Further Studies Using A Novel
Free Molecule Rocket Plume Model
Michael Woronowicz
Swales Aerospace
5050 Powder Mill Road
Beltsville, Maryland 20705, USA
Abstract. This paper describes some recent studies conducted using a set of analytic point source
transient free molecule equations generated to model behavior ranging from molecular effusion to rocket
plumes. These studies include comparisons to experimental data regarding steady flow from a sonic
orifice and generation of a thruster backflow environment, followed by a transient development of plumes
due to steady thruster operations and to a single pulse.
INTRODUCTION
As the plume from a chemical thruster or rocket engine expands into the high vacuum of space, the
gas quickly passes from the continuum regime to free molecule flow. However, because high velocity
levels and relatively high Mach numbers characterize flow in the high-density region near a nozzle exit,
where significant intermolecular collision rates occur, relative velocity levels may be low, and relatively
little thermal scattering occurs normal to the mainly radial motion. Under these circumstances, such
observations lead one to consider describing the plume using free molecule theory.
Such a model has been developed [1] and is being used to analyze design and operational issues for
satellite programs at NASA Goddard Space Flight Center. The purpose of this paper is to present
results from a validation effort involving two studies comparing model predictions to experimental data,
followed by a brief description of transient plume behavior due to steady thruster operations as well as a
single pulse.
MODEL FORMULATION
A transient solution of the collisionless Boltzmann equation was developed [1] to describe the
molecular distribution f(x,t) for flow from a point source step function Q1, where
∂f
∂f
+v⋅
= Q1 ;
∂t
∂x
Q1 =
(
2β4
δ ( x ) m& (t ) v ⋅ nˆ exp − β 2 (v − u e )2
A1 π
).
(1)
In Eq. (1), Q1 represents directed flow from a source with a Lambertian, single-temperature thermal
velocity distribution superimposed on a convective exit velocity ue at constant rate m& . The v ⋅ nˆ factor
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
588
emphasizes the directional constraint imposed by the rocket nozzle. Parameter β ≡ 1
a normalization factor defined as:
2 RT , and A1 is
2
2
A1 ≡ e − s cos φe + π s cos φ e (1 + erf (s cos φ e )) .
(2)
In Fig. 1, a schematic diagram is presented, describing the general relationship between important
geometric elements in this model.
axis along
exit normal
n̂
1
ue
x
φe
φ
θ
3
source location
2
FIGURE 1. Schematic representation of various quantities and angles used in analytic model.
Speed ratio s ≡ β u e can be regarded as a molecular Mach number. For thruster analyses, n̂
represents the nozzle exit plane. Generally, ue is not necessarily aligned with n̂. The angle between ue
and n̂ is described by φe. Angle φ is measured between variable position x (with velocity v) and n̂, and
angle θ is measured between ue and x. Generally, φ + φe ≠ θ in three dimensions, but for axisymmetric
conditions φe = 0 and φ = θ.
The particular solution of Eq. (1) is found using the approach outlined by Bird [2] and Narasimha
[3]. The steady-state density field generated in response to a step function in mass flow rate m& , with
constant properties across the nozzle exit, is given by [1]:
ρ ( x, t ) =
β m& cos φ
A1 π r
2
2 2
2

1

e w − s (α + w) e − z +  + w 2  π erfc z  ,
2



(3)
where z ≡ α − w, α ≡ β r t , and w ≡ s cos θ . Solving for successive velocity moments, one obtains
& , normal momentum flux (“pressure”) p⊥, and translational energy flux
expressions for mass flux Φ
q& TR [1]:
(
)
2 2
2
& ( x, t ) = m& cos φ x e w − s  α 2 + α w + w 2 + 1 e − z +  3 + w 2  π w erfc z  ,
Φ
2

A1 π r 2 r


589
(4)
m& cos φ
2 2 
2
5
3 
e w − s α 3 + α 2 w + α w 2 + w 3 + w + α  e − z
2
2 
β A1 π r


3

+  + 3 w 2 + w 4  π erfc z  ,
4


p ⊥ ( x, t ) =
2
m& cos φ
(5)
x w2 − s 2
e
2 β A1 π r r

7
9
 2
× α 4 + α 3w + α 2 w2 + α w3 + w4 + 2α 2 + α w + w2 + 2 e − z
2
2



 15

+  + 5w2 + w4  π w erfc z .
 4


q&TR ( x, t ) =
2
2
(6)
In addition, Eqns. (3) – (5) may be combined to obtain expressions for velocity v, translational
temperature TTR, and internal energy flux q& INT for polyatomic molecules with specific heat ratio γ:
v( x ) =
& (x)
Φ
;
ρ (x )
TTR ( x ) =

1  p⊥ (x )
− (v ( x ))2  ;

3R  ρ ( x )

 5 − 3γ
q& INT ( x ) = 
 γ −1
& (x)
Φ

.
 4β 2
(7)
The flowfield emanating from this point source is only valid where x ⋅ n̂ > 0 due to the velocity
constraint in Eq. (1). Also for a single source, the equations assume one can use constant, averaged
properties to describe the state of the gas issuing from the nozzle. A more realistic analysis would
incorporate the effects of viscous boundary layer development within the thruster to create locally
varying conditions across the exit plane. These local conditions would be used to create a network of
point sources to describe the expansion downstream [1]. An example using such refinement will be
presented later in this work.
CASE REVIEWS
The first comparison presented regards steady model results for the pressure of molecular nitrogen
under sonic conditions, whose angular distribution was measured experimentally at different mass flow
rates in a thermal vacuum chamber. The second comparison regards the steady angular distribution of
mass flux from 5-lbf thrusters at high angles, including upstream influence.
Sonic Orifice
An experimental investigation had been conducted at NASA Goddard Space Flight Center (GSFC)
to observe the angular distribution of molecular nitrogen venting through circular tubes of different
diameters and mass flow rates [4]. The test was conducted at T0 = 290 K using GSFC’s Thermal
Vacuum Chamber 238 (C238), with background pressure levels pC238 in the 10-4-10-3 torr range during
operations and driving pressures varying from 2-50 torr (Fig. 2). These conditions ensured that the N2
issuing from the end of the tube would achieve sonic conditions, and for the lower driving pressure
cases, pC238 should not have interfered with the sonic expansion.
Measurements were made using a Baratron capacitance manometer fitted to flexible tubing, the open
end of which was attached to a goniometer/crankwheel mechanism (Fig. 2). It appears that substantial
duct losses were not accounted for in the measurement system, and with the entrance to the measuring
590
tube only 5.1 cm downstream of vents having diameters ranging from 0.48-3.81 cm, [4] most of these
results should not be considered representative of flow from a point source.
FIGURE 2. Experimental arrangement for sonic orifice angular distribution measurements (from Ref. 4).
Nevertheless, it appeared that for cases having the lowest driving pressures ( m& N 2 below 0.02 g/s,
pC238 < 4 × 10-4 torr), data collected from the smallest cross-section duct (d = 0.48 cm) might provide a
suitable set of conditions for making comparisons with the plume model. Data for the four cases
meeting these criteria were normalized by their centerline values and plotted in Fig. 3 along with steady
results for Eq. (5) of the plume model.
DISTANCE PARALLEL TO ORIFICE [-]
0.5
Plume Model (M = 1)
p (mdot = 9.9e-3 g/s)
p (mdot = 1.04e-2 g/s)
p (mdot = 1.53e-2 g/s)
p (mdot = 2.03e-2 g/s)
0.4
0.3
0.2
p (avg.)
0.1
0
-0.1
-0.2
-0.3
-0.4
0
0.2
0.4
0.6
0.8
1
DISTANCE NORMAL TO ORIFICE [-]
FIGURE 3. Comparison of plume model to Ref. 4 data for steady nitrogen flow under sonic conditions.
For the data depicted in Fig. 3, the right and left sides of each distribution have been averaged, and
the average for all four cases is also plotted. The data still exhibit a wide degree of variation, but it
appears that Eq. (5) reproduces the angular behavior fairly well.
Unfortunately, more recently the data of Ref. 4 have been used erroneously to create a thruster
plume model of the following familiar form [5]:
& ( x ) = 2 m& cos3 θ .
Φ
s
π r2
591
(8)
The exponent of Eq. (8) was advocated [5], providing what the author of Ref. 4 considered to be the
best fit for his pressure data. In the Ref. 5 model, angular distributions of all velocity moments are
assumed to have the same form. A normalized comparison of this distribution to Eq. (5) of the free
molecule plume model (sonic) is presented below in Fig. 4.
DISTANCE PARALLEL TO SURFACE [-]
0.4
Scialdone (n = 3)
Plume Model (M = 1)
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
0.2
0.4
0.6
0.8
1
DISTANCE NORMAL TO SURFACE [-]
FIGURE 4. Comparison of Ref. 5 thruster model to free molecule sonic orifice flow for nitrogen, Eq. (5).
The same Eq. (5) sonic orifice pressure distribution is depicted in Figs. 3 & 4. Taken together, these
figures confirm that the Ref. 5 thruster model actually represents flow from a sonic orifice, and not the
highly supersonic expansion produced by a chemical thruster.
Thruster Backflow
It was suggested earlier that modeling thruster expansions with the current free molecule point
source plume model could be improved if one were to work with a network of sources on a starting
surface located beyond the nozzle exit, accounting for boundary layer (b.l.) expansion around the nozzle
lip. A recent opportunity to demonstrate this concept presented itself in a study of the General
Dynamics MR-106E monopropellant hydrazine (N2H4) thruster for the NASA STEREO project [6].
Although this thruster typically operates around 5-lbf, conditions were provided for operations at 6.9 lbf.
In this study, the following thruster core flow exit data had been provided by General Dynamics or
deduced from it [6]: core mass flow rate m& = 9.59 g/s, Mach number Me = 5.67, exit velocity Ve = 2231
m/s, static temperature Te = 185 K, and specific heat ratio γ = 1.37. Complete decomposition of N2H4 to
NH3 was assumed, along with a 59 percent dissociation of NH3 to N2 and H2. In addition, a
compressible turbulent b.l. model developed by Bartz had been used to describe properties near the
nozzle wall [7]. The b.l. momentum thickness was estimated to be 23 percent of the nozzle exit radius.
An approximate technique was developed to translate local conditions across the nozzle exit plane to
a cylindrical cap external to it. It was decided to define the axial length l of this cylinder by the position
where the b.l. edge had expanded radially to a position equal to the nozzle radius. Based on a
characteristic net created for studying the lip region of flow for an Aerojet AJ-5 5-lbf bipropellant
thruster [8], it was decided l = 0.978 cm, versus a nozzle exit radius Re = 1.645 cm. The nozzle exit b.l.
profile was subdivided into 23 rings having equal width spacing, further subdivided into 20 equal-angle
elements. Using local mass conservation along with constraints for isentropic, compressible expansion,
the gaseous state at each b.l. element was propagated to corresponding elements on the cylindrical
portion of the starting surface. Similarly, the core region was allowed to expand from the nozzle exit
plane to the disk of the starting surface. This disk was subdivided into 20 rings of equal width in 20
equal-angle segments.
592
These 860 elements comprised the starting surface for a network of point sources. Although the
thruster product chemical composition was assumed constant everywhere on the starting surface, free
molecule plumes were calculated separately for each species to account for the species separation effect
expected to occur through the expansion [1,9].
A number of MR-106E flowfield contour maps were produced from the model results. Of
particular interest for STEREO was the composite density field, which is presented below in Fig. 5. In
this figure, it is apparent that the boundary layer begins to affect the solution for angles greater than 25°.
log10 ρ
3
[g/cm ]
180
-6--5
150
90
60
-7--6
-8--7
Y [cm]
120
-9--8
-10--9
-11--10
-12--11
-13--12
30
-14--13
-15--14
0
-200
-160
-120
-80
-40
0
40
80
120
160
200
240
280
320
360
400
440
-16--15
480
X [cm]
FIGURE 5. MR-106E density contours (logarithmic scale). Thruster located at (x,y) = (0,0), facing right.
Because these are the first backflow calculations computed using this model, it was desired to
compare this approach to actual data. Results from a particularly detailed investigation were available
for the Aerojet AJ-5, 5-lbf MMH/N2O4 bipropellant thruster with measurements taken at high angles and
for background pressure levels below 10-4 torr [10].
Although the two thrusters make use of different chemical reactions, it was thought comparisons
could be made for composite mass flux distributions after adjusting for nozzle mass flux. Because the
AJ-5 has a higher specific impulse, its mass flux at the nozzle exit was only 0.639 g/cm2/s, while for the
MR-106E it would be 1.49 g/cm2/s.
MASS FLUX PER UNIT SOLID ANGLE I
[g/s/steradian]
1
AJ5 data fit
MR-106E model fit
0.1
0.01
I MR-106E = 8.12exp(-0.0728 θ )
0.001
I AJ5 = 9.49exp(-0.0752 θ )
0.0001
0
20
40
60
80
100
120
ANGLE OFF THRUSTER CENTERLINE θ [deg.]
140
160
FIGURE 6. Comparison of MR-106E model mass flux results (adjusted by 0.43×)
with experimental data for AJ-5 thruster, including backflow region.
In Fig. 6, a best fit exponential curve was used to compare adjusted model results with the best fit
exponential curve to the AJ-5 data used in Fig. 10 of Ref. 10. They tend to be quite similar throughout
the entire measured angular range. The model results are summations over all three species mass flux
distributions, each of which is characterized by a different speed ratio at each starting surface facet due
to variations in individual molecular masses.
593
TRANSIENT STUDY
As the number of cases demonstrating verisimilitude for this model continue to mount [1,6],
confidence in its capabilities continues to increase. While these capabilities require a continued
validation effort, it is of interest to explore the model’s transient capabilities.
Enhanced Formation Flying
This study was performed for NASA-GSFC’s Earth Observing-1 (EO-1) satellite, which orbits Earth
at a 705 km altitude, following Landsat 7 (L7) in the same ground track [11]. There were concerns
regarding the effects on EO-1 due to wake-directed 1-lbf N2H4 thruster firings from Landsat 7. Thruster
parameters assumed m& = 2.06 g/s, exit velocity Ve = 2157 m/s, static temperature Te = 156 K. Thruster
products assumed 99 % decomposition of MIL-P-26536 high-purity grade hydrazine to NH3, which
became 65% dissociated into H2 & N2.
Although currently EO-1 actually trails L7 by 1 min., separation times as short as 10 s were
considered. At an orbital velocity of 7.5 km/s, the distance between the two would be approximately 75
km. Two such cases are reproduced below in the absence of atmospheric interactions. One of these
predicts the density flowfield that develops for a thruster that is turned on and remains on (Eq. (3)). The
second is meant to describe the flowfield that develops in response to a single 1-s pulse.
When the constant mass rate m& in Eq. (1) is replaced by a Dirac Delta function mδ (t ) , a simpler set
of velocity moments is produced to describe pulse flowfields. Convolution of these equations at
constant strength reproduce Eqns. (3)-(6).
ρ ( x, t ) =
ρx
ρ r2
ρ r2x
2 mβ 4 r cos φ w 2 − s 2 − z 2 &
&
(
)
q
x
,
t
e
e
;
Φ
,
=
;
p
(
x
,
t
)
=
;
(
)
=
. (9)
x
t
⊥
TR
t
A1π
t4
2t3
t2
Composite density results are shown in Figs. 7 & 8 at 10, 20, 30, and 35 s after initiation. The step
response practically reaches steady state throughout Fig. 7 by 40 s. The figures show how the step and
pulse responses begin in similar fashion, but while the step response builds up to steady state, which it
approaches at timescales that increase with distance, the strength of the pulse response decays as it
diffuses downstream, as expected. The peak centerline density level experienced for the pulse case is
only about 2.5 × 10-19 g/cm3, while it is 2.3 × 10-18 g/cm3 for the steady burn.
Putting these numbers into perspective, the MSIS-86 Thermospheric Database estimates the ambient
atmospheric density is 2.3 × 10-16 g/cm3. So from a kinetic standpoint these plumes represent a few
exhaust molecules in a bath of ambient atmosphere, and atmospheric scattering would probably alter the
solution quite a bit. Combining this model with the Bhatnagar-Gross-Krook (BGK) technique [12]
would be a logical extension for such a study at this point.
CONCLUDING REMARKS
The plume model described herein continues to exhibit verisimilitude as the validation effort
continues. In addition, it appears the transient feature can offer insights beyond steady predictions. In
the final example above, it was shown that the peak centerline density due to a 1-s pulse was about one
ninth the value reached for steady operations. With greater separation distances, the disparity would
only increase.
594
12 km
9 km
6 km
21 km
18 km
15 km
12 km
9 km
6 km
3 km
3 km
15 km
30 km
45 km
0 km
75 km
60 km
0 km
15 km
30 km
45 km
0 km
75 km
60 km
24 km
b
18 km
15 km
12 km
9 km
6 km
24 km
d
21 km
21 km
LATERAL DISTANCE [km]
0 km
18 km
15 km
12 km
9 km
6 km
3 km
3 km
0 km
15 km
30 km
45 km
0 km
75 km
60 km
LATERAL DISTANCE [km]
15 km
LATERAL DISTANCE [km]
18 km
24 km
c
21 km
LATERAL DISTANCE [km]
24 km
a
0 km
15 km
30 km
45 km
0 km
75 km
60 km
DISTANCE ALONG CENTERLINE [km]
DISTANCE ALONG CENTERLINE [km]
log10 ρ [g/cm ]
log10 ρ [g/cm ]
3
3
-2.70E+01--2.60E+01
-2.60E+01--2.50E+01
-2.50E+01--2.40E+01
-2.40E+01--2.30E+01
-2.30E+01--2.20E+01
-2.20E+01--2.10E+01
-2.10E+01--2.00E+01
-2.70E+01--2.60E+01
-2.60E+01--2.50E+01
-2.50E+01--2.40E+01
-2.40E+01--2.30E+01
-2.30E+01--2.20E+01
-2.20E+01--2.10E+01
-2.10E+01--2.00E+01
-2.00E+01--1.90E+01
-1.90E+01--1.80E+01
-1.80E+01--1.70E+01
-1.70E+01--1.60E+01
-1.60E+01--1.50E+01
-1.50E+01--1.40E+01
-1.40E+01--1.30E+01
-2.00E+01--1.90E+01
-1.90E+01--1.80E+01
-1.80E+01--1.70E+01
-1.70E+01--1.60E+01
-1.60E+01--1.50E+01
-1.50E+01--1.40E+01
-1.40E+01--1.30E+01
FIGURE 7. Logarithmic density contours for steady operations of a single 1-lbf N2H4 thruster. Transient response
to step function at (a) 10 s, (b) 20 s, (c) 30 s, & (d) 40 s.
21 km
15 km
12 km
9 km
6 km
18 km
15 km
12 km
9 km
6 km
3 km
15 km
30 km
45 km
60 km
3 km
0 km
75 km
b
0 km
15 km
30 km
45 km
60 km
21 km
18 km
15 km
12 km
9 km
6 km
15 km
30 km
45 km
60 km
24 km
21 km
18 km
15 km
12 km
9 km
6 km
3 km
3 km
0 km
0 km
75 km
d
24 km
LATERAL DISTANCE [km]
0 km
21 km
0 km
75 km
0 km
15 km
30 km
45 km
60 km
0 km
75 km
FIGURE 8. Logarithmic density contours following 1-s pulse operation of a single 1-lbf N2H4 thruster. Transient
response to delta function at (a) 10 s, (b) 20 s, (c) 30 s, & (d) 35 s. (Same scale as Fig. 7.)
ACKNOWLEDGMENTS
The author gratefully acknowledges support from NASA Contract NAS5-01090.
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595
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LATERAL DISTANCE [km]
c
24 km
LATERAL DISTANCE [km]
a
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