Thermal buoyancy driven flows inside a differentially heated enclosure with porous fins of multiple morphologies attached to the hot wall

International Journal of Thermal Sciences 147 (2020) 106138

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Thermal buoyancy driven flows inside a differentially heated enclosure with porous fins of multiple morphologies attached to the hot wall Lei Wang a, b, c, Run-Zhe Liu a, b, c, Di Liu d, Fu-Yun Zhao a, b, c, *, Han-Qing Wang e a

Key Laboratory of Hydraulic Machinery Transients (Wuhan University), Ministry of Education, Wuhan, Hubei Province, China Hubei Key Laboratory of Waterjet Theory and New Technology (Wuhan University), Wuhan, Hubei Province, China c School of Power and Mechanical Engineering, Wuhan University, Wuhan, Hubei Province, China d College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao, Shandong Province, China e School of Civil Engineering, University of South China, Hengyang, Hunan Province, China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Enclosure natural convection Porous fins Electronic cooling morphology Numerical heat transfer and fluid flow

The application of porous media for electronic cooling process, instead of solid one, generally strengthens heat conduction while weakens convection insignificantly at high Darcy number. In the present work, effects of morphology and topology of porous fins on the laminar natural convection heat transfer were investigated in a differentially heated enclosure. Volume averaged Darcy-Forchheimer model was applied to solve the transport process within the porous media while the Navier-Stokes equations were employed within pure fluid region. Relevant governing parameters, including thermal Rayleigh number, Darcy number, thermal conductivity of the porous matrix, designs of porous fins, are sensitively varied to identify their effects and roles on the natural convection flows. Depending on thousands of numerical data, the correlation has been developed for all designs of porous fins. Numerical results illustrate that the adding porous fins with excellent permeability and heat conduction contribute to the remarkable heat transfer enhancement while the adding fins, acting like solid ones and having poor heat conduction, could result in an increase of thermal resistance and the deterioration of heat transfer. Numerical results further show that there exists an optimal design of porous fins to achieve the best performance of heat transfer if some conditions were satisfied. Overall, this study could benefit the electronic cooling by the installation of porous-alike materials.

1. Introduction The flow, heat and mass transports driven by thermal buoyancy forces are important phenomena in many scientific fields and industrial applications, such as thermal storage, environmental comfort, grain drying process, solar collectors, double-wall insulation, electric ma­ chinery, cooling system of electronic devices and geophysical systems and others [1–6]. During natural convection process, the thermal and hydrodynamic are coupled [4]. Engineering applications of natural convection demand solving coupled problems concerning heat transfer and fluid mechanism which depend on the thermo-physical properties, the temperature difference and the system geometry. Natural convection inside enclosures, considered as internal flows according to Bejan [7], presents the complicated interaction between a finite-size fluid system in thermal communication with all the walls that confine it. Heat transfer augmentation in enclosures by using solid fins and

obstacles has been paid significant attention in recent years [8–14]. Ho and Chang [15] conducted a numerical investigation of conjugate nat­ ural convection in enclosures divided by horizontal fins. They found that the feasibility and effectiveness of using fins to augment heat transfer across a vertical rectangular enclosure heavily depended on the aspect ratio, the Rayleigh number, the thermal conductivity and the number of fins. Further, Yu and Joshi [16] experimentally studied the natural convection heat transfer under the presence of pin-fin sinks. Results showed that heat transfer rate was controlled by the orientation of enclosure. Meanwhile, da Silve and Gosselin [17] proposed a numerical analysis of the laminar natural convection heat transfer in a 3D-dimen­ sional differentially heated cubic enclosure with a fin attached to the hot wall. They were interested in optimizing the heat transfer through modifying the aspect ratios of the fin. Besides, heat transfer enhance­ ment in cubical enclosures with vertical fins was numerically confirmed by Frederick [18]. Outcomes revealed that remarkable heat transfer enhancement over 40% was found and this augmentation decreased for

* Corresponding author. School of Power and Mechanical Engineering, Wuhan University, Dong-Hu South Road, 430072, Wuhan, Hubei Province, PR China. E-mail addresses: [email protected] (D. Liu), [email protected] (F.-Y. Zhao). https://doi.org/10.1016/j.ijthermalsci.2019.106138 Received 18 February 2019; Received in revised form 26 September 2019; Accepted 10 October 2019 Available online 22 October 2019 1290-0729/© 2019 Elsevier Masson SAS. All rights reserved.

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Nomenclature CF CP d1 d2 D1 D2 Da g H k K L lfin Lfin m n N Nu p P Pr Ra T Th

Tc (u,v) (U,V) wfin Wfin (x,y) (X,Y)

the inertia coefficient the isobaric heat capacity (J/kgK) vertical distance between two consecutive fins(m) vertical pitch between both top and bottom fins and horizontal walls (m) dimensionless vertical distance between two consecutive fins dimensionless vertical pitch between both top and bottom fins and horizontal walls Darcy number gravitational acceleration (m/s2) heat function thermal conductivity (W/mK) permeability (m2) dimensional length of the square enclosure(m) dimensional length of porous fins (m) dimensionless length of porous fins iteration number normal vector number of porous fins Nusselt number dimensional pressure (N/m2) dimensionless pressure Prandtl number Rayleigh number dimensional temperature (K) high temperature (K)

cooling temperature(K) dimensional velocity components (m⋅s 1) dimensionless velocity components dimensional width of porous fins (m) dimensionless width of porous fins Cartesian coordinates (m) dimensionless Cartesian coordinates

Greek Symbols thermal diffusivity(m2/s) β thermal expansion coefficient (1/K) δ stop criterion ε porosity θ dimensionless temperature λ thermal conductivity ratio between porous medium and air υ kinematic viscosity (m2/s) ξ total volume fraction of porous fins ρ density(kg/m3) ϕ stand for general variable Ψ stream function

α

subscripts fin f eff s sf 0

high Rayleigh numbers. Bocu and Altac [19] analyzed the laminar natural convection in three-dimensional rectangular enclosures with pin arrays attached to hot wall. They observed that the staggered arrange­ ment of pin was the best configuration to optimize the heat transfer, which showed that heat transfer rate increased positively with fin length. In addition, Elatar et al. [20] numerically investigated the laminar natural convection within a square enclosure with single hori­ zontal fin mounted at the heated wall. Results showed that the fin thickness had negligible effect on the heat transfer while the effective­ ness of fin was generally improved with an increase of the fin length. Besides the above-mentioned studies that focus on the steady natural convection, comprehensive investigations of the natural convection flow transitions within cavities with solid fins have been reported. The fin positioned at the heated wall could induce the transition from steady state regime to turbulent regime at lower critical Rayleigh numbers compared with those from natural convection within cavities without a fin [25]. The literatures [21–26] disclosed that the above-mentioned transition in a differentially heated enclosure, undergoing sudden heating and cooling, presented three distinct stages: an initial stage, a transitional stage and a fully developed stage. In the initial stage, this thin fin initially hindered the development of the vertical thermal boundary layer along the finned sidewall and enforced it to deviate from this wall and thus an intrusion front under the fin appeared. Then the intrusion under the fin reattached to the downstream of the sidewall after it passed the fin. In the transitional stage, a double-layer structure of the thermal boundary layer occurred and finally turned to be in the fully developed stage. These studies showed that the flow around the fin was oscillatory, which perturbed the downstream thermal boundary layer adjacent to the finned sidewall. The predictions of the time-average heat transfer rate were enhanced in comparison with those from situations without a fin. Further, Ma and Xu [27] investigated the unsteady natural convection and heat transfer in a differentially heated cavity with a fin in high Rayleigh number. The adiabatic fin hindered the

porous fins fluid effective porous matrix special fin reference state

development of thermal boundary layer and a thermal plume occurred around the tip of the fin, which caused that the average Nusselt number increased due to the unsteady natural convection induced by the fin. In addition, Dou and Jiang [28] numerically investigated the flow insta­ bility and heat transfer of natural convection in a differentially heated cavity with thin fins on the heated sidewall through energy gradient theory. They found that the heat transfer was optimized as the only one fin was fixed at the middle height of the cavity. Another effective option to optimize the heat transport is to use porous medium. Investigations concerning heat transfer in porous media are extensively documented [29–32]. Porous medium has extensive applications in terms of nuclear reactor, oil and gas extraction, thermal storage, chemical reaction engineering and so on [33–36]. Recently, M. Sheikholeslami et al. [37–43] investigated the coupling effects of nanofluids and porous medium on natural convection heat transfer within enclosures. There exist many studies concerning effects of porous fins in convective heat transfer exposed in open environments where different mechanisms of heat transfer are considered, including natural convection heat transfer, forced convection heat transfer and convective-radiative heat transfer [44–49]. However, effects of the porous fin on natural convection heat transfer within enclosures has received minimal attention. Laminar natural convection heat transfer in a differentially heated cavity with an inclined porous fin in touch with the hot wall was numerically studied by Khanafer et al. [50] for different domains of governing parameters (103 � Ra � 105 and 10 6 � Da � 10 2). They presented that the porous fin enhanced the average Nusselt number and should be horizontally placed either close to the bottom surface or in the middle height of the finned sidewall in order to achieve the optimum heat transfer. Alshuraiaan and Khanafer [51] also inves­ tigated the effect of shifting the heated thin porous fin on the laminar natural convection heat transfer in a differentially heated cavity. A nu­ merical investigation of effects of nanofluid and porous fins on natural convection and entropy generation in a cavity was performed by 2

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of 103-106. They found that the porous fins had better performance of heat transfer in comparison with that of the solid fins. Results showed that the heat transfer rate increased positively with the length of the fins. The application of porous fins could be an efficient way to maintain strong convection and achieve great heat conduction improvement simultaneously. However, to the best knowledge of authors, few works have been done involving porous fins, the analysis of their morphology and topology in an enclosure has not been investigated so far, consid­ ering wide ranges of properties of the porous medium. In the present study, natural convection heat transfer in a square enclosure with various designs of porous fins is analyzed numerically with wide ranges of the governing parameters. Effects of various parameters including Rayleigh number (105� Ra �107), Darcy number (10 6� Da �102), thermal conductivity ratio (100� λs � 103) and designs of the porous fins are analyzed. 2. Physical and mathematical models As shown schematically in Fig. 1, buoyancy-driven laminar natural convection within a square enclosure with porous fins attached to the heated sidewall is introduced. This square enclosure has length L where its left wall is subjected to a high temperature Th while the opposite one maintains a cooling temperature Tc, and horizontal walls are assumed to be adiabatic. For economic consideration, total volume fraction ξ of porous fins maintains small. The length and width of them are repre­ sented by lfin and wfin, respectively. The vertical distance between two consecutive fins is d1. The vertical pitch between both top and bottom fins and horizontal walls is d2. The saturated porous fins are assumed to be isotropic and homoge­ neous with constant thermal properties while the porous medium and fluid phases are considered to coexist under local thermal equilibrium. In addition, the saturated porous fins are assumed to be in the perfect contact with the heat source. Hypotheses of incompressible and laminar flow are applied. Further, working fluid (room air) is considered to be Newtonian fluid with a constant Prandtl number 0.71. Gravitational acceleration acts parallel to the vertical walls. Thermal properties of fluid phase are considered to be constant excluding density variation in the buoyancy force terms, which varies linearly with temperature, i.e., Boussinesq approximation is considered. In addition, effects of viscous heat dissipation and thermal radiation are assumed to be neglected. Within the porous region, Darcy-Forchheimer model is considered to be valid in the present work. Total volume fraction ξ of porous fins is defined as volume ratio of all fins to the whole enclosure. ξ¼

Nlfin wfin lfin wfin ¼N ¼ NLfin Wfin L2 L L

(1)

where N indicates quantities of porous fins, the total volume fraction ξ of porous fins is fixed at 0.12, Lfin represents the dimensionless length of fins and Wfin is the dimensionless width of fins. The dimensionless vertical distance D1 between two consecutive fins is calculated as the following expression, D1 ¼

Fig. 1. (a) Schematic of physical model. (b) A special porous fin structure of physical model (Wf ¼ 1.0).

d1 1 ¼ L

NWfin 1 ¼ N N

ξ NLfin

(2)

Dimensionless vertical pitch D2 between both top and bottom fins and horizontal walls is defined as,

Siavashi et al. [52] for Rayleigh number of 104-106, Darcy number of 10 4-10 1 and nanoparticle volume fraction of 0–4%. Numerical results illustrated that the presence of porous fins enhanced the heat transfer at high Darcy numbers whereas they hindered the heat transfer due to the weakness of convection flow for tiny Darcy numbers. They also found that an increase either in the length of or in the number of fins had insignificant effect on the average Nusselt number. A numerical analysis of low-pressure flows in an inclined square cavity with two fins placed at the heated wall was reported by Al-Kouz et al. [53] for Rayleigh number

D2 ¼

d2 D1 ¼ L 2

(3)

If Wfin was equivalent to 1.0, both D1 and D2 decrease to zero and corresponding physical schematic was exhibited in Fig. 1(b), which could be named by case Nsf [31–33,61], where the subscript ‘sf’ in­ dicates the special fin.

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Table 1 Grid independence tests for two typical designs of porous fins at Ra ¼ 107, λs ¼ 103, ε ¼ 0.9 and Pr ¼ 0.71.

�

Grid system Case Nsf

Case with N ¼ 10 and Lfin ¼ 0.6

Da 10 6 10 4 10 2 10� 102 10 6 10 4 10 2 10� 102

102 � 102 17.929 24.138 27.628 27.944 27.987 15.909 26.032 27.076 27.226 27.245

�

152 � 152 17.407 23.835 27.172 27.473 27.515 15.711 25.587 26.607 26.755 26.775

202 � 202 17.163 23.829 27.051 27.343 27.387 15.602 25.536 26.646 26.842 26.871

∂uT ∂vT ∂2 T ∂2 T þ þ ¼ αf ∂x ∂y ∂x2 ∂y2

252 � 252 17.032 23.767 26.973 27.262 27.306 15.577 25.356 26.372 26.518 26.538

De Vahl Davis [65] Fusegi [66] Barakos and Mitsoulis [67] Khanafer et al. [50] Present Work

Ra ¼ 104

Ra ¼ 105

Ra ¼ 106

1.118 1.105 1.114 1.115 1.117

2.243 2.302 2.245 2.226 2.246

4.519 4.646 4.51 4.505 4.536

8.798 9.012 8.806 8.778 8.922

� (7)

∂u ∂v þ ¼0 ∂x ∂y 1

�

ε2 1

�

ε2

(8) �

∂uu ∂vu þ ¼ ∂x ∂y �

∂uv ∂vv þ ¼ ∂x ∂y

þ gβf ðT

1 ∂p νf þ ρ0 ∂x ε 1 ∂p μf þ ρ0 ∂y ε

Present

1.520 2.337 3.097 6.383 12.980

1.588 2.415 3.183 6.522 13.188

Ra

10-4

10–2

105 106 107 5 � 107 103 104 105 5 � 105

keff ¼ εkf þ ð1

Present

Nithiarasu [68]

Present

1.067 2.55 7.81 13.82 1.01 1.408 2.983 4.99

1.065 2.600 7.880 14.050 1.000 1.361 2.900 5.010

1.071 2.725 8.183 15.567 1.015 1.53 3.555 5.74

1.068 2.719 8.235 15.850 1.011 1.490 3.447 5.780

θ¼

�

νf K

v

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi CF v2 þ u2 pffiffiffi v K (10)

� (11)

(12)

(13)

εÞks

ðT ðTh

gβf ðTh Tc ÞL3 T0 Þ υf K keff ks ; Da ¼ 2 ; λeff ¼ ; λs ¼ ; Pr ¼ ; Ra ¼ Tc Þ αf αf νf kf kf L

(14)

The aforementioned governing equations could be converted into the conservative non-dimensional forms with these dimensionless variables. Within the cavity,

The governing equations of natural convection heat transfer problem investigated in this study are presented as follows, Within the cavity [60, 62],

∂U ∂V þ ¼0 ∂X ∂Y

(15) �

�1=2 �

(4)

∂UU ∂VU ∂P Pr þ ¼ þ Ra ∂X ∂Y ∂X

(5)

∂UV ∂VV ∂P Pr þ ¼ þ Ra ∂X ∂Y ∂Y

�

�

∂uu ∂vu 1 ∂p ∂2 u ∂2 u þ þ ¼ þ vf ∂x ∂y ρ0 ∂x ∂x2 ∂y2

�

∂2 v ∂2 v þ ∂x2 ∂y2

(9)

x y uL vL p X ¼ ;Y ¼ ;U ¼ ;V ¼ ;P ¼ L L ρ0 gβΔTL αf ðPrRaÞ1=2 αf ðPrRaÞ1=2

2.1. Governing equations

∂u ∂v þ ¼0 ∂x ∂y

K

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi CF v2 þ u2 pffiffiffi u K

where ks represents thermal conductivity of the porous medium and kf indicates the thermal conductivity of fluid. To facilitate the dimensionless analysis, following non-dimensional variables were introduced,

ε ¼ 0.6

Nithiarasu [68]

u

whereas it is assumed that heat conduction in the porous medium and fluid phases occurs in parallel, we could obtain the overall conductivity keff through the weighted arithmetic mean of the conductivities of porous medium and fluid in the following format,

Nu

ε ¼ 0.4

νf

1:75 CF ¼ pffiffiffiffiffiffiffiffiffiffiffi 150ε3

Table 4 Numerical results for natural convection heat transfer in a differentially heated square porous cavity by using Darcy-Forchheimer model (Pr ¼ 1.0 and λeff ¼ 1.0). Da

�

�

where ε represents the porosity of the porous medium, CF indicates the inertia coefficient and keff represents the effective thermal conductivity of the saturated porous medium. Further, the inertia coefficient CF could be mathematically expressed as that proposed by Refs. [57,58].

Nu Leong et al. [69]

∂2 u ∂2 u þ ∂x2 ∂y2

∂uT ∂vT keff ∂2 T ∂2 T þ þ ¼ ∂x ∂y ðρCP Þf ∂x2 ∂y2

Table 3 Average Nusselt numbers compared with experimental results from the case of natural convection heat transfer of air in a differentially heated cubic cavity.

104 4 � 104 105 106 107

�

T0 Þ �

Ra

(6)

T0 Þ

In the aforementioned equations, ρ0, υf, αf, β and g represent the density at temperature T0, kinematic viscosity, thermal diffusivity, co­ efficient of volumetric thermal expansion and acceleration due to the gravity respectively. Meanwhile, the governing equations for mass, momentum and en­ ergy conservations within the porous medium are based on total representative element volume-averaging method and outlined in Refs. [57,58,60],

Table 2 Validation of the numerical code for the case of natural convection heat transfer in a differentially heated square enclosure with single phase fluid (air). Ra ¼ 103

�

∂uv ∂vv 1 ∂p ∂2 v ∂2 v þ þ gβf ðT þ ¼ þ νf ∂x ∂y ρ0 ∂y ∂x2 ∂y2

4

∂2 U ∂2 U þ ∂X 2 ∂Y 2

�1=2 �

� (16)

�

∂2 V ∂2 V þθ þ ∂X 2 ∂Y 2

(17)

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Fig. 2. Streamlines and isotherms of natural convection heat transfer within a square enclosure with a solid fin centrally mounted at the heated wall at Ra ¼ 104, Pr ¼ 0.71, ε ¼ 0, λeff ¼ 103, N ¼ 1 and Wfin ¼ 0.04.

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Fig. 3. Streamlines and isotherms of natural convection heat transfer in a tall enclosure of an aspect ratio 10 with solid fins at Pr ¼ 0.71, ε ¼ 0, λeff ¼ 30, N ¼ 10, Lfin ¼ 0.5 and Wfin ¼ 0.1.

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tions are presented here. Over the enclosure sidewalls, no slip is taken into consideration. (24)

U¼V ¼ 0 Temperature boundary conditions on the solid walls are

(25-a)

θð0; YÞ ¼ 1; θð1; YÞ ¼ 0 �

�

∂θ �� ∂θ � ¼ � ¼0 ∂Y �ðX;0Þ ∂Y �ðX;1Þ

(25-b)

However, fluid flow and heat transfer occur at these interfaces be­ tween pure fluid and saturated porous fins. Flow conditions at these interfaces could be expressed in the following formats [50,51,54–56], � � ∂U � ∂U � ¼ μf �� (26) Uporous ¼ Ufluid ; μeff �� ∂n porous ∂n fluid �

Vporous ¼ Vfluid ; μeff

Table 5 Average Nusselt numbers for the case of natural convection heat transfer in a differentially heated cavity with a thin porous fin attached to the hot wall (Ra ¼ 105, ε ¼ 0.9, λeff ¼ 100.0, Lfin ¼ 0.5 and N ¼ 1).

Khanafer et al. [50] Present Work

Da ¼ 10

5

Da ¼ 10

4

Da ¼ 10

3

Da ¼ 10

4.327

4.682

5.474

5.88

5.946

4.325

4.587

5.489

5.920

5.988

�

∂Uθ ∂Vθ 1 ∂2 θ ∂2 θ þ þ ¼ pffiffiffiffiffiffiffiffiffiffi ∂X ∂Y PrRa ∂X 2 ∂Y 2

2

(18)

ε2

�

�

�1=2 �

∂UU ∂VU ∂P 1 Pr þ ¼ þ ∂X ∂Y ∂X ε Ra

∂2 U ∂2 U þ ∂X 2 ∂Y 2

�

1 Da

1

ε

2

�

Streamlines and heatlines are the effective visualization of the unique features of fluid flow and heat transfer. ‘Heatline’ was first proposed by Kimura and Bejan [59] in 1983 for the effective heat flow visualization during convective heat flow. Such lines are described as, respectively, constant lines of stream function Ψ and heat function H, defined through their first derivatives. The stream function and heat function could be made dimensionless through the characteristic scales pffiffiffiffiffiffiffiffiffiffi αf RaPr and kf ΔT respectively,

rffiffiffiffiffi Pr U Ra

CF pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi V 2 þ U 2 U Da

∂UV ∂VV þ ∂X ∂Y

(20) �

� ¼

�1=2 �

∂P 1 Pr þ ∂Y ε Ra

∂2 V ∂2 V þ ∂X 2 ∂Y 2

�

1 Da

rffiffiffiffiffi Pr V Ra

C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiFffiffiffiffi V 2 þ U 2 V þ θ Da �

∂Uθ ∂Vθ λeff ∂2 θ ∂2 θ þ þ ¼ pffiffiffiffiffiffiffiffiffiffi ∂X ∂Y PrRa ∂X 2 ∂Y 2

∂Ψ ∂Ψ ¼ V; ¼U ∂X ∂Y (21)

εÞλs

(30)

∂H ¼ ðPrRaÞ1=2 Uθ ∂Y

�

λ

∂H ¼ ðPrRaÞ1=2 Vθ ∂X

(22)

∂θ ∂X λ

∂θ ∂Y

(31-a) (31-b)

Then, equations valid across the enclosure are obtained [61–63].

where dimensionless effective thermal conductivity λeff is given by λeff ¼ ε þ ð1

fluid

2.4. Streamlines and heatlines

(19)

�

(28-b)

porous

� ∂θ� ¼ �� ∂n

where values of λ are 1 in pure fluid or λeff within the saturated porous fins, respectively [50,51].

within the porous fins,

1

(28-a)

� ∂θ � λeff �� ∂n

In order to quantify effects of governing parameters on heat transfer, the average Nusselt number (Nu) along the heated wall could be expressed as, Z 1 � ��� ∂θ � Nu ¼ λ dY (29) ∂X �X¼0 0

�

∂U ∂V þ ¼0 ∂X ∂Y

θporous ¼ θfluid ¼ θinterface

2.3. Calculation of Nusselt number

Nu 6

(27)

At the interfaces between pure fluid and saturated porous fins, following continuities of temperature and heat flux should also been satisfied [50,51,54–56],

Fig. 4. Comparison of the present results with those of Yucel and Turhkoglu [70] for the case of natural convection heat transfer of an aspect ratio of 10 with solid fins mounted at the cooling wall at Pr ¼ 0.71, ε ¼ 0, λeff ¼ 30, Lfin ¼ 0.5 and Wfin ¼ 0.1.

Da ¼ 10

�

∂V �� ∂V � ¼ μf �� ∂n �porous ∂n fluid

2

∂ Ψ ∂2 Ψ ∂V þ þ ∂X 2 ∂Y 2 ∂X

(23)

2.2. Boundary conditions Boundary conditions of foregoing dimensionless governing equa­ 7

∂U ¼0 ∂Y

(32)

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International Journal of Thermal Sciences 147 (2020) 106138

Fig. 5. Streamlines, isotherms and heatlines plots (from left to right) for different designs of porous fins at Ra ¼ 107, Da ¼ 10

8

6

, λs ¼ 103 and Lfin ¼ 0.6.

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International Journal of Thermal Sciences 147 (2020) 106138

Fig. 6. Streamlines, isotherms and heatlines plots (from left to right) for different designs of porous fins at Ra ¼ 107, Da ¼ 102, λs ¼ 103 and Lfin ¼ 0.6.

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Fig. 7. Streamlines, isotherms and heatlines plots (from left to right) for different designs of porous fins at Ra ¼ 107, Da ¼ 10

10

6

, λs ¼ 103 and Lfin ¼ 0.3.

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International Journal of Thermal Sciences 147 (2020) 106138

Fig. 8. Streamlines, isotherms and heatlines plots (from left to right) for different designs of porous fins at Ra ¼ 107, Da ¼ 102, λs ¼ 103 and Lfin ¼ 0.3.

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Fig. 9. Streamlines, isotherms and heatlines plots (from left to right) for case Nsf at Ra ¼ 107 and λs ¼ 103.

�

�

�

� �

�

∂ 1 ∂H ∂ 1 ∂H ∂ 1 PrRa þ þ ∂X λ X λ ∂X Y λ ∂Y

!

�1=2 Vθ

∂ 1 PrRa ∂Y λ

!1=2

!

0

Uθ ¼ 0

Z 0

0

0

X ¼ 0; H @0; Y A ¼ H @0; 0A þ

Z Y� ðPrRaÞ1=2 Uθ 0

Z dY ¼ 0

Y

� ∂θ � λ �� dY ∂X X¼0

∂θ �� ∂X �X¼0

Y

� ðPrRaÞ1=2 Uθ

0

1

Z

X

� ðPrRaÞ1=2 Vθ

0 1

dX ¼ 0

��

λ

∂θ �� ∂X �X¼1

�

(35-c)

��

∂θ �� ∂Y �Y¼1

(35-d)

∂θ � λ �� dY ∂X X¼0

3. Numerical procedure and code validation 3.1. Numerical procedure The discretization of governing equations presents similarity to that of the previous study concerning heat transfer from porous/fluid coupled domains [60]. Former discretization depends on the Control Volume Method (CVM) and these governing equations with corre­ sponding boundary conditions could be numerically solved by SIMPLE algorithm [64]. The QUICK scheme and second-order central difference scheme are, respectively, implemented for the convection and diffusion terms to improve numerical accuracy and stability [61–63]. This itera­ tive procedure is conducted by line-by-line TDMA (Tri-Diagonal Matrix Algorithm) and SOR (Successive Over-Relaxation). In order to check whether the steady state solution is obtained or not, the maximum re­ siduals of variables at each nodal point between the consecutive itera­ tion steps are satisfied the following expression [50],

��

λ

1

Z

(34)

1

Z

Y ¼ 1; H @X; 1A ¼ H @0; 1A þ

Due to no-slip boundary condition, it is easy to obtain that boundary conditions of stream function are all zeros by integrating the stream function definition equation along the enclosure boundaries. For square enclosures illustrated in Fig. 1, boundary conditions of heat function are expressed in the following format, �� Z X� ∂θ �� Y ¼ 0; HðX; 0Þ ¼ Hð0; 0Þ þ dX ¼ 0 (35-a) ðPrRaÞ1=2 Vθ ∂Y �Y¼0 0 1

1

� ∂θ � λ �� dY ∂X X¼1

Y

dY ¼

where λ represents 1 in pure fluid or λeff within those saturated porous fins, respectively. Harmonic mean values obtained from the thermal conductivities of pure fluid and saturated porous fins are implemented at the interfaces between the pure fluid and porous fins. The Ψ and the H fields are determined by their first-order derivatives, being thus important only difference but not their levels. Thus it is free to choose the reference point and its value.

0

0

0

(33)

Ψ ð0; 0Þ ¼ Hð0; 0Þ ¼ 0

1

X ¼ 1; H @1; Y A ¼ H @1; 0A þ

(35-b)

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Fig. 10. Effects of the designs of porous fins on the average Nusselt number at Da ¼ 10

� P�� mþ1 � ϕmi;j � �ϕi;j P�� mþ1 �� � δ ϕi;j

0

Z 1 �pffiffiffiffiffiffiffiffiffiffi þ RaPrVθ 0

��

λ

∂θ �� dY ∂X �X¼1 �� ∂θ �� ∂Y �

Y¼1

dX

Z 1 �pffiffiffiffiffiffiffiffiffiffi RaPrUθ

Grid independence tests were presented in Table 1. Two typical de­ signs of porous fins were used to examine the grid independence at Ra ¼ 107, λs ¼ 103, ε ¼ 0.9 and Pr ¼ 0.71 under different magnitudes of Darcy number, i.e., case Nsf and the case with Lfin ¼ 0.6 and N ¼ 10. It was shown that these tests were satisfied within 1.4% in terms of average Nusselt number (Nu) for a grid size of 152 � 152, indicating that a (152 � 152)(X,Y) grid system ensured the grid independence of the solutions in terms of investigating the heat transfer and fluid flow. A refined grid system was performed in the vicinity of boundaries and the interfaces between pure fluid and porous fins to achieve high accuracy. In order to validate the present numerical procedure codes, four stages had been considered during the comparison of the present results and those of experimental and numerical studies, i.e., initially natural convection heat transfer in square enclosures filled with air, then natural convection heat transfer in a square porous cavity by using the DarcyForchheimer model, afterwards natural convection heat transfer in

0

Z 1 �pffiffiffiffiffiffiffiffiffiffi RaPrVθ 0

��

λ

∂θ �� dY ∂X �X¼0 ��

(No Fin: A reference case).

3.2. Code validation

(36)

where δ ¼ 10 6 is set here, ϕ stands for general variables and m repre­ sents the iteration step. For steady state natural convection, total energy balance is an additional accuracy control within this system as the following equation. Z 1 �pffiffiffiffiffiffiffiffiffiffi RaPrUθ

6

(37)

∂θ �� dX ¼ 0 ∂Y �Y¼0

In this equation, λ represents unity within the pure fluid region and λeff within the saturated porous region respectively. Energy balance in Eq. (37) is satisfied within 10 4. 13

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International Journal of Thermal Sciences 147 (2020) 106138

Fig. 11. Effects of the designs of porous fins on the average Nusselt number at Da ¼ 102 (No Fin: A reference case).

λeff ¼ 103 and Wfin ¼ 0.04. Figs. 3 and 4 presented the great agreement in terms of streamlines, isotherms and average Nusselt number for the case of natural convection in a tall enclosure of an aspect ratio 10 with solid fins at Pr ¼ 0.71, λeff ¼ 30, N ¼ 10 and Lfin ¼ 0.5. A final benchmark was conducted verse the results of Khanafer et al. [50] for a differentially heated enclosure with a porous fin (Lfin ¼ 0.5). Table 5 showed an excellent agreement. The maximum relative deviation was within 2.1%. Hence, it was confident to apply our self-developed numerical proced­ ures for the present research.

enclosures of different aspect ratios with solid fins and finally natural convection heat transfer in a square enclosure with a porous fin mounted at the heated wall. As shown in Table 2, the relative variations between the benchmark results and present solutions were within 1.6%. These comparisons exhibited good agreement in terms of average Nusselt number. Particularly, Table 3 presented the comparisons of average Nusslet number between the experimental results of Leong et al. [69] and those of the present study. As Ra increased, deviations of average Nusselt number between the present result and those of Leong et al. [69] decreased dramatically due to that effects of the sidewalls of this cubic cavity weakened continuously. The relative variation of average Nusselt number between the present work and that of Leong et al. [69] was 4.47% at Ra ¼ 104 while became 1.60%. at Ra ¼ 107. As illustrated in Table 4, natural convection heat transfer in a square porous cavity with Pr ¼ 1.0 was presented in order to validate non-Darcy flow model. These comparisons showed good agreement in Nu. Excellent agreement was observed as depicted in Fig. 2 for the case of natural convection heat transfer in a square enclosure with a solid fin at Ra ¼ 104, Pr ¼ 0.71,

4. Results and discussion In this section, contours for streamlines, heatlines, isotherms and heat transfer rates are presented and analyzed for various governing parameters, including spatial designs of porous fins, thermal conduc­ tivity of porous medium, Darcy number and thermal Rayleigh number. In the previous investigations [9–28,50–53], total volume fraction of all fins is small. For simplicity, total volume fraction of porous fins is fixed 14

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Fig. 12. Streamlines, isotherms and heatlines plots (from left to right) with different Ra at Da ¼ 102, λs ¼ 103, N ¼ 4 and Lfin ¼ 0.3.

Fig. 13. Streamlines, isotherms and heatlines plots (from left to right) with different Ra for case Nsf at Da ¼ 102 and λs ¼ 103. 15

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International Journal of Thermal Sciences 147 (2020) 106138

of the northwest corner. Meanwhile multiple core vortexes appear, due to the presence of these fins with excellent heat conduction, which leads to the horizontal temperature difference in the core region. Note that the lower vortex near the bottom initially expands and shifts towards the heated wall until N ¼ 2, then disappears if an increase in N was considered while the upper one weakens. Isotherms shrink downwards with an increase of the quantity of the porous fin, indicating that these fins have better performance in terms of heating air. As N increases, isotherms near the cooling wall present sharper temperature gradients, which illustrate that heat transfer rate increases. Heatlines disclose that as N increases, direct heat transfer rate from the heating wall to air decreases dramatically, which indicates that the heated fins play more important role in terms of heat transportation. Heatlines from Fig. 6(a) show that the presence of this separated vortex hinders the heat transfer and leads to the longer path followed by the heat. Fig. 8 presents that main circulation intensifies as N increases. The stronger main circulation contributes to the increase of heat transfer rate. As N increases, the lower vortex near the bottom weakens and fades away. For N > 2, these fins have minor effects on the distribution of isotherms near the cooling wall due to their shorter length under identical quantities of the porous fin, compared with isotherms from Fig. 6. This indicates that the heat transfer rate could not be boosted by using more fins. Heatlines show the similar patterns illustrated by those from Fig. 6. Fig. 9(b) presents the maximum absolute value of stream function. Isotherms show that tem­ perature difference within this porous layer maintains tiny, illustrating that this porous layer could achieve the excellent performance of heat­ ing air. As the length of these fins decreases, the total width of all porous fins increases according to Eq. (1). The vertical pitches from two consecutive fins and formed by both top and bottom fins and horizontal walls vanish under the limiting condition (Lfin ¼ 0.12 and Wfin ¼ 1.0). This indicates that streamlines, isotherms and heatlines from Figs. 6 and 8 would be similar to those from Fig. 9(b) as Lfin decreases further. Figs. 10 and 11 demonstrate the effects of various designs of these fins on the predictions of average Nusselt number (Nu) with different magnitudes of the thermal Rayleigh number, the Darcy number and the non-dimensional thermal conductivity of the porous medium. The case ‘No Fin’, representing natural convection heat transfer within the identical enclosure filled with air, is used to be a reference situation to evaluate the thermal performance of the adding fins. Fig. 10 shows that the adding porous medium results in the dramatic increase of flow resistance and effects of designs of the porous fin on the heat transfer are very complicated. For λs ¼ 1, the adding porous medium does not contribute to the heat conduction improvement and causes that the flow resistance increases dramatically, thus leads to the sharp decrease of heat transfer rate. For Ra ¼ 105, an increase in N presents insignificant effects on the heat transfer for N > 3. The longer fins result in the worst performance of the heat transfer for N > 2. For Ra ¼ 106, the longer fins also present the worst performance of heat transfer for N > 4 while case Nsf has the worst performance for N � 3. For Ra ¼ 107, an increase in N leads to the continuous decrease of heat transfer rate. And case Nsf shows the worst performance of heat transfer for N � 6, which is ascribed to the heat conduction within the porous layer. For λs ¼ 103, the adding porous medium leads to the great heat conduction improvement and the considerable increase of flow resis­ tance simultaneously, which causes the multiple patterns concerning the effects of the porous fin on the heat transfer. For case Nsf, the porous layer shows the high temperature level, thence the presence of this porous layer reduces the horizontal pitch between heat source and the cooling wall, which contributes to the better performance of the global heat transfer [71,72]. This could be observed from Fig. 10(d)–(f). For Ra ¼ 105, these longer fins present the worst performance of heat transport for N > 2. Nu from the case with Lfin ¼ 0.6 initially decreases with N, afterwards increases while that from the case with Lfin ¼ 0.3 increases continuously. For Ra ¼ 106, Nu from the case with Lfin ¼ 0.6 represents the best performance of heat transfer N � 2, reporting the improvement 5.37% over the ‘No Fin’ case at N ¼ 2 while shows the

Fig. 14. Effects of Ra on the average Nusselt number from two typical designs of the porous fin at Da ¼ 102, λs ¼ 103 and Lfin ¼ 0.3 (No Fin: A reference case).

at 0.12. Besides, Prandtl number and the porosity of porous medium are 0.71 and 0.9, respectively. 4.1. Effects of morphology and topology of porous finss Figs. 5–9 illustrate the effects of the designs of the porous fin on the flow and energy transportation structures at Ra ¼ 107, Da ¼ 102, λs ¼ 103 and 10 6. As shown in Figs. 5, 7 and 9(a), main circulation maintains weak and these fins lead to dramatic increase of flow resis­ tance. Heatlines show that heat conduction is in control of heat transport in the region dominated by these saturated fins. Due to tiny Darcy number, these fins hinder the fluid flow, acting like solid ones. The thermal boundary layer along the finned wall is initially interrupted by these fins, then enforced to deviate from this wall and thus intrusion fronts under porous fins appear. Finally, these intrusions under porous fins reattach to the downstream between two consecutive fins after they pass these fins, which result in the multi-C shaped flow structure. However, this pattern will fade away with an increase of the quantity of the porous fin. This is due to the shorter vertical pitches according to Eqs. (2) and (3). Figs. 5 and 7 show that main circulation initially strengthens with an increase in N, then weakens. Particularly, in this rectangular region (lfin � L), dominated by these fins, heat conduction turns to be in control of energy transport as the quantity of the porous fin increases. This phenomenon could be observed from isotherms and heatlines presented by Figs. 5 and 7, which also indicates that multiple fins with great heat conduction and poor permeability could lead to the deterioration of heat transfer. As shown in Fig. 9(a), isotherms present that the vertical stratification in the clear region could be observed. Heatlines indicate that the heat transfer within the porous layer is dominated by heat conduction. As Darcy number equals 102, main circulation from Figs. 6, 8 and 9 (b) maintains strong due to quite high Darcy (Da ¼ 102) and Rayleigh (Ra ¼ 107) numbers, where these porous fins have minor influences on the increase of flow resistance. Heat transfer rates increase dramatically due to the presence of these fins with great heat conduction (λs ¼ 103) proposed by Fig. 11(f). Fig. 6 shows that main circulation initially weakens with an increase of quantity of the porous fin, then strengthens. Fig. 6(a) shows that there coexist two vortexes close to the horizontal walls, induced by the inertia forces. As N increases, there exists an anticlockwise vortex driven by main circulation, owing to the baffling effect 16

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International Journal of Thermal Sciences 147 (2020) 106138

Fig. 15. Streamlines, isotherms and heatlines plots (from left to right) with different magnitudes of λs at Ra ¼ 107, Da ¼ 102, N ¼ 4 and Lfin ¼ 0.6.

worst performance for N > 3. The predictions of average Nusselt number from two typical cases classified by the length of these fins firstly reach their bottom values with an increase in N, then increase. For Ra ¼ 107, the case with the longer fins has the best performance of the heat transfer enhancement for N < 4 while presents the worst performance of the heat transfer for N > 6. Nu from the above case reports improvement 6.47% over the ‘No Fin’ case at N ¼ 3. Nu from the case with Lfin ¼ 0.3 initially increases to its peak value, showing improvement 2.4% over the ‘No Fin’ case, thereafter decreases. These curves, describing the corre­ lations between the average Nusselt number and the quantity of the porous fin, present multiple patterns, which could be related to the interruption, redevelopment and attachment of the thermal boundary layer along the finned wall as shown in Figs. 5 and 7. As observed from Fig. 11, heat transfer rates from two typical cases classified by the length of porous fins (Lfin ¼ 0.3 and 0.6 respectively) present the almost similar pattern regardless of various magnitudes of the thermal Rayleigh number. As N exceeds 4, an increase in N shows the

insignificant effect on the heat transfer rate. For λs ¼ 1, the adding porous medium leads to the deterioration of the heat transfer, caused by the increase of flow resistance. Note that the longer fins have the least effects on the heat transfer deterioration. Case Nsf (considered as the case with the shortest fin) has the worst performance of heat transfer. For λs ¼ 103, the adding porous medium contributes to the consid­ erable improvement of heat conduction, which leads to the remarkable increase of heat transfer rate presented by Fig. 11(d)–(f). The perfor­ mance of these fins with different lengths concerning the heat transfer enhancement heavily depends on the thermal Rayleigh number. For low Rayleigh number (Ra ¼ 105), case Nsf has the least performance of heat transfer enhancement while the deviations of heat transfer rate from two typical cases classified by the fin length are insignificant for N > 3. For intermediate Rayleigh number (Ra ¼ 106), the case with Lfin ¼ 0.3 shows the best performance of heat transfer enhancement for N > 1 while that with Lfin ¼ 0.6 presents the worst performance concerning the heat transfer enhancement for N � 3 and case Nsf has the similar trend for 17

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International Journal of Thermal Sciences 147 (2020) 106138

Fig. 16. Streamlines, isotherms and heatlines plots (from left to right) with different magnitudes of λs for case Nsf at Ra ¼ 107 and Da ¼ 102.

N > 4. At Lfin ¼ 0.3 and N ¼ 10, the prediction of average Nusselt num­ ber reports improvement 66.3% over the ‘No fin’ case, which is used as a reference case. For high Rayleigh number (Ra ¼ 107), case Nsf presents the best performance of the heat transfer enhancement while the case with Lfin ¼ 0.6 has the least performance. At Ra ¼ 107 and Da ¼ 102, Nu from case Nsf presents improvement 60.03% over the ‘No Fin’ case, which is used as a reference case. In a summary, for large thermal conductivity of the porous medium and quite high values of both Rayleigh and Darcy numbers, the porous layer (case Nsf) could be the optimal design to maximize the heat transfer. For both low Darcy number and high Rayleigh number, the longer fins present the better performance. For both high Darcy number and intermediate Rayleigh number, there exists an optimal length of the porous fin to maximize the heat transfer.

fins could have the best performance of heat transfer if both the mod­ erate Rayleigh number and the high Darcy number were considered simultaneously. Besides, case Nsf presents the most effective perfor­ mance of heat transfer enhancement for both high Rayleigh and Darcy numbers. Therefore Da ¼ 102, λs ¼ 103, N ¼ 4 and Lfin ¼ 0.3 are main­ tained to investigate the effects of the Rayleigh number on the flow and heat transportation structures for two typical designs of the porous fin as shown in Figs. 12 and 8(c), 13 and 9 (b). As Ra increases, it could be seen from Figs. 12 and 8(c) and 13 and 9(b) that there will coexist multiple vortexes near horizontal walls. Figs. 12 and 8(c) and illustrate that isotherms shrink downwards, indicating these porous fins have better performance in terms of heating air. This is also disclosed by isotherms with sharper temperature gradients along the cooling wall, owing to heat conduction improvement. As shown in Figs. 13 and 9(b) and , isotherms within the porous layer present tiny variations of temperature due to remarkable improvement of heat conduction caused by the porous medium. As Ra increases,

4.1.1. Effect of Rayleigh number According to the analysis from the above subsection, these shorter 18

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International Journal of Thermal Sciences 147 (2020) 106138

Fig. 17. Effects of λs on the average Nusselt number from two typical designs of the porous fin for different magnitudes of Da at Ra ¼ 107 (No Fin: A reference case).

isotherms along the cooling wall show sharper temperature gradients, which indicates that the heat transfer rate increases. In addition, iso­ therms shrink towards the bottom, which illustrates that the porous medium presents better performance in terms of heating air. Main cir­ culation from the case with Lfin ¼ 0.3 and N ¼ 4 is stronger than that from case Nsf for Ra ¼ 105 and 106 respectively while this pattern is opposite for Ra ¼ 107. This phenomenon agrees with results presented by Fig. 11(d)–(f). Heatlines from the above contours illustrate that convective heat transfer is in control of heat transportation across the enclosure. Fig. 14 illustrates the effects of the thermal Rayleigh number on average Nusselt number with two typical cases of the porous fin at Da ¼ 102 and λs ¼ 103. One could confirm that Nu increases positively with Ra. The additional porous fins could enhance heat transfer rates dramatically. For Ra > 106, the deviations of the average Nusselt numbers between case Nsf and the case with Lfin ¼ 0.3 and N > 4 are insignificant. Case Nsf presents the least performance of heat transfer enhancement for Ra < 4 � 105. As N increases, the corresponding design of these fins achieves a better performance of the effective heat transfer. Particularly, the prediction of the average Nusselt number from the case with N ¼ 8 and Lfin ¼ 0.3 reports improvement 71.69% over the ‘No Fin’ case at Ra ¼ 105 while that from case Nsf presents improvement 60.03% at Ra ¼ 107.

4.1.2. Effect of thermal conductivity of the porous medium As shown in Figs. 15 and 6(c), 16 and 9(b), contours of streamlines, isotherms and heatlines are depicted to illustrate the effect of the ther­ mal conductivity of the porous medium on the thermal and flow struc­ tures for two typical cases of the porous fin at Ra ¼ 107 and Da ¼ 102. Figs. 15 and 6(c) show that main circulation strengthens with an in­ crease of λs. And there two anti-clockwise vortexes near the horizontal walls coexist. As λs increases, the upper vortex shifts towards the cooling wall while the lower one disappears at λs ¼ 103. This is due to the var­ iations of temperature difference around the tips of these fins with increasing λs. The thermal boundary layer along the finned wall becomes thicker while that along the cooling wall becomes thinner. This could be easily observed from isotherms and ascribed to the great heat conduc­ tion improvement. Heatlines indicate that the presence of the upper vortex near the ceiling leads to the longer path followed by heat and hinders the heat transfer. This vortex should be avoided in the design of thermal system. As shown in Figs. 16 and 9(b). and main circulation presents the similar pattern of that from Figs. 15 and 6(c). For λs > 1, there coexist two vortexes close to both top and bottom walls. As λs increases, the lower vortex will be strengthened and move towards the heated wall while the upper one will weaken and shift towards the cooling wall. Isotherms within the porous layer show the lower temperature gradients 19

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International Journal of Thermal Sciences 147 (2020) 106138

Fig. 18. Streamlines, isotherms and heatlines plots (from left to right) with different magnitudes of Da at Ra ¼ 107, λs ¼ 103, N ¼ 4 and Lfin ¼ 0.6.

while those along the cooling wall present sharper temperature gradi­ ents. Heatlines along the heated wall show that the effective width of the path followed by heat increases, which indicates that the porous layer will present better performance of heating air. In a short summary, higher heat transfer rate could be achieved with an increase of the thermal conductivity of the porous medium. Fig. 17 presents the effect of the dimensionless thermal conductivity of the porous medium on the average Nusselt number with different magnitudes of Darcy number, concerning two typical cases of the porous fin at Ra ¼ 107. One could find that Nu increases positively with λs, due to better performance of heat conduction improvement. For case Nsf, it is interesting that rate of increase in Nu initially speeds up with an increase in λs, then drops. However, the rates of increase in predictions of the average Nusselt number from the case with Lfin ¼ 0.6 continuously in­ crease at Da ¼ 10 6 while are initially boosted and finally maintain almost unaffected for Da � 10 4. For the case with Lfin ¼ 0.6, the more quantities of the porous fin, the higher rate of increase in Nu if these fins

showed the positive effects on the heat transfer enhancement. For Da ¼ 10 6, these fins could be treated as solid baffles, which cause that natural convection within this enclosure weakens dramatically. This results in the remarkable deterioration of heat transfer, excluding that the porous medium presents excellent heat conduction improvement around λs ¼ 103. And an increase in N causes that the heat transfer de­ creases further. For Da ¼ 10 4, these fins have the positive effect on the heat transfer enhancement for λs > 25.0. At λs ¼ 103, the prediction of the average Nusselt number from the one with N ¼ 8 and Lfin ¼ 0.6 shows improvement 67.89% over the ‘No Fin’ case, which is used as a reference case. For Da � 10 2, case Nsf shows the best performance of the heat transfer enhancement for λs > 5 and presents that the prediction of the average Nusselt number from it reports improvement 60.03% over the ‘No Fin’ case at Da ¼ 102 and λs ¼ 103. Predictions of Nu from the case with Lfin ¼ 0.6 show that the quantity of the porous fin has insig­ nificant effects on the heat transfer for λs < 10. As shown in Fig. 17, case Nsf has the least performance of the heat transport for small λs. But there 20

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International Journal of Thermal Sciences 147 (2020) 106138

Fig. 19. Streamlines, isotherms and heatlines plots (from left to right) with different magnitudes of Da for case Nsf at Ra ¼ 107 and λs ¼ 103.

exist different domains of λs where case Nsf presents the best perfor­ mance of the heat transport for Da � 10 4 (see Fig. 18).

Isotherms shrink downwards while those along the cooling wall present sharper temperature gradients, indicating that the heat transfer rate increases. However, as Darcy number exceeds 10 2, it has the minor effect on the flow and thermal transportation structures. As Da ¼ 10 6, isotherms and heatlines show that heat conduction is in control of heat transportation within the region affected by these fins. For high Darcy number, heat transfer across this region will be dominated by convective heat transfer. For case Nsf (Figs. 9 and 19), there coexist two vortexes near the horizontal walls at Da ¼ 10 6. As Darcy number increases, both of them initially strengthen, then the lower vortex maintains almost unaffected for Da > 10 2 while the upper one weakens and moves towards the cooling wall before reaching to the unaffected region at Da ¼ 10� . Iso­ therms and heatlines within the porous layer show that the heat trans­ portation is determined by heat conduction at Da ¼ 10 6. Meanwhile, isotherms also present the vertical temperature stratification in the clear region. As an increase in Darcy number is considered, heat

4.1.3. Effects of Darcy number In order to investigate the effect of Darcy number on the flow and heat transportation structures for two typical designs of the porous fin, Ra ¼ 107 and λs ¼ 103 are maintained. As shown in Figs. 5(c) and 6(c) and 18, 9 and 19, main circulation strengthens with an increase in Da due to the continuous decrease of the flow resistance. For the case with Lfin ¼ 0.6 and N ¼ 4 (Figs. 5(c), 6(c) and 18), these fins are initially surrounded by streamlines owing to the tiny Darcy number and the thermal boundary layer presents the characteristics concerning inter­ ruption, redevelopment and reattachment. As Darcy number increases, streamlines will pass through these fins as a result of the weakened flow resistance and the multiple C-shaped flow structure will fade away. There will coexist three core vortexes. Besides, two anti-clockwise vor­ texes close to the northwest and the northeast corners also appear. 21

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International Journal of Thermal Sciences 147 (2020) 106138

Fig. 20. Effects of Da on the average Nusselt number from two typical designs of the porous fin for different magnitudes of λs at Ra ¼ 107 (No Fin: A reference case).

transportation through the porous layer will turn to be controlled by natural convection and isotherms also shrink towards the bottom and present sharper temperature gradients along the cooling wall, indicating that the porous layer has the excellent performance in terms of heating air and heat transfer rate increases. Note that isotherms along the interface that distinguishes the porous layer from the clear region pre­ sent insignificant variations for Da > 10 2. Fig. 20 illustrates the effect of Darcy number on the average Nusselt number from two typical cases of the porous fin for different magnitudes of λs at Ra ¼ 107. It is observed that Nu increases positively with Darcy number. Regardless of various designs of the porous fin, the rate of in­ crease in Nu initially is boosted with an increase in Darcy number, then decreases and finally could be considered to be zero for Da > 10 2. As the adding porous fins lead to the deterioration of heat transfer, an in­ crease in N causes that the heat transfer rate decreases further. However, an increase in N contributes to the better performance of heat transfer enhancement for λs � 102 if porous fins presented the positive effects. For λs ¼ 1.0, these fins contribute to a decrease in Nu. This is ascribed to the increase of the flow resistance and the absence of heat conduction improvement. Case Nsf has the least performance of the heat transport, which reports deterioration 68.50% over the ‘No Fin’ case at Da ¼ 10 6. In addition, the rates of decrease in Nu from the case with Lfin ¼ 0.6 are

boosted with an increase in N. This is particularly obvious for tiny Darcy number. For λs ¼ 10, as Darcy number increases, these fins present the positive effects on the heat transfer enhancement for Da > 10 3, owing to that the heat conduction improvement from the porous medium turns to be predominant. Case Nsf presents the best performance of the heat transport for Da > 5 � 10 4, where Nu from it reports improvement 16.21% over the ’ No Fin’ case at Da ¼ 102. For λs ¼ 102, heat conduc­ tion improvement from the porous medium strengthens further. One could find that these fins could achieve the heat transfer enhancement for tiny Darcy numbers (Da > 10 5). Case Nsf has the best performance of the effective heat transfer where Nu from it presents improvement 50.80% over the ‘No Fin’ case at Da ¼ 102. For λs ¼ 103, the effective dimensionless thermal conductivity of the saturated porous medium presents a higher value 100.9 based on Eq. (23). This indicates that the saturated porous fins could lead to remarkable heat conduction improvement. The one with N ¼ 8 and Lfin ¼ 0.6 achieves the maximum heat transfer for 10 5< Da <10 3 while case Nsf has the best perfor­ mance of heat transfer for Da > 10 3 and Nu from it reports improve­ ment 60.03% over the ‘No Fin’ case at Da ¼ 102. 4.1.4. Correlation of the average Nusselt number Finally, the correlation based on least-square method for the average 22

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International Journal of Thermal Sciences 147 (2020) 106138

designs and the permeability of the porous material. This correlation could be beneficial for the development of effective cooling methodol­ ogy for future electronics simultaneously of size-minimization and en­ ergy-maximization. 5. Conclusion In this study, buoyancy-driven laminar natural convection within a differentially heated enclosure with porous fins attached to the hot wall is investigated comprehensively. Results are presented vividly through streamlines, isotherms and heatlines. This investigation illustrates that the morphology and topology of porous fins have significant effects on the heat transfer through modifying the heat conduction improvement and baffling effects induced by these fins. Rates of increase in Nu initially are boosted with an increase in Darcy number, then decreases and finally could be considered to be zero for Da > 10 2. As the porous medium has high permeability and great heat con­ duction, these fins present the positive effects on the heat transfer. And an increase in number of porous fins leads to the heat transfer enhancement further while this effect is opposite for tiny Darcy number and small λs of the porous medium. For both low Darcy number and high Rayleigh number, the longer fins present the best performance of heat transfer enhancement while the shorter ones lead to largest increase of heat transfer rate for high Darcy number and intermediate Rayleigh number. As the porous medium leads to great heat conduction improvement, case Nsf presents the best performance of heat transfer enhancement for both high Darcy and Rayleigh numbers. Particularly, case Nsf shows that average Nusselt number reports improvement 60.03% over the ‘No Fin’ case at Ra ¼ 107, Da ¼ 10 2 and λs ¼ 103. Finally, case Nsf could be considered to have the shortest fin length, which indicates that there exist an optimal length of porous fins to achieve the maximum heat transfer if some conditions were satisfied. It has been presented that the average Nusselt number is an increasing function of Ra, Da and λs. The effects of quantity and length of porous fins are heavily affected by Darcy number. Meanwhile, average Nusslet number correlation, as a function of Rayleigh number, Darcy number, dimensionless thermal conductivity of porous medium, quan­ tity and length of porous fins, has been provided, which could be used to obtain the heat transfer coefficient in the electronic component. This research could benefit the electronic cooling process and thermal design.

Fig. 21. Average Nusselt number correlation of discrete heat transfer rates versus Eq. (38).

Nusselt number of natural convection heat transfer in a square enclosure with porous fins was developed for the employed ranges of various governing parameters. And this correlation is given as the following expressions, � 0:4783 þ 0:2793Da 0:1688 Nu ¼ 0:9212Ra0:2562 ð1:7589 þ 3:4011 sinðlogλs Þ þ 0:2036 cosðlogλs ÞÞ � (38) 0:3341 þ 0:9722ðlog DaÞN 0:0003ðlog DaÞN 2 � . . � 0:3796 0:0849 Lfin þ 0:0554 L2fin ; R2 ¼ 97:20% Domains of governing parameters from the above correlation are summarized as. Rayleigh number: 105 � Ra � 107 Darcy number: 10 6 � Da � 102 Dimensionless thermal conductivity of porous fins: 1 � λs � 103 Quantities of porous fins:1 � N � 10. Length of these fins: 0.12 � Lfin � 0.6. The above correlation together with the numerical data is illustrated in Fig. 21. Discrete numerical simulation results very fit with the correlated solutions. This correlation demonstrates that the most powerful parameter in affecting the energy transfer across the enclosure is Rayleigh number, followed by the dimensionless thermal conductivity of porous materials, then Darcy number, thereafter quantities of fins and finally length of the porous fin. In the domain of Rayleigh number used in this work, natural convection is basically in control of heat transport across the enclosure. Thus, the Rayleigh number presents the strongest effects on the heat transfer. And the improvement of heat conduction is determined by the magnitudes of thermal conductivity of porous ma­ terials. As the thermal conductivity of the porous materials increases, the thermal resistance between the heat source and sink decreases dramatically. Furthermore, the baffling effect of porous fins depends on Darcy number. For tiny Darcy number, these fins act like solid ones, while they present insignificant effects on the heat transfer for Da > 10 2. And effect of the number of porous fins is affected by the Darcy number. Finally, due to the fixed volume fraction of porous fins, these fins will act like the porous layer as the length of them continuously decreases. This established correlation could be used in thermal design applications to manipulate the heat transfer characteristics, for instance, within an enclosure by proper combination of Rayleigh number, fin

Declaration of competing interest All authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgements This research was financially supported by the National Key Research and Development Program of the Ministry of Science and Technology of China (Grant No. 2018YFC0705201, Grant No. 2018YFB0904200), Natural Science Foundation of China (NSFC Grant No. 51778504, Grant No. U1867221), Beijing Institute of Satellite Environmental Engineering (CAST-BISEE Grant No. CAST-BISEE2019025),Joint Zhuzhou - Hunan Provincial Natural Science Foundation (Grant No. 2018JJ4064), National Defense Research Funds for the Central Universities (Grant No. 2042018gf0031, Wuhan University), and Shandong Provincial Natural Science Foundation (Grant No. ZR2018MEE035). Both Prof. Fu-Yun Zhao and Prof. Han-Qing Wang would also like to acknowledge the support from the Collaborative Innovation Center for Building Energy Conservation and Environment Control, Hunan Prov­ ince, China.

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Appendix A. Supplementary data

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