Weakly-Ionized Air Plasma Theory  
For the plasma technologies here considered (aerodynamic flow control, MHD power generation, energy bypass in pulse detonation engines), the airflow is expected to be ionized artificially through electron beams, microwaves, or strong applied electric fields. Since the cost of ionization for the air molecules is rather large, only a small fraction of the gas can be ionized in order to keep the power requirements to a reasonable level. Hence why the plasma can be considered weakly-ionized. Similarly to strongly-ionized plasmas, weakly-ionized plasmas require the simultaneous solution of the mass, momentum, and energy equations for the neutrals and the charged species, as well as of the Maxwell equations for the electric and magnetic fields. However, because of the low ionization fraction of weakly-ionized plasmas, the electrical conductivity is expected to be quite small and the plasma becomes collision-dominated. Under such conditions, the governing equations take on a very different formulation as those describing strongly-ionized plasmas. A brief outline of the chemical model, of the charged species transport equations, of the neutrals transport equations and of the electric field potential equation applicable to weakly-ionized air is here given.
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Chemical Reactions
The degree of ionization of the air plasma as well as its chemical composition can be predicted using a finite rate nonequilibrium 8-species 28-reactions model as outlined below in Table 1. The model [2] is especially suited to air plasmas ionized by electron beams. Additionally to chemical reactions related to electron-beam ionization (see reactions 7a and 7b), the model also includes chemical reactions related to Townsend ionization (specifically reactions 1a and 1b).
 
table1.png

Townsend ionization consists of an electron accelerated by an electric field impacting the nitrogen or oxygen molecules and releasing in the process a new electron and a positive ion. This chemical reaction is the physical phenomenon that is at the origin of sparks and lighting bolts and that occurs in a weakly-ionized plasma whenever the electric field reaches very high values. It needs to be included in the chemical model when solving plasma aerodynamics in order to predict correctly the voltage drop within the cathode sheaths. Cathode sheaths are thin regions near the cathodes where the electric field is particularly high due to the current being mostly ionic.
Charged Species Transport Equations
The mass-conservation transport equations for the charged species must contain chemical source terms to account for ion and electron creation and destruction as well as other chemical reactions taking place in air: $$ \frac{\partial}{\partial t} \rho_k + \sum_{j} \frac{\partial }{\partial x_j} \rho_k \boldsymbol{V}_j^k = W_k $$ with $\rho_k$ the density of the $k$th species, $\boldsymbol{V}^k$ the velocity of the species under consideration including both drift and diffusion, and $W_k$ the chemical source terms. The chemical source terms are determined from the chemical reactions taking place in weakly-ionized air (see Table 1 above). The charged species velocity can be obtained from the momentum equation assuming negligible ion and electron inertia compared to the collision forces as follows (see Ref. [1] for details): $$ \boldsymbol{V}^{k}_i = \boldsymbol{V}^{\rm n}_i + \sum_{j=1}^3 s_k \tilde{\mu}^k_{ij} \left( \boldsymbol{E} + \boldsymbol{V}^{\rm n} \times \boldsymbol{B} \right)_j - \sum_{j=1}^3 \frac{\tilde{\mu}^{k}_{ij}}{|C_k| N_k} \frac{\partial P_k}{\partial x_j} $$ where $\boldsymbol{E}$ is the electric field, $\boldsymbol{V}^{\rm n}$ is the neutrals velocity including drift and diffusion, $N_k$ is the number density, $P_k$ is the partial pressure of species $k$, $s_k$ the sign of species $k$ (equal to +1 for the positive ions and to -1 for the electrons and negative ions), $C_k$ the charge of species $k$ (equal to $-e$ for the electrons, $+e$ for the positive ions, $-e$ for the negative ions, etc), and where $\tilde{\mu}$ is the tensor mobility equal to: $$ \tilde{\mu}^k \equiv\frac{\mu_k}{1+\mu_k^2|\boldsymbol{B}|^2} \left[\begin{array}{c} 1+\mu_k^2 \boldsymbol{B}_1^2 \\ \mu_k^2\boldsymbol{B}_1\boldsymbol{B}_2-s_k\mu_k\boldsymbol{B}_3 \\ \mu_k^2\boldsymbol{B}_1\boldsymbol{B}_3 +s_k\mu_k\boldsymbol{B}_2 \end{array} \right. \left.\begin{array}{c} \mu_k^2\boldsymbol{B}_1\boldsymbol{B}_2+s_k \mu_k \boldsymbol{B}_3 \\ 1+\mu_k^2\boldsymbol{B}_2^2 \\ \mu_k^2 \boldsymbol{B}_2\boldsymbol{B}_3-s_k\mu_k\boldsymbol{B}_1 \end{array}\right. \left. \begin{array}{c} \mu_k^2\boldsymbol{B}_1\boldsymbol{B}_3-s_k \mu_k \boldsymbol{B}_2 \\ \mu_k^2\boldsymbol{B}_2\boldsymbol{B}_3+s_k\mu_k\boldsymbol{B}_1 \\ 1+\mu_k^2\boldsymbol{B}_3^2 \end{array} \right] $$ where $\mu_k$ is the mobility of species $k$ and $\boldsymbol{B}$ the magnetic field vector.
Neutrals Mass Conservation Equation
The mass-conservation transport equations for the neutral molecules must contain chemical source terms to account for ion creation and destruction as well as other chemical reactions taking place in air: $$ \frac{\partial}{\partial t} \rho_k + \sum_{j} \frac{\partial }{\partial x_j} \rho_k \boldsymbol{V}^{\rm n}_j - \underbrace{\sum_j \frac{\partial}{\partial x_j}\left(\nu_k \frac{\partial w_k}{\partial x_j} \right)}_{\textrm{diffusion terms}} ={W_k} $$ with $\boldsymbol{V}^{\rm n}_j$ the bulk velocity of the neutrals, $w_k$ the mass fraction and $\nu_k$ the diffusion coefficient. The diffusion terms are here limited to the diffusion of the neutrals species within each other and neglect the diffusion of the neutrals within the charged species. Such is an excellent approximation as long as the plasma remains weakly-ionized (i.e, the ionization fraction should remain less than $10^{-4}$ or so).
Total Momentum Conservation Equation
The total momentum equation for the plasma is obtained by adding the momentum equations for the neutrals and the charged species: $$ \frac{\partial}{\partial t} \rho \boldsymbol{V}^{\rm n}_i + \sum_j \frac{\partial }{\partial x_j} \rho \boldsymbol{V}^{\rm n}_j \boldsymbol{V}^{\rm n}_i + \frac{\partial P}{\partial x_i} = \underbrace{\sum_j \frac{\partial \tau_{ji}}{\partial x_j}}_\textrm{viscous force} + \underbrace{\rho_{\rm c} \boldsymbol{E}_i }_{\rm EHD~force} + \underbrace{ \left(\boldsymbol{J} \times \boldsymbol{B}\right)_i}_{\rm MHD~force} $$ with $P$ the total pressure of the gas including the electron and ion partial pressures and $\tau_{ji}$ the shear stress tensor and $\rho_{\rm c}$ and $\boldsymbol{J}$ are the net charge density and current density defined as: $$ \rho_{\rm c} \equiv \sum_k N_k C_k $$ $$ \boldsymbol{J}_i\equiv \sum_k C_k N_k \boldsymbol{V}^k_i $$ Also known as the Lorentz force, the MHD force occurs as a result of the magnetic field acting on the charges in motion, and can hence only take place when a current is flowing within the gas. On the other hand, the EHD force occurs as a result of the electric field acting on a non-neutral region of the plasma. The momentum imparted to the charged particules by the MHD and EHD forces is then transferred to the bulk of the gas through collisions between the charged particules and the neutrals.
Vibrational Energy Conservation Equation
A particularity of the relatively-low-temperature weakly-ionized plasmas is the nonequilibrium of the electron temperature and vibrational temperature with respect to the translational temperature of the neutrals. An example of the large degree of thermal nonequilibrium near an electrode can be seen below in Fig. 1.

figure2.png  ./download/file.php?id=2214&sid=d808087ce01ca8afff72498f93245cd2  ./download/file.php?id=2214&t=1&sid=d808087ce01ca8afff72498f93245cd2
Figure 1. The large degree of thermal nonequilibrium typical of cold plasmas can here be seen through the temperature, vibrational temperature, and electron temperature near an electrode as computed by Parent et al. [2].

Because the nitrogen vibrational energy relaxation distance can reach several centimeters or even meters at the low translational temperatures typical of weakly-ionized air plasmas, it is necessary to solve a separate equation accounting for the transport of the nitrogen vibrational energy: [38,39] $$ \frac{\partial}{\partial t} \rho_{\rm N_2} e_{\rm v} + \sum_j \frac{\partial }{\partial x_j} \left( \rho_{\rm N_2} \boldsymbol{V}^{\rm n}_j e_{\rm v} -e_{\rm v} \nu_{\rm N_2} \frac{\partial w_{\rm N_2}}{\partial x_j} +q^{\rm v}_j \right)\\ = \eta_{\rm v} \underbrace{ Q_{\rm J}^{\rm e} }_{\begin{array}{l} \rm Joule\\ \rm Heating \end{array} } + {\frac{\rho_{\rm N_2}}{\tau_{\rm vt}}}\left( e_{\rm v}^0 -e_{\rm v} \right) + W_{\rm N_2} e_{\rm v} $$ where $e_{\rm v}$ is the nitrogen vibrational energy, $e_{\rm v}^0$ is the nitrogen vibrational energy that would be obtained should $T_{\rm v}=T$, $q^{\rm v}$ the vibrational energy heat flux, and $Q_{\rm J}^{\rm e}$ the Joule heating due to the electron velocity being different from the velocity of the bulk of the plasma. The fraction of the Joule heating consumed in the excitation of the vibration levels of the nitrogen molecule, $\eta_{\rm v}$, is obtained from the electron temperature as shown below in Fig. 2.

figure1.png  ./download/file.php?id=2213&sid=d808087ce01ca8afff72498f93245cd2  ./download/file.php?id=2213&t=1&sid=d808087ce01ca8afff72498f93245cd2
Figure 2. Fraction of energy consumed in the excitation of the vibration levels of the nitrogen molecule as a function of the electron temperature. From Ref. [36] and Ch. 21 of Ref. [37].

The nitrogen vibrational energy is hence seen to be highly dependent on the Joule heating especially when the electron temperature is in the range 7000-30,000 K. Because weakly-ionized air plasmas often exhibit an electron temperature in that range, a large amount of the Joule heating typically gets deposited in form of nitrogen vibrational energy. Because the relaxation time $\tau_{\rm vt}$ of the nitrogen vibrational temperature is quite low in air in typical flight conditions [38,39], there is not enough time for most of the Joule heating to be transferred from the vibrational energy modes to the translational energy modes. Then, the heating does not result in a significant decrease of the gas density. This is a desirable feature when solving MHD generator flowfields since it can limit the negative effects of large density gradients on the generator performance. However, this may not be a desirable feature when trying to perform aerodynamic flow control through heat deposition because the latter performs satisfactorily only if a density gradient is created by the heating process (which would occur only if the Joule heating is converted to translational energy of the neutrals).
Electron Energy Conservation Equation
Because the electron temperature is in significant non-equilibrium with the neutrals temperature ($T_{\rm e}$ is typically 10-100 times higher than $T$), it is necessary to solve an additional transport equation for the electron energy. The electron energy transport equation (as outlined in Ref. [62]) can be derived from the first law of thermo applied to the electron fluid and substituting the pressure gradient from the momentum equation for the electron species shown above. We thus obtain: $$ \frac{\partial }{\partial t} \left( \rho_{\rm e} e_{\rm e} \right) + \sum_{i} \frac{\partial }{\partial x_i} \left(\rho_{\rm e} h_{\rm e} \boldsymbol{V}_i^{\rm e} \right) + \sum_{i} \frac{\partial q_i^{\rm e}}{\partial x_i} = W_{\rm e} e_{\rm e}+C_{\rm e} N_{\rm e}\boldsymbol{V}^{\rm e} \cdot \boldsymbol{E} - \frac{3 e P_{\rm e} \zeta_{\rm e}}{2 m_{\rm e} \mu_{\rm e}} - Q_{\rm ei} $$ where $\rho_{\rm e}$ is the electron density, $e_{\rm e}$ the electron translational energy, $h_{\rm e}$ the electron enthalpy, $q_i^{\rm e}$ the electron heat flux, $\zeta_{\rm e}$ the electron energy loss function, $P_{\rm e}$ the electron partial pressure, $m_{\rm e}$ the mass of one electron, $\mu_{\rm e}$ the electron mobility, $e$ the elementary charge, and $Q_{\rm ei}$ the energy the electrons lose per unit time per unit volume in creating new electrons through Townsend ionization. In the latter, the kinetic energy of the electrons does not appear, which is a consequence of the electron momentum equation not including the inertia terms, a valid assumption as long as the plasma remains weakly-ionized. It is noted that this does not necessarily entail that the change in kinetic energy of the electrons is negligible compared to the change in internal energy. In fact, the kinetic energy of the electrons is not negligible within cathode sheaths even when the plasma is weakly-ionized. But, including the kinetic energy terms would not improve the accuracy of the simulation in this case. In fact, including them would result in increased physical error because when combined with the momentum equation in which the inertia terms are neglected, the electron energy equation would not satisfy the first law of thermo. Because the inertia terms part of the momentum equation are neglected, the kinetic energy terms should also be neglected for the energy transport equation to satisfy the first law.
Total Energy Conservation Equation
The neutrals and ions translational temperature can be determined through the total energy transport equation which can be derived by summing the energy equations for each species as obtained from the first law of thermo and then making some simplifications applicable to a weakly-ionized plasma. The following is thus obtained: $$ \begin{array}{l}\displaystyle \frac{\partial }{\partial t}\left(\rho_{\rm N_2} e_{\rm v}+\sum_k \rho_k (e_k+h_k^\circ)+\frac{1}{2}\rho|\boldsymbol{V}^{\rm n}|^2 \right) \\ \displaystyle + \sum_{j} \frac{\partial }{\partial x_j} \left(\rho_{\rm N_2} \boldsymbol{V}_j^{\rm N_2} e_{\rm v} + \sum_{k} \rho_k \boldsymbol{V}^k_j (h_k+h_k^\circ)+\frac{1}{2}\rho \boldsymbol{V}^{\rm n}_j|\boldsymbol{V}^{\rm n}|^2 \right)\\ \displaystyle = -\sum_{i} \frac{\partial q_i}{\partial x_i} +\sum_{i} \sum_{j} \frac{\partial }{\partial x_j} \tau_{ji} \boldsymbol{V}_i^{\rm n} + \boldsymbol{E}\cdot\boldsymbol{J} + Q_{\rm b} \end{array} $$ where $Q_{\rm b}$ corresponds to the energy deposited to the gas by an external ionizer (such as electron beams, microwaves, laser beams, etc.), where $q_i$ is the total heat flux from the charged species and the neutrals, where $h_k$ is the species enthalpy and where $h_k^\circ$ is the species heat of formation. The species energy and enthalpy contains the translational, rotational, vibrational, and electronic energies at equilibrium. For all heavy species (the heavy species here refer to all ions and neutrals but do not include electrons) except nitrogen the translational, rotational, vibrational, and electronic energies are assumed to be at equilibrium at the temperature $T$; for nitrogen, the vibrational energy is determined from a separate transport equation, as outlined previously; for the electrons, the translation energy is determined from the electron energy transport equation.
Electric Field Potential Equation
In the momentum and energy equations outlined in the previous section, the electric and magnetic fields appeared as part of the MHD force, the EHD force, or the Joule heating. The electric and magnetic fields must hence be determined simultaneously to the fluid flow equations to close the system of equations. This can be done by solving the Maxwell equations. The Maxwell equations are particularly complex to solve as they involve the solution of 3 transport equations for the magnetic field and 3 other transport equations for the electric field. However, they can be reduced to simpler form by making some assumptions applicable to a weakly-ionized plasma. Indeed, because of the low ionization fraction of weakly-ionized plasmas, the electrical conductivity is expected to be quite small, leading in turn to a very small magnetic Reynolds number. At a small magnetic Reynolds number, the induced magnetic field can be assumed to be negligible whether or not an external magnetic field (originating from a permanent or electro magnet) is applied. Then, the partial differential equations solving for the induced magnetic fluxes do not need to be solved. We can simplify further the physical model by assuming steady-state of the electromagnetic fields with respect to the fluid flow (the so-called “electrostatic” assumption). The assumption of a steady-state for the electromagnetic fields is an excellent one for a weakly-ionized plasma even when solving unsteady fluid flows, since the flow speed and sound speed are considerably less than the electromagnetic wave speed (the speed of light). At steady-state, the curl of the electric field would be zero, and an electric field potential would exist. Then, the 3 transport equations for the electric field can be dropped in favour of one equation: the electric field potential equation. For a quasi-neutral weakly-ionized plasma, the electric field potential equation can thus be obtained from Gauss's law as follows: $$ \sum_{j=1}^3 \frac{\partial^2 \phi}{\partial x_j^2} = -\frac{1}{\epsilon_0} \sum_k C_k N_k $$ from which the electric field can be obtained as $ \boldsymbol{E}_j = - {\partial \phi}/{\partial x_j}$.
Limitations of the Physical Model
The physical model outlined above is hence valid both in the non-neutral sheaths and the quasi-neutral regions of weakly-ionized plasmas, and can predict accurately physical phenomena such as ambipolar diffusion, ambipolar drift, cathode sheaths, dielectric sheaths, unsteady effects in which the displacement current is significant, etc. Nonetheless, it is noted that the physical model considered herein makes several assumptions: (i) the induced magnetic field is assumed negligible, (ii) the drag force due to collisions between charged species is negligible compared to the one originating from collisions between charged species and neutrals, and (iii) the forces due to inertia change are assumed small compared to the forces due to collisions. The mathematical expressions for the latter forces as well as the justification for neglecting them when simulating weakly-ionized plasmas can be found in Ref. [1]. Finally it is cautioned that, because the electric field is obtained from Gauss's law, the physical model outlined in this section can not be used to tackle problems where the electric field is a significant function of a time-varying magnetic field, such as in inductively coupled plasmas or microwave induced plasmas. In those cases, the electric field would cease to be a potential field and would need to be determined through the full or simplified Maxwell equations. More details on when Gauss's law can and can not be used to determine the electric field can be found in Refs. [62,63].
References
[1]  B Parent, MN Shneider, and SO Macheret, “Generalized Ohm’s Law and Potential Equation in Computational Weakly-Ionized Plasmadynamics,” Journal of Computational Physics, Vol. 230, No. 4, 2011, pp. 1439–1453.
[2]  B Parent, SO Macheret, MN Shneider, and N Harada, “Numerical Study of an Electron-Beam-Confined Faraday Accelerator,” Journal of Propulsion and Power, Vol. 23, No. 5, 2007, pp. 1023–1032.
[30]  B Parent, SO Macheret, and MN Shneider, “Electron and Ion Transport Equations in Computational Weakly-Ionized Plasmadynamics,” Journal of Computational Physics, Vol. 259, 2014, pp. 51–69.
[31]  NL Aleksandrov, EM Bazelyan, IV Kochetov, and NA Dyatko, “The Ionization Kinetics and Electric Field in the Leader Channel in Long Air Gaps,” Journal of Physics D Applied Physics, Vol. 30, 1997, pp. 1616–1624.
[32]  A Kossyi, AY Kostinsky, AA Matveyev, and VP Silakov, “Kinetic Scheme of the Non-Equilibrium Discharge in Nitrogen-Oxygen Mixtures,” Plasma Sources Science and Technology, Vol. 1, 1992, pp. 207–220.
[33]  EM Bazelyan and YP Raizer, Spark Discharge, CRC, Boca Raton, Florida, 1997.
[34]  YI Bychkov, YD Korolev, and GA Mesyats, Inzhektsionnaia Gazovaia Elektronika, Nauka, Novosibirsk, Russia, 1982, (Injection Gaseous Electronics, in Russian).
[35]  OE Krivonosova, SA Losev, VP Nalivayko, YK Mukoseev, and OP Shatalov, Khimiia Plazmy [Plasma Chemistry], edited by B. M. Smirnov, Vol. 14, Energoatomizdat, Moscow, Russia, 1987, p. 3.
[36]  NL Aleksandrov, FI Vysikailo, RS Islamov, IV Kochetov, AP Napartovich, and VG Pevgov, “Electron Distribution Function in 4:1 N2-O2 Mixture,” High Temperature, Vol. 19, No. 1, 1981, pp. 17–21.
[37]  IS Grigoriev and EZ Meilikhov, Handbook of Physical Quantities, CRC, Boca Raton, Florida, 1997.
[38]  SO Macheret, MN Shneider, and RB Miles, “Electron-Beam-Generated Plasmas in Hypersonic Magnetohydrodynamic Channels,” AIAA Journal, Vol. 39, No. 6, 2001, pp. 1127–1138.
[39]  SO Macheret, L Martinelli, and RB Miles, “Shock Wave Propagation and Structure in Non-Uniform Gases and Plasmas,” 1999, AIAA Paper 99-0598.
[62]  YP Raizer, Gas Discharge Physics, Springer-Verlag, Berlin, Germany, 1991.
[63]  YP Raizer, MN Shneider, and NA Yatsenko, Radio-Frequency Capacitive Discharges, CRC Press, U.S.A., 1995.
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