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In this case, the initial and boundary conditions are

T ð x; 0Þ ¼ 0

@T

@ x

x¼0

¼ rVL

k dðt Þ

T ð1; t Þ ¼ 0

Using the dimensionless quantities,

x ¼ d x; t ¼ ad2t ; T ¼ kd

I 1T ; . . . V ¼ V =ad and b

¼ rV L

I 1

The governing equations become

@2T

@ x2 þ V

@T

@ x þ 1ðt Dt Þe x ¼ @T

@t (6)

with the initial and boundary conditions,

T ð x; 0Þ ¼ 0

@T

@ x

x¼0

¼ bdðt Þ

T ð1; t Þ ¼ 0

Taking the Laplace transformation of Eq. (6) with respect to

time, one can get

@2 ¯ T

@ x2 þ V

@ ¯ T

@ x þ eDt s

s e x ¼ s ¯ T

T ð x; 0Þ (7)

where T ¯ * is the Laplace transform of temperature and s is theLaplace variable. After solving Eq. (7) in the Laplace domain and

transferring to the physical plane results in the temperature

distribution for a step input laser pulse; therefore, the resulting

equation for temperature distribution is [15]

T ð x; t Þ þ Z 1ðt Þ þ Z 2ðt Dt Þ1ðt Dt Þ þ Z 3ðt Dt Þ1ðt Dt Þ(8)

where Z 1(t*) is

Z 1ðt Þ ¼ beðV =2Þ x

eðV 2=4t Þ eð x2=4t Þ ffiffiffiffip

p ffiffiffiffit

p

V

2 eðV 2=4Þt þðV =2Þ x

erfc ffiffiffiffit

p V

2 þ x

2 ffiffiffiffit

p !!

and Z 2(t*) is

Z 2ðt Þ ¼ eðV =2Þ xeðV =4Þt l1

eð x2=4t Þ ffiffiffiffip

p ffiffiffiffit

p x1e xx1þt x2

1

"

erf x

2 ffiffiffiffit

p þ ffiffiffiffit

p x1

þ l2eð x2=4t Þ ffiffiffiffi

pp ffiffiffiffi

t p x1e

ð xx1Þþt x21erfc

x

2 ffiffiffiffit

p þ ffiffiffiffit

p x1

!

þ l3eð x2=4t Þ ffiffiffiffi

pp ffiffiffiffi

t p x2e

xx2þt x22erfc

x

2 ffiffiffiffit

p þ ffiffiffiffit

p x2

!

þ l4eð x2=4t Þ

ffiffiffiffip

p

ffiffiffiffit

p x2eð xx2Þþt x2

2erfc x

2

ffiffiffiffit

p þ ffiffiffiffit

p x2

!

þ l5e

ð x2=4t

Þ ffiffiffiffip

p ffiffiffiffit p

V

2 e xðV =2Þþt ðV 2

=2Þerf x

2 ffiffiffiffit p þ ffiffiffiffi

t p V

2 !

(9)

where

l1 ¼ 1

2x1ðx21 x

22Þðx1 þ x3Þ

l2 ¼ 1

2x1ðx21 x2

2Þðx1 x3Þ

l3 ¼ 1

2x2ðx21 x

22Þðx1 þ x3Þ

l4 ¼ 1

2x2

ðx

22

x

21

Þðx2

x3

Þl5 ¼ 1ðx2

1 x23Þðx2

2 x23Þ

and

x21 ¼ V 2=4; : x2

2 ¼ V 2

4 þ V 1 and x3 ¼ V =2

and Z 3(t *)

Z 3ðt Þ ¼ 1 et ð1þV Þ

1 þ V e x

(10)

Hence, since the functions Z 1(t *), Z 2(t *) and Z 3(t *) are obtained,

the temperature distribution is explicitly known (Eq. (10)). A

Mathematica software is used to compute temperature distribu-

tion (Eq. (9)).

ARTICLE IN PRESS

Time (t*)

I n t e n s i t y

(Unit Step Function - Shifted

Unit Step Function)

Unit Step Function

Step Input

IntensityShifted Unit Step Function

Fig. 2. Construction of laser step input intensity pulse from unit step function and shifted unit step function.

H. Al-Qahtani, B.S. Yilbas / Optics & Laser Technology 41 (2009) 931–937 933

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2.2. Time exponentially decaying pulse

Temporal variation of the laser pulse can be represented in

terms of two exponential functions [13]. This function can be

written as I 1ðeb1t eb2t Þ, where I 1 is the peak laser intensity, and

b1 and b2 are the laser pulse parameters. Fig. (3) shows the time

exponentially decaying complete laser pulse for b1 ¼10111/s and

b2 ¼ 5 10111/s. The solution of conduction equation (the Fourier

equation) for evaporative heating situation can be obtained foronly one exponential term exp(b1t ) of the laser heating pulse;

then, the solution for the second exponential term can be added

to the solution of for the first exponential term according to

the superposition rule. Consequently, temperature variation for

the complete laser heating pulse can be obtained. In this case, the

Fourier heat transfer equation due to time exponentially decaying

laser pulse for the first term b (b is used for the general purpose

and it will be replaced with b1 and b2 later in the mathematical

analysis) can be written as

k@2T

@ x2 þ rC pV

@T

@ x þ I oð1 r f Þ expðbt Þd expðd xÞ ¼ rC p @T

@t (11)

with the boundary conditions

@T @ x

x¼0

¼ rVLk

: T ð1; t Þ ¼ 0; and T ð x; 0Þ ¼ 0

where k is the thermal conductivity, C p is the specific heat

capacity, r is the density, V is the recession velocity, b is the pulse

parameter, L is the latent heat of evaporation, I o is the peak power

intensity, and r f is the surface reflectivity. The recession velocity of

the surface can be formulated from energy balance at the free

surface of the irradiated workpiece [14]. In this case the energy

flux at the free surface can be written as [14]

V ¼ I 1r½C pT s þ L (12)

where I 1 ¼ I o(1r f ) and T s is the surface temperature.

It should be noted that the peak power intensity does not varywith time. Since the surface temperature is time dependent, the

recession velocity varies with time. This results in non-linear form

of Eq. (11), which cannot be solved analytically by a Laplace

transform method. Moreover, there exists a unique value for the

recession velocity for a known surface temperature. Consequently,

an iterative method can be introduced to solve Eq. (11) analy-

tically. In this case, keeping the recession velocity constant in

Eq. (11) enables to determine the surface temperature analytically,

and after obtaining the surface temperature, the recession velocity

can be recalculated using Eq. (14). This procedure can be repeated

unless the surface temperature and recession velocity converge

correct results.

Eq. (11) can be written as

@2T

@ x2 þ V

a@T

@ x þ I 1

k expðbt Þd expðd xÞ ¼ 1

a@T

@t (13)

The Laplace transform of Eq. (13) with respect to t , after

substituting of initial condition T ( x, 0) ¼ 0, can be written as

@2 ¯ T

@ x2 þ V

a@ ¯ T

@ x þ I 1

d

k

1

ð p þ bÞ expðd xÞ ¼ 1

a½ p ¯ T (14)

where a is the thermal diffusivity, which is

a ¼ k

pC p

Introducing the dimensionless variables:

V

¼

1

ad

V : b

¼

1

ad2 b : t

¼ ad2

t : and x

¼ x

d

And after the lengthy algebra, temperature distribution yields

[13]:

T ð x; t Þ ¼ I 1kd

eðV 2=2Þð xþðV =2Þt Þ

eððV 2=4ÞðV 1ÞÞt

ðV ð1 þ bÞÞ eð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðV 2=4Þbp

Þ xh(

erfc ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiððV 2=4Þ b

Þt q

þ ð x=2 ffiffiffiffit

p ÞÞ

ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV 2 b

q þ V Þ

þeð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðV 2=4Þbp

Þ x erfc ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiððV 2=4Þ b

Þt q

þ ð x=2 ffiffiffiffit

p ÞÞ

ðV ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV 2 b

q

Þ

375

þ eððV 2=4

ÞðV

1ÞÞt

ðb ðV 1ÞÞ e ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðV 2=4ÞðV 1Þp h

erfc ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiðV 2=4Þ ðV 1Þt þ ð x=2

ffiffiffiffit

p Þ

q Þ

ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV 2 4ðV 1Þ þ V

q Þ

þe ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðV 2=4ÞðV 1Þp erfc ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiððV 2=4Þ ðV 1ÞÞt þ ð x=2

ffiffiffiffit

p Þ

q Þ

ðV 2 ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV 2 4ðV 1Þ

q ÞÞ

375

1

2ðbðV 0 1ÞÞ eðV 2=4Þt eðV 2=2Þ x

erfc V

2 t þ x

2 ffiffiffiffit

p

þ I 1kd

1

ððV 1Þ bÞ e x ðeb

t eðV 1Þt Þ

arL2k

eV xerfc

x V t

2 ffiffiffiffit

p

ð1 þ V x þ V 2t Þerfc x þ V t

2 ffiffiffiffit

p

þ

2V t ffiffiffiffipp eððV 2=4ÞðV x=2ÞþðV t =4ÞÞ (15)

Temperature distribution can be non-dimensionalized using the

relation

T ¼ T

I 1=ðkdÞ

Temperature distribution for the complete laser heating pulse,

including both exponential terms, is possible subtracting T*

obtained for b*2 from T* obtained from b*1. Therefore, solving

Eq. (15) for b*2 and b*1 and, then, mathematical subtraction of the

resulting temperatures provides the solution for temperature

distribution for the complete laser pulse. The Mathematica

software is used to compute dimensionless temperature distribu-

tion (Eq. (15)) for the complete pulse.

ARTICLE IN PRESS

0

0.15

0.3

0.45

0.6

0

TIME

I N T E N S I T Y

2 4 6 8

Fig. 3. Tim e ex ponent ially d ecay ing p ulse for comp lete p ulse

(I 1ðt Þ ¼ eb1 t eb2 t ).


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3. Results and discussions

Laser evaporative heating of surface is considered and effect of

laser pulse shape on the temperature rise is examined. Analytical

solutions are presented for laser non-conduction heating process.

Two different laser pulse shapes, namely exponential and step

input pulses, are accommodated in the analysis for the closed-

form solutions. Temporal variation of temperature distribution

is presented for different pulses with the same energy content.Table 1 gives the material properties used in the simulations.

Fig. 4 shows profiles of exponential and step input laser pulses,

provided that exponential pulse has three shapes having the same

energy content, while Fig. 5 shows corresponding temperature

profiles obtained from the closed-form solutions (Eqs. (9)

and (15)). Once the evaporation temperature is reached, due to

evaporative boundary at the surface, temperature reduces rapidly

and, then, rises above the evaporation temperature of the

substrate material as the heating progresses. Moreover, surface

temperature after reaching its maximum decays gradually for the

exponential pulses while sharp decay is observed for the step

input pulses. The rise of temperature in the solid phase, before the

evaporation, is rapid in the early heating period. This is more

pronounced for the exponential pulses with short pulse length

and high peak intensity than that of other pulse with slow

rising pulse intensity. Consequently, laser short pulse with highintensity results in rapid rise of temperature in the early heating

period. This can be attributed to the internal gain of the substrate

material in the surface region. Small decay of temperature after

reaching the evaporation temperature is because of the energy

taken during the evaporation process, which is considerably

high. However, temperature rise after the evaporation is rapid

for exponential pulse with the short pulse length. Moreover,

temperature rise is relatively slower for the step input pulse as

compared to that corresponding to exponential pulses. The

initiation of cooling cycle is more pronounced for step input

pulses than exponential pulses; in which case, temperature

decay is rapid onset of the pulse ending. Temperature gradient

developed in the surface region of the substrate material becomes

high towards the pulse ending. Once the pulse energy ceases,

diffusional energy transport from the surface region to the solid

bulk becomes the only energy transfer mechanism in the surface

region. This, in turn, rapidly lowers temperature in this region.

Although the intensity is low at the tail of the pulse, it provides

internal energy gain of the substrate material from the irradiated

field. Consequently, diffusional energy transport from surface

region to solid bulk results in gradual decay of temperature in the

surface region due to internal energy gain of the substrate

material from the irradiated field.

Fig. 6 shows temporal variation of exponential and step input

pulses, provided that step input pulse shape is varied while

keeping the energy content of the pulse constant. Fig. 7 shows

corresponding temperature rise at the surface. Temperature rise is

rapid for step input pulse having the highest peak intensity and

shortest pulse length. However, exponential pulse results in slowrise of temperature in the early heating period. The fast rise of

temperature in the early heating period for step input pulse is

because of internal energy gain of the substrate material in the

surface region from the irradiated field. Consequently, high rate of

energy absorption results in rapid rise of temperature in the early

heating period. Moreover, the rise of temperature after reaching

ARTICLE IN PRESS

1.2

1

0.8

0.6

0.4

0.2

0

0 2 4 6

TIME

I N T E N S I T Y

Step Pulse ∆t = 1

Step Pulse ∆t = 3

Step Pulse ∆t = 5

Exp. Pulse 1 = 1/2, 2 = 1

Fig. 4. Temporal variation of laser pulse shapes used in the simulations. The pulse

energy in each pulse is kept constant while pulse length of step input pulses is

varied.

0.5

0.4

0.3

0.2

0.1

0

0 2 4 6 8 10

TIME

T E M P E R A T U R E

Step Pulse ∆t = 1

Step Pulse ∆t = 3

Step Pulse ∆t = 5

Exp. Pulse 1 = 1/2, 2 = 1

Fig. 5. Temporal variation of surface temperature for laser pulses showing in Fig. 1.

0.5

0.4

0.3

0.2

0.1

00 2 4 6

TIME

I N T E N S I T Y

Exp. Pulse 1 = 1/2, 2 = 1

Exp. Pulse 1 = 2/3, 2 = 2

Exp. Pulse 1 = 5/6, 2 = 5Step Pulse ∆t = 3

Fig. 6. Temporal variation of laser pulse shapes used in the simulations. The pulse

energy in each pulse is kept constant while pulse length of time exponentially

decaying pulse is varied.

Table 1

Material properties used in the simulations.

r (kg/m3) C p (J/kg K) k (W/mK) d (m1) a (m2/s)

7880 460 80.3 6.17 10

7

2.22 10

4


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the evaporation temperature is highest for step input pulse with

shortest pulse length. However, the rise of temperature for

exponential pulse after reaching the evaporation temperature is

higher than that of other step input pulses. The decay rate of

temperature after reaching maximum is faster for step input pulse

with the highest intensity. This is because of the attainment of

high-temperature gradient in the surface region during the

heating period. Once the pulse ends, high-temperature gradient

causes energy diffusion from the surface region to the solid

bulk at a higher rate than that of the other pulses. Exponential

decay of the pulse intensity towards the pulse ending results ingradual decay of temperature. However, temporal variation of

temperature at the surfaces does not follow exactly the temporal

variation of laser pulse intensity. This is because of the diffusional

energy transfer from the surface region to the solid bulk, which

suppresses the internal energy gain from the irradiated field in the

surface region. This situation is more pronounced for step input

pulses.

Fig. 8 shows dimensionless temperature distribution inside the

substrate material for three exponential pulse parameters and one

step input pulse with dimensionless pulse length of 3 for the

dimensionless heating period of 1.5. Increasing b1 and b2 results

in attainment of high temperature in the surface region due to the

high peak power intensity (Fig. 6). Moreover, temperature attains

high values for the step input pulse as compared to that

corresponding exponential pulse of b1 ¼ 2/3 and b2 ¼ 2. It

should be noted that the peak intensity corresponding to

exponential pulse of b1 ¼ 2/3 and b2 ¼

2 is similar to that the

step input pulse peak intensity for pulse length of 3 (Fig. 6).

Consequently, the pulse intensity distribution with time has

significant effect on temperature distribution inside the substrate

material. Temperature decays sharply in the surface vicinity of the

substrate material, particularly for the step input pulse. The sharp

decay results in high-temperature gradient in this region

enhancing the conduction energy transfer from the irradiated

surface to the solid bulk. However, energy absorbed from the

irradiated field increases significantly internal energy gain of the

substrate material in the surface region; in which case, internal

energy gain dominates over the conduction losses from the

surface region while resulting high temperature at the surface. In

the case of exponential pulse, temperature decay is gradual in the

surface region and as the distance increases away from the

irradiated surface towards the solid bulk it decays sharply.

4. Conclusion

Laser evaporative heating of substrate surface is considered

and temperature rise due to time exponentially varying and step

input pulses is compared. The closed-form solutions obtained

for temperature rise due to both pulses are presented in the

non-dimensional form. Moreover, pulse intensities used for

temperature comparison have the same energy content. It is

found that the rise of temperature in the solid surface is rapid for

step input pulses due to high amount of energy gain of the

substrate material from the irradiated field in the surface region.

However, temperature rise beyond the evaporation temperatureof the substrate material is faster for exponential laser pulses than

that of step input pulses. In the cooling cycle of the step input

pulse, temperature decays rapidly immediately after the laser

pulse ends. This is because of the energy conducted from the

surface region to the solid bulk, i.e., high-temperature gradient in

the surface region enhances the energy diffusion from the surface

region to the solid bulk. In the case of exponential pulse,

exponential decay of laser pulse intensity provides heating of

the substrate material through the absorption; consequently, no

definite cooling cycle can be identified. Therefore, energy transfer

from the surface region to the solid bulk does not suppress

temperature rise and temperature decay with time becomes

gradual. Temporal distribution of laser pulse intensity at the

surface has significant effect on temperature decay in the

ARTICLE IN PRESS

0.25

0.2

0.15

0.1

0.05

0

0 1 2 3 4 5 6 7

TIME

T E M

P E R A T U R E

Exp. Pulse 1 = 1/2, 2 = 1Exp. Pulse 1 = 2/3, 2 = 2Exp. Pulse 1 = 5/6, 2 = 5

Step Pulse ∆t = 3

Fig. 7. Temporal variation of surface temperature for laser pulses showing in Fig. 4.

0.4

0.3

0.2

0.1

0

0 2 4 6 8 10

SPACE

T E M P E R A T U R E

Exp. Pulse 1 = 1/2, 2 = 1

Exp. Pulse 1 = 2/3, 2 = 2

Exp. Pulse 1 = 5/6, 2 = 5

Step Pulse ∆t = 3

Fig. 8. Temperature distribution inside the substrate material for three exponen-

tial pulses and a step input pulse for dimensionless heating period of 1.5.


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substrate material. In this case, temperature decay is sharp for the

step input pulse while it is gradual for the exponential pulse in the

surface region, despite the fact that the peak intensity corre-

sponding to exponential pulse of b1 ¼ 2/3 and b2 ¼ 2 is similar to

that the step input pulse peak intensity for pulse length of 3.

Acknowledgment

The authors acknowledge the support of King Fahd University

of Petroleum and Minerals, Dhahran, Saudi Arabia for this work.

References

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