Nominal shear rate

Figure 3.5 Calculated rheograms for a Bingham plastic with rj(oo) = 5 x 10 3 Pas and aB — 0.8 Pa. Note how the nominal shear rate is significantly lower than the actual value resulting from a given applied stress...

For this simple geometry the shear rate, 7, is equal to the difference between the velocity at the top of the element, U, and the velocity at the bottom of the element, zero, divided by the height of element H. The shear stress is again r = F/A, the element surface area divided by the force. The viscosity, q, is the ratio of shear stress, r, divided by shear rate, 7, at any shear rate, q = rjq. For Newtonian materials such as water, molasses, or gasoline at the nominal shear rates found in everyday life, the slope of the shear stress with shear rate curve is a constant and equal to the Newtonian viscosity. 

The data in Fig. 12 actually collapse onto a master curve when the wall stress o is rescaled by temperature T and the nominal shear rate y is normalized by a WLF factor aT [29]. Thus Eq. (6) for the critical stress oc is supported by the data in Fig. 12, where V does not change with T. Another feature of the transition is that the amplitude of the flow discontinuity does not vary with T. In other words, the extrapolation length bc, which is evaluated according to Eq. (4a) at the transition, is a constant with respect to T. Thus for a given surface, bc is more than just a material property such as the melt viscosity r. It essentially depends only... 

In drop-breakup experiments with polymers, the shear viscosity is typically shear-rate-dependent. So in plots such as Fig. 9-11, the viscosities are taken to be those of the melts at a nominal shear rate in the mixer. In the experiments of Sundararaj and Macosko, for example, the nominal shear rate y at a given motor speed is estimated from the linear drag flow that is assumed to exist in the narrowest gap of the mixer. 

Figure 9.13 Number averaged diameter of droplets d of polypropylene (M = 60,000) in polystyrene (M = 200,000) as a function of wt% polypropylene mixed in three different mixers at a nominal shear rate of around 65 sec and T — 200°C. The viscosities of the the PP and PS under these conditions are 840 and 950 Pa-s, respectively. The interfacial tension F is 5.0 dyn/cm. The error bars represent the distribution of droplet sizes, and they encompass one standard deviation in each direction from the mean. The deviation from the Taylor limit at low concentrations is attributed to non-Newtonian effects, while the increase in droplet size at higher concentrations is attributed to droplet coalescence. Note that similar droplet sizes are obtained in all three different mixers. (Reprinted with permission from Sundararaj and Macosko, Macromolecules 28 2647. Copyright 1995, American Chemical Society.)...

Figure 16. Wall shear stress vs. nominal shear rate for homogeneous and heterogeneous slurries. (Reproduced with permission from reference 50. Copyright 1991 Society of Petroleum Engineers.)...

The slurry velocity at which a particle bed forms is defined as critical deposition velocity, VD, and represents the lower pump rate limit for minimum particle settling. A further decrease in slurry velocity leads to increased friction loss, as indicated by a characteristic hook upward of curve A, and may also lead to pipe plugging. After shutdown, if flow rate over the settled solids is gradually increased, a response similar to curve A of Figure 16 is once again obtained. With increasing nominal shear rate, wall shear stress decreases until a minimum is reached and then increases rapidly thereafter. The fluid velocity that corresponds to this minimum stress value is the critical resuspension velocity, Vs. 

Here, (8V/D) is the wall shear rate for a Newtonian fluid and is referred to as the nominal shear rate for a non-Newtonian fluid which is identical to ecjuation (2.5) in Chapter 2. Alternatively, writing it in terms of the slope of logT - log(8V/D) plot s. 

First, the tube viscometer data will be converted to give the wall shear stress, and nominal shear rate, (8V/D) ... 

Attention is drawn to the fact that the values of m and n for use in turbulent region are deduced from the data in the laminar range at the values of (SVp/D) which is only the nominal shear rate at the tube wall for streamline flow, and thus this aspect of the procedure is completely empirical. Dziubinski [1995] stated that equation (4.26) reproduced the same experimental data as those referred to earlier with an average error of 15%, while equation (4.28) correlated the turbulent flow data with an error of 25%. Notwithstanding the marginal improvement over the method of Dziubinski and Chhabra [1989], it is reiterated here that both methods are of an entirely empirical nature and therefore the extrapolation beyond the range of experimental conditions must be treated with reserve. 

Although the early versions of capillary models, namely, the Blake and the Blake-Kozeny models, are based on the use of V = Vq/s and = LT, it is now generally accepted [Dullien, 1992] that the Kozeny-Carman model, using Vi = VoT/e provides a more satisfactory representation of flow in beds of particles. Using equations (5.36), (5.38) and (5.40), the average shear stress and the nominal shear rate at the wall of the flow passage (equations 5.34 and... 

For generalised non-Newtonian fluids, Kemblowski et al. [1987] postulated that the shear stress at the wall of a pore or capillary is related to the corresponding nominal shear rate at the wall by a power-law type relation ... 

The flow of viscoplastic fluids through beds of particles has not been studied as extensively as that of power-law fluids. However, since the expressions for the average shear stress and the nominal shear rate at the wall, equations (5.41) and (5.42), are independent of fluid model, they may be used in conjimction with any time-independent behaviour fluid model, as illuslrated here for the streamline flow of Bingham plastic fluids. The mean velocity for a Bingham plastic fluid in a circular tube is given by equation (3.13) ... 

As seen in Chapter 3, the quantity (8V/D) is the nominal shear rate at the wall (also see equation 5.35, for a circular tube, Dh = D, Kq = 2). Substituting for the nominal shear rate and wall shear stress from equations (5.41) and (5.42) in equation (5.48), slight re-arrangement gives ... 

Upper branch. Figure 12.13 shows laser-Doppler velocimetry data by Munstedt and co-workers on an HDPE during the two parts of the cycle at a nominal shear rate of 137 s the slip velocity on the upper branch is ten times that on the lower branch. 

For turbulent flow through rotor-stator devices with teeth, the aforementioned velocity field results indicate that flow stagnation on the leading edge of the downstream stator teeth provides a major energy field for emulsification and dispersion. It is not clear from these results what role is played by flow in the shear gap. The simulations indicate that the flow in the rotor-stator gap is not a simple shear flow but is more like a classical turbulent shear flow. Use of nominal shear rate may not be useful in scale-up. 

Vendors often design and scale-up rotor-stator mixers based on equal rotor tip speed, Vtip = kND, where N is the rotational speed of the rotor and D is the rotor diameter. This criterion is equivalent to equal nominal shear rate in the rotor-stator gap, y- In niost industrial rotor-stator mixers, the shear gap width S, remains the same on scale-up, making the two criteria equivalent. 

The nominal shear rate in the rotor-stator gaps is calculated as follows ... 

The classical way to evaluate k and n for a power law fluid is by using logio scales to plot the shear stress (in fps units, that is, poundal/ft or lb (mass)/ft s on the vertical axis, versus nominal shear rate (s ) on the... 

To interpret the dependence of ju, on E, Radko and Chrambach proceed by analogy with the shear rate induced by a falling ball, and propose for the fluid flow near an electrophoresing sphere a nominal shear rate... 

Shear Rate in the Melting Zone. Using the conditions given in Problem 8A.6, calculate the shear rate in the melt film and compare this with the nominal shear rate in the metering section. 

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