Optimization of the vane geometry

Abstract

The use of nonstandard geometries like the vane is essential to measure the rheological characteristics of complex fluids such as non-Newtonian fluids or particle dispersions. For this geometry which is of Couette type, there is no analytical simple model defining the relation between the shear stress and the torque or relating the angular velocity to the shear rate. This study consists on calibrating a nonstandard vane geometry using a finite volume method with the Ansys Fluent software. The influence of geometrical parameters and rheological characteristics of the complex fluids are considered. First, the Newtonian fluid flow in a rotative vane geometry was simulated and a parametric model is derived therefrom. The results show an excellent agreement between the calculated torque and the measured one. They provide the possibility to define equivalent dimensions by reference to a standard geometry with concentric cylinders where the relationships between shear stress (resp. shear rate) and the torque (resp. the angular rotation) are classical. Non-Newtonian fluid flows obeying a power law rheology with different indices were then simulated. The results of these numerical simulations are in very good agreement with the preceding Newtonian-based model in some ranges of indices. The absolute difference still under 5 % provided the index is below 0.45. Finally, this study provides a calibration protocol in order to use nonstandard vane geometries with various heights, gaps, and distance to the cup bottom for measuring the rheology of complex fluids like shear thinning fluids and concentrated suspensions.

Appendices

Appendix A: Data results

Table 8 Total torque $\overline \Gamma$ data for different gaps $\overline e$ and different immersions $\overline z$ with distance to bottom $\overline d =0.5$

Table 9 Total torque $\overline \Gamma$ data for different gaps $\overline e$ and distances to the bottom cup $\overline d$ with immersion $\bar z=3$

Table 10 [$\overline H_{eqv}-\overline H$] data for different gaps $\overline e$ and distances to the bottom cup $\overline d$. H is the geometric height of the vane

Table 11 The equivalent radius depends only on the gap $\overline e$ (see Eq. (11))

Appendix B: Linear fit of the experimental torque results

The relative errors on R _1eqv and H _eqv are then calculated from the relative error on the torque measurement.

Fig. 15

Dimensionless torque ($\overline \Gamma$) versus $\overline z$ with distance to bottom $\bar d=0.5$ and for a gap $\bar e=0.1$

Table 12 Linear interpolations of the experimental values of the torque for various vane geometries where e is the gap and d the distance to bottom cup

Appendix C: Variation of F _γ /F _γN versus R ₂/R ₁

Fig. 16

Variation of F _γ /F _γN with R ₂/R ₁ for various values of the fluid index n. The two dashed lines specify the range F _γ /F _γN = 1 with a deviation of ±5 %. Figure 16b is a enlargement of 16a