# Fundamentals and Applications of Nonthermal Plasma Fluid Flows: A Review

## Abstract

**:**

## 1. Introduction

## 2. Fundamental System of Equations for Heat Transfer in Plasma Fluids

#### 2.1. Plasma Fluid Concept

**x**. In other words, it is a feature of continuum dynamics to consider a minute volume as a point, define a position vector, and consider it with coordinates fixed to an inertial system. In this case, physical quantities such as the density and velocity of the plasma fluid at the position vector

**x**must be defined as the mean value of the particles in the minute volume element. For example, density ρ as a field quantity, momentum per unit volume, or mass flux density ρ

**u**can be defined as

_{i}and u

_{i}are the mass and velocity of individual particles, respectively. Physical quantities appearing in the basic equations described in Section 2.7, namely specific enthalpy, temperature, component concentrations, or quantities of electromagnetic fields such as magnetic field

**H**, magnetic flux density

**B**, electric field

**E**, etc., are scalar or vector quantities similar to density or momentum. Therefore, field quantities can be defined as the average value in the particle in the same way as Equations (1) and (2). Furthermore, it is difficult to consider the averaging operation for second-order tensor quantities such as stress, but if the normal vector of the acting surface at position

**x**is specified, it can be converted to the force vector by stress, which is a linear transformation. Therefore, the average can be defined inversely from the operation. On the other hand, if we assume an ideal gas, the temperature of fluid particles can be clearly defined, as discussed in the next section.

#### 2.2. Plasma Fluid Temperature

_{e}can be defined for electrons, the ion temperature for ions, and the normal gas temperature T

_{g}for gas molecules. As the mass of electrons is usually much smaller than that of heavy particles such as neutral particles, ion particles, and radical particles, their velocities can be large. This is nonequilibrium plasma (T

_{e}>> T

_{g}).

_{e}is explained. It is assumed that sufficient collisions between electrons and heavy particles occur inside the particle. Moreover, if their collisions are considered to be almost completely elastic collisions, their behavior can be approximated to that of an ideal gas. In this case, the electrons are moving randomly, but overall, they follow the following simple law:

_{1}, u

_{2}, u

_{3}) = (u

_{x}, u

_{y}, u

_{z}) of the particle velocity

**u**follows the probability density function of Equation (3), and the magnitude u follows the cumulative distribution function (Equation (4)). Here, electrons are used as an example, but heavy particles can be considered in the same way. Gas molecular kinetic theory is applied to gas molecules. In order to facilitate the demonstration of Equations (3) and (4), variable transformation is performed to introduce x

_{i}and x

_{u}.

_{i}) is symmetrical and is maximum at x

_{i}= 0. The distribution of F(x

_{u}) is maximum at x

_{u}= 1 and is not symmetrical. It becomes zero at x

_{u}= 0. Because u is the magnitude of velocity, u > 0 and x

_{u}is also positive. If the speed of the most probable electron with x

_{u}= 1 is u

_{p}, then,

#### 2.3. Kinetic Energy and Internal Energy

_{c}be the position vector of the center of gravity of the considered fluid particle, as shown in Figure 2. x

_{i}is the position vector of individual particle (heavy particle and electron) i in the fluid particle, and Δx

_{i}is the difference between x

_{i}and x

_{c}. Consider the total kinetic energy K, which is decomposed by x

_{i}= x

_{c}+ Δx

_{i}as follows:

_{c}is the kinetic energy of the center of gravity assumed to bear the total mass, and K

_{M}is the sum of the kinetic energies around the center of gravity. In other words, K

_{c}is the energy of the macroscopic motion of fluid particles, K

_{M}is the energy of the thermal motion of individual particles in it, and the internal energy U in the case of an ideal gas. The relationship between the distribution functions, i.e., Equations (3) and (8), is as follows:

#### 2.4. Thermal Conduction, Convection, and Radiation

#### 2.4.1. Thermal Conduction

#### 2.4.2. Heat Transfer

#### 2.4.3. Thermal Radiation

#### 2.5. Method to Express Inertia Term for Field Quantities

**A**/∂t of a physical quantity

**A**at a point

**x**is not the time variation of

**A**for one plasma fluid particle but for different plasma fluid particles passing through point

**x**one after another. In other words, the Euler expression overlooks the entire flow field at each time, and not the behavior of individual fluid particles. On the other hand, if x

_{0}is the position coordinate of a specific volume element at time t, the Lagrange equation is used to describe the behavior of the plasma thermal fluid with x

_{0}and t as independent variables. The relationship between the Lagrangian time derivative (left side of Equation (12)) and the Euler time derivative (right side of Equation (12)) of a certain physical quantity

**A**is as follows:

#### 2.6. Constitutive Equation

**w**and the field quantity, is as follows:

**τ**is proportional to

**ε**. This is called Hooke’s law of elasticity, and is written in the form

**E**is a fourth-order symmetric tensor, written using subscripts, e.g., E

_{ijkl}. This is also one of the quantities of fields.

**u**, can be given as

**τ**is proportional to

**e**. This can be assumed for ordinary fluids such as water, air, and plasma. This is called Newton’s law of viscosity and is written in the form

**C**’ and

**C**are tensors of the second- and fourth-orders, respectively. It is noted from Equations (13) and (15) that the dimensions of

**ε**and

**e**differ only by time. In Equation (14) for the elastic body, the stress is

**τ**=

**0**for

**ε**=

**0**. In other words, no internal stress exists when the strain

**ε**is

**0**. However, in fluids, it becomes a constant value of

**τ**=

**C**’ at

**e**= 0.

**C**’ is called pressure. From the viewpoint of classical mechanics of the field, fluid mechanics deals with fluids, solid mechanics the theory of elasticity, and the strength of materials deals with solids (elastic bodies). It is noted that the subject of this paper is limited to plasma as a “fluid”.

**C**’ represents the isotropic pressure and the symmetry of

**e**,

**e**=

**e**

^{T}is taken into account, and the fluid stress tensor equation (Equation (16)) finally becomes

_{V}is the volume viscosity (≈0), I is the unit tensor (=δ

_{ij}), and μ is the viscosity coefficient. This is called the stress constitutive equation. By substituting this constitutive equation into Cauchy’s equation of motion (Equation (18)), a general equation of motion for a continuum, the equation of motion (Equation (20)) can be obtained.

#### 2.7. Fundamental Equations for Plasma Heat Transfer Fluids

_{p}dT strictly holds between specific enthalpy h and heavy particle temperature T. Therefore, Equation (22) is expressed using T as follows:

**E**= −∇ϕ, and Poisson’s equation is obtained from Equation (36).

**u**: velocity, p: pressure, μ: viscosity coefficient, R: gas constant, T: absolute temperature, h: specific enthalpy, λ: thermal conductivity ψ

_{D}: dissipation energy loss, S

_{c}: Chemical reaction heat generation term, n

_{e}: electron number density, ε

_{k}: energy loss, K

_{k}: equilibrium constant, n

_{k}: number density of neutral particles, C

_{p}: specific heat at constant pressure, Y

_{i}: mass fraction, J

_{i}: mass flux, M

_{i}: molecular weight, ω

_{i}: molecular formation rate, D

_{i}= μ/(ρSc): diffusion coefficient, u

_{di}: drift velocity, J

_{ci}: mass flux, ν′

_{ij}and ν″

_{ij}: forward and reverse equivalence coefficients, respectively, N

_{R}: number of chemical reactions, N

_{s}: number of chemical species, c

_{m}: molar concentration, K

_{cj}: equilibrium constant, k

_{j}: reaction rate coefficient, T

_{j}(=T

_{e}or T): absolute temperature of species j, A

_{j}, n, E

_{j}: Coefficients related to chemical reaction for species j, σ

_{col}: collision cross section of species j (function of T

_{e}), f: electron energy distribution function (Maxwellian distribution),

**G**

_{e}: electron density flux vector, S

_{e}: chemical reaction, μ

_{e}: electron mobility (=e/(m

_{e}ν

_{e}), e: electron charge, m

_{e}: electron mass, ν

_{e}: collision frequency), D

_{e}: electron diffusion coefficient (=μ

_{e}T

_{e}), ϕ: potential, χ = 5 n

_{e}D

_{e}/2: thermal diffusion coefficient,

**J**: current density,

**E**: electric field, R

_{a}: radiation energy loss,

**H**: magnetic field,

**D**: electric flux density,

**B**: magnetic flux density, ε

_{r}: relative permittivity, ε

_{0}: vacuum permittivity, and ρ

_{e}: volume charge density.

#### 2.8. Boundary Conditions

**J**is continuous, and the tangential component of the electric field

**E**is continuous. Meanwhile, when an interface of two media with different permeability values μ exists inside the system, the condition that the normal component of the magnetic flux density

**B**is continuous and the tangential component of the magnetic field

**H**is continuous is imposed. Regarding (3), there are conditions such as the conservation of electron number density on the boundary and the conservation of number density in surface chemical reactions.

#### 2.9. Analysis Procedure for the System of Fundamental Equations

## 3. Characteristics of Plasma Fluid Heat Transfer

#### 3.1. Effect on Transport Coefficient

#### 3.2. Effect on Thermal Conductivity

^{−31}kg, and they are extremely light compared to molecules and atomic ions. Because the speed of thermal motion is inversely proportional to the square root of the mass, the speed of an electron with such a small mass becomes very high. As the transport coefficient is proportional to this thermal velocity, electrons act in a positive direction in transport phenomena, which is in contrast to the Coulomb collisions described earlier. However, because the mass of electrons is small, the momentum transfer is small, implying only a small contribution to the viscosity coefficient. Electrons make a large contribution to thermal conductivity as they carry the same energy as heavy particles and are actively moving; therefore, the presence of electrons can transfer a large amount of heat.

#### 3.3. Effects of Ionization and Chemical Reactions

#### 3.4. Effects of Electromagnetic Fields

#### 3.5. Effects of Joule Heating

_{j}generated per unit volume/unit time is given by q

_{j}= J

^{2}/σ = σ E

^{2}, where J is the current density, σ is the electrical conductivity, and E is the electric field strength. From the above, it is confirmed that Joule heat is transferred to fluid particles via charged particles such as electrons.

_{e}, and gas temperature T

_{g}. Since electrons impart only a small amount of energy to heavy particles in a single collision, the energy obtained from the electric field of electrons is only owing to the acceleration of the electrons, which tends to result in nonequilibrium plasma in which only the electron temperature is high.

#### 3.6. Meaning of Terms Characteristic of Plasma Heat Transfer in the Fundamental Equations

**J**×**B**: This body force acts perpendicular to both the current and magnetic field, which is the Lorentz force.- ρ
_{e}**E**: Electrostatic force. If there is a charge density, this body force is exerted by the electric field.

- S
_{c}: Generation term of chemical reaction heat including ionization reaction. - ${n}_{\mathrm{e}}{\displaystyle \sum _{k}{\epsilon}_{k}{n}_{k}{K}_{k}}$ represents the interphase energy transfer from electrons to heavy particles.

- In Equation (25) for chemical species, not only chemical reactions but also ionization reactions are considered.
- M
_{i}ω_{i}generation term. In reactive fluids, the components change due to reactions; thus, the conservation equation for the changing components requires terms for the generation and extinction rates associated with reactions. - Equations (29) and (30) regarding ionization reactions are newly introduced.

- These equations are newly introduced to represent the law of conservation of electron generation and vanishing.
- ∇●
**G**_{e}: Transfer term due to the electric field. This term is necessary in the electron transport equation because the electrons are forced to move by the electric field. - S
_{e}: An electron generation term. This term is required in the electron transport equation.

- Newly introduced as the law of conservation of energy for electrons.
- P
_{elec}=**J**●**E**: Joule heating. When current flows in plasma, Joule heat is generated per unit volume and is given by the product of current density and electric field strength. - ∇●(5/2)T
_{e}**G**_{e}: Electron enthalpy transfer due to current by drift flux model. - R
_{a}: Radiant energy. Plasma is hot and emits electromagnetic waves by several mechanisms. Higher densities and higher temperatures result in greater heat transfer.

#### 3.7. Boundary Conditions for the Effects of Current Flow in and out of Bodies and Heat Transfer

_{i}transferred per unit of time and unit of area is

_{i}is the ion current density, e is the unit charge, E

_{i}is the ionization energy, and ϕ is the work function. In contrast, when electrons are absorbed by the object (current outflows), an energy corresponding to the work function is transferred to the solid, and the heat flux q

_{e}is

_{e}is the electron current density. By analyzing the basic equation system, Equations (43) and (44) can be used to calculate, for example, the transfer of heat due to the inflow and outflow of current from the plasma to the object through the electrode, an example of which will be described later.

## 4. Analysis Example of Fundamental Equations Systems

_{4}[28] and the supersonic flow in nonequilibrium magnetohydrodynamic (MHD) generators [29]. From the viewpoint of computational cost, each analysis does not take into account all the terms in the basic equations, and approximation is performed without considering the terms having a small effect. Furthermore, the method of numerical analysis using a computer is completely different for nonequilibrium plasma, thermal equilibrium plasma, supersonic flow, and subsonic flow. Here, at the beginning of each section, we provide a self-contained description of the highlights of the analysis, including relevant information. Interested readers should refer to the respective papers for details.

#### 4.1. Heat Transfer and Heat Conduction to Metal Particles Emitted in a Plasma Jet (Equilibrium Plasma Heat Transfer)

_{p}is the particle density, c

_{p}is the specific heat of the particle, k

_{p}is the particle thermal conductivity, and r is the particle radial coordinate. The boundary condition of the particle temperature T = T

_{p}at the particle surface r = r

_{p}is given by

_{S}is the Stefan–Boltzmann constant, ξ is the emissivity, and T

_{o}is the core temperature. The heat transfer coefficient h is given by Equation (47) using the plasma thermal conductivity and kinematic viscosity at film temperature, $\overline{T}=\left(T+{T}_{\mathrm{p}}\right)/2$:

_{p}is the particle diameter, Re

_{p}is the particle Reynolds number, and Pr is the Prandtl number. The conditions at the particle center r = 0 and the solid–liquid interface r = r

_{s}inside the particle during melting are as follows:

_{L}is the latent heat of the particle. Since a thermal equilibrium plasma is treated, T = T

_{e}. As an analysis method, the flow field is divided into a number of control volumes, and a discretization method (control volume method) is adopted to obtain the turbulent thermal fluid flow field. At that time, in the direct simulation, the mesh in the vicinity of the substrate must be considered to be extremely small. Therefore, the analysis is performed using the k–ε model, which is a type of turbulence model [31]. Using the calculated velocity U and temperature T of the fluid field as input data, an equation of motion for a single metal particle is established. At that time, the forces acting on the particles, i.e., Stokes resistance force, Basset resistance force, Saffman force, rotational lift force, and forces due to turbulent flow are considered. It is noted that the Saffman force acts on the particle perpendicular to the flow when the particle exists in a shear flow field, rotational lift force acts on the particle when the particle is rotating, and Basset force is a resistance force that acts when the particle is in unsteady motion. A time-dependent analysis is performed using the fourth-order Runge–Kutta method. Then, an arbitrarily large value is set as the upper limit of the time step Δt. At this time, the upper limit value should be set considerably smaller than the time required for the particles to pass through the control volume, but if it is too small, the calculation time will increase accordingly. The obtained speed becomes the next initial speed; the temperature rise value of the particles in motion is obtained, and the correct temperature is judged using the latent heat. These steps are performed for each control volume until the particles reach the substrate.

_{2}O

_{3}) particles, where z is the axial length dimensionless with the nozzle radius (x < 15.75). Tc is the temperature at the particle center. Calculations are performed for two particle diameters, 10 and 50 μm. The nozzle exit velocity of the plasma flow is U

_{0}= 223.3 m/s, the temperature is T

_{0}= 10,000 K, and the initial particle ejection velocity is U

_{p0}= 0. As can be seen from Figure 5, even with a small particle diameter of 10 μm, the temperature difference between the surface and the center is approximately 60 K, which is quite large due to the relatively low thermal conductivity of alumina and the short residence time. In the case of 10 μm, the temperature difference exists only on the surface at z = 10 and z = 15 because the particles are in a phase change (melting) state.

_{2}O

_{3}), nickel (Ni), and tungsten (W) with a particle size of 10 μm. Here, the temperature change of each particle of Al

_{2}O

_{3}, Ni, and W is indicated. In addition, the values of their melting points (M.P.) and boiling points (B.P.) are shown by straight lines parallel to the horizontal axis, labeled as “A.M.P” and N.M.P” (Al

_{2}O

_{3}), “W.M.P” and “A.B.P” (W), and “N.B.P” and “W.B.P” (Ni). As can be seen from the figure, the surfaces of all particles are melted, but only alumina and nickel reach the metal boiling point. In the vicinity of the nozzle exit, the relative velocity between the particles and the surrounding air flow is large, and the particle Reynolds number is large; thus, the heat transfer coefficient also increases, as in Equation (47). Therefore, most of the temperature rise occurs near the nozzle exit.

#### 4.2. Thermal Fluid Dynamics of Streamers in Atmospheric Pressure Plasma Flow (Nonequilibrium Plasma Heat Transfer)

#### 4.2.1. Model for Analysis

#### 4.2.2. Analysis Procedure

^{+}, N

_{2}, N

_{2}

^{+}, N

_{2}(a

^{1}Σ

_{u}

^{+}), N

_{2}(A

^{3}Σ

_{u}

^{+}), N

_{2}(B

^{3}Π

_{g}), N

_{2}(X

^{3}Π

_{u}), N

_{3}

^{+}, N

_{4}

^{+}, O, O(

^{1}D), O(

^{1}S), O

^{+}, O

^{−}, O

_{2}, O

_{2}**, O

_{2}(a

^{1}Δ), O

_{2}(b

^{1}Σ), O

_{2}

^{+}, O

_{2}

^{−}, O

_{2}(v), O

_{3}, O

_{3}

^{−}, and O

_{4}

^{+}) of the air plasma (air components are N

_{2}and O

_{2}, N

_{2}:O

_{2}= 79:21), 197 gas-phase reactions of the species, and 21 surface reactions on the dielectric barrier surface are considered and incorporated from various literature references in the air plasma under atmospheric pressure based on the papers by Bogdanov et al. [9], Ségur and Massines [32], and NIST (National Institute of Standards and Technology) Atomic Spectra Database [33]. The chemical model or many reaction rates used are given as Supplemental data (Tables S1–S3.1–46). All calculation conditions and boundary conditions are determined based on experimental conditions. Based on the waveform measured in the experiment, the potential at the wire discharge electrode is assumed to change with respect to the ground potential, as shown in Figure 9 (nanosecond pulse high voltage, peak voltage 35 kV, width 600 ns). The above system of equations is solved under appropriate electromagnetic fields and flow boundary conditions at the interface. The initial conditions are T = 300 K, T

_{e}= 0.2 eV, and ϕ = 0. It is noted that T

_{e}= 0.2 eV was used as an initial value of T

_{e}for unsteady calculation, and it is a non-zero empirical value.

_{n}and E

_{t}represent the normal and tangential components of the electric field at the interface, and σ

_{s}represents the surface charge density.

^{−12}s = 6 ps, the plasma calculation diverges; therefore, this value is adopted. The total time step of the calculation is 100,000, and the calculation is performed up to one pulse at t = 600 ns. Although domain decomposition and adaptive mesh are possible in the plasma simulation [34], they are not carried out in the present study. The inflow condition is that a laminar flow with a parabolic velocity distribution at a temperature of 300 K is introduced at 5 L/min. The viscosity coefficient is calculated using Sutherland’s law, with Schmidt and Prandtl numbers of 0.7 and 0.707, respectively. The thermal conductivity, specific heat, and dielectric constant of the quartz dielectric barrier are set to 2.0 W/(m K), 1000 J/(kg K), and 3.5, respectively.

#### 4.2.3. Quasi-One-Dimensional Model Calculation

_{e}. A streamer with a high electron temperature head of approximately 3 eV develops from the wire pole and reaches the glass surface at t = 200 ns. T

_{e}increases as the applied voltage V increases and reaches approximately 1.0 eV (=11,600 K) at t = 300 ns, i.e., the time when V reaches its maximum. After that, T

_{e}decreases once but then increases again and becomes a nearly constant value of approximately 1.7 eV (=19,700 K).

_{e}is determined from the electron energy equation (Equation (33)). Therefore, in order to increase T

_{e}and activate the chemical reaction by plasma, it is necessary to improve the energy balance in Equation (33), but the value of T

_{e}hardly changes even if a large amount of power is applied. In general, it is not clear how to raise T

_{e}by generating nonequilibrium plasma heat transfer, which is an important issue to be solved in the future.

_{e}= 10

^{15}m

^{−3}at the tip of the streamer but exceeds 10

^{15}m

^{−3}at t = 300 ns. After that, ne decreases temporarily but increases again, reaching 10

^{15}m

^{−3}at the end of the pulse. Furthermore, near the surface of the dielectric barrier (r = 15 mm), ne increases and reaches 10

^{16}m

^{−3}or more. This relatively strong ionization causes charge accumulation and actively generates radicals or active species (O, N, and O

_{3}). During a single pulse, the O

_{3}concentration reached 40 ppm near the surface.

_{e}and n

_{e}are higher than these average values. The calculation was performed using a PC with a Pentium IV and a 3.2-GHz clock, and required approximately 12 h [25].

#### 4.2.4. Two-Dimensional Model Calculation Result

_{e}. In the figure, a primary streamer that is nonuniform in the z-direction propagates from the wire to the glass surface and disappears once at a time near t = 300 ns, corresponding to the peak voltage. Next, a secondary streamer with a shorter wavelength in the z-direction appears and disappears at the end of the applied voltage pulse without reaching the glass surface. The term “streamer” is used like this in this field and corresponds to electrically non-uniform discharges. The non-uniformity in the z-direction is considered to be caused by a type of unstable plasma wave, and it is evident that the wavelength is determined by the balance of the plasma and external load.

_{e}is averaged in the θ direction. However, these results are in qualitative agreement with experimental observations made using high-speed video on fast-pulsed corona discharges at the Eindhoven Institute of Technology [35]. In the calculation, the voltage peaks at t = 40 ns, but in the experiment, the secondary streamer occurs near t = 40 ns. This is an astonishing result obtained from a purely theoretical analysis. In the present simulation, the plasma was only simulated during the first period of the applied voltage because the calculation took approximately one month with a fast personal computer (PC) (Precision T5500 Workstation, Dell Japan Co., Tokyo, Japan) in 2015. Further detailed numerical results are reported by the author [27].

#### 4.3. Heat Transfer and Decomposition of Exhaust Gas CF_{4} from Semiconductor Manufacturing Equipment

_{4}by ICP are considered [28]. CF

_{4}makes a large contribution to global warming; therefore, it is preferable to decompose it into CO

_{2}before exhausting it. A coil is wound 17 times around the alumina tube shown in Figure 12. A gas mixture comprising CF

_{4}and O

_{2}is passed through this system, and a high-frequency current of 2 MHz is passed through the coil; then, plasma is generated in the tube, and CF

_{4}is decomposed.

#### 4.3.1. Analytical Model and Boundary Conditions

_{4}exhaust gas from semiconductor manufacturing equipment by ICP. We consider the same system as the author’s group’s experimental setup. Gas enters from the left and exits from the right under the action of plasma. For the basic Equations (1)–(33), we derive and use stationary equations obtained by time-averaging physical quantities with nonstationary terms set to 0. In addition, assuming an axisymmetric two-dimensional flow in a cylindrical coordinate system (r, θ, x), setting the velocity vector

**u**= (u

_{r}, 0, u

_{x}) and the gas component mass fraction Y

_{i}, the vector potential

**A**of the magnetic flux density

**B**is introduced by

**B**= ∇ × A, and the boundary conditions are given as follows:

- Inlet boundary condition (gas inlet):

- Outlet boundary condition (gas outlet):

- Conditions for the center line of the reactor:

- Inner wall boundary condition:

- Outer wall boundary condition:

- Side wall boundary condition:

- Horizontal wall boundary condition:

- Conditions for vertical walls:

#### 4.3.2. Calculation Conditions

_{4}and O

_{2}are the same as those in the experiment. The inlet gas temperature and outlet gas temperature are assumed to be equal to the ambient temperature. The viscosity coefficient is obtained from Sutherland’s law. The thermal conductivity is set to 3 W/(m K) to stabilize the calculation. The Schmidt number is set to 0.7. Analyses were performed using CFD-ACE+.

#### 4.3.3. Calculation Results and Discussion

_{4}decomposition can occur.

_{4}number density. It is observed that CF

_{4}is decomposed downstream; conversely, it is partially recombined near the exit of the reactor. Considering the complex systems studied, it is difficult to get an exact idea of the advantages of the methods used. For detailed results, please refer to the original paper [28].

#### 4.4. Thermo-Fluid Analysis in Nonequilibrium Plasma MHD Generator

#### 4.4.1. Model for Analysis

_{r}, and the Faraday current J

_{θ}, nonequilibrium ionization rapidly progresses in the nozzle. As a result, nonequilibrium plasma with a high electron temperature is formed and flows into the MHD channel to generate electricity. The electrical output is tapped by an external load resistor R

_{L}connected between A2 and C2. The calculations are performed for the same operating conditions as those in the experiments, which are listed in Table 2.

_{L}to the wall and pressure loss by wall friction P

_{L}are approximately considered by semi-empirical formulae.

#### 4.4.2. Calculation Procedure

#### 4.4.3. Initial and Boundary Conditions

#### 4.4.4. Calculated Results and Comparison with Experiments

_{e}/n

_{Cs}under the same operating conditions when the external load resistance is constant at R

_{L}= 0.125 Ω, and the three cases of higher seed rates (n

_{e}: electron number density, n

_{Cs}: total number density of cesium). In the upper visualization photograph, the first anode A1, the disk support rod, the second anode A2, and some measurement ports appear black as shown in the figure. In the lower figure of the calculation results, under conditions (a) and (b), the solution becomes stationary at approximately 500 steps from the start of the calculation; therefore, the steady solution is shown. A periodic solution in which the spiral structure rotates counterclockwise in 400 steps is obtained, and the instantaneous distribution of the solution is shown.

_{h}and the output current I

_{h}. The performance parameters of the disk-type MHD generator, such as the enthalpy extraction rate, output voltage, and output current, can be calculated with fairly good accuracy within the range of this calculation condition.

## 5. Conclusions

## Supplementary Materials

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Position vector

**x**

_{c}of the center of gravity of fluid particles, position vector

**x**

_{i}of individual particle i, and the difference between

**x**

_{i}and

**x**

_{c}, Δ

**x**

_{i}.

**Figure 3.**Relationship between plasma pressure, electron temperature T

_{e}, and gas temperature T

_{g}(nonequilibrium plasma, 1 Torr is 133.3 Pa) in DC discharge of mercury and rare gas mixture at different gas pressures and the same current [30].

**Figure 4.**Schematic diagram of plasma spraying [24].

**Figure 5.**Particle internal temperature distribution [24].

**Figure 6.**Average temperature distribution of particles [24].

**Figure 7.**Model for analysis [25].

**Figure 9.**Waveform of a nanosecond-pulse applied voltage [25].

**Figure 10.**Time-dependent radial change of ϕ, T

_{e}, and n

_{e}(quasi-one-dimensional calculation) [25]. (

**a**) Electrical potential ϕ. (

**b**) Electron temperature T

_{e}. (

**c**) Electron number density n

_{e}.

**Figure 11.**Time-dependent spatial change in distribution of electron number density n

_{e}(calculation of primary and secondary streamer propagation by two-dimensional calculation) [26].

**Figure 13.**Gas temperature distribution inside the reactor [28].

**Figure 14.**Distribution of electron temperature inside the reactor [28].

**Figure 15.**Distribution of electron number density inside the reactor [28].

**Figure 17.**Analysis model for numerical calculation. In the system, plasma is generated by a heat transfer from a power-source combustor to Ar gas with Cs seeding to obtain higher ionization degree: direct power generation without a turbine [29].

**Figure 18.**Comparison of photographs taken by a high-speed camera of discharge and calculation results of the degree of ionization n

_{e}/n

_{Cs}under the same operating conditions [29].

**Figure 19.**Calculation result of electron temperature distribution [29].

**Figure 20.**Calculated result of ionization degree distribution [29].

**Figure 21.**Relationship between output voltage and output current [29].

AC Voltage frequency, MHz | 2 | |

Atmospheric temperature, K | 293 | |

Absolute pressure, Pa | 80 | |

Power, kW | 2.0 | |

Mass flow rate of gas, g/s | 0.02107 | |

Initial mass fraction of species | CF_{4} | 0.5789 |

O_{2} | 0.42105 |

**Table 2.**Calculation conditions [29].

Working gas | Ar/Cs |

Stagnation temperature, K | 2480–2520 |

Stagnation pressure, MPa | 0.14 |

Seed fraction | (3.9–12.0) × 10^{−4} |

Ext. load resistance R_{L0}, Ω | 10 |

Ext. load resistance R_{L}, Ω | 0.075–0.215 |

Area ratio of channel | 4.2 |

Magnetic flux density, T | 2.7 (r = 82.5 mm) 0.39 (r = 270 mm) |

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**MDPI and ACS Style**

Okubo, M.
Fundamentals and Applications of Nonthermal Plasma Fluid Flows: A Review. *Plasma* **2023**, *6*, 277-307.
https://doi.org/10.3390/plasma6020020

**AMA Style**

Okubo M.
Fundamentals and Applications of Nonthermal Plasma Fluid Flows: A Review. *Plasma*. 2023; 6(2):277-307.
https://doi.org/10.3390/plasma6020020

**Chicago/Turabian Style**

Okubo, Masaaki.
2023. "Fundamentals and Applications of Nonthermal Plasma Fluid Flows: A Review" *Plasma* 6, no. 2: 277-307.
https://doi.org/10.3390/plasma6020020