Velocity profile cross-channel

The retention theory for focusing-FFF was developed for focusing-S-FFF but can be transferred to other focusing mechanisms [72,73]. Janca and Chmelik developed this theory for several shapes of fractionation channels [74] and found that it is advantageous to form axially asymmetrical velocity profiles in channels with modulated cross-sectional permeability [83]. 

Vfjp is the friction velocity and =/pVV2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu p/. . The universal velocity profile is vahd in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, = AP/4L where AP is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are vahd only near the pipe wall. 

Figure 8.4 illustrates pressure-driven flow between flat plates. The downstream direction is The cross-flow direction is y, with y = 0 at the centerline and y = Y at the walls so that the channel height is 2Y. Suppose the slit width (x-direction) is very large so that sidewall effects are negligible. The velocity profile for a laminar, Newtonian fluid of constant viscosity is... 

For laminar flow in channels of rectangular cross-section, the velocity profile can be determined analytically. For this purpose, incompressible flow as described by Fq. (16) is assumed. The flow profile can be expressed in form of a series expansion (see [100] and references therein), which, however, is not always useful for practical applications where often only a fair approximation of the velocity field over the channel cross-section is needed. Purday [101] suggested an approximate solution of the form... 

The mass transfer coefficient is usually obtained from correlations for flow in non-porous ducts. One case is that of laminar flow in channels of circular cross-section where the parabolic velocity profile is assumed to be developed at the channel entrance. Here the solution of LfivfiQUE(7), discussed by Blatt et al.(H>, is most widely used. This takes the form ... 

In FFF, separation is determined by the combined action of the nonuniform flow profile and transverse field effects. The classical configuration assumes the FFF channel as two infinite parallel plates (see Figure 12.4), of which the accumulation wall lies at x=0, where x is the cross-channel axis (directed upward from the accumulation wall). Inside the channel, the carrier fluid, assumed to have a constant viscosity, has a velocity profile u(x) that takes the form... 

The classical FEE retention equation (see Equation 12.11) does not apply to ThEEE since relevant physicochemical parameters—affecting both flow profile and analyte concentration distribution in the channel cross section—are temperature dependent and thus not constant in the channel cross-sectional area. Inside the channel, the flow of solvent carrier follows a distorted, parabolic flow profile because of the changing values of the carrier properties along the channel thickness (density, viscosity, and thermal conductivity). Under these conditions, the concentration profile differs from the exponential profile since the velocity profile is strongly distorted with respect to the parabolic profile. By taking into account these effects, the ThEEE retention equation (see Equation 12.11) becomes ... 

Consider the steady, laminar flow of an incompressible fluid in a long and wide closed conduit channel subject to a linear pressure gradient, (a) Derive the equation for velocity profile, (b) Derive the equation for discharge per unit width and cross-sectional mean velocity, and compare this with the maximum velocity in the channel, (c) Derive the equation for wall shear stress on both walls and compare them. Explain the sign convention for shear stress on each wall. 

Figure 5.17 illustrates the nondimensional axial (i.e., u) and cross-channel velocity (i.e., 0) profiles for several values of Rev- It is apparent that increasing the injection velocity (Reynolds number) from below skews the velocity profile toward the upper wall of the... 

In macroscopic reactors, knowledge of the velocity profile in the channel cross-section is a necessary and sufficient prerequisite to describe the material transport. In microscopic dimensions down to a few micrometers, diffusion also has to be considered. In fact, without the influence of diffusion, extremely broad residence time distributions would be found because of the laminar flow conditions. Superposition of convection and diffusion is called dispersion. Taylor [91] was among the first to notice this strong dominating effect in laminar flow. It is possible to transfer his deduction to rectangular channels. A complete fluid dynamic description has been given of the flow, including effects such as the influence of the wall, the aspect ratio and a chemical wall reaction on the concentration field in the cross-section [37]. 

If we combine the flow generated by the down channel and cross channel flows, a net flow is generated in axial or machine direction (w ) of the extruder, schematically depicted in Fig. 6.5. As can be seen, at open discharge, the maximum axial flow is generated, whereas at closed discharge, the axial flow is zero. From the velocity profiles presented in Fig. 6.5 we can easily deduce, which path a particle flowing with the polymer melt will take. 

Figure 6.5 Down channel, cross channel and axial velocity profiles for various situations that arise in a single screw extruder.

The integration of constants C and C2 is evaluated from the boundary conditions Vj (0) = 0 and vx(H) = Vt,x. Substituting these boundary conditions into Eq. 6.3-12 yields the cross-channel velocity profile... 

Thus we observe that the cross-channel gradient is proportional to screw speed and barrel diameter, and inversely proportional to the square of the channel depth. By substituting Eq. 6.3-15 into Eq. 6.3-13, we obtain the cross-channel velocity profile... 

A full analytical solution of the cross channel flow vx(x,y) and vy x, y), for an incompressible, isothermal Newtonian fluid, was presented recently by Kaufman (18), in his study of Renyi entropies (Section 7.4) for characterizing advection and mixing in screw channels. The velocity profiles are expressed in terms of infinite series similar in form to Eq. 6.3-17 below. The resulting vector field for a channel with an aspect ratio of 5 is shown... 

Fig. 6.10 Cross-channel velocity profile from Eq. 6.3-16. Note that melt circulates around a plane located at exactly two-thirds of the height.

A negative pressure flow for the positive pressure gradient led to the term back flow, namely, that the pressure drives the fluid opposite to the direction of the net flow. This term led to the erroneous concept that actual flow toward the feed end occurs in some part of the channel. However, it is important to note that under no condition does the melt flow backward along the screw axis.5 Fluid particles may move backward along the z direction, but not along the axial direction, /. Once a fluid particle passes a given axial location, it cannot cross this plane backward. This is evident from the velocity profile in the axial direction ... 

Fig. 6.14 Cross-channel, down-channel, and axial velocity profiles for various Qp/Qj values, in shallow square-pitched screws [Reproduced by permission from J. M. McKelvey, Polymer Processing, Wiley, New York, 1962.]...

The flow rate is obtained by integrating the velocity profile over the cross section of the channel to give... 

Let us next consider the simple isothermal drag flow (dP/dz = 0) of a shear-thinning fluid in the screw channel. The cross-channel flow, induced by the cross-channel component of the barrel surface velocity, affects the down-channel velocity profile and vice versa. In other words, the two velocity profiles become coupled. This is evident by looking at the components of the equation of motion. Making the common simplifying assumptions, the equation of motion in this case reduces to... 

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