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Journal of Thermal Analysis and Calorimetry

, Volume 135, Issue 2, pp 1047–1067 | Cite as

Conjugate natural convection of nanofluids inside an enclosure filled by three layers of solid, porous medium and free nanofluid using Buongiorno’s and local thermal non-equilibrium models

  • S. A. M. Mehryan
  • Mohammad GhalambazEmail author
  • Mohsen Izadi
Article

Abstract

The natural convective heat transfer of nanofluids was addressed inside a square enclosure filled by three different layers: solid, porous medium and free fluid. The behavior of the porous layer has been simulated using local thermal non-equilibrium model. The Buongiorno’s model was utilized to evaluate the distribution of nanoparticles inside the enclosure that arose from the thermophoresis and Brownian motion. The governing equations were solved by the Galerkin finite element method in a non-uniform grid. The governing parameters are Rayleigh number Ra = 103–106, porosity ε = 0.3–0.9, Darcy number Da = 10−5–10−2, interface parameter Kr = 0.1–10, H = 0.1–1000; ratio of wall thermal conductivity to that of the nanofluid, Rk = 0.1–10, dimensionless length of the heater B = 0.2–0.8; dimensionless centre position height of the heater Z = 0.3–0.7 and Lewis number Le = 10–100. A considerable concentration gradient of nanoparticles was found inside the enclosure. In some studied cases, the non-dimensional volume fraction of nanoparticles is about 10% higher than the average volume fraction of nanoparticles at the region near the cold wall. The variability of Darcy and the Rayleigh numbers indicated significant effects on heat transfer rate and the concentration patterns of the nanoparticles and inward the cavity. The increase in Le and Nr amplifies and decreases the heat transfer rates through fluid and solid phases, respectively. In addition, it can be seen that the increment in heat transfer rates with Le increases as Nr increases.

Keywords

Buongiorno’s model Local thermal non-equilibrium Porous medium layer Free fluid layer Solid layer 

List of symbols

Latin symbols

b

Length of the heater (m)

B

Dimensionless length of the heater

C

Nanoparticle volume fraction

C0

Ambient nanoparticle volume fraction

d

Wall thickness (m)

D

Dimensionless wall thickness

Da

Darcy number

DB

Brownian diffusion coefficient

DT

Thermophoresis diffusion coefficient

g

Gravitational acceleration vector (m s−2)

hnfs

Volumetric heat transfer coefficient between the nanofluid and solid porous matrix (W m−3 K−1)

H

Interface heat transfer coefficient parameter

k

Thermal conductivity (W m−1 K−1)

K

Permeability of the porous medium (m2)

Kr

Nanofluid to solid porous matrix thermal conductivity ratio parameter

L

Square cavity size (m)

Le

Lewis number

n

Normal vector (m)

N

Dimensionless normal vector

Nb

Brownian motion parameter

Nr

Buoyancy ratio parameter

Nt

Thermophoresis parameter

Nu

Local Nusselt number

\( \overline{Nu} \)

Average Nusselt number

p

Pressure (Pa)

P

Dimensionless pressure

Pr

Prandtl number

\( q_{\text{i}} \)

Total interfacial heat flux (W m−2)

Qw

Dimensionless local heat transfer through the wall

\( \overline{{Q_{\text{w}} }} \)

Dimensionless average heat transfer through the wall

Ra

Rayleigh number

Rk

Wall to nanofluid thermal conductivity ratio parameter

s

Porous layer thickness (m)

S

Dimensionless porous layer thickness

Sh

Local Sherwood number

T

Temperature (K)

u, v

Velocity components along x, y directions, respectively (m s−1)

U, V

Dimensionless velocity components along x, y directions, respectively

x, y

Cartesian coordinates (m)

X, Y

Dimensionless Cartesian coordinates

z

Center position height of the heater (m)

Z

Dimensionless center position height of the heater

Greek symbols

α

Effective thermal diffusivity (m2 s−1)

β

Thermal expansion coefficient of the fluid (K−1)

Δ

Difference value

ε

Porosity of the porous medium

θ

Dimensionless temperature

μ

Dynamic viscosity (kg m−1 s−1)

ν

Kinematic viscosity (m2 s−1)

ρ

Density (kg m−3)

(ρc)

Effective heat capacity (J K−1 m−3)

τ

Parameter defined by τ = (ρc)p/(ρc)nf

ϕ

Relative nanoparticle volume fraction

Ψ

Dimensionless stream function

Subscripts

0

Ambient property

c

Cold

eff

Effective

h

Hot

max

Maximum

nf

Nanofluid

p

Nanoparticle

s

Solid porous matrix

w

Wall

Introduction

Natural convection heat transfer arises from a change in the density of a working fluid, where there is no need for any external power or moving parts to induce a flow. Hence, the natural convection mechanism is simple and safe. In natural convection, the fluid motion is smooth and slow. Thus, there are very low noise levels in heat transfer devices manufactured based on such mechanism. Therefore, this heat transfer procedure inside a cavity has been the subject of many recent studies due to its important advantages. However, the main disadvantage of the natural convection is the complexity of the design and also its low capacity of heat transfer. Therefore, further investigations on the enhancement of the heat transfer via free convection mechanism are still demanded. Studies and applications of the natural convection can be seen in [1, 2, 3, 4].

One method to improve the heat transfer rate is utilization of extended surfaces. In recent years, some researchers have applied very thermal conductive metals such as aluminum and copper to develop a porous medium structure, metal foam, with both high porosity and effective surface area [5, 6, 7, 8, 9]. In general, two main approaches exist for modeling heat transfer through porous media: the local thermal equilibrium (LTE) and the local thermal non-equilibrium (LTNE) approach [10]. For LTE approach, it is assumed that the pore solid walls and the captured fluid are at the same temperature, and hence, only one temperature represents the temperature of both domains [11]. However, when the thermal interplay between the pore walls and connected fluid is low or when the thermal conductivity of solid matrix is far higher than the thermal conductivity of the fluid, the temperature of the solid matrix may significantly differ from that of the fluid captured the pores. In such cases, two different domains should be defined to investigate thermal behavior of porous media. One domain represents the temperature of the solid matrix, and the other one denotes the temperature of the fluid captured the pores. Hence, LTNE models are also propounded as two-equation models [12, 13]. Considering natural convection heat transfer in metal foams, the LTNE models are demanded due to fact that the thermal conductivity of a metal foam is much higher than that of typical working fluids. Moreover, in natural convection flows, the fluid velocity is low and a low thermal interplay between the fluid and solid matrix can be also expected.

There are numerous researches in the literature, addressing the natural convection heat transfer in porous enclosures using LTE and LTNE models [14, 15, 16]. The books by Nield and Bejan [10] as well as Vafai [17] provide a thorough review of these pieces of research. There are only few studies that have considered the conjugate conduction-natural convection heat transfer through porous media using LTNE model [18]. Indeed, by considering conjugate heat transfer using LTNE, there are difficulties in modeling and satisfying the temperature continuity and energy balance at the interface of different three layers located inside the cavity [19, 20].

Another way to enhance the heat transfer is using enhanced thermal conductivity of the working fluid. Experiments show that scattering a small volume fraction of nanoparticles in a conventional fluid can enhance the overall thermal conductivity of the resulted mixture [21, 22, 23, 24, 25, 26, 27, 28]. A synthesized stable mixture of nanoparticles and a conventional thermal fluid is known as a nanofluid. The nanofluids also seem promising for enhancing convection heat transfer across a wide range of industrial applications [29].

According to one perspective, there are two models for the study of heat transfer of nanofluids. One is the homogeneous mixture model, known as the static model, for which a uniform mixture of nanoparticles and the base fluid is assumed. In the homogeneous mixture, model no relative movement between the nanoparticles and base fluid is allowed [30, 31, 32, 33]. The other more advanced model, which can represent the behavior of the nanoparticles, is the mixture model, proposed by Buongiorno. Buongiorno’s model considers a relative movement between the nanoparticles and the base fluid [34, 35, 36, 37]. Buongiorno has concluded that the thermophoresis and Brownian forces, action on the nanoparticles in a nanofluid, are the important nanoscale mechanisms for nanofluids and can result in a significant concentration gradient of nanoparticles. The thermophoresis force usually carries the nanoparticles along a direction across from the temperature gradient. Conversely, the Brownian motion force typically homogenizes the nanoparticles inside the base fluid. Hence, a nanofluid which experiences a temperature gradient would experience a concentration gradient [38, 39, 40, 41, 42]. As a result, the concentration gradient of the heavy nanoparticles can affect the buoyancy forces and other aspects of flow and heat transfer in the nanofluids [43, 44].

Considering the homogeneous model of nanofluids, Alsabery et al. [45] have performed the conjugate conduction-natural convection heat transfer inside a trapezoidal enclosure. Alsabery et al. [45] reported that the average Nusselt number significantly increased with increasing nanoparticle volume fractions. This arises from higher thermal conductivity of the nanoparticles compared to base fluid. Alsabery et al. [46] have also studied the transient natural convection of nanofluids inside a trapezoidal cavity. Sheikholeslami and Ganji [47] also Sheikholeslami and Chamkha [48] have addressed the natural convection heat transfer of nanofluids in a collector enclosure with sine shape walls subjected to a magnetic field on. Bondareva et al. [49] have investigated the conjugate heat transfer of nanofluids inside a porous cavity. Considering the Buongiorno’s model, Sheikholeslami and Chamkha [50] as well as Reddy and Chamkha [51] and Reddy et al. [52] have analyzed the force convective heat transfer of nanofluids over confined. There are also very recent studies that have analyzed the natural convection heat transfer of nanofluids inside a cavity using the Buongiorno’s mathematic model such as [42, 53, 54]. As a summary, the convection heat transfer of nanofluids through porous media has been studied in recent researches of Kasaeian et al. [55] and Sheikholeslami [56, 57].

Systems that consist of multilayer media, a layer of solid shell wall, a layer of saturated porous medium and a free fluid layer are very common in industrial applications. For instance, an enclosure metal tank partially filled with grains is a common example of such a system, where the metal sheet of the tank shell is the solid layer, the grains are the porous medium layer, and the free space over the grains is the free layer. Although there are a wide range of industrial applications for such systems, there are only few studies, which have addressed the convective heat transfer in multilayer systems.

Considering a cavity partially saturated by a porous medium, the mixed convection of nanofluids in a partly layered porous enclosure with an internally revolving cylinder by Chamkha et al. [58] using the homogeneous model of nanofluids and assuming LTE model. Chamkha and Ismael [59] have addressed the natural convection heat transfer of nanofluids in a cavity partly saturated with a layer of porous medium filled by a nanofluid. The nanofluid was modeled as a homogeneous mixture, where the porous matrix and the nanofluid were considered to be in local thermal equilibrium. Hence, there are no concentration gradients of nanoparticles. In addition, no temperature difference between the fluid and the solid matrix is allowed. The natural convection of nanofluids inside a tilted trapezoid enclosure partly filled with a layer of porous medium has been examined by Alsabery et al. [60]. Further, in a very recent study [59], Ismael and Chamkha [61] have addressed the nanofluids’ convection heat transfer in a square cavity. The top and bottom walls of the cavity were well insulated, and the right vertical side wall was at the isothermal temperature of Th, while the left wall was at the isothermal cold temperature of Tc. There was a layer of solid over the hot wall, and then there was a layer of a porous medium over the solid layer, and finally there was a layer of free fluid between the cold wall and the porous layer. As with [59, 60], Ismael and Chamkha [61] also have utilized the homogeneous model of nanofluids using local thermal equilibrium (LTE) model.

Following the study of Ismael and Chamkha [61], in the present study, a non-homogeneous model of nanofluids, Buongiorno’s model, is adopted to capture the concentration gradients of nanoparticles. In addition, LTNE model is adopted to model the conjugate heat transfer inside the porous layer. The main physical contributions of the present study are (a) considering a concentration gradient for nanoparticles and (b) considering the temperature discrepancy between two elements of the porous medium layer. The aim of the study is to analyze the conjugate heat transfer of nanofluids inside a cavity filled by multilayers using LTNE and Buongiorno’s model.

Definition of the problem and mathematical formulation

Figure 1 demonstrates a schema of the physical model for conjugate natural convection heat transfer in a multilayer cavity. The studied square domain with the length of L was divided into three parts of solid, porous medium and nanofluid, where the width of the solid and the porous medium are denoted by s and d, respectively. Here, s and d were adopted as 0.1L and 0.45L, respectively. As pictured in Fig. 1, a flash heater with hot temperature Th and length b was embedded on the left wall with the center position height of z. The other parts of the left wall along with the entire top and bottom walls were well insulated. The entire right wall was held at the isothermal cold temperature Tc.
Fig. 1

Schematic view of the present problem

The following physical points have been taken into consideration in modeling the conjugate heat transfer in the cavity. The condition of no-slip boundary remained true on the solid surfaces. The porous medium included a solid matrix, and the pores were filled with an incompressible nanofluid such that all the pores of the porous region were occupied by the nanofluid. The solid matrix was assumed as isotropic and homogenous. In addition, the nanofluid was assumed Newtonian and its flow in the pores was laminar. Due to very low size of nanoparticles, the nanoparticles and the base fluid were assumed to be at the same temperature. However, there was a temperature difference between the porous matrix and the nanofluid inside the pores, and hence, the local thermal non-equilibrium model was used to consider heat transport through the porous medium. In the cases such as the metal foams, in which the thermal conductivity of the porous matrix is much higher than the thermal conductivity of the fluid inside the pores, the temperature of the porous matrix can significantly differ from the temperature of the fluid inside the pores due to the heat transfer through the porous matrix. This temperature difference can be boosted in a situation in which the convective heat transfer coefficient between the fluid and porous matrix structures is low. Such a low convective heat transfer coefficient can be seen in natural convective heat transfer where the fluid velocities are low. Hence, in such cases the LTNE model is more accurate and has been adopted in the present study. Moreover, it was assumed that the solid nanoparticles were always stable and suspended in the base fluid, meaning that there was no sedimentation and accumulation of the nanoparticles. Apart from density, all the thermophysical characteristics of the nanofluid were regarded as constant. The impact of the buoyancy volume force was considered using Boussinesq approximation model. The gravity acceleration vector acted in the direction of the negative y, as revealed in Fig. 1. In this study, the non-homogenous dispersion of solid nanoparticles in the fluid was modeled using Buongiorno’s model [34]. Buongiorno [34], using the scale analysis, has discussed several nanoscale forces which can affect the movement of nanoparticles in a nanofluid. Among these forces, the thermophoresis and Brownian motion forces found to be important. The thermophoresis force tends to move the nanoparticles from hot to the cold due to the difference in momentum of molecules in the hot and cold sides of a nanoparticle. In contrast, the Brownian motion tends to uniform the nanoparticles in the nanofluid. Since Buongiorno’s model [34] includes the migration effect of nanofluids, this model has been adopted in the present study to simulate the concentration gradients of nanoparticles as well as flow and heat transfer in the cavity.

Considering the assumptions mentioned above, the set of governing equation for the nanofluid flow and heat transfer in the free layer were written as follows [10, 34, 62]:
$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 $$
(1)
$$ \rho_{\text{nf}} \left( {u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}} \right) = - \frac{\partial p}{\partial x} + \mu_{\text{nf}} \left( {\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} u}}{{\partial y^{2} }}} \right) $$
(2)
$$ \begin{aligned} \rho_{\text{nf}} \left( {u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y}} \right) & = - \frac{\partial p}{\partial y} + \mu_{\text{nf}} \left( {\frac{{\partial^{2} v}}{{\partial x^{2} }} + \frac{{\partial^{2} v}}{{\partial y^{2} }}} \right) \\ & \quad + \left[ { - \left( {\rho_{{{\text{p}},0}} - \rho_{{{\text{f}},0}} } \right)\left( {C - C_{0} } \right) + \left( {1 - C_{0} } \right)\rho_{{{\text{f}},0}} \beta \left( {T_{\text{nf}} - T_{\text{c}} } \right)} \right]g \\ \end{aligned} $$
(3)
$$ \begin{aligned} u\frac{{\partial T_{\text{nf}} }}{\partial x} + v\frac{{\partial T_{\text{nf}} }}{\partial y} & = \frac{{k_{\text{nf}} }}{{\left( {\rho c} \right)_{\text{nf}} }}\left( {\frac{{\partial^{2} T_{\text{nf}} }}{{\partial x^{2} }} + \frac{{\partial^{2} T_{\text{nf}} }}{{\partial y^{2} }}} \right) \\ & \quad + \tau \left\{ {D_{\text{B}} \left( {\frac{\partial C}{\partial x}\frac{{\partial T_{\text{nf}} }}{\partial x} + \frac{\partial C}{\partial y}\frac{{\partial T_{\text{nf}} }}{\partial y}} \right) + \frac{{D_{\text{T}} }}{{T_{\text{c}} }}\left[ {\left( {\frac{{\partial T_{\text{nf}} }}{\partial x}} \right)^{2} + \left( {\frac{{\partial T_{\text{nf}} }}{\partial y}} \right)^{2} } \right]} \right\} \\ \end{aligned} $$
(4)
$$ u\frac{\partial C}{\partial x} + v\frac{\partial C}{\partial y} = D_{\text{B}} \left( {\frac{{\partial^{2} C}}{{\partial x^{2} }} + \frac{{\partial^{2} C}}{{\partial y^{2} }}} \right) + \frac{{D_{\text{T}} }}{{T_{\text{c}} }}\left( {\frac{{\partial^{2} T_{\text{nf}} }}{{\partial x^{2} }} + \frac{{\partial^{2} T_{\text{nf}} }}{{\partial y^{2} }}} \right) $$
(5)
where the nomenclature section provides variables’ definitions. The governing equations of the flow of nanofluid and heat transfer inside the porous medium layer are developed as follows:
$$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 $$
(6)
$$ \frac{{\rho_{\text{nf}} }}{{\varepsilon^{2} }}\left( {u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y}} \right) = - \frac{\partial p}{\partial x} + \frac{{\mu_{\text{nf}} }}{\varepsilon }\left( {\frac{{\partial^{2} u}}{{\partial x^{2} }} + \frac{{\partial^{2} u}}{{\partial y^{2} }}} \right) - \frac{{\mu_{\text{nf}} }}{K}u $$
(7)
$$ \begin{aligned} \frac{{\rho_{\text{nf}} }}{{\varepsilon^{2} }}\left( {u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y}} \right) & = - \frac{\partial p}{\partial y} + \frac{{\mu_{\text{nf}} }}{\varepsilon }\left( {\frac{{\partial^{2} v}}{{\partial x^{2} }} + \frac{{\partial^{2} v}}{{\partial y^{2} }}} \right) - \frac{{\mu_{\text{nf}} }}{K}v \\ & \quad + \left[ { - \left( {\rho_{{{\text{p}},0}} - \rho_{{{\text{f}},0}} } \right)\left( {C - C_{0} } \right) + \left( {1 - C_{0} } \right)\rho_{{{\text{f}},0}} \beta \left( {T_{\text{nf}} - T_{\text{c}} } \right)} \right]g \\ \end{aligned} $$
(8)
$$ \begin{aligned} \frac{1}{\varepsilon }\left( {u\frac{{\partial T_{\text{nf}} }}{\partial x} + v\frac{{\partial T_{\text{nf}} }}{\partial y}} \right) & = \alpha_{\text{nf}} \left( {\frac{{\partial^{2} T_{\text{nf}} }}{{\partial x^{2} }} + \frac{{\partial^{2} T_{\text{nf}} }}{{\partial y^{2} }}} \right) + \tau \left\{ {D_{\text{B}} \left( {\frac{\partial C}{\partial x}\frac{{\partial T_{\text{nf}} }}{\partial x} + \frac{\partial C}{\partial y}\frac{{\partial T_{\text{nf}} }}{\partial y}} \right)} \right. \\ & \quad + \left( {\frac{{D_{\text{T}} }}{{T_{\text{c}} }}} \right)\left. {\left[ {\left( {\frac{{\partial T_{\text{nf}} }}{\partial x}} \right)^{2} + \left( {\frac{{\partial T_{\text{nf}} }}{\partial y}} \right)^{2} } \right]} \right\} + \frac{{h_{\text{nfs}} \left( {T_{\text{s}} - T_{\text{nf}} } \right)}}{{\varepsilon \left( {\rho \,c} \right)_{\text{nf}} }} \\ \end{aligned} $$
(9)
$$ 0 = \alpha \,\left( {\frac{{\partial^{2} T_{\text{s}} }}{{\partial x^{2} }} + \frac{{\partial^{2} T_{\text{s}} }}{{\partial y^{2} }}} \right) + \frac{{h_{\text{nfs}} }}{{\left( {1 - \varepsilon } \right)\left( {\rho \,c} \right)_{\text{s}} }}\left( {T_{\text{nf}} - T_{\text{s}} } \right) $$
(10)
$$ \frac{1}{\varepsilon }\,\left( {u\frac{\partial C}{\partial x} + v\frac{\partial C}{\partial y}} \right) = D_{\text{B}} \left( {\frac{{\partial^{2} C}}{{\partial x^{2} }} + \frac{{\partial^{2} C}}{{\partial y^{2} }}} \right) + \left( {\frac{{D_{\text{T}} }}{{T_{\text{c}} }}} \right)\left( {\frac{{\partial^{2} T_{\text{nf}} }}{{\partial x^{2} }} + \frac{{\partial^{2} T_{\text{nf}} }}{{\partial y^{2} }}} \right) $$
(11)
Finally, the following equation can be stated for the solid impermeable wall:
$$ \frac{{\partial^{2} T_{\text{w}} }}{{\partial x^{2} }} + \frac{{\partial^{2} T_{\text{w}} }}{{\partial y^{2} }} = 0 $$
(12)
The corresponding boundary conditions for free nanofluid—porous interface, i.e. x = s + d, can be written as follows [63, 64]:
$$ \begin{aligned} & u_{{{\text{free}}\;{\text{nanofluid}}}} = u_{\text{porous}} ,\quad v_{{{\text{free}}\;{\text{nanofluid}}}} = v_{\text{porous}} \\ &\left. { \mu_{\text{nf}} \frac{\partial u}{\partial n}} \right|_{{{\text{free}}\;{\text{nanofluid}}}} = \left. {\mu_{{{\text{nf}},{\text{eff}}}} \frac{\partial u}{\partial n}} \right|_{\text{porous}} ,\quad \left. {\mu_{\text{nf}} \frac{\partial v}{\partial n}} \right|_{{{\text{free}}\;{\text{nanofluid}}}} = \left. {\mu_{{{\text{nf}},{\text{eff}}}} \frac{\partial v}{\partial n}} \right|_{\text{porous}} \\ &\left. { T_{\text{nf}} } \right|_{{{\text{free}}\;{\text{nanofluid}}}} = \left. {T_{\text{nf}} } \right|_{\text{porous}} = \left. {T_{\text{s}} } \right|_{\text{porous}} \\ &\left. { k_{\text{nf}} \frac{{\partial T_{\text{nf}} }}{\partial n}} \right|_{{{\text{free}}\;{\text{nanofluid}}}} = \left. {k_{{{\text{nf}},{\text{eff}}}} \frac{{\partial T_{\text{nf}} }}{\partial n}} \right|_{\text{porous}} + \left. {k_{{{\text{s}},{\text{eff}}}} \frac{{\partial T_{\text{s}} }}{\partial n}} \right|_{\text{porous}} = q_{i} \\ & C_{{{\text{free}}\;{\text{nanofluid}}}} = C_{\text{porous}} ,\quad \left. {\frac{\partial C}{\partial n}} \right|_{{{\text{free}}\;{\text{nanofluid}}}} = \left. {\frac{\partial C}{\partial n}} \right|_{\text{porous}} \\ \end{aligned} $$
(13)
where \( k_{\text{nf,eff}} = \varepsilon k_{\text{nf}} ,\quad k_{\text{s,eff}} = \left( {1 - \varepsilon } \right)k_{\text{s}} ,\quad \mu_{\text{nf,eff}} = \frac{{\mu_{\text{nf}} }}{\varepsilon }. \)

The third line of the above boundary conditions, Eq. (13), indicates that the temperature of the nanofluid inside the pores, porous matrix and the nanofluid in the clear flow are equal. This assumption can be true due to the fact that the interaction between the fluid and the porous matrix is very high in the interface of the porous medium and clear fluid region. The fourth line indicates that the amount of heat which reaches the clear fluid layer is equal to the amount of heat that carries on through the porous matrix medium layer and the amount of heat that reached the interface due to the fluid inside the pores.

In addition, the boundary conditions associated with the wall-porous interface, i.e. x = s, is given by [18, 64]:
$$ \begin{aligned} & \left. {T_{\text{w}} } \right|_{\text{wall}} = \left. {T_{\text{nf}} } \right|_{\text{porous}} = \left. {T_{\text{s}} } \right|_{\text{porous}} \\ &\left. { k_{\text{w}} \frac{{\partial T_{\text{w}} }}{\partial n}} \right|_{{\,{\text{wall}}}} = \left. {k_{\text{nf,eff}} \frac{{\partial T_{\text{nf}} }}{\partial n}} \right|_{\text{porous}} + \left. {k_{\text{s,eff}} \frac{{\partial T_{\text{s}} }}{\partial n}} \right|_{\text{porous}} = q_{i} \\ &\left. { D_{\text{B}} \frac{\partial C}{\partial n}} \right|_{\text{porous}} + \frac{{D_{\text{T}} }}{{T_{\text{c}} }}\left. {\frac{{\partial T_{\text{nf}} }}{\partial n}} \right|_{\text{porous}} = 0 \\ \end{aligned} $$
(14)
The first line of the above boundary conditions, Eq. (14), indicates that the temperature of the nanofluid inside the pores, porous matrix and the solid wall are equal. This assumption can be true due to the fact that both three layers have merged at the wall interface. Indeed, the fluid velocity at the wall surface is considered as zero, and the porous matrix is also unified with the wall. Hence, assuming a uniform equal temperature for the solid wall, porous matrix and the quiescent nanofluid in the pores next to the wall is a rational assumption. The second line demonstrates the energy balance at the interface surface of the solid wall and porous region. This equation indicates that the amount of heat which reaches the interface through the solid wall is equal to the amount of heat that reaches the porous matrix and the fluid in the pores.
Eventually, the boundary conditions applied to the external walls of the cavity are.
$$ \begin{aligned} & \left\{ {\begin{array}{*{20}l} {T_{\text{w}} \left( {0,y} \right) = T_{\text{h}} } \hfill & {z - b/2 \le y \le z + b/2} \hfill \\ {\frac{{\partial T_{\text{w}} \left( {0,y} \right)}}{\partial x} = 0} \hfill & {y \ge z + b/2\;{\text{and}}\;y \le z - b/2} \hfill \\ \end{array} } \right. \\ & u\left( {L,y} \right) = v\left( {L,y} \right) = 0,\quad T_{\text{nf}} \left( {L,y} \right) = T_{\text{c}} ,\quad D_{\text{B}} \frac{{\partial C\left( {L,y} \right)}}{\partial x} + \frac{{D_{\text{T}} }}{{T_{\text{c}} }}\frac{{\partial T_{\text{nf}} \left( {L,y} \right)}}{\partial x} = 0 \\ & u\left( {X,0} \right) = v\left( {X,0} \right) = 0,\quad \frac{{\partial T_{\text{w}} \left( {x,0} \right)}}{\partial y} = \frac{{\partial T_{\text{nf}} \left( {x,0} \right)}}{\partial y} = \frac{{\partial T_{\text{s}} \left( {x,0} \right)}}{\partial y} = 0,\quad \frac{{\partial C\left( {x,0} \right)}}{\partial y} = 0 \\ & u\left( {x,L} \right) = v\left( {x,L} \right) = 0,\quad \frac{{\partial T_{\text{w}} \left( {x,L} \right)}}{\partial y} = \frac{{\partial T_{\text{nf}} \left( {x,L} \right)}}{\partial y} = \frac{{\partial T_{\text{s}} \left( {x,L} \right)}}{\partial y} = 0,\quad \frac{{\partial C\left( {x,L} \right)}}{\partial y} = 0 \\ \end{aligned} $$
(15)
According to [41, 65], the boundary condition of \( D_{\text{B}} \frac{{\partial C\left( {L,y} \right)}}{\partial x} + \frac{{D_{\text{T}} }}{{T_{\text{c}} }}\frac{{\partial T_{\text{nf}} \left( {L,y} \right)}}{\partial x} \) represents the zero flux of nanoparticles on the wall surfaces. A homogeneous uniform concentration of nanoparticles (C0) is assumed in the cavity. As the walls are impermeable to the base fluid and nanoparticles, the following constraint should be hold true in all times, with the steady state solution: \( \frac{1}{{L^{2} }}\int\limits_{0}^{1} {\int\limits_{0}^{1} {C\;{\text{d}}x{\text{d}}y = C_{0} } } \). In order to non-dimensionalize the governing Eqs. (1)–(12) and the corresponding boundary conditions (13)–(15), the following dimensionless parameters are utilized:
$$ \begin{aligned} & X = \frac{x}{L},\quad Y = \frac{y}{L},\quad \, D = \frac{d}{L},\quad \, S = \frac{s}{L},\quad \, B = \frac{b}{L}, \, \quad \, Z = \frac{z}{L} ,\quad \, U = \frac{uL}{{\alpha_{nf} }},\quad \, V = \frac{vL}{{\alpha_{\text{nf}} }}, \\ & P = \frac{{pL^{2} }}{{\rho_{\text{nf}} \alpha_{\text{nf}}^{2} }}, \, \quad \, Pr = \frac{{\nu_{\text{nf}} }}{{\alpha_{\text{nf}} }}, \, Da = \frac{K}{{L^{2} }}, \, \phi = \frac{C}{{C_{0} }}, \, Ra = \frac{{\left( {1 - C_{0} } \right)\rho_{{{\text{f}},0}} g\beta {\kern 1pt} \Delta TL^{3} }}{{\alpha_{\text{nf}} \mu_{\text{nf}} }}, \\ & \theta_{\text{nf}} = {{\left( {T_{\text{nf}} - T_{\text{c}} } \right)} \mathord{\left/ {\vphantom {{\left( {T_{\text{nf}} - T_{\text{c}} } \right)} {\Delta T}}} \right. \kern-0pt} {\Delta T}} , { }\quad \theta_{\text{s}} = {{\left( {T_{\text{s}} - T_{\text{c}} } \right)} \mathord{\left/ {\vphantom {{\left( {T_{\text{s}} - T_{\text{c}} } \right)} {\Delta T}}} \right. \kern-0pt} {\Delta T}} , { }\quad \theta_{\text{w}} = {{\left( {T_{\text{w}} - T_{\text{c}} } \right)} \mathord{\left/ {\vphantom {{\left( {T_{\text{w}} - T_{\text{c}} } \right)} {\Delta T}}} \right. \kern-0pt} {\Delta T}} , { } \\ & Nr = \frac{{\left( {\rho_{\text{p,0}} - \rho_{\text{f,0}} } \right)C_{0} }}{{\rho_{\text{f,0}} \beta {\kern 1pt} \Delta T\left( {1 - C_{0} } \right)}} ,\quad Nb = \frac{{\tau D_{\text{B}} C_{0} }}{{\alpha_{\text{nf}} }} , { }Nt = \frac{{\tau D_{\text{T}} \Delta T}}{{\alpha_{\text{nf}} \,T_{\text{c}} }}, \\ & H = \frac{{h_{\text{nfs}} L^{2} }}{{k_{\text{nf}} }} , { }\quad K_{\text{r}} = \frac{{k_{\text{nf}} }}{{\left( {1 - \varepsilon } \right)k_{\text{s}} }}, \, \quad \, Le = \frac{{\alpha_{\text{nf}} }}{{D_{\text{B}} }} , { }R_{\text{k}} = \frac{{k_{\text{w}} }}{{k_{\text{nf}} }} \\ \end{aligned} $$
(16)
where \( \Delta T = T_{\text{h}} - T_{\text{c}} \). Based on the non-dimensional variables of Eq. (16), the dimensionless forms of Eqs. (1)–(12) are obtained as follows:
Nanofluid domain:
$$ \frac{\partial U}{\partial X} + \frac{\partial V}{\partial Y} = 0 $$
(17)
$$ U\frac{\partial U}{\partial X} + V\frac{\partial U}{\partial Y} = - \frac{\partial P}{\partial X} + Pr\left( {\frac{{\partial^{2} U}}{{\partial X^{2} }} + \frac{{\partial^{2} U}}{{\partial Y^{2} }}} \right) $$
(18)
$$ U\frac{\partial V}{\partial X} + V\frac{\partial V}{\partial Y} = - \frac{\partial P}{\partial Y} + Pr\left( {\frac{{\partial^{2} V}}{{\partial X^{2} }} + \frac{{\partial^{2} V}}{{\partial Y^{2} }}} \right) - Ra \cdot Pr \cdot Nr\left( {\phi - 1} \right) + Ra \cdot Pr \cdot \theta_{\text{nf}} $$
(19)
$$ U\frac{{\partial \theta_{\text{nf}} }}{\partial X} + V\frac{{\partial \theta_{\text{nf}} }}{\partial Y} = \frac{{\partial^{2} \theta_{\text{nf}} }}{{\partial X^{2} }} + \frac{{\partial^{2} \theta_{\text{nf}} }}{{\partial Y^{2} }} + Nb\left( {\frac{\partial \phi }{\partial X}\frac{{\partial \theta_{\text{nf}} }}{\partial X} + \frac{\partial \phi }{\partial Y}\frac{{\partial \theta_{\text{nf}} }}{\partial Y}} \right) + Nt\left[ {\left( {\frac{{\partial \theta_{\text{nf}} }}{\partial X}} \right)^{2} + \left( {\frac{{\partial \theta_{\text{nf}} }}{\partial Y}} \right)^{2} } \right] $$
(20)
$$ U\frac{\partial \phi }{\partial X} + V\frac{\partial \phi }{\partial Y} = \frac{1}{Le}\left( {\frac{{\partial^{2} \phi }}{{\partial X^{2} }} + \frac{{\partial^{2} \phi }}{{\partial Y^{2} }}} \right) + \frac{Nt}{Le \cdot Nb}\left( {\frac{{\partial^{2} \theta_{\text{nf}} }}{{\partial X^{2} }} + \frac{{\partial^{2} \theta_{\text{nf}} }}{{\partial Y^{2} }}} \right) $$
(21)
Porous medium layer:
$$ \frac{\partial U}{\partial X} + \frac{\partial V}{\partial Y} = 0 $$
(22)
$$ \frac{1}{{\varepsilon^{2} }}\left( {U\frac{\partial U}{\partial X} + V\frac{\partial U}{\partial Y}} \right) = - \frac{\partial P}{\partial X} + \frac{Pr}{\varepsilon }\left( {\frac{{\partial^{2} U}}{{\partial X^{2} }} + \frac{{\partial^{2} U}}{{\partial Y^{2} }}} \right) - \frac{Pr}{Da}U $$
(23)
$$ \frac{1}{{\varepsilon^{2} }}\left( {U\frac{\partial V}{\partial X} + V\frac{\partial V}{\partial Y}} \right) = - \frac{\partial P}{\partial Y} + \frac{Pr}{\varepsilon }\left( {\frac{{\partial^{2} V}}{{\partial X^{2} }} + \frac{{\partial^{2} V}}{{\partial Y^{2} }}} \right) - \frac{Pr}{Da}V - Ra \cdot Pr \cdot Nr\left( {\phi - 1} \right) + Ra \cdot Pr \cdot \theta_{\text{nf}} $$
(24)
$$ \begin{aligned} U\frac{{\partial \theta_{\text{nf}} }}{\partial X} + V\frac{{\partial \theta_{\text{nf}} }}{\partial Y} & = \varepsilon \left( {\frac{{\partial^{2} \theta_{\text{nf}} }}{{\partial X^{2} }} + \frac{{\partial^{2} \theta_{\text{nf}} }}{{\partial Y^{2} }}} \right) + Nb \cdot \varepsilon \left( {\frac{\partial \phi }{\partial X}\frac{{\partial \theta_{\text{nf}} }}{\partial X} + \frac{\partial \phi }{\partial Y}\frac{{\partial \theta_{\text{nf}} }}{\partial Y}} \right) \\ & + Nt \cdot \varepsilon \left[ {\left( {\frac{{\partial \theta_{\text{nf}} }}{\partial X}} \right)^{2} + \left( {\frac{{\partial \theta_{\text{nf}} }}{\partial Y}} \right)^{2} } \right] + H\left( {\theta_{\text{s}} - \theta_{\text{nf}} } \right) \\ \end{aligned} $$
(25)
$$ 0 = \frac{{\partial^{2} \theta_{\text{s}} }}{{\partial X^{2} }} + \frac{{\partial^{2} \theta_{\text{s}} }}{{\partial Y^{2} }} + H \cdot K_{\text{r}} \left( {\theta_{\text{nf}} - \theta_{\text{s}} } \right) $$
(26)
$$ U\frac{\partial \phi }{\partial X} + V\frac{\partial \phi }{\partial Y} = \frac{\varepsilon }{Le}\left( {\frac{{\partial^{2} \phi }}{{\partial X^{2} }} + \frac{{\partial^{2} \phi }}{{\partial Y^{2} }}} \right) + \frac{Nt \cdot \varepsilon }{Le \cdot Nb}\left( {\frac{{\partial^{2} \theta_{nf} }}{{\partial X^{2} }} + \frac{{\partial^{2} \theta_{nf} }}{{\partial Y^{2} }}} \right) $$
(27)
Solid wall layer:
$$ \frac{{\partial^{2} \theta_{\text{w}} }}{{\partial X^{2} }} + \frac{{\partial^{2} \theta_{\text{w}} }}{{\partial Y^{2} }} = 0 $$
(28)
In addition, the boundary conditions in dimensionless state are obtained as follows:
Free nanofluid-porous interface boundary conditions:
$$ \begin{aligned} & U_{{{\text{free}}\;{\text{nanofluid}}}} = U_{\text{porous}} ,\quad V_{{{\text{free}}\;{\text{nanofluid}}}} = V_{\text{porous}} \\ & \left. {\frac{\partial U}{\partial N}} \right|_{{{\text{free}}\;{\text{nanofluid}}}} = \left. {\frac{1}{\varepsilon }\frac{\partial U}{\partial N}} \right|_{\text{porous}} ,\quad \left. {\frac{\partial V}{\partial N}} \right|_{{{\text{free}}\;{\text{nanofluid}}}} = \left. {\frac{1}{\varepsilon }\frac{\partial V}{\partial N}} \right|_{\text{porous}} \\ & \left. {\theta_{\text{nf}} } \right|_{{{\text{free}}\;{\text{nanofluid}}}} = \left. {\theta_{\text{nf}} } \right|_{\text{porous}} = \left. {\theta_{s} } \right|_{\text{porous}} \\ & \left. {\frac{{\partial \theta_{\text{nf}} }}{\partial N}} \right|_{{{\text{free}}\;{\text{nanofluid}}}} = \left. {\varepsilon \frac{{\partial \theta_{\text{nf}} }}{\partial N}} \right|_{\text{porous}} + \left. {K_{\text{r}}^{ - 1} \frac{{\partial \theta_{\text{s}} }}{\partial N}} \right|_{\text{porous}} = Q_{\text{i}} \\ & \phi_{{{\text{free}}\;{\text{nanofluid}}}} = \phi_{\text{porous}} ,\quad \left. {\frac{\partial \phi }{\partial N}} \right|_{{{\text{free}}\;{\text{nanofluid}}}} = \left. {\frac{\partial \phi }{\partial N}} \right|_{\text{porous}} \\ \end{aligned} $$
(29)
where \( Q_{\text{i}} = q_{\text{i}} L/k_{\text{nf}} \Delta T \).
The wall-porous interface boundary condition is:
$$ \begin{aligned} & \left. {\theta_{\text{w}} } \right|_{\text{wall}} = \left. {\theta_{\text{nf}} } \right|_{\text{porous}} = \left. {\theta_{\text{s}} } \right|_{\text{porous}} \\ & \left. {R_{\text{k}} \frac{{\partial \theta_{\text{w}} }}{\partial N}} \right|_{\text{wall}} = \left. {\varepsilon \frac{{\partial \theta_{\text{nf}} }}{\partial N}} \right|_{\text{porous}} + \left. {K_{\text{r}}^{ - 1} \frac{{\partial \theta_{\text{s}} }}{\partial N}} \right|_{\text{porous}} = \frac{{q_{\text{i}} L}}{{k_{\text{nf}} \Delta T}} = Q_{\text{i}} \\ & \left. {Nb\frac{\partial \phi }{\partial N}} \right|_{\text{porous}} + Nt\left. {\frac{{\partial \theta_{\text{nf}} }}{\partial N}} \right|_{\text{porous}} = 0 \\ \end{aligned} $$
(30)
Other boundary conditions on the vertical and horizontal walls include:
$$ \begin{aligned} & \left\{ {\begin{array}{*{20}c} {\theta_{\text{w}} \left( {0,Y} \right){ = 1}} & {Z - B/2 \le Y \le Z + B/2} \\ {\frac{{\partial \theta_{\text{w}} \left( {0,Y} \right)}}{\partial X} = 0} & {Y \ge Z + B/2{\text{ or Y}} \le Z - B/2} \\ \end{array} } \right. \\ & U\left( {1,Y} \right) = V\left( {1,Y} \right) = 0 ,\quad \theta_{\text{nf}} \left( {1,Y} \right){ = 0,}\quad Nb\frac{{\partial \phi \left( {1,Y} \right)}}{\partial X} + Nt\frac{{\partial \theta_{\text{nf}} \left( {1,Y} \right)}}{\partial X} = 0 \\ & U\left( {X,0} \right) = V\left( {X,0} \right) = 0,\quad \frac{{\partial \theta_{\text{w}} \left( {X,0} \right)}}{\partial Y} = \frac{{\partial \theta_{\text{nf}} \left( {X,0} \right)}}{\partial Y} = \frac{{\partial \theta_{\text{s}} \left( {X,0} \right)}}{\partial Y} = 0,\quad \frac{{\partial \phi \left( {X,0} \right)}}{\partial Y} = 0 \\ & U\left( {X,1} \right) = V\left( {X,1} \right) = 0,\quad \frac{{\partial \theta_{\text{w}} \left( {X,1} \right)}}{\partial Y} = \frac{{\partial \theta_{\text{nf}} \left( {X,1} \right)}}{\partial Y} = \frac{{\partial \theta_{\text{s}} \left( {X,1} \right)}}{\partial Y} = 0,\quad \frac{{\partial \phi \left( {X,1} \right)}}{\partial Y} = 0 \\ \end{aligned} $$
(31)
The non-dimensional constraints of \( \int\limits_{0}^{1} {\int\limits_{0}^{1} {\phi {\text{d}}X{\text{d}}Y = 1} } \) for the overall volume fraction of nanoparticles should also hold true in the steady state solution. In this report, the physical quantities of interest are the heat transfer through the wall, Nusselt number of the nanofluid and solid phases in the interface boundary of the wall as well as the porous medium, and finally Nusselt number of the free nanofluid, which are, respectively, listed below:
$$ \begin{aligned} Q_{\text{w}} = \frac{{q^{\prime\prime}L}}{{k_{\text{w}} \Delta T}} = \frac{{ - k_{\text{w}} \left( {\frac{{\partial T_{\text{w}} }}{\partial x}} \right)_{\text{x = 0,d}} L}}{{k_{\text{w}} \Delta T}} = \left( { - \frac{{\partial \theta_{\text{w}} }}{\partial X}} \right)_{\text{X = 0,D}} \\ \left. {Nu_{\text{nf}} } \right|_{\text{porous}} = \frac{hL}{{k_{\text{nf}} }} = \frac{{q^{\prime\prime}L}}{{k_{\text{nf}} \Delta T}} = \frac{{ - k_{\text{nf}} \left( {\frac{{\partial T_{\text{nf}} }}{\partial x}} \right)_{\text{x = d,d + s}} L}}{{k_{\text{nf}} \Delta T}} = \left( { - \frac{{\partial \theta_{\text{nf}} }}{\partial X}} \right)_{\text{X = D,D + S}} \\ Nu_{\text{s}} = \frac{{q^{\prime\prime}L}}{{k_{\text{s}} \Delta T}} = \frac{{ - k_{\text{s}} \left( {\frac{{\partial T_{\text{s}} }}{\partial x}} \right)_{\text{x = d,d + s}} L}}{{k_{\text{s}} \Delta T}} = \left( { - \frac{{\partial \theta_{\text{s}} }}{\partial X}} \right)_{\text{X = D,D + S}} \\ \left. {Nu_{\text{nf}} } \right|_{{{\text{free}}\;{\text{nanofluid}}}} = \frac{hL}{{k_{\text{nf}} }} = \frac{{q^{\prime\prime}L}}{{k_{\text{nf}} \Delta T}} = \frac{{ - k_{\text{nf}} \left( {\frac{{\partial T_{\text{nf}} }}{\partial x}} \right)_{\text{x = d + s,L}} L}}{{k_{\text{nf}} \Delta T}} = \left( { - \frac{{\partial \theta_{\text{nf}} }}{\partial X}} \right)_{\text{X = D + S,1}} \\ \end{aligned} $$
(32)
$$ \overline{{Q_{\text{w}} }} = \int\limits_{0}^{1} {Q_{\text{w}} \,} {\text{d}}y, \, \overline{Nu} = \int\limits_{0}^{1} {Nu\,} {\text{d}}y $$
(33)
Eventually, using Eqs. (29) and (30):
$$ \overline{{Q_{\text{w}} }} = \varepsilon R_{\text{k}}^{ - 1} \left. {\overline{{Nu_{\text{nf}} }} } \right|_{\text{porous}} + R_{\text{k}}^{ - 1} K_{\text{r}}^{ - 1} \overline{{Nu_{\text{s}} }} \;{\text{at}}\;X = D $$
(34)
$$ \left. {\overline{{Nu_{\text{nf}} }} } \right|_{{{\text{free}}\;{\text{nanofluid}}}} = \varepsilon \left. {\overline{{Nu_{\text{nf}} }} } \right|_{\text{porous}} + K_{\text{r}}^{ - 1} \overline{{Nu_{\text{s}} }} \;{\text{at}}\;X = D + S $$
(35)
It is notable that the values of Sherwood number are not included in this investigation since according to the boundary conditions given in Eqs. (30) and (31), \( Sh = \frac{Nt}{Nb}Nu \). In addition, considering the steady state solution and based on the energy conservation, it can be deduced that \( \left| {\left( { - \frac{{\partial \theta_{\text{w}} }}{\partial X}} \right)_{{{\text{X}} = 0}} } \right| = \left| {\left( { - \frac{{\partial \theta_{\text{w}} }}{\partial X}} \right)_{\text{X = D}} } \right| \) and \( \left| {\left( { - \frac{{\partial \theta_{\text{nf}} }}{\partial X}} \right)_{\text{X = D + S}} } \right| = \left| {\left( { - \frac{{\partial \theta_{\text{nf}} }}{\partial X}} \right)_{\text{X = 1}} } \right| \). Thus, the results would only be reported for the Nusselt numbers at the interface of the free fluid and the porous layer at X = D + S. The other values of important heat transfer characteristics can be evaluated using the above relations.

Numerical approach, grid independence test and validation

The governing equations, Eqs. (17) and (18), are nonlinear and coupled to each other; hence, it is essential to apply a numerical approach to solve them, which are associated with the boundary conditions expressed as (29)–(31). Here, the Galerkin finite element method was utilized to solve the equations with the corresponding boundary conditions. This method has been explained in details in [66, 67, 68]. The quadrilateral structural elements were employed to discretize the computational domain such that the compression of the elements near the solid walls and internal interface was more than that of other regions. However, before starting calculations in order to obtain the results, the grid independence test was conducted to evaluate the solution sensitivity to the grid and ensure the accuracy of the results. The grid check was performed for the following set of non-dimensional parameters: Z = 0.5, B = ε = 0.6, Ra = 106, Da = 10−2, Nr = Nb = Nt = Kr = 0.1, Le = H = Rk = 10 and Pr = 6.2. As displayed in Table 1, the test was performed for five grids with different grid elements numbers. Evaluation of the variations of three quantities of \( \left. {\overline{{Nu_{\text{nf}} }} } \right|_{{{\text{free}}\,{\text{nanofluid}}}} \), \( \left. {\overline{{Nu_{\text{nf}} }} } \right|_{\text{porous}} \) and \( \left| \psi \right|_{ \hbox{max} } \) showed that error was less than 0.3% when the grid size was 90 × 90. Hence, according to the solution accuracy and the time required for convergence, the grid size 90 × 90 has been utilized to perform the calculations.
Table 1

Grid independency test when Z = 0.5, B = ε = 0.6, Ra = 106, Da = 10−2, Nr = Nb = Nt = Kr = 0.1, Le = H = Rk = 10 and Pr = 6.2

Grid

\( \left. {\overline{{Nu_{\text{nf}} }} } \right|_{{{\text{free}}\;{\text{nanofluid}}}} \)

Error/%

\( \left. {\overline{{Nu_{\text{nf}} }} } \right|_{\text{porous}} \)

Error/%

\( \left| \psi \right|_{\hbox{max} } \)

Error/%

30 × 30

9.8489

 

5.2138

 

25.553

 

50 × 50

9.7345

1.162

5.1124

1.945

24.590

3.769

70 × 70

9.6347

1.025

5.0557

1.109

24.268

1.309

90 × 90

9.6418

0.0737

5.0545

0.024

24.206

0.255

110 × 110

9.6475

0.0591

5.0550

9.89E−03

24.179

0.112

The results of the present numerical solution were compared with the findings presented in the literature [12, 13, 18]. Considering a regular fluid (Nt = 0) and setting D = 0, S = B = 1 and Z = 0.5 (the entire of the hot wall was at the hot temperature θw = 1, and the cavity was filled with a porous medium layer), the present study reduces to the study of Baytas and Pop [12]. Hence, the first validation consists of the comparison between average Nusselt number versus H for the results reported by Baytas and Pop [12] and those obtained here. The results of this comparison are illustrated in Fig. 2 when Ra × Da = 103 and Da = 10−4.
Fig. 2

Measurement of the obtained values of average Nusselt number versus H from the present study with the study conducted by Baytas and Pop [12]

Considering a triangular cavity saturated with a porous medium filled with a nanofluid (without a free fluid or solid layer), the results of our research can be compared with the results of Sheremet and Pop [13]. Hence, in another validation, Fig. 3 evaluates the results obtained for local Nusselt number with the data reported by Sheremet and Pop [13] when Nr = Nb = Nt = 0.1 and Le = 1.0.
Fig. 3

Comparison of local Nusselt number between the current study and work carried out by Sheremet and Pop [13]

Considering the conjugate heat transfer of a regular fluid in a cavity, the results of the present study can be compared with the results of Saeid [18]. In the study of Saeid, the bottom and top walls of the cavity were insulated, while the right and left walls were isothermal with temperature difference. Each of the cold and hot walls was covered with a solid layer with a thickness S. A layer of porous medium imbued with a regular base fluid was sandwiched between the two solid layers. Neglecting the nanoparticles effects, (Nt = 0), and setting B = 1, Z = 0.5, the present study can be reduced to [18]. By setting Ra ×  Da = 103, D = 0.1, Kr = H = 1 and Da = 10−5, Table 2 revels the values of four parameters \( \overline{Nu}_{\text{f}} \), \( \overline{Nu}_{\text{s}} \), \( \overline{Q}_{\text{w}} \) and \( \left| \psi \right|_{ \hbox{max} } \) for the various values of Rk within the range 0.1–10 in comparison with the results provided by Saeid [18]. In addition, as shown in Fig. 4, the patterns of the isotherms and streamlines concluded from the present modeling have been compared with the work conducted by Saeid [18]. As seen, there is an excellent agreement between the results provided by the present modeling and those represented by the published literature [12, 13, 18].
Table 2

Comparison of the results of the present numerical solution and those performed by Saeid [18] at Ra × Da = 103, Da = 10−5, D = 0.1 and Kr = H = 1

\( R_{\text{k}} \)

\( \overline{Nu}_{\text{f}} \)

\( \overline{Nu}_{\text{s}} \)

\( \overline{Q}_{\text{w}} \)

\( \left| \psi \right|_{ \hbox{max} } \)

\( \Delta \theta \)

Present results

     

0.1

0.343

0.113

4.557

3.611

0.03

1

2.903

0.429

3.332

8.006

0.05

10

9.727

1.016

1.074

15.821

0.05

Saeid

     

0.1

0.326

0.110

4.357

3.536

0.03

1

2.814

0.418

3.232

7.898

0.05

10

9.887

1.010

1.090

16.219

0.05

Fig. 4

Comparison of streamlines and isotherms related to a the current study and b those provided by Saeid [18] at Rk = 10, Ra × Da = 103, Da = 10−5 and Kr = H = 1

Results and discussion

Here, the findings of the present numerical investigation have been illustrated in the forms of streamlines, isotherms and isoconcentrations along with the mean Nusselt number of nanofluid at the right wall \( \left. {\overline{{Nu_{\text{nf}} }} } \right|_{{{\text{free}}\,\,{\text{nanofluid}}}} \), the average Nusselt number of nanofluid in the porous layer \( \left. {\overline{{Nu_{\text{nf}} }} } \right|_{\text{porous}} \) and solid phases of the porous medium \( \overline{{Nu_{\text{s}} }} \) at the solid wall-porous interface boundary. Calculations have been performed for various values of Ra = 103–106; B = 0.2–0.8; Z = 0.3–0.7; Le = 10–100; Kr = 0.1–10; Rk = 0.1–10, H = 0.1–1000, ε = 0.3–0.9 and Da = 10−5–10−2 and constant values of Nt = Nb = Nr = 0.1 and Pr = 6.2.

Effects of Rayleigh number

In order to investigate the impact of the buoyancy force on the flow, concentration fields, and temperature, the streamlines, isoconcentrations patterns and isotherms are demonstrated in Fig. 5 for various values of Ra, while the other parameters have been kept constant at B = 0.4; Z = 0.7; H = Rk = Le = 10; Kr = 1; ε = 0.6, Nr = Nb = Nt = 0.1 and Da = 10−2. When Ra is low (Ra = 103 and 104), the conduction mode is predominant, and a weak recirculating flow can be found in the cavity. As shown, an increase in Rayleigh number, representing the buoyancy force, increases the power of the flow circulation. Further, the distance between the streamlines in the porous region is increased. This increase indicates the reduction in the fluid velocity. As the Rayleigh number rises, the existence of the porous layer induces a more significant effect on the streamlines. Within the entire range of the studied Rayleigh numbers, comparatively symmetric patterns of streamlines density next to the vertical walls in the free fluid layer and the porous layer can be observed. However, the distance of the streamlines next to the solid porous interface, i.e. X = D, is higher than that of the cold wall at X = 1. In other words, the boundary layer in the free flow region is the main difference in the streamlines of the regions of the porous layer and free flow is in the core zones of the cavity, where the buoyancy forces become weak.
Fig. 5

Display of the streamlines, isotherms and isoconcentrations for aRa = 103, bRa = 104, cRa = 105 and dRa = 106 at fixed parameters B = 0.4; Z = 0.7; Rk = Le = H = 10; Kr = 1, ε = 0.6 and Da = 10−2

It is also clear that with elevation of Rayleigh number, the streamlines next to the vertical walls get closer, indicating that the convection flow is strengthened. At Ra = 106, the fluid velocity in the middle of the porous region is very low in comparison with the free flow region. This is because of the fact that the pressure drop in a porous medium which is due to the viscous resistance caused via Darcy term grows by the increase in the velocity. As mentioned, in the core region of the cavity, the buoyancy forces are weak and the difference between the streamlines of the porous layer and the free flow region is more obvious. Another result of increasing the pressure drop in the porous region is the shift of the core of the recirculation cell in the cavity toward the cold wall as shown in Fig. 5. When Ra = 103, the conduction mode is predominant, the isotherms of the solid phase (dash lines) and the nanofluid are very similar in the porous region, with the difference between those becoming more evident as Ra increases. As Ra grows, lines of isotherms of the nanofluid in the core zone of the cavity tend to possess a stratified pattern. This trend suggests that the dominance of the convection grows by increasing Ra, confirming the formation of horizontal jets at the top and bottom walls of the cavity.

In the temperature contours, the dashed lines reveal the temperature of the solid matrix (θs). The continuous lines in the solid region indicate the temperature of the solid wall (θw) in the solid layer, while they show the temperature of the nanofluid (θnf) in the porous layer and the free fluid layer. When Rayleigh number is small, i.e. Ra = 103, the patterns of the temperature in the solid matrix of the porous medium and the nanofluid are very close. In this case, the conduction mechanism is the dominant mechanism of the heat transfer, while the fluid motion is very smooth. In this studied case, Kr is adopted as unity (Kr = 1.0), revealing that the thermal conductivity of the porous matrix is the same as that of the nanofluid. Thus, neglecting the convection effects, the diffusion of the heat absorbed from the hot wall in the solid porous matrix and the nanofluid would be almost the same. In addition, as the fluid motion is low and there is an interactive mechanism between the solid porous matrix and the nanofluid (H = 10), the temperatures of the solid porous matrix and the nanofluid are close to each other and follow the same pattern. Nevertheless, as the Rayleigh number rises, the convection mechanism grows stronger and the velocity of the nanofluid also increases. Hence, the weak interface convection heat transfer mechanism (H) between the solid matrix phase and the nanofluid could not significantly affect the temperature patterns of the nanofluid inside the porous layer region, and hence, the temperature patterns of the nanofluid in the porous layer are almost the same as those in the free flow region.

Attention to the contours of the concentration of nanoparticles in the cavity shows that the patterns of this concentration almost follow the pattern of the streamlines. However, there are some deviations due to the Brownian and thermophoresis effects (diffusive terms). It can be seen that the nanoparticles concentration in the vicinity of the hot interface of the solid wall-porous layer is low compared to the cold vertical wall of the cavity. This effect is the result of the thermophoresis force. The thermophoresis force typically carries nanoparticles along the direction opposite to the temperature gradient (from hot regions to cold regions). Next to the hot wall, the temperature gradient is toward the hot wall (as the X increases, the temperature decreases), and hence, the nanoparticles tend to move away from the hot wall. Hence, as can be seen, the nanoparticles’ concentration is low on the hot wall. In contrast, the concentration of the nanoparticles at the cold wall is high. The core zones of the cavity have nearly a uniform nanoparticles’ concentration. In this region, the temperature gradients are smooth, and hence, the Brownian motion has effectively uniformed the nanoparticles in the cavity.

Effects of Kr

To display the effects of Kr on the flow, temperature and nanoparticles’ concentration fields, the isotherms, streamlines and isoconcentrations patterns are represented in Fig. 6 when Ra = 106; B = 0.5; Z = 0.6; Le = H=Rk = 10; ε = 0.6 and Da = 10−2. As shown, the increase in Kr decreases the solidity of the recirculation cell, formed inside the cavity. Kr indicates the ratio of the thermal conductivity of the nanofluid to the thermal conductivity of the solid porous matrix. A very low value of Kr represents the great thermal conductivity of the solid porous matrix compared to the nanofluid. This is the case which occurs for metal foams. In this case, the heat conductivity in the phase of the solid porous matrix is very powerful. Hence, the porous medium would channel the heat inside the solid matrix and distribute it in the nanofluid due to the interface convection heat transfer mechanism (H). Thus, the heat transfer can be enhanced and the circulation flows are also strong for the low values of Kr. Further, the temperature contours demonstrate that the temperature patterns of the solid matrix phase (θs) are almost parallel to the vertical hot wall when Kr = 0.1, which indicates the strong conduction mechanism. The deviation of the temperature curves (from the vertical direction) is due to the interaction between the nanofluid inside the porous medium and the solid porous matrix. As Kr increases, the temperatures in the porous and pure nanofluid region tend to converge. Moreover, a more uniform dispersion of nanoparticles inside the porous layer can be observed when Kr is high. Indeed, for high values of Kr, the temperature gradients inside the nanofluid decrease, and as a result, the thermophoresis force also declines. Thus, in the cavity, one can expect a more uniform nanoparticle distribution.
Fig. 6

Display of the streamlines, isotherms and isoconcentrations for two values of aKr = 0.1, bKr = 1 and cKr = 10 at B = 0.5; Z = 0.6; Ra = 106, Le = H=Rk = 10, ε = 0.6 and Da = 10−2

Effects of the length B and the position of the heating element Z

The effects of the length and the position of the heating element are discussed in this section for Ra = 105, Le = H = Kr = Rk = 10, ε = 0.6 and Da = 10−2. Figure 7 displays the effects of the length of the heating element, while its position has been fixed at Z = 0.5. As can be seen, the streamlines, isotherms and isoconcentrations fields have kept their general patterns constant with increasing the length of heating element from 0.2 to 0.8 with the step 0.2. Although the change in the length of the element reveals a smooth effect on the strength and shape of the recirculation cells, the patterns of the isotherms and isoconcentration contours are almost independent of the element length. This is due to the fact that the solid wall acts as a redistributor for the heat absorbed from element and quickly diffuses the absorbed heat inside itself.
Fig. 7

The effect of the element length (aB = 0.2, bB = 0.4, cB = 0.6 and dB = 0.8) on the streamlines, isotherms and isoconcentrations at Ra = 105, Z = 0.5, Le = H=Kr = Rk = 10, ε = 0.6, Nt = Nb = Nr = 0.1, Da = 10−2

Figure 8 depicts the effects of the position of the heating element on the streamlines, isotherms and isoconcentration patterns, while the other parameters have been kept constant at Ra = 105, B = 0.5, Le = H=Rk = 10, Kr = 1, ε = 0.6, Nt = Nb = Nr = 0.1, Da = 10−3. The shape and strength of the formed circulation cell within the cavity decline by the increase in Z. However, this elevation is not very obvious, and thus the changes in the isotherms and isoconcentration contours are not very significant. It can be observed that the isotherms in the solid layer follow the location of the element. However, the isotherms and isoconcentrations in both of the solid porous matrix and nanofluid phases are almost fixed as the position of the element changes.
Fig. 8

The effect of the element position (Z) on the streamlines, isotherms and isoconcentrations at aZ = 0.25, bZ = 0.5, cZ = 0.75, Ra = 105, B = 0.5, Le = H = Rk = 10, Kr = 1, ε = 0.6, Nt = Nb = Nr = 0.1, Da = 10−3

Effects of Darcy number

The effects of Darcy number on the streamlines, isotherms and isoconcentration patterns are illustrated in Fig. 9 for the fixed values of B = 0.5; Z = 0.25; Nr = Nb = Nt = 0.1, H = Le = Kr = Rk = 10, ε = 0.6, Ra = 106. According to the figure, the variation of Darcy number, Da, obviously affects the governing patterns of the nanofluid flow. This result arises from extreme variation of the pressure drop caused by the Darcy term in the porous region. Since the density of the streamlines depicts the velocity amplitude of the nanofluid, it can be concluded that when Da = 10−5, the solid matrix impedes the fluid motion and diminishes the velocity of the nanofluid substantially in the porous zone. In addition, as the decrease in the pressure drop or hydrodynamic resistance is the result of the rise of the Darcy term, the increase of Da causes augmentation of the size and strength of the recirculation cells formed in the whole of the cavity region. As Darcy number grows, the advection regime becomes stronger within the porous region. The increased difference between thermal fields, i.e. θs and θnf, corresponding to the different values of Da illustrates this fact. It can be seen that the thermal mixing of nanofluid is enhanced in the porous region when Darcy number rises. In addition, it is clear that the distribution of nanoparticles within the cavity entirely depends on the velocity field varying with Darcy number. When Da = 10−5, the comparison of isoconcentration lines in the saturated porous and single nanofluid regions clearly demonstrates the effect of the nanofluid velocity on the dispersion of the nanoparticles. As depicted, the nanoparticles’ mixing develops by promoting the fluid strength due to the increase in Da.
Fig. 9

Display of the streamlines, isotherms and isoconcentrations for various Darcy numbers (aDa = 10−5, bDa = 10−4, cDa = 10−3, dDa = 10−2) at B = 0.5; Z = 0.25; Nr = Nb = Nt = 0.1, H = Le = Kr = Rk = 10, ε = 0.6, Ra = 106

Figure 10a indicates that the mean Nusselt numbers for both the fluid and solid phases grow as Ra and Kr increase. Indeed, when Ra increases, the strength of the fluid flow is boosted due to the increase in the buoyancy force. Hence, the heat transfer resulting from the convection mode becomes predominant with respect to the conduction mode. Thus, the increase in the average Nusselt number for the nanofluid phase is more than that of the solid porous phase, and the difference between them is intensified as Ra increases. Moreover, as was previously discussed, the increase in Kr enhances the heat transfer and flow circulations in the porous layer, which results in augmentation of the mean Nusselt number for two phases of nanofluid and porous medium in the porous region. Since the \( \left. {\overline{{Nu_{\text{nf}} }} } \right|_{{{\text{free}}\,{\text{nanofluid}}}} \) value is a function of Kr, it is necessary to determine the value of \( \left. {\overline{{Nu_{\text{nf}} }} } \right|_{{{\text{free}}\,{\text{nanofluid}}}} \) separately. From Fig. 10b, it can be found that the Nusselt number in the right cold boundary has an ascending trend with the increase in Kr. According to Eq. (35), this result is not unexpected. Nonetheless, to justify this trend physically, it should be noted that the increase in Kr indeed represents the increase in the thermal conductivity of the nanofluid compared to that of the solid matrix phase. In the free fluid region, where there is no interface convection heat transfer between the porous matrix and the nanofluid, the growth of the thermal conductivity of the nanofluid would result in overall enhancement of the heat transfer. However, it would reduce the temperature gradient and the average Nusselt number, as the better the thermal diffusion, the more uniform temperature distribution will be. In addition, it is observed that for Kr > 4 and for all of the studied Rayleigh numbers, the variations of the average Nusselt number for the fluid phase in the wall-porous interface \( \overline{Nu}_{\text{nf}} \) and \( \left. {\overline{{Nu_{\text{nf}} }} } \right|_{{{\text{free}}\,{\text{nanofluid}}}} \) are negligible. However, when Ra is high (Ra = 105 and 106), the average Nusselt Number for the solid phase has a continuously increasing trend with increasing Kr, which is the result of the interface convection heat transfer mechanism between the solid porous matrix and the nanofluid inside the pores (H = 10).
Fig. 10

The variations of a\( \overline{Nu}_{\text{nf}} \) and \( \overline{Nu}_{\text{s}} \) and b\( \left. {\overline{{Nu_{\text{nf}} }} } \right|_{{{\text{free}}\;{\text{nanofluid}}}} \) versus Kr for the various values of Ra at B = Z = 0.5; Nr = Nb = Nt = 0.1, H = Le = Rk = 10, Da = 10−3, ε = 0.6

The influence of the porosity ε and the length of the heating element B on the mean Nusselt number of nanofluid and the porous medium is demonstrated in Fig. 11, while the other parameters have been kept constant such that Ra = 105, Da = 10−2, Nr = Nb = Nt = 0.1, Kr = H = Le = Rk = 10, Z = 0.5. This figure demonstrates that the increase in ε can strongly decrease both \( \overline{Nu}_{\text{nf}} \) and \( \overline{Nu}_{\text{s}} \), while the increase in B augments these two Nusselt numbers. The increase in these Nusselt numbers by increasing B is due to the augmentation of the strength of the fluid flow. Indeed, when the strength of the fluid flow rises, the natural convection can transfer more heat.
Fig. 11

The variations of a\( \overline{Nu}_{\text{nf}} \) and \( \overline{Nu}_{\text{s}} \) and b\( \left. {\overline{{Nu_{\text{nf}} }} } \right|_{{{\text{free}}\;{\text{nanofluid}}}} \) versus B for the various values of ε at Ra = 105, Nr = Nb = Nt = 0.1, Kr = H = Le = Rk = 10, Da = 10−2, Z = 0.5

Figure 12 demonstrates the effect of the position of the heating element and Darcy number on the average Nusselt number of nanofluid and solid phases at the porous region boundary, i.e. X = S + D. As seen, the augmentation of Da increases the average Nusselt numbers for both phases. This observed trend is due to the increase in the nanofluid velocity arising from elevation of the permeability of the porous region or reduction in the hydraulic fluid resistance coming from the Darcy term. In addition, it can be observed that the increase in the mean Nusselt number of the nanofluid phase is far larger than that of the porous matrix phase, and the difference between these two Nusselt numbers becomes larger as Da increases. In addition, as shown in Fig. 12, when Da value is low (Da = 10−5 and 10−4), the effect of the position of the heating element on the average Nusselt number for the nanofluid phase can be ignored. For all values of Da, the average Nusselt number for the solid phase negligibly varies with the variation in the element heating location. When Da value is high (Da = 10−3 and 10−2), the average Nusselt number for the fluid phase increases by the increment in Z from 0.3 to 0.4. Then, the mean Nusselt number stays almost constant when Z reaches 0.5. After this, a decreasing trend can be observed for this number with the growth of Z. In general, it can be deduced that the minimum value of \( \overline{Nu}_{\text{nf}} \) occurs when the heating element is located at its maximum position. In addition, it can be said that the maximum value of \( \overline{Nu}_{\text{nf}} \) occurs at Z = 0.4 or 0.5.
Fig. 12

The variations of \( \overline{Nu}_{\text{nf}} \) and \( \overline{Nu}_{\text{s}} \) versus Da for the various values of Z at Ra = 105, Nr = Nb = Nt = 0.1, Kr = H = Le = Rk = 10, B = ε = 0.6

Figure 13a, b displays the effect of Rk on the average Nusselt number for the fluid and the solid phases in the interface boundary of wall-porous region for different values of the interphase heat transfer coefficient H. As shown, the average Nusselt numbers rise by increasing Rk. Indeed, the augmentation of Rk means that more heat is transferred to the porous region through the solid wall. In addition, the graphs drawn in Fig. 13a, b depict that the Nusselt number of the fluid phase decreases with increase in H, while the increase in H extremely enhances the Nusselt number of the solid phase. A closer look at the results indicates that the average Nusselt number for the fluid phase remains almost constant with the variations in H when it is in the order of O(0.1) or O(100). In addition, the rise of the mean Nusselt number of the solid phase with the increase of Rk is more evident at the high values of H. According to the definition of \( \overline{Q}_{\text{w}} \), an opposite behavior can be observed for the variations of \( \overline{Q}_{\text{w}} \) with increasing Rk such that \( \overline{Q}_{\text{w}} \) diminish as Rk increases.
Fig. 13

The variations of \( \overline{Nu}_{\text{nf}} \) and \( \overline{Nu}_{\text{s}} \) at X = D versus Rk for the various values of H at Ra = 105, Nr = Nb = Nt = 0.1, Kr = Le = 10, Da = 10−2, B = 0.7, Z = 0.5, ε = 0.6

The variations in the Nusselt number for the fluid and solid phases at the interface of the wall-porous region as a function of the buoyancy ratio parameter, Nr, are demonstrated in Fig. 14 for the different values of Lewis number Le, while the other parameters have been kept constant. The growth of Nr reduces the mean Nusselt number for both phases. This result can be attributed to the fact that increasing Nr induces a buoyancy force which in some regions acts against the buoyancy force. Hence, elevation of Nr decreases the strength of the fluid flow, and in response to this reduction, the heat transfer rates declines. In addition, it can be deduced from Fig. 14 that the increment of Lewis number Le enhances the average Nusselt number of both phases. Indeed, the increase in the Lewis number means diminished thickness of the nanoparticles concentration boundary layers over the vertical walls. Consequently, with the reduction in the thickness of the concentration boundary layers, the nanoparticles can transfer the energy more effectively from high- to low-energy regions through their migration. Sheremet et al. [44] have also reported a decreasing trend of the average Nusselt number of nanofluids as a function of Nr for a cavity entirely filled with a LTE porous medium. They have also reported a growing trend for the mean Nusselt number of nanofluids as a function of Lewis number. Accordingly, the patterns of this research findings are in line with the results of [44].
Fig. 14

The variations of \( \overline{Nu}_{\text{nf}} \) and \( \overline{Nu}_{\text{s}} \) at X = D versus Nr for the various values of Le at Ra = 105, Nb = Nt = 0.1, Kr = H = Rk = 10, Da = 10−2, B = 0.7, Z = 0.5, ε = 0.6

Conclusions

In this research, the flow, heat and mass transfer of nanofluids were addressed in a multilayer cavity. The cavity was filled with a layer of solid wall, a layer of porous medium and a layer of free fluid. The layer of the porous medium was saturated by a nanofluid. The local thermal non-equilibrium model was employed for modeling thermal behavior of the porous medium layer. A drift flux of nanoparticles was considered because of the thermophoresis and Brownian motion effects via the Buongiorno’s model. The results for the streamlines, the temperature patterns and the concentration of nanoparticles were plotted and discussed. In addition, the Nusselt numbers for the solid porous matrix, the nanofluid inside the porous layer and the nanofluid in the free layer as the important heat transfer properties were also introduced. The impact of the different non-dimensional parameters on the heat transfer properties was discussed further. The main outcomes of this research can be summarized as follows:
  1. 1.

    The non-dimensional nanoparticles’ concentration next to the cold wall was about 1.1 which is 10% higher than the average concentration of nanoparticles in the cavity. The non-dimensional concentration of nanoparticles next to the hot wall was low and about 0.92 which is 8% lower than the non-dimensional mean concentration of nanoparticles. When Rayleigh number was small, i.e. Ra = 103, the concentration gradients were distributed in the cavity. However, as the convection mechanisms became stronger, the distribution of nanoparticles in the core zones of the cavity became uniform, and hence, the concentration gradients were negligible in the core regions of the cavity; however, strong concentration gradients can be seen in the vicinity of the walls.

     
  2. 2.

    The thermal conductivity ratio of nanofluid/porous matrix, Kr, induced a significant effect on the streamlines and concentration patterns inside the porous layer. The very low values of Kr, i.e. Kr = 0.1, resulted in a strong circulation cell in the free fluid layer. In addition, in this case, a considerable concentration gradient of nanoparticles could be observed in the center of the cavity inside the porous layer. The increase in Kr would smoothly reduce the average Nusselt number for the free nanofluid. It should be noticed that the increase in Kr may represent the decrease in the thermal conductivity of the solid porous matrix. In this case, the heat transfer by the solid porous matrix decreased, resulting in the diminished overall heat transfer.

     
  3. 3.

    Due to the presence of a high thermally conductive solid layer (Rk = 10), the length and position of the element did not show obvious effects on the temperature and concentration patterns inside the cavity. The maximum values of the average Nusselt number could be found about Z = 0.4 for both phases of the nanofluid and solid matrix. This means that mounting element slightly below the center of the cavity results in greater heat transfer. The increase in the size of the element would also slightly increase the average Nusselt number for the free nanofluid (Nufree nanofluid).

     
  4. 4.

    Darcy number is very important parameter which plays a significant role in the shape of the streamlines, temperature and concentration patterns. For very low values of Darcy number, i.e. Da = 10−5, the velocity in the porous layer significantly decreases. The circulation cells are mainly formed in the free nanofluid layer. The temperature contours in the porous layer would almost show a linear distribution, which confirms the diffusive dominant region of flow. In this case, the concentration gradients of nanoparticles even in the core region of the cavity inside the porous layer could be detected. The increase in Darcy number allowed the fluid to move more freely in the porous layer, and hence, the high values of Darcy number induced a convective heat transfer dominant regime in the porous layer. The increase in Darcy number significantly enhanced the mean Nusselt number for the free nanofluid.

     
  5. 5.

    The elevation of buoyancy ratio, Nr, resulted in the decrease in mean Nusselt number of the solid porous matrix and the nanofluid inside the porous layer. In contrast, the rise of Lewis number elevated the mean Nusselt number for the both phases at the interface.

     

Notes

Acknowledgements

Mohammad Ghalambaz is thankful to Dezful Branch Islamic Azad University of the financial support of the present study. The authors are tankful to Iran Nanotechnology Initiative Council (INIC) for its crucial support.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  • S. A. M. Mehryan
    • 1
  • Mohammad Ghalambaz
    • 2
    Email author
  • Mohsen Izadi
    • 3
  1. 1.Young Researchers and Elite Club, Yasooj BranchIslamic Azad UniversityYasoojIran
  2. 2.Department of Mechanical Engineering, Dezful BranchIslamic Azad UniversityDezfulIran
  3. 3.Mechanical Engineering Department, Faculty of EngineeringLorestan UniversityKhorramabadIran

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