THERMAL EFFECTS OF HIGH ENERGY AND
ULTRAFAST LASERS
A Thesis
Presented to
The Faculty of the Graduate School
University of Missouri
------
In Partial Fulfillment
of the Requirement for the Degree
Doctor of Philosophy
------
By
Nazia Afrin
Thesis Supervisor: Dr. Yuwen Zhang
December 2015
DECLARATION
The undersigned, appointed by the dean of the Graduate Faculty, have examined the thesis entitled
THERMAL EFFECTS OF HIGH ENERGY AND ULTRAFAST LASER
Presented by
NAZIA AFRIN A candidate for the degree of Doctor of Philosophy in Mechanical and Aerospace Engineering, and hereby certify that, in their opinion, it is worthy of acceptance.
Professor Dr. Yuwen Zhang
Professor Dr. Jinn-Kuen Chen
Professor Dr. Gary Solbrekken
Professor Dr. Matt Maschmann
Professor Dr. Stephen Montgomery-Smith
DEDICATION
I dedicate this thesis to my parents, Shamsun Nahar Islam and late F.K.M. Aminul Islam, my husband Zobayer Khizir, my sister Dr. Aneesa Islam Keya for their endless love and support.
ACKNOWLEDGEMENT
I am highly grateful to my supervisor Professor Dr. Yuwen Zhang, Chairman of
Department of Mechanical and Aerospace Engineering for his encouragement, support, patience and guidance throughout this research work also in daily life. This dissertation would not have been possible without guidance and help of him. I would like to thank the members of my thesis evaluation committee, Dr. J. K. Chen, Dr. Gary Solbrekken and Dr. Matt Maschmann and Dr.
Stephen Montgomery-Smith for giving the time to provide valuable comments and criticism.
Special thanks must be extended to Yijin Mao for his help.I would like to thank my all coworkers at my lab. It is really great time to work with them and I really enjoy their company in our lab.
I would like to express my gratitude to my parents, Shamsun Nahar Islam and Late F. K. M
Aminul Islam. My mother always gives me inspirations all the time about my study. Even though my father is not alive in this world, however, still I feel his contribution on my every success in my life. I also like to thanks my husband Zobayer Khizir for his support.
Support for this work by the U.S. National Science Foundation under grant number CBET-
1066917 and CBET- 133611 are gratefully acknowledged. The authors would like to thank the
Test Resource Management Center (TRMC) Test and Evaluation/Science & Technology
(T&E/S&T) Program for their support. This work is funded by the T&E/S&T Program through the US Army Program Executive Office for Simulation, Training and Instrumentation’s contract number W900KK-08-C-0002. Support for this work by the Air Force Research Lab under grant number STTR FA9451-12 is gratefully acknowledged.
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TABLE OF CONTENTS
ACKNOWLEDGEMENT ………………………………………………………………………ii
LIST OF FIGURES………………………………………………………………………………vi
LIST OF TABLE………………………………………………………...……………………… x
NOMENCLATURE……………………………………………………..…………………..….. xi
ABSTRACT…………………………………………………………………………………….xix
CHAPTER 1: Introduction ……………………………………………………………...….….. 1
CHAPTER 2 Duel-Phase Lag behavior of a gas-saturated porous-medium heated by a short pulse laser
2.1 Introduction ………………………………………………………………..………………. 4
2.2 Physical model ……………………………………………………………………….……. 7
2.3 Laplace transform solution …………………………………………………….………….. 11
2.4 Results and discussion ……………………………………………………….……………. 13
2.5 Conclusion ………………………………………………………………………………… 23
CHAPTER 3 Inverse estimation of front surface temperature of a locally heated plate with temperature-dependent conductivity via Kirchhoff transformation
3.1 Introduction ……………………………………………………………………..…………. 25
3.2 Mathematical and approximation model ………………………………………..…………. 27
3.3 Laplace transform solution …………………………………………………………..…….. 32
3.4 Simulation results ………………………………………………………….……….……… 34
3.5 Conclusion ………………………………………………………………………….……… 44
CHAPTER 4 Multicomponent gas particle flow and heat/mass transfer induced by a localized laser irradiation on a Urethane-Coated stainless steel substrate
4.1 Introduction ………………………………………………………………………...………. 46
4.2 Physical model ……………………………………………………………………………....48
4.2.1 Continuous phase………..………………………………………………………………...48
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4.2.2 Chemical reaction ……………………………………………………………….………...52
4.2.3 Discretized phase …………………………………………………………….………….. 56
4.3 Results and discussion ……………………………………………….……………………. 57
4.4 Conclusion ………………………………………………………………………………… 79
CHAPTER 5 Effects of beam size and laser pulse duration on the laser drilling process
5.1 Introduction ………………………………………………………………………...………. 80
5.2 Analytical model …………………………………………………………………...………. 82
5.2.1 Fluid flow ……………………………………….………………………………..………. 83
5.2.2 Heat transfer………………….………………………………………………..…………..84
5.2.3 Optical consideration ….…………………………………………………..………..……. 85
5.3 Numerical simulation …………………………………………………………….………… 88
5.3.1 Velocity and pressure calculation ……………………….…………………….…………. 88
5.3.2 Temperature calculation (solving energy equation) ………………..………….…………89
5.4 Results and discussion ………………………………………………………..……….…… 90
5.4.1 Effects of beam diameter…………………………………………………………………..93
5.4.2 Effects of laser pulse……………………………………………………………………....97
5.5 Conclusion ………………………………………………………………...……………….102
CHPTER 6 Uncertainty analysis of melting and resolidification of gold film irradiated by nano- to-femtosecond lasers using Stochastic method
6.1 Introduction………………………………………………………………..……………….103
6.2 Physical model……………………………………………………………………………...106
6.3 Stochastic modeling of uncertainty…………………………………………………………110
6.4 Results and discussions……………………………………………………………………..112
6.5 Conclusion……………………………………………………..………………………….. 134
7. CONCLUSION ……………………………………………………………………………...135
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REFERENCES ………………………………………………………………………….……. 138
VITA……………………………………………………………………………………………153
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LIST OF FIGURES
Fig. 2-1 Physical model ………………………………………………………………………….8
Fig. 2-2 Powder temperature (Ts) at the heating surface and the adiabatic surface with J =
5 2 1.25×10 J/m , tp = 100 ns, dp = 15µm (τT = τq =3.9 ns): (a) t/tp < 1 and (b) t/tp > 1 …….….15
5 2 Fig. 2-3 Temperature distribution over the powder layer with J = 1.25×10 J/m , tp = 1 ns, dp = 15
µm (τT = τq = 3.9 ns)……………………………………………………………………………..16
Fig. 2-4 Phase lag times (τT and τq) effects on the powder layer temperature: (a) t/tp < 1 and (b) t/tp > 1………………………………………………………………………………………..…..19
Fig. 2-5 Effects of laser fluence (J) on the temperature of powder layer with tp = 10 ns and dp =
15 µm: (a) t/tp < 1 and (b) t/tp > 1…………………………………………………………….…20
5 2 Fig. 2-6 Effects of porosity (φ) on the temperature of powder layer with J = 1.25×10 J/m , tp = 1 ns, and dp = 15 µm: (a) t/tp < 1 and (b) t/tp > 1………………………………………………..22
5 Fig. 2-7 Effects of pulse width (tp) on the maximum temperature of powder layer (J=1.25×10
J/m2)…………………………………………………………………………………………...…23
Fig. 3-1 Relationship between T and for stainless steel with Tr 318 K………..….35
Fig. 3-2 Schematic diagram of meshing on the back surface ……………………………..….36
Fig. 3-3 Comparison of front surface temperature contours for SS 304: Exact (left), CGM
(middle) and DCT/Laplace transformation solution (right)………………………………..….39
Fig. 3-4 Front surface temperature distributions along Y direction at different sensors location at time t=1.55s ………………………………………………………………………………….….40
Fig. 3-5 Comparison of front surface temperature between DCT/Laplace transformation and exact solution along Y direction at different sensors location at time t=1.55s ………………41
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Fig. 3-6 Comparison of CGM and DCT/Laplace transformation solutions of front surface
19L 19L temperature vs time at three sensor locations (center ( Y Y , Z z ), two off centers ( 40 40
25L 19L 29L 19L Y Y , Z z and (Y Y , Z z )) …………………………………..……….…42 40 40 40 40
Fig. 3-7 Comparison of the RMS values at different time steps for DCT/Laplace (reference temperature (Tr) as all average values of back surface temperatures and average of maximum and minimum front surface temperatures) and CGM method ………………………………….…..44
Figure 4-1 Node moving mechanism ……………………………………………………….….55
Figure 4-2 Illustration of mesh arrangement ………………………………………………….58
Figure 4-3 Maxmum temperatures in the paint vs three different mesh configurations…….…64
Figure 4 Temperature distribution across the middle cross section area of the gaseous domain at the end of simulation …………………………………………………………....66
Figure 4-5 Time history of temperatures at the center of laser heating spot for the six laser powers …………………………………………………………………………..….66
Figure 4-6 Density distributions across the middle cross section area of the gaseous domain at the end of simulation …………………………………………………………..…...68
Figure 4-7 Density variations at the center of the laser irradiation spot with time ……..……69
Figure 4-8 Velocity distributions across the middle cross section area of the gaseous domain at the end of simulation ……………………………………………………………….70
Figure 4-9 State of parcel flow and gaseous phase at different times …………………..……72
Figure 4-10 Time histories of the mass concentration of O2, H2O, CO2, NO2 at the center of laser heating spot …………………………………………………………..…………...... 75
Figure 4-11 Time histories of paint thickness removal for the six laser powers ……….…….76
Figure 4-12 Parcel and gaseous flow at the end of the simulation ………………………..……77
Figure 4-13 Comparison of paint removal between simulation and experiment ………..…...78
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Figure 5-1 Schematic diagram of laser drilling process……………………………………...... 83
Figure 5-2 Comparison of the fluid contour of the literature (top) and the current result (bottom) at different time sequence………………………………………………………………….…….93
Figure 5-3 Fluid contour at (a) 0; (b)40 ; (c) 50; (d) 60; (e) 90 and (f) 210 µs with ,
R= 508µm and ………………………………………………………...... 94 Figure 5-4 Temperature contours at (a) 0; (b)40 ; (c) 50; (d) 60; (e) 90 and (f) 210 µs with
, R= 508µm and …………………………….………………95
Figure 5-5 Fluid contour at (a) 0; (b)40 ; (c) 50; (d) 60; (e) 90 and (f) 210 µs with ,
R= 1.5 mm and …………………………………………………………..96 Figure 5-6 Temperature contours at (a) 0; (b)40 ; (c) 50; (d) 60; (e) 90 and (f) 210 µs with
, R= 1.5 mm and …………………………….….……….….96
Figure 5-7 Fluid contour at (a) 0; (b)20 ; (c) 25; (d) 30; (e) 45 and (f) 105 µs with ,
and …………………………………………………..…….98
Figure 5-8 Temperature contours at (a) 0; (b)20 ; (c) 25; (d) 30; (e) 45 and (f) 105 µs with
, and ………………………………….…….....99
Figure 5-9 Fluid contour at (a) 0; (b)10 ; (c) 12.5; (d) 15; (e) 22.5 and (f) 52.5 µs with
, and ………………………..…………………..…100
Figure 5-10 Temperature contours at (a) 0; (b)10 ; (c) 12.5; (d) 15; (e) 22.5 and (f) 52.5 µs with
, and …………………….…………………...101
Figure 6-1 Sample-based stochastic model……………………………………………..…110
Figure 6-2 Stochastic convergence analysis of mean value of the input parameters (a) GRT,
(b) , (c) η, (d) J and (e) tp…………………………………………………………………….115
Figure 6-3 Stochastic convergence analysis of standard deviation of the input parameters (a)
GRT, (b) , (c) η, (d) J and (e) tp………………………………………………………………..118
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Figure 6-4 Stochastic convergence analysis of mean value of the output parameters (a) s, (b) us, (c) Tl,I and (d) Te………………………………………………………………………….....120
Figure 6-5 Stochastic convergence analysis of standard deviation of the input parameters (a) s, (b) us, (c) Tl,I and (d) Te……………………………………………………………………...122
Figure 6-6 Typical distributions of the input parameters (a) GRT, (b) , (c) η, (d) J and (e) tp………………………………………………………………………………………………...125
Figure 6-7 Typical distributions of the output parameters (a) s, (b) us, (c) Tl,I and (d) Te…127
Figure 6-8 The IQRs of the output parameters with different COVs of the input parameters
(a) s, (b) us, (c) Tl,I and (d) Te………………………………………………………………….129
Figure 6-9 The IQRs of the output parameters with different values and COVs of J (a) s, (b) us, (c) Tl,I and (d) Te………………………………………………………………………….....132
Figure 6-10 The IQRs of the output parameters with different values and COVs of GRT (a) s,
(b) us, (c) Tl,I and (d) Te………………………………………………………………………...134
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LIST OF TABLES
Table: 2-1 Phase lag times for different particle diameter and porosity …………………....17
0 2 Table 4-1 Initial mass diffusivity between gas species D ij (m /s) …………………………59
Table 4-2 Specific heat capacity and absolute viscosity of gas species[79]………...... 59
Table 4-3 Material properties of solids…………………………………………………...... 59
Table 4-4 Initial and boundary conditions for the gaseous domain………………………...61
Table 4-5 Initial and boundary conditions for the solid domain……..……………………..62
Table 5-1 Thermophysical properties of the Hastelloy-X……………….………………….90
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NOMENCLATURE
a Polynomial coefficients
B Source term in the nonlinear energy equation
c Specific heat (J/Kg.K)
C0 Coefficient of the nonlinear term
cp specific heat, J/Kg K
dp diameter of the powder particle, m d Diameter of particles (m)
d0 Characteristic length
2 Di Mass diffusion coefficient for species i in the mixture (m /s)
Dij Mass diffusivity coefficient between species i and j
0 D ij Mass diffusion coefficient from species i to species j at 298K and 1atm
E Activation energy (KJ/mol)
E Enthalpy (J/kg)
F Volume of fluid function
F Force (N)
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i th F p Force acting on a single particle in a parcel located in the i cell f normalized heat flux, K
G coupling factor, W/m3 K
G Gravity force (N)
3 GRT Electron-lattice coupling factor (W/m K) g Gravitational acceleration (9.8 m/s2) h Enthalpy(J/kg)
hm Latent heat of fusion (J/kg)
2 hp heat transfer coefficient at the powder particle surface, W/m K
2 h0 heat transfer coefficient at the powder bed surface, W/m K
I Unit matrix
I Intensity of the laser beam (W/m2)
J heat source fluence/laser influence, J/m2 k Thermal conductivity (W/m K),
-1 kc Chemical reaction rate constant (s ),
kd Node diffusion coefficient k thermal conductivity, W/m K xii
L thickness of powder layer
lx slab length, m
ly slab width, m
lz slab thickness, m
L total length, m
Lv Latent heat of vaporization (J/g)
Lm Latent heat of melting (J/g)
Lx ratio of slab length to thickness, lx/lz
Ly ratio of slab width to thickness, ly/lz
M maximum mode number along the length direction m Mass of particle
M Molecular weight (g/mol)
N maximum mode number along the width direction
i th N p Number of particles in a parcel located in the i cell
Nu Nusselt number p Pressure (Pa)
P Laser power (W) xiii
pvap,0 Vaporization pressure (Pa)
Pr Prandtl number q heat flux, W/m2
R reflectivity
r0 Radius of laser beam (m)
R Universal gas constant (J/Kg K)
Re Reynold number r radial coordinate
R Gas constant(J/kg.K)
S0 Coefficient
Sc Schmidt number
Sh Source term in the energy equation
SU Source term in the momentum equation
S intensity of the internal heat source, W/m3 s Laplace transform variable
t, tf time, s
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tc characteristic time, s
Tr reference temperature, K
Ts, T temperature of slab, K t time, s
tp full-width at half maximum (FWHM) pulse width, s
T temperature, K t Time coordinate
T0 Temperature
T Temperature (K)
U Laplace transform of temperature
U Velocity of gas (m/s) u Velocity in x-direction(m/s)
V Volume of control volume (m3) v Velocity in y-direction (m/s)
Xi Molar fraction of species i x spatial coordinate variable along the length, m y spatial coordinate variable along the length, m xv
z spatial coordinate variable along the length, m z Axial coordinate
X dimensionless spatial coordinate variable along the length direction
Y dimensionless spatial coordinate variable along the width direction
Z dimensionless spatial coordinate variable along the thickness direction
Greek Symbols
α thermal diffusivity, m2/s
αab Absorptivity
3 ΔVrem Volume removed (m )
ε Emissivity
λ Thermal conductivity constant (W/mk)
μ Dynamic viscosity(kg/m s)
δ optical penetration depth of the powder layer, m
η Thermal conductivity constant
ρ density, kg/m3
φ porosity
σ Collision diameter (m) xvi
2 4 σrad Stefan-Boltzmann constant (W/m K )
ψ Compressibility(s2/m2)
η Dynamic viscosity (g/cm.s)
2 γST Surface tension (J/cm )
ƙ Thermal diffusivity of melt (m2/s)
ωi Mass fraction
τT phase lag time of the temperature gradient, s
τq phase lag time of the heat flux vector, s
τ dimensionless time
θ temperature, K
Θ Laplace transform of temperature
Φ Laplace transform of normalized heat flux
Subscript b back surface quantity c chemical reaction bd binder f front surface quantity xvii
g gas i Number of species j Number of species i initial m mode number along the length direction n mode number along the width direction p Particle pg pigment rad Radiation
∞ Ambient condition s solid phase (particle) e Electron
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ABSTRACT
Heat transfer describes the exchange of thermal energy, between physical systems depending on the temperature and pressure, by dissipating heat. The fundamental modes of heat transfer are conduction or diffusion, convection and radiation. Heat and mass transfer are kinetic processes that may occur and be studied separately or jointly. Studying them apart is simpler, but both processes are modeled by similar mathematical equation in the case of diffusion and convection.
There are complex problems where heat and mass transfer processes are combined with chemical reactions, as in combustion. The resulting behavior of heat transport in microscale will be very different from macroscale heat transfer based on the averages taken over hundreds of thousands of grains (in space) and collision (in time). From the microscopic point of view, the process of heat transport is governed by phonon-electron interaction in metallic films and by phonon scattering in dielectric films, insulators and semi-conductors. For extremely heated surfaces by high energy laser pulse, it is very difficult to measure temperature of flux at the heated surface because of the unendurable capacity of the conventional sensors. Laser is the tool of choice when drill holes ranging in diameter from several millimeters to less than one micro-meter. Instead of having advanced melting and resolidification modeling process recently, the inherent uncertainties of the input parameters can directly cause unstable characteristics of the output results which means the parametric uncertainties may influence the characteristics of the phase change processes (melting and resolidification) which will affect the predictions of interfacial properties i.e., temperature, velocity and mainly the location of solid-liquid interface. All of those processes can be considered under high energy laser interaction with materials.
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CHAPTER 1
INTRODUCTION
Heat transfer is the flow of thermal energy driven by thermal non-equilibrium (effect of a non- uniform temperature field) commonly measured as a heat flux vector (the heat flow per unit time) at a control surface. The exchange of kinetic energy of particles through the boundary between two systems which are at different temperature from each other or from their surroundings. Heat transfer always occurs from a region of high temperature to another region of lower temperature. Heat transfer changed the internal energy of both systems involved according to the first law of thermodynamics.
Conventional theories established on the macroscopic level, such as heat diffusion assuming
Fourier’s law, are not expected to be informative for microscale conditions because they describe macroscopic behavior averaged over many grains. This should be transient behavior at extremely short times, say of the order of picoseconds to femtoseconds which is a major concern. A typical example is the ultrafast laser heating in thermal processing of materials.
A two-temperature model can be applied to describe the heat transfer in gas precursors and powder particles. A two temperature model is a valuable tool to investigate ultrafast electron dynamics. In general two-temperature model describes the temporal and spatial evolution of the lattice and electrons temperature in the irradiated metal by two coupled nonlinear differential equations. In Chapter 2, A dual-phase lag (DPL) model is used to investigate the heat conduction in a gas-saturated porous medium subjected to a short-pulsed laser heating. The energy equations for the powder and gas phase are combined together to obtain a DPL heat conduction equation with temperature of the powder layer as the sole unknown. A perfect correlation obtained from
1
Laplace transformation is applied to analytically solve the DPL problem with internal heat source. The Riemann sum approximation is applied to find the inverse Laplace transform of the powder layer temperature distribution. Variations of powder temperature at heating and adiabatic surface and powder temperature distribution are studied. The results show that the analytical solutions are in a good agreement with the numerical solutions. The effects of phase lags times, pulse width, laser fluence, porosity on the DPL behavior of the gas-saturated powder layer are also investigated.
In this thesis in Chapter 3, by Kirchhoff transformation of the temperature variable, the temperature dependence of thermal conductivity is eliminated, thereby simplifying the 3- dimendsional heat conduction equation. Through Hadamard Factorization Theorem, transfer function relating the front and back surface temperature as infinite product of polynomial is established. The inverse Laplace transform of the polynomial provide the relationship for every mode in the time domain. The front surface temperature is revealed through iterative time domain operations from the data on the back surface. The comparison between direct solution,
Conjugate Gradient Method (CGM) and DCT/Laplace transform solutions are given. Root Mean
Square (RMS) of the errors at different time steps for DCT/Laplace solution and CGM method are also presented.
In Chapter 4 of this thesis, a three-dimensional numerical simulation is conducted for a complex process in a gas-solid system, which involves heat and mass transfer in a compressible gaseous phase and chemical reaction during laser irradiation on a urethane paint coated on a stainless steel substrate. A finite volume method (FVM) with a co-located grid mesh that discretizes the entire computational domain is employed to simulate the heating process. The results show that when the top surface of the paint reaches a threshold temperature of 560 K, the polyurethane
2
starts to decompose through chemical reaction. As a result, combustion products CO2, H2O and
NO2 are produced and chromium (III) oxide, which serves as pigment in the paint, is ejected as parcels from the paint into the gaseous domain. Variations of temperature, density and velocity at the center of the laser irradiation spot, and the concentrations of reaction reactant/products in the gaseous phase are presented and discussed, by comparing six scenarios for different laser powers
In this thesis, in Chapter 5, A two-dimensional axisymmetric transient laser drilling model, which includes heat transfer in terms of conduction and advection in the transient development of the flow, phase change phenomena in terms of melting, solidification and vaporization, and material removal results from the vaporization and melt ejection is used to analyze the effects of laser beam diameter and laser pulse duration in the laser drilling process. Firstly, this paper discusses the verification of the model with the available literature results. Secondly, the verified model is applied to study the effects of the laser beam size and pulse duration on the drilled geometry of hole, which is found to be significant factors. Contour plots of fluid and temperature are presented at different time sequence of the laser drilling process.
In Chapter 6, A sample-based stochastic model is presented to investigate the effects of uncertainties of various input parameters, including laser fluence, laser pulse duration, thermal conductivity constants for electron, and electron-lattice coupling factor, on solid-liquid phase change of gold film under nano- to femtosecond laser irradiation. Rapid melting and resolidification of a free standing gold film subject to nano- to femtosecond laser are simulated using a two-temperature model incorporated with the interfacial tracking method. The IQR analysis shows that the laser fluence and the electron-lattice coupling factor have the strongest influences on the interfacial location, velocity, and temperatures.
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CHAPTER 2
Dual-Phase Lag Behavior of a Gas-Saturated Porous-Medium Heated by a
Short-Pulsed Laser
2.1 Introduction:
Selective area laser deposition vapor infiltration (SALDVI) is a Solid Freedom Fabrication
(SFF) technique in which porous layers of powder are densified by infiltrating the pore spaces with solid material deposited from a gas precursor during laser heating [1]. SALDVI process combines Selective area laser deposition (SALD) process and the Chemical vapor infiltration
(CVI) process to directly fabricate ceramic and ceramic/metal structures and composites. Three dimensional object fabrications can be made in SFF from powder (Selective laser sintering;
SLS), gas (SALD) and combination of both (SALDVI). It is very important to obtain the temperature distribution in the SALDVI process to understand the effect of the various processing parameters on the quality of the final products. The relative density of the powder layer continuously changes with processing time until it reaches near full density during the
SALDVI process. Such continuous changes in the relative density cause the continuous changes in thermal conductivity of the SALDVI workpiece [2]. SALDVI utilizes Laser Chemical Vapor
Deposition (LCVD) technique which can be based on reactions pyrolytically, photolytically or a combination of both [3]. Mazumder and Kar [4] presented a very detailed literature review about theory and applications of LCVD. A 3-D transient thermal problem for LCVD of a moving slab was solved analytically in [5]. They introduced Kirchoff’s transformation to linearize the heat conduction equation to account for the temperature-dependent properties; the boundary conditions are linearized by an effective convective heat transfer coefficient.
4
The SALDVI has a great potential due to several inherent features like to produce fully dense shapes without post processing, can make wide materials selection and can overcome the dimensional constrains that present in traditional chemical vapor infiltration techniques. A significant change of porosity occurs during the SALDVI process and the properties of the powder layer structure are affected by the porosity change. The chemical reaction that occurs on the surface of the particles results the deposition on the surface of the powder and joining the powder particles together. The powder responds differently than a simple and fully dense material. Dai et al. [6] performed a numerical simulation of SALDVI using finite-element method. Before laser densification, the density of the powder layer was assumed to be 50% of its theoretical density and powder layer density become 100% of its theoretical value when the temperature of powder layer reaches to a maximum temperature. The same group also performed experiment using a closed loop control to achieve the constant temperature of powder layer by modifying the laser power from one time step to another [7]. They also improved the model that they used previously by introducing a densification model by vapor infiltration based on growth rate obtained experimentally [8].
Forced, mixed and free convection flows and heat transfer in fluid-saturated porous media are interesting topics to many researchers in geophysical and engineering applications [9-12].
Brouwers [13] investigated the heat and mass transfer between a permeable wall and a fluid saturated porous medium with the effects of wall suction or injection on sensible heat transfer.
They applied thermal correction factor to investigate free, mixed and forces convection flows along vertical and horizontal permeable walls. Non-Darcy and Darcy effects on flow in fluid saturated porous media were reported. The generalized non-Darcy approach was applied to investigate double diffusion natural convection in a fluid saturated porous media [14]. Effect of
5
surface fluctuation on the natural convection heat transfer of Darcian fluid saturated porous media was studied using finite element method [15]. Unsteady, laminar and 2-D hydro-magnetic natural convection in an inclined square filled with fluid saturated porous medium with transverse magnetic field was numerically investigated by Khanafer and Chamkha [16].
When the powder layer is heated by laser during SALDVI, the gas precursors are assumed to be transparent and the laser beam interacts only with the powder particles. Heat transfer occurs in two steps: first powder particles absorb the laser energy and then heat is transferred from the powder particles to the precursors. The time it takes that the temperatures of powder particles and gas phase reach to equilibrium is referred to as relaxation time. For long pulsed laser, the local thermal equilibrium assumption between powder particles and the gas is valid because the pulse duration is longer than the relaxation time. The short pulse laser has the advantage to control the porosity of the final product by controlling the LCVD on the powder layer surface via controlling pulse width and repetition rate. A non-equilibrium model for transport phenomena in the powder particles and gas needs to be developed if the laser pulse duration in the SALDVI is shorter than the relaxation time. Zhang [17] modeled the heat transfer in a gas saturated porous media with a short-pulsed volumetric heat source using a two-temperature model. The results showed that the degree of non-equilibrium in the process decreased with the increase of laser pulse width and become insignificant for the laser pulse width longer than 1µs.
In this paper, the DPL behavior of the gas-saturated porous medium heated by a short-pulsed laser will be studied. The two energy equations are combined using operator method to obtain one equation with the temperature of the powder particle as the sole unknown. The analytical solution is compared with the finite-volume method solution, and the effect of phase lags in terms of heat capacities and coupling factor are discussed.
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2.2 Physical Model
Figure 2-1 shows the physical model and coordinate system. A temporal Gaussian laser beam with a FWHM pulse width of tp is irradiated to a power layer with a thickness L and initial temperature Ti. Due to the porous nature of the powder layer, the laser can penetrate the powder layer, which results in absorption of laser energy within the layer instead of at the surface of the layer. It is assumed that heat transfer in one-dimensional along the thickness of the powder layer because the size of the laser beam is much larger than thickness of the powder layer. The effect of chemical reaction heat on heat transfer along the thickness of the powder layer is negligible
[18]. The porosity of the powder layer during irradiation of a single pulse is assumed to be constant due to the small amount of deposition in the duration of one short pulse. The convection effect in the gas phase is neglected since the pulse duration is very short. Under these assumptions, the problem becomes a simple heat conduction problem in a gas-saturated porous medium with an internal heat source.
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-tp/2 tp/2
0
L
x
Adiabatic
Figure 2-1 Physical model
A two-temperature model can be applied to describe the heat transfer in gas precursors and powder particles as they are not in thermal equilibrium. The energy equations of the powder particles (s) and the precursors (g) can be respectively expressed as:
TT (1 )(c )ss ( k ) S G ( T T ) (2.1) p st x seff x s g
T ()()cg G T T (2.2) p gt s g where kseff and G are the effective thermal conductivity of the powder layer and the coupling factor between powder particles and precursors, respectively. In arrival to Eq. (2), heat conduction in the gas phase has been neglected because the conductivity of the gas is several orders of magnitude lower than that of the powder material. Light intensity of the laser beam appears as the volumetric heat source term in Eq. (1):
8
x a t2 t p 1 R t S( x , t ) 0.94 J ( ) e p (2.3) t p where R is the reflectivity, δ is the optical penetration depth and a = 4ln(2) = 2.77. The particular form of the light intensity is used to facilitate the direct use of the Riemann sum approximation for the Laplace inversion [19].
Combining Eqs. (1) and (2), the following energy equation can be obtained:
2 3 2 TTTTs s11S s q s 2Tq 2 ()S 2 (2.4) x x t kseff t t t where the phase lag times of the heat flux vector and temperature gradient are:
C CC g , sg (2.5) T q G GCC()sg and
Ccs(1 ) s ps , Ccg g pg (2.6)
Cs and Cg are effective heat capacities of the solid and gas phases, respectively. The effective thermal diffusivity is
k seff (2.7) CCsg
Equation (4) describes the temperature response with lagging accommodating the first-order
effect of T and q . It captures several representative models in heat transfer as special cases.
This equation reduces to the diffusion equation in the absence of the two phase lags, Tq0 .
In the absence of the phase lag of the temperature gradient,T 0, it reduces to the CV wave model.
9
If Newton law of cooling can be used to describe heat transfer between powder particles and gas precursors, the coupling factor can be determined by
(1 )Ah G pp (2.8) Vp
(1 )A where p represents the specific interfacial area(m2/m3). If the porous particle is spherical Vp
6(1 )h in shape,the surface-area-to-volume ratio becomes p . The coupling factor becomes Vp
6(1 )h G p (2.9) d p
where dp and hp are the diameter of the powder particles and the heat transfer coefficient at the particle surface, respectively. The heat transfer coefficient at the particle surface can be calculated from the Nusselt number. In the absence of natural convection in gas phase, Nusselt number can be written as [20]
hd Nu pp 2 (2.10) kg
Substituting Eqs. (6) and (9)-(10) into Eq. (5), the phase lag times become:
2 2 gcd pg p gcd pg p T , q (2.11) 6(1 )Nukg 6Nukg [(1 ) g c pg / ( s c ps )]
The initial conditions for temperature of the powder particle are assume given below
Tsi(,) x T (2.12)
10
T s (x ,0) 0 (2.13) x
The boundary conditions are
Ts kseff h0 () T s T (2.14) x x0
T s 0 (2.15) x xL
It is evident from Eq. (4) that the dual phase lag process depends on the phase lag times and thermal diffusivity. The phase lag times, in turn, depend on the heat capacities of the solid and gas, porosity, diameter of the powder particles and heat transfer coefficient of the powder particles, as indicated by Eq. (11).
2.3 Laplace Transform Solution
The method of Laplace transform is especially suitable for the equations involving special structures in time. The Laplace transformation can be defines as
T(,)(,) x p T x t e pt dt (2.16) 0
Applying the above Laplace Transform to Eqs. (4), (15) and (16), the following second ordinary differential equation and boundary conditions can be obtained
22 TTss11 q 2 22Tp () S q pS pT s p T s (2.17) x x kseff
11
T 1 ks h() T (2.18) seffxp0 s x0
T s 0 (2.19) x xL
The solution of Eq. (17) with its associated boundary conditions, Eqs. (18) and (19), can be expressed as
x Dx Dx d Ts (,) x p Ae1 A 2 e A 3 e (2.20)
where A1, A2, and A3 are given by:
1 ()DL d 2a Ak3()()seff dh 0 AdDk 2 seff h 0 dhT 0 2DL A3 e S 0() C 2 e C 1 Sb A1, A 2 Ae 1 , A 3 d() h Dk dD 1 2 0 seff D d 2
2a 22ptpp pt e e e p(1 pq ) 1 p q q Sbp t[ ], D , C12 , C (2.21) ptaptap p (1 p T ) kp seff (1 T ) kp seff (1 T )
To perform the Laplace inversion, a special technique has been developed [19], which the
Fourier representation of the inverse Laplace transformation is called the Riemann sum approximation. Bromwich contour integration is the standard procedure for obtaining the inverse solution. The improper integrals involved in Bromwich contour must be calculated numerically.
This numerical procedure is approximated by the series of summations to be performed.
Recognizing the nature in the numerical approximations, a special technique of approximation –
Riemann sum approximation - is developed for the Laplace inversion [17]. This approach has been evaluated by a large class of Laplace solutions, including effects of finite media, thermomechanical coupling, and parabolic and hyperbolic two step models. Introducing a
12
variable transformation from p (complex) to ω (real), Riemann sum approximation of the
Laplace inversion can be expressed as
t N e1 in n Ts( x , t ) [ T s ( x , ) Re T s ( x , )( 1) ] (2.22) tt2 n1 where Re stands for the real part of the summation. The quantity γ is the real value in the
Bromwich cut from (γ- i∞) to (γ + i∞). For faster convergence, the value of γ satisfies the relation
t 4.7 (2.23)
Substituting Ts (,) x t from Eq. (20) into Eq. (22), the temperature of the solid, Ts (x, t) can be determined.
2.4 Results and Discussions
DPL behavior of a system with silicon carbide (SiC) powder and tetramethylsilane (TMS) gas under short-pulsed laser irradiation will be investigated. The thermophysical properties of the
3 3 SiC are: s 3.21 10 kg/m , cps 660 J/kg K, ks = 58.86 W/m-K. The effective thermal
conductivity for the solid phase is kkseff(1 ) s . The porosity of the powder layer set as 0.42, which is within the range of a randomly packed powder bed [21]. The specific heat of the TMS
3 is cpg 3,438/kg K at 1,273K [22]. The density of the TMS is ρg = 0.045 kg/m . The reflectivity of the laser beam is taken to be R = 0.6. The analytic solution is started from t = -5tp
with a time step of ttp /100 until t = 5tp. The time step is changed to ttp for 5tp < t <
100tp; after t > 100tp, the time step is further increased to tt100 p . The initial temperature of
13
the powder particles is taken as Ti = 300 K. The optical penetration depth (δ) is a function of the powder particle size, porosity and the absorptivity of the powder material and is taken as the twice of the powder particle diameter, δ = 2dp [23]. The particle sizes are taken as dp = 15, 20, and 25 µm, which are consistent with the particle size used in the SALDVI experiments [8, 24].
To validate the Laplace transform solution, a numerical solution [17] to the problem based on the two-temperature model described by Eqs. (1) - (2) are also carried out.
Figure 2-2 shows the comparison of the variation of powder temperature at the heated surface
5 2 (x = 0) and adiabatic surface (x = L) for J = 1.25×10 J/m , tp = 100 ns and dp = 15µm, which correspond to phase lag times of τT ≈ τq = 3.9 ns. The temperature variation for t/tp < 1 is shown in Fig. 2(a) while Fig. 2(b) shows the temperature variation for t/tp > 1. The powder temperature increases at the heating surface and it reaches to the maximum of 1650 K in analytical solution and 1630 K in numerical solution. The adiabatic surface powder temperature converge to the
6 heating surface temperature at t/tp> 1×10 , which means that the entire powder layer temperature becomes uniform after this time.
14
(a)
(b)
Figure 2-2 Powder temperature (Ts) at the heating surface and the adiabatic surface with J =
5 2 1.25×10 J/m , tp = 100 ns, dp = 15µm (τT = τq =3.9 ns): (a) t/tp < 1 and (b) t/tp > 1
15
Figure 2-3 shows the comparisons of powder temperature distributions between the analytical and numerical solutions. At the peak of laser pulse (t = 0) , the maximum powder layer temperatures at heated surface are 980 K (analytical) and 989 K (numerical) and decreases to the initial temperature at the adiabatic surface. At time t = tp, the maximum powder temperatures are
1665 K (analytical) and 1650 K ( numerical) at the heated surface where at the adiabatic surface temperatures are the same as the initial temperature.
5 2 Figure 2-3 Temperature distribution over the powder layer with J = 1.25×10 J/m , tp = 1 ns, dp =
15 µm (τT = τq = 3.9 ns)
It follows from Eq. (11) that phase lag times for temperature gradient (τT) and heat flux vector (τq) are functions of thermophysical properties like density, specific heat, porosity and coupling factor. Table 1 shows the phase lag times (τT and τq) for different diameters (dp) of powder particles and porosity (φ). It can be seen from Table 2-1 that the phase lag times for heat flux vector and temperature gradient are always the same because the heat capacity of the gas is
16
several orders of magnitudes smaller than that of the solid (see Eq. (11)). The phase lag times increase with the increasing particle diameter and porosity of the powder layer.
Table: 2-1 Phase lag times for different particle diameter and porosity
dp (µm) φ τT (ns) τq(ns)
15 0.26 1.91 1.91
15 0.32 2.53 2.53
15 0.36 3.01 3.01
15 0.42 3.96 3.96
20 0.26 3.38 3.38
20 0.32 4.54 4.54
20 0.36 5.44 5.44
20 0.42 6.93 6.93
25 0.26 5.21 5.21
25 0.32 7.14 7.14
25 0.36 8.53 8.53
25 0.42 10.01 10.01
17
Figure 2-4 shows the powder temperature response at different phase lag times (τT and τq). It is shown that the peak values of the temperature of the powder layer increase with increasing phase lag times (τT and τq). Increasing phase lag time from 1.9 ns to 4.9 ns implies that the responses of heat flux vector and temperature gradient increase, leading to an increase of the temperature of the powder. When the laser light impinges into the powder layer, with the larger phase lag times, heat is transferred into a deeper part of the powder layer with longer delay.
Consequently, the powder temperature increase more rapidly at the heating surface than the cases of shorter phase lags time. For the phase lag times, τT = τq= 1.9 ns, the powder particle
3 2 temperature starts decreasing from about t/tp > 10 while it decrease from about t/tp > 10 for phase lag times, τT = τq = 4.9 ns. The Dual phase lag effect reduce to the Fourier’s law as τq=τT but the results are different due to the presence of heat source.
(a)
18
(b)
Figure 2-4 Phase lag times (τT and τq) effects on the powder layer temperature: (a) t/tp < 1 and (b) t/tp > 1
Figure 2-5 shows the effect of laser fluence (J) on powder layer temperature under the same pulse width and powder diameter. Although the time when the peak temperature of the powder layer occurs does not change, it increases with the increase of laser influence. As the laser fluence increases from J = 1.5×105 J/m2 to 5×105 J/m2, the peak powder temperature increase from 1650 K to 3433 K.
19
(a)
(b)
Figure 2-5 Effects of laser fluence (J) on the temperature of powder layer with tp = 10 ns and dp =
15 µm: (a) t/tp < 1 and (b) t/tp > 1
20
The effects of porosity are shown in Fig. 2-6. The results are obtained for J = 1.5×105 J/m2.
The increment of porosity results in an increase of heat transfer coefficient as indicated by Eq.
(8); consequently, the powder temperature is increased.
(a)
21
(b) 5 2 Figure 2-6 Effects of porosity (φ) on the temperature of powder layer with J = 1.25×10 J/m , tp =
1 ns, and dp = 15 µm: (a) t/tp < 1 and (b) t/tp > 1
Figure 2-7 shows the effect of pulse width on the powder temperature under the same laser fluence (J = 1.5×105 J/m2) for different powder particle diameters. As the pulse width increases, the maximum temperature of the powder particles decrease because the same amount of energy is deposited in the powder layer in a longer period of time. Moreover, the powder temperature at the heated surface decrease with the increase of powder particle diameter as the optical penetration depth increases with the increasing particle diameter. The maximum surface powder temperature decreases from 2094 K for the 15 µm power particles to 1381K for the 25µm power particles.
22
5 Figure 2-7 Effects of pulse width (tp) on the maximum temperature of powder layer (J=1.25×10
J/m2)
2.5 Conclusion
A dual-phase lag model has been studied analytically and numerically for gas-saturated porous media subjected to short-pulsed laser heating. The powder layer temperature has been obtained analytically by applyng Laplace transform. The Riemann-Sum approximation of the
Fourier integral has been used to transform from the Laplace inversion integral. The distributions of the temperature of powder layer obtained by analytical and numerical methods have been compared. It is shown that the analytical solution shows a good agreement with the results obtained from the numerical simulation. It is also found that the peak temperature of the powder particles increases with the increasing phase lag times. The peak temperature of the powder layer increase with the increase of laser influence but the peak temperature occurring time does not.
23
The powder temperature increases with the increasing porosity. Under the same laser fluence, the maximum powder temperature decrease with the increasing pulse width and particle diameter.
24
CHAPTER 3
Inverse estimation of front surface temperature of a locally heated plate with temperature-dependent conductivity via Kirchhoff transformation
3.1 Introduction
Direct measurements of temperature and heat flux are very challenging for extremely heated surfaces by high energy laser (HEL) pulses because conventional sensors cannot withstand the intense heat. It is more convenient to place sensors away from direct HEL irradiation. For example, sensors can be placed on the back surface of a thin plate under HEL irradiation. In that case, the front surface temperatures and heat flux can be determined by solving an Inverse Heat
Conduction Problem (IHCP) based on the transient temperatures and heat flux measured at the back surface [25-34].
For solving the IHCP, researchers have recently shown that the availability of both the temperature data and heat flux data can increase the stability of the solution of the IHCP.
However, many researchers prefer solution methods that only require the temperature measurements on the back surface. The reason behind this preference of temperature measurements is that temperature can be measured with less uncertainty than heat flux measurements [35-38].
In our previous works, we have developed a method that uses Laplace transform to solve
IHCP [39-40]. For one dimensional IHCP, relationships between temperature and heat flux of the two surfaces in the form of transfer functions in the Laplace domain are obtained. The transfer functions are expressed as infinite products of simple polynomials using the Hadamard
Factorization Theorem. The inverse Laplace transforms of the simple polynomials led to relationships in the time domain involving time derivatives of the data on the back surface.
25
Savitzky-Golay method [41] is employed to achieve smoothing and numerical derivatives of the sampled temperature data even in the presence of sensor noise. The method is generalized to solve three dimensional problems over a finite slab. In particular, for a thin slab under HEL irradiation, temperature gradient across the thickness is much higher than in other directions. It is thus convenient to express the temperature distribution perpendicular to the thickness direction in terms of Fourier series for slabs. For rectangular geometries with no heat flux on the four sides, cosine series are used. Expressing the temperature distribution using cosine series amounts to discrete cosine transforms (DCT) resulting in one dimensional IHCP for each cosine mode. The IHCP for each mode is then solved using the Laplace transform approach describe above. We call the combined approaches the DCT/Laplace method for the IHCP.
The DCT/Laplace method has been shown to provide accurate solutions even when some noise is assumed to be present in the sensor data [40]. When compared with other methods such as Conjugate Gradient Method (CGM) [42], it has the advantage of computational time savings.
Therefore, it has the potential of being used to provide fast preliminary estimation of the front surface temperature and heat flux in actual HEL irradiation tests.
At present, the DCT/Laplace method has been developed for three dimensional slabs whose physical properties are assumed to be temperature independent. Since HEL irradiation often resulting in temperature rises to the extent over which significant changes in physical properties such as thermal conductivity and thermal capacity take place, we wish to generalize the DCT/Laplace method to treat these problems. In this paper, we try to overcome the limitation by introducing Kirchhoff transformation.
In the following section, we briefly summarize the mathematical formulation of the problem and introduce the Kirchhoff transformation. The Kirchhoff transform amounts to a
26
modification of the back surface temperature sensor data based on the known temperature dependence of the thermal conductivity of the slab. This paper assumes a two-dimensional square array of temperature sensors on the back surface of a thin slab. In section 3, we present the procedures of solving the IHCP using the DCT/Laplace method. In section 4, we provide results obtained in validating our method. A direct heat conduction problem with known front surface heat flux and temperature dependent thermal properties is first solved. This solution provides the back surface temperature data at the locations of a 20×20 sensor array.
Simultaneously, front surface temperatures are obtained from the direction solution; they are refers as the “exact solution”. The back surface temperature data are used by the DCT/Laplace method to calculate the front surface temperature. The same data are also used in CGM method.
Comparisons are made among the “exact solution”, the DCT/Laplace solution, and the CGM solution. Moreover, Root Mean Square (RMS) of the errors at different time steps for
DCT/Laplace solution and CGM solution [43, 44] are also calculated.
3.2 Mathematical and approximation model
3.2.1 Mathematical model The heat conduction equation with temperature dependent properties is:
TTTT cs()()() k s k s k s (3.1) s pst x s x y s y z s z where ρs, ks and cps are the density, thermal conductivity and specific heat of the solid respectively. The thermal conductivity (ks) and specific heat (cps) are in general functions of temperature. Only density (ρs) is assumed to be a constant property of the material. We consider adiabatic boundary conditions on the back surface as well as on the four surfaces on the edges of the thin plate:
27
Ts 0 for x =0 and x =lx (3.2) x
Ts 0 for y =0 and y =ly (3.3) y
T ks q(,,) x y t for z =0 (3.4) s z
T s 0 for z =lz (3.5) z with the dimensionless variables:
t=tc τ, x=lzX, y=lzY, z=lzZ
2 scl ps z where tc and dimensionless heat flux and temperature (K) are: ks
k qx,y,t s f (,,) X Y (3.6) lz
TXYZs x,y,t ( ,, ,) (3.7)
The normalized form of Eq. (1) is
2 2 2 (3.8) XYZ2 2 2 with boundary conditions:
0 for X =0 and X =Lx (3.9) X
0 for Y =0 and Y =Ly (3.10) Y
0 for Z =1 and (3.11) Z
28
l lx y f(,,) X Y for Z =0 where Lx and Ly (3.12) Z lz lz
Three dimensional heat conduction problem can be converted into one dimensional problem by expressing the temperature and heat flux as superposition of two-harmonic functions as follows [40]:
m X n Y (,,,)(,)X Y Z mn Z Cos Cos (3.13) mn, 0,1,.... LLxy
m X n Y f(,,)() X Y fmn Cos Cos (3.14) mn, 0,1,.... LLxy
Substituting Fourier series expansions i.e. Eqs. (13) and (14), one can derive the one dimensional heat equation as below:
2 mnc 2 mn 0 for m, n =0,1,2…. (3.15) Z 2 mn mn
2mn 2 2 where c mn ()() (3.16) LLxy and the boundary conditions on the front surface and back surface are:
mn f () for Z =0 (3.17) Z mn
mn 0 for Z =1 (3.18) Z
Here mn (,)Z is the modal temperature. This equation is similar to the one dimensional heat conduction problem representing fin with convection [45].
29
3.2.2Kirchhoff’s Transformation
If the thermal conductivity can be made independent of temperature (and therefore also independent of spatial position for homogeneous materials), then the heat conduction equation
(1) can be simplified to:
222 () (3.19) t x2 y 2 z 2 where
k α= c p (3.20) through Kirchhoff transformation:
T kd() (3.21) 0
Since the function k (T) is usually based on the curve fit of the data in the area of interest, we propose the following transformation:
1 T T k() d (3.22) r k r Tr
where Tr is the reference temperature which is determined as follows:
t 1 f LL T T(,,) y z t dydzdt r tL2 f 0 0 0 (3.23)
30
where L is the total length and tf is the final time and kr= k (Tr). Any value for the reference temperature Tr can be used; for simplicity, we choose Tr to be the average back surface temperature, Tr = 326.8362 K.
If the conductivity can be expressed as a quadratic in T,
2 k() T c0 c 1 T c 2 T (3.24)
It can be written as
2 k()()() T kr k12 T T r c T T r (3.25) where
k1 c 12 c 2 Tr (3.26)
Substituting Eq. (24) into Eq. (22), we obtain
1 kc1223 TTTTT [()()] rr (3.27) kr 23
The graph of the above relationship is that of a line with small curvature. We thus call the
“warped” temperature.
The DCT/Laplace procedures introduced above are now applied to Eq. (20) by assume that the thermal diffusivity is temperature independent. Specifically, we use the thermal
diffusivity at Tr given in (23). The sensor temperature data is first converted to , the so-called
“warped temperature” using (27). The standard heat equation is then solved using DCT/Laplace method. The solution is then converted to the real temperature using the inverse of Eq. (27). The
31
inverse of Eq. (27) can be obtained by finding the cubic roots of the equation. However, the following approximation is obtained if the cubic term is ignored in Eq. (27):
kr 2k1 T Tr [ 1 ( Tr ) 1] (3.28) k1 kr
For stainless steel material properties, we have found that the approximation provided by (28) is indistinguishable from the accurate inverse of (27). We call the process of recovering real temperature using (28) the “unwarping” process.
3.3 Laplace transformation solution
The analytic solution of Eq. (15) can be obtained by taking Laplace transform. Laplace transform of Eq. (15) which is a second order ordinary differential equation can be solved by two undetermined coefficients. These two undetermined coefficients can be obtained by any two boundary conditions of the two surfaces. Eventually the solution can be expressed as the linear relationships between the Laplace transformation of the temperature and heat flux on the front and back surface as below [40, 46]:
221 coshs cmn sinh s c mn mn(ss ,0) 2 mn ( ,1) sc mn (3.29) (ss ,0) ( ,1) mn 2 2 2 mn s cmnsinh s c mn cosh s c mn
where mn (s ,0) , mn (s ,0), mn (s ,1) and mn (s ,1) are the Laplace transform of the temperature and heat flux at front surfaces and back surfaces respectively. From Eq. (29), one can easily obtained the simplified relationships considering no back surface heat flux as follows:
2 mn(s ,0) cosh s c mn mn ( s ,1) (3.30)
32
22 mn(s ,0) s c mn sinh s c mn mn ( s ,1) (3.31)
Eqs. (30) and (31) are called transfer functions which represent the relationships between front surface temperature and heat flux with back surface temperature. Applying Hadamard
Factorization Theorem to transfer function and applying the inverse Laplace transform of the simple polynomials, we have the following relationships [40]:
s coshs c2 cosh c {1 } (3.32) mn mn k1 ck22[(2 1) ] mn 2
sc 2 sinh c s sinhmn mn [1 ] (3.33) 2 c c22 () k sc mn mnk1 mn
Substituting the values of Eqs. (32) and (33) into Eqs. (30) and (31), we have the Laplace transform of temperature and normalized heat flux in the time domain as follows:
s (s ,0) cosh c [1 ] ( s ,1) (3.34) mn mn mn k1 ck22[(2 1) ] mn 2
sinh c s (,0)s mn ( s c2 )[1 ](,1) s (3.35) mn mn 22 mn cmnk1 c mn () k
The corresponding equations in the time domain are:
sd ( ,0) coshc {1 } ( ,1) (3.36) mn mn mn k1 ck22[(2 1) ] d mn 2
sinh c d s d fc()mn ( 2 )[1 ](,1) (3.37) mn mn 22 mn cmn dk1 c mn () k d
33
where mn ( ,0) , mn ( ,1) and fmn () are the front and back surface modal temperatures and front surface modal heat flux respectively. The modal temperatures and heat flux relationship are exact and these exact relationships in the time domain show similarity with the result in paper
[47]. Burggraf [47] found an exact solution of the inverse problem by specified the boundary conditions at a single location where the surface conditions were unknown.
From Eqs. (36) and (37), it is found out that the coefficient of front of the derivative term becomes small as k becomes large. Therefore the approximation can be done by eliminating the infinite product from those equations. In an iterative procedure, starting form the back surface
temperature for a given mode i.e.mn ( ,1) , the initial modal front surface temperature and heat flux can be obtained from Eqs. (36) and (37). Number of the iteration is limited with the value of truncated tolerance.
In short, with the given back surface temperatures i.e. found from exact solution with data array, we first do temperatures warping using Eq. (24). Then, we solved the 3D inverse heat transfer problem and obtain the front surface temperatures. Moreover, front surface temperatures need to unwarp using Eq. (28) after obtaining from solution of 3D heat transfer problem.
3.4 Simulation Results
In this paper we used stainless steel (SS-304) as our target material. The following
3 thermophysical properties of the target material are used for this analysis: s =7900 kg/ m , c0 =
2 6 3 10.06453 (W/m K), c1 =0.01719 (W/m K ), c2 =1.85055 10 (W/m K ). The geometry of the slab: thickness (lz) = 0.00215m, length (lx) =0.1m and width (ly) =0.1 m. The initial temperature
34
maintained at 300K. The total time is used 2s with the time step 0.05s. The number of the grid in thickness is 12. For stainless steel (SS-304), thermal conductivity is expressed as below:
k( T ) 10.0645 0.01719 T 1.85055 1062 T (3.38)
The thermal conductivity formula is derived based on the curve fitting of the values of thermal conductivity at different temperatures [48] to find a mathematical formula for conductivity.
Therefore, this quadratic form of equation is really an accurate one exactly represent the thermal conductivity of SS 304.
Figure 3-1 shows the relationship between warped and unwarped temperature of stainless steel at reference temperature 318K. The sensor array with size of 20×20 is used in the direct problem and after solving the direct problem, this data array of 20×20 for the back surface temperature to be employed as measured data in the inverse problem. Sensors are evenly placed in the back surface. The sensors are distributed on the back surface as shown in Fig. 3-2. The coupon size is kept as 100 mm×100 mm.
1200
1000
800
600
400
300 400 500 600 700 800 900 1000 T
Figure 3-1 Relationship between T and for stainless steel with Tr 318 K
35
Figure 3-2 Schematic diagram of meshing on the back surface
Instead of conducting actual experiment, the measurement data of temperature are generated numerically from solving the direct problem described by the governing heat equation with temperature dependent properties. The heat flux on the front surface in the direct problem for generating measured data on back surface is dynamic with following form:
2 2 2 qxytq( , , )max exp{ [( xMyNw 0.5 ) ( 0.5 ) ]/ }(1 sin(2 ft )) (3.39)
where is surface absorptivity; qmax is the maximum heat flux at the center of the heating flux spot; w is 1/e radius of Gaussian laser beam; f is frequency . Their values are set to be: α = 0.05,
2 qmax = 5000 W/cm , w = 10.0 mm, and f = 2.0 Hz.
The result for the above stated problem at different time is shown in Fig. 3-3. Figures 3-3
(a), (b), (c), (d), (e), (f) and (g) show the contour plots for the front surface temperature at time t=0.4s, 0.8s, 0.85s,1.25s,1.5s,1.85s and 1.9s respectively. The left column is the exact solution of
36
the front surface temperature, middle column is obtained by CGM method and right column is obtained by using DCT/Laplace solution using reference temperature as the average of all the back surface temperatures. It can be seen from those figures that the results of CGM method and
DCT/Laplace transformation are in reasonable agreement with the exact solution as time increase. However, in the DCT solution, the central temperature does not reach as high as expected at time 0.8s. This deviation occurs due to the fact that, CGM method could handle temperature dependent thermophysical properties like thermal conductivity (k), specific heat (cp) and thermal diffusivity (α) where the DCT/Laplace method can only handle temperature dependent thermal conductivity (k).
(a) t =0.4s
(b) t =0.8s
37
(c) t =0.85s
(d) t =1.25s
(e) t = 1.55s
38
(f) t =1.85s
(g) t = 1.9s
Figure 3-3 Comparison of front surface temperature contours for SS 304: Exact (left), CGM
(middle) and DCT/Laplace transformation solution (right)
Figure 3-4 shows the front surface temperature distribution alone Y axis (mm) at
LL different group of 20 sensors correspond to a row with coordinate given by Zzz ( row 1) 40 20 at time t=1.55s. However because of the symmetry, we only plot the temperature distribution over half of the plate. It is found that front surface temperature reaches maximum point at the
19L sensors 181-200 with coordinate Z z ; where front surface temperature is minimum at the 40
L edge i.e., sensors location 1-20 with coordinate Z z . 40
39
Figure 3-4 Front surface temperature distributions along Y direction at different sensors location at time t=1.55s
Figure 3-5 shows the comparison of front surface temperature of DCT/Laplace transformation with exact solution along Y direction at time t= 1.55s. From this figure, it is shown that the maximum deviation occurs at the center of the slab where the front surface temperatures are maximum.
40
Figure 3-5 Comparison of front surface temperature between DCT/Laplace transformation and exact solution along Y direction at different sensors location at time t=1.55s
The comparison of front surface temperature w. r. t time for CGM and DCT/Laplace
19L 19L transform solutions in three different locations (center (Y Y , Z z ), two off centers ( 40 40
25L 19L 29LL 19 19L Y Y , Z z and (YZYz, , Z z ) is shown in Fig.3-6. In this figure, it is 40 40 40 40 40 shown that the fluctuations in the DCT solution are in phase with CGM solution, but with reduced amplitude.
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Figure 3-6 Comparison of CGM and DCT/Laplace transformation solutions of front surface
19L 19L temperature vs time at three sensor locations (center (Y Y , Z z ), two off centers ( 40 40
25L 19L 29L 19L Y Y , Z z and (Y Y , Z z )) 40 40 40 40
Since the CGM and DCT/Laplace show some deviations from the exact solution, we present the RMS value of error in this article. The Root Mean Square (RMS) of the errors at different time steps for DCT/Laplace solution and CGM method are given by the following equations:
N 1 2 RMSDCT/Laplace= RMSDCT/Laplace (TTDCT / Laplace exact ) n i1 (3.40)
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N 1 2 RMSCGM= ()TTCGM exact n i1 (3.41)
The RMS of errors is calculated over all the nodes for every time steps. Two different reference temperatures in the Kirchhoff transformation are used in the comparison of RMS of the errors.
One reference temperature is calculated by averaging all back surface temperatures and the other is found by averaging of the maximum and the minimum of back surface temperatures. These averages are 326.8362 K and 489.675 K respectively. Figure 3-7 shows the comparison between three RMS values at different time steps. From this figure, it is shown that the RMS value of error for DCT/Laplace transform method fluctuates with the same period as the front surface heat flux and the maximum value is approximately 27K where the RMS of CGM method is much less than that value. However, the calculation above for DCT/Laplace solution was completed within
4-5 s when implemented in MATLAB on Dell personal computer. On the other hand, CGM method required 1.5 /2 hrs to complete in FORTRAN on a 32-bit personal computer.
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Figure 3-7 Comparison of the RMS values at different time steps for DCT/Laplace (reference temperature (Tr) as all average values of back surface temperatures and average of maximum and minimum front surface temperatures) and CGM method
3.5 Conclusion
Kirchhoff transformation is introduced in the solution of three-dimensional inverse heat conduction problem. In this paper 3D heat transfer problem has been solved with a special geometry of a thin sheet. First the 3D heat conduction equation is simplified into a 1D hear conduction equation through modal representation. Then one dimensional problem is solved by previously developed model [10, 15, 16].
After solving the direct problem, the data array of 20×20 for the back surface temperature to be employed as measured data in the inverse problem. With the given back surface temperature,
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temperature warping is done using Kirchhoff transformation. Then the 3D inverse heat transfer problem is solved through simplifying into 1D heat conduction problem. Through a Laplace transformation, the relationships are obtained between front surface heat quantities with the same quantities on the back surface. Haramard Factorization Theorem has been applied to expand the transfer functions. Then time domain iteration has been used to calculate the front surface heat quantities i.e. temperature and heat flux. The front surface temperatures are then unwrapped to obtain the actual value using Eq. (28). Then the comparisons between the DCT/Laplace problem to exact solution and CGM are shown in this paper as well as Root Mean Square comparison for three cases are shown. From the comparisons, it is evident that DCT/Laplace transform solution shows a good agreement with the exact solution.
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CHAPTER 4
Multicomponent Gas-Particle Flow and Heat/Mass Transfer Induced by a Localized Laser
Irradiation on a Urethane-Coated Stainless Steel Substrate
4.1 Introduction
Because of the unique characteristics of coherency, hmonochromaticity and collimation, lasers have been widely used in various areas, such as etching and ablation of polyimide [49, 50], ablation of biological tissues [51, 52], and interaction with composite materials [53], to name a few. For many laser applications, scientists and engineers frequently encounter a situation that requires to couple multi-scales and multi-physics in solution of laser-material interaction. For example, laser cutting is one of the most important applications of laser in industry. To accomplish the task in terms of work quality and efficiency, a thoroughly understanding of the physics that are involved in the laser cutting process thus is of importance, including thermal transport across the object, change of material themophysical properties, phase change of melting and vaporization, chemical reaction in the material within or nearby the irradiated spot, and discretized particle ejection dynamics in the gaseous phase.
Numerous works on simulation of laser material processing has been published. For example,
Mazumder and Steen [54] developed a three-dimensional (3D) heat transfer model for laser material processing with a moving Gaussian heat source using finite difference method. The results showed that some of the absorbed energy dismissed by radiation and convection from both the top and bottom surfaces of the substrate. Lipperd [55] investigated laser ablation of polymers with designed materials to evaluate the mechanism of ablation. Zhou et al. [56] developed a numerical model to simulate the coupled compressible gas flow and heat transfer in a micro-channel surrounded by solid media. Kim et al. [57] studied the pulsed laser cutting using
46
finite element method, and found that there were some fixed threshold values in the number of laser pulses and power in order to achieve the predetermined amount of material removal and the smoothness of crater shape. As a follow-up work, Kim [58] further reported a 3D computational modeling of evaporative laser cutting process. Mahdukar et al. [59] investigated laser paint removal with a continuous wave (CW) laser beam as well as repetitive pulses. The specific energy, a measure of the process efficiency that is defined as the amount of laser energy needed to remove per unit volume of paint prior to the onset of substrate damage, was found to be dependent of laser processing parameters. The result also showed that for a CW mode, the specific energy reduced with increase of laser scanning speed, irradiation time, and laser power.
The study of simultaneous fluid flow and heat and mass transfer in a coated medium induced by laser heating is scant. The objective of this work is to investigate the effects of laser irradiation on heat transfer and mass destruction of a urethane paint-coated substrate using a 3D numerical simulation. The paint starts to decompose through chemical reaction when the paint’s hottest spot reaches a threshold temperature, 560 K. As a result, combustion products CO2, H2O and
NO2 are produced and chromium (III) oxide, which is buried (as pigment) in the paint, is ejected as parcels from the paint into the surrounding gaseous domain. The results, including the variation of temperature and species concentration in the gaseous phase, amount of mass loss from the coated paint, and irradiation time before the onset of melting in the substrate steel, will be analyzed and discussed in detail.
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4.2 Physical model
The entire process of laser irradiation to a paint coated on a steel substrate includes: (1) thermal transport in the paint and substrate, (2) chemical reaction in the paint, and (3) heat and mass transfer of reactant and products in a multi-component gaseous phase.
4.2.1 Continuous Phase
For the gaseous phase, the governing equations of mass, momentum and energy are given as follows [60]:
(a) Continuity equation:
U 0 (4.1) t where and U are density and velocity of the gaseous phase, respectively. For a multi- component compressible flow, the mass density is a component-weighted density.
(b) Momentum equation:
U 2 U U p g U trace U I Su (4.2) t 3 where p is pressure, g is gravitational acceleration, μ is dynamic viscosity, I is the unit matrix, and Su is the source term that accounts for interaction between generated parcels and gaseous phase in each cell and is expressed as:
N ppF S= p (4.3) V
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th where Np is the number of total particles in the p parcel (see Section 2.2)that locates in one cell,
Fp represents the force acting on a single particle in the parcel, and ΔV is the volume of the cell.
The calculation of Fp will be introduced later.
(c) Energy equation:
1122 Dp hsUUU h s h s S h (4.4) t22 Dt
Dp where hs is sensitive enthalpy, is enthalpy-type thermal diffusivity, is material derivative Dt of pressure p, and Sh is the source term that represents heat transfer between particles and gaseous phase and is expressed in the form of
Nhpp S = p (4.5) h V
th where hp is the enthalpy transferred between each individual particle in the p parcel and the surrounding gaseous phase.
(d) Species concentration equation:
i ()iU D i i (4.6) t
where , i and Di represent the mass density, mass fraction, and diffusion coefficient of specie i, respectively. The bulk density ρ of the system is estimated through
p (4.7)
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where ψ is the bulk compressibility which is averaged over all the gaseous species. Since all the gas species are considered as perfect gas, the bulk compressibility can be estimated by
N 1 i (4.8) RTi1 Mi
where Mi denotes molar mass of specie i, R is the gas universal gas constant, N is the total number of species in the gaseous phase.
It is noted that the energy equation is solved in the enthalpy form. However, temperature can be solved using the following thermodynamic relationship [61]，
h c s (4.9) p T where the specific heat capacity is estimated through a forth order polynomial function of temperature [62]:
2 3 4 cp Ra0 aTaT 1 2 aT 3 aT 4 (4.10)
The temperature in the computational domain is thus corrected through iterative method
th according to the hs-T diagram. In addition, the viscosity (μi) of the i specie is treated to be temperature dependent [63]
AT si, i (4.11) 1/TTsi,
in which As,i and Ts,i are both constants for gas in question and they can be found from many online resources [64].
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The bulk viscosity (μ) and thermal diffusivity () of the gaseous phase given in Eqs. (2) and (4) can be obtained by the molar-weight mean [65], which is similar to the bulk compressibility,
N i i (4.12) i1 Mi
N i i (4.13) i1 Mi
th The mass diffusivity of the i specie (Di) in Eq. (6) can be determined using the Maxwell-Stefan mass transport model [18] that considers the multi-species system as a special binary system,
1 X D i i (4.14) (/)XDi ij j, j i
where Xi denotes the molar fraction of specie i, and Dij are the mass diffusivity between species i and j that is temperature and pressure dependent [60]:
3 2 0 T (4.15) DDij ij p where D0 is the mass diffusivity from specie i to j at 300 K and 1 atm. It can be determined by ij combining Chapman-Enskog theory and the method introduced Bird et al. [66] as follows:
1.858 103TMM 3/2 1/ 1/ D0 ij (4.16) ij 2 p ij
where σij is the average collision diameter of species i and j, and Ω is a collision integral which is tabulated in [67].
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For the solid domain, it composes of two layers: the bottom layer is stainless steel (AISI 304L) and the top layer is the paint that is a homogeneous mixture of binder (40%) and pigment (60%).
Only heat conduction energy equation is solved in this domain:
cp T k T (4.17) t
4.2.2 Chemical Reaction
The urethane based paint is considered in this work because it is an industrial standard for automobile paint. In the past two decades it has mostly replaced acrylic paints as automakers’ preferred choice [68]. For this paint, the binder is polyurethane (C3H7NO2) and the pigment is
Chromium (III) oxide (Cr2O3). The overall chemical reaction occurring in the paint is as follows:
4CHNO3 7 2 19 O 2 12 CO 2 14 HO 2 4 NO 2 (4.18)
For the stainless steel, both thermal conductivity and heat capacity are temperature-dependent
[69].
The chemical reaction considered in this work is a complete-type and zero order level. As mentioned previously, polyurethane will start decomposing when the temperature at the top surface of the paint reaches the threshold temperature. The produced gas species (CO2, H2O and
NO2) then diffuse into the gaseous domain. At the same time, pigment (chromium (III) oxide) in the paint will be ejected into the gaseous domain from the paint surface. The reaction rate is approximated by Arrhenius equation [70],
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1/2 E/ RT (4.19) kcO RT/2 M e 2
where E is activation energy, and MO2 is molar weight of Oxygen. Alternatively, the reaction rate can be defined as the slope of the concentration-time plot for a specie divided by the stoichiometric coefficient of that specie. For consistency, a negative sign is added if the specie is a reactant. Thus, the reaction rate constant can be represented as [71],
[]C H NO 4[][][][]O 4 CO 4 H O 4 NO k 3 7 2 2 2 2 2 (4.20) c t19 t 12 t 14 t 4 t
The reactant consuming rate and the products generating rate can be estimated through Eq. (18).
Another important concept that should be pointed out is parcel. Since it is computationally exhaustive to capture all particles’ dynamic behaviors during the entire computational process due to the huge number of particles, a concept of parcel which is a collection of real particles
(pigment) is adopted to deal with solid particles in a fluid flow. In this sense, all the particles in one parcel share the same particle properties, i.e. size, velocity, temperature, etc. For the parcel generating mechanism, a field activation burning type of particle injection model that mimics the particle generation process is developed with the idea of introducing “injectors.” Specifically, mass destruction is considered as parcel injection from “injectors” which are buried at the center of the cells that are attached to the coupled boundary between the solid and gaseous domains.
The fields associated to those cells, namely temperature and oxygen concentration, determine whether injectors start or stop to work. In other words, when the temperature on the paint surface exceeds the threshold value and the oxygen concentration is sufficiently high, parcel will be generated and ejected into the gaseous domain according to the chemical reaction rate. A random diameter generator function is used to describe the particle size distribution for each injector, by
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fixing the maximum and minimum bound. The number of particles in each parcel generated in each time-step is estimated by
4 3 pg/ pg r N 3 (4.21) NVV rem rem bd// bd pg pg
N where ΔVrem is the volume removed from an individual “injector,” is number density of particles buried in paint, ρpm and ρbd are density of pigment and binder respectively, and r is the mean radius of particles in this parcel. It should be noted that a fractional number of particle is not allowed in this simulation. If the number is smaller than one, then the diameter will be rescale to a value to fit this number to unity.
According to the chemical reaction model, mass lost from paint can be determined; thus, the volume changed in the solid domain can also be known. In order to mimic the mass destruction, the mesh topology will be updated through moving the nodes shared by the solid and gaseous domains. In this work, it is proposed that the node displacement is approximated by the volume change of cell that is adjacent to the solid/gas interface as follows:
4 kc C O t M bd 19 2 (4.22) dremoval bd
where kc is chemical reaction rate constant, CO2 is molar concentration of oxygen. It should be noted that this expression is allowed only by assuming that the node move uniformly in all directions. Figure 4-1 is a two dimensional illustration that shows how the node displacements are applied to mimic paint mass destruction during the chemical reaction.
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Figure 4-1 Node moving mechanism
The yellow dots represent temperature and oxygen concentration obtained by solving the corresponding equations; while the red dots represent temperature and oxygen concentration interpolated from the yellow dots. Accordingly, the mass destruction can be more easily realized through moving nodes based on the temperature and oxygen concentration at those red dots. In this case, the temperature in the solid domain and the oxygen concentration in the gaseous domain are interpolated to the corresponding nodes (or points) of its own grids. If the temperature at a node is higher than the threshold value and the oxygen concentration at that node is sufficiently high, then the nodes in both grids will be moved by a certain displacement that is determined through Eq. (22). After that, the concentration of reactants and products will be updated in both domains as well. Once the locations of the nodes in the paint region are updated, the nodes that share the same location but in the neighbor (gaseous) domain will be updated in order to make the domain spatially continuous. In addition, the internal nodes in both regions will also be updated to guarantee the mesh quality during computation. This can be done by solving the following 1-D diffusion equation for each domain,
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Y 2 kYd (4.23) t
where ΔY represent the moving distance in the y-direction in each time-step, kd is a diffusion coefficient that is equal to 1×106 by trial and error. These equations are solvable, because the boundary conditions are known according to the chemical reaction at each time step; it is affordable due to its simplicity. By solving Eq. (23), the displacements of all the control volumes are known, and then a volume to point interpolation will be applied to obtain the displacement for each node.
4.2.3 Discretized Phase
In the aspect of parcels’ motion, they can be described using Newton’s 2nd law:
ma G Fdrag (4.24)
where m and a are mass and acceleration of a parcel. The right hand side of Eq. (24) accounts for gravitational force G and the drag force Fdrag due to the velocity difference from the gaseous phase. The drag force is estimated in consideration of particle’s sub-micro size [72] as follows
[73]:
3 24 2 FUUdragmd c p p p (4.25) 4 Cc
where c is molecular viscosity of the fluid, Up is the velocity of the parcel, and Cc is the
Cunningham correction to Stokes’ drag law [72]:
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1.1d 2 p Ce1 1.257 0.4 2 c (4.26) d p in which λ is the molecular mean free path. The gravitational force is calculated by
Ggm 1 (4.27) p
In addition, the simulation will be automatically terminated when the maximum temperature in the stainless metal reaches to its melting point (1,670 K) [60], since the current work only focuses on the physical process of paint removal before the phase change of stainless steel takes place, though the total simulation time and laser pulse irradiation time is set at 10 s. The simulation is performed by using the most recent OpenFOAM-2.3.0 framework with the incorporation of our newly developed solver.
4.3 Results and Discussions
Figure 4-2 illustrates the physical and geometric model of the problem under consideration. A solid (paint + stainless steel) with a size of l×w×h (length×width×height) in the x-, z- and y- directions is placed at the bottom of the entire simulation domain that has a size of L×W×H
(length×width×height).
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Figure 4-2 Illustration of mesh arrangement
The dimensions used in the simulation are 250×250×200 mm3 for the gaseous domain and
30×30×1.35 mm3 for the solid domain. The thickness is 0.150 mm for paint and the rest 1.2 mm for stainless steel (AISI 304L).The six Gaussian laser beams of the same radius ro = 13.4 mm have laser power ranging from 2.5 kW to 15.0 kW with an increment of 2.5 kW. The material absorptivity is assumed to be constant, 0.8 for the paint and 0.1 for stainless steel (AISI 304L).
The total laser irradiation time on the paint is set to be 10 s. The initial temperature of the entire domain is 300 K. As described previously, the mass diffusivities of all species are pressure and temperature dependent and their initial values are tabulated in Table 4-1.
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0 2 Table 4-1 Initial mass diffusivity between gas species D ij (m /s)
O2 N2 CO2 H2O NO2 -3 -3 -3 -3 O2 / 2.037×10 1.509×10 2.769×10 1.559×10 -3 -3 -3 -3 N2 2.037×10 / 1.499×10 2.69×10 1.556×10 -3 -3 -3 -3 CO2 1.509×10 1.499×10 / 2.079×10 1.109×10 -3 -3 -3 -3 H2O 2.769×10 2.69×10 2.079×10 / 2.177×10 -3 -3 -3 -3 NO2 1.559×10 1.556×10 1.109×10 2.177×10 /
The specific heat capacity and the absolute viscosity of each species are given in Table 4-2, and the density, specific heat and thermal conductivity of stainless steel in Table 4-3 [60,74].
Table 4-2 specific heat capacity and absolute viscosity of gas species
O2 N2 CO2 H2O NO2 cp (kJ/kg K) 0.918 0.807 0.846 1.86 1.04
μ (×10-5 Pa s) 2.06 1.78 1.51 1.23 1.33
The properties of paint are estimated based on the volume-weighted average over all components and are given in Table 4-3.
Table 4-3 Material properties of solids Heat of Chemical Conductivity Heat capacity Density Reaction (W/m K) (J/kg K) (Kg/m3) (J/Kg)
Pigment 32.94 781 5210 / Cr2O3
Binder 0.4 1755 1424 6.28×106 C3H7NO2
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AISI 304L 7.9318 0.023051T 426.7 0.17T 7900 / Stainless Steel 6.4166 1062T 5.2 1052T
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The threshold temperature for chemical reaction to take place is 560 K, the activation energy of the chemical reaction is 45 kJ/mol [75,76]. The particles have a mono-size distribution with a diameter of 750 nm. The initial velocity of each ejected parcel is zero. The initial and boundary conditions are summarized in Table 4 and Error! Reference source not found. for the gaseous and solid domain, respectively.
Table 4-4 Initial and boundary conditions for the gaseous domain
Xmin Xmax Ymin Ymax Zmin Zmax Air_to_Solid Internal
Velocity (m/s) (0,0,0) (0,0,0)
p Pressure (atm) 0dependent on instant local velocity 1 n
T Temperature (K) 0 coupled 300 n
O2 0.21
N2 0.79 C H2O 0 0 n
CO2 0
NO2 0
The laser beam source is applied as a heat flux right on the top boundary of the solid domain. In this work, the radiation effect is ignored due to its relatively small magnitude order in comparison other thermal terms. In Table 5, P and r0 represent the total power and radius of the laser beam respectively, αa is absorptivity, Uf and Sf are receding velocity and area of the irradiated surface respectively, and ΔHR is the heat of chemical reaction.
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Table 4.5 Initial and boundary conditions for the solid domain
Ymin Solid_to_Air Internal
2xz22 2P 2 USf f pai T a eHr0 Temperature 0 2 R 300K T r0 S f n yk
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A mesh independent study is first carried out before conducting the whole simulation in order to identify an optimal arrangement of mesh. To overcome the difficulty from creating an eligible mesh configuration caused by the large spatial difference between the solid and gaseous domain, the entire computational domain is decomposed into 18 small blocks, and grading hexahedral cells are applied to each block with the purpose of reducing numerical diffusion caused by non- orthogonality, skewness and smoothness [77]. Figure 4-3 shows the temperature variation after one time-step (Δt = 1.25×10-4 s) from three different meshes, with laser power of 15,000watts. It is found that the maximum temperatures do not show significant difference (< 0.02%) between the two finer meshes, 452.76 K vs. 452.68 K. The finest mesh that has a total of 194,400 cells, including 178,400 in the gaseous domain and 16,000 in the solid domain is employed in the following simulations. The maximum temperature in the paint is chosen here as a benchmark because of its importance in affecting properties across the entire domain. Additionally, it plays a key role in activating chemical reaction.
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Figure 4-3 Maxmum temperatures in the paint vs three different mesh configurations
In this work, six scenarios with different laser powers are setup to study how the laser power affects the chemical reaction rate and temperature, density, velocity and concentration of the resulting gas species from the heated paint. Figure 4 shows the temperature distribution across the middle cross-section area of the gaseous domain at the ends of simulation for all the six cases. The areas of hot region in the gaseous domain decrease with the increasing in laser power.
This trend is mainly attributed to the simulation time, 2.77 s, 1.18 s, 0.72 s, 0.51s, 0.38 s, and
0.30 s for the ascending six laser powers. According to the absorbed laser powers and heating times, the total laser energies deposited into the entire system at the ends of simulation are 6.925,
5.900, 5.400, 5.100, 4.750, and 4.500 kJ, respectively. The absorbed energy in the case of the lowest power is 1.53 times that of the highest power. The more the energy absorbed, the more the gas species are generated. In addition, a longer time for the species to move into the gaseous
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domain allows the hot species to spread over. Therefore, the lower the laser power is, the larger the area of hot region in the gaseous domain can be observed. It is also found from Figure 4 that the maximum temperatures of the gas species are 1720 K, 1780 K, 1820 K, 1880 K, 1910 K, and
1960 K, respectively. The trend that the maximum temperature increases with laser power is because a shorter heating time not only limits the spread area of the hot species, but prevents the thermal energy in the hot region from diffusing into the surrounding colder region. As a consequence, a higher maximum temperature adjacent to the center of the heated spot is expected for the higher laser power.
(a) P = 2.5 kW, time = 2.77 s (b) P = 5.0 kW, time = 1.18 s
(c) P = 7.5 kW, time = 0.72 s (d) P = 10.0 kW, time = 0.51 s
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(e) P = 12.5 kW, time = 0.38 s (f) P = 15.0 kW, time = 0.3s Figure 4-4 Temperature distribution across the middle cross section area of the gaseous domain at the end of simulation
Figure 4-5 reveals the time histories of temperature at the center of the laser heating spot on the top surface of paint. Apparently, a lower laser power leads to a lower maximum temperature and a longer simulation time, which is also confirmed by the results in Figure 4.
Figure 4-5 Time history of temperatures at the center of laser heating spot for the six laser powers
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Correspondingly to the temperature distributions shown above, Figure 4-6 shows the mass density distribution across the same cross-section area in the gaseous domain at the ends of simulation. Since the gas thermal state is determined by the ideal gas law which is a univalent function of temperature, the areas of lower mass density are the same as those distributions of the outflow gas temperature shown in Figure 4. Accordingly, the higher the temperature is, the lower the density will be. As seen in Figure 4-6, the minimum densities are 0.196, 0.190, 0.185, 0.180,
0.176, and 0.172 kg/m3 for the six laser powers, respectively.
(a) P = 2.5 kW, time = 2.77 s (b) P = 5.0 kW, time = 1.18 s
(c) P = 7.5 kW, time = 0.72 s (d) P = 10.0 kW, time = 0.51 s
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(e) P = 12.5 kW, time = 0.38 s (f) P = 15.0 kW, time = 0.30 s Figure 4-6 Density distributions across the middle cross section area of the gaseous domain at the end of simulation Figure 4-7 presents the time histories of mass density of the gaseous phase close to the center of the laser heating spot. All the curves show a decreasing trend, dropping from the initial value of
1.13 kg/m3 at room temperature. Due to the fact that the chemical reaction rate is proportional to temperature, a higher laser power would result in a quicker decline of the gaseous mass density as shown in Figure 4-7.
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Figure 4-7 Density variations at the center of the laser irradiation spot with time
Figure 4-8 plots the gas velocity distributions across the same cross section area of the gaseous domain at the ends of simulation. It is found that the maximum velocities are 0.898, 1.73, 0.901,
0.866, 0.813, and 0.800 m/s corresponding to the ascending laser powers at the end of the simulation. From the standpoint of energy conservation, the more the laser energy deposited into the system, the more kinematic energy can be absorbed by the gaseous phase.
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(a) P = 2.5 kW, Time = 2.77 s (b) P = 5.0 kW, Time = 1.18 s
(c) P = 7.5 kW, Time = 0.72 s (d) P = 10.0 kW, Time = 0.51 s
(e) P = 12.5 kW, Time = 0.38 s (f) P = 15.0 kW, Time = 0.30 s Figure 4-8 Velocity distributions across the middle cross section area of the gaseous domain at the end of simulation
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The change of gas velocity against laser power confirmed this conclusion, except the case with laser power of 2.5 kW. In order to exam a close look at the interaction between ejected parcels and gaseous phase, four snapshots, which are at time of 1.00 s, 2.00 s, 2.12 s and 2.77 s, are given in Figure 4-9. It can be seen that parcels (white dots) are lifted by the surrounding air as shown in Figure 4-9 (a). And the entire velocity field is symmetric. Figure 4-9 (b) shows the similar phenomena, but with more parcels and higher velocity. However, asymmetric velocity appear in Figure 4-9 (c) due to the momentum transfer between parcels and gaseous phase which is the leading factor of purtubating the entire velocity field in simulation domain. In addition, from the perspective of momentum conservation, the momentum exchange will lead to slow down the gas velocity and accelerate the parcels’ velocity if the drag force is larger than gravity. However, once the number of parcel is too huge, the momentum of gaseous phase will be completely drag down as a result of this momentum exchange. In other words, the gas velocity will be pulled downward the earth. The last snapshot, Figure 4-9 (d) shows that the velocity of the zone that closes the laser heating spot is directing to the earth. As a result, the parcels on the paint will be pushing to around as shown. A comparison between Figure 4-9 (c) and (d) shows that the velocity does not have a significant increase during the period of 0.57s due to large number increase of parcels which certainly consume a large amount of momentum that hold by gaseous phase. This explains why the lower maximum velocity of the case with laser power of 2.5kW than that with power of 5.0 kW as shown in Figure 4-8.
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(a) P=2.5 kW, time = 1.00 s (b) P=2.5 kW, time = 2.00 s ,
(c) P=2.5 kW, time = 2.20 s, (d) P=2.5 kW, time = 2.77 s, Figure 4-9 State of parcel flow and gaseous phase at different times Figure 4-10 shows the mass concentration variations for the species, namely, O2 as reactant and
H2O, CO2 and NO2 as products, adjacent to the center of the laser heating spot. It can be clearly seen in Figure 4-10(a) - (f) that the concentration of the reactant O2 keeps flat in the very beginning of heating and then decreases, accompanying the increasing of the produced species.
Apparently, the chemical reactions occur after these short time periods. For those laser powers higher than 10 kW shown in Figure 4-10 (g), (i) and (k), the concentrations of O2 fall down very quickly once the lasers heat the paint (in fact, those flat period is shorter than 0.01s which is the time interval for data presentation). For all the cases, the decreasing O2 concentration changes its course to increasing at a turning point where the mass concentration is about 1.7%. The generation of the three product species depends upon the reaction rate and the mass diffusion
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whose intensity is governed by the concentration gradient produced by the continuous chemical reaction. For example, all the product concentrations increase with time in Figure 4-10 (b), while the H2O concentration converts the increasing to decreasing in later time shown in Figure 4-10
(d) and (f) and to remaining almost no change in Figure 4-10 (h), (j) and (l). It is also observed that the increasing trends of all products can be categorized to the fast and slow zones. The fast zones appear immediately when the chemical reaction starts to take place, and the slow zones come out later. In view of the fact that mass concentration adjacent to the center of the heating spot is contributed from the two competing mechanisms: chemical reaction and mass diffusion, the relative strength of the two parts can explain the trends shown in these figures. For the products, chemical reaction would increase the mass concentration, while mass diffusion tends to decrease it. Therefore, the fast zones suggest that the intensity of chemical reaction is relatively stronger than that of the mass diffusion, while the slow zones show the opposite.
(b) P = 2.5 kW, chemical products’ mass (a) P = 2.5 kW, O mass concentration variation 2 concentration variation
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(d) P = 5.0 kW, chemical products’ mass (c) P = 5.0 kW, O mass concentration variation 2 concentration variation
(f) P = 7.5 kW, chemical products’ mass concentration (e) P = 7.5 kW, O mass concentration variation 2 variation
(h) P = 10.0 kW, chemical products’ mass (g) P = 10.0 kW, O mass concentration variation 2 concentration variation
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(j) P = 12.5 kW, chemical products’ mass (i) P = 12.5 kW, O mass concentration variation 2 concentration variation
(l) P = 15.0 kW, chemical products’ mass (k) P = 15.0 kW, O mass concentration variation 2 concentration variation Figure 4-10 Time histories of the mass concentration of O2, H2O, CO2, NO2 at the center of laser heating spot
Figure 4-11 gives the thickness history of paint removal. It can be seen that the lower the laser intensity is, the thicker the paint is removed. The removed thicknesses are 40.1, 21.4, 15.9, 12.4,
11.3, and 10.7 µm corresponding to the ascending laser powers of 2.5 kW – 15 kW. Similar to the kinematic energy in the gaseous phase, the trend of thickness reduction here can be explained from the energy conservation. In this case, the longer laser heating time can well compensate the energy loss due to the decrement in laser power. As a result of more energy absorbed, more paint would be removed by a lower power laser as expected.
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Figure 4-11 Time histories of paint thickness removal for the six laser powers
Figure 4-12 shows the behavior of pigments (each dot represents one parcel) after a partial portion of paint is removed by the laser heating. It can be seen that the all parcels are flowing upward due to laser heating caused natural convection. It can be seen that for the cases with laser power of 7.5 kW, 10.0 kW, 12.5 kW and 15.0 kW, the parcels are at the stage of gathering and moving upward before the simulations are completed. For the cases with power of 2.5 kW and
5.0 kW, it is found that parcels start blowing around by the downward velocity due to large number of generated parcels.
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(a) P = 2.5 kW, time = 2.77 s (b) P = 5.0 kW, time = 1.18 s
(c) P = 7.5 kW, time = 0.72 s (d) P = 10.0 kW, time = 0.51 s
(e) P = 12.5 kW, time = 0.38 s (f) P = 15.0 kW, time = 0.30 s Figure 4-12 Parcel and gaseous flow at the end of the simulation
This work also attempts to reveal the relationship for the amount of paint removal with laser power and irradiation time. Figure 4-13 compares the simulation and experiment results of laser power and irradiation time that are needed to remove a portion of 14 µm thick from the paint (for the cases with less than 14 µm paint removal, extrapolation is applied). It should be pointed out
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that the experiment reported in Reference [78] might be different from the conditions that are considered in the present study. However, the authors believe that the functional form obtained from that experimental work illustrated a general form of paint-removal rate versus beam intensity. Also due to lack of experimental data, only the result in Reference [79] is adopted for comparison here. It can be seen from Figure 4-13 that both the trends and the order of magnitude of the simulation results are in consistence with those experimental data. For the same laser power, the present simulation leads to a longer heating time for the paint removal.
Figure 4-13 Comparison of paint removal between simulation and experiment What causes the discrepancy is not clear at the time being; but a possible reason is that only chemical reaction is taken account for the paint removal in the present simulation model. In reality, each of vaporization and chemical reaction could yields a certain amount of material removal. From the comparison shown in Figure 4-13, it may be conjectured that chemical reaction is dominating in the paint removal for lower laser power while vaporization for higher laser power. Further model improvement by including more realistic physical process is suggested.
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4.4 Conclusions
A multi-physics problem that involves compressible gas flow, heat and mass transfer, and chemical reaction is numerically studied for a stainless steel substrate coated with paint under a high power laser heating. Six scenarios with laser powers of 2.5, 5.0, 7.5, 10, 12.5, and 15 kW are considered while the beam diameter is kept at 13.4 mm. A new solver is developed and incorporated into the current OpenFOAM-2.3.0 framework. The numerical simulations are stopped when the maximum temperature of the stainless steel reaches its melting point. The results reveal the effects of laser power on temperature, density, velocity, and species concentration of the gas species around the heated paint. It is found that a higher laser power leads to a shorter simulation time (2.77, 1.18, 0.72, 0.51, 0.38, and 0.30 s), a higher maximum temperature in the paint (1720, 1780, 1820, 1880, 1910, and 1960 K), a lower minimum mass density (0.196, 0.190, 0.185, 0.180, 0.176, and 0.172 kg/m3) and lower velocity (0.895, 1.730,
0.901, 0.866, 0.813, and 0.800 m/s) of gas species, and a smaller amount of paint removal (40.1,
21.4, 15.9, 12.4, 11.3, and 10.7 µm). The variation of species mass concentrations around the heat spot shows how it is affected by the chemical reaction and mass diffusion. It is also found that all the parcels are scattered over the paint surface when the numerical simulations stop with the current mean of calculating initial velocity of parcel. In comparison, the present chemical reaction model predicts the paint removal that is quantitatively consistent with published experimental result. Further model improvement by including more realistic physical process is suggested.
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CHAPTER 5
Effects of beam size and laser pulse duration on the laser drilling process
5.1 Introduction
Laser drilling (LD) can find its applications in automotive, aerospace, and material processing
[80-83]. The laser material interaction and its applications have undergone much study in recent years. In the laser drilling (LD) problem, a laser beam is produced and delivered to a target material which absorbs some fraction of the incident of laser energy. This energy is conducted into the target material and heating occurs, resulting in melting and vaporization of the target material. A time and position dependent vapor pressure exerts on the melt surface which results in a time and position dependent melt surface temperature at the liquid surface. The resultant surface pressure pushes the liquid out of the developing cavity and material removed by a combination of vaporization and melt expulsion. Laser drilling process includes heat transfer into the metal, thermodynamics of phase change and incompressible fluid flow due to the impressed pressure with a free boundary at the melt/vapor interface, and another one a moving boundary at the melt/solid interface due to the presence of melting and solidification process. The moving melt-solid interface and moving liquid-vapor interface result in a special type of problem called
Stefan problem with two moving boundaries, where Stefan boundary conditions are enforced.
Laser drilling process requires clear understanding of fundamental physics for better control and increasing the efficiency of the process. Due to the small size of the hole and melting region, even though the presence of the laser beam itself, it is almost impossible to measure regularly temperature, pressure as well as flow condition above the melt region. Moreover, vaporization, phase change and gas dynamics are important in LD process. Numerical simulation for LD process helps understanding the complex phenomena. Two-dimensional axisymmetric model
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was proposed by Ganesh et al. [84] to consider resolidification of the molten metal, transient drilled hole development and expulsion of the liquid metal in the LD process.
A number of studies of the laser drilling process can be found in the literature [85-89]. Most of these studies considered one dimensional and primarily based on thermal arguments. In 1976,
Von Allmen [85] used one dimensional theoretical model for rate of vaporization and liquid expulsion to calculate the velocity and the efficiency of laser drilling process as a function of absorbed intensity. Chan and Mazumder [86] invented a one dimensional steady-state model which provides close form of analytical solution for damage by liquid vaporization and expulsion. Kar and Mazumder [90] formulated a two dimensional axisymmetric model which neglected the fluid flow of the target material in melt layer.
The effects of fluid flow and convection were considered on the melted pools in welding [91,
92]. Chan et al. [93] developed a two-dimensional transient model where the solid-liquid interface was considered as a part of the solution and the surface of the melt pool is assumed to be flat to simplify the application of the boundary conditions. A Gaussian temperature flux boundary condition was imposed on the top surface, surface tension and buoyancy driving forces are accounted in Kou and Wang’s study [94]. In their early study with Sun [95] a conjugate heat transfer model considered enthalpy-temperature and viscosity-temperature relationships has been employed to obtained solid-liquid interface in the solution.
Two-dimensional axisymmetric transient development of LD problem considering conduction and advection heat transfer in the solid and liquid metal, free flow of liquid melt and its expulsion and the evolution of latent heat of fusion over a temperature range was modeled to track the solid –liquid and liquid-vapor interfaces with different thermo-physical properties [96-
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97]. Zhang and Faghri [98] developed a thermal model of the melting and vaporization phenomena in the laser drilling process considering energy balance analysis at the solid-liquid and liquid-vapor interfaces. The predicted material removal quantatively agreed very well with the experimental literature data. They found out that heat loss through conduction was insignificant on the vaporization where the locations of melting front is significant on conduction heat loss for low laser intensity and longer pulse.
There are many parameters that have influences on the laser drilling process and thereby the quality that can be achieved. Those parameters are called processing parameters. Laser wavelength, laser pulse width and peak power are most influential among of them. The objective of this paper is to model LD in order to understand the physical significances of the processing parameters such as laser diameter and pulse width.
5.2 Analytical Model
A schematic representation of the processes occurs in LD in shown in Fig.5-1. In this model, a laser beam is produced and directed towards a metal target which absorbs some fraction of the light energy results melting and vaporization eventually. Back pressure which is the result of vaporization pushes the liquid material away in the radial direction, which implies the material has been removed by the combination of the vaporization and liquid expulsion. This model includes heat conduction and convection, fluid dynamics of melting flow with free surface at the liquid-vapor interface, and vaporization at the melting surface and resulting melting surface temperature and pressure profiles.
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Figure 5-1 Schematic diagram of laser drilling process
5.2.1 Fluid flow
The hydrodynamical equations are applicable in the liquid regions. The non-dimensional governing equations for 2-D axisymmetric polar coordinate system are (dropping of the asterisk marks)
UVU 0 (5.1) RRR
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UUUUUUUP 22 1 UV Pr[ ] (5.2) TRZRZRRRR 2 2 2
VVVVVVP 22 1 UVG Pr[ ] (5.3) TRZRZRRR 22 where non-dimensional variables are
3 ud0 vd 0 gd 0 UVG,, 2 1 1 1
2 pd0 rz t1 P22, R , Z ,Pr , t (5.4) p1 d 0 d 0 1 d 0 where g, σ and Pr are the velocity acceleration due to gravitation, surface tension coefficient and the ratio of momentum to thermal diffusivities, respectively. Characteristics length (laser beam
diameter) and the thermal diffusivity of the liquid melt are represent by d0 and1 , respectively.
The energy equation is solved as an advection-diffusion equation which accounts phase change phenomena via temperature dependent heat capacity method. For a single time step, temperature field is obtained for a given fixed velocity. The important treatment of the LD problem is to consider melting surface as a free flexible surface. In volume of fluid (VOF) method, volume of fluid function, F is defined as unity for full fluid cell and null (zero) for the empty cell. A donor- acceptor flux approximation method is used to handle the VOF function (F) where finite volume method is incapable to solve that. The governing equation for F is given by
FFF UV 0 (5.5) t r z
5.2.2 Heat transfer
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The nondimensional thermal energy equation in cylindrical coordinate system is
(CT ) ( uCT ) ( vCT ) T T 1 ()()()K K KT B (5.6) t r z r r z z r r where
C sl ()TT 11 CTCTTT( ) (1 ) ( ) 22sl Ste T ()TT 1
S ()() uS vS B [] t r z
CTsl ()TT 11 STCTTTT( ) (1 ) ( ) 22sl Ste ()TT 1 CT sl Ste
Ksl ()TT (1KTT )( ) KTKTTT()() sl (5.7) sl 2T ()TT 1
The nondimensional variables are
TT00 S00 C k TSCK',,,m *** T0 T 0 c() T 0 T 0 c k h c11 h c (5.8) 00 c1() Th T c c s k s Ste,, Csl K sl L c1 kl
5.2.3 Optical considerations
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It is important to know the characteristics of the laser beam profile where laser is produced and applied during the LD process. Generally, the spatial dependence of the laser beam intensity is represented as either a top-hat profile or Gaussian distribution, while the temporal dependence may often be approximated as constant or Gaussian profile. The general laser beam intensity is represented as
22 ()()t tnc r r I( r , t ) I0 exp[ 22 ]exp[ ] (5.9) tr00
where I0 is the peak value of beam intensity. By integrating the intensity over the beam area in space and the pulse duration in time, one can found the total amount of energy delivered by the laser beam (where R is the beam radius) as follows
tR E 2 dt rdrI ( r , t ) (5.10) 00
In this study the target material is metal. As the beam penetrates into the target material, the electromagnetic energy is absorbed and resulting in damping of the intensity occurs over a very shallow depth of the material. The energy deposition is considered by assuming that all the energy is deposited into the top surface of the target material as a source on the surface. In this study, it is assumed that the surface temperature of the target material is high enough so that the reflectivity can be neglected.
Temperature is considered to be continuous across the melt/vapor region which is an extension of Von Allmen [98]. In a non-dimensional condition, the melt surface properties are determined from the conservation of mass, momentum and energy fluxes across the melt/vapor interface.
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The mass, momentum and energy balance across the melt/vapor interface with respect to a moving frame can be written as follows
muu m v v (5.11)
22 pm m u m p v v u v (5.12)
T Iabs L v v u v k 0 (5.13) n s/ melt
where Iabs is the rate of energy absorption. Some previous studies indicate that the gas velocity leaving from the surface is considered as sonic at the laser intensities typical of laser drilling.
The relation between surface pressure, temperature and intensity represents by Clausius-
Clapeyron equation considering ideal gas law
Lv 11 p( TS ) p vap,0 exp[ ( )] (5.14) RTTvap,0 S
pv R v T v (5.15)
Applying ideal gas law in the combined equation of the energy equation (13), Clausius-
Clapeyron equation (5.14), we get
1T Lv 1 1 RTs( I abs k ) p vap,0 exp[ ( )] (5.16) Lv ns/ melt R T vap,0 T S
Temperature gradient may be avoid for the high beam intensities due to the less conduction in
T I melt region where Eq. (16) can be approximate with abs due to the low vapor flow nks/ melt velocity.
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It is necessary to have the boundary conditions at mesh boundaries and at the free surfaces.
Layer of artificial cells is enforced to the different boundary conditions. Zero normal component of the velocity and zero normal gradients tangential velocity are considered. The left boundary is assumed to be a no-slip rigid wall which results zero tangential velocity component at the wall.
Normal stressed boundary condition is applied to the free surface. The surface cell pressure is calculated by a linear interpolation between impressed pressure on the surface and the pressure inside the fluid of the adjacent full fluid cell. Insulated boundary conditions are applied at the left, right and bottom boundaries. Stefan boundary condition is applied to solve the problem where the temperature of melt/vapor is unknown as priori. The target material is considered as ambient temperature at the beginning where the top surface of the substrate is considered as free surface liquid cells.
5.3 Numerical simulations
Volume of fluid (VOF) method is used to solve the continuity and momentum equations to find out the velocity and pressure for the melting region. The obtained melting velocity field is used to solve the energy equation to obtain the temperature field at the same time step. The velocity and pressure filed is solved for the free surface. The temperature field is solved by using control volume finite difference method [100] for the phase front as well.
5.3.1 Velocity and pressure calculation
VOF is a free surface modeling numerical technique which is used for tracking the free surface.
It refers to the Eulerian methods which are characterized by a mesh that is either stationary or moving in a certain manner to accommodate the shape of the interface. A rough shape of free surface is produced from the upstream and downstream values of F of the flux boundary. This
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shape is then used to calculate the boundary flux using pressure and velocity as a primary dependent variables. If the cell has nonzero value of F and at least one neighboring empty cell is defined as a free surface cell. The velocities for (n+1)th interval is calculated from the pressure occurring at the same (n+1)th interval. The pressure iteration from the continuity equation is carried out until it is satisfied the implicit relationship between pressure and velocity is satisfy for all the fluid cells as well as the pressures in all the free surface cells the applied surface pressure boundary condition required for gas dynamic model. The conservation of mass is maintained when applying free boundary condition for the free surface cells. The pressure which is the product of the surface tension coefficient and local curvature in each boundary cells is imposed on all the interfaces.
5.3.2 Temperature calculation
Finite volume approach is used to discretise the non linear energy equation. The iterative procedure requires for the solution as the energy equation is non linear due to the incorporation of phase change capability, The iteration procedure is first at each time step using VOF method; the velocity for fixed grid is obtained. Those velocities are used in the advection terms of the energy equation to obtain the temperature field for the same fixed grid. The location of the temperature field is at the center of the cell where velocity components are located at the middle of the grid points on the control volume in the staggered grids. VOF method is basically a finite difference method but to handle the donor-acceptor cell approximation a special function of F which results free surface location is used. So to handle both methods a combined single expression with a variable parameter which controls the relative amount of each is applied in the problem. It is shown that the location of the velocity variables in the control volume is same as
VOF method because the VOF method was developed precedes the development of the control
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volume finite difference method. As pressure and temperature goes together, the free surface temperature boundary condition resulting from the gas dynamics and the pressure boundary condition are applied in the free surface. The velocity at the solid-liquid interface is attained by defining the kinematic viscosity as a function of temperature. The value of kinematic viscosity at liquid region is defined as the value of fluid viscosity and then gradually increased through the mushy zone to a large value for the solid region. No slip velocity boundary conditions are applied implicitly at the solid-liquid interface which can be easily implemented in the solution algorithm.
5.4 Results and discussion
The LD model treats the coupled problem consists of convection and conduction heat transfer; phase change processes (melting, solidification and vaporization); time and position dependent temperature and pressure which develops at the melt/vapor interface; and incompressible laminar flow of the melt with a free surface. The computer code has been used to solve two dimensional axisymmetric LD simulation using Hastelloy-X as a target material.
Laser drilling on a Hastelloy-X workpiece is simulated and results are compared with the experimental data and calculated data from 2-D model in [97]. The thermal properties of
Hastelloy-X are given in Table 5-1.
Table 5-1: Thermophysical properties of the Hastelloy-X
Property Symbol Value Thermal conductivity of melt k 21.7 W/m.k
3 3 Density of melt ρm 8.4×10 kg/m
5 Vaporization Pressure, pvap,0 1.013 10 pa
Specific heat of melt cp/c 625 J/Kg.K
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Temp. of vaporization Tv,0 3100 K
Temp. of melt Tm 1510 K 6 Latent heat of vaporization Lv 6.44×10 J/g 5 Latent heat of melt Lm 2.31×10 J/g Molar mass M 76 g/mol Dynamic viscosity η 0.05 g/cm.s
2 Surface tension γST 0.0001 J/cm Prandtl number Pr 0.142 Schmidt number Sc 0.27 Gas constant R 109 J/kg.K Thermal diffusivity of melt ƙ 4.2×10-6 m2/s
The diameter of the laser beam is , which is also the length of the solid in the radial direction ( cells). In addition, there are cells of solid and cells of air (empty) in the axial direction. Therefore, the length of the contour (in the radial direction) is with cells. There are 25 empty cells located on the top of the solid cells. Each cell represents as
by 5.08µm square.
Figure 5-2 shows the fluid contour at different times where the fluid cells are marked by values ranging from 0 to 1. The sequence of fluid contour illustrates the radial movement of the melt caused by the pressure gradient and its ejection.
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(a) (b)
(c) (d)
(e) (f)
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Figure 5-2 Comparison of the fluid contour of the literature (top) and the current result (bottom) at different time sequence
5.4.1 Effects of laser beam diameter
Starting from the case discussed in Fig. 5-3, we considered several additional cases by changing the beam diameter from 508µm with the same Imax and some cases with the same beam diameter but changing the laser intensity for study the effect of beam size, laser intensity. Figure 5-3 shows fluid contour at different time sequence for the original case (d=508 µm and Imax= 1
MW/cm2). Figure 5-4 shows the temperature contour for the original case. Figure 5-5 and 5-6 represent the fluid and temperature contours for the case with laser diameter of 1.5 mm (3 times
2 to the original diameter) and the same maximum laser intensity (Imax= 1 MW/cm It is shown
t=0µs t=40µs
t=50µs t=60µs
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t=90µs t=210µs
Figure 5-3 Fluid contour at (a) 0; (b)40 ; (c) 50; (d) 60; (e) 90 and (f) 210 µs with ,
R= 508µm and
t=0µs t=40µs
t=50µs t=60µs
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t=90µs t=210µs Figure 5-4 Temperature contours at (a) 0; (b)40 ; (c) 50; (d) 60; (e) 90 and (f) 210 µs with
, R= 508µm and the figure 5-3 and 5-5 that the ablation effect decreases with the increase of the laser diameter under constant laser intensity and laser pulse. Although a deeper hole should observed due to the higher the laser power for the beam diameter D=1.5mm than the of the beam diameter D=508
µm, we found a shallow depth for the increased beam diameter. The reason behind is that under the same laser pulse width and the laser intensity, the increase of beam diameter results increased vaporization rate and then a thin layer of molten layer appeared. Another reason should be the validity of the application of Clausius/Clapeyron equation in this model. Under high pressure and near the critical point, Clausius/Clapeyron equation will give inaccurate results.
t=0µs t=40µs
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t=50µs t=60µs
t=90µs t=210µs
Figure 5-5 Fluid contour at (a) 0; (b)40 ; (c) 50; (d) 60; (e) 90 and (f) 210 µs with ,
R= 1.5 mm and
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t=0µs t=40µs
t=50µs t=60µs
t=90µs t=210µs
Figure 5-6 Temperature contours at (a) 0; (b)40 ; (c) 50; (d) 60; (e) 90 and (f) 210 µs with
, R= 1.5 mm and 5.4.2Effects of laser pulse
The effects of laser pulse are presented and discussed in this section. Fluid and temperature contours are shown in Figs.5-7 and 5-8 for the pulse duration of 105µs and maximum intensity of 2MW/cm2 with original beam diameter 508µm. It is shown from the figures that the
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penetration decreases as the pulse duration decreases. Figures 5-9 and 5-10 represent the fluid and temperature contours for the case with pulse duration of 52.5µs and maximum intensity of
4MW/cm2. Comparing the fluid contour plots in Figs 5-7 and 5-9, it is shown that the hole diameter decreases with the decrease of laser pulse.
Figure 5-7 Fluid contour at (a) 0; (b)20 ; (c) 25; (d) 30; (e) 45 and (f) 105 µs with ,
and
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Figure 5-8 Temperature contours at (a) 0; (b)20 ; (c) 25; (d) 30; (e) 45 and (f) 105 µs with
, and
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Figure 5-9 Fluid contour at (a) 0; (b)10 ; (c) 12.5; (d) 15; (e) 22.5 and (f) 52.5 µs with
, and
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Figure 5-10 Temperature contours at (a) 0; (b)10 ; (c) 12.5; (d) 15; (e) 22.5 and (f) 52.5 µs with
, and
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5.5 Conclusion
The generalized thermal laser drilling problem is compared with the literature to verify the code.
It is found out that the result shows a good agreement with the result present in the paper [18].
After verification of the code, the model is applied to study the effect of laser parameters like laser pulse width and beam diameter. The cases where the laser diameter changed from the original case (d=508µm) with the same maximum laser intensity are studied. It is shown that the hole depth increase with the decrease of beam diameter. The pulse duration effects with different laser intensities are also studied here. The pulse duration study concludes that when the laser pulse duration increases, the depth of the hole increases.
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CHAPTER 6
Uncertainty analysis of melting and resolidification of gold film irradiated by nano-to femtosecond lasers using Stochastic method
6.1 INTRODUCTION
At micro and nanoscales, ultra-fast laser material processing is a very important part in fabrication of some devices. Conventional theories established on the macroscopic level, such as heat diffusion assuming Fourier’s law, are not applicable for the microscopic condition because they describe macroscopic behavior averaged over many grains [19]. For ultrashort laser pulses, the laser intensities can be high as 1012 W/m2 or even higher up to 1021 W/m2. During the laser interaction with materials, those electrons in the range of laser penetration of a metal material absorb the energy from the laser light and move with the velocity of ballistic motion. The hot electrons diffuse their thermal energy into the deeper part of the electron gas at a speed much slower than that of the ballistic motion. Due to the electron-lattice coupling, heat transfer to the lattice also occurs and a nonequilibrium thermal condition exists [101]. The nonequilibrium of electrons and the lattice are often described by two-temperature models by neglecting heat diffusion in the lattice [102, 103]. The accurate thermal response is only possible when the lattice conduction is taken into account in the physical model, particularly in the cases with phase change. Chen and Beraun [104] proposed a dual hyperbolic model which considered the heat conduction in the lattice.
In the physical process, melting in the lattice could take place for laser heating at high influence. When the lattice is cooled, the liquid turns to solid via resolidification. The solid will be superheated in the melting stage, and the liquid will be undercooled in the resolidification stage. When the phase change occurs in a superheated solid or in an undercooled liquid, the
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solid-liquid interface can move at a very high velocity. Kuo and Qiu [105] investigated picoseconds laser melting of metal films using the dual-parabolic two-temperature model.
Chowdhury and Xu [106] modeled melting and evaporation of gold film induced by a femtosecond laser. During the melting stage, the solid is superheated to above the normal melting temperature. During resolidification, the liquid is undercooled by conduction and the solid-liquid interface temperature can be below melting point. The solid-liquid interface can move at a high velocity which implies that the phase change is controlled by the nucleation dynamics, rather than energy balance [107].
In the melting and resolidification model of metal under pico- to femtosecond laser heating, the energy equation for electrons was solved using a semi-implicit scheme, while the energy and phase change equations for lattice were solved using an explicit enthalpy model [105,106]. The explicit scheme is easier to implement numerically than the implicit scheme for the enthalpy model [108]. Zhang and Chen proposed a fixed grid interfacial method [109] and an interfacial tracking method [110] to solve rapid melting and resolidification during ultrafast short-pulse laser interaction with metal films. A nonlinear electron heat capacity obtained by Jiang and Tsai
[111, 112] and a temperature-dependent coupling factor based on a phenomenological model
[113] were employed in the two-temperature modeling [110]. The results showed that a strong electron-lattice coupling factor results in a higher lattice temperature which results a more rapid melting and longer duration of phase change.
Although the modeling of melting and resolidification of metal has significantly advanced in recent years, the inherent uncertainties of the input parameters can directly cause unstable characteristics of the output results. Among them, the laser fluence and pulse duration may fluctuate during the process. Moreover, the thermophysical properties of electrons and the lattice
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are not accurately determined at high temperatures. For example, the electron phonon coupling factor cannot be small but have a certain value instead [114]. These parametric uncertainties may influence the characteristics of the phase change processes (melting and resolidification) which will affect the predictions of interfacial location, temperature and velocity and also the electron temperature. In the selective laser sintering (SLS), the fluence and width of laser pulses and the size of metal powder particles may influence the characteristics of the final product [115-119].
Therefore, study of parametric uncertainty is vital in simulation of the phase change of metal particles under nano- to femtosecond laser heating.
Sample-based stochastic model has been proposed to analyze the effects of the uncertainty of the parameters in order to integrate the parametric uncertainty distribution. Stochastic models possess some inherent randomness where the same set of parameter values and initial condition will lead to ensemble of different outputs. The stochastic model was applied on the nonisothermal filling process to investigate the effect of the uncertainty of parameters [120]. An improved simulation stochastic model was used in the ASPEN process simulator by Diwekar and
Ruben [121]. The applications of the stochastic model in optical fiber drawing process [122,
123], thermosetting-matrix composite fabrication [124], sheetpile cofferdam design [125] and proton change membrane (PEM) fuel cells [126] were found in the open literature. The sample– based stochastic model was applied to study the phase change of metal particle under uncertainty of particle size, laser properties and initial temperature to investigate the influences of the output parameters in the solid-liquid-vapor phase change of metal under nanosecond laser heating
[127]. Convergence of variance (COV) was used to characterize the variability of the input parameters where the interquartile range (IQR) was used to measure the uncertainty of the output parameters.
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In this paper, the sample-based stochastic model will be applied to study the melting and resolidification of gold film irradiated by nano to femtosecond laser under certain electron- phonon coupling factor, laser fluence, laser pulse width and constants for electron thermal conductivity to reveal the different influences of those parameters in the interfacial location, interfacial velocity, and interfacial and electron temperatures.
6.2 PHYSICAL MODEL
A gold film with a thickness L and an initial temperature Ti is subjected to a laser pulse with a FWHM pulse width tp and fluence J from the left hand surface. The energy equations of the free electrons and the lattice are:
TT Cee()() k G T T S' (6.1) et x e x e l
TT Cll()() k G T T (6.2) lt x l x e l where C represents heat capacity, k is thermal conductivity, G is electron-lattice coupling factor and T is temperature. The heat capacity of electrons expressed as below is only valid for Te <
0.1TF with TF denoting Fermi temperature,
CBTe e e (6.3)
where Be is a constant. According to Chen et al. [128], the electron heat capacity can be approximated by the following relationship:
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BT, ee TF Te 2BTC' 2 e e e , TT 33 FFT 3 22e Ce ' (6.4) Ce NkB , T 3 3 F TT 2 eF 3Nk B 2 TTeF where
NkB BTeF 3 2 ' TT2 CBTFF () (6.5) e e22 e TF TF 2
The bulk thermal conductivity of metal at equilibrium can be represented as
keq k e k l (6.6)
At the nonequilibrium condition the thermal conductivity of electrons depends on both electron and lattice temperatures. For a wide range of electron temperature ranging from room temperature, the thermal conductivity of electron can be measured as follows:
(2 0.16) 5/4 ( 2 0.44) k e e e e 2 1/2 2 (6.7) (e 0.092) ( e l )
Te Tl where e and l are dimensionless temperature parameters and and are the two TF TF constants for the thermal conductivity of electrons. In general the values of those two constants for gold are = 353 W/mK and = 0.16. For the low electron temperature limit (ϑe << 1), the electron thermal conductivity can be expressed as
Te kke eq () (6.8) Tl
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Under high energy laser heating, the electron and lattice temperatures change significantly which results in a temperature-dependent coupling factor in the ultra-fast laser heating. Chen et al. [113] proposed a relationship between electron and lattice temperatures for the coupling factor as follows:
Ae GGTTRT[ ( e l ) 1] (6.9) Bl
where Ae and Bl are two material constants for the electron relaxation time; GRT is the room temperature coupling factor.
The heat source term in Eq. (1) can be represented as
1 R x t SJ'20.94 exp 2.77( ) (6.10) L/()b t tepb( )[1 ] bp
where R is reflectivity of the film, J is the laser fluence, δ is the optical penetration depth, and δb is the ballistic range. At equilibrium, the bulk thermal conductivity of metal is measured as the summation of the electron thermal conductivity (ke) and lattice thermal (kl) conductivity. Free electrons are dominated in the heat conduction as the conduction mechanism is defined by the diffusion of free electron. So, for gold, the lattice and electron thermal conductivities are taken as
1% and 99% of the bulk thermal conductivity, respectively [129].
The energy balance at the solid-liquid interface in the system is given as
TT kl,, s k l h u (6.11) l,, sxx l m s
108
where is the mass density of liquid, hm is the latent heat of fusion and us is the solid-liquid interfacial velocity. For a metal under superheating the velocity of solid–liquid interface is expressed as follows:
hm TTl, I m uVs 0[1 exp( )] (6.12) RTTg m l, I
where V0 is the maximum interfacial velocity, Tl,I is the interfacial temperature and Rg is the gas constant. The interfacial temperature could be higher than the normal melting temperature during melting and lower during solidification. The boundary conditions are given as
TTTT e e l l 0 (6.13) xx00 x x L x x x x L
The initial temperature conditions are
Te( x , 2 t p ) T l ( x , 2 t p ) T i (6.14)
The total computational domain is discretized with non-uniform grids. The implicit finite- difference equations are solved in each of the control volume (CV) and time step. The numerical solution starts from time -2tp. During the solving process, the lattice temperature is set as interfacial temperature for that control volume that contains solid-liquid interface location. The energy equation in terms of enthalpy form is applied and solved for the solid liquid interface CV.
The relationship of interfacial temperature and liquid fraction can be written by
T f T C()()() TlI, h kl G T T (6.15) l,, s l It m t x l x e l
109
where TlI, is the interfacial temperature, Cls, is the heat capacity at solid-liquid interface, and f is the liquid fraction in the system. The liquid fraction is related to the location of the solid-liquid interface [110]. Before onset of melting, Eqs. (1) and (2) are solved simultaneously to obtain electron and lattice temperatures until the lattice temperature exceeds the melting point. Once it exceeds, the lattice temperature is set as the melting temperature and phase change will be considered in the system. After melting starts, an iterative procedure is applied to find the interfacial temperature and the interfacial location at each time step [109].
6.3 STOCHASTIC MODELING OF UNCERTAINTY
Stochastic modeling is a process where the variability of the output parameters is evaluated based on the different combination of the input parameters [127]. In this paper, a sample-based stochastic model is used to study the melting and resolidification of the gold film under uncertain laser fluence, pulse width, coupling factor, and thermal conductivity of electrons to show the effects of the output parameters such as interfacial location, interfacial temperature, interfacial velocity and electron temperature. Figure 6-1 shows the detailed procedure of stochastic modeling.
Fig. 6-1 Sample-based stochastic model
110
In the stochastic modeling process, the first need is to quantify the degree to which the input parameters vary, and then to determine the appropriate number of combination of the input parameters to use with a stochastic convergence analysis. After determining the number of combination of input parameters, one need to calculate the uncertainties of the input parameters though the deterministic physical model that was previously established. Eventually, the variability of the output parameters is quantified based on the uncertainty of input parameters.
The coupling factor at room temperature between electron and lattice (GRT), laser fluence (J), electron thermal conductivity constants ( and η), and laser pulse duration (tp) are the input parameters whose uncertainties are going to be investigated. Due to the unavailability of experimental distribution of those uncertain parameters, it is acceptable to assume that all the input parameters follow Gaussian distributions of uncertainty [122]. The Gaussian distribution is defined by a mean value (µ) and a standard deviation (σ), where the mean value is expressed by the nominal value of uncertainty parameters and the standard deviation represents the uncertainty of the input parameters. The coefficient of variance (COV) is an important parameter which represents the degree of uncertainty of the input parameters. The COV is defined as
COV (6.16)
After determining the distributions of the input parameters, a commonly used sampling method called Monte Carlo Sampling (MCS) is used to obtain the combination of the input parameters. According to the MCS input parameters are randomly selected from their prescribed
Gaussian distributions and combined them together as one sample. Due to the high dependency on the number of the samples of input parameters on the variability of the output parameters, the exact number of samples of input parameters is determined carefully. In the stochastic
111
convergence process, when the number of the sample increases the mean value and the standard deviation of input parameters converge to the nominal mean value and standard deviation of the
Gaussian distribution. The mean value and standard deviation of the output parameters will also converge within a certain tolerance. After selecting the required number of samples for each input parameter, the physical model of melting and resolidification of gold film is solved. The effects of the input parameters variability on the output parameters uncertainty is evaluated by obtaining each output parameter’s set. The output parameters in this paper include interfacial location (s), interfacial temperature (Tl,I), interfacial velocity (us) and electron temperature (Te).
The probability distribution is calculated from the resulting set of the output parameters. The interquartile range (IQR) is a measurement of variability, based on dividing a data set into quartiles. It is defined as the difference between the 25th percentile and the 75th percentile,
IQR P75 P 25 (6.17)
6.4 RESULTS AND DISCUSSIONS
3 7 The thermophysical and optical properties of pure gold film are: Be = 70 J/m K, Ae =1.2×10
-2 -1 11 -1 -1 16 3 16 3 K s and Bl =1.23×10 K s , GRT = 2.2×10 W/m K (solid) and 2.6×10 W/m K (liquid), ρ
3 3 3 3 =19.30×10 kg/m (solid) and 17.28×10 kg/m (liquid) reflectivity, R = 0.6, δ = 20.6 nm, δb
4 4 =105 nm, Tm = 1336 K, TF = 6.42×10 K, hm = 6.373×10 J/kg, and V0 =1300 m/s. The sample- based stochastic model provides the output parameter distributions with respect to the uncertain input
112
(a)
(b)
113
(c)
(d)
114
(e)
Fig. 6-2 Stochastic convergence analysis of mean value of the input parameters (a) GRT, (b) , (c)
η, (d) J and (e) tp parameter distributions. A large number of input samples is required to get the real distribution of the output parameters. Due to the difficulty in prohibitively intensive computation, it is important to find a minimum number of input samples (N) with which steady necessary output distributions can be generated.
To find the required number of N, we assume the nominal mean values of GRT, , η, J and tp are
2.2×1016 W/m3K, 353 W/mK, 0.16, 0.3 J/cm2 and 20 ps, respectively. The coefficient of variance
(COV) of each input parameter is set to be 0.02. Figure 6-2 represents the stochastic convergence analysis of the mean value of the input parameters GRT, , η, J and tp. It is shown from this figure that when the number of samples is small, the mean values of the input parameters fluctuate significantly. For the value N = 200, the mean values of the input parameters oscillate in a smaller range, suggesting that a total of 200 samples should be sufficient for steady nominal
115
mean values of input parameters. Figure 6-3 represents the stochastic convergence analysis of standard deviation of the five input parameters. It is shown that although the mean values of input parameters converges for 200 samples, the standard deviation still fluctuate. The reason behind this is that the deviation is a higher order moment which allows converging slower than the mean value. From Figure 6-3, it may conclude that the minimum number of the input samples is 300.
After determining the minimum number of input samples, the stochastic convergence analysis for the mean value and standard deviation of the output parameters are obtained, as shown in
Figures 6-4 and 6-5. It can be seen that when the number of the samples is beyond 300, the mean values of all the output parameters fluctuate in a smaller range (2.5%). Therefore, the minimum number of samples N = 400 is selected and used to calculate the results.
(a)
116
(b)
(c)
117
(d)
(e)
Fig. 6-3 Stochastic convergence analysis of standard deviation of the input parameters (a) GRT,
(b) , (c) η, (d) J and (e) tp
118
(a)
(b)
119
(c)
(d)
Fig. 6-4 Stochastic convergence analysis of mean value of the output parameters (a) s, (b) us, (c)
Tl,I and (d) Te
120
(a)
(b)
121
(c)
(d)
Fig. 6-5 Stochastic convergence analysis of standard deviation of the output parameters (a) s, (b) us, (c) Tl,I and (d) Te
Figure 6-6 shows the typical distributions of the input parameters with the nominal mean
16 3 2 values of GRT, , η, J and tp being 2.2×10 W/m K, 353 W/mK, 0.16, 0.3 J/cm and 20ps respectively and the COV of each parameter being 0.02. Figure 6-7 gives the typical distribution
122
of the output parameters s, us, Tl,I and Te. In the histograms, the distributions of the output parameters are no longer Gaussian due to the nonlinear effect in the solid liquid interface.
The IQRs of the output parameters s, us, Tl,I and Te as functions of COV of the input parameters GRT, , η, J and tp are shown in the Fig. 6-8. When the COV of the one input parameter increases from 0.01 to 0.03 and the COVs of the other input parameters are kept constant at 0.01, the effect of that input parameter can be manifested. In the IQR analysis of the interfacial location (s), the IQR of s significantly increases from 1.5 nm to 4.5 nm when the COV of J increases from 0.01 to 0.03, from 1.5 nm to 2.75 nm with the change of COV of J, and from
1.5 nm to 2.5 nm with the change of COV of tp. On the contrary, the COV of the thermal conductivity constants is relatively less impact to the interfacial location.
(a)
123
(b)
(c)
124
(d)
(e)
Fig. 6-6 Typical distributions of the input parameters (a) GRT, (b) , (c) η, (d) J and (e) tp
125
(a)
(b)
126
(c)
(d)
Fig. 6-7 Typical distributions of the output parameters (a) s, (b) us, (c) Tl,I and (d) Te
127
(a)
(b)
128
(c)
(d)
Fig. 6-8 The IQRs of the output parameters with different COVs of the input parameters (a) s, (b) us, (c) Tl,I and (d) Te
The IQR analysis of the interfacial velocity (us) shows that the laser influence J is also most influential among the five input parameters. With the increment of COV of J from 0.01 to 0.03, the IQR of us increases from 8.8 m/s to 23.1 m/s. As shown in Fig. 6-8, the order of influence of
129
the COV of the five input parameters on the output parameters are J, GRT, tp, , and η. Figure 6-9 represents the IQRs of s, us, Tl,I and Te for different laser influences with different COVs. As previously described, the COV of J varies from 0.01 to 0.03 while the COVs of other parameters remain the same. It can be seen from Fig. 6-9 shows that for each laser influence the COV of J significantly affects the IQR of the output parameters. The larger the COV is, the more the IQR increases. Figure 6-10 represents the IQRs of s, us, Tl,I and Te at different electron-lattice
16 16 16 coupling factor (GRT) with different COVs. Three values of GRT, 2.1×10 , 2.2×10 and 2.3×10
W/cm3K, are considered with the COV ranging from 0.01 to 0.03, and the COVs of the other parameter remains the same. It is shown in Fig. 6-10 that for each GRT, its COV significantly affects in the IQR of the output parameters. The IQRs of s, us, Tl,I and Te increase as the COV increases. The reason is that with the increase of the electron-phonon coupling factor, the hot electron heated up faster than metal lattice, leading to a more severe superheating process.
Figures 6-9 and 6-10 indicate that the interfacial location, velocity and temperature and electron temperature greatly depends on the energy of laser and phonon-electron coupling factor.
(a)
130
(b)
(c)
131
(d)
Fig. 6-9 The IQRs of the output parameters with different values and COVs of J (a) s, (b) us, (c)
Tl,I and (d) Te
(a)
132
(b)
(c)
133
(d)
Fig. 6-10 The IQRs of the output parameters with different values and COVs of GRT (a) s, (b) us,
(c) Tl,I and (d) Te
6.5 CONCLUSION
The sample-based stochastic model was applied to analysis the influence of parametric uncertainty on melting and resolidification of gold film subjected to nano- to femtosecond laser irradiation. Rapid solid-liquid phase change was modeled using a two-temperature model with an interfacial tracking method. Temperature dependent electron heat capacity, thermal conductivity, and electron-lattice coupling factor were considered. The uncertainties of laser pulse fluence, pulse duration, electron-lattice coupling factor, and electron thermal conductivity on the results of solid-liquid interface temperature, interfacial velocity and location, and electron temperature were studied. The results show that the mean value and the standard deviation of laser influence and electron-lattice coupling factor have dominant effects on rapid phase change.
134
CHAPTER 7
7. CONCLUSIONS
Due to the unique characteristics like coherency and collimation, laser has been widely used in the various fields of science, medical and military. Laser cutting, drilling and printing are most popular applications of the laser in science. High energy laser provides naval platforms with highly effective and affordable defense capability. Appropriately developed and applied high energy laser systems can become key contribution in 21st century. High energy lasers have two characteristics that make them particularly valuable for effects-based application: they are extremely fast and extremely precise. The industrial implementation of laser ablation, cutting and drilling by use of ultrafast pulse has been a vision for more than 20 years. Absorption of a laser pulse in metals is basically an energy transfer from the laser pulse to the electrons of the material. The shorter the laser pulse, the faster the energy transfer to the electrons. For the short pulse laser, there is not enough time for temperature equilibrium condition for electron and lattice. After a characteristic time, the hot electron diffusion occurs to the surrounding lattice begins. At the same time scale but little bit delayed, a abrupt energy transfer between the hot electrons and lattice takes place which results a phase explosion. So the fundamental conclusion can be done that the duration of laser pulse must be short enough to prevent the temperature equalization between electron and lattice.
From macroscopic point of view, the process of heat transfer is governed by phonon-electron interaction in metallic films and by phonon scattering in dielectric film, insulator and semiconductor. Conventional theory established for macroscopic point of view is not applicable for microscopic view such as Fourier’ law. Fouriers’ law of heat conduction dictates immediate
135
response which results in an infinite speed of heat propagation. The Dual phase lag model aims is to remove the precedence assumption made in the thermal wave model. In porous medium like
SALDVI which is a fabrication process where the pore spaces of the layers are densified by the infiltration of the material from the gas precursor, the DPL model is the appropriate model to investigate the thermal behavior of the process and control the density of the output product.
Laser cutting is one of the most important applications of the high energy ultrafast laser. This process not only considered the thermal transport across the object but also it is more important to study the change of material thermophysical properties, phase change of melting and vaporization, effect of the chemical reaction in the material due to the reactant and product generation due to the chemical reaction in the continuum domain nearby the irradiated spot.
Laser drilling is another most important application of ultrafast laser. Laser drilling process requires clear understanding of fundamental physics for better control on the hole diameter and increasing the efficiency of the process. Sometimes it is difficult to attain small and accurate diameter of the holes on the workpiece.
At micro and nanoscales, ultra-fast laser material processing is a very important part in fabrication of some devices. The variation of the input parameters may effects the resulting output parameters in the fabrication process.
High energy application in microscopic level is the most interest area recent a days. In future, one direction of application of the ultrafast high energy laser in microscopic level would be the fabrication of ceramic and ceramic/metal/composite structures and developing 3D nano- structuring by polymerization, application of complex polarized shaped pulse, coherent control to control the chemical reaction introduced by ultrafast laser. Inverse heat transfer problems are
136
very important in science like microelectronics for the thermal testability of the integrated circuits. Grinding is another section where prediction of grinding temperature requires a detailed knowledge about the heat flux to the workpiece at the grinding zone. The inverse heat transfer is suitable analysis that can also be applied to control the motion of the solid liquid interface during solidification. Ultrafast may also be used for stimulated emission depletion microscopy which is one of the techniques that make super resolution microscopy.
137
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VITA
Nazia Afrin was born in Dhaka, Bangladesh. She completed her schoolwork in 1999 at Kamrun nesa Govt Girls’ High School in Dhaka, high school in 2001 at Ideal School and College,
Bangladesh and started her undergraduate studies in 2003 at the Bangladesh University of
Engineering and Technology (BUET), which is the most prestigious engineering university in
Bangladesh. Nazia received her Bachelor of Science in Mechanical Engineering from BUET in
2008 and started working as an Assistant Manager in stainless steel and water purification company “Prodhan Polymers Ltd”. She moved in Columbia, Missouri in 2009 and enrolled herself in the University of Missouri, Columbia for graduate studies. She completed her Master of Science (MSc.) and Doctor of Philosophy (PhD) in Mechanical and Aerospace Engineering in
2011 and 2015, respectively. In her doctoral research, she worked with high-energy laser materials interaction.
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