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9.4 The Analysis

The model is presented here with a minimal of mathematical details. However, emphasis is given to all the physical understanding of the phenomena. The interested reader can find more detailed discussions in several other sources [#!poro:genickthesis!#]. As before, the integral approach is employed. All the assumptions which are used in this model are stated so that they can be examined and discussed at the conclusion of the present chapter. Here is a list of the assumptions which are used in developing this model:

  1. The main resistance to the air flow is assumed to be in the venting system.

  2. The air flow in the cylinder is assumed one-dimensional.

  3. The air in the cylinder undergoes an isentropic process.

  4. The air obeys the ideal gas model, .

  5. The geometry of the venting system does not change during the filling process (i.e., the gap between the plates does not increase during the filling process).

  6. The plunger moves at a constant velocity during the filling process, and it is determined by the diagram calculations.

  7. The volume of the venting system is negligible compared to the cylinder volume.

  8. The venting system can be represented by one long, straight conduit.

  9. The resistance to the liquid metal flow, , does not change during the filling process (due to the change in the , or Mach numbers).

  10. The flow in the venting system is an adiabatic flow (Fanno flow).

  11. The resistance to the flow, , is not affected by the change in the vent area.

With the above assumptions, the following model as shown in Figure [*] is proposed. A plunger pushes the liquid metal, and both of them (now called as the piston) propel the air through a long, straight conduit.

Figure: A simplified model for the venting system

The mass balance of the air in the cylinder yields

    (9.1)

This equation ([*]) is the only equation that needed to be solved. To solve it, the physical properties of the air need to be related to the geometry and the process. According to assumption [*], the air mass can be expressed as
    (9.2)

The volume of the cylinder under assumption [*] can be written as
    (9.3)

Thus, the first term in equation ([*]) is represented by
    (9.4)

The filling process occurs within a very short period time [], and therefore the heat transfer is insignificant [*]. This kind of flow is referred to as Fanno flow4. The instantaneous flow rate has to be expressed in terms of the resistance to the flow, , the pressure ratio, and the characteristics of Fanno flow [#!poro:shapiro!#]. Knowledge of Fanno flow is required for expressing the second term in equation ([*]).

The mass flow rate can be written as

    (9.5)

where
    (9.6)

The Mach number at the entrance to the conduit, , is calculated by Fanno flow characteristics for the venting system resistance, , and the pressure ratio. is the maximum value of . In vacuum venting, the entrance Mach number, , is constant and equal to .

Substituting equations ([*]) and ([*]) into equation ([*]), and rearranging, yields:

    (9.7)

The solution to equation ([*]) can be obtained by numerical integration for . The residual mass fraction in the cavity as a function of time is then determined using the ``ideal gas'' assumption. It is important to point out the significance of the . This parameter represents the ratio between the filling time and the evacuation time. is the time which would be required to evacuate the cylinder for a constant mass flow rate at the maximum Mach number when the gas temperature and pressure remain at their initial values, under the condition that the flow is choked, (The pressure difference between the mold cavity and the outside end of the conduit is large enough to create a choked flow.) and expressed by
    (9.8)

Critical condition occurs when . In vacuum venting, the volume pushed by the piston is equal to the flow rate, and ensures that the pressure in the cavity does not increase (above the atmospheric pressure). In air venting, the critical condition ensures that the flow is not choked. For this reason, the critical area is defined as the area that makes the time ratio equal to one. This can be done by looking at equation ([*]), in which the value of can be varied until it is equal to and so the critical area is

    (9.9)

Substituting equation ([*]) into equation ([*]), and using the fact that the sound velocity can be expressed as , yields:
    (9.10)

where is the speed of sound at the initial conditions inside the cylinder (ambient conditions). The should be expressed by Eckert/Bar-Meir equation.


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