3.2. Governing equation
The fluid flow is modelled by using the Navier-Stokes equations for incompressible flows:
Where is the velocity, the pressure,the density,the kinematic viscosity, g the gravitational acceleration.
For a particle in a plasma jet, two characteristics are studied: motion (trajectory, velocity, acceleration) and thermal evolution (temperature, physical state, heat flux).When a particle and plasma are in relative motion, a drag force is given by the fluid to the particle. This force comes from current lines dissymmetry between particle upstream and downstream. This force is given by [10].
The convective and conductive heat transfer is described by:
is the velocity field predicted by the model incompressible Navier-Stokes. For transport by conduction and convection, the thermal flux vector is given by:
Fig .2 present the geometry and dimensions adopted in the present numerical study.
Where r is the radial coordinate,is the particle radius, T is its temperature, Represents the plasma gas.
  1. Results and discussion
In this paper, the numerical simulations of the H2-Ar75%, plasma gas flow and heat transfer around spherical particle have been carried out for different flow velocity. Numerical results are presented, in order to illustrate temperature distribution around particles. In a second part, the heat transfer rate between spherical particle and plasma gas has been estimated. Fig. 4 show simulation results obtained for plasma temperature, namely. In these figure, show that the particle is gradually heated through a boundary layer flow by heat convection phenomena due the fluid flow. Several works shows that in absence of chemical reactions on the surface, the heating by thermal conduction and convection in the boundary layer is the principal mechanism of heating particle [19].
Figures 5 show the heat transfer Nusselt number (Nu ) as a function of the inlet temperature, obtained numerically and also from some correlations in the literature, applied to empirical or semi empirical results. significant differences between the values obtained can be observed, showing the complexity of phenomena studied and the influence of gas velocity and the thermal spraying conditions on the phenomena involved. The correlations studied, those of Lewis-Gauvin, Fizsdon and kalganova seem to disagree with the other low on H2-Ar 75%. The correlations of Ranz and Marshall in their predictions up to t = 3700 k (ionization temperature), but widely diverge for higher temperatures,
Fig. 5 illustrates a general disagree in all-temperature range, especially for very high inlet temperature. Following this study, it can be seen that we have three families of behaviors, the first family it can be observed that values in the Ranz-Marshall data are higher than all the predicted values in the literature for all range-temperature. We note that this correlation is applicable for very low velocities and a temperature lower than the ionization temperature.
The second family (Lewis-Gauvin, Fiszdon and Kalganova) which are always lower than other data. And the third that we are interested (Lee-Pfender and present model). Good agreement between the two models is noticed, especially when increasing the inlet velocity. The maximum at T=3700K which is due to thermal conductivity and heat capacity behavior (see Fig 2).
Figure 5 indicates general agreement for Ranz-Marshal, Lee-Pfender and present model (Nu comsol), in the low-temperature range up to ionization temperature (3700 K). Opening from 4000 K larger deviations observed in behavior for all correlation without lee-Pfender and present model named Comsol ho save the same curve.
The valid range of the Ranz -Marshall correlation is a 3700 k temperature value [REFERENCE ??]. This is mainly due to the fact that the correlation is established for a spherical particle immersed in a cold fluid, our calculation are at high temperature.
The assumption of the constant Prandtl number for Lewis et al correlation is not valid, when ionization or dissociation occurs. Kalganova and Klubniki present an experimental investigation to the description of heat transfer to a sphere in an ionized gas flow. But they did not leave sufficient information to enable other researchers to use their experimental data.
  1. Conclusion
In this study, we investigate numerically the heat transfer between a spherical particle and our gas surrounding at high temperature. We show that the model has been validated after comparison between the present results and those of the literature. The model equations were solved by using Comsol Multiphysics; a solver for partial differential Navier-Stokes equations based on Finite Element Method.
Following this study, it appears that the interaction of spherical particle and H2-Ar 75% plasma spraying involves several complex mechanisms. Semi-empirical correlations for heat transfer between a spherical particle and plasma jet have been analyzed. Finally, the present results are in excellent agreement with the reported published results. The work progress to investigate the heat transfer for elliptical particle and with other gas monatomic, diatomic and ternary mixtures (Ar-He-N2).
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