Effects of Baffle on Entropy Generation in Separated Convection Flow Adjacent to Inclined Backward-Facing Step

Transcription

1 Journal of Eletronis Cooling and Termal Control,,, 53- ttp://dx.doi.org/.3/jet.. Pulised Online Deemer (ttp:// Effets of Baffle on Entropy Generation in Separated Convetion Flow Adjaent to Inlined Bakward-Faing Step Asad Barami, Seyyed Adolreza Gandjalikan Nassa, Malie Hasemipour Meanial Engineering Department, Sool of Engineering, Said Baonar University, Kerman, Iran Computer Department, Kerman Bran of Azad University, Kerman, Iran Reeived Novemer, ; revised Deemer, ; aepted Deemer 7, ABSTRACT Numerial simulations of a two-dimensional laminar fored onvetion flow adjaent to inlined akward-faing step in a retangular dut are presented to examine effets of affle on flow, eat transfer and entropy generation distriutions. Te main aim of using affles is to enane te value of onvetion oeffiient on te ottom wall. But te useful energy an e destroyed due to intrinsi irreversiilities in te flow y te affle. In te present work, te amount of energy loss is estimated y te omputation of entropy generation. Te values of veloity and temperature wi are te inputs of te entropy generation equation are otained y te numerial solution of momentum and energy equations wit loked-off metod using omputational fluid dynami tenique. Disretized forms of te governing equations in te (x, y) plane are otained y te ontrol volume metod and solved using te SIMPLE algoritm. Numerial expressions, in terms of Nusselt numer, entropy generation numer, Bejan numer and oeffiient of frition are derived in dimensionless form. Results sow tat altoug a affle mounted onto te upper wall inreases te magnitude of Nusselts numer on te ottom wall, ut a onsiderale inrease in te amount of entropy generation numer takes plae eause of tis tenique. For validation, te numerial results for te Nusselt numer and entropy generation numer are ompared wit teoretial findings y oter investigators and reasonale agreement is found. Keywords: Entropy Generation; Inlined Bakward Step; Baffle; Convetion Flow. Introdution Fored onvetion flows in annels wit arupt expansion or ontration are widely enountered in engineering appliations, su as ooling passages of turine lades, diffusers, omustors and eat exangers. Tese separated flows are intrinsially irreversile eause of visous dissipation, reattament and reirulation. Te flow over akward faing step (BFS) was studied y several investigators ot teoretially and experimentally to find te pysis of su separated flows. Most of resear works on BFS as een extensively arried out from fluid meanis and eat transfer perspetives. Armaly et al. [] analyzed laminar, transition, and turulent isotermal flows over a BFS ot experimentally and teoretially. Flow over a BFS wit fore onvetion eat transfer was onduted y oter investigators y different numerial teniques [-7]. Tese results sowed tat te flow separation and susequent reirulation result in very poor eat transfer performane in te region near te akward faing step. A review of resear on laminar mixed onvetion flow over forwardand akward-faing steps was done y Mulawe []. Te ompatness of new termal systems leads te importane of finding effetive means for enaning eat transfer. Using affles is one of an effetive metod for tis purpose. In a reent study y Nie et al. [9] aout onvetion flow over BFS, it was revealed tat a affle mounted onto te upper wall inreases onsideraly te magnitude of maximum Nusselt numer at te stepped wall. In tat study, te set of governing equations inluding ontinuity, momentum and energy equations for tree-dimensional onvetion laminar flow over BFS in a dut were solve numerially y CFD tenique. Te effets of affle loation and its eigt on flow and eat transfer distriutions were arried out. Reently, entropy generation analysis as een extensively applied in many fluid flows wit eat transfer in different geometries. Entropy generation analysis provides a useful tool to identify te irreversiilities in any termal system as well as to determine te optimum ondition for any proess. Heat transfer and visous dissipa- Copyrigt SiRes.

2 5 tion are te only soures of entropy generation in fore onvetion fluid flow. Tere are a few studies in wi te analysis of entropy generation due to fored onvetive flow over a BFS as een onduted [,]. Reently, investigation of entropy generation in a flow over a BFS under leeding ondition was done y Au-Nada []. In tat work, te set of governing equations were solved y te finite volume metod and te distriutions of entropy generation numer on solid surfaes at different onditions were alulated. Moreover, te effets of leed oeffiient for ot lowing and sution on te entropy generation numer and Bejan numer were presented. In te present work for te first time, te entropy generation analysis for onvetion flow over an inlined BFS in a dut wit affle is arried out. Considering inlined step makes te prolem more general in omparison to te researes in all of te aove referenes in wi te step as rigt angle. Te main aim is to identify te irreversiilities and entropy generation due to affle against its advantage wi is enaning te onvetion oeffiient in te termal system. Toward tis end, te set of governing equations onsisting te onservation of mass, momentum and energy and also te entropy generation equation are solved y CFD metod in te Cartezian oordinate system using loked-off metod. Finally, te effets of affle loation and its eigt on te distriutions of Nusselt numer, entropy generation numer and frition oeffiient are explored.. Teory For alulating te amount of entropy generation in te fore onvetion flow indiated in Figure, te veloity and temperature distriutions are primary needed. For two-dimensional inompressile, steady and laminar flow, te non dimensional forms of te governing equations wi are te onservations of mass, momentum and energy an e written as follows: U () Y U U U P U () Re Y Re Y U P (3) Re Y Re Y Y U () Pe Y Pe Y in Equations () to (), te following dimensionless groups are used: x y u v p l l Y,,, U,,, P l TT Pr, Re, T T were Pr and Re are te prandtl and Reynolds numers, respetively and l is te arateristi lengt wi is assumed to e te dut eigt after te step... Boundary Conditions Te oundary onditions for te ydrodynami prolem are impermeaility and non-sliping on te solid oundaries inluding te affle. Besides, at te inlet setion, uniform flow wit veloity is onsidered and at te outlet setion, zero axial gradients for veloity omponents are imposed. In solving te energy equation, te surfaes of te top wall wit te affle and ottom wall inluding te step are onsidered isoterm wit te onstant temperatures T and T, respetively. Besides, at te inlet setion, a uniform temperature T i is assumed and at te outlet setion, zero temperature gradient in x-diretion is employed... Regular Grid wit Bloked-Off Region We now desrie te manner in wi we treat te geometry of inlined step using te loking-off metod (Patanker [3]). By loked-off tenique, a CFD ased omputer program written for a regular grid an e improved to andle an irregularly saped omputational domain. Tis an e done y rendering inative some of te ontrol volumes of te regular grid so tat te remaining ative ontrol volumes form te desired irregular domain wit omplex oundary. Example is sown in Figure, were te saded areas denote te inative ontrol volumes. It is ovious tat aritrary geometries are approximated y a series of te retangular grids. It is wort mentioning tat using fine grids in te interfae region etween ative and inative zones auses to ave an approximated oundary wi is more similar to te true oundary. Idea of te loking-off operation onsists of estalising known values of te relevant dependent variales in te inative ontrol volumes. If te inative region represents a stationary solid zone as in te ase, te veloity omponents in tat region must e equal to zero, and if te region is regarded as isotermal one, te known temperature must e estalised in te inative ontrol volumes. In te present omputations, zero veloity in te inative zone is employed y use of a very large visosity for grid points in tat region wit onsidering no slip ondition on te nominal oundaries. In termal prolem, te nominal oundaries are onsidered isoterm and large value for termal ondutivity for te grid points inserted in te inative zone is employed. By Copyrigt SiRes.

3 55 Top wall D W T u(y) T i Bottom wall a y x ( L T H Figure. Pysial model. Figure. Bloked-off region in a regular grid. tis tenique, te grid points in te inative zone eome isoterm su tat teir temperatures are equal to te temperature onsidered on te nominal oundaries. 3. Entropy Generation In tis setion, a rief overview of te numerial metod used for te solution of te governing equations is presented. Finite differene forms of te ontinuity, momentum and energy equations are otained y integrating over an elemental ell volume wit staggered ontrol volumes for te x- and y- veloity omponents. Te disretized forms of te governing equations are numerially solved y te SIMPLE Algoritm of Patankar and Spalding [5] for te pressure orretion in te iteration proedure. Numerial solutions of te governing equa- To analyze te entropy generation in fored flow prolems, te analyzer needs to use equations from wi te entropy generation an e alulated. Tese equations an e derived y identifying te soures of entropy generations and ten applying te seond law of termodynamis. In te fored onvetion flow, te entropy generation is assoiated to te eat transfer and fluid flow frition. Bejan [] derived te overall general entropy generation in Cartesian oordinate system as follows:. k Sgen T T u v u v (5) T x y y x te non dimensional form of tis equation for alulating te entropy generation numer is given as []: Y () U U Y Y te following dimensionless quantities are defined:. gen k T Br S l T T,,, Br k T T were is te entropy generation numer, S gen te volume rate of entropy generation, Br te Brinkman numer and τ is te non-dimensional temperature differene. In Equation (), te first term represents entropy generation due to eat transfer ( ond ), wile te seond term represents te entropy generation due to te fluid visous effet ( vis ). Te Bejan numer wi is defined as follows is also omputed in te present analysis. ond Be (7) Aording to te definition of Bejan numer, te irreversiilities due to te visous effet are dominant wen Be. Wen Be, te eat transfer irreversiilities dominate te proess and if Be =.5, te entropy generation due to te visous effet and eat transfer are equal.. Solution Strategy ond vis Copyrigt SiRes.

4 5 tions togeter wit te oundary onditions are otained iteratively y te line-y-line metod y utilizing te TriDiagonal-Matrix Algoritm (TDMA). Iterations are terminated wen sum of te asolute residuals eomes less tan 5 for ea equation. As te result of grid tests for otaining te grid-independent solutions, an optimum grid of 5 wit lustering near te solid surfaes is determined in x- and y- diretions. After alulation of veloity and temperature fields, Equation () is used to solve for te entropy generation numer at ea grid point in te flow domain. Ten, te total entropy generation troug te flow domain is alulated as:, Y d () were is te volume of omputational domain. Also, te average Bejan numer is also alulated y te following equation: Beaverage Be, Y d Besides, te distriutions of frition oeffiient and Nusselt numer on te ottom wall are determined as follows: U f (9) Re Y Boundary Nu Y w m Boundary 5. alidation of Computational Results () Te present numerial implementation is validated y reproduing te teoretial results of two oter investigators. Along two different test ases, te results of Nusselt numer in onvetion flow over BFS wit affle are ompared wit tose presented y Nie et al. [9] and te results of entropy generation numer wit tat of te Au Nada []. In tese two works, te BFS was onsidered to e vertial to te stepped wall. It sould e noted tat in solving te governing equations for tese two test ases, te same values for parameters wi were used in Refs. [9,] are onsidered in te present analysis. Fiure 3 sows te variation of Nusselt numer along te ottom wall. It is seen tat te value of maximum Nusselt numer inreases due to affle effet. For instane, te maximum Nusselt numer for DH is aout tree times of te one for DH (witout affle). Also, te loation of Nu max moves downstream as te affle moves along te stream wise diretion. For anoter test ase in a onvetion flow over BFS witout affle, te variation of entropy generation numer along te ottom wall is sown in Figure. It is seen tat te minimum value of ours diretly at x = at te ottom step orner, were te fluid as no motion. Te maximum value of takes plae inside te reirulation zone and ten it drops sarply to a very low value at te reattament point after wi inreases and approaes to a onstant value far from te step. However Figures 3 and sow tat te general agreements etween te present results wit te teoretial findings y oter investigators are quite good and te values of minimum and maximum entropy generation numers and Nusselt numers and teir predited loations are reasonaly losed to ea oter.. Results and Disussion Te entropy generation in onvetion laminar air flow over an inlined akward step in a dut wit a affle Nu 5 3 = = Present work Teor. ( Nie et al. 9) Figure 3. Distriution of Nusselt numer on te ottom wall at two different affle loations, Re = 33, θ = 9. Witout affle Present work Teor. (Au Nada, ) 5 Figure. ariation of entropy generation numer along te ottom wall Re =, θ = 9. Copyrigt SiRes.

5 57 mounted onto te upper wall is alulated for te purpose of determining te amount of irreversiilities due to affle. Toward tis end, te ontinuity, momentum and energy equations are solved numerially y CFD metod. In te present omputations for te pysial domain sown in Figure, te expansion ratio ( ER H ) is set equal to in all of te test ases wen te distane of affle from te step depited y D, is onsidered to e varied from DH to DH. Te omputed domain in te x-diretion downstream of te step is restrited y x H and te distane etween te inlet setion and step depited y a set equal to H to ensure fully developed ondition for veloity and temperature distriutions efore te step. First in order to sow te flow pattern, te streamlines are plotted in Figure 5 for an inlined step wit θ = under te ondition of DH, W H and Re = 5. Te effets of step and affle on te flow are learly seen from te urvatures of streamlines and separated regions. Figure 5 sows tat five different reirulation zones are enountered for Re = 5 in te flow domain. Te primary reirulation region on te ottom wall ours adjaent to te step upstream te affle and te seondary one takes plae on te ottom wall down stream te affle, wereas tree oter reirulation zones our adjaent to te affle surfae. Sine, te main task of te present study is to investigate te effet of affle on te entropy generation in laminar fored onvetion flow, te following results sown in Figures to are due to air flows over inlined step wit for tree different loations of affle, DH, and 3, tree different affle s eigt, H.,. and.5, wen te affle s widt is equal to H/. In te omputations of tese figures, te Prandtl numer is kept onstant equal to.7 to guarantee onstant fluid pysial properties for moderate and small values of temperature differene T T. Te value of Reynolds numer is equal to 5 exept in Figures 9 and in wi te effet of Re on entropy generation and Bejan numer is studied. Besides, te values of non-dimensional temperatures for ottom wall (inluding te step), top wall (inluding te affle) and inlet fluid are set equal to, and.7, respetively. ariations of entropy generation numer along te ottom wall at different loations of affle and also at tree different affle s eigts are plotted in Figures (a) and (), respetively. Also, te variation in BFS flow witout affle is plotted in anoter oordinate system in Figure (a) eause of different orders in te values of. Tis figure sows tat te affle makes a different Y Figure 5. Streamlines in flow over inlined step in a dut Re = 5, θ =, H =.5, D H =. = Witout affle Witout affle / H=. / H=. / H=.5 = =3 (a) () Figure. ariation of entropy generation numers along te ottom wall Re = 5, θ =. (a) Effet of affle loation, =.5 ; () Effet of affle eigt, D H=. H Copyrigt SiRes.

6 5 trend for distriution along te ottom wall in omparison to BFS flow witout affle. In te asene of affle, te value of entropy generation numer dereases along te step lengt and reaes to zero on te ottom wall at te step orner. After tis point, inreases and te maximum value of ours inside te reirulation zone and ten it drops sarply to a very low value at te reattament point after wi inreases and approaes to a onstant value far from te step. Tis eavior an e explained y noting tat after flow separation, te vorties inrease dramatially inside te reirulation region tat auses to take plae maximum value of in tis zone. Besides, at te reattament point no sear stresses are taking plae and te entropy generation is totally due to ondution. But te presene of affle auses a different trend for entropy generation numer su tat te affle effet governs to te oter fators in variation of entropy generation. Aording to Figure (a), te maximum entropy generation numer ours on te ottom wall just elow te affle wit a value wi is very greater tan tat of takes plae for BFS flow exluding te affle. For example in te ase of DH, te maximum entropy generation numer is aout times of te one for DH (witout affle). Tis is related to te inreased temperature and veloity gradients elow te affle were te flow is pused toward te ottom wall. Also, it is seen from Figure (a) tat te value of max inreases as te affle moves toward te BFS (wit te derease of D). Te effet of affle eigt of te distriution of along te ottom wall is indiated in Figure (). Tis figure sows tat te affle eigt as an important effet on te variation of entropy generation umer su tat te value of inreases max dramatially y a small inrease in te eigt of affle. It is also due to inreasing in veloity and temperature gradients on te ottom wall y te affle. Besides, it is seen from Figure () tat te effet of reirulation region after te step on te variation of eomes important for small value of te affle eigt, su tat for iger values for, te variation of is only affeted y te affle effet instead of flow separation and reattament. It is wort mentioning tat in Figure and also in later similar figures for inlined step, te region.9 is due to te step projeted lengt after wi tere is te ottom wall. ariations of Nusselt numer on te ottom wall at different loations of te affle and also at different affle s eigts are plotted in Figures 7(a) and (), respetively. For onvetion flow over BFS exluding te affle ( DH), it is seen from Figure 7(a) tat Nu dereases along te step projeted lengt and te minimum value of Nu ours on te ottom wall adjaent to step orner and te maximum value at te reattament point after wit Nu approaes to a onstant value. In te presene of affle, similar trend is seen for te variation of Nu on te ottom wall, ut te maximum Nusselt numer takes plae just elow te affle wit a value wi is very grater tan of te one for DH. Besides, it is seen from Figures 7(a) and () tat te maximum Nusselt numer inreases as te affle moves toward te BFS and also y inreasing in te affle eigt. Distriutions of frition oeffiient along te ottom wall at different affle loations and also at different affle s eigt are illustrated in Figures (a) and (), respetively. It is seen tat f is negative inside te irulation zone due to te ak flow and te minimum 5 = = =3 5 / H=. / H=. / H=.5 Nu 3 Nu 3 (a) () Figure 7. ariation of Nusselt numer along te ottom wall. (a) Effet of affle loation, H =.5 ; () Effet of affle eigt, D H=. Copyrigt SiRes.

7 = = =3... / H =. / H =. / H =.5. f.. C f (a) -. Figure. ariation of frition oeffiient along te ottom wall Re = 5, θ =. (a) Effet of affle loation, /H =.5; () Effet of affle eigt, =. () value of frition oeffiient takes plae in tis region after wi te value of f eomes equal to zero at te point of reattament. If one noties to te f urves in te viinity of x = in detail, it is seen tat te value of frition oeffiient is negative along te step projeted lengt and ten eomes zero at te step orner on te ottom wall. Tis is related to te fat tat te surfae of inlined step is exposed to te reirulation flow as sown in Figure 5. Inreasing in te value of frition oeffiient eause of te affle is very onsiderale, su tat Figure (a) sows tat for te ase of DH, tat te maximum frition oeffiient is aout 5 times of te one in te asene of affle. Comparison etween te urves plotted in Figure sows tat te value of maximum frition oeffiient wi is takes plae on te ottom wall elow te affle inreases sarply as te affle moves toward te BFS and also y inreasing in te affle eigt. In te previous setion, it was disussed tat te amount of irreversiility in a fluid flow an e estimated y omputing te total entropy generation in te flow domain. In order to sow te effet of affle on, Total Figure 9 sows te variation of total entropy generation in te flow domain against te Reynolds numer for tree different affle loations and also for te onvetion flow over BFS witout affle. It is seen tat te affle as a onsiderale effet on te total entropy generation, su tat te value of tis parameter inreases sarply y te affle, espeially as te affle moves toward te BFS and at ig values of Reynolds numer. In a similar manner, te variations of average Bejan numer wit Re for tree different affle loations and also for te onvetion flow over BFS witout affle are plotted Total Entropy Generation = = = Re Figure 9. Total entropy generation v.s Reynolds numer, θ =, /H =.5. in Figure. As it was mentioned efore, te Bejan numer is te ratio of entropy generation due to ondution to te total entropy generation. Figure sows tat Be dereases as te affle moves toward te BFS. Beause under tis ondition, te value of veloity gradient on te ottom wall inreases tat auses an inrease in te value vis. Besides it is seen tat te average Bejan numer dereases wit inreasing in Reynolds numer. Tis is in onsistent wit te teoretial findings y Au Nada []. 7. Conlusion Te present resear deals te seond law analysis of Copyrigt SiRes.

8 Average Bejan Numer = = = Re Figure. ariation of average Bejan numer wit Reynolds numer, θ =, /H =.5. onvetion laminar flow over inlined akward-faing steps in a dut wit a affle mounted onto te upper wall. Te set of equations governs to te fluid flow, eattransfer and entropy generation is solved numerially y CFD teniques in te Cartesian oordinate system using loked-off metod. Results sow tat altoug te affle enanes te onvetion oeffiient on te ottom wall, ut it auses a onsiderale inrease in te value of entropy generation. Su tat more irreversiilities take plae in onvetion flow over BFS as te affle moves toward te step, espeially for ig value of te affle eigt and at ig Reynolds numer. REFERENCES [] B. F. Armaly, F. Durst, J. C. F. Pereira and B. Sonung, Experimental and Teoretial Investigation of Bakward-Faing Step Flow, Journal of Fluid Meanis, ol. 7, 93, pp doi:.7/s339 [] G. C. radis,. Outgen and J. Sanez, Heat Transfer over a Bakward-Faing Step: Solutions to a Benmark, Benmark Prolems for Heat Transfer Codes, ASME HTD, Anaeim, 99, pp [3] D. W. Pepper, K. L. Burton and F. P. Bruenkner, Numerial Simulation of Laminar Flow wit Heat Transfer over a Bakward-Faing Step, Benmark Prolems for Heat Transfer Code, ASME HTD, Anaeim, 99, pp. -. [] G. radis and L. an Nostrand, Laminar Coupled Flow Downstream an Asymmetri Sudden Expansion, Journal of Termopysis Heat transfer, ol., No., 99, pp doi:.5/3.357 [5] L. Kaiktsis, G. E. Karniadakis and S. A. Orszag, Onset of Tree-Dimensionality Equiliria, and Early Transition in Flow over a Bakward-Faing Step, Journal of Fluid Meanis, ol. 3, 99, pp doi:.7/s93 [] N. Tylli, L. Kaiktsis and B. Ineien, Side Wall Effets in Flow over Bakward-Faing Step: Experiments and Numerial Solutions, Pysis of Fluids, ol., No.,, pp doi:.3/.53 [7] D. Brakely, M.Gariela, M. Gomes and R. D. Henderson, Tree-Dimensional Instaility in Flow over a Bakward-Faing Step, Journal of Fluid Meanis, ol. 73,, pp [] H. I. A. Mulawe, A Review of Resear on Laminar Mixed Convetion Flow over Bakward- and Forward-Faing Steps, International Journal of Termal Sienes, ol., No. 9, 3, pp doi:./s9-79(3)- [9] J. H. Nie, Y. T. Cen and H. T. Hsie, Effets of a Baffle on Separated Convetion Flow Adjaent to Bakward-Faing Step, International Journal of Termal Sienes, ol., No. 3, 9, pp. -5. doi:./j.ijtermalsi..5.5 [] E. Au Nada, Numerial Predition of Entropy Generation in Separated Flows, Entropy, ol. 7, No., 5, pp doi:.339/e73 [] E. Au Nada, Entropy Generation Due to Heat and Fluid Flow in Bakward-Faing Step Flow wit arious Expansion Ratios, International Journal of Exergy, ol. 3, No.,, pp doi:.5/ije..3 [] E. Au Nada, Investigation of Entropy Generation over a Bakward-Faing Step under Bleeding Conditions, Energy Conversion and Management, ol. 9, No., 9, pp doi:./j.enonman.7..3 [3] S.. Patankar, Numerial Heat and Mass Transfer and Fluid Flow, Hemispere Pulising Corporation, Wasington DC, 9. [] A. Bejan, Entropy Generation troug Heat and Fluid Flow, Wiley, New York, 9. [5] S.. Patankar and B. D. Spalding, A Calulation Proedure for Heat, Mass and Momentum Transfer in Tree- Dimensional Paraoli Flow, International Journal of Heat Mass Transfer, ol. 5, No., 97, pp doi:./7-93(7)95-3 Copyrigt SiRes.

9 Nomenlature a distane etween inlet setion and step (m) W widt of affle (m) Be Bejan numer (x, y) oordinates (m) Br Brinkman numer (,Y) dimensionless oordinates f oeffiient of frition Greek Symols ER expansion ratio (H/) termal diffusivity (m /s) H annel eigt (m) step inlined angle step eigt (m) dynami visosity (N s m ) affle eigt (m) density (kg m 3 ) k termal ondutivity (W m K ) dimensionless temperature parameter L lengt of te annel (m) non-dimensional temperature arateristi lengt (m) kinemati visosity (m /s) entropy generation numer visous dissipation numer Nu Nusselt numer Susripts P dimensionless pressure affle p pressure (Pa) ond ondution Pr Prandtl numer vis visous Re Reynolds numer w wall " s gen volume rate of entropy generation (W m 3 K ) m mean ulk value T temperature (K) T i inlet fluid temperature (K) T temperature of old wall (K) T temperature of ot wall (K) (u, v) x- and y-veloity omponents (m/s) (U, ) dimensionless x- and y-veloity omponents. inlet fluid veloity (m/s) Copyrigt SiRes.