On the performance of static mixers: A quantitative comparison
Introduction
Numerous static mixers are extensively used in various homogenization processes in industrial operations like, e.g. polymer blending, chemical reactions, food processing, heat transfer, and in cosmetics and pharmaceutics, but also in waste-water treatment. They are moreover frequently applied in disposable applications, like in in situ mixing of two-component epoxy adhesives and sealants. The question posed in this paper is which static mixer is the best, in particular for systems where efficient mixing via turbulence is absent and mixing can only be achieved by chaotic advection, which is the repeated stretching and folding of material. Static mixers, also known as motionless mixers, are typically devices that contain static mixer elements in a cylindrical, or squared, housing. Elements and housing are made from metals or polymers, depending on particular applications like sustainable versus disposable.
The actual first patent on a static mixer goes back to 1874 where Sutherland describes a single element, multilayer, motionless mixer, used to mix air with a gaseous fuel [1]. Since the first application of static mixers in industry in the 1970s, a wealth of papers have been published that deal with various scientific and more technological questions. One of the first papers on static mixers is that of Thomas Bor discussing the application of static mixers as a chemical reactor [2]. Static mixers in process industry are initially developed for blending of fluids in laminar flow [3] and applications in heat transfer, turbulence and multiphase systems appear much later, although Nauman presents, also already in 1970s, a study on the enhancement of heat transfer and thermal homogeneity using static mixers [4]. Later he extends this work to studies on reactions taking place based on residence time distributions [5], [6]. Other simulations are conducted in the 1980s by Arimond and Erwin [7]. Since those seminal papers the number of publications have grown exponentially dealing with the applications, including liquid–liquid systems (e.g. liquid–liquid extraction), gas–liquid systems (e.g. absorption), solid–liquid systems (e.g. pulp slurries) and solid–solid systems (e.g. solids blending). The review paper of Thakur et al. [8] provides a nice summary that discusses in what applications static mixers are beneficial and to be preferred above stirred vessels and other conventionally agitated vessels. Over the years various groups work on mixing in those stirred and agitated vessels using both new experimental and computational methods, like for example Barrue, who focuses on particle trajectories in stirred vessels [9] and also studies mixing in the turbulent regime [10]. Important handbooks and textbooks that discuss fluid mixing and applications in process technology from a much wider perspective are those of Oldshue [11], Nienow et al. [12] and Kresta [13].
In general, in the evaluation of mixers for highly viscous liquids, two different criteria can be used to judge their efficiency: the first is energy consumption (measured, e.g. in terms of the dimensionless pressure drop) and the second is compactness (measured in terms of dimensionless length). Until now only a few studies have been reported that compare the performance of static mixers [14], [15], [16]. During the last decade, we developed in our group an efficient and accurate evaluation tool to analyze mixing in prototype flows, and later also in static and dynamic mixers, known as the mapping method. The method provides time or spatial resolved qualitative mixing profiles as well as a quantitative measure: the flux-weighted cross-sectional area-averaged intensity of segregation [17], see e.g. [18], [19], [20], [21]. One of the drawbacks of the original mapping formulation was the computational difficulty to compute the coefficients of the matrix. Later, our group and the group of Philippe Tanguy implemented alternative, simplified formulations [22], [23] both based on regular particle tracking. Using this new mapping method as a research tool, we evaluated and optimized various industrially relevant static mixers, like the Kenics mixer [24], the Ross Low-Pressure Drop (LPD) and Low-Low-Pressure Drop (LLPD) mixer [25], and the Sulzer SMX mixer including a recently developed new series of SMX mixers, the SMX(n) [26].
In this paper we mutually compare the global performance of all these mixers in the regime of laminar flow to find the optimum design using both criteria, i.e. pressure drop and length. We start with an overview on earlier approaches to quantify mixing in motionless mixers and most studied are the Kenics RL-180 mixer, with a right–left twist and an angle of blade twist of 180°, and the Sulzer SMX. The first is characterized by its relatively simple geometry, see Fig. 1a, which gives excellent mixing at low pressure gradients at the cost of long lengths. The second mixer, see Fig. 1c, represents the complete opposite and combines compactness with a complex geometry that needs large pressure gradients to sustain the flow. Apart from compactness and energy use, a third issue is whether mixer geometries can be (injection) molded which is of utmost importance for disposable applications, which represent a huge market. Since basically only simple geometries can be (de)molded, disposables generally suffer from too long lengths, which implies a large wasted volume. Also this aspect is part of this study and via a thorough analysis of the elementary working principles of the various designs a new geometry is proposed that combines compactness with simplicity of shape.
In the eighties it proved that even the simple geometry of the Kenics mixer was too complex for a detailed analysis and the so-called PPM, partitioned pipe mixer, geometry was proposed instead [27]. It consists of straight plates with length 1.5D, crossing each other under an angle of 90°, placed inside a rotating tube with diameter D. The transverse flow in the PPM is comparable to that in the Kenics mixer, allowing relevant mixing analyses, but this only holds for Newtonian fluid flows. When shear thinning is present, the difference in driving mechanisms, pressure flow in the Kenics versus drag flow in the PPM, yields significantly different velocity fields, and thus different mixing. In particular pressure-driven mixers are far less sensitive to changes in the rheology. From a theoretical point of view the rather simple PPM has several advantages and simplifying assumptions can be made. One of the main assumptions is that axial and radial flow can be decoupled and expressed in closed-form analytical relations; a second simplification is that transition zones from one blade to another are infinitely thin. With these assumptions, the theory and dynamical tools developed for prototype flows based on concepts of chaotic advection as introduced by Aref [28], [29] could be applied. Poincaré maps, three-dimensional islands separated from the main flow by KAM (Kolmogorov, Arnold and Moser) boundaries, were detected and minimalization of these KAM tubes formed the basis for mixing analyses and optimization by different groups, i.e. Khakhar et al. [27], Meleshko et al. [30], Muzzio [31], and Ottino [32]. Also the standard Kenics geometry allowed for numerical analyses, starting in 1995, see [33], [34], [35], [36]. Dynamical system techniques were applied to understand and optimize mixing, see [31], [32], [33], [34], [35], [36], [37], [38], and applied to the Kenics geometry [39], in the low [40], and high [41], [42], Reynolds number regime, all the way to turbulent flows [43], [44], [45], [46], and the use of the Kenics as reactor [47], [48], [49]. The group of Muzzio developed analytical and numerical methods to quantify stretching distributions in these types of mixers, which is far from trivial since interfaces grow exponentially in time. A somewhat related study was conducted by Galaktionov et al. [50], who extended the original mapping method to also map a microstructural variable, the area tensor, in time. One of the striking conclusions of that study was the importance in choice of the mixing measure while comparing different designs of a mixer. In particular, it was found that in pressure driven mixers, like static mixers, a mixing measure should be flux weighted. The distribution of material close to walls is far less interesting and important compared to the material in a zone of maximum velocity. In almost all studies the rheology was chosen to be Newtonian, where some groups also included the influence of shear thinning [51], both theoretically [52], and experimentally [53], as well as the influence of two phase flows [54], [55], [56]. Interestingly, the Kenics mixer is often used to promote heat exchange [57], and the optimum geometry for mixing a 50–50% equal viscosity fluid differs from that for exchange of wall fluid [21], which basically is a warning against too rapid conclusions. Finally, the availability of proper experimental data concerning quantification of mixing quality are scarce; the best data result from the PhD work of Jaffer [58].
Similar analyses of mixing in the SMX designs seriously suffered from its geometrical 3D complexity where meshing becomes non-trivial and details of the boundary conditions are important. Only few groups were capable of performing simulations, like Muzio [59], [60] and Tanguy [61]. Application of methods from dynamical systems theory is complicated due to the inherent numerical nature of the velocity field and any attempt of optimization remained a problem given the limited number of computations that could be performed in reasonable time [62]. Interestingly it was believed, even by the manufacturers of the Sulzer SMX, that the standard geometry, which we called the SMX (2, 3, 8) based on the number and structure of the crossing bars inside the mixer, performed optimal, since neither experimentally nor via modeling improvements were found [63]. The first proper analysis of mixing in the SMX geometry had to wait until the new way of computing the components of the mapping matrix was developed and incorporated as a post-processing operation added to standard CFD software [23]. Understanding mixing, even in these complex geometries, is based on investigating the optimum interface stretch in the cross section of the device, which is extensively dealt with in Section 2 below, and resulted not only in improvements in the design of the SMX but moreover in a complete new SMX series, the SMX(n) (n, Np, Nx) = (n, 2n − 1, 3n), see [26].
Two more, important application areas of motionless mixers exist. The first where the aim is producing stratified systems with uniform layer distributions and the prime example here is the beautiful Multiflux mixer, developed by Sluijters while working at Akzo, and originally meant to improve the melt temperature homogeneity in spin lines [64], [65]. The working principle of the Multiflux most closely realizes the perfect bakers transformation of stretching, cutting and stacking, see [66] and the overview Mixing of immiscible liquids in [67]. Modifications of the original design were the addition of so-called I and H elements to even improve the homogeneity in layer distribution [68], and splitting not in the middle, to realize a hierarchical layer thickness distribution [69]. The second area is that of microfluidics where, since the Reynolds numbers are small and the Péclet numbers are large, chaotic advection is again the only way to mix fluids in reasonable time, see the overview Scale-down of mixing equipment: microfluidics in [70] and the review paper by Hessel et al. [71]. Basically people working in microfluidics reinvented what was already rather well known in polymer processing, where not the small channel dimensions but the high viscosities require efficient mixing in the laminar flow regime. All designs realized on the micro-level have in common that they possess a simple geometry such that they can be realized, e.g. on the interface between bottom and top part of the device. Optimized mixers create crossing streamlines, the prerequisite for periodic points around which chaotic advection is realized, by generating two counter rotating vortices of unequal size in the cross section of the channel, that change position during axial flow. The prime example is the staggered herringbone mixer, see Stroock et al. [72] for precise and spectacular experiments, and later [73], [74] for computations and optimization. A second example is the barrier-embedded mixer [75], which uses slanted grooves at the bottom of the channel with a barrier placed at the top, see [76] for explanation and optimization. The third example is the serpentine channel. Originally, this one only worked at Reynolds numbers higher than a critical value, Re > 50, see [77] for design and [78] for calculations. Later, it was recognized that the original serpentine mixer relied on inertia to induce folding, and the splitting serpentine channel was developed, greatly improving its performance in the low Re number regime by completely changing the working principle [79], [80], [81]. Further optimizing the geometry resulted in even better layer distributions by bringing fluid from outwards inwards and vice versa. This design is in our lab also macroscopically realized notably on the two halves on a mold for injection molding, making the fabrication of multiple layers during the injection process possible. Finally based on these principles an interesting multiple flow splitting and recombining design was developed, also for use on the macroscopic scale [82].
After these excursions, partly to mixing on the microscale, we are back on the macroscale and start the analysis of the flow inside the most commonly used motionless mixers, in order to understand how they stretch interfaces, which is the basis of making better new designs. This is the topic of Section 2 presenting the qualitative results. Section 3 reports the quantitative results where the mapping method is applied, and the flux-weighted cross-sectional area-averaged discrete intensity of segregation is computed, that allows optimizing the mixers using the two different criteria, compactness and energy use. Given the relevance of disposable mixers with small volumes, also (de)molding issues are sometimes touched when appropriate.
Section snippets
Interfacial stretch
Fig. 1 shows the mixing devices that are compared in this paper, and clarifies in the legend the notation used, and Table 1 reveals their characteristic dimensionless length, L/D, and pressure drop ΔP* (=ΔP/ΔP0 where ΔP is the pressure drop per element and ΔP0 is the pressure drop in an empty pipe with the same diameter D), and their mixing efficiency expressed in the characteristic interfacial stretch generated. Nelem reflects the number of elements of the device. These characteristics form
Quantitative results
The qualitative results of the mutual comparison of the different motionless mixers of Fig. 1 in Section 2 gave understanding of their working principle which resulted in their mutual comparison using energy consumption and compactness as criteria, see Table 1 and Fig. 8, and in a new mixer design that combines all beneficial extra's: the SMX (1,1,4, θ = 135°). Here we compare all designs of Fig. 1 in a quantitative way, by computing the flux-weighted cross sectional area-averaged discrete
Conclusions
This study on the performance of static mixers is conclusive. Explained is why interface stretch can be optimal and, based thereupon, why some alternative versions in the same design family perform better. Based on the Mapping Method, the interface stretch in the different motionless mixers is visualized and quantified. Two surprises give better interface stretching than expected on first sight, based upon which a new mixer design is proposed that combines the two beneficial effects. The
Acknowledgement
The authors thank the Dutch Polymer Institute (DPI) for financial support (grant # 446).
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