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Solid-liquid mixing performance in a stirred tank with a double punched rigid-flexible impeller coupled with a chaotic motor
Accepted Manuscript Title: Solid-liquid mixing performance in a stirred tank with a double punched rigid-flexible impeller coupled with a chaotic motor Authors: Deyin Gu, Zuohua Liu, Chuanlin Xu, Jun Li, Changyuan Tao, Yundong Wang PII: DOI: Reference:
S0255-2701(16)30616-X http://dx.doi.org/doi:10.1016/j.cep.2017.04.018 CEP 6980
To appear in:
Chemical Engineering and Processing
Received date: Revised date: Accepted date:
19-11-2016 25-4-2017 27-4-2017
Please cite this article as: Deyin Gu, Zuohua Liu, Chuanlin Xu, Jun Li, Changyuan Tao, Yundong Wang, Solid-liquid mixing performance in a stirred tank with a double punched rigid-flexible impeller coupled with a chaotic motor, Chemical Engineering and Processinghttp://dx.doi.org/10.1016/j.cep.2017.04.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Graphic Abstract of the Manuscript
The double punched rigid-flexible impeller consists of two rigid impellers and six flexible connection pieces. A certain number of holes exist on the surface of the flexible connection pieces. The flexible connection pieces are under the interaction of agitating shaft and flowing medium, which can cause a series of disturbances on the flowing medium and transfer the impeller energy through the form of wave. With the existence of apertures, a series of high-speed jet flows and a plurality of vortices can be generated in the flow field. These conditions are beneficial to improve the mixing performance in the solid-liquid mixing process.
Highlights Largest Lyapunov exponent (LLE) was used to assess the mixing performance. Relative standard deviation (RSD) was applied for solid suspension quality evaluation. Double punched rigid-flexible impeller can improve solid-liquid homogeneous degree. Chaotic motor can improve solid suspension quality with co-reverse periodic rotation.
Solid-liquid mixing performance in a stirred tank with a double punched rigid-flexible impeller coupled with a chaotic motor Deyin Gua,b, Zuohua Liu a,b,*, Chuanlin Xua,b, Jun Lia, Changyuan Taoa,b, Yundong Wangc a School of Chemistry and Chemical Engineering, Chongqing University, Chongqing 400044, China b Chongqing Key Laboratory of Chemical Process for Clean Energy and Resource Utilization, Chongqing 400044, China c
Department of Chemical Engineering, Tsinghua University, Beijing 100084, China
Abstract The mixing performance of solid-liquid system was investigated in a stirred tank with a double rigid impeller, a double rigid-flexible impeller, a double punched rigid-flexible impeller, and a double punched rigid-flexible impeller coupled with a chaotic motor. The effects of the impeller types, the flexible connection piece width/length, the aperture size/ratio, and the variable/constant speed periods were investigated. The mixing performance was characterized by the largest Lyapunov exponent (LLE) and axial local solid concentration profile. The results indicated that the double punched rigid-flexible impeller and chaotic motor could enhance the LLE value of solid-liquid mixing system on the basis of the double punched rigid-flexible impeller and traditional motor under the same power consumption. The variation in solid homogeneous degree was in good agreement with that of LLE. It was found that the double punched rigid-flexible impeller coupled with a chaotic motor could further improve the solid suspension quality and energy efficiency compared with the double rigid impeller or double rigid-flexible impeller coupled with a traditional motor in the solid-liquid mixing process. Keywords Stirred tank; Solid suspension; Lyapunov exponent; Double punched rigid-flexible impeller; Chaotic motor 1. Introduction Stirred tanks are widely used in the chemical, biochemical, pharmaceutical, polymer, petrochemical, mineral processing, and many other applications. Solid-liquid mixing in stirred tanks is a vital aspect of mixing operations, and it is required to accomplish a vast range of process objectives, such as crystallization, polymerization, catalytic reactions, water treatment, blending, dissolution, leaching, and other kinds of processes [1-3]. The primary goals of solid-liquid mixing are to avoid solid particles stacking at the bottom of the stirred tank, to make solid particles uniformly dispersed in the liquid phase, to enhance the interface area between the solid and liquid phases, and to improve the overall rate of mass/heat transfer and chemical reaction. The solid-liquid homogeneous degree in the stirred tank is of vital importance, because inadequate or excessive agitation may lead to low product quality and high processing cost [4-7].
* Corresponding author. Tel.: +86-15922926287. E-mail address: [email protected].
Nomenclature H T D L W R r LLE Re Ch Cav z Pv m τ X1, X2, …, XK, …, XN Y(t0) Y0(t0) L0 ε Q M ti Wb Wb/D S Np K P N Greek Letters ρL ρS ρm
liquid height (m) stirred tank diameter (m) impeller diameter (m) length of the flexible connection piece (m) width of the flexible connection piece (m) stirred tank radius (m) radial coordinate (m) largest Lyapunov exponent Reynolds number local solid volume fraction at height of h average solid volume fraction height of sample connection (m) power consumption per unit volume (kw·m-3) embedded figure delay time (s) time series initial point nearest neighbor point distance between the initial point and the nearest neighbor point fixed value >0 end point of the time series total iterative number of the tracking evolution process tracking time (s) width of the blade (m) shape factor of the impeller projected area of blade in the rotation direction (m2) power number proportionality factor total power consumption (w·m-3) impeller agitation speed, s-1 liquid density (kg·m-3) solid density (kg·m-3) average density of the mixing system (kg·m-3)
The mixing performance of solid-liquid in stirred tanks has been studied extensively [8-12]. To obtain a good solid-liquid mixing performance, not only should the solid particles be completely distributed in the liquid phase, but the solid-liquid mixing process should be also induced in chaotic mixing state. The traditional method is to enlarge the power input (e.g., increasing impeller agitation speed); however, excessive agitation may lead to the formation of a stable symmetry flow field, and the shaft torque may exceed hardware capabilities [13,14]. In some processes (e.g., biochemical, polymerization, and pharmaceutical processes), the reaction substrates are sensitive to the shear force of the impeller, and the excessive shear force may result
in a failure reaction [15-18]. Several attempts have been made to improve the mixing performance in the solid-liquid mixing process. For example, Liu et al. [19] applied a single rigid-flexible impeller in the electrolytic metal manganese process, and found that the rigid-flexible impeller could enhance manganese leaching rate and shorten ore leaching time. Liu et al. [20] also applied a single rigid-flexible impeller to the dispersion of propene polymer particles in the glycerol system, and found that the rigid-flexible impeller could improve the mixing performance compared with the traditional rigid impeller. In other studies, Zhao et al. [21] applied a type of punched rigid impeller in the mixing process of phosphorite-water system, and found that the punched rigid impeller could reduce the agitation speed of suspending solid particles off the tank bottom. Chen et al. [22] applied computational fluid dynamics (CFD) method to study the flow field characteristics in a stirred tank with a punched rigid impeller, and found that the punched rigid impeller could increase the fluid velocity gradient and turbulent degree, and the axial fluid velocity of the punched rigid impeller was larger than that of the traditional rigid impeller. These findings suggest that the rational design of the rigid-flexible impeller and punched impeller can enhance the mixing performance and energy efficiency in the mixing process. In order to further improve the solid-liquid suspension quality, a new type of double punched rigid-flexible impeller is proposed in this work. With the development of chaos theory, chaotic mixing has been gradually recognized as an effective method to achieve an efficient and energy-saving mixing performance [23-25]. Liu et al. [26] reported that the mixing performance in aperiodic flows was better than that in periodic flows, because aperiodic perturbations generated widespread chaos, whereas periodic flows generated minimal or no chaos. Lamberto et al. [27] reported that the segregated regions in the flow field could be easily eliminated by using a controlled periodic fluctuation in the impeller rotation rate. Yao et al. [28] found that an unsteady rotating speed could obtain a high frequency of periodic co-reverse rotation and a large amplitude of time-periodic fluctuation to shorten the mixing time in the mixing process. Previous researches have suggested that the time-varying rotational agitator may be an effective approach to induce the mixing system into a chaotic state. The chaotic motion of the impeller can be achieved through a chaotic motor, which is controlled by an electric programmable logic controller. However, the time-varying rotational agitator has never been researched in the solid-liquid mixing process. Many different methods can be used to evaluate the mixing performance of solid-liquid system and a large number of related studies have been published, such as measurements of the solid concentration profile, cloud height and mixing time [8,11,29,30]. In recent years, spurred by the advances in chaos theory, chaotic characteristic parameters were conducted to analyze the mixing characteristics in the stirred tank [15,20,31]. The principal aims of the current study are to improve the mixing performance in the solid-liquid mixing process via optimizing a range of parameters, including the flexible connection piece width and length of double rigid-flexible impeller, the aperture diameter and ratio of double punched rigid-flexible impeller, and the variable and constant speed periods of chaotic motor. In addition, a typical chaotic characteristic parameter, namely, the largest Lyapunov exponent (LLE) was used to provide an indication of the chaotic mixing characteristics in the solid-liquid mixing process. The axial local solid concentration profile was measured to provide detailed information for the dispersion of solid particles in the stirred tank.
2. Experimental section 2.1. Experimental apparatus
Fig. 1. Schematic of the experimental apparatus. 1-Programmable Logic Controller; 2-Motor; 3-Decelerator; 4-Bracket; 5-Torque transducer; 6-Stirring shaft; 7-Baffle; 8-Rigid blades of impeller; 9-Silicone flexible connection piece; 10, 11-Data acquisition card; 12-Computer
(a)
(b)
(c)
Fig. 2. Structure diagram of the impellers. (a) Double rigid impeller, (b) Double rigid-flexible impeller, (c) Double punched rigid-flexible impeller.
A schematic of the experimental apparatus is shown in Figure 1. Experiments were conducted in a 0.48 m diameter (T) transparent cylindrical plexiglass vessel. The stirred tank has a flat bottom with four equally spaced vertical baffles having a width of 0.048 m (T/10). A double rigid impeller, a double rigid-flexible impeller and a double punched rigid-flexible impeller were used in the experiments. The double rigid impeller (Fig. 2a) consisted of a six-bladed pitched blade disc turbine (PBDT) impeller and a Rushton disc turbine (RDT) impeller fixed on the shaft. The two rigid impellers had a diameter of 0.20 m (D). The double rigid-flexible impeller (Fig. 2b) had six silicone flexible connection pieces on the blades of the impeller compared with the double rigid impeller. The double punched rigid-flexible impeller (Fig. 2c) had additional apertures on the flexible connection pieces compared with that of the double rigid-flexible impeller. The impeller off-bottom clearance was set at T/3, the distance between the upper and lower rigid impellers was set at T, and the height (H) of the liquid from the tank bottom was maintained at 0.80 m. A torque transducer (DaYang Company, Model: HX-90D) was used to measure the power consumption. A pressure transducer (Honewell Company, Model: 140PC) was employed to measure the fluctuation of liquid pressure. A programmable logic controller (Schneider Electric Company, Model: ATV12) was used to regulate the chaotic motion of the impeller. The axial local solid concentration was measured with the sample withdrawal method. For simplicity and versatility, the sample withdrawal method has been widely used in many studies [3,7,32-34]. When the solid particles were in a homogeneous suspension, the solid volume fraction in the sample was nearly coincident with the average solid volume fraction, Cav. In order to reduce the interference to the flow field in the mixing process, the sample tubes were installed at the radial position of r/R = 0.95. The samples were withdrawn at six axial positions of the stirred vessel, namely, z/H = 0.25, 0.375, 0.5, 0.625, 0.75, 0.875. In the experiment, tap water (ρL = 1000 kg·m-3) was used as liquid phase, and glass bead particles (ρS = 2470 kg·m-3) were used as solid phase, which the mean diameter is 120 μm. The solid loading was maintained at 5 % (v/v) in the experiment. 2.2. Experimental methods The Lyapunov exponent is an important parameter to evaluate the dynamic characteristics of a dynamic system. The existence of chaotic mixing in the mixing system can be judged intuitively from the value of the largest Lyapunov exponent (LLE). If LLE > 0, then the mixing system is in the chaotic state; and the value of LLE directly determines the degree of chaotic state [15]. In the
current work, the time series data of pressure fluctuation were processed using the C++ program in MATLAB, and LLE was calculated through using the Wolf algorithm [15,35]. The core procedures are stated below. If the time series are X1, X2, …, XK, …, XN, the embedded figure is m, the delay time is τ, and the reconstructing phase space can be expressed as follows:
Y (ti ) ( X (ti ), X (ti ),..., X (ti (m 1) ))
(i 1 , 2 , N ...,
)
(1)
If the distance between the initial point Y(t0) and the nearest neighbor point Y0(t0) is L0, and the time evolution of the two points is tracked until t1. When the distance is larger than a fixed value ε > 0 and L'0 = |Y(t1)-Y0(t1)| > ε, the point Y(t1) is retained and another point Y1(t1) adjacent to Y(t1) is found, which is satisfied with L1 = |Y(t1)-Y1(t1)| < ε, this process is iterated until Y(t) reaches the end point of the time series Q. The total iterative number of the tracking evolution process is up to M. The value of LLE can be obtained as follows:
1 t M t0
M
ln i 0
L'0 L1
(2)
3. Results and discussion 3.1. Effect of impeller types and agitation speed
Fig. 3. Effect of impeller types on LLE.
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(b)
Fig. 4. Effect of impeller types and agitation speed on axial solid concentration profile. (a) Effect of impeller types (N = 300 rpm), (b) Effect of agitation speed (Double rigid-flexible impeller).
The results showed that the impeller types and impeller agitation speed had significant effects on the mixing performance. Fig. 3 shows the variable relationship between LLE and Re with two different impeller types (double rigid impeller and double rigid-flexible impeller). The values of LLE were greater than zero, which indicated that the mixing system were entered into a chaotic state. As shown in Fig. 4b, at a low agitation speed, solid particles were partially suspended, and a stagnant layer of solid particles was stacked at the bottom of the vessel. Hence, the values of Ch/Cav at the higher part of the stirred vessel was low. As the agitation speed further increased, large quantities of solid particles were suspended and dispersed in the stirred vessel. However, when the agitation speed reached a certain level, the solid suspension quality increased slowly by continuing to increase the agitation speed. As shown in Fig. 3, the LLE value of the double rigid-flexible impeller was higher than that of the double rigid impeller at the same agitation speed. The solid-liquid homogeneity of the double rigid-flexible impeller was better than that of the double rigid impeller (Fig. 4a), although absolute homogeneity could not be achieved. Therefore, the double rigid-flexible impeller could improve the mixing performance and chaotic mixing degree compared with the double rigid impeller. This finding might be explained by the fact that the flexible connection pieces of the double rigid-flexible impeller were under the interaction of
agitating shaft and flowing medium in the mixing process. The flexible connection pieces could cause a series of disturbances on the flowing medium, and transfer the impeller energy to the fluid through the form of wave [19,20,35]. Therefore, the flexible connection pieces were beneficial to improve the energy dispersion in the flowing medium, and enhance the solid-liquid mixing performance. 3.2. Effect of flexible connection piece width and length
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(b)
Fig. 5. Effect of flexible connection piece width and length on LLE. (a) Effect of flexible connection piece width (L = 56cm), (b) Effect of flexible connection piece length (W = 2cm).
(a)
(b)
Fig. 6. Effect of flexible connection piece width and length on axial solid concentration profile (N = 300 rpm). (a) Effect of flexible connection piece width (L =56cm), (b) Effect of flexible connection piece length (W = 2 cm).
As illustrated in Figs. 5 and 6, the width and length of flexible connection piece both had significant influence on the chaotic mixing degree and solid suspension homogeneous degree. The LLE of the double rigid-flexible impeller increased with an increase in the width and length of flexible connection piece. By increasing the flexible connection piece width and length, the capability of the double rigid-flexible impeller to suspend solid particles increased, and the local solid concentration at the higher part of the stirred vessel subsequently increased. Specifically, the values of Ch/Cav at higher part became high. This phenomenon might be due to the fact that a narrow and short flexible connection piece has a limited capability to transfer impeller energy to the fluid, whereas a wide and long flexible connection piece has obvious perturbations on flowing medium through its swing [20,24]. A wide and long flexible connection piece can generate considerable vortices and enhance the chaotic mixing degree in the mixing process [24]. 3.3. Effect of aperture size and aperture ratio With the development of turbulence theory, the essence of solid suspension has been recognized, that is, the solid particle obtains energy from the vortex the size of which is similar to that of the solid particle [21,22,38]. Eddy diffusion can produce a large number of local small vortices, and the vortices are subsequently distributed to the entire vessel by the main convection diffusion. The hoisting capacity of the vortex increases with the increase of the fluid turbulent degree, which depends on the instantaneous velocity gradient [21,22,36-38]. Therefore, under the premise that adequate main convection exists in the stirred tank, the instantaneous velocity gradient should be improved, and the eddy diffusion should be strengthened.
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(b)
Fig. 7. Effect of aperture size and aperture ratio on LLE (L = 56 cm, W = 2 cm). (a)Effect of aperture size (aperture ratio = 12 %), (b) Effect of aperture ratio (aperture diameter = 8 mm).
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(b)
Fig. 8. Effect of aperture size and aperture ratio on axial solid concentration profile (N = 300 rpm, L = 56 cm, W = 2 cm). (a) Effect of aperture size (aperture ratio = 12 %), (b) Effect of aperture ratio (aperture diameter = 8 mm).
There are a certain number of holes on the surface of the punched rigid-flexible impeller. When the punched impeller rotates in the stirred tank, the fluid will flow through the orifices, and a series of high-speed jet flows are produced in the flow field. The jet flows can increase the fluid velocity gradient and form a plurality of vortices [36-38]. Owing to the jet flow, the length of the vortex tube can be further stretched, and a large number of small stable vortices can be formed in the flow field [21,22,36-38]. Fig. 7 shows the variation in LLE with Re in different aperture sizes and aperture ratios, and Fig. 8 illustrates the effects of aperture sizes and aperture ratios on the axial solid concentration profile. The aperture size of double punched rigid-flexible impeller had a great influence on the orifice flow velocity and the instantaneous velocity gradient. When the aperture size increased, the orifice flow velocity decreased, thereby forming a small velocity gradient and many large sized vortices in the flow field [22,36,38]. In addition, the shear force of the impeller became weak, which was negative on the local eddy diffusion. When the aperture size decreased, the circumference of the sheared edge increased, the friction power consumption increased, and the energy utilization rate decreased [22,36,38]. As shown in Figs. 7a and 8a, the aperture diameter of 8 mm was particularly suitable for the solid-liquid mixing process in this work. When the aperture size was constant, the number of aperture increased with the increase of the aperture ratio. This condition would affect the number of vortex and the fluid turbulent degree. When the aperture ratio was too large, jet flows were converted to concurrent flows, thus causing a significant kinetic energy loss and a negative effect on the main convection diffusion [22,36,38]. When the aperture ratio was too small, no enough vortices existed for the energy diffusion [22,36,38]. As shown in Figs. 7b and 8b, the aperture ratio of 12 % was particularly suitable for the solid-liquid mixing process in this work. As can be seen from Figs. 7 and 8, when the aperture diameter and aperture ratio of double punched rigid-flexible impeller were equal to zero, the impeller used in the experiment was the double rigid-flexible impeller. The LLE value of the double punched rigid-flexible impeller was higher than that of the double rigid-flexible impeller at the same agitation speed. The local solid concentration of the double punched rigid-flexible impeller at the higher part of the stirred vessel was also higher than that of the double rigid-flexible impeller. The aperture sizes and aperture ratios also had significant effects on the power consumption. Vusse et al. [39] reported the correlation between the shape factor of an impeller (Wb/D) or the projected area of blades in the rotation direction (S) and power number (Np), that is,
Np K (
Wb 0.3~0.4 ) KD (0.6~0.8) S (0.3~0.4) D
(3)
The power consumption (P) can be expressed as follows:
P N p m N 3 D5
(4)
As mentioned earlier, a certain number of holes exist on the surface of the punched rigid-flexible impeller. Such holes could lead to a decrease in the blade projected area (S) of the rotation direction, and subsequently a drop in the power number (Np) and power consumption (P).
In this work, power consumption was measured with a torque transducer. As illustrated in Fig. 9, the power consumption per unit volume (Pv) of the double punched rigid-flexible impeller was lower compared with that of the double rigid-flexible impeller. The Pv of 8 mm diameter hole was minimum, and the descent rate of Pv varied insignificantly when the aperture ratio was more than 12 %.
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Fig. 9. Effect of aperture size and aperture ratio on Pv (N = 300 rpm, L = 56cm, W = 2 cm). (a) Effect of aperture size (aperture ratio = 12 %), (b) Effect of aperture ratio (aperture diameter = 8 mm).
3.4. Effect of variable speed period and constant speed period The chaotic motor was used to provide an approach of changing the moving direction of the impeller, namely, co-reverse periodic rotation of the impeller, to impose a dynamic perturbation on the mixture system. The rotation mode of the chaotic motor is shown in Fig. 10. As can be seen, A and D were the accelerating processes, B and E were the constant speed processes, C and F were the decelerating processes, and A~F were entitled as a complete agitation cycle. The variable speed period (the accelerating period or the decelerating period) and constant speed period were investigated in this work.
Fig. 10. Rotation mode of the chaotic motor.
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(b)
Fig. 11. Effect of variable speed period and constant speed period on LLE (L = 56cm, W = 2 cm, aperture diameter = 8 mm, aperture ratio = 12 %). (a) Effect of variable speed period (constant speed period = 15 s), (b) Effect of constant speed period (variable speed period = 10 s).
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Fig. 12. Effect of variable speed period and constant speed period on axial solid concentration profile (N = 300 rpm, L = 56cm, W = 2 cm, aperture diameter = 8 mm, aperture ratio = 12 %). (a) Effect of variable speed period (constant speed period = 15 s), (b) Effect of constant speed period (variable speed period = 10 s).
Figs. 11 and 12 show the experimental results operated under steady rotation (i.e., with traditional motor) and co-reverse periodic rotation (i.e., with chaotic motor), in which the chaotic mixing characteristics is shown in the LLE, and the solid suspension quality is shown in the axial solid concentration profile. As shown in Figs. 11a and 12a, when the variable speed period was shorter than 30 s, the LLE value of the chaotic motor was larger than that of the traditional motor, and the solid-liquid homogeneous degree of the chaotic motor was also better than that of the traditional motor. This result was probably caused by the fact that the amplitude or the strength of the perturbation imposed by the chaotic motion of the impeller increased as the variable speed
period became short [27,28]. In addition, stronger dynamic perturbations on the mixture system occurred within a shorter variable speed period, which could easily cause chaotic mixing in the mixing process. However, according to the experimental evidence, although the co-reverse rotational impeller in the stirred tank could cause dynamic perturbations, some amounts of perturbations had an insignificant effect on improving the mixing performance. Only when the perturbation was stronger than some value could the mixing performance be improved. When the variable speed period was too long, the relative velocity between impeller and fluid became small, which would result in a small velocity gradient in the flow field. As illustrated in Figs. 11b and 12b, the LLE of the chaotic motor was larger than that of the traditional motor, and the value of Ch/Cav at the higher part of the stirred vessel was higher than that of the traditional motor. As discussed earlier, numerous dynamic perturbations were imposed on the mixture system by the chaotic motor, the fluid turbulent degree increased, thereby the mixing performance was also enhanced [27,28]. However, when the constant speed period was too long, the number of the variable speed process decreased in a certain amount of time and the number of perturbation subsequently decreased. As shown in Fig. 10, the value of impeller agitation speed in constant speed process was the largest in the mixing process, and a large velocity gradient could be obtained in constant speed process. Hence, maintaining a short constant speed period was not sensible. As shown in Fig. 11b, the constant speed period of 15 s was suitable for the solid-liquid mixing process in this work. The LLE value of the double punched rigid-flexible impeller coupled with a chaotic motor was higher than that of the double punched rigid-flexible impeller with a traditional motor at the same agitation speed. Moreover, the solid particle dispersion of the double punched rigid-flexible impeller coupled with a chaotic motor was also more homogeneous than that of the double punched rigid-flexible impeller with a traditional motor. 3.5. Relative Standard Deviation (RSD) The relative standard deviation (RSD) of the axial solid concentration profile is widely used to quantify the effect of the stirred vessel and impeller geometry, solid particle properties, and impeller agitation speed on the homogeneity of the solid-liquid mixing system [8,40]. The value of RSD can be calculated as follows:
1 1 n RSD (Ch Cav )2 Cav n 1 h 1
0.5
(5)
The value of RSD decreases as the solid particles dispersion becomes highly homogeneous. When the value of RSD is down to zero, the mixing system achieves absolute homogeneity. As can be seen from Fig. 13, the double punched rigid-flexible impeller could dramatically enhance the homogeneous degree of the solid-liquid mixing system compared with the double rigid-flexible impeller and double rigid impeller. The double punched rigid-flexible impeller coupled with a chaotic motor could further improve the solid suspension quality of the solid-liquid mixing system on the basis of the double punched rigid-flexible impeller with a traditional motor at the same power consumption.
Fig. 13. RSD of the axial solid concentration profile at different Pv.
3.6. Comparative experiment
Fig. 14. Comparison of power number Np at same Re (Single liquid phase).
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(b)
Fig. 15. Comparative study of LLE and Ch/Cav. (a) Comparison of LLE at same Pv, (b) Comparison of Ch/Cav at Pv= 0.5 kw·m-3.
As shown in Fig.14, the power numbers of the four cases (i.e. with a double rigid impeller, a double rigid-flexible impeller, a double punched rigid-flexible impeller, and a double punched rigid-flexible impeller coupled with a chaotic motor) were compared with each other at the same Re. It was observed that the power numbers of the double rigid-flexible impeller, double punched rigid-flexible impeller, and double punched rigid-flexible impeller coupled with a chaotic motor were larger than that of the double rigid impeller for single liquid phase. Therefore, it was necessary to compare the solid-liquid mixing performances of the four cases at the specific power consumption. As illustrated in Figs. 15 and 16, the mixing performances of the four cases were compared with each other at the same power consumption. As shown in Fig.15, the LLE value of double punched rigid-flexible impeller was larger than that of double rigid-flexible impeller and double rigid impeller at the same Pv, and double punched rigid-flexible impeller coupled with a chaotic motor could further enhance the LLE on the basis of double punched rigid-flexible impeller coupled with a traditional motor at the same Pv. Moreover, the solid-liquid homogeneous degree of double punched rigid-flexible impeller coupled with a chaotic motor was the best of the four cases at the same Pv. As seen in Fig. 16, the mixing processes of the four cases were presented with the help of photographs taken at the same power consumption and rotation time (i.e., 1 min). When the impeller power consumption was zero or low, a large heap of glass bead particles stacked at the bottom of the stirred tank, a few particles were dispersed into the liquid, and a clear liquid zone existed at the higher part of the vessel. With the increase of impeller power consumption, a large quantity of solid particles suspended and dispersed in the liquid phase, and the cloud height increased. The photographs of the four cases in Fig. 16 were in good agreement with the experimental results in Fig. 15, which indicated that the double punched rigid-flexible impeller and chaotic motor could effectively enhance the mixing performance and energy efficiency in the solid-liquid mixing process.
(a)
(b)
(c)
(d) Fig. 16. Comparison of mixing performance. (a) Double rigid impeller, (b) Double rigid-flexible impeller, (c) Double punched rigid-flexible impeller, (d) Double punched rigid-flexible impeller coupled with a chaotic motor.
4. Conclusions The double punched rigid-flexible impeller and chaotic motor were introduced as novel approaches in this wok to improve the mixing performance in the solid-liquid mixing process. The solid-liquid mixing performance was experimentally investigated using measurements of the largest Lyapunov exponent and axial local solid concentration profile in a stirred tank with a double rigid impeller, a double rigid-flexible impeller, a double punched rigid-flexible impeller, and a double punched rigid-flexible impeller coupled with a chaotic motor. The experimental results showed that the double punched rigid-flexible impeller and chaotic motor could enhance the LLE value of solid-liquid mixing system on the basis of the double punched rigid-flexible impeller and traditional motor under the same power consumption. The variation in solid homogeneous degree was in good agreement with that of LLE. It was found that the double punched rigid-flexible impeller coupled with a chaotic motor could effectively improve the mixing performance and energy efficiency compared with the double rigid impeller or double rigid-flexible impeller coupled with a traditional motor in the solid-liquid mixing process. A wide and long flexible connection piece were conductive to the solid-liquid mixing process. The aperture diameter of 8 mm and the aperture ratio of 12 % of the double punched rigid-flexible impeller and the variable speed period of 10 s and the constant speed period of 15 s of the chaotic motor were particularly suitable for the solid-liquid mixing process in this work. The mixing processes were presented with the help of photographs taken under the same operating conditions, and the images were in good agreement with the experimental results. Acknowledgements The study was supported by the National Natural Science Foundations of China (21576033, 21636004). References [1] A. Tagawa, N., Dohi; Y. Kawase, Dispersion of Floating Solid Particles in Aerated Stirred Tank Reactors: Minimum Impeller Speeds for Off-Surface and Ultimately Homogeneous Solid Suspension and Solids Concentration Profiles, Ind. Eng. Chem. Res. 45 (2006) 818-829. [2] T. Kumaresan, J. B. Joshi, Effect of Impeller Design on the Flow Pattern and Mixing in Stirred Tanks, Chem. Eng. Sci. 115 (2006) 173-193. [3] G. Baldi, R. Conti, E. Alaria, Complete Suspension of Particles in Mechanically Agitated Vessels, Chem. Eng. Sci. 33 (1978) 21-25. [4] R. W. Thring, M. F. Edwards, An Experimental Investigation into the Complete Suspension of Floating Solids in an Agitated Tank, Ind. Eng. Chem. Res. 29 (1990) 678-682. [5] N. Kuzmanic, E. M. Kessler, Continuous Sampling of Floating Solids Suspension from a Mixing Tank, Ind. Eng. Chem. Res. 36 (1997) 5015. [6] Y. Y. Bao, Z. G. Hao, Z. M. Gao, L. T. Shi, J. M. Smith, Suspension of Buoyant Particles in A Three Phase Stirred Tank, Chem. Eng. Sci. 60 (2005) 2283-2292. [7] N. Dohi, T. Takahashi, K. Minekawa, Y. Kawase, Power Consumption and Solid Suspension of
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[29] C. Carletti, G. Montante, T. Weterlund, A. Paglianti, Analysis of Solid Concentration Distribution in Dense Solid-Liquid Stirred Tank by Eletrical Resistance Tomography, Chem. Eng. Sci. 119 (2014) 53-64. [30] G. Montante, A. Paglianti, F. Magelli, Analysis of Dilute Solid–Liquid Suspensions in Turbulent Stirred Tanks, Chem. Eng. Res. Des. 90 (2012) 1448-1456. [31] H. Li, Y. Zhou, B. Sun, Y. Yang, Muti-Scale Chaotic Analysis of the Characteristics of Gas-Liquid Two-Phase Flow Patterns, Chin. J. Chem. Eng. 18 (2010) 880-888. [32] K. Takahashi, S. Sasaki, Complete Drawdown and Dispersion of Floating Solids in Agitated Vessel Equipped with Ordinary Impellers, J. Chem. Eng. Jpn. 32 (1999) 40-44. [33] S.-A. Xu, L.-F. Feng, X.-P. Gu, K. Wang, G.-H. Hu, Gas-Liquid Floating Particle Mixing in an Agitated Vessel, Chem. Eng. Technol. 23 (2000) 103-113. [34] A. Barresi, G. Baldi, Solid Dispersion in an Agitated Vessel, Chem. Eng. Sci. 42 (1987) 2949. [35] Z. H. Liu, C. Chen, R. L. Liu, C. Y. Tao, Y. D. Wang, Chaotic Mixing Enhanced by Rigid-Flexible Impeller in Stirred Vessel, CIESC J. 65 (2014) 61-70. [36] J. Liu, Y. F. Ou, N. Q. Wang, Agitation Characteristics of a New Type Punched Agitator Used in Solid-Liquid System, J. Si Chuan Union Uni. 3 (1999) 48-53. [37] Z. P. Zhao, S. Y. Jiao, C. H. Chen, Study of Mixing Mechanism for a New-Type Stirred Tank with Penetrated Baffle, Pro. Chem. Eng. 22 (2002) 85-88. [38] H. Yang, F. Wang, C. X. Wang, Optimizing Structural Parameters of Punched Impeller, J. Wuhan Inst. Tech. 37 (2015) 39-43. [39] J.G. Van de Vusse, Mixing by Agitation of Miscible Liquids, Chem. Eng. Sci. 45 (1955) 178-200. [40] P. Tervasmaki, J. Tiihonen, H. Ojamo, Comparison of Solids Suspension Criteria Based on Electrical Impedance Tomography and Visual Measurements, Chem. Eng. Sci. 116 (2014) 128-135.
Fig. 1. Schematic of the experimental apparatus. 1-Programmable Logic Controller; 2-Motor; 3-Decelerator; 4-Bracket; 5-Torque transducer; 6-Stirring shaft; 7-Baffle; 8-Rigid blades of impeller; 9-Silicone flexible connection piece; 10, 11-Data acquisition card; 12-Computer
(a)
(b)
(c)
Fig. 2. Structure diagram of the impellers. (a) Double rigid impeller, (b) Double rigid-flexible impeller, (c) Double punched rigid-flexible impeller.
Fig. 3. Effect of impeller types on LLE.
(a)
(b)
Fig. 4. Effect of impeller types and agitation speed on axial solid concentration profile. (a) Effect of impeller types (N = 300 rpm), (b) Effect of agitation speed (Double rigid-flexible impeller).
(a)
(b)
Fig. 5. Effect of flexible connection piece width and length on LLE. (a) Effect of flexible connection piece width (L = 56cm), (b) Effect of flexible connection piece length (W = 2cm).
(a)
(b)
Fig. 6. Effect of flexible connection piece width and length on axial solid concentration profile (N = 300 rpm). (a) Effect of flexible connection piece width (L =56cm), (b) Effect of flexible connection piece length (W = 2 cm).
(a)
(b)
Fig. 7. Effect of aperture size and aperture ratio on LLE (L = 56 cm, W = 2 cm). (a)Effect of aperture size (aperture ratio = 12 %), (b) Effect of aperture ratio (aperture diameter = 8 mm).
(a)
(b)
Fig. 8. Effect of aperture size and aperture ratio on axial solid concentration profile (N = 300 rpm, L = 56 cm, W = 2 cm). (a) Effect of aperture size (aperture ratio = 12 %), (b) Effect of aperture ratio (aperture diameter = 8 mm).
(a)
(b)
Fig. 9. Effect of aperture size and aperture ratio on Pv (N = 300 rpm, L = 56cm, W = 2 cm). (a) Effect of aperture size (aperture ratio = 12 %), (b) Effect of aperture ratio (aperture diameter = 8 mm).
Fig. 10. Rotation mode of the chaotic motor.
(a)
(b)
Fig. 11. Effect of variable speed period and constant speed period on LLE (L = 56cm, W = 2 cm, aperture diameter = 8 mm, aperture ratio = 12 %). (a) Effect of variable speed period (constant speed period = 15 s), (b) Effect of constant speed period (variable speed period = 10 s).
(a)
(b)
Fig. 12. Effect of variable speed period and constant speed period on axial solid concentration profile (N = 300 rpm, L = 56cm, W = 2 cm, aperture diameter = 8 mm, aperture ratio = 12 %). (a) Effect of variable speed period (constant speed period = 15 s), (b) Effect of constant speed period (variable speed period = 10 s).
Fig. 13. RSD of the axial solid concentration profile at different Pv.
Fig. 14. Comparison of power number Np at same Re (Single liquid phase).
(a)
(b)
Fig. 15. Comparative study of LLE and Ch/Cav. (a) Comparison of LLE at same Pv, (b) Comparison of Ch/Cav at Pv= 0.5 kw·m-3.
0 kw·m-3
0.1 kw·m-3
0.3 kw·m-3
0.5 kw·m-3
0.7 kw·m-3
0.9kw·m-3
(a)
0 kw·m-3
0.1 kw·m-3
0.3 kw·m-3
0.5 kw·m-3
0.7 kw·m-3
0.9kw·m-3
(b)
0 kw·m-3
0.1 kw·m-3
0.3 kw·m-3
0.5 kw·m-3
0.7 kw·m-3
0.9kw·m-3
0.5 kw·m-3
0.7 kw·m-3
0.9kw·m-3
(c)
0 kw·m-3
0.1 kw·m-3
0.3 kw·m-3 (d)
Fig. 16. Comparison of mixing performance. (a) Double rigid impeller, (b) Double rigid-flexible impeller, (c) Double punched rigid-flexible impeller, (d) Double punched rigid-flexible impeller coupled with a chaotic motor.
S0255-2701(16)30616-X http://dx.doi.org/doi:10.1016/j.cep.2017.04.018 CEP 6980
To appear in:
Chemical Engineering and Processing
Received date: Revised date: Accepted date:
19-11-2016 25-4-2017 27-4-2017
Please cite this article as: Deyin Gu, Zuohua Liu, Chuanlin Xu, Jun Li, Changyuan Tao, Yundong Wang, Solid-liquid mixing performance in a stirred tank with a double punched rigid-flexible impeller coupled with a chaotic motor, Chemical Engineering and Processinghttp://dx.doi.org/10.1016/j.cep.2017.04.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Graphic Abstract of the Manuscript
The double punched rigid-flexible impeller consists of two rigid impellers and six flexible connection pieces. A certain number of holes exist on the surface of the flexible connection pieces. The flexible connection pieces are under the interaction of agitating shaft and flowing medium, which can cause a series of disturbances on the flowing medium and transfer the impeller energy through the form of wave. With the existence of apertures, a series of high-speed jet flows and a plurality of vortices can be generated in the flow field. These conditions are beneficial to improve the mixing performance in the solid-liquid mixing process.
Highlights Largest Lyapunov exponent (LLE) was used to assess the mixing performance. Relative standard deviation (RSD) was applied for solid suspension quality evaluation. Double punched rigid-flexible impeller can improve solid-liquid homogeneous degree. Chaotic motor can improve solid suspension quality with co-reverse periodic rotation.
Solid-liquid mixing performance in a stirred tank with a double punched rigid-flexible impeller coupled with a chaotic motor Deyin Gua,b, Zuohua Liu a,b,*, Chuanlin Xua,b, Jun Lia, Changyuan Taoa,b, Yundong Wangc a School of Chemistry and Chemical Engineering, Chongqing University, Chongqing 400044, China b Chongqing Key Laboratory of Chemical Process for Clean Energy and Resource Utilization, Chongqing 400044, China c
Department of Chemical Engineering, Tsinghua University, Beijing 100084, China
Abstract The mixing performance of solid-liquid system was investigated in a stirred tank with a double rigid impeller, a double rigid-flexible impeller, a double punched rigid-flexible impeller, and a double punched rigid-flexible impeller coupled with a chaotic motor. The effects of the impeller types, the flexible connection piece width/length, the aperture size/ratio, and the variable/constant speed periods were investigated. The mixing performance was characterized by the largest Lyapunov exponent (LLE) and axial local solid concentration profile. The results indicated that the double punched rigid-flexible impeller and chaotic motor could enhance the LLE value of solid-liquid mixing system on the basis of the double punched rigid-flexible impeller and traditional motor under the same power consumption. The variation in solid homogeneous degree was in good agreement with that of LLE. It was found that the double punched rigid-flexible impeller coupled with a chaotic motor could further improve the solid suspension quality and energy efficiency compared with the double rigid impeller or double rigid-flexible impeller coupled with a traditional motor in the solid-liquid mixing process. Keywords Stirred tank; Solid suspension; Lyapunov exponent; Double punched rigid-flexible impeller; Chaotic motor 1. Introduction Stirred tanks are widely used in the chemical, biochemical, pharmaceutical, polymer, petrochemical, mineral processing, and many other applications. Solid-liquid mixing in stirred tanks is a vital aspect of mixing operations, and it is required to accomplish a vast range of process objectives, such as crystallization, polymerization, catalytic reactions, water treatment, blending, dissolution, leaching, and other kinds of processes [1-3]. The primary goals of solid-liquid mixing are to avoid solid particles stacking at the bottom of the stirred tank, to make solid particles uniformly dispersed in the liquid phase, to enhance the interface area between the solid and liquid phases, and to improve the overall rate of mass/heat transfer and chemical reaction. The solid-liquid homogeneous degree in the stirred tank is of vital importance, because inadequate or excessive agitation may lead to low product quality and high processing cost [4-7].
* Corresponding author. Tel.: +86-15922926287. E-mail address: [email protected].
Nomenclature H T D L W R r LLE Re Ch Cav z Pv m τ X1, X2, …, XK, …, XN Y(t0) Y0(t0) L0 ε Q M ti Wb Wb/D S Np K P N Greek Letters ρL ρS ρm
liquid height (m) stirred tank diameter (m) impeller diameter (m) length of the flexible connection piece (m) width of the flexible connection piece (m) stirred tank radius (m) radial coordinate (m) largest Lyapunov exponent Reynolds number local solid volume fraction at height of h average solid volume fraction height of sample connection (m) power consumption per unit volume (kw·m-3) embedded figure delay time (s) time series initial point nearest neighbor point distance between the initial point and the nearest neighbor point fixed value >0 end point of the time series total iterative number of the tracking evolution process tracking time (s) width of the blade (m) shape factor of the impeller projected area of blade in the rotation direction (m2) power number proportionality factor total power consumption (w·m-3) impeller agitation speed, s-1 liquid density (kg·m-3) solid density (kg·m-3) average density of the mixing system (kg·m-3)
The mixing performance of solid-liquid in stirred tanks has been studied extensively [8-12]. To obtain a good solid-liquid mixing performance, not only should the solid particles be completely distributed in the liquid phase, but the solid-liquid mixing process should be also induced in chaotic mixing state. The traditional method is to enlarge the power input (e.g., increasing impeller agitation speed); however, excessive agitation may lead to the formation of a stable symmetry flow field, and the shaft torque may exceed hardware capabilities [13,14]. In some processes (e.g., biochemical, polymerization, and pharmaceutical processes), the reaction substrates are sensitive to the shear force of the impeller, and the excessive shear force may result
in a failure reaction [15-18]. Several attempts have been made to improve the mixing performance in the solid-liquid mixing process. For example, Liu et al. [19] applied a single rigid-flexible impeller in the electrolytic metal manganese process, and found that the rigid-flexible impeller could enhance manganese leaching rate and shorten ore leaching time. Liu et al. [20] also applied a single rigid-flexible impeller to the dispersion of propene polymer particles in the glycerol system, and found that the rigid-flexible impeller could improve the mixing performance compared with the traditional rigid impeller. In other studies, Zhao et al. [21] applied a type of punched rigid impeller in the mixing process of phosphorite-water system, and found that the punched rigid impeller could reduce the agitation speed of suspending solid particles off the tank bottom. Chen et al. [22] applied computational fluid dynamics (CFD) method to study the flow field characteristics in a stirred tank with a punched rigid impeller, and found that the punched rigid impeller could increase the fluid velocity gradient and turbulent degree, and the axial fluid velocity of the punched rigid impeller was larger than that of the traditional rigid impeller. These findings suggest that the rational design of the rigid-flexible impeller and punched impeller can enhance the mixing performance and energy efficiency in the mixing process. In order to further improve the solid-liquid suspension quality, a new type of double punched rigid-flexible impeller is proposed in this work. With the development of chaos theory, chaotic mixing has been gradually recognized as an effective method to achieve an efficient and energy-saving mixing performance [23-25]. Liu et al. [26] reported that the mixing performance in aperiodic flows was better than that in periodic flows, because aperiodic perturbations generated widespread chaos, whereas periodic flows generated minimal or no chaos. Lamberto et al. [27] reported that the segregated regions in the flow field could be easily eliminated by using a controlled periodic fluctuation in the impeller rotation rate. Yao et al. [28] found that an unsteady rotating speed could obtain a high frequency of periodic co-reverse rotation and a large amplitude of time-periodic fluctuation to shorten the mixing time in the mixing process. Previous researches have suggested that the time-varying rotational agitator may be an effective approach to induce the mixing system into a chaotic state. The chaotic motion of the impeller can be achieved through a chaotic motor, which is controlled by an electric programmable logic controller. However, the time-varying rotational agitator has never been researched in the solid-liquid mixing process. Many different methods can be used to evaluate the mixing performance of solid-liquid system and a large number of related studies have been published, such as measurements of the solid concentration profile, cloud height and mixing time [8,11,29,30]. In recent years, spurred by the advances in chaos theory, chaotic characteristic parameters were conducted to analyze the mixing characteristics in the stirred tank [15,20,31]. The principal aims of the current study are to improve the mixing performance in the solid-liquid mixing process via optimizing a range of parameters, including the flexible connection piece width and length of double rigid-flexible impeller, the aperture diameter and ratio of double punched rigid-flexible impeller, and the variable and constant speed periods of chaotic motor. In addition, a typical chaotic characteristic parameter, namely, the largest Lyapunov exponent (LLE) was used to provide an indication of the chaotic mixing characteristics in the solid-liquid mixing process. The axial local solid concentration profile was measured to provide detailed information for the dispersion of solid particles in the stirred tank.
2. Experimental section 2.1. Experimental apparatus
Fig. 1. Schematic of the experimental apparatus. 1-Programmable Logic Controller; 2-Motor; 3-Decelerator; 4-Bracket; 5-Torque transducer; 6-Stirring shaft; 7-Baffle; 8-Rigid blades of impeller; 9-Silicone flexible connection piece; 10, 11-Data acquisition card; 12-Computer
(a)
(b)
(c)
Fig. 2. Structure diagram of the impellers. (a) Double rigid impeller, (b) Double rigid-flexible impeller, (c) Double punched rigid-flexible impeller.
A schematic of the experimental apparatus is shown in Figure 1. Experiments were conducted in a 0.48 m diameter (T) transparent cylindrical plexiglass vessel. The stirred tank has a flat bottom with four equally spaced vertical baffles having a width of 0.048 m (T/10). A double rigid impeller, a double rigid-flexible impeller and a double punched rigid-flexible impeller were used in the experiments. The double rigid impeller (Fig. 2a) consisted of a six-bladed pitched blade disc turbine (PBDT) impeller and a Rushton disc turbine (RDT) impeller fixed on the shaft. The two rigid impellers had a diameter of 0.20 m (D). The double rigid-flexible impeller (Fig. 2b) had six silicone flexible connection pieces on the blades of the impeller compared with the double rigid impeller. The double punched rigid-flexible impeller (Fig. 2c) had additional apertures on the flexible connection pieces compared with that of the double rigid-flexible impeller. The impeller off-bottom clearance was set at T/3, the distance between the upper and lower rigid impellers was set at T, and the height (H) of the liquid from the tank bottom was maintained at 0.80 m. A torque transducer (DaYang Company, Model: HX-90D) was used to measure the power consumption. A pressure transducer (Honewell Company, Model: 140PC) was employed to measure the fluctuation of liquid pressure. A programmable logic controller (Schneider Electric Company, Model: ATV12) was used to regulate the chaotic motion of the impeller. The axial local solid concentration was measured with the sample withdrawal method. For simplicity and versatility, the sample withdrawal method has been widely used in many studies [3,7,32-34]. When the solid particles were in a homogeneous suspension, the solid volume fraction in the sample was nearly coincident with the average solid volume fraction, Cav. In order to reduce the interference to the flow field in the mixing process, the sample tubes were installed at the radial position of r/R = 0.95. The samples were withdrawn at six axial positions of the stirred vessel, namely, z/H = 0.25, 0.375, 0.5, 0.625, 0.75, 0.875. In the experiment, tap water (ρL = 1000 kg·m-3) was used as liquid phase, and glass bead particles (ρS = 2470 kg·m-3) were used as solid phase, which the mean diameter is 120 μm. The solid loading was maintained at 5 % (v/v) in the experiment. 2.2. Experimental methods The Lyapunov exponent is an important parameter to evaluate the dynamic characteristics of a dynamic system. The existence of chaotic mixing in the mixing system can be judged intuitively from the value of the largest Lyapunov exponent (LLE). If LLE > 0, then the mixing system is in the chaotic state; and the value of LLE directly determines the degree of chaotic state [15]. In the
current work, the time series data of pressure fluctuation were processed using the C++ program in MATLAB, and LLE was calculated through using the Wolf algorithm [15,35]. The core procedures are stated below. If the time series are X1, X2, …, XK, …, XN, the embedded figure is m, the delay time is τ, and the reconstructing phase space can be expressed as follows:
Y (ti ) ( X (ti ), X (ti ),..., X (ti (m 1) ))
(i 1 , 2 , N ...,
)
(1)
If the distance between the initial point Y(t0) and the nearest neighbor point Y0(t0) is L0, and the time evolution of the two points is tracked until t1. When the distance is larger than a fixed value ε > 0 and L'0 = |Y(t1)-Y0(t1)| > ε, the point Y(t1) is retained and another point Y1(t1) adjacent to Y(t1) is found, which is satisfied with L1 = |Y(t1)-Y1(t1)| < ε, this process is iterated until Y(t) reaches the end point of the time series Q. The total iterative number of the tracking evolution process is up to M. The value of LLE can be obtained as follows:
1 t M t0
M
ln i 0
L'0 L1
(2)
3. Results and discussion 3.1. Effect of impeller types and agitation speed
Fig. 3. Effect of impeller types on LLE.
(a)
(b)
Fig. 4. Effect of impeller types and agitation speed on axial solid concentration profile. (a) Effect of impeller types (N = 300 rpm), (b) Effect of agitation speed (Double rigid-flexible impeller).
The results showed that the impeller types and impeller agitation speed had significant effects on the mixing performance. Fig. 3 shows the variable relationship between LLE and Re with two different impeller types (double rigid impeller and double rigid-flexible impeller). The values of LLE were greater than zero, which indicated that the mixing system were entered into a chaotic state. As shown in Fig. 4b, at a low agitation speed, solid particles were partially suspended, and a stagnant layer of solid particles was stacked at the bottom of the vessel. Hence, the values of Ch/Cav at the higher part of the stirred vessel was low. As the agitation speed further increased, large quantities of solid particles were suspended and dispersed in the stirred vessel. However, when the agitation speed reached a certain level, the solid suspension quality increased slowly by continuing to increase the agitation speed. As shown in Fig. 3, the LLE value of the double rigid-flexible impeller was higher than that of the double rigid impeller at the same agitation speed. The solid-liquid homogeneity of the double rigid-flexible impeller was better than that of the double rigid impeller (Fig. 4a), although absolute homogeneity could not be achieved. Therefore, the double rigid-flexible impeller could improve the mixing performance and chaotic mixing degree compared with the double rigid impeller. This finding might be explained by the fact that the flexible connection pieces of the double rigid-flexible impeller were under the interaction of
agitating shaft and flowing medium in the mixing process. The flexible connection pieces could cause a series of disturbances on the flowing medium, and transfer the impeller energy to the fluid through the form of wave [19,20,35]. Therefore, the flexible connection pieces were beneficial to improve the energy dispersion in the flowing medium, and enhance the solid-liquid mixing performance. 3.2. Effect of flexible connection piece width and length
(a)
(b)
Fig. 5. Effect of flexible connection piece width and length on LLE. (a) Effect of flexible connection piece width (L = 56cm), (b) Effect of flexible connection piece length (W = 2cm).
(a)
(b)
Fig. 6. Effect of flexible connection piece width and length on axial solid concentration profile (N = 300 rpm). (a) Effect of flexible connection piece width (L =56cm), (b) Effect of flexible connection piece length (W = 2 cm).
As illustrated in Figs. 5 and 6, the width and length of flexible connection piece both had significant influence on the chaotic mixing degree and solid suspension homogeneous degree. The LLE of the double rigid-flexible impeller increased with an increase in the width and length of flexible connection piece. By increasing the flexible connection piece width and length, the capability of the double rigid-flexible impeller to suspend solid particles increased, and the local solid concentration at the higher part of the stirred vessel subsequently increased. Specifically, the values of Ch/Cav at higher part became high. This phenomenon might be due to the fact that a narrow and short flexible connection piece has a limited capability to transfer impeller energy to the fluid, whereas a wide and long flexible connection piece has obvious perturbations on flowing medium through its swing [20,24]. A wide and long flexible connection piece can generate considerable vortices and enhance the chaotic mixing degree in the mixing process [24]. 3.3. Effect of aperture size and aperture ratio With the development of turbulence theory, the essence of solid suspension has been recognized, that is, the solid particle obtains energy from the vortex the size of which is similar to that of the solid particle [21,22,38]. Eddy diffusion can produce a large number of local small vortices, and the vortices are subsequently distributed to the entire vessel by the main convection diffusion. The hoisting capacity of the vortex increases with the increase of the fluid turbulent degree, which depends on the instantaneous velocity gradient [21,22,36-38]. Therefore, under the premise that adequate main convection exists in the stirred tank, the instantaneous velocity gradient should be improved, and the eddy diffusion should be strengthened.
(a)
(b)
Fig. 7. Effect of aperture size and aperture ratio on LLE (L = 56 cm, W = 2 cm). (a)Effect of aperture size (aperture ratio = 12 %), (b) Effect of aperture ratio (aperture diameter = 8 mm).
(a)
(b)
Fig. 8. Effect of aperture size and aperture ratio on axial solid concentration profile (N = 300 rpm, L = 56 cm, W = 2 cm). (a) Effect of aperture size (aperture ratio = 12 %), (b) Effect of aperture ratio (aperture diameter = 8 mm).
There are a certain number of holes on the surface of the punched rigid-flexible impeller. When the punched impeller rotates in the stirred tank, the fluid will flow through the orifices, and a series of high-speed jet flows are produced in the flow field. The jet flows can increase the fluid velocity gradient and form a plurality of vortices [36-38]. Owing to the jet flow, the length of the vortex tube can be further stretched, and a large number of small stable vortices can be formed in the flow field [21,22,36-38]. Fig. 7 shows the variation in LLE with Re in different aperture sizes and aperture ratios, and Fig. 8 illustrates the effects of aperture sizes and aperture ratios on the axial solid concentration profile. The aperture size of double punched rigid-flexible impeller had a great influence on the orifice flow velocity and the instantaneous velocity gradient. When the aperture size increased, the orifice flow velocity decreased, thereby forming a small velocity gradient and many large sized vortices in the flow field [22,36,38]. In addition, the shear force of the impeller became weak, which was negative on the local eddy diffusion. When the aperture size decreased, the circumference of the sheared edge increased, the friction power consumption increased, and the energy utilization rate decreased [22,36,38]. As shown in Figs. 7a and 8a, the aperture diameter of 8 mm was particularly suitable for the solid-liquid mixing process in this work. When the aperture size was constant, the number of aperture increased with the increase of the aperture ratio. This condition would affect the number of vortex and the fluid turbulent degree. When the aperture ratio was too large, jet flows were converted to concurrent flows, thus causing a significant kinetic energy loss and a negative effect on the main convection diffusion [22,36,38]. When the aperture ratio was too small, no enough vortices existed for the energy diffusion [22,36,38]. As shown in Figs. 7b and 8b, the aperture ratio of 12 % was particularly suitable for the solid-liquid mixing process in this work. As can be seen from Figs. 7 and 8, when the aperture diameter and aperture ratio of double punched rigid-flexible impeller were equal to zero, the impeller used in the experiment was the double rigid-flexible impeller. The LLE value of the double punched rigid-flexible impeller was higher than that of the double rigid-flexible impeller at the same agitation speed. The local solid concentration of the double punched rigid-flexible impeller at the higher part of the stirred vessel was also higher than that of the double rigid-flexible impeller. The aperture sizes and aperture ratios also had significant effects on the power consumption. Vusse et al. [39] reported the correlation between the shape factor of an impeller (Wb/D) or the projected area of blades in the rotation direction (S) and power number (Np), that is,
Np K (
Wb 0.3~0.4 ) KD (0.6~0.8) S (0.3~0.4) D
(3)
The power consumption (P) can be expressed as follows:
P N p m N 3 D5
(4)
As mentioned earlier, a certain number of holes exist on the surface of the punched rigid-flexible impeller. Such holes could lead to a decrease in the blade projected area (S) of the rotation direction, and subsequently a drop in the power number (Np) and power consumption (P).
In this work, power consumption was measured with a torque transducer. As illustrated in Fig. 9, the power consumption per unit volume (Pv) of the double punched rigid-flexible impeller was lower compared with that of the double rigid-flexible impeller. The Pv of 8 mm diameter hole was minimum, and the descent rate of Pv varied insignificantly when the aperture ratio was more than 12 %.
(a)
(b)
Fig. 9. Effect of aperture size and aperture ratio on Pv (N = 300 rpm, L = 56cm, W = 2 cm). (a) Effect of aperture size (aperture ratio = 12 %), (b) Effect of aperture ratio (aperture diameter = 8 mm).
3.4. Effect of variable speed period and constant speed period The chaotic motor was used to provide an approach of changing the moving direction of the impeller, namely, co-reverse periodic rotation of the impeller, to impose a dynamic perturbation on the mixture system. The rotation mode of the chaotic motor is shown in Fig. 10. As can be seen, A and D were the accelerating processes, B and E were the constant speed processes, C and F were the decelerating processes, and A~F were entitled as a complete agitation cycle. The variable speed period (the accelerating period or the decelerating period) and constant speed period were investigated in this work.
Fig. 10. Rotation mode of the chaotic motor.
(a)
(b)
Fig. 11. Effect of variable speed period and constant speed period on LLE (L = 56cm, W = 2 cm, aperture diameter = 8 mm, aperture ratio = 12 %). (a) Effect of variable speed period (constant speed period = 15 s), (b) Effect of constant speed period (variable speed period = 10 s).
(a)
(b)
Fig. 12. Effect of variable speed period and constant speed period on axial solid concentration profile (N = 300 rpm, L = 56cm, W = 2 cm, aperture diameter = 8 mm, aperture ratio = 12 %). (a) Effect of variable speed period (constant speed period = 15 s), (b) Effect of constant speed period (variable speed period = 10 s).
Figs. 11 and 12 show the experimental results operated under steady rotation (i.e., with traditional motor) and co-reverse periodic rotation (i.e., with chaotic motor), in which the chaotic mixing characteristics is shown in the LLE, and the solid suspension quality is shown in the axial solid concentration profile. As shown in Figs. 11a and 12a, when the variable speed period was shorter than 30 s, the LLE value of the chaotic motor was larger than that of the traditional motor, and the solid-liquid homogeneous degree of the chaotic motor was also better than that of the traditional motor. This result was probably caused by the fact that the amplitude or the strength of the perturbation imposed by the chaotic motion of the impeller increased as the variable speed
period became short [27,28]. In addition, stronger dynamic perturbations on the mixture system occurred within a shorter variable speed period, which could easily cause chaotic mixing in the mixing process. However, according to the experimental evidence, although the co-reverse rotational impeller in the stirred tank could cause dynamic perturbations, some amounts of perturbations had an insignificant effect on improving the mixing performance. Only when the perturbation was stronger than some value could the mixing performance be improved. When the variable speed period was too long, the relative velocity between impeller and fluid became small, which would result in a small velocity gradient in the flow field. As illustrated in Figs. 11b and 12b, the LLE of the chaotic motor was larger than that of the traditional motor, and the value of Ch/Cav at the higher part of the stirred vessel was higher than that of the traditional motor. As discussed earlier, numerous dynamic perturbations were imposed on the mixture system by the chaotic motor, the fluid turbulent degree increased, thereby the mixing performance was also enhanced [27,28]. However, when the constant speed period was too long, the number of the variable speed process decreased in a certain amount of time and the number of perturbation subsequently decreased. As shown in Fig. 10, the value of impeller agitation speed in constant speed process was the largest in the mixing process, and a large velocity gradient could be obtained in constant speed process. Hence, maintaining a short constant speed period was not sensible. As shown in Fig. 11b, the constant speed period of 15 s was suitable for the solid-liquid mixing process in this work. The LLE value of the double punched rigid-flexible impeller coupled with a chaotic motor was higher than that of the double punched rigid-flexible impeller with a traditional motor at the same agitation speed. Moreover, the solid particle dispersion of the double punched rigid-flexible impeller coupled with a chaotic motor was also more homogeneous than that of the double punched rigid-flexible impeller with a traditional motor. 3.5. Relative Standard Deviation (RSD) The relative standard deviation (RSD) of the axial solid concentration profile is widely used to quantify the effect of the stirred vessel and impeller geometry, solid particle properties, and impeller agitation speed on the homogeneity of the solid-liquid mixing system [8,40]. The value of RSD can be calculated as follows:
1 1 n RSD (Ch Cav )2 Cav n 1 h 1
0.5
(5)
The value of RSD decreases as the solid particles dispersion becomes highly homogeneous. When the value of RSD is down to zero, the mixing system achieves absolute homogeneity. As can be seen from Fig. 13, the double punched rigid-flexible impeller could dramatically enhance the homogeneous degree of the solid-liquid mixing system compared with the double rigid-flexible impeller and double rigid impeller. The double punched rigid-flexible impeller coupled with a chaotic motor could further improve the solid suspension quality of the solid-liquid mixing system on the basis of the double punched rigid-flexible impeller with a traditional motor at the same power consumption.
Fig. 13. RSD of the axial solid concentration profile at different Pv.
3.6. Comparative experiment
Fig. 14. Comparison of power number Np at same Re (Single liquid phase).
(a)
(b)
Fig. 15. Comparative study of LLE and Ch/Cav. (a) Comparison of LLE at same Pv, (b) Comparison of Ch/Cav at Pv= 0.5 kw·m-3.
As shown in Fig.14, the power numbers of the four cases (i.e. with a double rigid impeller, a double rigid-flexible impeller, a double punched rigid-flexible impeller, and a double punched rigid-flexible impeller coupled with a chaotic motor) were compared with each other at the same Re. It was observed that the power numbers of the double rigid-flexible impeller, double punched rigid-flexible impeller, and double punched rigid-flexible impeller coupled with a chaotic motor were larger than that of the double rigid impeller for single liquid phase. Therefore, it was necessary to compare the solid-liquid mixing performances of the four cases at the specific power consumption. As illustrated in Figs. 15 and 16, the mixing performances of the four cases were compared with each other at the same power consumption. As shown in Fig.15, the LLE value of double punched rigid-flexible impeller was larger than that of double rigid-flexible impeller and double rigid impeller at the same Pv, and double punched rigid-flexible impeller coupled with a chaotic motor could further enhance the LLE on the basis of double punched rigid-flexible impeller coupled with a traditional motor at the same Pv. Moreover, the solid-liquid homogeneous degree of double punched rigid-flexible impeller coupled with a chaotic motor was the best of the four cases at the same Pv. As seen in Fig. 16, the mixing processes of the four cases were presented with the help of photographs taken at the same power consumption and rotation time (i.e., 1 min). When the impeller power consumption was zero or low, a large heap of glass bead particles stacked at the bottom of the stirred tank, a few particles were dispersed into the liquid, and a clear liquid zone existed at the higher part of the vessel. With the increase of impeller power consumption, a large quantity of solid particles suspended and dispersed in the liquid phase, and the cloud height increased. The photographs of the four cases in Fig. 16 were in good agreement with the experimental results in Fig. 15, which indicated that the double punched rigid-flexible impeller and chaotic motor could effectively enhance the mixing performance and energy efficiency in the solid-liquid mixing process.
(a)
(b)
(c)
(d) Fig. 16. Comparison of mixing performance. (a) Double rigid impeller, (b) Double rigid-flexible impeller, (c) Double punched rigid-flexible impeller, (d) Double punched rigid-flexible impeller coupled with a chaotic motor.
4. Conclusions The double punched rigid-flexible impeller and chaotic motor were introduced as novel approaches in this wok to improve the mixing performance in the solid-liquid mixing process. The solid-liquid mixing performance was experimentally investigated using measurements of the largest Lyapunov exponent and axial local solid concentration profile in a stirred tank with a double rigid impeller, a double rigid-flexible impeller, a double punched rigid-flexible impeller, and a double punched rigid-flexible impeller coupled with a chaotic motor. The experimental results showed that the double punched rigid-flexible impeller and chaotic motor could enhance the LLE value of solid-liquid mixing system on the basis of the double punched rigid-flexible impeller and traditional motor under the same power consumption. The variation in solid homogeneous degree was in good agreement with that of LLE. It was found that the double punched rigid-flexible impeller coupled with a chaotic motor could effectively improve the mixing performance and energy efficiency compared with the double rigid impeller or double rigid-flexible impeller coupled with a traditional motor in the solid-liquid mixing process. A wide and long flexible connection piece were conductive to the solid-liquid mixing process. The aperture diameter of 8 mm and the aperture ratio of 12 % of the double punched rigid-flexible impeller and the variable speed period of 10 s and the constant speed period of 15 s of the chaotic motor were particularly suitable for the solid-liquid mixing process in this work. The mixing processes were presented with the help of photographs taken under the same operating conditions, and the images were in good agreement with the experimental results. Acknowledgements The study was supported by the National Natural Science Foundations of China (21576033, 21636004). References [1] A. Tagawa, N., Dohi; Y. Kawase, Dispersion of Floating Solid Particles in Aerated Stirred Tank Reactors: Minimum Impeller Speeds for Off-Surface and Ultimately Homogeneous Solid Suspension and Solids Concentration Profiles, Ind. Eng. Chem. Res. 45 (2006) 818-829. [2] T. Kumaresan, J. B. Joshi, Effect of Impeller Design on the Flow Pattern and Mixing in Stirred Tanks, Chem. Eng. Sci. 115 (2006) 173-193. [3] G. Baldi, R. Conti, E. Alaria, Complete Suspension of Particles in Mechanically Agitated Vessels, Chem. Eng. Sci. 33 (1978) 21-25. [4] R. W. Thring, M. F. Edwards, An Experimental Investigation into the Complete Suspension of Floating Solids in an Agitated Tank, Ind. Eng. Chem. Res. 29 (1990) 678-682. [5] N. Kuzmanic, E. M. Kessler, Continuous Sampling of Floating Solids Suspension from a Mixing Tank, Ind. Eng. Chem. Res. 36 (1997) 5015. [6] Y. Y. Bao, Z. G. Hao, Z. M. Gao, L. T. Shi, J. M. Smith, Suspension of Buoyant Particles in A Three Phase Stirred Tank, Chem. Eng. Sci. 60 (2005) 2283-2292. [7] N. Dohi, T. Takahashi, K. Minekawa, Y. Kawase, Power Consumption and Solid Suspension of
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Fig. 1. Schematic of the experimental apparatus. 1-Programmable Logic Controller; 2-Motor; 3-Decelerator; 4-Bracket; 5-Torque transducer; 6-Stirring shaft; 7-Baffle; 8-Rigid blades of impeller; 9-Silicone flexible connection piece; 10, 11-Data acquisition card; 12-Computer
(a)
(b)
(c)
Fig. 2. Structure diagram of the impellers. (a) Double rigid impeller, (b) Double rigid-flexible impeller, (c) Double punched rigid-flexible impeller.
Fig. 3. Effect of impeller types on LLE.
(a)
(b)
Fig. 4. Effect of impeller types and agitation speed on axial solid concentration profile. (a) Effect of impeller types (N = 300 rpm), (b) Effect of agitation speed (Double rigid-flexible impeller).
(a)
(b)
Fig. 5. Effect of flexible connection piece width and length on LLE. (a) Effect of flexible connection piece width (L = 56cm), (b) Effect of flexible connection piece length (W = 2cm).
(a)
(b)
Fig. 6. Effect of flexible connection piece width and length on axial solid concentration profile (N = 300 rpm). (a) Effect of flexible connection piece width (L =56cm), (b) Effect of flexible connection piece length (W = 2 cm).
(a)
(b)
Fig. 7. Effect of aperture size and aperture ratio on LLE (L = 56 cm, W = 2 cm). (a)Effect of aperture size (aperture ratio = 12 %), (b) Effect of aperture ratio (aperture diameter = 8 mm).
(a)
(b)
Fig. 8. Effect of aperture size and aperture ratio on axial solid concentration profile (N = 300 rpm, L = 56 cm, W = 2 cm). (a) Effect of aperture size (aperture ratio = 12 %), (b) Effect of aperture ratio (aperture diameter = 8 mm).
(a)
(b)
Fig. 9. Effect of aperture size and aperture ratio on Pv (N = 300 rpm, L = 56cm, W = 2 cm). (a) Effect of aperture size (aperture ratio = 12 %), (b) Effect of aperture ratio (aperture diameter = 8 mm).
Fig. 10. Rotation mode of the chaotic motor.
(a)
(b)
Fig. 11. Effect of variable speed period and constant speed period on LLE (L = 56cm, W = 2 cm, aperture diameter = 8 mm, aperture ratio = 12 %). (a) Effect of variable speed period (constant speed period = 15 s), (b) Effect of constant speed period (variable speed period = 10 s).
(a)
(b)
Fig. 12. Effect of variable speed period and constant speed period on axial solid concentration profile (N = 300 rpm, L = 56cm, W = 2 cm, aperture diameter = 8 mm, aperture ratio = 12 %). (a) Effect of variable speed period (constant speed period = 15 s), (b) Effect of constant speed period (variable speed period = 10 s).
Fig. 13. RSD of the axial solid concentration profile at different Pv.
Fig. 14. Comparison of power number Np at same Re (Single liquid phase).
(a)
(b)
Fig. 15. Comparative study of LLE and Ch/Cav. (a) Comparison of LLE at same Pv, (b) Comparison of Ch/Cav at Pv= 0.5 kw·m-3.
0 kw·m-3
0.1 kw·m-3
0.3 kw·m-3
0.5 kw·m-3
0.7 kw·m-3
0.9kw·m-3
(a)
0 kw·m-3
0.1 kw·m-3
0.3 kw·m-3
0.5 kw·m-3
0.7 kw·m-3
0.9kw·m-3
(b)
0 kw·m-3
0.1 kw·m-3
0.3 kw·m-3
0.5 kw·m-3
0.7 kw·m-3
0.9kw·m-3
0.5 kw·m-3
0.7 kw·m-3
0.9kw·m-3
(c)
0 kw·m-3
0.1 kw·m-3
0.3 kw·m-3 (d)
Fig. 16. Comparison of mixing performance. (a) Double rigid impeller, (b) Double rigid-flexible impeller, (c) Double punched rigid-flexible impeller, (d) Double punched rigid-flexible impeller coupled with a chaotic motor.