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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)
