Investigation of Magnetic Field Effect on Natural Convection Flow using Fast Ψ Ω Method

Transcription

1 International Research Journal of Applied and Basic Sciences 3 Aailable online at ISSN / ol, 4 (): Science Eplorer Publications Inestigation of Magnetic Field Effect on Natural Conection Flow using Fast Ψ Ω Method M.A. aghikhani*, H.Najafkhani. Department of Engineering, Imam Khomeini International Uniersit, Qazin, Iran *Corresponding Author taghikhani@eng.ikiu.ac.ir ABSRAC: he laminar stead magneto-hdrodnamics natural conection in a filled square enclosure with internal heat generation is inestigated b two-dimensional numerical simulation. he enclosure is heated b a uniform olumetric heat densit and walls hae constant temperature. A fied magnetic field is applied to the enclosure. he dimensionless goerning equations are soled numericall for the stream function, orticit and temperature using finite difference method for arious Raleigh (Ra) and Hartmann (Ha) numbers in MAAB software. he stream function equation is soled using fast Poisson's equation soler on a rectangular grid (POICAC function in MAAB), oricit and temperature equations are soled using red-black Gauss-Seidel and biconjugate gradient stabilized (BiCGSAB) methods respectiel. he proposed method is fast and there is no need to the under-relaation factors for ariables compared to commercial methods (e.g. SIMPE) and is applicable for high Raleigh numbers in stead state. he ratio of the orentz force to the buoanc force (Ha /Ra) is introduced as an inde to compare the contribution of natural conection and magnetic field strength on heat transfer. Kewords: Magnetohdrodnamics(MHD); Natural Conection; Square Cait; Stream Function; orticit; POICAC Function INRODUCION he set of equations which describe MHD are a combination of the Naier-Stokes equations of fluid dnamics and Mawell's equations of electromagnetism. hese differential equations hae to be soled simultaneousl, either analticall or numericall. In industrial problems and microelectronic heat transfer deices was used an electricall conducting fluid subjected to magnetic field, thus, the fluid eperiences a orentz force and its effect is to reduce the flow elocities which in turn affects in the heat transfer rate. aminar natural conection flows hae significant applications in man engineering areas including cooling of electronic equipment, nuclear reactor insulation, solar energ collection, and crstal growth in liquids and hae been inestigated b a number of researchers and a rich and ariet of numerical results hae been published due to this phenomenon. An analtical solution to the equations of magnetohdrodnamics is proposed in (Garandet and Alboussiere, 99) that can be used to model the effect of a transerse magnetic field on buoanc drien conection in a two-dimensional cait. he control olume algorithm is used in (Al-Najem et al., 998; Sarris et al., 5; Kandaswam et al., 8; Sheikhzadeh et al., ) to sole the two dimensional transient MHD equations with alternating direct implicit procedure (ADI). Finite difference method and Finite element method hae been deeloped in (Borghi et al., 996; Borghi et al., 4; erardi and Cardoso 998; erardi et al., ; erardi et al., ; Shadid et al., ) for the solution of two-dimensional stead state electrodnamic problem in magnetohdrodnamic flows. A mathematical model describing the dnamics of magnetic field influence on a conducting liquid in a square cait is presented in (Krzeminski et al., ) such that biharmonic mathematical model was used with stream function and the magnetic potential. A finite element method for the solution of 3D incompressible magnetohdrodnamic is presented in (Salah et al., ). he buoanc-drien magnetohdrodnamic flow in a liquid-metal filled cubic enclosure with internal heat generation is inestigated b three-dimensional numerical simulation in (Piazza and Ciofalo, ). An analsis has been performed (Mahmud et al., 3) to stud the first and second laws (of thermodnamics) characteristics of flow and heat transfer inside a ertical channel made of two parallel plates under the action of transerse magnetic field. A two-dimensional mathematical model has been deeloped to stud the interaction between graitational bod force and self-induced electromagnetic bod force in a Jouleheated liquid pool in a rectangular cait, with an aspect ratio of in (Sugilal et al., 5). Stead, laminar, natural-conection flow in the presence of a magnetic field in an inclined rectangular

2 Intl. Res. J. Appl. Basic. Sci. ol., 4 (), , 3 enclosure heated from one side and cooled from the adjacent side was considered in (Ece and Büük, 6; Ece and Büük 7) so that the goerning equations were soled numericall for the stream function, orticit and temperature using the differential quadrature method. A finite olume code based on PAANKAR s SIMPER method is utilized in (Pirmohammadi et al., 9; Pirmohammadi and Ghassemi 9; Pirmohammadi et al., ; Pirmohammadi et al., ) with constant Prandtl (Pr) number. he present stud inestigates the laminar stead conection in an enclosure in the presence of a magnetic field. he enclosure is filled with an electricall conducting fluid whose Prandtl number is.733. herefore, a two-dimensional numerical model is deeloped to sole the orticit, stream function and temperature goerning equations of buoanc-drien natural conection flow inside a cait. he stud pursues numerical solution to stud the effect of magnetic field strength and Ra number on the natural conection and heat transfer. Mathematical formulation Stead, laminar, natural-conection flow in the presence of a magnetic field in a square enclosure was considered. Dimensional coordinates with the -ais measuring along the bottom wall and -ais being normal to it along the left wall are used. he geometr and the coordinate sstem are schematicall shown in Fig.. Magnetic flu densit B is applied with respect to the coordinate sstem. he walls are kept at a constant temperature. Wall, Wall, g Wall, B Wall, Figure. Geometr and the coordinate sstem Continuit, change of linear momentum and energ equations are written as follows. g p µ q (. ) α ( ) J B () C p where, is the elocit ector, p is the pressure, is the temperature, g is the graitational acceleration, is the densit, µ is the iscosit, C p is the specific heat, α is the thermal diffusiit of the fluid and q is olumetric heat densit, respectiel, J is the current densit and B is magnetic field. he magnetic Renolds number was assumed to be small and the induced magnetic field due to the motion of the electricall conducting fluid was neglected. he current densit can be written as J σ ( φ B) (4) Where σ is the electrical conductiit of the fluid and φ is the electric potential. he conseration of the electric charge is J (5) From (4) and (5) is deried: φ (6) Since there is alwas somewhere around the cait an electricall insulating boundar, the unique solution is φ which means that the electric field anishes eerwhere. he goerning equations in scalar form under Boussinesq approimation are written as () (3) 94

3 Intl. Res. J. Appl. Basic. Sci. ol., 4 (), , 3 94 (7) p µ (8) B g p ) ( σ β µ (9) p C q α () Here β is the coefficient of thermal epansion of the fluid and is the densit of the fluid at temperature. Stream function and orticit are defined as follows ψ () ψ () (3) herefore the goerning equations reduce to ψ ψ (4) B g ) ( ) ( σ β µ (5) Dimensionless ariables used in the analsis are according to,, Y, α, α, α ψ Ψ, α Ω, q k. (6) where k is the thermal conductiit. Dimensionless numbers, the Prandtl, Grashof, Raleigh and Hartmann numbers are defined as follows, α µ Pr, 5 µ β k q g Gr, Gr Ra. Pr, µ σ B Ha (7) According to the equations (6) and (7), the goerning equations in this stud are gien in dimensionless form as Ω Ψ Ψ Y (8) Ha Ra Y Y Ω Ω Ω Ω Pr Pr ) Pr( ) ( (9) ) ( Y Y () Which hae to be soled subject to the boundar conditions Ψ and at all walls. he orticit alues at the wall is calculated using Jensen s formula (Erturk, 9) h h Ψ Ψ Ω () Where subscript refers to the points on the wall, refers to the points adjacent to the wall, refers to the second line of points adjacent to the wall, refers to the elocit of the wall with its alue being equal to on the moing wall, and on the stationar walls while h is the grid spacing. he elocities at a point (,) in the cait ma be approimated as follow(gupta and Kalita, 5):

4 Intl. Res. J. Appl. Basic. Sci. ol., 4 (), , 3 3 (, ) [ Ψ(, h) Ψ(, h)] [ (, h) (, h)] 4h 4 3 (, ) [ Ψ( h, ) Ψ( h, )] [ ( h, ) ( h, )] () 4h 4 Solution method he dimensionless goerning equations associated with the boundar conditions are soled for stream function, orticit and temperature numericall using the second order finite difference method. he hbridscheme, which is a combination of the central difference scheme and the upwind scheme, is used to discretize the conection terms. he sequence of algorithm is proided here:. Guess the elocit and the stream function fields.. Sole discretized temperature equation using Jacobi BiCGSAB method. 3. Calculate elocit field using stream function field (equation ). 4. Calculate orticit boundar condition using elocit and stream function fields (equation ). 5. Sole the discretized orticit equation using red-black Gauss-Seidel method. 6. Sole the discretized stream function equation using fast Poisson's equation soler on a rectangular grid (POICAC function) in MAAB. 7. Check error in temperature, orticit and stream function fields. If errors are below the specified tolerance then eit the loop otherwise return to step. Repeat the whole procedure till conerged solution is obtained. he tolerance of the conergence criterion used for all ariables is -6 : k k 6 (3) k k Ψ Ψ k Ψ k Ω Ω k Ω k k 6 6 (4) (5) Grid refinement check In order to determine the proper grid size for this stud, a grid independence test are conducted with Pr and Ra 5. he following si grid sizes are considered for the grid independence stud. hese grid densities are 3 3, 4 4, 64 64, 8 8, and 8 8. he maimum temperature ma and maimum stream function Ψ ma of the fluid in the cait are used as a sensitiit measure of the accurac of the solution and are selected as the monitoring ariables for the grid independence stud. able shows the dependence of the quantities ma and Ψ ma on the grid size. Considering the accurac of the numerical alues, the following calculations are performed with grid. able. Grid sensitiit check at Pr and Ra 5 Grid size Ha Ha ma Ψ ma ma Ψ ma Code alidation In order to erif the accurac of the numerical code, comparisons with the preiousl published results are necessar. he present numerical code is erified against a documented numerical stud. he code is benchmarked with a differentiall heated cait problem, with the right wall maintained at cooled condition. he left wall is hot whereas the two horizontal walls are under adiabatic condition. he goerning equations are soled on a uniform grid of 64 64, and for a Prandtl number, Pr.733. he solutions are obtained for two alues of Raleigh number 4, 5 and Hartmann number Ha. he Pr and Ra are chosen such that the direct comparison is possible with the benchmark solution (Pirmohammadi et al., 9) which is based on finite olume scheme. he isotherms and streamlines are compared with results of (Pirmohammadi et al., 9) in 94

5 Intl. Res. J. Appl. Basic. Sci. ol., 4 (), , 3 figures and 3. It is obsered that the present results agree well with preious numerical work. Present work Ra 4 Ra 5 Figure. Comparison of isotherms at arious Ra and Ha Present work Ra 4 Ra 5 Figure 3. Comparison of streamlines at arious Ra and Ha RESUS AND DISCUSSIONS Parametric inestigations are performed for a square cait in the following range of parameter alues: Raleigh number: 4 Ra 7 ; Prandtl number: Pr.733; Hartmann number: Ha 5. he influence of the Ha number on the streamlines and isotherms inside the cait at Ra 6 are shown in figures 4-5. able shows ariations of dimensionless maimum temperature ( ma ) and dimensionless maimum stream function (Ψ ma ) with Ha and Ra numbers. he maimum alue of stream function can be iewed as a measure of the intensit of natural conection in the cait. It is eident from the table that in absence of magnetic field, b increasing the Ra, the maimum alue of the stream function increases; this means that the flow moe faster as natural conection is stronger and the isotherm will be distorted. For the square cait, the maimum dimensionless temperature ( ma ) reduces with increasing Ra. his is because as Ra increases, heat transfer due to conection increases. It is also obsered that at low Ra number, the ma is at the centre of the cait and it shifts upwards with increase in Ra. For Ra 5 5 due to strong conectie rolls, two local temperature maima are obsered in the cait. hese maima shift upwards and towards side walls with the increase in Ra. he fluid circulates in the square cait as two smmetrical counter-rotating rolls, moing upwards at the center and downwards near the cold side walls. B appling a magnetic field we can suppress the natural conection so that the maimum alue of the stream function reduces as Ha number increases and ma is at the center of the cait, indicating that most of the heat transfer is b heat conduction. For high Raleigh number and for a weak magnetic field strength, conection is dominant heat transfer mechanism. From the streamlines pattern we see that as the Ha number increases, the streamlines will be parallel with the two side walls of the cait and the streamlines are elongated. On the other hand it is clear from table that for all Ra numbers b increasing the Ha number hae a pure conduction regime. Because orentz force interacts with the buoanc force and suppresses the 943

6 Intl. Res. J. Appl. Basic. Sci. ol., 4 (), , 3 conection flow b reducing the elocities. Furthermore, it is shown that as the Ra number is increased, the conectie heat transfer is increased, so that for suppression of conection is needed er high magnetic field. able.ariation of ma and Ψ ma with Ha and Ra numbers Ra Ha ma Ψ ma Figure 4. Isotherms of natural conection in a square cait for Ra 6 and Ha Figure 5. Streamlines of natural conection in a square cait for Ra 6 and Ha 944

7 Intl. Res. J. Appl. Basic. Sci. ol., 4 (), , Figure 6.Isotherms of natural conection in a square cait for Ra 6 and Ha Figure 7. Streamlines of natural conection in a square cait for Ra 6 and Ha Figure 8.Isotherms of natural conection in a square cait for Ra 6 and Ha Figure 9. Streamlines of natural conection in a square cait for Ra 6 and Ha 945

8 Intl. Res. J. Appl. Basic. Sci. ol., 4 (), , Figure.Isotherms of natural conection in a square cait for Ra 6 and Ha Figure. Streamlines of natural conection in a square cait for Ra 6 and Ha Figure. Isotherms of natural conection in a square cait for Ra 6 and Ha Figure 3. Streamlines of natural conection in a square cait for Ra 6 and Ha3 946

9 Intl. Res. J. Appl. Basic. Sci. ol., 4 (), , Figure 4.Isotherms of natural conection in a square cait for Ra 6 and Ha Figure 5. Streamlines of natural conection in a square cait for Ra 6 and Ha5 As was mentioned, natural conection of electricall conductie fluid in the enclosure is affected b buoanc and orentz forces. he buoanc force has an aiding effect on natural conection, but the orentz force has an opposing effect. Either of the two forces are important, when Ha /Ra. he buoanc force is dominant when Ha /Ra<< and the orentz force is dominant when Ha /Ra>>. ariations of the maimum temperature in terms of Ha /Ra for arious Ra are shown in Fig. 6. he figure 6 shows that at Ha /Ra>., the alue of maimum temperature is constant (conduction regime). While at low Ha /Ra (Ha /Ra<.5), the electromagnetic bod force can be ignored..75 Ha /Ra Inde.7 Maimum emperature Ra 4.45 Ra 5 Ra 6 Ra Ha /Ra Figure 6. he maimum temperature ersus Ha /Ra for arious Ra 947

10 Intl. Res. J. Appl. Basic. Sci. ol., 4 (), , 3 CONCUSION In this paper was inestigated the laminar stead conection in a cait in the presence of a magnetic field. he cait is filled with an electricall conducting fluid whose Prandtl number is.733. A twodimensional numerical model was deeloped to sole the orticit, stream function and temperature goerning equations of buoanc-drien natural conection flow inside the cait. he stream function equation was soled using fast Poisson's equation soler on a rectangular grid (POICAC function in MAAB software), orticit and temperature equations were soled using red-black Gauss-Seidel and bi-conjugate gradient stabilized (BiCGSAB) methods respectiel. It was obsered that the effect of the magnetic field is to reduce the conectie heat transfer inside the cait. Furthermore, it is shown that as the Ra number is increased, the conectie heat transfer is increased, so that for suppression of conection is needed er high magnetic field. he ratio of the orentz force to the buoanc force (Ha /Ra) was introduced as an inde to compare the contribution of natural conection and magnetic field intensit on heat transfer: a) hermall drien natural conection eists when Ha /Ra<.5. b) Electromagneticall drien flows occur when Ha /Ra>.. c) Natural conection is goerned b both electromagnetic bod force and graitational bod force when.5< Ha /Ra<.. Nomenclature B magnetic flu densit ector: Wb m C p specific heat: J.kg -.K - g graitational acceleration ector: m s h grid spacing: m J electric current densit ector: A m k thermal conductiit: W m K dimension of cait: m p pressure: N m q olumetric heat source densit: W.m -3 temperature: K elocit ector: m s,, z Cartesian coordinates: m Greek smbols thermal diffusiit: m s coefficient of olumetric epansion: K electric potential: dnamic iscosit: kg. m -.s densit: kg.m 3 electrical conductiit: mho.m orticit: s - stream function: m.s - Subscript reference alue ma maimum alue,, z component of a ector quantit Dimensionless quantities elocit ector Cartesian coordinate in direction Y Cartesian coordinate in direction orticit stream function temperature Dimensionless numbers Gr Grashof number Ha Hartmann number Pr Prandtl number Ra Raleigh number 948

11 Intl. Res. J. Appl. Basic. Sci. ol., 4 (), , 3 REFERENCES Al-Najem NM, Khanafer KM, El-Refaee MM Numerical Stud of aminar Natural Conection in ilted Enclosure with ranserse Magnetic Field ; Int J Num Method Heat Fluid Flow, 8(6): Borghi CA, Carraro MR, Cristofolini A. 4." Numerical Solution of the Nonlinear Electrodnamics in MHD Regimes with Magnetic Renolds Number near One", IEEE rans. Magn., 4(): Borghi CA, Cristofolini A, Minak G. 996." Numerical Methods for the Solution of the Electrodnamics in Magnetohdrodnamic Flows", IEEE rans. Magn., 3(3): -3. Ece MC, Büük E. 6. "Natural-Conection Flow under a Magnetic Field in an Inclined Rectangular Enclosure Heated and Cooled on Adjacent Wall", Fluid Dnam. Res., 38: Ece MC, Büük E. 7. Natural Conection Flow under a Magnetic Field in an Inclined Square Enclosure Differential Heated on Adjacent Walls, Meccanica, 4: Erturk E. 9." Discussions on Drien Cait Flow", Int. J. Num. Meth. Fluids, 6: Garandet JP, Alboussiere. 99." Buoanc Drien Conection in a Rectangular Enclosure with a ranserse Magnetic Field", Int J Heat Mass ransf, 35(4): Gupta MM, Kalita JC. 5." A new paradigm for soling Naier Stokes equations: stream function elocit formulation", J. Comput. Phs., 7: Kandaswam P, Sundari SM, Nithadei N. 8. Magnetoconection in an Enclosure with Partiall Actie ertical Walls, Int J Heat Mass ransf, 5: Krzeminski SK, Smiałek M, Wlodarczk M.." Finite Element Approimation of Biharmonic Mathematical Model for MHD Flow using Ψ- A Approach", IEEE rans. Magn., 36(4): Mahmud S, asnim SH, Mamun MAH. 3." hermodnamic Analsis of Mied Conection in a Channel with ranserse Hdromagnetic Effect", Int. J. hermal Sci, 4: Piazza ID, Ciofalo M.." MHD Free Conection in a iquid-metal Filled Cubic Enclosure. II. Internal Heating", Int J Heat Mass ransf, 45: Pirmohammadi M, Ghassemi M, Hamedi M.." Effect of Inclination Angle on Magneto-Conection Inside a ilted Enclosure", IEEE rans. Magn, 46(9): Pirmohammadi M, Ghassemi M, Keshtkar A.. "Numerical Stud of Hdromagnetic Conection of an Electricall Conductie Fluid with ariable Properties Inside an Enclosure", IEEE rans. Plasma Sci, 39(): Pirmohammadi M, Ghassemi M, Sheikhzadeh GA. 9. "Effect of a Magnetic Field on Buoanc-Drien Conection in Differentiall Heated Square Cait", IEEE rans. Magn, 45(): Pirmohammadi M, Ghassemi M. 9." Numerical Stud of Magneto-Conection in a Partitioned Enclosure", IEEE rans. Magn, 45(6): Salah NB, Soulaimani A, Habashi WG.." A Finite Element Method for Magnetohdrodnamics", Comput. Meth. Appl. M., 9: Sarris IE, Kakarantzas SC, Grecos AP, lachos NS. 5. MHD Natural Conection in a aterall and olumetricall Heated Square Cait, Int J Heat Mass ransf, 48: Shadid JN, Pawlowski RP, Banks JW, Chacon, in P, uminaro RS.. owards a Scalable Full-Implicit Full-Coupled Resistie MHD Formulation with Stabilized FE Methods, J. Comput. Phs., 9: Sheikhzadeh GA, Fattahi A, Mehrabian MA.. Numerical Stud of Stead Magneto-Conection around an Adiabatic Bod inside a Square Enclosure in ow Prandtl Numbers, Heat Mass rans, 47: Sugilal G, Wattal PK, Ier K. 5." Conectie Behaiour of a Uniforml Joule-Heated iquid Pool in a Rectangular Cait", Int. J. hermal Sci, 44: erardi S, Cardoso JR, Costa MC.." hree-dimensional Finite Element Analsis of MHD Duct Flow b the Penalt Function Formulation", IEEE rans. Magn., 37(5): erardi S, Cardoso JR "A Solution of wo-dimensional Magnetohdrodnamic Flow using the Finite Element Method", IEEE rans. Magn., 34(5): erardi S, Machado JM, Cardoso JR.." he Element-Free Galerkin Method Applied to the Stud of Full Deeloped Magnetohdrodnamic Duct Flows", IEEE rans. Magn., 38():