Down channel velocity

The down-channel velocity in the laboratory frame for very wide channels (N/W < 0.1) as a function of the height of the channel/is as follows ... 

The down-channel velocity due to screw rotation in the laboratory frame for a channel with a small aspect ratio is provided by Eq. 7.25 that is, H/Wis small and less than 0.1. 

The down-channel velocity in the transformed frame for/ /PFless than 0.1 is shown by Eq. 7.26. This equation is the same as the z-direction flow in the literature. 

Figure 6.3 Down-channel velocity profiles for different pumping situations with a single screw extruder.

From observation, it is clear that the solid bed has a local down channel velocity usz and a local velocity component into the melt film of usy. As before, we resolve the barrel surface velocity Ub into a down channel, v,bz, and a cross channel component, ubx. Using Tadmor s notation, we define the relative velocity between barrel surface and solid bed as... 

Figure 11.27 Down-channel velocity field for the unwrapped screw channel with a down-channel pressure gradient of 20MPa/m.

Fig. 6.12 Down-channel velocity distribution for pure drag flow from Eq. 6.3.17 for various HjW ratios. [Reprinted by permission from E. C. Bernhardt, Ed., Processing of Thermoplastic Materials, Reinhold, New York, p. 290 (1959).]...

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... 

The cross- and down-channel velocity profiles are (see Section 6.3) ... 

Thus, the operating conditions affect the down-channel velocity profile, but not the crosschannel velocity profile. At closed discharge conditions (Qp/Qd = — 1), both the down-channel and cross-channel velocities vanish at / 2/3, implying that the whole plane at... 

The change in size of the solid bed over a small down-channel increment will depend on the rate of melting at the solid bed-melt film interface. Consider a small differential volume element, perpendicular to the solid-melt interface (Fig. 9.32). The solid bed has a local down-channel velocity Vsz and a local velocity component into the melt film of Vsy. The barrel surface velocity Vb is resolved into down-channel and cross-channel components Vb . and Vbx. 

The down-channel velocity of the solid bed is obtained from Eq. 9.3-27... 

Melt conveying is the forward motion of the molten polymer through the extruder, due to the pumping action of the rotating screw. This simple drag flow Md is proportional to melt density, down-channel velocity, and cross-sectional area of the screw channel. In most cases, however, there is also a pressure gradient as the melt moves downstream, either... 

The infinite channel width assumption applies to shallow channels, channels with a width-to-depth ratio higher than 10 (W/H > 10). If the depth of the channel is large relative to the width of the channel, the effect of the flight flanks on the down-channel velocity profile has to be taken into account. Several reviews of the work on melt conveying in extruders have been written [101-106]. 

By integration of Eq. 7.196, the down-channel velocity as a function of normal distance y is obtained. By using boundary conditions V2(0) = 0 and v lH) = Vb, the foliowing expression is obtained ... 

By the same procedure employed to derive the down-channel velocity, the crosschannel velocity can be determined ... 

It can be seen that the cross channel velocity at y = 2H/3 is zero. Thus, the material in the top one-third of the channel moves towards the active flight flank and the material in the bottom two-thirds of the channel moves towards the passive flight flank. It is clear that in reality the situation becomes more complex at the flight flanks because normal velocity components must exist to achieve the circulatory flow patterns in the cross-channel direction. However, these normal velocity components will be neglected in this analysis. Normal velocity components were analyzed by Perwadtshuk and Jankow [129] and several other workers. The actual motion of the fluid is the combined effect of the cross- and down-channel velocity profiles. This is shown in Fig. 7.57. 

The shear stresses can be evaluated from Eqs. 7.223 and 7.224 and the equations for down-channel velocity profile, Eq. 7.197 or 7.216, and the-cross channel velocity profile, Eq. 7.211. The power consumption in the screw channel can be written as ... 

Equation 7.246 combined with Eq. 7.195 describes the basic problem. It is convenient to write the resulting equation in dimensionless form. For this purpose, the following dimensionless quantities are defined the dimensionless depth = y/H, the dimensionless down-channel velocity v = v /Vi,, and a reduced pressure gradient Fr. The reduced pressure gradient is defined as ... 

Velocity profiles and temperature profiles in extruder dies are intimately related because of the high polymer melt viscosity and because the melt viscosity is temperature dependent. It is important to understand and appreciate this interrelationship in order to understand the die forming process and the variables that influence this process. The relationship between velocity and temperature profiles can be illustrated by considering the down-channel velocity profile in a circular die. Typical velocity profiles are shown in Fig. 7.106 for several values of the power law index. 

If it is assumed that the down-channel velocity profile can be approximated with Eq. 10.28, then the volumetric throughput is approximately ... 

Rowell and Finlayson [1] solved the down channel velocity profile for a screw pump. 

Fig. 46. Down-channel velocity profiles for three pressure conditions.

The cross-channel fiow is shown in Figure 44. At the top of the channel, the material fiows to the left by drag flow and at the bottom of the channel, the material flows to the right by pressure flow The shear rate can be determined from the velocity profile the shear rate is equal to the slope of the velocity profile. This slope of the velocity profile is also called the velocity gradient. From the down-channel velocities, we can determine the down-channel shear rates these are shown in Figure 47. 

It can be shown that the change in striation thickness on mixing, and hence in the degree of laminar mixing, is a simple function of the total shear strain imposed on the system. However, at the end of the process the components of the mixture still exist as discrete components. The total shear strain exerted on the melt is a function of the residence time of the melt in the process. As a result of the complex velocity profile of the melt in the screw channel, the residence time of the melt in the channel varies as a function of the position of the melt in the screw channel as well as the down-channel velocity of the melt. 

Referring to the mixing process mechanisms described in Sections 8.2 and 8.3, the variables of screw geometry and flow restriction effect down channel velocity profile but not those of the cross channel. 

The solid bed is assumed to move as a plug with down channel velocity of Vsj, where... 

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