FLOW AND HEAT TRANSFER OF THREE IMMISCIBLE FLUIDS IN THE PRESENCE OF UNIFORM MAGNETIC FIELD

Transcription

1 THERMAL SCIENCE: Year 04, Vol. 8, No., pp FLOW AND HEAT TRANSFER OF THREE IMMISCIBLE FLUIDS IN THE PRESENCE OF UNIFORM MAGNETIC FIELD by Dragša D. NIKODIJEVI], vojn M. STAMENKOVI], Mloš M. JOVANOVI], Mloš M. KOCI], and Jelena D. NIKODIJEVI] Faculty of Mechancal Engneerng, Unversty of Ns, Ns, Serba Orgnal scentfc paper DOI: 0.98/TSCI4009N The magnetohydronamc flow of three mmscble fluds n a horzontal channel wth sothermal walls n the presence of an appled magnetc feld has been nvestgated. All three fluds are electrcally conductng, whle the channel plates are electrcally nsulated. The general equatons that descrbe the dscussed problem under the adopted assumptons are reduced to ordnary dfferental equatons and closed-form solutons are obtaned n three flud regons of the channel. Separate solutons wth approprate boundary condtons for each flud have been obtaned and these solutons have been matched at the nterface usng sutable boundary condtons. The analytcal results for varous values of the rtmann number, the rato of flud heghts and thermal conductvtes have been presented graphcally to show ther effect on the flow and heat transfer characterstcs. Keywords: mmscble fluds, heat transfer, rtmann number, magnetc feld Introducton The flow and heat transfer of electrcally conductng fluds n channels and crcular ppes under the effect of a transverse magnetc feld occurs n magnetohydronamc (MHD) generators, pumps, accelerators, and flowmeters, and t has applcatons n nuclear reactors, fltraton, geothermal systems and others. The nterest n effects of outer magnetc feld on heat-physcal processes appeared n the early seventes of the last century. One of the frst works n the feld of mass and heat transfer n the presence of magnetc feld was presented by Blum []. The flow and heat transfer of a vscous ncompressble electrcally conductng flud between two nfnte parallel nsulatng plates have been studed by many researchers [-6]. The problem of convectve MHD channel flow between two parallel plates subjected smultaneously to an axal temperature gradent and pressure gradent was studed numercally by Yang and Yu [7]. One decade ago, Bodosa and Borkakat [8] analyzed the problem of an unstea -D flow of vscous ncompressble and electrcally conductng flud between two parallel plates n the presence of a unform transverse magnetc feld, for the case of sothermal plates and one sothermal and other adabatc. Ghosh [9] has obtaned an analytcal soluton to the problem of stea and unstea hydromagnetc flow of vscous ncompressble electrcally con- Correspondng author; e-mal: zkas@masfak.n.ac.rs

2 00 THERMAL SCIENCE: Year 04, Vol. 8, No., pp ductng flud under the nfluence of constant and perodc pressure gradent n presence of nclned magnetc feld. There are many other nvestgatons, lke Borkakat and Chakrabarty's [0] nvestgaton of unstea free convecton MHD flow between two heated vertcal parallel plates n nduced magnetc feld, or Aydn and Avc's [] analytcal nvestgaton of lamnar heat convecton n a Couette-Poseulle flow between two parallel plates wth a smultaneous pressure gradent and an axal movement of the upper plate. Recently, Snga [] has gven an analytcal soluton to the problem of MHD free convectve flow of an electrcally conductng flud between two heated parallel plates n the presence of an nduced magnetc feld. All the mentoned studes refer to a sngle-flud model. Most of the problems relatng to the petroleum ndustry, geophyscs, plasma physcs, magneto-flud namcs and so forth nvolve multflud flow stuatons. Because of that, Shal [] studed rtmann flow of conductng flud and a non-conductng flud layer contaned n channel and hs results predcted that an ncrease of the order of 0% can be achved n the flow rate for sutable ratos of heghts and vscostes of the two fluds. Lohrasb and Saha [4] studed two-phase MHD flow and heat transfer n parallel plate channel wth the flud n one phase beng conductng. There have been some expermental and analytcal studes on hydronamc aspects of the two-flud flow reported n the recent lterature. Malashetty [5, 6] have studed the two flud MHD flow and heat transfer n an nclned channel, and flow n an nclned channel contanng porous and flud layer. Recently, Umavath [7, 8] have analyzed the MHD Poseulle-Couette flow and heat transfer of two mmscble fluds between nclned parallel plates. Due the mportance of the two flud flow models, n our prevous paper [9] flow and heat transfer of two mmscble fluds n the presence of a unform nclned magnetc feld was nvestgated. Whle most of the prevous studes manly consder two-flud flow, there s a model whch dscuss the combned effects of pressure gradent and electroosmoss for the three-flud flow [0]. Mult-layer flows occur ndustrally n three man settngs. Frstly, there are co-extruson processes, where a product s made of more than one layer smultaneously. Secondly, there are flm-coatng processes, where a layer s appled to a flud substrate. Thrdly, there are lubrcated transport processes, where a lubrcatng flud les n a layer between the wall of a duct and the transported flud []. The development of mcrofludcs platforms n recent years has led to an ncrease n the number of applcatons nvolvng the flow of multple mmscble layers of vscous electrolyte fluds []. Recent studes show that MHD flows can also be a vable opton for transportng weakly conductng fluds n mcroscale systems, such as flow nsde the mcrochannel networks of a lab-on-a-chp devce [, 4]. In mcrofludc devces, multple fluds may be transported through a channel for varous reasons. For example, an ncrease moblty of a flud can be acheved by stratfcaton of hghly moble flud or mxng of two or more fluds n transt may be desgned for heat and mass transfer applcatons. In that regard, magnetc feld-drven mcropumps are n ncreasng demand due to ther long-term relablty, absence of movng parts, low power requrement, flow reversblty and mxng effcency [5, 6]. The present work proposes a theoretcal model of three electrcally conductve flud flow under the nfluence of unform magnetc feld. Mathematcal model As alrea mentoned, the MHD flow of three mmscble fluds n a horzontal channel wth sothermal walls n the presence of an appled magnetc feld has been nvestgated. The fluds n the three regons have been assumed mmscble and ncompressble and the flow has

3 THERMAL SCIENCE: Year 04, Vol. 8, No., pp been stea, one dmensonal, and fully developed. All three fluds have dfferent vscostes m, m, and m and denstes r, r, and r. The analytcal solutons for veloctes and temperature dstrbuton have been obtaned and computed for dfferent values of the characterstc parameters. The physcal model, shown n fg., conssts of two nfnte parallel plates extendng n the and k-drecton. Both nfnte parallel plates are fxed. In the regon I h y h we have flud of vscosty m, electrcal conductvty s, thermal conductvty k, and specfc heat capacty c p, n the regon II 0 y h, whch has been flled by a layer of dfferent flud of vscosty m, flud propertes are electrcal conductvty s, thermal conductvty k, and specfc heat capacty c p, and n the last regon III h y 0, whch has been flled by a layer of flud of vscosty m, propertes are electrcal conductvty s, thermal conductvty k and specfc heat capacty c p. A unform magnetc feld of the strength B s perpendcular to the flud flow n j drecton Fgure. Physcal model and co-ordnate system and we are consderng -D flow n -drecton. UU () B B j () where B s the magnetc feld vector. The upper and lower plates have been kept at the two constant temperatures T w and T w, respectvely, and the plates are electrcally nsulated. We are consderng a statonary problem ( f/t=0). The descrbed MHD three flud flow problem s mathematcally presented wth a contnuty equaton: U V V 0, because of 0 we have U = U, ( y), () x y y momentum equaton (for all three flud layers) du p m sbu 0,,, (4) x energy equaton T J rcp UT kt mf (5) t s where current densty vector J s J= s(e+u B) (6) and E s the vector of appled electrc feld, whch s neglected n present stu. Energy equaton now has the followng form: d T du k BU 0 s (7) and, as we can see, the temperature changes only n j drecton, because the temperature on the plates s constant n drecton. Now, we can wrte the energy equaton for all three flud layers:

4 0 THERMAL SCIENCE: Year 04, Vol. 8, No., pp d T du k m sbu 0,,, (8) The flud and thermal boundary condtons for ths problem are represented by equa- U( h) 0, U() h U (), h U () 0 U (), 0 U ( h) 0 du du du du m m for yh, m m for y= 0 (9) tons: and T( h) TW, T() h T(), h T() 0 T(), 0 T( h) TW dt dt dt dt k k for yh, k k for y 0 (0) Now the followng transformatons have been used to transform prevous equatons to dmensonless form: y U h p T T y U W, ( y), P, q h U mu x TW T W () whereu s the average velocty n channel and after transformatons we obtan the followng momentum equaton: d U U P 0 () where s the rtmann number, defned as: Bh s m () The general soluton of eq. () s: U A cosh( y ) B snh( y ) P (4) Now we have to determne constants A and B usng the boundary condtons n dmensonless form: U () 0, U () U (), U () 0 U (), 0 U ( ) 0 U a d du m du du m for y, a, a for y (5) 0, a m m The soluton of transformed momentum equaton, wth boundary condtons has the followng form: P U Acosh( y ) Bsnh( y ) C, C (6) where P A ( M M N M M N ) tgh() ( )B tgh( ) (7) cosh( ) B M M N ( M M N ) B (8) A N NB (9)

5 THERMAL SCIENCE: Year 04, Vol. 8, No., pp and B a A N cosh( ) M 5 snh( ) N cosh( ) P P N N B B a B (0) () () M=[snh( ) tgh( )cosh( )] () M = M cosh( ) (4) M = M snh( ) (5) M M P P cosh( ) (6) cosh( ) a [cosh( ) tgh( )snh( )] (7) M 4 M 5 P P [cosh( ) ] cosh( ) (8) N [ M M snh( )] 4 (9) N =[M M 4 cosh ( )]N (0) ap snh( ) N MM4 N () cosh( ) Usng the same transformatons, we obtan the second energy equaton n the followng form: d q du Pr Ec Ec 0 d Pr U y () where cp Prandtl number Pr m () k U Eckert number Ec cp ( TW TW) The general soluton of eq. () s: q Pr Ec ( A B )cosh( y ) ABsnh( y ) 4 [ cosh( ) snh( C )] A y B y C y Dy E (5) The dmensonless boundary condtons that were used for the determnaton of the constants D and E are: (4)

6 04 THERMAL SCIENCE: Year 04, Vol. 8, No., pp dq q(), q() q(), q() 0 q(), 0 q( ) 0 dq k dq dq k b for y, b, b for y 0, b k k Usng prevous condtons, we have determned the constants D and E ( =,,)n the form: D C5 bc6 bb D (7) D C6 b D (8) C C C C4 C5 C6( b) D b bb (9) E =C D (40) E =C +C 4 +D (4) E =C 4 +D (4) and C A B A B Pr Ec ( )cosh( 4) snh( 4 ) 4 C [ A cosh( ) B snh( )] C (4) C Pr ( A B )cosh( ) A B snh( ) Ec 4 C[ Acosh( ) Bsnh( )] C (44) C Pr ( A B )cosh( ) A B snh( Ec ) 4 C[ Acosh( ) Bsnh( )] C (45) C =C C (46) C A B C A A B Pr Ec ( ) 4 Pr Ec ( 4 ) C A (47) (6) C4 A B A B Pr Ec ( )cosh( ) snh( ) 4 C[ Acosh( ) Bsnh( )] C (48) Q Pr Ec ( A B )snh( ) A B cosh( ) C [ A snh( ) B cosh( )] C (49) Q b Pr Ec ( A B A B )snh( ) cosh( ) C [ A shn( ) B cosh( )] C (50) C 5 =Q Q (5)

7 THERMAL SCIENCE: Year 04, Vol. 8, No., pp Results and dscusson C 6 = Pr Ec (A B + C B ) b Pr Ec (A B + C B ) (5) In the prevous secton, we have defned the mathematcal model for flow and heat transfer of three mmscble fluds n the presence of unform magnetc feld. Usng ths model, the graphs of velocty and temperature are generated for dfferent rtmann numbers and for dfferent ratos of flud vscosty m and thermal conductvty k. Fgures and llustrate the effect of the rtmann number on velocty and temperature profles. The nfluence of the rtmann number on the velocty profle was more pronounced n the channel regon III than n the regon II, and much more than n the regon I. The regon III contans the flud wth the greatest electrcal conductvty. It was found that for large values of rtmann number, flow can be almost completely stopped n the regon III, whle n the regon I, velocty decrease s sgnfcant. The effect of ncreasng the rtmann number on temperature profles n all three channel regons was n equalzng the flud temperatures. The effect of the rato of fluds vscostes n regons I, II, and III on the velocty profles s shown n fg. 4 to 7. In the case of parameter a alteraton, the change n velocty s sgnfcant n all three flud regons, whle n the case of parameter a velocty sgnfcantly changes only n the regon III, whle n other two regons remans almost constant. These results are gven for the same value of Lorentz force ntensty (rtmann number s constant) for all three fluds, and t can be concluded that the domnant effect of changes n flud vscosty occurs n regon III, whle ths nfluence n other two regons s much less pronounced. Fgures 5 and 7 also show the effect of the rato of vscosty of fluds n regons I, II, and III to temperature alteraton. In frst consdered case a sgnfcant change n temperature s notced n all three flud regons. Temperature rse n the mddle of the channel s a consequence of vscous dsspaton and large velocty gradents and constant Joule heatng effect. The total temperature ncreases due to mutual effects of fluds at the nterface. Fgure. Velocty profles for dfferent values of rtmann numbers Fgure. Temperature profles for dfferent values of rtmann numbers Fgure 4. Velocty profles for dfferent ratos of fluds vscostes a = m /m

8 Nkodjev}, D. D., et al.: Flow and Heat Transfer of Three Immscble n THERMAL SCIENCE: Year 04, Vol. 8, No., pp. Fgure 5. Temperature profles for dfferent ratos of fluds vscostes a = m /m Fgure 6. Velocty profles for dfferent ratos of fluds vscostes a = m /m Fgure 7. Temperature profles for dfferent ratos of fluds vscostes a = m /m Fgure 8. Temperature profles for dfferent ratos of thermal conductvtes b = k /k In second consdered case (change of a ) there s no sgnfcant change n temperature. Smaller temperature changes occur n the regons II and III, whle n the regon I, the temperature reman almost constant. Fgures 8 and 9 shows the temperature dstrbuton over the channel heght for dfferent values of the fluds thermal conductvtes ratos b and b. Decrease of thermal conductvtes ratos cause an ncrease of dmensonless temperature n all three flud regons. In the case when the parameter b sgnfcantly ncreases the conducton plays domnant role n heat transfer. In the case of ncreasng the parameter b vscous dsspaton and Joule heatng effects are stll present (fg. 8). Influence of Eckert and Prandtl number can be dsplayed together takng ther product (fg. 0). The product of these two numbers s usually called Brnkman number, and t represents the rato of the knetc energy dsspated n the flud and the conducton of heat n flud or from t. When the product of EcPr s sgnfcantly lower than, the energy dsspaton can be neglected n comparson wth the heat conducton. As ths Fgure 9. Temperature profles for dfferent ratos of thermal conductvtes b = k /k

9 THERMAL SCIENCE: Year 04, Vol. 8, No., pp product ncreases the energy dsspaton becomes an mportant parameter n the process of heat transfer and plays an mportant role n the temperature dstrbuton n the flud stream and the total heat transfer. Conclusons The problem of MHD flow and heat transfer of three mmscble fluds between parallel plates n the presence of an appled magnetc feld was nvestgated analytcally. All three fluds were assumed Newtonan and electrcally conductng. Closed form solutons for dmensonless velocty Fgure 0. Velocty profles for dfferent values of Eckert and Prandtl number and temperature of each flud were obtaned takng nto consderaton sutable nterface matchng condtons and boundary condtons. The results were numercally evaluated and presented graphcally for three fluds. Only the part of the results s presented for varous values of rtmann number and ratos of vscostes and thermal conductvtes. The obtaned results show that the control of flow and heat transfer for observed case can be realzed by changng the magnetc feld ntensty and defned ratos of flud propertes. Acknowledgments Ths paper s supported by the Serban Mnstry of Educaton, Scence and Technologcal development (Project No. TR 506; Research of MHD flow n the channels, around the bodes and applcaton n the development of the MHD pump). Nomenclature B magnetc feld vector, [T] X longtudnal co-ordnate, [m] B 0 strength of appled magnetc feld, [T] Y transversal co-ordnate, [m] c p specfc heat capacty, [Jkg K ] E electrc feld vector, [Vm ] Greek symbols rtmann number n regon a vscostes rato of fluds h regon heght, [m] b rato of thermal conductvtes J current densty vector, [Am ] q dmensonless temperature n regon k thermal conductvty of flud, m namc vscosty n regon, [kgm s ] [WK m ] r densty of flud n regon, [kgm ] p pressure, [Pa] s electrcal conductvty regon, [Sm ] T temperature, [K] dsspatve functon U flud velocty n regon, [ms ] References [] Blum, E. L., et al., Heat Exchange and Mass Exchange n Magnetc Feld, Znatne,, 967 [] Cramer, K. R., Pa, S. I., Magnetoflud Dynamcs for Engneers and Appled Physcsts, McGraw-Hll, New York, N. Y., USA, 97 [] Soundalgekar, V. M., Uplekar, A. G., ll Effects n MHD Couette Flow wth Heat Transfer, IEEE Transactons on Plasma Scence, PS-4, (986) 4, pp [4] Atta, H. A., ll Current Effects on the Velocty and Temperature Felds of an Unstea rtmann Flow, Canadan Journal of Physcs, 76 (998), 9, pp [5] Kyasatfar, M., et al., Investgaton of Thermal Behavor and Flud Moton n DC Magnetohydronamc Pumps, Thermal Scence, 8 (04), Suppl., pp. S55-S56

10 08 THERMAL SCIENCE: Year 04, Vol. 8, No., pp [6] Vyas, P., Srvastava, N., Radatve MHD Compressble Couette Flow n a Parallel Channel wth a Naturally Permeable Wall, Thermal Scence, 8 (04), Suppl., pp. S57-S585 [7] Yang, H. K., Yu, C. P., Combned Forced and Free Convecton MHD Channel Flow n Entrance Regon, Internatonal Journal of Heat and Mass Transfer, 7 (974), 6, pp [8] Bodosa, G., Borkakat, A. K., MHD Couette Flow wth heat Transfer between Two Horzontal Plates n the Presence of a Unform Transverse Magnetc Feld, Journal of Theoretcal and Appled Mechancs, 0 (00), pp. -9 [9] Ghosh, S. K., A Note on Stea and Unstea Hydromagnetc Flow n a Rotatng Channel n the Presence of Inclned Magnetc Feld, Internatonal Journal of Engneerng Scence, 9 (99) 8, pp [0] Borkakat, A. K., Chakrabarty, S., Unstea Free Convecton MHD Flow between Two Heated Vertcal Parallel Plates n Induced Magnetc Feld, Indan Journal of Theoretcal Physcs, 47 (999), pp [] Aydn, O., Avc, M., Lamnar Forced Convecton wth Vscous Dsspaton n a Couette-Poseulle Flow between Parallel Plates, Appled Energy, 8 (006), 8, pp [] Sngha, K. G., Analytcal Soluton to the Problem of MHD Free Convectve Flow of an Electrcally Conductng Flud between Two Heated Parallel Plates n the Presence of an nduced Magnetc Feld,, Internatonal Journal of Appled Mathematcs and Computaton, (009), 4, pp. 8-9 [] Shal, R., On Lamnar Two-Phase Flows n Magnetohydronamcs", Internatonal Journal of Engneerng Scence, (97), 0, pp [4] Lohrasb, J., Saha, V., Magnetohy-Dronamc Heat Transfer n Two-Phase Flow between Parallel Plates, Appled Scentfc Research, 45 (988),, pp [5] Malashetty, M. S., et al., Convectve Magnetohydronamc Two Flud Flow and Heat Transfer n an Inclned channel, Heat and Mass Transfer, 7 (00), -, pp [6] Malashetty, M. S., et al., Two Flud Flow and Heat Transfer n an Inclned Channel Contanng Porous and Flud Layer, Heat and Mass Transfer, 40 (004),, pp [7] Umavath,J.C.,et al., Oscllatory rtmann Two-Flud Flow and Heat Transfer n a Horzontal Channel, Internatonal Journal of Appled Mechancs and Engneerng, (006),, pp [8] Umavath, J. C., et al., Unstea Two-Flud Flow and Heat Transfer n a Horzontal Channel, Heat and Mass Transfer, 4 (005),, pp [9] Nkodjev}, D., et al., Flow and Heat Transfer of Two Immscble Fluds n the Presence of Unform Inclned Magnetc Feld, Mathematcal Problem n Engneerng, Vol. 0, do: 0.55/0/0 [0] wang,l., et al., Analytcal Model of Mxed Electroosmotc/Pressure Drven Three Immscble Fluds n a Rectangular Mcrochannel, Internatonal Journal of Heat and Mass Transfer, 5 (009), pp [] Hormoza, S., et al., Mult-Layer Channel Flows wth Yeld Stress Fluds, Journal of Non-Newtonan Flud Mechancs, 66 (0), 5-6, pp [] L, J., et al., Analyss of Mult-Layer Immscble Flud Flow n a Mcrochannel, Journal of Fluds Engneerng, DOI: 0.5/ [] m, H. B., et al., A Magneto-Hydronamcally Controlled Fludc Network, Sensors and Actuators B, 88 (00),, pp [4] Hussameddne, S. K., et al., Analytcal Predcton of Flow Feld n Magneto-Hydronamc-Based Mcrofludc Devces, Journal of Fluds Engneerng, DOI: 0.5/.950 [5] Y, M., et al., A Magnetohy-Dronamc Chaotc Strrer, Journal of Flud Mechancs, 468 (00), pp [6] Weston, M. C., et al., Magnetc Felds for Flud Moton, Analytcal Chemstry, 8 (00), 9, pp Paper submtted: February, 04 Paper revsed: June, 04 Paper accepted: June 4, 04