An analytical model for the prediction of temperature distribution and evolution in hybrid laser-waterjet micro-machining

Accepted Manuscript Title: An analytical model for the prediction of temperature distribution and evolution in hybrid laser-waterjet micro-machining Author: Shaochuan Feng Chuanzhen Huang Jun Wang Hongtao Zhu Peng Yao Zhanqiang Liu PII: DOI: Reference:

S0141-6359(16)30092-7 http://dx.doi.org/doi:10.1016/j.precisioneng.2016.07.002 PRE 6423

To appear in:

Precision Engineering

Received date: Revised date: Accepted date:

21-2-2016 10-6-2016 11-7-2016

Please cite this article as: Feng Shaochuan, Huang Chuanzhen, Wang Jun, Zhu Hongtao, Yao Peng, Liu Zhanqiang.An analytical model for the prediction of temperature distribution and evolution in hybrid laser-waterjet micro-machining.Precision Engineering http://dx.doi.org/10.1016/j.precisioneng.2016.07.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

An analytical model for the prediction of temperature distribution and evolution in hybrid laser-waterjet micro-machining

Shaochuan Feng1, Chuanzhen Huang* 1, Jun Wang2, Hongtao Zhu1, Peng Yao1, Zhanqiang Liu1

1

Center for Advanced Jet Engineering Technologies (CaJET), Key Laboratory of High-efficiency and Clean Mechanical Manufacture (Ministry of Education), School of Mechanical Engineering, Shandong University, Jinan, 250061, China 2

School of Mechanical and Manufacturing, The University of New South Wales (UNSW), Sydney, NSW 2052, Australia

*

Corresponding author. Tel. and Fax: +86 531 88396913. E-mail address: [email protected] (C. Z. Huang). 1

HIGHLIGHTS     

An analytical temperature field model in hybrid laser-waterjet machining is obtained. The interaction among laser, waterjet and material is investigated. 3-D temperature profiles in silicon carbide during the hybrid process are established. Comparison of dry laser ablation and the hybrid process are conducted. The relationships between the temperatures and the parameters are illustrated.

Abstract A hybrid laser-waterjet micro-machining technology was developed for near damage-free micro-ablation recently. It uses a new material removal concept where the laser-softened material is expelled by a pressurised waterjet. The temperature field in this hybrid machining process is an essential quantity for understanding the underlying material removal mechanism and optimizing the process conditions. This study presents a three-dimensional (3-D) analytical model for the temperature field in this hybrid laser-waterjet micro-machining process. The interaction among the laser, waterjet, and workpiece is considered in the model. The absorption of laser by water, the formation of laser-induced plasma in water, the bubble formation and the laser refraction at the air-water interface are discussed. DuHamel’s principle is used to determine a closed-form temperature equation and a solution in a variable separation form is obtained. A calculation for silicon carbide is conducted. The results are illustrated by a group of 3-D temperature profiles intuitively and visually. It is shown that the temperatures are below the melting point during the process due to the cooling action of waterjet. The almost damage-free micro-machining can be achieved. Besides, the maximum temperature increases with the increased average laser power and waterjet offset distance and decreased nozzle exit diameter where the average laser power takes a major action. Keywords Analytical model, Temperature field, Hybrid process, Micro-machining, Laser, Waterjet 2

Nomenclature ab

absorption coefficient of material q2

heat flux caused by surface

(1/m)

radiation (W/m2)

Al

laser-irradiated area (m2)

c

heat capacity at constant pressure Rw

Rf

targeted material reflectivity water reflectivity

of material (J/Kg K) cw

heat capacity at constant pressure Ref

average Reynolds number on the

of water (J/Kg K)

thin water film region

db

laser beam diameter (m)

df2

focused

laser

t

diameter

after T

time (s) temperature (K)

refracted (m) dw

nozzle exit diameter (m)

D

laser

beam

Tf

diameter

before u

surrounding temperature (K) temperature-dependent property

focused by a lens (m) Ep

laser pulse energy (J)

uc

temperature-independent property

f

laser pulse frequency (Hz)

v

traverse speed of laser (m/s)

F

focal length (m)

v0

waterjet velocity at nozzle exit (m/s)

ha

the

combined

heat

transfer x, y, Cartesian system coordinates

coefficient of air (W/m2 K) hw

z

heat transfer coefficient of water xw

waterjet offset distance (m)

(W/m2 K) I

laser beam intensity (W/m2)

zc

downward

distance

of

focal

position (m) I0

k

laser beam intensity at center α

thermal diffusivity of material

position (W/m2)

(m2/s)

thermal conductivity of material β

laser absorptivity of water

(W/m K) 3

kw

thermal conductivity of water γ1

divergence angle in air (deg)

(W/m K) la

light absorption length in water γ2

divergence angle in water (deg)

(m) ε

lw

water layer thickness (m)

Lf

characteristic length of thin water θ

surface emissivity waterjet inclination angle (deg)

film region (m) λ

M2R

laser beam quality factor in water

nw

refractive index of water relative μ

laser wavelength (m) viscosity of water (Pa s)

to air Nuf

average Nusselt number on the ρ

material density (kg/m3)

thin water film region Pi

hydrostatic pressure at nozzle ρw

water density (kg/m3)

inlet (Pa) Pa

average power of laser (W)

σ

Stefan-Boltzmann’s

constant

(5.6710-8 W/m2 K4) ΔP

pressure drop in nozzle (Pa)

τ0

relaxation time of material (s)

Pr

Prandtl number of water

τp

laser pulse duration (s)

Q

heat source (W/m3)

φ

laser pulse function in time domain

q1

heat flux caused by waterjet ψ

laser pulse function in space

impingement (W/m2)

domain

4

1. Introduction Lasers have been used in a variety of material processing applications especially for the hard and brittle materials that are difficult to machine. However, heat-affected zone (HAZ) is considered as a major drawback of the process. Attempts to reduce HAZ have been carried out by many researchers. For this purpose, liquid-assisted laser ablation processes have been developed in the processing of engineering materials such as silicon, silicon carbide and other thermal-sensitive materials. Some typical liquid-assisted laser ablation processes are under-water laser machining [1], waterjet-guided laser machining [2], and low-pressure waterjet assisted laser machining [3,4]. Nevertheless, the earlier developed water assisted laser machining processes remove material at its liquid status by locally heat and increase the material temperature. Tangwarodomnukun et al. [5] has proposed a novel hybrid laser-waterjet machining technology which used a new material removal concept where a waterjet is applied off-axially to expel the ‘‘softened’’ elemental material by laser radiation and cool the material for near damage-free micromachining. It combines the advantages of laser processing with those of waterjet cutting. The near damage-free micro-ablation of silicon [5], germanium [6] and silicon carbide [7] are achievable by using this technology. However, further investigations are essential to provide a deeper understanding of the process and optimize the process. As the thermal effect is utilized in the hybrid laser-waterjet micro-machining, the temperature distribution is a crucial intermediate result that would determine the characteristics of grooves and final surfaces machined. Thus establishing a model for the temperature field is considerable and necessary. Several comprehensive models of temperature field in laser dry ablation have been proposed. Three-dimensional (3-D) numerical models for steady temperature distributions [8,9] and for dynamic temperature distributions [10,11] in laser ablation process have been developed respectively using the finite element method (FEM). Marek Polák et al. used two numerical methods, i.e. FEM and finite difference method (FDM), for the calculation 5

of the temperature field in laser cutting and drilling [12]. The temperature field in laser pulse heating and phase change processes of the substrate material are formulated numerically using the energy method [13]. Chi-Kyung Kim has dealt with a 1-D analytical solution to transient heat conduction in the medium subjected to a moving heat source [14]. An analytical solution of temporary temperature field in half-infinite body caused by moving heat source tilted towards the direction of motion is presented [15]. Based on the two-temperature coupling theory, a numerical model of temperature field of electron and lattice in the heating process of femtosecond laser for metal has also been established by using FEM [16]. For the liquid-assisted laser machining, the processes are more complex than that in laser dry ablation because of the interaction among laser, waterjet and workpiece. The temperature field models for the waterjet guided laser grooving of silicon have been presented by Li et al. using FDM [17] and by Yang et al. using FEM [18]. Both of these models are assumed that the silicon was removed in its liquid status. Tangwarodomnukun et al. [19] has developed a mathematical model to predict the temperature field in the hybrid laser-waterjet micro-grooving process for a single crystalline silicon using an enthalpy-based FDM, without considering the plasma effect. The present study is to investigate the temperature field in the hybrid laser-waterjet micro-machining process using an analytical approach where the various physical phenomena associated with the process are discussed to give an insight into the process. Firstly, a governing equation and its boundary conditions will be proposed according to the problem under investigation. The coupled effects of laser heating and waterjet cooling associated with the process will then be discussed. The interference between laser and waterjet, including the absorption of laser by water, the formation of laser-induced plasma in water, the bubble formation and the laser refraction at the air-water interface, will be studied. Subsequently, an analytical solution to the temperature field model that was established previously will be carried out and the method of eigenequation and DuHamel’s theorem will be applied. In order to represent the results intuitively and visually, a group of 3-D temperature profiles 6

will be plotted via MATLAB. In addition, the relationship between the temperature and process parameters will also be obtained. The calculation for silicon carbide will be implemented as an example.

2. Problem description The laser beam and the pressurised waterjet beam are applied off-axially travel on the workpiece surface where the waterjet beam is exerted behind the laser beam at a fixed distance, i.e. waterjet offset distance, as shown in Fig. 1. The laser is used to locally heat and soften the material and the waterjet is applied to expel the laser-softened elemental material. Water also takes a cooling action. Hence, there is the complicated interaction between the laser, waterjet and work material under the coupled effect of laser heating and waterjet expelling and cooling.

The mechanical properties of most of the engineering materials, such as strength and hardness, are temperature-dependent. Further, these mechanical properties of the material will decrease obviously with the increase of the temperature [20,21]. Based on this principle, the work material could be removed by the pressurised waterjet with an appropriate pressure when it is heated to a softened status. By contrast, the work material with a surround room temperature cannot be removed by a waterjet with the same pressure. Consequently, there is an intrinsic correlation between the material removal and its temperature field. The temperature field is crucial for the hybrid cutting process. The modelling of the temperature field is a precondition for the research of cutting process.

3. Governing equations During the hybrid laser-waterjet micro-machining process, a complicated interaction occurs on the machining surface of workpiece as both laser irradiation and pressurised waterjet impingement are existing. In order to investigate the evolution and distribution of the temperature field during the process, a three-dimensional (3-D) 7

heat transfer model is applied. For nanosecond pulsed laser ablation of semiconductor materials such as silicon and silicon carbide, τp (10-7~10-8 s) >> τ0 (10-10~10-14 s), where τp, and τ0 are the laser pulse duration and relaxation time of material, respectively. τ0 is a physical parameter which indicates the time duration from the generation of temperature gradient to the establishment of heat flux [22]. It can be assumed reasonably that the heat flux establishes immediately when the temperature gradient generates. Consequently, the material relaxation is negligible. The Fourier heat conduction equation is applied in which the heating action of laser is treated as an inner heat source, cT   k 2T  Q t

(1)

where T, ρ, c, k, t and Q are temperature, material density, heat capacity at a constant pressure of material, thermal conductivity of material, time and heat source, respectively. In Eq. (1), the work material is assumed to be isotropic. Besides, the material properties are considered as temperature independent and obtained by

uc 

1 T  Tf

T u T dT T

(2)

f

where uc and u (T) are temperature-independent property and temperature-dependent property of material, respectively. u refers to the thermal properties of work material, e.g. heat capacity at constant pressure, density, and thermal conductivity. Eq. (1) is the governing equation of temperature field in hybrid laser-waterjet micro-machining. During the hybrid micro-machining process, the targeted material is heated to a soften status, which remains the solid status, by laser irradiation and then expelled by the waterjet impingement, melting of the material does not happen. Therefore, the melting latent should not be considered in the governing equation.

4. Initial and boundary conditions 4.1. Initial conditions To solve the temperature field, the initial condition and boundary conditions 8

should be determined. As the pulsed laser is a periodical heat source, the workpiece is heated and cooled in sequence during every period. The temperature field of the workpiece at the initial time of the micro-machining is uniform and equal to the surrounding room temperature, i.e.

t  0

T  Tf

(3)

where Tf is a surrounding temperature.

4.2. Boundary conditions The lengths of the workpiece are 2δ1 × 2δ2 × δ3 in x, y and z directions, as shown in Fig. 1. On the irradiated surface of the workpiece, i.e. the upper surface (z = 0), cooled by the waterjet impingement and surface radiation occur simultaneously. For the cooling of waterjet impingement, the heat flux towards to the surrounding can be calculated by Newton’s law of cooling, i.e. [23]

q1  hw T  Tf 

(4)

where q1 and hw are heat flux caused by waterjet impingement and heat transfer coefficient of water, respectively. For the cooling of surface radiation, the heat flux towards to the surrounding is obtained by the Stefan-Boltzmann law as [23]



q 2   T 4  Tf4



(5)

where q2, ε and σ are heat flux caused by surface radiation, surface emissivity and Stefan-Boltzmann’s constant (5.67 × 10-8 W/m2 K4), respectively. Hence, the total heat flux on the upper surface towards to the surrounding can be calculated by Fourier’s law of heat conduction as [23] k

T  q1  q 2 z

(6)

where k is the thermal conductivity of the material. On the right-hand side of Eq. (6), a plus means that heat flows out of the workpiece. Hence q1 and q2 makes a negative contribution to heating the workpiece. The rate of radiation emitted from the surface (no more than 105 W/m2, as shown 9

in Fig. 2), however, is far less than the rate of the forced convection by waterjet impingement (normally 108~1011 W/m2), q3 is neglected. As a result, the boundary condition of the upper surface can be simplified as,

z  0

k

T  hw T  Tf  z

(7)

Eq. (7) is the Robin boundary condition which defines the amount of heat transfer across the obtained boundary. .

As the bottom surface of workpiece (z = − δ3) is glued to the fixture, a large thermal resistance is exist at the interface between the work and the fixture. The bottom boundary is considered as a thermally insulated condition without the heat flow into and out of the surface or satisfying the adiabatic condition. Consequently, the Neumann boundary condition is applied, i.e.

z   3

T  0 z

(8)

Both of natural convection of air and radiation heat transfer act on the side surfaces of the workpiece (x = ± δ1, y = δ2). Thus the total heat transfer is determined by adding the contributions of both heat transfer mechanisms. For simplicity and convenience, this is done by defining a combined heat transfer coefficient ha that includes the effects of both convection and radiation. Then the total heat transfer rate from the surfaces by convection and radiation is expressed as [23] T  ha T  Tf  x

x  1

k

x  1

k

y  - 2

k

y  2

k

(9)

T  ha T  Tf  x

(10)

T  ha T  Tf  y

(11)

T  ha T  Tf  y

(12) 10

where ha is the combined heat transfer coefficient of air. Eq. (7) to Eq. (12) are the boundary conditions of the temperature field in hybrid laser-waterjet micro-machining. All of these boundary conditions are homogeneous and the variable separation approach can be applied to solve the model.

5. Interaction of multiphysics As mentioned in Section 2, during the hybrid laser-waterjet micro-machining process, the complicated interaction among the workpiece, laser and pressurised waterjet is occurred on the machining surface of the workpiece. For this hybrid micro-machining process, they are the crucial factors that will determine the temperature field distribution and the final surface quality. Accordingly, the study on this interaction is necessary and challenging. The interaction of multiphysics in this hybrid micro-machining process is mainly: a) workpiece surface absorbs the laser and heated by it; b) part of the laser is absorbed or diverged by water; and c) workpiece surface is impinged and cooled by the waterjet.

5.1. Heat source caused by laser Eq. (1) has a term of Q, which indicates the laser heat source absorbed by the workpiece. Before being absorbed by the material, the laser beam goes through a water layer upon the workpiece surface and is absorbed and reflected by water partially. In the next moment, a part of the laser beam intensity that arriving at the workpiece surface is reflected by the surface and the other is absorbed by the material and becomes the heat source in the model. Consequently, the laser beam intensity absorbed by the material is obtained by

Q x ,y ,z ,t   I x ,y ,t   1  R f   1  Rw   1 -    ab  expab z 

(13)

where I (x, y, t), Rf, Rw, β and ab are the laser beam intensity, the targeted material reflectivity, the water reflectivity, the laser absorptivity of water and optical absorption coefficient of material, respectively. β is defined to characterize the 11

interference between the laser and the water. A further study of β will be obtained in Section 5.3 subsequently. The laser beam intensity is a function of time and space in the form of

I x ,y ,t   I 0   x ,y    t 

(14)

where I0, ψ (x, y) and φ (t) are laser beam intensity at center position, laser pulse function in space domain and laser pulse function in time domain, respectively. In general, the laser pulse function in time domain is a polynomial-exponential function and fitted from the measured pulse shape profile provided by the laser machine manufacturer. For the purpose of calculation, it can be considered approximately as a periodical function with a normal distribution at the full-width at half-maximum (FWHM), obtained as

  exp    t        t   

 2 ln 2  t   p ln 100       p  

2

   

1  f 

 1  0  t   f   1 t   f 

(15)

where f is the laser pulse frequency. Eq. (15) is a piecewise function. It can be expressed as a Fourier series expansion, i.e.

 t   a0 



ai cos2ift   bi sin2ift   i 1

where,

a0  f 

 exp  

1

f

0

1

ai  2f  f 0

bi  2f 

1

f

0

 2 ln 2t   p ln 100       p  

2

 dt  

 exp  

 2 ln 2t   p ln 100       p  

2

  cos2ift dt  

 exp  

 2 ln 2t   p ln 100       p  

2

  sin2ift dt  

12

(16)

It indicates that the Fourier series with the first 150 terms entirely fits the original function where the laser works at a pulse duration of 150 ns and a frequency of 50 kHz, as shown in Fig. 3.

When the laser machine works at a transverse electromagnetic mode (TEM) of TEM00, the laser pulse function in space domain is distributed as the Gaussian function, i.e. [24]



 x ,y   exp 





8 x2  y2   d b2 

(17)

Regarding the Paraxial Helmholtz equation,

I0 

2E p 8Pa   p  Al  p d b2f

(18)

where Ep, Al, db and Pa are laser pulse energy, laser-irradiated area, laser beam diameter and average power of laser, respectively. Hence, the laser heat source is written as

Q x ,y ,z ,t  





 8 x2  y2         1  R  1  R  1    exp   f w  p d b2f d b2   8Paab

  expab z   a0  





1



ai cos2ift   bi sin2ift   i

(19)

The moving of laser beam along the x direction with a constant speed v needs to be considered. It is assumed that the center position of laser beam is located at (x0, y0) at the initial moment. The heat source caused by laser can be rewritten as

Q x ,y ,z ,t  

 1  R f   1  Rw   1 -    expab z 

8Pa ab

 p d b2f

2 2   x  x 0  vt   y  y 0     exp 8   d b2    

  a0  





1



ai cos2ift   bi sin2ift   i

where v is the traverse speed of laser. It can be directly considered as, 13

(20)

x 0  0  y 0  0

(21)

Eq. (21) indicates that the initial point locates at the origin of coordinates.

5.2. Heat transfer caused by waterjet In Section 3.2, the convection heat transfer coefficient of water hw is applied. This heat transfer coefficient is not a property of water, but rather depends on all the variables influencing the forced convection in the study. Thereby, to determine the value of hw, the properties of the pressurised waterjet impingement, e.g. nozzle geometry, water pressure, stand-off distance and inclination angle, should be considered. A free jet is formed when the water is ejected from the nozzle orifice. It is generally recognized that there exist three waterjet regions at free jet stage, i.e. the initial, main and final regions [25]. As the initial region, which is close to the nozzle exit, has tremendous kinetic energy, it is usually used to cut. In the initial region, the velocity inside the potential core is equal to the jet velocity at the nozzle exit. In addition, the divergence angle in this region is small, and it means the change of jet diameter is little. As a result, the diameter of waterjet is equal to the nozzle exit diameter approximately. Because of the high pressure (that means a high velocity at the nozzle exit), typical impinging waterjet conditions are well into the turbulent flow regime. For turbulent flow, the upper limit on the maximum velocity of a pure waterjet as it exits the nozzle is obtained by [26]

v0 

2Pi  P 

(22)

w

where v0, Pi, ΔP and ρw are waterjet velocity at nozzle exit, hydrostatic pressure at nozzle inlet, pressure drop in nozzle and water density, respectively. For well-designed and unworn nozzles, the pressure drop is determined as [26]

P  0.2Pi

(23) 14

On the stagnation region, the pressurised waterjet impinges rapidly onto the workpiece surface with the velocity decreasing to zero and then reflects, forming a parallel flow which has an initial velocity of v0 on the workpiece surface [27]. The water spreads out in a thin layer until a hydraulic jump occurs. The jump condition in vertical impingement is derived by E. J. Watson [28]. This is similar to the situation that the waterjet impinged obliquely. As the dimension of machined surface of the workpiece is much smaller than the radius of the standing wave, only the thin water layer on the inside of the jump is considered. The inclination angle has an effect on the radius of the hydraulic jump, but affects the initial velocity of the reflected water flow barely. Since the water in the thin film flows with a large Reynolds number, it is natural to apply the ideas of turbulent boundary-layer theory so as to discuss the forced heat convection on the thin water film region. On this water film region around the stagnation point, the effect of viscous stress is negligible and the water flow here is influenced by the stagnation point region observably. Therefore, the flow velocity over the substrate can be assumed as a constant and equals to the initial velocity at the stagnation point. The flow velocity is varied from 89.4 to 178.9 m/s for the water pressures of 5–20 MPa. It is assumed the air keyhole does not exist since the rapid flow of water layer. For the turbulent boundary-layer, the average Nusselt number on the thin water film region is determined by performing the integration on the relation for local Nusselt number and obtained by [29]

Nuf  0.037 Ref0.8 Pr1 3

(24)

where Nuf, Ref and Pr are average Nusselt number on the thin water film region, average Reynolds number on the thin water film region and Prandtl number of water, respectively. For the average Reynolds number of turbulent flow on the thin water film region, it can be considered as [29] Ref 

wv 0 Lf 

(25)

where Lf and μ are characteristic length of thin water film and viscosity of water, 15

respectively. And the Prandtl number of water can be determined as [29] Pr 

cw kw

(26)

where cw and kw are heat capacity at a constant pressure of water and thermal conductivity of water. In this model, the characteristic length of thin water film can be presented by

Lf 

 1 2 1   2

(27)

Therefore, the total convection heat transfer coefficient is the weighted average of the coefficients on these two regions, and the weight is the percentage of their action areas [30] 2

4

4

1

7

1  k  hw  w  Nu f  0.037kw3  Lf 5  w5  v 05  cw3   15 Lf

(28)

5.3. Interference between laser and waterjet In this section, the interference between laser and waterjet will be discussed. The absorption of laser by water, the formation of laser-induced plasma in water， the bubbles formation and the refraction at the inclined air-water interface will be considered.

5.3.1. Absorption of laser by water As the hybrid laser-waterjet micro-machining employed in this study, the water layer thickness is determined by the offset distance between the laser beam and the waterjet beam (shown in Fig. 4).

In front of the stagnation zone, the waterjet transits to the thin water layer at a critical point. If the offset distance is smaller than the critical distance, i.e. the laser spot is irradiated inside the stagnation zone, an interference is happened directly between the waterjet and the laser beam. Inversely, the laser beam goes through the thin water layer and is absorbed partially by it. 16

As discussed in Section 5.2, the flow velocity in the water film is assumed to be constant at v0 and the effect of water viscosity and surface tension are negligible. The cross-section of the oblique waterjet impingement on the workpiece surface is an ellipse. It is simplified as a circle, which is similar to the situation in vertical impingement. Considering a circle of radius xw with its center at the waterjet axis, the volume flow rate within the circle boundary should be equal to the volume flow rate of waterjet [28]. The thickness of the water film can be derived. Hence, the thickness of water film that the laser goes through can be calculated by

  dw   x w  tan     2 sin   lw   2  dw  8x  w

1   2   1  1  sin 2    4 sin    1   dw  1  1 - sin 2 2   4 sin   

xw  xw

dw

(29)

where dw, xw and θ are nozzle exit diameter, waterjet offset distance and waterjet inclination angle, respectively. Fig. 5 shows the water layer thicknesses under different waterjet offset distances and inclination angles with a nozzle exit diameter of 0.3 mm. When the waterjet offset distance varied from 0.2 mm to 0.6 mm, the water layer thickness varied from 56 μm to 19 μm, i.e. 0.19 to 0.06 times of the nozzle exit diameter.

The laser beam intensity absorbed by water can be determined by the Beer-Lambert law as [31]



  1  exp  

lw la

  

(30)

where lw, and la are water layer thickness and light absorption length in water, respectively. Light absorption length is most convenient for estimating the amount of light absorbed by a layer of water in a case of low absorption and determined by the wavelength of light. For the Yb:Glass nanosecond-pulsed fiber laser with a wavelength of 1080nm, its light absorption length is about 30mm [31]. 17

5.3.2. Formation of laser-induced plasma in water As material melting or vaporization must be avoided by properly selecting the process parameters in this technology, the plasma induced by workpiece vaporization is not concerned. The laser beam irradiates a thin water layer. The optical breakdown, which is also known as laser-induced breakdown, would be occurred and the plasma would form in the condensed water, once the laser irradiance exceeds the breakdown threshold. The threshold intensity of laser-induced breakdown for a wavelength of 1080 nm and a pulse duration more than 1 ns in water with seed electrons provided by impurities is 1010 W/cm2 [32]. When a laser beam is focused on the surface of the water layer, the peak intensity is calculated by Eq. (18), as shown in Fig. 6. It indicates that the peak laser intensity is less than the threshold intensity of laser-induced breakdown when a laser with a focused diameter of 18.34 μm and a maximum average power of 30 W is applied. Consequently, the laser-induced plasma could not form in the water layer.

5.3.3. Laser refraction at the air-water interface The laser will encounter a significant refraction at the inclined air-water interface thereby changing the focal position and the focused diameter in the process, as shown in Fig. 7. It is assumed that the laser is focused on the workpiece surface when the water isn’t exerted.

The divergence angle in water can be calculated by refraction law,

 sin 1 2   nw 

 2  2 arcsin 

(31)

where γ2, γ1 and nw are divergence angle in water, divergence angle in air and refractive index of water relative to air, respectively. The refractive index evolves as a function of water temperature and pressure, where water temperature plays a major role. For the water temperature ranging from 20 °C to 100 °C, the refractive index of water ranges from 1.326 to 1.312 under a 18

water pressure of 5 MPa and from 1.328 to 1.314 under a water pressure of 20 MPa [33-35]. This would affect the laser divergence angle and its irradiation diameter. It indicates that the laser irradiation diameter decreases with the water temperature is increased. However, this effect is considered minor when comparing to the effect of laser refraction and divergence caused by other factors. The divergence angle in air can be obtained by

 1  2 arctan

D 2F

(32)

where D and F are laser beam diameter before focused by a lens and focal length, respectively. The focused diameter after refracted changes into

df 2 

2M R2  tan 2 2

(33)

where df2, λ and M2R are focused diameter after refracted, laser wavelength and laser beam quality factor in water, respectively. Therefore, the laser beam diameter irradiated at the workpiece surface can be obtained by

db

  4z  d f 2 1   2c   d f 2 

   

2

   

12

(34)

where zc is the downward distance of focal position that can be calculated by

 tan 1 2  zc  lw   1  tan 2 2 

(35)

5.3.4. Bubble formation Bubbles in water can be formed during the process. It can optically disturb the incident laser by scattering and diffracting the laser beam, causing the change of laser beam quality factor (M2). With an increased water pressure, the turbulence in the water film is enhanced (implies a larger Reynolds number), which decrease laser beam quality (and increases the laser quality factor). 19

A laser beam analysis meter (HAAS LTI BA-CAM-200-1100-R2) was employed to measure the values of M2 of the incident laser and emergent laser. The results show that M2 in water can be up to 1.9 times that in air under a water pressure of 10 MPa, as a result of the increased divergence angle and laser diameter induced by the water layer. This in fact results in a decreased laser intensity and heating ability. According to Eqs. (31) to (33), in order to decrease the focused beam diameter to compensate for this beam divergence caused by water bubbles, a larger laser beam diameter before going through the focusing lens is needed by using a beam expander.

6. Solution of temperature field The model of temperature field in waterjet-laser hybrid micro-machining is determined by Eq. (1), Eq. (3), Eq. (7) to Eq. (12) and Eq. (20). The temperature field model is presented as,

c

T  k2T  Q x ,y ,z ,t  t

t  0

T  Tf

x  1

k

x  1

k

y   2

k

y  2

k

z   3

T  0 z

z  0

k

T  ha T  Tf  x T  ha T  Tf  x

T  ha T  Tf  y

(36)

T  ha T  Tf  y

T  hw T  Tf  z

The surplus temperature is defined by, 20

U  T  Tf

(37)

Thus the model is rewritten as,

  2U U  2U  2U     Q x ,y ,z ,t     2   t y 2 z 2  k  x

t  0

U  0

x  1

U h  aU  0 x k

x  1

U h  aU  0 x k

y   2

h U  aU  0 y k

y  2

U h  aU  0 y k

z   3

U  0 z

z  0

U h  w U  0 z k

(38)

A coordinate transformation is implemented by using

x  x   1  y  y   2 z  z   3 

(39)

Thereby, a temperature field problem for determining solution is derived as,

  2U U  2U  2U     Q x ,y ,z ,t     2   t y 2 z 2  k  x

t  0

U  0

x  0

U h  aU  0 x k

x  21

U h  aU  0 x k

21

y  0

U ha  U  0 y k

y  2 2

U h  aU  0 y k

z  0

U  0 z

z  3

U h  w U  0 z k

(40)

According to the DuHamel’s theorem [36], the method of Eigen function and its general form of solution [37], the problem (Eq. (40)) is solved with a solution in variable separation form, i.e.

    B mnl  exp mnl t m n l 

U x ,y ,z ,t ,  

 

1

1

1

t

2

0



 x      sin m   m   2 1 

 y  z  sin n   n   cos l d 3  2 2  where,

mnl 2

  2    2   2     m    n    l    2 1   2 2    3  

B mnl 

1

M mnl

3

2 2

21

0

0

0

  

 cos

l z dx dydz 3



sin m



m

M mnl   1 1  

 x   y    Q x ,y ,z ,   sin m   m   sin n   n  k  2 1   2 2 

3 

1  2 

tan  m 

km 2ha1

tan  n 

k n 2ha  2

   sin n  cosm  2 m    2 1  cosn  2 n  n   

sin 2 l 2 l

  

22

(41)

μm, νn and κl are the positive roots (positive zeros) of the transcendental equations. μm are the positive zeros of

 k 4h 2 2    x  a2 1  4ha1  k x 

g 1 x   cot x 

νn are the positive zeros of

g 2 x   cot x 

 4h 2 2   x  a2 2   k x  

k 4ha 2

and κl are the positive zeros of

g 3 x   cot x 

kx hw  3

Consequently, the temperature field in hybrid laser-waterjet micro-machining is

    bmnl  exp mnl t m n l   

T x ,y ,z ,t  

1

1

1

t

2

0

  x   1       sin  m  m  2 1  



 y   2    z   3   sin  n   n   cos l d  Tf 2 2 3  

(42)

where

bmnl 

1

0

2

1







M mnl       3

2

1

  x   1     Q x ,y ,z ,   sin  m  m  k 2 1  

 y   2    z   3   sin  n   n   cos l dxdydz 2   2 3  

7. Case study In this section, a case study is obtained where the temperature field of a silicon carbide wafer is calculated. The thermal and optical properties of silicon carbide and water used in this study are listed in Table 1. A Yb:Glass nanosecond-pulsed fiber laser with a wavelength of 1080 nm and a pulse duration of 150 ns at a frequency of 50 kHz was applied. Laser was focused on the upper surface of the workpiece before water is exerted. The settings of laser and waterjet parameters are listed in Table 2. The hybrid cutting head was traversed along the x direction with a speed of 1 mm/s. The initial and the surrounding temperatures 23

were 293.15 K. Eq. (42) is the analytical solution of the temperature field in hybrid laser-waterjet micro-machining process. It has a complex structure since a quadruple integral and a Fourier series are included. To express the results intuitively and visually, a group of 3-D temperature profiles are plotted via MATLAB, as shown in Fig. 8. The temperature fields at the upper surface and a cross-section are illustrated. The cross-section considered is located in the y-z plane at x = 0 position.

Figs. 8 (a) to (h) illustrate the evolution of the temperature distribution at the cutting front during the first laser pulse period at 100ns, 150ns, 200ns, 330ns, 500ns, 1000ns, 3500ns and 20000ns respectively. The times corresponding to Figs. 8 (a) to (h) are labelled as A to H in Fig. 3 so that a correspondence of the temperatures to the laser pulse locations can be obtained. It can be seen that the temperature at (0, 0, 0) point reaches the maximum value and decreases gradually along the depth direction and the radial direction of laser beam. The isotherm is a set of Gaussian curve. The maximum temperature (2726.5 K) during the whole laser pulse period is found at 330ns, as shown in Fig. 9. Subsequently, the workpiece is cooled down by the pressurised waterjet since its strong cooling effect. The heating and cooling processes are periodically alternating.

Fig. 8 (h) and Fig. 9 indicate that due to the waterjet cooling effect, heat accumulation between laser pulses is not expected to take place. During the ablation process, the maximum temperature at (0, 0) point on the workpiece occurs in the first laser pulse whose beam centre is directly on the point. As the heat source is being moved away, the heat energy that could be absorbed by this point reduces gradually until to the stage where no heat input is made to this point, as shown in Fig 10.

The workpiece temperature cannot be measured directly at the laser-irradiated area as water is not transparent in the infrared wavelength. Therefore, the measurement was conducted on the back side of wafer instead. An infrared camera 24

(NEC TH5104R) with a resolution of 50 μm was used and the average temperature over a period of time was obtained. The results derived from the measurement and the simulation are compared in Fig. 11. The measurement results are a little lower than the simulated temperatures. This may be due to the fact that a measuring window was opened on the fixture and the measured zone of bottom surface was exposed to the air directly. As a result, a small fraction of heat might be carried away by the air. Nevertheless, this error appears to be acceptable.

FEM simulation for only laser cutting was performed using the same laser parameters as in the hybrid laser-waterjet configuration, as shown in Fig. 12. The temperature exceeds the melting temperature (3100 K) in the third laser pulse due to the heat accumulation between laser pulses. Besides, cutting experiments were also implemented using the same parameters and the machined surface morphologies were observed and measured under a 3D laser microscope (Keyence VK-X200), as shown in Fig. 13. It can be seen that obvious HAZ and recast layer were formed in laser dry cutting. By contrast, the clean and straight edges and almost thermal-damage-free grooves were obtained using the hybrid laser-waterjet micro-machining technology.

The essence of the technology is to use a laser to heat and soften the local material and a waterjet to remove the laser-softened material in its solid state without melting or vaporization. If the pressure applied for the waterjet is too low, melting or vaporization can occur similar to laser dry cutting, which indicates an improper selection of the waterjet parameters. Similarly, if the laser cannot soften the material to the strength low enough for the waterjet to remove, there is no material removal takes place. For a given waterjet parameters, particularly the water pressure, there is a material ultimate strength below which the waterjet is able to remove the elemental material. This threshold material strength is related to the material temperature which in turn is related to the process parameters, not least the laser optical and waterjet parameters. There are several important parameters that influence the temperature field 25

during the machining process and the characteristics of the groove, e.g. average laser power, nozzle exit diameter and waterjet offset distance. Hereby, the temperatures at the central point of cutting front under different process parameters after 300 ns of laser irradiation are shown in Fig. 14 and Fig. 15. Fig. 14 illustrates the simulated temperatures at the central point of the cutting front and experimental cutting depth under different average laser powers after 300 ns of laser irradiation. The linear fitting of the relationship between the cutting depth and average laser power is derived. It can be seen from Fig. 14 (a) that the maximum temperature is proportional to the average laser pulse with a pretty large slope. Thus, the average laser power is a conclusive factor in the rise of temperature. When the waterjet parameters are kept constant, the cutting depth of grooves is approximatively positive linear correlated to the average laser power, as shown in Fig. 14 (b).

Fig. 15 illustrate the temperatures at the central point of the cutting front under different waterjet offset distances and nozzle exit diameters after 300 ns of laser irradiation. The temperatures as shown in Fig. 15 increase rapidly at the initial stage and then turns to a gentle rising. This indicates that interference between the laser and the waterjet takes place when the laser spot is irradiated inside the stagnation zone. Inversely, the change of waterjet offset distance has a smaller effect on the temperature since the thickness of the thin water layer that the laser goes through is small and stable. The turning point of the curve is the critical point of the waterjet offset distance. With the increase of nozzle exit diameter, the critical point moves away from the central point of waterjet beam, and the thickness of water layer would also increase. As a result, a larger amount of laser would be absorbed and cooling action of waterjet is more obvious. However, the influence of these two factors is not significant comparing with the average laser power. Since the variation of waterjet parameters causes change to not only the temperature field, but also the impinging force, there is a more complicated relationship among these parameters, the temperature and the cutting depth. More study on these aspects is required in the future. 26

8. Conclusions A 3-D analytical model for the temperature field in the hybrid laser-waterjet micro-machining has been developed using a parabolic heat conduction equation with a laser heat source inside the workpiece. The upper boundary was a Robin boundary where the complicated interaction among the multiphysics during this hybrid process was considered. The heat transfer caused by waterjet, and the interference between laser and waterjet, including the absorption of laser by water, the formation of laser-induced plasma in water, the bubble formation and the laser refraction at the air-water interface, were discussed. Furthermore, the Robin and Neumann boundary conditions were applied to the lateral and bottom boundaries, respectively. For deriving the solution of the heat conduction equation, the heat flux caused by laser on the upper surface was substituted by an equivalent body heat source. A coordinate transformation was also made. The material properties were assumed to be temperature-independent and isotropic. The calculation for silicon carbide was conducted as an example. A group of 3-D temperature profiles at different times during a laser pulse period were plotted via MATLAB. Furthermore, the temperatures at the central point of cutting front under different process parameters were studied. According to the results, the maximum temperature is below the melting point during the whole machining process. Heat accumulation between laser pulses is not expected to take place. The average laser power has a conclusive influence on the temperature field, and other process parameters, such as nozzle exit diameter and waterjet offset distance, also have a certain role. The temperature field is a vital intermediate result that would determine the characteristics of grooves and final surfaces. But it can’t be measured directly due to the water upon the workpiece. The temperature field on bottom surface was measured to compare with the simulation results. On the basis of the present study, a further work could be conducted to investigate the underlying mechanism of material removal in the hybrid laser-waterjet micro-machining process. 27

Acknowledgment This work is financially supported by National Natural Science Foundation of China (51375273) and Taishan Scholars Program (TS20130922).

28

References [1] Nath A, Khare A. Transient evolution of multiple bubbles in laser induced breakdown in water. Laser Part Beams 2011;29:1-9. [2] Richerzhagen B, Kutsuna M, Okada H, Ikeda T. Waterjet-guided laser processing. Proceedings of SPIE - The International Society for Optical Engineering, Osaka, 2002, 91-4. [3] Molian R, Neumann C, Shrotriya P, Molian P. Novel laser/water-jet hybrid manufacturing process for cutting ceramics. Journal of Manufacturing Science and Engineering, Transactions of the ASME 2008;130:310081-3100810. [4] Kalyanasundaram D, Shrotriya P, Molian P. Obtaining a relationship between process parameters and fracture characteristics for hybrid CO2 laser/waterjet machining of ceramics. Journal of Engineering Materials and Technology, Transactions of the ASME 2009;131:110051-1100510. [5] Tangwarodomnukun V, Wang J, Huang CZ, Zhu HT. An investigation of hybrid laser–waterjet ablation of silicon substrates. International Journal of Machine Tools and Manufacture 2012;56:39-49. [6] Zhu H, Wang J, Li WY, Li HZ. Microgrooving of germanium wafers using laser and

hybrid

laser-waterjet

technologies.

Advanced

Materials

Research

2014;1017:193-8. [7] Feng SC, Huang CZ, Wang J, Zhu HT, Yao P, Wang L. A comparison among dry laser ablation and some different water-laser co-machining processes of single crystal silicon carbide. Materials Science Forum 2016;861:3-8. [8] Kim MJ. 3D finite element analysis of evaporative laser cutting. Appl Math Model 2005;29:938-54. [9] Shuja SZ, Yilbas BS, Momin O. Laser heating of a moving slab: Influence of laser intensity parameter and scanning speed on temperature field and melt size. Opt Laser Eng 2011;49:265-72. [10] Han G, Zhao J, Li J. Dynamic simulation of the temperature field of stainless steel laser welding. Mater Design 2007;28:240-5. [11] Li X, Wang L, Yang L, Wang J, Li K. Modeling of temperature field and pool 29

formation during linear laser welding of DP1000 steel. J Mater Process Tech 2014;214:1844-51. [12] Polak M, Chmelickova H, Stranyanek M, Vasicek L. Modeling of laser cutting and drilling. Proceedings of SPIE - The International Society for Optical Engineering, Olomouc, 2002, 146-51. [13] Yilbas BS, Mansoor SB. Laser pulse heating and phase changes in the irradiated region: Temperature-dependent thermal properties case. Int J Therm Sci 2009;48:761-72. [14] Kim CK. An analytical solution to heat conduction with a moving heat source. J Mech Sci Technol 2011;25:895-9. [15] Parkitny R, Winczek J. Analytical solution of temporary temperature field in half-infinite body caused by moving tilted volumetric heat source. Int J Heat Mass Tran 2013;60:469-79. [16] Lu L, Shi Y, Xu C, Xu G, Wang J, Xu B. The influence of pulse width and energy on temperature field in metal irradiated by ultrashort-pulsed laser. Physics Procedia 2012;32:39-47. [17] Li CF, Johnson DB, Kovacevic R. Modeling of waterjet guided laser grooving of silicon. International Journal of Machine Tools and Manufacture 2003;43:925-36. [18] Yang LJ, Wang ML, Wang Y, Tang J, Chen YB. Numerical simulation on the temperature field of water-jet guided laser micromachining. Advanced Materials Research 2009;69-70:333-7. [19] Tangwarodomnukun V, Wang J, Huang CZ, Zhu HT. Heating and material removal process in hybrid laser-waterjet ablation of silicon substrates. International Journal of Machine Tools and Manufacture 2014;79:1-16. [20] Milman YV, Chugunova SI, Goncharova IV, Chudoba T, Lojkowski W, Gooch W. Temperature dependence of hardness in silicon–carbide ceramics with different porosity.

International

Journal

of Refractory Metals

and Hard

Materials

1999;17:361-8. [21] Mu Oz A, Martínez-Fernández J, Domínguez-Rodríguez A, Singh M. High-temperature compressive strength of reaction-formed silicon carbide (RFSC) 30

ceramics. J Eur Ceram Soc 1998;18:65-8. [22] Nasri F, Ben Aissa MF, Belmabrouk H. Microscale thermal conduction based on Cattaneo-Vernotte model in silicon on insulator and Double Gate MOSFETs. Appl Therm Eng 2015;76:206-11. [23] Çengel YA. Heat Transfer: A Practical Approach. 2nd ed. New York: McGraw-Hill; 2003. [24] Steen WM. Laser Material Processing. 3rd ed. London: Springer; 2003. [25] Leu MC, Meng P, Geskin ES, Tismeneskiy L. Mathematical modeling and experimental verification of stationary waterjet cleaning process. Journal of Manufacturing Science and Engineering 1998;120:571-9. [26] Ali YM, Wang J. Impact Abrasive Machining. In: Jackson MJ, Davim JP, editors. Machining with Abrasives, New York: Springer US; 2011, p. 385-419. [27] Shu JJ. Microscale heat transfer in a free jet against a plane surface. Superlattice Microst 2004;35:645-56. [28] Watson EJ. The radial spread of a liquid jet over a horizontal plane. J Fluid Mech 1964;20:481-99. [29] Incropera FP, DeWitt DP. Fundamentals of Heat Transfer. 1st ed. New York: John Wiley & Sons; 1981. [30] Kays WM, Crawford ME. Convection Heat and Mass Transfer. 2nd ed. New York: McGraw-Hill; 1980. [31] Kruusing A. Underwater and water-assisted laser processing: Part 1—general features, steam cleaning and shock processing. Opt Laser Eng 2004;41:307-27. [32] Noack J, Vogel A. Laser-induced plasma formation in water at nanosecond to femtosecond time scales: calculation of thresholds, absorption coefficients, and energy density. IEEE J Quantum Elect 1999;35:1156-67. [33] Schiebener P, Straub J, Levelt Sengers JMH, Gallagher JS. Refractive index of water and steam as function of wavelength, temperature and density. J Phys Chem Ref Data 1990;19:677. [34] Weiss L, Tazibt A, Tidu A, Aillerie M. Water density and polarizability deduced from the refractive index determined by interferometric measurements up to 250 MPa. 31

The Journal of Chemical Physics 2012;136:124201. [35] Weiss L, Tazibt A, Tidu A, Aillerie M. Erratum: “Water density and polarizability deduced from the refractive index determined by interferometric measurements up to 250 MPa” [J. Chem. Phys.136(12), 124201 (2012)]. The Journal of Chemical Physics 2014;141:149904. [36] Gu Q. Mathematical Methods for Physics. 1st ed. Beijing: Science Press; 2012. [37] Wang LQ, Zhou XS. Dual-phase-lagging Heat Conduction Equations. 1st ed. Jinan: Shandong University Press; 2000. [38] Duc DH, Naoki I, Kazuyoshi F. A study of near-infrared nanosecond laser ablation of silicon carbide. Int J Heat Mass Tran 2013;65:713-8. [39] Müller SG, Eckstein R, Fricke J, Hofmann D, Hofmann R, Horn R. Experimental and Theoretical Analysis of the High Temperature Thermal Conductivity of Monocrystalline SiC. Materials Science Forum 1998;264-268:623-6.

32

Traverse direction of hybrid cutting head

Laser Waterjet -δ

z

δ

y

2

δ

1

1

x

O -δ 0 -δ

2

3

Fig. 1. The schematic of waterjet and laser beams and workpiece geometry.

33

Fig. 2. Heat flux caused by surface radiation under different surface emissivity and temperatures

34

Fig. 3. The fitting of the laser pulse function in time domain using a Fourier series with its first 150 terms (pulse duration of 150 ns and frequency of 50 kHz)

35

Fig. 4. Setting parameter for the hybrid laser-waterjet micro-machining process.

36

Fig. 5. Water layer thicknesses under different waterjet offset distances and inclination angles with a nozzle exit diameter of 0.3 mm.

37

Fig. 6. Peak laser intensities under different average laser powers and laser pulse frequencies with a pulse duration of 150 ns and a focused beam diameter of 18.34 μm.

38

Laser

γ1 df1 Water layer Work material

db γ2

zc

df2

Fig. 7. Laser refraction at the inclined air-water interface.

39

(a) t = 100 ns

(b) t = 150 ns

40

(c) t = 200 ns

(d) t = 330 ns

41

(e) t = 500 ns

(f) t = 1000 ns

42

(g) t = 3500 ns

(h) t = 20000 ns Fig. 8. Temperature fields at different times during a laser pulse period (average laser power Pa = 30 W, laser beam diameter db = 31.7 μm, water pressure Pi = 10 MPa, nozzle exit diameter dw = 0.3 mm, waterjet offset distance xw = 0.4 mm and waterjet inclination angle θ = 50 °).

43

Fig. 9. Temperature evolution at the central point of cutting front (average laser power Pa = 30 W, laser beam diameter db = 31.7 μm, water pressure Pi = 10 MPa, nozzle exit diameter dw = 0.3 mm, waterjet offset distance xw = 0.4 mm and waterjet inclination angle θ = 50 °).

44

Fig. 10. Temperature evolution at (0, 0) point with respect to the number of laser pluses (average laser power Pa = 30 W, laser beam diameter db = 31.7 μm, water pressure Pi = 10 MPa, nozzle exit diameter dw = 0.3 mm, waterjet offset distance xw = 0.4 mm and waterjet inclination angle θ = 50 °).

45

Fig. 11. The comparison between computation and measurement of bottom surface temperature (average laser power Pa = 30 W, laser beam diameter db = 31.7 μm, water pressure Pi = 10 MPa, nozzle exit diameter dw = 0.3 mm, waterjet offset distance xw = 0.4 mm and waterjet inclination angle θ = 50 °).

46

Fig. 12. Temperature evolution at the central point of cutting front in laser dry ablation (average laser power Pa = 30 W and laser beam diameter db = 31.7 μm).

47

(a) Top view from laser dry cutting.

(b) Top view from hybrid laser-waterjet cutting.

(c) Comparison of the cross-sectional profiles from the two processes. Fig. 13. The comparison of machined surface morphologies from laser dry and hybrid laser-waterjet cutting (average laser power Pa = 30 W, laser beam diameter db = 31.7 μm, water pressure Pi = 10 MPa, nozzle exit diameter dw = 0.3 mm, waterjet offset distance xw = 0.4 mm and waterjet inclination angle θ = 50 °). 48

(a) Temperatures under different average laser powers.

(b) Cutting depth under different average laser powers. Fig. 14. Temperatures at the central point of the cutting front from simulation and the cutting depth from experiments under different average laser powers after 300 ns of laser irradiation (laser beam diameter db = 31.7 μm, water pressure Pi = 10 MPa, nozzle exit diameter dw = 0.3 mm, waterjet offset distance xw = 0.4 mm and waterjet inclination angle θ = 50 °).

49

Fig. 15. Temperatures at the central point of the cutting front under different waterjet offset distances and nozzle exit diameters after 300 ns of laser irradiation (average laser power Pa = 30 W, laser beam diameter db = 31.7 μm, water pressure Pi = 10 MPa, and waterjet inclination angle θ = 50 °).

50

Table 1. Thermal and optical properties of silicon carbide and water. Properties Melting temperature of SiC (K)

3100

[23]

Absorption coefficient of SiC (1/m)

6012

[38]

Thermal diffusivity of SiC (m2/s)

1.83×10-5

[39]

Thermal conductivity of SiC (W/m K)

52.93

[39]

Thermal conductivity of water (W/m K)

0.67

[19]

Heat capacity at a constant pressure of water (J/Kg K)

4.2×103

[19]

Water density (kg/m3)

1×103

[19]

Water viscosity (Pa s)

8.9×10-4

[19]

SiC reflectivity

0.15

[23]

Water reflectivity

0.02

[19]

Table 2. Settings of laser and waterjet parameters. Parameters Average laser power (W)

30

Laser beam diameter irradiated at the workpiece surface (μm) 31.7 Waterjet pressure (MPa)

10

Nozzle exit diameter (mm)

0.3

Waterjet offset distance (mm)

0.4

Waterjet inclination angle (deg)

50

51