Two Phase Flow Phase Change and Numerical Modeling Part 1 pot

TWO PHASE FLOW,
PHASE CHANGE AND
NUMERICAL MODELING

Edited by Amimul Ahsan













Two Phase Flow, Phase Change and Numerical Modeling
Edited by Amimul Ahsan


Published by InTech
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Publishing Process Manager Ivana Lorković

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Image Copyright alehnia, 2011. Used under license from Shutterstock.com

First published August, 2011
Printed in Croatia

A free online edition of this book is available at www.intechopen.com
Additional hard copies can be obtained from



Two Phase Flow, Phase Change and Numerical Modeling, Edited by Amimul Ahsan
p. cm.
ISBN 978-953-307-584-6

Contents

Preface IX
Part 1 Numerical Modeling of Heat Transfer 1
Chapter 1 Modeling the Physical Phenomena Involved by
Laser Beam – Substance Interaction 3
Marian Pearsica, Stefan Nedelcu, Cristian-George Constantinescu,
Constantin Strimbu, Marius Benta and Catalin Mihai
Chapter 2 Numerical Modeling and Experimentation on
Evaporator Coils for Refrigeration in Dry and Frosting
Operational Conditions 27
Zine Aidoun, Mohamed Ouzzane and Adlane Bendaoud
Chapter 3 Modeling and Simulation of the Heat Transfer Behaviour of
a Shell-and-Tube Condenser for a Moderately
High-Temperature Heat Pump 61
Tzong-Shing Lee and Jhen-Wei Mai
Chapter 4 Simulation of Rarefied Gas Between Coaxial Circular
Cylinders by DSMC Method 83
H. Ghezel Sofloo
Chapter 5 Theoretical and Experimental Analysis of Flows and
Heat Transfer Within Flat Mini Heat Pipe Including Grooved
Capillary Structures 93
Zaghdoudi Mohamed Chaker, Maalej Samah and Mansouri Jed
Chapter 6 Modeling Solidification Phenomena in the Continuous
Casting of Carbon Steels 121
Panagiotis Sismanis

Chapter 7 Modelling of Profile Evolution by Transport Transitions in
Fusion Plasmas 149
Mikhail Tokar
VI Contents

Chapter 8 Numerical Simulation of the Heat Transfer from a Heated
Solid Wall to an Impinging Swirling Jet 173
Joaquín Ortega-Casanova
Chapter 9 Recent Advances in Modeling Axisymmetric
Swirl and Applications for Enhanced Heat Transfer and
Flow Mixing 193
Sal B. Rodriguez and Mohamed S. El-Genk
Chapter 10 Thermal Approaches to Interpret
Laser Damage Experiments 217
S. Reyné, L. Lamaignčre, J-Y. Natoli and G. Duchateau
Chapter 11 Ultrafast Heating Characteristics in Multi-Layer Metal Film
Assembly Under Femtosecond Laser Pulses Irradiation 239
Feng Chen, Guangqing Du, Qing Yang, Jinhai Si and Hun Hou
Part 2 Two Phase Flow 255
Chapter 12 On Density Wave Instability Phenomena – Modelling
and Experimental Investigation 257
Davide Papini, Antonio Cammi,
Marco Colombo and Marco E. Ricotti
Chapter 13 Spray Cooling 285
Zhibin Yan, Rui Zhao, Fei Duan, Teck Neng Wong, Kok Chuan Toh,
Kok Fah Choo, Poh Keong Chan and Yong Sheng Chua
Chapter 14 Wettability Effects on Heat Transfer 311
Chiwoong Choi and Moohwan Kim
Chapter 15 Liquid Film Thickness in Micro-Scale Two-Phase Flow 341
Naoki Shikazono and Youngbae Han

Chapter 16 New Variants to Theoretical Investigations of
Thermosyphon Loop 365
Henryk Bieliński
Part 3 Nanofluids 387
Chapter 17 Nanofluids for Heat Transfer 389
Rodolphe Heyd
Chapter 18 Forced Convective Heat Transfer of Nanofluids
in Minichannels 419
S. M. Sohel Murshed and C. A. Nieto de Castro
Contents VII

Chapter 19 Nanofluids for Heat Transfer – Potential and
Engineering Strategies 435
Elena V. Timofeeva
Chapter 20 Heat Transfer in Nanostructures Using the Fractal
Approximation of Motion 451
Maricel Agop, Irinel Casian Botez,
Luciu Razvan Silviu and Manuela Girtu
Chapter 21 Heat Transfer in Micro Direct Methanol Fuel Cell 485
Ghayour Reza
Chapter 22 Heat Transfer in Complex Fluids 497
Mehrdad Massoudi
Part 4 Phase Change 521
Chapter 23 A Numerical Study on Time-Dependent Melting and
Deformation Processes of Phase Change Material (PCM)
Induced by Localized Thermal Input 523
Yangkyun Kim, Akter Hossain, Sungcho Kim and Yuji Nakamura
Chapter 24 Thermal Energy Storage Tanks Using Phase Change Material
(PCM) in HVAC Systems 541
Motoi Yamaha and Nobuo Nakahara

Chapter 25 Heat Transfer and Phase Change in Deep CO
2
Injector for CO
2

Geological Storage 565
Kyuro Sasaki and Yuichi Sugai












Preface

The heat transfer and analysis on laser beam, evaporator coils, shell-and-tube
condenser, two phase flow, nanofluids, and on phase change are significant issues in a
design of wide range of industrial processes and devices. This book introduces
advanced processes and modeling of heat transfer, flat miniature heat pipe, gas-solid
fluidization bed, solidification phenomena, thermal approaches to laser damage, and
temperature and velocity distribution to the international community. It includes 25
advanced and revised contributions, and it covers mainly (1) numerical modeling of
heat transfer, (2) two phase flow, (3) nanofluids, and (4) phase change.
The first section introduces numerical modeling of heat transfer on laser beam,

evaporator coils, shell-and-tube condenser, rarefied gas, flat miniature heat pipe,
particles in binary gas-solid fluidization bed, solidification phenomena, profile
evolution, heated solid wall, axisymmetric swirl, thermal approaches to laser damage,
ultrafast heating characteristics, and temperature and velocity distribution. The second
section covers density wave instability phenomena, gas and spray-water quenching,
spray cooling, wettability effect, liquid film thickness, and thermosyphon loop.
The third section includes nanofluids for heat transfer, nanofluids in minichannels,
potential and engineering strategies on nanofluids, nanostructures using the fractal
approximation, micro DMFC, and heat transfer at nanoscale and in complex fluids.
The forth section presents time-dependent melting and deformation processes of
phase change material (PCM), thermal energy storage tanks using PCM, capillary rise
in a capillary loop, phase change in deep CO
2 injector, and phase change thermal
storage device of solar hot water system.
The readers of this book will appreciate the current issues of modeling on laser beam,
evaporator coils, rarefied gas, flat miniature heat pipe, two phase flow, nanofluids,
complex fluids, and on phase change in different aspects. The approaches would be
applicable in various industrial purposes as well. The advanced idea and information
described here will be fruitful for the readers to find a sustainable solution in an
industrialized society.
The editor of this book would like to express sincere thanks to all authors for their
high quality contributions and in particular to the reviewers for reviewing the
chapters.
X Preface

ACKNOWLEDGEMENTS
All praise be to Almighty Allah, the Creator and the Sustainer of the world, the Most
Beneficent, Most Benevolent, Most Merciful, and Master of the Day of Judgment. He is
Omnipresent and Omnipotent. He is the King of all kings of the world. In His hand is
all good. Certainly, over all things Allah has power.

The editor would like to express appreciation to all who have helped to prepare this
book. The editor expresses the gratefulness to Ms. Ivana Lorkovic, Publishing Process
Manager InTech Open Access Publisher, for her continued cooperation. In addition,
the editor appreciatively remembers the assistance of all authors and reviewers of this
book.
Gratitude is expressed to Mrs. Ahsan, Ibrahim Bin Ahsan, Mother, Father, Mother-in-
Law, Father-in-Law, and Brothers and Sisters for their endless inspirations, mental
supports and also necessary help whenever any difficulty.

Amimul Ahsan
Department of Civil Engineering,
Faculty of Engineering,
University Putra Malaysia
Malaysia



Part 1
Numerical Modeling of Heat Transfer

1
Modeling the Physical Phenomena Involved by
Laser Beam – Substance Interaction
Marian Pearsica, Stefan Nedelcu, Cristian-George Constantinescu,
Constantin Strimbu, Marius Benta and Catalin Mihai
“Henri Coanda” Air Force Academy
Romania
1. Introduction
The mathematical model is based on the heat transfer equation, into a homogeneous
material, laser beam heated. Because transient phenomena are discussed, it is necessary to

consider simultaneously the three phases in material (solid, liquid and vapor), these
implying boundary conditions for unknown boundaries, resulting in this way analytical and
numerical approach with high complexity.
Because the technical literature (Belic, 1989; Hacia & Domke, 2007; Riyad & Abdelkader,
2006) does not provide a general applicable mathematical model of material-power laser
beam assisted by an active gas interaction, it is considered that elaborating such model,
taking into account the significant parameters of laser, assisting gas, processed material,
which may be particularized to interest cases, may be an important technical progress in this
branch. The mathematical methods used (as well the algorithms developed in this purpose)
may be applied to study phenomena in other scientific/technical branches too. The majority
of works analyzing the numerical and analytical solutions of heat equation, the limits of
applicability and validity of approximations in practical interest cases, is based on results
achieved by Carslaw and Jaeger using several particular cases (Draganescu & Velculescu,
1986; Dowden, 2009, 2001; Mazumder, 1991; Mazumder & Steen, 1980).
The main hypothesis basing the mathematical model elaboration, derived from previous
research team achievements (Pearsica et al., 2010, 2009; Pearsica & Nedelcu, 2005), are: laser
processing is a consequence of photon energy transferred in the material and active gas jet,
increasing the metal destruction process by favoring exothermic reactions; the processed
material is approximated as a semi-infinite region, which is the space limited by the plane
z0= , the irradiated domain being much smaller than substance volume; the power laser
beam has a “Gaussian” type radial distribution of beam intensity (valid for TEM
00
regime);
laser beam absorption at z depth respects the Beer law; oxidations occurs only in laser
irradiated zone, oxidant energy being “Gaussian” distributed; the attenuation of metal
vapors flow respects an exponential law. One of the mathematical hypothesis needing a
deeper analysis is the shape of the boundaries between liquid and vaporization, respectively
liquid and solid states, supposed as previously known, the parameters characterizing them
being computed in the thermic regime prior to the calculus moment.
The laser defocusing effect, while penetrating the processed metal is taken into

consideration too, as well as energy losses by electromagnetic radiation and convection. The

Two Phase Flow, Phase Change and Numerical Modeling

4
proposed method solves simultaneously the heat equation for the three phases (solid, liquid
and vapor), computing the temperature distribution in material and the depth of
penetration of the material for a given processing time, the vaporization speed of the
material being measurable in this way.
2. Analytical model equations
The invariant form of the heat equation for an isotropic medium is given by (1).

v
T
c
T
kt
∂
⋅ρ
=Δ
∂

(1)
where:
3
[k
g
/m ]ρ is the mass density;
11
v

c[Jk
g
K]
−−
⋅⋅ – volumetric specific heat; T[K] –
temperature;
11
k[W m K ]
−−
⋅⋅ – heat conductivity of the material; t[s] – time; Δ – Laplace
operator.
Because the print of the laser beam on the material surface is a circular one, thermic
phenomena produced within the substantial, have a cylindrical symmetry. Oz is considered
as symmetry axis of the laser beam, the object surface equation is z 0= and the positive
sense of Oz axis is from the surface to the inside of the object. The heat equation within
cylindrical coordinates
()
,r,zθ will be:

22
22 2
TT
11T1 T
r
Kt r rr r z

∂∂
∂∂ ∂
=+ +


∂∂θ∂∂∂


(2)
where:
2
K[m /s] is the diffusivity of the material.
Limit and initial conditions are attached to heat equation according to the particularly cases
which are the discussed subject. These conditions are time and space dependent. In time, the
medium submitted to the actions of the laser presents the solid, liquid and vapor state
separated by previously unknown boundaries. A simplifying model taking into consideration
these boundaries, by considering them as having a cylindrical symmetry, was proposed. By
specifying the pattern D, the temperature initial conditions and the conditions on D pattern
boundaries, one can have the solution of heat equation, T(x,y,z,t) for a certain substantial.
2.1 Temperature source modeling
The destruction of the crystalline network of the material and its vaporization, along the
pre-established curve, is completed by the energy of photons created inside the material,
and by the jet of the assisting gas (O
2
). This gas intensifies the material destroying action due
to the exothermic reactions provided. Dealing with a semi-infinite solid heated by a laser
beam uniform absorbed in its volume, it is assumed that Beer law governs its absorption at z
depth. It is considered a radial “Gaussian” distribution of the laser beam intensity, which
corresponds to the central part of the laser beam. It is assumed that photons energy is totally
transformed in heat. So, the heat increasing rate, owing the photons energy, at z depth
(under surface) is given by:

()
2
rz

dl
L
2
dQ P
hIr,z e
dt dV d l



−+








=ν⋅σ⋅ρ⋅ =
⋅π⋅

(3)

Modeling the Physical Phenomena Involved by Laser Beam – Substance Interaction

5
where:
3
dV[m ] and dt[s] are the infinitesimal volume and time respectively,
2

[m /k
g
]σ –
the absorption cross section,
1/ lσ=
ρ
⋅ ,
2
I(r t) [W /m ] – photons distribution in material
volume,
l[m] – the attenuation length of laser radiation,
L
P[W] – the laser power;
22
d[m]π
– irradiated surface,
r[m] – radial coordinate, and h[J]
ν
– the energy of one photon.
The vaporized material diffuses in oxygen atmosphere and oxidizes exothermic, resulting in
this way an oxidizing energy, which appears as an additional kinetic energy of the surface
gas constituents, leading to an additional heating of the laser processed zone. It is assumed
an exponential attenuation of the metal vapors flow and oxidizing is only inner laser
irradiated zone, the oxidizing energy being “Gaussian” distributed. The rate of oxidizing
energy release on the material is given by (4):

2
ox
2
zr

ld
ox
oox o s
dQ
nv e
dt dV M



−








ε
=η ⋅σ ⋅ρ⋅ ⋅ ⋅
⋅

(4)
where:
o
η
is the oxidizing efficiency,
[J]ε
– oxidizing energy on completely oxidized metal
atom,

2
ox
[m /k
g
]σ – effective oxidizing section,
2
3
o
n[m]
−
– oxygen atomic concentration,
s
v[m/s]– vaporization boundary speed, M[k
g
] – atomic mass of metal, and
ox
l[m] –
oxidizing length,
2
ox o ox
l1/(n )=⋅σ. In (4), z is negative outside the material, so the
attenuation is obvious. The full temperature source results as a sum of (3) and (4), and
assuming a constant vaporization boundary speed, the instantaneous expression of
temperature source is given by (Pearsica et al., 2010):

() () ()
2
s
s
ox

vtz
r
zvt
l
d
Ls
l
so s
2
ox
Pv
Sr,z e e hz v t e hv t z
dl Ml
⋅−

−⋅
−
−
−




ε⋅ρ⋅
=⋅ ⋅−⋅+η ⋅⋅−


π⋅ ⋅






(5)
where h(x) is Heaviside function. In temperature source expression, z origin is the same
with the vaporization boundary, which advance in profoundness as the material is drawn.
The spatial and temporal temperature distribution in material is governed by the full
temperature source and results by solving the heat equation.
2.2 Boundary and initial conditions for heat equation
a. Dirichlet conditions
Let
1
SS⊂ . For S
1
surface points it is assumes that the temperature T is known as a function
f(M,t), and the remaining surface, S, the temperature is constant, T
a
:

1
a1
f(M,t), M S
T(M,t)
T, M S\S
∈

=

∈


(6)
b. Neumann conditions
Let
2
SS⊂ . It is known the derivate in the perpendicular n direction to the surface S
2
:

()
()
2
TM,t
g
M,t , M S
n
∂
=∈
∂

(7)
c. Initial conditions
It is assumed that at
o
tt= time is known the thermic state of the material in D pattern:

Two Phase Flow, Phase Change and Numerical Modeling

6

()
oo
TM,t T(M), M D=∈

(8)
In time, successions the phases the object suffers while irradiate by the power laser beam are
the following:
-
phase 1, for
top
0tt≤< ;
-
phase 2, for
top vap
ttt≤< ;
-
phase 3, for
vap
tt≥ , where
top
t and
vap
t are the starting time moments of the melting,
respectively vaporization of the material.
The surfaces separating solid, liquid and vapor state are previously unknown and will be
determined using the conditions of continuity of thermic flow on separation surfaces of two
different substantial, knowing the temperature and the speed of separation surface
(Mazumder & Steen, 1980; Shuja et al., 2008; Steen & Mazumder, 2010).
The isotropic domain D is assumed to be the semi-space z 0≥ , so its border, S, is
characterized by the equation z 0= . The laser beam acts on the normal direction,

developing thermic effects described by (1). In the initial moment, t 0= , the domain
temperature is the ambient one, T
a
. If the laser beam radius is d and axis origin is chosen on
its symmetry axis, then the condition of type (7) (thermic flow imposed on the surface of the
processed material) yields:

()
222
S
222
z0
1
M,t , x
y
d,z 0
T
k
x
0, x y d , z 0
=

−
ϕ
+≤ =
∂

=

∂


+> =


(9)
where
()
2
S
M,t [W/m ]ϕ is the power flow on the processed surface, corresponding to the
solid state:

()
2
r
222
d
SL
S
2
AP
M,t e , r x
y
,z 0
d

−


⋅

ϕ
==+=
π

(10)
where:
S
A
is the absorbability of solid surface, and
L
P[W]
– the power of laser beam.
Regarding the working regime, two kinds of lasers were taken into consideration:
continuous regime lasers (P
L
= constant) and pulsated regime lasers (P
L
has periodical time
dependence, governed by a “Gaussian” type law). If the laser pulse period is
ponoff
tt t=+,
then the expression used for the laser power is the following:

() ()
()
2
on
p
on
t

tkt
2
t
1
4
ppon
L
poff p
Ce e ,t kt,kt t
P;k
0, t k 1 t t , k 1 t

−−


−

−









−∈+





=∈Ν







∈+ − +



(11)
where:
1/4
Lmax
CP e=⋅. Due to the cylindrical symmetry,
2
2
T
0
∂
=
∂θ
, so (2) changes to:

Modeling the Physical Phenomena Involved by Laser Beam – Substance Interaction

7

22
22
TT
11 TT
Kt rr r z
∂∂
∂∂
=++
∂∂∂∂

(12)
Equations (6) and (7) will be:


() ()
[
]
[
]
a
T r,z,0 T , r,z 0, r 0, r
∞∞
=∈×

(13)

()
()

1
r,0,t , r d
Tr,0,t
k
z
0, r d

−
ϕ
≤
∂

=

∂

>


(14)
Because it was assumed that the area of thermic influence neighboring the processing is
comparable to the processing width it may consider that
r6d
∞
≈ , and is valid the relation
(Dirichlet condition):

()
a
Tr,z,t T, z 0

∞
=>

(15)
In order to avoid the singularity in r 0
= it is considered that:


()
T0,z,t
0
r
∂
=
∂

(16)
The power flow on the processed surface corresponding to the solid state is given by the
relation (10).
As a result of laser beam action, the processed material surface heats, the temperature
reaching the melting value,
top
T at a certain moment of time. The heating goes on, so in
another moment of time, the melted material temperature reaches the vaporization value,
vap
T . That moment onward the vapor state appears in material. The equations (12), (13),
(14), and (15) still govern the heating process in all of three states (solid, liquid and vapor),
changing the material constants k and K, which will be denoted according to the state of the
point M(r,z), as it follows: k
1

, K
1
– for the solid state, k
2
, K
2
– for the liquid state, respectively
k
3
, K
3
– for the vapor state.
The three states are separated by time varying boundaries. To know these boundaries is
essential to determine the thermic regime at a certain time moment. If the temperature is
known, then the following relations describe the boundaries separating the processed
material states:
-
solid and liquid states boundary:

() ()()
top l
Tr,z,t T , r,z C t=∈

(17)
-
liquid and vapor states boundary:

() ()()
vap v
Tr,z,t T , r,z C t=∈

(18)
The material temperature rises from
top
T to
vap
T

between the boundaries
l
C(t) and
v
C(t).
The power flow on the processed surface corresponding to the liquid state is given by:

Two Phase Flow, Phase Change and Numerical Modeling

8

()
2
r
222
d
LL
L
2
AP
M,t e , r x
y

,z 0
d

−


⋅
ϕ
==+=
π

(19)
where
L
A is the absorbability on liquid surface.
The power flow on the processed surface corresponding to vapor state is given by:

()
2
V
r
d
222
VG f
M,t C e , r x
y
,z z

−




ϕ
=⋅ =+ =

(20)

where:
12
LovS
GG G
2
V
Pv
CC C
dM
η
⋅ε⋅
ρ
⋅
=+= +
π
(
V
d[m]

– radius of the laser beam on the
separation boundary between vapor state and liquid state and it is calculated with the
relation (21),
f

z
– z coordinate corresponding to the boundary between vapor state and
liquid state;
2
G
C is considered only in the vapor state, because the vaporized metal diffusing
in atmosphere suffers an exothermic air oxidation, thus resulting an oxidizing energy which
provides supplemental heating of the laser beam processed zone).

Vf
Dd
dd z
f
−
=+ ⋅

(21)
where:
D[m] is the diameter of the generated laser beam and f[m] is the focusing distance
of the focusing system.
In (14), the power losses through electromagnetic radiation,
2
r
[W /m ]ϕ and convection,
2
c
[W /m ]ϕ were taken into account (Pearsica et al., 2008a, 2008b):

()
44

rbvapa
TTϕ=σ − ,
()
cvapa
HT Tϕ= −

(22)
where:
b
σ is Stefan-Boltzmann constant, H – substantial heat transfer constant. The
emittance of irradiated area was considered as equal to 1.
2.3 Separating boundaries equations
To solve analytical the presented problem is a difficult task. The method described bellow is
a numerical one. An iterative process will be used to find the surfaces
l
C(t)
and
v
C(t)
. An
inverse method was applied, choosing the boundaries as surfaces with rotational symmetry,
ellipsoid type (Pearsica et al., 2008a, 2008b). Because the rotational ellipsoid is characterized
by a double parametrical equation:

22
22
rz
1+=
αβ

(23)
it’s enough to know the points
11
(r , z ) and
22
(r , z ) on the considered surface in order to
determine the parameters
α and
β
. The points
(r(t), 0)
and
(0, z(t))
, with
*
top
r(t ) r= and
*
top
z(t ) z= were chosen, where
top
t is the time moment when the temperature is
top
T.
On the surface
l
C(t) is known the equation relating temperature gradient and the surface
movement speed in this (normal) direction:

Modeling the Physical Phenomena Involved by Laser Beam – Substance Interaction

9

22
n
2
TL
v
nk
∂ρ
=−
∂

(24)
where:
11
2
k[Wm K ]
−−
⋅⋅ is the heat conductivity that belongs to liquid state,
2
L[J/k
g
] – the
latent melting heat,
3
2
[k
g
/m ]ρ – the mass density that belongs to liquid state, and

n
v[m/s] is the movement speed of the boundary surface,
l
C(t), in the direction of its
external normal vector
n .
The boundary at the t moment is supposed as known, respectively the points
(r(t), 0) and
(0, z(t)) on it. It is enough to determine the points
()
()
rt t,0+Δ and
()
()
0, z t t+Δ in order
to find
l
C(t t)+Δ . In the point(r(t), 0) , (24) yields:


22
r
2
TL
v
rk
∂ρ
=−
∂


2
r
22
T
k
v
Lr
∂
=−
ρ
∂

(25)
where:
()
top
Tr r T
T
rr
+Δ −
∂
=
∂Δ

It obtains:

()
()
r

rt t rt v t+Δ = + ⋅Δ

(26)
In
(0, z(t)) point, (24) yields:

22
z
2
TL
v
zk
∂ρ
=−
∂


2
z
22
T
k
v
Lz
∂
=−
ρ
∂

(27)

where:
()
top
Tz z T
T
zz
+Δ −
∂
=
∂Δ
. It results:

()
()
z
zt t zt v t+Δ = + ⋅Δ

(28)

The new boundary parameters,
(t t)α+Δ
and
(t t)
β
+Δ
, are returned by (26) and (28):

()()
ttrttα+Δ= +Δ,

()()
ttzttβ+Δ= +Δ

(29)
The moment
top
t is the first time when the above procedure is applied.
top
z(t ) 0= and
top
r(t ) 0= at this moment of time. Because the temperature gradient (having the z direction,)
is known in z 0= and r 0= :

()
L
l
z0
T
1
0,0,t
zk
=
∂
=− ϕ
∂


(30)
in (28) results:

()
()
L
top
22
0,0,t
zt t t
L
ϕ
+Δ = Δ
ρ

(31)

Two Phase Flow, Phase Change and Numerical Modeling

10
where
L
ϕ
is the power flow on the processed surface corresponding to the liquid state. In
these conditions, (26) becomes:

()
()
top
top 2
22
TTr
rt t k t

rL
−Δ
+Δ = ⋅Δ
Δ⋅ρ⋅

(32)
The same procedure is applied to find the
v
C(t) boundary, taking into account the latent
heat of vaporization
3
L[J/k
g
] , the mass density corresponding to vapor state
3
3
[k
g
/m ]ρ
and respectively the heat conductivity corresponding to vapor state
11
3
k[Wm K ]
−−
⋅⋅ .
2.4 Digitization of heat equation, boundary and initial conditions
The first step of the mathematical approach is to make the equations dimensionless
(Mazumder, 1991; Pearsica et al., 2008a, 2008c). In heat equation case it will be achieved by
considering the following (
r

∞
and
z
∞
are the studied domain boundaries, where the
material temperature is always equal to the ambient one):

2
a
1
r
r xr, z yr, T Tu, t
K
∞
∞∞
== ==τ

(33)
The heat equation (12) in the new variables
x
,
y
,
τ
, and u yields:

22
1
22
i

uu
1uuK
xx x y K
∂∂
∂∂
++=
∂∂ ∂ ∂τ
,
()
[
]
[
]
x,y 0,1 0,1∈×, 0τ≥ , and
i 1,2,3=


(34)
The initial and limit conditions for the unknown function, u yield:
-
phase 1, for
top
0tt≤<

u(x,
y
,0) 1= , (x,
y
)[0,1][0,1]∈×

(35)

u(1,
y
,) 1τ= ,
y
[0, 1]∈ ,
top
[0, ]τ∈ τ ,
1
top top
2
K
t
r
∞
τ=

(36)

u(x,1, ) 1τ= , x[0,1]∈ ,
top
[0, ]τ∈ τ

(37)
-
phase 2, for
top vap
ttt≤<

top top
u(0,0, ) uτ=

(38)

top 1 top
u(x,
y
,)u(x,
y
,)τ= τ,
(x,
y
)(0,1](0,1]∈×


(39)
where:
top
τ is the τ value when
top
uu,=
top top a
uT/T,= and
1top
u(x,
y
,)τ is the heat
equation solution in according to phase 1.
If

top vap
[, )τ∈ τ τ both solid and liquid phases coexist in material, occupying
s
D()τ and
l
D( )τ
domains respectively, which are separated by a time varying boundary,
l
C( )τ , so
top
u(x,
y
,) uτ= on it. The projection of the domain
l
D( )τ on
y
0= plane is the set
1
{x /x x }≤ . For x 1= and
y
1= respectively, the conditions are:

Modeling the Physical Phenomena Involved by Laser Beam – Substance Interaction

11
u(1,
y
,) u(x,1,) 1τ= τ=

(40)

Phase 2 is going on while
top vap
[, )τ∈ τ τ , where:
vap 1
vap
2
tK
r
∞
⋅
τ=
.
-
phase 3, for
vap
tt≥

vap vap
u(0,0, ) uτ=

(41)

vap 2 vap
u(x,
y
,)u(x,
y
,)τ= τ,
lvap
(x,

y
)D( )\(0,0)∈τ

(42)

vap 1 vap
u(x,
y
,)u(x,
y
,)τ= τ,
svap
(x,
y
)D( )∈τ

(43)
where
2vap
u(x,
y
,)τ is the heating equation solution from phase 2. In this temporal phase all
the three (solid, liquid and vapor) states coexist in material, occupying the domains:
s
D()τ
,
l
D( )τ and
v
D()τ , separated by mobile boundaries

l
C( )τ and
v
C()τ , on which
vap
u(x,
y
,) uτ= . The projection of the domains
l
D( )τ and
v
D()τ on plane
y
0= are the sets:
21
{x /x [x , x ]}∈
and
2
{x /x [0, x ]}∈
. According to phase 3, the conditions on
y
0=
surface
(Neumann type conditions) are:
a.
21
d
xx
r
∞

≤≤:

()
[]
() (
]
()
Vf rc 2
a3
L21
a2
S1
a1
r
x,
y
,,x0,x
Tk
r
x,0, , x x , x
Tk
u
y
rd
x,0, , x x ,
Tk r
d
0, x ,1
r
∞

∞
∞
∞
∞


−ϕτ−ϕ−ϕ∈


⋅



−ϕτ∈

⋅

∂
=

∂


−ϕτ∈



⋅






∈







(44)
b.
21
d
xx
r
∞
≤≤:

()
[]
()
Vf rc 2
a3
L2
a2
r
x,
y

,,x0,x
Tk
u
rd
x,0, , x x ,
yTk r
d
0, x ,1
r
∞
∞
∞
∞


−ϕτ−ϕ−ϕ∈


⋅




∂

=− ϕ τ ∈



∂⋅






∈








(45)

Two Phase Flow, Phase Change and Numerical Modeling

12
c.
2
d
x
r
∞
> :

()
Vf rc
a3

rd
x,y , , x 0,
Tk r
u
y
d
0, x ,1
r
∞
∞
∞




−ϕτ−ϕ−ϕ∈




⋅
∂



=

∂



∈







(46)
Similar Neumann type conditions are settled for temporal phases 1 and 2, accordingly to
their specific parameters.
For x 1
= , and
y
1= respectively, the conditions are given by (40).
2.5 Digitization of equations on separation boundaries
The speed of time variation of separation boundaries,
n
v
, is given by (47), where n is the
external normal vector of the boundary.

e
n
ee
T
k
v
Ln
∂

=−
ρ
⋅∂
,
e2,3=


(47)
For
y
0= and
f
xx= , it results:

eea
rf
ee ee
Tu
kkT
v(x,0)
Lr Lrx
∞
∂∂
⋅
=− =−
ρ
⋅∂
ρ
⋅⋅∂
, e2,3=

(48)
respectively,

1
rf
dr K dx
v(x,0)
dt r d
∞
==
τ


(49)
It results:

ea
ee 1
u
dx k T
dLKx
∂
⋅
=−
τ
ρ
⋅⋅ ∂
, e2,3=

(50)
The
α parameter of separation boundary at τ+Δτ moment is:

ea
ff f
ee 1
u
dx k T
x( ) x( ) x( )
dLKx
∂
⋅
α= τ+Δτ = τ + Δτ= τ − Δτ
τρ⋅⋅∂
, e2,3=

(51)
where:

kf
kf
u u(x ,0) u(x ,0)
xxx
∂−
≈
∂−


(52)

where
k
x ∈ digitization network. For
top
τ=τ and
vap
τ=τ respectively, (52) yields:

11 top
ftop
1
u(x,0) u
u
,x( )0
xx
−
∂
=τ=
∂


(53)

Modeling the Physical Phenomena Involved by Laser Beam – Substance Interaction

13

21 vap
fvap
1

u(x,0) u
u
,x( )0
xx
−
∂
=τ=
∂


(54)
For x 0
= and
f
yy
= , it results:

eea
zf
ee ee
Tu
kkT
v(0,y)
Lz Lr
y
∞
∂∂
⋅
=− =−
ρ

⋅∂
ρ
⋅⋅∂
, e2,3=

(55)
respectively,

1
zf
d
y
dz K
v(0,y)
dt r d
∞
==
τ


(56)
It results:

ea
ee 1
d
y
u
kT
dLK

y
∂
⋅
=−
τ
ρ
⋅⋅ ∂
, e2,3=

(57)
The
β
parameter of separation boundary at
τ+Δτ
moment is:

ea
ff f
ee 1
dy u
kT
y( ) y() y( )
dLKy
∂
⋅
β
=τ+Δτ=τ+Δτ=τ− Δτ
τρ⋅⋅∂
,
e2,3=

(58)
where:

kf
kf
uu(0,
y
)u(0,
y
)
yyy
∂−
≈
∂−


(59)
For
top
τ=τ it results:

Ltopftop
a2
u
r
(0,0, ),
y
()0

yTk
∞
∂
=−
ϕ
ττ=
∂⋅


(60)
For
vap
τ=τ it results:

Vtoprcfvap
a3
u
r
(0,0, ) ,
y
()0
yTk
∞
∂

=−
ϕ
τ−
ϕ
−

ϕ
τ=

∂⋅


(61)
3. Determination of temperature distribution in material
Using the finite differences method, the domain [0, 1] [0, 1]× is digitized by sets of
equidistant points on Ox and Oy directions (Pearsica et al., 2008a, 2008b).
3.1 Digitization of mathematical model equations
In the network points, the partial derivatives will be approximated by:

()
i1,
j
i1,
j
i,j
uu
u
x2x
+−
−
∂
≈
∂Δ
,
()
()

2
i1,
j
i,
j
i1,
j
2
2
i,j
u2uu
u
x
x
+−
−+
∂
≈
∂
Δ


(62)