Sage Journals: Discover world-class research

Abstract

Enhancement of buoyancy-driven convection heat transfer within vertical cavities containing nanofluids subjected to different side wall temperatures and various aspect ratios is investigated. The computations are based on an iterative, finitevolume numerical procedure (SIMPLE) that incorporates the Boussinesq approximation to simulate the buoyancy term. With the base fluid being water, three different nanoparticles (Cu, TiO₂, and Al₂O₃) are considered as the nanofluids. This study has been carried out for the pertinent parameters in the following ranges: the Rayleigh number, Ra_f = 10⁵–10⁷ and the volumetric fraction of nanoparticle between 0 and 5 percent. The results are presented for different length-to-height ratios varying from 0.1 to 1.0. The comparisons show that the mean Nusselt numbers and velocity magnitudes increase with volume fraction for the whole range of the Rayleigh numbers. The predictions show a noticeable heat transfer enhancement compared to pure fluid. It is also found that the heat transfer enhancement utilizing nanofluid is more pronounced at low aspect ratios than high aspect ratios. Moreover, the results depict that the addition of nanoparticles to the pure fluid has more effects at lower Rayleigh numbers.

1. Introduction

Natural convection is frequently encountered in various engineering applications such as cooling systems for electronic devices [1, 2], chemical vapor deposition instruments (CVD) [3], furnace engineering [4], solar energy collectors and building energy systems [5], non-Newtonian chemical processes [6, 7], and domains affected by electromagnetic fields [8, 9]. Heat transfer enhancement in these systems is an essential topic from an energy saving perspective. An innovative technique to improve heat transfer is by suspending nanoscale particles in the base fluid. Fluids with nanoparticles suspended in them are called nanofluids. Nanofluids introduce a unique opportunity for realizing more effective heat removal in thermal-fluid systems. Adoption of these fluids in a variety of processes and industries is anticipated since materials with sizes of nanometers possess unique physical and chemical properties. To date, theoretical, numerical, and experimental studies on nanofluids including thermal conductivity modeling [10, 11], viscosity estimation [12, 13] and boiling heat transfer and natural convection [14, 15] have appeared. As a pioneer, Masuda et al. [16] reported on thermal conductivity improvement of dispersed ultrafine (nanosize) particles in liquids. Soon thereafter, Choi [17] was the first to coin the term “nanofluids” for this new class of fluids with superior thermal properties. Khanafer et al. [18] investigated heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids for a range of Grashof numbers and volume fractions. It was found that the heat transfer across the enclosure was found to increase with the volumetric fraction of the copper nanoparticles in water at any given Grashof number. Ho et al. [19] investigated the influences of uncertainties due to adopting various relations for the effective thermal conductivity and dynamic viscosity of alumina/water nanofluid on the heat transfer characteristics. It was found that the uncertainties associated with different formulas adopted for the effective thermal conductivity and dynamic viscosity of the nanofluid have a strong bearing on natural convection heat transfer characteristics in the enclosure. Nnanna [20] performed an experimental investigation on the heat transfer behavior of buoyancy-driven Al₂O₃-water nanofluid in an enclosure. It was found that natural convection heat transfer was enhanced at low volume fractions in the range of 0.002 ≤ ϕ ≤ 0.02. However, for volume fractions above 0.2, natural convective heat transfer was decreased due to reduction in the Rayleigh number caused by an increase in kinematic viscosity. Ho and Lin [21] performed an experimental investigation of natural convection heat transfer of Al₂O₃-water nanofluid in vertical square enclosures of three different sizes. It was found that the systematic heat transfer degradation for the nanofluids containing nanoparticles of ϕ≥0.02 over the entire range of the Rayleigh number. They also reported that for the nanofluid containing much lower particle fraction of 0.001, heat transfer enhancement of around 18% compared with that of water was found to arise in the largest enclosure at sufficiently high Rayleigh number. Abu-Nada et al. [22] investigated the influences of nanoparticle on natural convection heat transfer enhancement in horizontal annuli with various nanoparticles and volume fractions, reporting an enhancement of heat transfer. Wang and Mujumdar [23] covered fluid flow and heat transfer characteristics of nanofluids in forced and free convection flows and potential applications of nanofluids. Khodadadi and Hosseinizadeh [24] investigated nanoparticles within conventional phase change materials such as water. Their findings show that nanoparticle-enhanced phase change materials (NEPCM) have a great potential for demanding thermalenergy storage applications. Although heat transfer predictions for pure fluids in closed cavities have been widely studied in the past, there has been little attempt to report on cases of thin cavities using nanofluids. Wang et al. [25] investigated free convection heat transfer of water/Al₂O₃ nanofluids in horizontal and vertical rectangular enclosures. They reported that the ratio of heat transfer coefficient of nanofluids to that of base fluid decreased as the size of nanoparticles increases. Hwang et al. [26] presented the numerical solution of natural convection in a Al₂O₃-water mixture in a rectangular cavity heated from below. They used various models to obtain the effective thermal conductivity and viscosity. Their results show that the ratio of heat transfer coefficient of nanofluids to that of base fluid decreased as the average temperature of nanofluids was lowered. Karimipour et al. [27] numerically investigated the mixed convection of a water-copper nanofluid inside a rectangular cavity. They observed that when Reynolds number is less than one, heat transfer rate is much greater than when Reynolds number is more than one. Moreover, they found that increasing the volume fraction of the nanoparticles increases the heat transfer rate.

Oztop and Abu-Nada [28] simulated the natural convection flow in a rectangular cavity by adding a heater placed at the right-hand side of the cavity. Their findings show that the Cu-water mixture has a better heat transfer enhancement compared to Al₂O₃-water mixture. Jahanshahi et al. [29] presented the numerical simulation of free convection based on experimental measured conductivity in a square cavity using water/SiO2 nanofluid. They investigated the influences of uncertainties due to adopting various formulas for the effective thermal conductivity of silica-water nanofluid on the heat transfer characteristics. They reported that the increase of heat transfer due to experimental formula is more than numerical formula. Also their results have shown that the heat transfer due to numerical formula decreases with increase in volume fraction. Manca et al. [30] numerically investigated on laminar mixed convection in a water-Al₂O₃ nanofluid, flowing in a triangular cross-sectioned duct. They survey the effects of different values of Richardson number and nanoparticle volume fractions on the convective heat transfer of nanofluid. They found that the average convective heat transfer increases by increasing values of Richardson number and nanoparticle volume fraction. Özerinç et al. [31] numerically analyzed the laminar forced convection heat transfer with temperature-dependent thermal conductivity of nanofluids and thermal dispersion inside a straight circular tube. They applied some recent correlations based on a thermal dispersion model for both constant wall heat flux and constant wall temperature boundary conditions. The correlations are according to single-phase approach assumption. The results show that the single-phase assumption is an accurate way of heat transfer enhancement analysis of nanofluids in convective heat transfer. Celli [32] applied a nonhomogenous model for investigating the spatial distribution of the nanoparticles dispersed inside a square cavity subject to different side wall temperatures using nanofluid for natural convection flow. The Brownian motion and the thermophoresis are considered as the leading physical transport mechanisms for the nanoparticles. They reported that for low Rayleigh number nonhomogenous method is appropriate for description the nanofluid systems and for high Rayleigh numbers (Ra≥102) homogenous method becomes reliable.

The main aim of the present study is the investigation of natural convection heat transfer utilizing nanofluids in vertical cavities. Three different nanoparticles were selected to compare the heat transfer enhancement variations due to the change of nanoparticles. Three different Rayleigh numbers up to the limit of laminar flow regime have been studied in order to elucidate the effect of buoyancy terms. The investigation covered low aspect ratios where conduction heat transfer is marked, up to an aspect ratios close to unity where the convection heat transfer is dominant. A very fine mesh distribution has been used in order to obtain the benchmark solutions for all aspect ratios. The results have been validated with those available in the literature for both pure and nanobased fluids. Finally, the average and maximum Nusselt numbers, streamlines and temperature fields for different values of volume fraction, Rayleigh number, and aspect ratio are illustrated.

2. Problem Statement and Boundary Conditions

Consider a two-dimensional enclosure of height (H) and width (L) with impermeable walls that is filled with nanofluid as shown in Figure 1. The top and the bottom walls are assumed to be insulated, whereas the two vertical walls are maintained constant but with different temperatures. Gravity acts parallel to the active vertical walls pointing toward the bottom wall. The nanofluid is treated as an incompressible and Newtonian fluid. Thermophysical/transport properties of the nanofluid are assumed to be constant, whereas the density variation in the buoyancy force term is handled by the Boussinesq approximation.

Figure 1:

Schematic diagram of the physical model.

The pertinent thermophysical/transport properties are given in Table 1.

Table 1:

Thermophysical/transport properties of different phases.

Property	Fluid phase (water)	Solid phase (Al₂O₃)	Solid phase (Cu)	Solid phase (TiO₂)

$c_{p} (J / kg K)$	4179	765	383	686.2
$ρ (kg / m^{3})$	997.1	3970	8954	4250
$k (W / m K)$	0.6	36	400	8.954
β × 10⁻⁵ (K⁻¹)	21	0.63	0.167	0.9
α × 10⁻⁷ (m²/s)	1.44	118.536	1166.391	30.703
μ × 10⁻⁴ (kg/m's)	8.9	—	—	—

3. Governing Equations

Considering the nanofluid as a continuous media with thermal equilibrium between the base fluid and the solid nanoparticles, the governing equations are as follows.

Continuity [29]:

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 .

(1)

X-momentum equation [29]:

u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \frac{1}{ρ_{nf}} (- \frac{\partial p}{\partial x} + μ_{nf} \nabla^{2} u + {(ρ β)}_{nf} g_{x} (T - T_{C})) .

(2)

Y-momentum equation [29]:

u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = \frac{1}{ρ_{nf}} (- \frac{\partial p}{\partial y} + μ_{nf} \nabla^{2} v + {(ρ β)}_{nf} g_{y} (T - T_{C})) .

(3)

Energy equation [29]:

u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \frac{\partial}{\partial x} [\frac{(k_{(eff) stagnant} + k_{d})}{{(ρ c_{p})}_{nf}} \frac{\partial T}{\partial x}] + \frac{\partial}{\partial y} [\frac{(k_{(eff) stagnant} + k_{d})}{{(ρ c_{p})}_{nf}} \frac{\partial T}{\partial y}] .

(4)

The density of the nanofluid is given by [29]:

ρ_{nf} = (1 - ϕ) ρ_{f} + ϕ ρ_{s},

(5)

whereas the heat capacitance of the nanofluid and part of the Boussinesq term are [29]:

\begin{matrix} {(ρ c_{p})}_{nf} = (1 - ϕ) {(ρ c_{p})}_{f} + ϕ {(ρ c_{p})}_{s}, \\ {(ρ β)}_{nf} = (1 - ϕ) {(ρ β)}_{f} + ϕ {(ρ β)}_{s} \end{matrix}

(6)

with ϕ being the volume fraction of the solid particles and subscripts f, nf, and s standing for base fluid, nanofluid, and solid, respectively. The viscosity of the nanofluid containing a dilute suspension of small rigid spherical particles is given by [29]:

μ_{nf} = \frac{μ_{f}}{{(1 - ϕ)}^{2.5}},

(7)

whereas the thermal conductivity of the stagnant nanofluid is [29]:

\frac{k_{(eff) stagnant}}{k_{f}} = \frac{k_{s} + 2 k_{f} - 2 ϕ (k_{f} - k_{s})}{k_{s} + 2 k_{f} + ϕ (k_{f} - k_{s})} .

(8)

The effective thermal conductivity of the nanofluid is [29]:

k_{eff} = k_{(eff) stagnant} + k_{d}

(9)

and the thermal conductivity enhancement term due to thermal dispersion is given by [29]:

k_{d} = C {(ρ c_{p})}_{nf} \sqrt{u^{2} + v^{2}} ϕ d_{p} .

(10)

The empirically determined constant C is evaluated following the work of Wakao and Kaguei [34].

The boundary conditions are

u = v = \frac{\partial T}{\partial y} = 0 at 0 \leq x \leq H, y = 0, H,

(11a)

u = v = 0, T = T_{H} at x = 0, 0 \leq y \leq H,

(11b)

u = v = 0, T = T_{C} at x = L, 0 \leq y \leq H .

(11c)

In order to write the nondimensional form of the governing equations, we have to introduce a number of references quantities. For example, dimensionless coordinates are defined as

\begin{matrix} X^{*} = \frac{x}{L}, \\ y^{*} = \frac{x}{L}, \end{matrix}

(12)

where L is the dimension of the chosen enclosure (Figure 1). Similarly, fluid properties and the flow variables can be nondimensionalized with respect to the reference quantities. This consideration yields [29]

\begin{matrix} ρ^{*} = \frac{ρ}{ρ_{f, \circ}}, μ^{*} = \frac{μ}{μ_{f, \circ}}, U^{*} = \frac{u}{U_{\circ}}, V^{*} = \frac{v}{U_{\circ}}, \\ κ^{*} = \frac{κ}{κ_{f, \circ}}, p^{*} = \frac{p}{ρ_{f, \circ} U_{\circ}^{2}}, T^{*} = \frac{T - T_{C}}{T_{H} - T_{C}} . \end{matrix}

(13)

The reference velocity is defined as [29]

U_{\circ} = \frac{α_{\circ}}{L}, α_{\circ} = \frac{k_{(eff) stagnant}}{ρ_{f, \circ} c_{p_{f}, \circ}} .

(14)

Using the above nondimensional variables/parameters, the nondimensional form of the governing equations can be written as

\begin{matrix} u^{*} \frac{\partial u^{*}}{\partial x^{*}} + v^{*} \frac{\partial u^{*}}{\partial y^{*}} - \frac{1}{ρ_{nf}^{*}} μ_{nf}^{*} \nabla^{2} u^{*} = \frac{1}{ρ_{nf}^{*}} (- \frac{\partial P^{*}}{\partial x^{*}}), \\ u^{*} \frac{\partial v^{*}}{\partial x^{*}} + v^{*} \frac{\partial v^{*}}{\partial y^{*}} - \frac{1}{ρ_{nf}^{*}} μ_{nf}^{*} \nabla^{2} v^{*} = \frac{1}{ρ_{nf}^{*}} (- \frac{\partial P^{*}}{\partial x^{*}}) + \frac{β_{nf}}{β_{f}} PrRa T^{*}, \\ u^{*} \frac{\partial T^{*}}{\partial x^{*}} + v^{*} \frac{\partial T^{*}}{\partial y^{*}} = \frac{\partial}{\partial x^{*}} [\frac{(k_{(eff) stagnant}^{*} + k_{d}^{*})}{{(ρ^{*} c_{p}^{*})}_{nf}} \frac{\partial T^{*}}{\partial x^{*}}] + \frac{\partial}{\partial y^{*}} [\frac{(k_{(eff) stagnant}^{*} + k_{d}^{*})}{{(ρ^{*} c_{p}^{*})}_{nf}} \frac{\partial T^{*}}{\partial y^{*}}] . \end{matrix}

(15)

The dimensionless groupings, that is, the Rayleigh and Prandtl numbers and the Boussinesq term, are given by

\begin{matrix} Pr = \frac{μ_{f} c_{p_{f}}}{k_{f}}, Ra = \frac{ρ_{f}^{2} β_{f} g L^{3} c_{p_{f}}}{μ_{f} k_{f}}, \\ β_{nf} = \frac{1}{ρ_{nf}} (1 - ϕ) {(ρ β)}_{f} + ϕ {(ρ β)}_{s} . \end{matrix}

(16)

The variations of the Nusselt numbers along vertical walls and averaged Nusselt number are derived as follows [29]:

\begin{matrix} N u_{y} = k_{eff}^{*} {\frac{\partial T^{*}}{\partial x^{*}} |}_{x^{*} = 0, y^{*}}, \\ N u_{avg} = \frac{1}{H} \int_{0}^{H} N u_{y} d y^{*} . \end{matrix}

(17)

Results of a grid independency check for the average Nusselt number are given in Table 2 and Figure 2. Since the difference between the numerical results for grid densities 201 × 201 and 301 × 301 at Ra = 10⁵ and 10⁶ and grid densities 301 × 301 and 401 × 401 at Ra = 10⁷ is very small, 201 × 201 and 301 × 301 were deemed adequate for the L/H = 1 case. Basically, these grid densities produce square cells inside the cavity. As for the other aspect ratios, the grid resolution in the x-direction was properly reduced to maintain the same square grid resolution inside the domain.

Table 2:

Values of the average Nusselt number for different mesh densities.

Grid	101 × 101	201 × 201	301 × 301	401 × 401

Ra = 10⁵	4.928	4.916	4.915	—
Ra = 10⁶	9.762	9.689	9.680	—
Ra = 10⁷	—	18.372	18.295	18.299

Figure 2:

Nusselt number distributions of Al₂O₃-water on the hot and cold walls for different grid densities (ϕ = 0.05, L/H = 1).

Finally, the following criterion was invoked to assure convergence of the solution:

Error = | \frac{2 (N u_{avg, H} - N u_{avg, C})}{N u_{avg, H} + N u_{avg, C}} | \leq 1 0^{- 5} .

(18)

This definition guarantees a balance between total energy in and out of the cavity.

4. Results and Discussion

The nanofluid in the cavity is chosen as suspension of Al₂O₃, Cu, and TiO₂ in the water. The thermophysical/transport properties of the base fluid and nanoparticles are given in Table 1. Many test cases were investigated at different moderate-to-high Rayleigh numbers of 10⁵, 10⁶, and 10⁷, six length-to-height aspect ratios of L/H = 0.1, 0.2, 0.25, 0.5, 0.75, and 1.0, and three volume fractions ϕ = 0.0, 0.025 and 0.05. The computations are based on an iterative, finitevolume numerical procedure (SIMPLE) using a staggered grid arrangement in combination with the QUICK differencing scheme. The present code for nanofluid has been validated for a water/SiO₂ nanofluid mixture [29] in a square cavity. Also, the predicted maximum axial and vertical velocities and the average and maximum Nusselt numbers are compared with those of de Vahl Davis [33] in Table 3. Moreover, the present numerical code was also validated against the results of Khanafer et al. [18] for natural convection in an enclosure filled with Cu-water nanofluid at Gr = 10⁴ and 10⁵, Pr = 6.2 and ϕ = 0.05, as shown in Figure 3. Based on these comparisons of the present results to well-established prior work, the accuracy of the computer code was validated.

Table 3:

The validation of the current results in a square cavity.

	U_max (% diff)	V_max (% diff)	Nu_max (% diff)	Nu_avg (% diff)

Ra = 10⁵
de Vahl Davis [33]	34.730	70.440	7.812	4.519
Present	34.980 (0.72)	68.581 (0.013)	7.739 (0.29)	4.529 (0.22)

Ra = 10⁶
de Vahl Davis [33]	64.630	219.360	17.925	8.800
Present	65.09 (0.60)	220.89 (0.69)	17.610 (1.76)	8.853 (0.60)

Figure 3:

Comparison of the temperature variation on the axial midline between the present and numerical results by Khanafer et al. [18] (Pr = 6.2, ϕ = 0.1).

Figure 4 illustrates comparison of the streamline patterns between the case of Al₂O₃-water nanofluid at ϕ = 0.05 and base water fluid for three Rayleigh numbers. The results show that regardless of the Rayleigh number and the type of fluid, streamlines are characterized by a recirculatory pattern for all the L/H ratios. The heated fluid next to the left wall rises vertically, thus replacing the fluid that travels horizontally toward the cold wall on the right side, and then sinks along the right wall. For a given L/H ratio, as the Rayleigh number is increased, the intensity of convection increases as evidenced by packing of the streamlines. The streamlines at L/H = 1 show that the central vortex breaks up into two, three, and four vortexes in the case of a base fluid at Ra = 10⁵, 10⁶, and 10⁷, respectively. In the case of a nanofluid at the same streamfunction, the central vortex does not break up at the three Rayleigh numbers. The streamlines also show that the central vortex of a nanofluid occupies a larger zone than that for pure fluid at higher aspect ratios and is smaller at lower aspect ratios. As a result of these discussions, presence of nanoparticles enhances convection heat transfer at higher aspect ratios and enhances conduction heat transfer at lower aspect ratios compared to corresponding cases with a base fluid.

Figure 4:

Comparison of streamlines for water (- - - -) and Al₂O₃-water nanofluid (—) for L/H = 1.0, 0.75, 0.5, 0.2, at 0.1 and various Rayleigh numbers (ϕ = 0.05).

Figure 5 illustrates comparison of isotherm contours between the nanofluid at ϕ = 0.05 and base fluid for various Rayleigh numbers. For the highest aspect ratio being the case of a square cavity, the isotherms at the center of the cavity are nearly horizontal and become vertical within the thermal boundary layers next to the vertical active walls. However, the isotherms are vertical within the central part of the thinnest cavity; that is, L/H = 0.1. This indicates that the role of conduction is dominant in the central region of the thin cavities. However, convection remains important at the top and bottom ends of the thin cavities. The results also show that the isotherms of nanofluid are vertical more than base fluid at whole aspect ratios. This indicates that the effect of conduction in nanofluid is more than base fluid.

Figure 5:

Comparison of the isotherms contours between Al₂O₃-water nanofluids (—) and base fluid (- - - -) for L/H = 1.0, 0.75, 0.5, 0.2, and 0.1 at various Rayleigh numbers (ϕ = 0.05).

Figure 6 presents a comparison of the values of dimensionless U_max determined at x = 0.5 against the L/H aspect ratio for the base fluid and nanofluid at ϕ = 0.05 for various Rayleigh numbers. For the base fluid, the magnitudes of U_max⁡,bf for L/H = 1 are 35.634, 80.832, and 181.52 for Ra = 10⁵, 10⁶, and 10⁷, respectively. To make the figure more informative, the magnitude of U_max for each Rayleigh number is nondimensionalized with the magnitude of U_max for the base fluid (U_max⁡,bf) for L/H = 1 and at the same Rayleigh number; that is, U_max^* = U_max/(U_max⁡,bf for L/H = 1). The data presented in Figure 6 show that the magnitude of U_max^* increases with increasing of the L/H ratio for different Rayleigh numbers and subsequently decreases with increasing of the L/H ratio. As it can be seen, when the Rayleigh number increases, the peaks of U_max^* distributions shift from middle L/H ratio to lower L/H ratios. The figure indicates that U_max^* becomes relatively independent of the L/H ratio for L/H>0.6 and Ra = 10⁶ and 10⁷. As it can be seen, in a nanoparticle suspension both of viscous terms and buoyant terms growth more compare to base fluid. The first one slows down the fluid flow and the second one speeds it up. At lower aspect ratios the first parameter is more dominant and for this we see the maximum velocity decreases when the nanoparticles are used. At higher aspect ratios, the maximum velocity increases because the buoyant terms are dominated.

Figure 6:

Comparison of U_max for Al₂O₃-water nanofluids and base fluid in terms of L/H ratio at various Rayleigh numbers (ϕ = 0.05).

Similarly, Figure 7 demonstrates the comparison of V_max determined at y = 0.5 in terms of the L/H ratio between the base fluid and nanofluid with ϕ = 0.05 for three different Rayleigh numbers. To make the figure more informative, the magnitude of V_max at each Rayleigh number is nondimensionalized with the magnitude of V_max for base fluid (V_max⁡,bf) at L/H = 1 at the same Rayleigh number; that is, V_max^* = V_max/(V_max⁡,bf at L/H = 1). The magnitudes of V_max⁡,bf for the case L/H = 1 are 73.825, 236.09, and 737.7 for Ra = 10⁵, 10⁶, and 10⁷, respectively. The figure shows that the magnitudes of V_max⁡,nf and V_max⁡,bf initially increase and then level off, becoming almost invariant at higher aspect ratios. The present results predict very low magnitude for V_max at lower aspect ratios for Ra = 10⁵ and 10⁶. The results also show that the magnitudes of V_max⁡,bf at each aspect ratio is more than V_max⁡,nf. This may happen by an increase in kinematic viscosity of nanofluid compared to base fluid.

Figure 7:

Comparison of V_max for Al₂O₃-water nanofluids and base fluid in terms of L/H ratio at various Rayleigh numbers (ϕ = 0.05).

Figure 8 presents the variation of Nu_max^* in terms of L/H ratio for different Rayleigh numbers and various volume fractions. The magnitudes of Nu_max at each Rayleigh number are nondimensionalized with the magnitude of Nu_max⁡,bf at L/H = 1 at the same Rayleigh number; that is, Nu_max^* = Nu_max/(Nu_max⁡,bf at L/H = 1). The magnitudes of Nu_max⁡,bf at L/H = 1 are 8.495, 19.696, and 45.163 for Ra = 10⁵, 10⁶, and 10⁷, respectively. For given geometries, the maximum values of Nusselt number are observed to increase consistently as the volume fraction of the nanoparticles is increased.

Figure 8:

Variation of Nu_max in terms of L/H ratio for Al₂O₃-water nanofluids for various volume fractions at (a) Ra = 10⁵, (b) Ra = 10⁶, and (c) Ra = 10⁷.

The variations of Nu_max with the L/H ratio at different Rayleigh numbers for the base fluid and nanofluid at ϕ = 0.05 are compared in Figure 9. As is observed, the maximum Nusselt number becomes more sensitive to L/H ratio when it decreases for whole range of Rayleigh numbers. The details show that the values of the maximum Nusselt number of nanofluid increase remarkably as compared with the base fluid at three Rayleigh numbers except from L/H = 0.25 to L/H = 0.4 at Ra = 10⁵ where values of the maximum Nusselt number of nanofluid decrease as compared with those of base fluid. The details show that the values of maximum Nusselt number at Ra = 10⁵ increase after their initial decrease and finally decrease with increasing of the aspect ratio. But the values of maximum Nusselt number at Ra = 10⁶ decrease after their initial increase with increasing of the aspect ratio, and finally for Ra = 10⁷, we see a fine decrease in whole aspect ratios.

Figure 9:

Comparison between the variation of Nu_max for Al₂O₃-water nanofluids and base fluid in terms of L/H ratio at various Rayleigh numbers (ϕ = 0.05).

Figure 10 shows the variation of Nu_avg in terms of L/H ratio at different Rayleigh numbers and various volume fractions. The magnitudes of Nu_avg at each Rayleigh number are nondimensionalized with the magnitude of Nu_{avg, bf} at L/H = 1 at the same Rayleigh number; that is, Nu_avg^* = Nu_avg/(Nu_{avg, bf} at L/H = 1). The magnitudes of Nu_{avg, bf} at L/H = 1 are 4.725, 9.247, and 17.386 for Ra = 10⁵, 10⁶, and 10⁷, respectively. As is observed, the values of average Nusselt number increase with increase in volume fraction. When the volume fraction increases, random movement of nanoparticles increases the thermal dispersion in the flow of nanofluid and consequently enhances the heat transfer rates in the enclosure. Also the values of Nu_avg decrease with increasing of the aspect ratio. The results show that the maximum and minimum values of Nu_avg at each volume fraction for Ra = 10⁵ and 10⁶ are seen at L/H = 0.1 and L/H = 1 and for Ra = 10⁷ at L/H = 0.2 and L/H = 0.1, respectively. Moreover, the results depict that more increase in Rayleigh number occurred at lower Rayleigh number. It shows that the addition of nanoparticles to the pure fluid has more effects at lower Rayleigh numbers.

Figure 10:

Variation of Nu_avg in terms of L/H ratio for Al₂O₃-water nanofluids at different volume fractions and (a) Ra = 10⁵, (b) Ra = 10⁶, and (c) Ra = 10⁷.

Figure 11 displays the Nusselt number distributions on the hot and cold walls for different aspect ratios (L/H = 1.0, 0.75, 0.5, 0.25, 0.2, and 0.1) and different volume fractions at Ra = 10⁶. As is observed, the figures are entirely symmetric in all six parts of the vertical cavities at each volume fraction. The results show that the value of the Nusselt number increases with increase in volume fraction.

Figure 11:

Nusselt number distributions on the hot and cold walls of a wide range of vertical cavities with different L/H ratios at various volume fractions (Ra = 10⁶).

Figure 12 shows the variation of Nu_avg^* in terms of L/H ratio using three different nanoparticles and Rayleigh numbers at volume fraction equal to 0.05 (ϕ = 0.05). The magnitudes of Nu_avg at each Rayleigh number are nondimensionalized with the magnitude of Nu_{avg, bf} at L/H = 1 at the same Rayleigh number; that is, Nu_avg^* = Nu_avg/(Nu_{avg, bf} at L/H = 1). As is observed, the lowest Nu_avg was obtained for TiO₂ due to domination of conduction mode of heat transfer since TiO₂ has the lowest value of thermal conductivity compared to Cu and Al₂O₃. However, the thermal conductivity of Al₂O₃ is approximately one-tenth of Cu, as given in Table 1, the values of Nu_avg for Al₂O₃ and Cu are close to each other specially at lowest Rayleigh number. However, a unique property of Al₂O₃ is its low thermal diffusivity compared with Cu. The reduced value of thermal diffusivity leads to higher temperature gradients and, therefore, higher enhancements in heat transfer. The Cu nanoparticles have high values of thermal diffusivity and, therefore, this reduces temperature gradients which will affect the performance of Cu nanoparticles.

Figure 12:

Variation of average Nusselt numbers in terms of aspect ratios for different types of nanoparticles: (a) Ra = 10⁵, (b) Ra = 10⁶, and (c) Ra = 10⁷. (ϕ = 0.05).

5. Conclusion

Heat transfer enhancement in a wide range of thin-to-thick vertical cavities subject to different side wall temperatures using nanofluid is studied numerically. The results are presented at different Rayleigh numbers, a wide range of vertical cavity aspect ratios, different volume fractions, and different types of nanoparticles. The present results illustrate that the suspended nanoparticles substantially increase the heat transfer rate at any given Rayleigh number and aspect ratio. In addition, the results illustrate that the average and maximum Nusselt number increase with an increase in volume fraction of nanoparticles. As is observed, the most enhancement of average Nusselt number is seen at Ra = 10⁵ and L/H = 0.1 at any volume fraction and also the results show that the average Nusselt number increases with decrease of aspect ratio except from L/H = 0.1 to L/H = 0.2 for Ra = 10⁷. The type of nanofluid is a key factor for heat transfer enhancement. The results illustrate that the highest values of Nusselt number are obtained when using Cu nanoparticles.

Footnotes

Nomenclature

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors wish to thank the support and encouragement of their good friend Dr. Seyed Farid Hosseinizadeh, God bless his soul.

References

Florio

L. A.

and Harnoy

, “Combination technique for improving natural convection cooling in electronics,” International Journal of Thermal Sciences, vol. 46, no. 1, pp. 76–92, 2007.

Alipanah

Hasannasab

Hosseinizadeh

S. F.

, and Darbandi

, “Entropy generation for compressible natural convection with high gradient temperature in a square cavity,” International Communications in Heat and Mass Transfer, vol. 37, no. 9, pp. 1388–1395, 2010.

Iwanik

P. O.

and Chiu

W. K. S.

, “Temperature distribution of an optical fiber traversing through a chemical vapor deposition reactor,” Numerical Heat Transfer A, vol. 43, no. 3, pp. 221–237, 2003.

Lim

K. O.

Lee

K. S.

, and Song

T. H.

, “Primary and secondary instabilities in a glass-melting surface,” Numerical Heat Transfer A, vol. 36, no. 3, pp. 309–325, 1999.

Singh

and Sharif

M. A. R.

, “Mixed convective cooling of a rectangular cavity with inlet and exit openings on differentially heated side walls,” Numerical Heat Transfer A, vol. 44, no. 3, pp. 233–253, 2003.

Patil

P. M.

and Kulkarni

P. S.

, “Effects of chemical reaction on free convective flow of a polar fluid through a porous medium in the presence of internal heat generation,” International Journal of Thermal Sciences, vol. 47, no. 8, pp. 1043–1054, 2008.

Lamsaadi

Naimi

Hasnaoui

, and Mamou

, “Natural convection in a vertical rectangular cavity filled with a non-newtonian power law fluid and subjected to a horizontal temperature gradient,” Numerical Heat Transfer A, vol. 49, no. 10, pp. 969–990, 2006.

Saha

L. K.

Hossain

M. A.

, and Gorla

R. S. R.

, “Effect of Hall current on the MHD laminar natural convection flow from a vertical permeable flat plate with uniform surface temperature,” International Journal of Thermal Sciences, vol. 46, no. 8, pp. 790–801, 2007.

Yan

Y. Y.

Zhang

H. B.

, and Hull

J. B.

, “Numerical modeling of electrohydrodynamic (EHD) effect on natural convection in an enclosure,” Numerical Heat Transfer A, vol. 46, no. 5, pp. 453–471, 2004.

10.

Maxwell

J. C. A.

, Treatise on Electricity and Magnetism, vol. 1, Clarendon Press, Oxford, UK, 2nd edition, 1881.

11.

Hamilton

R. L.

and Crosser

O. K.

, “Thermal conductivity of heterogeneous two-component systems,” Industrial and Engineering Chemistry Fundamentals, vol. 1, no. 3, pp. 187–191, 1962.

12.

Einstein

, “Eine neue Bestimmung der Molekuldimensionen,” Annalen der Physik, vol. 19, pp. 289–306, 1906.

13.

Brinkman

H. C.

, “The viscosity of concentrated suspensions and solutions,” The Journal of Chemical Physics, vol. 20, no. 4, p. 571, 1952.

14.

Xuan

and Li

, “Heat transfer enhancement of nanofluids,” International Journal of Heat and Fluid Flow, vol. 21, no. 1, pp. 58–64, 2000.

15.

Bang

I. C.

and Chang

S. H.

, “Boiling heat transfer performance and phenomena of Al₂O₃-water nano-fluids from a plain surface in a pool,” International Journal of Heat and Mass Transfer, vol. 48, no. 12, pp. 2407–2419, 2005.

16.

Masuda

Ebata

Teramae

, and Hishinuma

, “Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles,” Netsu Bussei, vol. 7, pp. 227–233, 1993.

17.

Choi

U. S.

, “Enhancing thermal conductivity of fluids with nanoparticles,” in Developments and Application of Non-Newtonian Flows, pp. 99–105, ASME, 1995.

18.

Khanafer

Vafai

, and Lightstone

, “Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids,” International Journal of Heat and Mass Transfer, vol. 46, no. 19, pp. 3639–3653, 2003.

19.

C. J.

Chen

M. W.

, and Li

Z. W.

, “Numerical simulation of natural convection of nanofluid in a square enclosure: effects due to uncertainties of viscosity and thermal conductivity,” International Journal of Heat and Mass Transfer, vol. 51, no. 17–18, pp. 4506–4516, 2008.

20.

Nnanna

A. G. A.

, “Experimental model of temperature-driven nanofluid,” Journal of Heat Transfer, vol. 129, no. 6, pp. 697–704, 2007.

21.

C. J.

and Lin

C. C.

, “Experiments on natural convection heat transfer of a nanofluid in a square enclosure,” in Proceedings of the 13th International Heat Transfer Conference, Sydney, Australia, 2006.

22.

Abu-Nada

Masoud

, and Hijazi

, “Natural convection heat transfer enhancement in horizontal concentric annuli using nanofluids,” International Communications in Heat and Mass Transfer, vol. 35, no. 5, pp. 657–665, 2008.

23.

Wang

X. Q.

and Mujumdar

A. S.

, “Heat transfer characteristics of nanofluids: a review,” International Journal of Thermal Sciences, vol. 46, no. 1, pp. 1–19, 2007.

24.

Khodadadi

J. M.

and Hosseinizadeh

S. F.

, “Nanoparticle-enhanced phase change materials (NEPCM) with great potential for improved thermal energy storage,” International Communications in Heat and Mass Transfer, vol. 34, no. 5, pp. 534–543, 2007.

25.

Wang

X. Q.

Mujumdar

A. S.

, and Yap

, “Free convection heat transfer in horizontal and vertical rectangular cavities filled with nanofluids,” in Proceedings of the International Heat Transfer Conference, Sydney, Australia, 2006.

26.

Hwang

K. S.

Lee

J. H.

, and Jang

S. P.

, “Buoyancy-driven heat transfer of water-based Al₂O₃ nanofluids in a rectangular cavity,” International Journal of Heat and Mass Transfer, vol. 50, no. 19–20, pp. 4003–4010, 2007.

27.

Karimipour

Nezhad

A. H.

Behzadmehr

Alikhani

, and Abedini

, “Periodic mixed convection of a nanofluid in a cavity with top lid sinusoidal motion,” Proceedings of the Institution of Mechanical Engineers C, vol. 225, no. 9, pp. 2149–2160, 2011.

28.

Oztop

H. F.

and Abu-Nada

, “Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids,” International Journal of Heat and Fluid Flow, vol. 29, no. 5, pp. 1326–1336, 2008.

29.

Jahanshahi

Hosseinizadeh

S. F.

Alipanah

Dehghani

, and Vakilinejad

G. R.

, “Numerical simulation of free convection based on experimental measured conductivity in a square cavity using water/SiO₂ nanofluid,” International Communications in Heat and Mass Transfer, vol. 37, no. 6, pp. 687–694, 2010.

30.

Manca

Nardini

Ricci

, and Tamburrino

, “Numerical investigation on mixed convection in triangular cross-section ducts with nanofluids,” Advances in Mechanical Engineering, vol. 2012, Article ID 139370, 13 pages, 2012.

31.

Özerinç Yazicioĝlu

A. G.

, and Kakaç

, “Numerical analysis of laminar forced convection with temperature-dependent thermal conductivity of nanofluids and thermal dispersion,” International Journal of Thermal Sciences, vol. 62, pp. 138–148, 2012.

32.

Celli

, “Non-homogeneous model for a side heated square cavity filled with a nanofluid,” International Journal of Heat and Fluid Flow, vol. 44, pp. 327–335, 2013.

33.

de Vahl Davis

, “Natural convection of air in square cavity: a benchmark solution,” International Journal for Numerical Methods in Fluids, vol. 3, no. 3, pp. 249–264, 1983.

34.

Wakao

and Kaguei

, Heat and Mass Transfer in Packed Beds, Gordon and Breach Science Publishers, New York, NY, USA, 1982.

Numerical Study of Natural Convection in Vertical Enclosures Utilizing Nanofluid

Abstract

1. Introduction

2. Problem Statement and Boundary Conditions

3. Governing Equations

4. Results and Discussion

5. Conclusion

Footnotes

Nomenclature

Conflict of Interests

Acknowledgment

References