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A. del Val et al.
For a dissociated, ionized gas, the system is not closed. Extra equations describing the species continuity and energy conservation including a diffusion term, must
be implemented to fully describe the system [25]
Species Continuity Equation
∂ρ i
∂t
+ ∇ · (ρ i v) + ∇ · (ρ i V i ) = 0, i ∈ S
(16.5)
with the additional constraint
j ∈S ρ j V j = 0
Species Energy Conservation Equations
∂
∂t
(ρ i e i )+∇·(ρ i e i v)+∇·(q i )−j i ·E’+ρ i V i ·
∂
∂t
v+P i : ∇v = 0, i ∈ S
(16.6)
Being V i the diffusion velocity of species i.
Simulating re-entry flows in inductively-coupled plasma facilities requires and
extra effort to model the electromagnetic field as seen in the previous equations.
In this case, the Maxwell equations [26] are solved to obtain the electromagnetic
field, playing a role in the flowfield through the effect of the Lorentz force and
Joule heating [27]. In Section 16.3.1.1, a particular computational model that
implements these continuity and conservation equations for a plasma wind tunnel
is introduced for its role in determining experimental conditions and other nonmeasured parameters of the flow field.
16.3.1.1 Resistive Magneto-Hydrodynamics (MHD) Model
In the particular case of inductively-coupled plasma facilities, the hydrodynamic and
electromagnetic equations (MHD) are discretized and solved. The physical description incorporates various kinetic, chemical, thermodynamic, and mathematical tools
developed in the previous chapters. The following hypothesis are assumed to be
satisfied in the plasma flow:
f << f p The plasma frequency f p = n
1/2
e q e /[2π(( 0 m e ) 1/2 ] represents the
oscillation frequency of the free electrons about their equilibrium positions [28].
The displacement current can be neglected in the plasma provided that the torch
frequency f is lower than the plasma frequency [25].
L C << λ EM The current in the inductor generates electromagnetic waves of
length λ EM = 1/[f (( 0 μ 0 ) 1/2 ]. When the coil length L C is smaller than the
electromagnetic wavelength, the displacement current in the inductor remains
negligible in the computation of the electromagnetic field.
LTE Local Thermodynamic Equilibrium is a strong hypothesis. The composition
of the mixture of the various species present in the plasma is considered to
instantaneously adapt itself to changes in the flow, such that the plasma can be
A. del Val et al.
For a dissociated, ionized gas, the system is not closed. Extra equations describing the species continuity and energy conservation including a diffusion term, must
be implemented to fully describe the system [25]
Species Continuity Equation
∂ρ i
∂t
+ ∇ · (ρ i v) + ∇ · (ρ i V i ) = 0, i ∈ S
(16.5)
with the additional constraint
j ∈S ρ j V j = 0
Species Energy Conservation Equations
∂
∂t
(ρ i e i )+∇·(ρ i e i v)+∇·(q i )−j i ·E’+ρ i V i ·
∂
∂t
v+P i : ∇v = 0, i ∈ S
(16.6)
Being V i the diffusion velocity of species i.
Simulating re-entry flows in inductively-coupled plasma facilities requires and
extra effort to model the electromagnetic field as seen in the previous equations.
In this case, the Maxwell equations [26] are solved to obtain the electromagnetic
field, playing a role in the flowfield through the effect of the Lorentz force and
Joule heating [27]. In Section 16.3.1.1, a particular computational model that
implements these continuity and conservation equations for a plasma wind tunnel
is introduced for its role in determining experimental conditions and other nonmeasured parameters of the flow field.
16.3.1.1 Resistive Magneto-Hydrodynamics (MHD) Model
In the particular case of inductively-coupled plasma facilities, the hydrodynamic and
electromagnetic equations (MHD) are discretized and solved. The physical description incorporates various kinetic, chemical, thermodynamic, and mathematical tools
developed in the previous chapters. The following hypothesis are assumed to be
satisfied in the plasma flow:
f << f p The plasma frequency f p = n
1/2
e q e /[2π(( 0 m e ) 1/2 ] represents the
oscillation frequency of the free electrons about their equilibrium positions [28].
The displacement current can be neglected in the plasma provided that the torch
frequency f is lower than the plasma frequency [25].
L C << λ EM The current in the inductor generates electromagnetic waves of
length λ EM = 1/[f (( 0 μ 0 ) 1/2 ]. When the coil length L C is smaller than the
electromagnetic wavelength, the displacement current in the inductor remains
negligible in the computation of the electromagnetic field.
LTE Local Thermodynamic Equilibrium is a strong hypothesis. The composition
of the mixture of the various species present in the plasma is considered to
instantaneously adapt itself to changes in the flow, such that the plasma can be