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Chaotic mixing created by object inserted in a vessel agitated by an impeller
chemical engineering research and design 8 7 ( 2 0 0 9 ) 386–390
Contents lists available at ScienceDirect
Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
Chaotic mixing created by object inserted in a vessel agitated by an impeller Koji Takahashi ∗ , Mitsunori Motoda Department of Chemistry and Chemical Engineering, Yamagata University, Yonezawa, 992-8510, Japan
a b s t r a c t In order to improve the mixing performance in an agitated vessel, a very unique method to insert an object into a vessel agitated by an impeller was proposed and an experimental investigation of mixing performance was conducted. The objects inserted move so as to destroy the isolated mixing regions above and below the impeller, which are usually observed for vessels stirred by ordinary small impellers under steady mixing conditions. As a result, it was demonstrated that the method proposed in this work has a significant enhancement of mixing efficiency at low Reynolds numbers. This method may bring a new innovational agitator system. © 2009 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Special chaotic mixing; Inserting object; Agitated vessel; Segregated regions; Mixing time; Laminar mixing
1.
Introduction
Laminar mixing is commonly encountered in many industries in processes involving physical and chemical changes. At low Reynolds numbers, the isolated well-mixed regions can be formed close to the impeller. Thus, very long mixing times and high-energy consumptions are required to achieve a total liquid homogenization. Increasing the rotational speed of the impeller can solve these problems, however, the liquid is submitted to high-shear rates, which can be inconvenient for shear-sensitive media. Lamberto et al. (1996) first demonstrated that the timedependent rotational speed could enhance the mixing in a stirred tank equipped with ordinary small impellers, but only qualitative results and conjectional explanations of experimental phenomena were provided. Recently, the several experimental studies have been undertaken to exhibit that the mixing performance can be remarkably improved by increasing the chaotic degree, in either temporal or spatial terms (Nomura et al., 1997; Yao et al., 1998; Hirata et al., 2006; Xiao and Takahashi, 2007). The spatial way can increase the chaotic degree by circumferential symmetry in a normal mixing equipment. The special measures in an agitated vessel concerned with the special chaotic mixing include baffles, offcenter mounting, and uncircumferentially symmetrical tank geometry. The baffles can enhance the mixing performance
∗
remarkably and have been widely used and the off-center mounting has also been investigated thoroughly (e.g., Karcs et al., 2005). The hydrodynamics performance of double planetary mixer (Tanguy et al., 1999) and a coaxial mixer (Rivera et al., 2006) were investigated numerically and experimentally with Newtonian and non-Newtonian fluids. The objective of the present work is to propose a new spatial chaotic mixing method, that is to eliminate the segregated regions, which are formed in an agitated vessel stirred by small impellers under steady mixing conditions, by putting objects into. The experimental investigation of mixing mechanism for this new system was carried out and compared with the results obtained by using time-dependent rotational speed, which is one of common temporal chaotic mixing.
2.
Experiment
The schematic diagram of experiment apparatus used in this work is shown in Fig. 1. The vessel was made of transparent acrylic resin, the diameter of which, T, was 0.240 m and four T/10 width baffles made of stainless steal were fitted on the vessel wall at 90◦ intervals. The impeller used was a Rushton disc turbine impeller. The diameter of impeller, D, was 0.120 m and thus an impeller to vessel diameter ratio of the impeller was 0.5.
Corresponding author. Tel.: +81 238 26 3156; fax: +81 238 26 3156. E-mail address: [email protected] (K. Takahashi). Received 26 October 2008; Received in revised form 18 December 2008; Accepted 5 January 2009 0263-8762/$ – see front matter © 2009 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2009.01.003
chemical engineering research and design 8 7 ( 2 0 0 9 ) 386–390
Nomenclature C D dv H N T tm Re w
off bottom clearance of impeller (m) diameter of impeller (m) diameter of spherical capsule or sphere equivalent diameter (m) liquid height without aeration (m) impeller speed (s−1 ) tank diameter (m) mixing time (s) Reynolds number baffle width (m)
The liquids used were aqueous solutions of corn syrup of viscosity 2.9–3.2 Pas and density 1394 kg/m3 . For convenience, the Reynolds number was usually kept 10. The objects mainly adopted were the spherical capsules of polypropylene, the diameters of which were 0.01, 0.02 and 0.03 m, which is shown in Fig. 2 with a Rushton disc turbine. To adjust the density of the obstacle, the solution of corn syrup and, if circumstances require, Barium sulfate were injected inside of the capsules. The other objects shown in Fig. 3 were also used to investigate the effect of shape on mixing performance. To investigate the mixing pattern and mixing time, the decoloration method by using a reaction between iodine and sodium thiosulfate was adopted. First, aqueous solution of corn syrup mixed with iodine was placed in the vessel. After
387
the air bubbles are thoroughly removed from the vessel, aqueous solution of corn syrup mixed with thiosulfate was poured into the vessel slowly from the same port in every experiment to eliminate the effect of the location of the addition of the material on mixing time. The optimum equivalent ratio, 1.4-fold sodium thiosulfate to iodine, suggested by several investigators (e.g., Takahashi et al., 1985) was adopted in all experiments. Meanwhile the system was operated. Photographs of the mixing patterns were taken with a camera for later analysis.
3.
Results and discussion
3.1.
Movement of spherical capsule
As a preliminary experiment, the movement of spherical capsules was observed. As well known, in a vessel agitated by Rushton turbine, the strong radial flow created by the impeller moves to vessel wall, separates to upward and downward at vessel wall, turned to inside near bottom and free surface and then return to the impeller. Therefore, the upper and lower circulation loops were formed. The spherical capsule could not move across the interface between these two circulation zones. Therefore, it would be imagine that the two spherical capsules should be inserted into above and below the impeller to remove those two isolated regions. To adjust the density of two spherical capsules to be 8% heavier and lighter than that of the liquid and then put into the vessel. Obviously, heavier and lighter capsules move to
Fig. 1 – Experimental setup.
Fig. 2 – Spherical capsule used.
388
chemical engineering research and design 8 7 ( 2 0 0 9 ) 386–390
Fig. 3 – Other objects used.
Fig. 4 – Positions of spherical capsule reached (a) and doughnuts rings formed without obstacles (b). bottom and free surface, respectively, at last and then the agitation was started. Both capsules moved according to the flow created by the impeller and at last reached to the region where the doughnut rings were formed (see Fig. 4). This is because the circulation loops were formed to create the doughnut rings in the center and thus the velocity of surface of doughnut ring was smaller than those in the area surrounding the doughnut rings (e.g., Spragg et al., 1985). Thus the spherical capsule rotate so as to move to the doughnut rings. Therefore, it will be concluded that the capsules may contribute to disappear the doughnut rings.
3.2.
shows that the doughnut ring when the spherical capsules does not reach to and (b) and (c) the fluid elements folded by the movement of spherical capsules, which are very similar to those reported for time-periodic fluctuation of RPM (Lamberto et al., 1996; Yao et al., 1998). It is clearly seen from Fig. 6(b) and (c), the solution including more iodine solution moved out of the doughnut ring and a new stream of fluid from the outer system, rich in sodium thiosulphate solution, penetrated and entrained into the doughnut ring. As a result, the spherical capsule disturbs the formation of doughnuts rings and exceeds the exchange flow between the segregated regions and the rest of the system. With an increase in the
Mixing time by inserting spherical capsules
The relation between dimensionless mixing time Ntm , where N is the rotational speed of impeller and tm the practical decolorization time, and the diameter of spherical capsules is shown in Fig. 5, where the dimensionless mixing time measured without any objects was 5.5 × 105 . As can be seen from the figure, by inserting spherical capsules, mixing time decreases drastically and the dimensionless mixing time decreases linearly with an increase in diameter of spherical capsules. The mechanism that the spherical capsules destroy the doughnut rings was investigated as follows. Without spherical capsules, the mixing was carried out in an agitated vessel and two toroidal regions, so-called the doughnut rings, were clearly formed both above and below the impeller. These regions remain segregated from the rest of the system and are not mixed by convective flow mechanisms. Then, the spherical capsule were put into the vessel and the process of disappearing the doughnut rings was observed. Fig. 6(a)
Fig. 5 – Mixing time measured.
chemical engineering research and design 8 7 ( 2 0 0 9 ) 386–390
389
Fig. 6 – Photographs of doughnut ring (a) and chaotic mixing pattern by inserting spherical capsule (b) and (c). volume of spherical capsule, the volume of exchange flow increases.
3.3.
Mixing time by inserting object of different shapes
The measured mixing times obtained for the objects of different shapes are also shown in Fig. 5, where the abscissa is the volume mean diameter. Any objects of different shapes adopted in this work could disturb the formation of the doughnuts rings. It is clearly seen from the figure, mixing times for objects of small sphericities, which are disk and ring, are smaller than those for spherical capsules but that for oval similar to sphere is almost the same with that of sphere at the same volume mean diameter because the movement of such objects may be more complicated. This assumption may supported by the fact that the mixing times for disk and ring which have completely different shapes from sphere are small. The power consumption was also measured for all experimental conditions with and without any kinds of objects when the Reynolds number was kept 10 and it is concluded that there is no significant difference in power consumption.
3.4. Comparison with results obtained for co-reverse periodic rotation of impeller In this work, the mixing times by inserting spherical capsules of 0.03 m diameter was compared with those for co-reverse
periodic rotation of impeller, which is one of the common temporal chaotic mixing at different Reynolds numbers. The period of co-reverse was kept 20 s and, to calculate the Reynolds number, the absolute value of rotational speed was used. The result is shown in Fig. 7. In this figure, the marked X indicates the critical Reynolds number when the spherical capsules were impacted on the impeller and thus the mixing times were not measured at the Reynolds number larger than the critical Reynolds number. The mixing times obtained by inserting spherical capsules are comparable to those for coreverse periodic rotation of impeller, which may indicate the spatial mixing proposed in this work is chaotic. The variation of data for co-reverse periodic rotation of impeller with Re is higher than exhibited by data for inserting spherical capsule. Therefore, at low Reynolds numbers less than 10, the mixing by inserting spherical capsules has advantage.
4.
Conclusions
A new spatial chaotic mixing method was proposed by inserting the objects into a vessel agitated by an impeller. The experimental results showed that mixing time can be significantly reduced, especially for larger objects. The comparison of this spatial mixing method with the temporal time dependent one showed that the method proposed in this work has an advantage at low Reynolds numbers. The method proposed in this work may diminish the isolated regions formed in an agitated vessel because the object inserted moves to the place where the velocity was the smallest.
References
Fig. 7 – Mixing time comparison of inserting spherical capsule and co-reverse periodic rotation of impeller.
Hirata, Y., Dote, T., Yoshoka, T., Komoda, Y. and Inoue, Y., 2006, Performance of chaotic mixing caused by reciprocating a disk in a cylinderical vessel, In 12th Euro. Conf. on Mixing Bologna, Italy, , pp. 655–662. Karcs, J., Cudak, M. and Szoplik, J., 2005, Stirring of a liquid in a stirred tank with an eccentrically located impeller. Chem. Eng. Sci., 60: 2369–2380. Lamberto, D.J., Muzzio, F.J., Swanson, P.D. and Tonkovich, A.l., 1996, Using time-dependent RPM to enhance mixing in stirred vessels. Chem. Eng. Sci., 51: 733–741. Nomura, T., Uchida, T. and Takahashi, K., 1997, Enhancement of mixing by unsteady agitation of an impeller in an agitated vessel. J. Chem. Eng. Jpn., 30: 875–879. Rivera, C., Foucault, S., Heniche, M., Espinosa-Solares, T. and Tanguy, P.A., 2006, Mixing analysis in a coaxial mixer. Chem. Eng. Sci., 61: 2895–2907. Spragg, A.J.P., Maskell, S.J. and Patrick, M.A., 1985, Mixing and flow in a spinning disc blender, In Proc. 5th Euro. Conf. on Mixing Wurzburg, Germany, , pp. 507–514.
390
chemical engineering research and design 8 7 ( 2 0 0 9 ) 386–390
Takahashi, K., Takahata, Y., Yokota, T. and Konno, H., 1985, Mixing of two miscible highly viscous Newtonian liquids in a helical ribbon agitator. J. Chem. Eng. Jpn., 18: 159–162. Tanguy, P.A., Thibault, F., Dubois, C. and Ait-Kadi, A., 1999, Mixing hydrodynamics in a double planetary mixer. Chem. Eng. Res. Des., 77: 318–324.
Xiao, H. and Takahashi, K., 2007, Mixing characteristics in the horizontal non-baffled stirred vessel in low viscosity fluid. J. Chem. Eng. Jpn., 40: 679–683. Yao, W.G., Sato, H., Takahashi, K. and Koyama, K., 1998, Mixing performance experiments in impeller stirred tanks subjected to unsteady rotational speeds. Chem. Eng. Sci., 53: 3031–3040.
Contents lists available at ScienceDirect
Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
Chaotic mixing created by object inserted in a vessel agitated by an impeller Koji Takahashi ∗ , Mitsunori Motoda Department of Chemistry and Chemical Engineering, Yamagata University, Yonezawa, 992-8510, Japan
a b s t r a c t In order to improve the mixing performance in an agitated vessel, a very unique method to insert an object into a vessel agitated by an impeller was proposed and an experimental investigation of mixing performance was conducted. The objects inserted move so as to destroy the isolated mixing regions above and below the impeller, which are usually observed for vessels stirred by ordinary small impellers under steady mixing conditions. As a result, it was demonstrated that the method proposed in this work has a significant enhancement of mixing efficiency at low Reynolds numbers. This method may bring a new innovational agitator system. © 2009 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Special chaotic mixing; Inserting object; Agitated vessel; Segregated regions; Mixing time; Laminar mixing
1.
Introduction
Laminar mixing is commonly encountered in many industries in processes involving physical and chemical changes. At low Reynolds numbers, the isolated well-mixed regions can be formed close to the impeller. Thus, very long mixing times and high-energy consumptions are required to achieve a total liquid homogenization. Increasing the rotational speed of the impeller can solve these problems, however, the liquid is submitted to high-shear rates, which can be inconvenient for shear-sensitive media. Lamberto et al. (1996) first demonstrated that the timedependent rotational speed could enhance the mixing in a stirred tank equipped with ordinary small impellers, but only qualitative results and conjectional explanations of experimental phenomena were provided. Recently, the several experimental studies have been undertaken to exhibit that the mixing performance can be remarkably improved by increasing the chaotic degree, in either temporal or spatial terms (Nomura et al., 1997; Yao et al., 1998; Hirata et al., 2006; Xiao and Takahashi, 2007). The spatial way can increase the chaotic degree by circumferential symmetry in a normal mixing equipment. The special measures in an agitated vessel concerned with the special chaotic mixing include baffles, offcenter mounting, and uncircumferentially symmetrical tank geometry. The baffles can enhance the mixing performance
∗
remarkably and have been widely used and the off-center mounting has also been investigated thoroughly (e.g., Karcs et al., 2005). The hydrodynamics performance of double planetary mixer (Tanguy et al., 1999) and a coaxial mixer (Rivera et al., 2006) were investigated numerically and experimentally with Newtonian and non-Newtonian fluids. The objective of the present work is to propose a new spatial chaotic mixing method, that is to eliminate the segregated regions, which are formed in an agitated vessel stirred by small impellers under steady mixing conditions, by putting objects into. The experimental investigation of mixing mechanism for this new system was carried out and compared with the results obtained by using time-dependent rotational speed, which is one of common temporal chaotic mixing.
2.
Experiment
The schematic diagram of experiment apparatus used in this work is shown in Fig. 1. The vessel was made of transparent acrylic resin, the diameter of which, T, was 0.240 m and four T/10 width baffles made of stainless steal were fitted on the vessel wall at 90◦ intervals. The impeller used was a Rushton disc turbine impeller. The diameter of impeller, D, was 0.120 m and thus an impeller to vessel diameter ratio of the impeller was 0.5.
Corresponding author. Tel.: +81 238 26 3156; fax: +81 238 26 3156. E-mail address: [email protected] (K. Takahashi). Received 26 October 2008; Received in revised form 18 December 2008; Accepted 5 January 2009 0263-8762/$ – see front matter © 2009 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2009.01.003
chemical engineering research and design 8 7 ( 2 0 0 9 ) 386–390
Nomenclature C D dv H N T tm Re w
off bottom clearance of impeller (m) diameter of impeller (m) diameter of spherical capsule or sphere equivalent diameter (m) liquid height without aeration (m) impeller speed (s−1 ) tank diameter (m) mixing time (s) Reynolds number baffle width (m)
The liquids used were aqueous solutions of corn syrup of viscosity 2.9–3.2 Pas and density 1394 kg/m3 . For convenience, the Reynolds number was usually kept 10. The objects mainly adopted were the spherical capsules of polypropylene, the diameters of which were 0.01, 0.02 and 0.03 m, which is shown in Fig. 2 with a Rushton disc turbine. To adjust the density of the obstacle, the solution of corn syrup and, if circumstances require, Barium sulfate were injected inside of the capsules. The other objects shown in Fig. 3 were also used to investigate the effect of shape on mixing performance. To investigate the mixing pattern and mixing time, the decoloration method by using a reaction between iodine and sodium thiosulfate was adopted. First, aqueous solution of corn syrup mixed with iodine was placed in the vessel. After
387
the air bubbles are thoroughly removed from the vessel, aqueous solution of corn syrup mixed with thiosulfate was poured into the vessel slowly from the same port in every experiment to eliminate the effect of the location of the addition of the material on mixing time. The optimum equivalent ratio, 1.4-fold sodium thiosulfate to iodine, suggested by several investigators (e.g., Takahashi et al., 1985) was adopted in all experiments. Meanwhile the system was operated. Photographs of the mixing patterns were taken with a camera for later analysis.
3.
Results and discussion
3.1.
Movement of spherical capsule
As a preliminary experiment, the movement of spherical capsules was observed. As well known, in a vessel agitated by Rushton turbine, the strong radial flow created by the impeller moves to vessel wall, separates to upward and downward at vessel wall, turned to inside near bottom and free surface and then return to the impeller. Therefore, the upper and lower circulation loops were formed. The spherical capsule could not move across the interface between these two circulation zones. Therefore, it would be imagine that the two spherical capsules should be inserted into above and below the impeller to remove those two isolated regions. To adjust the density of two spherical capsules to be 8% heavier and lighter than that of the liquid and then put into the vessel. Obviously, heavier and lighter capsules move to
Fig. 1 – Experimental setup.
Fig. 2 – Spherical capsule used.
388
chemical engineering research and design 8 7 ( 2 0 0 9 ) 386–390
Fig. 3 – Other objects used.
Fig. 4 – Positions of spherical capsule reached (a) and doughnuts rings formed without obstacles (b). bottom and free surface, respectively, at last and then the agitation was started. Both capsules moved according to the flow created by the impeller and at last reached to the region where the doughnut rings were formed (see Fig. 4). This is because the circulation loops were formed to create the doughnut rings in the center and thus the velocity of surface of doughnut ring was smaller than those in the area surrounding the doughnut rings (e.g., Spragg et al., 1985). Thus the spherical capsule rotate so as to move to the doughnut rings. Therefore, it will be concluded that the capsules may contribute to disappear the doughnut rings.
3.2.
shows that the doughnut ring when the spherical capsules does not reach to and (b) and (c) the fluid elements folded by the movement of spherical capsules, which are very similar to those reported for time-periodic fluctuation of RPM (Lamberto et al., 1996; Yao et al., 1998). It is clearly seen from Fig. 6(b) and (c), the solution including more iodine solution moved out of the doughnut ring and a new stream of fluid from the outer system, rich in sodium thiosulphate solution, penetrated and entrained into the doughnut ring. As a result, the spherical capsule disturbs the formation of doughnuts rings and exceeds the exchange flow between the segregated regions and the rest of the system. With an increase in the
Mixing time by inserting spherical capsules
The relation between dimensionless mixing time Ntm , where N is the rotational speed of impeller and tm the practical decolorization time, and the diameter of spherical capsules is shown in Fig. 5, where the dimensionless mixing time measured without any objects was 5.5 × 105 . As can be seen from the figure, by inserting spherical capsules, mixing time decreases drastically and the dimensionless mixing time decreases linearly with an increase in diameter of spherical capsules. The mechanism that the spherical capsules destroy the doughnut rings was investigated as follows. Without spherical capsules, the mixing was carried out in an agitated vessel and two toroidal regions, so-called the doughnut rings, were clearly formed both above and below the impeller. These regions remain segregated from the rest of the system and are not mixed by convective flow mechanisms. Then, the spherical capsule were put into the vessel and the process of disappearing the doughnut rings was observed. Fig. 6(a)
Fig. 5 – Mixing time measured.
chemical engineering research and design 8 7 ( 2 0 0 9 ) 386–390
389
Fig. 6 – Photographs of doughnut ring (a) and chaotic mixing pattern by inserting spherical capsule (b) and (c). volume of spherical capsule, the volume of exchange flow increases.
3.3.
Mixing time by inserting object of different shapes
The measured mixing times obtained for the objects of different shapes are also shown in Fig. 5, where the abscissa is the volume mean diameter. Any objects of different shapes adopted in this work could disturb the formation of the doughnuts rings. It is clearly seen from the figure, mixing times for objects of small sphericities, which are disk and ring, are smaller than those for spherical capsules but that for oval similar to sphere is almost the same with that of sphere at the same volume mean diameter because the movement of such objects may be more complicated. This assumption may supported by the fact that the mixing times for disk and ring which have completely different shapes from sphere are small. The power consumption was also measured for all experimental conditions with and without any kinds of objects when the Reynolds number was kept 10 and it is concluded that there is no significant difference in power consumption.
3.4. Comparison with results obtained for co-reverse periodic rotation of impeller In this work, the mixing times by inserting spherical capsules of 0.03 m diameter was compared with those for co-reverse
periodic rotation of impeller, which is one of the common temporal chaotic mixing at different Reynolds numbers. The period of co-reverse was kept 20 s and, to calculate the Reynolds number, the absolute value of rotational speed was used. The result is shown in Fig. 7. In this figure, the marked X indicates the critical Reynolds number when the spherical capsules were impacted on the impeller and thus the mixing times were not measured at the Reynolds number larger than the critical Reynolds number. The mixing times obtained by inserting spherical capsules are comparable to those for coreverse periodic rotation of impeller, which may indicate the spatial mixing proposed in this work is chaotic. The variation of data for co-reverse periodic rotation of impeller with Re is higher than exhibited by data for inserting spherical capsule. Therefore, at low Reynolds numbers less than 10, the mixing by inserting spherical capsules has advantage.
4.
Conclusions
A new spatial chaotic mixing method was proposed by inserting the objects into a vessel agitated by an impeller. The experimental results showed that mixing time can be significantly reduced, especially for larger objects. The comparison of this spatial mixing method with the temporal time dependent one showed that the method proposed in this work has an advantage at low Reynolds numbers. The method proposed in this work may diminish the isolated regions formed in an agitated vessel because the object inserted moves to the place where the velocity was the smallest.
References
Fig. 7 – Mixing time comparison of inserting spherical capsule and co-reverse periodic rotation of impeller.
Hirata, Y., Dote, T., Yoshoka, T., Komoda, Y. and Inoue, Y., 2006, Performance of chaotic mixing caused by reciprocating a disk in a cylinderical vessel, In 12th Euro. Conf. on Mixing Bologna, Italy, , pp. 655–662. Karcs, J., Cudak, M. and Szoplik, J., 2005, Stirring of a liquid in a stirred tank with an eccentrically located impeller. Chem. Eng. Sci., 60: 2369–2380. Lamberto, D.J., Muzzio, F.J., Swanson, P.D. and Tonkovich, A.l., 1996, Using time-dependent RPM to enhance mixing in stirred vessels. Chem. Eng. Sci., 51: 733–741. Nomura, T., Uchida, T. and Takahashi, K., 1997, Enhancement of mixing by unsteady agitation of an impeller in an agitated vessel. J. Chem. Eng. Jpn., 30: 875–879. Rivera, C., Foucault, S., Heniche, M., Espinosa-Solares, T. and Tanguy, P.A., 2006, Mixing analysis in a coaxial mixer. Chem. Eng. Sci., 61: 2895–2907. Spragg, A.J.P., Maskell, S.J. and Patrick, M.A., 1985, Mixing and flow in a spinning disc blender, In Proc. 5th Euro. Conf. on Mixing Wurzburg, Germany, , pp. 507–514.
390
chemical engineering research and design 8 7 ( 2 0 0 9 ) 386–390
Takahashi, K., Takahata, Y., Yokota, T. and Konno, H., 1985, Mixing of two miscible highly viscous Newtonian liquids in a helical ribbon agitator. J. Chem. Eng. Jpn., 18: 159–162. Tanguy, P.A., Thibault, F., Dubois, C. and Ait-Kadi, A., 1999, Mixing hydrodynamics in a double planetary mixer. Chem. Eng. Res. Des., 77: 318–324.
Xiao, H. and Takahashi, K., 2007, Mixing characteristics in the horizontal non-baffled stirred vessel in low viscosity fluid. J. Chem. Eng. Jpn., 40: 679–683. Yao, W.G., Sato, H., Takahashi, K. and Koyama, K., 1998, Mixing performance experiments in impeller stirred tanks subjected to unsteady rotational speeds. Chem. Eng. Sci., 53: 3031–3040.