster [26]. Under turbulent conditions, the relationship between dimensionless shedding frequency and string grille wire Reynolds number is as Eq. (33) The effective saturation of water on the plate refers to water flowing in the gap, accounts for the volume of the entire plate gap, and is calculated as follows: where w S is the water saturation, which refers to the degree to which the grille gap is filled by water mist, is the water-to-void volume ratio and is related to spray pressure. Water saturation can be obtained from the experimental data concerning spray pressure. where 0 w S is the irreducible saturation of water, part of the water in the gap does not flow owing to adsorption forces. This part of the water is called irreducible water. The irreducible water-to-wire void volume ratio is the irreducible saturation of water, which can be calculated by the static liquid holding capacity of the wire. In the process of water film dust removal, dust is considered to be captured as long as it contacts with water film. Therefore, the dust removal efficiency of water film ( w  ) can be expressed as follows: Combined Eq. (35) with Eq. (31), The dust removal efficiency of double wet string grille is composed of wire dust removal and water film dust removal. Therefore, the dedusting efficiency of the double-layer wet string grille is calculated as follows: According to the practice of wet string grille dust removal project, when the dust removal wind speed is set at 3 m/s, the string grille wire diameter is 0.23mm, the dust particle size is 10um, and the water irreducible saturation is 0.2, the string grille dust removal efficiency can be expressed as follows:  

Theoretical analysis of dust removal efficiency
Calculation Eq. (37) shows that the efficiency of dust removal is directly proportional to the water film area of the string grille. With the increase of the water film area, the efficiency of dust removal is increased. When the dust removal speed is 3m/s, the spray pressure is 0.01-1.1MPa, the string grille wire diameter is 0.23mm, the string grille wire distance is 0.26mm, the dust diameter is 10um, the water irreducible saturation is 0.2, and combining Eq. (9) (35) (37), the relationship between water film area and dust removal efficiency of double wet string grille is calculated, as shown in Fig. 4. The figure shows that when the dust removal speed, the string grille wire distance diameter ratio, the particle size of dust and the irreducible saturation of water are fixed, the area of water film and the efficiency of dust removal increase with the increase of spray pressure. Generally, it can be divided into three stages. In these three stages, the growth rate of water film area is 3-5 times higher than that of the dust removal efficiency. In particular, the spray pressure is from 0.01MPa to 0.20MPa. At this stage, the water film area and the dust removal efficiency increase faster, the average growth rate of the water film area can reach 3.05%, the efficiency of dust removal efficiency is 0.70%, and the spray pressure is 0.21MPa to 0.40MPa. At this stage, the growth rate of water film area and dust removal efficiency is slow, the average growth rate of water film area is only 0.48%, and the growth rate of dust removal efficiency is only 0.14%. When the spray pressure exceeds 0.40MPa, the water film area and dust removal efficiency keep increasing, but the growth rate is extremely slow, the average growth rate of water film area is only 0.17%, and the growth rate of dust removal efficiency is only 0.06%. Fig. 4 Relationship between water film area and dust removal efficiency under different spray pressure Fig. 5 is the change of the string grille wire distance diameter ratio, while the dust removal speed is 3m/s, the spray pressure is 0.8MPa, the string grille wire diameter is 0.23mm, the dust particle diameter is 10um, the water irreducible saturation is 0.2, and the Eq. (9) (35) (37) is combined to calculate the relationship between the water film area of the double-layer wet string grille and the dust removal efficiency. The graph shows that when the wind speed, spray pressure, string grille wire diameter, dust particle diameter and water irreducible saturation are constant, the water film area and dust removal efficiency increase with the increase of the string grille wire distance diameter ratio, and the growth of the water film area is slightly different from that of the dust removal efficiency, while the water film area grows slower and slower, while the dust removal efficiency increases first and then slows down. On the whole, the string grille wire distance diameter ratio has a certain influence on the water film area and dust removal efficiency. When the ratio increases from 0.47 to 0.84, the water film area increases from 0.158 m 2 to 0.226 m 2 , the total growth rate is 35.798%, the maximum growth rate is 1.438%, and the average growth rate is 0.968%; while the dust removal efficiency increases from 69.84% to 76.85%, the total growth rate is 9.576%, the maximum growth rate is 0.326%, and the average growth rate is 0.259%. At this time, the total growth rate of water film area is 3.74 times of the total growth rate of dust removal efficiency. When the ratio increases from 0.85 to 1.47, the area of water film increases from 0.227m 2 to 0.294 m 2 , the total growth rate is 26.575%, the maximum growth rate is 0.644%, and the average growth rate is 0.422%; while the dust removal efficiency increases from 77.10% to 91.79%, the total growth rate is 17.790%, the maximum growth rate is 0.325%, and the average growth rate is 0.282%. At this time, the total growth rate of water film area is 1.50 times of that of dust removal efficiency.
11 Fig. 5 Relationship between water film area and dust removal efficiency with different string grille wire distance diameter ratio Based on the theoretical Eq. (37) for calculating the dust removal efficiency of wet string grille, the dust removal speed, dust particle diameter, water irreducible saturation and string grille wire diameter are defined. Taking the spray pressure and the string grille wire distance diameter ratio as variables. Through the comparison, it is found that the effect of spray pressure on the water film area and the dust removal efficiency is slightly larger than that of the string grille wire distance diameter ratio on the water film area and the dust removal efficiency. In order to obtain ideal dust removal effect, spray pressure is the key. With the increase of spray pressure, the water film area also increases. The proportion of water film in the string grille wire gap is bigger and bigger. If the dust is not captured by string grille wire, the probability of being captured by water film is greatly increased, thus the dust removal efficiency is greatly improved.

Conclusion
(1) The adjacent string grille wire is subjected to adhesion forces between the water mist and wire surface, resulting in the formation of a water film. Based on the Bond number, the water film formation conditions include wire spacing of <0.34 mm. According to mechanical equilibrium conditions, the film thickness can be derived from the Kelvin equation. Combined with the Navier-Stokes equation and Hagen-Poiseuille law. The expression of the height of the capillary action of the water film in the gap of string grille wire is constructed.
(2) Assuming that the water film between string grille wire fills the gap of wire evenly. The effective area function of the water film is expressed by the effective saturation of the water mist, and the dedusting efficiency of double-layer wet string grille wire is calculated.
(3) From the theoretical expression of wet string grille dust removal efficiency, we conclude that the dedusting efficiency is proportional to the water film area. Larger water film area is associated with higher dedusting efficiency. When other factors are limited, only spray pressure and string grille wire distance diameter ratio are used as variables. The effect of spray pressure on water film area and dust removal efficiency is more significant.