Numerical Investigation of Natural Convection Heat Transfer from Square Cylinder in an Enclosed Enclosure Filled with Nanofluids

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1 Rochester Institte o echnolog RI Scholar Works Articles Nmerical Inestigation o Natral Conection Heat ranser rom Sqare Clinder in an Enclosed Enclosre Filled with Nanolids Ghalib Kahwaji Rochester Institte o echnolog Omar M. Ali Uniersit o Zakho Follow this and additional works at: Recommended Citation Kahwaji, Ghalib and Ali, Omar M., "Nmerical Inestigation o Natral Conection Heat ranser rom Sqare Clinder in an Enclosed Enclosre Filled with Nanolids" (0). International Jornal o Recent Adances in Mechanical Engineering (IJMECH), (), -7. Accessed rom his Article is broght to o or ree and open access b RI Scholar Works. It has been accepted or inclsion in Articles b an athorized administrator o RI Scholar Works. For more iormation, please contact ritscholarworks@rit.ed.

2 NUMERICAL INVESIGAION OF NAURAL CONVECION HEA RANSFERFROM SQUARE CYLINDER IN AN ENCLOSED ENCLOSURE FILLED WIH NANOFLUIDS Omar M. Ali and Ghalib Y. Kahwaji - Department o Mech. Eng., Uniersit o Zakho, Iraq, omarslian@gmail.com. - Department o Mech. Eng., Rochester Institte o echnolog-dbai, UAE.(gkcad@rit.ed) ABSRAC he enhancement o natral conection heat transer sing nanolids rom horizontal sqare clinder placed in a sqare enclosre is inestigated nmericall. Water-based C is sed as the working nanolid. he inestigation coered a range o Raleigh nmbers o 0-0, nanoparticles olme raction o (00.), enclosre width to clinder height ratio, W/H o.. he inestigation incldes the soltion o the goerning eqations in the Vorticit-Stream nction space with the aid o a bod itted coordinate sstem. Algebraic grid generation is sed in the initial transormations, ollowed b an elliptic transormation to complete the grid generation to comptational domain. he reslting discretized sstem o eqations is soled sing an ADI method. he bilt code is alidated and the reslts showed an increase in aerage Nsselt nmber with increasing the olme raction o the nanoparticles or the whole range o Raleigh nmber. he isotherms are nearl similar when the olme raction o nanoparticles is increased rom 0 to 0. or each Raleigh nmber bt a change in the streamlines is obsered. KEYWORDS: Heat ranser, Sqare Clinder, Sqare Enclosre, Nmerical. NOMENCLAURE Smbol Deinition Unit Smbol Deinition Unit d i,j Sorce term in eqation, eqn. (8). Greek Smbols Volme raction o nanolid. emperatre dierence. C h Conectie heat transer coeicient. W/m.C Viscosit o the air. kg/ms H Height o the clinder. m Coeicient o thermal epansion. /C J Jacobian. Vertical ais in comptational domain. k hermal condctiit- air. W/m.C Horizontal ais in comptational domain. N Aerage Nsselt nmber Dimensionless stream nction. p Pressre. N/m Vorticit. /s P Coordinate control nction. Dimensionless orticit. Pr Prandtl nmber, (/). Kinematic iscosit. m /s Q Coordinate control nction. Dimensionless temperatre. R Maimm absolte residal Volme raction o nanolid. Ra Ralieh nmber, (gh 3 /). Dependent ariable. t ime. seconds Stream Fnction. /sec. emperatre. C Sbscript Velocit in -direction. m/s s Clinder srace. Velocit in -direction. m/s Enironment. W Enclosre Width. cm X Deriatie in -direction. W Relaation actor. Y Deriatie in -direction. X Horizontal direction in phsical domain. Dimensionless horizontal direction in phsical domain. m D Deriatie in -direction. Circlar clinder diameter. Vertical ais in phsical domain. m Stream nction. Y Dimensionless ertical ais in phsical domain. emperatre.

3 . INRODUCION here are a nmber o practical applications o natral conection heat transer rom er long horizontal clinders o noncirclar sections. his sbject has receied onl limited attention in the literatre, Ali, [].Nanolids are deined as lids which consist o a base lid sch as water with nano-size particles (e.g. metal, metal oide, and carbon materials), sspended in it. he size o the nanoparticles is between -00 nm. he dispersion o highl-condctie nanoparticles into the base liqids is seen as a promising approach to improe the perormance o the engineered heat transer lids, Choi []. Zi-ao Y, et al. [3], reiewed the reported literatre bot laminar natral conection o nanolids in coined regions (sqare and rectanglar caities, horizontal annli and trianglar enclosres), or a ariet o combinations o base liqids and nanoparticles. Nanolids were considered as single phase lids and the presence o nanoparticles plas a role in modiing the macroscopic thermo-phsical properties o their base liqids. A large nmber o stdies hae dealt with the mechanism s o thermo-phsical properties o the nanolids, Zi-ao Y, et al, []. he reslts indicated a gradal decrease in Nsselt nmber with the decrease o the olme at constant Raleigh nmber. Natral conection heat transer in horizontal annli sing ariable properties o Al O 3 water nanolid is stdied nmericall b Eiad Ab-Nada [], where the heat transer enhancement in the annls is ealated sing dierent models o iscosit and thermal condctiit. It was obsered that the Ngen et al. data and Brinkman model gies completel dierent predictions at Ra 0 where the dierence in prediction o Nsselt nmber reached 30%. Hakan, et al, [], stdied the heat transer and lid low de to boanc orces in a partiall heated enclosre sing dierent tpes o nanoparticles. A heater is located to the let ertical wall with a inite length. he goerning eqations were soled sing inite olme techniqe. Dierent tpes o nanolids are sed with ariable range o Raleigh nmber, height o heater, location o heater, aspect ratio and olme raction o nanoparticles. Nsselt nmber increased with the olme raction o nanoparticles or the whole range o Raleigh nmber. Heat transer also enhances with increasing o height o heater. It was ond that the heater location aects the low and temperatre ields and the heat transer enhancement, sing nanolids, is more prononced at low aspect ratio than at high aspect ratio. he present work deals with nmerical inestigation o natral conection heat transer ora water-based C nanolid and a sqare horizontal clinder sitated in closed sqare cait. he work inestigates the eect o nanolids on the low and heat transer characteristics. he std ses dierent Raleigh nmbers, and dierent olme raction o nanoparticles.. M A HEMAIC AL FO RM L A ION Figre () displas a schematic diagram o the low between the heated horizontal sqare clinder and the enclosre. he lid is water containing nano-sized particles o copper. It is assmed that the lid is incompressible, the base lid (water) and nanoparticles are in thermal eqilibrim and no slip condition occrs between them. he goerning eqations were soled nder the assmptions that the low is laminar, no internal heat sorces, low is two-dimensional and Bossinesq approimation applies. he thermo-phsical properties, gien in table (), are assmed to be constant, []. able (): hermo-phsical properties o the pre lid and the nanoparticles, Hakan, et al, []. Phsical Properties Flid phase (water) Nanoparticles (C) C p (J/kg.K) (kg/m 3 ) k (W/m.K) (m /sec) (m /sec).7

4 3 Figre () Coigration o clinder-enclosre combination he stead, incompressible continit eqation is gien b, []: 0 () he momentm eqation is: g g p t () he momentm eqation is: g g p t (3) And the energ eqation is gien b: t () With Bossinesq approimations, the densit is constant or all terms in the goerning eqations ecept or the boanc orce term where the densit is a assmed a linear nction o the temperatre. o () where is the coeicient o thermal epansion. he stream nction () and orticit () are deined as ollows, Anderson [7], and Petroic [8]:, () (7) Or V Sbstittion in the goerning eqations ields: W W

5 Energ Eqation: t (8) Momentm Eqation: Ra t s s s s Pr Pr 0. (9) Continit Eqation: (0) Where = k k ( φ)+φ (ρc P ) s (ρc P ) () α = k e (ρc P ) () he eectie densit and heat capacitance o the nanolid are calclated rom: ρ = ( φ)ρ + φρ s (3) (ρc P ) = ( φ)(ρc P ) + φ(ρc P ) s () Assming that the nanoparticles are spherical, the eectie thermal condctiit o the nanolid is approimated b the Mawell Garnetts model: k k = k s+k φ(k k s ) k s +k +φ(k k s ) () he iscosit o the nanolid can be considered as that o a base lid containing dilte sspension o ine spherical particles, Brinkman [9]: μ = μ ( φ). () Introdcing the ollowing dimensionless ariables: H X, H Y, H U, H V, H t,, H, c (7) ransorms the goerning eqations to: (8) d b J t a

6 Where is an dependent ariable. he goerning eqations are obtained b replacing the dependent ariable in the three goerning eqations as ollow: a b d 0 Pr 0. s Ra Pr s s s 0 Note that b represents the nstead term, t J ;is the dision term, and d is the sorce term. is the conectie term,. Grid Generation: he initial comptational grid, generated sing an algebraic grid generation techniqe are ed into elliptic, Poisson eqations to generate the inal orthogonal comptational grid points: P, (9a) Q, (9b) Interchanging dependent and independent ariables or eqations (9a, and b), gies: J J P Q 0 P Q 0 (0a) (0b) Where,, and he coordinate control nctions P and Q are chosen to ilence the strctre o the grid, [0]. he soltion o these eqations is obtained sing Sccessie oer Relaation (SOR) method with relaation actor ale eqal to., [and ].he transormed comptational grid is shown in igre () below. Figre (): Phsical to comptational domains transormation sing elliptic grid generation.

7 . Soltion Procedre: he goerning eqations were conerted into algebraic eqations sing Finite Volme based Finite Dierence method, Ferziger [3]. he hbrid scheme (o the central and the pwind dierencing schemes) is sed to aoid the instabilit o the central dierencing scheme (second order or conectie term) at high Peclet nmber (Cell Renolds Nmber) and the inaccrac o the pwind dierencing scheme (irst order or conectie term). a P a P M a E a E W a W i, j i, j i, j i, j di, j N a N S a S o P o P () a P E W N S o P a a a a a () he reslting algebraic eqation is soled sing alternating direction implicit method ADI in two sweeps; in the irst sweep, the eqations are soled implicitl in -direction sing Cclic ri-diagonal Matri Algorithm (CDMA), becase o its cclic bondar conditions, and eplicitl in -direction. In the second sweep, the eqations are soled implicitl in -direction sing ri-diagonal Matri Algorithm (DMA) and eplicit in - direction. he soltion o the stream nction eqation was obtained sing Sccessie Oer-Relaation method (SOR). he initial conditions o the low between heated clinder and ented enclosre are: =0, = 0, = 0; or t = 0 (3) he temperatre bondar condition o the clinder srace was assmed as constant. m 0 ; atthe enclosre wall (a) Using nd order dierence eqation, the temperatre at the enclosre srace becomes: i, m i, m i, m (b) 3 3 Vorticit bondar conditions, Roache [], are i, m i, m at enclosre wall (a) J i, i, at clinder srace (b) J he stream nction o the clinder and the enclosre are assmed eqal to zero. he Nsselt nmber N is a non-dimensional heat transer coeicient that calclated in the ollowing manner: hd N k he heat transer coeicient is epressed as (7) h = q w H L he thermal condctiit is epressed as (8) k = q w θ n Sbstitting Eqations (), (), and (7) into Eqation (3), sing the dimensionless qantities, the Nsselt nmber on the let wall is written as: k N k 0 n (9) (30a)

8 he deriatie o the non-dimensional temperatre is calclated sing the ollowing ormla, Fletcher []: n const. J And = 0 at clinder srace J A compter program in (Fortran 90) was bilt to eecte the nmerical algorithm eplained aboe; it is general or a natral conection rom heated clinder sitated in an enclosre. (30b) 3. RESULS AND DISCUSSION he deeloped nmerical soltion is sed to sole the natral conection heat transer rom a sqare horizontal clinder placed in a sqare enclosre. he enclosre is illed with nano-lid with Prandtl nmber o.. he enclosre width to clinder characteristic length ratio W/H =., Raleigh nmbers o 0, 0, and 0, and olme ractions o nanolid are 0, 0.0, 0., 0. and 0.were stdied. he conergence criteria are chosen as R <0 -, R <0 - and R <0 - or, and respectiel. When all the three criteria are satisied, the conergent reslts are sbseqentl obtained. 3. Stabilit and Grid Independenc Std: he stabilit o the nmerical method is inestigated or the case Ra=0, W/D=., Pr = 0.7. hree time steps are chosen with ales 0 -, 0 -, 0 -. he maimm dierence between the ales o N with dierent time steps is %. he grid-independence o nmerical reslts is stdied or the case with Ra=0, and 0, W/D =., Pr =.. hree mesh sizes o 9, 8, and 9 0 were sed or the grid-independence std. It is noted that the total nmber o grid points or the aboe three mesh sizes is, 80, and 90 respectiel. Nmerical eperiments showed that when the mesh size is aboe 9, the compted N remain the same. 3. Validation est: he deeloped code alidation inclded nmerical inestigation o the natral conection problem or a low temperatre oter sqare enclosre and high temperatre inner circlar clinder. he aerage Nsselt nmbers and maimm stream nction ma are compared with the benchmark ales b Mokalled and Achara []. Comparisons are condcted or Prandtl nmber Pr=0.7, enclosre width to clinder diameter ratios (W/H=.) and Ra=0 and 0 as gien in table (). he reslts show a good agreement with Mokalled and Achara []. able (): Comparisons o Nsselt nmbers and maimm stream nction L/D Ra Present ma Mokalled and Achara [] Present N Mokalled and Achara [] Flow Patterns and Isotherms: he low patterns and isotherms displaed in igres (3-) are or olme raction range = 0 to 0.. Figre (3) shows a comparison o streamlines and isotherms between C-water nanolid (=0.) and pre lid (=0) or W/H=. with Raleigh nmber ales Ra=0, 0, and 0. At Ra=0 and 0, the isotherms o two cases are nearl identical. here are some dierences in isotherms between the two cases or Ra =0. he isotherms at the pper region aboe the sqare clinder or =0. are dierent as compared with pre lid. he width o the thermal plme or pre lid is narrower than those or 7

9 =0.. he same behaior occrs with thermal plmes at the corners o the sqare clinder. he isotherms are smmetrical arond ertical center line aboe the sqare clinder or pre lid, while, the isotherms appear as nearl asmmetrical arond ertical center line or =0.. he aspects o the streamlines are dierent or two cases. he low circlation or =0. is greater than those or pre lid or all Raleigh nmbers. able (3) displa the ales o maimm stream nction or pre lid and =0. with Ra = 0, 0, 0. able (3) Comparisons o maimm stream nctions between pre lid and =0. Maimm Stream Fnction Ra Pre lid = At Ra=0, the dominant heat transer is the condction, thereore, the streamlines o the two cases are nearl similar ecept that the sizes o the internal eddies are dierent. At Ra=0, the streamlines appear as nearl kidne-shaped or two cases. wo circlar tin eddies displa at the pper region near the center line aboe the sqare clinder. For =0., the low coer the most region between the sqare clinder and the enclosre, thereore; the stagnant area is er small. he coerage o the low redces or pre lid, thereore; the stagnant area increases. he densel package o the low or =0. is more than those or pre lid. At Ra=0, the low moes pward or two cases. he coerage o the low in the region between the sqare clinder and the enclosre or pre lid is more than those or =0.. he densel packed o the kidneshaped eddies and tin eddies or pre lid are more than those or =0.. Figre (3) Streamlines (on the top) and Isotherms (on the bottom) or C-water nanolids (- - -), pre lid ( ), W/D=., (a) Ra = 0, (b) Ra = 0 (c) Ra =0. 8

10 Figres (-) displa the streamlines and isotherms or W/H=., and nanolid olme ractions =0.0, 0., 0. and 0.. he strength o the low circlation aries with the ariation o the Raleigh nmber ales. he maimm stream nction ale aries between ma= 0.9 at Ra=0 and =0.0 to ma = 3 at Ra=0 and =0.. At Ra=0, the low circlation is weak, thereore; the maimm stream nction ale is small. he low is smmetrical abot the ertical line throgh center o the sqare clinder. he low patterns appear as a cred kidne-shaped single longitdinal edd. he edd core is small and the ertical bondar laers are thick, indicatie o the weak driing boanc. As the nanolid olme raction increases to =0., the low circlation enhances ( ma =.9) and the edd core becomes wider. he low strength rther increases or =0. ( ma =.) and 0. ( ma =.7), howeer, the core starts growing smaller indicatie o the hdrodnamic bondar laer growth with iscosit. At Ra=0, the strength o the low circlation becomes higher and two tin eddies appear at the pper region o the clinder near the ertical center line. he tin eddies appear as rings. he densel packed o the low or =0. and 0. are more than other lows. As Raleigh nmber increases to Ra=0, the low becomes stronger and the maimm stream nction increases or all cases ( ma= 7., 3., 37., and 3.0 or =0.0, 0., 0. and 0. respectiel). he low is asmmetrical abot the ertical center line. he streamlines near the bottom enclosre wall moe to the pward that lead to an increase in the stagnant area. he kernel edd size becomes more and it takes a trianglar shape. he low region moes pwards and the lower stagnant area enlarges with increasing the olme ractions o the nanolids. =0.0 =0. =0. =0. Figre (): Eect o olme raction o nanolids on streamlines at W/H =. and: (a) Ra = 0, (b) Ra = 0, (c) Ra = 0. he temperatre distribtions or W/H=. are presented b means o isotherms in igre (). he isotherms do not change with changing the olme ractions o the nanolids or all Raleigh nmbers. he isotherms are smmetrical abot ertical center line or Ra=0 and Ra=0 or all olme ractions. As Raleigh nmber increases, the thermal bondar laer adjacent to the clinder becomes thinner and thinner. At Ra=0, the isotherms are similar or all olme ractions o the nanolids. he mode o heat transer is condction and the eect o conection heat transer is er low. he isotherms displa as rings arond the 9

11 clinder. As Raleigh nmber increases to Ra=0, the isotherms distorts below the clinder de to the eect o the conection heat transer. A thermal plme appears on the top o the clinder. wo thermal plmes appear at the pper corners o the sqare clinder. he isotherms appear as cred below the clinder with low distortion de to the eect o the conection low. At Ra=0, the isotherms are nearl similar and independent o olme ractions o the nanolids. he conection becomes the dominant mode o heat transer. he width o the thermal plme at the middle o the clinder becomes narrow, and it impinging on the top o the enclosre. he widths o the two thermal plms at the pper corners become narrow. he isotherms appear asmmetrical abot the ertical center line. he thermal stratiication (nearl horizontal and lat isotherms) is ormed near the bottom region o the enclosre. =0.0 =0. =0. =0. Figre () Eect o olme raction o nanolids on isotherms at W/H =. and (a) Ra = 0, (b) Ra = 0, (c) Ra = Local and aerage Nsselt nmbers he distribtions o local Nsselt nmber arond the sqare clinder are presented in igres () or Raleigh nmbers Ra=0, 0 and 0 and = 0, 0.0, 0., 0. and 0.. he measred angle starts rom the middle o the right side o the enclosre and ends at the same point. he local Nsselt nmbers increase with increasing olme ractions o the nanolid or all Raleigh nmbers. he peak ales o the local N occr at the corners o the sqare clinder with all Raleigh nmbers and olme ractions o the nanolid. he maimm enhancement o the local N occrs at the lower corners o the sqare clinder. 0

12 N =0 =0.0 =0. =0. = N =0 =0.0 =0. = Ɵ N =0 =0.0 =0. =0. = Ais itle Figre () Eect o olme raction o Nanolids on the local Nsselt nmber or each enclosre width to sqare height W/D, (a) Ra=0, (b) Ra=0, (c) Ra=0.

13 he aerage Nsselt nmber is chosen as the measre to inestigate the heat transer rom the sqare clinder. he eect o olme raction o the nanolids on the aerage Nsselt nmbers with Ra=0, 0, and 0 or enclosre width to the clinder height W/H=. is presented in igre (7). he olme ractions,was aried as:0, 0.0, 0., 0., 0.. he Nsselt nmber increases with increasing the Raleigh nmber or all ales o. Nsselt nmber increases with increasing the olme raction o the nanolids. he enhancement o the Nsselt nmber de to increasing the nanolid olme raction is magniied with increasing Raleigh nmber as indicated b the increased slop o the N- cres. he maimm enhancement in the Nsselt nmber when the olme raction o nanoparticles is increased rom 0 to 0., sing Ra=0, is approimatel %, the maimm enhancement is arond 9% or Ra= 0, whereas the maimm enhancement is arond 8% or Ra= 0. his tells that the enhancement in heat transer, de to the presence o nanoparticles, is prononced or all Raleigh nmbers. he heat transer enhances with increasing the olme raction o the nanolids becase more particles sspended and the eect o thermal condctiit and iscosit o the nanolids on the heat transer. N Ra=0000 Ra=00000 Ra= Figre (7): Eect o olme raction o Nanolids on the aerage Nsselt nmber or each enclosre width to sqare height W/D, (a) Ra=0, (b) Ra=0, (c) Ra=0. 3. Flid low and Heat ranser Correlations he aerage Nsselt nmber and the maimm stream nction rom sqare clinder in a ented enclosre are correlated in terms o the Raleigh nmber in the range (0-0 ) and the nanolid olme ractions between 0-0., sing the reslts rom the present work. he correlation o the aerage Nsselt nmber can be epressed as: N =.9( ) (3) With R = he correlation o the maimm stream nction can be epressed as: ma = 0.03( )Ra 0.8 (3) With R = CONCLUSIONS Eect o the presence o the nanolids on the natral conection heat transer rom sqare horizontal clinder in a sqare enclosre was inestigated nmericall oer a airl wide range o Ra. he main conclsions o the present work can be smmarized as ollows:. he nmerical reslts show that the Nsselt nmber increases with increasing the Raleigh nmber or all cases.. he low patterns and isotherms displa the eect o Ra, and olme ractions o the nanolids on the thermal and hdrodnamic characteristics.

14 3. he Condction is the dominant o the heat transer at Ra=0 or all cases. he contribtion o the conectie heat transer increases with increasing the Raleigh nmber.. he reslts show that the isotherms are nearl similar when the olme raction o nanoparticles is increased rom 0 to 0. or each Raleigh nmber.. he streamlines are asmmetrical when the olme raction o nanoparticles is increased rom 0 to 0. or each Raleigh nmber.. he aerage Nsselt nmber enhances gradall when the olme raction o nanoparticles is increased rom 0 to 0. or each Raleigh nmber. 7. he correlation eqation o the aerage Nsselt nmber is: N =.9( ) 8. he correlation eqation o the maimm stream nction is: ma = 0.03( )Ra 0.8 REFERENCES. Ali O. M. (008), Eperimental and Nmerical Inestigation o Natral Conection Heat ranser From Clinders o Dierent Cross Section Clinder In a Vented Enclosre, Ph. D., hesis, College o Engineering, Uniersit o Mosl.. S.U.S. Choi, (99), "Enhancing thermal condctiit o lids with nanoparticles, in: D.A. Signer, H.P. Wang (Eds.)," Deelopments and Applications o Non-Newtonian Flows, ASME, New York, 3(), Zi-ao Y, X X, Ya-Cai H, Li-W Fan, Ke-Fa Cen, (0), " Nmerical std o transient boanc-drien conectie heat transer o water-based nanolids in a bottom-heated isosceles trianglar enclosre", International Jornal o Heat and Mass ranser, 3.. Zi-ao Y, X X, Ya-Cai H, Li-W Fan, Ke-Fa Cen, (0), "A nmerical inestigation o transient natral conection heat transer o aqeos nanolids in a horizontal concentric annls," International Jornal o Heat and Mass ranser, 8.. E. Ab-Nada, (009), "Eects o ariable iscosit and thermal condctiit o AlO3 water nanolid on heat transer enhancement in natral conection," Int. J.Heat Flid Flow 30, Hakan F. Oztop, Eiad Ab-Nada, (008), "Nmerical std o natral conection in partiall heated rectanglar enclosres illed with nanolids," International Jornal o Heat and Flid Flow 9, John D. Anderson Jr., (99), "Comptational Flid Dnamics, the Basics with Applications," McGraw Hill Book Compan. 8. Petroić Z., and Stpar S., (99), "Comptational Flid Dnamics, One," Uniersit o Belgrade. 9. Brinkman, H.C., (9), "he iscosit o concentrated sspensions and soltions," J. Chem. Phs. 0, homas P. D., and Middleco J. F., (980), "Direct Control o the Grid Point Distribtion in Meshes Generated b Elliptic Eqations," AIAA Jornal, 8, -.. Homann K. A., (989), "Comptational Flid Dnamics or Engineers, Engineering Edcation Sstem, USA.. hompson J. F., Warsi Z. U. A. and Mastin C. W., (98), "Nmerical Grid Generation: Fondations and Applications," Mississippi State, Mississippi. 3. Ferziger J. H. and Peric M., (00), "Comptational Methods or Flid Dnamics," Springer, New York.. Roache, P., J., (98), "Comptational Flid Dnamics," Hermosa pblishers.. Fletcher C., (988), "Comptational echniqes or Flid Dnamics," Springer, Verlag.. Mokalled F., Achara S., (99), "Natral conection in the annls between concentric horizontal circlar and sqare clinders," Jornal o hermo-phsics and Heat ranser, 0(3), Sh; and Zh, (00), "Eicient comptation o natral conection in a concentric annls between an oter sqare clinder and an inner circlar clinder," International Jornal or Nmerical Methods in Flids, 38, 9-. 3