## 4 Numerical Simulation Model
### Grid and Calculation  
The provided Figure 5 illustrates the grid calculation cloud diagram of the cross-sectional configuration within the internal meshing screw mixing section. This depiction encompasses the cross-sectional profiles of both the rotor and stator, delineating the fluid domain that lies between these two components.  
Employing Ansysy 2021 R1 software, heat transfer in the internal meshing screw mixing section was simulated. The geometric model employed in the simulation precisely matched the structural parameters of the mixer utilized in the experimental setup. The computational domain was saturated with fluid, mirroring the properties observed in the experiment. Throughout the investigation, the rotor speed remained below 90 rpm, and the Reynolds number did not surpass 109. It was assumed that the mixing process occurred within a laminar flow state. The modeling of the fluid domain was performed with Space Claim 2021 R1 software. The process involved importing mesh settings, designating the rotor section wall as the interior surface, labeling the stator's outer wall surface, and identifying the fluid domain in between as the liquid phase. The mesh of the fluid domain was subdivided into a structured triangular mesh, with a uniform unit size set at 1 mm for the overall mesh. Further refinement was applied to the rotor wall's grid, adjusting the unit size to 0.1 mm. Specify the mixing wall surfaces of the rotor and stator as no-slip boundary conditions, with the rotor wall surface configured as an adiabatic boundary. The simulation analysis  
Figure 5: Grid operation cloud diagram of the cross-section of the internal meshing screw mixing section.  
employed the material parameters described previously, considering the selected physical properties as representative of Newtonian fluids.  
This article examines the influence of two distinct modes on the heat transfer efficiency of the internal meshing screw mixer system. Mode 1 involves establishing a constant wall temperature of 50 \({}^{\circ}\)C while maintaining an internal temperature of 0 \({}^{\circ}\)C. Conversely, Mode 2 maintains an internal temperature of 50 \({}^{\circ}\)C while ensuring the wall is designed to achieve zero heat flux. This distinction between Mode 1 and Mode 2 is commonly referred to as 'external heat' and 'internal heat' configurations.  
The rotor's motion was enabled by implementing a user-defined function (UDF), which was subsequently compiled into Fluent 2021 R1 software. The moving mesh technique, accompanied by smoothing and mesh reconstruction methods, was employed. Given the eccentricity parameter, \(e=3\) mm, a moving mesh interior region was established. The selection of 'rigid body' as the type was made, along with the incorporation of the compiled UDF. The center of gravity position was defined at \(Y=3\) mm.  
The numerical solution technique employed in this study is based on the SIMPLE algorithm, a pressure-velocity coupling approach. The momentum equation and pressure discretization are both resolved using the second-order upwind method. The sub-relaxation factor is initially set to 0.5, a value selected to expedite the computational speed. Convergence thresholds for the residual values are specified at 10\({}^{-9}\) for each component, and a maximum of 30 iterations are allowed in a single time step. The computational process concludes upon achieving convergence.