Reynolds number scaling of flow in a Rushton turbine stirred tank. Part I—Mean flow, circular jet and tip vortex scaling

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Abstract

We consider scaling of flow within a stirred tank with increasing Reynolds number. Experimental results obtained from two different tanks of diameter 152.5 and 292.1 mm, with a Rushton turbine operating at a wide range of rotational speeds stirring the fluid, are considered. The Reynolds number ranges from 4000 to about 78,000. Phase-locked stereoscopic PIV measurements on three different vertical planes close to the impeller give phase-averaged mean flow on a cylindrical surface around the impeller. The scaling of θ- and plane-averaged radial, circumferential and axial mean velocity components is first explored. A theoretical model for the impeller-induced flow is used to extract the strength and size of the three dominant elements of the mean flow, namely the circumferential flow, the jet flow and the pairs of tip vortices. The scaling of these parameters with Reynolds number for the two different tanks is then obtained. The plane-averaged mean velocity scales with the blade tip velocity above a Reynolds number of about 15,000. However, parameters associated with the jet and tip vortices do not become Reynolds number independence until Re exceeds about 105. The results for the two tanks exhibit similar Reynolds number dependence, however, a perfect collapse is not observed, suggesting a sensitive dependence of the mean flow to the finer details of the impeller.

Introduction

Scaling of flow and mixing within a stirred tank reactor is of significant practical importance. For lack of a satisfactory understanding, the design of production-scale stirred tanks often evolves through a number of stages that iterate between laboratory experiments and pilot plants, a time-consuming and expensive step-by-step scale-up process. Part of the difficulty is that mean flow and turbulence quantities, such as rms fluctuation and dissipation, often scale differently with increasing tank size and operating speed. As a result the scaling of mixing within the tank can be complex.

The last three decades have seen several high quality experimental measurements of flow inside stirred tanks using modern techniques such as laser doppler anemometry (LDA) and particle image velocimetry (PIV) (Desouza and Pike, 1972, Van’t Riet and Smith, 1975, van der Molen and van Maanen, 1978, Kolar et al., 1984; Costes and Couderc, 1988; Dong et al., 1994, Sturesson et al., 1995, Stoots and Calabrese, 1995; Ducoste et al., 1997; Kemoun et al., 1998, Lamberto et al., 1999; Montante et al., 1999; Mahouast et al., 1989, Schaffer et al., 1997, Derksen et al., 1999, Escudie and Line, 2003, Escudie et al., 2004). In particular, several of these experimental investigations have considered flow and mixing inside stirred tanks of varying size operating over a range of speeds and, thus, have addressed, directly or indirectly, the scaling of mean and turbulence quantities with Reynolds number (Desouza and Pike, 1972, Van’t Riet and Smith, 1975, van der Molen and van Maanen, 1978, Kolar et al., 1984; Costes and Couderc, 1988; Dong et al., 1994, Sturesson et al., 1995, Stoots and Calabrese, 1995, Ducoste et al., 1997, Kemoun et al., 1998; Lamberto et al., 1999; Montante et al., 1999). These studies together cover a wide range of Reynolds number from 10 to 1.2×105, where Reynolds number is defined in terms of impeller blade-tip diameter, D, and the number of blade rotations per second, N, as Re=ND2/ν (ν is the kinematic viscosity of the fluid). However, the range of Re considered in each investigation is limited. At higher Reynolds numbers relevant to turbulent flow, the overall time-averaged mean flow and rms turbulent fluctuations appear to scale with the impeller blade-tip velocity.

Costes and Couderc (1988) have addressed the problem of scaling of mean flow and turbulence using two different sized tanks over three different Reynolds numbers. Their results suggest that velocity statistics such as, mean, rms, spectra and autocorrelation scale with the blade-tip velocity. However, higher order turbulence statistics, such as dissipation, do not show complete collapse when scaled appropriately in terms of impeller diameter and rotation rate. The results of Stoots and Calabrese (1995) on deformation rate over a Reynolds number range of 29,200–45,800, however, show a reasonable Reynolds number independence when nondimensionalized by the inverse time scale.

van der Molen and van Maanen (1978) emphasized the importance of blade-tip trailing vortices. They pointed out that the time-averaged flow in the laboratory frame of reference (as measured by a fixed probe) averages out the influence of tip vortices as they sweep past the probe along with the blades. The resulting time-averaged mean flow was observed to scale well with the blade-tip velocity. The effect of tip vortex pairs was carefully isolated through phase-average, and the tip vortex strength did not scale perfectly with blade-tip velocity. Some dependence on tank size was observed. Van’t Riet and Smith (1975) also investigated the scaling of blade-tip vortices over a wide Reynolds number range of 300–90,000. The vortex trajectory was observed to be Reynolds number dependent at lower, transitional Reynolds numbers, but Reynolds number independent, within experimental uncertainty, above a Reynolds number of 15,000. Other quantities such as vortex strength, when appropriately scaled, also tended towards Reynolds number independence at higher Reynolds numbers.

In spite of the above efforts the scaling of flow and mixing within a stirred tank with increasing tank size and impeller speed remains not fully understood. In this paper, we address the question of scaling with experiments performed in two different tanks with impeller speed varying over Reynolds numbers ranging from 4000 to 80,000. The two tanks were constructed to be geometrically similar. Both employ a lid at the top of the tank in order to prevent free surface (Froude number) effects. The flow is entirely driven by the rotating impeller and the operational speeds are such that Mach and Rossby numbers are irrelevant. Reynolds number is expected to be the only relevant parameter of the problem. The results for each tank, when appropriately nondimensionalized by the blade tip radius and velocity, show Reynolds number independence with increasing impeller speed at sufficiently high Reynolds number. However, surprisingly the results for the two different tanks do not exhibit a perfect collapse, suggesting sensitive dependence on small differences, especially in the geometric scaling of their impellers.

Stereoscopic PIV measurements were made on three different vertical planes within the tank. The instantaneous measurements are phase-locked with the blade position and ensemble averaging over many such realizations yields the phase-averaged mean velocity. The measurement on the three planes, which are located close to the impeller swept volume, are interpolated to obtain all three components of the phase-averaged mean velocity on a cylindrical plane of constant radius located just beyond the impeller tip. The velocity on this cylindrical plane is the impeller-induced inflow and it can be considered to dictate the flow over the entire tank at large (Yoon et al., 2002). Thus, here we consider the scaling of velocity measured over this plane as a proxy for scaling of flow over the entire stirred tank.

The phase-averaged mean flow within the tank stirred with a Rushton turbine can be considered to be made up of three different basic flow elements: circumferential flow, a jet flow and pairs of tip vortices associated with the impeller blades. This simple decomposition has been shown to be effective in modeling the impeller induced flow, particularly in the neighborhood of the impeller (see Yoon et al., 2001). Here we investigate the scaling of each of these elements individually with increasing tank size and impeller rotation rate. By looking at the scaling of parameters, such as the strength and size of the jet and tip vortex pairs individually, we hope to address the question of mean flow scaling in more detail. In particular, it is of interest to establish the minimum Reynolds number necessary for these parameters to be Reynolds number independent.

Here we will also investigate the scaling of both vorticity and dissipation for the two tanks and in particular, address when they become Reynolds number independent. If the scaling of the mean flow were to be uniform over the entire tank, we expect the scaling of these higher order quantities to follow that of the mean flow. However, it can be anticipated that the Reynolds number independence of the nondimensional vorticity and dissipation will be delayed to much higher Reynolds numbers, since these derivative quantities give more importance to the smaller scales of motion. Thus, the results on the scaling of vorticity and dissipation can be used to interpret the scale-dependence of mean flow scaling. The present work is limited to investigation of only the scaling of velocity field. Practical factors relevant to industrial mixing such as mixing time, heat and mass transfer, and multiphase flow are not considered in this study. However, the scaling properties of the flow to be discussed here will likely have an effect on these additional complex processes.

Section snippets

Apparatus

The experiments were an extension of more limited experiments first reported by Hill et al. (2000). A schematic overhead view of the set-up for the current stereoscopic PIV experiments is shown in Fig. 1. To assess the influence of geometric scaling, two geometrically similar test sections were considered. In both cases, an acrylic, unbaffled1

Mean flow scaling

The mean flow ensemble-averaged over all the realizations (ur, uθ, uz) and interpolated onto the cylindrical surface is shown in Fig. 3 for the small tank at the lowest rotation speed, corresponding to Re=4300. In Fig. 3 frame (a) shows the in-plane velocity vector plot and frame (b) shows the out-of-plane (radial) velocity contours. Only a 60 sector is shown, and the view is limited to the top half of the tank with the region below the center-plane obtained by symmetry. The θ-dependence

Conclusions

Experimental measurements of flow induced by a Rushton turbine in an unbaffled stirred tank have been performed over a wide range of operating speeds. Two different tank sizes were used with water as the working fluid to cover a Reynolds number range of 4000–80,000. Phase-locked stereoscopic PIV measurements were made on three different vertical planes near the impeller to obtain all three components of the impeller-induced flow. Instantaneous realizations were averaged to obtain the

Acknowledgements

This work was supported by a grant from National Science Foundation (CTS-9910543) and a gift from the Dow Chemical Company.

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