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8/6/2019 Two Phase Analysis of Heat Transfer and Dispersion of Nano Particles in a Micro Channel
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Proceedings of 2008 ASME Summer Heat Transfer ConferenceHT2008-56342
August 10-14, 2008, Jacksonville, Florida USA
Two phase Analysis of Heat Transfer and Dispersion of NanoParticles in a Microchannel
Seyed Mojtaba Mousavi NayinianK.N.Toosi University of Technology, Iran
Mehrzad ShamsK.N.Toosi University of Technology, Iran
Hossein Afshar K.N.Toosi University of Technology, Iran
Goodarz AhmadiClarkson University, USA
Abstract:
The effect of different parameters on dispersion of nanoparticles in a microchannel in slipflow regime is studied. The equations of particle motion and energy balance are solved
numerically and the effect of particle diameter, starting position of particles in microchannel,
and slip coefficient on dispersion of particles is discussed. Radiative heat flux in energyequation and drag force, Saffman lift force, Brownian force and gravitational force in
momentum equation are included. The results show that the Brownian force has considerableeffect on particle motion in microchannel. Particles temperature at the outlet can be
controlled by variation of their diameter and starting position in microchannel.
Keywords: Two Phase Flow, Nanoparticle, Microchannel, Brownian Force, Saffman LiftForce, Heat Transfer
Introduction
Two phase modeling of nanoparticles is of
ionterest for many medical and engineeringapplications including dust and aerosol
control, fire fighting systems, treating skindiseases and tumors. The nanoparticles
coated with the suitable antibodies are used
as labels to detect the malignant tumorsand to get adsorbed on the surface of thetumor cells. Many nanoparticles respond to
an externally applied field includingmagnetic field or focused light and others.
The nanoparticles convert the absorbed
energy of the external field to heat anddestroy the cell to which they are adsorbed
to [1]. Therefore, the tumor shrinks or aretotally destroyed under the controllable
power of the external field.
G. Aguilar et al. (2002), studied themodeling and characterization of cryogen
spray cooling for application to port wine
stain laser therapy [2]. The patients are
treated with laser pulses that induce permanent thermal damage to the target
blood vessels. However, absorption of laser energy by melanin causes localized
heating of the epidermis, which may result
in complications, such as hypertrophy,scarring, or dyspigmentation. By applyinga cryogen spurt in the form of very tiny
particles to the skin surface for anappropriately short period of time (10 to
100 milliseconds), the epidermis can be
precooled prior to the application of thelaser pulse and, therefore, reduce or
eliminate undesirable skin damage.Lin and Yang (2005) simulated the heat
transfer problem when the Alanine tissue is
heated by the gold nanoparticle in the fieldof molecular dynamics [3]. They
HT2008-56342
Proceedings of 2008 ASME Summer Heat Transfer ConferenceHT2008
August 10-14, 2008, Jacksonville, Florida USA
1 Copyright © 2008 by ASME
8/6/2019 Two Phase Analysis of Heat Transfer and Dispersion of Nano Particles in a Micro Channel
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2
investigated two kinds of problems. One isthe Alanine tissue heated by the constant
heat source and the other is by the time-varying heat source. The numerical results
show that a temperature jump exists
around the source and the temperature profiles drop to the environmental
temperature within a very short distance. Itconcludes that only a small region around
the nano-scale heat source is affected bythe heating process. Therefore, the results
of the nanoparticle-heating method could be applied to the clinical therapy of tumor,
and the normal cells are destroyed onlywithin a smaller region when compared
with those of chemotherapy or surgical
treatment.The classical continuum theory is not
suited for the analysis of nano-scale processes because the particle motion
becomes the major influencing actor in thenano-scale system. In addition, some
macro-scale processes do not existein thenano-scale regimes such as the no-slip
condition and the local heat equilibrium. Aspecial phenomenon that is observed in the
nano-scale is the thermal jump [4].Most reported investigations in the field of
fluid flow in microchannels, are concerned
with fluid flow and dispersion of nanoparticles in microchannels in
continuum regime. In particular, fluidflow and dispersion of nanoparticles in the
slip flow regime is not well understood. Inmost macro-scale applications, the fluid
flow in channels is in turbulent flowregime but in micro-scale and nano-scale
applications, most fluid flows are inlaminar regime.
The Knudsen number is the ratio of mean
free path over flow characteristic lengthwhich defines flow characteristics when
the flow dimensions approach themolecular mean free path. The Knudsen
number, which is a non dimensionalquantity, is defined as
L Kn
(1)
Where L is the flow characteristic length,(hydraulic diameter in a microchannel),
and is the molecular mean free path.Flow regime is defined according to the
value of Knudsen number.
Continuum Flow310kn Slip Flow13 1010 kn Transitional Flow1010 1 kn Free Molecular
Flow )10(Okn
Tian and Ahmadi (2007) reported the
results of their studies on transport anddeposition of nano- and micro-particles in
turbulent flow fields. They conducted aseries of numerical simulations to study the
transport and deposition of nano andmicro-particles in a turbulent duct flow
using different turbulence models [5].
For nanoparticles, the Brownian force isconsiderable. In addition, the Stokes drag
needs to be modified.Ahmadi et al. (1991) studied the Brownian
dispersion of submicrometer particles inthe viscous sublayer. The particles were
released from a point source in the viscoussublayer of a turbulent shear flow near a
smooth wall. The effect of particlesdiameter, distance of point source from the
wall and the particle-to-fluid density ratio
on dispersion of particles was studied [6].The Knudsen number for nanoparticles
traveling in air at standard atmosphericconditions, based on the particle diameter
is greater than 0.1 and the particles are intransitional regime. Therefore, the
correction factor for the Stokes drag should be considered.
Karr and Owen (2007) studied the dragforce of nanoparticles in the transitional
flow regime and introduced the proper
correction factor for the Stokes dragcoefficient [7].
In this study, an Eulerian-Lagrangianapproach is used and the airflow condition
and nano-particle dispersion inmicrochannels were studied. .
Governing Equations
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The momentum equation for particles in xand y directions are given as
)(1
1 t nuudt
du p f p
(2)
g m
Saffman F
t nvvdt
dv
p
L
p f p
)(
)(1
2
(3)
Here f u is the fluid velocity, pu is the
particle velocity, is the molecular mean
free path, Cc is the Cunningham correction
factor to the Stokes drag [7] given as
999.0,558.0,142.1
exp1
Kn
KnC c (4)
Here is the particle relaxation time and is
defined as
18
2d C c p (5)
The Saffman lift force is given by [8].
dy
duSgn
dy
du
uud saffman F
f f
p f
f L5.0
25.0 )(615.1)(
(6)
In this equation, f is the fluid density and
is the fluid kinematic viscosity.
In Eqs. (2) and (3), ni(t) is the Brownianrandom force that is evaluated at each time
step using Eqs. (7), (8) and (9) [9].
t
S Gt n ii
0)(
(7)
cC S d
kT S
2520
216
(8)
f
pS
(9)
Here Gi is a zero mean, unit variance
Gaussian random number, T is the fluid
temperature in Kelvin, and
K
jk 2310*38.1 is the Boltzmann
constant.
To calculate the particle velocity for achannel flow shown in figure 1 in each
time interval, the fluid velocity should beknown. From analytical solution, the
velocity profile for slip flow in a
microchannel is given as
Eq. (10) is the solution to the momentum
equation for particles in x and y directionsgiven by Eqs. (2) and (3). Here y is the
distance from channel center line and v is
the tangential momentum accommodation
coefficient (TMAC) which is set accordingto experimental data [10].
The volume fraction is set to be less than0.5% and it is assumed that the
temperature variation of particles will notcause any variation in the fluid
temperature.The energy equation for particles is givenas
)( 44
p f p p
p pc
p
p
p
T T A
T T Ahdt
dT cm
(11)
where C p is the particle specific heat, T p isthe particle temperature, hc is the
convection heat transfer, Tf is the fluid
temperature, p is the particle emissivity
coefficient and 81067.5 is the
Stefan-Boltzmann coefficient.The Reynolds number and the Prandtl
number of nanoparticles is less than unity,
so the Nusselt number is [11]
2 Nu (12)
The temperature of nanoparticles in each
time step is evaluated from
Knh
y
dx
dphu
v
v
f
281
2
2
2
(10)
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4
44
1 6
p
n
p p
n
p
p
f
p p p
n
p
n
p
T T T T T d
k Nu
d c
t T T
(13)
where k f is the fluid conductivitycoefficient.
Results
Dispersion of nanoparticles in a 2-Dmicrochannel with constant wall
temperature is studied. Air temperature isset at 800K. The particles-to-fluid density
ratio is assumed to be 9046. In this
analysis, channel height and length,
respectively, are 4 micron and 1mm. Thetemperature and velocity of nanoparticlesat channel entrance is set to 300K and 8.69
m/s, respectively. The pressure differencein microchannel is 100 KPa. The fluid flow
in the microchannel is assumed to be fullydeveloped. The Reynolds number based
on the microchannel hydraulic diameter and the mean velocity for all cases studied
is less than 10 so that the flow is laminar.
Dispersions of particles with diameters of
500, 200, 100 and 50 nanometers areshown, respectively, in figures 2, 3, 4 and5. In these cases, the tangential
momentum accommodation coefficient isset to unity. As the diameter of particles
decreases, the particle Brownian diffusionrapidly increases. Particles temperature at
the microchannel outlet is equal to the fluidtemperature of 800K. Figures 6 and 7
show the temperature variation of particles
as they travel in the microchannel. Thetemperature of 50nm and 100nm particles
reach the fluid temperature after travelingless than 4 micrometers but the 500 nm
particles reach the fluid temperature after traveling about 100 micrometers.
The effect of convection and radiation heattransfer on temperature variation of
particles with different diameters is shownin table 1. Obviously radiation effect on
larger particles is more than smaller ones
but after comparison of radiation effectwith convection effect, it is concluded that
radiation effect can be neglected for particles with less than 500nm in diameter.
Dispersion of 50 and 500 nanometer particles that are released from different
sources in the microchannel is shown in
figures 8 and 9, respectively. Since theinitial velocity of particles is more than the
local fluid velocity, the Saffman lift forcecauses 500 nanometer particles to travel
out of the shear flow near the wall but thisforce does not have a noticeable effect on
the trajectory of 50 nanometer particles.Variation of Saffman Lift force on
particles with 500 nanometers in diameter for TMAC=1 and TMAC=0.2 is shown in
figure 12. Reduction of TMAC from unity
to 0.2 causes an increase on the fluidvelocity in the microchannel according to
equation (10), so the particles velocity(which is constant and equal to 8.69 m/s) at
the entrance of the microchannel will beless than the local fluid velocity. This
velocity difference for 500nm particlesreleased near the upper wall causes a
positive Saffman Lift force and makesthem travel closer to the wall.
Trajectory of nanoparticles for TMACequal to 0.2 is shown in figures 10 and 11.
Reduction of TMAC causes more velocity
slip and thinner shear layer. So, 500nanometer particles can travel near the
wall. Comparison of figures 8, 9, 10 and 11shows that the reduction of TMAC makes
the particles to follow the stream lines.
Conclusions
In general, the Brownian force has
considerable effect on nanoparticlemotions in a microchannel. Particles
temperature at the outlet can be controlled
by variation of their diameter, starting position and slip coefficient in a
microchannel.
Other forces such as weight and Saffman
lift force can have important effectdepending on the flow conditions.
Nomenclature
λ Melocualr mean free pathL Charactristic length if the flow
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Kn Knudsen number u p Particle velocity in x direction
v p
Particle velocity in y directionuf Fluid velocity in x direction
vf
Fluid velocity in y direction
Particle relaxation timeni(t) Brownian force per unit of massm p Mass of the particle
Cc Correction factor for Stokes drag
f Fluid density
p Particle density
Fluid kinematic viscosity
μ Fluid dynamic viscosityd Particle diameter
Gi A random number between 0-1 with
Gaussian distributionT Fluid temperature in Kelvin
T p Particle temperature in Kelvin
h Half of microchannel widthy Distance from the center of channel
v Tangential momentum
accommodation coefficient (TMAC)
Stefan-Boltzmann coefficient
k Boltzmann coefficientk f Fluid thermal conductivity
coefficientC p Particle specific heathc Convection heat transfer
p Particle emissivity coefficient
Nu Nusselt number
References[1] I. Hilger, R. Hergt, W.A. Kaiser, IEEE
Proc. Nanobiotechnol. 152 (1) (2005)33–39.
[2] Guillermo Aguilar; "On the Modeling
and Characterization of Cryogen SprayCooling for Application to Port WineStain Laser Therapy"; 9th LatinAmerican Congress in Heat and Mass
Transfer, San Juan, Puerto Rico,October 20-22, 2002
[3] David T.W. Lin a, Ching-yu Yang, "Theheat transfer analysis of nanoparticleheat source in alanine tissue bymolecular dynamics", InternationalJournal of Biological Macromolecules36 (2005) 225–231
[4] Peng, X. F., and Wang, B. X., Proc. 10thInternational heat transfer conference,Brighton, UK, Aug, 14-18, pp.159-177,1994
[5] L. Tian, G. Ahmadi, "Particle depositionin turbulent duct flows—comparisons of different model predictions", AerosolScience 38, 377 – 397, 2007
[6] H. Ounis, G. Ahmadi, and J. B.McLaughlin; "Brownian Delusion of Sub micrometer Particles in the Viscous
Sublayer"; Journal of Colloid andInterface Science, 143(1):266{277,1991.
[7] Gerald Karr and Miles Owen, "Drag of Nano-Particles in the Transitional FlowRegime", Rarefied Gas Dynamics: 25-thInternational Symposium, 2007,1112-1127
[8] P.G.Saffman; "The Lift on a SmallSphere in a Slow Shear Flow"; J. FluidMech., 22:385{400, 1965
[9] A. Li and G. Ahmadi; "Dispersion and
Deposition of Spherical Particles fromPoint Sources in a Turbulent ChannelFlow"; Aerosol Science andTechnology, 1992
[10] Timothée Ewart, Pierre Perrier, Irina A.
Graur and J. Gilbert Méolans, "Massflow rate and tangential momentumaccommodation coefficient fromexperiments in a single micro tube",Rarefied Gas Dynamics: 25-th
International Symposium, 2007, 567-572
[11] "NANOFLUIDICS science andtechnology", S.K.Das, S.U.S.Choi,W.Yu, T.Pradeep, John Wiley & Sons,2007
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Figure 1. 2D M icrochannel
Figure 2. Trajectory of 500 nm particles Figure 3. Trajectory of 200 nm particles
200
300
400
500
600
700
800
0.0E+00
1.0E-06
2.0E-06
3.0E-06
4.0E-06
5.0E-06
6.0E-06
X (m)
P a r t i c l e T e p m e r a t u r e ( k )
50nm
100nm
Figure 6. Temperature variation of 50 and
100nm particles in the microchannel
200
300
400
500
600
700
800
0.0E+00
2.0E-05
4.0E-05
6.0E-05
8.0E-05
1.0E-04
X (m)
P a r t i c l e T e p m e r a t u r e ( k )
200nm
500nm
Figure 7. Temperature variation of 200 and
500nm particles in the microchannel
Figure 4. Trajectory of 100 nm particles Figure 5. Trajectory of 50 nm particles
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Particle Diameter Convection Effect (%) Radiation Effect (%)
50nm 99.9953 0.0047100nm 99.9896 0.0104200nm 99.9788 0.0212500nm 99.9468 0.0532
Figure 8. Dispersion of 50 nm particles in
the microchannel (TMAC=1)
Figure 10. Dispersion of 50 nm particles in
the microchannel (TMAC=0.2)
Table 1. Effect of convection and radiation on temperature variation of particles
Figure 9. Dispersion of 500 nm particles in
the microchannel (TMAC=1)
Figure 11. Dispersion of 500 nm particles
in the microchannel (TMAC=0.2)
-3.E-12
-2.E-12
-1.E-12
0.E+00
1.E-12
2.E-12
3.E-12
4.E-12
5.E-12
6.E-12
0.0E+00
2.0E-05
4.0E-05
6.0E-05
8.0E-05
1.0E-04
X (m)
S a f f m a n L i f t F o r c e ( N )
500nm-TMAC=1
500nm-TMAC=0.2
Figure 12. Variation of Saffman Lift
force for different TMAC
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