Pressure shear-thinning fluid

As indicated earlier, non-Newtonian characteristics have a much stronger influence on flow in the streamline flow region where viscous effects dominate than in turbulent flow where inertial forces are of prime importance. Furthermore, there is substantial evidence to the effect that for shear-thinning fluids, the standard friction chart tends to over-predict pressure drop if the Metzner and Reed Reynolds number Re R is used. Furthermore, laminar flow can persist for slightly higher Reynolds numbers than for Newtonian fluids. Overall, therefore, there is a factor of safety involved in treating the fluid as Newtonian when flow is expected to be turbulent. 

Because n < 1 for a shear thinning fluid, < 7 will be Jess than unity and a reduction in pressure drop occurs. The lower the value of n and the larger the value of b, the greater the effect will be. It will be noted that the effects of expansion of the air as the pressure falls have not been taken into account. 

In Chapter 5 of this book we derived the equations that govern the pressure flow between two parallel discs for a Newtonian fluid. In a similar fashion, we can derive the equations that govern flow rate, gate pressure, and pressure distributions for disc-shaped cavities filling with a shear thinning fluid. For the equations presented in this section, we assumed a power law viscosity. For the velocity distribution we have... 

The lubrication approximation was discussed in terms of Newtonian fluids. Considering a nearly parallel plate pressure flow (H = Ho — Az), where A is the taper, what additional considerations would have to be made to consider using the lubrication approximation for (a) a shear-thinning fluid flow, and (b) a CEF fluid ... 

The discussion so far has related to the drag reduction occurring when a gas is introduced into a shear-thinning fluid initially in streamline flow. A more general method is required for the estimation of the two phase pressure drop for mixtures of gas and non-Newtonian liquids. The well-known Lockhart-Martinelli [1949] method will now be extended to encompass shear-thinning liquids, first by using the modified Lockhart-Martinelli parameter, Xmod (equation 4.8). Figure 4.14 shows a comparison between... 

The flow rate Increases when b Increases. The temperature sensitivity increases when the power law index becomes smaller. This indicates that strongly shear thinning fluids will be more sensitive to temperature fluctuations than weakly shear thinning fluids. Based on the results with Newtonian fluids it can be expected that the sensitivity to temperature fluctuations will get worse with large positive values of the pressure gradient. 

As discussed earlier, the helix angle has a strong effect on the pressure drop. The minimum pressure drop occurs at a helix angle between 50 and 60°. Below a helix angle of 50° and above 60° the pressure drop increases quite rapidly. Thus, the proper value of the helix angle is around 50 to 60°. The optimum helix angle for shear thinning fluids is less than 50°. 

Motion in a capillary is described using cylindrical co-ordinates r, z and 6 L and R are the length and radius of the capillary tube and Pq — is the pressure drop along the tube. This is illustrated in Figure 3.16. Only shear thinning fluids are considered in this section. 

Example 5.6 illustrates the use of this model to estimate the pressure drop for the injection of a shear-thinning fluid into a reservoir. 

Viscosity is function of die temperature into the die. Lee et al (27) used an Arrhenius type function and Amstrom and Pipes (29) considered a non-Newtonian behavior of the polymer melt in order to calculate the pressure into the die for a shear-thinning fluid. The model of a non-newtonian fluid produces a relation between the pulling force and pulling velocity. 

Non-Newtonian fluids vary significantly in their properties that control flow and pressure loss during flow from the properties of Newtonian fluids. The key factors influencing non-Newtonian fluids are their shear thinning or thickening characteristics and time dependency of viscosity on the stress in the fluid. 

A highly concentrated suspension of flocculated kaolin in water behaves as a pseudo-homogeneous fluid with shear-thinning characteristics which can be represented approximately by the Ostwald-de Waele power law, with an index of 0.15. It is found that, if air is injected into the suspension when in laminar flow, the pressure gradient may be reduced even though the flowrate of suspension is kept constant, Explain how this is possible in slug flow and estimate the possible reduction in pressure gradient for equal volumetric flowrates of suspension and air. 

This disagreement between theory and practice must therefore partly be due to the non-Newtonian shear-thinning viscosity. This conclusion is supported by the work of Kiparissides and Vlacopoulos (35), who showed that for a Power Law model fluid, lower n values reduce the disagreement between theory and experiments, as illustrated in Fig. 6.27. They used the FEM for computing the pressure profile, which eliminates the geometrical approximations needed in the Gaskell model. 

Distributed Parameter Models Both non-Newtonian and shear-thinning properties of polymeric melts in particular, as well as the nonisothermal nature of the flow, significantly affect the melt extmsion process. Moreover, the non-Newtonian and nonisothermal effects interact and reinforce each other. We analyzed the non-Newtonian effect in the simple case of unidirectional parallel plate flow in Example 3.6 where Fig.E 3.6c plots flow rate versus the pressure gradient, illustrating the effect of the shear-dependent viscosity on flow rate using a Power Law model fluid. These curves are equivalent to screw characteristic curves with the cross-channel flow neglected. The Newtonian straight lines are replaced with S-shaped curves. 

Inspection of the data shows that the pressure difference increases less rapidly than the flowrate. Taking the first and the last entries in the table, it is seen that when the flowrate increases from 1 x 10 7 to 1 x 10 4 m3/s, that is by a factor of 1000, the pressure difference increases from 1 x 103 to 1 x 105 N/m2 that is by a factor of only 100. In this way, the fluid appears to be shear-thinning and the simplest model, the power-law model, will be tried. 

Information about the behavior of shear thinning melts in twin screws can be derived from Eq. 7.13, in addition to the approximation of the measured values. The concept of representative viscosity based on Frederickson [3] is used for this purpose. The representative viscosity is the viscosity that provides the same pressure generation as a Newtonian fluid under the specified conditions. If we combine Eqs. 7.13 and 7.2, we obtain for the representative viscosity ... 

A method for predicting pressure drop and volume fraction for non-Newtonian fluids in annular flow has been proposed by Eisen-berg and Weinberger (AlChE J., 25, 240-245 [1979]). Das, Biswas, and Matra (Can. J. Chem. Eng., 70,431 37 [1993]) studied holdup in both horizontal and vertical gas/liquid flow with non-Newtonian liquids. Farooqi and Richardson (Trans Inst. Chem. Engrs., 60, 292-305, 323-333 [1982]) developed correlations for holdup and pressure drop for gas/non-Newtonian liquid horizontal flow. They used a modified Lockhart-Martinelli parameter for non-Newtonian liquid holdup. They found that two-phase pressure drop may actually be less than the single-phase liquid pressure drop with shear thinning liquids in laminar flow. 

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