Reduced mixing time in stirred vessels by means of irregular impellers

Previous research has shown that using fractal-like blades instead of regular ones can result in a significant decrease in the power consumption of an unbaffled stirred vessel with a four-bladed radial impeller. In order to fully assess the mixing efficiency of a fractal-like or just irregular impeller with respect to a regular impeller, the mixing time required to homogenize an injected passive scalar was evaluated for both impeller types at $\text{Re}=320$ and 1600 using direct numerical simulations. It was observed that the irregular impeller can lead to a considerably shorter mixing time. This result was explained by the differences in characteristics of flow and scalar fields generated by the two impellers. We also assess the effect of Re in the transitional regime. Moreover, a simple mathematical model is proposed which can approximate the decay rate of the passive scalar fluctuations integrated over the tank volume.

Figure 1

Tank geometry, dimensions, and blade shapes: (a) vertical view along the axis, (b) horizontal view at the midheight, and (c) irregular (one-iteration fractal-like) (top) and regular (bottom) blades.

Figure 2

Left-hand side (LHS) and right-hand side (RHS) of Eq. (6) for the irregular impeller at $\text{Re}=1600$.

Figure 3

Variation of COV for five scalars shown for regular and irregular (fractal-like) impellers at both Re. The dashed horizontal line indicates $\text{COV}=5\%$. (a) Regular impeller, $\text{Re}=320$. (b) Fractal-like impeller, $\text{Re}=320$. (c) Regular impeller, $\text{Re}=1600$. (d) Fractal-like impeller, $\text{Re}=1600$.

Figure 4

Average COV curves for the two Re examined: (a) $\text{Re}=320$ and (b) $\text{Re}=1600$. The black dashed line indicates $\text{COV}=5\%$. Reg. stands for the regular and Fra. for the fractal-like (irregular) impeller. Vertical lines are error bars (plus or minus one standard deviation from the mean).

Figure 5

(a) Contour plot of $c/\overline{c}$ at the midheight plane. (b) Isoscalar surface $c=\overline{c}$. The regular impeller is at $\text{Re}=1600$, half a revolution after the start of injection.

Figure 6

(a) Contour plot of $c/\overline{c}$ on the vertical plane ${30}^{\circ }$ behind a blade (the cross section of the shaft is left vertically blank at the center of the plot). (b) Isoscalar surface $c=\overline{c}$. The regular impeller is at $\text{Re}=1600$, two revolutions after the start of injection.

Figure 7

Normalized area of isoscalar surfaces for five realizations in all cases. (a) Regular impeller, $\text{Re}=320$. (b) Fractal-like impeller, $\text{Re}=320$. (c) Regular impeller, $\text{Re}=1600$. (d) Fractal-like impeller, $\text{Re}=1600$.

Figure 8

Area of isosurfaces averaged for the five injections in all cases.

Figure 9

Running average of the normalized area of isosurfaces for the regular impeller at $\text{Re}=1600$.

Figure 10

(a) Rightmost term in Eq. (7) computed using four different values of $\mathrm{\Delta }c$ and (b) trend of ${\eta }^{*}$ against $t^{*}$, for scalar-1 of the regular impeller at $\text{Re}=320$.

Figure 11

Exact calculation of the decay rate (exact) and the model based on $\mathrm{\Delta }c/\overline{c}=5\%$ (model) as expressed in Eq. (8). The curves obtained with five realizations are averaged in all cases. The vertical dashed lines indicate the time when the injection stopped. (a) Regular impeller, $\text{Re}=320$. (b) Fractal-like impeller, $\text{Re}=320$. (c) Regular impeller, $\text{Re}=1600$. (d) Fractal-like impeller, $\text{Re}=1600$.

Figure 12

Ratio $K$ of the exact decay rate to the model value (9), for both impellers and Re.

Figure 13

Contours of $\langle g\rangle$ on a cut plane ${15}^{\circ }$ behind the blades, at $\text{Re}=320$ (the cross section of the shaft is left blank). (a) Regular impeller. (b) Fractal-like impeller.

Figure 14

Arrows show the direction of the mean velocity projected on the cut plane (${15}^{\circ }$ behind the blades) and the contours display its magnitude normalized with the blade tip speed, at $\text{Re}=320$ for both impeller types (the cross section of the shaft is left blank). (a) Regular impeller. (b) Fractal-like impeller.