Influence of permeable circular body and CuO−H2O nanofluid on buoyancy-driven flow and entropy generation

International Journal of Mechanical Sciences 166 (2020) 105240

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Influence of permeable circular body and 𝐶 𝑢𝑂 − 𝐻2 𝑂 nanofluid on buoyancy-driven flow and entropy generation T.R. Vijaybabu∗ Department of Mechanical Engineering, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Bengaluru, India

a r t i c l e

i n f o

Keywords: Natural convection Nanofluid Circular cylinder LBM Porous media Entropy

a b s t r a c t A numerical analysis has been performed to demonstrate the impression of an isothermal circular permeable body on free-convection and thermo-dynamic irreversibilities in a square cavity by employing lattice Boltzmann technique. CuO-water nanofluid is considered as a working fluid, and the dynamic values of effective properties are evaluated by Koo-Kleinstreuer-Li (KKL) model. Primary aim of the present work is to evaluate the effects of Darcy number (Da), Rayleigh number (Ra), and nanofluid volume fraction (Φ) on flow and heat transfer characteristics in and around the permeable zone. Also, variation of entropy generation and mean Bejan number (BeM ), thermal-mixing and its uniformity are analysed. The ranges of Da, and Ra considered in this analysis are, 10−6 ≤ Da ≤ 10−2 , and 103 ≤ Ra ≤ 106 , respectively, for the nanofluid volume fraction values of 0%-4%. It has been observed that the momentum of the fluid in the enclosure intensifies while increasing Da and nanofluid volume fraction. Also, the flow variation produced by different values of nanofluid volume fraction significantly modifies the thermal traits. Further, the location of maximum entropy generation depends on the permeability of the cylinder. It is found that the volume fraction of nanofluid governs the dominance of thermal and/or fluid-friction irreversibility. Permeability enrichment increases the thermal-mixing and reduces the temperature uniformity.

1. Introduction Owing to the overwhelming pragmatic importance of buoyancy driven hydrothermal dynamics in several industrial and practical applications, the natural convection finds its spot in the recent research. Nuclear reactor cooling, solar panel, building insulation, heat dissipation from electronic equipment, and heat exchanger are the few typical realtime examples in which natural convection mechanism occurs. In most of the thermal devices, the primary objective is to obtain rich heat dissipation characteristics. From literature, it is a common understanding that employing higher thermal conductivity fluid (i.e. nanofluid), and adopting higher surface area (i.e. porous medium) elevate heat transfer performances. Besides, consideration of grouped elements as a single porous structure fades the complexity of analysis and reduces the numerical expenses. Such instances can be found in nuclear reactor (group of fuel rods), arrangement of tubes in heat exchanger, pin-fin apparatus placed on electronic chips, and food-stuffs. In these applications, by providing exact porosity, and permeability values equivalent to the grouped bodies facilitate to consider these as a porous body. From this point, performing design experiment to achieve required thermal traits is easy, and hence, understanding the natural convection behaviour of porous bodies is crucial. In most of the aforementioned applications,

∗

different kinds of nanofluid are used for the purpose of elevating heat transfer performances. Thus, it is essential to understand the underlying physics on the amalgamation effect of nanofluid and porous body. The core idea of this study is to analyse the hydrothermal characteristics of circular porous body placed in a confined boundary with nanofluid. It is believed that understanding the impact of various influencing parameters on natural convection from solid circular body would aid the discussion of results obtained from the current study. Moukalled and Acharya [1] have analysed the effect of Rayleigh number (Ra) on natural convection characteristics of solid circular cylinder positioned in a square enclosure for different values of aspect ratio. They have mentioned that the dominance of convection at higher values of Ra promotes thermal stratification in the lower half of the enclosure. Cesini et al. [2] have conducted experimental and numerical analysis to understand the free-convection traits of isothermally cooled circular cylinder placed in a rectangular square cavity. The numerical results were in good agreement with experimental values in terms of mean Nusselt number. They have also indicated the increasing heat transfer coefficient trend with Ra enhancement. Further, the numerical study performed by Shu and Zhu [3] has suggested that a critical aspect ratio may exist at higher Ra values to distinguish flow and thermal behaviour. Literature [4–7] indicate that apart from aspect ratio (i.e. height of circular cylin-

Corresponding author. E-mail address: [email protected]

https://doi.org/10.1016/j.ijmecsci.2019.105240 Received 28 August 2019; Received in revised form 4 October 2019; Accepted 13 October 2019 Available online 14 October 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.

T.R. Vijaybabu

International Journal of Mechanical Sciences 166 (2020) 105240

Nomenclature Notations cF cs Cp Da dp dpp ei Fb fi 𝑓𝑖𝑒𝑞 F Fb Fi g 𝑔𝑖𝑒𝑞 gi G L k K kb Ma N Nu Pr Ra Rf Rk u v U V W wi x, y

Non-dimensional Forchheimer term Speed of sound, [𝑚𝑠−1 ] Specific heat at constant pressure, [J/kg K] Darcy number, K/W2 Particle diameter of CuO nanoparticle, [m] Particle diameter of porous medium, [m]

𝑠

Number of lattices on the enclosure Local Nusselt number, 𝜕𝜃 𝜕𝑛 Prandtl number, 𝛼𝜈

𝐿 Rayleigh number, 𝑔𝛽Δ𝑇 𝛼𝜈 Thermal interfacial resistance, [km2 /W] Thermal conductivity of porous and nanofluid layers, [ke /knf ] Non-dimensional x component velocity, [𝑚𝑠−1 ] Non-dimensional y component velocity, [𝑚𝑠−1 ] Actual velocity of porous medium, [𝑚𝑠−1 ] Auxiliary velocity, [𝑚𝑠−1 ] Diameter of the circular body Weighing factor in direction i Horizontal & vertical coordinates 3

𝑛𝑓

𝜖 𝛽 Φ 𝛼 𝜎 ϱ Ξ Γ

Mean value Lattice link direction Wall Nanofluid Solid particle of nanofluid Effective Cold Hot

Δ𝑥

Discrete lattice velocity in direction i, Δ𝑡𝑖 Boussinesq force term, [N] Particle density distribution function in direction i Equilibrium distribution function of density in direction i Body force due to the presence of the porous medium, [N] Boussinesq force term, [N] Total force term due to porous medium, [N] Gravitational acceleration, [𝑚𝑠−2 ] Equilibrium distribution function of temperature in direction i Temperature distribution function in direction i Body force due to gravity, [N] Enclosure height Thermal conductivity, [W/m K] Permeability of the material, [m2 ] Boltzmann constant, [m2 kg 𝑠−2 𝐾 −1 ] Mach number, 𝑐𝑢

Greek symbols 𝜌 Fluid density, [kg 𝑚−3 ] 𝜏 Dimensionless relaxation time for density 𝜏′ Dimensionless relaxation time for temperature 𝜏 Non-dimensional time 𝜇 Dynamic viscosity of fluid [N s 𝑚−2 ] 𝜇 Λ Viscosity ratio, 𝜇 𝑒 Δx Δt 𝜈 𝜃

Subscripts M i w nf s eff c h

Lattice space Time step, [s] Fluid kinematic viscosity, [m2 𝑠−1 ] 𝑇 −𝑇 Dimensionless temperature, 𝑇 −𝑇𝑐 ℎ

𝑐

Porosity Thermal expansion coefficient, [𝑜 𝐶 −1 ] Nanoparticle volume fraction, [in %] Thermal diffusivity, [m2 𝑠−1 ] Electrical conductivity, [S/m] Density ratio Thermal expansion ratio Electrical conductivity ratio

der to enclosure height), the position of cylinder plays crucial role on buoyancy driven hydrothermal characteristics. Results indicate that at lower convection rates (Ra = 103 & 104 ), while placing the hot circular cylinder near top wall, the rate of heat transfer can be elevated [5]. Further, the placement of hot cylinder causes the unsteady hydrothermal behaviour [6], which substantially enhances the mean Nusselt number of cylinder. While placing two hot cylinders in the enclosure, rich heat dissipation features have been reported by enlarging the gap between the cylinders [7]. FEM study performed by Pinto et al. [8] shows that the shape of circular cylinder geometry eases the recirculation in comparison to square cylinder, and hence, circular cylinder offers higher heat transfer characteristics. This reason induces the engineers to consider the circular configuration while designing thermal systems. Park et al. [9] have examined the impact of inclined square enclosure with a heated circular cylinder on thermal performance. They have indicated that the inclination angle of enclosure on mean Nusselt number is meagre. Cho et al. [10] have shown that introducing an elliptical cylinder with circular body boosts heat transfer performance. Buoyancy characteristics between triangular enclosure and hot circular body have been elucidated by Xu et al. [11]. It has been conveyed that the influence of enclosure angle and geometry of inner body is less on overall heat transfer rates in spite of notable flow pattern variation. It is a well-established theory that the thermal performance can be elevated by employing nanofluid in the system. Nanofluid is a dispersion of nanoparticles with higher thermal conductivity in a working fluid. Khanafer et al. [12] have shown the heat transfer augmentation while using Cu-water nanofluid in a differentially heated cavity. Ghasemi and Aminossadati [13] carried out a numerical study to understand the effect of enclosure inclination angle and nanofluid volume fraction over thermal characteristics. They have indicated the variation of temperature trends with the enclosure angle and optimum angle for each Rayleigh number for maximum Nusselt number. Abu-Nada et al. [14] have analysed natural convection heat transfer between horizontal concentric annuli different nanofluids such as Cu, Ag, Al2 O3 , and TiO2 . They have indicated that the heat transfer enhances while utilizing higher thermal conductivity nanofluid at higher values of Rayleigh number. At lower values of Ra, employing lower thermal conductivity nanofluid has produced adverse effect on heat transfer performance. It is to be noted that in all the above studies, single-phase with static model has been used to model the nanofluid numerically. In this case the sway of Brownian motion and temperature of nanoparticles have been ignored. This ideology has been later remonstrated by researches as it is deviating while calculating the effective properties of nanofluid [15]. Hence, different models have been proposed to eliminate the drawbacks, in which KKL model proposed and refined by Koo [16], Koo and Kleinstreuer [17], Li [18] has been widely used by various researchers [19]. Sheikholeslami et al. [20] have analysed the significance of Cu-water nanofluid on free convection between a circular cylinder and square enclosure subjected to uniform magnetic field. In their study, the effective thermal conductivity and viscosity of nanofluid have been evaluated by using KKL model. Increasing trend in overall heat transfer with nanoparticle volume fraction has been reported in their study. Recent study carried out by Mikhailenko et al. [21] indicates that the insertion of nano-sized particles can significantly regulate the energy transport enhancement.

T.R. Vijaybabu

Apart from changing the circulating fluid, heat dissipation traits can be augmented by using porous medium in thermal systems. For instance, numerical work performed by Malik and Nayak [22] indicates that the porous medium with lower permeability promotes the heat dissipation characteristics of a differentially heated cavity with discrete heat source. Numerical analysis on convective heat transfer of nanofluid over a porous wavy surface carried out by [23] have shown the combined influence of porous medium and nanofluid on heat transfer enhancement. In flat-plate solar heat collector and/or empty channels, placing porous blocks serves as a thermal storage device, as a result, performance of this thermal system improves [24,25]. Further, studies of natural convection in open cavity [26–28], and mixed convection in enclosure [29] with different properties of porous blocks showcase the enhancement in heat transfer characteristics. Besides, the forced and mixed convection heat transfer behaviour from isothermally heated blocks of various shapes has also been carried out by researchers [30–35]. It has been shown that the under forced convection condition the mean Nusselt number is directly proportional to the Darcy number (i.e. non-dimensional permeability), and at leeward surface of porous body the thick thermal boundary layer offers adverse effect on heat transfer [31,34]. Intensity of such hostile nature slices while the gravitational force is accountable [33,34]. Recently, Vijaybabu and Dhinakaran [36] have carried out numerical investigation on buoyancy driven flow and heat transfer features of a permeable triangular-shaped cylinder positioned in an enclosure filled with 𝐴𝑙2 𝑂3 − 𝐻2 𝑂 nanofluid. From this study, it is evident that the momentum of nanofluid increases while enriching the permeability. However, apart from this study no other studies have been performed to probe the natural convection behaviour of porous body. In any thermal system, to utilize the energy in efficient manner or to attain maximum possible thermo-dynamic efficiency it is crucial to evaluate the associated irreversibilities. This analysis serves engineers to look into the possibilities for the reduction of irreversibilities. Jery et al. [37] have analysed the variation of entropy generation with inclination angle of differentially heated cavity. It has been elucidated that while inclining cavity more than 30° , the propensity of total entropy reduces. Also, while enhancing Rayleigh number, the total entropy generation tends to increase owing to the domination of fluid-friction and thermal irreversibilities [38]. Entropy generation analysis performed by [39] indicates that the Prandtl number (Pr) increment energises the irreversibilities in the system due to the domination of fluid viscosity. Mun et al. [40] and Doo et al. [41] have assessed the thermo-dynamic irreversibility produced by natural convection in the enclosure with inner circular cylinder due to enclosure angle, and vertical position of inner cylinder. Mun et al. [40] have reported that the enclosure angle variation has meagre effect on the degree of irreversibility, however, the location of maximum or minimum entropy varies with enclosure angle. In addition, the entropy generation value slightly increases while shifting the inner cylinder near top wall of enclosure Shahi et al.[42] have investigated entropy generation variation with Cu-water nanofluid for the natural convection in an enclosure with a local heat source for different cases. It has been concluded that the irreversibilities tend to decline while enriching nanofluid volume fraction due to the proper mixing. On the other hand, while placing solid obstacles inside an enclosure filled with nanofluid, the entropy generation increases with nanofluid volume fraction [43]. Further, the sway of irreversibilities due to the inclination of differentially heated porous cavity has been presented by Baytaş [44]. Author has claimed that the large values of Darcy-modified Rayleigh number offer higher degree of irreversibilities, and an optimum condition for minimum entropy generation has also been proposed. Further, Bondarenko et al. [45] have also shown that the placement of heat source and moving wall significantly vary the associated irreversibilities. Hence, it is clear that performing entropy generation is necessary to obtain the optimum design of thermal system for higher efficiency. Sheikholeslami et al. [46] have assessed the entropy generation in the porous enclosure with heated solid cylinder filled with Ferro fluid. It has been demonstrated that the permeability enrichment increases total

International Journal of Mechanical Sciences 166 (2020) 105240

entropy generation without altering entropy generation due to temperature. Further, in case of channel filled with porous medium, reducing permeability promotes pressure gradient and fluid friction irreversibility [47]. Besides, the arrival of lattice Boltzmann method (LBM) has made a tremendous impact over the past three decades on complex numerical investigations. The simple kinetic approach of this technique offers the researchers to solve any kind of problem by just coupling the corresponding governing term with distribution functions. Studies [33– 36] highlight the simplicity of LBM on its implementation and various vivid advantages. Especially, LBM has proven its expertness while solving fluid and energy transport through porous medium. By considering the aforementioned benefits, this method has been implemented to solve the current problem. Most of the studies have concentrated either the case of enclosure filled with porous medium or adiabatic porous layer. An assessment of existing literature indicates that the natural convection traits of heated porous body have not been addressed to the research community in detail for different configurations. Also, in many real-time applications such as compact electronic devices coupled with porous medium, porous heat sinks and other instances of grouped elements reflects porous body where the heat transfer is mainly due to natural convection. Thus, it is important to realise the flow and heat transfer traits of porous body under free-convection condition. These factors have driven the author to perform the current investigation. Hence, the primary aim of the current investigation is to analyse the influence of permeability, nanofluid volume fraction and intensity of buoyancy force on free-convection characteristics of porous circular body. 𝐶𝑢𝑂 − 𝐻2 𝑂 nanofluid has been used in this study, and the effective properties have been calculated using KKL model [18], in which the Brownian motion and temperature of nanoparticles are considered. Two-distribution lattice Boltzmann method along with single relaxation has been employed in the present investigation[18], in which the Brownian motion and temperature of nanoparticles are considered. Two-distribution lattice Boltzmann method along with single relaxation has been employed in the present investigation.

2. Mathematical formulation 2.1. Problem description Fig. 1(a) reflects the physical model considered in the present analysis. It consists of a porous circular cylinder of diameter W and a square cavity of height L. The diameter of circular cylinder is equal to 0.4 L. The inner permeable circular cylinder is maintained at a constant high temperature (Th ) and the walls of square cavity are kept at low temperature (Tc ). The gravitational force due to acceleration is meant to act towards downward direction (i.e. negative y-direction) and the enclosure is completely occupied by 𝐶𝑢𝑂 − 𝐻2 𝑂 nanofluid. Few assumptions have been considered to turn the present analysis flexible for numerical simulations, and they are as follows: 1. The nanofluid flows in and around an isotropic, homogeneous porous matrix with constant permeability and porosity. Flow is twodimensional, steady and incompressible. 2. Fluid flow occurs owing to the density variation which is produced by the temperature difference amid hot circular porous body and cold enclosure wall. The effective thermal conductivity and viscosity of the nanofluid depend on the temperature, volume fraction, nanoparticle size and Brownian motion. Apart from the gravitational force which acts in y-direction, no other body forces act on the fluid. 3. At porous zone, local thermal equilibrium (LTE) condition has been considered. It is to be noted that this postulation is not legitimate in case of fast heating or cooling across the porous body. Besides, radiation from the porous body has also been neglected.

T.R. Vijaybabu

International Journal of Mechanical Sciences 166 (2020) 105240

Fig. 1. (a) Computational domain used in the analysis of natural convection of 𝐶𝑢𝑂 − 𝐻2 𝑂 nanofluid between porous circular cylinder and square enclosure; (b) curved boundary on square lattice nodes.

2.2. Conservation equations The non-dimensional macroscopic governing equations for the present problem are as follows: 𝜕𝑢 𝜕𝑣 + = 0. 𝜕𝑥 𝜕𝑦

(1)

( ( ) ) 𝜕𝑝 1 𝜕𝑢 1 𝜕𝑢 𝜕𝑢 1 𝜕2 𝑢 𝜕2 𝑢 + 𝑢 +𝑣 =− + Λ𝑃 𝑟 + 2 2 2 𝜖 𝜕𝜏𝑡 𝜖 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜖 𝜕𝑥 𝜕𝑦 √ 2 2 𝑢 +𝑣 𝑃𝑟 1.75 1 − 𝐶1 𝑢 − 𝐶2 √ × 𝑢. √ 𝐷𝑎 𝜖 3∕2 150 𝐷𝑎

(2)

( ) ) ( 𝜕𝑝 1 𝜕𝑣 1 𝜕𝑣 𝜕𝑣 1 𝜕2 𝑣 𝜕2 𝑣 𝑃𝑟 + 𝑢 +𝑣 =− + Λ𝑃 𝑟 + − 𝐶1 𝑣 2 2 2 𝜖 𝜕𝜏𝑡 𝜖 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜖 𝜕𝑥 𝐷𝑎 𝜕𝑦 √ 𝑢2 + 𝑣2 1.75 1 − 𝐶2 √ × 𝑣 + Ξ𝑃 𝑟𝑅𝑎𝜃. √ 𝜖 3∕2 150 𝐷𝑎 ) ( 2 ) ( 𝑅 1 𝜕𝜃 𝜕𝜃 𝜕 𝜃 𝜕2 𝜃 𝑢 +𝑣 = 𝑘 + . 𝜖 𝜕𝑥 𝜕𝑦 𝑃 𝑟 𝜕𝑥2 𝜕𝑦2

(3)

(4)

Here, C1 and C2 are the integer values which hold 0 and 1 at fluid and porous region, respectively. The non-dimensional variables are specified as: 𝑃 𝑟 = (𝜇𝐶𝑝 )𝑛𝑓 ∕𝑘𝑛𝑓 (Prandtl number), Λ = 𝜇𝑒 ∕𝜇𝑛𝑓 , (viscosity ratio), 𝜚 = 𝜌𝑓 ∕𝜌𝑛𝑓 (density ratio), Ξ = 𝛽𝑛𝑓 ∕𝛽𝑓 (thermal expansion ratio). The viscosity of porous medium to the fluid, and thermal conductivity of porous to the fluid are assumed to be 1 in the present analysis. The nondimensional governing equations (Eqs. 1 to 4) are obtained by using the following relationships: 𝑥=

𝑥∗ , 𝐿

𝑦=

𝑦∗ , 𝐿

𝑝=

𝑝∗ 𝐿2 , 𝜌𝛼 2

𝑢=

𝑢∗ 𝐿 , 𝛼

Note that the variable with superscript form of it.

𝑣= ∗

𝑣∗ 𝐿 , 𝛼

𝜃=

𝑇 − 𝑇𝑐 . 𝑇ℎ − 𝑇𝑐

indicates the dimensional

2.3. Lattice Boltzmann Method (LBM) In this analysis, lattice Boltzmann method with single Relaxation factor (SRT) has been used to capture the buoyancy driven flow and thermal characteristics. This method is otherwise termed as mesoscopic scale simulation in which finite amount of particles in a certain direction is

taken as a group. Such group of particles undergo collision streaming operations followed by boundary condition. The distribution function variation of grouped particles provides the macroscopic information of fluid flow. Obtained macroscopic data (i.e. density and velocity) further fed in to another distribution function through which the heat transfer traits can be obtained. The explicit approach of LBM offers many advantages over conventional techniques, particularly while dealing with fluid flows through porous media, multi-phase flow and complex flows. 2.3.1. LB governing equations for fluid flow In this technique, the grouped particles initially subjected to collision process and then this group moves (i.e. streaming step) to their adjacent lattices according to its direction. In order to obtain the hydrothermal characteristics of porous medium and buoyancy effect, corresponding force or source terms are included with the collision operator. It is to be noted that all the governing equations related to LBM are same as the literature [36]. The collision equation can be expressed as, ] 1[ 𝑓𝑖 (𝑥 + 𝑒𝑖 Δ𝑡, 𝑡 + Δ𝑡) − 𝑓𝑖 (𝑥, 𝑡) = − 𝑓𝑖 (𝑥, 𝑡) − 𝑓𝑖𝑒𝑞 (𝑥, 𝑡) + Δ𝑡𝐹𝑖 + Δ𝑡𝐹𝑏 , (5) 𝜏 where, fi is the particle distribution function of ith link, 𝑓𝑖𝑒𝑞 is the corresponding equilibrium distribution and ei are the velocity direction vectors of particles residing in a lattice. In this study, D2Q9 lattice model (refer Fig. 1(a)) has been used in which eight particles have motion in different direction and one particle has no motion. Velocity of these particles (Δxi /Δt) can be written as ⎧(0, 0), 𝑖=0 ⎪ 𝑖=1−4 𝑒𝑖 = ⎨(𝑐𝑜𝑠[(𝑖 − 1)𝜋∕2], 𝑠𝑖𝑛[(𝑖 − 1)𝜋∕2])𝑒, √ ⎪(𝑐𝑜𝑠[(2𝑖 − 9)𝜋∕4], 𝑠𝑖𝑛[(2𝑖 − 9)𝜋∕4]) 2𝑒, 𝑖 = 5 − 8. ⎩

(6)

The collision operator with singe relaxation time (SRT) can be seen in Eq. (5) (first term on the right side). Once after the collision operation the particles have to be relaxed towards its equilibrium condition. Hence, a single relaxation factor 𝜏 has been used and this value depends on the kinematic viscosity 𝜈. This can be expressed through ChapmanEnskog relationship as ( ) 1 2 𝜈= 𝜏− (7) 𝑐 Δ𝑡. 2 𝑠 In above equation, cs is termed as the speed of sound and in case of D2Q9 √ model, it can be related with particle velocity as 𝑒∕ 3. By coupling a force term Fi which comprises the effects of inertial and viscous forces

T.R. Vijaybabu

International Journal of Mechanical Sciences 166 (2020) 105240

of porous medium with collision operator, fluid flow traits of porous medium have been obtained. The porous force term that includes the porosity (𝜖) is expressed as ( ] )[ 1 9 3 𝐹𝑖 = 𝑤𝑖 𝜌 1 − (8) 3(𝑒𝑖 .𝐹 ) + (𝑒𝑖 .𝑈 )(𝑒𝑖 .𝐹 ) − (𝑈 .𝐹 ) . 2𝜏 𝜖 𝜖 Here, F is otherwise mentioned as Darcy-Forchheimer term, and it is expressed as 𝜖𝑐 𝜖𝜈 𝐹 = − 𝑈 − √ 𝐹 |𝑈 |𝑈 + 𝜖𝐺. (9) 𝐾 𝐾 √ Here, K is the permeability, cF = 1.75∕( 150 × 𝜖 3 ) is the non√ dimensional Forchheimer term [48], |𝑈 | = 𝑢2 + 𝑣2 in which u and v are velocities in x and y directions, respectively, and the body force due to gravity is indicated as G. The equilibrium distribution function (𝑓𝑖𝑒𝑞 ) can be written as ] [ 9 3 𝑓𝑖𝑒𝑞 = 𝑤𝑖 𝜌 1 + 3(𝑒𝑖 .𝑈 ) + (𝑒𝑖 .𝑈 )2 − (𝑈 .𝑈 ) . (10) 2𝜖 2𝜖 In Eq. (10), wi refers the weighing factor, and in case of D2Q9, w0 = 4/9, 𝑤1−4 = 1/9, and 𝑤5−8 = 1/36. Collision process is followed by the streaming operation and implementation of the boundary conditions. Note that up to this point of algorithm, the fluid information has been indicated in terms of distribution function only. However, in the next step of simulation the macroscopic information of fluid properties need to be inserted in the collision equation. Therefore, the macroscopic properties, such as velocity and density values are evaluated from distribution functions. The density and velocity values are obtained from the following equations: 𝜌=

8 ∑ 𝑖=0

𝑓𝑖 .

(11)

In the above equation, the auxiliary velocity V can be evaluated as 𝜌𝑉 =

8 ∑ 𝑖=1

𝑒𝑖 𝑓𝑖 +

Δ𝑡 𝜌𝐹 . 2

(12)

Following equation relates the auxiliary velocity and the actual velocity of porous medium: 𝑈=

𝑉 . √ 𝑐0 + 𝑐02 + 𝑐1 |𝑉 |

(13)

) Δ𝑡 𝜈 Δ𝑡 𝑐𝐹 1 𝑐0 = 1+𝜖 ; 𝑐1 = 𝜖 √ . 2 2 𝐾 2 𝐾

(

(14)

In case of free convection, fluid flow happens owing to the temperature difference, and it strongly relies on the gravitational force. Thus, the Boussinesq force term Fb has been included with the collision equation (in Eq. (5)). It is expressed as 𝐹𝑏 = 3𝑤𝑖 𝑔𝛽 𝜃 𝑒𝑖𝑦 .

(15)

In the above equation, eiy is the velocity values in y-direction. 2.3.2. LB governing equations for temperature To obtain the temperature field, another distribution function, g(x, t) has been introduced. However, this function requires the macroscopic properties of fluid. The governing equation for temperature can be expressed as ) 1( 𝑔𝑖 (𝑥 + 𝑒𝑖 Δ𝑡, 𝑡 + Δ𝑡) − 𝑔𝑖 (𝑥, 𝑡) = − ′ 𝑔𝑖 (𝑥, 𝑡) − 𝑔𝑖𝑒𝑞 (𝑥, 𝑡) . (16) 𝜏 In the above equation, the relaxation time for temperature 𝜏′ has been calculated from thermal diffusion coefficient (𝛼 = (𝜏 ′ − 1∕2)𝑐𝑠2 Δ𝑡). The equilibrium distribution function for temperature, 𝑔𝑖𝑒𝑞 can be written as 𝑔𝑖𝑒𝑞 = 𝑤𝑖 𝜃[1 + 3(𝑒𝑖 .𝑈 )].

𝜃=

8 ∑ 𝑖=0

𝑔𝑖 .

(18)

2.3.3. LB model for nanofluid Owing to the rich thermal conductivity of nanofluid, it offers greater efficiency in energy transport, and it acts differently than pure fluid from microscopic perspective the to the forces on the nanoparticles and inter-particle potentials. In this study, the nanofluid is considered to be a homogeneous fluid represented by the effective properties, such as density (𝜌nf ), thermal expansion ((𝜌𝛽)nf ), heat capacity ((𝜌Cp )nf ) and electrical conductivity ((𝜎)nf ). Such governing properties are defined in the following equations [12]: 𝜌𝑛𝑓 = 𝜌𝑓 (1 − Φ) + 𝜌𝑠 Φ,

(19)

(𝜌𝐶𝑝 )𝑛𝑓 = (𝜌𝐶𝑝 )𝑓 (1 − Φ) + (𝜌𝐶𝑝 )𝑠 Φ,

(20)

(𝜌𝛽)𝑛𝑓 = (𝜌𝛽)𝑓 (1 − Φ) + (𝜌𝛽)𝑠 Φ,

(21)

( 𝜎𝑛𝑓 𝜎𝑓

3 =1+ (

𝜎𝑠 𝜎𝑓

𝜎𝑠 𝜎𝑓

(17)

) −1 Φ

) ( ) . 𝜎 + 2 − 𝜎𝑠 − 1 Φ

(22)

𝑓

In the Eqs. (19)–(22), Φ denotes the volume fraction of the nanoparticles and subscripts s, f and nf indicate the solid, base fluid, and nanofluid respectively. The thermo-physical properties of H2 O and CuO nanoparticles [18] are presented in Table 1. Koo [16] argued that the Brownian motion of nanoparticles influences greatly on the effective thermal conductivity of the nanofluid. Hence, authors have included the Brownian motion part with static characteristics of nanoparticles. Their thermal conductivity model accounts the effects of nanoparticle size, types of particle and base fluid combinations, volume fraction and temperature variation. It can be written as 𝑘𝑛𝑓 = 𝑘static + 𝑘Brownian , (

The parameters c0 and c1 in the above equation can be expressed as

Here, 𝜃 is the non-dimensional temperature, and this value is calculated from the distribution function g(x, t).

𝑘static =1+ ( 𝑘𝑓 𝑘𝑝 𝑘𝑓

3

𝑘𝑝 𝑘𝑓

(23) ) −1 Φ

) ( ) . 𝑘 + 2 − 𝑘𝑝 − 1 Φ

(24)

𝑓

Here, kstatic is the static thermal conductivity based on Maxwells classical correlation. Recently, many researchers have emphasized that the interfacial thermal resistance plays a key role in weakening the effective thermal conductivity. Hence, by introducing a thermal interface resistance Rf = 4 × 10−8 km2 /W, the original nanoparticle thermal conductivity in Eq. (24) is replaced by a the term kp,eff , and it can be written as 𝑑𝑝 𝑑𝑝 𝑅𝑓 + = . (25) 𝑘𝑝 𝑘𝑝,eff Table 1 Thermo-physical properties of H2 O (water) basefluid and CuO nanoparticles at room temperature [18]. Property

Pure water

CuO

𝜌 (kg/m3 ) Cp (j/ kg k) k (W/m k) 𝛽 (1/K) dp (nm) 𝜎 (Ω m)−1

997.1 4179 0.613 21 × 105 0.05

6500 540 18 85 × 103 29 2.7 × 10−8

T.R. Vijaybabu

International Journal of Mechanical Sciences 166 (2020) 105240

Table 2 The coefficient values of 𝐶𝑢𝑂 − 𝐻2 𝑂 nanofluid [18]. Coefficient

𝐶𝑢𝑂 − 𝐻2 𝑂

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10

-26.593310846 -0.403818333 -33.3516805 -1.915825591 6.42185846658 × 10−2 48.40336955 -9.787756683 190.245610009 10.9285386565 -0.72009983664

• •

On the enclosure walls, the temperature boundary conditions are specified as follows: • • • •

Further, constant temperature boundary condition can be specified 𝑔𝑖 + 𝑔̃𝑖 = wall temperature × (𝑤𝑖 + 𝑤̃ 𝑖 ).

𝑔 ′ (𝑇 , Φ, 𝑑𝑝 ) = (𝑎1 + 𝑎2 ln(𝑑𝑝 ) + 𝑎3 ln(Φ) + 𝑎4 ln(Φ) ln(𝑑𝑝 ) + 𝑎5 ln(𝑑𝑝 )2 ) ln(𝑇 ) + (𝑎6 + 𝑎7 ln(𝑑𝑝 ) + 𝑎8 ln(Φ) (26)

In the above equation, the coefficients (a1 to a10 ) are based on the type of nanoparticles, and these values for CuO are given in Table 2. The KKL (Koo-Kleinstreuer-Li) correlation for Brownian motion thermal conductivity can be written as √ 𝑘𝑏 𝑇 ′ 4 𝑘Brownian = 5 × 10 Φ𝜌𝑓 𝑐𝑝,𝑓 𝑔 (𝑇 , Φ, 𝑑𝑝 ). (27) 𝜌𝑝 𝑑 𝑝 Here, kb (=1.38064852 × 10−23 m2 kg 𝑠−2 𝐾 −1 ) is the Boltzmann constant. Koo [16] further investigated on effective viscosity and they have proposed the following model: 𝜇eff = 𝜇static + 𝜇Brownian = 𝜇static +

𝜇𝑓 𝑘Brownian × , 𝑘𝑓 𝑃 𝑟𝑓

𝜇

2.4. LB boundary conditions The following boundary conditions have been specified to perform the current analysis:

•

On the walls of enclosure - no-slip boundary condition for velocity field (i.e. u = 0, and v = 0). Low temperature (i.e. 𝜃 = 0) for temperature field. In the porous cylinder - Enforcing porous force term (Eq. (9)) for velocity field. High temperature (i.e. 𝜃 = 1) for temperature field.

In this numerical technique, the boundary condition is specified in terms of distribution functions. Bounce-back boundary condition (i.e. no-slip) has been specified on the walls of square enclosure. According to this boundary specification, those particles move towards wall, comes back into the fluid region. For example at right wall of the enclosure, the distributions f3 , f7 and f6 fall inside the fluid domain, and hence, along right side wall the bounce-back condition can be specified as, f3 = f1 , f7 = f5 and f6 = f8 . Similar approach is followed to specify this boundary condition for other walls of the enclosure. •

Along left side wall, f1 = f3 , f5 = f7 and f8 = f6 .

(29)

This boundary condition has been implemented on circular porous region in which 𝜃 = 1. In Eq. (29), 𝑔̃𝑖 and 𝑤̃ 𝑖 indicate the opposite distribution function and weighing factor of lattice link of i, respectively. If the wall temperature is kept as zero, this boundary condition shifts to, 𝑔𝑖 = − 𝑔̃𝑖 . To treat the curvilinear wall of circular cylinder for velocity and temperature, the LB distribution functions near curvilinear boundary need to be modified. The detailed Implementation of this approach can be seen from the literature works done by Sheikholeslami et al. [49]. To perform this operation, the distance between boundary line and fluid node is required. In case of circular boundary case, the distances dx , dy , and d (refer Fig. 1(b)) have been obtained from the following equations: √ 𝑟2 − (𝑗 − 𝑦0 )2 − (𝑖 − 𝑥0 ), √ 𝑑𝑦 = 𝑟2 − (𝑖 − 𝑥0 )2 − (𝑗 − 𝑦0 ), √ 𝑑 = 𝑟 − (𝑖 − 𝑥0 )2 + (𝑗 − 𝑦0 )2 . 𝑑𝑥 =

(30) (31) (32)

Here, (i, j) is the coordinate positioned inside the curvilinear boundary and (xo , yo ) is the centre point of the circular cylinder. The dx , dy and d are the distances separating the coordinate (i, j) and curvilinear boundary in x, y, and diagonal directions, respectively. √ Further, r (=W/2) is the radius of the cylinder. dx /Δx, dy /Δy, and 𝑑∕ Δ𝑥2 + Δ𝑦2 can be compared with 0.5 and then, the corresponding distribution function modification can be implied.

(28)

𝑓 where, 𝜇static = (1−Φ) 2.5 is static viscosity of nanofluid. The KKL model predicts the effective properties of water based nanofluid closer to the various experimental results for relatively wide temperature range [16– 18]. Hence, this model has been used in the present investigation.

•

Along left side wall, g1 = -g3 , g5 = -g7 and g8 = -g6 . Along right side wall, g3 = -g1 , g7 = -g5 and g6 = -g8 . Along bottom wall, g2 = -g4 , g5 = -g7 and g6 = -g8 . Along top wall, g4 = -g2 , g7 = -g5 and g8 = -g6 .

as

The thermal conductivity is further enhanced due to the Brownian motion of particles together with the ambient fluid motion. Li [18] modified the Brownian thermal conductivity model of Koo and Kleinstreuer [17], which capture the effects of particle diameter, temperature and volume fraction. Their model has an empirical g′-function, which varies according to the type of nanofluid. For 𝐶𝑢𝑂 − 𝐻2 𝑂, this function is defined as

+ 𝑎9 ln(Φ) ln(𝑑𝑝 ) + 𝑎10 ln(𝑑𝑝 )2 ).

Along bottom wall, f2 = f4 , f5 = f7 and f6 = f8 . Along top wall, f4 = f2 , f7 = f5 and f8 = f6 .

2.5. Entropy generation In case of buoyancy driven flow, the irreversibilities occur owing to the heat transfer and fluid-friction. According to the local thermodynamic equilibrium of linear transport theory, the non-dimensional equations for entropy generation due to heat transfer (S𝜃 ) and fluid-friction (S𝜓 ) for two-dimensional heat and fluid flow in porous media can be expressed as [50] ( ) ( )2 ( )2 𝜕𝜃 𝜕𝜃 𝑆𝜃 = + , (33) 𝜕𝑥 𝜕𝑦 { 𝑆𝜓 = 𝜖

(

[ ( ) ( )2 ]} ( )2 ( )2 ) 𝜕𝑣 𝜕𝑢 𝜕𝑣 𝜕𝑢 + + + . 𝑢2 + 𝑣2 + 𝐷𝑎 2 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥 (34)

The total entropy generation is the addition of entropy generation due to heat transfer (S𝜃 ) and fluid-friction (S𝜓 ) over the computational domain (Ω). It can be written as 𝑆𝑇 =

∫Ω

𝑆𝜃 𝑑Ω +

∫Ω

𝑆𝜓 𝑑Ω = 𝑆𝜃,𝑇 + 𝑆𝜓,𝑇 .

(35)

2.6. Dimensionless parameters The non-dimensional parameters that are used in the present investigation are Rayleigh number (Ra), Darcy number (Da), and these are described as follows:

T.R. Vijaybabu

International Journal of Mechanical Sciences 166 (2020) 105240

2.6.1. Rayleigh number (Ra) In LBM, the Rayleigh number is written in the form of number of lattices (N) on characteristic height, it can be expressed as 𝑅𝑎 =

𝑔𝛽Δ𝑇 𝐿3 𝑔𝛽Δ𝑇 𝑁 3 = . 𝛼𝜈 𝛼𝜈

(36)

𝐾 . 𝑊2

(37)

The Darcy number and porosity of a packed bed of spheres can be linked through Carman-Kozeny relationship [51] given by 𝐷𝑎 =

2 𝜖 3 𝑑𝑝𝑝 1 . 2 180 𝑊 (1 − 𝜖)2

Da = 10−6 Mesh size

2.6.2. Darcy number (Da) Porosity (𝜖) and permeability (K) are the two basic properties of porous medium. The non-dimensional form of permeability, which is the ratio of permeability to the square of characteristic height of porous medium, is called as Darcy number (Da). In the present study, the characteristic height of porous medium is the diameter (W) of porous circular cylinder. Thus, Darcy number can be written as 𝐷𝑎 =

Table 3 Grid independence analysis on mean Nusselt number (NuM ) of the square enclosure with heated porous circular cylinder of Da = 10−6 & 10−2 for Ra = 106 and Φ = 4%.

(38)

Here, (dpp ) is the spherical particle diameter in the porous medium. This is to be noted that different amalgamations of permeability and porosity values can be obtained by varying the value of dpp . In this study, this value is taken as 1 cm for 1 m width of porous region.

150 250 300 400 #

× × × ×

150 250 300# 400

Da = 10−2

NuM

% Deviation

NuM

% Deviation

7.0094 7.1331 7.1394 7.1466

1.920 0.189 0.101 –

10.7247 10.9295 10.9396 10.9527

2.082 0.212 0.119 –

- Mesh size used in the study.

is noticed that the present numerical algorithm is easy to implement in comparison to conventional CFD techniques. The results obtained from the current numerical scheme are second-order accurate, which is evident from Eq. (10). Further, the computations are executed until the following convergence is satisfied. √ ]2 ∑ [ (𝑘) (𝑘−100) 𝑖,𝑗 𝜃𝑖,𝑗 − 𝜃𝑖,𝑗 ≤ 10−8 . (40) √ ∑ [ ( 𝑘 ) ]2 𝑖,𝑗 𝜃𝑖,𝑗 Where, 𝜃 i,j is the fluid temperature and k is the iteration level. 3.2. Grid analysis

2.6.3. Bejan number (Be) The significance of entropy generation due to heat transfer and/or fluid-friction can be calculated through mean Bejan number (BeM ), and it can be expressed as [52] 𝐵𝑒𝑀 =

𝑆𝜃,𝑇 𝑆𝜃,𝑇 + 𝑆𝜓,𝑇

=

𝑆𝜃,𝑇 𝑆𝑇

.

(39)

This is to be noted that if BeM > 0.5, then irreversibility is dominated by heat transfer, and in case of BeM < 0.5 the irreversibility is influenced more by fluid-friction. 3. Numerical procedure and validation

The grid independence studies have been performed for Da = 10−6 & 10−2 at Ra = 106 and Φ = 4% on the deviation of mean Nusselt number (NuM ) of the square enclosure. At Da = 10−6 , the viscous resistance offered by the porous medium will be higher, and at Da = 10−2 , the thermal gradient around the porous cylinder will become large which enhances the convection rate. Since, these factors greatly depend on the number of lattices; grid independence study has been carried out for these values of Darcy number. Table 3 indicates that 300 × 300 mesh size will produce results independent of grid number variation with less computational efforts. Hence, this mesh has been used in the present numerical simulation. In LBM, the time step (Δt) is equal to lattice space (Δx), and in the present code the Δt = 1.

3.1. Numerical method 3.3. Code validation To conduct the present numerical experiments, a lattice Boltzmann code on FORTRAN language has been developed. LB simulation begins with collision operation followed by streaming process and imposition of appropriate boundary conditions and is concluded by evaluation of macroscopic properties. Before entering into this main loop of computations, the initialization processes such as calculation of appropriate LB viscosity value and effective properties of nanofluid (static) have been evaluated. In the temperature collision equation the 𝜏′ value has been evaluated based on the effective thermal conductivity of nanofluid (i.e. Eqs. (23)–(27)). Eq. (5) which comprises the flow distribution function along with porous and buoyancy force terms has been applied to the whole computational domain. However, the porous force term (Fi ) in Eq. (5) is nullified in the clear fluid region by multiplying an integer value of 0. In the current study, single relaxation time (SRT) has been used in the BGK collision operator. The non-dimensional relaxation time for flow (𝜏) has been evaluated from the LB kinematic viscosity (𝜈). In KKL model, the effective properties of nanofluid vary with temperature field, volume fraction, and nanoparticle size. Accordingly, the thermal diffusivity (𝛼) at each lattices varies with the aforementioned parameters. Hence, the non-dimensional relaxation time for temperature (𝜏′) at each grid point has been calculated for each computational step. Further, the term g𝛽 has also been varied according the effective properties of nanofluid. √ To avoid the compressibility effect that is produced by LBM, the 𝑔 𝛽Δ𝜃𝑁 term should be less than or equal to 0.1, and hence, this value has been kept as 0.1 in the present study. On the whole, it

To verify the present LBM code, the obtained results from the code have been compared with literature under different conditions. The mean Nusselt number of square enclosure (NuM ) at Pr = 0.7 has been compared for different values of Ra for the case of isothermally heated circular cylinder placed in a square enclosure (refer Table 4). It can be seen that the percentage variation of present study results are comparable with [5]. In order to verify the implementation of KKL model on 𝐶𝑢𝑂 − 𝐻2 𝑂 nanofluid, the present results are compared with [53] which is shown in Table 5. The obtained NuM values of heater for different values of Rayleigh number and heater length have shown good agreement with the literature [53]. Further, the mean Nusselt numbers of the heated left side wall of the enclosure filled with porous media for different Darcy numbers and Rayleigh numbers have been compared with Table 4 Comparison of the mean Nusselt number of the square enclosure with literature [5] for the case of heated circular cylinder placed in an enclosure for different values of Ra at Pr = 0.7. Ra Present study [5] % Deviation

103

104

105

106

1.672 1.662 0.598

1.729 1.715 0.810

2.521 2.507 0.555

4.702 4.691 0.234

T.R. Vijaybabu

International Journal of Mechanical Sciences 166 (2020) 105240

Table 5 Comparison of calculated mean Nusselt number (NuM ) of the heater with literature [53] for the case of enclosure filled with 𝐶𝑢𝑂 − 𝐻2 𝑂 nanofluid for different values of heater length and Rayleigh number.

Table 6 Comparison of average Nusselt number of the heated left side wall of the square enclosure filled with porous medium at different values of Ra and Da with literature [54,55] while Pr = 1.

NuM Heater length

Ra

0.4

103 104 105 103 104 105 103 104 105

0.6

0.8

Da

Present study

[53]

% Deviation

2.3122 3.1139 5.9831 3.0063 4.0500 7.5625 4.0032 5.0190 8.7206

2.3152 3.1189 5.9861 3.0113 4.0570 7.5825 4.0132 5.0220 8.7316

0.130 0.160 0.050 0.166 0.173 0.264 0.249 0.060 0.126

10

−6

10−4

10−2

∗

Ra

Present study

[54]

[55]∗

% Deviation

1.086 2.980 1.082 3.045 1.084 2.238 5.421

1.080 3.000 1.080 3.000 1.080 2.300 5.580

1.070 3.090 1.070 2.970 1.060 2.280 5.550

1.473 3.691 1.109 2.463 2.214 1.877 2.380

7

10 108 105 106 103 104 105

Data used to calculate percentage deviation.

porous circular cylinder and square enclosure for the following range of parameters: literature [54,55]. The percentage deviation for this case (Table 6) is observed to be less except for the case of higher Ra value. Furthermore, the entropy maps due to natural convection between hot circular cylinder and square enclosure at Ra = 106 and Pr = 0.7 produced from present code have been verified with [40] for heat transfer, fluid friction and total entropy generation. The entropy maps are seen to be similar to that of the results shown by Mun et al. [40], and the variation of maximum entropy values are observed to be less (refer Fig. 2). Overall, the quantitative and qualitative results obtained from present LBM code for different problems have shown very good agreement with the literature.

• • • •

Rayleigh number, Ra = 103 , 104 , 105 and 106 . Darcy number, Da = 10−6 , 10−4 , 10−3 and 10−2 . Porosity, 𝜖 = 0.629, 0.935, 0.977 and 0.993. Volume fraction of nanofluid, Φ = 0% to 4%.

This study will deliver the knowledge on the sway of permeability, nanofluid volume fraction and buoyancy force on the hydrothermal characteristics of porous circular cylinder. Obtained results are designated into flow and heat transfer characteristics and are elucidated as follows: 4.1. Flow characteristics

4. Numerical results and discussion The two-dimensional numerical simulations are performed for the 𝐶𝑢𝑂 − 𝐻2 𝑂 nanofluid natural convection between isothermally heated

A detailed discussion on the effect of Rayleigh number (Ra), Darcy number (Da), and nanofluid volume fraction on flow behaviour in and around the porous circular cylinder is presented in this section through streamline patterns and vertical velocity profiles.

Fig. 2. Comparison of entropy maps with [40] for the case of natural convection flow around circular cylinder positioned in a square enclosure while Ra = 106 and Pr = 0.7.

T.R. Vijaybabu

International Journal of Mechanical Sciences 166 (2020) 105240

Fig. 3. Streamline contours in and around porous circular cylinder placed in an enclosure for different Darcy numbers at Ra = 103 and 106 . Left and right portion indicates nanofluid with 4% (——) and 0% (– · – · –) volume fraction, respectively.

4.1.1. Streamlines Fig. 3 presents the effects of Darcy number and 𝐶𝑢𝑂 − 𝐻2 𝑂 nanofluid volume fraction on flow patterns in and around the porous circular cylinder for Ra = 103 and 106 . In case of buoyancy driven flow, the convective fluid motion induces due to the density variation of the fluid. It is a well described fact that, at lower values of Rayleigh number (i.e. Ra = 103 ), the convection current is weak, and hence, the heat transfer is predominantly due to conduction of the fluid. As the buoyancy force increases (i.e. Ra > 104 ), the density of the fluid near hot portion drastically declines, as a result, the fluid moves against gravitational force. Similar phenomenon has been noticed in the present investigation while enriching the Rayleigh number. It is to be noted that in Fig. 3, the left half portion indicates the nanofluid with 4% volume fraction, whereas, right half portion shows nanofluid with 0% volume fraction. The flow patterns are seen to be symmetric for all the parameters embraced in the study. In case of Ra = 103 , the velocity values of fluid is less, and hence, even at higher permeability levels (i.e. Da = 10−3 ) less amount of fluid penetrates into it. In other words, the fluid velocity is not strong enough to deplete viscous resistance offered by the porous cylinder completely, and thus, only less volume of fluid oozes through the porous zone. At extreme Darcy number (i.e. Da = 10−2 ), porous cylinder offers less resistance to the fluid movement through it, and thus, more fluid travels with greater intensity through the porous circular cylinder. Consequently, the flow intensity of nanofluid outside the cylinder also intensifies, and this enhancement is seen to be meagre while Ra value is less. As mentioned earlier, while enhancing Ra, the flow intensity increases, which is evident from the shifting of vortex eye centre towards top wall of the enclosure. For all the values of Ra considered in the study, no fluid enters into the porous cylinder while Da = 10−6 . Fluid penetration to the porous cylinder increases while enhancing Da value, and the velocity in and around of the porous zone strongly depend on the permeability. The formation and intensity of secondary vortices, which form at bottom wall of the cylinder is evident for higher kinetic energy of the fluid. At Da = 10−2 and Ra = 106 , ternary vortices penetrate through the porous cylinder while the volume frac-

tion of nanofluid is 0%. However, such occurrence has not been noticed while Φ = 4%. Apparently, the higher temperature gradient of nanofluid intensifies the kinetic energy and therefore, the ternary vortices get suppressed. Overall, the increment is volume fraction of nanofluid enhances the fluid momentum apart from the permeability of the cylinder. 4.1.2. Vertical velocity profile In order to gain further information on the effects of Da, Ra, and Φ on the fluid momentum, the vertical velocity (i.e. v-velocity) values along x-direction of the enclosure are plotted at y = 0. Fig. 4(a) and (b) show the influence of Darcy number and nanofluid volume fraction for Ra = 103 and 106 , respectively. For all the values of Φ and Ra, the vertical velocity values in the porous zone is seen to be zero while Da = 10−6 . This cements the statement that at this value of permeability, the porous cylinder completely obstructs the fluid penetration. Further increment of permeability level enhances the velocity values, and this increment is prominent while shifting Da value from 10−3 to 10−2 . However, such pattern gets violated while the vortices penetrates through the cylinder, which occurs at higher values of Ra and Da (refer Fig. 3). Fig. 4(b) for Φ = 0% indicates such instance, where the velocity values of Da = 10−2 case fall below the 10−3 case. The kinetic energy gained by the fluid at this permeability shifts the position of primary vortex eye towards top. However, due to the less space between the eye position and boundary, one more vortex generates and it penetrates into the porous region from top portion. As a result of such mechanism, the fluid deviates further from the permeable cylinder. It is to be noted that since the vertical velocity values have been taken at y = 0, the variation of vertical velocity in the porous zone with Da is seen to be meagre. Further, such instance has not been observed while introducing nanoparticles in the base fluid. At higher volume fractions, the intensity of primary vortices further amplifies and suppresses the formation and penetration of ternary vortices. Besides, it can also be viewed from Fig. 4 that the permeability increment supplements the enhancement of momentum around the permeable body. The velocity values of Ra = 106 case is quite larger than the case of Ra = 103 , which indicates the dominance of convection current

T.R. Vijaybabu

International Journal of Mechanical Sciences 166 (2020) 105240

Fig. 4. Vertical velocity profile (v-velocity) along x-direction of the square enclosure at y = 0 for different values of Da at a) Ra = 103 , and b) Ra = 106 while Φ = 0% & 4%.

Fig. 5. Isotherm contours around porous circular cylinder placed in an enclosure for different Darcy numbers at Ra = 103 and 106 . Left and right portion indicates nanofluid with 4% (——) and 0% (– · – · –) volume fraction, respectively.

T.R. Vijaybabu

International Journal of Mechanical Sciences 166 (2020) 105240

Fig. 6. Effect of Rayleigh number on mean Nusselt number (NuM ) of enclosure for different values of Darcy number (10−6 ≤ Da ≤ 10−2 ) and 𝐶𝑢𝑂 − 𝐻2 𝑂 nanofluid volume fraction (0% ≤ Φ ≤ 4%).

at higher buoyancy force condition. At larger values of Ra, the velocity values adjacent to the porous zone increases while reducing Da value. The profound intensity of primary vortices deviate the fluid path in the porous zone, and hence, the fluid near porous zone lose its momentum while enhancing Da value. However, at lower values of Ra, due to the meagre penetration of fluid, velocity values adjacent to permeable region amplifies with Da increment. Furthermore, the nanofluid with higher volume fraction (Φ = 4%) has higher values of velocity in comparison to the base fluid (Φ = 0%), and such augmentation is notable at higher values of Ra. 4.2. Heat transfer characteristics In this section, the influence of Rayleigh number, Darcy number, and nanofluid volume fraction on heat transfer traits around the permeable cylinder is discussed in detail. The qualitative and quantitative results are presented in the form of isotherm patterns and mean Nusselt number of the enclosure (NuM ), respectively. 4.2.1. Isotherms The variation of isothermal lines around the permeable circular cylinder with Ra, Da, and nanofluid volume fraction has been illustrated in Fig. 5. The left and right portions of the plot indicate 4% and 0% nanofluid volume fraction, respectively. At Ra = 103 , the isotherms are noticed to be uniformly distributed around the porous circular cylinder irrespective to the variations of permeability and nanofluid volume frac-

tion. This shows the dominance of conduction mode of heat transfer. On the other hand, the enhancement of Darcy number has shown marginal variation on isothermal lines. The oozed fluid through the porous region carries trivial amount of heat away from the porous region. However, the less amount of fluid penetration and weak velocity of fluid at Ra = 103 fail to pick the heat transfer intensification advantage of porous medium. Besides, the increment in nanofluid volume fraction has shown denser isotherms around the permeable body. Rich thermal conductivity of nanofluid promotes the conduction phenomenon (at Ra = 103 ) and thus, the thermal gradient around the permeable body significantly increases. Further increment in Rayleigh number (i.e. Ra > 103 ) distorts the uniformity of isotherm distribution and the contours are seen to be more clustered around bottom portion of the cylinder. Due to the circular shape of cylinder, the thin and thick thermal boundary layer position vary according to the intensity of buoyancy force. Notable deviation in isothermal lines can be seen while enhancing the permeability of the circular cylinder, and this variation is insignificant while Darcy number magnifies from 10−6 to 10−4 even at higher values of Ra. This shows the dominance of viscous resistance offered by porous body over buoyancy force while Da = 10−4 . Further augmentation of Darcy number reduces the thermal gradient around top surface of circular cylinder, which can be witnessed from the stretched isotherm contours. As the fluid flow resistance relieved (i.e. Da ≥ 10−3 ), large volume of fluid enters from the bottom portion of the cylinder and carries the heat from it before it leave from top surface. As a result, the thermal inversion occurs around top surface of the permeable cylinder while enriching the permeability.

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Fig. 7. Total entropy generation map due to heat transfer and fluid-friction in and around the porous circular cylinder at Da = 10−3 , Ra = 103 , and Φ = 4%.

Nevertheless, the region of thick thermal boundary formation can be altered while ternary vortices penetrate through the hot porous region. The secondary vortices which hits over the top surface eventually turns cool, and hence, while it connect the top surface of porous cylinder, the thermal gradient prominently increases. The penetrated cold fluid diverts the path, which changes the exit portion of hot fluid, and therefore the position of thick thermal boundary layer shifts from top portion. Besides, the nanofluid introduction has shown significant clustering of isothermal lines. It is to be noted that at higher values of temperature the Brownian motion of nanoparticle increases which is directly proportional to the thermal conductivity (refer Eq. (27)). Hence, the thermal conductivity of nanofluid adjacent to the hot region is higher than that of lower temperature areas. Consequently, more isothermal lines cling over the hot surface of permeable circular cylinder. Furthermore, the thermal inversion effect which occurs at top surface of the cylinder due to the rich permeability has been suppressed while employing nanofluid. This can be noticed from the isothermal plots of Da = 10−3 and 10−2 cases while Ra = 106 . Hence, it can be realised that the heat transfer augmentation of nanofluid is higher than the permeability increment effect. However, amalgamation of volume fraction and permeability enhancement produces rich thermal dissipation characteristics.

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4.2.2. Mean Nusselt number (NuM ) The influence of nanofluid volume fraction, Darcy number and Rayleigh number on mean Nusselt number of the enclosure (NuM ) has been presented in Figs. 6(a)–(d). It is clear that the increment in Ra enriches the heat transfer for all the values of Da and Φ. Also, the enhancement trend of NuM with Ra for different values of permeability and volume fraction is observed to be similar. Obviously, quantitative influence of Da and Φ varies notably. The thermal dissipation is insignificant while Ra value shifts from 103 to 104 for all the other parameters embraced in the study. This is due to the marginal convection of the fluid at Ra = 104 in comparison to higher Ra values, and therefore, the dissipation of heat is insignificant. Further enhancement of Rayleigh number results the prominent increment in NuM value. For instance, the percentage increase in NuM for Ra = 104 , 105 , and 106 is 1.87%, 57.68%, and 182.19%, respectively, with reference to Ra = 103 while Da = 10−6 and 0% nanofluid volume fraction. Whereas, in case of 4% volume fraction of nanofluid, the percentage increment of NuM is 1.87%, 57.43%, and 181.46%, respectively, for aforementioned parameters. This suggests that the introduction of more nanoparticles is not affecting the thermal enhancement percentage with reference to buoyancy force. It is to be noted that in the present investigation, elevating nanofluid volume fraction has shown marginal effect on fluid flow except the penetration of ternary vortices at higher Da value. Owing to this, inclusion of nanoparticles has not shown significant heat transfer variation while enriching the Rayleigh number. However, the mean Nusselt number values of 4% volume fraction are higher than that of 0% volume fraction (around 58% enhancement). The pronounced thermal conductivity of nanoparticle, temperature, and volume of nanoparticles are the important factors for such enhancement. Besides, the impact of permeability on mean Nusselt number is found to be meagre at lower values of Rayleigh number (i.e. Ra = 103 & 104 ). For example, the percentage enhancement of NuM at Ra = 104 and Φ = 0% is 0.14%, 0.76%, and 3.85% for Da values of 10−4 , 10−3 , and 10−2 , respectively, with reference to Da = 10−6 . For the same parameters, the percentage enhancement of NuM at Ra = 106 is noticed to be 5.58%, 24.45%, and 62.87%. As indicated earlier, due to the insignificant velocity increment with permeability at lower values of Ra results trivial heat transfer augmentation. At higher values of Rayleigh number, more fluid penetrates into the porous zone with greater intensity of velocity even at lower permeability values. This results, the depletion of viscous resistance offered by the porous medium, and hence, the sway of Da pronounces. Also, the thermal dissipation effect is marginal while Darcy number changed from 10−6 to 10−4 . However, the magnitude slightly enhances while increasing the Rayleigh number (refer Figs. 6(a) and (b)). From Figs. 6(c) and (d), drastic increment in mean Nusselt number can be observed. The thinning of thermal boundary layer around bottom surface of porous cylinder strongly influences the low temperature gradient formed around the top surface of porous cylinder at higher permeability values. As a

Fig. 8. Total entropy generation maps due to heat transfer and fluid-friction in and around the porous circular cylinder for different values of Darcy number at Ra = 106 and Φ = 4%.

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result, the heat transfer amplifies while increasing Da values. Further, it is observed that the heat transfer enrichment offered by nanofluid with volume fraction of 4% is slightly lower than that of 0% case at Da = 10−2 with reference to Da = 10−6 . This clarifies that the formation and penetration of ternary vortices has the ability to change classical heat transfer behaviour of porous medium. Nevertheless, the mean Nusselt number magnitudes of nanofluid with higher volume fraction are higher than that of base fluid irrespective to the permeability and buoyancy force variation. 4.3. Entropy generation In this section, the impact of Rayleigh number, Darcy number, and 𝐶𝑢𝑂 − 𝐻2 𝑂 nanofluid volume fraction is discussed in terms of entropy generation. Results are illustrated in the form of entropy maps, total entropy and Mean Bejan number.

Fig. 9. Total entropy generation map due to heat transfer and fluid-friction in and around the porous circular cylinder at Da = 10−2 , Ra = 106 , and Φ = 0%.

4.3.1. Entropy maps The entropy contours for different values of Da, Ra, and volume fraction of nanofluid have been presented in Figs. 7–9. It is a well-known fact that the irreversibilites produced by any thermal system hampers its theoretical efficiency, and hence, analysing the impact of different parameters on irreversibilities offers the opportunities to enhance the

Fig. 10. Influence of Rayleigh number on total entropy generation (ST ) in the enclosure for different values of Darcy number (10−6 ≤ Da ≤ 10−2 ) and 𝐶𝑢𝑂 − 𝐻2 𝑂 nanofluid volume fraction (0% ≤ Φ ≤ 4%).

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International Journal of Mechanical Sciences 166 (2020) 105240

Fig. 11. Variation of mean Bejan number (BeM ) with Rayleigh number for different values of Darcy number (10−6 ≤ Da ≤ 10−2 ) and 𝐶𝑢𝑂 − 𝐻2 𝑂 nanofluid volume fraction (0% ≤ Φ ≤ 4%).

overall performance of thermal systems. The amount of irreversibilities is directly proportional to the entropy generation, and in buoyancydriven flows, the entropy generation is mainly due to the heat transfer and fluid-friction. Fig. 7 depicts the entropy maps in and around the permeable circular cylinder at Da = 10−3 , Ra = 103 , and Φ = 4%. It is seen that the entropy generation due to fluid-friction is negligible in comparison to the heat transfer at lower value of Ra. Also, the entropy (S) values are observed to be less, and the maximum entropy generation occurs at the centre right or left extreme points of the permeable cylinder. Since, the permeability variation at this value of Ra has not shown any variation, entropy map of Da = 10−3 case alone has been shown. However, at higher values of Ra, the impact of permeability is notable on the entropy generation which can be seen from Fig. 8. It is observed that the maximum value of entropy and its position vary with the permeability. At higher values of Da, owing to the dominance of fluid convection, the irreversibilities are predominantly due to the fluid-friction. For all the values of Ra, the fluid-friction irreversibility enhances with Da, and therefore the total entropy value augments. At Da = 10−6 , the maximum entropy value has been observed at right or left centre extreme point of the permeable cylinder, where the fluidfriction value is high. Further rise in permeability marginally reduces the fluid-friction around the permeable cylinder due to the depletion of viscous layer around the cylinder. However, the fluid gains more ki-

netic energy around the cylinder and thus, the fluid-friction enriches considerably at the middle of the enclosure side wall. Interestingly, the maximum entropy value of Da = 10−4 is noticed to be lesser than that lower permeability case at higher values of Ra. At Da = 10−2 , large thermal gradient occurs at bottom extreme point of permeable cylinder due to more volume of fluid penetration. Consequently, the irreversibility due to heat transfer enhances drastically, and the position of maximum value of entropy generation moves to the bottom point of the permeable cylinder. Besides, the permeability increment develops the higher entropy values around bottom region of the cylinder. This occurs due to the thinning of thermal boundary layer around this region while increasing the Da value. On the other hand, entropy generation in the porous region is seen to be less sensitive with the Da increment. It is to be noted that the entropy maps of nanofluid with 4% volume fraction are similar to that of other nanofluid volume fraction. However, the entropy values degrade while reducing the value of Φ. This indicates the ability of nanoparticles on the heat transfer enhancement. The amalgamation of fluid flow intensity and heat transfer enhancement at higher values of Φ significantly enhances the irreversibilities due to fluid-friction and heat transfer, respectively. At Φ = 0% and Da = 10−2 , the entropy map is observed to be slightly different in comparison to Φ = 4% case due to the formation and penetration of vortices through the permeable circular cylinder (refer Fig. 9). In this case, the entropy values at top region of

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International Journal of Mechanical Sciences 166 (2020) 105240

Fig. 12. Influence of Rayleigh number on Cup-mixing temperature (𝜃 cup ) for different values of Darcy number and nanofluid volume fraction.

porous zone significantly increases owing to the thin thermal boundary layer. The formation of ternary vortices furthers energies the primary vortices, and therefore, the fluid-friction irreversibility increases at right side wall of the enclosure. 4.3.2. Total entropy generation (ST ) The variation of total entropy generation (ST ) with Ra for different values of Φ and Da has been depicted in Figs. 10(a)–(d). It is to be noted that the total entropy generation is equal to the summation of entropy generation due to fluid-friction (S𝜓 ) and heat transfer (S𝜃 ). The trend of ST with Ra enhancement is noticed to be similar for all the values permeability and nanofluid volume fraction. The total entropy values increase with the Ra increment and this augmentation is found to be less at lower values of Ra. While boosting Ra value, the convection current is dominant and hence, the fluid-friction irreversibility along with heat transfer increases. Since the fluid momentum and heat transfer is less at lower values of Ra, the total entropy values are less. Further, Figs. 10(a)–(d) elucidate that addition of nanoparticles amplifies the total irreversibilities in the enclosure. It is seen that the nanofluid with higher volume fraction has higher velocity values and heat transfer rate in comparison to lower values of Φ. Thus, the fluid-friction and heat transfer irreversibilities induces more while introducing more nanoparticles. Besides, the permeability increment has shown insignificant effect on total entropy generation at lower values of Ra. Moreover, the ST val-

ues are less sensitive while Da values shifts from 10−6 to 10−4 . At Da = 10−2 , the total entropy values increase drastically in comparison to 10−3 case. 4.3.3. Mean Bejan number (BeM ) The dominance of thermal irreversibility and/or fluid-friction irreversibility on total entropy generation is quantified through mean Bejan number. If BeM > 0.5, then the fluid-friction irreversibility is insignificant in comparison to the heat transfer irreversibility, and the value of BeM less than 0.5 suggests that the dominance of fluid-friction irreversibility over thermal entropy generation. In the present study, for all the values of Da and Φ, the entropy generation due to heat transfer is predominant on total entropy generation for Ra ≤ 105 (refer Fig. 11). However, while enhancing Ra value, the mean Bejan number declines and this reduction is notable while Rayleigh number varies from 104 to 105 . At lower values of Ra, the weak fluid flow motion avoids the production of frictional irreversibility. Only at Ra = 106 , the fluid-friction irreversibility primarily drives the total entropy generation for all the permeability and volume fraction embraced in the present study. Further, the variation of mean Bejan number with Da and Φ is trivial while Ra ≤ 104 . On the other hand, it is evident from Fig. 11 that the volume fraction increment offers a slight increment in heat transfer entropy generation. The increment in effective thermal conductivity of nanofluid with higher Φ yields large thermal gradient, and hence, the irreversibil-

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International Journal of Mechanical Sciences 166 (2020) 105240

Fig. 13. Variation of Root Mean Square Deviation of Cup mixing temperature (RMSD(𝜃 cup )) with Rayleigh number for different values of Darcy number and nanofluid volume fraction.

ity due to heat transfer augments. Besides, in case of the Da number increment the entropy generation due to temperature marginally suppresses the impact of fluid-friction irreversibility. In addition, the BeM increment gap with the Φ increment reduces while promoting the permeability of porous cylinder while Ra ≥ 105 . Overall, the nanofluid volume and permeability augmentation increases the production of thermal irreversibility on total entropy generation. 4.3.4. Cup-mixing temperature (𝜃 cup ) In order to understand the influence of Ra, Da and Φ on the thermalmixing in the enclosure, the Cup-mixing temperature (i.e. velocity weighted average temperature) has been evaluated for all the parameters considered in the study. This is to be noted that the higher value of 𝜃 cup indicates the rich thermal-mixing. Figs. 12(a)–(d) depict the variation of 𝜃 cup with Rayleigh number for different values of Da and Φ. It can be seen that the thermal-mixing in the enclosure improves while increasing the Ra value irrespective to the variation of other parameters. The profound convection rate at higher Rayleigh number promotes the proper mixing of temperature. The thermal-mixing enriches marginally while enhancing the nanofluid volume fraction for all values of Da while Ra ≤ 105 . At Ra = 106 , the 𝜃 cup values of 0% volume fraction are found to be higher than that of other Φ values, and this effect is notable at higher values of permeability. The higher convection rate at this value

of Ra and profound thermal gradient at higher values of Φ degrade the proper mixing. 4.3.5. RMSD of 𝜃 cup In the thermal processing of food and molten materials, it is essential to maintain the uniform temperature and concentration in a closed domain. For this purpose, it is customary to position heat source at different locations either on the boundary or in the enclosure. The degree of temperature uniformity in the enclosure can be measured through root mean square deviation of cup-mixing temperature (RMSD(𝜃 cup )). Figs. 13(a)–(d) illustrate the variation of RMSD(𝜃 cup ) with Rayleigh number for different values of permeability and nanofluid volume fraction. The lower values of RMSD(𝜃 cup ) suggests that the temperature uniformity is high. It can be seen from Fig. 13 that the temperature uniformity starts to decline while enhancing the Ra values up to 105 , and this decrement is notable at higher values of Da. Owing to the intense convection of fluid while increasing Ra, more temperature distribution occurs towards top portion of the enclosure. As an illustration, bottom portion of enclosure contains cold or denser fluid. Due to this fact the temperature uniformity gets affected in the enclosure. In addition, the Da increment further amplifies the kinetic energy, and hence, the thermal distribution further enhances at top half portion of the enclosure. At Ra = 106 , the temperature distribution uniformity in the enclosure

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drastically increases. The side-wise spreading of temperature contours (refer Fig. 5) at this value of Ra increases the temperature uniformity. However, such increment reduces while enriching the permeability of the circular cylinder. Besides, the volume fraction of nanofluid has not shown any deviation in the RMSD(𝜃 cup ) irrespective to the variation of other parameters. 5. Concluding remarks A numerical study has been carried out to showcase the impact of Rayleigh number (103 ≤ Ra ≤ 106 ), Darcy number (10−6 ≤ Da ≤ 10−2 ), and nanofluid volume fraction (0% ≤ Φ ≤ 4%) on natural convection traits between a heated circular porous cylinder and square enclosure. 𝐶𝑢𝑂 − 𝐻2 𝑂 nanofluid has been considered as a working fluid and the effective properties were evaluated through KKL model. Lattice Boltzmann method has been employed to solve the present problem. A force term corresponds for the inertial and viscous effects of porous medium have been coupled with the BGK collision operator to obtain porous flow characteristics. The important findings of the current investigation are as follows: •

•

•

•

•

•

The intensity of fluid flow in and around the permeable circular cylinder increases while enhancing the Darcy number, and this effect is prominent at higher values of Rayleigh number (i.e. Ra > 104 ). Also, nanofluid with higher volume fraction aids for the further augmentation of kinetic energy of the fluid. Presence of more nanoparticles in the base fluid significantly improves the thermal gradient around the permeable cylinder. Whereas, permeability increment offers large values of thermal gradient around bottom surface of the porous cylinder and thickens thermal boundary layer around top surface of it. However, higher volume fraction of nanofluid suppresses the adverse effect of Darcy number increment on heat dissipation. The heat transfer enhancement produced by Ra value increment is observed to be similar irrespective to the variation of nanofluid volume fraction. The impact of Darcy number enrichment on thermal dissipation is found to be significant while Ra > 104 . Effect of nanofluid volume fraction on entropy generation maps and location of entropy maximum entropy are noticed to be trivial except while Da = 10−2 , Ra = 106 and Φ = 0% (due to the penetration of vortices through porous region). Rich permeable cylinder reduces the fluid-friction irreversibilities around it. However, at Da = 10−2 the maximum entropy situates at bottom extreme point of porous zone owing to the large thermal gradient. Darcy number and volume fraction of nanofluid enhancement offer more production of irreversibilities. However, the permeability effect on entropy generation is meagre at lower values of Ra. Besides, more nanoparticle inclusion augments the total entropy values considerably even at lower values of Ra and Da. The thermal-mixing increases slightly while improving Φ values for Ra ≤ 105 . The temperature uniformity has seen to be more at higher values of Ra, and it declines while augmenting the permeability.

Hence, from the present investigation it can be understood that the hydrothermal traits and entropy generation can be regulated by nanofluid volume fraction, permeability and intensity of buoyancy force. Declaration of Competing Interest I wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. I confirm that I have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property.

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