Heat transfer and second law analyses of forced convection in a channel partially filled by porous media and featuring internal heat sources

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1 Heat transfer and seond law analyses of fored onvetion in a hannel partially filled by porous media and featuring internal heat soures Mohsen Torabi *, a, Nader Karimi b, Kaili Zhang a a Department of Mehanial and Biomedial Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong b Shool of Engineering, University of Glasgow, Glasgow G1 8QQ, United Kingdom Abstrat This paper provides a omprehensive study on the heat transfer and entropy generation rates in a hannel partially filled with a porous medium and under onstant wall heat flux. The porous inserts are attahed to the walls of the hannel and the system features internal heat soures due to exothermi or endothermi physial or physiohemial proesses. Dary-Brinkman model is used for modelling the transport of momentum and an analytial study on the basis of loal thermal non-equilibrium (LTNE) ondition is onduted. Further analysis through onsidering the simplifying, loal thermal equilibrium (LTE) model is also presented. Analytial solutions are, first, developed for the veloity and temperature fields. These are subsequently inorporated into the fundamental equations of entropy generation and both loal and total entropy generation rates are investigated for a number of ases. It is argued that, omparing with LTE, the LTNE approah yields more aurate results on the temperature distribution within the system and therefore reveals more realisti Nusselt number and entropy generation rates. In keeping with the previous investigations, bifuration phenomena are observed in the temperature field and rates of entropy generation. It is, further, demonstrated that partial filling of the hannel leads to a substantial redution of the total entropy generation. The results also show that the exothermiity or endothermiity harateristis of the system have signifiant impats on the temperature fields, Nusselt number and entropy generation rates. Keywords: Entropy generation; Internal heat soures; Fored onvetion; Loal thermal non-equilibrium; Mathematial modelling * Corresponding author. s: Mohsen.Torabi@my.ityu.edu.hk (M. Torabi), Nader.Karimi@glasgow.a.uk (N. Karimi). 1

2 Nomenlature interfaial area per unit volume of porous media, a sf 1 m T w lower wall temperature, K Be Average Bejan number U f 1 dimensionless veloity of the fluid in the lear region Bi Biot number defined in Eq. (13) U f Br Brinkman number defined in Eq. (13) U m p speifi heat at onstant pressure, J Kg u -1-1 K f 1 dimensionless veloity of the fluid in the porous medium dimensionless mean veloity of the fluid defined in Eq. (17) veloity of the fluid in the porous medium, Da Dary defined in Eq. (13) u f veloity of the fluid in the lear region, h one half of the hannel height, m u m h one half of the thikness of the lear setion, m w f h sf k k ef k es N f 1 N f N s N t fluid-to-solid heat transfer oeffiient, W m K - -1 ratio of effetive solid thermal ondutivity to that of fluid effetive thermal ondutivity of the fluid ( ε k f ), W m K -1-1 effetive thermal ondutivity of the solid (( 1 ε ) ks ), W m K -1-1 dimensionless loal entropy generation rate within the lear fluid region defined in Eq. (6) dimensionless loal entropy generation rate within the fluid phase of the porous medium defined in Eq. (6) dimensionless loal entropy generation rate within the solid phase of the porous medium defined in Eq. (6) dimensionless total entropy generation rate within the medium defined in Eq. (7) -3 W m w X x Y Y y s Greek symbol s ε mean veloity of the fluid, -1 ms -1 ms -1 ms dimensionless energy soure in fluid phase per unit volume, W m -3 dimensionless energy soure in solid phase per unit volume, -3 W m dimensionless axial distane axial distane, m dimensionless vertial distane dimensionless one half of the thikness of the lear setion vertial distane, m porosity Nu Nusselt number defined in Eq. (9) γ ratio of the heat flux at porous-fluid interfae to that of hannel s wall Pe Pelet number defined in Eq. (13) κ permeability, m loal entropy generation rate within the lear fluid S f µ f region, W m K -1-1 fluid visosity, Kg m s loal entropy generation rate within the fluid phase of S f -3-1 µ eff the porous medium, W m K -1-1 effetive visosity of porous medium, Kg m s loal entropy generation rate within the solid phase of S s -3-1 the porous medium, W m K θ dimensionless temperature defined in Eq. (13) energy soure in fluid phase per unit volume, s f -3 θ f 1 W m dimensionless temperature of the fluid within lear region energy soure in solid phase per unit volume, dimensionless temperature of the fluid phase of the porous s θ s f medium T temperature, K θ f, m dimensionless mean temperature of the fluid defined in Eq. () T f 1 temperature of the fluid within lear region, K θ s T f temperature of the fluid phase of the porous medium, K dimensionless temperature of the solid phase of the porous medium

3 T f, m mean temperature of fluid, K ρ T s 1. Introdution temperature of the solid phase of the porous medium, K fluid density, -3 Kg m Energy hallenges are urrently amongst the most substantial issues faing the human ivilisation. The rapid inreases in energy onsumption along with the subsequent atastrophi environmental problems have led to a omplex global risis. A range of ativities are being undertaken to resolve this issue aross the world. These, hiefly, inlude more extensive use of renewable energy and improving the effiieny of the onventional energy generation tehnologies. Both of these two families of tehnology heavily involve thermal proesses. Optimisation of these proesses is essential for improving the performane of a wide range of energy generation tehnologies. Similarly, optimal thermal systems are entral to the effiient use of thermal energy. In priniple, there are two approahes to the problem of thermal optimisation. In the most onventional approah, the system is analysed on the basis of an energy balane or first law of thermodynamis. This approah is well known and has been in use for a long time. Despite all its pratial merits, this remains an entirely quantitative method and provides no information on the quality of energy. Importantly, the degradation of energy due to the existene of irreversibilities is totally ignored by this approah [1 3]. These negative points have led the researhers to onsider an alternative approah for optimisation of a thermal proess by onsidering both of the first and seond laws of thermodynamis. The seond law of thermodynamis provides a powerful tool to evaluate the energy quality degradation in a thermal system while the first law governs the energy balane. This method onstruts fundamental relations to alulate the generation of entropy within a system and aordingly realises the level of irreversibility. It has been pointed out, in the literature, that through using this method the system an be optimised from the energy quality prospet through minimisation of the entropy generation. This has been elaborated in details by Bejan in his seminal textbooks [,3]. Shifting from the energy quantity point of view to an energy quality perspetive in a thermal proess reveals the advantage of the seond law of thermodynamis over the first law. By employing the former it is possible to optimise the proess suh that less entropy is generated, and onsequently less exergy is destruted. In other words, the energy quality remains as high as possible. This onept has been, already, exploited in various ondutive [4 6], onvetive [7 9] and radiative [10] environments. Channels under fored onvetion are an important part of various thermal systems [11]. Reently, heat transfer and entropy generation analyses in horizontal hannels, fully or partially filled with porous media, have attrated onsiderable attention [11]. This is, primarily, due to the fat that utilising porous media an lead to signifiant improvements in heat transfer harateristis [1]. Energy analysis in porous media is usually on the basis of a fundamental assumption about the presene or absene of loal thermal equilibrium [11,1]. The former leads to loal thermal equilibrium (LTE) model, also known as one-equation energy model. The latter, however, is regarded as loal thermal non-equilibrium (LTNE), or two-energy equation model [11,1]. Although LTE model [11] has been used extensively in heat transfer analyses [11], LTNE model is reeiving an inreasing attention from the researh ommunity [13 16]. This stems from the fat that in the emerging fields suh as MEMS and biotehnology 3

4 as well as some lassial areas, suh as hemial and nulear engineering, the auray of the analysis is of primary importane [1,17]. Hene, LTNE modelling beomes the preferred option. This, however, signifiantly inreases the mathematial omplexity of the analysis and therefore the hoie of thermal model should be made mindfully. There have been a large number of publiations on onvetive heat transfer in porous media under LTNE onditions. One of the pioneering works in this area was done by Nield [13]. Following Nield s persuasive work, many sholars have tried to re-examine thermal porous systems from the LTNE perspetive [18 6]. Bortolozzi and Deiber [19] investigated natural onvetion in a fluid-saturated annular porous avity onsidering both LTE and LTNE onditions. The governing equations were numerially solved using vortiity-stream funtion sheme. Comprehensive omparison was made between the veloity and temperature fields for both models and onsiderable differenes were observed in some ases [19]. In a fundamental study Kim and Jang [0] validated their similarity solution for onvetion in porous media against a numerial solution within the framework of LTNE. They also ompared the results of LTNE with those of LTE model [0]. Similar to an earlier work of Bortolozzi and Deiber [19], Kim and Jang [0] observed that for some ertain thermophysial parameters the differene between LTE and LTNE models is non-negligible. Khashan et al. [1] revisited the lassial problem of steady state fluid flow and heat transfer in a porous pipe using SIMPLE algorithm under LTNE model. Influenes of Reynolds and Biot numbers on the temperature ontours and other thermal harateristis of the system were investigated in this work [1]. Chen and Tso [] used LTNE model in a hannel fully filled with porous media. They inorporated visous dissipation effets into the energy equation for the fluid phase of the porous medium and numerially investigated the variation of Nusselt number with a number of thermophysial parameters. In a separate study, these authors developed analytial expressions for Nusselt number [3]. Ouyang et al. [4] used three different fundamental LTNE models in a hannel. The analytial solutions for the flow and temperature fields were obtained and ompared with those predited by numerial simulations. Through using the lassial definition of the thermal entry length on the basis of Nusselt number, the dimensionless thermal entry length was predited [4]. Dehghan et al. [5,6] onsidered both Dary and Forhheimer terms in the momentum equation but negleted the visous dissipation in the energy equation. Perturbation tehnique was employed by these authors to takle the resultant nonlinear governing equations [5,6] and semi-analytial solutions for the temperature and Nusselt number were derived. Comparisons were, further, made with the previously published works and good agreements were observed [5,6]. Ohoa-Tapia and Whitaker [7] were, perhaps, the first sholars who onsidered a flow onduit partially filled with porous media. Partial filling is an effetive avenue to irumvent signifiant pressure drops in a porous system [8]. By adopting this approah the desirable thermal effets of a porous medium an be mostly ahieved. Yet, the pumping power and the onsequent expenses are maintained within a reasonable range. These attrative features have resulted in a signifiant number of published studies on the thermal aspets of partially filled systems, see for example [14 16,9]. In the partially filled systems, the thermal boundary ondition of the interfae of the porous material and lear fluid poses a fundamental diffiulty. A preise explanation of the heat distribution on suh interfaes is yet to be developed [1]. Nonetheless, phenomenologial thermal models are often used to provide the essential mathematial boundary onditions in the modelling works. Yang and Vafai [30,31] provided two different 4

5 main models for the interfae ondition together with their analytial solutions. They have disussed the limitations of eah model and gave illustrative figures regarding the variation of Nusselt number in eah model. Further examples of suh modelling efforts an be found in Refs. [14,15,9] whih used models A and B of Yang and Vafai [31] for the porous-fluid interfae. Model A assumes that the heat flux is divided between the two phases on the basis of their effetive ondutivities and temperature gradients. However, in model B equal amounts of heat flux are transferred into eah phase [31]. Xu et al. [3,33] analytially solved the flow field and energy equation for a parallel-plate hannel [3] and a pipe [33] partially filled with porous media attahed to the inner wall of eah geometry. Convetive boundary ondition was onsidered at the porous-fluid interfae. They illustrated the temperature distribution with different thermophysial parameters and showed that Nusselt number dereases if the hannel is fully filled with porous media. Later, this was also demonstrated by other researhers [34,35]. Yang et al. [36] examined the differenes between the thermal performane of a tube partially filled with a metal foam when it is attahed to the inner wall or plaed in the ore of the tube. They onsidered equal temperature for the metal foam and fluid phase at the porousfluid interfae and developed analytial solutions for the temperature fields and Nusselt number [36]. The studies, disussed so far, were solely onerned with heat transfer aspets of the problem and therefore belong to the first law approah. As argued earlier, the first law of thermodynamis remains silent on the quality of energy in a given thermal proess and any judgment on this requires a seond law investigation. A partially filled flow onduit with signifiant heat transfer involves major soures of irreversibility. These inlude heat transfer through a finite temperature differene and visous dissipation of the flow kineti energy. It is, therefore, expeted that the system involves a non-negligible level of entropy generation and experienes a drop of energy quality. As the irreversibility is partially due to non-equilibrium heat transfer, taking LTNE approah is essential in the evaluation of the seond low performane of the proess. However, there is, urrently, a dearth of thermodynami analyses of partially or fully filled systems under LTNE ondition and most of the existing works in this area are limited to loal thermal equilibrium [37 40]. Reently, Buonomo et al. [41] have onduted a study on porous filled miro hannel by using LTNE model. They investigated the hydrodynami and thermal proesses between two parallel plates filled with a porous medium [41]. Due to the mirosale size of the hannel and effets of rarefation of the gas flow under onsideration, the first order veloity slip and temperature jump onditions at the fluid-solid interfae were used. In this work, the veloity and temperature fields were analytially investigated, and loal and total entropy generation rates were alulated [41]. Most reently, Torabi et al. [4] utilised LTNE model and analysed heat transfer and entropy generation in a horizontal hannel partially filled with porous media. They onsidered asymmetri boundary onditions for the hannel and inorporated the visous dissipation term into the energy equations. For the first time, these authors reported a bifuration phenomenon for the loal entropy generation rate [4]. The urrent work onduts a omprehensive study on the heat transfer and entropy generation in a hannel under fored onvetion, whih is partially filled with a porous medium. Both LTE and LTNE models are applied and the outomes are ompared. The hannel is under onstant and equal heat fluxes from both top and bottom surfaes. The 5

6 Dary-Brinkman model is used to model the transport of momentum and internal heat generation or onsumption is inorporated into the energy equation. These internal soures represent the exothermi or endothermi physial and hemial reations ourring in various pratial proesses [9,43,44]. The rest of this paper has been organised in the following order. Setion gives the detailed speifiations of the problem. In this setion, the governing equations of heat and fluid flow together with the boundary onditions for the employed interfae models are desribed. Setion 3 provides the fundamental equations of entropy generation in the onfiguration under investigation. By introduing dimensionless boundary onditions, the available loal and total entropy generation relations are non-dimensionalised. Subsequently, in setion 4 the momentum and energy equations are solved analytially. Through inorporating the veloity and temperature solutions within the entropy generation relations, given in Setion 3, the loal and total entropy generation rates are alulated. Setion 5 inludes a series of figures regarding temperature, Nusselt number and, loal and total entropy generation rates. This setion further provides a omprehensive disussion on the effets of pertinent parameters on the temperature and entropy generation. The paper is finally onluded in Setion 6.. Problem statement Consider a retangular, two dimensional hannel subjeted to uniform and equal heat fluxes on the upper and lower surfaes. The hannel is partially filled with a porous medium suh that the porous material is attahed to the upper and lower walls, as shown in Fig. 1. The height of the hannel is h and the ore of the hannel, with the thikness h, is lear. Constant thermophysial properties for both solid and fluid phases are assumed. This study, further, assumes steady, laminar flow along with fully developed veloity and temperature fields and ignores radiative heat transfer and gravitational effets. Dary-Brinkman model is utilised to model the transport of momentum within the porous material, and homogeneous and isotropi harateristis are assumed for the porous struture. In the ourse of this study the fluid and solid thermal soure terms are assumed to have onstant values. Due to the symmetry of the problem under investigation only half of the onfiguration shown in Fig. 1 is onsidered..1. Governing equations Considering the aforementioned assumptions and the onfiguration shown in Fig. 1, the momentum and energy equations under LTNE model are written as follows. Momentum equations in the lear and porous regions are expressed by p u f 1 + µ f = 0 0 y h, (1a) x y p u f µ f + µ eff u f = 0 h y < h. (1b) x y κ Transport of thermal energy for the lear region, and fluid and solid phases of the porous region are respetively written as 6

7 Tf 1 Tf 1 ρu p f 1 = kf + s 0 f y h, (a) x y Tf Tf ρu p f = kef + h sf asf ( Ts Tf ) + sf h y< h, (b) x y Ts 0 = kes h sf asf ( Ts Tf ) + ss h y < h, () y where the µ µ f eff = is the effetive visosity and different terms and notations are defined in the nomenlature. ε The boundary onditions for the above system of equations are y u T = = = y y f 1 f 1 0 : 0, 0, (3a) uf 1 uf Tf 1 Tf T s y = h : uf 1 = uf, µ f = µ eff, qint = kf = kef + kes, Tf 1 = Tf = Ts, (3b) y y y y y T y = h : u = 0, T = T = T, q = k + k y f f f s w w ef es T s y. (3) The boundary onditions expressed by Eq. (3b) and (3) are equivalent to model A of Yang and Vafai [31], whih has been also used in the investigations of partially filled hannels [16,9]. It is emphasised here that previous works [14 16] have demonstrated that the hoie of porous-fluid boundary ondition has signifiant effets upon the thermal behaviour of the system. In order to make analytial progress with the energy equations (a) and (b), their left hand sides should be evaluated. By integrating Eq. (a) and with the help of boundary onditions at the upper side of the hannel and the interfae, the following relation is derived, T h 1 h 1 1d h f Tf ρ d p uf y = kf y + s d f y x 0 0 y 0 qint. (4) Adding Eqs. (b) and (), integrating the resultant equation, and inorporating the boundary onditions at the upper side of the hannel and the interfae yields 7

8 T h h h h f Tf Ts ρp uf dy = kef dy + k d es y + ( sf + ss ) dy x h h y h y h qw qint. (5) By adding Eqs. (5) and (4) and noting that in a fully-developed flow subjeted to onstant wall heat flux T T = = onstant, the left hand side of Eqs. (a) and (b) beomes x f f1 x ρ p qw + sf dy + ssdy Tf 1 Tf 0 h = ρp = x x hu h m h, (6) where 1 h h um = uf 1dy + uf dy h 0 h. (7) Inorporating Eq. (6) into the Eqs. (a) and (b), results in the following energy equations for the fluid flow in the lear and porous regions, h h q + s dy + s dy T u k s y h, (8a) w f s 0 h f 1 f 1 = f + 0 f hu m y h h q + s dy + s dy T u = k + h a T T + s h y < h. (8b) w f s 0 h f f ef sf sf ( s f ) f hu m y The Nusselt number at the lower wall of the hannel an be written as [3,33,35,36] 4h qw Nu = k T T ( 1,, ) f f w f m. (9) where 1 h h T = u T dy + u T dy. (10) f, m f 1 f 1 f f hu 0 h m When ratio of the thermal ondutivity of the two phases of the porous setion is near unity, the LTE model an be often used [45]. This is due to the fat that as the thermal ondutivities of the two phases approah eah other and Biot number is large enough, the temperature differene between the solid and fluid phases in the porous region 8

9 diminishes. Hene, the energy equations (b) and () an be ombined to form a single energy equation for the porous region. Considering LTE model, the energy equations for the fluid and porous regions an be written as Tf 1 Tf 1 ρu p f 1 = kf + s 0 f y h, (11a) x y Tf Tf ρu p f = ( kef + kes ) + s f + ss h y< h. (11b) x y It should be noted that as a result of LTE assumption in Eq. (11b), T f = T s. The thermal boundary onditions for this model are slightly different to those of LTNE model and are desribed by the following relations. y = = y T 1 0: f 0, (1a) T T y = h : q = k = ( k + k ), T = T y y f 1 f int f ef es f 1 f, (1b) T f y = h : Tf = Tw, qw = kef + kes y ( ). (1).. Normalisation To provide further physial insight, the following dimensionless variables are introdued. These inlude a wide range of thermophysial properties and will be used in the proeeding disussions. ( ) ( 1 ε ) u kes T Tw k k es s hsf asf h y x h κ U =, θ =, k = =, Bi =, Y =, X =, Y =, Da = u q h k ε k k h h h h r w ef f es µ u ρuh q s h sh k T Br = Pe = = w = w = B = q h k q q q q h f r, p r int f, γ, f, s s, ef w w ef w w w w (13) where u r h = µ f p x. Substituting the above parameters into the momentum Eqs. (1a) and (1b), energy equations (), (8a) and (8b), and boundary equations (3), results in the following set of non-dimensional governing equations and boundary onditions. Momentum Eqs. (1a) and (1b) are onverted to U f 1 1+ = 0 0 Y Y, (14a) Y 9

10 1 Uf U f 1+ = 0 Y 1 < Y. (14b) ε Y Da The dimensionless form of energy Eqs. (a, b and ) are AU 1 θ = + w f 0 Y Y U k Y f 1 f 1 m ε, (15a) AU 1 θ = + Bi ( θs θf ) + w f Y Y 1, (15b) U k Y f f m θs 0= Bi ( θs θf ) + w s Y Y 1. (15) Y Through non-dimensionalisation, the boundary onditions redue to the followings Y U : f θ 0, f 0 = = = Y Y, (16a) Uf 1 1 Uf 1 θf 1 1 θf θs Y = Y : Uf 1 = Uf, =, γ = = +, θf 1 = θf = θs, (16b) Y ε Y εk Y k Y Y 1 θ θ = 1: = 0, 1 = +, θ = θ = 0, (16) k Y Y f s Y U f f s where 1 1 f s 0, (17a) Y A = 1+ w dy + w dy Y 1 m = f 1d + f d 0. Y U U Y U Y (17b) Further, γ an be readily alulated through Eqs. (5), (4) and the dimensionless parameters (13), from the following relation 1 1 Y 1+ w f dy + w sdy U f 1dY 0 Y 0 Y w f dy U. 0 m γ = (18) It is urious to note that the boundary onditions related to the heat flux at the upper wall of the hannel and the adiabati ondition in the middle of the hannel ( y = 0 ) have been used in the derivation of Eqs. (5) and (4). These 10

11 will not be used to derive the onstant parameters of the energy equations. To deouple the energy equations of the fluid phase from that of the solid phase, i.e., Eqs. (15b) and (15), the seond derivatives of these equations are needed. Some straightforward algebrai manipulations turn these two equations into the followings, A U 1 θ AU 1 θ w U m k U k 4 f f f f f = + Bi w f w s + Y Y m Y Y, (19a) θ kau θ w 0= s f s s Bi 4 ( k ) kw f kw s Y U m Y Y. (19b) Now, by using Eqs. (15b) and (15) and their first derivatives, the following boundary onditions are developed. These are essential for the losure of the problem and are given by, Y θs + w 0 s = Y 1 θf + w 0 f = k Y = 1: A U 1 θ θ θ w 3 f f s f f = + Bi + 3 U m Y k Y Y Y Y 3 θs θs θf w s Bi + = 0 3 Y Y Y Y (0) Aordingly, the dimensionless Nusselt number is given by the following relation. 4ε k Nu =, (1) θ f, m where 1 θ θ θ Y 1 f, m= U f 1 f 1dY + U f f dy U 0 Y m. () Considering LTE model and using the dimensionless parameters given by Eq. (13), the dimensionless LTE energy equations an be written as AU 1 θ = + w f 0 Y Y U k Y f 1 f 1 m ε, (3a) AU 1 θ = w f + w s Y Y 1. (3b) U k Y f f m 11

12 One again, it is emphasised that due to LTE in Eq. (3b) θ f = θ s. The thermal boundary onditions for LTE model are slightly different from the thermal boundary ondition for LTNE model and are expressed by, Y θ = = Y f 1 0: 0, (4a) 1 θ 1 θ ε k Y k Y 1 : f f Y = Y γ = = + 1, θ f 1 = θ f, (4b) Y 1 θ = = + = k Y f 1:1 1, θf 0. (4) 3. Entropy generation It has been intuitively onsidered in the previous publiations that the heat generation implies its effets on the entropy generation thorough diffusive heat transfer part of the entropy generation formula [46 48]. Moreover, in many seond law analyses for ondutive media it has been mathematially proven that the entropy generation formula does not affeted by internal heat generation and this feature of the system input its impat on the temperature distribution and therefore into the entropy generation. This an be learly seen in reent publiations in this field [5,6,49]. However, sine the energy equations in onvetive systems are mainly partial differential equation with many terms, this mathematial endorsement has not been taken previously. In keeping with previous literature in the field, it has been assumed in this study that the internal heat generation/onsumption does not have diret effet on the entropy generation and implies its effets on the temperature distribution, i.e., on the diffusion term of entropy formula. Bearing the abovementioned information in mind, it is assumed in this study that the generation of entropy in the thermal system under investigation is due to heat transfer over a finite temperature differene and visous dissipation of the flow kineti energy. These mehanisms generate entropy in the solid and fluid phases within the porous regions and the fluid phase of the lear region. Under LTNE model the following relations hold for the volumetri rate of the loal entropy generation within the fluid phase of the lear region, fluid phase of the porous medium and solid phase of the porous medium, respetively. k f Tf 1 T f 1 f uf 1 S µ f 1 = 0 y h T f 1 x + +, (5a) y Tf 1 y ( ) k ef Tf T h sf asf Ts T f f f eff uf S µ µ f = u f h y h T f x <, (5b) y TT s f κtf Tf y 1

13 ( ) k es Ts T h s sf asf Ts Tf S s = h y h Ts x + + <. (5) y TT s f The detailed derivations of these equations an be found in [41,4,46,50], and are not repeated here. Inorporation of the dimensionless parameters introdued in Eq. (13) into Eqs. (5a, b and ) results in the dimensionless loal volumetri entropy generation rates, whih are expressed by 1 Y 1 d d + w f Y + w s Y 0 0 θf 1 + ( Pe k ) U Y U Br N = = + 0 Y Y m f 1 S f 1h Y f 1 kes εk f 1 ( θ + B) ( θf 1 + B), (6a) N m S f h BrU f = = + + es k ( θf + B) s f f U f Br Y + Y Y < 1 ε θ ( + B ) f 1 Y 1 d d + w f Y + w s Y 0 0 θ f + ( Pe k ) U Y Bi ( θs θf ) f ( θ + )( θ + ) ( θ + ) k B B Da B 1 Y 1 d d + w f Y + w s Y 0 0 θs + ( Pe k ) U Y Sh N s = = + Y Y < 1, k B B m s es ( θs + B ) s f Bi ( θs θf ) ( θ + )( θ + ) (6b) (6) Tk w es where the parameter B = depends on the thermophysial properties of the hannel. It is worth mentioning qh w that, assuming referene temperature for the denominators of Eqs. (5a, b and ) would derease the mathematial omplexity of the model and ould be used in this work similar to Refs. [51 53]. However, to predit the loal and total entropy generation rates more aurately, the loal temperature is used in the denominator of these equations. This approah has been taken in some reent works [10,54,55]. Aordingly, the dimensionless total entropy generation rate for the hannel is given by integrating the dimensionless form of the volumetri loal entropy generation relations, over the height of the hannel. That is Y 1 1d ( ) d 0. (7) Y N = N Y + N + N Y t f f s 13

14 The average Bejan number, i.e., Be, defined as the ratio between the total entropy generation due to heat transfer by the total entropy generation [51], is expressed as Be N N h =. (8) t when N h whih is the entropy generation rate due to heat transfer an be alulated from integration over the speifi boundary for the first terms of Eq. (6a), the first and seond terms of Eq. (6b) and both terms of Eq. (6). It is worth mentioning here that the heat transfer irreversibility is dominant when Be approahes to 1. When Be is less than 1 and approahes to zero, the irreversibility due to the visous effets dominates the proesses and if Be = 1the entropy generation due to the visous effets and the heat transfer effets are equal [53]. 4. Veloity and temperature fields This setion provides analytial solutions for the momentum and energy equations derived in Setion. and therefore reveals the veloity and temperature fields in the porous and lear regions. Solution of Eqs. (14a) and (14b) results in the following expressions for the veloity fields within the porous and lear regions, 1 Uf 1 = Y + CY 1 + C, (9a) Y Y U f = C 3sinh + C 4osh + Da, Da ε Da ε (9b) where the four onstant parameters C1 C4 are obtained from the veloity boundary onditions and expressed by C 1 = 0, (30a) C Y 1 Y Y 1 Y ε Da ε sinh + + Da osh Da Da ε Da ε =, (30b) Y 1 osh Da ε C 3 1 Y Y ε Da ε osh + Da sinh Da ε Da ε =, (30) Y 1 osh Da ε 14

15 C 4 1 Y Y ε Da ε sinh Da osh Da ε Da ε =. (30d) Y 1 osh Da ε 15

16 Solutions of the differential energy equations, Eqs. (19a) and (19b), provide the general temperature distributions within the porous regions and under LTNE model. Further, solving Eq. (15a) renders the temperature field in the lear region. These temperature fields are θ θ kε AY ACY 1 ACY Umw fy = + DY + D, f 1 1 U m ( ) sinh ( ) A kda A kda D osh ka Y D ka Y Y 3 Y 4 f = osh + sinh + + Y + DY 5 + D6 ( A1kDa + ε) ε Da ε ε Da ε ka1 ka1 A1 Da Y Y D D B θs = B osh + B sinh osh ( ) ( AY ) + sinh ( AY ) Y + DY + D ε B Da + ε Da ε Da ε B B B where U m DaU m U m DaU m U m A (31a), (31b), (31) BiAC AεC BiAC AεC BiADa A = Bi Bi k, A =, A =, A = Bi ( w f + w s ), (3a) BikAC BikAC BikADa B = Bi Bi k, B =, B =, B = Bik ( w f + w s ) U m U m U m (3b) 16

17 The ten unknown parameters D1 D10 are obtained numerially using mathematial software Maple. The orretness of the solution proedure has been verified repeatedly in our previous works [4,56]. The solution for the temperature distribution with LTE model is straightforward and muh simpler than the elaborated proedure taken for LTNE model and is therefore not reported here. 5. Results and disussion The alulated veloity, temperature, and loal and total entropy generation rates are presented in this setion. The results, further, inlude ratio of the interfae heat flux to the heat flux of the boundary ondition, i.e. γ, and Nusselt number. The urrent setion has been divided into two subsetions. In subsetion 5.1 the veloity and temperature fields are presented. Subsetion 5. provides a disussion on the loal and total entropy generation rates whih are pertinent to the seond law of thermodynamis. It is noted that the problem under investigation has not been takled previously in any theoretial and numerial work. Hene, a diret omparison of the urrent results with those of others is not possible. Nonetheless, it was observed that through inreasing the Biot number the urrent solutions of the temperature fields approahed those predited by LTE analysis. It is well established that in the limit of infinite Biot number, LTNE and LTE solutions are equivalent. Thus, this observation serves as a validation of the urrent results. As a general matter, in all proeeding graphs of temperature and loal entropy generation rates, the dash and solid lines are, respetively, in onnetion with the fluid and solid phases Veloity, temperature and Nusselt number Figures and 3 show the veloity distribution within the partially filled porous hannel. These figures show that in keeping with the findings of the previous investigations [3,33,4], thiker porous setions tend to magnify the maximum veloity in the lear region. Further, a omparison between Figs. and 3 reveals that generally lower Dary numbers result in more abrupt hanges in the behaviour of the fluid veloity profile around the porous-fluid interfae. This is apparent in these two figures at around Y = 0.3 for Y = 0.3. The veloity field hanges its general pattern from the lear region in the lower part of these figures to that within the porous setion in the upper setion of the figures. Due to the smoother hange of the veloity pattern from the porous setion to the lear setion at higher Dary number (see Fig. ), the maximum veloity of the lear flow in Fig. is smaller than that in Fig. 3. This behaviour is, also, due to the fat that by lowering the Dary number in Fig. 3 a smaller volumetri flow rate enters the porous region. Hene, the share of the lear region from the total flow inreases, whih results in a more signifiant peak in the flow veloity. Figures 4-8 depit the temperature distribution within the hannel with varying values of internal heat soures, Dary number, porosity and ondutivity ratio. Figure 4 shows the effets of thermal ondutivity parameter and internal heat soures on the solid and fluid temperature fields. In Fig. 4a the solid and fluid soure terms have 17

18 idential numerial values. Under this ondition, it is observed that when the thermal ondutivity ratio is equal to unity the differene between the solid and fluid phases temperatures is quite small. However, for the thermal ondutivity ratios different to one, the temperature differene between the two phases rises. Figure 4a, further, shows that depending upon the value of ondutivity ratio the temperature of the solid phase within the porous phase an be either lower or higher than that of the fluid phase of the porous setion. Limiting the heat generation to either fluid or solid phase in Figs. 4b and leads to signifiant deviation from the pattern observed in Fig. 4a. The temperature differenes, between solid and fluid phases, are now always onsiderable and feature less sensitivity to the thermal ondutivity ratio. Further, as expeted, the phase with internal heat generation features a higher temperature. Figure 5 shows the temperature distribution at the same values of dimensionless parameters as Fig. 4, - 4 with the exeption of Dary number whih has been lowered to Da = 10. Similar to Fig. 4, different ombinations of internal energy soure terms have been investigated. A omparison between these two figures indiates that the general trend observed in Fig. 4 remains unhanged at lower value of Dary number shown in Fig. 5. However, the differenes between the solid and fluid temperatures have signified in Fig. 5. Interestingly, in Fig. 5a and under the same strengths of solid and fluid soure terms, the solid and fluid temperature differene remains negligible. However, this is not the ase for the two other ases (Figs. 5b and ), in whih heat is generated only in one phase. Figures 6 and 7 demonstrate the influenes of the thikness of the porous insert upon the temperature distribution within the hannel. The values of thermal ondutivity, k, in Figs. 6 and 7 have been, respetively, set to 1 and 5. Various ombinations of the solid and fluid soure terms have been onsidered in these figures. A areful inspetion of these figures reveals that depending on the strength of the thermal soure terms, varying the thikness of the porous setion may swith the hottest phase within the system from solid to fluid or vie versa. However, this is not always the ase and under some irumstanes either of the solid or fluid phases remains always the hottest phase within the porous region. For instane, in Figs. 6a and the solid phase is always the hottest phase, while Fig. 6b shows that hanging the thikness of the porous region an hange the hottest phase from solid to fluid. The phenomenon of swapping the hot and old phases in the porous media is alled bifuration and has been, already, analysed with LTNE model in a number of onfigurations [9,31,4,57]. The bifuration phenomenon is also learly seen in Figs. 7a and b. There is bifuration in Fig. 7a under idential strengths of the solid and fluid thermal soure terms. In this figure, when the dimensionless thikness of the lear setion is 0.1, the solid phase of the porous setion has a higher temperature ompared with the fluid phase. However, when the non-dimensional height of the lear setion inreases to 0.3 or 0.5 this trend is reversed and the temperature of the fluid phase beomes higher than that of the solid phase. There is a similar trend in Fig. 7b. However, when heat generation is limited to the solid phase (as in Fig. 7) bifuration disappears. Further, there exists another important feature in Fig. 6. The dimensionless temperature for the entreline of the hannel, Y = 0, may inrease or derease by inreasing the value of Y. For example, in Fig. 6a by inreasing the lear setion s thikness from 0.1 to 0.3 the dimensionless temperature at the entreline of the hannel dereases. Nonetheless, further inreasing of the value of Y to 0.5 inreases the dimensionless temperature of the entreline. Sine the dimensionless temperature of the upper part of the hannel has been set to zero, under most irumstanes, this temperature an be regarded as the largest 18

19 temperature differene between the solid and fluid phases and the hannel s wall. It is should be noted that in ase of exessive internal heat generation, it is possible to have a region of the hannel with higher temperature ompared to that of the wall. However, suh extreme ases are not onsidered in this work. Figure 8 shows the effets of variations in internal heat soures on the temperature fields. In Fig. 8a solid and fluid thermal soure terms vary equally and always maintain the same value. However, the solid heat soure in Fig. 8b is set to zero and only the fluid soure term varies. Expetedly, the non-dimensional temperatures in the system orrelate with the level of exothermiity. In Fig. 8a, as the strength of internal exothermiity dereases and approahes the endothermi ase, the differene between the temperature of the solid and fluid phases within the porous setion dereases. This behaviour hanges in Fig. 8b, here the variation in exothermiity of the fluid phase auses a bifuration with signifiant temperature differenes between the solid and fluid phase. Similar trends have been reported in other porous systems with internal heat soures and sinks [9]. Figures 9-11 depit variations of the maximum temperature differene between the wall and the fluid in the lear region as a result of hanges in the thikness of the lear region and, for a given set of parameters. As disussed earlier, this temperature differene is represented by the dimensionless temperature on the entreline of the hannel. It is lear from Figs. 9-11, that variation of the porous thikness generates a minimum value of the dimensionless temperature of the entreline. This temperature is the highest ahievable temperature differene between the fluid and the wall. Starting from zero thikness of the lear setion, inreasing this thikness auses an inrease in the magnitude of the highest temperature differene. The temperature differene then reahes its maximum value and subsequently starts to derease. It is important to note that the desribed trend is the reverse of that of θ f1 (0) in Figs. 9-11, as θ f ~(T f T w ) (see Eq. 13) and therefore the values of θ f are always negative. Figure 9 indiates that the magnitude of the temperature differene dereases by dereasing the thermal ondutivity ratio. Further, the maximum value of the temperature differene, between the fluid and the hannel wall, inreases by dereasing the Dary number (Fig. 10), and by inreasing the porosity of the porous setion of the hannel (Fig. 11). The ratio of the interfae heat flux and heat flux of the hannel s wall, denoted by γ, versus the thikness of the lear region has been illustrated in Figs. 1 and 13. These figures show that as the thikness of the lear region inreases, the heat flux ratio rises and reahes a maximum. This is then followed by a deline of the heat flux ratio suh that it approahes unity in the limit of fully lear hannel, whih is an antiipated behaviour. Further, as the Dary number dereases, the maximum value of heat flux ratio dereases in value and shifts towards higher thiknesses of the lear region. It is lear from Fig. 1 that the heat flux ratio inreases with Dary number. The influenes of the internal heat soures upon the heat flux ratio have been investigated in Fig. 13. This figure shows that in exothermi ases as the thikness of the lear setion inreases, the parameter γ starts to inrease versus the lear region thikness. Similar to that disussed in Fig. 1, it reahes a maximum value and then deays. However, the neutral and endothermi ases ( w = w = 0 and -1) remain as exeptions to this trend, in whih the initial rise f s is followed by a plateau or the trend is totally reversed. The behaviour observed in Figs. 1 and 13 is qualitatively onsistent with those reported in other heat generating/onsuming porous systems under fored onvetion [9]. The 19

20 negative heat flux ratio for w = w =- 1 in Fig. 13 means that in this ase the heat flux at the interfae is from f s fluid to porous setion. This is due to the endothermi proesses ourring in both fluid and solid parts of the system. As the thikness of the porous setion is high when Y has a low numerial value, the thermal energy onsumption in the porous setion is higher than the energy onsumption in the lear setion. Hene, the heat flux at the interfae is towards the porous medium. This behaviour is related to the bifuration phenomenon disussed earlier, whih has been also deteted in other endothermi porous systems [9]. Table 1 ompares Nusselt numbers alulated under LTE and LTNE models with varying values of the porous thikness and porosity and for an exothermi ase. The tabulated results learly show that the differenes between the outomes of these two models are mostly non-negligible. Signifiant differenes between the values of Nusselt number under LTE and LTNE have been previously reported [33]. The data in Table 1 are presented to extend the existing datasets to the ases with internal exothermiity. They, further, provide a means of omparison and validation for the future theoretial and numerial results. Figures 14 to 17 put forward more detailed information about the behaviour of the Nusselt number due to the variations in the pertinent parameters. Figure 14 shows that, at high porous thiknesses the differene between the Nusselt numbers for LTE and LTNE is signifiant and annot be negleted. In other words, in this limit LTE results are highly inaurate. This remains true even for thermal ondutivity ratio of one. Figures depit the variation of Nusselt number versus the lear setion thikness. These figures show that, in general, as the thikness of the lear region inreases from zero the Nusselt number dereases sharply and reahes a minimum value. Further inrease in the thikness of the lear region reverses this trend and the Nusselt number starts to gain higher values. Therefore, when partial filling is implemented in these hannels to ompensate the pump osts, speial attention should be paid to avoid the partiular porous thikness whih minimises the Nusselt number. This statement is in keeping with the earlier findings of Maerefat et al. [58], who onduted numerial analysis on a onfiguration similar to Fig. 1 but onsidered only the LTE ondition. Figure 15 shows the effets of variations in Dary number upon the value of Nusselt number. It is lear from this figure that dereasing Dary number results in inreasing the value of Nusselt number. One again this mathes the earlier findings of the analyses under LTE assumptions [58]. It is, however, noted that this agreement is qualitative and as Table 1 and Fig. 14 show there ould be onsiderable differenes between the Nusselt numbers predited by LTE and LTNE models. Figures 16 and 17 show the effets of thermal ondutivity ratio and porosity on the Nusselt number. These figures indiate that Nusselt number dereases with inreasing the thermal ondutivity ratio and porosity, respetively. 5.. Loal and total entropy generation The effets of pertinent parameters on the loal and total entropy generation rates in the investigated onfiguration have been illustrated in Figs In partiular, the influenes of the modifiations in internal energy soure terms, upon the rate of entropy generation are investigated in these figures. Figure 18 shows the effet of varying the lear setion thikness on the loal entropy generation within the hannel for three different sets of thermal soure terms. It an be, learly, seen in this figure that hanging the lear setion thikness from 0.3 to 0.4, derease the loal 0

21 entropy generation. The hange of entropy generation in both lear and porous setion of the hannel is quite substantial, and this is the ase for all the investigated sets of the internal energy soure terms in Figs. 18a-. This behaviour is in aordane with the earlier results, whih illustrated the strong effets of the hannel onfiguration upon the hydrodynamis and heat transfer harateristis of the problem. Equations (5 a-) learly show that modifiations of temperature and veloity fields affet the generation of entropy within the hannel. A omparison of Figs. 18a- reveals that the loal generation of entropy is strongly affeted by the thermal energy soure terms. This is suh that the values of loal entropy generation for the ase of idential solid and fluid exothermiity strengths (Fig. 18a) are between two to three times more than those in the ase of exothermiity in solid only (Fig. 18). Further, variations in the thermal soure terms an hange the phase with higher entropy generation. This an be seen more learly from the insets in Figs. 18a- and an be regarded as a bifuration of entropy generation. Figure 19 illustrates the effet of thermal ondutivity ratio on the loal entropy generation. This figure shows that when the thermal ondutivity ratio is unity, the loal entropy generations in both solid and fluid phases of the porous setion are lose to eah other. Depending upon the state of the thermal soures either of fluid or solid phases an have the maximum rate of entropy generation while, the differene between the two remains negligibly small. However, as the thermal ondutivity ratio inreases to k = the value of the loal entropy generation rates in the solid and fluid phases of the porous setion are ompletely different. Under this ondition the entropy generation rate in the solid phase exeeds that in the fluid phase by a signifiant amount. The loal entropy generation rates under two different Dary numbers are ompared in Fig. 0. This figure shows that dereasing Dary number inreases the loal entropy generation rate within the thermal system. Three sample alulations regarding the total entropy generation rate within the system have been onduted. Figs. 1-3 show the outome of these alulations. These figures show the hanges in the total entropy generation versus the thikness of the lear region and for varying values of thermal ondutivity ratio, Pelet number and exothermiity. Generally, in all these figures the total entropy generation rate goes through a sharp derease by inreasing the lear setion thikness. It reahes a minimum value and then starts to inrease by a small amount. This learly shows the highly irreversible situation enountered when thik porous inserts are plaed in the hannel. It is, therefore, inferred from these figures that, for the investigated system and within the onsidered range of parameters, total filling is the worst ase from the view of the seond law of thermodynamis. However, with partial filling the total entropy generation rate an be minimised and an optimum values for parameter Y, whih has a diret onnetion with the porous thikness of the system, an be ahieved. The total entropy generation rate inreases with thermal ondutivity ratio (Fig. 1), dereases by Pelet number (Fig. ), and inreases with energy soures regarding exothermi or endothermi harateristi of the system (Fig. 3). Further inspetion of these figures show that the optimum value of the lear setion thikness for ahieving the minimum value of the total entropy generation rate shifts towards higher values of Y through hanging a number of parameters. These inlude inreasing the thermal ondutivity ratio (Fig. 1), dereasing Pelet number (Fig. ), and inreasing the internal energy soures (Fig. 3). That is to say, if the total entropy generation is higher than this value for a given set of parameters, the minimum total entropy generation rate an be ahieved by inreasing the thikness of the lear setion. 1

22 5.3. Average Bejan number Figures 4-6 illustrate the variation of average Bejan number with thikness of the lear setion and the thermal ondutivity ratio (Fig. 4), Pelet number (Fig. 5) and energy soures in both lear and porous setions (Fig. 6). In all of these figures it is seen that when the lear setion thikness is small, almost all of the entropy generation rate is due to heat transfer, i.e., Bejan number is near to unity. However, when the height of the lear setion thikness is large, Bejan number is muh less than 0.5, whih implies that the entropy generation due to visous dissipation overtakes the entropy generation due to heat transfer. It is interesting to note here that, from Eq. (6) the Pelet number has inverse effet on the entropy generation due to heat transfer and therefore to Bejan number. This an be learly seen in Fig. 5 when inreasing the value of Pelet number dereases Bejan number. Figure 6 shows the effet of energy soures on Bejan number. As expeted, inreasing the value of internal heat generation, inreases the rate of internal heat transfer within the system and onsequently it inreases the entropy generation rate due to heat transfer. 6. Conlusions A two dimensional, axisymmetri hannel with porous inserts attahed to the walls and under onstant wall heat flux, was onsidered. The hannel inluded a steady, laminar and fully developed flow of a onstant density fluid. It was assumed that the solid and fluid phases an feature internal heat soures and the system is under LTNE. Dary- Brinkman model of transport of momentum along with model A of Yang and Vafai [31] for the desription of porous-fluid interfae thermal boundary ondition, were utilised. The problems of fored onvetion and entropy generation were investigated in this onfiguration. Analytial solutions were developed for the veloity, temperature, Nusselt number and, loal and total entropy generation within the hannel. In keeping with the previous investigations, it was shown that the existene of internal heat soures an heavily affet the thermal equilibrium state and invalidate LTE approah. Further, it was shown that variations in the internal heat soures ould lead to a bifuration phenomenon in whih the hottest phase in the porous medium is exhanged between the fluid and solid parts of the system. Compared to the previous studies under LTE, onsideration of LTNE and the existene of internal heat soures appeared to have no major influene on the qualitative behaviour of the Nusselt number. However, the results showed that the numerial values of the predited Nusselt numbers under LTNE ould be markedly different to those obtained through LTE approah. Analysis of the loal generation of entropy revealed that this property of the system is heavily affeted by the onfiguration of the hannel. This was suh that an inrease in the thikness of the porous inserts signifiantly inreased the rate of entropy generation. Considering the total entropy generation in the hannel, an optimal value for the thikness of the porous insert was found and the influenes of pertinent parameters upon this optimal thikness were disussed. The results of this work provide a guide through the omplex physial behaviour of fluid onduits partially filled with porous media, whih inlude internal heat soures. They an be, further, used for the validation of numerial and other theoretial analyses. Aknowledgments

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27 Fig. 1. Configuration of the hannel partially filled with a porous material. 7