Transport Properties of Gases Under Plasma Conditions

Abstract

This chapter is devoted to the transport phenomena under plasma conditions. The calculation of the transport coefficients related to the elementary particles in a gas is quite complex, even in the rather simple case of ideal gases. It is therefore valuable to develop very simple approximate methods that yield physical insight into basic mechanisms. Simple approximations often lead to the correct dependence of all significant parameters (such as pressure or temperature), even if the numerical values sometimes differ by up to 50 % from the results of the rigorous calculation The first part of this chapter deals with the simplest approximate method that can be used to calculate the transport properties, i.e., the elementary kinetic theory. This is followed by a more rigorous treatment of these phenomena for equilibrium plasmas and gases in complete thermodynamic equilibrium (CTE).

Nomenclature and Greek Symbols

A_x,x

Coefficients for calculating the reactional thermal conductivity

$\overrightarrow{{\mathrm{A}}_{\upiota}}$

Vectorial coefficients for calculating the perturbation function ϕ_i

b

Impact parameter

${\overleftrightarrow{\mathrm{B}}}_{\mathrm{i}}$

Second-order tensor allowing calculation of the perturbation function ϕ_i

${\overrightarrow{\mathrm{v}}}_{\mathrm{c}}$

Velocity of center of mass (m/s)

c_pi

Molar specific heat of species at constant pressure (J/mol.K)

c_v

Specific heat at constant volume (J/mole.K)

$\overrightarrow{{\mathrm{d}}_{\mathrm{i}}^{,}}$

Diffusion forces

dA

Elementary surface (m²)

$\mathrm{d}\overrightarrow{\mathrm{r}}$

Elementary volume in ordinary space (m³)

$\mathrm{d}\overrightarrow{\mathrm{v}}$

Elementary volume in velocity space: $\left(\mathrm{d}\overrightarrow{\mathrm{v}}={\mathrm{dv}}_{\mathrm{x}}{\mathrm{dv}}_{\mathrm{y}}{\mathrm{dv}}_{\mathrm{z}}\right)\left({\mathrm{m}}^3/{\mathrm{s}}^3\right)$

D

Diffusion coefficient (m²/s)

D_i

Scalar coefficients for calculating the perturbation function ϕ_i

D_i,j

Ordinary diffusion coefficients between species i and j (m²/s)

D_iT

Thermal diffusion coefficients (m²/K.s)

$\overline{{\mathrm{D}}_{\mathrm{AB}}^{\mathrm{E}}}$

Combined electric field diffusion coefficient (m²/V.s)

$\overline{{\mathrm{D}}_{\mathrm{AB}}^{\mathrm{P}}}$

Combined pressure diffusion coefficient (m²/s)

$\overline{{\mathrm{D}}_{\mathrm{AB}}^{\mathrm{T}}}$

Combined temperature diffusion coefficient (kg/m.s)

$\overline{{\mathrm{D}}_{\mathrm{AB}}^{\mathrm{x}}}$

Combined ordinary diffusion coefficient (m²/s)

$\overline{{\mathrm{X}}_{\mathrm{B}}}$

Sum of the mole fractions of the species of gas B

Left hand side of Boltzman equation, Eq.45

Right hand side of Boltzman equation, Eq.45

$\overrightarrow{\mathrm{E}}$

Electric field (V/m)

E

Externally applied field (V/m)

f_i

Distribution function

f ⁱ₀

Equilibrium distribution function

F_x

Component of the force in the x-direction (N)

g₁₂

Relative velocity $\left({\mathrm{g}}_{12}=\left|\overrightarrow{{\mathrm{v}}_1}-\overrightarrow{{\mathrm{v}}_2}\right|\left(\mathrm{m}/\mathrm{s}\right)\right)$

h

Planck’s constant (6.6 × 10⁻³⁴ W.s²)

H_i

Molar enthalpy of species i (kJ/mol)

H ^D_N

Molar dissociation enthalpy of nitrogen molecule (kJ/mol)

${\mathrm{H}}_{{\mathrm{N}}^{+}}^{\mathrm{I}}$

Molar ionization enthalpy of nitrogen atom (kJ/mol)

${\overrightarrow{\mathrm{J}}}_{\mathrm{E}}$

Energy flux (W/m²)

${\overrightarrow{\mathrm{J}}}_{\mathrm{n}}$

Particle flux (part/m².s)

${\overrightarrow{\mathrm{J}}}_{\mathrm{px}}$

Momentum flux in x-direction

J_χ

Flux density of χ or net quantity of χ (particles number, transverse momentum, energy, charge) transported across per unit area and unit time

${\overrightarrow{\mathrm{J}}}_{\mathrm{x}}$

Flux of the quantity x

k

Boltzmann constant (k = 1.38 × 10⁻²³ J/K)

ℓ

Mean free path (m)

ℓ _z

Mean free path in z-direction (m)

${\overline{\mathrm{m}}}_{\mathrm{A}}$

Number–density–weighted average mass of the species present in gas A

${\overline{\mathrm{m}}}_{\mathrm{B}}$

Number–density–weighted average mass of the species present in gas B

m_i

Mass of the particle of chemical species i (kg)

M_i

Atomic mass of chemical species i (kg)

n

Number density (m⁻³ )

n_i

Number density of chemical species i (m⁻³ )

n ^′_i

Mole number of chemical species i

p

Pressure (Pa)

q_x

Heat flux due to temperature gradients in x direction (W/m²)

$\overline{\overline{\mathrm{p}}}$

Pressure tensor

${\overrightarrow{\mathrm{q}}}_{\mathrm{i}}$

Flux vector for the transport of kinetic energy of particles of species i (J/m².s)

${\overrightarrow{\mathrm{q}}}_{\mathrm{R}}$

Reactional heat flux vector (J/m².s)

${\overrightarrow{\mathrm{q}}}_{\mathrm{z}}$

Heat flux vector in z-direction (J/m².s)

q ^ls_ij

Bracket integral

${\overline{\mathrm{Q}}}_{\mathrm{ij}}^{\mathrm{ls}}$

Product of the reduced collision integral by the cross-sectional area (m²)

Q ^l_ij

Quantum collision cross section

${\mathrm{Q}}^{\mathrm{l}}\left(\overrightarrow{\mathrm{v}}\right)$

Elastic collision cross section at the degree 1 of the Legendre functions

$\overrightarrow{\mathrm{r}}$

Position vector

$\overrightarrow{{\mathrm{r}}^{*}}$

Relative position vector $\left(\overrightarrow{{\mathrm{r}}^{*}}={\mathrm{r}}_1-{\mathrm{r}}_2\right)$ (m)

r_m

Distance of closest approach (m)

R

Perfect gas constant (R = 8.32 J/K.mole)

T

Absolute temperature (K)

$\overrightarrow{\mathrm{u}}$

Local macroscopic velocity (m/s)

${\overrightarrow{\mathrm{U}}}_{\mathrm{i}}$

Peculiar velocity $\left(\overrightarrow{{\mathrm{U}}_{\upiota}}=\overrightarrow{{\mathrm{v}}_{\mathrm{i}}}-\overrightarrow{{\mathrm{v}}_0}\right)$ of particles of species i (m/s).

${\overrightarrow{\mathrm{U}}}_{\mathrm{ix}}$

Peculiar velocity in the x-direction

$<{\overrightarrow{\mathrm{U}}}_{\mathrm{i}}>$

Diffusion velocity of particles of species i

${\overrightarrow{\mathrm{v}}}_{\mathrm{i}}$

Velocity of particle of species i (m/s)

$\overrightarrow{\mathrm{v}}$

Mean velocity of particles (m/s)

${\overrightarrow{\mathrm{v}}}_0$

Mean flow velocity {m/s)

V(r)

Interaction potential

${\overrightarrow{\mathrm{V}}}_{\mathrm{ij}}$

Relative velocity before collision $\left({\overrightarrow{\mathrm{V}}}_{\mathrm{i}\mathrm{j}}={\overrightarrow{\mathrm{v}}}_{\mathrm{i}}-{\mathrm{v}}_{\mathrm{j}}\right)$ (m/s)

${\overrightarrow{\mathrm{V}}}_{\mathrm{ij}}^{\prime }$

Relative velocity after collision $\left({\overrightarrow{\mathrm{V}}}_{\mathrm{i}\mathrm{j}}^{\prime }={\overrightarrow{\mathrm{v}}}_{\mathrm{i}}^{\prime }-{\overrightarrow{\mathrm{v}}}_{\mathrm{j}}^{\prime}\right)$ (m/s)

x_i

Mole fraction of species i

Z_ij

Dimensionless coefficient to calculate the viscosity of a mixture

Z_j

Charge number

$\overline{\upvarepsilon}$

Mean kinetic energy of particles (J)

ζ

Inverse of the frequency of collision (s)

θ

Angle in spherical coordinates

θ′

Deviation angle

χ

Variable

〈χ_i〉

Mean value of χ_i

${\overline{\upchi}}_{\mathrm{i}}$

Mean value of χ_i

κ

Thermal conductivity (W/m.K)

$\overline{\upkappa}$

Mean integrated thermal conductivity, see Eq. 71 (W/m.K)

κ_int

Internal thermal conductivity (W/m.K)

κ_R

Reactional thermal conductivity (W/m.K)

κ_tr

Translational thermal conductivity (W/m.K)

μ

Gas viscosity (kg/m.s)

μ_e

Electron mobility (m²/V.s)

μ_m

Reduced mass $\left({\upmu}_{\mathrm{m}}={\mathrm{m}}_{\mathrm{i}}{\mathrm{m}}_{\mathrm{j}}/\left({\mathrm{m}}_{\mathrm{i}}+{\mathrm{m}}_{\mathrm{j}}\right)\right)$ (kg)

μ_mix

Molecular viscosity of a mixture, Eq. (72) (kg/m.s)

ξ

Perturbation parameter: 1/ξ measures the frequency of collisions

ρ

Specific mass (kg/m³)

ρ′

Collision cross section for particles with initial velocities between ($\overrightarrow{{\mathrm{v}}_1}$ and $\overrightarrow{{\mathrm{v}}_1} +$ $\mathrm{d}\overrightarrow{{\mathrm{v}}_1}$) and ($\overrightarrow{{\mathrm{v}}_2}$ and $\overrightarrow{{\mathrm{v}}_2} +$ $\mathrm{d}\overrightarrow{{\mathrm{v}}_2}$) and velocities after collision between (${\overrightarrow{\mathrm{V}}}_1^{\prime }$ and ${\overrightarrow{\mathrm{V}}}_1^{\prime }+\mathrm{d}{\overrightarrow{\mathrm{V}}}_1^{\prime }$) and (${\overrightarrow{\mathrm{V}}}_2^{\prime }+\mathrm{d}{\overrightarrow{\mathrm{V}}}_2^{\prime }$)

σ_e

Electrical conductivity (mhos/m)

σ₀

Total collision cross section (m²)

σ_en

Total collision cross section between electrons and neutral particles (m²)

σ_ij

Total collision cross section between particles of type i and j (m²)

$\sigma \left(\overrightarrow{\mathrm{v}}\right).\mathrm{d}{\varOmega}^{\prime }$

Collision cross section for particles emerging after collision with a relative velocity ${\overrightarrow{\mathrm{V}}}^{\prime }$ into a solid angle range dΩ′ about θ′ and φ′

τ_zx

Stress in a plane at z in direction x (N/m²)

φ

Angle in spherical coordinates

ϕ _i

Perturbation function for calculating the distribution function

Ω ^ls_ij

Hirschfelder’s collision integral for particles of species i and j (m²)

${\overline{\Omega}}_{\mathrm{ij}}^{\mathrm{ls}}$

Reduced collision integral for particles of species i and j (m²)

${\overrightarrow{\nabla}}_{\mathrm{r}}$

Position gradient vector, $\left(\frac{\partial }{\partial \mathrm{x}},\frac{\partial }{\partial \mathrm{y}},\frac{\partial }{\partial \mathrm{z}}\right)$

${\overrightarrow{\nabla}}_{\mathrm{v}}$

Velocity gradient vector, $\left(\frac{\partial }{\partial {\mathrm{v}}_{\mathrm{x}}},\frac{\partial }{\partial {\mathrm{v}}_{\mathrm{y}}},\frac{\partial }{\partial {\mathrm{v}}_{\mathrm{z}}}\right)$