Apparent shear rate, capillary

The correction factor for converting apparent shear rates at the wall of a circular cylindrical capillary to true shear rates is (3n + l)/4n, where n is the power law index of the polymer melt being extruded. 

Before the viscosity can be calculated from capillary data, as mentioned above, the apparent shear rate, 7 , must be corrected for the effect of the pseudoplastic nature of the polymer on the velocity profile. The calculation can be made only after a model has been adopted that relates shear stress and shear rate for this concept of a pseudoplastic shear-thinning material. The model choice is a philosophical question [11] after rheologlsts tried numerous models, there are in general two simple models that have withstood substantial testing when the predictions are compared with experimental data [1]. The first Is ... 

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]. 

The shear rate at the capillary wall is then calculated from the apparent shear rate at the wall as follows ... 

The pressure change is measured for each capillary at the apparent shear rate. Regression analysis is then used to obtain the slope and intercept for the function of pressure change (column two) in Table 3.5 with respect to the length divided by the radius (L/R, column six). The slope of the function is 0.376 MPa, and the intercept is 1.5 MPa. The regressed pressure change is obtained from the slope and the intercept, and the pressure change corrected for the end effects are as follows ... 

The melt Index test measures the flow property at a fixed wall shear stress In the capillary. The shear stress depends on the load specified for the condition and it is provided in Table 3.8. The apparent shear rate at the capillary wall that the resin experiences depends on the Ml value measured, and it can be calculated using a modification of Eq. 3.33 as follows ... 

Equations of interest are the relations between apparent shear rate Ds and volumetric output rate Q and between the shear stress aw at the large side wall and the mentioned pressure gradient dpjdz along the slit capillary. One obtains ... 

Melt flow rheology measurements were obtained on the MBAS polymer using an Instron capillary rheometer. The data reported were obtained using an 0.056-inch capillary, 90° included angle, with an L/D of 36. In Figure 5 the maximum shear stress (lb/in2) is plotted vs. the apparent shear rate (sec 1). The apparent viscosity (lb-sec/in2) vs. tem-... 

Corporation film-grade Ziegler-Natta linear low density polyethylenes will be presented. They are Resin E, Nova FP-015-A, MFI = 0.55, p = 0.9175 g/cc, and Resin C, Nova PF-120-F, MFI = 1.00, p = 0.9170 g/cc. Their capillary-flow behavior in terms of apparent shear stress vs. apparent shear rate are shown on Fig. 12.24. The melt fracture onset is also noted in Figure 12.24 and the data presented in the table below, indicate that resin E undergoes both sharkskin and gross melt fracture at lower apparent shear rates and stresses. 

Polyester BB1 was run twice in steady mode at 290°C (Figure 10), and shows that the orientational effect of the first run has a drastic effect on steady shear viscosity. In the first run the log viscosity vs. log shear rate had a slope of -0.92 (solid like behaviour, yield stress), but in the second run a pseudo-Newtonian plateau was reached from approx. 1 sec 1. Capillary viscosity values corresponded reasonably well with the second run steady shear data. The slope at high shear rates was close to -0.91 which corresponds nicely to the first-run steady shear run. All this could suggest, that this system is not completely melted, but still has some solid like regions incorporated. At 300°C capillary viscosity data showed an almost pseudo-Newtonian plateau. This corresponds quite well to the fact that fiber spinning as mentioned earlier was difficult and almost impossible below 290°C, but easy at 300°C. At an apparent shear rate of 100 sec 1, a die-swell was found to be approximately 0.95. 

In other words, apparent viscosity (as well as other apparent values in polymer rheology, snch as apparent shear rate and apparent shear stress) is a value calculated assuming Newtonian behavior and considering all pressure drops within the capillary (when using a capillary rheometer). A nonlinearity between shear rate and shear stress is typically observed for polymer melts. The fluid may behave like Newtonian at a very low shear rates to give a limiting viscosity iJq. 

Fig. 4.6. Dependence of tte shear stress of condis crystals of polytetrafluoroethylene (crystal form I) as a function of temperature. E)tawn after Ref. Apparent shear rate 8.9 s (capillary rheon ter, diameter 1.111 mm, length 19.0 mm)...

Here, and y pp are the shear rate at the wall of the capillary and the apparent shear rate, respectively n is the power-law index Q is the volumetric flow rate and R is the radius of the capillary. 

The measured values of polymer flow taken by capillary rheometers are often presented as plots of shear stress versus shear rate at certain temperatures. These values are called apparent shear stress and apparent shear rate at the tube wall. Corrections must be applied to these values in order to obtain true values. The corrected value of shear stress is determined by the Bagley correction [20]... 

Measured with a capillary rheometer. The apparent shear rate was about 8.9 s (high-temperature crystal). 

The wall shear stress versus the apparent shear rate behavior of suspension I in capillary flow is shown in Figure 5. For each capillary, flow instabilities were observed at low shear stress at the wall values. The ranges of the shear stresses at which the flow instabilities are observed in each capillary are shown in Figure 5. 

Figure 6. Apparent shear rate versus the reciprocal diameter at constant snear stress in capillary flow of suspension I.

An apparent shear rate can be defined for the porous media when the polymer is in the shear-thinning region. This apparent shear rate is given by Equation 11. Note the apparent shear rate in the rock is not the same as the shear rate at the capillary wall because shear rate is not constant across the capillary cross section. 

Apparent shear rates estimated from the experimental data using Equation 35 are always higher than those estimated from Equation 11 which are based on the capillary bundle model. The mobility of the polymer at the apparent shear rate is given by Equation 36. 

The apparent shear rate at the capillary wall can be determined from the flow rate V through the capillary. This can be determined from Eqs. 6 and 13 in Appendix 5.1. 

The terms apparent shear rate and apparent viscosity are used because Eq. 6.51 is valid only for Newtonian fluids. Therefore, if the fluid is non-Newtonian, the actual value of the shear rate at the capillary wall will be different. If the fluid behaves as a power law fluid with power law index n, the actual shear rate at the capillary wall is ... 

Figure 3.248 (right). Apparent shear rate at different positions of the piston during extrusion of molten PET trough capillary (L, = 40 mm = 20 mm/min T = 280 C Reoscope 1000). 

Mooney viscosities of MNR-ADS blends were found to be much higher than those of MNR-STR 5L blends. Rheological measurements were carried out the same method as described above.The apparent values of shear stress, shear rate and shear viscosity were calculated using the derivation of the Poiseuille law for capillary flow as shown in Equations (18.1) to (18.3). Plots of the apparent shear stress versus apparent shear rate for various blend compositions of MNR-STR/STR 5L and MNR-ADS/ADS were shown in Figures 18.32 and 18.33, respectively. Straight lines of the flow curve were observed for all sets of the test. The results corresponded to the power law equation proposed by Ostwald as shown in Equation 18.4. 

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