UDK 519.61/.64:536.2:620.3 ISSN 1580-2949 Original scientific article/Izvirni znanstveni članek MTAEC9, 49(1)87(2015) HEAT-TRANSFER CHARACTERISTICS OF A NON-NEWTONIAN Au NANOFLUID IN A CUBICAL ENCLOSURE WITH DIFFERENTIALLY HEATED SIDE WALLS ZNAČILNOSTI PRENOSA TOPLOTE NENEWTONSKE Au NANOTEKOČINE V KOCKASTEM OHIŠJU Z RAZLIČNO GRETIMA STRANSKIMA STENAMA Primož Ternik1, Rebeka Rudolf2 3, Zoran Zunic4 1Private 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: 2013-10-14; sprejem za objavo - accepted for publication: 2014-02-17 The present work deals with the laminar natural convection in a cubical cavity filled with a homogenous aqueous solution of carboxymethyl cellulose (CMC) based gold (Au) nanofluid obeying the power-law rheological model. The cavity is heated on the vertical and cooled from the adjacent wall, while the other walls are adiabatic. The governing differential equations were solved with the standard finite-volume method and the hydrodynamic and thermal fields are coupled using the Boussinesq approximation. The main objective of this study is to investigate the influence of the nanoparticle volume fraction on the heat-transfer characteristics of CMC-based Au nanofluid over a wide range of the base-fluid Rayleigh number. Accurate numerical results are presented in the form of dimensionless temperature and velocity variations, the mean Nusselt number and the heat-transfer rate. It is shown that adding nanoparticles to the base fluid delays the onset of natural convection. In addition, numerical simulations showed that, just after the onset of natural convection, adding nanoparticles reduces the mean Nusselt number value for any given base-fluid Rayleigh number. Keywords: natural convection, CMC-Au nanofluid, heat transfer, Nusselt number Prispevek obravnava naravno konvekcijo v kockastem ohišju, napolnjenem s homogeno nanotekočino karboksimetil celuloza (CMC)-zlato (Au), z reološkim vedenjem, opisanim s potenčnim zakonom. Ohišje je greto na navpični in hlajeno na priležni steni, medtem ko so druge stene adiabatne. Vodilne diferencialne enačbe so bile rešene s standardno metodo končnih prostornin, pri čemer sta hidrodinamično in temperaturno polje sklopljena z Boussinesqovo aproksimacijo. Glavni cilj prispevka je raziskati vpliv prostorninskega deleža nanodelcev na značilnosti prenosa toplote CMC-Au nanotekočine za široko območje vrednosti Rayleighjevega števila nosilne tekočine. Natančni rezultati so predstavljeni v obliki spreminjanja brezdimenzijske temperature in hitrosti, srednje vrednosti Nusseltovega števila in hitrosti prenosa toplote. Pokazano je, da dodajanje nanodelcev v nosilno tekočino zakasni začetek naravne konvekcije. Poleg tega so numerične simulacije pokazale, da takoj za pojavom naravne konvekcije dodajanje nanodelcev zmanjšuje srednjo vrednost Nusseltovega števila za katero koli vrednost Rayleighjevega števila nosilne tekočine. Ključne besede: naravna konvekcija, nanotekočina CMC-Au, prenos toplote, Nusseltovo število 1 INTRODUCTION consumption, a negligible operating noise and a high reliability of a system are the main concerns (e.g., a reIn recent years, nanosized particles dispersed in a duction of the cooling time, manufacturing cost and an base fluid, known as nanofluid1, have been used and improvement of the product quality in the injection- researched extensively to enhance the heat transfer in moulding industry3). Although various configurations of many engineering applications. While the presence of the enclosure problem are possible4-8, one of the most nanoparticles shows an unquestionable heat2-transfer studied cases (involving nanofluids) is a two-dimensio-enhancement in forced-convection applications2 there is nal square enclosure with differentially heated isothermal vertical walls and adiabatic horizontal walls911. Recent numerical studies12,13 illustrated that the sus- still a dispute on the influence of nanoparticles on the heat-transfer enhancement of a buoyancy-driven flow. Natural convection (i.e., a flow caused by temperature-induced density variations) is one of the most exten- pended nanoparticles substantially increase the heat-sively analysed configurations because of its fundamen- transfer rate for any given Rayleigh number. In addition, tal importance as the "benchmark" problem for studying it is shown that the heat-transfer rate in water-based convection effects (and comparing as well as validating nanofluids increases with an increasing volume fraction numerical techniques). In addition to the obvious of Al2O3, Cu, TiO2 or Au nanoparticles. On the other academic interest, a thermally driven flow is the strategy hand, an apparently paradoxical behaviour of the heat-preferred by heat-transfer designers when a small power transfer deterioration was observed in many experimen- tal studies: e.g., Putra et al.14 reported that the 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 decreases with an increase in the volume fraction of nanoparticles. This was later explained by the work of Ternik and Rudolf4. Regarding the natural convection in a 3D cubic cavity, most (if not all) of the work was done for the Newtonian fluid. For example, Tric et al.15 studied the natural convection in a 3D cubic enclosure using a pseudo-spectra Chebyshev algorithm based on the projection-diffusion method with the spatial resolution supplied by polynomial expansions. Lo et al.16 also studied the same problem under different inclination angles using the differential-quadrature method to solve the velocity-vorticity formulation of the Navier-Stokes equation employing higher-order polynomials to approximate differential operators. The above review of the existing literature shows that the problem of natural convection in a cubical cavity filled with a nanofluid is an issue still far from being tackled and solved. Framed in this general background, the purpose of the present study is to examine the effect of adding Au nanoparticles to a non-Newtonian base fluid on the conduction and convection heat-transfer characteristics in a differentially heated cubical cavity heated over a wide range of the base-fluid Rayleigh number (101 < Rabf ^ 106) and the volume fraction of nanoparticles (0 % < ^ < 10 %). 2 NUMERICAL MODELLING The standard finite-volume method 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. The coupling of the pressure and velocity is achieved using the SIMPLE algorithm. The convergence criteria were set to 10-8 for all the relative (scaled) residuals. 2.1 Governing equations For the present study, a steady-state flow of an incompressible non-Newtonian CMC-Au nanofluid is considered. It is assumed that both the fluid phase and nanoparticles are in thermal and chemical equilibrium. Except for the density, the properties of the nanoparticles and the fluid (presented in Table 1) are taken to be constant. The Boussinesq approximation is invoked for the nanofluid properties to relate the density changes to the temperature changes, and to couple the temperature field with the velocity field. The governing equations (mass, momentum and energy conservation) of such a flow are9: dv,. dx,. - = 0 d^ _d dx, ~dX: . I' (I rl dXj J dp ^ d =-äX7+("f - ^^)+ädT (2) (PCp ) nf VJ dT d dXJ dX: J [ (- )dVj ^ -f(^ dL dX.. (3) where the cold-wall temperature Tc is taken to be the reference temperature for evaluating the buoyancy term (pß)n{g(T - Tc) in the momentum-conservation equation. In the momentum-balance law (Equation 2) the constitutive (i.e., rheological) equation is required for the viscous function ^ "f (y ), which, for the power-law model, reads as: n-1 V n (I rl )=^ (4) where -- 2XX y ijy ji is the II invariant of the symmetrical rate-of-deforma-tion tensor with Cartesian components yij = (dVi/dXj) + (dVj/dXi), K is the consistency and n is the dimensionless power-law index with experimentally determined values (for a w = 0.4 % aqueous CMC solution) of K = 0.048 Pasn and n = 0.882. The relationships between the properties of the nanofluid (nf), the base fluid (bf) and pure solid (s) are given with the following empirical models4,8-13: • Dynamic viscosity: V nf - V bf /(1- • Density: Pnf - (1-Pbf + ^Ps • Thermal expansion: (Pß) nf - (1-^)(pß)bf + ^(Pß)s • Heat capacitance: (PCp ) nf - (1- Tc), whereas the other boundaries are considered to be adiabatic. All the velocity components (i.e., Vx, Vy and Vz) are identically zero on each boundary because of the no-slip condition and the impenetrability of the rigid boundaries. To study the heat-transfer characteristics due to the natural convection in nanofluids, the local Nusselt number (along the vertical hot wall) is defined as15: Nu(y, z) = Q nf, conv h nf L dT (y, z) Q nf, cond T - T -'h ^ C dx (5) where Nu(y,z) presents the ratio of the heat-transfer rate by convection to that by the conduction in the nanofluid in question, knf is the thermal conductivity and hnf is the convection heat-transfer coefficient of the nanofluid: h nf =-k„f dT (y, z) dx 1 T - T -'h ^ C (6) Finally, in the present study, the heat-transfer characteristics are analysed in terms of the mean Nusselt number: Nu = 1 L2 Nu(y, z)dydz (7) and the ratio of the nanofluid heat-transfer rate to the base-fluid one: Qf = k nf Nu nf Qb k bf Nu bf (8) at the same value of the base-fluid Rayleigh number. In order to investigate the influence of solid-particle volume fraction (p on the heat-transfer characteristics, the Rayleigh number (Ranf) and Prandtl number (Prnf) of the CMC-based nanofluid (obeying the power-law viscous behaviour) are expressed as follows: Ra nf = Pnf(PCp )Üf gßnf ATL^ k nf K Prnf = K / \ n-2 (9) Pnf Using Equation 9 we show that Ranf < Rabf (Figure 2a) and Prnf < Prbf (Figure 2b) for all the values of p. The ratio of the CMC-based nanofluid Rayleigh and Prandtl numbers to the base-fluid Rayleigh and Prandtl numbers decreases with the increasing volume fraction of the nanoparticles; i.e., for the fixed values of Rabf and Prbf, the value of Ranf and Prnf decrease when adding nanoparticles. 2.3 Grid refinement and numerical accuracy The grid independence of the present results was established with a detailed analysis using three different non-uniform meshes (the elements were concentrated towards each solid wall), the details of which are presented in Table 2. The table includes the numbers of the elements in a particular direction as well as the normalized minimum cell size. Figure 2: Variation of the dimensionless numbers of the CMC-based nanofluids with the volume fraction of nanoparticles: a) Rayleigh number and b) Prandtl number Slika 2: Spreminjanje brezdimenzijskih števil CMC nanoteko~in v odvisnosti prostorninskega deleža nanodelcev: a) Rayleighjevo število in b) Prandtlovo število L nf x = 0 With each grid refinement the number of the elements (i.e., the control volumes) in a particular direction is increased and the element size is reduced. Such a procedure is useful for applying Richardson's extrapolation technique (a method for obtaining a higher-order estimate of the flow value from a series of lower-order discrete values) encountered in many numerical studies8-12,17. Table 2: Computational-mesh characteristics Tabela 2: Značilnosti računskih mrež Nx X Ny X Nz ^min/L Mesh I 40 X 40 X 40 2.250 X 10-3 Mesh II 60 X 60 X 60 1.500 X 10-3 Mesh III 90 X 90 X 90 1.000 X 10-3 For a general primitive variable 0 the grid-converged value (i.e., extrapolated to the zero element size) is given as8-12: 0ext = 0M3 - (0M2 - 0M3)/(rP - 1) where 0m3 is obtained on the basis of the finest grid and 0m2 is the solution based on the next level of the coarse grid; r = 1.5 indicates the ratio between the coarse- and fine-grid spacings and p = 2 indicates the order of accuracy. The numerical error e = (0M2 - 0ext / 0e for the mean Nusselt number Nu is presented in Table 3. It can be seen that the differences in the grid refinement are exceedingly small and the agreement between Mesh II and the extrapolated value is extremely good (the discretisation error is below 0.20 %). Based on this estimation, the simulations in the remainder of the paper were conducted on Mesh II that provided a reasonable compromise between high accuracy and computational effort. Table 3: Effect of the mesh refinement upon the mean Nusselt number (

1) the 12.0 9.0 ^ 6.0 3.0 1.0 (p = 0% — cp = 5% —-cp = 10% ..... ■ 1 i i ; 1 1 1 1 (a) 10^ 10^ 10^ 10^ 10^ 10^ Rahf 12.0 9.0 ^ 6.0 3.0 1.0 I ■ 1 ) (p = 9 = 0% — 0% O _1___1_ 10^ 10^ 10^ 10^ 10^ 10^ (b) Ra nf Figure 4: Variation in the mean Nusselt number with the: a) base-fluid Rayleigh number and b) nanofluid Rayleigh number Slika 4: Spreminjanje srednjega Nusseltovega {tevila v odvisnosti z: a) Rayleighjevim {tevilom nosilne teko~ine in b) Rayleighjevim {tevi-lom nanoteko~ine Figure 5: Onset of the heat-transfer convection as a function of the: a) base-fluid Rayleigh number and b) nanofluid Rayleigh number Slika 5: Nastop konvektivnega prenosa toplote v odvisnosti od: a) Rayleighjevega {tevila nosilne teko~ine in b) Rayleighjevega {tevila nanoteko~ine hydrodynamic-boundary-layer thickness remains much greater than the thermal-boundary-layer thickness, and thus the transport characteristics are driven primarily by the buoyancy and viscous forces, which is reflected in the weak Prandtl-number dependence on the mean Nusselt number. 3.3 Heat-transfer rate Figure 6 shows the effect of the base-fluid Rayleigh number Räbt on the ratio of the heat-transfer rates for the CMC-based Au nanofluid for different values of the volume fraction. In the range of Räbf ^ 281 the heat transfer occurs due to pure conduction and the mean Nusselt number equals Nu =1. Consequently, the ratio of the heat transfer is equal to the ratio of thermal conductivities and it is constant and independent of the base-fluid Rayleigh number. For Ränf ^ 281 and Räbf >281 the nanofluid remains in the conductive-heat-transfer regime, while the convection appears in the base fluid. The heat transfer is more important in the base fluid than in the nanofluid and the ratio of the heat-transfer rates is on a decrease. After the onset of the convective-heat-transfer regime in the nanofluid, the ratio of the heat-transfer rates is on an increase but remains lower than the ratio obtained when both the nanofluid and the base fluid are in the conductive regime. Last but not least, the present conclusions can be extrapolated to the other CMC-based nanofluids (e.g., Cu, TiO2 and Al2O3) since their nanofluid Rayleigh- and Prandtl-number values are within the range of the present (i.e., CMC-based Au nanoparticles) Rayleigh-and Prandtl-number values. Figure 6: Heat-transfer rate variation with the base-fluid Rayleigh number and nanoparticle volume fraction Slika 6: Spreminjanje razmerja prenosa toplote z Rayleighjevim številom nosilne tekočine in prostorninskim deležem nanodelcev 4 CONCLUSIONS In the present study, a steady laminar natural convection of a non-Newtonian (i.e., CMC-based Au) nanofluid obeying the power-law rheological model in a cubical enclosure with differentially heated side walls, subjected to constant wall temperatures was analysed with numerical means. The effects of the base-fluid Rayleigh number (101 < Räbf < 106) and the solid volume fraction (0 % < (p < 10 %) on the momentum and heat-transfer characteristics were systematically investigated in detail. The influence of the computational grid refinement on the present numerical predictions was studied throughout the examination of the grid convergence at Räbf = 106 and p = 10 %. By utilizing extremely fine meshes, the resulting discretisation error for Nu is well below 0.2 %. The present numerical method was validated for the case of the natural convection of air in a cubical cavity, for which the results of other authors are available in the available literature. Remarkable agreement of the present results with the benchmark results yields sufficient confidence in the presented numerical procedure and results. Highly accurate numerical results allow some important conclusions such as: Just after the onset of the convection, there is more heat transfer in the base fluid than in the nanofluid. At a fixed value of the base-fluid Rayleigh number Räbf, the nanofluid Rayleigh number Ränf decreases with the volume fraction of the nanoparticles. Thus, the nano-particles delay the onset of the convection. In the convective-heat-transfer regime the mean Nusselt number Nu is found to increase with the increasing jvalues of the base-fluid Rayleigh number Räbf, but the Nu values obtained for the higher values of the nano-particle volume fraction p are smaller than those obtained in the case of the base fluid (p = 0 %) at the same nominal values of Räbf. The transition from the conductive to the convective heat-transfer regime occurs at the same value of the nanofluid Rayleigh number, i.e., Ränf < 281. 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