Axial Dispersion in Nanofluid Poiseuille Flows Stirred by Magnetic

Article pubs.acs.org/IECR

Axial Dispersion in Nanofluid Poiseuille Flows Stirred by Magnetic Nanoagitators M. Rolland,† F. Larachi,*,‡ and P. Hajiani‡ IFP Énergies nouvelles, Rond-point de l’échangeur de Solaize, BP 3, 69360 Solaize, France Department of Chemical Engineering, Laval University, Québec City, Québec G1 V 0A6, Canada

† ‡

ABSTRACT: A Taylor−Aris dispersion in laminar capillary flows of magnetic nanofluids submitted to transverse rotating magnetic fields (RMFs) was analyzed with a simple phenomenological mixing approach. The nanofluid residence time distributions (RTDs) measured under RMFs were used to quantify the deviations, with respect to field-free Poiseuille flows, of the axial dispersion induced by the rotating magnetic nanoparticles (MNPs) as a function of MNPs concentration and diameter, and RMF frequency and strength. The attenuation of axial dispersion due to the magnetic field was ascribed to an enhanced transverse diffusion coefficient that thrust tracer radial transport, owing to nanoconvective streams in the nanoparticle neighborhoods, beyond molecular diffusion capability. To estimate the enhanced transverse diffusion coefficient, a semiempirical model was developed in which the nanofluid domain was viewed as an array of identical cells each containing a magnetic nanoparticle at its center. Owing to the nanoparticle rotation in the magnetic field, each cell consisted of an inscribed perfectly mixed core confined in a stagnant shell where molecular diffusion prevailed. Two- and three-dimensional diffusion simulations of the two-zone cell were used to quantify, and link, the size of the mixed core to the measured axial dispersion coefficients under various experimental conditions.

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cell positioned inside a magnet bore.25 Water self-diffusion coefficients were indeed found to undergo up to a 200-fold increase upon application of RMFs depending on nanoparticles volume fraction, field strength, and frequency.25 Alternatively, to monitor changes in Aris−Taylor dispersions under various types of magnetic fields, axial dispersion coefficients were determined for capillary Poiseuille flows through a series of residence time distribution (RTD) measurements. RMFmediated nanoparticle stimulation contributed to reduced axial dispersion coefficients compared to field-free Poiseuille flows. Oscillating magnetic fields, unlike RMFs, led to a negligible impact on axial dispersions, whereas static magnetic fields were counterproductive in that sense that they inflated axial dispersions.24 In the above-reported mixing studies using various magnetic field stimulations,23,24 no attempt was made to quantitatively rationalize in terms of a simple modeling the axial dispersion coefficients obtained from residence time distribution measurements for the various magnetic field conditions. An emphasis will be placed in this contribution to stretch further the interpretation on the role of magnetic nanoparticles, subject to rotating magnetic fields, and their contribution to mitigate axial dispersions of nanofluid Poiseuille flows. A phenomenological framework will thus be developed in which a simple nanomixing mechanism will reduce axial dispersion. For quantification purposes, the RTD measurements obtained in our previous study23 will be used to evaluate the effect of magnetic field frequency and intensity and nanoparticle

INTRODUCTION Molecular transport and diffusion in liquids are central phenomena to a number of disciplines in science and engineering.1−3 Currently, a fecund effervescence is enlivening the quest for new nanoparticle-based approaches to enhance liquid-phase transport properties at scales or in configurations traditionally beheld as the chasse gardée of classical diffusion.2 For example, it has been stated that the presence of nanoparticles in liquids may affect heat4−6 and mass transport7−15 in the so-called nanofluids. Most remarkably, magnetic nanoparticles (MNPs) are being looked at in a few studies for the promotion of mass transfer both under applied external magnetic fields8,17 and field-free experiments.9,16 Despite certain modes of nanoparticle interference on fluid transport being still viewed as anomalous,18 the technique consisting of excitation of nanofluids with time-varying external magnetic fields to induce, remotely and noninvasively, desirable MNP momentum/heat/mass responses is foreseen to hold promise in various technical and medical applications, for example, translation, resonance, hyperthermia, etc.19−22 It has been recently shown that transport phenomena in nanofluids can be boosted by means of a nanoagitation mechanism prompted by guest magnetic nanoparticles stimulated under purposefully positioned external magnetic fields.23,24 MNPs in this technique are per se harnessed as nanoscale devices (or nanoagitators) to intensify nanofluid mixing in the nanoparticles’ neighborhood while the system is being excited with various magnetic field types such as uniform rotating magnetic fields (RMFs), and oscillating, and stationary magnetic fields. Manifestation of nanoconvective mixing upon application of magnetic fields was first diagnosed by measuring liquid selfdiffusion coefficients of dilute MNP suspensions in a diffusion © 2014 American Chemical Society

Received: Revised: Accepted: Published: 6204

November 20, 2013 March 7, 2014 March 10, 2014 March 10, 2014 dx.doi.org/10.1021/ie4039423 | Ind. Eng. Chem. Res. 2014, 53, 6204−6210

Industrial & Engineering Chemistry Research

Article

magnetic fields at moderate strength ( Dm,C ≈ Dm. However, the longitudinal diffusion coefficient term, Dm,C, would contribute only marginally to axial dispersion on account of the preponderant influence of advection in the second term of eq 2. The net result for this magnetic field configuration is a relatively small decrease of the dispersion coefficient. On the contrary, for a magnetic field rotating along the flow axis as shown in Figure 2b, mixing would occur in the plane perpendicular to the flow resulting in a greatly enhanced crosswise diffusion, Dm,C, which would be tantamount to a significant decrease of axial dispersion. Experimental data indeed indicate a diffusion enhancement by up to a factor of 5, meaning that the axial diffusion term in the Aris−Taylor equation is always negligible in comparison to the advectiondiffusion term, so that eq 2 can be simplified to:

(Figure 4a), as well as by increasing either the volume fraction of magnetic cores in the nanofluid or, for a given ϕ, by increasing the nanoparticle size (Figure 4b). The mixed zones, represented in the form of mixing spheroids in Figure 2b, are expected to swell and/or grow under the increase of one of the four operating variables of Figure 4 until a plateau is reached which corresponds to an overlap of the mixed zones altogether. Modeling of RMF Effect on Nanofluid RTD. Now that we have extracted from RTD measurements axial dispersion coefficients under rotating magnetic fields and have proposed a mechanism to relate nanoparticles rotation to mixing, let us next propose a simple model to account for our observations. The MNPs are assumed to be uniformly distributed in the nanofluid domain. This domain is split into an array of identical cells each of which contains a magnetic nanoparticle at its center (Figure 5). The cell size, b, can straightforwardly be estimated from knowledge of MNP volume fraction and diameter: ⎛ π ⎞1/3 b = ⎜ ⎟ dc ⎝ 6φ ⎠

(5)

In each cell, the nanoparticle acts as a stirrer, and it is assumed that there exists a core subvolume, of diameter a, inscribed in 6207

dx.doi.org/10.1021/ie4039423 | Ind. Eng. Chem. Res. 2014, 53, 6204−6210


Article

ΔcDm D ⎛ a ⎞n = m =1−⎜ ⎟ ⎝b⎠ bΦ Dm,C

(6)

where the best fit for n = 1.6 in 2D and n = 2.6 in 3D are shown as solid lines in Figure 6. The physical value of n should be bracketed within this range on account of an ellipsoidal (or elongated) shape of a cell’s well-mixed core. As can be seen in Figure 6, the limit a/b → 0 corresponds to nanofluid RTDs with no magnetic field, that is no difference whatsoever between molecular diffusion and Dm,C. Conversely, approaching the a/b → 1 limit signifies the nanofluid RTDs under RMF conditions undergo substantial crosswise mixing together with a fall off of the axial dispersion coefficient as eq 2 would suggest. Using eqs 4 and 5 and the formalism developed insofar, it would be immediate to reciprocate the RTD measured axial dispersion coefficient, Dax, with the mixed-core diameter a under magnetic field for the several parameters studied: Reynolds number, nanoparticle diameter and concentration, magnetic field intensity and frequency. For each parameter, the curves seem to reach an asymptote when dispersion reduction, eq 6, reaches a threshold value of ∼0.2 which would correspond to a diffusion enhancement factor of 5. This pattern reflects nothing else than the plateau behavior already discussed with Figure 4 above. In our framework, this occurs when the mixing zone diameter is close to the distance between nanoparticles, that is the mixing zones of neighboring nanoparticles will commence to interact. Therefore, keeping only the experiments lying far from this threshold, a power-law expression of the form: a = kf a1 H0a2dca3 dc (7)

Figure 5. Estimation of diffusion enhancement due to the presence of a mixing zone in the two-zone cell. Colors represent concentration level (Comsol 4.3a simulation).

the cell and within which stirring is perfect. The extent of this perfectly mixed subvolume, bounded by the sizes of cell and nanoparticle, is some function of magnetic field frequency and intensity and of nanoparticle size and concentration. Moreover, as long as each cell encompasses its mixing zone, cells remain independent from each other. The two-zone cell, sketched in Figure 5, consists of a vigorously stirred core that is artificially endowed with tremendous diffusion (∼108Dm) to warrant gradient-less concentration profiles throughout the mixed core. This region is surrounded by a fluid shell in which transport is assumed to only occur by molecular diffusion. Boundary conditions are imposed as concentrations (c = 0 and c = 1) on opposite cell faces and as symmetry on the others (Figure 5). The COMSOL 4.3a simulation package was used to solve the diffusion equation (no convection) in the two-zone cell and to estimate the steady-state diffusion flux, Φ, across the imposed concentration sides for various perfectly mixed cores. Though neither do we know nanoparticle shape or core shape, the actual mixed core would probably bear close resemblance to an ellipsoid (Figure 2b). COMSOL simulations have been thus performed in 2D and 3D as limiting cases such that the 2D case represents the mixing zone as a cylinder of height b, whereas the 3D case represents the mixing zone as a sphere. The reciprocal of diffusion enhancement within the cellwhich equivalently corresponds to the reduction of dispersionis defined as bΦ/(ΔcDm) and is plotted in Figure 6 as a function of the core-to-cell size ratio, a/b. Whether in the 2D or 3D case, the reduction of dispersion is found to be best fitted with the following equation:

was regressed to the measured axial dispersion reduction factors using eq 8 with both 2D and 3D fitted parameter values of n: Dax ⎛ a ⎞n =1−⎜ ⎟ ⎝b⎠ Dax ,0

(8)

COMSOL-simulation accessed parameter n in 2D and 3D, combined with eqs 7 and 8 for computing Dax/Dax,0 ratios describe reasonably well (R2 ≈ 0.85) the Dax/Dax,0 ratios directly obtained from the experimentally measured residence time distributions under RMF. The quality of fit can be judged from the parity plots shown in Figure 7 panels a and b, respectively, for the 2D and 3D modalities. The scatter, intrinsic to the accuracy of axial dispersion coefficient measurements, makes it difficult to further improve the correlation coefficients. The presented plots encompass variations in MNP concentration and diameter, magnetic field intensity, and frequency. Note that elongated mixing zones, requiring definition of more than one principal diameter, would best describe the data but would have required more elaborate computations which we spared by defining one equivalent diameter in 2D and 3D simulations. The reality would be that the actual n value would be intermediate between the 2D and 3D fitted values. Quite interestingly, the mixing zone diameter, a, bears a proportionality dependence to the nanoparticle magnetic diameter, dc, i.e., a3 = 0 both for the 2D and 3D modalities. It is also proportional to the square root of the magnetic field intensity H0 (a2 = 0.5). The frequency exponent is the most sensitive to parameter n and is quite small (a1 ≈ 0.18 and 0.25). Confidence in this result is fair as a2 and a1 are positive and

Figure 6. Reduction of dispersion (eq 6), (ΔcDm)/bΦ, as a function of core-to-cell size ratio, a/b. Lines represent best fits using eq 6 and corresponding exponent n. 6208



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Figure 7. Predicted versus experimental axial dispersion reduction. 2D or elongated mixed core: k = 1.2, a1 = 0.25, a2 = 0.5, a3 = 0 (a). 3D or spherical mixed core: k = 1.72, a1 = 0.18, a2 = 0.5, a3 = 0 (b) modalities.

Dm,C = cross-wise diffusion, m2/s Dm,S = stream-wise diffusion, m2/s dc = nanoparticle magnetic core diameter, m f = magnetic field frequency, Hz H0 = magnetic field intensity, A/m k = parameter in eq 7, L = RTD probe interdistance, m Ms = saturation magnetization, A/m Re = Reynolds number Re = 2ruρ/μ, r = capillary radius, m u = average (superficial) velocity, m/s

of acceptable order of magnitude. These parameters are coherent with the observed fact that the faster the nanoparticles rotates or the stronger is the magnetic field, the better is the mixing. So far, simple physical models based on an analogy to asynchronous electrical engines, impellers, or rotating objects in a fluid have failed to provide a rationale for the presented exponents.

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CONCLUSION Residence time distributions and axial dispersion coefficients were determined for a magnetic nanoparticle containing nanofluid in a capillary Poiseuille flow subject to a transverse rotating magnetic field. Viewing the individual nanoparticles as ideal and independent nanomixers, a power law correlation was derived to predict the mixing zone size in adequacy with axial dispersion coefficients as extracted from tracer responses for varying nanoparticle concentrations and diameters, and magnetic field rotating frequency and amplitude. In this approach, the diffusion enhancement was assumed to be prompted by cross-wind mixing induced at the nanoscale by the rotating nanoparticles and where, as a first approximation, dispersion by modifications of the velocity profile was ignored. On account of the quite large experimental scatters, the proposed mechanism can neither be ruled out nor adopted with certainty especially with regard to the velocity profile approximation. Future experimental works such as microparticle image velocimetry (μ-PIV) would greatly help ascertaining or modifying the proposed approach.

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Greek Letters

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Δc = concentration driving force, mol/m3 ϕ = volume fraction of nanoparticles, Φ = diffusion flux across cell, mol/m2/s χ0 = initial susceptibility, -

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AUTHOR INFORMATION

Corresponding Author

*Tel.: (418) 656-2131 ext. 3566. Fax: (418) 656-5993. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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NOTATION a = diameter of perfectly mixing cell core, m ai = parameters in eq 7, i = 1,3, b = nanoparticle interdistance, m Dax,0 = axial dispersion coefficient without magnetic field, m2/s Dax = axial dispersion coefficient under magnetic field, m2/s Dm = molecular diffusion, m2/s 6209



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Axial Dispersion in Nanofluid Poiseuille Flows Stirred by Magnetic

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