3.5.1 Governing equations

The filling of the mold cavity can be divided into two periods. In the first period (only fluid mechanics; minimum heat transfer/solidification) and the second period in which the solidification and dissipation occur. We discuses how to conduct experiments in die casting¹⁸. It has to stress that the conditions down-stream have to be understood prior to experiment with the die filling. The liquid metal velocity profile and flow pattern are still poorly understood at this stage. However, in this discussion we will assume that they are known or understood to same degree¹⁹.

The governing equations are given in the preceding sections and now we discuses put the numbers of governing equations the boundary conditions. The boundary condition at the solid interface for the gas/air and for the liquid metal are assumed to be ``no-slip'' condition which reads is it true for large Ma number discussion

(3.31)

where we use the subscript indicates the gas phase. It noteworthy to mention that this also applied to the case where liquid metal is mixed with air/gas and both are touching the surface. At the interface between the liquid metal and gas/air pressure jump is expressed as

(3.32)

where and are the principal radii of the free surface curvature, and, , is the surface tension between the gas and the liquid metal. The surface geometry is determined by several factors which include the liquid movement²⁰instabilities etc.

Now to the difficult parts, the velocity at gate has to be determined from the diagram or previous studies on the runner and shot sleeve. The difficulties arise due to fact that we cannot assign a specific constant velocity and assume only liquid flow out. It has to be realized that due to the mixing processes in the shot sleeve and the runner (especially in a poor design process and runner system, now commonly used in the industry) some portion at the beginning has a significant part which contains air/gas. There are several possibilities that the conditions can be prescribed. The first possibility is to describe the pressure variation at the entrance. The second possibility is to describe the velocity variation (as a function of time). The velocity is reduced during the filling of the cavity and is a function of the cavity geometry. here is the parabolic process is gone The change in the velocity is a sharp in the initial part of the filling due to the change from a free jet to an immersed jet. The pressure varies also at the entrance, however, the variations are more mild. Thus, it is better possibility²¹ to consider the pressure prescription. The simplest assumption is constant pressure just to get the average value and to explain how to get the function later

(3.33)

We also assume that the air/gas obeys the ideal gas model.

(3.34)

where is the air/gas constant and is gas/air temperature. Here we must insert the previous assumption of negligible heat transfer and further assume that the process is polytropic²². We define the dimensionless gas density as

(3.35)

The subscript denotes the atmospheric condition. check the subscript is systematic

The air/gas flow rate out the cavity is assumed to behave according to the model in Chapter . Thus, the knowledge of the vent relative area and are important parameters. For cases where the vent is well design (vent area is near the critical area or above the density, can be determine as was done by [#!poro:genickreac!#]).

To study the controlling parameters the equations are dimensionlessed. The mass conservation for the liquid metal becomes

(3.36)

where , , , , , and the dimensionless time is defined as , where .

Equation () can be the same simplified under the assumption of constant density to read

(3.37)

Please note that we cannot use this simplification for the gas phase. The momentum equation for the liquid metal in the x-coordinate assuming constant density and no body forces reads

			(3.38)

where and .

The gas phase continuity equation reads

(3.39)

The gas/air momentum equation²³ is transformed into

			(3.40)

Note that in this equation additional terms were added, .

The ``no-slip'' conditions are converted to:

(3.41)

The surface between the liquid metal and the air satisfy

(3.42)

where the , , and are defined as

The solution to equations has the from of

			(3.43)

If it will be found that equation () can be approximated²⁴ by

(3.44)

then the solution is reduced to

			(3.45)

At this stage we do not know if it the case and it has to come-out from the experiments. The density ratio can play a role because two phase flow characteristic in major part of the filling process.