Chapter 2 Conduction Heating of Solid Surfaces

Transcription

1 Chater 2 Condution Heating of Solid Surfaes Abstrat Laser ondution heating involves with a solid hase heating for a stationary or moving soures. In order to aommodate the absortion of irradiated laser energy, a volumetri soure onsideration should be inororated. Sine the analytial solution for suh heating situation is ossible, a losed form solution for the temerature field is rovided firstly. In order to assess the influene of duty yle on the temerature field, a numerial model is introdued. To study the thermal effets, two-dimensional axisymmetri solid is onsidered for a stationary soure and three-dimensional heating situation is inororated for a moving soure in the model studies. The laser heating is involved with the assisting gas roessing; therefore, the onvetion effet of the assisting gas is inororated in the analysis. Sine the heating duration is longer than the thermalization time of the substrate material for most of the surfae treatment roesses, the Fourier heating law is inororated in the analysis. In this hater, Laser ondution heating of solid surfaes is introdued and analytial aroahes for temerature field in the irradiated region is resented for the aroriate boundary and heating onditions. 2.1 Introdution Laser ondution heating involves with a solid hase heating for a stationary or moving soures. In order to aommodate the absortion of irradiated laser energy, a volumetri soure onsideration should be inororated. Sine the analytial solution for suh heating situation is ossible, a losed form solution for the temerature field is rovided firstly. In order to assess the influene of duty yle on the temerature field, a numerial model is introdued. To study the thermal effets, two-dimensional axisymmetri solid is onsidered for a stationary soure and three-dimensional heating situation is inororated for a moving soure in the model studies. The laser heating is involved with the assisting gas roessing; therefore, the onvetion effet of the assisting gas is inororated in the analysis. B. S. Yilbas and S. Z. Shuja, Laser Surfae Proessing and Model Studies, Materials Forming, Mahining and Tribology, DOI: 1.17/ _2, Ó Sringer-Verlag Berlin Heidelberg 213 5

2 6 2 Condution Heating of Solid Surfaes Sine the heating duration is longer than the thermalization time of the substrate material for most of the surfae treatment roesses, the Fourier heating law is inororated in the analysis. 2.2 Analytial Treatment of Laser Pulse Heating Closed form solution of the ulse laser heating of the solid surfaes enables to identify the affeting arameters through analytial exressions develoed between the temerature field and the arameters. The losed form solution for laser reetitive ulse heating, therefore, beomes fruitful when examining the ossibility of the steady heating at the surfae by the laser reetitive ulses. Laser ulse heating of the solid substrates requires formulation of laser ulses due to the fat that laser intensity varies with time. This results in different heating situations due to temoral laser ulse behavior. In this ase, two heating situations an be onsidered to aount for the temoral behavior of the laser ulse intensity. The first ase is involved with a single exonential ulse heating and the other ase is the heating due to the multi-ulses (reetitive ulses). The losed form solutions for eah ases is resented aording to the below sub-headings Exonential Pulse Heating The analytial formulation of laser ulse heating of solid substrate with a onvetive boundary ondition at the surfae is onsidered in the light of the revious study [1]. The time exonentially varying ulse intensity is inororated in the analysis rovided that two different ulse tyes are taken into aount. In the first ulse tye (half ulse), the intensity deays exonentially with time ði 1 exð b sþþ while the intensity distribution resembling a tyial atual laser ulse I 1 ½exð b sþ exð sþš is onsidered in the seond ulse tye (full ulse). The losed-form solution obtained from the resent study is omared with the revious formulations for the aroriate ulse and boundary onditions. The heating analysis is arried out for nanoseond laser ulses, whih is longer than the thermalization time of the metal substrates; therefore, the Fourier theory of heating is used when modeling the heating roess [2]. The heat transfer equation for a laser heating ulse an be written as: o 2 T ox 2 þ I 1d k ðe bt e t Þe dx ¼ 1 ot ð2:1þ a ot where I 1 is the ower intensity, d is the absortion deth, b and are the ulse arameters, and a is the thermal diffusivity.

3 2.2 Analytial Treatment of Laser Pulse Heating 7 The outut ulse from a laser an be formulated through subtrating two exonential funtions. Hene, the ower intensity distribution of time exonentially varying ulse an be written as: I ¼ I 1 e bt e t ð2:2þ where I 1 ¼ 1 r f I and r f is the refletion oeffiient and I o is the eak ower intensity, and arameters b and an be hosen to give the aroriate rise time for the ulse. Sine the governing equation of heat transfer is linear (Eq. 2.1) the losed form solution an be obtained for a half ulse first, then, the omlete solution an be ahieved by subtration of the solutions for the individual arts of the time exonential ulse (half ulse). It should be noted that for the solution of full ulse, the ambient temerature is onsidered as zero ðt ¼ Þ: This is neessary sine the full ulse solution satisfies the onvetive boundary ondition when T ¼ : The heat transfer equation for the half ulse beomes: o 2 T ox 2 þ I 1d k e ðbtþdxþ ¼ 1 ot ð2:3þ a ot The absortion deth of the substrate material is onsiderable smaller than the thikness of the substrate material. This allows one to onsider the semi-infinite solid body, initially at uniform temerature, with onvetive boundary ondition at the surfae. It should be noted that onvetive boundary ondition at the surfae resembles the assisting gas jet effet at the surfae of the solid substrate during the heating roess. Therefore, the initial and boundary onditions are: at time t ¼ Tðx; Þ ¼ ot at the surfae x ¼ ¼ h ox x¼ k ðtð; tþ T oþ and at x ¼1 Tðt; 1Þ ¼ The solution of Eq. 2.3 an be obtained ossibly through Lalae transformation method, i.e., with reset to t, the Lalae transformation of Eq. 2.3 yields: o 2 T ox 2 þ I 1d 1 k e dx þ b ¼ 1 T Tðx; Þ a ð2:4þ where T ¼ Tðx; Þ and Tðx; Þ ¼ due to the initial ondition. Using the initial ondition, Eq. 2.4 yields: o 2 T ox 2 T a ¼ I 1d 1 k e dx þ b ð2:5þ Let onsider k 2 ¼ a and H o ¼ I 1d k 1 ðþbþ ; then Eq. 2.5 beomes:

4 8 2 Condution Heating of Solid Surfaes o 2 T ox 2 k2 T ¼ H o e dx Equation 2.6 has homogeneous and artiular solutions T ; i.e.: T h T ¼ T h þ T ð2:6þ ð2:7þ The homogeneous solution is: T h ¼ C 1 e kx þ C 2 e kx ð2:8þ where C 1 and C 2 are the onstants to be determined from the boundary onditions. Similarly the artiular solution is: T ¼ A o e dx ð2:9þ where A o is the onstant. Equation 2.7 yields the solution: A o d 2 e dx k 2 A o e dx ¼ H o e dx ð2:1þ or A o ¼ H o ðd 2 k 2 Þ After the rearrangement, the artiular solution T results: H o T ¼ ðd 2 k 2 Þ e dx Therefore, the solution of Eq. 2.4 in the Lalae domain beomes: ð2:11þ ð2:12þ T ¼ Tðx; Þ ¼C 1 e kx þ C 2 e kx þ ðd 2 k 2 Þ e dx ð2:13þ The oeffiients in Eq an be obtained from the boundary onditions. Consider k ¼ ffiffi a [ and from the boundary ondition Tð1; tþ ¼ result in C 1 ¼ : Moreover, let H ¼ H ; Eq beomes: ðd 2 k 2 Þ T ¼ C 2 e kx þ H o e dx H o ð2:14þ In order to determine C 2 ; the boundary ondition at the surfae an be inororated, i.e.: ot ¼ h ox x¼ k Tð; Þ T o ð2:15þ

5 2.2 Analytial Treatment of Laser Pulse Heating 9 where T ; whih is an ambient temerature (same as the initial temerature), an be seified. Introduing Eq into Eq and knowing that Tð; Þ ¼ C 2 þ H o ; it yields: kc 2 dh o ¼ h k C 2 þ H o T o ð2:16þ Hene, C 2 beomes: C 2 ¼ H oðh þ kdþ ðh þ kkþ þ T oh ðh þ kkþ ð2:17þ Substituting C 2 and the values of H o ; H o and k into Eq. 2.14, it beomes: ffi a x ffi I 1 dðh þ kdþe Tðx; Þ ¼ kð þ bþ d 2 ffiffi þ T ohe a k a h þ ffiffia hþ ffiffi I 1 d k ffiffia kð þ bþ d 2 a x e dx ð2:18þ The mathematial arrangements of inversion of Eq is given in the revious study [1]. Therefore, the Lalae inversion of Eq beomes: 8 ffi ffiffiffi b b ie bt l 1 e a ix ffiffiffiffi erf bt i þ x ffiffiffi þ l 2 at 2 e b ffi a ix erf ffiffiffiffi bt i þ x ffiffiffi 9 2 at >< Tðx; tþ ¼a 1 þd ffiffiffi h a e ad 2 t l 3 e dx erf d ffiffiffiffi at þ x ffiffiffi l 2 at 4 edxerf d ffiffiffiffi at þ x ffiffiffi i >= 2 at w 1 l 5 e w1 x a ffi ffiffi e w2 1 t erf w 1 t þ x ffiffiffi >: >; 2 at a 2 þ 2t ðb þ ad 2 Þ ðead e bt Þe dx þ a 3 e w1 x ae ffi ffiffi w2 1 t x erf w 1 t þ w 1 2 ffiffiffiffi x þ erf at 2 ffiffiffiffi ð2:19þ at where 1 l 1 ¼ 2 ffiffiffi b iðb þ ad 2 Þð ffiffiffi b i þ w1 Þ : l 1 2 ¼ 2 ffiffiffi b iðb þ ad 2 ffiffiffi Þð b i þ w1 Þ 1 l 3 ¼ 2 ffiffi a dðb þ ad 2 Þðd ffiffi a þ w1 Þ : l 1 4 ¼ 2 ffiffi a dðb þ ad 2 Þð d ffiffi a þ w1 Þ 1 l 5 ¼ ðw 2 1 þ bþðw2 1 ad2 Þ ð2:2þ and a 1 ¼ I 1a ffiffi a dðh þ kdþ k 2 : a 2 ¼ I 1ad k : a 3 ¼ h ffiffi a k T o : w 1 ¼ h ffiffi a k ð2:21þ

6 1 2 Condution Heating of Solid Surfaes After inserting l 1 ; l 2 ; l 3 ; l 4 ; l 5 ; a 1 ; a 2 ; a 3 ; and w 1 ; Eq beomes: 8 e dx e ad2t x erf ffiffi ffiffiffi d at 2 at ffi 2ðbþad 2 Þ h a Tðx; tþ ¼ I 1a ffiffiffi k þd ffiffi edx e ad2t x erf ffiffi ffiffiffi 9 þd at 2 at ffi a 2ðbþad 2 Þ h a k d ffiffi a ffiffiffi b a dðh þ kdþ k 2 þ e bt e a xi x erf ffiffi ffiffiffiffi ffiffiffi ffi þ bti 2 at 2ðbþad 2 Þ h ffiffi a þ e bt e b a xi x erf ffiffi ffiffiffi >< bt 2 at i >= ffi k b i 2ðbþad 2 Þ h ffiffi a k þ b i þ ffiffi h e hx k e h2 k 2at x erf ffiffi ffiffiffi þ h a 2 at k at k k ffiffi a e dx ðe ad2t e bx Þ >: >; k h2 a k 2 þb h 2 a k 2 ad2 ðbþad 2 ÞðhþkdÞ x þ T o erf 2 ffiffiffiffi e hx h k e 2 x k 2at erf at 2 ffiffiffiffi þ h ffiffiffiffi at ð2:22þ at k The surfae temerature an be obtained when setting x equals to zero (x = ) in Eq The mathematial arrangements are given in the revious study [1]. Therefore, the resulting surfae temerature beomes: " rffiffiffiffiffi # I 1 ad Tð:tÞ ¼ ðb þ ad 2 Þðbk 2 þ ah 2 Þ ðadh ab ffiffiffiffi bkþe bt þ 2 ðh þ dkþfð bt Þ I 1 ad 2 þ 2t ðdk hþðb þ ad 2 Þ ead erf ðd ffiffiffiffi at Þ I 1 adhk T o ðbk 2 þ ah 2 e h2 at k Þðh dkþ 2 erf h ffiffiffiffi at þ T o k ð2:23þ In order to redue Eq in dimensionless form, the following non-dimensional arameters are introdued: b ¼ b ad 2 : h ¼ h dk : s ¼ ad2 t ð2:24þ After mathematial arrangement and inororating Eqs. 2.24, 2.23 yields: Tð; sþ ¼ I 1 1 dk ð1 þ b Þðb ðh b Þe bs þ 2 qffiffiffiffi qffiffiffiffiffiffi ffiffiffi b ð1 þ h ÞF b s þ h 2 Þ þ I 1 1 ffiffi dk ð1 h Þð1 þ b Þ es erf s T o I 1 h dk ðb e h2s erf h ffiffi s þ To ð2:25þ þ h 2 Þðh 1Þ

7 2.2 Analytial Treatment of Laser Pulse Heating 11 where FðvÞ is Dawson s integral, whih is: Z v FðvÞ ¼ e v2 e n2 dn ð2:26þ The omlete solution for the heating ulse (full ulse) an be obtained through inororating the rinile of suerosition; in whih ase, after subtrating the solutions of the temerature rofile for two exonential terms e bt and e t as used in the full ulse rofile, the omlete solution for the full ulse beomes: Tð; sþ ¼ I 1 1 dk ð1 þ b Þðb þ h 2 Þ ðh b Þe b s þ 2 qffiffiffiffi qffiffiffiffiffiffi ffiffiffi b ð1 þ h ÞF b s þ I 1 1 ffiffi dk ð1 h Þð1 þ b Þ es erf ð s Þ þ I 1 h dk ðb þ h 2 Þð1 h Þ eh2s erf ðh ffiffi s Þ I 1 1 dk ð1 þ Þð þ h 2 Þ ðh Þe s þ 2 ffiffiffi ffiffiffi ð1 þ h ÞFð sþ þ I 1 1 ffiffi dk ð1 h Þð1 þ Þ es erf ð s Þ þ I 1 h dk ð þ h 2 Þð1 h Þ eh2s erf ðh ffiffi s Þ ð2:27þ where ¼ : Equation 2.27 satisfies the onvetion boundary ondition for zero ad 2 ambient temerature ðt o ¼ Þ: Equations 2.25 and 2.27 an be used to omute the dimensionless surfae temerature rofiles at the surfae for half and full laser heating ulses Laser Reetitive Pulse Heating Laser reetitive ulse heating of the solid surfae with onvetive boundary ondition at the surfae is onsidered to aount for the assisting gas jet effet. A losed form solution for laser heating roess is obtained using a Lalae transformation method in line with the revious study [1]. The onditions for onstant temerature heating at the surfae are investigated and the ulse arameter resulting in ossible steady temerature attainment at the surfae is disussed. In order to aount for the reetitive ulse heating, the intensity rofile resembling the onseutive ulses should be inororated [3]. This an be ahieved introduing a time shift in the intensity funtion, whih is given in b

8 12 2 Condution Heating of Solid Surfaes Eq Therefore, the heat transfer equation emloying the onseutive ulses an be written as [3]: o 2 T ox 2 þ I 1d k ðe bt e t Þe dx Fðt t o Þ¼ 1 ot ð2:28þ a ot where Fðt t o Þ is a ste funtion, whih aids to resemble the onseutive ulses, i.e.: Fðt t o Þ¼ ; t\t o ð2:29þ 1; t t o The solution of Eq is idential to Eq rovided that the time term is relaed by ðs s o Þ and the solution is multilied by a non-dimensional ste funtion. The solution of Eq for the surfae temerature yields: 8 " # 9 I 1 1 ðh o b o Þe b oðs s o Þ ffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dk ð1þb o Þðb o þh 2 o Þ þ 2 ffiffi b oð1 þ ho ÞFð b o ðs s o ÞÞ þ I 1 1 dk ð1 h o Þð1þb o Þ es s o ffiffiffiffiffiffiffiffiffiffiffiffi erf ð s s o Þ >< þ I 1 h o dk ðb Tð; sþ ¼Fðs s o Þ o þh 2 o Þð1 h oþ eh2 o ðs soþ ffiffiffiffiffiffiffiffiffiffiffiffi erf ðh o s s o Þ >= " # þ I 1 1 ðh o o Þe oðs s o Þ dk ð1þ o Þð o þh 2 o Þ þ 2 ffiffi ffiffiffiffi oð1 þ ho ÞFð o ðs s o ÞÞ þ I 1 1 dk ð1 h o Þð1þ o Þ es s o ffiffiffiffiffiffiffiffiffiffiffiffi erf ð s s o Þ >: þ I 1 h o dk ð o þh 2 o Þð1 h oþ eh2 o ðs soþ ffiffiffiffiffiffiffiffiffiffiffiffi erf ðh o s s o Þ >; ð2:3þ where Fðs s o Þ¼ ; s\s o and s 1; s s o ¼ ad 2 t o : o Equation 2.3 is used to omute the dimensionless surfae temerature rofiles at the surfae for a omlete laser heating ulse. In order to determine the eak temerature differenes and the maximum temerature differene in the surfae temerature rofile due to reetitive ulses, the followings are introdued: DT 1 ¼ T 1 T 2 ð2:31þ where DT 1 is the first eak temerature differene, T 1 and T 2 are the eak surfae temeratures orresonding to the first and seond onseutive ulses, resetively. The seond eak temerature differene an be written as: DT 2 ¼ T 2 T 3 ð2:32þ where T 3 is the eak surfae temeratures orresonding to the third onseutive ulse. The maximum temerature differene in between the first and seond eaks of the temerature rofile is:

9 2.2 Analytial Treatment of Laser Pulse Heating 13 DT max1 ¼ T 1 T min1 ð2:33þ where T min1 is the minimum surfae temerature in between the first and seond onseutive ulses. The maximum temerature differene in between the seond and third eaks of the temerature rofile is: DT max2 ¼ T 2 T min2 ð2:34þ where T min2 is the minimum surfae temerature in between the seond and third onseutive ulses. 2.3 Effet of Duty Cyle on Heating: Numerial Treatment In ratial laser heating situations, laser ulses emloyed have single duty yle (energy in er ulse or energy ontent of a single ulse) and yle frequeny. It has been demonstrated that the reetition rate of laser ulses imroves the laser roessing effiieny. However, the laser ulses, in general, have a rise and a fall times, whih make the analytial solution diffiult to ahieve, artiularly for twodimensional axisymmetri heating situations. Therefore, numerial solution for suh heating situation beomes unavoidable. Sine the heating roess is transient, initial and boundary onditions beome imortant resembling the atual hysial roess. Consider the laser reetitive ulse heating of solid surfae, the governing equation of heat diffusion onsistent with the ondution limited heating situation an be written for a two-dimensional semi-infinite solid. Laser beam ower intensity distribution at the solid surfae an be assumed to be Gaussian with its sot enter at the enter of the oordinate system. The transient heat ondution equation for a solid substrate irradiated by a laser beam with a Gaussian intensity rofile at the surfae an be written as: qc ot ot ¼ k 1 o r or r ot or þ o2 T ox 2 þ S ð2:35þ where x is the axial diretion (along the laser beam axis), r is the radial diretion (normal to the laser beam axis) q is the density, k is the thermal ondutivity, C is the seifi heat and S is the volumetri soure term. The volumetri heat soure an be arranged to resemble the laser reetitive ulses; therefore, r 2 S ¼ I fðþd t 1 r f exð dxþ ex a 2 ð2:36þ where I ; d; r f and a are the ower intensity, absortion deth and the Gaussian arameter, resetively.

10 14 2 Condution Heating of Solid Surfaes Fig. 2.1 Laser ulses with five duty yles for the first three onseutive ulses

11 2.3 Effet of Duty Cyle on Heating: Numerial Treatment 15 The temoral variation of ulse intensity resembling laser reetitive ulses an be defined by the funtion fðþ; t whih is: fðþ¼ t 8 9 t t r at >< >= t r \t t 1 t \t t f bt >: >; t f \t\t ð2:37þ where t r and t f are the onstants defining the rise and fall of the laser ulse, t is the ulse length, and a and b are the onstants. The ulse shae used in the simulation is shown in Fig The initial ondition an be reresented as the substrate material is assumed to be initially (t = ) at a uniform temerature T whih an be seified. In order to reresent the assisting gas, a onvetive boundary ondition is assumed at the surfae (x = ). The heat transfer oeffiient used in the resent simulations is h ¼ 1 4 W/m 2 K [1]. Therefore, at the surfae, where x =, the boundary ondition yields ot ox ¼ ht ð s T Þ: At the symmetry axis (r = ) temerature is assumed to be maximum. Therefore, at the symmetry axis (r = ), the boundary ondition is ot or ¼ : At a distane far away from the surfae distane signifiantly larger than the absortion deth of the substrate material (d), a room temerature is assumed, i.e. as x!1t ¼ T : Equation 2.35 is simulated numerially to obtain thermal resonse of the substrate material due to the reetitive heating ulse. The numerial sheme emloying an imliit formulation is used. To disretize the governing equations, a ontrol volume aroah is introdued. The alulation domain is divided into grids and grid indeendene test is being erformed for different grid size and orientation. A non-uniform grid with mesh oints along x and r axes, resetively, is emloyed after seuring the grid indeendene. The finer grids are loated near the irradiated sot enter in the viinity of the surfae and grids beome ourser as the distane inreases towards the bulk of substrate material. A omuter rogram based on imliit sheme is used to omute the temerature field. 2.4 Disussions The omarison of the analytial solutions with the literature [4, 5] and temerature reditions from the analytial solutions are resented in line with the revious studies [1, 2, 6]. The findings in relation to analytial solutions and the numerial reditions are disussed aording to the sub-headings given below.

12 16 2 Condution Heating of Solid Surfaes Exonential Pulse Heating Case and Convetion Condition Resembling Assisting Gas at the Surfae The losed-form solution for the temerature rofiles due to time exonentially varying ulse is obtained. In order to aount for the ulse resembling a tyial atual ulse, two different time exonentially varying ulse intensities are onsidered. The heating involved with the time exonentially varying ulse, having a single time exonential term, is alled the half ulse ase while for the time exonentially varying ulse, having two time exonential terms, is alled the full ulse ase in the disussion setion. The losed-form solution obtained is omared with the analytial solutions reorted in the literature for different heating onditions [4, 6]. The data used in the solutions are given in Table 2.1. To dedue the losed-form solution derived from the resent analysis to the analytial solutions reorted in the literature, the following heating onditions are onsidered: (1) ste inut intensity ulse with onvetive boundary ondition at the surfae, and (2) time exonentially varying ulse with non-onvetive boundary ondition at the surfae. The analytial solution for the ste inut intensity ulse with onvetive boundary ondition was obtained reviously by Blakwell [4]. In order to introdue the ste inut intensity ulse in the losedform solution derived at resent, the ulse arameters ðb and Þ in the soure term of Eq. 2.2 or in the losed-form solution Eq should be set to zero. Moreover, the initial and ambient temeratures, and the symbol of absortion oeffiient used in Blakwell s solution are different than those emloyed in the resent analysis, i.e. the initial temerature is set to zero, the ambient temerature is denoted as T and d is used for the absortion oeffiient in the resent study. The resulting equation is: x Tðx; tþ ¼T o erf 2 ffiffiffiffi e h2 k 2atþh k x x erf at 2 ffiffiffiffi þ h ffiffiffiffi at at k 8 1 þ kd h erf x ffiffiffi 12 t dxþ 2 at eðad2 erf d ffiffiffiffi at þ x ffiffiffi 9 2 at þ I >< 1 1 ð h kd þ1þ 2 h kd ð tþdxþ erf d ffiffiffiffi at þ x ffiffiffi >= kd 1 Þeðad2 2 at ffiffiffiffi þ 1 h h ð h kd kd 1Þe k xþh2 k 2at erf h k at þ x ffiffiffi >: þ e dx ðe ad2t 1Þ >; 2 at ð2:38þ Equation 2.38 is idential to the analytial solution obtained by Blakwell [4]. Table 2.1 Data used in the solutions for exonential ulse heating Substrate d (1/m) a (m 2 /s) (J/kgK) q (kg/m 3 ) k (1/s) (1/s) I (W/m 2 ) Steel ,

13 2.4 Disussions 17 The analytial solution for the non-onvetive boundary ondition at the surfae and time exonentially varying intensity ulse was obtained by Yilbas [6] reviously. To introdue the non-onvetive boundary ondition at the surfae, the heat transfer oeffiient (h) in Eq is set to zero. In this ase, for a half ulse, Eq redues to: " I 1 ffiffi Tð; sþ ¼ kdð1 þ b Þ es erfð s 2 Þþ 1 qffiffiffiffiffiffi # ffiffiffi ffiffiffiffi F b s e b s ð2:39þ Equation 2.39 is idential to the analytial solution obtained reviously by Yilbas [6]. Figure 2.2 shows the temoral variation of the surfae temerature for the half ulse ase and onstant heat transfer oeffiient h ¼ 1 4 W/m 2 K at the surfae while ulse arameter ðb Þ is variable. In general, the rise of the temerature rofiles is similar during the initial eriod of heating. As b redues, temerature reahes its eak value earlier. This ours beause of the time variation of the heating ulse as deited in Fig In this ase, as b inreases, the ower intensity distribution leans towards the ulse beginning rovided that the area under the ulse intensity urve beomes less. Consequently, the energy ontent of the heating ulse redues as b inreases, whih in turn results in low eak temerature at the surfae of the substrate. The rate of surfae temerature rise and deay varies as b varies. This an also be seen from Fig. 2.4, in whih ot os with time ðþis s shown. ot deays raidly as s inreases from the ulse beginning. The sloe of ot os os urve redues to zero at time orresonding to the eak temerature. The duration at whih raid deay of ot os indiates the raid inrease of internal energy gain of the substrate, whih dominates over the ondution and onvetion energy transort. At the time of minimum ot os ; internal energy gain of the substrate beomes onsiderably small, sine the ulse energy orresonding to this heating b Fig. 2.2 Temoral variation of temerature for Bi = and b is variable β ' = T β ' = β ' = τ

14 18 2 Condution Heating of Solid Surfaes Fig. 2.3 Temoral variation of ower intensity distribution as b ; and b = are variable 1.8 I1.6.4 β ' =.3 β ' =.2 β ' β =.1 ' /γ ' = 1/3 β ' /γ ' = 2/ τ Fig. 2.4 Temoral variation of time gradient of temerature for Bi ¼ and different b values Bi = 2E-5 dt/dt β ' =.1 β ' =.2 β ' = τ time is very small. As ulse intensity eases, the ondution and onvetion beome the sole mehanisms in energy transort roess. In aordane with the revious study [7], the equilibrium time an be introdued at the oint of minimum ot os : In this ase, the energy balane attains among the internal energy gain, due to absortion of laser ulse, ondution, and onvetion ontribution of the energy transort in the surfae viinity of the substrate. The equilibrium time shifts lose to the ulse beginning as b redues. This is due to the temoral distribution of the laser ulse intensity as indiated earlier. Figure 2.5 shows the temoral variation of surfae temerature for different values of Biot number (B i ) and fixed b value. The effet of Bi on the temerature rofiles is not signifiant for Bi :2: This beause the energy absorbed by the substrate, whih is onsiderably high as omared to the energy transorted due to onvetion from the surfae of the substrate. However, as Bi inreases further, the

15 2.4 Disussions 19 Fig. 2.5 Temoral variation of temerature for different b ¼ :1 and Biot number β'=.1 Bi = 2E-5 Bi = 2E-4 Bi = 2E-3 T.8 Bi = 2E-2.4 Bi = τ Fig. 2.6 Temoral variation of temerature for b = ¼ 1=3 and Biot number is variable β'/γ' =1/3 Bi = 2E-5 Bi = 2E-4 Bi = 2E-3 Bi = 2E-2 T.6.3 Bi = 2E τ rate of rise and the value of eak temerature at the surfae redue. In this ase, energy transorted from the surfae beause of onvetion beomes onsiderable. The time orresonding to eak temerature moves lose to the ulse beginning for high Bi. In order to resemble the atual laser ulse, a full ulse is onsidered; in whih ase, the ulse rofile I ¼ I 1 ½exð b sþ exð sþš is emloyed in the soure term of Eq Figure 2.6 shows the temoral variation of temerature rofiles at the surfae obtained for full ulse rofile, in whih b ¼ :1 and ¼ :3 are emloyed. The rate of hange of temerature rofile in the ulse beginning is lower than that orresonding to a half ulse for b ¼ :1: This is beause the ower intensity orresonding to a half ulse is higher than its ounterart orresonding to a full ulse Fig As similar to the behavior of Bi for the half ulse, the effet of Bi on temerature rofiles is more ronouned as Bi C.3. The equilibrium time, as defined earlier, for a full ulse varies with hanging B i.

16 2 2 Condution Heating of Solid Surfaes The equilibrium time orresonding to low b ; b and high Bi is small. This indiates that early rise of the ulse intensity results in small equilibrium time. Moreover, high value of Bi redues the equilibrium time for half and full ulses. In this ase, onvetion ooling of the surfae suresses the internal energy gain of the substrate. Consequently, the domination of the internal energy gain in the energy transort roess eases earlier during the heating ulse Reetitive Pulse Heating Case and Convetion Condition Resembling Assisting Gas at the Surfae In order to obtain a onstant temerature heating ondition at the surfae, the laser reetitive ulses with onstrutively deaying eak intensities are onsidered. To aount for the ooling of the surfae, the Biot number (Bi) is varied in between.2 and , the ulse arameters is varied in between 2.5 and 6, and the ratios of eak ulse intensity of onseutive ulses are I 2 I 1 ¼ :25 and I 3 I 1 ¼ :625; resetively. Table 2.2 gives the data used in the solutions of temerature field. The seond ulse of the onseutive ulses starts one the first ulse ends or about to end. The overlaing of the onseutive ulses is minimal. This is, however, more ronouned when b redues as seen from Fig. 2.7, in whih the ower intensity distribution with time is shown. It should be noted that the energy ontent of the overlaing ortion of the reetitive ulses is only a fration of the resetive reetitive ulses, whih overla. Figure 2.8 shows the temoral variation of the non-dimensional surfae tem- erature T I =kd b for three onseutive ulses with b ¼ 4; and Bi as variable while Fig. 2.9 shows the time derivative of temoral variation of the non-dimensional surfae temerature for the same ondition of Fig The eak temerature inreases as Bi redues. This is beause the rate of onvetive ooling of the surfae, whih redues as Bi redues, i.e. the heat transfer oeffiient redues. The effet of Bi on temerature rofile is less ronouned as Bi redues to 1-3 and beyond. In this ase, the onvetive ooling of the surfae does not substantiate and the internal energy gain dominates over the onvetive ooling of the surfae. As Bi inreases, the onvetive ooling of the surfae beomes as imortant as the internal energy gain of the substrate material, whih in turn redues the eak temerature at the surfae. When omaring the trend of the temerature rofiles Table 2.2 Data used in the solutions of the reetitive ulse heating Bi d (1/m) a (m 2 /s) (J/ kgk) q (kg/ m 3 ) k (W/ mk) b (1/s) (1/s) I (W/m 2 ) s ,

17 2.4 Disussions 21 POWER INTENSITY β ο /γ ο = τ POWER INTENSITY.5.4 β ο /γ ο = τ POWER INTENSITY POWER INTENSITY βο/γ ο= τ β ο /γ ο = τ Fig. 2.7 Temoral variation of dimensionless intensity as b = variable with time due to onseutive ulses and different Bi, the surfae temerature shows an inreasing trend with time as Bi redues. The trend beomes almost steady with some flutuations aross the eak temeratures for Bi = The

18 dt/dt 22 2 Condution Heating of Solid Surfaes Fig. 2.8 Temoral variation of dimensionless surfae temerature for b = ¼ 4 and Bi variable TEMPERATURE Bi = 2x1-4 Bi = 2x1-2 Bi = 2x1-3 Bi = 2x1-1 βo/γo = τ Fig. 2.9 Temoral variation of time derivative of dimensionless surfae temerature for b = ¼ 4 and Bi variable Bi = 2x1-4 βo/γo = 4 Bi = 2x Bi = 2x1-2 Bi = 2x τ further redution in Bi results in dereasing trend with time in surfae temerature rofile. Consequently, the onvetive ooling does not lower the eak value of the surfae temerature only, but effets the temoral behavior of the surfae temerature resulted from the onseutive ulses. In the ase of b ¼ 6 Fig. 2.1, the surfae temerature due to onseutive ulses shows inreasing trends with time for Bi B 1-2, and the trend beomes almost steady with time for Bi =.2. Consequently, inreasing b results in inreasing eak temerature and the surfae temerature rofile due onseutive ulses shows an inreasing trend with time. In the ase of high Bi ðbi ¼ :2Þ; the surfae temerature attain almost steady trend with time. Figures 2.11, 2.12, and 2.13 show the first eak temerature differene DT 1 ; seond eak temerature differene DT 2 ; the maximum temerature differene in between the first and seond eak temeratures ðdt max1 Þ; and the maximum temerature differene in between seond and third eak temeratures ð Þ DT max2

19 2.4 Disussions 23 Fig. 2.1 Temoral variation of dimensionless surfae temerature for b = ¼ 6 and Bi variable TEMPERATURE Bi = 2x1-4 Bi = 2x1-2 Bi = 2x1-3 Bi = 2x1-1 βo/γo = τ Fig Dimensionless surfae temerature differene with b = and Bi ¼ TEMPERATURE DIFFERENCE.6.3 Bi = 2x1-4 First eak Tem. diff. Seond eak Tem. diff. First Tem. diff. Seond Tem. diff βο/γο Fig Dimensionless surfae temerature differene with b = and Bi ¼ TEMPERATURE DIFFERENCE.6.3 Bi = 2x1-2 First eak Tem. diff. Seond eak Tem. diff. First Tem. diff. Seond Tem. diff β ο/γ ο

20 24 2 Condution Heating of Solid Surfaes Fig Dimensionless surfae temerature differene with b = and Bi ¼ TEMPERATURE DIFFERENCE Bi = 2x1-1 First eak Tem. diff. Seond eak Tem. diff. First Tem. diff. Seond Tem. diff β ο /γ ο with b as Bi variable. DT 1 is ositive for b 4 and this reverses for b [ 4: This is also true for DT 2 as Bi 1 3 : This indiates that the first eak temerature is always greater than the seond eak temerature for b \4; i.e. the temerature rofile shows an inreasing trend with time. However, DT 1 beomes zero for b ¼ 4:5 for Bi ¼ and Bi In the ase of DT 2 ; it beomes zero at b ¼ 3:25 for Bi : Therefore, as Bi inreases b also inreases for the zero value of DT 1 and DT 2 : In the ase of Bi =.2, first eak temerature differene is always ositive for b and this is also true for DT 2 : Consequently, the surfae temerature has a dereasing trend with time. DT max1 attains large values as Bi redues. Moreover, DT max1 inreases with inreasing b : As b inreases, the time orresonding to the eak temerature moves towards the ulse beginning as seen from Fig Consequently, high eak intensity differenes results in large eak temerature and maximum temerature differenes in the surfae temerature rofiles. DT 2 is relatively smaller than DT 1 : In general, a steady temerature at the surfae is unlikely to our, sine DT max1 and DT 1 attain high values, whih in turn results in large flutuation in the temerature rofile. However, the magnitude of flutuation beomes less as b and Bi redue Effet of Duty Cyle on Heating and Convetion Condition Resembling Assisting Gas at the Surfae Table 2.3 gives the data used in the numerial simulations. Figure 2.14 shows temoral variation of surfae temerature for different duty yles of laser ulses. The rise of surfae temerature is high in the early heating eriod for all duty

21 2.4 Disussions 25 Table 2.3 Data used for the numerial reditions for the effet of the duty yle on temerature d (1/ m) (J/ kgk) q (kg/ m 3 ) k (W/ mk) b (1/ s) h (W/ m 2 K) a (1/ m) (1/ s) I (W/ m 2 ) , Fig Temoral variation of surfae temerature at symmetry axis for the first three ulses Fig Temoral variation of surfae temerature differene (DT) at the symmetry axis for the first ten ulses and different duty yles yles and as the ulse rogresses towards its ending, the rate of rise beomes steady. The high rate of temerature rise is assoiated with the internal energy gain and temerature gradient in the surfae region of the substrate material. In this ase, energy absorbed by the substrate material is high in the surfae region due to Lambert s Law, whih in turn, inreases the internal energy gain substantially in this region. In the early heating eriod, temerature gradient is low, and diffusional energy transfer, due to temerature gradient, from the surfae region to solid bulk beomes less. This enhanes the temerature rise. However, as the heating rogresses, temerature gradient inreases aelerating the diffusional energy transfer from surfae region to the solid bulk. Although, inreasing duty yle inreases the laser energy in the ulse, enhaning the magnitude of temerature, the rate of

22 26 2 Condution Heating of Solid Surfaes Fig Temerature variation along the symmetry axis inside the substrate material for different heating eriods and duty yles temerature rise in both early and late heating eriods hanges. This indiates that inreasing duty yle enhanes temerature rise in a short duration at the substrate surfae. Figure 2.15 shows temoral variation of temerature differene at the surfae for different duty yles. Temerature differene is alulated as the differene

23 2.4 Disussions 27 between the maximum temerature in a ulse and the temerature at the orresonding ulse ending. Temerature differene remains almost onstant with time desite the fat that maximum temerature at the surfae inreases. This indiates that the thermal resonse of the surfae to the heating ulse beomes the same with a given duty yle of the onseutive ulses. Moreover, as the duty yle inreases, the magnitude of temerature also inreases. One the duty yle is ket onstant for the ulses, the temerature differene remains almost onstant with time. The high magnitude of temerature differene results in yli thermal loading of the surfae. Consequently, thermal integration (steady rise of surfae temerature during reetitive ulses) at the surfae relaes with thermal loading of the surfae. This situation results in thermal fatigue of the surfae. Therefore, inreasing duty yle enables surfae temerature to rise at high rates; but, it auses yli thermal loading at the surfae due to magnified temerature differene. Figure 2.16 shows temerature distribution inside the substrate material for different duty yles and heating eriods when temerature is maximum at the surfae. Temerature rofiles are lotted when surfae temerature is maximum for a given time eriod. The temerature gradient attains large values, artiularly in the surfae region for small duty yles. The temerature gradient attains minimum at some deths below the surfae and as the heating duration rogresses, the loation of minimum temerature gradient moves away from the surfae. The energy balane is attained at the oint of minimum temerature gradient suh that internal energy gain from the irradiated field balanes the diffusional energy loss from the surfae region to the solid bulk. Consequently, the diffusional energy transfer dominates the energy transfer roess in the region beyond the loation of minimum temerature gradient. Moreover, inreasing duty yle does not alter the loation of minimum temerature gradient onsiderably in the substrate material. This indiates that the duty yle hange the magnitude of temerature gradient; however, the loation of maximum temerature gradient remains almost the same in the substrate material for different duty yles. Consequently, amount of energy transort in the surfae region enhanes with inreasing duty yle, but energy transort by diffusion beomes imortant at the artiularly loation inside the substrate material for all duty yles. Referenes 1. Kalyon, M., Yilbas, B.S.: Analytial solution for laser evaorative heating roess: time exonentially deaying ulse ase. J. Phys. Part D: Alied Physis 34, (21) 2. Kalyon, M., Yilbas, B.S.: Reetitive laser ulse heating analysis: ulse arameter variation effets on losed form solution. Al. Surf. Si. 252, (26) 3. Yilbas, B.S., Shuja, S.Z.: Laser short-ulse heating of surfaes. J. Phys. D Al. Phys. 32, (1999) 4. Blakwell, F.J.: Temerature rofile in semi-infinite body with exonential soure and onvetive boundary onditions, ASME. J. Heat Transfer 112, (199)

24 28 2 Condution Heating of Solid Surfaes 5. Shuja, S.Z., Yilbas, B.S., Shazli, S.Z.: Laser reetitive ulse heating influene of ulse duty. Heat Mass Transf. 43, (27) 6. Yilbas, B.S.: Analytial solution for time unsteady laser ulse heating of semi-infinite solid. Int. J. Mehanial Sienes 39(6), (1997) 7. Yilbas, B.S., Sami, M.: 3-Dimensional laser heating inluding evaoration a kineti theory aroah. Int. J. Heat and Mass Transf. 41/13, (1998)

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