1

A computer program for the calculation of three dimensional fluid flow and heat transfer during keyhole mode laser welding

R. Rai, H. Zhao, G. G. Roy and T. DebRoy Department of Materials Science and Engineering

The Pennsylvania State University

INDEX

1. Model Capabilities 2

2. Special features of the program 2

3. Input file 2

4. Output files 4

5. Calculation Procedure 6

6. Major components of the code 7 a. Calculation of keyhole geometry 7 Assumptions 7 Energy balance on keyhole wall 7 Consideration for plasma absorption and multiple reflections 9 b. Three dimensional fluid flow and heat transfer 11 Governing equations 11 Boundary conditions 12 Turbulence model 13 Solution methodology 14

Convergence criterion 15

7. Remarks about the code a. User modifiable sections of the code 15 b. Header.for 15 8. Case studies

Case 1: Tantalum welds made with 1900 W input power and 12.7 mm/s welding speed. 16 Case 2: Vanadium welds made with 1980 W input power at 25.4 mm/s welding speed. 23 Case 3: 304L stainless steel welds made with 1980 W input power at 19.1 mm/s welding speed. 30 Case 4: Ti-6Al-4V welds made with 1400 W input power at 16.9 mm/s welding speed. 37

9. Plotting of results 44 10. Concluding remarks 44 11. References 45

2

1. Model Capabilities The computer code can calculate the following parameters for the keyhole mode laser welding:

• Three dimensional keyhole geometry. • Three dimensional temperature fields in the workpiece. • Fluid velocities in the weld pool in three dimensions. • Cooling rates at user specified locations in the workpiece. • Effects of welding variables on temperature fields, fluid velocities, and cooling

rates in the workpiece.

2. Special features of the program • Laser beam characteristics, welding process parameters, material properties, and

grid information is specified by the user through a text file. • Properties of some common materials are stored in the program and need not be

specified by the user. • Output in the form of a text file shows the peak temperature, the maximum fluid

velocities in three directions, the heat loss at different workpiece surfaces, and weld pool depth and half-width every 100 iterations.

• The output also contains various data files for visualization of results. • Uses a turbulence model to estimate the enhanced heat and mass transfer due to

fluctuating components of velocity.

3. Input file The input file contains the user specified variables. The input is divided into 5 categories: Process parameters, material properties, numerical scheme parameters, boundary conditions, and grid parameters.

1. Process parameters include the input power, material absorption coefficient, plasma attenuation coefficient, welding speed, beam characteristics (distribution factor, spot radius, divergence).

2. A material index identifies the workpiece material as indicated in the input file

below. If the user specifies zero as the value for any material property (thermal conductivity, specific heat, viscosity, solidus, liquidus, boiling temperature, etc.), the code uses the appropriate values stored in its database. Alternatively, if a better estimate is available, the user can specify a non-zero value for any material property. Please note that the values from the database will be selected only if the the user provides zero as the input for any material property.

3. The numerical scheme parameters include the maximum number of iterations, the

under-relaxation for pressure, velocity, and enthalpy, and also the indices for saving and loading file. If the index for saving a file is 1, the u, v, w velocities, pressure, and temperature at each grid location is stored in a file tmp.sv. If the index for loading is 1, the u, v, w velocities, pressure and temperature at each grid

3

location is read from the file tmp.sv and taken as the starting value of calculations. However, the total number of grid locations in x, y and z directions must be kept the same as that in the tmp.sv file.

4. The boundary conditions contain the heat transfer coefficient at the 5 faces of the

workpiece (all except the symmetry face), the temperature at these faces, the initial temperature (or pre-heat temperature) of the work-piece, and ambient temperature. We allow three types of boundary conditions at east, west, bottom, top, and north surfaces. (I) surface temperature is given; (II) convective heat flux with hc calculated using equation:

25.05c )T8.1(C)103571.1(h Δ×′×= −

and (III) convective heat flux with hc supplied by the user. This is done by determining the value of heat transfer co-efficient hc in the input file as follows.

5. Fixed non-uniform rectangular grids are used for x, y, and z directions. The

geometrical parameters specifies by the user generate the mesh – the number of zones in each direction, length of each zone, number of control volumes in each zone, and the exponents to control the location of control volume interfaces. Finer grids are used near the heat source compared to regions further away.

!-----process parameters-------------- 1900.0 !input power (watt) 0.24 !material absorbtion coefficient 1.0 !plasma attenuation coefficient,(1/cm) 0.020 !beam radius at focal point(cm) 0.045 !change in beam radius with depth, cm/cm 0.0 !defocus (cm) 1.0 !power distribution factor 0.0 !arc current (amp) 0.0 !arc voltage (volt) 0.7 !arc efficiency 0.27 !arc radius (cm) 0.5 !arc power distribution factor 0 !emf calculation needed (1 = yes; 0 = file already available) 0.3 !starting location of beam, cm 0.7 !starting location of arc, cm 1.69 !welding speed, cm/s !-----material properties----------------------- 4 !1 -> 304L SS, 2 -> V, 3 -> Ta, 4 -> Ti-6Al-4V, 5 -> 21-6-9 steel 90.0 !wt % of a, Fe/V/Ta/Ti 6.0 !wt % of b, Cr/ / /Al 4.0 !wt % of c, Ni/ / /V 0.0 !wt % of d 4.0 !density of liquid (gm/cm3) 3.78 !density at boiling point (gm/cm3) 0.05 !molecular viscosity of liquid (gm/cm-sec) 1878. !solidus temperature (K) 1928. !liquidus temperature (K)

⎪⎩

⎪⎨

⎧

⎪ ⎩

⎪ ⎨ ⎧

<< <

>

)II()III(

)I(then

0 h 20 h 0

20 h If

c c

c

4

3315.0 !boiling temperature (K) 268.4 !enthalpy of solid at melting point (cal/gm) 355.6 !enthalpy of liquid at melting point (cal/gm) 0.143 !specific heat of solid (cal/gm-K) 0.167 !specific heat of liquid (cal/gm-K) 0.174 !specific heat of liquid at boiling point (cal/gm-K) 0.05 !thermal conductivity of solid,(cal/cm-sec-K) 0.07 !thermal conductivity of liquid, (cal/cm-sec-K) 0.072 !therma conductivity at boiling point,(cal/cm-sec-K) 8.0e-6 !coefficient of thermal expansion (1/K) 0.0 !emissivity of the material -0.26 !d(gamma)/dT of pure material (dynes/cm-K) 0.0 !concentration of surface active species (wt%) 0.13e-8 !surface excess at saturation (mole/cm2) -0.166e6 !enthalpy of segregation (cal/mole) 0.318e-2 !entropy factor !-----numerical scheme parameters--------------- 2000 !maximum number of iterations 0.6 !underrelaxation for u-velocity 0.6 !underrelaxation for v-velocity 0.6 !underrelaxation for w-velocity 0.8 !underrelaxation for pressure 1.0 !underrelaxation for enthalpy 0 !index for saving file (1 = save) 0 !index for loading file (1 = load) !-----boundary conditions-------------------------------------------- 100.0 !heat transfer coefficient at west face (cal/cm2-s-K) 100.0 !heat transfer coefficient at east face (cal/cm2-s-K) 100.0 !heat transfer coefficient at north face (cal/cm2-s-k) 0.01 !heat transfer coefficient at bottom face (cal/cm2-s-K) 0.0 !heat transfer coefficient at top face (cal/cm2-s-K) 298.0 !temperature at west face (K) 298.0 !temperature at east face (K) 298.0 !temperature at north face (K) 298.0 !temperature at bottom face (K) 298.0 !preheat temperature (K) 298.0 !ambient temperature (K) !-----geometrical parameters--------- 5 !number of x-zones 0.22 0.18 1.5 4. 0.1 !length of each x-zone (cm) 11 41 40 35 10 !number of control volumes in each x-zone -1.3 1.0 1.3 1.0 1.0 !exponents to locate control volume interfaces 4 !number of y-zones 0.1 0.18 1.0 0.10 !length of each y-zone (cm) 30 15 15 10 !number of control volumes in each y-zone 1.0 1.0 1.3 1.0 !exponents to locate control volume interfaces 3 !number of z-zones 0.1 0.25 0.3 !length of each z-zone (cm) 7 15 30 !number of control volumes in each z-zone 1.0 1.0 1.0 !exponents to locate control volume interfaces

4. Output files a) The output.txt file contains all the input parameters specified in the input file. It also contains the x, y and z-grid locations in the domain eg. the positions of x(i) and xu(i) as shown in figure below. A section of the output file generated by these is given below.

5

A section of the output file: i= 50 51 52 53 54 55 56 x= 3.846E-01 3.890E-01 3.934E-01 3.978E-01 4.062E-01 4.215E-01 4.411E-01 xu= 3.824E-01 3.868E-01 3.912E-01 3.956E-01 4.000E-01 4.124E-01 4.305E-01

Every 100 iterations, the code gives the peak temperature, maximum fluid velocities in three directions, heat loss from the workpiece surface, weld depth and half-width, and the average and maximum values of viscosity and thermal conductivity. iter time/iter res_enth res_mass res_u res_v res_w 2000 0.646 2.20E-06 2.62E-07 3.33E-04 1.42E-04 1.26E-04 Tmax umax vmax wmax length depth half-width 3315. 61.4 83.6 14.5 0.591 0.371 0.147 north south top toploss bottom west east hout hin ratio -23.7 0.0 -0.6 0.0 -0.3 256.7 -355.2 -122.5 119.6 1.02 muAv muM kAv kM 0.46 3.34 0.15 0.68

After the completion of calculations, the following summary of results in given which includes the time to cool, and the accompanying cooing rate, between two temperatures (1073 K and 773 K): Length of the pool (cm) 5.9082E-01 Depth of the pool (cm) 3.7123E-01 Half-width of the pool (cm) 1.4745E-01 Peak temperature (K) 3.3150E+03 x1073(cm), x773(cm), t8-5(s) 1.329 2.008 0.402 Maximum u-velocity (cm/s) 6.1372E+01 Maximum v-velocity (cm/s) 8.3635E+01 Maximum w-velocity (cm/s) 1.4497E+01 Rate of heat input (cal/s) 1.1965E+02 Rate of heat output (cal/s) -1.2252E+02 Ratio of heat input to heat output 1.0240E+00 Date: 2008- 1- 8 time: 16:44:15 Total time used: 0 hr 21 m 52 s

b) Tecout contains following variables in ordered form, obtained after the final iteration: "X", "Y", "Z", "U", "V", "W","T","P","VIS" X, Y and Z are the co-ordinates (in mm) of grid points. U, V and W are the velocities (in mm/s) at the grid point in x, y and z-direction. T is the temperature in K, P is the pressure in dyne/cm2, and VIS is the viscosity in kg/m-s. This file can be directly opened using Tecplot graphing program. It is very important for visualization of results obtained.

X(i+1) X(i)

XU(i) XU(i+1)

6

c) For all values of z, geometry.dat stores the y boundary of keyhole wall (or vapor-liquid interface), the weld pool, and 1000 K temperature. This data can be used to plot weld cross-section in visualization software like tecplot. 5. Calculation procedure The program calculates the three-dimensional temperature and velocity fields in weld pool for steady state keyhole mode laser welding. The program first calculates the keyhole profile based on point by point energy balance on the keyhole surface that is assumed to be at boiling temperature of the material. The keyhole profile thus generated is mapped on a three dimensional calculation domain for subsequent calculation of fluid flow and heat transfer in the liquid and the solid-liquid region. An outline of the calculation procedure is given in the Flow Chart (Fig. 1).

Fig. 1. Outline of the calculation procedure.

No

Run keyhole code

Calculate keyhole profile

Map the keyhole profile in 3-D fluid flow, heat

transfer domain

Calculate fluid flow and heat transfer iteratively

Are the residuals of u, v, w, and

enthalpy sufficiently low?

Terminate the process

Yes

7

Z

Ia - Iv

X θ

Ic keyhole melt

wall

6. Major Components of the code a. Calculation of keyhole geometry Assumptions

The following assumptions are made in the model: 1) Temperature on the keyhole wall is assumed to be equal to the boiling point of the alloy. Since the keyhole is exposed to the atmosphere, this assumption is justified. 2) Since the keyhole is oriented almost vertically, and the surface temperature on the keyhole wall is the boiling point of the alloy, heat is transported mainly along the horizontal planes. 3) Plasma in the keyhole is assumed to have a constant absorption coefficient independent of location. Although this assumption cannot be rigorously defended, it greatly simplifies calculations.

Three groups of input data are necessary to run the model: the material properties, welding parameters, and the computational and geometrical parameters. The output of the model includes geometry of the keyhole and three-dimensional temperature field of the weldment based on heat conduction in the workpiece. The keyhole geometry is calculated based on point-by-point energy balance on the keyhole wall. The model then calculates the three-dimensional temperature field in the weldment. The energy absorption by the plasma and the enhanced energy absorption by multiple reflections within the keyhole are also considered.

Energy balance on the keyhole wall

The laser energy is absorbed and transferred into the molten metal on the keyhole wall. As illustrated in the following figure, the local angle of the keyhole wall is considered to be determined by the balance between heat flux transferred into the keyhole wall, Ic, the locally absorbed beam energy flux, Ia, and the heat loss due to heat of evaporation, Iv. The heat balance on the keyhole wall requires (Ia - Iv) sinθ = Ic cosθ. Therefore, we have:

va

c

I-II

)(tan =θ (1)

Fig. 2 Energy balance at keyhole walls

The calculation of local keyhole angle θ requires the determination of Ic, Ia, and Iv. The calculation is done in two steps. In the first step, the effects of plasma absorption and

8

Base metal

Weld pool

Welding direction

z

x

y

r ϕ

o

multiple reflections are not considered. In the second step, these effects are taken into account based on the keyhole geometry obtained in the first set of calculations. Cartesian (x, y, z) and cylindrical (r, ϕ, z) coordinate systems are used alternatively in this document. In the Cartesian system, x is the coordinate in the welding direction, z the beam axis direction and y perpendicular to both x and z. In the cylindrical system, r and ϕ are the equivalent polar coordinates corresponding to x and y, z is also the coordinate in the beam axis direction. The co-ordinate system is explained in the following diagram:

Fig. 3 Co-ordinate system for keyhole profile calculation.

The heat flux conducted into the keyhole wall is deduced from a moving line source model developed by Rosenthal, which gives a solution for the temperature field in an infinite plate of certain thickness by:

ϕ

λϕ rcosPe'

0th

a r)e(Pe'K2π

P'T)T(r, −+= (2)

where r and ϕ are defined schematically in the coordinate system shown in the figure below, Ta is the ambient temperature, P’ is the strength of the line source, i.e., power per unit depth, λth is the thermal conductivity, K0( ) is the second kind and zeroth order solution to the modified Bessel function, and Pe’ is defined as Pe’ = v/(2κ), where v is the welding speed and κ is the thermal diffusivity. Assuming the heat flow in z direction to be negligible, Fourier’s law of heat conduction determines the heat flux in radial direction to be:

rTλ) (r,I thc ∂

∂−=ϕ (3)

The spatial gradient of temperature with respect to r is obtained from Eqn. (2) as:

[ ] ϕϕλ

rcosPe''00

th

e r)(Pe'Kr)cos(Pe'KPe'2π

P'rT −+−=

∂∂ (4)

where K0’( ) is the derivation of K0( ) and

9

K0’(x) = -K1(x) (5)

where K1(x) is the second kind and first order solution to the modified Bessel function. For the first set of calculations, only Fresnel absorption on the keyhole wall is

considered. Therefore, the absorbed laser beam energy flux at any point (r, ϕ, z) on the keyhole wall is given by Ia(r, ϕ, z) = α I0(r, ϕ, z) (6) where α is the Fresnel absorption coefficient, I0(r, ϕ, z) is the local power intensity of the beam, and is given by

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛= 2

f

22

f

f0f00 r

rexprr

Iz) , (r,I ϕ (7)

where If0 is the peak intensity at the focal point, given by 2P/(πrf02), P is the laser power,

rf0 is the beam radius at the focal point, rf is the local beam radius. The evaporative heat flux, Iv, on the keyhole wall is calculated from the following relation:

Iv = ∑=

Δn

1ii,vv HJ (8)

where n indicates the total number of alloying elements in the alloy, Jv is the evaporation flux of element i, and ΔHv,i is the heat of evaporation. The evaporation flux at very low pressures can be accurately calculated from the Langmuir equation. However, at one atmosphere pressure the Langmuir equation significantly over predicts the vaporization rate. Based on previous studies at Penn State, the calculated evaporation flux using the Langmuir equation is usually 5 to 10 times higher than the experimental results. In this study, a factor of 7.5 is used to calculate the evaporation flux from the modified Langmuir equation:

Jv = v

ivii T

M)T(Pa5.734.44 o (9)

where ai is the activity of element i in the liquid alloy, Pio (Tv) is the equilibrium vapor

pressure of element i over pure liquid at temperature Tv, Mi is the molecular weight of element i. The activity of each alloying element is taken as its atomic fraction in the alloy.

Consideration for plasma absorption and multiple reflections:

The above calculation considers only Fresnel absorption of the laser energy. The calculation also takes into account other absorption mechanisms such as plasma absorption due to inverse Bremsstrahlung and Fresnel absorption by multiple reflections. Part of the laser beam is absorbed by the plasma before the beam hits the keyhole wall. The remaining beam intensity after absorption of a beam of intensity 0I by the plasma of length l is given as

0l

t IeI β−= (10) where tI is the transmitted intensity and β is the inverse Bremsstrahlung absorption coefficient.

10

When the laser beam falls on the workpiece, part of its energy is absorbed by the workpiece and the remaining part is reflected. The absorbed energy is equal to the product of incident beam energy and the Fresnel absorption coefficient. Assuming the Fresnel absorption coefficient to be constant and equal to the value for normal incidence, the fraction of beam energy absorbed after nmr reflections is given by n

a )1(1α α−−= (11)

where α is the Fresnel absorption coefficient.

Inci

dent

bea

m

_2 θ

_θ

_θ

Reflected beamKeyhole wall

Inci

dent

bea

m

_2 θ

_θ

_θ

Reflected beamKeyhole wall

_2 θ

_θ

_θ

_2 θ

_θ

_θ

Reflected beamKeyhole wall

Fig.4 Multiple reflection on keyhole walls

Based on the keyhole profile calculated in the first round, an approximate mean angle

of θ between the both the front and rear keyhole wall and the beam axis can be estimated. The keyhole profile is approximated to be symmetrical with respect to the initial incident beam with a mean wall angle of θ . After nmr reflections, the angle θr between the reflected beam and the initial incident beam axis is θr = 2nmr θ (12) A limiting angle of π/2 can be define, above which the reflected beam leaves the keyhole. So the total number of reflections n, including the last reflection leaving the keyhole can be calculated from

14

12

2/1nn mr +θ

π=+

θπ

=+= (13)

Thus the total energy absorbed by the keyhole walls after multiple reflections is calculated by ( ) 0

)/(41la I)1(1eI θπ+β− α−−= (14)

When these absorption mechanisms are considered, Eqn. (5) can be modified as

tan(θ) = v0

)θπ/(41zα-c

va

c

I]Iα)-(1-[1e(x)I

II(x)I

p −=

− + (15)

where αp is the inverse Bremsstrahlung absorption coefficient and θ is the mean angle of the keyhole wall. Here the term π/(4θ ) is the total number of reflections.

11

Second round of calculation is performed taking into account the Fresnel absorption by the workpiece during multiple reflections and inverse Bremstrahlung absorption by plasma. The calculations from steps (1) through (7) are repeated using Eqn. (15). From these calculations, the final front and rear keyhole wall profiles on the x-z plane and the position of local line source, xs, are obtained.

b. Three dimensional fluid flow and heat transfer After the calculation of keyhole geometry, the temperature at all grid points is stored in temp3d.dat contains. The data from temp3d.dat is used to map the keyhole onto a different mesh for the calculation of three dimensional heat transfer and fluid flow calculation domain. Governing equations:

After calculating the keyhole profile, the fluid flow and heat transfer in the weld pool is modeled by solving the equations of conservation of mass, momentum, and energy in three dimensions. The molten metal is assumed to be an incompressible, laminar and Newtonian fluid. The liquid metal flow in the weld pool can be represented by the following momentum conservation equation:

( )j

i

j

ii

jij Sxu

xxuu

tu

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

μ∂∂

=∂

∂ρ+

∂∂

ρ (16)

where ρ is the density, t is the time, xi is the distance along the ith (i = 1, 2 and 3) orthogonal direction, uj is the velocity component along the j direction, μ is the effective viscosity, and Sj is the source term for the jth momentum equation and is given as:

( )j

jrefj3

L

2L

j

j

jjj x

uUTTgu

Bf)f1(C

xu

xxpS

∂

∂ρ−−βρ+⎟

⎟⎠

⎞⎜⎜⎝

⎛

+

−−⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂

∂μ

∂∂

+∂∂

−= (17)

where p represents pressure, U is the welding velocity, and β is the coefficient of volume expansion. The third term represents the frictional dissipation in the mushy zone according to the Carman-Kozeny equation for flow through a porous media, where fL is the liquid fraction, B is a very small computational constant introduced to avoid division by zero, and C is a constant accounting for the mushy zone morphology (a value of 1.6×104 was used in the present study). The fourth term is the buoyancy source term while the last term accounts for the relative motion between the laser source and the workpiece.

The following continuity equation is solved in conjunction with the momentum equation to obtain the pressure field.

( ) 0xu

i

i =∂ρ∂ (18)

In order to trace the weld pool liquid/solid interface, i.e., the phase change, the total enthalpy H is represented by a sum of sensible heat h and latent heat content ΔH, i.e.,

12

HhH Δ+= . The sensible heat h is expressed as ∫= dTCh p , where Cp is the specific

heat, and T is the temperature. The latent heat content ΔH is given as LfH L=Δ , where L is the latent heat of fusion. The liquid fraction fL is assumed to vary linearly with temperature for simplicity:

⎪⎪⎩

⎪⎪⎨

⎧

<

≤≤−−

>

=

S

LSSL

S

L

L

TT0

TTTTTTT

TT1

f (19)

where TL and TS are the liquidus and solidus temperatures, respectively. Thus, the thermal energy transportation in the weld workpiece can be expressed by the following modified energy equation:

( )h

ipii

i Sxh

Ck

xxhu

th

+⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

=∂

∂ρ+

∂∂

ρ (20)

where k is the thermal conductivity. The source term Sh is due to the latent heat content and is given as:

iii

ih x

HUxhU

x)Hu(

t)H(S

∂Δ∂

ρ−∂∂

ρ−∂

Δ∂ρ−

∂Δ∂

ρ−= (21)

Boundary conditions:

Top surface

At the top surface, the fluid velocities are assigned according to the gradient of surface tension.

0wyT

dTdf

zv

xT

dTdf

zu

L

L

=∂∂γ

=∂∂

μ

∂∂γ

=∂∂

μ

(22)

where u, v and w are the velocity components along the x, y and z directions, respectively, and dγ/dT is the temperature coefficient of surface tension.

The heat flux at the top surface is given as:

( ) )T(ThTTσεr

)yf(xexpπrfQη

zTk ac

4a

42

b

22

2btop

−−−−⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

∂∂ (23)

where rb is the beam radius, f is the power distribution factor, Q is the total laser power, η is the absorptivity, σ is the Stefan-Boltzmann constant, hc is the heat transfer coefficient, and Ta is the ambient temperature. In equation (8), the first term on the right

13

hand side is the heat input from the Gaussian heat source. The second and third terms represent the heat loss by radiation, and convection, respectively.

Symmetric plane

The boundary conditions are defined as zero flux across the symmetric surface, i.e., the vertical plane defined by the welding direction, as:

0yw,0v,0

yu

=∂∂

==∂∂ (24)

0yh

=∂∂ (25)

Keyhole surface boilhh = (26)

where hboil is the sensible heat of the different materials at their respective boiling points. The velocity component perpendicular to keyhole surface is assigned zero to represent no mass flux due to convection.

Turbulence model A turbulence model is used to estimate the enhanced values of viscosity and thermal conductivity to account for the enhanced heat and mass transfer due to the fluctuating components of velocity in the turbulent weld pool. The turbulence model used in the code is based on Prandtl’s mixing length hypothesis is used to estimate the turbulent viscosity:

tmt vlρ=μ (27)

where tμ is the turbulent viscosity, ml is the mixing length, and tv is the turbulence velocity. Turbulence velocity can be estimated from the turbulent kinetic energy. Assuming turbulent kinetic energy to be about 10% of the mean kinetic energy, the turbulent velocity is approximately 30% of the mean velocity. Thus, the turbulent viscosity becomes,

vl3.0 mt ρ=μ (28)

The mixing length for a point (i,j,k) is taken as the distance of the solid wall in the transverse direction. do jind=njm1,2,-1 if(t(i,jind,k).ge.tsolid) exit enddo dL_mix = y(jind+1)-y(j)

Turbulence viscosity is calculated as:

vMag=sqrt(u(i,j,k)**2+v(i,j,k)**2+w(i,j,k)**2)

14

visTNew=dens*dL_mix*0.3*vMag visT=vis(i,j,k)-viscos+(visTNew-vis(i,j,k)+viscos)*0.5

Diffusivity is calculated from the turbulent viscosity value using an appropriate Prandtl

number, T

pT

kc

Prμ

= (0.9 for fully turbulent flow for stainless steel). The effective

viscosity and diffusivity values are the sum of turbulent and laminar viscosity and diffusivity values, respectively.

diffT = visT/Prdtl vis(i,j,k) = visT+ viscos diff(i,j,k) = diffT + difl

Solution methodology The equations of conservation of momentum and energy are written in the following general form:

( ) ji

j

iji

i

Sxx

ux

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂

φ∂Γ

∂∂

=φρ∂∂ [29]

where, φ is the general dependent variable, Γ is the diffusion coefficient, and S is the source term. The indices i or j = 1, 2, and 3 represent the x, y and z direction respectively. Thus the governing momentum equation may be modified into the general form, as given by Eq. [16], to yield:

( )ju

i

j

iji

i

Sxu

xuu

x+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂

∂Γ

∂∂

=ρ∂∂

[30]

where the source term for the momentum equations can be given as:

i

j1i

j

i

iju x

uU

xu

xxpS

j ∂

∂δρ−⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂

μ∂∂

+∂∂

−= [31]

Source due to welding velocity is only in the x-direction, hence it is multiplied by δj1 which is non-zero only when i = 1. The governing equations are discretized using the control volume technique where the workpiece is divided into small rectangular volumes. The scalar variable is stored on the grid point which is located inside the control volume. The grid points for storing the vectors like the velocities in the x, y and z direction are staggered with respect to the scalar grid points to ensure stability. The discretized equations are formulated using the fully implicit power law technique. The final discretized equation at a grid point “P” takes the following form:

VSa)a(a U0P

0P

nbnbnbPP Δ+φ+φ=φ ∑ [32]

where, φ represents a general variable such as velocity or enthalpy, “a” represents the coefficient of the variables calculated based on the power law scheme, subscript nb represents the neighbors of the grid point P, ΔV is the volume of the control volume, 0

Pa and 0

Pφ are the coefficient and value of the general variable at the concerned grid point P

15

at the previous time step, respectively. The coefficient of φ at the point P is defined in terms of neighboring grid points as follows: VSaaa P

0P

nbnbP Δ−+= ∑ [33]

The terms SU and SP are the coefficients of the linearized source term, defined as: φ+= PU SSS [34] The grids that lie within the keyhole have temperature equal to the boiling point of the alloy. To ensure this, SP is assigned a very large negative value (SP = - great) and SU is taken equal to (great*hboil), where hboil is the enthalpy of the alloy at its boiling point. From equation [33], we see that aP becomes equal to great*ΔV. Putting this into equation [32], we get φP = hboil. Similarly, the velocity boundary conditions can be implemented in the keyhole region by modifying the coefficients. Convergence Criterion: The convergence is achieved when the residuals of enthalpy, mass, and u, v, and w velocities are less than 1e-4. 7. Remarks about the code User modifiable section of the code The user modifiable section of the code is in the file user.for. It contains the following subroutines:

1. Assigntask: controls the sequence of operations in calculation of heat transfer and fluid flow

2. Geom: creates the mesh for numerical calculation 3. KeyInput: reads the temperature data from the file temp3d.dat 4. Initialize_new: initializes the variables 5. Bound: prescribes the boundary conditions 6. Props: updates the physical properties 7. Output: generates the output files

Header.for This file stores the variables that are shared between various subroutines.

16

8. Case Studies Case 1. Tantalum welds made with 1900 W input power and 12.7 mm/s welding speed. Output file: ----------------------------------------------------------- Laser welding ----------------------------------------------------------- Process parameters ----------------------------------------------------------- Laser power (watt) 1.9000E+03 Laser absorption coefficient 3.2000E-01 Plasma attenuation coefficient 1.0000E+00 Laser beam radius (cm) 2.0000E-04 Defocus (cm) 0.0000E+00 Laser power distribution factor 2.0000E+00 Arc current (amp) 1.8000E+02 Arc voltage (volt) 0.0000E+00 Arc efficiency 7.0000E-01 Arc radius (cm) 2.7000E-01 Arc power distribution factor 5.0000E-01 EMF calculation needed (1=yes; 0=file exists) 0 Starting location of the laser beam 3.0000E-01 Starting location of the arc 1.3000E+00 Welding velocity (cm/sec) 1.2700E+00 Free surface calculation needed (1 = yes) 0 ----------------------------------------------------------- Material properties ----------------------------------------------------------- Density of liquid (gm/cm3) 1.5000E+01 Density at boiling point 1.5000E+01 Molecular viscosity of liquid (gm/cm-sec) 1.0000E-01 Solidus temperature (K) 3.2880E+03 Liquidus temperature (K) 3.2930E+03 Boiling point (K) 5.6430E+03 Enthalpy of solid at melting point (cal/gm) 1.2447E+02 Enthalpy of liquid at melting point (cal/gm) 1.5710E+02 Specific heat of solid (cal/gm-K) 4.5000E-02 Specific heat of liquid (cal/gm-K) 5.5000E-02 Thermal conductivity of solid (cal/cm-sec-K) 1.3000E-01 Molecular thermal cond. of liq (cal/cm-sec-K) 1.6000E-01 Coefficient of thermal expansion (1/K) 6.6000E-06 Emissivity of the material 0.0000E+00 d(gamma)/dT of pure material (dynes/cm-K) -2.5000E-01 Concentration of surface active species (wt%) 0.0000E+00 Surface excess at saturation (mole/cm2) 1.3000E-09 Enthalpy of segregation (cal/mole) -1.6600E+05 Entropy factor 3.1800E-03 ----------------------------------------------------------- Numerical scheme parameters ----------------------------------------------------------- Maximum number of iterations 3500 Underrelaxation for u-velocity 6.0000E-01 Underrelaxation for v-velocity 6.0000E-01 Underrelaxation for w-velocity 6.0000E-01 Underrelaxation for pressure 8.0000E-01 Underrelaxation for enthalpy 1.0000E+00 ----------------------------------------------------------- Boundary conditions ----------------------------------------------------------- Heat transfer coeff at west face (cal/cm2-s-K) 1.0000E+02 Heat transfer coeff at east face (cal/cm2-s-K) 1.0000E+02

17

Heat transfer coeff at north face (cal/cm2-s-K) 1.0000E+02 Heat transfer coeff at bottom face (cal/cm2-s-K) 1.0000E-02 Heat transfer coeff at top face (cal/cm2-s-K) 0.0000E+00 Temperature at west face (K) 2.9800E+02 Temperature at east face (K) 2.9800E+02 Temperature at north face (K) 2.9800E+02 Temperature at bottom face (K) 2.9800E+02 Preheat temperature (K) 2.9800E+02 Ambient temperature (K) 2.9800E+02 ----------------------------------------------------------- Geometrical parameters ----------------------------------------------------------- x direction number of zones 5 zone( 1) length (cm) 0.2000E+00 zone( 1) number of control volumes (CV) 10 zone( 1) exponent to locate CV interfaces -.1300E+01 zone( 2) length (cm) 0.2000E+00 zone( 2) number of control volumes (CV) 35 zone( 2) exponent to locate CV interfaces 0.1000E+01 zone( 3) length (cm) 0.8000E+00 zone( 3) number of control volumes (CV) 35 zone( 3) exponent to locate CV interfaces 0.1200E+01 zone( 4) length (cm) 0.6000E+01 zone( 4) number of control volumes (CV) 70 zone( 4) exponent to locate CV interfaces 0.1000E+01 zone( 5) length (cm) 0.7000E-01 zone( 5) number of control volumes (CV) 15 zone( 5) exponent to locate CV interfaces 0.1000E+01 y direction number of zones 4 zone( 1) length (cm) 0.1000E+00 zone( 1) number of control volumes (CV) 35 zone( 1) exponent to locate CV interfaces 0.1000E+01 zone( 2) length (cm) 0.2200E+00 zone( 2) number of control volumes (CV) 35 zone( 2) exponent to locate CV interfaces -.1100E+01 zone( 3) length (cm) 0.7400E+00 zone( 3) number of control volumes (CV) 20 zone( 3) exponent to locate CV interfaces 0.1300E+01 zone( 4) length (cm) 0.7000E-01 zone( 4) number of control volumes (CV) 15 zone( 4) exponent to locate CV interfaces 0.1000E+01 z direction number of zones 2 zone( 1) length (cm) 0.3000E+00 zone( 1) number of control volumes (CV) 20 zone( 1) exponent to locate CV interfaces 0.1000E+01 zone( 2) length (cm) 0.3000E+00 zone( 2) number of control volumes (CV) 30 zone( 2) exponent to locate CV interfaces 0.1000E+01 Number of grid points in x-direction (length) 167 Number of grid points in y-direction (width) 107 Number of grid points in z-direction (depth) 52 Length of the specimen (cm) 7.2700E+00 Width of the specimen (cm) 1.1300E+00 Height of the specimen (cm) 6.0000E-01 --------------------------------------------------------------------------------- x-grid (cm) i= 1 2 3 4 5 6 7

18

x= 0.000E+00 1.280E-02 3.798E-02 6.228E-02 8.563E-02 1.079E-01 1.290E-01 xu= 0.000E+00 0.000E+00 2.560E-02 5.036E-02 7.421E-02 9.705E-02 1.188E-01 i= 8 9 10 11 12 13 14 x= 1.487E-01 1.668E-01 1.826E-01 1.950E-01 2.029E-01 2.086E-01 2.143E-01 xu= 1.392E-01 1.582E-01 1.753E-01 1.900E-01 2.000E-01 2.057E-01 2.114E-01 i= 15 16 17 18 19 20 21 x= 2.200E-01 2.257E-01 2.314E-01 2.371E-01 2.429E-01 2.486E-01 2.543E-01 xu= 2.171E-01 2.229E-01 2.286E-01 2.343E-01 2.400E-01 2.457E-01 2.514E-01 i= 22 23 24 25 26 27 28 x= 2.600E-01 2.657E-01 2.714E-01 2.771E-01 2.829E-01 2.886E-01 2.943E-01 xu= 2.571E-01 2.629E-01 2.686E-01 2.743E-01 2.800E-01 2.857E-01 2.914E-01 i= 29 30 31 32 33 34 35 x= 3.000E-01 3.057E-01 3.114E-01 3.171E-01 3.229E-01 3.286E-01 3.343E-01 xu= 2.971E-01 3.029E-01 3.086E-01 3.143E-01 3.200E-01 3.257E-01 3.314E-01 i= 36 37 38 39 40 41 42 x= 3.400E-01 3.457E-01 3.514E-01 3.571E-01 3.629E-01 3.686E-01 3.743E-01 xu= 3.371E-01 3.429E-01 3.486E-01 3.543E-01 3.600E-01 3.657E-01 3.714E-01 i= 43 44 45 46 47 48 49 x= 3.800E-01 3.857E-01 3.914E-01 3.971E-01 4.056E-01 4.185E-01 4.339E-01 xu= 3.771E-01 3.829E-01 3.886E-01 3.943E-01 4.000E-01 4.112E-01 4.258E-01 i= 50 51 52 53 54 55 56 x= 4.506E-01 4.683E-01 4.869E-01 5.062E-01 5.260E-01 5.465E-01 5.673E-01 xu= 4.420E-01 4.592E-01 4.774E-01 4.964E-01 5.160E-01 5.361E-01 5.568E-01 i= 57 58 59 60 61 62 63 x= 5.887E-01 6.104E-01 6.326E-01 6.551E-01 6.779E-01 7.011E-01 7.245E-01 xu= 5.779E-01 5.995E-01 6.214E-01 6.437E-01 6.664E-01 6.894E-01 7.127E-01 i= 64 65 66 67 68 69 70 x= 7.483E-01 7.723E-01 7.965E-01 8.211E-01 8.458E-01 8.708E-01 8.960E-01 xu= 7.363E-01 7.602E-01 7.843E-01 8.087E-01 8.334E-01 8.583E-01 8.834E-01 i= 71 72 73 74 75 76 77 x= 9.215E-01 9.471E-01 9.730E-01 9.990E-01 1.025E+00 1.052E+00 1.078E+00 xu= 9.087E-01 9.342E-01 9.600E-01 9.859E-01 1.012E+00 1.038E+00 1.065E+00 i= 78 79 80 81 82 83 84 x= 1.105E+00 1.132E+00 1.159E+00 1.186E+00 1.243E+00 1.329E+00 1.414E+00 xu= 1.092E+00 1.118E+00 1.145E+00 1.173E+00 1.200E+00 1.286E+00 1.371E+00 i= 85 86 87 88 89 90 91 x= 1.500E+00 1.586E+00 1.671E+00 1.757E+00 1.843E+00 1.929E+00 2.014E+00 xu= 1.457E+00 1.543E+00 1.629E+00 1.714E+00 1.800E+00 1.886E+00 1.971E+00 i= 92 93 94 95 96 97 98 x= 2.100E+00 2.186E+00 2.271E+00 2.357E+00 2.443E+00 2.529E+00 2.614E+00 xu= 2.057E+00 2.143E+00 2.229E+00 2.314E+00 2.400E+00 2.486E+00 2.571E+00 i= 99 100 101 102 103 104 105 x= 2.700E+00 2.786E+00 2.871E+00 2.957E+00 3.043E+00 3.129E+00 3.214E+00 xu= 2.657E+00 2.743E+00 2.829E+00 2.914E+00 3.000E+00 3.086E+00 3.171E+00 i= 106 107 108 109 110 111 112 x= 3.300E+00 3.386E+00 3.471E+00 3.557E+00 3.643E+00 3.729E+00 3.814E+00 xu= 3.257E+00 3.343E+00 3.429E+00 3.514E+00 3.600E+00 3.686E+00 3.771E+00

19

i= 113 114 115 116 117 118 119 x= 3.900E+00 3.986E+00 4.071E+00 4.157E+00 4.243E+00 4.329E+00 4.414E+00 xu= 3.857E+00 3.943E+00 4.029E+00 4.114E+00 4.200E+00 4.286E+00 4.371E+00 i= 120 121 122 123 124 125 126 x= 4.500E+00 4.586E+00 4.671E+00 4.757E+00 4.843E+00 4.929E+00 5.014E+00 xu= 4.457E+00 4.543E+00 4.629E+00 4.714E+00 4.800E+00 4.886E+00 4.971E+00 i= 127 128 129 130 131 132 133 x= 5.100E+00 5.186E+00 5.271E+00 5.357E+00 5.443E+00 5.529E+00 5.614E+00 xu= 5.057E+00 5.143E+00 5.229E+00 5.314E+00 5.400E+00 5.486E+00 5.571E+00 i= 134 135 136 137 138 139 140 x= 5.700E+00 5.786E+00 5.871E+00 5.957E+00 6.043E+00 6.129E+00 6.214E+00 xu= 5.657E+00 5.743E+00 5.829E+00 5.914E+00 6.000E+00 6.086E+00 6.171E+00 i= 141 142 143 144 145 146 147 x= 6.300E+00 6.386E+00 6.471E+00 6.557E+00 6.643E+00 6.729E+00 6.814E+00 xu= 6.257E+00 6.343E+00 6.429E+00 6.514E+00 6.600E+00 6.686E+00 6.771E+00 i= 148 149 150 151 152 153 154 x= 6.900E+00 6.986E+00 7.071E+00 7.157E+00 7.202E+00 7.207E+00 7.212E+00 xu= 6.857E+00 6.943E+00 7.029E+00 7.114E+00 7.200E+00 7.205E+00 7.209E+00 i= 155 156 157 158 159 160 161 x= 7.216E+00 7.221E+00 7.226E+00 7.230E+00 7.235E+00 7.240E+00 7.244E+00 xu= 7.214E+00 7.219E+00 7.223E+00 7.228E+00 7.233E+00 7.237E+00 7.242E+00 i= 162 163 164 165 166 167 x= 7.249E+00 7.254E+00 7.258E+00 7.263E+00 7.268E+00 7.270E+00 xu= 7.247E+00 7.251E+00 7.256E+00 7.261E+00 7.265E+00 7.270E+00 --------------------------------------------------------------------------------- y-grid (cm) j= 1 2 3 4 5 6 7 y= 0.000E+00 1.429E-03 4.286E-03 7.143E-03 1.000E-02 1.286E-02 1.571E-02 yv= 0.000E+00 0.000E+00 2.857E-03 5.714E-03 8.571E-03 1.143E-02 1.429E-02 j= 8 9 10 11 12 13 14 y= 1.857E-02 2.143E-02 2.429E-02 2.714E-02 3.000E-02 3.286E-02 3.571E-02 yv= 1.714E-02 2.000E-02 2.286E-02 2.571E-02 2.857E-02 3.143E-02 3.429E-02 j= 15 16 17 18 19 20 21 y= 3.857E-02 4.143E-02 4.429E-02 4.714E-02 5.000E-02 5.286E-02 5.571E-02 yv= 3.714E-02 4.000E-02 4.286E-02 4.571E-02 4.857E-02 5.143E-02 5.429E-02 j= 22 23 24 25 26 27 28 y= 5.857E-02 6.143E-02 6.429E-02 6.714E-02 7.000E-02 7.286E-02 7.571E-02 yv= 5.714E-02 6.000E-02 6.286E-02 6.571E-02 6.857E-02 7.143E-02 7.429E-02 j= 29 30 31 32 33 34 35 y= 7.857E-02 8.143E-02 8.429E-02 8.714E-02 9.000E-02 9.286E-02 9.571E-02 yv= 7.714E-02 8.000E-02 8.286E-02 8.571E-02 8.857E-02 9.143E-02 9.429E-02 j= 36 37 38 39 40 41 42 y= 9.857E-02 1.035E-01 1.103E-01 1.172E-01 1.241E-01 1.309E-01 1.377E-01 yv= 9.714E-02 1.000E-01 1.069E-01 1.138E-01 1.207E-01 1.275E-01 1.343E-01 j= 43 44 45 46 47 48 49 y= 1.445E-01 1.513E-01 1.580E-01 1.647E-01 1.714E-01 1.781E-01 1.847E-01 yv= 1.411E-01 1.479E-01 1.546E-01 1.614E-01 1.681E-01 1.747E-01 1.814E-01

20

j= 50 51 52 53 54 55 56 y= 1.913E-01 1.979E-01 2.044E-01 2.109E-01 2.174E-01 2.238E-01 2.302E-01 yv= 1.880E-01 1.946E-01 2.011E-01 2.076E-01 2.141E-01 2.206E-01 2.270E-01 j= 57 58 59 60 61 62 63 y= 2.365E-01 2.428E-01 2.491E-01 2.553E-01 2.615E-01 2.676E-01 2.736E-01 yv= 2.334E-01 2.397E-01 2.460E-01 2.522E-01 2.584E-01 2.645E-01 2.706E-01 j= 64 65 66 67 68 69 70 y= 2.796E-01 2.855E-01 2.913E-01 2.969E-01 3.025E-01 3.079E-01 3.131E-01 yv= 2.766E-01 2.825E-01 2.884E-01 2.941E-01 2.998E-01 3.053E-01 3.106E-01 j= 71 72 73 74 75 76 77 y= 3.178E-01 3.275E-01 3.461E-01 3.700E-01 3.971E-01 4.267E-01 4.584E-01 yv= 3.156E-01 3.200E-01 3.351E-01 3.571E-01 3.828E-01 4.113E-01 4.421E-01 j= 78 79 80 81 82 83 84 y= 4.919E-01 5.269E-01 5.635E-01 6.013E-01 6.404E-01 6.805E-01 7.218E-01 yv= 4.747E-01 5.090E-01 5.449E-01 5.821E-01 6.205E-01 6.602E-01 7.009E-01 j= 85 86 87 88 89 90 91 y= 7.641E-01 8.073E-01 8.514E-01 8.964E-01 9.422E-01 9.888E-01 1.036E+00 yv= 7.427E-01 7.854E-01 8.291E-01 8.737E-01 9.191E-01 9.653E-01 1.012E+00 j= 92 93 94 95 96 97 98 y= 1.062E+00 1.067E+00 1.072E+00 1.076E+00 1.081E+00 1.086E+00 1.090E+00 yv= 1.060E+00 1.065E+00 1.069E+00 1.074E+00 1.079E+00 1.083E+00 1.088E+00 j= 99 100 101 102 103 104 105 y= 1.095E+00 1.100E+00 1.104E+00 1.109E+00 1.114E+00 1.118E+00 1.123E+00 yv= 1.093E+00 1.097E+00 1.102E+00 1.107E+00 1.111E+00 1.116E+00 1.121E+00 j= 106 107 y= 1.128E+00 1.130E+00 yv= 1.125E+00 1.130E+00 --------------------------------------------------------------------------------- z-grid (cm) k= 1 2 3 4 5 6 7 z= 0.000E+00 7.500E-03 2.250E-02 3.750E-02 5.250E-02 6.750E-02 8.250E-02 zw= 0.000E+00 0.000E+00 1.500E-02 3.000E-02 4.500E-02 6.000E-02 7.500E-02 k= 8 9 10 11 12 13 14 z= 9.750E-02 1.125E-01 1.275E-01 1.425E-01 1.575E-01 1.725E-01 1.875E-01 zw= 9.000E-02 1.050E-01 1.200E-01 1.350E-01 1.500E-01 1.650E-01 1.800E-01 k= 15 16 17 18 19 20 21 z= 2.025E-01 2.175E-01 2.325E-01 2.475E-01 2.625E-01 2.775E-01 2.925E-01 zw= 1.950E-01 2.100E-01 2.250E-01 2.400E-01 2.550E-01 2.700E-01 2.850E-01 k= 22 23 24 25 26 27 28 z= 3.050E-01 3.150E-01 3.250E-01 3.350E-01 3.450E-01 3.550E-01 3.650E-01 zw= 3.000E-01 3.100E-01 3.200E-01 3.300E-01 3.400E-01 3.500E-01 3.600E-01 k= 29 30 31 32 33 34 35 z= 3.750E-01 3.850E-01 3.950E-01 4.050E-01 4.150E-01 4.250E-01 4.350E-01 zw= 3.700E-01 3.800E-01 3.900E-01 4.000E-01 4.100E-01 4.200E-01 4.300E-01 k= 36 37 38 39 40 41 42 z= 4.450E-01 4.550E-01 4.650E-01 4.750E-01 4.850E-01 4.950E-01 5.050E-01 zw= 4.400E-01 4.500E-01 4.600E-01 4.700E-01 4.800E-01 4.900E-01 5.000E-01

21

k= 43 44 45 46 47 48 49 z= 5.150E-01 5.250E-01 5.350E-01 5.450E-01 5.550E-01 5.650E-01 5.750E-01 zw= 5.100E-01 5.200E-01 5.300E-01 5.400E-01 5.500E-01 5.600E-01 5.700E-01 k= 50 51 52 z= 5.850E-01 5.950E-01 6.000E-01 zw= 5.800E-01 5.900E-01 6.000E-01 --------------------------------------------------------------------------------- Date: 2007- 7- 6 time: 17 : 3 :18 iter time/iter res_enth res_mass res_u res_v res_w 100 0.330 3.08E-04 2.01E-02 1.26E-03 1.30E-03 8.84E-04 Tmax umax vmax wmax length depth half-width 5643. 179.4 73.6 12.3 0.114 0.192 0.071 north south top toploss bottom west east hout hin ratio -346.8 0.0 -20.5 0.0 3.2 -221.9 130.2 -435.2 175.4 2.48 muAv muM kAv kM 0.35 6.16 0.18 0.53 iter time/iter res_enth res_mass res_u res_v res_w 500 0.334 5.68E-05 3.38E-03 3.24E-03 3.08E-03 2.11E-03 Tmax umax vmax wmax length depth half-width 5643. 70.1 73.1 12.0 0.155 0.187 0.074 north south top toploss bottom west east hout hin ratio -136.2 0.0 -2.1 0.0 1.8 -159.7 130.3 -163.8 173.7 0.94 muAv muM kAv kM 0.53 6.33 0.19 0.54 iter time/iter res_enth res_mass res_u res_v res_w 1000 0.337 1.86E-05 2.78E-03 2.97E-03 4.00E-03 3.74E-03 Tmax umax vmax wmax length depth half-width 5643. 44.0 57.9 11.8 0.168 0.187 0.077 north south top toploss bottom west east hout hin ratio -120.8 0.0 -4.7 0.0 -2.3 -156.1 128.9 -150.3 172.8 0.87 muAv muM kAv kM 0.59 6.48 0.19 0.55 iter time/iter res_enth res_mass res_u res_v res_w 1500 0.338 8.53E-06 4.02E-04 3.27E-03 1.21E-03 1.55E-03 Tmax umax vmax wmax length depth half-width 5643. 38.0 57.5 11.9 0.170 0.187 0.078 north south top toploss bottom west east hout hin ratio -120.0 0.0 -4.0 0.0 -2.5 -155.6 124.1 -153.9 172.6 0.89 muAv muM kAv kM 0.61 6.46 0.19 0.55 iter time/iter res_enth res_mass res_u res_v res_w 2000 0.339 4.73E-06 3.64E-04 1.68E-03 8.88E-04 1.10E-03 Tmax umax vmax wmax length depth half-width 5643. 43.1 57.5 12.0 0.171 0.187 0.078 north south top toploss bottom west east hout hin ratio -120.7 0.0 -3.0 0.0 -2.0 -155.4 119.8 -158.3 172.5 0.92 muAv muM kAv kM 0.61 6.46 0.19 0.55 iter time/iter res_enth res_mass res_u res_v res_w 2500 0.339 2.94E-06 3.68E-04 1.67E-03 8.85E-04 1.09E-03 Tmax umax vmax wmax length depth half-width 5643. 43.0 57.4 12.0 0.171 0.187 0.078 north south top toploss bottom west east hout hin ratio -121.3 0.0 -2.1 0.0 -1.5 -155.4 116.1 -162.1 172.5 0.94

22

muAv muM kAv kM 0.61 6.46 0.19 0.55 iter time/iter res_enth res_mass res_u res_v res_w 3000 0.340 1.95E-06 8.61E-04 3.27E-03 2.12E-03 2.70E-03 Tmax umax vmax wmax length depth half-width 5643. 43.0 57.4 12.0 0.171 0.187 0.078 north south top toploss bottom west east hout hin ratio -121.4 0.0 -1.4 0.0 -1.1 -155.4 112.8 -165.1 172.5 0.96 muAv muM kAv kM 0.61 6.46 0.19 0.55 iter time/iter res_enth res_mass res_u res_v res_w 3500 0.340 1.32E-06 3.70E-04 1.67E-03 8.85E-04 1.09E-03 Tmax umax vmax wmax length depth half-width 5643. 43.0 57.4 12.0 0.171 0.187 0.078 north south top toploss bottom west east hout hin ratio -121.2 0.0 -0.9 0.0 -0.7 -155.4 110.0 -167.3 172.5 0.97 muAv muM kAv kM 0.61 6.46 0.19 0.55 Some important calculated parameters at the end of heating cycle Length of the pool (cm) 1.7135E-01 Depth of the pool (cm) 1.8694E-01 Half-width of the pool (cm) 7.7712E-02 Peak temperature (K) 5.6430E+03 x1073(cm), x773(cm), t8-5(s) 0.817 1.214 0.312 Maximum u-velocity (cm/s) 4.3042E+01 Maximum v-velocity (cm/s) 5.7447E+01 Maximum w-velocity (cm/s) 1.1967E+01 Rate of heat input (cal/s) 1.7247E+02 Rate of heat output (cal/s) -1.6727E+02 Ratio of heat input to heat output 9.6984E-01 Date: 2007- 7- 6 time: 17:23:23 Total time used: 0 hr 20 m 5 s Total time used: 0 hr 20 m 8 s

Fig. 5 Comparison of experimental and calculated weld cross-sectional geometry. The solid line represents the solidus and the dashed line represents the keyhole marked by boiling point temperature. The dotted line shows the experimentally observed weld pool boundary.

Y, mm

Z,m

m

-3 -2 -1 0 1 2 32

3

4

5

6

23

Fig. 6 Fluid flow in the weld pool. The narrow and deep region with no velocity vectors is the keyhole. Case 2: Vanadium welds made with 1980 W input power at 25.4 mm/s welding speed. Output file: ----------------------------------------------------------- Laser welding ----------------------------------------------------------- Process parameters ----------------------------------------------------------- Laser power (watt) 1.7770E+03 Laser absorption coefficient 2.8000E-01 Plasma attenuation coefficient 1.0000E+00 Laser beam radius (cm) 2.0000E-04 Defocus (cm) 0.0000E+00 Laser power distribution factor 2.0000E+00 Arc current (amp) 0.0000E+00 Arc voltage (volt) 0.0000E+00 Arc efficiency 7.0000E-01 Arc radius (cm) 2.7000E-01 Arc power distribution factor 5.0000E-01 EMF calculation needed (1=yes; 0=file exists) 0 Starting location of the laser beam 3.0000E-01 Starting location of the arc 7.0000E-01 Welding velocity (cm/sec) 2.5400E+00 Free surface calculation needed (1 = yes) 0 ----------------------------------------------------------- Material properties ----------------------------------------------------------- Density of liquid (gm/cm3) 5.7000E+00 Density at boiling point 5.2000E+00 Molecular viscosity of liquid (gm/cm-sec) 5.0000E-02 Solidus temperature (K) 2.1750E+03 Liquidus temperature (K) 2.1800E+03 Boiling point (K) 3.6830E+03 Enthalpy of solid at melting point (cal/gm) 2.4428E+02 Enthalpy of liquid at melting point (cal/gm) 3.2277E+02 Specific heat of solid (cal/gm-K) 1.7300E-01 Specific heat of liquid (cal/gm-K) 1.8600E-01 Thermal conductivity of solid (cal/cm-sec-K) 7.0000E-02 Molecular thermal cond. of liq (cal/cm-sec-K) 1.2000E-01 Coefficient of thermal expansion (1/K) 1.0000E-05 Emissivity of the material 0.0000E+00 d(gamma)/dT of pure material (dynes/cm-K) -3.1000E-01 Concentration of surface active species (wt%) 0.0000E+00 Surface excess at saturation (mole/cm2) 1.3000E-09 Enthalpy of segregation (cal/mole) -1.6600E+05

(d)

4.0

5.0

6.0

Z,mm

2.0 4.0 6.0 8.0

X , mm0.0

1.02.0

3.0Y, mm

250 mm/s

24

Entropy factor 3.1800E-03 ----------------------------------------------------------- Numerical scheme parameters ----------------------------------------------------------- Maximum number of iterations 2000 Underrelaxation for u-velocity 6.0000E-01 Underrelaxation for v-velocity 6.0000E-01 Underrelaxation for w-velocity 6.0000E-01 Underrelaxation for pressure 8.0000E-01 Underrelaxation for enthalpy 1.0000E+00 ----------------------------------------------------------- Boundary conditions ----------------------------------------------------------- Heat transfer coeff at west face (cal/cm2-s-K) 1.0000E+02 Heat transfer coeff at east face (cal/cm2-s-K) 1.0000E+02 Heat transfer coeff at north face (cal/cm2-s-K) 1.0000E+02 Heat transfer coeff at bottom face (cal/cm2-s-K) 1.0000E-02 Heat transfer coeff at top face (cal/cm2-s-K) 0.0000E+00 Temperature at west face (K) 2.9800E+02 Temperature at east face (K) 2.9800E+02 Temperature at north face (K) 2.9800E+02 Temperature at bottom face (K) 2.9800E+02 Preheat temperature (K) 2.9800E+02 Ambient temperature (K) 2.9800E+02 ----------------------------------------------------------- Geometrical parameters ----------------------------------------------------------- x direction number of zones 5 zone( 1) length (cm) 0.2000E+00 zone( 1) number of control volumes (CV) 10 zone( 1) exponent to locate CV interfaces -.1300E+01 zone( 2) length (cm) 0.2000E+00 zone( 2) number of control volumes (CV) 35 zone( 2) exponent to locate CV interfaces 0.1000E+01 zone( 3) length (cm) 0.1500E+01 zone( 3) number of control volumes (CV) 40 zone( 3) exponent to locate CV interfaces 0.1300E+01 zone( 4) length (cm) 0.8000E+01 zone( 4) number of control volumes (CV) 70 zone( 4) exponent to locate CV interfaces 0.1000E+01 zone( 5) length (cm) 0.7000E-01 zone( 5) number of control volumes (CV) 15 zone( 5) exponent to locate CV interfaces 0.1000E+01 y direction number of zones 4 zone( 1) length (cm) 0.1500E+00 zone( 1) number of control volumes (CV) 25 zone( 1) exponent to locate CV interfaces 0.1000E+01 zone( 2) length (cm) 0.2200E+00 zone( 2) number of control volumes (CV) 30 zone( 2) exponent to locate CV interfaces 0.1000E+01 zone( 3) length (cm) 0.2070E+01 zone( 3) number of control volumes (CV) 30 zone( 3) exponent to locate CV interfaces 0.1300E+01 zone( 4) length (cm) 0.7000E-01 zone( 4) number of control volumes (CV) 15 zone( 4) exponent to locate CV interfaces 0.1000E+01 z direction number of zones 2 zone( 1) length (cm) 0.1600E+00 zone( 1) number of control volumes (CV) 20 zone( 1) exponent to locate CV interfaces 0.1000E+01

25

zone( 2) length (cm) 0.1600E+00 zone( 2) number of control volumes (CV) 20 zone( 2) exponent to locate CV interfaces 0.1000E+01 Number of grid points in x-direction (length) 172 Number of grid points in y-direction (width) 102 Number of grid points in z-direction (depth) 42 Length of the specimen (cm) 9.9700E+00 Width of the specimen (cm) 2.5100E+00 Height of the specimen (cm) 3.2000E-01 --------------------------------------------------------------------------------- x-grid (cm) i= 1 2 3 4 5 6 7 x= 0.000E+00 1.280E-02 3.798E-02 6.228E-02 8.563E-02 1.079E-01 1.290E-01 xu= 0.000E+00 0.000E+00 2.560E-02 5.036E-02 7.421E-02 9.705E-02 1.188E-01 i= 8 9 10 11 12 13 14 x= 1.487E-01 1.668E-01 1.826E-01 1.950E-01 2.029E-01 2.086E-01 2.143E-01 xu= 1.392E-01 1.582E-01 1.753E-01 1.900E-01 2.000E-01 2.057E-01 2.114E-01 i= 15 16 17 18 19 20 21 x= 2.200E-01 2.257E-01 2.314E-01 2.371E-01 2.429E-01 2.486E-01 2.543E-01 xu= 2.171E-01 2.229E-01 2.286E-01 2.343E-01 2.400E-01 2.457E-01 2.514E-01 i= 22 23 24 25 26 27 28 x= 2.600E-01 2.657E-01 2.714E-01 2.771E-01 2.829E-01 2.886E-01 2.943E-01 xu= 2.571E-01 2.629E-01 2.686E-01 2.743E-01 2.800E-01 2.857E-01 2.914E-01 i= 29 30 31 32 33 34 35 x= 3.000E-01 3.057E-01 3.114E-01 3.171E-01 3.229E-01 3.286E-01 3.343E-01 xu= 2.971E-01 3.029E-01 3.086E-01 3.143E-01 3.200E-01 3.257E-01 3.314E-01 i= 36 37 38 39 40 41 42 x= 3.400E-01 3.457E-01 3.514E-01 3.571E-01 3.629E-01 3.686E-01 3.743E-01 xu= 3.371E-01 3.429E-01 3.486E-01 3.543E-01 3.600E-01 3.657E-01 3.714E-01 i= 43 44 45 46 47 48 49 x= 3.800E-01 3.857E-01 3.914E-01 3.971E-01 4.062E-01 4.215E-01 4.411E-01 xu= 3.771E-01 3.829E-01 3.886E-01 3.943E-01 4.000E-01 4.124E-01 4.305E-01 i= 50 51 52 53 54 55 56 x= 4.634E-01 4.878E-01 5.139E-01 5.415E-01 5.704E-01 6.004E-01 6.316E-01 xu= 4.517E-01 4.752E-01 5.005E-01 5.274E-01 5.556E-01 5.851E-01 6.157E-01 i= 57 58 59 60 61 62 63 x= 6.637E-01 6.968E-01 7.308E-01 7.656E-01 8.011E-01 8.375E-01 8.745E-01 xu= 6.474E-01 6.800E-01 7.136E-01 7.480E-01 7.832E-01 8.191E-01 8.558E-01 i= 64 65 66 67 68 69 70 x= 9.122E-01 9.506E-01 9.895E-01 1.029E+00 1.069E+00 1.110E+00 1.151E+00 xu= 8.932E-01 9.312E-01 9.699E-01 1.009E+00 1.049E+00 1.090E+00 1.131E+00 i= 71 72 73 74 75 76 77 x= 1.193E+00 1.236E+00 1.278E+00 1.322E+00 1.365E+00 1.410E+00 1.454E+00 xu= 1.172E+00 1.214E+00 1.257E+00 1.300E+00 1.343E+00 1.387E+00 1.432E+00 i= 78 79 80 81 82 83 84 x= 1.500E+00 1.545E+00 1.591E+00 1.638E+00 1.684E+00 1.732E+00 1.779E+00 xu= 1.477E+00 1.522E+00 1.568E+00 1.614E+00 1.661E+00 1.708E+00 1.755E+00 i= 85 86 87 88 89 90 91

26

x= 1.827E+00 1.876E+00 1.957E+00 2.071E+00 2.186E+00 2.300E+00 2.414E+00 xu= 1.803E+00 1.851E+00 1.900E+00 2.014E+00 2.129E+00 2.243E+00 2.357E+00 i= 92 93 94 95 96 97 98 x= 2.529E+00 2.643E+00 2.757E+00 2.871E+00 2.986E+00 3.100E+00 3.214E+00 xu= 2.471E+00 2.586E+00 2.700E+00 2.814E+00 2.929E+00 3.043E+00 3.157E+00 i= 99 100 101 102 103 104 105 x= 3.329E+00 3.443E+00 3.557E+00 3.671E+00 3.786E+00 3.900E+00 4.014E+00 xu= 3.271E+00 3.386E+00 3.500E+00 3.614E+00 3.729E+00 3.843E+00 3.957E+00 i= 106 107 108 109 110 111 112 x= 4.129E+00 4.243E+00 4.357E+00 4.471E+00 4.586E+00 4.700E+00 4.814E+00 xu= 4.071E+00 4.186E+00 4.300E+00 4.414E+00 4.529E+00 4.643E+00 4.757E+00 i= 113 114 115 116 117 118 119 x= 4.929E+00 5.043E+00 5.157E+00 5.271E+00 5.386E+00 5.500E+00 5.614E+00 xu= 4.871E+00 4.986E+00 5.100E+00 5.214E+00 5.329E+00 5.443E+00 5.557E+00 i= 120 121 122 123 124 125 126 x= 5.729E+00 5.843E+00 5.957E+00 6.071E+00 6.186E+00 6.300E+00 6.414E+00 xu= 5.671E+00 5.786E+00 5.900E+00 6.014E+00 6.129E+00 6.243E+00 6.357E+00 i= 127 128 129 130 131 132 133 x= 6.529E+00 6.643E+00 6.757E+00 6.871E+00 6.986E+00 7.100E+00 7.214E+00 xu= 6.471E+00 6.586E+00 6.700E+00 6.814E+00 6.929E+00 7.043E+00 7.157E+00 i= 134 135 136 137 138 139 140 x= 7.329E+00 7.443E+00 7.557E+00 7.671E+00 7.786E+00 7.900E+00 8.014E+00 xu= 7.271E+00 7.386E+00 7.500E+00 7.614E+00 7.729E+00 7.843E+00 7.957E+00 i= 141 142 143 144 145 146 147 x= 8.129E+00 8.243E+00 8.357E+00 8.471E+00 8.586E+00 8.700E+00 8.814E+00 xu= 8.071E+00 8.186E+00 8.300E+00 8.414E+00 8.529E+00 8.643E+00 8.757E+00 i= 148 149 150 151 152 153 154 x= 8.929E+00 9.043E+00 9.157E+00 9.271E+00 9.386E+00 9.500E+00 9.614E+00 xu= 8.871E+00 8.986E+00 9.100E+00 9.214E+00 9.329E+00 9.443E+00 9.557E+00 i= 155 156 157 158 159 160 161 x= 9.729E+00 9.843E+00 9.902E+00 9.907E+00 9.912E+00 9.916E+00 9.921E+00 xu= 9.671E+00 9.786E+00 9.900E+00 9.905E+00 9.909E+00 9.914E+00 9.919E+00 i= 162 163 164 165 166 167 168 x= 9.926E+00 9.930E+00 9.935E+00 9.940E+00 9.944E+00 9.949E+00 9.954E+00 xu= 9.923E+00 9.928E+00 9.933E+00 9.937E+00 9.942E+00 9.947E+00 9.951E+00 i= 169 170 171 172 x= 9.958E+00 9.963E+00 9.968E+00 9.970E+00 xu= 9.956E+00 9.961E+00 9.965E+00 9.970E+00 --------------------------------------------------------------------------------- y-grid (cm) j= 1 2 3 4 5 6 7 y= 0.000E+00 3.000E-03 9.000E-03 1.500E-02 2.100E-02 2.700E-02 3.300E-02 yv= 0.000E+00 0.000E+00 6.000E-03 1.200E-02 1.800E-02 2.400E-02 3.000E-02 j= 8 9 10 11 12 13 14 y= 3.900E-02 4.500E-02 5.100E-02 5.700E-02 6.300E-02 6.900E-02 7.500E-02 yv= 3.600E-02 4.200E-02 4.800E-02 5.400E-02 6.000E-02 6.600E-02 7.200E-02 j= 15 16 17 18 19 20 21

27

y= 8.100E-02 8.700E-02 9.300E-02 9.900E-02 1.050E-01 1.110E-01 1.170E-01 yv= 7.800E-02 8.400E-02 9.000E-02 9.600E-02 1.020E-01 1.080E-01 1.140E-01 j= 22 23 24 25 26 27 28 y= 1.230E-01 1.290E-01 1.350E-01 1.410E-01 1.470E-01 1.537E-01 1.610E-01 yv= 1.200E-01 1.260E-01 1.320E-01 1.380E-01 1.440E-01 1.500E-01 1.573E-01 j= 29 30 31 32 33 34 35 y= 1.683E-01 1.757E-01 1.830E-01 1.903E-01 1.977E-01 2.050E-01 2.123E-01 yv= 1.647E-01 1.720E-01 1.793E-01 1.867E-01 1.940E-01 2.013E-01 2.087E-01 j= 36 37 38 39 40 41 42 y= 2.197E-01 2.270E-01 2.343E-01 2.417E-01 2.490E-01 2.563E-01 2.637E-01 yv= 2.160E-01 2.233E-01 2.307E-01 2.380E-01 2.453E-01 2.527E-01 2.600E-01 j= 43 44 45 46 47 48 49 y= 2.710E-01 2.783E-01 2.857E-01 2.930E-01 3.003E-01 3.077E-01 3.150E-01 yv= 2.673E-01 2.747E-01 2.820E-01 2.893E-01 2.967E-01 3.040E-01 3.113E-01 j= 50 51 52 53 54 55 56 y= 3.223E-01 3.297E-01 3.370E-01 3.443E-01 3.517E-01 3.590E-01 3.663E-01 yv= 3.187E-01 3.260E-01 3.333E-01 3.407E-01 3.480E-01 3.553E-01 3.627E-01 j= 57 58 59 60 61 62 63 y= 3.824E-01 4.131E-01 4.525E-01 4.973E-01 5.462E-01 5.985E-01 6.538E-01 yv= 3.700E-01 3.949E-01 4.312E-01 4.737E-01 5.208E-01 5.715E-01 6.255E-01 j= 64 65 66 67 68 69 70 y= 7.117E-01 7.720E-01 8.345E-01 8.990E-01 9.654E-01 1.033E+00 1.103E+00 yv= 6.821E-01 7.413E-01 8.027E-01 8.663E-01 9.317E-01 9.990E-01 1.068E+00 j= 71 72 73 74 75 76 77 y= 1.175E+00 1.247E+00 1.322E+00 1.397E+00 1.474E+00 1.553E+00 1.632E+00 yv= 1.139E+00 1.211E+00 1.284E+00 1.359E+00 1.436E+00 1.513E+00 1.592E+00 j= 78 79 80 81 82 83 84 y= 1.713E+00 1.794E+00 1.877E+00 1.961E+00 2.046E+00 2.132E+00 2.219E+00 yv= 1.672E+00 1.753E+00 1.835E+00 1.919E+00 2.003E+00 2.089E+00 2.175E+00 j= 85 86 87 88 89 90 91 y= 2.307E+00 2.395E+00 2.442E+00 2.447E+00 2.452E+00 2.456E+00 2.461E+00 yv= 2.262E+00 2.351E+00 2.440E+00 2.445E+00 2.449E+00 2.454E+00 2.459E+00 j= 92 93 94 95 96 97 98 y= 2.466E+00 2.470E+00 2.475E+00 2.480E+00 2.484E+00 2.489E+00 2.494E+00 yv= 2.463E+00 2.468E+00 2.473E+00 2.477E+00 2.482E+00 2.487E+00 2.491E+00 j= 99 100 101 102 y= 2.498E+00 2.503E+00 2.508E+00 2.510E+00 yv= 2.496E+00 2.501E+00 2.505E+00 2.510E+00 --------------------------------------------------------------------------------- z-grid (cm) k= 1 2 3 4 5 6 7 z= 0.000E+00 4.000E-03 1.200E-02 2.000E-02 2.800E-02 3.600E-02 4.400E-02 zw= 0.000E+00 0.000E+00 8.000E-03 1.600E-02 2.400E-02 3.200E-02 4.000E-02 k= 8 9 10 11 12 13 14 z= 5.200E-02 6.000E-02 6.800E-02 7.600E-02 8.400E-02 9.200E-02 1.000E-01 zw= 4.800E-02 5.600E-02 6.400E-02 7.200E-02 8.000E-02 8.800E-02 9.600E-02 k= 15 16 17 18 19 20 21

28

z= 1.080E-01 1.160E-01 1.240E-01 1.320E-01 1.400E-01 1.480E-01 1.560E-01 zw= 1.040E-01 1.120E-01 1.200E-01 1.280E-01 1.360E-01 1.440E-01 1.520E-01 k= 22 23 24 25 26 27 28 z= 1.640E-01 1.720E-01 1.800E-01 1.880E-01 1.960E-01 2.040E-01 2.120E-01 zw= 1.600E-01 1.680E-01 1.760E-01 1.840E-01 1.920E-01 2.000E-01 2.080E-01 k= 29 30 31 32 33 34 35 z= 2.200E-01 2.280E-01 2.360E-01 2.440E-01 2.520E-01 2.600E-01 2.680E-01 zw= 2.160E-01 2.240E-01 2.320E-01 2.400E-01 2.480E-01 2.560E-01 2.640E-01 k= 36 37 38 39 40 41 42 z= 2.760E-01 2.840E-01 2.920E-01 3.000E-01 3.080E-01 3.160E-01 3.200E-01 zw= 2.720E-01 2.800E-01 2.880E-01 2.960E-01 3.040E-01 3.120E-01 3.200E-01 --------------------------------------------------------------------------------- Date: 2007- 7- 6 time: 17 :47 :44 iter time/iter res_enth res_mass res_u res_v res_w 100 0.230 2.82E-04 3.10E-02 1.05E-01 5.82E-02 7.53E-02 Tmax umax vmax wmax length depth half-width 3683. 67.6 66.9 16.4 0.208 0.212 0.069 north south top toploss bottom west east hout hin ratio -158.3 0.0 -154.4 0.0 -171.5 -935.5 935.6 -329.8 115.2 2.86 muAv muM kAv kM 0.50 3.90 0.21 0.92 iter time/iter res_enth res_mass res_u res_v res_w 500 0.240 3.96E-05 9.29E-03 3.76E-03 3.80E-03 4.00E-03 Tmax umax vmax wmax length depth half-width 3683. 117.0 160.5 15.0 0.271 0.212 0.072 north south top toploss bottom west east hout hin ratio -4.8 0.0 64.2 0.0 88.3 -935.4 724.0 -127.9 110.3 1.16 muAv muM kAv kM 0.40 3.22 0.19 0.78 iter time/iter res_enth res_mass res_u res_v res_w 1000 0.245 1.62E-05 9.15E-03 1.17E-02 1.26E-02 1.29E-02 Tmax umax vmax wmax length depth half-width 3683. 64.3 106.5 15.0 0.271 0.212 0.072 north south top toploss bottom west east hout hin ratio -14.4 0.0 -0.8 0.0 -1.8 -935.4 841.5 -110.2 110.3 1.00 muAv muM kAv kM 0.40 3.22 0.19 0.78 iter time/iter res_enth res_mass res_u res_v res_w 1500 0.249 1.46E-05 8.79E-03 1.18E-02 1.14E-02 1.25E-02 Tmax umax vmax wmax length depth half-width 3683. 64.1 66.8 15.0 0.271 0.212 0.072 north south top toploss bottom west east hout hin ratio -11.8 0.0 -1.5 0.0 -1.9 -935.4 839.1 -110.0 110.3 1.00 muAv muM kAv kM 0.40 3.22 0.19 0.78 iter time/iter res_enth res_mass res_u res_v res_w 2000 0.250 1.34E-05 9.37E-03 7.81E-03 7.82E-03 8.05E-03 Tmax umax vmax wmax length depth half-width 3683. 81.0 119.5 15.0 0.271 0.212 0.072 north south top toploss bottom west east hout hin ratio -7.8 0.0 -1.0 0.0 -1.3 -935.4 834.2 -110.3 110.2 1.00 muAv muM kAv kM 0.40 3.22 0.19 0.78

29

Some important calculated parameters at the end of heating cycle Length of the pool (cm) 2.7107E-01 Depth of the pool (cm) 2.1153E-01 Half-width of the pool (cm) 7.1916E-02 Peak temperature (K) 3.6830E+03 x1073(cm), x773(cm), t8-5(s) 0.956 1.585 0.248 Maximum u-velocity (cm/s) 8.1004E+01 Maximum v-velocity (cm/s) 1.1948E+02 Maximum w-velocity (cm/s) 1.5014E+01 Rate of heat input (cal/s) 1.1024E+02 Rate of heat output (cal/s) -1.1033E+02 Ratio of heat input to heat output 1.0007E+00 Date: 2007- 7- 6 time: 17:56:24 Total time used: 0 hr 8 m 40 s

Fig. 7 Comparison of experimental and calculated weld cross-sectional geometry. The solid line represents the solidus and the dashed line represents the keyhole marked by boiling point temperature. The dotted line shows the experimentally observed weld pool boundary.

Y, mm

Z,m

m

-1 0 10

1

2

3

30

1.0

2.0

3.0

Z,mm

2.0 4.0 6.0 8.0

X , mm0.0

1.02.0

3.0Y, mm

250 mm/s

Fig. 8 Fluid flow in the weld pool. The narrow and deep region with no velocity vectors is the keyhole. Case 3: 304L stainless steel welds made with 1980 W input power at 19.1 mm/s welding speed. Output file: ----------------------------------------------------------- Laser welding ----------------------------------------------------------- Process parameters ----------------------------------------------------------- Laser power (watt) 1.7770E+03 Laser absorption coefficient 3.0000E-01 Plasma attenuation coefficient 1.0000E+00 Laser beam radius (cm) 2.0000E-04 Defocus (cm) 0.0000E+00 Laser power distribution factor 2.0000E+00 Arc current (amp) 0.0000E+00 Arc voltage (volt) 0.0000E+00 Arc efficiency 7.0000E-01 Arc radius (cm) 2.7000E-01 Arc power distribution factor 5.0000E-01 EMF calculation needed (1=yes; 0=file exists) 0 Starting location of the laser beam 3.0000E-01 Starting location of the arc 7.0000E-01 Welding velocity (cm/sec) 1.9100E+00 Free surface calculation needed (1 = yes) 0 ----------------------------------------------------------- Material properties ----------------------------------------------------------- Density of liquid (gm/cm3) 7.0000E+00 Density at boiling point 5.8000E+00 Molecular viscosity of liquid (gm/cm-sec) 7.0000E-02 Solidus temperature (K) 1.6970E+03 Liquidus temperature (K) 1.7270E+03 Boiling point (K) 3.0925E+03 Enthalpy of solid at melting point (cal/gm) 2.8660E+02 Enthalpy of liquid at melting point (cal/gm) 3.0000E+02 Specific heat of solid (cal/gm-K) 1.7000E-01 Specific heat of liquid (cal/gm-K) 1.9000E-01 Thermal conductivity of solid (cal/cm-sec-K) 6.5000E-02

31

Molecular thermal cond. of liq (cal/cm-sec-K) 7.0000E-02 Coefficient of thermal expansion (1/K) 1.9600E-05 Emissivity of the material 0.0000E+00 d(gamma)/dT of pure material (dynes/cm-K) -4.9000E-01 Concentration of surface active species (wt%) 0.0000E+00 Surface excess at saturation (mole/cm2) 1.3000E-09 Enthalpy of segregation (cal/mole) -1.6600E+05 Entropy factor 3.1800E-03 ----------------------------------------------------------- Numerical scheme parameters ----------------------------------------------------------- Maximum number of iterations 2000 Underrelaxation for u-velocity 6.0000E-01 Underrelaxation for v-velocity 6.0000E-01 Underrelaxation for w-velocity 6.0000E-01 Underrelaxation for pressure 8.0000E-01 Underrelaxation for enthalpy 1.0000E+00 ----------------------------------------------------------- Boundary conditions ----------------------------------------------------------- Heat transfer coeff at west face (cal/cm2-s-K) 1.0000E+02 Heat transfer coeff at east face (cal/cm2-s-K) 1.0000E+02 Heat transfer coeff at north face (cal/cm2-s-K) 1.0000E+02 Heat transfer coeff at bottom face (cal/cm2-s-K) 1.0000E-02 Heat transfer coeff at top face (cal/cm2-s-K) 0.0000E+00 Temperature at west face (K) 2.9800E+02 Temperature at east face (K) 2.9800E+02 Temperature at north face (K) 2.9800E+02 Temperature at bottom face (K) 2.9800E+02 Preheat temperature (K) 2.9800E+02 Ambient temperature (K) 2.9800E+02 ----------------------------------------------------------- Geometrical parameters ----------------------------------------------------------- x direction number of zones 5 zone( 1) length (cm) 0.2000E+00 zone( 1) number of control volumes (CV) 10 zone( 1) exponent to locate CV interfaces -.1300E+01 zone( 2) length (cm) 0.2000E+00 zone( 2) number of control volumes (CV) 35 zone( 2) exponent to locate CV interfaces 0.1000E+01 zone( 3) length (cm) 0.1500E+01 zone( 3) number of control volumes (CV) 40 zone( 3) exponent to locate CV interfaces 0.1300E+01 zone( 4) length (cm) 0.8000E+01 zone( 4) number of control volumes (CV) 70 zone( 4) exponent to locate CV interfaces 0.1000E+01 zone( 5) length (cm) 0.1500E+00 zone( 5) number of control volumes (CV) 15 zone( 5) exponent to locate CV interfaces 0.1000E+01 y direction number of zones 4 zone( 1) length (cm) 0.1500E+00 zone( 1) number of control volumes (CV) 25 zone( 1) exponent to locate CV interfaces 0.1000E+01 zone( 2) length (cm) 0.2000E+00 zone( 2) number of control volumes (CV) 30 zone( 2) exponent to locate CV interfaces 0.1000E+01 zone( 3) length (cm) 0.2070E+01 zone( 3) number of control volumes (CV) 30 zone( 3) exponent to locate CV interfaces 0.1300E+01 zone( 4) length (cm) 0.1500E+00

32

zone( 4) number of control volumes (CV) 15 zone( 4) exponent to locate CV interfaces 0.1000E+01 z direction number of zones 3 zone( 1) length (cm) 0.3000E+00 zone( 1) number of control volumes (CV) 15 zone( 1) exponent to locate CV interfaces 0.1000E+01 zone( 2) length (cm) 0.3000E+00 zone( 2) number of control volumes (CV) 15 zone( 2) exponent to locate CV interfaces 0.1000E+01 zone( 3) length (cm) 0.3500E+00 zone( 3) number of control volumes (CV) 30 zone( 3) exponent to locate CV interfaces -.1200E+01 Number of grid points in x-direction (length) 172 Number of grid points in y-direction (width) 102 Number of grid points in z-direction (depth) 62 Length of the specimen (cm) 1.0050E+01 Width of the specimen (cm) 2.5700E+00 Height of the specimen (cm) 9.5000E-01 --------------------------------------------------------------------------------- x-grid (cm) i= 1 2 3 4 5 6 7 x= 0.000E+00 1.280E-02 3.798E-02 6.228E-02 8.563E-02 1.079E-01 1.290E-01 xu= 0.000E+00 0.000E+00 2.560E-02 5.036E-02 7.421E-02 9.705E-02 1.188E-01 i= 8 9 10 11 12 13 14 x= 1.487E-01 1.668E-01 1.826E-01 1.950E-01 2.029E-01 2.086E-01 2.143E-01 xu= 1.392E-01 1.582E-01 1.753E-01 1.900E-01 2.000E-01 2.057E-01 2.114E-01 i= 15 16 17 18 19 20 21 x= 2.200E-01 2.257E-01 2.314E-01 2.371E-01 2.429E-01 2.486E-01 2.543E-01 xu= 2.171E-01 2.229E-01 2.286E-01 2.343E-01 2.400E-01 2.457E-01 2.514E-01 i= 22 23 24 25 26 27 28 x= 2.600E-01 2.657E-01 2.714E-01 2.771E-01 2.829E-01 2.886E-01 2.943E-01 xu= 2.571E-01 2.629E-01 2.686E-01 2.743E-01 2.800E-01 2.857E-01 2.914E-01 i= 29 30 31 32 33 34 35 x= 3.000E-01 3.057E-01 3.114E-01 3.171E-01 3.229E-01 3.286E-01 3.343E-01 xu= 2.971E-01 3.029E-01 3.086E-01 3.143E-01 3.200E-01 3.257E-01 3.314E-01 i= 36 37 38 39 40 41 42 x= 3.400E-01 3.457E-01 3.514E-01 3.571E-01 3.629E-01 3.686E-01 3.743E-01 xu= 3.371E-01 3.429E-01 3.486E-01 3.543E-01 3.600E-01 3.657E-01 3.714E-01 i= 43 44 45 46 47 48 49 x= 3.800E-01 3.857E-01 3.914E-01 3.971E-01 4.062E-01 4.215E-01 4.411E-01 xu= 3.771E-01 3.829E-01 3.886E-01 3.943E-01 4.000E-01 4.124E-01 4.305E-01 i= 50 51 52 53 54 55 56 x= 4.634E-01 4.878E-01 5.139E-01 5.415E-01 5.704E-01 6.004E-01 6.316E-01 xu= 4.517E-01 4.752E-01 5.005E-01 5.274E-01 5.556E-01 5.851E-01 6.157E-01 i= 57 58 59 60 61 62 63 x= 6.637E-01 6.968E-01 7.308E-01 7.656E-01 8.011E-01 8.375E-01 8.745E-01 xu= 6.474E-01 6.800E-01 7.136E-01 7.480E-01 7.832E-01 8.191E-01 8.558E-01 i= 64 65 66 67 68 69 70 x= 9.122E-01 9.506E-01 9.895E-01 1.029E+00 1.069E+00 1.110E+00 1.151E+00 xu= 8.932E-01 9.312E-01 9.699E-01 1.009E+00 1.049E+00 1.090E+00 1.131E+00

33

i= 71 72 73 74 75 76 77 x= 1.193E+00 1.236E+00 1.278E+00 1.322E+00 1.365E+00 1.410E+00 1.454E+00 xu= 1.172E+00 1.214E+00 1.257E+00 1.300E+00 1.343E+00 1.387E+00 1.432E+00 i= 78 79 80 81 82 83 84 x= 1.500E+00 1.545E+00 1.591E+00 1.638E+00 1.684E+00 1.732E+00 1.779E+00 xu= 1.477E+00 1.522E+00 1.568E+00 1.614E+00 1.661E+00 1.708E+00 1.755E+00 i= 85 86 87 88 89 90 91 x= 1.827E+00 1.876E+00 1.957E+00 2.071E+00 2.186E+00 2.300E+00 2.414E+00 xu= 1.803E+00 1.851E+00 1.900E+00 2.014E+00 2.129E+00 2.243E+00 2.357E+00 i= 92 93 94 95 96 97 98 x= 2.529E+00 2.643E+00 2.757E+00 2.871E+00 2.986E+00 3.100E+00 3.214E+00 xu= 2.471E+00 2.586E+00 2.700E+00 2.814E+00 2.929E+00 3.043E+00 3.157E+00 i= 99 100 101 102 103 104 105 x= 3.329E+00 3.443E+00 3.557E+00 3.671E+00 3.786E+00 3.900E+00 4.014E+00 xu= 3.271E+00 3.386E+00 3.500E+00 3.614E+00 3.729E+00 3.843E+00 3.957E+00 i= 106 107 108 109 110 111 112 x= 4.129E+00 4.243E+00 4.357E+00 4.471E+00 4.586E+00 4.700E+00 4.814E+00 xu= 4.071E+00 4.186E+00 4.300E+00 4.414E+00 4.529E+00 4.643E+00 4.757E+00 i= 113 114 115 116 117 118 119 x= 4.929E+00 5.043E+00 5.157E+00 5.271E+00 5.386E+00 5.500E+00 5.614E+00 xu= 4.871E+00 4.986E+00 5.100E+00 5.214E+00 5.329E+00 5.443E+00 5.557E+00 i= 120 121 122 123 124 125 126 x= 5.729E+00 5.843E+00 5.957E+00 6.071E+00 6.186E+00 6.300E+00 6.414E+00 xu= 5.671E+00 5.786E+00 5.900E+00 6.014E+00 6.129E+00 6.243E+00 6.357E+00 i= 127 128 129 130 131 132 133 x= 6.529E+00 6.643E+00 6.757E+00 6.871E+00 6.986E+00 7.100E+00 7.214E+00 xu= 6.471E+00 6.586E+00 6.700E+00 6.814E+00 6.929E+00 7.043E+00 7.157E+00 i= 134 135 136 137 138 139 140 x= 7.329E+00 7.443E+00 7.557E+00 7.671E+00 7.786E+00 7.900E+00 8.014E+00 xu= 7.271E+00 7.386E+00 7.500E+00 7.614E+00 7.729E+00 7.843E+00 7.957E+00 i= 141 142 143 144 145 146 147 x= 8.129E+00 8.243E+00 8.357E+00 8.471E+00 8.586E+00 8.700E+00 8.814E+00 xu= 8.071E+00 8.186E+00 8.300E+00 8.414E+00 8.529E+00 8.643E+00 8.757E+00 i= 148 149 150 151 152 153 154 x= 8.929E+00 9.043E+00 9.157E+00 9.271E+00 9.386E+00 9.500E+00 9.614E+00 xu= 8.871E+00 8.986E+00 9.100E+00 9.214E+00 9.329E+00 9.443E+00 9.557E+00 i= 155 156 157 158 159 160 161 x= 9.729E+00 9.843E+00 9.905E+00 9.915E+00 9.925E+00 9.935E+00 9.945E+00 xu= 9.671E+00 9.786E+00 9.900E+00 9.910E+00 9.920E+00 9.930E+00 9.940E+00 i= 162 163 164 165 166 167 168 x= 9.955E+00 9.965E+00 9.975E+00 9.985E+00 9.995E+00 1.000E+01 1.001E+01 xu= 9.950E+00 9.960E+00 9.970E+00 9.980E+00 9.990E+00 1.000E+01 1.001E+01 i= 169 170 171 172 x= 1.002E+01 1.003E+01 1.005E+01 1.005E+01 xu= 1.002E+01 1.003E+01 1.004E+01 1.005E+01 --------------------------------------------------------------------------------- y-grid (cm)

34

j= 1 2 3 4 5 6 7 y= 0.000E+00 3.000E-03 9.000E-03 1.500E-02 2.100E-02 2.700E-02 3.300E-02 yv= 0.000E+00 0.000E+00 6.000E-03 1.200E-02 1.800E-02 2.400E-02 3.000E-02 j= 8 9 10 11 12 13 14 y= 3.900E-02 4.500E-02 5.100E-02 5.700E-02 6.300E-02 6.900E-02 7.500E-02 yv= 3.600E-02 4.200E-02 4.800E-02 5.400E-02 6.000E-02 6.600E-02 7.200E-02 j= 15 16 17 18 19 20 21 y= 8.100E-02 8.700E-02 9.300E-02 9.900E-02 1.050E-01 1.110E-01 1.170E-01 yv= 7.800E-02 8.400E-02 9.000E-02 9.600E-02 1.020E-01 1.080E-01 1.140E-01 j= 22 23 24 25 26 27 28 y= 1.230E-01 1.290E-01 1.350E-01 1.410E-01 1.470E-01 1.533E-01 1.600E-01 yv= 1.200E-01 1.260E-01 1.320E-01 1.380E-01 1.440E-01 1.500E-01 1.567E-01 j= 29 30 31 32 33 34 35 y= 1.667E-01 1.733E-01 1.800E-01 1.867E-01 1.933E-01 2.000E-01 2.067E-01 yv= 1.633E-01 1.700E-01 1.767E-01 1.833E-01 1.900E-01 1.967E-01 2.033E-01 j= 36 37 38 39 40 41 42 y= 2.133E-01 2.200E-01 2.267E-01 2.333E-01 2.400E-01 2.467E-01 2.533E-01 yv= 2.100E-01 2.167E-01 2.233E-01 2.300E-01 2.367E-01 2.433E-01 2.500E-01 j= 43 44 45 46 47 48 49 y= 2.600E-01 2.667E-01 2.733E-01 2.800E-01 2.867E-01 2.933E-01 3.000E-01 yv= 2.567E-01 2.633E-01 2.700E-01 2.767E-01 2.833E-01 2.900E-01 2.967E-01 j= 50 51 52 53 54 55 56 y= 3.067E-01 3.133E-01 3.200E-01 3.267E-01 3.333E-01 3.400E-01 3.467E-01 yv= 3.033E-01 3.100E-01 3.167E-01 3.233E-01 3.300E-01 3.367E-01 3.433E-01 j= 57 58 59 60 61 62 63 y= 3.624E-01 3.931E-01 4.325E-01 4.773E-01 5.262E-01 5.785E-01 6.338E-01 yv= 3.500E-01 3.749E-01 4.112E-01 4.537E-01 5.008E-01 5.515E-01 6.055E-01 j= 64 65 66 67 68 69 70 y= 6.917E-01 7.520E-01 8.145E-01 8.790E-01 9.454E-01 1.013E+00 1.083E+00 yv= 6.621E-01 7.213E-01 7.827E-01 8.463E-01 9.117E-01 9.790E-01 1.048E+00 j= 71 72 73 74 75 76 77 y= 1.155E+00 1.227E+00 1.302E+00 1.377E+00 1.454E+00 1.533E+00 1.612E+00 yv= 1.119E+00 1.191E+00 1.264E+00 1.339E+00 1.416E+00 1.493E+00 1.572E+00 j= 78 79 80 81 82 83 84 y= 1.693E+00 1.774E+00 1.857E+00 1.941E+00 2.026E+00 2.112E+00 2.199E+00 yv= 1.652E+00 1.733E+00 1.815E+00 1.899E+00 1.983E+00 2.069E+00 2.155E+00 j= 85 86 87 88 89 90 91 y= 2.287E+00 2.375E+00 2.425E+00 2.435E+00 2.445E+00 2.455E+00 2.465E+00 yv= 2.242E+00 2.331E+00 2.420E+00 2.430E+00 2.440E+00 2.450E+00 2.460E+00 j= 92 93 94 95 96 97 98 y= 2.475E+00 2.485E+00 2.495E+00 2.505E+00 2.515E+00 2.525E+00 2.535E+00 yv= 2.470E+00 2.480E+00 2.490E+00 2.500E+00 2.510E+00 2.520E+00 2.530E+00 j= 99 100 101 102 y= 2.545E+00 2.555E+00 2.565E+00 2.570E+00 yv= 2.540E+00 2.550E+00 2.560E+00 2.570E+00 --------------------------------------------------------------------------------- z-grid (cm)

35

k= 1 2 3 4 5 6 7 z= 0.000E+00 1.000E-02 3.000E-02 5.000E-02 7.000E-02 9.000E-02 1.100E-01 zw= 0.000E+00 0.000E+00 2.000E-02 4.000E-02 6.000E-02 8.000E-02 1.000E-01 k= 8 9 10 11 12 13 14 z= 1.300E-01 1.500E-01 1.700E-01 1.900E-01 2.100E-01 2.300E-01 2.500E-01 zw= 1.200E-01 1.400E-01 1.600E-01 1.800E-01 2.000E-01 2.200E-01 2.400E-01 k= 15 16 17 18 19 20 21 z= 2.700E-01 2.900E-01 3.100E-01 3.300E-01 3.500E-01 3.700E-01 3.900E-01 zw= 2.600E-01 2.800E-01 3.000E-01 3.200E-01 3.400E-01 3.600E-01 3.800E-01 k= 22 23 24 25 26 27 28 z= 4.100E-01 4.300E-01 4.500E-01 4.700E-01 4.900E-01 5.100E-01 5.300E-01 zw= 4.000E-01 4.200E-01 4.400E-01 4.600E-01 4.800E-01 5.000E-01 5.200E-01 k= 29 30 31 32 33 34 35 z= 5.500E-01 5.700E-01 5.900E-01 6.070E-01 6.209E-01 6.347E-01 6.484E-01 zw= 5.400E-01 5.600E-01 5.800E-01 6.000E-01 6.140E-01 6.278E-01 6.416E-01 k= 36 37 38 39 40 41 42 z= 6.620E-01 6.755E-01 6.889E-01 7.022E-01 7.153E-01 7.284E-01 7.413E-01 zw= 6.552E-01 6.688E-01 6.822E-01 6.956E-01 7.088E-01 7.219E-01 7.348E-01 k= 43 44 45 46 47 48 49 z= 7.540E-01 7.667E-01 7.792E-01 7.915E-01 8.037E-01 8.157E-01 8.276E-01 zw= 7.477E-01 7.604E-01 7.730E-01 7.854E-01 7.977E-01 8.098E-01 8.217E-01 k= 50 51 52 53 54 55 56 z= 8.392E-01 8.507E-01 8.619E-01 8.729E-01 8.837E-01 8.941E-01 9.043E-01 zw= 8.334E-01 8.450E-01 8.563E-01 8.675E-01 8.783E-01 8.890E-01 8.993E-01 k= 57 58 59 60 61 62 z= 9.140E-01 9.234E-01 9.322E-01 9.403E-01 9.470E-01 9.500E-01 zw= 9.092E-01 9.188E-01 9.279E-01 9.364E-01 9.441E-01 9.500E-01 --------------------------------------------------------------------------------- Date: 2007- 7- 6 time: 17 :32 :51 iter time/iter res_enth res_mass res_u res_v res_w 100 0.380 1.96E-04 8.36E-03 1.30E-01 3.98E-02 4.89E-02 Tmax umax vmax wmax length depth half-width 3093. 93.0 60.1 22.7 0.236 0.312 0.103 north south top toploss bottom west east hout hin ratio -112.9 0.0 -99.8 0.0 -30.7 1591.9 -1592.2 -143.8 123.3 1.17 muAv muM kAv kM 1.20 10.22 0.31 2.21 iter time/iter res_enth res_mass res_u res_v res_w 500 0.382 1.40E-05 1.54E-03 2.89E-04 3.40E-04 1.86E-04 Tmax umax vmax wmax length depth half-width 3093. 229.0 107.5 19.2 0.312 0.312 0.097 north south top toploss bottom west east hout hin ratio -13.1 0.0 2.3 0.0 1.4 1592.0 -1679.9 -99.6 100.7 0.99 muAv muM kAv kM 0.59 4.66 0.18 1.04 iter time/iter res_enth res_mass res_u res_v res_w 1000 0.383 5.84E-06 6.91E-04 6.84E-05 1.20E-04 2.67E-05 Tmax umax vmax wmax length depth half-width 3093. 229.1 107.3 19.2 0.306 0.312 0.097 north south top toploss bottom west east hout hin ratio

36

-8.3 0.0 -2.3 0.0 -0.1 1592.0 -1679.3 -95.7 100.8 0.95 muAv muM kAv kM 0.59 4.66 0.18 1.04 iter time/iter res_enth res_mass res_u res_v res_w 1500 0.383 3.31E-06 6.88E-04 6.61E-05 1.19E-04 2.54E-05 Tmax umax vmax wmax length depth half-width 3093. 229.1 107.3 19.2 0.306 0.312 0.097 north south top toploss bottom west east hout hin ratio -3.7 0.0 -1.3 0.0 0.0 1592.0 -1687.7 -99.4 100.8 0.99 muAv muM kAv kM 0.59 4.66 0.18 1.04 iter time/iter res_enth res_mass res_u res_v res_w 2000 0.383 1.50E-06 6.88E-04 6.61E-05 1.19E-04 2.54E-05 Tmax umax vmax wmax length depth half-width 3093. 229.1 107.3 19.2 0.306 0.312 0.097 north south top toploss bottom west east hout hin ratio -1.2 0.0 -0.5 0.0 0.0 1592.0 -1693.5 -102.7 100.8 1.02 muAv muM kAv kM 0.59 4.66 0.18 1.04 Some important calculated parameters at the end of heating cycle Length of the pool (cm) 3.0620E-01 Depth of the pool (cm) 3.1191E-01 Half-width of the pool (cm) 9.6781E-02 Peak temperature (K) 3.0925E+03 x1073(cm), x773(cm), t8-5(s) 0.885 1.367 0.252 Maximum u-velocity (cm/s) 2.2911E+02 Maximum v-velocity (cm/s) 1.0731E+02 Maximum w-velocity (cm/s) 1.9237E+01 Rate of heat input (cal/s) 1.0076E+02 Rate of heat output (cal/s) -1.0269E+02 Ratio of heat input to heat output 1.0192E+00 Date: 2007- 7- 6 time: 17:45:54 Total time used: 0 hr 13 m 3 s

Fig. 9 Comparison of experimental and calculated weld cross-sectional geometry. The solid line represents the solidus and the dashed line represents the keyhole marked by boiling point temperature. The dotted line shows the experimentally observed weld pool boundary.

Y, mm

Z,m

m

-2 -1 0 1 26

7

8

9

37

7.0

8.0

9.0

Z,mm

2.0 4.0 6.0 8.0

X , mm

1.02.0

3.0Y, mm

250 mm/s

Fig. 10 Fluid flow in the weld pool. The narrow and deep region with no velocity vectors is the keyhole. Case 4: Ti-6Al-4V welds made with 1400 W input power at 16.9 mm/s welding speed. Output file: ----------------------------------------------------------- Laser welding ----------------------------------------------------------- Process parameters ----------------------------------------------------------- Laser power (watt) 1.4000E+03 Laser absorption coefficient 2.8000E-01 Plasma attenuation coefficient 1.0000E+00 Laser beam radius (cm) 2.0000E-04 Defocus (cm) 0.0000E+00 Laser power distribution factor 2.0000E+00 Arc current (amp) 0.0000E+00 Arc voltage (volt) 0.0000E+00 Arc efficiency 7.0000E-01 Arc radius (cm) 2.7000E-01 Arc power distribution factor 5.0000E-01 EMF calculation needed (1=yes; 0=file exists) 0 Starting location of the laser beam 3.0000E-01 Starting location of the arc 7.0000E-01 Welding velocity (cm/sec) 1.6900E+00 Free surface calculation needed (1 = yes) 0 ----------------------------------------------------------- Material properties ----------------------------------------------------------- Density of liquid (gm/cm3) 4.0000E+00 Density at boiling point 3.7800E+00 Molecular viscosity of liquid (gm/cm-sec) 5.0000E-02 Solidus temperature (K) 1.8780E+03 Liquidus temperature (K) 1.9280E+03 Boiling point (K) 3.3110E+03 Enthalpy of solid at melting point (cal/gm) 2.6840E+02 Enthalpy of liquid at melting point (cal/gm) 3.5560E+02 Specific heat of solid (cal/gm-K) 1.4300E-01 Specific heat of liquid (cal/gm-K) 1.6700E-01 Thermal conductivity of solid (cal/cm-sec-K) 5.0000E-02 Molecular thermal cond. of liq (cal/cm-sec-K) 7.0000E-02

38

Coefficient of thermal expansion (1/K) 8.0000E-06 Emissivity of the material 0.0000E+00 d(gamma)/dT of pure material (dynes/cm-K) -2.6000E-01 Concentration of surface active species (wt%) 0.0000E+00 Surface excess at saturation (mole/cm2) 1.3000E-09 Enthalpy of segregation (cal/mole) -1.6600E+05 Entropy factor 3.1800E-03 ----------------------------------------------------------- Numerical scheme parameters ----------------------------------------------------------- Maximum number of iterations 2000 Underrelaxation for u-velocity 6.0000E-01 Underrelaxation for v-velocity 6.0000E-01 Underrelaxation for w-velocity 6.0000E-01 Underrelaxation for pressure 8.0000E-01 Underrelaxation for enthalpy 1.0000E+00 ----------------------------------------------------------- Boundary conditions ----------------------------------------------------------- Heat transfer coeff at west face (cal/cm2-s-K) 1.0000E+02 Heat transfer coeff at east face (cal/cm2-s-K) 1.0000E+02 Heat transfer coeff at north face (cal/cm2-s-K) 1.0000E+02 Heat transfer coeff at bottom face (cal/cm2-s-K) 1.0000E-02 Heat transfer coeff at top face (cal/cm2-s-K) 0.0000E+00 Temperature at west face (K) 2.9800E+02 Temperature at east face (K) 2.9800E+02 Temperature at north face (K) 2.9800E+02 Temperature at bottom face (K) 2.9800E+02 Preheat temperature (K) 2.9800E+02 Ambient temperature (K) 2.9800E+02 ----------------------------------------------------------- Geometrical parameters ----------------------------------------------------------- x direction number of zones 5 zone( 1) length (cm) 0.2000E+00 zone( 1) number of control volumes (CV) 10 zone( 1) exponent to locate CV interfaces -.1300E+01 zone( 2) length (cm) 0.2000E+00 zone( 2) number of control volumes (CV) 35 zone( 2) exponent to locate CV interfaces 0.1000E+01 zone( 3) length (cm) 0.1000E+01 zone( 3) number of control volumes (CV) 40 zone( 3) exponent to locate CV interfaces 0.1300E+01 zone( 4) length (cm) 0.8000E+01 zone( 4) number of control volumes (CV) 70 zone( 4) exponent to locate CV interfaces 0.1000E+01 zone( 5) length (cm) 0.1000E+00 zone( 5) number of control volumes (CV) 12 zone( 5) exponent to locate CV interfaces 0.1000E+01 y direction number of zones 4 zone( 1) length (cm) 0.1600E+00 zone( 1) number of control volumes (CV) 25 zone( 1) exponent to locate CV interfaces 0.1000E+01 zone( 2) length (cm) 0.2000E+00 zone( 2) number of control volumes (CV) 30 zone( 2) exponent to locate CV interfaces 0.1000E+01 zone( 3) length (cm) 0.2070E+01 zone( 3) number of control volumes (CV) 30 zone( 3) exponent to locate CV interfaces 0.1300E+01 zone( 4) length (cm) 0.1000E+00 zone( 4) number of control volumes (CV) 12

39

zone( 4) exponent to locate CV interfaces 0.1000E+01 z direction number of zones 3 zone( 1) length (cm) 0.4700E+00 zone( 1) number of control volumes (CV) 22 zone( 1) exponent to locate CV interfaces 0.1000E+01 zone( 2) length (cm) 0.5000E+00 zone( 2) number of control volumes (CV) 30 zone( 2) exponent to locate CV interfaces -.1200E+01 zone( 3) length (cm) 0.3000E+00 zone( 3) number of control volumes (CV) 30 zone( 3) exponent to locate CV interfaces 0.1000E+01 Number of grid points in x-direction (length) 169 Number of grid points in y-direction (width) 99 Number of grid points in z-direction (depth) 84 Length of the specimen (cm) 9.5000E+00 Width of the specimen (cm) 2.5300E+00 Height of the specimen (cm) 1.2700E+00 --------------------------------------------------------------------------------- x-grid (cm) i= 1 2 3 4 5 6 7 x= 0.000E+00 1.280E-02 3.798E-02 6.228E-02 8.563E-02 1.079E-01 1.290E-01 xu= 0.000E+00 0.000E+00 2.560E-02 5.036E-02 7.421E-02 9.705E-02 1.188E-01 i= 8 9 10 11 12 13 14 x= 1.487E-01 1.668E-01 1.826E-01 1.950E-01 2.029E-01 2.086E-01 2.143E-01 xu= 1.392E-01 1.582E-01 1.753E-01 1.900E-01 2.000E-01 2.057E-01 2.114E-01 i= 15 16 17 18 19 20 21 x= 2.200E-01 2.257E-01 2.314E-01 2.371E-01 2.429E-01 2.486E-01 2.543E-01 xu= 2.171E-01 2.229E-01 2.286E-01 2.343E-01 2.400E-01 2.457E-01 2.514E-01 i= 22 23 24 25 26 27 28 x= 2.600E-01 2.657E-01 2.714E-01 2.771E-01 2.829E-01 2.886E-01 2.943E-01 xu= 2.571E-01 2.629E-01 2.686E-01 2.743E-01 2.800E-01 2.857E-01 2.914E-01 i= 29 30 31 32 33 34 35 x= 3.000E-01 3.057E-01 3.114E-01 3.171E-01 3.229E-01 3.286E-01 3.343E-01 xu= 2.971E-01 3.029E-01 3.086E-01 3.143E-01 3.200E-01 3.257E-01 3.314E-01 i= 36 37 38 39 40 41 42 x= 3.400E-01 3.457E-01 3.514E-01 3.571E-01 3.629E-01 3.686E-01 3.743E-01 xu= 3.371E-01 3.429E-01 3.486E-01 3.543E-01 3.600E-01 3.657E-01 3.714E-01 i= 43 44 45 46 47 48 49 x= 3.800E-01 3.857E-01 3.914E-01 3.971E-01 4.041E-01 4.143E-01 4.274E-01 xu= 3.771E-01 3.829E-01 3.886E-01 3.943E-01 4.000E-01 4.083E-01 4.204E-01 i= 50 51 52 53 54 55 56 x= 4.423E-01 4.586E-01 4.759E-01 4.943E-01 5.136E-01 5.336E-01 5.544E-01 xu= 4.345E-01 4.501E-01 4.670E-01 4.849E-01 5.037E-01 5.234E-01 5.438E-01 i= 57 58 59 60 61 62 63 x= 5.758E-01 5.979E-01 6.205E-01 6.437E-01 6.674E-01 6.916E-01 7.163E-01 xu= 5.649E-01 5.867E-01 6.091E-01 6.320E-01 6.554E-01 6.794E-01 7.039E-01 i= 64 65 66 67 68 69 70 x= 7.415E-01 7.670E-01 7.930E-01 8.194E-01 8.462E-01 8.734E-01 9.009E-01 xu= 7.288E-01 7.541E-01 7.799E-01 8.061E-01 8.327E-01 8.597E-01 8.870E-01

40

i= 71 72 73 74 75 76 77 x= 9.288E-01 9.570E-01 9.856E-01 1.014E+00 1.044E+00 1.073E+00 1.103E+00 xu= 9.148E-01 9.428E-01 9.712E-01 9.999E-01 1.029E+00 1.058E+00 1.088E+00 i= 78 79 80 81 82 83 84 x= 1.133E+00 1.163E+00 1.194E+00 1.225E+00 1.256E+00 1.288E+00 1.320E+00 xu= 1.118E+00 1.148E+00 1.179E+00 1.210E+00 1.241E+00 1.272E+00 1.304E+00 i= 85 86 87 88 89 90 91 x= 1.352E+00 1.384E+00 1.457E+00 1.571E+00 1.686E+00 1.800E+00 1.914E+00 xu= 1.335E+00 1.368E+00 1.400E+00 1.514E+00 1.629E+00 1.743E+00 1.857E+00 i= 92 93 94 95 96 97 98 x= 2.029E+00 2.143E+00 2.257E+00 2.371E+00 2.486E+00 2.600E+00 2.714E+00 xu= 1.971E+00 2.086E+00 2.200E+00 2.314E+00 2.429E+00 2.543E+00 2.657E+00 i= 99 100 101 102 103 104 105 x= 2.829E+00 2.943E+00 3.057E+00 3.171E+00 3.286E+00 3.400E+00 3.514E+00 xu= 2.771E+00 2.886E+00 3.000E+00 3.114E+00 3.229E+00 3.343E+00 3.457E+00 i= 106 107 108 109 110 111 112 x= 3.629E+00 3.743E+00 3.857E+00 3.971E+00 4.086E+00 4.200E+00 4.314E+00 xu= 3.571E+00 3.686E+00 3.800E+00 3.914E+00 4.029E+00 4.143E+00 4.257E+00 i= 113 114 115 116 117 118 119 x= 4.429E+00 4.543E+00 4.657E+00 4.771E+00 4.886E+00 5.000E+00 5.114E+00 xu= 4.371E+00 4.486E+00 4.600E+00 4.714E+00 4.829E+00 4.943E+00 5.057E+00 i= 120 121 122 123 124 125 126 x= 5.229E+00 5.343E+00 5.457E+00 5.571E+00 5.686E+00 5.800E+00 5.914E+00 xu= 5.171E+00 5.286E+00 5.400E+00 5.514E+00 5.629E+00 5.743E+00 5.857E+00 i= 127 128 129 130 131 132 133 x= 6.029E+00 6.143E+00 6.257E+00 6.371E+00 6.486E+00 6.600E+00 6.714E+00 xu= 5.971E+00 6.086E+00 6.200E+00 6.314E+00 6.429E+00 6.543E+00 6.657E+00 i= 134 135 136 137 138 139 140 x= 6.829E+00 6.943E+00 7.057E+00 7.171E+00 7.286E+00 7.400E+00 7.514E+00 xu= 6.771E+00 6.886E+00 7.000E+00 7.114E+00 7.229E+00 7.343E+00 7.457E+00 i= 141 142 143 144 145 146 147 x= 7.629E+00 7.743E+00 7.857E+00 7.971E+00 8.086E+00 8.200E+00 8.314E+00 xu= 7.571E+00 7.686E+00 7.800E+00 7.914E+00 8.029E+00 8.143E+00 8.257E+00 i= 148 149 150 151 152 153 154 x= 8.429E+00 8.543E+00 8.657E+00 8.771E+00 8.886E+00 9.000E+00 9.114E+00 xu= 8.371E+00 8.486E+00 8.600E+00 8.714E+00 8.829E+00 8.943E+00 9.057E+00 i= 155 156 157 158 159 160 161 x= 9.229E+00 9.343E+00 9.404E+00 9.412E+00 9.421E+00 9.429E+00 9.438E+00 xu= 9.171E+00 9.286E+00 9.400E+00 9.408E+00 9.417E+00 9.425E+00 9.433E+00 i= 162 163 164 165 166 167 168 x= 9.446E+00 9.454E+00 9.462E+00 9.471E+00 9.479E+00 9.487E+00 9.496E+00 xu= 9.442E+00 9.450E+00 9.458E+00 9.467E+00 9.475E+00 9.483E+00 9.492E+00 i= 169 x= 9.500E+00 xu= 9.500E+00 --------------------------------------------------------------------------------- y-grid (cm)

41

j= 1 2 3 4 5 6 7 y= 0.000E+00 3.200E-03 9.600E-03 1.600E-02 2.240E-02 2.880E-02 3.520E-02 yv= 0.000E+00 0.000E+00 6.400E-03 1.280E-02 1.920E-02 2.560E-02 3.200E-02 j= 8 9 10 11 12 13 14 y= 4.160E-02 4.800E-02 5.440E-02 6.080E-02 6.720E-02 7.360E-02 8.000E-02 yv= 3.840E-02 4.480E-02 5.120E-02 5.760E-02 6.400E-02 7.040E-02 7.680E-02 j= 15 16 17 18 19 20 21 y= 8.640E-02 9.280E-02 9.920E-02 1.056E-01 1.120E-01 1.184E-01 1.248E-01 yv= 8.320E-02 8.960E-02 9.600E-02 1.024E-01 1.088E-01 1.152E-01 1.216E-01 j= 22 23 24 25 26 27 28 y= 1.312E-01 1.376E-01 1.440E-01 1.504E-01 1.568E-01 1.633E-01 1.700E-01 yv= 1.280E-01 1.344E-01 1.408E-01 1.472E-01 1.536E-01 1.600E-01 1.667E-01 j= 29 30 31 32 33 34 35 y= 1.767E-01 1.833E-01 1.900E-01 1.967E-01 2.033E-01 2.100E-01 2.167E-01 yv= 1.733E-01 1.800E-01 1.867E-01 1.933E-01 2.000E-01 2.067E-01 2.133E-01 j= 36 37 38 39 40 41 42 y= 2.233E-01 2.300E-01 2.367E-01 2.433E-01 2.500E-01 2.567E-01 2.633E-01 yv= 2.200E-01 2.267E-01 2.333E-01 2.400E-01 2.467E-01 2.533E-01 2.600E-01 j= 43 44 45 46 47 48 49 y= 2.700E-01 2.767E-01 2.833E-01 2.900E-01 2.967E-01 3.033E-01 3.100E-01 yv= 2.667E-01 2.733E-01 2.800E-01 2.867E-01 2.933E-01 3.000E-01 3.067E-01 j= 50 51 52 53 54 55 56 y= 3.167E-01 3.233E-01 3.300E-01 3.367E-01 3.433E-01 3.500E-01 3.567E-01 yv= 3.133E-01 3.200E-01 3.267E-01 3.333E-01 3.400E-01 3.467E-01 3.533E-01 j= 57 58 59 60 61 62 63 y= 3.724E-01 4.031E-01 4.425E-01 4.873E-01 5.362E-01 5.885E-01 6.438E-01 yv= 3.600E-01 3.849E-01 4.212E-01 4.637E-01 5.108E-01 5.615E-01 6.155E-01 j= 64 65 66 67 68 69 70 y= 7.017E-01 7.620E-01 8.245E-01 8.890E-01 9.554E-01 1.023E+00 1.093E+00 yv= 6.721E-01 7.313E-01 7.927E-01 8.563E-01 9.217E-01 9.890E-01 1.058E+00 j= 71 72 73 74 75 76 77 y= 1.165E+00 1.237E+00 1.312E+00 1.387E+00 1.464E+00 1.543E+00 1.622E+00 yv= 1.129E+00 1.201E+00 1.274E+00 1.349E+00 1.426E+00 1.503E+00 1.582E+00 j= 78 79 80 81 82 83 84 y= 1.703E+00 1.784E+00 1.867E+00 1.951E+00 2.036E+00 2.122E+00 2.209E+00 yv= 1.662E+00 1.743E+00 1.825E+00 1.909E+00 1.993E+00 2.079E+00 2.165E+00 j= 85 86 87 88 89 90 91 y= 2.297E+00 2.385E+00 2.434E+00 2.442E+00 2.451E+00 2.459E+00 2.467E+00 yv= 2.252E+00 2.341E+00 2.430E+00 2.438E+00 2.447E+00 2.455E+00 2.463E+00 j= 92 93 94 95 96 97 98 y= 2.476E+00 2.484E+00 2.492E+00 2.501E+00 2.509E+00 2.517E+00 2.526E+00 yv= 2.472E+00 2.480E+00 2.488E+00 2.497E+00 2.505E+00 2.513E+00 2.522E+00 j= 99 y= 2.530E+00 yv= 2.530E+00 --------------------------------------------------------------------------------- z-grid (cm)

42

k= 1 2 3 4 5 6 7 z= 0.000E+00 1.068E-02 3.205E-02 5.341E-02 7.477E-02 9.614E-02 1.175E-01 zw= 0.000E+00 0.000E+00 2.136E-02 4.273E-02 6.409E-02 8.545E-02 1.068E-01 k= 8 9 10 11 12 13 14 z= 1.389E-01 1.602E-01 1.816E-01 2.030E-01 2.243E-01 2.457E-01 2.670E-01 zw= 1.282E-01 1.495E-01 1.709E-01 1.923E-01 2.136E-01 2.350E-01 2.564E-01 k= 15 16 17 18 19 20 21 z= 2.884E-01 3.098E-01 3.311E-01 3.525E-01 3.739E-01 3.952E-01 4.166E-01 zw= 2.777E-01 2.991E-01 3.205E-01 3.418E-01 3.632E-01 3.845E-01 4.059E-01 k= 22 23 24 25 26 27 28 z= 4.380E-01 4.593E-01 4.800E-01 4.998E-01 5.196E-01 5.391E-01 5.586E-01 zw= 4.273E-01 4.486E-01 4.700E-01 4.899E-01 5.097E-01 5.294E-01 5.489E-01 k= 29 30 31 32 33 34 35 z= 5.779E-01 5.970E-01 6.159E-01 6.347E-01 6.534E-01 6.718E-01 6.901E-01 zw= 5.683E-01 5.875E-01 6.065E-01 6.254E-01 6.441E-01 6.626E-01 6.810E-01 k= 36 37 38 39 40 41 42 z= 7.081E-01 7.260E-01 7.436E-01 7.610E-01 7.782E-01 7.951E-01 8.117E-01 zw= 6.991E-01 7.171E-01 7.348E-01 7.524E-01 7.697E-01 7.867E-01 8.035E-01 k= 43 44 45 46 47 48 49 z= 8.281E-01 8.442E-01 8.599E-01 8.752E-01 8.902E-01 9.046E-01 9.186E-01 zw= 8.200E-01 8.362E-01 8.521E-01 8.676E-01 8.828E-01 8.975E-01 9.118E-01 k= 50 51 52 53 54 55 56 z= 9.319E-01 9.445E-01 9.561E-01 9.658E-01 9.750E-01 9.850E-01 9.950E-01 zw= 9.254E-01 9.385E-01 9.506E-01 9.616E-01 9.700E-01 9.800E-01 9.900E-01 k= 57 58 59 60 61 62 63 z= 1.005E+00 1.015E+00 1.025E+00 1.035E+00 1.045E+00 1.055E+00 1.065E+00 zw= 1.000E+00 1.010E+00 1.020E+00 1.030E+00 1.040E+00 1.050E+00 1.060E+00 k= 64 65 66 67 68 69 70 z= 1.075E+00 1.085E+00 1.095E+00 1.105E+00 1.115E+00 1.125E+00 1.135E+00 zw= 1.070E+00 1.080E+00 1.090E+00 1.100E+00 1.110E+00 1.120E+00 1.130E+00 k= 71 72 73 74 75 76 77 z= 1.145E+00 1.155E+00 1.165E+00 1.175E+00 1.185E+00 1.195E+00 1.205E+00 zw= 1.140E+00 1.150E+00 1.160E+00 1.170E+00 1.180E+00 1.190E+00 1.200E+00 k= 78 79 80 81 82 83 84 z= 1.215E+00 1.225E+00 1.235E+00 1.245E+00 1.255E+00 1.265E+00 1.270E+00 zw= 1.210E+00 1.220E+00 1.230E+00 1.240E+00 1.250E+00 1.260E+00 1.270E+00 --------------------------------------------------------------------------------- Date: 2007- 7- 6 time: 17 :18 :38 iter time/iter res_enth res_mass res_u res_v res_w 100 0.440 2.21E-04 4.60E-03 3.05E-02 1.67E-02 2.85E-02 Tmax umax vmax wmax length depth half-width 3311. 74.3 61.6 14.2 0.240 0.306 0.113 north south top toploss bottom west east hout hin ratio -206.8 0.0 -75.1 0.0 -38.9 920.4 -922.4 -247.8 87.5 2.83 muAv muM kAv kM 0.39 3.23 0.13 0.66 iter time/iter res_enth res_mass res_u res_v res_w 500 0.472 1.68E-05 6.08E-06 2.50E-04 1.45E-04 2.05E-04

43

Tmax umax vmax wmax length depth half-width 3311. 53.0 63.1 13.2 0.394 0.305 0.111 north south top toploss bottom west east hout hin ratio -7.6 0.0 8.0 0.0 5.1 921.8 -1021.8 -102.6 84.4 1.21 muAv muM kAv kM 0.38 3.11 0.13 0.64 iter time/iter res_enth res_mass res_u res_v res_w 1000 0.475 4.58E-06 3.97E-07 1.25E-04 9.39E-05 5.98E-05 Tmax umax vmax wmax length depth half-width 3311. 53.0 63.1 13.2 0.390 0.305 0.111 north south top toploss bottom west east hout hin ratio -11.5 0.0 -0.2 0.0 0.2 921.8 -994.7 -84.3 84.4 1.00 muAv muM kAv kM 0.38 3.11 0.13 0.64 iter time/iter res_enth res_mass res_u res_v res_w 1500 0.479 2.94E-06 3.89E-07 1.24E-04 9.39E-05 5.97E-05 Tmax umax vmax wmax length depth half-width 3311. 53.0 63.1 13.2 0.390 0.305 0.111 north south top toploss bottom west east hout hin ratio -9.0 0.0 -1.0 0.0 0.1 921.8 -996.2 -83.4 84.4 0.99 muAv muM kAv kM 0.38 3.11 0.13 0.64 iter time/iter res_enth res_mass res_u res_v res_w 2000 0.481 1.89E-06 3.95E-07 1.25E-04 9.40E-05 5.99E-05 Tmax umax vmax wmax length depth half-width 3311. 53.0 63.1 13.2 0.390 0.305 0.111 north south top toploss bottom west east hout hin ratio -6.0 0.0 -0.7 0.0 0.1 921.8 -1000.5 -84.7 84.4 1.00 muAv muM kAv kM 0.38 3.11 0.13 0.64 Some important calculated parameters at the end of heating cycle Length of the pool (cm) 3.8983E-01 Depth of the pool (cm) 3.0503E-01 Half-width of the pool (cm) 1.1110E-01 Peak temperature (K) 3.3110E+03 x1073(cm), x773(cm), t8-5(s) 0.990 1.444 0.269 Maximum u-velocity (cm/s) 5.2971E+01 Maximum v-velocity (cm/s) 6.3082E+01 Maximum w-velocity (cm/s) 1.3198E+01 Rate of heat input (cal/s) 8.4443E+01 Rate of heat output (cal/s) -8.4702E+01 Ratio of heat input to heat output 1.0031E+00 Date: 2007- 7- 6 time: 17:35:19 Total time used: 0 hr 16 m 41 s

44

Fig. 11 Comparison of experimental and calculated weld cross-sectional geometry. The solid line represents the solidus and the dashed line represents the keyhole marked by boiling point temperature. The dotted line shows the experimentally observed weld pool boundary.

10.0

11.0

12.0

Z,mm

0.0 2.0 4.0 6.0 8.0

X , mm

1.02.0

3.0Y, mm

250 mm/s

Fig. 12 Fluid flow in the weld pool. The narrow and deep region with no velocity vectors is the keyhole. 9. Plotting results The fluid velocities and temperature fields can be plotted using the data file tecout and commercially available plotting software Tecplot®. The weld pool cross-sections can be plotted using the data file geometry.dat while the keyhole and beam profiles can be plotted using the file geom.dat. 10. Concluding Remarks The calculated results depend on the accuracy of material properties and the beam characteristics specified by the user. Finer grids should be used in the region near the heat source or else errors may be encountered in the calculation of weld characteristics. The workpiece should be large enough to reduce edge effects, especially when the thermal

Y, mm

Z,m

m

-1 0 19

10

11

12

45

cycles in the workpiece are of interest. The under-relaxation parameters can also be adjusted to improved convergence. In case of issues that are not resolved by the above suggestions, the user may contact the following people for help: Mr. Rohit Rai rohit@psu.edu Dr. T. DebRoy debroy@psu.edu 11. References Results obtained using this program, and their discussion, can be obtained from the following papers which are available at http://www.matse.psu.edu/modeling/.

1. R. Rai, J.W. Elmer, T.A. Palmer, T. DebRoy, Heat transfer and fluid flow during keyhole mode welding of tantalum, Ti-6Al-4V, 304L stainless steel, and vanadium, Journal of Physics D: Applied Physics, 40, 5753-5766, 2007

2. R. Rai, G.G. Roy, T. DebRoy, A computationally efficient model of convective heat transfer and solidification characteristics during keyhole mode laser welding, Journal of Applied Physics, 101, 054909, 2007.

3. R. Rai, T. DebRoy, Tailoring weld geometry during keyhole mode laser welding using a genetic algorithm and a heat transfer model, Journal of Physics D: Applied Physics, 39, 1257-1266, 2006.