Heat and mass transfer effects on nanofluid past a horizontally inclined plate

Transcription

1 Journal o Phyic: Conerence Serie PAPER OPEN ACCESS Heat and ma traner eect on nanoluid pat a horizontally inclined plate To cite thi article: M Selva rani and A Govindarajan 08 J. Phy.: Con. Ser View the article online or update and enhancement. Related content - Heat and ma traner eect on MHD natural convection low pat a moving vertical plate Siva Reddy Sheri and J Anand Rao - Radiative luid low o a nanoluid over an inclined plate with non-uniorm urace temperature M Selva rani and A Govindarajan - Unteady MHD ree convective boundary layer low o a nanoluid pat a moving vertical plate T Sravan Kumar and B Ruhi Kumar Thi content wa downloaded rom IP addre on 6/08/08 at 00:39

2 National Conerence on Mathematical Technique and it Application (NCMTA 8) IOP Con. Serie: Journal o Phyic: Con. Serie (08) 07 doi :0.088/7-6596/000//07 Heat and ma traner eect on nanoluid pat a horizontally inclined plate. Introduction M Selva rani and A Govindarajan * Department o Mathematic, SRM Intitute o Science and Technology, Kattankulathur, Tamilnadu, India elvaraniilango@gmial.com, Govindarajan.a@ktr.rmuniv.ac.in *correponding author Abtract: In thi paper, the ree convective low pat a nanoluid over an inclined plate with heat and ma traner i invetigated. Dierent kind o nanoluid containing nano particle like Silver, Aluminum oxide and Copper were conidered with water a a bae luid. The governing equation were tranormed into the nonlinear equation. Then Solution are obtained uing a crank Nicolon method; an eicient, tri-diagonal, iterative implicit inite dierence method. The reult are analyzed and preented through Graph and table or variou parameter. Alo, the reult are in excellent agreement with the reult in the open literature. Keyword: Free Convection, Inclined plate, Nanoluid, Heat Traner, Ma Traner Nanoluid technical term introduced by Choi [995].Nanotechnology baed luid contain a upended ubmicronic nano-cale particle ha a main advantage that it enhance the bae luid thermal conductivity experienced by Mauda et al. [993]. Since or, many reearcher like Xuan and Li, Xue [000, 003] are intereted to do reearch in nanoluid in the lat ew year, becaue o longetablihed luid like water, ethylene, mineral oil have a low thermal conductivity compared with nanoluid, many publication are in earch o undertanding the behavior o nanoluid in indutrial and engineering ytem, Nuclear reactor, electronic, advanced nuclear ytem and alo in medicine too. The application carried out currently and alo to be ued in the uture, which involve nanoluid were given by Wong and Leon [00]. A amou Rayleigh problem, dealt with the impulively tarted ininite horizontal luid through which the vicou incompreible luid low through got olved by Stoke [85] uing mixed explicit implicit inite dierence method. Under uniorm urace temperature and concentration, heat and ma trnaer o ree convetive low pat a vertical plate wa olved by Somer [956] and Wilcox [96] uing an integral method. Followed them Gebhart and Pera [97] tudied the natural convection on a vertical plate with variable urace temperature and ma diuion uing imilarity variable. Soundalgkar [977] concerned with the problem o toke or the vertical plate under the eect o MHD. Later Soundalgekar and Ganean [98] employed implicit inte dierence method to explore the natural convection on an iothermal lat plate. Chen et al. [986] Gave an important reult o tudying the heat traner in horizontal, vertical and inclined plate with varying wall temperature and Content rom thi work may be ued under the term o the Creative Common Attribution 3.0 licence. Any urther ditribution o thi work mut maintain attribution to the author() and the title o the work, journal citation and DOI. Publihed under licence by Ltd

3 National Conerence on Mathematical Technique and it Application (NCMTA 8) IOP Con. Serie: Journal o Phyic: Con. Serie (08) 07 doi :0.088/7-6596/000//07 heat lux. Chamkha and Al-Mudha [005] numerically invetigate the unteady laminar heat and ma traner. Mohammed Ibrahim and Bakar Reddy [03] tudied the Heat and Ma Traner or natural convection over a moving vertical plate with internal heat generation and a convective boundary condition in the preence o thermal radiation vicou diipation, and chemical reaction. Upto the knowledge o the author, there i no reearch undertaken or the ree convective nanoluid pat an inclined plate under heat and ma traner.. Mathematical Analyi The preent tudy conider ree convective nanoluid low along an inclined pate with MHD and heat generation/aborption eect. The chematic diagram and Carteian coordinate ytem o the problem are hown in Figure. in which the x-axi i taken along the plate and the y-axi i taken normal to the ' ' plate. A hown in Figure.,the inclined angle along the plate i aumed to be φ. T, C i the temperature o both the luid and the plate at the initial tage, i.e. at time t 0.Then at time t > 0, the ' ' temperature o the plate i raied to T w, Cw The luid taken into account here i a nanoluid containing nanoparticle like ilver, aluminium oxide and copper. Alo, the nanoluid conidered to be an incompreible luid. All o the luid phyical propertie are aumed to be contant except the denity variation, which include the buoyancy orce in the momentum equation. The governing boundary layer equation o continuity, momentum and energy under the Bouineq approximation (Herman Schlichting [969]) are a ollow: u v + = 0 x y u u u ' ' n + u + v ' = g ( β ) co φ ( T T ) dy g ( β ) co φ ( C C ) dy n + n t x y x x y y ' + g β φ T T + g β φ C C + µ ' in in n n n u y () ()

4 National Conerence on Mathematical Technique and it Application (NCMTA 8) IOP Con. Serie: Journal o Phyic: Con. Serie (08) 07 doi :0.088/7-6596/000//07 ' ' ' ' T T T T c u v k n + + ' = n t x y y (3) ( p ) ' ' ' ' C C C C + u + v D ' = t x y y () Initial and boundary condition are ' ' ' ' ' t 0, u= 0, v= 0, T = T, C = C or all xand y > 0, = 0, = 0, =, = at = 0 ' ' ' ' ' t u v T T C C x u= 0, v= 0, T = T C = C at y = 0 ' ' ' ' w, w u v T T C C y ' ' ' ' 0, 0,, a (5) Non - Dimenional quantitie ued to dimenionalize the baic equation are ' ' ' x y ul vl νt T T X =, Y = Gr, U = Gr V = Gr, t = Gr, T =, ' ' L L ν ν L T T ( C p ) ( w ) β ( w ) C C gβl T T g L C C Gr C = Gr =, Gr =, N =, C C Gr ' ' 3 ' ' 3 ' ' * * ' ' w ν ν ν ν ν Pr = ( or)pr =, Sc = α k D (6) For nanoluid, the expreion o denity n, thermal expanion coeicient and heat capacitance ( c p ) are given by n µ k n = ( ϕ ) + ϕ ( β ) = ( ϕ )( β ) + ϕ ( β ) n ( cp) = ( ϕ)( cp) + ϕ( cp) n n µ = n ( ϕ ).5 k + k ϕ k k = k k + k + ϕ k k (7) w 3

5 National Conerence on Mathematical Technique and it Application (NCMTA 8) IOP Con. Serie: Journal o Phyic: Con. Serie (08) 07 doi :0.088/7-6596/000//07 Equation in Non-dimenional orm ater the impliication will be U U + = 0 X Y ( β ) ( β ) ( β ) ( β ) ϕ ϕ β φ + U U U + U + V = Gr co T dy t X Y X ϕ + ϕ Y ( β ) ( β ) ϕ + ϕ ϕ + ϕ + N Gr coφ C dy + T inφ X ϕ + ϕ Y ϕ + ϕ ϕ + ϕ β + NC inφ + ( ϕ ) ϕ + ϕ ϕ + ϕ.5 U Y (8) (9) k Pr ϕ + ϕ T T T n T + U + V = t X Y k Y ( cp ) ( cp ) (0) C C C C + U + V = t X Y Sc Y () Initial and boundary condition in non-dimenional orm i 0, = 0, = 0, = 0, = 0or all and ' t U V T C X Y > 0, = 0, = 0, = 0, = at = 0 ' t U V T C X U= 0, V = 0, T =, C= aty = 0 U 0, Y 0, T 0, C ay ()

6 National Conerence on Mathematical Technique and it Application (NCMTA 8) IOP Con. Serie: Journal o Phyic: Con. Serie (08) 07 doi :0.088/7-6596/000//07 By ubtituting 3 = Pr ϕ+ ϕ ( cp ) ( cp ) n, ( β ) ϕ+ ϕ β E=, E.5 =, ( ϕ ) ϕ+ ϕ ϕ+ ϕ E k k (3) In non-dimenional equation, the equation become U U + = 0 X Y U U U U Gr coφ T dy + N C dy + U + V = E X X + E Y Y t X Y Y + [ T + NC] inφ () (5) t X Y Y T T T T + U + V = E3 (6) C C C C + U + V = t X Y Sc Y (7) Local and Average Skin Friction, Nuelt Number, Sherwood Number are τ X 3 u = Gr Y ( ).5 ϕ Y = 0 (8) 3 u τ = Gr Y.5 ϕ 0 Y = 0 dx (9) k n T NuX = Gr X k Y = Y 0 (0) 5

7 National Conerence on Mathematical Technique and it Application (NCMTA 8) IOP Con. Serie: Journal o Phyic: Con. Serie (08) 07 doi :0.088/7-6596/000//07 k n T 0 Y Nu = Gr dx k Y = 0 Sh X C Y = Gr X C Y = 0 Y Y = 0 C 0 Y = 0 Y = 0 C Sh = Gr dx () () (3) 3. Numerical olution and dicuion The et o non-linear coupled equation along with the initial and boundary condition are olved uing a mot reliable, unconditionally table implicit inite dierence method o the Crank-Nicolon type. The reulting problem attained rom the partial dierence equation are olved uing a tridiagonal ytem which i olved by uing the Thoma Algorithm a decribed by Carnahan et al. [969] and the integral in the equation are olved uing the Newton-Cote ormula. The reult are illutrated graphically to dicu intereting eature o the problem. 6

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10 National Conerence on Mathematical Technique and it Application (NCMTA 8) IOP Con. Serie: Journal o Phyic: Con. Serie (08) 07 doi :0.088/7-6596/000//07. Concluion: On increment o the value o N, both Velocity and Concentration increae and temperature decreae On increment o the value o Gr, Velocity decreae and the temperature and concentration ha no eect. On increment o the value o Angle, Velocity increae and both the temperature and concentration decreae. On increment o the value o Sc, both Velocity and Concentration decreae and temperature increae On increment o the value o ϕ, Velocity, temperature and concentration increae and when the value o ϕ i mall, copper how the maximum reaction while the value i larger, Aluminum how the maximum reaction. Reerence: [] Choi SU Enhancing thermal conductivity o luid with nanoparticle, development and application o non-newtonian low. ASME, FED, MD, 3:pp. 99-I05 [] Mauda H, Ebata A, Teramae K, Hihinuma N. Alteration o thermal conductivity and vicoity o liquid by dipering ultraine particle (diperion o γ-alo3, SiO, and TiO ultra-ine particle), Netu Buei pp [3] Xuan Y, Li Q Heat traner enhancement o nanoluid. International Journal o heat and luid low. :pp [] Xue QZ Model or eective thermal conductivity o nanoluid. Phyic letter A. 307:pp.33-7 [5] Wong KV, De Leon O. 00. Application o nanoluid: current and uture. Advance in Mechanical Engineering. :59659 [6] Stoke GG. 85. On the eect o the internal riction o luid on the motion o pendulum. Cambridge: Pitt Pre [7] Somer EV Theoretical conideration o combined thermal and ma traner rom a vertical lat plate. J. Appl. Mech. 3:30 [8] Wilcox WR. 96. Simultaneou heat and ma traner in ree convection. Chemical Engineering Science. 3:pp. 3-9 [9] Gebhart B, Pera L. 97. The nature o vertical natural convection low reulting rom the combined buoyancy eect o thermal and ma diuion. International Journal o Heat and Ma Traner.:05-50 [0] Soundalgekar VM Free convection eect on the Stoke problem or an ininite vertical plate. Journal o Heat Traner. 99: pp

11 National Conerence on Mathematical Technique and it Application (NCMTA 8) IOP Con. Serie: Journal o Phyic: Con. Serie (08) 07 doi :0.088/7-6596/000//07 [] Soundalgekar VM, Ganean P. 98. Finite-dierence analyi o tranient ree convection with ma traner on an iothermal vertical lat plate. International Journal o Engineering Science.; 9 :pp [] Chen TS, Tien HC, Armaly BF Natural convection on horizontal, inclined, and vertical plate with variable urace temperature or heat lux. International journal o heat and ma traner. 9:pp [3] Chamkha AJ, Al-Mudha A Unteady heat and ma traner rom a rotating vertical cone with a magnetic ield and heat generation or aborption eect. International journal o thermal cience. :pp [] Ibrahim SM, Bhahar Reddy N. 03. Similarity olution o heat and ma traner or natural convection over a moving vertical plate with internal heat generation and a convective boundary condition in the preence o thermal radiation, vicou diipation, and chemical reaction. ISRN Thermodynamic.;03 [5] Schlichting H. Boundary-layer theory, 968 [6] Carnahan B, Luther HA. Applied numerical method