First- and second-law efficiencies for laser drilling of stainless steel

Energy Vol. 21, No. 3, pp. 197-203, 1995

Pergamon

o:~,o.s442(9s)ooo99-2

Copyright © 1996 Elsevier Science Lid Printed in Great Britain. All rights reserved 0360-5442/96 $15.00+0.00

FIRST- AND SECOND-LAW EFFICIENCIES FOR LASER DRILLING OF STAINLESS STEEL B. S. YILBAS,t M. SAMI, A. K. KAR, and A. Z. SAHIN Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

(Received 24 January 1995; received for publication 29 September 1995)

Abstract--The first- and second-law efficiencies for laser drilling have been investigated. These efficiencies improve at a given ambient gas pressure for a 1 mm thick workpiece and decrease with workpiece thickness.

1. INTRODUCTION

Studies of the laser-induced heating process have been published, L2 as have studies of heating mechanisms in laser drilling. 3.4 Laser drilling may be controlled by varying the electron number density in the plasma above the surface.l'5 Recently, Yilbas 6 showed that the use of oxygen as ambient gas during drilling increased the mean hole diameters of steel above those in air. The present study deals with the first- and second-law efficiencies of laser-hole drilling at sub-atmospheric pressures of ambient gases. The laser heat-transfer mechanism is modeled with phase-change processes, and first- and second-law efficiencies are computed for holes drilled in stainless steel workpieces of 0.5 and 1 mm thicknesses using theoretical predictions and experimental results obtained from a previous study. 6 2. MODELING OF THE LASER PROCESS

2.1. G o v e r n i n g e q u a t i o n s A heat-transfer model describing the evaporation process with laser heating has been investigated. 7 The liquid surface layer formed during a laser pulse moves into the metal at a rate determined by the quantity of vapor expelled. As the temperature of the liquid molecules is increased, the energy needed for evaporation decreases. The latent heat of vaporization of the liquid decreases with temperature until the critical temperature is reached where it remains at zero; it is taken to be the following elliptic function of temperature: 8 L(T) = Lo [1 - (TslT¢)Z] ~/2 ,

(1)

where Lo = latent heat of vaporization at the absolute zero. The rate of change with temperature is ~ (dL/dT) = ( L I T ) + (Cp2 - Cpl) - [ L l ( v 2 -

vl)][(Ov2/OT)p - (Ovl/0T)p] ,

(2)

where Cpl and Cp2 are specific heats at constant pressure, vz and v2 specific volumes, and the subscripts 1, 2 denote liquid and vapor states, respectively. It may be shown that little inaccuracy is involved in using the latent heat at the absolute zero. Both the specific volume of gas and its rate of change with T are much greater than those of condensed liquid. Therefore, (dL/dT) = ( L / T ) + (Cp2 - Cpl) - ( L I v 2 ) ( a v 2 / a T ) v • Applying the perfect gas law and differentiating with respect to T,

t T o whom all correspondence should be addressed. 197

(3)

198

B.S. Yilbas et al

(4)

(dL/dT) = (L/T) + (Cp2 - Cp,) - L / T = ACp.

Since Cp is very small up to room temperature, 9 little error will result in taking Lo as the latent heat at room temperature. According to Maxwell's law, the molecular velocity distribution is ~°

(5)

f(Vz)dV, = [m/(2 qrkBT)] 1:2 e x p [ - ( m ~ ) / ( 2 'rrkBT)] dVz,

where V~ -- velocity in the direction normal to the surface, T = temperature of the solid, liquid or gas, and m = mass of the atom. Also,

f ( Vz)dV~ -

number o f atoms with velocity Vz to V~+azper unit volume number o f atoms per unit volume

(6)

Only molecules with velocity greater than Vml, will escape. This minimum velocity in the z direction is (mVZm~n)/2= L(T).

(7)

If n is the number of atoms per unit volume, then the number of atoms with velocities Vz to Vz + dVz per unit volume is nJ(V~)dV~ and the number of atoms with these velocities passing a unit area per unit time is nf(V~)V~dVz. Atoms with V~ > Vmi, evaporate. The number of atoms evaporated per unit time per unit area is

G =

f

or

(8)

nf(Vz)VzdVz = n[(kaT)l(2qrm)] llz exp[-L(T)/(ksT)] . Vmin

3.0

2.5

~" 6"

2.0

o UJ o

1.5

1.0 Z

_z 0.5

0.0 0.(

I

I

I

I

I

I

0.2

0.4

O.O

0.8

1.0

1.2

TIME (ms) Fig. 1. The Nd-YAG l a i r pulse profile.

1.4

1.6

Efficiencies for laser drilling of stainless steel

199

If the atoms arc equally spaced within the lattice, a surface layer would consist of n 2/3 atoms with an evaporation time of (n2/3)/G. The average velocity of the surface is therefore

Vs = (1/n~/3)/( GIn 2/3) = [( kBTs)/( 2"trm ) ] 1/2 exp[-L( Ts)/( kBTs) ] .

(9)

The rate at which the evaporation front moves is determined by the rate of heat transfer per unit volume used in vaporization. It is given by

pCpVs(OT/Ox).

(10)

Incorporating this convection term in the previously derived energy equation ~ yields

dT(x,t)/dt = (loS)/(pCp) exp(-Sx) - [(fK)/(2hg)]T(x,t)

+ 070/(4x3)[f~T(s,t) exp(-[x+sllk)ds +

fiT( s,t )

exp(-lx sl/×)ds -

+ f [T(s,t) exp( +lx - sl/X )ds] + Vs(OT/Ox) .

(11)

Equation (11) is the kinetic theory model for laser conduction and convection. A numerical method employing an explicit scheme was used to obtain the temperature distribution on the surface and inside the material for a laser pulse. 2.2. First- and second-law analyses of the drilling process The Nd-YAG laser-output-pulse power as a function of time is shown in Fig. 1. The energy delivered to the workpiece surface by the laser beam is 21 J and the pulse duration is 1.48 msec.

First-law analysis The first-law efficiency is "ql = [energy required to form a hole (Qreq)]/[energy input

(Qinput)]

Focussing lens Laser beam r

kj'

1 2 3 4

Focal position Focal positions I 2 3 4

Intensity

(1012 W/m 2) 0.239 0.262 0.289 0.262

Fig. 2. The relationship between focal position and laser power intensity.

•

(12)

200

B.S. Yilbas et al

35 30 25

,,=, 20

ee

'~" 5 -

A M B I E N T GAS P R E S S U R E (torr) -A- 2 0 0 4 - 2 5 0 "~" 300 "0 350 -~" 760 I

0

1

2

3

4

FOCAL POSITION Fig. 3. Variation of the first-law efficiency withfoocarklpieP~c:itionat different gas pressures for a I mm thick

~2o~-

i---

•

.

0

1

.

.

.

2

3

4

FOCAL POSITION Fig. 4. Variation of the second-law efficiency with focal position at different gas pressures for a 1 mm thick workpiece.

Here,

Qreq = mhol,

ItLJT o

Cp dT +

f;

]

Cp d r + Lm + Lv ,

(13)

where mhome= mass of the material filling the hole, Cp = specific heat of the workpiece, To = room temperature, Tm= melting temperature, Tv = evaporation temperature, Lm = latent heat of melting, and L~ = latent heat of evaporation. The mass of the hole for a cylindrical geometry is

Efficiencies for laser drilling of stainless steel

201

(14)

mhol~ = ('nD2Z)/4,

where D = mean hole diameter and Z = thickness. Therefore,

"ql = mhole

(f"

Cp dT +

ro

f;

)

Cp dT + Lm + L~ /(El . . . .

(15)

tlmt . . . . gy).

Second-law analysis The second-law efficiency is

~qII= mhole

Cp d T -

ToCpln(Tm/To) +

ro

I;

Cp d T - ToCpln(Tv/To)

+ Lm(l - To/Tm) + L,(1 - To/T,,)]/(E~,,~ro~t,=~rgy).

(16)

It should be noted that the laser energy is considered to be the available energy at the surface, i.e. the temperature of the source of laser radiation is not the same as the temperature of the thermally radiating body. Temperature profiles for the integrals in Eqs. (15) and (16) are obtained from the heat-transfer analysis. "l~i, "f~nand mho,e are calculated numerically using data obtained from previous work. 6 3. DISCUSSION Figure 2 shows the relation between focal position and laser-power intensity. Figures 3 and 4 show the first- and second-law efficiencies for laser drilling of a stainless steel workpiece with 1 mm thickness. It is evident that the mean hole diameter changes with the setting of the focusing lens. This procedure provides for power variation at the focus which affects the size of the mean hole diameter. The ambient gas pressure used for drilling affects formation of the surface plasma which absorbs the incident laser beam, resulting in less laser power reaching the surface. The surface plasma acts as a heat source enhancing hole formation, m 3 Therefore, the ambient gas and its pressure affect the mean hole diameter and the first- and second-law efficiencies. In Fig. 3, the first-law efficiency is a maximum at 200 torr 15

u) u.

A M B I E N T GAS P R E S S U R E (torr) -i- 2 0 0 4 - 2 5 0 -4- 3 0 0 4)- 350 -~" 760 10 1

I 2

3

4

FOCAL P O S I T I O N

Fig. 5. Variation of the first-law efficiencywith focal position at different gas pressures for a 0.5 mm thick workpiece.

202

B. S. Yilbas et al 10

m

u. U. I1l

a z O o uJ

5

1

J

A M B I E N T GAS P R E S S U R E (torr) -4r 2 0 0 4 - 2 5 0 "~- 300 "0" 350 -~" 760

2

3

4

FOCAL POSITION Fig. 6. Vadafionof~e ~cond-law efficiencywi~ f~flposifionat~ffe~ntg~ p~ssures ~ra0.5mm~ick wo~pi~e.

pressure, which may be due to improved coupling of plasma heating to absorption. In general, as the laser power increases at the surface, the first-law efficiency increases due to the increased mean hole diameter. This argument is valid for some range of workpiece thickness. ~4 Figure 4 shows the secondlaw efficiency, which varies with ambient gas pressure, increases with laser power intensity at the surface and reaches a maximum of 20% at an ambient gas pressure of 200 torr. A comparison of Figs. 3 and 4 indicates that the first-law efficiency is greater than the second-law efficiency for the same setting of laser power and gas pressure. This result may be due to the fact that in the second-law analysis, the temperature at which heat is rejected is taken into account; therefore, the available heat for hole formation is decreased. Figures 5 and 6 show the first- and second-law efficiencies for laser drilling of a 0.5 mm thick workpiece. They vary with the pressure of the ambient gas. However, they are not significantly affected by the power intensity because, for thin samples, the cavity produced in the workpiece during laser interaction is reduced, which produces less recoil pressure. ~5 Therefore, the mass removal rate becomes independent of the power received by the workpiece, i.e. the mean hole diameter is relatively less affected by the laser power received at the surface. For a 0.5 mm thick workpiece, the first- and secondlaw efficiencies are reduced considerably and the maximum values are 14 and 10%, respectively. REFERENCES I. B. S. Yilbas, Mech. Engng Technol. 4, 5 (5 February 1986).

2. 3. 4. 5. 6. 7. 8. 9. 10. II. 12. 13. 14. 15.

B. S. Yilbas, J. Engng Mater. Technol. 109, 282 (1987). B. S. Yilbas, Int. Comm. Heat Mass Transfer 20, 545 (1993). B. S. Yilbas and Z. Yilbas, Opt. Laser Technol 20, 29 (1988). B. S. Yilbas, Opt. Laser Technol. 18, 27 (1986). B. S. Yilbas, J. Univ. Kuwait (Sci.) 15, 39 (1988). B. S. Yilbas and A. Z. Sahin, Int. Comm. Heat Mass Transfer 21, 509 (1994). B. S. Yilbas and K. Apalak, Egypt J. Phys. 18, 133 (1987). D. Tabor, Gases, Liquids and Solids, 2rid exl., Cambridge Univ. Press, London (1979). M. Galeev and A. Sudan, Plasma Physics, North Holland Publication, New York, NY (1983). B. S. Yilbas, Int. J. Engng $ci. 24, 1325 (1986). B. S. Yilbas and Z. Yilbas, Opt. Lasers Engng 7, 1 (1987). B. S. Yilbas, Proc. Inst. Mech. Engrs 202, 123 (1988). B. S. Yilbas, SPIE Proceedings, "Lasers In Motion For Industrial Applications", pp. 87-91 (1987). B. S. Yilbas and Z. Yilbas, Pramana J. Phys. 31, l (1988).

Efficiencies for laser drilling of stainless steel NOMENCLATURE ot = Thermal diffusivity (m2/sec) p = Density (kg/m 3) Cp = Specific heat (J/kg K) K = Thermal conductivity ( W / m K) 8 = Absorption coefficient (m -~ ) h = Mean free path of electrons (m) V = Electron mean velocity (m/sec) N = Electron number density (m -3) No = Number density of lattice site (m -3)

ks = Boltzmann constant (1.38 × 10-23 J/K) f = Fraction of excess energy exchange T(s,t) = Electron temperature (K) T(x,t) = Phonon temperature (K) Es.t = Electron energy (J) Ex.t = Phonon energy (J) Io = Laser power intensity (W/m 2) dx = Spatial increment (m) dt = Time increment (s)

203