# Type-Road Heat Transfer Coefficient Equation Proposal
## 3 The Tyre-Road Heat Transfer Coefficient \(Hc\) Equation Proposal  
As stated above, the \(Hc\) parameter should have a dependence upon the tyre rolling speed and the footprint length. In the current section, a formulation that takes into account such variables is proposed, starting from a simplification of the temperature gradient behaviour. The transient temperature gradient in the first infinitesimal type tread layer shows a shape like the one in Figure 2[31], which can be approximated by a simplified polygonal curve, as depicted in Figure 3.  
In the above Figure 3, \(p(t)\) [m] is defined as the \"propagation depth\" of the temperature gradient: \(p(t)\) is the \"knee\" of the real curve gradient, and it represents the depth over which the temperature is still \"undisturbed\" and can be considered the one of the surface layer. Such a parameter \(p(t)\) is a function of time, equal to 0 at time \(t_{0}=0\), that corresponds to the instant at which the surface \(\Delta S\) touches the ground, and then increasing monotonically as long as \(\Delta S\) 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. The heat stored inside the rubber is proportional to the reached temperature and the heat capacity. When the temperature decreases, it means that part of the stored heat is transmitted outside the tyre. 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. Taking into account the representation of the thermal gradient behaviour shown in Figure 3, it can be inferred that the amount of heat exchanged with the ground at time \(t\) [s], \(Q(t)\) [J], is proportional to the area of the triangle depicted in red in said figure. Thus:  
\[Q(t)\propto\frac{T2-T1}{2}\cdot p(t) \tag{2}\]  
Introducing the heat capacity per unit of volume, \(C\) [kJ/(m\({}^{3}\)K)], considered as a constant as in ref. [33], the energy lost by the rubber of the tyre surface \(\Delta S\) at time \(t\) and transferred to the ground can thus be written as:  
\[Q(t)=\Delta S\cdot C\cdot\frac{T2-T1}{2}\cdot p(t) \tag{3}\]  
Given the heat flux definition:  
\[\varphi=\frac{dQ}{dt} \tag{4}\]  
Equation (3) can be derived as follows:  
\[\varphi=\Delta S\cdot C\cdot\frac{T2-T1}{2}\cdot\frac{dp}{dt} \tag{5}\]  
The heat flux going from the tyre to the road, however, must equal the heat flux inside the tyre rubber expressed by the Fourier thermal conduction law, considering the temperature gradient, the depth \(p\) and the heat conductivity coefficient, \(\lambda\) [W/(mK)] (assuming, as said above, that the asphalt conductivity is one order of magnitude higher than the rubber one):  
\[\varphi=\lambda\cdot\Delta S\cdot\frac{T2-T1}{p} \tag{6}\]  
From the comparison of Equations (5) and (6) comes:  
\[\Delta S\cdot C\cdot\frac{T2-T1}{2}\cdot\frac{dp}{dt}=\lambda\cdot\Delta S \cdot\frac{T2-T1}{p}\Rightarrow\frac{C}{2}\cdot\frac{dp}{dt}=\frac{\lambda} {p}\Rightarrow\frac{C}{2}p\cdot dp=\lambda\cdot dt \tag{7}\]  
Integrating the above differential equation with the boundary conditions \(p(t)=0\) when \(t=0\), it is possible to obtain, for \(p(t)\), the following expression:  
\[p(t)=\sqrt{\frac{4\cdot\lambda\cdot t}{C}} \tag{8}\]  
Thanks to the above formulation, by comparing Equations (8) and (3), it is possible to express the amount of heat exchanged by the tyre surface \(\Delta S\) and the ground as a function of the contact time \(t\):  
\[Q(t)=\Delta S\cdot(T2-T1)\cdot\sqrt{\lambda\cdot C\cdot t} \tag{9}\]  
The above expression implies that \(Q(t_{0})=0\) because before any contact between the ground and tyre, no flux exchange with the ground happens.