Transp Porous Med (2015) 106:595–610 DOI 10.1007/s11242-014-0415-3

Free Convection in a Square Cavity Filled with a Porous Medium Saturated by Nanofluid Using Tiwari and Das’ Nanofluid Model M. A. Sheremet · T. Grosan · I. Pop

Received: 22 August 2014 / Accepted: 5 November 2014 / Published online: 19 November 2014 © Springer Science+Business Media Dordrecht 2014

Abstract Free convection in a square differentially heated porous cavity filled with a nanofluid is numerically investigated. The mathematical model has been formulated in dimensionless stream function and temperature taking into account the Darcy–Boussinesq approximation. The Tiwari and Das’ nanofluid model with new more realistic empirical correlations for the physical properties of the nanofluids has been used for numerical analysis. The governing equations have been solved numerically on the basis of a second-order accurate finite difference method. The developed algorithm has been validated by direct comparisons with previously published papers and the results have been found to be in good agreement. The results have been presented in terms of the streamlines, isotherms, local, and average Nusselt numbers at left vertical wall at a wide range of key parameters. Keywords Free convection · Square cavity · Porous media · Nanofluids · Numerical method

M. A. Sheremet Department of Theoretical Mechanics, Faculty of Mechanics and Mathematics, Tomsk State University, 634050 Tomsk, Russia e-mail: [email protected] M. A. Sheremet Institute of Power Engineering, Tomsk Polytechnic University, 634050 Tomsk, Russia T. Grosan · I. Pop (B) Department of Applied Mathematics, Babe¸s-Bolyai University, 400084 Cluj-Napoca, Romania e-mail: [email protected] T. Grosan e-mail: [email protected]

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1 Introduction Conventional heat transfer fluids such as water, oil, and ethylene glycol have low thermal conductivity, which is a primary limitation in enhancing the performance and the compactness of many engineering electronic devices. Therefore, there is a strong need to develop advanced heat transfer fluids with substantially higher conductivities. With recently introduced the term of nanofluids by Choi (1995), which are the fluids with suspended solid particles of higher thermal conductivity such as metals within it, the need mentioned before has been overcome. The nanofluids have many applications in the industry since materials of nanometer size have unique physical and chemical properties. Eastman et al. (2001) point out that nanofluids posses a substantially larger thermal conductivity than that of conventional ones. Keblinski et al. (2002) investigated for possible mechanisms for this anomalous increase: Brownian motion of the nanoparticles, molecular level layering of the liquid at the liquid-particle interface, the nature of heat transfer in the nanoparticles, and nanoparticle clustering. Keblinski et al. (2002) also showed that the Brownian motion is a negligible contributor, and that liquid layering around the nanoparticles could cause rapid conduction. However, it seems that there are no solid enough theories to predict the thermal conductivity of nanofluids. In some cases, the observed enhancement in thermal conductivity of nanofluids is orders of magnitude larger than predicted by well-established theories. Other results in this rapidly evolving field of nanofluids include a surprisingly strong temperature dependence of the thermal conductivity (Patel et al. 2003). Suspensions of metal nanoparticles are also being developed for other purposes, such as medical applications including cancer therapy. Yu and Choi (2003) proposed an alternative expression to predict the thermal conductivity of solid–liquid mixtures, mentioning that a structural model of nanofluids might consist of a bulk liquid, solid nanoparticles, and solid-like nanolayers. These authors have compared the results of their model for ratio of the nanolayer thickness to the original particle radius of 0.1 with existing experimental results in literature and obtained a reasonably good agreement. The prediction is effective, most particularly when the nanoparticles have a diameter of less than 10 nm. Convective heat transfer of nanofluids in cavities has been extensively studied by many researchers in recent years. Khanafer et al. (2003) investigated the heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids for various pertinent parameters and taking into account the solid particle dispersion. The effect of suspended ultrafine metallic nanoparticles on the fluid flow and heat transfer processes is analyzed and effective thermal conductivity enhancement maps are developed for various controlling parameters. It is shown that the variances within different models have substantial effects on the results. A heat transfer correlation of the average Nusselt number for various Grashof numbers and volume fractions is also presented. Jou and Tzeng (2006) numerically investigated the heat transfer performance of nanofluids inside the two-dimensional rectangular enclosures and found that increasing the volume fraction causes a significant increase in average heat transfer rate. Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids has been studied by Tiwari and Das (2007) using a simpler nanofluid model than that proposed by Khanafer et al. (2003). Santra et al. (2008) have conducted a similar kind of study for nanoparticle volume fraction parameter ϕ up to ϕ = 10 % using the thermal conductivity models proposed by Maxwell-Garnet (1904) and Bruggeman (1935). The results show that the Bruggeman model (1935) predicts higher heat transfer rates than the Maxwell-Garnett model (1904). Oztop and Abu-Nada (2008) investigated heat transfer and fluid flow in a partially heated enclosure and found that the heat transfer enhancement due to using a nanofluid is more pronounced at a low aspect ratio than at a high aspect ratio. Other

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studies are by Abu-Nada and Oztop (2009) on the application of nanofluids for heat transfer enhancement within an inclined two-dimensional enclosure. Ghasemi and Aminossadati (2010), studied the natural convection in a right triangular enclosure, with a heat source on its vertical wall and filled with a water-CuO nanofluid. The results show that when Brownian motion is considered in the analysis, the solid volume fraction, the heat source location and the enclosure aspect ratio affect the heat transfer performance differently at low and high Rayleigh numbers. They have proposed an optimum value for the solid volume fraction, which results in the maximum heat transfer rate. We mention also the papers by Ögüt (2009, 2010) who has studied the heat transfer of water-based nanofluids with natural convection in an inclined square enclosure. Finally, we point out the very interesting published paper by Celli (2013), where a non-homogeneous model for a side heated square cavity filled with a nanofluid has been numerically studied. The thermophysical properties of the nanofluid are assumed to be functions of the average volume fraction of nanoparticles dispersed inside the cavity. The definitions of the dimensionless governing parameters (Rayleigh number, Prandtl number, and Lewis number) are exactly the same as for the clear fluids. It has been found that the distribution of the nanoparticles shows a particular sensitivity to the low Rayleigh numbers. The average Nusselt number at the vertical walls is sensitive to the average volume fraction of the nanoparticles dispersed inside the cavity and it is also sensitive to the definition of the thermophysical properties of the nanofluid. Highly viscous base fluids lead to a critical behavior of the model when the simulation is performed in pure conduction regime. According to Godson et al. (2010), one of the main objectives of using nanofluids is to achieve the best thermal properties with the least possible (ϕ < 1 %) volume fraction of nanoparticles in the base fluid. Porous media heat transfer problems have several engineering applications such as geothermal energy recovery, crude oil extraction, ground water pollution, thermal energy storage, flow through filtering media, etc. Representative studies in this area may be found in the books by Nield and Bejan (2013), Pop and Ingham (2001), Ingham and Pop (2005), Vafai (2005, 2010), Vadasz (2008), Jansen (2013), Bagchi and Kulacki (2014), etc. There are many studies on the mechanism behind the enhanced heat transfer characteristics using nanofluids. The collection of papers on this topic is included in the books by Das et al. (2007) and Nield and Bejan (2013), and in the review papers by Kakaç and Pramuanjaroenkij (2009), Wong and Leon (2010), Godson et al. (2010), Saidur et al. (2011), Wen et al. (2011), Mahian et al. (2013), and many others. Above literature shows that the free convection in a square cavity filled with a porous medium saturated by a nanofluid using the mathematical nanofluid models proposed by Tiwari and Das (2007), and Celli (2013) has not been investigated yet. Therefore, this study aims to look at free convection inside a square porous enclosure suggesting new more realistic empirical correlations for the physical properties of the nanofluids (heat capacitance, thermal conductivity and thermal diffusivity). The solid matrix of the porous medium is aluminum foam and glass balls. Particular efforts have been focused on the effects of the Rayleigh number, the solid volume fraction parameter, and the porosity of the porous medium on flow field, temperature distribution, Nusselt number, streamlines, and isotherms. To our best of knowledge this problem has not been studied before.

2 Basic Equations Consider the steady free convection in a two-dimensional porous square cavity filled with a water-based nanofluid. It is assumed that nanoparticles are suspended in the nanofluid using

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Fig. 1 Physical model and coordinate system

either surfactant or surface charge technology. This prevents nanoparticles from agglomeration and deposition on the porous matrix (Kuznetsov and Nield 2010, 2013; Nield and Kuznetsov 2009, 2014). A schematic geometry of the problem under investigation is shown in Fig. 1, where x¯ and y¯ are the Cartesian coordinates and L is the height of the cavity. It is assumed that the left vertical wall is heated and maintained at the constant temperature Th , while the right vertical wall is cooled and has the constant temperature Tc . The horizontal walls are adiabatic (∂ T /∂ y¯ ) = 0, where T is the fluid temperature. The Darcy–Boussinesq approximation is employed. Homogeneity and local thermal equilibrium in the porous medium is assumed. Under these assumptions and using the model of the nanofluid proposed by Tiwari and Das (2007), the equations governing this problem are (Nield and Bejan 2013) ∇ ·V = 0

μmnf 0 = −∇ p − V − (ρβ)nf (T − Tc ) g K  2  kmnf ∂ T ∂2T  + , (V · ∇) T =  ∂ y¯ 2 ρCp nf ∂ x¯ 2

(1) (2) (3)

where V is the Darcian velocity vector, K is the permeability of the porous medium, g is the gravitational acceleration vector, p is the pressure, ρ is the density, β is the thermal expansion coefficient, μ is the dynamic viscosity, k is the thermal conductivity, and Cp is the specific heat at a constant pressure.   The physical properties of the nanofluid: viscosity μnf , heat capacitance ρCp nf , thermal conductivity knf , and buoyancy coefficient (ρβ)nf are given by (Oztop and Abu-Nada 2008) μf , (ρCp )nf = (1 − ϕ)(ρC p )f + ϕ(ρCp )p , μnf = (1 − ϕ)2.5 kp + 2kf − 2ϕ(kf − kp ) knf = (4) , (ρβ)nf = (1 − ϕ)(ρβ)f + ϕ(ρβ)p , kf kp + 2kf + ϕ(kf − kp ) where ϕ is the uniform concentration of the nanoparticles in the cavity and indices “nf”, “f”, and “p” refer to nanofluid, fluid, and (nano) particle, respectively. The dynamic viscosity of the nanofluid μnf can be approximated as viscosity of a base fluid μf containing dilute suspension of fine spherical particles and is given by Brinkman (1952). It is worth mentioning

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Table 1 Thermal-physical properties of fluid and Cooper (Cu) nanoparticles (Oztop and Abu-Nada 2008) and solid structure of the porous medium Physical properties

Base fluid (water)

Cu

Aluminum foam

Glass balls

C p (J · kg−1 · K−1 )

4179

385

897

840

ρ(kg · m−3 )

997.1

8933

2700

2700

k (W · m−1 · K−1 )

0.613

400

205

1.05

α × 10−7 (m2 · s−1 )

1.47

1163.1

846.4

4.63

β × 10−5 (K−1 )

21

1.67

2.22

0.9

that the expressions (4) are restricted to spherical nanoparticles, where it does not account for other shapes of nanoparticles. The thermophysical properties of the nanofluid and solid structure of the porous medium are given in Table 1. These thermophysical properties of the nanofluids were also used by Khanafer et al. (2003), Oztop and Abu-Nada (2008), Abu-Nada and Oztop (2009), and Muthtamilselvan et al. (2010). Further, it should be noticed that for kp  kf it results in the following limitation knf ≈ kf

1 + 2ϕ 1 − ϕ

(5)

which limits the present study to the value of knf given by this relation. On the other hand, we have (see Nield and Bejan 2013) (ρCp )m = ε(ρCp )f + (1 − ε)(ρCp )s , km = εkf + (1 − ε)ks ,

(6)

and using the relations (4) and (6), the physical properties (heat capacitance, thermal conductivity and thermal diffusivity) of the nanofluid saturated porous medium are given by   (ρCp )f − (ρCp )p , (ρCp )mnf = ε(ρCp )nf + (1 − ε)(ρCp )s = (ρCp )m 1 − εϕ (ρCp )m   3εϕkf (kf − kp )

, kmnf = εknf + (1 − ε)ks = km 1 − km kp + 2kf + ϕ(kf − kp ) αmnf =

kmnf , (ρCp )nf

(7)

where ε is the porosity of the porous medium and indices “mnf”, “s”, and “m” are related to nanofluid saturated porous medium, solid matrix of the porous medium, and clear fluid saturated porous medium. The enhancement of the thermal conductivity given in eq. (7) is in agreement with the enhancement reported by Yu et al. (2008) for a common porous medium material saturated by a water base nanofluid. Equations (1)–(3) can be written in Cartesian coordinates as ∂ u¯ ∂ v¯ + =0 ∂ x¯ ∂ y¯   ∂T ∂ v¯ μmnf ∂ u¯ − = −g (ρβ)nf K ∂ y¯ ∂ x¯ ∂ x¯  2  ∂T ∂ T ∂2T ∂T + v¯ = αmnf + u¯ ∂ x¯ ∂ y¯ ∂ x¯ 2 ∂ y¯ 2

(8) (9) (10)

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Further, we introduce the following dimensionless variables ¯ x = x/L, ¯ y = y¯ /L, u = u¯ L/αmnf , v = vL/α mnf , θ = (T − T0 )/(Th − Tc ), (11) where T0 = (Th + Tc )/2 is the mean temperature of heated and cooled walls. One can introduce a dimensionless stream function ψ defined by u=

∂ψ ∂ψ , v=− ∂y ∂x

(12)

so that Eq. (8) is satisfied identically. We are then left with the following equations ∂ 2ψ ∂θ ∂ 2ψ + = −Ra · H (ϕ) 2 ∂x ∂ y2 ∂x

(13)

∂ψ ∂θ ∂ 2θ ∂ 2θ ∂ψ ∂θ − = + ∂y ∂x ∂x ∂y ∂x2 ∂ y2

(14)

with the boundary conditions ψ = 0, θ = 0.5 on x = 0 ψ = 0, θ = −0.5 on x = 1 ∂θ ψ = 0, = 0 on y = 0 and y = 1 ∂y

(15)

Here Ra = gK (ρβ)f (Th − Tc ) L/(αm μf ) is the Rayleigh number for the porous medium and the function H (ϕ) is given by H (ϕ) =



      1 − ϕ + ϕ ρ β p /(ρ β)f 1 − ϕ + ϕ ρ C p p / ρ C p f 1 −

3εϕkf (kf − kp ) km [kp + 2kf + ϕ(kf − kp )]

(1 − ϕ)2.5 (16)

and it depends on the nanoparticles concentration ϕ and physical properties of the fluid, nanoparticles, and solid structure of the porous medium. It should be noticed that for ϕ = 0, that is H (ϕ) = 1 (regular fluid), Eqs. (13) and (14) reduce to those of Walker and Homsy (1978), Bejan (1979), Beckermann et al. (1986), Gross et al. (1986), Moya et al. (1987), Manole and Lage (1992), and Baytas and Pop (1999). The physical quantities of interest are the local Nusselt numbers N u l and N u r , which are defined as     kmnf ∂θ kmnf ∂θ , N ur = − (17) N ul = − kf ∂ x x=0 kf ∂ x x=1 and the average Nusselt numbers N u l and N u r , which are given by 1 N ul = 0

123

1 N u l dy, N u r =

N u r dy. 0

(18)

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Table 2 Comparison of the average Nusselt number of the hot wall Authors

Ra 10

100

1000

10000

Walker and Homsy (1978)

−

3.097

12.96

51.0

Bejan (1979)

−

4.2

15.8

50.8

Beckermann et al. (1986)

−

3.113

−

48.9

Gross et al. (1986)

−

3.141

13.448

42.583

Moya et al. (1987)

1.065

2.801

−

−

Manole and Lage (1992)

−

3.118

13.637

48.117

Baytas and Pop (1999)

1.079

3.16

14.06

48.33

Present results

1.079

3.115

13.667

48.823

Table 3 Comparison of the average Nusselt number in a porous triangular cavity filled with a nanofluid Ra

ϕ

Nu Sun and Pop (2011)

Chamkha and Ismael (2013)

Present

500

0

9.66

9.52

9.65

500

0.1

9.42

9.44

9.41

1000

0

13.9

13.6

14.05

1000

0.2

12.85

12.82

12.84

3 Numerical Method The partial differential Eqs. (13) and (14) with corresponding boundary conditions (15) were solved using the finite difference method with the second-order central differencing schemes (see Aleshkova and Sheremet 2010; Sheremet and Trifonova 2013; Sheremet et al. 2014; Sheremet and Pop 2014a, b). The solution for the corresponding linear algebraic equations was obtained through the successive over relaxation method. Optimum value of the relaxation parameter was chosen on the basis of computing experiments. The computation is terminated when the residuals for the stream function get bellow 10−10 . The present models, in the form of an in-house computational fluid dynamics (CFD) code, have been validated successfully against the works of Walker and Homsy (1978), Bejan (1979), Beckermann et al. (1986), Gross et al. (1986), Moya et al. (1987), Manole and Lage (1992), and Baytas and Pop (1999) for steady-state natural convection in a square porous cavity with isothermal vertical and adiabatic horizontal walls. Table 2 shows the values of the average Nusselt number computed for various Rayleigh numbers in the range 10 to 104 in comparison with other authors.The numerical methodology was coded in C++, and to check its validity, a comparison with selective data from the published literature was carried out. The performance of the nanofluid part of the model was tested against the results of Sun and Pop (2011) for steady-state free convection in a triangular enclosure filled with a nanofluid-saturated porous medium with heater length of 0.8 times of the vertical wall for Rayleigh numbers 500 and 1000. Table 3 shows the values of the average Nusselt number computed for various Rayleigh numbers and the solid volume fraction in comparison with other authors. For the purpose of obtaining grid-independent solution, a grid sensitivity analysis is performed. The grid-independent solution was performed by preparing the solution for

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Table 4 Variations of the average Nusselt number of the heat wall with the non-uniform grid Non-uniform grids

N ul

=

100 × 100

923.495

1.11

150 × 150

916.532

0.36

200 × 200

914.23

0.11

250 × 250

913.24

−

400 × 400

912.327

0.1

    N u li× j −N u l250×250  N u li× j

× 100 %

steady-state free convection in a square porous cavity filled with a Cu-water nanofluid at Ra = 1000, ϕ = 0.05, ε = 0.8, solid matrix of the porous medium is the aluminum foam. Five cases of non-uniform grid are tested: a grid of 100 × 100 points ( xmin = ymin = 0.00087 and xmax = ymax = 0.038), a grid of 150 × 150 points ( xmin = ymin = 0.00022 and xmax = ymax = 0.033), a grid of 200 × 200 points ( xmin = ymin = 0.00019 and xmax = ymax = 0.024), a grid of 250 × 250 points ( xmin = ymin = 0.00015 and xmax = ymax = 0.0194), and a much finer grid of 400 × 400 points ( xmin = ymin = 0.00004 and xmax = ymax = 0.0146). Table 4 shows an effect of the mesh on the average Nusselt number of the hot wall. On the basis of the conducted verifications the non-uniform grid of 250 × 250 points has been selected for the following analysis.

4 Results and Discussion Numerical investigations of the boundary value problem (13)–(15) has been carried out at the following values of key parameters: Rayleigh number (Ra = 10 − 1000), the solid volume fraction parameter of nanoparticles (ϕ = 0.0 − 0.05), the porosity of the porous medium (ε = 0.1 − 0.8), and the solid matrix of the porous medium (aluminum foam and glass balls). Particular efforts have been focused on the effects of these parameters on flow field, temperature distribution, and Nusselt number. Rayleigh number is a very important parameter that has effects on heat transfer within a porous medium. Figure 2 illustrates streamlines and isotherms for Cu-water nanofluid under different values of the Rayleigh number at ϕ = 0.05, ε = 0.5 and different materials for the solid matrix of the porous medium. Solid lines correspond to the aluminum foam as the solid matrix of the porous medium and dashed lines correspond to the glass balls as the solid matrix of the porous medium. Regardless of the Rayleigh number value and the material for the solid porous medium matrix a single circulation flow is formed in the clockwise direction inside the cavity. The main reason for an appearance of this circulation is an effect of the horizontal temperature difference. When the Rayleigh number increases, the flow convection is strengthened, the flow cell is extended along the horizontal axis, and the boundary layers become more significant. The former can be confirmed by maximum absolute values of the Ra=10 Ra=100 Ra=1000 stream function as the following |ψ|max = 0.71 < |ψ|max = 4.63 < |ψ|max = 20.2 in case of the aluminum foam as the solid matrix of the porous medium. Further it can be seen that for large Rayleigh numbers (Ra = 1000) there is a stratification of the flow and temperature. It is worth noting here that a decrease in the thermal conductivity of the solid matrix material leads to an attenuation of the convective heat transfer inside the cavity.

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Fig. 2 Streamlines ψ and isotherms θ for ϕ = 0.05, ε = 0.5: Ra = 10—a, Ra = 100—b, Ra = 1000—c (solid lines the aluminum foam as the solid matrix of the porous medium, dashed lines the glass balls as the solid matrix of the porous medium)

It is well known that in case of natural convection all changes in hydrodynamic structures are related to the modification of the temperature field. At Ra = 10 the dominated heat transfer mechanism is a heat conduction that defines weak heat transfer along the horizontal coordinate owing to an interaction of the left vertical temperature source and the right vertical temperature sink. An increase in Ra leads to a formation of ascending thermal wave from the heat source and descending thermal wave from the heat sink. The results in Fig. 3 present the distribution of local Nusselt number N u l along the left vertical wall for different values of Ra. An increase in the Raleigh number from 10 to 1000 leads to an increase in N u l . It should be noted that more intensive increment in N u l occurs close to the bottom wall of the cavity owing to an effect of the cold temperature wave from the right heat sink. This temperature wave deforms the thermal boundary layer which is extended along the vertical direction. As a result temperature gradient increases close to the bottom wall due to small thickness of the thermal boundary layer. A decrease in the thermal conductivity of the solid matrix material leads to an essential decrease in the local Nusselt number (Fig. 3b). Porosity of porous medium is another feature that has an influence on the heat transfer within the cavity. Fig. 4 demonstrates streamlines and isotherms under different values of the porosity at Ra = 1000, ϕ = 0.05 and different materials for the solid matrix of the porous medium. An increase in ε leads to insignificant changes in the convective flow intensity for high thermal conductivity material of solid matrix (solid lines for aluminum foam in Fig. 4). Such behavior can be confirmed by maximum absolute values of the stream ε=0.5 ε=0.8 function as following |ψ|ε=0.3 max = 20.21 > |ψ|max = 20.2 > |ψ|max = 20.19 in case of the aluminum foam as the solid matrix of the porous medium. At the same time in case of the glass balls as the solid matrix of the porous medium an increase in the porosity leads

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Fig. 3 Variation of the local Nusselt number with the Rayleigh number for ϕ = 0.05, ε = 0.5: the aluminum foam as the solid matrix of the porous medium—a, the glass balls as the solid matrix of the porous medium—b

Fig. 4 Streamlines ψ and isotherms θ for Ra = 1000, ϕ = 0.05: ε = 0.3—a, ε = 0.5—b, ε = 0.8—c (solid lines the aluminum foam as the solid matrix of the porous medium; dashed lines the glass balls as the solid matrix of the porous medium)

to essential attenuation of the convective flow inside the porous cavity in the following way ε=0.5 ε=0.8 |ψ|ε=0.3 max = 19.85 > |ψ|max = 19.56 > |ψ|max = 19.02. An effect of the porosity on the local Nusselt number N u l along the left vertical wall is presented in Fig. 5. An increase in ε from 0.3 to 0.8 leads to a decrease in the local Nusselt number. It should be noted that more essential reduction of N u l occurs in case of the high thermal conductivity material of solid matrix (Fig. 5a). Therefore, an increase in porosity leads to a decrease in the average Nusselt number regardless of the solid volume fraction

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Fig. 5 Variation of the local Nusselt number with the porosity of porous medium for Ra = 1000, ϕ = 0.05: the aluminum foam as the solid matrix of the porous medium—a, the glass balls as the solid matrix of the porous medium—b

Fig. 6 Dependence of average Nusselt numbers on porosity of porous medium ε for different solid volume fraction parameter ϕ for the aluminum foam as the solid matrix of the porous medium: Ra = 100—a, Ra = 1000—b

parameter (Figs. 6, 7). It should be noted that in case of the aluminum foam as the solid matrix of the porous medium an increase in ε leads to a decrease in an influence of the solid volume fraction parameter (Fig. 6). But in case of the glass balls this abovementioned effect is essential (Fig. 7). Solid volume fraction parameter ϕ is a key factor to study how nanoparticles affect the heat transfer of nanofluids. Fig. 8 presents streamlines and isotherms under different values of ϕ at Ra = 1000, ε = 0.5 and different materials for the solid matrix of the porous medium. An increase in ϕ leads to an attenuation of the convective flow inside the porous cavity regardless of the solid matrix material. At the same time changes in the solid volume fraction parameter ϕ do not lead to temperature differences. Therefore, an increase in the concentration of nanoparticles inside the cavity allows to conserve the main convective cell with upward flows along the left hot wall and downward flows close to the right cold wall.

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Fig. 7 Dependence of average Nusselt numbers on porosity of porous medium ε for different solid volume fraction parameter ϕ for the glass balls as the solid matrix of the porous medium: Ra = 100—a, Ra = 1000—b

Fig. 8 Streamlines ψ and isotherms θ for Ra = 1000, ε = 0.5: ϕ = 0.04—a, ϕ = 0.05—b (solid lines – the aluminum foam as the solid matrix of the porous medium; dashed lines – the glass balls as the solid matrix of the porous medium)

Fig. 9 demonstrates a decrease in the average Nusselt number with solid volume fraction parameter for the aluminum foam as the solid matrix of the porous medium regardless of the porosity of porous medium. It should be noted here that regardless of the Rayleigh number

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Fig. 9 Dependences of average Nusselt numbers on solid volume fraction parameter ϕ for different porosities of porous medium ε for the aluminum foam as the solid matrix of the porous medium: Ra = 10—a, Ra = 100— b, Ra = 1000—c

Fig. 10 Dependence of average Nusselt numbers on solid volume fraction parameter ϕ for different porosities of porous medium ε for the glass balls as the solid matrix of the porous medium: Ra = 10—a, Ra = 100— b, Ra =1000—c

values a decrease in N u l is insignificant with an increase in ϕ. On the basis of these figures it is possible to conclude that in case of the high thermal conductivity material of solid matrix (like aluminum foam) an insertion of nanoparticles inside the base fluid do not lead to essential changes in heat and mass transfer rate regardless of Ra and ε. Combined effect of the porosity and solid volume fraction parameter on the average Nusselt number for glass balls as the solid matrix of the porous medium is presented in Figs. 7 and 10. It is worth noting here that an increase in ϕ leads to a decrease in the average Nusselt number at ε ≤ 0.4 for Ra = 100 and at ε ≤ 0.3 for Ra = 1000 (Fig. 7). So an increase in the Rayleigh number leads to a decrease in a range for the porosity, where one can find a reduction of convective heat transfer rate with ϕ. A significant increase in the average Nusselt number at high values of ε with the solid volume fraction parameter occurs at high Rayleigh numbers (Figs. 7, 10c). On the basis of Fig. 10 it is possible to conclude that more essential effect of nanoparticles insertion inside the base fluid is in case of small Rayleigh values and high porosity, e.g. at ε = 0.8 and Ra = 10 an increase in ϕ from 0.0 to 0.05 leads to an increase in N u l up to 9.27 % while at ε = 0.8 and Ra = 1000 an increase in ϕ from 0.0 to 0.05 leads to an increase in the average Nusselt number at left hot wall up to 2.97 %.

5 Conclusions Natural convection in a square porous cavity with isothermal vertical walls and adiabatic horizontal ones filled with a water-based nanofluid has been studied. Particular efforts have been focused on the effects of the Rayleigh number, solid volume fraction parameter of

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nanoparticles, porosity of the porous medium and solid matrix of the porous medium on flow field, temperature distribution, and Nusselt number. It has been found that the average Nusselt number is an increasing function of the Rayleigh number and a decreasing function of the porosity of porous medium. An increase in the Rayleigh number for glass balls as the solid matrix of the porous medium leads to a decrease in a range for the porosity, where one can find a reduction of convective heat transfer with ϕ. It has been also shown that a significant increase in the average Nusselt number at high values of ε with the solid volume fraction parameter occurs at high Rayleigh numbers for glass balls as the solid matrix of the porous medium. A decrease in the thermal conductivity of the solid matrix material leads to an attenuation of the convective heat transfer inside the cavity. It has been found that more essential reduction of the average Nusselt number with increase in ε occurs for the high thermal conductivity material of solid matrix. Acknowledgments This work of M.A. Sheremet was conducted as a government task of the Ministry of Education and Science of the Russian Federation, Project Number 13.1919.2014/K.

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