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This study has focused on laminar mixed convection in an inclined square ventilated lid-driven cavity fi lled with a copper–water nanofl uid. The governing equations in the two-dimensional space are discretized by using the fi nite volume method with
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   Heat Transfer Research 45(4), 361–390 (2014) 1064-2285/14/$35.00 © 2014 by Begell House, Inc. 361  1. INTRODUCTION Heat transfer enhancement is one of the key issues of saving energies and of compact designs for mechanical and chemical devices and plants. Until today, people have ANALYSIS OF LAMINAR MIXEDCONVECTION IN AN INCLINED SQUARE LID-DRIVEN CAVITY WITH A NANOFLUID BY USING AN ARTIFICIAL NEURALNETWORK  M. R. Faridzadeh, 1  D. Toghraie Semiromi, 1,*   & Amirhossein Niroomand 2 1 Department of Mechanical Engineering, Islamic Azad University, Khomeinishahr Branch, Isfahan, Iran 2 Department of Mechanical Engineering, Kashan University, Kashan, Isfahan, Iran * Address all correspondence to D. Toghraie Semiromi E-mail: T  his study has focused on laminar mixed convection in an inclined square ventilated lid-driv-en cavity   fi lled with a copper–water nano   fl  uid. The governing equations in the two-dimensional  space are discretized by using the   fi nite volume method with the SIMPLER algorithm. The effects of independent parameters, including the Richardson number, Reynolds number, inclination angle, and the solid volume fraction of nanoparticles, on the streamlines, isotherm lines, and the average  Nusselt number along the heat source have been studied. It is found that both the inclination an- gle and solid volume fraction, especially the second one, have remarkable effects on the   fl  uid   fl  ow and heat transfer characteristics in the cavity. Arti   fi cial neural networks (ANN) used to extract a relation involve independent parameters for calculating the Nusselt number. The back propa- gation-learning algorithm with the tangent sigmoid transfer function is used to train the ANN.  Finally, analytical relations for the nano   fl  uid mixed convection in a lid-driven cavity are derived  from the available ANN. It is found that the coef    fi cient of multiple determinations (R 2  ) between the real values and ANN results is equal to 0.9999, the maximum error being less than 0.5829 and the mean square error being equal to 5.37 × 10  –5 . KEY WORDS:   lid-driven cavity, mixed convection, nano  fl uid, Richardson number, Reynolds number, arti  fi cial neural networks   Heat Transfer Research 362 Faridzadeh, Semiromi, & Niroomand made efforts to enhance convective heat transfer by means of surface enlargement using obstacles such as ribs and fi ns, and the increase in fl ow turbulence due to the interaction between the fl ow and the obstacles. However, this resulted in additional  pressure losses. The low thermal conductivity of conventional fl uids such as water, oil, and ethylene glycol mixture is a serious limitation on improving the performance and compactness of this engineering equipment. To overcome those disadvantages, NOMENCLATURE b  bias T  temperature, K  c  p speci fi c heat at constant pressure, u , v components of velocity, m ⋅ s  –1 J ⋅ kg  –1 ⋅ K   –1 U  , V  dimensionless velocityCGPPola–Ribiere conjugate gradientcomponentsalgorithm W  ij weight factor covcoef  fi cient of variation, %  x ,  y Cartesian coordinates, m  E  i weighted sum of the input  X  , Y  dimensionless coordinate  g  gravitational acceleration, m ⋅ s  –2  Z  value of each neuron cellGrGrashof number  Greek symbols  H  channel height, m  α thermal diffusivity, m 2 ·s  –1 k  thermal conductivity,  β coef  fi cient of volumeW ⋅ m  –1 ⋅ K   –1 expansion, K   –1  L length of the channel, m  γ rotation angleLMLevenberg–Marquardt algorithm  θ dimensionless temperatureMSEmean square error   μ dynamic viscosity, kg·m  –1 ·s  –1  NuNusselt number   ν kinematic viscosity, m 2 ·s  –1  P  dimensionless pressure  ρ density, kg·m  –3  p  pressure, Pa  φ solid volume fractionPrPrandtl number  Subscripts q "heat fl ux, W·m  –2  R 2 absolute fraction of varianceaveaverageRaRayleigh numberf   fl uidReReynolds numbernfnano fl uidRiRichardson numberpparticleRMSroot mean squaressolidSCGscaled conjugate gradientwwallalgorithm0reference state  363 Laminar Mixed Convection in an Inclined Square Lid-Driven Cavity Volume 45, Number 4, 2014 there is a strong motivation to develop advanced heat transfer fl uids with substantially higher conductivity. An innovative way of improving the thermal conductivities of fl uids is to suspend small solid particles in a fl uid. Nano fl uids are new heat transfer fl uids containing a small quantity of nano-sized particles that are uniformly and sta- bly suspended in a liquid. The dispersion of a small amount of solid nanoparticles in conventional fl uids remarkably changes their thermal conductivity. Compared to the existing techniques for enhancing heat transfer, the nano fl uids show a superior  potential for increasing heat transfer rates in a variety of cases. Polidori et al. (2007) investigated the natural convection heat transfer of Newtonian nano fl uids in a laminar external boundary layer using the integral formalism approach. They found that the natural convection heat transfer is not solely characterized by the nano fl uid effective thermal conductivity and that the sensitivity to the viscosity model used seems un-deniable and plays a key role in the heat transfer behavior. Akbarinia (2008) solved three-dimensional elliptic governing equations to investigate laminar mixed convec-tion of a nano fl uid consisting of water and Al 2 O 3 , buoyancy-affected, and heat trans-fer of a curved tube. He concluded that for a given heat fl ux at a given fl ow rate, the  Nusselt number is reduced by augmenting the nanoparticles concentration. Decreasing the Nusselt number for a given fl ow rate by increasing the nanoparticles concentration is a crucial issue that reduces the centrifugal force. In recent years, mixed convec-tion heat transfer inside enclosures has attracted the attention of many researchers. Investigation of mixed convection especially in lid-driven enclosures has varied ap- plications in engineering technologies such as electronic devices, furnaces, lubrication technologies, chemical processing equipment, drying technologies, etc. This study has investigated the nano fl uid mixed convection in a square lid-driven cavity with a heat source partially embedded on the bottom wall. Moallemi and Jang (1992) examined the mixed convection heat transfer in a lid-driven cavity. They investigated the effects of the Prandtl number on the fl ow fi eld. Mohamad and Viskanta (1991) examined the fl ow fi eld in a shallow lid-driven cavity heated from below. Ozotop and Dagtekin (2004) studied mixed convection in a vertical two-sided lid-driven partially heated square cavity. They found that both the Richardson number and direction of wall mo-tion would affect the fl uid fl ow. Abu-Nadaa and Chamkha (2010) studied numerical modeling of steady laminar mixed convection fl ow in a lid-driven inclined square enclosure fi lled with water–Al 2 O 3  nano fl uid. They found that signi fi cant heat transfer enhancement can be achieved due to the presence of nanoparticles. Mahmoodi (2011) investigated mixed convection fl uid fl ow and heat transfer in lid-driven rectangular enclosures fi lled with an Al 2 O 3  –water nano fl uid. He observed that the average Nusselt number of the hot wall of tall enclosures is higher than that of shallow enclosures. Billah et al. (2011) investigated the transport mechanism of mixed convection in an inclined lid-driven triangular enclosure subjected to cooling at the inclined right sur-face and simultaneous heating at the base surface fi lled with nano fl uids. They found that the tilt angle strongly affects the fl uid fl ow and heat transfer in the enclosure in   Heat Transfer Research 364 Faridzadeh, Semiromi, & Niroomand three convective regimes. Elhar  fi  et al. (2012) studied mixed convection in a shal-low lid-driven rectangular cavity fi lled with water-based nano fl uids and subjected to a uniform heat fl ux along the vertical sidewalls. They found that the addition of Cu nanoparticles to a pure water leads to the enhancement of heat transfer depending on the values of Re and Ri. Alinia et al. (2011) studied mixed convection of a nano fl u-id consisting of water and SiO 2  in an inclined enclosure. Their results revealed that addition of nanoparticles remarkably enhances heat transfer in the cavity and caus-es signi fi cant changes in the fl ow pattern. Besides, the effect of inclination angle is more pronounced at higher Richardson numbers. The thermal conductivity and effec-tive viscosity of a nano fl uid were calculated by the Patel et al. (1952) and Brinkman (1952) models, respectively. In addition, the ANN method is used for formulating the  Nusselt number. It is shown that the ANN method is a feasible and powerful method for investigating heat transfer in a lid-driven cavity. Arti fi cial neural networks have  been largely used in the previous decades. Madadi et al. (2008) have studied the op-timal location of three discrete heat sources inside a ventilated cavity. Santra et al. (2009) investigated heat transfer inside a differentially heated square cavity by using the ANN method and showed that the ANN predicts heat transfer correctly within the given range of training data. The purpose of this work is to enhance the discussion on the use of nano fl uids with the sole aim of increasing the heat transfer coef  fi cient in laminar mixed convection fl ows. The results obtained may have direct applications in industrial processes and technologies such as furnaces, lubrication technologies, drying technologies, chemical processing equipment, and others.  2. MODELING AND GOVERNING EQUATIONS2.1 Problem Statement We consider a steady two-dimensional fl ow of a nano fl uid contained in an inclined lid-driven square enclosure as shown in Fig. 1. The geometry considered in this study is a two-dimensional inclined square ventilated lid-driven cavity of height  H   and length  L  (in this study  H   =  L ). The cavity consists of an inlet and an outlet port. The nano fl uid enters with uniform velocity U  0  and emerges through the outlet port. The length of the inlet and outlet ports is equal to  H  /15. The top wall is moving with a uniform velocity U  0 . All the walls are insulated and a heat source of constant heat fl ux q ′  is embedded symmetrically on the bottom wall. In this study, we used a cop- per–water nano fl uid. Table 1 presents the thermophysical properties of water and cop- per at a reference temperature. The boundary conditions are de fi ned by Eqs. (1)–(5):1) left and right vertical walls   0,0,0, T  u v x ∂= = =∂  (1)  365 Laminar Mixed Convection in an Inclined Square Lid-Driven Cavity Volume 45, Number 4, 2014 2) top wall   0 0 , , 0,y T u U v ∂= = =∂  (2) 3) inlet port   0c , 0 , , U U v T T  = = =  (3) 4) outlet port   0 , 0 , 0, u T v x x ∂ ∂= = =∂ ∂  (4) 5) bottom horizontal wall FIG. 1:  Schematic of the problem geometry (enclosure) TABLE 1: Thermophysical properties of water and copper Property  c  p  (J/kg ⋅ K)  ρ  (kg/m 3 )  k   (kW/m ⋅ K)  β   (1/K)Water 4179 997.1 0.6 2.1 ⋅ 10  –4 Copper 383 8954 400 1.67 ⋅ 10  –5
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