### Abstract

- Top of page
- Abstract
- Introduction
- Materials and Methods
- Characterization of Mixing Performance Using PLIF Data
- Results and Discussion
- Conclusions
- Literature Cited

The performance of KM static mixers has been assessed for the blending of Newtonian and time-independent non-Newtonian fluids using planar laser induced fluorescence (PLIF). A stream of dye is injected at the mixer inlet and the distribution of dye at the mixer outlet is analyzed from images obtained across the pipe cross section. The effect of number of mixing elements, fluid rheology, and apparent viscosity ratio for two-fluid blending have been investigated at constant mixture superficial velocity of 0.3 m s^{−1}. Aqueous solutions of glycerol and Carbopol 940 are used as the working fluids, the latter possessing Herschel–Bulkley rheology. The PLIF images have been analyzed to determine log variance and maximum striation thickness to represent the intensity and scale of segregation, respectively. Conflicting trends are revealed in the experiments, leading to the development of an areal-based distribution of mixing intensity. For two-fluid blending, the addition of a high viscosity stream into the lower viscosity main flow causes very poor mixing performance, with unmixed spots of this component observable in the PLIF image. © 2013 The Authors AIChE Journal published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers *AIChE J*, 60: 332–342, 2014

### Introduction

- Top of page
- Abstract
- Introduction
- Materials and Methods
- Characterization of Mixing Performance Using PLIF Data
- Results and Discussion
- Conclusions
- Literature Cited

Many process industry sectors, including food, home and personal care, catalyst and plastic manufacture, are tasked with the blending of highly viscous or non-Newtonian materials, often incorporating multiple immiscible phases. Applications include the blending of concentrated solid–liquid slurries, polymerizations, and the dissolution of solids or surfactants into liquids to form gels or complex surfactant/fluid phases. Due to the high apparent viscosities of some of these materials, the blending is performed under conditions which are predominantly laminar, which presents difficulties due to the lack of eddy diffusion which would assist mixing operations if the flow was turbulent.[1]

Overcoming this challenge has led to development of mixing strategies which aim to introduce chaotic flow to improve the performance; these have been employed in both batch stirred vessels[2] and inline continuous static mixers[3] which have been in use since the 1950s. Due to the complexity of the resultant flow fields formed in stirred vessels, substantial experimental and numerical studies on chaotic mixing have been undertaken to illustrate its potential to improve mixing.[2, 4] Experimental work has focused on the use of optical flow diagnostic methods such as Particle Image Velocimetry (PIV) or (Planar) Laser Induced Fluorescence (PLIF)[4] on transparent systems, which have enabled the development of methods to quantify mixing performance as a function of the flow field and fluid viscosity. Modeling has involved direct numerical solution of the Navier–Stokes equations (DNS), as well as other forms of computational fluid dynamics (CFD).[5] More recent work has extended these approaches to consider the blending of non-Newtonian fluids in stirred vessels, focusing on yield stress fluids.[6] This approach has raised understanding from an empirical level, where the entire mixing quality is based upon a single measured or derived parameter, to a multi-dimensional problem which considers the spatial distribution of mixing quality as a function of the fluid flow field and rheology.

In contrast, despite the industry drive toward continuous processing due to its improved sustainability (reductions in inventory and plant footprint), there has been little effort in obtaining equivalent understanding of non-Newtonian blending within continuous inline static (motionless) mixers, though limited design information for the blending of Newtonian fluids is in the public domain.[3] The blending of non-Newtonian fluids is complicated by a nonlinear relationship between the applied shear stress and the shear rate obtained within the fluid. Newtonian design equations rely on the linear coupling between these quantities described by Newton's law of viscosity. Mixing quality relationships are expressed in terms of a pipe-averaged shear rate, , which for the flow of a Newtonian fluid can be determined as

- (1)

where *K* is a constant (equal to 8 for a plain pipe and 28 for a KM static mixer used later in this study), *V* is the superficial pipe velocity, and *D* is the pipe diameter. For different types of static mixer, equivalent values of *K* are quoted which compensate for the increased wetted perimeter due to the mixer internals; increased dissipation due to changes in flow pattern (fluid deformation, stretching, and folding) and changes in fluid drag forces cause increased pressure drop over a plain pipe.[3] This shear rate is thus related to the pressure drop per unit length, a measure of the energy input to the fluid to obtain the required mixing and *L*/*D*, where *L* is the length of static mixer and *D* the diameter. Clearly, this approach is fundamentally flawed for non-Newtonian systems as Eq. (1) is no longer valid and any extrapolation must be carefully checked.[7]

The above parameters are usually related to the mixing performance expressed in terms of a coefficient of variance, CoV, (or the log variance). The CoV is defined as

- (2)

where *σ* is the standard deviation and is the average of the property (e.g., concentration) used to characterize the mixing through the device. Literature correlations[3] may be found which relate CoV to the length of static mixer required

- (3)

where CoV_{0} is initial coefficient of variance in the unmixed material and CoV is the coefficient of variance required by the mixing duty. CoV_{r} is the ratio of these two quantities, thus expressing the reduction in CoV required by the process. *K*_{i} is 0.87 for a Kenics KM static mixer in Newtonian laminar flow.

The CoV is often used as the sole criterion for characterizing mixing efficiency or performance. However, the reality is much more complex as while CoV gives a measure of the range of a mixing property after a mixing operation, this is only one dimension of the problem. Kukukova et al.[8] proposed segregation, which may be thought of as the degree to which a material is unmixed, as being composed of three separate dimensions. The first dimension is the “intensity of segregation,” which can be quantified by the CoV or alternatively by the log variance (LogVa) of concentration[8]

- (4)

where *C* is the normalized mixing quantity and *N* is the number of instantaneous measurements made on the mixing system. The second dimension is the “scale of segregation,” a length scale which for a static mixer can be related to the thicknesses of the striations produced.[9] The third is the exposure or the potential to reduce segregation. Choice of which mixing criterion is most important is often dictated by the downstream process, for example, a downstream reactor may require minimization of concentration gradients to ensure adequate control of product quality, in which case control of CoV is of greatest importance. Conversely, creation of a pre-emulsion passing into a downstream emulsification process may rely on control of maximum particle size and therefore scale of segregation is most critical.

This multi-dimensional approach has not yet been applied to determine mixing quality for non-Newtonian flows in static mixers. Of the limited information available in the open literature, work has generally focused on pressure drop measurements for time independent[10-12] and viscoelastic[13] non-Newtonian fluids in static mixers with only a few recent studies examining them in more detail.[13]

In this paper, a PLIF-based method is used to characterize blending of non-Newtonian fluids in a Kenics KM mixer as function of number of mixer elements (6 and 12 elements) and fluid rheology. The transparent model fluids used are a Newtonian fluid (aqueous solution of glycerol) and two time-independent shear thinning fluids (aqueous solution of Carbopol 940 polymer) whose behavior may be described using the Herschel–Bulkley model. The blending of two fluids is explored via addition of a secondary flow at the mixer inlet which has a volumetric flow equal to ∼10% of the main flow, enabling the blending of fluids with different rheologies. As in previous work,[4] the PLIF method is performed by doping the secondary fluid phase with fluorescent dye at the mixer inlet; the mixing pattern is thus obtained from images taken from a transverse section across the outlet of the mixer. From the images obtained, the scale and intensity of segregation are determined via calculation of values of LogVa and striation thicknesses respectively. A new criterion based on areal analysis of regions in the image with the same mixing intensity is proposed which combines aspects of both intensity and scale of segregation. Examination of these areal-based distributions of mixing intensity enables a deeper understanding of the complexity of the mixing to be elucidated which has the potential to provide useful information for process designers.

### Conclusions

- Top of page
- Abstract
- Introduction
- Materials and Methods
- Characterization of Mixing Performance Using PLIF Data
- Results and Discussion
- Conclusions
- Literature Cited

Analysis of PLIF images has been performed to determine the mixing performance of KM static mixers using Newtonian and non-Newtonian aqueous solutions as a function of number of elements and viscosity ratio of the two fluids. Analysis of the data using log variance for intensity of segregation and striation thickness for scale of segregation has demonstrated the importance of considering both aspects in tandem for correct interpretation of the mixing performance; considering only a single measure is a known problem in the literature.[8, 9, 19] A method is presented which considers the distribution of the cross-sectional area with a given intensity of mixing, this areal analysis combines both intensity, in terms of log variance, and scale, in terms of the fraction of the cross section with a given intensity. The method shows promise for the evaluation of mixing performance and can be considered as an addition to conventional approaches. The analysis does also to some extent identify striations of similar intensity, but identification of individual contiguous striations would be a useful future development. The identification of areas in the pipe cross section with a given range of log variance enables identification of regions where the mixing is performed down to the micro-scale, but also unmixed or poorly mixed regions in the flow. The analysis of PLIF images allowed the detection of viscous stream filaments evident as spots when a fluid of higher viscosity was injected into a lower viscosity continuous phase, which is not predictable using conventional design approaches. This new method shows promise in unraveling the complexity of information-rich PLIF images, beyond a sole number-based mixing criterion.

#### Acknowledgments

FA is funded by an EPSRC DTA studentship and Johnson Matthey. The PIV equipment was purchased using funds from EPSRC grants GR/R12800/01 and GR/R15399/01.