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H*Optimum multiwire-to-plane EHD thrus ter design guide v2.0*LH*HcL Michel Jullian 2005, this version published 02072005 on Blazelabs Yahoo Group, 26062005 for version 1.0*L

H*Constants and parameters, all in SI units*Leo = 8.85 10^12 H permittivity of air in Fm L;k = 2.0 10^4 H ion mobility such that ion speed is kE , i n m s p e r V mL;E0 = 3.0 10^6 H air breakdown field in Vm L;Komwp = 0.49 H BorgSigmond "K" current density coefficient, dimensionless, for Optimum

MultiWire to Plane i.e. wire spacing 1d to 1.3d, Jullian & Borg 2005, cf Appendix L; H*

F is thrust in N

P is consumed power in W

A is total thruster area Hactual area times the number of stages if multistageL in m^2 rw is wire radius in m

Vmin is voltage in V for which plasma losses are 5% of total consumption

V is voltage in V

i is current in A

d is gap length in m

ww is wirewire spacing in m L;

H*Design steps

Husing Saviour'

s 100

cN payload weight 200

cN total thrust thruster as a running exampleL*L

H* 1 Choose F and P *LF = 200 10^2; P = 200;

H i.e. thrust to power ratio FP = 1cNW in this example, practical min value for multiwire toplane being 0.5cNW corresponding to an average field of 1kVmm; lower values of FP may not seemdesirable at first view, but they yield higher values for thrust per m^2 so they might be

needed, at least for short periods such as vertical takeoff L

[82]:= H* 2 Find A *LA= k ^ 2 HKomwpeoL F ^ 3 P ^ 2H* Determines required area from primary design goals F and P only, Jullian 2005 cf Appendix *L

t[82]= 1.84481

[83]:= H* 3 Choose rw and deduce Vmin *L

rw = 0.1 10^3H0.2mm diameterL; Vmin = 20E0 rwikjjjjj1 +

0.03!!!!!!!

rw

y{zzzzz LogA1 +

0.03!!!!!!!

rw

EHLog=Neperian Log =Ln L

t[83]= 33271.1

[84]:= H* 4 Choose V Vmin *LV= 46000.;

[85]:= H*

5 Find i*L

i = P Vt[85]= 0.00434783

[86]:= H* 6 Find d *Ld= k F i

t[86]= 0.092

(* 7/ Find ww *)ww = 1.3*d (*end of guide proper*)

t[91]=0.1196

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[92]:=

(*APPENDIX*)...;

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H Formulae this guide is based on:

Thrust: F = idkTrue for all geometries with a flat grid collector, whatever the

emitter HwireHsL,pointHsL,plane...L provided it is at a constantdistance d from the collector, and whatever the field distribution,

as shown by Sigmond 1989 cf files section of Blazelabs Yahoo group.

Power: P = Vi

Electrical power consumed by the thruster

Thrust to power ratio: FP = 1HkHVdLLEasily derived from the above two equations. Goes as the inverse of

average efield Vd: the lower the Vd, the higher the FP.

Average efield: Vd = 1HkHFPLLFrom above equation

Corona voltage drop: Vd = E0 rwJ1+ 0.03!!!!!!!!rw

N LogA1+ 0.03!!!!!!!!rw

EHLog = Mathematica's notation of NeperianLog a.k.a. LnL Easily derived from coronaradius.pdfHJullian 2004, in links section of Blazelabs siteL

Vmin = 20 E0 rwJ1+ 0.03!!!!!!!!rw

N LogA1+ 0.03!!!!!!!!rw

EMinimum voltage for which nonthrusting plasma

power loss Vdi is less than 5% of total power Vi

Optimum multiwireplane condition: ww=1.3d

Current density definition: J = iA

Generalized BorgSigmond equation for average current density at collector: J = KeokV^2

d^3

Derived by R.S. Sigmond 1982 Hcf Sigmond82.pdf in files section of Blazelabs Yahoo groupLfor saturated planeplane HK=1.13 i.e. 98L, linecylinderHK=1L, and pointsphereHK=0.38L,and generalized to all EHD geometries by Saviour Borg 2005 HBlazelabs Yahoo GroupL as a handyreplacement for my postulated geometrical factor G relative to planeplane, cf history below.

xtoplane EHD thruster area equation: A = k^2HKeoL F^3P^2HJullian 2005, derivation in history below L

H History

It all started from a design guide I made for hypothetical planetoplane EHD thrusters, for which

interest sprung up when we discussed air triodes on the Blazelabs group. That design guide wasbased on the following formula for planetoplane saturation current density:

Jmaxpp = 98eokV^2d^3

which I had rederived in the Planeplane Thruster calculations pdf paper I had posted to the

group, which BTW features AFAIK the first ever analytical solution for

emitter efield EH0L as a function of V, d and J, kindly solved for me by MathematicaHI just had to compact it from a 12 lines expression to a more manageable 3 lines!L.

To find the required thruster area A for a given thrust F at saturation current, I combined Jmaxpp

with the current density definition and the thrust equation F=idk :

A = iJ = iJmaxpp = FkdJmaxppThe guide used d, F and P as the only variables which I thought was already pretty neat,

but I soon was surprised to find that changing d while keeping F and P the same didn't change A,

which could only mean that A depended solely on primary design goals F and P!

I therefore developped the above equation looking for an F,P dependency:

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A = Fkd Jmaxpp= Fkd H98eokV^2d^3L= F8H9eoL d^2V^2= F8H9eoL HkHFPLL^2= 89 k^2eo F^3P^2 Hrequired area for planetoplaneL

Indeed, required area depended only on F and P, and required area per unit thrust AF evendepended solely on thrust to power ratio FP, how neat!!!

I then thought it would be nice to have a handy formula like this for existing multiwire

or multipoint to grid technologies, so I postulated that average current density at collector

J for any xtoplane geometry was simply that of planetoplane at same V and d divided by

some purely geometrical coefficient, which I first named N, and later GHas in GeometricalL :J = JmaxppG = 98eokV^2d^3 G

from which followed the nice handy formula I was looking for:

A = G 89 k^2eo F^3P^2 Hrequired area for xtoplaneL

Encouragingly, at least for the case of multiwireplane, I was happy to discover

that the same G value of 2.3 was able to account for both the 85 g thrust @ 30 kV Hno loadLand the 200 g thrust @ 46 kV Hfull loadL operating points documented by Saviour for his 100gpayload thruster in http:www.blazelabs.comeexp14.asp ! The wire spacing to gap lengthratio, which is of course what geometry boils down to in the case of multiwiretoplane,

was 1.3 for that thruster.

Saviour then found in a breakthrough experiment, using a small test setup where he kept a fixed

collector area and shared it between an increasingly large number of wires n, measuring thrust

vs n at constant gap 50mm and voltage 40kV, that the thrust reached a maximum for a wire

spacing to gap length ratio close to 1, for which by applying my formula for A he found a

G value of 2.3 again! This confirmed my postulate of a purely geometrical

dependance of the ratio G of planeplane current density Jmaxpp to current density J,

and fixed the value of 2.3 for optimum multiwireplaneHwire spacing between 1d and 1.3dL.

Saviour then wisely suggested we drop the 98 and G, and replace them by the K coefficientfrom the above Sigmond 1982 equation, by broadening its scope to all EHD geometries. So the

new generalized BorgSigmond coefficient K replaced 9H8GL in my J and A equationswhich thus took their final simple form:

J = KeokV^2d^3 Hgeneralized BorgSigmond equation for average current densityat collector for any EHD geometry, assumes emitter size is negligibleL

A = k^2HKeoL F^3P^2 HJullian EHD thruster area equation for xtoplane geometriesL

where K is a characteristic to be determined for the particular xtoplane geometry, e.g.

in the present case of Optimum MultiwirePlane K = Komwp has been found to be 9H82.3L = 0.49

L