Average shear rate

Obviously shear rate in different parts of a mixing tank are different, and therefore there are several definitions of shear rate (/) for average shear rate in the impeller region, oc V, the proportionaUty constant varies between 8 and 14 for all impeller types (2) maximum shear rate, oc tip speed (%NU), occurs near the blade tip (3) average shear rate in the entire tank is an order of magnitude less than case / and (4) minimum shear rate is about 25% of case 3. 

An example of liquid/liquid mixing is emulsion polymerization, where droplet size can be the most important parameter influencing product quality. Particle size is determined by impeller tip speed. If coalescence is prevented and the system stability is satisfactory, this will determine the ultimate particle size. However, if the dispersion being produced in the mixer is used as an intermediate step to carry out a liquid/liquid extraction and the emulsion must be settled out again, a dynamic dispersion is produced. Maximum shear stress by the impeller then determines the average shear rate and the overall average particle size in the mixer. 

The maximum particle size obtained in an experimental study using a stirred tank (Figure 8.15) and bubble column (Figure 8.11c) respectively is plotted against the estimated average shear rate in Figure 8.17. 

The injection and holding pressures were 100 MPa and 50 MPa, respectively. For thick (DIN 53455 Form 3) and thin (cut from the plates) dumbbell-shaped samples, an average shear rate of 230 s was kept constant. 

ND3, impeller speed (Hz) IftDfa, power NDpt, average shear rate Na 1/mixing time. 

Maximum shear rate, average shear rate and mixing time, impeller tip velocity, NDt. 

When in suspension however cells tend to move at the minimum relative velocity with respect to the surrounding fluid. This means that a spherical cell will rotate at the velocity gradient of the suspension, or in proportion to it. For a spherical cell of diameter, d, the average shear rate, Yavg> on its surface is given by ... 

Hydrodynamic effects on suspended particles in an STR may be broadly categorized as time-averaged, time-dependent and collision-related. Time-averaged shear rates are most commonly considered. Maximum shear rates, and accordingly maximum stresses, are assumed to occur in the impeller region. Time-dependent effects, on the other hand, are attributable to turbulent velocity fluctuations. The relevant turbulent Reynolds stresses are frequently evaluated in terms of the characteristic size and velocity of the turbulent eddies and are generally found to predominate over viscous effects. 

Fig. 2.8.11 Schematic constitutive relationship (shear stress versus shear rate) for a wormlike micelle system exhibiting constitutive instability. When the average shear rate exceeds yc the fluid subdivides into two coexisting shear bands residing on stable branches of the constitutive relationship.

Graham (10) flocculated porous silica particles, diameter 7.6 pm, in a paddle-stirred vessel with an average shear rate of... 

Fig. 11 presents the results of some 30 simulations for various conditions and two impeller types in terms of the mean agglomeration rate constant observed in the various simulations vs. the vessel-averaged shear rate found in the simulations. The simulations all started from the dotted curve for relating local agglomeration rate constant to local shear rate. A clear decrease in the maximum of /i0 as well as a shift toward higher average shear rates was found which are caused by the local nature of the nonlinear flow interactions only. These... 

Fig. 11. The discrepancy between the original kinetic relation due to Mumtaz et al. (1997) and the observed relation between mean agglomeration rate constant fl0 and volume-averaged shear rate. Symbols refer to individual numerical simulations (LES). RT stands for Rushton Turbine, PBT for Pitched Blade Turbine. Reproduced with permission from Hollander et al. (2001b).

Zj helical length of the channel at the barrel wall z(r) helical length of the channel at radial position r Zf, helical length of the metering channel at the barrel wall melt density of the fluid 7 average shear rate in the channel... 

The combined mass flow for rotation-driven and pressure-driven flow is given by Eq. 1.28 and is the expected rate of the process. The average shear viscosity is calculated using the average shear rate in the channel for screw rotation and the bulk temperature. This method is also known as the generalized Newtonian method. 

Two different calculation methods were used for the simulations (1) the generalized Newtonian method as developed above, and (2) the three-dimensional numerical method presented in Section 7.5.1. The generalized Newtonian method used a shear viscosity value that was based on the average barrel rotation shear rate and temperature in the channel. The average shear rate based on barrel rotation (7ft) is provided by Eq. 7.52. Barrel rotation shear rate and the generalized Newtonian method are used by many commercial codes, and that is why it was used for this study. 

Now that the temperature is known at the /c+1 position, the pressure gradient at the end of the volume can be estimated using Eq. 7.105 as follows by evaluating the viscosity at temperature T, i and the average shear rate in the channel using Eq. 7.41 ... 

Scale Agitator speed (rpm) Tip velocity (m/S) Average shear rate = (tip speed/ distance from tip to baffle) (1/s) VisiMix simulation shear rate in bulk volume (1/s) VisiMix simulation shear rate near impeller blade (1 /s) VisiMix simulation shear rate near baffle (1/s)... 

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