# Tyre–Road Heat Transfer Coefficient Equation Proposal

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## Featured Application

**Advanced thermo-mechanical tyre models, real time vehicle dynamic simulation, heat transfer calcualtion.**

## Abstract

## 1. Introduction

- (1)
- heat directly generated because of tyre deformations (mainly because of rolling resistance);
- (2)
- heat exchanged with external air by (forced) convection (tyre rolling at speed);
- (3)
- heat internally exchanged between rubber layers or other parts of the tyre;
- (4)
- heat directly generated in the surface layer by viscoelastic excitement of the polymer from rubber sliding over road asperities;
- (5)
- heat exchanged with the road at the contact patch, removed from the surface layer, which requires knowledge of the tyre–road heat transfer coefficient Hc.

^{2}] is the tyre contact patch area and ΔT [K] is the temperature difference between the road and the surface of the tyre tread.

^{4}W/m

^{2}K. Smith et al. [25] proposed one of the most used values, 1.2 × 10

^{4}W/(m

^{2}K), employed, as it is, also by [26,27,28], without any discussion about its validity in relation to the testing conditions. Hackl et al. [29] found results significantly lower and around 2 × 10

^{3}W/m

^{2}K. Taking into account the above discussion, the Hc range seems rather wide, suggesting that its expression should be more complex than a simple constant. To date, there is no evidence that the value 1.2 × 10

^{4}W/(m

^{2}K), even if it is the most used, is the correct one, because as far as the authors’ knowledge goes, there is not an extensive validation of such a number for all of the possible tyre use conditions. Farroni et al. [30] considered Hc a function of the thermal characteristics of the compound only, but as far as the authors’ knowledge goes, there is not a relation or an equation that permits calculating the coefficient on the basis of known parameters related to actual vehicle conditions. There is a lack of knowledge about the phenomenon.

- Section 2: An analysis of the theoretical heat transfer mechanism between the tyre surface and the road, based on energy conservation principles;
- Section 3: A physics-based equation describing the process;
- Section 4: A validation of the model on the basis of the few literature results available;
- Section 5: A discussion of the results;
- Section 6: Some considerations for future possible developments.

## 2. Heat Transfer between Road and Tyre—Thermophysical Aspects

^{2}] and what happens inside the first infinitesimal layer of rubber in terms of the thermal gradient and thermal flux, φ, exchanged with the ground, as can be seen in Figure 2.

_{1}[K], the road one, and T

_{2}[K], the tyre one. Nevertheless, as soon as ΔS is pressed into contact with the ground, because of the tyre rotation, at least the very first molecular layer substrate must reach thermal equilibrium. In that situation, a steep thermal gradient develops inside the rubber surface ΔS, so an intense thermal flux, φ, starts to flow from the tyre surface to the ground [31].

## 3. The Tyre–Road Heat Transfer Coefficient Hc Equation Proposal

_{0}= 0, that corresponds to the instant at which the surface ΔS touches the ground, and then increasing monotonically as long as ΔS remains in contact with the road. The approximation in Figure 3 is based on the hypothesis that, given the short time of contact for a tyre rolling at speed, the length p(t) is very short as well, in comparison to the whole rubber thickness, and that the entire gradient is developing (happening) well inside the surface layer of the tyre rubber.

^{3}K)], considered as a constant as in ref. [33], the energy lost by the rubber of the tyre surface ΔS at time t and transferred to the ground can thus be written as:

_{0}) = 0 because before any contact between the ground and tyre, no flux exchange with the ground happens.

_{c}− t

_{0}, where t

_{c}is the instant when the surface ΔS loses contact with the ground, it comes:

_{2}− T

_{1}):

_{c}as FL/v, and the final formulation of the real-time Hc heat transfer coefficient becomes:

## 4. Equation Validation

^{6}J/(m

^{3}K) 121, Equation (12), with a speed of 41.6 m/s, equal to 150 km/h, for an FL equal to 0.14 [24], gives:

^{3}W/(m

^{2}K). Again, the parameters are in good agreement with the testing conditions and the results are of the same order of magnitude as those found in ref. [29].

^{−5}m (v = 41.6 m/s, that is, 150 km/h) and 10

^{−4}m (v = 2.7 m/s, that is, 10 km/h). Given an average rubber layer thickness in the order of magnitude of 3 × 10

^{−2}m [35], the initial assumption that the whole process happens only at the tyre surface layer can be considered confirmed and therefore validated.

## 5. Discussion

^{4}W/(m

^{2}K) and that, in many models and papers, it is used without discussion about the boundary conditions. In particular, it is used without paying attention to the rolling speed of the tyre or to the wheel footprint length. It is worth noting that, on the basis of the equation analysis, as well as of the testing conditions of some of the users of such a number, it seems that the number should be suitable only for high speeds. In the graph in Figure 4, the Hc behaviour is parameterised as a function of the footprint length of 0.14 m and calculated as a function of the rolling speed, as expressed in kilometres per hour.

^{6}J/(m

^{3}K), it was possible to calculate the Hc values parametrised as a function of increasing the FL values, starting from 0.1 m, with an increment step of 0.02 m. The calculation was performed for increasing speeds in a range from 10 km/h to 300 km/h. The analytical results are reported in Table 2.

^{4}W/(m

^{2}K) remains suitable for high speeds: from 70 km/h, for the lightest footprint, related to very small cars, to 300 km/h, for the heaviest one. When low speeds are considered, it seems more adequate to use values one order of magnitude below. For example, it would be suitable to use 3.1 × 10

^{3}W/(m

^{2}K) for a speed range from 50 km/h to 10 km/h, representing typical city speeds. Nevertheless, if, for example, an FL of 0.18 is considered, the value of 130 km/h could be the lowest threshold where 1.2 × 10

^{4}W/(m

^{2}K) can be applied. Below such a speed, a lower Hc value should be considered. As a matter of fact, from the considerations above, an attentive Hc evaluation should not be a negligible step in the tyre thermal model definition. Actually, it must be remembered that the use of a completely wrong Hc provides, as a result, an unrealistic tyre behaviour, because such a coefficient should be able to offer an account of all of the heat discharged by the tyre through contact with the ground.

## 6. Further Developments

## 7. Conclusions

^{4}W/(m

^{2}K). Conversely, at lower speeds of 2.7 m/s (equivalent to 10 km/h), the equation yields a value of 3.1 × 10

^{3}W/(m

^{2}K). These results support the theoretical justification for employing the empirical values that are commonly found in the literature.

^{4}W/(m

^{2}K), is most suitable within the medium- to high-speed range, spanning from 100 km/h to 200 km/h, with the tire footprint length set at 0.14 m. However, when vehicle speeds fall within the 10 km/h to 50 km/h range, it is more fitting to halve the aforementioned Hc value. For higher speeds, employing the widely disseminated Hc value could potentially lead to an underestimation of the actual heat exchanged with the ground.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Acronyms | |

FEA | Finite Element Analysis |

FEM | Finite Element Method |

SUV | Sports Utility Vehicle |

TRT | Thermo Racing Tyre |

TRICK | Tire/Road Interaction Characterization and Knowledge |

Symbols | |

A | Contact area between tyre and road (m^{2}) |

C | Heat capacity per unit of volume (J/(kgK)) |

R | Radial rigidity of the tyre (N/mm) |

FL | Tyre footprint length (m) |

H_{c} | Tyre–road heat transfer coefficient (W/(m^{2}K)) |

p(t) | Propagation depth (m) |

Q(t) | Heat (J) |

P | Normal wheel load (N) |

S | Tyre surface (m^{2}) |

T | Dry bulb temperature (°C) |

t | Time (s) |

v | Tyre speed (m/s) |

∆z | Radial tyre deformation (mm) |

λ | Conductivity coefficient (W/(mK)) |

φ | Heat flux (W) |

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**Figure 2.**Thermal gradient evolution inside the tyre due to the heat flux passing through the tyre surface ΔS.

**Figure 3.**(

**a**) Temperature gradient approximation between the tyre tread surface and the road. (

**b**) Heat exchanged between the tyre tread surface and the road as a function of the temperature gradient penetration.

**Figure 4.**Hc behaviour as a function of the rolling speed (footprint length of 0.14 m). Red dots represent the literature empirical values.

Speed [m/s] | [km/h] | Hc Literature Values [W/(m ^{2}K)] | References | Hc Equation Results [W/(m ^{2}K)] |
---|---|---|---|---|

2.7 | 10 | 2 × 10^{3} | [29] | 3.1 × 10^{3} |

41.6 | 150 | 1.2 × 10^{4} | [25] | 1.22 × 10^{4} |

Footprint Length [m] | |||||
---|---|---|---|---|---|

0.1 | 0.12 | 0.14 | 0.16 | 0.18 | |

Speed [km/h] | Hc [W 10^{4}/(m^{2}K)] | ||||

10 | 0.372678 | 0.340207 | 0.31497 | 0.294628 | 0.277778 |

30 | 0.645497 | 0.589256 | 0.545545 | 0.51031 | 0.481125 |

50 | 0.833333 | 0.760726 | 0.704295 | 0.658808 | 0.62113 |

70 | 0.986013 | 0.900103 | 0.833333 | 0.779512 | 0.734931 |

90 | 1.118034 | 1.020621 | 0.944911 | 0.883883 | 0.833333 |

110 | 1.236033 | 1.128339 | 1.044639 | 0.97717 | 0.921285 |

130 | 1.34371 | 1.226633 | 1.135642 | 1.062296 | 1.001542 |

150 | 1.443376 | 1.317616 | 1.219875 | 1.141089 | 1.075829 |

170 | 1.536591 | 1.402709 | 1.298656 | 1.214782 | 1.145307 |

190 | 1.624466 | 1.482928 | 1.372924 | 1.284253 | 1.210805 |

210 | 1.707825 | 1.559024 | 1.443376 | 1.350154 | 1.272938 |

230 | 1.787301 | 1.631575 | 1.510545 | 1.412985 | 1.332175 |

250 | 1.86339 | 1.701035 | 1.574852 | 1.473139 | 1.388889 |

270 | 1.936492 | 1.767767 | 1.636634 | 1.530931 | 1.443376 |

290 | 2.006932 | 1.83207 | 1.696167 | 1.586619 | 1.495879 |

300 | 2.041241 | 1.86339 | 1.725164 | 1.613743 | 1.521452 |

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**MDPI and ACS Style**

Cattani, P.; Cattani, L.; Magrini, A.
Tyre–Road Heat Transfer Coefficient Equation Proposal. *Appl. Sci.* **2023**, *13*, 11996.
https://doi.org/10.3390/app132111996

**AMA Style**

Cattani P, Cattani L, Magrini A.
Tyre–Road Heat Transfer Coefficient Equation Proposal. *Applied Sciences*. 2023; 13(21):11996.
https://doi.org/10.3390/app132111996

**Chicago/Turabian Style**

Cattani, Paolo, Lucia Cattani, and Anna Magrini.
2023. "Tyre–Road Heat Transfer Coefficient Equation Proposal" *Applied Sciences* 13, no. 21: 11996.
https://doi.org/10.3390/app132111996