Analyses of the power generation performance and the non-equilibrium plasma of a disk MHD generator

Recently, closed cycle MHD power generation system studies have been focused on improving the isentropic efficiency and the enthalpy extraction ratio. By reducing the crosssection area ratio of the disk MHD generator, it is believed that a high isentropic efficiency can be achieved with the same enthalpy extraction. In this study, the results relating to a plasma state which takes into account the ionization instability of non-equilibrium seeded plasma is added to the theoretical prediction of the relationship between enthalpy extraction and isentropic efficiency. As a result, the electron temperature, which reaches the seed complete ionization state without the growth of ionization instability, can be realized at a relatively high seed fraction condition. However, the upper limit of the power generation performance is suggested to remain lower than the value expected in the low seed fraction condition. It is also suggested that a higher power generation performance may be obtained by implementing the electron temperature range, which reaches the seed complete ionization state at a low seed fraction.


INTRODUCTION
In recent years, closed cycle MHD power generation research has been placed on improving the isentropic efficiency in addition to the enthalpy extraction ratio with a great interest.
Enthalpy extraction ratio (EE) is defined as the ratio of the electrical output to the heat input while the isentropic efficiency (IE) is the ratio of |hi-hf|actual to |hi-hf|isentropic.Here hi, hf represent the total enthalpy of the initial state and the final state respectively, and IE represents the ratio of the enthalpy change in case of extracting the enthalpy isentropically to the actual enthalpy change.
By reducing the cross-section area ratio of the disk MHD generator (generator channel horizontal outlet section area / throat crosssection area), a high IE can be achieved experimentally with the same EE.It is also predicted the relationship between EE and IE agrees with a simple theoretical study that using the generator channel cross-sectional area ratio and an outlet Mach number as variables in a small-scale MHD generator.Furthermore, from

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the study of the relationship between the stagnation pressure and the power generation performance, the fluid machine characteristics is clearly revealed.
On the other hand, it is known that the affect of the state of non-equilibrium plasma used as a working fluid to the power generation performance is large, and the ionization instability that causes the spatial non-uniformity of the plasma is the main reason for the deterioration of power generation performance.This causes the effective decrease of Hall parameter and electrical conductivity.
In this study, the results relating to a plasma state which takes into account the ionization instability of non-equilibrium seeded plasma is added to the theoretical prediction of the relationship between EE and IE considering only in hydrodynamics.Thus, this study considers the plasma physical aspects that have not been sufficiently performed in the previous study [1], [2] for a generator with small cross-sectional area ratio.Here, instead of obtaining the numerical solutions of differential equations as in the conventional numerical simulation, a discussion based on the simple analytic steady local calculations has been carried out.The purpose is to further develop the analytical consideration on the power generation performance.

Electron energy transfer
Non-equilibrium plasma is composed of noble gas Argon atoms, argon ions, seed Cesium atoms, Cesium ions, and electrons.The temperatures of Argon atoms, Argon ions, Cesium atoms, Cesium ions (Tg) are equal.Only the electron temperature Te is different (Te>Tg) and a two-temperature model is used [3], [4].In the non-equilibrium seeded plasma generated in the disk-shaped MHD generator, the Joule heating is shown below and the energy transfer due to the collision are balanced in the steady local state.
From the generalized Ohm law, the Joule heating, which has an effect of increasing the electron energy, becomes the following equation.
Here, E=(Er, 0, 0) is an electric field strength vector, u=(ur, 0, 0) is the flow velocity vector, then the load factor Kh defined as Kh≡|Er/βeffurB|.j, B is the current density vector and the constant magnetic flux density to be applied to the vertical direction of the flow (r-θ-z coordinate system).The effective electrical conductivity σeff and the effective Hall parameter βeff will be explained in the next section.
The energy loss of electrons by collision A is shown by the following equation Here κ, me, ne is the Boltzmann constant, electron mass, electron number density, The electron energy equation balanced by Eq. ( 1) and (2) can be solved analytically for Te, therefore by using Te as a variable, the solutions on plasma quantities can be achieved.
In this paper, the collision phenomenon of power generation plasma will be described.Here, the effect of elastic collisions in the form of Eq.
(2), including the effects of inelastic collisions as ionization process described below, is taken into account.However in this case, because the change of Tg is considered very small with respect to the change of Te, Tg is kept constant.Furthermore, because the local in the mainstream of plasma fluid is a study object, the energy flow to the wall is not considered.
Although there are various processes in the ionization and recombination phenomena of plasma, they are in a balanced state in the MHD generator plasma, meaning that the ionization equilibrium is assumed.This means the Saha equation representing the equilibrium conditions on ionization is used, and the electron number density is calculated in a plasma with the local thermal equilibrium.In this case, the state equations of perfect gas are used.

Critical Hall parameter
In the linear theory on ionization instability of non-equilibrium MHD power generation plasma, the growth rate of small disturbances is considered, and the stability condition of the plasma which the disturbances do not grow is found out [3], [5].Here, the critical Hall parameter βcr have been defined.(5) βcr < β: By using the βeff and σeff, the effective quantities of non-equilibrium MHD seeded plasma that takes into account the ionization instability in the analytical calculation can also be represented.In other words, the Hall parameter and the spatial non-uniformity of electrical conductivity are difficult to analyze at the present time but they will become apparent from the measurement, and when estimating their effective values this calculation, to some degree, is expected to agree with.Thereafter, the Hall parameters and electrical conductivity that do not take into account the ionization instability are described as the β and σ to distinguish from the effective values.

Enthalpy extraction and isentropic efficiency
The enthalpy extraction (EE) is represented by the following equation.
Here p0e, p0i, γ, At, Ae and Me are the outlet total pressure, the inlet total pressure, specific heat ratio, throat cross-sectional area, outlet cross-sectional area, and the outlet Mach number.In addition, the polytropic efficiency ηp is represented by the following equation using the local Mach number M, βeff, Kh.
Trang 54 The relationship of EE and IE is shown as follows The plasma fluid expands isentropically from a stagnation point state of p0i=0.21MPa,total temperature T0=2500K, and the analytical calculation is carried out with respect to the plasma state in the local that has reached a certain Mach number M. However, the magnetic flux density in the experiment attenuates smoothly (nonlinearly) 0.3T in the vicinity of the generator channel outlet along the disk radial direction when B=3T at the disk generator center position.In this paper, the flux density in the channel radial middle flow area where the power generation MHD interaction is considered relatively large to act sufficiently is assumed typical values, and use B=2T as calculation condition.By using ηp obtained from the Eq. ( 10), the EE is determined by Eq. ( 9) and relates to the IE by Eq. ( 12).

Calculation method
A splitting scheme is applied to the governing system with the subsets of equations describing different physical processes being solved in sequence by appropriate program modules.The MHD system is solved by the generalized TVD Lax-Friedrichs scheme which was developed for the unstructured mesh applications.
A general monotonous reconstruction of mesh-defined functions is designed taking into account the dependence on two variables.For the case of a regular triangulation, this scheme ensures the second order approximation to spatial derivatives (the third order is possible with a special choice of the anti-diffusion limiters).The time integration is explicit, the second approximation order is reached due to the predictor -corrector procedure.The time step is restricted by the Courant criterion.The predictor and corrector steps are organized similarly but the numerical fluxes differ due to the different reconstruction of the functions.Namely, a nonmonotonic piecewise-linear continuous interpolation is used for the predictor, and special monotonic discontinuous reconstruction is created for the corrector.Thus, the corrector scheme not only improves the time-advance resolution but also appears to be a stabilizing procedure.For the solution of parabolic equations describing the conductive heat transfer, the author developed the finite-volume schemes constructed by analogy with mixed finite element method.
The electron energy transfer is described by the equation for spectral radiation intensity.Practical calculations are done via multigroup spectral approximation.The author solves the electron transport equation by means of semianalytical characteristic algorithm.The analytical solution along the characteristic direction is constructed by means of the backward-forward angular approximation to the photon distribution function.The two-group angular splitting gives an analytical expression for radiation intensity dependent on opacity and emissivity coefficients.The energy exchange between radiation field and the gas is taken into account via a radiative flux divergence, which is incorporated into the energy balance as a source function.
The governing Critical Hall parameter set of equations is solved numerically employing a fully implicit finite difference method.To obtain the correct solutions of this method, the MHD equations have to be rewritten into conservation form, and the numerical scheme also has to be conservative.Numerical schemes used in computational MHD are Lax-Wendroff scheme

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with artificial viscosity, finite differences with artificial viscosity, flux corrected transport, finite elements, spectral elements, discontinuous Galerkin.Furthermore, numerical diffusion can produce magnetic reconnection even in an ideal MHD model and the energy conservation will be violated by a non-conservative ideal MHD scheme because Joule heating is neglected.However, we should note that the reconnection is not very well modeled either by ideal or by resistive MHD.One needs to include more physics in MHD disk generator and/or go to really high grid resolution so that small scale instabilities can develop.

RESULTS AND DISCUSSION
Here the description of the static pressure in the channel and the outlet Mach number measurement is referred in [1],[2], and the spectroscopic measurement of electron temperature is performed by a multi-channel spectrometer [7].In this case, the exposure time of 50μs is set up to detect a sufficiently strong light.On the other hand, the staying time of the fluid in the generator is about 150μs.Thus, Te in this study has a meaning as average values with respect to time and space, therefore, it is impossible to get the local Te corresponding to the small plasma structures that may be present in the generator.

Plasma stability
In order to examine the plasma stability and the relationship between EE and Te, the Te dependence of σ, σeff, β, βcr, βeff, and EE is shown in Fig. 1(a,b,c).The calculation conditions are SF (seed fraction) =2×10 -4 , M=1.7, Me=1.7.
As seen from the figure (a), the ionization of cesium atom is mainly accelerated when 2000 ≤ Te ≤ 4350K, and because the degree of ionization increases with the increase of Te, σ increases.When Te≈4350K, σ reaches 250S/m.The β from Fig. (b) decreases when Te increases and it can be higher than 7 when Te<3500K.Thus, the σ and β that do not take into account the ionization instability is not a primary factor that Cesium weakly ionized plasma state degrades the power generation performance.
On the other hand, from the calculation which takes into account the ionization instability, the βcr that shows the ionization instability affect causing the Cesium weak ionization is very low when 2000 ≤ Te ≤ 4350K.For this reason, βeff and σeff greatly decrease.The solution of EE that reflects the plasma state of extremely low βeff and σeff does not exist (Fig. 1(c)).This result, by the steady local calculation, introduces the affect of ionization instability, and indicates that the spatial non-uniformity of plasma can be used to consider, to some extent, the affect to the power generation performance.Trang 56 When Te is 4650K to 6000K, the plasma is considered as a state that Cesium is almost completely ionized while Argon is nearly not ionized.In the seed complete ionization state, ionization degree of plasma matches the ionization degree of Cesium, and σeff is nearly independent of Te.As shown in Figs.1(a,b), the large βcr, the high βeff of 7 as well as the σeff=230~250S/m can be realized.The large βcr means that the plasma stability is high and the plasma is more uniform.Since the highest EE=28% is obtained when Te=5500K, the realization of the seed complete ionization state is a useful condition for the high power generation performance (see Fig. 1(c)).
On the other hand, in the Te ≥ 6000K which accelerated not only Cesium but also the Argon ionization, βcr is greatly reduced with the rise of Te.This indicates that the ionization instability, which causes the Argon partial ionization, is occurred.Particularly when Te is 6000~7100K, since βeff as well as σeff is very low, the EE can not be solved.
When Te is 7100~8200K as shown in Fig. 1(a), the relatively high σeff ≥ 75S/m is obtained.Furthermore, although βcr is less than β, the difference between the two is small compared to the difference found in Te ≤ 4350K as shown in Fig. 1(b).This suggests that the affect of ionization instability is relatively small.Therefore, there is a solution of EE even when Te is high.However in this case EE is less than 6%, and this value is low as compared to the value obtained in the seed complete ionization state.
The above results have suggested that the plasma state varies greatly depending on the change of Te and that state is a significant effect on the power generation performance.

Seed fraction
The dependence of seed fraction (SF) on the electron temperature Te is described experimentally in [8] and shown in Fig. 2. The channel load resistance RL, in this case, is 0.13Ω, 0.2Ω, 0.3Ω.However, the affect of load resistance value is not much, therefore, this paper focuses only on the change of SF.
From the MHD generation theory, the SF of 2~4×10 -4 has been suggested as a condition that the plasma fluid is expanded isentropically from the stagnation point state in the supersonic nozzle, and the Mach number of 1.7 at the nozzle outlet (or the MHD channel inlet) can be designed.On the other hand, a strong Lorentz force caused by the MHD interaction in the vicinity of the nozzle outlet works in the condition of 6~9×10 -4 , so the fluid has a large deceleration.The increasing of SF increases the σeff of plasma and has the effect of accelerating Joule heating.Moreover, as the σeff increasing, the Lorentz force acting on the fluid becomes stronger and this causes a rise in static pressure which increases the energy loss due to the collisions and the reduction in flow speed that leads to the decreasing of induced electromotive

Seed Fraction [%]
T e [K] force.Te of plasma is determined by the balance of both effects.The dependence of SF on Te in Fig. 2, as well as the setting value of SF, indicates that this is one of the main factors to control the plasma state.
Figure 3 shows the dependence of (a) EE, (b) βeff, (c) σeff on Te when the SF is changed in the range 2×10 -4 ~9.5×10 -4 .Here from the fluid behavior, the calculation conditions of M=Me=1.7 for SF=2~4×10 -4 and M=Me=1.1 for SF=6.5×10 - or 9.5×10 -4 are given.In addition, the plot points (,,) in the same Fig.3(a  From the calculation results with the condition of SF=2×10 -4 shown in Fig. 3(a), it can be seen that the solution of EE is obtained only in the range of Te=4560~6000K and 7100~8200K.
Similarly, EE is solved discontinuously only in the range of Te=4900~6300K and 6900~8300K with the condition of SF=2×10 -4 .Firstly, the calculated results are considered to compare with the experimental results when the range of Te is more than 6500K.The ionization instability due to Argon weak ionization may occur in the case of Te≥6500K, and from the calculation, in this case, it is considered that βeff is very low (nearly 1) as shown in Fig. 3(b).Here, the highest value of EE is as low as 6% and 8% when SF is 2×10 -4 and 4×10 -4 respectively.EE is also obtained relatively low value of 7~12% from the experiment in [8].However, there is a trend that the calculated value is slightly higher than the experimental value.The cause of the difference between these two values can be explained that the value of M depends on the condition.In other words, there is a possibility that the experimental flow speed is higher than the calculated flow speed is given by the condition of M=1.7.
Next, the range of Te=4600~6300K is considered.From the calculation, σeff ≥ 200S/m can be obtained even in the low SF condition, and a very high EE can be expected because a sufficient βeff is maintained.However, that power generation performance is not obtained experimentally.This is because the temperature Te which implements a seed complete ionization state cannot be achieved experimentally at a low SF.
From the calculation results with SF=6.5~9.5×10 - in Fig. 3, it is seen that the discontinuity does not appear in the EE-Te curve because the dependence of Te on βeff is relatively small.In addition, the EE obtained from the calculation in the range Te= 4600~4700K is maximum, and this maximum value tends to decrease from 19% to 15% with the increase of SF.In this case, the obtaining of EE is considered independent from low βeff to some extent because RL=0.13ΩRL=0.2ΩRL=0.3Ω M=Me=1.1 6.5×10 M=Me=1.1 6.5×10 -4 9.5×10 -4 4×10 -4 2×10 -4 M=Me=1.7 Trang 58 the σeff is very high.The phenomenon that EE decreases with the increase of SF is because βeff decreases with the increase of the collision frequency while σeff increases with the increase of SF in the seed complete ionization state.The decrease of βeff reduces the electrical conversion efficiency ηp, therefore, EE decreases.
As shown in Fig. 2 under the condition of SF=6.5~9.5×10 - , Te reaches 4500~5500K and the seed complete ionization state is implemented.In this case, as suggested by Fig. 3, it is believed that an appropriate σeff can be achieved with a high SF, and a relatively high power generation performance is also obtained.With the condition of SF=6.5×10 -4 and 9.5×10 -4 , the EE calculated here and the EE from reference [8] agree well.This suggests that the plasma is stable as discussed from the calculation in the case of Te implementing the seed complete ionization.

Plasma state
Fig. 4 shows the IE-EE relationship obtained from the calculation with a typical SF.Since the measured Me is in the range from 1.1 to 1.7, this relationship described by a theoretical curve of Eq. ( 12) is illustrated by the dotted line with Ae/At=4.25,Me=1.1 and 1.7. Figure (b) shows the calculated contour lines of Te on the IE-EE relationship with calculation conditions of SF=3×10 -4 , M=1.7, Me=1.1 and 1.7,and Fig. (c) with calculation conditions of SF=9.5×10 -4 , M=1.1, Me=1.1 and 1.7.
From the theoretical curve in Figs.4(a,b,c), it is shown that there is an achievable EE with the condition that supersonic flow is maintained (Me=1.1~1.7).The ratio of the EE and IE follows the macro hydrodynamic behavior.From Fig. 4(a) when increasing the SF, IE and EE increase.It is understood that the inclination IE/EE both values formed in this case may not deviate greatly the nearly linear slope by a given Me.
As already discussed in the previous section, the power generation performance can be best when implementing the seed complete ionization plasma.The calculation results in Figs.4(b,c) also indicate the highest power generation performance according to the conditions of SF and M when implementing the seed complete ionization plasma of Te=4500~5500K.Trang 59 be said that these calculation results have suggested an upper limit of the achievable power generation performance.Furthermore, when M decreases due to the high SF, it is understood that the upper limit of power generation performance is lower than the expected value in case of low SF, high M even when obtaining a seed complete ionization plasma.
The MHD power generation using nonequilibrium seeded plasma depends on the electron temperature and its characteristic is that the plasma behavior changes clearly by the seed partial ionization, seed complete ionization, and the noble gas weak ionization state.Thus, the calculation values of IE, EE (Figs. (b,c)) based on the experimental values of the seed fraction and electron temperature is somewhat agreed with the experimental results.

CONCLUSIONS
This paper has carried out the steady local analysis of non-equilibrium seeded plasma which takes into account the ionization instability and has added aspects relating to a plasma state in regard to performance prediction of the macro hydrodynamic side that considers the relationship of isentropic efficiency and enthalpy extraction.By the theory analysis, the analysis of power generation results was performed.
The electron temperature, which reaches the seed complete ionization state without the growth of ionization instability, can be realized at a relatively high seed fraction condition.At this time, the high enthalpy extraction and isentropic efficiency are achieved.As a result, the electrical conductivity with small effective Hall parameter can be high, and a high power generation performance is suggested with the conditions of low Mach number, high seed fraction.However, the upper limit of the power generation performance, which is considered to be achievable at a high seed fraction, is suggested to remain lower than the value expected in the low seed fraction condition, simultaneously.
On the other hand, since the electron temperature increases to the Argon weak ionization state in the small seed fraction experiment, the power generation performance is low.From the analysis result, it is suggested that a higher power generation performance may be obtained by implementing the electron temperature range, which reaches the seed complete ionization state at a low seed fraction.By adding the affect of ionization instability, the plasma state in the experiment can be predicted by a steady local calculation, and a study of power generation performance based on this can be carried out.
the heavy particles, nh is the h particle number density, eh Q is the average momentum transfer collision crosssection between h particle and electron, Plasma will be stable under the condition that the Hall parameter β does not exceed βcr.In this study, βeff and σeff are determined as[6]    βcr ≥ β: