Viscosity functions

Finally, a generalized viscosity function in the form of a weight fraction-dependent viscosity ratio, r]Q/r], could be derived as follows ... 

When other parameters are fixed, the viscosity function, Eq. (7) is a reducing function of TLCP weight fraction. Figure 1 shows the patterns. Generally, the blend viscosities are lower than the matrix viscosity (i.e., V Tjo < 1.0), even when a small amount of TLCP is added to the system and even when the TLCP itself has a viscosity higher than that of the matrix (e.g., when 8 = 10). The smaller the ratio 8 of the TLCP viscosity rji to the matrix viscosity tjo, the lower their blend viscosity T). 

Figure 2 Theoretical viscosity function patterns by varying the exponent e.

It is interesting to note that our derivation of the viscosity function Eq. (7) was not restricted within TP-TLCP systems. The negative deviation flow behav-... 

Figure 6 Theoretical viscosity function versus viscosity ratio 8 = 771/770 at different blending ratios .

The viscosity function of the natural gums is utilized in both oil in water and water in oil emulsions. Often the gums are referred to as emulsifying agents. They are considered not so much as emulsifiers, but rather as emulsion protectors or stabilizers. To a large extent, the function is to increase the viscosity of the aqueous phase so that it approaches, or slightly exceeds, that of the oil hence, there is less tendency for the two phases, once emulsified, to separate by mechanical slippage. 

If the properties of the fluid are such that the shear stress and shear rate are not directly proportional but are instead related by some more complex function, the fluid is said to be non-Newtonian. For such fluids the viscosity may still be defined as xyx/yyx, but it is no longer a constant. It is, instead, a function of either the shear rate or shear stress. This is called the apparent viscosity (function) and is designated by ip. 

The two viscous rheological properties are m, the consistency coefficient, and n, the flow index. The apparent viscosity function for the power law model in terms of shear rate is... 

Starting with the equations for r = fn(j>) that define the power law and Bingham plastic fluids, derive the equations for the viscosity functions for these models as a function of shear stress, i.e., rj = fn(r). 

The usual approach for non-Newtonian fluids is to start with known results for Newtonian fluids and modify them to account for the non-Newtonian properties. For example, the definition of the Reynolds number for a power law fluid can be obtained by replacing the viscosity in the Newtonian definition by an appropriate shear rate dependent viscosity function. If the characteristic shear rate for flow over a sphere is taken to be V/d, for example, then the power law viscosity function becomes... 

Intrinsic viscosity is related to the relative viscosity via a logarithmic function and to the specific viscosity by a simple algebraic relationship. Both of these functions can be plotted on the same graph, and when the data are extrapolated to zero concentration they both should predict the same intrinsic viscosity. The specific viscosity function has a positive slope and the relative viscosity function has a negative slope, as shown in Fig. 3.7. The molecular weight of the polymer can be determined from the intrinsic viscosity, the intercept of either function, using the Mark-Houwink-Sakurada equation. 

The following section will now focus on experimental methods for determining viscosity and how the viscosity function relates to analyzing single-screw extrusion processes. 

The viscosity activation energy is often evaluated to determine the temperature dependence for these viscosity functions. Viscosities at two temperatures are provided in Table 3.7. 

In this example the temperature viscosity function used is based on the viscosity, Tin, at a reference temperature which has units of Kelvin ... 

For the power law region at a shear rate of 20 1/s, the calculations lead to the temperature dependence of the power law viscosity function using the same method ... 

The Newtonian viscosity of some polymers increases essentially linearly with the weight average molecular weight, and for other polymers the Newtonian viscosity increases with an exponential power of the molecular weight. The exponential power is found to be about 3.4, but this power does deviate for some polymers. These two transitions, Newtonian to pseudo-plastic and linear to 3.4 power in the Newtonian range are often related to molecular structure as demonstrated in Fig. 3.31 [22]. The polystyrene data used to develop the Adams-Campbell viscosity function showed almost no shear thinning at [18]. That is why the power law slope, s, is a function of and M. At the slope is zero and the material would be essentially Newtonian. 

The results presented here are encouraging but only qualitative and have been produced using this first-order model. Current limitations of the model are the use of a constant-viscosity function independent of temperature and shear rate. Also, the dynamic local temperature of the barrel and screw (Section fO.lO) must be incorporated into the model they are currently set as constants. An enhanced model for the film thickness at both the barrel and screw surfaces should be added to the current model along with flows induced by pressure gradients. 

The base case pressure flow calculation and the above viscosity function require that the shear rate be calculated from the screw rotation equations using Eq. 7.41 ... 

The symbols Nt and N2 denote the normal stress functions in steady state shear flow. Symmetry arguments show that the viscosity function t](y) and the first and second normal stress coefficients P1(y) and W2(y) are even functions of y. In the... 

At > 2 the viscosity o/x in Maxwell s element grows unlimited with increase of x. When value mkWk is not small, the activation mechanism makes the viscosity function pass under stationary flow from x via a maximum, which was observed in experiments described in Ref.21). 

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