# Numerical Simulation of Polymer Solutions in a Single-Screw Extruder
## 3 Results and Discussion  
This section presents the numerical studies results of the 0.5% PAA and 1.5% CMC flows in a channel with a screw. The geometric characteristics of the channel (Figure 1) are as follows: the outer diameter of the pipe \(D_{2}\) = 0.039 m, the inner diameter \(D_{1}\) = 0.026 m, the length of the straight section \(L_{0}=1.5\cdot D_{2}\). The following values of the pitch of twist of the screw fins \(L/D_{2}\) = 1, 2, and 3 were considered. The screw is twisted counterclockwise, and the length of the channel with the screw depends on the pitch of twist and varies from \(3D_{2}\) to \(7.5D_{2}\) depending on the pitch of twist (\(L/D_{2}\)). The main calculations were performed for an average velocity of the fluid of 0.05 m/s. Figure 8 shows the lines of equal values of the dimensionless axial (Figure 8a) and radial (Figure 8b) velocity, as well as viscosity (Figure 8c) for 0.5% PAA flow in the channel with a screw. Designations \(A_{45}\), \(A_{90}\), \(A_{360}\) are cross-sections of the channel according to Figure 4. Figure 8c shows the dimensionless viscosity, which is the ratio of the actual viscosity to the average viscosity \(\eta_{inlet}\) calculated in the inlet cross-section of the channel. For 0.5% PAA \(\eta_{inlet}\) = 0.0509 Pa\(\cdot\)s. As can be seen from the figure, when polymer solution flows into the curved section, provided by the presence of a screw, the velocity and viscosity profiles are rearranged. Areas with the maximum axial velocity appear at the inlet to the screw, then as the fluid flows along the channel, the velocity profile becomes more flatted. At the average velocity of 0.05 m/s, it was revealed that a steady-state flow regime is achieved in the section up to cross-section \(A_{360}\). Besides, the steady-state flow regime of 0.5% PAA is characterized by the presence of a radial velocity (up to 11%) comparable to the average velocity. This indicates the presence of radial fluid flows. At a steady-state flow, the flatted velocity profile is accompanied by the presence of an area with very high viscosity (Figure 8c) (a double exceedance of viscosity with respect to the average viscosity at the channel inlet). Figure 7: Profile of the dimensionless axial velocity in the coaxial channel. Profiles of the dimensionless axial velocity in various cross-sections of the channel (from \(A_{0}\) to \(A_{450}\)) along a circle with a radius \(r=r_{3}\) (see Figure 4) were plotted to determine the length of the initial hydrodynamic section in the channel with a screw (Figure 9). Results are presented for the case when outer pipe (barrel) is rotated with speed \(N^{*}=0\), 30, 60 rpm. It was found that for 1.5% CMC, the hydrodynamic entrance length is bounded by the cross-section \(A_{180}\), while for 0.5% PAA, it is bounded by the cross-section \(A_{360}\). This is explained by the fact that at the same average velocity of two fluid, the corresponding Reynolds numbers differ by two orders of magnitude, namely Re (0.5% PAA) = 10.64, Re (1.5% CMC) = 0.87. It should be noted that the Reynolds number was calculated using the average viscosity (\(\eta_{\mathrm{inlet}}\)) in the inlet section of the channel and the hydraulic diameter \(D_{h}=4S/p\), where \(S\) is the cross-sectional area of the channel, \(p\) is the wetted perimeter. Figure 8: Lines of equal values of dimensionless axial (**a**) and radial (**b**) velocity, (**c**) viscosity (\(L/D_{2}=2\), \(N^{*}=0\)). Figure 9: Profiles of the dimensionless axial velocity in various cross-sections of the channel (\(V_{\mathrm{a}}=0.05\) m/s): 1—\(A_{0}\), 2—\(A_{45}\), 3—\(A_{90}\), 4—\(A_{135}\), 5—\(A_{180}\), 6—\(A_{360}\), and 7—\(A_{450}\) are dashed line in black. Figure 10 shows the axial velocity profiles for the steady-state flow of the studied fluids for various rotation velocities of the outer pipe (\(r=r_{3}\), see Figure 4). For both fluids, an increase in the outer tube rotation velocity leads to a slight deformation of the velocity profile, which is consistent with the results of the work [16].