status solidi Computer simulation of heat transfer through a ferrofluid

Transcription

1 physica pss status solidi basic solid state physics b Computer simulation of heat transfer through a ferrofluid omasz Strek and Hubert Jopek Institute of Applied Mechanics, Poznan University of echnology, ul. Piotrowo 3, Poznan, Poland Received 7 November 5, accepted 31 August 6 Published online 18 January 7 PACS a, Cb he viscous, two-dimensional, incompressible and laminar time dependent heat transfer flow through a ferromagnetic fluid is considered in this paper. he flow takes place in channel between two parallel flat plates under the influence of the magnetic dipole located below the channel. It is assumed that there are no electric field effects and the variation in the magnetic field vector, which could occur due to temperature gradients within the ferrofluid, is negligible. his magneto-thermo-mechanical problem is governed by dimensionless equations. Results are obtained using computational fluid dynamics code COMSOL with modifications to account for the magnetic term (the Kelvin body force or the thermal power) when needed. phys. stat. sol. (b) 44, No. 3, (7) / DOI 1.1/pssb.577 REPRIN

2 Original Paper phys. stat. sol. (b) 44, No. 3, (7) / DOI 1.1/pssb.577 Computer simulation of heat transfer through a ferrofluid omasz Strek * and Hubert Jopek ** Institute of Applied Mechanics, Poznan University of echnology, ul. Piotrowo 3, Poznan, Poland Received 7 November 5, accepted 31 August 6 Published online 18 January 7 PACS a, Cb he viscous, two-dimensional, incompressible and laminar time dependent heat transfer flow through a ferromagnetic fluid is considered in this paper. he flow takes place in channel between two parallel flat plates under the influence of the magnetic dipole located below the channel. It is assumed that there are no electric field effects and the variation in the magnetic field vector, which could occur due to temperature gradients within the ferrofluid, is negligible. his magneto-thermo-mechanical problem is governed by dimensionless equations. Results are obtained using computational fluid dynamics code COMSOL with modifications to account for the magnetic term (the Kelvin body force or the thermal power) when needed. 7 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction In this paper time dependent heat transfer through a ferrofluid in channel flow under the influence of magnetic dipole is considered and simulated. Results are obtained using standard computational fluid dynamics code COMSOL with modifications to account for the magnetic term (the Kelvin body force and the thermal power) when needed. COMSOL is a powerful interactive environment for equationbased multiphysics modelling and solving all kinds of scientific and engineering problems based on partial differential equations (PDEs) using the finite element method. During the last decades, an extensive research work has been done on the fluids dynamics in the presence of magnetic field. he effect of magnetic field on fluids is worth investigating due to its innumerable applications in wide spectrum of fields. he study of interaction of the magnetic field or the electromagnetic field with fluids have been documented e.g. among nuclear fusion, chemical engineering, medicine, high speed noiseless printing and transformer cooling [1]. One of the most exciting areas of technology to emerge in recent years is MEMS (micromechanical systems), where engineers design and build systems with physical dimensions in micrometers, e.g. MEMS-based biosensors or microscale heat exchangers. he transport of momentum and energy in miniaturized devices is diffusion limited because of their very low Reynolds numbers. Using ferrofluids in these applications and manipulating the flow of ferrofluids in these applications by external magnetic field can be a viable alternative to enhance convection in these devices. Another application is the auxetic foam coated with magnetorheological fluid (MR) or ferrofluid. An auxetic solid expands in all directions when pulled in only one, thus behaving in an opposite manner compared to classical solids []. he paper [3] deals with a comparative study on the mechanical, acoustic, and electromagnetic properties of a novel class of auxetic (negative Poisson s ratio) rigid polyurethane (PU) foam with a magnetorheological (MR) fluid coating. With regard to the foams electro- * Corresponding author: tomasz.strek@put.poznan.pl, Phone: ** hubert.jopek@put.poznan.pl 7 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

3 18. Strek and H. Jopek: Computer simulation of heat transfer through a ferrofluid magnetic properties, the auxetic structure and MR coating cause an increase in the refractive index and loss factor compared to the conventional foams. hese electromagnetic effects are due to the presence of a high relative density and metal particulates in the foam material. Ferrofluids are non-conducting fluids and the study of the effect of magnetisation has yielded interesting information. In equilibrium situation the magnetization property is generally determined by the fluid temperature, density and magnetic field intensity and various equations, describing the dependence of static magnetization on these quantities. he simplest relation is the linear equation of state, given by [4]. It can be assumed that the magneto-thermo-mechanical coupling is not only described by a function of temperature [4], but by an expression involving also the magnetic field strength [5]. his assumption permits us not to consider the ferrofluid far away from the sheet at Curie temperature in order to have no further magnetization. his feature is essential for physical applications because the Curie temperature is very high (e.g. 143 Kelvin degrees for iron) and such a temperature would be meaningless for applications concerning most of ferrofluids. So instead of having zero magnetization far away from the sheet, due to the increase of fluid temperature up to the Curie temperature this formulation allows us to consider whatever temperature is desired and the magnetisation will be zero due to the absence of the magnetic field sufficiently far away from the sheet. Moreover, ferrofluids are mostly organic solvent carriers having ferromagnetic oxides, acting as solute. Ferrofluids consist of colloidal suspensions of single domain magnetic nanoparticles [6]. hey have promising potential for heat transfer applications, since a ferrofluid flow can be controlled by using an external magnetic field [7]. However, the relationship between an imposed magnetic field, the resulting ferrofluid flow and the temperature distribution is not understood well enough. he references regarding heat transfer with magnetic fluids is relatively sparse (see e.g. [7 11]). An overview of prior research on heat transfer in ferrofluid flows e.g. thermomagnetic free convection, thermomagnetic forced convection and boiling, condensation and multiphase flow are presented in paper [7]. he natural convection of a magnetic fluid in a partitioned rectangular cavity was considered in paper [9]. It was found that the convection state may be largely affected by improving heat transfer characteristic at higher Rayleigh number when a strong magnetic field was imposed. he influence of a uniform outer magnetic field on natural convection in square cavity was presented in paper [1]. It was discovered that the angle between the direction of temperature gradient and the magnetic field influences the convection structure and the intensity of heat flux. Numerical results of combined natural and magnetic convective heat transfer through a ferrofluid in a cube enclosure were presented in paper [9]. he purpose of this work was to validate the theory of magnetoconvection. he magnetoconvection was induced by the presence of magnetic field gradient. he Curie law states that magnetization is inversely proportional to temperature. he effect of magnetic field on the viscosity of ferroconvection in an anisotropic porous medium was studied in paper [13]. It was found that the presence of anisotropic porous medium destabilizes the system, where as the effect of magnetic field dependent viscosity stabilizes the system. In this paper the investigated fluid was assumed to be incompressible having variable viscosity. Experimentally it has been demonstrated in prior research that the magneto viscosity has got exponential variation, with respect to magnetic field. As a first approximation for small field variation, linear variation of magneto viscosity has been used in paper [13]. Governing equations he governing equations of the fluid flow under the action of the applied magnetic field and gravity field are: Maxwell s equations, the mass conservation equation, the energy equation for temperature and the fluid momentum equation in the frame of Boussinesque approximation. he Maxwell s equations, simplified for a non-conducting fluid with no displacement current, become, B =, (1) H =, () 7 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

4 Original Paper phys. stat. sol. (b) 44, No. 3 (7) 19 where B is the magnetic induction, M is the magnetization vector and H is the magnetic field vector. Further the magnetic induction, the magnetization vector and the magnetic field vector are related by the constitutive relation: B = µ ( M + H ), (3) where µ is a magnetic permeability in vacuum. In this paper the magnetic dipole which gives rise to a magnetic field, sufficiently strong to saturate the fluid, is considered. In the magnetostatic case where there are no currents present it is possible to define a magnetic scalar potential by the relation H =- Vm. he scalar potential for the magnetic dipole is given by [4] γ x1 - a Vm( x ) = Vm( x1, x) =, (4) π ( x - a) + ( x -b) 1 so the magnitude of the magnetic field can be written in the form γ 1 H( x1, x) = H ( x1, x) =, (5) π ( x - a) + ( x -b) 1 where γ is the magnetic field strength at the source (of the wire) and ( ab, ) is the position where the source of the magnetic dipole is located (below the channel). he mass conservation equation for an incompressible fluid is: v =. (6) he momentum equation for magnetoconvective flow is modified typical natural convection equation by addition of a magnetic term ρ Ê v + v v = - p + S + αρg ( - ) k + ( M ) B, (7) Ë t where ρ is the density, v is the velocity vector, p is the pressure, is the temperature of the fluid, S is the extra stress tensor, k is the unit vector of the gravity force and α is the thermal expansion coefficient of the fluid. Extra stress tensor is commonly known as a tensor which is related to deformation rate by the constitutive equation. In general S = ηg, where G = v + ( v ) is the deformation rate tensor (the rate of strain tensor). For the Newtonian fluid viscosity is constant. he energy equation for an incompressible fluid which obeys the modified Fourier s law is: ρc Ê M + v = k + ηφ - µ (( v ) H ), (8) Ë t where η is the viscosity and ηφ is the viscous dissipation v1 v v v1 Φ Ê Ê Ê Ê Ê = Á Á Ë x Á Ë y Á Ë x y Ë Ë (9) he last term in the energy Eq. (8) represents the thermal power per unit volume due to the magnetocaloric effects. 7 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

5 13. Strek and H. Jopek: Computer simulation of heat transfer through a ferrofluid 3 Kelvin body force for the magnetoconvective flow he last term in the momentum Eq. (7) represents the Kelvin body force per unit volume f = ( M ) B, (1) which is the force that a magnetic fluid experiences in a spatially non-uniform magnetic field. In a narrow temperature range, the magnetization vector M of the magnetic fluid can be expressed as a linearized function of H and the fluid temperature. he relationship between the magnetization vector and magnetic field vector can be written in the form [3] where M = H, (11) χ χ m χ = χ ( ) =, (1) 1 + α ( - ) m m is the total magnetic susceptibility, χ is the differential magnetic susceptibility of the ferrofluid, α is the thermal expansion coefficient of the fluid and is the reference temperature. Using the constitutive relation Eq. (3) the magnetic induction vector can be written in the form B = µ ( 1 + χ m ) H. (13) he variation of the total magnetic susceptibility χ m is treated solely as being dependent on the temperature so the Kelvin body force defined by Eq. (1) can be written as Because f = µ ( χmh )(( 1 + χm) H ). (14) χ = χ χ ( ( x) ) =, 1 + α ( ( x) - ) m m (15) the Kelvin body force (14) can be represented by f = µχ ( 1+ χ )( H ) H + µχ H( H )( 1 + χ ). (16) m m m m Using Eq. () and ( H ) H = ( H H) -( H ) H the following equation can be written ( H ) H = (1/ ) ( H H ) = (1/ ) H. Finally the Kelvin body force can be represented by 1 f = µχ m ( 1 + χ m) ( H H) + µχ mh( ( H ) χm). (17) Consequently Eqs. (7) and (8) can be written in the form, respectively ρ Ê v + v v = - p + S + αρg( -) k Ë t 1 + µχ ( + χ ) ( H H) + µχ H( ( H ) χ ), (18) m 1 m m m and ( χm ) ρc Ê H + v = k + ηφ - µ (( v ) H ). (19) Ë t 7 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

6 Original Paper phys. stat. sol. (b) 44, No. 3 (7) Dimensionless equations For simplicity the preferred work choice is to work in non-dimensional frame of reference. Some dimensionless variables will be introduced in order to make the system much easier to study. his procedure is very important so that one can see which combination of parameters is more important than others. Non-dimensionalised variables and scales are defined as follows: time t = ( κ / h ) t; distance variable x = (1/ h) x; velocity v = v/ vr = ( h/ κ ) v; pressure p = p/ ρvr = ph / ρκ ; magnetic field strength H = H / H r ; magnitude of the magnetic field H = H/ Hr; temperature = /δ and viscosity η = η/ η, where κ = k/ ρc is the thermal diffusivity of the fluid, the expression κ /h is a typical thermal diffusion time over the distance h, η is the zero-shear-rate viscosity and vr = κ / h is a characteristic velocity. he next step is to put these variables into the Navier Stokes equation ρκ È v ρκ H p 3 3 ( αρg δ Ê ) ηκ µ r, 3 h Í + v v = t k S f () Î h Ë δ h h 3 and then multiply through by h / ρκ which gives v g δh H h + v v = - p + α Ê - k + η S + µ f, (1) t κ δ ρ κ ρ κ or after mathematical manipulations 3 r Ë v g δh H h + v v = - p + α ρ η Ê - k + η S + µ f. () t κη ρ κ δ ρ κ ρ κ 3 r Ë he dimensionless Kelvin body force is represented by 1 f = χm ( 1 + χm ) ( H H ) + χm H (( H ) χm ), (3) where χ χm = χm ( ( x )) =. 1+ ( α δ Ê ) ( x )- Ë δ (4) he dimensionless magnitude of the magnetic field defined by Eq. (5) is given by H( x1, x) H ( x1, x ) =, (5) H r where γ Hr = H( a, ) =, (6) πb is characteristic magnetic field strength. Hence Eq. (5) can be written in the form b hb H ( x1, x ) = h =. Ê a b ( hx1 - a) + ( hx1 -b) x1 - + Ê x - Ë h Ë h (7) 7 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

7 13. Strek and H. Jopek: Computer simulation of heat transfer through a ferrofluid he dimensionless variables are also applied to the thermal diffusion Eq. (19) ρ c δ κ Ê k δ H ( ) η κ µ κ r χ H + v = + ηφ + m 4 (( v ) H ) (8) h Ë t h h h and then multiply through by h / ρ cδκ which gives k δ h ηκ µ H r ( χ m H ) + v = + ηφ + (( v ) H ) (9) t h ρ cδκ h ρ cδκ ρ cδ or after mathematical manipulations k δ h + v = t h ρcδκ Hrh ηφ ρ δ κ ρκ ηκ µ κ ( m ) χ H + + ( ( v ) H ). (3) h c h cδ Moreover some of the dimensionless ratios can be replaced with well-known parameters: the Prandtl number Pr, the Rayleigh number Ra, the Eckert number Ec, the Reynolds number Re and the magnetic number Mn: Pr η ρκ =, Ra αρ gh δ ηκ 3 =, v r κ Ec = =, cδ cδh Re hρ v r η ρ κ η = =, µ H µ H h Mn = ρ =. (31) r r vr ρκ he dimensionless form of Navier Stokes () and thermal diffusion (3) equations are as follows: v + = - + Ê t Ë δ v v p Ra Pr k Pr S Mn f (3) and ( χm H ) + v = + Pr EcηΦ + Mn Ec (( v ) H ). (33) t Since now primes will not be written (old variables symbols will be used) but it is important to remember that they are still there and v Ê + v v = - p + Ra Pr - k + Pr S + Mn f, (34) t Ë δ where ( χmh ) + v = + Pr EcηΦ + Mn Ec (( v ) H ), t (35) 1 f = χm( 1 + χm) H + χmh( ( H ) χm), (36) 7 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

8 Original Paper phys. stat. sol. (b) 44, No. 3 (7) 133 and χ χ = χ ( ( x) ) =. 1+ ( αδ Ê ) ( x )- Ë δ m m 5 Numerical solution (37) he viscous, two-dimensional, incompressible and laminar time dependent heat transfer flow through a ferrofluid is considered in this paper. Flow takes place in channel between two parallel flat plates. he length of the channel is L and distance between plates is h. Below the channel the magnetic dipole is located at point ( ab., ) he fluid is assumed to be electrically nonconducting so the ferrofluid flow does not induce any electromagnetic current in it. Another assumption is that there are no electric field effects. able 1 Numerical values of variables and parameters of the flow. name variable unit dimensional non-dimensional density ρ 3 ÈÎ kg/m 118. kg/(ms) viscosity η [ ] thermal conductivity k [ J/(msK) ] 1. 1 heat capacity c [ J/(kgK) ] 14. thermal diffusivity k (diffusion coefficient) κ = ρc thermal expansion coefficient characteristic velocity r velocity magnetic permeability µ = 4π 1 of a vacuum ÈÎ m/s α [ 1/K ] υ Èm/s u -7 Î ÈÎ m/s N/A ÈÎ difference of temperatures δ [ K] 3 temperature [ K ] 3 1 temperature of upper wall u = + δ [ K] temperature of lower wall l = [ K ] 3 1 high h [ m ]. 1 length L [ m]. 1 centre of magnetic wire ( ab, ) [ ] magnetic field strength at the source characteristic magnetic field strength m (.5,.5) (.5,.5) γ [ Am ] 6 H [ ] r A/m Prandtl number Pr 1.4 Rayleigh number Ra Eckert number Ec Reynolds number Re.714 magnetic number Mn magnetic susceptibility χ WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

9 134. Strek and H. Jopek: Computer simulation of heat transfer through a ferrofluid able Dimensionless numbers for the four different values of magnetic field in ferrofluid. variable fluid A fluid B fluid C fluid D Hr Pr Ra Ec Re he variation in the magnetic field that could occur due to temperature gradients within the ferrofluid is negligible. Anisotropy in fluid properties is negligible too. his magneto-thermo-mechanical problem is governed by dimensionless Eqs. (34) (37). he following boundary conditions for dimensionless variables are assumed: For the upper wall ( x L, y = 1) : the upper wall temperature is kept at constant temperature u. he velocity is (no slip condition). For the lower wall ( x L, y = ): the lower wall temperature is kept at constant temperature l. he velocity is (no slip condition). For inlet (the left wall) ( x =, y 1): the temperature is varying linearly from l to u and is given by equation in = ((u - l ) / δ ) y + (l / δ ) where δ = u - l. here is a parabolic laminar flow profile given by equation uin = -4(u /ur ) y ( y - 1) for y Œ, 1Ò at the inlet end. For outlet (the right wall) ( x = L, y 1) : the convective flux is assumed for temperature, n ( - k ) =. Pressure outlet is also assumed, ( - pi + S ) n = - p n, where p is the dimensionless atmospheric pressure. Fig. 1 (online colour at: ) ime evolution of the dimensionless velocity field (surface) and the dimensionless temperature (contour) of the fluid B for (a) t =.1, (b) t =.1, (c) t =. and (d) t =.3. 7 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

10 Original Paper phys. stat. sol. (b) 44, No. 3 (7) 135 Fig. (online colour at: ) Comparison of the dimensionless velocity field (surface) and the dimensionless temperature (contour) for the different values of the magnetic number: (a) fluid A, (b) fluid B, (c) fluid C, (d) fluid D. he following initial conditions for dimensionless variables are assumed: the fluid is motionless, the pressure is zero and the temperature is varying linearly from lower to upper wall. he time-dependent flow is considered for dimensionless time t Œ,.5Ò. he problem governed by Eqs. (34) (37) is solved with finite element method COMSOL code using direct UMFPACK linear system solver. Relative and absolute tolerance used in calculations are.5 and.5, respectively. he numerical values of the variables and the dimensionless numbers of the flow fluid are presented in able 1. ime-dependent flow is considered in the ferrofluid whereas, four different values of magnetic field are imposed (so the fluid is called as follows: Fluid A, Fluid B, Fluid C and Fluid D). he dimensionless numbers for these fluids are presented in able. Notice that the flow field was distributed near the magnetic dipole such that certain points near the disturbances are exposed to a larger flow rate per unit area (online: close to the reddish area) whereas some parts are less exposed (online: blue areas near channel wall) (see Figs. 1 and ). he cooler ferrofluid flows in the direction of the magnetic field gradient and displaces hotter ferrofluid. he magnetic field induces the production of a local vortex near the cold wall where the magnetic source is located. he profiles of the dimensionless temperature and the magnitude of the velocity field along specific locations in the channel at time t =.5 for the different values of the magnetic field in fluid are shown in Figs. 3 and 4, respectively. It can be observed that the maximum value of the magnitude of the velocity field of the flow in the channel under the magnetic dipole increases due to the value of the magnetic number. he value of the magnitude of the velocity field at half of length of channel and time t =.5 is about three times bigger than the magnitude of the velocity profile at the left end of the rectangular channel (inlet) when the ferrofluid D is considered (see Figs. d and 4d) and only about 5% bigger when the ferrofluid A is considered (see Figs. a and 4a). 7 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

11 136. Strek and H. Jopek: Computer simulation of heat transfer through a ferrofluid Fig. 3 emperature profiles at the various vertical cross sections of channel at time t =.5 for the different values of the magnetic number: (a) fluid A, (b) fluid B, (c) fluid C, (d) fluid D. 6 Conclusions wo-dimensional heat transfer in ferrofluid channel flow under the influence of the magnetic field created by the magnetic dipole using the computational fluid dynamics code COMSOL based on finite element method was simulated. here was assumed a parabolic laminar flow profile at the left end of the rectangular channel. he upper plate was kept at constant temperature u and the lower at l. he flow was relatively uninfluenced by the magnetic field until its strength was large enough for the Kelvin body force to overcome the viscous force. It can be observed that the cooler ferrofluid flows in the direction of the magnetic field gradient and displaced hotter ferrofluid. his effect is similar to natural convection where cooler, denser material flows towards the source of gravitational force. Ferrofluids have promising potential for the heat transfer applications because a ferrofluid flow can be controlled using an external magnetic field. he Kelvin body force arises from the interaction between the local magnetic field within the ferrofluid and the molecular magnetic moments characterized by the magnetization. An imposed thermal gradient produces a spatial variation in the magnetization through the temperature-dependent magnetic susceptibility for ferrofluids and therefore renders the Kelvin body force non-uniform spatially. his thermal gradient induced inhomogeneous magnetic body force can promote or inhibit convection in a manner similar to the gravitational body force. 7 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

12 Original Paper phys. stat. sol. (b) 44, No. 3 (7) 137 Fig. 4 Velocity field magnitude profiles at the various vertical cross sections of the channel at the time t =.5 for the different values of the magnetic number: (a) fluid A, (b) fluid B, (c) fluid C, (d) fluid D. References [1] B. M. Berkovsky, V. F. Medvedev, and M. S. Krakov, Magnetic Fluids Engineering Applications (Oxford Science Publications, Oxford, 1993). [] K. W. Wojciechowski, in: Proceedings of the NAO Advanced Research Workshop on Properties and Applications of Nanocrystalline Alloys from Amorphous Precursors, Budmerice, Slovak Republic 9 15 June 3, edited by Bogdan Idzikowski, Peter Švec, and Marcel Miglierini (5). [3] F. Scarpa and F. C. Smith, J. Intell. Mater. Syst. Struct. 15(1), (4). [4] H. I. Andersson and O. A. Valnes, Acta Mech. 18, (1998). [5] H. Matsuki, K. Yamasawa, and K. Murakami, IEEE rans. Magn. 13, (1997). [6]. ynjälä, A. Hajiloo, W. Polashenski Jr., and P. Zamankhan, J. Magn. Magn. Mater. 5, (). [7] R. Ganguly, S. Sen, and I. K. Puri, J. Magn. Magn. Mater. 71, (4). [8] V. C. Loukopoulos and E. E. zirtzilakis, Int. J. Eng. Sci. 4, (4). [9] H. Yamaguchi, Z. Zhang, S. Shuchi, and K. Shimada, J. Magn. Magn. Mater. 5, 3 5 (). [1] L. Yang, J. Ren, Y. Song, J. Min, and Z. Gou, Int. J. Eng. Sci. 4, (4). [11] C.-Y. Wen, C.-Y. Chen, and S.-F. Yang, J. Magn. Magn. Mater. 5, 6 8 (). [1] M. S. Krakov and I. V. Nikiforov, J. Magn. Magn. Mater. 5, 9 11 (). [13] A. Ramanathan and G. Suresh, Int. J. Eng. Sci. 4, (4). [14] L. Yang, J. Ren, Y. Song, J. Min, and Z. Gou, Int. J. Eng. Sci. 4, (4). [15] C.-Y. Wen, C.-Y. Chen, and S.-F. Yang, J. Magn. Magn. Mater. 5, 6 8 (). 7 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim