Eur. Phys. J. Plus (2017) 132: 509 DOI 10.1140/epjp/i2017-11769-0

THE EUROPEAN PHYSICAL JOURNAL PLUS

Regular Article

Natural convection in a differentially heated enclosure having two adherent porous blocks saturated with a nanofluid Marina S. Astanina1 , Mikhail A. Sheremet1,2 , Hakan F. Oztop3,4,a , and Nidal Abu-Hamdeh4 1 2 3 4

Laboratory on Convective Heat and Mass Transfer, Tomsk State University, 634050, Tomsk, Russia Department of Nuclear and Thermal Power Plants, Tomsk Polytechnic University, 634050, Tomsk, Russia Department of Mechanical Engineering, Technology Faculty, Fırat University, Elazig, Turkey Department of Mechanical Engineering, King Abdulaziz University, Jeddah, Saudi Arabia Received: 9 October 2017 / Revised: 29 October 2017 c Societ` Published online: 7 December 2017 –  a Italiana di Fisica / Springer-Verlag 2017 Abstract. Laminar natural convection in a square cavity having two centered adherent porous blocks filled with an alumina/water nanofluid under the effect of horizontal temperature gradient is studied numerically. Each porous block has the unique values of the porosity and permeability. Water-based nanofluids with alumina nanoparticles are chosen for investigation. The control characteristics of this study are the Darcy number of the first porous block (10−7 ≤ Da1 ≤ 10−3 ), the dimensionless porous blocks size (0.1 ≤ δ ≤ 0.4) and nanoparticles volume fraction (0 ≤ φ ≤ 0.04). The developed computational code has been validated comprehensively using the grid independency test and experimental data of other authors. The obtained results revealed the heat transfer enhancement at the hot wall with the Darcy number, while a growth of the porous layers size reduces the heat transfer rate at this hot wall. The behavior of the average Nusselt number at the right cold wall is opposite.

Nomenclature Roman letters c Specific heat [J · kg−1 · K−1 ] Da Darcy number [–] g Gravitational acceleration vector [m · s−2 ] h Height of porous blocks [m] H1 , H2 , H3 , H4 , H5 , H6 Special functions [–] K1 Permeability of the porous block I [m2 ] K2 Permeability of the porous block II [m2 ] L Length and height of the cavity [m] l1 Length of porous block I [m] l2 Length of porous block II [m] Nu Local Nusselt number [–] Nu Average Nusselt number [–] p¯ Dimensional pressure [Pa] Pr Prandtl number [–] Ra Rayleigh number [–] T Dimensional temperature [K] t Dimensional time [s] Tc Right wall temperature [K] Th Left wall temperature [K] u, v Dimensionless velocity components [–] u ¯, v¯ Dimensional velocity components [m · s−1 ] x, y Dimensionless Cartesian coordinates [–] x ¯, y¯ Dimensional Cartesian coordinates [m] Greek symbols β Thermal expansion coefficient [K−1 ] a

e-mail: [email protected] (corresponding author)

γ Dimensionless height of the porous blocks [–] δ Dimensionless length of the porous blocks [–] 1 Porosity of the porous block I [–] 2 Porosity of the porous block II [–] η1 Heat capacitance ratio for the porous block I [–] η2 Heat capacitance ratio for the porous block II [–] θ Dimensionless temperature [–] λ Thermal conductivity [W · m−1 · K−1 ] μ Dynamic viscosity [Pa · s] ρ Density [kg · m−3 ] ρc Heat capacitance [J · m−3 · K−1 ] ρβ Buoyancy coefficient [kg · m−3 · K−1 ] τ Dimensionless time [–] φ Nanoparticles volume fraction [–] ψ Dimensionless stream function [–] ω Dimensionless vorticity [–] Subscripts c Cold f Fluid h Hot l Left max Maximum value mnf Porous medium saturated with nanofluid nf Nanofluid p (nano)particle r Right s Solid matrix of porous block

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Fig. 1. Physical model.

1 Introduction Porous baffles or blocks can help control natural convection or forced convection in cavities or ducts [1–9]. They have control parameters such as permeability and porosity in their construction. Thus, heat transfer, temperature distribution and flow field can be controlled via these passive elements. It should be noted that porous baffles or porous fins can be found in various engineering applications, such as heat exchangers, solar collectors, sediment retention ponds, electronic cooling devices, thermal regenerators, internal cooling systems of gas turbine blades, etc. [1–14]. Shuja et al. [9] solved a numerical problem on the flow inside the cavity with porous blocks under the influence of blocks sizes and porosity. They used the control volume approach to solve the governing equations. They observed that the presence of porous blocks results in the widening of the high-temperature zone over these blocks due to the natural convection and the fluid flow penetration in the porous medium. In another paper [15], they applied the heat flux as boundary condition to the same geometry. Guerroudj and Kahalerras [16] studied the mixed convection in a channel in the presence of heated porous blocks with various shapes. They found that optimal selection of control parameters can lead to heat transfer enhancement with a moderate growth of pressure drop. Huang et al. [17] investigated the pulsating convective cooling problem over two porous-covering heated blocks. They observed that the periodic modification of the recirculation flow structure in the inter-block area and behind the downstream block essentially amplifies the heat transfer rate at the block surfaces. Khanafer et al. [18] made a numerical study on free convection in an enclosure with a thin porous fin located on the hot wall of the cavity. They showed that the average Nusselt number is lower than the same in the case of without fins for very low Darcy numbers. Idan and Feldman [19] used porous blocks to control the heat transfer in constructions. Hassanien et al. [20] solved the transient flow and heat transfer problem in the presence of three-dimensional body embedded in a porous medium. Kiwan [21] studied a simple method for the effect of porous fins on natural convection using the Darcy model. He found that the higher heat transfer performance is obtained via porous fins. The number of works on the natural convective heat transfer in nanofluid filled closed spaces have been rapidly increasing after Choi coined the term “nano fluid” [22]. Also, the application of nanofluid flow and heat transfer in porous media is studied in the literature by Kasaeian et al. [23], Shekholeslami [24,25], EL-Dabe et al. [26], Zeeshan et al. [27], Sheremet et al. [28]. Furthermore, Sheikholeslami and Seyednezhad [29] made a computational work by using the lattice Boltzmann method for a CuO-water nanofluid flow in a porous enclosure with a hot obstacle. The main aim of this study is to explore the convective flow and heat transfer of an alumina-water nanofluid in a differentially heated cavity with a centered composite porous block. To the authors’ knowledge and the above literature, there is no work on such analysis. Thus, the obtained results will help engineers and contribute to understand the heat transfer and nanofluid flow of the studied system.

2 Basic equations Free convection in a square enclosure of length L filled with an alumina-water nanofluid having two centered porous blocks of different permeability and porosity has been investigated. Figure 1 shows the considered cavity with the boundary conditions. The enclosure includes two porous blocks of height h and lengths l1 and l2 in the central part

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of the cavity. The left wall (¯ x = 0) is kept at temperature Th , the right wall (¯ x = L) is maintained at temperature y = 0, y¯ = L) are adiabatic. The base fluid is water including solid spherical Tc (Th > Tc ), while the horizontal walls (¯ alumina nanoparticles and their physical properties can be found in [30,31]. Thermal equilibrium between water and nanoparticles is assumed. The Brinkman model is applied inside the porous blocks [32,33]. The partial differential equations based on the previous assumptions will be formulated as follows. – For the nanofluid [34]: v ∂u ¯ ∂¯ + = 0, ∂x ¯ ∂ y¯    2  ∂u ¯ ∂u ¯ ∂ p¯ ∂u ¯ ∂ u ¯ ∂2u ¯ +u ¯ + v¯ =− + μnf , ρnf + ∂t ∂x ¯ ∂ y¯ ∂x ¯ ∂x ¯2 ∂ y¯2    2  ∂¯ v ∂¯ v ∂ p¯ ∂¯ v ∂ v¯ ∂ 2 v¯ +u ¯ + v¯ =− + μnf ρnf + (ρβ)nf g(T − Tc ), + ∂t ∂x ¯ ∂ y¯ ∂ y¯ ∂x ¯2 ∂ y¯2  2  ∂T ∂T λnf ∂ T ∂2T ∂T +u ¯ + v¯ = . + ∂t ∂x ¯ ∂ y¯ (ρc)nf ∂ x ¯2 ∂ y¯2

(1) (2) (3) (4)

– For the porous block I [32,33]: v ∂u ¯ ∂¯ + = 0, ∂x ¯ ∂ y¯     u ¯ ∂u v¯ ∂ u ∂ p¯ μnf ∂ 2 u 1 ∂u μnf ¯ ¯ ¯ ¯ ∂2u ¯ + 2 + 2 =− + − + u ¯, ρnf ε1 ∂t ε1 ∂ x ¯ ε1 ∂ y¯ ∂x ¯ ε1 ∂x ¯2 ∂ y¯2 K1     u ¯ ∂¯ v¯ ∂¯ ∂ p¯ μnf ∂ 2 v¯ ∂ 2 v¯ 1 ∂¯ μnf v v v + 2 + 2 =− + + (ρβ)nf g(T − Tc ) − + v¯, ρnf ε1 ∂t ε1 ∂ x ¯ ε1 ∂ y¯ ∂ y¯ ε1 ∂x ¯2 ∂ y¯2 K1   ∂T ∂T λmnf 1 ∂ 2 T ∂2T ∂T +u ¯ + v¯ = . + η1 ∂t ∂x ¯ ∂ y¯ (ρc)nf ∂ x ¯2 ∂ y¯2

(5) (6) (7) (8)

– For the porous block II [32,33]: v ∂u ¯ ∂¯ + = 0, ∂x ¯ ∂ y¯     u ¯ ∂u v¯ ∂ u ∂ p¯ μnf ∂ 2 u 1 ∂u μnf ¯ ¯ ¯ ¯ ∂2u ¯ + 2 + =− + + 2 − u ¯, ρnf ε2 ∂t ε2 ∂ x ¯ ε22 ∂ y¯ ∂x ¯ ε2 ∂x ¯2 ∂ y¯ K2    2  u ¯ ∂¯ v¯ ∂¯ ∂ p¯ μnf ∂ v¯ ∂ 2 v¯ 1 ∂¯ μnf v v v + 2 + 2 =− + + 2 + (ρβ)nf g(T − Tc ) − v¯, ρnf ε2 ∂t ε2 ∂ x ¯ ε2 ∂ y¯ ∂ y¯ ε2 ∂x ¯2 ∂ y¯ K2  2  ∂T ∂T λmnf 2 ∂ T ∂2T ∂T +u ¯ + v¯ = . + η2 ∂t ∂x ¯ ∂ y¯ (ρc)nf ∂ x ¯2 ∂ y¯2

(9) (10) (11) (12)

Nanofluid characteristics can be used in the following form [30,31,35]: ρnf = (1 − φ)ρf + φρp ,

(ρβ)nf = (1 − φ)(ρβ)f + φ(ρβ)p ,

(ρc)nf = (1 − φ)(ρc)f + φ(ρc)p ,

while λnf = λf (1 + 2.944φ + 19.672φ2 ) and μnf = μf (1 + 4.93φ + 222.4φ2 ) for 1% ≤ φ ≤ 4% [35]. The presented governing equations (1)–(12) will be rewritten in dimensionless form exploiting the following parameters: x=x ¯/L, y = y¯/L,  τ = t gβ(Th − Tc )/L,

 u=u ¯/ gβ(Th − Tc )L,

 v = v¯/ gβ(Th − Tc )L,

θ = (T − Tc )/(Th − Tc )

and the stream function ψ (u = ∂ψ/∂y, v = −∂ψ/∂x) with vorticity ω = As a result, eqs. (1)–(12) will be presented in the following form.

(13) ∂v ∂x

−

∂u ∂y .

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– For the nanofluid: ∂2ψ ∂2ψ + = −ω, ∂x2 ∂y 2

(14)

   ∂θ P r ∂2ω ∂2ω ∂ω ∂ψ ∂ω ∂ψ ∂ω + − = H1 (φ) , + H2 (φ) + ∂τ ∂y ∂x ∂x ∂y Ra ∂x2 ∂y 2 ∂x  2  ∂ψ ∂θ ∂ψ ∂θ H3 (φ) ∂ θ ∂2θ ∂θ + − =√ . + ∂τ ∂y ∂x ∂x ∂y ∂y 2 Ra · P r ∂x2

(15) (16)

– For the porous block I: ∂2ψ ∂2ψ + = −ω, ∂x2 ∂y 2

ε1

(17)

   ∂θ ∂ω ∂ψ ∂ω ∂ψ ∂ω P r ∂2ω ∂2ω ω + − = ε1 H1 (φ) , + ε21 H2 (φ) + − ε 1 ∂τ ∂y ∂x ∂x ∂y Ra ∂x2 ∂y 2 Da1 ∂x   ∂ψ ∂θ ∂ψ ∂θ H4 (φ, ε1 , λs1 ) ∂ 2 θ ∂2θ ∂θ √ + − = + 2 . ∂τ ∂y ∂x ∂x ∂y ∂x2 ∂y Ra · P r

(18) (19)

– For the porous block II: ∂2ψ ∂2ψ + = −ω, ∂x2 ∂y 2

ε2

(20) 

  ∂θ ∂ω ∂ψ ∂ω ∂ψ ∂ω P r ∂2ω ∂2ω ω + − = ε2 H1 (φ) , + ε22 H2 (φ) + − ε 2 ∂τ ∂y ∂x ∂x ∂y Ra ∂x2 ∂y 2 Da2 ∂x   ∂ψ ∂θ ∂ψ ∂θ H4 (φ, ε2 , λs2 ) ∂ 2 θ ∂2θ ∂θ + − = √ . + ∂τ ∂y ∂x ∂x ∂y ∂x2 ∂y 2 Ra · P r

(21) (22)

The control conditions for eqs. (14)–(22) can be presented as τ =0:

ψ = ω = 0,

τ >0:

ψ = 0,

ψ = 0,

∂2ψ , ∂x2 ∂2ψ ω=− 2 , ∂y ω=−

ψ = 0, ⎧ ⎪ ⎨θnf = θporous-I ,

θ = 0.5; ω=−

∂2ψ , ∂x2

θ = 1 at x = 0;

θ = 0 at x = 1; ∂θ = 0 at y = 0 and y = 1; ∂y ⎧ ⎪ ⎨ψnf = ψporous-I ,

⎧ ⎪ ⎨ωnf = ωporous-I ,



at the interface between



porous block I ∂ψ ∂ω ∂θ ∂θ ∂ψ ∂ω ⎪ ⎩ nf = H5 (φ, ε1 , λs1 ) porous-I , ⎪ ⎩ nf = porous-I , ⎪ ⎩ nf = porous-I

and the nanofluid; ∂n ∂n ∂n ∂n ∂n ∂n ⎧ ⎧ ⎧

⎪ ⎪

at the interface between ⎪ ⎨ψnf = ψporous-II , ⎨ωnf = ωporous-II , ⎨θnf = θporous-II ,



∂θporous-II ∂θnf ∂ψnf ∂ψporous-II ∂ωnf ∂ωporous-II porous block II ⎪ ⎪ ⎪ ⎩ ⎩

and the nanofluid; ⎩ = H5 (φ, ε2 , λs2 ) , = , = ∂n ∂n ∂n ∂n ⎧ ∂n ⎧ ⎧ ∂n ⎪ ⎪ωporous-I = ωporous-II , ⎪ ⎨ψporous-I = ψporous-II , ⎨ ⎨θporous-I = θporous-II , at x = 0.5. ∂ψ ∂ω ∂θ ∂θ ∂ψ ∂ω ⎪ ⎩ porous-I = porous-II , ⎪ ⎩ porous-I = porous-II ⎩ porous-I = H6 (φ, ε, λs ) porous-II , ⎪ ∂x ∂x ∂x ∂x ∂x ∂x

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Table 1. Dependences of the average Nusselt numbers on the uniform grids. Δ=

|N uli×j − N ul200×200 | × 100% N uli×j

N ur

Δ=

|N ur i×j − N ur 200×200 | × 100% N ur i×j

Uniform grids

N ul

50 × 50

3.005

1.3

4.825

3.3

100 × 100

2.955

0.4

4.726

1.2

200 × 200

2.966

–

4.668

–

300 × 300

2.977

0.4

4.646

0.5

400 × 400

2.984

0.6

4.635

0.7

Here, P r = μf (ρc)f /(ρf λf ) is the Prandtl number, Ra = g(ρβ)f (ρc)f (Th − Tc )L3 /(μf λf ) is the Rayleigh number, and the subsidiary functions H1 (φ), H2 (φ), H3 (φ), H4 (φ, ε, λs ), H5 (φ, ε, λs ), and H6 (φ, ε, λs ) are given by H1 (φ) =

1 + 4.93φ + 222.4φ2 , 1 − φ + φρp /ρf

H2 (φ) =

1 − φ + φ(ρβ)p /(ρβ)f , 1 − φ + φρp /ρf

H3 (φ) =

1 + 2.944φ + 19.672φ2 , 1 − φ + φ(ρc)p /(ρc)f

ε(1 + 2.944φ + 19.672φ2 ) + (1 − ε)λs /λf λs /λf , H5 (φ, ε, λs ) = ε + (1 − ε) , 1 − φ + φ(ρc)p /(ρc)f 1 + 2.944φ + 19.672φ2 ε2 (1 + 2.944φ + 19.672φ2 ) + (1 − ε2 )λs2 /λf . H6 (φ, ε, λs ) = ε1 (1 + 2.944φ + 19.672φ2 ) + (1 − ε1 )λs1 /λf H4 (φ, ε, λs ) =

The integral defined parameters are the local Nusselt number N u along the vertical walls and the average Nusselt numbers N u at these walls



 1  1 λnf ∂θ

λnf ∂θ

· , N u = − · , N u = N u dy, N u = N ur dy. (23) N ul = − r l l r λf ∂x x=0 λf ∂x x=1 0 0

3 Numerical method and validation Equations (14)–(22) with control conditions were solved using an in-house computational code developed using the finite difference method of second-order accuracy [31,33,34,36]. The developed model and numerical method have been verified comprehensively using the numerical and experimental data of other authors [31,33,34,36]. The grid independency test has been performed for the present formulation (see fig. 1 and eqs. (14)–(22)) at Ra = 105 , P r = 6.82, Da1 = 10−3 , Da2 = 10−5 , 1 = 0.8, 2 = 0.2, φ = 0.03, l1 /L = l2 /L = δ = 0.2, h/L = γ = 0.4. During the analysis, the porous block I is the aluminum foam and porous block II is the glass balls. Five uniform grids are tested. Table 1 presents the influence of the grid elements on the average Nusselt numbers. Using the results presented in table 1, the uniform grid of 200 × 200 points has been chosen for the present study.

4 Results and discussion Numerical research has been implemented for the Rayleigh number (Ra = 105 ), the Prandtl number (P r = 6.82), the Darcy numbers (Da1 = 10−7 –10−3 , Da2 = 10−5 ), porosities (1 = 0.8, 2 = 0.2), dimensionless porous blocks lengths (δ = 0.1–0.4), dimensionless height of porous blocks (γ = 2δ), and nanoparticles volume fraction (φ = 0–0.04). The isolines of stream function and temperature as well as the local and average Nusselt numbers and nanofluid flow rate for the different control characteristics mentioned above are illustrated in figs. 2–9. Streamlines and isotherms for different values of the Darcy number Da1 , dimensionless porous blocks lengths δ and nanoparticles volume fraction φ are shown in figs. 2–5. In the case of δ = 0.1 (fig. 2), regardless of the Darcy number values, one can find the formation of one convective cell with two cores, as in the case of clear fluid for Ra = 105 [37,38]. The presence of centered porous blocks distorts the streamlines, namely, the left core is located near the porous block I, while the right core is located close to the porous block II. Isotherms are also distorted due to different thermal conductivity of the porous blocks. More essential heating occurs inside the porous block I, where we have the aluminum foam and less intensive heating is within the porous block II with the glass balls. Addition of aluminum nanoparticles leads to modification of convective cell cores. One can find that these cores displace to the porous blocks. At the same time, isotherms, in the case of a nanofluid, illustrate more vigorous heating of the left part of the cavity owing to the increasing effective thermal conductivity of the liquid. Moreover, the presence of the porous block I with high

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Fig. 2. Isolines of stream function ψ and temperature θ for δ = 0.1 and different Darcy numbers for porous block I: Da1 = 10−7 – a, Da1 = 10−5 – b, Da1 = 10−3 – c (solid lines for φ = 0 and dashed lines for φ = 0.04).

Fig. 3. Isolines of stream function ψ and temperature θ for δ = 0.2 and different Darcy numbers for porous block I: Da1 = 10−7 – a, Da1 = 10−5 – b, Da1 = 10−3 – c (solid lines for φ = 0 and dashed lines for φ = 0.04).

thermal conductivity allows to modify essentially the temperature field in the central part of the cavity. Variation of the Darcy number for the solid block I between 10−7 and 10−5 does not lead to essential modification of flow structure and temperature field. In the case of Da1 = 10−3 (fig. 2(c)), where we have more permeable material, streamlines penetrate inside this block and isotherm θ = 0.55 does not penetrate the porous blocks while, for lower values of the Darcy number, this penetration has been observed. An increase in the porous blocks length δ = 0.2 (fig. 3) reflects the beginning of dissipation of the left convective cell core, where this core becomes smaller. An effect of permeability of the porous block I is observed only in the case of Da1 = 10−3 (fig. 3(c)). At the same time, isotherms more essentially show a difference in the effective thermal conductivity of porous block I and porous block II, where, inside the porous block I, temperature changes rapidly. It is interesting to note that, for high values of the Darcy number, heating of the porous block I occurs not so intensively as for low permeability porous material. The main reason for such behavior is the more intensive circulation inside block I, in the case of high permeability, that leads to less intensive heating.

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Fig. 4. Isolines of stream function ψ and temperature θ for δ = 0.3 and different Darcy numbers for porous block I: Da1 = 10−7 – a, Da1 = 10−5 – b, Da1 = 10−3 – c (solid lines for φ = 0 and dashed lines for φ = 0.04).

Fig. 5. Isolines of stream function ψ and temperature θ for δ = 0.4 and different Darcy numbers for porous block I: Da1 = 10−7 – a, Da1 = 10−5 – b, Da1 = 10−3 – c (solid lines for φ = 0 and dashed lines for φ = 0.04).

Further growth of the porous blocks sizes (figs. 4 and 5) shows a disappearance of the left convective cell core, while the right one is located near the surface of porous block II. Disappearance of the left core and conservation of the right one are due to the reduction of the temperature differences between the left hot wall and the surrounding fluid. The latter can be explained by extension of porous block I with high effective thermal conductivity where the high temperature wave, from the left vertical wall, penetrates rapidly inside this block in the case of small gap between this block and walls. Such situation is clearly presented, for δ = 0.4, in fig. 5. At the same time, in the case of moderate gap (fig. 4) a rise in the Darcy number for porous block I leads to less intensive heating of this block. Such behavior has been explained above. The effect of aluminum nanoparticles, that is related with more essential heating of the cavity left part, has been described above and also in fig. 4, such influence can be observed.

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Fig. 6. Dependences of vertical velocity (a) and temperature (b) along y = 0.5 on the porous blocks size and Darcy number of the porous block I for φ = 0.02.

A detailed description of the effect of the considered governing parameters on velocity and temperature fields can be seen in fig. 6 for a nanofluid with φ = 0.02. Figure 6 shows vertical velocity (fig. 6(a)) and temperature (fig. 6(b)) profiles at middle cross-section, y = 0.5. First of all, one can find a formation of maximum vertical velocity on the left and right sides from the porous blocks and this huge velocity reduces with increase in the porous blocks sizes. Taking into account the above-mentioned features concerning a diminution of temperature difference between the left hot wall and the surrounding liquid, due to the decrease in the liquid gap with a growth of the porous blocks sizes, the maximum value of the vertical velocity in the left part is less than that in the right part. Moreover, for δ = 0.4, vertical velocity on the left side from the porous blocks is close to zero. It should be noted that the influence of the permeability of the porous block I is reflected on the left side from the porous block, where the increase in Da1 results in an increase in the vertical velocity while, on the right side from the porous block II, there are no changes in Da1 . The intensity of the motion is significantly lower inside the porous blocks in comparison with clear liquid parts. At the same time, temperature profiles illustrate the heating of the left part and cooling of the right part inside the cavity with respect to the porous blocks. A distortion of isotherms at internal interfaces between nanofluid and porous blocks and between porous blocks is due to the different effective thermal conductivity of these media. An increase in the porous blocks sizes leads to significant heating of the left cavity part with porous blocks. In the case of δ = 0.4, the temperature inside the left gap and porous block I is the same as in the hot vertical wall. All distributions shown in figs. 2–6 have been obtained for the steady state at τ = 200. This time moment can be defined as a steady state using the time behavior of the heat transfer rate and the nanofluid flow rate presented in fig. 7. Different values of the average Nusselt numbers at the left and right vertical walls can be explained by the effect of the porous blocks. It is seen that, for τ = 200, N ul , N ur and |ψ|max have constant values. An increase in the Darcy number of the porous block I leads to an increase in N ul and a reduction in N ur , while the nanofluid flow rate diminishes weakly. At the same time, a rise of the nanoparticles volume fraction leads to the heat transfer rate reduction at vertical walls and nanofluid flow rate decreases also. Profiles of the local Nusselt number along the left and right vertical walls are shown in fig. 8. Along the hot wall the local Nusselt number decreases due to the reduction of the temperature gradient with the height where the hot liquid is located while, along the right vertical wall, the local Nusselt number rises because the maximum value of the temperature gradient is located in the upper part of this wall. The increase in the porous blocks sizes results in a diminution of N ul while, along the right wall, this effect is non-monotonic. Especially this non-linear effect can be found for δ = 0.4. At the same time, an inclusion of the alumina nanoparticles illustrates a reduction of N ul and also non-monotonic variation of N ur . Figure 9 presents the dependences of the heat transfer rates and nanofluid flow rate on the Darcy number Da1 , porous block sizes δ and nanoparticle volume fraction φ. The influence of the Darcy number of the porous block I for small and large sizes of the porous blocks is non-significant, while for moderate values of this parameter, N ul increases, N ur decreases and |ψ|max decreases weakly. An increase in δ results in the diminution of the heat transfer rate at the hot wall and the heat transfer enhancement at the right wall, while the nanofluid flow rate decreases. Addition of nanoparticles leads to heat transfer rate reduction at left and right vertical walls for all values of δ, except for δ = 0.4, where one can find heat transfer enhancement for N ur with φ, while N ul does not change. At the same time, the nanofluid flow rate decreases with the nanoparticle volume fraction due to a rise in the nanofluid viscosity.

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Fig. 7. Dependences of the average Nusselt numbers at left wall (a), right wall (b) and fluid flow rate (c) on time, the Darcy number of the porous block I and nanoparticles volume fraction for δ = 0.2.

Fig. 8. Profiles of the local Nusselt numbers along the left vertical wall (a) and right vertical wall (b) versus the porous blocks size and nanoparticle volume fraction for Da1 = 10−3 .

Fig. 9. Dependences of the average Nusselt numbers at left wall (a), right wall (b) and nanofluid flow rate (c) on the Darcy number of the porous block I, porous block size and nanoparticle volume fraction.

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5 Conclusions The convective nanofluid flow and the heat transfer in an enclosure under the effect of two centered adherent porous blocks and horizontal temperature gradient have been studied. Experimental correlations for the effective viscosity and thermal conductivity of the alumina-water nanofluid have been applied. The effects of the Darcy number of the porous block I, porous blocks size and nanoparticle volume fraction on streamlines and isotherms as the local and average Nusselt numbers and nanofluid flow rate have been analyzed. The calculated results have shown that a growth of the porous layers sizes leads to the nanofluid flow attenuation, heat transfer rate reduction at the hot vertical wall, while the average Nusselt number at the right vertical wall increases. The increase in the Darcy number of the porous block I leads to an increase in N ul and a decrease in N ur for the moderate values of Da1 . Inclusion of alumina nanoparticles inside water can enhance the heat transfer only for high values of porous block sizes. It should be noted that the obtained data can be used for the heat transfer enhancement in heat exchangers and solar collectors using the porous insertion with nanoparticles. Moreover, the internal porous blocks can be considered as a control parameter for convective heat transfer in a closed volume under the temperature difference. The work of MAS was conducted as a government task of the Ministry of Education and Science of the Russian Federation (Project Number 13.6542.2017/6.7). Authors also wish to express their gratitude to the very competent reviewers for this valuable comments and suggestions.

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