# The Heat Partition Ratio during Braking in a Functionally Graded Friction Couple

^{*}

## Abstract

**:**

_{2}O

_{3}–Cu (first body) and ZrO

_{2}–Ti–6Al–4V (second body), under the conditions corresponding to a single braking with a constant deceleration. It was established that the vast majority (almost 90%) of heat that was generated by friction was absorbed by the first body in the selected couple. The possibilities of using the obtained results were discussed herein.

## 1. Introduction

## 2. Heating of the FGM Semi-Space by the Heat Flux with Constant Intensity

^{2}components of the series on the left-hand side of Equation (32) is equal to 0.248985.

## 3. Heating of the FGM Semi-Space by Heat Flux with the Intensity Linearly Decreasing in Time

## 4. The Heat Partition Ratio

## 5. Example of Calculation of the Heat Partition Ratio for an FGM Couple

_{2}O

_{3}(base, $m=0$) and copper Cu (core, $m=1$), and the other ($l=2$) contains zircon dioxide ZrO

_{2}(base, $m=0$) and titanium alloy Ti-6Al-4V (core, $m=1$). The thermo-physical properties of these materials are demonstrated in Table 1.

_{2}O

_{3}–Cu) is typical for the evolution of the friction surface temperature during braking with a constant deceleration—a rapid increase in the temperature at the beginning of braking, reaching its maximum value in the middle of the process, followed by a temperature reduction until the standstill. However, in the second element ($l=2$, ZrO

_{2}–Ti–6Al–4V) a rise of temperature on the heated surface is monotonic during the whole process. Such temperature behavior is decisively influenced by the thermo-physical properties of the component materials of each element. In the functionally graded friction couple under consideration, the materials of both of the components of the first element have a significantly greater ability to dissipate the heat from the heated surface than the materials of the second element (Table 1). Moreover, this is confirmed, by the values of the coefficients ${\alpha}_{1}=0.896$, ${\alpha}_{2}=0.104$, which prove that the first element absorbs almost $90\%$, and the second only slightly more than $10\%$ of the entire heat flux intensity $q$ (42). Due to the low thermal conductivity of the component materials of the second element, in particular the zircon dioxide, the temperature of the heated surface of this element continues to rise during heating, even with a linearly decreasing intensity of the heat flux.

_{l}, l = 1, 2 [2]. In general, for the determination of the temperature mode at the design stage of the brake, the Fourier numbers ${\tau}_{l,s},l=1,2$ are given, and the effective heating depths based on the relation (54) are calculated in the following form:

## 6. Conclusions

_{2}O

_{3}–Cu and ZrO

_{2}–Ti–6Al–4V. It was established that element Al

_{2}O

_{3}–Cu absorbs most of the heat that is generated due to friction (almost 90%). The maximum temperature on the friction surface of the Al

_{2}O

_{3}–Cu element is about 820 °C and is achieved in the middle of the braking time. However, the highest temperature of the friction surface of the ZrO

_{2}–Ti–6Al–4V element is achieved at the stop moment and amounts to 970 °C. Thus, the crucial influence on the evolution of the temperature in the functionally graded friction couple have the thermo-physical properties of the component materials.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$a$ | Effective depth of heat penetration ($\mathrm{m}$) |

$c$ | Specific heat capacity ($\mathrm{J}\hspace{0.17em}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$) |

${f}_{0}$ | Coefficient of friction |

${I}_{k}(\cdot )$ | Modified Bessel functions of the first kind of the kth order |

${J}_{k}(\cdot )$ | Bessel functions of the first kind of the kth order |

$k$ | Thermal diffusivity (${\mathrm{m}}^{2}{\mathrm{s}}^{-1}$) |

$K$ | Thermal conductivity ($\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$) |

${K}_{\epsilon}$ | Dimensionless coefficient of thermal activity of friction couple |

$p$ | Dimensionless parameter of the Laplace integral transform |

$q$ | Intensity of heat flux ($\mathrm{W}{\mathrm{m}}^{-2}$) |

${q}_{0}$ | Nominal intensity of the heat flux ($\mathrm{W}{\mathrm{m}}^{-2}$) |

$t$ | Time ($\mathrm{s}$) |

${t}_{s}$ | Braking time ($\mathrm{s}$) |

$T$ | Temperature (${}^{\circ}\mathrm{C}$) |

${T}_{0}$ | Initial temperature (${}^{\circ}\mathrm{C}$) |

$v$ | Volume fraction of the material phases (dimensionless) |

$z$ | Spatial coordinate in axial direction ($\mathrm{m}$) |

$\alpha $ | Heat partition ratio |

$\gamma $ | Parameter of material gradient (${\mathrm{m}}^{-1}$) |

${\gamma}^{\ast}$ | FGMs gradient ratio |

$\Theta $ | Temperature rise (${}^{\circ}\mathrm{C}$) |

${\Theta}^{\ast}$ | Dimensionless temperature rise |

${\Theta}_{0}$ | Scaling factor of temperature rise (${}^{\circ}\mathrm{C}$) |

$\rho $ | Density ($\mathrm{kg}{\mathrm{m}}^{-3}$) |

${\tau}^{*}$ | Dimensionless time |

${\tau}_{s}$ | Dimensionless braking time |

$\zeta $ | Dimensionless spatial coordinate in axial direction |

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**Figure 3.**Isolines of the heat partition ratio $\alpha $ (69) in the coordinate system (${\gamma}^{\ast}$, ${K}_{\epsilon}$).

**Figure 4.**Evolutions of temperature ${T}_{l}$, $l=1,2$ (55) and (56) on the heated surfaces $z=0$ of the considered semi-spaces made of FGMs.

**Figure 5.**Dependences of the heat partition ratio $\alpha $ (62) on: (

**a**) Fourier number ${\tau}_{1,s}$ for different values of ${\tau}_{2,s}$; (

**b**) Fourier number ${\tau}_{2,s}$ for different values of ${\tau}_{1,s}$.

**Table 1.**Thermo-physical properties of FGM components [32].

Element Subscript | Material Subscript, m | Material | Thermal Conductivity ${\mathit{K}}_{\mathit{l},\mathit{m}}^{(\mathbf{0})}\mathbf{,}\hspace{0.17em}{\mathbf{Wm}}^{\mathbf{-}\mathbf{1}}{\mathbf{K}}^{\mathbf{-}\mathbf{1}}$ | Specific Heat Capacity ${\mathit{c}}_{\mathit{l},\mathit{m}}^{(\mathbf{0})}\mathbf{,}\hspace{0.17em}\mathbf{J}\hspace{0.17em}{\mathbf{kg}}^{\mathbf{-}\mathbf{1}}{\mathbf{K}}^{\mathbf{-}\mathbf{1}}$ | $\mathbf{Density}{\mathit{\rho}}_{\mathit{l},\mathit{m}}^{(0)}\mathbf{,}\hspace{0.17em}{\mathbf{kgm}}^{\mathbf{-}\mathbf{3}}$ |
---|---|---|---|---|---|

$l=1$ | base, 0 | Al_{2}O_{3} | 37.24 | 727.29 | 3990.92 |

core, 1 | Cu | 402.65 | 147.35 | 8947.92 | |

$l=2$ | base, 0 | ZrO_{2} | 1.94 | 452.83 | 6102.16 |

core, 1 | Ti-6Al-4V | 6.87 | 538.08 | 4431.79 |

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Yevtushenko, A.; Topczewska, K.; Zamojski, P.
The Heat Partition Ratio during Braking in a Functionally Graded Friction Couple. *Materials* **2022**, *15*, 4623.
https://doi.org/10.3390/ma15134623

**AMA Style**

Yevtushenko A, Topczewska K, Zamojski P.
The Heat Partition Ratio during Braking in a Functionally Graded Friction Couple. *Materials*. 2022; 15(13):4623.
https://doi.org/10.3390/ma15134623

**Chicago/Turabian Style**

Yevtushenko, Aleksander, Katarzyna Topczewska, and Przemysław Zamojski.
2022. "The Heat Partition Ratio during Braking in a Functionally Graded Friction Couple" *Materials* 15, no. 13: 4623.
https://doi.org/10.3390/ma15134623