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
Janeza Trdine 9, 51000 Rijeka, Croatia

Copyright © 2011 InTech
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have the right to republish it, in whole or part, in any publication of which they 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 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

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.

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

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

∂∂
∂∂ ∂
=+ +


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

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,

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⋅ ⋅




(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


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
+≤ =
∂

=

∂

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



=∈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


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

, 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=∈

,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

−


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

(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

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
=−
∂


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)

()()
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:


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

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]∈×


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

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,

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
∞

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

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

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

=−
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=

(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)

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

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=

yTk
∞
∂
=−
ϕ
ττ=
∂⋅(60)
For
vap
τ=τ it results:

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

2
i1,
j
i,
j
i1,
j
2
2
i,j
u2uu
u
x
x
+−
−+
∂
≈
∂
Δ(62)