3. Dimensional Analysis

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^{1}.
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 presentation^{2}.
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 simulated^{3}.
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.

- 3.1 Introduction
- 3.2 The processes in die casting
- 3.2.1 Filling the shot sleeve
- 3.2.2 Shot sleeve
- 3.2.3 Runner system
- 3.2.4 Die cavity
- 3.2.5 Intensification period and after

- 3.3 Special topics
- 3.3.1 Is the flow in die casting is turbulent?
- 3.3.2 Dissipation effect on the temperature rise
- 3.3.3 Gravity effects

- 3.4 Estimates of the time scales in die casting
- 3.4.1 Utilizing semi dimensional analysis for characteristic time
- filling time
- Atomization time
- Conduction time in the liquid metal (solid)
- Solidification time
- Dissipation time
- 3.4.2 The ratios of various time scales

- 3.5 Similarity applied to Die cavity

- 3.6 Summary of dimensionless numbers
- Reynolds number
- Eckert number
- Brinkman number
- Mach number
- Ozer number
- Froude number
- Capillary number
- Weber number
- Critical vent area

- 3.7 Summary
- 3.8 Questions

copyright Dec , 2006

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