thermal performance analysis and geometrical optimization
TRANSCRIPT
i
Thermal Performance Analysis and Geometrical
Optimization of Automotive Brake Rotors
By
Zhongzhe Chi
A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of
Master of Applied Science
in
Mechanical Engineering
Faculty of Engineering and Applied Science
University of Ontario Institute of Technology
July 2008
© Zhongzhe Chi, 2008
ii
Abstract
The heat dissipation and thermal performance of ventilated brake discs strongly
depends on the aerodynamic characteristics of the air flow through the rotor
passages. In this thesis, the thermal convection is analyzed using an analytical
method, and the velocity distribution, temperature contours and Nusselt number are
determined. Then numerical models for different rotors, pillar post rotors and vane
rotors are generated and numerical simulations are conducted to determine the
desired parameters. To analyze more realistic vane and pillar post rotor models,
commercial CFD software packages, Fluent and Gambit, are used to simulate the
heat flux rate, air flow rate, velocity distributions, temperature contours, and
pressure distributions inside the rotors. Furthermore, sensitivity studies have been
performed, to determine the effects of a different number of vanes or pillar posts,
inner and outer radii and various angles of vanes. To automate the tedious and
repetitive design process of the disc rotor, a design synthesis framework, iSIGHT,
is used to integrate the geometrical modeling using GAMBIT and numerical
simulations based on FLUENT. Through this integrated design synthesis process,
the disc rotor geometrical optimization is performed using design of experiment
studies.
Keywords: Heat transfer, Brake rotor, Simulation, Geometrical optimization
iii
Acknowledgements
It is with tremendous appreciation that I wish to acknowledge my thesis supervisors,
Dr. Yuping He and Dr. Greg Naterer. Their advice, guidance and encouragement
have made this program rewarding and enjoyable.
Financial support of this research from the Natural Sciences and Engineering
Research Council of Canada is gratefully acknowledged.
Finally the great acknowledgement I would like to give to my parents, my wife
and my daughter for supporting and encouraging me.
iv
Contents
1 Introduction 1
1.1 Motivation ……………………………………………………………... 1
1.2 Literature review ……………………………………………………… 2
1.2.1 Thermal analysis of solid rotors …………………………………….. 3
1.2.2 Influence of rotor materials ………………………………………. 8
1.2.3 Other experimental and analytical work ……………………………… 9
1.3 Research outline ………………………………………………………. 10
2 Thermal Performance Analyses of Vented Rotors with an Analytical Method 14
2.1. Estimation of heat flux generation ……………………………………. 14
2.2 Heat dissipation ………………………………………………………. 15
2.2.1 Heat conduction ………………………………………………… 15
2.2.2 Radiation ……………………………………………………….. 15
2.2.3 Convection heat transfer ……………………………………….. 17
2.3. Convection heat transfer analysis ……………………………………. 17
2.3.1 The velocity distribution in a simplified model ………………... 18
2.3.1.1 Conservation of mass (continuity equation) ……………… 19
2.3.1.2 Conservation of momentum (Navier-Stroke equation) …… 20
2.3.1.3 Velocity distribution ………………………………………. 24
2.3.2 Heat transfer coefficient and Nusselt number ………………….. 26
2.3.2.1 Energy equation ………………………………………….. 26
v
2.3.2.2 Heat transfer coefficient and Nusselt number ……………. 31
2.3.3 Temperature distribution in 3 dimensions ……………………… 31
2.3.4 Velocity, temperature and Nusselt number equation analysis …. 35
2.3.4.1 Velocity equation ………………………………………… 35
2.3.4.2 Nusselt number equations ………………………………… 38
2.3.4.3 Maximum temperature equation …………………………. 40
2.4. Summary …………………………………………………………….. 41
3 Thermal Performance Analysis of Vane Rotors Based on Numerical Simulations
43
3.1 GAMBIT models ……………………………………………………. 43
3.2 Numerical Simulations using FLUENT …………………………….. 44
3.3 Results and discussion ………………………………………………. 48
3.3.1 Effects of vane numbers ………………………………………. 48
3.3.2 Effects of vane angles …………………………………………. 50
3.3.3 Effects of curved vanes ……………………………………….. 51
3.3.4 Effects of short-long vanes …………………………………… 54
3.4. Summary …………………………………………………………… 56
4 Thermal Performance Analysis of Pillar Post Rotors Based on Numerical
Simulation 58
4.1 GAMBIT models …………………………………………………… 58
4.2 Numerical simulations using FLUENT …………………………….. 60
4.3 Results and discussion ……………………………………………… 62
4.3.1 Effects of pillar post numbers ………………………………… 62
vi
4.3.2 Effects of different positions of middle pillar posts ………….. 63
4.3.3 Effects of modified entrance pillar posts …………………….. 65
4.3.4 Effects of increased pillar post sizes …………………………. 67
4.3.5 Comparison of pillar post rotors and vane rotors ……………. 68
4.4. Summary …………………………………………………………... 70
5 Geometrical Optimization with 2-D Models 72
5.1 Framework for geometrical optimization models ………………….. 72
5.2 Numerical simulations with CFD ………………………………….. 74
5.3 Design of experiments with iSIGHT ………………………………. 77
5.4 Results and discussion ……………………………………………… 80
5.4.1 Effects of vane numbers ……………………………………… 80
5.4.2 Effects of inner and outer radius ……………………………... 81
5.4.3 Effects of vane offset and angle ……………………………… 84
5.5 Summary ……………………………………………………………. 85
6 Geometrical Optimization in 3-D Models 86
6.1 Framework for geometrical optimization models ………………….. 86
6.2 Numerical simulations with CFD ………………………………….. 87
6.3 Design of experiments with iSIGHT ………………………………. 92
6.4 Results and discussion ……………………………………………… 93
6.4.1 Effects of vane numbers …………………………………….. 94
6.4.2 Effects of inner and outer radius ……………………………. 94
6.4.3 Effects of vane offset and vane angle ………………………. 95
6.4.4 Effects of rotor thickness …………………………………… 97
vii
6.5 Summary ………………………………………………………….. 97
7 Conclusions 98
References 101
viii
List of figures
1.1 A solid rotor …………………………………………………………………. 2
1.2 Temperature distributions with constant heat flux for a solid rotor ………… 3
1.3 Temperature distributions as a function of rotor thickness …………………. 4
1.4 Temperature distributions with constant deceleration for a solid rotor …….. 5
1.5 Surface temperature distributions with constant deceleration and heat flux … 6
1.6 Surface temperatures with different thicknesses at 5th second at a constant heat
flux …………………………………………………………………………. 7
1.7 Surface temperatures as a function of time for different materials ………….. 8
2.1: Radiative heat transfer coefficient versus rotor temperature ……………… 16
2.2: A pillar post rotor ………………………………………………………….. 17
2.3: A vane rotor ………………………………………………………………… 18
2.4: Straight duct (a simple model for disc rotors) with a rectangular section …. 18
2.5 Heat transfer analysis in a control volume ………………………………….. 30
2.6: Velocity/u0 vs. the rotor ratio D/a ………………………………………….. 36
2.7: Non-dimensional mean velocity vs. pillar post coefficient ………………… 37
2.8: The air flow rate versus revolutions of wheels …………………………….. 38
2.9: The rotor ratio versus Nusselt number ……………………………………... 39
2.10: Non-dimensional temperature change versus rotor ratio …………………. 41
3.1: 2-D mesh with 40-vane rotor mesh model …………………………………. 44
3.2: Velocity contours for 40-vane rotor ………………………………………… 46
ix
3.3: Static pressure contours for 40-vane rotor …………………………………. 46
3.4: Velocity vector distribution for 40-vane rotor …………………………….. 47
3.5: Turbulence distribution for 40-vane rotor …………………………………. 47
3.6: Vane numbers vs. heat transfer rate ……………………………………….. 49
3.7: Velocity field in 24, 32, 40, 50 and 60 vanes ……………………………… 49
3.8: Vane angles vs. heat transfer (32 vanes) …………………………………… 50
3.9: Vane angles vs. heat transfer (40 vanes) …………………………………… 51
3.10: Radii of curvature vs. heat transfer rate (32 vanes) ……………………… 52
3.11: Radii of curvature vs. heat transfer rate (40 vanes) ………………………. 52
3.12: Radii of curvature vs. heat transfer rate (40 vanes, 30 degree angle) …….. 53
3.13: Velocity distribution of 32-vane rotor at angular velocity of 120 rad/s …… 54
3.14: Short-long ratio vs. heat transfer rate (32 vanes) ………………………….. 55
3.15: Short-long ratio vs. heat transfer (40 vanes) ……………………………… 55
3.16: Velocity contours for a rotor with short-long vanes (40 vanes, ω = 88
rad/s) ……………………………………………………………………… 56
4.1: 2-D mesh with 160 pillar post rotor mesh model ………………………….. 59
4.2: Pillar post wall refinement of a 160 pillar post model ……………………… 59
4.3: Velocity contours of the 160 pillar post rotor ……………………………… 61
4.4: Static pressure contours of the 160 pillar post rotor ……………………….. 62
4.5: Pillar post numbers vs. heat transfer rate …………………………………... 63
4.6: Original positions of middle pillar posts vs. modified positions …………… 64
4.7: Velocity contours of disc rotors with 96, 160 and 224 pillar posts and modified
positions of middle pillar posts ……………………………………………. 64
x
4.8: Different middle pillar posts; angular orientations vs. heat transfer rate when
ω=44 ……………………………………………………………………… 65
4.9: Velocity contours of brake rotor with 160 pillar posts and modified entrance
posts (ω=44rad/s) …………………………………………………………. 66
4.10: Different triangle post entrance vs. heat transfer rate (ω=44, 88 and
120rad/s) …………………………………………………………………. 66
4.11: Velocity contours of brake rotor with enlarged pillar posts (number of posts =
160 and ω=44 rad/s) ……………………………………………………… 67
4.12: Pillar post number vs. heat transfer rate when the pillar posts have different
sizes and ω=44, 88 and 120 rad/s ………………………………………… 68
4.13: Comparison of pillar post rotors and vane rotors at various vane or pillar post
numbers (ω=44 rad/s) …………………………………………………….. 69
4.14: Comparison of pillar post rotors and vane rotors at various vane or pillar post
numbers (ω=88 rad/s) …………………………………………………….. 69
4.15: Comparison of pillar post rotors and vane rotors at various vane or pillar post
numbers (ω=120 rad/s) ……………………………………………………. 70
5.1: Schematic representation of the framework for automated design synthesis of
brake discs ………………………………………………………………… 73
5.2: 2-D mesh produced by GAMBIT ………………………………………….. 75
5.3 Full circular mesh produced by GAMBIT ………………………………….. 75
5.4: Velocity distribution in a section …………………………………………… 76
5.5: Pressure contours in a section ………………………………………………. 77
5.6: GAMBIT models with vane angles of 10° (left) and 30° (right) …………… 79
xi
5.7: GAMBIT models with offset of 0.1 (left) and 0.3 cm (right) ……………… 79
5.8: Vane numbers vs. heat transfer rate ……………………………………… 81
5.9: Inner radius vs. total heat transfer rate …………………………………… 82
5.10: Outer radius vs. total heat transfer rate ………………………………….. 82
5.11: Vane angle, vane offset vs. total heat transfer rate with 40 vanes ………. 83
5.12: Vane angle, vane offset vs. total heat transfer rate with 48 vanes ……….. 83
5.13: Vane angle, vane offset vs. total heat transfer rate with 56 vanes ……… 84
5.14: Vane angle, vane offset vs. total heat transfer rate with 64 vanes ……… 84
6.1: Schematic representation of the framework for automated design synthesis of
brake discs …………………………………………………………………….. 87
6.2: 3-D section mesh produced by GAMBIT ………………………………… 88
6.3: Outlet section of 3-D section mesh produced by GAMBIT ……………… 88
6.4: Inlet section of 3-D section mesh produced by GAMBIT ……………….. 89
6.5 Computational domain and boundary conditions of 3-D section mesh …… 90
6.6 Predicted velocity distributions in one section ……………………………. 91
6.7 Predicted static pressure distributions in one section ……………………… 91
6.8 Predicted turbulence distributions in one section ………………………….. 92
6.9 Vane numbers vs. heat transfer rate ……………………………………….. 94
6.10 Inner radius, outer radius vs. heat transfer rate …………………………… 95
6.11 Vane offset, vane angle vs. heat transfer rate at 40 vane numbers ……….. 96
6.12 Vane offset, vane angle vs. heat transfer rate at 56 vane numbers ……… 96
6.13 Rotor thickness vs. heat transfer rate at 56 vane numbers ……………… 97
xii
Notation
a Length of one section, m
A Area, m2
cp Specific heat, kJ/kgK
D Thickness of rotors, m
Convective heat transfer coefficient (W/m2k)
I Mass moment of inertia, kg m2
IR Inner radius
k Pillar post coefficient, thermal conductivity
KE Kinetic energy, kW
One-half rotor thickness (m)
m Mass, kg
numbers 1, 2, 3…
Nu Nusselt number
OR Outer radius
p Contact pressure, Pa
q” Heat flux, kW/m2
″)0(q Time-varying heat flux into the rotor at time t=0 (W/m2)
″0q Average heat flux into rotor (W/m2)
Q Thermal energy, kW
xiii
Qcond Conduction heat flow rate, kW
Qrad Radiation heat flow rate, kW
Time (s)
st Braking time to a stop (s)
T Temperature, K
, Transient temperature distribution in rotor due to a constant heat flux
(k)
TD Average disc surface temperature, K
Initial temperature (k)
T∞ Ambient air temperature, K
R Distance between node and rotor center, m
RT Rotor thickness
VA Vane angle
v0 Initial speed, m/s
VNu Vane number
VO Vane offset
Horizontal distance measured from midplane of rotor (m)
α Thermal diffusivity, m2/sec
Thermal diffusivity (m2/s)
ω Angular velocity, rad/s
ω0 Initial angular velocity, rad/s
σ Stefan-Boltzmann constant, W/m2K4
ε Emissivity
xiv
Initial temperature difference between brake and ambient (k)
, Relative temperature of brake resulting from constant heat flux (k)
( )tz,0θ Relative temperature of brake, resulting from a constant heat flux
(k)ρ Fluid density, kg/m3
p Pressure, Pa
/ λ
φ Dissipation function, 1/sec
μ Friction coefficient
x,y,z Cartesian coordinates
u,v,w x,y,z direction velocities, m/s
1
Chapter 1
Introduction
1.1 Motivation
A braking system is one of the most important safety components of an automobile.
It is mainly used to decelerate vehicles from an initial speed to a given speed. In
some vehicles, the kinetic energy is able to be converted to electric energy and
stored into batteries for future usage. These types of vehicles are known as electric
or hybrid vehicles. However, these kinds of vehicles still need a backup system due
to sometimes insufficient electric energy or failures which inevitably increase the
cost of the vehicles. So friction based braking systems are still the common device
to convert kinetic energy into thermal energy, through friction between the brake
pads and the rotor faces.
Excessive thermal loading can result in surface cracking, judder and high wear of
the rubbing surfaces. High temperatures can also lead to overheating of brake fluid,
seals and other components.
Based on the design configurations, vehicle friction brakes can be grouped into
drum and disc brakes. The drum brakes use brake shoes that are pushed in a radial
direction against a brake drum. The disc brakes use pads that are pressed axially
against a rotor or disc. Under extreme conditions, such as descending a steep hill
2
with a heavy load, or repeated high-speed decelerations, drum brakes would often
fade and lose effectiveness. Compared with their counterpart, disc brakes would
operate with less fade under the same conditions. An additional advantage of disc
brakes is their linear relationship between brake torque and pad/rotor friction
coefficient [15]. Advantages of disc brakes over drum brakes have led to their
universal use on passenger-car and light-truck front axles, many rear axles, and
medium-weight trucks on both axles. Thus, how to select better geometrical design
variables and improve thermal performance of automotive brake rotors is a task that
the vehicle designers and researchers are often confronted.
1.2 Literature review
Original disc brake rotors are solid rotors as shown in figure 1.1. They are still used
today in some applications.
Figure 1.1 Solid rotor
3
1.2.1 Thermal analysis of solid rotors
The temperature distribution of solid rotors has been investigated by Limpert using
Duhamel’s Theorem [15]. The derivation of the temperature equation is
accomplished with the assumption of a constant heat flux during constant-speed
downhill braking and a constant heat flux.
For solid rotors, the analytical solution for a constant heat flux is shown in equation
(1.1) [15]. Both sides of the rotor are heated by the heat flux q0"”, cooled by
convection (coefficient hR) and the conditions are based on a constant heat flux.
( ) ( )( ) ( ) ( )
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡+×
+⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−″
″= ∑
∞
=
−
10
00 1cos
cossinsin
12,2
nn
ta
nnn
nRi
R
zeLLL
L
q
hhq
tz nt λλλλ
λθθ λ (1.1)
Figure 1.2 Temperature distributions with constant heat flux for a solid rotor
4
As shown in figure 1.2, which is derived from equation (1.1), the highest
temperature occurs on the surface of the solid rotor. It can be also clearly seen from
Figure 1.2 that with an increase of time, the temperature increases almost linearly.
Figure 1.3 Temperature distributions as a function of rotor thickness
Limpert also derived the temperature equation when vehicles have constant
deceleration, in which case a linearly decreasing heat flux is assumed.
( ) ( )( ) ( ) ( )
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ −×
+−
″−″
″= ∑
∞
=
−
12
00
0
)0(0 cos1
cossinsin
2),(,2
nn
nt
ta
nnn
n
Rs
za
eLLL
Lt
htq
tzq
qtz
nt
λλλλλ
λθθ
λ
5
(1.2)
where,
Figure 1.4 Temperature distributions with constant deceleration for a solid rotor
Figure 1.4 shows that the highest temperature also occurs on the surfaces of the
solid rotors. At the 5th-6th second, the temperature reaches the peak point. Figure
1.5 illustrates a comparison of temperature distributions for constant deceleration
and constant heat flux.
6
Figure 1.5 Surface temperature distributions with constant deceleration and heat
flux
From the analytical solution, to achieve a low surface temperature of solid rotors, it
is useful to increase the thickness of the solid rotors. As shown in figure 1.6, when
the rotor thickness increases from 5 mm to 7 mm, the surface temperature greatly
reduces from 680 °C to 510 °C.
7
Figure 1.6 Surface temperatures with different thicknesses at the 5th second at
constant heat flux
1.2.2 Influence of rotor materials
The thermal properties of brake rotors are dependent on the temperature and
materials with different properties.
Newcomb [18] revealed that the thermal properties vary linearly with temperature
as indicated in the following equations.
k=ki(1+K1T)
cρ= ciρi(1+K1T) (1.3)
8
Newcomb [18] investigated the thermal properties of rotors with different materials
involving cant iron, steel, aluminum bronze and duralumin. Based on equation (1.2),
the temperature distributions can be calculated for various alloys as shown in figure
1.7.
Figure 1.7: Surface temperatures as a function of time for different materials
As shown in figure 1.7, the steel rotors have the lowest temperature among these
four materials and the duralumin rotors have the highest temperature.
1.2.3 Other experimental and analytical work
Much experimental and analytical work has been conducted to improve the thermal
performance of brake rotors. Limpert [14] compared solid and ventilated rotor
thermal performance at higher rotor speeds, wherein the internal cooling may
9
contribute as much as 50 or 60 percent to the total cooling. Parish and MacMauns
[20] revealed the effects of disc geometry and rotating speed on the mean flow,
passage turbulence intensity, and mass flow. The aerodynamic characteristics of the
mass flow were found to be reasonably independent of rotational speed, but highly
dependent upon rotor geometry. Johnson et al. [16, 13] used PIV (Particle Image
Velocimetry) to measure air velocities through a high solidity radial flow fan
utilized as a vented brake rotor. Sakamoto [23] analyzed the basic equations for
heat convection, with brake tests and measurements of the flow through the fins.
Repmann [22] examined a brake rotor geometry with straight angled cooling ducts
through numerical studies by CFD (Computational Fluid Dynamics). Gao and Lin
[10] presented an analytical model for the determination of the contact temperature
distribution on the working surface of a brake disc, using a transient finite element
technique. Choi and Lee [9] performed a transient thermoelastic analysis of disc
brakes in repeated braking applications, using a finite element method with
frictional heat generation. Voller et al. [25] studied automotive disc brake thermal
characteristics experimentally, using a specially developed spin rig.
Research work has also been conducted to disclose the effects of design variables
on thermal performance of disc brakes. Grieve et al. [11] performed parametric
sensitivity studies to define suitable design-material combinations for a disc brake
prototype. Sun [24] studied the effects of modifying the rotor, dust shield, wheel
and air deflector on the brake equilibrium temperature rise, under a cyclical braking
cycle in simulated mountain test cases.
10
Much attention has been focused on improving the thermal performance of brake
discs. Numerical simulations and Computational Fluid Dynamics (CFD) are
commonly applied to brake disc thermal performance analyses [7]. Many
experimental studies have also been conducted to measure the air flow and
temperature field inside the discs under braking operations. It has been
demonstrated that CFD simulation results have achieved good agreement with
those based on experimental studies.
Various CFD tools for a computer-aided design have been applied to the
development of brake discs. For example, FLUENT has been used by the
automotive industry to model and design various configurations of brake discs [7].
From a design point of view, the drawback of most commercial packages such as
FLUENT is that they only provide analysis of the brake discs whose design
variables have been specified. Design syntheses including parametric and
sensitivity analyses have been difficult. Instead, engineers must decide by trial and
error to change design variable values and re-perform the analysis until a set of
performance criteria becomes acceptable [21]. This “manual” process, often
accompanied by prototype testing, can be difficult and time-consuming for
complex systems, such as vehicle disc brakes.
1.3 Research outline
11
A literature survey reveals some limitations of previous work in geometrical
optimization of automotive brake rotors.
Unlike these past studies, the current research investigates the thermal
characteristics of brake discs with both analytical and numerical approaches. In this
thesis, analytical models and the corresponding solutions are presented first. Based
on the analytical solution, CFD sensitivity studies for more realistic brake disc
models are then conducted. The thesis then discusses and concludes the insightful
findings derived from the research.
In addition, past attention has been paid to the investigation of the effects of
geometrical parameters of pillar post rotors on vehicle brake thermal performance.
Moreover, there is an ongoing debate on which design configuration, between the
pillar post and vane rotors, is more effective for heat convection. Sun [7] observed
that the average heat transfer coefficients of the pillar post rotor are approximately
25% lower than those of the vane rotor at vehicle speeds of 24 km/h and 64 km/h.
In contrast, some manufacturers insist that pillar post rotors can provide better
thermal performance than vane rotors.
In this thesis, different pillar post rotor models are generated and the corresponding
numerical simulations are conducted, in order to investigate the effects of various
geometrical parameters on the thermal performance. To evaluate the effect of
12
geometrical configurations of the rotor passages on the thermal performance, the
pillar post and vane rotors are compared.
In past research, little attention has been paid to the potential of a comprehensive
automated design synthesis process of brake discs [21]. Numerical optimization
may help automate the design synthesis by altering variable values in a search to
optimize performance criteria, subject to constraints. Due to the complexity of the
CFD model of brake discs, even a single iteration of the optimization may take a
large amount of time. If these optimizations are implemented in a massively-
parallel computer system, the computation time could easily be reduced,
approximately by a factor equal to the total number of computers.
To explore the potential of an automated design synthesis process of brake discs, an
integrated design synthesis will be proposed and implemented. Commercial
software GAMBIT is used for geometrical modeling and automatic mesh
generation for vented discs. Then, the CFD package, FLUENT, is employed to
simulate the air flow through the vented disc. To automate the design process of the
disc rotor, a software framework, iSIGHT, is used to integrate the geometrical
modeling using GAMBIT and numerical simulations based on FLUENT. Through
this integrated design synthesis process, the disc rotor geometrical optimization is
performed using design of experiment (DOE) studies. The main design criterion of
the geometrical optimization is to maximize the convective cooling by increasing
the heat transfer rate inside the rotor passages.
13
Chapter 2 discusses the analytical solution to vented vane and pillar post rotors.
Sensitivity studies for vented vane rotors and pillar post rotors are conducted in
Chapter 3 and Chapter 4. Chapter 5 and Chapter 6 present geometrical optimization
for 2-D models and 3-D models. Finally conclusions and recommendations are
made in Chapter 7.
14
Chapter 2
Thermal Analysis of Vented Rotors with an
Analytical Method
2.1. Estimation of generated heat flux
The total heat flux generated by friction between the rotor and pad surfaces (on
each side of the disc rotor) is given by:
ωμpRq =′′ (2.1)
where μ: pad friction coefficient
p: contact pressure between rotor and pad surfaces
R: distance between node and rotor center
ω: angular velocity of the rotor
With equation (2.1), the heat flux can be calculated directly, but the distribution of
pressure is uneven. Therefore it is difficult to calculate the contact pressure.
Another method to estimate the heat flux can be used based on kinetic energy. A
moving vehicle has a certain amount of kinetic energy, and the brakes must
dissipate this energy in order to stop the vehicle. Each time, when we stop a vehicle,
the brakes convert the majority of kinetic energy into heat generated by friction
15
between the pads and the discs. Therefore, when a vehicle is braking from an
initial speed to zero, the heat generation can be estimated by:
20
20 2
121 ωImvKQ E +=Δ= (2.2)
where v0: the initial speed
I: the mass moment of inertia
ω0: initial angular velocity
2.2 Heat dissipation
2.2.1 Heat conduction
There are two paths of heat conduction from the discs, one through the bearing
assembly (which should be avoided) and another through the wheel carrier, which
is the major conductive path. The heat flow can be estimated by Fourier’s law of
heat conduction as follows:
conddTQ kAdx
= − (2.3)
The small area A and very low temperature difference (TD -TC) limits the amount of
power dissipated by conduction. Therefore, the heat conduction can become
negligible in brakes.
2.2.2 Radiation
The radiation heat dissipation is defined by:
( )44∞−= TTAQ DDradiationradiatin σε (2.4)
16
where Qradiation: the thermal energy dissipated by radiation
σ: the Stefan-Boltzmann constant, 3.56*10-5 Nm/m2K
ε: rotor surface emissivity ( for cast iron, ε=0.55 )
ADradiation: the surface area of the disc radiation heat
TD: the average disc surface temperature
T∞: the ambient air temperature
Figure 2.1: Radiative heat transfer coefficient versus rotor temperature [13]
As shown in figure 2.1, where the data were collected by experimentation [13], the
radiation cooling does not occur until a high brake temperature is attained.
However, for hot brakes rotating at a low speed, radiation cooling may be
predominant.
17
2.2.3 Convection heat transfer
The major aim of designing brake discs is to improve the convection dissipation of
disc braking systems. In operations of braking systems , convection is the most
important mode of heat transfer, dissipating the highest proportion of heat to
surrounding air. The current research focuses on heat convection of disc rotors.
2.3. Convection heat transfer analysis
In disc braking systems, there are two types of rotors, a pillar post rotor and a vane
rotor, shown in figure 4 and figure 5 respectively.
Figure 2.2: A pillar post rotor
18
Figure 2.3: A vane rotor
When the vehicles are moving at high speeds, the disc braking systems dissipate
the heat by passing the air from the inlet to outlet. In these two different rotors,
there is a difference in heat convection transfer, which will be addressed separately.
2.3.1 Velocity distributions
To simplify the complex motion of air in a rotor, a simplified duct model was
developed.
D
a y
x
z
q”
q”
u
19
Figure 2.4: Straight duct (a simplified model for disc rotors) with a rectangular
section
where
q”: constant heat flux at the sides of the duct
u: velocity of air flow from the left inlet to the right outlet
D: thickness of the vane or pillar post rotors
a: length of one vane or pillar post section
For this problem, the model is based on the following assumptions:
1) Continuum flow
2) Newtonian fluid
3) Steady state
4) Laminar flow
5) Constant properties (density, conductivity, specific heat, and viscosity)
6) Uniform surface heat flux
7) Negligible gravitational effect
2.3.1.1Conservation of mass (continuity equation)
The governing equation that describes the mass conservation is expressed as:
0)( =∂∂
+∂∂
+∂∂
+∂∂
+∂∂
+∂∂
+∂∂
zw
yv
xu
zw
yv
xu
tρρρρρ (2.5)
For an incompressible flow, ρ is constant. Thus
20
,0=∂∂
tρ ,0=
∂∂
xρ 0=
∂∂
zρ
In the case of parallel flow,
v=0, w=0.
Therefore, the continuity equation can be simplified as
0=∂∂
xuρ
0=∂∂
xu (2.6)
2.3.1.2. Conservation of momentum (Navier-Stokes equations)
For the system shown in figure 2.4, the conservation of y-momentum equation can
be described as
)()( 2
2
2
2
2
2
zv
yv
xv
ypg
zvw
yvv
xvu
tv
y ∂∂
+∂∂
+∂∂
+∂∂
−=∂∂
+∂∂
+∂∂
+∂∂ μρρ (2.7)
Based on the previous assumptions, this can be reduced to
0=∂∂
−yp (2.8)
This implies that pressure in the y direction remains constant.
For z direction momentum,
)()( 2
2
2
2
2
2
zw
yw
xw
zpg
zww
ywv
xwu
tw
z ∂∂
+∂∂
+∂∂
+∂∂
−=∂∂
+∂∂
+∂∂
+∂∂ μρρ (2.9)
With the assumption, equation (2.9) can be simplified as
0=∂∂
−zp (2.10)
21
which implies that the pressure in the z direction remains constant.
The conservation of momentum equation in the x direction can be expressed as
)()( 2
2
2
2
2
2
zu
yu
xu
xpg
zuw
yuv
xuu
tu
x ∂∂
+∂∂
+∂∂
+∂∂
−=∂∂
+∂∂
+∂∂
+∂∂ μρρ (2.11)
For steady state tu∂∂ =0. Assume there is no air flow in the y and z directions. Thus
the momentum equation can be simplified as
(μρρμ +∂∂
−=∂∂
xxu )2
2
2
2
2
2
zu
yu
xu
∂∂
+∂∂
+∂∂
For fully developed flow, xu∂∂ =0. So the above equation can be reduced to
)( 2
2
2
2
zu
yu
xp
∂∂
+∂∂
=∂∂ μ (2.12)
For two parallel in the y direction, equation (2.12) can be reduced to
)( 2
2
yu
xp
∂∂
=∂∂ μ (2.13)
If there are two parallel walls in the z direction, equation (2.12) can be reduced to
)( 2
2
zu
xp
∂∂
=∂∂ μ (2.14)
This approach would solve equations (2.13) and (2.14) separately, and then
combine them together.
In equation (2.13), on the left side, it is a function of x, and on the right side it is a
function of y. Thus this equation must be constant, as shown below.
22
tconsyu
xp tan2
2
=∂∂
=∂∂ μ
To solve it, we need to integrate it twice, as shown in equation (2.15)
22
121)( cycy
xyu ++⋅
∂∂
=ρ
μ (2.15)
Where, C1 and C2 are constant.
To determine the coefficients C1 and C2, the appreciate boundary conditions should
be selected as blow
(1) y=D/2 u(D/2)= 0
(2) y=-D/2 u(-D/2)= 0 (2.16)
The boundary conditions mean that on the upper and lower boundary layers, the
velocity is 0.
Substituting the boundary conditions into equation (2.15), yields the following
equations
(1) 22 )2/()4/(
210 1 cDcD
x++⋅
∂∂
=ρ
μ
(2) 22 )2/()4/(
210 1 cDcD
x+−⋅
∂∂
=ρ
μ (2.17)
Define coefficients C1 and C2
(1) C1=0
(2) C2=- x
D∂∂ρ
μ8
2
(2.18)
Substituting C1 and C2 into equation (2.15), yields
))2/
(1(8
)( 22
Dy
xpDyu −∂∂
−=μ
(2.19)
23
The mean velocity can be obtained by integrating the velocity equation over the
whole area, as shown below,
xpDdayDy
xp
aDudA
Ayu
D
DAC
C C∂∂
=⎥⎦
⎤⎢⎣
⎡−
∂∂
== ∫∫−
μμ 12))
2((
2111)(
22
2
22 (2.20)
The velocity ratio equation can be expressed by the velocity over the mean velocity
as follows,
2
22
2
22
))2/((6
12
))2
((21
)(D
Dy
xpD
Dyxp
uyu −
=
∂∂
−
−∂∂
=
μ
μ (2.21)
In the same way, equation (2.14) in the z-direction can be solved.
The parallel walls in the z-direction and the boundary conditions are:
(1) z=a/2 u(a/2)=us
(2) z=-a/2 u(-a/2)=us (2.22)
On the walls, the velocity is constant. For vane rotors, the wall velocity is zero, and
for pillar post rotors, the velocity is a constant value.
We will assume us=ku0 , where
k=0, for vane rotors
k=constant, for pillar post rotors.
Integrating equation (2.14) twice yields,
212
21)( czcz
xpzu ++∂∂
=μ
(2.23)
Where, C1 and C2 are constant coefficients.
Appling the boundary conditions yields
24
212
212
2)
2(
21
2)
2(
21
cacaxpu
cacaxpu
s
s
+−∂∂
=
++∂∂
=
μ
μ (2.24)
Solving the above equations,
(1) C1=0
(2) C2= su - 2)2
(21 a
xp∂∂
μ (2.25)
Substituting C1 and C2 into equation (2.23),
u(z)= ( ) ⎟
⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂∂
−−∂∂
−=∂∂
−+∂∂
xpa
ua
zxpaa
xpuz
xp s
s
μμμμ
82/
18
)2
(21
21
22
2222 (2.26)
Set u0=-xpa∂∂
μ8
2
as the centerline (peak) velocity, so the above equation becomes
u(z) =u0(1-( )2
2
2/az -k) (us=ku0 (0<k<1)) (2.27)
2.3.1.3 The velocity distribution:
From the previous analysis, the velocity equation can be obtained by combining
equations (2.19) and (2.27), yielding
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−= k
az
Dyuzyu
22
0 2/1
2/1, (2.28)
Where k=0, for vane rotors
k=constant, for pillar post rotors
This equation satisfies the boundary conditions:
25
(1) u=0 at y=±D/2
(2) u=us at y=0, z= ±a/2 (2.29)
To determine u0, substitute equation (2.28) into (2.14), yielding
⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛−+−−⎟
⎠⎞
⎜⎝⎛−=
∂∂
+∂∂
= )8(2/
1()8)(2/
1()( 2
2
2
2
02
2
2
2
aDy
Dk
azu
zu
yu
dxdp μμ (2.30)
Integrating 2.30 over the entire cross section gives
aDdxdp = dydz
aDy
Dk
azu
D
D
a
a ⎥⎥⎦
⎤
⎢⎢⎣
⎡−⎟
⎠⎞
⎜⎝⎛−+−−⎟
⎠⎞
⎜⎝⎛−∫∫
−−
)8(2/
1()8)(2/
1( 2
2
2
22
2
2
2
0μ
= ⎥⎦
⎤⎢⎣
⎡+⎟
⎠⎞
⎜⎝⎛ +−
aD
Daku
231
316
0μ (2.31)
From equation (2.31), the centerline equation can be expressed by
22
22
0
23116
3
Dak
Dadxdpu
+⎟⎠⎞
⎜⎝⎛ +
−=μ
(2.32)
Substituting equation (2.32) into (2.30) yields the following velocity distribution
equation
( ) ⎥⎦
⎤⎢⎣
⎡−−⎥⎦
⎤⎢⎣⎡ −
++⋅⋅−= )
2/1()
2/(1
)231(16
3),( 2
22
22
22
ka
zD
y
Dak
Dadxdpzyu
μ (2.33)
The mean velocity is obtained by integrating equation (2.33) over the entire cross
section.
)231(
94
))2/
(1()2/
(11),(1),(
0
2
2
220
2
2
ku
dydzka
zD
yuaD
dAzyuA
zyu
D
D
a
aAC
C C
−=
⎥⎦⎤
⎢⎣⎡ −−⎥⎦⎤
⎢⎣⎡ −== ∫∫∫
−− (2.34)
26
The velocity ratio is the velocity distribution equation over the mean velocity
equation.
)231(
)2/
(1)2/
(1
49
)231(
94
))2/
(1()2/
(1),(
22
0
220
k
az
Dy
ku
ka
zD
yu
uzyu
−
⎥⎦⎤
⎢⎣⎡ −⎥⎦⎤
⎢⎣⎡ −
=−
⎥⎦⎤
⎢⎣⎡ −−⎥⎦⎤
⎢⎣⎡ −
= (2.35)
2.3.2 Heat transfer coefficient and Nusselt number
2.3.2.1 Energy equation
The governing equation that describes the air flow energy equation is expressed as
μφρ +∂∂
+∂∂
+∂∂
=∂∂
+∂∂
+∂∂
+∂∂ )()( 2
2
2
2
2
2
zT
yT
xTk
zTw
yTv
xTu
tTc p (2.36)
where:
2222222 )(32)()()()()()(2
zw
yv
xu
zu
xw
yw
zv
xv
yu
zw
yv
xu
∂∂
+∂∂
+∂∂
−⎥⎦
⎤⎢⎣
⎡∂∂
+∂∂
+∂∂
+∂∂
+∂∂
+∂∂
+⎥⎦
⎤⎢⎣
⎡∂∂
+∂∂
+∂∂
=φ
(2.37)
The air flow is steady, so the temperature does not change with time.
tT∂∂ =0 (2.38)
There is no axial variation in the z-direction ( 0=∂∂z
) and it is a parallel flow (v=0).
Applying these assumptions to equation (2.36) gives
uc pρ μφ+∂∂
+∂∂
=∂∂ )( 2
2
2
2
yT
xTk
tT
(2.39)
where: 222 )(32)()(2
xu
yu
xu
∂∂
−∂∂
+∂∂
=φ
27
From the continuity equation, 0=∂∂
xu . Substituting this equation,
2)(yu∂∂
=φ (2.40)
Equation (2.39) can be reduced to
22
2
2
2
)()(yu
yT
xTk
xTucp ∂
∂+
∂∂
+∂∂
=∂∂ μρ (2.41)
Neglecting the viscous dissipation term μΦ, the energy equation becomes
)( 2
2
2
2
zT
yT
ck
xTu
p ∂∂
+∂∂
=∂∂
ρ
or
)( 2
2
2
2
zT
yT
xTu
∂∂
+∂∂
=∂∂ α (2.42)
To solve equation (2.42), the same method to solve the velocity equation can be
used here. So sub-divide equation (2.42) into two equations (2.43) and (2.44) and
then to combine them together.
xTu
yT
∂∂
=∂∂
α2
2
(2.43)
xTu
zT
∂∂
=∂∂
α2
2
(2.44)
To solve equation (2.43), the temperature distribution only depends on y, not x, so
xT∂∂ can be assumed constant.
Integrating equation (2.43) twice, gives
21
2
2)( cycy
xTuyT ++∂∂
=α
(2.44)
28
where C1 and C2 are constant coefficients
The temperature boundary conditions are shown as follows. On the surface of the
rotor, the temperature is constant Tw, and in the centerline of the duct, there is no
temperature gradient.
(1) y=D/2 T=Tw
(2) y=0 0=∂∂
yT (2.45)
Substituting the boundary conditions into equation (2.44), and solving the two
equations, yields
(1) c1=0
(2) c2= Tw- xTuD∂∂
α8
2
(2.46)
Introducing C1 and C2 into equation (2.44),
T(y) = ww TDyxTu
xTuDTy
xTu
+−∂∂
=∂∂
−+∂∂ ))
2(
2(
2822
222
ααα (2.47)
From the analysis above: if the temperature distribution is only analyzed in y
direction the equation is
))2/
(1(8
)( 22
Dy
xTuDTyT w −∂∂
−=α
(2.48)
In the same way, the equation (2.44) can be solved as
))2/
(1(8
)( 22
az
xTuDTzT w −∂∂
−=α
(2.49)
Therefore, the temperature distribution in the y and z directions can be expressed
by combining equations (2.48) and (2.49) together
29
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛ −−=
22
0 2/1)
2/(1),(
az
DyTTzyT w (2.50)
Therefore, if we substitute equation (2.50) into (2.42) and then integrate over the
entire cross section, the centerline temperature equation can be derived as follows,
dydzaD
yDa
zTxTuDa
D
D
a
a∫∫−−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛−+⎟
⎠⎞
⎜⎝⎛ −=
∂∂
⋅2
2
2
2
22
0
2
2
82/
18)2/
(1α
⎟⎠⎞
⎜⎝⎛ +=
Da
aDT
316
0 (2.51)
Rearranging equation (2.51), gives the following centerline temperature equation,
22
22
0 163
DaDa
xTuT
+∂∂
=α
(2.52)
Substituting equation (2.52) into (2.49), gives the following temperature
distribution equation,
⎟⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ −
+∂∂
−= 2222
22
)2/
(1)2/
(1163),(
az
Dy
DaDa
xTuTzyT w α
(2.53)
In equation (2.53), there is still an unknown dxTd To solve, a heat transfer analysis
in a control volume is introduced as shown in figure 2.5. In this control volume,
heat is added at the surface at a uniform flux qw”, then the input energy is Tcm p&
and the output energy is ( )dxTcmdxdTcm pp && + .
30
Figure 2.5 Heat transfer analysis in a control volume
Applying conservation of energy, yields
( )
( )Tcmdxdpq
or
dxTcmdxdTcmpdxqTcm
pw
ppwp
&
&&&
=′′
+=′′+
(2.54)
This equation can be simplified to
p
w
cmpq
dxTd
&
″= (2.55)
Introducing equation (2.55) into (2.53), gives
⎟⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ −
″⋅
+−= 22
22
22
)2/
(1)2/
(1163),(
az
Dy
cmpq
DaDauTzyT
p
ww &α
(2.56)
where p=2(a+D) and Daum ρ=& . Substituting into equation (2.55), we have
( ) ⎟⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ −
++″
−= 2222 )
2/(1)
2/(1)(
83),(
az
Dy
DacaDDaqTzyT
p
ww αρ
(2.57)
pdxqw′′
Tcm p&
dx
( )dxTcmdxdTcm pp && +
CV
31
The mean temperature distribution equation can be achieved by integrating the
temperature equation over the whole cross section as follows
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎠⎞
⎜⎝⎛ −⎟⎠⎞
⎜⎝⎛ −
″
+−== ∫ ∫∫
∫
− −
2
2
2
2
2222
22
)2/
(1)2/
(11631
a
a
D
D p
ww
A
A dydza
zD
ycm
pqDa
DauTuuaDudA
uTdAT
&αρ
ρρ
ρ
p
ww cm
pqDa
DauT&
″
+−= 22
22
121α
(2.58)
Substituting p=2(a+D) and Daum ρ=& , into above equation, gives
( )22
)(61
DakaDDaqTT w
w ++″
−= (2.59)
2.3.2.2 Heat transfer coefficient and Nusselt number
The heat transfer coefficient can be obtained by substituting equation (2.59) into
the heat transfer coefficient equation.
aDDaDak
TTq
hw
w
)()(6 22
++
=−′′
= (2.60)
The Nusselt number equation can be obtained from Nu(De)=hDe/k, where
De=equivalent diameter
DaaD
DaaD
pADe
+=
+== 2
)(244 (2.61)
Substituting De into the Nusselt number equation, yields
2
2
22
)(
21
112
2*)(
)(6*
⎟⎠⎞
⎜⎝⎛++
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
=++
+==
aD
aD
aD
DaaD
aDDakDak
kDehNu De (2.62)
2.3.3 Temperature distribution in 3 dimensions
32
Equation (2.56) gives the temperature distribution in the y and z directions, but
there is an unknown variable Tw, which is not constant and difficult to be
determined. Therefore, the temperature equation must be solved by another method.
Integrating equation (2.56)
p
w
cmpq
dxTd
&
″= gives
1)( cxcmpq
xTp
w +′′
=&
(2.63)
where, C1 is a constant coefficient.
The boundary conditions occur at the entrance of the duct, where the temperature is
the ambient air temperature.
x=0 inTT = (ambient air temperature)
Introducing the boundary conditions into equation (2.63),
inp
w Txcmpq
xT +′′
=&
)( (2.64)
where, Daum ρ=& and p=2(D+a). Substituting into equation (2.64),
inp
w TxDacu
aDqxT +
+′′=
ρ)(2
)( (2.65)
The governing energy equation is
xTu
yT
∂∂
=∂∂
α2
2
Integrating the equation twice, gives
21
2
2)( cycy
xTuyT ++∂∂
=α
(2.66)
33
Appling the boundary conditions (y=0, 0=∂∂
yT ) to equation (2.65), gives
)(2
)(2
xgyxTuyT y+∂∂
=α
(2.67)
where gy(x) represents the centerline temperature
In the same way, equation (2.44) can be solved to give
)(2
)(2
xgzxTuzT z+∂∂
=α
(2.68)
Combining equations (2.67) and (2.68), the temperature distribution equation
becomes
T=T0y2z2+g(x) (2.69)
Substituting equation (2.69) into energy equation (2.43) to find T0,
20
20 zTyT
xTu
+=∂∂
α (2.70)
Integrating equation (2.70) over the entire cross section
∫∫∫∫−−−−
+=∂∂ 2
2
220
2
2
2
2
2
2
)(
D
D
a
a
D
D
a
a
dydzzyTxTu
α
xTu
DaT
∂∂
+=
α33012 (2.71)
Substituting equation (5.71) back into (2.69), gives
)(12 2233 xgzy
xTu
DaT +
∂∂
+=
α (2.72)
Therefore, the mean temperature can be achieved by integrating the temperature
equation over the whole area.
34
)()(12
)(
)(
33
22
xgcmpqu
DaDaxT
udA
uTdAxT
p
w
A
A
+′′
+=
=∫
∫
&α
ρ
ρ
(2.73)
Combining equation (2.73) and (2.65), g(x) can be determined as
xaDcu
Daqc
qDaDaaDTxg
p
w
p
win ραρ
)(2)(6)()( 33
+′′+
′′++
−= (2.74)
Substituting equation (2.74) into (2.71), yields
p
w
p
w
p
win c
qDaDaaDx
aDcuDaq
zyc
qaDDa
DaTTαρραρ
′′++
−+′′
+′′
++
+=)(6)()(2
)()(
)(2433
2233
Rearranging the above expression gives
p
w
p
w
p
win c
q
aD
aD
aD
xcu
aD
qaD
Dayz
cq
aD
aD
aD
TTαρραρ
′′
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
⎟⎠⎞
⎜⎝⎛+
−′′⎟
⎠⎞
⎜⎝⎛ +
+⎟⎟⎠
⎞⎜⎜⎝
⎛′′
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
+=3
2
22
22
3
2
16
12
1
24
(2.75)
This is the final temperature distribution equation in the x, y and z directions.
The maximum temperature occurs at y=a/2 and z=D/2, so the temperature
distribution on the corners of the walls will be:
p
w
p
w
p
winw c
qDaDaaDx
aDcuDaq
cq
DaaDDaTT
αρραρ′′
++
−+′′
+′′
++
+=)(6)()(2
)(2)(3
3333max
xcu
aD
qaD
cq
aD
aD
aD
TTp
w
p
winw
ραρ
′′⎟⎠⎞
⎜⎝⎛ +
+′′
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
+=12
13
4
3
2
max (2.76)
35
where ⎟⎠⎞
⎜⎝⎛ −
+⎟⎠⎞
⎜⎝⎛ +
−= kDak
Dadxdpu
231
23112
122
22
μ
or ⎟⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛ +
⎟⎠⎞
⎜⎝⎛
−= k
aDk
aD
dxdpu
231
231
121
2
2
μ (2.35)
For vane rotors, k takes the value of zero. For pillar post rotors, k is constant
(0<k<1)
2.3.4 Velocity, temperature and Nusselt number analysis
2.3.4.1 Velocity equation
As shown in section (2.3.1.3), the velocity equation can be expressed as
⎟⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛ +
⎟⎠⎞
⎜⎝⎛
−= k
aDk
aD
dxdpu
231
231
121
2
2
μ (2.35)
where k is the pillar post coefficient, For vane rotors, k is zero, and for pillar rotors,
k is a constant coefficient.
A close observation indicates that the velocity depends on three factors involving
the rotor ratio D/a, pillar post coefficient k and dp/dx. These factors are examined
as follows.
36
Figure 2.6 shows that the velocity changes with rotor ratio, D/a, when other
variables remain constant.
Figure 2.6: Velocity/u0 vs. the rotor ratio, D/a
As shown in figure 2.6, the velocity increases with an increase of rotor ratio. This
means that a high speed in disc rotors benefits from increasing the rotor gaps of
disc brakes, or the number of vanes or pillar posts.
Figure 2.7 indicates that the non-dimensional mean velocity changes with the pillar
post coefficient, when other variables remain constant.
37
Figure 2.7: Non-dimensional mean velocity vs. pillar post coefficient
Figure 2.7 shows the variation of non-dimensional mean flow velocity with the
change of pillar post coefficient k. When k has a value of 0 which is the case of a
vane rotor, the maximum mean velocity occurs. With an increase of k, the mean
velocity drastically decreases, whereas the tangential air flow velocity increases.
Figure 2.7 indicates that vane rotors have better thermal performance than pillar
post rotors, since the former have a higher mean air flow velocity than the latter.
For the term dp/dx, it is different to be determined, because it changes with the
speed of the vehicle. The air velocity inside the rotors can be measured by
experimentations. The trend is linear as depicted by figure 2.8.
38
Figure 2.8: Air flow rate versus revolutions of wheels [5]
From figure 2.8, the air flow rate is increasing, when the speed of the vehicle is
increasing. The trend is given by
vu 49.0=
where: v (m/s) is the speed of the vehicle.
2.3.4.2 Nusselt number equations
The Nusselt number equation has been derived in section (2.3.2.2).
2
2
)(
21
112*
⎟⎠⎞
⎜⎝⎛++
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
==
aD
aD
aD
kDehNu De (2.45)
From the Nusselt number equation, the Nusselt number is only affected by the rotor
ratio, D/a.
0
20
40
60
80
100
120
140
0 500 1000 1500 2000 2500
Revolution (1/min)
Flow
rat
e (l/
s)
39
Figure 2.9: Rotor ratio vs. Nusselt number
Figure 2.9 shows the relation between Nusselt number and rotor ratio (D/a). To
achieve better thermal performance (larger Nusselt number), we need to increase
the rotor ratio. In practice, the thickness of rotors (D) does not change drastically
due to the limited space and stiffness of rotors. Therefore, to increase the Nusselt
number, we need to increase the width (a). But the width cannot be increased too
much, because of the limitation of rotor stiffness. So to increase the Nusselt number,
we should decrease the width of the rotor sections, in other words, increase the
number of vanes or pillar posts.
40
The types of rotors do not affect the Nusselt number. So pillar post and vane rotors
have the same Nusselt number.
2.3.4.3 The maximum temperature equation
The maximum temperature distribution equation has been solved in section (2.3.3).
xcu
aD
qaD
cq
aD
aD
aD
TTp
w
p
winw
ραρ
′′⎟⎠⎞
⎜⎝⎛ +
+′′
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
+=12
13
4
3
2
max (2.75)
where, ⎟⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛ +
⎟⎠⎞
⎜⎝⎛
−= k
aDk
aD
dxdpu
231
231
121
2
2
μ (2.45)
From experimentation, vu 49.0= and v is the speed of the vehicle.
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛⎟⎠⎞
⎜⎝⎛ +
+
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
′′+= x
vaD
aD
aD
aD
aD
cq
TTp
winw
49.0
12
13
4
3
2
max
αρ
(2.76)
where α=22.5*10-6, when T=300K, α<<0.49v/x.
Therefore the second item in the bracket can be neglected. Equation (2.76) can be
reduced to:
αρ
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
′′+=
3
2
max
13
4
aD
aD
aD
cq
TTp
winw (3.77)
41
Figure 2.10: Non-dimensional temperature change versus rotor ratio
Figure 2.10 shows the relationship between non-dimensional temperature changes
versus the rotor ratio. It can be concluded that when the rotor ratio takes the value
of 1, the maximum temperature on the surface of the wall is the highest. When the
rotor ratio is 0 or infinity, the maximum temperature on the surface of the wall is
lowest, which is the same as the results obtained from the Nusselt number equation.
2.4. Summary
Among the three modes of heat dissipation (heat conduction, heat radiation and
heat convection) heat conduction has the least effect on a rotor’s heat dissipation.
The heat radiation only plays an important role at high temperatures and low speeds.
The heat convection is considered the most important mode of heat dissipation.
42
From the air flow velocity analysis, it is observed that the velocity inside the discs
changes with pillar post coefficient and rotor ratio. The velocity in vane rotors is
higher than pillar post rotors. The velocity increases when the gap between the
discs increases or the numbers of vanes or pillar posts increases.
From the Nusselt number analysis, it was found that when the rotor ratio is 1, there
is a minimum Nusselt number of 6, while the maximum Nusselt number of 12
occurs when the rotor ratio is 0.
43
Chapter 3
Thermal Performance Analysis of Vane Rotors
Based on Numerical Simulations
The main limitation of the analytical model discussed in Chapter 2 is that it is based
on the assumption of laminar flow and a simplified geometry. In practice, the air
flow through the passage of vane rotors is complex turbulent flow. With an
analytical method, it is difficult to determine the effects of geometrical parameters
of rotors on thermal performance of disc brakes. These geometry parameters
include the vane angles, radii of curvature and short-long ratios. The short-long
ratio refers to the ratio of lengths of the short and long vanes.
3.1 GAMBIT models
In order to improve the accuracy of simulation results, a commercial CFD solver,
Fluent, was used to simulate the air flow inside different rotors and find the
optimized shape of rotors. The 2-D mesh was built with GAMBIT, using boundary
conditions of pressure at the inlet, pressure at the outlet and the wall vane, as shown
in figure. 3.1.
44
Figure 3.1: 2-D mesh model of a 40-vane rotor
The vane rotor has an inner radius of 6.5 cm and an outer radius of 17.1 cm, with a
vane thickness of 0.28 cm and a length of 5.65 cm.
3.2 Numerical Simulations Using FLUENT
In the CFD simulation, the following assumptions have been made:
• Steady state air flow
• Segregated solver and implicit formulation
• Standard k-epsilon viscous turbulence model
• Standard wall functions
• Moving reference frame at a constant velocity of 44 rad/s
• Vane- wall interface with a constant temperature of 900 K
45
• Momentum-Second Order Upwind Scheme
• Turbulence Kinetic Energy-Second Order Upwind Scheme
• Turbulence Dissipation Rate-Second Order Upwind Scheme
• Energy-Second Order Upwind Scheme
Steady-state conditions were assumed because the time dependent behaviour was
not needed. There are three different solvers in FLUENT, namely segregated,
coupled implicit and coupled explicit solvers. They differ in the ways that they
solve equations sequentially or simultaneously. The segregated solver traditionally
has been used for incompressible and mildly compressible flows. The coupled
approach, on the other hand, was originally designed for high-speed compressible
flows [29]. The air flow inside the vehicle is assumed incompressible, so the
segregated solver and implicit formulation were used. The standard - model is
a semi-empirical model based on model transport equations for the turbulence
kinetic energy ( ) and dissipation rate ( ) [29]. It is used for fully turbulent flows,
particularly in the inlet and outlet areas of the rotors. FLUENT can also model the
air motion inside rotors by using the moving reference frame at constant speed. The
non-equilibrium wall functions are recommended for use in complex flows
involving separation, reattachment, and impingement where the mean flow and
turbulence are subjected to strong pressure gradients and change rapidly [29].
Standard wall functions were selected. By using the second-order upwind scheme,
higher accuracy was achieved.
46
In the simulations, the heat transfer rate was found. The predicted velocity
distribution is shown in figure 3.2. The predicted pressure distribution is shown in
figure 3.3.
Figure 3.2: Velocity contours of a 40-vane rotor
Figure 3.3: Static pressure contours of a 40-vane rotor
47
The velocity vector distribution and turbulence distribution are shown in figures.
3.4 and 3.5, respectively.
Figure 3.4: Velocity vector distribution of a 40-vane rotor
Figure 3.5: Turbulence distribution of a 40-vane rotor
48
To increase the thermal performance of rotors, both the air velocity and heat
transfer rate should be increased. The models with different vane numbers, vane
angles and radii of curvature will be investigated in the following sub-sections.
3.3 Results and discussion
3.3.1 Effects of vane numbers
Five different rotors with 24, 32, 40, 50 and 60 vanes were chosen to evaluate the
effects of vane numbers on the change of heat transfer rate. If more heat is
transferred from the rotors, there will be better cooling performance. Figure 3.6
shows the relationship between vane numbers and the heat transfer rate at angular
velocities of 44, 88 and 120 rad/s, respectively.
As shown in figure. 3.6, with an increase of vane numbers, the heat transfer rate
increases as well. This observation is consistent with the result derived from the
previous analytical solution. The comparison of velocity distributions for five
different models with 24, 32, 40, 50, 60 vanes is shown in figure 3.7, from which it
is observed that the velocity distribution is uneven in each section of vane rotor.
The rotors are all rotating at a constant velocity in the clockwise direction. The
rotors with lower vane numbers have a larger non-uniformity of velocity
distribution than those with more vane numbers. This occurs because the direction
of air flow is not straight along the vanes, as shown in figure. 3.4. The air flow
49
vectors in rotors with fewer vanes have larger angles along the vanes than those
with more vanes.
Figure 3.6: Vane numbers vs. heat transfer rate
Figure 3.7: Velocity field for 24, 32, 40, 50 and 60 vanes
70
170
270
370
470
570
24 32 40 50 60Vane Numbers
Hea
t Tra
nsfe
r Rat
e (k
w)
angular velocity 44 angular velocity 88 angular velocity 120
50
3.3.2 Effects of vane angles
To investigate the influence of vane angles on the heat transfer rate, two sets of
rotor models with 32 and 40 vanes are developed. For each set of rotor models, the
vane angles take the values of 100, 200, 300 and 400, respectively. With these rotor
models, numerical simulations are conducted at angular velocities of 44, 88 and
120 rad/s, respectively.
Figure 3.8: Vane angles vs. heat transfer (32 vanes)
The simulation results, as shown in figures. 3.8 and 3.9, indicate that when the vane
number is 32, the vane angle does not contribute to a rise of the heat transfer rate.
On the contrary, the rate drastically decreases when the angle increases to 10
degrees and then remains almost constant until the angle reaches 30 degrees. With
40 vanes, the rate increases slightly when the angle is 10 degrees, and then slightly
drops until the angle rises to 30 degrees. Therefore, the heat transfer rate does not
100
150
200
250
300
350
400
0 10 20 30
Vane Angles(degrees)
Heat
Tra
nsfe
r Rat
e(kw
)
Angular Velocity 44 Angular Velocity 88 Angular Velocity 120
51
benefit from the vane angle rise at fewer vane numbers of rotors. However,
numerical experiments show that this rate does increase with a rise of angle and
more vane numbers.
Figure 3.9: Vane angles vs. heat transfer (40 vanes)
3.3.3 Effects of curved vanes
To allow the air flow through the vane ducts without blockage, a modified design
of rotors with curved (instead of straight) vanes was implemented. Two rotors of
curved vanes with radii of curvature of 114 cm and 57 cm are tested at various
angular velocities.
100
150
200
250
300
350
400
0 10 20 30
Vane Angles(degrees)
Heat
Tra
nsfe
r Rat
e(kw
)
Angular Velocity 44 Angular Velocity 88 Angular Velocity 120
52
Figure 3.10: Radii of curvature vs. heat transfer rate (32 vanes)
Figure 3.11: Radii of curvature vs. heat transfer rate (40 vanes)
100
150
200
250
300
350
400
∞ 114 57
Radius of Curvature of Vanes(cm)
Heat
Tra
nsfe
r R
ate(
kw)
ω=44 ω=88 ω=120
100150200250300350400450500
∞ 114 57
Radius of Curvature of Vanes(cm)
Heat
Tra
nsfe
r Rat
e(kw
)
ω=44 ω=88 ω=120
53
Figure 3.12: Radii of curvature vs. heat transfer rate (40 vanes, 30 degree angle)
In figure 3.10 for the case of 32 vanes, when the radius of curvature decreases to
114 cm, the heat transfer rate decreases. With the decrease of radius down to 57 cm,
the heat transfer rate rises. However, in the case of 40 vanes, as shown in figure
3.11, a decrease of the radius leads to an increase of the heat transfer rate. For
rotors with 40 vanes and a vane angle of 300, as shown in figure 3.12, the heat
transfer rate increases with a decrease of radius of curvature. This phenomenon can
be interpreted using the simulation result illustrated in figure 3.13, which shows the
velocity distribution of a rotor with 32 vanes at the angular velocity of 120 rad/s.
Since the direction of the curved vane is coordinated to the airflow vectors, there is
less resistance for the flow.
100150200250300350400450500
∞ 114 57
Radius of Curvature of Vanes(cm)
Heat
Tra
nsfe
r Rat
e(kw
)ω=44 ω=88 ω=120
54
Figure 3.13: Velocity distribution of 32-vane rotor at angular velocity of 120 rad/s
3.3.4 Effects of short-long vanes
To further improve the thermal performance of brake discs, two rotor models with
a short-long ratio of 0.83 at different angular velocities were generated and the
corresponding numerical simulations were conducted. Recall that the short-long
ratio is defined as the length ratio of the short and long vanes. A close observation
of figures. 3.14 and 3.15 discloses that the short-long vanes do not increase the heat
transfer rate inside the rotors. This is due to the unevenly distributed velocities in
different vane sections as shown in Figure 3.16.
55
Figure 3.14: Short-long ratio vs. heat transfer rate (32 vanes)
Figure 3.15: Short-long ratio vs. heat transfer rate (40 vanes)
100
150
200
250
300
350
400
0 0.83
Short-long Ratio
Hea
t Tra
nsfe
r R
ate(
kw)
ω=44 ω=88 ω=120
100
150
200
250
300
350
400
0 0.83
Short-long Ratio
Heat
Tra
nsfe
r Rat
e(kw
)
ω=44 ω=88 ω=120
56
Figure 3.16: Velocity contours for a rotor with short-long vanes (40 vanes, ω = 88
rad/s)
3.4. Summary
Various braking rotor models with different geometric parameters, such as vane
numbers, vane angles, radii of curvature and short-long ratios were studied
numerically. The results demonstrated that CFD is an effective method for
analyzing the heat transfer behavior of vented disc brakes with different vane
configurations. The simulation results indicate that an increase of vane numbers
drastically improves the thermal performance by 63.5%, if the vane numbers
increase from 32 to 60 at an angular velocity of 44 rad/s, by 67.9% at an angular
velocity of 88 rad/s, and 69.2% at an angular velocity of 120 rad/s. The vane angles
do not contribute to the improvement of thermal performance when the vane
57
number is 32. With a vane number of 40, the heat transfer increases by 16% at an
angular velocity of 44 rad/s, and 1.16% at an angular velocity of 120 rad/s. For the
curved vanes, with 32 vanes, the thermal performance does not increase for curved
vanes. But it does increase with 40 vanes by 27%, 30%, 31.6% at an angular
velocity of 44, 88 and 120 rad/s, respectively, when the radius of curvature
increases to 114 cm. From the simulation results, the short-long ratio does not
substantially contribute to the improvement of cooling performance.
58
Chapter 4
Thermal Performance Analysis of Pillar Post
Rotors Based on Numerical Simulations
Based on the thermal performance analysis of brake rotors with analytical methods
discussed in Chapter 3, it was concluded that there is a little difference between
vane and pillar post rotors. But it is difficult to determine the more detailed effects
of geometrical parameters of rotors on the thermal performance of pillar post rotors.
In this chapter, a parametric study of pillar post rotors will be conducted. These
geometrical parameters include the number of pillar posts, different pillar post
positions and various pillar post sizes.
4.1 GAMBIT models
To improve the accuracy of the predictions, a number of 2-D mesh models was
constructed with a commercial software GAMBIT. Due to the axial symmetric
configuration of the rotor, a partial cooling passage was introduced with periodic
boundary conditions applied to both sides. The computational domain and mesh are
shown in figure 4.1. Pressure inlet and pressure outlet boundary conditions were
applied to the bottom inlet and top outlet layers. The model size varies with
59
different configurations in the range of 9,314 to 11,840 cells, 14,260 to 17,905
faces and 4,943 to 6,132 nodes. The grids were built with triangular cells due to the
complication of the model. To refine the pillar post wall boundary conditions, 30-
40 nodes were employed for each pillar post side, as shown in figure 4.2.
Figure 4.1: 2-D rotor mesh model with 160 pillar post
Figure 4.2: Pillar post wall refinement of a rotor model (160 pillar posts)
60
4.2 Numerical Simulations Using FLUENT
A commercial CFD solver, Fluent, was used to simulate the air flow inside
different rotors and find the optimized design variables. The CFD simulation
assumptions are listed below:
• Steady state air flow;
• Segregated solver and implicit formulation;
• Standard k-epsilon viscous turbulence model;
• Standard wall functions;
• Moving reference frame at a constant velocity of 44 rad/s, 88 rad/s
and 120 rad/s respectively;
• Pillar post wall conditions at a constant temperature of 900 K;
• Momentum-Second Order Upwind Scheme;
• Turbulence Kinetic Energy - Second Order Upwind Scheme;
• Turbulence Dissipation Rate - Second Order Upwind Scheme;
• Energy - Second Order Upwind Scheme.
In the simulations, the heat transfer rate is an important design criterion to optimize
the configuration of the pillar post rotors. The numerical simulations were based on
2-D rotor models and heat transfer from the rotor surfaces was neglected. When the
pillar post number increases from 96 to 256, the heat transfer from the pillar post
walls drastically increases from 305 kW to 809 kW, by 166.7%. However, heat
transfer from the rotor surfaces decreases slightly from 832 kW to 746 kW by
61
10.34%. This indicates that the heat transfer variation largely depends on the
change of the pillar post walls, not the disk rotor surfaces. Thus, 2-D models are
reasonably accurate for the pillar post rotor design analysis. The ratio of heat
transfer from the disc surfaces vs. the pillar post walls varies for different models.
With the pillar post, when the number increases from 96 to 256, the ratio decreases
from 2.77 to 0.93. The predicted velocity distribution is shown in figure 4.3 and the
predicted pressure distribution is shown in figure 4.4. To increase the thermal
performance of rotors, the heat transfer rate should be increased. The models with
different pillar post numbers, different pillar post positions and various pillar post
sizes will be investigated in the following sub-sections.
Figure 4.3: Velocity contours of the rotor with 160 pillar posts
62
Figure 4.4: Static pressure contours of the rotor with 160 pillar posts
4.3 Results and discussion
4.3.1 Effects of pillar post numbers
Six different rotors with 96, 128, 160, 192, 224 and 256 pillar posts were chosen to
evaluate the effects of pillar post numbers on the change of heat transfer rate.
Figure 4.5 shows the relationship between pillar post numbers and the heat transfer
rate at an angular velocity of 44, 88 and 120 rad/s, respectively. As shown in figure
4.5, with the increase of pillar post numbers, the heat transfer rate increases as well.
However, when the pillar post number reaches 256, the heat transfer rate will
slightly decrease compared with the rotor of 224 pillar posts. This can be explained
since the air flow will be blocked by pillar posts, if the number is increased beyond
a certain value.
63
Figure 4.5: Pillar post numbers vs. heat transfer rate
4.3.2 Effects of different positions of middle pillar posts
To investigate the influence of different positions of middle pillar posts on the heat
transfer rate, three sets of rotor models with 96, 160 and 224 pillar posts were
developed. For each set of rotor models, the relative moving angle β (as shown in
figure 4.6) of a pillar post arranged on the two middle rings is 7.5°, 4.5° and 3.2°,
respectively, as shown in figure 4.7. With these rotor models, numerical
simulations were conducted at angular velocities of 44, 88 and 120 rad/s,
respectively. The simulation results, as shown in Figure. 4.8, indicate that when the
middle pillar posts are moved by an angle of 7.5°, 4.5° and 3.2°, respectively, the
heat transfer rate decreases slightly. The numerical simulations show that the heat
transfer dose not benefit from the middle pillar post’ angular orientation variation,
within the specified value range.
80
180
280
380
96 128 160 192 224 256
Pillar post number
Heat
tran
sfer
rate
(Kw
)
ω=44 ω=88 ω=120
64
Figure 4.6: Original positions of middle pillar posts vs. modified positions
Figure 4.7: Velocity contours of disc rotors with 96, 160 and 224 pillar posts and modified positions of middle pillar posts
β
Original positions Modified positions
65
Figure 4.8: Different middle pillar posts; angular orientations vs. heat transfer rate
when ω=44 radius/s
4.3.3 Effects of modified entrance pillar posts
To allow the air flow to pass through the ventilated ducts without blockage, a
modified design of rotors with inverse entrance pillar post triangles was
implemented, as shown in figure 4.9. In the case of the modified entrance pillar
posts, the rotors with 96, 160 and 224 posts were tested at an angular velocity of 44,
88 and 120 rad/s, respectively. As shown in figure 4.10, when the entrance pillar
posts are arranged in the form of reverse triangles, the heat transfer rate increases
progressively with an increase of the pillar post number, when ω=44 rad/s, ω=88
rad/s and ω=120 rad/s. Since the direction of the modified entrance pillar posts is
aligned with the airflow vectors, there is less resistance to the flow.
80
110
140
96 pillar posts 160 pillar posts 224 pillar posts
Different middle pillar posts positions
Hea
t tra
nsfe
r rat
e (K
w)
original modif ied
66
Figure 4.9: Velocity contours of brake rotor with 160 pillar posts and modified
entrance posts (ω=44rad/s)
Figure 4.10: Different triangle post entrance vs. heat transfer rate (ω=44, 88 and
120rad/s)
Different triangle entrance
80
130
180
230
280
330
380
96 pillar posts 160 pillar posts 224 pillar posts
Hea
t tra
nsfe
r rat
e(K
W)
ω=44 original
ω=44 reversed
ω=88 original
ω=88 reversed
ω=120 original
ω=120 reversed
67
4.3.4 Effects of increased pillar posts sizes
To further improve the thermal performance of brake discs, six rotor models with
an increase of pillar post size by 82% were generated and the corresponding
numerical simulations were conducted at different angular velocities, as shown in
figure 4.11. Figure 4.12 shows that the enlarged pillar post rotor increases the heat
transfer rate, at different angular velocities. This is due to more evenly distributed
air flow velocities in pillar posts sections, as shown in figure. 4.11.
Figure 4.11: Velocity contours inside brake rotor with enlarged pillar posts
(number of posts = 160 and ω=44 rad/s)
68
Figure 4.12: Pillar post number vs. heat transfer rate when the pillar posts have
different sizes and ω=44, 88 and 120 rad/s
4.3.5 Comparison of pillar post rotors and vane rotors
To compare the vane and pillar post rotors, several models for both rotors were
chosen, as shown in figure 4.13. Note that the vane rotor models are based on those
reported in chapter 4. The simulation results, as shown in figures 4.13, 4.14 and
4.15, indicate that the geometrical configuration, in the form of vane rotors, has
more significant effects on the thermal performance of disc brakes, than that of the
design in pillar post rotors. For the case of a small number of vanes and pillar posts,
these two geometrical configurations have minor differences, in terms of the heat
transfer rate. As the number of vanes and pillar posts increase, the vane rotors have
a much higher heat transfer rate than their counterparts, including the original pillar
post and modified pillar post rotors.
80
130
180
230
280
330
380
430
96 128 160 192 224 256
Pillar post numbers
Hea
t tra
nsfe
r ra
te(K
W)
ω=44 originalω=44 increasedω=88 originalω=88 increasedω=120 originalω=120 increased
69
Figure 4.13: Comparison of pillar post rotors and vane rotors at various vane or
pillar post numbers (ω=44 rad/s)
Figure 4.14: Comparison of pillar post rotors and vane rotors at various vane or pillar post numbers (ω=88 rad/s)
050
100150200250300
24vanes(96
pillorposts)
32vanes(128
pillorposts)
40vanes(160
pillorposts)
48vanes(192
pillorposts)
56vanes(224
pillorposts)
64vanes(256
pillorposts)
Vane (pillar post) numbers
Heat
tran
sfer
rat
e(K
w)
vane original pillar posts increased sized pillar posts
0
100
200
300
400
500
24vanes(96
pillorposts)
32vanes(128
pillorposts)
40vanes(160
pillorposts)
48vanes(192
pillorposts)
56vanes(224
pillorposts)
64vanes(256
pillorposts)
Vane or pillar post numbers
Heat
tran
sfer
rate
(Kw
)
vane original pillar posts increased sized pillar posts
70
Figure 4.15: Comparison of pillar post rotors and vane rotors at various vane or pillar post numbers (ω=120 rad/s)
4.4. Summary
Various pillar post braking rotor models with different geometrical parameters,
such as post numbers, modified entrance post triangles, various positions of posts
on middle rings and enlarged pillar posts, were investigated numerically in this
chapter. The results demonstrated that computational fluid dynamics (CFD) is an
effective tool for analyzing the heat transfer behavior of vented disc brakes, with
different geometrical parameters. The simulation results indicate that there are a
number of pillar posts, with which the disc brake will achieve optimal thermal
performance. By orienting the entrance triangle posts, the heat transfer rate can be
improved by 6%. By enlarging the pillar post size, the heat transfer rate can be
increased by 30%. However, the relative angular orientation between the middle
0
200
400
600
800
24vanes(96
pillorposts)
32vanes(128
pillorposts)
40vanes(160
pillorposts)
48vanes(192
pillorposts)
56vanes(224
pillorposts)
64vanes(256
pillorposts)
Vane or pillar post numbers
Heat
tran
sfer
rat
e(K
w)
vane original pillar posts increased sized pillar posts
71
ring’s pillar posts has little contribution to the improvement of thermal performance.
The numerical results demonstrate that the geometrical configuration, in the form
of vane rotors, has more significant effects on the thermal performance of disc
brakes, than that of the design in the form of pillar post rotors. This new insight can
provide useful guidelines to optimize the geometry of vented disc rotors of vehicle
brakes.
72
Chapter 5
Geometrical Optimization with 2-D Models
To explore the potential of an automated design synthesis process of brake discs, in
this chapter an integrated design synthesis will be proposed and implemented.
Commercial software GAMBIT is used for geometrical modeling and automatic
mesh generation for vented discs. Then, the CFD package, FLUENT, is employed
to simulate the air flow through the vented disc. To automate the design process of
the disc rotor, a software framework, iSIGHT, is used to integrate the geometrical
modeling using GAMBIT and numerical simulations based on FLUENT. Through
this integrated design synthesis process, the disc rotor geometrical optimization is
performed with design of experiment (DOE) studies. The main design criterion of
the geometrical optimization is to maximize the convective cooling by increasing
the heat transfer rate inside the rotor passages.
5.1 Framework for geometrical optimization models
The geometrical optimization was implemented by integrating geometrical
modeling and CFD numerical simulations. The framework for automated design
synthesis is depicted in figure 5.1. This framework consists of the following three
software packages: GAMBIT, FLUENT and ISIGHT.
73
Figure 5.1: Schematic representation of the framework for automated design
synthesis of brake discs
GAMBIT is a geometrical modeling and automatic mesh generation software
package. With various input parameters such as the inner radius, outer radius, vane
number, vane angle and vane offset, GAMBIT generates various mesh files for
CFD and other numerical simulations.
FLUENT is a reliable CFD solver and it is used to simulate the complex turbulent
air flow inside the rotor passages.
iSIGHT
Objective: Maximize the heat transfer rate Strategy: DOE
GAMBIT FLUENT
Output
Parametric and meshing model
Solver
Input
Inner radius Outer radius Vane number Angel of vanes Vane offset
Net heat transfer rate
Mesh file
74
iSIGHT provides a suite of visual and flexible tools to set up an automated design
synthesis platform for thoroughly exploring the design space and finding optimum
solutions, using techniques such as optimization, approximations and DOE. As an
integration and optimization tool, iSIGHT can integrate GAMBIT and FLUENT,
leading to optimal design parameters. With a given set of input parameters, such as
the inner radius, outer radius, vane numbers, angles of vanes and vane offset, from
iSIGHT, GAMBIT will build geometrical models and generate corresponding mesh
files. Then, FLUENT will call these meshing files to calculate specified
performance indices, such as heat transfer rates. The resulting performance indices
will be returned to iSIGHT. Through objective fitness evaluations, iSIGHT will
coordinate the trade-off relations among various design criteria and constraints and
identify a set of better design variables. The resulting design variable set will be
forwarded to GAMBIT as the new input parameters for the next iteration. The
above procedure will repeat until an optimal set of design variables is obtained.
5.2 Numerical Simulations with CFD
In order to simulate the complex air flow inside rotors, a 2-D mesh model was
constructed with the commercial software package GAMBIT. Due to the axial
symmetric configuration of the rotor, a partial cooling passage was generated with
periodic boundary conditions applied on both sides. The computational domain and
mesh are shown in figure 5.2 and figure 5.3. Pressure inlet and outlet boundary
conditions
varies with
grids were b
were applie
h different
built with tr
Fi
Fig
ed to the bo
configuratio
riangular ce
igure 5.2: 2-
gure 5.3 Ful
75
ottom inlet
ons in the
ells.
-D mesh pro
ll circular m
and top ou
range from
oduced by G
mesh produc
utlet layers.
m 3,500 to
GAMBIT
ced by GAM
The mode
6,300 cells
MBIT
el size
s. The
76
The vane rotor has an inner radius of 5.5 cm and an outer radius of 17.1 cm, with a
vane thickness of 0.56 cm and a length of 5.65 cm. The CFD simulation
assumptions are the same as those summarized in chapter 3.
In the simulations, the heat transfer rate was calculated. The predicted velocity and
pressure distributions are shown in figures. 5.4 and 5.5, respectively.
Figure 5.4: Velocity distribution in one section
77
Figure 5.5: Pressure contours in one section
5.3 Design of experiments with iSIGHT
iSIGHT provides various design synthesis tools, such as numerical optimization,
design of experiments (DOE), quality engineering methods (QEM), multi-criteria
trade-off analysis and approximation, etc. To avoid a long computational time, a
DOE study was applied as an optimization tool. iSIGHT is also featured with six
design analysis techniques, namely central composite design, data file, full-factorial,
orthogonal arrays and a parameter study. In the current study, the full-factorial was
selected, in which the combinations of all design variables at multiple levels are
evaluated. The design variables, involving inner radius, outer radius, vane numbers,
vane angles and vane offset, are divided evenly.
78
With the introduction of the periodic side boundary conditions, the whole disc rotor
can be modeled as a partial section, as shown in figure. 5.2. Since the vanes are
evenly arranged along the whole disc rotor, a given vane number corresponds to a
specified section angle. For example, a 9° angular section corresponds to a rotor
with 40 vanes. A comparison between two partial sections with different arc angles
is shown in figure 5.6.
The vane offset is used as a measurement for the curvature of a vane and it is
defined as the offset from the center of the vane, as shown in figure 5.7. The
optimization problem is formulated as follows.
Maximize Q (total heat transfer rate)
Subject to 24 ≤ VNu ≤ 76
6.84 ≤ IR ≤ 8.36 (cm)
12.69 ≤ OR ≤ 15.51 (cm)
0° ≤ VA ≤ 40°
0 ≤ VO ≤ 0.4 (cm)
Where VNu is the vane number, IR is inner radius, OR is outer radius, VA is vane
angle and VO is the vane offset.
79
Figure 5.6: GAMBIT models with vane angles of 10° (left) and 30° (right)
Figure 5.7: GAMBIT models with offset of 0.1 (left) and 0.3 cm (right)
80
5.4 Results and discussion
To improve the computational performance, a DOE study with the full-factorial
technique was used for the geometrical optimization of vented brake discs. The
specified ranges of design variables were divided evenly in terms of their
significance. Within the range from 24 to 76, the vane number may take 11 integers,
namely 24, 28, 32 …… 76. The design variable of the inner radius will be assigned
one of the five values through which the variable range from the lower bound to the
upper bound is evenly divided into four sections. This is also the case for the design
variable of the outer radius. The vane angles and vane offsets were split into 0°, 10°,
20°, 30°, 40° and 0, 0.1, 0.2, 0.3, 0.4 cm respectively. The resulting optimal design
variables are listed as follows.
Vane number: 64
Inner radius: 6.84 cm
Outer radius: 15.51 cm
Vane angle: 0°
Vane offset: 0.1 cm
5.4.1 Effects of vane numbers
Figure 5.8 shows the relation between the vane number and the total heat transfer
rate. When the vane number increases, the heat transfer rate increases. This occurs
81
because the rotors with less vanes have a larger non-uniformity of velocity
distribution than those with more vanes. With an increase of vane numbers, the heat
dissipation area increases as well. However, when the vane number exceeds 64, the
heat transfer rate decreases, because the narrow vane passages have blocked the air
flow.
Figure 5.8: Vane numbers vs. heat transfer rate
5.4.2 Effects of inner and outer radius
As shown in figures 5.9 and 5.10, the inner radius and outer radius vary nearly
linearly with heat transfer rate.
100000
150000
200000
250000
300000
24 28 32 36 40 44 48 52 56 60 64 68 72 76
Vane numbers
Tota
l hea
t(W)
82
Figure 5.9: Inner radius vs. total heat transfer rate
Figure 5.10: Outer radius vs. total heat transfer rate
5.4.3 Effects of vane offset and angle
For the vane angle and offset, they do not affect the heat transfer rate significantly.
As shown in figures. 5.11 – 5.14, the heat transfer rate has a maximum value when
100000
120000
140000
160000
6.84 7.22 7.6 7.98 8.36
Inner radius (cm)
Tota
l hea
t tra
nsfe
r rat
e (K
W)
130000
160000
190000
220000
12.69 13.4 14.1 14.8 15.51
Outer radius (cm)
Tota
l hea
t tra
nsfe
r rat
e (K
W)
83
the vane angle has a value within the range of 0° - 20°, whereas for the vane offset,
there is no deterministic relation with the heat transfer rate.
Figure 5.11: Vane angle and vane offset vs. total heat transfer rate with 40 vanes
Figure 5.12: Vane angle and vane offset vs. total heat transfer rate with 48 vanes
84
Figure 5.13: Vane angle and vane offset vs. total heat transfer rate with 56 vanes
Figure 5.14: Vane angle and vane offset vs. total heat transfer rate with 64 vanes
85
5.5 Summary
This chapter has presented an automated design synthesis approach to the
geometrical optimization of vented brake discs of automotive vehicles. The design
optimization was implemented using a software framework, iSIGHT, to integrate
the geometrical modeling by a commercial software package, GAMBIT, and
numerical simulations based on a computational fluid dynamics tool, FLUENT.
The effectiveness and efficiency of the automated design synthesis approach was
investigated by optimizing the geometrical parameters of a 2-D model of a vented
brake disc with a design of experiments (DOE) technique.
In the case of the geometrical optimization of vented brake discs, the parameter
studies revealed that by increasing the vane numbers from 40 to 64, the cooling
performance can be increased by 31.1%. Increasing the outer radius and decreasing
the inner radius can improve the heat transfer rate by 43.2% and 31.4%. However,
changing the vane angle and vane offset slightly increases the heat transfer rate
only by 0.1% and 0.2%, respectively.
86
Chapter 6
Geometrical Optimization with 3-D Models
In Chapter 6, the 2-D models were used for optimizing the geometry of ventilated
vane rotors. However, the 2-D models can also be used to investigate the effects of
other parameters, such as the thickness of disks, on the thermal performance of
rotors. Moreover, to achieve more accurate numerical simulation results, brake
rotor models with high fidelity are required. Thus, in this chapter, more realistic 3-
D rotor models will be presented and the corresponding geometrical optimization is
implemented.
6.1 Geometrical optimization models
The geometrical optimization was implemented by integrating geometrical
modeling and CFD numerical simulations. The framework for automated design
synthesis is depicted in Figure 6.1. This framework consists of the following three
software packages: GAMBIT, FLUENT and ISIGHT as shown in chapter 6. The
integrating process is the same as shown in the Chapter 6. The only differences are
that the 2-D models have been changed into 3-D models and the thickness variable
is included as shown in Figure 6.1.
87
Figure 6.1: Schematic representation of the framework for automated design
synthesis of brake discs
6.2 Numerical Simulations with CFD
In order to achieve more accurate results, a 3-D mesh model was constructed with
the commercial software package GAMBIT. Due to the axial symmetric
configuration of the rotor, a partial cooling passage was generated with periodic
boundary conditions applied on both sides, as shown in figures 6.2, 6.3 and 6.4.
iSIGHT
Objective: Maximize the heat transfer rate Strategy: DOE
GAMBIT FLUENT
Output
3-D parametric and meshing model
Solver
Input
Inner radius Outer radius Vane number Angel of vanes Vane offset Thickness
Net heat transfer rate
Mesh file
88
Figure 6.2: 3-D mesh produced by GAMBIT
Figure 6.3: Outlet section of a 3-D section produced by GAMBIT
89
Figure 6.4: Inlet of a 3-D section produced by GAMBIT
The computational domain and mesh are shown in figure 6.5. The boundary
conditions are as follows:
1. A pressure inlet was applied to the bottom inlet layers, as shown in a yellow
color;
2. Pressure outlet was applied to the top outlet layers, as shown in a red color;
3. The blue layers show the inlet and outlet periodic boundary conditions;
4. The wall boundary layers were depicted by a black color at the side of
section mesh.
90
Figure 6.5 Computational domain and boundary conditions of a 3-D section
The vane rotor has an inner radius of 6.5 cm and an outer radius of 17.1 cm, with a
vane thickness of 0.56 cm, a length of 5.65 cm and a thickness of 2.0cm. The CFD
simulation assumptions and parameters are the same as those described in Chapter
4.
In the simulations, the heat transfer rate was calculated. The predicted velocity,
static pressure and turbulence distributions are shown in figure 6.6, 6.7 and 6.8,
respectively.
91
Figure 6.6 Predicted velocity distributions in one section
Figure 6.7 Predicted static pressure distributions in one section
92
Figure 6.8 Predicted turbulence distributions in one section
6.3 Design of experiments with iSIGHT
The design of experiments (DOE) was selected as the optimization synthesis tool
and a full-factorial study by the design analysis technique was used as in chapter 6.
The design variables, involving inner radius, outer radius, vane numbers, vane
angles, vane offset and thickness are divided evenly.
The optimization problem is formulated as follows.
Maximize Q (total heat transfer rate)
Subject to 32 ≤ VNu ≤ 76
6.84 ≤ IR ≤ 8.36 (cm)
93
12.69 ≤ OR ≤ 15.51 (cm)
0° ≤ VA ≤ 30°
0 ≤ VO ≤ 0.3 (cm)
1.8 ≤ RT ≥ 2.2 (cm)
where VNu is the vane number, IR is inner radius, OR is outer radius, VA is vane
angle, VO is the vane offset and VT is rotor thickness.
6.4 Results and discussion
A DOE study with the full-factorial technique was used for the geometrical
optimization of vented brake discs. The specified ranges of design variables were
divided evenly in terms of their significance. Within the range from 32 to 72, the
vane number may take 11 integers, namely 32, 34, 38 …… 72. The design variable
of the inner radius will be assigned one of the five values, through which the
variable range from the lower bound to the upper bound was evenly divided into
four sections. This is also the case for the design variable of the outer radius. The
vane angles and vane offsets were split into 0°, 10°, 20°, 30° and 0, 0.1, 0.2, 0.3 cm
respectively. The rotor thickness is evenly divided into 1.8, 1.9, 2.0, 2.1 and 2.2 cm.
The resulting optimal design variables are listed as follows.
Vane number: 56
Inner radius: 8.36 cm
Outer radius: 15.51 cm
94
Vane angle: 0°
Vane offset: 0.2 cm
Rotor thickness: 2.2 cm
6.4.1 Effects of vane numbers
Figure 6.9 shows the relation between the vane number and the total heat transfer
rate. When the vane number increases, the heat transfer rate increases and then
reaches a peak value at a vane number value of 56. When the vane number is
higher, the heat transfer rate begins to decrease.
Figure 6.9 Vane numbers vs. heat transfer rate
6.4.2 Effects of inner and outer radius
The influence of inner radius and outer radius is shown in figure 6.10, from which
the heat transfer rate increases when the inner radius increases and outer radius
drastically rises.
5000
6000
7000
32 36 40 44 48 52 56 60 64 68 72
Vane Numbers
Hea
t Tra
nsfe
r Rat
e(w
)
95
Figure 6.10 Inner radius and outer radius vs. heat transfer rate
6.4.3 Effects of vane offset and vane angle
Regarding the vane angle and offset, they do not affect the heat transfer rate
significantly. As shown in figures 6.11 and 6.12, the heat transfer rate has a
maximum value when the vane angle takes a value within the range of 0° - 20°,
whereas for the vane offset, there is no deterministic relation with the heat transfer
rate.
96
Figure 6.11 Vane offset and vane angle vs. heat transfer rate for 40 vanes
Figure 6.12 Vane offset and vane angle vs. heat transfer rate for 56 vanes
97
6.4.4 Effects of rotor thickness
The influence of rotor thickness is shown in figure 6.13. The heat transfer rate
increases with an increase of rotor thickness.
Figure 6.13 Rotor thickness vs. heat transfer rate at 56 vane numbers
6.5 Summary
For 3-D geometrical optimization of vented brake discs, the parametric studies
revealed that by increasing the vane numbers from 40 to 56, the cooling
performance can be increased by 9.2%. Increasing the outer radius and the inner
radius can improve the heat transfer rate by 15.4% and 2.7% respectively. For rotor
thickness, the cooling performance increases by 17.6%, with an increase of rotor
thickness from 1.8 to 2.2 cm. However, changing the vane angle and vane offset
slightly increases the heat transfer rate by only 0% and 0.4%, respectively.
6000
7000
8000
1.8 1.9 2 2.1 2.2
Thickness(cm)
Hea
t tra
nsfe
r rat
e(W
)
98
Chapter 7
Conclusions
For solid rotors, the highest temperature occurs on the surfaces of the rotors. To
decrease the maximum temperature, the most effective way is to increase the
thickness of the rotors. However, the increase is limited by the pistons. Materials
also have effects on the rotor’s temperature. From this research, steel is a better
alloy to dissipate heat from the rotors. But in practical design problems, the thermal
performance is not the only requirement. From the perspective of stiffness, friction
resistance and cost, the cast iron material is common used in industry.
By the analytical solution of heat convection inside vented discs, using the velocity,
temperature, and Nusselt number analysis, the same conclusion can be drawn that
increasing the number of vanes or pillar posts can drastically decrease the surface
temperature of rotors. From the velocity analysis, the vane rotors have slightly
better cooling performance than pillar post rotors. But from the temperature and
Nusselt number analysis, the difference between them is minor.
The powerful simulation packages GAMBIT and FLUENT give more accurate
solutions to vane and pillar post rotors. From the sensitivity studies, the vane rotors
have better cooling performance than pillar post rotors. Moreover the investigation
99
of pillar post rotors reveals that the optimal pillar post number is 224. Rearranging
and resizing the pillar posts can greatly improve the cooling performance of rotors.
The optimal geometrical shape of vane rotors is revealed by the optimization tool
iSIGHT, using 2-D and 3-D models. The most important variables affecting
thermal performance are vane numbers, inner and outer radius and also thickness
for 3-D models. The vane offset and vane angles have little influence in enhancing
the cooling performance of vane rotors.
However, the results generated by 2-D and 3-D models are different. For the inner
radius, the 2-D models revealed that a smaller radius can give better cooling
performance. In contrast, with 3-D models, when decreasing the inner radius of
rotors, the heat transfer rate does not increase, but slightly decreases. This occurs
because with 2-D models, the thickness of the rotors is neglected, so when the
radius of the rotors decreases, the increased surface area gives better cooling
performance. However, in more realistic 3-D models, when the radius of the rotor
decreases, the entrance area decreases as well, which blocks the inlet air flow.
Regarding the optimal vane numbers, the 2-D models gave the value of 64, but
from the 3-D models, the optimal vane numbers is 56.
Future research is needed in this area. More detailed geometrical optimization of
pillar post rotors is desired. To shorten the computational time, parallel calculation
by using several computers should be utilized. If the computational time is
100
shortened enough, different optimization methods in iSIGHT could be used such as
SQP, Approximation and Monte Carlo Simulation. Furthermore, the other
optimization variables could be taken into account, such as various shapes of vanes
and different thickness of vanes.
101
References
[1] Bejan, Adrian, Convection heat transfer, 3rd edition. Hoboken, N.J. : Wiley,
c2004.
[2] S. Artus, V. Cocquempot, S. Hayat, M. Staroswiecki, Ph. De Larminat and C.
Covo, CHV’s Brake Discs Temperature Estimation: Result in Open Road Tests, the
8th International IEEE Conference on Intelligent Transportation Systems Vienna,
Austria, Sep 13-16, 2005.
[3] S. Aus Der Wiesche, Heat Transfer from a Rotating Disk in a Parallel Air
Crossflow, International Journal of Thermal Sciences 46(2007) 745-754.
[4] S. Aus Der Wiesche, Heat Transfer and Thermal Behavior of a Rotating Disk
Passed by a Planar Air Stream, Forschung im Ingenieurwesen 67(2002) 161-174,
Springer-Verlag 2002.
[5] Dipl.-Ing. C. Repmann, Volkswagen AG, Wolfsburg, Investigation of the
Thermal Behavior of Vented Disc Brakes By Means of CFD, SAE 2005 World
Congress, Detroit, Michigan.
[6] Z. Chi, G. F. Naterer and Y. He, Thermal Performance Analysis of Vented
Automotive Brake Discs, CSME Forum 2008, University of Ottawa, Ottawa,
Canada, 2008.
[7] Z. Chi, G. F. Naterer and Y. He (2008) Effects of Brake Disc Geometrical
Parameters and Configurations on Automotive Braking Thermal Performance,
CSME Transactions 2008, University of Ottawa, Ottawa, Canada, 2008
102
[8] Z. Chi, Y. He and G. F. Naterer, Geometrical Optimization of Vented Brake
Discs of Automotive Vehicles, submitted to CATS 2008, Toronto, Canada, 2008
[9] J. H. Choi, I. Lee, Finite Element Analysis of Transient Thermoelastic
Behaviors in Disk Brakes, Science Direct, Wear 257 (2004) 47-58.
[10] C.H. Gao and X.Z.Lin, Transient Temperature Field Analysis of a Brake in a
Non-axisymmetric Three-Dimensional Model, Journal of Materials Processing
Technology 129 (2002) 513-517.
[11] D. G. Grieve, D. C. Barton, D. A. Crolla, J. T. Buckingham, Design of a
lightweight Automotive Brake Disc Using Finite Element and Taguchi Techniques,
Institution of Mechanical Engineers; 1998; 212, 4; ProQuest Science Journal.
[12] Latif M. Jiji, Heat Convection, Berlin ; New York : Springer, c2006
[13] D. A. Johnson, B. A. Sperandei, R. Gilbert, Analysis of the Flow Through a
Vented Automotive Brake Rotor, Journal of Fluids Engineering, ASME, Nov. 2003,
Vol. 125/979
[14] R. Limpert, Cooling Analysis of Disc Brake Rotors, SAE paper 751014, 1975.
[15] R. Limpert, Brake Design and Safety, Second Edition, Warrendale, PA :
Society of Automotive Engineers, c1992
[16] A. D. Mcphee and D. A. Johnson, Experimental Heat Transfer and Flow
Analysis of a Vented Braked Brake Rotor, International Journal of Thermal
Sciences, Vol. 3, No. 6, 2007.
[17] M. Naji, M. Al-Ninr, S.Masoud, Transient thermal behavior of a cylindrical
brake system, Heat and Mass Transfer 36 (2000) 45-49, Springer-Verlag 2000.
103
[18] T. P. Newcomb and R. T. Spurr. Braking of road vehicles London : Chapman
and Hall, 1967.
[19] J. M. Owen, C. M. Haynes; F. J. Bayley, Heat Transfer from an Air-Cooled
Rotating Disk, Royal Society of London, Series A, Mathematical and Physical
Sciences, Vol. 336, No. 1607. (Feb. 19, 1974), pp. 453-473.
[20] D. Parish and D. G. MacManus, Aerodynamic Investigations of Ventilated
Brake Discs, the Inatitution of Mechanical Engineers, Wilson Applied Science &
Technology Abstracts, Vol. 4, pp 219, 2005..
[21] C. Qian, Aerodynamic Shape Optimization using CFD Parametric Model with
Brake Cooling Application, SAE 2002 World Congress, Detroit, Michigan, March
4-7, 2002.
[22] Dipl.-Ing. C. Repmann, Volkswagen AG, Wolfsburg, Investigation of the
Thermal Behavior of Vented Disc Brakes By Means of CFD, SAE 2005 World
Congress, Detroit, Michigan, 2005.
[23] H. Sakamoto, Heat Convection and Design of Brake Discs, institution of
Mechanical Engineering; Sep 2004; 218,3; Wilson Applied Science & Technology
Abstracts. Pg. 203
[24] H. Sun, Sensitivity Study on Brake Cooling Performance, SAE Paper 2006-
01-0694.
[25] G. P. Voller, M. Tirovic, R. Morris and P. Gibbens, Analysis of Automotive
Disc Brake Cooling Characteristics, Institution of Mechanical Engineers; 2003;
217,8; Wilson Applied Science & Technology Abstracts, pg. 657.
[26] http://www.engineous.com/iSIGHTFD.htm
104
[27] http://www.fluent.com/software/gambit/index.htm
[28] iSIFHT user’s manual.
[29] FLUENT user’s manual