# Numerical Simulation of Polymer Solutions in a Single-Screw Extruder
## 3 Results and Discussion
It can be seen that the greatest deformation is observed for 0.5% PAA since its apparent viscosity is less than that of 1.5% CMC. It should be noted that for 0.5% PAA the axial velocity profile has the inflection point near the left screw flight caused by the rotation velocity of the outer pipe. Thus, the obtained results indicate that the outer pipe rotation makes the velocity profiles more flatted. Figure 11 shows the axial velocity profiles for fully developed flow in the channels with the different pitch of twist of the screw (\(L/D_{2}\)). Results are presented for equal average velocity (Figure 11a) and equal Reynolds number (Figure 11b). The corresponding profiles were plotted along a circle with a radius \(r=r_{3}\). It can be seen (Figure 11a) that with an increase of the twist intensity (decrease of the \(L/D_{2}\) parameter), the velocity profile for 1.5% CMC becomes more elongated. Another behavior is observed for the flow of 0.5% PAA. With a decrease of the \(L/D_{2}\) parameter, the velocity profile takes on a more flatted form. So in the near-wall region the velocity of 0.5% PAA increases compared with 1.5% CMC. The observed increase of velocity in the near-wall region is explained by the presence of higher velocity gradients, which contribute to a decrease in viscosity. A similar result was obtained for the same Reynolds numbers (Figure 11b), in this case, the displacement of the axial velocity profile to the left wall is pronounced for 0.5% PAA, where the direction of displacement coincides with the direction of rotation of the outer pipe. Figure 11: Profiles of the dimensionless axial velocity in the cross-section \(A_{360}\) for 0.5% polyacrylamide (PAA) (red) and 1.5% carboxymethyl cellulose (CMC) (blue): 1 and 3—\(L/D_{2}\) = 1 (dashed line); 2 and 4—\(L/D_{2}\) = 3 (solid line). Figure 10: Profiles of the dimensionless axial velocity for different the rotational speed \(N^{*}\), (\(L/D_{2}\) = 2, \(v_{a}\) = 0.05 m/s). Figure 12 shows the dependence of the pressure drop on the number of revolutions of the outer pipe and average velocity at a fully developed flow of 0.5% PAA and 1.5% CMC in the metering zone of a single-screw extruder (distance between \(A_{360}\) and \(A_{450}\) cross-sections). It was found that with an increase in the \(L/D_{2}\) parameter (with an increase of the twist intensity), a decrease in the pressure drop due to an increase in the number of revolutions is observed for both fluids (Figure 12a). An increase in the average velocity leads to an increase in the pressure drop (Figure 12b). Processing the numerical results shown in Figure 12 revealed that the dependences of the pressure drop on the number of revolutions and average velocity could be approximated by a second-order polynomial with accuracy of approximation \(R^{2}\) = 0.999. For example, for 0.5% PAA, the following relationships were obtained:  
\[L/D_{2} = 1:\ dP/dz = -5787.6\cdot\left(N\star\right)^{2}+19708\cdot N\star+436.06 \left(R^{2}\ =\ 0.999\right)\] \[L/D_{2} = 2:\ dP/dz = -75389\cdot\left(N\star\right)^{2}+15599\cdot N\star-74.412\left( R^{2}\ =\ 0.999\right)\] \[L/D_{2} = 3:\ dP/dz = -54978\cdot\left(N\star\right)^{2}+11439\cdot N\star-88.396\left( R^{2}\ =\ 0.999\right)\]  
It can be seen from the figure (Figure 12a,b) that in both cases the highest values of the pressure drop correspond to 1.5% CMC (blue lines), which is due to its higher viscosity compared to that of 0.5% PAA (see Figure 3).