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Electronic Theses, Treatises and Dissertations
The Graduate School
2011
Multiphysics Analysis for Thermal
Management of Electromagnetic Launchers
a Coupled Electromagnetic and Thermal
Problem with Pulsating Heat Generation
Han Zhao
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THE FLORIDA STATE UNIVERSITY
COLLEGE OF ENGINEERING
MULTIPHYSICS ANALYSIS FOR THERMAL MANAGEMENT OF ELECTROMAGNETIC
LAUNCHERS – A COUPLED ELECTROMAGNETIC AND THERMAL PROBLEM WITH
PULSATING HEAT GENERATION
By
HAN ZHAO
A Dissertation submitted to the
Department of Mechanical Engineering
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Degree Awarded:
Summer Semester, 2011
The members of the committee approve the Dissertation of Han Zhao defended on June 17th,
2011.
_______________________________________
Juan C. Ordonez
Professor Directing Dissertation
_______________________________________
Simon Y. Foo
University Representative
_______________________________________
Ching-Jen Chen
Committee Member
_______________________________________
Ongi Englander
Committee Member
Approved:
_____________________________________
Chiang Shih, Chair, Department of Mechanical Engineering
_____________________________________
John Collier, Dean, FAMU-FSU College of Engineering
The Graduate School has verified and approved the above-named committee members.
ii
To my parents and sister Helen,
for giving me the encouragement to help me get where I am today.
iii
ACKNOWLEDGEMENTS
I would first like to express all my gratitude to Dr. Juan C. Ordonez, my advisor, for the
opportunity to work with him, for his ideas, inspiration, patience and research support. I would
also thank Dr. Jeferson A. Souza for giving me the guidance and assistance all along.
I would like to thank all my committee members: Dr. Ching-Jen Chen, Dr. Ongi
Englander, and Dr. Simon Y. Foo, for their time and constructive remarks.
I am very thankful to all the people from our group, especially Dr. Rob Hovsapian, Dr.
Alejandro Rivera-Alvarez, and Dr. Srinivas Kosaraju. I really enjoy working with them and look
forward towards future collaborations.
Finally, I am extremely grateful to my parents and sister for their unlimited support along
these years. Their patience and encouragements were priceless.
iv
TABLE OF CONTENTS
Table of Contents
Acknowledgements ........................................................................................................................ iv
table of contents .............................................................................................................................. v
List of Tables ............................................................................................................................... viii
List of Figures ................................................................................................................................ ix
Abstract ......................................................................................................................................... xii
Chapter One Introduction ............................................................................................................... 1
1.1 Introduction to Electromagnetic Launchers .......................................................................... 1
1.2 Design Parameters for a Long-range Naval Railgun ............................................................ 2
1.3 Research Objectives .............................................................................................................. 5
1.4 Literature Review.................................................................................................................. 6
Chapter Two Thermal Analysis .................................................................................................... 10
2.1 Computational Model ......................................................................................................... 10
2.2 Formalism ........................................................................................................................... 12
2.3 Boundary and initial conditions .......................................................................................... 17
2.4 2-D model verification ........................................................................................................ 19
2.5 3-D first shot simulation ..................................................................................................... 26
2.5.1 Material properties ...................................................................................................... 26
2.5.2 Mesh convergence........................................................................................................ 28
2.5.3 Time step and solver parameters ................................................................................. 31
2.5.4 Launch data ................................................................................................................. 32
Chapter Three Axial Cooling Analysis for Single Shot................................................................ 37
3.1 Introduction ......................................................................................................................... 37
v
3.2 No channels arrangement .................................................................................................... 43
3.3 Axial cooling channel assemble.......................................................................................... 47
3.3.1 Flow parameters and channel layout .......................................................................... 47
3.3.2 Governing equations .................................................................................................... 50
3.3.3 Boundary and initial conditions................................................................................... 51
3.3.4 Mesh convergence........................................................................................................ 52
3.4 Results and discussion ........................................................................................................ 54
Chapter Four Axial Cooling Analysis for Multiple Shots ............................................................ 60
4.1 Computational domain ........................................................................................................ 60
4.2 Governing equations, boundary, initial conditions and material properties ....................... 63
4.3 Simulation procedure .......................................................................................................... 63
4.4 New mesh for multi-shots scenarios ................................................................................... 63
4.5 Results and discussion ........................................................................................................ 67
4.5.1 3-centered case ............................................................................................................ 67
4.5.3 3-eccentric case ........................................................................................................... 74
4.5.3 Comparison of the two cooling arrangements ............................................................. 80
Chapter Five Vertical Cooling Analysis for Single Shot .............................................................. 85
5.1 Introduction ......................................................................................................................... 85
5.2 Computational domain ........................................................................................................ 86
5.3 Governing equations, boundary, initial conditions and material properties ....................... 86
5.4 Simulation procedure .......................................................................................................... 86
5.5 Mesh.................................................................................................................................... 86
5.6 Channel distribution ............................................................................................................ 87
5.7 Results and discussion ........................................................................................................ 89
Chapter Six Conclusions ............................................................................................................... 93
vi
References ..................................................................................................................................... 96
Biographical Sketch ...................................................................................................................... 99
vii
LIST OF TABLES
Table 1.1 Notional Naval Railgun Parameters ............................................................................... 4
Table 2.1 Parameters for the solver .............................................................................................. 31
Table 3.1 Physical properties of water .......................................................................................... 48
Table 3.2 Grid refinement text ...................................................................................................... 53
Table 3.3 Energy analysis of the first cooling period (values correspond to half-rail) ................ 56
Table 4.1 Energy deposited in each shot and removed in each cooling period ............................ 72
Table 4.2 Energy deposited in each shot and removed in each cooling period ............................ 79
Table 5.1 Channel distribution parameters ................................................................................... 89
viii
LIST OF FIGURES
Figure 1.1 Basic configuration of a railgun .................................................................................... 2
Figure 1.2 Sketch of battle field ( Precision Strike Armaments Technology Fire Power Forum,
2008) ............................................................................................................................ 3
Figure 1.3 Energy distribution after a single shot (Smith et al., 2007) ........................................... 5
Figure 2.1 Energy distribution after a single shot (Smith et al., 2007) ......................................... 10
Figure 2.2 Projection view of rails and armature with current path (not in scale) ....................... 11
Figure 2.3 Electric conductivity distribution of the armature domain .......................................... 12
Figure 2.4 Computational domain and boundary index................................................................ 18
Figure 2.5 x-y cross section of the rails (Powell, 2008) ............................................................... 20
Figure 2.6 Magnetic potential contour at t=5s .............................................................................. 22
Figure 2.7 Rail temperature contour at t=5s ................................................................................. 24
Figure 2.8 Results of σair=1 [S/m]................................................................................................. 25
Figure 2.9 Copper properties ........................................................................................................ 27
Figure 2.10 Schematic of partial mesh of the rail and the armature region and the results
comparison ................................................................................................................. 29
Figure 2.11 Schematic of cross-section mesh of the rail and armature region and results
comparison ................................................................................................................. 30
Figure 2.12 3-D launch results ...................................................................................................... 33
Figure 2.13 Temperature distribution in a section of the rail. (Upper half: z>0, and 0< x<0.5m) 34
Figure 2.14 Temperature response after the first shot .................................................................. 35
Figure 3. 1 2-D rail geometry ....................................................................................................... 38
Figure 3.2 Current distribution for two cases ............................................................................... 39
Figure 3.3 Temperature distribution for two cases ....................................................................... 41
Figure 3.4 Axial cooling assemble................................................................................................ 42
ix
Figure 3.5 Comparison of internal edge temperature ................................................................... 44
Figure 3.6 Rail internal edge temperature for the no-channels arrangement ................................ 45
Figure 3.7 Cross-section temperature profile at x = 0.3m. (a) t = 9ms; (b) t = 5s ....................... 46
Figure 3. 8 Axial cooling channel assembles................................................................................ 49
Figure 3.9 Boundary and initial conditions................................................................................... 51
Figure 3.10 Grid refinement test ................................................................................................... 52
Figure 3.11 Internal edge temperature profiles ............................................................................. 54
Figure 3.12 Cross-section temperature distributions at x=0.3m, t=5s .......................................... 57
Figure 3.13 Cross section temperature distribution at x = 9m and t = 5s ..................................... 59
Figure 4.1 Computational domain for launch simulation in multi-shots scenario........................ 61
Figure 4.2 Computational domain for cooling analysis in multi-shots scenario .......................... 62
Figure 4.3 Partial mesh of rail, channels and armature................................................................. 65
Figure 4.4 Cross section mesh of rail and channels ...................................................................... 66
Figure 4.5 Temperature distribution of 3-centered case after the first shot .................................. 68
Figure 4.6 Internal edge temperature of 3-centered case after each shot...................................... 69
Figure 4.7 Internal edge temperature of 3-centered case after each cooling period ..................... 70
Figure 4.8 Increase of Tmax between every two shots and every two cooling period ................. 71
Figure 4.9 Energy deposited and removed for each launch and cooling period ........................... 72
Figure 4.10 Temperature profile of 3-centered case at x=0.8m, t=25s ......................................... 73
Figure 4.11 Temperature distribution of 3-eccentric case after the first shot ............................... 75
Figure 4.12 Internal edge temperature of 3-eccentric case for 5 shots ......................................... 76
Figure 4.13 Internal edge temperature of 3-eccentric case after 5 cooling period ....................... 77
Figure 4.14 Increase of Tmax between every two shots and every two cooling period ............... 78
Figure 4.15 Energy deposited and removed for each launch and cooling period ......................... 79
x
Figure 4.16 Temperature profile of 3-eccentric case at x=0.8m, t=25s ........................................ 80
Figure 4.17 Internal edge temperature of two cases after 1st, 3rd, and 5th shots ......................... 82
Figure 4.18 Internal edge temperature of two cases after 1st, 3rd, and 5th cooling periods ........ 83
Figure 4. 19 Energy deposited and removed for 3-centered and 3-eccentric cases ...................... 84
Figure 5.1 Layout of vertical cooling channels ............................................................................ 85
Figure 5.2 Partial mesh for vertical cooling analysis .................................................................... 87
Figure 5.3 Layout of vertical cooling channels ............................................................................ 88
Figure 5.4 Temperature field for case1 at t=5s ............................................................................. 89
Figure 5.5 Internal edge temperature profile ................................................................................ 90
Figure 5.6 Internal edge temperature profile for three cooling cases ........................................... 91
Figure 5.7 Temperature profile of the center-plane for the first channel of case 1 ...................... 92
xi
ABSTRACT
An electromagnetic launcher is a device used to accelerate projectiles at velocities
exceeding those attained with conventional propelling systems. A magnetic field is generated
when current is sent through two conductive rails that are connected by a moving armature. The
current that passes through the rails exerts an electromagnetic force on the armature and causes it
to accelerate to high speeds. In this work, a three-dimensional model for coupled thermal and
electromagnetic solution of the launch process is presented. The model accounts for the
determination of the current and temperature profiles inside the rails, the magnetic force, and the
projectile movement.
Focus is given to the thermal management after the shot. For a single shot scenario, five
different axial cooling arrangements, one without cooling and four with cooling channels, are
studied. A 50% reduction in the peak temperature was obtained with the inclusion of cooling
channels. For multiple-shot scenario, two selected cooling arrangements are studied and
compared. Results show that by moving the cooling channels towards the hotter corners, a better
cooling performance could be obtained. A vertical cooling arrangement is also considered. The
temperature distributions after the cooling period for three different channel configurations are
compared. A heat reversal phenomenon that observed for long rails is discussed.
These analysis and comparisons provide some directions to optimize the cooling design
of EML rail conductors.
Keywords: Electromagnetic launcher (EML), electromagnetic analysis, thermal analysis,
thermal management, numerical modeling
xii
CHAPTER ONE
INTRODUCTION
1.1 Introduction to Electromagnetic Launchers
Electromagnetic launch (EML) is the acceleration of an object by electromagnetic forces
along a guideway to initiate subsequent flight (Fair, 1982). Comparing to conventional chemical
propellant launches, it has three major advantages: 1) high launch velocity – 7km/s at small scale
and 9MJ at 2-3km/s (McNab, 2003); 2) long range and shot time of flight; 3) Less expensive
than rocket-fueled shells, missiles and any other long-range strike options. Their uses include
civil applications like launch to space (Jayawant et al., 2008, McNab, 2003) and military
applications such as Naval catapults (Li et al., 2009) and railguns (Powell, J. D. and Zielinski, A.
E., 2008, Smith, A. N. et al., 2005, Schneider, M., 2009).
The electromagnetic railgun is the simplest configuration of the electromagnetic
launchers. It typically consists of two parallel conductors (rails) and a movable sliding-contact
conductor (armature), see Fig. 1.1. The space between the two rails forms the barrel and the
projectile is attached to the armature. During launching, the driving current flows in from one
end of the first rail, and a magnetic field surrounding this rail will be generated according to this
current flow. The armature works as a “bridge”, enabling the current to enter the second rail and
back to the energy source. There is also a magnetic field generated by the current in the second
rail. These two magnetic fields have approximately the same direction (upright) in between these
two rails.
1
Figure 1.1 Basic configuration of a railgun
From Faraday’s law, we know that a conducting wire carrying a current ⃗ subjected to a
magnetic field with a magnetic field flux density �⃗ , experiences a Lorentz force defined by Eq.
(1.1). The direction of the Lorentz force exerted on the conductor is perpendicular to the
magnetic field and to the current in the armature. In railgun, the direction of the current and
Lorentz force exerted on the conductor (the armature) is shown in Fig. 1.1. It is the Lorentz force
that accelerates the armature and the projectile. If the current is high enough, the railgun has the
potential capability to achieve hypersonic muzzle velocities (Ghassemi and Barsi, 2005).
⃗ = ⃗ × �⃗
(1.1)
1.2 Design Parameters for a Long-range Naval Railgun
The intermediate Navy railgun technology goal (McFarland and McNab, 2003; McNab
and Beach, 2007) is to establish the capability for a multishot rate of up to 12 rounds per minute,
the range of 300km-500km, and notional muzzle energy of 32MJ at 2500m/s for ship defense.
The sketch representing these parameters in the battle fields is shown in Fig. 1.2.
2
Figure 1.2 Sketch of battle field ( Precision Strike Armaments Technology Fire Power Forum,
2008)
From the muzzle energy and velocity, we know the flight mass (projectile mass) is
approximated 10kg. And since the flight mass (projectile) will be approximately 50% of the
overall launch mass (Walls et al., 1999), the launch mass in this study is chosen to be 20kg. For
2500m/s muzzle velocity, barrel length of 8.2m would be enough (McNab et al., 2001). In this
study, the barrel is initially set to be 10m. The dimensions for the cross-section of the rails and
bore are chosen to be the same as in (Smith et al., 2005) - 135mm high, 60mm thick and 135mm
wide for rail separation. The cross-section dimension of the armature is set to be the same as that
of the square bore for simplicity, and the thickness is 0.2m. The materials for the rails and the
armature are pure copper and pure aluminum, respectively. All the parameters used in this study
are listed in Table 1.1.
3
Table 1.1 Notional Naval Railgun Parameters
Launch Mass
20 kg
Rail Length
10 m
Rail Height
0.135 m
Rail Thickness
0.06 m
Bore Size
0.135 m×0.135 m
Firing Rate
12 RPM
Previous studies (Bernards et al., 2003, Webb et al., 2007, Smith et al., 2007) showed that
for a conceptual pulsed-power system working with total energy loads larger than 155MJ,
approximated 26% of the total energy is dissipated as heat in the rails and armature. The
remaining energy dissipates among other system components are 39% as kinetic energy, 29% in
shunt, and ~16% in cables, capacitors, inductors, diodes, and bus. The energy distribution after a
single shot (Smith et al., 2007) for specific parameters is shown in Fig. 1.3.
4
Figure 1.3 Energy distribution after a single shot (Smith et al., 2007)
1.3 Research Objectives
For single shot, due to the skin depth of the current, most of the core region of the rails
will be at the original temperature (see Fig. 2.13) and could be the heat sink to absorb heat from
high temperature region. To keep a high firing rate during long periods of time, effective thermal
management system of rail conductors holds a predominant role among all the challenging
technical issues because: i) the large amounts of heat dissipated into the structures puts at risk the
integrity of the materials; ii) the pulsating high heat flux nature of the problem represents a
challenge to traditional cooling systems and iii) the extent of energy dissipated into the rails
opens the opportunity for waste heat recovery and system integration. Prior to designing an
effective thermal management system, an accurate thermal analysis to quantify the energy
deposited as heat for the rail conductors is required. So the objectives for this study are listed as
follows:
•
Simulate launch processes to determine the temperature distribution in the rail
conductors as a function of time;
5
•
Find an effective cooling scheme, to avoid localized high temperature spots and
effectively remove heat from the rails.
1.4 Literature Review
Beginning in the 1980s, investigations on railgun heating and cooling analysis appear in
the literature. Because of the experimental costs, especially for multi-shots, and because the
experimental data is very difficult to obtain, most of the work is concentrated on theoretical
modeling and numerical simulations.
In 1984, Kerrisk (Kerrisk, 1984) developed a two-dimensional model to calculate current
and thermal diffusion in cross-sections for railgun conductors under high-frequency current
situation. He developed another two-dimensional model (Kerrisk, 1986) for a rail cross section
near the breech end in 1986, to further investigate the acceptable correlation for the cooling
channel distributions, the current pulse frequency, and the coolant heat transfer coefficient.
Drake and Rathmann (Drake and Rathmann, 1986) presented a closed-form solution of
the Maxwell equations in two dimensions to describe the current diffusion in EML rail
conductors. They compared this 2-D solution with previously commonly used 1-D model
solution, and indicated that the thermal analysis should be modified by using this twodimensional model solution. They also indicated a challenge to the thermal management of
EML: the rail heats on “electromagnetic” time scale and cools on a much slower “thermal” scale.
Auton et al. (Auton et al., 1989) developed a 2-D finite-element method to calculate the
joule heating in rail conductors for arbitrary launcher geometries and arbitrary driving voltage
waveforms. Unlike the previous model, the magnetic vector potential was introduced to solve the
diffusion equations. The surrounding region of the conductors was also included in the
calculations.
In 1989, Nearing and Huerta (Nearing and Huerta, 1989) derived a closed-form,
asymptotic solution of the current density in a simplified 2D rail geometry (assuming infinite rail
height) with arbitrary armature motions, to study the skin heating effects of railgun currents.
They found out that the rail current may reverse when total current decreases because of the eddy
current effects. In the same year, Johnson and Bauer (Johnson and Bauer, 1989) developed a 1-D
model to calculate the current diffusion, and applied this model to 2-D geometry by introducing
6
an equivalent rail height. They quantified that 25% energy is deposited in the rails as heat for the
one case they conducted.
From the 1990s, due to the improvement of computer speed, more and more 2-D and 3-D
computational results have been published. In 1991, Roger et al. (Roger et al., 1991) developed a
new finite element formulation for modeling 3-D transient eddy currents with moving
conductors. They implemented this formulation for 2-D and 3-D electromagnetic field analysis.
In the same year, Liu (Liu, 1991) published a 3-D rail cooling analysis for continuous shots with
20s intervals. The current diffusion model used by Liu to analyze the resistive heating during
current pulses was a 2-D model which was reported by Wu and Sun (Wu and Sun, 1989). Liu
found out that the heat reversal from coolant to rail will occur at the rear part of the cooling
channel if the coolant direction is from the breech towards the muzzle. He also reported that this
phenomenon can be eliminated if the coolant direction is reversed (from muzzle to breech) or if
shorter cooling channels were used. His thermal analysis of actively cooled rail under four
consecutive shots showed that energy deposition in the rail, heat removal from the rail, and
highest temperature can be stabilized after several shots.
In 1995, a Lagrangian formulation for coupled mechanical, thermal, and electromagnetic
diffusive process with moving conductors was presented by Hsieh (Hsieh, 1995). The
electromagnetic part of his formulation is based on quasi-static Maxwell’s equations in terms of
magnetic vector potential and electric scalar potential. The finite element code EMAP3D
(Electro-Mechanical Analysis Program in 3 Dimension) to solve these equations was
implemented based on this formulation.
An experimental study of the thermal management of sequentially fired small scale
railgun (30mm bore) was performed by the Air Force High Energy Railgun Integration
Demonstration (HERID) program. Jamison and Petresky (Jamison and Petresky, 1995) presented
the results. A simplified thermal time constant was determined in their study, and the results
from this time constant and experimental data are compared.
In 1999, Fish et al. (Fish et al., 1999) performed a 2-D finite element analysis for heat
dissipation between multi-shots for rails with and without cooling. In their study, heat diffusion
along the length of the rail was neglected because the temperature gradient is much lower in this
direction. The results of temperature history after first shot for no cooling case were compared to
experimental data. Among three selected positions, the one close to the corner showed an
7
obvious inconsistency and the other two inner points matched well. In addition, both temperature
dependent and constant rail properties were considered. The results showed that temperature
dependent case had an increased energy deposition.
In 2003, Satapathy and Persad (Satapathy and Persad, 2003) evaluated the thermal
stresses in rails with thin resistive overlays. 1-D coupled electromagnetic diffusion equation and
thermal diffusion equation were used to solve the electromagnetic field diffusion into the rail
structure and temperature distribution. The effects of different cooling channel locations- a) close
to the front, b) at the center of the rail, and c) at the back of the rail, on the temperature
distribution were compared. The numerical results showed that the central location cooling
channel enables faster heat removal and a more even temperature distribution in most of the
periods between firing, even though at very early time (a time scale much smaller than duration
between launches) is ineffective.
Smith et al. (Smith et al., 2005) presented a quasi-1-D model to calculate the penetration
of the magnetic field into the conductors, and then calculate the resistive heating and temperature
response. The effect of the cooling channels at the centerline of the rails was also investigated.
One-dimension thermal diffusion perpendicular to the rail surface was assumed to simplify the
calculation. For the chosen example, moving the cooling channels closer to the surface would
have better effects than the centerline channels.
In 2007, a 3-D numerical study of the first shot that included the armature motion was
conducted by Vanicek and Satapathy (Vanicek and Satapathy, 2005). Software package
EMAP3D was utilized to perform the calculations. The data of current, armature velocity and
position during the shot were compared between the numerical results and the actual shot data
from IAT’s (Institute of Advanced Technology) Medium Caliber Launcher. The numerical
results showed a very good correlation with the lab data. They found that about 80% of the heat
energy in the rails is within the first 2-m of the 7-m rails for their particular case, and thus
suggested that a very short section close to the breech needs to be cooled. The ratio between heat
energy stored after the shot and kinetic energy was also quantified.
Currently a systematic study for thermal analysis and thermal management system under
fully coupled 3-D electromagnetic, thermal, fluid mechanics and mechanical equations is
lacking. In this dissertation, the first step is to develop a 3-D transient thermal analysis for
obtaining the details of the launching process especially transient temperature distribution. After
8
this, the second step is focused on the active cooling process. The impact of the cooling channel
locations and the cooling channel directions on the cooling performance were studied for single
shot and multiple shots.
9
CHAPTER TWO
THERMAL ANALYSIS
In this chapter, a three-dimensional transient electromagnetic and thermal analysis is
conducted on a 10m long railgun. The temperature distribution in the rails is determined. The
railgun parameters are taken from those in Naval railgun design requirements (Table 1.1).
2.1 Computational Model
A sketch of the problem geometry is shown in Fig. 2.1. The launcher is represented by
two parallel rails and a moving conductor- the projectile, which is attached to the armature and
placed in the gap of the two rails. In this study, the set of the armature and projectile will be
referenced as armature only. The surrounding air is also included to account for the outside
magnetic field.
Figure 2.1 Energy distribution after a single shot (Smith et al., 2007)
10
In this configuration, the current flows in through the lower rail, passes the armature, and
flows out through the upper rail. Figure 2.2 is the projection views of the two rails and the
armature in horizontal and vertical planes with the current flow direction shown in it.
Current
0.06m
0.2m
y
Armature
0.135m
z
Figure 2.2 Projection view of rails and armature with current path (not in scale)
The dimensions of the rails and the armature are 10m x 0.06m x 0.135m and 0.2m x
0.135m x 0.135m (x, y, and z directions), respectively. The radius of the surrounding air cylinder
is 0.5m, and its length of this cylinder is equal to the length of the rail. Due to the symmetry of
this configuration (Hsieh, 1995), the computational domain is defined as the top-right quarter of
the geometry as shown in Fig. 2.1.
The armature is not modeled as a separate region but as a moving conductor distribution
that is interpolated on the fixed mesh. Figure 2.3a shows a sketch of the centerline between the
two rails along the rail length direction. The conductivity along this centerline is shown in Fig.
2.3b. The variable xa is the center position of the armature. The region with high conductivity
represents the armature; and the other regions the air. The width of this high conductivity
centering on xa represents the thickness of the armature. On the interface between the armature
11
and the air, the conductivity is taken the form of a smoothed Heaviside function to avoid infinite
spatial derivatives. When the armature is moving forward, its position variable xa will change
and the high conductivity region moves forward accordingly. Mathematically, this moving
conductivity distribution represents exactly the same movement of the armature.
(a) Sketch of the centerline of the armature
(b) Electric conductivity along the centerline
Figure 2.3 Electric conductivity distribution of the armature domain
2.2 Formalism
To model the electromagnetic launcher with moving conductors, three fields should be
involved and combined together. They are electromagnetics, rigid body motion, and thermal
diffusion in the conductors.
The electromagnetic problem is described by the Maxwell’s equations. Their differential
forms are given by Ampere’s law
�⃗ = ⃗ +
∇×�
� �⃗
�
(2.1)
12
Faraday’s law
�⃗ = −
∇×E
Gauss’s law
�⃗
∂B
∂t
(2.2)
��⃗ =
∇∙D
(2.3)
�⃗ = 0
∇∙B
(2.4)
and Gauss’s law for magnetism
�⃗ is the magnetic field intensity [A/m], ⃗J is the current density [A/m2], �D
�⃗ is the electric
where �H
flux density [C/m2], �E⃗ is the electric field intensity [V/m], �B⃗ is the magnetic flux density
[Wb/m2], t is the time [s] and
is the volume charge density [C/m3].
Other equations used in the present analysis are the charge continuity equation
∇ ∙ ⃗J = −
∂
∂t
(2.5)
and the constitutive equations given by
�B⃗ = µH
��⃗
(2.6)
⃗J = σE
�⃗
(2.8)
ɛ = ɛ ɛr
(2.10)
��⃗ = εE
�⃗
D
(2.7)
µ = µ µr
(2.9)
where µ is the permeability of the material [H/m], µ the permeability of free space [4 ×
10− H/m ], µr the relative permeability, ε the permittivity of the dielectric [F/m], ɛ the
permittivity of free space [8.854×10-12 F/m], ɛr the relative permittivity and σ the electric
conductivity [S/m].
13
The electrical conductivity of air is ~10-15S/m, however in this study it was set to σair = 1
S/m. This simplification was made to speed up convergence of the numerical solution.
Comparison of σair = 0 and σair = 1 S/m solutions for a 2-D problem was shown in section 2.4
and has resulted in variations of magnetic and temperature distribution smaller than 0.05%. The
electrical conductivity for the armature is σaluminum = 3.77 x 107 S/m and a function of the
temperature for the rail material copper (Eq. (2.20)).
In this study, all materials used for the launcher are assumed to be isotropic and nonferromagnetic. So the relative magnetic permeability is set to be 1 for all materials. Equation
(2.6) is simplified to take the form in Eq. (2.11).
�⃗ =
�⃗
�
(2.11)
Assuming that the dimensions of the studied launcher geometries are considerably
smaller than the electromagnetic wavelength (quasi-static approximation), the displacement
current term � ⁄� in Eq.(2.1) can be neglected (Cheng, 1991). Thus the Ampere’s law can be
simplified to be of the form
�⃗ = ⃗
∇×�
(2.12)
By taking time derivatives on both side of Eq. (2.3) and applying the “quasi-static
approximation”, we can get
�
=0
�
(2.13)
∇∙ ⃗= 0
(2.14)
Then we can get Eq. (2.14) by the combination of Eq. (2.5) and Eq. (2.13).

Let A and V be the magnetic vector potential [Wb/m] and electric scalar potential [V],
respectively. Their relations with magnetic flux density �⃗ and electric field intensity �⃗ are shown
in (2.15) and (2.16) (Ulaby, 2001),
�⃗ = −∇V −
E
�⃗
∂A
∂t
(2.15)
14
�⃗ = ∇ × �A⃗
B
(2.16)
By combining Eqs. (2.8), (2.11), (2.12), and (2.15) and (2.16) together, the governing
equation for the magnetic vector potential has the form:
�
�⃗ 1
+ �∇ × ∇ × ⃗� + �∇ = 0
�
(2.17)
The quasi-static approximation also implies that the changes in time are slowly
(COMSOL Multiphysics, 2005). Thus, a scalar potential equation can be obtained by the
combination of Eqs. (2.8), (2.14), (2.15) and the hypothesis that � ⃗⁄� ~0. The final form for
this equation is given by
∇ ∙ (�∇ ) = 0
(2.18)
Equations (2.15) and (2.17) form the governing equations for electromagnetic fields. To
determine a unique magnetic vector potential ⃗ the Coulomb gauge constraint ∇ ∙ ⃗ = 0 is
imposed (COMSOL Multiphysics, 2005).
In this study, the heat generation is only considered in the rails and only due to the
electrical resistivity. The friction between the armature and the rails, the air resistance, and the
heat generation in the armature are all neglected. Based on these simplifications, the energy
equation, which is solved only for the rails, can be written as:
⃗∙ ⃗
�
= ∇ ∙ (�∇ ) +
�
�
⃗
��
where ⃗ = � �−∇ − �,
�
(2.19)
is the density [kg/m3]; CP is the specific heat [kJ/ (kg∙K)]; k
is the thermal conductivity [W/(m∙K)]; T is temperature [K].
In Eq. (2.19), cp, σ, and k are properties of copper and temperature dependent, where σ is
given by Eq. (2.20) (Cheng, 1989), k and cp are interpolated from thermodynamic tables
�=
5.64 × 10
1 + 0.0039 × ( − 300)
(2.20)
15
� = 424.26667 − 0.072286
(2.21)
= 342.5333 + 0.144
(2.22)
where the density is set to be constant and equal to 8933 kg/m3 (cooper) (Frank, 2001).
The set of equations for coupled electromagnetic and thermal problems are given by Eqs.
(2.17), (2.18), and (2.19). They are summarized in Eq. (2.23).
�⃗ 1
+ �∇ × ∇ × ⃗� + �∇ = 0 ( )
�
( )
∇ ∙ �∇ = 0
⃗∙ ⃗
�
= ∇ ∙ (�∇ ) +
( )
�
�
�
(2.23)
The commercial software COMSOL (COMSOL Multiphysics, 2005) is used to solve
simultaneously the system of differential equations given by Eq. (2.23), auxiliary equations (Eqs.
(2.20), (2.21), and (2.22)) and boundary conditions (Eqs. (2.28-2.33)). The dependent variables
to be solved are ⃗, V and T.
After obtaining the solution of Eq. (2.23) it is possible to determine the position and
velocity of the armature during the launch. For the armature movement, rigid body motion
restricted to x direction is assumed. The armature is accelerated through the rails due to the
Lorentz force (Eq. (2.24)) and its position (xa) is determined by Newton's second Law (Eq.
(2.25)).
�F⃗ = � ⃗J × �B⃗dv
(2.24)
∀
�
=
(2.25)
In Eqs. (2.24) and (2.25), ⃗ is the Lorentz force [N], m the mass of the armature and
projectile combined [kg], xa the armature position in the length of the rail and ∀the armature
volume [m3].
16
Combining Eqs. (2.24) and (2.25), the position and velocity of the armature (plus
projectile) during the launch are given by
�
�
=
�⃗dv
= �⃗ ∙ � ⃗J × B
(2.26)
∀
�
(2.27)
2.3 Boundary and initial conditions
For the magnetic vector potential ⃗ (Eq. (2.23a)), the magnetic insulation boundary
condition (Eq. (2.28)) is applied to the boundaries confining the region with air (surfaces 1 to 3
in Fig. 2.4), the symmetry surface where the magnetic field is known to be tangential to the
boundary (Cheng, 1989) (surface 5 and 9), and the surfaces which are interfaces between the
conductor and the air (surfaces 6 and 7). This boundary condition sets the tangential component
of the magnetic potential to zero (COMSOL Multiphysics, 2005). For the symmetry plane, which
separates the rail in two parts (surface 4 and 8), the electric insulation boundary condition (Eq.
(2.29)) is applied. This boundary condition sets the normal component of the electric current to
zero (COMSOL Multiphysics, 2005).
17
5 2
z
x
Internal
rail edge
3
1
9
y
7
8
4
6
Figure 2.4 Computational domain and boundary index
�⃗ × ⃗ = 0
(2.28)
�⃗ = 0
�⃗ × �
(2.29)
For the electric scalar potential (Eq. (2.23b)), a prescribed voltage (Eq. (2.30)) is applied
to the current inlet boundary (surface 6). Ground (Eq. (2.31)) is applied to the armature
symmetry boundary (surface 9). For all other boundaries the current insulation (Eq. (2.32)) is
applied.
= ( )
(2.30)
=0
(2.31)
�⃗ × ⃗ = 0
(2.32)
18
where �⃗ is the outward normal direction vector.
For the energy equation (Eq. (2.23c)), adiabatic boundary condition (Eq. (2.33)) is set to
all rail exterior boundary surfaces and symmetry surface. Due to the short launch period (less
than 10 ms), it was assumed that it is possible to neglect the heat convection on the exterior
boundaries and consider the worst case condition where all generated heat remains inside the
rails.
�⃗ ∙ ∇ = 0
(2.33)
For the electromagnetic problem, zero voltage (V = 0V) and zero magnetic field ( ⃗ =
0 Wb ⁄m) are specified for all regions. Constant initial temperature (T = 300K) is specified in the
rails for the thermal problem. The armature’s position is
�
= 0.2
and its velocity is 0m/s.
2.4 2-D model verification
Due to the lack of full detailed 3-D numerical or experimental results, the current model
is first compared with a two-dimensional study investigated by Powell and Zielinski (Powell,
2008). To make a convenient comparison, in this section, the coordinate system is changed to be
the same as that in (Powell, 2008), which is different from that in section 2.1.
The two-dimensional computational domain in (Powell, 2008) is shown in Fig. 2.5. The
two assumptions taken in (Powell, 2008) are: (a) ⃗ = ⃗�, �⃗ = �⃗ �, ⃗ = ⃗; and (b) all variables
depend only on coordinate x and y and not on z.
19
Figure 2.5 x-y cross section of the rails (Powell, 2008)
The governing equations in (Powell, 2008) are shown in Eqs. (2.34) and (2.35), where ∅
is the scalar electric potential, the same as V in Eq. (2.23b); e is the specific internal energy.
�
�⃗ 1
1
= ∇ ⃗ − ∇ � � × ∇ × ⃗ + �� ⃗ × ∇ × ⃗� − �∇∅
�
�
+
�
�
+
�
�
�
=�
�
�
+�
�
�
+
�� �
�� �
+
+
� �
� �
�
(2.34)
(2.35)
In this two-dimensional model, the armature’s motion is not considered, so the term
�� ⃗ × ∇ × ⃗�is actually not presented. By removing this term and expanding the cross-product
term in Eq. (2.23a), the two equations are the same but in different expressions. For the same
reason, variables u and w in Eq. (2.33) are zero; so Eq. (2.33) is a 2-D expression of Eq. (2.23c).
20
Also because of the 2-D nature and based on the two assumptions, ∅ can only be the
function of time. So Eq. (2.23b) is not solved, and an appropriate specification of the term �∇∅
is needed to determine ⃗.
Now we can see that by setting the identical boundary conditions expressed in (Powell,
2008), we can verify our model in 2-D with the results from (Powell, 2008). Due to the two
symmetry lines, only one quarter (left-top) of the total geometry in Fig. 2.5 is modeled. Figures
2.6a and 2.7a were taken from (Powell, 2008) and Figs. 2.6b and 2.7b are our 2-D results. These
figures validate, in a qualitative way, the solution obtained with the present model. A
quantitative validation was not possible since only graphical information is available in (Powell,
2008).
Figure 2.6a shows the solution (including the half rail and the surrounding air) obtained
in (Powell, 2008) for the magnetic potential. This figure is compared with Fig. 2.6b obtained
with the current solution. A qualitative analysis shows that the gradients in both figures are of the
same order of magnitude and that the maximum values of the magnetic potential are also in good
agreement.
21
(a) Results in (Powell, 2008)
(b) Current model results
Figure 2.6 Magnetic potential contour at t=5s
22
In Figs. 2.7a and 2.7b the temperature fields are compared (only the half rail). In this
comparison, the important features to be analyzed are the maximum temperature (365K in both
solutions) and the two hot spots on the right and left-top corners of the rail.
(a) Results in (Powell, 2008)
23
(b) Current model results
Figure 2.7 Rail temperature contour at t=5 s
The impact of air electric conductivity on the magnetic field and temperature response
within the current 3-D study is also evaluated in this 2-D model. The previous results shown in
Figs. 2.6b and 2.7b are obtained by setting the �� = 0S/m. The results of the temperature and
magnetic potential shown in Fig. 2.8 are obtained by setting the �� = 1S/m and keep all the
other parameters the same. The comparison between Figs. 2.8a and 2.6b shows that the magnetic
potential has the same magnitude and the same distribution in the two cases. The same
conclusion for temperature can be made through the comparison between Fig. 2.8b and 2.7b. So
in the 3-D thermal analysis the air electric conductivity is set to be 1S/m instead of its exact
value of 10-15S/m. By doing so, the convergence speed can be greatly improved and the
computation time is greatly reduced.
24
(a) Magnetic potential
(b) Temperature
Figure 2.8 Results of σair=1 [S/m]
25
2.5 3-D first shot simulation
In the simulation of the 3-D real launch process, by adjusting the boundary condition
variable V(t), different launch processes could be obtained. The chosen V(t) to satisfy the design
requirements listed in section 1.2 is shown in Fig. 2.12a in subsection 2.5.3. Among the
simulation results, the temperature distribution after the launch process is our initial temperature
for the cooling analysis.
2.5.1 Material properties
The materials used for the rails and the armature are pure cooper and pure aluminum
respectively. The properties used in the simulation are addressed in section 2.2. The temperaturedependent copper properties are calculated and interpolated, and are shown in Fig. 2.9.
(a) Electric conductivity
26
(b) Thermal conductivity
(c) Specific heat
Figure 2.9 Copper properties
27
2.5.2 Mesh convergence
Since the armature is modeled as moving conductor distribution, the grid element
resolution in the moving direction (x direction) is much more important comparing to that in the
other two directions. Figure 2.10a shows a partial mesh for the rail and the armature. To keep up
with a sufficiently accurate solution, a comparison of the rail internal edge temperature between
two 3-meter cases of different element resolutions in the x direction is conducted and shown in
Fig. 2.10b.
(a) Partial mesh
28
(b) Internal edge temperature for two cases
Figure 2.10 Schematic of partial mesh of the rail and the armature region and the results
comparison
The two element resolutions are 6.67mm (450 elements) and 5mm (600 elements) in x
direction and the other mesh parameters are kept the same. In Fig. 2.9b, we can see that along the
rail internal edge, the biggest discrepancy occurs at the same position as where the maximum is.
The maximum temperatures for these two cases are 1352.85K and 1343.46K respectively. The
discrepancy is
.
.
−
.
−
= 0.89%, which less than one percent. Based on this comparison,
the 6.67mm case is adopted in this study.
The cross-section mesh resolution is another factor that should be checked. The crosssection mesh for the rail (red part) and the armature region (grey part) is shown in Fig. 2.11a.
The symbols a1, a2 and a3 represent number of the elements in the three edges. Two cases are
compared – “8×8×2” (a1×a2×a3) and “10×10×3”. Figure 2.11b shows this comparison. The
discrepancy of the maximum temperature for the two cases is
29
. −
. −
.
= 2.6% . This
discrepancy is a bit higher than the normally acceptable value 1%. But in this 3-D study, under
the available 12GB memory, this is the best grid we can afford for the 10m-length model.
(a) Cross-section mesh
(b) Internal edge temperature for two cases
Figure 2.11 Schematic of cross-section mesh of the rail and armature region and results
comparison
30
After the two comparisons, the mesh resolution adopted for the rail and armature of the
10m model is 8×8×2×1500 (6.67mm resolution in the x coordinate). The total mesh elements
including the mesh in the surrounding air are 279000.
2.5.3 Time step and solver parameters
The time step determines the discrete spatial jump size of the armature; and the position
of the armature determines the current. The current composes the source term in energy
generation. So the time step is very important for the temperature response. It is obvious that the
temporal resolution will increase if the time step is decreased. Another crucial thing that should
be considered is the total computation time. To keep the computation in a reasonable amount of
time as well as sufficiently accurate results, in this study, we fixed the tolerance and maximum
time step and let the solver to determine the time step for each iteration. These solver parameters
are listed in Table 2.1. Under these parameters, the total amount of calculation time is about 2-3
days for one shot process.
Table 2.1 Parameters for the solver

A (Wb/m)
10 −7
V (V)
10 −2
T (K)
10 −2
x0 (m)
10 −2
Absolute tolerance
Relative tolerance
10 −2
Maximum time step (s)
1 × 10 −5
31
2.5.4 Launch data
The chosen voltage, resulted current, and armature position and velocity are shown in
Figs. 2.12a and 2.12b. In Fig. 2.12a, the voltage corresponds to the boundary condition Eq.
(2.30) which is applied to the current inlet.
(a) Inlet voltage and current
32
(b) Armature’s position and velocity
Figure 2.12 3-D launch results
From Fig. 2.12b we can see that the launch period duration is about 9ms and the muzzle
velocity is about 2200m/s, which satisfies the muzzle velocity for a long-range railgun
mentioned in section 1.2. In Fig. 2.12a, the inlet current is the integral of the current density on
the half rail cross-section; so it is the half value of the total inlet current. The total inlet current
has its maximum value of 6.8 MA and its average value 4.38 MA.
The temperature response after the shot for the first 0.5m of the half rail is shown in Fig.
2.13. Due to the skin effect, the current concentrates on the two edges and the adjacent regions,
especially the internal edge (see Fig. 2.4). Consequently, the heat generation concentrates on the
same regions, and there is almost no heat generation in the center region. During the short launch
period, there is no enough time for the heat to diffuse into the center region, resulting in a highly
non-uniform temperature distribution on the rail cross section. Along the rail length direction, the
temperature distribution is also non-uniform.
33
Figure 2.13 Temperature distribution in a section of the rail. (Upper half: z>0, and 0< x<0.5m)
Based on the two non-uniform features of the temperature distribution, let’s look at the
temperature on the cross section at x=0.3m and along the internal edge in Fig. 2.14.
34
a) Cross-section temperature at x=0.3m
b) Temperature along the rail internal edge
Figure 2.14 Temperature response after the first shot
35
In Fig. 2.14b, along the internal edge, the temperature increases fast close to the breech
end, reaches the highest value at x=0.3, and remains above 900K along most of the edge length.
The peak value does not occur at the inlet because the initial position of the armature is slightly
away from the inlet (x0=0.2m). Figure 2.14a shows the cross section temperature distribution at
x=0.3m. As can be seen in Fig. 2.14b, the maximum temperature exceeds 1200K. The hottest
temperature is close to the melting point of copper (1358K) and should be avoided in an EML
design. It is important to keep in mind that the results correspond to the worst case scenario as
the rail has been treated as insulated from the surrounding area.
36
CHAPTER THREE
AXIAL COOLING FOR SINGLE SHOT
3.1 Introduction
In this study, the thermal management is focused on the active cooling method that is the
use of water cooling channels passing through the rails. The main goal is to propose a strategy to
lower the maximum temperature and remove as much deposited energy as possible. The
dimensions and locations of the cooling channels and the mass flow rate are the primary
parameters to be investigated.
From Fig. 2.14a, it can be seen that the highest temperature profile along the rail length
direction occurs on the internal edges of the rail. The core of the rails is still at initial
temperature. Based on this situation, in the first investigation, the thermal analysis for the single
shot was performed by first assuming a solid rail and solving the electromagnetic and thermal
problem in a coupled way during the shot period (~9ms). After that, for the next 5s before the
following shot (the required 12RPM in Table 1.1), the cooling channels were added to the rails.
That is assuming the cooling channels’ existence in the rails during the launch process have
trivial effect on the temperature response. The 2-D model in section 2.4 was used to evaluate this
effect. Two cases were compared- one is solid rail, the other has 3 channels in the rail. The two
geometries are shown in Fig.3.1.
37
(a) Solid rail
(b) Solid rail and 3 channels
Figure 3. 1 2-D rail geometry
The dimensions of these two geometries are changed to be the same as those in the Fig.
2.2. The physical settings here are all kept the same as those in section 2.4 except for the input
current, which is adjusted to make the temperature response at the same order of magnitude with
the 3-D result. The current distributions for the two cases are shown in Fig.3.2. To compare them
easily, the contour levels are set to be the same for the two figures.
38
(a) Solid rail
(b) Rail with 3 channels
Figure 3.2 Current distribution for two cases
39
The current concentrates on the corner regions in both cases. Both the distribution and the
value are the same. At the core part, the current distribution is different in the channels and the
adjacent regions. However, the order of the magnitude of the current in the core part is very
small comparing to that in the corner regions. So essentially, in the whole rail the existence of
the channels does not affect the current distribution. Since the current corresponds to the heat
generation term in the energy equation, it can be expected that the temperature distribution
should have trivial difference in the rail for these two cases.
The temperature distribution for the two cases is shown in Fig. 3.3. As was expected, the
core part is still at the initial temperature for the two cases. The maximum temperature is slightly
different in the two cases,
. −
. −
.
.
= 1.05%. So the impact of the existence of the channels
on the temperature field is mostly less than 1%.
The comparisons of the current distribution and temperature response prove that it is
appropriate to use solid rail in the launch process and add channels into the rails in the following
cooling analysis. With this simplification it was possible to reduce the calculation time and get
the results much faster. And during cooling period, the remnant magnetic fields are also
neglected. So only the thermal problem was solved in this period (from 9ms to 5s).
40
(a) Solid rail
(b) Rail with 3 channels
Figure 3.3 Temperature distribution for two cases
41
This insertion approach will not be applied to the cooling analysis for the following shots
in the next chapter. Because the electric conductivity is temperature dependent and diminish with
the increasing temperature (Fig. 2.9a), according to Eq. (3.1), the skin depth will become larger
due to the smaller electric conductivity. And with the larger skin depth in the following shots, the
temperature of core part will increase more than the first shot.
� =
1
�
(3.1)
�
In axial cooling assemble, the layout of the cooling channels are chosen to be symmetric
about the horizontal midplane of the rail (see Fig. 3.4). Then the midplane is also the symmetry
plane for the temperature field. Due to symmetry, only the top half of one rail (including the
channels) is modeled and calculated in axial cooling assemble.
Figure 3.4 Axial cooling assemble
42
In the following sections five cooling assembles are considered: one with no cooling
channels and four with axial channels in different configurations. The mechanical strength is
assumed to be satisfied for all cases. So the mechanical strength analysis will not be included.
3.2 No channels arrangement
This is the simplest possible configuration and it is a comparison to the cooling assembles.
For the first shot, the internal region can be used as a heat sink where the generated heat can
diffuse. The governing equation for this arrangement is the heat conduction equation given by,
�
= ∇ ∙ (�∇ )
�
(3.2)
The thermal boundary conditions are set to be adiabatic for all solid boundaries to
consider the worst case situation in the cooling period. Under this boundary condition and
without the cooling channels, the deposited energy will remain in the rails. The comparison of
internal edge temperature for with- and without free convection is shown in Fig. 3.5. The free
convection heat transfer coefficient is chosen to be a typical vale of 20[W/(m2∙K)]. In this plot, it
can be seen that the two lines nearly coincide with each other and it illustrates that for the short
cooling period after the first shot, the effect of free convection on boundary surface is trivial. For
the same reason, the adiabatic boundary condition will be set on all exterior solid surfaces for
with-channel cases. The additional benefit to neglect the exterior convection effects is that we
can concentrate wholly on comparing the effectiveness of different layouts of the cooling
channels, which are the only exit for the deposited heat.
43
Figure 3.5 Comparison of internal edge temperature
The temperature along the internal rail edge, where the peak temperatures concentrate, is
shown in Fig. 3.6 at different times (0s, ~9ms and 5s). As can be seen in Fig. 3.4, during the 5s,
the period between the first and the second shot, the heat diffuses to the center of the rail and the
temperature in the internal rail edge drops to values below 400K. This comparison shows that
due to the high temperature gradient caused by highly non-uniform temperature distribution after
the first shot, the stored energy can dissipate fast into the internal regions of the rail.
44
Figure 3.6 Rail internal edge temperature for the no-channels arrangement
The hottest position shifted slightly towards the breech after the shot. Initially (t = ~9ms),
the peak temperature is located at x = 0.3m. The cross-section temperature profiles are plotted at
this location (x = 0.3m) for t = 9ms (Fig. 3.7a) and t = 5s (Fig. 3.7b). The high temperature
gradient observed at t = ~9ms becomes a much more homogeneous distribution at t = 5s. The
hottest position (left top corner) remains the same, however the temperature at this point drops
from 1281K to 367K during the 5s period. As can be seen from Fig. 3.7, no cooling is necessary
for a one shot scenario.
45
(a)
(b)
Figure 3.7 Cross-section temperature profile at x = 0.3 m. (a) t = 9 ms; (b) t = 5 s
46
In a multiple shot scenario, if the energy generated during the launch period is not
effectively removed, the rail temperature will start increasing and quickly will reach values
higher than the material melting point. For this reason an efficient cooling strategy must be
added to the system.
3.3 Axial cooling channel assemble
Starting from the common solution for the first shot (0 < t < 9ms) used in the previous
case, the flow and energy equations are solved for cooling period (9ms < t < 5s) in a geometry
that includes only the rails with the channels. The initial temperature field is interpolated into the
mesh corresponding to this domain.
3.3.1 Flow parameters and channel layout
In the following studies in this chapter, the channels are aligned in the rail length
direction. Single phase liquid cooling is assumed with water chosen as the coolant. The mass
flow rate is estimated by
ṁ =
�
,
�
(3.3)
�
where the Etotal = 2.562 MJ is the energy deposited in the half rail during the first shot, ∆t = 5s,
and cp,w is given in Table 3.1. The coolant temperature change is assumed to be ∆T = 40K. With
these values, the calculated mass flow rate is approximated to 3 kg/s.
47
Table 3.1 Physical properties of water
Density
997 kg/m3
Specific heat
4200 J/(kg∙K)
Thermal conductivity k
0.6 W/(m∙K)
Kinematic viscosity
0.001003 Pa∙s
Prandtl number Pr
7.437
Four cooling assembles shown in Fig. 3.8 were studied. The average coolant velocity for
all cases is kept constant and equal to 6.4m/s and the cross section area of the channels changes
to keep the same total mass flow rate in all simulations.
The four cases can be grouped by two characteristics – channel location and number of
channels. In cases a and b, the channels are aligned along the rail’s mid-plane in y-direction; so
they are named “centered”. In cases c and d, at least one of the channels is not in the rail’s midplane so they are named “eccentric” accordingly. The prefix number represents the number of
channels. The detailed dimensions are shown in Fig. 11. The Reynolds number can now be
calculated as 1.12 x 105 and 1.37 x 105, respectively for the two-channel and three-channel
assembles.
48
case a. 3-centered
case b. 2-centered
case c. 3-eccectric
case d: 2-eccentric
Figure 3. 8 Axial cooling channel assembles
49
3.3.2 Governing equations
The coupled heat and fluid flow problem was solved with the software FLUENT (Ansys,
2009). The flow is assumed incompressible and turbulent (Re~105) with constant physical
properties (density and viscosity). The turbulence is solved with the κ -ε model. The set of
equations for the problem is given by the continuity equation
∂
∂
�U⃗) = 0
+∇∙(
(3.4)
is the water density, �⃗ is the mean velocity vector,
where
The Reynolds-averaged Navier-Stokes equation
�(
�
�⃗)
�⃗ �⃗� = −∇� −
+∇∙�
2
T
�⃗ + �∇U
�⃗� � − ∇ ∙ (
∇�∇ ∙ �U⃗� + µ∇ ∙ �∇U
3
where P is the pressure [Pa], µ the absolute viscosity [Pa s], and
′
��������
u′ ⨂u′ )
(3.5)
the fluctuating velocity
vector, the last item in the right side of Eq. (3.5) is the Reynolds stresses.
The turbulent kinetic energy equation
�(
�
�)
+∇∙�
� �⃗� = ∇ ∙ �� +
�
� ∇�� +
1
2
�
�∇ �⃗ + �∇ �⃗� � −
�
(3.6)
where κ is the turbulent kinetic energy [J], ε the rate of dissipation of turbulent kinetic energy
[J/s],
is the rate-of-deformation tensor,
=
is the turbulent viscosity defined by
The turbulent dissipation equation
�(
�
�)
where
+∇∙�
�, �
∂(
, �� ,
,
∂
� �⃗� = ∇ ∙ �� +
�,
)
� are
��
� ∇ε� +
1
2
�
�
�
�
�∇ �⃗ + �∇ �⃗� � −
�
�
�
� �
.
(3.7)
adjustable constants, and the energy equation for the coolant flow
+ �⃗ ⋅ ∇(
,
) = ∇ ⋅ �� +
,
��
∇ �
(3.8)
where Prt is the turbulent Prandtl number (Ansys, 2009). The energy equation for the solid
50
copper is given by Eq.(3.3) in previous subsection. The physical properties used in this work
were given in table 3.1.
3.3.3 Boundary and initial conditions
For the fluid flow inside the channels the boundary conditions are prescribed mass flow
rate Eq. (3.3) at the inlet section and fully developed flow at the outlet section. No slip condition
is assumed on the channels walls.
The boundary conditions for the energy equation (Eq. (3.8)) in the flow direction (x axis)
are set as prescribed temperature (T=300 K) at the inlet section and zero derivative (∂ / ∂ = 0)
at the outlet section. The advection heat flux in the fluid equal to conduction heat flux in the solid
boundary is applied at the pipe walls.
u
∂T
=0
∂x
T=300K
u= u0
Figure 3.9 Boundary and initial conditions
51
3.3.4 Mesh convergence
Figure 3.10 illustrates the rail mesh used during the cooling period. The mesh includes
now the cooling channels and it is built in two steps: i) a mesh is created in the breech section
and ii) the mesh is swept along the x axis. Due to the higher temperature gradient in the region
close to the internal edge, the cross section distribution of the mesh elements is non-uniform.
Actually, two different domains exist: solid and fluid. Quadrilateral elements, excepted for the
center elements, are used for the channel while triangular elements are used for the solid region.
The grid is more refined in the left superior corner (internal edge) and close to the channel wall.
Figure 3.10 Grid refinement test
For the fluid flow inside the channel, two parameters were tested: the number of divisions
in the radial direction (Nr) and the number of divisions in the x (normal to the cross section)
direction (Nx). In the x direction, it was not possible to obtain a solution with a grid with less than
52
200 divisions. The number of divisions in the radial direction was then kept constant (equal to 10)
and three grids, with 200, 300 and 400 elements were compared. The maximum and minimum
temperatures in both fluid and solid regions and the maximum velocity in the fluid channel were
used to compare the different grids. Table 3.2 shows that when Nr equals to 10, the difference
between the calculated temperatures and velocities are smaller than 1% for all cases. Based on
this, the grid with 300 divisions in the x direction was chosen.
Table 3.2 Grid refinement text
Nr
10
10
10
5
10
15
Nx
200
300
400
300
300
300
�
[K]
310.24
310.26
310.36
310.22
310.26
310.26
�
[K]
324.64
324.80
324.87
324.85
324.80
324.84
�
[m/s]
7.22
7.26
7.32
7.26
7.26
7.23
In the next step, the number of divisions in the radial direction is tested. Values for Nr
equal to 5, 10 and 15 are shown in Table 3.2. Again, the difference between the calculated values
is lower than 1%, thus the grid with Nx = 300 and Nr = 10 was selected for all simulations.
The cross section grid in the solid was tested based on the different between the total
energy ( = ∫
) in the initial (t = 9ms) and final (t = 5s) time steps. These solutions were
obtained for an insulated solid rail with no cooling channels. The imposed restriction was that
(Et=9ms – Et=5s)/Et=9ms × 100 < 1%. The selected grid has a total of 234600 elements (782 × 300)
and is show in Fig. 3.10.
53
3.4 Results and discussion
Temperature profiles along the rail’s internal edge of all four cooling and the no-channels
assemble, at t = 5 s, are shown in Fig. 3.11.
Figure 3.11 Internal edge temperature profiles
The no-channels assemble has the highest peak temperature and remains above all other
curves up to approximately 8m in the x direction. In this case, only diffusion occurs from the rail
edges to its center and no energy is removed from the rail.
With the two centered cooling channel assembles the maximum temperature at t = 5 s is
considerably reduced. From Fig. 3.11 it can be seen that the peak temperature is reduced from
~380 K to ~360 K. This happens because part of the energy is transferred to the water and
removed from the rail.
Finally it can be seen in Fig. 3.11 that the eccentric assembles are even more efficient in
54
reducing the peak temperature. This is due to the fact that in these assembles the cooling
channels are positioned closer to the internal rail edge (see Fig. 3.8) where the temperature (and
temperature gradients) are higher. In these two cases, part of the energy removed from the first
few meters of the rail returns to it downstream due to a heat reversal phenomenon. The heat
reversal phenomenon has been previously observed (Liu, 1991), and it basically consists of an
inversion in the heat flow direction. In the initial sections of the rail the heat flows from the rail
to the water in the channels, but by the exit, the heat flow direction changes and heat flows from
the water to the rails. In this way, the water acts as a carrier of energy from the rail entrance
towards the rail ends.
The crossing of the lines in Fig. 3.11 shows that the rail temperature (after ~8m) is higher
for the eccentric cases than for the no channels one. This is only possible if heat is being
transferred from the water to the rail. The same phenomena also occurs in the two centered
assembles, however it is shown in Fig. 3.11 that the no-channel and centered lines crosses
themselves only at the end of the rail (~9.5m) and for this reason the amount of heat that returns
to the rail is much smaller.
The energy deposited into the rail during launching can be obtained by
=�
, =
|
∀
−�
=
|
∀
=
(3.9)
The amount of energy removed from the rails is evaluated by the volume integral given
by
∆ =�
∀
|
=
−�
|
∀
and summarized in Table 3.3.
55
=
(3.10)
Table 3. 3 Energy analysis of the first cooling period (values correspond to half-rail)
�
at t=5s
, =
∆
∆
, =
0
× 100%
No-channels
367.3
2.56MJ
0
3-centered
359.5K
2.53MJ
0.610MJ
24.10%
2-centered
357.2K
2.51MJ
0.665MJ
26.51%
3-eccentric
330.4k
2.46MJ
0.831MJ
33.74%
2-eccentric
324.8K
2.42MJ
0.944MJ
38.99%
Table 3 lists for each assemble the maximum temperature at the end of the 5s cooling
period, the deposited energy, the amount of energy removed(∆ ), and the percentage of the
energy removal. The amounts of energy removed from the rail are evaluated by the volume
integral given by Eq. (3.10). All the values are for half rail. It should be noticed that the amounts
of energy deposited in the rail are slightly different in all the five cases. This happens because, as
described before, the initial temperature field is obtained for a solid rail after the electromagnetic
problem solution and the cooling channels are only added during the thermal management
analysis.
The volume occupied by the cooling channels is the same in both the two-channel and
three-channel assembles. The two “eccentric” assembles remove more energy than the two
“centered” assembles and they exhibit lower peak temperatures after 5s (
�
).
The maximum temperature value occurs at a position close to the channel entrance,
where the water entering the channels is at 300K. Figure 3.12 shows the cross section
temperature profiles at x = 0.3m and t = 5s. Only half of the rail is shown in these five figures
and the same contour levels are applied for all five plots.
56
(a). Sketch of the cross section position
(b). no-cooling
�
= 367.5
(c). 3-centered (
�
= 355.8 )
(d). 2-centered (
�
= 354.3 )
(e). 3-eccentric (
�
= 330.1 )
(f). 2-eccentric (
�
= 324.2 )
Figure 3.12 Cross-section temperature distributions at x=0.3m, t=5 s
57
It can be seen in Figure 13 that the temperature inside the channels is practically
unchanged and remains equal to the inlet temperature of 300K up to the x=0.3m cross section.
This feature accelerates the cooling process. The lowest maximum temperature was obtained
with the 2-eccentric channel assemble.
Results in Fig. 3.12 show that the maximum temperature can be reduced by positioning
the cooling channels as close as possible to the internal rail edge. However, there is a physical
limit of how close to the internal edge the channel can be positioned and how large its diameter
can be due to structural limitations. It is clear that optimizing the size and position of the cooling
channels is an important topic to be addressed in the thermal management of electromagnetic
launchers.
The heat reversal can be observed in Fig. 3.13 which shows the
cross section
temperature profiles at x = 9m and t = 5s. At this position, the no channels assemble (Fig. 3.13b)
shows the lowest maximum temperature. This is an indication that heat reversal is occurring. In
Fig. 3.13c-f it can be seen that the water is at a higher temperature than the rail and heat is
flowing from the water to the rail at this cross section in all four cases.
58
(a) sketch of the cross section position
(b) no-cooling (Tmax = 309.8 K)
(c) 3-centered (Tmax = 310.6 K)
(d) 2-centered (Tmax = 310.0 K)
(e) 3-eccentric (Tmax = 314.2 K)
(b) 2-eccentric (Tmax = 313.7 K)
Figure 3.13 Cross section temperature distribution at x = 9 m and t = 5 s
59
CHAPTER FOUR
AXIAL COOLING FOR MULTIPLE SHOTS
Based on the analysis of the effect of channel locations on the cooling effectiveness for
single shot, the next step is to extend the single shot solutions to multiple shots to see if the same
conclusion can be made. Two cooling configurations will be considered for five consecutive
shots, 3-centered (Fig. 3.8a) and 3-eccentric (Fig. 3.8c).
4.1 Computational domain
The dimensions of the armature are kept the same as those in Chapter 3. However, the
rail length is set to be 1.2m to save computational time and the armature mass was reduced to
10kg. The computational domains for the 3-centered and 3-eccentric cases for launch simulation
are shown in Fig. 4.1. The channel positions and diameters for the two cases are the same as
those shown in Fig. 3.8.
60
(a) 3-centered
(b) 3-eccectric
Figure 4.1 Computational domain for launch simulation in multi-shots scenario
61
In cooling periods, only the rail and the channels are included, see Fig. 4.2. The remnant
magnetic field is neglected.
(a) 3-centered
(b) 3-eccentric
Figure 4.2 Computational domain for cooling analysis in multi-shots scenario
62
4.2 Governing equations, boundary, initial conditions and material properties
The governing equations are the same as those used in Chapters 2 and 3. In launch
simulation, the boundary condition for the channel inlet and outlet is set as magnetic insulation
(Eq. (2.28)). For the symmetry plane of the middle channel, the electric insulation (Eq. (2.29)) is
applied. In cooling period, the inlet coolant velocity is changed to be 2m/s. All the other
boundary conditions for launch period and cooling period are kept the same with those in
chapters 2 and 3. The initial conditions for the first shot of total five shots are also kept the same
with those in chapter 3.
During launch simulation, the water inside the channels is treated as dielectric. The
electric conductivity is set to be 1 instead of its real value about 1×10-4 to make the convergence
faster. It is already proven in section 3.1 that this simplification will only cause limited effects on
the current and temperature field. The other material properties are kept be to the same.
4.3 Simulation procedure
Each shot simulation consists of two steps – the launch and the subsequent cooling. The
cooling period between two consecutive shots is ~5s, a time period dictated by the design
parameters (Table 1.1). Starting from the second shot, for the launch simulation, the initial
temperature field of the rail is the resulting temperature of the previous cooling period. Since the
launch period is very short (~2.7ms), the heat convection inside the channels in this step is
neglected and the coolant temperature distribution is assumed to remain constant during the
2.7ms duration of launch. That is, there is no heat transfer at the rail-channel interface in this
period. The rail is thermally insulated from the surrounding air, the same way discussed in
previous chapters.
4.4 New mesh for multi-shots scenarios
For the launch simulation, the mesh of the rail and armature for the two cases is shown in
Fig. 4.3. Because of the existence of the channels, on the cross-section both quadrilateral and
triangular mesh are included. The channels and adjacent core part are meshed with triangular
63
mesh. For the regions close to the edges, the quadrilateral mesh is used. Meshing in this way
could save the memory usage and computation time by assigning more elements at high current
density regions and fewer elements at low current density regions. For the surrounding air,
triangular mesh is adopted. The mesh on the inlet cross section is swept along the length
direction with 240 layers (resolution of 0.005m) to generate the whole mesh.
(a) 3-centered
64
(b) 3-eccentric
Figure 4.3 Partial mesh of rail, channels and armature
There is interpolation error generated in the process of transferring temperature from
COMSOL to FLUENT. In multi-shots scenario, this error will accumulate and could reach an
unacceptable value. To control this error, in the cooling analysis the mesh is different from that is
used in launch simulation. The mesh on the inlet cross section for the two cases is shown in Fig.
4.4. Swept method with the same layers used in launch simulation is also adopted to generate the
mesh for the whole geometry. Due to this more refined mesh, the interpolation error of the total
energy is less than 1% for each shot.
65
(a) 3-centered
(b) 3-eccectric
Figure 4.4 Cross section mesh of rail and channels
66
As can be seen in Figs. 4.3 and 4.4, the meshes for multi-shots scenario are more refined
that those in Chapter3 (Figs. 2.11a and 3.10). Therefore the mesh convergence test is omitted
here.
4.5 Results and discussion
4.5.1 3-centered case
The temperature distribution of 3-centered case after the first shot is shown in Fig. 4.5. In
comparison with the 10m case (Figs. 2.13 and 2.14), the same non-uniform features occur with
lower maximum temperature.
(a) Whole rail
67
x=0.3m, t=0.0027ms
Figure 4.5 Temperature distribution of 3-centered case after the first shot
The internal edge temperature profiles of 3-centered case after each shot and each cooling
period are shown in Figs. 4.6 and 4.7, respectively. Figure 4.6b is created by focusing on the
region between x=0.24 m and x=0.48 m.
68
(a) Whole edge
(b) Zoomed-in figure
Figure 4.6 Internal edge temperature of 3-centered case after each shot
(1st: t=2.7ms; 2nd: t=5s+2.7ms; 3rd: t=10s+2.7ms; 4th: t=15s+2.7ms; 5th: t=20s+2.7ms)
69
Figure 4.7 Internal edge temperature of 3-centered case after each cooling period
(1st: t=5s; 2nd: t=10s; 3rd: t=15s; 4th: t=20s; 5th: 25s)
In Fig. 4.6a, the temperature falls sharply at x=1m because the current is shut down at
this time t=0.0027ms. With more shots proceeding, the temperature profile goes up because not
all the deposited heat is removed from the rail. Another feature that could be found in Fig. 4.6b is
that the amplitude of the profile increment will become smaller with increasing shot numbers.
This phenomenon also occurs in Fig. 4.7, which shows temperature profile after each cooling
period.
The maximum temperature increase between two consecutive shots and between two
cooling periods is shown in Fig. 4.8. This tendency demonstrates that the 3-centered cooling
arrangement could stabilize the peak temperature after a number of shots.
70
Figure 4.8 Increase of Tmax between every two shots and every two cooling period
In Fig. 4.8 the term ∆
�
−∆
−
�
represents the difference in the maximum
temperature between two consecutive shots and between two consecutive cooling periods.
The energy deposited into the rail in the launch period of the 3-centered case for each
shot is integrated by Eq. (4.1),
∆
=�
|
∀
=[( − )∗ + .
]
−�
|
∀
=( − )∗
(4.1)
where n is the shot number and ∀ is the volume of the rail.
The energy removed in each cooling period in the 3-centered case is found by:
∆
=�
∀
|
=[( − )∗ + .
]
−�
∀
|
= ∗
(4.2)
The results for five shots of the 3-centered case are listed in Table 4.1 and plotted in Fig.
4.9.
71
Table 4.1 Energy deposited in each shot and removed in each cooling period
Shot number
1
∆
[MJ]
0.1275
∆
[MJ]
0.02256
∆
∆
× 100%
17.7%
2
0.1306
0.04549
34.8%
3
0.1327
0.06390
48.2%
4
0.1342
0.07889
58.8%
5
0.1353
0.09102
67.3%
Figure 4.9 Energy deposited and removed for each launch and cooling period
72
In Table 4.1, the energy deposited during each launch period is less than 0.15MJ, which
is much less than that in the 10m case in Chapter 3. This is due to the lower armature mass and
shorter rail length.
The amount of energy deposited during each launch period keeps increasing slowly with
additional shots (see Fig. 4.9). From Fig. 4.7, we can see that the temperature profile of internal
edge will also increase. So the bulk rail temperature will increase. With the increased bulk rail
temperature, the resistive heat generation will be higher.
The amount of energy removed during each cooling period goes up continuously for the
five shots. In Fig. 4.9, it can be seen that the amount of energy removed during each launch
period increases much faster than the deposited energy. So the energy removal efficiency will
keep increasing and the net deposited energy during each shot (launch and cooling period) will
decrease. We expect that this tendency will finally cause a stable energy level in the rail, and as a
result of it, the bulk temperature will reach a stable value after a number of shots. However, a
study with additional shots could be necessary to verify this claim.
The temperature profile at x=0.8m after the fifth cooling period is shown in Fig. 4.10.
Due to the short length of the rail, no heat reversal occurs.
Figure 4.10 Temperature profile of 3-centered case at x=0.8m, t=25s
73
4.5.3 3-eccentric case
The temperature distribution after the first shot is shown in Fig. 4.11. It is similar to the
3-centered case (see Fig. 4.5). A non-uniform temperature distribution occurs on both the length
direction and cross section.
(a) Whole rail
74
(b) x=0.3m,t=0.0027ms
Figure 4.11 Temperature distribution of 3-eccentric case after the first shot
The internal edge temperature profiles after each shot and each cooling period are shown
in Figs. 4.12 and 4.13. The Fig. 4.12b is created by focusing on between x=0.24 m and x=0.48
m.
75
(a) Whole edge
(b) Zoomed-in figure
Figure 4.12 Internal edge temperature of 3-eccentric case for 5 shots
76
Figure 4.13 Internal edge temperature of 3-eccentric case after 5 cooling period
In Fig. 4.12a, with more shots proceeding, the temperature profile goes up continuously.
The same feature shown in Fig. 4.6b is also found in Fig. 4.12b - the amplitude of profile
increment will decrease with increasing shot numbers until the fifth shot. The maximum
temperature increase between the 5th and 4th shots is slightly larger than that between the 4th and
3rd shots. In Fig. 4.13, just like we have found in Fig. 4.7, the increase of the temperature profile
between two consecutive cooling periods becomes smaller with the increasing shot numbers.
The change of maximum temperature between two consecutive launch periods and two
consecutive cooling periods is shown in Fig. 4.14. It is clear that the increase of maximum
temperature decreases with the increasing shot numbers except for the Tmax after the 5th shot. A
study with additional shots would be necessary to extrapolate the behavior.
77
Figure 4.14 Increase of Tmax between every two shots and every two cooling period
In Fig. 4.14 the term ∆
�
−∆
−
�
represents the difference in the maximum
temperature between two consecutive shots and between two consecutive cooling periods.
The energy deposited during each launch period and removed in each cooling period is
calculated by Eqs. (4.1) and (4.2). The results are shown in Table 4.2 and plotted in Fig. 4.15.
78
Table 4. 2 Energy deposited in each shot and removed in each cooling period
Shot number
1
∆
[MJ]
0.1280
∆
[MJ]
0.03591
∆
∆
× 100%
28.1%
2
0.1309
0.05856
44.7%
3
0.1324
0.07483
56.5%
4
0.1331
0.08726
65.6%
5
0.1338
0.09725
72.7%
Figure 4.15 Energy deposited and removed for each launch and cooling period
79
Just like the 3-centered case, the deposited energy keeps increasing but in a slower speed.
The amount of energy removed during each cooling period and the energy removal efficiency for
each shot goes up continuously for the five shots. So the net energy deposited in the rail during
each complete shot period (launch and cooling period) will be reduced with the increasing shot
numbers. Therefore, after a number of shots, the rail energy level and bulk temperature will
become stable.
The temperature profile at x=0.8m after the fifth cooling period is shown in Fig. 4.16.
Also there is no heat reversal.
Figure 4.16 Temperature profile of 3-eccentric case at x=0.8m, t=25s
4.5.3 Comparison of the two cooling arrangements
Based on the foregoing analysis, both the 3-centered and 3-eccentric arrangements can
stabilize the energy level and bulk and peak temperature after a number of shots. In Chapter 3,
for 10m rail, the 3-eccentric case has a better cooling performance. In this subsection,
80
comparison between the two cases is conducted to see the cooling performance in multi-shots
scenario.
The internal edge temperature after the 1st, 3rd, and 5th shots for both cases is shown in
Fig. 4.17. Figure 4.17b is created by enlarging Fig. 4.17a around x=0.3 m. In these two figures,
the solid lines represent the 3-centered case, and dashed lines the 3-eccentric case. Red, green
and blue represents the 1st, 3rd, and 5th shots, respectively. In Fig. 4.17b, for the first shot, the 3eccentric case has a higher temperature profile than 3-centered case. It is probably because in 3eccentric case, the top channel is closer to the internal edge and will cause the current to be more
focused on the edge region. After the 3rd shot, the two cases have almost the same edge
temperature. And after the 5th shot, the 3-eccentric case has a much lower temperature profile
than 3-centered case. This demonstrates that in the sense of controlling the peak temperature, the
3-eccentric arrangement is better than 3-centered arrangement; although in the beginning it will
have a higher maximum temperature.
(a) Whole edge
81
(b) Enlarged figure
Figure 4.17 Internal edge temperature of two cases after 1st, 3rd, and 5th shots
The internal edge temperature after the 1st, 3rd, and 5th cooling periods for both cases is
shown in Fig. 4.18. The line type and color has the same meaning with those in Fig. 4.17. For all
the cooling periods, the 3-eccentric arrangement has a lower temperature profile at the front part
(high temperature region) and higher temperature profile at the rear part close to the end (low
temperature region). That means the 3-eccentric case controls the maximum temperature better
and gets a more even temperature distribution along the whole rail length.
82
Figure 4.18 Internal edge temperature of two cases after 1st, 3rd, and 5th cooling periods
The energy deposited and removed for each launch and each cooling period is shown in
Fig. 4.19. The solid line is for the 3-centered arrangement and dashed line for the 3-eccentric
arrangement. The red and green colors are for the deposited and removed energy, respectively.
The energy deposited in the 1st shot has a higher value in the 3-eccentric case. That means
moving the cooling channels towards the corner will cause a higher heat generation, which is a
factor that we don’t want to expect in a cooling design. However, the 3-eccentric arrangement
has higher energy removal efficiency (green lines in Fig. 4.19). The two cases almost reach the
same energy deposited values in the 3rd shot. For the 4th and 5th shot, the 3-centered case has
more deposited energy than the 3-eccentric case. So, by combining these two factors (more heat
generation and higher heat removal efficiency), it is the 3-eccentric arrangement that has a better
performance. Because of this, the 3-eccentric arrangement will finally have a lower bulk rail
temperature.
83
Figure 4. 19 Energy deposited and removed for 3-centered and 3-eccentric cases
84
CHAPTER FIVE
VERTICAL COOLING ANALYSIS FOR SINGLE SHOT
5.1 Introduction
Besides the traditional way of putting the cooling channels in the rail length direction,
another layout of the cooling channels - the cooling channels in the radial direction of the rail
(see Fig. 5.1) - will also be considered. In this study, it is called vertical cooling. Only one shot
and the subsequent cooling analysis are conducted.
The idea of this cooling arrangement comes from the non-uniform feature of temperature
distribution in rail length direction (see Fig. 2.14b). With this configuration, the cooling channels
could be assigned more densely in the high temperature region (front part) and less densely in the
low temperature region (rear part). Along rail height direction, the rail’s top and bottom surfaces
have the same temperature. Therefore the vertical cooling channels have the inherent capability
to avoid heat reversal.
Figure 5.1 Layout of vertical cooling channels
85
5.2 Computational domain
For the launch simulation, the same computational domain as that in chapter 2 (see Fig.
2.1) is adopted. All dimensions are kept the same as those shown in Fig. 2.2 except for the
length, which is changed to 1.2m. The rail is modeled as a solid block without cooling channels
in it in this step.
For the cooling analysis, the cooling channels are added to the rail. In this channel
configuration, the temperature field will no longer be symmetric about the middle x-y (horizontal
plane in Fig. 3.4) plane. So the whole rail and the cooling channels are included in the
computational domain is this step.
5.3 Governing equations, boundary, initial conditions and material properties
The governing equations and material properties are the same as those used in chapters 2
and 3. In launch simulation, the boundary and initial conditions are also the same of those used
in chapter 2. In cooling analysis, all the boundary conditions are kept the same as those used in
chapter 3 except for inlet velocity, which is set to be 2m/s.
5.4 Simulation procedure
Two steps are included - launch simulation and cooling simulation. The cooling period is
5s. The temperature field obtained after launch simulation will be the common initial
temperature for cooling analysis.
5.5 Mesh
The mesh used for launch simulation is the same as shown in Fig. 2.9a. For the cooling
simulation, because the channels are in rail height direction, the mesh cannot be obtained by
meshing the inlet and then sweep along the rail length. Partial mesh is shown in Fig. 5.2.
86
Figure 5.2 Partial mesh for vertical cooling analysis
The mesh is generated by first mesh the top surface and then sweep along the rail height
direction. To assign enough mesh elements in high temperature regions, the rail is divided into
three parts along the width direction, the left, middle, and right parts. The cooling channels are
included in the middle part. In the left and right parts, quadrilateral mesh is used. In middle part,
both the quadrilateral and triangular mesh is used.
Because this mesh in more refined than those used in chapter 3, the mesh convergence is
omitted.
5.6 Channel distribution
In the y-direction (rail width) all channels are in the middle. Figure 5.3 shows a uniform
distribution of the total 8 channels.
87
Figure 5.3 Layout of vertical cooling channels
Three different channel distributions were studied to evaluate the impact of the
distribution on the cooling effectiveness. The relative position of the channels in the x-direction
was selected using MATLAB (The MathWorks, 2008) function Geospace (x0, xN, N, G). This
function generates N points between x1 and xN with a spacing that is G times bigger than the
previous step. The parameters used in these three cases are listed in table 5.1. The case 1 is a
uniform distribution with the first channel located on x0=0.1333m. In case 2 and case 3, the first
channel was kept on the same location, whereas the following channels take the different
distribution by changing the G value. Larger G-values results in the channels being more
concentrated towards the front part of the rail.
88
Table 5.1 Channel distribution parameters
G
x1
x2
0.1333 0.2667
x3
0.4
x4
x5
0.5333 0.6667
x6
0.8
x7
x8
1
1
0.9333 1.0667
2
1.3
0.1333 0.1780 0.2361 0.3117 0.4099 0.5376 0.7036 0.9194
3
1.5
0.1333 0.1550 0.1874 0.2362 0.3093 0.4189 0.5833 0.8300
5.7 Results and discussion
Case 1 in Table 5.1 corresponds to an even distribution of 8 channels. The temperature
field after 5s’ cooling period is shown in Fig. 5.4. From this figure, it is clear that each channel
can control the adjacent temperature.
Figure 5.4 Temperature field for case1 at t=5s
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The hotter internal edge temperature is shown in Fig. 5.5 together with the no cooling
channel arrangement. As was mentioned in previous section, the temperature field is no longer
symmetric about the middle x-y plane. So the two internal edges had different temperature
profiles. The internal edge on the top surface had a higher temperature than the one on the
bottom surface. In Fig.5.5, the hotter internal edge corresponds to the edge on the top surface.
Figure 5.5 Internal edge temperature profile
The temperature in the regions next to the cooling channels are significantly reduced,
however for regions far away from the channels, the temperature shows not much difference
from the no-cooling case. The channels are more effective in reducing the temperature in the
hotter regions (towards x=0), while the channels towards the rail end (x =1.2) (channels 7th and
8th) produce very small temperature drops when compared to the no-cooling situation.
The hotter internal edge temperature for all the three cases at t=5s is shown in Fig. 5.6.
As expected, the concentration of channels towards the higher temperature region lowers the
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maximum temperature. Nevertheless, the high concentration of channels in higher temperature
region will cause less channels in the lower temperature region, and thus in some regions the
cases 2 and 3 will have a higher temperature than the case 1. However, the goal in this work is to
control the maximum temperature to protect the rail from melting and in this sense the uneven
channel distribution is beneficial.
Figure 5.6 Internal edge temperature profile for three cooling cases
The temperature profile at x=0.1333m, t = 5s (center plane of the first channel) for the
case 1 is shown in Fig. 5.7. The water inside the channel has lower temperature than the solid.
The heat reversal is avoided as we expected.
91
Figure 5.7 Temperature profile of the center-plane for the first channel of case 1
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CHAPTER SIX
CONCLUSIONS
A three-dimensional model for the thermal analysis of electromagnetic launchers has
been proposed and presented in detail. The model accounts for the electromagnetic and thermal
phenomena and determines the projectile’s position and velocity, rail temperature rise and the
energy deposited during the launch process. Two features of rail temperature distribution were
found:
•
On cross sections, the high temperature regions are concentrated on the two corners
(half rail) with the peak temperature on the internal corner; the core region is
essentially at initial temperature.
•
Along the rail length direction, the temperature distributions are also non-uniform.
The internal edge is a representative line to show the rail temperature distribution.
In the thermal management for single shot 10m-rail problem, five different assembles,
with and without cooling channels were considered. The following conclusions can be obtained
by analyzing and comparing the five cases:
•
Solely conduction is effective in reducing the rail’s peak temperature for single shot
situations because the rail core is essentially at the initial temperature. But it will not
work in multiple shot scenarios when the energy accumulates in the rail.
•
Comparing to the no channel case, the cooling channel can reduce the peak
temperature from 18% to 64% (
���� −����
���� −
× 100% ). The energy removal
percentage for all cases ranges from 24% to 39%.
•
Moving the channels towards the hotter edge can improve the energy removal
efficiency and lower the peak temperature.
•
There is a heat reversal phenomenon in long rail assembles with axial cooling
channels. In the front part of the channels heat is transferred from the solid to the
coolant and in the rear part of the channels, heat is transferred back into the solid.
93
Under these conditions, the water acts both as a coolant and as a carrier of energy
towards the cooler end part of the rail.
To find whether the above conclusions can be extended to multiple shots or not, two
cases were selected for the thermal management of multiple-shot scenario for a shorter 1.2m rail.
They are 3-centered and 3-eccentric cases. A total of 5 shots and their subsequent cooling were
simulated. Results showed that:
•
The maximum temperature will increase with increasing shots. But for the chosen
two cooling arrangements, the amplitude of the maximum temperature increment will
become smaller with increasing shots. So, after a number of shots (more than 5), the
peak temperature will become stable.
•
The amount of heat deposited in the rails for the two arrangements (3-centered, 3eccentric) keeps increasing slowly for all 5 shots. The amount of heat removed during
each cooling period increases continuously and in a comparatively faster speed. So
the net deposited heat in a complete shot period (launch and cooling) will decrease
with the increasing shots and heat removal efficiency will increase as well. It can be
concluded that the deposited heat and bulk temperature will reached a stabilized level
after a number of shots (more than 5).
•
The 3-eccentric case has higher energy removal efficiency during all the five cooling
periods in comparison with the 3-centered case. So, although it has a higher
maximum temperature initially, after five shots its maximum temperature is lower
than the 3-centered case.
•
For both the maximum temperature control and energy removal efficiency, the 3eccentric arrangement has better performance than 3-centered arrangement. Moving
the cooling channels towards the high temperature region will improve the cooling
efficiency for single shot and for multiple shots.
•
No heat reversal occurs in a short rail (1.2m), but it appears in the long rail (10m).
An alternative vertical cooling configuration was also considered. By examining three
vertical cooling cases and the no-cooling case, it was found that the vertical cooling scheme
avoids heat reversal, and it is effective controlling the maximum rail temperature. Additionally,
94
the effectiveness of the vertical cooling scheme was higher when the channels were concentrated
in the high temperature region.
So there should be an optimal geometry that can produce a rail design which at the same
time reduces the maximum temperature inside the rail, removes most of the deposited energy,
and also guarantees that it will not have its structural strength reduced to a point on which the
generated electromagnetic force will cause the system to fail (break). Also the mass flow rate is
required to be optimized as well to achieve the optimal geometry design.
95
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BIOGRAPHICAL SKETCH
Han Zhao was born in Shenyang, Liaoning Province, China. He got his BS degree in
Mechanical Engineering from Hebei University of Technology (Tianjin, China) in 2000. Under
the advisement of Dr. Wei Cao, he then earned his MS degree in 2004 from Tianjin University
(Tianjin China) in 2004 majoring in fluid mechanics and heat transfer. In 2004, he entered
Florida State University to pursue his doctorate degree and completed his PhD, under the
supervision of Professor Juan C. Ordonez, in the summer of 2011. His research interests include
computational fluid dynamics (CFD), mathematical modeling, thermal analysis, and thermal
management.
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