*5.3. Heat Dissipation Capability*

The heat dissipation capability was investigated using the method presented in reference [45]. The Nusselt number is an important non-dimensional quantity used in heat transfer, which relates the convective heat transfer to the conductive heat transfer by a fluid across a surface. In this study, local and average Nusselt numbers were utilized to describe the heat dissipation capability under different conditions. The local and the average Nusselt numbers on the surface of the stationary flat disk are respectively defined as:

$$Nu\_l = \frac{qr}{(T\_W - T)k} \tag{15}$$

$$Nu\_{av} = \frac{qr\_2}{(T\_{Wav} - T)k} \tag{16}$$

where *q* denotes the heat flux, *TW* denotes the local temperature of the stationary disk surface, *T* is the temperature of the two-phase flow, *k* is the effective thermal conductivity of the fluid, and *TWav* indicates the average temperature of the stationary disk surface.

Figure 13 shows the radial distribution of the local Nusselt number along the center line of the stationary disk. At a certain angular velocity, it is noted from the figure that the local Nusselt number was high close to the inner radius position of the disk. It dropped rapidly and fluctuated remarkably outward along the radial direction. Near the outer diameter position of the disk, the Nusselt number increased rapidly to a great value. The reasons for the variation of the Nusselt number along the radial direction were as follows. The inner radius region was filled with the continuous oil phase. The oil temperature was low, and the heat convection of the oil was much stronger than the air, leading to a high local Nusselt number near the inlet. In the middle radius region, the oil volume fraction decreased quickly around the interface of oil film and two-phase flow. The heat convection of the two-phase flow was much weaker than the pure oil phase. The heat dissipation was weakened, and the fluid temperature rose. Furthermore, compared with the flow within the full oil phase zone, the flow of the air–oil two-phase flow was much more unstable around the interface of the oil film and the two-phase flow, which led to drastic fluctuations of the Nusselt number. Near the outer diameter position of the disk, though the oil volume fraction was very small, the velocity of the air flow was relatively much higher than the air–oil flow. Besides, the temperature of the air phase was lower, thus the cooling intensity of the disk surface was higher. This made the local Nusselt number increase near the outlet boundary.

With the angular speed of the disk increasing, it was concluded from Figure 13 that the fluctuation of the local Nusselt number moved towards the inner diameter, and its range notably increased. This was because the oil volume fraction decreased at higher speeds, and the area of the air phase expanded towards the inner diameter, which resulted in an increase in the mixing area of the two-phase flow. The higher the speed of the disk was, the more unstable the two-phase flow became, and the higher the flow velocity was. Thus, the fluctuation was aggravated. In the regions near the inner and the outer diameter, a higher local Nusselt number was attributable to a greater shear rate of the oil phase and the air phase, respectively, which led to stronger heat dissipation.

**Figure 13.** Results of the local Nusselt number on the stationary disk surface along the radial direction.

Lastly, the variation of the average Nusselt number on the stationary disk surface with increasing angular velocity is shown in Figure 14. The change of the average Nusselt number with angular velocity could be divided into three stages. For stage I, the angular velocity of the rotating disk was small, and the gap was filled with the oil phase. As the disk speed increased, the shear rate of the oil phase in the gap increased, resulting in stronger heat dissipation. Therefore, the average Nusselt number rose remarkably in stage II. In stage II, the oil volume fraction decreased rapidly due to the aeration effect. The heat dissipation capability of the mixture was weaker than that of the full oil phase, which led to a rapid decrease of the average Nusselt number. In stage III, the oil volume fraction in the flow field decreased much more slowly. However, the speed of the air–oil two-phase flow and the pure air phase inside the flow field increased with a higher disk speed. The mixture flow around the inner diameter region and the pure air phase flow around the outer diameter region played major roles in the heat convection. Thus, the average Nusselt number increased gradually with the angular velocity of the disk.

**Figure 14.** Curve of the average Nusselt number on the stationary disk surface at different angular velocities.
