Internatinal Jurnal f Mechanical & Mechatrnics Engineering IJMME-IJENS Vl:3 N:0 46 Buyancy Effect n the Fully Develped Air Flw in Cncentric Inclined Annulus Asmaa Ali Hussein Lecturer Fundatin Of Technical Educatin/Iraq-Baghdad Abstract-- Numerical investigatin have been perfrmed a hrizntal annulus f 0.5 radius rati by examining the systematically t study the mixed cnvectin heat transfer and air flw patterns in the simultaneusly fully develped regin f inclined annulus with cnstant heat flux fr inner cylinder and adiabatic uter cylinder. The time dependent partial differential equatins representing the vrticity, the stream functin and the temperature were apprximated by finite difference equatins and slved n the cmputer. The values f ayliegh number (a) are varied frm 0 3 t 0 6 and the value f radius rati is 0.5. The results shw that the transitin frm single-eddy pattern t the duble-eddy pattern appears t ccur at a between 0 5 and 0 6. Large temperature gradients and higher lcal Nusselt number attain at the bttm except at vertical psitin in which the angular variatin f the Nusselt number remains cnstant because f symmetry abut vertical axis. similarity cnditin f fully develped laminar flws f fluid (Pr=) ver the range f 0 4 Gr 0 6. The finite difference equatins btained were slved with the quadratic upstream differencing methd t stabilize the cnvectin terms with sufficient accuracy. esults shw that the thermal bundary cnditin f cnstant wall temperature gives large values f the Nu m than thse fr the case f cnstant heat flux. Nieckele and Patankar [], presented a numerical study fr the fully develped regin f the buyancy affected flw with an axial laminar flw in annular pipe f the range f 0. r /r 0.6 and 0 4 a 0 7,, 5,0 and, and a/(e.pr) <. The inner wall was heated isthermally while the uter wall was insulated. esults shw that at a given a, the effect f buyancy n the heat transfer cefficient was strnger at higher Pr, and the lcal heat transfer cefficients n the inner Index Term-- Heat Transfer, Cmbined (Mixed) Cnvectin, Cncentric Annulus. cylinder became highly nn unifrm at high a, the lwest value being at the tp f the cylinder. Kaviany [3], analyzed numerically steady state, fully develped velcity and I. INTODCTION Cmbined frced and free cnvectin has received cnsiderable attentin frm the therm fluid-dynamically pint f view as well as the practical interest f its engineering applicatins. Since the assciated fluid mtin is largely interacted with the energy equatin. Much theretical and experimental wrk has been published n this subject. Ktake and Hattri [], studied numerically the mixed cnvectin in temperature fields in mixed cnvectin thrugh an annulus (r /r =.5), with a prescribed cnstant heat flux n the inner cylinder and an adiabatic utside cylinder ver the range f 0 5 a 0 9, and, 7 and 70, using difference apprximatins. esults f the inner surface temperature, develpment f axial velcity prfiles, and the effect f the buyancy n the radial temperature distributin alng the annulus were calculated. Barletta [4] presented an 3380-6767-IJMME-IJENS @ February 03 IJENS
Internatinal Jurnal f Mechanical & Mechatrnics Engineering IJMME-IJENS Vl:3 N:0 47 investigatin f mixed cnvectin and flw reversal in a vertical annular duct with reference t laminar and fullydevelped flw f a pwer-law fluid. The bundary surfaces were suppsed t be isthermal, with unequal temperatures. Analytical expressins which allw the determinatin f the dimensinless velcity and f the frictin factrs have been btained. Nazrul Islam et al [5] submitted a numerical investigatin f steady laminar mixed cnvectin heat transfer in a hrizntal cncentric annulus using air and water as the wrking fluid. Special attentin has been paid t the entrance regin effect. The thermal bundary cnditin chsen is that f unifrm heat flux at the inner wall and an adiabatic uter wall. The basic numerical prcedure used was the SIMPLE algrithm which used the finite difference methd fr slving the mmentum and energy equatins. The investigatins revealed that the Nusselt numbers fr air were cnsiderably greater than the crrespnding values fr fully develped mixed cnvectin values ver a significant prtin f the annular duct. Habib and Negm [6] studied numerically steady, laminar, mixed cnvectin in the fully develped regin f hrizntal cncentric annuli fr the case f nnunifrm circumferential heating. It was fund that bttm heating arrangement gave rise t a vigrus secndary flw, with the result that the average Nusselt numbers were much higher than thse fr pure frced cnvectin. Esmail and Maged [7] used a biplar mdel and a numerical algrithm fr slving this mdel t investigate the prblem f cmbined cnvectin in vertical eccentric annuli with simultaneusly sufficiently large, aiding free cnvectin culd develp t vercme the fluid frictin and the eccentric annular channel eventually wrks as a diffuser. Numerical investigatin f the thermal cnvectin fr a thermdependent nn-newtnian fluid in an annular space between tw caxial rtating cylinders was presented by Zeraibi et al [8]. The gverning equatins are slved using mixed finite elements methd. The influence f the temperature n the structure f the dynamic and thermal fields is examined. Mhammed [9] carried ut a numerical investigatin f duble-diffusive laminar mixed cnvectin within a tw-dimensinal, hrizntal annulus. The inner cylinder was cnsidered t rtate in an anti-clckwise directin t intrduce the frced cnvectin effect. In additin, the slutal and thermal buyancy frces are sustained by maintaining the inner and uter cylinders at unifrm temperatures and cncentratins, but their values fr the inner are higher than the uter. The predicted results fr bth average Nusselt and Sherwd numbers were crrelated in terms f Lewis number, thermal ayleigh number and buyancy rati. Velcity vectrs in a vertical caxial dubleduct heat exchanger fr parallel ascending flw f water under cnditins f laminar mixed cnvectin had been determined experimentally and numerically by Thierry Mare et al [0]. The measured velcity distributins fr large annular flw rates, resulting in an essentially isthermal envirnment fr the stream in the inner tube, were in very gd agreement with crrespnding numerical predictins. Fr flw rates f the same rder f magnitude in the inner tube and the annulus, and develping hydrdynamic and thermal bundary layers under crrespnding temperature differences f abut 0 C, thermal bundary cnditins f having ne f the annulus bundaries at a cnstant temperature while the ther bundary was insulated. esults shwed that after a distance frm the channel entrance and prvided that the value f Gr/e was experimental bservatins shwed that flw reversal ccurs simultaneusly in bth streams ver large axial distances fr bth heating and cling f the flw in the inner tube. The cupled phenmenn f ppsing mixed cnvectin and 3380-6767-IJMME-IJENS @ February 03 IJENS
Internatinal Jurnal f Mechanical & Mechatrnics Engineering IJMME-IJENS Vl:3 N:0 48 radiatin within differentially heated eccentric hrizntal cylindrical annulus had been numerically simulated by Anjan present additinal examples where the results fr the buyancy-influenced cnvectin in an annulus are useful. Sarkar et al []. The walls f the annulus were cnsidered t be paque, diffuse and gray. In the study it had been bserved that the ichardsn number had a small effect n the ttal II. ANALYSIS Fr an air flw in an inclined annulus, bth free and Nusselt number in mixed cnvectin heat transfer with r withut radiatin. The influence f radiative parameters n the interactin phenmenn had been delineated thrugh istherm and streamline pattern. Fatthi et al [] submitted Lattice Bltzmann mdel (LBM) t simulate numerically the mixed cnvectin heat transfer in eccentric annulus which was based n multi-distributin functin duble-ppulatin apprach. The effect f eccentricity n heat transfer at varius lcatins was examined at a=0 4. The results shwed that the average Nusselt number increases when the inner cylinder mves dwnward regardless f radial psitin. The present paper describes a numerical study f the laminar mixed cnvectin in an inclined annular duct flw subjected t heating frm the inner cylinder. The situatin is encuntered in practice when the fluid in a pipe is heated by frced cnvectin effect in axial, radial, and circumferential directins are presented. A three- dimensinal mdel can be used t describe the mixed cnvectin heat transfer in an inclined annulus at angle α frm hrizntal psitin with inner radius r and uter radius r which has a cnfiguratin shwn in Fig.(). As shwn in this figure, the directin f flw is frm bttm t tp ( i.e., aiding flw). The fllwing assumptins are used in the mdeling: - steady three dimensinal laminar airflw. - hydrdynamically and thermally fully develped aiding flw. 3- n internal heat generatin and heat dissipatin. 4- The physical prperties are cnstant except the density in the buyancy term f mmentum equatins which varies accrding t Bussinesq, s apprximatin. passing it ver a central heater rd r electrical resistr. Similarly, gas-cled electrical cables require the cnsideratin f mixed cnvectin in an annular flw. Duble pipe heat exchangers and the cling f nuclear fuel rds 3380-6767-IJMME-IJENS @ February 03 IJENS
Internatinal Jurnal f Mechanical & Mechatrnics Engineering IJMME-IJENS Vl:3 N:0 49 v, r u, z z r w, r r g α Fig.. Three-Dimensinal Annular Gemetry. The time dependent partial differential equatins representing the vrticity, the stream functin and the temperature were apprximated by finite difference equatins and slved n the cmputer. In these calculatins all velcities are nrmalized with respect t the bulk average velcity u, uˆ / r lengths, with the radius r, pressures with respect t where ˆ is the thermal diffusivity f the fluid. The average heat flw t the fluid per unit area q and the velcity u are nt varying with time s the bulk average temperature t b is cnstant. A dimensinless temperature T is defined as the difference between the fluid temperature t and the bulk averaged temperature nrmalized with respect t qr / where is the thermal cnductivity f the fluid. A slutin is sught fr which the wall temperature t w is nt varying arund the utside circumference f the inner cylinder, but is varying with time. Therefre, althugh the average heat flux is cnstant the lcal heat flux varies with circumferential psitin and with time. If the axial density gradients are neglected and the fully develped regin f the heat transfer sectin is cnsidered the dimensinless time dependent equatins describing the vrticity, the stream functin, the axial velcity and the temperature are as fllws: cs Pr Pr. a T sint cs () () (3) T (4) P Pr Gsin T T T ˆ Anther equatin is needed t define the pressure gradient P Z. Since axial density gradient are neglected, the pressure gradient is cnstant ver the crss sectin f the annular gap and eq. (3) can be divided by P Z t give : Û (5) G Û Û Pr Û sin Z P Z 3380-6767-IJMME-IJENS @ February 03 IJENS
Internatinal Jurnal f Mechanical & Mechatrnics Engineering IJMME-IJENS Vl:3 N:0 50 where The values f,,, T, and P Z can be calculated = Û P Z (6) Since the average velcity remains cnstant the fllwing fr given values f a & Pr by slving 7) using bundary cnditins eq.(8). eqs. (,, 4, 5 and integral defines P Z : The vrticity and stream functin were set equal t P Z ˆ ˆ Û d 0 d zer thrughut the whle field at =0. Initially the temperature field is zer every were except at the inner wall and the heat flux is nt varying arund the circumference. (7) The hydrdynamic and thermal bundary cnditins are: (, ) (ˆ, ) Û(, ) Û(ˆ, ) 0 (8.a) (,0) (, ) (,0) (, ) 0 (8.b) Û (,0) Û (, ) 0 (8.c) (, ) (, ) (8.d) ( ˆ, ) (ˆ, ) (8.e) T (, ) (8.f) T (ˆ, ) T (,0) T (, ) 0 Frm the definitin f the dimensinless temperature T/ at the inner wall fr =0. The elliptic stream functin equatin is numerically slved by using central difference and Gauss -Seidel eliminatin methd in all the iteratin prcess t find stream functin. The finite difference equatins are written in cylindrical crdinates with variable mesh spacing in the radial directin. A fine radial grid spacing was used near the heated wall where the temperature and velcity gradients are large and a carse spacing near the uter adiabatic wall where the gradients are small. The slutin prcess starts with first iteratin and is based n the initial cnditins t calculate the stream functin. When the slutin is cmpleted and the values f the ther parameters are fund, the values f lcal Nusselt number in directin and its average value which are cnsidered as a bundary cnditin fr the fllwing iteratin prcess are calculated. (8.g) The lcal and mean Nusselt number are respectively given as: III. ESLTS AND DISCSSION Nu Nu (ˆ ) / T w (9) (0) (ˆ ) / m T w Istherms and Streamlines Figs. (-9) display the temperature distributin and the secndary flw pattern in the crss-sectin f annulus. Because f the symmetry abut the vertical diameter, the istherms are pltted in the left half and the s treamlines in the right half. The value f max is a measure f the secndary 3380-6767-IJMME-IJENS @ February 03 IJENS
Internatinal Jurnal f Mechanical & Mechatrnics Engineering IJMME-IJENS Vl:3 N:0 5 flw: max is the maximum value f the dimensinless stream functin. Fig.(- a & b) shws, fr =0 (hrizntal),, and a=0 5 & 0 6 ; respectively, the effect f the ayleigh number n the temperature and streamline patterns. Fr a=0 5, the secndary flw is weak ( max =4.036) and frms a symmetrical eddy rtating in the clckwise directin (i.e., there is ne cell nly in each side). The istherms are nearly circular and thus little affected by the secndary currents. As the ayleigh number increases, the secndary mtin becmes strnger ( max =4.055). At a=0 6 as seen in Fig.(-b), a small secnd eddy is seen t have frmed belw the main eddy, which is pushed upward and rendered unsymmetrical. The istherms tend t becme hrizntal, especially in the regins away frm and under the inner cylinder, apprximating the temperature distributin in a stably stratified fluid. It is expected that the heat transfer prcess in hrizntal psitin is better than ther angles f inclinatins because f the strnger secndary flws assciated with free cnvectin which behave s as t reduce temperature difference in the annulus. In general, in hrizntal & inclined psitins, the fluid flws up alng the inner wall t frm vrtices having their center in the upper part f the annulus. The vrtex strength increases with ayleigh number, and finally the vrtex breaks up int tw r mre vrtices. At small ayleigh number, the vrtex circulatin is weak, s it is expected that ne cell will be frmed in each side abut the vertical line f annulus. In vertical psitin, the main and secndary flws are in the same directin, s the vrtex strength diminishes. Fig.(3-a&b) and Fig. (4-a&b) shw the effect f ayleigh number n the heat and fluid flw patterns in the cncentric annulus fr,, and a=0 5 & 0 6 at these figures that, there is a slight effect n the streamlines behavir and mre prnunced effect n istherm lines especially at a=0 6. The istherms tend t be mre hrizntal as angle f inclinatin deviates frm hrizntal twards vertical psitins. In vertical psitin, the velcities due t buyancy frces are parallel t the directin f the frced mtin; thus, rtatinal symmetry is retained. This situatin leads t ne cmpnent f the velcity due t buyancy frces in the same directin f axial velcity because there are n cmpnents f buyancy frces in (r, ) directin cmpared with the hrizntal and inclined psitins in which three cmpnents f velcity in (r,,z) directins are frmed. Thus, there is n tangential velcity(w) and radial velcity (v), and the value f stream functin in terms f these tw velcities is equal t zer. As can be seen frm this figure, it is impssible t represent the secndary flw by pltting a diagram describes streamlines since stream functin is equal t zer. This is a guide fr accuracy f the numerical methd used in slutin f the gverning equatins f flw. Fig. (5) shws -fr bth branches- n the left hand side the istherms cntur that are nearly circular and have the same center lcated exactly at the center f annulus, further indicate little influence f the cnvective flw n heat transfer. It is nticed that, there is n prnunced effect fr increasing the value f a n the heat transfer prcess. The istherms lines seem t be clser t each ther near the heated inner wall because the natural cnvectin is weak and have a slight effect n the flw field cmpared with the frced cnvectin. This effect is limited t accelerating the fluid velcity near the heated wall. Since the study is achieved in the regin f thermal and hydraulic fully develped, then the istherms lines remain in a frm f circular lines which have the same center lcated at angles f inclinatin (30 & 60 ); respectively. It is clear frm 3380-6767-IJMME-IJENS @ February 03 IJENS
Internatinal Jurnal f Mechanical & Mechatrnics Engineering IJMME-IJENS Vl:3 N:0 5 the center f annulus and distributed between the temperature Fig.(7-a) & Fig.(7-b) represent clearly the influence f f heated inner wall and adiabatic uter wall. azimuthal angle n the axial velcity prfile fr a=0 3 & 0 6 ; respectively. It is nticed that, there is n significant Axial Velcity Prfile The lngitudinal velcity prfiles fr several azimuthal angles and ayleigh numbers fr,, and =0 (hrizntal) are given in Figs. (6 7). The influence f the ayleigh number n the axial velcity prfile with the radial psitin at three different angular lcatins =0 (tp), 90 (center), and 80 (bttm) is shwn in Fig. (6-a), Fig. (6-b), and Fig. (6-c); respectively. The axial velcity prfile fr pure frced cnvectin (i.e., a=0) is als presented as a reference. It is nticed frm Fig.(6-a) & Fig.(6-c) that, the maximum value f the axial velcity at the same radial psitin decreases and deviates twards uter adiabatic wall at the tp ( =0 ) as a increases, while the ppsite ccurs at the bttm ( =80 ) withut any distrtin in the parablic prfile. On the ther hand, and as shwn in Fig.(6-b) that the axial velcity prfile at the center ( =90 ) takes the same behavir btained in Fig.(6-a) but the distrtin is mre prnunced near the inner heated wall at a=0 6. It can cncluded frm these three figures that, the lateral buyancydriven cnvectin decreases the lngitudinal flw rate in the upper prtin f the annulus and increases it in the lwer prtin. This is due t the higher circulatin strength assciated with the upper cell which ccurs at a=0 6, and with the upper part f single cell (ne fr each side) at a 0 5 effect fr changing f the angular lcatin n the axial velcity prfile at lw ayleigh number (a=0 3 ) as shwn in Fig.(7-a), and the values f the axial velcity seem t be clser t each ther in the regin bunded between ˆ =.55 & (uter wall), and diverges slightly and increases as angular lcatin change frm tp t bttm psitin in the regin bunded between ˆ = (inner wall) &.55. The effect f changing f angular lcatin n the axial velcity prfile is mre prnunced at high ayleigh number (a=0 6 ) as shwn in Fig.(7-b). As shwn in the figure, the maximum axial velcity ccurs at =80 (at bttm) & ˆ =.5 (center f annular gap), then decreases & deviates tward the inner heated wall at =90 (center) then decreases further and deviates tward the uter wall at =0 (tp). N available investigatins exist abut the mixed cnvectin f air () at thermally and hydrdynamically fully develped regin f a vertical annulus with a heated inner cylinder and adiabatic uter cylinder because f n appreciable effect fr any parameter n the behavir f heat and fluid flw prcess. As shwn in Fig.(8), the axial parablic prfile is the same fr all values f ayleigh numbers, because f n cellular mtin at vertical psitin =0, and angular lcatins, because f symmetry abut vertical axis. ; causes higher resistance t the lngitudinal flw that increases as a increases and decreases as the angular lcatin deviates frm tp ( =0 ) twards the angular line that separates the upper cell frm the lwer cell (90 < <80 ) which have lwer circulatin strength causes lwer resistance Angle f Inclinatin Effect n the Axial Velcity Prfile The effect f angle f inclinatin n the axial velcity prfile fr & are shwn in Figs. (9-0). Figs. (9- a, b, and c) which represent the effect f angle f t the lngitudinal flw that decreases as a increases. inclinatin n the axial velcity prfile fr a=0 3 (lw 3380-6767-IJMME-IJENS @ February 03 IJENS
Internatinal Jurnal f Mechanical & Mechatrnics Engineering IJMME-IJENS Vl:3 N:0 53 ayleigh number) at three different angular lcatins =0 (tp), 90 (center), and 80 (bttm); respectively. There is a slight effect fr changing the angle f inclinatin at tp and bttm f the annular gap, and n effect at center because f dminating frced cnvectin in the heat and fluid flw prcess. If ayleigh number increases t 0 6, anther insight will be btained as shwn in Figs. (0- a, b, and c) which represent the effect f angle f annulus inclinatin n the axial velcity prfile at three different angular lcatins =0 (tp), 90 (center), and 80 (bttm); respectively. The value f maximum axial velcity at vertical psitin ccurs at center f annular gap and is higher than that at ther angles f inclinatin which seem t be clser t each ther and deviates tward the uter adiabatic wall at angular lcatin =0 (tp) as shwn in Fig.(0-a) and twards the inner heated wall at =90 (center) as shwn in Fig.(0-b). On the ther hand, and as shwn in Fig.(0-c), the maximum value f the axial velcity at angular lcatin =80 (bttm) fr all angles f inclinatin ccurs at the centerline annular gap and increases as annulus psitin deviates frm vertical t hrizntal because f dminant natural cnvectin in the heat and fluid flw prcess which reaches a maximum value at angular lcatin =80 (bttm). As explained befre, the cellular mtin at hrizntal psitin is greater than ther angles f inclinatin except vertical psitin in which the cellular mtin diminishes. variatin f Nu Φ with the circumferential psitin arund the inner cylinder at varius angles f inclinatin and fr,, a=0 3 & 0 6 ; respectively. In general, since clder fluid tends t cllect belw the inner cylinder and htter fluid abve it, large temperature gradients and higher lcal Nusselt numbers are t be expected at the bttm except at vertical psitin in which the angular variatin f the Nusselt number remains cnstant because f symmetry abut vertical axis. This is clearly seen fr bth cases f a=0 3 & 0 6. Fig.(-a) shws that, the heat transfer prcess enhances in the angular regin bunded between =0 (tp) and 60 as angle f inclinatin deviates frm hrizntal t vertical psitin. Then, the values f Nu Φ at varius angles f inclinatin seem t be clser t each ther in the angular regin bunded between 60 & 80. After =80, the behavir will be reversed, and the heat transfer prcess will be better as angle f inclinatin deviates frm vertical twards hrizntal psitin. As explained befre, the higher circulatin strength assciated with the upper part f single cell (ne fr each side) causes higher resistance t the lngitudinal flw (i.e., higher hearting) that decreases as angular lcatin deviates frm tp ( =0 ) t bttm ( =80 ) leads t accelerate the lngitudinal flw at this lcatin (i.e., lwer heating and better heat transfer prcess). As a increases t 0 6, tw cells will be frmed (fr each side) and the circulatin strength assciated with the upper cell increases and causes very higher resistance t the lngitudinal flw that Lcal Angular Nusselt Number Nu Φ Since the velcity and temperature patterns encuntered in mixed cnvectin are far frm being symmetrical (except vertical psitin), it fllws that the lcal heat transfer cefficient alng the circumference f the inner cylinder will be highly nnunifrm. Figs.(- a &b) shw the decreases as angular lcatin deviates frm tp t bttm leading t better heat transfer prcess, as shwn in Fig.(-b). IV. VEIFICATION OF ESLTS The present results agree with the available literature results. Fig.() reveals the effect f ayliegh number n the 3380-6767-IJMME-IJENS @ February 03 IJENS
Internatinal Jurnal f Mechanical & Mechatrnics Engineering IJMME-IJENS Vl:3 N:0 54 temperature and stream patterns perfrmed by Nieckele and Patankar []. The istherms are pltted in the left half and the streamlines in the right half because f the symmetry abut the vertical diameter. As shwn in this figure, the maximum dimensinless streamline max fr,, and a=0 5 & 0 6 are 3.95 and 0.3, respectively; while Ktake and Hattri [] gave a predicated value f max =4.787 & 3.7 and in the present wrk are 4.036 & 4.055, respectively. The little difference amng the values f max in each wrk fr bth values f ayliegh number may be referred t the difference f the assumptins and methd f slutin between them. 5. The maximum value f the axial velcity at the same radial psitin decreases and deviates twards uter adiabatic wall at the tp ( =0 ) as a increases, while the ppsite ccurs at the bttm ( =80 ) withut any distrtin in the parablic prfile. On the ther hand, the axial velcity prfile at the center ( =90 ) takes the same behavir btained at the tp ( =0 ) but the distrtin is mre prnunced near the inner heated wall at a=0 6. 6. Large temperature gradients and higher lcal Nusselt numbers are t be expected at the bttm except at vertical psitin in which the angular variatin f the Nusselt number remains cnstant because f symmetry abut vertical axis. 7. Fr future wrk, mixed cnvectin can be studied in the fully develped r entrance regin f inclined annulus with rtating inner and/r uter cylinder. V. CONCLSIONS. Fr high a (a 0 6 ), and at any angle f inclinatin, there are tw cells n each side f the annulus. The strnger cell extends slightly further upward as a increases. The shrter path length (resulting frm the presences f mre than ne cell), fr the cellular cnvectin, means that the fluid reaching the tp may have a lwer temperature than wuld have been expected if there was nly ne cell (n each side). Hwever, this depends n the lateral mass flw rate and the extent f lateral mixing.. The cellular mtin at hrizntal psitin is greater than ther angles f inclinatin except vertical psitin in which the cellular mtin diminishes. 3. There is n significant effect fr changing f the angular lcatin n the axial velcity prfile at lw ayleigh number. 4. The transitin frm single-eddy pattern t the duble-eddy pattern appears t ccur at a between 0 5 and 0 6. EFEENCES [] Ktake, S., and Hattri, N.,"Cmbined frced and free cnvectin heat transfer fr fully develped laminar flw in hrizntal annuli", Int. J. Heat Mass Transfer, Vl.8, N., (985) 3-0. [] Nieckele, A. O., and Patankar, S. V.,, "Laminar mixed cnvectin in a cncentric annulus with hrizntal axis", Transactins f the ASME, 07 (Nvember 985), 90-909. [3] Kaviany, M., "Laminar cmbined cnvectin in a hrizntal annulus subject t cnstant heat flux inner wall and adiabatic ut er wall", Transactins f the ASME, 08 (May 986), 39-397. [4] Barletta, A. " Cmbined frced and free flw f a pwer-law fluid in a vertical annular duct", Internatinal Jurnal f Heat and Mass Transfer 43 (000) 3673-3686. [5] Nazrul Islam, Gaitnde,.N., Sharma, G.K., "Mixed cnvectin heat transfer in the entrance regin f hrizntal annuli", Internatinal Jurnal f Heat and Mass Transfer 44 (00) 07-0. 3380-6767-IJMME-IJENS @ February 03 IJENS
Internatinal Jurnal f Mechanical & Mechatrnics Engineering IJMME-IJENS Vl:3 N:0 55 [6] Habib, M. A. and Negm, A. A. A., " Laminar mixed cnvectin in hrizntal cncentric annuli with nn- unifrm circumferential heating", Heat and Mass Transfer 37 (00) 47-435. [7] Esmail M. A. Mkheimer and Maged A. I. El-Shaarawi, "Develping mixed cnvectin in vertical eccentric annuli", Heat Mass Transfer (004) 4: 76 87. [8] Zeraibi N., Amur M., Benzaui A., and Gareche M.," Numerical study f a thermdependent nn-newtnian fluid flw between vertical cncentric cylinders", Internatinal Cmmunicatins in Heat and Mass Transfer 34 (007) 740-75 [9] Mhamed A. Teamah, " Numerical simulatin f duble diffusive laminar mixed cnvectin in a hrizntal annulus with ht, slutal and rtating inner cylinder", Internatinal Jurnal f Thermal Sciences 46 (007) 637-648 [0] Thierry Mare, Niclas Galanis, Inut Vicu, Jacques Miriel, and Ousmane Sw, " Experimental and numerical study f mixed cnvectin with flw reversal in caxial duble-duct heat exchangers", Experimental Thermal and Fluid Science 3 (008) 096 04 [] Anjan Sarkar, Mahapatra S. K. and Sarkar A.,"Oppsing mixed cnvectin and its interactin with radiatin inside eccentric hrizntal cylindrical annulus",int. J. Numer. Meth. Fluids 009; 6:9 30 [] Fattahi E., Farhadi M., and Sedighi K., "Lattice Bltzmann simulatin f mixed cnvectin heat transfer in eccentric annulus", Internatinal Cmmunicatins in Heat and Mass Transfer 38 (0) 35 4 NOMENCLATE g, acceleratin f gravity ; G = gr ˆ u N, radius rati Pr, Prandtl number= P Z = P/ Z ; / c ; q, rate f heat transfer t the fluid per unit area; r, radius f inner cylinder; r, radius f uter cylinder;, radial distance nrmalized with respect t r ; a, ayleigh number= g t, temperature f the fluid ; t w, temperature at the wall ; 4 ; qr t b, bulk averaged temperature f the fluid ; t =t w t b ; T =(t w t b )/(qr / ) ; u, bulk averaged velcity ;, axial velcity nrmalized with respect t u ; Û = /P z ;, cefficient f thermal expansin;, angle measure frm the tp f the annulus;, thermal cnductivity f fluid;, angle f inclinatin f annulus; ˆ, thermal diffusivity f fluid;, viscsity;, kinematic viscsity;, density ;, vrticity nrmalized with respect t ˆ /, time nrmalized with respect t r / ˆ ; r ;, stream functin nrmalized with respect t ˆ. Nu, Lcal Nusselt number alng angular lcatin ; P, pressure nrmalized with respect t u / r ; ˆ 3380-6767-IJMME-IJENS @ February 03 IJENS
0.3845.04937 0.653075 Internatinal Jurnal f Mechanical & Mechatrnics Engineering IJMME-IJENS Vl:3 N:0 56 0.93095.9885 0.66075 0.46305 0.58535 3.04605 0.93095 0.4075.45973.578 0.60766.88 0.60 8.33744 6.4475 0.3407 0.4075.9885 0.8375 0.09605.45973 0.93095 0.4075 0.554337 0.44795 0.3456 0.3575 4.4906.56938 0.646688-0.38508 0.8787-0.97835 a- a=0 5 b- a=0 6 Fig.. Stream Functin and Istherm Lines Cntur Fr, 0 (Hrizntal). 0.468563 0.59875 3.3093.8395 0.604763.88 0.60 8.33744 6.4475 0.3455 0.938 0.09865-0.046875 0.398075 0.8835.33878.8536.36843 0.54553 0.43795 0.330388 0.85 0.563-0.97835 4.4906.56938 0.646688-0.38508 a- a=0 5 b- a=0 6 Fig. 3. Stream Functin and Istherm Lines Cntur Fr, 30 0.5 0.63875.3994 0.55633 0.584459 0.65989 8.33744 8.33744 6.4475 4.4906 0.5787 0.3035 0.003365.069 0.6795.7068.3645 0.45378 0.3539 0.54567 0.55 4.4906.56938 0.646688-0.38508 0.33656 0.0558556-0.97835 a- a=0 5 b- a=0 6 Fig. 4. Stream Functin and Istherm Lines Cntur Fr,. 60. 0.68033 0.54755 0.68033 0.54755 0.406583 0.477067 0.406583 0.477067 0.68857 0.68857 a- a=0 5 b- a=0 6 Fig. 5. Stream Functin and Istherm Lines Cntur Fr, 90 (Vertical). 3380-6767-IJMME-IJENS @ February 03 IJENS
Internatinal Jurnal f Mechanical & Mechatrnics Engineering IJMME-IJENS Vl:3 N:0 57.9 ^.8.6.4 =0 =0 a=0 a=0 3 a=5 0 3 a=0 5 a=0 6 ^.8.7.6.5.4.3 =0 a=0 3 =0 =90 =80... 0 0. 0.4 0.6 0.8..4.6 0 0. 0.4 0.6 0.8..4.6 (a) =0 (a) a=0 3.9.8.8.6 =0 =90.7.6 =0 a=0 6 ^.4. a=0 a=0 3 a=5 0 3 a=0 5 a=0 6 ^.5.4.3. =0 =90 =80. 0 0. 0.4 0.6 0.8..4.6 0 0. 0.4 0.6 0.8..4.6.8 (b) =90 (b) a=0 6 Fig. 7. Effect f Angular Lcatin n the Axial Velcity fr,,.8.6 =0 =80.8, =0 (Hrizntal). ^.4. a=0 a=0 3 a=5 0 3 a=0 5 a=0 6 ^.6.4 =90 =0-80 a=0-0 6 0 0. 0.4 0.6 0.8..4.6.8. (c) =80 0 0. 0.4 0.6 0.8..4.6 Fig. 6. Effect f ayleigh Number n the Axial Velcity fr, =0 (Hrizntal). Fig. 8. Effect f ayleigh Number n the Axial Velcity fr,, =0-80, =90 (Vertical). 3380-6767-IJMME-IJENS @ February 03 IJENS
^ ^ ^ ^ ^ ^ Internatinal Jurnal f Mechanical & Mechatrnics Engineering IJMME-IJENS Vl:3 N:0 58.8.6.4 =0 3 a=0 =0 (hrizntal) =90 (vertical).8.6.4 =0 6 a=0 =0 (hrizntal) =90 (vertical).. 0 0. 0.4 0.6 0.8..4.6 0 0. 0.4 0.6 0.8..4.6 (a) =0 (a) =0.8.8.6 =90 3 a=0.6 =90 6 a=0.4 =0 (hrizntal) =90 (vertical).4 =0 (hrizntal) =90 (vertical).. 0 0. 0.4 0.6 0.8..4.6 0 0. 0.4 0.6 0.8..4.6 (b) =90 (b) =90.8.6.4 =80 3 a=0 =0 (hrizntal) =90 (vertical).8.6.4 =80 6 a=0 =0 (hrizntal) =90 (vertical).. 0 0. 0.4 0.6 0.8..4.6 0 0. 0.4 0.6 0.8..4.6.8 (c) =80 (c) =80 Fig. 9. Effect f Angle f Inclinatin n the Axial Velcity fr, a=0 3,. Fig. 0. Effect f Angle f Inclinatin n the Axial Velcity fr, a=0 6,. 3380-6767-IJMME-IJENS @ February 03 IJENS
Internatinal Jurnal f Mechanical & Mechatrnics Engineering IJMME-IJENS Vl:3 N:0 59 3.6 9 3. =0 =90 8 7 =0 =90 6 Nu.8.4 Nu 5 4 a=0 5 3 a=0 3.6 0 40 80 0 60 00 (degree) 0 0 0 40 60 80 00 0 40 60 80 (degree) (a) a=0 3 (b) a=0 6 Fig.. Circumferential Variatin f the Lcal Nusselt Number fr Varius Angle f Inclinatin,,. Fig.. Streamlines and Istherms Cntur []. 3380-6767-IJMME-IJENS @ February 03 IJENS