Calculation of Active Nucleation Site Density in Boiling Systems

In this study, realizing the significant of active nucleation site as an important parameter for predicting heat transfer, the formula of calculating active nucleation site density is given by taking into account the statistical property of active nucleation site on boiling surfaces. In the present approach, the proposed model is explicitly related to the contact angle and wall superheat. It is found that the total number of nucleation sites increases with increasing of the wall superheat. This model contains less empirical constants than the conventional correlations. No additional empirical constant is introduced. An excellent agreement between the proposed model predictions and experimental data is found. The validity of the present model for active nucleation site density is thus verified. The proposed mathematical model can reveal the physical mechanisms of heat transfer in boiling systems.


INTRODUCTION
The physical mechanisms of heat transfer in boiling systems have hotly been debated in the past decades and the physical nature is still far from being well understood because of its complexity and diversity (Basu et al., 2002;Xiao et al., 2009Xiao et al., , 2010;;Jiang et al., 2012).The active nucleation site density is important for predicting the heat transfer in boiling systems.The density of active sites on heater surfaces is affected by the interaction of several parameters such as heater and liquid sides, distributions of cavities on heater surfaces and liquid-solid contact angles.At the same time, the transport properties of the heater affect the thermal interaction among the cavities, causing activation and deactivation of individual cavities.It has been shown that the density or number of active nucleate sites (N a ) on heated surfaces has the great effect on boiling heat transfer.Several studies have been performed on N a , which give the functional dependence of N a on heat flux (q) and wall superheat (∆T w ).However, from a mechanistic viewpoint, although the influences of some parameters such as the heat flux and wall superheat have been discussed in literature (Xiao et al., 2010), an overall mechanistic description is still unavailable.The calculation of active nucleate sites was one of the most controversial subjects for heat transfer of pool boiling in the past.It is a challenge to predict the number of active nucleate sites analytically due to the extremely complicated mechanisms of heat transfer as well as the interrelationship between active nucleation site density and the heat flux and wall superheat.So, the exact mechanism of heat transfer is still unclear in boiling systems.This study focuses on the heat flux nucleate boiling region and attempts to develop a mathematical model for active nucleate sites.

SEVERAL CORRELATION FOR PREDICTING ACTIVE NUCLEATION SITE DENSITY
In what follows, some recent correlations for predicting the active nucleation site density are briefly summarized.Here some notable studies are discussed in this field.Gaertner and Westwater (1960) employed a novel technique in which nickel salts were dissolved in water and the copper-surface acted as one of the electrodes.By counting the numbers of holes in the deposited layer, they found the functional dependence of active nucleation site density on wall heat flux to be: (1) Mikic and Rohsenow (1969) might be the first to relate the active nucleation site density to the sizes of cavities present on the heated surfaces and expressed the functional dependence of active nucleation site density on cavity for commercial surfaces as: where, D c, max = The diameter of the largest cavity present on surfaces m = An empirical constant (= 6.5) D c = Given by: Bier et al. (1978), on the other hand, expressed N a as a functional of cavity size from heat transfer data.The expression is given by: where, N max = The value corresponding to D c = 0 The value of the exponent m was found to depend on the surface preparation procedure.Cornwell and Brown (1978) made a systematic study on active nucleation site density of water boiling at 1.013 bars on a copper surface, with surface condition varying from smooth to rough and related the dependence of active site density on wall superheat as: They justified their observed functional dependence on wall superheat by assuming only conical cavities existed on surfaces and that vapor needed to be trapped in cavities before any nucleation could occur.They also related the cavity size to the total number of cavities presenting on the surface from the cavity size data obtained using an electron microscope and N a, tot is expressed as: Kocamustafaogullari and Ishii (1983) correlated existing active nucleation site density data by means of parameter study, developed a relation for active nucleation site density in pool boiling.They assumed that the active nucleation site density in pool boiling,   * , was influenced by both the surface conditions and the thermo-physical properties of the liquid.Their correlation expressed active nucleation site density as a function of dimensionless minimum cavity size and density ratio.The correlation for system pressures from 1.0 to 198.0 bars is: Paul and Abdel-Khalik (1983) conducted their experiments on the pool boiling of saturated water at 1 atm along an electrically heated horizontal platinum wire.Using high-speed photography, they measured active nucleation site density and bubble departure diameter up to 70%.They found that the active nucleation site density of N a can be represented by the linear relationship with the boiling heat flux as follows: 3 1.207 10 15.74 Yang and Kim (1988) made the first attempt to quantitatively predict the active nucleation sites from knowledge of the size and cone angle distribution of cavities that are actually present on the surface.Using a scanning electron microscope and a differential inference contrast microscope, they established the dependence of the nucleation site density on the characteristic of a boiling surface with the aid of statistical analysis approach.The size distribution was found to fit a Poisson distribution while a normal distribution was used for cone half angle β.They used Bankoff (1958) criteria to determine which cavities will trap gas.This condition is given by φ>2β.By combining the probability distribution functions and this criterion, they related N a to the average  � on the surface as: where,  ̅ = The mean value of cone half angle λ, s = Statistical parameters These parameters are dependent upon the surface preparation procedure and the material of surface.Wang and Dhir (1993) might be the first to perform a systematic study of the effect of contact angle on the density of active nucleation sites.The correlated cavity size D c was related to the wall superheat for nucleation as given by Eq. (2b).It was found that there was a strong influence of wettability on active nucleation site density.For surfaces with 18 deg ≤ φ ≤ 90 deg, they correlated N a with D c as: Benjamin and Balakrishnan (1997) conducted the experiment on the nucleation site density in nucleate pool boiling of saturated pure liquids at low-tomoderate heat flux.During the boiling phenomena they examined the surface-liquid interaction and its effects on the nucleation site density.They utilized the arithmetic average roughness, r a , defined as the average values of the peaks and valleys to characterize the surface micro-roughness on the surface.So they found that the nucleation site density depended on the surface micro-roughness, the surface tension of the liquid, thermo-physical properties of the heating surface and the wall superheat.The expression is given by: 1.63 0.4 3 1 218.8Pr where, Pr j = The Prandtl number of fluid defined by: Pr l l l ν α = η = The surface-liquid interaction parameter defined by: C pw , ρ w and k w = The wall specific heat at constant pressure The wall density and the wall thermal conductivity, respectively.The dimensionless surface roughness parameter, Θ, is given by: 0.4 14.5 4.5 a a r P r P The range of parameters covered in developing and validating the correlation were 1.7< Pr l <5, 4.7<η<93, 0.02 mm<r a <1.17mm, 13×10 -3 N/m < σ <59×10 -3 N/m, 5K< ΔT w <25K, 2.2< Θ<14.Basu et al. (2002) performed subcooled boiling experiment at 1 atm and proposed an empirical correlation including the effect of contact angle on the active nucleation site density during forced convective boiling of water on a vertical surface.They changed the wettability of the surface by controlling the degree of oxidation of the surface in the experiment.The correlation is given by: 2.0 0.34(1 cos ) where, ΔT ONB = The wall superheat at the onset of the nucleate boiling It is clear that so far the available models for N a and q are usually correlated with several empirical constants which have no physical meanings and the mechanisms behind these constants are still not clear until now.Therefore, a complete mechanistic description for N a and q is desirable from a mechanistic viewpoint.The next section will discuss the statistical property of nucleation sites present on heat surfaces.

MODEL DESCRIPTION
Based on the recent observations, this study will aim at developing a new theoretical model to correlate the active nucleation site density.It is anticipated that the nucleation site density may vary with the cavity size, which is a function of wall superheat.In Eq. ( 8) the units of N a and D c (D c = 2R c ) are Sites/cm 2 and μm, respectively.Thus, in the SI unit based on Eq. ( 8) the cumulative size distribution of nucleation site density can be expressed as: where, R c, min = The minimum active cavity radius R c, max = The maximum active cavity radius The number of nucleation site density from Eq. ( 12) becomes infinite as R c →0.This is consistent with physical phenomena.From Eq. ( 12) as R c → R c, min the number of nucleation site density is close to the largest number, N a, max , is expressed as: From Eq. ( 12), the number of nucleation site density of sizes lying between R c and R c +dR c can be obtained as: where, dR c >0 and -dN a .Eq. ( 14) indicates that the nucleation site number decreases with the increase of diameter of active cavity.Divided by Eq. ( 13), Eq. ( 14) becomes: ( ) where, f(R c ) = 6 =  ,

6
−7 is the probability density function of the nucleation site.Obviously, Eq. ( 15) satisfies: Patterned after the probability theory, the probability density function f(R c ) of the nucleation site should satisfy the following relationship: The integration result is expressed as: The integration result of Eq. ( 18) shows that Eq. ( 18) holds if and only if : is satisfied.Eq. ( 19) implies that R c, min <<R c, max must be satisfied for the probability theory.In general, R c, min /R c, max ≤10 -2 , Eq. ( 18) holds.The total number of nucleation sites from the minimum active cavity (R c, min ) to the maximum active cavity (R c, max ) can be obtained from Eq. ( 14) as: Eq. ( 20) suggests that the total number of nucleation sites is a function of contact angle.Eq. ( 20) takes account of the effect of contact angle.In this model, no new empirical constant is introduced.If the distribution of the nucleation site satisfies the probability theory, Eq. ( 20) can be further reduced to: 29 6 , ,min 7.8125 10 (1 cos ) Eq. ( 21) suggests that the total number of nucleation sites is a function of contact angle and the maximum active cavity radius.
The minimum active cavity radius R c, min and the maximum active cavity radius R c, max could be predicted by Hsu (1962) where, with  being the contact angle of the fluid and the heater material.δ is the thermal boundary layer thickness which can be usually expressed as: where h nc is the average heat transfer coefficient for natural convection which is given by Han and Griffith (1965): where, R a is Rayleigh number for turbulent range where, 2×10 7 <R a <3×10 10 and for laminar range where, 10 5 <Ra<2×10 7 , h nc is given by: where, A is the area of heating surface.Hsu (1962) pointed out that a cavity can be active in the range of R c, min ≤R c ≤R c, max .A cavity can be ineffective at low wall temperature (or low heat flux).

RESULTS AND DISCUSSION
Figure 1 is a plot of ratio (R c, min / R c, max ) versus wall superheat (ΔT w ) of liquid at p = 1.013 bar and Figure 1 shows that the ratio (R c, min / R c, max ) decreases as the wall superheat increases.It was found that R c, min / R c, max <10-2 when the wall superheat ∆T w >10°C.This means that the probability density function f(R c ) of the nucleation site satisfied Eq. ( 18) if they are in the wall superheat as specified above.It can be seen from Fig. 1 that the R c, min / R c, max decreases with increasing of the wall superheat.Basu et al. (2002) performed experiment at 1 atm for boiling of water on a vertical surface.Figure 2 shows a comparison between the predictions from Eq. ( 20) and experimental data (Basu et al., 2002) at contact angle  = 30°.The results show that prediction for the number of nucleation site from Eq. ( 20) is in excellent agreement with experimental observation (Basu et al., 2002).The deviation between the predicted results and experiment data (Basu et al., 2002) is very small.Further comparisons between the predictions from Eq. ( 20) and some existing data are presented in Fig. 3. Zeng and Klausner (1993) conducted experiments in a forced convective R-113 in a 25×25 mm test section with a nichrome heating strip.In the literature since the contact angle was not given, the contact angle of 0 38 was used, which is a typical value for water boiling.It is seen that fair agreement between the predictions from Eq. ( 20) and the experimental data (Zeng and Klausner, 1993) is again found.It can be seen from Fig. 2 and 3 that the total number of nucleation sites increases with wall superheat.This is expected because the high wall superheat may cause the increase of number of nucleation sites in fluids.These are all expected and consistent with the practical physical phenomena.

CONCLUSION
In this study, the active nucleation site density is predicted based on based on statistical property of active nucleation site in boiling systems.The proposed model is expressed as a function of wall superheat and the contact angle.Every parameter of the proposed formulas on the active nucleation site density has clear physical meaning.No additional empirical constant is introduced.This model contains less empirical constants than the conventional correlations.The model predictions are compared with the existing experimental data and excellent agreement between the model predictions and experimental data is found.The validity of the present model is thus verified.The proposed model can reveal the physical mechanisms of heat transfer of boiling systems.Based on the parametric effect analysis, we conclude that the total number of nucleation sites is positively correlated with the wall superheat.