Low Reynolds Number Performance Comparison of an
Underexpanded Orifice and a DeLaval Nozzle
Andrew J. Jamison† and Andrew D. Ketsdever‡
†University of Southern California, Department of Aerospace and Mechanical Engineering
Los Angeles, CA USA
‡Air Force Research Laboratory, Propulsion Directorate
Edwards AFB, CA USA
Abstract. Low thrust propulsion systems are required for microspacecraft operations. For thruster systems that utilize
gas expansion through micronozzle geometries, the operation at low Reynolds number is required. Cold gas flows are
compared for a thin-walled, underexpanded orifice and a conical nozzle as a function of Reynolds number. The orifice
diameter and the conical nozzle throat diameter are both 1.0 mm. The Reynolds numbers investigated range from below 1
to nearly 400. The nozzle to orifice thrust ratio is found to be below unity for Reynolds numbers below 70 for helium,
argon and nitrogen propellants. For a thrust ratio below unity, the orifice is shown to have a higher propulsive efficiency
when compared to the conical nozzle. The thrust measured in this study ranges from several hundred nano-Newtons to
almost 1 mN. The thrust data is shown to transition from a free molecule solution at the low thrust range to nearly the
continuum, inviscid solution at the high thrust range. Thrust data is presented for the same nozzle geometry over almost
four orders of magnitude in thrust.
NOMENCLATURE
A = area
a = speed of sound
d = diameter
γ = ratio of specific heats
M& = mass flow rate
p = pressure
ρ = density
u = velocity
µ = viscosity
subscripts
e = exit
n = nozzle
o = orifice
s = stagnation
t = nozzle throat
superscript
* = sonic condition
INTRODUCTION
Low thrust propulsion systems are required for orbital maneuvers on microspacecraft. In principle, the geometric
scaling of nozzles for low thrust applications appears favorable. The availability of micromachined nozzle
geometries with throat diameters on the order of several microns (µm) creates the possibility for the development of
efficient low thrust propulsion systems by scaling thrust with throat area instead of with pressure. For a given
stagnation pressure, the thrust decreases with the square of the throat diameter while the Reynolds number, which
gives a measure of the nozzle efficiency in terms of viscous losses, only decreases linearly with the throat diameter.
In order to maintain the same efficiency in a micronozzle as in a large-scale nozzle (i.e. maintain the Reynolds
number), the stagnation pressure must be increased as the throat diameter decreases. In practice, however, the
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
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limitations on microspacecraft volume and power indicate that micronozzle propulsion systems may be required to
operate at relatively low pressures.[1]
To achieve a factor of 10 decrease in thrust while maintaining the same Reynolds number, the diameter could be
decreased by a factor of 10 while the pressure is increased by the same factor. High pressure gas storage to this level
may not be consistent with volume or mass limitations on microspacecraft. Also, the relatively large amount of
power required to vaporize and pressurize propellant stored as a liquid or solid may be prohibitive on
microspacecraft.[1] Previous studies have indicated that micronozzles will most likely operate at Reynolds numbers
on the order of 10 to 1000, [1,2] which suggests significant viscous losses may result in typical micronozzle flows.
The Reynolds number at a nozzle throat or at the exit plane of a thin-walled orifice is given by
ρ * u * dt
(1)
Re * =
µ*
For flow in a micronozzle, low Reynolds numbers can result from low pressure operation, small throat diameter,
and/or high stagnation temperature as with chemical, resistojet or arcjet thrusters. At sufficiently low Reynolds
number, the viscous losses within a micronozzle geometry will be large enough that the specific impulse along the
nozzle centerline will decrease from the nozzle throat to the exit plane, making the concept of a nozzle expansion
useless. At some point in the low Reynolds number regime, a thin-walled sonic orifice may outperform, or at least
perform as well as, a typical micronozzle geometry.
Although a significant amount of work has been done on numerical modeling of low Reynolds number nozzle
and micronozzle flows,[3-7] there has been little experimental effort to compliment these simulations.[8,9] More
experimental work is necessary to validate these models especially in the transitional regime where the NavierStokes solutions may not be valid and Direct Simulation Monte Carlo is computationally intensive.
This study seeks to evaluate the performance differences between a thin-walled, underexpanded orifice and a
typical conical nozzle geometry. The purpose is to find the approximate Reynolds number where below which, the
concept of a nozzle is no longer beneficial to micropropulsion systems. This study provides the first look at
comparisons between orifice and nozzle performance at low Reynolds number. The orifice used in this study has the
same diameter as the conical nozzle throat allowing for direct comparison of the thrust measurements obtained from
each of the geometries.
THEORY
Assuming a uniform velocity across the exit plane of an orifice equal to the speed of sound, the theoretical thrust
from a sonic orifice is given by
To = M& a + p * Ao
(2)
Assuming an inviscid flow, the theoretical thrust can be re-written as
γ



 2  γ −1 

To = (γ + 1)
(3)
 p s Ao
 γ + 1




The isentropic thrust from a nozzle is given by
1
γ +1

 2
γ −1  p e 
 2  2 


Tn = γ 
p s At + p e Ae
1 −

p s 

 γ − 1  γ + 1 


Ae
For an infinite expansion ratio, ε =
= ∞ where pe=0, Eqn. (4) reduces to
At
(4)
1
γ +1

 2
γ −1 
 2  2 


Tn = γ 
p s At

 γ − 1  γ + 1 



558
(5)
From Eqn. (3) and Eqn. (5), the ratio of thrust from an infinite expansion ration nozzle to a sonic orifice at the
same stagnation pressure with Ao=At can be given by
1
 2
Tn 
γ2

=
(6)
To  (γ + 1)(γ − 1) 


which is equal to 1.43 for γ=1.40 and 1.25 for γ=1.67.
Taking into consideration an expansion ratio ε=62.4, as is the case for the conical nozzle under investigation in
T
is equal to 1.37 for γ=1.40 and 1.24 for γ=1.67. Therefore the thrust ratio is
this study, the thrust ratio n
To
expected to asymptote to these values for large Reynolds numbers where viscous losses can be neglected.
EXPERIMENT
The thrust for the thin-walled, underexpanded orifice and the conical DeLaval nozzle was measured on the nanoNewton Thrust Stand (nNTS) shown in Fig. 1. The nNTS, which has been described in detail by Jamison, et. al [10],
was installed in Chamber-IV of the Collaborative High Altitude Flow Facility (CHAFF-IV). CHAFF-IV is a 3 m
diameter by 6 m long stainless steel vacuum chamber that was pumped by a 1 m diameter diffusion pump with a
pumping speed of 25,000 L/s for nitrogen and 42,000 L/s for helium. The ultimate facility pressure was
approximately 10-6 Torr with operational pressures up to 10-4 Torr. Previous studies have indicated that the facility
background pressure should not significantly affect the thrust measurements at the relatively high thrust levels
measured in this study at the background pressures maintained.
Flexure Pivots
Orifice
LVDT
Inverted
Cylinder
Baratron
Gas Inlet
High Viscosity Oil
(Gas Seal / Damping)
FIGURE 1. Nano-Newton Thrust Stand installed in a vacuum chamber.
The nozzle used in this study is shown schematically in Fig. 2. The conical nozzle geometry was scaled down to
a 1 mm throat diameter from the nozzle of Rothe.[8] This geometry was chosen because of its history of being both
experimentally [8] and numerically [2] investigated. The nozzle has a throat diameter of 1.0 mm and an exit diameter
of 7.9 mm which gives an expansion ratio of ε = 62.41. The nozzle is conventionally machined from aluminum. The
nozzle is attached to a 1.3 cm x 2.3 cm cylindrical plenum assembly shown photographed in Fig. 3.
The thin-walled orifice used in this study is shown schematically in Fig. 4 attached to an aluminum plenum. The
orifice has the same 1.0 mm diameter as the throat of the conical nozzle. The orifice is machined in a 0.015 mm
thick tantalum shim, which gives a thickness to diameter ratio (t/d) of 0.015. A thin-walled geometry was chosen so
that the viscous effect in the orifice geometry is minimized as opposed to a long tube.
559
10.7 mm
20°
30°
1.0 mm
7.9 mm
FIGURE 2. Schematic of DeLaval, conical nozzle geometry.
Plenum
Thrust stand arm
Nozzle
FIGURE 3. Nozzle and plenum connected to the nano-Newton thrust stand.
The thrust level is calibrated using an analytical solution to the orifice flow in the free molecule range. [10] This
approach has been verified by the Direct Simulation Monte Carlo numerical technique. [11] The orifice and nozzle
flows at higher stagnation pressures are calibrated using the orifice data in the free molecule range.
For both the nozzle and the orifice, the stagnation pressures are measured directly in each plenum respectively
using MKS™ pressure transducers. The propellant is introduced into the stagnation chamber of the thruster through
an adjustable needle valve located downstream of Omega™ mass flow meters. In the experimental configuration,
the mass flow meters were operated in the continuum regime throughout the pressure range investigated. In the
current configuration, the stagnation pressure for both the orifice and nozzle is limited to about 17 Torr. The
stagnation temperature was measured to be 295 K (cold gas flow). This gives a Reynolds number maximum of about
350 for argon and nitrogen and about 150 for helium. Argon, nitrogen, and helium propellants were used in this
study.
560
Aluminum Plenum
65 mm H x 35 mm W
x 13 mm T
Orifice
(Dt = 1.0 mm)
Tantalum Shim
(t = 0.015 mm)
Gas Inlet
(I.D. = 10 mm)
FIGURE 4. Schematic of thin-walled, sonic orifice geometry.
RESULTS
The thrust from the underexpanded orifice is shown for a helium propellant in Fig. 5 as a function of Reynolds
number. The thrust follows the free molecule thrust solution at Reynolds numbers less than unity (high Knudsen
number) and transitions to the continuum solution from Eqn. (3) as the Reynolds number increases. A similar plot is
shown in Fig. 6 for the nozzle as a function of the throat Reynolds number. Again, the thrust follows the free
molecule analytical solution for low Reynolds number and transitions to the continuum solution of Eqn. (4) at higher
Reynolds number. Comparing Fig. 5 and Fig. 6 at the larger Reynolds numbers indicates that the orifice has
transitioned to a continuum thrust near a Reynolds number of 25. However, the nozzle, which still exhibits a large
1.0E-03
1.E-03
Continuum
1.0E-04
Continuum
1.E-04
Free molecule
Thrust (N)
Thrust (N)
Free molecule
1.0E-05
1.E-05
1.E-06
1.0E-06
1.E-07
1.0E-07
0
5
10
15
20
25
0
5
10
15
20
25
Re *
Re *
FIGURE 5. Orifice thrust as a function of Re for helium with
theoretical lines for free molecule and continuum flow.
561
FIGURE 6. Nozzle thrust as a function of throat Re for
helium.
boundary layer in the expansion region, has not fully transitioned to the corresponding inviscid, continuum solution
by a Reynolds number of 25. This is not unexpected due to the thin walled geometry of the orifice in which the
effect of any boundary layer would be negligible.
Figure 7 shows the thrust deflection as a function of Reynolds number for the orifice and nozzle using argon and
helium propellants. As expected, the thrust at a given Reynolds number is larger for helium than for argon. This is
due to the higher stagnation pressure required for helium to achieve the same Reynolds number as argon or nitrogen.
Figure 8 shows a similar plot for nitrogen. The data in Figures 7 and 8 were used to obtain the thrust ratios shown in
Fig. 9. The thrust ratio is less than one for helium and argon for Reynolds numbers below approximately 70. This
value is consistent between the helium and argon data to within the experimental error. A similar plot is shown in
Fig. 10 for nitrogen propellant compared with similar data for argon. As seen in Fig. 9, the argon data at relatively
large Reynolds number appears to asymptote to the theoretical value of 1.24 for γ = 1.67.
3
Thrust Stand Deflection (mm)
2.5
2
1.5
helium nozzle
helium orifice
1
argon orifice
argon nozzle
0.5
0
0
50
100
150
200
250
300
350
400
*
Re
FIGURE 7. Thrust stand deflection versus Re for helium and argon propellants.
3
Thrust Stand Deflection (mm)
2.5
2
1.5
orifice
nozzle
1
0.5
0
0
50
100
150
200
250
300
350
Re*
FIGURE 8. Thrust stand deflection versus Re for nitrogen.
562
400
Tn
is above the asymptote limit for γ = 1.67; however, it has not quite
To
reached the asymptote value expected for γ = 1.4. Fig. 10 shows that the thrust ratio for nitrogen is also below unity
for Reynolds numbers below about 70. For a thrust ratio less than unity, the orifice would have a higher relative
specific impulse compared to the nozzle, and visa versa as shown by data obtained in this study. Again the
propulsive efficiency of the conical nozzle is higher than that of the orifice for Reynolds numbers greater than about
70.
In Fig. 10, the nitrogen thrust ratio
1.4
γ = 1.67
1.2
Tn/To
1
0.8
Argon
Helium
0.6
0.4
0.2
0
0
100
200
300
400
500
Re*
FIGURE 9. Thrust ratio for argon and helium propellants.
1.6
γ = 1.4
1.4
γ = 1.67
1.2
Tn /To
1
0.8
Nit rogen
Argon
0.6
0.4
0.2
0
0
100
200
300
400
Re *
FIGURE 10. Thrust stand ratio for argon and nitrogen propellants.
563
500
CONCLUSIONS
The data obtained for the thin-walled, underexpanded orifice and the conical nozzle shows that the thrust
transitions from a free molecule solution to a continuum solution as expected. The orifice appears to transition to the
orifice continuum solution by a Reynolds number of about 25. However, the nozzle is still below the nozzle
continuum solution at this same Reynolds number indicating that the viscous losses in the nozzle are still significant,
and the subsonic boundary layer still occupies a reasonable fraction of the expanding section of the nozzle. The
ratio of thrust measured between the nozzle and the orifice was found to be below unity for Reynolds numbers
below about 70 for helium, argon, and nitrogen propellants. This indicates that a thin-walled orifice should be
considered for gas expansion in micropropulsion systems where the Reynolds number is expected to be below 70.
Although this study only considers one nozzle geometry (conical configuration), it shows the general trend for low
Reynolds number flows.
T
At Reynolds numbers above 350, the thrust ratio n
appears to asymptote to the theoretical value for an
To
argon propellant. The helium propellant appears to asymptote to the theoretical value, but has not yet done so for the
range of Reynolds number obtained in this study. The nitrogen data does not quite reach its asymptotic value for a
Reynolds number of 330, which indicates that increased losses are still evident for molecular propellant flows. For a
thrust ratio (as a function of mass flow rate) less than unity, the orifice shows a higher propulsive efficiency as
measured by the specific impulse. For a thrust ratio greater than unity, the nozzle is more efficient.
ACKNOWLEDGMENTS
This work was supported by the Missile and Space Propulsion Division of the Air Force Research Laboratory
(AFRL/PRS). The authors wish to thank Mr. Joshua Cripps for his assistance with this research.
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