1. Introduction

The quasi one-dimensional MHD theory [1-2] tells us that performance characteristics of a diagonal-type, nonequilibrium MHD generator are similar to the performance characteristics of a Faraday-type MHD generator. However, the quasi one-dimensional MHD theory is only relevant when talking about the boundary conditions of the working gas plasma, and does not accurately reflect the spatial nonuniformity effects in a cross-section of the generator’s channel. Moreover, so-called end effects arise in the exit regions of the MHD generators, degrading the total electrical performance.

Through numerical analysis, the author has investigated the electrical performance in the central part of the diagonal type non-equilibrium MHD generator channel as described in [3].

In addition, while the end effects in the Faraday-type generator have been analyzed in fair detail [4-6], the end effects in the diagonal-type require further analysis [7-9]. Thus, the author has investigated influences of the output electrodes’ arrangement and the attenuation of the magnetic induction along the generator channel on the current, as well as potential distributions near the entrance and exist exit of the diagonal-type MHD channel when physical quantities in the channel are assumed to be uniform. The author’s examinations demonstrate that the variation of the arrangements of output electrodes has little effect on current distribution, etc. [10]

In this paper the author studies the end effects in the diagonal-type, non-equilibrium MHD generator through numerical analysis. Section 2, introduces basic equations and boundary and subsidiary conditions, and then shows the configurations of gas velocity and applied magnetic induction adopted throughout the paper. In section 3, numerical calculations are made, to investigate the effect of the magnetic induction attenuation on current and potential distributions, internal resistance, etc., in the exit regions of the generator

 

2. Basic Equations

2.1 Basic equations for current distribution

In the analysis of end effects in a diagonal-type MHD generator, the following assumptions are made: 1) the electric quantities, such as the current, electric field etc., vary with x and y, where x and y are the coordinates as shown in Fig. 1, and 2) the gas velocity and temperature depend only on y according to (9) and (10) which will be presented later, and 3) the pressure is kept constant.

In order to evaluate the current distribution in the generator channel, the author introduce the conventional stream function Ψ defined by:

where Jx and Jy are the x and y components of current density, and the z component Jz is assumed not to exist

Fig. 1.Coordinate system and generator channel geometry

Then, it is assumed that the magnetic induction and the gas velocity have only the z component B and the x component u, respectively. From the Maxwell and the generalized Ohm's law Eqs. (1) and (2) in [3], the following partial differential equation can be derived:

where,

in which e is the electron charge, pe=nekTe is the electron partial pressure, ne is the electron density, k is the Boltzmann constant, Te is the electron temperature, β is the Hall parameter for electron, βi is the Hall parameter for an ion, and σ the scalar electrical conductivity of the plasma. In addition, σ=μnee, β=eB/nemeQce are explained in detail in [3], where μ is electron mobility, me is electron mass, Q is the collision cross section, and ce is the velocity of individual electron.

2.2 Boundary and subsidiary conditions

First, the boundary conditions on the electrode surfaces is defined as:

where, Ex is the x component of the electric field.

The boundary condition on the insulating wall surface is defined as:

Using (1), these conditions (4) and (5) are transformed to:

Next, in the diagonal-type generator, the potential difference between the anode Ai and cathode Ci must be zero as shown in Fig. 1. Therefore, the first subsidiary condition is obtained as:

where, E is the electric field intensity vector, ds is the line element vector of an optional integral path from Ai to Ci, and Vi is the potential difference between Ai and Ci

As the load current I runs through an arbitrary surface Si, it crosses the insulating wall surfaces A'i and C'i, the second subsidiary condition is written as

where dS is the element vector of the surface Si.

Finally, let us assume that the electric quantities vary periodically in the period of the electrode pitch s along the gas flow behind the n-th electrode pair An and Cn. Then the condition for the current density J(x) is given by

By (1, 8) is transformed into

where is the current flowing into An.

The current distributions in the diagonal type generator can be found by numerically solving (2) under the conditions (4′)~(7) and (8′) (see section 3).

2.3 Calculation of potential

With the obtained numerical solution of Ψ, (2) can be solved numerically using conditions (4′)~(7) and (8′), and the electric field E at the optional point can be evaluated using (1) and the generalized Ohm's law. Then, the potential at any point can be calculated by the numerical line integration of E along an arbitrary integral path from a reference point to the point in consideration.

2.4 Gas velocity and temperature distributions

As assumed in section 2, the velocity u has only the x component u, and u and T vary only in the y-direction according to the following relation [11]

where, h is the channel height, u0 is the gas temperature and T0 is the velocity at the center of flow, namely y=h/2 and Tw is the wall temperature.

2.5 Configuration of applied magnetic induction

For effective use of the applied magnetic flux density B, the MHD generator channel should be arranged in the attenuation region of B. Thus, in order to investigate the influence of the configuration of B on the current distribution in the exit regions of the diagonal type generator, the intensity of B is assumed to be constant in the central region and decreases linearly from the left edge of the j-th electrode in the exit regions of the generator. In this this numerical analysis, the author assumes the six configurations of B as plotted in Fig. 2, where g is the gradient of B and j=5.

Fig. 2.Configuration of applied magnetic induction

 

3. Numerical Method for Subsidiary Conditions

In a diagonal generator, the solution of (3) is required to satisfy the two subsidiary conditions (6) and (7). From (1) and (7), the author can derive the following equation:

where and are the values of Ψ on the insulating wall surfaces and respectively, and w is the channel width in the z direction.

First, if the values of I and w are assumed and are given plausible values, the values of are decided by (11). When (2) is digitally solved with these values of and and using the appropriately assumed values of u, σ and β, the author can obtain the numerical solution of Ψ. By applying the solution to (1) and the generalized Ohm's law, the author can find the values of Ex and Ey. Furthermore, by substituting the values of Ex and Ey into the integral in (6), the author can decide the value of Vi. Then the value of Vi obtained is not necessarily equal to zero.

Let us consider the resistance between the electrodes Ai and Ci

where h, c and θ are the channel height, the electrode width and the angle of inclination to the y axis of the lines joining the equipotential electrodes, respectively, and the author assume that an imaginary current defined by

flows through the resistance Ri′. To make Vi zero, it is needed to flow the inverse current –Ii through Ri. Then it is required to increase by –Ii the value of w(), which gives the current running into the anode Ai

Again beginning with the new modified values of , the author must repeat the above mentioned calculation process. When Vi becomes adequately small after the many repetitions of the above mentioned process, at last the author can obtain the satisfactory numerical solution of Ψ.

In connection, the other parts of numerical calculation processes are explained in [12].

 

4. Numerical Calculation

4.1 Numerical conditions

Numerical analysis is carried out for the diagonal type MHD generator with the cesium seeded helium in nonequilibrium ionization using the following parameters:

where εs is the seed fraction of Cs, B0 is the magnetic induction in the central region of generator channel, and δ is the collision loss factor. These conditions are assumed with respect to a generator of the pilot plant [13]. The load current I is assumed to flow equally into two output electrodes E1 and E2 through a ballast resistance Rb defined by (see Fig. 1)

4.2 Calculation results

In Figs. 3(a)~(c), the current distributions are plotted in the case of g=0, 6 and 10T/m, respectively, B0=4T and I=70A, where the contour interval of current streamlines is 1/20 of the load current I. In the figures, 〈J〉el = 0.583A/cm2, 〈σ〉=1.84 mho/m, 〈β〉=2.01 and βcrit=2.48, where 〈J 〉 el is the average current density on the output electrodes, 〈σ〉 and 〈β〉 are the average electrical conductivity and Hall parameter in the center of flow, respectively, and β crit is the critical Hall parameter [14].

Fig. 3(a) shows that the current concentration at the edges of the output electrodes is very intensive when B does not attenuate. On the other hand, Figs. 3(b) and (c) indicate that the concentration weakens as the attenuation of B increases, since β becomes small in the area suffering a spatial reduction of B. Also it is seen that the current flowing into a diagonally connected electrode pair reduces with an increasing gradient of magnetic induction in the entrance region of channel. For instance, the currents of about 60, 25 and 15% of I flow into C, when g=0, 6 and 10T/m, respectively. Also the figures denote that the eddy current is not induced when the output electrodes are disposed in the region of the attenuating magnetic induction [5], and that arranging the output electrodes within the attenuation region of B does not have a great influence on the current distribution in the central part of generator channel.

Fig. 3.Current distributions

Next, Fig. 4 shows the variation of the potential difference between the electrode pairs A1-C1~A8-C8 and the electrode E1. From the figures it is seen that the relatively large potential difference arises between the two output electrodes E1 and E2 when B does not attenuate, namely g=0. On the other hand, the potential difference become smaller as g becomes larger, it almost vanishes for g=6, and the inverse difference appears for g>7. Also Fig. 4 demonstrates that the potential differences in the central part of generator channel are influenced little by the decrease of the magnetic induction.

Next, for estimation of the end effects of the generator, the author evaluates the internal resistance Ri of the exit regions and the grade of the current concentration on the output electrodes given by the relations where V0 and V are no-load and load potential difference between the output electrode E1 and the n-th electrode, respectively, and Jpeak is the maximum current density on the output electrodes. In this connection, Jpeak/〈J〉el=1 means the state of no current concentration and Jpeak/〈J〉el >>1 does the intensive current concentration at an electrode edge.

Fig. 4.Variation of potential difference for B0=4

Fig. 5.Influence of g on Ri/Ri0, Rb/Rb0 and Jpeak/〈J〉el when B0=4

Now Fig. 5 shows the variations of Ri/Ri0, Rb/Rb0 and Jpeak/〈J〉el by g, where Ri0 and Rb0 are Ri and Rb for g=0, respectively. From the figure, it is seen that Ri decreases with g, for instance the value of Ri for g=6.0 becomes about 80% of the one of Ri0, and that Jpeak/〈J〉el decreases from g=0 to 8T/m, reaches the minimum value 1.90 and increases again. This fact shows that the current concentration at the edges of the output electrodes is almost diminished when g=8T/m. Accordingly, arranging the output electrodes in the attenuation area of the magnetic flux density is useful to guard the output electrodes. Also Fig. 5 tells that Rb/Rb0 decreases with g, becomes almost zero for g=6.5 and then increases with g. Therefore, it is shown that many output electrodes will require large ballast resistors when B does not attenuate or exceeds 8, but they can be used without large ballast resistors in the range of g=6~7T/m.

In Fig. 6, the current distribution is plotted when g=6T/m, B0=5T and I=150A, 〈J〉el =1.25A/cm2, 〈σ〉=2.85 mho/m, 〈β〉=2.48 and βcrit=1.90. The figure indicates that the streamer is induced in the central part of generator, while the current distribution becomes successively uniform as B attenuates along the generator channel and the current concentration is almost swept away near the output electrodes. This demonstrates that arranging the output electrodes within the attenuating region of B is effective for the case where the streamer is generated in the central region of generator channel, too.

Fig. 6.Current distribution for g=6 and B0=5

 

5. Conclusions

The main conclusions derived from the above described numerical calculation are as follows:

A suitable distribution of the magnetic flux density can make the current distribution very uniform near the exit region of generator channel, both when the streamer is not induced and when it is induced in the central region.

Disposing the output electrodes within the attenuation area of magnetic flux density has little influence on the current distribution in the central part of generator channel.

When the output electrodes are disposed in the region with a suitably reduced magnetic flux density, the potential difference and the ballast resistance between two output electrodes become very small. Accordingly it is thought that many output electrodes can be used without large ballast resistors.

The internal resistance in the exit region of the generator channel decreases as the magnetic flux density attenuates.

The current concentration at the edges of output electrodes can be fairly eliminated by attenuating magnetic flux density.

As mentioned above, it is made clear that the output electrodes of the diagonal type non-equilibrium plasma MHD generator should be arranged in the region of the attenuating magnetic flux density, since arranging them in the region of the decreasing magnetic flux density become useful for the improvement of the electrical performance of the generator