Power-law index values

The rheological properties of pastes of emulsion PVC having k-values of 69,70 and 75 for coating fabrics were investigated using a coaxial viscometer and the influence of the content and type of plasticiser (dioctyl phthalate and dioctyl adipate) on these properties evaluated. Consistency index and power law index values for the various formulations were determined and the results obtained analysed statistically using shear stress as the variable for each paste. 14 refs. 

Also, on the same run, values for the up part of the ramping procedure as well as the down part have been averaged. In essence, then, the values reported correspond to averages of four data points. In general, the power-law index values differed by no more than 0.03 on the duplicate experiments. 

Another interesting point which is noted from the plot of the power-law index versus weight percent displayed in Figure 1 is that the power-law index values reach a maximum at the 50/50 composition blend. The power-law value of about 0.7 obtained for this blend is higher than for either of the component materials. This result indicates that the blend has a flow curve more closely approximating Newtonian in shape than either of the component materials. 

One can see that in Run 1 a mixture of two HDPE was used, both of which had rather low power-law index values (0.40 and 0.42), and the second one had a significantly lower value of the critical shear rate (300 s ). What is worse, the regrind, used in the amount of 15% in the final mix, had an exceptionally low critical shear rate (50 s ) along with a rather low value of the power-law index (0.32). This means that the processing window for this combination was quite narrow, and a slight increase in a flow rate would result in melt fracture, board roughness, sharkskin, and other profile defects (Figs. 17.10-17.14, and 17.19-17.23). 

The presence of calcium ion also affects the power-law index. Figure 12 shows that higher power-law index values were obtained with the calcium ion, i.e., divalent ions enhance Newtonian behavior. It should be mentioned that there was no phase separation due to the presence of the calcium ion [9,55]. This is presumably due to the temperature (20°C) and the degree of hydrolysis of the polymer examined (0.28). 

Figure 7.157 Velocity profiles for various power law index values...

Closed form solutions of Eqs. 8.114 and 8.115 are only possible for a few specific values of the power law index n, namely those for which (l-n)/2n is an integer. Worth [50] derived a solution for a power law index value n = 1/3. This is a usefui case because many of the high-volume commodity polymers have a power law index close to 1/3 see also Table 6.1. When n = 1/3, the throughput as a function of pressure can be written as ... 

From Fig. 9.2 it is clear that even small differences in thickness can result in large velocity differences, particularly for small values of the power law index. Even with thickness differences of 50% the velocity difference can be as high as 5 1 up to 10 1 for power law index values between 0.2 and 0.4. Most high-volume polymers have power law index values in this range. Velocity differences of 10 1 will cause severe distortion in the extrudate, which means that balancing the flow will be absolutely necessary. 

The shear stress vs. rate plots of molten PES are shown in Figure 2. As e q)ected, due to the strong non-Newtonian behavior, the variations of log (shear rate) with log(shear stress) are not linear. This suggests the power law index, n, will change with shear rate. The power law index values calculated are reported in Table 1. 

Figure 2 Power law index values for various polymers as summarized from Extrusion Processing Data (6)...

Where, K2 is called consistency and n is power law index. A typical polymer melt has power law index value less than 0.5. The viscosity of glass-filled PBT 13K follows the power-law (Power law index of 0.42) in whole shear rate indicates that glass fibers begin to orient even at the lowest shear rate. 

The described algorithm may not yield a converged solution in particular for values of power law index less than 0.5. To ensure convergence, in the iteration cycle (h + 1) for updating of the nodal pressures, an initial value found by... 

The effect of power-law index on the velocity profile is seen by plotting equation 3.134 for various values of n, as shown in Figure 3.39. 

Yooi24) has proposed a simple modification to the Blasius equation for turbulent flow in a pipe, which gives values of the friction factor accurate to within about 10 per cent. The friction factor is expressed in terms of the Metzner and Reed(I8) generalised Reynolds number ReMR and the power-law index n. 

A liquid w hose rheology can be represented by the power law model is flowing under streamline conditions through a pipe of 5 mm diameter. If the mean velocity of flow in I nt/s and the velocity at the pipe axis is 1.2 m/s, what is the value of the power law index n ... 

The latter form is required to reflect the fact that the direction of the shear stress must reverse when the shear rate is reversed, and to overcome objections such as y , and therefore r, having imaginary values when y is negative. The power n is known as the power law index or flow behaviour index, and K as the consistency coefficient. 

Here, K is sometimes referred to as the consistency index and has units that depend on the value of the power law index, n—for example, N-s"/m. The power law index is itself dimensionless. Typical values of K and n are listed in Table 4.4. In general, the power law index is independent of both temperature and concentration, although fluids tend to become more Newtonian (n approaches 1.0) as temperature increases and concentration decreases. The consistency factor, however, is more sensitive to temperature and concentration. To correct for temperature, the following relationship is often used ... 

If some or all of this curve is present, the models used to fit the data are more complex and are of two types. The first of these is the Carreau-Yasuda model, in which the viscosity at a given point (T ) as well as the zero-shear and infinite-shear viscosities are represented. A Power Law index (mi) is also present, but is not the same value as n in the linear Power Law model. A second type of model is the Cross model, which has essentially the same parameters, but can be broken down into submodels to fit partial data. If the zero-shear region and the power law region are present, then the Williamson model can be used. If the infinite shear plateau and the power law region are present, then the Sisko model can be used. Sometimes the central power law region is all that is available, and so the Power Law model is applied (Figure H. 1.1.5). 

Garcia et al. plotted Ap2/Ap as a function of for various values of pressure dependence b. They used a power law index n = 0.3 and two typical low and high injection pressures of 40 MPa (400 bar) and 100 MPa (1000 bar), respectively. Figure 4.17 presents a plot of the pressure increase as a function of the down-scaled thickness, (R. As expected, the pressure increases with a decrease in thickness and with an increase in the pressure dependence coefficient, b. The process that already begins with a high pressure requirement of 1000 bars,... 

We can solve for Q by trial and error after setting the limit in eqn. (6.199) to i. Figure 6.40 presents the value for Q as a function of power law index. As can be seen, the shear thinning effect is not very large. 

Sample balancing problem. Let us consider the multi-cavity injection molding process shown in Fig. 6.54. To achieve equal part quality, the filling time for all cavities must be balanced. For the case in question, we need to balance the cavities by solving for the runner radius R2. For a balanced runner system, the flow rates into all cavities must match. For a given flow rate Q, length L, and radius R, solve for the pressures at the runner system junctures. Assume an isothermal flow of a non-Newtonian shear thinning polymer. Compute the radius R2 for a part molded of polystyrene with a consistency index (m) of 2.8 x 104 Pa-s" and a power law index (n) of 0.28. Use values of L = 10 cm, R = 3 mm, and Q = 20 cm3/s. 

In addition, the viscosity behaviour of many polymers follows the power law. In many cases, the power law index n is only constant over a few decades in the shear rate and in general, decreases with increasing shear rate. Table 15.9 gives this extreme decrease in n-values for some typical polymers. 

The effect of the mixing mode on the power law index is also shown in Table 3. The polymer chains dissolved in a relatively mild impeller mixer have been mechanically broken the least consequently, these solutions have large [1-n] values. The molecules that have been subjected to severe shearing conditions (380 psi across the orifice) exhibit a Newtonian behavior indicating that the polymer chains have been broken down into short segments. 

Figure 17.7 shows that shear rate and frequency are almost identical for neat polymers that is, the Cox-Merz rule (see above) is valid for these systems. The power-law index n, calculated from the slope of the two viscosity curves in Figure 17.7, would be the same (excluding data at very low frequencies, at so-called Newtonian plateau). But for filled plastics this rule is not applicable, and one will get different values of the power-law index from the two viscosity curves. In other words, one will obtain much higher viscosity data from a parallel plate rheometer. 

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