Heat Conduction in a Collision of Elastic Spheres

In a collision between two spheres of different temperatures, heat conduction occurs at the interface. The contact area is usually negligibly small compared to the cross-sectional area of the spheres. Since the duration of the impact is also very short, the temperature change of the colliding particles is confined to a small region around the contact area. Therefore, the heat conduction between the two particles can be treated as that between two semiinfinite media. It is also assumed that there is no thermal resistance between the contact surfaces. Hence, the temperature and heat flux distributions are continuous across the contact area. The surfaces outside the contact area are assumed to be flat and insulated. For general information on collision mechanisms of solids, readers may refer to Chapter 2. 

For simplicity, it is assumed that the impact is a Hertzian collision. Thus, no kinetic energy loss occurs during the impact. The problem of conductive heat transfer due to the elastic collision of solid spheres was defined and solved by Sun and Chen (1988). In this problem, considering the heat conduction through the contact surface as shown in Fig. 4.1, the change of the contact area or radius of the circular area of contact with respect to time is given by Eq. (2.139) or by Fig. 2.16. In cylindrical coordinates, the heat conduction between the colliding solids can be written by 

When the contact time is very short, the heat penetration by conduction from one medium to the other would not be very deep from the contact surface. Therefore, one-dimensional 

for a small Fourier number, the one-dimensional heat transfer yields the following equation 

Since the heat flux across the interface is continuous, we have 

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