7.4.2.4 Plunger area/diameter effects

explain what we trying to achieve here = 120 true mm

Figure: Schematic of the plunger and piston balance forces

The pressure at the plunger tip can be evaluated from a balance forces acts on the hydraulic piston and plunger as shown in Figure . The atmospheric pressure that acting on the left side of the plunger is neglected. (why? perhaps to create a question for the students) Assuming a steady state and neglecting the friction, the forces balance on the rod yields

(7.29)

In particular, in the stationary case the maximum pressure obtains

(7.30)

The equation () is reduced when the rode area is negligible; plus, notice that to read

(7.31)

= 0.4

Figure: Reduced liquid metal pressure at the plunger tip and reduced gate velocity as a function of the reduced plunger diameter

Rearranging equation () yields

(7.32)

²¹

The gate velocity relates to the liquid metal pressure at plunger tip according to the following equation combining equation () and () yields

(7.33)

perhaps, discuss Under the assumption that the machine characteristic is ,

more discussion on the meaning of the results

= 0.4

Figure: A general schematic of the control volume of the hydraulic piston with the plunger and part of the liquid metal

A control volume as it is shown in Figure is constructed to study the effect of the plunger diameter, (which includes the plunger with the rode, hydraulic piston, and shot sleeve, but which does not include the hydraulic liquid or the liquid metal jet). The control volume is stationary around the shot sleeve and is moving with the hydraulic piston. Applying the first law of thermodynamics, when that the atmospheric pressure is assumed negligible and neglecting the dissipation energy, yields why? should be included in the end.

(7.34)

In writing equation (), it should be noticed that the only change in the control volume is in the shot sleeve. The heat transfer can be neglected, since the filling process is very rapid. There is no flow into the control volume (neglecting the air flow into the back side of the plunger and the change of kinetic energy of the air, why?), and therefore the second term on the right hand side can be omitted. Applying mass conservation on the control volume for the liquid metal yields

(7.35)

The boundary work on the control volume is done by the left hand side of the plunger and can be expressed by

(7.36)

The mass flow rate out can be related to the gate velocity

(7.37)

Mass conservation on the liquid metal in the shot sleeve and the runner yields

(7.38)

Substituting equations (-) into equation () yields

(7.39)

Rearranging equation () yields

(7.40)

Solving for yields

(7.41)

Or in term of the maximum values of the hydraulic piston

(7.42)

When the term is neglected ( for liquid metal)

(7.43)

Normalizing the gate velocity equation () yields

(7.44)

= 0.5

Figure: The gate velocity, as a function of the plunger area,

The expression () is a very complicated function of . It can be shown that when the plunger diameter approaches infinity, (or when ) then the gate velocity approaches . Conversely, the gate velocity, , when the plunger diameter, . This occurs because mostly and . Thus, there is at least one plunger diameter that creates maximum velocity (see figure ). A more detailed study shows that depending on the physics in the situation, more than one local maximum can occur. With a small plunger diameter, the gate velocity approaches zero because approaches infinity. For a large plunger diameter, the gate velocity approaches zero because the pressure difference acting on the runner is approaching zero. The mathematical expression for the maximum gate velocity takes several pages, and therefore is not shown here. However, for practical purposes, the maximum velocity can easily (relatively) be calculated by using a computer program such as DiePerfect.