NUMERICAL STUDY OF A DC ELECTROMAGNETIC LIQUID METAL PUMP: LIMITS OF THE MODEL

Transcription

1 NUMERICAL SUDY OF A DC ELECROMAGNEIC LIQUID MEAL PUMP: LIMIS OF HE MODEL Ndltcho Kandv Institut d chch d'hydo-qubc (IREQ) Qubc, Canada 600, av d la Montagn, Shawinigan, Qubc, G9N 7N5, Canada kandv.ndltcho@iq.ca Abstact his wok psnts th sults of a 3D numical magnto-hydodynamic (MHD) simulation of an lctomagntic DC pump fo liquid mtal using a ctangula mtal flow channl subjctd to an xtnally imposd tansvsal inhomognous magntic fild. In this study, 3D numical simulation basd on th finit lmnt mthod was caid out using th comput packag COMSOL Multiphysics 3.5a. Fo this simulation, th liquid aluminum is usd as an lctically conductiv fluid. h application and th limits of th lctomagntic and th hydodynamic modls a discussd hin. h sults of two typical xampls a summaizd h, including bak flow and pumping conditions fo both lamina and tubulnt mtal flow. hs simulations accuatly psnt th fomation of an M shapd vlocity pofil of liquid mtal und th influnc of th imposd nonunifom magntic fild and a consistnt with th sults of cntly publishd xpimntal and thotical woks. Kywods: DC magntic pump, MHD flow, finit lmnt mthod. 1. Intoduction h concpt of lctomagntic pumping of liquid mtals was dvlopd in th nintn svntis fist fo zinc and aluminum and lat fo oth moltn mtals. Elctomagntic pumps hav many advantags ov mchanical pumps including pcis mtal flow contol without any moving pats, ducd ngy consumption and lss doss fomation. A wll-known xampl of th industial applications of th EMP pincipl is th lctomagntic bak usd in th continuous casting of lag stl stabs to suppss th liquid mtal motion within th mold. Anoth xampl of possibl industial application is as a vlocity-mt fo moltn mtal by masuing th Lontz baking foc [hss, Votaykov and Kolsnicov (2006)]. wo diffnt concpts of lctomagntic pumps fo moltn mtals hav bn dvlopd ov th last foty yas: a) a lina induction lctomagntic (AC) pump and b) a dict cunt (DC) lctomagntic pump. h most fquntly usd is th lina induction EM pump. h main advantag of this AC concpt is that no dict contact with th moltn mtal is ncssay. Howv, th lina induction AC pump can tansf a moltn mtal just hoizontally without any mtal had. h dict cunt (DC) lctomagntic pincipl has bn usd mainly to dvlop lctomagntic mico-pumps fo biomdical and chmical applications fo pcis contol of small volum of fluids in mico-channls [Jaim H. L. Paada and B.J. William (2007), Pi-Jn Wang, Chia-Yuan Chang and Ming-Lang Chang (2004) and Jang J. and S. S. L (2000)]. Ou cnt invstigation [N.P. Kandv V. Kagan and A. Daoud (2010), A. Daoud and N.P. Kandv (2008)] showd that th DC lctomagntic pump can b usd succssfully in many liquid mtal nvionmnts such as xtusion billt casting, mtal finy fo tanspoting moltn mtals at diffnt hads, alloys poduction tc. Elctomagntic DC pumps fo aluminum with capacity up to 30 /h w dvlopd in 2008 and commcializd in th USA fo diffnt industial applications. h gomty of th MHD modl considd in this wok is shown schmatically in Figu 1. A ctangula flat channl with lctically and thmally insulatd boundais is filld with liquid mtal and submittd to a tansvsal magntic fild B z vaying along th x and y axs. his non unifom magntic fild can b simulatd by using two pmannt magnts on th bottom and top walls as shown in Figu 1 o, by using a 3D paamtic function to impos th spac dpndnt vaiabls B z (x) and B z (y) insid th ctangula flat channl. A pai of lctods is intoducd on th vtical latal walls of th channl at a ight angl to th magntic fild. hy supply an xtnal lctostatic fild E with dsid magnitud and diction in th moltn mtal. h axial hoizontal plan (x y) of th

2 channl (at z = 0) is in th half distanc of th magntic gap. 1. Elctomagntic pat of th poblm J = σ ( φ + u B) (1) J = 0 (2) 2. Fluid dynamics fo lamina flow u ρ = P + ηρ 2 u + F L (3) t u = 0 (4) Figu 1. Simplifid schma of th MHD modl. Channl lngth L = 0.3 m, hight H = 0.02 m and width W = 0.1 m. Magnt lngth Lm = 0.05 m [N.P. Kandv V. Kagan and A. Daoud (2010)] h psnt complx MHD poblm involving tubulncs and a stongly inhomognous magntic fild is not ncssaily symmtical and, thfo, only a fully 3D high solution simulation could gasp all spatial aspcts of th lctomagntic and fluid dynamics phnomna. h dict numical simulation (DNS) would b th bst physical appoach fo tating ths MHD poblms, spcially fo th bounday lays gions ducd to vy small scals pmitting th highst solution. Howv, in many a DSN is not th idal solution sinc it quis nomous tim solution and mmoy spac. Oth numical appoachs mntiond in cnt woks [Votaykov and Zinick (2007)] a Lag ddy cunt simulations (LES) and Rynolds avagd stss modls (RASN). hs numical appoachs should b capabl of catching tubulnt fatus of th flow fo th Hatmann lays na th boundais, howv th application of ths appoachs fo inhomognous magntic filds povs that th a still sious tchnical difficultis. h usual thotical fomulation of th magnto hydodynamic modl fo lctically conducting and Nwtonian incompssibl fluid has bn divd fom Ohm s law fo moving mdia, coupld with th Navi-Stoks quations fo lamina flow with Lontz foc givn by th coss poduct F L = J B. h quations usd fo th 3D MHD modl can b summaizd as follows: h lctomagntic pat of th poblm is psntd by Ohm s law in quation (1) whby th lctical scala potntial φ has to b dtmind by solving 2 th Poisson quation: φ = ( u B). H u is th vlocity of th fluid, J is th cunt dnsity and σ is th lctical conductivity. Equation (2) dnots th consvation of th lctical cunt. Habitually, th xtnal non-unifom magntic fild B is simulatd by using a 3D paamtic function to impos th spac-dpndnt vaiabls B z (x) and B z (y) insid th ctangula flat channl. An xampl of such 3D function with maximum valu of B z (x=0, y=0) = 0.7 is givn in Figu 2. B, Figu 2. Exampl of 3D function simulating an xtnally imposd inhomognous magntic fild B h poblms a solvd using stationay fomulation fo th lctomagntic pat with inducd Lontz cunt dnsity tm in th moving mdium (th inducd cunt dos not mov with th moving mdium and dos not poduc magntic fild). his quasi-static appoximation is valid und th assumption that th inducd magntic fild is infinitly small in compaison to th xtnally imposd magntic fild. z y

3 Ou pvious studis [N.P. Kandv, V. Kagan and A. Daoud (2010)] showd that this assumption is absolutly accptabl fo th classical bak flow cas, wh th maximal magntic fild gnatd by th inducd cunts is small by a facto of about 10-5 in compaison to th xtnal magntic fild. h fluid dynamics pat of th poblm is dtmind by quation (3) psnting th consvation of momntum of th fluid in motion, wh P dnots th pssu, ρ is th dnsity and thη is th kinmatic viscosity of th fluid, wh th consvation of mass is givn in quation (4). h coupling btwn th lctomagntic modl and th fluid modl is achivd by intoducing th Lontz foc F L as a body foc in th consvation of momntum and th us of th fluid vlocity, calculatd by th fluid modl, in Ohm s law. h abov dscibd modl can povid a satisfactoy solution fo th lamina bak flow cas only, not fo th lctomagntic pumping cas. 2. Numical modling In ou simulations w hav usd th comput packag Comsol-Multiphysics which is basd on th finit lmnt mthod. h main advantag of using Comsol is that it is not ncssay to wit all intnal souc cods sinc th basic xpssions a alady built-in. Moov, with Comsol it is possibl to us coupling of diffnt physical moduls to cay out MHD simulations, fo xampl: Magntostatics, Conductiv Mdia DC, Navi-Stoks flow modl tc. In this study, wh th modl must psnt an actual pump fo aluminum, woking at diffnt opating conditions, liquid aluminum at 700 C is usd as an lctically conductiv fluid with dnsity ρ = 2385 kg/m 3, lctic conductivity σ = 56 S/m and kinmatic viscosity η = m 2 /s. In all simulatd cass in this study th lctomagntic domain is dlimitd by an ai sph. On th xtnal boundais of this ai domain th magntic and lctic conditions a fixd to: n A = 0 and n. J = 0, wh n is a nomal vcto to th bounday. h intio boundais btwn th pmannt magnts systm, th channl and th ai assum continuity, cosponding to a homognous Numann condition. Elctically insulatd boundais of th channl in th psnc of two lctods on th vtical latal walls of th channl at a ight angl to th magntic fild a considd. h poblms a solvd using a stationay sggatd solv. his mthod of solution was chosn bcaus it was mo stabl than th fully coupld mthod (only on goup of vaiabls). 2.1 Lamina MHD flow h fist cas involvs simulating a low lamina channl flow at lativly low magntic fild and without any xtnal DC cunt ( J = 0 ). h goal of this simulation is to validat th MHD bak flow hypothsis at a low Rynolds numb and to compa ths sults with thos of oth simila publishd woks. h fomulation of this modl in Comsol has bn divd fom th Maxwll-Amp quation, also using Ohm s law, coupld with th Navi- Stoks quations fo lamina flow with Lontz foc F L = J ( A) by intoducing th magntic vcto potntial A wh B = ( A). A = σ ( φ + u ( A)) (5) µ J = σ ( φ + u ( A)) (5-a) J = 0 (6) ρ ( u ) u η u + u = P + F (7) [ ( ( ) )] L u = 0 (8) H Maxwll-Ampè s law in quation (5) includs Ohm s law (5-a). h constants in quations (5) a th lctical conductivity σ and th pmability µ. H th spac dpndnt vaiabls a A (x, y, z), J (x, y, z) and φ (x, y, z). Coupling of th Magntostatics (mqav), and Incompssibl Navi-Stoks (chns) lamina modls is usd to cay out this simulation. h coupling btwn th lctomagntic modl and th fluid modl is achivd by intoducing th Lontz foc F L as a body foc in th consvation of momntum and th us of th fluid vlocity, calculatd by th fluid modl, in Ohm s law. h MHD ffct dpnds on th lctical conductivity, th dnsity and th viscosity of th liquid mtal. It is chaactizd by th Hatmann numb H B h σ a =, and by th intaction 0 ρη 2 paamt N = H a / R. H R is th Rynolds numb, givn by R = U h / 0 η, U 0 is th man vlocity of th liquid mtal and B 0 is th man magntic flux dnsity. h Hatmann and th Rynolds numbs a dfind h with th half hight of th channl (h=h/2=0.01m).

4 Fo th xampl givn blow th bounday conditions of th fluid modl a dtmind by th imposd inlt man vlocity U o = 0.02 m/s, th man magntic flux dnsity B 0 = and th outlt pssu P outlt = 0. As sult, th Rynolds numb is R= 367, th Hatmann numb is Ha = 57 and th intaction paamt N = No-slip vlocity conditions w considd on th sids, top and bottom walls of th channl. Fo this xampl, th shap of th xtnally imposd tansvsal magntic flux dnsity B z (x) along th x axis fo z = 0 and y = 0 is plottd in Figu 2a. h vlocity fild in th cntal hoizontal plan (z = 0) of th studid channl is givn on Figu 2b. dvloping a typical M shap, which is a dmonstation of MHD bak flow pocss. Figu 3: Fluid vlocity pofil u x (y) x-axis: Inlt lamina flow (solid cuv), and outlt vlocity (dashd cuv) Figu 2a: Magntic flux dnsity B z (x) along th x axis fo z = 0 and y = 0 h maximum magntic flux dnsity is 0.14 and is locatd in th vtical axis of th magntic gap. Notic that all MHD paamts a dfind by th ovall man magntic flux dnsity B 0 = Figu 2b: Vlocity fild in th cntal hoizontal plan (z = 0) of th channl. B 0 = 0.092, R=367, N=8.85, U 0 = 0.02 m/s. h fluid vlocity pofil at diffnt fixd positions along th x-axis in th cntal hoizontal plan (z=0) of th channl u x (y) is shown in Figu 3. hs sults show that und th opposit to th flow diction Lontz foc, th inlt lamina vlocity pofil is distubd in th magntic gion, his xampl shows that th simulation of lamina channl flow of liquid mtal coctly psnts th fomation of an M shapd vlocity pofil and concus with th sults of cntly publishd woks [2, 3 and 4]. 2.2 ubulnt MHD flow As mntiond abov, a tubulnt stuctu has to b applid wh th modl simulats a al MHD pump woking at high mtal flow at and high Rynolds numb. o cay out this simulation with Comsol, a tubulnt modl instad of th lamina flow modl, as wll as a DC conductiv mdia modl must b addd to simulat th xtnal DC cunt poducing th pumping Lontz foc. Actually, th diffnt modls Magntostatics (mqav), DC conductiv mdia (mdc) and k-ε ubulnc modl (chns) w coupld fo this simulation. h lctomagntic modl, showd abov in quations (5) and (6), was modifid in tms of th lctical fild intnsity E = φ, and xtnally applid lctical fild intnsity σ E = J as follows: A = σ E + u A + σe µ ( ( )) (9) J = σ ( E + u ( A)) + (10) J J = 0 (11)

5 Notic that in this cas, th Lontz foc F L = J ( A), intoducd as a body foc in th tubulnt flow modl, consids cunt intnsity J as a vcto sum of th intnal σ ( E + u ( A)) and th xtnal J cunt dnsity s quation (10). Actually, a DC lctical potntial diffnc is applid in th (mdc) modl btwn th two lctods to impos an xtnal lctostatic fild E in th moltn mtal with dsid magnitud and diction. In pumping opating conditions, th xtnal lctostatic fild E dominats th inducd lctomotiv fild and th sulting cunt is divn though th liquid mtal in th positiv y-diction, thus poducing lctomagntic foc acting in th diction of th flow. h tubulnt flow modl is dtmind by quations (12), (13) and (14), psnting consvation of momntum, tubulnc kintic ngy (k) and dissipation at (ε) of th fluid spctivly. Equation (15) is usd to calculat th kinmatical tubulnt viscosity ( η ) and quation (16) psnts th consvation of mass. [( η + η )( ( ) )] u + u = P + FL ρ ( u ) u (12) 2 ( u + ( u ) ) ρε η ρ η + = η σ 1 u k k (13) k 2 condition fo th channl walls to psnt th tubulnt bounday lay. As an xampl, DC potntial diffnc of 0.36V was imposd btwn th lctods, thus gnating xtnal cunts of 1794A. Notic that in this cas th magntic fild is about fiv tims gat (B 0 = 0.46 with maximum of 0.7 ) than that of th lamina flow cas. h bounday conditions of th fluid modl a dtmind by th imposd inlt constant pssu P inlt = Pa and outlt pssu P outlt = 0. his ngativ pssu psnts th pssu dops as a function of th flow at in th al mtal tansf cicuit usd in pvious aluminum pumping tsts. It was dtmind by calculating th hydodynamic losss of th xpimntal duct at man flow at of 0.88m/s [N.P. Kandv, V. Kagan and A. Daoud (2010)]. hs imposd conditions gnatd a man fluid vlocity of 0.88 m/s, thus giving Rynolds numb R=16110, Hatmann numb Ha = 280 and intaction paamt N = h dvlopd xtnal lctomagntic foc is 69.7 N and th intnal Lontz baking foc F Lb = N. Figu 5 illustats th shap of th vlocity fild in th cntal hoizontal plan (z = 0) of th channl fo this cas and Figu 6 shows th fluid vlocity pofil at diffnt positions along th x-axis u x (y). 2 ( u + ( u ) ) 2 η u 1 ε ρ ε η ε C 1 η ρc ε + = ε ε2 σ (14) ε 2 k k 2 = k ρc (15) ε η µ u = 0 (16) h constants a fixd as follows: C µ =0.09, C ε1 =1.44, C ε2 =1.92, σ k =1 and σ ε =1.3. In this cas of tubulnt flow, two goups of dpndnt vaiabls a chosn: th fist goup includs th fluid vlocity componnts u(x,y,z) and th pssu P(x,y,z); th scond goup includs th tubulnc componnts (log(ε) and log(k)). h main advantag of using this complx k-ε ubulnc modl built into Comsol Multiphysics is that th solving algoithms a optimizd and th modl cofficints hav bn tund by comput optimization using availabl xpimntal data. h domain of th flow is givn by a ctangula flat channl shown in Figu 1. In th cas of tubulnt flow, th inlt and outlt fluid bounday conditions a P=P inlt and P outlt =0 spctivly. A logaithmic wall function is applid as a bounday Figu 5: Vctos of th vlocity fild in th axial hoizontal plan (z = 0) of th channl. B 0 = 0.46, R=16110, N=5.05, U 0 = 0.88 m/s. h simulation sults show that th MHD quilibium is achd at a man vlocity of 0.88 m/s whn th xtnal DC lctomagntic diving focs F compnsats compltly th hydodynamic losss of th fluid tansf cicuit and th Lontz baking foc F Lb his sult shows also a significant tansfomation of th oiginally dvlopd tubulnt flow pofil into M o U chaps in th magntic zon.

6 Figu 8 dpicts th vctos of th total Lontz foc dnsity in th axial hoizontal plan (z = 0) of th channl. Fo this cas th total lctomagntic diving foc in th x diction is F = F +F Lb =30.77 N. Figu 6: Fluid vlocity pofil on y-axis u(y) at diffnt positions along th x-axis: Inlt flow (solid cuv), 0.25 L fom th inlt (dashd dottd cuv), Middl (dashd cuv) and Outlt vlocity h maximum to man vlocity atio FD = U max /U 0 in this cas is 3.23, dmonstating a significant distotion of th inlt tubulnt flow pofil by tavling though th magntic gion. At a quat of th distanc fom th inlt of th channl (0.25L), th vlocity pofil xhibits lativly low dfomation, but it is compltly distotd in th nxt quat lngth by nting in th high magntic fild aa. h vctos of th inducd cunt dnsity a plottd in Figu 7. Figu 7: Inducd cunt dnsity in th liquid aluminum on th axial hoizontal plan (z = 0) h appaanc of two coupls of small cunt loops locatd on both sid of th magntic pols is vidnt. hs amazing cunt tubulncs a locatd in th zon of dcasing magntic fild wh th lctical fild changs its sign. Figu 8: Vcto plot of th total Lontz foc in th axial hoizontal plan (z = 0) his figu indicats that th lctomagntic diving foc at high diving DC cunts and high flow vlocity is significantly non-unifom and is locatd in th magntic gion clos to th latal walls. 3. Conclusions On th whol, th simulation of lamina and tubulnt channl flow of liquid aluminum psntd in this study accuatly psnts th fomation of an M shapd vlocity pofil and cosponds with th sults of cntly publishd xpimntal and thotical woks [1, 2, 3 and 5]. Fo both lamina and tubulnt flow cass th lctomagntic domain was dlimitd by an ai sph and th non-unifom magntic fild was simulatd by using two pmannt magnts fixd on th bottom and top walls of th channl and coupld by a stl yok. his modl utilizs not only Ohm s law, but also th Maxwll - Amp quation, thus psnting mo pcisly a al EM pump, spcially fo tubulnt pumping conditions involving high diving DC cunts and high flow vlocity. Actually, fo th tubulnt pumping cas, ou fully 3D simulation vald th appaanc of two coupls of small cunt loops locatd on both sid of th magntic pols nsuing fom th complx spatial intaction of lctomagntic and hydodynamic phnomna.

7 4. Rfncs: 1. Kandv N., V. Kagan and A. Daoud: Elctomagntic DC Pump of Liquid Aluminium: Comput Simulation and Expimntal Study, Jounal of FDMP, Vol. 6, No 3, pp , Votaykov Evgny V., Egbt A. Zinick: Numical study of liquid mtal flow in a ctangula duct und th influnc of a htognous magntic fild. FDMP, vol.1, pp , Paada Jaim H. L. and William B.J. Zimman: Numical simulation of a magntohydodynamic DC micodvic. Multiphysics Modlling with Finit lmnt mthods Univsity of Shffild,U.K. Vol.18, pp , Jang J. and S. S. L: hotical and xpimntal study on MHD mico-pump. Sns. Actuatos A, Vol 80, PP , Andv O., Yu. Kolsnikov and A. hss: Expimntal study of liquid mtal channl flow und th influnc f a nonunifom magntic fild. Phys Fluids, Vol 18, , Holoyd Richad J.: An xpimntal study of th ffcts of wall conductivity, non-unifom magntic fild and vaiabl-aa ducts on liquid mtal flow at high Hatmann numb (Pat 1: Ducts with nonconducting walls. J. Fluid Mch, vol 93, pat 4, pp , Hughs M., K. A. Piclous and M. Coss: h Numical modlling of DC lctomagntic pump and bak flow. Appl. Math. Modlling, Vol 19, PP , Dc