Mean shear rate

The dynamics of reactor flow is also important for its effect on the crystal agglomeration, since the intensity of turbulent shear dominates the orthoki-netic mechanism for both processes of aggregation and disruption. The mean shear rate is estimated as (see Harnby etai, 1992)... 

The mean shear rate produced by an agitator in a mixing tank ym is proportional to the rotational speed of the agitator N [Metzner and Otto (1957)]. 

Since the mean shear rate in a mixing tank ym is given by equation 5.1... 

Cs = Cb - Co, Cb = 1, and Cd = 3 (Fox 1995).36 Note that at spectral equilibrium, Vp = p, % = To = p( I - i/i)), and (with Sc = 1) R = Rq. The right-hand side of (4.117) then yields (4.114). Also, it is important to recall that unlike (4.94), which models the flux of scalar energy into the dissipation range, (4.117) is a true small-scale model for p. For this reason, integral-scale terms involving the mean scalar gradients and the mean shear rate do not appear in (4.117). Instead, these effects must be accounted for in the model for the spectral transfer rates. 

Fig. 8.22 Fraction of naphthalene concentration due to colloidal entrainment at (a) G = 5 and (b) = 20 s , for naphthalene and naphthalene compounds containing 1,2, or 3 C as side chains, where is the mean shear rate. Reprinted with permission from Sterhng Jr MC, Bonner JS, Page CA, Ernest ANS, Auteniieth RL (2003) Partitioning of crude oil polycyclic aromatic hydrocarbons in aquatic systems. Environ Sci Technol 37 4429 434. Copyright 2003 American Chemical Society...

Relationship between the Mean Strain and the Mean Shear Rate Show that, for continuous, single-valued viscometric shear flows, the following relationship... 

Note that the mean shear rate in the channel is given by... 

From the preceding equations, we can now select a barrel diameter, calculate the optimum channel depth, then the screw speed, the mean shear rate, and the power per unit volume. In the following table, the results are shown with barrel diameters from 8 in to 16 in. 

At first sight, based on the results we have, any of these extruders would be able to do the job. The 8-in extruder would need 7.5 turns to build up the required pressure, whereas the 16-in one will only need 3.75 turns. We note that both mean shear rate and specific power input per... 

Figure 14.5 Mean shear rate over speed for different machine series and shear rate distribution in the center of the channel and in the intermeshing zone...

Metzner and Otto (1957) were the first to examine the power consumption of a turbine stirrer in liquids whose viscosities are adequately described by the power law. They found the mean shear rate to be proportional to the speed of rotation of the stirrer y = kN, k = 13. Therefore, the Reynolds number becomes Re = N2 mdip/K, and the power characteristic can be expressed in a dimensionless form Ne = /(Re, m). 

An analytical solution to this integro-partial differential equation is not possible without some simplifying assumptions. In the sections that follow, anal5d ical solutions are presented for particle growth in a CSTR and batch precipitation reactors. For systems in which shear is the dominant collision mechanism and not Brownian difhision, the birth and death functions can be rewritten in terms of the mean shear rate, y, as follows [104]. 

The mean shear rate obtained by averaging the absolute value of shear rate over a period and over the membrane area which, for VSEP, is an annular region with radii Ri and R2, can be formulated as... 

Problem 8-13. Lateral Migration of a Deformable Drop in 2D Poiseuille Flow. A viscous drop of viscosity fi and density p is carried along in the unidirectional motion of an incompressible, Newtonian fluid of viscosity p and density/) = p between two infinite plane walls. The radius of the undeformed drop is denoted as a, and the distance between the walls is d. We assume that the capillary number, Ca = apXl/rr, is small so that the drop deformation is also small. Here, a is the interfacial tension, and G is the mean shear rate of the undisturbed flow. 

Metzner [8] determined that the mean shear rate, y (s ), for a stirred tank in the transitional to laminar regime can be determined by the Metzner-Otto equation ... 

Impeller Type Mean Shear Rate Constant, K... 

The calculation of power follows methods described previously, using the estimated weight average density, p, in the expression for Reynolds number and power. The viscosity for a Newtonian system is best measured by using a paddle-type rotating bob viscometer. The Metzner-Otto method is commonly used to determine the Re for a non-Newtonian system, where viscosity is a function of shear rate. This method consists of determining the mean shear rate from y = KN, where N is the stirring speed in rps, K is 10 for a propeller, and y is the mean shear rate is in s. Viscosity will not influence the calculation of power when Re > 200. 

Jeffrey remarks that Kohlman (KlOa) also confirmed the correctness of Eq. (253) for Re,(a/ffo) < 0.25. (This is a particle Reynolds number based on sphere radius and mean shear rate, 1VJR , in the tube.)... 

When the concept of the mean shear rate predicted by Metzner (1961) was applied, an agitator speed of 700 rpm used in Midler s experiment should have an effea equivalent to only 120 see . However, comparing this with the results in the Couette type device, the equivalent shear rate in those turbulent conditions should be 10-100 times those predicted by Metzner. Therefore, Midler and Finn suggested that maximum rather than average shear stresses must be considered in the presence of high local velocity gradients. 

The mean shear rate obtained by averaging the absolute value of shear rate over a period and over the membrane area. 

The mean shear rate in laminar flow depends on the kinematic viscosity, v of the fluid, and the specific power dissipation, e, expressed in W kg T... 

For efificient viscosimetric measurements, only ISO cups (ISO 2431) should be used. The length of the capillary is 20 mm. The shear rates lie—for efflux times f=30-100 s—between y=60 and 1500 s" They depend on the capillary radius Ry which can be varied through interchangeable outlet nozzles. The calculation of the mean shear rate is made using the Hagen-Poiseuille-Law ... 

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