Mixing Of Newtonian And Viscoelastic Fluids Using “Butterfly” Impellers Of Newtonian And Viscoelastic Fluids Using “Butterfly” Impellers,

The mixing of high viscosity and viscoelastic fluids may be performed in industry using a “butterfly” impeller, which has received scant attention. This paper describes the characterisation of these impellers using both Newtonian and viscoelastic (Boger) fluids with an identical base viscosity under laminar flow conditions in batch tanks without baffles. Measurements are made using two rotational speeds ( N = 30 or 60 rpm) over a wide range of impeller to tank diameter ratios from 0.53 to 0.98. Particle Image Velocimetry (PIV) and Planar Laser Induced Fluorescence (PLIF) have been used to determine local velocity fields, shear rates, flow numbers and mixing times respectively, with impeller power number Po obtained from torque measurements. A down-pumping flow pattern is observed, with maximum normalised velocities of 25% of the impeller tip speed observed in the axial plane, increasing to 60% in the rotational plane due to solid body rotation. Angle resolved measurements found these to be slightly increased as the impeller passes the image plane, with viscoelastic fluids displaying a greater uniformity of velocity across the image planes due to elastic energy storage. Maximum shear rates are greatest between the impeller and the wall, yet remain low i.e. O (10) s -1 . Mixing times obtained from PLIF show that mixing times in viscoelastic fluids show a slight reduction compared to the Newtonian equivalent. The impeller power number was determined as Po = 0.6 in the turbulent regime, while the laminar constant K p and the Metzner-Otto constant k s were equal to 122 and 16.0 respectively. rpm.  Mixing times longer in viscoelastic fluids


Introduction
Agitated stirred tanks have been the subject of considerable investigation in the open literature.
Research has been carried out into the flows of various different fluids with both Newtonian and non-Newtonian rheology in vessels with many different impeller geometries, often with the primary aim of characterising the effectiveness of new blade designs (Ihejirika and Ein-Mozaffari, 2007;Szalai et al., 2004) and the development of scaling rules (Paul et al., 2003). The formulation of fluids with complex rheology is often carried out in a batch configuration; this includes complex multiphase products such as paints, inks and ceramic pastes which often require agitation to achieve the homogeneous dispersion of solids throughout a liquid matrix. At high solids loadings, these fluids display highly viscous or viscoelastic behaviour, and therefore mixing takes place in laminar flow. It is commonly vital to minimise or prevent the entrainment of air or other gases in order to ensure product consistency, as often it is exceedingly difficult to remove entrained gases from the viscous medium afterwards (Paul et al., 2003). An agitator often referred to as a "butterfly" impeller is often employed in batch systems for such mixing duties by manufacturers such as Westerlins (Malmö, Sweden) and VMA-Getzmann GmbH (Reichshof, Germany); an example of such a design is shown in Figure 1 with three hollow triangular blades attached to a cylindrical hub.

Figure 1: Typical butterfly impeller
Despite the myriad of impeller geometries subject to academic scrutiny there is, surprisingly, no work available describing the performance of butterfly impellers in any geometry. These impellers are thought to be advantageous by causing minimal entrainment of air during mixing, as well as the belief that they provide a uniform shear environment when operated in a close-clearance configuration, with impeller to tank diameter ratios D/T typically in excess of 0.9. However, much of this knowledge is based on operational experience and has minimal grounding in rigorous scientific investigation. Thus the mechanisms these impellers use to achieve mixing are poorly understood, which makes the design α and optimisation of processes using these impellers difficult to implement, which has implications for process efficiency and overall product quality.
The complexity of the mixing operation is driven primarily by the highly nonlinear constitutive (viscoelastic) behaviour of the fluid products, which is a function of the polymeric liquids and multiple phases (particles, bubbles, drops) present (Paul et al., 2003). Whilst viscoelastic fluids have been the subject of much investigation over the years, beginning with Weissenberg's observation of rod-climbing phenomena (Weissenberg, 1947), there has been little in the way of investigation into their effect on processes, although stirred tank geometries are the most well developed area. The majority of previous research has focussed on bulk mixing effects such as mixing times or power draw, with only a few select studies observing the fluid hydrodynamics within the fluid itself (e.g. Ozcan-Taskin and Nienow, 1995). Little research has been performed into observing flow structures or their relation to level of fluid elasticity, with Stokes' (Stokes, 1998) work into swirling flow being a notable exception. Both works characterised the elasticity through determination of the Elasticity number El, which Stokes found to be the controlling parameter in whether flows are elastically or inertially driven. Where flow dynamics have been investigated, most work has been performed using dye decolourisation (Lehwald et al., 2010;Shervin et al., 1991) or Particle Image Velocimetry (PIV) (Adrian, 1991;Chung et al., 2009). The fluids used in these investigations vary, however they are all optically transparent. A class of fluids called "Boger" fluids, which have a constant fluid viscosity and measurable elastic component, have been used in some studies as they allow for relatively simple separation of viscous and elastic effects (Boger and Yeow, 1992).
Despite Planar Laser Induced Fluorescence (PLIF) having been used to assess mixing in many systems (Adams and Barigou, 2007;Alberini et al., 2014;Guillard et al., 2000;Hu et al., 2010;Hu et al., 2012;Law and Wang, 2000;Szalai et al., 2004) the application of the technique to viscoelastic fluids is limited. Stokes & Boger (Stokes et al., 2001) studied the mixing performance of swirling flows using PLIF, highlighting the presence of different elastic regimes in a simple geometry.
Further, though the calculation of mixing times using PLIF has received some attention (Busciglio et al., 2014;Hu et al., 2012;Zadghaffari et al., 2009;Zhang et al., 2013), little is available on the application of this method with viscoelastic fluids. Ozcan-Taskin & Nienow (1995) investigated flow fields and calculated mixing times for low elasticity fluids using a fibre-optic technique, additionally calculating mixing times in viscoelastic media, whilst a colorimetric method was adopted by Chhabra et al. (2007).
The application of close-clearance impeller geometries in the blending of highly viscous or viscoelastic fluids has been previously performed with both helical ribbons (Ihejirika and Ein-Mozaffari, 2007) and the Maxblend design (Fontaine et al., 2013). These geometries are almost universally employed in the blending of complex fluids, as they all possess low power draw in laminar flow. Direct torque measurement has been performed to obtain power curves in laminar flow (Fradette et al., 2007;Zhang et al., 2013;Fontaine et al., 2013), whilst flow measurements have been found using both intrusive methods such as Laser Doppler Anemometry (Jahangiri, 2007) and nonintrusive methods investigating opaque systems (Patel et al., 2014;Ihejirika and Ein-Mozaffari, 2007) and transparent varieties (Stobiac et al., 2014;Chhabra et al., 2007;Shervin et al., 1991). PIV measurements have become prevalent as the method of choice for non-intrusive determination of flow fields, owing to their non-intrusive nature and ease of application.
This work seeks to characterise the mixing of "butterfly" impellers via investigation of their mixing performance. Flow dynamics have been determined from PIV measurements, power draw and energy consumption from torque measurements and mixing times from PLIF. Their behaviour has been examined using fluids of increasing complexity using transparent Newtonian and Boger fluids with the same base viscosity. Key performance parameters such as impeller power number and pumping capacity, local shear rates and mixing times have been evaluated over a range of industrially relevant process conditions.

Experimental Set-Up
Experiments were performed in an unbaffled batch stirred tank constructed from poly(methyl methacrylate) using a range of impeller diameters D between D = 80 mm and D = 148 mm and two rotational speeds N (N = 30 rpm & N = 60 rpm) at a tank diameter T = 152 mm. The impellers were constructed of stainless steel, with three hollow triangular blades affixed to a cylindrical hub with a diameter of 18 mm. The impellers were driven by a Heidolph RZR-2102 laboratory stirrer (Heidolph UK). Other geometric parameters were kept constant, and are listed in Table 1 below. Figures 2a & b displays the full experimental set-up. Observations made using PIV were taken in the axial (r, z) plane through the vessel diameter, and also at five rotational (r, θ) planes at constant z as shown on Figure   2b. The range of experimental conditions selected is typical of the industrial rotational speeds and impeller-to-tank diameter D/T ratios that butterfly impellers are typically employed in. Reynolds numbers are such that flows are in the laminar and low transitional region; owing to the polymeric turbulent drag reduction and the unbaffled nature of the vessel it is expected that the laminar regime persists beyond Re = 10, and therefore the effect of the onset of turbulence is expected to be minimal.
Further, these values are based on the impeller tip speed which may provide an overestimation of Reynolds number; see Figure 4 and the ensuing discussion for further justification.

Fluid Formulation
Fluids were formulated using a bench-scale laboratory mixer. Dilute polymer solutions of polyacrylamide (Sigma Aldrich, UK) and carboxymethyl cellulose (CMC) (Sigma Aldrich, UK) in aqueous glycerol (ReAgent, UK) were formulated, with water used to make up the remainder. Sodium chloride salt was added to aid dissolution of the polymers. Boger fluid formulations were selected in order to provide viscosities and elasticities similar to a range of industrially relevant viscoelastic materials. Rheological characterisation was performed using a Discovery Hybrid HR-2 and an AR-2000 rheometer (TA Instruments, USA) using a 40 mm 4° cone and plate geometry over the range of shear rates ̇ between 0.1 and 100 s -1 . This shear rate range was selected as an approximation of the tank-averaged shear rate using the Metzner-Otto correlation ( ̇ ) found the value to be 11.5 s -1 for a down-pumping pitched blade turbine (PBTd), which as an initial approximation is within the measured range. Normal stress differences N 1 , which are viscometric material properties arising as a result of differences in normal components of the stress tensor (Phan-Thien, 2002), were obtained during the acquisition of shear stress versus shear rate data on the same instrument through axial force measurement.
Relaxation times were then calculated from normal stress difference data and values were fitted to a power law model (Ozcan-Taskin, 1993) such that: where ψ 1 is the first normal stress coefficient, η the fluid apparent viscosity, N 1 the first normal stress difference, ̇ the shear rate, while a and b are constants. The compositions of the model fluids used and their rheological parameters can be found in Table 2   As Figure 3a shows, both Boger fluids possess similar viscosities, indicating that glycerol concentration controls overall solution viscosity and thus allows investigation of the varying elasticities of each fluid independently. These elastic responses are shown in Figure 3b: the outlined symbols show the response of first normal stress difference N 1 with shear rate whilst the filled symbols display the relaxation time. For both materials N 1 increases with increasing shear rate, however the rate of increase decreases at higher shear rates, indicating that the expected quadratic response of a "true" Boger fluid is only valid at low values of . However, it can be seen that over the whole measured range, Boger A displays lower values of N 1 , and thus lower levels of elasticity. This is also observed in the relaxation time, λ, trends. Furthermore, it can be seen that the relaxation time follows the expected trend of a power law model, as such fitting parameters are shown in Table 2.
Note that normal stress measurement is highly sensitive, and as such the fitting parameters possess large standard deviations.
At the experimental conditions used for PIV and PLIF, experimental Reynolds numbers Re are laminar and vary between 20 and 134, as defined by: ( where ρ is the fluid density.

Power Curve Determination
Power number curves were determined through direct torque measurement from the impeller shaft.
Power draw is related to torque through the relationship: where P is power, N is impeller rotational speed and M the torque on the impeller shaft. Additionally, power can be determined through assessment of the drag caused by the fluid using the impeller power number: where ρ is the fluid density and Po the impeller Power number, a form of drag coefficient. Equating (4) and (5) allows for calculation of Power number from direct torque measurement. An optical TorqSense RWT-420 (Sensor Technology, UK) sensor was attached to the impeller shaft which was driven by a Heidolph RZR-2120 laboratory stirrer motor (Heidolph, UK). Data was acquired and analysed using TorqView software (Sensor Technology, UK).
Measurements were taken over a range of impeller speeds using different Newtonian and shearthinning fluids enabling coverage of the laminar, transitional and turbulent flow regimes (0.4 < Re < 50 000). Water, glycerol, 12 500 cSt silicone oil and a shear-thinning solution of aqueous carboxymethyl cellulose were used, with torque data linearised and plotted graphically in order to determine the parameters Po, laminar gradient K p = Re.Po and Metzner-Otto constant k s respectively according to the method described in Rodgers et al. (2009). The systematic torque M 0 was also directly measured in a fluid-free system to determine the veracity of the curve fit, with a good degree of agreement found. Determination of the power number curve for a standard six-blade Rushton disc turbine (RDT) in a baffled vessel was also performed across all flow regimes to provide a comparison using a known geometry.

Particle Image Velocimetry (PIV)
PIV was performed on all stirred tank geometries (Table 1)  Neutrally buoyant silvered glass spheres of 10 μm diameter (Dantec Ltd., UK) were utilised within the flow. A laser sheet with a thickness less than 0.5 mm at the focal distance of the focussing lens used (500 mm) was employed to illuminate the observation plane, with the brightest portion of the sheet (at the focal distance) aligned with the centre of the field of view. The impeller blades and shaft were painted with a matte black coating to eliminate laser light reflections from surfaces. 500 image pairs were obtained for each experimental condition. The data were analysed as an ensemble and by separating successive images to obtain flow fields resolved by the position of the blade (angleresolved). Flow fields were obtained using a recursive Nyquist grid, with initial and final interrogation areas of 64 x 64 pixels and 32 x 32 pixels respectively, with a resolution of 55 μm pixel -1 used for each image. The time between each laser pulse ΔT was set to 1 200 μs; this value was selected in order to observe the fastest moving particles in the flow moving no more than one quarter of an interrogation region, based on a maximum rotational speed of 60 rpm and a corresponding impeller tip speed of 0.46 m s -1 , i.e. the largest possible velocity in any flow in this study.
Additionally, this value of ΔT was sufficiently low to ensure that particles did not move out of the image plane during capture. PIV measurements have previously been found to give velocities accurate to 1-2 % of the maximum (Adrian, 1991). Plots were generated from bespoke MATLAB code, with local spatial gradients being determined using a central difference method. All velocity data are displayed in a normalised form, with local velocities u shown as a fraction of impeller tip speed u t .
PIV data was used to determine impeller pumping number N Q : where Q is the impeller pumping capacity calculated from velocity data using the standard approaches described and used by Gabriele et al. (2009), amongst others. . N Q is a measure of the effectiveness of the impeller at circulating fluid within its vessel, and thus allows for comparison between different process conditions and impeller types. The flow rate from the impeller was split into two components, the axial flow N Q,z and the radial component N Q,r . These were calculated from: Time-averaged PIV data was used for this analysis, as this automatically provides the integral across all impeller angular positions θ. The remaining integral was calculated using bespoke MATLAB code using the trapezium rule, owing to the equal spacing of PIV velocity data after data post-processing.
This provides a minimum of 42 individual velocity measurements in for the determination of Q z and 68 data points for calculating Q r . The ratio of radial and axial pumping numbers, ⁄ , was calculated to provide an assessment of impeller pumping type, i.e. radial flow, axial flow or mixed flow. The integral limits, i.e. the geometrical extent of the impeller cross-section through the flow, was selected in accordance with standard procedures assuming that the impeller sweeps a cylindrical surface during rotation.
Shear rates have been calculated using time-averaged velocity fields according to Bird et al. (1960): where v is the velocity and subscripts x, y and z represent the principle axes in Cartesian co-ordinates whilst r, z and θ represent the radial, axial and rotational components of the flow respectively in cylindrical co-ordinates. Owing to the two-dimensional nature of PIV data acquisition, several components of the overall shear rate cannot be directly calculated from the two image planes separately. The missing components have been neglected based on the assumption of uniformity of rotational velocity gradients in the r-z plane, and negligible axial gradients in the r-θ plane. As the laser light sheet possesses a narrow thickness compared to the size of the PIV interrogation cell for data processing, it is assumed that the through-the-plane velocity components are uniform across the whole image and thus any spatial gradients are negligible.
Additionally, these shear rates can then be coupled to the local relaxation times, providing a map of local Weissenberg number magnitude which ratio of timescales of material and process deformation timescales, as described by: where Wi is the Weissenberg number and λ the fluid relaxation time. Wi has been determined both locally and as a tank average using a Metzner-Otto relationship ̇ (Ozcan-Taskin & Nienow 1995). The Weissenberg number can also be used to determine bulk elastic effects and to elucidate different elasticity regimes within process geometries. This is achieved through the Elasticity number, El: Owing to the difficulty of selecting an appropriate length scale for determining local Reynolds numbers, tank-averaged Weissenberg and Reynolds numbers have been implemented to determine average Elasticity numbers.

Planar Laser-Induced Fluorescence (PLIF)
PLIF was implemented using the same apparatus as PIV using the vertical imaging plane only with Rhodamine 6G (Sigma Aldrich, UK) as the fluorescent tracer added. Images were captured at a rate 0.5 Hz. A 545 nm cut-off filter lens was added to the camera in order to eliminate all but the light emitted by the fluorescing dye (λ = 560 nm). Mixing times for the D/T = 0.98 condition were calculated from the logarithm of the local concentration variance (Brown et al., 2004): where σ is the concentration variance, θ is the current time, n is the maximum number of data points (in this case the total number of locations), and C' a,i is defined as: where C a,i ' is the normalised concentration of component a at location i, C a,i is the concentration at location i, and the subscripts 0 and ∞ indicate the initial and fully-mixed conditions respectively (Grenville and Tilton, 1996). 90% and 95% confidence values were selected to determine final mixing time , corresponding to values of log σ 2 of -2.0 and -2.6 respectively. Three separate virtual probe locations were utilised during image analysis in order to determine final mixing time, each probe consisting of 40 (20 x 20) pixels; these are shown on Figure 2b. A calibration of the local greyscale value to dye concentration was performed to ensure all recorded greyscale values were linearly related to concentration; see Hall et al. (2005) for further details. Raw data images were preprocessed to eliminate the effect of diminishing laser light intensity across the geometry using a pixelby-pixel calibration and then analysed using a bespoke MATLAB code.
In order to compare performance of "butterfly" impellers to other geometries, the use of theoretical mixing time correlations have been implemented. The dimensionless mixing time has been evaluated for all conditions, as it has been observed that this value is constant for turbine-type impellers in Newtonian fluids in laminar flow (Grenville & Nienow, 2004).

Power Draw Measurements
The constructed Reynolds number Re versus Power number Po graph is displayed in Figure 4. Error bars display the standard deviation of the power draw at the given Reynolds number.  (Paul et al., 2003), providing confidence in the accuracy of the torque sensor method. The "butterfly" impeller possesses a turbulent Power number of 0.6; this value is consistent with the trends reported for other unbaffled systems (Hall et al., 2004). from literature (Hall, 2005) included for comparison. Comparing the values with that expected of a similar impeller such as a PBTd, the value of turbulent Po is noticeably lower (Po ~ 0.8 for a PBTd). This can be attributed to the minimal cross-section of the "butterfly" impeller through the flow due to the large hollow within the blade, reducing the value of Po in the turbulent regime. The laminar gradient K p is greater than that of a RDT, indicating increased energy efficiency in the laminar regime. The value for a butterfly impeller is similar to that of a Maxblend impeller, albeit slightly lower due the butterfly geometry's low surface area, indicating suitability for application in viscous media. The calculated impeller Metzner-Otto constant k s is slightly higher than that expected of a pitched-blade design, however is of a similar magnitude to RDT and PBT geometries and significantly lower than those of close-clearance impellers such as anchors and helical ribbons (Paul et al., 2004). Additionally, it can be noted that the data presented are for all the impeller diameters listed in Table 1, and can be seen to collapse onto one master curve. This is consistent with previous work showing Power number to be a weak function of impeller-totank diameter D/T ratio (Harnby, Edwards & Nienow, 1992).

Effect of Impeller to Tank Diameter Ratio (D/T)
The time-averaged velocity fields within the glycerol system in the axial plane as a function of D/T are shown in Figure 5. there is a region of very low velocity indicating a very low level of radial pumping. In addition, the greatest velocities can be seen at the upper and lower impeller blade edges. Also, the impeller appears to make a significant contribution to the velocity in this plane, with large velocities observed.
As can be seen, as the value of D/T approaches unity, the bulk flow discharged by the impeller becomes increasingly confined by the wall and as such bulk fluid velocity magnitudes decrease. This phenomenon has been observed in other unbaffled geometries (Hall, 2005), owing to the dominance of solid body rotation in the impeller region within these flows. However, "butterfly" impellers are typically used operationally at close wall clearances, i.e. D/T > 0.9. Therefore, this study shall hereafter focus on results obtained in the D/T = 0.98 geometry (also referred to as the close-clearance scenario), as this is typical of the industrial scenario and therefore of greatest relevance to this study.

Effect of Fluid Rheology
The time-averaged normalised velocity fields for the fluids Glycerol, Boger A and Boger B are shown in the axial plane in Figure 6, with rotational planes shown in Figure 7 taken at the image planes indicated in Figure 2b. Please note the difference in colour scales between these figures; these have been selected for clarity based on the maximum normalised velocity within the image plane.
Additionally, the rotational sections do not include any transects taken at z = 150 mm owing to exceptionally low observed velocities (< 1% of the impeller tip speed). midway between the impeller shaft (r = 0 mm) and the vessel wall (r = D/2 mm). Radial velocity components have been found to match those found in axial plane measurements, indicating that the increase in observed velocity in this region is due to the rotational velocity component. The region of greatest normalised velocity is found directly beneath the impeller in the Newtonian system, while it clearance C) in the viscoelastic systems. This change is due to the aforementioned shift from axial to radial impeller discharge.

(a) Glycerol (b) Boger A (c) Boger B
It is known that viscoelastic fluids, due to their relaxation time, display a time dependence that may not be visible in a time-averaged flow field. It is known that instantaneous velocities and shear rates can be greater than the average of the field, and as such must be investigated. Therefore it is prudent to observe the angle-resolved data in parallel with time-averaged data, as illustrated in Figures 8 & 9.
All angle-resolved data is shown at position θ = 0° relative to the impeller, as shown in Figure 2b.  further corroborating the shift in change in impeller discharge type. It can also be observed that for all angle-resolved measurements, the local velocities displayed by Boger B are greater than those for Boger A. The increase is also seen in time averaged measurements but not to the same extent, indicating that during the impeller passage local velocities are greater in the more elastic fluid, which is consistent with the increased energy storage capacity these materials display.

Impeller Pumping Numbers
Impeller pumping numbers in the axial plane, calculated as described by Equation (6), are displayed in Table 4. A small amount of elastic component displays no significant change to the overall impeller pumping capacity. However, as polymer concentration increases there is a significant reduction in impeller pumping capacity as the fluid becomes more elastic. This can be attributed to the increased storage of energy within the fluid, as expected with a viscoelastic medium, compared to the Newtonian equivalent. This reduces the kinetic energy, and thus momentum, transferred to the fluid and therefore lowers the fluid velocity.
As observed in the velocity fields, an increase in the magnitude of fluid elasticity leads to an increase in the ratio of radial to axial pumping number N Qr /N Qz , it appears that the presence of an elastic fluid component drives the flow from a down-pumping axial impeller discharge to a radial pattern. This phenomenon is also observed in the base velocity data, where the flow pattern changes from a single circulation to two radial flow loops. Note that this effect is diminished under the confined flow of D/T ratios of greater than 0.8, as wall effects prevent complete development of purely impellergoverned flow patterns in these geometries. The axial motion reduction can be attributed to the generation of secondary flows within viscoelastic media, generated by normal stresses as the material deforms. These flows act contrary to the principle flow generated by the impeller, causing a reversal of flow element direction, albeit at a significantly lower magnitude than the primary flow. This exhibits itself as a combination effect, creating a shift from axial to radial flow within the impeller.
This is expected to have a significant on the mixing performance of viscoelastic media, as a reduction in axial mixing may lead to longer mixing times as the amount of material transfer across the height of the vessel is diminished.

Shear Rates
In addition to the velocity fields, it is also necessary to assess the shear performance of the impeller geometries. Of critical importance is the maximum shear rate within the vessel, as this dictates the maximum deformation rate the material is subjected to. Figure 11 displays the time-averaged shear rate in the axial plane, and the rotational plane at z = C which displays the region of greatest rotational shear rate.
As the maximum shear rates observed are an order of magnitude lower than this value, is likely that true Couette flow does not occur within the impeller clearance. This suggests that the effect of a hollow blade significantly reduces shear rates within the vessel owing to the reduction in flow confinement at the wall. As a possible explanation, it may be that the impeller tip speed is not the most suitable choice of characteristic speed, nor the impeller diameter characteristic of the distance required for calculation of the flow Reynolds number or shear rate in between the impeller and vessel wall. This will in turn affect the determination of impeller performance, as it is expected that Reynolds numbers are currently significantly over-predicted therefore leading to lower impeller effectiveness at a given Reynolds number. A caveat is that this maximum shear rate will only be observed for a very short time as the impeller immediately passes the observation plane. Additionally, the resolution of the velocity field is of the order of the wall clearance (i.e. ~2 mm), and thus may underestimate the local values of velocity gradient and thus shear rate due to PIV grid resolution.
It should be noted that from the shear rate field, the local elastic effects within a viscoelastic material can be determined through the local Weissenberg number Wi. The results of this calculation are shown in Figure 12.

PLIF Results
Having assessed steady-state performance, it is necessary to move to a transient system in order to investigate mixing performance over time. This is achieved using PLIF and calculating mixing times.
Mixing performance plots in pseudo-colour plots for the greyscale range G = 0 -2000 are shown in Figure 13 below. Local concentration variance has been assessed to elucidate mixing times, with the logarithm of this value plotted against time in Figure 14, with final mixing times displayed Table 5. Theoretical mixing times calculated from Equation (13) are also displayed.  linear trend as shown in Figure 15, with a steeper gradient for the 95% mixed condition compared to 90%. This suggests that the overall mixing performance is governed by the comparative levels of fluid inertia and elasticity exhibited by the fluid, rather than simply fluid motion or elasticity in isolation.
However, it can be seen that for the highest elasticity number, corresponding to fluid Boger B at 30 rpm, the values fall significantly below this linear trend. This may be due to the greyscale values for the fully mixed condition being close to the background values, thus reducing the reliability of the MATLAB programme in calculating true local concentration variance values in this fluid.

Conclusions
The performance of "butterfly" impellers has now been assessed and their interaction with viscoelastic fluids has been observed across a wide range of geometries and experimental conditions.
The effect of fluid elasticity was isolated from viscous effects through the use of Boger fluids, which possess constant viscosity of approximately 0.18 Pa s and a measurable elastic response, with fluid relaxation times λ fitting a power law model. From an energy consumption perspective, these impellers perform well compared to other more commonplace geometries such as Rushton disc turbines and down-pumping pitched blade turbines, with the "butterfly" geometry possessing a turbulent Power number Po of 0.6 compared to 4.9 for a RDT and 0.8 for a PBTd. Additionally, a value of laminar power gradient K p of 122 indicates a slightly greater efficiency than a RDT's value of 71. However, despite the improved energy performance, the "butterfly" impeller performs poorly in other areas.
PIV experiments have yielded a wealth of data regarding flow fields and shear rates generated by "butterfly" impellers within Newtonian and viscoelastic fluids. Although impeller pumping numbers N Q are of a similar magnitude to other geometries, low velocity magnitudes (no more than 25% of the impeller tip speed) in the axial (r-z) plane indicate poor axial mixing performance. Viscoelasticity appears to increase the local velocity magnitudes within the impeller blade hollow whilst causing the flow field to become more uniform, reducing local velocity gradients. Additionally, in the rotational (r-θ) plane velocity magnitudes are maximum 50% of the impeller speed, indicating that the majority of energy input from the impeller goes into purely rotational motion. Further, maximum shear rate magnitudes of 20 s -1 indicate poor mixing performance, with fluid elasticity inhibiting shear in the axial plane. Additionally, torque measurements found the Metzner-Otto constant k s to equal 16, which suggests that shear rates in the impeller region as found by PIV are significantly lower than predicted by the k s parameter. Maximum shear occurs in the region between the impeller blade and the vessel wall, as expected of close-clearance impeller geometries. However, viscoelasticity appears to reduce shear rates, indicating poor shear performance in the materials that these impellers are designed to work with.
PLIF experiments have yielded apparently conflicting results, with mixing times not appearing to follow the expected trend of decreasing mixing time with increasing impeller rotation rate. However, further analysis of the data shows that fluid Elasticity number El is the governing parameter, with increasing values of El leading to increased mixing times as fluids exhibit more solid-like behaviour inhibiting mixing. At high values of El there appears to be a divergence from this behaviour, which may indicate a change in elastic regime. However, the current data cannot conclusively determine whether this is the case.
Future work with "butterfly" impeller geometries should henceforth focus on fluids with a greater range of elasticity, in particular with regard to the assessment of fluid mixing times in order to determine the presence of differing elastic regimes. Also, the application of novel methods of assessing PLIF images such as the areal mixing distribution should be applied to the stirred tank system to further elucidate the performance of these impellers. Additionally, the assessment of impeller performance with complex multiphase systems would also be highly recommended, as these particular systems are the principal applications of "butterfly" impeller geometries in the process industries.    (i) z = C + W/2, (ii) z = C, (iii) z = C -W/2, (iv) z = 0 mm