Numerical Investigation of Material and Structural Influence on Transient Temperature Behavior in Disc Brakes During Single-Stop Braking

This section briefly represents the history of heat flux in the disc brake. The heat flux is considered the frictional heat input load between the brake disc and the pad, which is applied to the friction surface of the brake disc during the simulation process. The following assumptions are made before calculating: The material properties of the brake disc are isotropic and independent of temperature. The brake pad pressure remains constant during the braking process. The friction coefficient between the brake disc and the pad is constant during braking. The convective heat transfer coefficient of the brake disc surface is constant. Radiation heat transfer is neglected due to the short braking time and low temperature. All kinetic energy at the disc brake surface is converted into frictional heat, which is uniformly distributed to the surface of the brake disc through heat flux.

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Figure 1. Disk brake assembly with a rotor and a caliper

Figure 1 illustrates a typical automotive disc brake, which includes a disc rotor and a caliper. The caliper holds two pads and presses them against both sides of the disc rotor. Each brake pad consists of a steel backing plate with friction material bonded to its surface. During braking, the friction material on both pads contacts the disc rotor, generating friction that slows down or stops the vehicle. The sliding contact model of the friction pair is depicted in Figure 2.

During braking, the pads are forced onto both sides of the disc, generating heat in this action. Two models have been proposed to calculate the heat generation in this process: the Macroscopic model and the Microscopic model [18]. This study uses the Microscopic model. The following is a brief overview of this method.

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Figure 2. A schematic contact model of the pad-disc braking system

The quantity of heat generated in the contact zone between the disc and pad is directly proportional to the friction force. When the brake pads press against the rotating disc, friction between the surfaces converts kinetic energy into thermal energy. This heat is absorbed by the disc, which accounts for approximately 90% of the heat [4, 5], and by the pads and other components of the brake system. Two models have been proposed to describe thermal contact: The perfect contact and the imperfect contact models [19]. This study uses the imperfect contact model. In this model, a third body of detached particles is introduced between the disc and pad surfaces. When employing this model, the heat flux on the disc and pad is calculated as following [18].

First, the thermal effusivity of the disc ($\xi_{\mathrm{d}}$) and thermal effusivity of the pad ($\xi_{\mathrm{p}}$) were defined as:

$\xi_d=\sqrt{k_d \rho_d c_d}$                                                  (1)

$\xi_p=\sqrt{k_p \rho_p c_p}$                                                  (2)

where, the subscripts d and p denote the disc and the pad, respectively; k represents thermal conductivity; ρ denotes density; and c stands for specific heat.

The contact surfaces area of the disc (S_d) and pad (S_p) are determined using the following equations:

$S_d=2 \pi \int_{r_d}^{R_d} r d r$                                                  (3)

$S_p=\varphi_0 \int_{r_p}^{R_p} r d r$                                                  (4)

Second, the coefficient of heat partition is calculated using Eq. (5) [20]:

$\gamma=\frac{\xi_d S_d}{\xi_d S_d+\xi_p S_p}$                                                  (5)

As previously mentioned, we assumed that all friction power is converted into thermal energy. Consequently, the heat generation rate resulting from friction between the contact zones of the disc and pad surfaces is computed using the following equations:

$d \dot{E}=d P_{b r}=V d F_f=r \omega \mu p \varphi_0 r d r$                                                  (6)

$d \dot{E}=d \dot{E}_p+d \dot{E}_d$                                                  (7)

$d \dot{E}_p=(1-\gamma) d P_{b r}=(1-\gamma) \mu p \omega \varphi_0 r^2 d r$                                                  (8)

$d \dot{E}_d=\gamma d P_{b r}=\gamma \mu p \omega \varphi_0 r^2 d r$                                                  (9)

where, $d \dot{E}$ is the rate of heat generated at the contact zone of disc-pad, P_br is the brake power, V is the relative sliding velocity (V = r.w) and dF_f is the friction force, $d F_f=\mu d F_n=\mu p d A$ (Figure 3), where F_n is the normal force and m is the coefficient of friction [3, 21]. $d \dot{E}_p$ and $d \dot{E}_d$ are the heat absorbed by the pad and disc, respectively. Furthermore, this work assumes that the vehicle's velocity decelerates at a constant rate during the braking process. Therefore, the wheel speed can be determined in Eq. (10):

$\omega(t)=\omega_0\left(1-\frac{t}{t_b}\right)$                                                 (10)

where, t_b is the braking time.

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Figure 3. Components of the contact surface element - (a) the disk, and (b) the pad

Then, the heat flux to the pad surface q₁(r,t) and to the disc surface q₂(r,t) were calculated by dividing $d \dot{E}_p$ and $d \dot{E}_d$ by dS_p and dS_d, respectively.

$q_1(r, t)=\frac{d \dot{E}_p}{d S_p}=(1-\gamma) \mu p r \omega(t)$                                                 (11)

$q_2(r, t)=\frac{d \dot{E}_d}{d S_d}=\frac{\varphi_0}{2 \pi} \gamma \mu p r \omega(t)$                                                 (12)

The pressure distributions have been suggested in two models: Uniform pressure and uniform wear. This study used for the second one. Hence, the pressure distribution can be expressed using following equation:

$p=p_{\max } \frac{r_p}{r}$                                                 (13)

where, p_max represents the peak pressure distributed across the pad, p is pressure at radial position r, and r_p is the inner radius in pad (Figure 3). When entered $p=p_{\max } \frac{r_p}{r}$ and w(t) into Eq. (13), the result of heat flux on the disc is:

$q_2(t)=\frac{d \dot{E}_d}{d S_d}=\frac{\varphi_0}{2 \pi} \gamma \mu p_{\max } r_p \omega_0\left(1-\frac{t}{t_b}\right)$                                               (14)

4.1 Comparison with existing studies

This section aims to validate the developed finite element model by comparing its calculations with existing literature. For this purpose, we used a current numerical method to reproduce the results of Dubale et al. [30].

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Note: Temperature trends over time from [30] (red solid line with circular dots) and current study (blue solid line with square dots)

Figure 7. Verification of temperature

Figure 7 illustrates the relationship between maximum temperature and time, showing a significant agreement between the results obtained from the numerical method in this work and previously published numerical results.

4.2 The influence of materials on disc brake transient temperature profiles

This section investigates the thermal characteristics of three disc models, with each model being fabricated from three different materials: GCI, Inconel 718, and AISI 420.

The ANSYS transient thermal model was employed to determine the maximum temperature over time and the temperature distribution contours. Figure 8 illustrates the maximum temperatures for three disc models, each made from different materials. These results are consistent with existing literature [31, 32]. As expected, differences in material properties lead to significant variations in thermal behaviour. The thermal conductivity of GCI (k_GCI = 50 W.m^-1. K^-1)) is more than three times higher than that of AISI 420 (K_AISI420 = 16 W.m^-1. K^-1) and over 4.5 times higher than that of Inconel 718 (K_{Inconel 718} = 11 W.m^-1. K^-1). Consequently, the Inconel 718 disc brake reaches the highest maximum temperature over time, while the GCI disc brake has the lowest maximum temperature. The lower thermal conductivity of Inconel 718 limits heat dispersion, resulting in higher thermal energy retention compared to the GCI disc brake. Furthermore, the maximum temperature observed in these results is affected by the mass and specific heat capacity of the material used to fabricate the disc, as indicated in Eq. (20):

$Q=m c_p \Delta T$                                      (20)

where, Q(J) is amount of heat absorbed by the disc, m (kg) is the mass of disc, c_P (J.kg^-1) is the specific heat capacity of the material used to make the disc and DT is the temperature variation.

As illustrated in Table 3, the heat rates supplied to GCI, AISI 420, and Inconel 718 differ due to differences in the heat partition coefficient. Nonetheless, they can be considered equivalent for comparative purposes. Consequently, by setting Eq. (20) equal to GCI, AISI 420, and Inconel 718, the following relationships are obtained:

$\left.\mathrm{mc}_{\mathrm{p}} \Delta \mathrm{T}\right|_{\mathrm{GCI}}=\left.\mathrm{mc}_{\mathrm{p}} \Delta \mathrm{T}\right|_{\text {AISI } 420}=\left.\mathrm{mc}_{\mathrm{p}} \Delta \mathrm{T}\right|_{\text {Inconel } 718}$                                      (21)

$\frac{\left.m c_p\right|_{G C I}}{\left.m c_p\right|_{A I S I 420}}=\frac{\left.\Delta T\right|_{A I S I 420}}{\left.\Delta T\right|_{G C I}}$                                   (22)

$\frac{\left.m c_p\right|_{ {AISI420 }}}{\left.m c_p\right|_{ {Inconel718 }}}=\frac{\left.\Delta T\right|_{{Inconel718 }}}{\left.\Delta T\right|_{ {AISI420 }}}$                              (23)

Substitute the values of mass of disc (note that m = $\rho$V, where $\rho$ is the density of body and V is the volume of body) and c_p from the Table 1 into these equations, we can draw the following relationship as below:

$\left.\rho \mathrm{V} c_p\right|_{ {Inconel718 }}<\left.\rho \mathrm{V} c_p\right|_{ {AISI420 }}<\left.\rho \mathrm{V} c_p\right|_{G C I}$                                    (24)

$\Delta T_{G C I}<\Delta T_{ {AISI420 }}<\Delta T_{ {Inconel718 }}$                                     (25)

It can thus be observed that the results from Figure 8 follow this relationship successfully.

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Figure 8. Maximum temperature of three disc models made from different materials: Gray cast iron (abbreviated as GCI), Inconel 718, and AISI 420

4.3 The influence of structure on disc brake transient temperature distributions

The simulation results of the thermal performance of three kinds of discs with the same material, grey cast iron, are presented in this section. In general, the heat of disk brake is dissipated by radiation, conduction, and convection. In this work, radiation is neglected, as in many other studies in the literature [3, 13]. The thermal energy generated at the friction surfaces is stored in the disc brake through heat conduction. Subsequently, this heat is conducted to the surrounding components of the brake system. However, these surrounding components, such as the wheel cylinders, brake fluid, and wheel bearings, are constrained by the maximum allowable temperature to maintain optimal working conditions. Therefore, the heat stored in the disc must be dissipated as quickly as possible [13]. The primary method for dissipating heat from the discs is through convection, which occurs in two ways: Along the sides of the disc and through the cooling fins. The convective terms are defined as the following general formula [33].

$Q=A h\left(T_d-T_{\infty}\right)$                                       (26)

where, Q is the rate of heat transfer out of the disc (W), h is heat transfer coefficient (W.m^-2K^-1), A is the surface area (m²), T_d is the surface temperature of the disc (℃), and $\mathrm{T}_{\infty}$ is the temperature of the environment (℃).

Eq. (26) reveals that with an increase in the surface area, the heat transfer out of the disc also rises. To investigate this further, simulations were conducted to explore the thermal behavior of three discs made of the same material, gray cast iron, but differing in surface area (A): Specifically, A_S < A_V42 < A_V32. According to the theoretical analysis, it's expected that the disc brake's temperature will decrease as the surface area increases, i.e., T(solid) > T(ventilated disc with 42 fins) > T(ventilated disc with 32 fins). However, an interesting observation arises from Figure 9. In the initial second, the rate of surface temperature increase is identical across all three types of brake discs.

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Figure 9. The evolution of maximum temperature over time for variants of grey cast iron discs

Initially, the maximum temperature increases on the surface of the three disc types is similar for about the first second. Subsequently, the maximum temperature of the ventilated disc brake exceeds that of the solid disc over time, i.e., T(solid) < T(ventilated disc with 42 fins) < T(ventilated disc with 32 fins). As shown in Table 1, the mass of the solid disc is greater than that of the ventilated disc. Therefore, the thermal capacity (C_T = c_p.m_d) of the solid disc is higher than that of the ventilated disc. This finding clarifies the earlier observation, indicating that the higher maximum temperature of the ventilated disc compared to the solid disc is due to its lower thermal capacity.

Figures 10-15 illustrate the temperature variation on the surface and through the thickness of a quarter of the disc brake. Utilizing gray cast iron, renowned for its high thermal conductivity (as mentioned in section 4.2), in crafting disc brakes promotes efficient heat conduction toward the interior of the disc. Consequently, the peak temperature reached on the disc's surface remains lower compared to discs made from materials with lower thermal conductivity. Moreover, when employing gray cast iron, the surface temperature of the ventilated disc brake exceeds that of the solid disc brake. This is because ventilated brake discs have reduced thermal capacity, hindering the efficient conduction of heat from the surface to the disc's interior. Conversely, when using other materials with low thermal conductivity coefficients, the maximum surface temperatures of solid and ventilated disc brakes are identical. This phenomenon can be explained by the limited time available for heat conduction, coupled with the material's low thermal conductivity, resulting in the retention of heat near the surface.

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Figure 10. Temperature variation on the surface and through the disc thickness of the solid disc for a material of (a) GCI, (b) AISI 420, and (c) INCONEL 718 when surface reaches the maximum temperature

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Figure 11. Temperature variation on the surface and through the disc thickness of the ventilated disc with 42 fins for a material of (a) GCI, (b) AISI 420, and (c) INCONEL 718 when surface reaches the maximum temperature

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Figure 12. Temperature variation on the surface and through the disc thickness of the ventilated disc with 32 fins for a material of (a) GCI, (b) AISI 420, and (c) INCONEL 718 when surface reaches the maximum temperature

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Figure 13. Temperature variation on the surface and through the disc thickness of the solid disc for a material of (a) GCI, (b) AISI 420, and (c) INCONEL 718 at t = 4s

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Figure 14. Temperature variation on the surface and through the disc thickness of the ventilated disc with 42 fins for a material of (a) GCI, (b) AISI 420, and (c) INCONEL 718 at t = 4s

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Figure 15. Temperature variation on the surface and through the disc thickness of the ventilated disc with 32 fins for a material of (a) GCI, (b) AISI 420, and (c) INCONEL 718 at t = 4s

Figure 16 illustrates the temperature distribution of the brake disc at the mean sliding radius and different axial positions. The surface temperature of the brake disc is consistently higher than the internal axial temperature. Notably, this temperature difference decreases towards the end of the braking process for the GCI disc (Figures 16 (b), 16(e), and 16(h)). GCI has higher thermal conductivity than AISI 420 and INCONEL 718 (Table 1). This superior conductivity allows for better heat transfer from the source to the brake disc, leading to a smaller increase in temperature for the same amount of heat. As a result, the surface temperature of the GCI disc is lower than that of the AISI 420 and INCONEL 718 discs. Additionally, Figure 16 shows that for the same material but different configurations, the surface temperature of the solid disc is lower than that of the ventilated disc. This can be explained by the volumetric heat capacity. The volumetric heat capacity (ρVc_p) of the solid disc is higher than that of the ventilated disc. A higher volumetric heat capacity improves heat conduction efficiency, resulting in a smaller increase in surface temperature for the same heat input.

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Figure 16. Temperature variation curves in the axial direction for various materials of three types of brake discs

The authors wish to thank the Thai Nguyen University of Technology for their support.