## 5 Heat Transfer Analysis of Internal Meshing Screw Mixer
### Effect of Process Parameters on Heat Transfer  
Investigate the influence of varying rotational speeds (\(\omega\) = 30, 60, and 90 rpm) on the heat transfer efficiency within the internal meshing mixing device. Figure 10a,b, presented below, provides an overview of the impact of rotational speed on \(T_{m}\), \(MI_{T}\), and \(CI\) in both operational modes. In the context of Figure 11a, corresponding to Mode 1, it is evident that the \(T_{m}\) of \(\omega\) = 30 and 60 rpm consistently surpasses that of \(\omega\) = 90 rpm throughout a complete cycle. Notably, within the time interval \(t\) = 0-1 s, the \(T_{m}\) curves of \(\omega\) = 30 rpm and \(\omega\) = 60 rpm exhibit alternating leadership. This alternation is rooted in the fact that 1 s represents one full cycle of rotor rotation, where a cycle involves movement from the upper apex to the lower apex and back to the starting position. It is imperative to highlight that, prior to 0.4 s, the \(T_{m}\) of \(\omega\) = 30 rpm exceeds that of \(\omega\) = 60 rpm. This phenomenon can be attributed to the heat transfer mechanism in Mode 1, which entails heat transfer from the outer wall of the stator to the interior. Consequently, when the rotor initially descends from the upper vertex, the top region of the stator's heat transfer wall is the closest point with the largest heat transfer area. Therefore, at lower rotational speeds, more heat is absorbed from the outer wall. As evidenced in Figure 11, the trend becomes apparent; at lower rotational speeds, the duration of the increasing \(T_{m}\) period is prolonged, thereby causing the alternating patterns between \(\omega\) = 60 rpm and \(\omega\) = 30 rpm. Due to the variance in the time required for a full cycle of rotation at different speeds, the moment corresponding to the maximum \(T_{m}\) value within one cycle also varies with speed. In Figure 10a, the \(MI_{T}\) curve exhibits a characteristic pattern characterized by an initial decrease, followed by the establishment of dynamic stability, which resembles the shape of a \(\sin\) x function. It is discernible that the \(MI_{T}\) variations at \(\omega\) = 90 rpm appear relatively smooth, whereas those at \(\omega\) = 60 rpm and 30 rpm are marked by a more convoluted profile. This phenomenon underscores that higher rotational speeds contribute to a more uniform temperature distribution within the system. Notably, with reference to the \(T_{m}\) diagram, \(\omega\) = 90 rpm is associated with the lowest \(T_{m}\) value, resulting in minimal temperature fluctuations, enhanced temperature uniformity, and elevated temperature uniformity values. Figure 9: Heat transfer index curves under two modes: (**a**) average temperature and temperature uniformity curves in two modes; and (**b**) heat transfer comprehensive index curves under two modes. Analyzing Pattern 1 in Figure 10b reveals that, under the influence of three different rotational speeds, there is an overall upward trend in the \(CI\) curve. The \(CI\) curve of \(\omega\) = 90 rpm follows a similar trajectory to that of \(CI\), exhibiting a uniform periodic increase. Notably, the change observed at \(\omega\) = 60 rpm is particularly significant, with the peak node within each cycle surpassing the value of \(\omega\) = 90 rpm, indicating that, at certain time points, its \(CI\) is higher than that of \(\omega\) = 90 rpm. Figure 10b also highlights that the \(CI\) curve adheres to a specific pattern. This pattern is characterized by different time requirements for a full cycle of rotation at varying speeds. For instance, \(\omega\) = 90 rpm experiences a complete cycle in approximately 0.5 s, while \(\omega\) = 60 rpm completes a cycle in 1 s, and \(\omega\) = 30 rpm takes 2 s for one cycle. Consequently, by evaluating the \(CI\) value of the peak node within each cycle, it becomes evident that, overall, increasing the rotational speed leads to higher \(CI\) values. Notably, \(\omega\) = 60 rpm stands out as having the highest heat transfer efficiency within the system.