(1e) Mixing Times in Multiple Shaft, Multiple Impeller (MSMI) Stirred Tanks with Anchors in the Laminar and Transitional Regime

1. Introduction

Complex rheological fluids can develop problematic behaviours like ‘dead zones’ and caverning during stirred tank mixing due to microstructure rearrangement, resulting in uneven mixing. Such effects are common amongst formulated products like hair conditioner, paint and mayonnaise. Multiple shaft, multiple impeller (MSMI) stirred tanks with an anchor offer advantages for mixing these fluids due to their multifaceted approach. Many different impeller types may be installed for different purposes, operated at different heights and speeds, and used at different process stages, e.g., pumping, shearing/aggregate break-up or elimination of dead zones. The flexibility of the MSMI mixer with an anchor makes it suitable for a wide range of mixing applications. However, there is limited literature on optimising its performance.

Insights from studies on similar multiple impeller mixers can provide some guidance. Coaxial-type stirred tanks (bottom-driven anchor and top-driven agitator) shouldn't operate in counter-rotation mode as it increases power requirements without enhancing mixing [1]. For the inner impeller choice, radial impellers are preferred over axial ones because they generate more circulation loops which are disrupted by the anchor and contribute to mixing. Dual inner impeller setups, compared to single inner impeller designs, are found to reduce mixing time for a given power density, P_v, and improve efficiency [2]. In a dual-shaft setup without an anchor (e.g., Rushton-Rushton), optimal mixing involves significant height and eccentricity differences between the agitators, along with a combination of radial and axial impeller. However, counter-rotation is more effective in this arrangement [3]. The dynamics of an MSMI with an anchor are likely to incorporate heuristics from both mixer types.

2. Methods

The present work aims to develop the understanding of MSMI stirred tanks with anchors by visualising the mixing process via the acid-base decolourisation technique. This technique leverages pH indicator colour change to observe mixing time and dead zones via stoichiometric addition of an acid or base as a passive scalar. In this experiment, a 93% v/v glycerol solution (μ ≈ 0.4 Pa.s) chosen as the viscous, Newtonian fluid containing 0.3% w/w bromocresol purple was used in a 19L flat-bottomed tank (T = 289 mm) equipped with a top-driven anchor (D_a = 270 mm) and an eccentric impeller (D_e = 60 mm). Both Rushton and down-pumping pitched blade turbine (PBT_d) impellers were examined. Torque and power measurements were recorded using strain gauge transducers (TorqSense SGR510, Sensor Technology). Figure 1 highlights the relative simplicity of this technique.

To initiate each experiment, the glycerol solution was set to purple by adding aq. NaOH (10M). Bromocresol purple changes from purple (pH > 6.8) to yellow (pH < 5.2) when acid (aq. 10M HCl) is added. To ensure observation of macromixing rather than micromixing, an acid-to-base ratio of at least 2 was maintained upon addition. The acid was pre-mixed with a small sample of the glycerol solution to minimise viscosity/density differences between the acid and bulk solution. Videos were captured using a Fujifilm X-T3 digital camera (1080p and 50 fps) and processed using the Python image processing library OpenCV. The post-processing script identifies frames where the anchor is squarely oriented relative to the camera, applies a mask and calculates mixedness as the ratio of pixels above a threshold ‘mixed’ image to the total coloured pixels. A comprehensive exploration of this technique can be found in the study of Cabaret et al. (2007) [5].

3. Results

A 20-run, 2-level screening Design of Experiments (DoE) (resolution IV) was conducted to explore the main effects of 7 variables of the MSMI and ascertain their significance. Eccentricity (e/R = 0.38-0.55) and clearance (C/T = 0.35-0.50) of the eccentric impeller, eccentric impeller type (Rushton or PBT_d), rotation mode (co- and counter-rotation), acid addition point (r/R = 0.35-0.90) and anchor and eccentric impeller speeds (20-60 and 300-750 rpm, respectively) were studied. Therefore, both the laminar and early-transitional regimes were traversed (Re_anchor = 75-230 and Re_eccentric = 55-140).

The results show t95 values ranging from 66.26-600.00 s (capped due to poor mixing). Figure 2 demonstrates the spread as a function of power density, P_v. Unsurprisingly, t95 was shown to decrease with P_v but the general trend did not imply efficient mixing. Two of the fastest mixing runs, with t95 values of 77.55 s and 84.57 s, require power density inputs of 632.5 W.m^-3 and 294.2 W.m^-3, respectively, evidencing the potential for fast mixing with lower energy requirements through greater understanding of MSMI operation. The most influential factors on P_v were the anchor and eccentric impeller speeds which, in turn, appeared as the only statistically significant factors influencing t95, i.e., p-value < 0.05. However, given the nature of 2-level, resolution IV DoE, it is assumed that the relationship between impeller speed and t95 is linear and interaction factors (e.g., anchor speed*eccentric impeller speed) are unimportant since two-factor interactions are aliased with each other. It is highly unlikely that this is true and is one of the pitfalls to this type of experimentation.

Additionally, the results echoed those of coaxial stirred tank literature such that the counter-rotation mode increased anchor power demand [1]. In this mode, the two impellers appear to be working against each other which results in an increased demand for the anchor only (N.B., anchor speed had no significant effect on eccentric impeller power). Rotation mode was not deemed a statistically significant factor for predicting t95 so it is likely that the co-rotation mode is optimal for MSMI mixers as well. The two other statistically significant factors relating to power demand were impeller type and eccentricity. Of course, the Rushton impeller had greater power consumption compared to the PBT_d which agrees with power curve data [5], however, the eccentricity of the Rushton or PBT_d was seen to decrease anchor power requirements and further investigation is required to understand the mechanics behind this effect.

Neither eccentric impeller clearance nor injection point were found to be statistically significant relative to t95 or power consumption and will be dropped from further study. This is not to say that they have no effect but compared to the other 5 variables (eccentricity, eccentric impeller type, rotation mode and anchor and eccentric impeller speeds) their impact is insignificant. Since it is likely that non-linear relationships and interaction effects are present, a higher resolution DoE is necessary so it is beneficial to reduce the number of variables studied to minimise the total number of runs required for future study.

4. Conclusions

The present work lays the foundation for further, more detailed study of MSMI stirred tanks with anchors in the laminar and early-transitional regime for viscous, Newtonian fluids. Mixing time, t95, and individual impeller power requirements were observed and it was seen that eccentricity, eccentric impeller type, rotation mode and anchor and eccentric impeller speeds were statistically significant contributors to MSMI performance. However, the screening-type DoE carried out in this work assumes a linear relationship between these variables and outputs and disregards any potential interaction effects which is unlikely. Further work, using a higher resolution DoE (resolution V or greater), is required to determine the truth of the above statement. It is interesting to note the similarity between MSMI and coaxial mixers, i.e., co-rotation mode reduces power demand with no t95 penalty, but further work is required to understand other effects such as the impact of impeller eccentricity on the anchor power draw (where no inference can be drawn from coaxial literature). 2 of the 7 studied variables, eccentric impeller clearance and injection point, were found to be insignificant factors in terms of t95 and power draw and will be dropped from future study. Therefore, a reduced number of variables will be taken forward in the future work which sets out to explore MSMI operation to greater degree of detail, accounting for non-linear relationships and interaction effects, and study the impacts of more complex, non-Newtonian fluid rheology on its operation and performance.

References

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[2] A. Kazemzadeh, F. Ein-Mozaffari, A. Lohi and L. Pakzad, “A new perspective in the evaluation of the mixing of biopolymer solutions with different coaxial mixers comprising of two dispersing impellers and a wall scraping anchor”, Chemical Engineering Research and Design, vol. 114, pp. 202–219, Oct. 2016. doi:10.1016/j.cherd.2016.08.017

[3] S. Wang et al., “CFD simulation of hydrodynamics and mixing performance in dual shaft eccentric mixers”, Chinese Journal of Chemical Engineering, vol. 62, pp. 297–309, Oct. 2023. doi:10.1016/j.cjche.2023.03.004

[4] F. Cabaret, S. Bonnot, L. Fradette and P. A. Tanguy, “Mixing Time Analysis Using Colorimetric Methods and Image Processing”, Industrial & Engineering Chemistry Research, vol. 46, no. 14, pp. 5032–5042, Jun. 2007. doi:10.1021/ie0613265

[5] R. L. Bates, P. L. Fondy and R. R. Corpstein, “Examination of Some Geometric Parameters of Impeller Power”, Industrial & Engineering Chemistry Process Design and Development, vol. 2, no. 4, pp. 310–314, Oct. 1963. doi:10.1021/i260008a011