- Email: [email protected]
Non-homogeneous model for a side heated square cavity filled with a nanofluid
International Journal of Heat and Fluid Flow 44 (2013) 327–335
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Non-homogeneous model for a side heated square cavity filled with a nanofluid Michele Celli ⇑ Dipartimento di Ingegneria Industriale, Alma Mater Studiorum, Università di Bologna, Viale Risorgimento 2, I-40136 Bologna, Italy
a r t i c l e
i n f o
Article history: Received 4 February 2013 Received in revised form 24 June 2013 Accepted 4 July 2013 Available online 27 July 2013 Keywords: Non-homogeneous model Nanofluid Side heating Square cavity Rayleigh number Buongiorno’s model
a b s t r a c t A side heated two dimensional square cavity filled with a nanofluid is here studied. The side heating condition is obtained by imposing two different uniform temperatures at the vertical boundary walls. The horizontal walls are assumed to be adiabatic and all boundaries are assumed to be impermeable to the base fluid and to the nanoparticles. In order to study the behavior of the nanofluid, a non-homogeneous model is taken into account. The thermophysical properties of the nanofluid are assumed to be functions of the average volume fraction of nanoparticles dispersed inside the cavity. The definitions of the nondimensional governing parameters (Rayleigh number, Prandtl number and Lewis number) are exactly the same as for the clear fluids. The distribution of the nanoparticles shows a particular sensitivity to the low Rayleigh numbers. The average Nusselt number at the vertical walls is sensitive to the average volume fraction of the nanoparticles dispersed inside the cavity and it is also sensitive to the definition of the thermophysical properties of the nanofluid. Highly viscous base fluids lead to a critical behavior of the model when the simulation is performed in pure conduction regime. The solution of the problem is obtained numerically by means of a Galerkin finite element method. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction In the last few decades an increasing attention has been focused on the heat transfer performances of a particular kind of colloids called nanofluids. A nanofluid is a suspension of metallic nanoparticles (or nanotubes) dispersed inside a base fluid. These colloids have been proposed as highly-effective heat transfer media (Choi and Eastman, 1995; Lee and Choi, 1996; Nguyen et al., 2007). Experimentally, great efforts have been spent in measuring the thermophysical properties of the nanofluids. Different kind of base fluids and dispersed nanoparticles have been tested in order to measure the thermal conductivity (Eastman et al., 1997; Lee et al., 1999; Wang et al., 1999; Xuan and Li, 2000; Eastman et al., 2001), the viscosity (Li et al., 2002; Prasher et al., 2006; Kwak and Kim, 2005) and the convective heat transfer coefficient (Xuan and Li, 2003; Wen and Ding, 2004; Heris et al., 2006). The thermophysical properties of the nanofluids have also been deeply studied from the theoretical point of view. A number of correlations have been proposed and employed in order to model the thermophysical properties (Das and Choi, 2006; Wang and Mujumdar, 2007). While the heat transfer performances of nanofluids have been widely studied, relatively few efforts have been undertaken with
⇑ Tel.: +39 051 2098330. E-mail address: [email protected] 0142-727X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.07.002
respect to the investigation of the sensitivity of the nanoparticles distributions to the heat transfer processes. In this contribution, a two dimensional square cavity filled with a nanofluid and subjected to side heating is studied. A nanofluid composed of Water as base fluid, and Alumina as nanoparticles dispersed into the base fluid, is investigated. The cavity walls are assumed to be impermeable to the base fluid and to the nanoparticles. The lower and upper boundary walls are assumed to be adiabatic. The side heating conditions are obtained by imposing two different temperatures at the vertical boundary walls. The model that is most frequently employed to simulate the nanofluids behavior is the homogeneous model. This model considers the nanofluid as a clear fluid with the only difference that the thermophysical properties of the nanofluid itself are modified in their values as functions of the average volume fraction of nanoparticles dispersed inside the base fluid. Magyari (2011) pointed out that, with a little rescaling effort, the homogeneous model produces the same results already obtained with the clear fluids models. Moreover, the assumption of homogeneity of the nanoparticles distribution may not hold when particle migration phenomena occur (Ding and Wen, 2005; Wen et al., 2009; Kang et al., 2007). The arguments just presented lead to the choice of employing a nonhomogeneous model for the present analysis. The non-homogeneous model prescribes a dedicated mass balance equation for the dispersed nanoparticle. The mathematical model is thus characterized by four balance equations: the mass
328
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335
Nomenclature c DB dp DT F(U0) g G(U0) H(U0) I(U0) j k kB L MES n N P r S T Tc Th T0 v x
specific heat (J/(kg K)) Brownian motion coefficient (m2/s) nanoparticles diameter (m) thermophoresis coefficient (m2/s) nondimensional function gravity acceleration (m/s2) nondimensional function nondimensional function nondimensional function nanoparticles mass flux (m/s) thermal conductivity (W/(mK)) Boltzmann constant (J/K) lenght of the cavity side (m) nondimensional maximum size of the grid elements normal unit vector number of grid elements nondimensional pressure radius of the neighborhood of / = 1 nondimensional cavity surface nondimensional temperature cold wall temperature (K) hot wall temperature (K) reference temperature (K) nondimensional velocity field (u, v) nondimensional position vector (x, y)
balance equation for the base fluid, the momentum balance equation for the nanofluid, the energy balance equation for the nanofluid and the mass balance equation for the nanoparticles, Buongiorno (2006). Moreover, the thermophysical properties of the nanofluid are here expressed by means of phenomenological correlations as functions of the average volume fraction of the nanoparticles. The thermophysical properties of the base fluid are, in fact, unavoidably modified by the presence of the dispersed nanoparticles. On the other hand, the nondimensional governing parameters (Rayleigh number, Prandtl number and Lewis number) are defined exactly as for clear fluids. This choice allows an easier comparison between the results obtained by the mathematical model here employed and the results for clear fluids found in the literature. The contribution of the average volume fraction of nanoparticles, coming from the definitions of the thermophysical properties, is thus taken into account by means of a number of ad hoc nondimensional parameters. The non-homogeneous model is here studied for different range of values of the nondimensional parameters involved. The main goals of this study are looking for possible non-homogeneities of the nanoparticles distribution and investigating the heat transfer performances of the nanofluid at the cavity side walls. Particular attention is focused on the low Rayleigh numbers regimes. The nanoparticles distribution shows indeed a strong sensitivity to the heat transfer processes for low Rayleigh numbers. The non-homogeneous model is here also tested for a particular highly viscous base fluids (Propylene Glycol) in the limit case of pure conduction and pure diffusion regime. The numerical solution of the problem is obtained by Galerkin’s finite element method.
Greek symbols a thermal diffusivity (m2/s) b thermal expansion coefficient (K1) bT nondimensional constant DT reference temperature jump (K) l dynamic viscosity (Pa s) q density (kg/m3) / rescaled volume fraction of nanoparticles U local volume fraction of nanoparticles U0 average volume fraction of nanoparticles Nondimensional numbers Le Lewis number NBT nondimensional number Nu average Nusselt number Pr Prandtl number Ra Rayleigh number Subscripts, Superscripts – dimensional quantity rescaled quantity f, p, nf fluid, nanoparticle, nanofluid max maximum value
side heating condition is obtained by imposing two different temperatures at the walls: the hot wall, Th, is assumed to be on the left vertical boundary and the cold wall, Tc, is assumed to be on the right vertical boundary. The cavity walls are subjected to the noslip condition. The boundary conditions are shown in their dimensional form in Table 1. A nanofluid composed of Water as base fluid, and Alumina (Al2O3, Auerkari, 1996) as nanoparticles dispersed inside the base fluid, is here studied. In order to analyze the nanoparticles distribution, a non-homogeneous model is employed, Buongiorno (2006). The following hypotheses are assumed: Non-homogeneous nanofluid model. Brownian motion and thermophoresis as leading physical transport mechanisms for the nanoparticles diffusion. Thermophysical properties of the nanofluid are expressed as functions of the average volume fraction of nanoparticles dispersed inside the base fluid.
2. Mathematical model The two dimensional side heated square cavity here investigated is sketched in Fig. 1. The cavity is assumed to be impermeable and the horizontal walls are assumed to be adiabatic. The
Fig. 1. Sketch of the system.
329
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335 Table 1 Dimensional boundary conditions.
ð3aÞ
v
T
U
jp ; qp v $ U ¼ $
ð3bÞ
v ¼ 0
T ¼ Th
n jp ¼ 0
v P þ l r 2v ¼ $ þ qnf bnf ðT T 0 Þg; qnf v $ nf
ð3cÞ
v ¼ 0
T ¼ 0 n$
n jp ¼ 0
T ¼ knf r 2 T cpjp $ T; $ ðqcÞnf v
ð3dÞ
v ¼ 0
T ¼ Tc
n jp ¼ 0
v ¼ 0
T ¼ 0 n$
n jp ¼ 0
(Th Tc)/T0 1. Steady analysis. Oberbeck-Boussinesq approximation. Negligible viscous dissipation. Local thermal equilibrium between nanoparticles.
base
fluid
and
Here the reference temperature T0 is defined as T0 = (Th + Tc)/2. The thermophysical properties of the nanofluid such as the density qnf, the specific heat cnf and the thermal expansion coefficient bnf, are defined by means of analytical expressions
qnf ¼ ð1 U0 Þqf þ U0 qp ; ðqcÞnf ¼ ð1 U0 ÞðqcÞf þ U0 ðqcÞp ; ðqbÞnf ¼ ð1 U0 ÞðqbÞf þ U0 ðqbÞp ;
ð1Þ
where U0 is the average volume fraction of nanoparticles dispersed inside the cavity, f stands for fluid, p stands for nanoparticles and nf stands for nanofluid. The dynamic viscosity lnf and the thermal conductivity knf are defined by means of empirical correlations, respectively the Brinkman model and the Maxwell model, namely
lf
lnf ¼
v ¼ 0; $
; ð1 U0 Þ2:5 kp þ 2kf 2U0 ðkf kp Þ : knf ¼ kf kp þ 2kf þ U0 ðkf kp Þ
ð2Þ
is the dimenwhere the overlines refer to dimensional quantities, v sional velocity field, T is the dimensional temperature field, P is the dimensional pressure field, g is the gravity acceleration, U is the local volume fraction of nanoparticles and jp is the dimensional nanoparticles mass flux. From Buongiorno (2006), the dimensional nanoparticles mass flux is defined by
! jp ¼ q DB $ U þ D T $T ; p T
where DB = (kBT0)/(3 plfdp) is the Brownian motion coefficient and DT = (bT lfU0)/qf is the thermophoresis coefficient. The symbol dp is the diameter of the nanoparticles and bT is a proportionality factor defined by McNab and Meisen (1973). Eq. (4) introduces the leading transport mechanisms for the diffusion of the nanoparticles inside the cavity: the first term on the right-hand side refers to the Brownian motion contribution and the second term on the righthand side refers to the thermophoresis contribution. Among the boundary conditions of Table 1, the fourth column shows the cavity walls impermeability boundary condition with respect to the nanoparticles. The impermeability is expressed by means of assuming a vanishing orthogonal flux of nanoparticles at the walls. The unit vector n is directed outward with respect to the walls. 3. Nondimensional formulation In order to proceed with the analysis, a nondimensional treatment of the governing equations is needed. The following scalings allow one to rewrite the system of Eqs. (3) in a nondimensional form
v ¼
¼ Lx; x
The behavior of lnf and knf as a function of U0 is shown in Fig. 2. It is worth noting that an increasing average volume fraction of nanoparticles dispersed inside the cavity, i.e. increasing values of U0, yields, with respect to the clear fluids, higher values of thermal conductivity and of dynamic viscosity. Since two mass balance equations (one for the base fluid and one for the nanoparticles), a momentum balance equation and an energy balance equation are taken into account, the dimensional set of governing equations are defined as
ð4Þ
T ¼ T DT þ T 0 ;
anf L
v;
P¼
lnf anf L2
P;
ð5Þ
U ¼ U0 /;
where L is the length of the cavity sides, DT = Th Tc and anf = knf/ (q c)nf is the thermal diffusivity of the nanofluid. The value of U, that is already nondimensional, is here rescaled by its average value over the cavity, U0, for a better interpretation of the oncoming results. On assuming the approximation DT/T0 1 the definition of the nanoparticles mass flux Eq. (4) may be simplified as
jp ¼ q DB $ U þ DT DT $ T : p T0
ð6Þ
On using Eqs. (5) and (6), the system of Eqs. (3) becomes
1.30
$ v ¼ 0;
ð7aÞ !
1.25
v $/ ¼
1.20
1 r2 T r2 / þ ; Lef FðU0 Þ NBT
v $v ¼ $P þ r2 v þ Raf HðU0 ÞT; Prf GðU0 Þ 1 $T $T ; v $T ¼ r2 T þ $/ $T þ Lef IðU0 Þ NBT
1.15 1.10 1.05 1.00 0.00
0.02
0.04
0.06
0.08
Fig. 2. Behavior of lnf/lf and knf/kf as functions of U0.
0.10
ð7bÞ ð7cÞ ð7dÞ
where Raf is the Rayleigh number relative to the fluid, Prf is the Prandtl number relative to the fluid, Lef is the Lewis number relative to the fluid and NBT is a nondimensional parameter that measures the relative strength of the Brownian motion contribution with respect to the thermophoresis contribution. The Rayleigh number arises
330
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335
from the buoyancy term that is present in the momentum balance equation under the Oberbeck–Boussinesq approximation. This nondimendional number describes the ratio of energy released by the buoyancy forces compared to the energy dissipated by heat conduction and viscous drag. The Prandtl number and the Lewis number are numbers dependent only on the fluid physical characteristics of the system and they do not depend on the geometry or on the flow variables. As a result, the Prandtl number describes the competition between two physical mechanisms, the viscous diffusion and the thermal diffusion. As for the Prandtl number, the Lewis number describes the competition between two diffusion mechanisms: the thermal diffusion and the mass diffusion. The Rayleigh, Prandtl and Lewis numbers are defined exactly in the same way as for clear fluids. This choice allows an easier comparison of the oncoming results with the literature on clear fluids. The nondimensional numbers just introduced are defined as follows:
Raf ¼ Lef ¼
lf qf bf ðT h T c ÞL3 g ; Prf ¼ ; lf af qf af af U0 T 0 DB DB
;
NBT ¼
DT DT
ð8Þ
:
The functions F(U0), G(U0), H(U0) and I(U0) in Eqs. (7) are a consequence of defining the nondimensional parameters of Eq. (8) in the same way as for clear fluids. The functions F(U0), G(U0), H(U0) and I(U0) thus gather the contribution of the average volume fraction of nanoparticles, U0, coming from the definitions of the thermophysical properties, Eqs. (1) and (2).
lnf kf qf ðqcÞnf ; lf knf qnf ðqcÞf lf kf ðqbÞnf ðqcÞnf knf ðqcÞf HðU0 Þ ¼ ; IðU0 Þ ¼ : lnf knf ðqbÞf ðqcÞf kf ðqcÞp U0
FðU0 Þ ¼
knf ðqcÞf ; kf ðqcÞnf
GðU0 Þ ¼
ð9Þ
The nondimensional boundary conditions for the square cavity are described in Table 2. 4. Code validation The system of governing Eqs. (7) together with the relative boundary conditions, Table 2, are here solved numerically by means of Galerkin’s finite element method implemented through the software package Comsol Multiphysics (Ó Comsol, Inc.). In order to validate the results obtained, a numerical test of mesh independence is performed. The cavity is meshed with an unstructured grid of triangular elements with Lagrange quadratic basis functions. A key parameter of the grid is the maximum size of the elements within the mesh (MES). For a given geometry, every value of MES corresponds to a fixed number of grid elements (N). Table 3 shows, as functions of MES, the behavior of two parameters: the average Nusselt number over the hot wall and the integral over the cavity of the absolute value of the nanoparticles distribution gradient. The average Nusselt number is defined as follows:
Table 2 Nondimensional boundary conditions. T
/
v=0
T ¼ 12
@/ @x
¼ @T=@x N BT
v=0
@T @y
@/ @y
¼0
v=0
T ¼ 12
@/ @x
¼ @T=@x N BT
v=0
@T @y
@/ @y
¼0
¼0
MES
N
Nu
0.03 0.025 0.02 0.015 0.01
2894 4040 6282 11,748 24,910
1.0362 1.0362 1.0362 1.0362 1.0362
R
S
j$/j dS
0.1002 0.0995 0.0994 0.0993 0.0993
Table 4 Benchmark validation of the average Nusselt number ðNuÞ, the maximum horizontal velocity (umax) at the mid-width (x = 0.5) and the maximum vertical velocity (vmax) at the mid-height (y = 0.5) for Raf = 103. Raf = 103
Present analysis
Ref. Davis and cavity (1962)
Ref. Manzari (1999)
Ref. Wan et al. (2001)
Ref. Barletta et al. (2006)
Nu umax y(umax)
1.118
1.12
1.074
1.117
1.118
3.653 0.813 3.701 0.177
3.634 0.813 3.679 0.179
3.68 0.817 3.73 0.183
3.489 0.813 3.686 0.188
– – – –
vmax x(vmax)
Nu ¼
Z 0
1
@T dy: @x
ð10Þ
The values shown in Table 3 are obtained for Raf = 103, U0 = 10% and NBT = 1. For the following numerical analysis, the value MES = 0.015 is chosen as the best value for the maximum element size. For a further validation of the code, a comparison of the numerical results with the benchmark solutions found in the literature is here performed. The limit U0 ? 0, i.e. the case of a fluid clear of nanoparticles, is considered. Moreover, the air is chosen as working fluid. For a side heated square cavity filled with air, the literature provides a number of benchmark results. The benchmark values here used are taken from Davis and cavity (1962), Manzari (1999), Wan et al. (2001) and Barletta et al. (2006) and refer to the case of Raf = 103. The choice of taking air as working fluid is due to the fact that it yields more reliable checks since the Prandtl number of air is less sensitive to the reference temperature with respect to the Prandtl number of water. Table 4 shows the present analysis results and the benchmark solutions for the value of Nu, the maximum value of umax and its respective coordinate y(umax), the maximum value of vmax and its respective coordinates x(umax). The results here obtained show a good agreement with the literature. The results obtained for other values of the Rayleigh number display again a good agreement with the literature but they are not reported here for the sake of brevity. 5. Results The system of governing Eqs. (7) is characterized by a relatively large set of nondimensional parameters: Raf, Prf, Lef, NBT and U0. The aim of this paper is investigating the relationship among the Rayleigh number, the volume fraction of nanoparticles and the heat transfer performances of the nanofluid. It is thus necessary
Table 5 Maximum values of jvj for different Rayleigh numbers, Raf and two different average volume fraction of nanoparticles, U0.
v
¼0
Table 3 Mesh independence test.
Raf = 100 Raf = 101 Raf = 102 Raf = 103
U0 = 1%
U0 = 10%
0.004 0.037 0.369 3.483
0.002 0.020 0.203 1.988
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335
331
Fig. 3. Nanoparticles distributions / for a fixed value of U0 = 1%. The column on the left refers to NBT = 1, the column on the right refers to NBT = 10. The first row from the top refers to Raf = 0, the second row refers to Raf = 5 and the third row refers to Raf = 10.
to fix the values of Prf, Lef and NBT. Once the basic fluid (Water), the nanoparticles material (Alumina), the nanoparticles size (dp = 108 m) and the reference temperature (T0 = 300 K) are chosen, the value of Prf and Lef are already set up. The value of dp refers to the effective average size of the nanoparticles dispersed inside the cavity. In fact, particles and agglomerations of particles with different sizes are dispersed in the base fluid as well shown in
Figs. 5–7 of Timofeevana et al. (2007). The choice to disregard the real distribution of the nanoparticle sizes present inside the cavity has been taken in order to simplify the treatment of a problem, where several governing parameters are already present. The value of NBT depends now only on DT, Eq. (8). Within the assumptions just made, the range of variability of NBT is 101 < NBT < 101 for a temperature jump between the vertical walls of 1 K < DT < 10 K.
332
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335
Fig. 4. Nanoparticles distributions / for a fixed value of U0 = 10%. The column on the left refers to NBT = 1, the column on the right refers to NBT = 10. The first row from the top refers to Raf = 0, the second row refers to Raf = 5 and the third row refers to Raf = 10.
In the following the cases NBT = 1 and NBT = 10 are studied. Eventually, the case of a relatively small value of NBT is investigated. The typical average volume fraction of nanoparticles varies between U0 = 1% and U0 = 10%. If a value of U0 10% is assumed, the system has to be modeled as a fluid saturated porous medium. In the following, both the cases, U0 = 1% and U0 = 10%, are analyzed.
On focusing the attention on the groups Lef F(U0) and Lef I(U0) of Eqs. (7), one may notice that they have an order of magnitude that varies in the range 103–105. These two groups of parameters tend, in most of the cases, to dominate the Brownian motion and thermophoresis contributions included in Eqs. (7b) and (7d). When, on the other hand, low velocities are taken into account,
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335
333
Fig. 5. Nanoparticles distributions / for fixed value of U0 = 10% and for NBT = 1 (left) and for NBT = 10 (right). Both frames refer to Raf = 100.
the Brownian motion and thermophoresis contributions become not negligible. If low velocities are involved, the convective terms of Eqs. (7b) and (7d) become indeed comparable with the Brownian motion and thermophoresis contributions. The cases of low velocity correspond to weak convection phenomena and thus, for the system here analyzed, to low Rayleigh numbers. In the following, low Rayleigh numbers are investigated. Table 5 displays the maximum values of jvj as functions of the Rayleigh number, Raf, and of the average volume fraction of nanoparticles, U0. While the velocity maxima increase with the Rayleigh number, they decrease as the average volume fraction of nanoparticles increases. A reason for the latest behavior has to be sought into the definitions of dynamic viscosity and thermal conductivity in Eq. (2). Brinkman’s and Maxwell’s models, in fact, prescribe that, when the number of suspended nanoparticles is increased, the dynamic viscosity and the thermal conductivity of the nanofluid increase as well. The increasing dynamic viscosity and thermal conductivity tend indeed to depress the strength of the convection phenomena (for fixed temperature jump, DT). Fig. 2 shows the enhancement of dynamic viscosity and thermal conductivity for increasing values of U0. Fig. 3 shows the distributions of the nanoparticles volume fraction inside the square cavity for a fixed value of the average volume fraction of nanoparticles, U0 = 1%, and for different values of the Rayleigh number and of the parameter NBT. Each row of Fig. 3 refers to a different Rayleigh number: the first row from the top refers to Raf = 0, the second row refers to Raf = 5 and the third row refers to Raf = 10. The left column of Fig. 3 refers to simulations performed with NBT = 1 and the right column refers to simulations performed with NBT = 10. The gray shaded areas drawn inside the nanoparticles distributions frames of Fig. 3 refer to those areas where the nanoparticles volume fraction values belong to the neighborhood of the value / = 1. This neighborhood is characterized by a radius r = 0.01. When the gray shaded area occupies a small or a negligible part of the cavity, the distribution of nanoparticles has to be considered as non-homogeneous. On the other hand, when the gray shaded area occupies most part of the cavity, the distribution of the nanoparticles can be considered as homogeneous. The case of pure conduction (Raf = 0, first row of Fig. 3) shows, as expected, distributions of nanoparticles that are symmetric with respect to the temperature distribution. The conduction regime enhances the effect of the thermophoresis phenomenon and one can note that the nanoparticles distribution is highly non-homogeneous. The pure conduction regime frame, in fact, displays a deviation up to 50% from the average value / = 1 relative to the case of
NBT = 1, and the gray shaded area occupies a relatively small area of the cavity. For positive values of the Rayleigh number, the second and third row from the top of Fig. 3, the mixing due to the thermal convection amplifies the area of the neighborhood of the average value / = 1. Eventually, Fig. 3 highlights the key role of the parameter NBT: a low value of NBT leads to a less homogeneous distribution of nanoparticles. When the average volume fraction of nanoparticles is increased to U0 = 10% (Fig. 4) the nanoparticles distributions and temperature fields obtained are not dramatically different with respect to the case U0 = 1%. As for Fig. 3 the first row from the top refers to Raf = 0, the second row refers to Raf = 5 and the third row refers to Raf = 10. Again, the left column refers to simulations performed with NBT = 1 and the right column refers to simulations performed with NBT = 10. It is worth noting that, with respect to Fig. 3, the homogeneous area of the frames relative to the same Rayleigh number decreases in size, second and third row of Fig. 4. It has already been pointed out that increasing values of U0 tend to inhibit the convective motion. When the convection is weaker, the mixing due to this phenomenon is less effective and this is the reason why the well mixed and homogeneous area is reduced in size. Fig. 5 shows the nanoparticles distributions for fixed values of the Rayleigh number, Raf = 100, of the average volume fraction of nanoparticles U0 = 10% and two different values of NBT: the left frame refers to NBT = 1 and the right frame refers to NBT = 10. In fact, if the Rayleigh number is high enough, the non-homogeneous areas become more and more confined close to the boundaries and the gray shaded area, the homogeneous area, increases in size until it occupies most part of the cavity. For values of Raf P 100 the nonhomogeneities are then totally confined nearby the boundaries and the nanoparticles distribution may be considered as homogeneous. The nanoparticles distributions relative to the case U0 = 1% is not shown here for the sake of brevity and because it does not add any further information. Fig. 6 describes the temperature behavior along the diagonal that goes from the upper left corner to the lower right corner of the square cavity for fixed values of NBT = 10 and U0 = 1%. The different curves are drawn for different values of the Rayleigh number. Only the case (NBT = 10, U0 = 1%) is displayed instead of all the possible permutations of the pair (NBT, U0) shown in Figs. 4 and 3 because Fig. 6 is representative of the system behavior. One can note that, as expected, low values of the Rayleigh number lead to a purely horizontal temperature distribution, i.e. the pure conduction regime for Raf = 0.
334
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335
the momentum balance equation can be solved without any contribution of the temperature field because the buoyancy term is negligible. The velocity field for the pure conduction regime is constant and equal to zero. Since the fluid is still, Eqs. (7) can be simplified
r2 T þ r2 / þ
1 $T $T ¼ 0; $/ $T þ Lef IðU0 Þ NBT
r2 T NBT
ð11aÞ
¼ 0:
ð11bÞ
The temperature field can now be rescaled by the parameter NBT
T : Te ¼ NBT
ð12Þ
The system of Eqs. (11) can be rewritten as
1 $/ $ Te þ $ Te $ Te ¼ 0; Lef IðU0 Þ r2 / þ r2 Te ¼ 0:
r2 Te þ
Fig. 6. Temperature distribution along the diagonal of the cavity (from the upper left angle to the lower right angle) for different values of Raf and fixed values of NBT = 10 and U0 = 1%.
1.10
Nu
1.08 1.06 1.04 1.02 1.00 0
200
400
600
800
1000
Raf Fig. 7. Nu as a function of Raf for different values of average volume fraction of nanoparticles U0.
Fig. 7 shows the behavior of the average Nusselt number over the hot wall, Eq. (10), as a function of the Rayleigh number for different values of the average volume fraction of nanoparticles U0 = 0%; 1%; 10%. From Table 5, the higher is U0 the more the convection phenomena are inhibited. If the heat transfer processes at the walls are more effective when the convection phenomena are stronger, thus high values of U0 mean a less efficient heat transfer at the wall. The reasoning just made agrees with the results shown in Fig. 7 where the value of the Nusselt number decreases as U0 increases (for a given value of Raf). 5.1. Limiting case Raf ? 0: pure conduction and pure diffusion regime The limiting case Raf ? 0 is characterized by the pure conduction regime for the temperature field and by the pure diffusion regime for the nanoparticles distribution. This limiting case may be achieved by means of imposing very small temperature difference between the vertical walls or by means of considering systems of very small characteristic length. In the pure conduction regime
ð13bÞ
Now an order of magnitude analysis is performed with the aim of simplifying Eqs. (13). Once the nanoparticles material and the reference temperature T0 are set, the parameter NBT becomes a function of the base fluid properties (qf, l) and a function of the temperature jump between the vertical walls DT, namely
NBT /
1.12
ð13aÞ
qf DT l 2
ð14Þ
:
Since the possible base fluids are characterized by densities of, approximately, the same order of magnitude, the value of NBT is mainly driven by the term DT l2. Now one can assume that the limit Raf ? 0 is obtained by imposing a very small characteristic length of the system. If, moreover, a highly viscous base fluid is considered, Propylene Glycol, one can obtain, for a given DT, a NBT significantly small (Eq. (14) and Table 6). Once the characteristic length L, the temperature jump DT and the average volume fraction of nanoparticles U0 are set (Table 6) the order of magnitude of the group Lef I(U0) is evaluated. Since the group Lef I(U0) has an order of magnitude of 106, one can neglect the second term of the left-hand side of Eq. (13a) and Eqs. (13) can be simplified again to obtain the pure conduction and pure diffusion regime. The simplified system and the relative boundary conditions are
d Te N1 N1 x ¼ 0; Te ð0Þ ¼ BT ; Te ð1Þ ¼ BT ; 2 dx 2 2 2 d / d Te d Te ¼ 0; /ð0Þ ¼ ; /ð1Þ ¼ : dx dx dx2 2
ð15aÞ ð15bÞ
The solutions of Eqs. (15) are
x þ 1=2 Te ¼ ; NBT d/ ¼ N 1 BT : dx
ð16aÞ ð16bÞ
If NBT 103, a critical large nanoparticle volume fraction gradient is obtained, Eq. (16b). The latest conclusion highlights a possible critical behavior of Boungiorno’s model for nanofluids based on
Table 6 Set up for the Propylene Glycol – Alumina nanofluid simulation in the limit of pure conduction. Order of magnitude of NBT and Lef I(U0). L (m)
DT (K)
U0
NBT
Lef I(U0)
104
1
10%
103
106
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335
highly viscous base fluids in the limit of Raf ? 0. A clear example of this critical behavior arises when the case of a nanofluid composed of Propylene Glycol as base fluid, a fluid much more viscous than water, and Alumina as dispersed nanoparticles (Table 6) is taken into account. If one simulates the distribution of the nanoparticle volume fraction for fixed U0 = 10% and Raf = 0 a nanoparticle distribution that increases linearly as a function of x from 100 to 100 for 0 < x < 1 is obtained. It is worth noting that the value of NBT 103 is, as expected, a few orders of magnitude smaller with respect to the case of the water based nanofluid. A value of NBT so small is the reason of the critical range of variation of the nanoparticles distribution / obtained in the Propylene Glycol base fluid case. 6. Conclusions A two dimensional square cavity subjected to side heating and filled by a nanofluid is here studied. The boundary walls are assumed to be impermeable to the base fluid and to the nanoparticles. In order to investigate the spatial distribution of the nanoparticles dispersed inside the cavity, a non-homogeneous model is taken into account. A mass balance equation for the dispersed nanoparticles is considered in addition to the mass balance equation for the base fluid, the momentum balance equation for the nanofluid and the energy balance equation for the nanofluid. Every thermophysical property of the nanofluid is expressed as a function of the average volume fraction of the nanoparticles. The Brownian motion and the thermophoresis are considered as the leading physical transport mechanisms for the nanoparticles. The main tasks of this study are the investigation of the possible non-homogeneity of the distribution of the nanoparticles dispersed inside the cavity and the investigation of the heat transfer performances of the nanofluid subjected to a natural convection regime. The following conclusions are achieved: Low Rayleigh numbers may yield a markedly nonhomogeneous distribution of the nanoparticles inside the cavity. In this range of the Rayleigh number a non-homogeneous model is more appropriate for the description of the system. Relatively high Rayleigh numbers confine the non-homogeneities nearby the boundaries leaving a fairly homogeneous distribution in the core of the cavity. The homogeneous model become reliable if Raf P 102. On employing the Maxwell model and the Brinkman model to define the dynamic viscosity and the thermal conductivity of the nanofluid, the heat transfer at the vertical boundaries becomes less efficient as the average volume fraction of nanoparticles increases. A critical behavior of Boungiorno’s model is found for highly viscous base fluids in the limit of pure conduction and pure diffusion.
Acknowledgements This work was financially supported by Italian government, MIUR Grant PRIN-2009KSSKL3.
335
The author is grateful to one of the anonymous Reviewers for the helpful suggestions that improved the quality of the paper. References Auerkari, P., 1996. Mechanical and Physical Properties of Engineering Alumina Ceramics. VTT Technical Research Center of Finland. Barletta, A., Nobile, E., Pinto, F., di Schio, E.R., Zanchini, E., 2006. Natural convection in a 2d-cavity with vertical isothermal walls: cross-validation of two numerical solutions. International Journal of Thermal Sciences 45, 917–922. Buongiorno, J., 2006. Convective transport in nanofluids. ASME Journal of Heat Transfer 128, 240–250. Choi, S.U.S., Eastman, J.A., 1995. Enhancing thermal conductivity of fluids with nanoparticles. In: International Mechanical Engineering Congress and Ewxhibition, San Francisco, CA, USA. Das, H.P.S.K., Choi, S.U.S., 2006. Heat transfer in nanofluids – a review. Heat Transfer Engineering 37, 3–19. Davis, G.D.V., 1962. Natural convection of air in a square cavity, a benchmark numerical solution. International Journal for Numerical Methods Fluids 3, 249– 264. Ding, Y., Wen, D., 2005. Particle migration in a flow of nanoparticle suspensions. Powder Technology 149, 84–92. Eastman, J., Choi, S., Li, S., Thompson, L., Lee, S., 1997. Enhanced thermal conductivity through the development of nanofluids. Materials Research Society Symposium – Proceedings 457, 3–11. Eastman, J., Choi, S., Li, S., Yu, W., Thompson, L., 2001. Anomalously increased effective thermal conductivities of ethylene glycol-based nano-fluids containing copper nanoparticles. Applied Physics Letters 78, 718–720. Heris, S., Etemad, S.G., Esfahany, M., 2006. Experimental investigation of oxide nanofluids laminar flow convective heat transfer. International Communications in Heat and Mass Transfer 33, 529–535. Kang, H., Kim, W., Kim, S.H., 2007. Effect of particle migration on the heat transfer of nanofluid. Korea–Australia Rheology Journal 19, 99–107. Kwak, K., Kim, C., 2005. Viscosity and thermal conductivity of copper oxide nanofluid dispersed in ethylene glycol. Korea–Australia Rheology Journal 17, 35–40. Lee, S., Choi, S.U.S., 1996. Application of metallic nanoparticle suspensions in advanced cooling systems. In: International Mechanical Engineering Congress and Exhibition. Lee, S., Choi, S., Li, S., Eastman, J., 1999. Measuring thermal conductivity of fluids containing oxide nanoparticles. Journal of Heat Transfer 121, 280–289. Li, J.M., Li, Z.L., Wang, B.X., 2002. Experimental viscosity measurements for copper oxide nanoparticle suspensions. Tsinghua Science and Technology 7, 198–201. Magyari, E., 2011. Comment on the homogeneous nanofluid models applied to convective heat transfer problems. Acta Mechanica 222, 381–385. Manzari, M.T., 1999. An explicit finite element algorithm for convective heat transfer problems. International Journal for Numerical Methods Heat Fluid Flow 9, 860–877. McNab, G.S., Meisen, A., 1973. Thermophoresis in liquids. Journal of Colloid and Interface Science 44, 339. Nguyen, C.T., Roy, G., Gauthier, C., Galanis, N., 2007. Heat transfer enhancement using al2o3-water nanofluid for an electronic liquid cooling system. Applied Thermal Engineering 27, 1501–1506. Prasher, R., Song, D., Wang, J., Phelan, P., 2006. Measurements of nanofluid viscosity and its implications for thermal applications. Applied Physics Letters 89, 133108. Timofeevana, E.V., Gavrilov, A.N., McCloskey, J.M., Tolmachev, Y.V., Sprunt, S., Lopatina, L.M., Selinger, J.V., 2007. Thermal conductivity and particle agglomeration in alumina nanofluids: experiment and theory. Physical Review E 76, 061203. Wan, D.C., Patnaik, B.S.V., Wei, G.W., 2001. A new benchmark quality solution for the buoyancy-driven cavity by discrete singular convolution. Numerical Heat Transfer, Part B: Fundamentals 40, 199–228. Wang, X., Mujumdar, A., 2007. Heat transfer characteristics of nanofluids: a review. International Journal of Thermal Sciences 46, 1–19. Wang, X., Xu, X., Choi, S.U.S., 1999. Thermal conductivity of nanoparticle fluid mixture. Journal of Thermophysics and Heat Transfer 13, 474–480. Wen, D., Ding, Y., 2004. Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions. International Journal of Heat and Mass Transfer 47, 51–81. Wen, D., Zhangi, L., He, Y., 2009. Heat and Mass Transfer 45, 1061–1067. Xuan, Y., Li, Q., 2000. Heat transfer enhancement of nanofluids. International Journal of Heat and Fluid Transfer 21, 58–64. Xuan, Y., Li, Q., 2003. Investigation on convective heat transfer and flow features of nanofluids. Journal of Heat Transfer 125, 151–155.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
Non-homogeneous model for a side heated square cavity filled with a nanofluid Michele Celli ⇑ Dipartimento di Ingegneria Industriale, Alma Mater Studiorum, Università di Bologna, Viale Risorgimento 2, I-40136 Bologna, Italy
a r t i c l e
i n f o
Article history: Received 4 February 2013 Received in revised form 24 June 2013 Accepted 4 July 2013 Available online 27 July 2013 Keywords: Non-homogeneous model Nanofluid Side heating Square cavity Rayleigh number Buongiorno’s model
a b s t r a c t A side heated two dimensional square cavity filled with a nanofluid is here studied. The side heating condition is obtained by imposing two different uniform temperatures at the vertical boundary walls. The horizontal walls are assumed to be adiabatic and all boundaries are assumed to be impermeable to the base fluid and to the nanoparticles. In order to study the behavior of the nanofluid, a non-homogeneous model is taken into account. The thermophysical properties of the nanofluid are assumed to be functions of the average volume fraction of nanoparticles dispersed inside the cavity. The definitions of the nondimensional governing parameters (Rayleigh number, Prandtl number and Lewis number) are exactly the same as for the clear fluids. The distribution of the nanoparticles shows a particular sensitivity to the low Rayleigh numbers. The average Nusselt number at the vertical walls is sensitive to the average volume fraction of the nanoparticles dispersed inside the cavity and it is also sensitive to the definition of the thermophysical properties of the nanofluid. Highly viscous base fluids lead to a critical behavior of the model when the simulation is performed in pure conduction regime. The solution of the problem is obtained numerically by means of a Galerkin finite element method. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction In the last few decades an increasing attention has been focused on the heat transfer performances of a particular kind of colloids called nanofluids. A nanofluid is a suspension of metallic nanoparticles (or nanotubes) dispersed inside a base fluid. These colloids have been proposed as highly-effective heat transfer media (Choi and Eastman, 1995; Lee and Choi, 1996; Nguyen et al., 2007). Experimentally, great efforts have been spent in measuring the thermophysical properties of the nanofluids. Different kind of base fluids and dispersed nanoparticles have been tested in order to measure the thermal conductivity (Eastman et al., 1997; Lee et al., 1999; Wang et al., 1999; Xuan and Li, 2000; Eastman et al., 2001), the viscosity (Li et al., 2002; Prasher et al., 2006; Kwak and Kim, 2005) and the convective heat transfer coefficient (Xuan and Li, 2003; Wen and Ding, 2004; Heris et al., 2006). The thermophysical properties of the nanofluids have also been deeply studied from the theoretical point of view. A number of correlations have been proposed and employed in order to model the thermophysical properties (Das and Choi, 2006; Wang and Mujumdar, 2007). While the heat transfer performances of nanofluids have been widely studied, relatively few efforts have been undertaken with
⇑ Tel.: +39 051 2098330. E-mail address: [email protected] 0142-727X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.07.002
respect to the investigation of the sensitivity of the nanoparticles distributions to the heat transfer processes. In this contribution, a two dimensional square cavity filled with a nanofluid and subjected to side heating is studied. A nanofluid composed of Water as base fluid, and Alumina as nanoparticles dispersed into the base fluid, is investigated. The cavity walls are assumed to be impermeable to the base fluid and to the nanoparticles. The lower and upper boundary walls are assumed to be adiabatic. The side heating conditions are obtained by imposing two different temperatures at the vertical boundary walls. The model that is most frequently employed to simulate the nanofluids behavior is the homogeneous model. This model considers the nanofluid as a clear fluid with the only difference that the thermophysical properties of the nanofluid itself are modified in their values as functions of the average volume fraction of nanoparticles dispersed inside the base fluid. Magyari (2011) pointed out that, with a little rescaling effort, the homogeneous model produces the same results already obtained with the clear fluids models. Moreover, the assumption of homogeneity of the nanoparticles distribution may not hold when particle migration phenomena occur (Ding and Wen, 2005; Wen et al., 2009; Kang et al., 2007). The arguments just presented lead to the choice of employing a nonhomogeneous model for the present analysis. The non-homogeneous model prescribes a dedicated mass balance equation for the dispersed nanoparticle. The mathematical model is thus characterized by four balance equations: the mass
328
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335
Nomenclature c DB dp DT F(U0) g G(U0) H(U0) I(U0) j k kB L MES n N P r S T Tc Th T0 v x
specific heat (J/(kg K)) Brownian motion coefficient (m2/s) nanoparticles diameter (m) thermophoresis coefficient (m2/s) nondimensional function gravity acceleration (m/s2) nondimensional function nondimensional function nondimensional function nanoparticles mass flux (m/s) thermal conductivity (W/(mK)) Boltzmann constant (J/K) lenght of the cavity side (m) nondimensional maximum size of the grid elements normal unit vector number of grid elements nondimensional pressure radius of the neighborhood of / = 1 nondimensional cavity surface nondimensional temperature cold wall temperature (K) hot wall temperature (K) reference temperature (K) nondimensional velocity field (u, v) nondimensional position vector (x, y)
balance equation for the base fluid, the momentum balance equation for the nanofluid, the energy balance equation for the nanofluid and the mass balance equation for the nanoparticles, Buongiorno (2006). Moreover, the thermophysical properties of the nanofluid are here expressed by means of phenomenological correlations as functions of the average volume fraction of the nanoparticles. The thermophysical properties of the base fluid are, in fact, unavoidably modified by the presence of the dispersed nanoparticles. On the other hand, the nondimensional governing parameters (Rayleigh number, Prandtl number and Lewis number) are defined exactly as for clear fluids. This choice allows an easier comparison between the results obtained by the mathematical model here employed and the results for clear fluids found in the literature. The contribution of the average volume fraction of nanoparticles, coming from the definitions of the thermophysical properties, is thus taken into account by means of a number of ad hoc nondimensional parameters. The non-homogeneous model is here studied for different range of values of the nondimensional parameters involved. The main goals of this study are looking for possible non-homogeneities of the nanoparticles distribution and investigating the heat transfer performances of the nanofluid at the cavity side walls. Particular attention is focused on the low Rayleigh numbers regimes. The nanoparticles distribution shows indeed a strong sensitivity to the heat transfer processes for low Rayleigh numbers. The non-homogeneous model is here also tested for a particular highly viscous base fluids (Propylene Glycol) in the limit case of pure conduction and pure diffusion regime. The numerical solution of the problem is obtained by Galerkin’s finite element method.
Greek symbols a thermal diffusivity (m2/s) b thermal expansion coefficient (K1) bT nondimensional constant DT reference temperature jump (K) l dynamic viscosity (Pa s) q density (kg/m3) / rescaled volume fraction of nanoparticles U local volume fraction of nanoparticles U0 average volume fraction of nanoparticles Nondimensional numbers Le Lewis number NBT nondimensional number Nu average Nusselt number Pr Prandtl number Ra Rayleigh number Subscripts, Superscripts – dimensional quantity rescaled quantity f, p, nf fluid, nanoparticle, nanofluid max maximum value
side heating condition is obtained by imposing two different temperatures at the walls: the hot wall, Th, is assumed to be on the left vertical boundary and the cold wall, Tc, is assumed to be on the right vertical boundary. The cavity walls are subjected to the noslip condition. The boundary conditions are shown in their dimensional form in Table 1. A nanofluid composed of Water as base fluid, and Alumina (Al2O3, Auerkari, 1996) as nanoparticles dispersed inside the base fluid, is here studied. In order to analyze the nanoparticles distribution, a non-homogeneous model is employed, Buongiorno (2006). The following hypotheses are assumed: Non-homogeneous nanofluid model. Brownian motion and thermophoresis as leading physical transport mechanisms for the nanoparticles diffusion. Thermophysical properties of the nanofluid are expressed as functions of the average volume fraction of nanoparticles dispersed inside the base fluid.
2. Mathematical model The two dimensional side heated square cavity here investigated is sketched in Fig. 1. The cavity is assumed to be impermeable and the horizontal walls are assumed to be adiabatic. The
Fig. 1. Sketch of the system.
329
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335 Table 1 Dimensional boundary conditions.
ð3aÞ
v
T
U
jp ; qp v $ U ¼ $
ð3bÞ
v ¼ 0
T ¼ Th
n jp ¼ 0
v P þ l r 2v ¼ $ þ qnf bnf ðT T 0 Þg; qnf v $ nf
ð3cÞ
v ¼ 0
T ¼ 0 n$
n jp ¼ 0
T ¼ knf r 2 T cpjp $ T; $ ðqcÞnf v
ð3dÞ
v ¼ 0
T ¼ Tc
n jp ¼ 0
v ¼ 0
T ¼ 0 n$
n jp ¼ 0
(Th Tc)/T0 1. Steady analysis. Oberbeck-Boussinesq approximation. Negligible viscous dissipation. Local thermal equilibrium between nanoparticles.
base
fluid
and
Here the reference temperature T0 is defined as T0 = (Th + Tc)/2. The thermophysical properties of the nanofluid such as the density qnf, the specific heat cnf and the thermal expansion coefficient bnf, are defined by means of analytical expressions
qnf ¼ ð1 U0 Þqf þ U0 qp ; ðqcÞnf ¼ ð1 U0 ÞðqcÞf þ U0 ðqcÞp ; ðqbÞnf ¼ ð1 U0 ÞðqbÞf þ U0 ðqbÞp ;
ð1Þ
where U0 is the average volume fraction of nanoparticles dispersed inside the cavity, f stands for fluid, p stands for nanoparticles and nf stands for nanofluid. The dynamic viscosity lnf and the thermal conductivity knf are defined by means of empirical correlations, respectively the Brinkman model and the Maxwell model, namely
lf
lnf ¼
v ¼ 0; $
; ð1 U0 Þ2:5 kp þ 2kf 2U0 ðkf kp Þ : knf ¼ kf kp þ 2kf þ U0 ðkf kp Þ
ð2Þ
is the dimenwhere the overlines refer to dimensional quantities, v sional velocity field, T is the dimensional temperature field, P is the dimensional pressure field, g is the gravity acceleration, U is the local volume fraction of nanoparticles and jp is the dimensional nanoparticles mass flux. From Buongiorno (2006), the dimensional nanoparticles mass flux is defined by
! jp ¼ q DB $ U þ D T $T ; p T
where DB = (kBT0)/(3 plfdp) is the Brownian motion coefficient and DT = (bT lfU0)/qf is the thermophoresis coefficient. The symbol dp is the diameter of the nanoparticles and bT is a proportionality factor defined by McNab and Meisen (1973). Eq. (4) introduces the leading transport mechanisms for the diffusion of the nanoparticles inside the cavity: the first term on the right-hand side refers to the Brownian motion contribution and the second term on the righthand side refers to the thermophoresis contribution. Among the boundary conditions of Table 1, the fourth column shows the cavity walls impermeability boundary condition with respect to the nanoparticles. The impermeability is expressed by means of assuming a vanishing orthogonal flux of nanoparticles at the walls. The unit vector n is directed outward with respect to the walls. 3. Nondimensional formulation In order to proceed with the analysis, a nondimensional treatment of the governing equations is needed. The following scalings allow one to rewrite the system of Eqs. (3) in a nondimensional form
v ¼
¼ Lx; x
The behavior of lnf and knf as a function of U0 is shown in Fig. 2. It is worth noting that an increasing average volume fraction of nanoparticles dispersed inside the cavity, i.e. increasing values of U0, yields, with respect to the clear fluids, higher values of thermal conductivity and of dynamic viscosity. Since two mass balance equations (one for the base fluid and one for the nanoparticles), a momentum balance equation and an energy balance equation are taken into account, the dimensional set of governing equations are defined as
ð4Þ
T ¼ T DT þ T 0 ;
anf L
v;
P¼
lnf anf L2
P;
ð5Þ
U ¼ U0 /;
where L is the length of the cavity sides, DT = Th Tc and anf = knf/ (q c)nf is the thermal diffusivity of the nanofluid. The value of U, that is already nondimensional, is here rescaled by its average value over the cavity, U0, for a better interpretation of the oncoming results. On assuming the approximation DT/T0 1 the definition of the nanoparticles mass flux Eq. (4) may be simplified as
jp ¼ q DB $ U þ DT DT $ T : p T0
ð6Þ
On using Eqs. (5) and (6), the system of Eqs. (3) becomes
1.30
$ v ¼ 0;
ð7aÞ !
1.25
v $/ ¼
1.20
1 r2 T r2 / þ ; Lef FðU0 Þ NBT
v $v ¼ $P þ r2 v þ Raf HðU0 ÞT; Prf GðU0 Þ 1 $T $T ; v $T ¼ r2 T þ $/ $T þ Lef IðU0 Þ NBT
1.15 1.10 1.05 1.00 0.00
0.02
0.04
0.06
0.08
Fig. 2. Behavior of lnf/lf and knf/kf as functions of U0.
0.10
ð7bÞ ð7cÞ ð7dÞ
where Raf is the Rayleigh number relative to the fluid, Prf is the Prandtl number relative to the fluid, Lef is the Lewis number relative to the fluid and NBT is a nondimensional parameter that measures the relative strength of the Brownian motion contribution with respect to the thermophoresis contribution. The Rayleigh number arises
330
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335
from the buoyancy term that is present in the momentum balance equation under the Oberbeck–Boussinesq approximation. This nondimendional number describes the ratio of energy released by the buoyancy forces compared to the energy dissipated by heat conduction and viscous drag. The Prandtl number and the Lewis number are numbers dependent only on the fluid physical characteristics of the system and they do not depend on the geometry or on the flow variables. As a result, the Prandtl number describes the competition between two physical mechanisms, the viscous diffusion and the thermal diffusion. As for the Prandtl number, the Lewis number describes the competition between two diffusion mechanisms: the thermal diffusion and the mass diffusion. The Rayleigh, Prandtl and Lewis numbers are defined exactly in the same way as for clear fluids. This choice allows an easier comparison of the oncoming results with the literature on clear fluids. The nondimensional numbers just introduced are defined as follows:
Raf ¼ Lef ¼
lf qf bf ðT h T c ÞL3 g ; Prf ¼ ; lf af qf af af U0 T 0 DB DB
;
NBT ¼
DT DT
ð8Þ
:
The functions F(U0), G(U0), H(U0) and I(U0) in Eqs. (7) are a consequence of defining the nondimensional parameters of Eq. (8) in the same way as for clear fluids. The functions F(U0), G(U0), H(U0) and I(U0) thus gather the contribution of the average volume fraction of nanoparticles, U0, coming from the definitions of the thermophysical properties, Eqs. (1) and (2).
lnf kf qf ðqcÞnf ; lf knf qnf ðqcÞf lf kf ðqbÞnf ðqcÞnf knf ðqcÞf HðU0 Þ ¼ ; IðU0 Þ ¼ : lnf knf ðqbÞf ðqcÞf kf ðqcÞp U0
FðU0 Þ ¼
knf ðqcÞf ; kf ðqcÞnf
GðU0 Þ ¼
ð9Þ
The nondimensional boundary conditions for the square cavity are described in Table 2. 4. Code validation The system of governing Eqs. (7) together with the relative boundary conditions, Table 2, are here solved numerically by means of Galerkin’s finite element method implemented through the software package Comsol Multiphysics (Ó Comsol, Inc.). In order to validate the results obtained, a numerical test of mesh independence is performed. The cavity is meshed with an unstructured grid of triangular elements with Lagrange quadratic basis functions. A key parameter of the grid is the maximum size of the elements within the mesh (MES). For a given geometry, every value of MES corresponds to a fixed number of grid elements (N). Table 3 shows, as functions of MES, the behavior of two parameters: the average Nusselt number over the hot wall and the integral over the cavity of the absolute value of the nanoparticles distribution gradient. The average Nusselt number is defined as follows:
Table 2 Nondimensional boundary conditions. T
/
v=0
T ¼ 12
@/ @x
¼ @T=@x N BT
v=0
@T @y
@/ @y
¼0
v=0
T ¼ 12
@/ @x
¼ @T=@x N BT
v=0
@T @y
@/ @y
¼0
¼0
MES
N
Nu
0.03 0.025 0.02 0.015 0.01
2894 4040 6282 11,748 24,910
1.0362 1.0362 1.0362 1.0362 1.0362
R
S
j$/j dS
0.1002 0.0995 0.0994 0.0993 0.0993
Table 4 Benchmark validation of the average Nusselt number ðNuÞ, the maximum horizontal velocity (umax) at the mid-width (x = 0.5) and the maximum vertical velocity (vmax) at the mid-height (y = 0.5) for Raf = 103. Raf = 103
Present analysis
Ref. Davis and cavity (1962)
Ref. Manzari (1999)
Ref. Wan et al. (2001)
Ref. Barletta et al. (2006)
Nu umax y(umax)
1.118
1.12
1.074
1.117
1.118
3.653 0.813 3.701 0.177
3.634 0.813 3.679 0.179
3.68 0.817 3.73 0.183
3.489 0.813 3.686 0.188
– – – –
vmax x(vmax)
Nu ¼
Z 0
1
@T dy: @x
ð10Þ
The values shown in Table 3 are obtained for Raf = 103, U0 = 10% and NBT = 1. For the following numerical analysis, the value MES = 0.015 is chosen as the best value for the maximum element size. For a further validation of the code, a comparison of the numerical results with the benchmark solutions found in the literature is here performed. The limit U0 ? 0, i.e. the case of a fluid clear of nanoparticles, is considered. Moreover, the air is chosen as working fluid. For a side heated square cavity filled with air, the literature provides a number of benchmark results. The benchmark values here used are taken from Davis and cavity (1962), Manzari (1999), Wan et al. (2001) and Barletta et al. (2006) and refer to the case of Raf = 103. The choice of taking air as working fluid is due to the fact that it yields more reliable checks since the Prandtl number of air is less sensitive to the reference temperature with respect to the Prandtl number of water. Table 4 shows the present analysis results and the benchmark solutions for the value of Nu, the maximum value of umax and its respective coordinate y(umax), the maximum value of vmax and its respective coordinates x(umax). The results here obtained show a good agreement with the literature. The results obtained for other values of the Rayleigh number display again a good agreement with the literature but they are not reported here for the sake of brevity. 5. Results The system of governing Eqs. (7) is characterized by a relatively large set of nondimensional parameters: Raf, Prf, Lef, NBT and U0. The aim of this paper is investigating the relationship among the Rayleigh number, the volume fraction of nanoparticles and the heat transfer performances of the nanofluid. It is thus necessary
Table 5 Maximum values of jvj for different Rayleigh numbers, Raf and two different average volume fraction of nanoparticles, U0.
v
¼0
Table 3 Mesh independence test.
Raf = 100 Raf = 101 Raf = 102 Raf = 103
U0 = 1%
U0 = 10%
0.004 0.037 0.369 3.483
0.002 0.020 0.203 1.988
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335
331
Fig. 3. Nanoparticles distributions / for a fixed value of U0 = 1%. The column on the left refers to NBT = 1, the column on the right refers to NBT = 10. The first row from the top refers to Raf = 0, the second row refers to Raf = 5 and the third row refers to Raf = 10.
to fix the values of Prf, Lef and NBT. Once the basic fluid (Water), the nanoparticles material (Alumina), the nanoparticles size (dp = 108 m) and the reference temperature (T0 = 300 K) are chosen, the value of Prf and Lef are already set up. The value of dp refers to the effective average size of the nanoparticles dispersed inside the cavity. In fact, particles and agglomerations of particles with different sizes are dispersed in the base fluid as well shown in
Figs. 5–7 of Timofeevana et al. (2007). The choice to disregard the real distribution of the nanoparticle sizes present inside the cavity has been taken in order to simplify the treatment of a problem, where several governing parameters are already present. The value of NBT depends now only on DT, Eq. (8). Within the assumptions just made, the range of variability of NBT is 101 < NBT < 101 for a temperature jump between the vertical walls of 1 K < DT < 10 K.
332
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335
Fig. 4. Nanoparticles distributions / for a fixed value of U0 = 10%. The column on the left refers to NBT = 1, the column on the right refers to NBT = 10. The first row from the top refers to Raf = 0, the second row refers to Raf = 5 and the third row refers to Raf = 10.
In the following the cases NBT = 1 and NBT = 10 are studied. Eventually, the case of a relatively small value of NBT is investigated. The typical average volume fraction of nanoparticles varies between U0 = 1% and U0 = 10%. If a value of U0 10% is assumed, the system has to be modeled as a fluid saturated porous medium. In the following, both the cases, U0 = 1% and U0 = 10%, are analyzed.
On focusing the attention on the groups Lef F(U0) and Lef I(U0) of Eqs. (7), one may notice that they have an order of magnitude that varies in the range 103–105. These two groups of parameters tend, in most of the cases, to dominate the Brownian motion and thermophoresis contributions included in Eqs. (7b) and (7d). When, on the other hand, low velocities are taken into account,
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335
333
Fig. 5. Nanoparticles distributions / for fixed value of U0 = 10% and for NBT = 1 (left) and for NBT = 10 (right). Both frames refer to Raf = 100.
the Brownian motion and thermophoresis contributions become not negligible. If low velocities are involved, the convective terms of Eqs. (7b) and (7d) become indeed comparable with the Brownian motion and thermophoresis contributions. The cases of low velocity correspond to weak convection phenomena and thus, for the system here analyzed, to low Rayleigh numbers. In the following, low Rayleigh numbers are investigated. Table 5 displays the maximum values of jvj as functions of the Rayleigh number, Raf, and of the average volume fraction of nanoparticles, U0. While the velocity maxima increase with the Rayleigh number, they decrease as the average volume fraction of nanoparticles increases. A reason for the latest behavior has to be sought into the definitions of dynamic viscosity and thermal conductivity in Eq. (2). Brinkman’s and Maxwell’s models, in fact, prescribe that, when the number of suspended nanoparticles is increased, the dynamic viscosity and the thermal conductivity of the nanofluid increase as well. The increasing dynamic viscosity and thermal conductivity tend indeed to depress the strength of the convection phenomena (for fixed temperature jump, DT). Fig. 2 shows the enhancement of dynamic viscosity and thermal conductivity for increasing values of U0. Fig. 3 shows the distributions of the nanoparticles volume fraction inside the square cavity for a fixed value of the average volume fraction of nanoparticles, U0 = 1%, and for different values of the Rayleigh number and of the parameter NBT. Each row of Fig. 3 refers to a different Rayleigh number: the first row from the top refers to Raf = 0, the second row refers to Raf = 5 and the third row refers to Raf = 10. The left column of Fig. 3 refers to simulations performed with NBT = 1 and the right column refers to simulations performed with NBT = 10. The gray shaded areas drawn inside the nanoparticles distributions frames of Fig. 3 refer to those areas where the nanoparticles volume fraction values belong to the neighborhood of the value / = 1. This neighborhood is characterized by a radius r = 0.01. When the gray shaded area occupies a small or a negligible part of the cavity, the distribution of nanoparticles has to be considered as non-homogeneous. On the other hand, when the gray shaded area occupies most part of the cavity, the distribution of the nanoparticles can be considered as homogeneous. The case of pure conduction (Raf = 0, first row of Fig. 3) shows, as expected, distributions of nanoparticles that are symmetric with respect to the temperature distribution. The conduction regime enhances the effect of the thermophoresis phenomenon and one can note that the nanoparticles distribution is highly non-homogeneous. The pure conduction regime frame, in fact, displays a deviation up to 50% from the average value / = 1 relative to the case of
NBT = 1, and the gray shaded area occupies a relatively small area of the cavity. For positive values of the Rayleigh number, the second and third row from the top of Fig. 3, the mixing due to the thermal convection amplifies the area of the neighborhood of the average value / = 1. Eventually, Fig. 3 highlights the key role of the parameter NBT: a low value of NBT leads to a less homogeneous distribution of nanoparticles. When the average volume fraction of nanoparticles is increased to U0 = 10% (Fig. 4) the nanoparticles distributions and temperature fields obtained are not dramatically different with respect to the case U0 = 1%. As for Fig. 3 the first row from the top refers to Raf = 0, the second row refers to Raf = 5 and the third row refers to Raf = 10. Again, the left column refers to simulations performed with NBT = 1 and the right column refers to simulations performed with NBT = 10. It is worth noting that, with respect to Fig. 3, the homogeneous area of the frames relative to the same Rayleigh number decreases in size, second and third row of Fig. 4. It has already been pointed out that increasing values of U0 tend to inhibit the convective motion. When the convection is weaker, the mixing due to this phenomenon is less effective and this is the reason why the well mixed and homogeneous area is reduced in size. Fig. 5 shows the nanoparticles distributions for fixed values of the Rayleigh number, Raf = 100, of the average volume fraction of nanoparticles U0 = 10% and two different values of NBT: the left frame refers to NBT = 1 and the right frame refers to NBT = 10. In fact, if the Rayleigh number is high enough, the non-homogeneous areas become more and more confined close to the boundaries and the gray shaded area, the homogeneous area, increases in size until it occupies most part of the cavity. For values of Raf P 100 the nonhomogeneities are then totally confined nearby the boundaries and the nanoparticles distribution may be considered as homogeneous. The nanoparticles distributions relative to the case U0 = 1% is not shown here for the sake of brevity and because it does not add any further information. Fig. 6 describes the temperature behavior along the diagonal that goes from the upper left corner to the lower right corner of the square cavity for fixed values of NBT = 10 and U0 = 1%. The different curves are drawn for different values of the Rayleigh number. Only the case (NBT = 10, U0 = 1%) is displayed instead of all the possible permutations of the pair (NBT, U0) shown in Figs. 4 and 3 because Fig. 6 is representative of the system behavior. One can note that, as expected, low values of the Rayleigh number lead to a purely horizontal temperature distribution, i.e. the pure conduction regime for Raf = 0.
334
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335
the momentum balance equation can be solved without any contribution of the temperature field because the buoyancy term is negligible. The velocity field for the pure conduction regime is constant and equal to zero. Since the fluid is still, Eqs. (7) can be simplified
r2 T þ r2 / þ
1 $T $T ¼ 0; $/ $T þ Lef IðU0 Þ NBT
r2 T NBT
ð11aÞ
¼ 0:
ð11bÞ
The temperature field can now be rescaled by the parameter NBT
T : Te ¼ NBT
ð12Þ
The system of Eqs. (11) can be rewritten as
1 $/ $ Te þ $ Te $ Te ¼ 0; Lef IðU0 Þ r2 / þ r2 Te ¼ 0:
r2 Te þ
Fig. 6. Temperature distribution along the diagonal of the cavity (from the upper left angle to the lower right angle) for different values of Raf and fixed values of NBT = 10 and U0 = 1%.
1.10
Nu
1.08 1.06 1.04 1.02 1.00 0
200
400
600
800
1000
Raf Fig. 7. Nu as a function of Raf for different values of average volume fraction of nanoparticles U0.
Fig. 7 shows the behavior of the average Nusselt number over the hot wall, Eq. (10), as a function of the Rayleigh number for different values of the average volume fraction of nanoparticles U0 = 0%; 1%; 10%. From Table 5, the higher is U0 the more the convection phenomena are inhibited. If the heat transfer processes at the walls are more effective when the convection phenomena are stronger, thus high values of U0 mean a less efficient heat transfer at the wall. The reasoning just made agrees with the results shown in Fig. 7 where the value of the Nusselt number decreases as U0 increases (for a given value of Raf). 5.1. Limiting case Raf ? 0: pure conduction and pure diffusion regime The limiting case Raf ? 0 is characterized by the pure conduction regime for the temperature field and by the pure diffusion regime for the nanoparticles distribution. This limiting case may be achieved by means of imposing very small temperature difference between the vertical walls or by means of considering systems of very small characteristic length. In the pure conduction regime
ð13bÞ
Now an order of magnitude analysis is performed with the aim of simplifying Eqs. (13). Once the nanoparticles material and the reference temperature T0 are set, the parameter NBT becomes a function of the base fluid properties (qf, l) and a function of the temperature jump between the vertical walls DT, namely
NBT /
1.12
ð13aÞ
qf DT l 2
ð14Þ
:
Since the possible base fluids are characterized by densities of, approximately, the same order of magnitude, the value of NBT is mainly driven by the term DT l2. Now one can assume that the limit Raf ? 0 is obtained by imposing a very small characteristic length of the system. If, moreover, a highly viscous base fluid is considered, Propylene Glycol, one can obtain, for a given DT, a NBT significantly small (Eq. (14) and Table 6). Once the characteristic length L, the temperature jump DT and the average volume fraction of nanoparticles U0 are set (Table 6) the order of magnitude of the group Lef I(U0) is evaluated. Since the group Lef I(U0) has an order of magnitude of 106, one can neglect the second term of the left-hand side of Eq. (13a) and Eqs. (13) can be simplified again to obtain the pure conduction and pure diffusion regime. The simplified system and the relative boundary conditions are
d Te N1 N1 x ¼ 0; Te ð0Þ ¼ BT ; Te ð1Þ ¼ BT ; 2 dx 2 2 2 d / d Te d Te ¼ 0; /ð0Þ ¼ ; /ð1Þ ¼ : dx dx dx2 2
ð15aÞ ð15bÞ
The solutions of Eqs. (15) are
x þ 1=2 Te ¼ ; NBT d/ ¼ N 1 BT : dx
ð16aÞ ð16bÞ
If NBT 103, a critical large nanoparticle volume fraction gradient is obtained, Eq. (16b). The latest conclusion highlights a possible critical behavior of Boungiorno’s model for nanofluids based on
Table 6 Set up for the Propylene Glycol – Alumina nanofluid simulation in the limit of pure conduction. Order of magnitude of NBT and Lef I(U0). L (m)
DT (K)
U0
NBT
Lef I(U0)
104
1
10%
103
106
M. Celli / International Journal of Heat and Fluid Flow 44 (2013) 327–335
highly viscous base fluids in the limit of Raf ? 0. A clear example of this critical behavior arises when the case of a nanofluid composed of Propylene Glycol as base fluid, a fluid much more viscous than water, and Alumina as dispersed nanoparticles (Table 6) is taken into account. If one simulates the distribution of the nanoparticle volume fraction for fixed U0 = 10% and Raf = 0 a nanoparticle distribution that increases linearly as a function of x from 100 to 100 for 0 < x < 1 is obtained. It is worth noting that the value of NBT 103 is, as expected, a few orders of magnitude smaller with respect to the case of the water based nanofluid. A value of NBT so small is the reason of the critical range of variation of the nanoparticles distribution / obtained in the Propylene Glycol base fluid case. 6. Conclusions A two dimensional square cavity subjected to side heating and filled by a nanofluid is here studied. The boundary walls are assumed to be impermeable to the base fluid and to the nanoparticles. In order to investigate the spatial distribution of the nanoparticles dispersed inside the cavity, a non-homogeneous model is taken into account. A mass balance equation for the dispersed nanoparticles is considered in addition to the mass balance equation for the base fluid, the momentum balance equation for the nanofluid and the energy balance equation for the nanofluid. Every thermophysical property of the nanofluid is expressed as a function of the average volume fraction of the nanoparticles. The Brownian motion and the thermophoresis are considered as the leading physical transport mechanisms for the nanoparticles. The main tasks of this study are the investigation of the possible non-homogeneity of the distribution of the nanoparticles dispersed inside the cavity and the investigation of the heat transfer performances of the nanofluid subjected to a natural convection regime. The following conclusions are achieved: Low Rayleigh numbers may yield a markedly nonhomogeneous distribution of the nanoparticles inside the cavity. In this range of the Rayleigh number a non-homogeneous model is more appropriate for the description of the system. Relatively high Rayleigh numbers confine the non-homogeneities nearby the boundaries leaving a fairly homogeneous distribution in the core of the cavity. The homogeneous model become reliable if Raf P 102. On employing the Maxwell model and the Brinkman model to define the dynamic viscosity and the thermal conductivity of the nanofluid, the heat transfer at the vertical boundaries becomes less efficient as the average volume fraction of nanoparticles increases. A critical behavior of Boungiorno’s model is found for highly viscous base fluids in the limit of pure conduction and pure diffusion.
Acknowledgements This work was financially supported by Italian government, MIUR Grant PRIN-2009KSSKL3.
335
The author is grateful to one of the anonymous Reviewers for the helpful suggestions that improved the quality of the paper. References Auerkari, P., 1996. Mechanical and Physical Properties of Engineering Alumina Ceramics. VTT Technical Research Center of Finland. Barletta, A., Nobile, E., Pinto, F., di Schio, E.R., Zanchini, E., 2006. Natural convection in a 2d-cavity with vertical isothermal walls: cross-validation of two numerical solutions. International Journal of Thermal Sciences 45, 917–922. Buongiorno, J., 2006. Convective transport in nanofluids. ASME Journal of Heat Transfer 128, 240–250. Choi, S.U.S., Eastman, J.A., 1995. Enhancing thermal conductivity of fluids with nanoparticles. In: International Mechanical Engineering Congress and Ewxhibition, San Francisco, CA, USA. Das, H.P.S.K., Choi, S.U.S., 2006. Heat transfer in nanofluids – a review. Heat Transfer Engineering 37, 3–19. Davis, G.D.V., 1962. Natural convection of air in a square cavity, a benchmark numerical solution. International Journal for Numerical Methods Fluids 3, 249– 264. Ding, Y., Wen, D., 2005. Particle migration in a flow of nanoparticle suspensions. Powder Technology 149, 84–92. Eastman, J., Choi, S., Li, S., Thompson, L., Lee, S., 1997. Enhanced thermal conductivity through the development of nanofluids. Materials Research Society Symposium – Proceedings 457, 3–11. Eastman, J., Choi, S., Li, S., Yu, W., Thompson, L., 2001. Anomalously increased effective thermal conductivities of ethylene glycol-based nano-fluids containing copper nanoparticles. Applied Physics Letters 78, 718–720. Heris, S., Etemad, S.G., Esfahany, M., 2006. Experimental investigation of oxide nanofluids laminar flow convective heat transfer. International Communications in Heat and Mass Transfer 33, 529–535. Kang, H., Kim, W., Kim, S.H., 2007. Effect of particle migration on the heat transfer of nanofluid. Korea–Australia Rheology Journal 19, 99–107. Kwak, K., Kim, C., 2005. Viscosity and thermal conductivity of copper oxide nanofluid dispersed in ethylene glycol. Korea–Australia Rheology Journal 17, 35–40. Lee, S., Choi, S.U.S., 1996. Application of metallic nanoparticle suspensions in advanced cooling systems. In: International Mechanical Engineering Congress and Exhibition. Lee, S., Choi, S., Li, S., Eastman, J., 1999. Measuring thermal conductivity of fluids containing oxide nanoparticles. Journal of Heat Transfer 121, 280–289. Li, J.M., Li, Z.L., Wang, B.X., 2002. Experimental viscosity measurements for copper oxide nanoparticle suspensions. Tsinghua Science and Technology 7, 198–201. Magyari, E., 2011. Comment on the homogeneous nanofluid models applied to convective heat transfer problems. Acta Mechanica 222, 381–385. Manzari, M.T., 1999. An explicit finite element algorithm for convective heat transfer problems. International Journal for Numerical Methods Heat Fluid Flow 9, 860–877. McNab, G.S., Meisen, A., 1973. Thermophoresis in liquids. Journal of Colloid and Interface Science 44, 339. Nguyen, C.T., Roy, G., Gauthier, C., Galanis, N., 2007. Heat transfer enhancement using al2o3-water nanofluid for an electronic liquid cooling system. Applied Thermal Engineering 27, 1501–1506. Prasher, R., Song, D., Wang, J., Phelan, P., 2006. Measurements of nanofluid viscosity and its implications for thermal applications. Applied Physics Letters 89, 133108. Timofeevana, E.V., Gavrilov, A.N., McCloskey, J.M., Tolmachev, Y.V., Sprunt, S., Lopatina, L.M., Selinger, J.V., 2007. Thermal conductivity and particle agglomeration in alumina nanofluids: experiment and theory. Physical Review E 76, 061203. Wan, D.C., Patnaik, B.S.V., Wei, G.W., 2001. A new benchmark quality solution for the buoyancy-driven cavity by discrete singular convolution. Numerical Heat Transfer, Part B: Fundamentals 40, 199–228. Wang, X., Mujumdar, A., 2007. Heat transfer characteristics of nanofluids: a review. International Journal of Thermal Sciences 46, 1–19. Wang, X., Xu, X., Choi, S.U.S., 1999. Thermal conductivity of nanoparticle fluid mixture. Journal of Thermophysics and Heat Transfer 13, 474–480. Wen, D., Ding, Y., 2004. Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions. International Journal of Heat and Mass Transfer 47, 51–81. Wen, D., Zhangi, L., He, Y., 2009. Heat and Mass Transfer 45, 1061–1067. Xuan, Y., Li, Q., 2000. Heat transfer enhancement of nanofluids. International Journal of Heat and Fluid Transfer 21, 58–64. Xuan, Y., Li, Q., 2003. Investigation on convective heat transfer and flow features of nanofluids. Journal of Heat Transfer 125, 151–155.