SECOND-LAW ANALYSIS OF LAMINAR NON- NEWTONIAN GRAVITY-DRIVEN LIQUID FILM ALONG AN INCLINED HEATED PLATE WITH VISCOUS DISSIPATION EFFECT

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1 Brazilia Joural of Chemical Egieerig ISSN -663 Prited i Brazil Vol. 6, No., pp. 7 -, April - Jue, 9 SECOND-LAW ANALSIS OF LAMINAR NON- NEWTONIAN GRAVIT-DRIVEN LIQUID FILM ALONG AN INCLINED HEATED PLATE WITH VISCOUS DISSIPATION EFFECT S. Saouli * ad S. Aïboud-Saouli Faculty of Scieces ad Egieer s Scieces, Phoe: +3 () 7 8, Fax: +3 () , Uiversity Kasdi Merbah, 3, Ouargla, Algeria. sveralgk@yahoo.fr Professioal Traiig Istitute, Saïd Otba, 3, Ouargla, Algeria. isamho@hotmail.fr (Submitted: February, 7 ; Accepted: Jue, 7) Abstract - A secod-law aalysis of a gravity-drive film of o-newtoia fluid alog a iclied heated plate is ivestigated. The flow is assumed to be steady, lamiar ad fully-developed. The upper surface of the liquid film is cosidered to be free ad adiabatic. The effect of heat geeratio by viscous dissipatio is icluded. Velocity, temperature ad etropy geeratio profiles are preseted. The effects of the flow behaviour idex, the Brikma umber ad the group parameter o velocity, temperature ad etropy geeratio umber are discussed. The results show that velocity profile depeds largely o the flow behaviour idex. They are flat ear the free surface for pseudoplastic fluids ad liear for dilatat fluids. Temperature profiles are higher for higher flow behaviour idex ad Brikma umber. The etropy geeratio umber icreases with Brikma umber ad the group parameter because of the heat geerated by the viscous dissipatio effect. For pseudoplastic fluids, the irreversibility is domiated by heat trasfer, whereas, for dilatat fluids, irreversibility due to fluid frictio is more domiat. Keywords: Etropy geeratio; No-Newtoia; Power-law fluids; Secod-law; Viscous dissipatio. INTRODUCTION Fluid flow ad heat trasfer characteristics i fallig liquid films alog iclied plates at differet boudary coditios are oe of the fudametal researches i egieerig. Studies of simpler systems are useful to uderstad some importat features of complex combiatios formig processes i may fields of sciece ad techology. These basic geometries are commo i may egieerig applicatios as sole uits or as a global etity. Etropy geeratio is closely associated with thermodyamic irreversibility, which is ecoutered i all heat trasfer processes. Differet sources are resposible for the geeratio of etropy, such as heat trasfer across fiite temperature gradiet, characteristic of covective heat trasfer, viscous effect, etc. Beja (98; 996) focused o the differet reasos behid etropy geeratio i applied thermal egieerig. Beja (979) preseted a simplified aalytical expressio for the etropy geeratio rate i a circular duct with imposed heat flux at the wall. This aalysis is the exteded by calculatig the optimum Reyolds umber as a fuctio of the Pradtl umber ad the duty parameter. Sahi (998) itroduced the secod-law aalysis to a viscous fluid i a circular duct with isothermal boudary coditios. I aother paper, Sahi (999) preseted the effect of variable viscosity o the etropy geeratio rate for a heated *To whom correspodece should be addressed

2 8 S. Saouliad S. Aïboud-Saouli circular duct. A comparative study of the etropy geeratio rate iside ducts of differet shapes (circular, triagular, square etc.) ad the determiatio of the optimum duct shape subject to isothermal boudary coditio for lamiar flow were carried out by Sahi (998). Narusawa (998) gave a aalytical ad umerical aalysis of the secodlaw for flow ad heat trasfer iside a rectagular duct. I a more recet paper, Mahmud ad Fraser (3) applied the secod-law aalysis to fudametal covective heat trasfer problems. They aalysed the secod-law characteristics of heat trasfer ad fluid flow due to forced covectio of steady-lamiar flow of icompressible fluid iside a chael with circular cross-sectio ad chael made of two parallel plates. Differet problems are discussed with their etropy geeratio profiles ad heat trasfer irreversibility characteristics. I each case, aalytical expressios for etropy geeratio umber ad Beja umber are derived i dimesioless form usig velocity ad temperature profiles. I aother paper, Mahmud ad Fraser () ivestigated aalytically the first ad secod law characteristics of fluid flow ad heat trasfer iside a chael havig two parallel plates with fiite gap betwee them. Fully developed forced covectio is cosidered. Fluid is assumed to be o-newtoia ad follow the power law model. Aalytical expressios for dimesioless etropy geeratio umber, irreversibility distributio ratio ad Beja umber are determied as a fuctio of dimesioless distace, Peclet umber, Eckert umber, Pradtl umber, dimesioless temperature differece ad fluid behaviour idex. Spatial distributio of etropy geeratio umber, irreversibility ratio ad Beja umber are preseted graphically. The same authors (Mahmud ad Fraser, ) reported, i terms of local ad average etropy geeratio, the iheret irreversibility of fluid flow ad heat trasfer for o-newtoia fluids i a pipe ad a chael made of two parallel plates. They assumed the flow to be fully developed with a uiform heat flux at the duct wall. They applied the first ad the secod laws of thermodyamics to develop expressios for dimesioless etropy geeratio umber, irreversibility ratio ad Beja umber as fuctio of geometric, fluid ad flow parameters. However, i these aalyses cocerig o- Newtoia fluids, the ifluece of viscous dissipatio is omitted. The preset paper aims at aalysig the mechaism of etropy geeratio i a gravity-drive lamiar film of a o-newtoia fluid alog a iclied heated plate, takig care of the presece of viscous dissipatio effect. ANALSIS The physical cofiguratio is illustrated schematically i Fig.. The fallig liquid, drive by gravity, flows a dow flat heated plate iclied at a agle θ to the horizotal. It is assumed that the flow is lamiar ad fully developed. The liquid surface is waveless, free ad adiabatic. The o-newtoia fluid used i this study is the power-law model (Ostwald-de Waele fluid). Such fluids are characterized by the followig rheological law: u y τ= K () where is the flow behaviour idex ad K is the cosistecy of the fluid. A fluid is pseudoplastic whe, Newtoia whe =, ad dilatat whe. y Q O δ θ Figure : Schematic diagram of the problem uder cosideratio. x Brazilia Joural of Chemical Egieerig

3 Secod-Law Aalysis of Lamiar No-Newtoia Gravity-Drive Liquid Film 9 Neglectig the iertia terms i the mometum equatio compared with the body force term, the mometum equatio reduces to the followig form: u y K + ρgsiθ= The associated boudary coditios are: No-slip coditio () u = (3a) Free surface coditio u δ = (3b) The velocity distributio is obtaied by itegratig Eq. () ad usig the boudary coditios give by Eqs. (3a) ad (3b). It may be writte: y u( y) = u m δ where u m is the velocity at the free surface: () subject to the followig boudary coditios: Ilet coditio T T,y Wall heat flux = (a) T x, λ = q Adiabatic surface T x, δ = (b) (c) The equatio of eergy ca be trasformed ito a dimesioless form by itroducig the followig dimesioless variables: ax X = u = δ, mδ, y u( y) =, ( X,) U u m Δ T = qδ λ () T T x,y Θ = ΔT The trasformatio yields: () u m ρgsiθ = δ K (5) Θ X, Θ U() X, Br U() = + X (3) The mass flow rate of the liquid is: δ Q= ρ u( y) dy (6) from which the liquid film thickess may be computed: + Q δ= ρu m (7) Combiig Eqs. (5) ad (7), we obtai a expressio for the liquid film thickess: { } { ( ) Q } ( gsi K) δ= + ρ θ The goverig eergy equatio is: T x,y T x,y K u(y) u( y) = a x ρc P ( + ) (8) (9) + KQ Br = qρ δ is the Brikma umber. The trasformed boudary coditios are: Θ (, ) = (a) Θ Θ ( X,) ( X,) = = (b) (c) To get a solutio of Eq. (3), a separatio of variables solutio is assumed i the followig form (Arpaci ad Larse, 98): ( X,) ( X) ( X) Θ =Θ Θ +Θ +Θ (5) Brazilia Joural of Chemical Egieerig Vol. 6, No., pp. 7 -, April - Jue, 9

4 S. Saouliad S. Aïboud-Saouli The first term o the right-had side of Eq. (5) is sigificat for decayig iitial trasitio ad etrace effects, the secod term is sigificat for axial temperature rise due to accumulated wall heat flux ad the third term is sigificat for trasverse temperature variatio due to wall heat flux ito fluid. Neglectig the etrace effect ad assumig that the system already passed the decayig iitial trasitio, the the first term at the right-had side of Eq. (5) will disappear (Mahmud ad Fraser ; 3). Combiatio of Eq. (5) ad Eq. (3) leaves two separate ordiary equatios coected by a scalar costat α : Θ X X =α =α ( ) Θ ( ) Br (6) (7) Itegratig Eqs. (6) ad (7) ad applyig boudary coditios described i Eq. (), the expressio for the dimesioless temperature is obtaied i the followig form: α Θ ( X,) =α X + α 3 ( ) Br + C + C where α ad C are: + Br + α= 3 (8), C = α (9) To obtai the costat of itegratio C, we use the mea bulk temperature, defied as: Θ b ( X) = Θ ( X,) da = Θ( X,) d A () A Sice Eq. (a) requires itegratio is: α C = + ( + )( 3 + )( + ) 3 Br α ( + )( + )( + ) 3 Θ b =, the costat of ENTROP GENERATION RATE () The etropy geeratio rate accordig to Mahmud ad Fraser () is: S λ T( x,y) T( x,y) = + x G T K u y + T The etropy geeratio umber may be defied as: N S () λt = SG (3) q Usig the defiitios of dimesioless velocity ad temperature, the followig expressio is obtaied for the etropy geeratio umber: N S ( X,) Θ( X,) Θ = + Pe X Br U + = N + N + N Ω C F I the above equatio, Pe = + QCP λ () is the Peclet umber, which determies the relative importace betwee covectio ad diffusio, Br is the Brikma umber, which determies the relative importace betwee viscous dissipatio effects ad fluid coductio, Ω = ΔTT is the dimesioless temperature differece. O the right-had side of Eq. (), the first term represets the etropy geeratio by heat trasfer due to axial coductio, the secod term accouts for etropy geeratio due to the trasverse directio ad the third is the part of the etropy geeratio due to the fluid frictio. Brazilia Joural of Chemical Egieerig

5 Secod-Law Aalysis of Lamiar No-Newtoia Gravity-Drive Liquid Film RESULTS AND DISCUSSION Dimesioless axial velocity profiles are plotted as a fuctio of dimesioless trasverse distace i Fig. for five differet values of the flow behaviour idex. For pseudoplastic fluids ( ), velocity profiles remai flat ear the free surface ad this flatess decreases with the icrease of the low behaviour idex. For Newtoia fluids ( = ), the dimesioless axial velocity shows the usual semiparabolic shape. For dilatat fluids ( ), velocity profiles approach a liear shape as the flow behaviour idex icreases. Dimesioless temperature profiles are plotted i Fig. 3 for the same rage of the flow behaviour idex. For the preset boudary coditio, temperature is maximum at the wall where a heat flux is imposed ad miimum at the free surface whatever the value of the flow behaviour idex is. For a particular trasverse distace, the temperature is higher for a higher flow behaviour idex. This meas that dilatat fluids heat more easily tha pseudoplastic fluids. The axial variatios of the dimesioless temperature profiles are plotted i Figs. ad 5 for pseudoplastic fluids ( =.) ad dilatat fluids ( = 5.). I all cases, the temperature icreases i the axial directio because of the cotiuous heatig of the wall. The effect of the Brikma umber o the temperature is illustrated i Figs. 6 ad 7 for pseudoplastic fluids ( =.) ad for dilatat fluids ( = 5.). The temperature icreases as the Brikma umber icreases either for pseudoplastic fluids or dilatat fluids. As the Brikma umber, which determies the relative importace betwee viscous dissipatio effects ad fluid coductio, icreases, more heat is geerated by the viscous dissipatio effect i the fluid. This geerated heat by viscous dissipatio effect results i higher temperature profiles.. = X=. Br=. U(). Θ (X,). = Figure : Velocity profiles. 3. =..5 Br=..... Figure 3: Temperature profiles. 3. =5. Br= Θ(X,)...5 X=..... Figure : Temperature profiles..5 Θ(X,)..5 X=..... Figure 5: Temperature profiles..5. =. X= 3 =5. X=.5 Θ(X,)..5. Br=. Θ (X,). Br=..... Figure 6: Temperature profiles.... Figure 7: Temperature profiles. Brazilia Joural of Chemical Egieerig Vol. 6, No., pp. 7 -, April - Jue, 9

6 S. Saouliad S. Aïboud-Saouli I Figs. 8 ad 9, the etropy geeratio umber is plotted as a fuctio of the dimesioless trasverse distace for differet values of the Brikma umber for pseudoplastic fluids ( =. ) ad dilatat fluids (= 5.). I all cases, o etropy is geerated at the free surface where both velocity ad temperature are maximum (or miimum), which cause zero velocity ad temperature gradiets, leavig o cotributio to the etropy geeratio umber (secod ad third term of Eq. ()). For a particular trasverse distace, the magitude of the etropy geeratio umber is higher for higher Brikma umbers because of the heat geerated by the viscous dissipatio effect. I the case of pseudoplastic fluids ( =.), the etropy geeratio umber decreases alog the trasverse distace to reach zero at the free surface. This ca be explaied by the fact that, for pseudoplastic fluids, the velocity profile is flat ear the free surface leavig o cotributio of fluid frictio o etropy geeratio. Therefore, the irreversibility is maily domiated by heat trasfer. For dilatat fluids ( = 5. ), for a particular trasverse distace, the etropy geeratio umber shows a maximum ear the wall as the Brikma umber icreases. Accordig to Fig., the velocity profile is early liear (high velocity gradiet);, this meas that the cotributio of fluid frictio to etropy geeratio umber icreases. Thus, for dilatat fluids, the irreversibility is domiated by fluid frictio. Figs. ad show the distributio of the etropy geeratio umber as fuctio of the trasverse distace at differet values of group parameter ragig from. to. No etropy geerates at the free surface where both velocity ad temperature are maximum (or miimum) which cause zero velocity ad temperature gradiets, leavig o cotributio to the etropy geeratio umber (secod ad third term of Eq. ()) for all values of group parameter. For a particular trasverse distace, the etropy geeratio umber is higher for higher group parameter. For pseudoplastic fluids ( =. ), the etropy geeratio umber decreases with the trasverse distace ad does ot show maxima except for the case where ( BrΩ =. ). This meas that the irreversibility is domiated by heat trasfer ad the wall acts as a strog cocetrator of irreversibility. For dilatat fluids ( = 5.), the cotributio of fluid frictio o etropy geeratio umber is domiat, the etropy geeratio umber shows maxima ear the wall. Comparig the magitude of etropy geeratio umber for pseudoplastic ad dilatat fluids, the results show that irreversibility is more proouced for pseudoplastic fluids. 8 =. BrΩ - =. Pe= 5 =5. BrΩ - =. Pe= N S 6 Br=.. 3 N S Br=..... Figure 8: Etropy geeratio umber. =. Br=. 8 Pe=... Figure 9: Etropy geeratio umber. 5 =5. Br=. Pe= N S 6 BrΩ - =.. N S 3 BrΩ - =..... Figure : Etropy geeratio umber.... Figure : Etropy geeratio umber. Brazilia Joural of Chemical Egieerig

7 Secod-Law Aalysis of Lamiar No-Newtoia Gravity-Drive Liquid Film 3 CONCLUSION The secod-law aalysis is applied to a gravitydrive, lamiar, o-newtoia liquid film with free ad adiabatic surface. The heat geeratio by viscous dissipatio is icluded i the aalysis. Aalytical expressios for velocity ad temperature withi the film are provided as a fuctio of the flow behaviour idex ad the Brikma umber. The effects of the flow behaviour idex, the Brikma umber ad the group parameter o etropy geeratio umber are discussed. From the results the followig coclusios could be draw: a) Velocity profile depeds largely o the flow behaviour idex. They are flat ear the free surface for pseudoplastic fluids ad liear for dilatat fluids. b) Temperature profiles shift to higher temperatures with a icreasig flow behaviour idex. c) For pseudoplastic fluids ad dilatat fluids, temperature profiles icrease with the axial distace because of the cotiuous heatig of the wall. d) As the Brikma umber icreases, the temperature profile icreases because of the heat geerated by the viscous dissipatio effect. e) The etropy geeratio umber icreases with the Brikma umber ad the group parameter. This is due to the heat geerated by the viscous dissipatio effect. f) For pseudoplastic fluids, the irreversibility is domiated by heat trasfer, whereas for dilatat fluids, irreversibility due to fluid frictio is more domiat. Nevertheless, it is ecessary to carry out further aalyses ad calculatios for differet geometries ad o-newtoia fluids other tha those obeyig the power-law model. NOMENCLATURE a thermal diffusivity m /s A area m Br Brikma umber C costat of itegratio C costat of itegratio C specific heat J/kg.K P g gravitatioal acceleratio m/s K cosistecy of the fluid Pa.s flow behaviour idex N etropy geeratio, axial C N F coductio etropy geeratio, fluid frictio N etropy geeratio umber, S total N etropy geeratio umber, trasverse coductio Pe Peclet umber q wall heat flux W/m Q liquid mass flow rate kg/m.s S G etropy geeratio rate W/m 3.K T temperature K u axial velocity m/s U dimesioless axial velocity x axial distace m X dimesioless axial distace y trasverse distace m dimesioless trasversetace Greek Symbols α scalar costat δ thickess of the liquid film m Δ T referece temperature K differece λ thermal coductivity W/m.K θ icliatio agle rad Θ dimesioless temperature Ω dimesioless temperature differece ρ desity of the fluid kg/m 3 τ shear stress Pa Subscripts b bulk value m maximum value ilet value, referece value REFERENCES Arpaci, V. S., Larse, P. S., Covectio heat Trasfer. Pretice-Hill. Egelwood Cliffs, New Jersey (98). Beja, A., Secod-law aalysis i heat trasfer ad thermal desig, Adv. Heat Trasfer, 5, pp. -58 (98). Beja, A., Etropy geeratio miimizatio. CRC Press, Boca Rato, New ork (996). Beja, A., A study of etropy geeratio i fudametal covective heat trasfer. J. Heat Trasfer,, pp (979). Mahmud, S., Fraser, R. A., The secod law aalysis i fudametal covective heat trasfer problems. It. J. Therm. Sci.,, pp (3). Brazilia Joural of Chemical Egieerig Vol. 6, No., pp. 7 -, April - Jue, 9

8 S. Saouliad S. Aïboud-Saouli Mahmud, S., Fraser, R. A., Thermodyamic aalysis of flow ad heat trasfer iside chael with two parallel plates. Exergy,, pp. -6 (). Mahmud S., Fraser, R. A.,, Iheret irreversibility of chael ad pipe flows for o- Newtoia fluids. It. Comm. Heat Mass Trasfer, 9, pp (). Narusawa, U., The secod-law aalysis of mixed covectio i rectagular ducts. Heat Mass Trasfer, 37, pp (998). Sahi, A. Z., Secod law aalysis of lamiar viscous flow through a duct subjected to costat wall temperature. J. Heat Trasfer,, pp (998). Sahi, A. Z., Effect of variable viscosity o the etropy geeratio ad pumpig power i a lamiar fluid flow through a duct subjected to costat heat flux. Heat Mass Trasfer, 35, pp (999). Sahi, A. Z., A secod law compariso for optimum shape of duct subjected to costat wall temperature ad lamiar flow. Heat Mass Trasfer, 33, pp. 5-3 (998). Brazilia Joural of Chemical Egieerig