Shear factor

The properties of large pads can be tested in hydraulic presses such as those used to cure elastomers. Any shear factors can be added by using side... 

The influence of the shear factor can be seen from the following table, which shows the derived thickness (8v) as a function of Sv, for a velocity V of 2m/s, equivalent to 300 rpm ... 

Literature reports have used the following reactor parameters to correlate the effects of agitation intensity with cell injury in bioreactors agitator rpm, impeller tip speed, integrated shear factor and Kolmogorov eddy size. Additional parameters have been used for microcarrier bioreactors (discussed below). All correlations of cell injury with a bioreactor parameter should be used only qualitatively. These correlations are, at present, indicative of various trends or mechanistic hypotheses and should not be used for quantitative bioreactor scale-up. In addition, such correlations are applicable to the specific cell type, because different cell types are likely to exhibit different responses to fluid forces. 

The integrated shear factor (ISF), which is assumed to be (incorrectly, strictly speaking) a measure of the strength of the shear field between the impeller and the vessel wall, and may be somewhat more useful for scale-up purposes, is defined as ... 

Equation 20 shows that a porous medium is permeative, that is, a shear factor exists to account for the microscopic momentum loss. Our preliminary study recently reveals that, however, a porous medium is not only permeative but dispersive as well. The dispersivity of a porous medium has been traditionally characterized through heat transfer (in a single- or multifluid flow) and mass transfer (in a multifluid flow) studies. For an isothermal single-fluid flow, the dispersivity of a porous medium is characterized by a flow strength and a porous medium property-de-pendent apparent viscosity. For simplicity, we discuss the single-fluid flow behavior in this chapter without considering the dispersivity of the porous medium. 

For the simple case of creeping flow, that is, no noticeable inertial effects, the shear factor is simply the inverse of the intrinsic permeability,... 

Permeability. Permeability is the hydraulic conductance of a medium defined with direct reference to Darcy s law. In a somewhat more general sense, the shear factor is the hydraulic resistivity of the medium. When the term permeability is used, one normally refers to linear flow systems (no inertial effects). 

Shear Factor, F. The shear factor F is a generalized hydrodynamic resistivity of porous media. It appears in the momentum equation 63 and is needed to solve the problem of single-phase flow in porous media. The shear factor F can be related to the pressure drop of a unidirectional flow without bounding wall effects, that is, in a one-dimensional medium through equation 21. In this section, we give a detailed account for the derivation of the expressions for fv and F. 

Toward the end of this chapter, the shear factor for the consolidated media is given. 

The shear factor can now be defined based on the above results as... 

Hence, we have formed the closure for the single-phase flow in porous media using a model of the shear factor for both consolidated and granular media. For the term closure, we mean that the shear factor F of equation 63, which was introduced through averaging procedures, is now defined. 

The shear factor corrected for wall effects based on the model of Liu et al. (32) is given by... 

Correspondingly, the shear factor based on the Ergun equation is... 

Multidimensional Effects. In the previous section, we studied the wall effect on the shear factor. To give a full account of the wall effects, we now look at the no-slip flow effect posed by the containing wall (multidimensional effect) on the total pressure drop. For simplicity, let us rewrite the normalized pressure drop factor, fv, based on the permeability of the medium rather than the particle diameter,... 

For simplicity, we assumed that F does not vary with u, that is, the flow is in the Darcy s flow regime and the second term of the shear factor is zero. Because the variable Y is expected to have a finite value at the center of the pipe, the solution of equation 114 gives... 

Steady Flow in Packed Beds of Monosized Spherical Particles. Steady incompressible fully developed flow in porous media confined in a circular pipe can be treated with a single differential equation as given by equation 111. The inertial effects are only reflected in the shear factor term. Two purposes are served in this section to verify the integrity of the models presented earlier, including the passage model on shear factor and wall effects on the flow, and to show the flow behavior itself. The flow problem is solved numerically with a central difference method. An abundance of experimental data are available in the literature. However, we confine ourselves to the laminar flow regime for a packed bed of spherical particles. We make use of the latest available data presented by Fand et al. (110) for a packed bed with weak wall effects and the experimental data of Liu et al. (32). 

The coefficient for multidimensional effect, equation 121 or 123, is strictly speaking valid for Darcy s flow only. However, it is possible to estimate the multidimensional effect. Because the multidimensional effect is only significant near the wall, the shear factor used for evaluating the multidimensional coefficient may take an average value in the boundary layer. Let... 

We then discussed the modeling for single-fluid phase flow in porous media. In particular, the shear factor and permeability model of Liu et al. (32) is discussed in detail. The bounding wall effects are presented. This section completed the modeling requirements for single-phase incompressible flow in porous media. We showed how to solve the governing equations for flow in porous media and an approximate solution of the pressure drop for an incompressible flow through a cylindrically bounded porous bed was constructed. 

For the case of multidimensional or composite flows, the complete averaged Navier-Stokes equation needs to be solved. They are given by equation 19 and 20. The shear factor F is given by either equation 107 or 109. The evaluation of ds follows that for the case of one-dimensional flow above. 

Work well with materials that are highly cohesive or have very high shear factors... 

Tonnage = Perimeter (Land Length, inches) x material Thickness, inches X Shear factor... 

The warp and weft yams are positioned without being interlaced (Figure 21.13). A second set of finer warp and weft yarns binds them together, but does not contribute to the mechanical performance of the fabric. This eliminates the crimp and shear factor. 

Wind and seismic design standards such as ASCE have base shear factors that are a function of the POV. This makes sense because the response of the vessel is... 

Larger droplets with diameters greater than dTcnt break down into smaller droplets. Cov-elH, Miilli [58] presented the following theory for the shear factor Co in Eq. (2-2) ... 

---