ARTICLE IN PRESS Optics & Laser Technology 41 (2009) 931–937 Contents lists available at ScienceDirect Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec Comparison of temperature field due to laser step input and time exponentially varying pulses H. Al-Qahtani, B.S. Yilbas  KFUPM Box 1913, Dhahran 31261, Saudi Arabia a r t i c l e in fo abstract Article history: Received 13 May 2008 Received in revised form 25 March 2009 Accepted 2 April 2009 Available online 12 May 2009 Laser evaporative heating of the solid surface is considered and the effect of temporal variation of laser pulse shape on temperature rise is examined. In the analysis, time exponentially varying and step input pulses are employed and closed-form solutions for temperature rise are presented. Comparison of temporal variation of surface temperature is carried out for various laser pulse parameters of exponential and step input pulses. The pulse energies are kept the same for all pulses used in the comparison. It is found that temperature distributions corresponding to pulses used in the simulations are different and temperature decay in cooling cycle (after ending of the laser pulse) is clearly evident for step input pulses; however, this is not clearly identified for exponential pulses. & 2009 Elsevier Ltd. All rights reserved. Keywords: Laser Pulse Temperature 1. Introduction High-intensity laser pulse heating of solids result in evaporation at the surface. Depending on the laser pulse temperature and pulse energy, evaporation rate from the surface increases and melt phase between the solid and the vapor phases becomes negligible [1]. This is because of the high magnitude of latent heat of evaporation, which is higher than the latent heat of melting of most of the metals. Moreover, temporal variation of the laser pulse intensity on temperature rise in the solid as well as evaporation rate from the surface is significant, since the material response to a heating pulse changes drastically with time [2]. Consequently, investigation into the effect of laser pulse shape on temperature profile becomes essential. Considerable research studies were carried out to examine the laser pulse heating process. Analytical solution for constant intensity laser pulse heating was introduced by Ready [3]. Blackwell [4] presented the closed-form solution for temperature field after considering the convective boundary at the surface. Yilbas [5] and Yilbas and Kalyon [6] presented analytical solutions for temperature rise due to the laser pulses resembling the pulses used in the practical applications of laser heating. However, solutions presented were limited to solid heating and surface evaporation was ignored due to complexity of the problem. Moreover, Lu [7] examined a square-shaped temperature distribution due to a laser beam with Gaussian intensity profile at the workpiece surface. The phase change processes were omitted in  Corresponding author. Tel.: +966 3 860 4481; fax: +966 3 860 2949. E-mail address: bsyilbas@kfupm.edu.sa (B.S. Yilbas). 0030-3992/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2009.04.009 the study. Modest and Abaikans [8] studied analytically the temperature rise in the substrate material due to laser conduction heating and moving semi-infinite workpiece. They omitted the absorption of the laser beam and phase changes in the workpiece during the heating process. Laser heating and phase change process was investigated by Shi et al [9]. They indicated that the laser power intensity had significant effect on temperature rise and melting rate at the surface. An analytical model for inverse pulse laser heating was presented by Morozov et al. [10]. They predicted the molten thickness and compared with the experimental findings. Laser vapor plume interaction during laser heating of surfaces was studied by Gurasov and Smurov [11]. They indicated that the model proposed under estimated the amount of energy absorbed and with the ablated surface. Finite-element analysis of laser evaporative cutting was presented by Kim [12]. He predicted the groove shapes and temperature distributions in the irradiated region. Yilbas and Kalyon [13] introduced a closedform solution for laser evaporative heating process for time exponentially varying pulse. They introduced an expression for the evaporation front velocity, which was accommodated in the analysis [14]. Although, the predictions gave good results with the experimental findings, temporal variation of laser pulse was limited with the exponential form. In the actual laser heating processes, laser pulses can also be in the form of step input pulses [15]. This modifies the heating situation and the closed-form solution obtained for the time exponentially varying laser pulse is not applicable for temperature calculations due to the step input pulse. Consequently, the comparison of the rise of temperature field in the substrate material due to exponential and step input pulses is fruitful for the practical laser heating applications. ARTICLE IN PRESS 932 H. Al-Qahtani, B.S. Yilbas / Optics & Laser Technology 41 (2009) 931–937 Nomenclature time (s) pulse length of the laser pulse (s) t* dimensionless time ( ¼ ad2t) Dt* dimensionless pulse length of laser pulse V recession velocity (m/s) V* dimensionless recession velocity x distance (m) x* dimensionless distance ( ¼ xd) a thermal diffusivity (m2/s) b, b1, b2 laser pulse parameters (1/s) b, b1, b2 dimensionless Laser pulse parameters d absorption coefficient (1/m) r density (kg/m3) t Dt Cp C1 I1 Io k P(t) s rf T(x,t) T* To T*o specific heat capacity (J/kgK) intensity multiplication factor power intensity (Io(1rf)) (W/m2) power intensity (W/m2) thermal conductivity (W/mK) laser pulse Laplace variable reflection coefficient temperature (K) dimensionless temperature ( ¼ T(x,t)kd/I1) ambient temperature (K) ( ¼ Tokd/I1) dimensionless ambient temperature In the present study, the closed-form solutions for temperature rise due to the non-conduction limited heating situation are presented for laser step input and exponential pulses. Temperature fields corresponding to different pulse parameters of exponential and step input pulses are predicted and compared. The pulse energy is kept the same for all the pulses employed in temperature comparison. 2. Mathematical analysis Heat transfer in the radial direction during the laser heating of solid surfaces is significantly less than that of in the axial direction (laser beam direction). This, in turn, reduces the heating situation into one-dimensional heating of the solid substrate in the irradiated region [16]. Consequently, laser heating can be assumed as one-dimensional with reasonable accuracy. The laser energy is absorbed within the absorption depth of the substrate material. The absorption depth for metals is in the order of 108 m, which is significantly smaller than the thickness of the substrate material, which in the order of 103 m. This leads the assumption of semi-infinite solid for the heat transfer media in the heating analysis. Therefore, the mathematical arrangements of the heating situation can be simplified and the closed-form solution for temperature rise during the laser heating pulse can achieved. Fig. 1 shows schematic view of the laser heating situation. Since the laser pulse shape depends on the type of laser and its operation, two types of common pulse shapes are considered in the analysis, namely time step input pulse and time exponentially decaying pulse. Moreover, the analysis associated with each pulse differs; therefore, mathematical analysis will be given under the appropriate pulse type. Laser Beam 2.1. Time step input pulse Laser pulse heating 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 as shown in Fig. 2. The construction of the step intensity pulse can be achieved through subtraction of two unit step functions as shown in Fig. 2, i.e., the first unit step pulse starts at time t ¼ 0 while the second unit step pulse (shifted unit step pulse) starts at time t+Dt. The difference in both pulses results in the step intensity pulse, i.e., [15]: (1) PðtÞ ¼ 1½t  1½t  Dt 1½t ¼ 1; t40 0; to0  and 1½t  Dt ¼ 1; t4Dt 0; toDt ! (2) and P(t) is the step intensity pulse with a unit intensity. Heat transfer equation governing the laser evaporative heating situation can be written as @2 T @T @T þ rC p V þ I0 ð1  r f ÞPðtÞ ¼ rC v @x @t @x2 (3) with the initial condition at time t ¼ 0: T(x, 0) ¼ 0. The boundary conditions: Evaporation of the substrate material is considered at the free surface: x¼0:  @T  rVL ¼ @x x¼0 k The distance far away from the surface the substrate material is assumed to have the initial temperature, i.e., at infinity: x ¼ 1 : Tð1; tÞ ¼ 0 The recession velocity of the surface can be written as [14] Evaporating surface V¼ x = 0 : Free Surface x - axis Recessing surface Fig. 1. Schematic view of laser heating and x-axis location. I1 rðC p T s þ LÞ (4) where I1 ¼ I0(1rf) Since, the heat transfer equation is linear, one can consider the half-pulse response and then apply the superposition principle to obtain the complete solution. Therefore, one can write @2 T V @T I1 d 1 @T 1ðt  DtÞedx ¼ þ þ k a @t @x2 a @x (5) ARTICLE IN PRESS 933 H. Al-Qahtani, B.S. Yilbas / Optics & Laser Technology 41 (2009) 931–937 Intensity Unit Step Function Step Input Intensity Shifted Unit Step Function Time (t*) (Unit Step Function - Shifted Unit Step Function) Fig. 2. Construction of laser step input intensity pulse from unit step function and shifted unit step function. and Z2(t*) is In this case, the initial and boundary conditions are Tðx; 0Þ ¼ 0  @T  rVL ¼ dðtÞ @x  k Z 2 ðt  Þ ¼ eðV x¼0  erf Tð1; tÞ ¼ 0 Using the dimensionless quantities, þ l2 kd L  x ¼ dx; t  ¼ ad t; T ¼ T; . . . V  ¼ V=ad and b ¼ rV I1 I1 2  þ l3 The governing equations become 2  @ T @T  @T   þ V  þ 1ðt   Dt  Þex ¼  2 @x @t @x þ l4 (6) þ l5 with the initial and boundary conditions, T  ðx ; 0Þ ¼ 0 l1 ¼ T ð1; t Þ ¼ 0 Taking the Laplace transformation of Eq. (6) with respect to time, one can get   @ T̄ @T̄ eDt s x  e þ V  þ ¼ sT̄  T  ðx; 0Þ @x s @x2 (7) where T̄* is the Laplace transform of temperature and s is the Laplace 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] Tðx ; t  Þ þ Z 1 ðt  Þ þ Z 2 ðt   Dt  Þ1ðt   Dt  Þ þ Z 3 ðt   Dt  Þ1ðt   Dt  Þ (8) l1 pffiffiffiffi x pffiffiffiffi þ t  x1 2 t  2  eðx =4t Þ  2  pffiffiffiffipffiffiffiffi  x1 ex x1 þt x1 p t    !  2 pffiffiffiffi 2 eðx =4t Þ x ðx x1 Þþt x1  erfc pffiffiffiffi þ t x1 pffiffiffiffipffiffiffiffi  x1 e p t 2 t   ! ðx2 =4t Þ pffiffiffiffi 2 e x x x2 þt x2  p p ffiffiffiffi ffiffiffiffi  þ t x e erfc x pffiffiffiffi  2 2 p t 2 t  !  2 pffiffiffiffi eðx =4t Þ  2 x  pffiffiffiffipffiffiffiffi  x2 eðx x2 Þþt x2 erfc pffiffiffiffi þ t  x2 p t 2 t   !  ðx2 =4t Þ pffiffiffiffi V   2 e V x ðV =2Þþt ðV =2Þ x  pffiffiffiffi þ t e erf pffiffiffiffipffiffiffiffi  2 2 p t 2 t l2 ¼ l3 ¼ l4 ¼ l5 ¼ 1 2  eðx =4t Þ Z 1 ðt Þ ¼  b e e pffiffiffiffipffiffiffiffi p t !! pffiffiffiffi   t V V ðV 2 =4Þt þðV  =2Þx x  e þ pffiffiffiffi erfc 2 2 2 t  ðV  =2Þx ðV 2 =4t  Þ 2 2 2 2 2 2 2 2 2x1 ðx1  x2 Þðx1 þ x3 Þ 1 2x1 ðx1  x2 Þðx1  x3 Þ 1 2x2 ðx1  x2 Þðx1 þ x3 Þ 1 2x2 ðx2  x1 Þðx2  x3 Þ 1 2 2 2 2 ðx1  x3 Þðx2  x3 Þ and x21 ¼ V 2 =4; : x22 ¼  V 2 þ V   1 and x3 ¼ V  =2 4 and Z3(t*) where Z1(t*) is  " where  2  e (9)  @T    ¼ b dðt  Þ @x x ¼0  =2Þx ðV  =4Þt Z 3 ðt  Þ ¼   1  et ð1þV Þ x e 1 þ V (10) Hence, since the functions Z1(t*), Z2(t*) and Z3(t*) are obtained, the temperature distribution is explicitly known (Eq. (10)). A Mathematica software is used to compute temperature distribution (Eq. (9)). ARTICLE IN PRESS 934 H. Al-Qahtani, B.S. Yilbas / Optics & Laser Technology 41 (2009) 931–937 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 I1 ðeb1 t  eb2 t Þ, where I1 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 ¼ 1011 1/s and b2 ¼ 5  1011 1/s. The solution of conduction equation (the Fourier equation) for evaporative heating situation can be obtained for only 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 @2 T @T @T þ rC p V þ Io ð1  r f Þ expðbtÞd expðdxÞ ¼ rC p @x @t @x2 (11) with the boundary conditions  @T  rVL ¼ : Tð1; tÞ ¼ 0; and Tðx; 0Þ ¼ 0 @x x¼0 k @2 T V @T I1 1 @T þ þ expðbtÞd expðdxÞ ¼ a @t @x2 a @x k I1 @2 T̄ V @T̄ d 1 1 expðdxÞ ¼ ½pT̄ þ þ I1 k ðp þ bÞ a @x2 a @x a¼ k pC p Introducing the dimensionless variables: 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]: I1 ðV 2 =2Þðx þðV  =2Þt Þ e kd ( 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ððV =4ÞðV  1ÞÞt h   2 e eð ðV =4Þb Þx   ðV   ð1 þ b ÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi  erfcð ððV 2 =4Þ  b Þt  þ ðx =2 t  ÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ð V 2  b þ V  Þ 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi erfcð ððV 2 =4Þ  b Þt  þ ðx =2 t  ÞÞ 7 ð ðV 2 =4Þb Þx q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi þe 5  ðV   V 2  b Þ Tðx; tÞ ¼  where I1 ¼ Io(1rf) and Ts is the surface temperature. It should be noted that the peak power intensity does not vary with 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) analytically. In this case, keeping the recession velocity constant in Eq. (11) enables to determine the surface temperature analytically, 2  eððV =4ÞðV 1ÞÞt h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e ðV =4ÞðV 1Þ  ðb  ðV   1ÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi ffi erfcð ðV 2 =4Þ  ðV   1Þt  þ ðx =2 t  ÞÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð V 2  4ðV   1Þ þ V  Þ 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi erfcð ððV 2 =4Þ  ðV   1ÞÞt  þ ðx =2 t  ÞÞ 7  ðV 2 =4ÞðV  1Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þe 5 ðV 2  ð V 2  4ðV   1ÞÞÞ    2 2 1  V  x  eðV =4Þt eðV =2Þx erfc t þ pffiffiffiffi    2 2ðb ðV 0  1ÞÞ 2 t  I1 1 x b t ðV  1Þt þ e ðe  e Þ kd ððV   1Þ  b Þ          arL x  V  t x þ V  t pffiffiffiffi pffiffiffiffi  ð1 þ V  x þ V 2 t  Þerfc  eV x erfc 2k 2 t 2 t  2     2V  t  þ pffiffiffiffi eððV =4ÞðV x =2ÞþðV t =4ÞÞ (15) 2   þ 0.6 0.45 INTENSITY (14) where a is the thermal diffusivity, which is (12) r½C p T s þ L (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 V ¼ where k is the thermal conductivity, Cp 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, Io is the peak power intensity, and rf 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] V¼ 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 p Temperature distribution can be non-dimensionalized using the relation 0.3 T ¼ 0.15 0 0 2 Fig. 3. Time exponentially     (I1 ðt  Þ ¼ eb1 t  eb2 t ). 4 TIME decaying pulse 6 for complete 8 pulse T I1 =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 distribution (Eq. (15)) for the complete pulse. ARTICLE IN PRESS 935 H. Al-Qahtani, B.S. Yilbas / Optics & Laser Technology 41 (2009) 931–937 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 closedform 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 Table 1 Material properties used in the simulations. r (kg/m3) Cp (J/kg K) k (W/m K) d (m1) a (m2/s) 7880 460 80.3 6.17  107 2.22  104 1.2 Step Pulse ∆t = 1 Step Pulse ∆t = 3 Step Pulse ∆t = 5 Exp. Pulse 1 = 1/2, 2 = 1 INTENSITY 1 0.8 0.6 0.4 0.2 0 0 2 4 6 TIME 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. 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 high intensity 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 slow rise 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 0.5 0.5 0.4 Step Pulse ∆t = 1 Step Pulse ∆t = 3 Step Pulse ∆t = 5 Exp. Pulse 1 = 1/2, 2 = 1 0.3 INTENSITY TEMPERATURE 0.4 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 0.3 0.2 0.2 0.1 0.1 0 0 0 0 2 4 6 8 10 TIME Fig. 5. Temporal variation of surface temperature for laser pulses showing in Fig. 1. 2 4 6 TIME 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. ARTICLE IN PRESS 936 H. Al-Qahtani, B.S. Yilbas / Optics & Laser Technology 41 (2009) 931–937 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 TEMPERATURE 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 6 7 TIME Fig. 7. Temporal variation of surface temperature for laser pulses showing in Fig. 4. 0.4 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 TEMPERATURE 0.3 0.2 0.1 0 0 2 4 6 8 10 SPACE Fig. 8. Temperature distribution inside the substrate material for three exponential pulses and a step input pulse for dimensionless heating period of 1.5. 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 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 in gradual 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 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 temperature of 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 H. Al-Qahtani, B.S. Yilbas / Optics & Laser Technology 41 (2009) 931–937 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 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. Acknowledgment The authors acknowledge the support of King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for this work. References [1] Yilbas BS, Mansour SB. Laser evaporative heating of surface: simulation of flow field in the laser produced cavity. J Phys D Appl Phys 2006;39(17): 3863–75. [2] Yilbas BS, Shuja SZ. Laser non-conduction limited heating and prediction of surface recession velocity in relation to drilling. Proc Inst Mech Eng, Part C: J Mech Eng Sci 2003;217:1067–76. [3] Ready JF. Effects due to absorption of laser radiation. J Appl Phys 1963;36: 462–70. [4] Blackwell FJ. 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