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Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions
G Model IJLEO-58526; No. of Pages 7
ARTICLE IN PRESS Optik xxx (2016) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions Guibo Chen ∗ School of Science, Changchun University of Science and Technology, Changchun 130022, PR China
a r t i c l e
i n f o
Article history: Received 30 September 2016 Accepted 23 November 2016 Keywords: Long pulsed laser heating Convective boundary conditions Analytical solutions Modeling
a b s t r a c t In this paper, we present an axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions. The two-dimensional axisymmetric physical model of the temperature rise problem for long pulse laser heating material with convective boundary conditions is established, and analytical solutions of the governing heat transfer equation are obtained by using the integral transformation method. It is found that the results obtained from the analytical solutions agree well with the existing finite element method. Temperatures of Silicon bulk for different convective boundary conditions are modeled, and effects of heat transfer coefficient and ambient temperature on the temperature distributions of Silicon are analyzed. © 2016 Elsevier GmbH. All rights reserved.
1. Introduction In recent years, long pulsed laser heating has become a research hotspot in the fields of laser processing and laser damage, because of its larger energy and higher heating efficiency [1–5]. Modeling of interaction between long pulsed laser and material is an important method to find physical mechanism and characteristics. Moreover, modeling can not only reduce the experimental cost and minimizes the experimentation time, but also able to provide results of laser heating for sufficient conditions even if in the environment that traditional experiment cannot achieve [6–8]. The modeling of physical problems can be divided into numerical modeling and analytical modeling. Analytical modeling establishes a direct functional relationship between the parameters and the laser heating process, which can provide very useful information for revealing mechanism of the laser irradiation effects and parameters optimization of the laser. Considerable research studies were carried out to solve the laser heating using analytical modeling. A closed form analytical solution for temperature rise inside solid substrate due to time exponentially varying laser pulse was obtained using Laplace transform by Yilbas [9]. Bi et al. presented a 2-D axisymmetric physical model to express the temperature and thermal stress in damage process of Silicon plate irradiated by annular millisecond laser, and analytical solutions of 2-D heat conduction equation and thermo-elastic equations are obtained using integral transform method [10]. Bi and Chen studied the surface damage of thermal decomposition to GaAs induced by a 532 nm millisecond pulse laser using a semi-analytical method [11]. An analytical solution for 2-D modeling of repetitive long pulse laser heating material was given using the method of separation of variables combining with Laplace transform by Chen et al. [12]. Yilbas introduced the analytical modeling methods and the related results of laser heating in the application field systematically [6].
∗ Correspondence to: No. 7089 of Weixing Road, Chaoyang District, Changchun City, Jilin Province, PR China. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijleo.2016.11.110 0030-4026/© 2016 Elsevier GmbH. All rights reserved.
Please cite this article in press as: G. Chen, Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions, Optik - Int. J. Light Electron Opt. (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.110
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Fig. 1. Schematic diagram of laser irradiation material surface.
In the current studies, most of boundary conditions are assumed to be adiabatic. But in the practical applications of laser processing or laser damage, it often encounters the convective boundary conditions [13,14]. In this paper, the convective boundary conditions are used to study the analytical modeling of long pulsed laser heating material. It is organized as follows. In Section 2 we introduce the theoretical model of long pulsed laser heating with convective boundary conditions, and analytical solutions of the governing equation for temperature are obtained. Results of temperature for different convective boundary conditions are presented in Section 3. Our main conclusions are summarized in Section 4. 2. Theoretical model The classical Fourier heat transfer equation for a long pulsed laser heating with a 2-D axisymmetric form can be written as [15]: 2
2
∂ T (r, z, t) 1 ∂T (r, z, t) ∂ T (r, z, t) Q (r, z, t) 1 ∂T (r, z, t) = + + + r ˛ k ∂r 2 ∂r ∂z 2 ∂t
(1)
where, k is the thermal conductivity of the material, ˛ = k/c is the thermal diffusivity, is the mass density and c is the heat capacity of the material. The temperature T is defined here as a function of (r, z, t), and variable ranges of the positional arguments r, z are 0 < r ≤ R, 0 < z ≤ H respectively (as shown in Fig. 1). In Eq. (1), Q (r, z, t) is the source function of laser heating. If we assume that the laser intensity is Gaussian distribution, and the energy gain mechanism of the material to the laser is the body absorption, then Q (r, z, t) can be expressed as:
Q (r, z, t) = I0 1 − rf ı exp −ız exp −r 2 /r02 g (t)
(2)
where, rf is the reflection coefficient, ı is absorption coefficient of the material, r0 is waist radius, and g (t) is time distribution function of laser intensity and yields:
g (t) =
1 for t ≤ p
(3)
0 for t > p
where p is pulse width. It is assumed that the boundary conditions of Eq. (1) are as:
∂T + h1 T = f1 ∂r
r=R:
k
z=0:
−k
z=H:
−k
(4)
∂T + h2 T = f2 ∂z
(5)
∂T =0 ∂z
(6)
where, h1 is the heat transfer coefficient of profile at r = R, h2 is the heat transfer coefficient of surface at z = 0, f1 is the ambient temperature of profile at r = R, and f2 is the ambient temperature of surface at z = 0. We assume that it is adiabatic at z = H. The initial condition yields: T (r, z, t) |t=0 = T0
(7)
Please cite this article in press as: G. Chen, Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions, Optik - Int. J. Light Electron Opt. (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.110
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In order to solve Eqs. (1)–(6), forward and inverse transforms for T (r, z, t) about r are introduced based on integral transformation method [15] and written as: T (r, z, t) =
∞ J0 (n r) · T¯ (n , z, t)
(8)
C1 (n )
n=1
R
r J0 n r T r , z, t dr
T¯ (n , z, t) =
(9)
0
where, J0 (n r) is the zero order Bessel function of the first kind, n (n ≥ 0, n = 1, 2, 3, · · ·) is the n-th positive root of equation
h1 J k 0
(n R) − n J1 (n R) = 0, and C1 (n ) is the normalization integral as: R2 J02 (n R)
C1 (n ) =
22n
2n + h1 /k
2
(10)
Eq. (9) is used to transform every term in Eq. (1) and the initial Condition (7), then using the Green theorem and the boundary Condition (4) at r = R, one can obtain that: −2n T¯ (n , z, t) +
2 ∂ T¯ (n , z, t) 1 ∂T¯ (n , z, t) + A (n , z, t) = 2 ˛ ∂z ∂t
(11)
where, Q¯ (n , z, t) R + J0 (n R) f1 k k
A (n , z, t) =
R
(12)
r J0 n r Q r , z, t dr
Q¯ (n , z, t) =
(13)
0
Transform of boundary Conditions (5) and (6) can be written as:
∂T¯ + h2 T¯ = f¯2 ∂z
(14)
∂T¯ =0 ∂z
(15)
z=0:
−k
z=H:
−k
where,
R
r J0 n r f2 dr
f¯2 =
(16)
0
Transform of initial Condition (7) is:
R
r J0 n r T0 dr
T¯ (n , z, t) |t=0 = T¯ 0 =
(17)
0
Next, the forward and inverse transforms for T¯ (n , z, t) about z are written as: T¯ (n , z, t) =
∞ cos [m (H − z)]
C2 (m )
m=1
T˜¯ (n , m , t) =
H
· T˜¯ (n , m , t)
cos m H − z
(18)
T¯ n , z , t dz
(19)
0
where, m (m ≥ 0, m = 1, 2, 3, · · ·) is the m-th positive root of equation m tan (m H) − h2 /k = 0, C2 (m ) is the normalization integral as:
C2 (m ) =
H
cos 0
2
m H − z
dz =
1 2
H 2m + h2 /k
2
2m + h2 /k
+ h2 /k
2
(20)
Eq. (19) is used to transform each term of Eq. (11), and considering the boundary Conditions (14) and (15), we can obtain that:
dT˜¯ (n , m , t) + ˛ 2n + 2m T˜¯ (n , m , t) = B (n , m , t) dt
(21)
Please cite this article in press as: G. Chen, Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions, Optik - Int. J. Light Electron Opt. (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.110
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where, R ˛Q˜¯ (n , m , t) ˛ + J0 (n R) f˜1 + cos (m H) f¯2 k k k
B (n , m , t) =
(22)
Using Eq. (13), double integral Q˜¯ n , j , t in Eq. (22) can be written as:
Q˜¯ n , j , t =
= I0 1 − rf
0
R
Q r , z , t dr dz
0
(23)
ıg (t ) ˝1 ˝2
cos m H − z 0
r J0 n r cos m H − z
H
˝1 =
=
H
exp −ız dz
ı + e−ıH m sin ıH − ı cos (m H)
(24)
2m + ı2
R
r J0 n r
˝2 =
exp −r /r02 dr 2
(25)
0
and,
H
cos m H − z
f˜1 =
f1 dz =
0
R
f1 sin (m H) m
(26)
r J0 n r f2 dr = f2 ˝3
f¯2 =
(27)
0
where,
R
r J0 n r dr
˝3 =2
(28)
0
The Eq. (21) is an ordinary differential equation about T˜¯ (n , m , t), which is easy to be solved. We substitute the solution of Eq. (21) into Eq. (18), solutions of T¯ (n , z, t) can be obtained. Furthermore, using T¯ (n , z, t) and Eq. (8), solutions of T (r, z, t) can be written as: T (r, z, t) =
˛ k
∞ ∞ J0 (n r) · cos [m (H − z)]
C1 (n ) · C2 (m )
n=1 m=1
t
exp ˛
2n
+ 2m
t
· exp −˛ 2n + 2m t ×
· B n , m , t
(29)
dt + T˜¯ 0 (n , m )
0
where,
T˜¯ 0 (n , m ) =
H
R
r J0 n r cos m H − z 0
0
T0 dr dz
(30)
= ˝3 T0 sin (m H) /m It notes that ˝2 in (25) needs to be calculated by numerical integration, the Gauss–Kronrod quadrature method is used to handle it [16]. Substituting Eqs. (22)–(28) into Eq. (29), the temperature T (r, z, t) of any location (r, z) in the material can be calculated at any time t. 3. Results and discussions Temperature of Silicon bulk irradiated by a long pulsed laser considering the effects of convective boundary conditions is modeled using above analytical solutions. The thermo-physical and geometric parameters of Silicon bulk are given in Table 1. We assume that the laser wavelength is 1.06 m, the laser peak power intensity is 1 × 109 W/m2 , and the initial temperature of Silicon is 0 ◦ C. All of the following calculated examples are using the above computational parameters. 3.1. Method validation It is assumed that the long pulsed laser is irradiated to the centre of the Silicon surface (as shown in Fig. 1), the waist radius of laser is 3 mm, and the pulse width of laser is 1 ms. The heat transfer coefficients h1 and h2 are equal to 100, and the ambient temperature f1 and f2 are equal to 0 ◦ C. Please cite this article in press as: G. Chen, Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions, Optik - Int. J. Light Electron Opt. (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.110
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Table 1 Thermo-physical and geometric parameters of Silicon. Parameters
Values
density/(kg/m3 ) specific heat/(J/kg K) coefficient of heat conductivity/(W/m K) reflection coefficient absorption coefficient/m radius of material/mm thickness of material/mm
2330 712 156 0.33 5000 5 2
Fig. 2. Temporal distributions of temperature for different N and M.
Fig. 3. Radial distributions of temperature for different N and M.
Temperature distributions for different irradiation time and different radial locations were calculated by using the analytical Solutions (29) of this paper and the finite element method (FEM) of the published literature [8] as shown in Figs. 2 and 3. We note that in Expression (29), only the finite N,M terms in the series of n,m need to be evaluated, while the rest of the terms can be omitted. It can be seen that the analytical solutions of this paper converges quickly with the gradual increase of N,M and the calculated results of the analytical solutions and the FEM are very close when N,M are equal to 40. Therefore, we can conclude that the presented method is correct and effective. Taking into account the calculation accuracy and computational efficiency, the following examples in this paper are calculated based on N = M = 50. In addition, we have also done a lot of validation samples, and results show that the analytical method is correct. Due to limited space, we cannot present all the validation results. 3.2. Results and discussions 3.2.1. Effects of surface convection In order to study the effects of surface convective boundary conditions on temperature distributions, we assume that the pulse width of laser is 1 ms, the waist radius is 4 mm, the heat transfer coefficient h1 is 0, and the ambient temperature f1 is 0 ◦ C. Fig. 4 gives axial temperature distributions of Silicon surface centre for f2 = 0 ◦ C and h2 = 100, 10 000, 100 000, 1 000 000 respectively. It can be seen that the heat transfer coefficient has a significant effect on the temperature distributions, especially the temperature of material surface. The greater the heat transfer coefficient, the lower the surface temperature. Please cite this article in press as: G. Chen, Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions, Optik - Int. J. Light Electron Opt. (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.110
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Fig. 4. Axial distributions of temperature for different h2.
Fig. 5. Axial distributions of temperature for different f2.
Fig. 6. Axial distributions of temperature for different h1.
With the increase of depth, the influence of heat transfer coefficient on the temperature distributions becomes weaker and weaker. When it reaches a certain depth, the heat transfer coefficient has little influence on the temperature distributions. Next, we investigated the effects of ambient temperature on the temperature distributions of material. The axial temperature distributions of material for h2 = 1 000 000 and f2 = −100, 0, 100, 300, 500 ◦ C are given as shown in Fig. 5. It can be seen that the ambient temperature has a significant effect on the temperature distributions of material, especially the temperature of material surface. The lower the ambient temperature, the lower the temperature of material surface. With the increase of depth, the influence of ambient temperature on the temperature distributions becomes weaker and weaker. When it reaches a certain depth, the ambient temperature has little influence on the temperature distributions of material. 3.2.2. Effects of profile convection In order to study the effects of profile convective on temperature distributions, we still assume that the pulse width of laser is 1 ms and the waist radius is 4 mm. The heat transfer coefficient h2 is 0, and the ambient temperature f2 is 0 ◦ C. Fig. 6 gives axial temperature distributions of Silicon surface centre for f1 = 0◦ C and h1 = 100, 10 000, 100 000, 1 000 000 respectively. It can be seen that the heat transfer coefficient has a significant effect on the temperature distributions, especially in the area near r = R. The greater the heat transfer coefficient, the lower the surface temperature. With the decrease of the radial Please cite this article in press as: G. Chen, Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions, Optik - Int. J. Light Electron Opt. (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.110
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Fig. 7. Axial distributions of temperature for different f1.
distance, the influence of heat transfer coefficient on the temperature distributions becomes weaker and weaker. When it reaches a certain distance, the heat transfer coefficient has little influence on the temperature distributions. Next, we investigated the effects of ambient temperature on the temperature distributions of material. The axial temperature distributions of material for h1 = 1 000 000 and f1 = −200, −100, 0, 100, 200 ◦ C are given as shown in Fig. 7. It can be seen that the ambient temperature has a significant effect on the temperature distributions of material, especially in the area near r = R. The lower the ambient temperature, the lower the temperature in this area. With the decrease of the radial distance, the influence of ambient temperature on the temperature distributions becomes weaker and weaker. When it reaches a certain distance, the ambient temperature has little influence on the temperature distributions of material. 4. Conclusions (1) Analytical solutions of modeling for temperature of material irradiated by a long pulsed laser considering the effects of convection boundary conditions are obtained by using 2-D integral transform method. (2) The heat transfer coefficient and ambient temperature have significant effect on the temperature distributions of material. The bigger the heat transfer coefficient, and the larger the difference between the ambient temperature and the initial temperature of material, the greater the influence of convection boundary conditions on the temperature distributions of material. Acknowledgement This work was supported by the Changchun University of Science and Technology Young Scientists Project under Grant No. XQNJJ-2014-03. References [1] Z.W. Li, H.C. Zhang, Z.H. Shen, X.W. Ni, Time-resolved temperature measurement and numerical simulation of millisecond laser irradiated silicon, J. Appl. Phys. 114 (2013) 033104. [2] Z.W. Li, X. Wang, Z.H. Shen, X.W. Ni, Numerical simulation of millisecond laser-induced damage in silicon-based positive-intrinsic-negative photodiode, Appl. Opt. 51 (2012) 2759–2766. [3] Z.W. Li, X. Wang, Z.H. Shen, J. Lu, X.W. Ni, Mechanisms for the millisecond laser-induced functional damage to silicon charge-coupled imaging sensors, Appl. Opt. 54 (2015) 378–388. [4] X. Wang, Z.H. Shen, J. Lu, X.W. Ni, Laser-induced damage threshold of silicon in millisecond, nanosecond, and picosecond regimes, J. Appl. Phys. 108 (2010) 033103. [5] Y.X. Pan, X.M. Lv, H.C. Zhang, J. Chen, B. Han, Z.H. Shen, J. Lu, X.W. Ni, Millisecond laser machining of transparent materials assisted by a nanosecond laser with different delays, Opt. Lett. 41 (2016) 2807–2810. [6] B.S. Yilbas, Laser Heating Applications: Analytical Modeling, Elsevier, Amsterdam, 2012. [7] J. Bi, X.H. Zhang, X.W. Ni, G.Y. Jin, C.L. Li, Numerical simulation of thermal damage effects on gallium arsenide (GaAs) induced by a 0.53 (m wavelength long pulsed laser, Lasers Eng. 22 (2011) 37–46. [8] G.B. Chen, X.Y. Gu, J. Bi, Numerical analysis of thermal effect in aluminum alloy by repetition frequency pulsed laser, Optik 127 (2016) 10115–10121. [9] B.S. Yilbas, A closed form solution for temperature rise inside solid substrate due to time exponentially varying pulse, Int. J. Heat Mass Transf. 45 (2002) 1993–2000. [10] J. Bi, G.B. Chen, G.Y. Jin, X.H. Zhang, Analytical modelling of annular millisecond laser-induced damage in Silicon, Lasers Eng. 29 (2014) 175–187. [11] J. Bi, G.B. Chen, Characteristic analysis of 532 nm millisecond pulse laser-induced surface damage in the shape of thermal decomposition to GaAs by means of a Semi-analytical method, Lasers Eng. 32 (2015) 99–117. [12] G.B. Chen, Y.D. Wang, J.J. Zhang, J. Bi, An analytical solution for two-dimensional modeling of repetitive long pulse laser heating material, Int. J. Heat Mass Transf. 104 (2017) 503–509. [13] B.S. Yilbas, M. Kalyon, Repetitive laser pulse heating with a convective boundary condition at the surface, J. Phys. D: Appl. Phys. 34 (2001) 222–231. [14] B.S. Yilbas, M. Kalyon, Parametric variation of the maximum surface temperature during laser heating with convective boundary conditions, J. Mech. Eng. Sci. 216 (2002) 691–700. [15] M.N. Özisik, Heat Conduction, John Wiley and Sons, New York, 1993. [16] D.P. Laurie, Calculation of Gauss-Kronrod quadrature rules, Math. Comput. 66 (1997) 1133–1145.
Please cite this article in press as: G. Chen, Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions, Optik - Int. J. Light Electron Opt. (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.110
ARTICLE IN PRESS Optik xxx (2016) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions Guibo Chen ∗ School of Science, Changchun University of Science and Technology, Changchun 130022, PR China
a r t i c l e
i n f o
Article history: Received 30 September 2016 Accepted 23 November 2016 Keywords: Long pulsed laser heating Convective boundary conditions Analytical solutions Modeling
a b s t r a c t In this paper, we present an axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions. The two-dimensional axisymmetric physical model of the temperature rise problem for long pulse laser heating material with convective boundary conditions is established, and analytical solutions of the governing heat transfer equation are obtained by using the integral transformation method. It is found that the results obtained from the analytical solutions agree well with the existing finite element method. Temperatures of Silicon bulk for different convective boundary conditions are modeled, and effects of heat transfer coefficient and ambient temperature on the temperature distributions of Silicon are analyzed. © 2016 Elsevier GmbH. All rights reserved.
1. Introduction In recent years, long pulsed laser heating has become a research hotspot in the fields of laser processing and laser damage, because of its larger energy and higher heating efficiency [1–5]. Modeling of interaction between long pulsed laser and material is an important method to find physical mechanism and characteristics. Moreover, modeling can not only reduce the experimental cost and minimizes the experimentation time, but also able to provide results of laser heating for sufficient conditions even if in the environment that traditional experiment cannot achieve [6–8]. The modeling of physical problems can be divided into numerical modeling and analytical modeling. Analytical modeling establishes a direct functional relationship between the parameters and the laser heating process, which can provide very useful information for revealing mechanism of the laser irradiation effects and parameters optimization of the laser. Considerable research studies were carried out to solve the laser heating using analytical modeling. A closed form analytical solution for temperature rise inside solid substrate due to time exponentially varying laser pulse was obtained using Laplace transform by Yilbas [9]. Bi et al. presented a 2-D axisymmetric physical model to express the temperature and thermal stress in damage process of Silicon plate irradiated by annular millisecond laser, and analytical solutions of 2-D heat conduction equation and thermo-elastic equations are obtained using integral transform method [10]. Bi and Chen studied the surface damage of thermal decomposition to GaAs induced by a 532 nm millisecond pulse laser using a semi-analytical method [11]. An analytical solution for 2-D modeling of repetitive long pulse laser heating material was given using the method of separation of variables combining with Laplace transform by Chen et al. [12]. Yilbas introduced the analytical modeling methods and the related results of laser heating in the application field systematically [6].
∗ Correspondence to: No. 7089 of Weixing Road, Chaoyang District, Changchun City, Jilin Province, PR China. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijleo.2016.11.110 0030-4026/© 2016 Elsevier GmbH. All rights reserved.
Please cite this article in press as: G. Chen, Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions, Optik - Int. J. Light Electron Opt. (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.110
G Model
ARTICLE IN PRESS
IJLEO-58526; No. of Pages 7
G. Chen / Optik xxx (2016) xxx–xxx
2
Fig. 1. Schematic diagram of laser irradiation material surface.
In the current studies, most of boundary conditions are assumed to be adiabatic. But in the practical applications of laser processing or laser damage, it often encounters the convective boundary conditions [13,14]. In this paper, the convective boundary conditions are used to study the analytical modeling of long pulsed laser heating material. It is organized as follows. In Section 2 we introduce the theoretical model of long pulsed laser heating with convective boundary conditions, and analytical solutions of the governing equation for temperature are obtained. Results of temperature for different convective boundary conditions are presented in Section 3. Our main conclusions are summarized in Section 4. 2. Theoretical model The classical Fourier heat transfer equation for a long pulsed laser heating with a 2-D axisymmetric form can be written as [15]: 2
2
∂ T (r, z, t) 1 ∂T (r, z, t) ∂ T (r, z, t) Q (r, z, t) 1 ∂T (r, z, t) = + + + r ˛ k ∂r 2 ∂r ∂z 2 ∂t
(1)
where, k is the thermal conductivity of the material, ˛ = k/c is the thermal diffusivity, is the mass density and c is the heat capacity of the material. The temperature T is defined here as a function of (r, z, t), and variable ranges of the positional arguments r, z are 0 < r ≤ R, 0 < z ≤ H respectively (as shown in Fig. 1). In Eq. (1), Q (r, z, t) is the source function of laser heating. If we assume that the laser intensity is Gaussian distribution, and the energy gain mechanism of the material to the laser is the body absorption, then Q (r, z, t) can be expressed as:
Q (r, z, t) = I0 1 − rf ı exp −ız exp −r 2 /r02 g (t)
(2)
where, rf is the reflection coefficient, ı is absorption coefficient of the material, r0 is waist radius, and g (t) is time distribution function of laser intensity and yields:
g (t) =
1 for t ≤ p
(3)
0 for t > p
where p is pulse width. It is assumed that the boundary conditions of Eq. (1) are as:
∂T + h1 T = f1 ∂r
r=R:
k
z=0:
−k
z=H:
−k
(4)
∂T + h2 T = f2 ∂z
(5)
∂T =0 ∂z
(6)
where, h1 is the heat transfer coefficient of profile at r = R, h2 is the heat transfer coefficient of surface at z = 0, f1 is the ambient temperature of profile at r = R, and f2 is the ambient temperature of surface at z = 0. We assume that it is adiabatic at z = H. The initial condition yields: T (r, z, t) |t=0 = T0
(7)
Please cite this article in press as: G. Chen, Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions, Optik - Int. J. Light Electron Opt. (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.110
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In order to solve Eqs. (1)–(6), forward and inverse transforms for T (r, z, t) about r are introduced based on integral transformation method [15] and written as: T (r, z, t) =
∞ J0 (n r) · T¯ (n , z, t)
(8)
C1 (n )
n=1
R
r J0 n r T r , z, t dr
T¯ (n , z, t) =
(9)
0
where, J0 (n r) is the zero order Bessel function of the first kind, n (n ≥ 0, n = 1, 2, 3, · · ·) is the n-th positive root of equation
h1 J k 0
(n R) − n J1 (n R) = 0, and C1 (n ) is the normalization integral as: R2 J02 (n R)
C1 (n ) =
22n
2n + h1 /k
2
(10)
Eq. (9) is used to transform every term in Eq. (1) and the initial Condition (7), then using the Green theorem and the boundary Condition (4) at r = R, one can obtain that: −2n T¯ (n , z, t) +
2 ∂ T¯ (n , z, t) 1 ∂T¯ (n , z, t) + A (n , z, t) = 2 ˛ ∂z ∂t
(11)
where, Q¯ (n , z, t) R + J0 (n R) f1 k k
A (n , z, t) =
R
(12)
r J0 n r Q r , z, t dr
Q¯ (n , z, t) =
(13)
0
Transform of boundary Conditions (5) and (6) can be written as:
∂T¯ + h2 T¯ = f¯2 ∂z
(14)
∂T¯ =0 ∂z
(15)
z=0:
−k
z=H:
−k
where,
R
r J0 n r f2 dr
f¯2 =
(16)
0
Transform of initial Condition (7) is:
R
r J0 n r T0 dr
T¯ (n , z, t) |t=0 = T¯ 0 =
(17)
0
Next, the forward and inverse transforms for T¯ (n , z, t) about z are written as: T¯ (n , z, t) =
∞ cos [m (H − z)]
C2 (m )
m=1
T˜¯ (n , m , t) =
H
· T˜¯ (n , m , t)
cos m H − z
(18)
T¯ n , z , t dz
(19)
0
where, m (m ≥ 0, m = 1, 2, 3, · · ·) is the m-th positive root of equation m tan (m H) − h2 /k = 0, C2 (m ) is the normalization integral as:
C2 (m ) =
H
cos 0
2
m H − z
dz =
1 2
H 2m + h2 /k
2
2m + h2 /k
+ h2 /k
2
(20)
Eq. (19) is used to transform each term of Eq. (11), and considering the boundary Conditions (14) and (15), we can obtain that:
dT˜¯ (n , m , t) + ˛ 2n + 2m T˜¯ (n , m , t) = B (n , m , t) dt
(21)
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where, R ˛Q˜¯ (n , m , t) ˛ + J0 (n R) f˜1 + cos (m H) f¯2 k k k
B (n , m , t) =
(22)
Using Eq. (13), double integral Q˜¯ n , j , t in Eq. (22) can be written as:
Q˜¯ n , j , t =
= I0 1 − rf
0
R
Q r , z , t dr dz
0
(23)
ıg (t ) ˝1 ˝2
cos m H − z 0
r J0 n r cos m H − z
H
˝1 =
=
H
exp −ız dz
ı + e−ıH m sin ıH − ı cos (m H)
(24)
2m + ı2
R
r J0 n r
˝2 =
exp −r /r02 dr 2
(25)
0
and,
H
cos m H − z
f˜1 =
f1 dz =
0
R
f1 sin (m H) m
(26)
r J0 n r f2 dr = f2 ˝3
f¯2 =
(27)
0
where,
R
r J0 n r dr
˝3 =2
(28)
0
The Eq. (21) is an ordinary differential equation about T˜¯ (n , m , t), which is easy to be solved. We substitute the solution of Eq. (21) into Eq. (18), solutions of T¯ (n , z, t) can be obtained. Furthermore, using T¯ (n , z, t) and Eq. (8), solutions of T (r, z, t) can be written as: T (r, z, t) =
˛ k
∞ ∞ J0 (n r) · cos [m (H − z)]
C1 (n ) · C2 (m )
n=1 m=1
t
exp ˛
2n
+ 2m
t
· exp −˛ 2n + 2m t ×
· B n , m , t
(29)
dt + T˜¯ 0 (n , m )
0
where,
T˜¯ 0 (n , m ) =
H
R
r J0 n r cos m H − z 0
0
T0 dr dz
(30)
= ˝3 T0 sin (m H) /m It notes that ˝2 in (25) needs to be calculated by numerical integration, the Gauss–Kronrod quadrature method is used to handle it [16]. Substituting Eqs. (22)–(28) into Eq. (29), the temperature T (r, z, t) of any location (r, z) in the material can be calculated at any time t. 3. Results and discussions Temperature of Silicon bulk irradiated by a long pulsed laser considering the effects of convective boundary conditions is modeled using above analytical solutions. The thermo-physical and geometric parameters of Silicon bulk are given in Table 1. We assume that the laser wavelength is 1.06 m, the laser peak power intensity is 1 × 109 W/m2 , and the initial temperature of Silicon is 0 ◦ C. All of the following calculated examples are using the above computational parameters. 3.1. Method validation It is assumed that the long pulsed laser is irradiated to the centre of the Silicon surface (as shown in Fig. 1), the waist radius of laser is 3 mm, and the pulse width of laser is 1 ms. The heat transfer coefficients h1 and h2 are equal to 100, and the ambient temperature f1 and f2 are equal to 0 ◦ C. Please cite this article in press as: G. Chen, Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions, Optik - Int. J. Light Electron Opt. (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.110
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Table 1 Thermo-physical and geometric parameters of Silicon. Parameters
Values
density/(kg/m3 ) specific heat/(J/kg K) coefficient of heat conductivity/(W/m K) reflection coefficient absorption coefficient/m radius of material/mm thickness of material/mm
2330 712 156 0.33 5000 5 2
Fig. 2. Temporal distributions of temperature for different N and M.
Fig. 3. Radial distributions of temperature for different N and M.
Temperature distributions for different irradiation time and different radial locations were calculated by using the analytical Solutions (29) of this paper and the finite element method (FEM) of the published literature [8] as shown in Figs. 2 and 3. We note that in Expression (29), only the finite N,M terms in the series of n,m need to be evaluated, while the rest of the terms can be omitted. It can be seen that the analytical solutions of this paper converges quickly with the gradual increase of N,M and the calculated results of the analytical solutions and the FEM are very close when N,M are equal to 40. Therefore, we can conclude that the presented method is correct and effective. Taking into account the calculation accuracy and computational efficiency, the following examples in this paper are calculated based on N = M = 50. In addition, we have also done a lot of validation samples, and results show that the analytical method is correct. Due to limited space, we cannot present all the validation results. 3.2. Results and discussions 3.2.1. Effects of surface convection In order to study the effects of surface convective boundary conditions on temperature distributions, we assume that the pulse width of laser is 1 ms, the waist radius is 4 mm, the heat transfer coefficient h1 is 0, and the ambient temperature f1 is 0 ◦ C. Fig. 4 gives axial temperature distributions of Silicon surface centre for f2 = 0 ◦ C and h2 = 100, 10 000, 100 000, 1 000 000 respectively. It can be seen that the heat transfer coefficient has a significant effect on the temperature distributions, especially the temperature of material surface. The greater the heat transfer coefficient, the lower the surface temperature. Please cite this article in press as: G. Chen, Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions, Optik - Int. J. Light Electron Opt. (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.110
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Fig. 4. Axial distributions of temperature for different h2.
Fig. 5. Axial distributions of temperature for different f2.
Fig. 6. Axial distributions of temperature for different h1.
With the increase of depth, the influence of heat transfer coefficient on the temperature distributions becomes weaker and weaker. When it reaches a certain depth, the heat transfer coefficient has little influence on the temperature distributions. Next, we investigated the effects of ambient temperature on the temperature distributions of material. The axial temperature distributions of material for h2 = 1 000 000 and f2 = −100, 0, 100, 300, 500 ◦ C are given as shown in Fig. 5. It can be seen that the ambient temperature has a significant effect on the temperature distributions of material, especially the temperature of material surface. The lower the ambient temperature, the lower the temperature of material surface. With the increase of depth, the influence of ambient temperature on the temperature distributions becomes weaker and weaker. When it reaches a certain depth, the ambient temperature has little influence on the temperature distributions of material. 3.2.2. Effects of profile convection In order to study the effects of profile convective on temperature distributions, we still assume that the pulse width of laser is 1 ms and the waist radius is 4 mm. The heat transfer coefficient h2 is 0, and the ambient temperature f2 is 0 ◦ C. Fig. 6 gives axial temperature distributions of Silicon surface centre for f1 = 0◦ C and h1 = 100, 10 000, 100 000, 1 000 000 respectively. It can be seen that the heat transfer coefficient has a significant effect on the temperature distributions, especially in the area near r = R. The greater the heat transfer coefficient, the lower the surface temperature. With the decrease of the radial Please cite this article in press as: G. Chen, Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions, Optik - Int. J. Light Electron Opt. (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.110
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Fig. 7. Axial distributions of temperature for different f1.
distance, the influence of heat transfer coefficient on the temperature distributions becomes weaker and weaker. When it reaches a certain distance, the heat transfer coefficient has little influence on the temperature distributions. Next, we investigated the effects of ambient temperature on the temperature distributions of material. The axial temperature distributions of material for h1 = 1 000 000 and f1 = −200, −100, 0, 100, 200 ◦ C are given as shown in Fig. 7. It can be seen that the ambient temperature has a significant effect on the temperature distributions of material, especially in the area near r = R. The lower the ambient temperature, the lower the temperature in this area. With the decrease of the radial distance, the influence of ambient temperature on the temperature distributions becomes weaker and weaker. When it reaches a certain distance, the ambient temperature has little influence on the temperature distributions of material. 4. Conclusions (1) Analytical solutions of modeling for temperature of material irradiated by a long pulsed laser considering the effects of convection boundary conditions are obtained by using 2-D integral transform method. (2) The heat transfer coefficient and ambient temperature have significant effect on the temperature distributions of material. The bigger the heat transfer coefficient, and the larger the difference between the ambient temperature and the initial temperature of material, the greater the influence of convection boundary conditions on the temperature distributions of material. Acknowledgement This work was supported by the Changchun University of Science and Technology Young Scientists Project under Grant No. XQNJJ-2014-03. References [1] Z.W. Li, H.C. Zhang, Z.H. Shen, X.W. Ni, Time-resolved temperature measurement and numerical simulation of millisecond laser irradiated silicon, J. Appl. Phys. 114 (2013) 033104. [2] Z.W. Li, X. Wang, Z.H. Shen, X.W. Ni, Numerical simulation of millisecond laser-induced damage in silicon-based positive-intrinsic-negative photodiode, Appl. Opt. 51 (2012) 2759–2766. [3] Z.W. Li, X. Wang, Z.H. Shen, J. Lu, X.W. Ni, Mechanisms for the millisecond laser-induced functional damage to silicon charge-coupled imaging sensors, Appl. Opt. 54 (2015) 378–388. [4] X. Wang, Z.H. Shen, J. Lu, X.W. Ni, Laser-induced damage threshold of silicon in millisecond, nanosecond, and picosecond regimes, J. Appl. Phys. 108 (2010) 033103. [5] Y.X. Pan, X.M. Lv, H.C. Zhang, J. Chen, B. Han, Z.H. Shen, J. Lu, X.W. Ni, Millisecond laser machining of transparent materials assisted by a nanosecond laser with different delays, Opt. Lett. 41 (2016) 2807–2810. [6] B.S. Yilbas, Laser Heating Applications: Analytical Modeling, Elsevier, Amsterdam, 2012. [7] J. Bi, X.H. Zhang, X.W. Ni, G.Y. Jin, C.L. Li, Numerical simulation of thermal damage effects on gallium arsenide (GaAs) induced by a 0.53 (m wavelength long pulsed laser, Lasers Eng. 22 (2011) 37–46. [8] G.B. Chen, X.Y. Gu, J. Bi, Numerical analysis of thermal effect in aluminum alloy by repetition frequency pulsed laser, Optik 127 (2016) 10115–10121. [9] B.S. Yilbas, A closed form solution for temperature rise inside solid substrate due to time exponentially varying pulse, Int. J. Heat Mass Transf. 45 (2002) 1993–2000. [10] J. Bi, G.B. Chen, G.Y. Jin, X.H. Zhang, Analytical modelling of annular millisecond laser-induced damage in Silicon, Lasers Eng. 29 (2014) 175–187. [11] J. Bi, G.B. Chen, Characteristic analysis of 532 nm millisecond pulse laser-induced surface damage in the shape of thermal decomposition to GaAs by means of a Semi-analytical method, Lasers Eng. 32 (2015) 99–117. [12] G.B. Chen, Y.D. Wang, J.J. Zhang, J. Bi, An analytical solution for two-dimensional modeling of repetitive long pulse laser heating material, Int. J. Heat Mass Transf. 104 (2017) 503–509. [13] B.S. Yilbas, M. Kalyon, Repetitive laser pulse heating with a convective boundary condition at the surface, J. Phys. D: Appl. Phys. 34 (2001) 222–231. [14] B.S. Yilbas, M. Kalyon, Parametric variation of the maximum surface temperature during laser heating with convective boundary conditions, J. Mech. Eng. Sci. 216 (2002) 691–700. [15] M.N. Özisik, Heat Conduction, John Wiley and Sons, New York, 1993. [16] D.P. Laurie, Calculation of Gauss-Kronrod quadrature rules, Math. Comput. 66 (1997) 1133–1145.
Please cite this article in press as: G. Chen, Axisymmetric modeling of long pulsed laser heating with convective boundary conditions using analytical solutions, Optik - Int. J. Light Electron Opt. (2016), http://dx.doi.org/10.1016/j.ijleo.2016.11.110