NATURAL CONVECTION OF NANOFLUIDS ABOUT A VERTICAL CONE EMBEDDED IN A POROUS MEDIUM: PRESCRIBED HEAT GENERATION/ABSORPTION SOURCE

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1 5 th International Conerene on Aliation o Porous Media Roania 3 NAURAL CONVECION OF NANOFLUIDS ABOU A VERICAL CONE EMBEDDED IN A POROUS MEDIUM: PRESCRIBED HEA GENERAION/ABSORPION SOURCE Ainreza Noghrehabadi Eail: noghrehabadi@su.a.ir Ali Behseresht Eail: behseresht.ali@gail.o Mohaad Ghalabaz Eail:.ghalabaz@gail.o *Corresonding author: Ainreza Noghrehabadi Ph.D. Assistant roessor at Deartent o Mehanial Engineering Shahid Charan University o Ahvaz Ahvaz Iran. Eail: noghrehabadi@su.a.ir Mobile: ell: Et Fa: Setion: Natural and ored onvetion in orous edia Abstrat he analysis and siulation o natural onvetion in saturated orous edia have any iortant engineering and geohysial aliations. In this aer natural onvetion heat transer over a vertial one in a Dary orous ediu saturated ith a nanoluid subjet to heat generation/absortion is studied. he odel used or the nanoluid inororates the eets o Bronian otion and therohoresis. he governing artial dierential equations are transored into a set o ordinary dierential equations using aroriate siilarity transorations and then nuerially solved using the Runge-Kutta ethod. he inluene o araetri variation o the heat generation/absortion araeter on the to iortant araeters o heat and ass transer redued Nusselt nuber and redued Sherood nuber is investigated. he results sho that an inrease in the heat generation/absortion araeter dereases the redued Nusselt nuber hereas inreases the redued Sherood nuber. Keyords: Nanoluids Natural Convetion Porous Media Heat generation/absortion Bronian otion herohoresis. Noenlature D B D Le Nb Nr Nt P Bronian diusion oeiient therohoreti diusion oeiient resaled nano artile volue ration Leis nuber Bronian otion araeter buoyany ratio therohoresis araeter ressure Ra loal Rayleigh nuber S diensionless strea untion Greek sybols µ visosity o luid α theral diusivity o orous edia β voluetri eansion oeiient o luid γ one angle η diensionless distane θ diensionless teerature τ araeter deined by equation (6) Introdution he study o natural onvetion heat transer in a orous ediu is gaining a lot o attention beause o its any engineering aliations suh as theral energy storage groundater systes lo through iltering edia and rude oil etration []. Aong these aliations the theral onvetion in a luid saturated orous ediu ith internal energy soures is very iortant in the theory o theral ignition and in robles dealing ith heial reations and those onerned ith dissoiating luids []. Possible heat generation eets in the orous sae ay alter the teerature distribution; onsequently the artile deosition rate in nulear reators seiondutor aers and eletroni his [3]. he roble o natural onvetion around a one ebedded in a orous ediu at high Rayleigh nubers has been nuerially analyzed by Cheng et al. [4]. hen Yih [5] nuerially investigated the eet o unior lateral ass lu on natural onvetion around a one ebedded in a saturated orous ediu using Keller bo ethod. Hoever the lo and heat transer o natural onvetion over ebedded bodies in orous ediu has been studied in large aount o aers but

2 e aers onsidered the nanoluid lo and heat transer. he nanoluid lo and heat transer in orous ediu is oletely a ne roble. Hene only has e orks been done in this area. he onsideration o additional heat transer ehaniss in the onvetive heat transer robles as taken one ste urther by Buongiorno in 6 [6]. He introdued sli ehaniss or nanoartiles. Fro this oint o vie the natural onvetive boundary-layer lo over a vertial late ebedded in a orous ediu hih is saturated by a nanoluid has been investigated by Nield and Kuznetsov [7]. he ied onvetive boundary layer lo over a vertial edge ebedded in a orous ediu saturated ith a nanoluid subjet to natural onvetion doinated regie has been analyzed by Gorla et al. []. Later Rashad et al. [8] investigated the natural onvetion boundary layer o a non-netonian luid about a ereable vertial one ebedded in a orous ediu saturated ith a nanoluid. hey negleted the heat generation/absortion eet in their odel. In the resent study the eet o heat generation/absortion araeter on the heat and ass transer over a vertial one ebedded in a orous ediu saturated ith a nanoluid has been theoretially eained. o the best o author s knoledge this is the irst tie hih the eet o heat generation/absortion in the resene o dynai eets o nanoluids is analyzed. Matheatial Forulations Consider the to-diensional roble and steady natural onvetion boundary layer lo ast along a vertial one hih is laed in a orous ediu saturated ith nanoluid in the resene o heat generation/absortion. he oordinate syste is hosen suh that the -ais is aligned ith the lo over the one. he shee o hysial odel and the oordinate syste have been shon in Figure. Although there are three distint boundary layers naely hydrodynai boundary layer theral boundary layer and nanoartile onentration boundary layer over the one only one boundary layer has been shon in this igure to avoid ongestion. It is assued that the teerature at the surae o the one is onstant and also the nanoartile volue ration (φ) at the all surae (y=) takes the onstant value o φ. In the orous ediu the heat is either generated or absorbed ith the rate o Q here Q is negative in the ase o heat absortion and ositive in ase o heat generation. he abient values o and φ are denoted by and φ resetively. he lo in the orous ediu ith orosity o ε and ereability o K is onsidered as Dary lo. It is also assued that the nanoluid and orous ediu are hoogeneous and in loal theral equilibriu. By eloying the Oberbek-Boussinesq aroiation and alying the standard boundary layer aroiations the basi steady onservation o total ass oentu theral energy and nanoartiles or nanoluids in the Cartesian oordinate syste o and y are as ollos [8] and [9] ru y u K rv y P g os g os D y y y y u v DB Q D u v DB y y y α and τ in the above equations are deined as: k () () (3) (4) (5) (6) Based on the roble desrition the boundary onditions are: v at y (7) u v at y (8) Equations () and (3) are siliied using rossdierentiation and the ontinuity equation ill also be satisied by introduing a strea untion (ψ): u v r y r (9)

3 Here an loal Rayleigh nuber Ra be introdued as: Ra gk os By introduing the siilarity variable η as () y Ra () and the diensionless siilarity quantities S θ and ƒ as S r Ra and also by onsidering () r sin (3) Eqs. ()-(5) an be ritten as olloing three ordinary dierential equations S Nr. (4) 3... S Nb Nt (5) 3 Nt Le. S. (6) Nb subjet to the olloing boundary onditions here Nr Nb Nt Le (7) : S= = : S (8) Q K.os. g D B D B D (9) In the above equations here Q varies ith the inverses o osition (/) the obtained equation are in the or o ully siilarity solution or else they are in the or o loal siilarity. o iortant heat transer araeters loal Nusselt nuber (Nu ) and loal Sherood nuber (Sh ) are deined as [7]: Nur Nu Ra r Sh Ra Sh () Solution Consider the syste o equations (4)-(6) subjet to the boundary onditions (7) and (8). he syste o equations is nuerially solved using an eiient iterative ourth-order Runge-Kutta ethod starting ith an initial guess. In this ethod every n th -order equation is onverted to n irst order equations. hereore the syste o ordinary dierential equation is onverted into a syste o irst order equations. hen an iteration ethod is alied on the entioned irst order dierential equations syste. he ritial value o this ethod is the seletion o aroriate asytoti value o η. In the resent study η is taken to be hih ensures the asytoti onvergene o the solution. hen the η diretion is divided into 3 nodal oints so that the results beoe aurate. A aiu relative error o -5 is used as the stoing riteria or the iterations. As a test o the auray o the solution the value o Nur is oared ith the reorted value by Cheng et al. [4] and Yih [5] in table hen the eets o nanoluid araeters and heat generation/absortion araeter are negleted in this study. able : Coarison o results or redued Nusselt nuber Cheng et Present Results Yih [5] al. [4] (Nt=Nb=Nr=) Results and disussion Figure shos the eet o heat generation /absortion on the veloity and nanoartile onentration roiles or ied values o Nr Nb Nt and Le. As seen an inrease in the heat generation/ absortion araeter inreases the veloity roile hereas dereases the nanoartile onentration roile. Figure 3 deonstrates the variation o teerature roiles or dierent values o heat generation/absortion araeter. he results deit that inrease o heat generation/absortion araeter inreases the value o teerature roile. It is orth entioning that by integrating Eq. (4) the obtained equation is learly shon a diret deendeny beteen the veloity teerature and nanoartile onentration. Nu qw q Sh k D W B W () here q and q are the all heat lu and ass lu resetively. Using siilarity transors introdued in Eq. () the redued Nusselt nuber -θ'() and redued Sherood nuber -'() are obtained as ollos: 3

4 Figure 4: Variation o Nu r as a untion o Nt and λ Figure : Veloity and nanoartile onentration roiles or various values o λ Figure 5: Variation o Sh r as a untion o Nt and λ Figure 3: eerature roiles or various values o λ Figures 4 and 5 are lotted or dierent values o λ (i.e. -.3<λ <.) and ied values o Nr Nb Nt and Le to dislay obviously the eet o heat generation araeter (λ) on the redued Nusselt nuber and the redued Sherood nuber. Figures 4 and 5 reveal that inrease o heat generation/absortion araeter dereases the agnitude o redued Nusselt nuber hereas it inreases the agnitude o redued Sherood nuber. Inrease o heat generation araeter dereases the agnitude o teerature gradient on the all surae hih results in the derease o redued Nusselt nuber. Also inrease o heat generation/absortion araeter inreases the agnitude o onentration gradient on the all and dereases the onentration boundary layer thikness hih these results in the inrease o redued Sherood nuber. Conlusion In this aer a obined siilarity and nuerial aroah have been used to theoretially investigate the natural onvetion ro a vertial one hih is ebedded in a nanoluid-saturated orous ediu in the resene o internal heat generation/absortion. In the odeling o nanoluid therohoresis and Bronian otion eets have been taken into aount. he results sho that inrease o heat generation/absortion araeter dereases the redued Nusselt nuber hereas inreases the redued Sherood nuber. he inrease o therohoresis araeter dereases the redued Nusselt and Sherood nubers. Reerenes [] Gorla R.S.R. Chakha J. and Rashad A.M.. onvetive boundary layer lo over a vertial edge ebedded in a orous ediu saturated ith a nanoluid: Natural Convetion Doinated Regie. Nanosale Researh Letters [] Hady F.M. Ibrahi F.S. Abdel Gaied S.M. and Eid M.R.. Eet o heat generation/ absortion on natural onvetive boundary-layer lo ro a vertial one ebedded in a orous 4

5 ediu illed ith a non-netonian nanoluid. International Couniations in Heat and Mass ranser [3] Ferdousi A. Ali M.A.. Natural onvetion lo ro a orous vertial late in resene o heat generation. Journal o siene and tehnology 5() [4] Cheng P. Le.. and Po I Natural onvetion o a darian luid about a one. Internationl Couniations in Heat and Mass ranser [5] Yih K.A he eet o unior lateral ass lu on ree onvetion about a vertial one ebedded in a saturated orous edia. Internationl Couniations in Heat and Mass ranser [6] Buongiorno J. 6. Convetive ransort in Nanoluids. Journal o Heat ranser [7] Nield D.A. and Kuznetsov A.V. 9. he Cheng Minkoyz roble or natural onvetive boundary layer lo in a orous ediu saturated by a nanoluid. International Journal o Heat and Mass ranser [8] Rashad A.M. El-Hakie M.A. and Abdou M.M.M.. Natural onvetion boundary layer o a non-netonian luid about a ereable vertial one ebedded in a orous ediu saturated ith a nanoluid. Couters and Matheatis ith Aliations [9] Nield D.A. and Bejan A. 6. Convetion in orous edia. Sringer Neyork. 5