Drag on a metallic or nonmetallic particle exposed to a rarefied plasma flow

Drag force on a metallic or nonmetallic spherical particle exposed to a plasma flow is studied for the extreme case of a free-molecule regime. Analytical expressions are derived for the drag components due to, respectively, atoms, ions, and electrons and for the total drag on the whole sphere due to all the gas species. It has been shown that the drag is proportional to the square of the particle radius or the drag coefficient is independent of the particle radius. At low gas temperatures with a negligible degree of ionization, the drag is caused mainly by atoms and could be predicted by using the well-known drag expression given in ordinary-temperature rarefied gas dynamics. On the other hand, the drag is caused mainly by ions at high plasma temperatures with a great degree of ionization. The contribution of electrons to the total drag is always negligible. Ignoring gas ionization at high plasma temperatures would overestimate the particle drag. There is a little difference between metallic and nonmetallic spheres in their total drag forces, with a slightly higher value for a metallic sphere at high plasma temperatures, but usually such a small difference could be neglected in engineering calculations. The drag increases rapidly with increasing gas pressure or oncoming speed ratio. For a two-temperature plasma, the drag increases at low electron temperatures but decreases at high electron temperatures with the increase in the electron/heavy-particle temperature ratio.Nomenclature C _d Drag coefficient - e Elementary charge - f _D,F _D Local and total drag (N/m ² andN) - f ^– Velocity distribution function for incident gas particles - f ⁺ Velocity distribution function for reflected gas particles - k Boltzmann's constant - m Gas particle mass (kg) - n Number density of gas species (m ^–3) - P ^–,P ⁺ Surface pressure due to incident and reflected gas particles - R ₀ Sphere radius (m) - S Speed ratio,S _j=U/(2kT _j/m_j)^1/2 - T _e,T _h Electron and heavy-particle (atom, ion) temperature - T _w Wall temperature - U Oncoming plasma flow velocity - v _x, v_y, v_z Velocity components of gas particles in thex, y, andz directions (m/sec) - v Thermal motion speed of gas particles,v _j =(8kT _j /m _j )^1/2 - v _ze Smallestv _z of electrons which could reach the sphere surface,v _ze=(2e/m _e)^1/2 (m/sec) - v _zw Value ofv _z of ions or electrons as arriving at the sphere surface (m/sec) - Center angle - Gas density (kg/m³) - Shear stress (N/m²) - Absolute value of the floating potential (V) - , Local and total particle fluxes incident to the surface - a Atoms - e Electrons - h Heavy particles - i Ions - j jth gas species - m Metallic sphere - mn Nonmetallic sphere A preliminary version of this paper was presented at the Eighth International Symposium on Plasma Chemistry held in Tokyo, September 1987.     

Thermophoretic force on a metallic or nonmetallic particle immersed into a rarefied Plasma

Analytical results of the thermophoretic force on a metallic or nonmetallic spherical particle immersed into a rarefied plasma with a heat flux within the plasma are presented for the extreme case of free-molecule regime and thin plasma sheath. It has been shown that the thermophoresis is predominantly caused by atoms at low plasma temperatures with negligible gas ionization, while it is mainly due to ions and electrons at high plasma temperatures with great degree of ionization. The ion flux incident to a particle is constant on the whole sphere surface, while the electron flux to the metallic sphere is dependent on the -position with slightly greater value at the fore stagnation point. Consequently, there is a small difference between the metallic and nonmetallic spheres in their -distributions of the floating potential on the surface, which causes the thermophoretic force on a nonmetallic sphere to be appreciably greater than that on a metallic sphere at high plasma temperatures. Expressions for the total thermophoretic force on a metallic sphere and its components due to, respectively, atoms, ions, and electrons have been given in a closed form. Calculated results are also presented on the effects of pressure and of electron/heavy-particle temperature ratio. These results can be understood based on the variation of atom, ion, and electron thermal conductivities with the gas pressure, the temperature, and the temperature ratio.     

Heat transfer to nonspherical particles in a rarefied plasma flow

The interaction of a nonspherical metallic or nonmetallic particle with a rarefied thermal plasma flow is considered. Heat transfer to a particle of arbitrary shape with an extremely thin plasma sheath due to, respectively, gas molecules, electrons, and ions is described. Analytical expressions are derived for charge and heat fluxes in the particular case of a spheroidal metallic or nonmetallic particle in a subsonic plasma flow. It has been shown that the intensity of heat exchange is greatly influenced by gas ionization, charge transfer processes, and particle shape, velocity, and orientation in the plasma flow.     

Unsteady heating of metallic particles in a rarefied plasma

Analytical results are presented concerning the unsteady heating of a metallic spherical particle innnersed in a rarefied plasma. The results show that the tinte periods required for the solid-phase heating, melting, liquid-phase heating, and evaporation are all proportional to the particle radius. For estimating the time needed for the solid-phase heating and that for the melting, the additional heat transfer rmechanism due to the thermionic emission front the particle surface is usually negligible since the surface temperatures of the particle heated in the plasma are, in general, compartively low during those heating steps. Thermionic emission assumes its effect only as the higher surface temperatures of the heated particle are involved (e.g., higher than 4000 K), while radiation loss shows its effects at much lower wall temperatures. As the plasma temperature is comparatively low, radiation heat loss may restrict the surface temperature of a particle to such a low value that the effect of thermionic emission on the overall heating time can he neglected and complete evaporation of refractor y metallic particles becomes impossible. The uncertainty in the calculation of the effect of thermionic emission is associated with the choice of the value of the effective work function for the particle material.     

Effect of particle charging on momentum and heat transfer from rarefied plasma flow

The features of interaction of a spherical metallic particle with a rarefied thermal plasma flow due to the presence o charges-electrons and ions in the gaseous phase-are considered. Analytical expressions describing charge, momentum, and energy exchange between the plasma and the particle für the cases of strong and weak Debye screening are obtained. It is illustrated that the efficiency of particle heating in the plasma considerably grows as compared with a hot molecular gas due to participation of electrons and ions in file transfer processes.     

Drag force acting on an evaporating particle immersed in a rarefied plasma flow

Analytical expressions are presented for the drag force acting on an evaporating or nonevaporating particle immersed in a plasma flow for the extreme case of free-molecule flow regime and thin plasma .sheath. It is shown that the drag force on a spherical particle is proportional to the square of the particle radius and to the relative velocity between the particle and the bulk plasma at low speed ratios. The existence of a relative velocity between the particle and the plasma results in a nonuniform heat flux distribution with its rnaximum value at the frontal stagnation point of tire sphere. This nonuniform distribution of the local heat fux density causes a nonuniforrn distribution of the local evaporated-mass flux and vapor reaction force around the surface of an evaporating particle, and thus induces an additional force on the particle. Consequently, the drag force acting on art evaporating particle is always greater than that on a nonevaporating one. This additional drag force due to particle evaporation is more significant for nonmetallic particles and for particle materials with lower latent heat of evaporation and lower vapor molecular mass. It increases with increasing plasma temperature and with decreasing gas pressure at the high plasma temperatures associated with appreciable gas ionization. The drag ratio increases with increasing electron/heavy-particle temperature ratio at high electron temperatures for a two-temperature plasma.     

Thermophoretic force acting on an evaporating particle suspended in a rarefied plasma

Analytical results of the thermophoretic force on an evaporating spherical particle immersed in a rarefied plasma with a large temperature gradient are presented for the extreme case of free-molecule regime and thin plasma sheath. It has been shown that the existence of a temperature gradient in the plasma causes a nonuniform distribution of the local heat flux density on the sphere surface with its maximum value at the fore-stagnation point of the sphere, although the total heal flux to the whole particle is independent of the temperature gradient existing in the plasma. This nonuniform-distribution of the local heat flux density causes a nonuniform distribution of the. local evaporated-mass flux and related reaction force around the surface of an evaporating particle, and thus causes an additional force on the particle. Calculated results show that the thermophoretic force on an evaporating particle may substantially exceed that on a nonevaporating one, especially for the case of a metallic particle (with infinite electric conductivity). The effect of evaporation on the thermophoretic force is more pronounced as the evaporation latent heat of the particle material is comparatively low and as high plasma temperatures are involved.     

Heat transfer to a particle under plasma conditions with vapor contamination from the particle

Heat transfer to a copper particle immersed into an argon plasma is considered in this paper, including the effects of contamination of the plasma (transport coefficients) by copper vapor from the particle. Except for cases of high plasma temperatures, the vapor content in the plasma is shown to have a considerable influence on heat transfer to a nonevaporating particle, and, to a lesser extent, on heat transfer to an evaporating particle. Evaporation itself reduces heat transfer to a particle substantially as shown in a previous paper [Xi Chen and E. Pfender, Plasma Chem. Plasma Process.,2, 185 (1982)]. Comparisons of the calculated results with those based on a method suggested in the above reference show that the simplified assumptions employed, i.e., that the surface temperature is equal to the boiling point and that plasma properties based on a fixed composition are applicable, can be employed to simplify calculations for many cases. This study reveals that a considerable portion of a particle must be vaporized before a steady concentration distribution is established around the particle.Nomenclature C _p specific heat at constant pressure - D diffusion coefficient of copper in the mixture - D _a diffusion coefficient of copper atoms in the mixture - D _i ambipolar diffusion coefficient of copper ions in the mixture - f mass fraction of copper in the mixture - f _a mass fraction of copper atoms in the mixture - f _i mass fraction of copper ions in the mixture - f mass fraction of copper in the plasma far away from the particle - f _s mass fraction of copper at the particle surface - G total mass flow rate due to evaporation - G _a mass flow rate of copper atoms - G _i mass flow rate of copper ions - H function defined in Eq. (19) - h specific enthalpy - h _s specify enthalpy at the particle surface - h specific enthalpy corresponding toT andf - k thermal conductivity - L latent heat of evaporation - M ₁ molecular weight of argon (M ₁=39.99) - M ₂ molecular weight of copper (M ₂=63.55) - p ₀ pressure of the gas mixture - p _s partial pressure of copper vapor at the particle surface - Q ₀ heat flux to a particle without evaporation - Q ₁ heat flux to a particle with evaporation - R gas constant - r radical coordinate - r _s particle radius - S heat conduction potential defined in Eq. (4) - S _s surface value ofS, corresponding toT _s andf _s - S free-stream value ofS, corresponding toT andf - T temperature - T _b boiling temperature of particle material - T _s particle surface temperature - T plasma temperature - density - T temperature step for numerical integration     

Heat transfer to a single particle exposed to a thermal plasma

This paper is concerned with an analytical study of the heat and mass transfer process of a single particle exposed to a thermal plasma, with emphasis on the effects which evaporation imposes on heat transfer from the plasma to the particle. The results refer mainly to an atmospheric-pressure argon plasma and, for comparison purposes, an argon-hydrogen mixture and a nitrogen plasma are also considered in a temperature range from 3000 to 16,000 K. Interactions with water droplets, alumina, tungsten, and graphite particles are considered in a range of small Reynolds numbers typical for plasma processing of fine powders. Comparisons between exact solutions of the governing equations and approximate solutions indicate the parameter range for which approximate solutions are valid. The time required for complete evaporation of a given particle can be determined from calculated values of the vaporization constant. This constant is mainly determined by the boiling (or sublimation) temperature of the particles and the density of the condensed phase. Evaporation severely reduces heat transfer to a particle and, in general, this effect is more pronounced for materials with low latent heat of evaporation.     

Heat transfer from a transferred-arc plasma to a cylindrical enclosure

Hear-transfer rates from an axially enclosed transferred arc to a surrounding water-cooled cylindrical sleeve, 15 cm high, were measured. The arc (argon or nitrogen) was struck between a movable cathode within the sleeve and a bath of molten copper below the sleeve, serving as anode. The distance from the bottom of the sleeve to the surface of the molten copper (L _o) was constant. Variables studied were the diameterD of the sleeve (5, 7.5. and 10 cm), the length of the arc within the sleeveL (5, 10, and 15 cm), the currentI (200, 250, and 300 A) and a tangential flow of gas or vortex within the sleeve (0, ?0, and 50 liters/min). The total power transferred to the sleeve,P _s was measured caloronetrically and was the sure ofP _r the effective power radiated by the arc of lengthL within the sleeve.P _a, the power radiated into the sleeve from the arc of length L_o below the sleeve, andP _o , the power radiated from the melt surface (a constant of small value), minusP _a , the power lost by convection from the sleeve (negligible, except for a strong vortex). BothP _r andP _o were found to be equal to the product of the Joule heat released within their respective arc lengths, IV_gL and IV_g0L₀ (where V_g and V_g0 are the voltage gradients), and dimenonless efliciency factors, η_r and η₀. which are functions ofL/D andL ₀ /D, respectively, for each gas, regardless of the geometry of the sleeve, the current, and the strength of the vortex.     

Heat transfer in RF plasma sintering: Modeling and experiments

The mechanisms of heat transfer from an argon RF plasma, generated in a water-cooled quartz tube, to a sintering sample immersed into the plasma and to the walls of the plasma torch have been studied both analytically and experimentally for pressures from 1 to 50 torr. The model, based on the assumption of chemical equilibrium in a two-temperature plasma with rotational symmetry, includes the influence of the magnetic field and of the Knudsen number on the thermal conductivity of the plasma. At pressures below 20 torr heat transfer to the sintering sample is enhanced compared to heat transfer to the wall of the plasma torch. This nonsymmetry is attributed to the Hall parameter and Knudsen number effect. The relative importance of the two effects is a function of the pressure. A comparison with experiments, based on calorimetric and indirect heat transfer measurements for a range of pressures and power levels, indicates satisfactory agreement with analytical predictions, with the exception of larger discrepancies at higher power levels and relatively low pressures. For pressures below 5 torr, the chemical equilibrium assumption becomes questionable, i.e., the sintering model underestimates the heat transfer to the sintering sample.     

A lubricated stagnation point flow of nanofluid with heat and mass transfer phenomenon: Significance to hydraulic systems

The improved thermal association of heat transfer is considerably observed due to interaction of nanoparticles in recent days. The lubrication phenomenon with heat and mass transfer effects plays a key role in the hydraulic systems. In current research, the thermal impact of nanofluid over a lubricated stretching surfaces near a stagnation point analytical has been studied. A thin layer of lubricating fluid with a variable thickness provides lubrication. The inspection of thermophoresis and Brownian motion phenomenon is illustrated via Boungrino model. The analytical finding of refurbished boundary layer ordinary differential equations is obtained by a reliable and proficient technique namely variational iteration method (VIM). The Lagrange Multiplier is a potent tool in proposed technique to reduce the computational work. In addition, a numerical comparison is presented to show the effectiveness of this study. The range of flow parameters is based on theoretical flow assumptions. Physical inspection of involved parameters on velocities, temperatures, concentrations, and other quantities of interest when lubrication is presented. The current results present applications in polymer process, manufacturing systems, heat transfer and hydraulic systems.