Experimental Study on Chaotic Mixing Created by a New Type of Mixer with Rotational Blades

Abstract

The aim of this paper is an experimental investigation of laminar mixing in a new type of chaotic mixer, which has been proposed by Hwu (2008), by means of material line deformation. The mixer is a circular cavity with two rotational blades which move along a semicircular path and drive the fluid motion. The flow visualization is carried out by marking of the free surface of the flow with a tracer in working fluid. In the present study the effects of length and rotational speed of blades on mixing efficiency are evaluated by measuring of the area covered by the tracer. As a result, it is demonstrated that the goodness of mixing increases as rotational speed of blades increases. Also, it is detected that the mixing efficiency strongly depends on the lengths of rotating blades.

1. Introduction

Rapid mixing is an important topic in almost all types of chemical process operation. The turbulent mixing is commonly utilized in low viscous fluids such as water, air, and light oils. However, numerous fluids employed in chemical processes have high viscosity and the use of turbulent flow into mixing procedures is not feasible for these fluids due to very energy consumption. On the other hand, the blades with high rotational speed can result in damaging the structure of high-viscous fluids. Chaotic advection is a new method which has been developed as a remedy for this problem. Aref [1, 2] reported that even laminar flow can cause an efficient mixing while the lagrangian particle path is chaotic.

Chien et al. [3] experimentally investigated the laminar mixing in several kinds of two-dimensional cavity flow. Their apparatus could be adjusted to create Reynolds number in the range of 0.01–100 and different types of flow field using various boundary conditions. They reported that the mixing efficiency depends on velocity of walls and there is an optimum value for it.

A lot of researches [4–6] investigated numerically and experimentally the laminar mixing in flow filled between two eccentric cylinders which were driven periodically. The results showed that mixing efficiency improves as cylinders are periodically rotated with large period.

In another configuration, the flow can be stirred in a rectangular cavity with a circular cylinder placed in its center. Galaktionov et al. [7] investigated numerically and experimentally such problem and reported that the presence of a rotational cylinder can be a simple technique to remove the dead zone.

Takahashi and Motoda [8] proposed a new spatial chaotic mixer by inserting the objects into a vessel agitated by an impeller. The experimental results demonstrated that mixing time decreases remarkably as larger objects are used. They compared their spatial mixing method with the temporal time-depended one and reported that spatial mixing method has a benefit at low Reynolds number and reduces the dead region where the fluid velocity is less.

Zhang and Chen [9] discussed a new approach to the design and implementation of a liquid mixer which can work under different impeller/tank velocity control schemes. They used Chua's circuit as a chaotic signal source for generating chaotic perturbations. The experimental results showed that mixing time can be reduced dramatically with the use of chaotic perturbation. Hwu [10] suggested a new conceptual apparatus that can be utilized to mix high-viscous fluids at low velocities. It comprised a circular cavity equipped with two blades arranged along perimeter. The numerical studies showed that the mixing efficiency depends on length and rotational speed of blades.

The objective of this work is to present a new apparatus, based on the idea suggested by Hwu [10], which permits one to create a chaotic advection in a circular cavity. The experimental investigation of mixing is carried out using several tests and the effects of blade length and rotational speed on mixing efficiency are reported by means of line deformation method.

2. Experiment

The liquid mixer is a circular container with two blades which are rotated along a semicircular path. The experimental apparatus is designed to accomplish two chief purposes. The first is to be able to move two blades at various speeds in the semicircular paths. The second is to be able to photograph the free surface of flow by means of a camera. The main set-up used in this work is shown in Figure 1. The upper surface of fluid is open to the atmosphere. The vessel diameter (D) and height (H) are 200 mm and 160 mm, respectively. The Radius of blades ( $R_b=D_b/2$ ) is equal to 85 mm for all cases and thus the distance between blades and vessel wall is 15 mm. Furthermore, the distance between blades and the bottom of vessel is set 5 mm. The Schematic diagram of experimental apparatus and sketch of top view of circular tank are illustrated in Figures 2 and 3, respectively. The experiments are carried out by the blades with various lengths ( $L=(\pi /6)R_b,(\pi /3)R_b$ and $(\pi /2)R_b$ ) and two identical step motors are utilized in order to rotate the blades. The required signals to drive motors are supplied using two similar drivers. The signals’ input to drivers is generated using a microcontroller ATMega32 which is programmed by a code written in Bascom. The computer code is modified with respect to rotational speed and length of blades in each experiment. The working fluid used is glycerin which fills the entire tank and the maximum speed of blades is set 30 rpm. In order to evaluate mixing pattern, deformation of material line, which is made up of a polymeric dye, is photographed through the top with a Fuji camera F770EXR every 10 seconds. The value of diffusion coefficient of dye is quite small (about 10⁻⁸ cm²/s) [3]. So the effect of diffusion on spreading of the dye in glycerin is approximately negligible. Another important point that we seek in this study is the ability to create two-dimensional flow. To achieve this purpose the material line of tracer is injected approximately 2–5 mm below the upper surface, as suggested by Chien et al. [3]. Each photo is analyzed using combination of a computer and processor code.

Figure 1:

Experimental set-up.

Figure 2:

Schematic diagram of experimental apparatus.

Figure 3:

Sketch of circular container with two rotational blades.

3. Results

Possible parameters that can affect the mixing efficiency include (1) the rotational speed of the blades, (2) the length of blades, (3) initial position of the blades, (4) the range of which the blades are moved, (5) protocol of the rotational speed of the blades, and (6) the distance between the blades and tank wall [10]. Taking into account the time-consuming and expensive nature of studying all the above-mentioned factors, the current research has only focused on the effects of the length and rotational speed of the blades. According to each rotational speed ω, two revere times corresponded to two blades are calculated as follows:

$\begin{array}{c}{T}_{\mathrm{1}}=\frac{\pi -L_{\mathrm{1}}}{\omega },T_{\mathrm{2}}=\frac{\pi -L_{\mathrm{2}}}{\omega }.\end{array}$

(1)

With respect to the above equations, a separate revere time for each blade with specific length is applied. This can generate chaotic advection in mixer under study [10].

3.1. Qualitative Analysis

In order to qualitatively evaluate the mixing, the free surface of working fluid was photographed at various times (t = 0, 10, 30, 60, and 100 s). The produced images can show the structure of dye during the mixing process. The experimental images were changed into black-white processed those by means of Matlab to have a more convenient comparison. This was carried out by assigning black and white colors to surface covered by dye and surface void of dye, respectively. A typical photo of the actual image and its corresponding processed image are shown in Figures 4(a) and 4(b), respectively.

Figure 4:

Typical photo of (a) The actual one and (b) The processed one ( $L_{\mathrm{1}}=(\pi /\mathrm{6})R_b$ , $L_{\mathrm{1}}=(\pi /\mathrm{3})R_b$ , ω = 30, t = 30 s).

For the purposes of discussion in this section, as suggested by Swanson and Ottino [4], the regular regions are the empty spaces of dye and chaotic region is the region over where dye spreads. As the blades’ rotational speeds have an important effect on flow and thus on the mixing in the tank, preliminary experiments were carried out to observe the deformation of material line in the tank at various rotational speeds (ω = 5, 10, 20, and 30 rpm), as shown in Figure 5. The deformation of material line for ω = 10 rpm is illustrated in Figure 5(a). As one can see in this figure, the material line gradually moves in the tank and only is stretched until $t=60\text{s}$ . But, as the experiment time increases, the material line can be folded and two material lines are generated, as illustrated in Figure 5(a) at t = 100 s. Figure 5(b) shows the structural changes of material line at ω = 10 rpm. It is clearly seen from this figure that the material line is only stretched until t = 30 s and between t = 30 and $60\text{s}$ folding of material line begins. The deformation of material line at ω = 20 rpm is illustrated in Figure 5(c). One can see from this figure that the area of free surface is colored more in comparison with previous tests. This is more remarkable in t = 100 s. Figure 5(d) illustrates the changes of material line at ω = 30 rpm. It is obviously seen that the material line can be divided more as the experiment time is equal to 30 s; consequently, more regions of free surface are covered by tracer and the area of regular region becomes less. Thus, the most area of colored surface can be detected at ω = 30 rpm and t = 100 s among all photos. It will be concluded that the higher rotational speeds of blades lead to disappear of regular region. This is due the fact that more strong radial flow is created by the blades with higher rotational speed which can improve the mixing performance in the tank. As we are interested to evaluate the effect of blade length on deformation of material line, Figures 6(a)–6(c) show the structure of material line for various lengths of blades. Figure 6(a) demonstrates deformation of material line for blades with $L_1=(\pi /6)R_b$ and $L_2=(\pi /2)R_b$ . One can clearly see that the material line and colored surface move towards a special side. This can be explained as follows. The more difference between lengths of blades causes more difference of forces exerted on working fluid by two blades. This causes that the tracer moves towards the blade with longer length. Figure 6(b) shows the structural changes of material line for blades with $L_1=(\pi /3)R_b$ and $L_2=(\pi /2)R_b$ at various times. It is seen that the tracer can distribute into container more uniformly in comparison to the previous case. Figure 6(c) illustrated deformation of material line for blades with $L_1=(\pi /6)R_b$ and $L_2=(\pi /3)R_b$ . This Figure is similar to Figure 5(d) which shows a good mixing. This is due the fact that the blades length can affect mixing by two mechanisms. On the one hand, the longer lengths can generate more rotation boundary conditions, but on the other hand, it leads lower reverse time. The first mechanism can improve mixing performance while the second one has an adverse effect on mixing. With respect to observations, the effect of second mechanism is more in the mixing under study and so the best performance occurs for blades with $L_1=(\pi /6)R_b$ and $L_2=(\pi /3)R_b$ , where one blade has middle length and the other has the smallest length.

Figure 5:

Deformation of material line for various ω; (a) ω = 5 rpm, (b) ω = 10 rpm, (c) ω = 20 rpm, and (d) ω = 30 rpm ( $L_1=(\pi /6)R_b$ and $L_2=(\pi /3)R_b$ ).

Figure 6:

Deformation of material line for various values of blade length; (a) $L_1=(\pi /6)R_b$ and $L_2=(\pi /2)R_b$ , (b) $L_1=(\pi /3)R_b$ and $L_2=(\pi /2)R_b$ , (c) $L_1=(\pi /6)R_b$ and $L_2=(\pi /3)R_b$ (ω = 30 rpm).

3.2. Quantitative Analysis

As we are interested in quantifying the mixing performance for various rotational speeds of blades, Figure 6 shows the colored surface (CS) parameter as a function of the mixing time at various values of ω. The CS parameter is defined as follows:

$\begin{array}{}\text{CS}=\frac{\text{Number of colored pixels}}{\text{Number of total pixels}}\times \mathrm{100}.\end{array}$

(2)

As one can clearly see from Figure 7, the blade rotational speed has a significant influence on the mixing and CS increases as rotational speed of blades increases. So the best value about 89% can be detected at ω = 30 rpm and t = 100 s. Results also reveal that the rate of the increase in colored surface is more remarkable for higher rotational speeds. This is due the fact that the radial flow is stronger in experiments with higher rotational speed which permits tracer to distribute in tank more rapidly. At the lower rotational speed, the movement of tracer is regular and the chaotic region cannot be produced in short times. Therefore, the amount of colored surface approximately remains constant at lower rotational speed.

Figure 7:

Effect of blade rotational speed on CS parameter.

Figure 8 illustrates the variations of CS as a function of time for various considered blades at ω = 30 rpm. One can clearly observe that the best CS is created by rotating two blades with lengths $(\pi /\mathrm{6})R_b$ and $(\pi /\mathrm{3})R_b$ .

Figure 8:

Effect of blade length on CS parameter.

Furthermore, although the experiment with blades of lengths $(\pi /\mathrm{3})R_b$ and $(\pi /\mathrm{2})R_b$ produces better tracer distribution in comparison to one with lengths $(\pi /\mathrm{6})R_b$ and $(\pi /\mathrm{2})R_b$ , the difference in amount of colored surface is not remarkable. One may claim that the initial size and shape of tracer fluid may affect the way of mixing with respect to time, or similar experiments show different results. In order to respond to this claim, it must be explained that the amount of colored surface in chaotic systems with similar conditions may be different with respect to time because of high-sensitive chaos phenomena, but the averages of the amount of colored surface are approximately similar [11]. So the average of the amount of colored surface is more important than the value of amount of colored surface in a specific time.

4. Conclusion

In this study, we have discussed a novel approach to the design and manufacturing of a high-viscous liquid mixer by applying two rotational blades in a circular container. The experimental results have shown that the mixing performance can be improved dramatically with increasing blade rotational speed due to chaotic perturbation. Furthermore, the comparable experiments have determined that the lengths of blades have a remarkable effect on mixing efficiency and the greatest colored surface (CS) can be generated by rotating two blades of lengths $(\pi /\mathrm{6})R_b$ and $(\pi /\mathrm{3})R_b$ .