click to edit master subtitle style professor, department of aerospace engineering, university of...

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Professor, Department of Aerospace Engineering, University of Pisa, Italy Chairman, Alta S.p.A, Via A. Gherardesca 5, 56121 Ospedaletto, Pisa, Italy

e-mail: [email protected]

IEPC-2005-187

Francesco Battista and Pietro Piliero

Mariano Andrenucci

Hall Thruster Scaling Methodology

mailto:[email protected]

2


29th International electric propulsion Conference - Princeton University, October 31 - November 4, 2005

Introduction

• Expansion of human activities in space will certainly pose the need to develop high power propulsion systems

• HETs will certainly play a major role in that scenario• But power range of devices developed so far relatively narrow

• limited power levels affordable in laboratory experiments• modest power levels typical of past and current space systems

• At the opposite end of the power gamut, growing case emerging for the application of HET technology to very low power levels (less than 200 W)

• This would grant access to the mini/micro-satellite market, a potentially large market already expressing a significant demand for thrusters in the performance range suitable for Mini-HETs

• Issue:How to extrapolate sizing criteria worked out for thrusters developed to date to different scales, or different operating conditions

3



Introduction

• sequel to previous paper

• mainly focussing on methodology

• more refined modelling of processes involved

4



General approach

• We are concerned with the way in which the different parameters characterizing a device of a given family will vary as a result of a change of size (defined by the value of any suitable parameter)

• This will result from the interplay of different physical processes, which will generally obey different scaling laws

• Each scaling law will consist of a simple power-law relation between the involved parameters

• Other physical magnitudes resulting from the combination of different processes will be calculated as a function of those pertaining to each of the costituent processes

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Scaling Model

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Scaling modes

7



Scaling mode algebra

8



Calculation of scaling parameters

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HET scaling Model: efficiency

• Standard description adopted

VA =VD −ΔV ηV =VA

VD

ID =I j + Ie i =Ie

ID

ηi =I j

ID

= 1− i

&m j =I j

M i

e &m = &mj + &mn =

&mj

ηm ηm =

&m j

&mtot

ηε =

ui2

v* 2 =v j

2

v* 2 =VA

VD

= 1−ΔV

VD v* =

2e

M i

VD

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• Loss factor

• Exhaust velocity from discharge voltage and loss factor

• and therefore

ηε =1 − ε L εL =ΔV

VD

= εW + ε a + ε i

vj

2 = 1−εL( )v*2 = 1−εL( )2eM i

VD =ηε2eM i

VD

T = &mjvj = &mj

2eVA

mi ve =

T&m

=&m jv j

&m j ηm=ηmvj =ηm

2eVA

mi

ηT = T 2

2 &mtot PD

= η mη j η ε

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• Other effects considered

• plume divergence

• spread of ion velocities

• Overall thrust efficiency

• usually assumed

η β ≈ cosβ 2 ≈uiz

2

ui2

η v ≈ui

2

ui2

ηT = η ε η J η m η β ηv

η β ≥0.9 η v ≥0.9η i ≈ 0.7 ÷ 0.8 η m≥ 0.98

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HET scaling Model: other factors

• ionization

• diffusion length

• wall losses

• anode and ionization losses

• lifetime, heat loads, etc. (see paper)

λi =Li

L=

uaz

σ i ue n L~

uaz

n LT −3 2

λdiff =

LAD

L:

1

L

Te

32

n B

εw ~n Te

3 2d L

n uaz d bVD

=Te

3 2

uaz VD

L

b

εa =Pa

PD

~ nTe3 2 d b

PD

=Te

3 2

uaz VD εi =

Pi

PD

=Ei

e

I j

IDVD

:η j

VD

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Scaling Exponent Matrix

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• Reference thruster: SPT-100

• Suppose we want to design a thruster with• 50 kw power• 369 mm average diameter• 88 mm channel width

• With respect to the reference thruster this meansP’/P=37.037 dm’/dm=4.341 b’/b=5.867

orln(P’/P)=3.612 ln(dm’/dm)=1.468 ln(b’/b)=1.769

• We need to use three independent transformations; we choose to use a combination of SL, L, R.

(1)Use of the model

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Use of the model

• From the general exponent matrix we extract

which means that

so that by matrix inversion we find

SL L R

P 3 1 1

dm

1 1 0

b 1 0 1

ln P '/ P( )

ln(dm'/ dm)

ln(b'/ b)

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

=3 1 11 1 01 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

lnςSL lnς L lnςR⎡⎣ ⎤⎦

lnςSL

lnς L

lnςR

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

1 −1 −1−1 2 1−1 1 2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

ln P'/ P( ) ln(dm'/ dm) ln(b'/ b)⎡⎣ ⎤⎦

(2)

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Use of the model

• We thus obtain

• so that the values of the scaling factors for the involved transformations are

• We can hence determine the equivalent overall scaling factor

and the relative weights of the component transformations

lnςSL =0.374 lnς L =1.094 lnςR1.395

ςSL =1.454 ς L =2.985 ςR =4.034

lnξ = lnς j =2.863

j∑ ⇒ ξ = ς j

j∏ =17.513

pj=

lnς j

lnς jj∑

⇒ pSL =0.131 pL =0.382 pR =0.487

(3)

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Use of the model

• We can thus generate the exponent vector for the equivalent transformation, where

and accordingly determine all other parameterssuch as

γi = α i jj

∑ p j

Isp =2398 s &m=127.145 mg/s T =2.991N ηT =0.622

VD =436 V VA =379V ID =114 A IA =92.85 A

etc.

(4)

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50 kW Thruster50 kW Thruster ---------------------------------------------------------- Scaling modes: SL,L,R ---------------------------------------------------------- Power scaling factors ---------------------------------------------------------- P(SL)= 3.0755 P(GL)= 1 P(L)= 2.9852 P(R)= 4.0342 ---------------------------------------------------------- Geometry ---------------------------------------------------------- L(mm)= 31.993 d_m(mm)= 369 b(mm)= 88 thickness_BN(mm)= 58.667 d_inner (mm)= 163.67 ---------------------------------------------------------- INPUT ---------------------------------------------------------- B_max(G)= 200 V_D(V)= 436.27 mdot (mg/s)= 127.09 ---------------------------------------------------------- Parameters ---------------------------------------------------------- lambda_i= 0.12378 lambda_diff = 0.42634 epsilon_w = 0.033813 epsilon_i = 0.068866 epsilon_anode = 0.028932 ---------------------------------------------------------- Performance ---------------------------------------------------------- V_A(V)= 378.85 v_e(m/s)= 23588 J_d(A)= 114.61 I_sp(s)= 2404.4 T(mN)= 2997.7 ETA_TOT = 0.62537 t_{life}[hr] = 23467 Th_l_{anode}(W/cm^2)= 4.4549 Th_l_{w} (W/cm^2)= 18.843

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5kW variants

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5 kW Thruster5 kW Thruster ---------------------------------------------------------- Scaling modes SL,GL ---------------------------------------------------------- Power scaling factors ---------------------------------------------------------- P(SL)= 3.1852 P(GL)= 1.1628 ---------------------------------------------------------- Geometry ---------------------------------------------------------- L(mm)= 37.639 d_m(mm)= 145.42 b(mm)= 25.663 thickness_BN(mm)= 17.109 d_inner(mm)= 85.543 ---------------------------------------------------------- INPUT ---------------------------------------------------------- B_max(G)= 172 V_D(V)= 441.4 mdot(mg/s)= 12.561 ---------------------------------------------------------- Parameters ---------------------------------------------------------- lambda_i= 0.12234 lambda_diff = 0.42139 epsilon_w = 0.13482 epsilon_i = 0.068066 epsilon_anode = 0.028596 --------------------------------------------------------- Prestazioni ---------------------------------------------------------- V_A(V)= 339.22 v_e(m/s)= 22320 J_d(A)= 11.328 I_sp(s)= 2275.2 T(mN)= 280.36 ETA_TOT = 0.55345 t_{life}[hr] = 7957.5 Th_l_{anode}(W/cm^2)= 3.8312 Th_l_{w} (W/cm^2)= 16.205

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Alta’s 5 kW HET

Operating power: nominal 5 kW, maximum efficiency 7 kW. Axially symmetric magnetic coil arrangement, three separate

windings Maximum recorded efficiency: 49% at 4.1 kW, including cathode

and magnetic circuit Thrust: 200 mN, Isp: 1850 s.

22



100 W variants

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100 W Thruster

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100 W Thruster ---------------------------------------------------------- Scaling criteria GL,SL,A ---------------------------------------------------------- Power scaling factors ---------------------------------------------------------- P(SL)= 0.024444 P(GL)= 0.87945 P(A)= 3.4457 ---------------------------------------------------------- Geometry ---------------------------------------------------------- L(mm)= 19.348 d_m(mm)= 21.695 b(mm)= 5.1114 thickness_BN(mm)= 3.3592 d_inner(mm)= 9.8652 ---------------------------------------------------------- INPUT ---------------------------------------------------------- B_max(G)= 227.41 V_D(V)= 300 mdot(mg/s)= 0.36963 ---------------------------------------------------------- Parameters ---------------------------------------------------------- lambda_i= 0.18 lambda_acc = 0.2 epsilon_w = 0.68352 epsilon_i = 0.10015 epsilon_anode = 0.042074 --------------------------------------------------------- Performance ---------------------------------------------------------- V_A(V)= 52.278 v_e(m/s)= 8762 J_d(A)= 0.33333 I_sp(s)= 893.17 T(mN)= 3.2387 ETA_TOT = 0.12549 t_{life}[hr] = 1777.2 Th_l_{anode}(W/cm^2)= 5.0655 Th_l_{w} (W/cm^2)= 21.426

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Alta’s HT-100 Mini-HET

25



Comparison with data

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Power levels > 5 kW

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Power levels > 5 kW

27






Power levels > 5 kW

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Power levels < 500 W

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Different operating points

30



Conclusions

• Despite simplicity of model adopted, fairly good approximation of the scaling behaviour of experimental devices easily obtained by careful combination of a few basic scaling modes

• Useful preliminary design tool, allows quick evaluation

of impact of design choices or operating conditions

• Possibility to improve predictive capability by more refined modelling of involved processes

• Intrinsic interest of methodology

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