Gas Flow–Particle Interaction

Abstract

In the thermal processing of powders for spraying of protective coatings or free-standing bodies, the proper control of particle or droplet (suspension or solution) trajectories and their temperature and momentum histories in the gas flow represents one of the critical aspects on which the overall success of the operation strongly depends. In fact, a slight deviation from near optimal conditions can easily lead to poor results due to either the lack of melting of particles, or insufficient impact velocities, or the modification of their composition due to particle evaporation or unwanted chemical reactions. The first parts of the chapter deal with a single particle trajectory and heating, including heat propagation, with mass transfers and chemical reactions for liquid or gaseous phases. Then ensemble of particles is considered with the injection problems and possible loading effects. The last section is devoted to the interaction between a high-energy gas and a liquid, which fragmentation and then vaporization cools the hot gases. Moreover liquid fragmentation and vaporization is drastically modified by arc root fluctuations for plasmas produced by direct current torches

Nomenclature

Units are indicated in parentheses, when no unit is indicated the parameter is dimensionless

a :

Accommodation coefficient

a _i :

Sound velocity (m/s)

A :

Particle projected surface area perpendicular to the flow (A = πd _p ²/4) (m²)

Bi:

Biot number $\left(\frac{\kappa }{\kappa_{\mathrm{p}}}\right)$

c :

The molar density of the bulk gas (mol/m³)

C _D :

Drag coefficient (F _D/A)/(0.5ρv ²)

c _i :

Mass fraction of metal vapor at the location i

c _pi :

Specific heat at constant pressure in the state i (J/kg.K)

c _vi :

Specific heat at constant volume in the state i (J/kg.K)

d _d :

Drop diameter (m)

d _l :

Drop or liquid jet diameter (m)

d _p :

Particle diameter (m)

D _vg :

Diffusion coefficient of metal vapor through the surrounding gas (m²/s)

F _B :

Basset history term (N)

F _c :

Coriolis force (N)

F _b :

Body force per unit particle mass (N)

F _d :

Force related to the pressure jump in the detonation wave ${F}_{\mathrm{d}}=\frac{\pi {d}_{\mathrm{p}}^2}{4}\Delta p$ (N)

F _D :

Drag force exerted by the fluid on the particle (N)

F _g :

Gravity force (N)

F _i :

Inertia force (m _p.γ _p) (N)

F _p :

Pressure gradient force (N)

F _S :

Surface tension force (N)

F _T :

Thermophoresis force (N)

g :

Gravity acceleration (9.81 m/s²)

h :

Heat transfer coefficient (W/m²⋅K)

h _g :

Enthalpy (J/m³)

h′_i :

Specific enthalpy of the plasma calculated at i (J/kg)

I :

Arc current (A)

I(T):

Heat conduction potential $\left({\displaystyle {\int}_{T_{300}}^T\kappa (s)\cdot ds}\right)$ (W/m)

Kn:

Knudsen number $\left(\raisebox{1ex}{$\lambda $}\!\left/ \!\raisebox{-1ex}{$ dp$}\right.\right)$

k _d :

Mass transfer coefficient (m/s)

L :

Mixing length in turbulent model (m)

m ^o_cg :

Carrier gas mass flow rate (kg/s)

m _p :

Particle mass (kg)

M :

Molecular weight (kg/mol)

Ma:

Mach number (v _g/a _g)

N _max :

Maximum molar flux of vaporization from a droplet (m²/s)

Nu:

Nusselt number (hd/κ)

N _v :

Molar flux of vapor (mol/m²⋅s)

p :

Total pressure (Pa)

p _i :

Partial pressure of species i (Pa)

p _o :

Saturation vapor pressure of a liquid (Pa)

P :

Torch power (kW)

P _l :

Splat parameter (m)

P _eff :

Torch effective power (kW)

Pr:

Prandtl number (μ⋅c _p/κ)

q :

Heat flux (W/m²)

Q :

Heat transferred to a particle in 1 s (W)

r :

Plasma or particle radius (m)

r _d :

Liquid droplet radius (m)

r _s :

Initial liquid drop radius (m)

R :

Plasma torch internal or particle radius (m)

R′:

Universal ideal gas constant (R′ = 8.32 J/K ⋅ mol)

Re:

Reynolds’ number (ρ⋅U _R.d _p/μ)

R _inj :

Internal radius of injection tube (m)

S _p :

Surface area of the particle (πd ²_p /4) or of the splat (π⋅D ²_p /4) (m²)

S :

Cross section of injection tube (m²)

Sc:

Schmidt number (ν/D _v,g)

Sh:

Sherwood number (k _d.d _p/D _v,g)

St:

Stokes number St = ρ _p d ²_p v _p/(μ _g l _BL)

Ste:

Stephan number; Ste = c _pi(T _m − T _s)/ΔH _m

t _d :

Drop fragmentation time (s)

t _r :

Time of flight of particles (s)

t _s :

Vaporization time (s)

T _a :

Ambient temperature (K)

T _f :

Mean film temperature ((T _s + T _∞)/2) (K)

T _s :

Particle surface temperature (K)

T _∞ :

Temperature of the flow out of the boundary layer around the particle (K)

U :

Mean arc voltage (V)

U(t):

Transient arc voltage (V)

u :

Gas velocity component in the axial direction (m/s)

u _p :

Particle velocity component in the axial direction (m/s)

u _R :

Relative velocity between the particle and its surrounding (m/s)

u _r :

Relative velocity gas–liquid (m/s)

v :

Gas velocity component in the radial direction (m/s)

v _p :

Particle velocity component in the radial direction (m/s)

V :

Volume (m³)

V _p :

Particle volume (m³)

V _s :

Liquid drop volume (m³)

w _s :

Work resulting from the drag force (J)

We:

Weber number ($\mathrm{We}=\frac{\rho_{\mathrm{g}}\times {u}_{\mathrm{r}}^2\times {d}_{\mathrm{l}}}{\sigma_{\mathrm{l}}}$)

Z :

Ohnesorge number $\left(Z=\frac{\mu_{\mathrm{l}}}{\sqrt{\rho_{\mathrm{l}}\times {d}_{\mathrm{l}}\times {\sigma}_{\mathrm{l}}}}\right)$

γ :

Isentropic coefficient (c _p/c _v)

δ :

Boundary layer thickness (m)

ΔE _s :

Variation of surface energy (J)

ΔH _m :

Latent heat of fusion (J/kg)

ΔH _v :

Latent heat of vaporization (J/kg)

ε :

Particle emissivity (integrated over all wave lengths)

ϕ(r):

Function representing temperature, enthalpy, velocity

γ _p :

Particle acceleration (m/s²)

η _th :

Torch thermal efficiency (%)

κ :

Thermal conductivity of the fluid (W/m K)

κ _p :

Thermal conductivity of the particle (W/m K)

$\overline{\kappa}$ :

Mean integrated thermal conductivity $\left(\frac{1}{ T\mathit{\infty}-{T}_{\mathrm{s}}}{\displaystyle {\int}_{T_{\mathrm{s}}}^{T\mathit{\infty}}\kappa (s)\cdot ds}\right)$ (W/m K)

λ :

Plasma mean free path (m)

μ _i :

Fluid viscosity (Pa⋅s)

ν _cg :

Carrier gas velocity (m/s)

ν _i :

Kinematic viscosity (μ/ρ) (m²/s)

ρ _g :

Gas mass density (kg/m³)

ρ _i :

Fluid density or specific mass (kg/m³)

σ :

Droplet surface tension (N/m)

σ _s :

Stephan–Boltzmann constant (5.670×10⁻⁸ W/m²⋅K⁴)

ξ :

Ratio of the cylindrical splat to droplet diameter (ξ = D _p/d _p)

i:

Index that relates to temperature at which the property is evaluated, or to indicate a specific species

i = s:

Property evaluated at surface temperature

i = ∞ or g:

Property evaluated at plasma temperature

i = g:

Gas

i = p:

Particle

i = l:

Liquid