EFFECTS OF NAVIER SLIP AND WALL PERMEABILITY ON ENTROPY GENERATION IN UNSTEADY GENERALIZED COUETTE FLOW OF NANOFLUIDS WITH CONVECTIVE COOLING

Transcription

1 U.P.B. Sci. Bull., Serie D, Vol. 77, I. 4, 5 ISSN EFFECTS OF NAVIER SLIP AND WALL PERMEABILITY ON ENTROPY GENERATION IN UNSTEADY GENERALIZED COUETTE FLOW OF NANOFLUIDS WITH CONVECTIVE COOLING Michael H MKWIZU, Oluole D MAKINDE, Ya NKANSAH-GYEKYE 3 Preent or invetigate the eect o generalized Couette lo ith convective cooling, Navier li and ermeable all on entroy generation in an unteady o ater baed nanoluid containing Coer (Cu and Alumina (Al O 3 a nanoarticle. Both irt and econd la o thermodynamic are alied to analye the roblem. The nonlinear governing equation o momentum and energy are olved numerically uing a emi dicretization inite dierence method together ith Runge-Kutta Fehlberg integration cheme. Grahical reult on the eect o dierent arameter variation on velocity, temerature, in riction, Nuelt number, entroy generation rate, and Bejan number are reented and dicued. Keyord: Channel; Nanoluid; Couette lo; Entroy generation; ermeability; Navier li. Introduction Thermodynamic irreveribility in the lo ytem rovide inormation on the energy and oer loe in the ytem. Minimization o entroy generation in the lo ytem enable the arametric otimization o the ytem oeration. With the groing demand or eicient cooling ytem, more eective coolant are required to ee the temerature o heat generating engine and engineering device uch a electronic comonent belo ae limit. In recent time, the ue o nanoluid ha rovided an innovative technique to enhance heat traner. The interet here i the tudy o the eect o heat traner, ermeable all and Navier li on entroy generation beteen to arallel late, one o hich i moving relative to the other and reure gradient introduced non a generalized Couette lo hich i motivated by everal imortant roblem in Mr.,School o Comutational and Communication Science and Engineering, Nelon Mandela Arican Intitution o Science and Technology,(NM-AIST Tanzania, mizum@nmait.ac.tz and Mr.,Det o Biometry and Mathematic, Faculty o Science, Sooine Univerity o Agriculture, mizumh@gmail.com Pro.,Faculty o Military Science, Stellenboch Univerity South Arica, mainded@gmail.com 3 PhD, School o Comutational and Communication Science and Engineering, Nelon Mandela Arican Intitution o Science and Technology,(NM-AIST, mizum@nm- ait.ac.tz, ya.nanah-gyeye@nm- ait.ac.tz

2 Michael H Mizu, Oluole D Mainde, Ya Nanah-Gyeye engineering and indutrie. Nanotechnology ha been idely ued in engineering and indutry ince nanometer ize material oe unique hyical and chemical roertie. The addition o nanocale article into the conventional luid lie ater, engine oil, ethylene glycol, etc., i non a nanoluid and a irtly introduced by Cho []. Hydromagnetic blood lo through a uniorm channel ith ermeable all covered by orou media o inite a done by Ramarihnan and Shailendhra []. They ound that the axial velocity o the luid i reduced by orou arameter and Hartmann number. Theuri and Mainde [3] conidered thermodynamic analyi o variable vicoity MHD unteady generalized Couette lo ith ermeable all. They ound that the decreae in luid vicoity increae Bejan number hile an increae grou arameter decreae Bejan number. A Reynold number increae, Bejan number rie at the loer ixed late and it all at the uer moving late. The ituation i revered ith increaing magnetic ield. Ozto and Abu-Nada [4] conidered natural convection in artially heated encloure having dierent aect ratio and illed ith nanoluid. They ound that the heat traner a more ronounced at lo aect ratio and high volume raction o nanoarticle. Wang and Mujumdar [5] reented a comrehenive revie o heat traner characteritic o nanoluid. Detail reort on convective tranort in nanoluid can be ound in Buongiorno [6], Tiari and Da [7]. Meanhile, in the nanoluid lo, the imrovement o the heat traner roertie caue the reduction in entroy generation. The oundation o noledge o entroy roduction goe bac to Clauiu and Kelvin tudie on the irreverible aect o the econd la o thermodynamic. Since then the theorie baed on thee oundation have raidly develoed, ee Bejan [8, 9]. Hoever, the entroy roduction reulting rom heat and ma traner couled ith vicou diiation in nanoluid ha remained untreated by claical thermodynamic, hich motivate many reearcher to conduct analye o undamental and alied engineering roblem baed on econd la analyi ith reect to nanoluid. Baed on the concet o eicient energy ue and the minimal entroy generation rincile, otimal deign o thermodynamic ytem have been idely rooed by the thermodynamic econd la []. It i oible to imrove the eiciency and overall erormance o all ind o lo and thermal ytem through entroy minimization technique. The analyi o energy utilization and entroy generation ha become one o the rimary objective in deigning a thermal ytem. Several tudie have thoroughly dealt ith conventional luid lo irreveribility due to vicou eect and heat traner by conduction [, ]. In the reent tudy, e analye the eect o convective cooling, Navier li and ermeable all on entroy generation rate in unteady generalized Couette lo channel o ater bae nanoluid.

3 Eect o Navier li and all [...] Couette lo o nanoluid ith convective cooling 3. Mathematical Model Conider unteady laminar lo o vicou incomreible nanoluid containing Coer (Cu and Alumina (Al O 3 a nanoarticle through a ermeable all Couette lo channel. It i aumed that the luid i injected uniormly into the channel at the loer late hile the uniorm luid uction occur at the moving uer late a deicted in Fig belo; T u = U, v = V, n = h( T T y = a (Suction Nanoluid y = (Injection μ u = βu, v = V, T = T Fig: Schematic diagram o the roblem under conideration The governing equation or the nanoluid momentum and energy in one dimenion ith aumtion above can be ritten a ollo [7-9] u u P μn u + V = +, ( t ρ x ρ n T T T α n μn u + V = α n +, ( t n here u i the nanoluid velocity in the x -direction, T i the temerature o the nanoluid, P i the nanoluid reure, t i the time, a i the channel idth, T i the loer tationary all temerature, i the dynamic vicoity o the nanoluid, n i the nanoluid thermal conductivity, ρ n i the denity o the nanoluid and [3, 9] α n n μ n i the thermal diuivity o the nanoluid hich are given by

4 4 Michael H Mizu, Oluole D Mainde, Ya Nanah-Gyeye μ α n n ( ρc μ = ( ϕ = ( ρc n n n.5 ( ρc, τ = ( ρc = ( ϕ( ρc, ρ n = ( ϕ ρ, n + ϕ( ρc + ϕρ, ( = (. + + ϕ ( + ϕ(, (3 The nanoarticle volume raction i rereented by ϕ ( ϕ = correond to a bae luid, ρ and and reectively, ( ρ are the denitie o the bae luid and the nanoarticle reectively, are the thermal conductivitie o the bae luid and the nanoarticle ρc and ( ρc are the heat caacitance o the bae luid and the nanoarticle reectively. It orth mentioning that the ue o the above exreion or n, i retricted to herical nanoarticle and doe not account or other hae o nanoarticle. Alo, aroximation ha been emloyed to aroximate the eective vicoity o the nanoluid μn a vicoity o a bae luid μ containing dilute uenion o ine herical article. The initial and boundary condition are given a ollo: u y, =, T ( y, = T, (4 ( u μ (, t = βu(, t, T (, t = T, (5 u T = (6 ( a t U, ( a, t = h( T ( a, t T., n The table belo reent thermo hyical roertie o ater, coer and alumina at the reerence temerature. Table Thermohyical roertie o the luid hae (ater and nanoarticle [4, 7, 8, and 9] Phyical roertie Fluid hae (ater Cu Al O 3 c (J/g K ρ(g/m The dimenionle variable and arameter are introduced a ollo:

5 Eect o Navier li and all [...] Couette lo o nanoluid ith convective cooling 5 T T u tv μ Pa θ =, W =, t =, υ =, P =, T U a ρ μ U P x y μ c U A =, X =, =, Pr =, Ec =, (7 X a a cp Ta ( ρc ( + + ϕ( Va μ τ =, m =, Re =, α = = c + ( c a, ( ρ ( ϕ ( υ ρ β The dimenionle governing equation together ith the aroriate initial and boundary condition can be ritten a: W A W W = + (8.5 t Re ϕ + ϕρ / ρ Re ϕ + ϕρ / ρ ϕ ( ( ( θ θ Ec W θ = +.5 t m Pr Re( ϕ + ϕτ Re( ϕ ( ϕ + ϕτ (9 ith initial and boundary condition W (, =, θ (, = ( W βa (, t = W (. t, θ (, t = μ ( W (, t = θ (, t = mbi θ (, t ( here Pr i the Prandtl number, Ec i the Ecert number, Re i the Reynold number, A i the reure gradient arameter, β i the coeicient o liding raction and i li arameter. Other hyical quantitie o ractical interet in thi roblem are the in riction coeicient C and the local Nuelt number Nu hich are deined a aτ aq C =, Nu =, μ U T (3 here τ i the all hear tre and q i the heat lux at the channel all given by u T τ = μn, q = n y= a y= a (4 Subtituting equation (4 into (3 and uing dimenionle variable, e obtain C W θ =, Nu = at =..5 ( ϕ m (5

6 6 Michael H Mizu, Oluole D Mainde, Ya Nanah-Gyeye 3. Entroy Analyi The econd la o thermodynamic i an imortant tool to crutinize the irreveribility eect due to lo and heat traner. Thermodynamic irreveribility i cloely related to entroy roduction. Convection roce involving channel lo o nanoluid i inherently irreverible due to the exchange o energy and momentum, ithin the nanoluid and at olid boundarie. Folloing Wood [], the local volumetric rate o entroy generation i given by n T μ n u S = + (6 T T The irt term in equation (6 i the entroy generation due to heat traner hile the econd term i the entroy generation due to nanoluid riction. Uing dimenionle variable rom equation (7, e exre the entroy generation number in dimenionle orm a, a S θ Br W N = = + (7.5 m ( ϕ Where Br = Ec Pr i the Brinman number, the Bejan number Be i deine a N Be = = (8 N + Φ Where, N = N + N. N θ = (The entroy generation due to heat m Br W traner, N = (The entroy generation due to luid riction.5 ( ϕ The irreveribility ditribution ratio i deine a Φ = N /N. Heat traner irreveribility dominate or Φ < and luid riction irreveribility dominate hen Φ >. The contribution o both irreveribilitie to entroy generation i equal hen Φ =.Equation (8 ho that the Bejan number range rom to. The zero value o the Bejan number correond to the limit here the irreveribility i dominated by the eect o luid riction hile one value o Bejan number i the limit here the irreveribility due to heat traner dominate the lo ytem. The contribution o both heat traner and luid riction to irreveribility are the ame hen Be = Numerical Procedure Uing a emi-dicretization inite dierence method, the nonlinear initial boundary value roblem (IBVP in equation (8-( can be olved numerically. We artition the atial interval into equal art and deine grid ize and grid

7 Eect o Navier li and all [...] Couette lo o nanoluid ith convective cooling 7 oint. The irt and econd atial derivative in equation (8 and equation (9 are aroximated ith econd-order central inite dierence. dwi A ( Wi+ W i + Wi Wi+ Wi = +.5 dt Re ( ϕ + ϕρ / ρ Re ( ϕ + ϕρ / ρ ( ϕ ( Δ Δ (9 dθ i θi+ θ i + θ i Ec Wi+ Wi θi+ θ i = +.5 dt m Pr Re ( ϕ + ϕτ ( Δ Re ( ϕ + ϕτ ( ϕ Δ Δ ( ith initial condition and boundary condition W θ =, i N + ( ( i ( ( = i W W =, θ =, WN + =, θ N + = θ N ( mbiδ ( Δ Conidering equation (9-( e can ee that, they are irt order ordinary dierential equation ith non initial condition. So they can be eaily olved iteratively uing Runge-Kutta Fehlberg integration technique imlemented on comuter uing Matlab. The in-riction coeicient and the Nuelt number in equation (5 are alo ored out and their numerical reult are reented. 5. Reult and Dicuion For undertanding the dynamic o thi hyical roblem, Fig -3 ho the numerical comutation or the rereentative velocity ield, temerature ield, in riction, Nuelt number, entroy generation rate and Bejan number. Some arbitrary choen eciic value to variou thermohyical arameter controlling the lo ytem have been aigned. The detailed dicuion and grahical rereentation are reorted in thi ection. 5. Eect o arameter variation on velocity roile It i noted that, the velocity increae ith time or a given et o arameter value until a teady tate roile i achieved a hon in Fig. Fig 3 ho an intereting obervation that, alumina-ater nanoluid tend to lo ater than coer-ater nanoluid. Thi reult may be due to the high denity o coer nanoarticle a comared to alumina nanoarticle. An increae in nanoarticle volume raction caue a light decreae in the velocity roile ee Fig 4. Thi may be due to the denity, the dynamic vicoity o the nanoluid, li condition, uction and injection o the luid. Looing to Fig 5, it can be noted that the nanoluid velocity increae ith an increae in reure gradient. The ooite eect i oberved in Fig 6 here increaing Reynold number caue a

8 8 Michael H Mizu, Oluole D Mainde, Ya Nanah-Gyeye decreae in velocity roile. Thi haen becaue the vicou orce increae ithin the lo ytem Alumina-ater Alumina-ater Bi=5, A=., Ec=., Re=, φ=.3, =., Pr=6. t =.,.3,.5, Bi=5, A=., Ec=., Re=, φ=.3, =., Pr=6. =.,.5,.7, t Fig : Nanoluid velocity roile ith roile ith increaing time Fig 3: Nanoluid velocity ith increaing time.9.8 Bi=5, A=., Ec=., Re=, =., Pr=6., t= φ =,.,., Bi=5, Ec=., Re=, φ=.3, =., Pr=6., t=.7.7 A =,,, Fig 4: Nanoluid velocity roile ith increaing φ Fig 5: Nanoluid velocity roile ith increaing A

9 Eect o Navier li and all [...] Couette lo o nanoluid ith convective cooling Bi=5, A=., Ec=., φ=.3, =., Pr=6., t= Re =,, 3, Fig 6: Nanoluid velocity roile ith increaing Re 5. Eect o arameter variation on temerature roile The tranient eect on the nanoluid temerature roile are clearly revealed in Fig 7-. From Fig 7 it i noted that the temerature increae ith time near the loer all it then decreae toard the uer moving all or a given et o arameter value until a correonding teady tate roile i achieved. Thi behaviour may be attributed by the li condition and injection at the loer all, uction and moving uer all. Interetingly, the temerature o Cu-ater nanoluid rie higher than that o Al O 3 -ater nanoluid a hon in Fig 8. Moreover, ith increae in time the alternating temerature i oberved. The temerature roile attain it maximum value at the uer all and minimum value at the loer all. Fig 9- illutrate the eect o arameter variation on the temerature roile ith Cu-ater a the oring nanoluid. It i oberved that the temerature roile increae ith an increae in the nanoarticle volume raction a hon in Fig 9. Similar trend o increae in temerature i noticed ith increae in li arameter, thi behaviour may be attributed by liery at the all a hon in Fig. Ooite behaviour i oberved in Fig that, nanoluid temerature all ith an increae in Biot number. It i noted that ith an increae in Ecert number the temerature roile rie a hon in Fig. Thi increae in temerature may be attributed by vicou diiation.

10 Michael H Mizu, Oluole D Mainde, Ya Nanah-Gyeye θ Alumina-ater Bi=5, A=., Ec=., Re=, φ=.3, =., Pr=6. =.6,.7,.8, t Fig 7: Nanoluid temerature roile ith increaing time θ Alumina-ater Bi=5, A=., Ec=., Re=, φ=.3, =., Pr=6. t =.,.3,.5, Fig 8: Nanoluid temerature roile ith increaing time θ.8 θ =,.3,.6,.9.4. φ =,.,.,.3 Bi=5, A=., Ec=., Re=, =., Pr=6., t=.4. Bi=5, A=., Ec=., Re=, φ=.3, Pr=6., t= Fig 9: Nanoluid temerature roile roile ith increaing φ Fig : Nanoluid temerature ith increaing Bi=5, A=., Re=,φ=.3, =., Pr=6., t= θ Bi =, 4, 6, 8 A=., Ec=., Re=, φ=.3, =., Pr=6., t= θ Fig : Nanoluid temerature roile roile ith increaing Bi. Ec =.,.,.3, Fig: Nanoluid temerature ith increaing Ec

11 Eect o Navier li and all [...] Couette lo o nanoluid ith convective cooling 5.3 Sin riction and Nuelt number Fig 3 and Fig 4 illutrate the eect o arameter variation on in riction and Nuelt number uing Cu-ater a a oring nanoluid. In ig 3, it i oberved that the in riction increae ith an increae in nanoarticle volume raction. Thi may be due to movement o the uer all, injection at the loer all and uction at the uer all. It i alo noted that the in riction i mall at the injection all and increae toard the uction all. Moreover intereting reult i oberved, that the in riction increae ith an increae in Reynold number and the light increae i oberved hen increae in li arameter but decreae ith increae in reure gradient =,.6, Re = 5, 7, A =.3,.5,.7 Bi=5, Ec=., Pr=6. C φ.65 Fig 3: Sin riction ith increaingφ, A, Re and Meanhile, the Nuelt number increae ith an increae in nanoarticle volume raction, Reynold number and li arameter but decreae ith an increae in reure gradient a illutrated in Fig 4. Thi may be attributed by Couette lo at the uer all, injection and uction =,.6, Re = 5, 7, A =.3..5,.7 Bi=5, Ec=., Pr=6. Nu φ Fig 4: Nuelt number ith increaingφ, A, Re and

12 Michael H Mizu, Oluole D Mainde, Ya Nanah-Gyeye 5.4 Eect o arameter variation on entroy generation rate Fig 5 ho that the entroy generation rate increae ith time at the loer all and decreae ith time acro the uer all. Thi behaviour may be due to li and injection at the loer all, and Couette lo together ith uction. The entroy generated by Al O 3 -ater nanoluid i higher near the loer all comared ith that generated by Cu-ater nanoluid and revere a it aroache the uer all. A rie in an entroy generation rate i oberved at the all ith an increae in nanoarticle volume raction a illutrated in Fig 6. Similar obervation i noted in Fig 7 ith an increae in li condition. Fig 8 ho that the entroy generation rate decreae at the loer all ith an increae in Biot number and revere it trend a it aroache the uer all. N Alumina-ater Bi=5, A=., Ec=., Re=, φ=.3, =., Pr=6. t =.3,.5, N Bi=5, A=., Ec=., Re=, =., Pr= φ =,.,., Fig 5: Entroy generation rate ith rate increaing time Fig 6: Entroy generation ith increaing φ Bi=5, A=., Ec=., Re=, φ=.3, Pr=6., t=.75 A=., Ec=., Re=, φ=.3, =., Pr=6., t=.7 =,.3,.6,.9.7 Bi = 5, 7, 9, N.65 N Fig 7: Entroy generation rate ith increaing Fig 8: Entroy generation rate ith increaing Bi

13 Eect o Navier li and all [...] Couette lo o nanoluid ith convective cooling Eect o arameter variation on Bejan number Fig 9 illutrate the tranient eect on the Bejan number acro the channel. The Bejan number increae ith time near the channel loer all but the light decreae i noted at the channel centreline. Thi can be becaue o rie in the dominant eect o luid riction irreveribility ithin the channel centreline region, the heat traner irreveribility at the channel all, injection, uction, li condition and the Couette lo. The Bejan number roduced by Cu-ater nanoluid eem to be higher than that o Al O 3 -ater nanoluid. Fig ho an increae in the Bejan number at the all ith an increae in nanoarticle volume raction. Thi imlie that increae in nanoarticle volume raction caue domination eect o luid riction irreveribility. Furthermore, it i oberved in Fig that, the Bejan number decreae at loer all and at the centreline o the channel, but increae a it aroache the uer all. Dierent trend i oberved in Fig, hen increaing Ecert number the Bejan number increae in both all. Meanhile, the decreae in Bejan number i oberved at the loer all and centre but increae a it aroache the uer all ith an increae in reure gradient a hon in Fig 3. Generally, thi behaviour may be attributed by li condition, injection, uction, convective cooling and uer moving all Alumina-ater Bi=5, A=., Ec=., Re=, φ=.3, =., Pr=6. t =.8,, Bi=5, A=., Ec=., Re=, =., Pr=6., t= φ =,.,.,.3 Be Be Fig 9: Bejan number ith increaing time Fig : Bejan number ith increaing φ

14 4 Michael H Mizu, Oluole D Mainde, Ya Nanah-Gyeye. Be A=., Ec=., Re=, φ=.3, =., Pr=6., t= Bi = 5, 7, 9, Fig : Bejan number ith increaing Bi Be Bi=5, A=., Re=, φ=.3, =., Pr=6., t=.35.3 Ec =.,.,.3, Fig : Bejan number ith increaing Ec Coer- ater Bi=5, Ec=., Re=, φ=.3, =., Pr=6., t= A =,,, 3 Be Fig. 3: Bejan number ith increaing A 6. Concluion Comutational model and thermodynamic analyi o the eect o Navier li and all ermeability on entroy generation in unteady generalized Couette lo o nanoluid containing Coer (Cu and Alumina (Al O 3 a nanoarticle i reented. Uing a emi-dicretization method together ith Runge-Kutta Fehlberg integration cheme the tranient roblem i numerically tacled. Some o the reult obtained can be ummarized a ollo:

15 Eect o Navier li and all [...] Couette lo o nanoluid ith convective cooling 5 An increae in nanoarticle volume raction and Reynold number caue a decreae in the velocity roile. Meanhile nanoluid velocity roile increae ith an increae in reure gradient. The temerature roile increae ith an increae in the nanoarticle volume raction, li arameter and Ecert number. But a decreae in temerature roile i noticed ith an increae in Biot number. Sin riction increae ith an increae in nanoarticle volume raction, li arameter and Reynold number. But decreae ith an increae in reure gradient. The ame reult are obtained or the Nuelt number. A rie in an entroy generation rate i oberved ith an increae in nanoarticle volume raction and li arameter. It all near the loer all and rie near the uer all ith an increae in Biot number. The Bejan number increae ith time at the loer and uer all but light decreae at the channel centreline. It increae at the all ith an increae in nanoarticle volume raction. A Biot number and reure gradient increae, Bejan number decreae near the loer all and at the centre, but increae a it aroache the uer all. Ecert number caue the increae in Bejan number at the loer and uer all. R F E R E N C E S [] S. U. S. Choi, Enhancing thermal conductivity o luid ith nanoarticle, In: Proc. ASME Int. Mech. Engng. Congre and Exoition, San Francico, USA, ASME, FED 3/MD 66, 995, [] K. Ramarihnan, K. Shailendhra, Hydromagnetic Blood Flo through a Uniorm Channel ith Permeable Wall Covered by Porou Media o Finite, Journal o Alied Fluid Mechanic, Vol. 6, no.,3, [3] O. D. Mainde, D. Theuri, Thermodynamic analyi o variable vicoity MHD unteady generalized Couette lo ith ermeable all, Alied and Comutational Mathematic, 3(, 4, -8 [4] H. F. Ozto, E. Abu-Nada, Numerical tudy o natural convection in artially heated rectangular encloure illed ith nanoluid, Int. J. Heat Fluid Flo, 9, 8, [5] X. Q. Wang, A. S. Mujumdar, Heat traner characteritic o nanoluid: a revie, Int. J. Thermal Science, vol. 46, 7,. -9. [6] Buongiorno, J, Convective tranort in nanoluid, J. Heat Traner, vol. 8,6, [7] R. K. Tiari, M. K. Da, Heat traner augmentation in a to-ided lid-driven dierentially heated quare cavity utilizing nanoluid, Int. J. Heat Ma Tran., 5, 7, 8. [8] A. Bejan, Second-la analyi in heat traner and thermal deign, Adv. Heat Traner, 5, 98, 58.

16 6 Michael H Mizu, Oluole D Mainde, Ya Nanah-Gyeye [9] A. Bejan, Entroy Generation Minimization, CRC, Boca Raton, NY, 996 [] L.C. Wood, Thermodynamic o Fluid Sytem, Oxord Univerity Pre, Oxord, 975 [] U. Naruaa, The econd-la analyi o mixed convection in rectangular duct, Heat Ma Traner, 37, 998, [] A. Z. Sahin, Second la analyi o laminar vicou lo through a duct ubjected to contant all temerature, Journal o Heat Traner,, 998,