THEORETICAL MODEL TO
DETERMINE THE THERMAL
CONDUCTIVITY OF NANOFLUIDS
P. C. MUKESH KUMAR 1,* , J. KUMAR 2 , S. SENDHILNATHAN1
1
Anna University Tiruchirappalli, Pattukkottai Campus, Raja madam- 614 701 , Tamilnadu, India.
2
Erode Builders Educational Trust Group of Institutions , Kangeyam- 638 108, Tamilnadu, India.
*Corresponding author Email: pcmukeshkumar1975@gmail.com. Abstract
The thermal conductivity is the key role in presenting the unique thermal properties of nanofluids. In this paper, a new thermal conductivity model is proposed based on the combination of statistical mechanism model and Brownian motion with the inclusion of particle critical size. This model is compared with Al2O3/water and CuO
/water based nanofluids of spherical particles using the well known thermal conductivity models and the experimental results available in the open literature. This model is found fits well with the existing Brownian motion theoretical model and experimental results. It concludes that thermal conductivity is enhanced due to the effect of shape, nanolayer and Brownian motion of the particles. The Brownian motion contribution is significant only when the particle size is less that that of critical size and optimum particle volume fraction.
Key words: Nanofluids, Thermal conductivity, Optimum particle volume fraction, Brownian motion, Particle critical size,
1. Introduction
Nanofluid is the mixture of continuous base fluid and discontinuous suspended nanoparrticles in the base fluids. The thermal conductivity of heat transfer fluids contributes to the development of energy –efficient heat transfer equipments. Since all other cooling options have been exhausted nanofluids is the only option remaining with the possibility of increased heat transfer capability of the present systems. Therefore developing a novel heat transfer fluids becomes an urgent need in order to meet out the high capacity cooling requirements. S.U.S. Choi 1 proposed a heat transfer fluids, called nanofluids, as a fluids with suspended nanoparticles in the base fluids. S.U.S.Choi 1 reported the nanofluids have superior thermal properties to the conventional fluids.Nanofluids are proposed for a wide variety of industries, ranging from transportation to energy production and medical applications. The theoretical models to predict thermal conductivity of heat transfer nanofluids can be grouped into two models, such as static model and dynamic model. The static model accounts for the particle shape , nanolayer thickness and thermal conductivity of base fluids and particle volume fraction. The dynamic model accounts for the movement of particles, particle size, concentration and temperature.
1.1 Classical models
well with the Maxwell model but not specified the particular shape of nanoparticles. This model is commonly used for convection problems.
All the classical models depend on the particle shape, particles volume fraction, and base fluids and particle thermal conductivities in formulating .The classical models were found to be unable to predict the anomalously high thermal conductivity of nanofluids. This is because classical models do not include the effect of temperature dependant nanofluids, effects of particle size, interfacial layer of the particle fluids, nanoparticles cluster / aggregate and Brownian motion of particles. It is reported that the conduction is the mode of heat transfer for anomalous thermal conductivity of nanofluids.
1.2 Extension of existing conduction models 1.2.1 Static models
The models were formulated by assuming that the nanoparrticles are stationary in the base fluids and thermal transport is due to conduction. These models were developed by incorporating the other influential parameters which did not included into the classical models. Maxwell – Garnett’s 6 presented the model by considering the nanoparrticles are isolated in the base medium and there is no interaction between the nanoparticles. This model was developed based on the effective medium theory and two components systems of spherical particle and base fluids particles. Pak and Cho 7 developed a thermal conductivity model under the assumptions that the convective heat transfer enhancement is mainly due to dispersion of suspended nanoparticles. From the literature review of static model, there are eight influencing parameters that affect the thermal conductivity of nanofluids: particle volume fraction , particle material , particle size, particle shape, base fluids, temperature, effect of interfacial layer, base fluid thermal conductivity and nanolayer.
1.2.2. Dynamic models
These models are based on the fact that the nanoparticles have lateral and random motion of in the base fluids. The movement of particles causing the collision between nanoparticles, nanoconvection and resulting the enhanced thermal conductivity. The movement of nanoparticles is believed to be responsible for enhanced thermal conductivity of nanofluids. Xuan et al. 8 were the first to develop a dynamic model that takes into account the effects of Brownian motion. This model cannot exactly predict the strong temperature dependant nanofluids thermal conductivity data obtained by Das et al 9 and Patel et al 10.
Jang and Choi 11 presented a model which is based on conduction and convection caused by Brownian motion. This model takes the effect of four important modes .a) Collision between the base fluids molecules which represents the base fluids thermal conductivity b) Thermal diffusion in the base fluids with the effect of Kapitza resistance. c) Collision between the nanoparticles due to Brownian motion. d) Thermal interactions of dynamic or dancing nanoparticles .This model is able to predict the size dependant, temperature dependant and concentration dependant. They claimed that the nanoconvection is the key role in enhancing thermal conductivity of nanofluids. Bao Yang 12 developed a model based on the diffusive conduction and Brownian model and reported that the Brownian motion is the cause for improved thermal conductivity of nanofluids .Ravikanth et al 13 investigated the effect of Brownian motion along with conduction model and reported the Brownian motion contributes to the enhanced thermal conductivity of nanofluids. This model failed to include the effect of viscosity of fluids and the critical radius of particles. M. S. Murshed et al. 14 presented a model based on the combination of static and dynamic mechanism. This model is formulated by Brownian motion and DLVO potential. William et al 15 argued that the contribution of Brownian motion to the thermal conductivity of the nanofluids is very small and cannot be responsible for the extra ordinary thermal transport properties of nanofluids.
number of nanoparticles .The effect of nanolayer thickness was also taken into account. They reported that this model has relatively good agreement with the experimental data . This model did not include the effect Brownian motion.Yu and Choi 21 proposed that the nanolayer impact is significant for small particles (r – h). They claimed that the three to eight fold increase in the enhancement of thermal conductivity when the particle size is less than the critical size (10nm). Therefore a new optimum level is reached by replacing more volume fraction of particles with the particle s of critical size .Shukla et al 18 model was modified by Chandrasekar et al 22. and developed the model noted in equation 1. They included the cumulative effect of nanolayer , Brownian motion, and particles volume fraction. It has two terms; first term represents the contributions to the macroscopic Maxwell model 2 : the second term represents the contribution to the Brownian motion of nanoparticles. They revealed that the 30 % of nanofluids thermal conductivity enhancement is due to 21% by particles shape, 6% is due to nanolayer thickness and 3% is due to Brownian motion of particles. They reported that the non-spherical shaped particle have greater influence than the spherical shaped particles on the thermal conductivity of nanofluids.
4 3 3 ) ( ) ( ) 1 ( ) 1 ( ) ( ) 1 )( 1 ( ) 1 ( ka To T C k k k n k k k n k n k k k f p f p f p f p f eff
( 1 )
The model proposed , in equation 1, by Chandrasekar et al. 22 set the lower limit for of temperate for Brownian motion . They reported that the Brownian motion is enhanced when the particle size is decreased .They did not mention the optimum particles radius (critical size) for having higher velocity brownian motion. Moreover they did not inter relate the particle size and Brownian motion. Therefore the particle radius ‘a’ proposed by Chandrasekar et al. 22 is replaced by the critical radius rc of nanoparticlesin the proposed model.
The second term of Chandrasekar et al. 22 model contains the influencing parameters on Brownian motion such as particles volume fraction, particle size and viscosity of base fluids.According to the second term of Chandrasekar
22
the possibility of Brownian movement is limited. Because the viscosity of nanofluids increases when the particle volume fraction is increased. As a result the particles hardly move in the base fluids according the Linear viscosity model over the particle volume fraction. Brownian motion is suppressed when particle volume fraction is increased. Therefore the optimum particle volume fraction is essential for significant motion of particles and little agglomeration of nanoparticles. Lee et al. 23 reported that when particle size decreases, the surface area of the particle decreases as the square of the length dimension and volume decreases as the cube of the length dimension. As a result of this square /cube law, the surface –area to volume ratio is three orders of magnitude grater than that of larger size particles. Due to that dramatic enhancement of thermal conductivity is expected because of desirable situation of higher Brownian motion velocity of particle in the base fluids. The particle stability and the size can be related by the Stokes law
V
R
(
p l)
g
9
2
2
,Where V is the sedimentation velocity of particles: R is theradius of the spherical particles::
(
p
l)
is the density difference between the nanoparticles and the base fluids : g is the acceleration due to gravity .According to this equation, the sedimentation velocity is minimum when the particle size is reduced and viscosity of the base fluids is increased.S. P. Jang et al. 24 discussed various parameters on nanofluids thermal conductivity and reported that Brownian motion also contributes to the enhanced thermal conductivity of nanofluids .They related the random motion velocity CBM of particles with the particle size Do and viscosity as
BF BM
l
Do
C
, Where Do is given by Stokes-Einstein 25 law
nano B o
d
T
K
D
3
,where Do is the Einstein diffusion coefficient (m2/s), T is the temperature,l
BF isthe base fluid men free path and d nano .According to this equation the Brownian motion velocity is higher when the
viscosity is lower and particles diameter is smaller. The random velocity of nanoparticles was developed by Uhlenbeck et al 26 and can be measured by V α T 0.5 / dp1.5 . As per this equation, the particle motion velocity V is inversely proportional to the particle radius dp .Where T is the temperature .Therefore an optimum effective particle
size is needed for achieving the significant Brownian motion and kinetic stability.
4 ) 0
(
)
(
)
1
(
)
(
)
1
(
)
1
(
c f nf e p f e f p p f e f p f effr
k
T
T
C
k
k
k
n
k
k
k
n
k
n
k
k
k
( 2)The effective volume fraction
3
1
c er
h
where h is the liquid layer thickness rc is the critical particle size.The lesser the particle size, the higher the liquid layer thickness. In particular the nanolayer thickness is equal to radius of the particle when the nanolayer impact is higher. The value of constant C and temperature T are set to equal to 7x 10-36 and 21oC respectively. The lower limit of temperature is set by freeing point of nanofluids where there is no Brownian motion of nanoparticles.
3. Results and Discussion
Fig.1 Variation of thermal conductivity ratio of Al2O3 / water nanofluids with particle volume fraction.
3.1. Effect of nanolayer thickness
The proposed model is validated by the experimental results of Das et al. 9 and they used 38nm of particle size of Al2O3 /water and CuO /water nanofluids. It is seen from fig.1 & 2 the present theoretical model closely approach
the experimental results of Das et al. 9.The deviation between them may be due to particle size, temperature of nanofluids and the dispersion technique. The present model also compared the theoretical models presented by HC model 3 and Chandrasekhar et al. 22 .Fig.1 & 2 illustrate the wide deviation between the present model and the classical HC model 3. This is because of the HC Model 3 did not consider the size of the particle, nanolayer thickness and did not include the effect of nanoconvection into account for calculating thermal conductivity. Therefore it is clear that the particle volume fraction, shape and thermal conductivities are not the only depending factors of enhanced thermal conductivity of nanofluids. But also the random motion of the particle at low volume fraction, temperature, viscosity and critical size of particles also influence the enhanced thermal conductivity of nanofluids. The results in Fig.2 can be interpreted as the maximum enhancement is 20% at the particle volume fraction of 0.05 is the sum of effect of particle shape, the effect of nanolayer thickness and the effect of Brownian motion of nanoparticles.
0.01 0.02 0.03 0.04 0.05
1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18 1.20 1.22 Therm al conductivity r atio k ef f /k f
Particle volume fraction
H C M odel [3]
C handrasekar et al M odel [22] D as et al [9]
Fig.2 Variation of Thermal conductivity ratio of CuO /water nanofluids with particle volume fraction
Fig .3. Variation of particles size with thermal conductivity ratio due to Brownian motion
3.2. Effect of Critical size on Brownian motion
Fig.3 represents the effect of particle size on contribution to the enhancement of thermal conductivity of nanofluids. It shows that the Brownian motion contribution is higher at lesser particles size when particle volume fraction is fixed. It is seen that when particle volume fraction is increased the effect of Brownian motion is significant under the condition of critical size of the particle. From fig.3 the changes in brownian motion is in the range of 10 nm to 15nm and this range is the critical size. The critical size of particles can be defined as the size at which the brownian motion is considerable when the particle volume fraction and temperature are fixed. The brownian motion
0 .0 1 0 .0 2 0 .0 3 0 .0 4 0 .0 5 1 .0 0
1 .0 5 1 .1 0 1 .1 5 1 .2 0
The
rm
al c
onduc
tivity
ratio
k ef
f
/k f
P a rtic le v o lu m e fra c tio n
H C M o d el [3 ]
C h an d rasekar et al [2 2 ] D as et al [9 ]
P resen t M o d el
5 1 0 1 5 2 0 2 5
1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2
Enh
ance
m
ent
o
f k
ef
f
/
k f
d
u
e
to
B
ro
w
n
ia
n
m
o
ti
o
n
e
ff
ec
t(
%
)
P a r tic le r a d iu s r ( n m )
is suppressed when particle size and particle volume fraction is increased. This may be due to the fact that the brownian motion velocity is inversely proportional to the particles diameter.
4. Conclusion
This investigation presents a new thermal conductivity theoretical model for nanofluids. This model is based on the statistical mechanism, and the effect of brownian motion based on the critical radius of particles. The existing models important influencing parameters such as particle size , particle are replaced by critical radius of particle and effective particle volume fraction respectively. This model is validated with the experimental results of Al2O3/water and CuO/water nanofluids and found that it fits well with the experimental results. On comparing the
available Brownian motion models, this model optimizes the particle size by which the Brownian motion takes place. Found that that the effect Brownian motion on heat transports is higher when the particle size is less than that of critical size. This model is applicable for particles with equal or less than the critical size and valid for the optimum volume fraction. Further work is needed to quantitatively optimize the particle volume fraction and velocity of brownian motion of particles at different temperature of nanofluids.
Nomenclature
a Radius of particle ,m Subscripts f Base fluids nf Nanofluids p Particle f Base fluid rc Critical size eff Effective
T Temperature, K
k Thermal conductivity ,W/mK Greek letters
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