Two-phase mixture modeling of mixed convection of nanofluids in a square cavity with internal and external heating

doi:10.1016/j.powtec.2015.02.015

Powder Technology

Volume 275, May 2015, Pages 304-321

https://doi.org/10.1016/j.powtec.2015.02.015 Get rights and content

Highlights

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Mixed convection of nanofluids in a cavity is studied using Two-phase mixture model.
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The results show that, at low Ri, the distribution of the solid particles remains almost uniform.
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There is an optimum volume fraction of nanoparticles for maximum Nusselt number.

Abstract

Steady state mixed convection heat transfer of nanofluid in a two-sided lid driven cavity with several pairs of heaters and coolers (HACs) inside is investigated numerically using two-phase mixture model. The governing equations have been discretized using the finite volume method while the SIMPLE algorithm has been introduced to couple the velocity–pressure. The influences of volume fraction, diameter and type of the nanoparticles, Richardson number, number of the Heaters and Coolers (HACs), external and internal heating and moving direction of the cavity walls on flow structure, the heat transfer rate and distribution of nanoparticles are investigated. The results of this investigation illustrate that, at low Richardson number by increasing number of the HACs, the heat transfer rate increases. On the other hand, at high Ri, a saturated number of HACs exists which beyond that the value of mean Nusselt number does not changes significantly. In addition, the results reveal that by reducing the diameter of the nanoparticles and Ri, the heat transfer rate increases. It is also observed that at high Richardson numbers, distribution of nanoparticles with d_p ≥ 145 nm is fairly non-uniform while at low Richardson numbers particle distribution remains almost uniform. Moreover, it is found that by changing direction of the moving walls the heat transfer rate changes significantly.

Graphical abstract

Introduction

Mixed convection occurs in a great number of industrial applications such as indoor ventilation with radiators, cooling electrical components, cooling reactors and heat exchangers [1], [2]. In all of these applications engineers are constantly looking for ways to improve the overall heat transfer efficiency by implementing a wide spectrum of technics, from design optimization to use of novel materials like nanofluids. In this process, numerical simulation as a relatively inexpensive design and research tool, has been extensively used in recent decades to provide close insights to the heat transfer phenomenon. Since complex industrial geometries can be computationally expensive and case specific, simplified geometries like square cavity are widely used to isolate and demonstrate effect of design parameters on the heat transfer phenomenon. Work of Oztop et al. [3] can be mentioned as example of such studies, in which they numerically studied mixed convection in square cavities with two moving walls. Their results suggest that when the vertical walls move upwards in the same direction, the heat transfer decreases significantly compared to when the vertical walls move in opposite directions. In a similar work, Islam et al. [4] performed a numerical simulation on a lid-driven cavity with an isothermally heated square blockage. Their results showed that Richardson number, size and location of the heater eccentricities affect the average Nusselt number of heater. Qi-Hong Deng [5] studied laminar natural convection in a two dimensional square enclosure with two and three source–sink pairs on the vertical side walls. They reported that averaged Nusselt number values indicate that heat transfer between heaters and coolers, is one to one in a reversed manner.

The heat transfer can be improved when nanofluid is used instead of pure fluids (e.g. water, oil, ethylene glycol). Nanofluids are mixtures of nano-sized solid particles (usually smaller than 150 nm) dispersed in a base fluid. The most important characteristic of nanofluids is their high thermal conductivity relative to the pure fluids, which can be achieved even at very low volume fraction of nanoparticles. The heat transfer characteristics of nanofluids depend on the size, volume fraction, shape and thermo-physical properties of nanoparticles as well as the base fluid properties [6], [7], [8], [9].

In general, numerical simulation of the velocity field, temperature distribution and heat transfer rate of nanofluids can be performed using two main approaches, namely single-phase and two-phase methods. The former is based on continues phase assumption and assumes nanoparticles to be in thermal equilibrium with base fluid and move with the same velocity, and have been used to simulate both natural convection [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] and mixed convection [20], [21], [22], [23], [24] in nanofluids. Corcione [10] and Garoosi et al. [11] investigated the natural convection of nanofluid at different geometries using the model proposed in [6] to estimate the effective viscosity and thermal conductivity of nanofluid. Their results illustrated that there is an optimum volume fraction of nanoparticles, where the maximum heat transfer rate occurs. Mahmoodi [12], studied free convection of Cu-water nanofluid in L-shaped cavity. He observed that, at all Rayleigh numbers, the heat transfer rate increases when the aspect ratio of the cavity and the volume fraction of the nanofluid increase. Mejri et al. [13] investigated the effects of magnetic field on entropy generation in a square cavity with sinusoidal heating on both side walls. The results indicated that the average Nusselt number and fluid flow decline with Hartmann number and increase with Rayleigh number and volume fraction of the nanoparticles. Sheikholeslami et al. [14] have performed a numerical study of magnetic field effects on natural convection around a horizontal circular cylinder inside a square enclosure filled with nanofluid. They found that the heat transfer rate is an increasing function of nanoparticle volume fraction as well as the Rayleigh number, while it is a decreasing function of the Hartmann number (Ha). In addition, their results indicated that for Ha < 20 the enhancement in average Nusselt number at Ra = 10⁴ is greater than at other Rayleigh numbers. Sheikholeslami et al. [15] studied the effects of magnetic field on natural convection flow in a cavity filled with nanofluid. They used the Brinkman [7] and Maxwell–Garnetts [8] models to estimate the effective viscosity and thermal conductivity of the nanofluid. They stated that the increase in volume fraction of nanoparticles and Rayleigh numbers enhances the heat transfer rate. In a similar work, Sheikholeslami et al. [19] studied the effects of magnetic field on ferrofluid flow and heat transfer in a semi annulus enclosure. They reported that increasing Magnetic number, Rayleigh number and volume fraction of the nanoparticles leads to augmentation of the heat transfer rate but the average Nusselt number decreases with the increase of Hartmann number and Radiation parameter. Kalteh et al. [20] investigated laminar mixed convection of nanofluid in a lid-driven square cavity with a triangular heat source. Their results showed that the increase in nanoparticle diameter and Richardson numbers leads to a decrease in the heat transfer rate. Pishkar and Ghasemi [21] conducted a numerical simulation to investigate the problem of mixed convection of the Cu–water nanofluid in a horizontal channel with two fins. They found that by increasing the distance and the thermal conductivity of the fins, the mean Nusselt number increases significantly. The influence of the solid volume fraction on the increase of heat transfer is more noticeable at higher values of the Reynolds number. Talebi et al. [22] investigated and reported mixed convection of nanofluids inside a differentially heated cavity using models of Brinkman [7] and Patel et al. [9] for estimating the effective viscosity and thermal conductivity of the nanofluid. They concluded that the heat transfer rate has a direct relationship with Rayleigh number and solid particle concentration. Moreover, their results showed that for a given Reynolds number with increasing volume fraction of the solid particles, the stream function increases particularly at the higher Rayleigh number.

Khanafer et al. [23] studied mixed convection in a lid driven cavity with a circular body inside. They found that the presence of the cylinder results in an increase in the heat transfer rate compared with the case with no cylinder. In addition, their results indicated that for dominant mixed convection, the average Nusselt number increased with an increase in the radius of the cylinder for various Richardson numbers. Khorasanizadeh et al. [24] numerically investigated the fluid flow, heat transfer and entropy generation of Cu–water nanofluid mixed convection in a square cavity. The results show that the minimum entropy generation occurs in pure fluid at low Rayleigh and low Reynolds numbers.

Experimental studies, however, question the validity of the single-phase assumption for nanofluids [25]. As the slip velocity between the base fluid and particles may not be zero, more advanced methods, such as two-phase mixture models are developed [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39] to reproduce the real phenomenon more accurately. In a pioneering work, Manninen et al. [29] considered drag and gravity as two important primary slip mechanisms between solid and liquid phases. Alinia et al. [26] performed a numerical simulation on mixed convection of a nanofluid (SiO₂–water) in an inclined enclosure using two-phase mixture method. Their results show that at low Richardson number the particle distribution is fairly uniform while by increasing the Richardson number, a degree of non-uniformity can be observed. Lotfi et al. [28] studied the heat transfer of a nanofluid in a horizontal circular tube by using single-phase, two-phase mixture and two-phase Eulerian models. The results of this study confirmed that the two phase mixture method is more accurate than others. Mirmasoumi et al. [30], [31] studied laminar mixed convection of Al₂O₃–water in a horizontal tube by using two phase mixture method. They discovered that the concentration of the nanoparticles at the top of the tube is lower than that of the bottom and it is more concentrated near the wall. They also found that by increasing the nanoparticle diameter, their distribution becomes more non-uniform. Mokhtari Moghari et al. [32] investigated the effects of volume fraction and diameters of the nanoparticle on the thermal and hydrodynamic parameter of the laminar mixed convection in an annulus. Their results indicated that, reducing mean diameter of nanoparticles enhances the heat transfer rate while it does not have any significant effect on the friction factor. In addition, they found that, increasing the diameter of nanoparticles increases the non-uniformity of nanoparticle distribution at any cross section of annulus. Moraveji et al. [33] compared the heat transfer rate of the Al₂O₃–water in mini-channel heat sink by using single-phase and two-phase method. They found that, the two phase models represented a better approximation of the experimental data compared to the single phase model. Moreover, their results show that the increase in the Reynolds number (Re) and volume fraction of the nanoparticles leads to the enhancement of heat transfer rate. A similar study has been done by Shariat et al. [34], [35] who simulated laminar mixed and force convection of the nanofluid in an elliptic duct using two phase mixture model. The results of this study suggest that nanoparticle distribution is higher near the bottom wall and lower near the top wall of the tube. Akbarinia et al. [36] numerically investigated laminar mixed convection of a nanofluid in horizontal curved tubes using two phase mixture model. The results show that for a given Re, at low Gr, nanoparticle concentration does not have a significant effect on the skin friction reduction. However, at the larger value of the Grashof numbers, by increasing the volume fraction of the nanoparticles, the skin friction decreases. Goodarzi et al. [38] carried out a numerical analysis of nanofluid mixed convection in a shallow cavity. They concluded that, the effects of the volume fraction on turbulent kinetic energy, turbulence intensity, skin friction and wall shear stress are insignificant. In addition, they found that, at constant Grashof number, the average Nusselt number enhances with a decrease in the Richardson number.

Considering the accepted validity of the two-phase model, this research intends to explore the mixed convection heat transfer rate, in a square cavity using two phase mixture model [29]. The effects of the volume fraction (0 ≤ φ ≤ 0.05), diameter (25 nm ≤ d_p ≤ 145 nm) and type (Cu, Al₂O₃, TiO₂) of the nanoparticles, Richardson number (0.01 ≤ Ri ≤ 1000), number of the heaters and coolers (HACs), external and internal heating and moving direction of the cavity walls on the heat transfer rate and distribution of nanoparticles are investigated. The square cavity with internal heater and cooler can be observed in heat exchangers, in order to prevent freezing or condensing of fluid in pipelines [1], [2]. In the following sections, the investigated cases will be defined and the applied computational method will be discussed in detail.

Section snippets

Problem statement

The physical models under consideration are mixed convection inside a square cavity with multiple HACs flush-mounted on the walls or inside the cavity (Fig. 1). Seven different cases are considered in which the horizontal or vertical walls of the cavity move to create mixed convection conditions. The flow is assumed to be incompressible, two dimensional, steady, and laminar with the Boussinesq approximation used for the fluid density. The thermo-physical properties of the nanofluid in this

Mathematical formulation

Two-dimensional continuity, momentum, energy and volume fraction equations are as follows [29], [30]: $\nabla \cdot {\vec{V}}_{m} = 0,$ $\begin{array}{l} \nabla ρ_{m} {\vec{V}}_{m} {\vec{V}}_{m} = - \nabla P + \nabla . μ_{m} \nabla {\vec{V}}_{m} + \nabla . [\sum_{k = 1}^{n} φ_{k} ρ_{k} {\vec{V}}_{d r, k} {\vec{V}}_{d r, k}] \\ + {(ρ β)}_{m} (T - T_{c}) g \end{array}$ $\nabla . [\sum_{k = 1}^{n} (ρ_{k} C_{p k}) φ_{k} {\vec{V}}_{k} T] = \nabla . k_{m} \nabla T$ $\nabla . (φ_{p} ρ_{p} {\vec{V}}_{m}) = - \nabla . (φ_{p} ρ_{p} {\vec{V}}_{d r, p}),$ where T and P are respectively the dimensional temperature and pressure field. φ is the volume fraction of the solid or liquid phases. ${\vec{V}}_{m}$ is mass average velocity: ${\vec{V}}_{m} = \frac{\sum_{k = 1}^{n} φ_{k} ρ_{k} {\vec{V}}_{k}}{ρ_{m}} .$

In Eq. (2), ${\vec{V}}_{d r, k}$ is the drift velocity for the secondary phase k, i.e. the nanoparticles in the

Numerical implementation

A finite volume formulation, given by Patankar [43] on a staggered grid, is applied for discretization of governing equations and boundary conditions. The coupled pressure–velocity equation is solved by SIMPLE algorithm and hybrid differencing scheme of Spalding [44] is employed for the convective terms. The TDMA method [44] is used for solving the system equation iteratively until sum of the residuals became less than 10^− 6. The numerical values of the thermo-physical properties of nanofluid

Results and discussion

Mixed convection heat transfer of the nanofluid in the square cavity for seven cases is simulated. The key parameters in this study are the volume fraction (0 ≤ φ ≤ 0.05), diameter (25 nm ≤ d_p ≤ 145 nm) and type (Cu, Al₂O₃, TiO₂) of the nanoparticles, Richardson number (0.01 ≤ Ri ≤ 1000), number of the heaters and coolers (HACs), external and internal heating and moving direction of the cavity walls. To adjust the Richardson number, the Grashof number is fixed at 10⁴ while the Reynolds number is varying from

Conclusions

This study investigates the mixed convection of nanofluids in a two-dimensional square cavity containing several pairs of heater and cooler (HACs) using the two-phase mixture model of Manninen [29]. The effects of various design parameters on the heat transfer rate and distribution of nanoparticles, such as nanoparticles volume fraction (0 ≤ φ ≤ 0.05), diameter (25 nm ≤ d_p ≤ 145 nm) and type (Cu, Al₂O₃, TiO₂) of the nanoparticles, Richardson number (0.01 ≤ Ri ≤ 1000), number of the heaters and coolers

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Two-phase mixture modeling of mixed convection of nanofluids in a square cavity with internal and external heating

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Problem statement

Mathematical formulation

Numerical implementation

Results and discussion

Conclusions

Appl. Therm. Eng.

Int. J. Heat Mass Transf.

Int. J. Heat Mass Transf.

Int. J. Heat Mass Transf.

Energy Convers. Manag.

Renew. Sustain. Energy Rev.

Int. J. Heat Fluid Flow

Int. J. Heat Mass Transf.

Int. J. Therm. Sci.

Int. Commun. Heat Mass Transfer

Powder Technol.

J. Mol. Liq.

Powder Technol.

Adv. Powder Technol.

J. Taiwan Inst. Chem. Eng.

Powder Technol.

Int. J. Therm. Sci.

Int. Commun. Heat Mass Transfer

Int. J. Heat Mass Transf.

Eur. J. Mech. B. Fluids

Int. J. Heat Mass Transf.