HEAT TRANSFER IN NATURAL CONVECTION FLOW OF NANOFLUID ALONG A VERTICAL WAVY PLATE WITH VARIABLE HEAT FLUX. Irfan MUSTAFA 1, Tariq JAVED 2

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1 HEAT TRANSFER IN NATURAL CONVECTION FLOW OF NANOFLUID ALONG A VERTICAL WAVY PLATE WITH VARIABLE HEAT FLUX Iran MUSTAFA Tariq JAVED Department o Matematics Allama Iqbal Open Universit H-8 Islamabad Paistan Department o Matematics and Statistics FBAS International Islamic Universit Islamabad Paistan Abstract: Te present analsis is concerned to eamine te enancement o eat transer in natral convection low o nanolid trog a vertical wav plate assmed at variable eat l. Te rate o eat transer in nanolid low as compared to pre water can be increased de to increase te densit o nanolid wic depends on te densit and concentration o nanoparticles. For tis analsis Tiwari and Das model [] is sed b considering two nanoparticles i.e. Al O 3 (Almina) and C (Copper) are sspended in a base lid (water). A ver eicient implicit inite dierence tecniqe converges qadraticall is applied on te concerning partial dierential eqations or nmerical soltion. Te eects o pertinent parameters namel volme raction parameter o nanoparticle wav srace amplitde Prandtl nmber and eponent o variable eat l on streamlines isotermal lines local sin riction coeicient and local Nsselt nmber are sown trog graps. In tis analsis a maimm eat transer rate is noted in C-water nanolid trog a vertical wav srace as compared to Al O 3 -water and pre water. Kewords: Natral convection low Nanolid Heat transer Variable eat l Nmerical soltion. Introdction Investigation o natral convection low trog a vertical sinsoidal wav plate as signiicant importance in completel analzing te indstrial and engineering problems. Sc lows are controlled b te densit dierence and tereore te low as impact on eat transer. A low on te rog srace is distrbed b te srace and ten low and eat transer rates are canged. Tese ind o rogened sraces ave man applications in real lie wic are taen into accont sc are lat plate condensers eat transer collectors eat ecanger and rerigerators. A well-nown eample o te mecanism o eat ecanger is a radiator wic is commonl sed in veicle. In tis mecanism te generation o eat in engine can be sited to air lowing trog radiator. Yao [] was te pioneer wo stdied eat transer penomena trog vertical wav plate in natral convection low. He obtained nmerical soltion o te problem and ond tat eat transer rate become doble along wav srace. Molic and Yao [] etended te wor o Yao [] b inclding ree stream velocit in mied convection low trog a wav seet. Te conclded tat eat l trog a lat plate is greater tan tat o wav plate. Bavnani and Bergles [3] perormed an eperimental std in natral convection low trog wav plate and noted tat eat transer rate increases b abot % wen ratio between amplitde and wave Corresponding ator Tel.: address: iranmstaa983@aoo.com (I. Mstaa)

2 lengt is 0.3. In anoter article Yao [4] eamined ree convection low trog a comple vertical wav plate representing sinsoidal natre. He observed tat eat transer rate trog a lat seet was smaller as compared to comple wav srace. Jang et al. [] eamined ree convection low trog a vertical wav plate and analsis bot important penomena o eat and mass transer rates b calclating nmerical soltion. Jang and Yan [6] investigated transient aptitdes in natral convection low or bot eat and mass transer trog a comple vertical wav plate. Te applied a coordinate transormation or converting te wav seet to lat seet and nmerical soltion was calclated. Few representative stdies on natral convection low trog a wav seet can be seen in res [7 - ]. Flow along rogened sraces wit wall eat l can be applied or cooling prpose o electronic devices and nclear components []. Molic and Yao [3] reported ree convection low trog a rog vertical srace b considering CHF (Constant Heat Fl). Tastos and Irsaid [4] considered a variable eat l on a orizontal wav srace. In tis std separation points were calclated b considering amplitde o a wav srace as 0. and obtained soltions against wen eponent o variable eat l is greater tan -0.. Tis wor was etended b Tastos and Odat [] b introdcing magnetic ield to te low and ond tat te eat transer rate is directl proportional to te increasing magnetic ield strengt. Rees and Pop [6] stdied ree convection low trog a vertical wav seet in poros medim b considering niorm eat l. Te wor o Rees and Pop [6] etended b Molla et al. [7] introdcing comple wav srace and te ond tat additional armonic sbstantiall canges te temperatre distribtion and low ield near te srace. Raman et al. [8] investigated ree convection low trog a vertical wav cone b considering temperatre dependent viscosit and constant eat l. Neag [9] presented te analsis o te ree convection low trog vertical wav wall satrated in poros medim wit eat and mass les and obtained sstem was solved nmericall. A std o enancement o eat transer rate in lids sc as engine oil etlene glcol and water wit low termal condctivit as become an obstacle in engineering and indstrial problems. To overcome tis problems man researcers wored to rise termal condctivit o tese ind o lids. Eperimental std reveals tat termal condctivit o tese ind o lids can be increased wen nanoparticles are sspended wit size p to 00nm in tese lids and te reslting mitre is nown as nanolid. Te eat transer enancement in Nanolids are ver sel in man indstrial problem sc as parmacetical ood processing el cells microelectronics brid-powered engines and man oters. Nanoparticles in nano-materials are made b sing metals (C Al A Ag Fe) metal carbides (SiC) oides (CO Al O 3 TiO ) and nitrids (SiN AIN) etc. In te presence o tese nanoparticles termal condctivit o nanolid become greater as compared to base lid. Coi [0] was te irst wo introdced te term o nanolid. Te analsis o eat transer in nanolids were investigated b ew researcers i.e. Ab Nada [] Tiwari and Das [] Maiga et al. [3] Oztop and Ab Nada [4] Nield and Kznetsov [6] Jalria et al. [7]. A bondar laer analsis in natral convection low o a nanolid trog an isotermal vertical wav srace was presented b Gorla and Kmari [8]. Te considered Bongiorno s model and solved obtained eqations nmericall sing implicit sceme. Mad and Amed [9] stdied ree convection laminar low o nanolid trog a vertical wav seet satrated in poros medim. Srinivasacara and Kmar [30] discssed mied convection low o a nanolid on an inclined wav seet wit radiation eect satrated in poros medim and solved nmericall sing cebsev spectral collocation tecniqe. Te conclded tat local eat transer rate increases in bot opposing and adding boanc low cases wit te increase o radiation. Habiba et al. [3] investigated a nmerical std on ree convection low o C-water

3 nanolid trog a vertical wav seet wit niorm eat l. Srinivasacara and Kmar [3] investigated ree convection low o a nanolid along an inclined wav srace satrated in a poros medim in presence o wall eat l and observed tat mass and eat transer rates in nanolid increase b increasing te inclination o wav srace. Te aim o crrent std is to investigate te rise o eat transer rate in natral convection low o nanolids b considering Al O 3 and C nanoparticles trog a vertical rog seet wit wall eat l as a nction o. Te soltion o obtained eqations is calclated nmericall sing Keller Bo inite dierence tecniqe and eects o emerging parameters on sin riction coeicient streamlines isoterm lines and local Nsselt nmber are presented grapicall. Matematical ormlation Te investigation on two dimensional stead natral convection low o an incompressible nanolid containing nano-size particles trog a vertical wav plate is carried ot. Te srace o wav is considered as a sinsoidal nction in term o sine is deined as sin / L were and L are amplitde o te waviness and caracteristic lengt o wav plate respectivel sown in Fig. A wav srace is taen place along ais and ais is taen normal to it. Te srace is considered to be eated de to variable l q q m w 0 and applied beind te wav srace wic was considered b Merin and Mamood [33] were q o is constant and / L is dimensionless variable. Figre. : Te psical model and coordinate sstem. Table : Material properties o nanoparticles and base lid [4]. Properties Flid Al O 3 C c p (Jg - K - ) ρ(gm -3 ) (Wm - K - ) (K - ) Two inds o nanoparticles namel Al O 3 and C niorm size and sperical sape [] are sspended in a base lid. Te material properties o nano-size particles and base lid are presented in Table. In tese properties all psical properties ecept densit o nanolid do not depend on temperatre. Te governing partial dierential eqations in nanolid low model 3

4 presented b Tiwari and Das [] in presence o Bossinesq approimations are deined as v 0 () p n ( ) ss v gt T () n n n v v p (3) n v v n n T T v n T (4) were te velocit components and v are taen along and directions respectivel is dnamic viscosit o nanolid described b Brinman[34] p is pressre and s s are densities and termal epansion coeicients o water base lid and nanoparticles respectivel n is termal condctivit and n is densit n is termal disivit o nanolid g is gravit is volme raction parameter o nanoparticle T is temperatre o nanolid niorm ambient temperatre c p n cp and water base lid and nanoparticle respectivel. Te relations o n n n c p s are described as ollows [4] s s n n. s s n T are eat capacities o nanolid c p n n n s n cp cp cp. n s c p n and n In above epression and are sed to represent termal condctivit and dnamic viscosit o water base lid. Te bondar conditions related to present low geometr are given b qw v 0 n T at n () 0 T T p p as were p is pressre at ambient temperatre and n is nit normal to srace. Upon introdcing non-dimensional variables as considered b Pop et al. [3] / L / L 4/ L Gr v Gr v p Gr p p T T / / d Gr Gr qw( ) L L L L L d into Eqs. (-4) and neglecting te term o small order o Gr (Graso nmber) te Eqs. (-4) in dimensionless orm are written as is 4

5 3 / p Gr p v (6) / p Gr v (7). Pr 4 m v n (8) Te vales o Gr 3 and 4 are given b. 3 s s s 4 4. p w s p Gr c g q L c Te inviscid low soltion is sed or calclating te pressre gradient along - ais wic entails / 0. p In Eq. (7) te term on L.H.S is o ) ( O tereore order o p / wold be ). ( / Gr O From Eqs. (6) and (7) te term p / are eliminated we get. 3 v (9) Te bondar conditions in Eq. () tae te new orm as. as at 0 p v n (0) Te ollowing transormation are introdced or redcing te governing partial dierential eqations (8 9) into a convenient orm / 4 / / () in wic represents stream nction and epressed as / and. / v Te above eqations tae orm as () 4 0 Pr 4 m n (3) and bondar conditions are as at 0 n (4)

6 were prime signs denote partial derivative w.r.t.. Te sin riction coeicient nmber N are deined as wit wall sear stress w qw C N U Tw T w and eat l q w are deined as n q n T. 6 C and Nsselt w n 0 w n 0 (6) Te dimensionless orm o sin riction coeicient and Nsselt nmber are obtained b sbstitting Eqs. () and (6) into Eq. () we get C Gr / /. ( 0) N Gr / / 4/ 0. Reslts and Discssion Te soltion o obtained sstem o non-linear PDEs () and (3) sbject to te bondar conditions (4) are calclated nmericall sing Keller Bo implicit inite dierence tecniqe. For detailed std o te metod reader is reerred to Cebeci and Bradsaw [36]. Te compted reslts b present sceme or crrent low problem are sown in Table and compared wit tat o previos reslts o Pop et al. [3] and Siddiqa et al. [0] or varios vales o and ond in good agreement wic sows te eectivit and accrac o te metod. In tis std Al O 3 -water and C-water nanolids are considered. Eects o pertinent parameters namel volme raction parameter o nanoparticle Prandtl nmber Pr amplitde o wav srace and eponent o variable eat l m on C Gr / (local sin riction coeicient) and N Gr -/ (local Nsselt nmber) are presented trog graps. In igres te solid and dased lines are sed to represent te beavior o C-water and Al O 3 -water nanolids respectivel. Also te streamlines and isotermal lines are drawn wic describes te complete mecanism o te present low problem. Figs. and 3 illstrate te variations in C Gr / and N Gr -/ against volme raction parameter or bot C-water and Al O 3 -water nanolids along vertical lat srace ( = 0) and wav srace ( = 0.). It is observed tat bot C Gr / and N Gr -/ increase in bot nanolids wit increasing te vale o wic is de to te reason tat te viscosit o nanolid increases de to te enancement o nanoparticle volme raction. It is clear tat te vales o bot C Gr / and N Gr -/ are iger in C-water nanolid Table:. Nmerical reslts o ( 0) or dierent vales o wit = m = 0.0 and Pr =.0. = 0.0 = 0. Present Pop et al. [3] Siddiqa et al. [0] Present Pop et al. [3] Siddiqa et al. [0] ()

7 / Figre : Te variation o C Gr (local sin riction coeicient) against volme raction parameter wen m = =.0 and Pr = 6. are ied. / Figre 3: Te variation o N Gr (local Nsselt nmber) against volme raction parameter wen m = =.0 and Pr = 6. are ied. tan te vales o Al O 3 -water nanolid de to te igest viscosit and termal condctivit o C-water nanolid. It is also observed tat low and eat transer rates in bot nanolids are 7

8 larger or vertical lat srace ( = 0) as compared to tat o rog wav srace ( = 0.). Figs. 4 and sow te variations o C Gr / and N Gr -/ or bot nanolids against along a wav srace or some vales o amplitde / Figre 4: Te variation o C Gr (local sin riction coeicient) against wen m =.0 Pr = 6. and = 0. are ied. Figre : Te variation o 6. and = 0. are ied. N Gr / (local Nsselt nmber) against wen m =.0 Pr = parameter. It is seen tat C Gr / and N Gr -/ increase along a wav srace in pstream direction also bot decrease wit increase in te amplitde o wav srace. Tis is becase iger rogness o srace enances te viscos eects witin te bondar laer and conseqentl low and eat transer rate are 8

9 / Figre 6: Te variation o C Gr = 6. and = 0. are ied. (local sin riction coeicient) against wen = 0. Pr Figre 7: Te variation o and = 0. are ied. N Gr / (local Nsselt nmber) against wen = 0. Pr = 6. redced. Also low and eat transer rates in C-water nanolid is noted greater tan tat o Al O 3 -water. Eects o eponent o variable eat l m on C Gr / and N Gr -/ are presented in igs. 6 and 7 respectivel. It is seen tat C Gr / decreases and N Gr -/ increases against wit te increasing vales 9

10 / Figre 8: Te variation o C Gr (local sin riction coeicient) amplitde wen m =.0 Pr = 6. and =.0 are ied. Figre 9: Te variation o 6. =.0 and m =.0. N Gr / (local Nsselt nmber) against amplitde wen Pr = o m along a wav srace. Tis is becase decrease o lid temperatre reslts in increase o eat transer rate at wav srace. It is also observed tat te amplitde o te waves decreases te magnitde o C Gr / and increases te magnitde o N Gr -/. Te vales o C-water nanolid in bot C Gr / and N Gr -/ is greater tan te Al O 3 -water nanolid becase densit and termal condctivit o C-water nanolid is larger tan Al O 3 -water nanoid. Figs. 8 and 9 sow te variation in C Gr / and N Gr -/ against amplitde o wav srace or pre water and bot nanolids. It is seen tat as te amplitde o wav plate increases C Gr / and N Gr -/ 0

11 decrease in pre water and or bot nanolids. Te reason or tis decrease is tat wit increase in amplitde te rogness o wav srace enances te viscos eects and ence cases a decrease in low and eat transer rates. Figs. 0(a-) and igs (a-) demonstrate te eecto nanoparticle volme raction parameter on te streamlines and isotermal lines or ied vales o Pr = 6. and = 0.. Fig. 0(a-c) and ig. (a-c) are plotted or niorm eat l i.e. m = 0 and ig. 0(d-) and ig. (d-) are plotted or variable eat l i.e. m = 0. or bot pre water and nanolids. In streamlines te maimm vales o in niorm eat l and variable eat l are sown as ma ( = 0 pre water) ( = 0. AlO3) 3.39 ( = 0. C) and ( = 0 pre water) 3.46 ( = 0. AlO3).899 ( =0. C). It is observed tat velocit decreases in nanolids as compared to pre water tis is becase de to te enricment o viscosit o nanolid increases. In addition to tis in nanolids it is also observed tat velocit in C-water is smaller tan AlO3-water de to large densit o C nanoparticle. In case o variable eat l te vales o become smaller as

12 Figre 0(a-): streamlines plot or m = 0.0 (a) = 0.0 (b) = 0. (AlO3) (c) = 0. (C) and m = 0. (d) = 0.0 (e) = 0. (AlO3) () = 0. (C) wit Pr = 6. and = 0..

13 Figre (a-): Isotermal lines plot or m = 0.0 (a) = 0.0 (b) = 0. (Al O 3 ) (c) = 0. (C) and m = 0. (d) = 0.0 (e) = 0. (Al O 3 ) () = 0. (C) wit Pr = 6. and = 0.. compare to te vales o niorm eat l. Wic is becase velocit redces in variable eat l. In Fig. (a-) it is seen tat isotermal lines epands or bot Al O 3 -water and C-water nanolids as compared to water base lid. It reveals tat temperatre in nanolids become iger de to larger termal condctivit o nanolids as pre water. It is also seen tat isotermal lines contracts in case o variable eat l i.e. m = 0. as compared to niorm eat l i.e. m = 0. It reveals tat temperatre decreases witin te bondar laer in presence o variable eat l. Conclsions In tis article we analzed te wa ow one can enance te rate o eat transer in natral convection low trog a sinsoidal wav plate. For tis action we incorporated two nano-size particles i.e. Al O 3 and C in a water base lid. Te soltion o obtained eqations is calclated nmericall sing Keller Bo inite dierence tecniqe and eects o involving parameters namel eponent o variable eat l m amplitde o wav srace volme raction parameter o nanoparticle and Prandtl nmber Pr on C Gr / and N Gr -/ are eamined trog graps. Te core inding o te present analsis are mentioned below:. Te vales o C Gr / and N Gr -/ or eat transer rate or bot C-water and Al O 3 -water nanolids along wav plate are iger tan tat o water base lid.. For increasing concentration o nanoparticle te vales o C Gr / and N Gr -/ or C-water nanolid is larger tan Al O 3 -water nanolid or bot lat and wav plates bt tese vales are smaller in case o wav plate as compared to lat plate. 3. Te vales o C Gr / and N Gr -/ redce wit increasing amplitde o wav seet. 4. Wit increasing te eponent o variable eat l te vales o C Gr / decreases and N Gr -/ increases.. Velocit and temperatre o nanolid decreases and increases witin te bondar laer as compared to water base lid. Acnowledgement: Ators are gratel to editor/anonmos reviewers or teir valable comments. Tis researc material is based on wor spported b te Higer Edcation Commission o Paistan trog National Researc Program or Universities (NRPU) nder Project No 07 and Dr. Tariq Javed is ver tanl or it. 3

14 Reerences [] Yao L. S. Natral convection along a vertical wav srace Jornal o Heat Transer 0 (983) pp [] Molic S. G. Yao L. S. Mied convection along wav srace Jornal o Heat Transer (989) pp [3] Bavnani S. H. Bergles A. E. Natral convection eat transer rom sinsoidal wav sraces Heat and Mass Transer 6 (99) pp [4] Yao L. S. Natral convection along a vertical comple wav srace International Jornal o Heat and Mass Transer 49 (006) pp [] Jang J. H. et al. Natral convection eat and mass transer along a vertical wav srace International Jornal o Heat and Mass Transer 46 (003) pp [6] Jang J. H. Yan W. M. Transient analsis o eat and mass transer b natral convection over a vertical wav srace International Jornal o Heat and Mass Transer 47 (004) pp [7] Mad A. et al. MHD ree convection low along a vertical wav srace wit eat generation or absorption eect International Commnications in Heat and Mass Transer 33 (006) pp [8] Naraana P. A. L. Sibanda P. Soret and Dor eects on ree convection along a vertical wav srace in a lid satrated Darc poros medim International Jornal o Heat and Mass Transer 3 (00) pp [9] Alim M. A. et al. Heat generation eects on MHD natral convection low along a vertical wav srace wit variable termal condctivit American Jornal o Comptational Matematics (0) pp [0] Siddiqa S. et al. Natral convection low wit srace radiation along a vertical wav srace Nmerical Heat Transer Applications 64 (03) pp [] Parveen N. et al. Viscos dissipation eect on natral convection low along a vertical wav srace Procedia Engineering International Conerence on Mecanical Engineering 90 (04) pp [] S J. J. Pop I. On termal bondar laers on a lat plate sbjected to a variable eat l International Jornal o Heat and Flid Flow 9 (998) pp [3] Molic S. G. Yao L. S. Natral convection along a wav srace wit niorm eat l Jornal o Heat Transer (989) pp [4] Tastos B. Irsaid E. A. Heat and lid low rom a wav srace sbjected to a variable eat l Acta Mecanica (00) pp. -8. [] Tastos B. Al-Odat M. Magnetic ield eect on eat and lid low over a wav srace wit a variable eat l Jornal o Magnetism and Magnetic Materials 68 (004) pp [6] Rees D. A. Pop I. Free convection indced b a vertical wav srace wit niorm eat l in a poros medim Jornal o Heat Transer 7 (99) pp [7] Molla M. et al. Natral convection low along a vertical comple wav srace wit niorm srace eat l Jornal o Heat Transer 9 (007) pp [8] Raman A. et al. Natral convection low along te vertical wav cone in case o niorm srace eat l were viscosit is an eponential nction o temperatre International Commnications in Heat and Mass Transer 38 (0) pp

15 [9] Neag M. Free convective eat and mass transer indced b a constant eat and mass les vertical wav wall in a non-darc doble stratiied poros medim International Jornal o Heat and Mass Transer 4 (0) pp [0] Coi S. U. S. Enancing termal condctivit o lids wit nanoparticles Proceedings International Mecanical Engineering Congress. San Francisco USA ASME FED 3/MD 66 (99) pp [] Ab-Nada E. Application o nanolids or eat transer enancement o separated lows encontered in a bacward acing step International Jornal o Heat and Flid Flow 9 (008) pp [] Tiwari R. J. Das M. K. Heat transer agmentation in a two sided lid driven dierentiall eated sqare cavit tilizing nanolids International Jornal o Heat and Mass Transer 0 (007) pp [3] Maiga S. E. B. et al. Heat transer enancement b sing nanolids in orced convection lows International Jornal o Heat and Flid Flow 6 (00) pp [4] Oztop H. F. Ab-Nada E. Nmerical std o natral convection in partiall eated rectanglar enclosres illed wit nanolids International Jornal o Heat and Flid Flow 9 (008) pp [] Nield D. A. Kznetsov A. V. Te cengeminowcz problem or natral convective bondar laer low in a poros medim satrated b a nanolid International Jornal o Heat and Mass Transer (009) pp [6] Kznetsov A. V. Nield D. A. Natral convective bondar laer low o a nanolid past a vertical plate International Jornal o Termal Science 49 (00) pp [7] Jalria Y. et al. Heat transer in nanolid Advances in Mecanical Engineering Article ID (0) pp.. [8] Gorla A. S. Kmari M. Free convection along a vertical wav srace in a nanolid Jornal o Nanoengineering and Nanosstems (0) pp [9] Mad A. Amed S. E. Laminar ree convection over a vertical wav srace embedded in a poros medim satrated wit a nanolid Transport in Poros Media 9 (0) pp [30] Srinivasacara D. Kmar P. V. Mied convection over an inclined wav srace in a nanolid satrated non-darc poros medim wit Radiation Eect International Jornal o Cemical Engineering Article ID 9708 (0) pp.. [3] Habiba F. et al. Natral convection low o C-H O nanolid along a vertical wav srace wit niorm eat l International Conerence on Mecanical Engineering (0) pp [3] Srinivasacara D. Kmar P. V. Free convection o a nanolid over an inclined wav srace embedded in a poros medim wit wall ea l Procedia Engineering International Conerence on comptational Heat and Mass Transer 7 (0) pp [33] Merin J. H. Mamood T. On te ree convection bondar laer on a vertical plate wit prescribed srace eat l Jornal o Engineering Matematics 4 (990) pp [34] Brinman H. C. Te viscosit o concentrated sspensions and soltion Jornal o Cemical Psics 0 (9) pp [3] Pop I. et al. Free convection abot a vertical wav srace wit prescribed srace eat l in a micropolar lid Tecnisce Mecani 8 (998) pp [36] Cebeci T. Bradsaw P. Psical and comptational aspects o convective eat transer

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