Throttle ratio

The LMDE was one of the more notable achievements of the pintle injector technology. This engine used a translating sleeve to ccmtrol both annular and slot flow areas to provide the 10 1 throttling ratio (from 10,000 to 1,000 Ibf of thrust) demanded of this application [4,5]. Figure 28.2 provides a photo of the engine and a... 

The first term on the right-hand side of the equalation is the drag flow term. The second term is the pressure flow term. The ratio of pressure flow to drag flow is termed the throttle ratio, r (see Section 7.4.1.3) ... 

Equation 7 is not valid for rpumping efficiency can become close to 100%. This is considerably better than the single screw extruder where the optimum pumping efficiency is 33% at a throttle ratio value of 0.33 (r[Pg.44]

Equation 7.236 is valid if the clearance is negligible. The ratio of pressure flow to drag flow is often referred to as the throttle ratio (Drossel quotient in German). It enables a compact expression for the combined drag and pressure flow ... 

The optimum pumping efficiency can be determined by setting the first derivative of the pumping efficiency with respect to the throttle ratio equal to zero ... 

This yields the following expression for the optimum throttle ratio r ... 

Optimum throttle ratio and pumping efficiency versus helix angle... 

Figure 7.67 shows the velocity profiles for b = 1.0 and several values of the throttle ratio r. 

Figure 7.69 compares the velocities at b-values of -1, 0, and 1 at a throttle ratio of r = 0. It is clear that the temperature effect is significant even in pure drag flow (r = 0). The effect is greater for higher values of the throttle ratio. This is shown in Fig. 7.70. 

Figure 7.71 Dimensionless flow rate versus b for several values of the throttle ratio (r)...

The flow rate increases with b for positive values of the throttle ratio. For r = -0.2 and r = -0.4 the curves are fairiy flat and the effect of temperature is siight. At positive values of the throttle ratio the effect of temperature is significant and increases with the throttle ratio. For positive r-values the flow rate increases with b. This means increasing the screw temperature at constant barrel temperature will increase the flow rate. This points to a positive effect of internal screw heating. At constant screw temperature, reducing the barrel temperature will increase the flow rate. 

The relative flow rate can be defined at the ratio of F(b)/T(b = 0). Figure 7.72 shows how the relative flow rate varies with b for several values of the throttle ratio. 

The condition Ns = 0 corresponds to adiabatic conditions at the screw surface. As a result, the temperature gradient at the screw surface ( = 0) is zero. The condition Nb = 1000 corresponds to almost isothermal conditions at the barrel surface ( = 1). As a result, the temperature at = 1 equals unity for all curves. The screw temperature increases when the throttle ratio reduces as a result of higher shear rates at the screw surface. 

From this equation, the various parameters that influence the temperature rise by viscous dissipation can be cleariy distinguished. The temperature rise increases with consistency index (m), the length (L), the screw diameter (D), the screw speed (N), and the power law index (n). The temperature rise reduces with melt density (p), the specific heat (Cp), the channel depth (H), the throttle ratio (rt), and the helix angle (cp). 

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. 

Figure 7.92 shows how the adiabatic temperature development is affected by the value of the throttle ratio and the screw speed. The top curve is actually two curves almost completely overlapping. One curve corresponds to a screw speed of 1.67 rev/ sec (100 rpm) and a throttle ratio of 0.4, while the other curve corresponds to a screw speed of 9.3 rev/sec (558 rpm) and a throttle ratio of 0. The bottom curve is also two curves that are almost completely overlapping. One curve corresponds to a screw speed of 1.67 rev/sec (100 rpm) and a throttle ratio of -0.5, and the other curve corresponds to a screw speed of 0.42 rev/sec (25 rpm) and a throttle ratio of 0. 

Figure 7.92 Temperature vs. distance for different values of screw speed and throttle ratio...

Figure 7.128 shows the cross-channel shear strain as a function of normal distance for one value of the throttle ratio r = 0.1, assuming that ttDN/H = 1. 

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. 

With this expression the distributive mixing process in the melt conveying zone of a single screw extruder can he evaluated quantitatively. The total shear versus normal distance for several values of the throttle ratio is shown in Fig. 7.130. 

The total strain depends strongly on both the normal distance and on the throttle ratio. Fluid elements close to the wall experience a high level of shear strain and, thus, will be well mixed. Elements further away from the wall experience a lower level of strain and will not be mixed as well as elements close to the wall. 

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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