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Title:

Principle

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Date of

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Grantee

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FINAL REPORT

Theoretical Study on The Effect of the Design of

Small (milli-newton) Thruster Jets on Molecular

Contamination for the Space Station.

Ben R. Riley

Department of Physics

University of Evansville

Evansville, IN 47722

(812)-479-2696

November 15, 1935

to

June 30, 1986

University of Evansville

1800 Lincoln Avenue

Evansville, IN 47722

NAG 3-674

|NASa-CR-177263} 1'HEOBETICAL SfODI OH THE N86-26358

EPPEC1 OF ;TB£ JBBSIGB Of SI9S1L (KILLIrSESlOS)

THRUSIER JETS 'G.H.. HOLECOtAB C08TAMIN&TIOH FOR

THE SPACE SIA1IGH Final Report, 15 Hoy.,, Uncias

1985 - 30 Jan. 1986« JEvacsville Univ,4 44 p 63/18 43606
PART 1
PRELIMINARY INFORMATION REPORT

SPACE STATION SYSTEM DIRECTORATE

LEWIS RESEARCH CENTER

Title: Molecular Column Densities Resulting from a Point Source

Model

Author (s) Ben R. Riley

Present

Address:

Physics Department

University of Evansville

1800 Lincoln Avenue

Evansville, IN 47722

Division: Advanced Programs and Planning Office

Date: May 22, 1986

Distribution: T. Labus

D. Schultz

D. Hoffman

G. Barna

P. Penko

J. Serafini

R. Tacina
INTRODUCTION

The self-induced on-orbit molecular contamination around the Space Station

(SS) is of interest to anyone (user) anticipating doing experiments on or in

the vicinity of the SS. Aerospace engineers need to design a SS propulsion

system that keeps the SS in a stable orbit and at the same time does not allow

the propellant gases to interfere'with the experiments of the user. Another

engineering consideration that has to be addressed in the SS design is a

mechanism for venting waste gases.

One scenario that might accomplish the above requirements is to use an

electrothermal propulsion system, resistojet, that wil l thrust continuously in

the hundreds of tni11i-newton range which wil l provide a constant altitude for

the SS with a low g environment. As a first attempt to understand the

contamination from such a propulsion system, a point source model was

developed in the summer of 1985 by Riley at NASA - Lewis Research

Center. This paper gives the numerical results of the point source model.

Number column densities for C02 are presented in Table 1 as a function of

direction of observation (line of sight), temperature of the exit gas, and

mean exit velocity. One set of number column densities for N. is shown in

Table 1 to contrast the effect of molecular weight. All the results are for a

constant exhaust rate of 5,000 kg/year.

THE MODEL

Propellant gases from the jet nozzle are assumed to come from the jet exit

plane with a mean macroscopic velocity"!? relative to the nozzle. The gases

2
are assumed to be exhausted into a vacuum. Thus, there are no collisions with

ambient molecules. At a distance of a few nozzle exit diameters from the

nozzle, the gases wil l appear to come from a point.

The mathematical model developed (using kinetic theory of gases) allows

one to calculate at any position and time the number density around a point

source that has a Maxwellian velocity distribution.

Using the Boltzmann Equation one can write (see appendix A for

nomenclature).

(1)

where

(2)

and

(3)

The assumptions in writing down equation (1) above are: (a) no external

forces are exerted on the molecules, <b) the self-interaction terms are

neglected, and (c) the source expands into a vacuum. ,
Using a constant flux source Q(t)=Q and the Fourier-Laplace integral

transform method, the result for the number density is

vfe iff

where fU<2- ~ is defined by

_»

where r is the location of the point that the number density n is to be

calculated and u is the mean exit velocity. Both quantities are relative to

the point source nozzle.

CALCULATIONS

In order to make the number density calculations as a function of 9,

defined above, the erfc function in equation (4) was approximated (2) by

(5)

for small values of the erfc argument, and by

(6)

for large values of the erfc argument. The transfer from equation 5 to

equation 6 was made when x was greater than 1.68.
The point of observation was assumed to be 50 meters directly above the

point source (see figure 1). Once one knows n(r,t), the number density, it

is a simple matter to calculate the number column density (NCD~) by integrating

numerically along a particular line of sight (LOS).

The number column densities as a function of the LOS were calculated for

different temperatures at the exit plan (temperature of the gas at the point

source) and different magnitudes of the mean exit velocity u. The

calculations were performed for the steady state situation for continuous

firing. Therefore, the r/ut terms in equation (4) were assumed to be zero.

The number of CO- molecules per second, Q , exiting from the nozzle

21 21

was assumed to be 2.2x10 . For N2 Q =3.5x10 . These values were

obtained by assuming Q = 5,000 kg/year.

RESULTS

t

Table 1 gives the results for the" NCD calculations as would be seen by an

observer located 50 meters above the point source as indicated in fig. 1. The

observation angle, <>, 1s defined in fig. 1 to be the angle between the 'line

joining the observation point and the source and the line of sight. (For

example, an angle of 270° for $ would mean a line of sight in the direction

parallel to "u.)

The results of the NCD calculations are given in table 1 as a function of

the temperature of the source, the mean exit velocity, and the line of sight.
All calculations were done in the plane containing a lin e connecting the

source and observation point and the mean exit velocity vector.

Figures 2, 3, 5, 5, 6, and 7 respectively are graphical representations of

the NCD's as a function of observation angle for the data presented as Case 1,

2, 3, 4, 5, and 6 respectively in Table 1. The exit temperatures are the same

for figures 2 and 3 with the mean exit velocity being 4,000m/sec. and

l,000m/sec. respectively. Upon comparing the NCD's as a function of LOS in

figures 2 and 3, one observes that higher column densities are prevalent at

larger values of the angle between the LOS and the mean exit velocity vector

which is at 270 ° in figures 2 and 3. This implies that high mean exit

velocities are necessary to increase the acceptable viewing solid angle. The

mean exit velocities are the same for figures 3, 4, and 5 with the exit

temperatures being 1600°K, 800°K, and 100°K respectively.

When one compares figures 3, 4, and 5, the effect of the exit temperatures

on the NCD's is clearly seen. Low exit temperatures tend to keep the exit

plume from expanding in a direction perpendicular to the mean exit velocity.

This is clear from the fact that the NCD's are larger for observation angles

above 270° (which are for lines of sight through the exit plume) for the

lower temperature cases. This implies that low exit temperatures are

important in order to increase the acceptable viewing solid angle around the

15 2 SS. If one assumes the acceptable value of 1x10 /m for NCD's, then

figures 6 and 7 show that any LOS would have unacceptable values for NCD's for

a zero mean exit velocity. These zero mean velocity cases could represent a

point source leak.
To gain better Insight into how the number density function at different

positions around the nozzle contributes to the LOS number column densities,

equidensity contours as a function of the angle are given in f-igures 8, 9, and

10 for an exit temperature of 1600°K and a mean exit velocity of 4,000m/sec.

(Case 1 in Table 1). Figures 11, 12, and 13 are similar plots except the mean

exit velocity is l,000m/sec. (Case 2 in Table 1). Figures 14, 15, and 16 are

also a similar set except for a mean exit velocity of l,000m/sec. and a

temperature of 100°K (Case 4 in Table 1). Figures 17, 18, and 19 are

equidensity plots for zero mean exit velocity (Case 5 in Table 1).

Figures 20, 21, and 22 are presented to show how the equidensity contours

are elongated (for a fixed mean exit velocity) as the temperature approaches

zero. All three plots show how the gas expands in a direction perpendicular

to the mean exit velocity (positive x-axis). Notice the maximum distance (y

value) occurs at larger values of x for the lower exit temperatures. These

plots are not to imply that CO- wil l remain a gas at these low temperatures

as it exits from a real nozzle, but suggests how an ideal gas plume wil l

behave as the temperature approaches -zero with this model.

DISCUSSION

If the allowable limi t for the NCD's for C02 is assumed to be IxlO1 5

O /•%

molecules/nr, then any observation angle greater than 270 (parallel to

the mean exit velocity vector u) would be unacceptable for all the cases shown

in Table 1. A study of Table 1 and the figures presented reveal, as one might
expect from the kinetic theory point of view, the need to have high mean exit

velocities at low exit temperatures to confine the plume in such a manner as

to maximize the allowable viewing solid angle.

For a mean exit velocity of 2000m/sec. and a temperature of 100°K, a

maximum observation angle limi t of 260° is suggested by the data displayed

in figure 23. For lines of sight at an observation angle greater than 260 ,

the NCD's wil l be greater than Ixl015/m2.
TABLE 1

ANGLE

(deq)

<P

0

30

60

90

120

150

180

210

240

270

300

330

360

0

30

60

90

120

150

ISO

210

240

270

300

330

360

CASE 1

5.7E+03

3.0E+03

2.4E+03

2.6E+03

4.OE+03

1 . 2E+04

6.6E+06

3.8.E+12

1.5E+16

3.6E+16

6.3E+16

CASE 4

2.3E+04

1.2E+04

9.8E+03

1. 1E+04

1.6E+04

4.9E+04

2.7E+O7

1.5E+13

5.8E+16

1.5E+17

2.6E+17

COLUMN DENSITIES

(par-tic! es/mA2)

CASE 2

1.6E+15

7.3E+14

5.4E+14

5. 1E+14

6. 1E+14

8.5E+14

2.0E+15

5.7E+15

1.5E+16

2.9E+16

5.9E+16

CASE 5

6. 1E+16

2.8E+16

1.8E+16

: 1.4E+16

1.2E+16

. l.OE+16

1.2E-H6

1.4E+16

1.8E+16

2.8E+16

6. lE-t-16

CASE 3

3.4E+14

1.6E+14

1.2E-H4

1. 1E+14

1.4E+14

2.3E+14

7.8E+14

4.7E+15

2. 1E+16

4.6E+16

8.8E+16

CASE 6

8.0E+16

3.5E+16

2.3E+16

1.7E+16

1.5E+16

1.3E+16

1.5E+16

1.7E+16

2.3E+16

3.5E+16

8.0E+16

CASE l: (C02), TEMPERATURE 1600 K,

CASE 2: (C02), TEMPERATURE 16OO K,

CASE 3: (C02), TEMPERATURE 800 K,

CASE 4: (C02), TEMPERATURE 100 K,

CASE 5: (C02), TEMPERATURE 300 K,

CASE 6: (N2) , TEMPERATURE 300 K,

VELOCITY 4000 M/S.

VELOCITY 1000 M/S.

VELOCITY 1000 M/S.

VELOCITY 1QOO .M/S..

VELOCITY '"- C" '

VELOCITY

 p^0
.Point of Observation

Direction of Motion '

i 270°

of Space Station »

( \ /

X

Line of Sight dp""**

, , .% / /

/" /

' Observation

Angle (

^

i

50

i, 1

^

meters

\

B

Line of Sight

ju Mean Exit Velocity

Point Source

-Space Station

End of SS Nearest to the Earth

Figure 1
<N

D \i'l

••-"UJ

II

2

d

8

100

90

SO

70

GO

50

40

50

20

10

0

-B BFIGURE 2

CQ2; TEMP=1&OO K, VELQCnY=4000 M/5

-B- -B- -B- -Hf90 150

OBSERVATION AN3LE

270 3GO
o in

&T-

"•-- uj

^

IJ1 «}

8

7O

SO

50

40

30

2O

10

c>

FIGURE 3

CQ2: TEMP=16vO K, VELOCITY=1000 M/5

ISO

OBSERVATION ANGLE (dcg)

27O
3ft» O =^

c-"

o in

••^uj

uj •=

.•-i i—

_'

_i

8

I CO

90

50

70

40 -

0 -

10 ~

-gFIGURE: 4

C02; TEMP=SOO K, VELQCfTY=lWQ M/5

0

-¥-

90

-e- •&

270

OBSERVATION ANGLE (dcg)
\

CO2; TEMP=100 K, VELCCnT=1000 M/5

"iR

£

N

ti .-.

o IT/

•-'uj

'jT ii

*

fcTM>

 ^ uj —

^-'"

1

8

'240 -

:> -"' • 220 -

200 ~

160 ~

1 60 -

140 -

1 2O -

1 00 -

60 -

60 ~

40 -

20 ~

, I r

".' / "f

; \ .1 / J, t*'

/

i

r

/

/

d

/

/

/

/

Z

"

0 90 160 270 tt

OBSERVATION Ah-K3LE (dcg)

II*.>
8

1OO

go

50

70

60 -

50

40

30

10 -

V

0

FIGURE 6

C02; TCMP=300 K, VELOCITT=0 M/5

150 270

OBSERVATION ANGLE' (deg)

36O

iVV.M#*JV. it!-'
CM

£

«

t: ..-,

D If!

CL,-

SI

100

go

50

70

6O

50

40

30

20

10

0

0

FIGURE 7

N2: TEMP=300 K, VELOCITT=0 M/5

90 150

OBSERVATION ANGLE (deg)

270 360

A •*aa&tlM
UJ

O

il

10

0

-10 ~

-20

\

-20

I

20

FIGURE 8

EQUIDENSITY PLOT (10" 14/rn^

C02; TCMP=1600 K, VEUOCrrr=400O M/5

III ) i n

140

T I

150 220 250

X DISTANCE (metere)
ll

r~

LU

O

Cfl

0

-10 -

-20

-20

FIGURE 9

EQUi D ENS ITY PLOT (10"-15/rn ~ 3)

C02; TCMP=1&W K, VELOC(TT =400C- M/5

_J r

20

I I I TT

KX> 140

1 I

150 220

X (meters)

„*,.. •:-..-*.-* ", .
uj

Ijl

Q

V

50

20 -

1 0 -

0

- 1 0 ~

-Zv -

-50 I I

2O 20

FIGURE 10

EQUiDENSITY PLOT (1CT1 6/m~3)

C02; TEMP=160O K, VELOCITY =4000 M/5

SO TOO 140

X DISTANCE (meters)

150 220 26Q
UJ

o

cn

o

4O -

2O

10 H

10 -

20 -

30 -

40 -

-so

/

-1O

y

FIGURE 11

EQ U I D ENS 1TY P LOT ( 1 (T 1 4/m ~ 3 }

CQZ; TEMF=1SOO K, VELOCITY =1CCQ M/5

0 .30 50 70

X DISTANCE (meters)

I

90 no

•-., i.-t- -Aui>. »»•-. J"">
l!

UJ

Q

2

Cl

c

*~-*

40 ~

20

1O

0

-TO -

-20 ~

-30

-40

-10

FIGURE 12

EQUIDENSITY PLOT (1<ri5/m~3)

CQZ; 7EMP=1SW K, VELOCnFT = l OOO M/5

1

10

1

-50

I

-50

1

70 90

X DISTANCE (meters)

110

,_.. .. *.... S£>M; ijj k- *;**:v-. ^ .
UJ

o

1

'JT

/

FIGURE 13

EQUI DENSITY PLOT (1CT1 6/rrT 3}

TEMP=1SOO K, VELQCmr«10QQ M/5

40 ~

•30 -

20 -

10 -

0

-10 H

20 H

~

-40 H

-10 10 30 50 70

X DISTANCE (meters)

90 11O
LU

O

I

on

Cl

-50

FIGURE 14

EQUI DENSITY PLOT (I0~14/irr3)

CO2; TEMP=100 K, VELOCnT=10OO M/5

50 550 450

X DISTANCE (meters)
FIGURE 15

EQUI DENSITY PLOT (I0~15/rn~3)

CQZ; TEMP=100 K, VELQCrrr=1000 M/5

40 ~

i

20 -

"jj

tj 10 -

UJ ,-

0

tfl -10 -

d

-20 ~

-40 ~

-50 -

_[

50

'•'.- ''?V -^

.;-. •-: Ji ..;^y

^^^" — — \

A A \ / vx y

i i i 1. 1 i i i

5O 150 250 350 -45

X DISTANCE (meters)

x'jt-UitHt IHItll J
FIGURE 16

UJ

O

1

50

4O -

20

10

O

-10

-20

-30

-40

-50

EQ UIDENSITY PLOT (1CT1 6/m ~ 3)

CQ2; TEMP=100 K, VELQCfTY=lQQQ M/5

!

50 150 250

X DISTANCE (meters)

550 45O

'. ., Jh 'till1 V --••tL'A.
0

<

OT

Q.

FIGURE 17

EQUIDENSI7Y PLOT (icri4/m~3)

C02; TEMP=-3vO K, VELOCITT=0 M/5

X DISTANCE (meters)
FIGURE 18

£

UJ

o

2

<T

icn

a

100

. 90

60

. 70

•**«"

- GO

40 -

30 -

20 -

10 ~

0

100

-10 -

•20

•30

-40

•50

- 50

-70

•50

-90

EQUIDENSITY PLOT

C02; TEMP=300 K, VELOQTY=O M/5

i I i i I I I I I

-100 -50 -60 -40 -20 {-•-

\ \ I i I . I I

20 40 60

'"• '•'•4'

 :

-,.t •*;*•• •...-.

 • • '5• -W*

I I

50 100

X DISTANCE (meters)
FIGURE 19

EQUIDENSITY PLOT

C02; TEMP=300 K, VELQQTr=0 M/5

1

£

UJ

o

2!

in

Ci

1 vV

go -

! • • 70 -

40 -

30 ~

20 -

10 -

o -

-10 ~

-20 -

-4O -

-50 ~

- 60 -

-70 -

-so -

-go -

i /"•••"•.

r

L

"\

J

1 vv

 i i i i i i i i . i i • i i i .1 i i i i i

-100 -50 -60 -40 -20 0 20 40 60 50 1<X>

'•'"ii"1

X DISTANCE ..(meters)

L,.-kyMJh. ,-.*Aw?i.«js.si.,- .».;.. ,:, ,.A«L«jk,. *4.:t.
UJ

o

en

20 ~

O

-10 -

-20 -

1

0

FIGURE 20

EQUIDENSITY PLOT (1 <T 14/rrT 3}

C02; TEMP=100 K, VELOCITT=4<X>0 M/5

I

zoo 600

X DISTANCE (meters)

^50O

.aw., ..,>4&*iitoa.*^sj***.h h-^-LyaU- 'JudbM* •-'••• - * -»
FIGURE 21

flj

UJ

o

(fl

a

O

-10 -

-20 -

•3V

EQUI DENSITY PLOT (10~14/rrr3)

C02; TEMP=1Q K, VELOaTT=40<X> M/5

Q,4 0,5

i i

1,2 hell'Thousands')

X DISTANCE- (meters)

2.4 2.5
FIGURE 22

EQUIDENSITY PLOT (1 (T14/rrT 3)

CQ2: TEMP= 1 K, VELOCITY=4OW M/5

UJ

O

Q

20

10 -

\

-10 -

- ?c.- -

-&>

O 4 | 6

X DISTANCE (meters)

6

!!*•>•'
60

FIGURE 23

COLUMN DENSITY vs. OBSERVATION ANGLE

C02; TEMP=1OC> K, VELOCmr=20QO M/5

M

i

O.T2:

UJ

g

40 -

10 ~

O

26-2 264

OBSERVATION ANGLE (degrees)

26-5 270
Appendix A

Nomenclature

f=f(?,v",t) = phase space particle density

? = position relative to source

v" = velocity of particle in phase space

6 = delta function

m = mass of molecule

k = Boltzmann's constant

T = Kelvin temperature

/i = mean molecule velocity

n=n(r",t) = number density

Q0(t) = number flux rate from a source

-* -4

Cos 6 = 11 T

t = time source is on

->

^r = del operator (coordinate)

$ = velocity distribution function

erfc = complimentary error function
REFERENCES

1. B. Riley, "Initial Efforts at NASA-Lewis to Model the Self-Induced

Molecular Contamination of the Space Station," NASA-ASEE Case-Lewis Summer

Faculty Fellowship Program Final Report, 1985 (D123).

2. M.L. BOAS, Mathematical Methods in the Physical Sciences (John Wiley &

Sons, New York 1966), pp. 468 & 469.

ACKNOWLEDGEMENTS

The author would like to thank Kent Scheller who programmed the Computer

generated figures in this report.
PART 2
0293b

INTRODUCTION

The self-induced molecular contamination around the Space Station could have

... " .£'

adverse effects on Space Station components (for example solar panels) as well

as scientific experiments (for example experiments involving photons) that

might be done on or near the Space Station. One such source of self-induced

contaminates is the propulsion system. Consideration is presently being given

to continuously or semi-continuously firing small resistojets (in the hundreds

of millinewtons thrust range) to overcome on-orbit frictional drag.

This report outlines an initial effort to develop a mathematical model to

study the effect of nozzle design on the induced molecular environment around

the Space Station produced by simple gas propel!ants. The mathematical model

would allow one to follow the expansion of the gas from the throat of a nozzle

to the nozzle exit plane and then into the space external to the nozzle. A

description of the proposed two-dimensional model and the mathematical

calculations (by numerical methods) that could be performed in a rather

complex computer code that must be written to obtain actual predictions are

outlined in the remainder of this report.

BASIC APPROACH

Because of the expected relatively low densities of the propellant gas in the

region of the exit plane of the nozzle, the fluid flow will be considered from

%^%>r ;.*,V

the particle (molecular) nature of the gas using the methods^pfj-•.•'/<••

..•5?'.?%?s'»*v «

kinetic-statistical theory. The starting point for describing the gas flow

will be the Boltzmann* ' equation for binary collision. In particular, the

Jt-
model described below will be based on the steady state formulation of the BGK

t?) (3)

approximation ' or "first iterate equation ' of the Boltzmann

equation. Then by an iterative process a solution to the Boltzmann equation

is obtained.

MATHEMATICAL FORMULATIONS

For the steady state the first iterate equation in differential form is

(1)

A

where Maxwellian molecules have been assumed for describing the

interparticle collisions. (See Appendix for equation symbol meanings.) With

the appropriate boundary conditions, the first order solution to equation (1)

can be obtained. Then the following first moments can be obtained:

The value of ^ , jm » T > and -f can be the starting values for a second

iteration, etc.

Using the method of characteristics equation (1) for can be written in

integral form as follows:

4.
The /Y\g, term is to be determined from one dimensional isentropic flow.

SPECIFIC NOZZLE GEOMETRY AND BOUNDARY CONDITION ASSUMPTIONS

The following section describes the suggested nozzle geometry and boundary

.conditions for^f which can be used to try to simulate the flow of a propel!ant

through a nozzle as well as the flow of the plume. Figure 1 is a diagram of a

two-dimensional nozzle and accompanying physical boundaries where 7~(r< must

&

be specified.

The following boundary conditions (see Figure 1) for -j- seem appropriate:

7~. =£> for the inward component of velocity

f ea_ =. o

h, =*

diffuse scattering

where //?» and g are number densities on the respective boundaries. In

Figure 1, the outer boundary Rj is assumed to be approaching infinity thus

the number of particles with an inward component of velocity 1s zero. The

number of particles on the outer surfaced(&3 -f %*-) of this nozzle is assumed

&>:;. ..'4 ;-.

to be zero. The value of -7- at P^ is assumed to be that obtained from

3.
isentropic flow conditions at the throat of the nozzle. In Figure 1 the

inside nozzle wall boundary conditions are for complete diffuse scattering,

but spectral conditions could also be applied.

DISCUSSION

Within the limitation of this model, one would be able to predict from throat

conditions the number density at all points in space. Thus, with the model

one can generate number densities at all points around the nozzle as well as

number densities near the lip of the nozzle where back flow may be

significant.
----- - - x

FIGURE 1
APPENDIX

A = constant

ds = length in characteristic direction

fCO'V/ = phase space particle density

TB = phase space particle density at boundary

fe = Maxwellian velocity distribution

k = Boltzmann's constant

m = mass of molecule

= number density

= number density at boundary

TO

= space position relative^nozzle

.*

*c = location of boundary

T~ = local temperature

TB = temperature at boundary

"Tj^ = temperature of gas at throat

-*

•^ = local mean velocity

^« = mean velocity at nozzle throat

/lf-*

~ = velocity

= number density at boundary

= del operator (coordinate)
1. Patterson, Gordon N., Introduction to the Kinetic Theory of Gas Flows,

University of Toronto Press, 1971. pp 29-55.

2. Anderson, Donald G., On the steady Krook kinetic equation: part 1, J.

Fluid Mech. 26, 17 (1966).

3. Patterson, Gordon H., Introduction to the Kinetic Theory of Gas Flows,

University of Toronto Press, 1971. p 81.

4. Kogan, Mikhail N., Rarefield Gas Dynamics, Plenum Press, New York, 1969.

p 41.