Cross-channel velocity

Cross-channel velocity in the transformed (Lagrangian) frame ... 

Equation 7.21 is the literature expression for motion in the x direction for barrel rotation physics. The boundary conditions here are = 0 aty = 0 (screw root) and Erf = Kx aty = // (flight tip). Cross-channel velocity in the laboratory (Eulerian)... 

Consider the two-dimensional flow in a channel formed by parallel plates, through which fluid may enter or leave the channel (Fig. 5.13). The similarity analysis of this situation is facilitated by assuming the form of the cross-channel velocity. With an assumed crossstream velocity, the axial-momentum equation can be reduced to an ordinary differential equation for a scaled axial velocity. 

In search of a similarity reduction of the governing equations, two principal assumptions are made. First, assume that the cross-channel velocity t is a function of y alone, which is the coordinate across the channel. Second, assume that a nondimensional axial velocity that is scaled by W is a function of y alone,... 

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... 

From the above data, wecanfirstcomputethebarrel velocity to be Ub = tiDN=(). 995 m/s. Note that this speed is on the low end of realistic speeds used in industry for low density polyethylene. PE-LD usually can have screw speeds that lead to velocities up to of 1 m/s. Other polymers can take up to 0.5 m/s, and PVC about 0.2 m/s. The down channel and cross channel velocities become... 

Figure 11.28 Cross-channel velocity field for the unwrapped screw channel with a down-channel pressure gradient of 20MPa/m.

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... 

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.

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 residence time of a fluid particle at a given height in one cycle is obtained by dividing the channel width with the local cross-channel velocity given in Eq. 9.2-18. The residence time at is different, of course, from that at c,.. It is easy to show that the... 

The SSE is an important and practical LCFR. We discussed the flow fields in SSEs in Section 6.3 and showed that the helical shape of the screw channel induces a cross-channel velocity profile that leads to a rather narrow residence time distribution (RTD) with crosschannel mixing such that a small axial increment that moves down-channel can be viewed as a reasonably mixed differential batch reactor. In addition, this configuration provides self-wiping between barrel and screw flight surfaces, which reduces material holdback to an acceptable minimum, thus rendering it an almost ideal TFR. 

Observe the experimental performance of the mixer from Figure 10.35. The experimental cross channel temperature profiles are given however, to determine the CO Vs of the inlet and outlet stream we would also need the cross channel velocity profile. We would need the velocity profile because proper averaging must be done on a flowing enthalpy basis that is, the averaging would have to be done as follows ... 

The boundary conditions are zero velocity at the walls and zero slope at any planes of symmetry. Analytical solutions for the velocity profile in square and rectangular ducts are available but cumbersome, and a numerical solution is usually preferred. This is the reason for the transient term in Equation 16.7. A flat velocity profile is usually assumed as the initial condition. As in Chapter 8, is assumed to vary slowly, if at all, in the axial direction. For single-phase flows, u can vary in the axial direction due to changes in mass density and possibly to changes in cross-sectional area. The continuity equation is just AcUp = constant because the cross-channel velocity components are ignored. 

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 ... 

Similar problems occur in the melt conveying zone. A polymer element at about 2/3 of the height of the channel will have no cross-channel velocity component and as a result will have a short residence time in the melt conveying section and little mixing... 

The time that a fluid element spends in the upper portion of the channel (t ) is determined by the width of the channel and the cross-channel velocity ... 

The shear strain reaches -°o at the screw and barrel surface. The shear strain becomes zero at y = 0.98H and = 0.16H this corresponds to the streamline where the shear strain in region A is canceled exactly by the opposite shear in region Aj. The cross-channel shear strain reaches a maximum at y = (2/3)H this is where the cross-channel velocity becomes zero. The value of the maximum shear increases with increasing throttle ratio r, because the residence time increases when the throttle ratio increases. 

The two expressions above were derived for a Newtonian fluid using the flat plate approximation considering both down- and cross-channel velocity components. Figure 7.159 shows the RTD for a single screw extruder as well as for pressure flow of a Newtonian fluid between flat plates. 

In order to properly determine the RTD of an extruder we have to consider not only down- and cross-channel velocity components but also normal velocity components that occur at the flight flanks. This will require a numerical analysis either FDA, FEA, or BEA. Even though the depth of the channel is usually quite small compared to the channel width, the residence time at the flight flanks is substantial because... 

The temperature field is shown in Chapter 12, Fig. 12.7. The barrel surface is set at 175°C and the screw surface is taken as adiabatic (zero heat flux). The melt temperatures at any point in the channel are considerably higher than the barrel temperature. The highest temperatures occur at about two-thirds of the channel height this is where the cross-channel velocities are the lowest. The highest temperatures are about 31 °C above the barrel temperature. This agrees well with experimental results published earlier [83]. 

Substituting the expression for dp/dx back into Eq. 8.95, we obtain the cross-channel velocity profile ... 

Thus, Q is directly proportional to the degree of All and the screw speeed, N. The cross-channel velocity profile, Vx, is... 

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