thermal effects of high energy and ultrafast lasers
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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.
ii
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.
iii
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
iv
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
v
REFERENCES ………………………………………………………………………….……. 138
VITA……………………………………………………………………………………………153
vi
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 =
1.25×105 J/m
2, tp = 100 ns, dp = 15µm (τT = τq =3.9 ns): (a) t/tp < 1 and (b) t/tp > 1 …….….15
Fig. 2-3 Temperature distribution over the powder layer with J = 1.25×105 J/m
2, 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
Fig. 2-6 Effects of porosity (φ) on the temperature of powder layer with J = 1.25×105 J/m
2, tp = 1
ns, and dp = 15 µm: (a) t/tp < 1 and (b) t/tp > 1………………………………………………..22
Fig. 2-7 Effects of pulse width (tp) on the maximum temperature of powder layer (J=1.25×105
J/m2)…………………………………………………………………………………………...…23
Fig. 3-1 Relationship between T and for stainless steel with 318rT 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
vii
Fig. 3-6 Comparison of CGM and DCT/Laplace transformation solutions of front surface
temperature vs time at three sensor locations (center (19
40
YLY ,
19
40
zLZ ), two off centers (
25
40
YLY ,
19
40
zLZ and (
29
40
YLY ,
19
40
zLZ )) …………………………………..……….…42
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
viii
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
ix
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
x
LIST OF TABLES
Table: 2-1 Phase lag times for different particle diameter and porosity …………………....17
Table 4-1 Initial mass diffusivity between gas species D0
ij (m2/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
xi
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
Di Mass diffusion coefficient for species i in the mixture (m2/s)
Dij Mass diffusivity coefficient between species i and j
D0
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)
xii
Fip Force acting on a single particle in a parcel located in the i
th cell
f normalized heat flux, K
G coupling factor, W/m3 K
G Gravity force (N)
GRT Electron-lattice coupling factor (W/m3K)
g Gravitational acceleration (9.8 m/s2)
h Enthalpy(J/kg)
hm Latent heat of fusion (J/kg)
hp heat transfer coefficient at the powder particle surface, W/m2 K
h0 heat transfer coefficient at the powder bed surface, W/m2 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),
kc Chemical reaction rate constant (s-1
),
kd Node diffusion coefficient
k thermal conductivity, W/m K
xiii
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
Nip Number of particles in a parcel located in the i
th cell
Nu Nusselt number
p Pressure (Pa)
P Laser power (W)
xiv
,0vapp 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
xv
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
xvi
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
ΔVrem Volume removed (m3)
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)
xvii
σrad Stefan-Boltzmann constant (W/m2K
4)
ψ Compressibility(s2/m
2)
η Dynamic viscosity (g/cm.s)
γST Surface tension (J/cm2)
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
xviii
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
xix
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.
1
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
2
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
3
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.
4
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.
5
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
6
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.
7
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.
8
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:
(1 )( ) ( ) ( )
s sp s seff s g
T Tc k S G T T
t x x (2.1)
( ) ( )g
p g s g
Tc G T T
t
(2.2)
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):
L
x
0
tp/2 -tp/2
Adiabatic
9
2
1( , ) 0.94 ( )
p
p
a t tx
t
p
RS x t J e
t
(2.3)
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
2 2 2
1 1( )
qs s s sT q
seff
T T T TSS
x x t k t t t (2.4)
where the phase lag times of the heat flux vector and temperature gradient are:
g
T
C
G
,
( )
s g
q
s g
C C
G C C
(2.5)
and
(1 )s s psC c , g g pgC c (2.6)
Cs and Cg are effective heat capacities of the solid and gas phases, respectively. The effective
thermal diffusivity is
seff
s g
k
C C
(2.7)
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, 0T q .
In the absence of the phase lag of the temperature gradient, 0T , it reduces to the CV wave
model.
10
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 ) p p
p
A hG
V
(2.8)
where (1 ) p
p
A
V
represents the specific interfacial area(m
2/m
3). If the porous particle is spherical
in shape,the surface-area-to-volume ratio becomes 6(1 ) p
p
h
V
. The coupling factor becomes
6(1 ) p
p
hG
d
(2.9)
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]
2p p
g
h dNu
k (2.10)
Substituting Eqs. (6) and (9)-(10) into Eq. (5), the phase lag times become:
2
6(1 )
g pg p
T
g
c d
Nuk
,
2
6 [(1 ) / ( )]
g pg p
q
g g pg s ps
c d
Nuk c c (2.11)
The initial conditions for temperature of the powder particle are assume given below
( , )s iT x T (2.12)
11
( ,0) 0sTx
x
(2.13)
The boundary conditions are
0
0
( )sseff s
x
Tk h T T
x
(2.14)
0s
x L
T
x
(2.15)
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
0
( , ) ( , ) ptT x p T x t e dt
(2.16)
Applying the above Laplace Transform to Eqs. (4), (15) and (16), the following second ordinary
differential equation and boundary conditions can be obtained
2 22
2 2
1 1( )
qs sT q s s
seff
T Tp S pS pT p T
x x k
(2.17)
12
0
0
1( )s
seff s
x
Tk h T
x p
(2.18)
0s
x L
T
x
(2.19)
The solution of Eq. (17) with its associated boundary conditions, Eqs. (18) and (19), can be
expressed as
1 2 3( , )x
Dx Dx dsT x p Ae A e A e
(2.20)
where A1, A2, and A3 are given by:
1( )
23 0 2 0 0 2 3 0 2 1
1 2 1 320
2
( ) ( ) ( ), ,
1( )
D Lad
seff seff DL b
seff
A k dh A d Dk h dh T A e S C e C SA A Ae A
d h Dk dDD
d
2 22
1 2
(1 ) 1[ ], , ,
(1 ) (1 ) (1 )
p ppt ptaq q q
b p
p p T seff T seff T
p p pe e eS t D C C
pt a pt a p k p k p
(2.21)
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
13
variable transformation from p (complex) to ω (real), Riemann sum approximation of the
Laplace inversion can be expressed as
1
1( , ) [ ( , ) Re ( , )( 1) ]
2
t Nn
s s s
n
e inT x t T x T x
t t
(2.22)
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
4.7t (2.23)
Substituting ( , )sT 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
SiC are: 33.21 10s kg/m
3, 660psc J/kg K, ks = 58.86 W/m-K. The effective thermal
conductivity for the solid phase is (1 )seff sk k . 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
is 3,438pgc /kg K at 1,273K [22]. The density of the TMS is ρg = 0.045 kg/m3. 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 /100 pt t until t = 5tp. The time step is changed to pt t for 5tp < t <
100tp; after t > 100tp, the time step is further increased to 100 pt t . The initial temperature of
14
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
(x = 0) and adiabatic surface (x = L) for J = 1.25×105 J/m
2, 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
heating surface temperature at t/tp> 1×106, which means that the entire powder layer temperature
becomes uniform after this time.
15
(a)
(b)
Figure 2-2 Powder temperature (Ts) at the heating surface and the adiabatic surface with J =
1.25×105 J/m
2, tp = 100 ns, dp = 15µm (τT = τq =3.9 ns): (a) t/tp < 1 and (b) t/tp > 1
16
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.
Figure 2-3 Temperature distribution over the powder layer with J = 1.25×10
5 J/m
2, 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
17
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
18
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
temperature starts decreasing from about t/tp > 103 while it decrease from about t/tp > 10
2 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)
19
(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/m
2 to 5×10
5 J/m
2, the peak powder temperature increase
from 1650 K to 3433 K.
20
(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
21
The effects of porosity are shown in Fig. 2-6. The results are obtained for J = 1.5×105 J/m
2.
The increment of porosity results in an increase of heat transfer coefficient as indicated by Eq.
(8); consequently, the powder temperature is increased.
(a)
22
(b)
Figure 2-6 Effects of porosity (φ) on the temperature of powder layer with J = 1.25×105 J/m
2, 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/m
2) 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.
23
Figure 2-7 Effects of pulse width (tp) on the maximum temperature of powder layer (J=1.25×10
5
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.
24
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.
25
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.
26
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
27
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:
( ) ( ) ( )s s s ss ps s s s
T T T Tc k k k
t x x y y z z
(3.1)
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:
28
0sT
x
for x =0 and x =lx (3.2)
0sT
y
for y =0 and y =ly (3.3)
( , , )ss
Tk q x y t
z
for z =0 (3.4)
0sT
z
for z =lz (3.5)
with the dimensionless variables:
t=tc τ, x=lzX, y=lzY, z=lzZ
where
2
s ps z
c
s
c lt
k
and dimensionless heat flux and temperature (K) are:
q x, y, t ( , , )s
z
kf X Y
l (3.6)
x, y, t ( , , , )sT X Y Z (3.7)
The normalized form of Eq. (1) is
2 2 2
2 2 2X Y Z
(3.8)
with boundary conditions:
0X
for X =0 and X =Lx (3.9)
0Y
for Y =0 and Y =Ly (3.10)
0Z
for Z =1 and (3.11)
29
( , , )f X YZ
for Z =0 where xx
z
lL
l and
y
y
z
lL
l (3.12)
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]:
, 0,1,....
( , , , ) ( , )mn
m n x y
m X n YX Y Z Z Cos Cos
L L
(3.13)
, 0,1,....
( , , ) ( )mn
m n x y
m X n Yf X Y f Cos Cos
L L
(3.14)
Substituting Fourier series expansions i.e. Eqs. (13) and (14), one can derive the one
dimensional heat equation as below:
22
20mn mn
mn mncZ
for m, n =0,1,2…. (3.15)
where 2 2 2( ) ( )mn
x y
m nc
L L
(3.16)
and the boundary conditions on the front surface and back surface are:
( )mnmnf
Z
for Z =0 (3.17)
0mn
Z
for Z =1 (3.18)
Here ( , )mn Z is the modal temperature. This equation is similar to the one dimensional heat
conduction problem representing fin with convection [45].
30
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:
2 2 2
2 2 2( )
t x y z
(3.19)
where
α= p
k
c (3.20)
through Kirchhoff transformation:
0
( )
T
k d (3.21)
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( )
r
T
r
r T
T k dk
(3.22)
where Tr is the reference temperature which is determined as follows:
2
0 0 0
1( , , )
ft L L
r
f
T T y z t dydzdtt L
(3.23)
31
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
0 1 2( )k T c c T c T (3.24)
It can be written as
2
1 2( ) ( ) ( )r r rk T k k T T c T T (3.25)
where
1 1 22 rk c c T (3.26)
Substituting Eq. (24) into Eq. (22), we obtain
2 31 21[ ( ) ( ) ]
2 3r r
r
k cT T T T T
k (3.27)
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 rT 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
32
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):
]1)(2
1[ 1
1
r
r
rr T
k
k
k
kTT (3.28)
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]:
2 2
2
2 2 2
1cosh sinh( ,0) ( ,1)
( ,0) ( ,1)sinh cosh
mn mnmn mn
mn
mn mn
mn mn mn
s c s cs ss c
s ss c s c s c
(3.29)
where ( ,0)mn s , ( ,0)mn s , ( ,1)mn s and ( ,1)mn s 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( ,0) cosh ( ,1)mn mn mns s c s (3.30)
33
2 2( ,0) sinh ( ,1)mn mn mn mns s c s c s (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]:
2
2 21
cosh cosh {1 }
[(2 1) ]2
mn mn
kmn
ss c c
c k
(3.32)
2
2 221
sinhsinh [1 ]
( )
mn mn
kmn mnmn
s c c s
c c ks c
(3.33)
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:
2 21
( ,0) cosh [1 ] ( ,1)
[(2 1) ]2
mn mn mn
kmn
ss c s
c k
(3.34)
2
2 21
sinh( ,0) ( ) [1 ] ( ,1)
( )
mnmn mn mn
kmn mn
c ss s c s
c c k
(3.35)
The corresponding equations in the time domain are:
2 21
( ,0) cosh {1 } ( ,1)
[(2 1) ]2
mn mn mn
kmn
s dc
dc k
(3.36)
2
2 21
sinh( ) ( ) [1 ] ( ,1)
( )
mnmn mn mn
kmn mn
c d s df c
c d c k d
(3.37)
34
where ( ,0)mn , ( ,1)mn and ( )mnf 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. ( ,1)mn , 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
thermophysical properties of the target material are used for this analysis: s =7900 kg/ m3, c0 =
10.06453 (W/m K), c1 =0.01719 (W/m K2), c2 =
61.85055 10 (W/m K3). The geometry of the
slab: thickness (lz) = 0.00215m, length (lx) =0.1m and width (ly) =0.1 m. The initial temperature
35
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:
6 2( ) 10.0645 0.01719 1.85055 10k T T 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.
Figure 3-1 Relationship between T and for stainless steel with 318rT K
300 400 500 600 700 800 900 1000
400
600
800
1000
1200
T
36
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
max( , , ) exp{ [( 0.5 ) ( 0.5 ) ] / }(1 sin(2 ))q x y t q x M y N w ft (3.39)
where is surface absorptivity; maxq 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,
maxq = 5000 W/cm2, 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
37
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
38
(c) t =0.85s
(d) t =1.25s
(e) t = 1.55s
39
(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
different group of 20 sensors correspond to a row with coordinate given by ( 1)40 20
z zL LZ row
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
sensors 181-200 with coordinate 19
40
zLZ ; where front surface temperature is minimum at the
edge i.e., sensors location 1-20 with coordinate 40
zLZ .
40
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.
41
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
transform solutions in three different locations (center (19
40
YLY ,
19
40
zLZ ), two off centers (
25
40
YLY ,
19
40
zLZ and (
29 19,
40 40
Y zL LY Z ,
19
40
zLZ ) is shown in Fig.3-6. In this figure, it is
shown that the fluctuations in the DCT solution are in phase with CGM solution, but with
reduced amplitude.
42
Figure 3-6 Comparison of CGM and DCT/Laplace transformation solutions of front surface
temperature vs time at three sensor locations (center (19
40
YLY ,
19
40
zLZ ), two off centers (
25
40
YLY ,
19
40
zLZ and (
29
40
YLY ,
19
40
zLZ ))
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:
RMSDCT/Laplace=2
DCT/Laplace /
1
1RMS ( )
N
DCT Laplace exact
i
T Tn
(3.40)
43
RMSCGM= 2
1
1( )
N
CGM exact
i
T Tn
(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.
44
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,
45
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.
46
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
47
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.
48
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:
0Ut
(4.1)
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:
2
3u
UU U g U U I Sp trace
t
(4.2)
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:
F
S=
p p
p
N
V
(4.3)
49
where Np is the number of total particles in the pth
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:
2 21 1
2 2U U Us s s h
Dph h h S
t Dt
(4.4)
where hs is sensitive enthalpy, is enthalpy-type thermal diffusivity, Dt
Dp is material derivative
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
=
p p
p
h
N h
SV
(4.5)
where hp is the enthalpy transferred between each individual particle in the pth parcel and the surrounding
gaseous phase.
(d) Species concentration equation:
( )Uii i iD
t
(4.6)
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)
50
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
1
1 Ni
i iRT M
(4.8)
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],
sp
hc
T
(4.9)
where the specific heat capacity is estimated through a forth order polynomial function of
temperature [62]:
2 3 4
0 1 2 3 4pc R a a T a T a T a T (4.10)
The temperature in the computational domain is thus corrected through iterative method
according to the hs-T diagram. In addition, the viscosity (μi) of the ith
specie is treated to be
temperature dependent [63]
,
,1 /
s i
i
s i
A T
T T
(4.11)
in which isA , and isT , are both constants for gas in question and they can be found from many
online resources [64].
51
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,
1
Ni
i
i iM
(4.12)
1
Ni
i
i iM
(4.13)
The mass diffusivity of the ith 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
( / )
ii
i ij
j j i
XD
X D
(4.14)
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
20
ijij
TD D
p
(4.15)
where 0
ijD is the mass diffusivity from specie i to j at 300 K and 1 atm. It can be determined by
combining Chapman-Enskog theory and the method introduced Bird et al. [66] as follows:
3 3/2
0
2
1.858 10 1/ 1/
ij
i j
ij
T M MD
p
(4.16)
where σij is the average collision diameter of species i and j, and Ω is a collision integral which is
tabulated in [67].
52
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:
pc T k Tt
(4.17)
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:
3 7 2 2 2 2 24 19 12 14 4C H NO O CO H O NO
(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],
53
2
1/2// 2 E RT
c Ok RT M e
(4.19)
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],
3 7 2 2 2 2 2[ ] [ ] [ ] [ ] [ ]4 4 4 4
19 12 14 4c
C H NO O CO H O NOk
t t t t t
(4.20)
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
54
fixing the maximum and minimum bound. The number of particles in each parcel generated in
each time-step is estimated by
34/
3
/ /
pg pgN
rem rem
bd bd pg pg
r
N V V
(4.21)
where ΔVrem is the volume removed from an individual “injector,” N 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:
2
4
19c O bd
removal
bd
k C t M
d
(4.22)
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.
55
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,
56
2
d
Yk Y
t
(4.23)
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:
a G Fdragm
(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]:
23 24
4F U Udrag c p p p
c
m dC
(4.25)
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]:
57
1.1
22
1 1.257 0.4pd
c
p
C ed
(4.26)
in which λ is the molecular mean free path. The gravitational force is calculated by
1G gp
m
(4.27)
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).
58
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.
59
Table 4-1 Initial mass diffusivity between gas species D0
ij (m2/s)
O2 N2 CO2 H2O NO2
O2 / 2.037×10-3
1.509×10-3
2.769×10-3
1.559×10-3
N2 2.037×10-3
/ 1.499×10-3
2.69×10-3
1.556×10-3
CO2 1.509×10-3
1.499×10-3
/ 2.079×10-3
1.109×10-3
H2O 2.769×10-3
2.69×10-3
2.079×10-3
/ 2.177×10-3
NO2 1.559×10-3
1.556×10-3
1.109×10-3
2.177×10-3
/
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
Conductivity
(W/m K)
Heat capacity
(J/kg K)
Density
(Kg/m3)
Heat of Chemical
Reaction
(J/Kg)
Pigment
Cr2O3 32.94 781 5210 /
Binder
C3H7NO2 0.4 1755 1424 6.28×10
6
60
AISI 304L
Stainless Steel 6 2
7.9318 0.023051
6.4166 10
T
T
5 2
426.7 0.17
5.2 10
T
T
7900 /
61
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)
Pressure (atm) 0p
n
dependent on instant local velocity 1
Temperature (K) 0T
n
coupled 300
O2
0C
n
0.21
N2 0.79
H2O 0
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.
62
Table 4.5 Initial and boundary conditions for the solid domain
Ymin Solid_to_Air Internal
Temperature 0T
n
2 2
20
2
i
2
0
2 U S
S
x z
f f paraR
f
Pe H
rT
y k
300K
63
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.
64
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
65
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
66
(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
67
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
68
(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.
69
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.
70
(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
71
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.
72
(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
73
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.
(a) P = 2.5 kW, O2 mass concentration variation (b) P = 2.5 kW, chemical products’ mass
concentration variation
74
(c) P = 5.0 kW, O2 mass concentration variation (d) P = 5.0 kW, chemical products’ mass
concentration variation
(e) P = 7.5 kW, O2 mass concentration variation (f) P = 7.5 kW, chemical products’ mass concentration
variation
(g) P = 10.0 kW, O2 mass concentration variation (h) P = 10.0 kW, chemical products’ mass
concentration variation
75
(i) P = 12.5 kW, O2 mass concentration variation (j) P = 12.5 kW, chemical products’ mass
concentration variation
(k) P = 15.0 kW, O2 mass concentration variation (l) P = 15.0 kW, chemical products’ mass
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.
76
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.
77
(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
78
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.
79
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.
80
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
81
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-
82
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.
83
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)
0U V U
R R R
(5.1)
84
2 2
2 2 2
1Pr[ ]
U U U U U U U PU V
T R Z R Z R R R R
(5.2)
2 2
2 2
1Pr[ ]
V V V V V V PU V G
T R Z R Z R R R
(5.3)
where non-dimensional variables are
3
0 0 0
2
1 1 1
, ,ud vd gd
U V G
2
0 1
2 2
1 0 0 1 0
, , ,Pr ,pd tr z
P R Z tp d d d
(5.4)
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
0F F F
U Vt r z
(5.5)
5.2.2 Heat transfer
85
The nondimensional thermal energy equation in cylindrical coordinate system is
( ) ( ) ( ) 1( ) ( ) ( )
CT uCT vCT T TK K KT B
t r z r r z z r r
(5.6)
where
( )1 1
( ) (1 ) ( )2 2
( )1
sl
sl
CT T
C T C T T TSte T
T T
( ) ( )[ ]
S uS vSB
t r z
( )1 1
( ) (1 ) ( )2 2
( )1
sl
sl
sl
C TT T
S T C T T T TSte
T T
C TSte
( )(1 )( )
( ) ( )2
( )1
sl
slsl
KT T
K T TK T K T T T
TT T
(5.7)
The nondimensional variables are
0 0 0 0* * *
0 0 0 0
1 1
0 0
1
1
' , , ,( )
( ), ,
m
h c h c
h c s ssl sl
l
T T S C kT S C K
T T c T T c k
c T T c kSte C K
L c k
(5.8)
5.2.3 Optical considerations
86
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
2 2
0 2 2
0 0
( ) ( )( , ) exp[ ]exp[ ]n ct t r r
I r t It r
(5.9)
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
0 0
2 ( , )
t R
E dt rdrI r t (5.10)
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.
87
The mass, momentum and energy balance across the melt/vapor interface with respect to a
moving frame can be written as follows
m m v vu u (5.11)
2 2
m m m v v vp u p u (5.12)
/
0abs v v v
s melt
TI L u k
n
(5.13)
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
,0
,0
1 1( ) exp[ ( )]v
S vap
vap S
Lp T p
R T T (5.14)
v v vp R T (5.15)
Applying ideal gas law in the combined equation of the energy equation (13), Clausius-
Clapeyron equation (5.14), we get
,0
/ ,0
1 1 1( ) exp[ ( )]v
s abs vap
s meltv vap S
LTRT I k p
L n R T T
(5.16)
Temperature gradient may be avoid for the high beam intensities due to the less conduction in
melt region where Eq. (16) can be approximate with /
abs
s melt
IT
n k
due to the low vapor flow
velocity.
88
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
89
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
90
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
Density of melt ρm 8.4×103 kg/m3
Vaporization Pressure, ,0vapp 51.013 10 pa
Specific heat of melt cp/c 625 J/Kg.K
91
Temp. of vaporization Tv,0 3100 K
Temp. of melt Tm 1510 K
Latent heat of vaporization Lv 6.44×106 J/g
Latent heat of melt Lm 2.31×105 J/g
Molar mass M 76 g/mol
Dynamic viscosity η 0.05 g/cm.s
Surface tension γST 0.0001 J/cm2
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.
92
(a) (b)
(c) (d)
(e) (f)
93
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
to the original diameter) and the same maximum laser intensity (Imax= 1 MW/cm2It is shown
t=0µs t=40µs
t=50µs t=60µs
94
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
95
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
96
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
97
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
98
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
99
Figure 5-8 Temperature contours at (a) 0; (b)20 ; (c) 25; (d) 30; (e) 45 and (f) 105 µs with
, and
100
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
101
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
102
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.
103
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 10
21 W/m
2. 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
104
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
105
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.
106
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:
'( ) ( )
e ee e e l
T TC k G T T S
t x x (6.1)
( ) ( )l ll l e l
T TC k G T Tt x x
(6.2)
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,
e e eC B T (6.3)
where Be is a constant. According to Chen et al. [128], the electron heat capacity can be
approximated by the following relationship:
107
'
'
,
2,
3 3
,3
3
2
e e
e e e
ee
B
B
B T
B T C
C CNk
Nk
2
2 2
2
3
3
Fe
F Fe
Fe F
e F
TT
T TT
TT T
T T
(6.4)
where
2
'
2 2
2
32
( )
e FB
F Fe e e
FF
B TNk
T TC B T
TT
(6.5)
The bulk thermal conductivity of metal at equilibrium can be represented as
eq e lk k k (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 5/4 2
2 1/2 2
( 0.16) ( 0.44)
( 0.092) ( )
e e ee
e e l
k (6.7)
where ee
F
T
T and l
lF
T
T are dimensionless temperature parameters and and are the two
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
( ) ee eq
l
Tk k
T (6.8)
108
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:
[ ( ) 1] eRT e l
l
AG G T T
B (6.9)
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
' 2
/( )
10.94 exp 2.77( )
( )[1 ]bLb pp b
R x tS J
tt e
(6.10)
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
, ,, ,
l s ll s l m s
T Tk k h u
x x (6.11)
109
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:
,0
,
[1 exp( )]
l I mm
sg m l I
T Thu V
R T T (6.12)
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
0 0
0
e e l l
x x L x x L
T T T T
x x x x (6.13)
The initial temperature conditions are
( , 2 ) ( , 2 ) e p l p iT x t T x t T (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
,, ,( ) ( ) ( )
l I ll s l I m l e l
T TfC T h k G T T
t t x x (6.15)
110
where ,l IT is the interfacial temperature,
,l sC 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
111
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
112
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 75
th percentile,
75 25 IQR P P (6.17)
6.4 RESULTS AND DISCUSSIONS
The thermophysical and optical properties of pure gold film are: Be = 70 J/m3K, Ae =1.2×10
7
K-2
s-1
and Bl =1.23×1011
K-1
s-1
, GRT = 2.2×1016
W/m3K (solid) and 2.6×10
16 W/m
3K (liquid), ρ
=19.30×103 kg/m
3 (solid) and 17.28×10
3 kg/m
3 (liquid) reflectivity, R = 0.6, δ = 20.6 nm, δb
=105 nm, Tm = 1336 K, TF = 6.42×104 K, hm = 6.373×10
4 J/kg, and V0 =1300 m/s. The sample-
based stochastic model provides the output parameter distributions with respect to the uncertain
input
113
(a)
(b)
114
(c)
(d)
115
(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/cm
2 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
116
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)
117
(b)
(c)
118
(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
119
(a)
(b)
120
(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
121
(a)
(b)
122
(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
values of GRT, , η, J and tp being 2.2×1016
W/m3K, 353 W/mK, 0.16, 0.3 J/cm
2 and 20ps
respectively and the COV of each parameter being 0.02. Figure 6-7 gives the typical distribution
123
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)
124
(b)
(c)
125
(d)
(e)
Fig. 6-6 Typical distributions of the input parameters (a) GRT, (b) , (c) η, (d) J and (e) tp
126
(a)
(b)
127
(c)
(d)
Fig. 6-7 Typical distributions of the output parameters (a) s, (b) us, (c) Tl,I and (d) Te
128
(a)
(b)
129
(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
130
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
coupling factor (GRT) with different COVs. Three values of GRT, 2.1×1016
, 2.2×1016
and 2.3×1016
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)
131
(b)
(c)
132
(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)
133
(b)
(c)
134
(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.
135
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
136
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
137
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.
138
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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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