Numerical investigation of natural convection of Al2O3 nanofluid in ...

Heat Mass Transfer (2009) 46:15–23 DOI 10.1007/s00231-009-0540-7

ORIGINAL

Numerical investigation of natural convection of Al2O3 nanofluid in vertical annuli Omid Abouali Æ Ahmad Falahatpisheh

Received: 3 June 2009 / Accepted: 8 September 2009 / Published online: 17 September 2009 Ó Springer-Verlag 2009

Abstract This paper presents the numerical study of internal free convection of Al2O3 water nanofluid in vertical annuli. Vertical walls are maintained at constant temperatures and horizontal walls are adiabatic. Results are validated by experimental data. Effect of nanofluids on natural convection is investigated as a function of geometrical and physical parameters and particle fractions for aspect ratio of 1 B H/L B 5, Grashof number of 103 B Gr B 105 and concentration of 0 B / B 0.06. More than 330 different numerical cases are investigated to develop a new correlation for the Nusselt number. This correlation is presented as a function of Nusselt number of base fluid and particle fraction which is a linear decreasing function of particle fraction. The developed correlation for annuli is also valid for the natural convection of Al2O3 water nanofluid in a square cavity. Furthermore, the effect of the viscosity and conductivity models on the Nusselt number of nanofluids in cylindrical cavities are discussed. List of symbols A Aspect ratio H/L A Area B Constant of Kapitza resistance C Constant in Nusselt numbers ratio C1 Constant in thermal conductivity model cp Specific heat (J/kg K) O. Abouali (&) Mechanical Engineering Department, Shiraz University, Shiraz, Iran e-mail: [email protected] A. Falahatpisheh Mechanical Engineering Department, University of South Carolina, Columbia, SC, USA e-mail: [email protected]

CR:M D D0 g GrL H K kb k L l p Pr Q Rednano Ra r T V X, r

Random motion velocity of nanoparticles Diameter Diffusion coefficient Gravity acceleration (m/s2) Grashof number gbf DTL3 =m2f Height of annulus Radius ratio of annulus Boltzmann constant 1.3807 9 10-23 J/K Thermal conductivity (W/m K) Gap (ro - ri) Mean free path Static pressure (Pa) Prandtl number mf/af Heat flux (W) Reynolds number based on particle diameter Rayleigh number = GrLPr Radius Temperature (°C) or (K) Velocity Axial and radial coordinate

Greek symbols a Thermal diffusivity (m2/s) b Volumetric thermal expansion coefficient (1/K) / Nanoparticle fraction l Dynamic viscosity (kg/m s) m Kinematic viscosity (m2/s) h Non dimensional temperature q Density (kg/m3) Subscript bf Base fluid C Cold eff Effective property f Properties at film temperature H Hot

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16

i nf o Particle R.M r x

Heat Mass Transfer (2009) 46:15–23

Inner Nanofluid Outer Nanofluid particle Random motion Radial component Axial coordinate

1 Introduction Since traditional heat transfer fluids such as water, oil, and ethylene glycol mixtures have poor heat transfer characteristics, one way to enhance heat transfer is to increase thermal conductivity of fluid [1]. The thermal conductivity of solids is typically higher than that of liquids. For instance, thermal conductivity of copper is 700 times greater than that of water and 3,000 times greater than that of engine oil. The newly born method called nanofluid, a term proposed by [1], is supposed to enhance the performance of heat transfer by means of suspending nanoparticles in the range of sizes (10–50 nm) in clear and saturated liquids [2]. Undoubtedly, nanoparticles make the base fluid have a higher thermal conductivity; this can achieve even at very low volumetric fraction of particle such as 1% [1, 3–7]. There are several models for thermal conductivity of nanofluids. Recently used models are Wasp’s model [8] and Jang and Choi’s model [5]. Jang and Choi [5] showed that the Brownian motion of nanoparticles at the molecular and nanoscale level is a key mechanism governing the thermal behavior of nanofluids. Their model not only captures the fraction and temperaturedependent conductivity, but also predicts strongly sizedependent conductivity. There are several models that predict the viscosity of nanofluids. Einstein [9], Brinkman [10], Brownian [11], and Pak and Cho’s model [12] are among the important ones. Recently, Nguyen et al. [13] experimentally calculated viscosity of Al2O3 in sizes 36 and 47 nm. Their results clearly revealed the existence of a critical temperature beyond which the particle suspension properties seem to be severely changed, which, in turn, has triggered a hysteresis phenomenon. The critical temperature is strongly dependent on both particle fraction and size. The hysteresis phenomenon has raised serious concerns regarding the reliability of using nanofluids for heat transfer enhancement purposes. Their data have also shown that the Einstein’s formula and some other ones originated from the classical linear fluid theory seem to be limited to nanofluids with low particle fractions. They showed that for particle fraction lower than 4% the values of relative viscosity are approximately identical for the particle sizes

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studied and remain constant independent of temperature. However, for a higher particle fraction, it has been found that the relative viscosity is not only dependent on the temperature but also on the particle-size as well. It has been determined that none of formulas cited would be appropriately used for the considered nanofluids [14]. In fact, even for a relatively low particle fraction, the Einstein’s formula and the ones proposed by Brinkman [10], Lundgren [15] and Batchelor [16] have all underestimated nanofluid viscosity; and models of Pak and Cho [12] predicts a relatively larger value under fixed particle fraction. For natural convection heat transfer of nanofluid, there are some numerical and experimental researches. Khanafer et al. [17] investigated natural convection of nanofluid in a 2-D rectangular cavity. They used Wasp’s model [8] for thermal conductivity and Brinkman model [10] for viscosity of nanofluid. They showed heat transfer enhancement is proportional to particle fraction at any fixed Grashof number. Contrary to this result, Putra et al. [18] and Wen and Ding [19] experimentally found that there will be no improvement in natural convection heat transfer by using nanofluids. Putra [18] utilized a horizontal cylinder to experimentally explore the behavior of nanoparticles of Al2O3 and CuO suspended in water. Cylinder is heated from one end and is cooled from the other. Unlike the results of force convection, they observed deterioration of heat transfer which is proportional to particle fraction. Hwang et al. [2] theoretically investigated heat transfer characteristics of natural convection in a rectangular cavity heated from below. Nanoparticles of Al2O3 are suspended in water. They used thermal conductivity model of Jang and Choi [5] and various models for the effective viscosity. In addition, they compared the results with the data of Putra [18]. It is shown that the experimental results are put between two theoretical lines derived from Jang and Choi’s conductivity model [5] used along with both Einstein’s model [9] and Pak and Cho’s correlation [12] for viscosity. Finally, they theoretically showed that the ratio of heat transfer coefficient of nanofluids to that of base fluid is decreased as the size of nanoparticles increases, or the average temperature of nanofluids is decreased. In addition, they stated that the effective thermal conductivity models used by Kim et al. [20] and Khanafer [17] cannot predict the effect of the size of nanoparticles and the average temperature of nanofluids on the enhancement of the effective thermal conductivity, although the models are able to predict the effect of the particle fraction of nanoparticles on the enhancement of the effective thermal conductivity of nanofluids. Recently Ho et al. [21] presented a numerical identification for the effects due to uncertainties in effective dynamic viscosity and thermal conductivity of nanofluid on


laminar natural convection heat transfer in a square enclosure with alumina–water nanofluid. They found that the significant difference in the effective dynamic viscosity enhancement of the nanofluid calculated from Brinkman [10] and empirical formula of Maı¨ga et al. [22] formulas plays a major role and leads to contradictory results concerning the heat transfer efficacy of using nanofluid in the enclosure. There is no numerical work on the nanofluid heat transfer of annulus in the literature. The aim of this research is to explore the heat transfer behavior in annuli with different geometrical and physical parameters. When the inner radius ri ? ?, regardless of the notation used in cylindrical coordinate, governing equations are the same as those in Cartesian rectangular coordinate. Accordingly, a square cavity is studied once again with new experimentally proved thermal conductivity and viscosity models. The major contribution of this study is developing a new correlation for Nusselt number of nanofluids in annuli in terms of aspect ratio, Rayleigh number, radius ratio and particle fraction. It is shown that the general developed correlation for the annuli is also valid for Nusselt number of square cavity. The developed correlation is compared with other correlations presented earlier in literatures.

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solid particles are in thermal equilibrium with the base fluid and they are at the same velocity. Vertical Walls of annulus are in constant temperature and horizontal walls are adiabatic. Furthermore, inner wall is at a higher temperature (TH) and outer wall is at a lower temperature (TC). In this research, TC is 20°C and TH varies to obtain different Grashof numbers. In order to avoid hysteresis [13], maximum Grashof number is considered to be 105 and this corresponds to TH of 41°C. It should be noted that in the validation section of this paper a 3-D experimental case for natural convection of nanofluid in a horizontal cylinder was used. But for the brevity of the paper the details of the governing equations for 3-D numerical model were not mentioned here. The thermophysical properties such as density, thermal conductivity of base fluid and nanoparticle, viscosity, and Prandtl number are considered to be a function of temperature and are summarized in Table 1. Boussinesq approximation has not been used in free convection heat transfer and instead, density variation with temperature has been applied directly. The gap is: L ¼ ro ri

ð1Þ

and the aspect ratio is defined as height to gap: A ¼ H=L

ð2Þ

2 Governing equations The studied annulus is shown in Fig. 1. Due to gravity direction and geometry, the problem will be considered axi-symmetric and therefore it is two dimensional with respect to axial and radial coordinates. Annulus contains water and nanoparticles of Al2O3. These particles are assumed to be in the same size and shape. In addition,

Annulus gap is held constant and equals to 1 cm and the height of annulus varies in the range of 1–5 cm. Also, the particle diameter is assumed to be 36 nm. Boundary conditions for steady state conditions are as follows: Vx ¼ V r ¼

oT ¼0 ox

at x ¼ 0; L and ri r ro

ð3Þ

T ¼ TH ; Vx ¼ Vr ¼ 0

at r ¼ ri and 0 x H

ð4Þ

T ¼ TC ; Vx ¼ Vr ¼ 0

at r ¼ ro and 0 x H

ð5Þ

The governing equations for nanofluid in axi-symmetric annulus are continuity, momentum, and energy equations with their density, thermal conductivity, and viscosity modified for nanofluid application. Continuity for axisymmetric coordinate is as follows: o o q Vr ðqnf Vx Þ þ ðqnf Vr Þ þ nf ¼ 0 ox or r

ð6Þ

Momentum equation is:

Fig. 1 Geometry and boundary conditions of annulus

1o 1o ðrq Vx Vx Þ þ ðrqnf Vr Vx Þ r ox nf r or op 1 o oVx 2 ! rlnf 2 r: V ¼ þ ox r ox 3 ox 1o oVx oVr þ rlnf þ qnf g r or or ox

ð7Þ

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18


Table 1 Material properties of fluid and nanoparticle [2, 23] Temperature

290

300

310

320

330

Al2O3 Thermal conductivity (W/mK)

43.70

42.34

40.99

39.62

38.26

Water Thermal conductivity (W/mK)

0.598

0.613

0.628

0.64

0.65

Mean free path (nm)

0.735

0.739

0.745

0.747

0.748

Viscosity (Ns/m2) 9 10-3

1.08

Density (kg/m3)

0.855

999.0

Prandtl number

997.0

7.56

Volumetric expansion coeff. [1/K] 9 10-6

5.83

174.0

276.1

1o 1o ðrq Vx Vr Þ þ ðrqnf Vr Vr Þ r ox nf r or op 1 o oVr oVx rl þ ¼ þ or r ox nf ox or ð8Þ 1o oVr 2 ! rlnf 2 ðr: V Þ þ r or or 3 Vr 2 l ! 2lnf 2 þ nf ðr: V Þ 3 r r ! oVr Vr x where r: V ¼ oV ox þ or þ r . The axi-symmetric form of energy equation is as follows: oðq TÞ oðq TÞ cPnf Vr nf þ Vx nf or ox ð9Þ o oT 1o oT knf rknf ¼ þ ox ox r or or As mentioned before, nanofluid properties are combinations of base fluid and particle properties. The effective density of nanofluid is predicted by mixing theory: qnf ¼ ð1 /Þqbf þ /qparticle

ð10Þ

Density of the base fluid is a function of temperature while the density of solid is assumed to be constant in the range of temperatures considered in this paper. Specific heat is also defined by mixing theory: cPnf ¼ ð1 /ÞcPbf þ /cPparticle

ð11Þ

Regarding thermal conductivity of nanofluid, model of Jang and Choi [5] has been applied: knf ¼ kbf ð1 /Þ þ Bkparticle / þ 3C1

dbf kbf Re2dnano Pr/ dnano ð12Þ

6

in this relation, C1 = 6 9 10 , kbf is the thermal conductivity of base fluid, / is particle volumetric fraction, kparticle is the thermal conductivity of particle, B = 0.01 and is a constant related to Kapitza resistance, dbf is the diameter

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0.695 993.0 4.62 361.9

0.577 989.1 3.77 436.7

0.489 984.3 3.15 504.0

of base fluid molecule, and dnano is the diameter of particle molecule. Reynolds number is defined by: CR:M dnano 2D0 Rednano ¼ ; CR:M ¼ ð13Þ m lbf CR:M is the random motion velocity of nanoparticles, D0 = kbT/3pldnano is diffusion coefficient, lbf is mean free path, m is kinematic viscosity of base fluid, and kb = 1.3807 9 10-23 J/K is Boltzmann constant. The effective viscosity model of Nguyen [13] for 36 nm particles has been applied to the problem. This model has been experimentally obtained and is as follows: lnf ¼ 1:0 þ 0:025/ þ 0:015/2 ð14Þ lbf In this relation, contrary to the Eq. 12, / is the number of fraction in percent. It should be noted that the variation of the viscosity with temperature is hidden in lbf. The experimental work of Nguyen et al. [13] showed that the temperature dependency for the viscosity of Al2O3 nanofluid for particle volume fraction less than 7% is nearly similar to water. The correlation presented by Hagen [24] was used in the present work for viscosity variation of water with temperature. The Grashof number are defined as follow, GrL ¼

gbf ðTH TC ÞL3 m2f

ð15Þ

As was mentioned before, since the fluid properties were considered a function of temperature, all properties are calculated at film temperature Tf = (TH ? TC)/2 for Grashof number. The Prandtl number is variable with temperature and is defined as, mf Pr ¼ ð16Þ af Average Nusselt number on the inner wall is defined as: Nui ¼

Qconvection Qconvection ¼ Qconduction 2pri keff ðTH Tc ÞA

ð17Þ


keff is thermal conductivity of nanofluid at film temperature based on Eq. 12. The stagnant thermal conductivity used for conduction heat flux in Nusselt number. Qconvection was computed with integration of heat flux at the inner wall surface. Numerical simulation is based on finite volume formulation. The governing equations are integrated over each control volume to obtain a set of linear algebraic equations. These equations were solved by employing SIMPLE algorithm for the pressure correction processes, and convective and diffusive terms are discretized by upwind and central difference schemes, respectively. To find a gridindependent solution for each configuration, several attempts were made to find a constant flow field and an average Nusselt number with reasonable accuracy. Number of grids is different for various geometries because of a wide range for the annulus sizes. For example, the size of the grid for an annulus with the aspect ratio of unity and radius ratio of equal to two is 120 9 120. The convergence of the numerical solution is based on the residuals of governing equations that were summed over all cells in the computational domain. Convergence was achieved when the summation of residuals decreased to less than 10-8 for all equations.

3 Results The heat transfer performance in the form of natural convection in annulus has been investigated in this paper. Different aspect ratios 1 B A B 5, Grashof number in the range 103 B GrL B 105, as well as particle fraction of 0 B / B 6% are studied. The existing experimental data is for a laminar natural convection of water and nanofluid in a 3-D horizontal cylinder [18]. To investigate the accuracy of the viscosity and thermal conductivity models for nanofluids, available experimental data of Putra [18] for 3-D horizontal cylinder with nanofluids are used for comparison. Thermal conductivity models of Wasp [8] and Jang and Choi [5] were used in numerical model and results are compared with experimental data of Putra [18] (Fig. 2) in which particle fraction is 4% and aspect ratio of the cylinder is 1. As can be seen, thermal conductivity model of Wasp [8] considerably overestimates the Nusselt number and shows an increment of Nusselt number compared to base fluid, whereas the thermal conductivity model of Jang and Choi [5] with Pak and Cho viscosity model [12] shows a deterioration of Nusselt number in the cylinder. Therefore, Jang and Choi model [12] will be a good predictor model of heat transfer in annulus and consequently has been selected in this paper.

19

Fig. 2 Comparison of thermal conductivity models of Jang and Choi [5] and Wasp [8] in predicting nanofluid Nusselt number of 3-D horizontal cylinder of Putra et al. [18]

Figure 3 represents the comparison of different viscosity models used with thermal conductivity model of Jang and Choi [5] for 3-D horizontal cylinder of Putra et al. [18]. These models are Einstein [9] and Nguyen et al. [13] correlations. Since sizes of particles in Nguyen model [13] are 36 and 47 nm while the particle size in Putra experiment [18] is 131.2 nm, a model based on extrapolation of Nguyen [13] is developed. The data of Putra [18] lie between viscosity model of Nguyen correlation and model of Einstein [9]. Einstein’s model presents much higher values compared to experimental data. Figures 4, 5 and 6 illustrate Nusselt number as a function of aspect ratio for different particle fractions and for GrL = 103, 104, and 105 respectively. Obtained data represent the fact that Nusselt number as a function of aspect ratio has a maximum for a specified Grashof number (defined based on annulus gap). This maximum Nusselt number happens at aspect ratio of 1.8 and 1.2 for GrL = 103, 104 respectively. For 105 this maximum happens where A \ 1 (not shown here). Obviously, the aspect ratio of maximum Nusselt number decreases when GrL increases. Also, for GrL greater than 103 Nusselt number is a descending function of aspect ratio when A C 1.8. For very large aspect ratios (H/L ? ?) enclosure geometry suppresses the fluid circulation. In these cases, convective heat transfer contribution vanishes (Nu ? 1) as the buoyancy-driven loop is changed to a counter flow whose

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Fig. 3 Comparison of different viscosity models in nanofluid Nusselt number for 3-D horizontal cylinder of Putra et al. [18] with thermal conductivity model of Jang and Choi [5]

Fig. 4 Nusselt number for GrL = 103, K = 2 and different particle fractions as a function of aspect ratio in vertical annulus

two branches are in excellent thermal contact. Also the results show as the particle fraction increases Nusselt number decreases monotonically. In a constant particle fraction, as Grashof number increases the Nusselt number increases as it was expected. It should be noted that in higher Grashof number the decrease of Nusselt number of nanofluid compared to the base fluid is more noticeable. For all studied Grashof numbers and aspect ratios, it is seen (Fig. 7) that for a fixed particle fraction the ratio of Nusselt number of nanofluid to that of the base fluid is approximately the same. In other words, this ratio for a given particle fraction is almost independent of Grashof number and aspect ratio. In addition, it decreases as the particle fraction increases. This ratio is correlated as follows,

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Fig. 7 Nusselt number ratio of nanofluid to base fluid vs. particle fraction for A C 1 and K = 2 in vertical annulus for all studied Grashof number and aspect ratio


Nunf ¼ 2:41/ þ 1 Nubf

8 0:0 / 0:06 > > > > 3 5 > > < 10 GrL 10 where 5:35 Pr 6:91 > > > > 1A5 > > : K¼2

21

ð18Þ

The above equation has a maximum error less than 5.8% and an average error less than 1.6% for the specified ranges. It should be emphasized that correlation of (18) is valid when radius ratio is K = 2. The same analysis for radius ratio of K = 5 (not shown here) reveals this fact that the ratio of Nunf/Nubf changes with particle fraction similar to the case of K = 2 and Eq. 18 can be applied for other radius ratios like K = 5. The correlation (18) has a maximum error of 4% for the case of K = 5. Therefore, having the Nusselt number of base fluid, Nusselt number of nanofluid can be obtained easily using this correlation. Fortunately some correlations can be found in the literature for the Nusselt number of base fluid such as the correlation of Kumar and Kalam [25]. They presented a correlation for vertical annulus with the vertical walls to be in constant temperature as follows [25]: NuL ¼ 0:18Ra0:278 K ð0:329=kÞþ0:34 A0:122 ; L 8 3 103 RaL 106 > > < 2 K 15 > > : 1 A 10

ð19Þ

In this research, Kumar and Kalam’s correlation [25] has been used for the Nusselt number of base fluid. Figure 8 shows the comparison of Nusselt number from present work and correlation of Kumar and Kalam [25] which

shows a general agreement. Therefore, Nusselt number of nanofluid would be as follows: h i ð0:329=KÞþ0:34 0:122 Nunf ¼ 0:18Ra0:278 K A ð2:41/ þ 1Þ L 8 0:0 / 0:06 > > > > 3 5 > > < 10 GrL 10 where 5:35 Pr 6:91 > > > > 1A5 > > : 2 K 15 ð20Þ As mentioned before, Khanafer et al. [17] studied the effect of nanofluid in free convection of rectangular cavity (1 cm 9 1 cm). By the use of thermal conductivity of Wasp [8] and viscosity of Brinkman [10], they observed enhancement of natural convection inside the cavity, whereas Putra et al. [18] experimentally showed there is deterioration in free convection and as the particle fraction increases this deterioration becomes worse. Recently Ho et al. [21] showed that the under prediction of Brinkman model for the viscosity of nanofluid results to a false increase for Nusselt number of natural convection in a square cavity. Also in present work it was shown in Fig. 2 that use of thermal conductivity of Wasp [8] results to a wrong prediction for Nusselt number of natural convection in a horizontal cylinder. So in this study, a square cavity (A = 1) has been re-explored. Nanoparticles of size 36 nm as well as thermal conductivity model of Jang and Choi [5] and viscosity model of Nguyen [13] are used to investigate the behavior of nanofluid in buoyancy driven flows. Figure 9 shows the ratio of Nunf/Nubf as a function of particle fraction. As expected, there will be no increment in Nusselt number of free convection. Here again, properties are temperature dependent according to Table 1. This trend was observed in numerical work of Ho et al. [21]. As the Fig. 9 shows, the developed correlation for annuli is also valid for square cavity and has a maximum error of less than 6.9%. Therefore, by substituting the relation for Nubf for square cavities from the present work (Nubf = 0.124 Ra0.318) following correlation can be developed as: Nunf ¼ Nubf ð2:41u þ 1Þ Nunf ¼ 0:124Ra0:318 ð2:41u þ 1Þ

Fig. 8 Comparison of present numerical work for annulus with base fluid in different Grashof numbers with data of Kumar and Kamal [25]

ð21Þ

The above correlation is valid in the range of 103 B RaL B 105, and 5.35 B Pr B 6.91. It should be noted that the correlation for Nubf in square cavity has a maximum error of 2%. In addition, the presented correlation for square cavity is in good agreement to the correlations of Catton [26] and Kumar and Kalam [25] when K = 1 and A = 1. The developed correlation is compared with correlations of Ho et al. [21] and Khanafer et al. [17] in Fig. 10.

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22


Fig. 9 Nusselt number of nanofluid for a square cavity compared to developed correlation for annulus Eq. 18 at different Grashof numbers

Khanafer [17] presented following correlation for natural convection of nanofluid in a square cavity, Nu ¼ 0:5163ð0:4436 þ u1:0809 ÞGr0:3123

ð22Þ

It should be noted that Khanafer et al. [17] used the base fluid conduction heat transfer in definition of Nusselt number and they justified that increasing of their defined Nusselt number with particle fraction is because of using qconduction of base fluid in denominator of Nusselt number definition. So we modified their correlation with multiplication of kbf/knf to above definition. So with this modification the nanofluid heat conduction would be in denominator of Nusselt number definition. This modified correlation was shown in Fig. 10. Also the correlation of Ho et al. [21] is shown in Fig. 10. This correlation is as follow: Nu ¼ 0:148ð1 þ uÞ0:561 Ra0:298

ð23Þ

As the figure shows the correlation of Khanafer et al. [17] predicts an increasing with nano particle fraction even after modification; and as it was mentioned before, this

Fig. 11 Convection and conduction heat transfer for different particle fractions in annulus A = 5, K = 2 and GrL = 104

trend is against the experimental data which shows a decreasing for Nusselt number of free convention in a cavity. But the correlation of Ho et al. [21] and present correlation show a linear decreasing with nano particle fraction. Their predicted rate of decrease in Nusselt number is different for present correlation and Correlation of Ho et al. [21]. This might be because of using a different empirical viscosity model and also ignoring the variation of water viscosity with temperature in [21]. An important point should be emphasized that the developed correlation, Eq. 18, is valid for 36 nm Al2O3 particle and is based on experimental correlation of Nguyen et al. [13] for viscosity of nanofluid. But it is known that for other nanoparticle diameters, the viscosity variations with particle fractions would be different; and different models predicts even different values for viscosity of nanofluid with the same particle diameter. Therefore, two other different viscosity models presented by Pak and Cho [12] for 13 nm Al2O3 particles and Williams [27] for 46 nm Al2O3 particles were used in this research (the details are not shown here). The results show that for u 6%, ratio of Nunf/Nubf again shows a monotonically linear decreasing function with particle fraction but with different slopes. Therefore, the general correlation for Nusselt number of nanofluids is: Nunf =Nubf ¼ 1 c/

Fig. 10 Comparison of different correlations for Nusselt number of natural convection in rectangular cavity for various nano particle fractions

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ð24Þ

In above correlation c is a constant dependent on viscosity model and particle diameter. In this study, c = 2.41 for 36 nm Al2O3 nanoparticle and for Nguyen viscosity model [13]. This correlation is a good representative for variation of Nunf/Nubf. The above statement is also correct for natural convection in square cavity and the correlation 24 can be used in this regard. In the present work, the conductive heat transfer used in definition of Nusselt number is the conduction of nanofluid. So the decrease in Nusselt number with particle fraction


might be due to the increase in conduction of nanofluid. Therefore, the changes in convection and conduction of nanofluid are investigated separately. Figure 11 shows the convection and conduction heat transfer in annulus for A = 5 and GrL = 104 as an example. It can be seen that convection heat transfer decreases while conduction heat transfer increases as the particle fraction increases; and as a result, the Nusselt number decreases.

4 Conclusion The main purpose of this research was investigating free convection of nanofluid inside a vertical annulus numerically. Parameters related to the geometry of annulus as well as different physical parameters were applied to the problem. In addition, different nano particle fractions were studied. It was shown that a similar pattern exists for decreasing of Nunf/Nubf independent of Grashof number, aspect ratio and radius ratio in annulus when A C 1. A correlation for this ratio was developed which is only a monotonically linear decreasing function of particle fraction, Nunf ¼ Nubf ð1 cuÞ, where constant c is dependent on particle diameters and how viscosity varies with particle fraction. Having Nubf, the Nusselt number of nanofluids in vertical annulus can be obtained easily in various Rayleigh number, aspect ratio, radius ratio and particle fraction of nano particles. In addition, it was shown that thermal conductivity model of Wasp greatly overestimates the Nusselt number of experimental data of Putra et al. for 3-D horizontal cylinder; and the thermal conductivity of Jang and Choi agrees well with the experiment and shows deterioration of natural convection heat transfer. Furthermore, a specific correlation is also developed for Nusselt number of natural convection of nanofluid in a square cavity and it was compared with other correlations presented earlier in the literature and discussed.

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