Cross equation viscosity measurements

Viscosity models are sometimes used to estimate the zero-shear viscosity when no experimental data are available at shear rates sufficiently low that the viscosity is constant. However, this is an unreliable procedure, as there is no fundamental basis for any of these equations, and the resulting value of t/q should be deemed at best a rough estimate. For example, Kataoka and Ueda [70] found that the Cross equation yielded extrapolated values of t]q that were about 50% less than measured values. 

The most common instrument used for viscosity measurements in low-viscosity liquids is the capillary viscometer. In this instrument, a liquid is made to flow under its own potential head through a narrow capillary with a weU-defined length and cross section. The volumetric flow rate Qota simple Newtonian liquid undergoing laminar flow in a capillary of diameter d and length / is given by Poiseuille s equation ... 

The modification of the surface force apparatus (see Fig. VI-4) to measure viscosities between crossed mica cylinders has alleviated concerns about surface roughness. In dynamic mode, a slow, small-amplitude periodic oscillation was imposed on one of the cylinders such that the separation x varied by approximately 10% or less. In the limit of low shear rates, a simple equation defines the viscosity as a function of separation... 

In the capillary method, the time required for a liquid to flow through a capillary tube is determined. The melt under investigation flows with a constant rate through a tube with a small, definite cross-sectional area, such as a cylindrical capillary. The viscosity can be measured in an absolute way from the pressure drop. This method can yield the most reliable absolute data, the viscosity being given by a modified Hagen-Poiseuille equation ... 

Here x is a phenomenological parameter measuring the chirality and / is a size scale factor. Since here the Reynolds number is small ( 10 s), the Stokes equation can be used to get r = DS2. where D is the hydrodynamic drag coefficient and 2 is the rotational speed. The drag coefficient for a cylindrical object rotating about its axis with cross-sectional radius r and length L is D = 4ztT)r2L, where tj is the viscosity of the medium [19]. Therefore, D /3 and the rotational speed 2 of the rotor will scale as... 

From the experimental data we were able to determine both the intramolecular and intermolecular relaxation rates as a function of pressure and temperature. The availability of shear viscosities and self-diffusion coefficients of EHB, which were measured earlier in our laboratory, provided the opportunity to test the dependence of the experimental cross-relaxation rates on viscosity and/or diffusion of EHB. The reorientational correlation time Tc describing overall molecular motion is coupled to the rj/T term through the Debye equation, which in a modified form is ... 

The Cross eqnation (Cross, 1965) is a versatile equation that is based on measure ments of viscosity, fig at zero shear rate and at infinite shear rates. 

The use of cone-and-plate rheometers for polymer melts is limited to relatively low shear rates by the onset of flow instabilities, typically occurring not far beyond the onset of shear-rate dependence for t](y) and the a iVi crossing point. A capillary rheometer is sketched in Fig. 3.22. Stable operation at much higher shear rates is possible, but usually t]o cannot be determined because of instrumental limitations at low shear rates. The steady-state viscosity, however, can be obtained from measurements of the volumetric flow rate, Q, and the pressure drop, AP = P — Po, Po being the ambient pressure. For long tubes (L/D 1), the following equation applies for Newtonian liquids ... 

In describing the mechanical response of gel networks at long times, it is necessary to distinguish between conditions where the cross-links are. stable and where they may be forming and dissociating during the period of the experiment. In the former case, the behavior may resemble that of permanently cross-linked networks as discussed in Section B4 of Chapter 14. The relaxation of stress beyond the pseudoequilibrium value, or the additional creep beyond the pseudoequilibrium compliance, is very slight, however. Some creep measurements have been fitted to an equation of the Andrade form, equation 28 of Chapter 13 (with infinite viscosity). 

Model III relies on constant eddy viscosity (which is a function of gas velocity and column diameter) but predicts the power law form of the gas holdup profile. It accomplishes that by using both gas and liquid continuity equations, and be establishing that the cross-sectional integral of the slip velocity is a constant multiple of the superficial gas velocity. The agreement of model III with the velocity data of Hills is excellent as is its prediction of Hills s measured voidage profiles. 

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