SynopsisThe dependence of the viscosities of highly concentrated suspensions on solids concentrations and particle size distributions is investigated by using an orifice viscometer. Based on the extensive amount of data on pertinent systems, an empirical equation which correlates the relative viscosities of suspensions (or relative moduli of filled polymeric materials) as a function of solids concentrations and particle size distributions is proposed. The equation has a constant which characterizes size distributions of spherical particles and can be determined experimentally without measuring viscosities. For uniform-size spherical particles, it reduces to the well-known Einstein equation at dilute solids concentrations.
Terminal velocity drag coefficients CD were determined for cylinders, prisms, disks, and spheres in air and water a t N R~ from 1,000 to 300,000, the regime where particles rotate and/or oscillate. These and other similar data show that CD is a function of particle and fluid densities pp and pf, as well as shape and N R~.By considering CD a function of particle moment of inertia and the rotational moment generated by circulation (or alternatively the field force and the lift), one can deduce that a length ratio, NRE . This relationship correlates the data for pp = 1.2 to 8.3 and pf = 0.1 to 1.3 g./cc. to within iz 10%.Data on the fluid dynamic forces acting on bodies moving through fluids are of importance in the design and operation of chemical process equipment such as crystallizers, classifiers, centrifuges, dust collectors, pneumatic and hydraulic conveyors, and rocket engines.It would be desirable, given the shape and physical properties of a particle, the physical properties of the fluid, and the magnitude and direction of the force field, to be able to predict the motion of the particle by use of equations or graphs. This goal has been partially attained.It has been shown both theoretically and experimentally that, if fluid compression is negligible, the resisting force R acting on a body moving freely in a fluid depends on the velocity of the body relative to the fluid ti, an area A, the fluid density pf, and the fluid viscosity p. The area and velocity are usually arbitrarily chosen as the projected area and the relative velocity in the direction of the force field. By dimensional analysis and after the arbitrary substitution of u2p,/2 for u2pf, the following relationship for these variables can be obtained ( 1 3 ) :The exact function f is complicated and usually presented by plotting experimental values for the drag coefficient C, [the dimensionless group of the left side of Equation ( 1) ] vs. the Reynolds number N R , on log-log graph paper. In the case of steady state free fall in a gravitational forceNumerous free settling rate data have been collected for spherical bodies and, in general, it has been possible to represent these data by a single, continuous curve by plotting C, vs. N,,, usually on log-log paper (19). Extensive data, C, vs. N K e curves, and a useful correlation have been reported for isometric particles including spheres, cubes, octahedrons, cube octahedrons, and tetrahedrons for laminar flow conditions (Re > 0.05), transitional flow conditions ( N R , from 0.05 to about 100 to 1,000 depending on particle shape), and for turbulent flow conditions to NRa = 22,000 (3, 2 0 ) . In the case of more or less turbulent flow (NRa > 100 to 1,000, depending on particle shape) the previously published free settling data for well-defined, nonisometric particles are limited to those reported by Duplieh (9) for plates and some three-dimensional shapes in air and water, by Pernolet ( 1 8 ) and Krumbein ( 1 2 ) for particles of only one volume and density but of varying shape settling in water, by W...
The processing of non-Newtonian fluids is of continually increasing industrial importance. These fluids are characterized by a nonlinear rheogram or shearing stress-shearing rate relationship. Emulsions, slurries, and polymeric melts, solutions, and dispersions are often importantly non-Newtonian. Of the many types of non-Newtonian fluids considered in the literature (see for example references 1, 23, 27) pseudoplastic fluids are most commonly encountered and are the principal concern of this study. The rheograms for most of these pseudoplastic fluids are Quite accurateh represented by the eq;ation suggestgd b> Eyring et al. ( 2 2 ) :The occurrence of the shear rate in both the linear and inverse hyperbolic sine terms of Equation (1) makes this relationship somewhat cumbersome for many engineering purposes. Consequently the empirical Ostwald-deWaele (23) or power law equation(2) has often been used as an approximate representation of pseudoplastic rheology (reference 3 for example). In some cases Equation (2) has been found to Et rheological data as well as or better than Equation (1) (19).Since most non-Newtonian fluids have relatively high viscosities, these and highly viscous Newtonian fluids are often processed while in laminar flow. A study of the literature indicated that methods for predicting heat transfer to pseudoplastic fluids in laminar flow did not adequately incorpo-S. E. Craig, Jr., is with Arizona State University, Tempe, Arizona. Page 154rate the effect of the viscosity-temperature dependency. The major purpose of this investigation was to determine a realistic viscosity-temperature dependency and, based on this, to develop accurate methods for the prediction of heat transfer coefficients for the heating of pseudoplastic fluids in laminar flow in tubes of circular cross section and with constant wall temperature. Graetz: LAMINAR-FLOW HEATPlug flow ( n = 0)
Viscosities, first and second normal-stress differences, and related material functions were determined for steady-state and oscillatory shearing of polyethylene oxide and polyacrylamide solutions by use of a Weissenberg R-17 Rheogoniometer with cone-and-plate shearing geometry. Pressure transducers—located along the radius of the plate, with their 0.04-in-O.D. pressure-sensing membranes flush with the plate surface—were used to determine local values of the normal stress. The sensitivity of the transducers was 20–30 dynes/cm2 for steady-state and 10–15 dynes/cm2 for unsteady-state measurements. The ratio of the secondary to the primary normal-stress difference was negative, was as large as −0.2 for the solutions studied, and appeared to decrease with increasing shear rates over the shear-rate range investigated. Complex viscosity and displacement and amplitude functions for both the primary and secondary oscillatory normal-stress differences were determined as functions of the frequency. A significant improvement in the agreement predicted by the “analogies” between the oscillatory primary-normal-stress-difference functions and the complex viscosity functions was achieved in comparison with previous results by reduction of instrument compliance. The Spriggs four-constant and similar models, as well as the Carreau and Bogue-Chen models, were compared with our steady-state and oscillatory data, and these fit the data fairly well; but overall, the Bogue-Chen model appears to represent the data somewhat better.
The general equations of motion were solved numerically for the laminar isothermal flow of Newtonian fluids from a large tube of circular cross section through an abrupt contraction into a coaxial tube of smaller diameter and through the flow‐development region of the smaller tube. The ratio of the diameter of the large tube to that of the smaller tube was varied from one to eight (the latter in one case). Solutions were obtained for the case where the larger tube is real, with no slip at the wall, and for the case where it is a frictionless “stream” tube. The results are presented as charts giving excess pressure losses attributable to contracted and developing flow in terms of equivalent smaller‐tube diameters as functions of the tube‐contraction ratio and the Reynolds number, which was varied from 0.01 to as high as 500 in one case. Both radial‐ and axial‐velocity profiles are presented. The computed results are shown to be in satisfactory agreement with some experimental data. The results are presented in a manner convenient for use in the design of equipment in which contracted Newtonian flow occurs, such as fiber spinnerettes and heat exchangers, and in the analysis of experimental data for contracted flow.
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