3. Dimensional Analysis

questionFile[dimensional] The shear, S, at the ingate is determined by the average velocity, U, of the liquid and by the ingate thickness, t. Dimensional analysis shows that is directly proportional to (). The constant of proportionality is difficult to determine, Murray, CSIRO Australia

One of the important tools to understand and to design in the die casting process is dimensional analysis. Fifty years ago this method transformed the fluid mechanics/heat transfer into an ``uniform'' understanding. In this book I am attempting to introduce to the die casting field this established method<sup>1</sup>. Experimental studies will be ``expended/generalized'' as it was done in convective heat transfer. It is hoped that as a result, separate sections for aluminum, zinc and magnesium will not exist in anymore die casting conferences. This chapter is based partially on Dr. Eckert's book, notes and article on dimensional analysis applied to die casting. Several conclusions are derived from this analysis and they will be presented throughout this chapter. This chapter is intent for a reader who want to know why the formulation in the book is in the dimensionless form. It also can bring a great benefit to researchers who want to built their research on a solid foundation. For those who are dealing with the numerical research/calculation, it can be useful to learn when some parameters should be taken into account and why. Considerable amount of physical explanation is provided in this Chapter.

In dimensional analysis, the number of the effecting parameters is reduced to a minimum by replacing the dimensional parameters by dimensionless parameters. Some researchers point out that the chief advantages of this analysis are ``to obtain experimental results with a minimum amount of labor, results in a form having maximum utility'' [#!fluid:hansen!#, pp. 395]. The dimensional analysis has several other advantages which include 1)increase of understanding, 2) knowing what is important, and 3)compacting the presentation². should we include a discussion about advantages of the compact of presentation

Dimensionless parameters are parameters that represent a ratio that do not have a physical dimension. In this chapter only things related to die casting are presented. The experimental study assists to solved problem when the solution of the governing equation can not be solved To achieve this, we design experiments that are ``similar'' to the situation that we simulating. This method is called the similarity theory in which the governing differential equations needed to solve are defined and design experiments with the same governing differential equations. This does not necessarily means that we have to conduct experiments exactly as they were in reality. An example how the similarity is applied to the die cavity is given in the section . Casting in general and die casting in particular, I am not aware of experiments that utilize this method. For example, after the Russians [#!poro:firstWateranalogy!#] introduced the water analogy method (in casting) in the 40's all the experiments (known to the author such by Wallace's group, CSIRO etc) conducted poorly design experiments. For example, experimental study of Gravity Tiled Die Casting (low pressure die casting) performed by Nguyen's group in 1986 comparing two parameters and We. The flow is ``like'' free falling for which the velocity is a function of the height ( ). Hence, the equation should lead only to and not to any function of . The value of is actually constant for constant for height ratio. Many other important parameters which controlling the governing equations are not simulated [#!poro:nguyten!#]. The governing equations in that case include several other important parameters which have not been controlled, monitored and simulated³. Moreover, the number is controlled by the flow rate and the characteristics of the ladle opening and not as in the pressurized pipe flow as the authors assumed.