Viscosity limiting

The other models can be appHed to non-Newtonian materials where time-dependent effects are absent. This situation encompasses many technically important materials from polymer solutions to latices, pigment slurries, and polymer melts. At high shear rates most of these materials tend to a Newtonian viscosity limit. At low shear rates they tend either to a yield point or to a low shear Newtonian limiting viscosity. At intermediate shear rates, the power law or the Casson model is a useful approximation. 

The power law model can be extended by including the yield value r — Tq = / 7 , which is called the Herschel-BulMey model, or by adding the Newtonian limiting viscosity,. The latter is done in the Sisko model, 77 +. These two models, along with the Newtonian, Bingham, and Casson... 

The Cross equation assumes that a shear-thinning fluid has high and low shear-limiting viscosity (16) (eq. 4), where a and n are constants. 

Dilute Polymer Solutions. The measurement of dilute solution viscosities of polymers is widely used for polymer characterization. Very low concentrations reduce intermolecular interactions and allow measurement of polymer—solvent interactions. These measurements ate usually made in capillary viscometers, some of which have provisions for direct dilution of the polymer solution. The key viscosity parameter for polymer characterization is the limiting viscosity number or intrinsic viscosity, [Tj]. It is calculated by extrapolation of the viscosity number (reduced viscosity) or the logarithmic viscosity number (inherent viscosity) to zero concentration. 

The viscosity ratio or relative viscosity, Tj p is the ratio of the viscosity of the polymer solution to the viscosity of the pure solvent. In capillary viscometer measurements, the relative viscosity (dimensionless) is the ratio of the flow time for the solution t to the flow time for the solvent /q (Table 2). The specific (sp) viscosity (dimensionless) is also defined in Table 2, as is the viscosity number or reduced (red) viscosity, which has the units of cubic meters per kilogram (m /kg) or deciUters per gram (dL/g). The logarithmic viscosity number or inherent (inh) viscosity likewise has the units m /kg or dL/g. For Tj g and Tj p, the concentration of polymer, is expressed in convenient units, traditionally g/100 cm but kg/m in SI units. The viscosity number and logarithmic viscosity number vary with concentration, but each can be extrapolated (Fig. 9) to zero concentration to give the limiting viscosity number (intrinsic viscosity) (Table 2). 

The relative viscosity of a dilute dispersion of rigid spherical particles is given by = 1 + ft0, where a is equal to [Tj], the limiting viscosity number (intrinsic viscosity) in terms of volume concentration, and ( ) is the volume fraction. Einstein has shown that, provided that the particle concentration is low enough and certain other conditions are met, [77] = 2.5, and the viscosity equation is then = 1 + 2.50. This expression is usually called the Einstein equation. 

Curves for the viscosity data, when displayed as a function of shear rate with temperature, show the same general shape with limiting viscosities at low shear rates and limiting slopes at high shear rates. These curves can be combined in a single master curve (for each asphalt) employing vertical and horizontal shift factors (77—79). Such data relate reduced viscosity (from the vertical shift) and reduced shear rate (from the horizontal shift). 

For lubricated plug cocks, the lubricant must have limited viscosity change over the range of operating temperature, must have low solubility in the fluid handled, and must be applied regularly. There must be no chemical reaction between the luBricant and the fluid which... 

With heavy fuels, the ambient temperature and the fuel type must be considered. Even at warm environmental temperatures, the high viscosity of the residual could require fuel preheating or blending. If the unit is planned for operation in extremely cold regions, the heavier distillates could become too viscous. Fuel system requirements limit viscosity to 20 centi-stokes at the fuel nozzles. 

This is a rough measure of a limiting viscosity. At temperature above 2.5°C (6.5°F), oil ceases to flow when the vessel in which it has been cooled is held horizontally for 5 seconds. The pour point is a guide to behavior and care should always be taken that the operating temperatures are above the figure specified by the oil manufacturer as the pour point of a given oil. 

Viscosity, intrinsic Also called limiting viscosity number. For a plastic, it is the limiting value of an infinite dilution. It is the ratio of the specific viscosity of the plastic solution to its concentration in moles per liter. 

The viscosities of most real shear-thinning fluids approach constant values both at very low shear rates and at very high shear rates that is, they tend to show Newtonian properties at the extremes of shear rates. The limiting viscosity at low shear rates mq is referred to as the lower-Newtonian (or zero-shear /x0) viscosity (see lines AB in Figures 3.28 and 3.29), and that at high shear rates Mo0 is the upper-Newtonian (or infinite-shear) viscosity (see lines EF in Figures 3.28 and 3.29). 

When the fluid behaviour can be described by a power-law, the apparent viscosity for a shear-thinning fluid will be a minimum at the wall where the shear stress is a maximum, and will rise to a theoretical value of infinity at the pipe axis where the shear stress is zero. On the other hand, for a shear-thickening fluid the apparent viscosity will fall to zero at the pipe axis. It is apparent, therefore, that there will be some error in applying the power-law near the pipe axis since all real fluids have a limiting viscosity po at zero shear stress. The procedure is exactly analogous to that used for the Newtonian fluid, except that the power-law relation is used to relate shear stress to shear rate, as opposed to the simple Newtonian equation. 

Having obtained the value of the limiting viscosity number, we can calculate relative molar mass using the semi-empirical equation ... 

In the following sections, synthesis of the anionic polymers, copolymer molecular weight, limiting viscosity number, electrolyte effects, solution shear thinning, screen factor, polymer radius of gyration, and solution aging will be discussed and data on the copolymers presented. 

Assays. Nitrogen assays to determine 1-amidoethylene unit content were done by Kjeldahl method. Limiting viscosity numbers were determined from 4 or more viscosity measurements made on a Cannon-Fenske capillary viscometer at 30°C. Data was extrapolated to 0 g/dL polymer concentration using the Huggins equation(44) for nonionic polymers and the Fuoss equation(45) for polyelectrolytes. Equipment. Viscosities were measured using Cannon-Fenske capillary viscometers and a Brookfield LV Microvis, cone and plate viscometer with a CP-40, 0.8° cone. Capillary viscometers received 10 mL of a sample for testing while the cone and plate viscometer received 0.50 mL. 

Limiting Viscosity Number. Limiting viscosity numbers for the polymers in distilled water are given in Table 3 Limiting viscosity number increases with increasing copolymer molecular weight. 

Further, after 12 to 14 percent hydrolysis, limiting viscosity number of the derived, partially-hydrolyzed copolymer is 3 to 10 times larger than that of its nonionic precursor. The ratio of [n] after... 

TABLE 3 Limiting Viscosity lumbers for Complex Copolymers in Vater and Electrolyte Solution... 

Limiting Viscosity Number Limiting Viscosity Number... 

Comparison of the limiting viscosity numbers determined in deionized water with those determined in 1 molar sodium nitrate shows a 20 per cent decrease in copolymer intrinsic viscosity in the saline solution. These results are consistent with previous studies using aqueous saline solutions as theta solvents for 2-propenamide polymers(47) Degree of hydrolysis controls the value of limiting viscosity number for the hydrolyzed copolymers in distilled water. 

Figure 14 gives limiting viscosity numbers for hydrolyzed copolymer 11 as a function of shear rate. Since limiting viscosity number is a function of molecular size, these data show that solution pseudoplasticity occurs because of compaction of the solvated polymer with increasing shear. 

Figure 14. Limiting viscosity numbers for hydrolyzed poly(starch g (2 propenamide)), sample 11.

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