In this study a circular confined and submerged jet impinging on a horizontal hot plate is numerically simulated. Water and 36nm Al2O3-water nanofluid with various particle volume fractions are used as a working fluid for cooling the hot plate. Both laminar and turbulent impinging jets in various nozzle to plate distances and Reynolds numbers are considered. For laminar cases Navier-Stokes and energy equations and for turbulent cases RANS and time averaged energy equations were solved numerically to obtain the flowfield and temperature distribution. The turbulence effect was considered with a two equations model. The properties of nanofluid such as thermal conductivity, viscosity and density are modified using the appropriate models. The present study reports the Nusselt number on the hot plate for investigated cases. Temperature difference between the inlet fluid and the hot plate are obtained for different mass flow rates and particle volume fractions and are compared with experimental data for turbulent jets. The results show that using Al2O3 nano-particles in laminar jets enhances the heat transfer but for the turbulent jets Al2O3-water nanofluid has a lower performance for heat removal compared with clear fluid.
INTRODUCTION Jet impingement of liquids and gases on a hot surface is a cooling method for lots of applications such as quenching, turbine blade cooling and recently, cooling the electronic and computer devices [1]. Impinging fluids on a hot plate makes different zones on it and the maximum heat transfer will occur in the stagnation zone [2].
Both circular and slot jets are included in the literature experimentally and numerically in order to examine the cooling performance of jets. To achieve the purpose of a better cooling different methods are examined and utilized. One of those methods is geometrical considerations which have been an important concern in the impinging jet heat removal
performance. San and Shiao [1] investigated an experimental research in order to obtain a correlation for the stagnation Nusselt number as a function of jet plate size and plate spacing (the space between the jet nozzle and the hot plate). They also found the effect of geometrical changes on Reynolds and Nusselt numbers. Baonga et al. [2] experimentally studied the effects of geometry on stagnation Nusselt number. They also were more concerned about the hydraulic jump and its radius made by an impinging jet. Other researchers showed that fluid temperature and mass flow rate and even the plate material are also important in heat transfer behavior at stagnation point [3]. Mass flow rate or in other words, Reynolds number, has a significant role in heat transfer of the coolant. Since the Reynolds number specifies the regime to become steady or unsteady, amount of heat transfer will vary by changing the Reynolds number. Lee et al. [4, 5] numerically found the critical Reynolds number in which this transition happens.
In the literature sometimes it is stated that in addition to geometry and Reynolds number, inlet conditions are effective on the performance of the jet but it mostly affects free jet regions [6].
On the other hand, jet impingement type plays, in turn, an important role in cooling performance. Some of researchers showed that a confined submerged jet impinging on a hot surface has better heat transfer compared to a free jet [7], while others doubted this result in their paper in which a confined submerged impinging jet was experimentally investigated [8].
Although lots of methods are employed to find the best performance of impinging jets, a new technology which becomes widespread recently, has not attracted many of scientists to itself and this is nanotechnology. In recent decades adding small particles of metals or alloys to a base fluid has opened a new horizon to cooling industry. Addition of these particles to a base fluid such as water, oil or ethylene glycol increases the conductivity of the fluid and causes more heat transfer [9]. But the problem was that after a while these
Proceedings of the ASME 2009 7th International Conference on Nanochannels, Microchannels and Minichannels ICNMM2009
June 22-24, 2009, Pohang, South Korea
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particles settled down and agglomerated in special geometries. Hence, scientists decided to decrease the size of these particles to nanometer in order to achieve better solubility of particles. This leaded to make a new generation of fluids named nanofluids [10].
Although use of nanoparticles resolved the problem of agglomeration and also increased thermal conductivity of fluid, some other difficulties are yet left. Adding particles to a base fluid increases the viscosity of fluid and this leads to more pumping power. Also, addition of nanoparticles causes a decrease in specific heat capacity. However it is proved that using nanofluid in lots of problems has increased the heat transfer performance.
Thermophysical properties of nanofluids, depending on continuum or non-continuum assumption, can be obtained from theoretical formulas existing in the literature. In most of researches continuum assumption is used since both the conditions of simplicity and correctness will be satisfied in lots of cases [11]. Some of thermophysical properties are obtained by mixing theory but others have been always a controversial issue. Lots of experiments and numerical or analytical studies were performed to find appropriate correlation in order to define these properties in such a way that results be close to experimental data. One of the most important properties which should be determined is conductivity. Particle volume fraction, base fluid conductivity and particle conductivity are always included in correlations presented in the literature. Primary guesses of Maxwell were based on only these three factors [12]. Later, Hamilton and Crosser [12] added a shape factor for particles to Maxwell’s formula. Keblinsli et al. [13] added more details about nanoparticles size in 2002. New model was presented [14] which treated a nanofluid similar to a porous media which was completed by Khanafer et al.[15] in 2003. This model was based on a new factor named as thermal dispersion. In most of models presented conductivity was introduced with a static quantity which was not dependent on temperature or velocity fields. But later scientists considered conductivity as a dynamic quantity in such a way that even very small percentages of nanoparticles can change the conductivity of nanofluid considerably [16]. Jang and Choi [17] in their paper claimed that they had brought a new model which is not also dependent on particle volume fraction and conductivity of both materials. It is also related to the temperature and even size of nanoparticles. This model which was named as Brownian motion model made a revolution in nanofluid conceptions and was accepted as one of the best models presented for conductivity.
Another property of nanofluid which has been considered by researchers is nanofluid viscosity. However studies in this field are less that conductivity. All the studies show the same result and that is increasing nanoparticles percentage increases the viscosity but its rate differs in various references.
Most of researchers introduced viscosity as a function of particle volume fraction. But some others related that to other parameters such as temperature [18], particle size [19] and shear rate [20]. But Nguyen et al. [19] achieved an interesting result. They found out that increasing temperature of nanofluid to a critical point makes hysteresis viscosity in the nanofluid. Other thermophysical properties of nanofluids have often specified correlations and here are not discussed
NOMENCLATURE
A Area of the hot plate
MRC . random velocity of nano-particle
Cf Skin friction coefficient ))2/1(( 2
jetw Vρτ
Cp Pressure coefficient ))2/1()((
2
jetjetx Vpp ρ−
D Nozzle diameter or width
D0 Diffusion coefficient
ϕ,f Particle volume fraction
G Gravitational acceleration (m/s2)
H Nozzle to plate distance
K Thermal conductivity (W/Km)
Kb Boltzmann constant 1.3807×10-23J/K
P Pressure (pa)
Pr Dimensionless number Prandtl (α ν )
Q Heat flux (W)
Re Reynolds number
T Temperature (K or �C)
V Velocity in y direction
X Vertical direction
R Radial direction
Greek symbols
α Thermal diffusivity (m2/s)
β Thermal expansion coefficient (1/K)
β Kapitza resistance constant
µ Dynamic viscosity (kg/m.s)
ν kinematic viscosity (m2/s)
ρ Density (kg/m3)
Subscripts
i Inlet
f Fluid
nano Nano-particle
nf Nanofluid
R Radial direction
ref At the reference temperature (303K)
surf On the surface of hot plate
und Under the surface of hot plate
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As mentioned before application of nanofluids in impinging jets is not so common. One of papers which have mentioned nanofluid as the coolant in impinging jets is Nguyen et al’s [8] one. They used water and Al2O3-water nanofluid as the coolant for a circular confined and submerged impinging jet experimentally to study the heat transfer performance of nanofluids compared to pure fluid in cooling a horizontal hot plate. They examined a range of Reynolds numbers and different nozzle to plate spaces for some percentages of nanofluid and also pure water. They concluded that for some cases nanofluid with special particle volume fractions show better heat transfer performance, however this was not correct for all the studied cases.
In this study Al2O3-water nanofluid is investigated numerically for heat removal purpose. Effect of using nanofluids on heat transfer will be discussed. It is shown that Nusselt numbers are decreased using nanofluids for the turbulent impinging jets and increased for laminar jets. Skin friction coefficient is increased in nanofluid compared to clear fluids.
MODEL DESCRIBTION In this study two types of the impinging jets are studied.
One type is a confined submerged circular turbulent jet which is investigated by an axisymmetric model. The second type is a slot laminar jet which is modeled two dimensionally. For the turbulent cases a 30 millimeters diameter aluminum hot disk placed horizontally perpendicular to the jet is cooled by water and 36nm Al2O3-water nanofluid for various mass flow rates. Nanofluid is used with two percentages of 2.8% and 6%. Nozzle diameter is 3 millimeters and Vertical distance between the nozzle and hot surface is 5 and 15 millimeters. Aluminum disk is heated by 200W heat source from below. Inlet temperature of fluid is set at 20°C. Mass flow rates are in the range of turbulent flow. Uniform heat flux is assumed in the hot plate. For 2-D slot laminar jet the width of the nozzle is 6 millimeters and the distance of the plate to the nozzle is 12 and 30 mm.
For the case of the circular jet assuming the symmetry of the nozzle and plate an axisymmetric model is chosen to solve the problem. 25000 cells are used for grid generation after the grid study. Turbulence effect was considered with a two layer
zonal ε−k model in which the one equation model of the
Wolf-Stein is employed for the layer near the wall. Physical and computational domains are shown in figures 1 and 2. For the case of the slot jet the computational domain does not include the nozzle.
GOVERNING EQUATIONS Since the nanofluid is assumed to be a continuum single
phase fluid, conservation equations are the same for common Newtonian fluids and Navier-Stocks equations are satisfied in all directions. The only difference between nanofluids and a pure fluid is that the thermophysical properties used in the conservation equations should be substituted with nanofluid properties which are obtained from existing correlation in the literature as mentioned before. In following only governing equations for the circular jet are presented and to brief the paper the governing equation for 2-D slot jet are not presented. Hence, continuity and momentum equations will be written as:
(a)
(b)
Fig1. Schematic pictures of a) physical b)computational
domains Continuity equation:
(1) 0=+∂
∂+
∂
∂
r
v
r
v
x
vrrx
Radial- Momentum equation:
(2)
∇+∂
∂−=
∂
∂+
∂
∂
rnf
nf
rx
rr
vr
p
x
vv
r
vv
21µ
ρ
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There is an additional term of buoyancy in axial- momentum equation which corresponds to the gravitational force. Although in such forced convective problems, the buoyancy term does not play an important role and it can be neglected but for more accuracy this term is also included.
And the energy equation is as follows:
It should be noted that for turbulent flow, turbulent viscosity and turbulent conductivity are added to viscosity and conductivity of nanofluid in above equations. K-ε model of turbulence was used in this research. Details of governing differential equations for K and ε can be found in [21].
The model selected for conductivity here is Jang and Choi’s [17] model in which the Brownian motion of the particles is included. Also the conductivity is a function of temperature and particle volume fraction in the mixture.
6
1106×=C is an empirical constant, MR
C. , 0
D , fl and
ν are random velocity of nanoparticles, diffusion coefficient
given by Einstein [22] as nanob
dTKD πµ30= , mean free path
and the viscosity of the main fluid respectively. Kb is the
Boltzmann constant and its value is 1.3807×10-23
J/K. β is a
constant related to Kapitza resistance and is estimated to be 0.01. Pr is the Prandtl number of the base fluid.
This correlation is consisted of three parts. The first part is due to the base fluid conductivity, the second part is responsible for the particles conductivity and the third one is the effect of Brownian motion. As it is obvious the conductivity of the nanofluid will vary with the temperature since the diffusion coefficient is a function of temperature.
The model presented for viscosity is a temperature and particle size dependent model. For 36nm Al2O3 in the temperature range used in this problem the model of Nguyen et al. [19] is used in which volume fraction must be in percent.
For all of the cases a temperature dependent viscosity, conductivity and also density for the base fluid were used.
In order to obtain the density of nanofluid mixture theory was used as follows:
Specific heat capacity is also introduced by correlation
presented by Bergman [23] and for thermal expansion coefficient the simplest and most famous model is as follow which is used in this study.
RESULTS A) Laminar impinging jets Firstly the results for 2-D laminar slot jet are validated by
comparison with the experimental data for Nu number distribution in hot plate. Results are illustrated in figure 2 for two different Reynolds numbers in laminar regime. The working fluid here is air. Reynolds number for this laminar jet is defined based on the hydraulic diameter of the nozzle which is twice of the nozzle width. For these two Reynolds number the impinging jet is steady. In this figure, good agreement is seen between present numerical results and experimental work of Chiriac and Ortega [24]. The ratio of the nozzle to plate distance over nozzle width is equal five for these cases.
0
2
4
6
8
10
12
14
16
18
20
0 0.02 0.04 0.06 0.08
Nu
nu
mb
er
distance from centerline(m)
present study Re=250
experimental data [24] Re=250
present study Re=500
experimental data [24] Re=500
Fig2. Comparison of present numerical data with experimental data for a laminar air slot jet at H/D =5
The effect of adding Al2O3 nano-particles to water on the heat removal performance of the laminar jets are examined in two different Reynolds numbers and H/D. The results are shown in figures 3-6. As the figures show using nanofluids can enhance noticeably the heat transfer rate in the laminar impinging jets. This enchantment is more for smaller H/D and Reynolds number. The results show that the stagnation Nusselt number can even be doubled using nanofluid with 6% volume fraction of Al2O3. This positive effect of nano-particles on heat removal performance of the laminar jet was not reported before in the literature experiment and needed to be validated with experimental data which is not available in the literature for laminar jets.
(4) ( ) ( )TKx
Tv
r
TvC nfxrnfP ∇∇=
∂
∂+
∂
∂.ρ
(5)
PrRe3
)1(
2
1 nanodf
nano
f
nanofnf
kd
dC
fkfkk ++−= β
(6)
f
MRnanoMR
dl
DC
dCnano
0.
. 2,Re ==
ν
(7) 2015.0025.01 ff
f
nf ++=µ
µ
(8) ff nanofnf ρρρ +−= )1(
(9)
ff
fCfCC
nanof
nanopfp
nfp ρρ
ρρ
+−
+−=
)1(
)()1()(,
(10) ff
nf −=1β
β
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Fig3. Comparison of Nusselt number for pure fluid and nanofluid for 2-D slot jets
0
10
20
30
40
50
60
70
80
90
0 0.02 0.04 0.06 0.08
Nu
nu
mb
er
distance from centerline (m)
H/D=2 Re=500water
nanofluid 2.8%
nanofluid 6%
Fig4. Comparison of Nusselt number for pure fluid and nanofluid for 2-D slot jets
It should be noted that in smaller H/D ratio the maximum
Nusselt number is not in the center of the plate and it occurs in a distance from centerline.
B) Turbulent impinging jets. For the validation of the numerical model for the turbulent
impinging jets, results for the temperature difference between the inlet fluid and hot surface on the centerline is compared with experimental data of Nguyen et al. [8]. Temperatures in the experimental work were measured 4.5 mm below the hot surface but they are obtained on the surface in numerical study. By using the assumption of one directional heat flux, Fourier’s law can be used to calculate the temperature difference of the hot plate surface and the place of measurement in the experiments.
0
10
20
30
40
50
60
0 0.02 0.04 0.06 0.08
Nu
nu
mb
er
distance from centerline (m)
H/D=5 Re=250water
nanofluid 2.8%
nanofluid 6%
Fig5. Comparison of Nusselt number for pure fluid and nanofluid for 2-D slot jets
0
10
20
30
40
50
60
0 0.02 0.04 0.06 0.08
Nu
nu
mb
er
distance from centerline(m)
H/D=5 Re=500water
nanofluid 2.8%
nanofluid 6%
Fig6. Comparison of Nusselt number for pure fluid and nanofluid for 2-D slot jets
Using the equation (11) a temperature difference of about
6.2 degrees will be obtained. This amount is added to the computed temperature differences from numerical solution and graphs of figure 7 are drawn by these modified values for temperature differences. Although the numerical results are not in complete agreement with experimental data but the general trend is predicted by numerical model. It should be emphasized there is an unreliability with the experimental data of nanofluids especially for the impinging jets which only one experimental work is available.
The temperature difference between the inlet fluid and hot plate is changing using nanoparticles. Since the use of nanoparticles increases the thermal conductivity of fluid, decrease in temperature difference is expected. On the other hand specific heat is decreased using nanoparticles and this increases the temperature difference. In fig 8 a comparison between clear fluid and nanofluids with various particle volume fractions can be seen. As the figure shows, increase in nanoparticles volume fraction increases the temperature difference and this trend is more noticeable in higher mass flow rates. This trend was observed in the experimental work of Nguyen et al. [8] too.
(11)
kA
xQTT undsurf
.
.)(
∆=−
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Fig7. Temperature difference between inlet fluid and hot plate with the coolant of a) water b) nanofluid 2.8% c) nanofluid 6%
6
7
8
9
10
11
12
13
14
15
0 0.05 0.1 0.15 0.2 0.25
Ts-
Tf,
i
mass flow rate (kg/s)
water
nanofluid 2.8%
nanofluid 6%
Fig8. Comparison of temperature difference between inlet
fluid and hot plate for clear fluid and nanofluids
The heat transfer performance of the impinging jets can be investigated with examination of the Nusselt number over the plate. Figure9 depicts obviously that increase of nano-particles volume fraction has made a noticeable decrease in surface Nusselt numbers (from 800 to 550) for the investigated cases in turbulent regime. This shows that using nanofluids in this special case has deteriorated the convective heat transfer process. The other important point is the noticeable distance between the place of maximum Nusselt numbers and centerline of the plate.
Fig10 shows the same trend for the case of H/D=5. This shows that the deterioration of the heat transfer in turbulent impinging jets happens even for higher H/D ratios. It is interesting to note that place of maximum Nusselt numbers moves inward on the disk to the plate centerline as H/D increases.
Figure 11 shows the skin friction coefficient distribution on the hot plate. The skin friction coefficient is increased by adding nanoparticles which is another defect of using nanofluids in impinging jets. This trend could be expected as increasing particle volume fraction increases the viscosity of fluid and subsequently friction and skin friction coefficients are increased. The issue which is worth noting is that the increasing particle volume fraction from zero to 2.8% has made more effect on Cf (from 0.015 to 0.019) than changing it from 2.8 % to 6% (from 0.019 to 0.02). This may be because of temperature increase in higher fractions of nanoparticles which leads to decrease of base fluid viscosity.
0
100
200
300
400
500
600
700
800
900
0 0.005 0.01 0.015
Nu
sse
lt n
um
be
r
distance from centerline (m)
H/D=5/3
nanofluid 6%
nanofluid 2.8%
water
Fig9. Comparison of Nusselt number on the hot plate
between clear fluid and nanofluids for skgm /125.0=�
0
100
200
300
400
500
600
700
800
900
0 0.005 0.01 0.015
Nu
nu
mb
er
distance from centerline(m)
H/D=5water
nanofluid 2.8%
nanofluid 6%
Fig10. Comparison of Nusselt number on the hot plate
between clear fluid and nanofluids for skgm /125.0=�
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Fig11. Comparison of Cf on the hot plate between clear
fluid and nanofluids for of skgm /125.0=� and H/D=5/3
CONCLUSION In this study laminar and turbulent confined impinging jets
were numerically investigated for various Reynolds number and nozzle to plate distances. Pure water and also 36 nm Al2O3-water nanofluids with 2.8% and 6% particle volume fractions were used as coolant. In laminar regime using nano-particles has shown a positive effect on heat removal performance of the jet. Nusselt number is increasing on the plate for nanofluids compared to clear fluid. Even a double increase in stagnation Nusselt number was observed for the case of H/d=2. This phenomenon is reversed for the turbulent regime and the Nusselt number is decreased for Al2O3-water nanofluid compared with clear water. Skin friction coefficients showed a growth by increasing nano-particles percentage because of increased viscosity.
Confinement of the impinging jets plays an important role for both the value and location of the maximum Nusselt number on the plate especially for the turbulent regime.
REFERENCES [1] J-Y. San, W-A. Shiao, Effects of jet plate size and
plate spacing on the stagnation Nusselt number for a confined circular air jet impinging on a flat surface, International Journal of Heat and Mass Transfer 49 (2006) 3477–3486.
[2] J.B. Baonga, H. Louahlia-Gualous, M. Imbert, Experimental study of the hydrodynamic and heat transfer of free liquid jet impinging a flat circular heated disk, Applied Thermal Engineering 26 (2006) 1125–1138.
[3] F. Xu, M. S. Gadala, Heat transfer behavior in the impingement zone under circular water jet, International Journal of Heat and Mass Transfer 49 (2006) 3785–3799.
[4] H.G. Lee, M.Y. Ha, H.S. Yoon, A numerical study on the fluid flow and heat transfer in the confined jet flow in the presence of magnetic field, International Journal of Heat and Mass Transfer 48 (2005) 5297–5309.
[5] H.G. Lee, H.S. Yoon, M.Y. Ha, A numerical investigation on the fluid flow and heat transfer in the confined impinging slot jet in the low Reynolds
number region for different channel heights, International Journal of Heat and Mass Transfer 51 (2008) 4055–4068.
[6] Z. Xu, H. Hangan, Scale, boundary and inlet condition effects on impinging jets, Journal of Wind Engineering and Industrial Aerodynamics, doi:10.1016/j.jweia.2008.04.002.
[7] B. P. Whelan, A. J. Robinson, Nozzle Geometry Effects in Liquid Jet Array Impingement, Applied Thermal Engineering (2008), doi: 10.1016/j.applthermaleng.2008.11.003.
[8] C. T. Nguyen, N. Galanis, G. Polidori, S. Fohanno, C. V. Popa, A. L. Bechec, An experimental study of a confined and submerged impinging jet heat transfer using Al2O3-water nanofluid, International Journal of Thermal Sciences doi: 10.1016/j.ijthermalsci .2008.10.007.
[9] L.J. Eastman, S.U.S. Choi, S. Li, W. Yu, Thompson, Anomalously increased effective thermal conductivities of ethylene glycol-based nanofluids containing copper nanoparticles, Applied Physics Letter 78 (2001) 718–720.
[10] X-Q. Wang, A.S. Mujumdar, Heat transfer
characteristics of nanofluids: a review, International Journal of Thermal Sciences 46 (2007) 1–19.
[11] V. Trisaksria, S. Wongwises, Critical review of heat transfer characteristics of nanofluids, Renewable and Sustainable Energy Reviews 11 (2007) 512–523.
[12] M. Bahrami, M.M. Yovanovich, J.R. Culham, Assessment of Relevant Physical Phenomena Controlling Thermal Performance of Nanofluids, Proceedings of IMECE 2006 ASME International Mechanical Engineering Congress and Exposition.
[13] P. Keblinski, S.R. Phillpot, S.U.S. Choi, J.A. Eastman, Mechanisms of heat flow in suspensions of nano-sized particles (nanofluids), journal of heat and mass transfer 45 (2002) 855-866.
[14] Y. Xuan, W. Roetzel, Conceptions for heat transfer correlation of nanofuids, International Journal of Heat and Mass Transfer 43 (2000) 3701-3707.
[15] K. Khanafer,K. Vafai, M.Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, International Journal of Heat and Mass Transfer 46 (2003) 3639–3653.
[16] H. Patel, S. Das, T. Sundararajan, A. Sreekumaran, B. George, T. Pradeep, Thermal conductivities of naked and monolayer protected metal nanoparticle based nanofluids: Manifestation of anomalous enhancement and chemical effects, Applied Physics Letters, 83 (14) 2003 2931–2933.
[17] S.P. Jang, S.U.S. Chi, Cooling performance of a microchannel heat sink with nanofluids, Applied Thermal Engineering 26 (2006) 2457–2463.
[18] J. Lee, I. Mudawar, Assessment of the effectiveness of nanofluids for single-phase and two-phase heat transfer in micro-channels, International Journal of Heat and Mass Transfer 50 (2007) 452–463.
Downloaded From: http://proceedings.asmedigitalcollection.asme.org/ on 04/03/2014 Terms of Use: http://asme.org/terms
[19] C.T. Nguyen, F. Desgranges, G. Roy, N. Galanis, T. Mare, Temperature and particle-size dependent viscosity data for water-based nanofluids – Hysteresis phenomenon, Int. J. Heat Fluid Flow doi:10.1016/j.ijheatfluidflow.2007.02.004.
[20] N. utra, W. Roetzel, S.K. Das, Natural convection of nano-fluids, Heat and Mass Transfer 39 (2003) 775–784.
[21] C. W. Wilcox, Turbulence Modeling for CFD, third edition, 2003.
[22] A. Einstein, Investigation on the Theory of Brownian movement, Dover, New York, 1956.
[23] T.L. Bergman, Effect of reduced specific heats of nanofluids on single phase, laminar internal forced convection, International Journal of Heat and Mass Transfer 52 (2009) 1240-1244.
[24] V.A. Chiriac, A. Ortega, A numerical study of the unsteady flow and heat transfer in a transitional confined slot jet impinging on an isothermal surface, International Journal of Heat and Mass Transfer 45 (2002) 1237–1248.
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