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Semi-Analytical Correlation for Pool Boiling Heat Transfer Using Nanofluids Based on Fractal Theory

On the basis of fractal dispersion of active nucleation sites on boiling surface, a revised fractal model for nucleate pool boiling of nanofluids was established incorporating microlayer evaporation, nature convection and transient conduction. It has fewer empirical constants than that of traditional ones. Results of model using pure fluids and nanofluids conform to data from previous investigations. Thus, accuracy of model is verified. Effects of liquid properties and contact angle on pool boiling heat transfer coefficient (HTC) using nanofluids are discussed according to the present model. It is found that the HTC of nanofluid and the fractal dimension of nucleation sites decrease with the decreasing contact angle. The effect of liquid properties enhancement due to the presence of nanoparticles, has ignorable influence on HTC of dilute nanofluids while the decreased contact angle plays a more important role. It is also found that the surface with lower contact angle had a higher natural convection component.


Introduction
Boiling heat transfer is broadly applicated in nuclear reactors, electricity generating, and electronic component cooling. But the low heat conductivity of traditional thermal transfer fluids, which primarily to limit the HTC and thermal performance of above-mentioned systems. Conventional method such as fins has reached its limit. Thus, to explore newly developed techniques to improve the thermal behavior of traditional heat transfer fluids, great endeavor has been made.
Conflicting experimental results in table 1 indicated either an enhancement [3,4,5,6], or deterioration [7][8][9][10][11][12][13][14], or in some cases, no effect [2,9,15] on the HTC due to the presence of nanoparticles. Boiling of nanofluid refers to changes in liquid thermophysical performance as well as surface conditions in comparison with pure liquid [16]. Nanoparticle deposition on boiling surface that changes the contact angle during microlayer evaporation period, is thought to be the major reason for the changes of HTC [10]. Table 1. Summary of some investigations on nanofluid HTC.
To date, comparing with pure liquids, models for heat flux and nucleation site density of nanofluid boiling are very fewer. Rohsenow correlation [17] is widely used to represent the pool boiling since it was proposed in 1952, as shown Csf represents surface constant which depends on combination of surface and liquid performance, and s and R are the constants which depend on the combination of liquid surfaces. Taylor and Phelan [18] indicated nearly all experiment results of nanofluids boiling could be consistent with (1) while Csf changed between 0.0065 and 0.018. These phenomena indicated that since Csf is just related to surface status, surface modification due to nanoparticle deposition is the main factor determining the HTC of nanofluids.
Nevertheless, this does not mean that Eq. (1) can be easily used to calculate the pool boiling HTC of nanofluids because Csf must be determined through experimental. Rohsenow correlation is useful from the application point of view. Meanwhile, the reason for selecting the value of Csf is not given in the Rohsenow correlation, and Csf has a strong impact on the calculated results.
In theory, according to Bankoff's [19] bubble nucleation mechanism, the correlation formula proposed by Yang and Kim's [20] is considered to be a useful mechanism method to determine the density of active nucleation center.   (2) in which N represents density of cavity number on heating surface, f (β) and f (Dc) represent the Following similar approach to Ganapathy and Sajith [22], model for active site density was obtained by Li and Liu [21].
Unfortunately, the application scope of equation (3) is restricted by experimental conditions. There are few empirical constants with no clear physical meanings contained in each model.
As can be seen in the above-mentioned, the mechanism of nanofluid boiling still remains unclear. Thus, further theoretical research on pool boiling HTC using nanofluid has important value in both theory and practice. In present work, a revised fractal model incorporating the main thermal transfer theory in boiling process and fractal dispersion of nucleation sites on heating surfaces is obtained. The description of nucleation site density based on the fractal dispersion of nucleation sites provides a mean to investigate influence of surface properties on HTC for pool boiling of nanofluids. Contact angle is used as a parameter to define surface characteristics in present work.

Effective Properties of Nanofluid
Heat conductivity of nanofluids is obtained with equation as follows [23]: According to the mixture rule, the density of nanofluid is expressed as [24]: Likewise, specific heat and heat expansion coefficient of nanofluid can be expressed [24] as: The dynamic viscosity can be expressed as [24]:

Fractal Analysis of Active Sites on Boiling Surface
In the pool boiling process, carefully characterize the distribution of active sites is crucial. Because of the complex dispersion of active sites, actual values can be hard to be obtained by traditional method. Fractal theory provide an important means to study the irregular and complicated phenomenon or objects in the nature, and has been widely employed to predict the transport properties and the thermal properties of porous medium. Similar to pores in fractal porous media, Yu et al. [25] founded that dimension dispersion of nucleation center on heating surface obey fractal power law. The number of active cavities with size higher than or equal to Dc can be expressed as in which df is the fractal dimension, Dc,max and Dc,min represents the highest and lowest diameters of nucleation sites, respectively. By the differential forms of (10), the number of active sites with sizes of infinitesimal interval from Dc to Dc+dDc can be obtained as The total number of active sites with diameter ranging from Dc,min to Dc,max is obtained as Chu and Yu [26] proposed the expression for fractal size df of nucleation sites in pool boiling and is given as where c,max D represents mean value over all maximal active sites: Hsu et al. [27] suggested that the highest and lowest diameters of active cavity can be predicted by where hnc is the average HTC of natural convection which can be calculated as [ (2) Heat consumed by regeneration of heat boundary layer after the bubble leaves (qtc); (3) natural convection of base fluid in the region of the surface uninfluenced by bubble cycle (qnc); Considering the period of bubble grow and departure, the total heat flow is obtained by time average expression me g tc w tot n g w c q τ q τ q q τ τ where τg represents time for bubble growth, and τw represents waiting time for bubble regeneration.
The total heat flux of bubble growth for all nucleation sites within ranging from Dc,min to Dc,max is given as Equation (23) where K is the influence area factor, which is identified as ratio between the area of bubble influence to the projected area of bubble at departure time, which is suggested to be 4 by various researchers [18,24]. Thus, the heat flux during the bubble during the waiting time is given as Coupling the internal influence of bubble growth and departure, the growth time of bubble is obtained as Van Stralen et al. [30] suggested that the waiting time is triple of growing time, as shown where Anc is the area which is not affected by the bubble cycle per unit heater surface. According to the method of Benjamin and Balakrishnan [28], the area not influenced by the bubbles for flat plate surfaces should be written as Yu et al. [25] obtained the volume (area) fraction of nucleation sites, given as Equation (36) indicates that the total heat flux is a function of thermal physical properties of nanofluid, wall superheat, fractal dimension, the total number of nucleation sites, the smallest and largest sizes of active nucleation sites and contact angle.

Results and Discussion
A commonly feature in the boiling of nanofluids is nanoparticle deposition. To verify this present model, calculated results are in comparison with experimental results from different literatures [8,10,31,32]. Details of experimental condition can be seen from table 2. Table 2. Details of experiment conditions.
It is believed that the higher HTC of nanofluids is induced by the enhancement in thermophysical properties of liquid, by adding nanoparticle [1,2,7]. However, currently these hypotheses have been questioned and confuted for that the changes in the transport and thermal properties of nanofluid had nearly no effect on the boiling curve, especially for dilute nanofluid (<0.1vol %) [21]. To investigate influence of liquid properties on boiling thermal transfer for nanofluid, boiling curves of Al2O3-water nanofluids with varied volume concentration (0%, 0.02 vol%, 0.05 vol%, 0.1 vol.%, 0.5 vol.%, and 1 vol%) which obtained from (40) and experimental data of Shahmoradi et al. [12] are plotted in figure 3. For convenience, assuming the contact angles of all particle concentration are equal to 45° under the condition that only consider the liquid properties.
As can be seen from figure 3 that the predicted HTC has nearly no significant enhancement when particle concentration increasing from 0% (pure water) to 0.1 vol. %. However, predication result by considering two factors of thermal physical performance and contact angle, still conforms well with results of experiment. This comparison in figure 3 shows that thermophysical performance of nanofluids sole, is still not enough to increase dilute nanofluid HTC. Therefore, it is evidence that the decrease of active nucleation is more obvious than properties changes, although the thermal conductivity increases by adding nanoparticles, thus the heat flux still decreases.    Figure 4 depicts that fractal size of active nucleation sites rises with superheat increasing, while reduces as the contact angle going down from 80° to 30°. According to equations (16)- (20), as the contact angle decreases, more cavities on boiling surface are flooded by liquids. Thus, the decrease in nucleation sites results in decreasing of the fractal dimension of active nucleation sites. It is indicated that df is a function of the wall overheat and contact angle, which is proportion to nucleation sites density and HTC.
It is shown in figure 5 that the relative contribution of transient conduction flux, microlayer evaporation flux and natural convection flux to the total heat flux. The microlayer evaporation component increases with wall superheat increasing and account for 77.66-79.39% as heat flux closes to CHF. From figure 5, we can see that component of natural convection, which dominates the total heat flux at lower superheats, gradually decreases with increasing wall superheat and reach 9.4-11.38% when approaching the CHF. Nonetheless, the natural convection term should not be ignored even near CHF. It is important to note that surface with smaller contact angle has a higher natural convection component. This can be explained that the lower contact angle, fewer active nucleation sites present on boiling surfaces due to bubble cycle phenomena; thus, a bigger size of surface can be used for natural convection. Similar trend was found by Ganapathy and Sajith [22].

Conclusion
A fractal model to predict the pool boiling HTC using nanofluids is established according to fractal theory as well as heat partition method in this paper. Model data conforms well with the available experiment results. Conclusions in this study are listed as follows: (1) HTC of nanofluid decreases with the decreasing contact angle. For dilute nanofluids, Thermal conductivity and other thermal properties are heightened even caused by presence of nanoparticles, boiling performance of nanofluids was more dominantly influenced by the reduction of active nucleation sites that induced by nanoparticle deposition.
(2) Fractal size of active nucleation sites are associated with wall overheating and contact angle. It is also proportion to nucleation sites density and wall superheat while reduces as the contact angle reduces.
(3) Thermal transfer contribution caused by microlayer evaporation grows with wall overheat. Transient conduction component reached a peak, then slightly went down. Component of natural convection gradually decreases with increasing wall superheat and reach 9.41-11.38% at high heat flux.