two phase analysis of heat transfer and dispersion of nano particles in a micro channel

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

[email protected]

Mehrzad ShamsK.N.Toosi University of Technology, Iran

[email protected]

Hossein Afshar K.N.Toosi University of Technology, Iran

[email protected]

Goodarz AhmadiClarkson University, USA

[email protected]

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

mailto:[email protected]












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

Copyright © 2008 by ASME



3

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)




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




5

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




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




7

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)


the microchannel (TMAC=0.2)

Table 1. Effect of convection and radiation on temperature variation of particles


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


two phase analysis of heat transfer and dispersion of nano particles in a micro channel

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