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 concerned 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./1,000 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.

In spite of the great range of the available experimental data, further work is necessary in the transition and turbulent-flow regions. No data at all were available on thixotropic, rheopectic, and dilatant fluids, and extension of the correlation to these materials should prove most illuminative from both theoretical and practical viewpoints.

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A theoretical analysis for turbulent flow of non-Newtonian fluids through smooth round tubes has been performed for the first time and has yielded a completely new concept of the attending relationship between the pressure loss and mean flow rate. In addition, the analysis has permitted the prediction of non-Newtonian turbulent velocity profiles, a topic on which the published literature is entirely silent.To confirm the theoretical analysis, experimental data were taken on both polymeric gels and solid-liquid suspensions under turbulent-flow conditions. Fluid systems with flow-behavior indexes between 0.3 and 1.0 were studied at Reynolds numbers as high as 36,000. All the fully turbulent experimental data supported the validity of the theoretical analysis. The final resistance-law correlation represents a generalization of von Karman's equation for Newtonian fiuids in turbulent flow and is applicable to all non-Newtonians for which the shear rate depends only on shear stress, irrespective of rheological class& cation. All the turbulent experimental data for the non-Newtonian systems were correlated by this _.. .-..-ship with a mean deviation of 1.9%. THEORYNon-Newtonian fluids are defined as materials which do not conform to a direct proportionality between shear stress and shear rate. Because of this negative definition of non-Newtonian behavior, essentially an infinite number of possible rheological relationships exist for this class of fluids, and as yet no single equation has been proved to describe exactly the shear-stress-shearrate relationships of all such materials over all ranges of shear rates. If such a general equation could be developed, it seems probable that its complexity would be too great for general engineering utility. Although it is desirable that the following theoretical analysis be universally applicable to all time-dependent, nonelastic fluids, irrespective of any arbitrary rheological classifications such as Bingham plastic, pseudoplastic, or dilatant, this consideration is much too broad to be handled in the initial phases of the development. Therefore a particular rheological model will be selected for use initially, and application of the resulting development to fluids deviating from the assumed model will be considered subsequently.I t has been found experimentally (7, 11,19,20) that the relationship between shear stress and shear rate for a great many non-Newtonian fluids, possibly the majority, may be represented closely over wide ranges of shear rate (ten-to one-thousand-fold) by a twoconstant power function of the form D. W. Dodge is with E.

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The free energy of polymer solutions must depend upon the conformation of the macromolecules, and hence upon the deformation state imposed on the system, as well as upon the more familiar thermodynamic state variables of temperature and composition. As one consequence of the importance of this additional thermodynamic state variable, the precipitation temperature (cloud point) of polymer solutions may be increased by several tens of degrees Centigrade by imposition of steady shearing at low deformation rates. As a second consequence, the precipitated phase is sometimes found to be a solid of new morphology, and one which is quite refractory to re-solution. In this work a quantitative theoretical analysis of solubility phenomena in deforming solutions is given. The free energy of macromolecules in stagnant solutions is obtained from the Flory-Huggins theory, and changes with deformation state are computed from Marrucci's analysis for dilute solutions of elastic dumbbells. The parameters in the theoretical analysis were evaluated, using solutions of polystyrene in dioctyl phthalate, by measurements of the thermodynamic interaction parameter and of the rheological properties. The theory, which contains no adjustable parameters, was used to make a priori predictions of the change in cloud point with deformation state. The experimentally observed changes, of 3-28 °C, were predicted with a mean deviation of about 3 °C.

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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