Agitation of non‐Newtonian fluids

Abstract

Since the shear rate of a non-Newtonian fluid is of importance in fixing the rheological or viscometric behavior of such a material, the present study has been conoerned with the development of a general relationship between impeller speed and the shear rate of the fluid. The resulting relationship was then used to interpret and correlate power-consumption data on three non-Newtonian fluids by use of a generalized form of the conventional power-number-Reynolds-number plot for Newtonians.Flat-bladed turbines from 2 to 8 in. in diameter were used exclusively. Tank diameters ranged from 6 to 22 in. and power inputs from 0.5 to 176 hp./l,OOO gal. The study encompassed a 130-fold range of Reynolds numbers in the laminar and transition regions. The results to date indicate that power requirements for the rapid mixing of non-Newtonian fluids are much greater than for comparable Newtonian materials. REVIEW OF PRIOR ARTIn the study of the agitation or mixing of fluids, the system which has received the most attention consists of a single impeller centered in a cylindrical tank, as shown in Figure 1. The results of Newtonian powerconsumption studies are presented in terms of dimensionless groups involving power (Pgc/D5N3p) and a mixing Reynolds number (D2Np/p) or modifications of these. Below a Reynolds number of 300 the Froude number (DNZ/g), which measures the variation in flow due to changes in the free surface, was not important (22). The effects of geometrical parameters other than the impeller diameter [such as ( C / D ) , (TID), and (BID)] were not important within the wide ranges specified by previous workers. The entire power-number-Reynolds-number curve (Figure 2) has been divided into three sections which are directly analogous to the familiar three regions of flow in a circular pipe, i.e., the turbulent region in which the power number (or friction factor) is not greatly affected by Reynolds number, the laminar region where power number and friction factor are inversely proportional to Reynolds number, and the intermediate or transition region. However, unlike the case of flow in a round tube, the transition from laminar to turbulent flow does not occur over the narrow rangc of Reynolds numbers between 2,000 and 4,000 (6) but extends over the large range from 10 to 1,000, as shown in Figure 2.To date only three papers have been concerned with the agitation-power requirements of non-Newtonian fluids. Brown and Petsiavas (2) presented a power-number plot for a Bingham-plastic type* of nonNewtonian that makes use of the Binghamplastic Reynolds and Hedstrom numbers (4, IS) t o correlate the data. Their viscometric data, taken with a Brookfield viscometer, indicated that their fluids deviated appreciably from Bingham-plastic behavior, but for those few fluids which closely approach the ideal Bingham plastic this method of attack can be reworked into *Appendix A presents and discusses the classical types of non-Newtonian behavior.

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“…The shear rate for apparent viscosity determination is found by the approach of Metzner and Otto, k s N, where k s is 11 (Metzner and Otto, 1957). The k L a is determined by the direct method (Villadsen et al, 2011) also assuming a log mean driving force, as described in equation 7.…”

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“…where the average shear rate γ depended on the rotational speed of the impeller [26,27], as follows:…”

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“…Equation 5 applies to laminar, turbulent and transitional flows. In laminar ranges, γ is linearly related to the rotational speed of the rotor (N) [31]. In turbulent flow, it is nonlinearly related to N [30].…”