(171j) Experimental Analysis of Power Consumption and the Metzner-Otto Constant for Highly Shear-Thinning Fluids

In this research, the power consumption and Metzner-Otto constant were studied, taking Newtonian and non-Newtonian fluids stirred by paddle, pitched, and anchor impellers with different geometrical characteristics in a cylindrical vessel by means of experiments to investigate the effects of impeller geometry as well as the effects of fluid rheological properties in the laminar regime. This study establishes a complete correlation of the power consumption and Reynolds number with impeller geometrical characteristics and rheological properties of fluid through linear regression analysis.

Introduction:

Mixing of non-Newtonian fluids is an important unit operation in the chemical and related industries. Most non-Newtonian fluids are pseudoplastic and exhibit shear thinning characteristics. The rheological properties of these shear-thinning fluids may cause various problems, the most important being changes in the viscosity during processing. For example, the viscosity differs at different locations in a stirred tank and varies with the shear rate. In such mixing processes, it is challenging to predict the power consumption and average shear rate of non-Newtonian fluids accurately. The apparent viscosity required to calculate the impeller Reynolds number depends on the shear rate, which in turn depends on the flow parameters. Metzner and Otto (1957) proposed a method for calculating the apparent viscosity of non-Newtonian fluids, which became a universal method for evaluating the power consumption required by mixing of non-Newtonian fluids. The concept assumes that the average shear rate γ^·_avg is proportional to the impeller rotational speed N, γ^·_avg = K_sN (Eq. (1)), where the constant of proportionality K_s is the Metzner-Otto constant. The power curve of non-Newtonian fluids can be obtained using the Metzner-Otto method, which is consistent with the power curve of Newtonian fluids. In predicting the power consumption of non-Newtonian fluids, the K_s value becomes a key factor. It is widely agreed that K_s is a function of the impeller geometry; however, there are some conflicting results present in the literature regarding the relationship between K_s and the fluid rheological index n. In this study, measurements of the power consumption and average shear rate with rheological complex fluids using different impeller shapes, stages, and sizes in the laminar regime were carried out to understand the effects of the impeller geometry and fluid rheological properties in a stirred tank.

Experimental Method:

For the experiments, a cylindrical flat-bottom vessel without baffles was used as stirring tank (tank diameter T = 0.20 m). Three different types of impellers with varying D/T ratios, i.e., a six-blade paddle P_a (ratio of impeller diameter D to T, D/T = 0.25-0.75), four-blade pitched blade impeller P_c (D/T = 0.35-0.45) with different stages, and an anchor impeller (D/T = 0.675, 0.9) were selected (Figure 1). The liquid depth H was set to H = T = 0.20 m. For stirring, liquid starch syrup and polyglycerine (PGL-S, Sakamoto Orient chemical Corporation) were used as Newtonian fluids, and an aqueous solution of hydroxyethyl cellulose (HEC, SE600, Daicel) was used as pseudoplastic fluid. The HEC concentration varied between 1.75 wt. % and 3.0 wt.% and the flow behavior index n was in the range of 0.62-0.48. A rheometer (ONRH-1, Ohna Tech Inc.) was used to determine the rheological properties of the non-Newtonian fluids. Torque measurements were conducted using a torque meter (TM320, UNIPULSE). Using the power number-Reynolds number curves of Newtonian fluids in the laminar regime, as shown in Eqs. (2), (3) & (4): N_pRe = B (2); Power number N_p = P/(ρN³D⁵) (3); Reynolds number Re = (ρND²)/η (4); where N is the impeller speed and B is the constant of proportionality known as the power constant, the apparent viscosity η_a of non-Newtonian fluids at different impeller speeds was calculated as: η_a = P/(BN²D³) (5). The Carreau model was selected for calculating the rheological parameters of the shear thinning fluid: η = η_∞ + (η₀ − η_∞){1+(λγ^·)²}^{(n−1)/2}
(6), where η_∞,η₀, λ are the viscosity at infinite shear rate, viscosity at zero shear rate and characteristic time, respectively. With the help of the calculated apparent viscosity, the average shear rate γ^·_avg was estimated and further, K_s was calculated from the linear relationship between γ^·_avg and N at different concentrations of HEC.

Results and Discussion

Torque measurements as a function of the rotational speed were transformed into a power input, which was then converted into a dimensionless form of the power number N_p . The Reynolds Number Re was computed considering the impeller geometry, rotational speed, fluid viscosity, and fluid flow parameters. Subsequently, the power constant B was estimated using the linear relationship between the power data. Figure 2 shows the power consumption data for the single-stage 1S-P_a and double-stage six-blade paddle impeller 2S-P_a with D/T = 0.5. N_p decreased linearly with increasing Re for a given impeller-vessel system in the laminar regime. Rotation by 2S-P_a required double the power that 1S-P_a required. The power constant B value is also double for 2S-P_a as compared with 1S-P_a. Furthermore, a power curve was obtained for all impellers used in the study with Newtonian fluid, and B was evaluated for each geometry.

Four different concentrations of HEC were used, and Figure 3 shows the power consumption data for 1S-P_a and 2S-P_a with D/T = 0.5 for non-Newtonian fluids. Based on the apparent viscosity method, all power curves of the shear-thinning fluids coincide with those of the Newtonian fluid used. The same results were obtained for all geometries used in the experiment. This implies that the power constant B is a function of the impeller geometry. With the calculated η_a from Eq. (5), γ^·_avg was calculated using the Carreau model at varying concentrations for all geometries used. Figure 4 shows the linear relationship between γ^·_avg and N for the four-blade pitched blade impeller P_c (D/T = 0.45) at different concentrations of HEC. It can be observed that γ^·_avg varies linearly with N, but that the value of K_s, which is equivalent to the slope of γ^·_avg vs. N, decreases with a decrease in the value of n, that is, it decreases with an increase in the shear-thinning property. For 2S-P_c, the value of K_s is almost similar to that of the single-stage impeller. Similar results were obtained for all impeller geometries used in this study. Figure 5 shows the results of the K_s value with (1-n) for the anchor impeller. For the anchor impeller, the value of K_s decreases with the clearance ratios C/T, n, and the D/T ratio. These results imply that for any impeller geometry and for highly shear-thinning fluids, K_s depends on the flow behavior index n and the geometrical parameters of the impeller-vessel system. Based on the dependence of K_s on n and the geometrical parameters, the experimental results were used to propose a correlation for calculating the Reynolds number for a non-Newtonian fluid.

Conclusion:

Three types of impellers with different D/T ratios, shapes, and stages were experimentally investigated in terms of power consumption. From these results, B was found to be dependent only on the impeller geometry. The Metzner-Otto concept can be applied numerically by modifying the average shear rate equation, incorporating K_s as a function of n and the impeller-vessel geometry. It is now possible to calculate the Reynolds number Re’ for a non-Newtonian fluid with the proposed correlation as follows: Re' = (ρND²)/[η_∞ + (η₀ − η_∞){1+(λ a(1−n)^b N)²}^{(n−1)/2}]. The Re’ correlation relates the flow properties of a shear-thinning fluid, impeller speed, and geometrical parameters of the impeller-vessel system.

Future scope:

The distribution of the shear rate and the viscous dissipation energy will be analyzed in the same tank as used in these experiments using multi-purpose CFD software (Fluent 2022; ANSYS, Inc.). A scale-up study will be conducted with the geometrical similarity of impeller-vessel system to understand the effects of fluid rheology and geometry of the impeller, and to confirm the reliability of this study with other non-Newtonian fluids. Experiments will be conducted with a helical ribbon impeller, owing to its high mixing efficiency, to check the relationship between K_s and n.

Acknowledgements

This study received partial support from a Grant-in-Aid for Scientific Research (No. 22K04799) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan.