The dissipation rate of turbulent kinetic energy and its relation to pumping power in inline rotor-stator mixers

Rotor-stator mixers are applied for mixing or dispersion of multiphase systems, for instance, cosmetic, food and pharmaceutical industries. They can be operated in batch, semi batch and inline modes. However, there are increased industrial interests in transitioning from batch to inline operation owing to the economic benefits that come with this continuous operation.

However, insufficient theoretical understanding and additional complications that come with the rotor-stator mixers, are the main problems facing their application and design. Therefore, the current knowledge on how mixing and dispersion depend on operating and design parameters needs some improvement. Associate Professor Andreas Håkansson from Kristianstad University and Professor Fredrik Innings at Lund University in Sweden described the scaling of dissipation rate of turbulent kinetic energy in the effective region of mixing and dispersion with design and operating parameters. They set out to understand how dissipation power depends on the terms of the shaft power draw correlation. Their work is now in published in journal, Chemical Engineering and Processing.

The dissipation volume may be computed from geometric parameters after the effective region has been chosen. The assumptions on the position of the effective region, and how dissipation links to the terms in the power draw correlation, describe the different suggested models of the dissipation rate of the turbulent kinetic energy in inline rotor-stator mixers. By applying an energy balance, the total turbulent dissipation can be readily measured.

No expressions of how pumping power relates to the operating factors for inline rotor-stator mixers have been developed. However, inline rotor-stator mixers are identical to centrifugal pumps. For this reason, ideal centrifugal pumping model can be adopted as an initial step towards determining how pumping power relates to the operating parameters as well as design.

The pumping power may be split into a static and a dynamic component. The difference in dynamic pressure, and the dynamic pumping power, are however not reported in most experimental results, i.e. averaged fluid velocities are not measured. However, computational dynamic simulations provide a way to estimate the necessity for this factor.

By analyzing previously published data, the authors concluded that the pumping power of three different inline mixers is directly proportional to the flow-term of the power draw correlation multiplied with a linear function of the flow number. When compared to the centrifugal pump theory, this type of linear scaling is expected based on conservation of angular momentum in the rotor.

From the concept of conservation of energy, pumping power and total turbulent dissipation power are related. Moreover, the empirically observed pumping power as a function of flow number is not in agreement with the previous models for the dissipation rate of turbulent kinetic energy. A new correlation for the dissipated energy in an inline rotor-stator mixer was, therefore, suggested based on two main propositions. First, dissipation is the loss-free shaft power subtracted with the pumping power. Secondly, there’s a linear relationship between pumping power and the flow-term in the power draw correlation multiplied with a linear function of flow number.