Effects of two-phase nanofluid model and localized heat source/sink on natural convection in a square cavity with a solid circular cylinder

Abstract

In the present study, natural convection heat transfer of ${\mathrm{Al}}_2{\mathrm{O}}_3$-water nanofluid inside a square cavity with a solid circular cylinder is investigated numerically. For numerical computations, the finite element method is used by taking into consideration Buongiorno’s two-phase model. Parts of the vertical surfaces of cavity are kept at constant temperature (left wall $T_h$ and right wall $T_c$) while the other walls (horizontal walls and the remaining of the vertical walls) are taken as adiabatic. The effects of some pertinent parameters such as the Rayleigh number $(10^3\le Ra\le 10^6)$, nanoparticle volume fraction ($0\le \varphi \le 0.04$), thermal conductivity of the solid cylinder ($k_w=0.28$, 0.76, 1.95, 7 and 16), radius of solid cylinder ($0.1\le R\le 0.4$), heat source/sink length ($0.2\le D\le 0.8$), and the heat source/sink position ($0.2\le B\le 0.8$) on the fluid flow and heat transfer characteristics are investigated. The obtained numerical results are depicted graphically and discussed in detail from the point of view of the streamlines, isotherms, nanoparticle volume fractions and the local and average Nusselt number $Nu$. It is indicated that the heat transfer is enhanced with an increase in the nanoparticle volume fraction for all studied Rayleigh numbers. Furthermore, the thermal conductivity, solid circular cylinder size, $D$ and $B$ parameters are the key factors to control and optimize the heat transfer inside the cavity that is partially heated and cooled. The proposed method is found to be in good agreement between previously published experimental and numerical results.

Introduction

Energy is a very important issue in today’s societies. The main topic of this issue is to improve energy efficiency and decrease the energy consumption in all engineering systems and equipment. There are several ways to save energy. Improving the heat transfer rate is one way for reducing energy consumption. In many engineering systems, natural convection heat transfer in enclosures is a significant process due to its widespread applications such as heating and cooling nuclear systems of reactors, thermal design of buildings and heat exchangers, lubrication technologies, storage tanks, ventilation of rooms with radiators, double pane windows, etc. [[1], [2], [3], [4], [5]].

In these systems, heat transfer fluids are very important. The traditional heat transfer fluids (water, ethylene glycol, engine oil etc.) have been used for many years. In addition, heat transfer enhancement efforts via traditional technologies like fins and microchannels have reached its limits and these fluids have low thermal conductivity. Because of this, new materials have been sought and the suspensions were started to be studied. Firstly, metallic solids were used in base fluids because they have higher thermal conductivity values. Thus, solid metallic particles and metal oxides have been used in conventional heat transfer fluids. With the development of engineering equipment, smaller particles have been used in heat transfer applications. Millimeter and then micrometer sized particles have been used in suspensions for heat transfer studies. Some disadvantages raised in time such as erosion and clogging in the channels, high pressure drop, rapid settling of particles. Due to newly developed technology, nano-sized particles can be available and so nanofluids are suitable for industrial operations since they were firstly introduced by Choi and Eastman [6]. Nanofluids, which are suspended particles ($<100\mathrm{nm}$) in a base fluid represent an innovative way for heat transfer enhancement due to their higher thermal conductivity than conventional heat transfer fluids. Excellent books were made on the fundamental topics of convective flow and heat transfer in cavities filled with nanofluids by [[7], [8], [9]].

Based on this fact, nanofluids are used in many sectors such as transportation, electronic components, medical applications, energy storage etc. (see details in [[10], [11], [12], [13]]). Generally, metal or metal oxide solid particles (Cu, Ag, CuO, ${\mathrm{Al}}_2{\mathrm{O}}_3$, TiO${}_2$, SiO2, ZnO) are used in nanofluids. There are several studies [[14], [15], [16], [17], [18], [19]] done experimentally or theoretically to investigate the nanofluid flow and its heat transfer characteristics. Natural convection in a square enclosure that is partially heated from one side and cooled from the other side is a significant phenomenon in several engineering applications. Many research papers [[20], [21], [22], [23], [24], [25]] related to the natural convection of nanofluids in cavities have been published up to now. Öğüt [26] studied the natural convection problem in inclined square cavity. She reported that heat transfer increases with the increase in the nanoparticle concentration and the Rayleigh number. Natural convection problems of some water-based nanofluids flow in a partially heated and cooled cavity shaped as different geometries were studied by Guestal et al. [27] and Garoosi et al. [28].

In order to analyze the fluid flow, the temperature distribution and heat transfer characteristics of the considered nanofluid, two methods have been employed, namely (i) single-phase, and (ii) two phase model. When employing the single-phase model, the main presumption is that there is thermal equilibrium between the solid particles and the base fluid [4]. Some studies using the single-phase method are given here. Natural convection in a square cavity filled with a Cu–water nanofluid has been considered numerically by Boualit et al. [29] to determine the improvement of the coolant performance of fuel cell stacks. The results of the study indicated that the increase in nanoparticle volume fractions has led to a heat transfer enhancement for all $Ra$ numbers. Ismael et al. [30] investigated fluid flow and heat transfer in a lid-driven square cavity filled with a CuO–water nanofluid and heated by a corner heater. The authors’ results indicated that the impact of nanoparticles on heat transfer improvement is not meaningful for low and intermediate values of the Richardson number whereas it is meaningful for high Richardson numbers. Solomon et al. [31] experimentally investigated the aspect ratio ($AR$) effect of a rectangular cavity filled with nanofluids on natural convection characteristics. Three different cavities, having different $ARs$ were manufactured and tested to find the heat transfer performance. They reported that the $AR$ is a very important parameter on the heat transfer coefficient and the $Nu$ number as well as the $Ra$ number. Bhuiyana et al. [32] used the Galerkin method to study the natural convection of water-based nanofluids in a square cavity. In their study, the $Ra$ number and the nanoparticle concentrations were varied between $10^3$ and $10^6$, 0 and 0.2, respectively. From the study, it was concluded that the rate of heat transfer was enhanced by the increase in the nanoparticle concentration for the whole range of $Ra$ numbers. Garbadeen et al. [33] experimentally investigated natural convection of MWCNT-water nanofluids in a square cavity. The nanoparticle volume fraction was changed from 0 to 1% for a constant $Ra$ number ($Ra=10^8$). The maximum enhancement in the thermal conductivity was found as 6% for a particle volume fraction of 1.0%. It was also concluded from the study that the enhancement in heat transfer was found as 45% at the particle volume fraction of 1.0%. The mixed convection problem of a Cu–water nanofluid was studied numerically by Rajarathinam et al. [34] through an inclined porous cavity using three different directions of the moving wall. The impacts of some non-dimensional parameters such as the Richardson number, Darcy number, inclination angle, and the solid volume fraction on the fluid flow and heat transfer were studied. They concluded that the direction of moving walls played an important role. Natural convection problem in a square cavity filled with a ${\mathrm{Al}}_2{\mathrm{O}}_3$ nanofluid was also investigated by Cho [35] taking the entropy generation into consideration. It considered the effects of parameters such as the wavy-surface geometry, Rayleigh number and the nanoparticle concentration on the Bejan number, total entropy generation and mean Nusselt number. The results indicated that the mean $Nu$ number was an increasing function of the nanoparticle volume concentration; whereas the total entropy generation was a decreasing function. The phenomenon on how the natural convection of ${\mathrm{Al}}_2{\mathrm{O}}_3$-water nanofluid in a square cavity is affected by a heterogeneous heating was studied by Rashidi et al. [36]. Several different situations were evaluated in their study to determine the heat transfer characteristics.

In the above mentioned studies, a homogeneous model was used; but recently, researchers turned to study nanofluid problems using a two-phase model. For this reason; two-phase models are better and more accurate compared to the homogeneous (single phase) model to evaluate the heat transfer and nanofluid flow since it takes friction, Brownian effect and slip velocity between the base fluid and the particles into consideration [[15], [37]]. Apart from the studies, there are some numerical studies using the single-phase model [[38], [39], [40], [41], [42]] and others using different two-phase models (Eulerian–Eulerian, Eulerian–Lagrangian, two-phase mixture and VOF). For example, Emami et al. [43] used a mixture model to study the effect of the Rayleigh number, Darcy friction factor, and the volume fraction of nanoparticles on the natural convection heat transfer characteristics of a Cu–water nanofluid. It was concluded that the selection of a heater configuration and the inclination angle affected the heat transfer enhancement. Buongiorno [44] suggested a non-homogeneous equilibrium model taking into account the impact of the Brownian motion and thermophoresis as two prominent slip mechanisms in a nanofluid. In his study, seven slip mechanisms were offered between the base fluid and the nanoparticles. In that study, he also improved a two-component non-homogeneous equation in nanofluids stressing the superiority of these two effects with respect to the remaining transport mechanisms. Sheremet and Pop [45] reported the effect of a two-phase nanofluid model on natural convection heat transfer in a square porous cavity with non-uniform temperature distributions on the vertical walls. Sheremet et al. [46] used a two-phase Buongiorno’s model to solve the natural convection heat transfer problem inside two triangular cavities filled with a nanofluid. Motlagh and Soltanipour [47] considered Buongiorno’s two-phase model employing the Finite Volume Method (FVM) and the SIMPLE algorithm for natural convection of ${\mathrm{Al}}_2{\mathrm{O}}_3$-water nanofluid in an inclined cavity. In their research study, the right and left vertical walls of the cavity were kept at a constant temperature; whereas the other walls were thermally isolated. The Rayleigh number varied between $10^2$ and $10^6$ and the particle volume fractions ranged from 0.01 to 0.04. Their results showed that the enhancement percentages of heat transfer and $Nu$ number are almost constant for low Rayleigh numbers with increasing inclination angles whereas they increased with high $Ra$ numbers. Sheremet and Pop [48] studied the natural convection problem in a square porous cavity filled with a nanofluid by using Buongiorno’s two phase model. The fluid flow and heat transfer characteristics were examined in terms of the Brownian motion, thermophoresis effect and some dimensionless parameters such as the Rayleigh and Lewis numbers. Another study using the Buongiorno’s model was performed by Kefayati [49]. In his study; in a porous cavity, natural convection of a non-Newtonian nanofluid problem with entropy generation was solved by using the finite difference lattice Boltzmann method.

Placement of a solid block inside the cavities with various shapes filled with nanofluids or pure fluids is an important application due to its ability of heat transfer control as a passive element. This application can be seen in many applications such as building design, thermal management of solar energy systems, electronics and heat exchanger [50]. Mahmoodi and Sebdani [51] considered the natural convection problem inside a square cavity filled with a Cu–water nanofluid and adiabatic square block placed inside the cavity. The developed code depending on FVM was used in their numerical study. The impacts of some parameters such as the size of the adiabatic square block, Rayleigh number and the volume fraction of nanoparticles on the fluid flow and heat transfer characteristics were examined. They determined that the average $Nu$ number increased with the boosting of the nanoparticle volume fraction for all $Ra$ numbers, except $Ra=10^4$. At the same time it was found that the square body size significantly affected the heat transfer rate. Garoosi et al. [[52], [53]] studied numerically the natural and mixed convection problems for different nanofluids by considering Buongiorno’s two-phase model. They reported that thermophoretic effects could be negligible for nanoparticles which have high thermal conductivities and their studies concluded that the increasing thermal conductivity ratio and the Rayleigh number improved the value of heat transfer significantly. Furthermore, recently the conjugate natural and mixed convection heat transfer problem were studied numerically by Garoosi and Talebi [4]. They used many pairs of cylinders having cold and hot surfaces. For different nanofluids, Buongiorono’s two phase model was also achieved in their study successfully. Alsabery et al. [54] considered the conjugate mixed convection in a cavity including a solid inner body filled with ${\mathrm{Al}}_2{\mathrm{O}}_3$-water nanofluid, using the Buongiorno’s two phase model. They found that the nanoparticles addition into water caused the $Nu$ number to increase at low Reynolds numbers and high Richardson numbers. In addition, it was found from their study that the solid body size is a key parameter to increase heat transfer. According to the study of Alsabery et al. [5], it was concluded that the thermal conductivity and size of the solid block were ideal parameters to optimize heat transfer inside the cavity. Very recently, Selimefendigil and Öztop [55] studied the mixed convection of various water-based nanofluids in three-dimensional cavity with two adiabatic inner rotating cylinders. They concluded that the highest heat transfer rate at the highest value of $Ra$ number occurs in the situation of Cu–water nanofluid with the 4% nanoparticle volume fraction. An enhancement in the average heat transfer was found as 38.1% for a Cu–water nanofluid when compared to the base fluid. Alsabery et al. [56] performed a numerical investigation on the MHD natural convection and heat transfer in a discretely-heated square cavity in the presence of a conductive inner block using the two-phase nanofluid model.

Depending on the above research literature, and according to the best of the authors’ knowledge, although there are many papers on natural convection in a square cavity filled with nanofluids, there have been no studies considering the natural convection on different types of solid circular cylinder placed inside a square cavity filled with ${\mathrm{Al}}_2{\mathrm{O}}_3$-water nanofluid using the two-phase Buongiorno’s model, despite its importance in many engineering systems. Therefore, according to the authors, the study is considerable and has novelty. For this reason; the main purpose of this study is to numerically investigate the impacts of some parameters such as the Rayleigh number $(10^3\le Ra\le 10^6)$, volume fraction of nanoparticles ($0\le \varphi \le 0.04$), thermal conductivity of solid cylinder ($k_w=0.28$, 0.76, 1.95, 7 and 16), radius of solid cylinder ($0.1\le S\le 0.4$), heat source/sink length ($0.2\le D\le 0.8$), and the heat source/sink position ($0.2\le B\le 0.8$) on the fluid flow and heat transfer characteristics using Buongiorno’s two-phase model.