UNSTEADY MIXED CONVECTION WITH HEATING EFFECTS IN A HORIZONTAL ANNULUS WITH CONSTANT ROTATING OUTER CYLINDER AND CONSTANT MAGNETIC FIELD

Transcription

1 UNSEADY MIXED CONECION WIH HEAING EFFECS IN A HORIZONAL ANNULUS WIH CONSAN ROAING OUER CYLINDER AND CONSAN MAGNEIC FIELD H. R. MOZAYYENI*,ASGHAR B. RAHIMI** * Gaduate student ** Pofesso, Faculty of Engineeing, Fedowsi Univesity of Mashhad, Mashhad, Ian (Coesponding autho, ahimiab@yahoo.com) Abstact: In this study the poblem of mixed convection of a fluid in the fully developed egion between two hoizontally concentic cylindes with infinite lengths which ae held at diffeent unifom tempeatues is investigated numeically. he foced flow is induced by the cold otating oute cylinde in slowly constant angula velocity with its axis at the cente of annulus. Buoyanci effect also consideed with Boussinesq appoximation. Investigations ae made fo vaious combinations of non-dimensional goup numbes, Reynolds (Re), Rayleigh (Ra), Hatmann (Ha) and Ecket (Eck). We will show how the incease of Ra, Re and P values affect the steamline and isothems and also Nusselt distibutions on the inne and oute cylindes. Anothe pupose of this study is to investigate effects of applying adially constant magnetic field as well as heat geneation due to viscous dissipation tems on the chaacteistics of heat and fluid flow. It is also found that the flow oscillations in the annulus can be suppessed effectively by imposing an extenal MHD foce in adial diection. Moeove, it is shown that viscous dissipation deceases heat tansfe ate on the inne cylinde and inceases this phenomenon on the oute one. Keywods: Mixed convection, constant otation, constant magnetic field

2 - INRODUCION: Recently, the lamina in eithe vetical o hoizontal concentic annulus has eceived attentions by vaious investigatos because of its wide applications such as heat exchanges, cooling systems in electical devices, sola collectos, cooling of tubine otos and cooling of high speed gas beaings. Up to date, most woks fo mixed-convection poblems in otating systems have been pefomed fo the flows in vetical cylindical annuli [-3]. Relatively few studies, howeve, have been made fo the flows in hoizontal annuli. A few authos [4-6] have studied mixed-convective flows within a hoizontal annulus with a heated otating inne cylinde. he flow pattens in otating cylindes categoized into thee basic types accoding to the numbe of eddies by J.S.Yoo [7]. Abu-Hijleh and Heilen [8], has poposed a coelation fo the aveage Nusselt numbe as a function of Reynolds numbe and buoyancy paamete fom a otating cylinde. Mixed heat and fluid flow in otating annuli has been investigated numeically by Lee [9]. It is shown that the effects of viscous dissipation enhance the effects of buoyancy and vice vesa in an inclined channel by Baletta and Zanchini [0]. Also it is shown with using electomagnetic fields, heat tansfe and fluid flow ove a non-isothemal hoizontal cylinde in poous medium can be contolled in []. Non-unifom cicumfeential heating in hoizontal concentic annuli [] and categoization of the flow egimes accoding to the numbe of eddies established on the Ra-Re plane fo vaious numbes [3] ae anothe eseaches in this subject. A study of double-diffusive mixed convection of a binay fluid within a two-dimensional hoizontal annulus with cooled otating oute cylinde has been investigated by Al-Amii and Khanafe [4]. A study to povide an invese appoach fo estimating the viscosity of fluid and themal behavio of the concentic cylindes has been pesented in [5]. Mixed convection with a tempeatue-dependent viscosity and its effects on the dimensionless pessue and velocity is anothe study that has been investigated by Zanchini [6]. In the newest study, Baletta et al. [7], investigated combined foced and fee flow of an electically conducting fluid in a vetical annula poous medium suounding a staight cylindical electic cable with adially vaying magnetic field. he objective of the pesent study is to investigate numeically the poblem of mixed convection of a fluid in the fully developed egion between two hoizontally concentic cylindes with infinite lengths which ae held at diffeent unifom tempeatues. he foced flow is induced by the cold otating oute cylinde in slowly constant angula velocity with its axis at the cente of annulus. Buoyanci effect also consideed with Boussinesq appoximation. Investigations ae made fo vaious combinations of non-dimensional goup numbes, Reynolds (Re), Rayleigh (Ra), Hatmann (Ha) and Ecket (Eck). We will show that how inceasing of Ra, Re and P values affect the steamline and isothems and also Nu distibutions on the inne and oute cylindes. Anothe pupose of this study is to investigate effects of applying adially constant magnetic field as well as heat geneation due to viscous dissipation tems on the chaacteistics of heat and fluid flow. It is also found that the flow oscillations in the annulus can be suppessed effectively by imposing an extenal MHD foce in adial diection. Moeove, it is shown that viscous dissipation deceases heat tansfe ate on the inne cylinde and inceases this quantity on the oute one. NOMENCLAURES:

3 B 0 Constant magnetic field D Chaacteistic length of annulus, (Ro-R i ) Eck Ecket numbe g Gavity acceleation (m/s ) Ha Hatmann numbe Nu Nusselt numbe of pue conduction state Cond. Nu i, Nu o Local Nusselt numbe at the inne and oute cylinde Mean Nusselt numbe at the inne cylinde Mean Nusselt numbe at the oute cylinde P Hydodynamic pessue P * Pessue P Pandtl numbe Q Net ciculation of fluid in the diection of cylinde s otation, Dimensionless adial coodinate i Dimensionless adii of the inne cylinde o Dimensionless adii of the oute cylinde Radial coodinate R i, R o Radius of the inne and oute cylindes, m Ra Rayleigh numbe Re Reynolds numbe empeatue i, o empeatue of the inne and oute cylindes Refeence tempeatue m t *, ime, s Dimensionless adial and cicumfeential velocity components *, * Radial and cicumfeential velocity components, m/s Geeks olumetic themal expansion coefficient (K - ) Dimensionless tempeatue Cicumfeential coodinate iscous dissipation alues of the steam function at the inne and oute cylinde, espectively. - Angula velocity of oute cylinde, s Kinematic viscosity, m - s Electical conductivity of fluid o Ratio of the inne cylinde diamete to the gap width, o =R i /(Ro-R i ) Fluid density, kg/m 3 - PROBLEM FORMULAION: he poblem is fomulated in a pola coodinate system, as in Fig.. he nondimensional full Navie-Stokes and enegy equations fo mixed convection in the annulus 3

4 4 R o between hoizontal concentic cylindes with heated otating oute cylinde with constant velocity ae: Fig.- Geomety of the cylindes Continuity equation: 0 () - Momentum: Re cos P Re Re Ha Ra P t () - Momentum: sin Re P Re Ra P t (3) Enegy Equation: R i

5 5, Re Re P whee Ek t (4) Whee the non-dimensional paametes ae: R o D t t o R R o D R o P P o i o R o D Re P 3 D g Ra o i o i p o c R Ek D B Ha 0 (5) he dimensionless Nusselt numbe in pue conduction in the absence of fluid motion is: (6) he local Nusselt numbe is defined as the actual heat flux divided by Nu Cond. : at = i (7) at = o and the mean Nusselt numbes on the inne and oute cylindes ae given by: (8) (9) (0) Also, the net ciculation of fluid in the diection of cylinde s otation is defined by: ()

6 3- NUMERICAL PROCEDURES Finite volume scheme consisting of ti-diagonal matix algoithm (DMA) is used to solve these equations by SIMPLE algoithm. Convegence citeia fo the numeical solution of the above equations ae obtained by compaing the esidue of two epeated sequences of each of 9 the momentum and enegy equations in which Residual (n+) - Residual (n) < 5x 0. Mesh independency is shown in the following able whee IM and JM ae the numbe of nodes in - and diections fo specific values of non-dimensional goups of Re=00, P=0.7, Eck=0.000, Ra= and Ha=0.0: able. Mesh Independency IM JM Q PRESENAION OF RESULS A wide ange of petinent paametes such as Reynolds numbe, Rayleigh numbe, Pandtl numbe, Hatmann numbe and Ecket numbe ae consideed in this study in the ange of Re 00, 0 Ra 50000, P 0, 0 Ha 00, 0.00 Eck 0. and the atio of the inne cylinde diamete to the annulus gap width is taken between 0.5 Ro R5. he effect of Reynolds numbe on the steamlines and isothems is shown in Fig. fo a stationay inne cylinde whee othe paametes ae constant. Fo small Reynolds numbe the flow field within the annulus is chaacteized by two symmetic kidney-shaped eddies with espect to the vetical axis. his patten in the fluid flow is mainly due to the influence of the buoyancy foce. As the Reynolds numbe inceases due to the otation of the oute cylinde, thee diffeent flow pattens ae illustated in Fig. which ae chaacteized by two eddies, one eddy and no eddy depending on the speed of the oute cylinde. he flow field induced by pue buoyancy foce at vey small Reynolds numbe consists of two kidney-shaped eddies which ae symmetic with espect to the vetical axis = 0. When Reynolds numbe is inceased slightly, the symmety of flow pattens beaks down and speed of the eddy in the egion of is educed. As a esult of stong induced flow by the otating oute cylinde, the eddy in the ight hand side of the annulus disappeas at Re=00. his effect becomes moe ponounced at highe Reynolds numbes (Re=00). he isothems pesented in Fig. also show the above chaacteistics well. As it s shown, the isothems when Re= is vey simila to the case of 6

7 natual convection because of high aspect atio between Ra to Re but with incease of the speed of otation, the effect of natual convection disappeas gadually. Also it can be found that at Re=00, thee is no oscillation of flow within the annulus, theefoe isothems have completely cicle shapes with the same cente simila to foced flow patten. he effects of the Pandtl numbe on the steamlines and isothems fo Re=00.0, Eck=0.0007, Ra= and Ha=0.0 ae shown in Fig. 3. By using the above constant values and with consideing the fixed value fo the Ra numbe, thee is noticeable aspect atio between Gashof numbe and Re fo lowe P numbes. heefoe, buoyancy-induced flow plays moe impotant ole on the flow pattens and this causes the existence of two-eddy flow patten fo P=0.5. As the value of P numbe inceases, the aspect atio between Gashof numbe and Re deceases with the Ra numbe emaining at the fixed value and this tends to enhance the influence of foce-induced flow. he above easons along with foced flow being opposed by buoyancy-induced flow in the ight hand side of the annulus cause eduction of the stength of the eddy in that egion and disappeaance of the eddy at P=, eventually. he eddy emained in the left hand side within the annulus is disappeaed with inceasing the value of P numbe, too. In simila way, the effects of high values of P numbe on the statifying the tempeatue field in the adial diection can be simply obseved in this Figue. It is inteesting to note that the themal plumes nea the uppe potion of the inne cylinde ae tilted to the diection of cylinde s otation at P=. Steamlines Isothem Steamlines Isothems Re= P=0.5 Re=0 P= 7

8 Re=50 P= Re=00 P=5 Re=00 P=0 Fig.. Effect of Reynoldz numbe on the steamlines Fig. 3. Effect of Pandtl numbe on the steamlines and Isothems (P=,Eck=0.0007,Ra= and Isothems (Re=00.0,Eck=0.0007,Ra=0000.0,Ha=0.0),Ha=0.0) o see the effects of Ra numbe on the local heat fluxes, the distibutions of local Nusselt numbes at the inne and oute cylindes fo Re=50, P=, Eck=0.00 and Ha=0 ae shown in Figs 4 and 5. As it has been illustated, with incease of the value of Ra numbe, local heat tansfe ate inceases specially at the points of maximum local Nusselt numbe at both of the inne and oute sufaces when is about 80 degees fo the inne cylinde and is about 0.0 degees fo the oute cylinde. On the othe hand, thee s no pominent change at the minimum points of local heat fluxes with vaying the values of Ra numbe at the both of cylindes. It must be noticed that when Ra=0.0, thee is no effect of natual convection within the concentic annulus. In this case, the local values of Nusselt numbe in both of the sufaces ae about one. 8

9 Fig.4. Effect of Ra numbe on the local Nu numbe Fig. 5. Effect of Ra numbe on the local Nu numbe distibutions at the inne cylinde (Re=50, P= distibutions at the oute cylinde (Re=50, P=, Eck=0.00, Ha=0.0), Eck=0.00, Ha=0.0) he effects of viscous dissipation tems on the ate of heat tansfe ae poposed in Figs. 6 and 7. With inceasing the value of Ecket numbe, the influences of heat geneation due to the viscous dissipations in the concentic annulus ae enhanced, too. It is so clea that with enhancing the amount of heat geneation within the annulus, the aveage tempeatue of the fluid at the cicumfeential aea sections is inceased. Accoding to non-dimensional pocedue, the dimensionless tempeatues of the inne and oute cylindes ae held at.0 and 0.0, espectively. It is obvious that with inceasing the amount of aveage tempeatue at evey cicumfeential aea sections, the ate of heat tansfe between the fluid and the inne cylinde is deceased because of educing the tempeatue diffeences between them and on the contay with this, the heat tansfe ate between the fluid and the oute cylinde is inceased because of enhancing the tempeatue diffeences between them. he Figs. 6 and 7, which ae obtained with vaiation of Ecket numbe fom 0.00 to 0., confim the above explanations well. Also it can be easily found fom Fig. 6 that the fluctuations of local Nu distibutions ae suppessed by inceasing the value of Ecket numbe gadually. 9

10 Fig.6. Effect of Eck numbe on the local Nu numbe Fig. 7. Effect of Eck numbe on the local Nu numbe distibutions at the inne cylinde (Re=00, P=0 distibutions at the oute cylinde (Re=00, P=0, Ra=0000.0, Ha=0.0), Ra=0000.0, Ha=0.0) Figs. 8 and 9 show the effects of imposing the constant magnetic field on the flow and tempeatue distibutions fo Re=00, P=, Eck=0.0 and Ra=0000 at =0. In the absence of magnetic field (Ha=0.0), the cicumfeential velocity and tempeatue distibutions include high-gadient slope, especially nea the cold cylinde. It is inteesting to note that as the stength of the magnetic field inceases (Ha=50.0), the vaiation of tempeatue and flow field become moe unifom than befoe. As Hatmann numbe is futhe inceased (Ha=00), these distibutions ae nealy conduction like and this is due to the suppession of convection by the magnetic field. Also, it can be seen that the flow field becomes completely stable and conduction like at Ha=00. Fig.8. Effect of Ha numbe on the cicumfeential Fig.9. Effect of Ha numbe on the tempeatue velocity distibution at =0.0 (Re=00, P= distibution at =0.0 (Re=00, P=, Ra=0000.0, Eck=0.0), Ra=0000.0, Eck=0.0) 0

11 Up to this point, the case of o = has been pesented. he steamline and isothems fo the othe geometic configuations of annuli ae shown in the figue below. As it s illustated, the vaiation of geometic paamete of the annulus o altes the fluid flow and the tempeatue pattens within the annulus. As the atio of the inne cylinde diamete to the gap width inceases ( o =5.0), the fluid has a less space to move within the annulus which esults a significant decease in the heat tansfe ate. But fo o =0.5, the convective fluid has moe space to move aound and the fluid motion and heat tansfe chaacteistics tend to be moe convective like. Steamlines Isothem o =0.5 o = o =.5 o =5 Fig.0. Effect of the inne cylinde diamete to the gap

12 width on the steamlines and isothems (Re=00, P=, Eck=0.0, Ra=0000, Ha=0) Refeences: []Z. Y., Gue and C. M., Zhang, "hemal dive in centifugal fields - mixed convection in a vetical otating cylinde", Intenational Jounal of Heat and Mass ansfe, 35, , 99. []M. A., Saaawi, and M., Khamis,"Induced flow in unifomly heated vetical annuli with otating inne walls", Numeical Heat ansfe,, pp , 987. [3]M. A., Hessami, G.. D., Davis, E., Lenonadi, and J. A., Reizes, "Mixed convection in vetical cylindical annuli", Intenational Jounal of Heat and Mass ansfe, 30, pp.5-64, 987. [4]., Fusegi, B., Faouk, and K. S., Ball, "Mixed-convection flows within a hoizontal concentic annulus with a heated otating inne cylinde", Numeical Heat ansfe, 9, pp , 986. [5]. S., Lee, "Lamina fluid convection between concentic and eccentic heated hoizontal otating cylindes fo low-pandtl-numbe fluids", Intenational Jounal fo Numeical Methods in Fluids, 4, pp , 99. [6]. S., Lee, "Numeical computation of fluid convection with ai enclosed between the annuli of eccentic heated hoizontal otating cylindes", Computes and Fluids,, pp , 99. [7]J. S., Yoo, "Mixed convection of ai between two hoizontal concentic cylindes with a cooled otating oute cylinde", Int. J. Heat Mass ansfe, 4, pp , 998. [8]B.A., Abu-Hijleh, W.N., Heilen,"Coelation fo lamina mixed convection fom a otating cylinde", Int. Comm. Heat and Mass ansfe, 5, pp , 998. [9]. S., Lee, "Numeical study of mixed convection heat and fluid flow in annuli of heated otating cylindes", Intenational Jounal of Computational Fluid Dynamics, 9, pp. 5-63, 998. [0]A. Baletta, E. Zanchini, "Mixed convection with viscous dissipation in an inclined channel with pescibed wall tempeatues", Intenational Jounal of Heat and Mass ansfe, 44, pp , 00. []M. F. El-Amin, "Combined effect of viscous dissipation and joule heating on MHD foced convection ove a non-isothemal hoizontal cylinde embedded in a fluid satuated poous medium", Jounal of Magnetism and Magnetic Mateials, 63, pp , 003. []M. A. Habib, A.A.A. Negm, "Lamina mixed convection in hoizontal concentic annuli with nonunifom cicumfeential heating", Heat and Mass ansfe, 37, pp , 00. [3]K.h., Khanafe, A. J., Camkha, "Mixed convection within a poous heat geneating hoizontal annulus", Intenational Jounal of Heat and Mass ansfe, 46, pp , 003. [4]A. M., Al-Amii, K. M., Khanafe, "Numeical simulation of double-diffusive convection within a otating hoizontal annulus", Intenational Jounal of hemal Sciences, 45, pp , 006. [5]P.., Hsu, "he invese estimation of the themal behavio and the viscosity of fluid between two hoizontal concentic cylindes with otating inne cylinde", Applied hemal Engineeing, pp , 008.

13 [6]E., zanchini, "Mixed convection with vaiable viscosity in a vetical annulus with unifom wall tempeatues", Intenational Jounal of Heat and Mass ansfe, 5, pp , 008. [7]A. Baletta, S. Lazzai, E. Magyai, and I. Pop, "Mixed convection with heating effects in a vetical poous annulus with a adially vaying magnetic field", Intenational Jounal of Heat and Mass ansfe, 5, pp ,