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Analytical Prediction of the Temperature Field in Laser Assisted Machining
Available online at www.sciencedirect.com
ScienceDirect Procedia CIRP 46 (2016) 575 – 578
7th HPC 2016 – CIRP Conference on High Performance Cutting
Analytical prediction of the temperature field in laser assisted machining Mostafa M. Kashania, Mohammad R. Movahhedya,*, Mohammad T. Ahmadiana, Reza S. Razavib a
Department of Mechanical Engineering, Sharif University of Technology, Azadi Ave., Tehran, IRAN b Malek-Ashtar University, Shahinshahr, Isafahan, IRAN
* Corresponding author. Tel.: +98-21-66165505; fax: +98-21-66000021. E-mail address: [email protected]
Abstract Laser assisted machining (LAM) is a promising technology for machining hard to cut materials. In most experimental cases, LAM is undergone in two stages; first, the temperature at the material removal point (T mr ) is tuned by adjusting laser parameters and next, the cutting tool is engaged. Introduction of highly localized heat energy to the workpiece makes the modeling of this process very complicated. Hence, an analytical model for the first stage of the process—rather than multiple experiments or FE modelling—would be beneficial. This article presents an analytical solution to the transient, temperature field in a rotating cylinder subject to a localized laser heat source. The analytical model utilizes Green’s functions to find an exact solution for the temperature distribution field. © Authors. Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license Authors. Published by Elsevier B.V. © 2016 2016The The (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the International Scientific Committee of 7th HPC 2016 in the person of the Conference Chair Prof. Peer-review under responsibility of the International Scientific Committee of 7th HPC 2016 in the person of the Conference Chair Matthias Putz. Prof. Matthias Putz
Keywords: Laser assisted machining; Hybrid machining; temperature field; Analytical modelling.
1. Introduction Laser assisted machining (LAM) is a hybrid process that is often used for machining of hard to cut materials. Lasers with powers as high as 2500W are now employed to various machining processes [1] where “hard-to-machine materials” are mostly intended. Titanium alloys [2], hardened and tool steels [3] and various ceramics [4] and composites [5] are among the cases where LAM promises to be useful. Predictive modelling of the machining process has always been of interest. When laser enters the process, improper selection of tool and settings of the machining and laser parameters, not only distorts the optimum cutting conditions, but may result in microstructural damages to the machined surface. LAM is usually performed in two consecutive steps. In the first step, the workpiece is subjected to laser beam without the engagement of the cutting tool. During this stage, laser and machining parameters are tuned so that the workpiece temperature at the material removal point ( T mr ) reaches a predefined value (see e.g. [6]). That is, a series of trial and errors are performed to reach an initial guess for the settings of laser
and machining parameters. In the second stage, the cutting tool is engaged. An initial assessment of transient 3D thermal solution of the LAM was conducted by Rozzi [7, 8] where he introduced an approximate formulation besides his numerical solution for the temperature field. Rozzi defined a layer-wise model that could predict the surface temperature with good accuracy; however subject to significant errors at internal points. Other people such as Tian and Shin [6] had performed numerical modellings on complex features. However, none of the mentioned works can be considered as a complete analytical model. On the other hand, researchers working on the problems of laser welding and laser heat treatment had gone through a more systematic solution using the Green’s function method. For example, Nguyen [9] and Fachinotti [10] presented analytical solutions for heat sources with double-ellipsoidal and Gaussian power densities moving in semi-infinite bodies. In this study, an analytical model is developed to calculate the point-wise temperature in a rotating cylinder subject to a localized heat source. The solution is an integral function that can be quickly computed. Note that solving the same problem using FE methods requires several hours of computation.
2212-8271 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the International Scientific Committee of 7th HPC 2016 in the person of the Conference Chair Prof. Matthias Putz doi:10.1016/j.procir.2016.04.071
576
Mostafa M. Kashani et al. / Procedia CIRP 46 (2016) 575 – 578
The end faces of the workpiece ( z 0 , z L ) are subject to natural convection which is a negligible effect. Therefore, the end faces are considered to be at adiabatic condition:
2. Modelling Procedure and Formulation 2.1. Basic governing equations Eq. (1) is the general heat equation governing heat transfer problem in the cylindrical coordinates r , M , z :
1 w § wT · 1 w § wT ¨k ¨ kr ¸ r wr © wr ¹ r 2 wM © wM
· w ¸ ¹ wz
§ wT · ¨k ¸ © wz ¹
w Uc pT wt
(1)
where k is the coefficient of thermal conductivity, U is density and c P represents the specific heat capacity of the material, considered to be independent of temperature. 2.2. Initial and boundary conditions
wT wr
D ls q''l qtcc gen qtcccond qccc adv q fccconv r R
wT wz
(4)
0 z 0, z L
2.3. Solution of the temperature field The general approach to solve Eq. (1) in this article is through the application of the appropriate Green’s function. For the heat transfer problem, it gives the temperature increase at position r , M , z , t due to implication of a unit impulsive heat flux applied at U , T ,] ,W as given by Eq. (5): 'T r , M , z , t q lcc U , T , ] , W G r , U ; M T ; z ] ; t W (5)
At the beginning of the process, the workpiece is at room temperature; that is T r , M , z , 0 T f . Modelling of the LAM process is dominated by various BCs including the heat flux from the laser source, heat generation at the tool engagement location, conduction by the cutting tool, advection of heat by chip removal, forced convective heat transfer by the protective gas jet, natural convection from the surfaces and end sections and, possibly, conduction at machine jaws. These conditions are expressed on the RHS of Eq. (2), respectively, as follows: k
k
(2)
cc q ccj cond q nccconv q rad
Integrating the convolution of the Green’s function and heat flux function over time and space results in the temporalspatial evolution of the temperature field as given in Eq. (6). t
'T r , M , z , t
³³³³ 0
f
f
q''l U , T , ] ,W
(6)
G r U , M T , z ] , t W r dr d M dz dt
Due to rotation of workpiece and translation of the laser head, axial and circumferential variables as replaced as below:
T o T 0 Z t , ] o ] 0 f Z 2S t
(7)
where T 0 and ] 0 are initial circumferential and axial locations of the laser, Z is the angular velocity of the spindle and f is the machining feed. It is proved that the Green’s function of a multidimensional problem can be found as the product of the Green’s functions in each direction if the directions are linearly independent [13]. That is, in cylindrical coordinates: G 'r , 'M , 'z , 't G1 'z , 't u G 2 'r , 'M , 't
Fig. 1. Configuration of workpiece, pyrometer, laser head and cutting tool
Previous studies and experiments [11, 12] have proven that the laser source, heat generation by the cutting tool, advection of heat by chip removal and forced convection by protective gas jet are dominant thermal effects. Noting that the laser heat flux ( D ls q "l ) is a function of the laser absorption coefficient ( D ls ) and axial and circumferential positions of different surface points with respect to the laser center, Eq. (2) reduces to:
where in Eq. (8), the first function is the fundamental (1D) response for a finite body having a unit impulsive input of heat flux at ] ,W , and the second function is a two dimensional disk, experiencing a unit impulsive input of heat flux at U ,T ,W ; both having adiabatic boundary conditions. G 1 .
f m 1ª «1 2¦e L« m 1 ¬
G 2 .
2 R2
f
wT k wr
D ls q''l M , z , t r R
(3)
(8)
S 2D t W L2
§ mS z u cos ¨ © L
· § m S] ¸ cos ¨ ¹ © L
·º ¸» ¹ »¼
(9)
f 1 ª1 ¦ « cos n M T ® 2 S n 0 ¬S ¯
2 2 § E mn D t W · E mn J n E mn r R J n E mn U / R º ½° (10) »¾ ¸ 2 ¸ R E mn2 n 2 J n2 E mn © ¹ ¼» ¿°
¦exp ¨¨
m 1
2
Mostafa M. Kashani et al. / Procedia CIRP 46 (2016) 575 – 578
In Eq. (10), for n 0 , S inside the summation should be replaced by 2S . R, represents the outer radius of the cylinder, L is the total length and D is thermal diffusivity coefficient of the material, and E mn ’s are positive roots of the characteristic Eq. (11). J nc E mn
0 , m
1, 2, }, & n
0,1, }
(11)
In this equation, prime superscript represents differentiation with respect to the independent variable of the Bessel function. 3. Experiments
3.1. Material and instruments In order to verify the results of the analytical model, experiments were performed on samples made of DIN 1.7225 carbon steel with a diameter of 32mm and a length of 120mm . Samples were turned and the surface was sand blasted. Effective material properties are as given in Table 1. Table 1 Thermal properties of the test samples Item Specification Value Thermal conductivity ( k )
45
W m .K
2
Specific heat capacity ( c P )
450
J kg .K
3
Density ( U )
7820
Thermal diffusivity ( D )
12 u 10
First it should be noted that whereas the temperature field from analytical simulation is given at every point, the pyrometer reports an average temperature for the region in its focus. For the pyrometer in use, this region is a circle with a diameter of 5mm . Therefore, in order to be able to compare analytical results with the output of experiments, analytical solution was repeated for five points including the center of the measurement, and four other points located at equal distances ' 2mm at the sides of the central point. As seen in Fig. 2, the trend of the temperature history in all of these points is similar which is of course as predicted.
m2 s
3.2. Experimental Procedure During the experiments, the pyrometer has been fixed in its location while laser head traveled about 50mm along the workpiece length. Table 2 Machine tool and laser parameters Value
Specification
4.1. Adaptation of the experiments and analytical model
kg m 3 6
A lamp-pumped Nd:YAG laser with a maximum power of 700W was used. Local temperature measurement was performed using an infrared pyrometer from MICROEPSILON with an spectral range of 8 14 P m . Laser power was transferred to the machining region using fiber optics and through a lens to provide a spot size of 4 mm . The pyrometer is located in the horizontal plane and the laser head is located IL 73q above this plane, as shown in Fig. 1. The pyrometer was axially fixed in a position 20mm from the end face of the workpiece. Absorption of the laser emission by the surface of the workpiece is one of significant parameters of the problem and strongly depends on surface quality and laser type. This coefficient for similar steels being emitted by Nd:YAG laser is reported (see e.g.[14-17]) to be D ls 50 r 4 % .
Item
4. Results
Dimension
1
4
Experiments were performed for two cases during which the parameters were as listed in Table 2. Considering experimental constraints, these parameters were selected through a series of experiments to reach circumstances with the highest temperature values, while avoiding surface melting.
Exp. 1
Exp. 2
300
115
Dimension
1
Rotation speed of the machine ( N )
2
Machine feed ( f )
0.1044
0.1044
mm rev
3
Average laser power ( )
300
275
W
rpm
Fig. 2. Adapting the measurement output with the analysis
4.2. Comparison between the results of experiments and analytical model Experiments were performed in two cases and the analytical results are compared to the experimental data acquired. Due to the uncertainty in the absorption coefficient, lower and upper limits of the analytical solution are calculated for each case (Fig. 3). It can be observed that the analytical and experimental results are in good agreement. Differences between these results can be classified into three zones; before the laser arrives at the pyrometer location (Entrance Zone), when the laser is around the location of pyrometer (Engagement Zone) and the time laser is leaving (Exit Zone). In the Entrance Zone, the main source of deviation is that the experiment and analysis are close to their temporal and spectral boundaries. That is, when the laser is entering and passing the edge of the workpiece, some irregular emission occur. Reading the temperature in the Engagement Zone—that contains the most important piece of information—is prone to errors due to averaging of the larger
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Mostafa M. Kashani et al. / Procedia CIRP 46 (2016) 575 – 578
temperature gradients. On the other hand, augmented errors of the Entrance Zone are integrated to form a cumulative deviation in this zone. In the Exit Zone, one can distinguish different trends; that is, experimental results show a slow descending curve while analytical result exhibit a slow ascending curve. This is the result of neglecting the effects of natural convection and conduction from the workpiece contact with the machine jaws. This behavior can be interpreted as a measure of the significance of these effects.
cutting process, it would be easily possible to enter the heat fluxes originated from plastic deformation, tool-workpiece friction or advection of heat by chip removal in the thermomechanical model. Acknowledgment
The authors would like to acknowledge the assistance by Mr. M. Barkat, Mr. H. Roostaei and Mr. A. H. Mahmoudi in preparing the test setup and performing experiments. References
Fig. 3. Comparison between experimental and analytical results
Also, the analytical temperature of the center point is plotted in Fig. 3 to provide a better understanding of the real (point-wise) temperature of the points being heated by laser. The temperature of the center point at N 115rpm is 5% higher than the average temperature and at N 300rpm the difference reaches 10% . It would be expected that with the increase of the turning velocity, temperature gradients become larger. This large gradient is one of the indication of the sensitivity, and thus effectiveness, of the laser assisted machining process and the benefits that would be anticipated from a valid analytical model. 5. Conclusion
The main goal of this study was to create a predicting model for the first phase of the laser assisted machining process. Therefore, an analytical model was developed for the process utilizing Green’s functions as the major tool. Good agreement between the trend of analytical and experimental results was observed, with the largest difference in the peak temperature value 'T Exp 'T Anl / 'T Exp being less than 10% of the temperature increase of the workpiece. This conformity between the analytical and experimental results verifies the analytical approach, providing a basis for creating a comprehensive thermal model for the whole laser assisted machining process. The model can now be applied to design experimental parameters so that the T mr is increased to the predefined value prior to actual experiments. During this study, a systematic method was used to model the temperature distribution in the rotating part being emitted by a laser source. In order to model the input laser heat, infinitesimal flux elements were integrated over the surface. It means that any other distribution of input heat flux can be modeled through this process. Hence, when dealing with the
[1] Y. C. Shin, S. Lei, F. E. Pfefferkorn, P. Rebro, J. C. Rozzi, and F. P. Incropera, "Thermally (laser and plasma) assisted machining: Its potential and future," Abrasives, pp. 16-23, 2000. [2] C. R. Dandekar, Y. C. Shin, and J. Barnes, "Machinability improvement of titanium alloy (Ti-6Al-4V) via LAM and hybrid machining," International Journal of Machine Tools and Manufacture, vol. 50, pp. 174-182, // 2010. [3] H. Ding and Y. C. Shin, "Laser-assisted machining of hardened steel parts with surface integrity analysis," International Journal of Machine Tools and Manufacture, vol. 50, pp. 106-114, // 2010. [4] F. E. Pfefferkorn, Y. C. Shin, Y. Tian, and F. P. Incropera, "Laser-assisted machining of magnesia-partially-stabilized zirconia," Journal of Manufacturing Science and Engineering, Transactions of the ASME, vol. 126, pp. 42-51, 2004. [5] J. C. Rozzi and M. D. Barton, "The laser-assisted edge milling of ceramic matrix composites," 2009, pp. 845-852. [6] Y. Tian and Y. C. Shin, "Thermal modeling for laser-assisted machining of silicon nitride ceramics with complex features," Journal of manufacturing science and engineering, vol. 128, pp. 425-434, 2006. [7] J. C. Rozzi, Pfefferkorn, F. E., Incropera, F. P., Shin, Y. C., "Transient thermal response of a rotating cylindrical silicon nitride workpiece subjected to a translating laser heat source, part I: Comparison of surface temperature measurements with theoretical results," Journal of Heat Transfer, vol. 120, pp. 899-906, 1998. [8] J. C. Rozzi, F. P. Incropera, and Y. C. Shin, "Transient thermal response of a rotating cylindrical silicon nitride workpiece subjected to a translating laser heat source, part II: Parametric effects and assessment of a simplified model," Journal of Heat Transfer, vol. 120, pp. 907-915, // 1998. [9] N. Nguyen, A. Ohta, K. Matsuoka, N. Suzuki, and Y. Maeda, "Analytical solutions for transient temperature of semi-infinite body subjected to 3-D moving heat sources," WELDING JOURNAL-NEW YORK-, vol. 78, pp. 265-s, 1999. [10] V. D. Fachinotti, A. A. Anca, and A. Cardona, "Analytical solutions of the thermal field induced by moving doubleϋellipsoidal and doubleϋ elliptical heat sources in a semiϋinfinite body," International Journal for Numerical Methods in Biomedical Engineering, vol. 27, pp. 595-607, 2011. [11] C. R. Dandekar, Shin, Y.C., Barnes, J., "Machinability improvement of titanium alloy (Ti-6Al-4V) via LAM and hybrid machining," International Journal of Machine Tools and Manufacture, vol. 50, pp. 174-182, 2010. [12] J. C. Rozzi, F. E. Pfefferkorn, F. P. Incropera, and Y. C. Shin, "Transient, three-dimensional heat transfer model for the laser assisted machining of silicon nitride: I. Comparison of predictions with measured surface temperature histories," International Journal of Heat and Mass Transfer, vol. 43, pp. 1409-1424, // 2000. [13] D. G. Duffy, Green's functions with applications. Boca Raton, Fla.: Chapman & Hall/CRC, 2001. [14] G. E. Totten, Steel heat treatment : equipment and process design, 2nd ed. Boca Raton, FL: Taylor & Francis, 2007. [15] D. Bergstrom, J. Powell, and A. F. Kaplan, "Absorptance of nonferrous alloys to Nd:YLF and Nd:YAG laser light at room temperature," Appl Opt, vol. 46, pp. 1290-301, Mar 10 2007. [16] R. Patel and M. Brewster, "Gas-assisted laser-metal drillingExperimental results," Journal of thermophysics and heat transfer, vol. 5, pp. 26-31, 1991. [17] J. C. Ion, Laser processing of engineering materials : principles, procedure and industrial application. Amsterdam ;: Boston : Elsevier/Butterworth-Heinemann, 2005.
ScienceDirect Procedia CIRP 46 (2016) 575 – 578
7th HPC 2016 – CIRP Conference on High Performance Cutting
Analytical prediction of the temperature field in laser assisted machining Mostafa M. Kashania, Mohammad R. Movahhedya,*, Mohammad T. Ahmadiana, Reza S. Razavib a
Department of Mechanical Engineering, Sharif University of Technology, Azadi Ave., Tehran, IRAN b Malek-Ashtar University, Shahinshahr, Isafahan, IRAN
* Corresponding author. Tel.: +98-21-66165505; fax: +98-21-66000021. E-mail address: [email protected]
Abstract Laser assisted machining (LAM) is a promising technology for machining hard to cut materials. In most experimental cases, LAM is undergone in two stages; first, the temperature at the material removal point (T mr ) is tuned by adjusting laser parameters and next, the cutting tool is engaged. Introduction of highly localized heat energy to the workpiece makes the modeling of this process very complicated. Hence, an analytical model for the first stage of the process—rather than multiple experiments or FE modelling—would be beneficial. This article presents an analytical solution to the transient, temperature field in a rotating cylinder subject to a localized laser heat source. The analytical model utilizes Green’s functions to find an exact solution for the temperature distribution field. © Authors. Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license Authors. Published by Elsevier B.V. © 2016 2016The The (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the International Scientific Committee of 7th HPC 2016 in the person of the Conference Chair Prof. Peer-review under responsibility of the International Scientific Committee of 7th HPC 2016 in the person of the Conference Chair Matthias Putz. Prof. Matthias Putz
Keywords: Laser assisted machining; Hybrid machining; temperature field; Analytical modelling.
1. Introduction Laser assisted machining (LAM) is a hybrid process that is often used for machining of hard to cut materials. Lasers with powers as high as 2500W are now employed to various machining processes [1] where “hard-to-machine materials” are mostly intended. Titanium alloys [2], hardened and tool steels [3] and various ceramics [4] and composites [5] are among the cases where LAM promises to be useful. Predictive modelling of the machining process has always been of interest. When laser enters the process, improper selection of tool and settings of the machining and laser parameters, not only distorts the optimum cutting conditions, but may result in microstructural damages to the machined surface. LAM is usually performed in two consecutive steps. In the first step, the workpiece is subjected to laser beam without the engagement of the cutting tool. During this stage, laser and machining parameters are tuned so that the workpiece temperature at the material removal point ( T mr ) reaches a predefined value (see e.g. [6]). That is, a series of trial and errors are performed to reach an initial guess for the settings of laser
and machining parameters. In the second stage, the cutting tool is engaged. An initial assessment of transient 3D thermal solution of the LAM was conducted by Rozzi [7, 8] where he introduced an approximate formulation besides his numerical solution for the temperature field. Rozzi defined a layer-wise model that could predict the surface temperature with good accuracy; however subject to significant errors at internal points. Other people such as Tian and Shin [6] had performed numerical modellings on complex features. However, none of the mentioned works can be considered as a complete analytical model. On the other hand, researchers working on the problems of laser welding and laser heat treatment had gone through a more systematic solution using the Green’s function method. For example, Nguyen [9] and Fachinotti [10] presented analytical solutions for heat sources with double-ellipsoidal and Gaussian power densities moving in semi-infinite bodies. In this study, an analytical model is developed to calculate the point-wise temperature in a rotating cylinder subject to a localized heat source. The solution is an integral function that can be quickly computed. Note that solving the same problem using FE methods requires several hours of computation.
2212-8271 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the International Scientific Committee of 7th HPC 2016 in the person of the Conference Chair Prof. Matthias Putz doi:10.1016/j.procir.2016.04.071
576
Mostafa M. Kashani et al. / Procedia CIRP 46 (2016) 575 – 578
The end faces of the workpiece ( z 0 , z L ) are subject to natural convection which is a negligible effect. Therefore, the end faces are considered to be at adiabatic condition:
2. Modelling Procedure and Formulation 2.1. Basic governing equations Eq. (1) is the general heat equation governing heat transfer problem in the cylindrical coordinates r , M , z :
1 w § wT · 1 w § wT ¨k ¨ kr ¸ r wr © wr ¹ r 2 wM © wM
· w ¸ ¹ wz
§ wT · ¨k ¸ © wz ¹
w Uc pT wt
(1)
where k is the coefficient of thermal conductivity, U is density and c P represents the specific heat capacity of the material, considered to be independent of temperature. 2.2. Initial and boundary conditions
wT wr
D ls q''l qtcc gen qtcccond qccc adv q fccconv r R
wT wz
(4)
0 z 0, z L
2.3. Solution of the temperature field The general approach to solve Eq. (1) in this article is through the application of the appropriate Green’s function. For the heat transfer problem, it gives the temperature increase at position r , M , z , t due to implication of a unit impulsive heat flux applied at U , T ,] ,W as given by Eq. (5): 'T r , M , z , t q lcc U , T , ] , W G r , U ; M T ; z ] ; t W (5)
At the beginning of the process, the workpiece is at room temperature; that is T r , M , z , 0 T f . Modelling of the LAM process is dominated by various BCs including the heat flux from the laser source, heat generation at the tool engagement location, conduction by the cutting tool, advection of heat by chip removal, forced convective heat transfer by the protective gas jet, natural convection from the surfaces and end sections and, possibly, conduction at machine jaws. These conditions are expressed on the RHS of Eq. (2), respectively, as follows: k
k
(2)
cc q ccj cond q nccconv q rad
Integrating the convolution of the Green’s function and heat flux function over time and space results in the temporalspatial evolution of the temperature field as given in Eq. (6). t
'T r , M , z , t
³³³³ 0
f
f
q''l U , T , ] ,W
(6)
G r U , M T , z ] , t W r dr d M dz dt
Due to rotation of workpiece and translation of the laser head, axial and circumferential variables as replaced as below:
T o T 0 Z t , ] o ] 0 f Z 2S t
(7)
where T 0 and ] 0 are initial circumferential and axial locations of the laser, Z is the angular velocity of the spindle and f is the machining feed. It is proved that the Green’s function of a multidimensional problem can be found as the product of the Green’s functions in each direction if the directions are linearly independent [13]. That is, in cylindrical coordinates: G 'r , 'M , 'z , 't G1 'z , 't u G 2 'r , 'M , 't
Fig. 1. Configuration of workpiece, pyrometer, laser head and cutting tool
Previous studies and experiments [11, 12] have proven that the laser source, heat generation by the cutting tool, advection of heat by chip removal and forced convection by protective gas jet are dominant thermal effects. Noting that the laser heat flux ( D ls q "l ) is a function of the laser absorption coefficient ( D ls ) and axial and circumferential positions of different surface points with respect to the laser center, Eq. (2) reduces to:
where in Eq. (8), the first function is the fundamental (1D) response for a finite body having a unit impulsive input of heat flux at ] ,W , and the second function is a two dimensional disk, experiencing a unit impulsive input of heat flux at U ,T ,W ; both having adiabatic boundary conditions. G 1 .
f m 1ª «1 2¦e L« m 1 ¬
G 2 .
2 R2
f
wT k wr
D ls q''l M , z , t r R
(3)
(8)
S 2D t W L2
§ mS z u cos ¨ © L
· § m S] ¸ cos ¨ ¹ © L
·º ¸» ¹ »¼
(9)
f 1 ª1 ¦ « cos n M T ® 2 S n 0 ¬S ¯
2 2 § E mn D t W · E mn J n E mn r R J n E mn U / R º ½° (10) »¾ ¸ 2 ¸ R E mn2 n 2 J n2 E mn © ¹ ¼» ¿°
¦exp ¨¨
m 1
2
Mostafa M. Kashani et al. / Procedia CIRP 46 (2016) 575 – 578
In Eq. (10), for n 0 , S inside the summation should be replaced by 2S . R, represents the outer radius of the cylinder, L is the total length and D is thermal diffusivity coefficient of the material, and E mn ’s are positive roots of the characteristic Eq. (11). J nc E mn
0 , m
1, 2, }, & n
0,1, }
(11)
In this equation, prime superscript represents differentiation with respect to the independent variable of the Bessel function. 3. Experiments
3.1. Material and instruments In order to verify the results of the analytical model, experiments were performed on samples made of DIN 1.7225 carbon steel with a diameter of 32mm and a length of 120mm . Samples were turned and the surface was sand blasted. Effective material properties are as given in Table 1. Table 1 Thermal properties of the test samples Item Specification Value Thermal conductivity ( k )
45
W m .K
2
Specific heat capacity ( c P )
450
J kg .K
3
Density ( U )
7820
Thermal diffusivity ( D )
12 u 10
First it should be noted that whereas the temperature field from analytical simulation is given at every point, the pyrometer reports an average temperature for the region in its focus. For the pyrometer in use, this region is a circle with a diameter of 5mm . Therefore, in order to be able to compare analytical results with the output of experiments, analytical solution was repeated for five points including the center of the measurement, and four other points located at equal distances ' 2mm at the sides of the central point. As seen in Fig. 2, the trend of the temperature history in all of these points is similar which is of course as predicted.
m2 s
3.2. Experimental Procedure During the experiments, the pyrometer has been fixed in its location while laser head traveled about 50mm along the workpiece length. Table 2 Machine tool and laser parameters Value
Specification
4.1. Adaptation of the experiments and analytical model
kg m 3 6
A lamp-pumped Nd:YAG laser with a maximum power of 700W was used. Local temperature measurement was performed using an infrared pyrometer from MICROEPSILON with an spectral range of 8 14 P m . Laser power was transferred to the machining region using fiber optics and through a lens to provide a spot size of 4 mm . The pyrometer is located in the horizontal plane and the laser head is located IL 73q above this plane, as shown in Fig. 1. The pyrometer was axially fixed in a position 20mm from the end face of the workpiece. Absorption of the laser emission by the surface of the workpiece is one of significant parameters of the problem and strongly depends on surface quality and laser type. This coefficient for similar steels being emitted by Nd:YAG laser is reported (see e.g.[14-17]) to be D ls 50 r 4 % .
Item
4. Results
Dimension
1
4
Experiments were performed for two cases during which the parameters were as listed in Table 2. Considering experimental constraints, these parameters were selected through a series of experiments to reach circumstances with the highest temperature values, while avoiding surface melting.
Exp. 1
Exp. 2
300
115
Dimension
1
Rotation speed of the machine ( N )
2
Machine feed ( f )
0.1044
0.1044
mm rev
3
Average laser power ( )
300
275
W
rpm
Fig. 2. Adapting the measurement output with the analysis
4.2. Comparison between the results of experiments and analytical model Experiments were performed in two cases and the analytical results are compared to the experimental data acquired. Due to the uncertainty in the absorption coefficient, lower and upper limits of the analytical solution are calculated for each case (Fig. 3). It can be observed that the analytical and experimental results are in good agreement. Differences between these results can be classified into three zones; before the laser arrives at the pyrometer location (Entrance Zone), when the laser is around the location of pyrometer (Engagement Zone) and the time laser is leaving (Exit Zone). In the Entrance Zone, the main source of deviation is that the experiment and analysis are close to their temporal and spectral boundaries. That is, when the laser is entering and passing the edge of the workpiece, some irregular emission occur. Reading the temperature in the Engagement Zone—that contains the most important piece of information—is prone to errors due to averaging of the larger
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temperature gradients. On the other hand, augmented errors of the Entrance Zone are integrated to form a cumulative deviation in this zone. In the Exit Zone, one can distinguish different trends; that is, experimental results show a slow descending curve while analytical result exhibit a slow ascending curve. This is the result of neglecting the effects of natural convection and conduction from the workpiece contact with the machine jaws. This behavior can be interpreted as a measure of the significance of these effects.
cutting process, it would be easily possible to enter the heat fluxes originated from plastic deformation, tool-workpiece friction or advection of heat by chip removal in the thermomechanical model. Acknowledgment
The authors would like to acknowledge the assistance by Mr. M. Barkat, Mr. H. Roostaei and Mr. A. H. Mahmoudi in preparing the test setup and performing experiments. References
Fig. 3. Comparison between experimental and analytical results
Also, the analytical temperature of the center point is plotted in Fig. 3 to provide a better understanding of the real (point-wise) temperature of the points being heated by laser. The temperature of the center point at N 115rpm is 5% higher than the average temperature and at N 300rpm the difference reaches 10% . It would be expected that with the increase of the turning velocity, temperature gradients become larger. This large gradient is one of the indication of the sensitivity, and thus effectiveness, of the laser assisted machining process and the benefits that would be anticipated from a valid analytical model. 5. Conclusion
The main goal of this study was to create a predicting model for the first phase of the laser assisted machining process. Therefore, an analytical model was developed for the process utilizing Green’s functions as the major tool. Good agreement between the trend of analytical and experimental results was observed, with the largest difference in the peak temperature value 'T Exp 'T Anl / 'T Exp being less than 10% of the temperature increase of the workpiece. This conformity between the analytical and experimental results verifies the analytical approach, providing a basis for creating a comprehensive thermal model for the whole laser assisted machining process. The model can now be applied to design experimental parameters so that the T mr is increased to the predefined value prior to actual experiments. During this study, a systematic method was used to model the temperature distribution in the rotating part being emitted by a laser source. In order to model the input laser heat, infinitesimal flux elements were integrated over the surface. It means that any other distribution of input heat flux can be modeled through this process. Hence, when dealing with the
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