Abstract
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.
Introduction
The impressive heat transfer enhancement revealed experimentally in nanofluid suspensions by Eastman et al. [1], Lee et al. [2], and Choi et al. [3] conflicts apparently with Maxwell's [4] classical theory of estimating the effective thermal conductivity of suspensions, including higherorder corrections and other than spherical particle geometries developed by Hamilton and Crosser [5], Jeffrey [6], Davis [7], Lu and Lin [8], 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. [11] and Li and Peterson [12], confirming at least partially the results presented by Eastman et al. [1], Lee et al. [2], and Choi et al. [3], while others, such as Buongiorno and Venerus [13], Buongiorno et al. [14], show in contrast results that are in agreement with Maxwell's [4] effective medium theory. All these experiments were performed using the TransientHotWire (THW) experimental method. On the other hand, most experimental results that used optical methods, such as the "optical beam deflection" [15], "alloptical thermal lensing method" [16], and "forced Rayleigh scattering" [17] 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 [18] and Prasher et al. [19] 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. [20] showed that hyperbolic heat conduction also provides a viable explanation for the latter, although their further research and comparison with laterpublished experimental data presented by Vadasz and Govender [21] led them to discard this possibility.
Vadasz [22] derived theoretically a model for the heat conduction mechanisms of nanofluid suspensions including the effect of the surface areatovolume 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 nonhomogeneous systems. This timedependent 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. [23] reveal three important new results that allow the comparison of Vadasz's [22] theoretical model with experiments. The first important new result presented by Liu et al. [23] 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 timedependent "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 [4] 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 [23] and validates Maxwell's [4] method of estimating the effective thermal conductivity of suspensions. The theoretical results derived in this article are combined with experimental data [23] to conclude that, while there is no improvement in the effective thermal conductivity of nanofluids beyond the Maxwell's effective medium theory [4], 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 [4] effective medium theory, the experiments using optical methods, such as Putnam et al. [15], Rusconi et al. [16] and Venerus et al. [17], 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, k_{f} is the effective fluid phase thermal conductivity, a dimensional parameter that appears without an asterisk subscript without causing ambiguity.
Problem formulation
The theoretical model derived by Vadasz [22] to investigate the transient heat conduction in a fluid containing suspended solid particles by considering phaseaveraged equations will be presented only briefly without including the details that can be obtained from [22]. The phaseaveraged equations are
where t_{*} is time, T_{f} (r_{*},t_{*}), and T_{s} (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), γ_{s} = ερ_{s}c_{s} and γ_{f} = (1  ε)ρ_{f}c_{p} represent the effective heat capacity of the solid and fluid phases, respectively; with ρ_{s} and ρ_{f} are the densities of the solid and fluid phases, respectively; c_{s} and c_{p} are the specific heats of the solid and fluid phases, respectively; and ε is the volumetric solid fraction of the suspension. Similarly, k_{f} is the effective thermal conductivity of the fluid that may be defined in the form {k}_{\text{f}}=f(\epsilon ,\kappa ){\stackrel{~}{k}}_{\text{f}}, where {\stackrel{~}{k}}_{\text{f}} is the thermal conductivity of the fluid, \kappa ={\stackrel{~}{k}}_{\text{s}}/{\stackrel{~}{k}}_{\text{f}} 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 solidfluid interface as a volumetric heat source/sink within an REV. It is assumed to be independent of time, and its general relationship to the surfaceareatovolume ratio (specific area) was derived in [22]. Note that T_{s}(r_{*},t_{*}) is a function of the space variables represented by the position vector {\mathit{r}}_{*}={x}_{*}{\stackrel{\wedge}{\mathit{e}}}_{x}+{y}_{*}{\stackrel{\wedge}{\mathit{e}}}_{y}+{z}_{*}{\stackrel{\wedge}{\mathit{e}}}_{z}, in addition to its dependence on time, because T_{s}(r_{*},t_{*}) depends on T_{f}(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 ∂T_{s}/∂t_{*} = ∂T_{f}/∂t_{*} = 0, leading to local thermal equilibrium between the solid and fluid phases, i.e. T_{s}(r) = T_{f}(r).
For the case of a thin hot wire embedded in a cylindrical container insulated on its top and bottom one can assume that the heat is transferred in the radial direction only, r_{*}, rendering Equation (2) into
In a homogeneous medium without solidsuspended 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, T_{C}, within the whole domain, an ambient constant temperature, T_{C}, at the outer radius of the container and a constant heat flux, q_{0}, over the fluidwire interface that is related to the Joule heating of the wire in the form q_{0} = 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 [22] showed that the problem formulated by Equations (1) and (3) subject to appropriate initial and boundary conditions represents a particular case of DualPhaseLagging heat conduction (see also [24–28]).
An essential component in the application of the THW method for estimating experimentally the effective thermal conductivity of the nanofluid suspension is the assumption that the nanofluid suspension behaves basically like a homogeneous material following Fourier law for the bulk. The THW method is well established as the most accurate, reliable and robust technique [29] for evaluating the thermal conductivity of fluids. A very thin (580 μm in diameter) platinum (alternatively tantalum) wire is embedded vertically in the selected fluid and serves as a heat source as well as a thermometer (see [22] for details). Because of the very small diameter and high thermal conductivity of the platinum wire, it can be regarded as a line heat source in an otherwise infinite cylindrical medium. The rate of heat generated per unit length (l_{*}) of platinum wire due to a step change in voltage is therefore {\dot{q}}_{l*}=iV/{l}_{*} W m^{1}. Solving for the radial heat conduction due to this line heat source leads to an approximated temperature solution in the wire's neighbourhood in the form
provided a validity condition for the approximation is enforced, i.e. {t}_{*}>>{t}_{0*}={r}_{\text{w}*}^{2}/4\alpha, where r_{w*} is the radius of the platinum wire, \alpha ={\stackrel{~}{k}}_{\text{f}}/{\rho}_{\text{f}}{c}_{\text{p}} is the fluid's thermal diffusivity, and γ_{0} = 0.5772156649 is Euler's constant. Equation (4) reveals a linear relationship, on a logarithmic time scale, between the temperature and time. Therefore, one way of evaluating the thermal conductivity is from the slope of this relationship evaluated at r_{*} = r_{w*}. For any two readings of temperature, T_{1} and T_{2}, recorded at times t_{1*} and t_{2*} respectively, the thermal conductivity can be approximated using Equation (4) in the form:
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 t_{0*} ~ 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 twophase systems such as fluid suspensions is still under development, and no reliable validity conditions for the latter exist so far (see Vadasz [30] for a discussion and initial study on the latter). As a result, one needs to refer to the twoequation model presented by Equations (1) and (3), instead of the one Fourier type equation that is applicable to homogeneous media.
Two methods of solution are in principle available to solve the system of Equations (1) and (3). The first is the elimination method while the second is the eigenvectors method. By means of the elimination method, one may eliminate T_{f} from Equation (1) in the form:
and substitute it into Equation (3) hence rendering the two Equations (1) and (3), each of which depends on both T_{s} and T_{f}, into separate equations for T_{s} and T_{f}, respectively, in the form:
where the index i takes the values i = s for the solid phase and i = f for the fluid phase, and the following notation was used:
In Equation (8), τ_{q} and τ_{T} are the heat flux and temperaturerelated time lags linked to DualPhaseLagging [22, 24–27, 31], while α_{e} is the effective thermal diffusivity of the suspension. The resulting Equation (7) is identical for both fluid and solid phases. Vadasz [22] used this equation in providing the solution. The initial conditions applicable to the problem at hand are identical for both phases, i.e. both phases' temperatures are set to be equal to the ambient temperature T_{C}
The boundary conditions are
where r_{0*} is the radius of the cylindrical container. Equation (7) is secondorder in time and secondorder in space. The initial conditions (9) provide one such condition for each phase while the secondorder Equation (7) requires two such conditions. To obtain the additional initial conditions, one may use Equations (1) and (3) in combination with (9). From (9), it is evident that both phases' initial temperatures at t_{*} = 0 are identical and constant. Therefore, {\left({T}_{\text{f}}\right)}_{{t}_{*}=0}={\left({T}_{\text{s}}\right)}_{{t}_{*}=0}={T}_{\text{C}}=\text{constant}, leading to {\left({T}_{\text{f}}{T}_{\text{s}}\right)}_{{t}_{*}=0}=0 and {\left[\partial /\partial {r}_{*}\left({r}_{*}\partial {T}_{\text{f}}/\partial {r}_{*}\right)\right]}_{{t}_{*}=0}=0 to be substituted in (1) and (3), which in turn leads to the following additional initial conditions for each phase:
The two boundary conditions (10) and (11) are sufficient to uniquely define the problem for the fluid phase; however, there are no boundary conditions set for the solid phase as the original Equation (1) for the solid phase had no spatial derivatives and did not require boundary conditions. To obtain the corresponding boundary conditions for the solid phase, which are required for the solution of Equation (7) corresponding to i = s, one may use first the fact that at r_{*} = r_{0*} both phases are exposed to the ambient temperature and therefore one may set
Second, one may use Equation (6) and taking its derivative with respect to r_{*} yields
In Equation (14), the spatial variable r_{*} plays no active role; it may therefore be regarded as a parameter. As a result, one may present Equation (14) for any specified value of r_{*}. Choosing r_{*} = r_{w*} where the value of {\left(\partial {T}_{\text{f}}/\partial {r}_{*}\right)}_{{r}_{\text{w}*}} is known from the boundary condition (11), yields from (14) the following ordinary differential equation:
At steady state, Equation (15) produces the solution
where T_{s,st} is the steadystate solution. The transient solution T_{s,tr} = T_{s}  T_{s,st} satisfies then the equation:
subject to the initial condition
because {\left[\partial {T}_{\text{s}}/\partial {r}_{*}\right]}_{{t}_{*}=0}=0 for all values of {r}_{*}\in \left[{r}_{\text{w}*},{r}_{\text{0}*}\right] given that according to (9) at t_{*} = 0: {\left(\phantom{\rule{0.5em}{0ex}}{T}_{\text{s}}\right)}_{{t}_{*}=0}={\left({T}_{\text{f}}\right)}_{{t}_{*}=0}={T}_{\text{C}}=\text{constant}. Equation (17) can be integrated to yield
which combined with the initial condition (18) produces the value of the integration constant A = 0 and therefore the transient solution becomes
The complete solution for the solid temperature gradient at the wire is therefore obtained by combining (20) with (16) leading to
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. T_{s} (t_{*},r_{*}) = T_{f} (t_{*},r_{*}). This, however, may not happen because then T_{f}  T_{s} = 0 leads to conflicting results when substituted into (1) and (3). The result obtained here is identical to Vadasz [32] who demonstrated that a paradox revealed by Vadasz [33] 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 steadystate 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 "ramptype" 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 [22]; the fluid or the solid phase temperature?
By applying the eigenvectors method as presented by Vadasz [32], one may avoid the paradoxical solution and obtain both phases temperatures. The analytical solution to the problem using the eigenvectors method is obtained following the transformation of the equations into a dimensionless form by introducing the following dimensionless variables:
where the following two dimensionless groups emerged:
representing a heat flux Fourier number and a temperature Fourier number, respectively. The ratio between them is identical to the ratio between the time lags, i.e.
Equations (1) and (3) expressed in a dimensionless form using the transformation listed above are
where the following additional dimensionless groups emerged:
where Ni_{f} is the fluid phase Nield number. The solutions to Equations (25) and (26) are subject to the following initial and boundary conditions obtained from (9), (10) and (11) transformed in a dimensionless form:
The boundary conditions are
No boundary conditions are required for θ_{s}. The solution to the system of Equations (25)(26) is obtained by a superposition of steady and transient solutions θ_{i,st}(r) and θ_{i,tr}(t,r), respectively, in the form:
Substituting (33) into (25)(26) yields to the following equations for the steady state:
leading to the following steady solutions which satisfy the boundary conditions (31) and (32):
The transient part of the solutions θ_{i,tr}(t,r) can be obtained by using separation of variables leading to the following form of the complete solution:
Substituting (37) into (25)(26) yields, due to the separation of variables, the following equation for the unknown functions R_{on} (r):
subject to the boundary conditions
and the following system of equations for the unknown functions S_{in} (t), (i = s,f), i.e.
where
and where the separation constant {\kappa}_{n}^{2} represents the eigenvalues in space.
Equation (38) is the Bessel equation of order 0 producing solutions in the form of Bessel functions
Where J_{0}(κ_{ n }r) and Y_{0}(κ_{ n }r) are the order 0 Bessel functions of the first and second kind, respectively. The solution (43) satisfies the boundary condition (39) as can easily be observed by substituting r = 1 in (43). Imposing the second boundary condition (40) yields a transcendental equation for the eigenvalues κ_{ n }in the form:
where J_{1}(κ_{ n }r_{w}) and Y_{1}(κ_{ n }r_{w}) are the order 1 Bessel functions of the first and second kind, respectively, evaluated at r = r_{w}. The compete solution is obtained by substituting (43) into (37) and imposing the initial conditions (30) in the form
At t = 0, both phases' temperatures are the same leading to the conclusion that
Multiplying (45) by the orthogonal eigenfunction R_{ om }(κ_{ m },r) with respect to the weight function r and integrating the result over the domain [r_{w},1], i.e. {\displaystyle {\int}_{{r}_{\text{w}}}^{1}(\u2022){R}_{om}({\kappa}_{m},r)}\phantom{\rule{0.5em}{0ex}}r\phantom{\rule{0.5em}{0ex}}dr yield
The integral on the righthand side of (47) produces the following result due to the orthogonality conditions for Bessel functions:
where the norm N(κ_{ n }) is evaluated in the form:
The integral on the lefthand side of (47) can be evaluated using integration by parts and the equation for the eigenvalues (44) to yield
Substituting (48) and (50) into (47) yields the values of S_{in} at t = 0, i.e. S_{ no }= S_{ sn }(0) = S_{fn}(0)
that need to be used as initial conditions for the solution of system (41)
to produce the explicit solutions in time. With the initial conditions for S_{in} evaluated (i = s,f), one may turn to solving system (41) that can be presented in the following vector form:
where the matrix A is explicitly defined by
with the values of a,c and d_{ n }given by Equation (42), and the vector S_{ n }defined in the form S_{ n }= [S_{ sn },S_{ fn }]^{T}. The eigenvalues λ_{ n }corresponding to (52) are obtained as the roots of the following quadratic algebraic equation:
leading to
which upon substituting a,c and d_{ n }from Equation (42) yields
The following useful relationship is obtained from (55) and (56):
The corresponding eigenvectors υ_{1}_{ n }and υ_{2}_{ n }are evaluated in the form:
leading to the following solution:
and explicitly following the substitution of (58) and the initial conditions S_{in} (i = s,f), at t = 0, i.e. S_{ sn }(0) = S_{fn}(0) = S_{ no }with the values of S_{ no }given by Equation (51)
Substituting (57) into (60) and (61) and the latter into the complete solution (37) yields
where B_{ n }is
Comparing the solutions obtained above with the solution obtained by Vadasz [22] via the elimination method, one may conclude that the latter corresponds to the solid phase temperature θ_{s}.
The Fourier solution is presented now to compare the solution obtained from the DualPhaseLagging model to the former. The Fourier solution is the result obtained by solving the thermal diffusion equation
subject to the boundary and initial conditions
where the same scaling as in Equation (22) was applied in transforming the equation into its dimensionless form, hence the reason for the coefficient 1/β in the equation. The Fourier solution for this problem has then the form [34]
where
and the eigenvalues κ_{ n }are the solution of the same transcendental Equation (44) and the eigenfunctions R_{on}(r) are also identical to the ones presented in Equation (43). The relationship between the Fourier coefficient C_{ n }and the DualPhaseLagging model's coefficient B_{ n }is
Correction of the THW results
When evaluating the thermal conductivity by applying the THW method and using Fourier law, one obtains for the effective thermal conductivity the following relationship [22]:
where the temperature difference [T_{w}(t)  T_{C}] is represented by the recorded experimental data, and the value of the heat flux at the fluidplatinumwire interface q_{0} is evaluated from the Joule heating of the hot wire. In Equation (72) f\left(t\right)={\displaystyle {\sum}_{n=1}^{\infty}{C}_{n}{R}_{\text{on}}\left({r}_{\text{w}}\right)\mathrm{exp}\left({\kappa}_{n}^{2}t\right)}, 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 [22]. 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.
On the other hand, for nonhomogeneous materials, by means of the solutions (62) and (63) applicable to fluid suspensions evaluated at r = r_{w}, one obtains
where k_{f,act} is the actual effective thermal conductivity, T_{sw} (t) and T_{fw} (t) are the solid and fluid phases temperatures "felt" by the wire at the points of contact with each phase, respectively, and the functions g_{s} (t) and g_{f} (t) obtained from the solutions (62) and (63) evaluated at r = r_{w} take the form
When the wire is exposed partly to the fluid phase and partly to the solid phase, there is no justification in assuming that the wire temperature is uniform: on the contrary the wire temperature will vary between the regions exposed to the fluid and solid phases. Assuming that some solid nanoparticles are in contact with the wire in a way that they form approximately "solid rings" around the wire, then the "effective" wire temperature can be evaluated as electrical resistances in series. By defining the relative wire area covered by the solid nanoparticles as a_{s} = A_{s}/A_{tot} = A_{s}/2πr_{w*}l_{*} its corresponding wire area covered by the fluid is a_{f} = A_{f}/A_{tot} = 1  a_{s}, then from the relationship between the electrical resistance and temperature accounting for electrical resistances connected in series, one obtains an expression for the effective wire temperature (i.e. the temperature that is evaluated using the wire's lumped electrical resistance in the THW Wheatstone bridge) T_{w} in the form:
Substituting (73) and (74) into (77) yields
One may then use (78) to evaluate the actual nanofluid's effective thermal conductivity k_{f,act} from (78) in the form
When using the single phase Fourier solution (72) applicable for homogeneous materials to evaluate the effective thermal conductivity of nonhomogeneous materials like nanofluid suspensions instead of using Equation (79), one obtains a value that differs from the actual one by a factor of
where k_{f,app} is the apparent effective thermal conductivity obtained from the single phase Fourier conduction solution while k_{f,act} is the actual effective thermal conductivity that corresponds to data that follow a DualPhaseLagging conduction according to the derivations presented above. The ratio between the two provides a correction factor for the deviation of the apparent effective thermal conductivity from the actual one. This correction factor when multiplied by the ratio {k}_{\text{f,act}}/{\stackrel{~}{k}}_{\text{f}} produces the results for \sigma ({k}_{\text{f,act}}/{\stackrel{~}{k}}_{\text{f}})={k}_{\text{f,app}}/{\stackrel{~}{k}}_{\text{f}}, where {\stackrel{~}{k}}_{\text{f}} is the thermal conductivity of the base fluid without the suspended particles, and k_{f,act} is the effective thermal conductivity evaluated using Maxwell's [4] theory, which for spherical particles can be expressed in the form:
where k_{f,act} is Maxwell's effective thermal conductivity, \kappa ={\stackrel{~}{k}}_{\text{s}}/{\stackrel{~}{k}}_{\text{f}} 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 {k}_{\text{f,app}}/{\stackrel{~}{k}}_{\text{f}} can be compared with the experimental results presented by Liu et al. [23].
Results and discussion
The results for the solid and fluid phases' temperature at r = r_{w} as a function of time obtained from the solutions (62) and (63) are presented in Figures 1, 2 and 3 in comparison with the singlephase Fourier solution (69) for three different combinations of values of Fo_{ q }and a_{s}, and plotted on a logarithmic time scale. While the quantitative results differ amongst the three figures, there are some similar qualitative features that are important to mention. First, it is evident from these figures that the fluid phase temperature is almost the same as the temperature obtained from the singlephase Fourier solution. Second, it is also evident that the solid phase temperature lags behind the fluid phase temperature by a substantial difference. They become closer as steadystate conditions approach. It is therefore imperative to conclude that the only way, an excessively higher effective thermal conductivity of the nanofluid suspension as obtained by Eastman et al. [1], Lee et al. [2] and Choi et al. [3] could have been obtained even in an apparent form, is if the wire was excessively exposed to the solid phase temperature. The latter could have occurred if the electric current passing through the wire created electric fields that activated a possible mechanism of electrophoresis that attracted the suspended nanoparticles towards the wire. Note that such a mechanism does not cause agglomeration in the usual sense of the word, because as soon as the electric field ceases, the agglomeration does not have to persist and the particles can move freely from the wire's surface. Therefore, testing the wire's surface after such an experiment for evidence of agglomeration on the wire's surface may not necessarily produce the required evidence for the latter.
Liu et al. [23] used a very similar THW experimental method as the one used by Eastman et al. [1], Lee et al. [2] and Choi et al. [3] 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 [23] and used in the following presentation to compare our theoretical results. Three specimen data are presented in Figure 3 [23] 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.
The comparison between the theoretical results presented in this article with the experimental data [23] is presented in Figures 4, 5 and 6. The separation of these results into three different figures aims to better distinguish between the different curves and avoid overlapping as well as presenting the results on their appropriate scales. Figure 4 presents the results that are applicable to specimen No. 4 in Liu et al. [23] and corresponding to values of Fo_{ q }= 1.45 × 10^{2} and a_{s} = 0.45 in the theoretical model. Evaluating Maxwell's [4] effective thermal conductivity for specimen No. 4 leads to a value of 0.6018 W/mK, which is higher by 0.3% than that of the base fluid (water), i.e. {k}_{\text{f,act}}/{\stackrel{~}{k}}_{\text{f}}=1.003. From the figure, it is evident that the theoretical results match very well with the digitized experimental data. Furthermore, the steadystate result for the ratio between the effective thermal conductivity and that of the base fluid was estimated from the digitized data to be {k}_{\text{f,act}}/{\stackrel{~}{k}}_{\text{f}}=1.003\pm 0.001 clearly validating Maxwell's [4] predicted value. The results applicable to specimen No. 5 in Liu et al. [23] and corresponding to values of Fo_{ q }= 1.1 × 10^{2} and a_{ s }= 0.55 in the theoretical model are presented in Figure 5. The very good match between the theory and the digitized experimental data is again evident. In addition, the ratio between the effective thermal conductivity and that of the base fluid was estimated from the digitized data to be {k}_{\text{f,act}}/{\stackrel{~}{k}}_{\text{f}}=1.004\pm 0.001 again validating Maxwell's [4] predicted value of {k}_{\text{f,act}}/{\stackrel{~}{k}}_{\text{f}}=1.003. The last result is presented in Figure 6, which corresponds to specimen No. 9 in Liu et al. [23] and to values of Fo_{ q }= 6 × 10^{3} and a_{ s }= 0.35 in the theoretical model. The results are presented on an appropriately scaled vertical axis and show again a very good match between the theory presented in this article, and the experimental data as digitized from Liu et al. [23]. Since the volumetric solid fraction for this specimen was 0.2%, its corresponding Maxwell's [4] effective thermal conductivity for this specimen leads to a value of 0.6036 W/mK, which is higher by 0.6% than that of the base fluid (water), i.e. {k}_{\text{f,act}}/{\stackrel{~}{k}}_{\text{f}}=1.006. The steadystate result for the ratio between the effective thermal conductivity and that of the base fluid was estimated from the digitized data to be {k}_{\text{f,act}}/{\stackrel{~}{k}}_{\text{f}}=1.0059\pm 0.002 validating again Maxwell's [4] predicted value.
It should be mentioned that Liu et al. [23] explain their timedependent effective thermal conductivity by claiming that it was caused by nanoparticle agglomeration, a conclusion that is consistent with the theoretical results of this study.
Conclusions
The theoretical results derived in this article combined with experimental data [23] lead to the conclusion that, while there is no improvement in the effective thermal conductivity of nanofluids beyond the Maxwell's effective medium theory [4], 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.
Abbreviations
 REV:

representative elementary volume
 THW:

transienthotwire.
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Vadasz, P. Heat transfer augmentation in nanofluids via nanofins. Nanoscale Res Lett 6, 154 (2011). https://doi.org/10.1186/1556276X6154
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DOI: https://doi.org/10.1186/1556276X6154
Keywords
 Effective Thermal Conductivity
 Base Fluid
 Constant Heat Flux
 Effective Medium Theory
 Transient Heat Conduction