Numerical Investigation of Flow in a New DC Pump MHD

Transcription

1 Jounl of pplied Fluid Mechnics, Vol., No., pp. 3-8, 9. vilble online t ISSN Numeicl Investigtion of Flow in New DC Pump MHD N. Bennecib, S. Did nd R. bdessemed 3 Univesity Mentoui Constntine, City of Ein El bey, Constntine, 5, lgei L.S.P.I.E Resech Lbotoy, Electicl engineeing deptment; Univesity of Btn, 3 L.E.B Resech Lbotoy, Electicl engineeing deptment; Univesity of Btn, Emil: lzned@yhoo.f, s_did@yhoo.f (Received June, 8; ccepted Decembe 7, 8) BSTRCT Electomgnetic pumps hve sevel dvntges to mechnicl pumps. They offe mneuvebility by diectionl thust long with quietness nd e conceived with n im of eliminting ll moving pts, being lso fee fom poblems of we nd tiedness of use. The flow field in the chnnel is teted s stedy stte, incompessible nd fully developed lmin flow conditions. Ou numeicl code DCPMHD uses cylindicl coodintes (,ϕ, z) nd solves the incompessible MHD equtions fo mgnetic vecto potentil nd fluid velocity V. Using finite volume method fo numeicl clcultion. The numeicl esults of the pefomnce chcteistics of DC electomgnetic pump e discussed nd show tht ou new concept is cpble to delive bi-diectionl ctivtion nd hve stisfctoy contollbility, becuse of its popotionl output foce nd input cuent eltionship. Keywods: Design, Fluids mechnics, Finite volume method, Mgneto-hydodynmic pumps, Nvie-stokes equtions, Sewte pumps. Β D E i ex mgnetic vecto potentil mgnetic induction electic flux density (displcement electic) electic field electic conduction cuent density induced cuent density electicl cuent souce density electicl cuent density injected by electodes NOMENCLTURE P V υ σ pessue flow velocity kinemtic viscosity coefficient electic conductivity µ pemebility mgnetic ξ (,z) voticity vecto ψ (,z) vecto potentil hydodynmic v,v z components of the velocity c ω component of voticity β i pojection function Ω domin in element. INTRODUCTION Mgneto hydodynmics (MHD) coves esech on the genetion of phenomen (such s stiing, mixing, septing nd moving) unde mgnetic ction exeted on fluid. The utiliztion of MHD effects is multiple: mong those, the MHD pump effect is one of the impotnt effects. Conty to the odiny mechnicl pump necessily equipped with oty moving pt like blde, the MHD pump is notble fo the complete bsence of the blde- lke pt; Chi-yun (4,b). To poduce such n MHD pump effect, it is equied fom the pumping mechnism on imposition of constnt mgnetic field on fluid which is cossed by DC cuent to cete fluid movement. Hence, these pumps e designed without using ny movble pt nd thus e fee of we nd ftigue poblems cused by pessue-dop coss the mechnicl pts. Figue epesents new DC electomgnetic pump design. It consists of mgnetic in the tous shpe, two winds, fou electodes nd chnnel (Bennecib et l. 7). It is ssumed tht fluid is n incompessible, lmin nd tht mteil popeties such s kinemtics viscosity nd

2 B. Nedjou et. l / JFM, Vol., No., pp. 3-8, 9. density e constnts; Hughes et l. (995). The fluid in this cse is sewte pue o seeded by NCl to enhnce the conductivity. In this ppe, solution ws obtined fom both Nvie-stokes nd Mxwell equtions unde some ssumptions which hve cceptble physicl menings. Chnnel Mgnetic Cicuit The pesent ppe shows the esults of the DC pump MHD numeicl simultion tht we cied out. The poposed pump poduces n xil flow. s consequence, mgnetic field cn be obtined befoe clculting of flow field. Electodes ) Pump Concept Winds. NUMERICL NLYSIS OF FLOW ND ELECTROMGNETIC FIELD The xisymmetic poblem descibing mgneto hydodynmic devices is obtined fom the electomgnetic eqution in tems of the mgnetic vecto potentil. ot ot σ µ ( v ot ) = J ex + J whee J ex, J, µ nd σ e the cuent density in the exciting coil, the cuent density injected by electodes, the mgnetic pemebility nd the electic conductivity, espectively. In this study, electomgnetic field is consideed to be constnt. In the xisymmetic nlysis the electic cuent density hs only the ϕ -component which is independent ofϕ, so the esulting mgnetic vecto potentil hs only the ϕ - component. Using D cylindicl, z coodintes; Eq. () is developed s ϕ ϕ + µ µ ϕ σ v = J ex J If we intoduce the tnsfomtion = ϕ (3) Eq. () becomes v = J ex J z z + σ (4) µ µ Let Ω be the study domin enclosed by Γ = Γu Γq with boundy conditions = on Γ u, = q on Γ n q. nd q e the pescibed unknown potentil nd noml mgnetic fluxes, espectively, on the essentil boundy Γu nd on the flux boundy Γq nd n is the unit outwd noml diection to the boundy Γ. The motion of the liquid in the mgnetic field cn be descibed s Bhdi nd bbsov (5) q () () b) Font view nd top sight with n xis cut ( ) Fig.. Poposed DC MHD pump configution divv = (5) V F + ( V. ) V = gd P + υ V + (6) t ρ ρ Whee V is the velocity vecto, P is the pessue, ρ is the density of the liquid, υ is the kinemtic viscosity nd F is Lplce foces which e given by ( + ) Β F = J i J (7) Whee J i is the induced cuent density nd J is the electic cuent density injected by the electode. The coupled velocity V nd mgnetic induction field B vi Lplce foces is developed. Next, we will pply method which uses the voticity vecto ξ (, z) nd two vecto potentils: hydodynmic ψ (, z) nd mgnetic (, z).. (ψ,ξ, ) Model We intoduce two vecto potentils (, ψ ), the voticity vectoξ, nd using the following eltionships (Kzeminski et l. 996; Femigie 999). B = ot, V = ot ψ, ξ = ot V (8) v ψ =, v z ψ = (9) 4

3 B. Nedjou et. l / JFM, Vol., No., pp. 3-8, 9. v v z w = () Whee v nd v z e the components of the velocity, w is the component of voticity. ccoding to the configution of the pump poposed cylinde with y nd infinite length long xis (Oz). Unde these conditions, the guge condition is ntully checked. Using the new dependent vibles eqution 6 we obtin: w w w w w w w υ + + = + v + v + z t () v Fz + w + ρ ψ ψ + = w (). Poisson eqution fo pessue In ode to detemine the pessue pplying the (.) opeto to Eq. (6) with the condition divv = We get div ( V. ) = P ρ We used the Eq. (9) nd the following expession vz v div( V. ) V = z (3) p WP = nc Wnc + b (6) In which p tems e the ttctive coefficients on W nd nc implies summtion ove the neighboing nodes of "P" fo two dimensionl computtions nd b is the souce tems. Finlly, the finite volume method is employed fo solving the velocity pofile coss the chnnel nd fo study of two dimensionl electomgnetic model in dynmic mode. Pmetes fo the pplied mgnetic fields nd electic cuents plus specific electode length nd its povision hve been djusted fo ssessment on the pefomnce of the DC pump MHD. The computtion fluxogm of esolution lgoithm is pesented in Fig.. The pmetes of the pump The initil vlues Electomgnetic model Test of No Convegen- -ce Yes Electomgnetic thust Coupling E.m Hyd t = t + t The pessue is obtined s: Ρ Ρ Ρ ρ ψ ψ + + =. ρ ψ ψ Ρ =. (4) Hydodynmic model No Test of Convegen- -ce Yes Coupling Hyd E.m 3. NUMERICL METHOD ND RESULTS In finite volume method, ech pincipl node "P" is suounded by fou nodes close tht to Noth "N", the South "S", the Est "E", nd the West "W". By pojection of the diffeentil Eq. (4) on bsis of pojection functionsβ i, nd by integtion of this sme eqution on the volume of contol, coesponding to the node "P", we obtin (Ptnk, 98): P P = e E + w W + n N + s S (5) ' ' n s s v z + d The mtix fom of this system of eqution is witten in the fom: M + vl Α = F (6) [ ]{ } { } Whee [ M + v L] : Coefficients mtix, { Α } Unknown vecto nd { F } Souce vecto. The esolution of the electomgnetic model is elized by n itetive method Wit (979). fte tht the hydodynmic poblem will be solved by the sme method. The gid emins the sme, once the locted nodes, we intoduce souce tem which llows the coupling between the two equtions fo electomgnetic nd flow. The integtion of the Eq. () on the contol volume gives: lgoithm: If ( t > tf ) End Yes No Fig.. Computtion lgoithm Step : Initil, boundy conditions nd the pump dt s e given. Step : Fist, mgnetic potentil nd mgnetic induction e evluted. Step 3: Computtion of the eddy cuents nd the foces in chnnel. Step 4: The foces e injected in the Nvie-stokes equtions s volumetic momentum souces. Step 5: Computtion of the velocity by esolution of the Nvie-stokes equtions. Step 6: Test, if the time is less o equl the finl time go to Step 7. If the time is get thn the finl time go to Step. Step 7: End 5

4 B. Nedjou et. l / JFM, Vol., No., pp. 3-8, 9. N s convegence citeion, ( ) e = V V ), N i i i= N is the totl numbe of nodes used by the FVM in the studied domin, V i is the pst solution of ech node nd V i is the pesent pst solution. Figues 6, 7 nd 8, lso show tht the flow development is quicke. This is becuse diving foces of lge mgnitude e concentted ne the unde nd bottom electode. The nodes numbes used fo computtion e 3. The question of ccucy nd stbility of numeicl methods is extemely impotnt if ou solution is to be elible nd useful. ccucy hs to do with the closeness of the ppoximte solution to exct solutions (ssuming they exist). Stbility is the equiement tht the scheme does not incese the mgnitude of the solution with incese in time, υ. t which implies tht fo stbility, whee υ is the kinemtic viscosity coefficient. Tble pesents petinent tbulted mteil popeties of sewte. In ddition the geometic pmetes nd pplied fields needed fo the numeicl simultions e listed in Tble. Tble Petinent popeties of sewte Sewte Density, ρ (kg.m -3 ) Conductivity, σ 5 (S/m) Viscosity, µ N.s.m - Reltive pemebility z(m) Fig. 3. Mgnetic potentil vecto Owing to the geometic symmety, only hlf domin ws tken ccount. The Diechlet nd Newmnn boundy V V conditions e (V=, = nd = ). z n Tble Pmetes fo numeicl simultion pmete - Chnnel length L - Chnnel dius R - Electode length l - Electicl cuent souce density coils J ex - Electicl cuent density injected by electodes J vlue,4m,3m,8m,. 7 /m,5. 7 /m Figues 3 nd 4 show mgnetic potentil vecto nd its contous in the chnnel nd on the level of ech coil. The bsolute vlue of the mgnetic potentil vecto is less significnt t the inlet thn on the outlet side of the electode nd too wek long the electode, this is explined by the equi-potentil e vey concentted ne the outlet of the electode. Theefoe, s shown in Fig. 5, the xil MHD thust become much gete inceses in the outlet of the electode leding to push the se wte (m) Fig. 4. Mgnetic potentil vecto distibution in the DC pump MHD xil MHD Thust (bs (Fel)) (N/m3) x z(m).5 Fig. 5. MHD thust (m) 6

5 B. Nedjou et. l / JFM, Vol., No., pp. 3-8, 9. It should be noted lso tht fo the vious positions; t the enty of the chnnel (Fig. 6), the unde nd bottom electode (Fig. 7) nd the exit of the chnnel (Fig. 8), tnsient stte then velocity is stbilized. Figue 9 depicts tht the negtive velocity will occu t negtive mgnitude of MHD thust fo (=.3m, z=.5m); this chnge of sign velocity cusing the system to ct s bke x Fig. 9. Velocity t vious times fo (=.3m, z=.5m)..8 w [/s] Fig. 6. Velocity t vious times in the enty of the chnnel.74 z(m) Fig. 7. Velocity t vious times ne the unde nd bottom electode Fig. 8. Velocity t vious times in the outlet of the chnnel (m) Fig.. Distibution of voticity The voticity distibution in the duct is shown in Fig.. It is confimed tht the voticity cn be ceted by MHD foces Figue shows the velocity vious times in diect nd invese mode. The velocity chnged t.5 sec ccoding with the electodes cunt diection Fig.. Velocity vious times in diect nd invese mode 4. CONCLUSION It is suggested tht the bsic effects of the theoy of populsion MHD, which e usully concened s specil popeties of pumping fluids, cn lso be found fo the DC pump MHD

6 B. Nedjou et. l / JFM, Vol., No., pp. 3-8, 9. The configution poposed fo the DC pump MHD lgely simplifies the considetion. In pticully, it becomes woth noting tht the new design cn be employed in othe ppliction like MHD engine. Thee is need fo n electomgnetic pump tht woks without ny moving pts nd poduces lge thust. The esults obtined e in pefect geement with wht ws expected on the bsis of theoeticl considetions nd lso of esults epoted by othe uthos but obtined in diffeent woking conditions Hughes et l. (995); Wng et l. (4) nd Tked, Ysuki (5). REFERENCES Bhdi,.R. nd T. bbsov (5). numeicl investigtion of the liquid flow velocity ove n infinity plte which is tking plce in mgnetic field. Intentionl jounl of pplied electomgnetic nd mechnics, -. Bennecib, N., S. Did, nd R. bdessemed (7). Numeicl simultion of sewte flow poduced by conduction MHD pump. Intentionl eview of electicl engineeing (IREE) (3), Chi-Yun, C. (4). nlysis of Meso-scle Het Exchnges with mgneto hydodynmic pumps. Ph.D. thesis, Ntionl Tsing Hu Univesity. Femigie, M. (999). Hydodynmique physique. Poblèmes ésolus vec ppels de cous, collection sciences sup. physique, edition Dunod. Hughes, M., K.. Peicleous, nd M. Coss (995, Decembe). The numeicl modelling of DC electomgnetic pumps nd bke flow. ppl.mth.modelling 9. Kzeminski, S.K.,. Cl, nd M. Smilek (996, My). Numeicl simultion of D MHD flows ψ ξ Α method. IEEE, Tns-mgnet 3(3). Ptnk, S.V. (98). Numeicl Het Tnsfet nd Fluid. Seies in computtionl methods in mechnics theml sciences, Hemisphee Publishing Copotion. Pei-Jen, W., C. Chi-Yun nd C. Ming-Lng (4). Simultion of two-dimensionl fully developed lmin flow fo mgneto-hydodynmic (MHD) pump. Biosensos nd bioelectonics Elsevie, 5-. Tked, M. nd O. Ysuki (5). Fundmentl studies of helicl-type sewte MHD genetion system. IEEE Tns. pplied supeconductivity 5(). Wit, R. (979). The numeicl solution of lgebic equtions. Wiley-intescience publiction. 8