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Numerical analysis of magnetohydrodynamic accelerator performance with diagonal electrode connection Makbul Anwari *, Nobuomi Sakamoto, Triwahju Hardianto, Jun-ichi Kondo, Nobuhiro Harada Department of Electrical Engineering, Nagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka 940-2188, Japan Received 2 April 2005; accepted 2 October 2005 Available online 16 November 2005

Abstract Investigation of application of magnetohydrodynamic (MHD) acceleration process for advanced propulsion system has signiﬁcantly increased. The diagonal MHD accelerator is possible for this acceleration technique because of its minimum weight, size and complexity. In this paper, the fundamental performance of a diagonal type MHD accelerator under chemical and thermal equilibrium conditions is described. A one dimensional numerical simulation is performed to characterize the acceleration along the speciﬁed channel designed and developed at the NASA Marshall Space Flight Center. In order to solve the set of diﬀerential equations with MHD approximations, the MacCormack scheme is employed. In the present calculation, a working gas of air-plasma composed of diatomic molecules of nitrogen and oxygen seeded with potassium is considered in order to actualize application of the MHD accelerator propulsion system. Numerical results show the gasdynamic characteristics (ﬂow velocity, gas pressure and gas temperature), electrical properties (electrical conductivity, Hall parameter and Faraday current density) and electrical eﬃciency of the MHD accelerator. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: MHD accelerator; Diagonal electrode connection; Accelerator performance; Equilibrium plasma; Electrical eﬃciency

1. Introduction This paper describes a recent result of research on an MHD accelerator for space application by the authors. MHD accelerator research has been conducted since the past four years with segmented Faraday MHD accelerators. Faraday type MHD accelerators are well known. In such devices, a linear channel is combined with a magnet and a series of electrode-pairs, segmented in order to obtain a more homogeneous electric discharge in the channel. In such accelerators, the air-plasma is moved through the channel by Lorentz forces. Thus, it would be possible to substitute MHD accelerators in advanced space propulsion systems.

* Corresponding author. Address: Department of Electrical Engineering, University of Tanjungpura, Jalan Jendral Ahmad Yani, Pontianak 78124, Indonesia. Tel./fax: +62 561 740186. E-mail addresses: [email protected], [email protected] (M. Anwari).

0196-8904/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2005.10.008

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Nomenclature A Af B E Es Ex Ey I j jx jy me ne p pe P Pp PL QL T t u x Ue

cross sectional area of channel, m2 slanted area (= A tanh) magnetic ﬁeld, T electric ﬁeld, V/m total energy of working gas per unit volume, J/m3 x component of electric ﬁeld, V/m y component of electric ﬁeld, V/m total current, A current density, A/m2 x component of current density, A/m2 y component of current density, A/m2 mass of electron, kg number density of electrons, m3 static pressure, Pa electronic pressure, Pa power density, W/m3 push power pressure loss heat loss temperature, K time, s ﬂow velocity, m/s axial distance, m internal energy of electron, J/m3

Greek b q r D h u ga

Hall parameter gas density, kg/m3 electrical conductivity, S/m voltage drop diagonal angle electric ﬁeld direction (tan h) electrical eﬃciency of accelerator

In previous works by the MHD laboratory of Nagaoka University of Technology, Harada et al. [1,2] have evaluated the fundamental performance of a linear Faraday type MHD accelerator using one dimensional numerical simulation with an inert gas as the working plasma for simplicity at the ﬁrst step. As a result of this analysis, the acceleration of gas ﬂow by MHD eﬀects was conﬁrmed. For a certain case, however, the gas was decelerated by the applied voltage at the inlet region even as the Lorentz force pushed the gas downstream. The electric power not only accelerated the gas ﬂow but also increased the gas pressure downstream. Harada et al. [3] have conducted numerical analysis on the performance of a MHD accelerator with a Hall and diagonal electrode conﬁguration. In this numerical simulation, with a working gas of argon seeded with potassium, the MHD acceleration eﬀect was conﬁrmed with a Hall and diagonal conﬁguration. An optimum input electric power level existed, which corresponds to the thermal input to the accelerator. The acceleration eﬀect decreased signiﬁcantly with the decrease of the Hall parameter for both the Hall and diagonal connection, although almost the same acceleration eﬀect as that by the segmented Faraday conﬁguration could be achieved. The performance of a linear Faraday type MHD accelerator under the thermal and chemical equilibrium condition has been studied using air-plasma as the working gas [4]. The ﬂow velocity of the air-plasma

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working gas increased along the axial distance of the channel at constant current density conditions. In this case, higher acceleration than with the argon gas was obtained. We also conﬁrmed in this numerical simulation that the air-plasma gave high electrical conductivity for about 3000 K gas temperature. The authors have done numerical simulation to compare and clarify the MHD accelerator performance with diﬀerent electrode connections [5]. In this case, the performance is characterized using one dimensional numerical simulation with an applied magnetic ﬁeld. The Faraday type has the best acceleration eﬃciency and the Hall type gave a result that falls remarkably compared with the other two connection systems. Moreover, the diagonal type could also acquire almost the same acceleration eﬃciency as a Faraday one. The objective of this paper is to study the performance of a MHD accelerator with diagonal connection for advanced space propulsion system application. In actual application as a space propulsion system, it has been indicated that a diagonal type accelerator is optimum because of its minimum weight, size and complexity [3,6]. The optimal MHD accelerator conﬁguration is determined ultimately by the application needs. From a performance standpoint, the Hall conﬁguration is more eﬀective for low density ﬂows, whereas the Faraday conﬁguration is superior for high density ﬂows. The disadvantage of the Faraday conﬁguration, however, is the separate power conditioning required for each electrode pair, which leads to a complex and expensive system. In many cases, especially ﬂight applications, multi-terminal loading is not practical. The diagonal electrode conﬁguration as an alternative two terminal loading scheme has been proposed to avoid the multi-terminal complications. 2. Numerical model 2.1. The equations of magnetohydrodynamics The ﬂuid dynamic equations for heavy particles are given by the conservation equations for mass, momentum and energy including MHD eﬀects as follows: oqA oðquAÞ þ ¼0 ot ox oðquAÞ oðquuAÞ þ ¼ ðj B rpÞA AP L ot ox oðEs AÞ o þ ½ðEs þ pÞuA ¼ Aj E AQL ot ox

ð1Þ ð2Þ ð3Þ

The basic MHD approximation equations for the electron gas used in this numerical simulation are expressed in the following continuity, momentum and energy equations: o þ o þ n A þ n uA ¼ n_ þ ð4Þ i A ot i ox i b 1 rpe jþ jB¼r EþuBþ ð5Þ B ene oU e oðU e uÞ j2 ¼ ðCollÞ QL þ ox ot r

ð6Þ

The following simpliﬁed Maxwell equations are incorporated with the set of MHD equations. rE ¼0

rj¼0

ð7Þ

Then, the MacCormack scheme incorporated with the two temperature model [7] are used to solve these sets of equations with artiﬁcial viscosity under the initial conditions of isentropic ﬂow. 2.2. The equations of diagonal type MHD accelerator Diagonal electrode connection of the accelerator introduces consideration of external loading conditions and should be discussed in conjunction with the electrical model. In the case of a diagonal MHD device,

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the electric ﬁeld is forced to be perpendicular to the diagonal angle due to the hard wiring of the external or internal connection (Fig. 1). The diagonal angle due to the wiring of a pair of shorted electrodes is Ey ¼ tan h ¼ u Ex

ð8Þ

where h is the electric ﬁeld angle and u is the electric ﬁeld orientation. The two terminal load current I for an MHD device with diagonally conﬁgured electrode pairs is given by Z I¼ j dAf ¼ j n Af ð9Þ Af

where the integration is over the entire slanted area Af and n is the direction of Af. In component form I ¼ ðjx þ tan hjy ÞA ¼ ðjx þ ujy ÞA

ð10Þ

In all cases, the current density and electric ﬁeld intensity are related through the generalized OhmÕs Law j ¼ rðE þ u BÞ ðb=BÞðj BÞ

ð11Þ

Combining Eqs. (7)–(9) yields a set of equations governing diagonally connected accelerator operation in terms of the applied current jx ¼

ð1 buÞI þ AruBð1 DÞu Að1 þ u2 Þ

ð12Þ

jy ¼

ðu þ bÞI AruBð1 DÞ Að1 þ u2 Þ

ð13Þ

Ex ¼

ð1 þ b2 ÞI AruBð1 DÞðb uÞ rAð1 þ u2 Þ

ð14Þ

Ey ¼ uEx

ð15Þ

where jx, jy, Ex, Ey are the current density and electric ﬁeld components, respectively, A is the ﬂow cross sectional area, is the plasma conductivity, u is the ﬂow velocity, B is the magnetic ﬁeld intensity and b is the Hall parameter. In this calculation, the voltage drop D = 0.28 is assumed, which is considered in Ref. [8]. The independent variable in the above expression is current, I, which is the current through the primary power circuit as tapped end-to-end across the length of the accelerator.

Fig. 1. Diagonal electrode connection.

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The Hall current neutralized condition occurs by setting jx = 0 in Eq. (12). In this condition, the load current and electric ﬁeld direction have values of AruBð1 DÞu ð1 buÞ I u¼ bI AruBð1 DÞ I ruBð1 DÞ jy ¼ ¼ uA ð1 buÞ

ð16Þ

I ¼

ð17Þ ð18Þ

Eliminating Ey by using Eq. (7) yields I qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ P ¼jE¼ E2x þ E2y Af qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ I IEx 2 2 1 þ ðtan hÞ P¼ E2x þ ðtan hÞ E2x ¼ Af Af qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Because Af ¼ A 1 þ ðtan hÞ2 , the expression can be rewritten in the following form: IEx A The streamwise Lorentz force component at any cross section is deﬁned as

ð19Þ ð20Þ

P¼

ð21Þ

ðj BÞx ¼ jy B

ð22Þ

and the push power associated with this Lorentz force is determined by: P p ¼ u.ðj BÞx ¼ ujy B

ð23Þ

The electrical eﬃciency of an accelerator is deﬁned as the ratio of the push power to the applied power ga ¼

ujy B P p u ðj B Þx ¼ ¼ P jE IEx =A

ð24Þ

2.3. Numerical conditions The numerical simulation for the MHD accelerator analysis corresponds to the conditions of the magnetohydrodynamic augmented propulsion experiment (MAPX) facility. Fig. 2 shows a schematic of the accelerator channel designed and developed by the NASA Marshall Space Flight Center, which is described in Ref. [8]. This channel can be installed at the end of the arc heater, which is used for the plasma production. The channel height and width diverge along the ﬂow direction with a divergence angle of 1°. The inlet height and width are 1.56 cm, and the exit height and width are 3.56 cm. The channel has 65 total electrodes with an electrode width of 1 cm.

Fig. 2. Schematic of MHD accelerator channel.

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Table 1 Calculating conditions Working gas Magnetic ﬁeld Inlet Mach number Stagnation pressure Stagnation temperature Wall temperature Seed fraction Applied current

Air 2.0 T (peak) 1.35 8.8 atm. 3341 K 1000 K K (1%) 300 A

The conditions for the present numerical simulation are summarized in Table 1. In the present calculation, the composition of the simulated air-plasma consists of two species, N2 and O2, with concentrations of 1 mol and 0.284 mol, respectively. The mass fraction of these species is approximately 28.0134 g and 9.09 g, respectively. The working gas is seeded with potassium, and the seed fraction is 1% by weight to all the gas, which corresponds to 0.949% in mole ratio. 3. Results and discussion The analysis of the MHD accelerator with diagonal electrode connection was based on performance calculations with the one dimensional numerical simulation. The characteristic gasdynamic properties (ﬂow velocity, gas temperature and gas pressure) are shown in Figs. 3–5 and the electrical properties (Hall parameter, electrical conductivity and Faraday current density) along the channel are shown in Figs. 6–8 for normal operation and Hall current neutralised condition in terms of the applied current. For the diagonally connected accelerator at normal operation, constant diagonal angle setting is varied as 45°, 55°, 60°, 70° and 80°, which are within the realistic range of possibilities for electrode connection. The distribution of the gasdynamic parameters along the channel is shown in Figs. 3–5. Fig. 3 shows the velocity distribution in the axial distance. The velocity increases along the MHD accelerator channel for normal operation, except the diagonal angle of 45° case. These calculation results indicate a highest total velocity increase of about 190% can be achieved by setting the 60° diagonal angle for normal operation. The gas temperature increases slightly for all results other than the 45° diagonal angle case, as shown in Fig. 4. In the case of the 45° diagonal angle, signiﬁcant ﬂuctuation occurred near the outlet region. The static pressure at the end

4000 3500

θ = 45

o

θ = 55

o

Velocity, m/s

θ = 60

o

3000

θ = 70

o o

2500

θ = 80 jx= 0

2000 1500 1000 500 0 0.0

0.2

0.4

0.6

0.8

Axial distance, m Fig. 3. Flow velocity along the channel.

1.0

M. Anwari et al. / Energy Conversion and Management 47 (2006) 1857–1867 7000 6000

o

θ = 55

o

θ = 60

5000

Temperature, K

θ = 45

4000

1863

o

θ = 70

o

θ = 80 jx= 0

o

3000 2000 1000 0 0.0

0.2

0.4

0.6

0.8

1.0

Axial distance, m Fig. 4. Static temperature along the channel.

4.0 3.5

θ = 45

o

θ = 55

o

Gas pressure, atm

θ = 60

o o

3.0

θ = 70

2.5

θ = 80 jx= 0

o

2.0 1.5 1.0 0.5 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Axial distance, m Fig. 5. Static pressure along the channel.

of the accelerator is about 0.3 atm. for the diagonal angles 55–80°. These results are similar to the static pressure under Hall current neutralised condition. For the 45° setting case, the static pressure at the outlet is 0.2 atm. lower than that of the other cases. In Figs. 6 and 7, the distribution of the average Hall parameter and electrical conductivity for normal operation and Hall current neutralized condition are shown. As already seen in the calculation results above, the 45° setting case under normal operation shows ﬂuctuation near the channel outlet region. Here, the diﬀerent characteristic in the gasdynamic and electrical properties is thought to be because the accelerator operates in the generator mode, which can be studied in Fig. 8. With the present results, the Faraday current density decreases signiﬁcantly to a negative value near the outlet region. The numerical simulation results above also illustrate a major diﬀerence for the overall distribution near the outlet region. With the present simulation, there is a peculiar behavior in the distribution of the gasdynamic and, particularly, the electrical properties in the 45° diagonal angle case results.

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Hall parameter

5.5

θ = 45

o

θ = 55

o

5.0

θ = 60

4.5

θ = 70

o

4.0

θ = 80 jx= 0

o

3.5

o

3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 0.0

0.2

0.4

0.6

0.8

1.0

Axial distance, m Fig. 6. Distribution of Hall parameter.

700

Electrical conductivity, S/m

600

θ = 45

o

θ = 55

o

θ = 60

o

θ = 70

o

500

o

400

θ = 80 jx= 0

300 200 100 0 0.0

0.2

0.4

0.6

0.8

1.0

Axial distance, m Fig. 7. Distribution of electrical conductivity.

The details of the MHD accelerator performance characteristics are summarized in Table 2. The electrical conductivity, Hall parameter and electrical eﬃciency of the accelerator are average terms across the accelerator. As indicated in this table, the highest electrical eﬃciency of the MHD accelerator in this present work can be obtained with the Hall current neutralized condition. A numerical simulation result for normal operation, where the diagonal angle is set constant, is shown in Figs. 9 and 10. Fig. 9 shows the MHD accelerator electrical eﬃciency against the diagonal angle of the electrode connection under normal operation. The eﬃciency of the MHD accelerator increases for diagonal angle settings as 15–55° and decreases signiﬁcantly for diagonal angle settings as 60–80°. We can see clearly that the optimum diagonal angle was obtained for a diagonal angle of 55°, which provides the maximum eﬃciency of 57.8%. In order to know the eﬀect of Hall current on the MHD accelerator eﬃciency, we plot the Hall current against diagonal angle in Fig. 10. The optimum diagonal angle obtained in the illustration of Fig. 9, which

M. Anwari et al. / Energy Conversion and Management 47 (2006) 1857–1867

1865

o

100

θ = 45

80

θ = 55 o θ = 60o θ = 70

Current density jy, A/cm2

o

60

o

θ = 80 jx= 0

40 20 0 -20 -40 -60 -80 0.0

0.2

0.4

0.6

0.8

1.0

Axial distance, m Fig. 8. Distribution of current density.

Table 2 Summary of MHD accelerator performance characteristics

Inlet pressure (atm.) Inlet temperature (K) Inlet velocity (m/s) Exit pressure (atm.) Exit temperature (K) Exit velocity (m/s) Electrical conductivity (S/m) Hall parameter Eﬃciency (%)

h = 45°

h = 55°

h = 60°

h = 70°

h = 80°

jx = 0

3.1 2645 1338 0.2 2919 3690 266 2.04 56.2

3.1 2645 1338 0.3 3365 3848 203 2.8 57.8

3.1 2645 1338 0.27 3246 3921 185 2.9 48.7

3.1 2645 1338 0.29 3145 3848 146 2.9 33.6

3.1 2645 1338 0.34 2953 2698 85 3 23.9

3.1 2645 1338 0.3 3248 3710 196 2.84 58.7

Comparison between calculation with normal operation and Hall current neutralized condition.

70 60

Efficiency, %

50 40 30 20 10 0 0

10

20

30

40

50

60

70

Diagonal angle, degree Fig. 9. Eﬃciency against diagonal angle.

80

90

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Hall current jx (absolute), A/cm2

50

40

30

20

10

0 0

10

20

30

40

50

60

70

80

90

Diagonal angle, degree Fig. 10. Hall current against diagonal angle.

can provide the maximum eﬃciency; agrees well with that diagonal angle that gives minimum Hall current. We understand that for this case, the Joule dissipation due to Hall current is minimum, and therefore, a larger Jy can be applied, which contributes to push the gas ﬂow downstream and maximize the MHD accelerator eﬃciency. 4. Conclusions Numerical simulation has been conducted for investigating the performance of a MHD accelerator channel with diagonal electrode conﬁguration using an air-plasma working gas. These calculations were conducted by setting the Hall current neutralized condition and in terms of an externally applied current. At the ﬂow velocity of 3710 m/s in the outlet, the electrical eﬃciency of the MHD accelerator with Hall current neutralized condition is 58.7%, which is higher than that of the performance produced by setting only a constant diagonal angle. Under normal operation, the best performance is obtained by setting a constant diagonal angle of 55°, which can provide a maximum electrical eﬃciency of 57.8%. As a result, this optimum diagonal angle shows a minimum Joule dissipation due to the Hall current to produce a maximum electrical eﬃciency of the MHD accelerator. Acknowledgements The authors wish to thank Dr. John T. Lineberry, President, LyTec LLC. and Dr. Ron J. Litchford, NASA Marshall Space Flight Center, for valuable discussions on the MHD accelerator with diagonal electrode connection and the MAPX designed and developed by the George J. Marshall Space Flight Center, National Aeronautics and Space Administration. References [1] Harada Nob. MHD Acceleration Studies at Nagaoka University of Technology. AIAA paper 2001–2744; 2001. [2] Harada Nob, Ikewada J, Terasaki Y. Basic studies on an MHD accelerator. AIAA paper 2002–2175; 2002. [3] Harada Nob, Takahashi S, Lineberry JT. Comparative study of electrode connections of an MHD accelerator. AIAA paper 2003– 4288; 2003. [4] Anwari M, Takahashi S, Harada Nob. Performance study of a magnetohydrodynamic accelerator using air-plasma as working gas. J Energy Conversion Manage 2005;46(15–16):2605.

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Anwari M, Takahashi S, Harada Nob. ‘‘Numerical simulation for performance of an MHD accelerator. AIAA paper 2004–2363; 2004. Litchford RJ. Performance theory of diagonal conducting wall MHD accelerators. AIAA paper 2003–4284; 2003. Rosa RJ. Magnetohydrodynamic energy conversion. McGraw Hill; 1963. Litchford RJ, Cole JW, Lineberry JT, Chapman JN, Schmidt HJ, Lineberry CW. Magnetohydrodynamic augmented propulsion experiment: I. Performance analysis and design. AIAA Paper 2002–2184; 2002.