ENTROPY GENERATION IN RECTANGULAR DUCTS WITH NONUNIFORM TEMPERATURE ON THE CONTOUR

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1 ENROPY GENERAION IN RECANGULAR UCS WIH NONUNIFORM EMPERAURE ON HE CONOUR Cesar A. Marcelino Matoso Instituto ecnológico de Aeronáutica Ezio Castejon Garcia Instituto ecnológico de Aeronáutica Sergio Mourão Saboya Instituto ecnológico de Aeronáutica Abstract. Rectangular tubes are widely used in eat transer devices. e objective o tis work is to investigate te eects o te entropy generation associated wit eat transer and luid riction by using a computational model. is investigation allows to evaluate te entropy generation wic is inluenced by te temperature distribution in te contour, luid low type, properties o te luid and geometric parameters. e study is made or laminar regime, wit developed ydrodynamic and termal proiles. First, te model calculates te velocity ield, ten te energy equation is solved obtaining te luid temperature ield or a given distribution imposed in te contour, not necessarily uniorm. Known te velocity and te temperature ields, te entropy generation per unit volume is calculated. e entropy generation per lengt unit is calculated by a double integration in te analysed section. e computational model consistency is veriied by te mes reining and by te tolerance decreasing. e metod validations used are done or te velocity and temperature ields calculations. Keywords. Entropy Generation, Forced Convection, Rectangular ucts 1. Introduction ermal analysis o lows inside rectangular ducts is a very important subject o researc in termal science and engineering. Literature about te subject is extremely wide, but tere are comparatively less Second Law based results reported. Second Law analysis is te basis o te so-called "Entropy Minimization Generation Metod" (EMGM). is metod basically consists in searcing situations in wic termal processes could be "less irreversible". Examples o suc studies are given in ejan (1979), ejan (198), ejan (1996), Gerdov (1996) and Saboya (00). e EMGM, tereore, looks or pysical and geometrical conigurations in termal systems tat render te inevitable irreversibilities associated to eat transer and. luid low to a minimum. is minimization leads to more eicient equipment design, allowing uel saving, lesser pollution, etc. e purpose o tis paper is te computation o entropy generation rates in ully termal and ydrodynamically developed laminar lows in ducts wit rectangular cross sections and nonuniorm temperature in wall perimeter. is computation gives te necessary inormation or minimization mentioned above. Heat excangers wit rectangular cross sections ducts are widely used in cemical processes industries air conditioning and lubrication equipment and oter applications, wic require compact eat excangers. e low, in muc o tese applications, occurs at low velocities and small dimensions, resulting in low Reynolds numbers. us, te low tends to be laminar and, because o tis, laminar lows will be studied in tis paper Figure (1) and Fig. () sow te scematic o te studied duct. L is te duct eigt, is its widt and te duct lengt is considered ininite. emperature distribution at te wall duct perimeter is variable. section A dp/ Figure 1. Perspective view o te rectangular duct studied.

2 axis Y surace 4 - (x,l) surace 1 - (0,y) surace 3 - (L,y) surace - (x,0) L axis X Figure. uct cross-section and variable wall temperature distribution scematics.. Matematical Formulation.1. Velocity Field e low is considered steady, laminar wit a ully developed velocity proile. e work due to viscous tensions is neglected. ereore, te only nonzero velocity component is in te longitudinal direction. Let "u" te longitudinal velocity, "µ" te dynamical viscosity and "dp/dx" te axial gradient pressure (constant). us, te momentum equation is: u u y 1 dp. µ (1) e boundary conditions are zero velocity at te wall (nonslip condition). It is useul to deine te mean velocity at section wit area "A" (see Fig. (1)): 1 U. u. dx. dy A () Equation (1), wit its boundary conditions, was solved numerically using a inite dierence metod. etails o te numerical procedures can be ound in Matoso (1998) and Garcia (1996)... emperature ield e termal proile is considered developed. Heat conduction in te low direction is considered muc smaller tan tat in te transversal direction. Natural convection is also neglected. ecause te velocity proile is ully developed te transversal low velocities are zero. Hence, te energy equation is: y u. a z (3) were " " and "a" are te luid temperature and diusivity, respectively. e luid bulk temperature is: 1. u.. dx dy AU.. (4) Wit te purpose o deining a dimensionless variable tat will become te termal proile invariant in te longitudinal direction (caracteristic o developed proiles) it is necessary beore to deine te mean temperature on te duct wall:

3 1 ( y) dy ( x ) dx ( ) ( ) L. L L 0,. 1 y dy x L dx., L.,. 0 3., (5) were 1,, 3 e 4 are te temperatures on te suraces 1,, 3 and 4, respectively, as sown in Fig. 1. As te wall temperature distributions are invariant in te longitudinal direction " W " is constant in tat direction. e luid will be eated up or will be cooled asymptotically. ereore te present ormulation will be similar to tat ound in Clark and Kays (1953) tat studied rectangular ducts wit uniorm temperatures bot in te duct cross section and in te longitudinal direction. Making a similar development as presented in te quoted reerence, Eq. (3) becomes: y u. a d. (6) o complete te ormulation urter deinitions o dimensionless variables will be needed and it will be presented now: x X (7) y Y (8) [ ] d..( ) φ (9) U were " " te ydraulic diameter or te perimeter P e. e deinition o " " is: 4A P e. L. L (10) Substituting Eq. (7) to Eq. (10) into Eq. (6) te dimensionless energy equation is obtained: φ φ X Y u U. φ φ (11) In Equation (11) U φ is te value o φ, deined in Eq. (9), or replaced by, tat is: [ ] d..( ) φ (1) Equation (1) may be rewritten as: d or ( ) (13) U..φ. U d.. φ (14) a Equation (9) gives:

4 d φ. U..( ) (15) a Substituting Eq. (15) into Eq. (4) results: d.. u. φ. dx dy (16) A. Substituting Eq. (16) in Eq. (1), and using Eq. (7) and Eq. (8), it is obtained te dimensionless orm o te bulk temperature: φ udxdy (17) AU e dimensionless boundary conditions are: [ 1 ] φ( 0, Y) d U..( ) [ ] ( X,0) U..( φ (19) d ) [ ] (, Y ) U..( φ (0) 3 d [ ] ( X, L ) U..( 4 d ) ) φ (1) ese boundary conditions present a special case tat it sould be noted. I te temperatures on te walls are constant and te same, Eq. (18) to Eq. (1) are equal to zero and tey will not be unctions o, wic is an unknown. is simpliied case as been studied by Patankar (1991). Equation (11), Eq. (1) and Eq. (13), wit te boundary conditions, orm a dierential equation system wose unknowns are φ and. e determination o tese unctions allows te computation o te luid temperature, wic is needed to calculate te entropy generation. 3. Entropy Generation e entropy generation rate per unit volume is calculated by means o (ejan, 198): k u u S& µ ger () y z y y According to Clark and Kays (1953): d (18) d d (3) Hence: k d u u S& µ ger (4) y x y

5 Integrating over te duct cross-section te entropy generation rate per unit lengt: S& S& ger ger dxdy (5) 4. Numerical Metod and Computational Model e dierential equation system described in sections and 3 was solved numerically using a inite dierence metod. e coupling presented between equations and boundary conditions in te eat transer section o tis system requires an iterative treatment tat will be described bellow. Wit guessed values o φ and d / te dimensionless orm o energy equation, Eq. (11), is solved. en, an improved value o φ is calculated using Eq. (17), orming a irst iterative loop, as it is sown in Fig. (3). Ater te convergence o φ a better value o d / is calculated by Eq. (13). e boundary conditions are ten recalculated and te energy equation is again solved, orming a second iterative loop (Fig. (3)).Wen convergence is acieved, is computed using Eq. (15). Hence, te necessary inormation to determine te entropy generation rates, Eq. 4 and Eq. 5, is obtained. is procedure is scematized in Fig. (3). In tis igure "tol" represents te convergence criteria or d /" and or te dimensionless temperature ield. Start Velocity Field Mean Velocity oundary Conditions imensionless emperature Field >tol >tol Ø <tol d / <tol Entropy Generation Figure 3 - Computational Model Flowcart. 5. Results and iscussion able presents te input parameters o a solution example obtained using te computational model described. e luid is air. able 1. Input parameters End escription Value Unity uct eigt 0.01 m uct widt 0.01 m Fluid termal conductivity W/m 0 C Fluid density Kg/m 3 Fluid termal diusivity.x10-5 m /s Pressure gradient 16.0 Pa/m Fluid dynamic viscosity 1.853x10-5 Ps

6 Figure (4) sows two variable wall temperature distributions investigated. e temperature as been considered as a sine unction o te position on te wall. is unction was cosen because it represents well te numerical metod capacity to deal wit several type boundary conditions. istribution A istribution Figure 4. Wall temperature distributions. Figure (5) sows te velocity ield. is ield is te same or bot temperature distributions because te velocity ield does not depend on temperature. Figure 5 - Velocity Field. Figure 6 presents te temperature distributions obtained. It is observed te strong termal boundary condition inluence in te temperature distributions.

7 istribution A istribution emperature (K) Figure 6 - emperature Field. For convenience te entropy generation rate will be presented in two parts. e irst one is te entropy generation due to viscous low. It corresponds to te last bracket in Eq. (4). e oter bracket represents te entropy generation rate caused by eat transer. e entropy generation results due viscous low are presented in Fig. (7). It is observed tat tere is little dierence between te proiles resulting rom te two distributions. istribution A istribution Entropy Generation Figure 7. Entropy generation per lengt unit due to viscous low. e corresponding results or entropy generation rate due to eat transer are in Fig. (8). Here, as it was expected,, tere is a strong inluence rom luid temperature distributions.

8 istribution A istribution Figure 8 - Entropy generation rate per unit lengt due to eat transer. Entropy Generation Comparing Fig. (7) and Fig. (8) it is seen tat tere is little contribution o viscous low to te total entropy generation rate. is is also expected and it is almost a rule in laminar lows. 6. Conclusions A computational metod as been built or entropy generation rates in ully developed laminar lows in rectangular ducts. e metod allows te imposition o any temperature distribution on duct wall. e model proved to be numerically stable and consistent and it is intended to be a tool in engineering design, using te EGMM approac, o termal systems. 7. Reerences ejan, A.,1979, "Entropy Generation in Fundamental Convective Heat ranser", Journal o Heat ranser, Vol. 101, pp ejan, A., 198 "Entropy Generation troug Heat and Fluid Flow", Jon Wiley, New York, USA, 48 p. ejan, A., 1996, "Entropy Generation Minimization", CRC Press, oca Raton, USA, 36 p. Clark, S. H.and Kays, W. M.,1953, "Laminar-Flow Forced Convection in Rectangular ubes", rans. ASME, Vol.10, pp Garcia, E. C., 1996, "Condução, Convecção e Radiacão Acopladas em Coletores e Radiadires Solares", ese de outoramento, IA, São José dos Campos, rasil. Gerdov, G., 1996, "Second Law Analysis o Convective Heat ranser in Flow roug a uct wit Heat Flux as a Function o uct Lengt", HVAC&R Researc, vol N o, pp Matoso, C. A. M., 1998,"Geração de Entropia em uto Retangular com emperatura Variavel no Contorno", ese de Mestrado, IA, Sào José dos Campos, rasil. Patankar, S. V., 1991, "Computation o Conduction and uct Flow Heat ranser", Innovative Researc, Maple Grove, USA, 354 p. Saboya, S. M., 00, "Análise aseada na a Lei da ermodinâmica da ranserência de Calor e do Escoamento urbulento em utos Anulares Pinados", Proceedings o te 19t razilian Congress o ermal Engineering and Sciences", Caxambu, razil, paper CI