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Heat transfer model for cw laser material processing
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INTRODUCTION
The laser is finding increasing commercial use as a
welding, 1-4 cutting,2.5 or surface treatmene..s tool. However,
for its use to be consolidated it is necessary to understand
how it works in these processes and to predict how it would
work in novel situations or on novel materials.
In understanding how it works the experimenter is
faced with a multiparameter problem which is difficult to
solve without extensive factorial experimentation. The principal
variables are the substrate thermal and optical properties,
the laser beam's total power, power distribution and
diameter, and the traverse speed. Alternatively, as described
here, an assumed physical picture of the process can be modelled
mathematically and the model's results compared to
experimental results to prove the model's validity and thus
by inference the physical model.
A model capable of predicting experimental results
means that previously unmeasurably parameters can be estimated.
In metal welding Borland and 10rdan9 summarized
the most important data to be (1) thethermal cycle at each
location in the fusion and heat affected zone (HAZ), which
defines the extent of any phase change, (2) the peak temperature
distribution, and (3) the cooling rates which would affect
the formation of metastable structures such as martensite.
Without a mathematical model the values of these
parameters are very difficult to obtain
DERIVATION OF A MATHEMATICAL MODEL
In order to develop the mathematical model the process
is physically defined as follows:
A laser beam having a defined power distribution
strikes the surface of an opaque substrate having finite width
and thickness and infinite length moving in the positive x
direction (along the length) with a uniform velocity, Some of
the incident radiation is reflected. The rest is absorbed, but
the reflectivity is considered to be zero if the temperature
exceeds the boiling point since it is then assumed that a keyhole
is produced by vaporization which acts as a blackbody,
Some of the absorbed energy is lost by radiation and convection
from the surface while the rest is conducted into the
substrate. The convective heat transfer is enhanced on the
upper surface due to a concentric gas jet used for shielding,
The three-dimensional system is assumed to be in a quasi-
steady-state condition after the keyhole initiation period,
which for most practical purposes may be regarded as instantaneous21
and will only occur at the beginning of a weldmg
run.
Mathematical definition of the process to be modelled
Method of solution
A large number of numerical techniques have been reported
to solve partial differential equations.22
-
28 Since the
present problem concerns a given steady-state situation, a
"relaxation" technique was chosen.29
For computing stability it was found that at least five
grid points must lie within the incident laser beam diameter.
Since this diameter is approximately 250,um, the only way
the program could fit a computer with a finite store and yet
model a slab of reasonable size was to have an exponentially
expanding grid.29
The heat balance on an asymmetric control volume
within the body of the exponential grid, as shown in Fig. 2,
can be stated in Cartesian coordinates as
OPERATION OF THE MODEL
The mathematical model could be used for the following
purposes: (1) to predict the temeprature profile, (2) to
predict maximum welding speeds, (3) to predict the heat
affected zone, (4) to predict the thermal cycle at any location
or speed, (5) to predict the effect of thickness or any other
parameter (e.g., reflectivity, thermal conductivity, etc.), or
(6) to predict the effect of supplementary heating or cooling.
The results from the model can either be represented in
real units to simulate a particular run or shown as generalized
curves using dimensionless groups. The groups identified
as of principal importance were
CONCLUSION
It is possible by the numerical technique illustrated here
to predict the fusion, the heat affected zone, and thermal
947 J. Appl. Phys., Vol. 51, No.2, February 1980
cycles in the neighborhood of a laser/surface interaction;
and from these figures to calculate the conditions such as the
maximum welding speed as a function oflaser power, or
substrate thickness.