# Type-Road Heat Transfer Coefficient Equation Proposal
## 1 Introduction
[20] developed a model, based on finite element analysis (FEA), able to predict the thermal distribution inside the tyre at steady state. [PERSON] et al. [21] developed a simulation procedure that combines dynamic mechanical tests and material tests in order to evaluate, through computer simulation, the temperature distribution. Even solid rubber tyres for aircraft have been studied. For instance, [PERSON] et al. [22] developed a particular model from the thermo-mechanical point of view, acknowledging that the high heating of such parts of the airplane can threaten safety during the landing procedures. All of the above cited models, which depend upon the number of sub-systems into which the tyre is subdivided, calculate the heat transfer between the various parts by means of convection and conduction. It is worth mentioning that, usually, even advanced tyre models do not take into account any radiation thermal coefficient towards the external environment because it is considered negligible, as reported by [PERSON] et al. [12]. The said heat transfer mechanisms considered are usually (Figure 1) represented by:  
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 \(Hc\). Whereas the first four heat transfer mechanisms are well studied and discussed in the existing thermodynamic tyre models, the heat exchanged with the ground is vastly overlooked because information about the _Hc_ coefficient is scarce and its actual value, which was obtained only in empirical ways, is not easy to find in the literature. The _Hc_ coefficient is important because it takes into account, in a macroscopic manner, the conduction heat transfer during the transitory time of contact between the tyre and the ground. By means of the said coefficient, the expression of the heat flux, \(\varphi\) [W], exchanged with the ground can be calculated with the Newton formula as follows:  
\[\varphi=Hc\cdot A\cdot\Delta T \tag{1}\]  
where \(A\) [m\({}^{2}\)] is the tyre contact patch area and \(\Delta\)T [K] is the temperature difference between the road and the surface of the tyre tread. Usually, in the literature as well as in practice, the tyre-road heat transfer coefficient _Hc_ is proposed as an empirical constant, as declared by [PERSON] et al. in ref. [23]. However, the (few) values published in the literature are very different one from another, suggesting that the actual value of _Hc_ may depend upon the testing conditions. For example, [PERSON] [24] suggested a value of \(3\times 10^{4}\) W/m\({}^{2}\)K. [PERSON] et al. [25] proposed one of the most used values, \(1.2\times 10^{4}\) 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. [PERSON] et al. [29] found results significantly lower and around \(2\times 10^{3}\) 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 \(1.2\times 10^{4}\) 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. [PERSON] 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.