NASA/TP—2004–213173

May 2004

Performance Theory of Diagonal ConductingWall Magnetohydrodynamic AcceleratorsR.J. LitchfordMarshall Space Flight Center, Marshall Space Flight Center, Alabama

https://ntrs.nasa.gov/search.jsp?R=20040171926 2020-07-30T07:33:14+00:00Z

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NASA/TP—2004–213173

R.J. LitchfordMarshall Space Flight Center, Marshall Space Flight Center, Alabama

May 2004

National Aeronautics andSpace Administration

Marshall Space Flight Center • MSFC, Alabama 35812

Performance Theory of Diagonal ConductingWall Magnetohydrodynamic Accelerators

ii

Available from:

NASA Center for AeroSpace Information National Technical Information Service7121 Standard Drive 5285 Port Royal RoadHanover, MD 21076–1320 Springfield, VA 22161(301) 621–0390 (703) 487–4650

Acknowledgments

The author expresses his thanks to John T. Lineberry, President, LyTec, LLC; Prof. Y.C.L. Susan Wu (retired), Universityof Tennessee Space Institute; and Prof. Valentin A. Bityurin, Head of MHD Programs, Institute of High Temperatures

(IVTAN) of the Russian Academy of Sciences for valuable discussions and for clarifying basic conceptual ideasunderlying MHD performance theory.

iii

TABLE OF CONTENTS

1. INTRODUCTION ........................................................................................................................ 1

2. CROSSED-FIELD ACCELERATOR CONFIGURATIONS ...................................................... 3

3. GENERALIZED OHM’S LAW ................................................................................................... 4

3.1 Mean Ohm’s Law .................................................................................................................. 5

4. PERFORMANCE MODEL ......................................................................................................... 6

4.1 Diagonal Loading Constraint................................................................................................. 64.2 Electrical Parameter Relationships ........................................................................................ 64.3 Mean Power Parameters ........................................................................................................ 7 4.3.1 Power Density ............................................................................................................. 7 4.3.2 Push Power .................................................................................................................. 7 4.3.3 Dissipated Power ......................................................................................................... 7 4.3.4 Electrical Efficiency .................................................................................................... 8 4.3.5 Power Relationships .................................................................................................... 84.4 Generalized Vector Diagram ................................................................................................. 84.5 Current-Dependent Voltage Drop .......................................................................................... 9 4.5.1 Generator Mode ........................................................................................................... 10 4.5.2 Accelerator Mode ........................................................................................................ 10

5. PERFORMANCE DIAGRAMS .................................................................................................. 12

5.1 Dimensionless Performance Parameters ............................................................................... 125.2 Performance Map .................................................................................................................. 14

6. ELECTRODE BOUNDARY LAYER EFFECTS ........................................................................ 18

7. PRACTICAL DESIGN CONSTRAINTS .................................................................................... 21

8. CONCLUSIONS .......................................................................................................................... 22

REFERENCES ................................................................................................................................... 23

iv

LIST OF FIGURES

1. Composite graphic of alternative cross-field MHD accelerator configurations:(a) Linear Hall accelerator, (b) segmented Faraday accelerator, (c) series connecteddiagonal accelerator, and (d) DCW accelerator.................................................................... 26

2. Orientation of field vectors in a diagonally connected cross-field accelerator .................... 27

3. Vector diagrams for the crossed-field diagonal MHD devices: (a) Open-circuit generator,(b) jx-neutralized generator, (c) short-circuit generator, (d) open-circuit accelerator,(e) zero-efficiency accelerator, and (f) jx-neutralized accelerator. Each diagram depictsthe relative orientation of the current density vector and electric field vectors based

on the generalized Ohm’s law, j j B

EΣ

ΩΣ

+ ×

= ′

B, and the generalized electrical field

relation, ′ = + × −E E u B Ed ................................................................................................. 28

4. Variation of performance parameters based on representative values for flow/fieldconditions, material functions, and wall angle. Assumes that ϕ =±1, B =4 T, Σ=100,Ω =2, u =3 km/s, ∆o=–0.1, and ∆s =–0.02 .............................................................................. 29

5. Performance map for DCW MHD devices with current-dependent voltage drop effect ..... 30

6. Illustration of current transport and boundary layer dimensions for aDCW cross-field accelerator ................................................................................................ 31

7. Variation of ξc and VΣ /2δ as a function of jy / Σ′0. Assumes Ec= 40 kV/cm,n=7, m=18, and a=10 ........................................................................................................... 32

v

LIST OF TABLES

1. Summary of electrical parameter relationships for diagonalizedcross-field MHD devices ...................................................................................................... 25

2. Summary of mean power relationships for diagonalizedcross-field MHD devices ...................................................................................................... 25

3. Reference conditions for variable ∆ ..................................................................................... 25

vi

NOMENCLATURE

A cross-sectional area (=hw)

Af slanted area (=Atanθ )

a conductivity power law exponent

B magnetic field strength

b ion slip factor

E electric field in the laboratory frame

E′ generalized electric field intensity relative to axes moving with the gas

E dimensionless electric field

Ec critical electric field strength

Ed equivalent electric field of near-electrode voltage drop

Ex streamwise component of electric field

Ex dimensionless streamwise electric field

Ex,lim limiting streamwise electric field

Ey transverse component of electric field

Ey dimensionless transverse electric field

h channel height

I two-terminal load current

I′ load current at first reference condition

I′′ load current at second reference condition

I dimensionless total current

vii

NOMENCLATURE (Continued)

In load current density for jy-neutralized accelerator condition

Is load current for short-circuit generator condition

Iz load current for zero-power accelerator condition

j local current density

j dimensionless current density

jx dimensionless streamwise current density

jx,o open-circuit valve of jx

jx,s short-circuit valve of jx

jx,E=0 valve of jx when Ex=Ey=0

jy transverse current density

jy dimensionless transverse current density

jy,s short-circuit valve of jy

jy,E=0 valve of jy when Ex=Ey=0

jz current density along magnetic field lines

L channel length

m thermal boundary layer power law exponent

n unit vector

n velocity boundary layer power law exponent

P power density

P dimensionless power density

Pmax optimal value of P

viii

NOMENCLATURE (Continued)

Pp push power

Pp dimensionless push power

Pηa=0 value of P at jy-neutrelized condition

R load resistance

T temperature

T0 temperature in homogeneous core flow region

Tc critical temperature corresponding to electrical breakdown

u gas velocity

u streamwise component of gas velocity

u0 velocity in homogeneous core flow region

u×B induced motional emf

Va anode voltage fall

Vc cathode voltage fall

Vd effective electrode voltage drop

Vg gasdynamic voltage drop

Vu velocity dependent component of Vg

VΣ conductivity dependent component of Vg

x streamwise coordinate

y transverse coordinate parellel to sidewalls

z transverse coordinate perpendicular to sidewalls

α Hall angle (=tan_1 Ω)

ix

NOMENCLATURE (Continued)

βe electron Hall parameter

βi ion Hall parameter

∆ dimensionless voltage drop (=Vd /uBh)

∆′ value of ∆ at first reference condition

∆′′ value of ∆ at second reference condition

∆max value of ∆ at maximum power density

∆n value of ∆ at jy-neutralized condition

∆o value of ∆ at open-circuit condition

∆s value of ∆ at short-circuit condition

∆z value of ∆ at zero-power condition

δ boundary layer thickness

δa anode boundary layer thickness

δc cathode boundary layer thickness

ϕ electric field direction

ηa accelerator electrical efficiency

ηg generator electrical efficiency

Σ effective electrical conductivity

Σ0 value of Σ in homogeneous core flow region

Σc critical value of Σ corresponding to electrical breakdown

σ scalar electrical conductivity

θ electric field angle (=tan_1 ϕ)

x

NOMENCLATURE (Continued)

θw wall angle or diagonalization

Ω effective Hall parameter

Ω0 value of Ω in homogenious core flow region

ω conductivity ratio

ωc critical conductivity ratio corresponding to electrical breakdown

ξ dimensionless height parameter (=y/δ )

ξc critical value of ξ corresponding to electrical breakdown

1

TECHNICAL PUBLICATION

PERFORMANCE THEORY OF DIAGONAL CONDUCTING WALLMAGNETOHYDRODYNAMIC ACCELERATORS

1. INTRODUCTION

Historically, interest in magnetohydrodynamic (MHD) devices has centered on their use as electri-cal generators in commercial central power plants and mobile burst power systems. The primary attractionfor central power is associated with the attainment of higher peak cycle temperatures, which point tosignificant improvements in overall plant efficiency. The attraction for mobile pulse power, particularlyairborne military needs, derives from their intrinsic high power-density characteristics. It should be recog-nized that equally important though less noted attractions exist for accelerator configurations, as well.

Two major identifiable uses for crossed-field MHD accelerators are (1) propulsive devices and(2) hypersonic aerodynamic test facilities. Litchford et al. discuss the operational attributes that are ofparticular significance to these applications and provide an indepth historical perspective of their techno-logical development.1

From a fundamental point of view, the essential argument favoring the utilization of a Lorentz forceacceleration mechanism is the ability to avoid the inherent physical limitations encountered with purethermal approaches (viz, material thermal limits and ionization/dissociation losses). Simply put, it is moreeffective to transfer electrical energy into directed kinetic energy rather than first degrading it to thermalenergy. Crossed-field acceleration is of special interest in these cases because of its unique capacity forprocessing large amounts of power under conditions of high mass throughput.

Past emphasis on generator configurations has led to a substantial theoretical and experimentalbasis for understanding their performance and operational nuances. However, a similar level of under-standing has yet to be acquired for accelerator configurations despite a strong growing interest in theirpotential aerospace applications.

The standard theoretical approach for describing the interaction of an electrically conducting gaswith applied electric and magnetic fields relies on the application of a Cowling-Schlueter-type generalizedOhm’s law for a partially ionized, electrically neutral gas. Powers et al., for instance, used a cross-planeaveraged form of the generalized Ohm’s law and developed a graphically based methodology for describ-ing the general performance characteristics of MHD accelerators and generators.2

This classical model has proved to be extremely useful in illuminating the basic operational behav-ior of MHD devices, but its practical utility is hampered by a critical limitation. That is, the theory assumes

2

a constant near-electrode voltage drop, whereas experience has shown that the magnitude of the near-electrode voltage drop exhibits a significant load-current dependency.

Various theoretical refinements of this nature have been forthcoming but solely within the contextof generator configurations. Wu, for example, introduced an effective voltage drop parameter, accountedfor its load current dependency, and examined the resulting impact on diagonal conducting wall (DCW)generator performance.3 The MHD research group at the Max-Planck-Institut fuer Plasmaphysik made asimilar attack on the problem using a cross-plane averaged Ohm’s law, specialized for a DCW generatorconfiguration.4

It is widely recognized, of course, that the current and electric field structure in MHD devices isinherently three-dimensional and that spatial separation of physical processes is not warranted in general.As a matter of recourse, one could mount a brute force attack on the problem and perform a complete three-dimensional numerical analysis of the flow and electrical structure in the duct, assuming adequate comput-ing resources are available. Alternatively, one could invoke an infinite segmentation assumption (i.e., thestreamwise variation in electrical properties is small in comparison to transverse variations) and couple anapproximate cross-plane electrical model with a three-dimensional flow analysis as exemplified forgenerator configurations by Bityurin et al. and Ahluwalia et al.5,6 Each approach has merits and draw-backs, and experience has shown that each level of approximation is useful in fulfilling certain design andanalysis needs.

The purpose of this Technical Publication is to reexamine the classical theory and extend it forcross-field accelerator configurations with inclusion of a current-dependent near-electrode voltage dropmodel. This particular refinement of the classical theory, while generally recognized as a straightforwardtheoretical extension, has not explicitly appeared in the literature, and it is believed that the resultingdevelopment can yield practical insight into the basic operational characteristics of these importantdevices. Moreover, it is hoped that this work may serve as a convenient and compact resource for futuredesign practitioners.

3

2. CROSSED-FIELD ACCELERATOR CONFIGURATIONS

Various linear crossed-field MHD accelerator configurations are conceivable (fig. 1) although theoptimal configuration depends on the ultimate application need. The Hall configuration, for example, isgenerally more effective for low-density flows (i.e., very high Hall parameter condition) whereas thesegmented Faraday configuration yields superior performance for high-density flows. A significant disad-vantage of the Faraday configuration, however, is the separate power conditioning that is required for eachelectrode pair, which leads to a heavy, complex, and expensive system.

Alternative two-terminal loading schemes have been proposed to avoid the complications ofmultiterminal connections while also attempting to compensate for the effects associated with a finite Hallparameter. De Montardy, for instance, suggested the series connected scheme in which a segmentedFaraday channel is externally diagonalized.7 Dicks later extended this approach to DCW configuration inwhich slanted window frame-like electrode elements are stacked with thin insulators to form a completechannel.8,9 The DCW configuration not only simplifies fabrication and improves strength but providessuperior performance to the series connected device by allowing current to flow to the sidewalls.3

Hence, theoretical development is focused on the DCW accelerator configuration, since it has thegreatest practical relevance. The performance characteristics of the alternative crossed-field configura-tions can be subsequently deduced as special or extreme cases of the DCW configuration.

4

3. GENERALIZED OHM’S LAW

The local current density, j, in a partially ionized, electrically neutral gas may be determined froma Cowling-Schlueter-type generalized Ohm’s law.10–14 For application to typical MHD devices, it is usefulto adopt the simplified form derived by Rosa, which includes ion slip but neglects electron pressuregradients:15

j E j B j B B= ′ − × + ×( ) ×σ β β βe e iB B2 . (1)

Here, σ is the scalar electrical conductivity; E′′′′′ is the generalized electric field intensity relative to axesmoving with gas velocity, u; B is the magnetic field strength; and βe and βi are the electron and ion Hallparameters, respectively. The Hall effect and ion slip are introduced by the second and third terms on theright-hand side of equation (1).

As a practical matter, we define the generalized electric field, E′′′′′, in terms of the electric field in thelaboratory frame, E; the motional emf, u×B; and an equivalent electric field, Ed, which accounts for thenear-electrode voltage drop, Vd (i.e., Ed =Vd /h), inside the boundary layer,

′ = + × −E E u B Ed . (2)

Note that the equivalent electric field is applied only in a direction transverse to the magnetic field vectorand the streamwise velocity vector. It is always directed in opposition to the transverse current, jy .

For the special case in which no current flows along the magnetic field lines, ( jz= 0), it is readilyshown that equation (1) reduces to the compact form

j E j B= ′ − ×ΣΩ

B

, (3)

where we have introduced two new parameters:

Σ =+

=+

σβ β

σ1 1e i b

(4)

and

Ω =+

=+

ββ β

βe

e i

eb1 1

. (5)

5

In many cases, the ion slip factor, b=βeβi, is negligible, and further simplification is obtained as Σ→σ andΩ→βe.

3.1 Mean Ohm’s Law

Many of the interesting phenomena that occur in high-interaction MHD devices, such as velocityovershoots in the sidewall boundary layer, flow asymmetries, and generation of intense secondary flows inthe cross plane, are three-dimensional in nature. Therefore, high-fidelity performance predictions ulti-mately require three-dimensional numerical analyses of the combined flow and electrical structure. On theother hand, simplified approaches based on spatial decoupling or reduced spatial dimensions can also yielduseful results if all critical physical effects are accounted for in an appropriate fashion.

Exact theoretical treatment requires consideration of the fundamental equations of MHDand Maxwell’s electromagnetic equations, thereby leading to a system of coupled differential equations.Here, we invoke the infinite segmentation assumption and assume that all parameters in the generalizedOhm’s law can be effectively represented by their cross-plane averaged values and are dependent on thex-coordinate, only.

The dimensionless voltage drop parameter, ∆=Ed /uB=Vd /uBh, which incorporates all voltage lossesassociated with the electrode boundary layer, may be introduced into the generalized Ohm’s law to obtainthe component relations,

j E jx x y= −Σ Ω (6)

and

j E uB jy y x= − +( ) +Σ Σ ∆ Ω 1 . (7)

Equations (6) and (7) may also be combined to obtain the inverse expressions for jx and jy in terms of thelocal electric field components,

j E E uBx x y=+

−[ ] ++

+( )ΣΩ

ΩΩΩ

Σ ∆1 1

12 2 (8)

and

j E E uBy x y=+

+[ ] −+

+( )ΣΩ

ΩΩ

Σ ∆1

1

112 2 . (9)

6

4. PERFORMANCE MODEL

4.1 Diagonal Loading Constraint

Completion of the electrical model requires the consideration of external loading conditions. Inthe case of a diagonally linked device (fig. 2), the electric field is forced to align perpendicularly to thediagonal short so that

E Ey x = =tan ,θ ϕ (10)

where θ is the electric field angle and ϕ is the electric field direction. It follows that the diagonalization orwall angle is given by θw=π /2–θ. The sign of ϕ depends on the mode of operation. For an accelerator, u×Bopposes jy, 0<θ< π /2, and ϕ >0. For a generator, u×B aligns with jy, –π/2<θ <0, and ϕ <0.

The two-terminal load current, I, for an MHD device with diagonally linked electrode pairs is given by

I d AfA ff

= ⋅ = ⋅∫ j A j n , (11)

where the integration is over the entire slanted area, Af . In component form,

I j j A j j Ax y x y= +( ) = +( )tan .θ ϕ (12)

4.2 Electrical Parameter Relationships

A complete set of equations now exists for determining the cross-plane electrical characteristics. Ingeneral, either I or Ex may be specified, allowing the remaining unknown electrical parameters to bededuced from equations (6), (7), (10), and (12) in conjunction with appropriate material functions.

The resulting performance relationships for DCW accelerators are summarized in table 1 using Exand I as independent variables. The governing relations for generator performance are recovered whenboth ϕ and ∆ are less than zero and I =–Ex L /R, where R is the load resistance and L is the channel lengthover which the load is applied. Note also that the governing relations for a linear Hall channel configura-tion are recovered in the extreme case ϕ =0.

7

4.3 Mean Power Parameters

Ultimately, we seek to deduce suitably averaged power parameters from the resulting cross-planeelectrical model. The quantities j⋅E and (j×B)x, for instance, may be used to determine the power density,push power, dissipated power, and electrical efficiency of a device.

4.3.1 Power Density

The electrical power density at any cross section of the duct is defined by

PI

AE E

fx y= ⋅ = +j E 2 2 . (13)

Eliminating Ey using equation (10) yields

PI

AE E

IE

Afx x

x

f= + = +2 2 2 21ϕ ϕ , (14)

but given A Af = +1 2ϕ , the working form becomes

PIE

Ax= . (15)

4.3.2 Push Power

The streamwise Lorentz body force component at any cross section is defined as

j B×( ) =x yj B , (16)

and the push power associated with this Lorentz body force is given by

P uj Bp x y= ⋅ ×( ) =u j B . (17)

4.3.3 Dissipated Power

The power density of an MHD device must exceed the Lorentz force push power since the internalresistance to current flow leads to ohmic heating. This can be demonstrated by taking the dot product of jwith the generalized Ohm’s law, as defined by equation (3), which yields

8

j j j E j u B j E j j B⋅ = ⋅ + ⋅ ×( ) − ⋅[ ] − ⋅ ×( )ΣΩ

d B . (18)

Because jz = 0, the last term is identically zero, and we obtain

j jj E u j B j E

⋅= ⋅ − ⋅ × − ⋅ = − −

Σ∆d p yP P j uB , (19)

where j⋅j/Σ is the power dissipated in the core flow region, and j⋅Ed =jy Ed =jy uB∆ represents the powerdissipated in the electrical boundary layers. Therefore, the power dissipated over the entire cross plane isgiven by

P j uB P Pd y p=⋅

+ = −j jΣ

∆ . (20)

4.3.4 Electrical Efficiency

The electrical efficiency of an accelerator is simply the ratio of the push power to the appliedpower:

ηap x y

x

P

P

uj B

IE A= =

⋅ ×( )⋅

=u j B

j E . (21)

It follows that the generator efficiency is defined as the reciprocal of the accelerator efficiency, ηg=1/ηa.

4.3.5 Power Relationships

By combining the results in table 1 with the relationships above, it is possible to express the meanpower parameters in terms of the independent variables Ex or I. These results are summarized in table 2.

4.4 Generalized Vector Diagram

It is generally instructive to examine the qualitative behavior of the internal electric fields andcurrent densities under the assumption that MHD effects have only a slight effect on flow properties. In thiscase, the generalized Ohm’s law can be used to construct representative vector diagrams for diagonalizeddevices, as shown in figure 3 for fixed values of ϕ.

To construct these diagrams, the generalized Ohm’s law defined by equation (3) is put into the form

j j BE

ΣΩ

Σ+ ×

= ′

B , (22)

9

where we immediately recognize that the two terms on the left-hand side of equation (22) must be orthogo-nal. Thus, each of these terms represents one side of a right triangle with the hypotenuse, E′,′,′,′,′, inclined at anangle of tan–1 Ω. The generalized electric field must simultaneously satisfy the vector relation defined byequation (2).

Figures 3(a)–(c) illustrate the load current dependence of a diagonalized generator (ϕ <0). In theopen-circuit case (a), there is no current flow to the load, and the internal current density vector must alignwith the diagonal link, which defines planes of constant electric potential. Thus, the current is closedthrough the diagonal short, and Ey exactly compensates for the combined u×B and Ed induced potentials.Note that jx is directed upstream under these conditions. As the load current is increased, the currentdensity vector rotates out of the plane of the diagonal link until it aligns with the u×B vector. At this point,jx = 0 and we arrive at the Hall current neutralized condition (b). If the load current is increased further, thecurrent vector continues its rotation, and the direction of Hall current flow is reversed. Ultimately, thecurrent density vector becomes perpendicular to the plane of the diagonal link as the electric field goes tozero at the short-circuit condition (c).

Figures 3(d)–(f) illustrate the load current dependence of a diagonalized accelerator (ϕ > 0). Whenthe applied axial electric field is zero and there is no load current, the open-circuit accelerator loadingcondition is obtained (d). The internal current density vector corresponds to the generator short-circuitcase, under these unique circumstances. Note that jy< 0 (braking regime) and that the current flow is closedthrough the diagonal short. With increasing applied axial electric field, the current density vector rotatesout of the plane of the diagonal link. The device remains in a braking regime, however, until the appliedfield becomes large enough to force jy to zero at which point we attain the zero-efficiency acceleratorcondition (e). Further increases in applied current lead to continued rotation of the current density vectorand increasing values of jy . Eventually, one reaches a jx neutralized-accelerator condition (f). Increasingthe applied current beyond this point leads to a reversal in the direction of axial current flow while themagnitude of the current density vector continues to grow.

4.5 Current-Dependent Voltage Drop

The previously developed performance relationships hold for any fixed value of ∆. However,extensive experimental research with MHD generators has clearly established that the near-electrode volt-age drop is approximately linearly dependent on the transverse current density.16–18 That is, any increasein load current leads to a proportionate increase in voltage drop. This behavior is assumed to extend toaccelerator operation, as well.

The linear relationship between Vd and jy permits the construction of a simple two-parameter model,

VV V

I II I Vd

d dd=

′′ − ′′′ − ′

− ′( ) + ′ , (23)

where the prime and double-prime superscripts indicate two distinct loading conditions. It is generallymore convenient to express this relation directly in terms of ∆:

10

∆∆ ∆

∆=′′ − ′

′′ − ′

− ′( ) + ′

I II I . (24)

Selection of appropriate reference load conditions depends on the mode of operation. For a genera-tor, it is most natural to adopt the open-circuit (∆′=∆o) and short-circuit (∆′′=∆s) conditions on the extremeends of the operational load line. For an accelerator, it is convenient to adopt the zero-power condition,which corresponds to a short-circuit generator (∆′=∆z=∆s), and the jy neutralized condition (∆′′=∆n=0).The selected reference conditions are summarized in table 3. The implications of this model are nowexamined for the generator and accelerator modes.

4.5.1 Generator Mode

At the short-circuit generator condition, Ex = 0, I ′′= Is, and ∆′′=∆s . From table 1, we therefore deduce

′′ = =+( ) −( )+

I IuBA

ssΣ ∆ Ω

Ω1

1 2ϕ

. (25)

Elimination of I ′′ using equation (24) immediately yields an expression for ∆ in terms of I for a generator:

∆ ∆Ω ∆ ∆

Ω ∆ Σ= −

+( ) −( )−( ) +( )

o

o s

s

I

uBA

1

1

2

ϕ . (26)

This relationship may be used to eliminate ∆ from the performance relations of tables 1 and 2.

4.5.2 Accelerator Mode

The zero-power open-circuit accelerator condition corresponds to the short-circuit generator con-dition with ϕ > 0. In this case, Ex= 0, I′= Iz, and ∆′=∆z = ∆s, and we deduce that

′ = =+( ) −( )+

I IuBA

zsΣ ∆ Ω

Ω1

1 2ϕ

. (27)

The point of transverse current-density neutralization corresponds to a zero-efficient accelerator wherejy = 0, I ′′= In, and ∆′′= ∆n = 0. This implies

′′ = =+

I IuBA

nΣΩ ϕ

. (28)

11

Eliminating I ′ and I ′′ in equation (24) gives an expression for ∆ in terms of I for an accelerator,

∆ ∆Ω Ω

ΣΩ ∆

Ω Ω ∆= −

+( ) +( ) − −( ) +

+( ) − −( ) +

ss

s

I

uBA11 1

1 1

2 2 2

2 2 2

ϕ ϕ

ϕ

( )

( ) . (29)

This relationship may be used to eliminate ∆ in the performance relations of tables 1 and 2.

12

5. PERFORMANCE DIAGRAMS

It is possible to construct an informative performance map for DCW MHD devices following thegeneral methodology first outlined by Powers et al., who assumed a constant voltage drop in the electrodesheath layer.2 The significant modifications associated with a variable effective voltage drop model werelater noted and described by Wu for the DCW generator case.3 Here, Wu’s analysis is explicitly extendedfor DCW accelerator operation, as well.

5.1 Dimensionless Performance Parameters

As a first step, the governing performance relationships are simplified by defining the followingdimensionless quantities:

juB

_=

jΣ

, EuB

_=

E , II

uBA

_=

Σ , P

P

u B

_ .=

Σ 2 2(30)

Substitution into the component form of the generalized Ohm’s law then yields

j E Ex x y_ _ _

=+

+ + +( )

1

112Ω

Ω Ω ∆ (31)

and

j E Ey x y_ _ _

.=+

+ − +( )

1

112Ω

Ω ∆ (32)

It also follows that the electric field components can be stated explicitly in terms of current density:

E j jx x y

_ _ _= + Ω (33)

and

E j jy x y

_ _ _ .= − + + +( )Ω ∆1 (34)

An expression for the dimensionless load current is obtained from the substitution of equation (30) intoequation (12):

I j jx y

_ _ _ ,= + ϕ (35)

13

and the dimensionless laboratory power density follows directly from the substitution of equation (30) intoequation (13):

P j E j Ex x y y_ _ _ _ _ _

.= ⋅ = +j E_ (36)

Eliminating Ex and Ey using equations (33) and (34) yields the more convenient form,

P j j jx y y

_ _ _ _= + + +( )2 2 1 ∆ (37)

Furthermore, we may define the dimensionless push power as

PP

u B

uj B

u B

j

uBjp

p y yy

_ _ ,= = = =

Σ Σ Σ2 2 2 2(38)

and the electrical efficiency may be expressed in terms of dimensionless parameters by combiningequations (37) and (38):

ηηa

p y

x y yg

P

P

j

j j j

= =+ + +( )

=

_

_

_

_ _ _ .2 2 1

1

∆(39)

It is readily apparent that the relationships in tables 1 and 2 can all be put in dimensionless form usingequation (30); the results are obvious and are not tabulated. The current-dependent effective voltage dropeffect is accounted for by using either equation (26) or (29) in dimensionless form.

As a prelude to the construction of the graphical performance map, it is instructive to examineperformance parameter variations as a function of the dimensionless axial electric field, Ex. Figure 4, forexample, summarizes calculated parameter variations assuming representative values for electrical/flowconditions, material functions, and wall angle. Starting from the open-circuit generator condition and gradu-ally reducing the load impedance, we observe the sequential occurrence of all of the loading conditionsdepicted in figure 3.

For an open-circuit generator, I=0 and equation (35) yield the constraint jx,o=–ϕ jx,o. We thereforeinfer that jx,o< 0 since ϕ < 0 and jy,o< 0 for the assumed magnetic field direction. In this case, the internalcurrent is forced parallel to the wall angle and is completely shorted through the diagonal linkage. As theload impedance is decreased below a critical level, however, the total current becomes nonzero, and themagnitude of the negatively directed axial electric field begins to fall.

If the load impedance is decreased all the way to zero, we arrive at the short-circuit generatorcondition corresponding to Ex = 0. Here, application of equation (33) yields the constraint jx,s=– Ω jy, s, which implies jx,s> 0 since Ω > 0 and jy,s< 0. In this case, the internal current flow is forcedperpendicular to the wall angle, and no current flows through the diagonal linkage.

14

Consideration of these open- and short-circuit constraints clearly implies the existence of a Hallcurrent neutralized condition (i.e., jx =0) on the generator load line. It is also clear that maximum powerextraction and electrical efficiency must also occur at some point between the open- and short-circuitconditions. This point may or may not coincide with the Hall current neutralized condition. Note that themagnitude of ∆ is largest at open-circuit loading where the magnitude of jy is greatest.

Simply reversing the wall angle (ϕ >0) at the generator short-circuit condition yields the open-circuit accelerator condition. If power is then gradually applied such that a positive axial electric field isimpressed on the accelerator, we enter a regime where the transverse voltage is insufficient to overcomethe induced u×B potential, and jy remains negatively directed. This is commonly referred to as the brakingregime, where all applied power goes into joule heating of the working fluid. Eventually, the appliedvoltage becomes high enough to neutralize the transverse current at which point ∆→0.

Increasing the applied electric field beyond this point yields positive push work and flow accelera-tion. The transverse current and total current grow in magnitude, more power is delivered to the device,and ∆ steadily increases. At the same time, axial current flow gradually decreases until the jx neutralizationcondition is reached. Additional increases in applied field and power beyond this point simply reverse thedirection of jx . It is important to note that optimal accelerator electrical efficiency occurs at an appliedelectric field magnitude much less than that required to neutralize axial current flow.

5.2 Performance Map

The operational attributes of MHD devices can be exhibited in the most vivid manner on a perfor-mance map in the plane of dimensionless current density and electric field. Following the methodology ofPowers et al.,2 as it was modified by Wu for current-dependent voltage drop,3 we complete the square inequation (37) to obtain the family of circles defined by

j j Px y

_ _ _ .2

2 212

12

+ ++

= +

+

∆ ∆ (40)

These circles are centered at jx= 0, jy=(1+∆)/2 with a radius of P+[(1+∆)/2]21/2.

Transformation to the Ex, Ey axes is defined by equations (33) and (34), and the origin of the Ex, Eyaxes in the jx, jy plane follows directly from equations (31) and (32):

j jx E y E

_

,

_

,_ _ .= ==

+( )+

= −+( )

+0 2 0 21

1

1

1

Ω ∆Ω

∆Ω

(41)

The locus of points defining this origin can be expressed independently of Ω by combining equations (33)and (34) under the condition Ex=Ey= 0:

j jx E y E

_

,

_

,_ .= =+ +

+

=

+

0

20

2 212

12

∆ ∆ (42)

15

Note that the circles defined by equation (40) reduce to the loci defined by equation (42) at zero-powerdensity. Thus, the generator boundary coincides with the locus of the origin of the Ex, Ey axes, and allpower generating circles lie inside this boundary since P<0 for a generator. Further insight may be gainedthrough the examination of extreme generator loading conditions.

At generator short circuit, the Hall field goes to zero, and equation (33) gives the constraint

j j j jx s y s y s x s

_

,

_

,

_

,

_

,– – .= ⇒ ( ) =Ω Ωtan–1 1 (43)

Because Ω >0 and jy <0, given the assumed magnetic field orientation, the short-circuit operating domainresides in the fourth quadrant of the current density plane.

At generator open-circuit conditions, the total current falls to zero and equation (35) gives theconstraint

j j j jx o y o y o x o

_

,

_

,

_

,

_

, .= − ⇒ −( ) =ϕ ϕtan–1 1 (44)

Because ϕ < 0 and jy< 0 for the generator mode, the open-circuit operating domain resides in the thirdquadrant of the current density plane.

Clearly, the power density vanishes at both open and short circuit conditions, but the criticaldiscriminating factor to note is that ∆ varies with current over the entire load range (–1< ∆o< ∆s< 0). There-fore, the radius of the zero-power circle depends on the load. The zero-power loci for short- and open-circuit loading follows from equation (40) with P =0:

j jx s y ss s

_

,

_

,2

2 212

12

+ ++

=

+

∆ ∆ (45)

and

j jx o y oo o

_

,

_

, .22 21

21

2+ +

+

=

+

∆ ∆ (46)

Using these expressions, it is possible to construct the short- and open-circuit half-circles. Theshort-circuit half-circle is centered at jx =0, jy =(1+∆s)/2 with radius (1+∆s)/2, and the open-circuit half-circle is centered at jx =0, jy =(1+∆o)/2 with radius (1+∆o ) /2.These loci are illustrated in figure 5 where theload line connects in accordance with the constraints defined by equations (43) and (44). Because ∆o< ∆s,the radius of the open-circuit half-circle is smaller than the radius of the short-circuit half-circle, and thecenter points are shifted.

The origin of the Ex, Ey axes falls on the short-circuit half-circle at the points defined by equation(41). Inspection of equation (34) indicates that the Ey axis passes through the origin of the jx, jy axes since∆= 0 at this condition. The various points of interest are indicated in figure 5.

16

Combining the DCW electric field constraint defined by equation (10) with equations (30), (33),and (34) yields an expression for the operating load line in the form

Ω Ω ∆+( ) + −( ) = +ϕ ϕj jx y

_ _ .1 1 (47)

It is now possible to eliminate ∆ using either equation (26) for a generator or equation (29) for an accelera-tor. In both cases, one ends up with an equation for a line in the jx, jy plane since ∆ is linearly related to jxand jy. The resulting operating lines are illustrated in figure 5.

In the generator case, the operating line passes through the short- and open-circuit load conditions,as determined by equations (43) and (44), respectively. In the accelerator case, the operating line originatesfrom the generator short-circuit loading point on the zero-power half-circle. This line also projects througha point on the half-circle defined by the value of ϕ.

The P circles for accelerator operation are obtained from equation (40) using equation (29) toeliminate ∆. Note that the accelerator operates in the dissipative braking regime between open-circuit andzero-efficiency conditions. Thus, positive push work is obtained only when the power density exceeds thatassociated with the zero-efficiency condition where jy=ηa=∆n=0 so that

P j Ea x xn

_ _ _ .η ϕ ϕ

= = =++

=+( )0

2

21 1∆Ω Ω

(48)

Portions of the constant-P circles lying within the positive push work regime are illustrated infigure 5. Inspection of equation (40) reveals that the radii of these circles are larger than those that wouldoccur for an ideal device in which there is no parasitic voltage drop (∆=0). Thus, additional power must beapplied in a real device to achieve a desired intensity of acceleration.

It is important to note that the optimum accelerator efficiency occurs with a finite axial current.Increasing the accelerator power density until jx vanishes will inevitably reduce the electrical efficiencybelow the optimal value. In fact, to accelerate plasma efficiently (with little heat production), it is desirableto maintain gentle acceleration levels by keeping the back emf only slightly less than the applied electricfield. In practice, the added length and weight associated with gentle acceleration must be traded againstthe electrical inefficiencies encountered with high push power.

For a generator, the maximum power condition can differ considerably from the maximum effi-ciency condition. Furthermore, maximum generator efficiency operation also occurs with a finite axialcurrent as observed in figure 4. It should be pointed out that achieving optimal power generation for agiven value of Hall parameter depends on careful selection of the wall angle. This is examined for the sakeof completeness.

By definition, the dimensionless power density may be expressed in the form

PP

u B

IE A

u BI j jx

x x y x_ _ _ _ _ _

.= = = = +

Σ Σ2 2 2 2 E Eϕ (49)

17

If we now use equation (26) to eliminate I in the appropriate expressions for jx, jy, and Ex, we obtain thedimensionless generator power density in terms of ∆

Ps o

s o

o

s o s

_ .=

−( ) +( )+( ) +( )

−−

−−

−++

Ω ∆

Ω

∆ ∆∆ ∆

∆ ∆∆ ∆

∆∆

ϕ

ϕ

2 2

2 2

1

1 1

11

(50)

The maximum power density is defined by the constraint ∂P /∂∆ = 0, which yields the relation

∆∆ ∆

max .=+o s2

(51)

Substituting this result into equation (50) then yields

P s o_

max .=− −( )

+( ) +( )+( ) +( )Ω

Ω

∆ ∆ϕ

ϕ

2

2 21 1

1 1

4(52)

Thus, Pmax depends on ϕ, and the optimal wall angle for a given Hall parameter is determined by theconstraint ∂Pmax/∂ϕ = 0, which leads to the design criterion,

Ωϕ = −1 . (53)

If this criterion is enforced, we find the maximum possible generator power density to be

P s o_

max – .Ω∆ ∆

ϕ=( ) = −+( ) +( )

11 1

4(54)

18

6. ELECTRODE BOUNDARY LAYER EFFECTS

The effective voltage drop arises from cold-wall boundary layer effects including gasdynamic varia-tions, discharge constriction, and electrode falls. Because fluid temperature and velocity vary rapidlyapproaching the wall, the conductivity and induced voltage also exhibit strong variations in the thermaland momentum boundary layers, respectively. Furthermore, the rapid decrease in temperature near theelectrode surface leads to breakdown and a sudden switch from diffuse current transport to constrictedarcs. Current attachment to cold electrodes almost always occurs through a thin layer of short arcs.

The basic boundary layer features are depicted in figure 6 for an accelerator configuration. Forsimplicity, the thermal and momentum boundary layers are both assumed to have a thickness, δ. Inaddition, current constriction is indicated at the electrode surface where attachment occurs through shortarcs spanning the anode thickness, δa, and the cathode thickness, δc. Note that the illustration shows aconcentration of current due to the Hall effect at the upstream end of the cathode and the downstream endof the anode in accordance with experimental observations for accelerators.19,20

The effective electrode voltage drop, Vd, is defined as the potential difference by which the voltagebetween opposite electrodes is increased (due to boundary layer effects) with respect to the fully homoge-neous case. The increase in Vd is quantified through an integration of the transverse electric field defectacross the electrode boundary layers. For convenience, Vd is split into separate components,

V V V Vd a c g= + + , (55)

where the anode voltage fall, Va; the cathode voltage fall, Vc; and the gasdynamic voltage drop, Vg, aredefined by

V E E dy

V E E dy

V E E dy E E dy

a y y

c y yh

h

g y y y yh

h

a

c

a

c

= −( )= −( )= −( ) + −( )

∫

∫

∫ ∫

00

0

0 0

δ

δ

δδ

δδ

–

–

– .

(56)

Note that the zero subscript refers to the homogeneous core flow conditions.

Factors contributing to the anode and cathode voltage falls include the arc attachment region andthe potential associated with electron emission from cold electrode material. In general, the voltage drop inthe arc columns is negligible since the arc length is small compared to the boundary layer thickness, andthe anode electrode fall is negligible since it does not have to emit electrons. Thus, Va≈ 0. The cathode doesemit electrons, however, and the voltage drop associated with this process can only be approximated in anempirical fashion. For most materials, Vc is approximately 10 to 20 V.

19

The gasdynamic voltage drop may be evaluated by applying the generalized Ohm’s law in theboundary layer to eliminate Ey . In this particular case, the appropriate form for Ey may be obtained fromequation (9) by neglecting ∆:

E j E uBy y x=+

− +

1 2ΩΣ

Ω . (57)

Assuming Ω is invariant, Vg takes the form

Vj

uBj

u B dy

juB

ju B dy

gy y

y y

h

h

a=

++

−+

+

++

+

−+

+

∫ Σ Ω Σ Ω

Σ Ω Σ Ω

1 1

1 1

02

0 02 0

02

0 02 0

δδ

δδ

–

– cc∫ .

(58)

And, it is convenient to split Vg into two parts

Vj

dyy

aΣ Σ

ΣΣ

=′

′′

−

∫

21

0

0δδ (59)

and

V u Bu

udyu

c= −

∫2 10

0δδ

, (60)

where VΣ and Vu account separately for the decrease of Σ and induced voltage in the boundary layer, andΣ′=Σ /(1+Ω0

2). These integrals may be further simplified by defining ξ = y/δ and assuming δa = δc to yield

Vj

dy

cΣ Σ

ΣΣ

=′

′′

−

∫

21

0

01δξ

ξ(61)

and

V u Bu

udu

c= −

∫2 10

0

1δ ξ

ξ . (62)

The lower limit of integration ξc= δc/δ is the distance from the electrode surface where the currentbecomes constricted.

20

Evaluation of these integrals is facilitated by the introduction of model boundary layer profiles forvelocity and temperature.21 For example, the velocity profile in a turbulent boundary layer is well approxi-mated by the power law u/u0=(y/δ)1/n, where n≈7. If this relation is substituted into the integral for Vu andξc=0 is used as the lower limit of integration (since u goes to zero at the wall), we obtain

V u B nu ≈ +( )2 10 δ . (63)

Evaluation of the integral for VΣ is complicated by the fact that Σ′0 /Σ′ increases steeply near thecold electrode surface, which makes the integral extremely sensitive to the lower limit of integration. It isassumed that electrical breakdown leading to arc attachment occurs when jy /Σ′ exceeds the critical electricfield strength, Ec . In this case, the limit, ξc, corresponds to a critical conductivity ratio, ωc, where

ωcc y cj E

= ′′

=′

ΣΣ Σ0 0

. (64)

Experience shows that very near the wall the thermal boundary layer may be approximated by thepower law T/T0 =(y/δ )1/m, where m =18–25. Furthermore, it has also been established that the electricalconductivity can be approximated by the simple power law Σ′/Σ′0 =(T/T0)a(p/p0)b, where a =10–20 andb=–1/2. Thus, for negligible pressure variations, the integral for VΣ can be expressed in the simplified form

Vj

dy a m

cΣ Σ

=′

−[ ]∫2

10

1δξ ξ

ξ– . (65)

The calculation procedure is as follows. For a given value of jy /Σ′0 and a specified Ec, it is possibleto compute ωc, which may be used in turn to compute Tc/T0 =ωc

1/a. Then, the lower limit of integrationmay be determined from the relation ξc =(Tc /T0)m and the integral of equation (65) may be evaluated. Forillustrative purposes, calculated values for ξc and VΣ /2δ are shown in figure 7 as a function of jy /Σ′0,assuming Ec=40 kV/cm, n =7, m =18, and a =10. Note that VΣ∝jy in accordance with empirical observations.

21

7. PRACTICAL DESIGN CONSTRAINTS

In a real device, practical design constraints limit the available operating range. The axial electricfield, for instance, must be limited due to breakdown considerations. If the axial electric field becomes toolarge, inter-electrode arcing can occur, and the resulting shorts can degrade device performance. Further-more, high heat dissipation in these concentrated arc discharges can cause severe erosion of wall materialand dramatic reduction in channel operating life.

Experience has shown that a realistic upper limit for the axial electric field is no more than100 V/cm. The existence of such a design constraint is illustrated by the dashed lines marked ± Ex,lim on theperformance map of figure 5. This implies that DCW accelerators operating with large Ω are confined tolow power-density operation. High power densities are attainable with low Ω at the expense of decreasedefficiency.

An additional design constraint arises from the observation that excessive heat generation andmaterial erosion will occur if the current density entering the electrode surface becomes too large. Theactual limiting value depends on electrode material and geometry, but experience indicates good channeldurability for values up to about 10 A /cm2. For high power-density applications having short lifetimerequirements, this value can be exceeded considerably.

22

8. CONCLUSIONS

The classical cross-plane averaged performance theory for DCW MHD accelerators has beenextended to include a current-dependent sheath layer voltage drop. This approach yields analytical perfor-mance relationships and diagrams that can be used to illuminate the rudimentary behavior of these devicesand underscore the fundamental interplay of basic parameters. This simplified theoretical treatment is notintended for detailed performance prediction of practical devices but rather to aid in developing an intui-tive understanding of device operation of general value to analysts, designers, and experimentalists, alike.Despite its limitations, the theory can be extremely useful in defining anticipated performance rangeswhile accounting for critical nonideal effects.

23

REFERENCES

1. Litchford, R.J.; Cole, J.W.; Lineberry, J.T.; Chapman, J.N.; Schmidt, H.J.; and Lineberry, C.W.: “Mag-netohydrodynamic Augmented Propulsion Experiment: I. Performance Analysis and Design,” AIAAPaper No. 2002–2184, 33rd AIAA Plasmadynamics and Lasers Conference, Maui, HI. May 20–23,2002.

2. Powers, W.L.; Dicks, J.B.; and Snyder, W.T.: “A Graphical Presentation of MHD Accelerator andGenerator Performance Characteristics,” AIAA J., Vol. 5, No. 12, pp. 2232–2236, December 1967.

3. Wu, Y.C.L.: “Performance Theory of Diagonal Conducting Wall MHD Generators,” AIAA J., Vol. 14,No. 10, pp. 1362–1368, October 1976.

4. Buende, R.; Muntenbruch, H.; Raeder, J.; Volk, R.; and Zaukl, G.: MHD Power Generation: SelectedProblems of Combustion MHD Generators, J. Raeder (ed.), Springer-Verlag, Berlin, Germany, 1975.

5. Bityurin, V.A.; Zatelepin, V.N.; and Lyubimov, G.A.: “A Consideration of Some Three-DimensionalEffects in MHD Channels,” Proceedings of 16th Symposium on Engineering Aspects of Magnetohy-drodynamics (SEAM), Pittsburgh, PA, May 1977.

6. Ahluwalia, R.K.; Vanka, S.P.; Im, K.H.; and Zwick, S.A.: “Formulation and Assessment of a Cross-Plane Electrical Model for Magnetohydrodynamic Channels,” J. Energy, Vol. 6, No. 5, pp. 314–322,October 1982.

7. De Montardy, A.: “MHD Generator with Series-Connected Electrodes,” Proceedings of the Interna-tional Symposium on MHD Electrical Power Generation, Newcastle upon Tyne, England, PaperNo. 19, September 1962.

8. Dicks, J.B.: “Design and Operation of Open Cycle Hall Current Neutralized MHD Accelerators andGenerators with Diagonal Conducting Strip Walls,” Proceedings of 5th Symposium on EngineeringAspects of Magnetohydrodynamics (SEAM), Massachusetts Institute of Technology, Cambridge, MA,April 1964.

9. Dicks, J.B.: “Improvements in Design of MHD Accelerator Channels for Aerodynamic Purposes,” inArc Heaters and MHD Accelerators for Aerodynamic Purposes, AGARDograph 84: SupplementalVolume, Proceedings of AGARD Specialists Meeting, North Atlantic Treaty Organization, AdvisoryGroup for Aeronautical Research and Development, pp. 127–174, September 1964.

10. Schlueter, A.: “Dynamik des Plasmas I,” Z. Naturforschung, Vol. 5a, p. 72, 1950; Vol. 6a, pp. 73–78,1951.

24

11. Cowling, T.G.: Chapter 6 in Magnetohydrodynamics, Interscience Publishers, Inc., New York, NY,p. 107, 1957.

12. Sherman, A.: “The Generalized Ohm’s Law for a Partially Ionized Gas,” Internal Report DF59FPD–807, General Electric Company, October 1959.

13. Polovin, R.V.; and Cherkasova, K.P.: “Magnetohydrodynamic Description of a Plasma,” Soviet Phys-ics—Technical Physics, Vol. 7, No. 6, pp. 475–479, December 1962.

14. Liubimov, G.A.: “On the Form of Ohm’s Law in Magnetohydrodynamics,” Soviet Mathematics:Applied Mathematics and Mechanics, Vol. 25, No. 4, pp. 913–929, February 1962.

15. Rosa, R.J.: “Hall and Ion Slip Effects in a Nonuniform Gas,” Phys. Fluids, Vol. 5, No. 9, pp. 1081–1090, September 1962.

16. Wu, Y.C.L.; Dicks, J.B.; Crawford, L.W.; Muchlhauser, J.W.; Scott, M.A.; and Snood, N.: “Theoreti-cal and Experimental Studies of MHD Power Generation With Char,” Proceedings of 12th Symposiumon Engineering Aspects of Magnetohydrodynamics (SEAM), Argonne National Laboratory, Argonne,IL, pp. II.1.1–II.1.14, March 1972.

17. Wu, Y.C.L.; Rajogopal, G.; and Dicks, J.B.: “Investigation of Open-Cycle MHD Power Generation,”AIAA Paper No. 74–175, 12th American Institute of Aeronautics and Astronautics AerospaceSciences Meeting, Washington, D.C., January 1974.

18. Shanklin, R.V., III; and Nimmo, R.A.: “Report on the Status and Results of the KIVA–I Open CycleMHD Generator System,” Proceedings of 14th Symposium on Engineering Aspects of Magnetohydro-dynamics (SEAM), University of Tennessee Space Institute, Tullahoma, TN, pp. I.8.1–I.8.5,April 1974.

19. Demetriades, S.T.; and Lenn, P.D.: “Electrical Discharge Across a Supersonic Jet of Plasma in Trans-verse Magnetic Field,” AIAA J., Vol. 1, pp. 234–236, 1963.

20. Denison, M.R.; and Zeimer, R.W.: “Investigation of the Phenomena in Cross-Field Plasma Accelera-tors,” in Physico-Chemical Diagnostics of Plasmas, Proceedings of the Fifth Biennial Gas DynamicsSymposium, T.P. Anderson, R.W. Springer, and R.C. Warder (eds.), NorthwesternUniversity Press, Evanston, IL, pp. 201–232, 1964.

21. Schlichting, H.: Boundary Layer Theory, Sixth Edition, McGraw-Hill, New York, NY, 1968.

25

Table 3. Reference conditions for variable . ∆

Mode ′∆ ′′∆ ′I ′′I

Generator ∆o ∆s Io = 0 Is

Accelerator ∆ ∆= ∆ n= 0 Iz Inz s

Table 2. Summary of mean power relationships for diagonalized cross-field MHD devices.*

Power Parameter In terms of applied electric field, Ex In terms of applied electric current, I

P = ⋅j E Σ

Ω ∆Ω

EuB E

xx2

2

2

1 1

1

+( ) + −( ) +( )+

ϕ ϕ I

A

uB A I2

2

2

2

1

11

1

1ΣΩ Ω Σ ∆+( ) + −( ) +( )

+

ϕϕ

Pp x= ⋅ ×( )u j B Σ

Ω ∆Ω

E uBuB E

xx+( ) − +( )

+

ϕ 1

1 2

uBI

A

uB A IΩ Σ ∆+( ) − +( )+

ϕϕ

1

1 2

ηηa

x

g

=⋅ ×( )

⋅=

u j B

j E1 uB

E

uB E

uB Ex

x

x

Ω ∆Ω ∆

+( ) − +( )+( ) + −( ) +( )

ϕϕ ϕ

1

1 12

Σ Ω Σ ∆Ω Ω Σ ∆

uB

I A

uB A I

uB A I

+( ) − +( )+( ) + −( ) +( )

ϕϕ

12

*The sign of ϕ and ∆ depends on the mode of operation. Generator mode: ϕ <0 and accelerator mode: ϕ>0 and ∆≥0.

1≤∆≤0;

Table 1. Summary of electrical parameter relationships for diagonalized cross-field MHD devices.*

Electrical Parameter In terms of applied electric field, Ex In terms of applied electric current, I

jx1 1

1 2

−( ) + +( )+

Ω Σ ΩΣ ∆Ω

ϕ E uBx

1 2

−( ) + +( )+

Ω Σ ∆ϕ ϕϕ

I A uB

jyΩ Σ Σ ∆

Ω+( ) − +( )

+ϕ E uBx 1

1 2

Ω Σ ∆+( ) − +( )+

ϕϕ

I A uB 1

1 2

Ex Ex1

2

2

+( ) + −( ) +( )+( )

Ω Ω Σ ∆Σ

I A uBϕϕ

Ey ϕEx ϕEx

I1

2

2

+( ) + −( ) +( )+

ϕ ϕΣ Ω Σ ∆Ω

E A uB Ax I

*The sign of ϕ and ∆ depends on the mode of operation. Generator mode: ϕ <0 and accelerator mode: ϕ>0 and ∆≥0.

11

1≤∆≤0;

1 1

1 1

26

(a) L

inea

r Hal

l Acc

eler

ator

(b) S

egm

ente

d Fa

rada

y Ac

cele

rato

r

(c) S

erie

s Co

nnec

ted

Diag

onal

Acc

eler

ator

(d) D

CW A

ccel

erat

or

I

uB

j y

j x

appl

ied

volta

ge

I

I

I

appl

ied

volta

ge

u B

j y

appl

ied

volta

ge

u

uxB

BB

j yj y

j x

j x

appl

ied

volta

ge

θθ

θ w

Fig

ure

1. C

ompo

site

gra

phic

of

alte

rnat

ive

cros

s-fi

eld

MH

D a

ccel

erat

or c

onfi

gura

tions

: (a)

Lin

ear

Hal

l acc

eler

ator

,(b

) se

gmen

ted

Fara

day

acce

lera

tor ,

(c)

seri

es c

onne

cted

dia

gona

l acc

eler

ator

, and

(d)

DC

W a

ccel

erat

or.

27

Diag

onal

Lin

k

θθ

δ δ

θw

= /

2–π

B

E

u

u×B

Figu

re 2

. O

rien

tatio

n of

fie

ld v

ecto

rs in

a d

iago

nally

con

nect

ed c

ross

-fie

ld a

ccel

erat

or.

28

(a)

Open

-Circ

uit G

ener

ator

(b)

j x-Ne

utra

lized

Gen

erat

or(c

) Sh

ort-C

ircui

t Gen

erat

or

(d)

Open

-Circ

uit A

ccel

erat

or(e

) Ze

ro-E

ffici

ency

Acc

eler

ator

(f) j

x-Ne

utra

lized

Acc

eler

ator

θ

θθ

θ

E′

E′

E′

E′

E′

ΩΣj

BB×

()

ΩΣj

BB×

()

ΩΣj

BB×

()

ΩΣj

BB×

()

E d

E d

E d

E d–E

d

E

EE

Eu×

B

u×B

u×B

u×B

=u×B

φ=

cons

tant

φ=

cons

tant

φ=

cons

tant

φ=

cons

tant

φ=

cons

tant

φ=

cons

tant

αΩ

= ta

n–1

αΩ

= ta

n–1

αΩ

= ta

n–1

αΩ

= ta

n–1

αΩ

= ta

n–1

αΩ

= ta

n–1

Σ j/Σ j/

Σ j/

E′–E

d=u

×B

Σ j/

Σ j/

Σ j/

ΩΣ

BB×

()

ΩΣ

BB×

()

j

j

Figu

re 3

. V

ecto

r di

agra

ms

for

the

cros

sed-

fiel

d di

agon

al M

HD

dev

ices

: (a)

Ope

n-ci

rcui

t gen

erat

or, (

b) j x

-neu

tral

ized

gene

rato

r, (c

) sh

ort-

circ

uit g

ener

ator

, (d)

ope

n-ci

rcui

t acc

eler

ator

, (e)

zer

o-ef

fici

ency

acc

eler

ator

, and

(f)

j x-n

eutr

aliz

ed a

ccel

erat

or. E

ach

diag

ram

dep

icts

the

rela

tive

orie

ntat

ion

of th

e cu

rren

t den

sity

vec

tor

and

elec

tric

fie

ld v

ecto

rs b

ased

on

the

gene

raliz

ed O

hm’ s

law

, j

jB

EΣ

ΩΣ

+×

=′

B, a

nd th

e ge

nera

lized

elec

tric

al f

ield

rel

atio

n,′=

+×

−E

Eu

BE

d.

29

1.0

0.8

0.6

0.4

0.2

–0.2

–0.4

–0.4

0.2

0.4

0.5 1.0 1.5 2.0

0.6

0.8

1.0

–0.2

–0.5–1.0–1.5

0.5 1.0 1.5 2.0–1.0–1.50

–0.5

Generator Mode Accelerator Mode( <0, < 0)∆φ ( >0, > 0)∆φ

∆∆o∆

n=0∆

jy-neutralized accelerator

jx-neutralizedaccelerator

open-circuitgenerator

short-circuitgenerator

open-circuitaccelerator

Ex

Ijxjy∆

PPpη

Ex=s z

Brak

ing

Regi

me

Brak

ing

Regi

me

Figure 4. Variation of performance parameters based on representative valuesfor flow/field conditions, material functions, and wall angle. Assumesthat ϕ =±1, B =4 T, Σ =100, Ω =2, u =3 km/s, ∆o=–0.1, and ∆s=–0.02.

30

jy

DCW AcceleratorLoad Line

jx -neutralizedaccelerator

Ex,lim

x,lim–E

short-circuithalf-circle

0.5

1

2–2

–1

–0.5

DCW GeneratorLoad Line

open-circuithalf-circle

gφ Ω φor a

tan–1 (–1/ )Ωtan–1 (–1/ )φ

jo

js Ω

φaφg

jy -neutralizedaccelerator

=constantP

IncreasingAcceleratorPower

Ex

Ey

jx

Braking Regime

Figure 5. Performance map for DCW MHD devices with current-dependent voltage drop effect.

31

Fig

ure

6. I

llust

ratio

n of

cur

rent

tran

spor

t and

bou

ndar

y la

yer

dim

ensi

ons

for

a D

CW

cro

ss-f

ield

acc

eler

ator

.

uB

u × B

j

Cath

ode

Anod

e+

–V

E

δδ c δ a

δ

h

32

10–5

10–6

10–7

10–8

10–9

10–1

0

10–1

1

10–1

2

025

5075

1000102040506070809010

0

110

120

130

140

30

δΣV 2

Σj y

/ 0′

ξ c

ξ c

E c =

40

kV/c

m

V /2

δΣ

(V/c

m)

(V/c

m)

Figu

re 7

. V

aria

tion

of ξ

c an

d V

Σ/2δ

as

a fu

nctio

n of

j y /Σ

′ 0. A

ssum

es E

c=40

kV

/cm

, n=

7, m

=18

, and

a=

10.

34

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May 2004 Technical Publication

Performance Theory of Diagonal Conducting Wall Magnetohydrodynamic Accelerators

R.J. Litchford

George C. Marshall Space Flight CenterMarshall Space Flight Center, AL 35812 M–1107

44

National Aeronautics and Space AdministrationWashington, DC 20546–0001 NASA/TP—2004–213173

Unclassified-UnlimitedSubject Category 20 Availability: NASA CASI (301) 621–0390

The theoretical performance of diagonal conducting wall crossed-field accelerators is examined on the basis of an infinite segmentation assumption using a cross-plane averaged generalized Ohm’s law for a partially ionized gas, including ion slip. The desired accelerator performance relationships are derived from the cross-plane averaged Ohm’s law by imposing appropriate configuration and loading constraints. A current-dependent effective voltage drop model is also incorporated to account for cold-wall boundary layer effects, including gasdynamic variations, discharge constriction, and electrode falls. Definition of dimensionless electric fields and current densities leads to the construction of graphical performance diagrams, which further illuminate the rudimentary behavior of crossed-field accelerator operation.

Unclassified Unclassified Unclassified Unlimited

Prepared by the Advanced Space Transportation Program, Space Transportation Directorate

magnetohydrodynamics, accelerator, electric propulsion