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

References

1. B.R. Birmingham, J. V. Tomplins and H. L. Marcus, Silicon carbide shapes by selected area

laser deposition vapor infiltration. In: Bourell DL, Beaman JJ, Marcus HL, Crawford RH,

Barlow JW, editors. Solid Freedom Fabrication Proceedings 1994. Austin: The University of

Texas, Sept. 1994, pp. 348-355.

2. J.J. Beaman, J. W. Barlow, D. L. Bourell, R. H. Crawford, H. L. Marcus and K. P. McAlea,

Solid Freedom Fabrication: A New Direction in Manufacturing, Kluwer, Dorchrecht, The

Netherlands, 1997.

3. H.L. Marcus, G. Zong, P.K. Subramanian, Residual stresses in laser processed solid freedom

fabrication, in: E.V. Barrera, I. Dutta (Eds), Residual stresses in composites: Measurement,

Modeling and Effects on Thermomechanical Properties, TMS, 1993.

4. J. Mazumder, A. Kar, Theory and Application of Laser Chemical Vapor Deposition, Plenum

Press, New York, 1995.

5. A. Kar, J. Mazumder, Three-dimensional transient thermal analysis for laser chemical vapor

deposition on uniformly moving finite slabs, J. of Applied Physics 65 (1989), pp. 2923-2934.

6. K. Dai, J. Crocker, L. Shaw, and H. Marcus, Finite element analysis of the SALDVI process,

Proc. 11th Solid Freeform Fabrication Symp., Austin, TX, 2000, pp. 393-398

7. K, Dai, J. Crocker, L. Shaw, and H. Marcus, Modeling of selective area laser deposition

vapor infiltration (SALDVI) of silicon carbide, Proc. 12th Solid Freeform Fabrication Symp.,

2001, pp. 392-399.

8. K. Dai, J. Crocker, L.Shaw, and H. Marcus, Thermal modeling of selective area laser

deposition vapor infiltration (SALD) and SALD vapor infiltration of silicon carbide, Rapid

Prototyp. J., 9 (2003), pp. 231-239.

139

9. M. Kaviany, Principle of Heat Transfer in porous media, 2d ef., Springer-Vergal, New York,

1995.

10. D. B. Ingham and I. Pop, Transport Phenomena in porous media, Elsevier, Oxford, UK,

1998.

11. D. B. Ingham and I. Pop, Transport phenomena in porous media II, Elsevier, Oxford, UK,

2002.

12. D. A. Nield and A.Bejan, Convection in Porous media, second ed., Springer-Verlag, New

York, 1999.

13. H. J. H. Brouwes, Heat transfer between a fluid-saturated porous medium and a permeable

wall with fluid injection or withdrawal, Int. J. Heat Mass Transfer, 37 (6) (1994), pp. 989-

996.

14. P. Nithiarasu, K. N. Seetharamu, and L. Sundararaian, Double-diffusion natural convection

in an enclosure filled with fluid saturated porous medium: A Generalized Non-Darcy

approach, Numer. Heat Transfer A, 30 (1996), pp. 413-426.

15. P.V. S. N. Murphy, B. V. R. Kumar, and P. Singh, Natural Convection Heat Transfer from a

Horixontal Wavy Surface in a porous enclosure, Numer. Heat Transfer A, 33 (1996), pp.

207-221.

16. K. M. Khanafer and A.J. Chamkha, Hydromagnatic Natural convection from an inclined

porous square enclosure with heat generation, Numer. Heat Transfer A, 33 (1998), pp. 891-

910.

17. Y. Zhang, Non-equilibrium modeling of heat transfer in a gas-saturated powder layer subject

to a short-pulsed heat source, Numerical Heat Transfer, Part A 50 (2006), pp. 509-524.

140

18. Y. Zhang, A. Faghri, Thermal modeling of selective area laser deposition of titanium nitride

on a finite slab with stationary and moving laser beams, Int. J. Heat and Mass Transfer, 43

(2000), pp. 3835-3846.

19. D. Y. Tzou, Macro to Micro scales Heat Transfer-The Lagging behavior, Taylor & Francis,

Bristol, PA.

20. S. W. Churchill, Free convection around immersed bodies,in E.U . Schlunder (ed.), Heat

Exchanger Design Handbook, Hemisphere, New York, 1983.

21. M. Kaviany, Principle of heat transfer in porous media, second ed., Springer-Verlag, New

York, 1995.

22. W. M. Chase, JANAF Thermochemical Tables, third ed., J. Phys. Chem. Ref. data, vol. 14,

suppl. 1, 1986.

23. T. Alexandre, J. Giovanola, S. Vaucher, O. Beffort, and U. Vogt, Layered manufacturing of

porous ceramic parts from SiC powders and preceramic polymers, Proc. 4th

Laser Assisted

Net Shape Engineering (LANE), Erlangen, Germany, 2004, pp. 497-505.

24. J. E. Crocker, L. L. Shaw, and H. L. Marcus, Powder effects in SIC matrix layered structures

fabricated using Selective Area Laser Deposition Vapor Infiltration, J. Mater. Sci., vol 37,

pp. 3149-3154, 2002.

25. O. M. Alifanov, Inverse Heat Transfer Problems. Spinger-Verlag, Berlin/Heidelberg, 1994.

26. E. M. Sparrow, A. Haji Sheikh, T.S. Lundgren, The inverse problem in transient heat

conduction. ASME J. Appl. Mech. 86 (1964) 369-375.

27. C. Y. Yang, C-K. Chen, The boundary estimation in two-dimensional inverse heat

conduction problems, J. Phys. D: Appl. Phys. 29 (1996) 333-339.

141

28. M. N. Özisik, H.R.B. Orlande, Inverse Heat Transfer: Fundamentals and Applications.

Taylor & Francis, New York, 2000.

29. A. F. Emery, A. V. Nenarokomov, T. D. Fadale, Uncertainties in parameter estimation: the

optimal experimental design, Int. J. Heat Transfer 43 (2000) 3331-3339.

30. M. Monde, H. Arima, Y. Mitsutake, Estimation of surface temperature and heat flux using

inverse solution for one-dimensional heat conduction, ASME J. Heat Transfer 125 (2003)

213-223.

31. X. Xue, R. Luck, J. K. Berry, Comparisons and improvements concerning the accuracy and

robustness of inverse heat conduction algorithms, Inverse Prob. Sci. Eng. 13 (2) (2005) 177-

199.

32. J. Zhou, Y. Zhang, J. K. Chen, Z. C. Feng, Inverse heat conduction using measured back

surface temperature and heat flux, J. Thermophys. Heat Transfer 24 (2010) 95-103.

33. J. Zhou, Y. Zhang, J. K. Chen, Z. C. Feng, Inverse heat conduction in a composite slab

with pyrolysis effect and temperature-dependent thermophysical properties, ASME J. heat

Transfer 132 (2010) 034502-1.

34. Z. C. Feng, J. K. Chen, Y. Zhang, S. Montgomery-Smith, Temperature and heat flux

estimation from sampled transient sensor measurements, Int. J. of Thermal Sciences 49

(2010) 2385-2390.

35. T.E. Diller, Advances in heat flux measurements. Advances in Heat Transfer 23 (1993)

279-368.

142

36. P.R.N. Childs, J. R. Greenwood, C. A. Long, Heat flux measurements techniques.

Proceedings of the institute of Mechanical Engineers. Part C: Journal of Mechanical

Engineering Science 213(C7) (1999) 655-677.

37. P.R.N. Childs, Advances in temperature measurements. Advances in Heat Transfer 36 (2002)

111-181.

38. A. Tong, Improving the accuracy of temperature measurements. Sensor Review 21 (3)

(2001) 193-198.

39. Z. C. Feng, J. K. Chen, Y. Zhang, Real-time solution of heat conduction in a finite slab for

inverse analysis, Int. J. of Thermal Sciences 49 (2010) 762-768.

40. Z. C. Feng, J. K. Chen, Y. Zhang, J. L. Griggs Jr., Estimation of front surface temperature

and heat flux of a locally heated plate from distributed sensor data on the back surface, Int.

J. of Heat and Mass Transfer 54 (2011) 3431-3439.

41. A. Savitzky, M. J. E. Golay, Smoothing and differentiation of data by simplified least

squares procedures, Anal. Chem. 36(1964) 1627-1639.

42. J. Zhou, Y. Zhang, J. K. Chen, Z. C. Feng, Inverse estimation of surface heating condition in

a three-dimensional object using conjugate gradient method, Int. J. of Heat and Mass

Transfer 53 (2010) 2643-2654.

43. J. Zhou, Y. Zhang, J.K. Chen, Z.C. Feng, Inverse estimation of surface heating condition in

three dimensional object using conjugate gradient method, Int. Journal of Heat and Mass

Transfer, 53 (2010) 2643-2654.

143

44. C.-Y. Yang, C.-K. Chen, A three-dimensional inverse heat conduction problem in

estimating surface heat flux by conjugate gradient method, Int. Journal of Heat and Mass

Transfer, Int. J. Heat and Mass Transfer, 42 (1999) 3387-3405.

45. D.Q. Kern, A. D. Kraus, Extended Surface Heat Transfer, McGraw-Hill, New York, 1972.

46. D. Maillet, S. Andre, J.C. Batsale, A. Degiovanni, C. Moyne, Thermal Quadrupoles:

Solving the Heat Equation through Integral Transformations, John Wiley & Sons, New

York, 2000.

47. O. R. Burggraf, An exact solution of inverse Laplace problem in heat conduction theory

and application, J. Heat Transfer 86c (1964) 373-382.

48. A. Faghri, Y. Zhang, J.R. Howell, Advanced Heat and Mass Transfer, Global Digital Press,

Columbia, MO, 2010.

49. Brannon, J. H., Lankard, J. R. , Baise, A. I. , Burns, F., Kaufman, J. , "Excimer Laser

Etching of Polyimide," Journal of Applied Physics, 1985; 5(58): pp. 2036-2043.

50. Sobehart, J. R., "Polyimide Ablation Using Intense Laser Beams," Journal of Applied

Physics, 1993; 4(74): pp. 2830-2833.

51. Lane, R. J., Wynne, J. J., Gernonemus, R.G., "Ultraviolet Laser Ablation of Skin:

Healing Studies and a Thermal Model," Lasers in surgery and medicine, 1986; 6: pp.

504-513.

52. Lecartentier, G. L., Motamedi, M., McMath, L. P., Rasregar, S., Welch, A. J.,

"Continuous Wave Laser Ablation of Tissue: Analysis of Thermal and Mechanical

Events," IEEE Transactions on Biomedical Engineering, 1993; 40: pp. 188-200.

144

53. Chen, J. K., Perea, A., Allahdadi, F. A., "A Study of Laser/Composite Material

Interactions," Composite Science and Technology, 1995; 54: pp. 35-44.

54. Mazumder, J., Steen, W. M., "Heat Transfer Model for Cw Laser Material Processing,"

Journal of Applied Physics, 1979; 51: pp. 941-947.

55. Lipperd, T., "Laser Application in Polymers," Advances in Polymer Science, 2004; 168:

pp. 51-246.

56. Zhou, J. H., Zhang, Y. W., Chen, J. K. , "Numerical Simulation of Compressible Gas

Flow and Heat Transfer in a Microchannel Surrounded by Solid Media," Int Journal of

Heat and Fluid Flow, 2007: pp. 1484-1491.

57. Kim, M. J., Zhang, J., "Finite Element Analysis of Evaporative Cutting with a Moving

High Energy Pulsed Laser," Applied Mathematical Modeling, 2001; 25: pp. 203-220.

58. Kim, M. J., "Three Dimentional Finite Element Analysis of Evaporative Laser Cutting,"

Applied Mathematical Modeling, 2005; 29: pp. 938-954.

59. Madhukar, Y. K., Mullick, S. , Shukla, D. K. , Kumar, S. , Nath, A.K. , "Effect of Laser

Operating Mode in Paint Removal with a Fiber Laser," Applied Surface Science, 2013;

264: pp. 892-901.

60. Faghri, A., Zhang, Yuwen ,Howell, John R. . Advanced Heat and Mass Transfer. MO,

USA: Global Digital Press; 2010.

61. Bejan, A. Advanced Engineering Thermodynamics. Hoboken, New Jersey: Wiley & Sons

Inc.; 2006.

62. Gardiner, W. C. Gas-Phase Combustion Chemistry. 2 ed. New York: Springer; 2000.

63. Sutherland, W., "The Viscosity of Gases and Molecular Forces," Philosophical Magazine

Series 5, 1893; 36(223): pp. 507-531.

145

64. Izzi, U. Function Gas.C of the Program Stim.Exe. 2002.

http://zigherzog.net/stirling/simulations/DHT/ViscosityTemperatureSutherland.html.

65. Zhang, Y., Fgahri, A. , "Thermal Modeling of Selective Area Laser Deposition of

Titanium Nitride on a Finite Slab with Stationary and Moving Laser Beams,"

International Journal of Heat and Mass transfer, 2000; 43: pp. 3835-3846.

66. Bird, R. B., Stewart, W. E., Lightfoot, E. N. Transport Phenomena. New York: John

Wiley & Sons; 1960.

67. Hirschfelder, J. O., Curtiss, C.F., Bird, R. B. Molecular Theory of Gases and Liquids:

Wiley; 1964.

68. Lu, Q.-W., Hernandez-Hernandez, M. E., Macoska, C. W. , "Explaining the Abnormally

High Flow Activation Energy of Thermoplastic Polyurethanes," Polymer, 2003; 44: pp.

3309-3318.

69. Graves, R. S., Kollie, T. G., McElroy, D. L., Gilchrist, K. E., "The Thermal Conductivity

of Aisi 304l Stainless Steel," International Journal of Thermophysics, 1991; 12(2): pp.

409-415.

70. Jyoti, M., Aravinda, K. Theory and Application of Laser Chemical Vapor Deposition.

New York: Plenum Press; 1995.

71. Petrucci, R. H., Harwood, W. S., Herring, G. E. Madura, J. General Chemistry: Principles

and Modern Application, 9th Edition. 9th ed: Prentice Hall; 2006.

72. Ounis, H., Ahmadi, G., McLaughlin, J. B., "Brownian Diffusion of Submicrometer

Particles in the Viscous Sublayer," Journal of Colloid and Interface Science, 1991;

143(1): pp. 266-277.

146

73. Haider, A., Levenspiel, O., "Drag Coefficient and Terminal Velocity of Spherical and

Nonspherical Particles," Powder Technology, 1989; 58(1): pp. 63-70.

74. Graves, R. S., Kollie, T. G., McElroy, D. L., Gilchrist, K. E., "The Thermal Conductivity

of Aisi 304l Stainless Steel," International Journal of Thermophysics, 1991; 12(2): pp.

409-415.

75. Hentschel, T., Münstedt, H. , "Kinetics of the Molar Mass Decrease in a Polyurethane

Melt: A Rheological Study," Polymer, 2001; 42: pp. 3195-3203.

76. Chang, W. L., "Decomposition Behavior of Polyurethanes Via Mathematical Solution,"

Journal of Applied Polymer science, 1994; 53: pp. 1759–1769.

77. Jasak, H. Error Analysis and Estimation for the Finite Volume Method with Applications

to Fluid Flows. London: Imperial College.

78. Liu, K., Garmire, E., "Paint Removal Using Lasers," Appl Opt, 1995; 34(21): pp. 4409-

4415.

79. Petker, I., Mason, D. M., "Viscosity of the N2o4-No2 Gas System," Journal of Chemical

and Engineering DATA, 1964; 9(2): pp. 280-281.

80. Bass, M., Laser Materials Processing. North-Holland Publishing Company, 1983.

81. Luxon, J. T. and Parker, D. E., Industrial Lasers and their Applications, 2nd

edition,

Prentice Hall, 1991.

82. Steen, W. M., Laser Material Processing. Springer, Berlin 1991.

83. Ready, J. F., Effects Due to Absorption of Laser Radiation, Journal of Applied Physics,

vol 36, 462 (1965).

147

84. Ganesh, R. K., Bowley, W. W., Bellantone, R. R. and Hahn, Y., A model for laser hole

drilling in metals, Journal of Computational Physics, 125 pp 161-176 (1996)

85. Von Allmen, M., Laser Drilling Velocity in Metals, Journal of Applied Physics 47 (12),

pp 5460, 1976.

86. Chan, C. L. and Mazumder, J., One- dimensional steady-state model for damage by

vaporization and liquid expulsion due to laser-material interaction, Journal of Applied

Physics 62 (11), pp 4579 1987.

87. Anisimov, S. I., Vaporization of Metal Absorbing Laser Radiation, Sov. Physics JETP

27 (1), 182-183 (1968)

88. Knight, C. J., Theoretical modeling of rapid surface vaporization with back pressure¸

AIAA Journal, 17 pp 519 (1979)

89. Chun, M. K. and Rose, K., Interaction of High-Intensity Laser Beams with Metal, Journal

of Applied Physics 41 614 (1979)

90. Kar, A., and Mazumder, J., Two dimensional model for laser-induced material damage

due to melting and vaporization during laser irradiation, Journal of Applied Physics, 68

3884 (1990)

91. Mazumder, J., Overview of melt dynamics in laser processing, Opt. Eng. 30 (8), pp 1208

(1991)

92. Zacharia, T., Eraslan, A. H., Audun, D. K. and David, S. A., 3-Dimensional transient

model for arc welding process, Metall. Trans. B 20 20 pp 645 (1989)

148

93. Chan, C., Mazumder, J. and Chen, M. M., A two dimension transient model for

convection in a laser melted pool, Metall. Trans. A 15, pp 2175 (1984)

94. Kou, S. and Wang, Y. H., Three-Dimensional Convection in Laser Melted Pools, Metall.

Trans. A 17, pp 2265 (1986)

95. Kou, S. and Sun, D. K., Fluid Flow and Weld Penetration in Stationary Arc Welds,

Metall. Trans. A 16, pp 203 (1985)

96. Ganesh, R. K. and Faghri, A., A generalized thermal modeling for laser drilling process-

I. Mathematical modeling and numerical methodology, Int. J. Heat and Mass Transfer 14

pp 3351-3360 (1997)

97. Ganesh, R. K. and Faghri, A., A generalized thermal modeling for laser drilling process-

II. Numerical simulation and results, Int. J. Heat and Mass Transfer 40 pp 3361-3373

(1997)

98. Zhang, Y., and Faghri, A., Vaporization, melting and heat conduction in the laser drilling

process, International Journal of Heat and Mass Transfer 42 (1999) pp. 1775-1790.

99. Von Allmen, M., Laser drilling velocity in metals, Journal of Applied Physics, 47 (12), pp

5460-5463 (1976)

100. Patankar. S. V., Numerical heat transfer and fluid flow, McGraw-Hill, New York, 1980

101. Hohlfeld, J., Wellershoff, S. -S., Güdde, J., Conrad, U., Jähnke, and V., Matthias, E.,

2000, “Electron and Lattice Dynamics Following Optical Excitation of Metals,”

Chemical Physics, 251, pp. 237-258.

149

102. Anisimov, S. I., Kapeliovich, B. L., and Perel’man, T. L., 1974, “Electron Emission from

Metal Surfaces Exposed to Ultrashort Laser,” Sov. Phys. Journal of Experimental and

Theoretical Physics, 39, pp. 375-377.

103. Qiu, T. Q., and Tien, C. L., 1993, “Heat Transfer Mechanism During Short-Pulse Laser

Heating of Metals,” Journal of Heat Transfer. 115, pp. 835-841.

104. Chen, J. K., and Beraun, J. E., 2001, “Numerical Study of Ultrashort Laser Pulse

Interactions with Metal Films,” Numerical Heat Transfer, A 40(1), pp. 1-20.

105. Kuo, L. S., and Qiu, T., 1996, “Microscale Energy Transfer During Picosecond Laser

Melting of Metal Films,” ASME Natl. Heat Transfer Conf. 1, pp 149-157.

106. Chowdhury, I. H., and Xu, X., 2003, “Heat Transfer in Femtosecond Laser Processing of

Metal,” Numerical Heat Transfer. A 44 (3), pp. 219 -232.

107. Von Der Linde, D., Fabricius, N., Denielzik, B., and Bonkhofer, T., 1987, “Solid Phase

Superheating During Picoseconds Laser Melting of Gallium Arsenide,” Materials Research

Society Proceedings, 74, pp. 103-108.

108. Voller, V.R., 1997, “An Overview of Numerical Methods for Solving Phase Change

Problems,” Advances in Numerical Heat Transfer, 1 ed. by W. J. Minkowycz, E. M.

Sparrow (Taylor& Francis), Basingstoke, pp. 341-379.

109. Zhang, Y., and Chen, J. K., 2007, “An Interfacial Tracing Method for Ultrashort Pulse

Laser Melting and Resolidification of a Thin Metal Film,” ASME-JSME Thermal

Engineering Conf. Summer Heat Transfer Conf., HT2007-32475.

110. Zhang, Y., and Chen, J. K., 2007, “Melting and Resolidification of Gold Film Irradiated by

Nano-to Femtosecond Lasers,” Applied Physics, A 88, pp. 289-297.

150

111. Jiang, L., and Tsai, H.-L., 2005, “Improved Two-Temperature Model and Its Application in

Ultrashort Laser Heating of Metal Films,” ASME Journal of Heat and Mass Transfer, 127

(10), pp.1167-1173.

112. Jiang, L., and Tsai, H.-L., 2005, “Energy Transport and Material Removal in Wide

Bandgap Materials by a Femtosecond Laser Pulse,” International Journal of Heat and Mass

Transfer, 48 (3), pp.487-499.

113. Chen, J. K., Latham, W. P., and Beraun, J. E., 2005,” The Role of Electron-Phonon

Coupling in Ultrafast Laser Heating,” Journal of Laser Application, 17 (1), pp.63-68.

114. Lin, Z., Zhigilei, L. V., and Celli. V., 2008, “Electron-Phonon Coupling and Electron Heat

Capacity of Metals under Condition of Strong Electron-Phonon Equilibrium,” Physical

Review B 77 075133.

115. Beaman, J. J., Barlow, J. W., Bourell, D. L., Crawford, R.H., Marcus, H. L., and McAlea,

K. P., 1997, Solid Freedom Fabrication: A New Direction in Manufacturing, Kluwer,

Dordrecht, The Netherlands.

116. Chen, T., and Zhang, Y., 2006, “Three-Dimensional Modeling of Selective Laser Sintering

of Two-Component Metal Powder Layers,” ASME Journal of Manufacturing Science and

Engineering, 128 (1), pp. 299-306.

117. Chen, T., and Zhang, Y., 2007, “Three-Dimensional Modeling of Laser Sintering of a Two-

Component Metal Powder Layer on Top of Sintered Layers,” ASME Journal

Manufacturing Science and Engineering, 129 (3), pp. 575-582.

118. Bin, B., and Zhang, Y., 2008, “Numerical Simulation of Direct Metal Laser Sintering of

Single-Component Powder on Top of Sintering Layers,” ASME Journal of Manufacturing

Science, 130 (4), p. 041002

151

119. Fischer, P., Romano, V., Blatter, A., and Weber, H. P., 2005, “High Precision Pulsed

Selective Laser Sintering of Metallic Powders,” Laser Physics Letters, 2 (1), pp. 48-55.

120. Padmanabhan, S. K., and Pitchumani, R., 1999, “Stochastic Modeling of Nanoisothermal

Flow during Resin Transfer Molding Processes, “International Journal of Heat and Mass

Transfer, 42 (16), pp. 3057-3070.

121. Diwekar, U.M., and Rubin, E. S., 1991, “Stochastic Modeling of Chemical Processes,”

Computers Chemical Engineering, 15 (2), pp. 105-114.

122. Marwadi, A., and Pitchumani, R., 2008, “Numerical Simulations of an Optical Fiber

Drawing Process under Uncertainty”, Journal of Lighwave Technology, 26 (5), pp. 580-

587.

123. Myers, M. R., 1989, “A Model for Unsteady Analysis of Perform Drawing,” AICheE J. 35

(4), pp. 592-602.

124. Marwadi, A., and Pitchumani, R., 2004, “Cure Cycle Design for Thermosetting-Matrix

Composites Fabrication under Uncertainty,” Annals Operations Research, 132, pp. 19-45.

125. Campbell, J. E., and Rao, B. S. R., 1987, “An Uncertainty Analysis Methodology Applied

to Sheetpile Cofferdam Design,” Journal of Hydraulic Engineering, 41, pp. 36-41.

126. Marwadi, A., and Pitchumani, R., 2006, “Effect of Parameter Uncertainty on the

Performance Variability of Proton Exchange Membrane (PEM) Fuel Cells,’ J. Power

Sources, 160, pp. 232-245.

152

127. Pang, H., Zhang, Y., and Pai, P. F., 2013, “Uncertainty Analysis of Solid-Liquid-Vapor

Phase Change of a Metal Particle Subject to Nanosecond Laser Heating,” Journal of

Manufacturing Science and Engineering, 135 (2), 021009.

128. Chen, J. K., Beraun, J. E., and Tzou, D. Y., 2006, “A Semiclassical Two-Temperature

Model for Ultrafast Laser Heating,” International Journal of Heat and Mass Transfer, 49

(1-2), pp.307-316.

129. Klemens, P.G., And Williams, R.K., 1986. “Thermal Conductivity of Metals and Alloys,”

International Materials Reviews, 31, pp. 197-215.

153

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.