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Numerical Simulation of the Solidification Process of Nanofluids

Y. M. F. El Hasadi * and J. M. Khodadadi *

*Mechanical Engineering Department, Auburn University

207 Ross Hall, Auburn, Alabama 36849-5341

Tel: (334) 844-3333, Fax: (334) 844-3307; E-Mail: khodajm@auburn.edu

.

ABSTRACT

The effects of mass transfer of the nanoparticles on the

morphology of the solid-liquid interface and evolving

concentration field during solidification of colloids have been

reported. The numerical method that has been used was based

on the one-fluid-mixture model. The model takes into

account the thermal as well as the solutal convection effects.

A differentially-heated square cavity was used in the

simulation. The colloid was composed of a suspension of

copper nanoparticles in water. The temperature difference

between the hot and cold sides was 5 degrees centigrade and

the loadings of the nanoparticles that have been used in the

simulation were 1%, 5%, and 10% by mass. The solid-liquid

interface for the case of nanofluid with 10 wt% of

nanoparticles evolved from a planar shape at the beginning of

the solidification process to a dendritic shape as the

solidification process proceeds in time.

Keywords: Nanofluids, colloidal , solidification , mushy

zone, dendrites .

1 INTRUDUCTION

Investigations of nanofluids which can be considered as a

class of colloidal suspensions have been reported by the

scientific community for the past few years. The improved

thermophysical properties of nanofluids has promoted these

colloids to be considered as serious candidates to replace

conventional heat transfer fluids. More recently, due to the

great demand for improvement of thermal energy storage

systems, Khodadadi and Hosseinizadeh [1] proposed the

idea of suspending nano-size particles in phase change

materials (PCM), in order to improve their properties. This

concept is also known as nanoparticle enhanced phase

change materials (NEPCM).

Disclaimer: This report was prepared as an account of work sponsored by an

agency of the United States Government. Neither the United States

Government nor any agency thereof, nor any of their employees, makes any

warranty, express or implied, or assumes any legal liability or responsibility

for the accuracy, completeness, or usefulness of any information, apparatus,

product, or process disclosed, or represents that its use would not infringe

privately owned rights. References herein to any specific commercial product,

process, or service by trade name, trademark, manufacturer, or otherwise does

not necessarily constitute or imply its endorsement, recommendation, or

favoring by the United States Government or any agency thereof. The views

and opinions of authors expressed herein do not necessarily state or reflect

those of the United States Government or any agency thereof.

Ideally, NEPCM can go through cycles of melting and

solidification through their life span. Modeling of those

processes will enable one to better predict their

performance. Generally, there are two class of methods

that are found in the literature. The first one considers the

interaction of a single particle with the solid-liquid interface

such as [2] and other one considers the particles as a

concentration field and uses the same methods that are used

to simulate multi-component mixtures such as binary

alloys, and aqueous solutions such as [3]. Peppin et al. [3]

developed a similarity solution for the volume fraction and

the temperature ahead of the planar interface in the case of

unidirectional solidification for hard sphere colloidal

suspensions by considering the particles as a concentration

field. The same method has been used to solve the

solidification of binary mixtures. They developed relations

for the mass diffusivity of the particles as a function of the

volume fraction using the classical theory of colloidal

suspensions. They assumed also that the growth velocity

was very small and all the particles have been rejected.

They found that for very small particles, from the solid

Brownian diffusion is important and the concentration and

temperature profiles resemble those observed during alloy

solidification. Also, in certain cases they found that the

interface can become unstable due to the constitutional

supercooling. However, for large particles Brownian

diffusion is weak or absent, and a porous medium is formed

against the freezing front. Due to the fact that the porous

medium is supercooled, it allows for the morphological

instability of the interface.

To the best knowledge of the authors there is no study in

the literature that considers the solidification of colloidal

suspensions, which takes into account the effects of the

thermal and solutal convections. So, the objective of the

present paper is to employ the one-fluid mixture model,

which will account for the complicated convection and

phase change process during the dendritic solidification of a

NEPCM colloidal suspension. For this purpose, the

physical system of a square cavity with vertical sides kept

at uniform temperatures has been selected. Initially, the

cavity is occupied with water/copper nanoparticles. The

suspension is solidified by lowering the right wall

temperature below the liquidus temperature that

corresponds to the initial concentration.

2 MATHEMATICAL MODEL

The model equations are obtained by integrating the

microscopic conservation equations over a small volume

element. The volume element is occupied simultaneously by

the liquid and solid phases of the colloidal suspension. The

model is based on the one-fluid mixture model as described in

[4] and implemented by the commercial code FLUENT

which uses the enthalpy method, and calculates liquid fraction

explicitly. The averaged equations in non-dimensional form

that are valid in the liquid and solid regions as well as the

mushy zone, can be summarized as the following:

Continuity:

,0

y

v

x

u (1)

x-direction Momentum:

u

L

C

u

x

P

y

u

v

x

u

u

u

3

2

2)1(

Pr)(

(2)

y-direction Momentum:

v

L

C

Rav

y

P

y

v

v

x

v

u

v

3

2

2)1(

PrPr)(

(3)

Thermal Energy:

))()()

)

1

(

(

2

)( Ste

v

y

Ste

u

x

Ste

y

v

x

u

(4)

Species:

]

11

.[)(

T

w

www LeLey

v

x

u (5)

By denoting fluid and particles with subscripts f and p,

respectively, the thermal conductivity of the NEPCM can

be computed from the following relation:

(6a)

(6b)

(6c)

The Brownian diffusivity is calculated from:

,

d)z(d

)1(

d3

Tk

D6

p

B

B

(7a)

And for spherical nanoparticles we have:

64.0

1

)

64.010

18()

64.04

10()

64.0 1

4(1

)(z

32

(7b)

Equations (7a-b) are obtained from[ 3]. The thermophoretic

diffusivity is computed from following relation :

,

kT

D (8)

The rest of the thermal physical properties of the NEPCM

are determined by using a mixture law.

At the liquid-solid interface, the following relation holds

with subscripts l and s referring to the liquid and solid

phases, respectively:

interface0interface k)()( wlws

(9)

The geometry that has been selected is shown in Fig 1. The

computational domain consisted of 10,000 equally-spaced

cells and the time step varied from 0.01 to 0.1 sec. The

total number of time steps was 20,000 steps which were

sufficient to cover the time span in which all the

solidification phenomenon’s occurs. The convergence

criteria for each time step was that the residuals of the

continuity and momentum reached a value below 10-5 and

those for solute and thermal energy were below 10-7.

Figure 1 The geometry of the physical model

L

L

TC TH

g

)(2

)(22

1pffp

pffp kkkk

kkkk

k

pp dvuCKk

22

2)(

21 kkk

Benchmarking of the model was conducted by comparing

to the detailed one-component melting study by Hannoun et

al. [5] of a differentially-heated square cavity. The

interface locations after 100 and 200 seconds of initiation of

melting are shown in Fig 2. Very good agreement between

the results of the model used here and the computational

results of [5] is observed.

0.00 0.02 0.04 0.06 0.08 0.10

0.009

0.010

0.011

0.012

0.013

Ref[12], t=100 sec

Current model, t= 100 sec

Ref[12], t = 200 sec

Current model t = 200 sec

Interface location (m)

height of the cavity (m)

Figure 2 Comparison between the predicted values of the

solid-liquid interface and those of Ref [5] after t = 100 and

200 sec.

3 RESULTS

The numerical investigations were carried out for the case of

unidirectional solidification of water with copper

nanoparticles NEPCM contained within a square cavity with

the following operational parameters ( H

= 1, C

= 0, and,

with initial

= 0.2) and with the following nanoparticles

mass fractions 1%, 5% and 10%. Copper was chosen since

a similar colloid was studied earlier [1]. The diameter of

the spherical nanoparticles was 10 nm and the segregation

coefficient was set to ko = 0.1. As for thermal conductivity

of copper, a value for bulk copper was used and quantum

effects were not considered.

As shown from Fig. 3, the interface tends to be planar at the

early stages of the solidification process, however at later

times the solid-liquid interface assumes a dendritic shape

due to the phenomena of constitutional supercooling which

has been observed experimentally during colloidal

solidification [3]. Fig. 4 shows the development of

concentration field. It is clearly shown that at t = 100s,

concentration is nearly uniform throught the domain expect

near the interface. However, at t = 1000 sec, the

nanoparticles move away from the interface due to the

solutal convection and segregate in the space between the

dendrites.

Figure 3 Development of the liquid fraction field at

different time instants: (a) 100 s and (b) 1,000 s for an

initial concentration field of 10 percent.

Figure 4 Development of the nanoparticle concentration

field at different time instants (a) 100 s, and (b) 1,000s, for an

initial concentration field of 10 percent.

(a)

(b)

(a)

(b)

Figure 5 Comparison between the contours of the liquid

fraction for the case of (a) w

= 1% and (b) w

= 5% and

(c) w

= 1% after t=1000 s..

As shown in Fig 5, as the initial nanoparticles concentration

increases the shape of the solid-liquid interface changes

from a planar shape to a dendritic shape.

4 CONCLUSIONS

The solidification process of copper–water nanofluid has

been investigated. It has been observed that the movement

of the nanoparticles significantly affect of the shape of the

interface. Also it has been shown that solutal convection

plays a role in the distribution of the nanoparticlers.

ACKNOWLEDGEMENTS

This material is based upon work supported by the US

Department of Energy under Award Number DE-SC0002470.

The first author acknowledges the support of the Department

of Mechanical Engineering at Auburn University through

providing a teaching assistantship.

NOMENCLATURE

L Length of enclosure

(m) Ra Rayleigh number

3

CH L)TT(g

DB Brownian Difusivity

(m2s-1) Lh Latent heat

Ste Stefan number ,

h

CHp L

)TT(c

Le Lewis number

(B

D/

) Pr Prandtl number,

k

cp

LeT )D/( T

u uL/

T temperature v vL/

u Velocity in x

direction

(T-TC)/(TH-TC)

v Velocity in y

direction

2

L

t

Thermal

diffusivity

Thermal

expention

coefficent

REFERENCES

1. Khodadadi, J. M. and Hosseinizadeh, S. F.,

Nanoparticle-Enhanced Phase Change Materials

(NEPCM) with Great Potential for Improved

Thermal Energy Storage, International

Communications in Heat and Mass Transfer, Vol.

34, pp. 534-543, 2007.

2. Garvin, J. W. and Daykumar, H. S., Particle–

Solidification Front Dynamics Using a Fully

Coupled Approach, Part I: Methodology, Journal

of Crystal Growth, Vol. 252, pp. 451-466, 2003.

3. Peppin, S. S. L., Elliott, J. A. W., and Worster, M.

G., Solidification of Colloidal Suspensions, J.

Fluid Mechanics, Vol. 554, pp. 147-166, 2006.

4. Voller, V. R., Brent, A. D., and Prakash, C., The

Modeling of Heat, Mass and Solute Transport in

Solidification Systems, International Journal of

Heat and Mass Transfer, Vol. 32, pp. 1719-1731,

1989.

5. Hannoun, N., Alexiades, V., and Mai, T.Z., A

Reference Solution for Phase Change with

Convection, Int. J. Numer. Meth. Fluids, Vol. 48, pp.

1283-1308, 2005.

(a) (b)

(c)