# Type-Road Heat Transfer Coefficient Equation Proposal
## 5 Discussion  
On the basis of the discussion in Section 4, it is possible to state that the equation provides results in the same order of magnitude as the values found in the literature in similar conditions. Thus, the study, as far as the authors' knowledge goes, for the first time offers a theoretical justification for the use of the empirical values normally found in the literature and provides an explanation, mainly driven by the tyre rolling, for the wide range of them. It is important to remember, even in this section, that one of most adopted values for \(Hc\) is 1.2 \(\times\) 10\({}^{4}\) 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. Analysing the graph above, it is clear that the most used value of Hc could be employed for a speed range between 100 km/h and 200 km/h, if a total error of 40% (\(\pm\)20%) can be considered acceptable. Such a range is certainly suitable for racing cars and, as a matter of fact, it must be recognised that many tyre models are focused on such a target. Nevertheless, if passenger cars and city speeds are considered, it seems more appropriate to halve the Hc values employed. It is interesting to analyse, as well, what happens if the tyre load changes, influencing the footprint length. Regarding the \(FL\) evaluation, [PERSON] et al. [37] proposed an interesting work. The paper starts from the Hedekel formula reported below, which is derived from geometric considerations: \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Speed** **[m/s]** & **[km/h]** & **\(Hc\) Literature Values** **[W/(m\({}^{2}\)K)]** & **References** **\(Hc\) Equation Results** **[W/(m\({}^{2}\)K)]** \\ \hline
2.7 & 10 & \(2\times 10^{3}\) & [29] & \(3.1\times 10^{3}\) \\ \hline
41.6 & 150 & \(1.2\times 10^{4}\) & [25] & \(1.22\times 10^{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison between the \(Hc\) literature values and \(Hc\) equation results. Figure 4: \(Hc\) behaviour as a function of the rolling speed (footprint length of 0.14 m). Red dots represent the literature empirical values. \[FL=2\sqrt{\Delta z(2R-\Delta z)} \tag{13}\]  
In the equation above, \(\Delta z\) represents the radial tyre deformation [mm], while \(R\) is the free wheel radius [mm]. The gradient of \(z\) is given by the ratio between the normal wheel load \(P\) [N] and the radial rigidity of the tyre, R, [N/mm], as written below:  
\[\Delta z=\frac{P}{R} \tag{14}\]  
In said paper, [PERSON] et al., proposed a correction coefficient for Equation (13) in order to obtain the best fit between the empirical findings and the calculated ones. One of the most interesting parts of the work is that, in order to evaluate such a coefficient, several experiments were carried out, concerning both various types of passenger cars as well as trucks. For the current work, the very experimental results reported in Table 2 of ref. [37] were taken into account. Only passenger vehicles, characterised by a normal wheel load that goes from 1863 N to 3923 N, were considered. Here, 1863 N is about 750 kg, a typical mass for a small car, while 3923 N is about 1568 kg, a mass ascribable to a station wagon or small sports utility vehicle (SUV) [38]. The average free wheel radius was 0.3 m.  
Table 2 of ref [37] permits inferring an \(FL\) range that goes from 0.1 to 0.18 for the said car range.