Applied Surface Science 252 (2006) 2242–2250
www.elsevier.com/locate/apsusc
Repetitive laser pulse heating analysis: Pulse parameter
variation effects on closed form solution
M. Kalyon a, B.S. Yilbas a,b,*
a
ME Department, King Fahd University of Petroleum and Minerals (KFUPM), KFUPM Box 1913, Dhahran 31261, Saudi Arabia
b
AE Department, Istanbul Technical University, Istanbul, Turkey
Received 19 February 2005; received in revised form 13 March 2005; accepted 1 April 2005
Available online 13 June 2005
Abstract
Laser conduction limited heating finds wide application in surface processing industry. Modelling of the heating process
gives insight into the physical processes involved in conduction limited heating. Moreover, analytical solutions provide
functional relation among the parameters that influence the heating process. In the present study, repetitive laser pulse heating of
a solid substrate is considered. A closed form solution for the temperature rise including the cooling cycle is obtained using a
Laplace transformation method. It is found that the results obtained from the closed form solution agree well with the numerical
predictions. The maximum surface temperature rises rapidly once the cooling period between the consecutive pulses reduces.
# 2005 Elsevier B.V. All rights reserved.
Keywords: Laser; Conduction; Pulse heating; Closed form
1. Introduction
Laser repetitive pulse heating process finds wide
application in industry. Since lasers deliver high
intensity beam onto the substrate surface, a local
thermal processing is possible. Irradiation of a
surface with a single laser pulse does not provide
sufficient energy deposited onto the surface. Consequently, repetitive pulse irradiation becomes
inevitable during the laser heating process. More* Corresponding author. Tel.: +966 3 860 4481;
fax: +966 3 860 2949.
E-mail address: bsyilbas@kfupm.edu.sa (B.S. Yilbas).
over, a closed form solution for the temperature rise
provides useful information on the heating parameters and laser pulse properties. In addition, model
studies reduce the experimental cost and minimizes
the experimentation time.
Laser pulse heating falls into conduction and nonconduction limited heating processes. In the case of
conduction limited heating process, the substrate
material remains in solid phase while phase change
occurs in non-conduction limited heating process.
Considerable research studies were carried out to
explore the laser conduction heating process. An
analytical solution for a laser step input pulse intensity
was obtained by Ready [1]. Simon et al. [2] studied the
0169-4332/$ – see front matter # 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.apsusc.2005.04.032
M. Kalyon, B.S. Yilbas / Applied Surface Science 252 (2006) 2242–2250
Nomenclature
Cp
I1
I0
k
rf
s
t
t1
t2
t
t1
specific heat (J/kg K)
power intensity (I0 (1- rf )) (W/m 2 )
laser peak power intensity (W/m 2 )
thermal conductivity (W/m K)
reflection coefficient
Laplace variable
time (s)
time initiation of first pulse (s)
time initiation of first pulse (s)
dimensionless time (= ad2 t)
time corresponding to first pulse
beginning
t2
time corresponding to second pulse
beginning
dt1
pulse length of first consecutive
pulse (s)
dt2
pulse length of second consecutive
pulse (s)
dt1
pulse length of first pulse
dt1
pulse length of second pulse
Tðx; tÞ temperature (K)
T ð0; t Þ dimensionless surface temperature
(= Tð0; tÞ kd
I1 )
T0
ambient temperature (K)
T̄
temperature in Laplace domain (K)
x
spatial coordinates corresponding to the
x-axis (m)
x
dimensionless distance (= xd)
Greek letters
a
thermal diffusivity (m 2 /s)
d
absorption coefficient (1/m)
D
shift in time
r
density (kg/m 3 )
heat conduction in deep penetration welding and
showed that the time modulated laser beam had little
effect on the resulting heat affected zone. Laser
heating of a two-layer system was studied by AlAdawi et al. [3] using a Laplace transformation
method. They indicated that the time to reach the
melting of the substrate material was dependent highly
on the thermal properties. Blackwell [4] studied
analytically the temperature rise due to a laser heating
2243
pulse and convective boundary condition at the surface.
He obtained a closed form solution for the temperature
rise inside the substrate material. Thermal analysis of a
surface transformation hardening due to laser pulse
heating was carried out by Woodard and Dryden [5]. An
analytical solution for the axisymmetric heat conduction was presented. Thermal analysis of laser heat
treatment of engineering alloys was studied by Yilbas
et al. [6]. They obtained a closed form solution for
temperature rise and developed a relation between
temperature rise and equilibrium time. An analytical
solution for a laser step input pulse heating was
obtained by Yilbas and Shuja [7]. They introduced a
power relation between the dimensionless temperature
and distance. An analytical solution for a repetitive laser
pulse heating with a convective cooling boundary
condition at the surface was obtained by Yilbas and
Kalyon [8]. They introduced a time exponentially
varying pulse in the analysis. When formulating the
laser conduction limited heating process, the cooling
cycle onset of laser pulse ending should be taken into
account. Moreover, in repetitive laser pulse heating
process, the period (cooling period) between the
successive pulses should be considered. Consequently,
the analysis of repetitive laser pulse heating employing
the cooling period between the consecutive pulses
becomes necessary.
In the present study, an analytical solution for the
laser repetitive pulse heating process is considered. A
closed form solution for the temperature rise due to
laser repetitive pulses is obtained using a Laplace
transformation method. The results obtained from the
closed solution is compared with the numerical
predictions. It should be noted that in order to obtain
a closed form solution for the temperature distribution
independent of material and laser pulse properties, the
Fourier heat transfer equation is non-dimensionalized
and the results are presented in dimensionless form.
Consequently, the results obtained can be changed into
the dimensional form through accommodating the
material properties and pulse intensity in the
dimensionless time, space and temperature.
2. Mathematical modelling
The heat transfer equation for a laser heating pulse
can be written as
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M. Kalyon, B.S. Yilbas / Applied Surface Science 252 (2006) 2242–2250
@2 T I1 d dx
1 @T
e ðC1 P1 þ C2 P2 Þ ¼
þ
@x2
k
a @t
(1)
where
I1 ¼ ð1 rf ÞI0
and
P1 ¼ 1½t t1 1½t ðt1 þ dt1 Þ, P2 ¼ 1½t t2 1½t ðt2 þ dt2 Þ.
1½t ¼
1; t > 0
;
0; t < 0
1½t t0 ¼
1; t > t0
0; t < t0
The non-dimensional initial and boundary conditions become:
At time t ¼ 0 : T ðx ; 0Þ ¼ 0; At the surface x ¼ 0 :
@T
¼ 0 and at x ¼ 1 : Tð1; t Þ ¼ T0
@x x ¼0
(2)
where rf is the reflection coefficient and I0 is the peak
power intensity. P1 and P2 are the step input intensity
functions for the first and second consecutive pulses,
1½t is the unit step function and 1½t t0 is the shifted
unit step function (shifted by t0 ), t1 and t2 are first
and second pulse initiations, dt1 and dt2 are the first
and second pulse lengths, and C1 and C2 are the
constants defining the peak amplitude of pulses
(0 C1 1 and 0 C2 1) (Fig. 1a). Therefore,
the amount of time shift between two consecutive
pulses is t2 ðt1 þ dt1 Þ.
The initial and boundary conditions are:
The solution of Eq. (3) can be obtained through
Laplace transformation method, i.e., with respect to
t , the Laplace transformation of Eq. (3) yields:
@2 T̄
þ ½C1 P1 ðsÞ þ C2 P2 ðsÞ ex ¼ ½sT̄ T0 @x2
or
@2 T̄
sT̄ ¼ ½C1 P1 ðsÞ þ C2 P2 ðsÞ ex T0
@x2
The homogenous solution of Eq. (5) can be written as
T̄h ¼ K1 e
At time t ¼ 0 : Tðx; 0Þ ¼ T0 ; At the surface x ¼ 0 :
@T
¼ 0 and at x ¼ 1 : Tð1; tÞ ¼ T0
@x x¼0
In order to generalize the solution independent of
material and laser pulse properties, which could be
applicable to any material, non-dimensionalizing of
the parameters in the Fourier heat transfer equation
(Eq. (1)) are essential. After introducing the dimensionless parameters as
2
t ¼ ad t;
x ¼ dx;
kd
T ¼T
I1
T̄p ¼
pffi sx
(7)
Therefore, the solution of dimensionless temperature
in the Laplace domain becomes
pffi sx
þ K2 e
T̄ ¼ K1 e
pffi sx
C1 P1 ðsÞ þ C2 P2 ðsÞ x T0
e þ
s1
s
"
(3)
@T @x
¼ K 2 e
(8)
pffi pffiffi
sx
s
x ¼0
C1 P1 ðsÞ þ C2 P2 ðsÞ x
e
s1
where
P1 ¼ 1½t t1 1½t t1 þ dt1 ;
P2 ¼ 1½t t2 1½t t2 þ dt2 (6)
The boundary condition Tð1; t Þ ¼ T0 results in
K1 ¼ 0. The boundary condition at the surface yields
@ T
@T
þ ½C1 P1 þ C2 P2 ex ¼ @x2
@t
þ K 2 e
C1 P1 ðsÞ þ C2 P2 ðsÞ x T0
e þ
s1
s
Eq. (1) becomes
2 pffi sx
and the particular solution for Eq. (5) is
þ
(5)
(4)
where t1 ¼ ad2 t1 , t2 ¼ ad2 t2 , dt1 ¼ ad2 dt1 , and
dt2 ¼ ad2 dt2 .
which results
K2 ¼ C1 P1 ðsÞ þ C2 P2 ðsÞ
pffiffi
sðs 1Þ
#
¼0
x ¼0
M. Kalyon, B.S. Yilbas / Applied Surface Science 252 (2006) 2242–2250
2245
Fig. 1. (a) Construction of a step input intensity pulse. (b) Consecutive two pulses used in the analysis.
(
Therefore, Eq. (8) becomes
pffi pffi C1 e sx t1 s C1 e sx ðt1 þdt1 Þs
pffiffi
pffiffi
T̄ ¼ s sðs 1Þ
s sðs 1Þ
)
pffi pffi C2 e sx t2 s C2 e sx ðt2 þdt2 Þs
pffiffi
þ pffiffi
s sðs 1Þ
s sðs 1Þ
(
C1 ex t1 s C1 ex ðt1 þdt1 Þs C2 ex t2 s
þ
þ
sðs 1Þ
sðs 1Þ
sðs 1Þ
)
C2 ex ðt2 þdt2 Þs
T
(10)
þ 0
sðs 1Þ
s
T̄ ¼ þ
C1 P1 ðsÞ þ C2 P2 ðsÞ pffisx
pffiffi
e
sðs 1Þ
C1 P1 ðsÞ þ C2 P2 ðsÞ x T0
e þ
ðs 1Þ
s
Noting from Eq. (4) that:
P1 ðsÞ ¼
et1 s eðt1 þdt1 Þs
;
s
s
P1 ðsÞ ¼
et2 s eðt2 þdt2 Þs
s
s
Hence Eq. (9) becomes:
(9)
Noting that:
1
1
1
¼
sðs 1Þ s 1 s
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M. Kalyon, B.S. Yilbas / Applied Surface Science 252 (2006) 2242–2250
The inversion of Laplace transforms can be written as
pffi pffisx pffisx e sx
e
1
1
1 e
pffiffi
pffiffi
pffiffi ;
L
L
¼L
sðs 1Þ
s sðs 1Þ
s s
L1
x
x
x
e
e
e
¼ L1
L1
sðs 1Þ
ðs 1Þ
s
Consequently,
Laplace
h pffisx i
1 p
effi
L
yields [9]:
L1
transformation
of
s sðs1Þ
L1
pffi
sx
t
"
L
ex
¼ ex ½et 1ðt Þ ¼ HB
sðs 1Þ
(12)
Ds
e
ex
¼ ex ½et 1ðt DÞ
sðs 1Þ
Moreover, we note from Eqs. (10) and (11) that:
pffi e s x
pffiffi
L e
¼ Term1A ¼ HA ðDÞjD¼t ;
1
s sðs 1Þ
pffi e sx
1 ðt1 þdt1 Þs
pffiffi
L e
¼ Term2A
s sðs 1Þ
1 t1 s
¼ HA ðDÞjD¼t þdt ;
1
1
pffi
s x
e
pffiffi
¼ Term3A ¼ HA ðDÞjD¼t ;
2
s sðs 1Þ
ffi
p
e sx
L1 eðt2 þdt2 Þs pffiffi
¼ Term4A
s sðs 1Þ
Consider the following Laplace operation:
L1 et2 s
L1 eDs FðsÞ ¼ f ðt DtÞ
where L1 FðsÞ ¼ f ðtÞ.
Therefore
pffi e sx
1 Ds
pffiffi
L e
¼ f ðx ; t Þ
s sðs 1Þ
¼ HA ðDÞjD¼t þdt
2
More explicitly:
"
#
pffi e s x
1
Ds
pffiffi
e
L
s sðs 1Þ
3
pffiffiffiffiffiffiffiffiffiffiffiffiffi
x
e Erfc t D þ pffiffiffiffiffiffiffiffiffiffiffiffiffi 7
et Dt 6
2 t D 7
6
pffiffiffiffiffiffiffiffiffiffiffiffiffi
7
¼
6
5
2 4 x
x
e Erfc
t D þ pffiffiffiffiffiffiffiffiffiffiffiffiffi
2 t D
x
" pffiffiffiffiffiffiffiffiffiffiffiffiffi
2 t D x2 =4ðt DÞ
pffiffiffi
e
p
#
x
x Erfc pffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ HA
2 t D
where 1ðt Þ is a unit step function. Similarly,
1
pffiffiffiffi
e
e
x
pffiffi
¼
ex Erfc t þ pffiffiffiffi
2
s sðs 1Þ
2 t
#
pffiffiffiffi
x
ex Erfc
t þ pffiffiffiffi
2 t
" pffiffiffiffi
2 t
2
pffiffiffi ex =4t
p
#
x
x Erfc pffiffiffiffi
2 t
2
where D is amount of shift in time due to two consecutive pulses.
h x i
e
yields
Laplace transformation of L1 sðs1Þ
2
(13)
Similarly, the following terms can be written:
x e
1 t1 s
L e
¼ Term1B ¼ HB ðDÞjD¼t ;
1
sðs 1Þ
x e
L1 eðt1 þdt1 Þs
¼ Term2B ¼ HB ðDÞjD¼t þdt ;
1
1
sðs 1Þ
x e
L1 et2 s
¼ Term3B ¼ HB ðDÞjD¼t ;
2
sðs 1Þ
x e
L1 eðt2 þdt2 Þs
¼ Term4B ¼ HB ðDÞjD¼t þdt
2
2
sðs 1Þ
(14)
(11)
Using the terms in equations (13) and (14), dimensionless temperature (Laplace inversion of Eq. (10))
becomes:
M. Kalyon, B.S. Yilbas / Applied Surface Science 252 (2006) 2242–2250
Table 1
Pulse properties used in the present study
t1
t2
dt1
dt2
C1
C2
Fig. 1b, i.e., the first unit step input pulse starts at time
t1 while the second unit step input pulse starts at time
t1 þ dt1 . The difference in both pulses results in the
first pulse of the consecutive pulses, i.e.:
0
5–15
5
5
1
1
P1 ðtÞ ¼ 1½t t1 1½t ðt1 þ dt1 Þ
where 1½t t1 and 1½t ðt1 þ dt1 Þ represent the
shifted unit step functions with shifted amounts of
t1 and dt1 , respectively. These unit step functions are
defined in Eq. (2). Similar arguments are applied to the
second pulse of the consecutive pulses when constructing it, i.e.:
"
T
¼T0
þ C1 ðTerm1B Term1A Þ
#
ðTerm2B Term2A Þ
P2 ðtÞ ¼ 1½t t2 1½t ðt2 þ dt2 Þ
"
1ðt t2 Þhas the value of zero at t < t2 and 1ðt ðt2 þ
dt2 ÞÞ has the value of zero when t < ðt2 þ dt2 Þ. The
time shift between two consecutive pulses is
þ C2 ðTerm3B Term3A Þ
#
ðTerm4B Term4A Þ
2247
(15)
Eq. (15) is developed for two consecutive pulses
with (t2 ðt1 þ dt1 Þ) apart and having pulse lengths
dt1 and dt2 with amplitude multiplication factors of C1
and C2 as shown in Fig. 1.
The pulse properties employed in the computation
are given in Table 1.
3. Results and discussions
Laser repetitive heating of solid substrate is
modelled and closed form solution for temperature
rise due to laser pulse is obtained. The governing
equation of heat conduction is non-dimensionalized
and three pulse types are considered when presenting
the results. In the closed form solution, two
consecutive pulses are taken into account. The first
pulse starts at t1 with a pulse length of dt1 and the
second pulse initiates at t2 with a pulse length of dt2 .
The first pulse consists of two cycles, namely heating
and cooling cycles. The heating cycle starts with the
initiation of the pulse and ends when the pulse
intensity reduces to zero. The cooling cycle starts after
the pulse ends as shown in Fig. 1b. The mathematical
construction of the first pulse of consecutive pulses is
achieved through subtraction of two unit step
functions (infinitely extending pulses) as shown in
t2 ðt1 þ dt1 Þ
Moreover, the pulse intensity of consecutive pulses
can be varied by introducing the intensity multiplication factors for each pulse of the consecutive pulses,
i.e.:
C1 P1 ðtÞ þ C2 P2 ðtÞ
where C1 and C2 are intensity multiplication factors
where 0 < C1 < 1 and 0 < C2 < 1. Table 1 gives the
properties of different pulse types. In order to obtain a
general solution for the temperature rise, the Fourier
heating equation is non-dimensionalized. This provides the solution independent of material properties
and laser pulse intensity. Consequently, the closed
form solution for temperature distribution can be used
for any substrate material. One of the examples of such
situation is given in Fig. 2, i.e. temporal variation of
the surface temperature distribution for steel is shown
for a single pulse with pulse length 1:4 108 s and
pulse intensity 2 1011 W/m 2 .
Fig. 2 shows the temporal variation of surface
temperature obtained from the closed form solution
and numerical simulations. The temperature profiles
obtained from both analysis are in excellent agreement.
Fig. 3a shows the temporal variation of nondimensional temperature distribution at different
locations inside the substrate material. Temperature
rises rapidly in the beginning of the heating pulse and
as the heating pulse progresses the rate of temperature
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M. Kalyon, B.S. Yilbas / Applied Surface Science 252 (2006) 2242–2250
Fig. 2. Temporal variation of dimensionless temperature at the
surface obtained from closed form and numerical solutions for
pulse type 1.
profile rise reduces slightly. This can also be seen from
Fig. 3b, in which time derivative of temperature
gradient is shown. This indicates that in the early
Fig. 3. (a) Temporal variation of dimensionless temperature at
different x-axis locations for pulse type 1. (b) Temporal variation
of time derivative of dimensionless temperature at different x-axis
locations for pulse type 1.
heating period, the internal energy gain of the
substrate material due to absorption of irradiated
energy dominates over the conduction energy
transport from surface vicinity to the solid bulk.
As the heating period progresses, temperature
gradient increases and heat conduction from the
surface vicinity to the solid bulk enhances. The
material response to a heating pulse at some depth
below the surface differs. In this case, temperature
profiles becomes almost similar in depths similar to
absorption depth of the substrate material (x < 1).
As the depth increases further from x ¼ 1, the
energy gain due to absorption becomes almost
negligible and energy transport by conduction is
almost the sole mechanism governing the temperature rise in this region. Consequently, material
response to a heating pulse slows down and the rate
of temperature rise reduces in this region. The
influence of the second pulse on temperature profiles
is similar at the surface as well as inside the substrate
material, i.e. magnitude of temperature rise in the
surface region is similar to that corresponding to
some depth below the surface, provided that x < 1.
Moreover, the domination of internal energy gain
over the conduction energy transport in the surface
region during the early heating period is not
observed at some depth below the surface. This is
because of the energy transfer mechanism, in which
case, absorption replaces with the conduction
heating in this region.
Figs. 4 and 5 show temporal variation of nondimensional temperature profiles corresponding to
Fig. 4. Temporal variation of dimensionless temperature at different
x-axis locations for pulse type 2.
M. Kalyon, B.S. Yilbas / Applied Surface Science 252 (2006) 2242–2250
2249
4. Conclusions
Fig. 5. Temporal variation of dimensionless temperature at different
x-axis locations for pulse type 3.
other two pulse types employed in the simulations
(Table 1). It should be noted that the duration
(cooling period) between two pulses is reduced for
pulse type 2 and the cooling period is set to zero for
pulse type 3. The later (pulse type 3) provides the
basis for comparison of temperature profiles due to a
single pulse with double pulse intensity and pulse
type 3. Temperature profiles show similar trends to
those shown in Fig. 3a, provided that the magnitude
of maximum temperature increases considerably
after the second pulse arrives. This occurs because of
the cooling period between the pulses, which is short
for pulse type 2, i.e. temperature inside the substrate
material remains high after the first pulse when the
second pulse is initiated. Consequently, high base
temperature results in the attainment of a high
maximum temperature after the arrival of the second
pulse. In the case of non-existing cooling period
(pulse type 3), temperature profile rises continuously, i.e. the second pulse is triggered immediately
after the first pulse ends. Although the material
response to the second pulse results in gradual rise of
temperature during the early heating period of the
second pulse, the rise of temperature is not
discontinued, i.e. the rate of temperature rise
changes gradually due to two pulses. Material
response to a single pulse with double intensity
becomes significant when comparing its counterpart
due to two pulses with zero cooling period. In this
case, the rate of temperature rise is considerably
higher than the consecutive two pulses with no
cooling period between the pulses.
An analytical solution for repetitive laser pulse
heating of solid substrate is considered and a closed
form solution for temperature rise inside the material
is obtained using a Laplace transformation method.
The governing equation of heat transfer is nondimensionalized to obtain a closed form solution
independent of material and laser pulse properties and
the influence of the period (cooling period) between
two successive pulses on the temperature rise is
investigated. It is found that the magnitude of the
maximum surface temperature is influenced by the
cooling period of two successive pulses. The rapid
response of material to heating pulses is more
pronounced in the region just below the surface
(x < 1). The specific conclusions derived from the
present study can be listed as follows:
1. Temperature profiles obtained from the closed
form solution agrees well with the numerical
predictions. Analytical solution provides the
temperature distribution in the heating cycle as
well as in the cooling cycle of the heating pulse
(after the pulse ends).
2. The rate of temperature rise in the early heating
period is higher than that corresponding to longer
heating periods. In this case, internal energy gain
dominates over the conduction energy transfer
from surface vicinity to the solid bulk of the
substrate material.
3. Temperature rise due to consecutive tow pulses
without having the cooling period between them is
less than that corresponding to a single pulse with
double intensity.
Acknowledgment
The authors acknowledge the support of King Fahd
University of Petroleum and Minerals, Dhahran, Saudi
Arabia for this work.
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