Dimensional Analysis For Planetary Mixer: Mixing Time And Reynolds Numbers
Oct 19, 2023
Abstract
Mixing time number is a convenient parameter to characterize mixing performance of stirred tanks. This dimensionless number is now well established for agitated vessels equipped with vertically and centrally mounted impeller for Newtonian as well as for non-Newtonian fluids. To our knowledge, there is more ambiguity concerning its definition for planetary mixers especially when they have dual motion (around two perpendicular axes) to achieve homogenization. In this study, dimensional analysis of mixing time and reliability of the modified Reynolds and mixing time numbers are proposed for such a planetary mixer particularly named as
system. These two numbers are based on the maximum tip speed of mixer as the characteristic velocity. Modified dimensionless numbers are consistent with the definition of conventional Reynolds and mixing numbers (when only one revolving motion around the vertical axis of the mixing device occurs in the vessel).
Mixing time experiments with
mixer for highly viscous Newtonian fluids showed that the proposed modified Reynolds and mixing time numbers succeeded to obtain a unique mixing curve irrespective of the different speed ratio chosens. This agreement proves that the proposed modified dimensionless numbers can be well adapted for engineering purposes and they can be used to compare the mixing performance of planetary mixers.
Introduction
In the second half of the 20th century, systematic use of dimensional analysis to investigate mixing processes has allowed this field to evolve from "arts into sciences". Today the whole field of classical stirring technology (here the word "classical" refers to impellers vertically and centrally mounted in the tank) has been examined, as a result the definition of significant dimensionless groups has been well established. Thus, for most of the mixing operations for heat transfer, homogenization, suspension, emulsification, etc., depending on the flow regime and mixing systems investigated, numerous correlations involving various dimensionless numbers (power, Reynolds, mixing time, Weber, Archimedes, Nusselt, Prandtl, etc.) have been proposed by the scientific community. These works allow a deeper process understanding and provide design guidelines and scale-up rules for conventional mixing operation. Unfortunately, this is not yet the case for off-centered double agitator, coaxial and planetary mixers. However, these mixing equipments are taking a big sweep in process industry fulfilling the need of consumers for complex products with specific functionalities. Analysis of their performance characteristics have only recently appeared in open literature.
Pioneer and more intensive works on non-conventional mixers have been conducted by Tanguy and coworkers, however if we focus only on planetary mixer the literature is still scarce (Tanguy et al., 1996, Tanguy et al., 1999; Landin et al., 1999, Zhou et al., 2000, Jongen, 2000; Delaplace et al., 2004, Delaplace et al., 2005). Numerical and experimental approaches carried out by these works have highlighted the tricky problems and their respective response for optimal design and process parameters, and scale-up of this non-conventional mixer. These papers have also highlighted the difficulties to compare the performances of the planetary mixer with those of well-established mixers. Therefore, it is needed to modify the dimensional analysis established for classical mixers, and to adapt them for non-conventional mixers. For planetary mixer, the problematic issue is to define characteristic speed and length. Delaplace et al. (2005) have recently investigated the power draw through dimensional analysis for planetary mixers. They proposed modified Reynolds and power numbers. These numbers involve the maximum impeller tip speed as the characteristic velocity and a dimension perpendicular to the vertical axis of revolution as the characteristic length. They showed experimentally for a particular planetary mixer which combines dual revolutionary motions that such dimensionless numbers allow to obtain a unique power characteristic of the mixing system, regardless of a variation in speed ratio. Moreover, the modified dimensionless numbers proposed are consistent with the definition of classical Reynolds and power number. It is verified for the condition when the impeller is forced to perform only one motion around the vertical axis in the tank (as in the case of a classical mixing system), the modified numbers take the form as obtained for classical systems.
Unfortunately, to evaluate the efficiency of planetary mixer both modified mixing time and power numbers are required. To the best of our knowledge, the mixing time number is not well established for planetary mixer which has two revolving motions.
