HEAT TRANSFER IN FERROFLUID IN CHANNEL WITH POROUS WALLS

Transcription

1 HEAT TRANSFER IN FERROFLUID IN CHANNEL WITH POROUS WALLS T. Strek Institute of Applied Mechanics, Poznan University of Technology, Poland Abstract The viscous, two-diensional, incopressible and lainar tie dependent heat transfer flow through a ferroagnetic fluid is considered in this paper. Flow takes place in channel between two porous walls under the influence of the agnetic dipole located beyond the channel. It is assued that there is no electric field effects and the variation in the agnetic field vector that could occur within the ferrofluid is negligible. This agneto-thero-echanical proble is governed by diensionless equations. Results are obtained using standard coputational fluid dynaics code COMSOL with odifications to account for the agnetic ter when needed. Introduction In this paper tie dependent heat transfer through a ferrofluid in channel flow under the influence of the agnetic dipole is considered and siulated. Results are obtained using standard coputational fluid dynaics code COMSOL with odifications to account for the agnetic ter when needed. During the last decades, an extensive research work has been done on the fluids dynaics in the presence of agnetic field. The effect of agnetic field on fluids is worth investigating due to its innuerable applications in wide spectru of fields. The study of interaction of the agnetic field or the electroagnetic field with fluids have been docuented e.g. aong nuclear fusion, cheical engineering, edicine, high speed noiseless printing and transforer cooling. One of the ost exciting areas of technology to eerge in recent years is MEMS (icroechanical systes), where engineers design and build systes with physical diensions in icroeters, e.g. MEMS-based biosensors or icroscale heat exchangers. The transport of oentu and energy in iniaturized devices is diffusion liited because of their very low Reynolds nubers. Using ferrofluids in these applications and anipulating the flow of ferrofluids in these applications by external agnetic field can be a viable alternative to enhance convection in these devices. Ferrofluids are non-conducting fluids and the study of the effect of agnetisation has yielded interesting inforation. In equilibriu situation the agnetization property is generally deterined by the fluid teperature, density and agnetic field intensity and various equations, describing the dependence of static agnetization on these quantities. The siplest relation is the linear equation of state, given by [And998]. It can be assued that the agneto-thero-echanical coupling is not only described by a function of teperature [And998], but by an expression involving also the agnetic field strength [Mat997]. This assuption perit us not to consider the ferrofluid far away fro the sheet at Curie teperature in order to have no further agnetization. This feature is essential for physical applications because the Curie teperature is very high (e.g. 43 Kelvin degrees for iron) and such a teperature would be eaningless for applications concerning ost of ferrofluids. So instead of having zero agnetization far away fro the sheet, due to the increase of fluid teperature up to the Curie teperature this forulation allows us to consider whatever teperature is desired and the agnetisation will be zero due to the absence of the agnetic field sufficiently far away fro the sheet [Tzi3]. Moreover, ferrofluids are ostly organic solvent carriers having ferroagnetic oxides, acting as solute. Ferrofluids consist of colloidal suspensions of single doain agnetic nanoparticles. They have proising potential for heat transfer applications, since a ferrofluid flow can be controlled by using an external agnetic field [Gan4]. However, the relationship between an iposed agnetic field, the resulting ferrofluid flow and the teperature distribution is not understood well enough. The literature regarding heat transfer with agnetic fluids is relatively sparse.

2 An overview of prior research on heat transfer in ferrofluid flows e.g. theroagnetic free convection, theroagnetic forced convection and boiling, condensation and ultiphase flow are presented in paper [Gan4]. Many researchers are seeking new technologies to iprove the operation of existing oil-cooled electroagnetic equipent. One approach suggested in literature is to replace the oil in such devices with oil-based ferrofluids, which can take advantage of the pre-existing leakage agnetic fields to enhance heat transfer processes. In paper [Tan999] authors present results of an initial study of the enhanceent of heat transfer in ferrofluids in agnetic fields which are steady but variable in space. Finite eleent siulations of heat transfer to a ferrofluid in the presence of a agnetic field are presented for flow between flat plates and in a box. The natural convection of a agnetic fluid in a partitioned rectangular cavity was considered in paper [Ya]. It was found that the convection state ay be largely affected by iproving heat transfer characteristic at higher Rayleigh nuber when a strong agnetic field was iposed. The influence of a unifor outer agnetic field on natural convection in square cavity was presented in paper [Kra]. It was discovered that the angle between the direction of teperature gradient and the agnetic field influences the convection structure and the intensity of heat flux. Nuerical results of cobined natural and agnetic convective heat transfer through a ferrofluid in a cube enclosure were presented in paper [Sny3]. The purpose of this work was to validate the theory of agnetoconvection. The agnetoconvection is induced by the presence of agnetic field gradient. The Curie law states that agnetization is inversely proportional to teperature. That is way the cooler ferrofluid flows in the direction of the agnetic field gradient and displaces hotter ferrofluid. This effect is siilar to natural convection were cooler, ore dense aterial flows towards the source of gravitational force. Results were obtained using standard coputational fluid dynaics codes with finite eleent ethod. The effect of agnetic field on the viscosity of ferroconvection in an anisotropic porous ediu was studied in paper [Ra4]. It was found that the presence of anisotropic porous ediu destabilizes the syste, where as the effect of agnetic field dependent viscosity stabilizes the syste. In this paper the investigated fluid was assued to be incopressible having variable viscosity. Experientally it has been deonstrated in prior research that the agneto viscosity has got exponential variation, with respect to agnetic field. As a first approxiation for sall field variation, linear variation of agneto viscosity has been used in paper [Ra4]. The effect of agnetic field dependent (MFD) viscosity (agnetoviscosity) on ferroconvection in a rotating sparsely distributed porous ediu has been studied in paper [Vai]. The effect of MFD viscosity on therosolutal convection in ferroagnetic fluid has been considered for a ferroagnetic fluid layer heated and soluted fro below in the presence of a unifor vertical agnetic field [Sun5]. Using the linearized stability theory and the noral ode analysis ethod, an exact solution was obtained for the case of two free boundaries. One of the probles associated with drug adinistration is the inability to target a specific area of the body. Aong the proposed techniques for delivering drugs to specific locations within the huan body, agnetic drug targeting [Vol, Rit4] surpasses due to its non-invasive character and its high targeting efficiency. Although the ethod has been proposed alost 3 years ago, the technical probles obstruct possible applications. It was the ai of the paper [Vol] to classify the eerging probles and propose satisfactory answers. A general phenoenological theory was developed and a odel case was studied, which incorporates all the physical paraeters of the proble. A hypothetical agnetic drug targeting syste, utilizing high gradient agnetic separation principles, was studied theoretically using FEMLA siulations in paper [Rit4]. This new approach uses a ferroagnetic wire placed at a bifurcation point inside a blood vessel and an externally applied agnetic field, to agnetically guide agnetic drug carrier particles through the circulatory syste and then to agnetically retain the at a target site. The governing equations. Magnetostatic and quasi-static fields Under certain circustances it can be helpful to forulate the probles of electroagnetic analysis in ters of the electric scalar potential V and the agnetic vector potential A. They are given by the equalities [Kov99]

3 = A, A E = V (.) The defining equation for the agnetic vector potential is a direct consequence of the agnetic Gauss law. The electric potential results fro Faraday s law. Using the definitions of the potentials µ H + M, Apère s law can be rewritten as and the constitutive relation ( ) = A σ + e ( µ A M) σv ( A) + σ V = J. (.) The equation of continuity, which is obtained by taking the divergence of the above equation, gives us the equation A e σ σv ( A) + σ V J = (.3) These two equations give us a syste of equations for the two potentials A and V. In the static case we have the equations The ter v ( A) static agnetic field, static electric field, e ( v ( A) + σ V J ) = σ, (.4) e ( µ ) A M σv ( A) + σ V = J. (.5) σ represents the current generated otion with a constant velocity in a E e J = σ v. Siilarly the ter σ V represents a current generated by a e J = σe. If v = the equations decouple and can be solved independently. The other forulation is the single equation ~ e ( A M ) = J µ. (.6) The conductivity cannot be zero anywhere when the electric potential is part of the proble, as the dependent variables would then vanish fro the first equation. Siplifying to a two-diensional proble with perpendicular currents that are it should be noted that this iplies that the agnetic vector potential has a nonzero coponent only perpendicularly to the plane Apère s law can be rewritten as ( ) A =,, A z. (.7) ( M ) = µ. (.8) A z Along a syste boundary reasonably far away fro the agnet we can apply a agnetic insulation boundary condition A z =. Solving equation (.8) together with the constitutive relation we can get agnetic vector potential. Using = A and the constitutive relation we can get Az Az =,,, H = M. (.9) y x µ. The agnetic field intensity In this paper the considered flow is influenced by agnetic dipole. We assued that the a,b. The agnetic dipole gives agnetic dipole is located at distance b below the sheet at point ( ) rise to a agnetic field, sufficiently strong to saturate the fluid. In the agnetostatic case where there are no currents present, Maxwell-Apere s law reduces to H =. When this holds, it is also

4 possible to define a agnetic scalar potential by the relation agnetic dipole is given by [And998] V ( ) = V ( x,x ) γ x a H = V and its scalar potential for the x =. ( x ) ( ) (.) a + x b where γ is the agnetic field strength at the source (of the wire) and (,b) source is located. a is the position were the.3 Heat transfer and fluid flow The governing equations of the fluid flow under the action of the applied agnetic field and gravity field are: the ass conservation equation, the fluid oentu equation and the energy equation for teperature in the frae of oussinesque approxiation. The ass conservation equation for an incopressible fluid is v =. (.) The oentu equation for agnetoconvective flow is odified fro typical natural convection equation by addition of a agnetic ter ρ + v v = p + S + αρg( T T ) k + ( M ) (.) where ρ is the density, v is the velocity vector, p is the pressure, T is the teperature of the fluid, S is the extra stress tensor, k is unit vector of gravity force and α is the theral expansion coefficient of the fluid. The energy equation for an incopressible fluid which obeys the odified Fourier s law is T M ρ c + v T = k T + ηφ µ T (( v ) H) (.3) T where k is the theral conductivity, η is the viscosity and η Φ is the viscous dissipation Φ = x + y + + x y. (.4) The last ter in the energy equation represents the theral power per unit volue due to the agnetocaloric effects..4 The Kelvin body force for agnetoconvective flow The last ter in the oentu equation represents the Kelvin body force per unit volue f = ( M ), (.5) which is the force that a agnetic fluid experiences in a spatially non-unifor agnetic field. We have established the relationship between the agnetization vector and agnetic field vector M = χ H. (.6) Using the constitutive relation (relation between agnetic flux density and agnetic field vector) we can write the agnetic induction vector in the for = µ ( + χ )H. (.7)

5 The variation of the total agnetic susceptibility χ is treated solely as being dependent on teperature [Gan4] ( T ) Finally, the Kelvin body force can be represented by χ χ = χ =. (.8) + α( T T ) = µ H ( + χ ) ( H H) + µ χ H( ( ) χ ) f χ Using equation (.9) we can write Eq. (.) and (.3) in the for, respectively and ρ + v v = p + S + αρg + µ χ ( T T ) k + ( + χ ) ( H H) + µ χ H( ( H ) χ ) ( χ H). (.9) (.) T c + v T = k T + ηφ µ T (( v ) H). (.) T ρ.5 The rinkan equations for porous edia flow Fluid and flow probles in porous edia have attracted the attention of industrialists, engineers and scientists fro varying disciplines, such as cheical, environental, and echanical engineering, geotheral physics and food science. There has been a increasing interest in heat and fluid flows through porous edia [Ing5]. The rinkan equations describe flow in porous edia where oentu transport by shear stresses in the fluid is of iportance. The odel extends Darcy s law to include a ter that accounts for the viscous transport, in the oentu balance, and introduces velocities in the spatial directions as dependent variables. The flow field is deterined by the solution of the oentu balance equations in cobination with the continuity equation η ρ = p + S + v + F (.) k where η is the viscosity, k p is the pereability of the porous structure (unit: ). The rinkan equations applications is of great use when odelling cobinations of porous edia and free flow. The coupling of free edia flow with porous edia flow is coon in the field of cheical engineering. This type of probles arises in filtration and separation and in cheical reaction engineering, for exaple in the odelling of porous catalysts in onolithic reactors. Flow in the free channel is described by the Navier-Stokes equations and the ass conservation equation described in previous sections. In the porous doain, flow is described by the rinkan equations according η ρ = p + S + v p (.3) k p and v =. (.4)

6 3 The diensionless equations For siplicity the preferred work choice is to work in non-diensional frae of reference. Now soe diensionless variables will be introduced in order to ake the syste uch easier to study [Str5]. Moreover soe of the diensionless ratios can be replaced with well-known paraeters: the Prandtl nuber Pr, the Rayleigh nuber Ra, the Eckert nuber Ec, the Reynolds nuber Re, the Darcy nuber Da and the agnetic nuber Mn, respectively: η Pr =, ρ κ αρ gh vr κ Ra =, Ec = =, η κ c ch 3 H r vr H r ρκ µ µ h Mn = =. ρ hρ v η r Re = =, ρ κ η h Da =, Since now pries will not be written (old variables sybols will be used) but it is iportant to reeber that they are still there. The diensionless for of Navier-Stokes (.) and theral diffusion (.) equations are as follows: and where and k p (3.) T + v v = p + Ra PrT - k + Pr S + Mn f (3.) ( χ H) T + v T = T + Pr EcηΦ + Mn Ec T T f = χ H ( + χ ) + χ H( ( ) ) H χ χ χ = χ ( T( x )) =. T + ( α ) T( x) Diensionless rinkan equations are as follows = p + Pr S In the presents of agnetic field Kelvin body force is added = p + Pr S + Pr Da v + Pr Da v + (( v ) H) (3.3) (3.4) (3.5). (3.6) Mn f. (3.7) 4 Nuerical solution and conclusions In this section we present nuerical siulation results of heat transfer in ferrofluid. The flow takes place in channel and in channel with porous walls. The two-diensional tie dependent flows are assued viscous, incopressible and lainar. Above the channel agnetic dipole is located. The fluid is assued to be electrically nonconducting. It is assued also that there is no electric field effects. This agneto-thero-echanical proble is governed by diensionless equations (3.-3.7). 4. Heat transfer in ferrofluid in channel Considered flow takes place in channel between two parallel flat plates. The length of the channel is L and distance between plates is h. The corresponding boundary conditions for diensionless variables are assued:

7 For the upper wall ( x L, y = ) teperature T u / δ T. The velocity is (no slip condition). For the lower wall ( x L, y = ) teperature T l / δ T. The velocity is (no slip condition). For inlet (the left wall) ( =, y ) T u / δ T and is given by equation T : the upper wall teperature is kept at constant : the lower wall teperature is kept at constant x : the teperature is varying linearly fro T l / δ T to lainar flow profile given by equation ( ) in Tu Tl Tl = y + where δ T = T u Tl. There is a parabolic u u = 4 in y y for y, at the inlet end. u For outlet (the right wall) ( x = L, y ) n ( k T ) =. Pressure outlet is also assued, ( p + S) n = p n diensionless atospheric pressure. r : the convective flux is assued for teperature, I, where p is the The following initial conditions for diensionless variables are assued: the fluid is otionless, the pressure is zero and the teperature is varying linearly fro lower to upper wall. The tie-dependent flow is considered for diensionless tie t,. 5. The proble is solved with COMSOL code using direct UMFPACK linear syste solver. Relative and absolute tolerance used in calculations are.5 and.5, respectively. The following values of teperatures are assued T l = T, T u = T + where T = 3K and δ T = 3K. TALE THE QUANTITIES FOR FERROFLUID FLOWS Quantity Flow A Flow Flow C Flow D H r Mn.696e e+7.6e e+8 Pr Ra.57e+7.57e+7.57e+7.57e+7 Ec.8e-.8e-.8e-.8e- R e It can be observed that the axiu value of the agnitude of the velocity field of the flow in the channel under the agnetic dipole increases due to the value of the agnetic nuber. The flow was relatively uninfluenced by the agnetic field until its strength was large enough for the Kelvin body force to overcoe the viscous force. It can be observed that the cooler ferrofluid flows in the direction of the agnetic field gradient and displaced hotter ferrofluid (Fig.,). This effect is siilar to natural convection where cooler, ore dense aterial flows towards the source of gravitational force. Ferrofluids have proising potential for the heat transfer applications because a ferrofluid flow can be controlled by using an external agnetic field.

8 a b c d Figure. Tie evolution of diensionless velocity field (surface) and teperature contour of flow for (a) t=. (b) t=. (c) t=. C (d) t=.3. a b c d Figure. Coparison of diensionless velocity field (surface) and teperature contour for the different values of agnetic nuber: (a) flow A (b) flow (c) flow C (d) flow D. 4. Heat transfer in ferrofluid in channel with porous walls Flow takes place in channel between two parallel porous doains. The length of the channel is L and distance between porous doains is h. The length of porous doains are L and the height are 4h /.

9 For free-porous structure interface - the upper and the lower wall ( ) The corresponding boundary conditions for diensionless variables in channel flow are assued: x L,y =, y =, freeporous structure interface: p = p. The expression for the pressure at the boundary between the channel and the porous doain states that the pressure is continuous across this interface. x =, y : the teperature is varying linearly fro T l / δ T to For inlet (the left wall) ( ) T u / δ T and is given by equation T lainar flow profile given by equation ( ) in Tu Tl Tl = y + where δ T = T u Tl. There is a parabolic u u = 4 in y y for y, at the inlet end. u For outlet (the right wall) ( x = L, y ) n ( k T ) =. Pressure outlet is also assued, ( p + S) n = p n diensionless atospheric pressure. r : the convective flux is assued for teperature, I, where p is the The corresponding boundary conditions for diensionless variables in porous doain are assued: For free-porous structure interface: v = v. These conditions iply that the coponents of the velocity vector are continuous over the interface between the free channel and the porous doain. For the upper doain walls: the teperature is kept at constant teperature T u / δ T. The velocity is (no slip condition). For the lower doain walls: the teperature is kept at constant teperature T l / δ T. The velocity is (no slip condition). The following initial conditions for diensionless variables are assued: the fluid is otionless, the pressure is zero and the teperature is T l / δ T. The tie-dependent flow is considered for diensionless tie t,.. The proble is solved with COMSOL code using direct UMFPACK linear syste solver. Relative and absolute tolerance used in calculations are.5 and.5, respectively. The following values of teperatures are assued T l = T, T u = T + where T 3K δ T = 3K. The heat transfer in ferrofluid flowing in channel with porous walls is considered in four different flows with different agnetic susceptibility, inlet velocity or pereability of the porous structure. The ost interesting exaple of flow we can observe in the last considered flow (flow H). In this case the agnetoconvection is observe (Figure 4-5). We observe vortex created near the centre of agnetic dipole. Each vortex is oving fro left to right where the agnetic field intensity is getting saller. The intensity of agnetic field is plotted on each figure presented in this subsection as contour lines. TALE THE QUANTITIES FOR FERROFLUID FLOWS Quantity Flow E Flow F Flow G Flow H H r χ u 5e-3 5e-4 5e- 5e-4 Mn.8846e e e e+8 Pr Ra.57e+7.57e+7.57e+7.57e+7 Ec.8e-.8e-.8e-.8e- R e Da

10 a b c d Figure 3. Coparison of diensionless velocity field for the different flows in channel with porous walls: (a) flow E (b) flow F (c) flow G (d) flow H for tie t =.. a b

11 c d Figure 4. Tie evolution of diensionless velocity field (surface) of flow H for (a) t=.5 (b) t=.5 (c) t=.75 C (d) t=.. a b c d Figure 5. Tie evolution of diensionless teperature (surface) of flow H for (a) t=.5 (b) t=.5 (c) t=.75 C (d) t=..

12 5 References [Gan4] Ganguly R., Sen S., Puri I.K., Heat transfer augentation using a agnetic fluid under the influence of a line dipole, Journal of Magnetis and Magnetic Materials, 7 (4) pp.63-73, 4. [Ing5] Ingha D., Pop I., Transport Phenoena in Porous Media III, Elsevier Science, 5. [Kov99] Kovetz A., The Principles of Electroagnetic Theory, Cabridge University Press, 99 [Mat997] Matsuki H., Yaasawa K., Murakai K., Experiental considerations on a new autoatic cooling device using teperature sensitive agnetic fluid, IEEE Transactions on Magnetics, 3 (5), pp.43-45, 997. [Ra4] Raanathan A., Suresh G., Effect of agnetic field dependent viscosity and anisotropy of porous ediu on ferroconvection, International Journal of Engineering Science, 4 (4) pp.4-45, 4. [Rit4] Jaes A. Ritter,Arin D. Ebner,Karen D. Daniel,Krystle L. Stewart, Application of high gradient agnetic separation principles to agnetic drug targeting, Journal of Magnetis and Magnetic Materials, 8 (4) pp.84, 4. [Sny3] Snyder S.M., Cader T., Finlayson.A., Finite eleent odel of agnetoconvection of a ferrofluid, Journal of Magnetis and Magnetic Materials, 6 (3) pp.69-79, 3. [Str5] Strek T., Ferrofluid channel flow under the influence of agnetic dipole, International Journal of Applied Mechanics and Engineering, vol., 5, pp [Tan999] Tangthieng C., Finlayson.A., Maulbetsch J., Cader T., Heat transfer enhanceent in ferrofluids subjected to steady agnetic fields, Journal of Magnetis and Magnetic Materials, (999) pp.5-55, 999. [Tzi3] Tzirtzilakis E.E., Tanoudis G.., Nuerical study of bioagnetic fluid flow over a stretching sheet with heat transfer, International Journal of Nuerical Methods for Heat and Fluid Flow, Vol.3 No.7, pp , 3. [Ya] Yaaguchi H., Zhang Z., Shuchi S., Shiada K., Heat transfer characteristic of agnetic fluid in a partitioned rectangular box, Journal of Magnetis and Magnetic Materials, 5 () pp.3-5,. [Vai] Vaidyanathan G., Seker R., Vasanthakuari R., Raanathan A., The effect of agnetic field dependent viscosity on ferroconvection in a rotating sparsely distributed porous ediu, Journal of Magnetis and Magnetic Materials, 5 () pp.65-76,. [Vol] P.A. Voltairas P.A., Fotiadis D.I., Michalis L.K., Hydrodynaics of agnetic drug targeting, Journal of ioechanics, 35 () 83 8,. Author Dr Toasz Strek Institute of Applied Mechanics Poznan University of Technology, ul. Piotrowo 3, Poznan, Poland eail: toasz.strek@put.poznan.pl.