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We investigate the flow in planar microscale nozzles and find that design and analysis paradigms based on the assumption of a dominant isentropic core with moderate viscosity corrections are not valid. Instead, the flow downstream of the throat is dominated by boundary layers that may choke the flow to subsonic velocities. The geometrical expansion ratio is found to be essentially irrelevant, instead, the length from throat to exit plane is found to be a much more important design parameter. Full 3D simulations are required to predict the flow topology; thermophysical modeling of the expanding gas has a noticeable impact on predicted performance. An analytical estimation of the Knudsen number in the expanding flow is given, allowing to determine its values from the expansion pressure ratio. An axial thrust analysis suggest truncation of the nozzle, resulting in a predicted 20% increase in thrust and 30% increase in specific impulse compared to the baseline configuration. The work has been carried out within the European Commission co-funded PRECISE project which was focused on designing and testing a micro chemical propulsion system thruster prototype using catalytically decomposed hydrazine as propellant.
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Flow characteristics of monopropellant
micro-scale planar nozzles
Daniel T. Banutia,
, Martin Grabea, Klaus Hannemanna
aGerman Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology,
Spacecraft Department, G¨ottingen, Bunsenstr. 10, Germany
Abstract
We investigate the flow in planar microscale nozzles and find that design and
analysis paradigms based on the assumption of a dominant isentropic core with
moderate viscosity corrections are not valid. Instead, the flow downstream of
the throat is dominated by boundary layers that may choke the flow to subsonic
velocities. The geometrical expansion ratio is found to be essentially irrelevant,
instead, the length from throat to exit plane is found to be a much more im-
portant design parameter. Full 3D simulations are required to predict the flow
topology; thermophysical modeling of the expanding gas has a noticeable impact
on predicted performance. An analytical estimation of the Knudsen number in
the expanding flow is given, allowing to determine its values from the expansion
pressure ratio. An axial thrust analysis suggest truncation of the nozzle, result-
ing in a predicted 20% increase in thrust and 30% increase in specific impulse
compared to the baseline configuration. The work has been carried out within
the European Commission co-funded PRECISE project which was focused on
designing and testing a micro chemical propulsion system thruster prototype
using catalytically decomposed hydrazine as propellant.
Keywords: MEMS, rocket engine, hydrazine, cube sat, satellite, propulsion
Corresponding author, email: daniel@banuti.com; Currently at Caltech, Pasadena, CA
91125, USA
Preprint submitted to Elsevier October 20, 2018
Contents
1 Introduction 2
2 Methods 4
3 Preliminary analysis 7
3.1 Thruster configuration . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Rocket performance fundamentals . . . . . . . . . . . . . . . . . 8
3.3 Thermochemical analysis . . . . . . . . . . . . . . . . . . . . . . 9
3.3.1 Hydrazine decomposition . . . . . . . . . . . . . . . . . . 10
3.3.2 Heat loss and composition . . . . . . . . . . . . . . . . . . 10
3.4 Analytical assessment of nozzle flow properties . . . . . . . . . . 12
3.4.1 Nozzle Knudsen number . . . . . . . . . . . . . . . . . . . 13
4 Simulations 15
4.1 Conditions............................... 15
4.2 Effects of geometry and composition . . . . . . . . . . . . . . . . 16
4.3 Analysis of the baseline nozzle . . . . . . . . . . . . . . . . . . . 20
4.4 Truncatednozzle ........................... 22
5 Conclusion 24
6 Acknowledgments 24
1. Introduction
Interest in micro chemical propulsion systems (µCPS) is growing with the
beginning commoditization of small scale satellites (Marcu [28]). This is fur-
ther facilitated by the the recent surge in cheap launch systems that allow for
more launch opportunities, e.g. SpaceX, and by companies that develop launch-
ers particularly for the small satellite market, e.g. Vector, RocketLab, Virgin
Galactic.
The European Commission co-funded project PRECISE (Gauer et al. [9,
10]) focused on designing and testing a µCPS thruster for application in a for-
mation flying mission. The thruster was developed to use MEMS technology
2
to etch µCPS nozzles and combustion chambers into silicon wafers like com-
puter chips, allowing for cheap and scalable mass production. The prototype
was designed to use catalytically decomposed monopropellant (hydrazine1) to
achieve high performance using a simple propulsion system. Owing to the man-
ufacturing method, such engines exhibit characteristics which set them apart
from macro scale nozzles. The two-dimensional geometry of high aspect ratio
etched geometries differs radically from classical rotationally symmetric nozzles,
impacting design procedures and performance evaluation.
A number of small scale engines and nozzles have been investigated experi-
mentally and numerically [16, 21, 1, 26], in 2D [35, 30, 13] and 3D [16, 35, 1, 26].
Rarefaction is of prime importance outside of the nozzle, for example for plume-
structure interaction [26]. Its effect is found to be negligible for integral perfor-
mance measures, such as thrust [13, 7, 26, 25]. Exit pressures exceeding 1 kPa
are used to ensure the validity of continuum models using Navier-Stokes equa-
tions with no-slip boundary conditions [13, 7]; these models are even used at
exit pressures of 50 Pa [19] and 0 Pa [6].
However, analyses are typically based on evaluation of constant property
pure ideal gases, thus the impact of variable properties has not been assessed.
Furthermore, a Knudsen number analysis can typically only be performed a
posteriori [13].
In the present paper, we extend these studies to the case of a monopropellant.
In a preliminary study, we demonstrate the impact of the composition and the
thermo-chemical modeling of the exhaust gas. We show how gas temperature
affects the monopropellant decomposition, and derive a closed form estimation
of the nozzle core flow Knudsen number. Finally, we evaluate 3D nozzle flow and
introduce an improved nozzle that outperforms the baseline contoured nozzle
by 30%.
1While hydrazine is considered hazardous, this choice was a boundary condition in the
PRECISE project due to its performance and the prior experience of the project partners
with this propellant.
3
2. Methods
A combination of methods is used for the present study. The preliminary
analysis in Sec. 3 is carried out using a combination of numerical and analytical
approaches. Specifically, equilibrium chemistry is evaluated using CEA [29, 12].
The simulation results discussed in Sec. 4 were obtained using the DLR TAU
code, a hybrid grid, finite volume, compressible CFD solver developed by the
German Aerospace Center (DLR). For the flow conditions investigated herein
(chamber pressure 10 bar, exit pressure ratio of p0/pe= 30) we assume that
continuum flow governed by the Navier-Stokes equations with no-slip boundary
conditions is applicable, consistent with [7, 13, 6, 19].
The DLR TAU code has been validated for various space propulsion cases,
including high-pressure gaseous combustion [17], cryogenic transcritical combus-
tion [2], combustion instabilities [3], and scramjets [20, 22]. Detailed descriptions
of the solver can be found in Gerhold [11], Schwamborn et al. [31], Mack and
Hannemann [27], Hannemann [14], and Karl [20].
The system of governing equations is solved using a Godunov type finite
volume method. Second order spatial accuracy is obtained using MUSCL re-
construction at cell interfaces [34]. We use Liou’s AUSMDV Riemann solver
[24]. The Spalart-Allmaras [32] turbulence model is used in a RANS framework.
A mixture of chemically reacting ideal gases is governed by the Navier-Stokes
equations. Reynolds decomposition to total and partial densities and Favre de-
composition to the remaining flow variables yields the set of equations to solve
numerically. In integral form, they read
∂t ZZZ
V
~
UdV+ZZ∂V
~
Finvd~
A=ZZ∂V
~
Fviscd~
A+ZZZV
~
SdV . (1)
Here, ~
Uis the vector of conservative variables
~
U=
ρs
ρ~uT
ρE
.(2)
4
The inviscid Euler fluxes can be written as
~
Finv =
ρs~uT
ρ~u~uT
ρH~u
+
0
pI
0
,(3)
accompanied by the viscous terms
~
Fvisc =
(ρD)eff Tρs
ρ
P
keff TT+ (ρD)eff PshsTρs
ρ+ (P ~u)T
.(4)
The viscous stress tensor Pis modeled according to the Stokes hypothesis,
P=µ~uT+ (~uT)T2
3µ(T~u)I. (5)
Using the Spalart-Allmaras turbulence model, an additional equation for eddy
viscosity µtis solved. The effective species diffusion (ρD)eff is modeled using
the laminar (without subscript) and turbulent (subscript t) Schmidt numbers
(ρD)eff =µ
Sc +µt
Sct
.(6)
The effective thermal conductivity is similarly calculated using the Prandtl num-
ber
keff =k+kt=k+Pr
Prt
µt
µk. (7)
The Dufour effect (thermal diffusion due to concentration gradient) is taken into
account. The source terms ~
Sinclude the chemical contribution to the species
transport equations ωs,
~
S=
ωs
~
0
0
.(8)
Here, as we only regard frozen flow, ωs0 and no sub grid scale filters are
used.
Chemical equilibrium calculations are carried out using CEA (Gordon and
McBride [29, 12]). Fluid models for N2H4, NH3, N2, H2have been identified and
5
integrated into the TAU solver. We chose the NASA 7-coefficient polynomial
expressions [5] as our caloric fluid models, as given by
¯c0
P,i/R0=a1,i +a2,iT+a3,i T2+a4,iT3+a5,i T4,(9)
¯
h0
i/R0T=a1,i +a2,iT
2+a3,iT2
3+a4,iT3
4+a5,iT4
5+a6,i/T . (10)
Blottner et al. curve fits [4] are used to calculate species laminar viscosity,
µs= 1 Ns
m2exp(Cs)T(Asln(T)+Bs).(11)
The individual values are then combined using Wilke’s mixing rule [36]
µ=X
s
nsµs
Pmnsφs,m
(12)
with
φ=1
81 + Ms
Mm1/2"1 + µs
µm1/2Mm
Ms1/4#2
.(13)
Thermal conductivity is computed using a modification of the Eucken correction
by Hirschfelder et al. [15]
k=µs5
2(cv)t
s+(cv)rot
s+ (cv)vib
s+ (cv)e
s
Sc .(14)
The mixture heat conductivity is then determined following Zipperer and Hern-
ing [37],
k=X
s
nsks
PmnspMm/Ms
.(15)
The diffusion fluxes are calculated from viscosity and a constant Schmidt num-
ber using Fick’s law. Then, the diffusion coefficient Dreads
D=µ
ρ
1
Sc (16)
Caloric data were available for all concerned species. Transport coefficients
were not available for N2H4. Due to its apparent similarity to water, data of
the latter have been used for gaseous hydrazine. Validation has been carried
out by comparison with the NIST database [23]2. Again, no data were available
for hydrazine.
2Data are not available for the whole temperature range. Upper limits in the database for
hydrogen, nitrogen, and ammonia are 1000 K, 2000 K, and 700 K, respectively.
6
Figure 1: Ratio of specific heats temperature dependence for hydrazine decomposition prod-
ucts hydrogen, nitrogen, and ammonia. γ= 1.4 corresponds to calorically perfect hydrogen
and nitrogen; γ= 1.33 corresponds to calorically perfect ammonia.
Figure 1 compares the CEA based TAU model with NIST reference data.
Lowering of γfor increasing temperature is due to the successive excitation of
molecular degrees of freedom. In the case of ammonia, a larger discrepancy has
been found between data obtained from CEA and data from NIST (open green
triangles in Fig. 1). For lower pressures (1 bar) the agreement is found to be
much better (open circles in Fig. 1). Due to the ideal gas modeling within TAU,
high pressure effects might not be captured. However, the discrepancy arises at
conditions of simultaneous high pressures and low temperatures, which we do
not expect to encounter: the high pressure, high temperature gas in the com-
bustion chamber expands in the nozzle to a low pressure, low temperature state,
and will hence not affect actual performance analysis of the µCPS thruster.
3. Preliminary analysis
In this section, we will perform a brief initial investigation of the µCPS. First,
we introduce the geometry and operating conditions; second, we review briefly
rocket performance metrics; third, we study the possible variation of exhaust
7
gas properties and their impact on performance; fourth, we derive expressions
for nozzle Knudsen and Reynolds numbers.
3.1. Thruster configuration
Figure 2 is a schematic of the thruster. The geometry is etched into silicon,
resulting in a high aspect ratio, rectangular cross section. The exit width is
1.070 mm, the distance from throat to exit is 2.250 mm, the throat width is
0.09 mm, the depth of the nozzle is 0.07 mm. Liquid hydrazine is supplied
through the feedline and vaporized in the vaporization chamber (1). The vapor
is then distributed and injected into the chamber (2), where it passes through
a catalyst bed (3). We assume the flow is frozen afterwards. The present paper
focuses on the expansion of the flow from the plenum (4), through the throat
(5) and the parabolic nozzle (6), until the nozzle exit (7). Expansion of the
plume (8) is not considered here.
Figure 2: Schematic of µCPS thruster.
3.2. Rocket performance fundamentals
The standard measures of rocket performance [33] will be briefly introduced,
before we proceed to analyze the µCPS. The thrust Fis the force that the engine
exerts on the spacecraft. In integral form, it is a function of axial velocity u,
the pressure p, and a reference area A,
F=ZZA
(ρuu +p)dA. (17)
The specific impulse, Isp , is a measure of global efficiency, relating fuel mass
flow ˙mto thrust F, using the standard acceleration g0,
Isp =F
˙mg0
.(18)
8
As common in engineering, the global efficiency can be expressed as the product
of the inner efficiencies, i.e. quality of combustion measured as chamber pressure
p0acting on the nozzle throat area A,
c=p0A
˙m,(19)
and the thrust coefficient, measuring the quality of expansion through the noz-
zle,
cF=F
p0A,(20)
with
Isp =cFc
g0
.(21)
The exit velocity uecan be calculated assuming 1D isentropic flow with constant
properties[33],
ue=v
u
u
t2γ
γ1
RT
M"1pe
p0γ1
γ#,(22)
using the exit pressure pe, the ratio of specific heats γ, the molar mass M, and
the universal gas constant R. The maximum velocity that can be achieved for
pe0 is
umax =s2γ
γ1
RT
M(23)
Equations (22,23) show that fluid properties affect thruster performance via
the isentropic exponent γand the molar mass Mof the exhaust gas. We will
now proceed to investigate the range of γand Mwe expect in our thruster, and
the corresponding impact on performance.
In the following, we consider the thrust and the specific impulse as our
measures of performance. Due to the essentially constant mass flow rates, both
are proportional and can be considered synonymous in this study.
3.3. Thermochemical analysis
We can determine the exhaust gas composition and properties assuming
chemical equilibrium from the CEA [29, 12] package.
9
γ1.344
µ0.4462e-4 Pa s
ρ0.9590 kg/m3
T1340.15 K
M10.685 g/mol
p10.0 bar
Table 1: Conditions for decomposed hydrazine in chemical equilibrium at 10 bar.
3.3.1. Hydrazine decomposition
The micro liquid propellant rocket engine is designed to use hydrazine as
a monopropellant; the reaction is initiated in a catalyst bed. In a first step,
hydrazine decomposes exothermally into NH3and N2[8],
3N2H44NH3+ N2336,280 J.(24)
Subsequently, NH3decomposes endothermally into N2and H2:
4NH32N2+ 6H2+ 184,400 J.(25)
The global reaction can be characterized by the reaction scheme
3N2H44(1 ξ)NH3+ (1 + 2ξ)N2+ 6ξH2(26)
where ξdenotes the degree of ammonia decomposition. As ammonia dissocia-
tion is an endothermic reaction, ξeffectively controls the attained temperature.
The exhaust gas composition is determined by the degree of hydrazine decom-
position.
At the nominal chamber pressure of 10 bar, chemical equilibrium yields the
fluid properties compiled in Table 1.
3.3.2. Heat loss and composition
Given the extreme volume-to-surface ratios encountered in planar micro-
scale configurations, heat transfer to the structure can have a large impact
on the flow conditions [26]. In the case of the µCPS, the walls will heat up
during operation. Initially, this will lead to a substantially reduced temperature
in the combustion chamber, until a sufficiently heated structure reduces heat
10
flux. Removing heat from the chamber will influence the reaction and thus the
mixture composition.
Figure 3: CEA chemical equilibrium calculation of hydrazine decomposition; products de-
pending on temperature (10 bar chamber pressure).
Figure 3 shows how the equilibrium composition changes at a nominal cham-
ber pressure of 10 bar over a temperature range from 200 K to 1500 K. Depletion
of hydrazine is complete for all equilibrium states. If the chamber walls are adi-
abatic, a temperature of 1340.15 K is reached. As can be seen, this operation
point corresponds to completely dissociated ammonia. All other states shown
in Fig. 3 can be arrived at by subsequent removal or addition of heat. The equi-
librium composition remains unchanged for temperatures exceeding 1000 K,
recombination of NH3occurs for lower temperatures until hydrogen is depleted.
Note that N2H4will not be formed again in equilibrium once it is decomposed.
Exit velocity and thus efficiency depend on molar mass and reservoir temper-
ature, as shown in Fig. 4. The mean molar mass decreases monotonically with
decreasing NH3fraction, as more and more H2molecules are formed with rising
temperature. The isentropic exponent behaves in a more complex manner. The
small isentropic exponent of NH3is reduced with rising temperature. At the
same time, the mass fraction of NH3reduces as it is consumed and H2and N2,
with their relatively high isentropic exponents, are formed. Once no NH3is left,
11
Figure 4: Molar mass M, isentropic exponent γ, and resulting Isp for equilibrium composition.
γreduces monotonous with rising temperature. Finally, these effects combine
to a specific impulse that grows monotonously with rising temperature.
It can be seen that heat loss in the combustion chamber which leads to lower
temperatures will give rise to a change in gas composition and a reduction in
efficiency. Vice versa, if the gas is heated beyond the adiabatic flame temper-
ature, the specific impulse can be increased (e.g. electrothermally enhanced
hydrazine thruster).
Figure 5 shows the sensitivity of the maximum exhaust velocity on γ. The
molar mass is chosen exemplarily to cover the extremes and the mid-range value
achieved for hydrazine decomposition (see Fig. 4), the chamber temperature
corresponds to equilibrium hydrazine decomposition at 10 bar T0= 1340.15 K,
determined using CEA.
3.4. Analytical assessment of nozzle flow properties
The goal of this section is to derive simple expressions that allow us to esti-
mate the core flow Knudsen and Reynolds numbers from the chamber, through
the nozzle throat, to the nozzle exit. This can be useful in determining the
numerical models to be used for CFD simulations.
12
Figure 5: Influence of ratio of specific heats γon theoretical maximum exhaust velocity for
hydrazine decomposition at 1340.15 K.
3.4.1. Nozzle Knudsen number
The Knudsen number is a nondimensional measure of the deviation from
continuum behavior in a flow. It is defined as the ratio of the molecular mean
free path λand a characteristic length scale Lin the flow,
Kn = λ
L.(27)
The mean free path can be calculated as a function of the viscosity µ, the density
ρ, the temperature T, the molar mass M, and the gas constant R,
λ=µ
ρrπM
2RT .(28)
Then, the ratio of two Knudsen numbers in frozen flow can be expressed as
Kn0
Kn1=µ0
µ1ρ1
ρ0T1
T01
2L1
L0.(29)
In order to interpret this Knudsen number ratio in a nozzle flow, we express each
of the terms in Eq. (29) as a function of the pressure ratio assuming isentropic
expansion. Kinetic theory suggests that the viscosity ratio can be expressed in
terms of the temperature ratio [18] and thus a pressure ratio,
µ0
µ1=T1
T02
3
=p1
p02
3
γ1
γ
.(30)
13
γ1.2 1.35 1.4 1.67
p0/p1.77 1.86 1.89 2.05
λ01.58 1.54 1.53 1.47
Table 2: Ratios of pressure and mean free path at nozzle throat.
The density ratio yields
ρ1
ρ0=p1
p01
γ
,(31)
and the temperature ratio becomes
T1
T01
2
=p1
p01
2
γ1
γ
.(32)
Then, the ratio of the mean free paths can be written as
λ0
λ1=p1
p07γ
6γ
,(33)
and the Knudsen number ratio is
Kn1
Kn0=L0
L1p0
p17γ
6γ
.(34)
At the nozzle throat, denoted by , (T0/T ) = 1
2(γ+ 1), so that
Kn
Kn0=L0
Lγ+ 1
27γ
6(γ1)
.(35)
For a γof 1.35 this can readily be evaluated to
Kn
Kn0= 1.54 L0
L.(36)
Figure 6 shows the evolution of the ration of mean free paths from the
pressure ratio. With knowledge of the geometry, a local Knudsen number can
be determined. Throat conditions are compiled in Table 2.
This method allows to estimate the Knudsen number a priori using Table 1
with Eqs. (27), (28), (34), and (35). Here, we calculate the chamber Knudsen
number as Kn0= 0.0258, the throat Knudsen number as Kn= 0.0399, and
the exit Knudsen number as Kne= 0.2806. These results are consistent with
values from the literature obtained from simulations [7, 30].
14
Pressure ratio p0/p1
Ratio of mean free paths 1/ 0
100101102
5
10
15
20
25
30
= 1.2
= 1.35
= 1.4
= 1.67
4
1
2
3
Figure 6: Ratio of mean free paths in expanding nozzle flow from Eq. (33) (solid lines); throat
conditions from Eq. (35) (symbols); exit condition for µCPS (vertical dashed line).
4. Simulations
4.1. Conditions
The nominal engine discussed in [9, 10] has a mass flow of ˙m= 5 g/s and
a thrust of 10 mN. We calculate a number of test cases to study the impact
of the geometrical and thermo-chemical models, and to see whether the initial
design goals were met. Figure 7 shows the computational mesh; a quarter of the
physical domain is calculated, taking advantage of symmetries. The chamber
pressure is 10 bar with an exit pressure ratio of p0/pe= 30, ensuring continuum
flow, consistent with [7, 13, 6, 19].
Figure 7: Mesh
We consider two compositions, one in chemical equilibrium, and one with
ξ= 0.5. Computations of chemical equilibrium composition and properties have
15
ξ= 0.5 Equilibrium
Chemistry Frozen Frozen
YN2 0.5833 0.8738
YH2 0.0625 0.1257
YNH3 0.3542 0.0005
Table 3: Composition of the two considered states.
Case Geometry Properties Composition
I 2D planar inviscid equilibrium
II 2D rotational inviscid equilibrium
III 2D planar viscous equilibrium
IV 2D rotational viscous equilibrium
V 3D quarter viscous equilibrium
VI 3D quarter viscous, perfect gas equilibrium
VII 3D quarter viscous ξ= 0.5
Table 4: Overview of computational cases.
been carried out using the CEA code by Gordon and McBride [29, 12]. Table 3
shows species mass fractions of both states.
Table 4 gives an overview of the test matrix. In all cases, the composition
does not change. In order to assess the impact of composition, the chamber
temperature is held constant to the equilibrium temperature. To account for
wall heating during operation, a constant wall temperature of 800 K is set.
Table 5 summarizes these conditions.
The differences between the 2D planar, 2D rotational, and 3D quarter config-
urations are illustrated in Fig. 8, along with the respective boundary conditions.
In all cases, the contoured wall is a no-slip wall. Symmetry boundary condi-
tions are used based on the expected symmetric averaged flow field in a RANS
approach.
4.2. Effects of geometry and composition
Figure 9 compares flowfields for rotational 2D, planar 2D, and 3D compu-
tations, comparing viscous and inviscid results. It is impressive how any of
the simplified approaches Figs. 9(b) to 9(c) grossly overestimates the achieved
16
pressure outlet
total pressure
no-slip wall
symmetry planesymmetry plane
no-slip wall
(symmetry plane)
(a) 3D quarter and 2D planar. The
bottom is a no-slip wall in 3D, and a
symmetry plane in the 2D planar case.
no-slip wall periodic
axisymmetry axis
total pressure
pressure outlet
(b) Rotational. Computed is a 1
slice of the whole domain using pe-
riodic boundary conditions in az-
imuthal direction.
Figure 8: Boundary conditions.
Propellant N2H4(g)
Chemistry Frozen
T01340.15 K
Twall 800 K
p010.0 bar
p0/pe30.0
Table 5: Flow conditions.
velocity compared to the appropriate 3D computation, Figs. 9(e) and 9(f).
The difference is not merely quantitative: using a more realistic representa-
tion, the whole topology of the flow field is qualitatively changed, it no longer
expand continuously towards the exit. Instead, the axial velocity reaches a local
maximum after a quarter nozzle length downstream of the throat. Boundary
layers grow from all walls, a separation bubble with a recirculation zone forms
in the nozzle.
The effect of composition modeling shown in Fig. 10 is less pronounced but
noticeable. Figure 5 shows that the exit velocity will increase with decreasing γ
17
(a) I: 2D planar, inviscid.
(b) II:2D rotational, inviscid.
(c) III: 2D planar, viscous.
(d) IV: 2D rotational, viscous.
(e) V: 3D viscous, center plane.
(f) V: 3D isometric view of quarter nozzle.
(g) Axial velocity scale in m/s.
Figure 9: Comparison of geometrical model influence on velocity distributions.
18
and M. The ξ= 0.5 case exhibits poor performance, consistent with its higher
molar mass caused by hydrogen still being bound in the ammonia molecule. The
perfect gas case VI exhibits reduced performance compared to the base case V
because of the higher isentropic exponent, see Fig. 1.
(a) V: equilibrium (b) VI: perfect gas (c) VII: ξ= 0.5
(d) Axial velocity scale in m/s.
Figure 10: Influence of composition on velocity distribution.
Table 6 compares key parameters such as mass flow, thrust, specific impulse,
and axial exit velocity for cases I through VII. In addition, the overall maximum
axial velocity umax and the position x(umax) where it is reached are given. None
of the cases reaches the nominal mass flow of 5 g/s; only the 2D cases reach
thrusts exceeding 10 mN. This is likely caused by the unobstructed expansion,
allowing to reach the maximum velocity towards or at the end of the nozzle.
Notably, the viscous rotationally symmetric case develops substantial boundary
layers that interact when growing inwards, while they grow independently in
the planar case.
However, when the realistic 3D geometry with boundary layers growing from
all walls is taken into account, the predicted thrust decreases by 30%, reaching
a mere 7 mN in the equilibrium composition cases. Clearly, 2D simulations are
not suited for performance predictions of micro scale planar nozzles. We will
continue with a more in-depth analysis of case V, the highest fidelity model.
19
Case ˙m F Isp ueumax x(umax)
in mg/s in mN in s in m/s in m/s in mm
I 4.40 11.0 256.0 2457.3 2457.3 2.25
II 4.69 12.4 269.9 2665.5 2665.5 2.25
III 4.31 10.4 246.7 2423.6 2423.6 2.25
IV 4.61 10.5 233.3 2567.6 2568.4 2.08
V 4.39 7.09 164.8 1153.5 2013.7 0.49
VI 4.30 6.91 163.8 1115.2 1992.9 0.47
VII 4.73 4.20 148.2 1036.8 1854.0 0.55
Table 6: Results of test matrix computations.
4.3. Analysis of the baseline nozzle
Figure 11 shows the baseline 3D result of case V in more detail for further
analysis. The height of the nozzle is vertically stretched twentyfold to aid clarity.
Contours on the 5 vertical slices perpendicular to the nozzle axis are again axial
velocity. The arrows show the velocity vector. The red convoluted plane marks
the Mach 1 isosurface.
Following the flow from the reservoir (left) to the nozzle exit (right), we
see how the initial homogeneous velocity distribution is accelerated towards
the throat. Before passing through the throat, the flow is subsonic. At the
throat, the boundary layer acts to prevent the wall near part of the flow to
reach supersonic velocities. Downstream, the boundary layers grow, stronger
from the planar bottom plane than from the contoured side wall. Less than
half the nozzle height exhibits the undisturbed maximum velocity at the first
slice after the throat. Further downstream, this core is diminished until it has
disappeared in the last two slices. The velocity distribution does not change
after the core is gone, resembling fully developed pipe flow. At the mid of the
nozzle, the bottom boundary layer has expanded to take up more than half of
the nozzle height. Close to the nozzle exit, they grow together, almost choking
the whole cross section. Finally, the flow expands right at the exit towards the
boundary condition.
How does this qualitative view translate into actual nozzle performance? As
20
Figure 11: 3D view of quarter nozzle. Bottom and back planes are viscous walls, top and front
planes are symmetry axes. Contours are axial velocity in m/s. The red plane is the Mach 1
isosurface. The plot is vertically stretched twentyfold to aid clarity.
shown above, the acceleration of the flow is our ultimately desired action, as it
determines thrust. Contrary to classical macro scale rotational nozzles, the flow
in the small scale high aspect ratio nozzle does not accelerate all the way towards
the exit but reaches its velocity maximum closer to the throat. This lends itself
to the idea that a local evaluation along the central axis of the nozzle might
show a local thrust maximum, too. To do this, we integrate thrust according to
Eq. (17) across various cross sections of the nozzle. Figure 12 shows the result,
along with axial velocity, pressure, and the nozzle contour. Note that the thrust
scale does not start at zero. We see that the flow accelerates after the throat as
the pressure drops, until about x= 0.5 mm. Further downstream, the pressure
reaches a plateau, the velocity drops again. In the last third of the nozzle, the
pressure drops again, causing an inflection point in the velocity distribution:
the deceleration slows down and turns into acceleration at x= 2.0 mm.
The resulting flow is very different from classical nozzles. For both, thrust
increases after the nozzle throat is passed. However, lacking a further expansion
and acceleration, the thrust roughly stagnates at 0.4 mm <x<1.5 mm in the
planar micro nozzle. Towards the exit, the thrust further reduces to values below
the thrust at the throat–expansion through a mere pinhole could outperform a
21
Figure 12: Development of velocity, pressure, and resulting internal thrust along the central
axis.
nozzle designed after classical paradigms.
4.4. Truncated nozzle
The results shown in Fig. 12 suggest that a truncated nozzle might be the op-
timal solution in this case. Due to the large subsonic flow domains, a significant
upstream influence can be expected, so this might not be case after all. How-
ever, this is sufficient motivation to investigate this question with an additional
CFD computation on a new, truncated mesh. While the global thrust maxi-
mum occurs at x= 1.2 mm, a more compact nozzle with only minor penalty in
performance can be designed when cut at X= 0.5 mm.
Figure 13 shows the resulting flowfield. We see that the boundary layers
have not yet developed to the extent that they threaten a choking of the flow.
However, comparing the slice after the throat with the corresponding slice in
the full nozzle, Fig. 11, we see that the reached axial velocity is reduced.
In order to allow for a quantitative assessment of the relative quality of
both nozzles, Eqs. (17) to (20) have been evaluated and compiled in Tab. 7.
Nominal chamber pressure and mass flow have been used, hence camounts
22
Figure 13: Quarter flowfield of truncated nozzle. Bottom and back planes are viscous walls,
top and front planes are symmetry axes. Contours are axial velocity in m/s. The red plane is
the Mach 1 isosurface. The plot is vertically stretched twentyfold to aid clarity.
F Isp cFc rot ARARe
Nozzle in mN in s - in m/s - - - -
Contoured 7.1 144.8 1.07 1330 11.3 127.0 1.4 15.3
Truncated 9.2 187.6 1.38 1330 5.0 25.1 1.4 6.8
Table 7: Comparison of contoured full length and truncated nozzle.
to the identical value for both cases. Truncation of the nozzle has significantly
improved the performance of the engine. The expansion ratio =Ae/Ais
shown for both nozzles and compared to the ratio rot if the contour had been
used for a rotational nozzle instead. For expansion into vacuum, rot of the
contoured nozzle is a value one would traditionally choose. Finally, the aspect
ratio AR at throat and exit plane is defined as the ratio of width and depth.
23
5. Conclusion
Nozzles in µCPS, i.e. high aspect ratio, planar, micro nozzles, behave fun-
damentally different than classical, macro scale axisymmetric nozzles. Basic
assumptions of mostly isentropic flow with moderate boundary layers are re-
versed. Instead, a boundary layer dominated flow is found, essentially choking
the nozzle.
An analytical derivation of the core flow Knudsen number has been given,
allowing for an a priori estimation based on the pressure ratio which yields plau-
sible results. However, it may be more suitable in cases with a more substantial
isentropic core.
The classical recipe of adapting a nozzle for expansion into vacuum - i.e.
high expansion ratio realized over the required nozzle length to ensure only
moderate deflection - loses its significance. Instead, avoiding the axial build-up
of the boundary layers has become the dominant design constraint, calling for
short nozzles. In a way this is fortunate: as the etched depth is fixed, the cross
section only grows linearly with nozzle width, not with the square as is the case
for rotational nozzles, preventing the realization of high nozzles. Thus, new
approaches are needed for design and optimization.
We demonstrate that accurate modeling of variable exhaust gas properties
has a noticeable impact on performance prediction, and how heat loss to the
walls changes engine performance due to changing equilibrium composition.
Finally, we have to note that the analytically estimated Knudsen numbers
appear contradictory to the assumption in the literature that an exit pressure
of 1 kPa ensures continuum flow.
6. Acknowledgments
The research leading to these results has received funding from the Eu-
ropean Community’s Seventh Framework Programme (FP7/2007-2013) under
grant agreement No. 282948. Further information on PRECISE can be found
on www.mcps-precise.com.
24
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