Argon–hydrogen plasma has important applications in many thermal plasma devices [1–3]. The supersonic plasma spraying using internal feeding way could prepare higher quality coatings than traditional plasma spraying using external feeding way . In-flight particle behavior in plasma jet directly affects the coating quality . However, particle trajectories are difficult to obtain in heating and acceleration processes through experimental methods. In order to essentially investigate the interactions of particles and flow, numerical calculation method should be used to analyze the particle properties in the plasma spraying process. From the past several years, various two-dimensional and three-dimensional models have been published on the fluid flow and heat transfer characteristics of the plasma jet and plasma–particle interactions [6–8]. Feng  analyzed particle velocity and temperature distributions in plasma jet using the Runge–Kutta method. Yang  analyzed the two-phase flow of high concentration powder using two-fluid model. Wang  tracked the supersonic flame spraying particles and analyzed the particle velocity and temperature by establishing a discrete model of inert particles.
However, these studies did not consider coupling of plasma spraying multi-physical fields and ignored the thermophoretic force. These simulation calculations mainly concentrated in the study of plasma spraying process in low-pressure environment. No related reports exist on transient calculation for supersonic spray gun internal and external flow field combinations.
Hence, in this article, we establish a particle heating and acceleration calculation model in plasma jet and study internal and external flow characteristics of supersonic plasma spray gun based on the analysis of plasma spraying multi-physical fields. Drag, thermophoretic, and pressure-gradient forces are considered in this study. Flow field calculation and in-flight particle analysis are separated. The flow field calculation results are used as the initial conditions of particle computation. The heating and acceleration behavior of particles in the flow field are analyzed. In-flight particles are also monitored by Spray Watch and compared with the calculated value.
Two-phase flow calculations are performed by solving mass, momentum, and energy conservation equations under a Lagrangian coordinate system. The conservation equations are expressed as follows:
Mass conservation equation (1)
Momentum conservation equation (2)
Energy conservation equation (3)
The calculations in this article assume the following:
Particle is isothermal sphere with constant diameter
Average intensity of plasma fluctuation velocity is constant under turbulent conditions
The influences of boundary layer and flow source term are neglected
The evaporation of particles is neglected
The flow drag force and gravity are applied on particles in the movement process and can be represented as follows: (4)where is the particle mass, is the particle velocity vector (), is the drag coefficient, , , and are density, velocity, and gas pressure, respectively, and is the particle windward area.
The drag coefficient is a function of the Reynolds number : (5)The pressure-gradient force of the particle is written as  (6)The thermophoretic force of the particle is written as (7)where is the Knudsen number, is the gas thermal conductivity/W m−1 K−1, is the particle thermal conductivity/W m−1 K−1, is a constant (1.17), is a constant (1.14), is a constant (2.18) .
Heat transfer inside the particles can be written as follows: (8)where is the radial distance of particle/m, H is the enthalpy/J kg−1, and is the thermal conductivity/W m−1 K−1.
Heat balance item of the particle surface can be written as : (9)where h is the coefficient of heat conduction, is the Boltzmann constant, is the emission coefficient, is the particle surface temperature/K, is the plasma temperature/K, and is the energy taken away by particle evaporation/J.
Convective heat transfer coefficient can be written as follows: (10)where is the Nusselt number.
The mass loss ratio caused by particle evaporation can be written as follows : (11)where is the total mass conductivity coefficient, is the concentration of particle surface evaporative substance/g · m−3, and is the vapor concentration/g · m−3.
The concentration of particle surface evaporative substance can be written as (12)where M is the molar mass of steam/g · mol−1, is the steam pressure/Pa.
The total mass conductivity coefficient can be written as follows: (13)where is the Langmuir evaporation rate coefficient and is the mass conduction coefficient.
The heat conduction caused by evaporation can be written as follows : (14)where is the enthalpy of steam/J · kg−1.
Subscripts v, l, and s represent steam, liquid, and solid states, respectively.
Particles involved in this study are ZrO2 with density of 5.89 g/cm3, specific heat capacity of 580 J/(kg · K), melting temperature of 2,950 K, and thermal conductivity of 2.0 W/(m · K). The flow field calculation and in-flight particle analysis are separated. The structural mesh is used in flow field calculation model. Mesh density is higher in the border of fluid and solid. Mesh density in fluid region is greater than that in solid region. The whole flow field calculation domain contains 384,867 nodes and 332,024 elements. The particle calculation model is shown in Figure 1.
During the simulation, turbulent model is adopted and the finite volume method is used to solve the N-S equations.
The main calculation condition is based on the parameters of the practical spraying process. The main process parameters of supersonic plasma spraying and the particle parameters are shown in Tables 1 and 2.
Particle calculation boundary condition is based on the experimental data. The boundary conditions of flow field calculation model are shown in Table 3.
The flow field calculation results are used as the initial conditions of particle computation.
Results and discussion
The heating and acceleration processes of the particles in supersonic jet are calculated. The velocity and temperature distributions of particles in the flow field are shown in Figure 2.
After high-speed jet flowed into the atmosphere, a large amount of air entered into the jet (Figure 2). The plasma jet deforms, particularly in the tail. Particles affect the flow stability near the delivering powder gas entrance. Particles are sent into the center of the supersonic jet because the internal feeding way is used. The drag force to the particles is more intense. Therefore, the particles mainly distribute along the axis. The average particle velocity and temperature in different spraying distances are calculated and compared with flow velocity and temperature (Figure 3).
The flight process of particles can be divided into three stages: acceleration, constant speed, and deceleration (Figure 3(a)). At 80 mm away from the nozzle exit, the flow velocity is higher than the particle velocity; thus, the particles are accelerated. At 80–100 mm away from the nozzle exit, the particle velocity approaches the flow velocity and reaches the maximum. At 100 mm away from the nozzle exit, the particle velocity is higher than the flow velocity, so the particles are decelerated.
The heating process of particles can also be divided into three stages (Figure 3(b)). At 80 mm away from the nozzle exit, the flow temperature is higher than the particle temperature; thus, the particles are heated. At 80–100 mm away from the nozzle exit, the particle and flow temperatures are close and particles are molten. At 100 mm away from the nozzle exit, the particle temperature is higher than the flow temperature, so the particles are cooled. Therefore, the workpiece should be placed around 80–100 mm away from the nozzle exit to make particles to have the highest velocity and temperature when hitting the substrate.
The drag force of particles can be calculated by eq. (1). The drag force of particles in various spraying distances for both supersonic plasma spraying and traditional plasma spraying are shown in Figure 4.
The drag force of particles in supersonic plasma spraying is greater than that in traditional plasma spraying (Figure 4); thus, the particle velocity in supersonic plasma spraying is higher than that in traditional plasma spraying. Consequently, the molten particles have great momentum when hitting the substrate, and molten droplets spread out completely. The coating bonding strength, density, and porosity are improved. In addition, when the spraying distance is less than 90 mm, the drag force of particles is greater than 0, thus particles are accelerated. When the spraying distance is around 90 mm, the drag force approaches 0. When the spraying distance is greater than 100 mm, the drag force is less than 0, thus particles are decelerated. The results match the conclusions in Figure 3(a).
Weber number can be used for characterizing the droplet crushing process. A larger Weber number indicates a greater broken degree of droplets [18, 19]: where is the droplet surface tension. Weber numbers of supersonic plasma spraying and traditional plasma spraying are shown in Figure 5.
The Weber number of supersonic plasma spraying reaches the maximum at the nozzle exit, and the value is greater than that of traditional plasma spraying (Figure 5). The results clearly show that particle breakage mainly occurs near the nozzle exit, and particles in supersonic plasma spraying are more likely to refine. The Weber number is also approximately equal to 0 at 80–100 mm away from the nozzle exit because particle velocity approaches flow velocity.
Particles are monitored by Spray Watch (Figure 6(a)), which is installed at the side of the plasma spray gun. We measured the particle trajectories length using the charge-coupled device (CCD) sensor and divided the length by the exposure time to obtain the velocity. The average velocity was calculated through the statistics of 1,000 particle trajectories. The focal length is 185 mm. The exposure time is 5 μs (Figure 6(b)).
The contrast curves of particle temperature and velocity between measurements and computations are shown in Figure 7.
The computation of particle temperature is slightly higher than the measurement (Figure 7(a)). Both parameters have the same change trend. The maximum error is 10.7%. The evaporation effect of the particles has not been considered in particle computing model, which mainly resulted in error.
The acceleration process of particles with small diameter is consistent with the measurements (Figure 7(b)). The error between computations and measurements is within 10%. This finding is ascribed to the refinement that occurs under the interaction between molten particles and high-speed flow in the supersonic plasma spraying process. The acceleration effect of these small particles is stronger in the flow.
Particle refinement process can be shown by comparing the size of the original particles and that of the collected particles in the jet. The morphology of original particles and that of the collected particles at the spraying distance of 90 mm are shown in Figure 8 . The refinement occurs obviously from the original particle to that in the jet. The analysis results are consistent with that in Figure 5.
After considering particle internal phase change, particle temperature distribution is more reasonable. Particle temperature increases with time. The velocity and temperature of particles approach those of flow at 80–100 mm away from the nozzle exit by analyzing particle heating and accelerating processes of supersonic plasma spraying. Particle velocity and temperature reach the maximum to attain ideal heating and accelerating effects. Therefore, substrate should be placed in this range. This conclusion has industrial value and guiding significance in practical application.
Particles in supersonic plasma spraying are more likely to refine near the spray gun exit, which conform to experimental observations. The velocity calculation of particles with small diameter is consistent with the measurement. These conclusions can improve the understanding of the in-flight particles in supersonic plasma spraying process and provide technical guidance for high-quality coating.
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