Timmy_Cheung_Master_Thesis.

Timmy_Cheung_Master_Thesis.
Nitrogen Catalytic Recombination
on Copper Oxide in Tertiary Gas
Mixtures
T. M. Cheung
April 27, 2015
Copyright © T. M. Cheung
All rights reserved. No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by
any information storage and retrieval system, without the prior permission of the author.
Delft University of Technology
Faculty of Aerospace Engineering
Nitrogen Catalytic Recombination
on Copper Oxide in Tertiary Gas
Mixtures
Master Thesis
For obtaining the degree of Master of Science in Aerospace
Engineering at Delft University of Technology
Timmy Man-Choung Cheung
April 27, 2015
Preface
This master thesis represents the work that I have conducted at KAIST (Republic of Korea) from July
2013 until December 2013 and August 2014 until March 2015. During these period I have studied the
field of aerothermodynamics, learned the necessary steps to conduct experiments individually, explored
and studied various numerical programs to fulfill the goal of this research. This thesis is the final step
in the journey to acquire the degree of Master of Science in Aerospace Engineering at Delft University
of Technology.
This report consists of three main parts. One part discusses the shock tube and the fabrication of
heat transfer gauges. The second part introduces the numerical programs which are used to simulate
the conditions inside the shock tube and around the model, whereas the third part covers the theory
regarding heat flux and catalycity. These three parts together form a chain of required steps to acquire
the desired results for this thesis.
I would like to express my sincere gratitude towards my two supervisors: Assistant Professor Ferry
Schrijer (TU Delft) and Assistant Professor Gisu Park (KAIST). Both have guided me through this
thesis with their priceless experience and knowledge concerning hypersonic flows and experimental aerodynamics. Moreover, I would like to thank Professor Chul Park (KAIST) for his valuable knowledge in
the theoretical aspect of hypersonic aerothermodynamics. Without the support of these professors the
achievements made in this thesis would have never been realizable.
Additionally, I would like to thank KAIST for providing me the opportunity to use their facilities for
this project, while they covered for all the costs necessary to conduct the experiments. To my parents
and friends, thank you for all the great support and the unforgettable moments in the Netherlands and
in Korea.
This work has been financially sponsored by the aerodynamics group of TU Delft and the Universiteitsfonds Delft (Ufd) with application number #24.14.
Timmy Cheung
Delft, April 27th 2015
i
List of Symbols
Roman Symbols
A
a
Be
c
c
c
c0
cp
cv
D
DA
De
D0
Da
E
E
f
f
G
g
ḡ
H
h
h̄
hD
hR
J
j
K
k
kB
kf
kw
Le
l
l
l
l
M
M
Cross-sectional area of arc-jet nozzle throat
Speed of sound
Rotational constant
Average molecular speed
Thermal capacity
Local light speed
Light speed in vacuum
Specific heat at constant pressure
Specific heat at constant volume
Diffusion coefficient
Dissociation energy in the adsorbed phase
Rotational constant
Dissociation energy
Damköhler number
Surface activation energy
Electronic state energy
Dimensionless stream function
Friction factor
Statistical weight for molecule
Statistical weight for atom
Dimensionless frozen stagnation enthalpy
Total enthalpy
Enthalpy
Frozen enthalpy
Dissociation enthalpy
Heat of recombination
Total electronic angular momentum quantum number
Net mass flux of diffused atoms
Gladstone-Dale constant
Thermal conductivity
Boltzmann’s constant
Forward reaction rate constant
Catalytic rate
Lewis number
Length
Current
Chapman-Rubesin constant
Angular momentum quantum number
Mach number
Average molecular weight of gas mixture
[m2 ]
[m/s]
[−]
[m/s]
[J/(kg · K)]
[m/s]
[m/s)]
[J/(kg · K)]
[J/(kg · K)]
[m2 /s]
[J/mol]
[−]
[J]
[−]
[J]
[J]
[−]
[−]
[−]
[−]
[−]
[J/kg]
[J/kg]
[J/kg]
[J/kg]
[J/kg]
[−]
[kg/(m2 · s)]
[m3 /kg]
[W/(m · K)]
[J/K]
[m3 /(mol · s)]
[m/s]
[−]
[m]
[A]
[−]
[−]
[−]
[kg/mol]
ii
Mi
m
m
mi
ṁ
NA
n
n
n
n
Nu
p
Pr
Pr
Q
Q
q
R
R
R
Runi
Re
r
r0
re
S
s
s
Sc
St
T
T?
Te
t
u
V
v
v
Xi
x
x
y
z
z(0)
iii
Molecular weight of species i
Average mixture mass
Order of catalytic reaction
Particle mass of species i
Mass flow rate in arc-jet
Avogadro’s constant
Number density
Refractive index
Concentration
Principle quantum number
Nusselt number
Pressure
Prandtl number
Frozen Prandtl number
Partition function
Cumulative heat input
Heat flux
Characteristic distance or nose radius
Resistance
Specific gas constant for air
Universal gas constant
Reynolds number
Radius
Radial distance from axis of symmetry
Internuclear distance
Entropy
Reduced streamwise distance parameter
Spin quantum number
Schmidt number
Stanton number
Temperature
Normalized temperature
Electronic energy level or molecules
Time
Velocity along wall
Voltage
Velocity perpendicular to wall
Fundamental vibrational frequency
Molar fraction of species i
Distance along the wall from stagnation point
Distance along the shock tube
Distance perpendicular to wall measured from wall
Ratio of mass fraction
Dimensionless atomic concentration at wall
[kg/mol]
[kg]
[−]
[kg]
[kg/s]
[mol−1 ]
[m−3 ]
[−]
[mol/m3 ]
[−]
[−]
[N/m2 ]
[−]
[−]
[−]
[J/m2 ]
[W/m2 ]
[m]
[Ω]
[J/(K · kg)]
[J/(K · mol)]
[m−1 ]
[m]
[m]
[Å = 10−10 m]
[J/K]
[−]
[−]
[−]
[−]
[K]
[−]
[cm−1 ]
[s]
[m/s]
[V]
[m/s]
[Hz]
[−]
[m]
[m]
[m]
[−]
[−]
Greek Symbols
α
α
αe
β
βe
βR
χ
δ
δ
γ
γ
η
Λ
µ
Ω?ij (1,1)
(2,2)
Ω̄ij
ωe
ωe xe
ωe ye
ψ
ρ
σ
σ
τ
θr
ϕ1
ϕ2
ξ
Mass fraction
Thermal diffusivity
Rotational constant
Stream velocity gradient
Rotational constant
Coefficient of resistivity
Variable parameter for ϕ2
Boundary Layer Thickness
Penetration depth
Catalytic efficiency
Specific heat ratio
Internal energy
Potential parameter
Boundary layer coordinate normal to wall
Compressibility correction
Dynamic viscosity
Reduced average collision cross-section
Average collision integral
Vibrational constant
Vibrational constant
Vibrational constant
Stream function
Density
Stefan-Boltzmann constant
Collision diameter
Shear stress
Rotational constant
Correction factor for catalytic effects of binary gas mixture
Correction factor for catalytic effects of tertiary gas mixture
Reduced streamwise distance parameter
[−]
[m2 /s]
[−]
[s−1 ]
[−]
[K−1 ]
[−]
[m]
[m]
[−]
[−]
[J]
[−]
[−]
[−]
[kg/(m · s)]
[−]
[m2 ]
[−]
[−]
[−]
[−]
[kg/m3 ]
[W/(m2 · K4 )]
[m]
[N/m2 ]
[−]
[−]
[−]
[−]
Superscripts
0
Derivative with respect to η
iv
Subscripts
af ter
C
D
e
el
eq
i
j
j
pre
rot
s
se
tr
vib
w
0
1
1
2
2
3
3
4
∞
v
Behind the standing shock
Conduction
Diffusion
Boundary layer edge (In Goulard’s theory: outside the boundary layer)
Electronic
Equilibrium
ith component of mixture
j th component of mixture
j th electronic energy level
Before the standing shock
Rotation
Stagnation
Boundary layer edge
Translation
Vibration
Wall
Settling chamber of arc-jet
Test gas prior to primary shock
Nitrogen atoms
Test gas after primary shock
Nitrogen molecules
Driver gas after expansion waves
Krypton molecules
Driver gas prior to expansion waves
Freestream
Contents
Preface
i
List of Symbols
ii
Summary
1
1 Introduction and Relevance
1.1 Historical Perspective . . . . . . .
1.2 Flow Conditions . . . . . . . . . .
1.3 Relevance of Heat Flux . . . . . .
1.4 Aspects Affecting Heat Flux . . . .
1.5 Research Question and Objectives
1.6 Report Outline . . . . . . . . . . .
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2 Heat Flux and Catalycity
2.1 Heat Flux . . . . . . . . . . . . . . . . . . . .
2.2 Fundamentals of Catalycity . . . . . . . . . .
2.3 Theoretical Research in Catalycity . . . . . .
2.4 The Flow Conditions at the Stagnation Point
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3 Experimental Research in Catalycity
3.1 Side-arm Reactor . . . . . . . . . . . . . . . . . . .
3.2 Shock Tube/Tunnel . . . . . . . . . . . . . . . . .
3.3 Arc-jet Facility . . . . . . . . . . . . . . . . . . . .
3.4 Usage and Uncertainties of Copper Calorimeters in
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11
11
11
14
21
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Arc-jet Facilities
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27
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4 Experimental Equipment
37
4.1 Shock Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Heat Transfer Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Shadowgraph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 Determination of Flow Conditions
54
5.1 Shock Tube Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2 Equilibrium Composition Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3 Flow Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6 Data Reduction
6.1 Pressure Measured by Piezo-electric Sensor . . . . . . . . . . .
6.2 Surface Heat Flux Measured By Thin-Film Gauge . . . . . . .
6.3 Diffusive and Convective Heat Flux for Tertiary Gas Mixtures .
6.4 Comparison Between Binary and Tertiary Gas Mixture Theory
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59
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64
vi
7 Results
65
7.1 Pressure Measurements and Shadowgraph Visualization . . . . . . . . . . . . . . . . . . . 65
7.2 Heat Flux Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
7.3 qD /qC vs. γw Diagram and Catalytic Efficiency . . . . . . . . . . . . . . . . . . . . . . . 69
8 Discussion
72
8.1 Flow Condition A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
8.2 Flow Conditions B & C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9 Sensitivity Analysis & Uncertainties
75
9.1 Pitot Pressure and Total Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.2 Collision Integral and Binary Diffusion Coefficient . . . . . . . . . . . . . . . . . . . . . . 76
10 Conclusions
78
10.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
10.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Bibliography
A Thermochemical Equilibrium and Boundary Layer
A.1 Species Concentrations . . . . . . . . . . . . . . . . .
A.2 Flow Conditions . . . . . . . . . . . . . . . . . . . .
A.3 Distance and Damköhler Number . . . . . . . . . . .
A.4 Boundary Layer Thickness . . . . . . . . . . . . . . .
80
Thickness
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83
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B Shock Tube Program Comparison
88
B.1 Stube9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
B.2 Comparison Between L1d2 and Stube9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
C Equilibrium Composition Program Comparison
90
C.1 Therm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
C.2 Comparison between CEA and Therm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
C.3 Comparison Between Therm and Eqtab . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
D Heat Flux Conversion Comparison
93
E Heat Flux Profile Measurements
95
F Alternative Pressure Model
vii
101
Summary
A reentry vehicle returning back to Earth experiences a significant temperature increase. This temperature increase is a result of the enormous heat flux the vehicle is exposed to, which originates from
the transfer of kinetic energy to heat. Moreover, the high temperature results in the dissociation of air
molecules which complicates the situation. The recombination of these atoms is an exothermic process
which releases an additional amount of energy. This recombination requires a third body which can
either be another air particles or the surface particles of the vehicle.
To properly design these vehicles it is essential to test them in ground facilities. One of these widely
used facilities is an arc-jet, which is capable of reaching temperatures of several thousands Kelvin and
relatively high velocities for an extensive period of time (i.e. in the order of seconds). The problem
that arises with arc-jets is the variation of the enthalpy from the centerline towards the walls. It is
this enthalpy that dictates the flow condition the model experiences, since enthalpy is a quantity that
specifies energy. To determine the centerline enthalpy, the easiest and often applied method is by using
copper slug calorimeters.
A copper calorimeter is essentially a slug made of copper with a connection from which the temperature is acquired. The temperature is converted to heat flux values and with the aid of stagnation heat
transfer theories is the enthalpy determined. To use these stagnation heat transfer theories, a parameter
called “catalytic efficiency” is required. This parameter specifies the ratio of the mass flux of formed
molecules to the mass flux of the atoms hitting the surface.
The copper slug however experiences oxidation (hereby forming copper oxide) and roughens when
exposed to the flow in the arc-jet. It is because of these changes that there is a large discrepancy between catalytic efficiencies in the literature. The focus of most literature is either on the catalytic effect
of oxygen with respect to copper or copper oxide. Since one of the main components of air is nitrogen
it is essential to also investigate this case. However, so far only copper has been covered for nitrogen.
In this thesis the catalytic efficiency of nitrogen is determined on copper and copper oxide in a tertiary gas mixture. The tertiary gas mixture is created by mixing the nitrogen with krypton which is an
inert gas. This is done to be able to achieve sufficient amount of dissociation inside the shock tube.
The shock tube, that is used, is located at the hypersonic lab in KAIST (South Korea). The experiments are conducted using helium as driver gas and a test gas consisting of 65% krypton and 35%
nitrogen. Three different flow conditions are covered which are denoted with the letters A, B, and C,
with respectively 1%, 33.3%, and 54.3% dissociation. In the test section of the shock tube a flat-disk
cylinder model with a diameter of 12 mm was used, which contains a flush-mounted thin-film gauge.
The thin-film gauges are manufactured in this thesis and contain a thin strip of platinum with a
width of approximately 0.5 mm. After manufacturing these gauges they are mounted inside the models
and calibrated. The probe is then coated using the electron beam method which vaporizes the source
material and condensates on the sample. The probes are separated in four batches, each with a different
set of coating:
1. 1000 nm SiO2
2. 1000 nm SiO2 and 180 nm black CuO
3. 1000 nm SiO2 and 180 nm brown CuO
4. 1000 nm SiO2 and 180 nm Cu
The difference between the black CuO and brown CuO is the amount of copper atoms present in the
mixture, the latter being richer in copper.
1
Using this experimental setting the different coatings were exposed to the three flow conditions. The
temperature data acquired from these experiments were converted to heat flux values with the onedimensional heat diffusion equation.
These heat flux values were then matched with the stagnation heat transfer theories mentioned earlier
to acquire the catalytic efficiencies. The necessary parameters for these theories are determined numerically, with the exception of the pitot pressure which is measured using a piezo-electric sensor. A series
of numerical programs are used to acquire the flow parameters: a Lagrangian quasi-1D flow solver called
“L1d2” from the CFCFD group of the University of Queensland (Australia) is used for determining the
flow conditions inside the shock tube, an equilibrium composition calculator called “Eqtab” of Prof. Chul
Park of KAIST to determine the flow conditions at the boundary layer edge of the model behind the
standing shock, and a chemical equilibrium program called “CEA” of NASA to determine the conditions
at the wall of the model.
Combining the heat flux measurements with the stagnation heat transfer theories the catalytic efficiencies were found to be between 6.10 · 10−4 and 1.36 · 10−3 for black CuO, 9.50 · 10−4 for brown CuO,
and 5.29 · 10−2 for Cu. Each of these values have a relatively large uncertainty not only from the fluctuations in the heat flux measurements, but also from the changing sensitivity of the curve that relates
the heat flux ratio and the catalytic efficiency. For Cu the heat flux value is located in a very insensitive
region of the curve hence it is rather safer to conclude that its efficiency is between 3.07 · 10−3 and
5.29 · 10−2 .
Additionally all the values potentially suffer from uncertainties in the numerical values. From the
sensitivity analysis it was found that a 10% deviation of the measured pitot pressure changes the catalytic efficiencies of copper oxide with less than 10%. However, a 10% discrepancy in the total enthalpy
results in a 50% increase of the catalytic efficiencies of copper oxide. A combination of 10% error for
both parameters together can even yield a 61% error in the catalytic efficiency of copper oxide.
Further research regarding the catalytic efficiency of copper oxide should focus on the roughness effect
of the model, which has been neglected here by polishing the thin-film gauges during the manufacturing
process. Moreover, the wall remained at temperatures near room temperature during the shock tube experiments, which is not representative of the temperature that is reached during real flight. Although it
is known that the catalytic efficiency increases with temperature, this relation has not been investigated
quantitatively for copper or copper oxide.
The last recommendation for further research concerns the coupling between oxygen and nitrogen
for copper and copper oxide. This requires the stagnation heat transfer theory to be rewritten for a
gas mixtures consisting of five or more species. Only in this case the catalytic efficiency of air can be
determined properly for copper oxide.
2
1
Introduction and Relevance
1.1
Historical Perspective
The sky has always had an important role in the history of mankind. Astonished by the beauty of the
night sky, famous scientists – like Galileo Galilei, Johannes Kepler, and Isaac Newton – started to observe
and study it. Not only stars, but also comets, meteoroids and meteors were observed. With the limited
technology available, scientists attempted in finding patterns by approaching it mathematically.
The interest in space did not cease as time passed by. On the contrary, mankind started to fantasize
about interplanetary travel by escaping from the Earth. Unfortunately the technology required for
accomplishing this was still absent. It was only after the accomplishment of the Wright brothers in 1903
that the sky was within reach. Not long after that the First World War was a fact, resulting in the first
large scale application of aircrafts. However, the role of these aircrafts was mere and mainly used for
observations and coordinations of other weapons such as artillery.
After the First World War efforts were made to improve different aspects of aircrafts, and speed was
one of these. Better aerodynamic shapes were found and larger (propeller) engines were constructed.
It was only after the invention of the jet engine in 1939, that led to the first aircraft which is powered
purely by a turbojet: the Heinkel He 178 (Fig. 1.1). The development of the jet engine accelerated the
development of aircrafts resulting in aircrafts that could fly faster than before.
During the Second World War the air force had a more important role compared to the First World
War. The outcome of the battlefield was often dependent on the strength of the air force. Nazi Germany
was heavily investing in the development of long-distance rockets which could be used as weapons. The
infamous V-2 rockets were used by Nazi Germany at the end of the war to attack London, cruising at
5,760 km/h (=1,600 m/s) or Mach 6.
It was only after the war that the United States and the Soviet Union started their own missile
program, using the knowledge of the German scientists who fled to either one of the two countries.
Their knowledge formed the foundation of the two programs. The political and the military atmosphere
became coarse and the cold war was a fact where the two superpowers were involved in an competition
to become the strongest and most powerful country.
This battle did not only take place on Earth. Space became another arena for their competition which
is known as the Space Race. The Soviet Union won the first round on October 4 1957 by successfully
launching the first artificial satellite in the history of mankind: Sputnik 1 (Fig. 1.2). The batteries only
lasted until October 26 1957 and on January 4 1958 – after 92 days – the orbit of Sputnik 1 has decayed
so much that it reentered the atmosphere and burned up completely. While Sputnik 1 was still orbiting
the Earth, the Soviet Union was already preparing for Sputnik 2. On November 3 1957 Sputnik 2 was
launched carrying a dog. Although the capsule orbited the Earth for 162 days until it reentered the
Earth’s atmosphere, the dog survived only for a few hours.
In the time that the Soviet Union was working on the Sputnik, the United States were attempting
to launch their own satellite. Only after the launch of Sputnik 2, the United States managed to have a
success on February 1 1958 with the Explorer 1.
The race continued with highlights such as the launch of the Luna 2 of the Soviet Union in 1959
which was the first probe to hit the moon, and the Soviet cosmonaut Yuri Gagarin who became the first
person to orbit the Earth in April 1961. The adventure only took 108 minutes at an altitude of 327 km.
The United States were getting anxious and shocked by the rapid achievements of the Soviet Union,
hence on May 25 1961 President John F. Kennedy held the famous speech where he announced that the
United States had set the goal to land a man on the moon and return safely by 1970, resulting in the
well-known Apollo program.
3
1.1. Historical Perspective
Figure 1.1: Heinkel He 178, the first turbojet air- Figure 1.2: Sputnik 1, the first man-made object in
craft. [1]
space. [2]
The United States eventually succeeded with Apollo 11 where Neil Armstrong was the first person to
set foot on the moon in July 20 1969. The previous Apollo missions were used to test the necessary technology and to explore the moon. The Apollo program followed up the Mercury and Gemini programs,
all focused on human spaceflight. The experience and knowledge acquired from these two programs were
of great value to the Apollo program.
All these three programs shared the aspect that reentry was required. The reentry vehicle was an
aerodynamically stable capsule shaped as a blunt body, for example see Fig. 1.3 and 1.4. Because of
this shape it had a low lift-over-drag ratio which led to a ballistic trajectory. Although this trajectory
resulted in a fast reentry, the capsule had to face enormous heat loads since deceleration started to occur
at lower altitudes where the air density is higher. However, due to the short reentry time it was possible
to simply increase the protective layer at the blunt part of the body which would function as a heat sink.
Figure 1.3: The Mercury-Atlas 9 capsule called
Faith 7. [3]
Figure 1.4: The Apollo 11 command module. [4]
The Soviet Union also developed reentry vehicles based on the same principle for manned programs.
One of the vehicles was the Vostok reentry module used by Yuri Gagarin. A more famous vehicle is the
Soyuz module which fourth generation is still in operation nowadays.
In the years after Apollo 11 the United States had a few more Apollo programs where they landed
again on the moon. The Soviet Union dismantled its lunar program and on July 17 1975 both superpowers declared an end to the space race. Despite this the space technology continued its development
although at a lower pace.
After the Second World War the National Aeronautics and Space Administration (NASA) started working on a large number of secret projects to develop new technologies and aerodynamic concepts, which
4
Chapter 1. Introduction and Relevance
were called X-planes. A good example is the X-15 (Fig. 1.5) which had its first flight in 1959. It was
the first manned hypersonic aircraft being able to fly at Mach 6.7. Another important project was the
X-20 Dyna-Soar (Fig. 1.6), which introduced an revolutionary design to reentry vehicles compared with
the blunt bodies.
Figure 1.5: The North American X-15. [5]
Figure 1.6: An artist impression of the X-20 DynaSoar. [6]
In contrast with the previously used capsule shaped bodies, the idea with these X-planes was to use
a lifting body which would be able to reach high lift-over-drag ratios. This would enhance the range of
the reentry vehicle hereby increasing the flexibility of missions since there would be more control over
the vehicle. Moreover, the lifting body would be able to follow a gliding reentry trajectory which would
significantly lower the heat loads compared to the capsule shaped bodies, because the vehicle would
start to reduce its velocity at higher altitudes where the air density is lower. A disadvantage is that the
heating time would be increased due to this type of trajectory.
The problem faced with this concept was that the geometry of the vehicle became more complex since
lift has to be generated by the body, and control surfaces (such as flaps or rudders) were introduced.
This increase of complexity raises the risk of failure of the vehicle in the harsh environment experienced
during reentry. Moreover, although the overall heat load might be lower for these vehicles, locally – such
as at sharper edges – the heat load could increase tremendously. These local heat loads would have to
be considered carefully to prevent failure of the vehicle.
Although the X-20 has never been built, the acquired knowledge had an important role in the development of future reentry vehicles. The most famous program, the Space Transportation System, is
a result of this knowledge, featuring a reusable reentry spacecraft launched for the first time on April
12 1981: the Space Shuttle Orbiter. It is considered to be the most successful spacecraft because it has
been launched 135 times in total, distributed among five vehicles.
Shown in Fig. 1.7 is the Space Shuttle Atlantis from which it can be seen that it had a delta wing
shape. The belly of the vehicle was covered with insulting tiles which served as a thermal protection
system. By reentering the atmosphere with a high angle of attack, the effective surface area would be
large, hereby slowing down the vehicle. Despite the experience and knowledge gained from the Apollo
and the X-program which resulted in more accurate calculations of the heat loads, it was – according
to Iliff and Shafer [7] – witnessed after the first few flights that the values were larger than initially
predicted. Luckily the heat loads were still within the limits. Because replacement of the heat tiles was
necessary after each flight the costs were a lot higher than initially anticipated, resulting in the end of
the program in 2011 after 30 years of service.
The Space Shuttle Orbiter was capable of bringing a payload of almost 25,000 kg to low earth orbit.
Not only a large number of experiments, supply, and parts were delivered to the International Space
Station (ISS), also the Hubble Space Telescope was brought in orbit by the Space Shuttle.
While the Space Shuttle Orbiter was mainly used as a transportation system, there were a number of
vehicles designed for the mere goal of understanding the reentry conditions better. As witnessed in
the design of the Space Shuttle Orbiter, the conditions during reentry could not be calculated correctly
beforehand hence validation data is necessary.
A vehicle that fits in this group is the European eXPErimental Reentry Testbed (EXPERT) vehicle
of European Space Agency (ESA), shown in Fig. 1.8. Functioning as a testbed it can carry different
experiments on-board to obtain aerothermodynamic data during each re-entry, which could be used for
validation purposes of computer models and experiments.
14 difference experiments were chosen to be carried by the vehicle, which can be grouped in four
scientific disciplines. The surface and gas-phase chemistry, and the hypersonic flow phenomena are two
of these disciplines which are crucial to understand the conditions experienced during reentry. Moreover,
any transition of the flow from laminar to turbulent is worthwhile to investigate since it would change the
nature of the flow. And at last, different materials can be tested in the high temperature environment.
5
1.2. Flow Conditions
Figure 1.7: The Space Shuttle Orbiter Atlantis landing on the 121st shuttle flight. [8]
Although it was planned to be launched in 2011, it has currently still not been launched due to
problems with the availability of Russian launchers.
Another project that has been launched successfully by ESA is the Intermediate eXperimental Vehicle
(IXV) for the same purpose as the EXPERT. While the EXPERT has the well-known conical shape, the
IXV is a lifting body with no wings but with thrusters and aerodynamic surfaces (Fig. 1.9). Not only
can IXV acquire aerothermodynamic data which will eventually be used to understand the hypersonic
regime better, the complex structure has thermal protection materials and complex design solutions to
cope with the heat which need to be validated.
Figure 1.8: A replica of the EXPERT vehicle. [9]
1.2
Figure 1.9: An artist drawing of the IXV. [10]
Flow Conditions
Although each reentry vehicle has its own distinctive shape and features, they all have in common that
they experience all flight regimes starting with the hypersonic regime and ending with the subsonic
regime. An altitude-velocity diagram is shown in Fig. 1.10 where the differences between the high and
low lift trajectories are shown. As can be seen, the former stays for a longer time decelerating at the
higher altitudes, whereas the latter penetrates deeper into the atmosphere before it starts to decelerate.
Also the Mach numbers are indicated in this diagram. At the beginning of the reentry, the Mach
number can be as large as 30 and it reduces when the altitude drops. The constant Mach number lines
are however not perfectly vertical which is caused by the varying speed of sound due to the variation of
the air temperature with altitude.
In Fig. 1.11 the unit Reynolds number is also shown in the altitude-velocity diagram. The Reynolds
number increases from a high altitude to a lower altitude, because the density increases with decreasing
altitude. Although the vehicle decelerates, this is significantly smaller than the density increase, hereby
effectively increasing the Reynolds number.
The transition Reynolds number forms the boundary between the laminar and turbulent flow expe5
−1
as shown in Fig. 1.11. From this figure it can
rienced by the vehicle. This occurs around Re
l = 10 m
also seen that it would occur around an altitude of 50 km for a 1 m long vehicle.
6
Reynolds number per meter, obtained from [81]. Note that the higher-altitude
portions of the flight trajectories experience combined conditions of high Mach
number and low Reynolds number—conditions that accentuate the effects of
hypersonic viscous flows. Indeed, for most of the reentry trajectory a hypersonic
vehicle is going to experience important Reynolds-number effects. Also note that
a purely arbitrary transition Reynolds number of 10^ is assumed, so that regions
of purely laminar flow and of turbulent flow for a 10-m-long vehicle are identified
on the right of Fig. 6.3. The main thrust of Fig. 6.3 is to indicate that viscous
effects are important in hypersonic flight; such viscous effects are the subject
of Part 2. Again, emphasis is made that only the purely viscous effects of viscosity and thermal conduction are highlighted in Part 2; the effects of high
Chapter 1. Introduction and Relevance
HYPERSONIC SHOCK AND EXPANSION-WAVE
RELATIONS 37
km I
Lifting reentry from orbit
0
1
2
3
4
5
6
7
8
9
10
km/s
V
Figure 1.10: An altitude-velocity diagram indicat- Fig. 6.3 Velocity-altitude map, with superimposed lines of constant unit Reynolds
number (from
Koppenwallner
181]).
ing the Mach numbers and the differences between Figure
1.11:
An altitude-velocity
diagram indicata high and low lift trajectory. [11]
ing the magnitude of the Reynolds numbers. [11]
In the hypersonic limit
Generally the Mach number is used to split up the trajectory in different flight regimes:
• Subsonic: M < 0.8.
• Transonic: 0.8 < M < 1.2.
• Supersonic: 1.2 < M < 5.0.
• Hypersonic: 5.0 < M .
Comparing this with Fig. 1.10 it canIn thebehypersonic
concluded
that a reentry vehicle spends most of its flight in
limit
and earlier
for small 0
the hypersonic regime. Although the
mentioned boundary between supersonic and hypersonic
flow at M = 5 seems as a hard line, this is absolutely not true. The difference between supersonic and
hypersonic flow is more than just the Mach number: the temperature is another important quantity.
Fig. 2.2
Oblique shocli-wave geometry.
When a vehicle moves through the air the molecules are pushed away. This introduces an energy transfer
from kinetic to internal energy, hereby increasing the temperature of the particles. During operations
below a Mach number of 5 this increase in temperature can be neglected, however, in the hypersonic
speed regime this temperature rise is so severe that the behavior of the air particles changes. Not only the
molecules start to vibrate substantially (according to Park [12] hereby increasing the degree of freedom
of the diatomic molecules from 5 to 7) which causes the air to be no longer a calorically perfect gas, also
dissociation will take place in which molecules will be split up into atoms. At around 2500 K oxygen
starts to dissociate, and at around 4000 K nitrogen dissociation starts, as illustrated in Fig. 1.12.
Figure 1.12: The ranges of oxygen and nitrogen dissociation for air at 1 atm pressure. [11]
These phases are shown in Fig. 1.13 from which it can be seen that a reentry vehicle is exposed to
different types of flow during its descent. These changes in the flow composition are one of the many
factors that make the computation for the hypersonic regime complex.
7
equa
with
press
also
1.3. Relevance of Heat Flux
Figure 1.13: An altitude-velocity diagram indicating the flight regimes where chemical reactions occur. [11]
1.3
Relevance of Heat Flux
From the previous section (Section 1.2) it was elaborated that the high temperature has a serious effect
on the thermodynamical properties of the flow, and thus on the flow field. Because of the high Mach
number, a thermal boundary layer is present which introduces a heat transfer to the body due to the
high temperature.
This heat flux causes many problems in the design of reentry vehicles. As witnessed with the Space
Shuttle Orbiter where the heat flux could not be determined accurately enough. This resulted in an
enormous increase of the operational costs caused by the constant replacement of the heat tiles.
The Space Shuttle Orbiter was fortunate enough that the inaccuracy of the heat flux calculations did
not result in a failure of the vehicle. What could have happened was shown in the disaster of the Space
Shuttle Columbia in 2003. During the launch of the shuttle a piece of foam insulation broke off and hit
the leading edge of the left wing. The damage allowed the hot gases to penetrate inside the structure of
the wing hereby weakening it. Since it was chosen for a cold structure philosophy for the Space Shuttle
Orbiter, the structure could not cope with the loads and eventually it resulted in the destruction of the
entire vehicle.
1.4
Aspects Affecting Heat Flux
The example of the Space Shuttle Columbia in the previous section shows the importance of properly
determining the heat flux transferred to the vehicle. However, the heat flux dependents on many variables.
In this section the aspects affecting the heat flux will be elaborated and discussed.
The shape of the vehicle has a large influence on the heat flux. A sharp nose or sharp edges on a
vehicle experience a higher heat flux than blunt bodies, since in the latter case the heat can be distributed
over a larger area. Fay and Riddell [13] presented the relation between the heat flux and the nose radius
as: q ∝ R1 . This will be discussed further in Section 2.3.
The state of the boundary layer also has a great influence on the heat flux. Not only there is a velocity
boundary layer, also there is a thermal boundary layer as mentioned in Section 1.3. Whether or not
these boundary layers have the same thickness depends on the Prandtl number which is a dimensionless
number being the ratio of the kinematic viscosity and the thermal diffusivity.
The thickness of the boundary layer can vary depending on the Reynolds number as described by
Fig. 1.11. Especially when the boundary layer has changed from laminar to turbulent the heat flux can
be three times larger [11]. For this reason it is desired to know where transition occurs in the flow.
However, this is a highly complex phenomenon [11] which depends on the Mach number, disturbances
in the environment, Reynolds number, angle of attack, nose bluntness, wall temperature, and wall
roughness.
Another aspect of the boundary layer is related to the shape of the vehicle: separation. When a flow
separates and reattaches downstream, it will locally introduce a shock wave. This “shock wave boundary
layer interaction” is a topic where a lot of research is conducted to truly understand and simulate that
situation. This high interest is due to the enormous increase of heat flux locally where reattachment
8
Chapter 1. Introduction and Relevance
occurs. The exact location of reattachment and the magnitude of this increase of heat flux cannot be
determined accurately yet, especially due to flow unsteadiness.
The third and last aspect that influences the heat flux is the surface of the body, often also referred
to as the wall. The wall can partially absorb the energy of the dissociated air particles which can result
in the recombination of these particles. This phenomenon is known as catalycity. Because recombination
is an exothermic process there will be an additional increase of the heat to the wall which increases the
heat flux.
The amount of recombination is affected by the composition of the wall material. It is known that
metals have a positive effect on the amount of recombination whereas ceramics have a negative effect. [14]
Moreover, the roughness of the wall also has a prominent role. A rougher wall results in a larger amount
of recombination compared to a smooth one. More can be found in Section 2.2.
The increase in additional heat flux is denoted as the diffusive heat flux. It is caused by the concentration gradient near the wall that is created by the catalytic recombination. As will be seen in Section 2.3
the diffusive heat flux can have a considerable effect on the total heat flux which cannot be neglected.
1.5
Research Question and Objectives
The concept of catalycity has been mentioned at the end of Section 1.4. Although it will be further
discussed in Chapter 2, it is essential to know that this principle affects the heat flux significantly. The
magnitude of this effect is controlled by the amount of atoms that recombine at the wall: the catalytic
recombination efficiency.
In Chapter 3 the experimental research in catalycity is introduced. An important test facilities for
this purpose is discussed in this chapter: the arc-jec facility. As will be discussed in Section 3.4 the usage
of copper slug calorimeters is essential to determine the total enthalpy which dictates the flow condition
a sample experiences. The total enthalpy is typically measured by means of copper calorimeters which
tend to oxidize when used. However, the catalytic efficiency of copper oxide is not the same as copper. It
is therefore essential to know the catalytic efficiency of copper oxide to accurately determine the enthalpy.
In Section 3.4.3 the literature regarding the catalytic efficiencies of copper and copper oxide is presented. While a large number of experiments have been conducted for oxygen on copper oxide, it is
noticed that values for nitrogen on copper oxide is still missing. Since one of the main components of
air is nitrogen it is essential to investigate also this case.
The method of Park [15] is followed in this work using a shock tube, thin-film gauges and the theory
for tertiary gas mixtures. The theory is discussed in Chapter 2 and the experimental equipment is discussed in Chapter 4.
From this the research question for this thesis project can be formulated as:
“What is the catalytic recombination efficiency of copper oxide when exposed to nitrogen?”
To answer this research question there are several sub-goals, called objectives, that need to be accomplished. These objectives are as follows:
1. Define the initial conditions for the shock tube such that nitrogen dissociation at varying degrees
is present in the test section.
2. Construct and calibrate the thin-film heat transfer gauges.
3. Analysis of the tertiary gas mixture theory composed by Park [15] (Section 2.3.2.2), necessary to
construct the diffusion-conduction heat transfer ratio vs. catalytic efficiency diagram.
4. Measurement of the heat transfer rate by using a heat transfer gauge which is mounted in a model
with no coating.
5. Measurement of the heat transfer rate by using a heat transfer gauge which is mounted in a model
and coated with silicon dioxide.
6. Measurement of the heat transfer rate by using a heat transfer gauge which is mounted in a model
and coated with copper oxide.
7. Measurement of the heat transfer rate by using a heat transfer gauge which is mounted in a model
and coated with copper.
8. Determination of the catalytic efficiency by combining the theory with the measured heat transfer
rate.
The flow conditions used in the shock tube have to be determined using numerical programs (objective 1). Additionally, the thin-film gauges need to be manufactured and calibrated which will be used
for the measurements (objective 2).
9
1.6. Report Outline
Objective 3 is the required theory to translate the heat transfer rate to the catalytic efficiency. Hence
it forms a bridge between objectives 4 to 7 with objective 8. Objective 5 concerns a silicon dioxide
coating only, which acts as a reference since it is considered to be a non catalytic material. Because
objectives 6 and 7 are catalytic surfaces, these values can be compared with the former.
Not only different types of coating are investigated, also three different flow conditions are considered:
one with almost no dissociation (around 1% dissociation), one with a moderate amount of dissociation
(around 30% dissociation), and the last with a high amount of dissociation (around 50% dissociation).
These flow conditions are given the letters A, B, and C respectively.
Condition A functions as a check to compare the measured heat transfer rate between no coating and
silicon dioxide (objective 4 and objective 5 respectively). Theoretically the difference between the two
should be negligible. Condition B and C are used to determine the catalytic efficiency and theoretically
they should yield about the same value. However, to achieve a higher amount of dissociation at condition
C the test time might be shorter hereby increasing the uncertainty of the results.
1.6
Report Outline
The first three chapters of this report functions as a literature review to introduce the concept of catalycity. This section closes off Chapter 1 in which the background information regarding hypersonic vehicles
has been presented together with the introduction to heat flux. The next chapter continues on the heat
flux and extends it to the different aspects that have an influence. This eventually leads to the discussion
regarding the theory of catalycity and the flow conditions at the stagnation point. Chapter 3 focuses on
the experimental research that has been conducted regarding catalycity, from which it can be concluded
that measurement data on the combination of nitrogen and copper oxide is still absent.
After finishing the literature review the experimental equipment is discussed in Chapter 4 by presenting the shock tube and the entire manufacturing procedure of the thin-film gauges. Chapter 5 then
presents the numerical programs used to determine the basic flow parameters around the model. It
is then followed up by the discussion of the post-processing procedure, in which all the variables are
presented that are required to compute the catalycity.
In Chapter 7 the heat flux and pressure results are presented together with the shadowgraph visualization images. This chapter also presents the catalytic efficiency values found from the experiments.
Discrepancies between the results and the values predicted by numerical programs or theory are discussed in Chapter 8. In Chapter 9 a sensitivity analysis is conducted from which it can be concluded
which parameters have the largest affect on the accuracy of the results. This report is then closed by a
conclusion and recommendations shown in Chapter 10.
10
2
Heat Flux and Catalycity
In the previous chapter the concept of heat flux has been discussed. In this chapter the basic principles
of heat transfer are listed (Section 2.1) from a macroscopic perspective. However, when dissociation
occurs an additional heat flux mode arises: the diffusive heat flux. This heat flux originates from the
diffusion of atoms which is caused by catalycity at the wall. To fully grasp the concept of catalycity the
fundamentals are presented in Section 2.2.
The effect of catalycity can be expressed mathematically by the theory presented in Section 2.3,
which relates the heat flux with the catalytic efficiency for both a binary and tertiary gas mixture at the
stagnation point. This chapter then finishes by presenting the flow conditions at the stagnation point
which requires the introduction of partition functions, which is a mathematical concept used to perform
calculations based on the physical behavior of atoms and molecules.
2.1
Heat Flux
In Section 1.4 the aspects that affect the heat flux were listed. To fully understand the influence of
catalycity on the heat flux it is necessary to present more details regarding the latter.
Heat flux is by definition the rate at which heat is transferred through a surface per unit area, hence
the SI units are W/m2 . There are several different modes by which the heat transfer can take place [16]:
• Conduction: The heat transfer between objects that have a physical contact is called conductive
heat flux. It is required that a temperature gradient exists (e.g. in the form of a temperature
difference between two objects) which results in a flow of energy in the form of heat from the
high temperature to the lower temperature. This energy is stored in the form of translational,
rotational, and vibrational motion of the molecules.
• Convection: The convective heat flux is caused by a physical displacement of a medium which
transfers a part of its internal energy to another medium. Similar to the conductive heat transfer
a temperature difference needs to exist between the two media.
• Radiation: An electromagnetic process where energy is transfered by means of radiation. A single
medium can emit radiation without the need of a second medium.
For classical macroscopic heat transfer problems, it is sufficient to consider these three modes. However, in hypersonic applications especially in high enthalpy cases the conductive heat transfer will be
larger than determined using the macroscopic method. To cover for this increase a fourth heat transfer
mode has to be introduced.
• Atomic diffusion: The atoms resulting from dissociated molecules translate due to a concentration
gradient. This gradient results in a net transport of these atoms to a certain direction which is also
known as diffusion. The recombination of these atoms releases energy that adds on the conductive
heat flux. This additional heat flux is called diffusive heat transfer.
It has been shown by Herdrich et al. [17] that this diffusive heat transfer could increase the heat flux
experienced by a vehicle by a factor of 3. To fully understand this diffusive heat transfer it is necessary
to explain catalycity first which will be tackled in the following section.
2.2
Fundamentals of Catalycity
This section starts with the gas surface interaction (Section 2.2.1) which covers the adsorption of the
gas on the surface. These adsorbed atoms can move from one stable site to another at the surface and
11
2.2. Fundamentals of Catalycity
collide, possibly resulting in the recombination of two atoms. The surface hereby acts as a catalyst
(Section 2.2.2). The probability of a successful recombination increases when imperfections are present
in the surface or when the roughness increases (Section 2.2.3). Moreover, the type of material dictates
the catalytic behavior which can be categorized in three groups (Section 2.2.4).
2.2.1
Gas Surface Interaction
In a solid, the atoms are ordered in a closed-packed structure. Inside the solid the atoms have neighbors
on all sides, however, in the most outer layer the atoms are missing neighbors on one side. This layer is
called the surface layer.
As described by Park [12], when a gas atom is brought close to the surface there will be an attractive
force felt by this particle. The gas particle will therefore have the intention to stick to the surface
(i.e. adsorbed by the surface). This force can either be created by an “induced-dipole induced-dipole”
interaction between atoms (also known as van der Waals interaction) which is called physisorption, or
by a chemical bonding called chemisorption.
As the gas atom moves closer, the attractive force will become lower and the acceleration decreases.
This principle is shown in Fig. 2.1. At a certain distance the attractive force changes to a repulsive force.
However, because the surface atom has moved by a fraction due to the force exerted by the gas atom
during the attractive part, some of the energy is absorbed by the surface atom and transferred to its
neighboring atoms. The gas atom has therefore a lower energy than initially before it interacted with the
surface, hence it cannot reach the same velocity as before. The gas atom becomes trapped by the surface
layer and vibrates until the motion damps out by slowly losing its kinetic energy to the surface. [18]
Figure 2.1: Intermolecular force as a function of the distance. [19]
This adsorbed atom will vibrate on the surface at a frequency proportional to the local temperature.
Due to this vibration the kinetic energy increases which could lead to desorption (i.e. atom escaping
from the surface layer) if it is larger than the adsorption energy. It should be noted that not all atoms
become adsorbed when they hit the surface layer, the probability that an atom is adsorbed is expressed
by the adsorption coefficient. [12]
2.2.2
Wall Recombination
Not only desorption can occur to the adsorbed atom, there is also the possibility that recombination
takes place. Molecules can be formed from the adsorbed atoms by two different modes: the catalytic
mode and the reactive mode. [12]
The catalytic mode follows the so called three-body atomic recombination process. When two atoms
collide they first have an attractive potential, hence they move towards each other. However, after a
certain point this potential will become repulsive and because during this process energy is preserved
they will return back to the original conditions. Therefore, the two atoms need the aid of a third atom to
recombine to one molecule. This third atom will absorb partly or all kinetic energy of the two colliding
atoms when they interact.
12
Chapter 2. Heat Flux and Catalycity
Normally this process occurs when two gas atoms are flowing in the air and collide. The third atom
is then pushed away by the two atoms, which results in an energy drop for those two respective atoms
and they become trapped in each others potential fields, hereby forming a molecule.
This process can also occur with an adsorbed atom which has enough kinetic energy to move from
one stable site (of having four bonds, shown in Fig. 2.2(a)) to the neighboring one (Fig. 2.2(b)). The
adsorbed atom can then move freely on the surface layer and collide with another atom. The surface
atoms function as a third body that transfers the energy to the neighboring atoms. However, not every
collision is successful and the two atoms can just bounce off each other. [12]
(a) The initial stable site of the adsorbed atom.
(b) The neighboring stable site of the adsorbed atom.
Figure 2.2: The translation of an adsorbed atom shown in gray from one stable site to a neighboring
one. [12]
In this catalytic mode the catalytic efficiency (γ) can be defined as the ratio of the mass flux of the
formed molecules (J4 ) to the mass flux of the atoms hitting the surface (J1 ) [12]:
γ=2
J4
1
q
=
E−DA
J1
1 + 0.25 θTr e kB T
(2.1)
The factor 2 is the result of two atoms combining into a single molecule. This efficiency depends on the
temperature (T ), the dissociation energy in the adsorbed phase (DA ), and the surface activation energy
(E). There are also two constants: a rotational constant (θr ), and the Boltzmann’s constant (kB ).
This catalytic efficiency is strongly related to the catalytic rate (kw ). In order to explain this relation,
the arrival rate has to be introduced which can be derived in the following way:
The number of particles crossing one particular area is the number density multiplied by the velocity,
hence nc. But from the other side there are also particles crossing that area, hence only half of the
total particles are effectively hitting one side. However, the particles do not necessarily hit the surface
perpendicularly hence the normal velocity component needs to be taken of all angles which gives another
factor half:
nc
Arrival Rate =
(2.2)
4
The arrival rate is thus the number of atoms hitting the surface per unit time and unit area. It depends
on the number density of the atoms one mean free path away from the surface (n) and the average
molecular speed (c). This speed is dependent on the temperature (T ) and the molar mass (m):
r
8kB T
c=
(2.3)
πm
The number of molecules formed through catalytic recombination is the multiplication of the catalytic
efficiency and the arrival rate. This same number of formed molecules can also be gained by multiplying
the catalytic rate with the number density, hence the catalytic rate can be expressed as:
kw = γ
c
4
(2.4)
The reactive mode is the second mode at which molecules can be formed. In this mode a surface molecule
gives up one of its atom and donates it to the adsorbed atom to form a diatomic molecule. The resulting
cavity will be filled up again by adsorbing another gas atom hence the composition of the surface molecule
does not change.
The molecules formed at the surface layer are in general only held at their place by physisorption
using van der Waals forces, which is considered to be a relatively weak force. For this reason the formed
molecules will desorb almost instantly. With this, the wall recombination process is completed.
13
2.3. Theoretical Research in Catalycity
2.2.3
Imperfections and Roughness
Until now it has been assumed that the surface layer is perfectly smooth. However, in reality the surface
has a certain roughness dictated by imperfections. Gas atoms are more likely to become trapped in these
imperfections. In general one can discern different types of imperfections: [20]:
• Point imperfection: There is a vacancy in the surface layer hence an atom is missing. Another
possibility is when interstitialcy occurs hence there is an extra atom.
• Line imperfection: There is a line of atoms is missing in the surface, or an extra line of atoms is
present in the surface.
• Surface imperfection: A portion of the surface atoms are missing, or an extra portion of surface
atoms is present.
• Volume imperfection: A volume of atoms (void) is missing or there is an extra volume of atoms
present.
These imperfections have an effect on the gas atoms hitting the surface layer. Especially when atoms
are missing there will be a larger attractive potential to the incoming gas atom.
Not all incoming atoms will be adsorbed when they hit the surface, so a number of atoms will
simply bounce back in the reversed direction (Fig. 2.3(a)). However, the roughness of the surface has
an important effect on this interaction, because an atom that is bounced back will experience a few
more collisions before “escaping” from the surface atoms (Fig. 2.3(b)). And with each collision there is
a possibility that it will be adsorbed.
(a) A smooth surface will result in a single bounce if the (b) A rough surface will result in several bounces if the
atom is not adsorbed.
atom is not adsorbed.
Figure 2.3: The differences between a smooth and a rough surface.
2.2.4
Catalytic Materials
Materials can be distinguished by their catalytic behavior. The activation energy (E) present in Eq. (2.1)
defines the tendency of the surface atoms to absorb the energy of two adsorbed atoms to make them
recombine, hence they directly influence the catalytic behavior. [12]
A material that does not allow wall recombination by the catalytic mode are non-catalytic materials,
and they have a catalytic efficiency (γ) of 0. These materials do not adsorb the incoming atoms nor
make them recombine by absorbing their energies.
On the other hand, materials that have wall recombination only by catalytic mode are fully catalytic
materials, and thus they have a catalytic efficiency of 1. They adsorb the incoming atoms very easily
and make them recombine at an infinite rate.
Materials with a finite recombination rate are called catalytic materials and have a catalytic efficiency
between 0 and 1. [11, 21]
2.3
Theoretical Research in Catalycity
In the previous sections the fundamentals of heat flux and catalycity were discussed, which are necessary
to understand the theoretical background of catalycity. This section starts with the fundamental research
done by Fay and Riddell regarding the stagnation point heat transfer. In this work the catalycity has
14
Chapter 2. Heat Flux and Catalycity
been neglected in the sense that they only consider a fully or non-catalytic material. In the second part
of this section the heat transfer theory is introduced that includes catalycity.
2.3.1
Stagnation Point Heat Transfer
In 1958 Fay and Riddell [13] introduced a theory to determine the stagnation point heat transfer in
dissociated air. Starting from the momentum, species continuity, and the energy equation they managed
to develop a set of boundary layer equations, where the diffusion and atomic recombination in the boundary layer are included. At the stagnation point these equations could be reduced to nonlinear ordinary
differential equations. From this Fay and Riddell provided the solutions for a frozen and equilibrium
boundary layer, which correspond to zero recombination rates and infinitely large recombination rates
respectively.
Two of these results are presented in Fig. 2.4 for Le = 1.4 and Pr = 0.71. The Lewis number is a
non-dimensional number which is defined as the ratio of thermal diffusivity and mass diffusivity:
Le =
α
D
(2.5)
whereas the Prandtl number is the ratio between momentum diffusivity and thermal diffusivity:
Pr =
cp µ
k
(2.6)
It was stated by Fay and Riddell that both the Lewis and Prandtl numbers do not vary significantly
with temperature. While the Prandtl number of air is often taken as 0.71, the Lewis number value was
estimated as 1.4.
(a) Enthalpy profiles at the stagnation point for a frozen (b) Temperature profiles at the stagnation point for a
and equilibrium boundary layer for a fully catalytic frozen and equilibrium boundary layer for a fully catwall. [13]
alytic wall. [13]
Figure 2.4: Enthalpy and temperature profiles for a frozen and equilibrium boundary layer.
h+ u
2
In Fig. 2.4(a) the enthalpy ratio (g = hs2 ) is shown as a function of the non-dimensional y-distance
(η) from the stagnation point in the direction of the flow. This is expressed by the non-dimensional
coordinate system:
Z x
ξ = ξ(x) =
ρw µw ue r2 dx
0
Z y
(2.7)
rue
η = η(x, y) = √
ρdy
2ξ 0
For a given value of η, g is higher for the equilibrium case than for the frozen case which matches the
fundamental difference between the two cases. In case of an equilibrium boundary layer all the atoms will
recombine inside the boundary layer which brings along the additional energy release during this process.
For a frozen boundary layer there is no change in the species concentrations (hence it is “frozen”) which
results in a smaller g. This same applies to the temperature ratio shown in Fig. 2.4(b).
Moreover, in Fig. 2.4(b) also the atom mass fraction is shown for the two cases. For the equilibrium
case the mass fraction changes due to the recombination of the atoms inside the boundary layer. For
the frozen case this is due to the fully catalytic wal,l that forces all the atoms to recombine at the wall.
There is however a change in the mass fraction away from the wall which is caused by the diffusion of
the atoms in the boundary layer towards the wall.
15
2.3. Theoretical Research in Catalycity
Fay and Riddell [13] provided closed form solutions for both the equilibrium shown in Eq. (2.8a) and
frozen boundary layer (with fully catalytic wall shown in Eq. (2.8b) and non-catalytic wall shown in
Eq. (2.8c)) cases. It should be noted that the wall – fully catalytic or non-catalytic – has no effect for
an equilibrium boundary layer since all the atoms have recombined when they reached wall.
s
due
h
D
−0.6
0.1
0.4
0.52
(hs − hw )
(2.8a)
Eq. BL : q = 0.76P r
(ρw µw ) (ρs µs )
1 + (Le
− 1)
hs
dx s
s
hD
due
−0.6
0.1
0.4
0.63
Fr. BL, fully cat. wall : q = 0.76P r
(ρw µw ) (ρs µs )
1 + (Le
− 1)
(hs − hw )
hs
dx s
(2.8b)
s
due
hD
Fr. BL, non cat. wall : q = 0.76P r−0.6 (ρw µw )0.1 (ρs µs )0.4 1 −
(2.8c)
hs
dx s
Comparing Eq. (2.8a) and (2.8b) more closely, it can be seen that the only difference is in the exponential
of the Lewis number. This indicates that there is almost no difference between an equilibrium and a
frozen boundary layer for a fully catalytic wall. While for the latter case all the recombination occurs at
the wall hereby releasing the heat, for the former most of the heat that is released inside the boundary
layer is conducted to the wall.
Fay and Riddell [13] also calculated the heat transfer for both catalytic and non-catalytic walls at
different recombination rates, shown in Fig. 2.5. The frozen case corresponds to a negligibly small
recombination rate whereas equilibrium is at very large rates. It can be seen that for the catalytic wall
the heat transfer only changes slightly when the recombination rate in the boundary layer is varied. At
the far left (i.e. frozen) the diffusive heat flux is dominant since the conductive part is small, whereas at
the far right (i.e. equilibrium) the opposite is shown since the conductive heat transfer curve overlaps
the total heat transfer curve. This matches with the observation made earlier when comparing Eq. (2.8a)
and (2.8b).
Figure 2.5: Stagnation point heat transfer for fully catalytic and non-catalytic walls at different recombination rates. [13]
The curve for the non-catalytic wall shows the dependency of the heat transfer on the recombination rate, corresponding to Eq. (2.8c). The heat transfer is substantially smaller for frozen conditions
compared to the catalytic wall, while at equilibrium conditions the heat transfer is identical as identified
earlier. In the region between frozen and equilibrium, called non-equilibrium, the heat transfer is still
significantly smaller and cannot be ignored.
Considering Eq. (2.8a) and (2.8b) it can be seen that the enthalpy (at both the stagnation point and
the wall) has the largest effect on the heat flux. The velocity gradient, which is directly related to the
radius of the model, is another parameter that affects the heat flux. Additionally, the conditions at the
wall and stagnation point also influence the heat flux.
2.3.2
Stagnation Point Heat Transfer With Catalycity
The theory provided by Fay and Riddell [13] was only limited to fully catalytic and non-catalytic walls. In
this section the theoretical work of Goulard [14] is introduced for a binary gas mixture valid for catalytic
walls. In the second part of this section an extension of Goulard’s work for a tertiary gas mixture as
developed by Park [15] is presented.
16
Chapter 2. Heat Flux and Catalycity
2.3.2.1
Binary Gas Mixture
In the same year as the publication of Fay and Riddell [13], Goulard [14] extended their theory by
including the catalytic effect. It was noted by Goulard that the case of a fully catalytic wall investigated
by Fay and Riddell gives a rate of recombination that is only possible with extremely active catalysts.
From a more practical perspective a material with a recombination rate that is between fully catalytic
and non-catalytic is more relevant for hypersonic vehicles. For this reason Goulard [14] developed the
theory for stagnation heat transfer of finite catalytic recombination rates in a frozen flow.
Goulard [14] manipulated the conservation equations (i.e. atom mass, mixture mass, momentum,
and energy equations) with basically the same similarity transformation as Fay and Riddell, shown by:
Z x
s=
ρe µe ue r02 dx
0
Z y
(2.9)
ue
r0 ρdy
η=√
2s 0
Moreover the concept of a frozen stagnation enthalpy (he ) was introduced which represents the enthalpy
without including the chemical energy from the recombination of the atoms. A dimensionless frozen
stagnation enthalpy (g = hhs ) and a dimensionless atom concentration (z = ααe ) were also introduced.
se
Additionally, a stream function (ψ) was used:
ψy = ρur0
ψz = −ρvro
The dimensionless stream function was defined as f = √ψ2s which means that uue =
the coordinate system and flow parameters at the stagnation point are shown.
(2.10)
∂f
∂η
= f 0 . In Fig. 2.6
Figure 2.6: The stagnation point parameters defined by Goulard. [14]
With these parameters, the entire similarity transformation can be applied and the conservation
equations can be rewritten to represent the momentum, energy, and atom mass conservation, respectively:
2s ∂ue ρe
02
(lf ) + f f +
−f
=0
ue ∂s
ρ
0
0
l 0
µ2e
1
0
0 00
fg +
g
+
2l 1 −
ff
=0
Pr
Pr
2hse
0
l 0
∂z
z
=0
2s f 0 − f z 0 −
∂s
Sc
00 0
17
00
(2.11a)
(2.11b)
(2.11c)
2.3. Theoretical Research in Catalycity
By assuming that the wall temperature is a lot lower than the temperature at the boundary layer
edge (i.e. Tw /Tse 1) the boundary conditions can be introduced for both the momentum and energy
equations:
f (0) = f 0 (0) = 0
f (∞) = 1
g(0) 1
g(∞) = 1
(2.12)
Equation (2.11) is problematic to solve hence several simplifications are necessary. One of the assumption has been already shown in Eq. (2.12) as g(0) 1 (which corresponds to hs hse ), using which
it can be proven that the coupling between the momentum and the energy equations can be dropped.
More details can be found in Goulard’s paper [14].
and P r are only dependent on the temperature. Goulard
As a result of these simplifications, l = µµρ
e ρe
makes a further simplification by assuming that these values are relatively insensitive to the temperature,
hence they can be considered to be constant (i.e. P r = 0.715 and Chapman-Rubesin constant l = 1).
With these assumptions the momentum and the energy equations are reduced to the form:
f 000 + f f 00 = 0
00
(2.13a)
0
g + P rf g = 0
(2.13b)
Solutions for this type of equations are well known and the following solutions can be found: f 00 (0) = 0.47
and g 0 (0) = 0.47P r
1/3
.
Regarding the atom mass conversation (Eq. (2.11c)) there is a nonsimilar term in s, which is caused
by the dependency on the catalytic process at the wall. Neglecting thermal diffusion and by assuming
that the catalytic reaction rate is proportional to the mth power of the atom concentration at the wall,
the net mass flux of atoms diffusing in the direction of the wall can be expressed by Fick’s law:
∂α
j w = ρw D w
= kw (αw ρw )m
(2.14)
∂y w
where m is a value between 1 and 2. This is the boundary condition for Eq. (2.11c) at the wall.
Furthermore, by introducing the local approximations for the stagnation point shown by:
r0 (x) = x
ue = βx
the nonsimilar part of the boundary condition (i.e.
√
2s
r0 ue )
(2.15)
can be substituted by
q
µe ρe
2β .
Because the Chapman-Rubesin constant l has been assumed to be equal to 1, the product of µe ρe can
be replaced by µse ρse . Moreover, since thermochemical equilibrium has been assumed the atom mass
fraction is constant, and it is furthermore assumed that m = 1 for a cold wall. This results in the final
boundary condition:
r
µes ρes kw
z 0 (0) =
z(0)
2β ρw Dw
(2.16)
z(∞) = 1
By dropping the nonsimilar term in s for Eq. (2.11c) and solving this using the boundary conditions,
the dimensionless concentration at the wall can be found:
z(0) = q
1
µse ρse
kw
2β 0.47Sc1/3 ρw Dw
(2.17)
+1
The Schmidt number is present in this equation which is a non-dimensional number defined as the ratio
of momentum diffusivity and mass diffusivity.
With the expression for the dimensionless atomic concentration at the wall, Goulard gives the expression
for the diffusive heat flux as:
p
qD = 0.47Sc−2/3 2βµes ρes hR αe ϕ1
(2.18)
Some terms were clustered by Goulard in a correction factor ϕ1 which is defined as:
ϕ1 =
1
1+
√
0.47Sc−2/3 2βµse ρse
ρw kw
(2.19)
18
Chapter 2. Heat Flux and Catalycity
The conductive heat flux is determined using the solution for g 0 (0) and is given as:
p
−2/3
qC = 0.47 2βµse ρse P rw hse
(2.20)
When analyzing Eq. (2.18) together with Eq. (2.19), it can be seen that the catalytic rate (kw ) has a
profound effect on the diffusive heat flux (qD ). Since the catalytic efficiency is related to the catalytic
rate by Eq. (2.4) it can be stated that the influence of the efficiency is large. The Schmidt number,
defined as the ratio between the viscosity and the mass diffusivity, is not only affecting ϕ1 but also
qD directly. Additionally, the atomic mass fraction is also proportional to qD , while the effect of the
viscosity, density, and the velocity gradient is relatively small due to the square roots. The majority of
the variables are required at the boundary layer edge.
In Eq. (2.20) the frozen enthalpy has the largest influences on the conductive heat flux (qC ). Similar
to the diffusive heat flux, the majority of the parameters are required at the boundary layer edge. Another observation that can be made is the relatively simple structure of the equation and the low amount
of necessary parameters, compared to the diffusive heat flux. Moreover, the catalytic efficiency has no
influence on the conductive heat flux.
Goulard introduced several catalytic efficiencies (Fig. 2.7) for different materials which were acquired
mostly from experiments conducted by Linnett and Marsden [22, 23]. He also provided some plots to
indicate the effect of including the catalytic effect in the heat flux calculations. The reduced heat transfer (q) expresses the heat flux with respect to the heat flux of an infinitely fast catalyst, for a range of
catalytic rates. A representative example is shown in Fig. 2.8 which is for an altitude of 60960 m, a
nose temperature of 700 K, and a nose diameter of 100 cm. Each of the curves correspond to a different
flight velocity given in feet/second (indicated as fps). From this it can be seen that the heat flux varies
depending on the catalytic rate which is dependent on the material type.
Figure 2.7: The catalytic efficiencies for different materials exposed to atomic oxygen or nitrogen. [14]
2.3.2.2
Tertiary Gas Mixture
In 2013 Park [15] extended Goulard’s theory for a tertiary gas mixture where there is not only the atomic
and molecular particles of a gas but also a third (inert) gas species. Park follows the same assumptions
as Goulard which can be summarized as:
• Thermochemical equilibrium at the boundary layer edge.
• Frozen flow inside the boundary layer.
19
2.3. Theoretical Research in Catalycity
Figure 2.8: The reduced heat transfer against a range of catalytic rates. [14]
• Chapman-Rubesin constant is equal to 1 across the boundary layer, i.e. l =
ρµ
ρe µe
= 1.
• Constant Prandtl number and Schmidt number across the boundary layer.
Moreover, the subscript convention of Goulard and Park differs slightly. Goulard denotes e as outside
the boundary layer and se as the boundary layer edge at the stagnation point, whereas Park uses e
for the boundary layer edge. Park also denotes the atomic particles, molecular particles, and the inert
particles as 1, 2 and 3 respectively. In the derivation of Park, it has been assumed that z30 is equal to
zero which corresponds to a chemically inert gas. Moreover, it is also assumed that the coefficient of
self-diffusion Dii is equal to zero which neglects the diffusive effects of each species on themselves.
By following the same assumption as Goulard [14], Park [15] derived again the dimensionless concentration at the wall:
1
z(0) = q
(2.21)
µe ρe
kw
+1
2β m2 ρ D
φ3 m
w
w21
2
Compared to Eq. (2.17) the difference is rather small: instead of having “0.47Sc1/3 ” there is “ φm3 m
”
where φ3 is defined as:
12 Sc31
Z ∞ 00 mm ScSc
32
f (η) 1
φ3 =
dη
(2.22)
00
f (0)
0
1
For a binary gas mixture a numerical value for φ3 was available for air (i.e. 0.47Sc
1/3 ) hence this value
was substituted. For a tertiary gas mixture such numerical value is not available hence φ3 has to be
calculated using Eq. (2.22).
Using Eq. (2.21) and the assumption of a constant Schmidt number across the boundary layer, Park
presented the conductive and diffusive heat flux respectively as:
p
−2/3
(2.23a)
qC = 0.47 2βµe ρe P rw he
p
qD = χ 2βµe ρe hR αe1 ϕ2
(2.23b)
A similar correction factor as Goulard’s (Eq. (2.19)) is also present in this equation defined as:
ϕ2 =
1
1+
√
χ 2βµe ρe
ρw kw1
(2.24)
In addition to that, Park gathered several parameters in a variable parameter χ to ease the comparison
with the equations for a binary gax mixture:
χ=
1 m2 1
φ3 m Sc21
(2.25)
20
Chapter 2. Heat Flux and Catalycity
From these equations it can be seen that for a binary gas mixture χ corresponds to 0.47Sc1/3 .
Park [15] presented in his paper a comparison between the dimensionless concentration at the wall
using the assumption of both a binary and tertiary gas mixture. In his work where he considered a test
gas consisting of 21% oxygen and 79% argon it was concluded that the difference is small as can be seen
in Fig. 2.9. At lower catalytic efficiencies there is no difference between the two theories, whereas at
552 difference.
PARK
higher values there is a slight
1
Tertiary gas mixture
Binary gas mixture
variable parameter χ. For the case of the binary
to 0.47Sc−2∕3 .
B. Comparison with Experiment
D C
The Flow Conditions at the Stagnation Point
q /q
2.4
Downloaded by KOREA ADVANCED INST OF SCIENCE on July 3, 2013 | http://arc.aiaa.org | DOI: 10.2514/1.A32312
z1(0)
Figure 20 shows the dependence of heat-transfe
catalytic efficiency for conditions B and C.
exhibited low temperature behind the bow sho
0.5
chemical reaction does not occur noticeably. He
presented here. In the figure qC and qD denote h
conduction and diffusion, respectively. The qC w
the relation defined in Eq. (8). In the calculation, t
edge values were again obtained from using the th
the code does not calculate viscosity, the viscosity
the CEA program [24] with ps and T e as inp
0
−4
−3
−2
−1
0
conditions there were very small differences in the
10
10
10
10
10
and the species compositions between the two
γw
Prandtl number at the wall Prw was obtain
Fig. 19 Dependence of z1 (0) on surface catalytic efficiency γ w .
equilibrium value at the boundary-layer edg
Figure 2.9: The comparison between the dimensionless concentration at the wall (z(0))
between a binary
program.
Z ∞ 0 0 m∕m ·Sc Sc ∕Sc The
qD was calculated using Eqs. (9) and (44)
and tertiary gas mixture for a test gas consisting
of
21%
oxygen
and
79%
argon.
[15]
1
12
31
32
f η
ϕ3 dη
(40)
the
tertiary
gases, respectively. For the binary-g
00
f 0
0
Goulard’s assumption of Sc 0.485 was used.
the required diffusion-collision-integral values fo
Combining Eqs. (37) and (39), and after some rearranging, yields
binary Schmidt number Scij were obtained from
Determination of the effective nose radius of f
1
(41)
z1 0 p
required when calculating the stagnation
μ ρe ∕2βkw1 ∕m2 ∕ϕ3 mρw Dw21 1
β du
The data of
In the previous sections the diffusive heate transfer
and catalycity have been discussed.
The
influence
ofZoby and Sullivan [35
e ∕dx.
flows
and
at
high
Mach
numbers.
Because the pre
these phenomena differs per
location
onthe
the
hypersonic
during
re-entry.
the stagnation point
Figure
19 shows
dependence
of thevehicle
ratio of mass
fraction
at the At on
the O2 -Ar gas mixture and at low Mach numb
wallare
withthe
respect
the boundary-layer
edge flow
z1 0 technically
on surface comes
the high-temperature effects
mostto prominent
since the
tobelieved
an haltthat
there,
author
the use of the Zoby and
corresponds
the flow condition
w. The
hereby inducing all of its catalytic
kinetic efficiency
energy γto
the result
body.
For thisto reason
the stagnation
point situation
determining
the velocity is
gradient is somewhat no
B. Distributions for other conditions were qualitatively similar. The
considered here. This is also
in
line
with
the
theory
that
has
been
discussed
in
Section
2.3.
Goulard’s binary-gas solution is included for comparison.
The theory discussed in Section
requires
a large number
have to
1.0be determined.
When the2.3
wall
is noncatalytic,
say at a of
veryvariables
low valuewhich
of
Tertiary gas mixture
γ w ≃ 10−5to
, z1understand
0 for both thethe
binary
and the tertiary
become 1. point better which
To accomplish this it is necessary
situation
at thegases
stagnation
will
be
Binary gas
mixture
−5
−3
In the section
noncatalytic
wall region,
say parts:
at about first
10 <the
γ w general
≤ 10 , the
discussed in this section. This
consists
of two
situation at the stagnation
differences between the two seem almost negligible. Beyond this
point is discussed together region
with they
the both
assumptions
that can be made, afterwards the method of determining
show a gradual decrease to nearly 0 when the wall is
the flow conditions at the fully
boundary
edge
presented.
catalyticlayer
at γ w 1. Inisthe
finite-catalytic wall region, say at
about 10−3 < γ w ≤ 10−1 , some differences can be seen, but they are
still not large.
0.5
2.4.1 Standing Shock
Layercan be considered to be a
Heatand
transferBoundary
to the wall by diffusion
product of energy release due to recombination of atoms to molecules
The shape of the shock wave
induced
by anand
object
fasterofthan
the
speed of sound, depends on
hR for
atomic oxygen
the ratetraveling
of mass diffusion
species
i J w1
qD/qC
the geometry of the object: a sharp nose causes a sharp shock whereas a blunt body induces a bow shock.
qD hR Jw1tosuch
hR kw1an
αe1 ρextent
It is this shock wave that increases the enthalpy
air molecules dissociate.
The
w z1 0 that the (42)
Experiment (CuO)
sharp nose situation is not discussed further since it is not relevant for the present investigation.
Combining Eqs. (41) and (42)
0
For a blunt nose the bow shock stays at a finite distance from the wall. For spherical shapes,
especially
−4
−3
−2
10
10
10
γ
at the stagnation point, this distance is rather small, whereas
for a flat-disk cylinder the distance is
1
w
q h kw1 αe1 ρw p
relatively larger. In Fig. 2.10D the Rbow
shock μ
ise ρshown
for both a spherical and flat-disk cylinder model.
e ∕2βkw1 ∕m2 ∕ϕ3 mρw Dw21 1
1.0
While various body diameters are investigated, the average shock stand-off distance
for a spherical
model
Tertiary gas
mixture
(43)
gas mixture
(Fig. 2.10(a)) is only half of the distance corresponding to the flat-disk cylinder model (Fig.Binary
2.10(b)).
Assuming
the Chapman-Rubesin
constant
l 1, that
and after
This larger shock distance
is necessary
for the existing
theory
has some
been discussed in Section 2.3.
rearranging, Eq. (43) becomes
One of the main assumptions in that theory is thermochemical equilibrium at the boundary layer edge
p
located between the standing shock and the
wall,
as eillustrated
qD χ 2βμ
ρe hR αe1 φ2 in Fig. 2.11.
(44)This basically means that
the dissociation has been completed when the particles reach the boundary layer edge.
To reach this
0.5
Experiment (Cu)
where
equilibrium, the distance between
the standing shock and the boundary layer edge should be sufficiently
large, which has a higher probability to be achieved using
a flat-disk cylinder model.
1
p large to ensure (45)
φ2 is sufficiently
In order to make sure that this distance
thermochemical equilibrium,
1 χ 2βμe ρe ∕ρw kw1 the conditions between the standing shock and the wall have to be calculated. It is however challenging
Experiment (CuO)
to determine this accurately at every location at the stagnation point. A method is introduced in
m
1
Appendix A where a collection of assumptions
has1 been
· 2 · made to attain some
(46)first order approximations.
χ
0
−4
−3
−2
ϕ3 m Sc21
10
While this is a preliminary method it is possible to determine the variation of the flow parameters
with 10 γ 10
w
respect to distance.
It is found that the present correction factor φ2 is quite similar to that
Fig. 20 Dependence of heat-transfer rate on
of the Goulard’s φ1 defined in Eq. (10), but there is an additional
21
efficiency for the O-O2 -Ar interaction.
th
2.4. The Flow Conditions at the Stagnation Point
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c
(a) The standing shock in front of a sphere at various (b) The standing shock in front of a flat-disk cylinder at
body diameters indicated by the different symbols. [24] various body diameters indicated by the different symbols. [24]
Figure 2.10: The difference in standing shock distance between a sphere and a flat-disk cylinder at the
same flow condition.
Figure 2.11: The standing shock wave and the boundary layer around the model.
22
Chapter 2. Heat Flux and Catalycity
There is however a more accurate method to determine the conditions at the boundary layer edge.
This method relies on the thermochemical equilibrium assumption and cannot show the variation of the
flow parameters with respect to the distance. The bow shock is interpreted as a normal shock wave
at the stagnation point. Through this shock wave the total enthalpy is assumed to be constant since
the flow is adiabatic. Using the known quantities before the standing shock (called region 2) this total
enthalpy can be determined:
1
(2.26)
H2 = h2 + u22
2
The boundary layer is assumed to be frozen, which means that the reaction rates within this layer is zero
which corresponds to a constant chemical composition. This assumption implies that the recombination
of the atoms occurs at the wall. A catalytic wall will result in a higher number of successful recombinations
compared to a non-catalytic wall.
Since the static enthalpy (h2 ) and the flow velocity (u2 ) can be determined using numerical programs
(for shock tubes), it is possible to determine the total enthalpy. Moreover, since the pitot pressure can
also be determined by such numerical programs, the total pressure is known. It is however necessary to
assume that the total pressure through the boundary layer is constant.
2.4.2
Conditions at the Boundary Layer Edge
At the boundary layer edge there is a gas mixture between atoms and molecules of all the present gas
species. The exact composition of this gas is necessary to determine the remaining flow parameters, since
until this stage only the total enthalpy and the total pressure are known (as discussed at the end of the
previous section).
Traditional perfect gas relations will yield inaccurate results since the fundamental assumption of a
perfect gas has been violated. To acquire the remaining thermodynamic gas properties such as temperature, density, and the composition of the gas, the concept of partition functions is used which originates
from the field of statistical mechanics. To understand this concept the theory of quantum numbers is
discussed.
2.4.2.1
Quantum Numbers
The basic structure of an atom is well known: it consists of a nucleus and electrons orbiting around it.
For such a seemingly simple structure it is however challenging to determine the location of a electron at
a certain moment, hence it is more common to express it in a probability by means of a wave function.
The differential equation which governs this wave function and is derived from energy conservation is
called the Schrödinger equation. The exact form and derivation of this equation is beyond the scope
of this thesis, however in summary it makes sure that the sum of kinetic and potential energies of all
particles is preserved.
There are three sets of eigenvalues of the Schrödinger equation [12]:
1. Principal quantum number, n: The orbit of the electron around the nucleus. The first orbit (n = 1)
is denoted as K-shell followed by L-shell (n = 2) etc.
2. Angular momentum quantum number, l: The angular momentum of the electron which is numbered
with integers starting from zero. These orbits can also be referred to by the letters s, p, d, f, g,
h, etc. While the principal quantum number dictates the distance between the electron and the
nucleus, the angular momentum quantum number controls the shape and the orientation of the
orbit.
3. Spin quantum number, s: The spin of the electron which can either be 0.5 (the spin is aligned with
the angular momentum) or -0.5 (the spin is opposite to the angular momentum).
Each electron has its own combination of these quantum numbers. According to the Pauli exclusion
principle it is impossible to have two electrons in an atom or molecule with the same combination of
these three quantum numbers; hence they cannot occupy the same orbit. [19]
For all atoms the energy levels are known and available in the NIST Atomic Spectra Database. [25]
The first four levels of atomic nitrogen are shown as an example in Table 2.1. The first level is the
ground state and functions as a reference for all the other levels. In Fig. 2.12 the nomenclature of the
configuration is explained. As can be seen the number before the letter refers to the principal quantum
number and the superscript refers to the number of electrons. Since the most inner electrons (1s2 ) never
changes, they are omitted by NIST from the table. [12]
Combining Table 2.1 with Fig. 2.12 it is clear that the electrons occupy the first and second orbits
for the first three levels; only at the fourth level an electron has moved for the 2p orbit to the 3s orbit.
To reach this state, energy has to be applied to the electron hereby increasing its energy level.
23
2.4. The Flow Conditions at the Stagnation Point
Table 2.1: NIST Atomic Spectra Database levels data for the first four levels of atomic nitrogen. [25]
Configuration Term
4 ◦
2s2 2p3
S
2
3
2 ◦
2s 2p
D
2s2 2p3
2
P◦
2s2 2p2 3s
4
P
Ji [-]
3/2
5/2
3/2
1/2
3/2
1/2
3/2
5/2
Level [cm−1 ]
0.000
19 224.464
19 233.177
28 838.920
28 839.306
83 284.070
83 317.830
83 364.620
Figure 2.12: The meaning of the electron configuration.
The second column in the table represents the angular momentum quantum number together with
the spin and is not essential for the overall understanding of this thesis topic. For this reason it will not
be further discussed.
What is essential to understand is the third column which consists of values for the total electronic
angular momentum quantum number (J). This is used to determine the statistical weight (g) which has
an important role in the computation of partition functions. [12]
The configurations with more than one total electronic angular momentum quantum number (J)
have to be combined. The energy values (denoted by i) are averaged using the statistical weight and the
total statistical weight is simply the summation:
X
g=
(2Ji + 1)
(2.27)
i
Hence for the second energy level this would result in g = 10 and an energy of 19227.95 cm−1 .
For molecules there is a similar system. However, instead of covering the atomic behavior, this data
covers the vibrational and rotational behavior of the molecule. From the NIST Chemistry WebBook [26]
the parameters of molecular nitrogen have been presented in Table 2.2 for the first three levels. The
electronic states of L=1, 2, 3, etc. have changed from s, p, d to Σ, Π, ∆, etc. The superscript in front
of this is an indication of the statistical weight (g). For Σ states the indicated number is equal to the
statistical weight, for all other states this number has to be doubled.
Moreover, the letter in front of this symbol is an designator for the most famous states. The remaining
super- and subscripts are not of importance for this thesis topic hence will not be discussed.
Table 2.2 shows the parameters that are essential to determine the partition functions for molecules.
The second column (Te ) represents the energy of that particular level. Moverover, three vibrational
constants are shown as ωe , ωe xe , and ωe ye , together with four other parameters – Be , αe , De , andβe –
representing the rotational constants. The last entry is the internuclear distance (re ). [26] To understand
how these variables are used, it is necessary to present the theory and equations of partition functions.
Table 2.2: The NIST Chemistry WebBook data for the first three levels of molecular nitrogen. [26]
State
X 1 Σ+
g
A3 Σ+
u
B 3 Πg
Te [cm−1 ]
0
50203.6
59619.3
ωe [-]
2358.57
1460.64
1733.39
ωe xe [-]
14.324
13.87
14.122
ωe ye [-]
−2.26 · 10−3
0.0103
-0.0569
Be [-]
1.99824
1.4546
1.6374
αe [-]
0.017318
0.0180
0.0179
De [-]
5.76·10−6
6.15·10−6
5.9 · 10−6
re [Å]
1.09768
1.2866
1.2126
βe [-]
0
0
0
24
Chapter 2. Heat Flux and Catalycity
2.4.2.2
Partition Functions
Having discussed the quantum numbers it is now possible to focus on the concept of partition functions.
Partition functions are used to determine how the particles are partitioned among the energy groups. [12]
Electrons do not necessarily have to stay in the lowest orbits but can be excited to higher orbits or to
an orbit with a different shape or orientation.
There are in total four groups of partition functions: electronic, translational, vibrational, and rotational. [12, 19]
• Electronic partition function: This focuses on the internal energy of a particle. The mathematical
expression is given by:
X
Ej
(2.28)
Qel =
gj e− kT
j
From this equation several parameters can be recognized: the statistical weight (g) and the energy
of each electronic state (E). By summing this for all the energy levels the electronic partition
function can be determined.
• Translational partition function: The translational energy of a particle is covered by this partition
function which is mathematically expressed as:
Qtr =
2πmkT
h2
3/2
(2.29)
In this equation the only value that depends on the type of particle is the mass.
• Vibrational partition function: The vibrational energy of a particle is included in this partition
function which is only present in molecules. For atoms this value is equal to zero. Using the
three vibrational constants (ωe , ωe xe , ωe ye ) shown in Table 2.2 the vibrational energy level can be
computed depending on the fundamental vibrational frequency (v) using:
2
3
1
1
1
G(v) = ωe v +
− ωe xe v +
+ ωe ye v +
2
2
2
(2.30)
These energy levels can then be used to determine the vibrational partition function as shown by:
X G(v)
Qvib =
e− kT
(2.31)
v
• Rotational partition function: The rotational energy of a particle is taken care of by this partition
function which is also only present in molecules. Part of the entry values shown in Table 2.2 is
required for the computation of this partition function (i.e. Be , αe , De , βe ). As can be seen in
Eq. (2.32) the vibrational frequency is required, which indicates the coupling between the vibrational and rotational energy, to determine the rotational energy. Please note that the J is the
rotational quantum number in this equation.
F (J) = BJ(J + 1) − DJ 2 (J + 1)2
1
1
where B = Be − αe (v + ); D = De + βe (v + )
2
2
(2.32)
The rotational partition function can then be determined using:
Qrot = (2J + 1)e−
F (J)
kT
(2.33)
The vibrational and rotational partition functions are tackled together due to their entanglement. The
maximum rotational quantum number, depending on the potential curve of a energy level of the molecule,
is hereby determined. The vibrational-rotational partition number can then be determined:
!
X
X
Qvib,rot =
Qvib
Qrot
(2.34)
v
J
The translational and electronic partition numbers can also be included in a similar way as expressed
by:
"
!#
X
X
X
Qmolecule,tot = Qtr
Qel
Qvib
Qrot
(2.35)
j
25
v
J
2.4. The Flow Conditions at the Stagnation Point
The resulting partition function of a molecule is dependent on the temperature. By taking the derivative
of the total partition function with respect to temperature, the internal energy can be determined [12, 19]:
= kT 2
d
[ln(Qmolecule,tot )]
dT
(2.36)
For an atom the electronic and translational partition functions are multiplied in the same form as
Eq. (2.35) and the total internal energy can also be determined as in Eq. (2.36). Adding the internal
energy of all these particles together results in the internal energy of the whole gas mixture. This can
then be used to acquire the specific heat at constant volume [19]:
∂e
cv =
(2.37)
∂T
In case of dissociation the number density can also be determined using partition functions [12] as
expressed by:
Qatom1,tot Qatom2,tot − D0
natom1 natom2
=
e kT
(2.38)
nmolecule
Qmolecule,tot
In this expression the molecule consists of atom 1 and atom 2 which are separated due to dissociation.
At this point it is clear that the concept of partition functions tackles a problem from a real gas perspective. The specific heat ratio (γ) is not constant and is changing depending on the temperature and
the gas composition.
By applying this to the situation at the boundary layer edge, and together with the known total
enthalpy and total pressure (Section 2.4.1), it is possible to determine the temperature, density, and
other flow parameters.
26
30
G. Herdrich et al. / Progress in Aerospace Sciences 48–49 (2012) 27–41
of measured data is a future item, (see also Section 4). The first
four of the described methods are dominated by concentration
measurements while the methods that use plasma wind tunnel
facilities are dominated by energy balances. However, it is not in
the ambition of the authors to categorize too strictly as some of
the methods combine the two categories or could at least have an
improvement e.g. if concentration data for the plasma wind
tunnel based methodologies would be measured.
2.1. Side-arm method
decrease of measured temperature
side-arm reactor was used by Linne
Greaves and Linnett [10,11,13].
When the material sample i
heating elements the recombinatio
can be obtained.
In the experimental setup in Fi
form of a hollow cylinder was ins
area (lining in position). If the conc
is relatively low and the temperat
ment area is isotherm, the cataly
following equation:
3
The side-arm method was first developed by Smith [9] in 1943
and was widely used in the 1950s and 1960s. Later the original
2
lnðaA a1
B Þ
measurement technique was modified and used by many others
g¼K
:
Dx
e.g. 1959 by Greaves and Linnett [10,11] or 1964 by Dickens and
Sutcliffe [3]. It was one of the first procedures for the experiHere, aA and aB are the atomic
mental determination of recombination coefficients.
the material sample and Dx is th
main part
experimental
setup
consistsThis
on awas
quartz
In Chapter 2 the effect ofThe
catalycity
on of
thethe
heat
flux has been
explored.
followedparameter
by the theories
K can be described a
tube with
a gas supply
at the one
side
and a vacuum
pump efficiency
on the
that are available regarding
catalycity.
To utilize
these
theories
the catalytic
has
to be known.
circumference
and S as the crossother
side. Closeby
to different
the vacuum
pump connection
a second
tube
In the past this has been
determined
researchers
using a variety
of experimental
is the average velo
The facilities
size Vav for
diverged
from the main pipe. This second tube or
coefficient of diffusion.
a wide range of materials
and orthogonally
gases.
side-arm gave the name for this kind of apparatus. Between the
Although
the side-arm method
The first three sections will discuss the facilities that are typically used: the side-arm reactor,
shock
main pipe and second pipe the gas passes a plasma generator
used
tube/tunnel, and arc-jet. Section 3.4 then focuses on the application of copper slug techniques
calorimeters
in for determine the
which generates dissociated and ionized plasma. In the literature
is still in use even today
arc-jets. In this section the current problems and uncertainties of these calorimeters are materials,
discusseditwhich
side-arm reactors with electrodes and electrode-free plasma
[14–16].
forms the direct motivation
for this thesis project.
generators using high-frequency coils can be found.
The disadvantages of this metho
For achieving higher degrees of dissociation the gas was often
the knowledge of the coefficients of
mixed with e.g. water vapor which of course represents a
temperature range (up to 1000 1C
contamination of the plasma which was done by Dickens and
apparatus. The catalytic property of
Sutcliffe [3].
and side-arms
must be known or
The side-arm reactor is one
of
the
first
facilities
used
to
study
the
catalytic
behavior
of
different
materials
By passing the side-arm the plasma diffuses towards the material
before
installing the material sam
and was widely used sample
in the 1950s
and
1960s.
This
facility
consists
of
a
quartz
tube
which
is
connected
in the measurement area of the side-arm. Depending on the
level
must tube)
be low to minimize th
to a gas supply at one
side
and a vacuum
at the other
end. of
A the
“side-arm”
a second
type
of side-arm
reactorpump
the isothermal
properties
measure- (i.e. atomic
species in order to reduce th
is connected perpendicular
to the
tubebenear
A plasma
(e.g. high
ment area
in quartz
length can
up the
to avacuum
couple pump.
of decimeters.
Thegenerator
flow. Typical pressure levels are 4 P
frequency coils) is attached
to
the
quartz
tube
which
dissociates
and
ionizes
the
gas
into
a
plasma.
recombination coefficient will be determined by measuring the
cient g depends on temperature and
When this plasmadecreasing
passes the
side-arm,
it diffusesatoms
towards
that is located inside. By
amount
of dissociated
alongthe
thespecimen
material sample
reactors are not representative for c
measuring the amount
of dissociated
before
and after the
the e.g.
amountNevertheless
of recombined
under
inspection. atoms
The species
concentration
canspecimen,
be measured
these values are im
massefficient
spectroscopy,
laser induced
fluoresatoms is determined using
from Pirani-Manometer,
which the catalytic
is calculated.
Several
methods
can be catalytic
used models.
theoretical
cence
other techniques.
scheme of a side-arm
reactorfluorescence, Piranito measure the amount
ofordissociated
atoms:A principle
mass spectroscopy,
laser induced
can
be seen in Fig. 6 but represents only one possible configuration.
manometer, and more.
[17]
Anotherreactor
methodis isshown
to install
thermocouples
In Fig. 3.1 the side-arm
thatseveral
was used
by Greaveswith
and catalytic
Linnett in 1959
[27,
28] and
Effusion method
coatings.
Theby
greater
the and
distance
to thein
main
pipe
is 23].
the less
is the and 2.2.
a similar configuration
was used
Linnett
Marsden
1956
[22,
Greaves
Linnett [27, 28]
concentration
of the
atoms in
sidenumber
arm, hence,
less atoms
experimentally determined
the catalytic
efficiency
of the
a large
of materials
when exposed
oxygen.Method was introd
ThetoEffusion
recombine at the surface of the catalytic coatings. This leads to a
A selection is shown in Table 3.1. A lot of these values from Linnett et al. have been used[17].
by Goulard
Hereby[14]
the gas streams th
to make Fig. 2.7.
evacuated tank in which the ma
pressure and density should also b
aperture the gas become dissociat
coil. Behind the aperture the mat
and orthogonal to the flow direc
species recombines to molecules a
Fig. 7 shows a scheme of the ar
[17]. The sample under investigati
with the catalytic coating applied
known flow characteristics the tota
particles can be calculated. The pa
similar to the techniques mentio
side-arm method. Together with t
sample and the total number of im
nation coefficient can be determ
resistance thermometers which w
of the Pyrex disk. Like all measure
mine the catalysis by measuring th
heat flux respectively, the recom
energy accommodation factor can
Fig. 6. Side-arm reactor of Greaves and Linnett [10].
the effective recombination coeffic
Experimental Research in Catalycity
3.1
Side-arm Reactor
Figure 3.1: The side-arm reactor used by Greaves and Linnett in 1959. [27]
27
3.1. Side-arm Reactor
Table 3.1: Catalytic efficiencies for various materials at room temperature exposed to oxygen. [27, 29]
Material
Gold
Silver
Magnesium
Nickel
Iron
Copper
γ[-]
5.2 · 10−3
2.4 · 10−1
2.6 · 10−3
2.8 · 10−2
3.6 · 10−2
1.7 · 10−1
Boric oxide
Aluminium oxide
Lead oxide
Magnesium oxide
Ferric oxide
Nickel oxide
Cupric oxide (CuO)
Zinc oxide
Silicon oxide
6.3 · 10−5
2.1 · 10−3
6.3 · 10−4
2.5 · 10−2
5.2 · 10−3
8.9 · 10−3
4.3 · 10−2
4.4 · 10−4
1.6 · 10−4
On the other hand, Linnett and Marsden [22, 23] investigated the catalytic efficiency for a glass surface
at elevated temperatures. Using heating elements the temperature of the sample could be increased which
showed an increase in the catalytic efficiency, as presented in Fig. 3.2. For this reason it is important to
know the temperature of the material when comparing catalytic efficiencies.
Figure 3.2: Catalytic efficiency variation with wall temperature. [22]
Although the focus of most research has been done using oxygen, also nitrogen has been used in
side-arm reactors. In 1971 Rahman and Linnett [30] did research on a range of metal surfaces which
results are summarized in Table 3.2. The same setup as for the oxygen cases has been used.
Table 3.2: Catalytic efficiencies for various metals at room temperature exposed to nitrogen. [30]
Material
Silver
Nickel
Iron
Copper
γ[-]
2.4 · 10−1
6.7 · 10−5
2.1 · 10−3
6.8 · 10−2
Another very similar facility to the side-arm reactor is the diffusion tube. Instead of having a side-arm
the diffusion simply takes place in the main tube. This will not be further discussed since it works on
the same principle as the side-arm reactor.
The use of side-arm reactors however brings significant uncertainties that may affect the results. This
is caused by the side-arm through which the dissociated flow diffuses. When the flow turns around the
corner it is prone to become less steady. Increasing the length of the side-arm could solve this problem
but due to the added length a lot of the atoms will be recombined.
To investigate catalytic materials it is important to have a sufficient amount of dissociated atoms in
order to be able to measure the differences in front and behind the specimen. In order to achieve a high
amount of dissociation some researchers, such as Dickens and Sutcliffe [31], added traces of water vapor
to the gas. It is suspected that the increase of molecular weight due to the water vapor is causing this
increase in dissociation. However, a disadvantage of this method is the contamination of the test gas.
28
Chapter 3. Experimental Research in Catalycity
3.2
Shock Tube/Tunnel
During the 1960s not only side-arm reactors were utilized for catalycity research. In 1966 Reddy [32]
used catalytic probes to measure the atom concentration in a hypersonic flow using a shock tunnel. A
shock tunnel consists of a driver and driven section which is connected by a nozzle and a test section
(and a dump tank). With this facility Reddy was able to reach high enthalpies. Shown in Fig. 3.3(a)
are the driver and driven sections. The driven section is then connected to a nozzle system illustrated
in Fig. 3.3(b).
(a) The driver and driven tube of the shock tunnel.
(b) The nozzle system connected to the driven tube of
the shock tunnel.
Figure 3.3: The shock tunnel used by Reddy [32] for his catalytic research of oxygen on silver oxide.
The shock tunnel used by Reddy is fired using an active way. The driver section is filled with a
stoichiometric mixture of hydrogen and oxygen which is diluted with helium. A tungsten wire suspended
along the centerline of the tube is heated instantaneously to ignite the gas mixture. Different types of
stainless steel diaphragms were used to control the bursting pressure. After bursting, a primary shock
wave travels from the diaphragm into the driven section.
Inside the driven section the test gas (i.e. oxygen) is located. After the primary shock hits the
test gas, the temperature and pressure increase instantaneously. Moreover, the gas propagates in the
same direction as the shock. Thereafter, the gas enters the nozzle and accelerates to a Mach number
between 15 and 17. Then it enters a test section with a probe that holds thin-film gauges using which
the temperature change resulting from the stagnation heat flux is measured.
The thin-film gauges used by Reddy had a small strip of platinum with a known resistance. This
resistance changes as a function of temperature, hence by flowing a constant current through the platinum
the voltage change is measured which represents the temperature change. Reddy derived a relation
between the heat transfer and this voltage change to acquire the heat flux.
The gauges were first coated with a 1000 nm thick layer of silicon monoxide followed by a 200 nm
layer of silver. This silver was converted to silver oxide by running the shock tunnel several times so it
would be exposed to oxygen. It was believed that silver oxide is responsible for the surface recombination
of oxygen rather than silver.
The catalytic efficiency determined by Reddy was 0.81 for a axisymmetric probe and 0.89 for a
two-dimensional probe, which are shown in Fig. 3.4.
Not only for oxygen the shock tunnel was used. In 2008 Maclean and Holden [33] performed experiments using nitrogen, air and carbon dioxide on stainless steel. The enthalpy was varied from 5 to
15 MJ/kg and a variety of test models were used: axisymmetric spherical capsule, axisymmetric spherical
cone, and a two-dimensional cylinder. The catalytic efficiency found for nitrogen was in the order of
10−3 whereas for both air and carbon dioxide the catalytic efficiency was found to be infinitely large (i.e.
super-catalytic). It was acknowledged by Maclean and Holden that the excessive catalytic efficiencies
are not correct and are caused by the lack of sufficient understanding of the freestream and shock-layer
state of the gas.
The result and conclusion of Maclean and Holden [33] is a clear example of a large disadvantage of
using a shock tunnel. The dissociated flow in the driven section is further accelerated to higher Mach
numbers in the nozzle. During this acceleration recombination can occur which alters the species concentration that reaches the model. Due to the many unknowns in the process, it is complicated and
therefore inaccurate to determine the exact amount of recombination.
Reddy [32] tried to minimize this effect by having a flow condition such that the gas would have a low
pressure and a high temperature, after it is being hit by the primary shock wave. In this condition the
gas is assumed to be frozen when it travels through the nozzle, hence a significant amount of dissociated
species reaches the test section.
In order to overcome the problem of dissociation in the nozzle, a shock tube can be used. Consisting of only a driver and driven tube (and potentially a dump tank) it is however not able to reach as
29
3.3. Arc-jet Facility
Figure 3.4: The
(upper)
and
two-dimensional
probe
used by Reddy. [32]
AXISYMMETRIC
AND
TWODIMI:NSIONAl PROBES
{d3/d(lower)
FIG.axisymmetric
5. 1
2 =O.6}
high Mach number and enthalpies as using a shock tunnel. However, the simplicity of this facility makes
it possible to more accurately control the flow the model experiences.
In 2013 Park [15] utilized a shock tube for this purpose. The goal of the research was to find the
catalytic efficiency of oxygen on copper oxide (and copper). The enthalpy of the flow ranged from 1.92
to 4.24 MJ/kg, which is significantly lower than for the shock tunnel used by Maclean and Holden [33].
In order to achieve a high amount of dissociation in the flow the oxygen was mixed with argon which is
an inert gas. Since argon does not dissociate, less energy is needed to elevate its temperature. Moreover,
the inert gas increases the molecular weight of the gas mixture which slows down the shock speed, hereby
having a slightly positive effect on the test time. Also, the total pressure increases significantly – while
the total enthalpy drops slightly – which increases the amount of dissociation.
The consequence of this change from a binary gas mixture (O - O2 ) to a tertiary gas mixture (O
- O2 - Ar) was that the theory had to be rewritten which was presented in Section 2.3.2.2. The heat
flux is measured using a similar method as Reddy [32]. With the aid of thin-film gauges coated with a
500 nm thick layer of silicon dioxide and a layer of 180 nm copper oxide the heat flux was determined
experimentally. The same has been repeated for copper, however, only one shot was successful. The
catalytic efficiency determined by Park [15] for copper oxide was 2.6 · 10−3 to 3.2 · 10−3 and for copper
1.6 · 10−2 .
3.3
Arc-jet Facility
The third major group of facilities that are used to investigate catalycity are arc-jets. These facilities use
an electric discharge (an arc) to accelerate the flow to high velocities (i.e. supersonic freestream [34, 35])
at high temperatures (i.e. several thousands Kelvin). This is achieved using two direct current power
supplies, vacuum system, cooling system, high pressure gas system, and some other additional systems
to operate the arc-jet. A schematic of an arc jet complex is shown in Fig. 3.5.
The two electrodes are connected to the direct current power supply and this creates an electric
discharge which converts the gas to a plasma. This plasma consists of dissociated and potentially ionized
particles, which is accelerated through the nozzle towards the model. Arc-jets are capable of providing
test times in the order of seconds, minutes or even as long as an hour. An illustration of a specimen
being testing in an arc-jet is shown in Fig. 3.6.
In 1973 Anderson [34] utilized three different arc-jets at the NASA Ames research center, each with
a different range in total enthalpy ranging from 3.2 MJ/kg to 120 MJ/kg, to investigate the catalytic
30
Chapter 3. Experimental Research in Catalycity
Figure 3.5: The working principle of the arc jet complex at NASA Ames Research Center. [36]
Figure 3.6: Material being tested in an arc-jet at the NASA Ames Research Center. [36]
efficiencies of several materials when exposed to nitrogen. For a limited amount of cases also air was
tested.
Both flat face and hemisphere models were used, as shown in Fig. 3.7. With the aid of slug calorimeters
mounted at the center of the models, the heat flux was measured. Using Goulard’s theory (Section 2.3.2.1)
the catalytic efficiency was determined. Different coatings were applied on top of the samples (nickel,
platinum, gold, chromium, copper, and silicon dioxide).
The catalytic efficiency results of Anderson [34] are shown in Fig. 3.8. Not only the different surface
materials show a different value, also increasing the temperature of the model results in a change.
Figure 3.7: The models used by Anderson for arc-jet experiments. [34]
The high enthalpy and temperature that can be reached using an arc-jet, make it possible to find
catalytic efficients of thermal protection materials which is relevant for re-entry vehicles. The total
enthalpy and long test time that can be reached using an arc-jet surpasses the capabilities of a side-arm
reactor or shock tube/tunnel. Pidan et al. [35] used in 2005 an arc-jet to produce a 34 MJ/kg flow to
test sintered silicon carbide (SSiC) when exposed to dissociated oxygen. A double probe was used to
measure the heat flux using a calorimeter and the pitot pressure sensor, shown in Fig. 3.9.
While an arc-jet is a formidable facility to achieve a high temperature, a high enthalpy, a high Mach
number, and a long test time, it is not capable of simulating the pressure that is experienced during
re-entry. The freestream static pressure in an arc-jet is in the order of several tens or hundreds pascals
(e.g. Pidan et al. [35] reported 40 Pa and one of world’s most powerful arc-jet SCIROCCO [37] can reach
290 Pa). This makes it impossible to properly mimic the stagnation pressure in real flight, which results
in an inaccurate catalytic efficiency when used in Goulard’s theory (Eq. (2.18)-(2.20)) since the Schmidt
number might be incorrect.
31
3.3. Arc-jet Facility
567
PIDAN ET AL.
lues of γSiC(O) are about 10−3 (quite low), typical
ic materials.
ents the investigation of five materials regardbehavior in air plasma and pure oxygen plasma,
icon carbide (SSiC), chemical-vapor-deposited
VD-SiC) coating on the carbon/carbon silicon
sample, yttrium silicate coating on the C/C-SiC
ly produced magnesium spinel, and oxidized aluly, investigations on SSiC, CVD-SiC coating, ytng, and spinel, referring to their spectral emissivplasma, are described. The investigations of the
with CVD-SiC and yttrium silicate coatings were
he German space research program Ausgewählte
nologien für zukünftige Raumtransportsystem-
where the fully catalytic heat flux can be calculated. The dissociation
degree ξ O in the oxygen plasma flow at the measurement position
has been determined to be very close to unity.7
In the next step, the heat flux measured on the investigated materials was used to calculate the recombination rate constant on the
material surface from Eqs. (2) and (3) and eventually to calculate
the recombination coefficient by using Eq. (1).
All plasma parameters (gas constant, density, etc.) at the
boundary-layer edge and at the sample wall have been calculated
under the assumption that the plasma at the boundary-layer edge
and at the wall is in thermochemical equilibrium. Admittedly, this
simplification results in erroneous values of some of the terms of
Eqs. (2) and (3), which lead to discrepancies in values of recombination coefficients. However, the assessment of the influence of the
equilibrium assumption for calculation of plasma parameters leads
periments have been conducted with the magneto values of recombination coefficients of the same order of magnigenerator RD53,4 (qualitative comparison tests)
tude. Schmidt and Lewis numbers and viscosity values have been
calculated
according to paper
Ref. 8. [34] showing his catalytic efficiency results.
tunnel PWK2 and with the inductively
heated
Figure
3.8:
A
copy
of Anderson’s
For the calculation of the local stagnation enthalpy at boundaryIPG33,4 (quantitative comparison tests) in the
layer edge h se , the following equation was used7 :
l PWK3 of the Institute of Space Systems (IRS)
of Stuttgart. The planned catalytic experiments
Rpl
2
ogen and air plasmas will be conducted in the
q̇(x, 0)
q̇(x, y)
h se (x, 0) Rpl
≈
(4)
y dy
h tot
2
ppitot (x, 0)
ppitot (x, y)
0
nvestigations were conducted in the IRS PWK2
, which is equipped with the RD5 magnetoplaswhere Rpl = 100 mm.
tor.
The total flow enthalpies h were measured with a cavity
tot
ermination of the Recombination
cients and Spectral Emissivity
cribes the methods for determining the recombin oxygen plasma flows and in situ determination
sivities of the investigated materials.
of Recombination Coefficients
calorimeter7 and the radial profiles of the heat flux and pitot pressure were measured with the double probe for the heat flux and pitot
pressure measurements (Fig. 1).7
To calculate plasma speed u ∞ , Mach number Ma∞ and the effective isentropic exponent κ∞ are needed. Their values were obtained
from the pitot and ambient pressure measurements and from the
measurements with the wedge probe7 (Fig. 2). The measurements
y used for determining the recombination coefstigated materials is similar to the methodology
effective recombination coefficients in nitrogen
ed by Scott,5 with the exception that the experiuantitative comparison of the catalytic behavior
materials have been conducted in pure oxygen
mination of the recombination coefficients has
ased on the Goulard’s theory6 and heat fluxes
the investigated materials in the stagnation point
ow. The recombination rate constant kw (and the
ficient γo ) on the material surface can be detero of the fully catalytic heat flux to the measured
flux and under the assumption of complete ache released chemically energy (i.e., the energy
efficient, β = 1).
on of the fully catalytic heat flux, copper oxide
d as reference material, because other materials
Fig. 1 Double probe for the heat flux (calorimetric) and pitot-pressure
y catalytic and for which Figure
the literature
of the
3.9:data
The
double measurements.
probe used by Pidan et al. [35] to measure the heat flux and pitot pressure.
ion coefficients can be found were not available
ic heat flux q̇full cat was calculated with the heat
measured on a cooled oxidized copper sample.
ygen recombination coefficient of CuO, one can
mbination rate constant kwCuO(O) for the oxygen
CuO surface using
2γCuO(O)
CuO(O) =
2 − γCuO(O)
0.665
u ∞ µe ρe
Tw
2π M O
0.5 ρ∞
(1)
ρ∞
0.25 32
absorbed oxygen – that is to say that the rate is infinite initially followed by steady decline as the surface becomes
more saturated. According to the literature, the copper surface oxidizes (Cu2O) within seconds after exposure to
oxygen18.
Chapter 3. Experimental Research in Catalycity
The literature also indicated that exposure of copper to oxygen at temperatures above 300oC would result in the
formationMoreover,
of cupric since
oxidethe
(CuO)
rather
than that
Cu2O
– this was
with
oven tests (described
in the next
electric
current
discharges
theconfirmed
flow cannot
be continuous,
the flow properties
section).
We
mention
this
because
in
practice
is
often
difficult
to
avoid
raising
the
slug’s
temperature
above
300oC.
are fluctuating over time. This non-uniform flow is highly undesired and gives uncertainties in the actual
That flow
was conditions.
not the casePobst
for this
in thisthat
study
deliberately
minimized
the exposure
time so
that the
et al.study,
[38] noted
for awe1 kW
arc-jet the
temperature
could fluctuate
between
o
C
during
their
exposure
to
the
arc
jet.
temperature
of
the
slugs
would
stay
below
300
4950 K and 6000 K.
In thisDue
study
we harsh
discovered
that exposing
slugs toisdissociated
wouldof change
the surface
to the
environment
in whichcopper
the specimen
placed the oxygen
composition
the material
can
composition
from
cuprous
oxide,
Cu
O,
to
cupric
oxide,
CuO.
This
was
observed
to
be
the
case
for
all
levels of
2
change. Metals may for example oxidate which changes the catalytic efficiency completely. It was noticed
enthalpy
tested,[39]
regardless
of the
insertion
The calorimeters
surface analysis
indicated
thathave
all slugs
arc jet
by Park
in 2006 that
the
surface time.
of copper
discolored
after
beingexposed
used in to
an the
arc-jet.
showed more
than
80%
of
the
probed
volume
(50-100Å
depth,
1400µm
x
300
µm
area)
consisted
of
CuO,
the
rest
In 2013 Nawaz et al. [40] confirmed that exposing the copper calorimeters to a flow inside an arc-jet
O (neglecting
the
organic
materials
that
accumulate
naturally
on
the
slug
after
the
run
due
to
contaminants
being for
Cu2only
2-3 sec is sufficient to create an oxidized layer, this is shown in Fig. 3.10. The oxide layer causes
in the the
air).red color, hence a darker red color corresponds to a thicker layer. It was noted by Nawaz et al. [40]
It that
must even
be noted
thatthe
surface
color at
does
not seem
to not
be ashow
reliable
for these
depths
of oxidation.
though
slug used
5 MJ/kg
does
anyindicator
color change,
afterlow
XPS
analysis
it was
Whilefound
it is safe
to
associate
a
discoloration
with
change
in
surface
composition,
the
reverse
argument
cannot be
that oxidation has taken place.
made. Figure
11 shows
after to
thethe
tests
in AHF. The
5MJ/kg condition
in particular
very closely
resembles
Another
effect the
thatslugs
is linked
occurrence
of oxidation
is the increase
in roughness
of the surface.
the surface
color
of
the
unexposed
slug,
falsely
leading
to
the
conclusion
that
no
oxidation
took
place.
XPS
As explained in Section 2.2.3 the roughness may have a significant effect on the catalytic efficiency. analysis
Since
showed
the surface
of all slugs
that
wereenvironment
tested to consist
CuO
(Fig
three slugs
on the
left). The higher
thethat
samples
are exposed
to the
harsh
for a of
long
time
the11surface
is prone
to roughen.
enthalpy conditions seem to change the oxide layer thickness, which in turn is responsible for the change in color.
Figure 11 Slug surfaces after AHF tests.
Figure 3.10: Copper slugs used in the calorimeters
before and after arc-jet experiments. [40]
12
American Institute of Aeronautics and Astronautics
3.4
Usage and Uncertainties of Copper Calorimeters in Arc-jet
Facilities
In Section 3.3 copper calorimeters have been mentioned on several occasions. In this section the concept
of a slug calorimeter will be discussed together with the application in arc-jet facilities. From this the
uncertainties and problems will be discussed that arises in an arc-jet. Section 3.4.3 then focuses on the
catalytic efficiencies of copper and copper oxide from which the main reason for this master thesis project
arises.
3.4.1
Concept of Copper Slug Calorimeter
A slug calorimeter is used to determine the heat transfer rate to the surface by measuring the time rate
of the temperature change. [41] The temperature is measured using a thermocouple junction which is
mounted to a slug. The voltage change across the two metals in the thermocouple is measured. [39] Typically the slug is thin to approach the thin-skin approximation where there is no significant temperature
gradient across the slug. Moreover, it is also necessary to use a highly conductive material as slug to
minimize the temperature gradient. The basic concept of a slug calorimeter is shown in Fig. 3.11.
Copper is a common material used for the slugs due to its highly conductive properties and its low
price. Examples of copper slugs are shown in Fig. 3.10.
The temperature measurements can be converted to heat flux using [41]:
qs = ρcl
dT
dt
(3.1)
where the density (ρ), specific heat (c) and the length (l) of the slug are required.
Copper calorimeters are mostly applied in arc-jet facilities to determine the centerline enthalpy which
will be discussed more in the next section. This is possible because arc-jets have a run time in the order
of seconds, sufficient to conduct the heat through the copper slug. These calorimeters are however not
suitable for impulse facilities such as shock tube or shock tunnel due to the short run time.
33
3.4. Usage and Uncertainties of Copper Calorimeters in Arc-jet Facilities
Figure 3.11: Copper calorimeter used by Park et al. [39] to measure the heat flux.
3.4.2
Measurement of Centerline Enthalpy of Arc-jet
In an arc-jet the enthalpy is an important parameter to properly determine the flow condition a sample
experiences. Such a sample could for example be a panel of a re-entry vehicle. The enthalpy is higher at
the centerline of the nozzle compared to the nozzle walls. Since the sample is exposed to the flow from
the centerline of the nozzle, it is necessary to determine this accurately. In general there are several ways
to determine the centerline enthalpy: [39]
• Mass-averaged enthalpy measurement, heat balance method: The total power is determined by
subtracting the cooling water loss from the electrical power input. The average enthalpy can then
be determined experimentally by dividing the total power by the mass flow rate of the test gas.
The centerline enthalpy is then assumed to be equal to the average enthalpy.
• Mass-averaged enthalpy measurement, sonic throat method: The average enthalpy is determined
for air using [39]:
2.519
0.293Ap0
H=
(3.2)
ṁ
This method is based on the principle that the flow is choked at the throat of the nozzle. Also here
the centerline enthalpy is assumed to be equal to the averaged enthalpy.
• Centerline enthalpy measurement, heat transfer method: This method is used to determine the
stagnation point heat flux for a cold wall. The tool that is usually used for an arc-jet is a copper
slug calorimeter, which had been discussed in Section 3.4.1. Using Goulard’s theory (Section 2.3.2.1,
Eq. (2.18) and (2.20)) the enthalpy can be determined. Hereby the catalytic efficiency is necessary.
For copper it is usually taken to be equal 0.17 (Fig. 2.7).
• Centerline enthalpy measurement, spectrometric method: Using a spectrometer the radiation emitted from the shock layer is captured. This device is able to identify the wavelengths of which this
radiation consists of, hereby making a spectra. The wavelengths with high intensities can be linked
with certain species.
Using CFD a theoretical spectra can be made that is depending on the centerline enthalpy. By
simply matching the theoretical spectra to the measured one the enthalpy can be found.
• Mass-averaged/Centerline enthalpy measurement, CFD: Using CFD the flow in an arc-jet can be
simulated. With this it is possible to calculate both the mass-averaged and the centerline enthalpy.
Since this is out of the scope of this thesis, it will not be discussed further. More literature can be
found in the paper of Kim et al. [42].
While the heat balance method is in practice the easiest method, it yields large errors. Shown by
Park et al. [39] the centerline enthalpy can be 1.41 times larger than the average enthalpy. On the other
hand, the spectrometric method is considered to be a more accurate method to determine the centerline
enthalpy. However, it requires an optical setup, a spectrometer, knowledge regarding the wavelengths of
the species and CFD to reproduce the spectra.
34
Chapter 3. Experimental Research in Catalycity
Compared to the spectrometric method it is easier to use a copper calorimeter to determine the
centerline enthalpy. However, for this the catalytic efficiency has to be known accurately. It was discussed
in Section 3.3 that the copper oxidizes and roughens in the harsh environment of the arc-jet. This
influences the catalytic efficiency and could yield inaccurate results. It was shown by Park et al. [39]
that simply using γw = 0.4 (Fig. 2.7) which is the value of copper determined by Greaves and Linnett [29]
using a side-arm reactor, the centerline enthalpy would only be 0.75 of the spectroscopically determined
value.
Hence by assuming that the calorimeter is highly catalytic (i.e. copper) instead of moderately catalytic
(i.e. copper oxide) would result in a lower centerline enthalpy. This underestimation of the environment
could lead to the conclusion that a certain material that is tested in an arc-jet is less resistant than
it actually is. This forces the engineers to choose for a more conservative design (e.g. increasing the
thickness) than necessary.
3.4.3
Copper and Copper Oxide Catalytic Efficiency
From the previous section the importance of the catalytic efficiency of copper was discussed. Not only
does this value depend on the composition of the copper (i.e. the amount of oxidation), the roughness
of the surface, and the temperature of the surface, also the type of gas has an effect on the catalytic
efficiency.
A significant amount of research has been conducted on the catalytic efficiency of copper and copper
oxide, summarized in Table 3.3 which has been based mainly on the table made by Nawaz et al. [40].
It should be noted that several authors only mentioned “room temperature” in their papers. For these
a temperature of 300 K has been taken. The values in the table are also shown graphically in Fig. 3.12
with the uncertainties – if they were noted by the authors.
In this table the distinction is made between the two types of copper oxides: cupric oxide (CuO) and
cuprous oxide (Cu2 O). The former is black in color whereas the latter is red. This distinction is rather
important since copper can oxidize in either of the two with different catalytic behavior as discussed by
Nawaz et al. [40].
Table 3.3: Summary of catalytic efficiencies of copper and two types of copper oxide (CuO and Cu2 O).
γCu
γCuO
γCu2 O
0.17
0.15
0.063
0.015
0.016
-
0.02
0.043
0.045
0.01
0.11
0.025
0.0026-0.0032
-
Test Twall
Gas [K]
O2 300
O2 300
O2 300
O2 300
O2 300
O2 300
O2 313
O2 350
O2 300
0.03
0.07
0.08
0.068
0.10
0.4
-
-
N2
N2
N2
N2
N2
N2
0.03
0.01
-
0.01-0.03
-
Air 350
Air 480
Air 373
314
300
350
300
500
1000
Facility
Year
Authors
Side-arm Reactor
Side-arm Reactor
Side-arm Reactor
Glow Discharge Tube1
Effusion Tube2
Diffusion Tube
Side-arm Reactor
Shock Tube
Side-arm Reactor
1958
1959
1964
1965
1967
1971
1998
2013
2013
Greaves, Linnett [29]
Greaves, Linnett [27]
Dickens, Sutcliffe [31]
Hartunian et al. [43]
May, Linnett [44]
Melin, Madix [45]
Cauquot et al. [46]
Park [15]
Nawaz et al. [40]
Diffusion Tube
Glow Discharge Tube1
Arc-jet
Side-arm Reactor
1961
1965
1968
1971
Prok [47]
Hartunian et al. [43]
Pope [48]
Rahmann, Linnett [30]
Arc-jet
1973
Anderson [34]
Arc-jet
Arc-jet
Arc-jet
1968
2006
2013
Pope [48]
Park [39]
Nawaz et al. [40]
When considering the catalytic efficiencies of copper and copper oxide for oxygen in Table 3.3, it can
be seen that there is a significant variation in the values. The values for copper show a variation up to
one order of magnitude (Greaves, Linnett [29] vs. Park [15]), whereas for Cu2 O the variation can be even
1 A glow discharge tube creates a discharge using high voltage and translates it towards the sample, similar to a diffusion
tube.
2 An effusion tube is similar to a diffusion tube but it has an orifice with openings smaller than the mean free path of
the gas through which the gas has to effuse through.
35
3.4. Usage and Uncertainties of Copper Calorimeters in Arc-jet Facilities
two orders of magnitude (Dickens, Sutcliffe [31] vs. Park [15]). All these values are determined around
300 K.
This difference in catalytic efficiency could be explained by several reasons. First of all, the surface
activation energy (E) of copper and copper oxide differ which influences the catalytic efficiency directly,
as shown by Eq. (2.1). Moreover, the roughness which has a significant effect on the catalytic efficiency,
is not the same for all cases shown in the table.
The copper oxide results of Park [15] is at least an order of magnitude smaller than all the other
results, which could potentially indicate the inaccuracy of the values determined using facilities different
than a shock tube.
The results of Park [15] using a shock tube indicates that the copper oxide catalytic efficiency – whether
it is CuO or Cu2 O – has to be determined again using a shock tube, where the uncertainty on the
roughness and composition is minimum. While it is desirable to determine this for air, it brings complications regarding the number of species that are present in the flow. Covering air not only includes the
dissociation of oxygen, the dissociation of nitrogen, but also the coupling between the two (e.g. NO).
This results in a mixture consisting of five or more species.
An important step towards this goal is to determine the copper oxide catalytic efficiency for nitrogen
which has not be done before, as can be seen in Table 3.3. Having the catalytic efficiency of both oxygen
and nitrogen separately, will function as a preliminary indication of the value belonging to copper oxide
in air.
1200
Air
Cu
CuO
Anderson
1000
Tw [K]
1000
Tw [K]
Nitrogen
Cu
1200
800
800
600
600
Park
Anderson
Nawaz et al.
400
400
Pope
Prok
Pope
200 −3
10
−2
10
−1
γw [−]
10
(a) Air catalytic efficiency.
1200
0
10
Rahman, Linnett
200 −5
10
−4
10
−3
10
−2
γw [−]
10
Hartunian et al.
−1
10
(b) Nitrogen catalytic efficiency.
Oxygen
Cu
CuO
Cu2O
800
Greaves,Linnett
w
T [K]
1000
Dickens,Sutcliffe
600
Park
400
Greaves,Linnett
Cauquot et al.
May et al.
Nawaz et al. Melin et al. Greaves,Linnett Hartunian et al.
200 −3
−2
−1
0
10
10
10
10
γw [−]
(c) Oxygen catalytic Efficiency.
Figure 3.12: Catalytic efficiencies of air, nitrogen and oxygen against temperature from literature.
36
4
Experimental Equipment
For this project the facilities at the Korea Institute of Science and Technology (KAIST) in Daejeon
(Republic of Korea) are utilized. The goal of this chapter is to introduce the principles of a shock tube
(Section 4.1) and the sensors used for the experiments (Section 4.2). With the basic understanding of
the shock tube it is possible to investigate the effect of various gases in the driver and driven tube, which
is valuable when choosing the flow conditions in Chapter 5.
Regarding the sensors a discussion is made to consider several options to measure the heat transfer
rate. The thin-film gauge proves to be the ideal sensor for the experiments in the shock tube. Not
only does Section 4.2 discuss the manufacturing procedure of a thin-film gauge, also the calibration and
coating of the probes are covered. The coating of copper oxide introduced some difficulties which resulted
in two variations of copper oxide coatings, referred as “black CuO” and ”brown CuO”.
This chapter ends with the shadowgraph setup which is used to perform flow visualizations (Section 4.3). The goal of the images is to visualize the shape of the standing shock, determine the shock
stand-off distance, and to measure the shock radius at the stagnation point. These parameters are
necessary in further calculations which will be discussed in Chapter 6.
4.1
Shock Tube
In Section 4.1.1 the characteristics of the shock tube are presented with a qualitative discussion of the
working principles. This is then followed by the introduction of the test models and the test section. In
Sections 4.1.3 and 4.1.4 fundamental calculations of the shock tube are presented based on a perfect gas
to fully grasp the physics behind this facility. With the analytical equations presented in Section 4.1.4 a
comparison is made between various gases used as driver and driven gas.
4.1.1
Shock Tube Characteristics
This approximately 4.65 m long tube consists of several parts: a driver tube, a transition piece, and a
driver tube (Fig. 4.1), having respectively a length of approximately 0.80 m, 0.25 m, and 3.60 m. The
driver and the driven tube have an internal diameter of 6.8 cm and 4.75 cm respectively. As shown in
Fig. 4.2(a) the driver tube and the transition piece are separated by a 0.35 mm thick diaphragm made
of polyethylene. The driver pressure can be doubled using two diaphragms pressed on each other.
A rotary pump is connected to both sides of the diaphragm by which a pressure below 8 Pa can be
reached inside the tube. Followed by this, the test gas is introduced in the driven tube to the desired
pressure. Consecutively the driver gas is allowed into the driver tube until the diaphragm burst due to
the pressure difference between the driver and driven sections. Two piezo-electric pressure sensors are
flush-mounted to the wall of the shock tube to not only measure the static pressure, but also the velocity
of the primary shock at 3.45 m and 3.95 m, respectively. At the end of the driven tube the test section
is located which will be elaborated upon in the next section.
The situation after the bursting of the diaphragm is shown in Fig. 4.2(b). The gas from the driver
tube has moved to the driven tube section and pushes the test gas towards the model. A primary shock
wave propagates through the test gas and compresses the gas. This is followed by a contact surface which
forms the boundary between the driver and test gases. In the ideal case the test time ranges from the
moment the primary shock wave reaches the model until the arrival of the contact surface, as indicated
in the x-t diagram (Fig. 4.2(c)).
Some time is required for the flow to establish and to form a standing shock in front of the model.
When the primary shock wave reaches the model it reflects locally at the surface, opposite to the flow
37
4.1. Shock Tube
Figure 4.1: The shock tube of KAIST.
(a) Schematic of the shock tube before the rupture of the diaphragm.
(b) Schematic of the shock tube after the rupture of the diaphragm.
(c) The x-t diagram.
Figure 4.2: Schematic of the shock tube at KAIST before and after the rupture of the diaphragm.
38
e contact
the
contactsurface.
surface.The
Thedensity
densitybehind
behindthethebow
bowshock
shockwas
wasmuch
much
gher than
higher
thanthetheinitial
initialcharging
chargingdensity
densitybecause
becausethe
thestanding
standingshock
shock
ow
flowhad
hadgone
gonethrough
throughthethetwo
twonormal
normalshock
shockwaves.
waves.Therefore,
Therefore,it itis is
latively easy
relatively
easy toto obtain
obtain anan equilibrium
equilibrium flow
flow behind
behind the
the
anding
standingshock.
shock.
Chapter 4. Experimental Equipment
Fig.
3 3 Typical
time
history
ofof
the
measured
stagnation
pressure.
Fig.
Typical
time
history
the
measured
stagnation
pressure.
direction. After a finite moment this reflective wave steadies
and
becomes
a standing
shock
at a certain
distance (i.e. shock stand-off distance) from the model.
4.1.2
Test Models and Test Section
Two different flat disk-cylinder models Sapphire
are
utilized
for the experiments. One model is used to flushSapphire
window
window
mount the thin-film gauges (presented in Section 4.2) to measure the surface heat transfer rate at the
stagnation point, whereas the other one is used to flush-mount a piezo-electric sensor to measure the
stagnation pressure. Both models are made of stainless steel SUS 630.
The heat transfer model is shown in Fig. 4.3(a). It has a length of 15 mm, an outer diameter of
12 mm, and the edges have a fillet of 1.5 mm. The model can be attached to a 400 mm long sting with
a diameter of 9 mm. This sting is then fixed to the end wall of the shock tube.
The pressure model is shown in Fig. 4.3.
It is slightly
Experimental
model larger with a length of 40 mm, an outer
Experimental
model
diameter of 20 mm, and the edges have a fillet of 3.0 mm. The sting is slightly thicker (14 mm) and is
with 383 mm slightly shorter than the previous sting. Similar to the previous sting it can be attached
to the end wall of the shock tube.
Pressure
model
Heat-transfer
model
Pressure
model
Heat-transfer
model
(a) The dimensions of the flat disk-cylinder
heat transfer
(b)
The flat disk-cylinder
Fig.
Schematic
the
experimental
models.pressure model used to mount
Fig.
2 2 Schematic
ofof
the
experimental
models.
model. [15]
the piezo-electric sensor. [15]
Figure 4.3: The flat disk-cylinder model used to host the heat transfer gauge and pressure sensor.
As introduced in Section 2.4.1 it is necessary to have a model that induces a standing shock with
a sufficiently large stand-off distance to comply with the thermochemical equilibrium assumption. A
spherical model cannot fulfill this requirement while a flat-disk cylinder model does. This is verified by
an analysis discussed in Appendix A.
The corner radius of a flat disk-cylinder model influences the stagnation point velocity gradient. By
increasing the radius the gradient increases which results in a larger heat transfer rate as shown by Zoby
and Sullivan [49]. However, increasing the radius too much would alter the general shape of the body to
such a degree that the surface is no more flat. In practice the corner radius is taken to be a quarter of
the body radius [50] which is also true for the heat transfer and pressure models presented in this section.
The test section of the shock tube is shown in Fig. 4.4(a). On each side two openings are located
where sapphire windows can be placed. The diameter of the window ports is approximately 14.5 mm.
In Fig. 4.4(a) the port is indicated and corresponds to the location which is used in the experiments.
The presence of the windows allows the use of visualization techniques to capture the flow around the
model, which will be elaborated in Section 4.3. In Fig. 4.4(b) a heat transfer model is located inside the
test section seen through the window.
(a) The test section of the shock tube.
(b) The heat transfer model located inside the test section.
Figure 4.4: The test section located at the end of the shock tube.
39
4.1. Shock Tube
4.1.3
Riemann Problem
In this section the situation inside the shock tube will be analyzed more closely for a perfect gas. These
calculations are valid for flow conditions where the temperature is low enough that the molecules are not
vibrationally excited.
The diaphragm bursts in a passive way by the pressure difference between the driver and driven tube. At
this moment a shock wave is created starting from this diaphragm and compresses the test gas. While
the test gas was initially at rest, this compression results in an acceleration of the flow in the direction
of the test section. In the opposite direction of the diaphragm an expansion wave is created.
From the x-t diagram shown in Fig. 4.2(b) it can be concluded that it is a 1-D unsteady non-linear
Riemann problem. The different regions are numbered from 1 to 4. Region 4 is located in the driver tube
hence the pressure is a lot higher than in region 1 which is located in the driven tube. Because of the
expansion wave traveling to the left the pressure will drop in region 4, expressed by the Poisson curve
as shown in Fig. 4.5. In this figure it is also shown that because of the shock wave traveling into region
1 the pressure increases, expressed by the Hugoniot curve. The intersection of the two curves forms the
situation of regions 2 and 3. [51]
As can be seen from this pressure-velocity diagram, the velocity increases when the gas travels from
region 1 to 2 or from 4 to 3. Hence the pressure and velocities in regions 2 and 3 are the equal, but the
densities, entropies and internal energies are different. This is caused by the different gases in the two
regions.
Figure 4.5: Pressure-velocity diagram with the expansion and shock wave.
4.1.4
Quantitative Analysis of Shock Tube
In the previous section the situations in regions 2 and 3 have been described qualitatively. A quantitative
analysis is presented in this section which is based on the equations presented by Glass and Patterson [52]
to determine the four flow conditions in the shock tube. While Glass and Patterson also present equations
for the reflected shock wave and reflected expansion wave, they are omitted in this section.
The strength of the shock wave, dictated by P21 = pp21 , depends on the initial conditions by:
P14
"
#1/β4
r
1
β4 E14
=
1 − (P21 − 1)
P21
α1 P21 + 1
(4.1)
This equation is written explicitly for the pressure ratio between the driven and driver gas (P14 = pp41 )
and can be solved using the Newton’s method to find P21 . As concluded in Section 4.1.3 is the pressure
in region 2 identical to region 3. The internal energy across the diaphragm is dependent on the gases
used in the driver and driven tube by:
Cv1 T1
(4.2)
E14 =
Cv4 T4
40
Chapter 4. Experimental Equipment
Constants that occur frequently in shock tube equations are collected in the constants αi and βi :
γi + 1
γi − 1
γi − 1
βi =
2γi
αi =
(4.3a)
(4.3b)
With the acquired P21 it is possible to calculate the remaining flow parameters:
• The density ratio: The density ratio (Γij =
ρi
ρj )
can be determined using:
1/γ4
Γ34 = (P14 P21 )
1 + α1 P21
Γ21 =
α1 + P21
• Speed of sound and temperature ratio: The speed of sound ratio (Aij =
(Tij =
Ti
Tj )
(4.4a)
(4.4b)
ai
aj )
and temperature ratio
are related to each other and can be calculated using:
p
β
T34 = (P14 P21 ) 4
s
p
P21 (α1 + P21 )
= T21 =
1 + α1 P21
A34 =
(4.5a)
A21
(4.5b)
• Shock wave velocity or Mach number: The Mach number or the velocity of the shock wave can be
determined using:
p
ushock
(4.6)
Mshock =
= β1 (1 + α1 P21 )
a1
• Contact surface or gas slug velocity: The velocity of the flow in regions 2 and 3 can be determined
using:
U21 =
U34
P21 − 1
p
γ1 β1 (α1 P21 + 1)
1
=
[1 − (P14 P21 )β4 ]
γ4 β4
(4.7a)
(4.7b)
where the normalization Uij = auji has taken place. When substituted with realistic initial conditions
for the shock tube it is witnessed that the difference between u2 and u3 is negligible which confirms
the conclusion of Section 4.1.3.
• Velocity of heat and tail of expansion wave: The heat (C44 ) and the tail (C34 ) of the expansion
wave can be determined using:
C44 = −1
1
C34 =
[1 − (P14 P21 )β4 ] − (P14 P21 )β4
γ4 β4
where the normalization Cij =
ui
aj
(4.8a)
(4.8b)
has taken place.
• Local Mach numbers: The Mach number of regions 2 and 3 can be calculated using:
1
[(P14 P21 )−β4 − 1]
β 4 γ4
P21 − 1
M2 = p
γ1 β1 P21 (α1 + P21 )
M3 =
The speed of sound in regions 1 and 4 can be determined using
r
p
Runi
ai = γi Ri Ti = γi
Ti
Mi
(4.9a)
(4.9b)
(4.10)
and the density is acquired from the perfect gas law:
ρi =
41
pi
Runi
Mi Ti
(4.11)
4.1. Shock Tube
With Eq. (4.1)-(4.11) all the flow parameters can be determined for regions 1, 2, 3, and 4, with the
exception of the expansion wave. While the head and the tail of the expansion wave are known from
Eq. (4.8), Glass and Patterson [52] presented an additional series of equations to determine the flow
conditions across the expansion wave:
n(x, t) + 1
a(x, t) = 1 −
a4
(4.12a)
α4
2
n(x, t) + 1
T4
(4.12b)
T (x, t) = 1 −
α4
γ 1β
4 4
n(x, t) + 1
ρ4
(4.12c)
ρ(x, t) = 1 −
α4
β1
4
n(x, t) + 1
p4
(4.12d)
p(x, t) = 1 −
α4
2(a4 − a)
u(x, t) =
(4.12e)
γ4 − 1
u(x, t)
M (x, t) =
(4.12f)
a(x, t)
where n(x, t) is defined as:
n(x, t) =
x
a4 t
(4.13)
In Fig. 4.6 the results are shown at two instances for a condition where the helium is used as driver
gas and air as driven gas. The driver and driven pressure are 1.65 MPa and 5000 Pa respectively, and
the initial temperature is 300 K for both gases. The length of the shock tube discussed in Section 4.1.1
is used to display the results, where x = 0 m corresponds to the location of the diaphragm.
The primary shock wave is not only clearly visible as a sudden drop in Fig. 4.6(a) at x = 0.21 m for
t = 0.1 ms, also this abrupt change is visible in the remaining figures. The more gradually transition
before the shock is the expansion wave (Fig. 4.6(a) from x = −0.13 m to x = 0.51 m, for t = 0.1 ms). The
contact discontinuity is visible as a jump between the expansion wave and the shock wave (Fig. 4.6(c)
at x = 0.15 m for t = 0.1 ms). The flow behind the shock wave has reached Mach 2 (Fig. 4.6(d)) and a
temperature of almost 1500 K (Fig. 4.6(e)).
For the given situation the shock speed is 1576.0 m/s and the contact surface travels at 1249.6 m/s.
At the location of the sample in the shock tube (presented in Section 4.1.1) the test time is 566.8 µs.
To investigate the effect of the gases used in the driver and driven tube on the flow conditions, it is
important to compare several cases. The gases that are considered are presented in Table 4.1 together
with the necessary properties for the calculations. Hydrogen and helium are considered as driver gases
whereas oxygen, air, nitrogen, argon and krypton are covered as driven gases. Argon and krypton are
included to investigate the effect of a heavy gas particle on the results.
Table 4.1: Properties of the gases considered in the shock tube.
Gas
Hydrogen
Helium
Nitrogen
Air
Oxygen
Argon
Krypton
γ [−]
1.405
1.667
1.4
1.4
1.395
1.667
1.67
Cv [kJ/(kg · K)]
10.16
3.12
0.743
0.718
0.659
0.312
0.151
M [g/mol]
2.016
4.003
28.014
28.970
31.998
39.948
83.798
The results are summarized in Table 4.2. When the driver gas is kept constant it can be seen that
the static pressure, static temperature, and test time increases together with the molecular weight of the
driven gas. A larger test time is advantageous since it offers more time to the flow to become steady.
However, the velocity decreases which is generally also observable in the shock speed and Mach number.
The shock speed and the Mach number decrease since the larger weight of the test gas accelerates more
difficultly than a lighter gas. Nonetheless, the heavier gas has a higher amount of energy stored and this
is released when accelerated. The increased amount of energy is expressed by the higher temperatures
that is reached. This is also expressed by the decrease of Cv with the molecular weight (Table 4.1) which
42
Chapter 4. Experimental Equipment
1800
1400
t = 0.1 ms
t = 0.5 ms
1600
t = 0.1 ms
t = 0.5 ms
1200
1400
1000
u [m/s]
p [kPa]
1200
1000
800
600
800
600
400
400
200
200
0
−1
0
1
2
3
0
−1
4
0
1
2
3
4
x [m]
x [m]
(a) Pressure-location diagram.
(b) Velocity-location diagram.
2.5
3
t = 0.1 ms
t = 0.5 ms
t = 0.1 ms
t = 0.5 ms
2.5
2
1.5
M [−]
ρ [kg/m3]
2
1.5
1
1
0.5
0.5
0
−1
0
1
2
3
4
0
−1
0
1
2
3
4
x [m]
x [m]
(c) Density-location diagram.
(d) Mach-location diagram.
1500
t = 0.1 ms
t = 0.5 ms
T [K]
1000
500
0
−1
0
1
2
3
4
x [m]
(e) Temperature-location diagram.
Figure 4.6: Results of the shock tube calculations at two instances using helium as driver gas and air as
driven gas. High temperature effects are not included.
43
4.2. Heat Transfer Gauge
Table 4.2: Shock tube results for various combinations of driver and driven gas. High temperature effects
are not included.
Driver
Hydrogen
Driven
Nitrogen
Air
Oxygen
Argon
Krypton
p2 [kPa]
203.59
207.34
218.16
259.02
365.64
u2 [km/s]
1.692
1.680
1.644
1.524
1.257
ρ2 [kg/m3 ]
0.295
0.306
0.343
0.299
0.637
M2 [−]
1.721
1.724
1.744
1.268
1.284
T2 [K]
2326.4
2363.9
2449.5
4165.1
5785.3
ushock [km/s]
2.090
2.074
2.022
2.082
1.708
ttest [µs]
384.94
386.84
389.40
601.52
717.31
Helium
Nitrogen
Air
Oxygen
Argon
Krypton
116.95
119.37
126.41
153.78
229.70
1.257
1.250
1.228
1.156
0.987
0.270
0.280
0.316
0.286
0.619
1.614
1.619
1.643
1.221
1.255
1459.1
1483.4
1540.8
2585.7
3738.0
1.587
1.576
1.541
1.607
1.355
565.93
566.81
565.73
828.77
939.46
indicates that the heavier particles reaches a higher temperature when absorbing the same amount of
energy.
Although the calculation become inaccurate at high temperatures because γ and Cv are considered to
be constant, it is still valuable to compare the results for nitrogen and a heavy gas as krypton specifically.
The molecular weight of krypton is three times larger than nitrogen which results in a reduction of the
shock speed with a factor 0.85. The temperature however increase with a factor 2.5 and the test time
with factor 1.66.
When comparing the driver gases it can be observed that the shock speed increases significantly which
is caused by the higher speed of sound of the driver gas. Since the shock speed directly affects the remaining flow parameters, the increase of the parameters shown in Table 4.2 meets the expectations. Apart
from having the risk of decreasing the test time too much which could result in measurements before the
flow has completely reached a steady state, there is also some safety concern with hydrogen as driver gas.
From these results it is clear that the shock tube is able to create a high velocity field in the test
section by having a large initial pressure difference between the driver and driven tube. By choosing an
even higher initial pressure difference and by potentially mixing the test gas with a heavy inert gas, the
temperature can rise to such values that dissociation can take place. This cannot be done using these
analytical equations since they are based on the perfect gas assumption. In Chapter 5 the flow conditions
in the shock tube is chosen with the aid of more powerful programs.
4.2
Heat Transfer Gauge
The heat transfer gauge – a thin-film sensor – is selected to measure the heat transfer rate. Although
these sensors are very fragile they have a high sensitivity which is advantageous for the measurements.
During the manufacturing process a large number of gauges failed and had to be discarded. In this
chapter the steps will be presented which were followed to successfully produce a heat transfer gauge.
It should be noted that the presented steps are the result of a long iterative process where many
different approaches were attempted. The main difficulties were to make sure that the wires stayed
properly connected to the gold bridges during the static calibration (above 45◦ C the heat shrink becomes
a little soft hence the connection breaks), while at the same time the gauge should be able to be flushmounted in the model, and the model can be connected to the sting without the gauge blocking it.
This section starts with the working principles of thin-film gauges and the reasons for choosing
this type of sensor are presented. It then continues with the steps to manufacture a thin-film gauge
(Section 4.2.3- 4.2.4). In Section 4.2.5 the gauge has been successfully produced and is mounted in
the model, hereby forming a probe which is then calibrated. The probes that survived this calibration
process were coated (Section 4.2.6). The probes were split in four groups, each receiving a different type
of coating. The ones that eventually also survive the coating process were used for experiments in the
shock tube.
4.2.1
Working Principles and Characteristics of the Heat Transfer Gauge
It is essential to clarify the working principles of the heat transfer gauge to fully understand the application method in the shock tube. A thin-film gauge works on the principle that the resistance of
the metal (in this case the platinum) rises with temperature. Since the width of the platinum bridge
is approximately 0.5 mm the measurements are conducted near the stagnation point. The gold bridges
44
Chapter 4. Experimental Equipment
simply function as a conductor and is attached to the wires. Both the gold bridges and the wires have
ideally no resistance.
These wires are connected to a resistance temperature detector (RTD) using which a constant current
Ic of 9.88 mA is applied. Inside the RTD there is a Wheatstone bridge which increases the voltage during
a temperature increase. This RTD is attached to an oscilloscope by which the voltage can be plotted.
This oscilloscope has a trigger function which activates when a certain value is surpassed. The voltage
values acquired from the oscilloscope are then post-processed. This process is discussed in Chapter 6.
One might wonder why thin-film gauges are chosen for this experiment. This is related to a useful
feature of these gauges: the sensitivity. Compared to other candidates, such as a thermocouple gauge,
the sensitivity is superior. Schultz and Jones [41] presented the sensitivity of some common thermocouples and some metals. The temperature sensitivity of platinum is 3.01 mV/k whereas for some common
thermocouples – such as copper-constantan – is in the order of µV/k. A larger voltage output per Kelvin
is preferred since it will be easier to detect. The lower sensitivity of the thermocouples might be too low
for this application.
A big disadvantage of thin-film gauges is the fragility. The platinum bridge is very thin hence any
small damage could disrupt the closed circuit hereby making the gauge useless. Thermocouple gauges
do not have this problem. Moreover, while the copper wires are being kept in place using heat shrink
and Araldite, that connection is still sensitive. Especially during the calibration process on average half
of the sensors would die due to connection failure.
There is however one more aspect that makes the thermocouple disadvantageous compared to thinfilm gauges. While thin-film gauges measure the resistance change, thermocouples measure the voltage
change between the two metals. This junction between the two metals can be affected by the coatings
and hereby adapting the voltage change.
Moreover, the connection between the cables and the thermocouple introduces error due to the “cold
junction effect” that causes a voltage change. Those cables are also connected to a device to supply
electricity which again is sensitive to the same effect. While the latter can be corrected for, the cold
junction effect at the thermocouple is unknown and its effect is also not known. For this reason the
KAIST Hypersonics laboratory does not use thermocouples for their measurements.
As introduced in Section 3.4 the slug calorimeter is another probe that can be used to measure the
heat flux. It is based on a thin-skin principle and it requires a thermocouple connection to acquire the
temperature profile at the bottom on the slug. The thermocouple connection is also prone to the “cold
junction effect”.
However, the major disadvantage of a thin-skin sensor is the time that is required to conduct the
temperature through the slug. While in an arc-jet the test time is in the order of seconds, in a shock
tube the test time is in the order of hundreds of microseconds as presented in Section 4.1.4. The test
time could be too short hence the temperature rise measured at the bottom of the slug is significantly
smaller than at the top surface of the slug.
Moreover, in order to determine the catalytic efficiency of copper oxide it is necessary to have a slug
that is made entirely of this material since the heat has to be conducted through the entire material.
Copper oxide usually occurs as an oxidation layer at the surface of copper. To have a slug made of copper
oxide, oxygen has to be added to molten copper such that the oxygen is distributed evenly. Moreover,
this process should be performed in a high vacuum to prevent other contamination in the copper. It is
therefore extremely difficult to have a slug of pure copper oxide. Additionally, it is not possible to use
silicon dioxide as the slug material due to its low thermal conductivity hence there is no reference to
compare the results with.
All in all, thermocouples and slug calorimeters are not suitable sensors to measure the heat flux for
this research. The advantages of thin-film gauges outweigh the disadvantages. The main disadvantage
of a thin-film gauge (i.e. the fragility) can be solved by manufacturing a large number of gauges.
4.2.2
Preparation of Quartz Rod and Models
Quartz is a material that has a low thermal conductivity (approximately 1.4 W/(m · K) [41]) and a low
porosity hereby suitable to be used as a substrate for the gauge. The molecular structure is a continuous
repetition of SiO4 in a shape of a tetrahedron where each connection shares an oxygen atom. The quartz
is bought commercially in the form of rods with a diameter of 2 mm and a length of around 10 mm.
Since the surface roughness has a large influence on the catalytic efficiency as described in Section 2.2.3
it is essential to polish the rods. Handling them individually is not only a very tedious process it is also
difficult to properly orientate them on the grinder. Using acrylic cement and an acrylic ring the rods
are attached all in the same orientation to form a block. This block of quartz rods is then polished on
both sides using SiC sandpaper of increasing grits: 400, 800, 1500, and 2000 which corresponds to an
45
4.2. Heat Transfer Gauge
average particle diameter of 35.0, 21.8, 12.6, and 10.3 µm. At each grit the polishing machine is set at
a rotational speed of 350 rpm on which the block is polished for approximately 15 minutes and using
1-µm-particle-size filtered water, waste is removed. This step is then concluded by polishing the block
using diamond suspension with an average particle diameter of 6.0 µm for approximately 20 minutes.
Afterwards acetone is used to dissolve the acrylic cement which only leaves the individual quartz rods.
The models in which the gauges will be mounted (Fig. 4.3(a)) also need to be prepared. The polishing
process follows the same steps as for the quartz rods but without the grinder. Moreover, instead of 20
minutes the models are only polished by hand for a few seconds.
4.2.3
Platinum and Gold Bridge
On the rods that are evenly polished a platinum bridge can be applied. Since quartz is electrically
non-conducting this bridge will make the surface conductive. The platinum paste that has been used is
a platinum metallo-organic conductor. By trimming a size 0 paint brush until a single hair, the paste is
applied to the upper surface with a single stroke, from one end to another through the center. Practice
is required to put a consistent stroke on all rods.
A muffle furnace – with a 13x11x15 cm large chamber – is used to heat up the quartz rods with the
platinum strip starting from room temperature to cure the platinum paste. By placing a steel block in
the furnace, the rods can be placed vertically with the painted surface pointing upwards. The furnace is
set to the desired temperature of 850◦ C, and is kept at this temperature for 30 minutes after which the
furnace is turned off.
When the furnace is cooled down the quartz rods are taken out and the platinum bridges are tested
individually using a multimeter. The desired range of resistance is between 50 and 400 Ω. A too small
resistance results in a sensor which is not sensitive enough to be useful, whereas a too high resistance
could lead to a failure of the bridge after one utilization in the shock tube due to a too high voltage. The
ones which meet this requirement are again sorted on their lay-out. This is because the goal is to have
platinum bridges with a uniform width of approximately 0.5 mm to minimize the differences between
different sensors.
On the rods meeting these requirements two gold bridges are made on the side. Each of these gold
bridges start at the far end of the platinum bridge, go around the corner, and continues longitudinally
until the end of the rod. The gold paste used is a gold metallo-organic conductor. It should be made
sure that the gold is touching the platinum bridge slightly so a connection will be established. The rods
are again placed in the furnace at 850◦ C for 25 minutes. The rods are again tested with a multimeter to
determine whether all the connections are successful. The final result is shown in Fig. 4.7.
Figure 4.7: The top view of a thin-film gauge with the platinum and gold bridges.
Those who failed are not discarded yet. A common problem was that the gold bridge was not
continuous over the corner on the top surface. If the platinum bridge is still intact, then a second
attempt can be made in the furnace with some additional gold paste. The temperature is lowered to
700◦ C and the time is set to 25 minutes, which improves the survival probability of the platinum bridge.
46
Chapter 4. Experimental Equipment
4.2.4
Attachment of Wires
Wires are connected to the gold bridges in order to connect them to the a power source. Prior to this
the sensor length should be reduced to 6.5 mm using sandpaper to ensure that everything fits when it is
installed in the model.
Two pieces of enameled copper wires with a thickness of 0.2 mm are used to connect to the gold
bridges. Each wire is pressed against one of the two gold bridges at the side of the rod (near the lower
part of the rod). Using heat shrink with a diameter of 2 mm which is made of flexible Polyolefin capable
of shrinking to a ratio of 2:1, the wires are kept in place and it covers the entire rod.
To prevent even the slightest movement of the wire attachments, Araldite is applied to the bottom
of the rod where the two wires are sticking out and until halfway the sensor so the entire region of the
attachments is covered. Using a heat gun the glue is hardened immediately hereby. The result is shown
in Fig. 4.8.
(a) Top view of the sensor.
(b) Front view of the sensor.
Figure 4.8: Sensor with two copper wires held at its place by heat shrink and Araldite.
The last step to complete the sensor is by attaching golden connectors to the end of the wires. A
male and female connector are shown in Fig. 4.9(a). On both wires a male connector is soldered and
using a heat shrink of 3 mm the soldered region is covered.
The completed sensor is then mounted in the model with the aid of Araldite to keep it in place. It
is made sure that the front face is level with the surface of the model (flush-mounted). This resulting
product is referred as a probe which is shown in Fig. 4.9(b) for illustration. During this entire process
the resistance is checked to make sure that the gauge is still intact.
(a) The two different connectors. The top one was used
on the sensors.
(b) A finished probe.
Figure 4.9: The gold connectors and the finished probe.
4.2.5
Calibration
Using a convection oven the probes are calibrated. The temperature is increased starting from room
temperature with increments of 5◦ C until 65◦ C. This range corresponds to the temperature increase
47
4.2. Heat Transfer Gauge
expected during the experiments. At each step the temperature is kept constant for about 10 minutes to
ensure equilibrium. The resistance is then recorded of each probe. From this a resistance-temperature
diagram is made for each probe, see for example the diagram shown in Fig. 4.10.
The relationship between the resistance and the temperature is linear as shown by Kinnear and
Lu [53], hence a least square method is used to filter out uncertainties during the calibration process.
This results in the dashed line shown in Fig 4.10.
95
Measurements
Lease Square Method
94
93
Resistance [Ω]
92
91
90
89
88
87
86
20
25
30
35
40
45
Temperature [K]
50
55
60
65
Figure 4.10: An example of a resistance-temperature diagram for a probe.
The slope ( dR
dT ) of each probe is required to convert the measurements to temperature. This will
be further discussed in Section 6.2. It is common to plot this slope against the resistance at room
temperature for all probes which should yield a linear relationship (Fig. 4.11(a)).
Having a low deviation among the probes is desired hence several outliers from the linear trend
were discarded. From Fig. 4.11(a) it can be seen that the R2 value, which denotes how close the linear
fit matches the data points, is 0.939. Such a value indicates that the probes can be considered to be
consistent. The majority of the probes show a resistance between 40 and 120 Ω hence for clarity an
enlarged version is shown in Fig. 4.11(b).
This diagram is useful to correct dR
dT in case there is a change of the resistance at room temperature.
It has been experienced that a change of less than 10% is common due to daily temperature fluctuations
and especially after the coating process which will be discussed in the next section. Ideally a calibration
curve should be made using the probes that have been coated. However, the risk of breaking the probe
during calibration was too high hence it was assumed that the effect of the coating on the calibration
curve is negligible.
The correction is done by calculating the difference using the slope (i.e. 0.0025197) of the linear fit
and correct the measured value at room temperature of that particular probe.
4.2.6
Coating and SEM Analysis
The coating of the probes was performed by a company called “Thin Films and Fine Materials” (THIFINE)
located in Incheon (Seoul, South Korea). The probes were taped together in a batch and the golden
connectors were also covered with tape for protection. An example is shown in Fig. 4.12. Two rounds of
probes were sent to the company for coating and in each round there were three batches:
1. 1000 nm SiO2
2. 1000 nm SiO2 and 180 nm CuO
3. 1000 nm SiO2 and 180 nm Cu
On all three batches a layer of silicon dioxide is applied. Only for the second and third batch a second
coating is applied on top of this SiO2 . The SiO2 functions as a insulator to prevent contact between
the copper (oxide) and the platinum or gold. The thickness of SiO2 , CuO, and Cu were taken from the
experiments conducted by Park [15] which is also very similar to the thicknesses applied by Reddy [32]
as presented in Section 3.2.
In an attempt to prevent the short-circuiting problem experienced by Park [15] with the copper coating a small amount of Araldite was applied on the gold bridge for only that particular batch.
48
Chapter 4. Experimental Equipment
Sensor Models
Least Square Method
2014−11
1
2014−13
dR/dT [Ω/K]
0.8
2013−62014−17
2014−27
0.6
2013−1
2013−3
2014−15
2014−42
0.4
2014−16 2014−33
2014−48
2014−4
2014−2
2014−38
2014−43
2013−8
2014−44
2014−31
2014−28
2013−9
2014−39
2014−9
2014−25
2014−29
2014−6
2014−47
2014−35
2014−30
2014−36
2014−37
2013−7
2014−20
2014−46
0.2
0
0
50
100
150
dR/dT = 0.0025197*R0 + 0.00010137
R−Squared = 0.93913
Standard Deviation= 0.058488
200
250
300
350
400
R [Ω]
0
(a) Resistance slope vs. resistance diagram.
Sensor Models
Least Square Method
0.35
2014−16
2014−4
dR/dT [Ω/K]
0.3
0.25
2014−28
2014−39
2014−48
2014−2
2014−38 2013−8
2014−43
2014−44
2014−31
2013−9
0.2
0.15
2014−9
2014−29 2014−25
2014−47
2014−35
2014−30
2014−362014−37
2013−7
2014−20
2014−46
2014−6
0.1
0.05
40
50
60
70
80
R0 [Ω]
90
100
110
120
(b) Close up of the resistance slope vs. resistance diagram between 40 and
120 Ω.
Figure 4.11: Resistance slope vs. resistance diagram att room temperature (20◦ C) for all probes used in
this thesis.
Figure 4.12: The probes are grouped together prior to the coating process.
49
4.2. Heat Transfer Gauge
The coating is applied using an electron beam physical vapor deposition method (E-beam) which evaporates the source material using an electron beam that has a purity higher than 99%. The vapor of
this material touches the upper layer of the models and it condenses forming the desired layer. The
deposition rate was around 0.3 nm/s. [54] This deposition was conducted under high vacuum conditions
(i.e. 8.5 · 10−5 Torr = 0.011 Pa). The accuracy of the 1000 nm SiO2 is 50 nm whereas for the 180 nm
CuO or Cu layer it is 20 nm.
While the application of the SiO2 and Cu layers went without problems, the CuO was troublesome.
During the first round THIFINE attempted to perform the E-beam method on the CuO source. However
it was noticed by the coating engineer that the electron beam could not evaporate the source material.
Hence, instead of using an electron beam a resistor was used to evaporate the material. The CuO layer
had a black color hence it is referred as “black CuO”.
After analyzing the samples from the first round it was decided to process the second round differently. By evaporating the Cu source material using the E-beam method and by injecting oxygen gas
in the vacuum chamber copper oxide was formed. With this method it could not be guaranteed that
the ratio between the copper and monatomic oxygen was 1:1. The CuO layer had a brown color and is
referred as “brown CuO”.
In Fig. 4.13 the five different types of samples are shown: uncoated, SiO2 , black CuO, brown CuO,
and Cu respectively. It can be seen in Fig. 4.13(b) that SiO2 introduces a slight greenish glow to the
surface when compared to Fig. 4.13(a). The CuO shown in Fig. 4.13(c) is significantly darker compared
to SiO2 . One might even claim that there is a blueish/greenish glow, possibly from the SiO2 layer
underneath and/or the oxygen present in the layers.
In general the color of copper oxide is either black (i.e. CuO) or red (i.e. Cu2 O). Based on inspection the black CuO (Fig. 4.13(c)) would correspond to a mixture of the black from CuO and the
green/transparent color of SiO2 . The brown CuO (Fig. 4.13(d)) would correspond to a mixture of CuO
and Cu2 O or Cu which corresponds to a color mixture of black and brown or red.
When comparing the brown CuO with Cu (Fig. 4.13(e)) it is clear that the latter is lighter and more
orange compared to the brown of the former.
(a) Uncoated.
(b) SiO2 .
(c) Black CuO.
(d) Brown CuO.
(e) Cu.
Figure 4.13: Five different types of surfaces.
Between the two rounds it was decided to perform a Scanning Electron Microscope (SEM) analysis on
the CuO models to investigate the quality of the coating. Using an electron microscope the surface
can be investigated by measuring the secondary electrons (SE), backscattered electrons (BSE), and by
conducting elemental analysis X-ray spectroscopy (EDS).
With SE the contour of the surface can be made visible whereas with BSE the differences in atomic
species can be made visible through contrast. A lighter color means a higher atomic number whereas a
darker color corresponds to a lighter element. An example is shown in Fig. 4.14 for a probe coated with
SiO2 and another probe with black CuO.
From Fig. 4.14(a) and 4.14(c) it can be seen that the coatings applied to the platinum bridge is
not perfectly smooth. Both coatings show a similar amount of irregularity. Away from the platinum
bridge the amount of irregularities drops significantly, since the quartz rods were polished extensively.
Therefore it can be concluded that the irregularities visible on top of the platinum bridge is caused by
the roughness of the dried platinum.
From Fig. 4.14(b) and 4.14(d) it is can be observed that there is no significant contrast variation
in the images which means that the coatings have a high purity. When combined with the SE contour
images, it is suspected that the darker areas in the BSE images are caused by flakes on top of the coating.
50
Chapter 4. Experimental Equipment
(a) SE image on a heat flux probe coated with SiO2 .
(b) BSE image on a heat flux probe coated with SiO2 .
(c) SE image on a heat flux probe coated with black (d) BSE image on a heat flux probe coated with black
CuO.
CuO.
Figure 4.14: Secondary electron method and backscattered electron method taken of a the platinum
bridge of a heat flux probe coated with SiO2 and another probe with black CuO.
With the EDS method it is possible to measure the elemental composition of the surface. The CuO
models of the first round were tested using this method. A representative example is shown in Fig. 4.15.
The ordinate of the figure can be interpreted as the amount of particles present at the surface while the
abscissa is the wavelength corresponding to a particular species.
From this it can be seen that mainly Si and O are measured while the presence of Cu is minuscule.
Since the EDS is capable of penetrating the surface for several hundreds nanometers, the presence of
Si in the spectrum corresponds to the expectations. The small concentration of Cu in the spectrum
indicates that the amount of Cu atoms is low which can either correspond to a very oxygen rich copper
oxide coating or the thickness is a lot thinner than expected.
SEM analysis was not conducted on the brown CuO probes since it was clear from visual inspection
that copper was present.
Figure 4.15: The EDS spectrum of the surface of a black CuO coated probe.
51
4.3. Shadowgraph
4.3
Shadowgraph
As mentioned before in Section 4.1.2 there are openings at the sides of the test section in which sapphire
windows can be placed. Using the shadowgraph visualization method the flow – especially the standing
shock – is captured. In this section a short overview is presented of this technique.
The shadowgraph technique is a non-intrusive method that has a short measurement time. It is based
on the principle that light is refracted differently by different densities. Through a shock wave there is a
large density change hence the jump in density can be made visible. The refraction principle is known
as the Gladstone-Dale relation and is dictated by the refractive index (n):
n=
c0
= 1 + Kρ
c
(4.14)
where the local density (ρ) dictates the refractive index (n). The Gladstone-Dale constant (K) is dependent on the composition of the gas and the wavelength of the light. [55]
Since different substances have a different refractive index, light is bend differently according to Snell’s
law:
n1 sin α1 = n2 sin α2
(4.15)
As dictated by the combination of Eq. (4.14) and (4.15), light bends towards the substance with a higher
density. Since the density increases through a shock wave, the light is deflected towards the region behind
the shock, creating a bright spot at the recording plane. A dark spot is located upstream due to the
light that has been bent (Fig. 4.16).
2
2
The shadowgraph method measures the ∂∂xn2 + ∂∂yn2 which corresponds to ∇2 ρ, hence it is not possible to acquire quantitative values for the density.
The setup used at KAIST is shown in Fig. 4.17. A green colored laser with a wavelength of 532 nm
is used as the light source. At a distance of 2.4 cm a biconcave lens is placed with a focus length of
-30.0 mm and a diameter of 20.0 mm. This lens diverges the laser and at a distance of 81.5 cm a 15x15 cm
square mirror (i.e. mirror 1) is placed which changes the orientation of the beam. 123 cm from there a
converging mirror with a diameter of 20 cm is situated that changes the diverging laser to a collimated
beam. This beam travels towards the model located at 261 cm from the converging mirror and passes
through the two optical windows towards another 15x15 cm square mirror (i.e. mirror 2). The distance
between the model and this mirror is 102 cm. At last the laser is reflected towards a high-speed camera
located at 168 cm from the mirror. In the figure the path of the laser is indicated by a red dotted line.
An 80-400 mm 1:4.5-5.6D Nikkor lens has been mounted to the camera. The lens was used at its
longest focal length of 400 mm with an aperture of 5.6. The focus was located at 2.3 m. The high speed
camera was set at 200,000 fps.
52
Chapter 4. Experimental Equipment
Figure 4.16: The refraction principle of light due to Figure 4.17: The shadowgraph setup used for this
a density difference. [55]
thesis.
53
5
Determination of Flow Conditions
In Chapter 4 the shock tube has been introduced together with an analysis of the effect of varying
the driver and driven gas. However, there are more parameters that can be changed to alter the flow
condition. The pressure in the driver tube can be adjusted by changing the thickness of the diaphragm
and also the pressure until which the test gas is introduced in the driven tube has to be chosen. Moreover,
the composition of this test gas also has to be determined.
In this chapter the focus is on the numerical determination of the flow conditions which will be used
for the actual experiments. Two sets of programs are necessary: one for determining the flow conditions
inside the shock tube including the high temperature effects (Section 5.1), another one for determining
the condition behind the standing shock at the boundary layer edge (Section 5.2). Only with these two
programs the flow conditions and the amount of dissociation are known. Additionally a third program
is introduced in Section 5.2 which is used to determine the conditions at the wall of the model. These
conditions are necessary for the theory presented in Chapter 6.
At the end of this chapter the exact procedure of determining the flow conditions is presented together
with the values of the chosen flow conditions.
5.1
Shock Tube Program
To numerically determine the flow conditions it is necessary to have an accurate shock tube program.
Two different programs have been considered: L1d2 from the Compressible-Flow CFD group of the University of Queensland in Australia [56] and stube9 from Prof. Chul Park of KAIST. After a comparison
L1d2 has been chosen to be better suited for the present application (see Appendix B). For this reason
this section will focus on L1d2 only.
L1d2 is a Lagrangian quasi-1D flow solver for shock and expansion tubes. Although there is only one
spatial coordinate, gradual changes of the cross-sectional area are possible. It is capable of simulating
the transient part of the flow which occurs after a sudden change such as a shock wave. These changes
can either be introduced by a moving piston or a bursting diaphragm. Not only the dynamics of the
piston and the gas slugs are considered, also the viscous effects including the heat transfer effects are
included.
Different gas slugs can be simulated which are either independent (e.g. separated by a diaphragm)
or interacting (e.g. when the diaphragm bursts). It also has the option to include several pistons and
diaphragms. These slugs are simulated in such a way that each of them are split up into a number of
control masses of which the positions are followed. The viscous effects are simulated using the correlations
for friction and heat transfer in a pipe flow to take into account the effect of compressibility. [57]
Because shock tubes are used for conditions where shock waves are created it is essential that L1d2
can simulate those. These shocks are not explicitly identified nor tracked but are included using a shock
capturing scheme. This scheme functions in such a way that the same set of equations is used to compute
the motion of the gas, whether or not there is a shock. This simplifies the code in cases where shocks,
compression waves or contact surfaces are interacting with each other. [57]
In the core of the program the predictor-corrector method is used to propagate the values to the next
54
Chapter 5. Determination of Flow Conditions
time step. [57] This scheme can be expressed as:
dX (n)
dt
(n)
= X + ∆X (1)
∆X (1) = ∆t
X (1)
dX (1)
dt
1
(1)
=X +
∆X (2) − ∆X (1)
2
(5.1)
∆X (2) = ∆t
X (n+1)
The superscript (1) represents the predictor stage, and together with superscript (2) it shows the intermediate results to acquire the value at the next time step (n + 1).
X can be replaced by any variable that has to be propagated to the next time step which are: interface
positions, cell momentum, cell energy, piston velocity and piston position.
To determine the position of the interfaces in both the predictor and corrector phase, the Riemann
problem is solved (introduced in Section 4.1.3). The pressure and the velocity of the interface can then
be determined. This method is similar to the Godunov’s scheme which is a finite volume method where
the Riemann problem is solved at each cell interface. [58]
In Eq. (5.1) the time derivative is necessary. This is determined using the differences of a certain
variable (e.g. momentum, energy) between two adjacent cells together with the losses, hence the finite
difference scheme.
After the corrector step the chemical species are updated by considering an equilibrium gas mixture
because the chemical reactions are relatively fast with respect to the flow time scales. For most gases
the thermodynamic constants (also know as JANAF coefficients) are stored in a table for three different
temperature ranges: 200 K - 1000 K, 1000 K - 6000 K, and 6000 K - 20,000 K. It uses NASA’s CEA
program to determine the composition of the mixtures. This program is discussed in Section 5.2.1.
The pitot pressure is also determined by L1d2 using the Rayleigh pitot formula:
γ
γ−1
(1 − γ) + 2γM12
(γ + 1)2 M12
pt,2
(5.2)
=
p1
4γM12 − 2(γ − 1)
γ+1
where subscript 1 is before the standing shock and 2 after the shock. Although with different numbers,
this principle was illustrated in Fig. 2.11. Since the chemical species distribution is known the specific
heat ratio is available.
5.2
Equilibrium Composition Program
The shock tube program (discussed in Section 5.1) determines the conditions behind the primary shock
wave. With the exception of the pitot pressure the program does not determine the conditions behind
the standing shock induced by the model such as temperature, density, enthalpy, and gas composition.
As discussed in Section 2.4.1 the total enthalpy is assumed to be constant through the standing shock
(determined by L1d2) and the total pressure is assumed to be constant through the boundary layer at
the stagnation point (measured using a pressure sensor and calculated by L1d2). These two quantities
function as the input variables for the equilibrium composition program.
In this section two equilibrium composition programs are presented, both used for different purposes.
The first program called “CEA” (Section 5.2.1) is used to determine the conditions at the wall of the
model where the temperature is low. Moreover, CEA is capable of determining the frozen Prandtl number
and the dynamic viscosity which are necessary parameters at the wall (Section 2.3.2.2). The second
program named “Eqtab” calculates the equilibrium conditions at the boundary layer edge (Section 5.2.2).
5.2.1
CEA
Chemical Equilibrium and Applications (CEA) is a program that is not only capable of determining
the chemical equilibrium compositions for the assigned thermodynamic states, it can also be used to
calculate the performance of rockets and detonations. [59] The option that is used is the calculations
of the chemical composition with the following thermodynamic state: temperature and pressure. These
parameters are known at the wall since the temperature is measured by the heat flux probe and the
pressure is measured using the piezo-electric sensor.
CEA uses empirical equations to fit the thermodynamic equations. The parameters that are determined by this method are Cp , H, and S of each species:
Cp
a1
a2
= 2+
+ a3 + a4 T + a5 T 2 + a6 T 3 + a7 T 4
R
T
T
55
(5.3)
5.3. Flow Conditions
a2
H
a1
T
T2
T3
T4
a8
=− 2 +
ln(T ) + a3 + a4 + a5
+ a6
+ a7
+
RT
T
T
2
3
4
5
T
(5.4)
a2
S
1 a1
T2
T3
T4
−
=−
+ a3 ln(T ) + a4 T + a5
+ a6
+ a7
+ a9
2
R
2T
T
2
3
4
(5.5)
C
where seven constants are necessary for Rp and nine integration constants for the enthalpy and entropy. [59]
These input constants are called JANAF coefficients and are divided in three temperature ranges:
200 - 1000 K, 1000 - 6000 K, and 6000 - 20,000 K. While it is claimed by the authors of the program
that it is capable of calculating the conditions accurately for high temperatures, in the past versions it
has been found that starting from 6000 K the results tend to become inaccurate1 . For this reason CEA
is only used at low temperatures which is at the wall.
A stripped down version of CEA, which has been improved for higher temperatures, called “Therm”
has been considered for the calculation of the equilibrium composition at the boundary layer edge. It
was however found that the results were less accurate compared to the results acquired by Eqtab (see
next section). It was noticed that the accuracy of the JANAF coefficients for high temperatures drops
significantly resulting in large errors. More details can be found in Appendix C.
5.2.2
Eqtab
“Eqtab” is the program used to determine the equilibrium composition of a gas mixture at the boundary
layer edge. It is based on the concept of partition functions which has been introduced in Section 2.4.2.2.
For a specific pressure and enthalpy combination, Eqtab calculates the temperature, density, internal
energy, and the species composition of the gas.
The composition is given by the parameter “spgam” in the units [mol/kg] which can be converted to
molar fractions by taking the fractions of the total amount of each species. The mass fractions can be
acquired by multiplying “spgam” with the molecular weight.
5.3
Flow Conditions
In sections 5.1 and 5.2.2 the programs “L1d2” and “Eqtab” were introduced which are used together to
determine the flow conditions inside the shock tube and behind the standing shock. Three flow conditions
are chosen: an almost no dissociation case, a moderate amount of dissociation case and a high amount
of dissociation case which are referred as A, B, and C respectively.
Condition C has the highest amount of dissociation hence it will be the upper limit. Therefore it
is important make sure that there is a sufficient amount of dissociation for this condition. While it is
preferred to have 100% dissociation it was noticed that this is not possible using a shock tube. Therefore, the aim is to have approximately 50% dissociation of nitrogen. It is believed that this is a sufficient
amount of dissociation when compared to the 67% dissociation used by Park [15] for a O2 -Ar gas mixture.
During the first stage of choosing the flow conditions it was observed that dissociation cannot be reached
when pure nitrogen is used as test gas. Although lowering the driven pressure results in a stronger and
faster shock wave, hence it is theoretically possible to choose a pressure that would result in a high
amount of dissociation, there is a physical limit of what can be achieved during the experiments. Since
the shock tube could only be vacuumed until 8 Pa, it was decided that the initial pressure in the driven
section should be at least 100 Pa which corresponds to an error in the pressure of less than 10%.
In Section 4.1.4 it was found that a lighter driver gas and a heavier test gas result in a stronger shock
wave. Due to safety concerns the usage of hydrogen as driver gas was not considered hence helium was
selected for this purpose. Therefore it was necessary to adapt the test gas to reach flow conditions were
dissociation is achieved. Similar to literature (Section 3.2) the nitrogen can be mixed with an inert gas.
A tertiary gas mixture is then formed when nitrogen dissociates.
At first, argon was considered in various ratios with the nitrogen. The amount of energy in the flow,
expressed by the enthalpy, was not high enough to dissociate a significant amount of nitrogen. Similarly
to the minimum pressure of the test gas, a minimum volume mixture ratio was selected of at least 20%
nitrogen to ensure a sufficient amount of nitrogen. The amount of dissociation found with this test gas
for various mixture ratio were all below 8% which is not sufficient. The calculations were also conducted
for a driver pressure that is doubled by using two diaphragms pressed against each other, however, this
only increased the dissociation to values below 10%.
Instead of argon it was decided to use krypton as the inert gas. It was found in Section 4.1.4 that
krypton – that has a molecular weight of krypton that is twice as high compared to argon – results in a
1 Private
communication with Prof. Chul Park
56
Chapter 5. Determination of Flow Conditions
temperature that is a factor 1.44 larger than using argon. The increased weight by mixing nitrogen with
krypton results in a higher enthalpy which causes a higher amount of dissociation.
After several iterations it was found that a mixture of 35% nitrogen and 65% krypton resulted in
flow conditions with sufficient amount of dissociation when a double diaphragm is used. Flow condition
B can then be selected by using one diaphragm and a driven pressure that results in approximately 30%
dissociation, and flow condition A is chosen freely by varying the driven pressure to ensure almost no
dissociation.
However, the flow condition cannot solely be chosen based on the amount of dissociation. In Chapter 6 the steps regarding data reduction is covered. The final goal is to generate a qD /qC vs. γw diagram
which is used to read off the catalytic efficiency from the measured heat flux. The diagram made by
Park [15] for a O-O2 -Ar gas mixture is shown in Fig. 5.1. This curve has an S-shape and it should
be made sure that the intersection between the experimentally determined heat flux ratio (horizontal
lines in Fig. 5.1) with the curve is not located in flattened regions. In Fig. 5.1 these flattened regions
correspond to γw < 10−3 and 10−1 < γw . If this is poorly chosen the error in the measured heat flux
would be amplified significantly when expressed as an uncertainty of the catalytic efficiency.
Figure 5.1: Heat transfer ratio vs. catalytic efficiency for the O-O2 -Ar interaction. [15]
To decrease the size of the flattened regions the qD /qC value at which it occurs should be increased.
The increase of qD /qC is achieved by increasing the amount of nitrogen in the gas mixture. However,
increasing the amount of nitrogen translates to a smaller amount of dissociation which is undesired. A
balance has to be found between the two.
The flow conditions are summarized in Table 5.1. The values are the average within the steady regime
that is chosen in Section 7.1.1. From this table it can be seen that for all flow conditions a driven
gas of 35% N2 and 65% Kr is used in combination with helium as the driver gas. Not only the driven
pressure is varied for all three conditions, but for condition C the driver pressure is doubled by using
two diaphragms which are pressed against each other.
Both the shock speed and the total enthalpy increase when comparing flow conditions A to C. The
total enthalpy is an indication of the total amount of energy in the flow. Together with the pitot
pressure it determines the amount of nitrogen that dissociates. Condition A is a case with essentially no
dissociation. The temperature behind the standing shock at the boundary layer edge is calculated to be
around 4800 K, whereas flow condition C has a temperature of almost 7000 K. The combination of all
these results in a dissociation of 1%, 33.3%, and 54.3% respectively.
57
5.3. Flow Conditions
Table 5.1: The three flow conditions predicted using L1d2 and Eqtab. Shock speed and pitot pressure
shown here are experimentally determined.
Driver Gas
Driven Gas
pdriver [kPa]
pdriven [Pa]
ushock [km/s]
Flow Condition
B
C
He
35% vol. N2 - 65% vol. Kr
1650
1650
3300
2000
150
100
1.563
2.358
2.660
M2 [-]
T2 [K]
p2 [kPa]
H2 [MJ/kg]
ρ2 [kg/m3 ]
u2 [km/s]
1.69
3192
117.9
2.012
0.286
1.310
1.85
6175
19.90
4.411
0.025
1.988
1.89
7550
16.72
5.756
0.017
2.255
ppitot [kPa]
Te [K]
ρe [kg/m3 ]
XN [-]
XN2 [-]
XKr [-]
αN [-]
αN2 [-]
αKr [-]
Mass fraction N2 Dissociation
446.0
4874
0.706
0.007
0.345
0.648
0.001
0.151
0.847
0.010
100.9
6425
0.109
0.209
0.209
0.582
0.051
0.102
0.847
0.333
94.23
6874
0.089
0.319
0.134
0.546
0.083
0.070
0.847
0.543
A
Driver & Driven
Tube
Driven Tube
Freestream
Behind Standing
Shock at BL edge
58
6
Data Reduction
Regardless of the flow condition the raw data of the measurements has voltage as unit. For the static and
pitot pressure measurements the voltage has to be converted to pressure values whereas for the heat flux
measurements it has to be converted to heat flux. The steps to accomplish this are covered in Section 6.1
and 6.2.
To convert the heat flux to the catalytic efficiency it is necessary to use the theory introduced in
Section 2.3.2. The theory for a tertiary gas mixture requires a large number of variables. Although part
of these variables have been determined by L1d2 and Eqtab in Chapter 5, the remaining variables will
be covered in Section 6.3 by introducing several theories. In Section 6.4 the comparison is made between
the input variables of the binary and tertiary gas mixture theories.
6.1
Pressure Measured by Piezo-electric Sensor
As discussed in Section 4.1 two piezo-electric sensors were flush-mounted in the wall of the shock tube to
measure the static pressure and shock speed, and one sensor is used in the pressure model (Fig. 4.3(b))
to measure the pitot pressure.
Since these sensors were bought commercially, a calibration sheet was provided with the corresponding
sensitivity. Using this sensitivity it was possible to convert the raw data from voltage to pressure values.
6.2
Surface Heat Flux Measured By Thin-Film Gauge
Each thin-film gauge located in a probe was calibrated, as discussed in Section 4.2.5 which resulted in a
resistance-temperature diagram. Since the current (Ic ) applied to the sensor is constant and known (i.e.
9.88 mA) the voltage measured using the oscilloscopes (V (ti )) can be translated to resistance. With the
aid of the results from the static calibration the temperature can be determined [15]:
T (ti ) =
V (ti )
V (ti )
V (ti )
=
=
dR
Vsupply · βR
Ic · R0 · R10 dR
I
c · dT
dT
(6.1)
From this equation it can be seen that the resistance variation with the temperature ( dR
dT ) is required
from the calibration, while the initial resistance at room temperature (R0 ) cancels out. dR
dT is corrected
for any variations at room temperature as discussed in Section 4.2.5.
It was noticed that the temperature profiles acquired from the experiments had some noise instead of
a smooth pattern as shown by Fig. 6.1(a). Since any noise in temperature is amplified when converted
to heat flux it was decided to apply a Savitzky-Golay filtering to the temperature. This filter is a low
pass filter and it fits a low degree polynomial using the linear least square method between the data
points. [60] The filter has been set to a third order polynomial and a frame size of 5. It was witnessed
that this would filter away the noise but not distort the original shape of the profile (Fig. 6.1(b)).
59
50
50
40
40
30
30
∆ T [K]
∆ T [K]
6.3. Diffusive and Convective Heat Flux for Tertiary Gas Mixtures
20
20
10
10
0
0
−10
0
20
40
60
80
100
−10
0
20
40
60
80
100
t [µs]
t [µs]
(a) Original temperature profile.
(b) Filtered temperature profile.
Figure 6.1: Temperature profile prior and after the Savitzky-Golay filter.
To convert this temperature to the desired output value, the heat flux, assumptions have to be made
regarding the thin-film gauges. Based on the semi-infinite principle discussed by Schultz et al. [41] the
assumption of one-dimensional heat transfer is made. This transfer is in the form of conduction through
a thin metallic film, which is in this case the platinum bridge. It is assumed that the semi-infinite
substrate (i.e. the quartz rod) defines the heat capacity of the entire sensor while the metallic film has
no heat capacity. Moreover, due to the small width of the metallic film it is assumed that it does not
affect the temperature of the substrate surface.
From these assumptions the substrate defines the properties of the gauge. To meet the one-dimensional
theory the quartz rod has to be of sufficient length to become similar to a semi-infinite solid. Kinnear and
Lu [53] investigated the substrate base temperature to surface temperature ratio for a thin-film gauge
and concluded that the substrate should be at least 3.0 mm even for a long test time of 1 s. In this case
the substrate base temperature to surface temperature ratio is less than 1%. Since the quartz rod has a
total length of 6.5 mm it is more than sufficient.
With the temperature profile the heat flux can be determined using proper physical models. Each
ot these methods give a different heat flux profile where some are more prone for noise than others.
Several examples are the methods presented by Cook and Felderman [61]:
r
n
ρck X
Ti − Ti−1
√
qs,Cook&F elderman = 2
(6.2)
√
π i=1 tn − ti + tn − ti+1
or Kendall et al. [62]:
r
Qs,Kendall
qs,Kendall
et al.
et al.
=
=
n
ρck X
Ti + Ti−1
√
∆t
√
π i=1 tn − ti + tn − ti+1
dQs,Kendall
dt
et al.
(6.3a)
(6.3b)
The method of Cook and Felderman [61] was selected to converted the temperature to the stagnation
heat flux. [41, 61] A comparison between different heat flux conversion methods is made in Appendix D
from which it was chosen to use this method.
The properties of the substrate are assumed to be constant physical properties
taken at room tem√
perature, expressed by three variables: ρ, c, and k. The root of this product ( ρck) is called the thermal
product. According to Park [15] its value is 1510 ± 100 Ws0.5 /(m2 · K), which is similar to the value
determined by Schultz et al. [41].
6.3
Diffusive and Convective Heat Flux for Tertiary Gas Mixtures
The theory required to calculate the diffusive and convective heat flux for a tertiary gas mixtures was
introduced in Section 2.3.2.2. For clarity the equations are repeated here again:
p
−2/3
qC = 0.47 2βµe ρe P rw he
(2.23a)
p
qD = χ 2βµe ρe hR αe1 ϕ2
(2.23b)
60
Chapter 6. Data Reduction
1
ϕ2 =
1+
(2.24)
√
χ 2βµe ρe
ρw kw1
1 m2 1
(2.25)
φ3 m Sc21
12 Sc31
Z ∞ 00 mm ScSc
32
f (η) 1
φ3 =
dη
(2.22)
00
f (0)
0
These equations have a large number of variables that need to be determined, which will be discussed
in this section. It is essential to note that some of these variables have a subscript that corresponds to
one of the three gas components in the tertiary gas mixtures: 1 corresponds to N, 2 corresponds to N2 ,
and 3 to Kr.
χ=
6.3.1
Determination of the Dynamic Viscosity and Density
The dynamic viscosity shows up in calculations of the conductive and diffusive heat fluxes (Eq. (2.23)).
The viscosity at the boundary layer edge is computed using:
µ=
n
X
i=1
mi Xi
n
P
(2)
Xj ∆ij
(6.5)
j6=i
(2)
where ∆ij is defined as [63]:
(2)
∆ij
16
=
5
s
2mi mj
(2,2)
π Ω̄
πkT (mi + mj ) ij
(6.6)
To use these equations the particle mass per species is required which can be derived as:
mi =
ρi
αi ρ
ρkB T
αi ρkB T
= pi = αi
=
ni
pXi
Xi p
kB T
(6.7)
In this equation the mass fraction (αi ), molar fraction (Xi ), and temperature, and pressure are known
quantities given in Section 5.3.
(2)
(2,2)
Additionally, ∆ij requires the average collision integral Ω̄ij that is expressed by:
2
2
X
σij
αij ζijz Γ(4 − ξijz ) kB T ξijz
(2,2)
Ω̄ij =
(6.8)
π(αij + 1)(αij + 2)
ij
z=1
This latter equation is a method suggested by Kim et al. [64]. In his paper five collision parameters
are given for common species combinations. Since krypton is an uncommon case it was not present in
this paper. After consulting for Kim’s help it was suggested that the behavior of krypton is similar to
argon since they are both inert gases, hence it was assumed that the values for argon could be used for
krypton. The collision parameters are given in Table 6.1.
Table 6.1: Collision parameters of N, N2 , and Kr combinations. [64]
Species Combination
N-N2 / N2 -N
N-Kr / Kr-N
N2 -Kr / Kr-N2
ζ1
ζ2
ξ1
ξ2
α
−4.2367·10−1
−6.5575·10−1
−1.0200·10−1
7.6369
5.3756
3.3400
−2.9284·10−2
−4.3998·10−2
−2.9123·10−1
2.4099 · 10−1
1.5531 · 10−1
8.2679 · 10−2
1.7826
1.5596
1.3678
The second parameter that is discussed in this section is the density at the boundary layer edge.
Since the molar masses and the molar fractions of the species are known the molar mass of the entire
mixture can be determined. This can be implemented in the perfect gas equation of state:
Pn
p i=1 Mi Xi
pMmix
ρ=
=
(6.9)
Runi T
Runi T
which gives the density. This method is compared with the value given by Eqtab and the difference is
negligible.
The equations presented for the density and dynamic viscosity were only used at the boundary layer
edge. The viscosity at the wall is determined using CEA which has been introduced in Section 5.2.1. The
density at the wall is determined using the Chapman-Rubesin constant which was used in the derivation
of both the binary and tertiary catalytic theory (Section 2.3.2), which is expressed as l = ρρµ
= 1.
e µe
61
6.3. Diffusive and Convective Heat Flux for Tertiary Gas Mixtures
6.3.2
Determination of ϕ2
The factor ϕ2 corrects for catalytic effects and influences the diffusive heat flux as shown by Eq. (2.23b).
The correction factor depends on the variable parameter χ which will be discussed in the next section
and the velocity gradient (β) (discussed in Section 6.3.4). Moreover, ϕ2 also depends on the catalytic
velocity kw .
The catalytic velocity can be determined using:
r
γw 8kB Tw
(6.10)
kw1 =
4
πm1
which was shown in Section 2.2.2. [15] This equation relates the catalytic efficiency at the wall γw to the
diffusive heat flux.
6.3.3
Determination of χ
A major difference between the equations derived by Park [15] for a tertiary gas mixtures compared to
the binary case is the variable parameter χ. This parameter depends on several variables as expressed
by Eq. (2.25). Since the mass per particle is known from Eq. (6.7) it is possible to acquire the average
mixture mass by combining it with the mass fractions. However, the two remaining parameters – φ3 and
the Schmidt number Scij – are more complicated to be determined.
6.3.3.1
Determination of φ3
The mathematical expression for φ3 is shown in Eq. (2.22). In this equation the reverse of the mass
fraction can be recognized, as well as the Schmidt number. Since the Schmidt number will be discussed
in the next section it will be considered to be known. Hence, the remaining values required to acquire
φ3 is expressed by the stream function f (η). This function is used in the similarity reduced momentum
equation which was first introduced by Eq. (2.13a) in Section 2.3.2.1. The boundary conditions can be
included which is very similar to the well-known Blasius equation (f 000 + 12 f f 00 = 0):
f 000 + f f 00 = 0
BC : f (0) = f 0 (0) = 0; f 0 (∞) = 1
(6.11)
Solving Eq. (6.11) is done using the shooting method combined with the Euler method for an ordinary
differential equation. By first assuming a particular value for f 00 (0) the values at the next time step can be
determined in combination with the first two boundary conditions. What happens is that this boundary
value problem changes into an initial value problem since all initial values are known. At infinity for
which 10 is chosen as a sufficiently large value, the result is compared with the third boundary condition.
Using the Newton’s method the next guess for f 00 (0) is determined which requires the slope at the
previous guess:
f 00 (0)new = f 00 (0)old −
g(∞)
g 0 (∞)
(6.12)
g(∞) = 1 − f 0 (∞)
The progression towards the next step is done using the Euler method which can be expressed as:
fi+1 = fi + fi0 · ∆η
0
fi+1
= fi0 + fi00 · ∆η
00
fi+1
= fi00 + (−fi · fi00 ) · ∆η
(6.13)
It can be seen that in Eq. (6.13) the first two equations are linked using Euler method. The last expression
is the rewritten form of Eq. (6.11).
Using this method Eq. (6.11) was solved for 0 ≤ η ≤ 10 with step size ∆η = 10−6 . The iteration was
terminated when the difference between the third boundary condition and the actual value is smaller
than 10−10 .
From this it is found that f 00 (0) = 0.4696. After 6 iterations of the initial condition the convergence
was sufficient enough to meet the termination criterion. The convergence is shown in Fig. 6.2(a). In
Fig. 6.2(b) the distribution of f 00 (η) is shown. It can be seen that it drops rapidly with increasing distance normal to the wall.
Having acquired the values for this stream function all the values can be normalized which is exactly the
f 00 (η)
f 00 (0) part of Eq. (2.22). Using the Riemann integral method which approximates the integral by the
sum of a large number of rectangles, Eq. (2.22) can be solved numerically. Since η ranges from 0 to 10,
the integral is performed until 10.
62
Chapter 6. Data Reduction
0.5
0.5
0.45
0.45
0.4
0.4
0.35
0.3
f" [−]
f"(0) [−]
0.35
0.3
0.25
0.2
0.25
0.15
0.2
0.1
0.15
0.1
1
0.05
2
3
4
Number of iteration
5
0
6
(a) The convergence of f 00 (0) starting from 0.1.
0
2
4
η [−]
6
8
10
(b) The distribution of f 00 (η).
Figure 6.2: Convergence and distribution of f 00 (η).
6.3.3.2
Determination of the Schmidt Number
The Schmidt number is defined as the ratio between the viscosity and the mass diffusivity. Since there is
a tertiary gas mixture, there are in total nine different binary mass diffusion cases which results in nine
Schmidt numbers:
µ
(6.14)
Scij =
ρDij
However, not all combinations are required as can be seen in Eq. (2.22).
The binary diffusion coefficient is determined using [18]:
q
Mi +Mj
2.628 · 10−7 T 3 2000M
i Mj
Dij =
(6.15)
(1,1)
p
?
10
2
· 1020 )
101325 (σij · 10 ) (Ωij
The collision diameter (σ) of each species is given in Table 6.2, and since the diffusion coefficients are
considered for a combination of two species, the average has to be taken of these collision diameters:
σi + σj
(6.16)
σij =
2
Table 6.2: Characteristics of the species. [64, 65]
Species
N
N2
Kr
σ [m]
3.246 · 10−10
3.681 · 10−10
3.675 · 10−10
kB
[K]
49.43
102.0
170.0
(1,1)
The diffusion-collision integral Ω̄ij is a necessary parameters to model the collision of gas particles.
When this integral is normalized by the rigid sphere collision integral it results in the reduced diffusioncollision integral Ω?ij (1,1) which can be computed using [18, 66]:
A
C
E
G
+ DT ? + F T ? + HT ?
(6.17)
e
e
e
T ?B
This is based on the Lennard-Jones potential theory which describes the interaction between two particles
with a repulsive term and an attractive term (introduced in Section 2.2.1). Depending on the distance
between these two particles the repulsive or the attractive term will be dominant. The coefficients
A, B, ..., H are given by Neufeld [66] as: 1.06036, 0.1561, 0.193, 0.47635, 1.03587, 1.52996, 1.76474,
3.89411.
T ? is the temperature normalized by the potential parameter:
(1,1)
Ω? ij
=
T? =
T kB
ij
(6.18)
These potential parameters differ for all species (Table 6.2), and since the binary diffusion coefficient is
required the binary potential parameter [18] is necessary:
√
ij = i j
(6.19)
63
6.4. Comparison Between Binary and Tertiary Gas Mixture Theory
6.3.4
Determination of Frozen Enthalpy and Stream Velocity Gradient
The only variables that have not been discussed in this section are the frozen enthalpy at the boundary
layer edge, used in Eq. (2.23a), and the stream velocity gradient, present in Eq. (2.23) and (2.24).
The total enthalpy at the boundary layer edge is used to determine the frozen enthalpy at the
boundary layer edge. The summation of the frozen enthalpy and the energy that is released when all the
nitrogen atoms recombine is equal to the total enthalpy:
h̄e = he − hR α1
(6.20)
Since the total enthalpy, the heat of recombination of nitrogen (i.e. 33844376 J/kg [14]), and the mass
fraction of N are known, it is possible to determine this frozen enthalpy.
e
The final parameter is the stream velocity gradient β = du
dx at the boundary layer edge. It is complicated to determine this value since it cannot be measured directly. Two different methods are used
which can be compared to each other:
1. By performing shadowgraph measurements the standing shock is made visible around the model.
The radius of the shock at the stagnation point is measured and the gradient is determined using:
s
2pstag
du
1
≈
(6.21)
dx stag
ref f
ρstag
where the pitot pressure and density at the boundary layer edge were presented in Table 5.1. [49]
This equation introduced by Zoby and Sullivan [49] is based on several important assumptions.
Firstly, the theory behind the equation is based on the assumption of a Newtonian flow which is
only applicable in a flow with a Mach number larger than 3.5. The flow conditions, presented in
Table 5.1, have a Mach number below 2 which is significantly lower than required.
Moreover, in the equation an effective radius is required. This radius corresponds to a hemisphere
that creates the same velocity gradient at the stagnation point. Zoby and Sullivan [49] presented a
correction for a wide range of body radii and corner radii with respect to a hemisphere. For the flat
disk-cylinder models used in this thesis (where the body radius is 6 mm, corner radius is 1.5 mm,
and nose radius is infinite) the correction is: rrefBf = 0.365. Since rB = 6 mm the effective radius
is known.
The equation based on modified Newtonian flow includes the freestream pressure as presented by
Fay and Riddell [13]. Including this to Eq. (6.21) results in:
s
du
1
2pstag − p∞
≈
(6.22)
dx stag
ref f
ρstag
2. The second method is based on the assumption that silicon dioxide is a fully non-catalytic material.
Although in reality this is not completely true, the diffusive heat flux (Eq. (2.23b)) is assumed to be
zero. With this the measured heat flux is equal to the conductive heat flux (Eq. (2.23a)). Since all
the remaining parameters are known using the programs discussed in Chapter 5 and the equations
presented in this chapter, the velocity gradient can be calculated.
6.4
Comparison Between Binary and Tertiary Gas Mixture Theory
In Section 6.3 the focus was on the theory for a tertiary gas mixture. For a binary gas mixture, discussed
in Section 2.3.2.1, the resulting equation were as follows:
p
(2.18)
qD = 0.47Sc−2/3 2βµes ρes hR αe ϕ1
ϕ1 =
1
√
0.47Sc−2/3 2βµse ρse
ρw kw
1+
p
−2/3
qC = 0.47 2βµse ρse P rw hse
(2.19)
(2.20)
Most of these parameters were discussed in Section 6.3, while for several parameters some elaboration is
required.
Since in the case of a binary mixture there is no inert gas, the mass fraction (αe ) was taken to be
equal to α1 of the tertiary mixture. Moreover, Sc12 was taken to be the Schmidt number for this binary
case.
The values at the boundary layer edge were substituted in the density, dynamic viscosity, and frozen
enthalpy with subscript se.
64
7
Results
In this chapter the results of the pressure (Section 7.1) and heat flux measurements (Section 7.2) are
presented and compared to theory. A comparison is also made between the flow conditions. Explanations
for the discrepancies witnessed in the comparison are discussed in Chapter 8. The chapter ends with the
catalytic efficiencies for the different coatings (Section 7.3).
7.1
Pressure Measurements and Shadowgraph Visualization
In this section the pressure measurements are shown and compared to the predicted values by L1d2.
This comparison is essential since L1d2 is used to simulate the flow conditions and hereby has a large
influence on the theoretical calculations (presented in Chapter 6). Furthermore, in combination with
the shadowgraph visualization images (presented in this section) the steady regime is determined which
dictates the test time of each flow condition. This steady regime is crucial for determining the heat flux
measurements in Section 7.2.
7.1.1
Pressure Measurements
Static and pitot pressure measurements are conducted for all flow conditions. These graphs are shown
in Fig. 7.1. Four static measurements are shown at each flow condition in Fig. 7.1(a), 7.1(c), and 7.1(e)
respectively. Two clear jumps can be seen, the first corresponds to arrival of the primary shock wave at
the first pressure sensor (at t = 0 µs) and the second to the arrival at the second pressure sensor (e.g.
around t = 300 µs in Fig. 7.1(a)), which is separated by 50 cm. The shock speeds deducted from the
experiments are shown in Table 7.1. Compared to L1d2 the error is less than 3%.
Table 7.1: Shock speed, static pressure and pitot pressure of flow conditions A, B, and C compared to
L1d2.
Shock Speed
L1d2 [m/s]
Experiment [m/s]
Relative error [%]
A
1.632·103
1.600·103
-1.96
Pitot Pressure
L1d2 [N/m2 ]
Experiment [N/m2 ]
Relative error [%]
5.286·105
4.460·105
-15.64
Flow Condition
B
2.410·103
2.370·103
-1.66
1.022·105
1.009·105
1.27
C
2.696·103
2.625·103
-2.63
8.753·104
9.423·104
7.66
The pitot pressure measurements are shown in Fig. 7.1(b), 7.1(d), and 7.1(f) respectively. The steady
pitot pressure is shown in the corner of each figure. These values are determined by taking the mean
of the average pressure of each measurement curve within the steady regime. The range of the steady
regime will be discussed in more detail later in this section. The variation from the mean with a 95%
confidence interval of each curve is determined. The largest value is then printed together with the mean
value in the figure. The pitot pressure values are compared to L1d2 in Table 7.1 from which it can be
seen that there is a relatively large error in the value of flow condition A compared to the other conditions.
The presented values are based on a steady regime that requires an additional explanation. Since a
65
7.1. Pressure Measurements and Shadowgraph Visualization
200
1200
1000
150
Pitot Measurement 1
Pitot Measurement 2
Pitot Measurement 3
L1d2 pitot
ppitot,st = 445.971±12.36
ppitot [kPa]
p2 [kPa]
800
100
0
0
400
Run 1
Run 2
Run 3
Run 4
L1d2
50
100
200
300
400
Time [µs]
500
600
200
0
0
700
(a) Static pressure measurements of case A.
60
400
50
350
300
ppitot [kPa]
30
20
Run 1
Run 2
Run 3
Run 4
L1d2
10
0
−10
0
100
200
300
400
Time [µs]
500
600
50
100
150
Time [µs]
200
250
(b) Pitot pressure measurements of case A.
40
p2 [kPa]
Steady
600
Pitot Measurement 1
Pitot Measurement 2
Pitot Measurement 3
L1d2 pitot
ppitot,st = 100.904±4.51
250
200
Steady
150
100
50
0
0
700
(c) Static pressure measurements of case B.
20
40
60
Time [µs]
80
100
120
(d) Pitot pressure measurements of case B.
100
400
350
80
Pitot Measurement 1
Pitot Measurement 2
L1d2 pitot
ppitot,st = 94.234±4.22
300
ppitot [kPa]
p2 [kPa]
60
40
Run 1
Run 2
Run 3
Run 4
L1d2
20
0
0
100
200
300
400
Time [µs]
500
600
(e) Static pressure measurements of case C.
700
250
200
Steady
150
100
50
0
0
20
40
60
Time [µs]
80
100
(f) Pitot pressure measurements of case C.
Figure 7.1: Pressure measurements at flow conditions A, B, and C.
66
Chapter 7. Results
finite amount of time is required for the standing shock to establish which is visible as relatively large
oscillations, the start of the steady regime is chosen to be after those oscillations. The regime ends right
before the standing shock is perturbed. This perturbation is visible as a second peak at around 70 µs for
condition B (Fig. 7.1(d)) and 45 µs for condition C (Fig. 7.1(f)). The shadowgraphs results (discussed
in Section 7.1.2) were also used in combination with these pitot measurements to determine the steady
regime. The resulting steady periods are presented in Table 7.2.
Table 7.2: Steady time periods of flow conditions A, B, and C. t = 0 s is equal to the arrival of the
primary shock.
Flow Condition
A
B
C
7.1.2
Steady start [µs]
50
20
10
Steady end [µs]
125
60
30
Steady time period [µs]
75
40
20
Shadowgraph Visualization Results
The shadowgraph images are taken at an interval of 5 µs. The images corresponding to flow condition B
from 0 µs until 25 µs are shown in Fig. 7.2. The incoming primary shock wave travels from left to right.
Comparing the different images in a chronological order it can be seen that the incoming primary
shock (Fig. 7.2(a)) causes a reflective wave when it hits the model wall (Fig. 7.2(b)). This reflective wave
travels in the opposite direction of the flow and it becomes a bow shock (Fig. 7.2(c)-7.2(d)). At 20 µs
the standing shock is fully developed and moves barely, which can be concluded by comparing Fig. 7.2(e)
and 7.2(f).
(a) 0 µs.
(b) 5 µs.
(c) 10 µs.
(d) 15 µs.
(e) 20 µs.
(f) 25 µs.
Figure 7.2: Shadowgraph images of flow condition B.
By overlapping several shadowgraph images of the steady regime, the bow shock can be made clearer.
The resulting images are shown in Fig. 7.3 for each flow condition. The first thing that is noticeable is the
stand-off distance which is the largest for condition A and it becomes smaller with the flow conditions.
Since the real distance corresponding to each pixel is known this distance can be measured. The results
are shown in Table 7.3.
From Fig. 7.3 also the shock radius can be determined at the stagnation point. This is done by
overlapping the centerpiece of the shock with a circle of which the radius is measured. The radius of
each flow condition is shown in Table. 7.3. Using Eq. (6.22) the velocity gradient is calculated which is
also presented in this table. Also the velocity gradient is shown which is calculated using the heat flux
values for SiO2 as introduced in Section 6.3.4.
67
7.2. Heat Flux Measurements
(a) Shadowgraph measurements of con- (b) Shadowgraph measurements of con- (c) Shadowgraph measurements of condition A.
dition B.
dition C.
Figure 7.3: Summation of several shadowgraph pictures of the standing shock (indicated with red line)
of conditions A, B, and C.
Table 7.3: Standing shock characteristics and velocity gradient.
δshock [mm]
rshock [mm]
β [s−1 ] (Shadowgraph)
β [s−1 ] (SiO2 )
7.2
A
4.64
18.74
1.878 · 104
1.939 · 104
Flow Condition
B
2.56
15.36
2.898 · 104
7.447 · 104
C
2.24
16.24
2.960 · 104
1.614 · 105
Heat Flux Measurements
The heat flux measurements are summarized in Table 7.4. In Appendix E all the measurements can
be found. Similar to the pitot pressure, the 95% confidence region of each curve is calculated and the
largest is shown as the overall uncertainty in Table. 7.4. The flow conditions are discussed separately:
Table 7.4: Summary of heat flux measurements.
Coating Type
Uncoated
SiO2
Black CuO
Brown CuO
Cu
Flow Condition A
qs [MW/m2 ] # tests
4.375 ± 0.40
7
5.200 ± 0.27
7
-
Flow Condition B
qs [MW/m2 ] # tests
9.974 ± 0.56
2
5.918 ± 0.36
6
6.568 ± 0.33
11
6.451 ± 0.51
4
7.203 ± 0.21
7
Flow Condition C
qs [MW/m2 ] # tests
8.821 ± 0.62
10
9.437 ± 0.43
4
7.174 ± 1.24
2
-
• Flow Condition A: As initially planned in Section 1.5 this flow condition functions as a check of the
coating, since it should in principle not affect the measured heat flux. Because this flow condition is
a low enthalpy case, the dissociation is very low hence experiments with the copper (oxide) coated
probes were not conducted.
Comparing the heat flux values of the uncoated and SiO2 cases, it is clear that in the latter the
value is higher. This result is highly unexpected since the coating should not have an effect on the
heat flux. In Section 8.1.3 several possibilities are considered to clarify these measurements.
• Flow Condition B: The heat flux values for the five different coatings vary significantly. SiO2
shows the lowest heat flux which is expected for a non-catalytic material. Copper oxide (both
black CuO and brown CuO) are around 1.5 MW/m2 higher than SiO2 . And copper – which
is a more catalytic material than copper oxide – is 2.3 MW/m2 higher than SiO2 . The results
corresponding to the uncoated probes show the highest heat flux, since catalytic recombination is
occurring at the platinum and/or the gold.
• Flow Condition C: The heat flux experiments for only three different coatings are available. The
uncoated probes were not tested at this flow condition after witnessing the results of flow condition
B. Also the copper results are absent since no probes were intact after conducting the experiments
at condition B.
The heat flux of SiO2 has increased significantly when comparing conditions B and C, which is also
the case for the black CuO. The latter heat flux is larger than the value of SiO2 which meets the
expectations of a catalytic material against a non-catalytic material.
68
Chapter 7. Results
The brown CuO results are however significantly lower than the values of SiO2 . Moreover, the
deviation is larger than any other measurement (i.e. 1.24 [MW/m2 ]). When the models were
visually inspected it was noticed that the brown CuO layer was damaged significantly at large
portions of the surface, hereby exposing the underlying layers (Fig. 7.4). For this reason only two
tests have been made with this coating. However, the surface was already damaged to such a
degree that the results are not useful.
Figure 7.4: The damaged surface of a brown CuO coated probe.
7.3
qD /qC vs. γw Diagram and Catalytic Efficiency
Using the heat flux values presented in Section 7.2 and the velocity gradients calculated with the SiO2
heat flux measurements, the diffusive and conductive heat flux ratio are expressed in a qD /qC vs. γw
diagram. The two curves are calculated using the binary and tertiary gas mixture theories (presented
in Section 2.3.2) and the necessary values were gathered from L1d2 (Section 5.1), CEA (Section 5.2.1),
Eqtab (Section 5.2.2) and the miscellaneous theories presented in Section 6.3.
The results for flow condition B are shown in Fig. 7.5. As can be seen the binary and tertiary gas
mixtures agree well at low catalytic efficiencies and both show no diffusive heat transfer and thus a zero
heat transfer ratio when the wall is non-catalytic. As the catalytic efficiency increases, the ratio increases
also. The rise of the binary theory is steeper than for the tertiary mixture. Eventually both curves reach
a plateau at high efficiencies (around γw = 10−1 ).
In Fig. 7.5 the heat transfer ratio is also shown for the different coatings by the horizontal lines. The
uncertainties, that were calculated with a confidence region of 95%, are shown by the error bars. By
determining the intersection of the curve for a tertiary gas mixture and the experimental values, the
catalytic efficiencies are determined. These are shown in Table 7.5.
Table 7.5: Surface catalytic efficiency results.
Coating
Black CuO
Brown CuO
Cu
Black CuO
Flow Condition
B
B
B
C
γ [-]
1.36 · 10−3
9.50 · 10−4
5.29 · 10−2
6.10 · 10−4
γlower bound [-]
4.10 · 10−4
1.00 · 10−5
5.75 · 10−3
1.34 · 10−4
γhigher bound [-]
4.40 · 10−3
6.18 · 10−3
1.30 · 10−3
The intersections corresponding to the two copper oxides take place in a region with a high sensitivity
since the curve is steep. However, the copper heat ratio intersects at a location where the curved has
reach a plateau. This region has a low sensitivity and introduces a large uncertainty in the catalytic
efficiency.
The qD /qC vs. γw diagram for flow condition C is shown in Fig. 7.6. Only the experimental data
of black CuO is shown since experimental data for Cu is absent and the data for brown CuO have been
dropped. The figure shows the same shape as Fig. 7.5 but the curves reach their plateaus at higher heat
flux ratios. Using the same method as flow condition B, the catalytic efficiency has been determined and
is shown in Table 7.5.
69
7.3. qD /qC vs. γw Diagram and Catalytic Efficiency
1
0.9
Condition B
Tertiary Gas Mixture
Binary Gas Mixture
0.8
0.7
qD/qC
0.6
0.5
0.4
0.3
Experiment(Cu)
0.2
0.1
0 −5
10
Experiment(Black CuO)
Experiment(Brown CuO)
−4
10
−3
10
−2
γw
10
−1
10
0
10
Figure 7.5: Heat transfer ratio against black CuO, brown CuO, and Cu surface catalytic efficiency for N
- N2 - Kr interaction at flow condition B.
From Table 7.5 it can be seen that the catalytic efficiency of Cu is the highest. The black and brown
copper oxides give a similar catalytic efficiency and both are significantly smaller than the value for
Cu. This overall phenomenon matches with the oxygen case (Fig. 3.12(c)) where in general the copper
efficiencies are higher than the copper oxides.
The catalytic efficiencies for flow condition B and C of black CuO are within the same range and
within each others uncertainties. This same order of magnitude was also found by Park [15] for oxygen
(i.e. γw = 2.6 · 10−3 and 3.2 · 10−3 ).
Because the Cu line intersects the tertiary gas mixture curve at a plateau there is a large uncertainty
in the catalytic efficiency. When the binary curve is used for Cu it is found that the efficiency is 3.07·10−3
which is 17 times smaller.
When the results of this thesis are put in perspective with the literature (Fig. 3.12), Fig. 7.7 is made.
Since the catalytic efficiencies for copper oxide are unique there is no comparison possible. Only the
efficiency for Cu acquired using the tertiary curve can be compared and it is within the same range as
Prok [47], Hartunian et al. [43], Rahman and Linnett [30], and Pope [48].
70
Chapter 7. Results
1
Condition C
Tertiary Gas Mixture
Binary Gas Mixture
0.9
0.8
0.7
qD/qC
0.6
0.5
0.4
0.3
0.2
0.1
Experiment(Black CuO)
0 −5
10
−4
−3
10
−2
10
γ
−1
10
0
10
10
w
Figure 7.6: Heat transfer ratio against black CuO surface catalytic efficiency for N - N2 - Kr interaction
at flow condition C.
Nitrogen
Cu
CuO
1200
Anderson
Tw [K]
1000
800
600
Anderson
Present Data
400
Pope
Prok
200 −5
10
Rahman, Linnett
−4
10
−3
10
−2
γw [−]
10
Hartunian et al.
−1
10
Figure 7.7: Nitrogen catalytic efficiency including the present work.
71
8
Discussion
In the previous chapter the results of the measurements were presented and compared with each other.
Some of these results (such as the pressure and heat flux) did not match with the expectations from theory
or numerical programs. If these discrepancies remained undiscussed, the reliability of the experiments
could be questioned. Therefore, in this chapter these discrepancies are discussed for the three flow
conditions. Since flow conditions B and C both show a large number of similarities in their results, they
are considered together in a section.
8.1
Flow Condition A
For flow condition A there were some discrepancies regarding the pitot pressure, velocity gradient, and
the heat flux. Possible reasons for these differences are given in this section.
8.1.1
Pitot Pressure
From Table. 7.1 it can be seen that there is a 15% misalignment between the measured pitot pressure
and the one predicted by L1d2, while the error is significantly smaller for the other conditions. The
overestimation of L1d2 is suspected to be caused by an improper prediction of the viscous effects. It
is speculated that the viscous data of krypton contains an error that causes the misalignment. This
is because condition A has essentially a binary gas mixture since the presence of atomic nitrogen is
negligible. On the other hand, the atomic nitrogen in conditions B and C has a much larger effect on the
viscosity which hides the possible error caused by krypton. Strong conclusions however cannot be made.
8.1.2
Velocity Gradient
When the velocity gradients are compared (Table 7.3) it can be found that the values only have a 3.3%
discrepancy for flow condition A between the shadowgraph visualization and the SiO2 method. With
the shadowgraph visualization method the error is caused by the modified Newtonian flow assumption
which is not valid due to the low Mach number of the flow. However, in the SiO2 method it is assumed
that SiO2 is a completely non-catalytic material hence the influence of the small amount of diffusive heat
flux has been neglected. Nonetheless, because the amount of dissociation is negligible it is believed that
the SiO2 method is more accurate than the shadowgraph visualization method.
8.1.3
Heat Flux
The heat flux measured by the uncoated probe was smaller than the heat flux determined by the SiO2
coated probe. This measurement is highly unexpected since the coating should not affect the measured
heat flux. If there is an effect, it should insulate the platinum from the flow, hereby resulting in a lower
heat flux compared to the uncoated probe.
One possibility that could explain the excessive heat flux at the SiO2 surface is a chemical reaction
between the SiO2 and the nitrogen molecules. Since the coated probes are not heated up in an oven
to make the transition from the amorphous SiO2 to a strong crystalline structure of SiO2 , there are
holes within the structure of SiO2 . By considering the chemical reaction of this coating with nitrogen
molecules it was found, using the formation energies of each species, that the reaction is endothermic
which would lower the heat flux. Since the measured heat flux is larger, the possibility of a chemical
reaction was ruled out.
72
Chapter 8. Discussion
The last possible theoretical explanation for the increased heat flux for SiO2 is adsorption. This
phenomenon can occur when a gas particle is trapped in the amorphous SiO2 layer. The tendency of
this phenomenon dependents on the gas species and is expressed by the liquefaction temperature. This
liquefaction temperature of krypton is -153 ◦ C and -195.8 ◦ C for nitrogen molecules. In the experiments
of Park [15] a test gas consisting of argon and oxygen was used, which resulted in identical heat flux
measurements using the uncoated and SiO2 coated probes. The liquefaction temperature of argon and
oxygen are -185.8 ◦ C and -183 ◦ C respectively. When comparing these temperature it can be seen that
krypton has the highest temperature, which means that it tends to be adsorbed easier than the others
species. Since adsorption is an exothermic process, it is therefore believed that the krypton gas could
have caused the increase in heat flux seen with the SiO2 measurements.
8.2
Flow Conditions B & C
Flow conditions B and C showed similar discrepancies in the static pressure, pitot pressure, and velocity
gradient. While krypton made it possible to reach flow conditions with a significant amount of dissociation (33.3% for condition B and 54.3% for condition C), it is suspected that the presence of this gas
caused unexpected effects on the boundary layer of the shock tube and the stability between the driver
and driven gas.
8.2.1
Static Pressure
From Fig. 7.1(c) and 7.1(e) it can be seen that there is a large disparity between the measured static
pressure and L1d2. Moreover, when compared with Fig. 7.1(a) it can be seen that there is a significant
increase in oscillations of the static pressure measurements. These oscillations could be caused by velocity
fluctuations from a turbulent boundary layer at the wall of the shock tube, which cannot be simulated
by L1d2.
A turbulent boundary layer is thicker than a laminar boundary layer. [67] Hence, it decreases the
effective cross-section behind the primary shock wave which increases the velocity of the gas slug behind
the shock wave. The effect of a boundary layer growth on the velocity of the contact surface has been
investigated throughly by several authors [68, 69]. The increase in velocity yields a decrease in static
pressure which propagates through the boundary layer (since the static pressure is constant through the
boundary layer). This justification potentially explains the lower static pressure (and the oscillations)
seen in the measurements.
8.2.2
Pitot Pressure
In Fig. 7.1(d) and 7.1(f) a perturbation of the standing shock is shown as a second peak starting from
approximately t = 70 µs and t = 45 µs, respectively. This second peak does not correspond to the
arrival of the contact surface (predicted by L1d2) nor the reflective shock wave. Efforts were made to
postpone or lower the effect of this second peak to increase the steady regime. One method was by
designing and manufacturing a pressure model with the same size as the heat flux model (as introduced
in Section 4.1.2). Due to some leakage problems resulting from this new model, the results were less
accurate. However, it was possible to conclude that the second peak is almost unaffected by the model
size. More can be found in Appendix F.
The occurrence of the second peak can be explained by a phenomenon called Rayleigh-Taylor instability. [70, 71] This happens at the interface between two gases which in this case is helium and the
N2 -Kr mixture. Since there is a normal pressure gradient from the helium to the test gas, the helium
can push into the latter at the centerline, ahead of the contact surface. It was found by Youngs [70] that
a large density ratio between the two gases will result in a larger mixed region. This principle is shown
in Fig. 8.1. It is suspected that the second peak is the arrival of helium ahead of the contact surface and
reflected shock wave.
Another possibility is the decrease in effective cross-section due to the boundary layer growth as
described in Section 8.2.1. This effect could drastically decrease the test time as was found by Musgrove
and Appleton [69], since the contact surface arrives significantly faster than determined in the ideal case
(without the effect of a thick boundary layer).
73
negative, the sign of the perturbation is reversed.
hows the variation of the width of the
The computer simulation shows that for this shock
on with time for a calculation with no
perturbation. The
graph
shows
the
8.2. Flow Conditions B & A
CU = - 2 2 0 mm/ms. After the shock has passed
the velocity at the interface due to shock
meshes were used in the x-direction in
(a.)
region. The helium and the air were
perfect gases with adiabatic constants
0 msl~
d ~ = 1.63, respectively. The density
Air
Helium
at the interface remained in the interval
imensional calculation was carried out
lowing initial amplitude perturbation at
ce:
i
(b)
0.4
2ny
os 2 '
.....
ii!!!i
(c)
and
2=50mm.
0.8
th of helium
on, mm
/First shock
/
ms
/
ii!
~ k
" t ~
dkReflected shocks
//I \
N /
I\\
....
.... i ,!ii ........
d)
~
iii iii!i
iii
!!ii
~
....
ms
!!31i
............... iili
1.2 ms
.2 0.3 0.4 0.5 0.6 0.70.8 0.9 1.0 1.1
Time, ms
k tube experiment; computer simulation with no
urbation.
8.2.3
Fig. 7. Shock tube experiment with single wavelength initial
perturbation
at theRayleigh-Taylor
helium/air interface.instability. [70]
Figure 8.1:
Velocity Gradient
While the velocity gradient of flow condition A determined by the shadowgraph visualization and the
SiO2 method showed a small difference, conditions B and C show an error up to 445%. Since the modified
Newtonian flow theory does not consider dissociation, it is expected that the discrepancy increases with
the flow condition. Moreover, similar to flow condition A, the Mach number is not sufficient to fulfill
the Newtonian flow assumption. For this reason the discrepancy for conditions B and C is expected.
It is believed that the error caused by the modified Newtonian method is significantly larger than the
error caused by SiO2 method hence the velocity gradient determined using SiO2 were used for further
calculations.
74
9
Sensitivity Analysis & Uncertainties
In Chapter 7 the uncertainties of both the pitot pressure and heat flux measurements were expressed by
the 95% confidence interval of the curves. While the indicated uncertainty of the catalytic efficiency in
Fig. 7.5 and 7.6 only depends on the uncertainty of the heat flux measurements, this does not include
the uncertainties in the tertiary gas mixture curve. To create this curve a large number of variables are
required which are gathered from L1d2, Eqtab, and the theories presented in Section 6.3.
This chapter will discuss the different uncertainties of the curve corresponding to the tertiary gas
mixture by performing a sensitivity analysis on various parameters. Since almost all catalytic efficiency
values are from flow condition B the analysis is tuned towards this case.
In Section 9.1 the sensitivity analysis is performed on the pitot pressure and total enthalpy. These
variables dictate the flow behind the primary shock wave of the shock tube. Any variation in these values
causes a chain effect on the remaining calculations.
Section 9.2 covers the sensitivity analysis on the average collision integrals and the binary diffusion
coefficients. These parameters have an essential role in the calculation of the heat flux ratio.
9.1
Pitot Pressure and Total Enthalpy
Both the pitot pressure and total enthalpy are input variables for Eqtab. Uncertainties in these parameters result in a different flow condition at the boundary layer edge. These parameters are necessary to
create the qD /qC vs. γw curve.
The pitot pressure is used at three locations: determination of the species particle mass (Eq. (6.7)),
calculation of the density at the boundary layer edge (Eq. (6.9)), and determination of the binary diffusion
coefficient (Eq. (6.15)). The total enthalpy is used in the expression of the frozen enthalpy (Eq. (6.20)).
The two variables are analyzed together as a combination. In Table 9.1 the results of having a 10%
variation is shown. The fluctuations experienced during the pitot measurements are well within this
percentage (Fig. 7.1). The same percentage has been chosen for the total enthalpy for consistency.
Table 9.1: Sensitivity analysis on pitot pressure and total enthalpy.
10%↑ ppitot
10%↓ ppitot
10%↑ H
10%↓ H
10%↑ ppitot & ↑ H
10%↑ ppitot & ↓ H
10%↓ ppitot & ↓ H
10%↓ ppitot & ↑ H
Te
XN
XN2
XKr
↑0.42%
↓0.47%
↑2.52%
↓2.74%
↑2.97%
↓2.35%
↓3.18%
↑2.02%
↓0.62%
↑0.66%
↑17.89%
↓18.46%
↑17.22%
↓19.05%
↓17.82%
↑18.57%
↑0.43%
↓0.44%
↓12.07%
↑12.46%
↓11.62%
↑12.87%
↑12.03%
↓12.52%
↑0.07%
↓0.08%
↓2.09%
↑2.15%
↓2.01%
↑2.22%
↑2.07%
↓2.17%
N2
diss.
↓0.62%
↑0.82%
↑20.50%
↓20.11%
↑19.72%
↓20.75%
↓19.43%
↑21.28%
β
γblack
↓9.86%
↑12.11%
↓3.22%
↑3.56%
↓12.81%
↓6.60%
↑16.16%
↑8.57%
↓7.35%
↑8.82%
↓24.26%
↑49.26%
↓30.15%
↑38.97%
↑61.03%
↓16.91%
Cuo
γbrown
Cuo
↓7.37%
↑9.47%
↓21.05%
↑38.95%
↓27.37%
↑29.47%
↑51.58%
↓13.68%
γCu
↑59.78%
↓23.97%
↓85.78%
↓86.16%
↓85.21%
When considering the individual influence of ppitot and H, it is clear from Table. 9.1 that the former has
a small influence on the conditions at the boundary layer edge, while the latter has a large influence.
A good example is the amount of dissociation that varies less than 1% when changing ppitot , whereas
changing H yields a 20% change in dissociation.
75
9.2. Collision Integral and Binary Diffusion Coefficient
The catalytic efficiencies show the same pattern where changing H results in a larger variation than
changing ppitot . Having an increase of ppitot and/or H yields a decrease of catalytic efficiencies, and vice
versa. The catalytic efficiency of Cu shows at some occasions a different behavior which can be clarified
by the fact that the Cu heat transfer ratio intersects the curve at a plateau. Any small changes which
result in a variation of the curve result in a large change in catalytic efficiency. For this reason some
values are absent when there is no intersection between the curve and the Cu heat transfer ratio.
When considering the combination of ppitot and H, it can be observed that the influence of H
dominates. ppitot strengthens the effect of H when they both change by 10% in the opposite direction.
9.2
Collision Integral and Binary Diffusion Coefficient
While the inaccuracies in the pitot pressure is a result of the pressure experiments and the inaccuracies
in the total enthalpy are caused by L1d2, there is also a source of uncertainty in the theory which is used
(2,2)
to create the qD /qC vs. γw curve. This is mainly caused by the average collision integrals (Ω̄ij ) and
the binary diffusion coefficients (Dij ).
The collision integrals are used in the calculations of the dynamic viscosity (Eq. (6.6) and (6.8)). In
these calculations the collision parameters of krypton were assumed to be equal to the ones of argon
(Table 6.1). Moreover, Kim et al. [64] compared the average collision integrals from literature and found
a large variation among them for a range of species combinations. This variation is estimated in this
thesis to be 20% which is used for the sensitivity analysis.
The binary diffusion coefficients are used to calculate the Schmidt number (Eq. (6.14)). Since the
Schmidt number is used directly in Eq. (2.22) and (2.25), it has a great effect on the results. To be
consistent with the average collision integral variation, the same percentage of variation is taken for the
sensitivity analysis on the binary diffusion coefficients. Additionally, since the Schmidt number combines
the dynamic viscosity and the binary diffusion coefficient, it is relevant to investigate the coupling effect
of both parameters on the results. The results of the sensitivity analysis are shown in Table 9.2.
Table 9.2: Sensitivity analysis on the collision integrals and binary diffusion coefficients.
(2,2)
Ω̄12
(2,2)
Ω̄12
(2,2)
Ω̄13
(2,2)
Ω̄13
(2,2)
Ω̄23
(2,2)
Ω̄23
20%↑
20%↓
20%↑
20%↓
20%↑
20%↓
20%↑ D12
20%↓ D12
20%↑ D13
20%↓ D13
20%↑ D23
20%↓ D23
(2,2)
20%↑ Ω̄23 & ↑ D12
(2,2)
20%↓ Ω̄23 & ↓ D12
(2,2)
20%↑ Ω̄23 & ↓ D12
(2,2)
20%↓ Ω̄23 & ↑ D12
(2,2)
(2,2)
(2,2)
20%↑ Ω̄12 , Ω̄13 , Ω̄23
& ↓ D12 , D13 , D23
(2,2)
(2,2)
(2,2)
20%↓ Ω̄12 , Ω̄13 , Ω̄23
& ↑ D12 , D13 , D23
µe
↓0.33%
↑0.36%
↓6.69%
↑7.85%
↓11.00%
↑14.31%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
↓11.00%
↑14.31%
↓11.00%
↑14.31%
Sc12
↓0.33%
↑0.36%
↓6.69%
↑7.85%
↓11.00%
↑14.31%
↓16.67%
↑25.00%
0.00%
0.00%
0.00%
0.00%
↓25.84%
↑42.88%
↑11.25%
↓4.75%
Sc23
↓0.33%
↑0.36%
↓6.69%
↑7.85%
↓11.00%
↑14.31%
0.00%
0.00%
0.00%
0.00%
↓16.67%
↑25.00%
↓11.00%
↑14.31%
↓11.00%
↑14.31%
Sc13
↓0.33%
↑0.36%
↓6.69%
↑7.85%
↓11.00%
↑14.31%
0.00%
0.00%
↓16.67%
↑25.00
0.00%
0.00%
↓11.00%
↑14.31%
↓11.00%
↑14.31%
β
↑0.33%
↓0.36%
↑7.17%
↓7.28%
↑12.36%
↓12.51%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
↑12.36%
↓12.51%
↑12.36%
↓12.51%
γblack Cuo
0.00%
0.00%
↑2.21%
↓2.21%
↑4.41%
↓3.68%
↓10.29%
↑18.38%
↑6.62%
↓6.62%
↓5.88%
↑8.09%
↓4.41%
↑17.65%
↑20.59%
↓15.44%
γbrown Cuo
0.00%
0.00%
↑4.21%
↓4.21%
↑7.37%
↓6.32%
↓7.37%
↑11.58%
↑4.21%
↓4.21%
↓4.21%
↑5.26%
0.00%
↑6.32%
↑17.89%
↓14.74%
γCu
↓7.39%
↑9.47%
↓60.31%
↓70.58%
↓81.19%
↓73.38%
↓69.39%
↓85.61%
↓60.28%
↓16.67% ↑4.17% ↑4.17% ↑4.17% ↑20.00% ↑22.79%
↑22.11%
-
↑25.00% ↑4.17% ↑4.17% ↑4.17% ↓20.00% ↓17.65%
↓18.95%
-
(2,2)
(2,2)
(2,2)
Three different average collision integrals are used in the calculations: Ω̄12 , Ω̄13 , and Ω̄23 . From
(2,2)
Table 9.2 it can be found that the effect of Ω̄12 is negligibly small, which corresponds to the collision
(2,2)
(2,2)
of nitrogen atoms and nitrogen molecules. Changing Ω̄13 affects the results more but it is Ω̄23
that has the largest effect on the results. This average collision integral corresponds to the collision of
nitrogen molecules and krypton, which concerns the heaviest particles in the mixture. It can be seen that
increasing the collision integral reduces the dynamic viscosity and Schmidt numbers while the velocity
gradient and catalytic efficiencies increase, and vice versa.
76
Chapter 9. Sensitivity Analysis & Uncertainties
While the collision integrals affect all the Schmidt numbers through the dynamic viscosity, this is not
the case for the binary diffusion integrals. It can be seen that increasing a binary diffusion coefficient
results in a 16.67% decrease of the corresponding Schmidt number, and when decreasing the diffusion
coefficient the Schmidt number increases with 25%. Moreover, when comparing the effects of the diffusion
coefficients on the catalytic efficiencies, it can be found that D12 has the largest effect which indirectly
means that Sc12 has the largest effect among the different Schmidt numbers.
Comparing the effect of the collision integrals with the diffusion coefficients, it can be seen that the
latter has a larger effect on the catalytic efficiencies. This however does not take the coupling of the two
into account.
(2,2)
When Ω̄23 and D12 are combined it can be found that the effects on the parameters increases substan(2,2)
tially. The Schmidt numbers and catalytic efficiencies are greatly affected by Ω̄23 and D12 . In the case
(2,2)
of increasing Ω̄23 by 20% while D12 is reduced by 20% the catalytic efficiency can change with 20%.
At last, when all the collision integrals and diffusion coefficients are increased or decreased, it can be
seen that not only the catalytic efficiencies are greatly affected, also the dynamic viscosity and velocity
gradients changes drastically.
All in all, when comparing the influence of the average collision integrals and binary diffusion coefficient with the pitot pressure and total enthalpy, it can be concluded that the total enthalpy is the most
crucial parameter for the catalytic efficiency results. This value was determined by L1d2 and is assumed
to be constant at the stagnation point.
77
10
Conclusions
10.1
Conclusions
The research question of this thesis was stated as:
“What is the catalytic recombination efficiency of copper oxide when exposed to nitrogen?”
To obtain the answer to this research question, the shock tube located at KAIST (South Korea) has
been used in combination with heat transfer gauges (i.e. thin-film gauges). These gauges were highly
polished and flush-mounted in a flat-disk cylinder model with a diameter of 12 mm to form a probe.
The probe was coated using the E-beam evaporation technique. The probe received one of the following
coating combination:
1. 1000 nm SiO2
2. 1000 nm SiO2 and 180 nm black CuO
3. 1000 nm SiO2 and 180 nm brown CuO
4. 1000 nm SiO2 and 180 nm Cu
These coated models were used to conduct experiments at different flow conditions: flow condition A
has a low enthalpy (2.0 MJ/kg) with a mass fraction dissociation of 1%, B has a moderate amount of
enthalpy (4.4 MJ/kg) and a dissociation of 33.3%, whereas C has a high enthalpy (5.8 MJ/kg) and a
dissociation of 54.3%. Flow conditions B and C were used to determine the catalytic efficiency. All
coatings have been tested in flow condition B, whereas only SiO2 and black CuO could be tested with
successful results in condition C.
The temperature measurements acquired from each probe were first filtered using a Savitzky-Golay
filter and then converted to heat flux using the Cook & Feldermann [61] method. The heat flux profiles
for flow condition C showed in general more oscillations than flow condition B. Moreover, due to the
brittleness of the brown CuO the coating was more prone for damage which resulted in larger fluctuations
in the heat flux profile.
Using the binary and tertiary gas mixture theory, the diffusive and convective heat flux ratio was
related to the catalytic efficiency in an S-shaped curve, from which it was possible to read off the value
based on the experimentally determined heat flux. To create this S-curve the velocity gradient was required, which was determined using the shadowgraph measurements post-processed using the modified
Newtonian theory, and using the SiO2 measurements. The difference in velocity gradients between the
two methods for condition A was insignificant hereby validating the latter method as an appropriate
method. For flow conditions B and C the difference was large due to the presence of dissociation which
cannot be tackled by the modified Newtonian theory.
With this the catalytic efficiency was determined to be between 6.10 · 10−4 and 1.36 · 10−3 for black
CuO, 9.50 · 10−4 for brown CuO, and 5.29 · 10−2 for Cu. These values however have a relatively large
error margin. This error is caused by the fluctuations of the heat flux measurements, the uncertainties
in the numerical programs, and the uncertainties caused by the assumptions in the binary and tertiary
theories. For both copper oxides, the difference in catalytic efficiencies between the binary and tertiary curves was small. Since Cu is a highly catalytic material, the difference between the two curves
is excessive. For this reason it is more appropriate to conclude that the catalytic efficiency for Cu is
between 3.07 · 10−3 and 5.29 · 10−2 . Due to all the uncertainties, it is therefore pertinent to state that
the determined efficiencies for CuO and Cu are more indications of the order of magnitude rather than
very accurate values.
78
Chapter 10. Conclusions
10.2
Recommendations
• Roughness variation: During reentry (and also in arc-jet facilities) the harsh environment affects
the surface of the body. Chemical reaction and ablation occur simultaneously which alter the
structure of the material. This increase in roughness has a great impact on the number of collisions
an incoming test gas experiences as discussed in Section 2.2.3, hence a rougher surface results
in a higher catalytic efficiency. The magnitude of the roughness effect is not investigated and is
unknown, but is important to be known for a better determination of the catalytic efficiency.
In this thesis project the quartz rods were highly polished with an average particle diameter of
6.0 µm. Although it is possible to, for example, omit the last step of the polishing process, it is not
know what the effect of a rougher surface has on quality and survivability of the platinum paste.
A rougher surface could increase the difficulty for the platinum particles to bond properly to form
a continuous and uniform platinum bridge.
• Wall Temperature variation: As described by Fig. 3.2, the catalytic efficiency varies depending on
the wall temperature for glass. In this thesis the wall temperature was around 320 K due to the
short run time of a shock tube . Since the wall temperature increases to temperature above 1000 K
in both arc-jets and real reentry flights, it is important to investigate the wall temperature effect
on the catalytic efficiency of copper and copper oxide.
• Coupling of oxygen and nitrogen: Using the same shock tube, Park [15] conducted experiments in
2013 for oxygen on copper oxide. Together with the conclusions made in this thesis, the preliminary
conclusion is that copper oxide (whether it is CuO or Cu2 O) has a catalytic efficiency that is
significantly lower than presented in literature.
To represent real-flight conditions more, air has to be used as a test gas in the shock tube. For
this reason the next step should focus on the catalytic efficiency of air on copper and copper oxide,
which has not been investigated thoroughly yet as seen from Fig. 3.12(a).
However, tackling this problem for air brings an even more complicated gas mixture situation. Not
only will there be O, O2 , N, and N2 , but a coupling exists between O and N which creates NO and
possibly also NO2 and N2 O. This results in a gas mixture with at least five species. Such theory
is not available and has to be derived using the same method as for the binary and tertiary gas
mixtures.
79
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82
A
Thermochemical Equilibrium and Boundary Layer Thickness
The theory presented in Section 2.3.2 is based on the assumption that thermochemical equilibrium is
reached at the boundary layer edge. This means that the dissociation of the molecules has finished
between the standing shock and the boundary layer. While the method presented in Section 2.4.2 is
more accurate than the theory presented in this section, the latter gives the possibility to show the
time and distance dependency of the flow conditions in the region between the standing shock and the
boundary layer.
In this appendix the species concentrations are first calculated against time in Section A.1 which is
necessary to determine the flow conditions which is tackled in Section A.2. In Section A.3 the final results
are presented where it is computed how much distance is required to reach thermochemical equilibrium.
This appendix closes off with the calculations of the boundary layer thickness in Section A.4 which
functions as a final check that thermochemical equilibrium is indeed established before reaching the
boundary layer edge.
A.1
Species Concentrations
Assuming that the freestream conditions before the standing shock (called region 2 in Fig. 2.11) are known
it is possible to determine the flow parameters right behind the shock using normal shock relations [72]:
2γM 2 − (γ − 1)
γ+1
[2γM 2 − (γ − 1)][(γ − 1)M 2 + 2]
Taf ter = Tpre
(γ + 1)2 M 2
(γ + 1)M 2
ρaf ter = ρpre
(γ − 1)M 2 + 2
2 + (γ − 1)M 2
uaf ter = upre
(γ + 1)M 2
paf ter = ppre
(A.1a)
(A.1b)
(A.1c)
(A.1d)
The specific heat ratio is determined using:
Pn
i fi Xi
(f
i − 1)Xi
i
γ = Pn
(A.2)
This is for a gas mixture consisting of n species and fi is a constant which is 3.5 for monotonic gases
and 2.5 for diatomic molecules.
The values determined using Eq. (A.1) are only the ones directly behind the standing shock. To
determine the values of each of these parameters at the next time instance, it is necessary to calculate
the concentration of each species over time.
The total concentration of a gas mixture can be calculated using:
n=
p
kB T NA
(A.3)
In this equation the Avogadro’s constant (NA ) is present since n has the unit [ mol
m3 ] instead of the usual
[m−3 ]. Using the molar fractions that is known before the standing shock the concentration of each
species can be determined. These are then the species concentrations directly after the standing shock.
83
A.2. Flow Conditions
The dissociation of a gas can be determined using the rate coefficient. The equation depends on the
type of gas that is considered. For nitrogen [12] this is:
(N2 + N2 → N + N + N2 ) : kf = 7 · 1015 T −1.6 e
−113200
T
(A.4)
It is hereby assumed that the dissociation only takes place by collision of two nitrogen molecules. This
rate coefficient is then used to determine the rate at which the concentration changes for N2 and N
respectively:
dnN2
= −kf · nN2
dt
dnN
= 2kf · nN2
dt
It is assumed that the concentration of N2 decays exponentially, hence it is fitted using:
dnN
2
dt
nN2 = nN2t=0 · e
nN
(A.5a)
(A.5b)
·t
(A.6)
2t=0
The amount at which N2 decreases can be used to determine the amount at which N increases. Since
N2 splits into two atoms, the increase of N is twice as large as the decrease of N2 . For flow condition B
(Table. 5.1) the results are shown in Fig. A.1. Both Fig. A.1(a) and Fig. A.1(b) show an asymptotic
behavior as time passes by: N2 asymptotically approaches 0 whereas N approaches 0.6 mol/m3 .
0.7
0.35
0.6
0.3
0.5
nN [mol/m3]
2
nN [mol/m3]
0.25
0.2
0.15
0.4
0.3
0.1
0.2
0.05
0.1
0
0
50
100
150
t [µs]
200
250
300
0
0
50
100
150
t [µs]
200
250
300
(a) The exponential decay of N2 over time between the (b) The increase of N over time between the standing
standing shock and the wall.
shock and the wall.
Figure A.1: The decrease and increase of N2 and N between the standing shock and the wall for flow
condition B.
Because of this exponential behavior it is common to define an e-folding time which is the time that
is required to grow by a factor e for N (this corresponds to 1 − 1e of the asymptotic value), and it is
also common to define the 99% time which corresponds to 99% of the asymptotic value for N. For flow
condition B this corresponds to te−f olding = 49.3 µs and t99% = 217.8 µs.
A.2
Flow Conditions
With the known concentration of N2 and N over time it is possible to determine the behavior of the
remaining flow parameters. At each time instance cp is calculated:
cp =
2.5kB n1 + 3.5kB n2 + 2.5kB n3
ρ
(A.7)
This is then used in
H = cp T + hD αN
(A.8)
where hD is the dissociation energy of N2 (946 J/mol) [73]. Using the initial temperature from Eq. (A.1b)
the total enthalpy can be determined since at that moment there is no dissociation (αN = 0).
The total enthalpy stays constant hence when dissociation occurs the temperature drops. Using
Eq. (A.8) the temperature is then determined at each time instance. By assuming a constant pressure
the density is also calculated using:
p
ρ=
NA
(A.9)
kB T M
The resulting curves for the temperature and density for condition B are shown in Fig. A.2(a) and
A.2(b), respectively.
84
Appendix A. Thermochemical Equilibrium and Boundary Layer Thickness
11000
0.2
10000
0.18
9000
0.16
ρ [kg/m3]
T [K]
8000
7000
6000
0.14
0.12
0.1
5000
0.08
4000
0.06
3000
2000
0
50
100
150
t [µs]
200
250
0.04
0
300
50
100
150
t [µs]
200
250
300
(a) Temperature against time between the standing (b) Density against time between the standing shock and
shock and the wall.
the wall.
Figure A.2: The change in temperature and density between the standing shock and the wall for flow
condition B.
A.3
Distance and Damköhler Number
The results calculated in the previous sections are expressed against time. Time is however a relative
quantity since the velocity varies hence it is necessary to express the time as function of the distance.
The distance (y) starts from the standing shock and ends at the wall and the shock stand-off distance
(ys ) has been measured in Section 4.3. With these parameters the time can be computed by:
y
y
y
y
= s
t= =
−1
(A.10)
u
us ys
us ysy−y
s
Using this equation the distance and the velocity can be expressed against time, shown in Fig. A.3 for
flow condition B.
3
900
800
2.5
700
600
u [m/s]
y [mm]
2
1.5
1
500
400
300
200
0.5
100
0
0
50
100
150
t [µs]
200
250
300
0
0
50
100
150
t [µs]
200
250
300
(a) Distance against time between the standing shock (b) Velocity against time between the standing shock
and the wall.
and the wall.
Figure A.3: The change in distance and velocity between the standing shock and the wall for flow
condition B.
From Fig. A.3(a) it can be seen that the same asymptotic behavior occurs at large time values. Since
the shock stand-off distance was determined to be 2.56 mm (Section 4.3) the curve approaches this
value. This is explained by the velocity curve shown in Fig. A.3(b) where the velocity approaches zero
asymptotically.
The e-folding and 99% times can be translated to distances: ye−f olding = 0.049 mm and y99% =
0.219 mm. These values can be expressed by the Damköhler number which is the ratio between the
shock stand-off distance and the e-folding distance [12]:
Da =
δshock
yeq
This corresponds to Dae−f olding = 51.8 and Da99% = 11.7, respectively.
85
(A.11)
A.4. Boundary Layer Thickness
Having such large values means that equilibrium is reached very quickly. In order to conclude whether
this state is established before the boundary layer edge is reached, it is necessary to determine the
boundary layer thickness.
A.4
Boundary Layer Thickness
To derive the boundary layer thickness it is necessary to use the coordinate transformation (Eq. (2.9))
shown in Section 2.3.2.1. In this case the subscipt e is taken to be the boundary layer edge. Moreover,
so far in this appendix the distance (y) is assumed to originate from the standing shock. This made the
figures more intuitive since the flow is followed. In the remainder of the thesis the coordinate system has
been taken as shown in Fig. 2.6 and the non-dimensional coordinate system is based on this. Since this
latter system is necessary again, the orientation is reversed which means that x=0 corresponds to the
wall.
First the s coordinates need to be rewritten:
Z x
Z x
1
(A.12)
s=
ρe µe r2 ue dx = ρe µe r2 β
xdx = ρe µe r2 βx2
2
0
0
using which η can be expressed as:
rue
η=√
2s
Z
0
y
rβx
ρdy = q
2 21 ρe µe r2 βx2
Z
s
y
ρdy =
0
β
ρe µe
Z
y
ρdy
To do this Eq. (2.15) has been used. Additionally, the derivative of η is calculated:
s
β
dη =
ρdy
ρe µe
0
At last, the following is defined: uue = ∂f
∂η = f .
The displacement boundary layer thickness can be determined using:
Z ∞
ρu
δ=
1−
dy
ρe ue
0
(A.13)
0
(A.14)
(A.15)
The original equation for an incompressible flow did not include the density ratio, but since the flow is
dissociated inside the boundary layer it has been included. [74] Using the definition of f 0 and Eq. (A.14)
the boundary layer thickness can be expressed in the non-dimensional coordinate as:
r
Z ∞
r
Z ∞
Z ∞
Z ∞
ρ ∂f
ρ ∂f
ρe µe 1
µe
ρe
δ=
1−
dy =
1−
dη =
dη −
f 0 dη
(A.16)
ρe ∂η
ρe ∂η
β ρ
ρe β 0 ρ
0
0
0
The necessary values at the boundary layer edge (i.e. µe , ρe , and β) are determined using the method
presented in Section 6.3. The value for β that is determined using the SiO2 measurements is chosen over
the shadowgraph measurements.
Moreover, the computation of f 0 has been conducted in Section 6.3.3.1. The result is shown in
Fig. A.4. By choosing the boundary layer edge to be at the location where f 0 = uue = 0.99 it is found
that η = 3.471. Hence the upper limit of the two integrations corresponds to this value. The integration
of f 0 can then be done using the trapezoidal method.
The density values first need to be transferred to y coordinates by combining Fig. A.2(b) and A.3(a).
Afterwards Eq. (A.13) is used to transform it to η coordinates using the trapezoidal method to perform
the integration. At the end the density ratio is integrated over η.
For flow condition B where µe = 7.16 · 10−4 kg/(m · s), ρe = 1.09 · 10−1 kg/m3 , and β = 7.45 · 104 s−1 ,
this would result in a boundary layer thickness of 1.1 mm. Since the shock stand-off distance was determined to be 2.56 mm and thermochemical equilibrium is reached at around 2.34 mm there is still around
1.24 mm left between the moment equilibrium is reached and the boundary layer edge.
While the situation has been simplified by assumptions which introduce uncertainties in the representation of the reality, the results indicate that thermochemical equilibrium is established relatively far
from the boundary layer edge. Even in the pessimistic case where the errors of this calculation influences the results negatively, it is believed that equilibrium is ensured since the velocity approaches zero
asymptotically near the wall (Fig. A.3(b)).
86
Appendix A. Thermochemical Equilibrium and Boundary Layer Thickness
1
0.8
f‘ [−]
0.6
0.4
0.2
0
0
2
4
η [−]
6
8
Figure A.4: The behavior of f 0 against the non-dimensional coordinate η.
87
B
Shock Tube Program Comparison
In Section 5.1 the shock tube program L1d2 was introduced. However, in this thesis project another
shock tube program was considered which was given by Prof. Chul Park of KAIST. This program will
be introduced in Section B.1 and compared with L1d2 in Section B.2.
B.1
Stube9
The alternative shock tube program that was considered is called “stube9”. It uses the equations which
are based on the perfect gas assumption from Glass and Patterson [52] as estimated values which have
been presented in Section 4.1.4.
These estimated values are then submitted into different subroutines which determines the real gas
solution using the concept of partition function which was discussed in Section 2.4.2. It should be noted
that stube9 calculates the flow conditions for a given shock speed. Moreover, this program is only capable
of presenting the steady values hence the transient behavior of the flow parameters is absent.
B.2
Comparison Between L1d2 and Stube9
From Sections 5.1 and B.1 it can be understood that there is a significant difference in nature between the
two shock tube programs. L1d2 uses the JANAF coefficients and stube9 is based on partition functions,
which requires the data regarding the electronic states of atoms and molecules, found in the NIST Atomic
Spectra Database. [25]
By comparing both programs to experimentally determined values the superior one can be chosen.
Using the shock tube the following flow condition was reproduced: an absolute driver pressure of 1.65 MPa
helium and an absolute driven pressure of 150 Pa with 21% volume N2 and 79% volume Kr.
As indicated in Fig. 4.2 two piezo-electric pressure sensors are used which are exactly 0.5 m separated
from each other. From these the shock speed and the static pressure behind the shock wave (p2 ) can
be measured. These values are then compared to numerical results from L1d2 and stube9. Since stube9
needs the shock speed as input value to calculate the remaining flow parameters, the measured shock
speed has been taken while L1d2 calculates it by itself based on the initial conditions.
The results are summarized in Table B.1 from which it can be seen that there is a 10% difference
between the predicted shock speed by L1d2 and the measured value. L1d2 predicts a static pressure that
is around 15% higher than measured while stube9 predicts a significantly lower pressure.
Table B.1: Validation of L1d2 and stube9 using two shock tube runs.
Run 1
Run 2
Experiment
Shock speed [m/s]
p2 [Pa]
2.17 · 103
2.21 · 104
2.16 · 103
2.48 · 104
L1d2
Shock speed error [%]
10.08
11.04
p2 error [%]
15.88
16.01
stube9
p2 error [%]
-16.77
-27.06
The first run is also shown in Fig. B.1. Since the results of L1d2 are available with respect to the
time it has been shown in this figure as the circled dashed lines. The steady part of the experimental
data has been chosen after the initial peak, hence this same part is taken from L1d2 to determine the
steady static pressure.
88
Appendix B. Shock Tube Program Comparison
120
100
Pressure Sensor 1
L1d2
stube9
p2 [kPa]
80
60
40
20
0
−20
0
100
200
300
400
Time [µs]
500
600
700
Figure B.1: Static pressure vs time containing the results of the experiment, L1d2, and stube9.
It is important to note that the experiment itself has flaws which can be seen in the differences in
measured static pressure between the two runs. At the moment of conducting these experiments, the
shock tube has some leakage problems which means that the lowest pressure which could be reached
with the rotary pump was around 10 Pa and it leaked at a rate of approximately 20 sec/Pa. Therefore,
the test gas was contaminated and the driven pressure could not be exactly 150 Pa. This problem was
solved in a later phase of the thesis which made it possible to pump the tube consistently below 8 Pa for
the real experiments. However, at that phase the choice between L1d2 and stube9 was already made.
Taking these things into account it could explain the differences between the experimental data and
the results of L1d2. However, since the differences in static pressure of stube9 are larger and the results
cannot be presented against time, it is less powerful than L1d2. The error of stube9 is possibly caused
by the absence of the viscous effect. At a pressure of 150 Pa the viscous effect can have an effect that
cannot be neglected.
Based on this validation the choice was made to use L1d2 for the shock tube calculation rather than
stube9.
89
C
Equilibrium Composition Program Comparison
In Section 5.2 the equilibrium composition program “Eqtab” was introduced which was used to determine
the conditions at the boundary layer edge. Also CEA was presented which was used to calculate the
conditions at the wall. The program that is presented in this appendix is called “therm” and is based
on the same principles as CEA. It used to be the improved version of the old version of CEA.
In Section C.1 this program will be presented which is then followed by a comparison with CEA in
Section C.2. This appendix ends with a comparison between Therm and Eqtab from which the latter
was chosen.
C.1
Therm
The program “therm” can determine the equilibrium composition of a gas mixture. To use therm
properly an input file is required with all the species present in the gas. Each species has (next to its
molar weight) a set of six coefficients for different temperature ranges: 500 K until 3000 K, 3000 K until
6000 K, and above 6000 K. These coefficients are called JANAF thermodynamic coefficients and they
represent the following variables: Hf298 , Sc , Hc , a1 , a2 , and a3 . [75] These variables represent respectively
the heat of formation at 298.13 K, the reference entropy (Eq. (C.1)), the reference enthalpy (Eq. (C.2)),
and the last three values are coefficients based on the curve fits.
Z Ts
Cp
dT
(C.1)
Sc =
T
0
Z Ts
Hc =
Cp dT
(C.2)
T0
With these six coefficients the Cp , H, and S of each species can be determined as shown by Eq. (C.3)(C.5). The reference temperature T0 is set to 298.13 K whereas the split temperature Ts is 3000 K.
Cp = a1 + a2 T +
a3
T2
H = Hf298 + ∆H
(C.3)
(C.4a)
2
∆HT = A + BT + CT + DT
3
(C.4b)
Ts2
a3
a2
; C = a1 ; D =
Ts
2
T
a3
1
1
S = Sc + a1 ln
+ a2 (T − Ts ) −
− 2
Ts
2 T2
Ts
A = −a3 ; B = Hc − a1 Ts − a2
2
+
(C.4c)
(C.5)
Cp , H, and S describe the thermodynamic properties of the species. The species enthalpy is directly
related to the species energy by:
∆H = ∆E + (∆n)RT
(C.6)
which is derived by substituting the ideal gas law with the definition of enthalpy. Since the initial molar
fraction between the different species and the initial temperature are all input values, the species energy
can be determined.
With the given pressure and enthalpy as input values the temperature is varied to find the equilibrium
state of all species. This is done by varying the species concentration with a slight amount and determine
90
Appendix C. Equilibrium Composition Program Comparison
the change in the species energy. This slope has to be zero (or very near zero) to represent the equilibrium
position. At the end of the iteration the species concentration, temperature and the thermodynamic
properties are presented. Since the conditions are matched for the pressure and enthalpy, these are also
known. Based on these the density and the molecular weight of the mixture are determined.
C.2
Comparison between CEA and Therm
In Table C.1 a comparison has been made between CEA and therm. By matching the temperature and
the pressure for a certain gas mixture the differences are expressed. From this table it is clear that
the differences are small. There is less than 2% discrepancy between the two enthalpies, whereas the
densities and molar masses are essentially identical. This verification proves that there is basically no
significant difference between the two programs since they are based on the same principles.
Table C.1: Comparison between CEA and therm for a binary gas mixture consisting of 42.8% in volume
N2 & 57.2% in volume Kr.
Variable
T [K]
p [N/m2 ]
h [J/kg]
ρ [kg/m3 ]
Mmix [kg/mol]
C.3
CEA
500
9.869 · 10−1
8.2254 · 104
1.4416
5.9931 · 10−2
Therm
500
9.869 · 10−1
8.375 · 104
1.441
5.993 · 10−2
Difference
1.819%
-0.042%
-0.002%
Comparison Between Therm and Eqtab
The performance of therm and eqtab have to be compared to see the accuracy of the JANAF coefficients
for high temperatures. Since the equilibrium composition program needs to be applied at the stagnation
boundary layer edge located behind the standing shock, the temperature could surpass the maximum
limit of the JANAF coefficients. Moreover, for lower temperature cases eqtab can be verified using the
results of therm.
Both codes give the results based on a chosen pressure and enthalpy. A total of ten different cases
have been chosen where not only the aforementioned two variables are varied but also the gas mixture
composition. These ten test cases are shown in Table C.2.
Test cases 1 to 3 show a possible gas mixture composition for the actual experiments of this thesis.
The pressure and enthalpy are varied to such a degree that there is a no dissociation, a moderate amount
of dissociation, and a high amount of dissociation, respectively. For cases 1 and 3 the differences in
density and temperature between the two programs are ranging from 5 to 8%, while the differences in
dissociation are small. For case 2 the differences in density and temperature are small but the dissociation
differs.
From these three cases it is rather impossible to conclude whether therm or eqtab is the more accurate program. For this reason the following cases cover either an almost purely krypton mixture or an
almost purely nitrogen mixture. Both programs were written for a binary gas mixture hence it was still
necessary to have a very small amount of the other gas.
Cases 4 to 6 cover the almost purely krypton case. There was a suspicion that the JANAF coefficients for krypton were old and inaccurate. In the past these mistakes were reported by the users to the
JANAF committee but krypton is an uncommon gas, hence it is possible that the coefficients have never
been updated.
The first thing that is noticed from these three cases is that the error ranges from 5 to 15%, which is
significantly larger than in cases 1 to 3. Case 4 has a low temperature hence the results between therm
and eqtab should match, which is definitely not true here. The amount of dissociation differs significantly
but the accuracy of both program is rather doubtful, since there is basically no nitrogen in the mixture.
Since the gas mixture mainly consists of a monatomic molecule and more importantly it is inert, the
c
temperature can be calculated analytically. Since it is known that Rp = 2.5, hence multiplying this by
J
the universal gas constant results in the heat capacity at constant pressure. However, the units are mol·K
J
which can be converted to kg·K using the molar mass of krypton (i.e. 0.083798 kg/mol). Using this cp
value of krypton the temperature can be determined since H = cp T . From this the temperature can be
91
C.3. Comparison Between Therm and Eqtab
determined analytically for cases 4 to 6 which results in respectively: 2051.7 K, 4031.4 K, and 8062.9 K.
It can be seen that the temperatures calculated by eqtab are a lot closer to these values than therm.
Cases 7 to 10 are for an almost purely nitrogen case. The error for these four cases are ranging from
0.25 to 2%. Not only the density and temperature match very well, also the amount of dissociation is very
similar. Even case 10 which has a very high amount of dissociation and a temperature of almost 8000 K,
the results of therm matches well with eqtab. This means that the JANAF coefficients of nitrogen have
been updated to such a degree that it is relatively accurate for the higher temperature range.
This, however, does show that both programs perform very well and give almost the same results
while they are fundamentally different. Moreover, this also proves that the JANAF coefficients of krypton
are inaccurate and outdated. The differences seen in the gas mixture (cases 1 to 3) are mostly caused
by this accuracy of the JANAF values of krypton.
From this it is concluded that eqtab is superior relative to therm for the equilibrium composition
calculations.
Table C.2: The ten test cases to compare the performance of therm and eqtab.
1
Gas mixture &
no diss.
21% N2
79% Kr
1.17 · 102
1.00 · 106
Therm
Eqtab
3.06·10−4
3.32·10−4
3.32·103
3.06·103
Mass
fraction
N2 Diss.
0.00
0.00
Gas mixture &
medium diss.
21% N2
79% Kr
1.17 · 103
3.88 · 106
Therm
1.55·10−3
5.66·103
0.73
2
−3
3
0.70
3
1.00
3
0.99
3
0.00
3
0.00
3
0.73
3
0.40
3
1.00
3
0.00
3
0.12
3
4.87·10
0.11
Case Description
3
4
5
6
Gas mixture &
high diss.
Kr & low temp.
Kr & medium
temp.
Kr & high temp.
7
N2 & low diss.
8
N2 & medium
diss.
9
10
N2 & high
diss.
N2 & very
high diss.
Gas composition
p [N/m2 ]
H [J/kg]
Program ρ [kg/m3 ]
21% N2
79% Kr
Eqtab
1.17 · 104
0.03% N2
1.17 · 103
99.97% Kr
0.03% N2
1.17 · 103
99.97% Kr
0.03% N2
1.17 · 103
99.97% Kr
5.88 · 106
Eqtab
5.00 · 105
1.00 · 106
Therm
Eqtab
2.00 · 106
Therm
Eqtab
1.00 · 107
99.997% N2
1.17 · 104
0.003% Kr
2.50 · 107
99.997% N2
1.17 · 104
0.003% Kr
Therm
Eqtab
99.997% N2
1.17 · 103
0.003% Kr
99.997% N2
1.17 · 104
0.003% Kr
Therm
3.50 · 107
9.52·10
−3
1.00·10
−2
5.10·10
−3
5.85·10
−3
2.73·10
−3
2.93·10
−3
1.41·10
−3
1.49·10
−3
5.61·10
8.81·10
8.32·10
2.31·10
2.02·10
4.32·10
4.02·10
8.34·10
7.91·10
7.19·10
−4
Eqtab
7.30·10
−4
Therm
4.21·10−3
6.30·103
0.49
Eqtab
4.24·10−3
6.29·103
0.48
−3
3
0.74
3
0.74
3
0.97
3
0.96
Therm
Therm
Eqtab
4.50 · 107
1.57·10
T [K]
Therm
Eqtab
3.33·10
3.35·10
−3
2.56·10
−3
2.61·10
−3
4.90·10
6.80·10
6.77·10
7.85·10
7.72·10
92
D
Heat Flux Conversion Comparison
In Section 6.2 two different heat flux conversion methods were mentioned: Cook and Feldermann [61]
(Eq. (6.2)) and Kendall et al. [62] (Eq. (6.3)). The latter is less prone for noise which will be shown in
this appendix using actual experimental data.
In Fig. D.1 the temperature profile is shown for a probe that was coated with black CuO and tested
at flow condition B. This is analyzed by the Cook and Feldermann method and Kendall et al. method.
Additionally, the method that has been used in this thesis, is shown where the Savitzky-Golay filter was
used for the temperature values prior to the Cook and Feldermann method. These are shown in Fig. D.2.
50
40
∆ T [K]
30
20
10
0
−10
0
20
40
60
80
100
t [µs]
Figure D.1: Temperature change with respect of time measured by the heat transfer gauge.
From Fig. D.2 it can be seen that the Cook and Feldermann method shows a large amount of
fluctuations. However, the Kendall et al. method shows a significant reduction of these fluctuations
while maintaining the same overall shape.
The interesting part is when the comparison is made between these methods and the current method
used for this thesis. It can be seen that the Savitzky-Golay filter has cleaned the signal to such a degree
that the fluctuations are very similar to the Kendall et al. method. This comparison indicates that the
filter does not alter the results negatively while the Cook and Feldermann method is still used.
93
14
Cook and Feldermann
Kendall et. al
Current Method
12
10
q [MW/m2]
8
6
4
2
0
−2
−4
0
20
40
60
80
100
t [µs]
Figure D.2: Comparison between Cook and Feldermann method, Kendall et al. method, and the method
used for this thesis.
94
E
Heat Flux Profile Measurements
In this appendix all the heat flux measurements are shown. Figures E.1-E.2 correspond to flow condition
A, whereas Fig. E.3-E.7 represents condition B. The results of condition C are shown in Fig. E.8-E.10.
In the legend the name of each probe is shown which is used for identification. Also the number of
run is shown. In the right upper corner of each graph the average is shown of all the results inside the
steady regime. Also the largest error among all the curves with a 95% confidence interval is shown which
represents the uncertainty.
22
20
18
16
Uncoated
Uncoated 2014−48 Run 1
Uncoated 2014−46 Run 1
Uncoated 2014−46 Run 2
Uncoated 2014−46 Run 3
Uncoated 2014−47 Run 1
Uncoated 2014−47 Run 2
Uncoated 2014−47 Run 3
qs,st = 4.375±0.40
qs [MW/m2]
14
12
10
8
Steady
6
4
2
0
0
50
100
150
200
250
Time [µs]
Figure E.1: Heat flux profile measurements of the uncoated probes at flow condition A.
95
22
20
18
16
SiO2
SiO2 2014−13 Run 2
SiO2 2014−13 Run 3
SiO2 2014−38 Run 1
SiO2 2014−38 Run 2
SiO2 2014−38 Run 3
SiO2 2014−35 Run 1
SiO2 2014−35 Run 2
qs,st = 5.200±0.27
qs [MW/m2]
14
12
10
Steady
8
6
4
2
0
0
50
100
150
200
250
Time [µs]
Figure E.2: Heat flux profile measurements of the SiO2 probes at flow condition A.
22
20
Uncoated
Uncoated 2014−23 Run 3
Uncoated 2014−23 Run 4
qs,st = 9.974±0.56
18
16
Steady
qs [MW/m2]
14
12
10
8
6
4
2
0
0
20
40
60
Time [µs]
80
100
120
Figure E.3: Heat flux profile measurements of the uncoated probes at flow condition B.
96
Appendix E. Heat Flux Profile Measurements
22
20
18
SiO2
SiO2 2013−1 Run 1
SiO2 2013−1 Run 2
SiO2 2013−1 Run 3
SiO2 2014−13 Run 2
SiO2 2014−13 Run 3
SiO2 2014−13 Run 4
qs,st = 5.918±0.36
16
qs [MW/m2]
14
12
10
Steady
8
6
4
2
0
0
20
40
60
Time [µs]
80
100
120
Figure E.4: Heat flux profile measurements of the SiO2 probes at flow condition B.
22
20
18
16
qs [MW/m2]
14
12
Black CuO
Black CuO 2014−15 Run 1
Black CuO 2014−15 Run 2
Black CuO 2014−15 Run 3
Black CuO 2014−15 Run 4
Black CuO 2014−15 Run 5
Black CuO 2014−16 Run 1
Black CuO 2014−16 Run 2
Black CuO 2014−16 Run 3
Black CuO 2014−9 Run 2
Black CuO 2014−9 Run 3
Black CuO 2014−9 Run 4
qs,st = 6.568±0.33
Steady
10
8
6
4
2
0
0
20
40
60
Time [µs]
80
100
120
Figure E.5: Heat flux profile measurements of the black CuO probes at flow condition B.
97
22
20
18
Brown CuO
Brown CuO 2014−28 Run 1
Brown CuO 2014−28 Run 2
Brown CuO 2014−28 Run 3
Brown CuO 2014−25 Run 1
qs,st = 6.451±0.51
16
qs [MW/m2]
14
12
Steady
10
8
6
4
2
0
0
20
40
60
Time [µs]
80
100
120
Figure E.6: Heat flux profile measurements of the brown CuO probes at flow condition B.
22
20
18
16
Cu
Cu 2014−20 Run 3
Cu 2014−20 Run 4
Cu 2014−17 Run 1
Cu 2014−17 Run 2
Cu 2014−17 Run 3
Cu 2014−17 Run 4
Cu 2014−17 Run 5
qs,st = 7.203±0.21
qs [MW/m2]
14
12
Steady
10
8
6
4
2
0
0
20
40
60
Time [µs]
80
100
120
Figure E.7: Heat flux profile measurements of the Cu probes at flow condition B.
98
Appendix E. Heat Flux Profile Measurements
22
20
18
16
qs [MW/m2]
14
SiO2
SiO2 2014−35 Run 1
SiO2 2014−35 Run 2
SiO2 2014−35 Run 3
SiO2 2014−36 Run 1
SiO2 2014−36 Run 2
SiO2 2014−36 Run 3
SiO2 2014−36 Run 4
SiO2 2014−37 Run 1
SiO2 2014−37 Run 2
SiO2 2014−37 Run 3
12
qs,st = 8.821±0.62
Steady
10
8
6
4
2
0
0
10
20
30
40
Time [µs]
50
60
70
80
Figure E.8: Heat flux profile measurements of the SiO2 probes at flow condition C.
22
20
18
Black CuO
Black CuO 2014−9 Run 1
Black CuO 2014−15 Run 1
Black CuO 2014−15 Run 3
Black CuO 2014−15 Run 4
qs,st = 9.437±0.43
16
qs [MW/m2]
14
Steady
12
10
8
6
4
2
0
0
10
20
30
40
Time [µs]
50
60
70
80
Figure E.9: Heat flux profile measurements of the black CuO probes at flow condition C.
99
22
20
Brown CuO
Brown CuO 2014−28 Run 1
Brown CuO 2014−28 Run 2
qs,st = 7.174±1.24
18
16
qs [MW/m2]
14
12
Steady
10
8
6
4
2
0
0
10
20
30
40
Time [µs]
50
60
70
80
Figure E.10: Heat flux profile measurements of the brown CuO probes at flow condition C.
100
F
Alternative Pressure Model
From the pitot pressure experiments it was found that there is was second peak after the arrival of the
standing shock, as seen in Fig. 7.1(d) and 7.1(f). Initially it was tried to prevent or lowering this second
peak to increase the steady regime.
Since these experiments were conducted with the pressure model with a 20 mm diameter, it was
decided to design and create a model with the same dimensions as the heat flux model (i.e. 12 mm
diameter) which would fit a piezo-electric sensor. This is necessary to determine how prominent the
second peak is for the smaller model, since that is eventually the model that is used to measure the heat
flux to acquire the catalytic efficiency.
The engineering drawing that was made is shown in Fig. F.1(a) from which it can be seen that the
front part of the model has the same dimensions as the heat flux model (Fig. 4.3(a)). Since the piezoelectric sensor has a certain length, the length of the model had to be increased. Moreover, a new sting
had to be manufactured to fit the screws that were chosen for this model and also to compensate for
the increase in length of the model. The final product mounted on the sting with a piezo-electric sensor
inside is shown in Fig. F.1(b).
(a) Engineering drawing of the alternative pressure model.
(b) Picture of the alternative pressure model.
Figure F.1: The pressure model with the same dimensions as the heat flux model.
With an o-ring around the piezo-electric sensor and also at each screw it was attempted to prevent
leakages. This same method worked for the original pressure sensor with a diameter of 20 mm. Unfortunately, leakages did occur which could not be solved hence instead of pumping the shock tube to
pressure values below 8 Pa only 13 Pa was possible.
The results are shown in Fig. F.2. When comparing the absolute pressure values for the different
flow conditions it can be seen that they are basically the same for both models in flow condition A
(Fig. F.2(a)).
However, for flow conditions B and C there is a noticeable difference as can be seen from Fig. F.2(b)
and F.2(c), respectively. The pressure values for the new model is significantly lower than the ones
corresponding to the previous model. This difference can be justified by the leakage problem that has
relatively a larger impact on the purity of the test gas used for flow conditions B and C compared to A.
The focus is on the peak and the moment of arrival of the second peak. Both Fig. F.2(b) and F.2(c)
show a slightly postponed arrival of the second peak compared to the old model, however the impact is
101
still the same. Since the test gas is more contaminated due to the leakages it cannot be excluded that
the observed delay is caused by that.
The problem of the second peak was eventually evaded by ending the steady period before the arrival
of the peaks which has been confirmed by shadowgraph measurements.
1000
ppitot [kPa]
800
600
400
Pitot Measurement 1 (Old)
Pitot Measurement 2 (Old)
Pitot Measurement 3 (Old)
Pitot Measurement 4 (New)
Pitot Measurement 5 (New)
Pitot Measurement 6 (New)
L1d2 pitot
Steady
Pitot Measurement 1 (Old)
Pitot Measurement 2 (Old)
Pitot Measurement 3 (Old)
Pitot Measurement 4 (New)
Pitot Measurement 5 (New)
Pitot Measurement 6 (New)
L1d2 pitot
Steady
350
300
ppitot [kPa]
1200
250
200
150
400
100
200
0
0
50
50
100
150
Time [µs]
200
250
(a) Pitot pressure measurements of case A.
400
350
ppitot [kPa]
300
250
0
0
20
40
60
Time [µs]
80
100
120
(b) Pitot pressure measurements of case B.
Pitot Measurement 1 (Old)
Pitot Measurement 2 (Old)
Pitot Measurement 3 (New)
Pitot Measurement 4 (New)
Pitot Measurement 5 (New)
L1d2 pitot
Steady
200
150
100
50
0
0
20
40
60
Time [µs]
80
100
(c) Pitot pressure measurements of case C.
Figure F.2: Pitot pressure measurements of case A, B, and C for both the old (i.e. 20 mm) model and
new (i.e. 12 mm) model.
102
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