This investigation is reported in two parts. Necessary background information is introduced in Part 1. The converging flow investigation proper is described in Port I1 (18).In Part I, viscosity data are presented for polymer solutions used in the converging flow experiment. These data ore fitted with o new three-parameter viscosity model which fits the data better than previous three-parameter models. (The viscosity model parameters are used in Port I1 to characterize rheologicol behavior of the polymer solutions in the converging flow experiment.) The corresponding relationship between flow rote and pressure drop for laminar flow in cylindrical tubes is derived. (In Part I1 this relationship is used in deriving an analogous relationship for slow non-Newtonian flow in conical sections.)The primory purpose of Part 1 is to provide background information for Port 11. However, the new viscosity model and the tube flow relotionship are of some interest in themselves. The new viscosity model should prove useful for describing viscosity data of a variety of polymer solutions ond polymer melts. A simple procedure for fitting the model to viscosity data is described. The tube flow relationship can be used for predicting pressure losses once the viscosity model parameters hove been determined. Conversely, it can be used for determining the viscosity model parameters from tube flow data.
JournalJanuary, 1966 1 = q m + ( T o -9 m ) [ ' Equation (10) is strictly applicable only to Newtonian fluids. Natrosol solutions are non-Newtonian. But for low shear rates (as in the falling sphere measurement) they behave very nearly like Newtonian fluids with viscosities near v0. Vol. 12, No. 1 A.1.Ch.E. Journal Page 65 --dP/dZ. (A similar procedure for the Powell-Eyring model is described in reference 5.) Determination of Viscosity Model Parameters from Tube
Flow DataSuppose now that Q vs. dP/dZ data are available and that it is desired to determine parameters qo, A, and B. As described in the preceding paragraph, a master plot of
The test fluids were three aqueous solutions of Natrosol 250 H hydroxyethyl cellulose. Their nominal concentrations were 0.3, 0.5, and 0.7%. Viscosity data for each solution wee filled with a generalized Newtonian viscosity model which properly describes the zero-shear viscosity. The test geometries were two conical sections. Their vertex angles were approximately 14 and 21°. Laminar flow rate vs. pressure drop data were taken for each Natrosol solution in each conical section. Approximate expressions relating flow rate and pressure drop were derived for the limiting cases of very low and very high flow rates. The low flow rate (non-Newtonian flow) expression was of form: pressure drop=function of (flow rate, geometry, viscosity model parameters). The high flow rate (inviscid flow) expression was of form: pressure drop=function of (flow rate, geometry, fluid density). These two expressions were in excellent agreement with data at low and high flow rates. The sum of these two expressions was in good agreement with data over the entire range of flow rates. This superposition expression in no way accounts for normal stresses, time-dependent elastic effects, or the effect of the third invariant on viscosity. Its success in describing the data implies that these phenomena were not important. For engineering purposes generalized Newtonian viscosity models will probably be adequate for characterizing the flow of dilute polymer solutions in conical sections.
The classical expression for determining viscosity from sphere fall velocity is Stokes' law. Stokes' law applies in the absence of wall and inertial effects. In the usual experimental apparatus there are wall effects and there may be inertial effects. The objective of this paper is to establish a correlation of wall and inertial corrections to Stokes' law in the range appropriate for falling sphere viscometry, and to present this correlation in a manner convenient for application. The desired correlation is presented in Table II and Figure 5 of the text. The new correlation should be useful in determining the viscosity of Newtonian fluids and in determining the zero shear rate viscosity of polymer solutions.
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