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Seridonio fachini conem_draft Seridonio fachini conem_draft Document Transcript

  • Heat dissipation in a fuel drop with magnetic nanoparticles in the presence of a circular polarized eld and vorticity A. C. Seridonio and F. F. Fachini INPE-Instituto Nacional de Pesquisas Espaciais, Rodovia Presidente Dutra Km 40, CEP 12630-000, Cachoeira Paulista, SP, Brazil This work investigates the thermal response of an external magnetic eld on magnetic nanoparti- cles embedded in a fuel drop. We adopt as theoretical approach the well-established magnetization equation of ferrohydrodynamics proposed by M. I. Shiliomis et al, which describes the motion of free dipoles in a viscous uid. Taking into account vorticity eects, we show that new features arise when a circular polarized eld is considered. The interplay between the Brown and Néel mechanisms of relaxation times also presents an important role in the heating process of the drop. Thus the friction of the particle rotation, due to magnetic torques, with the viscous medium of the drop, constitutes an extra source of heat. Our treatment allows the determination of the drop temperature prole as a function of the model parameters, which is necessary to characterize completely its thermal properties. Such prole is of fundamental interest in this emerging eld of research of fuel drops with magnetic nanoparticles. I. INTRODUCTION The use of alternating magnetic elds on uids with magnetic nanoparticles (MNPs) allows a production of heatin the host system [1]. Such power loss is due to a friction eect mediated by the viscous and magnetic torques onthe nanoparticles, being the former opposite to the latter. We propose that this mechanism of heating in ferrouidscan be also used to heat fuel drops with MNPs. To that end, a theoretical study must be done to establish thebasis of future applications. As the ferrouid community has restricted the determination of the system heatingrate to quiescent uids with linearly polarized magnetic elds [13], we investigate this quantity for rotating eldsin uids with nite ow vorticities. Taking into account these eects in our calculations of the heating rate function, we show that the heatingprocess is favored and becomes more ecient when performed with circularly polarized magnetic elds. This paperis organized as follows. In Sec. II, the phenomenological model of M. I. Shiliomis et al [46] for ferrouids isdiscussed and particular attention is given to the relaxation mechanisms of Brown and Néel [7]. In Sec. III, usingthe formalism of the complex frequency dependent susceptibility [1, 8, 9], we determine the net magnetization inthe system due to a circular counterclockwise polarization. The heating rate function for an adiabatic process isderived in Sec. IV, and in Sec. V, numerical results for the frequency dependence of this function are presented.We summarize our ndings in Sec. VI. II. THEORETICAL MODEL FOR FERROFLUIDS The phenomenological model of M. I. Shliomis et al [46] describes the net magnetization M of a viscous uidwith rotating magnetic MNPs characterized by a macroscopic angular velocity Ωp due to an applied magnetic eldH. Magnetic and viscous torques on these particles embedded in a uid with vorticity ow Ω are considered in thismodel. The external eld interacts with the MNP dipoles m forcing an alignement with the eld direction. As theparticles are in contact with a thermal bath of nite temperature T, the equilibrium magnetization ϕ ξ M0 = |m| L ξ (1) V ξis reached. The constant ϕ is the volume fraction of the MNPs, V is the MNP single volume and L ξ is theLangevin function L (ξ) = coth ξ − ξ −1 (2)that depends on the magnitude of
  • 2 |m| H ξ= . (3) kB TThis ratio measures the competition between the magnetic energy |m| H of the oriented MNPs and the thermalenergy kB T of the ferrouid. Due to a mechanism of relaxation for the dipoles, a magnetization rate of changeappears until the equilibrium magnetization is established. In the rotating frame of the MNPs, this relaxationprocess is assumed to be governed by the equation [7] dM 1 =− M − M0 , (4) dt τwhose solution is M = [1 − exp (−t/τ )] M0 (5)to ensure the expected value M = M0 for t τ. In the laboratory frame we have to add to the right side of theEq. (4) the contribution ΩP × M due to the magnetic torque τmag = M × H (6)to derive dM 1 = ΩP × M − M − M0 (7) dt τas the equation of motion for the out-of-equilibrium magnetization. The relaxation parameter τ consists of twoprocesses in parallel [1, 7], which is expressed as 1 1 1 = + , (8) τ τB τNwith 3ηV τB = (9) kB Tas the Brownian time of rotational diusion and exp (KVM /kB T ) τN = τ0 (10) KVM /kB Tthe Néel relaxation. The parameters are the uid viscosity η, the hydrodynamic volume V of a single MNP, theconstant of proportionality τ0 , the anisotropy constant K and the magnetic volume VM of a MNP. In the Brownianrelaxation, the magnetic dipoles rotate together with the particles. In the Néel case, the MNPs are frozen and themagnetic dipoles rotate exclusively in respect to the crystal axes of the particles. Thus considering the viscosity ηof the uid, the viscous torque τvisc = Γ ΩP − Ω (11)on the rotating MNPs is opposite to the magnetic torque of Eq. (6) and is determined by the moment of inertia perunit time Γ = 6ϕη, the macroscopic angular velocity ΩP and the vorticity Ω. As a result, the macroscopic angularacceleration for the MNPs obeys the following rotational equation dΩp I = τmag − τvisc , (12) dt
  • 3where I is the total rotation moment of inertia. The model considered here treats the steady state of Eq. (12),which gives 1 ΩP = Ω + M ×H (13) 6ϕηas the macroscopic angular velocity for the MNPs. Thus the magnetization rate of change in the laboratory frameis given by dM 1 1 =− M − M0 + Ω × M − M × M ×H . (14) dt τ 6ϕηAccording to Ref. [6], the dynamics of Eq. (14) agrees very well with numerical results extracted from a Focker-Planck analysis for the limit Ωτ < 1. To apply this model, this paper investigates the heating process in suchrange. III. THE NET MAGNETIZATION AND THE CIRCULAR POLARIZATION The heating rate of the system is determined by the magnetization of the ferrouid due to an alternate magneticeld. In particular, we choose a circular counterclockwise polarization H = H0 cos ωtˆ + H0 sin ωty, x ˆ (15)where H0 and ω are the amplitude and the angular frequency of the eld, respectively. The solution of Eq. (14)for the magnetization can be obtained numerically, but a guessed solution based on the expected behavior of themagnetization can be used instead . To that end, we remind the role of the relaxation process in the system. Inthe absence of a relaxation mechanism, the magnetization is in-phase with the alternating magnetic eld, which isdescribed by the instantaneous version of Eq. (1). Thus, a nite relaxation time introduces a phase shift in themagnetization, i.e., M = M cos (ωt − δ) x + M sin (ωt − δ) y . ˆ ˆ (16)Making the denitions χ´ = M cos δ, (17) ¨ χ = M sin δ, (18)the magnetization can be presented in terms of the components of the frequency dependent complex susceptibilityχ (ω) = χ´ − iχ¨ [1, 8, 9], M = χ´ cos ωt + χ¨ sin ωt x + χ´ sin ωt − χ¨ cos ωt y. ˆ ˆ (19)To determine the unknown variables of Eqs. (17) and (18) as functions of the model parameters, we consider thelow-amplitude approximation for the magnetic eld |m| H /kB T 1, which leads to the instantaneous equilibriummagnetization M0 (t) = χ0 H (t) , (20) ϕ 2with χ0 = V |m| /3kB T as the equilibrium susceptibility. Performing the substitutions of Eqs. (15) and (16) inEqs. (7) and (13), we show that M = χ0 H0 cos δ (21)
  • 4and tan δ = τ (ω − Ω) . (22)Combining these results, the complex susceptibility for the system becomes χ0 χ (ω) = = χ´ − iχ¨ , (23) 1 + iτ (ω − Ω)with the magnetization χ0 H0 M= [cos (ωt − δ) x + sin (ωt − δ) y ] ˆ ˆ (24) 2 1 + τ 2 (ω − Ω)phase shifted to the external magnetic eld by {χ (ω)} δ = arctan − . (25) {χ (ω)} IV. THE HEATING RATE According to the rst law of thermodynamics dE = δQ − δW per unit volume, the dissipation energy densityE in a cycle T = 2π/ω for an adiabatic process δQ = 0 is the work density δW = −H.dB done by the magneticeld on the ferrouid. Using the constitutive equation B = µ0 H + M for the magnetic induction and the cycle ¸condition H.dH = 0, we derive for the limit Ωτ < 1, the following dissipation energy density ˆ T dM 2 E = µ0 dtH. = −2πµ0 H0 {χ (ω)} , (26) 0 dtwhere µ0 is the magnetic permeability of the free space. Taking into account Eq. (23) in Eq. (26), this energydensity becomes 2 τ (ω − Ω) E = 2πµ0 χ0 H0 2. (27) 1 + τ 2 (ω − Ω) For a ow vorticity Ω = 0, our expression for the density has the same functional dependence as the result derivedby [1] for a magnetic eld with a linear polarization, except for a factor of two. This factor arises from the degreesof freedom of the alternating magnetic eld, while the circular polarization has two components in the xOy plane,the linear case is xed to only one direction. However, both cases introduce a shift of Ω in the frequency dependenceof the energy density for Ω = 0. Thus, our expression allows an enhanced heating process for external elds thatsatisfy the condition ω > Ω. The characterization of the system temperature prole for a constant relaxation can be performed dening theheating rate function [1, 2] ∆T ωE = , (28) ∆t 2πρcwhere ∆T and ∆t are the temperature rise and the duration of the heating, respectively. For the ferrouid, ρ is thedensity and c the specic heat. According to Eq. (26), the heating rate is governed by the behavior of the minuspart of the complex frequency dependent susceptibility given by Eq. (23). So this quantity becomes important toinvestigate the features of the heating process as a function of the model parameters. To explicit such dependencies,we prefer to dene the dimensionless variables ∆ = τ (ω − Ω) and ε = Ωτ to derive
  • 5Figure 1: Heating rate HC due to a circular polarization as a function of the dimensionless relative frequency ∆ = τ (ω − Ω)with dierent dimensionless vorticities ε = Ωτ. For external frequencies near the ow vorticity, the heating rate displays alinear behavior. A crossover region occurs until a saturation value is reached for large relative frequencies. Constant heatingrates dependent on the uid vorticity appear. A comparison between the linear and the circular heating for a quiescent uid(Ω = 0) is presented in the inset. As it can be observed, the circular case is more ecient. τ ρc ∆T ∆2 ∆ HC = 2 = +ε (29) µ 0 χ 0 H0 ∆t 1 + ∆2 1 + ∆2as the renormalized heating rate due to the circular polarization. Keeping ε < 1 to ensure the validity of the model,Eq. (29) allows the knowledge of the dependence of the ferrouid heating rate in terms of the dimensionless relativeeld frequency and the ow vorticity given by and ε, respectively. According to Ref. [1], the heating rate due toa linear polarization is the half of a circular, 1 HL = HC . (30) 2 V. RESULTS In Fig. (1) we present simulations for the heating rate function. They show a linear behavior for small deviationsof the relative frequency , which is ow vorticity dependent. For intermediate deviations, this behavior disappearsand the slopes of the curves start to decrease. Such decrease stabilizes and constant heating rates occurs for largedeviations, whose intensities are determined by low vorticities. Ecient heating rates for circular polarization areproduced, see the comparison between the circular with linear cases in the inset. These behaviors are consequenceof the Eqs. (29) and (30). VI. CONCLUSIONS Based on a phenomenological model for ferrouids, we derived the heating rate function due to a magnetic eldwith a circular counterclockwise polarization. For a xed relaxation parameter, we demonstrated that the heating
  • 6process is more ecient for large dierences between the external and vorticity frequencies with a saturation valuedetermined by low vorticities. This mechanism can be applied to heat fuel drops with MNPs. Acknowledgments This work was supported by the brazilian agency CAPES.[1] R. E. Rosensweig, J. Magn. Magn. Mater 252, 370 (2002).[2] S. Maenosomo, S. Saita, IEEE Trans. Magn. 42, 1638 (2006).[3] F. F. Fachini, COBEM 2009.[4] M. I. Shliomis, T. P. Lyubimova, D. V. Lyubimov, Chem. Eng. Comm. 67, 275 (1988).[5] M. I. Shliomis, K. I. Morozov, Phys. Fluids 6, 2855 (1993).[6] M. I. Shliomis, Phys. Rev. E 64, 060501 (2001).[7] R. E. Reosensweig, Ferrohydrodynamics, Dover Publications (1985).[8] P. C. Fannin, T. Relihan, S. W. Charles, Phys. Rev. B 55, 14423 (1997).[9] P. C. Fannin, J. Magn. Magn. Mater 258, 446 (2003).