Transport in Porous Media

, Volume 126, Issue 1, pp 223–247 | Cite as

Multi-layered Porous Foam Effects on Heat Transfer and Entropy Generation of Nanofluid Mixed Convection Inside a Two-Sided Lid-Driven Enclosure with Internal Heating

  • Sasan Asiaei
  • Ali Zadehkafi
  • Majid Siavashi


Mixed convection of Cu-water nanofluid inside a two-sided lid-driven enclosure with an internal heater, filled with multi-layered porous foams is studied numerically and its heat transfer and entropy generation number are evaluated. Use of multi-layered porous media instead of homogeneous ones is capable of heat transfer enhancement, by weakening flow where does not impose a pivotal role on heat transfer and amplifying the flow in regions where have more effects on the heat transfer. Eight different arrangements of porous layers are considered and the two-phase mixture model is implemented to simulate nanofluid mixed convection inside the cavity. Results are presented in terms of stream functions, isotherms, Nusselt and entropy generation number for the eight cases considering various Richardson numbers (Ri = 10−4 to 103) and nanofluid concentrations (φ = 0 to 0.04). Results indicate that using the multi-layered porous material can confine flow vortices in the vicinity of the moving walls and could enhance the heat transfer up to 17 percent (with respect to the case using homogeneous porous material with the highest permeability), such that this enhancement is more in lower Ri values (stronger convective effects). Entropy generation number also increases by nanofluid volume fraction increment and Ri decrement. Cases with a higher heat transfer rate also have the higher entropy generation number. In addition, an increase of volume fraction decreases the relative entropy generation number (S*) for low Ri number, while contrary fact observed for high Ri values.


Nanofluid mixed convection porous media multi-layered entropy generation 

List of symbols


Acceleration (m s−2)


Drag coefficient


Specific heat (J kg−1 K−1)


Darcy number


Nanoparticles diameter (m)


Drag function


Acceleration due to gravity (m s−2)


Grashof number


Cavity length


Heat transfer coefficient (W m−2 K−1)


Thermal conductivity (W m−1 K−1)


Boltzmann constant (J K−1)


Nusselt number


Pressure (Pa)


Prandtl number

\( S_{gen,F}^{'''} \)

Friction entropy generation rate (W m−3 K)

\( S_{gen,T}^{'''} \)

Thermal entropy generation rate (W m−3 K)

\( S_{gen,tot}^{'''} \)

Total entropy generation rate (W m−3 K)


Temperature (K)


Ambient temperature


Cold wall temperature


Hot wall temperature

\( \vec{V} \)

Velocity vector (m s−1)

Greek symbols

\( \alpha \)

Thermal diffusivity (m2 s−1)

\( \beta \)

Thermal expansion coefficient (K−1)

\( \varepsilon \)


\( \varphi \)

Volume fraction

\( \kappa \)

Permeability of porous medium (m2)

\( \mu \)

Dynamic viscosity (kg m−1 s−1)

\( \nu \)

Kinematic viscosity (m2 s−1)

\( \rho \)

Density (kg m−3)





Cold wall






Hot wall


Mixture (nanofluid)



1 Introduction

The heat transfer due to the movement of the fluid flow is known as convection heat transfer that can be divided into three main categories: natural convection, forced convection and mixed convection. The mixed convection heat transfer is used in many industries and equipment, including cooling of the electronic equipment and reactors (Boutina and Bessaïh 2011), heat exchangers (Ghorbani et al. 2010; Sillekens et al. 1998; Wang et al. 2014b), air-conditioning equipment (Mossolly et al. 2009), electromagnetic pumps (Yousofvand et al. 2017), food and chemical industries (Forson et al. 2007), and solar heaters (Cha and Jaluria 1984). An ongoing challenge in heat transfer is increasing heat transfer rate per surface area (Toosi and Siavashi 2017) and due to its significance, it has been the subject of many studies (Dehghan et al. 2015; Li et al. 2014; Hatami and Ganji 2014; Wang et al. 2014a; Jiang et al. 2014; Amani et al. 2017). In addition to the conventional methods for increasing heat transfer rate, such as using fins and baffles, the use of nanofluids and porous media has elicited the attention of heat engineers in the recent years. An effective method to increase heat transfer is using high conductive nanoparticles within the base fluid (Mahian et al. 2012b). Nanoparticles improve the thermo-physical properties of the base fluid, because of their high thermal conductivity with subsequent heat transfer enhancement (Xuan and Li 2003). Alternatively, use of the porous medium adds up the heat exchange area, also, causes fluid particles to move around the irregular posts and loops in the porous medium, thereby, augments the heat transfer (Lu et al. 2006).

There are a lot of studies on using nanoparticles in different applications (Kumar and Sharma 2018; Ramezanpour and Siavashi 2018; Zhang and Jacobi 2016; Siavashi et al. 2018a; Siavashi et al. 2018b; Amani et al. 2018) and also to enhance heat transfer (Xuan and Li 2000; Das et al. 2003; Xie et al. 2002; Ghasemi and Siavashi 2017; Yaghoubi Emami et al. 2018). The obtained results indicate that use of a small volume fraction of nanoparticles can lead to a high heat transfer enhancement (Wen and Ding 2004; Khanafer et al. 2003; Aminossadati and Ghasemi 2009). Investigating the mixed convection in a square cavity, Mansour et al. (2010) showed that using nanofluid would increase the heat transfer. Also, Tiwari and Das (2007) demonstrated that in mixed convection, the direction of moving walls and also Richardson number are effective on the heat transfer. Garoosi et al. (2015) showed that the heat transfer rate decreases in mixed convection in a square cavity by reducing the nanoparticles diameter and Richardson number. Selimefendigil and Öztop (2015) by studying mixed convection concerning the rotation of a cylinder in a nanofluid show that the heat transfer rate will be increased by increasing the external Rayleigh number, volume fraction of nanoparticle and the value of cylinder’s angular velocity.

There has been a lot of works done in the field of heat transfer in the porous media. Extensive applications of porous media have attracted the attention of many researchers in fields such as geology, water engineering, oil and gas reservoirs (Whitaker 1986; Siavashi et al. 2014), double-diffusive mass transfer (Siavashi et al. 2017b; Mchirgui et al. 2012), cooling of electronic components and heat exchangers (Siavashi et al. 2018d; Kefayati 2016; Kasaeian et al. 2017; Varol et al. 2008; Jamarani et al. 2017). Sheikholeslami and Ganji (2017) analyzed magneto-hydrodynamic CuO-water nanofluid inside a porous enclosure with consideration of the shape factor. In another study by Sheikholeslami et al. (Sheikholeslami et al. 2017), transport of MHD nanofluid inside a porous medium was studied. Jiang et al. (2004) investigated the numerical and experimental results of the forced convection inside a porous channel and showed that the porous channel can increase the heat transfer coefficient for 4 to 8 times. Mealey and Merkin (2009) studied the convective heat transfer in a porous medium with internal energy production. Siavashi et al. (2018d) studied the effects of implementation of porous fins on natural convection of nanofluid inside a cavity. A review on nanofluid flow and heat transfer has been done by Kasaeian et al. (2017).

In recent years, the use of partially porous media has attracted the attention of many researchers due to its high efficiency (Yang et al. 2012; Hajipour and Dehkordi 2012). Mahmoudi and Maerefat (2011) have examined the heat transfer of laminar flow in partially porous pipes and channels. The analytical and numerical results confirm that by increasing the heat transfer coefficient of the porous material, which is situated close to the wall, the heat transfer rate can be increased. Upon placing porous portions in the center of the channel, heat transfer increases as the flow rate increases. Siavashi et al. (2015, 2017a) studied the effect of partial filling of an annulus with porous media and also using of porous ribs on forced convection heat transfer of nanofluid. They showed that an optimal porous layer thickness and also an optimal porous rib arrangement exist in order to maximize the heat transfer rate. Nimvari et al. (2012) also investigated the heat transfer of partially porous channels in turbulent flows, with two porous arrangements in the boundary and the channel center. Their results show that depending on Darcy number, optimal thicknesses for the porous region exist in order to reach the maximum Nusselt number. Studying the natural heat convection in a partially porous cavity shows that the optimum thickness for maximum heat transfer changes for different Rayleigh and Darcy numbers (Toosi and Siavashi 2017). A similar study for non-Newtonian nanofluids has been conducted by Siavashi and Rostami (2017) in an annulus partially or completely filled with porous media. Additionally, there are some studies on the gradient porous materials, including Wang et al. (2015) research, in which they have studied the convective heat transfer in porous heat sinks and observed that the domain with gradient porous material can improve the hydraulic and thermal performance of the heat sink in comparison with a homogeneous porous heat sink. They also showed that in pipes with a gradient porosity, heat transfer increases in line with the flow direction. Xu et al. (2018) have studied the heat transfer in tubes sintered with gradient metal foams (GMFs). The obtained result shows that the pressure drop and the heat transfer rate strongly depend on pore density, porosity and GMF thickness gradients. Furthermore, Wang et al. (2018) used porous fins instead of solid fins. The obtained results reveal that the new design reduces pumping power with same Reynolds numbers. In addition, the thermal resistance is decreased by about 10% in a wide range of Reynolds numbers with the same pumping power. Maghsoudi and Siavashi (2018) found an optimal pore size arrangement for mixed convection inside a lid-driven cavity. Siavashi et al. (2018c) have also optimized the permeability of porous layers inside a heat exchanging tube to maximize the performance evaluation criterion.

Entropy generation study has also found a lot of attention finding the optimal working condition of thermal systems. This method has been widely used by many researchers and among them the works done by (Moghaddami et al. 2011; Moghaddami et al. 2012; Mahian et al. 2012a; Mahian et al. 2014; Hayat et al. 2018; Chakravarty et al. 2017; Datta et al. 2016; Mchirgui et al. 2012; Maskaniyan et al. 2018; Nouri et al. 2018) can be cited.

Although there are benevolent studies on partially and gradient porous foams, few of them has comprehensively investigated the impact of a multilayer porous in nanofluid flow in mixed convection conditions. The advantages of multilayer porous materials are controlling the flow and strengthening the vortices in desired locations which are more effective in heat transfer. In fact, fluid flow can be weakened in parts which do not play an important role in heat transfer and boost the flow in regions that have a significant role in heat transfer, thus, increasing the heat transfer rate. In this paper, the impact of multilayer porous media with different porous arrangements on heat transfer and entropy generation of mixed convection of nanofluid inside a porous square cavity is investigated numerically. The two-phase mixture model is implemented to model nanofluid and Darcy-Brinkman-Forchheimer model is chosen to find pressure drop through the porous region. Results are presented in terms of Nusselt and entropy generation numbers for 8 different porous arrangements and a wide range of Richardson (Ri = 10−4 to 103) and various nanofluid concentrations (φ=0 to 0.04). It will be shown that use of a multilayer porous media instead of a simple homogeneous porous material can enhance the heat transfer rate and also reduce the entropy generation.

2 Problem Statement

In this article, the mixed convection of Cu-water nanofluid flow and its heat transfer inside a cavity filled with a multilayer porous material and including an internal heater is numerically investigated. The schematic of the problem is shown in Figure 1. The horizontal walls are moving in opposite directions and the vertical walls are stationary. The properties of water and copper nanoparticles are summarized in Table 1. The temperature difference between the cavity and heater walls, along with the movement of the horizontal walls of the cavity cause mixed convection inside the cavity. The nanofluid and the porous region are assumed to be in thermal equilibrium. The flow inside the cavity is considered to be laminar and the density of the nanofluid is assumed to be changed according to the Boussinesq hypothesis. The porous material is constructed from different layers, however, each layer is isotropic, homogeneous and all layers have the porosity of 0.8. These configurations are suggested to strengthen the flow in some regions and weaken that in some other regions. In order to accentuate mixed convection characteristics on heat transfer, the thermal conductivity of porous region is considered to be equal to that of base fluid (ksolid = kfluid) to preserve conduction and convection in the same order. The flow of fluid in the porous medium follows the Darcy-Brinkman-Forchheimer model and two-phase mixture model is chosen to simulate nanofluid flow. According to Kozney-Carmen equation, the permeability of the porous medium depends on the average diameter of its grains known as pore diameter. In this paper, the maximum and the minimum pore diameters, listed in Table 2, and their mean value are considered. To increase the heat transfer rate, the impact of pore diameters on the heat transfer coefficient are compared. The arrangement of porous layers are changed in different ways (as exhibited in Figure 2) and the effect of various arrangements of porous layers (for 8 cases), Richardson number (Ri = 10−4 to 103) and the nanofluid volume fraction (φ = 0 to 0.04) on Nusselt and entropy generation numbers are investigated.
Fig. 1

Schematic of the studied problem

Table 1

Thermo-physical properties of water and Cu-nanoparticles


\( \rho \) (kg/m3)

\( C_{p} \) (J/kg.k)

\( k \) (W/mk)

\( \beta \times 10^{5} \) (k−1)

\( \mu \times 10^{5} \) (kg/ms)

\( d_{p} \) (nm)















Table 2

The maximum, minimum and mean values of pore diameters





Pore diameter (m)




Fig. 2

Different arrangements of porous layers

3 Governing Equations

Assuming an unsteady, incompressible, Newtonian and laminar two-dimensional flow, the governing equations (continuity, momentum, energy and volume fraction equations) for two-phase mixture nanofluid model flowing through a porous region are shown as follows (Manninen et al. 1996; Ishii and Hibiki 2010):
$$ \nabla .(\varepsilon \rho_{m} \,\vec{V}_{m} ) = 0 $$
$$ \begin{aligned} &\nabla .(\varepsilon \rho_{m} \,\vec{V}_{m} \,\vec{V}_{m} ) = \hfill - \varepsilon \nabla P + \nabla (\varepsilon \mu_{m} \,(\nabla \vec{V}_{m} + \nabla \vec{V}^{T}_{m} )) \\ &\quad + \nabla .\left[ {\sum\limits_{k = 1}^{2} {\varepsilon \varphi_{k} \,\rho_{k} \,\vec{V}_{dr,k} \vec{V}_{dr,k} } } \right] + (\varepsilon \rho \beta )_{m} (T - T_{c} )\vec{g} - F_{p} \hfill \\ \end{aligned} $$
$$ \nabla .\left[ {\sum\limits_{k = 1}^{2} {(\,\phi_{k} \vec{V}_{k} \rho_{k} (C_{p} )_{k} )\,\,T} } \right] = \nabla .(k_{eff} \nabla T) $$
$$ \nabla .(\varepsilon \varphi_{np} \,\vec{V}_{m} ) = - \nabla .(\varepsilon \varphi_{np} \,\vec{V}_{dr,p} ) $$
where \( \varepsilon \) is the porosity and \( \varphi_{k} \) is the volume fraction of phase \( {\text{k}}. \overrightarrow {{V_{m} }} {\text{and }}\rho_{m} , \) are known respectively as the mixture velocity and mixture density which are computed in (Toosi and Siavashi 2017).
The mixture heat capacity and thermal expansion coefficient are calculated by the following equations:
$$ (\rho \,C_{p} )_{m} = (1 - \varphi_{p} )(\rho \,C_{p} )_{f} + \varphi_{p} (\rho \,C_{p} )_{p} $$
$$ (\rho \,\beta )_{m} = (1 - \varphi_{p} )(\rho \,\beta )_{f} + \varphi_{p} (\rho \,\beta )_{p} $$
Using Corcione’s model (Corcione 2011), the mixture thermal conductivity and its dynamic viscosity are determined:
$$ \mu_{m} = {{\mu_{f} } \mathord{\left/ {\vphantom {{\mu_{f} } {\left( {1 - 34.87({{d_{p} } \mathord{\left/ {\vphantom {{d_{p} } {d_{f} }}} \right. \kern-0pt} {d_{f} }})^{ - 0.3} \,\varphi_{p}^{1.03} } \right)}}} \right. \kern-0pt} {\left( {1 - 34.87({{d_{p} } \mathord{\left/ {\vphantom {{d_{p} } {d_{f} }}} \right. \kern-0pt} {d_{f} }})^{ - 0.3} \,\varphi_{p}^{1.03} } \right)}} $$
$$ \frac{{k_{m} }}{{k_{f} }} = 1 + 4.4\text{Re}_{B}^{0.4} \,\Pr^{0.66} \,\left( {\frac{T}{{T_{fr} }}} \right)^{10} \left( {\frac{{k_{p} }}{{k_{f} }}} \right)^{0.03} \varphi_{p}^{0.66} $$
$$ \text{Re}_{B} = \frac{{\rho_{f\,} u_{B\,} d_{p} }}{{\mu_{f} }} $$
$$ u_{B} = \frac{{2k_{b} T}}{{\pi \,\mu_{f\,} d_{p}^{2} }} $$
and \( k_{eff} \) is the effective thermal conduction coefficient and is evaluated by:
$$ k_{eff} = (1 - \varepsilon )k_{s} + \varepsilon k_{m} $$
In the momentum equation (Eq. 2), \( F_{p} \) is the drag force caused by the porous domain and is calculated by Darcy-Brinkman-Forchheimer model as follows:
$$ F_{p} = \sum\limits_{k = 1}^{2} {\phi_{k} } \left( \frac{{\mu_{k} }}{{K_{q} }}\vec{V}_{k} + \frac{{C_{d,p} \rho_{q} }}{{\sqrt {K_{q} } }}\left| {\vec{V}_{k} } \right|\vec{V}_{k} \right) $$
where \( C_{d} \) and K are the inertia coefficient and permeability of the porous medium, respectively, and are evaluated from Kozney-Carmen relation and are considered to be the same for the two phases:
$$ C_{d} = \frac{1.75}{{\sqrt {150} \,\varepsilon^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }} $$
$$ K = \frac{{d_{m}^{2} \varepsilon^{3} }}{{175\left( {1 - \varepsilon } \right)^{2} }} $$
To investigate the mixed convection heat transfer, the non-dimensional Richardson number is used and is defined as follows:
$$ {\text{Ri}} = \frac{Gr}{{\text{Re}^{2} }} $$
$$ {\text{Gr}} = \frac{{g\beta (T_{h} - T_{c} )H^{3} }}{{\nu_{f}^{2} }} $$
$$ {\text{Re}} = \frac{{U_{0} H}}{{\nu_{f} }} $$
Nusselt number is also defined as:
$$ {\text{Nu}}_{ave,wall} { = }\frac{hH}{{k_{f} }}{ = }\frac{{q^{\prime\prime}_{wall} H}}{{k_{f} (T_{h} - T_{c} )}} \, $$
Low Richardson numbers indicate the high wall velocity and thereby increasing the contribution of the forced convection in total heat transfer. Nevertheless, high Richardson numbers mean low wall velocity and cause a change in heat transfer mechanism from forced convection to natural convection. The volumetric entropy generation rate is calculated as follows:
$$ S^{\prime\prime\prime}_{gen,tot} = S^{\prime\prime\prime}_{gen,T} + S^{\prime\prime\prime}_{gen,F} $$
In Eq. (19), \( S^{\prime\prime\prime}_{gen,T} \) represents the entropy generation rate caused by the heat transfer irreversibility and is given by Eq. (20):
$$ S^{\prime\prime\prime}_{gen,T} = \frac{{k_{m} }}{{T^{2} }}\left[ {\left( {\frac{\partial T}{\partial x}} \right)^{2} + \left( {\frac{\partial T}{\partial y}} \right)^{2} } \right] $$
\( S^{\prime\prime\prime}_{gen,F}\) is the entropy generation rate associated with fluid friction and is calculated by:
$$ S^{\prime\prime\prime}_{gen,F} = \frac{{\mu_{m} }}{{T_{0} }}\left\{ {2\left[ {\left( {\frac{\partial u}{\partial x}} \right)^{2} + \left( {\frac{\partial v}{\partial y}} \right)^{2} } \right] + \left( {\frac{\partial u}{\partial y} + \frac{\partial u}{\partial y}} \right)^{2} } \right\} $$
The non-dimensional entropy (entropy generation number) is defined as:
$$ \overline{S} = S^{\prime\prime\prime}_{gen,tot} \frac{{T_{0}^{2} H}}{{k_{bf} (T_{h} - T_{c} )^{2} }} $$

In this paper, \( S^{*} \) is defined as the relative entropy generation number which is the ratio of the porous media entropy generation number to non-porous media entropy generation number.

Boundary conditions used according to what is shown in Figure 1, are:

The cold horizontal walls:
$$ \begin{aligned} & y = 0{\text{ }},\quad 0 \le x \le H:\quad u = U_{0} {\text{ }},{\text{ }}v = 0{\text{ }},\quad T = 310 \\ & y = H,\quad 0 \le x \le H:\quad u = - \;U_{0} ,{\text{ }}v = 0{\text{ }},\quad T = 310 \\ \end{aligned} $$
The cold vertical walls:
$$ \begin{aligned} x = 0 \, ,\,\,0 \le y \le H \, \,:\,\,\,\, \, u = v = 0, \, \,\,\, \, T = 310 \hfill \\ x = H,\,\,0 \le y \le H \, \,:\,\,\,\, \, u = v = 0, \, T = 310 \hfill \\ \end{aligned} $$
The heater wall:
$$ u = v = 0, \, \,\,\, \, T = 315 $$

4 Numerical Solution

A structured tetrahedral grid is generated for the computational domain and governing equations are discretized and solved employing finite volume method. The SIMPLE algorithm is implemented to handle the pressure and velocity coupling (Patankar 1980). In order to achieve the solution that is independent of the computational grid, the problem is solved for different meshes and its results are presented in Table 3. As can be seen, the grid with 4215 nodes could predict the heat flux on the hot wall with an acceptable accuracy.
Table 3

Mesh independency

Grid size






q’’hot wall (W)






To validate the numerical results of the solver, two different cases are used. First, a single-phase, natural convection in a porous cavity is considered and the fluid flow is modeled using Darcy-Brickman-Forchheimer relation. Then the results are compared with those of Nithiarasu et al. (1997). Table 4 represents the results of this comparison which indicates a good match. Next, in order to validate the two-phase mixture model used in this article, a mixed convective heat transfer of water and copper nanofluid in a square cavity with 25 nm nanoparticles is considered and compared with the results of Garoosi et al. (2015). Furthermore, free convection of the nanofluid is compared with the results obtained by Khanafer et al. (2003) and Jahanshahi et al. (2010) in terms of temperature variation on a horizontal line at the middle of the cavity. Accordingly, the numerical simulation solver is validated (see Figs. 3 and 4).
Table 4

Nusselt number comparison of the present work and (Nithiarasu et al. 1997) with Pr=1


\( \varepsilon = 0.6 \)

\( \varepsilon = 0.9 \)

Nithiarasu et al. (1997)

Present study

Nithiarasu et al. (1997)

Present study

Da = 10−4

Ra = 105






Ra = 106





Da = 10−2

Ra = 104






Ra = 105





Fig. 3

Comparison between results of the present work and those of Garoosi et al. (2015) for Ri = 1, Ri = 0.1, and Ri = 0.01

Fig. 4

Comparison of the temperature on axial midline between results by Khanafer et al. (2003) and Jahanshahi et al. (2010) with the present results (Pr = 6.2, ∅ = 0.1, Gr = 105)

5 Results and Discussions

In this section, the results of mixed convection and entropy generation number of the problem described in Sect. 2 are presented. The Grashof number is considered to be fixed at 104. Results are presented in two subsections: first, the heat transfer and fluid flow characteristics of the problem is analyzed in Sect. 5.1, and after that in Sect. 5.2, results are presented and investigated from the viewpoint of the second law of thermodynamics.

5.1 Heat Transfer and Fluid Flow Study

5.1.1 Porous Layers Arrangement Effects on Streamlines & Isotherms

In this section, the effect of different arrangements of porous layers on forming the streamlines, vortices, and isotherms are studied. The increase of vortices intensity and their number, and also stretching out the vortices and pushing them close to walls can lead to the increment of the heat transfer rate. Figure 5 shows the streamlines in each 8 arrangement of the porous layers, for Ri = 0.0001, 0.001 and 0.01, while the volume fraction of nanoparticles is zero. Regardless of the porous layer arrangements, vortices are created in the vicinity of the moving walls in low Ri values. By increasing Ri, these vortices are formed near the stationary walls, while the vortices strengths decrease, and thus, the heat transfer is expected to decrease.
Fig. 5

Effect of different porous layer arrangements on non-dimensional streamlines in Ri = 0.0001, 1 and 1000

Figure 5 also illustrates the effect of different arrangements of porous layers on the streamlines and vortices formation. In Ri = 1000, the arrangement of the layers in the cases 2 and 3, generate four weak vortices while in other cases, just two vortices are formed, whose their strength, elongation and distance from walls are different from each other. In Ri = 0.0001, the strength of the formed vortices in case 1 is greater than the cases 2 and 3, however, the generated vortices in cases 2 and 3 are more concentrated between the cold and hot walls, and it’s likely to provide higher heat transfer rates with respect to case 1. Accordingly, the arrangement of porous layers is highly effective in the heat transfer values.

To further study the effect of different porous layer arrangements on heat transfer characteristics, isotherms of the 8 different arrangements of porous layers in three Richardson numbers of 0.0001, 1 and 1000, for the pure water are depicted in Figure 6. The heat transfer rate increases for more severe temperature gradients in denser and more compact isotherms. Heat transfer paths are formed perpendicular to the isotherms. Since the pure conduction occurs in the radial direction, isotherms are formed in circular shapes, therefore, stretching isotherms and getting out of the circular shapes, means the stronger convective heat transfer mechanism. In low Ri values, isotherms get out of the circular shape and stretch in the direction of the square’s diameter, which means convection is intensified, and thereupon, the total heat transfer rate increases. It can also be understood from that regardless of the arrangements of the porous layers, increasing the Richardson number would lead to the decrease in condensation of isotherms, and thus, reduction of heat transfer rate.
Fig. 6

Effect of different porous layer arrangements on isothermal lines (K) in Ri = 0.0001, 1 and 1000

5.1.2 Impact of Volume Fraction and Richardson on Nusselt Number

Nusselt number is augmented by increasing the volume fraction and decrease of Richardson number. Figure 7 illustrates the variation of Nu due to volume fraction and Richardson number in different arrangements of porous layers. As observed in this figure, in all cases, the Nusselt number decreases by increasing Richardson number and transformation of heat transfer mechanism from mixed convection to natural convection. The reason for this reduction can be figured out by investigating the formation of streamlines and vortices. According to Figure 5 and Figure 6, as the Richards number increases, the strength of the formed vortices and the temperature gradient decrease, and consequently, the heat transfer is weakened. Moreover, increasing the volume fraction would raise the thermal conductivity, and improves the heat transfer, which is more pronounced in low Richardson numbers. This means that in low Richardson values, increasing the volume fraction for increasing the heat transfer is more effective compared to higher Richardson values.
Fig. 7

The effect of the volume fraction and Richardson Number on Nusselt Number in 8 cases

5.1.3 Impact of Porous Layers Arrangement on Nusselt Number

In this section, in order to determine the best arrangement with the highest heat transfer performance, the 8 cases with different Richardson numbers have been compared with each other. Figure 8 compares the variation of Nu against nanofluid volume fraction for various Ri values. It is observed that in Ri = 0.0001, 0.001 and 0.01, the cases 2, 3 and 7 have the greater Nusselt numbers rather than case 1, which has the biggest pore diameter (or the highest permeability). The reason of this fact is related to the arrangement of the porous layers. According to Figure 5, it can be stated that since in low Ri figures, vortices have been formed near the moving wall, the arrangement with a bigger pore diameter near the moving wall and smaller pore diameter near the stationary wall increases the heat transfer rate due to the amplification and elongation of the formed vortices. This means that guiding the flow and boosting the vortices in particular areas of cavities, that have a great role in heat transfer, is more effective than increasing the pore diameter in improving the heat transfer. In cases 2 and 3 the top and bottom vortices are separated with a low permeable region, in the middle of the cavity, that confines flow transfer between these two regions and strengthens these two vortices. It is also observed that in volume fractions less than 0.02, Nu of case 3 is greater than that of case 7, however for φ > 0.02, Nu of case 7 becomes greater than case3’s. Similarly, after case 1, cases 5, 4, 6 and 8, respectively have the highest Nu values.
Fig. 8

Comparison of Nusselt values between the 8 different arrangements of porous layers in Richardson numbers of 0.0001, 0.001, 0.1, 1 and1000

In Ri = 1, Nu of case 3 is smaller compared to case 1, but Nu in cases of 2 and 7 are still higher than that of case 1. Furthermore; in Ri = 1000, the highest Nu number belongs to the case 1 and cases 5, 4, 7, 6, 2, 8 and 3 are in the next ranks, respectively. Actually in high Ri numbers (natural convection is dominated), unlike the low Ri values (forced convection is dominated), the cases with greater pore diameter close to the stationary wall attains the greater Nusselt value compared to the other cases. This can be also explained due to the location of the formed vortices in the cavity. In accordance with Figure 5, vortices are formed near the stationary wall, and thus, rising the pore diameter in these areas can cause amplification of vortices, which leads to an increase in the heat transfer rate.

Based on the results of this section, it can be concluded that use of layered porous materials with a specific arrangement of layers can substantially increase the heat transfer rate, and by increasing the forced convection effects this increase could become more and more. It was observed that for case 2 at Ri = 0.0001 about 17% increment of Nu number could be obtained with respect to the homogeneous porous medium with the highest permeability.

5.2 Entropy Generation Study

Entropy generation analysis can be used as a tool to find the optimal working condition of systems, considering the second law of thermodynamics. In this section, results of the 8 different mentioned cases will be analyzed using the entropy generation concept. It is worth noting that entropy generation values are presented in the dimensionless form.

5.2.1 Impact of the Volume Fraction and Richardson on Entropy Generation Number

The effect of the volume fraction and Richardson number in different arrangements of porous layers on the entropy generation number has been illustrated in Figure 9. As it can be seen, the entropy generation number, decreases by increasing the Richardson number due to reduced heat transfer and weakening of vortices, regardless of the arrangement of the porous layers. In addition, it can be observed that increasing the volume fraction, increases the entropy generation number, which is higher in low Richardson values, so it is reasonable to consider that the entropy generation number in high Richardson numbers is independent of the volume fraction. The reason is that for low Ri values the effect of forced convection and consequently the share of frictional entropy generation number in total entropy generation number is higher than high Ri numbers. By increasing Ri, forced convection will be weakened and the majority of entropy generation number is caused by thermal irreversibility and the share of frictional irreversibility in total entropy generation number becomes negligible. Increasing nanofluid concentration increases the frictional entropy generation number – which its share in the total irreversibility of low Ri numbers is very low – due to viscosity promotion and hence the impact of nanofluid volume fraction on the total entropy generation number is ignorable in high Ri values.
Fig. 9

The effect of the volume fraction and Richardson number on relative total entropy generation number in the 8 different arrangements of porous layers

Table 5, 6 and 7 show the thermal and frictional entropy generation number in Ri = 0.0001, 1 and 1000 for the case 1. As it can be observed, the entropy generation number caused by heat transfer is far more than the entropy generation number due to the viscosity effects, so the viscous effect can be ignored. The insignificant effect of viscosity on the generation of entropy is due to the small velocity gradients in the cavity compared to extremely high-temperature gradients. Figure 10 shows the thermal entropy generation number as well as frictional entropy generation number in the cavity. As it can be seen, thermal entropy generation number is high near the hot wall and frictional entropy generation number has high value in the neighbourhood of the moving walls. It is worthy to say, increasing the Richardson Number leads to a decrease in the wall velocity and mixed convection turned to natural convection. As a result, the fluid velocity is decreased significantly and afterwards, the entropy generation number due to viscosity effect declines.
Table 5

Entropy generation number due to thermal and frictional effects in case1 at Ri = 0.0001

Volume fraction

Thermal entropy generation number

Frictional entropy generation number
















Table 6

Entropy generation number due to thermal and frictional effects in case1 at Ri = 1

Volume fraction

Thermal entropy generation number

Frictional entropy generation number
















Table 7

Entropy generation number due to thermal and frictional effects in case1 at Ri = 1000

Volume fraction

Thermal entropy generation number

Frictional entropy generation number
















Fig. 10

a) Thermal entropy generation number, b) Frictional entropy generation number for case1 in Richardson Number of 0.0001 at zero volume fraction

5.2.2 Impact of Porous Layers Arrangement on Entropy Generation Number

Figure 11 compares the relative total entropy generation number (S*) of the 8 different arrangements of porous layers in Richardson numbers of 0.0001, 0.001, 0.1, 1 and 1000. It should be mentioned that the relative entropy generation number (as mentioned before in section 3) is defined as the ratio of the total entropy generation number of the porous cavity to that of the non-porous cavity. As it can be observed, cases 7, 3 and 2, which have the greater Nusselt (higher heat transfer rate) in comparison with case 1, have more entropy generation number compared to case 1. Although fluid motion is limited in some parts of the cavity in cases 7, 3 and 2 due to the reduction of the pore diameter (and thus reduction of permeability) and as a result, the entropy generation number caused by frictional effects is decreased, but, as can be seen in Tables 5, 6, and 7, the entropy generation number due to the viscosity effect is insignificant and its reduction against the growth of thermal entropy generation number is negligible and cannot reduce the total entropy generation number. Also in high Richardson numbers, case 1 has greater Nu and higher entropy generation number in comparison with the other cases. An interesting point of Figure 11 is that for low Ri values S* is higher than 1 and decreases by increasing nanofluid volume fraction, however, by increasing Ri, the relative entropy generation number becomes lower than 1 and its trend of variation with volume fraction becomes ascending.
Fig. 11

Comparison of the relative entropy generation number between the 8 different arrangements of porous layers in Ri = 0.0001, 0.001, 0.1, 1 and 1000

6 Conclusions

This paper investigates the impact of multi-layer porous media on mixed convection heat transfer and entropy generation number of nanofluid flow, through a lid-driven enclosure with an internal heater. Cu-water nanofluid flow has been simulated numerically using the two-phase mixture model. Flow through the porous medium obeys Darcy-Brinkman-Forchheimer relation. Eight different porous layer arrangements have been considered. The results indicated that using a multi-layered porous medium could enhance the heat transfer rate up to 17 percent compared to the case using a uniform porous medium with the largest pore diameter (the highest permeable medium). Heat transfer strongly depends on the arrangement of porous layers. The proper arrangement of porous layers in low Richardson numbers (dominated forced convection) is achieved by setting the maximum pore diameter close to the moving walls and the minimum pore diameter near the stationary walls. Nonetheless, in high Richardson values (dominated natural convection), a uniform porous medium with the largest pore diameter could lead to the highest heat transfer rate. Increasing the nanofluid volume fraction in low Richardson values is more effective in the growth of the heat transfer rate rather than high Richardson numbers. Comparison of the 8 different cases from the second law perspective reveals similar behavior to the heat transfer rate behavior. Contrary trends observed for variations of the relative entropy generation number (porous to the non-porous cavity) with respect to nanofluid volume fraction increase. For low and high Ri values the trends are descending and ascending, respectively. In mixed convection problems, application of multi-layered porous materials with specified design, instead of homogeneous ones, can substantially enhance the heat transfer rate, especially in the cases with stronger forced convection forces.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Sensors and Integrated Bio-Microfluidics/MEMS Laboratory, School of Mechanical EngineeringIran University of Science and TechnologyTehranIran
  2. 2.Applied Multi-Phase Fluid Dynamics Lab, School of Mechanical EngineeringIran University of Science and TechnologyTehranIran
  3. 3.School of Mechanical EngineeringIran University of Science and TechnologyNarmak, TehranIran

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