Created in COMSOL Multiphysics 5.

Heat Generation in a Disc Brake

This model is licensed under the COMSOL Software License Agreement 5.4.
All trademarks are the property of their respective owners. See www.comsol.com/trademarks.
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 application, 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 application (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:

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

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.

4 | HEAT GENERATION IN A DISC BRAKE

Figure 3: Acceleration profile of the disc.

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

P 1 dv
P b = ---- = – --- m v ------
8 8 dt

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

Pb
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

At the end of the computation, produced and dissipated heat can be recovered using the
relations

5 | HEAT GENERATION IN A DISC BRAKE

 t0
W prod = 0 Qprod dt
(1)
t0
W diss = 0 Qdiss dt

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 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 = 3.8 s).

6 | HEAT GENERATION IN A DISC BRAKE

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
shown in Figure 5. The result is displayed in Figure 6. You can see that the maximum
temperature is approximately 430 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

8 | HEAT GENERATION IN A DISC BRAKE

(z = 0.013 m) as a function of time.

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 (W) 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 produced heat (solid line) and dissipated heat (dashed).

The results of this application 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.

9 | HEAT GENERATION IN A DISC BRAKE

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

Application Library path: Heat_Transfer_Module/

Thermal_Contact_and_Friction/brake_disc

Modeling Instructions
From the File menu, choose New.

NEW
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 General Studies>Time Dependent.
6 Click Done.

GLOBAL DEFINITIONS
Define the global parameters by loading the corresponding text file provided.

1 In the Model Builder window, under Global Definitions click Parameters 1.

2 In the Settings window for Parameters, locate the Parameters section.
3 Click Load from File.
4 Browse to the model’s Application Libraries folder and double-click the file
brake_disc_parameters.txt.

GEOMETRY 1

Cylinder 1 (cyl1)
1 In the Geometry toolbar, click Cylinder.

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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 In the Geometry toolbar, click Build All.

Cylinder 2 (cyl2)
1 In 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 In the Geometry toolbar, click Build All.

Work Plane 1 (wp1)

1 In 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.
4 Click Show Work Plane.

Work Plane 1 (wp1)>Bézier Polygon 1 (b1)

1 In the Work Plane toolbar, click 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.
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.

11 | HEAT GENERATION IN A DISC BRAKE

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 In the Work Plane toolbar, click Build All.

Work Plane 1 (wp1)

In the Model Builder window, under Component 1 (comp1)>Geometry 1 click
Work Plane 1 (wp1).

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

Distances (m)
0.0065

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4 In the Geometry toolbar, click Build All.
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.

DEFINITIONS

Explicit 1
1 In 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 In 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 In the Definitions toolbar, click Explicit.
2 In the Settings window for Explicit, type Contact Faces in the Label text field.

13 | HEAT GENERATION IN A DISC BRAKE

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 in the Graphics toolbar.
5 Select Boundary 11 only.
6 Click the Wireframe Rendering button in the Graphics toolbar again to return to the
original state.

Explicit 4
1 In 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.

Integration 1 (intop1)
1 In 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.

Integration 2 (intop2)
1 In 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.

Now, define the velocity and acceleration of the car through these two piecewise and
analytic functions.

Piecewise 1 (pw1)
1 In 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.

14 | HEAT GENERATION IN A DISC BRAKE

6 In the Size of transition zone text field, type 0.2.
The function definition expects nondimensional quantities for the interval starts and
ends, and the function values. The function definition below uses unit conversions to
do so.
7 Find the Intervals subsection. In the table, enter the following settings:

Start End Function

0 t_brake_start[1/s] v0[s/m]
t_brake_start[1/s] t_brake_end[1/s] v0[s/m]+a0*(t[s]-
t_brake_start)[s/m]
t_brake_end[1/s] 12 v0[s/m]+a0*(t_brake_end-
t_brake_start)[s/m]

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

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

Analytic 1 (an1)
1 In 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.

15 | HEAT GENERATION IN A DISC BRAKE

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

8 Click Plot.

MATERIALS

Material 1 (mat1)
1 In the Materials toolbar, click 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 Variable Value Unit Property

group
Thermal conductivity k_iso ; kii = 82 W/(m·K) Basic
k_iso, kij = 0
Density rho 7870 kg/m³ Basic
Heat capacity at constant Cp 449 J/(kg·K) Basic
pressure

16 | HEAT GENERATION IN A DISC BRAKE

Material 2 (mat2)
1 In the Materials toolbar, click Blank Material.
2 In the Settings window for Material, type Pad in the Label text field.
3 Select Domain 3 only.
4 Locate the Material Contents section. In the table, enter the following settings:

Property Variable Value Unit Property

group
Thermal conductivity k_iso ; kii = 8.7 W/(m·K) Basic
k_iso, kij = 0
Density rho 2000 kg/m³ Basic
Heat capacity at constant Cp 935 J/(kg·K) Basic
pressure

HEAT TRANSFER IN SOLIDS (HT)

Solid 1
In the Model Builder window, under Component 1 (comp1)>Heat Transfer in Solids (ht) click
Solid 1.

Translational Motion 1
1 In 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 In 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.

17 | HEAT GENERATION IN A DISC BRAKE

7 In the U text field, type v(t).
8 In the Text text field, type T_air.

Thermal Contact 1
1 In 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.
4 In the p text field, type ht.tc1.Qb/(mu*v(t)).
5 In the Hc text field, type 800[MPa].
6 Locate the Thermal Friction section. Click the Heat rate button.
7 In the Pb 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.

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

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

18 | HEAT GENERATION IN A DISC BRAKE

Symmetry 1
1 In the Physics toolbar, click Boundaries and choose Symmetry.
2 Select Boundary 3 only.
To compute the produced dissipated heats, 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.

Global Equations 1
1 In the Physics toolbar, click Global and choose Global Equations.
2 In the Settings window for Global Equations, locate the Units section.
3 Click Select Dependent Variable Quantity.
4 In the Physical Quantity dialog box, type energy in the text field.
5 Click Filter.
6 In the tree, select General>Energy (J).
7 Click OK.
8 In the Settings window for Global Equations, locate the Units section.
9 Click Select Source Term Quantity.
10 In the Physical Quantity dialog box, type power in the text field.
11 Click Filter.
12 In the tree, select General>Power (W).
13 Click OK.
14 In the Settings window for Global Equations, locate the Global Equations section.
15 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- 0 0 Produced heat
intop1(ht.tc1.Qb)
W_diss W_disst+ 0 0 Dissipated heat
(intop2(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.Qb) and intop2(ht.q0+

19 | HEAT GENERATION IN A DISC BRAKE

 ht.rflux) correspond to Q_prod and Q_diss. The table thus defines the first-order
ODEs corresponding to Equation 1, so that W_prod and W_diss host the produced and
dissipated heats. The initial values follow from setting t = 0.

MESH 1

Free Triangular 1
1 In the Mesh toolbar, click Boundary and choose Free Triangular.
2 Click the Transparency button in the Graphics toolbar.
3 Select Boundaries 4, 7, and 11 only.
4 Click the Transparency button in the Graphics toolbar again to return to the original
state.

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.

Distribution 1
1 In the Mesh toolbar, click Swept.
2 In the Mesh toolbar, click Distribution.
3 In the Settings window for Distribution, locate the Distribution section.
4 In the Number of elements text field, type 2.
5 In the Model Builder window, click Mesh 1.

20 | HEAT GENERATION IN A DISC BRAKE

6 In the Settings window for Mesh, click Build All.
The complete mesh consists of roughly 5,700 elements.

STUDY 1

Step 1: Time Dependent

1 In the Model Builder window, under Study 1 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).

4 In the Study toolbar, click Show Default Solver.

Solution 1 (sol1)
1 In the Model Builder window, expand the Solution 1 (sol1) node, then click Time-
Dependent Solver 1.
2 In the Settings window for Time-Dependent Solver, click to expand the Absolute Tolerance
section.
3 From the Tolerance method list, choose Manual.
4 In the Absolute tolerance text field, type 1e-4.

21 | HEAT GENERATION IN A DISC BRAKE

5 Click to expand 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.
6 In the Study toolbar, click Compute.

RESULTS
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 step just before
releasing the brake.

Temperature (ht)
1 In the Model Builder window, under Results click Temperature (ht).
2 In the Settings window for 3D Plot Group, locate the Data section.
3 From the Time (s) list, choose 3.8.
4 In the Temperature (ht) toolbar, click Plot.
Compare the result to the plot shown in Figure 4.

To compare the total produced heat and the dissipated heat, as done in Figure 7, follow
the steps given below.

1D Plot Group 3
1 In the Home toolbar, click Add Plot Group and choose 1D Plot Group.
2 In the Settings window for 1D Plot Group, type Dissipated and Produced Heats 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).

Point Graph 1
1 In the Dissipated and Produced Heats 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. 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.

22 | HEAT GENERATION IN A DISC BRAKE

8 In the table, enter the following settings:

Legends
log10(W_prod+1), produced heat

Point Graph 2
1 Right-click Point Graph 1 and choose Duplicate.
2 In the Settings window for Point Graph, locate the y-Axis Data section.
3 In the Expression text field, type log10(W_diss+1).
4 Locate the Coloring and Style section. Find the Line style subsection. From the Line list,
choose Dashed.
5 Locate the Legends section. In the table, enter the following settings:

Legends
log10(W_diss+1), dissipated heat

Dissipated and Produced Heats

Finally, follow the steps below to reproduce the plot in Figure 6.

Cut Line 3D 1
1 In 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, y to 0.1316, and z to 0.013.

23 | HEAT GENERATION IN A DISC BRAKE

5 Click Plot.

Parametric Extrusion 1D 1
1 In the Results toolbar, click More Data Sets and choose Parametric Extrusion 1D.
2 In the Settings window for Parametric Extrusion 1D, locate the Data section.
3 From the Time selection list, choose From list.
4 Click and shift-click in the list to select all time steps from 1.5 s through 5 s.
5 In the Results toolbar, click 2D Plot Group.

2D Plot Group 4
1 In the Model Builder window, under Results click 2D Plot Group 4.
2 In the Settings window for 2D Plot Group, type Temperature Profile vs Time in the
Label text field.

Surface 1
1 In the Temperature Profile vs Time toolbar, click Surface.
2 In the Settings window for Surface, locate the Coloring and Style section.
3 From the Color table list, choose ThermalLight.

Height Expression 1
1 In the Temperature Profile vs Time toolbar, click Height Expression.
2 Click Plot.

24 | HEAT GENERATION IN A DISC BRAKE

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