Finite Element Analysis of Ferrofluid Cooling of Heat Generating Devices

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1 Excerpt rom the Proceeding o the COMSOL Conerence 8 Hannover Finite Element Analyi o Ferroluid Cooling o Heat Generating Device omaz Strek Intitute o Applied Mechanic, Poznan Univerity o echnology, ul. Piotrowo 3, Poznan, Poland, tomaz.trek@put.poznan.pl Abtract: he vicou, two-dimenional, laminar and incompreible erromagnetic luid low i conidered in thi paper. Flow take place in channel between two parallel lat plate. here are rectangular block (heat-generating device) below the upper wall.in thi paper the conidered erroluid low i inluenced by magnetic dipole. he magnetic dipole give rie to a magnetic ield. Ferroluid have promiing potential or heat traner application becaue a erroluid low can be controlled by uing an external magnetic ield. A trong magnet placed near the device which produce heat will alway attract colder erroluid toward it more than warmer. Keyword: erroluid, Kelvin orce, magnetic dipole, magnetic calar. 1. Introduction During the lat decade, an extenive reearch work ha been done on the luid dynamic in the preence o magnetic ield (magnetorheological luid - MR, erroluid - FF, electrorheological luid - ER and certain type o polymeric gel). he eect o magnetic ield on luid i worth invetigating due to it innumerable application in wide pectrum o ield. he tudy o interaction o the magnetic ield or the electromagnetic ield with luid have been documented e.g. among nuclear uion, chemical engineering, medicine, high peed noiele printing and tranormer cooling. Ferroluid are indutrially prepared magnetic luid which conit o table colloidal upenion o mall ingle-domain erromagnetic particle in uitable carrier liquid. Uually, thee luid do not conduct electric current and exhibit a nonlinear paramagnetic behavior. he variety o ormulation available or erroluid permit a great number o application, rom medical to atellite and vacuum technologie [3-5]. An external magnetic ield impoed on a erroluid with varying uceptibility, e.g., due to a temperature gradient, reult in a nonuniorm magnetic body orce, which lead to a orm o heat traner called thermomagnetic convection. hi orm o heat traner can be ueul when conventional convection heat traner i inadequate, e.g., in miniature microcale device or under reduced gravity condition. A good undertanding o the relationhip between an impoed magnetic ield, the reulting erroluid low, and the temperature ditribution i a prerequiite or the proper deign and implementation o application involving thermomagnetic convection.. Governing equation In thi paper the conidered erroluid low i inluenced by magnetic dipole. We aumed that the magnetic dipole i located at ditance a, b. he magnetic b below the heet at point ( ) dipole give rie to a magnetic ield, uiciently trong to aturate the luid. In the magnetotatic cae where there are no current preent, Maxwell-Ampere law reduce to H =. When thi hold, it i alo poible to deine a magnetic calar potential by the relation H = V m and it calar potential or the magnetic dipole i given by [1] x1 a V m ( x ) = γ (1) π x a + x b ( ) ( ) 1 where γ i the magnetic ield trength at the ource (o the wire) and ( a,b) i the poition were the ource i located. he governing equation o the luid low under the action o the applied magnetic ield and gravity ield are: the ma conervation equation, the luid momentum equation and the energy equation or temperature in the rame o Bouineque approximation. he ma conervation equation or an incompreible luid i v =. ()

2 he momentum equation or magnetoconvective low i modiied rom typical natural convection equation by addition o a magnetic term v ρ + v v = p + η v + t (3) + αρ g k + M ( ) ( )B where ρ i the denity, v i the velocity vector, p i the preure, i the temperature o the luid, η i the vicoity, k i unit vector o gravity orce and α i the thermal expanion coeicient o the luid. he energy equation or an incompreible luid which obey the modiied Fourier law i ρc + v = k + t (4) M + ηφ µ (( v ) H) where k i the thermal conductivity and η Φ i the vicou diipation v1 v v v1 Φ = (5) x y x y he lat term in the energy equation repreent the thermal power per unit volume due to the magnetocaloric eect. he lat term in the momentum equation repreent the Kelvin body orce per unit volume = ( M )B, (6) which i the orce that a magnetic luid experience in a patially non-uniorm magnetic ield. We have etablihed the relationhip between the magnetization vector and magnetic ield vector M = χ m H. (7) Uing the contitutive relation (relation between magnetic lux denity and magnetic ield vector) we can write the magnetic induction vector in the orm B = µ ( 1 + χm )H. (8) he variation o the total magnetic uceptibility χ m i treated olely a being dependent on temperature [] χ χ m = χm ( ) =. (9) 1 + α ( ) Finally, the Kelvin body orce can be repreented by 1 = µ χm( 1 + χm ) ( H H) +. (1) + µ χmh( ( H ) χm ) Uing equation (7-9) we can write Eq. (3) and (4) in the orm, repectively v ρ + v v = p + η v + t + αρg( ) k + (11) 1 + µ χm( 1 + χm ) ( H H) + + µ χmh( ( H ) χm ) and ρc + v = k t. (1) ( χ H) + ηφ µ m (( v ) H) For implicity the preerred work choice i to work in non-dimenional rame o reerence [8]. Now ome dimenionle variable will be introduced in order to make the ytem much eaier to tudy. Moreover ome o the dimenionle ratio can be replaced with wellknown parameter: the Prandtl number Pr, the Rayleigh number Ra, the Eckert number Ec, the Reynold number Re and the magnetic number Mn, repectively: 3 η αρ gh δ Pr =, Ra =, ρ η vr Ec = =, cδ cδh r vr hρ v η r Re = =, µ H µ H h Mn = =. ρ 3. Numerical reult r ρ ρ η (13) hi cae examine the temperature ield in the erroluid and in the electronic component with heat ource. he erroluid tranport heat by convection and conduction. Finally, to approximate the electronic component that require cooling, the model ue a rectangular block with a given volume heat ource. he electronic component tranport thermal energy by pure conduction. he vicou, two-dimenional, laminar and incompreible erromagnetic luid low i conidered in thi paper. Flow take place in

3 channel between two parallel lat plate. here are rectangular block (heat-generating device) below the upper wall. he length o the channel i L = 1 and ditance between plate i h = 1. Outide the channel the magnetic dipole i located at point ( a,b). hi magneto-thermo-mechanical problem conidered in thee example i governed by ollowing dimenionle equation: momentum equation or magnetoconvective low (Navier-Stoke equation) with Kelvin orce; ma conervation equation or an incompreible erroluid; thermal diuion equation; heat traner by conduction or olid domain i the heat equation. he ollowing boundary condition or dimenionle variable are aumed: For upper wall: he velocity i (no lip condition). Inulation condition or heat traner by conduction (in olid domain) n q = n ( k ) = and or heat traner by conduction and convection (in luid domain) n q = n k + c u = ( ) ρ peciie where the domain i well inulated. For lower wall: he velocity i (no lip condition). Inulation condition or heat traner by conduction and convection (in luid domain) n q = n ( k + ρ cu) =. For inlet (let wall): he temperature i l where δ = u l. At the inlet δ boundary there i a parabolic laminar low proile given by equation u uin = 4 y( y 1) or y, 1. u r For outlet (right wall): he convective lux i aumed or temperature, n =. Preure outlet i alo ( k ) aumed, ( p + Sn ) = p n I, where p i the dimenionle atmopheric preure. For olid-luid interace: he velocity i (no lip condition). Continuity equation or heat traner equation n q q = where q = and q ( ) = k + ρ c u. he ollowing initial condition or dimenionle variable are aumed: luid i motionle (velocity i zero), preure i zero and l temperature i or whole domain (with luid δ and olid). able 1. Quantitie o luid Quantity Variable Unit Value kg Denity ρ m Vicoity η kg m hermal J conductivity k m K k.8.6 J Heat capacity c 4 kg K hermal diuivity k m = (diuion 1.1e-7 ρ c coeicient) hermal expanion α 5.6e-3 coeicient Magnetic χ - 6e- uceptibility able. Quantitie o olid (heat generating device) Quantity Variable Unit Value kg Denity ρ m hermal conductivity Heat capacity hermal diuivity (diuion coeicient) k c J m K J kg K k m = ρ c J Heat ource Q 3 m e-4 8.e+8

4 able 3. Flow parameter Quantity Variable Unit Value Velocity u [ m ] 5.e-3 Characteritic velocity v r [ m ] 1.1e-7 Magnetic 7 permeability µ = 4π 1 o a vacuum [ N A ] 1.56e-6 Dierence o δ [ K] 3 temperature emperature [ K ] 3 emp. o upper wall u = + δ [ K] 33 emp. o lower wall l = [ K ] 3 High, length h, L [ ] Centre o magnetic wire Magnetic ield trength at the ource Ratio o thermal diuivitie ( a, b) [ m ] m 1e-3, 1e- γ [ A m] 1 (e-3,-3e-3) ratio = able 4. Quantitie o low A and B Quantity Flow A Flow B a b H r e e+5 Pr 56 Ra.7938 Ec e-13 Re.1786 Mn.718e e+9 Qn V avg (luid) avg (luid) avg (all) able 5. Quantitie o low C and D Quantity Flow C Flow D a b H r e e+5 Pr 56 Ra.7938 Ec e-13 Re.1786 Mn 1.159e e+1 Qn V avg (luid) avg (luid) avg (all) In all table in thi chapter quantitie V avg, avg and dom are calculated uing ollowing ormula: V = uda avg p, (14) = Ap avg p, (15) = Ap A da da dom, (16) where A p = 8. 8 denote area o erroluid and A = 1 denote area o channel (luid and olid). In the low A, B,C and D magnetic dipole i placed below the channel on the line perpendicular to the horizontal wall o channel in ditance rom let wall. When the ditance rom ource o magnetic dipole to bottom channel wall i decreae, it can be oberved that, due to the value o the characteritic value o magnetic ield H r, the maximum value o: the magnitude o the velocity ield o the low increae and the temperature decreae (ee able 4 and 5).

5 (a) (a) Figure 1. Comparion o intenity o magnetic ield in channel: (a) low A, low B, low C and low D or time t=.5. Figure. Comparion o temperature in channel and heat generating device or the dierent low in channel: (a) low A, low B, low C and low D or time t=.5.

6 (a) (a) Figure 3. Comparion o velocity ield in erroluid or the dierent low in channel: (a) low A, low B, low C and low D or time t=.5. Figure 4. emperature evolution in channel in low A or time: (a).1,.1,.3.5.

7 able 6. Quantitie o low E and F Quantity Flow E Flow F a b H e e+5 r Pr 56 Ra.7938 Ec e-13 Re.1786 Mn.718e e+9 Qn V avg (luid) avg (luid) avg (all) able 7. Quantitie o low G and H Quantity Flow G Flow H a b H r e e+5 Pr 56 Ra.7938 Ec e-13 Re.1786 Mn 1.159e e+1 Qn V avg (luid) avg (luid) avg (all) In the low E, F, G and H magnetic dipole i placed below the channel with erroluid on the line perpendicular to the bottom wall o channel at ditance 4 rom let wall. Again we oberved that, due to the value o the characteritic value o magnetic ield, the maximum value o: the magnitude o the velocity ield o the low increae and the temperature decreae (ee able 6 and 7). (a) Figure 5. Comparion o temperature in channel and heat generating device or the dierent low in channel: (a) low E, low F, low G and low H or time t=.5.

8 4. Concluion We have imulated two-dimenional heat traner in erroluid channel low under the inluence o the magnetic ield created by magnetic dipole uing computational luid dynamic code COMSOL baed on inite element method. At the let end o rectangular channel there wa aumed a parabolic laminar low proile. he upper plate wa kept at contant temperature u and the lower at l. he low wa relatively uninluenced by the magnetic ield until it trength wa large enough or the Kelvin body orce to overcome the vicou orce. he magnetoconvection wa induced by the preence o magnetic ield gradient. We oberved that the cooler erroluid low in the direction o the magnetic ield gradient and diplaced hotter erroluid. Ferroluid have promiing potential or heat traner application becaue a erroluid low can be controlled by uing an external magnetic ield [6-7]. he Kelvin body orce arie rom the interaction between the local magnetic ield within the erroluid and the molecular magnetic moment characterized by the magnetization. An impoed thermal gradient produce a patial variation in the magnetization through the temperature-dependent magnetic uceptibility or erroluid and thereore render the Kelvin body orce non-uniorm patially. hi thermal gradient induced inhomogeneou magnetic body orce can promote or inhibit convection in a manner imilar to the gravitational body orce. A trong magnet placed near the device which produce heat will alway attract colder erroluid toward it more than warmer erroluid thu orcing the heated erroluid away, toward the heat ink. hi i an eicient cooling method which require no additional energy input. [3] Odenbach S., Magnetovicou Eect in Ferroluid (Lecture Note in Phyic), Springer (). [4] Odenbach S. (red.), Ferroluid: Magnetically Controllable Fluid and heir Application, Springer (). [5] Roenweig R.E., Ferrohydrodynamic, Cambridge Univerity Pre, Cambridge (1985). [6] Strek., Ferroluid heat traner in channel with porou wall, in: COMSOL Conerence 6, CD-ROM, ISBN: , (6). [7] Strek., Jopek H., Computer imulation o heat traner through a erroluid, Phyica Statu Solidi (B), 44, No. 3, (7). [8] Strek., Finite Element Simulation o Heat raner in Ferroluid, in: Recent Advance in Modelling and Simulation, Edited by: G. Petrone and G. Cammarata, I-ech Education and Publihing, Vienna (8). 5. Reerence [1] Anderon H.I., Valne O.A., Flow o a heated erroluid over a tretching heet in the preence o a magnetic dipole, Acta Mechanica, 18, (1998). [] Ganguly R., Sen S., Puri I.K., Heat traner augmentation uing a magnetic luid under the inluence o a line dipole, Journal o Magnetim and Magnetic Material, 71, pp (4).