Mixing Technology Naked-eye Visualization of Unyielded Zones for a Close-Clearance Impeller Agitating Viscoplastic Fluids

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The mixing of a viscoplastic fluid with an anchor impeller is studied visually. Three different types of experiments, documented through videos and still images, demonstrate the existence of an unyielded region, rotating like a rigid body. These results are in good agreement with previous workers’ numerical predictions, and dramatically show the influence of yield stress on mixing effectiveness, even with a well-designed agitation system. The unyielded region is the equivalent, for a close-clearance impeller, of the stagnant region outside the well-known ‘cavern’ forming around a small-diameter impeller. Furthermore, for such systems the presence of motion is not by itself a criterion by which quality of mixing can be assessed.

Transport phenomena are of great industrial importance. Processing of fluids often requires mixing to homogenize them but also to provide adequate heat and mass transfer [1-4]. Careful equipment design is thus required, leading to a great variety of commercially available impeller systems, and accompanying science. Process development cannot be performed directly at industrial scale for cost and convenience reasons. Scale-up rules are thus necessary to transpose small-scale results, using theoretical understanding and empirical findings, to full commercial scale by setting parameters such as the agitation rate and rate of addition to maintain representativeness across different sizes.

The basic physical property governing flow is rheology. Appreciating the importance and practical consequences thereof is therefore critical to chemical engineers. Newtonian fluids exhibit the simplest possible rheology: constant viscosity at all shear rates. Amongst more complex rheologies, viscoplastic fluids are particularly common. Such fluids exhibit no motion until the stress upon them is larger than a critical value, the so-called yield stress. Once flowing, they also appear to become thinner at larger shear rates. These two defining characteristics are expressed mathematically together by expressions such as the Herschel-Bulkley equation (see Fig.6, Picture Gallery).

Gallery

Gallery with 6 images

Yield stress fluids are common in everyday life: for instance, desserts (e.g., chocolate mousse) are often found to exhibit a yield stress; as a possible sign of cosmic equilibrium, so does tooth paste. In the fine chemicals industry, a class of fluids often exhibiting a yield stress are slurries, such as those encountered in crystallizations. Despite such examples, viscoplasticity is not well understood/appreciated. Indeed, the fundamental existence of yield stress has been vigorously debated [6-7].

Viscoplastic fluids present some special challenges to ensure that they remain in motion: stagnant zones must be ‘eradicated’ to ensure homogeneity. Early mixing studies showed that small diameter impellers only moved fluid in a limited zone close to them, termed a ‘cavern’. Outside this zone, the fluid was insufficiently stressed and remained motionless; [10] and [11] provide both theoretical and empirical equations for cavern behavior. To mix viscoplastic fluids, close-clearance impellers are often used.

Later work has focused on Computational Fluid Dynamics (CFD), with limited experimental work and interpretation thereof. The experimental work available largely concerns itself with mixing times or power measurements rather than the more difficult to characterize flow fields, except for [8] where visualization of central caverns is performed for multiple scales and fluids to propose a new dimensionless correlation for the cavern diameter. [2] calculated flow patterns, which were interpreted to show the existence of a central ‘unyielded zone’, which would rotate like a rigid body. Most recent CFD work has focused on characterizing the mixing efficiency of different impellers: helical screw, ribbon, anchors and coaxial impeller systems [12-17].

Parameter sensitivity studies have also been carried out [18] [19]. [20] recognized limits of existing literature, and emphasized the importance and interpretation of the unyielded zone. In the context of these studies, coaxial mixers are believed to offer the best combination of flexibility and efficiency for mixing viscoplastic fluids [17, 21-22]. By contrast to the case of central caverns, simple equations are not available to calculate the extent of unyielded zone for close-clearance impellers, nor the dependence thereof on scale and operating parameters.

The essential novelty of the present work is visually confirming, and documenting, the formation of unyielded zones for close-clearance impellers agitating viscoplastic fluids. These experiments can easily be adapted to a teaching demonstration to impart to fluid mechanics students a visceral, naked-eye appreciation for the importance of non-Newtonian behavior and its consequences. The authors found such a visual demonstration equally helpful to embed the concept of scale-up [23].

Materials and Methods

The agitation was carried out in an un-baffled single-jacketed torispherical based glass vessel of 19 L nominal volume. The vessel was fitted with an anchor impeller of D/T = 0.87, rotated by an overhead electrical motor with a digital speed controller. The exact dimensions of the vessel and impeller are detailed in Table 1, and illustrated in Figure 2 (see Gallery).

A model fluid displaying a mild yield stress (<10 Pa) and shear thinning behavior was desired, so as to mimic a manufacturing fluid for a scale-down study. A non-hazardous suspension in deionized water was thus formulated, containing 50 g/l Bentopharm (clay, Wilfrid Smith) and 3.6 g/l Rhodopol 23 (Xanthan gum, Solvay). All the ingredients were added simultaneously to a large vessel (50 L) and mixed with a high-shear Silverson mixer (approximately 60 min at 600 rpm) until no traces of powder were visible anymore.

The rheology of the fluid could be described well by the Herschel-Bulkley model, see Figure 7.

In other words, it exhibits shear-thinning behavior. The yield stress of 7.6 Pa is relatively low compared to values for ketchup (~15 MPa) and mayonnaise (~90 MPa). The suspension is opaque and light grey in color.

Experimental procedure

Agitation

The experimental procedure was kept very simple by intent, to enable a visceral appreciation of the phenomena. First the material was agitated at the desired agitation rate (48 rpm or 90 rpm) for approximately 60 minutes to allow the material to form its intermolecular structure. Three different motion experiments were then performed relying on different physical phenomena: bulk motion, solids motion, addition of a second liquid. In all cases, a digital camera was used to record the motion of the main fluid and of the added materials. Measurements of relative distances were performed on still images along their left-right axis.

Floating solid addition

Two colored, transparent plastic A4 folders were cut into square pieces with an edge of roughly 1 cm. These squares were then dropped onto the surface of the agitated fluid, 5 cm (red square; r/T of 1/6, where r denotes the radial coordinate) or 10 cm (blue square; r/T of 1/3) from the center of the vessel.

Dye addition

Drops of colored blue food dye (Sainsbury’s, London, UK) were added to the surface of the agitated fluid via a pipette at different vessel locations (approximately: 1/4, 1/2 and 2/3 radii from the center of the vessel).

Results and Discussion

Videos of the experiments can be viewed in the text below. They demonstrate most clearly the results and behavior of the fluid and system. The results are also presented as sequences of still images below. Table 2 lists the observations recorded, and their interpretation is described in the sections below.

Qualitative motion

Prior to any addition, the fluid visually shows on the surface a marked central discontinuity, indicated with a striped arrow in Figure 1, and visible in all videos. The central region width is approximately one-fifth to one-quarter tank diameter. Based on visual observations, it was hypothesized that the discontinuity separated flowing bulk material from unyielded material near the center of the vessel where the material is rotating like a rigid body.

By virtue of these distinct behaviors, the discontinuity appears to acts like a barrier to flow. The CFD predictions and discussion of this behavior by [20] were only found after completion of the experimental work. While [20] predicts multiple unyielded zones, the resolution of the present work only allows the validation of the existence of an unyielded zone, not of its fine structure. Two more formal experiments were performed to test this hypothesis using different physical phenomena. First, however, we consider the quantitative results with respect to motion alone.

Quantitative analysis

Measurements on the images show asymmetric behavior in the vessel, with the discontinuity extending further from the center towards the right side (of the images) than on the left. This asymmetry cannot be ascribed to the baffling, which is symmetric. Possible explanations include: imperfect alignment of the impeller; a slightly bent shaft; or asymmetry of the impeller upon fabrication. Some vibration of the shaft can be observed in the video recordings.

To minimize the impact of such imperfections on the experimental results, the full width w of the central zone was therefore measured, so as to avoid introducing directional bias. The fraction of the surface rotating like a rigid body was then calculated as (w/T)², and is shown in Table 3.

Since both fluid and equipment used here are different from those of [20], direct comparisons are not possible. Nonetheless, the fraction of the rigid zone decreases with faster stirring, as qualitatively expected. The fact that it decreases only by 23 % for an 88 % increase in agitation rate suggests that it could be quite difficult to totally eradicate the unyielded zone in the case of incorrect mixing system design and/or incorrect formulations (with too high a yield stress).

This far from linear behavior was unexpected, highlighting the lack of information in the literature about unyielded central zones, and how their extent depends on composition and operating conditions. Indeed, not even general recommendations of mixing parameters to eradicate or reduce the central cavern were found.

Floating solids addition

Figure 3 (see Gallery) shows the trajectories of the two colored squares. The first two frames show the addition of each plastic piece. Thereafter, each frame is spaced by 20 seconds. Both colored squares slowly spiral towards the center of the vessel, until they reach the discontinuity surrounding the central zone, in the fifth (red, inner square) and eighth (blue, outer square) frame. Upon reaching this discontinuity the squares continue to rotate tangentially, but do not move further in the radial direction. This behavior is consistent with the central zone rotating like a rigid body.

Video 1 also shows this motion, in a more inherently dynamic fashion.

Dye addition

Figures 4 and 5 (see Gallery) show the movement of dye droplets over different timescales, 1.9 and 13.2 s respectively. Figure 4 presents the initial phase with each frame separated by 100 ms. The drops, progressively elongated by shear forces, move towards the center of the vessel until they reach the discontinuity. Figure 5, with each frame separated by 1.2 s, demonstrates that at the discontinuity the drops form a (partial) annular shape and over time slowly diffuse into the center. This behavior is also consistent with the central zone rotating like a rigid body.

Video 2 also shows this motion dynamically, with the addition occurring at timepoint 01:02.

Interpretation

The experiments described here, considered together, visually show in short succession that: (i) for a yield stress fluid stirred with a close-clearance impeller, a discontinuity can appear, at least on the surface; (ii) both solids and liquids behave differently when they reach this zone: outside it they move towards the center, but the bulk radial motion stops when reaching the discontinuity. Together these findings imply that the bulk motion of the surface inside the discontinuity is similar to that of a rigid body – in excellent qualitative agreement with the explicit predictions and interpretation of ‘unyielded zones’ ‘certainly very bad for mixing’ in [20], following on earlier work such as [2]. The importance of considering yield stress in the agitation of viscoplastic fluids, even for close-clearance impellers, is thus visually demonstrated since the unyielded state is detrimental to good mixing.

The behavior described here is encountered in industrial practice. For concentrated slurries in crystallizers, close-clearance impellers, especially anchors, are often observed to lead to poor movement close to the centre. In such cases, the walls can be exempt from fouling while the central shafts simultaneously build up heavy solid deposits at the centre. Consider a plant with a multi-m³ vessel, mixing with an anchor. At the relatively slow rotation rates typically practiced for anchors, the tip speed may be relatively high, but the tangential speed at the shaft surface will be very low.

The size of the central unyielded zone does not scale linearly with agitation rate, suggesting scale-up challenges: specifically, the need to operate an anchor excessively fast to eradicate the unyielded zone completely – if even possible. Estimating the corresponding tip speed or power-per-unit-volume value is thus of great practical relevance. Especially so, if the required mixing conditions are found to exceed the ratings of the motor, gear box, or shaft materials. While the experimental setup is no longer available to the authors of this study, such a follow-up study is a natural extension of the work reported here.

For CFD studies, this means that the absence of motion is not the only condition corresponding to an unyielded (and hence very poorly mixed) state. Velocity magnitudes are not a an exclusive criterion. A second possible condition for a cylindrical tank is the absence of gradient in the angular coordinate, i.e (Fig.8).

This criterion could prove quite informative in CFD studies comparing the effectiveness of different types of impellers agitating viscoplastic fluids.

Conclusions

Visual and video observations of an opaque suspension of a viscoplastic fluid confirm the existence of an unyielded zone in the center of a tank when using a close-clearance impeller. Such regions are detrimental to homogeneity and must be avoided, e.g., by (much) faster agitation or the use of a different geometry. The work reported here can be used to demonstrate the importance of understanding non-Newtonian rheology, as well as the practical implications thereof: namely, significant fouling issues at industrial scale. The materials are safe and the discontinuity is clearly visible. The quantitative dependence of the size of the unyielded zone on various parameters, such as agitation rate, can be characterized by image analysis. This dependence, especially upon scale-up, will be important to characterize for many industrial applications. Indeed, there is a significant gap in mixing knowledge with respect to yield stress fluids with close clearance impellers.

In addition to the well-known cavern phenomenon observed for small-diameter impellers, the results shown here for close-clearance impellers support the notion that yield strength is a realistic and useful engineering description of the physical world, in good accordance with [6]. Further work in the area should include systematic explorations of the effect of various parameters on the extent of the unyielded zone. The use of transparent fluids, such as those in [8], is highly recommended, as this would allow the quantification of not just the radius but also volume of the unyielded zone.

The authors would like to thank Rob Lind (Syngenta) for help with video and image processing, and Professor Christos Markides (Imperial College) for commenting on an early draft of the manuscript.

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* P. M. Piccione and A. Quarmby used to work at Syngenta at the time of this work: Process Studies Group, Syngenta, Bracknell, UK.

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