# 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

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## Abstract

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 10^{3}) 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.

## Keywords

Nanofluid mixed convection porous media multi-layered entropy generation## List of symbols

*a*Acceleration (m s

^{−2})*C*_{d}Drag coefficient

*C*_{p}Specific heat (J kg

^{−1}K^{−1})- Da
Darcy number

*d*_{p}Nanoparticles diameter (m)

*f*_{drag}Drag function

*g*Acceleration due to gravity (m s

^{−2})- Gr
Grashof number

*H*Cavity length

*h*Heat transfer coefficient (W m

^{−2}K^{−1})*k*Thermal conductivity (W m

^{−1}K^{−1})*k*_{b}Boltzmann constant (J K

^{−1})- Nu
Nusselt number

*p*Pressure (Pa)

- Pr
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)*T*Temperature (K)

*T*_{0}Ambient temperature

*T*_{c}Cold wall temperature

*T*_{h}Hot wall temperature

- \( \vec{V} \)
Velocity vector (m s

^{−1})

## Greek symbols

- \( \alpha \)
Thermal diffusivity (m

^{2}s^{−1})- \( \beta \)
Thermal expansion coefficient (K

^{−1})- \( \varepsilon \)
Porosity

- \( \varphi \)
Volume fraction

- \( \kappa \)
Permeability of porous medium (m

^{2})- \( \mu \)
Dynamic viscosity (kg m

^{−1}s^{−1})- \( \nu \)
Kinematic viscosity (m

^{2}s^{−1})- \( \rho \)
Density (kg m

^{−3})

## Subscripts

*avg*Average

*c*Cold wall

*dr*Drift

*f*Fluid

*h*Hot wall

*m*Mixture (nanofluid)

*p*Nanoparticles

## 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 10^{3}) 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

_{solid}= k

_{fluid}) 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 10

^{3}) and the nanofluid volume fraction (

*φ*= 0 to 0.04) on Nusselt and entropy generation numbers are investigated.

Thermo-physical properties of water and Cu-nanoparticles

\( \rho \) (kg/m | \( C_{p} \) (J/kg.k) | \( k \) (W/mk) | \( \beta \times 10^{5} \) (k | \( \mu \times 10^{5} \) (kg/ms) | \( d_{p} \) (nm) | |
---|---|---|---|---|---|---|

water | 993 | 4178 | 0.628 | 36.2 | 695(Pr=4.62) | 0.3385 |

Cu | 8933 | 385 | 401 | 1.67 | - | 25 |

The maximum, minimum and mean values of pore diameters

Max. | Mean | Min. | |
---|---|---|---|

Pore diameter (m) | 0.000900 | 0.000675 | 0.000225 |

## 3 Governing Equations

*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:

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:

## 4 Numerical Solution

Mesh independency

Grid size | 672 | 1872 | 3324 | 4215 | 5549 |
---|---|---|---|---|---|

q’’ | 4.4276 | 4.4520 | 4.4531 | 4.4576 | 4.4591 |

*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).

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

## 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 10^{4}. 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

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.

#### 5.1.2 Impact of Volume Fraction and Richardson on Nusselt Number

#### 5.1.3 Impact of Porous Layers Arrangement on Nusselt Number

*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.

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

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 |
---|---|---|

0 | 0.00056 | 1.78E-05 |

0.01 | 0.000611 | 1.93E-05 |

0.02 | 0.000646 | 2.11E-05 |

0.03 | 0.000677 | 2.35E-05 |

0.04 | 0.000707 | 2.64E-05 |

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

Volume fraction | Thermal entropy generation number | Frictional entropy generation number |
---|---|---|

0 | 0.000211 | 7.70E-10 |

0.01 | 0.000228 | 8.38E-10 |

0.02 | 0.000238 | 9.20E-10 |

0.03 | 0.000247 | 1.02E-09 |

0.04 | 0.000255 | 1.16E-09 |

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

Volume fraction | Thermal entropy generation number | Frictional entropy generation number |
---|---|---|

0 | 0.00014 | 2.67E-12 |

0.01 | 0.000149 | 2.59E-12 |

0.02 | 0.000154 | 2.51E-12 |

0.03 | 0.000158 | 2.44E-12 |

0.04 | 0.000162 | 2.39E-12 |

#### 5.2.2 Impact of Porous Layers Arrangement on Entropy Generation Number

^{*}) 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.

## 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.

## References

- Amani, M., Ameri, M., Kasaeian, A.: The experimental study of convection heat transfer characteristics and pressure drop of magnetite nanofluid in a porous metal foam tube. Transp. Porous Media
**116**(2), 959–974 (2017). https://doi.org/10.1007/s11242-016-0808-6 Google Scholar - Amani, P., Amani, M., Ahmadi, G., Mahian, O., Wongwises, S.: A critical review on the use of nanoparticles in liquid–liquid extraction. Chem. Eng. Sci.
**183**, 148–176 (2018). https://doi.org/10.1016/j.ces.2018.03.001 Google Scholar - Aminossadati, S., Ghasemi, B.: Natural convection cooling of a localised heat source at the bottom of a nanofluid-filled enclosure. Eur. J. Mech.-B/Fluids
**28**(5), 630–640 (2009)Google Scholar - Boutina, L., Bessaïh, R.: Numerical simulation of mixed convection air-cooling of electronic components mounted in an inclined channel. Appl. Therm. Eng.
**31**(11–12), 2052–2062 (2011)Google Scholar - Cha, C., Jaluria, Y.: Recirculating mixed convection flow for energy extraction. Int. J. Heat Mass Transf.
**27**(10), 1801–1812 (1984)Google Scholar - Chakravarty, A., Datta, P., Ghosh, K., Sen, S., Mukhopadhyay, A.: Thermal Non-equilibrium heat transfer and entropy generation due to natural convection in a cylindrical enclosure with a truncated conical. Heat-Gener. Porous Bed. Transp. Porous Media
**116**(1), 353–377 (2017). https://doi.org/10.1007/s11242-016-0778-8 Google Scholar - Corcione, M.: Empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids. Energy Convers. Manag.
**52**(1), 789–793 (2011)Google Scholar - Das, S.K., Putra, N., Thiesen, P., Roetzel, W.: Temperature dependence of thermal conductivity enhancement for nanofluids. J. Heat Transf.
**125**(4), 567–574 (2003)Google Scholar - Datta, P., Mahapatra, P.S., Ghosh, K., Manna, N.K., Sen, S.: Heat transfer and entropy generation in a porous square enclosure in presence of an adiabatic block. Transp. Porous Media
**111**(2), 305–329 (2016). https://doi.org/10.1007/s11242-015-0595-5 Google Scholar - Dehghan, M., Valipour, M.S., Saedodin, S.: Temperature-dependent conductivity in forced convection of heat exchangers filled with porous media: a perturbation solution. Energy Convers. Manag.
**91**, 259–266 (2015)Google Scholar - Forson, F., Nazha, M., Akuffo, F., Rajakaruna, H.: Design of mixed-mode natural convection solar crop dryers: application of principles and rules of thumb. Renew. Energy
**32**(14), 2306–2319 (2007)Google Scholar - Garoosi, F., Rohani, B., Rashidi, M.M.: Two-phase mixture modeling of mixed convection of nanofluids in a square cavity with internal and external heating. Powder Technol.
**275**, 304–321 (2015)Google Scholar - Ghasemi, K., Siavashi, M.: MHD nanofluid free convection and entropy generation in porous enclosures with different conductivity ratios. J. Magn. Magn. Mater.
**442**, 474–490 (2017). https://doi.org/10.1016/j.jmmm.2017.07.028 Google Scholar - Ghorbani, N., Taherian, H., Gorji, M., Mirgolbabaei, H.: Experimental study of mixed convection heat transfer in vertical helically coiled tube heat exchangers. Exp. Therm. Fluid Sci.
**34**(7), 900–905 (2010)Google Scholar - Hajipour, M., Dehkordi, A.M.: Analysis of nanofluid heat transfer in parallel-plate vertical channels partially filled with porous medium. Int. J. Therm. Sci.
**55**, 103–113 (2012)Google Scholar - Hatami, M., Ganji, D.: Thermal and flow analysis of microchannel heat sink (MCHS) cooled by Cu–water nanofluid using porous media approach and least square method. Energy Convers. Manag.
**78**, 347–358 (2014)Google Scholar - Hayat, T., Khan, M.I., Qayyum, S., Alsaedi, A.: Entropy generation in flow with silver and copper nanoparticles. Colloids Surf. A
**539**, 335–346 (2018). https://doi.org/10.1016/j.colsurfa.2017.12.021 Google Scholar - Ishii, M., Hibiki, T.: Thermo-fluid dynamics of two-phase flow. Springer Science & Business Media, (2010)Google Scholar
- Jahanshahi, M., Hosseinizadeh, S., Alipanah, M., Dehghani, A., Vakilinejad, G.: Numerical simulation of free convection based on experimental measured conductivity in a square cavity using water/SiO
_{2}nanofluid. Int. Commun. Heat Mass Transf.**37**(6), 687–694 (2010)Google Scholar - Jamarani, A., Maerefat, M., Jouybari, N.F., Nimvari, M.E.: Thermal performance evaluation of a double-tube heat exchanger partially filled with porous media under turbulent flow regime. Transp. Porous Media
**120**(3), 449–471 (2017). https://doi.org/10.1007/s11242-017-0933-x Google Scholar - Jiang, L., Ling, J., Jiang, L., Tang, Y., Li, Y., Zhou, W., Gao, J.: Thermal performance of a novel porous crack composite wick heat pipe. Energy Convers. Manag.
**81**, 10–18 (2014)Google Scholar - Jiang, P.-X., Si, G.-S., Li, M., Ren, Z.-P.: Experimental and numerical investigation of forced convection heat transfer of air in non-sintered porous media. Exp. Therm. Fluid Sci.
**28**(6), 545–555 (2004)Google Scholar - Kasaeian, A., Daneshazarian, R., Mahian, O., Kolsi, L., Chamkha, A.J., Wongwises, S., Pop, I.: Nanofluid flow and heat transfer in porous media: a review of the latest developments. Int. J. Heat Mass Transf.
**107**, 778–791 (2017). https://doi.org/10.1016/j.ijheatmasstransfer.2016.11.074 Google Scholar - Kefayati, G.: Heat transfer and entropy generation of natural convection on non-Newtonian nanofluids in a porous cavity. Powder Technol.
**299**, 127–149 (2016)Google Scholar - Khanafer, K., Vafai, K., Lightstone, M.: Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids. Int. J. Heat Mass Transf.
**46**(19), 3639–3653 (2003)Google Scholar - Kumar, R.S., Sharma, T.: Stability and rheological properties of nanofluids stabilized by SiO
_{2}nanoparticles and SiO_{2}–TiO_{2}nanocomposites for oilfield applications. Colloids Surf. A**539**, 171–183 (2018). https://doi.org/10.1016/j.colsurfa.2017.12.028 Google Scholar - Li, Z., Haramura, Y., Kato, Y., Tang, D.: Analysis of a high performance model Stirling engine with compact porous-sheets heat exchangers. Energy
**64**, 31–43 (2014)Google Scholar - Lu, W., Zhao, C., Tassou, S.: Thermal analysis on metal-foam filled heat exchangers. Part I: metal-foam filled pipes. Int. J. Heat Mass Transf.
**49**(15), 2751–2761 (2006)Google Scholar - Maghsoudi, P., Siavashi, M.: Application of nanofluid and optimization of pore size arrangement of heterogeneous porous media to enhance mixed convection inside a two-sided lid-driven cavity. J. Therm. Anal. Calorim. (2018). https://doi.org/10.1007/s10973-018-7335-3 Google Scholar
- Mahian, O., Kianifar, A., Sahin, A.Z., Wongwises, S.: Entropy generation during Al
_{2}O_{3}/water nanofluid flow in a solar collector: effects of tube roughness, nanoparticle size, and different thermophysical models. Int. J. Heat Mass Transf.**78**, 64–75 (2014)Google Scholar - Mahian, O., Mahmud, S., Heris, S.Z.: Analysis of entropy generation between co-rotating cylinders using nanofluids. Energy
**44**(1), 438–446 (2012a)Google Scholar - Mahian, O., Mahmud, S., Heris, S.Z.: Effect of uncertainties in physical properties on entropy generation between two rotating cylinders with nanofluids. J. Heat Transf.
**134**(10), 101704 (2012b)Google Scholar - Mahmoudi, Y., Maerefat, M.: Analytical investigation of heat transfer enhancement in a channel partially filled with a porous material under local thermal non-equilibrium condition. Int. J. Therm. Sci.
**50**(12), 2386–2401 (2011)Google Scholar - Manninen, M., Taivassalo, V., Kallio, S.: On the mixture model for multiphase flow. In. Technical Research Centre of Finland Finland, (1996)Google Scholar
- Mansour, M., Mohamed, R., Abd-Elaziz, M., Ahmed, S.E.: Numerical simulation of mixed convection flows in a square lid-driven cavity partially heated from below using nanofluid. Int. Commun. Heat Mass Transf.
**37**(10), 1504–1512 (2010)Google Scholar - Maskaniyan, M., Nazari, M., Rashidi, S., Mahian, O.: Natural convection and entropy generation analysis inside a channel with a porous plate mounted as a cooling system. Therm. Sci. Eng. Progr.
**6**, 186–193 (2018). https://doi.org/10.1016/j.tsep.2018.04.003 Google Scholar - Mchirgui, A., Hidouri, N., Magherbi, M., Brahim, A.B.: Entropy generation in double-diffusive convection in a square porous cavity using Darcy–Brinkman formulation. Transp. Porous Media
**93**(1), 223–240 (2012). https://doi.org/10.1007/s11242-012-9954-7 Google Scholar - Mealey, L., Merkin, J.: Steady finite Rayleigh number convective flows in a porous medium with internal heat generation. Int. J. Therm. Sci.
**48**(6), 1068–1080 (2009)Google Scholar - Moghaddami, M., Mohammadzade, A., Esfehani, S.A.V.: Second law analysis of nanofluid flow. Energy Convers. Manag.
**52**(2), 1397–1405 (2011)Google Scholar - Moghaddami, M., Shahidi, S., Siavashi, M.: Entropy generation analysis of nanofluid flow in turbulent and laminar regimes. J. Comput. Theor. Nanosci.
**9**(10), 1586–1595 (2012)Google Scholar - Mossolly, M., Ghali, K., Ghaddar, N.: Optimal control strategy for a multi-zone air conditioning system using a genetic algorithm. Energy
**34**(1), 58–66 (2009)Google Scholar - Nimvari, M., Maerefat, M., El-Hossaini, M.: Numerical simulation of turbulent flow and heat transfer in a channel partially filled with a porous media. Int. J. Therm. Sci.
**60**, 131–141 (2012)Google Scholar - Nithiarasu, P., Seetharamu, K., Sundararajan, T.: Natural convective heat transfer in a fluid saturated variable porosity medium. Int. J. Heat Mass Transf.
**40**(16), 3955–3967 (1997)Google Scholar - Nouri, D., Pasandideh-Fard, M., Javad Oboodi, M., Mahian, O., Sahin, A.Z.: Entropy generation analysis of nanofluid flow over a spherical heat source inside a channel with sudden expansion and contraction. Int. J. Heat Mass Transf.
**116**, 1036–1043 (2018). https://doi.org/10.1016/j.ijheatmasstransfer.2017.09.097 Google Scholar - Patankar, S.: Numerical heat transfer and fluid flow. CRC press, (1980)Google Scholar
- Ramezanpour, M., Siavashi, M.: Application of SiO
_{2}–water nanofluid to enhance oil recovery. J. Therm. Anal. Calorim. (2018). https://doi.org/10.1007/s10973-018-7156-4 Google Scholar - Selimefendigil, F., Öztop, H.F.: Mixed convection in a two-sided elastic walled and SiO
_{2}nanofluid filled cavity with internal heat generation: effects of inner rotating cylinder and nanoparticle’s shape. J. Mol. Liq.**212**, 509–516 (2015)Google Scholar - Sheikholeslami, M., Ganji, D.D.: Numerical modeling of magnetohydrodynamic CuO–Water transportation inside a porous cavity considering shape factor effect. Colloids Surf. A
**529**, 705–714 (2017). https://doi.org/10.1016/j.colsurfa.2017.06.046 Google Scholar - Sheikholeslami, M., Ziabakhsh, Z., Ganji, D.D.: Transport of magnetohydrodynamic nanofluid in a porous media. Colloids Surf. A
**520**, 201–212 (2017). https://doi.org/10.1016/j.colsurfa.2017.01.066 Google Scholar - Siavashi, M., Bahrami, H.R.T., Saffari, H.: Numerical investigation of flow characteristics, heat transfer and entropy generation of nanofluid flow inside an annular pipe partially or completely filled with porous media using two-phase mixture model. Energy
**93**, 2451–2466 (2015)Google Scholar - Siavashi, M., Bahrami, H.R.T., Saffari, H.: Numerical investigation of porous rib arrangement on heat transfer and entropy generation of nanofluid flow in an annulus using a two-phase mixture model. Numer. Heat Transf. Part A: Appl.
**71**(12), 1251–1273 (2017a). https://doi.org/10.1080/10407782.2017.1345270 Google Scholar - Siavashi, M., Blunt, M.J., Raisee, M., Pourafshary, P.: Three-dimensional streamline-based simulation of non-isothermal two-phase flow in heterogeneous porous media. Comput. Fluids
**103**, 116–131 (2014)Google Scholar - Siavashi, M., Bordbar, V., Rahnama, P.: Heat transfer and entropy generation study of non-Darcy double-diffusive natural convection in inclined porous enclosures with different source configurations. Appl. Therm. Eng.
**110**, 1462–1475 (2017b)Google Scholar - Siavashi, M., Ghasemi, K., Yousofvand, R., Derakhshan, S.: Computational analysis of SWCNH nanofluid-based direct absorption solar collector with a metal sheet. Sol. Energy
**170**, 252–262 (2018a). https://doi.org/10.1016/j.solener.2018.05.020 Google Scholar - Siavashi, M., Rasam, H., Izadi, A.: Similarity solution of air and nanofluid impingement cooling of a cylindrical porous heat sink. J. Therm. Anal. Calorim. (2018b). https://doi.org/10.1007/s10973-018-7540-0 Google Scholar
- Siavashi, M., Rostami, A.: Two-phase simulation of non-Newtonian nanofluid natural convection in a circular annulus partially or completely filled with porous media. Int. J. Mech. Sci.
**133**, 689–703 (2017)Google Scholar - Siavashi, M., Talesh Bahrami, H.R., Aminian, E.: Optimization of heat transfer enhancement and pumping power of a heat exchanger tube using nanofluid with gradient and multi-layered porous foams. Appl. Therm. Eng.
**138**, 465–474 (2018c). https://doi.org/10.1016/j.applthermaleng.2018.04.066 Google Scholar - Siavashi, M., Yousofvand, R., Rezanejad, S.: Nanofluid and porous fins effect on natural convection and entropy generation of flow inside a cavity. Adv. Powder Technol.
**29**(1), 142–156 (2018d). https://doi.org/10.1016/j.apt.2017.10.021 Google Scholar - Sillekens, J., Rindt, C., Van Steenhoven, A.: Developing mixed convection in a coiled heat exchanger. Int. J. Heat Mass Transf.
**41**(1), 61–72 (1998)Google Scholar - Tiwari, R.K., Das, M.K.: Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf.
**50**(9), 2002–2018 (2007)Google Scholar - Toosi, M.H., Siavashi, M.: Two-phase mixture numerical simulation of natural convection of nanofluid flow in a cavity partially filled with porous media to enhance heat transfer. J. Mol. Liq.
**238**, 553–569 (2017)Google Scholar - Varol, Y., Oztop, H.F., Pop, I.: Numerical analysis of natural convection for a porous rectangular enclosure with sinusoidally varying temperature profile on the bottom wall. Int. Commun. Heat Mass Transf.
**35**(1), 56–64 (2008)Google Scholar - Wang, B., Hong, Y., Wang, L., Fang, X., Wang, P., Xu, Z.: Development and numerical investigation of novel gradient-porous heat sinks. Energy Convers. Manag.
**106**, 1370–1378 (2015)Google Scholar - Wang, F., Tan, J., Shuai, Y., Gong, L., Tan, H.: Numerical analysis of hydrogen production via methane steam reforming in porous media solar thermochemical reactor using concentrated solar irradiation as heat source. Energy Convers. Manag.
**87**, 956–964 (2014a)Google Scholar - Wang, Q., Lei, H., Wang, S., Dai, C.: Natural convection around a pair of hot and cold horizontal microtubes at low Rayleigh numbers. Appl. Therm. Eng.
**72**(1), 114–119 (2014b)Google Scholar - Wang, S.-L., Li, X.-Y., Wang, X.-D., Lu, G.: Flow and heat transfer characteristics in double-layered microchannel heat sinks with porous fins. Int. Commun. Heat Mass Transf.
**93**, 41–47 (2018)Google Scholar - Wen, D., Ding, Y.: Experimental investigation into convective heat transfer of nanofluids at the entrance region under laminar flow conditions. Int. J. Heat Mass Transf.
**47**(24), 5181–5188 (2004)Google Scholar - Whitaker, S.: Flow in porous media I: a theoretical derivation of Darcy’s law. Transp. Porous Media
**1**(1), 3–25 (1986)Google Scholar - Xie, H., Wang, J., Xi, T., Liu, Y., Ai, F., Wu, Q.: Thermal conductivity enhancement of suspensions containing nanosized alumina particles. J. Appl. Phys.
**91**(7), 4568–4572 (2002)Google Scholar - Xu, Z., Qin, J., Zhou, X., Xu, H.: Forced convective heat transfer of tubes sintered with partially-filled gradient metal foams (GMFs) considering local thermal non-equilibrium effect. Appl. Therm. Eng.
**137**, 101–111 (2018)Google Scholar - Xuan, Y., Li, Q.: Heat transfer enhancement of nanofluids. Int. J. Heat Fluid Flow
**21**(1), 58–64 (2000)Google Scholar - Xuan, Y., Li, Q.: Investigation on convective heat transfer and flow features of nanofluids. J. Heat Transf.
**125**(1), 151–155 (2003). https://doi.org/10.1115/1.1532008 Google Scholar - Yaghoubi Emami, R., Siavashi, M., Shahriari Moghaddam, G.: The effect of inclination angle and hot wall configuration on Cu–water nanofluid natural convection inside a porous square cavity. Adv. Powder Technol.
**29**(3), 519–536 (2018). https://doi.org/10.1016/j.apt.2017.10.027 Google Scholar - Yang, C., Nakayama, A., Liu, W.: Heat transfer performance assessment for forced convection in a tube partially filled with a porous medium. Int. J. Therm. Sci.
**54**, 98–108 (2012)Google Scholar - Yousofvand, R., Derakhshan, S., Ghasemi, K., Siavashi, M.: MHD transverse mixed convection and entropy generation study of electromagnetic pump including a nanofluid using 3D LBM simulation. Int. J. Mech. Sci.
**133**, 73–90 (2017). https://doi.org/10.1016/j.ijmecsci.2017.08.034 Google Scholar - Zhang, F., Jacobi, A.M.: Aluminum surface wettability changes by pool boiling of nanofluids. Colloids Surf. A
**506**, 438–444 (2016). https://doi.org/10.1016/j.colsurfa.2016.07.026 Google Scholar