Numerical Simulation of Melting and Solidification

Int. J. Computational Materials Science and Surface Engineering, Vol. 1, No. 6, 2007 717

Copyright © 2007 Inderscience Enterprises Ltd.

Numerical simulation of melting and solidification in laser welding of mild steel

M. Sundar* School of Laser Science and Engineering, Jadavpur University, USIC Building, Kolkata 700032, India E-mail: [email protected] *Corresponding author

A.K. Nath Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur – 721302, India E-mail: [email protected]

D.K. Bandyopadhyay, S.P. Chaudhuri, P.K. Dey and D. Misra School of Laser Science and Engineering, Jadavpur University, USIC Building, Kolkata 700032, India E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

Abstract: Melting and solidification which is mainly responsible for weld pool geometry in a laser welding process is analysed following two different transient models: (1) without fluid flow, that is, pure conduction and (2) with surface tension driven flow considering various beam power, beam diameter and scanning speed. Based on these models, numerical simulations for laser welding of mild steel has been carried out using commercial software, fluent. By solving the conservation equations of energy, momentum and mass, the temperature distribution, liquid fraction and flow fields has been predicted and the fusion zone and Heat Affected Zone (HAZ) were obtained. Using predicted results, this present study investigates the effects of laser beam power, beam diameter and scanning speed on heat transfer and the weld pool geometry. Analysis shows that the effect of consideration of fluid motion has pronounced effects on the thermal field and weld pool geometry. [Received 20 January 2007; Accepted 14 September 2007]

Keywords: laser welding; modelling; Marangoni force; numerical simulation; surface tension force.

718 M. Sundar et al.

Reference to this paper should be made as follows: Sundar, M., Nath, A.K., Bandyopadhyay, D.K., Chaudhuri, S.P., Dey, P.K. and Misra, D. (2007) ‘Numerical simulation of melting and solidification in laser welding of mild steel’, Int. J. Computational Materials Science and Surface Engineering, Vol. 1, No. 6, pp.717–733.

Biographical notes: M. Sundar is a Research Scholar in the School of Laser Science and Engineering, Jadavpur University, Kolkata, India. He received a Masters in Mechanical Engineering from Jadavpur University, Kolkata. His area of interest includes material processing, laser cutting, laser welding and numerical simulation of material processing.

A.K. Nath is the Head of Industrial CO2 laser section RRCAT, Indore, India.

He received his PhD from the University of Bombay. He is working on the indigenous development of high power CW and Pulsed CO

2 laser. He is

working extensively on various laser material processing including laser welding, laser cutting.

D.K. Bandyopadhyay is a reader in the Department of Mechanical Engineering, at Jadavpur University, Kolkata, India. He received a Masters in Mechanical Engineering from Jadavpur University, Kolkata, India. He has 38 years of teaching experience and his area of research includes laser safety, laser material processing and machine tool design.

S.P. Chaudhuri is a Professor in the Department of Mechanical Engineering, at Jadavpur University, Kolkata, India. He has 38 years of teaching experience and his area of research includes material science and technology, laser material processing and its applications.

P.K. Dey is a Reader in the Department of Mechanical Engineering, at Jadavpur University, Kolkata, India. He received his Masters from Jadavpur University and PhD from Indian Institute of Science, Bangalore. He has 24 years of teaching experience and his area of research includes fluid mechanics, heat transfer and laser material processing.

D. Misra is a Lecturer in the Department of Mechanical Engineering, at Jadavpur University, Kolkata, India. He received his Masters from Jadavpur University. He has 22 years of teaching experience and his area of research includes fluid mechanics, heat transfer, numerical simulation of material processing and laser material processing.

1 Introduction

In laser welding process the parts to be joined are locally melted by an intense laser beam, followed by a solidification process as the beam moves away. The mechanical strength and microstructure of such joints are strongly dependent on the thermal history in the weld zone and the nearby Heat Affected Zone (HAZ). One of the main advantages of laser welding is its low HAZ. To keep track of the HAZ and other properties, it is necessary to have a thorough understanding of the fluid flow and heat transfer mechanisms in the weld pool. Since molten metals are opaque, flow visualisation of the weld pool is difficult. Hence, it is necessary to have an appropriate mathematical model for clear understanding of the transport mechanisms in the weld pool.

Numerical simulation of melting and solidification in laser welding 719

Figure 1 is a schematic representation of the laser welding process considered. A continuous laser beam of sufficient intensity is incident upon the work piece, which is moving at a constant velocity (scanning speed). A significant fraction of the incident energy is absorbed by the work piece leading to the formation of weld pool. As the work piece passes the intense laser beam, the weld pool travels along the scanning direction and starts solidifying when it crosses the laser beam.

Figure 1 Schematic diagram of laser welding process

The research on welding heat source models dates back to early 1940s when Rosenthal (1941) first proposed a mathematical model of the moving heat source under the assumptions of quasi-stationary state and concentrated point heating in the 3D analysis. Numerical studies of weld pool in laser welding process have been studied by various researches (Carmignani, 1999; Hanbin et al., 2004) and several mathematical models have been developed to investigate the nature of heat transfer in the laser welding process. However, most investigations have focused on modelling laser weld pool behaviour in a quasi-steady state, and little attention has been paid to the transient dynamics of the weld pool, especially the transient heating period after the laser beam moves away. Mahrle and Schmidt (2002) numerically predicted temperature and velocity distributions as well as weld pool geometry using a steady-state model for transport phenomena in the fusion zone of deep penetration laser beam welded joints.

Many works done in the past based on traditional approach, the so called Stefan’s problem, assumes a heat conduction-controlled process; that is, there exist no motion of liquid in the molten weld pool during melting (Crowley, 1978; Nedja, 2002). A two-dimensional model has been used by Sluzalec (1988) to study the flow of metal subjected to a laser beam, in which changes of the laser beam intensity in time was taken into account. Recently a three-dimensional model was developed by Ye and Xi (2002) to study heat transfer and fluid flow in the full-penetration laser welding.

The hydrodynamics of the liquid metal arising in laser welding is a very complex process, which is still the subject of studies. Experimental investigations show melt flow velocities of a few metres per second (Fabbro et al., 1998; Ki et al., 2000). Two physical mechanisms can generate such motion, namely convection driven by the gradient of surface tension (Marangoni effect) and the bouncy force. As the workpiece melts, the absorbed energy induces a predominantly surface tension driven flow from the top, which redistributes the fluid momentum as well as the thermal energy in the weld pool.


This along with scanning speed strongly affects the temperature distribution as well as the final shape and size of the weld pool and it is obvious that a correct modelling of the melt flow has to be conducted in order to build a numerical tool for the study of laser welding.

The physical effects during laser welding involves heat transfer into the material, radiation and convection for the boundary conditions, thermo-dynamics of phase- changes, moving boundary at the melt/solid interface, fluid flow caused by surface tension and buoyancy and mass transportation within the molten body. In order to model the transient dynamic development of the weld pool, transient analysis (Wu and Yan, 2004) of the thermal and the fluid flow must be considered. Antony and Cline (1977) have carried out a quantitative analysis and proposed that the flow in molten pool is created by the surface tension gradient. Srinivasan and Basu (1986) studied the surface tension flow during laser melting and have shown that the effect of buoyancy force is negligible compared to the surface tension force.

In this present method, the energy equation is generally written in terms of the sensible enthalpy, and an appropriate formulation of the latent heat function play a pivotal role in ensuring that the results from the energy equation are consistent with phase-change considerations. The most recent effort to solve this problem was by Chakraborty and Dutta (2001), who outlined generalised formulation for evaluation of latent heat functions that addressed wide variety of phase-change situations.

This study involves the simulation of laser welding process for investigating the effects of fluid flow, laser power, beam diameter and scanning speed in the weld pool. The transient heat and fluid flow phenomena of the laser weld pool is studied and the time needed for attaining the quasi-steady state in laser welding is found out. To this end, this work considers two models: (I) Model with pure conduction, (II) Model involving Marangoni force and compares the results in terms of transient temperature distribution, velocity distribution and the weld pool geometry.

2 Formulation

Numerical simulation is carried out by the finite volume based code Fluent. The mathematical model used in this work is based on the Navier-Stokes equations with the Reynolds method of averaging the time-dependent equations (RANS).

Continuity and momentum equation

( ) ( ) ( ) 0u v w

t x y z

ρρ ρ ρ

∂ ∂ ∂ ∂+ + + =

∂ ∂ ∂ ∂ (1)

( ) ( ) ( ) ( )

x u

u uu uv uwt x y z

p u u ug s

x x x y y z z

ρ ρ ρ ρ

µ µ µ ρ

∂ ∂ ∂ ∂+ + +

∂ ∂ ∂ ∂

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= − + + + + −⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠∂ ∂ ∂ ∂ ⎝ ∂ ⎠ ∂ ∂

(2)

( )2

m3

(1 )( )us A u

ββ ξ−

=+


( ) ( ) ( ) ( )

y y

v vu vv vwt x y z

p v v vg s

y x x y y z z

ρ ρ ρ ρ

µ µ µ ρ

∂ ∂ ∂ ∂+ + +

∂ ∂ ∂ ∂

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= − + + + + −⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠∂ ∂ ∂ ∂ ⎝ ∂ ⎠ ∂ ∂

(3)

( )2

m3

(1 )( )vs A v

ββ ξ−

=+

( ) ( ) ( ) ( )

z w

w wu wv wwt x y z

p w w wg s

z x x y y z z

ρ ρ ρ ρ

µ µ µ ρ

∂ ∂ ∂ ∂+ + +

∂ ∂ ∂ ∂

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= − + + + + −⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠∂ ∂ ∂ ∂ ⎝ ∂ ⎠ ∂ ∂

(4)

( )2

m3

(1 )( )ws A w

ββ ξ−

=+

The phase change model makes use of the melt/freeze enthalpy-porosity technique described in Voller and Prakash (1987) and Voller and Swaminathan (1991). The melt/solid interface is not explicitly tracked; instead a liquid fraction value is used for each cell in the computational domain. Hence, the mushy zone is a region where liquid-fraction values lie between zero and unity and its temperature ranges between the ones of the liquid and the solid. The energy equation used in heat transfer calculations, is expressed in terms of enthalpy h at the temperature of the melt:

ref

ref p, dT

T

H h H h h C T= + ∆ = + ∫ (5)

The liquid fraction can be defined as

( ) ( )1

s 1 s s 1

s

1

/

0

T T

T T T T T T T

T T

β>⎧ ⎫

⎪ ⎪= − − ≤ ≤⎨ ⎬⎪ ⎪< ⎭⎩

(6)

The energy equation based on enthalpy is as follows:

( ) ( ) ( ) ( )

HuH vH wH

t x y z

T T TK K K

x x y y z z

ρρ ρ ρ

∂ ∂ ∂ ∂+ + +

∂ ∂ ∂ ∂

⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠∂ ∂ ∂ ⎝ ∂ ⎠ ∂ ∂

(7)

2.1 The boundary conditions are as follows

At the top surface: Heat flux input with heat loss due to convection and radiation (Li et al., 2002) can be

expressed as,


( )c( , )T

K q x y h T Tn ∞

∂= − −

∂ (8)

The laser beam can be modelled as a heat source as follows (Mazmumder and Steen, 1980)

( ), f( , ) 1x yq x y P r= − (9)

The value of Px,y for TEM00 mode structure is given in the form of Gaussian power distribution as

total, 2 2

b b

2expx y

P rP

r rπ⎛ ⎞

= −⎜ ⎟⎝ ⎠ (10)

where r = (x2 + y2)1/2 is the radial distance from the beam centre m, Ptot is the total incident power in laser beam W and rb is the Gaussian beam radius defined as the radial distance at which the power density falls to 1/e2 the central value m.

The combined coefficient due to convection and radiation is given by

4 1.61c 24.1 10h Tε−= × (11)

The shear stresses caused by the variation of surface tension due to temperature is given by

sTT

στ

∂= ∇∂

(12)

where ∂σ/∂T = Surface tension gradient and ∆sT = Surface gradient. At all other surfaces, except at the top, heat loss due to convection and radiation is

given by

( )c

TK h T T

n ∞

∂= −

∂ (13)

3 Numerical procedure and grid structure

The equations are solved using commercial finite different code Fluent, version 6.11 (Fluent 6.1 User Guide (1993). The discretisation scheme used is second order upwind for momentum and energy equations, and the SIMPLE scheme for pressure-velocity coupling. The equations are solved using the segregated solver. As the governing equations are non-linear and coupled, several iterations of the solution loop is performed before a converged solution is obtained. The convergence criterion for all the simulations was 10−6.

The simulation is transient with initial temperature of 300 K. For simplicity, we have taken constant thermophysical properties for the present analysis, except for the variation of surface tension. Because of the presence of surface-active elements such as sulphur and manganese, the temperature coefficient of surface tension is usually positive for the case of steel. It is assumed that the surface of weld pool is flat, the flow is Newtonian & laminar and the distribution of laser energy density is parabolic.


The computational domain with dimensions: length, L = 15 mm, width, (W) 7.5 mm and thickness (B) 5 mm has been considered for computation. The model consists of 300 × 50 × 75 grids. In order to reduce the computational time and to increase the accuracy, moving mesh method is considered, that is, the length of the plate is assumed to be infinity and the analysis is carried out till steady state is attained along the length. A grid system of variable spacing has been utilised with fine grid near the heat source and course grid away from the heat source. Minimum grid size is 0.1 mm and maximum grid size is 0.5 mm the minimum grid size is used all along the length of the workpiece. The workpiece is assumed to be symmetric with respect to the X-Y plane.

In order to regulate the heat flux and boundary conditions applied on the surface, an interpreted User Defined Function (UDF) file was introduced to the prepared FLUENT case file for the problem. Another compiled UDF is used to take care of moving mesh condition. The UDFs are written in the C programming language and linked to the Fluent solver. Table 1 summarises the values of different physical properties of the work piece considered in this study.

Table 1 Physical properties of the work piece

Physical properties Value

Density 7400 Kg/m3

Thermal conductivity 59 w/m-K

Specific heat 586 J/Kg-K

Viscosity 0.00578 Kg/m-sec

Temperature coefficient of surface tension 0.0005 N/m-K

Latent heat of fusion 272,000 J/Kg

Solidus temperature 1523 K

Liquidus temperature 1723 K

4 Results and discussion

For getting a better insight of the mechanisms of fluid and heat flow in the weld pool and to find the effects of speed, beam diameter and power, two Models have been considered with various parameters as given in Table 2.

Model I: The Model involving pure conduction, with out any fluid flow.

Model II: The Model involving Marangoni convection.

The numerical results of the two Models with two different speeds have been initially compared and particularly the contours of temperature distribution, liquid fraction and weld pool velocity have been discussed in detail. All the contours presented here correspond to time level of 0.4 sec with scanning speed 20 mm/sec.


Table 2 Process parameters used for study

Case No. Beam diameter (mm) Power (kW) Speed (mm/sec)

1 1 1 5, 10, 20, 30

2 1 2 5, 10, 20, 30

3 1 4 5, 10, 20, 30

4 2 1 5, 10, 20, 30

5 2 2 5, 10, 20, 30

6 2 4 5, 10, 20, 30

4.1 Temperature distribution

To highlight the transport phenomena vis-à-vis the Models considered, first the results are prepared in the form of temperature contours. Analysis shows that the isotherms expand as time progresses after welding starts. The isotherms move with the beam along the x-direction. It is observed that, due to motion of the workpiece relative to the laser beam, as advection heat flow occurs in the scanning direction, the peak temperatures at the workpiece surface appears near the trailing edge of the laser beam rather than at the beam centre.

The temperature distribution for scanning speeds 20 mm/s is shown, in Figures 2 and 3. Model II that includes fluid flow (Figure 3) predicts low maximum temperature, high melting pool depth and narrower HAZ than Model I (Figure 2). The isotherms in the case of Model I appear to be much more spherical, as compared to those obtained from Model II. This is because of an isotropic nature of the heat flow in case of Model I, which partially nullifies the asymmetric and directional effects imposed by the laser beam scanning. Due to fluid flow the isotherms elongate along the scanning speed in Model II.

Figure 2 Temperature contour for Model I, case 5 with scanning speed 20 mm/s, beam diameter 2 mm and laser power 2 kW (a) top surface and (b) mid-cross section


Figure 2 Temperature contour for Model I, case 5 with scanning speed 20 mm/s, beam diameter 2 mm and laser power 2 kW (a) top surface and (b) mid-cross section (continued)

Figure 3 Temperature contour for Model II, case 5 with scanning speed 20 mm/s, beam diameter 2 mm and laser power 2 kW (a) top surface and (b) mid-cross section

The maximum temperature for Model I is 2860 and for Model II it is 2170. We observe a lower peak temperature in Model II (Figure 3) in comparison to Model I (Figure 2). This is due to the fact that fluid motion in Model II tends to homogenise the temperature in the weld pool to a large extent.


4.2 Velocity field

The velocity vector for Model II corresponding speed of 20 mm/s is shown in Figure 4. Since the surface tension coefficient is taken positive (0.0005), the fluid element located in the vicinity of the laser source has a higher surface tension than other fluid elements located away from the laser source. This differential of surface tension gives rise to a radially inward flow at the free surface. The forces, which influence the molten pool dynamics, are primarily the surface tension force, viscous force and buoyancy force. Amongst all these forces, surface tension force is predominant and the most important one, since it initiates the flow within the small molten pool. The maximum velocity corresponds to scanning speed of 20 mm/sec is about 0.554 m/sec. The maximum flow velocity appears on the surface of molten pool where there is the maximum temperature gradient.

Figure 4 Velocity vector for Model II, case 5 with scanning speed 20 mm/s, beam diameter 2 mm and laser power 2 kW (a) top surface; (b) mid-cross section and (c) on the plane along the beam movement


Figure 4 Velocity vector for Model II, case 5 with scanning speed 20 mm/s, beam diameter 2 mm and laser power 2 kW (a) top surface; (b) mid-cross section and (c) on the plane along the beam movement

The circular motion of the weld pool from top to bottom is seen in the trailing edge of the laser beam. Owing to the Marangoni convection, the high temperature fluid penetrates deeper into the work piece, which results in high depth in Model II than Model I. This Marangoni convection also lowers the temperature gradient in Model II. The molten metal is pulled from the relatively cooler pool edges towards the centre of the pool, resulting in counter-rotating vortices as shown in Figure 4. The weld pool is deeper and narrower when convection due to Marangoni effect is considered.

4.3 Shape of the weld pool

The liquid fraction contour with scanning speeds 20 mm/s for Models I and II are shown, through Figures 5 and 6, respectively. Model II which includes the fluid flow (Figure 6) shows low weld pool width, low weld pool length and high weld pool depth than those without fluid flow (Figure 5). It indicates that the Marangoni flow plays an important role in transferring the heat from the top surface of the workpiece to the interior region of the domain. The heat is quickly transferred to the solid–liquid boundary at the bottom of the of weld pool. Moreover, the molten metal in the rear of pool becomes cold and it solidifies more quickly due to dissipation of heat. Hence, the pool becomes deeper and narrower when Marangoni effect is considered. Figure 6(b) shows the molten pool profile of the cross section considering fluid flow.

Figure 5 Liquid fraction contour for Model I, case 5 with scanning speed 20 mm/s, beam diameter 2 mm and laser power 2 kW (a) top surface and (b) mid-cross section


Figure 5 Liquid fraction contour for Model I, case 5 with scanning speed 20 mm/s, beam diameter 2 mm and laser power 2 kW (a) top surface and (b) mid-cross section (continued)

Figure 6 Liquid fraction contour for Model II, case 5 with scanning speed 20 mm/s, beam diameter 2 mm and laser power 2 kW (a) top surface and (b) mid-cross section

(a)

(b)


The simulation without considering flow (Figure 5) does not predict the weld pool profile in agreement with the results observed in practice. Especially, the simulated weld pool widths are too high with shallow depth. In Model I higher heat transfer takes place along the horizontal direction and in Model II higher heat transfer takes place along the vertical direction. The maximum pool depth for Model I is 0.55 mm and for Model II it is 1.15 mm corresponding to the speed of 20 mm/sec.

4.4 Effect of speed, laser power and laser beam diameter on weld pool length

From the simulation it is well understood that, only Model II, which considers fluid flow is realistic. Hence, the effects of variation of speed, power and beam diameter on weld pool length, width and depth are studied only for Model II. Figure 7 shows the variation of weld pool length for various welding speed, laser power and laser beam diameter. It is found from the figure that, the increase in power leads to increase in length of the weldment; this is due to increased supply of heat by the laser beam. An increase of power from 2 to 4 kW tends to an increase in weld pool length from 2 to 4 mm, respectively for a speed of 10 mm/sec and beam diameter 1 mm. Increase in speed results in decrease of length of the weld pool. For low speed, the length of weld pool is high and it reduces rapidly as the speed increases. Finally, it reaches a quasi steady state and further increases in speed makes the welding insignificant. As the diameter of the beam increases, the length of the weld pool decreases, this is due to the fact that increase in beam diameter leads to reduction of power supply per unit area. The effect of speed and laser beam diameter is marginally low compared to the effect of laser power on the weld pool length.

4.5 Effect of speed, laser power and laser beam diameter on weld pool width

Width of the weld pool is measured in the transverse direction (direction perpendicular to the length). This gives a direct indication of dimensions of the HAZ. Figure 8 shows the weld pool width for various speed, diameter and power. An increase in welding speed (within the optimal range) leads to a decrease in weld pool width, that is, lower HAZ. This is because, the increase in scanning speed reduces the interaction time of laser beam with the job. Therefore, the heat input reduces, which leads to less volume of the base metal being melted. The results indicate that the welding speed and beam diameter are the most important factors affecting the welded pool width. Weld pool width increases with increase in laser power and decrease in beam diameter. This trend was expected since laser power density is inversely proportional to the square of the beam diameter.

Figure 7 Variation of length with respect to speed


Figure 8 Variation of width with scanning speed for Model II

4.6 Effect of speed, laser power and laser beam diameter on weld pool depth

Depth of the weld pool gives direct indication of the material thickness that can be welded effectively with the available power. Figure 9 shows the depth of the weld pool with various speed, power and beam diameter. From the figure, it is clear that all the three parameters (speed, laser power and beam diameter) significantly affect the depth of weld pool. These effects are due to the following reasons. The increase in laser power leads to an increase in the heat input; therefore, more molten metal and consequently higher depth is predicted. However, the effect is reversed in the case of welding speed effect, as lesser energy is delivered at higher scanning speed. Increase in beam diameter reduces the depth of the weld pool. A small beam radius results in increasing the power density. This means that the heating will be localised and intense in a small metal portion resulting in a higher depth. Moreover, high beam diameter, which means wide laser beam results in spreading the same laser power onto wide area. Therefore, wide area of the base metal will be heated, also, the heat loss due to convection radiation and conduction will be higher, which leads to decrease of weld pool depth. Increase in scanning speed results in reducing the weld pool depth.

Figure 9 Variation of depth with scanning speed for Model II

5 Conclusions

A comprehensive fluid and heat flow model has been developed for simulating the weld pool geometry in laser welding. From the simulation, it is observed that the fluid flow plays a significant role in deciding the temperature distribution, final shape and size of the weld pool. The calculated results show that due to fluid flow the molten pool


becomes deeper and narrower. Whereas, in the Model with pure conduction, the weld pool is more or less of spherical shape. Models including the fluid flow show lower maximum temperature, narrower HAZ and higher depth than those without fluid flow. Further, it is observed that with the increase in scanning speed, the weld pool length, width and depth decreases. Increase in laser power results in wider HAZ and higher depth. As the diameter of the laser beam increases length and depth of weld pool decreases and width of weld pool increases The proposed model can be effectively used for predicting the weld pool shape in laser welding with varying process parameters such as speed, laser power and beam diameter. These Models can be used as a tool for a parametric examination of the laser welding process, which will lead to optimised values of the welding parameters.

Acknowledgements

The authors gratefully acknowledge the financial support (Sanction No. 2004/34/3-BRNS/275) provided by the Board of Research in Nuclear Sciences, DAE, India, for carrying out this research work.

References

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Nomenclature

Am Mushy Zone Constant

B Thickness

Cp Specific heat

g Acceleration due to gravity

h Sensible enthalpy

hc Combined transfer coefficient for radiative and convective boundary conditions

href Reference enthalpy

H Specific enthalpy

HL Latent heat of fusion

∆H Latent Heat = βHL

K Conductivity

L Length

P Static pressure

P Laser Power

Q Heat flux

rf Reflectivity of the workpiece

s Momentum sink

T Temperature

Tl Liquidus temperature


Ts Solidus temperature

T∞ Temperature of radiation source

u, v, w Velocity component along x, y and z directions

U Scanning Velocity of Laser Beam

W Width

x, y, z Coordinates

t Time

Greek symbols

β Liquid fraction

ε Emissivity

µ Molecular viscosity

ρ Density

ξ A constant to prevent division by zero

Documents

Numerical Simulation of Melting and Solidification