Heat transfer augmentation in nanofluids via nanofins
© Vadasz; licensee Springer. 2011
Received: 11 September 2010
Accepted: 18 February 2011
Published: 18 February 2011
Theoretical results derived in this article are combined with experimental data to conclude that, while there is no improvement in the effective thermal conductivity of nanofluids beyond the Maxwell's effective medium theory (J.C. Maxwell, Treatise on Electricity and Magnetism, 1891), there is substantial heat transfer augmentation via nanofins. The latter are formed as attachments on the hot wire surface by yet an unknown mechanism, which could be related to electrophoresis, but there is no conclusive evidence yet to prove this proposed mechanism.
The impressive heat transfer enhancement revealed experimentally in nanofluid suspensions by Eastman et al. , Lee et al. , and Choi et al.  conflicts apparently with Maxwell's  classical theory of estimating the effective thermal conductivity of suspensions, including higher-order corrections and other than spherical particle geometries developed by Hamilton and Crosser , Jeffrey , Davis , Lu and Lin , Bonnecaze and Brady [9, 10]. Further attempts for independent confirmation of the experimental results showed conflicting outcomes with some experiments, such as Das et al.  and Li and Peterson , confirming at least partially the results presented by Eastman et al. , Lee et al. , and Choi et al. , while others, such as Buongiorno and Venerus , Buongiorno et al. , show in contrast results that are in agreement with Maxwell's  effective medium theory. All these experiments were performed using the Transient-Hot-Wire (THW) experimental method. On the other hand, most experimental results that used optical methods, such as the "optical beam deflection" , "all-optical thermal lensing method" , and "forced Rayleigh scattering"  did not reveal any thermal conductivity enhancement beyond what is predicted by the effective medium theory. A variety of possible reasons for the excessive values of the effective thermal conductivity obtained in some experiments have been investigated, but only few succeeded to show a viable explanation. Jang and Choi  and Prasher et al.  show that convection due to Brownian motion may explain the enhancement of the effective thermal conductivity. However, if indeed this is the case then it is difficult to explain why this enhancement of the effective thermal conductivity is selective and is not obtained in all the nanofluid experiments. Alternatively, Vadasz et al.  showed that hyperbolic heat conduction also provides a viable explanation for the latter, although their further research and comparison with later-published experimental data presented by Vadasz and Govender  led them to discard this possibility.
Vadasz  derived theoretically a model for the heat conduction mechanisms of nanofluid suspensions including the effect of the surface area-to-volume ratio of the suspended nanoparticles/nanotubes on the heat transfer. The theoretical model was shown to provide a viable explanation for the excessive values of the effective thermal conductivity obtained experimentally [1–3]. The explanation is based on the fact that the THW experimental method used in all the nanofluid suspensions experiments listed above needs a major correction factor when applied to non-homogeneous systems. This time-dependent correction factor is of the same order of magnitude as the claimed enhancement of the effective thermal conductivity. However, no direct comparison to experiments was possible because the authors [1–3] did not report so far their temperature readings as a function of time, the base upon which the effective thermal conductivity is being evaluated. Nevertheless, in their article, Liu et al.  reveal three important new results that allow the comparison of Vadasz's  theoretical model with experiments. The first important new result presented by Liu et al.  is reflected in the fact that the value of "effective thermal conductivity" revealed experimentally using the THW method is time dependent. The second new result is that those authors present graphically their time-dependent "effective thermal conductivity" for three specimens and therefore allow the comparison of their results with the theoretical predictions of this study showing a very good fit as presented in this article. The third new result is that their time dependent "effective thermal conductivity" converges at steady state to values that according to our calculations confirm the validity of the classical Maxwell's theory  and its extensions [5–10].
The objective of this article is to provide an explanation that settles the conflict between the apparent enhancement of the effective thermal conductivity in some experiments and the lack of enhancement in other experiments. It is demonstrated that the transient heat conduction process in nanofluid suspensions produces results that fit well with the experimental data  and validates Maxwell's  method of estimating the effective thermal conductivity of suspensions. The theoretical results derived in this article are combined with experimental data  to conclude that, while there is no improvement in the effective thermal conductivity of nanofluids beyond the Maxwell's effective medium theory , there is nevertheless substantial heat transfer augmentation via nanofins. The latter are formed as attachments on the hot wire surface by a mechanism that could be related to electrophoresis and therefore such attachments depend on the electrical current passing through the wire, and varies therefore amongst different experiments. Also since the effective thermal conductivity does not increase beyond the Maxwell's  effective medium theory, the experiments using optical methods, such as Putnam et al. , Rusconi et al.  and Venerus et al. , are also consistent with the conclusion of this study.
In this article, a contextual notation is introduced to distinguish between dimensional and dimensionless variables and parameters. The contextual notation implies that an asterisk subscript is used to identify dimensional variables and parameters only when ambiguity may arise when the asterisk subscript is not used. For example t* is the dimensional time, while t is its corresponding dimensionless counterpart. However, kf is the effective fluid phase thermal conductivity, a dimensional parameter that appears without an asterisk subscript without causing ambiguity.
where t* is time, Tf (r*,t*), and Ts (r*,t*) are temperature values for the fluid and solid phases, respectively, averaged over a representative elementary volume (REV) that is large enough to be statistically valid but sufficiently small compared to the size of the domain, and where r* are the coordinates of the centroid of the REV. In Equations (1) and (2), γs = ερscs and γf = (1 - ε)ρfcp represent the effective heat capacity of the solid and fluid phases, respectively; with ρs and ρf are the densities of the solid and fluid phases, respectively; cs and cp are the specific heats of the solid and fluid phases, respectively; and ε is the volumetric solid fraction of the suspension. Similarly, kf is the effective thermal conductivity of the fluid that may be defined in the form , where is the thermal conductivity of the fluid, is the thermal conductivity ratio, and ε is the solid fraction of suspended particles in the suspension. In Equations (1) and (2), the parameter h, carrying units of W m-3 K-1, represents an integral heat transfer coefficient for the contribution of the heat conduction at the solid-fluid interface as a volumetric heat source/sink within an REV. It is assumed to be independent of time, and its general relationship to the surface-area-to-volume ratio (specific area) was derived in . Note that Ts(r*,t*) is a function of the space variables represented by the position vector , in addition to its dependence on time, because Ts(r*,t*) depends on Tf(r*,t*) as explicitly stated in Equation (1), although no spatial derivatives appear in Equation (1). There is a lack of macroscopic level conduction mechanism in Equation (1) representing the heat transfer within the solid phase because the solid particles represent the dispersed phase in the fluid suspension, and therefore the solid particles can conduct heat between themselves only via the neighbouring fluid. When steady state is accomplished ∂Ts/∂t* = ∂Tf/∂t* = 0, leading to local thermal equilibrium between the solid and fluid phases, i.e. Ts(r) = Tf(r).
In a homogeneous medium without solid-suspended particles, Equation (1) is not relevant and the last term in Equation (3) can also be omitted. The boundary and initial conditions applicable are an initial ambient constant temperature, TC, within the whole domain, an ambient constant temperature, TC, at the outer radius of the container and a constant heat flux, q0, over the fluid-wire interface that is related to the Joule heating of the wire in the form q0 = iV/(πd w *l*), where d w * and l* are the diameter and the length of the wire respectively, i is the electric current and V is the voltage drop across the wire. Vadasz  showed that the problem formulated by Equations (1) and (3) subject to appropriate initial and boundary conditions represents a particular case of Dual-Phase-Lagging heat conduction (see also [24–28]).
Equation (5) is a very accurate way of estimating the thermal conductivity as long as the validity condition is fulfilled. The validity condition implies the application of Equation (5) for long times only. However, when evaluating this condition to data used in the nanofluid suspensions experiments, one obtains that t0* ~ 6 ms, and the time beyond which the solution (5) can be used reliably is therefore of the order of hundreds of milliseconds, not so long in the actual practical sense.
Two methods of solution
While the THW method is well established for homogeneous fluids, its applicability to two-phase systems such as fluid suspensions is still under development, and no reliable validity conditions for the latter exist so far (see Vadasz  for a discussion and initial study on the latter). As a result, one needs to refer to the two-equation model presented by Equations (1) and (3), instead of the one Fourier type equation that is applicable to homogeneous media.
producing the second boundary condition for the solid phase, which is identical to the corresponding boundary condition for the fluid phase. One may therefore conclude that the solution to the problem formulated in terms of Equation (7) that is identical to both phases, subject to initial conditions (9) and (12) that are identical to both phases, and boundary conditions (10), (11), and (13), (21) that are also identical to both phases, should be also identical to both phases, i.e. Ts (t*,r*) = Tf (t*,r*). This, however, may not happen because then Tf - Ts = 0 leads to conflicting results when substituted into (1) and (3). The result obtained here is identical to Vadasz  who demonstrated that a paradox revealed by Vadasz  can be avoided only by refraining from using this method of solution. While the paradox is revealed in the corresponding problem of a porous medium subject to a combination of Dirichlet and insulation boundary conditions, the latter may be applicable to fluids suspensions by setting the effective thermal conductivity of the solid phase to be zero. The fact that in the present case the boundary conditions differ, i.e. a constant heat flux is applied on one of the boundaries (such a boundary condition would have eliminated the paradox in porous media), does not eliminate the paradox in fluid suspensions mainly because in the latter case the steady-state solution is identical for both phases. In the porous media problem, the constant heat flux boundary condition leads to different solutions at steady state, and therefore the solutions for each phase even during the transient conditions differ.
The elimination method yields the same identical equation with identical boundary and initial conditions for both phases apparently leading to the wrong conclusion that the temperature of both phases should therefore be the same. A closer inspection shows that the discontinuity occurring on the boundaries' temperatures at t = 0, when a "ramp-type" of boundary condition is used, is the reason behind the occurring problem and the apparent paradox. The question that still remains is which phase temperature corresponds to the solution presented by Vadasz ; the fluid or the solid phase temperature?
and where the separation constant represents the eigenvalues in space.
Comparing the solutions obtained above with the solution obtained by Vadasz  via the elimination method, one may conclude that the latter corresponds to the solid phase temperature θs.
Correction of the THW results
where the temperature difference [Tw(t) - TC] is represented by the recorded experimental data, and the value of the heat flux at the fluid-platinum-wire interface q0 is evaluated from the Joule heating of the hot wire. In Equation (72) , where the coefficient C n is defined by (70) and the eigenvalues κ n are defined by Equation (44). Note that the definition of C n here is different than in . The results obtained from the application of Equation (72) fit extremely well the approximation used by the THW method via Equation (5) within the validity limits of the approximation (5). Therefore, the THW method is extremely accurate for homogeneous materials.
where kf,act is Maxwell's effective thermal conductivity, is the ratio between the thermal conductivity of the solid phase and the thermal conductivity of the base fluid, and ε is the volumetric solid fraction of the suspension. Then, these results of can be compared with the experimental results presented by Liu et al. .
Results and discussion
Liu et al.  used a very similar THW experimental method as the one used by Eastman et al. , Lee et al.  and Choi et al.  with the major distinction being in the method of producing the nanoparticles and a cylindrical container of different dimensions. They used water as the base fluid and Cu nanoparticles as the suspended elements at volumetric solid fractions of 0.1 and 0.2%. Their data that are relevant to the present discussion were digitized from their Figure 3  and used in the following presentation to compare our theoretical results. Three specimen data are presented in Figure 3  resulting in extensive overlap of the various curves, and therefore in some digitizing error which is difficult to estimate when using only this figure to capture the data.
It should be mentioned that Liu et al.  explain their time-dependent effective thermal conductivity by claiming that it was caused by nanoparticle agglomeration, a conclusion that is consistent with the theoretical results of this study.
The theoretical results derived in this article combined with experimental data  lead to the conclusion that, while there is no improvement in the effective thermal conductivity of nanofluids beyond the Maxwell's effective medium theory , there is nevertheless the possibility of substantial heat transfer augmentation via nanofins. Nanoparticles attaching to the hot wire by a mechanism that could be related to electrophoresis depending on the strength of the electrical current passing through the wire suggests that such attachments can be deliberately designed and produced on any heat transfer surface to yield an agglomeration of nanofins that exchange heat effectively because of the extremely high heat transfer area as well as the flexibility of such nanofins to bend in the fluid's direction when fluid motion is present, hence extending its applicability to include a new, and what appears to be a very effective, type of heat convection. A quantitative estimate of the effectiveness of nanofins requires, however, an extension of the model presented in this article to include heat conduction within the nanofins.
representative elementary volume
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