Wall shear rate, function

Values of n and k for the suspensions used are given in Table 5.2. Experimental results are shown in Figure 5.8 as wall shear stress R as a function of wall shear rate (dn /dyfr o using logarithmic coordinates. 

Tu, as a function of Newtonian wall shear rate, F, for a number of common polymers. Consider the melt spinning of PS at a volumetric flow rate of 4.06 x 10 cm /s through a spinneret that contains 100 identical holes of radius 1.73 X 10 cm and length 3.46 x 10 cm. Assume that the molecular weight distribution is broad. 

FIGURE 3.11 Extmdate swell as a function of die wall shear rate for polypropylene at 180°C. (From Minoshima, W. White, J.L. Spruiell, J.E. J. Appl. Polym. Sci., 1980, 25, 287. With permission.)... 

A 18% iron oxide slurry (density 1170kg/m ) behaves as a Bingham plastic fluid with Tq = 0.78 Pa and yu-s = 4.5 mPa s. Estimate the wall shear stress as a function of the nominal wall shear rate (8F/D) in the range 0.4 < V < 1.75 m/s for flow in a 79 mm diameter pipeline. 

If the function /(r) is unknown and the Newtonian result (eqn 7.17) cannot be assumed, how may the wall shear-rate be found experimen-ta% Fortunately the unknown dependence of y on r controls both Q and dQ/dAP, and it is possible to show that the wall shear rate is... 

This is the Rabinowitsch equation. Thus, determining Q and dQ/dAP as a function of AP, it is possible to determine y, the shear rate at the wall (eqn 7.19) the shear stress at the wall is also known (from eqn 7.13). It is then possible to plot wall shear stress versus wall shear rate and to determine the apparent viscosity. 

Figure 2 Different extrudate distortions for PP as function of flow rate. The experiment was performed on a Gottfert 2003 capillar rheometer using a die of dimensions L D = 20 I at 180°C. The extrudates are shown in order of increashig distortion and are obtained at apparent wall shear rates of 144,360,540, 720 and 12800 (from left to right).

Tube or pipe viscometers take many forms, but they should all be able to give the pressure-drop P as a function of flow rate Q for situations where the tube is long enough to be able to neglect entrance and end effects, say L/a > 50. In this case we can calculate the viscosity as a function of the wall shear rate, y, which is given by... 

Figure 18 The apparent wall shear rate as a function of inverse diameter for various wall shear stresses.

A.3 Viscosity from Capillary Rheometer Data for LLDPE. From values of the apparent wall shear stress, and the apparent wall shear rate, )>a, given in Appendix A.3.3 calculate the viscosity as a function of shear rate and compare your results to those which can be obtained directly from the data in Appendix A.3.3 (i.e., To and Yc where and are the corrected wall shear stress and rates, respectively). In particular, use the values of at each LID to obtain AFent at each shear rate. Correct the values of AFtot to obtain x. Determine )>w by correcting )>a for the nonparabolic velocity profile using Eq. 3.3-7. 

Capillary viscometer data on a food emulsion are given as follows in terms of the wall shear stress as a function of the volumetric flow rate. Use Eq. (14.4.9) to compute the wall shear rate corresponding to each datum point and thus obtain a graph of the shear viscosity versus the shear rate. 

Figure 9 Shear viscosity as a function of shear rate at the wall at 200°C, (O) Resin A, ( ) Resin B, (O) Resin C, (A) Resin D, (A) Resin E. (Refer to Table 2 for symbols code.) Source Ref. 56.

Equation (15) indicates that the wall shear stress changes with position on the cell surface and at each position the stress changes with time. Figures 9 and 10 show the distributions of the fluid shear rate in the eddy and the cell surface shear stress as a function of the dimensionless radius, R, and time, yt, respectively. 

Fig. 4.2.2 Dimensionless shear rate and viscosity as a function of radius for a power-law fluid under the conditions shown in Figure 4.2.1. For a highly shear thinning material, the shear rate is large near the wall and close to zero near the center. The viscosity can vary by several orders of magnitude in the pipe.

Figure 6. Viscosity as a function of concentration for PTF and BTF. Solvent n-hexane temperature 25°C. Number near each point indicates approximate shear rate at the wall.

By solving Eqs. (4) and (7) simultaneously, the mass flux can be calculated provided the wall shear stress is known as a function of particle superficial volume flow rate. Botterill and Bessant (1973) have proposed several relationships for shear stress, however, these are not general. LaNauze (1976) also proposed a method to measure this shear stress experimentally. 

Figure 3.18 Apparent shear rate as a function of the wall stress (tJ. The first derivative of the function is used to perform the Weissenberg-Rabinowitsch correction. The data are for the HDPE resin at 190°C as shown in Fig. 3.17...

The calculation of the shear rate at the capillary wall, 7 , is computed from the function slope of Fig 3.18 and the apparent shear rate using Eq. 3.36. The derivative of the function appears relatively constant over the shear stress range for Fig. 3.18. Many resin systems will have derivatives that vary from point to point. The corrected viscosity can then be obtained by dividing the shear stress at the wall by the shear rate i ,. Equation 3.36 is known as the Weissenberg-Rabinowitsch equation [9]. 

Figure 3.19 Calculation of power law index n from the shear stress at the wall (tJ as a function of the apparent shear rate ()...

There would be a minimum of 80 data sets needed to generate this data for one temperature. Because of the time involved, usually about 10 to 15 shear rate data points are generated at each temperature. The plot of the viscosity as a function of shear rate at 270°C is presented in Fig. 3.22. The viscosity below a shear rate of 5 1/s would be best taken using a cone and plate rheometer. The wall friction for the capillary rheometer between the piston and the rheometer cylinder wall would likely cause a force on the piston of the same order as the force due to the flow stress. 

Mooney clearly showed that the relationship between the shear stress at the wall of a pipe or tube, DAP/4L, and the term 8V/D is independent of the diameter of the tube in laminar flow. This statement is rigorously true for any kind of flow behavior in which the shearing rate is only a function of the applied shearing stress.1 This relationship between DAP/4L and 8VJD may be conveniently determined in a capillary-tube viscometer, for example. Once this has been done over the range of... 

If one considers fluid flowing in a pipe, the situation is highly illustrative of the distinction between shear rate and flow rate. The flow rate is the volume of liquid discharged from the pipe over a period of time. The velocity of a Newtonian fluid in a pipe is a parabolic function of position. At the centerline the velocity is a maximum, while at the wall it is a minimum. The shear rate is effectively the slope of the parabolic function line, so it is a minimum at the centerline and a maximum at the wall. Because the shear rate in a pipe or capillary is a function of position, viscometers based around capillary flow are less useful for non-Newtonian materials. For this reason, rotational devices are often used in preference to capillary or tube viscometers. 

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