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7.2 The ``common'' diagram

The injection phase is (normally) separated into three main stages which are: slow part, fast part and the intensification (see Figure [*]). In the slow part the plunger moves in the critical velocity to prevent wave formation and therefore expels maximum air/gas before the liquid metal enters the cavity. In the fast part the cavity supposed to be filled in such way to prevent premature freezing and to obtain the right filling pattern. The intensification part is to fill the cavity with additional material to compensate for the shrinkage porosity during the solidification process. The diagram deals with the second part of the filling phase. = 90 true mm
Figure: A typical trace on a cold chamber machine

In the diagram, the solution is determined by finding the intersecting point of the runner/mold characteristic line with the pump (die casting machine) characteristic line. The intersecting point sometime refereed to as the operational point. The machine characteristic line is assumed to be understood to some degree and it requires finding experimentally two coefficients. The runner/mold characteristic line requires knowledge on the efficiency/discharge coefficient, , thus it is an essential parameter in the calculations. Until now, has been evaluated either experimentally, to be assigned to specific runner, or by the liquid metal properties ( ) [#!poro:dcrf!#] which is de facto the method used today and refereed herein as the ``common'' diagram2. Furthermore, is assumed constant regardless to any change in any of the machine/operation parameters during the calculation. The experimental approach is arduous and expensive, requiring the building of the actual mold for each attempt with average cost of $5,000-$10,000 and is rarely used in the industry3. A short discussion about this issue is presented in the Appendix [*] comments to referee 2.

Herein the ``common'' model (constant ) is constructed. The assumptions made in the construction of the model as following

  1. assumed to be constant and depends only the metal. For example, NADCA recommend different values for aluminum, zinc and magnesium alloys.

  2. Many terms in Bernoulli's equation can be neglected.

  3. The liquid metal is reached to gate.

  4. No air/gas is present in the liquid metal.

  5. No solidification occurs during the filling.

  6. The main resistance to the metal flow is in the runner.

  7. A linear relationship between the pressure, and flow rate (squared), .

According to the last assumption, the liquid metal pressure at the plunger tip, , can be written as

    (7.1)

Where:

 the pressure at the plunger tip 

the flow rate
maximum pressure which can be attained by the die casting machine
in the shot sleeve
maximum flow rate which can be attained in the shot sleeve

The and values to be determined for every set of the die casting machine and the shot sleeve. The value can be calculated using a static force balance. The determination of value is done by measuring the velocity of the plunger when the shot sleeve is empty. The maximum velocity combined with the shot sleeve cross-sectional area yield the maximum flow rate,

    (7.2)

where represent any possible subscription e.g.

Thus, the first line can be drawn on diagram as it shown by the line denoted as 1 in Figure [*]. The line starts from a higher pressure () to a maximum flow rate (squared). A new combination of the same die casting machine and a different plunger diameter creates a different line. A smaller plunger diameter has a larger maximum pressure () and different maximum flow rate as shown by the line denoted as 2.

The maximum flow rate is a function of the maximum plunger velocity and the plunger diameter (area). The plunger area is a obvious function of the plunger diameter, . However, the maximum plunger velocity is a far-more complex function. The force that can be extracted from a die casting machine is essentially the same for different plunger diameters. The change in the resistance as results of changing the plunger (diameter) depends on the conditions of the plunger. The ``dry'' friction will be same what different due to change plunger weight, even if the plunger conditions where the same. Yet, some researchers claim that plunger velocity is almost invariant in regard to the plunger diameter4. Nevertheless, this piece of information has no bearing on the derivation in this model or reformed one, since we do not use it.



= 90 true mm

Figure: The ``common'' version

= 90 true mm

Figure: and as a function of the plunger diameter according to ``common'' model.
A simplified force balance on the rode yields (see more details in section [*] page [*])
    (7.3)

where subscript denotes the actuator.



In the ``common'' diagram is defined as

    (7.4)

Note, therefore is also defined as a constant for every metal5. Utilizing Bernoulli's equation6.
    (7.5)

The flow rate at the gate can be expressed as
    (7.6)

The flow rate in different locations is a function of the temperature. However, Eckert poro:ecksimilarity7demonstrated that the heat transfer is insignificant in the duration of the filling of the cavity, and therefore the temperature of the liquid metal can be assumed almost constant during the filling period (which in most cases is much less 100 milliseconds). As such, the solidification is insignificant (the liquid metal density changes less than 0.1% in the runner); therefore, the volumetric flow rate can be assumed constant: to make question about mass balance
    (7.7)

Hence, we have two equations ([*]) and ([*]) with two unknowns ( and ) for which the solution is
    (7.8)





insert a discussion in regards to the trends insert the calculation with respect to and


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