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Abstract

Effect of surface undulation on convective flow in a porous enclosure is investigated in this present article. The porous enclosure is filled with a water-base nanoliquid containing Cu solid nanoparticles, and the porous layer is modeled using the Darcy law. The governing equations are solved numerically using the built-in finite element method of COMSOL. The investigation was carried out for several parameters: the nanoparticle concentration, $ϕ$ ; the amplitude of undulations, $A$ ; the number of undulations, $N$ ; and the Rayleigh number, $Ra$ . It is concluded that the strength of the nanoliquid circulation increases with increasing the amplitude or number undulations. It is found that the heat transfer rates were sensitive to the variation of undulation property, convection intensity, and nanoparticle concentration.

Keywords

Nanoliquid natural convection porous medium wavy enclosure finite element method

Introduction

Free convection in enclosures has been extensively studied both experimentally and theoretically, being of considerable interest in industrial and science applications, such as solar collector, multi-pane windows, crystal growth in liquids, building insulation, electronic cooling, and geothermal energy systems. In free or natural convection, the thermal flow or buoyancy is induced by temperature variation between the horizontal or vertical walls.

Natural convection in enclosures filled with porous medium has been studied extensively in recent years, which involves moving ground water system, heat exchangers, food processing, energy harvesting, to name of a few. Walker and Homsy¹ studied the convective flow through the porous material using a number of different techniques. Other solution methods were also presented by Manole and Lage² and Saeid and Pop.³ Bilgen and Mbaye⁴ studied heating from below. Two convective solutions branches were found and bifurcate from the zero solution in the direction of increasing Rayleigh number.

Natural convection by heating from the vertical walls for the empty enclosure filled with the new fluids or called as nanoliquid has been analyzed by many authors. Dispersing solid nanoparticles in base liquid would substantially improved the liquid properties. Free convection in enclosures filled with copper nanoparticles in water was conducted by Khanafer et al.,⁵ Jou and Tzeng,⁶ and Das and Ohal.⁷Öğüt⁸ considered dispersed argentum in water, while Rashmi et al.⁹ engineered the alumina–water nanoliquid.

The porous enclosures filled with nanoliquid having different configurations from above were studied by Chamkha and Ismael.¹⁰ They found that the thermal performance advanced with the enhancement of the solid concentration at weak convective intensity. Chamkha and Ismael¹¹ studied a differentially heated partially porous layered cavities filled with a nanofluid, and Ghalambaz et al.¹² extended to the viscous dissipation and radiation effects. Mehryan et al.¹³ applied the hybrid Al₂O₃-Cu water nanoliquid in a differentially heated porous cavity. Sheikholeslami and Shehzad¹⁴ applied finite element method to solve nanoliquid migration in a porous medium for Darcy model. Sheikholeslami and Shehzad¹⁵ and Chamkha et al.¹⁶ simulated the convective nanoliquid flow in a square porous enclosure via non-equilibrium model.

All of the authors considered that the hot and cold surfaces are flat and straight. The thermal performance of the nanoliquid for the case of surface undulations has not been previously considered. The surface undulations were found to decrease the heat transfer¹⁷ for the porous enclosure filled with pure liquid. Kumar¹⁸ reported that the high frequency improves the natural convection from the undulated wall with uniform heat flux. Kumar and Shalini¹⁹ concluded that partial heat flux has a complex periodical pattern in the non-Darcy model case equivalent to the side undulated surface. Misirlioglu et al.²⁰ examined their results with those reported in the open literature for a square enclosure with straight walls. Misirlioglu et al.²¹ concluded that the convective flow is sensitive to the surface waviness for the enclosure orientation lower than 45° at high convective intensity. Khanafer et al.²² found that the amplitude and the number of undulations surface modified the thermal performance. Cheong et al.²³ included the combustion with non-uniform wall heating. Sheremet et al.²⁴ considered a porous wavy enclosure filled with a nanoliquid using Buongiorno’s formulation with thermal dispersion effect. Very recently, Hoghoughi et al.²⁵ included the effect of enclosure configurations on free convection in a porous wavy enclosure filled with a nanoliquid using Buongiorno’s mathematical model. This work focuses on studying the effect of surface waviness on the flow structure and thermal performance for various solid nanoparticles concentration using Tiwari and Das²⁶ nanoliquid mathematical model. Previously, the effect of surface waviness on the flow structure and thermal performance was investigated by Alkhalidi et al.²⁷ They concluded that the wavy cavities could be used as thermal insulation bodies.

Mathematical formulation

A schematic diagram under consideration is presented in Figure 1. It is considered in this study that the right wall has a low temperature $T_{c}$ , while the left undulated wall has a high temperature $T_{h}$ . The bottom and top walls are maintained adiabatic. The profile of the left and right walls follows the pattern

f (y) = A \cos (2 N π y)

(1)

where $A$ and $N$ are amplitude and number of undulations, respectively.

Figure 1.

Schematic representation of the physical domain.

Temperature variation between the left and right walls lead to a natural convection problem. In the porous medium, Darcy’s law is considered valid and the Oberbeck–Boussinesq approximation is applied. Following on these considerations, the continuity, the momentum, Darcy flow, and the energy equations with adopting Tiwari and Das²⁶ nanoliquid mathematical model are

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0

(2)

\frac{μ_{nl}}{K} u = - \frac{\partial p}{\partial x}

(3)

\frac{μ_{nl}}{K} v = - \frac{\partial p}{\partial y} + [ϕ ρ_{sp} β_{sp} + (1 - ϕ) ρ_{bl} β_{bl}] g (T - T_{c})

(4)

u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = α_{nl} (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}})

(5)

The boundary conditions are

u = v = 0 and T = T_{h} at x = - AL \cos (2 N π y / L)

(6)

u = v = 0 and T = T_{c} at x = L + AL \cos (2 N π y / L)

(7)

u = v = 0 and \frac{\partial T}{\partial y} = 0 at y = 0, L

(8)

where subscript $nl$ is nanoliquid; $u$ and $v$ are the velocity components along $x$ and $y$ axes, respectively; $T$ is the liquid temperature; $K$ is the permeability; $g$ is the magnitude of the acceleration due to gravity; $ρ$ is the density; and $μ$ is the dynamic viscosity. The flow is considered to be slow enough so that the Forchheimer quadratic drag could be negligible in equations (3) and (4). The dynamic viscosity of the nanoliquid formulated as Brinkman²⁸

\frac{μ_{nl}}{μ_{bl}} = \frac{1}{{(1 - ϕ)}^{2.5}}

(9)

where $ϕ$ is the nanoparticles concentration. Thermal diffusivity of the nanoliquid is

α_{nl} = \frac{k_{nl}}{{(ρ Cp)}_{nl}}

(10)

where the heat capacitance of the nanoliquid given is

(ρ Cp)_{nl} = (1 - ϕ) (ρ Cp)_{bl} + ϕ (ρ Cp)_{sp}

(11)

and $k_{nl}$ refers to the effective thermal conductivity of nanoliquid subject to spherical nanoparticles, estimated by the Maxwell–Garnetss (MG) model

\frac{k_{nl}}{k_{bl}} = \frac{k_{sp} + 2 k_{bl} - 2 ϕ (k_{bl} - k_{sp})}{k_{sp} + 2 k_{bl} + ϕ (k_{bl} - k_{sp})}

(12)

Thermophysical properties of pure water with Cu are tabulated in Table 1.

Table 1.

Thermophysical properties of pure water with Cu.

Physical properties	Pure water	Cu
$C_{p} (J / kg K)$	4179	383
$ρ (kg / m^{3})$	997.1	8954
$k (W m^{- 1} K^{- 1})$	0.6	400
$β \times 10^{- 5} (1 / K)$	21	1.67

Source: Khanafer et al.⁵

Equations (2)–(5) can be transformed into the stream function $ψ$ formulated as $u = \partial ψ / \partial y$ and $v = - \partial ψ / \partial x$ accompany with the following non-dimensional variables and parameter

\begin{matrix} Ψ = \frac{ψ}{α_{bl}}, Θ = \frac{T - T_{c}}{Δ T}, X = \frac{x}{L}, Y = \frac{y}{L} \\ Ra = \frac{g β_{bl} ρ_{bl} Δ TKL}{μ_{bl} α_{bl}} \end{matrix}

(13)

Then, the final equations are obtained

\begin{matrix} \frac{\partial^{2} Ψ}{\partial X^{2}} + \frac{\partial^{2} Ψ}{\partial Y^{2}} = - Ra (1 - ϕ)^{2.5} \\ [(1 - ϕ) + ϕ \frac{ρ_{sp}}{ρ_{bl}} \frac{β_{sp}}{β_{bl}}] \frac{\partial Θ}{\partial X} \end{matrix}

(14)

\frac{\partial^{2} Θ}{\partial X^{2}} + \frac{\partial^{2} Θ}{\partial Y^{2}} = \frac{α_{bl}}{α_{nl}} (\frac{\partial Ψ}{\partial Y} \frac{\partial Θ}{\partial X} - \frac{\partial Ψ}{\partial X} \frac{\partial Θ}{\partial Y})

(15)

Together with the boundary conditions

Ψ = 0, Θ = 1 at X = - A \cos (2 N π Y)

(16)

Ψ = 0, Θ = 0 at X = 1 + A \cos (2 N π Y)

(17)

Ψ = 0, \frac{\partial Θ}{\partial Y} = 0 at Y = 0, 1

(18)

The physical quantity of interest in this problem is the thermal performance across the enclosure. This quantity is evaluated by employing Fourier’s law at the hot wall, that is

q = \frac{k}{2} \sqrt{{(\frac{\partial T}{\partial x})}^{2} + {(\frac{\partial T}{\partial y})}^{2}}

(19)

The resulting non-dimensional heat transfer rate is

Nu = \frac{k_{nl}}{k_{bl}} \frac{1}{2} \sqrt{{(\frac{\partial Θ}{\partial X})}^{2} + {(\frac{\partial Θ}{\partial Y})}^{2}}

(20)

The surface-averaged heat transfer rate, $\bar{Nu}$ , on the hot wall is calculated as

\bar{Nu} = \frac{1}{S} \int_{0}^{s} Nu ds

(21)

where $s$ is the arc position of the hot surface and $S$ is the surface area of the undulated wall.

Computational methodology

The governing equations along with the boundary condition are modeled and solved numerically by the COMSOL. COMSOL is a general-purpose solver of interlinked partial differential equation (PDE) based on the Galerkin finite element method (GFEM). This program contains state-of-the-art numerical algorithms and visualization tools bundled together with an easy-to-use interface. We consider Poisson’s equation (poeq) for equations (14) and (15). In this study, mesh generation on an enclosure is made using triangles. The triangular mesh distribution calibrates for fluid dynamics condition.

Several grid sensitivity tests were conducted to determine the sufficiency of the mesh scheme and to ensure that the results are grid independent. We use the COMSOL default mesh settings, type physics-controlled mesh sizes, that is, extremely coarse, extra coarse, coarser, coarse, normal, fine, finer, extra fine and extremely fine. In the tests, we consider the parameters $Ra = 500$ and $A = 0.2$ , $ϕ = 0.05$ , and $N = 4$ as tabulated in Table 2. Considering accuracy, an extremely fine mesh size was selected for all the computations done in this article. As a validation, the computed streamlines and isotherms are verified with that obtained by Kumar¹⁸ for case the left undulation, $N = 3$ (a) and $N = 4$ (b) at $ϕ = 0.0$ , $Ra = 500$ , $A = 0.05$ as shown in Figure 2. The verification shows good agreement with the literature result. These validations efforts provide evidence for the robustness and accuracy of the current work.

Table 2.

Grid sensitivity check at $Ra = 500$ and $A = 0.2$ , $N = 4$ , and $ϕ = 0.05$ .

Predefined mesh size	Mesh elements	$Ψ_{\min}$	$\bar{Nu}$	CPU time (s)
Extremely coarse	480	$- 18.2286$	7.9443	3
Extra coarse	682	$- 15.2243$	7.7039	3
Coarser	949	$- 14.6259$	7.8599	2
Coarse	1140	$- 14.3518$	7.6870	2
Normal	1500	$- 13.7115$	7.4924	1
Fine	1719	$- 13.6601$	7.4689	2
Finer	2479	$- 13.5243$	7.4380	2
Extra fine	6662	$- 13.4478$	7.6550	4
Extremely fine	24,764	$- 13.4047$	8.1815	15

CPU: central processing unit.

Figure 2.

Verification of computed streamlines and isotherms with literature results for $N = 3$ (a) and $N = 4$ (b) at $ϕ = 0.0$ , $Ra = 500$ , $A = 0.05$ .

Results and discussion

The analysis in this studies is conducted in the following domain of the associated dimensionless groups: the solid volume fraction of Cu nanoparticles, $0.0 \leq ϕ \leq 0.1$ ; the amplitude of undulations, $0 \leq A \leq 0.3$ ; the number of undulations, $3 \leq N \leq 6$ ; and the Rayleigh number, $1 \leq Ra \leq 1000$ .

Figure 3 shows the evolutions of the streamlines and isotherms in the porous enclosure filled with nanoliquid $(ϕ = 0.05)$ or pure water at different amplitudes, $A$ , at $N = 4$ and $Ra = 500$ . The fluid motion as shown in Figure 3 is explained as follows. The hot, undulated plate transfers heat to the fluid and raises the level of fluid thermal particles adjoining the left wall. When the temperature increases, the fluid starts moving from the undulated plate to the rigid plate and falling toward insulated plate, then rising again at the undulated plate, creating a counter, single clockwise rotating cells in the enclosure as displayed in the streamline patterns. The strength of the flow circulation increases with increasing $A$ . This is related to the fluid acceleration response to the hot surface area variation from increasing the undulation amplitude. The isotherms are more curved by increasing the amplitude. The change of amplitude of undulation provides a visual alteration of the core vortex. We observed that the nanoliquid cell circulation is smaller than the water cell circulation, especially at the core of the enclosure. The strength of the flow circulations of the nanoliquid weaker than the water (known from the $Ψ_{\min}$ values). This is related to the viscosity enhancement from increasing the Cu concentration that inhibits the flow circulations. It notes that the liquid volume inside the enclosures for the different amplitudes are kept equivalent.

Figure 3.

Fluid flow and temperature distributions (nanoliquid (solid lines) and pure water (dashed lines)) for different amplitudes, $A$ , at $N = 4$ and $Ra = 500$ : (a) A = 0.0, (b) A = 0.1, (c) A = 0.2, and (d) A = 0.3.

Figure 4 shows the evolutions of the streamlines and temperature distribution in the porous enclosure filled with 5% Cu nanoparticles or pure water for different number of undulations $N$ at $Ra = 500$ and $A = 0.2$ . The strength of the flow circulation increases slightly by increasing the number of undulations for the both pure fluids and Cu–water nanoliquid. The shape of eye vortices is also modified by varying the shape of undulated wall. The strength of the flow movements of the nanoliquid is weaker than the pure water for the considered undulated plate shape. The isotherms for the both pure fluids and Cu–water nanoliquid are almost unchanged by varying the undulated number. It notes that the liquid volume for each enclosure is kept equivalent.

Figure 4.

Fluid flow and temperature distributions (nanoliquid (solid lines) and pure water (dashed lines)) for different number of undulations, $N$ , at $Ra = 500$ and $A = 0.2$ : (a) N = 3, (b) N = 4, (c) N = 5, and (d) N = 6.

Figure 5 shows the evolutions of the streamlines in the porous enclosure filled the pure water or dispersing 5% Cu nanoparticles for different Rayleigh numbers with $A = 0.0$ (left) and $A = 0.2$ (right) at $N = 4$ . Obviously, the strength of the flow circulation rises with increasing the Rayleigh number. The vortex of the flow circulations widens by intensifying the convection for the straight wall and the wave wall cases. The reduction in the streamlines strength of nanoliquid for the both enclosure types is due to the effect because of the viscosity increasing and conductivity enhancement by adding the nanoparticle. This influence is displayed separately in the next figure, where the absolute minimum value of the liquid circulation is depicted for various values of nanoparticle concentrations.

Figure 5.

Streamlines (nanoliquid (solid lines) and base liquid (dashed lines)) for different Rayleigh numbers with $A = 0.0$ (left) and $A = 0.2$ (right) at $N = 4$ : (a) Ra = 10, (b) Ra = 50, (c) Ra = 200, and (d) Ra = 1000.

Figure 6 displays the evolutions of the isotherms in the porous enclosure filled the pure water or dispersing 5% Cu nanoparticles for different Rayleigh numbers with $A = 0.0$ (left) and $A = 0.2$ (right) at $N = 4$ . The isotherms change considerably by varying the $Ra$ . The isotherms are more distorted by intensifying the convection. At the fixed Rayleigh number, the isotherms of pure water are more distorted than the isotherms of nanoliquid. Denser boundary layers at the isothermal walls were formed by increasing the convection intensity. At $Ra = 1000$ , the thermal boundary layer follows the wave walls shape, especially at the left, lower part and at the right, upper part of the enclosure.

Figure 6.

Isotherms (nanoliquid (solid lines) and pure water (dashed lines)) for different Rayleigh numbers with $A = 0.0$ (left) and $A = 0.2$ (right) at $N = 4$ : (a) Ra = 10, (b) Ra = 50, (c) Ra = 200, and (d) Ra = 1000.

Figure 7 shows the increasing absolute values of maximum clockwise flow circulation by increasing the Rayleigh number or the heating intensity. The flow strength was suppressed by increasing Cu nanoparticles volume fraction at moderate and strong Rayleigh numbers. It indicates that the water flow is stronger than nanoliquid flow. This result contradicts the previous result for the non-porous enclosure as given by Khanafer et al.⁵ and Fattahi et al.²⁹ They found an increase in flow circulation with increasing solid volume fraction, in particular at moderate and strong Rayleigh numbers. At moderate and strong Rayleigh numbers, the suppression of the flow strength due to the increase in the viscosity (which is the result of increasing Cu concentration) is greater than the heat transfer improvement due to the presence of nanoparticles. Moreover, the suppression of the flow strength by the presence of nanoparticles is more considerable for the case of a porous matrix with low permeability or a porous medium with high thermal conductivity.

Figure 7.

The $| Ψ_{\min} |$ against Rayleigh number for different $ϕ$ values at $N = 4$ and $A = 0.2$ .

The effects of the nanoparticles volume fraction and the Rayleigh number on the average Nusselt number at $A = 0.2$ and $N = 4$ are shown in Figure 8. Higher solid concentration gives higher $\bar{Nu}$ values for sufficiently weak Rayleigh number and higher solid concentration gives lower $\bar{Nu}$ values for sufficiently strong Rayleigh number. About $Ra = 50$ , a balance action between the effects of viscous and buoyancy forces on one side, and the effects of thermal conductivity on the different side. At weak Rayleigh number, the convection is small or raising the inertial and viscous forces do not have much influence on the gravitational force which is already low, so that the enhancement in the liquid conductivity will take its role in boosting the heat transfer. In contrast with result at strong Rayleigh number, the buoyancy force will be powerful and hence raising the inertial and viscous forces will have a contra effect on buoyancy and overcome the enhancement in the liquid conductivity that suppresses the heat transfer as shown in Figure 8. This result is equivalent to finding given by Chamkha et al.¹⁶ for the thermal non-eq22 modeling. Thus, the selected parameter values are all found to have favorable effects to satisfy the local thermal equilibrium and thermal non-equilibrium assumptions.

Figure 8.

Effect of the nanoparticles concentration and the Rayleigh number on the average Nusselt number at $A = 0.2$ and $N = 4$ .

The effects of the nanoparticles volume fraction and the amplitude undulation on the average Nusselt number at $Ra = 500$ and $N = 4$ are shown in Figure 9. It is clear that the undulated walls have an influence on the geometrical form taken by the cell as it is noticed on the different streamline patterns as shown in Figure 3. In this figure, we study influence of the undulated wall and the concentration to the heat transfer rate. Higher concentration gives lower $\bar{Nu}$ values for the all amplitude values considered here. It shows that the nature of the response to the decrement of the concentration is same for all amplitude points. The $\bar{Nu}$ values were observed minimum about $A = 0.1$ . The reason behind this behavior is that, at this amplitude, the normal velocity component leaves the streamline to the wall, so that suppresses the heat transfer rate. The heat transfer rate seems unsensitive to the rise of the undulation amplitude for $A > 0.12$ .

Figure 9.

Effects of the nanoparticles volume fraction and the amplitude undulation on the average Nusselt number at $Ra = 500$ and $N = 4$ .

The effects of the number of undulation and the amplitude undulation on the average Nusselt number at $Ra = 500$ with 10% Cu nanoparticles are shown in Figure 10. In general, higher undulation number produces lower heat transfer performance for the considered undulated amplitude. For the configuration with three and four undulations, a growth of $A$ from 0.05 to 0.2 shows that there is a reduction in the average Nusselt number, while on increasing the undulation amplitude further, an improvement in heat transfer is achieved. For the configuration with five and six undulations, increasing the amplitude undulation from $A > 0.07$ is aesthetically to decrease the thermal performance. This is related to the plate frequency response to the heat exchange area variation for the different wall amplitudes.

Figure 10.

Effects of the number of undulation and the amplitude undulation on the average Nusselt number at $Ra = 500$ and $ϕ = 0.1$ .

Conclusion

In this numerical simulation, we studied the natural convection induced by a temperature difference between a hot left wall and a cold right wall. The governing equations are solved numerically using the built-in finite element method of COMSOL. Detailed numerical results for liquid flow and temperature distributions and the heat transfer rate have been illustrated and tabulated. The amplitude of the undulated wall, solid concentration and Rayleigh number influenced the geometrical of cells circulation, temperature distribution and heat transfer performance. The main findings of this invest are as follows:

The strength of the nanoliquid flow circulation increases with increasing the amplitude or number corrugations. The change of amplitude and number of undulations provide a visual alteration of the core vortex.

Higher solid concentration gives higher heat transfer for sufficiently weak Rayleigh number and higher solid concentration gives lower heat transfer for sufficiently strong Rayleigh number.

The maximum heat transfer suppression happened in the heat transfer occurs at strong Rayleigh number and high concentration. The reduction is more significant for moderate undulation number and about $A = 0.1$ .

For the configuration with three and four undulations, a growth of $A$ from 0.05 to 0.2 shows that there is a reduction in the heat transfer, while on increasing the undulation amplitude further, an improvement in heat transfer is achieved.

Footnotes

Handling Editor: James Baldwin

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Habibis Saleh

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Natural convection induced by undulated surfaces in a porous enclosure filled with nanoliquid

Abstract

Keywords

Introduction

Mathematical formulation

Computational methodology

Results and discussion

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References