Fundamentals of Combustion and Thermal Plasma

Abstract

Except for Cold Spray, thermal spray processes are based either on combustion or thermal plasmas. This chapter recalls the basic phenomena involved allowing understanding how the high temperature jets are generated and what are their properties. First the bases of combustion are presented with flames, detonation, and explosions: their stability limits, their temperatures and velocities. Then bases of thermal plasmas are discussed with a short presentation of how are calculated their compositions, specific masses, enthalpies, viscosities, and electrical and thermal conductivities with finally the results for the main gases used in thermal spraying. At last the basic concepts in modeling are presented: conservation equations (continuity, momentum, and energy), electromagnetic field equations, and at last laminar and turbulent flows. Bases presented in this chapter are then used in the different chapters related to the various spray processes.

Nomenclature

In the following when no unit is given in parenthesis, it means that the quantity is dimensionless

a :

Thermal diffusivity of the combustion wave (κ/ρ c _p) (m²/s)

B :

Magnetic induction (T)

c _k :

Concentration of species of type k

c _p :

Specific heat at constant pressure (J/K kg)

D :

Detonation velocity (m/s)

e :

Elementary charge (C)

E :

Electric field strength (V/m) or energy (J)

E _I :

Ionization energy (J or eV)

F :

External forces per unit mass (N/kg)

F _k :

Represents external forces (N)

F _kj :

Rate of momentum transfer between species k and other species j (kg/m² s²)

F/A :

Molar ratio of fuel to oxidizer

g :

Statistical weight

h :

Planck’s constant (6.6 × 10⁻³⁴ J s)

h _m :

Enthalpy per unit mass (J/kg)

k _B :

Boltzmann constant (1.13 × 10 J/kg particle)

k _R :

Specific reaction rate constant

I _k :

Mass flux of particles of type k (kg/m² s)

I _p :

Heat flux potential: ${I}_{\mathrm{p}}={\displaystyle {\int}_{T_{\mathrm{o}}}^{T_{\mathrm{p}}}\kappa (T)\mathrm{d}T}\left(\mathrm{W}/\mathrm{m}\right)$

I _s :

Heat flux potential: ${I}_{\mathrm{s}}={\displaystyle {\int}_{T_{\mathrm{o}}}^{T_{\mathrm{s}}}\kappa (T)\mathrm{d}T}\left(\mathrm{W}/\mathrm{m}\right)$

j :

Current density (A/m²)

k _f :

Forward reaction rate, k _f = A × T ^Β × exp(−E/kT) (m³/part s)

k _r :

Reverse reaction rate, for a binary reaction (m³/part s)

K _x :

Molar equilibrium constant (K _x = k _f/k _r)

M :

Arbitrary specification of all chemical species

M _e :

Mach number

(M _i ):

Concentration of species i (m⁻³)

m :

Mass (kg)

N _i :

Number species i

n′:

Total number of compounds involved in a chemical reaction

n :

Number density of particles (m⁻³)

n _i :

Number density of species i: n _i  = N _i /V (m⁻³)

p _i :

Pressure of species i (Pa)

p :

Total pressure (Pa)

q :

Chemical energy release at constant pressure (J/kg)

Q :

Partition function

R :

Perfect gas law constant (J/K kg)

R′:

Equivalence ratio or richness: R = (F/A)/(F/A)_{stoichiometry}

R*:

Radical

RR:

Reaction rate (units depend on the species number involved)

r :

Radial coordinate (m)

S :

Entropy (J/K)

S_L :

Flame velocity (m/s)

S _R :

Optically thin radiation losses (W/m³)

S ^∘_u :

Combustion wave velocity (m/s)

T :

Temperature (K)

t :

Time (s)

u :

Axial velocity component (m/s)

V :

Volume (m³)

v :

Radial velocity component (m/s)

v _g :

velocity of the center of mass (m/s)

v _k :

Velocity of particles of type k (m/s)

W :

Heat flux (W/m²)

x :

Coordinate (m)

z :

Axial coordinate (m); ion charge number

∇:

Spatial derivative (m⁻¹)

$\frac{\partial }{\partial t}$ :

Time derivative (s⁻¹)

α′:

Multiplication factor of radicals

α :

Thermal diffusion factor (A/V m)

ΔE _I :

Lowering of the ionization energy (J or eV)

ε _o :

Dielectric constant (A s/V m)

Φ :

Electric potential (V)

Φ :

Dummy variable

Φ :

Time averaged dummy variable

Φ′ :

Time fluctuating dummy variable

Γ _k :

Mass generation rate of species of type k (kg/m³ s)

γ :

Ratio of specific heats

κ :

Thermal conductivity (W/m K)

κ′:

Integrated mean thermal conductivity ${\kappa}^{\prime }=\left[1/\left({T}_{\mathrm{p}}-{T}_{\mathrm{s}}\right)\right]{\displaystyle {\int}_{T_{\mathrm{s}}}^{T_{\mathrm{p}}}\kappa (T)\dot{c}\mathrm{d}T\left(\mathrm{W}/\mathrm{m}\;\mathrm{K}\right)}$

κ _total :

Thermal conductivity also noted κ (W/m K)

κ ^h_tr :

Translational thermal conductivity of heavy species (W/m K)

κ ^e_tr :

Translational thermal conductivity of electrons (W/m K)

κ _R :

Reactional thermal conductivity (W/m K)

μ :

Molecular viscosity (Pa s)

ν′ _j :

Stoichiometric coefficient of the reactants

ν″ _j :

Stoichiometric coefficient of the products

ρ _i :

Specific mass (kg/m³)

ρ :

Total mass density (kg/m³)

ρ _el :

Space charge density (C/m³)

σ :

Electrical conductivity (S/m)

${\overline{\overline{\tau}}}_k$ :

Stress tensor of species k (Pa)

ζ :

Dummy variable (m)

a:

Atoms

amb:

Ambipolar

b:

Burned gas

e:

Electrons

eff:

Effective value

i:

Ions

j :

Summation index

k :

Summation index

LES:

Large Eddy Simulation

max:

Maximum

NS:

Navier–Stokes

o :

Reference value

r :

Radial direction

u:

Unburned gas

w:

Wall

z :

Axial direction

φ :

Circumferential direction