# Effect of Convective Cooling on the Temperature in a Friction System with Functionally Graded Strip

^{*}

## Abstract

**:**

_{2}—Ti-6Al-4V) strip in combination with the cast iron semi-space. The influence of the convective cooling intensity (Biot number) on the temperature field in the considered friction system was investigated. The developed mathematical model allows for a quick estimation of the maximum temperature of systems, in which one of the elements (FGM strip) is heated on the friction surface and cooled by convection on the free surface.

## 1. Introduction

## 2. Statement to the Problem

## 3. Exact Solution

## 4. Some Special Cases of Solution

## 5. Numerical Analysis

_{2}, and on the second component of material the titanium alloy Ti-6Al-4V was selected. Properties of these materials at the initial temperature ${T}_{0}=20\xb0\mathrm{C}$ are given in Table 1. For the same volume fraction of ZrO

_{2}and Ti-6Al-4V, effective specific heat and density of the strip material amounted to ${c}_{1}=495.55\hspace{0.17em}\mathrm{J}\hspace{0.17em}{\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$ and ${\rho}_{1}=\hspace{0.17em}5266.98\mathrm{kg}\hspace{0.17em}{\mathrm{m}}^{-3}$, respectively. By means of Equation (8), the dimensionless gradient of selected FGM was also established ${\gamma}^{\ast}=1.26$. The rest dimensionless input parameters for the calculations are spatial variable $\zeta $, Fourier number $\tau $ and Biot number $Bi$ (9).

## 6. Conclusions

_{2}—Ti-6Al-4V), sliding against the cast iron half-space (ChNMKh). The following was established:

- (1)
- Applying of FGM for one element of the friction couple (strip) allows for a decrease in the temperature on the contact surface in comparison to the case of the homogeneous strip (zirconium dioxide);
- (2)
- A convective heat exchange with the environment on the free surface of the strip causes a decrease in the temperature on the contact surface at the values of the Biot number $0\le Bi\le 10$. However, the greatest drop in temperature on the free surface of the strip occurs in the range of changes $0\le Bi\le 60$;
- (3)
- Most part of the frictional heat is absorbed by the cast iron semi-space ($\approx 85\%$) in the initial stage of the heating process. With the elapse of the slipping time and the increase in the cooling intensity of the free surface of strip, the amount of heat absorbed by the half-space decreases to 73% for $\tau =1$ and $Bi=100$. The amount of heat directed to the FGM strip increases accordingly;
- (4)
- Obtained asymptotic solutions for small and large values of the Fourier number $\tau $ can be used to quickly estimate the temperature of both elements of the system, with high accuracy. At the same time, the solution for large values of the Fourier number is useful for determining the temperature at any time during the friction heating process at $\tau >0$;
- (5)
- Convective cooling of the FGM strip allows for a reduction in the effective depth of heating, i.e., the distance from the contact surface at which the temperature of each element reaches significant values;
- (6)
- The space–time distribution of isotherms in the strip and semi-space depends on the time profile of the specific friction power. With a constant friction power during sliding, the temperature monotonically increases with the increasing heating time (Fourier number $\tau $). However, in the case of braking with constant deceleration, the temperature of the friction surface reaches its maximum value around half the stopping time ${\tau}_{s}$.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$Bi$ | Biot number |

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

$d$ | Thickness of the strip ($\mathrm{m}$) |

$f$ | Coefficient of friction |

$h$ | Coefficient of convective heat exchange ($\mathrm{W}\hspace{0.17em}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$) |

${\mathrm{I}}_{\mathrm{n}}(\cdot )$ | Modified Bessel functions of the nth order of the first kind |

${\mathrm{J}}_{\mathrm{n}}(\cdot )$ | Bessel functions of the nth order of the first kind |

${\mathrm{K}}_{\mathrm{n}}(\cdot )$ | Modified Bessel functions of the nth order of the second kind |

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

$K$ | Thermal conductivity ($\mathrm{W}\hspace{0.17em}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$) |

$p$ | Parameter of the Laplace integral transform |

${p}_{0}$ | Nominal pressure on the contact surface (Pa) |

$q$ | Specific power of friction ($\mathrm{W}\hspace{0.17em}{\mathrm{m}}^{-2}$) |

${q}_{0}$ | Nominal value of the specific friction power ($\mathrm{W}\hspace{0.17em}{\mathrm{m}}^{-2}$) |

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

${t}_{s}$ | Stop moment of the process ($\mathrm{s}$) |

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

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

${V}_{0}$ | Sliding velocity ($\mathrm{m}\hspace{0.17em}{\mathrm{s}}^{-1}$) |

${\mathrm{Y}}_{\mathrm{n}}(\cdot )$ | Bessel functions of the nth order of the second kind |

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

${\gamma}^{\ast}$ | Gradient parameter of FGM |

$\mathsf{\Lambda}$ | Temperature rise scaling factor (${}^{\circ}\mathrm{C}$) |

$\epsilon $ | Dimensionless coefficient of thermal activity |

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

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

$\rho $ | Density ($\mathrm{kg}\hspace{0.17em}{\mathrm{m}}^{-3}$) |

$\tau $ | Dimensionless time |

${\tau}_{s}$ | Fourier number |

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

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**Figure 2.**Evolutions of dimensionless temperature rise ${\Theta}^{\ast}$ for selected values of Biot number: (

**a**) on the contact surface $\zeta =0$; (

**b**) on the free surface of the strip $\zeta =1$.

**Figure 3.**Dependence of dimensionless temperature rise ${\Theta}^{\ast}$ on the Biot number $Bi$ on the surfaces $\zeta =0$ and $\zeta =1$ for $\tau =1$.

**Figure 4.**Evolutions of dimensionless intensities of a ${q}_{l}^{\ast}(\tau )$ heat fluxes absorbed by FGM strip ($l=1$) and homogeneous half-space ($l=2$) for selected values of the Biot number $Bi$.

**Figure 5.**Evolutions of dimensionless temperature rise ${\Theta}^{\ast}$ for different values of dimensionless spatial variable $\zeta $ for (

**a**) $Bi=0.01$; (

**b**) $Bi=1$; (

**c**) $Bi=10$; and (

**d**) $Bi=100$. The solid curves are the exact solution (35)–(44), and the dashed curves are the asymptotic solution (64) and (65) for small values of the Fourier number $\tau $.

**Figure 6.**Evolutions of dimensionless temperature rise ${\Theta}^{\ast}$ for different values of dimensionless spatial variable $\zeta $ for (

**a**) $Bi=0.01$; (

**b**) $Bi=1$; (

**c**) $Bi=10$; and (

**d**) $Bi=100$. The solid curves are the exact solution(35)–(44), and the dashed curves are the asymptotic solution (74) and (75) for large values of the Fourier number $\tau $.

**Figure 7.**Isotherms of dimensionless temperature rise ${\Theta}^{\ast}(\zeta ,\tau )$ during sliding with specific friction surface: (

**a**,

**b**)—constant; (

**c**,

**d**)—linearly increasing in time for (

**a**,

**c**)—$Bi=1$; (

**b**,

**d**)—$Bi=100$. Solid lines—strip made of FGM ZrO

_{2}—Ti-6Al-4V, dashed lines—homogeneous strip made of ZrO

_{2}.

**Table 1.**Materials’ properties [36].

Material | Thermal Conductivity ${\mathbf{Wm}}^{-1}{\mathbf{K}}^{-1}$ | Specific Heat $\mathbf{J}\hspace{0.17em}{\mathbf{kg}}^{-1}{\mathbf{K}}^{-1}$ | Density $\hspace{0.17em}\mathbf{kg}\hspace{0.17em}{\mathbf{m}}^{-3}$ |
---|---|---|---|

ZrO_{2} | ${K}_{1,1}=1.94$ | ${c}_{1,1}=\hspace{0.17em}452.83$ | ${\rho}_{1,1}=\hspace{0.17em}6102.16$ |

Ti-6Al-4V | ${K}_{1,2}=6.87$ | ${c}_{1,2}=\hspace{0.17em}538.08$ | ${\rho}_{1,2}=\hspace{0.17em}4431.79$ |

ChNMKh | ${K}_{2}=52.17$ | ${c}_{2}=\hspace{0.17em}444.6$ | ${\rho}_{2}=\hspace{0.17em}7100$ |

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**MDPI and ACS Style**

Yevtushenko, A.; Kuciej, M.; Topczewska, K.; Zamojski, P.
Effect of Convective Cooling on the Temperature in a Friction System with Functionally Graded Strip. *Materials* **2023**, *16*, 5228.
https://doi.org/10.3390/ma16155228

**AMA Style**

Yevtushenko A, Kuciej M, Topczewska K, Zamojski P.
Effect of Convective Cooling on the Temperature in a Friction System with Functionally Graded Strip. *Materials*. 2023; 16(15):5228.
https://doi.org/10.3390/ma16155228

**Chicago/Turabian Style**

Yevtushenko, Aleksander, Michał Kuciej, Katarzyna Topczewska, and Przemysław Zamojski.
2023. "Effect of Convective Cooling on the Temperature in a Friction System with Functionally Graded Strip" *Materials* 16, no. 15: 5228.
https://doi.org/10.3390/ma16155228