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3D time-dependent simulations are performed using a computational method suitable for thermal plasma flows to capture a turbulent field induced by a thermal plasma jet and steep gradients in nanopowder distributions. A mathematica l model with a simple form is developed to describe effectively simultaneous processes of growth and transport of nanopowder in/around a thermal plasma flow. This growth-transport model obtains the spatial distributions of the number density and mean diameter of nanopowder with a lower computational cost. The results show that an argon thermal plasma jet induces multi-scale vortices even far from itself. A double-layer structure of high-temperature thicker vortex rings surrounded by low-temperature thinner vortex rings is generated in the upstream region. As the vortex rings flow downstream, the high-temperature thicker vortex rings deform largely whereas the low-temperature thinner vortex rings break up into smaller vortices. Nanopowder is generated at the fringe of plasma and transported widely outside the plasma region. The nanopowder grows up collectively by coagulation decreasing particle number as well as homogeneous nucleation and heterogeneous condensation. When a uniform magnetic field is applied in the axial direction, a longer and straighter thermal plasma jet is obtained because of Lorentz force and Joule heating. Larger nanopowder is produced around the plasma because turbulent diffusions of silicon vapor and nanoparticles by vortices are suppressed as well.

Thermal plasmas have been expected as a promising tool for high-speed production of nanopowder because thermal plasmas offer a very-high-temperature field with steep gradients at their fringes where many small nanoparticles are generated rapidly from the material vapor [

Concerning these backgrounds, here are four problems as follows.

• Simultaneous growth and transport of nanopowder produced by a thermal plasma flow are still poorly understood because it is difficult to investigate them experimentally due to technological limitations.

• Theoretical and computational studies will be powerful approaches. However, after the experiment by Pfender et al. [

• Up to now, simultaneous growth and transport of nanopowder in/around thermal plasma flows have been simulated only under 2D steady conditions which are oversimplified assumptions [

• A magnetic pinch effect on a thermal plasma jet has not been studied from the viewpoint of turbulent vortices using 3D time-dependent simulation. That effect on nanopowder production is still unknown as well.

To break through these problems, this study performs 3D time-dependent simulations using a computational method suitable for thermal plasma flows [

Thermal plasma which is generated under atmospheric pressure is described by the thermofluid approximation with several typical assumptions: 1) the entire fluid region including plasma and non-ionized gas is in a local thermodynamic equilibrium state, 2) the plasma is optically thin, and 3) buoyancy due to density variation is negligible. The governing equations are given as the conservations of mass, momentum and energy:

∂ ρ ∂ t + ∇ ⋅ ( ρ u ) = 0 (1)

∂ ( ρ u ) ∂ t + ∇ ⋅ ( ρ u u ) = − ∇ p + ∇ ⋅ { η [ ( ∇ u ) + ( ∇ u ) t r − 2 3 ( ∇ ⋅ u ) U ] } + J × B (2)

∂ ( ρ h ) ∂ t + ∇ ⋅ ( ρ u h ) = ∇ ⋅ ( λ C ∇ h ) + ∂ p ∂ t + u ⋅ ∇ p − q r a d + q c o n + Φ + | J | 2 σ (3)

where ρ is the density of fluid, t is the time, u is the velocity vector, p is the pressure, η is the viscosity, U is the unit matrix, B is the magnetic flux density vector, h is the enthalpy, λ is the thermal conductivity, C is the specific heat at constant pressure, q r a d is the radiation loss, q c o n is the heat generation due to condensation, Φ is the viscous dissipation, and σ is the electrical conductivity. The superscript t r means transposition. The momentum exchange at nanopowder generation is negligible because the mass flow rate of raw material is small in typical conditions of thermal plasma processing. The terms J × B and | J | 2 / σ are the Lorentz force and the Joule heating which are electromagnetic effects. J is the electric current density vector which is given from the generalized Ohm’s law as

J = σ ( E + u × B ) (4)

since thermal plasma is regarded as a collision-dominated plasma. E is the electric field vector.

In engineering time- and spatial-scales, the aerosol dynamics approach effectively describes the growth and transport processes of nanopowder produced by thermal plasma with several assumptions: 1) nanopowder consists of spherical nanoparticles, 2) electric charge effects are neglected, 3) nanoparticle temperature is identical to the fluid temperature, and 4) material vapor is treated as an ideal gas. Extending the previous model [

ρ ∂ ∂ t ( n p ρ ) + ρ u ⋅ ∇ ( n p ρ ) = ∇ ⋅ [ ρ D p ∇ ( n p ρ ) ] + I − 2 2 β 0 n p 11 / 6 f 1 / 6 − ∇ ⋅ ( K t h η n p ρ ∇ ln T ) (5)

ρ ∂ ∂ t ( f ρ ) + ρ u ⋅ ∇ ( f ρ ) = ∇ ⋅ [ ρ D p ∇ ( f ρ ) ] + I g c + β 0 ( n v − n s ) n p 1 / 3 f 2 / 3 − ∇ ⋅ ( K t h η f ρ ∇ ln T ) (6)

ρ ∂ ∂ t ( n v ρ ) + ρ u ⋅ ∇ ( n v ρ ) = ∇ ⋅ [ ρ D v ∇ ( n v ρ ) ] − I g c − β 0 ( n v − n s ) n p 1 / 3 f 2 / 3 (7)

where n is the number density, D is the diffusion coefficient, and T is the temperature. The subscripts p , v , and s denote particle, vapor, and saturated state, respectively. The variable f is defined as f = n p g . D p is the diffusion coefficient of nanoparticles derived from [

D p = k B T 3 π η d v ( g − 1 / 3 + 3 .314 l d v g − 2 / 3 ) (8)

where k B is the Boltzmann constant, d is the diameter, g is the average monomer number in a nanoparticle, l is the mean free path. D v is the diffusion coefficient of material vapor obtained from the molecular theory by Hirschfelder et al. [_{0} is the parameter related to collision frequency given as [

β 0 = ( 3 v v 4 π ) 1 / 6 6 k B T v v m v (9)

where v is the volume and m is the mass. K t h is the thermophoresis coefficient [

q c o n = m v H v β 0 ( n v − n s ) n p 1 / 3 f 2 / 3 (10)

where H ν is the latent heat of vaporization.

This growth-transport model obtains the spatial distributions of the number density n p and the mean diameter 〈 d p 〉 = d v ( f / n p ) 1 / 3 of nanoparticles in spite of its much simpler mathematical form than the often-used model which is known as the method of moment [

The present computation is addressed by an approach of large eddy simulation (LES). For LES, the governing equations are typically transformed using the filtering operations. The turbulent features at the sub-grid scale (SGS) are treated by the coherent structure model [

To solve the governing equations, a computational method “Method-III” proposed in [

The geometries of the domain are ( x , y , z ) = (0.0 - 255.8 mm, −51.2 - 51.1 mm, −51.2 - 51.1 mm). For the downstream outlet boundary, the unsteady outflow conditions based on mass conservation considering variable density [

The thermodynamic and transport properties of argon thermal plasma and non-ionized argon gas exhibit large variations with one or two orders of magnitude. This study takes account of those characteristics by implementing the temperature-dependent data [

^{−}^{2} which was normalized by the mean velocity of 160 m/s and the diameter of 8.0 mm at the nozzle exit. Many vortices are generated even far from the plasma jet cores. These visualized vortex structures are similar to the Schlieren photograph by Pfender et al. [

fluctuation causing vortices. The velocity fluctuations also induce electric current. The electric current components perpendicular to the magnetic field produce the Lorentz force in the direction perpendicular to both the electric current and the magnetic field. This direction of the Lorentz force is the opposite direction to the velocity fluctuation. In consequent, the velocity fluctuations are suppressed as described in Equation (2).

^{18} m^{−}^{3}, 1.0 × 10^{19} m^{−}^{3}, and 5.0 × 10^{19} m^{−}^{3}. ^{18} m^{−}^{3}.

3D time-dependent simulations were performed using a computational method suitable for thermal plasma flows to capture a turbulent field induced by a thermal plasma jet and steep gradients in nanopowder distributions. A mathematical model with a simple form was developed to describe effectively simultaneous processes of growth and transport of nanopowder in/around a thermal plasma flow. This growth-transport model obtained the spatial distributions of the number density and mean diameter of nanopowder with a lower computational cost. The major findings are enumerated as follows.

• An argon thermal plasma jet induces multi-scale vortices even far from itself. A double-layer structure of high-temperature thicker vortex rings surrounded by low-temperature thinner vortex rings is generated in the upstream region. As the vortex rings flow downstream, the high-temperature thicker vortex rings deform largely whereas the low-temperature thinner vortex rings break up into smaller vortices.

• Nanopowder is generated at the fringe of plasma and transported widely outside the plasma region. The regions where the nanopowder has larger size coincide with the regions where the nanopowder exhibits smaller number density. This result indicates that nanopowder grows up significantly by coagulation decreasing particle number as well as homogeneous nucleation and heterogeneous condensation.

• When a uniform magnetic field is applied in the axial direction, a longer and straighter thermal plasma jet is obtained because the induced Lorentz force suppresses turbulent vortices and the Joule heating is also generated in the plasma. Larger nanopowder is produced around the plasma because turbulent diffusions of silicon vapor and nanoparticles by vortices are suppressed as well.

This work was partially supported by “Joint Usage/Research Center for Interdisciplinary Large-scale Information Infrastructures” in Japan. The numerical results were obtained using supercomputing resources at Cyberscience Center, Tohoku University. This work was also partially supported by KAKENHI (Grant No. 16K13737). The author is grateful to Mr. Takeshi Yamashita of Tohoku University, Mr. Takashi Soga and Mr. Kenta Yamaguchi of NEC Solution Innovators, Ltd. for improving the solver code.

Shigeta, M. (2018) Numerical Study of Axial Magnetic Effects on a Turbulent Thermal Plasma Jet for Nanopowder Production Using 3D Time-Dependent Simulation. Journal of Flow Control, Measurement & Visualization, 6, 107-123. https://doi.org/10.4236/jfcmv.2018.62010