Entropy ISSN

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1 Entrop 3, 5, Entrop ISSN Entrop Generation Dring Flid Flow Between wo Parallel Plates With Moving Bottom Plate Latife Berrin Erba 1, Mehmet Ş. Ercan, Birsen Sülüş 3, M. Mrat Yalçın 4 1 Osmangazi Universit, School of Engineering and Architectre 648 Batı Meselik, Eskişehir rke. el: , Fa: , lberba@og.ed.tr Ford - Otosan İnönü Plant, 614 Eskişehir rke. mercan@ford.com.tr 3 Osmangazi Universit, School of Engineering and Architectre 648 Batı Meselik, Eskişehir rke. el: , Fa: , bsls@og.ed.tr 4 EI - USAŞ Engine Indstries, Inc. Çevre Yol Eskişehir rke. mrat.alcin@tei-tr.com Received: Jl 3 / Accepted: 31 December 3 / Pblished: 31 December 3 Abstract: wo dimensional nmerical analsis of entrop generation dring transient convective heat transfer for laminar flow between two parallel plates has been investigated. he flid is incompressible and Newtonian and the flow is the hdrodnamicall and thermall developing. he plates are held at constant eqal temperatres higher than that of the flid. he bottom plate moves in either parallel or in inverse direction to the flow. he governing eqations of the transient convective heat transfer are written in two-dimensional Cartesian coordinates and solved b the finite volme method with SIMPLE algorithm. he soltions are carried for Renolds nmbers of 1, 51 and 1 3 and Prandtl nmber of 1. After the flow field and the temperatre distribtions are obtained, the entrop vales and the sites initiating the entrop generation are investigated. he reslts have indicated that the nmber of the entrop generation has its highest vale at the highest Renolds and Br/Ω vales, which is obtained at conter motion of the lower plate. he lowest average nmber of the entrop generation on the bottom plate is obtained in parallel motion. he corners of the channel plates at the entrance pla the role of active sites where the generation of entrop is triggered. Kewords: Laminar flow, parallel plate, entrop generation

2 Entrop 3, 5, Nomenclatre Br Brinkman nmber D hdralic diameter, m h film coefficient, W/m -K H height, m k thermal condctivit, W/m-K L length, m N s dimensionless entrop generation nmber N SL average dimensionless entrop generation N Nsselt nmber N L average Nsselt nmber P dimensionless pressre P pressre, N/m Pr Prandtl nmber Re Renolds nmber S & entrop generation, W/m 3 -K gen temperatre difference, K dimensionless temperatre in inlet temperatre, K wall wall temperatre, K t temperatre, K time, s dimensionless horizontal velocit component v v horizontal velocit component, m/s inlet velocit, m/s dimensionless vertical velocit component vertical velocit component, m/s, coordinates, m, dimensionless coordinates Greek Letters α thermal diffsivit, m /s β coefficient of thermal epansion, 1/K ρ dimensionless densit ρ reference densit, kg/m 3 ρ densit, kg/m 3 ν kinematic viscosit, m /s τ dimensionless time Φ viscos dissipation fnction, s - µ dnamic viscosit, N-s/m φ irreversibilit distribtion fnction Ω dimensionless temperatre difference Introdction One of the fndamental flow geometries encontered in engineering processes is the channel between two parallel plates. When a viscos flid flows in a parallel plate channel, a velocit bondar laer develops along the inner srfaces of the channel. If the plates have different thermal conditions the heat transfer starts from the inlet of the channel and temperatre profile develops simltaneosl along the inside dct srfaces. he convection is treated as the combined hdrodnamic and thermal entr length problem which can be also referred to as the simltaneosl developing region problem in the literatre. Althogh the analsis of the heat transfer in sch a flow sstem is more comple de to the variation of the velocit distribtion in all directions, the problem has attracted attention and investigators have given nmerical soltions nder varios constraints. A comprehensive review of sch soltions is addressed b Kas and Crawford [1], Kakaç and Yener [,3] and Bejan [4]. he improvement of thermal sstems has gained a growing interest de to the relations with the problems of material processing, energ conversion and environmental effects. Efficient energ tilization dring the convection in an flid flow is one of the fndamental problems of the

3 Entrop 3, 5, engineering processes to improve the sstem. One of the methods sed for the prediction of performance of the engineering processes has been the second law analsis. he second law of the thermodnamics is applied to investigate the irreversibilities in terms of the generation of entrop. he method was introdced b Bejan [5-7] and followed b man investigators. Stdies on the entrop generation rates and the irreversibilities for the basic convective heat transfer arrangements can be fond in the literatre [8-11]. Since the entrop generation is the measre of the destrction of the available work of the sstem, the determination of the active sites motivating the entrop generation is also important in pgrading the sstem performances. De to the interest to the efficient tilization of energ and the wide applications of the flows in the parallel plate channel, related stdies can be fond in the literatre. Nag and Kmar [1] carried ot a second law analsis on convective heat transfer from a flid flowing in a dct with constant heat fl and fond an optimm vale of the initial temperatre difference for minimm entrop generation. he thermodnamics of laminar viscos flow throgh a dct sbjected to constant heat fl was stdied analticall b Şahin [13]. o determine the optimm dct geometr, Şahin [13] sed entrop generation for laminar viscos flow throgh varios geometries sbjected to constant heat fl. He considered hdralic diameters for sqare, eqilateral trianglar, rectanglar and sinsoidal geometries and made comparisons with respect to the dimensionless entrop generation and the pmping power to heat transfer ratio with respect to Renolds nmber. he entrop generation in trblent liqid flow throgh a smooth dct sbjected to constant temperatre was stdied b Şahin [14]. Narsawa [15] eamined the rate of entrop generation both theoreticall and nmericall for forced and mied convection in a rectanglar dct heated at the bottom. Şahin [16] made a thermodnamic analsis of trblent flid flow throgh a smooth dct sbjected to constant heat fl and considered the temperatre dependence of the viscosit. Mahmd and Fraser [17] investigated analticall the first and the second law characteristics of flid flow and heat transfer inside a channel having two parallel plates with finite gap between them and considered fll developed forced convection and non- Newtonian flid. Mahmd and Fraser [18] analzed the mechanism of entrop generation and its distribtion throgh flid flows in basic channel configrations inclding two fied plates and one fied and one moving plates b considering simplified or approimate analtical epressions for temperatre and velocit distribtions and derived analticall general epressions for the nmber of entrop generation and Bejan nmber. In this std, the second law analsis is applied to the simltaneosl developing transient laminar flow between two parallel plates. he irreversibilities within the channel are presented b the entrop generation nmber. he transient soltions of the set of the governing eqations for mass, momentm, energ, and entrop generation for two-dimensional Cartesian coordinates are obtained nmericall. Special attention has been given to the effect of the motion of the bottom plate on the entrop generation b considering the Prandtl nmber of 1. and the Renolds nmbers 1, 5 and 1. hree cases were considered at the bottom plate; stationar plate, moving plate in the parallel and reverse directions with the flow. he plates are held at eqal constant temperatre.

4 Entrop 3, 5, he reslts of the std provide valable fndamental information on the phsics of the simltaneosl developing transient laminar convection in a parallel plate channel with moving bottom plate to improve the corresponding engineering applications. he designers nder the responsibilities for the design and optimization of corresponding thermal sstems can emplo the reslts given abot the entrop generation to redce the loss of available work. Mathematical Formlation he phsical sstem nder consideration in the present std is shown schematicall in Fig.1. he origin of the coordinate sstem is placed at the left corner of the lower plate. he channel is composed of two parallel plates separated b a distance H in -direction. he length of the channel is L1H throgh -ais. he channel walls are at the same niform temperatre and impermeable no-slip bondaries. he phsical properties of flid are taken as constant. 1 v P not known (,), v, 1, P not known H 1 or -1, v, 1, P not known not known P not known Figre 1. he schematic of the phsical sstem. L he initial, inlet, otlet and the bondar conditions are given b, v, at τ 1, v, P not known, at inlet v,, P not known, not known at otlet, 1, -1, v, P not known, 1 at bottom plate, v, P not known, 1 at top plate Non-dimensional governing eqations are obtained as ρ ( ρ ) ( ρ v) τ ( ρ ) ( ρ ) ( ρ ) τ v ( ρ v) ( ρ v) ( ρ v) τ v v τ 1 RePr P P 1 Re 1 Re v v (1) () (3) (4)

5 Entrop 3, 5, where ν D Re, α ν Pr, D, D, D t τ, v v, ρ ρ ρ P P ρ, in wall in (5) where the hdralic diameter D becomes twice the plate spacing. he local entrop generation eqation is given b following Bejan [7] as in gen k S & µ in v v (6) B sing the same dimensionless parameters given in Eq.(5), Eq.(6) takes the following dimensionless form s N φ v v (7) where gen s k D S N Ω &, Ω φ Br, k Br µ in Ω, in wall (8) Soltion And Benchmarking he Reslts he present combined hdrodnamic and thermal entr length problem is solved b the finite volme method with SIMPLE algorithm. he soltions are started from qiescent conditions proceeded throgh the transient p to the stead-state case. A compter program was developed and benchmarked with the velocit and the thermal bondar laers for the soltions obtained b those of reported b Kakaç and Yener [] nder the case of fied channel. A grid sensitivit analsis was carried ot. he grid strctre of 1 and the dimensionless time step of.5 were sed. he rectanglar side lengths are eqall divided. his choice was determined from the vales of N nmber for the sstem. he present and benchmark reslts are smmarized in able 1. As the nmber of grid elements was increased, the convergence of the N nmber was observed. otal Nsselt nmber is calclated b sing the average Nsselt nmbers for the bottom and the pper plates, as follows L H L d L 1 N (9)

6 Entrop 3, 5, able 1. he Smmar of the present and benchmark reslts for Nsselt vales Grid Calclated Nsselt nmber Benchmark Nsselt nmber vales [19] % Ma Deviation he average nmber of entrop generation is obtained over at all vales b sing L 1 N SL N sd (1) L Reslts And Discssion he irreversibilit contors have been derived b sing Eqation (7) concerning the effect of the Renolds and Prandtl nmbers. he std is restricted to the flids of Prandtl nmber 1. and stdied for Renolds (Re) nmbers of 1, 51 and 1 3. he effect of Br/Ω on the entrop generation in the combined entrance length was investigated b solving the problem for Br/Ω.1, 1, and 1. Limited nmber of contors was sed to describe the strctre of the flow and temperatre fields. he variations of the average nmber of entrop generation were drawn on the N SL - coordinates for eas nderstanding. Figre illstrates the time vise variation of the velocit contors for each case corresponding to τ1, 3, 5, and the stead-state considering the Renolds nmber 1. he formation of the smmetric strctre observed with respect to mid-plane is onl observed in the fied plate case as it is seen in Fig.a. At the entire flow channel the stratified strctre of the velocit contors are observed. When the lower plate moves in parallel to the flow direction, the contors place heavil inside the pper plate. he maimm is obtained below the smmetr ais at the lower half of the channel as seen in Fig.b. Fig.c smmarizes the effect of conter motion of the lower plate on the velocit distribtion. he dense strctre of the velocit contors on the inner srface of the lower plate ields the reslt of increased friction, high shear stresses, and hence distrbed smmetr.

7 Entrop 3, 5, τ1 τ3 τ5 Stead state (a) (b) (c) Figre. ransient change in the contors of velocities from τ1 p to stead-state for the cases of lower plate as a) stationar, b) moving in parallel and c) inverse direction to the flow (Re1 ). τ1 τ3 τ5 Stead state (a) (b) (c) Figre 3. ransient change in the contors of isotherms from τ1 p to stead-state for the cases of lower plate as a) stationar, b) moving in parallel and c) inverse direction to the flow (Re1 ).

8 Entrop 3, 5, stationar parallel motion inverse motion 5 stationar parallel motion inverse motion (a) (b) Figre 4. Local Nsselt nmbers in the channel for a) lower and b) pper plate at stead-state. he distribtion of the isotherms considering all cases at initial steps of transient process for τ1, 3, 5, and at the stead-state are shown in Fig.3. he combination of Pr1. and Re1 is sed. At the initial step, thermal bondar laer is created immediatel on the inner srfaces of the plates having the same characteristic distribtion at all cases. he isotherms remain attached on the inner plate srfaces throghot the process. At the stationar case given in Fig.3a, the imposed thermal bondar condition indces smmetric thermal bondar laers along the inner srfaces of the parallel plate channel, and covers the entire channel in time. Fig.3b ields the strctre of the contors heavil inside the pper plate in the flow direction de to the effect of velocit profile obtained at the parallel motion of the lower plate. It seen in Fig.3c since the flows in the case of conter motion of the bottom plate create high shear stresses and enlarge the velocit profile inside the channel, the flow tend to lift the isotherms far from the lower plate. he average Nsselt nmbers, N L, at the stead- state are calclated 7.866, 7.871, and 7.63 for the parallel plate channel with stationar, forward and backward moving lower plates, respectivel (see Fig.4). Figre 5 shows the contors of the predicted entrop generation nmbers at different time steps for the combination of Renolds nmber 1 and Br/Ω1. he effect of transient changes p to the stead state is observed on the entrop generation nmber. he figre ields clearl the place of the active cites representing the spots of triggering irreversibilities in the parallel plate channel nder consideration. It is seen that the strctre of the contors of the entrop generation nmber follows the velocit and temperatre contors and is affected b the transient changes in the combined developing region. Fig.5 ields a general idea abot the active cites representing the spots of triggering irreversibilities in the channel nder consideration. he maimm vale of N SL is 45 at the entrance corners of the channel plates at the fied case. For the motion parallel in the flow direction the maimm N SL is 5 at the entrance corner of the moving bottom plate whereas 45 at the pper entrance. Dring the conter motion of the lower plate the maimm entrop generation nmber is obtained as 115 at the entrance corner of the plate.

9 Entrop 3, 5, τ1 τ3 τ5 Stead state (a) (b) (c) Figre 5. ransient change in the contors of entrop generation from τ1 p to stead-state for the cases of lower plate as a) stationar, b) moving in parallel and c) inverse direction to the flow (Re1 and Br/Ω1). N s d/d N s d/d N s friction N s (a) (b) (c) Figre 6. he components of the entrop generation at stead-state for the cases of lower plate as a) stationar, b) moving in parallel and c) inverse direction to the flow (Re5 and Br/Ω1).

10 Entrop 3, 5, Figre 6 demonstrates the partitioning of the nmber of entrop generation according to the sorces of irreversibilities. he contors of N s d/d accont the entrop generation de to the aial condctive heat transfer. he contors of N s d/d represent the entrop generation de to the heat transfer in normal direction. he entrop generation vales calclated from the terms in the second bracket on the RHS of the Eq.(7) illstrate the contribtion of flid friction, N s friction which is presented at the third line of the Fig.6. he contors of N s friction stretch along the inside srface of the pper and the lower plates in the similar manner with the velocit contors. he N s friction vales are ver small nder the present conditions. At the last lines of the Fig.6 (a), (b), and (c), the contors of the total vale of the entrop generation nmber are given. heir vales, the last line of the Fig.6, is mch the same with the reslts shown in the second line. Figre 7 smmarizes the effect of Br/Ω on the problem considered. It is seen that an increase in the Brinkman nmber which determines the relative importance between dissipation effects and flid condction ield strong increase in the entrop generation nmber. he relative motion of the lower plate with respect to the pper plate cases a deviation from the smmetric strctre of N SL. he shifted profile throgh pper plate is obtained in the case of parallel motion. he highest vales of the nmber of entrop generation are realized in the last case. Figre 8 is prepared to show the effect of Renolds nmber on the entrop generation dring the developing flow nder consideration. hree vales of the Renolds nmber 1, 51, and 1 3 are considered for Pr1 and Br/Ω 1. he nmber of the entrop generation increases with increasing Renolds nmber. he strong effect of Re nmber is realized on the inside srfaces of the channel plates. In Fig.8a, the smmetric distribtion is clearl observed at the fied channel case. he parallel motion of the bottom plate not onl redced the highest N SL vale bt also decreased the entrop generation on the pper plate (Fig 8b) with respect to fied case. It is seen in Fig.8c the highest vale of N SL is obtained at the highest Re nmber on the srface of the lower plate moving in conter direction to the flow. he obtained reslts addressing all cases are as follows: he formation of the parallel strctre of the entrop generation as well as the velocit and thermal bondar laers is obtained at the stationar plate case. Close thermal laers are fond on the heated srfaces. he strctre of the N s contors keeps the same distribtion at the transient steps. he N s has its highest vale at the highest Re and Br/Ω vales, which is obtained at conter motion of the lower plate. he lowest average nmber of the entrop generation on the bottom plate is obtained at the case of the plate moving in parallel direction to the flow. he pper and the lower corners of the channel plates at the entrance pla the role of active sites where the generation of entrop is triggered.

11 Entrop 3, 5, Br /Ω1 Br /Ω1 Br /Ω.1 15 Br/Ω1 Re1 Re5 Re (a) (a) Br/Ω1 Br/Ω1 Br/Ω.1 15 Br/Ω1 Re1 Re5 Re (b) (b) Br /Ω1 Br /Ω1 Br /Ω Br/Ω1 Re1 Re5 Re (c) Figre 7. he effect of irreversibilit distribtion fnction in terms of Br/Ω on the average entrop generation nmber for the cases of lower plate as a) stationar, b) moving in parallel and c) inverse direction to the flow (Re5, stead-state) (c) Figre 8. he effect of Renolds nmber on the average entrop generation nmber for the cases of lower plate as a) stationar, b) moving in parallel and c) inverse direction to the flow (Br/Ω1, stead-state).

12 Entrop 3, 5, Conclsion De to new challenges eisting continosl in each field of technolog entrop generation dring convective heat transfer becomes a distinct interest for the design engineers. he present std investigates the entrop generation indced b the transient laminar forced convection in the combined entrance region between two parallel plates. Considering the importance of entrop generation in terms of the energ loss, the sstem is evalated for epecting the reslts to be helpfl in pgrading the sstem performances. References 1. Kas, W.M. and Crawford, M.E. Convective Heat and Mass ransfer, McGraw-Hill Book C., nd ed.; (ISBN ) Kakaç, S. and Yener, Y. Convective Heat ransfer, CRC Press, nd ed. (ISBN ) 1995; Chapter 8, p Kakaç, S. and Yener, Y. Laminar Force Convection in the Combined Entrance Region of Dcts, Hemisphere Pbl. Corp., In Low Renolds Nmber Flow Heat Echangers, Kakaç, S.; Shah, R.K.; Bergles, A.E., Ed.; (ISBN ) 1995; pp Bejan, A. Convective Heat ransfer, John Wile & Sons. Inc., nd ed. (ISBN ) 1995; Chapter Bejan, A. Second law analsis in heat transfer. Energ - he Int. J. 198, 5, Bejan, A. Entrop Generation Minimization, CRC Press: USA, Bejan, A. Entrop Generation hrogh Heat and Flid Flow, John Wile & Sons. Inc.: Canada, 1994; Chapter 5, p Krane, R. J. A. Second law analsis of the optimm design and operation of thermal energ storage sstems. Int. J. Heat Mass ransfer 1987, 3, Arpacı, V. S. Radiative entrop prodction - lost heat into entrop. Int. J. Heat Mass ransfer 1993, 36, satsaronis, G. Design optimization of thermal sstems sing eerg - based techniqes. Proc. In Second Law Analsis: owards the 1 st Centr, Scibba, E. and Moran, M. J., Ed.; Roma, 1995; pp Erba, L.B.; Altaç, Z.; Sülüş, B. Entrop Generation in a Sqare Enclosre Heated From a Vertical Lateral Wall. Proceedings of the 15 th International Smposim on Efficienc, Costs, Optimization, Simlation and Environmental Aspects of Energ Sstems: ECOS, Jl 3-5, satsaronis, G.; Moran, M.J.; Cziesla, F.; Brckner,.,Ed.; Berlin, German, ; Vol.III, pp Nag, P.K. and Kmar, N. Second law optimization of convective heat transfer throgh a dct with constant heat fl. Int.J.Energ Research 1989, 13 (5),

13 Entrop 3, 5, Şahin, A.Z. Irreversibilities in varios dct geometries with constant wall heat fl and laminar flow. Energ,he International J. 1998, 3 (6), Şahin, A.Z. Entrop generation in trblent liqid flow throgh a smooth dct sbjected to constant wall temperatre. Int. J. Heat and Mass ransfer, 43, Narsawa, U. he second law analsis of mied convection in rectanglar dcts. Heat and Mass ransfer 1, 37, Şahin, A.Z. Entrop generation and pmping power in a trblent flid flow throgh a smooth pipe sbjected to constant heat fl. Eerg, an International Jornal,, Mahmd, S. and Fraser, R. A. hermodnamic analsis of flow and heat transfer inside channel with two parallel plates, Eerg, an International Jornal,, Mahmd, S. and Fraser, R. A. he second law analsis in fndamental convective heat transfer problems. Int. J. of hermal Sciences, 4 (), Genceli, O. Çözümlü Isı aşınımı Problemleri, Birsen Yaınevi, İstanbl, (ISBN ) ; Bölüm 3, s b MDPI. ( Reprodction for noncommercial prposes permitted.