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8.2.1 Garber's model

= 0.5
Figure: A schematic of wave formation in stationary coordinates
The description in this section is based on one of the most cited paper in the die casting research [#!poro:garber!#]. Garber's model deals only with a plug flow in a circular cross-section. In this section, we ``improve'' the model to include any geometry cross section with any velocity profile6.

Consider a duct (any cross section) with a liquid at level and a plunger moving from the left to the right, as shown in Figure [*]. Assuming a quasi steady flow is established after a very short period of time, a unique height, , and a unique wave velocity, , for a given constant plunger velocity, are created. The liquid in the substrate ahead of the wave is still, its height, , is determined by the initial fill. Once the height, , exceeds the hight of the shot sleeve, , there will be splashing. The splashing occurs because no equilibrium can be achieved (see Figure [*]a). For smaller than , a reflecting wave from the opposite wall appears resulting in an enhanced air entrainment (see Figure [*]b). Thus, the preferred situation is when (in circular shape ) in which case no splashing or a reflecting wave result. = 2.5in

Figure: A schematic of reflecting wave formation in sub and supper critical velocity

It is easier to model the wave with coordinates that move at the wave velocity, as shown in Figure [*]. With the moving coordinate, the wave is stationary, the plunger moves back at a velocity , and the liquid moves from the right to the left. Dashed line shows the stationary control volume. = 0.4

Figure: A schematic of the wave with moving coordinates

Mass conservation of the liquid in the control volume reads:

    (8.1)

where is the local velocity. Under quasi-steady conditions, the corresponding average velocity equals the plunger velocity:
    (8.2)



Assuming that heat transfer can be neglected because of the short process duration7. Therefore, the liquid metal density (which is a function of temperature) can be assumed to be constant. build a question about what happens if the temperature changes by a few degrees. How much will it affect equation [*] and other parameters? Under the above assumptions, equation ([*]) can be simplified to
    (8.3)

Where in this case can take the value of 1 or 2. Thus,
    (8.4)

where is a dimensionless function. Equation ([*]) can be transformed into a dimensionless form:

where .

Assuming energy is conserved (the Garber's model assumption), and under conditions of negligible heat transfer, the energy conservation equation for the liquid in the control volume (see Figure [*]) reads:
    (8.5)

where
    (8.6)



The shape factor, , is introduced to account for possible deviations of the velocity profile at section 1 from a pure plug flow. Note that in die casting, the flow is pushed by the plunger and can be considered as an inlet flow into a duct. The typical number is , and for this value the entry length is greater than 50, which is larger than any shot sleeve by at least two orders of magnitude.

The pressure in the gas phase can be assumed to be constant. The hydrostatic pressure in the liquid can be represent by [#!jump:rajaratnam65!#], where is the center of the cross section area. For a constant liquid density equation ([*]) can be rewritten as:

    (8.7)

Garber (and later Brevick) put this equation plus several geometrical relationships as the solution. Here we continue to obtain an analytical solution. Defining a dimensionless parameter as
    (8.8)

Utilizing definition ([*]) and rearranging equation ([*]) yields
    (8.9)

Solving equation ([*]) for the latter can be further rearranged to yield:
    (8.10)

Given the substrate height, equation ([*]) can be evaluated for the , and the corresponding plunger velocity ,. which is defined by equation ([*]). This solution will be referred herein as the ``energy solution''.
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