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Helical ribbon impellers are widely used in chemical and process industries for the mixing of pseudoplastic fluids of high viscosity. The design of such impellers is based on an assumed linear relation between shear rate and the rotation speed of the impeller. A number of computational fluid dynamics (CFD) simulations of the flow field have been carried to verify this hypothesis. It is shown that while the shear rate varies greatly within the mixing vessel, there does exist a linear relationship between the impeller speed and the local shear rate near the tip of the impeller. The proportionality constant Ks associated with this linear relation is found to be dependent on the geometric parameters of the system, but is largely independent of the flow behavior index. Based on these results, a new correlation, applicable to both Newtonian and power-law fluids for power consumption, is proposed.

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... For viscoelastic liquids the role of vicoelasticity appears to be ambigious [5,14,15]. As far as I am aware, no prior results about local shear rate of helical impeller on the mixing of viscoelastic liquids based on LDA velocity measurements are available in the literature [1][2][3]9,20]. The originality and novelty of this work is to investigate local shear rate values of helical ribbon impeller and correlate them with viscoelastic liquid properties and flow regime in the mixing tank. The objective of this work is two folds: (i) to obtain the local shear rate of helical ribbon impeller in mixing viscoelastic liquids directly from LDA velocity measurements in the transition region, (ii) to investigate the possibility of applying an appropriate correlation between local shear rate and impeller rotational speed. ...

... In other words, the Metzner-Otto coefficient, k s , in this work is directly calculated from local shear rate based on LDA velocity measurements with high accuracy. However, the Metzner-Otto coefficient, k s , in the literature may be obtained indirectly such as power consumption or electrochemical methods [14,16,20]. Table 1 shows that k s which was calculated directly from local shear rate in this work was consistent with literature. ...

... Thus the use of the improved correlation, eqn (6), instead of the Metzner and Otto relationship, eqn (1), could not lead to erroneous results and inadequate designs and this is in agreement with literature [4,5,7]. Values of b' in Z1-Z5 region can be taken as the local value at the Hall et al. [35] Rieger et al. [36] Nagata [37] Yap et al. [38] Yap et al. [38] Shamlou et al. [40] Kuriyama et al. [39] Carreau et al. [7] Cheng et al. [4] Cheng et al. [43] Brito-De la Fuente et al. [44] Brito-De la Fuente et al. [30] Brito-De la Fuente et al. [34] Shekhar et al. [20] Delaplace et al. [9] Montante et al. [42] The work (outer impeller blade tip) vicinity of inner and outer impeller tip for using in eqn (6). Variations of b' in the improved correlation, eqn (6), for different concentrations of PAA the solutions are shown in Figure 5. ...

O ne of the basic problems in the mixing of non-Newtonian fluids and especially diluted polymer solutions is the determination of the prevailing shear rates dur-ing the mixing process. The significant method of Metzner and Otto for calcula-tion of the effective shear rate is limited to the laminar region and it is not valid in the transition region. In this article, the local shear rate and the Metzner-Otto method for helical ribbon impeller have been studied using laser Doppler anemometry (LDA) for viscoelastic liquids. It is also shown that the variation of the local shear rate against the impeller speed is better correlated by a power equation, i.e., γ = k .N bín the transition region, i.e., 70 < Re < 6700. In addition, a correlation between the improved coefficient, k, and the elasticity number of the viscoelastic liquid is given that can be helpful in designing the mixing of viscoelastic as well as inelastic non-Newtonian fluids by means of relating the rheological properties to the kinematical and dynamical parameters of the mixing process.

... where K s is independent of the fluid rheological properties and geometric parameters, and decreases rapidly with distance from the impeller (Metzner and Otto, 1957;Paul et al., 2004;Torrez and André, 1998). The validity of Eq. (2) has been shown and extended to other impellers (Biardi et al., 1976;Carreau et al., 1993;Hiraoka et al., 1979;Metzner and Otto, 1957;Shekhar and Jayanti, 2003;Torrez and André, 1998;Wu et al., 2006). Rieger and Novák (1974) proposed an alternative scheme to that of Metzner and Ottó s, which has been applied numerically to several impellers (Bertrand et al., 1996;Luan et al., 2014;Tanguy et al., 1996) but the latter is more commonly used. ...

... For a double helical ribbon impeller, for example, _ c av is calculated by using the areaweighted average or the apparent viscosity around the impeller from CFD simulations and Eq. (1) (Zhang et al., 2008), and from the maximal shear rate at each rotational speed evaluated radially (Shekhar and Jayanti, 2003). For axial and radial flow impellers, it is common practice to calculate _ c av from the flow rate passing through a plane within the impeller diameter periphery (Wu et al., 2006). ...

The best-known paper on non-Newtonian fluids in mixing systems is the Metzner and Otto's work. A central idea of their contribution is based on the assumption that in laminar flow there exists an average shear rate (γ̇av) around the impeller (whose exact location and geometric shape is not clearly specified), and that it is proportional to the impeller speed (N), i.e., γ̇av=KsN, where Ks is the Metzner-Otto constant. In this work, an in-depth investigation of this assumption is carried out by using computational fluid dynamics (CFD) to calculate the three-dimensional flow induced by a Rushton turbine (RT). It was found that the volume swept by the blades is the region where Ks may be computed explicitly as Ks=γ̇av/N directly from non-Newtonian flow simulations. The obtained Ks values in this region were found in good agreement with reported data. Furthermore, power number measurements and data from literature were used to validate the simulations. The CFD method developed in this study can be used to readily and reliably evaluate Ks of industrial mixing impellers without resorting to power data of Newtonian fluids.

... Though LDA is being increasingly applied to stirred vessels, no LDA velocity measurements have been reported about viscoelastic fluids with helical ribbon impellers [16,17] . Most studies are concerned with LDA velocity measurements of Rushton turbines for Newtonian and non–Newtonian fluids18192021. In this paper, the variations of local velocity components on a horizontal plane passing nearly through the middle height of the helical ribbon impeller via LDA are measured. Efforts have been made to establish a relationship between the helical ribbon impeller velocity and the elasticity of polyacrylamide (PAA) solutions through dimensionless groups such as the elasticity and Weissenberg num- bers. ...

... Indeed, the results reveal that no considerable radial velocities could be measured . These results are in agreement with the findings of Cheng et al. [11] and Carreau et al. [14] and are different from those of Shekhar et al. [21]. ...

V elocity profiles are helpful for the confident design of mixing tanks in the transi-tion region. In this article, velocity profiles for helical ribbon impeller have been studied using laser Doppler anemometry for viscoelastic liquids with rheological properties typical of those found in polymer processes. Local tangential and radial velocities were measured at radial positions on a horizontal plane passing through the middle of the helical ribbon impeller. New correlation is suggested for dimensionless local tangential velocity profiles in the transition region, i.e. 70<N Re <6700. Experimental results show that the magnitudes of dimensionless radial velocities are much less than dimensionless mean tangential velocities such that no considerable radial flow could be detected.

... Therefore, pseudoplasticity become the basic reason of determining some mixing parameters such as impeller geometry, numbers of impellers, mixing time, impeller speed, type and maximum power of mixer. The optimation of those parameters leads to a better mixing performance [13], [14], [15], [16], [17] . ...

The modification of the percentage of aluminium is necessary to obtain certain specific impulse. But, it affects the pseudoplasticity of propellant in elapsed time that is important in casting. Therefore, this research attempts to investigate the pseudoplasticity of propellant slurry with varied aluminium contents and as time elapsed, the range of percentage of aluminium and time that allows propellant slurry to be well processed. The methods include measuring the viscosity of propellant slurries that contain 6, 8, 10, 12, 14, 16 and 18% of aluminium at varied shear rates until 40 minutes after mixing by using Brookfield viscometer. The graphs of viscosity versus shear rate were made to determine pseudoplasticity index. After that, the graph volume fraction versus pseudoplasticity index were made to be investigated. It is concluded that the more aluminium contents, the slurries with 6 to 12% aluminium contents exhibit more pseudoplastic behaviour, but the slurries with 12 to 16% aluminium exhibit less pseudoplastic. While, slurry of 18% aluminium exhibit high pseudoplasticity. In the correlation with the time, the slurry compositions of 6, 8, 14, 16% aluminium become more pseudoplastic as time elapsed. While, for compositions of 10, 12 and 18% aluminium, the trend becomes contrary. Based on the pseudoplasticity index, propellant slurries that contain 10 and 14% of aluminium are suitable for pressure casting. While for slurries with 6, 8 and 16% of aluminium are also suitable for vacuum casting. All of those suitability are possesed until 40 minutes after mixing. While, the composition of slurries that contain 12 and 18% of aluminium need to be modified to enhanced its castability.

... These rheological properties can lead to many operational problems such as low heat transfer, gel formation, fouling, and higher power consumption and mixing time [52,3,39,22,23]. To mitigate these problems, researchers have investigated the performances of various stirred systems such as a single impeller and a combination of different impellers in the agitation of non-Newtonian fluids [7,26,57,20,21,50,60,4,61,9,47,1,29,[31][32][33][34]13,19,25,8,55,36,63,43]. The findings of these studies have demonstrated that the abovementioned agitated systems are not efficient for the mixing of highly viscous non-Newtonian fluids. ...

The performance of different configurations of coaxial mixers composed of a wall scraping anchor impeller in combination with two different or identical central high speed impellers in the agitation of the shear-thinning fluids with yield stress (e.g. xanthan gum solution) was investigated. These multi-impeller mixers are more compact for the larger scale of mixing operations. The coaxial mixers employed in this study were the Scaba-Scaba-anchor, Scaba-Rushton-anchor, Rushton-Scaba- anchor, Scaba-pitched blade-anchor, and pitched blade-Scaba-anchor. The quality of mixing achieved by these five coaxial mixers was assessed for the Reynolds numbers in the range of the laminar to the transitional regime in the co-rotating mode through computational fluid dynamic (CFD) and electrical resistance tomography (ERT) techniques. A new correlation was introduced for these complex configurations of the coaxial mixers by incorporating the Metzner-Otto constants (Ks) of the different types of the central impellers into the Reynolds number. The experimental and CFD data were employed to evaluate the mixing intensification attained by these five coaxial mixers in regard to the power consumption, mixing time, velocity profiles, shear strain rate profiles, flow number, power number, and pumping effectiveness. The analysis of the collected data indicated that the intensification of mixing of the highly viscous non-Newtonian fluids achieved by the Scaba-pitched blade-anchor coaxial mixer was the highest among the mixing systems explored in this study.

... where is a constant value determined experimentally and is the impeller's rotational speed. Ayazi Shamlou and Edwards [19] proposed a correlation to estimate that was verified by Shekhar and Jayanti [20]: ...

When the mass concentration exceeds 7%, pulp suspensions stop flowing and act like a solid. To investigate the fluidization characteristics of medium-high-consistency pulp suspensions and achieve pulp fluidization and pumping, experiments were carried out with waste tissue pulp and unbleached kraft pulp. The objectives of this paper were to study the rheology of medium-high-consistency pulp and to determine accurate parameters for the physical Herschel-Bulkley model. To validate this model, computational fluid dynamics (CFD) results were compared to experimental data. The simulation values were very similar and were in agreement with experimental results.

... Rheological properties are an indication of product's structure and quality therefore reliable data are needed, in particular for heterogeneous systems such as chocolate masses. An alternative for getting better rheological data for complex systems such as chocolate mass is the mixing rheometry, where impellers such as the helical ribbon-type agitator (HR) have been used to measure the response from a fluid under deformation conditions similar to those found in actual flow processes [9][10][11][12] . Using a helical ribbon impeller as a measuring device accurate power prediction data and thus process viscosities for highly shear thinning fluids and concentrated suspensions were obtained by Brito et al. 13) . ...

The evolution of rheological properties of dark chocolate mass during conching was studied using a mixing rheometry approach. A helical ribbon impeller fitted to a rheometer was used to estimate viscous properties as a function of power process parameters. Torque and rotational velocity data were transformed into power consumption curves and therefore into process viscosities. In addition, scanning electron microscopy (SEM) was used to assess structural changes in the mass. Conching was conducted at different temperatures (40, 60, 80°C) and rotor speeds (0.66, 1 s-1) in order to find the best conditions for producing Mexican dark chocolate. Mass samples showed a strong shear thinning behaviour which increased with conching time under all conching conditions tested. Conching at 60 °C instead of 40 °C produced less viscous and more pseudoplastic masses improving flow properties of dark chocolate. Increasing speed from 0.66 to 1 s-1 did not affect shear thinning behaviour but contributed to decrease the consistency index of the mass during the first half of conching. SEM showed the progress of de-agglomeration of sugar and covering of particles with cocoa fat. Micrographs revealed that chocolate mass structure became smoother and more homogeneous as conching progressed under all analyzed conditions. Increasing temperature and speed reduced conching time and increased smoothness in the final product.

... It was found that the corotating mode is more efficient than the counterrotating mode in terms of energy consumption, pumping rate and mixing time. The finite-volume software FLUENT was employed to analyze the mixing of pseudoplastic fluids with a helical ribbon impeller by both Ihejirika and Ein-Mozaffari (2007) and Shekhar and Jayanti (2003). The multiframe of reference is used to cope with the rotation of the impeller. ...

The main concern of this study is to investigate the flow mixing generated by helical ribbon blade impellers and to show that with the help of CFD the performance of the mixing system can be significantly improved by optimizing the geometric configuration of the impeller. To fulfill this objective, a numerical model is developed to solve the Navier-Stokes equations for the flow field. However, difficulties arise due to the rotation of the impeller in the vessel. In order to ease the problem, the velocity field is assumed to be in a quasi-steady state and the multiframe of reference is adopted to tackle the rotation of the impeller. For discretization the fully conservative finite volume method, together with unstructured grid technology, is incorporated. It is shown that the flow in the mixer can be regarded as a flow in an open channel with a wall moving at an angle with respect to the channel. The influences of the blade pitch, the blade width, and the clearance gap between the blade and the surrounding wall are examined. The mechanism to cause these effects is delineated in detail. It is demonstrated that after optimization of the blade geometry, the circulating flow rate induced by the impeller is largely increased, leading to significant reduction in mixing time. In addition, the power demand is reduced. It is also evidenced that by enlarging the clearance, it is difficult for the fluid in this region to be mixed.

... L'Eq.( 63 ) montre aussi que cette constante adimensionnelle est indépendante du comportement rhéologique du fluide (en considérant les hypothèses évoquées). Ce résultat a été observé expérimentalement [131, 132, 133, 134] et théoriquement par approchenumérique[135]. ...

In order to limit the handling of pyrotechnical products presenting a risk with respect to the mechanical and thermal shocks, new energetic formulations have emerged. Among these Extremely Insensitive Detonating Substances, the industrial partner Nexter Munitions implements a formulation labeled XF13333 which has the property of being recyclable.
Before being in solid state, XF13333 explosives are in the form of suspensions whose solid volume fraction is about 55%. The flow ability of these pastes is governed by the rheology of concentrated suspensions. The method of preparation used by the industrial partner being the melt-cat process, the aim of the work presented in this manuscript is to propose a model that can predict the time required for an energetic concentrated suspension to flow by gravity from an agitated mixing reactor to an initially empty shell body.
The study of the influence of the solid volume fraction on the rheological behaviour of model inert suspensions has been performed. Two grains sizes scales have been considered: micrometric and nanometric. Depending on the particles size, pastes adopt different rheological behaviours: the suspension containing micrometric particles adopts a Newtonian behaviour, while the suspension containing nanoscaled particles evolves from a Newtonian to a yield stress fluid (Bingham or Herschel-Bulkley) behaviour.
Then, we investigated the influence of the state of organization of the solid phase, which can be described by the maximum volume fraction of solid. We have studied the influence of this parameter on the rheological behaviour of suspensions. Given the strong behavioural analogy between a paste of explosives and cement, we took benefit from researches conducted over past years in civil engineering. In particular, we used the De Larrard model to calculate the maximum volume fraction of solid from the size distribution of the grains of the mixture.
To characterize the rheology of XF13333suspensions, the use of a Couette rheometer is not appropriate because the particle sedimentation and the destabilization of the emulsion – which constitutes the suspending fluid – make the suspension inhomogenous. To maintain its usage function (i.e. homogeneous), we developed an unconventional rheometer and applied the Couette analogy. It emerged from this experimental study that the suspensions are Ostwal fluids.
The modelling of the flow time for XF13333 suspensions in industrial configuration is based on the Quemada viscosity relationship. The theoretical values have been compared with the experimental values of emptying time of a certain amount of XF13333 suspensions. It appears that the proposed flow time model matches the reality correctly.

... Kelly and Gigas [13] extended the study to transition flow regime. Shekhar and Jayanti [14] performed CFD simulations of pseudo plastic fluids of high viscosity in the laminar flow regime. Montante et al. [15] calculated the mixing time in a stirred tank agitated with 45°pitched blade turbines. ...

A simulation study reported on the velocity fields and the entropy generation of non-Newtonian fluids in a baffled stirred tank with Rushton turbine impeller. Two shear thinning fluids, carboxymethylcellulose (CMC) and xanthan gum (XG), with flow index (n) varying in the range from 0.64 to 0.85 are used as the working fluid. The steady state multiple reference frame and the realizable k–ε turbulence model is used for numerical simulation. The predicted velocities for a CMC solution with n = 0.85 compared with literature data, and overall good agreement observed. The effect of flow index on Reynolds number similarity, power number and flow number also studied. The present work also determines the entropy generation due to the fluid flow. The CFD simulation is applied to predict the effect of size of the impeller blade, impeller clearance, fluid flow behavior index and rotations of impeller on the spatial distribution of the total entropy generation.

... L'Eq.( 63 ) montre aussi que cette constante adimensionnelle est indépendante du comportement rhéologique du fluide (en considérant les hypothèses évoquées). Ce résultat a été observé expérimentalement [131, 132, 133, 134] et théoriquement par approchenumérique[135]. ...

Afin de rendre plus sûre la manipulation de produits pyrotechniques présentant par nature un risque d'explosion accidentelle vis-à-vis des agressions mécaniques et thermiques extérieures, de nouvelles formulations ont vu le jour. Parmi ces « Matières Détoniques Extrêmement Peu Sensibles », le partenaire industriel Nexter Munitions met en œuvre une formulation labellisée XF13333 qui a par ailleurs la propriété d'être recyclable.
Avant d'être à l'état solide, les explosifs de type XF13333 se présentent sous forme de suspensions dont la fraction volumique solide est de l'ordre de 55%. L'aptitude à l'écoulement de ces pâtes est donc gouvernée par la rhéologie des suspensions concentrées. Le procédé de préparation utilisé par le partenaire industriel étant de type coulée-fondue, l'objectif des travaux présentés dans ce manuscrit est de proposer un modèle prédictif du temps que met une suspension concentrée en matériaux énergétiques à s'écouler gravitairement depuis un réacteur agité de mélange vers un corps d'obus initialement vide.
L'étude de l'influence de la fraction volumique solide sur le comportement rhéologique de suspensions modèles inertes a d'abord été envisagée. Deux échelles de taille de grains ont été considérées, l'une micrométrique et l'autre nanométrique. En fonction de la taille des particules, la pâte adopte différents comportements rhéologiques : la suspension constituée de particules micrométriques adopte un comportement newtonien tandis que la suspension constituée de particules nanométriques passe d'un comportement newtonien à un comportement de fluide à seuil, de type Bingham ou Herschel-Bulkley.
Nous nous sommes ensuite intéressés à l'influence de l'état d'organisation de la phase solide que l'on peut décrire par la fraction volumique maximale de solide. Nous avons étudié l'influence de ce paramètre sur le comportement rhéologique des suspensions. Etant donné la forte analogie de comportement entre une pâte d'explosifs et le ciment, nous nous sommes intéressés aux travaux de recherche menés depuis de nombreuses années dans le secteur du génie civil. En particulier, nous avons utilisé le modèle de De Larrard pour calculer fraction volumique maximale de solide à partir des distributions en taille des grains constituant le mélange.
Pour caractériser la rhéologie des suspensions XF13333, l'utilisation d'un rhéomètre de Couette n'est pas adaptée car la sédimentation des particules et la déstabilisation de l'émulsion formant le fluide suspendant rendent la suspension inhomogène. Pour la maintenir dans sa fonction d'usage (i.e. homogène), nous avons développé un rhéomètre non conventionnel et appliqué l'analogie de Couette. Il est ressorti de cette étude expérimentale que ces suspensions complexes sont des fluides d'Ostwald.
La modélisation du temps d'écoulement des suspensions XF13333 en configuration industrielle s'est fondée sur l'expression de la viscosité d'une suspension concentrée de Quemada. Les valeurs théoriques ont été comparées aux valeurs expérimentales de temps de vidange d'un certain volume de suspension concentrée XF13333.. Il est ressorti que le modèle de temps d'écoulement proposé représente correctement la réalité.

The concept of Metzner and Otto was initially developed for correlating power measurements in stirred vessels for shear-thinning fluids in the laminar regime with regard to those obtained for Newtonian liquids. To get this overlap, Metzner and Otto postulated and determined an “effective shear rate” which was proportional to the rotational speed of the impeller Although it was not based on a strong theoretical background, it was rapidly admitted as a practical engineering approach and was extended for seeking out a “Newtonian correspondence” with non-Newtonian results (i.e. different classes of fluids). This was applied in a variety of tank processes even for predicting heat transfer or mixing time, which stretches far away from the frame initially envisaged by Metzner and Otto themselves. This paper aimed to show how dimensional analysis offers a theoretically founded framework to address this issue without the experimental determination of effective quantities. This work also aimed to enlarge the underlying questions to any process in which a variable material property exists and impacts the process. For that purpose, the pending questions of Metzner and Otto concept were first reminded (i.e. dependence of the Metzner–Otto constant to rheological parameters, physical meaning of the effective shear rate, etc.). Then, the theoretical background underlying the dimensional analysis was described and applied to the case of variable material properties (including non-Newtonian fluids), by introducing in particular the concept of material similarity. Finally, two examples were proposed to demonstrate how the rigorous framework associated with the dimensional analysis is a powerful method to exceed the concept of Metzner and Otto and can be adapted beyond the Ostwald–de Waele power law model to a wide range of non-Newtonian fluids in various processes, without being restricted to batch reactor and laminar regime.

Xanthan gum solutions with different mass concentrations were used to study the chaotic characteristics induced by the impeller of perturbed six-bent-blade turbine (6PBT) in a stirred vessel. Based on the velocity time series obtained by the experiment of particle image velocimetry (PIV), with the software MATLAB (R2016a), the distributions of the largest Lyapunov exponent (LLE) and Kolmogorov entropy (K entropy) of the system, as two important parameters for characterizing the chaotic degree, were investigated respectively. Results showed that both of the LLE and K entropy increased with the increasing speed at the beginning. As the speed was up to 200 rpm, the two parameters reached the maximal values meanwhile, corresponding to 0.535 and 0.834, respectively, which indicated that the chaotic degree of the flow field was up to the highest level. When the speed was increased further, both of the LLE and K entropy decreased on the contrary, which meant that the chaotic degree was decreasing. It was also observed that the chaotic characteristics of flow field were hardly affected by the fluid rheology and the detecting positions. The research results will enhance the understanding of the chaotic mixing mechanism and provide a theoretical reference for optimizing impeller structure.

Ce travail est consacré à une étude approfondie sur le procédé anaérobie pour les traitements des eaux usées et des boues d’épuration afin d’améliorer la production de biogaz. Tout d’abord, l’hydrodynamique dans un réacteur de type circulation interne (IC) a été caractérisée en maquette froide. La vitesse de circulation du liquide dépend davantage de l’écoulement de la phase gazeuse que de la phase solide, l’influence du débit de liquide entrant étant la plus faible des trois. La taille de bulles influe non seulement sur la vitesse de cisaillement mais aussi sur les types d’interactions bulles-granules. L’effet de la vitesse de cisaillement moyenne a ensuite été étudié dans une cuve agitée en maquette chaude. Lorsque la vitesse de cisaillement moyenne augmente, le débit de biogaz passe par un maximum pour une valeur de 6,8 s-1, tandis que le pourcentage en méthane diminue continument. La déformation des granules est provoquée par cisaillements et collisions, à masse volumique constante. A micro-échelle, une déformation locale d’un cratère à l’extrémité du canal d’acheminement du biogaz est découverte pour la première fois. Le diamètre de ce cratère est proportionnel à la taille du pore, et la pression de compression de la microbulle estimée par la loi de Laplace est comparable à la résistance mécanique du granule mesurée par la pénétrométrie. Enfin, l’étude s’attache aux propriétés rhéologiques de boues digérées hautement concentrées. Un comportement rhéofluidifiant à seuil de contrainte, et un comportement viscoélastique sont caractérisés. L’effet de la concentration en solide est beaucoup plus significatif que celui de la température. En outre, des expériences d’impact par une bille sur des boues digérées hautement concentrées sont réalisées pour révéler sa dynamique transitoire. Un modèle simplifié de la trainée est établi pour estimer le module de l’élasticité ainsi que la viscosité d’impact

Medium-consistency technology is known as the process with high efficiency and low pollution. The gas distribution was simulated in the medium-consistency pump with different degas hole positions. Rheological behaviors of pulp suspension were obtained by experimental test. A modified Herschel-Bulkley model and the Eulerian gas-liquid two-phase flow model were utilized to approximately represent the behaviors of the medium-consistency pulp suspension. The results show that when the relative position is 0.53, the gas volume ratio is less than 0.1% at the pump outlet and 9.8% at the vacuum inlet, and the pump head is at the maximum. Because of the different numbers of the impeller blades and turbulence blades and the asymmetric volute structure, the gas is distributed unevenly in the impeller. In addition, the pump performance was tested in experiment and the results are used to validate computational fluid dynamics outcomes.

Experiments on dispersion of floating solids into continuous liquid medium were carried out with low density microspheres with particle density of 680 kg/m3 and size 100, 230 and 325 μm in different liquids having density varying from 778 to 1830 kg/m3 and kinematic viscosity varying from 0.3 to 19 cP. A four-bladed radial impeller and axial paddle impellers with upward and downward flow directions were employed. The critical impeller speed (Ncrit) required for uniform dispersion of particles throughout the continuous medium was experimentally determined for various conditions. Experimental results indicate that the radial impeller with impeller location near the surface required minimum impeller speed for uniform dispersion. Strong effect of the density difference between the particles and the liquid and submergence of the impeller was noted. Based on the experimental results, a correlation in terms of relevant dimensionless groups was proposed for calculating Ncrit for the three impellers. The overall correlation indicates that the critical impeller speed is not significantly affected by liquid viscosity and particle diameter but that it is strongly influenced by the impeller diameter and the density difference. These results are in agreement with trends reported in the literature.

Sludge as a kind of non-Newtonian fluid, its rheological characteristics has a significant effect on mass and heat transfer, material mixing and transport in the process of anaerobic digestion. The rheology properties, such as influencing factors, rheological models and measurement of sludge rheological characterization were summarized. Due to the complexity of flow distribution in anaerobic digestion reactors and the opaqueness of sludge, numerical modelling is a much effective approach to describe the flow distribution by combining computational fluid dynamics (CFD) technology with the rheological parameters, instead of traditional experimental methods. Thereby, it's of great value to demonstrate the current research situation concerning the application of CFD technology on the anaerobic digestion and the choice of rotating blades handling during the mixing process. Besides, the problems of numerical simulation applied in anaerobic digestion were also discussed. With further study, the numerical modelling of biochemical reaction on the basis of sludge rheological and computational fluid mechanics would provide more comprehensive and important technical measures for the process control of the sludge anaerobic digestion.

The macromixing and flow characteristics of a low-density polyethylene autoclave reactor stirred with 32 impellers were investigated by experiments and computational fluid dynamics simulations. The simulation results were validated using the experimental data. The mixing curves show an excellent agreement with the experimental results. The flow field, distribution of velocity, and mass flux obtained from the simulations were further analyzed, and the results show that the reactor can be divided into two reaction zones. In each reaction zone, two axial flows form a big circulation loop. The downward flows near the reactor wall decrease along the flow direction, while the upward flows near the shaft increase. In each big circulation loop there are many small radial circulation loops that are similar to those encountered in continuous stirred-tank reactors. Finally, a more accurate macromixing model was proposed, which is helpful to reveal the multizone phenomenon and flow characteristics in reactors.

Drilling fluids are multicomponent emulsions and/or suspensions, which normally show non-Newtonian behavior, i.e. yield stress, shear-thinning and thixotropy, and strong thermal and pressure dependence. The rheological characterization of drilling fluids using conventional geometries can be a difficult task due to the inherently heterogeneous nature of these systems. These problems may be overcome using nonconventional geometries, such as helical ribbons and blade turbines, which maintain the homogeneity of the system during the measurement. The overall objective of this work was to evaluate the use of mixing geometries, such as helical ribbons and blade turbines, for characterizing the flow behavior of drilling fluids as a function of pressure. From the experimental results it may be concluded that, using the Metzner-Otto approach, both four-blade and helical ribbon type-geometries are suitable for this purpose. This study shows that the Metzner-Otto constant, for a four blade turbine geometry, is practically independent of the flow index at atmospheric pressure, which shows a linear dependence at higher pressures. On the contrary, for the helical ribbon geometry, an exponential dependence of the Metzner-Otto constant on the flow index is observed independently of the measured pressure. From the experimental results obtained, it can be concluded that both nonconventional geometries can be used to measure the influence of pressure on the rheological parameters of non-Newtonian fluids. These tools extend the experimental shear-rate window covered by the coaxial cylinders conventional geometry to lower values, allowing the measurement of important engineering parameters, such as, for instance, yield stress.

A standard configuration of a co-axial mixer is a combination of a close clearance impeller rotating at low speed, and a central impeller rotating at higher speed to achieve efficient and instantaneous bulk and shear flows within the mixing tank. The type of the central impeller has a significant effect on the mixing quality achieved by a co-axial mixer. Thus, in this study the electrical resistance tomography (ERT) and computational fluid dynamics (CFD) were utilized to analyze the mixing of xanthan gum solution (an opaque pseudoplastic fluid with yield stress) with the different types of central impellers. The following three co-axial mixers were employed: the anchor–Rushton turbine (a radial-flow impeller), the anchor–A200 (an axial-flow impeller), and the anchor–ARI (an axial–radial-flow impeller). An ERT system with a five-plane assembly of peripheral sensing rings, each containing 16 stainless steel electrodes, was utilized to measure the mixing time for these three co-axial mixers. The sliding mesh (SM) technique with the modified Herschel–Bulkley model was applied to simulate the impeller rotation and the rheological behavior. The CFD results for the power consumption and mixing time were compared to the experimental data for the model validation purposes. The CFD and ERT data were employed to investigate the effect of central impeller type, xanthan gum concentration, anchor impeller speed, and central impeller speed on the mixing time and the specific power consumption of the coaxial impellers. A statistical-based experimental design with RSM (response surface methodology) was applied to evaluate the individual and interactive effects of the design parameters and operating conditions.

Using the multiple reference frames (MRF) impeller method, the three-dimensional non-Newtonian flow field generated by a double helical ribbon (DHR) impeller has been simulated. The velocity field calculated by the numerical simulation was similar to the previous studies and the power constant agreed well with the experimental data. Three computational fluid dynamic (CFD) methods, labeled I, II and III, were used to compute the Metzner constant ks. The results showed that the calculated value from the slop method (method I) was consistent with the experimental data. Method II, which took the maximal circumference-average shear rate around the impeller as the effective shear rate to compute ks, also showed good agreement with the experiment. However, both methods suffer from the complexity of calculation procedures. A new method (method III) was devised in this paper to use the area-weighted average viscosity around the impeller as the effective viscosity for calculating ks. Method III showed both good accuracy and ease of use.

A continuous-flow mixer was designed and built to study the mixing of xanthan gum solution, a pseudoplastic fluid possessing yield stress. The extent of flow nonideality was quantified using a dynamic model that incorporated two parameters: channeling and fully mixed volume in the vessel. Dynamic experiments were made using the frequency-modulated random binary input of a brine solution to determine the magnitude of nonideal flow parameters. The same experiments were simulated using a computational fluid dynamics (CFD) package (Fluent 6.2). CFD flow fields were used to obtain the system dynamic response to a tracer injection applied at conditions identical to the experimental ones. The extents of channeling and effective mixed volume were determined using the CFD model and then compared with the parameters obtained experimentally. Validated CFD flow fields enabled us to effectively monitor the effect of various operating conditions on flow nonideality, to relate flow pattern and cavern dimension to flow nonideality, to compare the efficiency of impellers, and to provide a pictorial synopsis of continuous-flow mixing operation.

A commercial CFD package was used to simulate the 3D flow field generated in a cylindrical tank by a helical ribbon impeller. The study was carried out using a pseudoplastic fluid with yield stress in the laminar mixing region. Ultrasonic Doppler velocimetry (UDV), a noninvasive fluid flow measurement technique for opaque systems, was used to measure xanthan gum velocity. From flow field calculations and tracer homogenization simulations, power consumption and mixing time results were obtained. The torque and power characteristics remain the same for upward and downward pumping of the impeller, but the mixing times are considerably longer for the downward pumping mode. Overall, the numerical results showed good agreement with experimental results and correlations developed by other researchers. From the power and mixing time results, two efficiency criteria were utilized to determine the best pumping mode of the impeller.

Rotating wheel air classifiers are often used in many process industry applications. The internal geometry of these equipment is quite complicated and has not been investigated in detail. In the present study, the flow field inside a rotating wheel air classifier has been calculated using CFD techniques, taking full account of the internal geometrical features. The predicted overall flow rate and the flow pattern are in good agreement with measurements and flow visualization studies. The calculations show that the induced flow depends strongly on geometric parameters such as the location of the inlet and outlet ports and the type of shutters used. Trajectory calculations of single particles show that the particle motion is influenced principally by centrifugal force, air drag, and wall-rebound characteristics. The wall rebound is possibly one of the means of how large particles enter the fines stream, leading to low efficiency at high speeds or large particles. Experiments of the classification using angular and radial shutter vanes show distinct range of operability of each type. These results have been interpreted coherently in the light of the flow pattern and particle trajectory calculations. © 2005 American Institute of Chemical Engineers AIChE J, 51:776–790, 2005

The overall objective is to present a procedure based on a Couette analogy to quantitatively analyse torque/rotor speed data and extract viscosity/shear-rate curves using a non-conventional geometry. Diphasic flows of energetic concentrated suspensions of melt-cast insensitive explosives exhibit particular rheological properties. The characterization of these complex fluids may be a challenging task when conventional rheometers are used. Placing these tense suspensions in a classic cylindrical geometry may lead to a partial destruction of the internal fluid structure. To prevent that, a "RheoXF" a mixer-type rheometer has been developed : it consists of a mixing device with quite a complex geometry rotating in a cylindrical tank.

Power consumption measurements are presented for helical ribbon mixers which had two scales and a range of stirring speeds and ribbon sizes, and which operated in Newtonian and pseudoplastic, viscoelastic polymer solutions These measurements were correlated and interpreted using the analogy between the helical ribbon and the coaxial cylinder viscometer. The power consumption of a ribbon was equal to or less than that of solid coaxial cylinders, and the analogy provided a more general model for correlating all thepublished data than any other. This correlation included the viscous flow properties, stirring speed, scale, and the effects of pitch and wall clearance. The transition point marking the end of the laminar flow region was also defined.

Power input measurements are reported for helical ribbon impellers for two scales; a 0.15 m diameter and a 0.4 m diameter tank. Data for viscous Newtonian and non-Newtonian fluids are brought together by use of the average apparent viscosity concept and the following equation: where ks is the shear rate constant, c is the clearance between vessel wall and impeller with diameter D.Power measurements from this work combined with relevant information extracted from the published literature indicate that impeller geometry has a profound effect upon power requirement, particularly in the laminar region, where the reported data can be described by: where Kp is a geometric constant and all the other symbols have their usual significance. Theoretical models which fail to allow for system geometry and fluid properties give values which may be seriously in error.

Power consumption measurements for helical ribbon impellers of different geometrical variables were carried out under laminar flow conditions in pseudoplastic liquid, which is the most typical non-Newtonian liquid. On the basis of A. B. Metzner and R. E. Otto's method, an empirical correlation for prediction of the power consumption of helical ribbon impellers in pseudoplastic liquid is proposed.

A new power correlation for both anchor and helical ribbon impellers in highly viscous Newtonian liquids is proposed on the basis of a physical model developed from an analytical approximate expression for the drag of a plate in viscous liquids bounded by a plane wall. The correlation, obtained by inserting the empirical factor of geometrical variables in the above expression, shows good agreement with experimental data of power consumption of anchor and helical ribbon agitators obtained in this work and other literature.

Of necessity, the mixing process is sometimes restricted to the laminar regime, although turbulent mixing is generally more desirable. Common examples of laminar mixing are found when the fluid has a very high viscosity, or when one of the mixture components is shear sensitive. It has been pointed out that the helical ribbon agitator (HRA) is admirably suited to the low Reynolds number mixing process. This work derives a model to predict the power consumption of the HRA. The model has been developed with the aid of experimental data and tested extensively using literature data. For a wide range of mixer geometries and sizes, it predicts power consumption with an average deviation of 13%.
The concept of relative efficiency of mixers is also described as an aid to comparing different HRA mixer geometries. Finally, the problem of scale-up of different HRA configurations is discussed.

Batch mixing of viscous fluids with helical‐ribbon agitators in 2.4 liter and 13 liter vessels has been studied for agitator speeds up to 200 RPM. Seven different agitators of different dimensions were employed in this work. Mixing times were measured using a decoloration technique and circulation times were determined by the tracer bead method. In addition, velocity profiles were obtained from streak photographs using selective illumination of the vessel and PVC powder as tracer particles.
It was found that the mixing times of Newtonian fluids, which agreed with previously published data, were considerably (3 to 7 times) shorter than those of the viscoelastic fluids. The mixing time was strongly affected by the fluids' elasticity; increasing as the fluid elasticity increased. The velocity profiles were qualitatively similar for all the fluids but showed decreased axial circulation and increased circumferential flow as fluid elasticity increased. However, mixing is not only a function of the axial circulation (impeller pumping rate) but also is a function of the perturbations superimposed on the main flow. A simple, first approximation model based on the impeller geometry and flow patterns is proposed to correlate the circulation capacity and mixing time data for the various geometries studied.

The objective of this article is to study the mixing efficiency of an industrial reactor fitted up with three different helical ribbon impellers. To this end, criteria that seem complementary will be used: dispersion of tracers, length stretch values and their associated Lyapunov exponents, and finally dispersive mixing efficiency coefficients. All these measures are based on flow simulations with finite-element software POLY3D from RheoTek Inc.

In 1957 Metzner and Otto proposed a simplified way of calculating the power consumption in impeller-driven systems for power-law type of fluids. Assuming a proportionality between the shear rate and rotational speed of the impeller, they reported that the proportionality constant was independent of the geometric and fluid properties of the mixing system. This was later contested by a number of researchers, and different correlations have been given to evaluate the proportionality constant. An in-depth investigation of the Metzner–Otto concept is discussed with the help of computational fluid dynamics (CFD) techniques to calculate the flow field created by an anchor impeller. Fundamentally, both of the assumptions are broadly valid for typical operating parameters used in anchor-driven impeller systems. A new correlation, based on the Metzner–Otto concept and making use of the present CFD simulations, was developed to calculate power consumption in anchor-driven mixing systems.

The influence of shear thinning and viscoelasticity on the power required for the mixing of viscous liquids using six different helical ribbon agitators has been investigated. Four Newtonian and 12 non-Newtonian fluids prepared using several polymers dissolved in varying concentrations in different solvents cover a wide range of rheological properties. By a careful choice of test media, the specific and combined effects of shear thinning and viscoelasticity on the power requirement have been examined. Simple models are proposed to predict the effective shear rate in the tank from the knowledge of the torque or power number. The effective shear rate predictions compared with the effective shear rate estimated using the scheme of Metzner and Otto (1957) show that they slightly depend on the shear thinning properties. Fluid's elasticity increases appreciably the power requirement, and departures from the generalized Newtonian power curve in the laminar regime are observed at smaller Reynolds numbers for viscoelastic fluids. Bottom wall resistance of the mixing vessel makes a negligible contribution to the power consumption.

Batch mixing of rheological complex fluids using helical ribbon and helical ribbon screw impellers was investigated in the laminar mixing region. Eight models and one polysaccharide fermentation broth (gelkn) were used. These fluids exhibited different rheological properties: Newtonian viscosity, viscoelasticity, and pseudoplasti-city. The Ostwald-de Waele model (power law) described adequately the shear viscosity of pseudoplastic fluids, including the gellan broth (n ranged from 0.14 to 0.65).Power consumption decreased as pseudoplasticity increased and a model was developed to predict the deviations from Newtonian power input due to pseudoplasticity. A dimensionless, unique representation of the power data using the predictions of the model, showed a shift of the upper limit for the laminar mixing region towards higher values of Re (Re ≈ 100). The model predictions were found to be in good agreement with those obtained by using the classical approach of Metzner and Otto1. The constant of proportionality between the average shear rate and the impeller rotational speed, K s, was found to be a function of the power law index, especially for high shear thinning fluids.It was found that power consumption increased as the impeller blade pitch decreased or as the impeller blade width increased. Both impeller pitch and blade width do not have significant influence on KS.

Since the shear rate of a non-Newtonian fluid is of importance in fixing the rheological or viscometric behavior of such a material, the present study has been concerned with the development of a general relationship between impeller speed and the shear rate of the fluid. The resulting relationship was then used to interpret and correlate power-consumption data on three non-Newtonian fluids by use of a generalized form of the conventional power-number–Reynolds-number plot for Newtonians.
Flat-bladed turbines from 2 to 8 in. in diameter were used exclusively. Tank diameters ranged from 6 to 22 in. and power inputs from 0.5 to 176 hp./1,000 gal. The study encompassed a 130-fold range of Reynolds numbers in the laminar and transition regions. The results to date indicate that power requirements for the rapid mixing of non-Newtonian fluids are much greater than for comparable Newtonian materials.

The purpose of this paper is to extend basic information on mixing and circulation times for mixing systems equipped with
helical ribbon impellers.
In a first part, from a survey of existing literature, we have reviewed the different effects of the geometrical parameters
of the helical mixing systems (clearance-wall, pitch size ratio, blade width, number of blades) on the mixing process. From
this qualitative analysis, useful guidelines are provided on the influence of the helical impeller geometry on the mixing
effectiveness.
In a second part, the homogenization effectiveness of an atypical helical ribbon impeller was investigated and compared with
the performance of classical helical ribbons impellers.

An Analysis of the Flow Produced by Helical Ribbon Impellers

- Bourne J. R.

Power Consumption by Helical Ribbon Impellers

- Hall K. R.

Numerical Analysis of Reaction Process of Highly Viscous Liquids in a Stirred Vessel Equipped with a Double Helical Ribbon Impeller

- Kaminoyama M.

Mixing of Non-Newtonian Viscous Fluids with Helical Impellers: Experimental and Three-Dimensional Numerical Studies

- P. A. Tanguy
- R. Lacroix
- F. Bertrand
- L. Choplin
- E. Brito de la Fuente

On the Effect of Shear‐Thinning Behavior on Mixing with Helical Ribbon Impeller

- Brito de la Fuente E.

Mixing of Non‐Newtonian Viscous Fluids with Helical Impellers: Experimental and Three‐Dimensional Numerical Studies

- Tanguy P. A.