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CONVERGENCE OF TIME EXPONENTIAL DECAYING PULSE TO INTENSITY STEP INPUT PULSE FOR LASER HEATING OF SEMI-INFINITE BODY
Int. C-.
Pergamon
@
Hec.uMaw Tran@er, VOL24, No.6, pp.78S-791, 1997 Copyright @ 1997Elsevier Sci.mceLtd Printed intheUSA.Allrightsreserved 0735-1933/97S17.00+ .00
PII S0735-1933(97)00065-1
CONVERGENCE OF TIME EXPONENTIAL DECAYING PULSE TO INTENSITY STEP INPUT PULSE FOR LASER HEATING OF SEMI-INFINITE BODY
Bekir Sami Yilbss and Muhammad Sarni Department of Mechanical Engineering King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia.
(Communicated by J.P. Hartnett and W.J. Mirdcowycz)
ABSTRACT The analytical formulation of lsser heating mechanism is useful when exploring the lsser machining process. The present study covers the analytical formulation of laser pulse heating process for time exponential decaying and intensity step input pulses. The convergence of both solution is introduced in relation to pulse parameters. The analytical expression for the convergence is omitted due to complex nature of the mathematics involved, however, computational method is attempted to identify the pulse parameter for which both results converge. It is found that the surface temperature profiles obtained for both pulses merge when @ approaches to zero. @ 1997ElsevierScienceLtd Introduction Lasers are widely used in industry due to their precision of operation and attainment of considerable end product quality. In order to improve the laser machining process, investigation into laser heating mechanism becomes essential. Many researchers [1, 2, 3, 4] examined the case in detail through theoretical models and experimental studies. Guassian-shaped laser output beam was considered by Lu [5] demonstrating that the square shape of the temperature profile was almost of’ the same size as the original Guassian beam. Neto and Limo [6] computed the temperature profiles inside the material for high intensity short pulses. They showed that the model developed predicts the temperature profiles in agreement with the results of experiment. In laser heating process, the actual lzser pulse has the property of time dependent power intensity. However, the pulse power intensity distribution may be simplified by either introducing the time exponentially decaying pulse or intensity step input pulse. Hence, the analytical approach governing the unsteady laser heating
785
B.S. Yilbas and M. Sanri
1.6
Vol. 24, No. 6
‘ ‘-‘ ‘‘‘ ‘‘‘‘ -
1.4
-
I
+
~
1.2
-
_ _
~=o, S1.pinputp.lse
1 0.8 0.6 0.4 0.2 0 o
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0.0008
0.0009
0.001
Time(s) FIG. 1 Pulse Profiles.
of solids using both time exponentially decaying pulse and intensity step input pulse may become fruitful for practical laser heating process.
The present study is carried out to develop analytical approach for laser heating process employing time exponentially decaying and intensity step input pulses. The study is extended to include the convergency of the results obtained due to both pulses.
Mathematical
Modeling
The laaer pulses considered in the present analysis are shown in Figure 1. The time exponentially decaying pulse may be obtained from COZ laser radiation and may be written as: I = IOe-@ In the csse of Nd YAG lsser, the output pulse is made up of multiple individual spikes of only a few microseconds in duration. The spikes may exhibit to form an envelop shape which may be approximated as intensity step input pulse. In formulating the heat transfer process, some useful assumptions may be made including: 1. The laser energy is instantaneously converted to heat at the point the beam was absorbed. Therefore, conventional heat transfer analysis can be possible.
Vol. 24, No. 6
LASER HEATING OF SEMI-INFINITEBODY
787
2. Thermal properties of the workpiece are assumed to be independent of temperature. 3. Radiative heat losses from the heated spot are neglected. 4. The surface of the workpiece is assumed to be a perfect absorbant at the incident laser beam wavelength. Time Exponential
Decaying
Pulse
The Fourier equation governing the time exponential decaying pulse may be written as:
a2T + Qe.(6=+&) = Lz ~k a at
(1)
The relevant boundary conditions are: (2)
T(z,o) =o
(3)
t) = o
(4)
T(co,
The solution of equation 1 is visible in the Laplace domain. Laplace transformation of equation 1 gives: (5)
with ~ = ‘T(z,
p), g2= p/x and p is the transformed variable.
The complete solution of equation 5 can be obtained after using the appropriate boundary conditions, therefore, the solution is: T=
1
e–6z ~e-9z –Ilc$ k(p+ /3) [g(g2– 62) – (g2– 62)
—
The inversion of equation 6 is visible using expansion of functions into partial fractions. lengthy mathematics, the inverted solution yields:
(6)
After
788
B.S. Yilbas and M. Sami
–exp(–h)
erfc(
vol. 24,No. 6
-J& - 6JZ)] + 2 ezp(-6z)[ezp(d2t) - ezp(-~t)]]
(7)
which on using the relationship, erfc(-z)=2-erfc (z), and using the notation for the complex error function:
and w(iz) = ezzer~c(z) it becomes:
‘(z’)=%%%] [ -~)-e’’’-(’’+’’)’l ‘8) The surface temperature can be obtained by setting x=O in the above equation: w(itifi)
‘(Z’) =Y[*I[
– f5fi@w(-@)
– exp(–@)]
(9)
Since ‘(z)
‘e-z’
%’-”
/ O’ec’d<
then @w(–@)
= –~e-ot @o
@
ec2d(
I
Hence,
T(o, t)= @
~
k [1~ + cd2 [
ezp(dzt) erfc(tifi)
+ 26fiF(/@)
-
ezp(-@)]
(lo)
where F’(z) is Dawson’s integral, F(z) =
ezp(–z2) o’ ezp(<2)d< /
Equation 10 gives the surface temperature, however, in the limit it is expected that this solution converges to intensity step input pulse. Consequently, the surface temperature profiles relevant to intensity step input pulse needs to be derived.
Intensity
Step Input Pulse
Fourier heat conduction equation relevant to intensity step input pulse may be written -as:
(11)
Vol. 24, No. 6
LASER HEATING OF SEMI-INFINITEBODY
789
with boundary conditions:
~lz=o =o T(z, O) = O and T(rm, t) = O The solution of equation 10 is visible by using Laplace transformation, i.e.: (12) where T =
T(z, p)
Substitution of boundary conditions gives the solution of equation 12 as:
~ = IOabe-6x IObe-qz 1 1 kp(p – d2) – —2kpq [— q–15 – q+6
–1
(13)
However, inversion of equation 13 yields the solution as:
T(z, t) = ~&
ierfc
I – ~e-h’
& ()
+& ezp(ff62t – 6x)erfc +&
(
?iAi
– ~ 2/Z7 )
exp(n$2t+ 6x)erjc 6fi+ (
A 2a
(14)
)
The surface temperature may be obtained when x=O in equation 14, i.e.
T(O,t) = fi
[ 26
~
Discussion
+ exp(a62t) erjc(6@)
(15)
– 1
and Conclusions
The solutions to heat transfer equation appropriate to laser heating are developed for time exponential decaying and intensity step input pulses. Computational technique is employed for the convergence of surface temperatures obtained for both time exponential decaying and intensity step input pulses allowing different pulse parameters for the time decaying pulse. Figure 2 shows the dimensionless surface temperature with dimensionless time
(cr62t)for all
pulses of different ~ values. It is evident that as @ approaches to zero, the temperature profiles corresponding to all pulses tend to merge. The influence of ~ on the temperature profiles becomes less significant for low values of @. Consequently, both analytical solutions become identical at limit when ~ approaches zero. Surface temperature rises rapidly in the early part of the heating
790
B.S. Yilbas and M. Sami
Vol. 24, No. 6
4000 3500 3000 2500 ~ a =2000 = & 1500 1000 500 0 (JE+I)O
IE+06 2E+06
3E+OIj
4E+lJIj
5E+06
fjE+lJlj
7E+C16
8E+06
9E+06
a6zt
FIG. 2 Dimensionless Surface Temperature with /3.
0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001
0 8.41 E+05
2.52E+06
4.20E+06
5.88E+06
7.56E+06
a62t
FIG. 3 Dimensionless First Derivative of Surface Temperature with Dimensionless Time for Different ,6.
Vol. 24, No. 6
LASER HEATJNG OF SEMI-JNFINITEBODY
791
pulse and reaches the steady rise as pulse progresses. This may indicate that the energy gain through absorption is dominant as compared to conduction losses in the early part of the pulse. However, conduction losses and internal energy gain balance each other which in turn results in steady temperature rise ~ the pulse progresses. In this caae, it may be visible to introduce an equilibrium rise time at which the internal energy gain balances the conduction losses. This may correspond to a point on the temperature curve where steady rise starts. Therefore, the equilibrium time is:
c r= ai$zt where c is a constant. It is evident that C varies with the pulse shape, however, it becomes constant at low values of /3. This is also evident from Figure 3 in which dimensionless derivative of surface temperature with dimensionless time is given. In this case, the shape of the curve becomes almost constant after the equilibrium point on the curve.
Acknowledgements The authors acknowledge the support of King Fahd University of Petroleum and Minerals. Dhahran, Saudi Arabia for this work. References 1. T. Qin and L. Tieu, ht. J. Heat Mass Transfer_,35(3),719
(1992). (Journal Article)
2. B. S.Yilbas, A. Sahin and R. Davies, J. Machine Tools and Mfg., 35(7), Article)
1047 (1995). (Journal
3. W. Schulz, D. Becker, J. Franke, R. Kernmerling and G. Herziger, J. F&w-. 1357 (1993). (Journal Article)
D: Appl. Phys, 26,
4. B. F. Blackwell, ASME .J. of Heat Transfer, 112, 567 (1990). (Journal Article) 5. Y. Lu, Appl. Surface Science, 81, 357 (1994). (Journal Article) 6. D. Neto and C. Lima, J. Phys D: Appl. Phys, 27, 1795 (1994). (Journal Article)
Received April 23, 1997
Pergamon
@
Hec.uMaw Tran@er, VOL24, No.6, pp.78S-791, 1997 Copyright @ 1997Elsevier Sci.mceLtd Printed intheUSA.Allrightsreserved 0735-1933/97S17.00+ .00
PII S0735-1933(97)00065-1
CONVERGENCE OF TIME EXPONENTIAL DECAYING PULSE TO INTENSITY STEP INPUT PULSE FOR LASER HEATING OF SEMI-INFINITE BODY
Bekir Sami Yilbss and Muhammad Sarni Department of Mechanical Engineering King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia.
(Communicated by J.P. Hartnett and W.J. Mirdcowycz)
ABSTRACT The analytical formulation of lsser heating mechanism is useful when exploring the lsser machining process. The present study covers the analytical formulation of laser pulse heating process for time exponential decaying and intensity step input pulses. The convergence of both solution is introduced in relation to pulse parameters. The analytical expression for the convergence is omitted due to complex nature of the mathematics involved, however, computational method is attempted to identify the pulse parameter for which both results converge. It is found that the surface temperature profiles obtained for both pulses merge when @ approaches to zero. @ 1997ElsevierScienceLtd Introduction Lasers are widely used in industry due to their precision of operation and attainment of considerable end product quality. In order to improve the laser machining process, investigation into laser heating mechanism becomes essential. Many researchers [1, 2, 3, 4] examined the case in detail through theoretical models and experimental studies. Guassian-shaped laser output beam was considered by Lu [5] demonstrating that the square shape of the temperature profile was almost of’ the same size as the original Guassian beam. Neto and Limo [6] computed the temperature profiles inside the material for high intensity short pulses. They showed that the model developed predicts the temperature profiles in agreement with the results of experiment. In laser heating process, the actual lzser pulse has the property of time dependent power intensity. However, the pulse power intensity distribution may be simplified by either introducing the time exponentially decaying pulse or intensity step input pulse. Hence, the analytical approach governing the unsteady laser heating
785
B.S. Yilbas and M. Sanri
1.6
Vol. 24, No. 6
‘ ‘-‘ ‘‘‘ ‘‘‘‘ -
1.4
-
I
+
~
1.2
-
_ _
~=o, S1.pinputp.lse
1 0.8 0.6 0.4 0.2 0 o
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0.0008
0.0009
0.001
Time(s) FIG. 1 Pulse Profiles.
of solids using both time exponentially decaying pulse and intensity step input pulse may become fruitful for practical laser heating process.
The present study is carried out to develop analytical approach for laser heating process employing time exponentially decaying and intensity step input pulses. The study is extended to include the convergency of the results obtained due to both pulses.
Mathematical
Modeling
The laaer pulses considered in the present analysis are shown in Figure 1. The time exponentially decaying pulse may be obtained from COZ laser radiation and may be written as: I = IOe-@ In the csse of Nd YAG lsser, the output pulse is made up of multiple individual spikes of only a few microseconds in duration. The spikes may exhibit to form an envelop shape which may be approximated as intensity step input pulse. In formulating the heat transfer process, some useful assumptions may be made including: 1. The laser energy is instantaneously converted to heat at the point the beam was absorbed. Therefore, conventional heat transfer analysis can be possible.
Vol. 24, No. 6
LASER HEATING OF SEMI-INFINITEBODY
787
2. Thermal properties of the workpiece are assumed to be independent of temperature. 3. Radiative heat losses from the heated spot are neglected. 4. The surface of the workpiece is assumed to be a perfect absorbant at the incident laser beam wavelength. Time Exponential
Decaying
Pulse
The Fourier equation governing the time exponential decaying pulse may be written as:
a2T + Qe.(6=+&) = Lz ~k a at
(1)
The relevant boundary conditions are: (2)
T(z,o) =o
(3)
t) = o
(4)
T(co,
The solution of equation 1 is visible in the Laplace domain. Laplace transformation of equation 1 gives: (5)
with ~ = ‘T(z,
p), g2= p/x and p is the transformed variable.
The complete solution of equation 5 can be obtained after using the appropriate boundary conditions, therefore, the solution is: T=
1
e–6z ~e-9z –Ilc$ k(p+ /3) [g(g2– 62) – (g2– 62)
—
The inversion of equation 6 is visible using expansion of functions into partial fractions. lengthy mathematics, the inverted solution yields:
(6)
After
788
B.S. Yilbas and M. Sami
–exp(–h)
erfc(
vol. 24,No. 6
-J& - 6JZ)] + 2 ezp(-6z)[ezp(d2t) - ezp(-~t)]]
(7)
which on using the relationship, erfc(-z)=2-erfc (z), and using the notation for the complex error function:
and w(iz) = ezzer~c(z) it becomes:
‘(z’)=%%%] [ -~)-e’’’-(’’+’’)’l ‘8) The surface temperature can be obtained by setting x=O in the above equation: w(itifi)
‘(Z’) =Y[*I[
– f5fi@w(-@)
– exp(–@)]
(9)
Since ‘(z)
‘e-z’
%’-”
/ O’ec’d<
then @w(–@)
= –~e-ot @o
@
ec2d(
I
Hence,
T(o, t)= @
~
k [1~ + cd2 [
ezp(dzt) erfc(tifi)
+ 26fiF(/@)
-
ezp(-@)]
(lo)
where F’(z) is Dawson’s integral, F(z) =
ezp(–z2) o’ ezp(<2)d< /
Equation 10 gives the surface temperature, however, in the limit it is expected that this solution converges to intensity step input pulse. Consequently, the surface temperature profiles relevant to intensity step input pulse needs to be derived.
Intensity
Step Input Pulse
Fourier heat conduction equation relevant to intensity step input pulse may be written -as:
(11)
Vol. 24, No. 6
LASER HEATING OF SEMI-INFINITEBODY
789
with boundary conditions:
~lz=o =o T(z, O) = O and T(rm, t) = O The solution of equation 10 is visible by using Laplace transformation, i.e.: (12) where T =
T(z, p)
Substitution of boundary conditions gives the solution of equation 12 as:
~ = IOabe-6x IObe-qz 1 1 kp(p – d2) – —2kpq [— q–15 – q+6
–1
(13)
However, inversion of equation 13 yields the solution as:
T(z, t) = ~&
ierfc
I – ~e-h’
& ()
+& ezp(ff62t – 6x)erfc +&
(
?iAi
– ~ 2/Z7 )
exp(n$2t+ 6x)erjc 6fi+ (
A 2a
(14)
)
The surface temperature may be obtained when x=O in equation 14, i.e.
T(O,t) = fi
[ 26
~
Discussion
+ exp(a62t) erjc(6@)
(15)
– 1
and Conclusions
The solutions to heat transfer equation appropriate to laser heating are developed for time exponential decaying and intensity step input pulses. Computational technique is employed for the convergence of surface temperatures obtained for both time exponential decaying and intensity step input pulses allowing different pulse parameters for the time decaying pulse. Figure 2 shows the dimensionless surface temperature with dimensionless time
(cr62t)for all
pulses of different ~ values. It is evident that as @ approaches to zero, the temperature profiles corresponding to all pulses tend to merge. The influence of ~ on the temperature profiles becomes less significant for low values of @. Consequently, both analytical solutions become identical at limit when ~ approaches zero. Surface temperature rises rapidly in the early part of the heating
790
B.S. Yilbas and M. Sami
Vol. 24, No. 6
4000 3500 3000 2500 ~ a =2000 = & 1500 1000 500 0 (JE+I)O
IE+06 2E+06
3E+OIj
4E+lJIj
5E+06
fjE+lJlj
7E+C16
8E+06
9E+06
a6zt
FIG. 2 Dimensionless Surface Temperature with /3.
0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001
0 8.41 E+05
2.52E+06
4.20E+06
5.88E+06
7.56E+06
a62t
FIG. 3 Dimensionless First Derivative of Surface Temperature with Dimensionless Time for Different ,6.
Vol. 24, No. 6
LASER HEATJNG OF SEMI-JNFINITEBODY
791
pulse and reaches the steady rise as pulse progresses. This may indicate that the energy gain through absorption is dominant as compared to conduction losses in the early part of the pulse. However, conduction losses and internal energy gain balance each other which in turn results in steady temperature rise ~ the pulse progresses. In this caae, it may be visible to introduce an equilibrium rise time at which the internal energy gain balances the conduction losses. This may correspond to a point on the temperature curve where steady rise starts. Therefore, the equilibrium time is:
c r= ai$zt where c is a constant. It is evident that C varies with the pulse shape, however, it becomes constant at low values of /3. This is also evident from Figure 3 in which dimensionless derivative of surface temperature with dimensionless time is given. In this case, the shape of the curve becomes almost constant after the equilibrium point on the curve.
Acknowledgements The authors acknowledge the support of King Fahd University of Petroleum and Minerals. Dhahran, Saudi Arabia for this work. References 1. T. Qin and L. Tieu, ht. J. Heat Mass Transfer_,35(3),719
(1992). (Journal Article)
2. B. S.Yilbas, A. Sahin and R. Davies, J. Machine Tools and Mfg., 35(7), Article)
1047 (1995). (Journal
3. W. Schulz, D. Becker, J. Franke, R. Kernmerling and G. Herziger, J. F&w-. 1357 (1993). (Journal Article)
D: Appl. Phys, 26,
4. B. F. Blackwell, ASME .J. of Heat Transfer, 112, 567 (1990). (Journal Article) 5. Y. Lu, Appl. Surface Science, 81, 357 (1994). (Journal Article) 6. D. Neto and C. Lima, J. Phys D: Appl. Phys, 27, 1795 (1994). (Journal Article)
Received April 23, 1997