MATHEMATICAL MODELING FOR LASER TREATMENT PROCESSES
B. Medres, M. Bamberger, L. Shepeleva
Surface Technologies Ltd. Israel
ABSTRACT. This paper presents a simplified theoretical model for estimating the operating parameters of laser surface alloying and cladding by the direct injection of powder into the molten bath. It is shown that use of the Mie-theory takes into account the influence of the injected particles on laser beam energy transfer to the treated surface. Due to the reasonably good agreement between the calculated and the experimental data, this analytical model is applicable to the development of laser surface alloying and cladding by direct powder injection.
Keywords: laser treatment, laser surface alloying and cladding, direct injection of powder, and influence of the injected particles.
Introduction
Laser alloying and cladding are used to improve the mechanical properties and corrosion resistance of surface layers of materials [1]. If the mass of the molten bath is larger than that of the alloying elements, the process is named "Laser surface alloying" (LSA). If the mass of the molten bath is smaller than that of the alloying elements, the process is termed "Laser cladding" (LC) [1-3]. One of the most important problems of the LSA process is how to control the concentration of alloying elements within the molten bath [1-3]
The ratio between the mass of the alloying material which has entered the molten bath and that of the molten bath itself determines the composition of the alloyed layer. Thus, a method to estimate the concentration of alloying elements in the molten bath is important for laser processing. The ability to analytically determine these parameters would aid in optimizing the process. This is especially important in the development of new applications of LSA and LC.
Two variants of LSA and LC are currently used. In the first variant the alloying elements ( as a powder or paste) are spread over the surface which is to be treated. The conditions of the laser treatment can be determined by solving the problem of heating and melting a two-layered semi-infinite body by a source of heat moving on its surface [2, 3].
The other variant, and in our view the more technological one, involves the direct injection of powdered alloying elements into the molten bath [4]. In this case, interaction between the laser beam and the powder particles takes place.
In a number of publications, it was shown that the presence of a large quantity of particles in the impact area of the laser beam affects the way the laser radiation passes to the surface [4, 5, 6]. The influence of small particles on the passage of laser radiation to the treated surface was studied in investigations on laser cutting and welding [5, 6]. In these processes, the radiation is scattered by extremely fine particles, which are products of the condensation of the evaporated substrate. The particles are smaller than the wavelength of the laser radiation [3,6] . In LSA and LC, however, the laser radiation must pass through the powder stream, where the size of the particles can be ~100µm and more, i.e. much larger than the wavelength of laser radiation. Obviously, the intensity of the laser beam transmitted to the surface through a stream of powder particles having sizes in excess of the wavelength will be reduced. In this work we attempt to determine this intensity via a simplified theoretical model.
The theoretical model
In the LSA and LC process the powder particles are injected into the molten bath underneath the laser beam. The space in which the laser-material interaction and the dissolution of the added particles takes place, will be referred to hereafter as the "Laser Operation Volume" ( LOV ). The presence in the LOV of a sufficient amount of particles of sizes dplL, where lL is the wavelength of the laser beam, affects the characteristics of the virtual heat-source, such as its effective power and spatial energy distribution.
The model assumes uniform distribution of the laser power across the laser beam and one-dimensional heat flow into the workpiece. Thus the time-dependent heat conduction in the space underneath the irradiated surface is described by the equation [3,4] :
¶T ¶2T
(1) rc ¾¾ = k ¾¾
¶ t ¶z2
the boundary conditions being :
¶T
k ¾¾ (z = o) = (1-R)·P/S
¶z
T(z= ¥) = T0
Where -
r -density
c -specific heat
k -thermal conductivity
T -temperature
z -depth
P -laser power
R -reflectivity of the sample surface
d -beam diameter
v -laser scan speed
S -area irradiated by the laser
The solution of this equation is [3]
( 2 )
where "a " is the thermal diffusivity.
The maximum local temperature is reached during the laser-workpiece interaction, i.e. at a time equal to d/v. Thus, the highest-temperature profile across the substrate is given by :
( 2a )
From Eq. (2a) the maximum temperature of the irradiated surface can be written as :
( 3 )
But in laser alloying, in which the powder is fed into the molten bath at the point at which the beam impinges on the substrate, the actual power density reaching the sample surface is lower than the apparent laser power by a factor quantifying the transmission coefficient, j. Thus
(4)
where P0 is the apparent laser power and j may also be described as the attenuation factor.
This interaction takes place in the space above the point of impact of the laser beam on the substrate in the characteristic LOV. Based on Mie theory it was shown [7] that
½lnj½ = N· d· K· p· r2
so
(5) N=½lnj½/d· K· p· r2
where N is the density of powder particles in the LOV, r is the radius of an individual particle, and K is a constant [7]. According to the Mie theory, K = 5 when the wavelength of the laser radiation is smaller than the mean particle radius.
From the definition of N it is obvious that
n
N= ¾¾
VL
where n is the number of powder particles in the LOV and VL is the LOV volume.
In [7] it was found that the LOV can be considered as a cylinder of which the diameter equals the beam diameter and the height is 5 times its diameter. Knowing the powder feed rate, Q, and the density of the particles, mp, the number of the particles in the LOV can be calculated as
Q· G · tint
n = ¾¾¾¾¾
mp
where G is the fraction of the particles entrapped in the molten bath (G = 0.1) [7] and tint is the laser-workpiece interaction time, given by d/v, as mentioned before. Thus
and
(6)
Substituting Eq. 6 into Eqs. 3 & 4 yields :
(7)
Differentiation with respect to time gives the cooling rate as a function of the laser parameters.
The cooling rate on the upper surface is given by :
( 8 )
and elsewhere in the treated layer by
The effective beam diameter is defined as the section in which the surface temperature is above or equal to the melting temperature, Tm, of the substrate. Replacing T(z=0) in Eq. 7 by Tm and determining the effective beam diameter deff yields
(9)
Since the surface temperature cannot exceed the evaporation temperature of the workpiece, the maximum depth of the molten bath is attained when the surface temperature reaches the evaporation point. The thickness of the alloyed layer can then be calculated as :
(10) Zmax = 2·
where Tb is the evaporation temperature of the workpiece.
Using Eq. 9 and assuming the molten bath underneath the laser to be a hemisphere, the mass of the bath and the concentration of the alloying elements in it can be calculated as :
p· d3eff.
(11) mb = ¾¾¾ · r
12
and
12· Q· G· d/v
(12) C = ¾¾¾¾¾¾
p · d3eff.· r
where r is the density of the substrate.
The calculations and the experimental results
In this investigation the process parameters involved in the laser alloying of AISI 1045 steel by Ni2B and CrB powders and the laser cladding of copper to Al2O3 plate were obtained through the calculation of the analytical equations (see Table 1 for the thermophysical properties). The powders used were of a particle size in the range of 100-150mm. The steel samples were alloyed with Ni2B and CrB powder using a CW-CO2 laser of 2250W operating power. The scanning velocities were 1, 2, 3, 5, 10 cm/s and the laser beam diameter, df=0.21cm. Powder consumption was Q1=0.016 gr/s.
The processing of the ceramic samples was carried out with a CW-CO2 laser with an operating power in the range 200-800W; scanning velocities 0.66, 1.33, 2.0, and 4.0 cm/s, laser beam diameter, df=0.2 cm and powder consumption Q3=0.125 gr/s.
The model allows us to assess and predict the geometrical dimensions, e.g., the width and depth of the molten bath and the concentration of the alloying elements in the molten bath (Eq .12). The calculated and experimentally determined track width as a function of the scanning velocity are given in Fig. 1, while Fig. 2 shows the calculated and measured depth of the melt pool.
Fig.1. The dependence of effective laser beam diameter on scanning velocity
These results were received for a powder consumption of Q = 0.016 gr/s. Using Eq. 12, the volume ratio between the added copper and the molten bath is given by
( 13 )
The variations of the calculated and experimentally determined volume ratio with the laser beam energy density are given in Fig.3. From these data, it is clear that the discrepancy between the experimental and calculated results is less than 10 - 12% for the track width (Fig. 1), and depth (Fig. 2), but a 20% discrepancy was found between the calculated and measured volume ratio ( Fig. 3). The comparison between experimental and calculation results shows a reasonable agreement, so that despite the simplified model, it can be used to determine the LSA and LC process parameters.
Fig.2. The dependence of the depth of the melted zone on scanning velocity
Fig.3. A comparison between calculated and experimental volume ratios as a function of the energy density of the laser beam
Table 1: Thermophysical properties of the substrates
Property / Steel / Al2O3Melting temperature (K) / 1808 / 2319
Boiling temperature (K) / 3070 / 3253
Density (kg/m3) / 7190 / 3970
Specific heat (J/kg· K) / 400 / 862
Thermal diffusivity (m2/s) / 8· 10-6 / 1· 10-6
Thermal conductivity (W/m· K) / 24 / 3.8
Absorption coefficient / 0.18 / 0.77
Conclusions
The results show a good correlation between experimental and calculated data. The proposed model is thus a reasonable description of these processes. It enables an estimation of the operating parameters of laser alloying and cladding by means of direct alloying powder injection. Based on the proposed model, the process can be optimized with respect to minimizing the powder feed rate required to meet the predetermined composition in the molten bath. Also, the number of experiments necessary to set up the laser power and scan speed can be reduced.
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