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Uncertainties in modeling thermal conductivity of laminar forced convection heat transfer with water alumina nanofluids
International Journal of Heat and Mass Transfer 68 (2014) 78–84
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Uncertainties in modeling thermal conductivity of laminar forced convection heat transfer with water alumina nanofluids Alina Adriana Minea ⇑ Technical University ‘‘Gheorghe Asachi’’ from Iasi, Bd. D. Mangeron no. 63, Iasi 700050, Romania
a r t i c l e
i n f o
Article history: Received 1 June 2013 Received in revised form 9 August 2013 Accepted 11 September 2013
Keywords: Nanofluid Alumina Laminar flow Thermal conductivity Heat transfer coefficient
a b s t r a c t At this stage of nanofluids development, their thermal conductivity it is not yet known precisely and the judgment of their true potential is difficult. This fact was illustrated by analyzing their heat transfer performance for laminar fully developed forced convection in a tube with two zones: one adiabatic and one with uniform wall heat flux. Forced convective of a nanofluid that consists of water and Al2O3 in horizontal tubes has been studied numerically. Three different models from the literature are used to express the thermal conductivity in terms of particle loading and they led to different qualitative and quantitative results in a classical problem of replacement of a simple fluid (water) by a nanofluid in a given situation. In particular, the heat transfer coefficient of water-based Al2O3 nanofluids is increased by 3.4–27.8% under fixed Reynolds number compared with that of pure water. Also, the enhancement of heat transfer coefficient is larger than that of the effective thermal conductivity at the same volume concentration. Moreover, the effect of uncertainties in modeling nanofluids properties was noticed. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction With an ever-increasing thermal load due to trends toward smaller devices, greater power output for engines, cooling of such devices and related systems is a very important problem in hightech industries. The conventional approach for increasing cooling and heating rates is the use of extended surfaces such as fins and microchannels. However, new designs have already expanded this approach to its limits. Therefore, there is an urgent need for new and innovative concepts to achieve ultra-high performance processes. Choi [1] has pioneered ultra-high-thermal conductivity fluids, called nanofluids, by suspending nanoparticles in conventional coolants. Dispersing solid particles into liquids to improve the physical properties of liquids is hardly new, since the idea can be traced back to James Clerk Maxwell’s theoretical work [2]. Despite numerous studies for more than a century on the thermal conductivity of traditional solid/liquid suspensions containing nano- or micrometer-sized particles, the rapid settling of these particles in fluids has been a major barrier to developing suspensions for practical applications. In contrast, two significant features – well-suspended particles and high thermal conductivities far above those of traditional solid/liquid suspensions – make nanofluids strong candidates for the next generation of coolants for thermal systems.
⇑ Tel.: +40 723455071; fax: +40 232213575. E-mail address: [email protected] 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.09.018
Nanofluids offer theoretical challenges because the measured thermal conductivity of a nanofluid containing a low concentration of nanoparticles is one order of magnitude greater than that predicted by existing theories [3]. This discovery clearly suggested that conventional heat conduction models for solid-in-liquid suspensions are inadequate. Although liquid molecules close to a solid surface are known to form layered structures [4], little is known about the connection between this nanolayer and the thermal properties of solid/liquid suspensions. Yu and Choi [5] propose that the solid-like nanolayer acts as a thermal bridge between a solid nanoparticle and a bulk liquid and so is key to enhancing thermal conductivity. From this thermally bridging nanolayer idea, they suggested a structural model of nanofluids that consists of solid nanoparticles, bulk liquid and solid-like nanolayers. The thermal conductivity of the nanolayer on the surface of the nanoparticle is not known. However, because the layered molecules are in an intermediate physical state between a bulk liquid and a solid, the solid-like nanolayer of liquid molecules would be expected to lead to a higher thermal conductivity than that of the bulk liquid. Based on this assumption, they have modified the Maxwell equation for the effective thermal conductivity of solid/liquid suspensions to include the effect of this ordered nanolayer, as it will be shown later on in this study. It is important to note that the actual amount of experimental data regarding the nanofluid properties, in particular thermal conductivity, remains quite limited. Therefore, in order to estimate such properties, researchers often turned to available formulas either derived from the classical theory of two-phase mixtures or
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Nomenclature c D g h k L Nu p r R Pr Re T q x, y v
a
specific heat hydraulic diameter gravitational acceleration heat transfer coefficient thermal conductivity channel length Nusselt number pressure radius tube radius Prandtl number Reynolds number temperature wall heat flux Cartesian coordinates velocity
b
u q l
Subscripts 0 refers to bf refers to eff effective f fluid nf refers to p particle r refers to w wall m mean exit exit
Greek symbols
based on semi-empirical models. Most often, the latter were proposed for liquid suspensions containing larger (i.e. millimeter and micrometer) size particles. Such approximations induce important discrepancies in the determination of nanofluids thermal properties (see in particular, [6]) and can cause considerable errors when assessing the performance of nanofluids (heat transfer enhancement) in various thermal applications. This is especially true when a search for optimized operational conditions is pursued (see for example [7]). As a conclusion, there are a lot of models to predict thermal conductivity of nanofluids and their equations are obtained on theoretical or experimental background. In this article three common models were selected and the present work is an investigation of the effect of these models used to predict nanofluid thermal conductivity for laminar forced convection in a two zone tube: one with an isothermal domain and another one with uniform heat flux at the wall. The nanofluid under study is water–cAl2O3 mixture. 1.1. Thermal conductivity models 1.1.1. Maxwell equation To show a new connection between the nanolayers and thermal conductivity increase in nanofluids, Yu and Choi [5] assumed that the thermal energy transport in nanofluids is diffusive. This allowed them to use classical models to show the effect of the nanolayer. The Maxwell equation takes into account only the particle volume concentration and the thermal conductivities of particle and liquid. Other classical models include the effects of particle shape [8], particle distribution [9], and particle/particle interaction [10]. However, all of these models predict almost identical enhancements at the low volume concentrations (<4%) of interest in this nanofluid study. Therefore, the Maxwell model is used in this study as representative of all classical models. Particle size and the nanolayer have not been accounted for in any classical models. Based on Maxwell’s work [2], the effective thermal conductivity of a homogeneous suspension can be predicted as
knf kp þ 2kbf þ 2u kp kbf ¼ kbf kp þ 2kbf u kp kbf
thermal diffusivity ratio of the nanolayer thickness to the original particle radius volume fraction of particles density fluid dynamic viscosity
ð1Þ
where kp is the thermal conductivity of the dispersed particles, kbf is the thermal conductivity of the dispersion liquid, and u is the particle volume concentration of the suspension.
the reference (inlet) condition base-fluid
nanofluid property ‘‘nanofluid/base-fluid’’ ratio
1.1.2. Nanolayer impact Many studies have focused on the effect of a solid/solid interface on effective thermal conductivity [11–13]. Because of the imperfect contact of the solid/solid interface, the interface resistance is a barrier to heat transfer and lowers the overall effective thermal conductivity. In contrast, this solid/solid contact resistance phenomenon is not dominant at the solid/liquid interface of particle-in-liquid suspensions. In order to include the effect of the liquid layer, Yu and Choi [5] considered a nanoparticle-in-liquid suspension with monosized spherical particles of radius r and particle volume concentration u. Based on the above discussion, the Maxwell equation (1) was modified into
3 knf kp þ 2kbf þ 2u kp kbf ð1 þ bÞ ¼ kbf kp þ 2kbf u kp kbf ð1 þ bÞ3
ð2Þ
where b is the ratio of the nanolayer thickness to the original particle radius. The nanolayer impact is significant for small particles, as Yu and Choi [5] demonstrated. In this study, the value of b has been fixed to 0.1. 1.1.3. Hamilton and Crosser model and its modifications In his study, Maiga et al. [14] introduced Eq. (3) that have been obtained using the model proposed by Hamilton and Crosser [8] and this, assuming spherical particles. Such a model, which was first developed based on data from several mixtures containing relatively large particles i.e. millimetre and micrometer size particles, is believed to be acceptable for use with nanofluids, although it may give underestimated values of thermal conductivity. This model has been adopted in this study because of its simplicity as well as its interesting feature regarding the influence of the particle form itself.
knf kp þ ðn 1Þkbf ðn 1Þu kbf kp ¼ kbf kp þ ðn 1Þkbf þ u kbf kp
ð3Þ
In the equation n is the shape factor and is equal to 3 for spherical nanoparticles. Zhang et al. [15] have shown that this correlation accurately predicts the thermal conductivity of nanofluids. Details and discussion regarding the procedure of computing the physical properties of nanofluids considered have been presented elsewhere [14,16]. It is important to mention that the data employed for the nanofluids considered were obtained at fixed reference temperatures, that is to say that the influence of the
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temperature on fluid thermal properties has not been clearly established to date. Finally, for most of the nanofluids of engineering interest including the ones considered in the present study, the amount of experimental data providing information on their physical properties remain, surprisingly, rather scarce if not to say quasi-non-existing for some. Hence, much more research works will be, indeed, needed in this field.
knf ¼ 4:97u2 þ 2:72u þ 1 kbf
ð4Þ
Eq. (4) have been obtained by Maiga [14–16] using the well-known model proposed by Hamilton and Crosser [8]. It is very interesting to note that such a model, although originally being derived for a mixture with millimeter and micrometer size particles, appears appropriate for use with nanoparticles.
Table 1 Thermophysical properties of base fluid and nanoparticles at 293 K. Property
Base fluid (water)
Nanoparticle (Al2O3)
Specific heat (J/kg K) Density (kg/m3) Thermal conductivity (W/m K) Viscosity (kg/ms)
4179 1086.27 0.613 9.93 104
765 3880 42.64 –
2.1. Physical properties of the nanofluids The thermophysical properties of the base fluid (water) and the solid nanoparticles (Al2O3) used in the present study are specified in Table 1. By assuming the nanoparticles are well dispersed within the base fluid, the effective physical properties of the mixtures studied can be evaluated using some classical formulas.
qnf ¼ uqp þ ð1 uÞqbf 2. Problem statement
cnf ¼
The problem under consideration consists in a numerical study of steady, forced flow and heat transfer of a nanofluid inside a two zone tube: one zone is isothermal and the next one is uniformly heated, as seen in Fig. 1. The flow is assumed fully developed and was considered laminar. From a practical viewpoint, most nanofluids used for heat transfer enhancement are constituted of very fine particles, usually under 40 nm. Because of such reduced dimensions, it has been suggested that they may be easily fluidized and consequently, can be considered to behave like a fluid. Furthermore, by assuming a uniform distribution of the nanoparticles within the base fluid as well as negligible motion slip and thermal equilibrium conditions between the solid particles and the continuous liquid phase, the resulting mixture can be considered as a conventional homogeneous single-phase fluid, even if there are studies that considered the mixture model. Its effective thermophysical properties are functions of those of the constituents as well as of their respective concentration [17]. As a consequence, an extension from the conventional fluid to the nanofluid appears quite realistic, and all the equations for a conventional single-phase fluid can then be applied to a nanofluid as well. Although more data will likely be needed in order to definitely accept this assumption, it seems to be validated, through the recent experimental works by Pak and Cho [18] and Li and Xuan [19], in which correlations of the form similar to that of the well-known Dittus–Boelter formula [20] have been proposed to characterize the heat transfer of nanofluids. In the present study, in concurrence with the arguments stated above, the ‘single-phase fluid’ approach in order to be able to study the thermal behavior of nanofluids have been adopted. For the application considered here, we also assume that the nanofluid is Newtonian, incompressible with constant properties that are evaluated at a reference state.
uqp cp þ ð1 uÞqbf cbf qnf
lnf ¼ 123u2 þ 7:3u þ 1 lbf
ð5Þ ð6Þ
ð7Þ
Eq. (5) is a general relationship used to compute the density for a classical two-phase mixture. Specific heat of nanofluids are calculated by using the formulas summarized by Buongiorno [21] and presented as Eq. (6). The dynamic viscosity of nanofluids has been calculated through Eq. (7), which is obtained, by Maiga et al. [16] performing a least-square curve fitting of some experimental data available for the considered nanofluid. 2.2. Test cases For the purposes of the present study three combinations of relations for the calculation of the nanofluid properties were considered, as illustrated in Table 2. Density, specific heat and viscosity remained the same for each model and the variable is the thermal conductivity that was selected from the most popular relations used through latest research studies [17–19,21,22]. Case B is the one considered by Mansour et al. [23] and identified as the BMGN combination after the authors. Also, Case C is similar with the one considered by Manca et al. [24] in his research work. As one can observe in Fig. 2, these three models give substantially different results for the nanofluid thermal conductivity, knf, especially if it compares case B with cases A and C. The substantial differences between the predictions of these different expressions for thermophysical properties can be attributed to the fact that none (except Eq. (4)) was specifically developed for nanofluids. Furthermore they are all based on the assumption that nanoparticles are uniformly distributed throughout the base fluid while in reality there are considerable uncertainties with respect to their
Fig. 1. Tube geometry and boundary conditions.
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A.A. Minea / International Journal of Heat and Mass Transfer 68 (2014) 78–84 Table 2 Case studies. Case A
Case B
Case C
uq c þð1uÞqbf cbf cnf ¼ p p q nf
uq c þð1uÞqbf cbf cnf ¼ p p q nf
cnf ¼
qnf ¼ uqp þ ð1 uÞqbf
qnf ¼ uqp þ ð1 uÞqbf
qnf ¼ uqp þ ð1 uÞqbf
knf kbf
¼
kp þ2kbf 2uðkbf kp Þ kp þ2kbf þuðkbf kp Þ
lr ¼ llnfbf ¼ 123u2 þ 7:3u þ 1
knf kbf
¼
kp þ2kbf þ2uðkp kbf Þð1þbÞ3 kp þ2kbf uðkp kbf Þð1þbÞ3
lr ¼ llnfbf ¼ 123u2 þ 7:3u þ 1
knf kbf
uqp cp þð1uÞqbf cbf qnf
¼ 4:97u2 þ 2:72u þ 1
lr ¼ llnfbf ¼ 123u2 þ 7:3u þ 1
Fig. 2. Comparison of thermal conductivity models.
concentration. Finally, these expressions do not account for the effects of the size disparity between the nanoparticles. 2.3. Numerical method The single-phase model is employed in the simulation and a nanofluid composed of water and Al2O3 nanoparticles flowing in a two zone tube with uniform heating at the wall boundary of the second zone is considered. The set of nonlinear differential equations was solved by control volume approach. Control volume technique converts the governing equations to a set of algebraic equations that can be solved numerically. For the convective and diffusive terms, a second order upwind method was used. Pressure and velocity were coupled using Semi Implicit Method for Pressure Linked Equations ⁄⁄⁄[SIMPLE]. In the case studies, the selected grid for the present calculations is considered for 100 and 180 nodes for r-direction and x-direction, respectively. Other combinations of nodes were also tested in the present work, but all of them gave similar values of velocity and temperature as the outlet (the differences were maximum 5%). Therefore the mentioned nodes were accepted as the optimal ones. In order to obtain the required accuracy with minimum number of nodes, the nodes are concentrated at the entrance region and near the tube walls where temperature gradients are high. 3. Results and discussions 3.1. Validation of the present discussion The case studies presents the hydrodynamic and thermal behaviors of laminar forced convective flow of a conventional fluid
inside a circular tube with two zones: an isothermal zone and a zone with constant heat flux. As was stated before, the tube has a diameter of 0.12 m and a length of 8.64 m. The fluid enters the tube with a constant inlet temperature of 300 K and with uniform axial velocity. The Reynolds number was varied from 500 to 2300. A uniform heat flux of 10000 W/m2 was subjected to the walls in the second zone. Before the application of the numerical solution to the case of nanofluid heat transfer, the numerical solution was verified by considering the flow of pure water inside the flow configuration described in Fig. 1. In order to demonstrate the validity and also precision of the model and the numerical procedure, a comparison with the previously published traditional expressions have been done. A traditional expression for calculation of heat transfer in fully developed laminar flow in smooth tubes is that recommended by Shah, for constant wall heat flux [25]:
1 D 3 Nu ¼ 1:953 RePr L
ð8Þ
The equation is valid for fully developed laminar flow in tubes for fluids with RePr DL P 33:3. Fig. 3 displays the comparison of Nusselt Number from Shah correlation and computed values from the present study for water, with Prandtl number of 6.77. The figure shows that the numerical results of the present analysis and the predictions of the Shah solution are in complete agreement.
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Fig. 3. Comparison of Nu number from Shah formula and computed values for water.
3.2. Effect of uncertainties in thermal conductivity on heat transfer enhancement After confirming that the computational model is generating correct results, nanofluids were analyzed at various Reynolds numbers. The case study presents the hydrodynamic and thermal behaviors of forced convective flow of a nanofluid inside a circular tube with two zones: an isothermal one and a constant heat flux one. The nanofluid consists of Al2O3 nanoparticles with an average diameter of 38 nm. The tube has a diameter of 0.12 m and a total length of 8.76 m. The isothermal zone is at fluid entrance and is of 5.76 m, followed by the heating zone of 2.88 m. The fluid enters the tube with a constant inlet temperature of 300 K and with uniform axial velocity. The Reynolds number was varied from 500 to 2300. The heat transfer performance of flowing nanofluids was defined in terms of the following convective heat transfer coefficient (h) and the Nusselt number (Nu):
q h¼ Tw Tm
ð9Þ
Table 3 Fully developed Nusselt number and heat transfer coefficient values obtained from the numerical solution for pure water and Al2O3/water nanofluid with different particle volume fractions for Re = 500 and test case A. Fluid
Nu
Nu enhancement (Nunf/Nubf)
h
h enhancement (hnf/hbf)
Water 1% 2% 3% 4%
7.13 7.17592 7.26524 7.39705 7.51121
– 1.00644039 1.01896774 1.03745442 1.05346564
36.4436 37.7208 39.2883 41.0994 42.9569
– 1.035046 1.078058 1.127754 1.178723
Nu ¼
hD hð2RÞ ¼ knf knf
ð10Þ
where q is the heat flux, Tw is the pipe wall temperature at a given location along the pipe and Tm is the mean temperature in the pipe at the location where Tw is defined, D is the tube diameter, knf is the fluid thermal conductivity. For this study, the Nu number and the
Table 4 Summary of heat transfer enhancement for the numerical tests. Nanofluid volume fraction, %
Heat transfer enhancement for case A Re = 500
Re = 1000
Re = 1500
Re = 2000
Re = 2300
0 1 2 3 4
1 1.035 1.078 1.128 1.179
1 1.036 1.080 1.132 1.189
1 1.086 1.133 1.189 1.248
1 1.041 1.082 1.141 1.199
1 1.040 1.077 1.125 1.182
0 1 2 3 4
Heat transfer enhancement for case B 1 1 1.042 1.042 1.092 1.093 1.150 1.153 1.209 1.218
1 1.093 1.147 1.211 1.278
1 1.048 1.095 1.162 1.228
1 1.046 1.090 1.146 1.210
0 1 2 3 4
Heat transfer enhancement for case C 1 1 1.034 1.035 1.076 1.079 1.127 1.131 1.177 1.187
1 1.085 1.132 1.187 1.246
1 1.040 1.081 1.140 1.198
1 1.039 1.075 1.124 1.180
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A.A. Minea / International Journal of Heat and Mass Transfer 68 (2014) 78–84 Table 5 Thermal conductivity enhancement depending on selected cases. Nanofluid volume fraction, u [%]
1 2 3 4
Thermal conductivity enhancement, kr Case A
Case B
Case C
1.029 1.059 1.088 1.120
1.039 1.079 1.119 1.161
1.028 1.056 1.086 1.117
Fig. 4. Variation of heat transfer coefficient with nanofluid volume fraction at different Re numbers.
Fig. 5. Variation of wall temperature on tube exit for case B.
convective heat transfer coefficient were evaluated at the tube exit. The mean exit temperature was calculated with:
RR
v Tð2prÞdr ¼ v ð2prÞdr 0
T mexit ¼ R0 R
RR
v rTdr v rdr 0
0
RR
ð11Þ
In Table 3, fully developed Nusselt numbers and associated heat transfer coefficients are listed for nanofluids with particle volume fractions ranging between 1.0% and 4.0% (Re = 500). When the table is examined, it is seen that Nusselt number increases with the addition of nanoparticles to the working fluid. The reason of this enhance-
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ment can be the flattening in the radial temperature profile which is the result of the variation of thermal dispersion in radial direction. Table 4 is the summary of numerical results for heat transfer enhancement for all three cases and for five Reynolds numbers. The heat transfer enhancement was estimated with:
hr ¼
hnf hbf
ð12Þ
Table 5 contains the thermal conductivity enhancement of nanofluids compared with the base fluid on considered cases. One can see that the results for cases A and C are similar and case B is considering a higher thermal conductivity enhancement. Moreover, if it compares with Table 4 one can notice that the heat transfer enhancement is higher than the conductivity enhancement. Thermal conductivity enhancement was calculated as:
knf kr ¼ kbf
ð13Þ
If it looks at Tables 4 and 5 one can notice that the heat transfer coefficient and thermal conductivity are increasing with the nanofluid volume fraction. Moreover, a thermal conductivity enhancement of 2.9–16.1% can be observed along with a heat transfer enhancement that varies between 3.4–27.8%. Fig. 4 illustrates the development of the fluid heat transfer coefficient profile comparing the three considered cases for two Reynolds numbers. The profiles shown clearly indicate the discussed differences in fluid thermal performances and the similarities between cases A and C. As one can see from Fig. 4, case B offers better thermal performances for both illustrated Re numbers and the increase of Re number and/or nanofluid volume fraction generates a higher heat transfer coefficient. As with the wall temperature on tube exit, Fig. 5 shows comparatively the nanofluids for the case B. As is clearly seen, the wall temperature on exit obtained in this case is considerably lower as the volume fraction and Re number increases. Although the use of nanoparticles in traditional cooling fluids can be expected to increase the shear stresses and pressure losses inside any cooling application [26], it is believed that the heat transfer benefits of such fluids in certain engineering applications will outweigh this side-effect, especially in micro-sized applications in which high heat transfer, low temperature tolerances and small component size are required. 4. Conclusion In this paper, the effect due to the uncertainty in the values of the physical properties of water–cAl2O3 nanofluid on their thermohydraulic performance for laminar fully developed forced convection in a two zones tube was investigated. In order to demonstrate the validity and also precision of the model and the numerical procedure, comparison with the previously published traditional expressions has been done. Nusselt numbers from the present numerical analysis for forced convection flow are compared with the equations given by Shah formula. In addition this article clearly presents that the nanoparticles suspended in water enhance the convective heat transfer coefficient in the thermally fully developed regime, despite low volume fraction between 1% and 4%. Analyzing the three cases presented in Table 2 one can notice that case B have the most increased values for heat transfer coefficient, compared with cases A and C that offer similar results. In particular, the heat transfer coefficient of waterbased Al2O3 nanofluids is increased by 27.8% at case B and 4 vol% under the fixed Reynolds number (Re = 1500) compared with that of pure water. The smallest increase in heat transfer coefficient
(3.4%) was noticed for case C at Re = 500 for 1% volume fraction. Also, the convective heat transfer coefficient of water-based Al2O3 nanofluids is increased with volume fraction of Al2O3 nanoparticles and the enhancement of the heat transfer coefficient is larger than that of the effective thermal conductivity at the same volume concentration for all considered cases. Finally, by analyzing a classical problems of replacement of a simple fluid by a nanofluid in a given installation, it have illustrated that the heat transfer efficiency vary significantly with the thermal conductivity of the nanofluid. Since the effects of certain nanofluid characteristics (such as average particle size and spatial distribution of nanoparticles) on this property is not presently known precisely, it is quite difficult to conclude on the presumed advantages of nanofluids over conventional heat transfer fluids. References [1] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: D.A. Siginer, H.P. Wang (Eds.), Developments and Applications of NonNewtonian Flows, vol. 66, ASME, New York, 1995, pp. 99–105. [2] J.C. Maxwell, Electricity and Magnetism, Clarendon Press, Oxford, UK, 1873. [3] S. Özerinç, S. Kakaç, A.G. Yazıcıog˘lu, Enhanced thermal conductivity of nanofluids: a state-of-the-art review, Microfluid. Nanofluid. 8 (2010) 145–170. [4] S. Kondaraju, E.K. Jin, J.S. Lee, Effect of the multi-sized nanoparticle distribution on the thermal conductivity of nanofluids, Microfluid. Nanofluid. 10 (2010) 133–144. [5] W. Yu, S.U.S. Choi, The role of interfacial layers in the enhanced thermal conductivity of nanofluids: a renovated Maxwell model, J. Nanopart. Res. 5 (2003) 167–171. [6] Y.D. Liu, Y.G. Zhou, M.W. Tong, X.S. Zhou, Experimental study of thermal conductivity and phase change performance of nanofluids PCMs, Microfluid. Nanofluid. 7 (2009) 579–584. [7] L. Gosselin, A. da Silva, Combined heat transfer and power dissipation optimization of nanofluid flow, Appl. Phys. Lett. 85 (2004) 4160–4162. [8] R.L. Hamilton, O.K. Crosser, Thermal conductivity of heterogeneous twocomponent systems, I & EC Fundam. 1 (1962) 187–191. [9] S.C. Cheng, R.I. Vachon, The prediction of the thermal conductivity of two and three phase solid heterogeneous mixtures, Int. J. Heat Mass Transfer 12 (1969) 249–264. [10] D.J. Jeffrey, Conduction through a random suspension of spheres, Proc. R. Soc. London Ser. A 335 (1973) 355–367. [11] A. Plesßca, High breaking capacity fuses with improved cooling, Int. J. Therm. Sci. 70 (2013) 144–153. [12] L.C. Davis, B.E. Artz, Thermal conductivity of metalmatrix composites, J. Appl. Phys. 77 (1995) 4954–4960. [13] A. Devpura, P.E. Phelan, R.S. Prasher, Size effect on the thermal conductivity of polymers laden with highly conductive filler particles, Microscale Thermophys. Eng. 5 (2001) 177–189. [14] S.E.B. Maïga, C.T. Nguyen, N. Galanis, G. Roy, Heat transfer behaviours of nanofluids in a uniformly heated tube, Superlatt. Microstruct. 35 (2004) 543– 557. [15] X. Zhang, H. Gu, M. Fujii, Effective thermal conductivity and thermal diffusivity of nanofluids containing spherical and cylindrical nanoparticles, J. Appl. Phys. 100 (2006) 1–5. [16] S.E.B. Maiga, S.J. Palm, C.T. Nguyen, G. Roy, N. Galanis, Heat transfer enhancement by using nanofluids in forced convection flows, Int. J. Heat Fluid Flow 26 (2005) 530–546. [17] Y. Xuan, W. Roetzel, Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transfer 43 (2000) 3701–3707. [18] B.C. Pak, Y.I. Cho, Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles, Exp. Heat Transfer 11 (1998) 151–170. [19] Q. Li, Y. Xuan, Convective heat transfer performances of fluids with nanoparticles, in: Proc. 12th Int. Heat Transfer Conference Grenoble, France, 2002, pp. 483–488. [20] F. Dittus, L.M.K. Boelter, Heat transfer in automobile radiators of tubular type, Univ. California (Berkeley) Publ. Eng. 2 (1930) 443–461. [21] J. Buongiorno, Convective transport in nanofluids, J. Heat Transfer 128 (2006) 240–250. [22] D. Lelea, The performance evaluation of Al2O3/water nanofluid flow and heat transfer in microchannel heat sink, Int. J. Heat Mass Transfer 54 (2011) 3891– 3899. [23] R.B. Mansour, N. Galanis, C.T. Nguyen, Effect of uncertainties in physical properties on forced, convection heat transfer with nanofluids, Appl. Therm. Eng. 27 (2007) 240–249. [24] O. Manca, S. Nardini, D. Ricci, Enhancement of forced convection in ribbed channels by nanofluids, Appl. Therm. Eng. 37 (2012) 280–292. [25] R.K. Shah, A.L. London, Laminar flow forced convection in ducts, Advances in Heat Transfer, Academic Press, New York, 1978. Supplement 1. [26] G. Roy, C.T. Nguyen, P.R. Lajoie, Numerical investigation of laminar flow and heat transfer in a radial flow cooling system with the use of nanofluids, Superlatt. Microstruct. 35 (2004) 497–511.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Uncertainties in modeling thermal conductivity of laminar forced convection heat transfer with water alumina nanofluids Alina Adriana Minea ⇑ Technical University ‘‘Gheorghe Asachi’’ from Iasi, Bd. D. Mangeron no. 63, Iasi 700050, Romania
a r t i c l e
i n f o
Article history: Received 1 June 2013 Received in revised form 9 August 2013 Accepted 11 September 2013
Keywords: Nanofluid Alumina Laminar flow Thermal conductivity Heat transfer coefficient
a b s t r a c t At this stage of nanofluids development, their thermal conductivity it is not yet known precisely and the judgment of their true potential is difficult. This fact was illustrated by analyzing their heat transfer performance for laminar fully developed forced convection in a tube with two zones: one adiabatic and one with uniform wall heat flux. Forced convective of a nanofluid that consists of water and Al2O3 in horizontal tubes has been studied numerically. Three different models from the literature are used to express the thermal conductivity in terms of particle loading and they led to different qualitative and quantitative results in a classical problem of replacement of a simple fluid (water) by a nanofluid in a given situation. In particular, the heat transfer coefficient of water-based Al2O3 nanofluids is increased by 3.4–27.8% under fixed Reynolds number compared with that of pure water. Also, the enhancement of heat transfer coefficient is larger than that of the effective thermal conductivity at the same volume concentration. Moreover, the effect of uncertainties in modeling nanofluids properties was noticed. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction With an ever-increasing thermal load due to trends toward smaller devices, greater power output for engines, cooling of such devices and related systems is a very important problem in hightech industries. The conventional approach for increasing cooling and heating rates is the use of extended surfaces such as fins and microchannels. However, new designs have already expanded this approach to its limits. Therefore, there is an urgent need for new and innovative concepts to achieve ultra-high performance processes. Choi [1] has pioneered ultra-high-thermal conductivity fluids, called nanofluids, by suspending nanoparticles in conventional coolants. Dispersing solid particles into liquids to improve the physical properties of liquids is hardly new, since the idea can be traced back to James Clerk Maxwell’s theoretical work [2]. Despite numerous studies for more than a century on the thermal conductivity of traditional solid/liquid suspensions containing nano- or micrometer-sized particles, the rapid settling of these particles in fluids has been a major barrier to developing suspensions for practical applications. In contrast, two significant features – well-suspended particles and high thermal conductivities far above those of traditional solid/liquid suspensions – make nanofluids strong candidates for the next generation of coolants for thermal systems.
⇑ Tel.: +40 723455071; fax: +40 232213575. E-mail address: [email protected] 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.09.018
Nanofluids offer theoretical challenges because the measured thermal conductivity of a nanofluid containing a low concentration of nanoparticles is one order of magnitude greater than that predicted by existing theories [3]. This discovery clearly suggested that conventional heat conduction models for solid-in-liquid suspensions are inadequate. Although liquid molecules close to a solid surface are known to form layered structures [4], little is known about the connection between this nanolayer and the thermal properties of solid/liquid suspensions. Yu and Choi [5] propose that the solid-like nanolayer acts as a thermal bridge between a solid nanoparticle and a bulk liquid and so is key to enhancing thermal conductivity. From this thermally bridging nanolayer idea, they suggested a structural model of nanofluids that consists of solid nanoparticles, bulk liquid and solid-like nanolayers. The thermal conductivity of the nanolayer on the surface of the nanoparticle is not known. However, because the layered molecules are in an intermediate physical state between a bulk liquid and a solid, the solid-like nanolayer of liquid molecules would be expected to lead to a higher thermal conductivity than that of the bulk liquid. Based on this assumption, they have modified the Maxwell equation for the effective thermal conductivity of solid/liquid suspensions to include the effect of this ordered nanolayer, as it will be shown later on in this study. It is important to note that the actual amount of experimental data regarding the nanofluid properties, in particular thermal conductivity, remains quite limited. Therefore, in order to estimate such properties, researchers often turned to available formulas either derived from the classical theory of two-phase mixtures or
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Nomenclature c D g h k L Nu p r R Pr Re T q x, y v
a
specific heat hydraulic diameter gravitational acceleration heat transfer coefficient thermal conductivity channel length Nusselt number pressure radius tube radius Prandtl number Reynolds number temperature wall heat flux Cartesian coordinates velocity
b
u q l
Subscripts 0 refers to bf refers to eff effective f fluid nf refers to p particle r refers to w wall m mean exit exit
Greek symbols
based on semi-empirical models. Most often, the latter were proposed for liquid suspensions containing larger (i.e. millimeter and micrometer) size particles. Such approximations induce important discrepancies in the determination of nanofluids thermal properties (see in particular, [6]) and can cause considerable errors when assessing the performance of nanofluids (heat transfer enhancement) in various thermal applications. This is especially true when a search for optimized operational conditions is pursued (see for example [7]). As a conclusion, there are a lot of models to predict thermal conductivity of nanofluids and their equations are obtained on theoretical or experimental background. In this article three common models were selected and the present work is an investigation of the effect of these models used to predict nanofluid thermal conductivity for laminar forced convection in a two zone tube: one with an isothermal domain and another one with uniform heat flux at the wall. The nanofluid under study is water–cAl2O3 mixture. 1.1. Thermal conductivity models 1.1.1. Maxwell equation To show a new connection between the nanolayers and thermal conductivity increase in nanofluids, Yu and Choi [5] assumed that the thermal energy transport in nanofluids is diffusive. This allowed them to use classical models to show the effect of the nanolayer. The Maxwell equation takes into account only the particle volume concentration and the thermal conductivities of particle and liquid. Other classical models include the effects of particle shape [8], particle distribution [9], and particle/particle interaction [10]. However, all of these models predict almost identical enhancements at the low volume concentrations (<4%) of interest in this nanofluid study. Therefore, the Maxwell model is used in this study as representative of all classical models. Particle size and the nanolayer have not been accounted for in any classical models. Based on Maxwell’s work [2], the effective thermal conductivity of a homogeneous suspension can be predicted as
knf kp þ 2kbf þ 2u kp kbf ¼ kbf kp þ 2kbf u kp kbf
thermal diffusivity ratio of the nanolayer thickness to the original particle radius volume fraction of particles density fluid dynamic viscosity
ð1Þ
where kp is the thermal conductivity of the dispersed particles, kbf is the thermal conductivity of the dispersion liquid, and u is the particle volume concentration of the suspension.
the reference (inlet) condition base-fluid
nanofluid property ‘‘nanofluid/base-fluid’’ ratio
1.1.2. Nanolayer impact Many studies have focused on the effect of a solid/solid interface on effective thermal conductivity [11–13]. Because of the imperfect contact of the solid/solid interface, the interface resistance is a barrier to heat transfer and lowers the overall effective thermal conductivity. In contrast, this solid/solid contact resistance phenomenon is not dominant at the solid/liquid interface of particle-in-liquid suspensions. In order to include the effect of the liquid layer, Yu and Choi [5] considered a nanoparticle-in-liquid suspension with monosized spherical particles of radius r and particle volume concentration u. Based on the above discussion, the Maxwell equation (1) was modified into
3 knf kp þ 2kbf þ 2u kp kbf ð1 þ bÞ ¼ kbf kp þ 2kbf u kp kbf ð1 þ bÞ3
ð2Þ
where b is the ratio of the nanolayer thickness to the original particle radius. The nanolayer impact is significant for small particles, as Yu and Choi [5] demonstrated. In this study, the value of b has been fixed to 0.1. 1.1.3. Hamilton and Crosser model and its modifications In his study, Maiga et al. [14] introduced Eq. (3) that have been obtained using the model proposed by Hamilton and Crosser [8] and this, assuming spherical particles. Such a model, which was first developed based on data from several mixtures containing relatively large particles i.e. millimetre and micrometer size particles, is believed to be acceptable for use with nanofluids, although it may give underestimated values of thermal conductivity. This model has been adopted in this study because of its simplicity as well as its interesting feature regarding the influence of the particle form itself.
knf kp þ ðn 1Þkbf ðn 1Þu kbf kp ¼ kbf kp þ ðn 1Þkbf þ u kbf kp
ð3Þ
In the equation n is the shape factor and is equal to 3 for spherical nanoparticles. Zhang et al. [15] have shown that this correlation accurately predicts the thermal conductivity of nanofluids. Details and discussion regarding the procedure of computing the physical properties of nanofluids considered have been presented elsewhere [14,16]. It is important to mention that the data employed for the nanofluids considered were obtained at fixed reference temperatures, that is to say that the influence of the
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temperature on fluid thermal properties has not been clearly established to date. Finally, for most of the nanofluids of engineering interest including the ones considered in the present study, the amount of experimental data providing information on their physical properties remain, surprisingly, rather scarce if not to say quasi-non-existing for some. Hence, much more research works will be, indeed, needed in this field.
knf ¼ 4:97u2 þ 2:72u þ 1 kbf
ð4Þ
Eq. (4) have been obtained by Maiga [14–16] using the well-known model proposed by Hamilton and Crosser [8]. It is very interesting to note that such a model, although originally being derived for a mixture with millimeter and micrometer size particles, appears appropriate for use with nanoparticles.
Table 1 Thermophysical properties of base fluid and nanoparticles at 293 K. Property
Base fluid (water)
Nanoparticle (Al2O3)
Specific heat (J/kg K) Density (kg/m3) Thermal conductivity (W/m K) Viscosity (kg/ms)
4179 1086.27 0.613 9.93 104
765 3880 42.64 –
2.1. Physical properties of the nanofluids The thermophysical properties of the base fluid (water) and the solid nanoparticles (Al2O3) used in the present study are specified in Table 1. By assuming the nanoparticles are well dispersed within the base fluid, the effective physical properties of the mixtures studied can be evaluated using some classical formulas.
qnf ¼ uqp þ ð1 uÞqbf 2. Problem statement
cnf ¼
The problem under consideration consists in a numerical study of steady, forced flow and heat transfer of a nanofluid inside a two zone tube: one zone is isothermal and the next one is uniformly heated, as seen in Fig. 1. The flow is assumed fully developed and was considered laminar. From a practical viewpoint, most nanofluids used for heat transfer enhancement are constituted of very fine particles, usually under 40 nm. Because of such reduced dimensions, it has been suggested that they may be easily fluidized and consequently, can be considered to behave like a fluid. Furthermore, by assuming a uniform distribution of the nanoparticles within the base fluid as well as negligible motion slip and thermal equilibrium conditions between the solid particles and the continuous liquid phase, the resulting mixture can be considered as a conventional homogeneous single-phase fluid, even if there are studies that considered the mixture model. Its effective thermophysical properties are functions of those of the constituents as well as of their respective concentration [17]. As a consequence, an extension from the conventional fluid to the nanofluid appears quite realistic, and all the equations for a conventional single-phase fluid can then be applied to a nanofluid as well. Although more data will likely be needed in order to definitely accept this assumption, it seems to be validated, through the recent experimental works by Pak and Cho [18] and Li and Xuan [19], in which correlations of the form similar to that of the well-known Dittus–Boelter formula [20] have been proposed to characterize the heat transfer of nanofluids. In the present study, in concurrence with the arguments stated above, the ‘single-phase fluid’ approach in order to be able to study the thermal behavior of nanofluids have been adopted. For the application considered here, we also assume that the nanofluid is Newtonian, incompressible with constant properties that are evaluated at a reference state.
uqp cp þ ð1 uÞqbf cbf qnf
lnf ¼ 123u2 þ 7:3u þ 1 lbf
ð5Þ ð6Þ
ð7Þ
Eq. (5) is a general relationship used to compute the density for a classical two-phase mixture. Specific heat of nanofluids are calculated by using the formulas summarized by Buongiorno [21] and presented as Eq. (6). The dynamic viscosity of nanofluids has been calculated through Eq. (7), which is obtained, by Maiga et al. [16] performing a least-square curve fitting of some experimental data available for the considered nanofluid. 2.2. Test cases For the purposes of the present study three combinations of relations for the calculation of the nanofluid properties were considered, as illustrated in Table 2. Density, specific heat and viscosity remained the same for each model and the variable is the thermal conductivity that was selected from the most popular relations used through latest research studies [17–19,21,22]. Case B is the one considered by Mansour et al. [23] and identified as the BMGN combination after the authors. Also, Case C is similar with the one considered by Manca et al. [24] in his research work. As one can observe in Fig. 2, these three models give substantially different results for the nanofluid thermal conductivity, knf, especially if it compares case B with cases A and C. The substantial differences between the predictions of these different expressions for thermophysical properties can be attributed to the fact that none (except Eq. (4)) was specifically developed for nanofluids. Furthermore they are all based on the assumption that nanoparticles are uniformly distributed throughout the base fluid while in reality there are considerable uncertainties with respect to their
Fig. 1. Tube geometry and boundary conditions.
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A.A. Minea / International Journal of Heat and Mass Transfer 68 (2014) 78–84 Table 2 Case studies. Case A
Case B
Case C
uq c þð1uÞqbf cbf cnf ¼ p p q nf
uq c þð1uÞqbf cbf cnf ¼ p p q nf
cnf ¼
qnf ¼ uqp þ ð1 uÞqbf
qnf ¼ uqp þ ð1 uÞqbf
qnf ¼ uqp þ ð1 uÞqbf
knf kbf
¼
kp þ2kbf 2uðkbf kp Þ kp þ2kbf þuðkbf kp Þ
lr ¼ llnfbf ¼ 123u2 þ 7:3u þ 1
knf kbf
¼
kp þ2kbf þ2uðkp kbf Þð1þbÞ3 kp þ2kbf uðkp kbf Þð1þbÞ3
lr ¼ llnfbf ¼ 123u2 þ 7:3u þ 1
knf kbf
uqp cp þð1uÞqbf cbf qnf
¼ 4:97u2 þ 2:72u þ 1
lr ¼ llnfbf ¼ 123u2 þ 7:3u þ 1
Fig. 2. Comparison of thermal conductivity models.
concentration. Finally, these expressions do not account for the effects of the size disparity between the nanoparticles. 2.3. Numerical method The single-phase model is employed in the simulation and a nanofluid composed of water and Al2O3 nanoparticles flowing in a two zone tube with uniform heating at the wall boundary of the second zone is considered. The set of nonlinear differential equations was solved by control volume approach. Control volume technique converts the governing equations to a set of algebraic equations that can be solved numerically. For the convective and diffusive terms, a second order upwind method was used. Pressure and velocity were coupled using Semi Implicit Method for Pressure Linked Equations ⁄⁄⁄[SIMPLE]. In the case studies, the selected grid for the present calculations is considered for 100 and 180 nodes for r-direction and x-direction, respectively. Other combinations of nodes were also tested in the present work, but all of them gave similar values of velocity and temperature as the outlet (the differences were maximum 5%). Therefore the mentioned nodes were accepted as the optimal ones. In order to obtain the required accuracy with minimum number of nodes, the nodes are concentrated at the entrance region and near the tube walls where temperature gradients are high. 3. Results and discussions 3.1. Validation of the present discussion The case studies presents the hydrodynamic and thermal behaviors of laminar forced convective flow of a conventional fluid
inside a circular tube with two zones: an isothermal zone and a zone with constant heat flux. As was stated before, the tube has a diameter of 0.12 m and a length of 8.64 m. The fluid enters the tube with a constant inlet temperature of 300 K and with uniform axial velocity. The Reynolds number was varied from 500 to 2300. A uniform heat flux of 10000 W/m2 was subjected to the walls in the second zone. Before the application of the numerical solution to the case of nanofluid heat transfer, the numerical solution was verified by considering the flow of pure water inside the flow configuration described in Fig. 1. In order to demonstrate the validity and also precision of the model and the numerical procedure, a comparison with the previously published traditional expressions have been done. A traditional expression for calculation of heat transfer in fully developed laminar flow in smooth tubes is that recommended by Shah, for constant wall heat flux [25]:
1 D 3 Nu ¼ 1:953 RePr L
ð8Þ
The equation is valid for fully developed laminar flow in tubes for fluids with RePr DL P 33:3. Fig. 3 displays the comparison of Nusselt Number from Shah correlation and computed values from the present study for water, with Prandtl number of 6.77. The figure shows that the numerical results of the present analysis and the predictions of the Shah solution are in complete agreement.
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Fig. 3. Comparison of Nu number from Shah formula and computed values for water.
3.2. Effect of uncertainties in thermal conductivity on heat transfer enhancement After confirming that the computational model is generating correct results, nanofluids were analyzed at various Reynolds numbers. The case study presents the hydrodynamic and thermal behaviors of forced convective flow of a nanofluid inside a circular tube with two zones: an isothermal one and a constant heat flux one. The nanofluid consists of Al2O3 nanoparticles with an average diameter of 38 nm. The tube has a diameter of 0.12 m and a total length of 8.76 m. The isothermal zone is at fluid entrance and is of 5.76 m, followed by the heating zone of 2.88 m. The fluid enters the tube with a constant inlet temperature of 300 K and with uniform axial velocity. The Reynolds number was varied from 500 to 2300. The heat transfer performance of flowing nanofluids was defined in terms of the following convective heat transfer coefficient (h) and the Nusselt number (Nu):
q h¼ Tw Tm
ð9Þ
Table 3 Fully developed Nusselt number and heat transfer coefficient values obtained from the numerical solution for pure water and Al2O3/water nanofluid with different particle volume fractions for Re = 500 and test case A. Fluid
Nu
Nu enhancement (Nunf/Nubf)
h
h enhancement (hnf/hbf)
Water 1% 2% 3% 4%
7.13 7.17592 7.26524 7.39705 7.51121
– 1.00644039 1.01896774 1.03745442 1.05346564
36.4436 37.7208 39.2883 41.0994 42.9569
– 1.035046 1.078058 1.127754 1.178723
Nu ¼
hD hð2RÞ ¼ knf knf
ð10Þ
where q is the heat flux, Tw is the pipe wall temperature at a given location along the pipe and Tm is the mean temperature in the pipe at the location where Tw is defined, D is the tube diameter, knf is the fluid thermal conductivity. For this study, the Nu number and the
Table 4 Summary of heat transfer enhancement for the numerical tests. Nanofluid volume fraction, %
Heat transfer enhancement for case A Re = 500
Re = 1000
Re = 1500
Re = 2000
Re = 2300
0 1 2 3 4
1 1.035 1.078 1.128 1.179
1 1.036 1.080 1.132 1.189
1 1.086 1.133 1.189 1.248
1 1.041 1.082 1.141 1.199
1 1.040 1.077 1.125 1.182
0 1 2 3 4
Heat transfer enhancement for case B 1 1 1.042 1.042 1.092 1.093 1.150 1.153 1.209 1.218
1 1.093 1.147 1.211 1.278
1 1.048 1.095 1.162 1.228
1 1.046 1.090 1.146 1.210
0 1 2 3 4
Heat transfer enhancement for case C 1 1 1.034 1.035 1.076 1.079 1.127 1.131 1.177 1.187
1 1.085 1.132 1.187 1.246
1 1.040 1.081 1.140 1.198
1 1.039 1.075 1.124 1.180
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A.A. Minea / International Journal of Heat and Mass Transfer 68 (2014) 78–84 Table 5 Thermal conductivity enhancement depending on selected cases. Nanofluid volume fraction, u [%]
1 2 3 4
Thermal conductivity enhancement, kr Case A
Case B
Case C
1.029 1.059 1.088 1.120
1.039 1.079 1.119 1.161
1.028 1.056 1.086 1.117
Fig. 4. Variation of heat transfer coefficient with nanofluid volume fraction at different Re numbers.
Fig. 5. Variation of wall temperature on tube exit for case B.
convective heat transfer coefficient were evaluated at the tube exit. The mean exit temperature was calculated with:
RR
v Tð2prÞdr ¼ v ð2prÞdr 0
T mexit ¼ R0 R
RR
v rTdr v rdr 0
0
RR
ð11Þ
In Table 3, fully developed Nusselt numbers and associated heat transfer coefficients are listed for nanofluids with particle volume fractions ranging between 1.0% and 4.0% (Re = 500). When the table is examined, it is seen that Nusselt number increases with the addition of nanoparticles to the working fluid. The reason of this enhance-
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ment can be the flattening in the radial temperature profile which is the result of the variation of thermal dispersion in radial direction. Table 4 is the summary of numerical results for heat transfer enhancement for all three cases and for five Reynolds numbers. The heat transfer enhancement was estimated with:
hr ¼
hnf hbf
ð12Þ
Table 5 contains the thermal conductivity enhancement of nanofluids compared with the base fluid on considered cases. One can see that the results for cases A and C are similar and case B is considering a higher thermal conductivity enhancement. Moreover, if it compares with Table 4 one can notice that the heat transfer enhancement is higher than the conductivity enhancement. Thermal conductivity enhancement was calculated as:
knf kr ¼ kbf
ð13Þ
If it looks at Tables 4 and 5 one can notice that the heat transfer coefficient and thermal conductivity are increasing with the nanofluid volume fraction. Moreover, a thermal conductivity enhancement of 2.9–16.1% can be observed along with a heat transfer enhancement that varies between 3.4–27.8%. Fig. 4 illustrates the development of the fluid heat transfer coefficient profile comparing the three considered cases for two Reynolds numbers. The profiles shown clearly indicate the discussed differences in fluid thermal performances and the similarities between cases A and C. As one can see from Fig. 4, case B offers better thermal performances for both illustrated Re numbers and the increase of Re number and/or nanofluid volume fraction generates a higher heat transfer coefficient. As with the wall temperature on tube exit, Fig. 5 shows comparatively the nanofluids for the case B. As is clearly seen, the wall temperature on exit obtained in this case is considerably lower as the volume fraction and Re number increases. Although the use of nanoparticles in traditional cooling fluids can be expected to increase the shear stresses and pressure losses inside any cooling application [26], it is believed that the heat transfer benefits of such fluids in certain engineering applications will outweigh this side-effect, especially in micro-sized applications in which high heat transfer, low temperature tolerances and small component size are required. 4. Conclusion In this paper, the effect due to the uncertainty in the values of the physical properties of water–cAl2O3 nanofluid on their thermohydraulic performance for laminar fully developed forced convection in a two zones tube was investigated. In order to demonstrate the validity and also precision of the model and the numerical procedure, comparison with the previously published traditional expressions has been done. Nusselt numbers from the present numerical analysis for forced convection flow are compared with the equations given by Shah formula. In addition this article clearly presents that the nanoparticles suspended in water enhance the convective heat transfer coefficient in the thermally fully developed regime, despite low volume fraction between 1% and 4%. Analyzing the three cases presented in Table 2 one can notice that case B have the most increased values for heat transfer coefficient, compared with cases A and C that offer similar results. In particular, the heat transfer coefficient of waterbased Al2O3 nanofluids is increased by 27.8% at case B and 4 vol% under the fixed Reynolds number (Re = 1500) compared with that of pure water. The smallest increase in heat transfer coefficient
(3.4%) was noticed for case C at Re = 500 for 1% volume fraction. Also, the convective heat transfer coefficient of water-based Al2O3 nanofluids is increased with volume fraction of Al2O3 nanoparticles and the enhancement of the heat transfer coefficient is larger than that of the effective thermal conductivity at the same volume concentration for all considered cases. Finally, by analyzing a classical problems of replacement of a simple fluid by a nanofluid in a given installation, it have illustrated that the heat transfer efficiency vary significantly with the thermal conductivity of the nanofluid. Since the effects of certain nanofluid characteristics (such as average particle size and spatial distribution of nanoparticles) on this property is not presently known precisely, it is quite difficult to conclude on the presumed advantages of nanofluids over conventional heat transfer fluids. References [1] S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in: D.A. Siginer, H.P. Wang (Eds.), Developments and Applications of NonNewtonian Flows, vol. 66, ASME, New York, 1995, pp. 99–105. [2] J.C. Maxwell, Electricity and Magnetism, Clarendon Press, Oxford, UK, 1873. [3] S. Özerinç, S. Kakaç, A.G. Yazıcıog˘lu, Enhanced thermal conductivity of nanofluids: a state-of-the-art review, Microfluid. Nanofluid. 8 (2010) 145–170. [4] S. Kondaraju, E.K. Jin, J.S. Lee, Effect of the multi-sized nanoparticle distribution on the thermal conductivity of nanofluids, Microfluid. Nanofluid. 10 (2010) 133–144. [5] W. Yu, S.U.S. Choi, The role of interfacial layers in the enhanced thermal conductivity of nanofluids: a renovated Maxwell model, J. Nanopart. Res. 5 (2003) 167–171. [6] Y.D. Liu, Y.G. Zhou, M.W. Tong, X.S. Zhou, Experimental study of thermal conductivity and phase change performance of nanofluids PCMs, Microfluid. Nanofluid. 7 (2009) 579–584. [7] L. Gosselin, A. da Silva, Combined heat transfer and power dissipation optimization of nanofluid flow, Appl. Phys. Lett. 85 (2004) 4160–4162. [8] R.L. Hamilton, O.K. Crosser, Thermal conductivity of heterogeneous twocomponent systems, I & EC Fundam. 1 (1962) 187–191. [9] S.C. Cheng, R.I. Vachon, The prediction of the thermal conductivity of two and three phase solid heterogeneous mixtures, Int. J. Heat Mass Transfer 12 (1969) 249–264. [10] D.J. Jeffrey, Conduction through a random suspension of spheres, Proc. R. Soc. London Ser. A 335 (1973) 355–367. [11] A. Plesßca, High breaking capacity fuses with improved cooling, Int. J. Therm. Sci. 70 (2013) 144–153. [12] L.C. Davis, B.E. Artz, Thermal conductivity of metalmatrix composites, J. Appl. Phys. 77 (1995) 4954–4960. [13] A. Devpura, P.E. Phelan, R.S. Prasher, Size effect on the thermal conductivity of polymers laden with highly conductive filler particles, Microscale Thermophys. Eng. 5 (2001) 177–189. [14] S.E.B. Maïga, C.T. Nguyen, N. Galanis, G. Roy, Heat transfer behaviours of nanofluids in a uniformly heated tube, Superlatt. Microstruct. 35 (2004) 543– 557. [15] X. Zhang, H. Gu, M. Fujii, Effective thermal conductivity and thermal diffusivity of nanofluids containing spherical and cylindrical nanoparticles, J. Appl. Phys. 100 (2006) 1–5. [16] S.E.B. Maiga, S.J. Palm, C.T. Nguyen, G. Roy, N. Galanis, Heat transfer enhancement by using nanofluids in forced convection flows, Int. J. Heat Fluid Flow 26 (2005) 530–546. [17] Y. Xuan, W. Roetzel, Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transfer 43 (2000) 3701–3707. [18] B.C. Pak, Y.I. Cho, Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles, Exp. Heat Transfer 11 (1998) 151–170. [19] Q. Li, Y. Xuan, Convective heat transfer performances of fluids with nanoparticles, in: Proc. 12th Int. Heat Transfer Conference Grenoble, France, 2002, pp. 483–488. [20] F. Dittus, L.M.K. Boelter, Heat transfer in automobile radiators of tubular type, Univ. California (Berkeley) Publ. Eng. 2 (1930) 443–461. [21] J. Buongiorno, Convective transport in nanofluids, J. Heat Transfer 128 (2006) 240–250. [22] D. Lelea, The performance evaluation of Al2O3/water nanofluid flow and heat transfer in microchannel heat sink, Int. J. Heat Mass Transfer 54 (2011) 3891– 3899. [23] R.B. Mansour, N. Galanis, C.T. Nguyen, Effect of uncertainties in physical properties on forced, convection heat transfer with nanofluids, Appl. Therm. Eng. 27 (2007) 240–249. [24] O. Manca, S. Nardini, D. Ricci, Enhancement of forced convection in ribbed channels by nanofluids, Appl. Therm. Eng. 37 (2012) 280–292. [25] R.K. Shah, A.L. London, Laminar flow forced convection in ducts, Advances in Heat Transfer, Academic Press, New York, 1978. Supplement 1. [26] G. Roy, C.T. Nguyen, P.R. Lajoie, Numerical investigation of laminar flow and heat transfer in a radial flow cooling system with the use of nanofluids, Superlatt. Microstruct. 35 (2004) 497–511.