# The Mutual Influence of Thermal Contact Conductivity and Convective Cooling on the Temperature Field in a Tribosystem with a Functionally Graded Strip

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- Formulating and obtaining an exact solution to the appropriate boundary-value thermal conductivity problem.
- (2)
- Verification of the received solution.
- (3)
- Obtaining asymptotic solutions for small and large values of the Fourier number.
- (4)
- Numerical analysis for selected materials of the friction pair.
- (5)
- Summary and conclusions.

## 2. Materials and Methods

## 3. Results and Discussion

^{−4}.

_{r}of the results obtained by means of an asymptotic solution (85) for the FGM strip at high values $\tau $ is visible in Figure 8a. The difference in the results obtained by means of exact (30) and asymptotic (86) solutions for $B{i}_{r}=0.01;1$ was insignificant at $\tau \ge 5$, and for $B{i}_{r}=10;100$, it was permissible not only at large but also at small values $\tau $. The asymptotic solution (86) for semi-space at high values of the Fourier number could be used at $\tau \ge 0$ (Figure 8b).

_{c}= 100) convective cooling of the free surface of the strip for two values of the Biot number Bi

_{r}is demonstrated in Figure 9. First of all, it should be noted that the temperature of the strip made of FGM was lower compared to the case of applying a layer of zirconium dioxide to the material. Secondly, just like during sliding with constant specific friction power, the jumps of isolines were also visible when passing through the contact surface in the case of Bi

_{r}= 1 (Figure 9a). However, for Bi

_{r}= 100, the temperature of the friction surfaces of the strip and semi-space were the same (Figure 9b). At low contact thermal conductivity (Bi

_{r}= 1), the temperature of the strip was higher than the semi-space temperature (Figure 9a). The increase in parameter Bi

_{r}equalized the temperature of both elements of the friction pair (Figure 9b). The effect of the linearly decreasing time profile of the specific friction power was visible primarily in the achievement of the maximum temperature of both elements at a fixed distance from the contact surface, not at the stop moment $\tau ={\tau}_{s}=1$ (as it was in the case of sliding with constant specific friction power), but within the time interval $0<\tau <{\tau}_{s}$.

## 4. Conclusions

_{r}and Bi

_{c}, defined by Formula (9), on the temperature of such a friction system was examined. The first of them (Bi

_{r}) was directly proportional to the thermal contact conductivity h

_{r}, which in turn was inversely proportional to the thermal resistance of the friction surface of the strip and semi-space. Thus, at a fixed value of the contact pressure, the higher the roughness of these surfaces, the greater their thermal resistance and the lower the thermal contact conductivity.

_{r}or in the dimensionless form of the Biot number Bi

_{r}. Both of these conditions created the so-called conditions of imperfect thermal contact of friction. A characteristic of the solutions of appropriate heat problems of friction, obtained under such boundary conditions, is the temperature jump on the contact surfaces of the sliding bodies. It should be noted that during the sliding of smooth surfaces, the thermal resistance became negligible and the value of the parameter Bi

_{r}was large. In the limit case, $B{i}_{r}\to \infty $, the temperature difference in the friction surfaces disappeared. So, it can be said the generation of heat takes place in conditions of perfect thermal friction contact.

_{c}characterized the intensity of convective cooling on the exposed surface of the strip by means of the heat transfer coefficient h

_{c}. The influence of the parameter Bi

_{c}on the temperature of the system operating in conditions of perfect thermal contact of friction was examined in article [48].

- Regardless of the value Bi
_{c}, the perfect thermal contact of friction occurs when Bi_{r}≅ 100. - The use of the functionally graded material on a strip reduces its temperature compared to the case of homogeneous material. The greater the decrease in temperature, the smaller the parameter Bi
_{r}and the higher the parameter Bi_{c}is. The effect of the gradient nature of the strip material on lowering the temperature of the semi-space is insignificant. - Most of the heat generated during friction is absorbed by the cast iron semi-space. With a fixed value of the Biot number Bi
_{c}, the greatest changes in the intensity of heat fluxes, directed along the normal to the contact surface between the strip and the semi-space, occur in the range of 0 ≤ Bi_{r}≤ 10. As the parameter Bi_{c}increases, the impact of the gradient of the strip material on the intensity of heat fluxes becomes more noticeable. - The obtained asymptotic solutions at small and large values of the Fourier number τ can be used with sufficient accuracies to estimate the temperature of the considered friction pair. Increasing the parameter Bi
_{r}causes the widening of the interval of the parameter τ, in which the temperature of the strip obtained from asymptotic solutions slightly differs from the temperature determined by the exact solution. Such an effect was not noticed when determining the temperature on the friction surface of the semi-space: the appropriate asymptotic solutions allow one to obtain satisfactory results in the entire range of the Fourier number changes. - With a linearly decreasing time profile of the specific friction power, the spatial–temporal temperature distribution in the strip and semi-space is non-uniform. At each set distance from the contact surface, the temperature reaches its maximum value within the heating time interval. In a strip made of FGM, the moment of reaching the maximum temperature approaches the stop moment as the distance from the contact surface increases. However, when using a homogeneous layer and half-space materials, the maximum temperature is reached in approximately half the heating time.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$B{i}_{c}$ | Biot number on the free surface of the strip |

$B{i}_{r}$ | Biot number through the contact surface |

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

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

${h}_{c}$ | Coefficient of convective heat exchange on the free surface of the strip |

${h}_{r}$ | Coefficient of contact heat conductivity through the surfaces |

${\mathrm{I}}_{\mathrm{n}}(\cdot )$ | 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 ($\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$) |

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

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

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

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

$T$ | Temperature (${}^{\xb0}\mathrm{C}$) |

${T}_{0}$ | Initial temperature (${}^{\xb0}\mathrm{C}$) |

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

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

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

$\Lambda $ | Temperature rise scaling factor (${}^{\xb0}\mathrm{C}$) |

$\epsilon $ | Dimensionless coefficient of thermal activity |

$\Theta $ | Temperature rise (${}^{\xb0}\mathrm{C}$) |

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

${\widehat{\Theta}}^{\ast}$ | Dimensionless temperature rise during sliding |

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

$\tau $ | Dimensionless time |

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

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

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**Figure 2.**Dependences of dimensionless temperature rise ${\Theta}^{\ast}$ on the friction surface of the strip ($\zeta ={0}^{+}$) and the semi-space ($\zeta ={0}^{-}$) from the Biot number $B{i}_{r}$ for $\tau =1$ at four values of the Biot number $B{i}_{c}$: (

**a**) 0.01; (

**b**) 1; (

**c**) 10; (

**d**) 100. The continuous curves—the FGM strip; the dashed curves—the strip made of zirconium dioxide.

**Figure 3.**Dependences of dimensionless heat fluxes intensities ${q}_{i}^{\ast}$, $i=1,2$ on the Biot number $B{i}_{r}$ for $\tau =1$, at four values of the Biot number $B{i}_{c}$: (

**a**) 0.01; (

**b**) 1; (

**c**) 10; (

**d**) 100. The continuous curves—the FGM strip, the dashed curves—the strip made of zirconium dioxide.

**Figure 4.**Evolutions of dimensionless temperature rise ${\Theta}^{\ast}$ on the friction surface of the strip ($\zeta ={0}^{+}$) and the semi-space ($\zeta ={0}^{-}$) for different values of the Biot number $B{i}_{r}$ at four values of the Biot number $B{i}_{c}$: (

**a**,

**b**) 0.01; (

**c**,

**d**) 1; (

**e**,

**f**) 10; (

**g**,

**h**) 100. The continuous curves—the FGM strip; the dashed curves—the strip made of zirconium dioxide.

**Figure 5.**Evolutions of dimensionless intensities of the heat fluxes ${q}_{i}^{\ast}$, i = 1, 2 for different values of the Biot number Bi

_{r}at four values of the Biot number Bi

_{c}: (

**a**) 0.01; (

**b**) 1; (

**c**) 10; (

**d**) 100. The continuous curves—the FGM strip; the dashed curves—the strip made of zirconium dioxide.

**Figure 6.**Change in dimensionless temperature rise ${\Theta}^{\ast}$ with dimensionless distance $\zeta $ from the contact surface $\zeta =0$ at the final moment of time $\tau =1$ of the heating process for selected values of Biot number $B{i}_{r}$ at four values of the Biot number $B{i}_{c}$: (

**a**) 0.01; (

**b**) 1; (

**c**) 10; (

**d**) 100. The continuous curves—the FGM strip; the dashed curves—the strip made of zirconium dioxide.

**Figure 7.**Evolutions of the dimensionless temperature rise ${\Theta}^{\ast}$ on the surfaces of friction: (

**a**) FGM strip ($\zeta ={0}^{+}$); (

**b**) homogeneous semi-space ($\zeta ={0}^{-}$) for $B{i}_{c}=1$ at four values of the Biot number $B{i}_{r}=0.01;1;10;100$. The solid curves—the exact solutions (36) and (37); the dotted curves—the asymptotic solutions (80) and (81) at small values of the Fourier number $\tau $.

**Figure 8.**Evolutions of the dimensionless temperature rise ${\Theta}^{\ast}$ on the friction surfaces: (

**a**) FGM strip ($\zeta ={0}^{+}$); (

**b**) homogeneous semi-space ($\zeta ={0}^{-}$) for $B{i}_{c}=1$ at four values of the Biot number $B{i}_{r}=0.01;1;10;100$. The solid curves—the exact solutions (29) and (30); the dotted curves—the asymptotic solutions (85) and (86) at large values of the Fourier number $\tau $.

**Figure 9.**Isolines of the dimensionless temperature rise ${\widehat{\Theta}}^{\ast}\left(\zeta ,\tau \right)$ during braking for $B{i}_{c}=100$, ${\tau}_{s}=1$ at two values of the Biot number $B{i}_{r}$: (

**a**) 1; (

**b**) 100. The continuous curves—the FGM strip; the dashed curves—the strip made of zirconium dioxide.

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**MDPI and ACS Style**

Yevtushenko, A.; Topczewska, K.; Zamojski, P.
The Mutual Influence of Thermal Contact Conductivity and Convective Cooling on the Temperature Field in a Tribosystem with a Functionally Graded Strip. *Materials* **2023**, *16*, 7126.
https://doi.org/10.3390/ma16227126

**AMA Style**

Yevtushenko A, Topczewska K, Zamojski P.
The Mutual Influence of Thermal Contact Conductivity and Convective Cooling on the Temperature Field in a Tribosystem with a Functionally Graded Strip. *Materials*. 2023; 16(22):7126.
https://doi.org/10.3390/ma16227126

**Chicago/Turabian Style**

Yevtushenko, Aleksander, Katarzyna Topczewska, and Przemysław Zamojski.
2023. "The Mutual Influence of Thermal Contact Conductivity and Convective Cooling on the Temperature Field in a Tribosystem with a Functionally Graded Strip" *Materials* 16, no. 22: 7126.
https://doi.org/10.3390/ma16227126