MODELING OF MHD POWER GENERATION ON BOARD REENTRY VEHICLES

Sergey O. Macheret,* Mikhail N. Shneider,**

Princeton University, Department of Mechanical and Aerospace Engineering macheret@princeton.edu

and Graham V. Candler#

University of Minnesota, Department of Aerospace Engineering and Mechanics

Abstract

The possibility of MHD power generation on board of reentry vehicles is theoretically explored. The focus is on the ability to extract large amounts of electric power, at least several hundred kilowatts from square meter of the surface. Estimates of electrical conductivity in unseeded and alkali-seeded air and MHD performance, taking into account Hall and ion slip effects, are followed by the development of a coupled model of nonequilibrium hypersonic flow with MHD effects and power extraction. Simulations indicate that, with modest (~1%) amount of alkali seed and modest values of magnetic field (0.1-0.2 Tesla), it should be possible to extract substantial amounts electric power, from several hundred kilowatts to a few megawatts per square meter of the surface, from a 3-5 cm thick boundary layer at altitudes of 45-60 km and flight velocities of 6-7 km/s. One possible use of the extracted power would be to heat the flow upstream of the vehicle in order to reduce the drag. In the non-optimized example case, the drag was reduced and the L/D ratio increased by 15%, and the reduction in drag power was 41.5 times the power extracted from the boundary layer and added upstream of the nose. Further exploration of this use of MHD-generated power could potentially result in significant enhancement of L/D of reentry vehicles. Additionally, controlled off-axis power deposition can be expected to result in considerable aerodynamic lifting or turning moments.

1. Introduction In atmospheric reentry of vehicles with velocities of 5-8 km/s, temperature in the shock layer reaches as high as 10,000 – 20,000 K. Air in the shock layer is ionized, with the electrical conductivity at or near the stagnation point reaching several hundred mho/m. This ionized airflow can interact with magnetic fields. It has been suggested that magnetic field at the vehicle’s nose can change the position of the bow shock and reduce the heat flux at the surface.1-14 Early studies also recognized that Hall effect, resulting from low gas density, can significantly reduce the expected performance of magnetic control of bow shocks.4,12 In principle, the ionization in shock layer or in the boundary layer near the vehicle surface can also be used for MHD power generation. Most recently, MHD power harvesting from the boundary layer downstream of the nose region was conceptually considered, and a multipole magnetic field configuration suggested for this purpose, by V.A. Bityurin, A.N. Bocharov, and J.T. Lineberry.14 In the present work, we explore the possibilities of MHD power generation on board of reentry vehicles. The focus is on the ability to generate large amounts of electric power, at least several hundred kilowatts per square meter of the surface. Estimates of electrical conductivity in unseeded and potassium-seeded hypersonic boundary layers and an analysis of Hall and ion slip effects are followed by the development of * Senior Research Scientist, Associate Fellow AIAA ** Research Staff Member, Senior Member AIAA # Professor, Associate Fellow AIAA

1

42nd AIAA Aerospace Sciences Meeting and Exhibit5 - 8 January 2004, Reno, Nevada

AIAA 2004-1024

Copyright © 2004 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

a coupled model of nonequilibrium hypersonic flow with MHD effects and power extraction. If successful, the MHD power harvesting during reentry could allow the operation of power-intensive on-board devices and to possibly use the generated power for aerodynamic control. The aerodynamic control can be accomplished by, e.g., deploying plasma upstream of the vehicle to reduce drag. In principle, off-axis plasma heating of the flow or the use of asymmetric magnetic fields would result in asymmetric aerodynamic pressure distribution and, therefore, in lifting and turning forces. 2. Preliminary analysis and estimates To compute the flowfield around reentry blunt bodies, we used a thermo-chemical nonequilibrium flow code with 11-species air chemistry and two-temperature internal energy model.15 The fast and flexible parallel implicit multi-block solver15 allowed us to perform 2-D and axisymmetric calculations. A number of computations were done for various shapes and sizes of blunt bodies, altitudes, and velocities. As an example, Figure 1 shows the predicted static temperature and electrical conductivity around a blunt body flying at an altitude of 60 km with a speed of 7 km/s. From the computations, it became clear that at high altitudes, the vibrational and electron temperatures near the nose stagnation region are lower than the gas temperature. Therefore, the electron density and conductivity in the shock layer are considerably below their equilibrium values. In the boundary layer, viscous heating combined with electron convection from the stagnation region result in an electrical conductivity on the order of 10 mho/m near the vehicle surface downstream of the nose. Thus, it would be interesting to explore MHD power extraction from the flow downstream of the nose, close to the surface (within perhaps 3-10 cm). Since the conductivity on the order of 10 mho/m would not allow the generation of megawatt-level power from a square meter of the surface (see the estimates below), to increase the conductivity to several hundred mho/m, seeding the flow with easily ionized species such as alkali metals would be warranted. Additionally, to reduce the negative influence of Hall and ion slip effects (see the estimates below), lower flight altitudes (perhaps 40-45 km) where electron and ion collision rates are higher would be desirable for MHD power generation.

30

20

4050

10

x (m)0 0.1 0.2 0.3 0.4

1000

2000 5000

4000

x (m)0 0.1 0.2 0.3 0.4

σ (mho/m)

T (K)

Fig. 1. Predicted static temperature and electrical conductivity around a blunt body flying at an altitude of 60 km with a speed of 7 km/s. Prior to developing a fully coupled model of nonequilibrium hypersonic flow with MHD effects, estimates were performed of the electrical conductivity and MHD power generation, taking into account Hall and ion slip effects, for typical high-altitude flight conditions.

2

The scalar electrical conductivity is:16

( )2

e

en ei

e nm

σν ν

=+

, (1)

where e and m are the electron charge and mass, the electron-neutral collision frequency enν is proportional to the number density n of neutrals, and the electron-ion Coulomb collision frequency is proportional to the number density of ions, equal to the electron number density ne.

When the ionization fraction is low, 310 10enn

2− −≤ − , the scalar conductivity is proportional to the

ionization fraction: 52.7 10 mho/men

nσ ≈ × (2)

It is in this regime that the conductivity of unseeded air in the boundary layer at T=4,000 – 5,000 K reaches ~10 mho/m. At ionization fractions of about 10-2 or higher, Coulomb collisions dominate over electron-neutral collisions, and the scalar conductivity saturates and does not depend explicitly on the ionization fraction:16

3/ 2

lneTconstσ = ×

Λ (3)

where Te is the electron temperature, and ln Λ is the Coulomb logarithm. It is this regime that will exist in the boundary layer Te=T=4,000 – 5,000 K seeded with ~1% of alkali vapor. Calculations show that at this temperature alkali atoms will be almost fully ionized, the ionization fraction on the order 10-2 will be reached, and the scalar conductivity of about 600-1000 mho/m will be achieved. The maximum power that can be generated per unit volume with a Faraday MHD device is:17

2 2max

14 effP uσ= B (4)

where u is the flow velocity, B is the magnetic field induction, and effσ is the effective conductivity, different from the scalar conductivity due to Hall and ion slip effects. The specific expressions for effσ are different for continuous-electrode Faraday generators, where the Hall current is allowed to flow, and for ideal segmented-electrode Faraday generators, where there is no Hall current:17

2

for ideal Faraday generators

= for continuous-electrode Faraday generators1+

effσ σ

σ

=

Ω

(5)

Here σ and are the scalar conductivity and Hall parameter corrected for ion slip effects:Ω 17

, 1 1

e

e i e i

σσ Ω= Ω =

+ Ω Ω + Ω Ω (6)

eΩ and are the electron and ion Hall parameters: iΩ

( ), e i

en ei in

eB eBm Mν ν ν

Ω = Ω =+

, (7)

where M is the ion mass, and inν is the ion-neutral collision frequency. Inserting Eqs.(5) and (6) into Eq. (4), we obtain:

( )

2 2

max

2 222

1 for ideal Faraday generators4 1

11 = for continuous-electrode Faraday generators4 1

e i

e i

e e i

u BP

u B

σ

σ

=+ Ω Ω

+ Ω Ω

Ω + + Ω Ω

(8)

As can be seen from Eq. (8), several regimes can exist depending on the values of B and the electron and ion collision frequencies.

3

Regime 1. Weak Hall effect: . The values of extracted power in both ideal and

continuous-electrode Faraday generators are then close to each other and scale as B

e1, 1e iΩ Ω Ω

2: 2 2max

14

P uσ= B .

Regime 2. Strong Hall effect, weak ion slip effect: e1, 1e iΩ Ω Ω ≤ . While ideal Faraday generators

perform as before, 2 2max

14

P uσ= B , the performance of continuous-electrode generators reaches a plateau

and is virtually independent on B:

( )2

22 2 2 2 2max 2 2

1 1 1 14 4 4en ei

e

mP u B ue

σ ν ν σ≈ = +Ω

u Bσ (9)

Regime 3. Strong Hall and ion slip effects: e1, 1e iΩ Ω Ω . The performance of ideal Faraday generators reaches saturation:

( )2 2 2max 2

1 1 14 4 en ei in

e i

mMP u Be

uσ ν ν ν σ≈ = +Ω Ω

(10)

while the performance of continuous-electrode generators increases slowly with B: ( )2 2 2 2

max1 14 4

en eii

e in

mP u B

Mν ν

σν+Ω

≈ =Ω

u Bσ

∼

(11)

Estimates for the viscous boundary layer region with T=3,000-5,000 K, at flight altitudes of 46-60 km, with ~1-10% alkali seed, show that both Hall and ion slip effects are quite significant at B=1 Tesla. Indeed, at h=60 km, ; at h=46 km, 10, 10e e iΩ Ω Ω∼ 2 3, 1e e iΩ ≈ − Ω Ω .

0 2 4 6 8 10 12 14101

102

103

104

Pmax,cont

Pmax,segm

P max

,seg

m, P

max

,con

t, M

W/m

3

B, T

0.0 0.2 0.4 0.6 0.80

100

200

300

400

500

Pmax,cont

Pmax,segm

B, T

Fig. 2a. Maximum extracted power per unit volume of the boundary layer with ideal segmented-electrode ( ) and continuous-electrode Faraday generators ( ) versus magnetic field. Flight altitude is h=46 km (150 kft), flight velocity is 7 km/s; the boundary layer has a peak static temperature of T=5000 K and is seeded with 1% potassium, resulting in the scalar conductivity peaking at about 500 mho/m; the flow velocity at the point of peak temperature is about u=3500 m/s. Electron temperature is assumed to be close to the gas temperature, T T .

max,segmP max,contP

e =

4

0 2 4 6 8 10 12 1410-5

10-4

10-3

10-2

10-1

100

101

102

Ωe>>1; ΩeΩi>1Ωe>>1; ΩeΩi<1Ωe<<1

Ωe/(1+ΩeΩi)

ΩeΩi

ΩeΩ

e, Ω

e/(1+

ΩeΩ

i), Ω

eΩi

B, T

Fig. 2b. Hall and ion slip parameters versus magnetic field. Flight altitude is h=46 km (150 kft), flight velocity is 7 km/s; the boundary layer has a peak static temperature of T=5000 K and is seeded with 1% potassium, resulting in the scalar conductivity peaking at about 500 mho/m; the flow velocity at the point of peak temperature is about u=3500 m/s. Electron temperature is assumed to be close to the gas temperature, T T . e = The maximum extracted power per unit volume predicted by Eq. (8) is plotted in Fig. 2a for a typical case: flight at an altitude of h=46 km (150 kft), at 7 km/s; the boundary layer has a peak static temperature of T=5000 K and is seeded with 1% potassium, resulting in the scalar conductivity peaking at about 500 mho/m; the flow velocity at the point of peak temperature is about u=3500 m/s. Electron temperature is assumed to be close to the gas temperature, Te T= . The three regimes are clearly seen in Fig. 2a and in the accompanying Fig. 2b. At B=0.2 Tesla, regime 1 is realized, and the maximum extracted power with alkali-seeded air in the viscous boundary layer at a flight velocity of 7 km/s is about 50 MW/m3. With a 2-3 cm thick hot ionized region near the surface, this translates into 1.0-1.5 MW per square meter of the vehicle surface. 3. Computational model To develop a fully coupled model, we added the relevant equations to the thermo-chemical nonequilibrium two-temperature model. The model consists in the following. 3.1. Continuity and momentum (Navier-Stokes) equations The governing equations (in divergence form) are:

( )ss sj s sj s

j

u vt x

ρ ρ ρ∂ ∂+ +

∂ ∂ϖ= (12)

( ) ( )ii j ij ij i j k k j

j

u u u p j B j B j Bt x

ρ ρ δ τ∂ ∂+ + + = × = −

∂ ∂ (13)

where ρ is density, u is a velocity component, x is a os tion component, p is pressure, δ is the Kronecker delta function, τ is the viscous stress tensor, and the

p ij B× term adds a body force to the standard18 Navier-

Stokes momentum equation.

5

At low density expected during reentry, the ion slip effect has to be taken into account. The ion slip results in the effective velocity of charged species (ions and electrons) in the direction normal to the B field being lower than the bulk gas velocity:17, 19, 20

, 1 (1 )e ie i

uuk

=+ − Ω Ω

(14)

where Ωe is electron Hall parameter, Ωi is the ion Hall parameter, and k is the load factor. 3.2. Energy equations The equations for the heavy particles (molecules and ions) have to be modified in the presence of MHD effects:

2 32

1( ) / ( )

sn

j ij j j s sj s VT e e en m rsj

e e p u u q v h Q j E j n T Tt x

τ ρ σ ν δ +=

∂ ∂+ + + + + = + ⋅ − + − ∂ ∂

∑ (15)

(e+p) are the total energy (e refers to energy, not electron charge, in all these energy equations), q is the heat transfer, sjv

i

is the mass diffusion velocity of species s, ne is the electron density, and Te is electron temperature. Again, this is a standard7 energy equation with source terms added for MHD, Joule heating, and vibrational relaxation effects. is the vibrational-translational (VT) energy transfer rate;VTQ 19,20

is the MHD power extracted to the external load; 3

1i

ij E j E

=

⋅ = ∑ 2 /j σ is the power spent on heating of the

electrons;20 σ is the scalar conductivity; and 32 ( )e e en mn T T rν δ +− is the direct power transfer from electrons

to neutrals, where ~ 2 /m r m Mδ + . The energy equation for the electrons is:

( ) 25 32 2/ / ( ) (e

e e ej e e j e e en m r e ev e vj

e n T u T x j n T T n Nk T Tt x

λ σ ν δ +

∂ ∂+ − ∂ ∂ = − − −

∂ ∂)− (16)

where 5 5

2 2e e e e en D n Teλ µ= = is the electron thermal conductivity, and (e ev e vn Nk T T )− − is the rate of energy transfer from electrons to vibrational mode of molecules. The energy equation for the vibrational mode of the molecules is:

1

( )( )

snv jv

VT e ev e v vj sj vssj

e ue Q n Nk T T q v et x =

∂∂+ = − + − + +

∂ ∂ ∑ . (17)

Here, is the vibrational energy per unit volume, and the terms q and ve vj1

sn

sj vss

v e=

∑ correspond to

vibrational “heat” conduction and vibrational energy transport with mass diffusion, respectively. Note that, although the full model uses the separate equations for electron and vibrational energies, Eqs. (16) and (17), in the present paper a simplified modeling was conducted, assuming Te Tv= . In fact, the modeling showed that in the ionized boundary layer, both temperatures are close to the gas temperature, so that the gas is almost in a local thermal equilibrium. In the Lagrangian frame moving with the ionized gas, we obtain from Eqs. (15)-(17):

( )e vd e e e j Edt

+ += ⋅ . (18)

The generalized Ohm’s law, taking both Hall and ion slip effects into account, is:17,21

6

2( ) e ie

j Bj E u B j BB B

σ Ω Ω×= + × − Ω + × × B . (19)

The current continuity equation is:

0j∇ ⋅ = , (20) The electric field strength is related to the potential: E ϕ= −∇ . (21) From Eq. (19), we can find the current density components:

( , , , , , , , , )i i j k i j k i jj F E E E u u u B B B= k

.

, (22) where F are functions to be found from writing Eq. (19) in the appropriate coordinate system. From Eqs. (20)-(22), we can get an implicit form of Poisson equation for the electric potential:

( , , , , , , , ) 0i j k i j ku u u B B Bϕ ϕ ϕ∆ + Ψ ∇ = (23) where Ψ is a function to be found from Eqs. (20)-(22). 3.3. Boundary conditions and extracted power At a dielectric surface, there is no current normal to the surface:

0n j⋅ = (24) The electrodes are assumed to be continuous strips of equal length and finite width flush with the surface. Around the electrodes, the surface is dielectric. The flow is along x (see Fig. 3).

H

y z x

Fig. 3. Electrode configuration The difference in the potentials of the electrodes, V, should be an input to the calculation. The value of V should be selected so that V u . The entire surface of each electrode has a common potential. max( )x zB< An additional condition is imposed on the electric field:

/xE x 0ϕ= −∂ ∂ = (25)

7

The extracted electric power is computed as:

el

MHDA

P V j nd= ⋅∫ A , (26)

where is the normal vector to anode or cathode surface, and An el is the electrode surface area. 3.4. Conductivity, collision frequency, and collision cross-sections The electron and ion number densities are computed using the Saha equation22 for potassium seed atoms. The conductivity is then computed as e ee nσ µ= (e being the electron charge), where the electron mobility is:

0

( , )( )( ) ( )

ee e

m c

n TeTm

ε dµ εν ε ν ε

∞

=+∫ ; (27)

where ,m cν ν are transport electron-neutral and Coulomb collision frequencies, respectively, and ε is electron energy. The electron Hall parameter is computed as:

0

( , )( )( ) ( )

ee e

m c

n TeBTm

ε dεν ε ν ε

∞

Ω =+∫ . (28)

Here 3/ 2

2( , ) exp( / )ee

n T d T dT

εeε ε ε

π= − ε

0

( , ) 1en T dε ε∞

, =∫ (29)

is a Maxwellian distribution, and Te is in eV. Coulomb collision frequency is:16,22

4

2 2 30

4 ln(v) ,1/(4 ) v

e ic

n Z e sm

πνπε

Λ= (30)

where in our range of conditions, ε1iZ = o is the permittivity of free space, and ln Λ is the Coulomb logarithm. The energy-dependent Coulomb collision frequency is then:

( )( ) ( )

4

32 2 5

4 ln

4 5.93 10e

c

o

n e

m

πν επε ε

Λ≈

× (in units of 1/s) (31)

where the Coulomb logarithm and the Debye length (λD) are:

312ln ln e D

i

nZ

π λ Λ ≈

;

1/ 2

3

, [ ]5250, [ ]

eD

e

T eVn m

λ −

=

(in meters) (32)

The transport electron-neutral collision frequency is:

5,( ) v ( ) 5.93 10 ( )m i m i i

i i

N ,m iNν ε σ ε ε σ= = ×∑ ε∑ (33)

where ,m iσ are collisional cross sections. For the simplest air model, the N2 : O2 ratio = 4 : 1 , so that:

2 2

5, ,( ) 5.93 10 [0.8 ( ) 0.2 ( )]m m NN m Oν ε ε σ ε σ= × + ε /N p kT; = . (34)

8

The convenient interpolation formulas for electron-neutral scattering cross sections that we have developed based on experimental data from the literature23,24 are:

2

2 3

,

2 3

4 5

2.81968E-20+2.16541E-19 -2.15945E-19 +7.37124E-20 <1.75 eV

( )

1.2512E-17-2.75658E-17 +2.36387E-17 -9.78269E-18 + 1.75< <3 eV1.96648E-18 -1.54515E-19

m N

εε ε ε

σ εε

ε ε εε ε

=

(in m2) (35)

2

2 3

4

,

2 3

4

5.56971E-20+8.01476E-20 -2.57882E-19 +3.67888E-19 -1.51673E-19

( )-1.60979E-19+6.50015E-19 -

5.40574E-19 +1.72964E-19 -1.95107E-20

m O

εε εε

σ εε

ε εε

=

<1.15 eV

1.15< <3 eV

ε

ε

(in m2) (36)

Because Te<<3 eV, it is sufficient to do integration in formulas (27) and (28) from 0ε = to 3ε = eV. The ion Hall parameter is calculated as:

/i eB M inνΩ = (37) where the ion-neutral collision frequency is , 1/s. 5

[Pa] [K]2.58 10 p (300/T )inν = ×

4. Simplified models

4.1. Cylindrical geometry The cylindrical geometry for the model is shown schematically in Figures 4 and 5 below.

Fig. 4. Side view Fig. 5. Front view Input to the computations in this geometry:

9

[0, ( , ),0]( , , ) ( , , )

( ,0, )

( ,0, )( , )

r

x r

x

x

B B x ru u u u u v w

j j j

E E Ex r

ϕ

ϕ

ϕ

σ σ

== ≡

=

=

=

(38)

2( ) ee

j Bj E u B j BB B

σ +Ω Ω×= + × − Ω + × × B

wj

u B kuB iBw× = − ; (39) xj B kj B ij Bϕ× = −2 2

xj B B kj B ij Bϕ× × = − From Eq. (39):

( )x x e e i x

e x e i

j E j j Bj E uB j

ϕ

ϕ ϕ ϕ

σ σ

σ

= + Ω − Ω Ω −

= + − Ω − Ω Ω (40)

and ( )

( )x x

e x

j E Bw

j E uBe j

jϕ

ϕ ϕ

σ

σ

= − + Ω

= + − Ω,

Here σ and are the scalar conductivity and Hall parameter corrected for ion slip effects and determined by Eq. (6), and

Ω

2

2

[( ) ( )]1

[ ( ) ( )1

x ex

e

e x

e

E wB E uBj

E wB E uBj

ϕ

ϕϕ

σ

σ

− + Ω +=

+ Ω

−Ω − + +=

+ Ω]

. (41)

For the case with continuous electrodes, 0xE = , and, taking into account, that u w>> ,

2

2

( )1

( )1

ex

e

e

E uBj

E uBj

ϕ

ϕϕ

σ

σ

Ω +=

+ Ω

+=

+ Ω

. (42)

Determination of Eϕ is accomplished with the following procedure, illustrated in Fig.6.

Fig. 6. Geometry for determination of Eϕ

10

( , ) ( , ) ( , ) ( , )j x r x r u x r B x r Eϕ σ = ϕ + . (43) Total current:

elS

I jdS= ∫ , (44)

where is the current collecting electrode surface area. elSInserting Eq. (43) into Eq. (44):

[ ] ( )

( )el bMHDS x r x

I jdS uB E dxdr uBdxdr E dxdrϕσ σ∞

= = + = +∫ ∫ ∫ ∫∫ ∫∫ϕ σ . (45)

The Ohm’s law for the closed circuit for a given current tube: 2 2 /ruB rj IRϕπ π σ= + . (46) Substituting Eqs. (43) and (44) into Eq. (46), we obtain:

2

R uBdxdrE

r R dxdrϕ

σ

π σ= −

+∫∫

∫∫.

Since the MHD region is in the boundary layer, we have for the current tube radius: ( ) 1b

b

r rr

− << , and we

can set r . If , then br= ( )b br r x=

( )2 ( )b

R uBdxdrE x

r x R dxdrϕ

σ

π σ≈ −

+∫∫

∫∫. (47)

The resistance of the nth elementary current tube is:

,2

( , )pl nrR

x r dxdrπ

σ= .

All the current tubes are parallel to each other, therefore,

,

1 1 ( , )2npl pl n

x r dxdrR R r

σπ

= =∑ ∫∫ . (48)

The load factor is (by definition):

1pl

RkR R

=+

< . (49)

From Eq. (49), the load resistance, at known k, is 2

1kR dxdrk

r

πσ=

− ∫∫. (50)

Inserting Eq. (50) into Eq. (47), we have:

/ ( / )( )

( )(1 ) / ( / )b

k uBdxdr r dxdrE x

r x k k dxdr r dxdrϕ

σ σ

σ σ≈ −

− +∫∫ ∫∫

∫∫ ∫∫. (51)

At and ( )br x const= ( ) 1b

b

r rr

− << ,

11

uBdxdrE k

dxdrϕ

σ

σ= − ∫∫

∫∫. (52)

4.2. Wedge geometry The second model geometry is a wedge:

Fig. 7. Wedge geometry The equations for this geometry are:

[0,0, ( , )]( ,0, ) ( ,0, )( , ,0)

( , ,0)

( , )

z

x z

x y

x y

B B x zu u u u wj j j

E E E

x zσ σ

== ≡

=

=

=

(53)

u B juB× = − ; y xj B ij B jj B× = −2 2

x yj B B ij B jj B× × = − −

2

2

[ (1

[ (1

x e yx

e

e x yy

e

E E uBj

E E uBj

σ

σ

− Ω −=

+ Ω

Ω + −=

+ Ω

)]

)]. (54)

For the case with continuous electrodes, 0xE = :

2

2

( )1

( )1

e yx

e

yy

e

E uBj

E uBj

σ

σ

Ω −= −

+ Ω

−=

+ Ω

. (55)

At given B, jy and u directions, when ( )x yj B j B× = < 0 0yE kuB= > . The extracted power per unit volume is:

0y yw j E j E= ⋅ = < . The exact expression for : constyE =

12

,

,

x zy

x z

k uBdxdE

dxdz

σ

σ=

∫∫

∫∫

z. (56)

The external parameters of the MHD circuit, current I and voltage V, can be calculated easily:

( ),

1x z

I k uBdxσ= − ∫∫ dz (57)

,

,

x zy

x z

k uBdxdV E H H

dxdz

σ

σ= =

∫∫

∫∫

z (58)

5. MHD simulations for a blunted wedge: power generation and drag reduction The simulations were performed for a 12o half-angle wedge with nose bluntness radius of 10 cm and a length of 3 meters. The wall temperature was assumed to be 1000 K, and the magnetic field was assumed to decrease exponentially with the distance r from the surface: ( ) (0 exp /10 cmB r B r= − ) , where

. The electron temperature was assumed to be equal to the vibrational temperature. Both unseeded and seeded cases were computed. In the cases with potassium seed, the potassium at mass fraction of 0.1 – 1% was assumed to be injected upstream of the nose uniformly from the axis down to 3.0 cm below the axis at a distance 2 nose radii (20 cm) upstream of the nose. The Saha equation for potassium atoms was used to determine the electron density.

0 0.1, 0.2 TB =

Fig. 8 shows the temperatures in the boundary layer at x=1 m for the cases of flight at 46 km, 7 km/s, with 1% seed and B0=0.1 T. As seen in the figure, temperatures of about 5000 K and higher are reached in the 3-5 cm thick region. Fig. 9 shows the seed mass fraction and ionization fraction for the same case. As seen in Fig. 9, full ionization of the seed is reached. The corresponding electron number density is plotted in Fig. 10 and compared with that in the absence of seed. The ionization enhancement due to the seed is almost two orders of magnitude. The MHD generated power per square meter of the surface is plotted in Fig. 11 for unseeded cases. As seen in this figure, for a 24o wedge, only about 25-117 kW/m2 can be generated at h=46 km, u=6-7 km/s without seed. The dramatic effect of seeding is illustrated in Fig. 12: with 1% seed and B0=0.2 T, 1-2 MW/m2 can be generated. The dependence of generated power on the value of B for flight at 46 km, 7 km/s, with 1% seed, is shown in Fig. 13. The results, for 3 cm effective thickness of the ionized region in the boundary layer, are very close to those estimated in section 2 and plotted in Fig. 2. The chart in Fig. 14 summarizes the results on power generation in cases with seed and B0=0.2 T. Note that the simulations assumed a constant seed mass flow rate, thus, although the seed fraction is 1% at h=46 km, it is higher at higher altitudes. Overall, Fig. 14 demonstrates that power ranging from several hundred kilowatts to a few megawatts per square meter of the surface can be generated with seed injection and B0=0.2 T. It should be noted that the seed injection is highly idealized and has not been optimized in the present model. A complete analysis, including seed injection, penetration into the shock layer, and mixing needs to be conducted to determine the feasibility of MHD power extraction. For example, it may be possible to design an injection system that allows the seed to penetrate farther from the surface than shown in Fig. 9, resulting in a larger volume of ionized gas and higher power levels.

13

Distance from Surface (cm)

Tem

pera

ture

(K)

0 5 10 15 200

1000

2000

3000

4000

5000

6000 No MHDMHDHeat Addition

Fig. 8. Static temperature and electron-vibrational temperature in the boundary layer at x=1.0 m for the cases of flight at 46 km, 7 km/s, with 1% seed and B0=0.1 T.

Distance from Surface (cm)

Pot

assi

umM

ass

Frac

tion

(%)

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8No MHDMHDHeat Addition

Distance from Surface (cm)

Pot

assi

umIo

niza

tion

Frac

tion

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Fig. 9. Seed mass fraction and ionization fraction for the conditions of Fig. 8.

14

Distance from Surface (cm)

Ele

ctro

nN

umbe

rDen

sity

(cm

-3)

0 5 10 15 20109

1010

1011

1012

1013

1014

1015

1016

Total (seed + air)Air

Fig. 10. Electron number density in the boundary layer for the conditions of Fig. 8.

Wedge Angle (deg)

Pow

erEx

tract

ed(k

W)

10 15 20 2510-2

10-1

100

101

102

60 km, 6 km/s

60 km, 7 km/s

46 km, 6 km/s

46 km, 7 km/s

Fig. 11. Power extraction, in kW per 1 m2 of the surface, for various flight conditions and wedge angles. The flow is unseeded.

15

Seed Percent

Pow

erEx

tract

ed(k

W)

0 0.25 0.5 0.75 10

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

MHD; Bo = 0.1THeat Addition at 2Rn; Bo = 0.1TMHD; Bo = 0.2THeat Addition at 2Rn; Bo = 0.2THeat Addition at 4Rn; Bo = 0.1T

Seed Percent

L/D

0 0.25 0.5 0.75 11.3

1.35

1.4

1.45

1.5

1.55

MHD; Bo = 0.1THeat Addition at 2Rn; Bo = 0.1TMHD; Bo = 0.2THeat Addition at 2Rn; Bo = 0.2THeat Addition at 4Rn; Bo = 0.1T

Fig. 12. Effect of seed fraction on the generated power (in kW per 1 m2 of the surface) and on L/D for various cases of flight at 46 km, 7 km/s.

Bo, Magnetic Field (T)

Pow

erE

xtra

cted

(MW

/m)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Fig. 13. The dependence of generated power on the value of B for flight at 46 km, 7 km/s, with 1% seed.

16

1.41.31.21.1

10.9

0.80.7

0.60.5

0.4

0.3

0.2

Flight Speed (km/s)

Alti

tude

(km

)

5.5 6 6.5 740

45

50

55

60

65

70 1.51.41.31.21.11.00.90.80.70.60.50.40.30.20.10.0

MW/m

Fig. 14. MHD generated power per 1 m2 of surface at various altitudes and flight velocities. The body is a 24o blunted wedge. The magnetic field at the surface is 0.2 T. Note that the simulations assumed a constant seed mass flow rate, thus, although the seed fraction is 1% at h=46 km, it is higher at higher altitudes.

4000

5000

5000

3000

4000

3000

x (m)

y(m

)

0 0.5 1

-0.5

0

0.5

4000

4000

20003000

4000

x (m)

y(m

)

0 0.5 1

-0.5

0

0.5

Fig. 15. Static temperature contours for the case of flight at 46 km, 7 km/s, B0=0.2 T, 1% seed fraction, with MHD on and no heat addition (left plot) and with MHD on and heat addition of the entire extracted power 20 cm (2 nose radii) upstream of the nose (right plot).

17

Axial Distance (m)

Pre

ssur

e(P

a)

0 1 2102

103

104

105

No MHDMHDHeat Addition

Axial Distance (m)

Pre

ssur

e(P

a)

102

103

104

105

-0.1 0 0.1 0.2 0.3 0.4

No MHDMHDHeat Addition

Fig. 16. Surface pressure versus axial distance for the case of flight at 46 km, 7 km/s, B0=0.1 T, 1% seed fraction. One possible use of the MHD generated power might be to heat the flow upstream of the nose in order to reduce surface pressure and drag. In this work, a few simulations were performed, where the power was put into the flow in a concentrated region located at the axis at a distance of 20 cm (2 nose radii) upstream of the nose. While neither location not shape of the heat addition region was optimized, the simulations can be useful in that they may provide a general idea of the efficiency of such flow and drag control. Figs. 15 and 16 illustrate static temperature and pressure profiles with MHD power extraction, with and without the energy addition upstream of the nose. As seen in Figs. 16, upstream energy addition results in substantial decrease of the nose surface pressure. This should lead to decrease in drag. Indeed, in this case 800 kW was extracted from the boundary layer and added upstream of the nose. The reduction in drag power was 33.2 MW – about 15% of the total drag power of 220 MW. Correspondingly, the lift-to-drag ratio (L/D) was increased by 15%. Therefore, using the extracted power to reduce drag is very efficient: the ratio of drag power reduction to the power added is 41.5. If the 800 kW were used for propulsion, the effect would have not been higher than 800 kW. Note that conditions in the boundary layer and MHD-generated power depend on whether the upstream energy addition is on, as seen in Figs. 8 and 12. Thus, only coupled calculations of the flowfield with energy addition and MHD power generation would be correct, and the calculations were coupled in this work. 6. Concluding remarks Simulations performed in this work indicate that, with modest (~1%) amount of alkali seed and modest values of magnetic field (0.1-0.2 Tesla), it would be possible to extract substantial amounts electric power, from several hundred kilowatts to a few megawatts per square meter of the surface, from a 3-5 cm thick boundary layer at altitudes of 45-60 km and flight velocities of 6-7 km/s. One possible use of the extracted power would be to heat the flow upstream of the vehicle in order to reduce the drag. In the non-optimized example case, the drag was reduced and the L/D ratio increased by 15%, and the reduction in drag power was 41.5 times the power extracted from the boundary layer and added upstream of the nose. Further exploration of this use of MHD-generated power could potentially result in significant enhancement of L/D of reentry vehicles. Additionally, controlled off-axis power deposition can be expected to result in considerable aerodynamic lifting or turning moments.

18

Further development of the model should include three-dimensional computations with full coupling of the Poisson equation for electric potential; studies of electrode sheaths effects; calculations of ionization and recombination kinetics to replace the Saha equilibrium assumption; full solution of electron and vibrational energy equations; and investigation of the effects of induced magnetic field at finite magnetic Reynolds numbers. In addition, an extensive study of the seed injection process needs to be performed to establish its feasibility and to optimize the distribution of the seed material.

Acknowledgement

The work was supported by DARPA, Defense Science Office.

References

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19

20

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