Double Diffusive Convection from a Permeable Horizontal Cylinder of Elliptic Cross Section in a Saturated Porous Medium

Transcription

1 Proceedigs o the 4th WSEAS It. Co. o HEAT TRANSFER, THERMAL ENGINEERING ad ENVIRONMENT, Elouda, Greece, August -3, 6 (pp357-36) Double Diusive Covectio rom a Permeable Horizotal Cylider o Elliptic Cross Sectio i a Saturated Porous Medium CHING-YANG CHENG Departmet o Mechaical Egieerig Souther Taiwa Uiversity o Techology, Natai Street, Yugkag 7 TAIWAN Abstract: - The double diusive covectio ear a permeable horizotal cylider o elliptic cross sectio with uiorm wall temperature ad cocetratio i a luid-saturated porous medium are umerically studied. A coordiate trasormatio is employed to trasorm the goverig equatios ito odimesioal osimilar boudary layer equatios. The obtaied boudary layer equatios are the solved by the cubic splie collocatio method. The iluece o the traspiratio parameter ad the eccetricity o the heat ad mass traser characteristics ear a permeable horizotal cylider o elliptic cross sectio i a luid-saturated porous medium is examied as the maor axis o the elliptic cylider is vertical (sleder orietatio) ad horizotal (blut orietatio). Icreasig the traspiratio parameter teds to decrease the boudary layer thickess ad thus ehaces the heat ad mass traser rates betwee the luid ad the wall. Moreover, the heat ad mass traser rates o the cylider with sleder orietatio are higher tha those o the cylider with blut orietatio. Key-Words: - Heat ad mass traser, Permeable, Natural covectio, Porous medium, Elliptic cylider, Cubic Splie collocatio method, Coordiate trasormatio Itroductio Coupled heat ad mass traser drive by combied thermal ad solutal buoyacy orces i a luid-saturated porous medium is o great importace i geophysical, geothermal ad idustrial applicatios, such as the extractio o geothermal eergy, the dispersio o chemical cotamiats through water-saturated soil ad the migratio o moisture through air cotaied i ibrous isulatios. ea ad Khair [] used Darcy s law to study the eatures o atural covectio boudary layer low drive by temperature ad cocetratio gradiets. Lai ad Kulacki [] studied the atural covectio boudary layer alog a vertical surace with costat heat ad mass lux icludig the eect o wall iectio. Yih [3] studied the heat ad mass traser characteristics i atural covectio low over a trucated coe subected to variable wall temperature ad cocetratio or variable heat ad mass lux embedded i porous media. Cheg [4] uses itegral approach to study the magetic eects o heat ad mass traser by atural covectio rom a vertical plate i a luid-saturated porous medium. Similarity solutios or atural covectio heat traser o a horizotal cylider i a saturated porous medium have bee preseted by Merki [5]. Fad et al. [6] examied experimetally the heat traser characteristics by atural covectio rom a horizotal cylider embedded i porous media. Yücel [7] studied the heat ad mass traser about a vertical cylider with costat wall temperature ad cocetratio i a porous medium. Yih [8] examied the heat ad mass traser by atural covectio rom a permeable horizotal cylider i a porous medium with costat wall temperature ad cocetratio. Merki [9] studied the atural covectio boudary layer low o cyliders o elliptic cross sectio i a porous medium. Pop et al. [] examied the atural covectio heat traser about cyliders o elliptic cross sectio i a porous medium. Motivated by the works above, this article applied the coordiate trasormatio ad the cubic splie collocatio method to aalyze the heat ad mass traser by atural covectio alog a permeable horizotal cylider o elliptic cross sectio embedded i luid saturated porous media with costat wall temperature ad cocetratio. The results obtaied herei are compared with the similarity solutios or horizotal cyliders obtaied by Merki [5] ad by Yih [8] to check the accuracy. The iluece o the traspiratio parameter ad the eccetricity o the heat ad mass traser characteristics ear a permeable horizotal cylider o elliptic cross sectio i a luid-saturated porous

2 Proceedigs o the 4th WSEAS It. Co. o HEAT TRANSFER, THERMAL ENGINEERING ad ENVIRONMENT, Elouda, Greece, August -3, 6 (pp357-36) medium is examied whe the maor axis is horizotal (blut orietatio) ad vertical (sleder orietatio). Problem Formulatio Cosider the eect o traspiratio o the combied heat ad mass traser by ree covectio ear a buried horizotal cylider o elliptic cross sectio with blut orietatio embedded i a homogeeous luid-saturated porous medium, as show i Fig., where a is the legth o semi-maor axis ad b is the legth o semi-mior axis or the elliptical cylider. I this igure, A represets the agle made by the outward ormal rom the cylider with the dowward vertical ad is the eccetric agle. It should be oted that or cyliders o elliptic cross sectio there are two orietatios to cosider: the orietatio is blut whe the maor axis is horizotal, as show i Fig., ad the orietatio is sleder whe the maor axis is vertical. The surace o the cylider is held at a costat temperature T w which is higher tha the ambiet porous medium temperature T. I additio, the cocetratio o a certai costituet i the solutio that saturates the porous medium varies rom C w o the luid side o the surace o the cylider to C suicietly ar rom the surace o the cylider. The traspiratio velocity is uiorm. The luid properties are assumed to be costat except or desity variatios i the buoyacy orce term. With itroducig the boudary layer ad oussiesq approximatios, the equatios goverig the steady-state coservatio o mass, mometum, eergy ad costituet or Darcia low through a homogeeous porous medium ear the surace o the horizotal cylider o elliptic cross sectio ca be writte i two-dimesioal Cartesia coordiates ( x, y) as [] u v + = x K si A u = ρgβ t T T + ρgβ c C C µ [ ( ) ( )] () () T T T u + v = α x (3) C C C u + v = D x (4) The appropriate boudary coditios are: v = V w, T = Tw, C = Cw o y = (5) u =, T = T, C = C as y (6) g a Fig.. Physical model ad coordiates or a elliptic cylider o blut orietatio. Here u ad v are the volume-averaged velocity compoets i the x -directio ad y -directio, respectively. T ad C are the volume-averaged temperature ad cocetratio, respectively. Property µ is the dyamic viscosity o the luid, K is the permeability o the porous medium, ad ρ is the luid desity. Furthermore, α ad D are the equivalet thermal ad mass diusivity o the saturated porous medium, respectively. β t ad β c are the coeiciets or thermal expasio ad or cocetratio expasio o the saturated porous medium, respectively, ad g is the gravitatioal acceleratio. V w is the uiorm traspiratio velocity. Ater itroducig the stream uctio ψ to satisy the relatios: u = ad v x, we the deie the odimesioal variables: = x / a, η = ( y a), ψ = ψ ( α ), θ = ( T T ) ( Tw T ), = ( C C ) ( Cw C ). Equatios ()-(6) become the ollowig equatios: = ( θ + NC) si A (7) θ θ θ = (8) = (9) Le The associated boudary coditios are V w a, θ =, = o η = () α =, θ =, = as η () b x A y

3 Proceedigs o the 4th WSEAS It. Co. o HEAT TRANSFER, THERMAL ENGINEERING ad ENVIRONMENT, Elouda, Greece, August -3, 6 (pp357-36) I the above equatios, = Kβ t ga ( Tw T ) ( αν ) is the Darcy-yleigh umber, Le = α D is the Lewis umber ad N = β C C ) [ β ( T T )] c ( w t w is the buoyacy ratio. A urther trasormatio is eeded or bodies with rouded lower eds because si A approaches a costat value as approaches zero []. The ew odimesioal variable is deied as (, η) = ψ () Substitutig Eq. () ito Eqs. (7)-(9), we obtai the ollowig boudary-layer goverig equatios: si A = ( θ + N ) (3) θ θ + θ = θ (4) + = Le (5) The boudary coditios are = w, θ =, = o η = (6) =, θ =, = as η (7) I terms o the ew variables, the Darcia velocities i x- ad y- directios ca be expressed as u = α a (8) α v + a (9) Here primes deotes partial derivatio with respect to η. As b / a =, N = ad w =, Eqs. (3)-(4) become the equatios or the Darcy atural covectio heat traser ear a horizotal circular cylider i a luid-saturated porous medium preseted by Merki [5]. ( V a) α w = w is the traspiratio parameter. Note that w < whe V w > (the case o blowig), ad w > whe V w < (the case o suctio). Here ad si A ca be give i terms o the eccetric agle by the relatios: () For blut orietatio: ( e si ) d = () b si si A = () a ( e si ) () For sleder orietatio: ( e cos ) d = () Table. The value o θ F i, G i, S i, F i, G i, S i, θ F i,, i G,, ad S i,. + τ il τ θ + i τ + τ il τ i τ Le Table. Compariso o values o Nu or N = ad b / a = betwee the preset results with the solutios reported by Merki [5] ad Yih [8]. Merki Yih [8] Preset [5] results si si A = (3) ( e cos ) where e deotes the eccetricity expressed as e = ( b a ) ad b/a is the aspect ratio o the elliptic cylider. Whe approaches zero, as show I Eqs. ()-(3), the value o si A approaches the aspect ratio b / a or the elliptic cylider with blut orietatio while the value o si A

4 Proceedigs o the 4th WSEAS It. Co. o HEAT TRANSFER, THERMAL ENGINEERING ad ENVIRONMENT, Elouda, Greece, August -3, 6 (pp357-36) approaches the value o a b or the elliptic cylider with sleder orietatio. The local Nusselt umber ca be writte as Nu θ (,) (4) The local Sherwood umber ca be give by Sh (, ) (5) I Eqs. (4)-(5), Nu = ha k ad Sh = a D where h ad are the local heat traser coeiciet ad the local mass traser coeiciet, respectively. The average Nusselt umber or the elliptic cylider ca be derived as: () For blut orietatio, )( e si ) ( e si ) d θ ( d Nu m (6) () For sleder orietatio, )( e cos ) ( e cos ) d θ ( d Nu m (7) The average Sherwood umber or the elliptic cylider ca be give by: () For blut orietio, )( e si ) ( e si ) d ( d Sh m (8) () For sleder orietatio, )( e cos ) ( e cos ) d ( d Sh m (9) Note that Num = hma k ad Shm = ma D where h m ad m are the average heat traser coeiciet ad the average mass traser coeiciet or the elliptic cylider, respectively. 3 Problem Solutio The goverig dieretial equatios, Eqs. (4) ad (5), ad the appropriate boudary coditios, Eqs. (6) ad (7), ca be solved by the cubic splie collocatio method [, ]. The Simpso s rule or variable grids is used to calculate the value o at every positio rom Eq. (3) ad boudary coditios (6) ad (7). Variable grids with grid poits are used i the η directio. The miimum step size is.. The value o the edge o the boudary Nu Fig.. Eects o traspiratio parameter o the local Nusselt umber. Sh sleder blut sleder blut w=-.3 w= w=.3 b/a=.6 N= Fig. 3. Eects o traspiratio parameter o the local Sherwood umber. layer η is about. Moreover, a grid with 5 grid poits is used i the directio. At every grid poit, the iteratio process cotiues util the covergece 6 criterio or all the variables,, is achieved. The preset calculatio or Eqs. (3)-(7) ca be perormed rom the bottom up to the top o the elliptic cylider without ecouterig a sigularity. Here by usig the cubic splie collocatio method [, ], Eqs. (4) ad (5) ca be discretized by usig the alse trasiet techique, as w=-.3 w= w=.3 b/a=.6 N=

5 Proceedigs o the 4th WSEAS It. Co. o HEAT TRANSFER, THERMAL ENGINEERING ad ENVIRONMENT, Elouda, Greece, August -3, 6 (pp357-36) θ + θ θ + il il + θ τ lθ = Lθ (3) + + il il + τ + L + = + l Le where = i i, η = η i ηi ϑ ϑ ϑ = = ϑ i + or i = 3ϑ 4ϑ ϑ i or i (3) l ϑ =, L ϑ = (3) Note that ϑ reers to θ ad, ad the quatity + τ = τ τ represets the alse time step. Ater some arragemet, Eqs. (3) ad (3) ca be writte i the ollowig splie approximatio orm: ϑ = Fi, + G lϑ + S Lϑ (33) The quatities F, G, ad S are kow coeiciets evaluated at previous time steps (Table ). y usig the cubic splie relatios [, ], Eq. (33) may be writte i the ollowig tridiagoal orm as A ϑ + ϑ + C ϑ + = D (34) Here Eq. (34) ca be easily solved by usig the Thomas algorithm. I order to check the accuracy o the preset method, the local Nusselt umber Nu / or b a = ad N = obtaied i the curret study uder Darcia assumptios or a horizotal circular cylider are compared with the solutios reported by Merki [5] ad Yih [8]. As show i Table, the preset results are oud to be i excellet agreemet with the results o Merki [5] ad Yih [8]. Figs. ad 3 plot the variatio o the local Nusselt umber Nu / ad the local Sherwood umber Sherwood umber Sh / as uctios o the eccetric agle o the elliptical cylider or various traspiratio parameters ( w = -.3,,.3), N=, ad b/a=.6. For the elliptical cylider with blut orietatio, the local Nusselt umber ad the local Sherwood umber irst icrease with distace rom the stagatio poit, reach a maximum, Nu m Fig. 4. Eects o aspect ratio o the average Nusselt umber. Sh m b / a b / a w=.3 N= sleder blut w=.3 N= sleder blut Fig. 5. Eects o aspect ratio o the average Sherwood umber. ad the decrease to zero at the top o the elliptical cylider. For a elliptical cylider with sleder orietatio, the local Nusselt umber ad the local Sherwood umber decrease mootoically rom the lower ed o the cylider to the upper ed o the cylider; that is due to the icrease i boudary layer thickess. Comparig the curves i Figs. ad 3, we ca deduce that icreasig the traspiratio parameter teds to decrease the boudary layer thickess ad thus icreases the heat ad mass traser rates betwee the luid ad the wall. The results show that b

6 Proceedigs o the 4th WSEAS It. Co. o HEAT TRANSFER, THERMAL ENGINEERING ad ENVIRONMENT, Elouda, Greece, August -3, 6 (pp357-36) the use o blowig ( w < ad V w > ) teds to decrease the heat ad mass traser rate while the use o suctio ( w > ad V w < ) icreases the heat ad mass traser rate. Figs. 4 ad 5 show the average Nusselt umber Nu m / ad the average Sherwood umber Sh m / as a uctio o the aspect ratio b/a or w =.3, ad N=. The average Nusselt umber ad the average Sherwood umber o the elliptic cylider with sleder orietatio are higher tha those o the elliptical cylider with blut orietatio or ay aspect ratio b/a smaller tha oe. Whe the aspect ratio b/a is icreased (i.e., the eccetricity is decreased), the average Nusselt umber ad the average Sherwood umber or the elliptic cylider o sleder orietatio ted to decrease while those or the elliptic cylider o blut orietatio ted to icrease, ad ially the average Nusselt umber ad the average Sherwood umber o sleder orietatio equal to those o blut orietatio whe the aspect ratio b/a equals to oe. Thereore, the elliptic cyliders o sleder orietatio are oud to be superior to the elliptic cyliders o blut orietatio rom the viewpoit o the heat ad mass traser rates i luid-saturated porous media. 4 Coclusio The coupled heat ad mass traser by atural covectio o a permeable horizotal cylider with elliptic cross sectio has bee studied. Here a coordiate trasormatio is employed to trasorm the goverig equatios ito odimesioal osimilar boudary layer equatios. The obtaied boudary layer equatios are the solved by the cubic splie collocatio method. The eects o the traspiratio parameter ad the aspect ratio o the Nusselt ad Sherwood umbers or the permeable elliptical cyliders o blut ad sleder orietatios have bee studied. The results show that icreasig the traspiratio parameter teds to decrease the boudary layer thickess ad thus ehaces the heat ad mass traser rates betwee the luid ad the wall. Moreover, the heat ad mass traser rates o the cylider with sleder orietatio are higher tha those o the cylider with blut orietatio. Ackowledemets This work was supported by Natioal Sciece Coucil o Republic o Chia uder the grat o. NSC 94--E-8-6. Reereces: [] A. ea, K.R. Khair, Heat ad Mass Traser by Natural Covectio i a Porous Medium, Iteratioal Joural o Heat ad Mass Traser, Vol. 8, 985, pp [] F.C. La F.A. Kulack Coupled Heat ad Mass Traser by Natural Covectio rom Vertical Suraces i Porous Media, Iteratioal Joural o Heat ad Mass Traser, Vol. 34, 99, pp [3] K.A. Yih, Coupled Heat ad Mass Traser by Free Covectio over a Trucated Coe i Porous Media: VWT/VWC or VHF/VMF, Acta Mechaica, Vol. 37, 999, pp [4] C.Y. Cheg, Eect o a Magetic Field o Heat ad Mass Traser by Natural Covectio rom Vertical Suraces i Porous Media-a Itegral Approach, Iteratioal Commuicatios i Heat ad Mass Traser, Vol. 6, 999, pp [5] J.H. Merki, Free Covectio oudary Layers o Axi-Symmetric ad Two-Dimesioal odies o Arbitrary Shape i a Saturated Porous Medium, Iteratioal Joural o Heat ad Mass Traser, Vol., 979, pp [6] R.M. Fad, T.E. Steiberger, P. Cheg, Natural Covectio Heat Traser rom a Horizotal Cylider Embedded i a Porous Medium, Iteratioal Joural o Heat ad Mass Traser, Vol. 9, 986, pp [7] A. Yücel, Natural Covectio Heat ad Mass Traser alog a Vertical Cylider i a Porous Medium, Iteratioal Commuicatios i Heat ad Mass Traser, Vol. 33, 99, pp [8] K.A. Yih, Coupled Heat ad Mass Traser by Natural Covectio Adacet to a Permeable Horizotal Cylider i a Saturated Porous Medium, Iteratioal Commuicatios i Heat ad Mass Traser, Vol. 6, 999, pp [9] J.H. Merki, Free Covectio oudary Layers o Cyliders o Elliptic Cross Sectio, Joural o Heat Traser, Vol. 99, 977, pp [] I. Pop, M. Kumar G. Nath, Free Covectio about Cyliders o Elliptic Cross Sectio Embedded i a Porous Medium, Iteratioal Joural o Egieerig Sciece, Vol. 3, 99, pp [] S.G. Rubi, R.A. Graves, Viscous Flow Solutio with a Cubic Splie Approximatio, Computers ad Fluids, Vol. 3, 975, pp [] P. Wag, R. Kahawita, Numerical Itegratio o Partial Dieretial Equatios Usig Cubic Splie, Iteratioal Joural o Computer Mathematics, Vol. 3, 983, pp