## Abstract

A numerical study on natural convection in an inclined channel partially filled with open metal foam is investigated in two-dimensional laminar and incompressible and steady-state condition. The upper wall is heated at uniform heat flux, and the lower wall is adiabatic. The governing equations for the metal foam are written assuming the Brinkman–Forchheimer-extended Darcy model and the local thermal equilibrium (LTE) hypothesis. The inclination angle with respect to the horizontal direction ranges between 0 deg and 90 deg. The metal foam structure is homogenous and isotropic and the thermophysical properties are assumed constant with the temperature. The main aim of the present study is to analyze the effect of inclination angle, pore density, and porosity to improve the knowledge on the thermal behaviors of natural convection in partially filled channel with metal foam. Moreover, this study aims to evaluate the possible improvement with respect to the clean inclined channel. Results are presented in terms of velocity and temperature fields, and both temperature and velocity profiles at different significant sections are shown, to obtain a description of the natural convection inside the inclined channel. Finally, average Nusselt number values are evaluated. The presence of metal foam affects the fluid flow in the channel with different motion behaviors. The heat transfer rate increases with the increase in inclination angle, and the average Nusselt number shows different trends for lower and higher Rayleigh numbers, Ra. A significant enhancement is detected for the higher Ra and it depends on the porosity and pore density.

## 1 Introduction

Natural convection in parallel plates and channels has been extensively investigated due to its significant applications in several modern systems and components [1–4]. Many of these applications are related to electronic cooling [5], solar collectors [6], and ventilated roofs [7]. It should be underlined that along with the clear advantages of natural convection, such as simplicity, low costs, and reliability due to the absence of moving mechanical components, there is a critical limitation related to the low heat transfer rate [1,2]. However, there is a renewed strong research interest in inclined channels [8–10] for their applications in renewable energy systems. Moreover, there is the need to improve the heat transfer rate, and the passive techniques can be employed to enhance the heat transfer inside the parallel plates. One of these is the use of metal foam as underlined by Li et al. [11]. In fact, metal foams such as aluminum foams are porous media with solid matrix with high thermal conductivity and great surface area to volume ratio together with strong flow mixing inside, as pointed out by Zhao [12].

The effect of increase in heat transfer in natural convection by means of metal foams was well highlighted in several investigations. One of the first studies on natural convection in metal foams was accomplished by Phanikumar and Mahajan [13]. The study on a heat sink with rectangular base and heated from below was accomplished both experimentally and numerically. A metal foam disk with the lower surface at higher temperature with respect to the upper surface was studied both experimentally and numerically by Zhao et al. [14]. An experimental investigation on metal foam heat sink was given by Bhattacarya and Mahajan [15]. The investigation was accomplished on two configurations with and without fins with the metal foam between the fins both for horizontal and vertical position. The results pointed out that heat sink with metal foam inside fins presented the highest heat transfer coefficients. The better heat transfer conditions improved for the lowest pore density.

Cheng and Pao [16], numerically, and Hetsroni et al. [17] and Kathare et al. [18], experimentally, confirmed that in natural convection the use of metal foams on a heated surface can augment the heat transfer rate. An experimental investigation of the inclination effect on a heated plate with metal foams was proposed by Qu et al. [19]. The heat transfer rate in the metal foam was improved both by the heat conduction and buoyancy effect. Moreover, the results suggested that to enhance the heat transfer a metal foam with reduced pore density and porosity was better. The effect of inclination on heated surface with and without metal foams was experimentally investigated by Paknezhad et al. [20]. They found that for the heated surface with metal foam the vertical position presented the maximum cooling efficiency.

A heat sink with fins made up of metal foam was investigated experimentally by De-Schampheleire et al. [21] and numerically by De-Schampheleire et al. [22]. The experimental results given in Ref. [21] pointed out that the main parameters that affected the system was the pore density, the fin height and the method employed to bond the foam. The numerical analysis in Ref. [22] highlighted that the radiative heat transfer was an important component of the global heat transfer.

The inclined channel partially filled with metal foam was numerically studied by Piller and Stalio [23] considering the laminar fully developed natural convection condition. The analysis was carried out in local thermal equilibrium considering the inclined channel heated from the lower wall. The results showed that the inclination angle does not affect the Nusselt number whereas a dependence on the thermal conductivity ratio between solid and fluid and the metal foam thickness was detected. An experimental and numerical investigation on natural convection in horizontal channel heated from below and partially filled with metal foam was provided by Buonomo et al. [24]. The numerical simulation was developed in a simplified two-dimensional steady-state model with extended computational domain. The comparison between the channel with and without porous metal showed that the configuration with foam enhances the heat transfer. The horizontal channel partially filled with metal foam and heated from upper wall was numerically studied both with local thermal equilibrium (LTE) and local thermal nonequilibrium (LTNE) by Buonomo et al. [25,26]. The simulations were carried out for assigned porosity value equal to 0.93 with two external lateral reservoirs. For both models, the results highlighted that the low pore density value improves the heat transfer. An extension of the previous studies to investigate the effect of porosity was given by Buonomo et al. [27]. The results indicated that at lower Rayleigh numbers, Ra, the average Nusselt numbers for partially filled open cavity are lower than the ones for clear configurations. The contrary was detected for higher Ra value and the Nusselt number was higher for the horizontal channel partially filled with the metal foam. Moreover, for the horizontal channel partially filled the Nusselt numbers decrease for lower porosity and increase for higher pore density. Andreozzi et al. [28] carried out a numerical study on natural convection in air in a vertical channel filled with metal foam assuming the LTNE. A channel symmetrically heated at uniform heat flux was considered. The effects of metal foam pore density and porosity as well as the conductivity ratio between the solid and fluid and the channel aspect ratio on the air velocity and temperature, solid temperature and heat transfer rate were examined. The results showed that both the solid dimensionless temperature and global heat transfer coefficient increased decreasing the porosity and increasing the pore density.

After the short literature review, it seems that the natural convection in inclined channels partially filled with metal foam presents a lack of knowledge although its possible extended areas of applications. In fact, it can be particularly useful in many applications, such as thermal control in electronic cooling [11], material processing and manufacturing [29], solar energy components [30–32] and ventilated roofs [7].

In the present numerical study, natural convection in an inclined channel partially filled with metal foam is examined in two-dimensional steady-state conditions. The inclined channel is asymmetrically heated with the upper surface heated at assigned uniform wall heat flux and the unheated lower wall assumed adiabatic. The metal foam is considered on the upper wall. The metal foam is modeled assuming the Brinkman–Forchheimer-extended Darcy model in LTE. The analysis is carried out for different inclination angles from 0 deg, horizontal channel, to 90 deg, vertical channel, with aluminum foam inside the channel or without metal foam for a channel aspect ratio equal to 10. The effects of Rayleigh number and aluminum foam porosity and pore density on thermal and fluid dynamic characteristics are evaluated to highlight the differences between the channel with and without metal foam.

## 2 Problem Description and Mathematical Modeling

Incompressible and steady-state laminar free convection in an inclined parallel plates channel with aluminum foam insert in contact with heated upper wall is numerically investigated. The geometric configuration under investigation together with a local Cartesian coordinate system is shown in Fig. 1.

The origin of the local coordinate system is located at the inlet of the bottom plate, Fig. 1, and *x*, *y* axes denote the tangential and normal directions of the inclined channel, respectively. A constant uniform heat flux, $q\u02d9$ is imposed on the heated upper plate, while the lower plate is considered adiabatic. The gap between the parallel plates of the channel is *b*, the length plate is *L* and the thickness of the aluminum foam insert is *s*. The aspect ratio, *L*/*b*, of the channel and dimensionless thickness, *s*/*b*, of the porous insert are equal to 10 and 0.5 respectively for all numerical simulations. Various inclination angles, *α*, of the channel with respect to the horizontal direction in the range from 0 to 90 deg are considered. Two large reservoirs of size *L _{x}* ×

*L*placed upstream and downstream of the channel have been employed to consider the freestream conditions of the flow far away from the heated plate, as proposed in Ref. [33]. The working fluid is air and all its thermo-physical properties except the density are assumed constant and evaluated at the temperature of 300 K of the surrounding environment. The convective heat transfer of the air through aluminum foam is modeled using the Brinkman–Forchheimer-extended Darcy equation for mass transport and the energy equation under the LTE assumption for energy transport.

_{y}In the present problem, an estimation of the order of magnitude of the local temperature difference between foam, solid matrix, and air with respect to the total temperature difference can be made by means of the scale analysis. In the considered geometry, the increase of air temperature in the channel represents the total or global temperature differences and it is given by, as order of magnitude, $m\u02d9cp\Delta TG$ with $m\u02d9$ the air mass flowrate, *c _{p}* the specific heat at constant pressure, and Δ

*T*the global increase of air temperature. In the metal foam, the local heat transfer

_{G}*A*

_{sf}

*h*

_{sf}Δ

*T*determines the increase in air enthalpy. The terms

_{l}*A*

_{sf}is the total heat transfer surface inside the metal foam,

*h*

_{sf}is the local convective heat transfer coefficient, and Δ

*T*is the local temperature difference between the air and metal foam. These two values have the same order of magnitude and it results $m\u02d9cp\Delta TG$∼

_{l}*A*

_{sf}

*h*

_{sf}Δ

*T*. Consequently, Δ

_{l}*T*/Δ

_{l}*T*∼ $m\u02d9cp$/

_{G}*A*

_{sf}

*h*

_{sf}.

*A*

_{sf}is equal to $Vt\u02d9asf$ = bLWa

_{sf}with $Vt\u02d9$ the total volume of metal foam and

*a*

_{sf}the specific surface of metal foam. The mass flow rate is equal to

*b*Wu and $\Delta Tl\Delta TG\u223cubbL\nu cp\nu asfhsf=Ub2(Lb)cp\nu asfhsf$ with a minimum value of

*a*∼750 m

_{sf}^{2}/m

^{3}estimated by formula in Ref. [34],

*h*

_{sf}∼50 Wm

^{−2}K

^{−1}and, for air at 300 K,

*c*= 1000 Jkg

_{p}^{−1}K

^{−1}, ν = 1.7 × 10

^{−5}m

^{2}s

^{−1},

*b*= 4.0 cm,

*L*/

*b*= 10 and a maximum dimensionless velocity,

*U*, in the porous zone is found equal to about 300, for the highest considered Rayleigh number in the investigation equal to 1.2 × 10

^{7}, it is $\Delta Tl\Delta TG=8.5\xd710\u22123\u223c1.0\xd710\u22122$. The local temperature difference is negligible with respect to the global temperature difference and the local thermal equilibrium can be assumed in the present analysis.

The main parameters of the aluminum foam such as the permeability and Forchheimer coefficient are assumed to be direction independent and constant throughout the domain. The momentum and thermal dispersion in aluminum foam are assumed to be negligible. Furthermore, the viscous dissipation effects in the energy equation, due to the small velocities induced by the buoyancy force, are neglected for both the clear and the porous regions. The density of air in all terms of the governing equations is assumed to be constant except for the buoyancy force in the momentum equation where it is assumed to be linear with temperature.

Considering the above hypothesis and using the binary index *η* = {0,1} the dimensional form of the governing equations for both clear and aluminum foam regions can be written as

*x*—component of momentum equation

*y*—component of momentum equation

*T*

_{air}=

*T*

_{Al-foam}=

*T*) is

where *u*, *v* are the tangential and normal components of Darcy velocity. Furthermore, *ρ _{f}*,

*μ*,

_{f}*β*, and

_{f}*c*denote, respectively, the density, dynamic viscosity, thermal expansion coefficient and heat specific at constant pressure of the air.

_{p,f}*ε*,

*K*, and

*C*are the porosity, permeability, and Forchheimer coefficient of the aluminum foam, respectively. Finally,

_{F}*η*= 1 in the porous region, while in the clear zone

*η*= 0,

*ε*= 1, and the Darcy and Forchheimer terms are set to zero.

*ε*, the fiber diameter,

*d*, and pore diameter,

_{f}*d*, are correlated with the expression proposed by Calmidi [34]

_{p}*K*, and Forccheimer coefficient,

*C*, as a function of

_{F}*ε*and of

*d*/

_{f}*d*are evaluated considering the correlations suggest by Calmidi [34]

_{p}The pore diameter *d _{p}* is defined as the equivalent circle diameter with an area equal to the sum of the cell window areas representative of the aluminum foam microstructure.

where *k _{f}* and

*k*are the thermal conductivities of air and aluminum foam. The best fit of experimental data by expression (8) is obtained by setting the weighting coefficient A to 0.35 as suggest by Bhattacharya [35]. It is interesting to observe that for

_{s}*ε*= 0

*k*

_{eff}=

*k*and for

_{s}*ε*= 1

*k*

_{eff}=

*k*.

_{f}Table 1 summarizes the thermo-physical properties of the air and aluminum referred to an ambient temperature *T*_{0} = 300 K. Various samples of aluminum foam with pore densities of 10, 20, and 40 PPI, and porosities of 0.90 and 0.97 are investigated. The fiber diameter *d _{f}*, permeability

*K*, Forchheimer coefficient,

*C*, effective thermal conductivity,

_{F}*k*

_{eff}, are estimated by expressions (5)–(8) for different samples, and the values are shown in Table 2. The Darcy number values, Da =

*K*/

*b*

^{2}, are evaluated assuming

*b*=

*0.04 m, and are reported in the last column of Table 2.*

Properties | Air | Aluminum |
---|---|---|

Density (kg/m^{3}) | 1.20 | 2700 |

Thermal conductivity (W/(mK)) | 0.026 | 204 |

kinematic viscosity (m/s^{2}) | 1.59 × 10^{−6} | — |

Specific heat (J/(kg K)) | 1006 | 900 |

Properties | Air | Aluminum |
---|---|---|

Density (kg/m^{3}) | 1.20 | 2700 |

Thermal conductivity (W/(mK)) | 0.026 | 204 |

kinematic viscosity (m/s^{2}) | 1.59 × 10^{−6} | — |

Specific heat (J/(kg K)) | 1006 | 900 |

PPI | ε | d (mm)_{p} | d (mm)_{f} | K × 10^{−7} (m^{2}) | C_{F} | k_{eff} (Wm^{−1}K^{−1}) | Da × 10^{−6} |
---|---|---|---|---|---|---|---|

10 | 0.90 | 2.96 | 0.38 | 1.060 | 0.082 | 6.55 | 2.65 |

10 | 0.95 | 3.10 | 0.37 | 1.462 | 0.099 | 3.68 | 3.65 |

20 | 0.90 | 2.58 | 0.34 | 0.776 | 0.077 | 7.16 | 1.94 |

20 | 0.95 | 2.70 | 0.32 | 1.083 | 0.099 | 3.68 | 2.70 |

40 | 0.90 | 1.94 | 0.25 | 0.453 | 0.082 | 6.55 | 1.13 |

40 | 0.95 | 1.98 | 0.24 | 0.592 | 0.099 | 3.47 | 1.48 |

PPI | ε | d (mm)_{p} | d (mm)_{f} | K × 10^{−7} (m^{2}) | C_{F} | k_{eff} (Wm^{−1}K^{−1}) | Da × 10^{−6} |
---|---|---|---|---|---|---|---|

10 | 0.90 | 2.96 | 0.38 | 1.060 | 0.082 | 6.55 | 2.65 |

10 | 0.95 | 3.10 | 0.37 | 1.462 | 0.099 | 3.68 | 3.65 |

20 | 0.90 | 2.58 | 0.34 | 0.776 | 0.077 | 7.16 | 1.94 |

20 | 0.95 | 2.70 | 0.32 | 1.083 | 0.099 | 3.68 | 2.70 |

40 | 0.90 | 1.94 | 0.25 | 0.453 | 0.082 | 6.55 | 1.13 |

40 | 0.95 | 1.98 | 0.24 | 0.592 | 0.099 | 3.47 | 1.48 |

The boundary conditions for temperature and velocity fields associated with Eqs. (1)–(4) are indicated in Table 3. Furthermore, the continuity of the velocity, pressure, and shear stress are imposed at the interface between air and aluminum foam.

Boundary (Fig. 1) | u | v | T |
---|---|---|---|

AH, BO, CE, and DN | 0 | 0 | $\u2202T\u2202x=0\u2009$ |

CD | 0 | 0 | $\u2212keff\u2202T\u2202y=q\u02d9w\u2009$ |

GH and OP | 0 | $pt=101,325Pa$ | T = T_{0} |

FE and NM | 0 | $ps=101,325\u2009Pa$ | $for\u2009exit\u2009flow\u2202T\u2202y=0for\u2009inlet\u2009flowT=T0$ |

GF and PM | $for\u2009exit\u2009flow:pe=101,325\u2009Pa\u2009for\u2009inlet\u2009flow:pt=101,325\u2009Pa\u2009$ | 0 | $for\u2009exit\u2009flow:\u2202T\u2202x=0for\u2009inlet\u2009flow:T=T0$ |

Boundary (Fig. 1) | u | v | T |
---|---|---|---|

AH, BO, CE, and DN | 0 | 0 | $\u2202T\u2202x=0\u2009$ |

CD | 0 | 0 | $\u2212keff\u2202T\u2202y=q\u02d9w\u2009$ |

GH and OP | 0 | $pt=101,325Pa$ | T = T_{0} |

FE and NM | 0 | $ps=101,325\u2009Pa$ | $for\u2009exit\u2009flow\u2202T\u2202y=0for\u2009inlet\u2009flowT=T0$ |

GF and PM | $for\u2009exit\u2009flow:pe=101,325\u2009Pa\u2009for\u2009inlet\u2009flow:pt=101,325\u2009Pa\u2009$ | 0 | $for\u2009exit\u2009flow:\u2202T\u2202x=0for\u2009inlet\u2009flow:T=T0$ |

with *T _{w}* the temperature of the heated upper plate.

## 3 Numerical Model

The numerical solutions of the mathematical problem described by Eqs. (1)–(4) and the associated boundary conditions, reported in Table 3, are obtained using the finite volume method (FVM) adopted by the ansysfluent code [36]. The pressure-based algorithm with the steady-state option is employed to simulate the laminar incompressible natural flow. The coupled method is chosen to solve the coupling problem between velocity and pressure in the momentum and continuity equations. The equations of momentum, continuity, and energy are solved simultaneously when the combination of the pressure-based algorithm and coupled method is set. A pseudo-unsteady method [37] for computation of steady-state solutions starting from unsteady governing equations is used. For the discretization of the various terms present in the governing equations, the following techniques are used: a second order upwind scheme for all convective terms, a second order central scheme for all diffusive terms and last squares cell based for the computation of the gradient of a scalar variable.

A range of values to choose the relaxation factor is tried, and it is found that the appropriate value which allows simulations without excessive computational time, as indicated in Ref. [38], was less than 1. The under-relaxation factors are set as 0.75 for the energy equation, 0.5 for the pressure and momentum equations. The normalized residuals to end the iterative numerical procedure are set as 10^{−5} for the continuity and velocity components, 10^{−8} for energy equation. A further criterion to stop the iterative solver was represented by the achievement of the steady-state of the average temperature on the upper heated plate and of the mass flow on the channel. The dimensions of two large reservoirs present in the computational domain were chosen equal to *L _{x}* =

*L*and

*L*= 11b , respectively, to guarantee the good compromise between the calculation times and the accuracy of the numerical solutions, as suggested by Andreozzi and Manca [33].

_{y}Five rectangular meshes (n_{x} × n_{y}) 41 × 15, 81 × 30, 161 × 60, 321 × 120 and 641 × 320 were analyzed to evaluate the grid independence of the numerical solutions. The results of the grid analysis, for *α* = 0 deg, Ra = 4.80 × 10^{6}, Da = 2.65 × 10^{−6}, PPI = 10 and *ε* = 0.90, in terms of tangential velocity profiles, average Nusselt number and average temperature on the heated plate are shown in Fig. 2.

It is observed, in Fig. 2(a), that the tangential velocity profiles are completely overlapping except for the profile corresponding to coarse mesh 41 × 15 where the maximum percentage error with respect to the finer grid is about 16%. The Fig. 2(b) shows that the average Nusselt number increases as the number of grid nodes increases until up to 161 × 60 and then decreases reaching the asymptotic value of 5.532. An opposite trend, indicated by the circle symbol in Fig. 2(b), is observed for average temperature of the heated upper plate and reaches the asymptotic value of 0.1814. The percentage error of the monitored variables such as average Nusselt number and temperature is evaluated with respect to the asymptotic value. For the coarse mesh 41 × 15, the percentage errors are of −2.22% and 2.43% for the average Nusselt number and average temperature of the heated plate, respectively. They are less than 0.25% for the grid 321 × 120. The mesh 321 × 120 is employed for all numerical simulations, because it ensured a good compromise between the machine computational times and the high accuracy of the solution.

The results of the numerical model realized with ansys Fluent code are validated by comparing them with those obtained numerically by Andreozzi and Manca [33] and experimentally by Onur and Aktaş [39] and Azevedo and Sparrow [40]. Natural convection in the horizontal parallel plate channel with constant heat flux upper plate has been considered in Ref. [33], and the input parameters Ra = 10^{3} and 10^{6}, *L*/*b* = 5, and 10 are considered to validate the present numerical model. The present data in terms of the temperature profiles along the upper and bottom plates, Fig. 3(a), showed a good agreement with those illustrated in Ref. [33]. In the second validation test, the experimental results in Refs. [39,40] on free convection between inclined parallel plates with isothermal top plate are compared with the results of the present numerical model under the same input parameters and boundary conditions. It is noted a good agreement between the present numerical data and those obtained in Ref. [39] in terms of the average Nusselt number for *α* = 45 deg and 90 deg, while for *α* = 45 deg there are some discrepancies between the average Nusselt number profile, in Ref. [39], and those given in the present model. Another comparison with the numerical results given by Nithiarasu et al. [41] was accomplished to validate the porous medium model developed with ansysfluent. In Ref. [41], natural convection inside a square cavity with porous medium was studied numerically using Galerkin's finite element method. The comparison among the average Nusselt number profiles as a function of Ra obtained in the present work and in Ref. [41] for Da = 10^{−2} and Da = 10^{−6} are shown in Fig. 3(c), and the differences are very small.

## 4 Results and Discussion

Temperature and stream function fields, depicted in Fig. 4, for the assigned Rayleigh number equal to 1.2 × 10^{5}, show the main differences of the thermal and fluid dynamic behaviors between the two configurations with and without the metal foam, on the left and right sides. The vertical channel, with inclination angle *α* = 90 deg in Fig. 4(a), without metal foam presents a heated zone of the channel close to the heated wall whereas the unheated fluid is extended for large part of the channel. However, for the considered aspect ratio equal to 10, it is observed that the chimney effect determines a vertical hot plume outside the channel with the suction of fluid from the unheated wall. A consequent depression close to the unheated wall is present and a flow from the external incomes from the outlet section and goes down along the unheated surface and the cold inflow (downflow) extends up to 30% of the channel height from the outlet section. These results are in according to Refs. [42,43]. The decrease in the inclination angle, *α* = 45 deg in Fig. 4(b), determines an increase of the cold downflow penetration along the adiabatic surface, with the plume from the heated wall flowing almost adjacent to the upper adiabatic wall of the external reservoir. For lower angle, *α* = 2 deg in Fig. 4(c), the downflow presents a lower penetration due to the opposite backflow at the inlet section. This causes a higher velocity inside the inlet section and along the adiabatic surface reducing the effect of downflow. The mass flowrate decreases significantly causing an increase of temperature along the heated wall as show in Fig. 4(c). The maximum wall temperature starting close to the outlet section for the vertical channel, in Fig. 4(a), moves toward the center of the heated wall for the horizontal channel, as noted in Fig. 4(d). In fact, for *α* = 0 deg in Fig. 4(d), a symmetry with respect to the middle plane of the channel is reached and the flow presents the classical “C loop” in agreement to the experimental [44] and numerical [33,45] results. For the configurations with metal foam, in Figs. 4(e)–4(h), the presence of metal foam determines a significant increase of the effective thermal conductivity with lower maximum wall temperature from *α* = 2 deg to 90 deg whereas the horizontal channel reaches a higher maximum wall temperature due to the significant decrease of the mass flowrate in the clean channel.

For the vertical channel in Fig. 4(e), the presence of metal foam up to the half of the gap makes the passage narrow with higher air velocity inside the clean zone. Due to the higher air velocity and the lower gap, the downflow in the outlet section is not present. The part with metal foam exhibits a quasi-uniform temperature along the transversal sections with some temperature gradients along the axial coordinate. This allows to have the temperature at the interface between the metal foam and the clean channel almost equal to the value of the heated wall. The buoyancy due to the chimney effect is quite similar to the one related to the clean channel inducing a higher air flow velocity. For *α* = 45 deg in Fig. 4(f), the fluid flow is quite parallel along the clean part of the channel up to the exit section where the fluid flows adjacent to the fluid coming from the metal foam along the adiabatic wall in the upper reservoir. Also, for this configuration, the downflow is not detected in the outlet section. For *α* = 2 deg in Fig. 4(g), the downflow in the outlet section is detected. In the inlet section, the flow is different from the clean case and the back flow is not detected but the metal foam generates a plume along the inlet section related to the porous medium zone. In fact, the heated metal foam warms up the air close to the inlet section and the plume starts adjacent to the adiabatic wall. For the horizontal channel, *α* = 0 deg, in Fig. 4(h), the motion presents the characteristic C-loop, as for the clean case, but with an evident penetration inside the metal foam from the clean zone.

Dimensionless *U*-component velocity profiles at Ra = 1.20 × 10^{5} and for different angles *α* are reported in Fig. 5, for the clean case (Figs. 5(a) and 5(b)) and for the porous case with PPI = 10 and *ε *= 0.90 (Figs. 5(c) and 5(d)). The profiles are pertinent to the transversal sections *X* = 0 and *X* = 10. For the clean case at *X* = 0, Fig. 5(a), the dimensionless *U*-component velocity profiles show positive values in a large area starting from the lower adiabatic wall and negative values near the upper heated wall for *α* = 0 deg and 2 deg. The inversion of the velocity sign is observed at *Y*≈ 0.7 for *α* = 0 deg and at *Y* ≈ 0.8 for *α* = 2 deg. Similar trends but with opposite sign are observed in the outlet section *X* = 10, Fig. 5(b), for *α* = 0 deg and 2 deg. Particularly, for *α* = 0 deg the velocity profiles are hemi-symmetrical; this indicates the establishment of a C-Loop motion in the horizontal channel as already observed thanks to the stream function fields shown in Figs. 4(c) and 4(d). For *α* = 2 deg, the dimensionless *U*-component velocity profiles at *X* = 0 and *X* = 10 are no more hemi-symmetrical, even if they are like the ones for the horizontal channel (*α* = 0 deg). In fact, the inversion point of the sign of the *U* velocity at the outlet section *X* = 10, Fig. 5(b), is *Y* ≈ 0.55, that is about 30% lower than the one at the inlet section. This indicates the presence of a positive net mass flow entering the system when the channel is slightly inclined. For *α* > 2 deg, the profiles show increasing positive values as the angle of inclination increases at *X* = 0 (Fig. 5(a)); so, at the inlet section the backflow is no longer present for *α* > 2 deg due to the increase in the *X* component of the buoyancy force with the increase in the channel inclination angle. At *X* = 10, Fig. 5(b), for all the analyzed *α* values, the profiles show negative values close to the adiabatic wall and positive values close to the heated wall. The transversal section at which *U* is equal to zero decreases at increasing *α*, from *Y* ≈ 0.7 for *α* = 0 deg to *Y* ≈ 0.30 for *α* = 90 deg. The *U*-component velocity profiles for the porous case with PPI = 10 and *ε* = 0.90, Figs. 5(c) and 5(d), show in the clean region similar trends with respect to the configuration without porous medium at *X* = 0 and *X* = 10. In fact, in this region it is observed a complete C-loop for *α* = 0 deg and a partial C-loop for *α* = 2 deg. Moreover, in the porous region, the profiles are flat with mean values of two orders of magnitude smaller than those in the clear zone. This is due both to the high frictional resistance between fluid and the porous medium and to the buoyancy effects due to smaller temperature gradients in the porous area, as observed in Figs. 4(e) and 4(h).

Dimensionless temperature profiles at Ra = 1.20 × 10^{5} and for different angles *α* are reported in Fig. 6, for the clean case (Figs. 6(a) and 6(b)) and for the porous case with PPI = 10 and *ε* = 0.90 (Figs. 6(c) and 6(d)). The profiles are pertinent to the transversal sections *X* = 0 and *X* = 10. At *X* = 0, for the clean case, Fig. 6(a), the dimensionless temperature is almost equal to zero from the lower adiabatic plate to *Y* ≈ 0.45, for all the analyzed inclination angles. For *Y* > 0.45, the fluid dimensionless temperature increases at increasing *Y* reaching the maximum value close to the heated upper wall (*Y* = 1). Moreover, it is observed that the temperature is higher for the horizontal channel, *α* = 0, and it decreases at increasing *α*. In fact, the C-loop presence for the horizontal channel configuration and the partial C-loop presence for *α* = 2 deg, shown by the stream function and temperature fields, involves that the cold air enters the channel moving along the adiabatic wall, then rises toward the heated wall and finally heats up moving toward the outlet section, For *α* > 2 deg the C-Loop motion disappears completely and the heating of the air in the inlet section close to the overheated wall is due only to the thermal diffusion between the plate and the environment. For *α* = 0 deg at *X* = 10, Fig. 6(b), the temperature profile can be superimposed on the temperature profile at the inlet section, Fig. 6(a), thanks to the complete C-loop present in the channel. For *α* = 2 deg the penetration of the vortex in the outlet section is less than in the case *α* = 0 deg and this entails a higher temperature of the fluid close to the heated plate. For *α* > 2, the maximum value of the dimensionless temperature decreases at increasing the channel inclination angle up to *α* = 45 deg, whereas the maximum temperature values for *α* = 45 deg and *α* = 90 deg are approximately equal.

In the case of a channel partially filled with a porous medium, Figs. 6(c) and 6(d), the temperature decreases as the angle of inclination of the channel increases, due to the weak infiltration of cold air in the porous region and to the intensification of convective motions in the clean region with increasing of *α*. Moreover, for all the analyzed *α* values, the dimensionless temperature profiles are linear with *Y* with a small slope in the porous region that becomes very large at the interface between the porous and clean regions. This is due to the thermal diffusion dominance, associated with the high effective thermal conductivity of the porous region, on the weak infiltration motions of the fluid in the same region. Furthermore, the fluid flow with low thermal conductivity on the interface between the clean and porous regions causes a strong reduction of the temperature in this region. Figs. 6(c) and 6(d) show that the temperature profiles in the clean region are very similar to those observed in the channel in the absence of the porous medium.

The dimensionless wall temperature profiles are shown in Fig. 7, for the clean case and partially filled channel with metal foam with a porosity of 0.90, at Ra = 1.2 × 10^{5} and 1.2 × 10^{7}. The highest temperature values are attained for the horizontal channel in all configurations, both with and without metal foam. As expected, the direct consequence of the increase in inclination angle is the decrease in wall temperature and the asymmetric profiles with the inlet region at lower temperature. The maximum wall temperature moves from the center of the upper plate toward the outlet region. The sharp increase and decrease at the inlet and the outlet present for the clean channel, in Figs. 7(a) and 7(c), disappears with the presence of metal foam, as observed in Figs. 7(b) and 7(d). This is due to the higher diffusive effects related to the metal foam which determines lower temperature gradients along the heated wall.

Moreover, in the clean cases, for the lowest inclination angle (*α *≤ 5 deg) a decrease in the mass flowrate is also present and there is the presence of back flow (*α* ≤ 2 deg). For Ra = 1.2 × 10^{5}, the maximum temperature is attained for the case with metal foam, in Fig. 7(b), at *α* ≤ 2 deg for *X* = 5. For Ra = 1.2 × 10^{7}, in Fig. 7(c), the wall temperatures for the clean cases are higher than the ones for the cases with metal foam except for the inlet zone for *α* ≥ 15 deg. It is interesting to observe for the case with metal foam that in the outlet zone for *X* > 8 the wall temperature for α = 45 deg is lower than the one for *α* = 90 deg, the vertical channel. This is determined to the cold fluid that at the outlet section flows from the clean zone along the porous medium cooling the metal foam, according to Fig. 4(f). For the vertical channel the fluid motion in the outlet section presents a vertical plume that does not interact with the horizontal upper surface of the metal foam. The wall temperature profiles point out that the presence of metal foam allows to obtain an improved heat transfer for higher Ra values. This can be employed in the design of thermal control systems to realize a more efficient cooling effect. For lower Ra values the higher wall temperature attained in the horizontal case with the presence of metal foams suggests their use in thermal storage. However, the presence of metal foams determines in all cases lower temperature gradient along the heated plate.

The dependence on inclination angle of average Nusselt number is highlighted in Figs. 8 and 9 for clean case and partially filled cases. In all cases, the angle increase determines the increase in Nu_{avg} with a greater increase from 0 deg to 45 deg. For the greater angles the increase is low and mainly for the clean case. At Ra = 1.2 × 10^{5}, in Figs. 8(a) and 8(c), the comparison between the clean values and the cases with foam at a porosity equal to 0.90 and 0.95 shows very small differences between all cases. The differences increase for the higher inclination angle and for porosity value of 0.95 the differences are greater. The highest values are attained for a pore density equal to 10 PPI. For Ra = 1.2 × 10^{7}, in Figs. 8(b) and 8(d), the differences among the cases with and without metal foam is significantly greater than the previous cases at lower Ra. The presence of foam determines a relevant increase in terms of pore density manly for inclination angle greater than 30 deg. In fact, for *ε* = 0.90, the percentage increase for 10 PPI respect to the clean case is about 38% for *α* = 45 deg and about 42% for the vertical channel (*α* = 90 deg) whereas for 40 PPI the values are about 21% for both the angle. For *ε* = 0.95, the percentage increases are for 10 PPI 45% and 55%, for *α* = 45 deg and 90 deg, respectively, and for 40 PPI 25% and 26% for 45 deg and 90 deg, respectively. It is interesting to observe that decreasing the pore density the average Nusselt number increases, and the clean case presents the lowest values. This can indicate that could exist an optimal pore density due to the opposite forces linked with the buoyancy force and with the viscous force. The last force increases increasing with the pore density due to the increase of contact surface between the fluid and the solid matrix of the metal foam which determines a greater viscous friction. These aspects give significant indications for the thermal design to realize a more efficient system in terms of heat transfer coefficient.

The effect of porosity value on average Nusselt number is shown in Fig. 9. It remarks the previous considerations about the dependence on Rayleigh number with a weak variation for lower Ra = 1.2 × 10^{5}, in Fig. 9(a). For lower Ra, the diffusion effect is dominant, and a weak chimney effect is present. For higher Ra value, equal to 1.2 × 10^{7}, in Fig. 9(b), the average Nusselt number strongly increases with presence of metal foam and a greater increase is obtained for higher porosity value. The increase in average Nusselt number due to the porosity increase for the case at higher Ra values allows to design configurations for thermal control. For lower Ra the porous the porous media does not determine a significant increase but it can be employed to design a sensible thermal energy storage evaluating the more convenient porosity value.

Both trends with respect to the pore density and porosity point out that the use of metal foam in an inclined channel presents optimal configurations with respect to the convective heat transfer coefficient.

## Conclusions

Natural convection in an inclined channel with and without open metal foam was numerically studied in two-dimensional steady-state regime. The upper wall of the inclined channel was heated at uniform heat flux and the lower wall was adiabatic. The convective heat transfer in the metal foam was modeled assuming the Brinkman–Forchheimer-extended Darcy equation and the local thermal equilibrium. The results were accomplished to compare the inclined channel with and without metal foam and the analysis highlighted the following behaviors:

For the cases with and without metal foam the increase of inclination angle from the horizontal configuration determined the heat transfer increase with the higher increases from 0 deg to 60 deg.

For lower Ra value, 1.2 × 10

^{5}, the increase in average Nusselt number was contained in a narrow range both with respect to the porosity and pore density.For higher Ra value, 1.2 × 10

^{7}, the metal foam allowed to significantly improve the heat transfer from the heated wall in terms of average Nusselt number and it increased increasing the porosity and decreasing the pore density.In any cases with metal foam the temperature gradients along the heated wall are lower due to the higher diffusivity of metal foam.

These main behaviors for the inclined partially filled channel with and without the open metal foam allow to give some indications for their use in different applications such as the thermal control and thermal energy storage. For lower Ra some indications for the design of sensible thermal energy storage system for high temperature solar energy systems or ventilated roofs can be provided. For higher Ra value the results can be useful in the design of thermal control systems such as in electronic cooling. Moreover, the average Nusselt number trends with respect to the porosity and pore density point out the possible evaluation of optimal geometric configurations in terms of heat transfer rate density.

## Acknowledgment

This research was funded by Università degli Studi della Campania “Luigi Vanvitelli” with the grant number D.R.N. 138 under Nano-TES project—V:ALERE program 2020.

## Funding Data

Ministero dell'Istruzione, dell'Università e della Ricerca (Award No. PRIN-2017F7KZWS; Funder ID: 10.13039/501100003407).

Università degli Studi della Campania “Luigi Vanvitelli” (Award No. Nano-TES project—V:ALERE program 2020).

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*a*_{sf}=the specific surface of metal foam (m

^{−1})*A*=weighting coefficient Eq. (8)

*A*_{sf}=total heat transfer surface inside the metal foam (m

^{2})*b*=distance between parallel plates (m)

*C*=_{F}Forchheimer coefficient

*c*=_{p}specific heat at constant pressure (J kg

^{−1}K^{−1})- Da =
Darcy number

*d*,_{f}*d*=_{p}fiber and pore diameter (mm)

*g*=acceleration due to the gravity (m s

^{−2})- Gr =
Grashof number

*h*_{sf}=local convective heat transfer coefficient

*k*=thermal conductivity (Wm

^{−1}K^{−1})*K*=permeability (m

^{2})*L*=plate length, m

*L*=_{x}width of the reservoir, m

*L*=_{y}height of the reservoir, m

- $m\u02d9$ =
mass flow rate (kg s

^{−1})- Nu =
Nusselt number

*p*=pressure (Pa)

- PPI =
pores per inch

- Pr =
Prandtl number

- $q\u02d9$ =
heat flux (W m

^{−2})- Ra =
Rayleigh number

*s*=metal foam thickness (mm)

*S*=dimensionless metal foam thickness

*T*=temperature (K)

*u*,*v*=velocity components (m s

^{−1})*U*,*V*=dimensionless velocity components

*V*=_{t}total volume of metal foam (m

^{3})*W*=plate width (m)

*x*,*y*=Cartesian coordinates (m)

*X*,*Y*=dimensionless Cartesian coordinates

*α*=inclination angle (deg)

*α*=_{f}thermal diffusivity (m

^{2}s^{−1})*β*=thermal expansion coefficient (K

^{−1})*η*=binary parameter

*θ*=dimensionless temperature

*ρ*=density (kg m

^{−3})*ε*=porosity

*μ*=dynamic viscosity (kg m

^{−1}s^{−1})*ν*=kinematic viscosity (m

^{2}s^{−1})*ψ*=dimensionless stream function