Heat Generation in a Disc Brake

Cars need 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, temperature overload can change the material properties of the brake causing it to fade. Braking power already starts to fade 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?

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:

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.

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

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

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

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

(1)

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 “hotspot” 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).

To investigate the position of the hotspot 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 hotspot is positioned close to the radially outer edge of the pad. The highest temperature occurs approximately 1 s after engaging the brake.

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

Figure 6: Temperature profile along the line indicated in Figure 5 at the disc surface (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 been dissipated. In other words, in order to cool down the system sufficiently the brake needs to remain disengaged for a much 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 modify the parameters affecting the heat dissipation and investigate their influence.

Reference

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

Heat_Transfer_Module/Thermal_Contact_and_Friction/brake_disc

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

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

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file brake_disc_parameters.txt.

Geometry 1

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.14.

In the text field, type 0.013.

In the toolbar, click

Cylinder 2 (cyl2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.08.

In the text field, type 0.01.

Locate the section. In the text field, type 0.013.

In the toolbar, click

Work Plane 1 (wp1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.013.

Click

Work Plane 1 (wp1)>Cubic Bézier 1 (cb1)

In the toolbar, click

and choose .

In the window for , locate the section.

In row , set to 0.135.

In row , set to 0.02, and to 0.135.

In row , set to 0.05, and to 0.13.

In row , set to 0.04, and to 0.105.

Locate the section. In the text field, type 2.5.

Work Plane 1 (wp1)>Cubic Bézier 2 (cb2)

In the toolbar, click

and choose .

In the window for , locate the section.

In row , set to 0.04.

In row , set to 0.105, and to 0.03., and to 0.08.

In row , set to 0.035, and to 0.09., and to 0.09.

Work Plane 1 (wp1)>Cubic Bézier 3 (cb3)

In the toolbar, click

and choose .

In the window for , locate the section.

In row , set to 0.09.

In row , set to -0.035, and to 0.09.

In row , set to -0.03, and to 0.08.

In row , set to -0.04, and to 0.105.

Work Plane 1 (wp1)>Cubic Bézier 4 (cb4)

In the toolbar, click

and choose .

In the window for , locate the section.

In row , set to -0.04.

In row , set to -0.05, and to 0.09., and to 0.13.

In row , set to -0.02, and to 0.135.

In row , set to 0.135.

Locate the section. In the text field, type 2.5.

In the toolbar, click

Work Plane 1 (wp1)>Convert to Solid 1 (csol1)

In the toolbar, click

and choose .

Click in the window and then press Ctrl+A to select all objects.

In the toolbar, click

Extrude 1 (ext1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Distances (m)
0.0065

In the toolbar, click

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

Disc Faces

In the toolbar, click

In the window for , type Disc Faces in the text field.

Locate the section. From the list, choose .

Select Boundaries 1, 2, 4–6, 8, 13–15, and 18 only.

Pad Faces

In the toolbar, click

In the window for , type Pad Faces in the text field.

Locate the section. From the list, choose .

Select Boundaries 9, 10, 12, 16, and 17 only.

Contact Faces

In the toolbar, click

In the window for , type Contact Faces in the text field.

Locate the section. From the list, choose .

To select the contact surface boundary, it is convenient to temporarily switch to wireframe rendering.

Click the

button in the toolbar.

Select Boundary 11 only.

Click the

button in the toolbar again to return to the original state.

External Surfaces

In the toolbar, click

In the window for , type External Surfaces in the text field.

Locate the section. Select the check box.

Locate the section. From the list, choose .

These instructions make you select the external boundaries of the wheel and the pad.

Integration 1 (intop1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Integration 2 (intop2)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

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

Piecewise 1 (pw1)

In the toolbar, click

In the window for , type v in the text field.

Locate the section. In the text field, type t.

From the list, choose .

In the text field, type 0.2.

The function definition expects nondimensional quantities for the interval bounds, and the function values. The function definition below uses unit conversions to do so.

Find the 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]

Locate the section. In the text field, type s.

In the text field, type m/s.

Click

Analytic 1 (an1)

In the toolbar, click

In the window for , type a in the text field.

Locate the section. In the text field, type d(v(t),t).

In the text field, type t.

Locate the section. In the table, enter the following settings:


Argument	Unit
t	s

In the text field, type m/s^2.

Locate the section. In the table, enter the following settings:


Argument	Lower limit	Upper limit	Unit
t	0	10	s

Click

Materials

Disc

In the toolbar, click

In the window for , type Disc in the text field.

Locate the section. In the table, enter the following settings:


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

Pad

In the toolbar, click

In the window for , type Pad in the text field.

Select Domain 3 only.

Locate the section. In the table, enter the following settings:


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

Heat Transfer in Solids (ht)

Solid 1

In the window, under click .

Translational Motion 1

In the toolbar, click

and choose .

Select Domains 1 and 2 only.

In the window for , locate the section.

Specify the utrans vector as


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

Heat Flux 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

From the list, choose .

In the L text field, type 0.14.

In the U text field, type v(t).

In the Text text field, type T_air.

Thermal Contact 1

In the toolbar, click

and choose .

Select Boundary 11 only.

In the window for , locate the section.

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

In the Hc text field, type 800[MPa].

Locate the section. Click the button.

In the Pb text field, type -m_car*v(t)*a(t)/8.

Initial Values 1

In the window, click .

In the window for , locate the section.

In the T text field, type T_air.

Surface-to-Ambient Radiation 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the ε list, choose . In the associated text field, type 0.28.

In the Tamb text field, type T_air.

Surface-to-Ambient Radiation 2

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the ε list, choose . In the associated text field, type 0.8.

In the Tamb text field, type T_air.

Symmetry 1

In the toolbar, click

and choose .

Select Boundary 3 only.

To compute the produced and dissipated heats, integrate the corresponding heat rate variables, Q_prod and Q_diss, over time. For this purpose, define two ODEs using a node.

Click the

button in the toolbar.

In the dialog box, in the tree, select the check box for the node .

Click .

Global Equations 1

In the toolbar, click

and choose .

In the window for , locate the section.

Click

In the dialog box, type energy in the text field.

Click

In the tree, select .

Click .

In the window for , locate the section.

Click

In the dialog box, type power in the text field.

Click

In the tree, select .

Click .

In the window for , locate the section.

In the table, enter the following settings:


Name	f(u,ut,utt,t) (W)	Initial value (u_0) (J)	Initial value (u_t0) (W)	Description
W_prod	W_prodt-intop1(ht.tc1.Qb)	0	0	Produced heat
W_diss	W_disst+(intop2(ht.q0+ht.rflux))	0	0	Dissipated heat

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+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

In the toolbar, click