Solved with COMSOL Multiphysics 5.

Heat Generation in a Disc Brake

Introduction
Cars need a brakes for obvious reasons, and you do not want these to fail. Brake failure
can be caused by many things, one of which is the overheating of the brake’s disc. This
example models the heat generation and dissipation in a disc brake of an ordinary car
during panic braking and the following release period. When the driver is pressing
down on the brakes, kinetic energy is transformed into thermal energy. If the brake
discs overheat, the brake pads cease to function through brake fade where the material
properties of the brake change due to the temperature overload. Braking power starts
to fade already at temperatures above 600 K. This is why it is so important during the
design-stages to simulate the transient heating and convective cooling to figure out
what the minimum interval between a series of brake engagements is.

In this model, an 1,800 kg car is traveling at 25 m/s (90 km/h or about 56 mph),
until the driver suddenly panic brakes for 2 seconds. At that point the eight brake pads
slow the car down at a rate of 10 m/s2 (assuming the wheels do not skid against the
road). Upon braking for two seconds the driver releases the brake, leaving the car
traveling at 5 m/s for eight seconds without engaging the brakes. The questions to
analyze with the model are:

• How hot do the brake discs and pads get when the brake is engaged?
• How much do the discs and pads cool down during the rest that follows the
braking?

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Model Definition
Model the brake disc as a 3D solid with shape and dimensions as in Figure 1. The disc
has a radius of 0.14 m and a thickness of 0.013 m.

Figure 1: Model geometry, including disc and pad.

The model also includes heat conduction in the disc and the pad through the transient
heat transfer equation. The heat dissipation from the disc and pad surfaces to the
surrounding air is described by both convection and radiation. Table 1 summarizes the
thermal properties of the materials used in this model (Ref. 1).
TABLE 1: MATERIAL PROPERTIES

PROPERTY DESCRIPTION DISC PAD AIR

ρ (kg/m3) Density 7870 2000 1.170

Cp (J/(kg·K)) Heat capacity at constant pressure 449 935 1100
k (W/(m·K)) Thermal conductivity 82 8.7 0.026
ε Surface emissivity 0.28 0.8 -

After 2 s, contact is made at the interface between the disc and the pad. Neglecting
drag and other losses outside the brakes, the brakes’ retardation power is given by the
negative of the time derivative of the car’s kinetic energy:

2
P = – d  ----------- = – m v ------
mv dv
dt 2  dt

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Here m is the car’s mass (1800 kg) and v denotes its speed. Figure 2 shows the profile
of v and Figure 3 shows the corresponding acceleration profile.

Figure 2: Velocity profile of the disc.

Figure 3: Acceleration profile of the disc.

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At one of the eight brakes, the frictional heat source is:

P 1 dv
P fric,tot = ---- = – --- m v ------
8 8 dt

The contact pressure between the disc and the pad is related to the frictional heat
source per unit area, Pfric, according to:

P fric
p = ----------
μv

where the friction coefficient μ is here equal to 0.3.

The disc and pad dissipate the heat produced at the boundary between the brake pad
and the disc by convection and radiation. This example models the rotation as
convection in the disc. The local disc velocity vector is

v
v d = ---- (– y, x)
R

Results and Discussion

The surface temperatures of the disc and the pad vary with both time and position. At
the contact surface between the pad and the disc the temperature increases when the
brake is engaged and then decreases again as the brake is released. You can best see
these results in COMSOL Multiphysics by generating an animation. Figure 4 displays
the surface temperatures just before the end of the braking. A “hot spot” is visible at
the contact between the brake pad and disc, just at the pad’s edge. This is the area that
could overheat to the point of brake failure or fade. The figure also shows the

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temperature decreasing along the rotational trace after the pad. During the rest, the
temperature becomes significantly lower and more uniform in the disc and the pad.

Figure 4: Surface temperature of the brake disc and pad just before releasing the brake
(t = 1.8 s).

To investigate the position of the hot spot and the time of the temperature maximum,
it is helpful to plot temperature versus time along the line from the center to the pad’s
edge depicted in Figure 5. The result is displayed in Figure 6. You can see that the
maximum temperature is approximately 410 K. The hot spot is positioned close to the
radially outer edge of the pad. The highest temperature occurs approximately 1 s after
engaging the brake.

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Figure 5: The radial line probed in the temperature vs. time plot in Figure 6.

t
r (m)

Figure 6: Temperature profile along the line indicated in Figure 5 at the disc surface
(z = 0.013 m) as a function of time.

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To investigate how much of the generated heat is dissipated to the air, study the surface
integrals of the produced heat and the dissipated heat. These integrals give the total
heat rate (J/s) for heat production, Qprod, and heat dissipation, Qdiss, as functions of
time for the brake disc. The time integrals of these two quantities, Wprod and Wdiss,
give the total heat (J) produced and dissipated, respectively, in the brake disc. Figure 7
shows a plot of the total produced heat and dissipated heat versus time. Eight seconds
after the driver has stopped braking, a mere fraction of the produced heat has
dissipated. In other words, in order to cool down the system sufficiently the brake
needs to remain disengaged for a lot longer period than these eight seconds (100
seconds, in fact).

Figure 7: Comparison of total heat produced (solid line) and dissipated (dashed).

The results of this model can help engineers investigate how much abuse, in terms of
specific braking sequences, a certain brake-disc design can tolerate before overheating.
It is also possible to vary the parameters affecting the heat dissipation and investigate
their influence.

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Reference
1. J.M. Coulson and J.F. Richardson, Chemical Engineering, vol. 1, eq. 9.88;
material properties from appendix A2.

Model Library path: Heat_Transfer_Module/

Thermal_Contact_and_Friction/brake_disc

Modeling Instructions
From the File menu, choose New.

NEW
1 In the New window, click Model Wizard.

MODEL WIZARD
1 In the Model Wizard window, click 3D.
2 In the Select physics tree, select Heat Transfer>Heat Transfer in Solids (ht).
3 Click Add.
4 Click Study.
5 In the Select study tree, select Preset Studies>Time Dependent.
6 Click Done.

DEFINITIONS

Parameters
1 On the Model toolbar, click Parameters.
Define the global parameters by loading the corresponding text file provided.
2 In the Settings window for Parameters, locate the Parameters section.
3 Click Load from File.
4 Browse to the model’s Model Library folder and double-click the file
brake_disc_parameters.txt.

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GEOMETRY 1

Cylinder 1 (cyl1)
1 On the Geometry toolbar, click Cylinder.
2 In the Settings window for Cylinder, locate the Size and Shape section.
3 In the Radius text field, type 0.14.
4 In the Height text field, type 0.013.
5 Click the Build Selected button.

Cylinder 2 (cyl2)
1 On the Geometry toolbar, click Cylinder.
2 In the Settings window for Cylinder, locate the Size and Shape section.
3 In the Radius text field, type 0.08.
4 In the Height text field, type 0.01.
5 Locate the Position section. In the z text field, type 0.013.
6 Click the Build Selected button.

Work Plane 1 (wp1)

1 On the Geometry toolbar, click Work Plane.
2 In the Settings window for Work Plane, locate the Plane Definition section.
3 In the z-coordinate text field, type 0.013.

Plane Geometry
Right-click Component 1 (comp1)>Geometry 1>Work Plane 1 (wp1) and choose Show
Work Plane.

Bézier Polygon 1 (b1)

1 On the Geometry toolbar, click More Primitives and choose Bézier Polygon.
2 In the Settings window for Bézier Polygon, locate the Polygon Segments section.
3 Find the Added segments subsection. Click Add Cubic.
4 Find the Control points subsection. In row 1, set yw to 0.135.
5 In row 2, set xw to 0.02 and yw to 0.135.
6 In row 3, set xw to 0.05 and yw to 0.13.
7 In row 4, set xw to 0.04 and yw to 0.105.
8 Find the Added segments subsection. Click Add Cubic.
9 Find the Control points subsection. In row 2, set xw to 0.03 and yw to 0.08.

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10 In row 3, set xw to 0.035 and yw to 0.09.

11 In row 4, set xw to 0 and yw to 0.09.
12 Find the Added segments subsection. Click Add Cubic.
13 Find the Control points subsection. In row 2, set xw to -0.035.
14 In row 3, set xw to -0.03 and yw to 0.08.
15 In row 4, set xw to -0.04 and yw to 0.105.
16 Find the Added segments subsection. Click Add Cubic.
17 Find the Control points subsection. In row 2, set xw to -0.05 and yw to 0.13.
18 In row 3, set xw to -0.02 and yw to 0.135.
19 Click Close Curve.
To complete the pad cross section, you must make the top-left and top-right corners
sharper. Do so by changing the weights of the Bézier curves.
20 Find the Added segments subsection. In the Added segments list, select Segment 1
(cubic).
21 Find the Weights subsection. In the 3 text field, type 2.5.
22 Find the Added segments subsection. In the Added segments list, select Segment 4
(cubic).
23 Find the Weights subsection. In the 2 text field, type 2.5.
24 Click the Build Selected button.

Extrude 1 (ext1)
1 On the Geometry toolbar, click Extrude.
2 In the Settings window for Extrude, locate the Distances from Plane section.
3 In the table, enter the following settings:

Distances (m)
0.0065

4 Click the Build Selected button.

The model geometry is now complete.
Next, define some selections of certain boundaries. You will use them when defining
the settings for component couplings, boundary conditions, and so on.

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DEFINITIONS

Explicit 1
1 On the Definitions toolbar, click Explicit.
2 In the Settings window for Explicit, type Disc faces in the Label text field.
3 Locate the Input Entities section. From the Geometric entity level list, choose
Boundary.
4 Select Boundaries 1, 2, 4–6, 8, 13–15, and 18 only.

Explicit 2
1 On the Definitions toolbar, click Explicit.
2 In the Settings window for Explicit, type Pad faces in the Label text field.
3 Locate the Input Entities section. From the Geometric entity level list, choose
Boundary.
4 Select Boundaries 9, 10, 12, 16, and 17 only.

Explicit 3
1 On the Definitions toolbar, click Explicit.
2 In the Settings window for Explicit, type Contact faces in the Label text field.
3 Locate the Input Entities section. From the Geometric entity level list, choose
Boundary.
To select the contact surface boundary, it is convenient to temporarily switch to
wireframe rendering.
4 Click the Wireframe Rendering button on the Graphics toolbar.
5 Select Boundary 11 only.
6 Click the Wireframe Rendering button on the Graphics toolbar.

Explicit 4
1 On the Definitions toolbar, click Explicit.
2 In the Settings window for Explicit, type External surfaces in the Label text field.
3 Locate the Input Entities section. Select the All domains check box.
4 Locate the Output Entities section. From the Output entities list, choose Adjacent
boundaries.
These instructions make you select the external boundaries of the wheel and the
pad.

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Integration 1 (intop1)
1 On the Definitions toolbar, click Component Couplings and choose Integration.
2 In the Settings window for Integration, locate the Source Selection section.
3 From the Geometric entity level list, choose Boundary.
4 From the Selection list, choose Contact faces.
5 Select Boundary 11 only.

Integration 2 (intop2)
1 On the Definitions toolbar, click Component Couplings and choose Integration.
2 In the Settings window for Integration, locate the Source Selection section.
3 From the Geometric entity level list, choose Boundary.
4 From the Selection list, choose External surfaces.
Define the velocity and acceleration of car through these two piecewise and analytic
functions.

Piecewise 1 (pw1)
1 On the Definitions toolbar, click Piecewise.
2 In the Settings window for Piecewise, type v in the Function name text field.
3 Locate the Definition section. In the Argument text field, type t.
4 From the Smoothing list, choose Continuous second derivative.
5 From the Transition zone list, choose Absolute size.
6 In the Size of transition zone text field, type 0.2.
7 Find the Intervals subsection. In the table, enter the following settings:

Start End Function

0 t_brake_start v0
t_brake_start t_brake_end v0+a0*(t-t_brake_start)
t_brake_end 12 v0+a0*(t_brake_end-t_brake_start)

8 Locate the Units section. In the Arguments text field, type s.

9 In the Function text field, type m/s.

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10 Click the Plot button.

Analytic 1 (an1)
1 On the Definitions toolbar, click Analytic.
2 In the Settings window for Analytic, type a in the Function name text field.
3 Locate the Definition section. In the Expression text field, type d(v(t),t).
4 In the Arguments text field, type t.
5 Locate the Units section. In the Arguments text field, type s.
6 In the Function text field, type m/s^2.
7 Locate the Plot Parameters section. In the table, enter the following settings:

Argument Lower limit Upper limit

t 0 10

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8 Click the Plot button.

MATERIALS

Material 1 (mat1)
1 In the Model Builder window, under Component 1 (comp1) right-click Materials and
choose Blank Material.
2 In the Settings window for Material, type Disc in the Label text field.
3 Locate the Material Contents section. In the table, enter the following settings:

Property Name Value Unit Property group

Thermal conductivity k 82 W/ Basic
(m·K)
Density rho 7870 kg/m³ Basic
Heat capacity at constant Cp 449 J/ Basic
pressure (kg·K)

Material 2 (mat2)
1 In the Model Builder window, right-click Materials and choose Blank Material.
2 In the Settings window for Material, type Pad in the Label text field.
3 Select Domain 3 only.

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4 Locate the Material Contents section. In the table, enter the following settings:

Property Name Value Unit Property group

Thermal conductivity k 8.7 W/ Basic
(m·K)
Density rho 2000 kg/m³ Basic
Heat capacity at constant Cp 935 J/ Basic
pressure (kg·K)

H E A T TR A N S F E R I N S O L I D S ( H T )

Translational Motion 1
1 On the Physics toolbar, click Attributes and choose Translational Motion.
2 Select Domains 1 and 2 only.
3 In the Settings window for Translational Motion, locate the Translational Motion
section.
4 Specify the utrans vector as

-y*v(t)/r_wheel x
x*v(t)/r_wheel y
0 z

Heat Flux 1
1 On the Physics toolbar, click Boundaries and choose Heat Flux.
2 In the Settings window for Heat Flux, locate the Boundary Selection section.
3 From the Selection list, choose All boundaries.
4 Locate the Heat Flux section. Click the Convective heat flux button.
5 From the Heat transfer coefficient list, choose External forced convection.
6 In the L text field, type 0.14.
7 In the Uext text field, type v(t).
8 In the Text text field, type T_air.

Thermal Contact 1
1 On the Physics toolbar, click Boundaries and choose Thermal Contact.
2 Select Boundary 11 only.
3 In the Settings window for Thermal Contact, locate the Contact Surface Properties
section.

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4 In the p text field, type ht.tc1.Qfric/(mu*v(t)).

5 In the Hc text field, type 800[MPa].
6 Locate the Thermal Friction section. Click the Overall heat transfer rate button.
7 In the Pfric,tot text field, type -m_car*v(t)*a(t)/8.

Initial Values 1
1 In the Model Builder window, under Component 1 (comp1)>Heat Transfer in Solids (ht)
click Initial Values 1.
2 In the Settings window for Initial Values, locate the Initial Values section.
3 In the T text field, type T_air.

Diffuse Surface 1
1 On the Physics toolbar, click Boundaries and choose Diffuse Surface.
2 In the Settings window for Diffuse Surface, locate the Boundary Selection section.
3 From the Selection list, choose Disc faces.
4 Locate the Surface Emissivity section. From the ε list, choose User defined. In the
associated text field, type 0.28.
5 Locate the Ambient section. In the Tamb text field, type T_air.

Diffuse Surface 2
1 On the Physics toolbar, click Boundaries and choose Diffuse Surface.
2 In the Settings window for Diffuse Surface, locate the Boundary Selection section.
3 From the Selection list, choose Pad faces.
4 Locate the Surface Emissivity section. From the ε list, choose User defined. In the
associated text field, type 0.8.
5 Locate the Ambient section. In the Tamb text field, type T_air.

Symmetry 1
1 On the Physics toolbar, click Boundaries and choose Symmetry.
2 Select Boundary 3 only.
To compute the heat produced and the heat dissipated, integrate the corresponding
heat rate variables, Q_prod and Q_diss, over time. For this purpose, define two
ODEs using a Global Equations node.
3 In the Model Builder window’s toolbar, click the Show button and select Advanced
Physics Options in the menu.

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Global Equations 1
1 On the Physics toolbar, click Global and choose Global Equations.
2 In the Settings window for Global Equations, locate the Units section.
3 Find the Dependent variable quantity subsection. From the list, choose Energy (J).
4 Find the Source term quantity subsection. From the list, choose Power (W).
5 Locate the Global Equations section. In the table, enter the following settings:

Name f(u,ut,utt,t) (W) Initial value Initial value Description

(u_0) (J) (u_t0) (W)
W_prod W_prodt-intop1( 0 0 Produced heat
ht.tc1.Qfric)
W_diss W_disst+(intop2 0 0 Dissipated heat
(ht.q0+ht.rflux
))

Here, W_prodt (resp. W_disst) is COMSOL Multiphysics syntax for the time
derivative of W_prod (resp. W_diss). The quantities intop1(ht.tc1.Qfric) and
intop2(ht.q0+ht.rflux) correspond to Q_prod and Q_diss. The table thus
defines the two uncoupled initial value problems.
·
W prod/diss ( t ) = Q prod/diss ( t )
W prod/diss ( 0 ) = 0

To obtain the first-order ODEs, take the time derivative of the integrals
t

W prod/diss ( t ) =  Qprod/diss ( t' ) dt'

0

The initial values follow from setting t = 0.

MESH 1

Free Triangular 1
1 In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and
choose More Operations>Free Triangular.
2 Click the Transparency button on the Graphics toolbar.
3 Select Boundaries 4, 7, and 11 only.
4 Click the Transparency button on the Graphics toolbar again to return to the original
state.

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Size
1 In the Model Builder window, under Component 1 (comp1)>Mesh 1 click Size.
2 In the Settings window for Size, locate the Element Size section.
3 From the Predefined list, choose Extra fine.

Swept 1
In the Model Builder window, right-click Mesh 1 and choose Swept.

Distribution 1
1 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Swept
1 and choose Distribution.
2 In the Settings window for Distribution, locate the Distribution section.
3 In the Number of elements text field, type 2.
4 In the Model Builder window, right-click Mesh 1 and choose Build All.
The complete mesh consists of roughly 5700 elements.

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STUDY 1

Step 1: Time Dependent

1 In the Model Builder window, expand the Study 1 node, then click Step 1: Time
Dependent.
2 In the Settings window for Time Dependent, locate the Study Settings section.
3 In the Times text field, type range(0,0.5,1.5) range(1.55,0.05,3)
range(3.2,0.2,5) range(6,1,12).

Solution 1
1 On the Study toolbar, click Show Default Solver.
2 In the Model Builder window, expand the Study 1>Solver Configurations node.
3 In the Model Builder window, expand the Solution 1 node, then click Time-Dependent
Solver 1.
4 In the Settings window for Time-Dependent Solver, click to expand the Absolute
tolerance section.
5 Locate the Absolute Tolerance section. In the Tolerance text field, type 1e-4.
6 Click to expand the Time stepping section. Locate the Time Stepping section. From
the Steps taken by solver list, choose Intermediate.
This setting forces the solver to take at least one step in each specified interval.
7 On the Study toolbar, click Compute.

RESULTS

Temperature (ht)
The first of the two default plots displays the surface temperature of the brake disc and
pad at the end of the simulation interval. Modify this plot to show the time just before
releasing the brake.

1 In the Model Builder window, click Temperature (ht).

2 In the Settings window for 3D Plot Group, locate the Data section.
3 From the Time (s) list, choose 3.800.
4 On the 3D plot group toolbar, click Plot.
Compare the result to the plot shown in Figure 2.
To compare the total heat produced and the heat dissipated, as done in Figure 5,
follow the steps given below.

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1D Plot Group 3
1 On the Model toolbar, click Add Plot Group and choose 1D Plot Group.
2 In the Settings window for 1D Plot Group, type Dissipated and produced heat
in the Label text field.
3 Click to expand the Title section. From the Title type list, choose None.
4 Click to collapse the Title section. Locate the Plot Settings section. Select the x-axis
label check box.
5 In the associated text field, type Time (s).

Dissipated and produced heat

1 On the 1D plot group toolbar, click Point Graph.
2 Select Point 1 only.
3 In the Settings window for Point Graph, locate the y-Axis Data section.
4 In the Expression text field, type log10(W_prod+1).
5 Click to expand the Coloring and style section. Locate the Coloring and Style section.
Find the Line style subsection. From the Color list, choose Blue.
6 Click to expand the Legends section. Select the Show legends check box.
7 From the Legends list, choose Manual.
8 In the table, enter the following settings:

Legends
log10(W_prod+1), heat produced

9 Right-click Results>Dissipated and produced heat>Point Graph 1 and choose

Duplicate.
10 In the Settings window for Point Graph, locate the y-Axis Data section.
11 In the Expression text field, type log10(W_diss+1).
12 Locate the Coloring and Style section. Find the Line style subsection. From the Line
list, choose Dashed.
13 Locate the Legends section. In the table, enter the following settings:

Legends
log10(W_diss+1), heat dissipated

Finally, follow the steps below to reproduce the plot in Figure 3.

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Data Sets
1 On the Results toolbar, click Cut Line 3D.
2 In the Settings window for Cut Line 3D, locate the Line Data section.
3 In row Point 1, set z to 0.013.
4 In row Point 2, set x to -0.047.
5 In row Point 2, set y to 0.1316.
6 In row Point 2, set z to 0.013.
7 Click the Plot button.

8 On the Results toolbar, click More Data Sets and choose Parametric Extrusion 1D.
9 In the Settings window for Parametric Extrusion 1D, locate the Data section.
10 From the Time selection list, choose From list.
11 Click and Shift-click in the list to select all time steps from 1.5 through 5 s.

2D Plot Group 4
1 On the Results toolbar, click 2D Plot Group.
2 In the Settings window for 2D Plot Group, type Temperature profile vs time
in the Label text field.

Temperature profile vs time

1 Right-click Results>2D Plot Group 4 and choose Surface.

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2 In the Settings window for Surface, locate the Coloring and Style section.
3 From the Color table list, choose ThermalLight.
4 Right-click Results>Temperature profile vs time>Surface 1 and choose Height
Expression.

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