UDK 519.61/.64:620.3:536.2 Original scientific article/Izvirni znanstveni članek ISSN 1580-2949 MTAEC9, 47(2)211(2013) NUMERICAL STUDY OF RAYLEIGH-BENARD NATURAL-CONVECTION HEAT-TRANSFER CHARACTERISTICS OF WATER-BASED Au NANOFLUIDS NUMERIČNA ANALIZA PRENOSA TOPLOTE NANOTEKOČIN VODA-Au V RAZMERAH REYLEIGH-BENARDOVE NARAVNE KONVEKCIJE Primož Ternik1, Rebeka Rudolf2'3, Zoran Žunic4 iPrivate Researcher, Bresterniška ulica 163, 2354 Bresternica, Slovenia 2University of Maribor, Faculty of Mechanical Engineering, Smetanova 17, 2000 Maribor, Slovenia 3Zlatarna Celje, d. d., Kersnikova ul. 19, 3000 Celje, Slovenia 4AVL-AST, Trg Leona Štuklja 5, 2000 Maribor, Slovenia pternik@pt-rtd.eu Prejem rokopisa — received: 2012-10-01; sprejem za objavo - accepted for publication: 2012-10-15 The present work deals with the natural convection in a square cavity filled with a water-based Au nanofluid. The cavity is heated from the lower and cooled from the adjacent wall, while the other two walls are adiabatic. The governing differential equations have been solved with the standard finite volume method and the hydrodynamic and thermal fields have been coupled using the Boussinesq approximation. The main objective of this study is to investigate the influence of the nanoparticles' volume fraction on the heat-transfer characteristics of Au nanofluids at a given base-fluid (i.e., water) Rayleigh number Rabf. Accurate results are presented over a wide range of the base-fluid Rayleigh numbers (102 < Rabf < 105) and the volume fraction of Au nanoparticles (0 % < p < 10 %). It is shown that adding nanoparticles to the base fluid delays the onset of convection. Contrary to what is argued by many authors, we show, with numerical simulations, that the use of nanofluids can reduce the heat transfer instead of increasing it. Keywords: Rayleigh-Benard natural convection, water-Au nanofluid, heat transfer, numerical modelling V prispeveku obravnavamo naravno konvekcijo v kvadratni kotanji, napolnjeni z nanotekočino voda-Au. Kotanja je bila greta s spodnje in hlajena s priležne zgornje stene, preostali dve steni sta bili adiabatni. Vodilne diferencialne enačbe smo reševali s standardno metodo končnih prostornin. Hidrodinamično in temperaturno polje sta bila sklopljena z uporabo Boussinesqove aproksimacije. Glavni cilj prispevka je raziskati vpliv prostorninskega deleža nanodelcev na značilnosti prenosa toplote Au-nanotekočine pri podani vrednosti Rayleighjevega števila nosilne tekočine (vode) Rabf. Natančni rezultati so predstavljeni za široko območje vrednosti Rayleighjevega števila nosilne tekočine (102 < Rabf < 105) in prostorninskega deleža Au-nanodelcev (0 % < p < 10 %). Pokazali smo, da dodajanje nanodelcev v nosilno tekočino zakasni začetek naravne konvekcije. V nasprotju s trditvami mnogih avtorjev smo z numeričnimi simulacijami pokazali, da lahko uporaba nanodelcev prenos toplote zmanjša in ne poveča. Ključne besede: Rayleigh-Benardova naravna konvekcija, nanotekočina voda-Au, prenos toplote, numerično modeliranje 1 INTRODUCTION Buoyancy-induced flow together with the associated heat transfer is an important phenomenon found in many engineering applications (e.g., selective laser melting process1, cooling of electronic devices2). An enhancement of heat transfer in such systems is crucial from the energy-saving point of view. In recent years, nanosized particles dispersed in a base fluid, known as nanofluid, has been used and researched extensively to enhance the heat transfer. The presence of nanoparticles shows an unquestionable heat-transfer enhancement in forced convection applications3. However, with respect to the buoyancy-driven flow, there is still a dispute on the effect of nanoparticles on the heat-transfer enhancement. Several researchers have been focused on the numerical modelling of buoyancy-induced flows. Recent numerical studies by Ternik et al.4, Ternik and Rudolf5, Oztop et al.6 and Abu-Nada and Oztop7 illustrated that the suspended nanoparticles substantially increase the heat-transfer rate for any given Rayleigh number. In addition, they showed that the heat-transfer rate in water-based nanofluids increases with an increasing volume fraction of AFO3, Cu, TiO2 and Au nanoparti-cles. On the other hand, an apparently paradoxical behaviour of the heat-transfer deterioration was observed in many experimental studies8-10. For example, Putra et al.8 reported that a presence of Al2O3 nanoparticles in a base fluid reduces the natural convective heat transfer. However, they did not clearly explain why the natural convective heat transfer is decreased with an increase in the volume fraction of nanoparticles. The above review of the existing literature shows that the problem of natural convection in a bottom-heated horizontal cavity filled with a nanofluid is an issue still far from being completely solved. Framed in this general background, the purpose of the present study is to examine the effect of adding Au nanoparticles to the base fluid at the conduction and convection heat-transfer rates in a square cavity heated from below (Rayleigh-Bénard configuration) over a range of base-fluid Rayleigh numbers 102 < Rabf < 105 and volume fractions 0 % < p < 10 %. 2 NUMERICAL MODELLING The standard finite-volume method, successfully used in many recent studies,11-13 is used to solve the coupled conservation equations of mass, momentum and energy. In this framework, a second-order central differencing scheme is used for the diffusive terms and a second-order upwind scheme for the convective terms. Coupling of the pressure and velocity is achieved using the SIMPLE algorithm. The convergence criteria were set to 10-9 for all the relative (scaled) residuals. 2.1 Governing equations For the present study, a steady-state flow of an incompressible water-based Au nanofluid is considered. It is assumed that both the fluid phase and nanoparticles are in thermal equilibrium. Except for the density, the properties of the nanoparticles and fluid (presented in Table 1) are taken to be constant. The Boussinesq approximation is invoked for the nanofluid properties to relate density changes to temperature changes, and to couple the temperature field with the velocity field. The governing equations (mass, momentum and energy conservation) of such a flow are:4 5 dv; = 0 dxi . d ( dv i ) Pnf V i^Tn* — U dv. nf J dx; dx dp i -i Cnf ^ J (1) (2) dx +( ^ ^ 7-r C) + dXT dv ^ 'nf sX~ dT (PCP ) nf V i dx _d_ dx. ic ^T1 i -i I nf dxJ J (3) where the cold-wall temperature Tc is taken to be the reference temperature for evaluating the buoyancy term (pj3)n/g(T - Tc) in the momentum conservation equation. Relationships between the properties of the nanofluid (nf) and those of the base fluid (bf) and pure solid (s) are given with the following empirical models45: • Dynamic viscosity: _ V bf V nf _ (1-p)2 5 • Density: Pnf _ (1-P) Pbf + PPs • Thermal expansion: (PP) nf _ (1-P)(PP) bf + P(PP) s • Heat capacitance: (PCp ) nf _ (1-P)(PCp ) bf + P(PP) s • Thermal conductivity: k = k nf Kbf ks + 2kbf - 2 Tc), whereas the other boundaries are considered to be adiabatic. Both velocity components (i.e., v* and Vy) are identically zero on each boundary because of the no-slip condition and the impenetrability of the rigid boundaries. In the present study, the heat-transfer characteristics are presented in terms of the mean Nusselt number: — 1 L J0' Nu = y J Nu( x)dx (4) Figure 1: Schematic diagrams of the simulation domain Slika 1: Shematski prikaz območja simulacije Table 1: Thermo-physical properties of the Au nanofluid4'5 Tabela 1: Toplotno-fizikalne lastnosti Au-nanotekocine4'5 V (Pa s) p (kg/m3) Cp (J/kg K) k (W/m K) /3 (1/K) Pure water 1.003 x 10-3 997.1 4179 0.613 2.1 x 10-4 Au / 19320 128.8 314.4 1.416x 10-7 and the ratio of the nanofluid heat-transfer rate to the base-fluid one: Qnf _ knfNunf _ Qbf kbf Nubf kbf (5) where hnf and hbf are the convection heat-transfer coefficients of the nanofluid and the base fluid. In order to investigate the influence of volume fraction p on the heat-transfer characteristics, the Rayleigh (Ranf) and the Prandtl numbers (Prnf) for the nanofluids are expressed as follows: COp) nfkbf (PCP ) nf V bf Ra, COp) bf knf (PCp ) bf n nf Ra Pr _ nf n nfCp„fkbf (6) n bfcp, Pr Using equation (6) we show that Ranf < Rabf (Figure 2a) and Prnf < Prbf (Figure 2b) for all the values of 1) the relative balance between viscous and buoyancy forces is modified, so that the heat transport in the thermal boundary layer gets only marginally affec-ted14. This marginal modification is reflected in a weak Prandtl-number (and therefore nanoparticles' volume fraction) dependence of the mean Nusselt number. Figure 4 shows the effect of the base-fluid Rayleigh number on the ratio of the heat-transfer rate for the water-based Au nanofluid for different values of the volume fraction. In the range Rabf < 2586 the heat transfer occurs by pure conduction, so the ratio of heat transfer is equal to the ratio of thermal conductivities and is constant and independent of Rabf. For Ranf < 2586 and Rabf > 2586 the nanofluid remains in the conductive regime, while convection appears in the base fluid. The heat transfer is more important in the base fluid than in the nanofluid and the ratio Qnf/Qbf is on a decrease until Ranf ^ 2586. From this point onwards (i.e., the transition from conduction to convection) the ratio QnfQbf is on an increase and its value becomes higher than 1, but remains lower than the ratio that is obtained when both the nanofluid and the base fluid are in the conductive regime. When the ratio Qnf/Qbf > 1, the heat-transfer rate in the nanofluid becomes higher than that in the base fluid. Finally, in Figure 4, we observe that the heat transfer can decrease or increase depending on the value of the base-fluid Rayleigh number. For example, for a water-based Au nanofluid and for < = 10 %, the ratio of the heat-transfer rate becomes higher than 1 when the base-fluid Rayleigh number reaches the value of around 9500, so we obtain an enhancement of the heat transfer only after this value (Rabf > 9500). Therefore, adding nano-particles increases the heat transfer only for a given value of the temperature difference. 4 CONCLUSIONS In the present study, the steady laminar natural convection of water-based Au nanofluids in a square enclo- Figure 3: Variation of the mean Nusselt number along the hot wall with the: a) base-fluid Rayleigh number and b) nanofluid Rayleigh number Slika 3: Spreminjanje srednjega Nusseltovega števila vzdolž tople stene z: a) Rayleighjevim številom nosilne tekočine in b) Rayleighjevim številom nanotekočine Figure 4: Effect of the base-fluid Rayleigh number on the ratio of heat transfer Slika 4: Vpliv Rayleighjevega števila nosilne tekočine na razmerje prenosa toplote sure with differentially heated horizontal walls and with the bottom wall at a higher temperature has been numerically analysed. The effects of the base-fluid Rayleigh number (102 < Rab/ < 105) and the solid volume fraction (0 % < < < 10 %) on heat-transfer characteristics have been systematically investigated in detail. The influence of a computational grid refinement on the present numerical predictions was studied throughout the examination of the grid convergence at Rabf = 105 and < = 10 %. By utilizing extremely fine meshes the resulting discretisation error for Nu is well below 0.2 %. The numerical method was validated for the case of Rayleigh-Benard natural convection in a square cavity, for which the results are available in the open literature. A remarkable agreement of the present results with the benchmark results of Turan et al.14 yields sufficient confidence in the present numerical procedure and its results. Highly accurate numerical results pointed out some important points such as: • In the classical Rayleigh-Benard configuration, just after the onset of convection, there is more heat transfer in the base fluid than in the nanofluid. For a fixed value of the base-fluid Rayleigh number Rabf, the nanofluid Rayleigh number Ranf decreases with the volume fraction of nanoparticles. Thus, the nano-particles delay the onset of convection. • In the convective heat-transfer regime the mean Nusselt number Nu is found to increase with the increasing values of the base-fluid Rayleigh number Rabf, but the Nu values obtained for the higher values of the nanoparticles' volume fraction < are smaller than those obtained in the case of the base fluid (< = 10 %) with the same numerical values of Rabf. • The transition from the conductive to convective heat-transfer regime occurs at the same value of the nanofluid Rayleigh number, i.e., Ranf > 2586. • The values of the mean Nusselt number at a given Ranf are practically independent of the nanoparticles' volume fraction. • The heat transfer can decrease or increase depending on the value of the Rayleigh number. So, an addition of nanoparticles increases the heat transfer only for the given values of the temperature difference. 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