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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 Pshs∇Tρs

ρ+ (P ~u)T

.(4)

The viscous stress tensor Pis modeled according to the Stokes hypothesis,

P=µ∇~uT+ (∇~uT)T−2

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, ωs≡0 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

Mm−1/2"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

γ−1

RT

M"1−pe

p0γ−1

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

pe→0 is

umax =s2γ

γ−1

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],

3N2H4→4NH3+ N2−336,280 J.(24)

Subsequently, NH3decomposes endothermally into N2and H2:

4NH3→2N2+ 6H2+ 184,400 J.(25)

The global reaction can be characterized by the reaction scheme

3N2H4→4(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

T0−2

3

=p1

p0−2

3

γ−1

γ

.(30)

13

γ1.2 1.35 1.4 1.67

p0/p∗1.77 1.86 1.89 2.05

λ∗/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

γ−1

γ

.(32)

Then, the ratio of the mean free paths can be written as

λ0

λ1=p1

p07−γ

6γ

,(33)

and the Knudsen number ratio is

Kn1

Kn0=L0

L1p0

p17−γ

6γ

.(34)

At the nozzle throat, denoted by ∗, (T0/T ∗) = 1

2(γ+ 1), so that

Kn∗

Kn0=L0

L∗γ+ 1

27−γ

6(γ−1)

.(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 c∗amounts

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 AR∗ARe

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/A∗is

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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