Entropy ISSN

Entropy 006, 8[], 50-6 50 Entropy ISSN 099-4300 www.mdpi.org/entropy/ ENTROPY GENERATION IN PRESSURE GRADIENT ASSISTED COUETTE FLOW WITH DIFFERENT THERMAL BOUNDARY CONDITIONS Abdul Aziz Deprtment of Mechnicl Engineering, Gonzg University, Spokne, WA 9958, USA. Emil:ziz@gonzg.edu Received: 9 Februry 006 / Accepted: 3 My 006 / Published: 4 My, 006 Abstrct: The present pper is concerned with n nlyticl study of entropy genertion in viscous, incompressible Couette flow between sttionry plte nd moving plte. The flow induced by the moving plte is ssisted by constnt pressure grdient long the flow direction. Four different combintions of therml boundry conditions re investigted: () pltes t different tempertures, (b) sttionry plte t fixed temperture nd moving plte subjected to constnt het flux, (c) sttionry plte t fixed temperture nd convection t the moving plte, nd (d) convection t both pltes. Besides the velocity nd temperture profiles, dimensionless results re presented for the entropy generted due to het trnsfer, the entropy generted due to viscous dissiption, nd the totl entropy genertion. These results illustrte the effect of pressure grdient, temperture symmetry, het flux, convection Biot numbers, nd mbient tempertures. For certin combintions of therml vribles, the totl entropy generted is minimized. Keywords: Entropy, Couette flow, pressure grdient, therml boundry conditions. Introduction Since the pioneering work of Bejn [] on entropy genertion in convective het trnsfer, it is now widely recognized tht convective het trnsfer problems tht were previously studied using the first lw of thermodynmics be reexmined in the light of the second lw of thermodynmics so tht

Entropy 006, 8[], 50-6 5 therml systems cn be designed with the objective of minimizing thermodynmic irreversibility. This design methodology, known s entropy genertion minimiztion (EGM), is comprehensively covered in the book by Bejn []. The populrity of EGM cn be guged from the number of ppers on entropy genertion in convective flows tht continue to pper in the het trnsfer literture e.g. [3, 4]. We will refer only to the literture tht dels with entropy genertion for flow between prllel pltes which is of immedite relevnce to the present study. The flow of viscous, incompressible flow induced by plte moving prllel to sttionry plte nd ssisted by fvorble pressure grdient sitution, known s generlized Couette flow,is useful in mny engineering pplictions nd hs been studied extensively both from fluid mechnics nd het trnsfer perspectives. However, studies of entropy genertion in such flows hve been limited. Ibnez et l [5] hve studied the entropy genertion due to pressure grdient ssisted viscous flow between two prllel sttionry pltes when the pltes re subjected to convective boundry conditions. Their nlysis ssumes tht the convective environment temperture is the sme for both pltes but the het trnsfer coefficients nd consequently the Biot numbers re different. They concluded tht minimum volumetric entropy genertion rte cn be chieved for certin combintions of Biot numbers. This work ws lter extended to the generlized Couette flow [6]. The present pper is n nlyticl study of entropy genertion in generlized Couette flow with focus on the effect of therml boundry conditions imposed on the pltes. Four different combintions of therml boundry conditions re investigted: () both pltes t different tempertures, (b) sttionry plte t fixed temperture nd moving plte subjected to constnt het flux, (c) sttionry plte t fixed temperture nd convection t the moving plte, nd (d) convection t both pltes. The cse (d) is refinement of the nlysis presented in [6] becuse it llows both the environment tempertures nd the convective het trnsfer coefficients to be different. Anlyticl results for the velocity nd temperture profiles, entropy genertion due to het trnsfer, entropy genertion due to viscous dissiption, nd the totl entropy genertion re presented nd discussed. Hydrodynmic nd Therml Anlysis Fig. illustrtes the Couette flow of viscous, incompressible fluid confined between two prllel pltes seprted by distnce. The fluid viscosity, µ, is ssumed to be constnt. The bottom plte is sttionry while the top plte moves with uniform V. The fluid flow is lso ssisted by fvorble pressure grdient, dp/dx <0. V y=/ moving plte y y=-/ sttionry plte Fig.. Couette flow between prllel pltes

Entropy 006, 8[], 50-6 5 The momentum eqution for the velocity u long the x direction my be written in dimensionless form s d U + P = 0 () dy dp where U = u / V, Y = y /, nd P = is the dimensionless pressure grdient. μv dx The boundry conditions for eqution () re Y = /, U = 0 () Y = /, U = (b) The solution of equtions (-) is esily obtined s U = + Y + P( 4Y ) (3) 8 The derivtive of U with respect to Y which will be needed lter in the entropy genertion nlysis is du = PY (4) dy With the inclusion of het conduction in the y direction nd the viscous dissiption term, the energy eqution in dimensionless form is expressible s d du + = 0 (5) dy dy where = kt / μv is the dimensionless temperture nd k is the therml conductivity of the fluid. The use of dimensionless temperture in this form elimintes the Eckert number Ec nd Prndtl number Pr which pper in trditionl nlyses. The solution of eqution (5) depends on the therml boundry conditions imposed t the pltes. We consider the solutions for four combintions of boundry conditions. Constnt Pltes Tempertures The sttionry plte is ssumed to be t constnt temperture nd the moving plte t constnt temperture T. In terms of, the boundry conditions become Y = /, = (6) Y = /, = (6b) The solution of equtions (5, 6) is P P P 3 P 4 = ( + ) + Y Y + Y Y 8 + + 4 (7) 3 The derivtive of eqution (7) with respect to Y which will be needed lter in entropy genertion nlysis is d P P 3 = Y + PY Y (8) dy T

Entropy 006, 8[], 50-6 53 Constnt Temperture t Sttionry Plte nd Constnt Het flux t the Moving Plte In this cse, the boundry condition (6b) chnges to d Y = /, = Q (9) dy whereq = q / μv is the dimensionless het flux. q is the het flux t the moving plte. The solution of eqution (5) subject to equtions (6) nd (9) is 4 3 5P 6P + 7P + 6 + 9P + 96P Q = 9P 3 4 P 6P + P 8 + 4PQ ( PY ) ) + Y (0) 9P P Constnt Temperture t Sttionry Plte nd Convection t the Moving Plte In this cse, the boundry condition t the moving plte my be written s d Y = /, + Bi( ) = 0 () dy where in Biot number Bi = h / k, h is the convection het trnsfer coefficient nd in = kt / μv, T is the convection environment temperture. The solution of eqution (5) subject to equtions (6) nd () is = [ ] 4 3 ( PY ) P P ( Bi + 3) + P( Bi + Bi ) 8( Bi + ) + P 4P( + Bi) Y + 3 ( Bi + 5) 6P + 4P ( 4Bi + 4Bi + Bi + 8 + 3) + 6( + Bi) 9P ( Bi + ) 4 P Convection t both pltes () The boundry conditions for this cse re d Y = /, + Bi (, ) = 0 (3) dy d Y = /, + Bi (, ) = 0 (4) dy where Bi = h / k, Bi = h / k,, = kt, / μv,, = kt, / μv nd h, T, nd h, T, re the het trnsfer coefficients nd convection environment tempertures t the sttionry nd moving pltes, respectively. The solution of eqution (5) subject to the boundry conditions (3) nd 4) is 4 3 = P Y + PY Y + CY + C (5) 3

Entropy 006, 8[], 50-6 54 where C = P ( Bi + Bi ) + P( 3Bi 3Bi Bi Bi ) + ( Bi + Bi ) + 4Bi Bi ( ) 4 ( Bi + Bi Bi Bi ), (6), C = P ( 5Bi 5Bi + BiBi 6) 6P( Bi + Bi ) + 7( Bi Bi ) 9( Bi + Bi Bi Bi ) BiBi + ( 4 4 + ) + 8( ( Bi Bi )) + 3( Bi Bi ),, 8 ( Bi + Bi Bi Bi ),, 8 (7) Entropy Genertion Rte Adpting the generl result for the locl volumetric entropy genertion rte ( W / m 3 K) provided by Bejn [,] to the present convective flow sitution, we hve k dt du S & μ gen = + = S& gen, h + S& gen, f (8) T dy T dy which my be expressed in dimensionless form s S& gen d du = + = S h + S f (9) k dy dy Both in equtions (8) nd (9), the first term represents the entropy genertion due to het conduction (subscript h ) nd the second term the entropy genertion due to viscous or fluid friction effect (subscript f). Using eqution (4) for du / dy nd the pproprite results (depending on the therml boundry conditions) for d / dy, we cn integrte eqution (9) fromy = / to Y = / nd obtin the entropy genertion rte cross the gp between the pltes. Results nd Discussion The dimensionless velocity distribution given by eqution (3) is plotted in Fig. for P = 0 (red), (green), 4 (yellow), 6 (blue), 8(pink), nd 0 (qu blue). For P = 0 i.e. no xil pressure grdient, the velocity distribution is liner giving constnt velocity grdient du / dy cross the gp. As consequence, the entropy genertion due to fluid friction occurs t ll loctions cross the gp. As the pressure grdient P increses, the velocity profiles exhibit mxim indicting zero velocity grdients t certin loctions cross the gp nd consequently zero entropy genertion due to friction t those loctions. The loction of zero entropy genertion due to friction moves closer to the center line Y = 0 s the pressure grdient increses.

Entropy 006, 8[], 50-6 55 Fig.. Effect of pressure grdient P on velocity profiles Constnt Pltes Tempertures The effect of pressure grdient on the temperture distribution is illustrted in Fig.3 for the cse of constnt plte tempertures ( = 0, = 5 ). In the rnge from P = 0-0, the temperture distribution chnges only slightly. Since the temperture profiles do not exhibit ny mxim, the entropy genertion due to het conduction occurs t ll loctions cross the gp. However, distinct mximum in temperture is observed t P =50 nd indictes zero entropy genertion due to het conduction t tht loction. Fig.3. Effect of pressure grdient P on temperture profile

Entropy 006, 8[], 50-6 56 S h S f Fig.4. Locl entropy genertion due to het conduction Fig.4b.Locl entropy genertion due to fluid friction For fixed vlues of = 0 nd P =, the effect of vrying on the entropy genertion rte is depicted in Fig.4. Fig. 4 shows the entropy generted due to het conduction i.e. S& gen, h / k (denoted by S h ) while Fig. 4b shows tht due to fluid friction i.e. S& k (denoted by ). The temperture is gen, f / vried from =0 (red curve) in increment of 0 to 70 (qu curve). For = 0, the locl entropy genertion due to het conduction (Fig 4) is negligible. However, the entropy genertion due het conduction increses shrply s is incresed prticulrly t the sttionry plte where the lrgest temperture grdients occur. The effect of on the locl entropy genertion t the moving plte is rther smll becuse the chnges in result in comprtively smller chnges in the temperture grdients t the moving plte. The locl entropy genertion due to fluid friction (Fig. 4b) is mximum t the sttionry plte nd minimum t the moving plte. The highest entropy genertion due to fluid friction occurs t the lowest vlue of (red curve). This is due to the presence of in the denomintor of the fluid friction term in eqution (9). A comprison of the results in Fig. 4 nd 4b nd the corresponding results for other pressure grdients revels tht the entropy genertion due to fluid friction is much smller thn tht due to het conduction for the rnge of P investigted i.e. from 0 to 50. Fig.5 shows the integrted (cross the gp) totl entropy genertion rte, denoted in the figure by S = S h + S f for convenience, s function of the pressure grdient P for = 0, 40, 60, nd 80 with fixed t vlue of 0. The contribution of S is much smller thn S. The totl entropy genertion rte increses s P increses nd/or increses. This is consequence of the higher velocity nd temperture grdients cused by the increse in P nd, respectively. From system design perspective, the totl entropy genertion is minimized only when the pltes re held t identicl tempertures nd the pressure grdient is zero, tht is, when the flow is driven solely by the motion of the upper plte. f S f h

Entropy 006, 8[], 50-6 57 3 S.5.0.5.0 0.5 0 = 0 = 40 = 60 = 80 0 4 6 8 0 P Fig.5. Effect of pressure grdient P nd on totl entropy genertion. Constnt Temperture t Sttionry Plte nd Constnt Het flux t the Moving Plte For this cse, the temperture t the sttionry plte i.e. ws fixed t vlue of 0. The effect of vrying the pressure grdient P nd the het flux Q t the moving plte is illustrted in Fig. 6. The totl entropy genertion rte increses significntly with increse in pressure grdient but is comprtively less ffected by the chnges in het flux t the moving plte. The increse in entropy genertion rte is due to the increse in velocity nd temperture grdients tht ccompny the increse in P ndq. It my be noted tht in this cse the moving plte is colder thn the sttionry plte nd extrcts energy from the fluid. The entropy genertion is minimized only when the moving plte is insulted, the pressure grdient is zero nd the flow is induced solely by the motion of the plte. A comprison of Figs.5 nd 6 shows tht by extrcting het t constnt rte from the moving plte, the entropy genertion in the process cn be significntly reduced. For the cse where the moving plte is hotter thn the sttionry plte, the moving plte delivers energy to the fluid. Fig.7 shows the results for this scenrio. For low vlues of pressure grdient P, the entropy genertion rte increses rpidly s the het flux increses beyond Q 8. It is interesting to observe tht t high vlues of the het flux, the entropy curves for low vlues of P cross over the curves for high vlues of P. Thus, within the exmined intervls of P ndq, the highest entropy genertion is encountered t P = 6 nd Q =6. However, t higher pressure grdients, the increse in Q ffects the entropy genertion rte modertely. Furthermore, t P = 8 nd 0, both curves exhibit minimum in the neighborhood of Q = 8 (discernible with numericl results). This result is importnt in flows driven by high pressure grdients in which the moving plte supplies het to the fluid. If system is to be designed under these operting conditions, the mximum energy input from the moving plte should be close to Q = 8 to ensure minimum entropy genertion.

Entropy 006, 8[], 50-6 58 0.6 0.5 Q=0 0.4 Q= Q= S 0.3 Q=3 0. Q=4 0. Q=5 0 4 6 8 0 P Fig.6. Effect of pressure grdient P nd het flux Q on totl entropy genertion when the moving plte delivers het to the fluid. 3.5 S.5 P=6 P=8 P=0 P= P=4 P=6 P=8 P=0 0.5 0 0 4 6 8 0 4 6 Q Fig.7. Effect of pressure grdient P nd het flux Q on totl entropy genertion rte when the moving plte extrcts het from the fluid. Constnt Temperture t the Sttionry Plte nd Convection t the Moving Plte We choo se = 0 nd = 5 nd illustrte in Fig.8 the effect of Biot number on the totl entropy genertion rte for selected vlues of the pressure grdient. For ech vlue of P, the totl entropy genertion rte increses s Bi increses i.e. the convection t the moving plt e gets stronger. The increse in het removl from the fluid by the moving plte results in enhnced entropy genertion rte. For fixed Bi, the higher the pressure grdient nd hence the lrger the velocity grdients, the higher the entropy genertion rte.

Entropy 006, 8[], 50-6 59.8.6.4..0 P= P=4 S P=6 0.8 P=8 P=0 0.6 0.4 0. 0 3 4 5 Bi Fig. 8.Vrition of entropy genertion rte with Biot number Bi nd pressure grdient P. The effect of vrying the convection environment tempertur e on the entropy genertion rte is depicted in Fig. 9 for selected vlues of the pressure grdient P. This figure ws generted by fixing = 0 nd Bi =. As increses, the convective het removl from the moving plte decreses which leds to reducti on in entropy genertion rte..4. 0.8 S 0.6 0.4 P= P=4 P=6 P=8 P=0 0. 0 4 6 8 0 Fig.9. Entropy genertion rte s function of nd P with = 0 nd Bi=.

Entropy 006, 8[], 50-6 60 Convection t both pltes The vribles in this cse re P,,,,, Bi nd Bi. We fix the first three t P =,, = 0, = 0 nd study the effect of vrying Bi nd Bi. In Fig. 0, the entropy gene rtion rte is plotted s, function of Bi for prmetric vlues of Bi. At Bi=, the entropy genertion increses shrply s Bi increses. However t Bi =, 3, 4 nd 5, the increse in entropy genertion with Bi is moderte. Also the effect of Bi is significntly ttenuted beyond Bi =3..8.4.0 S 0.6 Bi= Bi= Bi=3 Bi=4 Bi=5 0. 0.0 3 4 5 Bi Fig.0. Entropy genertion rte s function of Bi nd Bi with P =,, = 0,, = 0. It hs been poin ted by Ibnez et l [ 6] tht for certin combintions of Bi nd Bi, the entropy genertion ttins minimum. A close exmintion of their Fig. 3 for G= (P = in present work) indictes the vrition in entropy genertion s Bi ( Bi in present work) increses from 0 to 6 is only bout 0. percent for the three vlues of Bi ( Bi in present work ) used nmely 0, 5 nd 30 with the minimum entropy genertion occurring t Bi ( Bi in present work) = 0.5. Becuse of the vrition of 0. percent, the minimum could not be identified grphiclly without lrge mgnifiction of S xis. We use the vlues of P,,,,, Bi nd Bi on w hich their Fig. 3 is bsed nd present our results in Fig.. It cn be observed tht in the rnge of Bi from 0.5-3.0 nd Bi, the entropy genertion rte is virtully minimum. A distinct minimum cn be identified by mgnifying the S xis or exmining the numericl results but the exct determintion of the minimum would pper to be of little prcticl use. The sme conclusion ws reched with the results for other vlues of P,,, nd,. The results of symmetric convective cooling of the pltes provide opportunity for minimum entropy design of Couette flow systems. For minimum entropy genertion design, the cooling of the moving plte must provide minimum Biot number of for the operting condition of P =,

Entropy 006, 8[], 50-6 6, = 0,, = 0.The designer then hs the flexibility of providing cooling t the sttionry plte within Biot numbers in the rnge 0.5-3. 0.5 0.4 S 0.3 0. 0. 0 0.5.0.5.0.5 3.0 Bi Bi= Bi= Bi=3 Bi=4 Bi=5 Conclusions Fig.. Entropy genertion rte s function of Bi nd Bi w ith P =,, = 0,, = 0. An nlyticl study of entropy genertion in plne Couette flow with fvorble pressure grdient nd viscous dissiption hs been presented for four therml boundry conditions: () constnt tempertures t both sttionry nd moving pltes, (b) constnt temperture t the sttionry plte nd constnt het flux t the moving plte, (c) constnt temperture t the sttionry plte nd convection t the moving plte, nd (d) convection t both pltes. The effect of pressure grdient nd therml prmeters on the entropy genertion rte hs been presented nd discussed in detil. For cse (), minimum entropy design cn be relized only when the pltes re mintined t identicl tempertures nd the pressure grdient is zero i.e. the flow is driven solely by the motion of the top plte. For cse(b), it is found tht significnt reduction in entropy genertion compred with the cse of isotherml pltes cn be chieved by extrcting het through the moving plte t constnt rte. If the moving plte is to supply het to the fluid, then there is n optimum vlue of het flux tht minimizes the entropy genertion. For cses (c) nd (d) there exists optimum combintions of therml prmeters tht result in minimum entropy genertion. These optimum combintions hve been identified. For the cse of convection t both pltes, i.e. cse (d), the present nlysis extends the work of Ibnez et l [6] by llowing the coolnt temperture t the moving plte to be different from the coolnt temperture t the sttionry plte. For identicl coolnt tempertures s ssumed by Ibnez et l [6], the present results vlidte their conclusions tht minimum entropy genertion cn be relized with symmetric convective cooling of the pltes.

Entropy 006, 8[], 50-6 6 References. Bejn, A., A study of entropy genertion in fundmentl convective het trnsfer, ASME Journl of Het Tnsfer, 0, 78-75 (979).. Bejn, A., Entropy Genertion Minimiztion, CRC Press, New York, (996). 3. Mhmud, S,nd R. A.Frzer, The scond lw nlysis of fundmentl convective het trnsfer problems, Interntionl Journl of Therml Sciences, 4,77-86 (003). 4. Ebry, L.B., M.S.Ercn, B.Sulus, nd M.M.Ylcin, Entropy genertion during fluid flow between two prllel pltes, Entropy,5, 506-58 (003). 5. Ibnez, G., S. Cuevs, nd M.L. de Hro, Minimiztion of entropy genertion by symmetric convective cooling,interntionl Journl of Het nd Mss Trnsfer, 46,3-38 (003). 6. Ibnez, G., S. Cuevs, nd M.L. de Hro, Het trnsfer in symmetric convective cooling nd optimized entropy genertion rte, Revist Mexicn De Fisic, 49,338-343 (003). 006 by MDPI (htpp:/www.mdpi.org). Reproduction for noncommercil purposes permitted.