Drag flow rate

Equations 7.57 and 7.58 that are developed above use the as the velocity component as shown for screw rotation physics. As previously discussed, the classic drag flow model [8] assumed that the equivalent flow rate will be obtained if the barrel is rotated in the direction opposite to that of the screw as long as the linear velocity of the unwound barrel Is numerically equal to the linear velocity of the screw core. For this classic barrel rotation model, is used as the velocity component instead of V(,z-Since is less than the drag flow rate would be reduced. It Is Interesting to note here that the classical model using reduced the drag flow rate such that It provided a better estimate of the actual rotational flow rate, but for the wrong reason. 

Equation E2.5-9 further indicates that, in the absence of a pressure drop, the net flow rate equals the drag flow rate. Note that qp is positive if Pq > PL and pressure flow is in the positive z direction and negative when Pp > Po- The net flow rate is the sum or linear superposition of the flow induced by the drag exerted by the moving plate and that caused by the pressure gradient. This is the direct result of the linear Newtonian nature of the fluid, which yields a linear ordinary differential equation. For a non- Newtonian fluid, as we will see in Chapter 3, this will not be the case, because viscosity depends on shear rate and varies from point to point in the flow field. 

Figure E3.6c plots the dimensionless flow rate q/qd, where qd is the drag flow rate, namely, the flow rate with zero pressure gradient, versus the dimensionless pressure gradient G. The figure shows that, whereas for Newtonian fluids, as expected, there is a linear relationship, non-Newtonian fluids deviate from linearity. The more non-Newtonian the fluid is, the greater is the deviation. Of particular interest is the inflection point indicating, for example, that in screw extruders, even for the isothermal case, increasing die resistance brings about somewhat unexpected changes in flow rate.

The screw extruder is equipped with a die, and the flow rate of the extruder as well as the pressure rise at a given screw speed are dependent on both, as shown in Fig. 6.16. The screw characteristic line at a given screw speed is a straight line (for isothermal Newtonian fluids). This line crosses the abscissa at open discharge (drag flow rate) value and the ordinate at closed discharge condition. The die characteristic is linearly proportional to the pressure drop across the die. The operating point, that is, the flow rate and pressure value at which the system will operate, is the cross-point between the two characteristic lines, when the pressure rise over the screw equals the pressure drop over the die. 

The increase in the drag-flow rate has a profound effect on performance. For a given flow rate q, the optimum gap size in SMP and JMP configurations can be obtained by differentiating Eqs. E6.10-2 and E6.10-4, respectively, with respect to H to give Hopt = 3q/Vo and Hopt = 3q/2Vo- Next we substitute these values in Eqs. E6.10-2 and E6.10-4 and find that the ratio of pressure rises is... 

Example 6.11 Drag Flow Rate in a CDP Pump Consider a 20-cm-diameter disk CDP with a = 0.5 and H — 1 cm rotating at 240 rpm. Calculate the volumetric flow rate. 

Volumetric flow rate Drag flow rate... 

Equation 9.2-5 can be represented by plotting the flow rate Qs versus the pressure rise APs. Such plots, called screw characteristics, appear in Fig. 9.4. The intersection with the ordinate gives the drag-flow rate value and that with the abscissa, the maximum pressure at closed discharge. For isothermal flow of a Newtonian fluid in the absence of leakage flow,... 

Example 9.3 Nonisothermal Drag Flow of a Power Law Model Fluid Insight into the effect of nonisothermal conditions, on the velocity profile and drag flow rate, can he obtained by analyzing a relatively simple case of parallel-plate nonisothermal drag flow with the two plates at different temperatures. The nonisothermicity originates from viscous dissipation and nonuniform plate temperatures. In this example we focus on the latter. 

The velocity profile in a parallel channel given in Eq. 6.6-23 can be written in dimensionless form in terms of the pressure and drag flow rates as follows (52) ... 

The NPD is given by Eq. 9.2-44 (which is equivalent to Eq. El 1.3-3) where X = tft is given in Eq. 9.2-47, and it is the ratio of residence time in the extruder (given by the ratio of free volume of the screw to SSE volumetric flow rate, V/Q) to the mean circulation time over the flight zone (given by the ratio of the free volume of the screw to the total (drag) flow rate... 

Qd Volumetric drag flow rate in a screw extruder (6.3-22)... 

Another interesting observation is that the volumetric drag flow rate is independent of any fluid property for Newtonian fluids. Thus, the drag flow rate for water will be the same as oil, molten nylon, etc. The drag flow rate is directly proportional to screw speed because Vb = riDNcoscp, where N is the screw speed. 

The clearance reduces the drag flow rate. The drag flow rate is reduced by a factor 5/H. The corrected drag flow becomes ... 

D) is about 0.01% of the drag flow rate at four times the normal clearance, the leakage flow is about 1% of the drag flow rate. Thus, the leakage flow becomes significant only when the clearance is larger than about four times its normal value. 

The dimensionless flow rate is the actual flow rate divided by the drag flow rate, thus ... 

The shape of the velocity profiles with one adiabatic wall is considerably different from the shape with two isothermal walls. In the latter case the velocity profile has a typical s-shape while in the former case there is monotonic reduction in the slope of the curves when A > 1. The velocities with an adiabatic screw are higher than with an isothermal screw. As a result, the flow rate is increased considerably compared to the isothermal drag flow rate. When A = 1 the flow rate equals 0.5 the flow rate increases with the value of A. When A becomes very large, the flow rate approaches unity and the velocity profile approaches plug flow. 

When the throttle ratio is zero, the mass flow rate equals the drag flow rate in this case the axial pressure gradient is zero. When the throttle ratio is positive, the axial pressure gradient is positive and the mass flow rate is less than the drag flow rate. This results in longer residence times and higher melt temperatures. 

If the flow rate through the mixing section is assumed to be about two-thirds of the drag flow rate of the preceding screw section and the axial length is about two diameters (L , = D), then Eq. 8.144 can be simplified to the following form ... 

The pumping capacity can be checked by using Eq. 7.291. The drag flow rate can be written as ... 

Since drag flow occurs as a result of polymer melt adhering to the barrel surface, the reduced barrel circumference will affect the drag flow rate. The drag flow per revolution is found by moving the barrel with respect to the screw over a distance of ttD. With a full barrel circumference, this results in a volume per revolution equal to ... 

This results in the familiar drag flow rate equation ... 

Vj = BH7tD--(l-f)BH7rD The actual drag flow rate becomes ... 

This is the screw characteristic for the Newtonian case, and as shown in the simple one-dimensional flat plate model described at the start of this section, it consists of drag and pressure flow terms. The ratio of pressure to drag flow rates (this is sometimes called the throttle ratio) is... 

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