Nucleation site density in pool boiling of saturated pure liquids: Effect of surface microroughness and surface and liquid physical properties

ELSEVIER

Nucleation Site Density in Pool Boiling of Saturated Pure Liquids: Effect of Surface Microroughness and Surface and Liquid Physical Properties R. J. Benjamin A. R. Balakrishnan Department of Chemical Engineering, Indian Institute of Technology Madras, Madras 600 036, India

• An experimental investigation on the nucleation density during nucleate pool boiling of saturated pure liquids at low to moderate heat fluxes is described. The surface-liquid interaction during the boiling phenomena and its effect on the nucleation site density and thereby the heat flux is examined. Stainless steel and aluminum with different surface finishes obtained by polishing the surfaces with different grades of emery paper was used in the study. A parameter, Ra, called the arithmetic average roughness or the centerline average was used to characterize the surface microroughness. The parameter R a is defined as the average values of the peaks and valleys on the surface. This was measured experimentally by using a profilometer. The liquids used in the study were distilled water, carbon tetrachloride, n-hexane, and acetone. The nucleation site density at different heat fluxes for various surface-liquid combinations was measured by using high-speed photography. The data showed that the nucleation site density depends on the surface microroughness, the surface tension of the liquid, the thermophysical properties of the heating surface and the liquid, and the wall superheat. A correlation in terms of the wall superheat, the Prandtl number, a surface-liquid interaction parameter (the ratio of the thermal conductivity, density, and specific heat of the solid to the liquid), and a dimensionless surface roughness parameter has been proposed. The correlation proposed matches data obtained in the present study. The correlation also matches data in the literature obtained on copper and nickel surfaces of various surface finishes, further validating the correlation and the mechanism suggested in this study. © Elsevier Science Inc., 1997

Keywords: nucleation site density, pool boiling, surface finishes and properties, liquid properties INTRODUCTION Nucleate boiling, by definition, is characterized by the formation of vapor bubbles at certain preferred locations known as "nucleation sites," when the heating surface is maintained at a temperature above the saturation temperature of the liquid with which it is in contact. This "nucleation site density" is one of the key parameters in nucleate boiling. With the utilization of this characteristic, much work has been directed through the years toward increasing the number of such sites by polishing, etching, sintering, and using coatings of various types on the heating surface. Moreover, it is generally accepted that the

nucleation site density depends not only on the surface physical properties and the surface finish, but also on the liquid physical properties and the wall superheat. One of the best-known correlations for nucleate pool boiling heat transfer, proposed by Rohsenow [1] in 1952, related the heat flux to the wall superheat, liquid properties, and a parameter Csf. The parameter Csf is a constant whose values are tabulated for various surface-liquid combinations by using experimental data. This parameter, Csf, apparently takes into account the contact angle, the surface microroughness, and their interaction in determining the nucleation site density. Vachon et al. [2] analyzed boiling data in the literature and concluded that the

Address correspondence to Professor A. R. Balakrishnan, Department of Chemical Engineering, Indian Institute of Technology Madras, Madras 600 036, India.

Experimental Thermaland Fluid Science 1997; 15:32-42 © Elsevier Science Inc., 1997 655 Avenue of the Americas, New York, NY 10010

0894-1777/97/$17.00 PII S0894-1777(96)00168-9

Nucleation Site Density in Pool Boiling 33 boiling flux for a given wall superheat is dependent on the surface preparation as well as the surface-liquid combination. Although there are a number of experimental studies reported in the literature to examine the nucleation site density and its dependence on surface and liquid properties, most of them pertain only to specific surfaces and liquids, and it is difficult to generalize the conclusions. Jakob and Linke [3] found a linear relation between the nucleation site density and the heat flux for water at low heat fluxes by visually counting the number of nucleation sites. Corty and Foust [4] and Griffith and Wallis [5] indicated that the microroughness of the boiling surface affects the position and slope of the nucleate boiling curve. Kurihara and Myers [6] conducted boiling experiments with different pure liquids on a flat copper plate polished with different grades of emery paper. They found that the number of nucleation sites on the surface increased with an increase in the surface microroughness. Gaertner [7] found that the nucleation site density fits the Poisson distribution, using the data of Gaertner and Westwater [8]. This was later confirmed by Sultan and Judd [9] and Del Valle and Kenning [10]. Mikic and Rohsenow [11] related the number of nucleation sites to the cavity mouth radius. They used the Clausius-Clapeyron equation, together with the ideal gas law, to relate the cavity mouth radius to the liquid properties and the wall superheat. However, the nucleation site density expression contained two constants that need to be evaluated for each surfaceliquid combination. The advantage of the method is that, when the constants have been evaluated for a surfaceliquid pair, the equation can be used to estimate the nucleation site density at any pressure. For water boiling on a copper surface polished with a 4 / 0 grade emery paper, Gaertner [12] related the nucleation site density to the wall temperature and liquid properties. Bier et al. [13] followed the approach of Mikic and Rohsenow [11] and related the nucleation site density to the corresponding cavity mouth radius. Shoukri and Judd [14] performed an experimental investigation on the nucleation site density for water boiling on a copper surface with different finishes. Their experimental data validated the correlation for the nucleation site density of Brown [15] and the expression for minimum cavity radius suggested by Griffith and Wallis [5]. Singh et al. [16] investigated the dependence of nucleation site density on the size, shape, and population of the cavities for different fluids. The boiling experiments were carried out on deep artificial cavities made by a laser drilling technique. Kopp [17] estimated the number of nucleation sites from a statistical description of a real surface, which compared well with his N / A data of mercury on a stainless steel surface with different roughness values. Judd and Hwang [18] obtained nucleation site density data for dichloromethane on a glass surface by using high-speed photography and incorporated the N / A values in their heat transfer model. They found that the total heat flux has a significant dependence on the nucleation site density. Kocamustafoagullari and Ishii [19] proposed a correlation for nucleation site density in terms of the boiling heat transfer coefficient, the liquid properties, the bubble departure diameter, and the wall superheat by using the experimental nucleation site density data of Kurihara and Myers [6] and Gaertner and

Westwater [8]. Their correlation, however, did not take into account the surface preparation and properties. Sgheiza and Myers [20] obtained experimental nucleation site density data by using a high-speed infra-red camera on a horizontal stainless steel surface for water and four organic liquids. Their study indicated that, for water, about 30% of the active sites nucleated consistently. They conducted that the heating surface consisted of "transient" and '"permanent" nucleation sites. Yang and Kim [21] developed an analytical expression for the nucleation site density as a product of two distribution functions--one each for the cavity mouth radius and the cavity angle--and compared it with their own data on water boiling on a stainless steel surface. Barthau [22] used an optical method for counting the active nucleation sites on a plain horizontal tube using R-114 as the boiling fluid. Wang and Dhir [23], from experiments on boiling water on a copper surface, found that, as wettability increases, the fraction of the cavities that nucleate decreases. Their correlation for the nucleation site density was in terms of cavity mouth diameter and contact angle. Hong et al. [24] measured contact angles on copper and aluminum surfaces. The surface finish was varied by polishing, oxidizing, and roughening the surfaces. The contact angle data obtained by using different pure liquids (water, organic, and refrigerant fluids) indicated that the contact angle depends on the surface microroughness and that the contact angle decreases as the surface microroughness increases. Kolev [25], using literature data on the boiling of water on different surfaces and surface finishes, concluded that the contact angle was perhaps the single most important factor in determining the nucleation site density and the heat flux and is responsible for the large spread in the experimental data on boiling heat transfer. The present study focuses on an experimental program to determine the nucleation site density with a variety of surface finishes on materials such as aluminum and stainless steel by using distilled water, acetone, n-hexane, and carbon tetrachloride in pool boiling at low to moderate heat fluxes and high speed photography. SURFACE CHARACTERIZATION A number of investigators, such as Griffith and Wallis [5], Mikic and Rohsenow [11], Bier et al. [13], Shoukri and Judd [14], Brown [15], Singh et al. [16], Yang and Kim [21], Wang and Dhir [23], and Eddington and Kenning [26], among others, have related the nucleation site density to the cavity mouth radius. Most of these investigators used the cavity mouth radius, r c, defined by Griffith and Wallis [5] as

2~r~ rc

p~A(Tw - Ts)

(1)

in their analysis. This r c is the minimum cavity mouth radius for nucleation to occur at a nucleation site. Equation (1), however, does not take into consideration the surface profile or the surface properties, which also affect the minimum cavity mouth radius for nucleation to occur. Furthermore, the expression for minimum cavity mouth radius is based on the assumption that the cavity is conical and that the mouth of the cavity is circular in shape. This assumption

34 R.J. Benjamin and A. R. Balakrishnan does not hold when the heating surface is etched chemically or polished by using emery paper. On the other hand, surface characterization has also been performed with the use of a profilometer by a number of investigators, such as Corty and Foust [4], Kurihara and Myers [6], Sultan and Judd [9], Shoukri and Judd [14], Kopp [17], Judd and Hwang [18], Hong et al. [24], Nishio and Chandratilleke [27], Hatton and Hall [28], Hsu and Schmidt [29], Nix et al. [30], Raad and Myers [31], Torikai et al. [32] and Gorenflo [33], among others.

In the present study, because the heating surfaces were polished with different grades of emery paper, the heating surfaces are characterized in terms of the microroughness by using a profilometer rather than the cavity mouth radii of the nucleation sites. Many parameters are available to define the microroughness of the heating surface. But the centerline average R a (Fig. la) and the average roughness depth R z (Fig. lb) are the most commonly used and approved by the International Standards Organization (ISO).

Z1 y

Z3

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+

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1 ZI * Z z . Z 3+ Z& * Z s ) Rz= "~'-(

CENTRE LINE AVERAGE R o

AVERAGED ROUGHNESS DEPTH Rz

IS THE ARITHMATICAL AVERAGE VALUE oF ALL ABSOLUTE DISTANCES OF THE ROUGHNESS PROFILE R FROM THE CENTRE LINE WITHIN T H E MEASURING LENGTH I m

IS THE AVERAGE VALUE OF THE INDIVIDUAL ROUGHNESS DEPTHS IN FIVE SEQUENTIAL SINGLE M E A S U R I N B LENGTHS I e

(b)

(a)

)

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COPPER SURFACE POLISHED WITH Ra = 0.0?lJ.m, Rz = 0.61 ]J.m

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t

I GRADE EMERY

PAPER

re)

tI ALUMINIUM SURFACE POLISHED WITH Ro = 1.t"/ kt.m, R z : 8.27 IJ, m

210

GRADE EMERY PAPER

(d) Figure 1. Definitions of R a and R z.

Nucleation Site Density in Pool Boiling

35

EXPERIMENTAL A schematic representation of the experimental setup is shown in Fig. 2. It consists of a double-walled glass column 25 cm in height with the annular space between the two walls maintained under vacuum to minimize heat loss. The inner diameter of the column is 93 mm. The column is held in place between two stainless steel slabs, which have circular grooves cut in them to fit the column exactly. Oil seals and tie rods help hold the column rigidly and prevent leakage between the two stainless steel slabs. The bottom slab has a hole in the center through which a hollow Teflon cylinder is fitted. Through this cylinder is introduced the heating block. Two different heating blocks were used in the present study: one made of aluminum and the other made of stainless steel. The bottom of the block is heated by a 3-kW plate heater and the sides are fully insulated, resulting in the top surface of the block with a diameter of 25 mm serving as the heating surface. Temperatures were measured at four points along the axis of the block with the use of platinum resistance thermometers (of resolution 0.1 C °) and, with these measurements, the heat flux and the temperature of the heating surface at steady state were determined. The heating surface was polished with different grades of emery paper to generate different microroughness values for the experiments. The surface microroughness was characterized by the centerline average, R a. After the surface was polished with a particular grade of emery paper, the R a and R z values were measured by using a profilometer (Perthen). The tip of the probe of the profilometer (which is a conical diamond bit and tip radius of the order of 0.1 ~m and with a resolution of 0.01 ~m) was placed at approximately the center of the surface. The traversing distance was 5.6 mm, and four sweeps were taken for each surface finish in different directions. Although both R a (centerline average) and R z (averaged roughness depth) were measured in the present study, the R a value was used to correlate the N / A data because the four R a values obtained in different directions for each surface finish were very close to each other numerically (variation of 0.05/xm), whereas Rz values gave a lot more variation. Further, R a has also been extensively used in the literature as a measure of surface roughness by many investigators, such as Corty and Faust [4], Sultan and Judd [9], Shoukri and Judd [14], Kopp [17], Barthau [22], Nishio and Chandratilleke [27], and Hatton and Hall [28]. In a recent study, Gorenflo [33] characterized the surface mi-

®

C) 1. Dimmerstnt, 2. Plate heQter, 3. Insulation, 4. Heating block, S. Teflon hollow cylinder, 6. Liquid, 7. Double-walled glass vessel, B. Cooling coil, 9. Pressure gauge, 10. Relief valve. Figure 2. Schematic of the experimental setup.

croroughness by using the R a value while developing a correlation for the heat transfer coefficient. The centerline average, Ra, is specified by ISO 4287/1:1984 [34] for characterization of surface microroughness. The experimental procedure consisted of first preparing the heating surface by polishing it with a particular grade of emery paper and then measuring the R a value. The materials, the grades of emery paper, and the R a values obtained experimentally are tabulated in Table 1. Although aluminum and stainless steel were used in the present study, copper and nickel surfaces were polished

Table 1. Summary of Surfaces and Liquids Used in Developing Present Correlation

. Surface Copper Copper Copper Copper Copper Nickel Stainless steel Aluminum Aluminum Aluminum

Surface Finish 3/0 emery paper, R a = 0.14/zm 4 / 0 emery paper, R a = 0.07/xm 4 / 0 emery paper, R a = 0.07/zm 4 / 0 emery paper, Ra = 0.07/zm Mirror finish, R a < 0.02/xm 4 / 0 emery paper, R a = 0.045/zm 1/0 emery paper, R~ = 0.2/xm 2/0 emery paper, R~ = 1.17/xm 3 / 0 emery paper, R~ = 0.89/~m 4 / 0 emery paper, R~ = 0.52/xm

Liquid Distilled water Acetone, n-hexane, and CC14 Aqueous solution of nickel salts Distilled water Distilled water n-Pentane Distilled water, Cfl4, acetone, and Distilled water, CC14, acetone, and Distilled water, CC14, acetone, and Distilled water, CC14, acetone, and

Source of Data

n-hexane n-hexane n-hexane n-hexane

Griffith and WaUis [5] Kurihara and Myers [6] Gaertner and Westwater [8] Gaertner [12] Wang and Dhir [23] Zuber [35] Present study Present study Present study Present study

36

R.J. Benjamin and A. R. Balakrishnan

with the appropriate grade emery paper to get the R a value for comparison with experimental data reported in the literature (see Table 1), which were in terms of grade of emery paper alone. Boiling experiments were conducted on the different surfaces with varying R a values by using distilled water, carbon tetrachloride, acetone, and n-hexane at atmospheric pressure and standard gravity conditions. The test liquids were degassed before each run. All the experiments were conducted under steadystate conditions. A steady-state condition was assumed when the temperature along the axis of the block did not vary by more than 0.1 K in about 2 min. The nucleation site density was measured for different heat fluxes by photographing the heating surface with the use of a highspeed still camera. The flash duration used was 1/25000 s (4 X 10 s s) although the camera shutter speed was only 1/250 s (0.004 s). The photograph was projected on a screen to get sufficient magnification, and the bubbles on the surface were counted. The experimental uncertainty in counting is _+ 10% with a confidence interval of 90%. This is the expected error in identifying and counting the number of bubbles on the surface. This is done about

three times and the mean obtained. This level of uncertainty was also noticed by Hui and Thome [36]. The bubble counting technique used by Hui and Thome [36] was also used in the present study to recheck the calculated nucleation site density values. For a given test fluid, the experimental procedure was repeated, and photographs were taken at the same heat fluxes. The nucleation site densities were estimated again from the photographs. The nucleation site density variation from the photographs (for a particular heat flux, liquid, and surface finish) was negligible. The bubbles nucleating on the surface and those that had detached and were rising in the liquid were differentiated by their shapes. Figure 3 shows sample photographs of the boiling of n-hexane. Figure 3a is on a stainless steel surface with a n R a value 0.2 /~m and a flux of 2.816 x 104 W / m 2. Figures 3b-3d are on an aluminum surface with Ra values of 0.52 /zm 0.89 /zm, and 1.17/zm, respectively, and heat flux values of 2.138 x 104 W / m z, 2.088 × 104 W / m 2, and 3.007 X 104 W / m 2, respectively. A preliminary examination of these photographs shows that, a s R a increases, the number of bubbles decreases and then increases. Stainless steel was

m

c

d

Figure 3. Sample photographs of the heating surface during boiling of n-hexane: (a) stainless steel surface, R a = 0.20 p~m, q = 2.816 X 10 4 W / m 2 ; (b) aluminum surface, R a = 0.52 /zm, q = 2.138 x 10 4 W / m 2 ; (c) aluminum surface, R a 0.89/zm, q = 2.088 X 10 4 W / m 2 ; ( d ) aluminum surface, R a = 1.17/zm, q = 3.007 x 10 4 W / m 2.

Nucleation Site Density in Pool Boiling 37 used in Fig. 3a to demonstrate the effect of a very small R a value, which was not possible with aluminum, which gives larger values of R a.

RESULTS Nucleation site density as a function of the heat flux for various surface finishes for each of the four liquids used in this study are shown in Fig. 4. The solid circles represent data obtained on stainless steel; the other data points are on aluminum. It can be seen from Fig. 4 that, as the microroughness increases, the nucleation site density decreases and then increases for a given heat flux. A similar trend is obtained when, instead of N/A values, the heat flux is plotted against the wall superheat (Fig. 5) for

various surface finishes for each of the four liquids used in the present study. This kind of decreasing-increasing behavior of the nucleation site density and the heat flux with increasing surface microroughness has also been noticed previously. Bereson's [37] data show that the heat transfer was higher for a lapped surface than for an emerypaper-finished surface (rougher surface). Thus the heat transfer decreased with an apparent increase in the surface microroughness. Raad and Myers [31] observed that the number of nucleation sites decreased and then increased with an increase in the surface microroughness. Vachon et al. [38] found that, on chemically etched and polished stainless steel surfaces, increasing surface microroughness caused an increase, then a decrease, followed by an increase in heat transfer rates. A similar trend is discernible from the results of Nishio and Chandratilleke

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Figure 4. Nucleation site density with heat flux: (a) n-hexane; (b) distilled water; (c) acetone; (d) carbon tetrachloride.

10

38

R.J. Benjamin and A. R. Balakrishnan II

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o Figure 5. Heat flux with wall superheat: (a) n-hexane; (b) distilled water; (c) acetone; (d) carbon tetrachloride. (Note: Ordinate for distilled water is an order of magnitude larger than that for the other three liquids.)

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[27], which were obtained in the pool boiling of saturated helium at atmospheric pressure. On the other hand, the experimental data of Torikai et al. [32] on the boiling of water on a copper surface show that, as the lattice depth increases, the heat flux for a given wall superheat increases and then decreases. Kant and Weber [39] found that the ratio of the cavity depth to diameter decreases and then increases with cavity diameter at different wall superheats for the subcooled boiling of water on a simulated cylindrical glass nucleation site. The reason for the data of Torikai et al. [32] showing a trend different from the present study and from the data of Nishio and Chandratilleke [27], Raad and Myers [31], Berenson [37], Vachon et al. [38], and Kant and Weber [39] is perhaps that Torikai et al. were studying boiling on machined surfaces, whereas the others were concerned with polishing the heating surface with different grades of emery paper.

i 20

25

To take into account the decreasing-increasing trend of with R a noticed by the previous investigators and in the present study, a dimensionless surface roughness parameter (0) was defined as

N/A

O=A + B ( - - ~

+C

(2)

where A, B, and C are constants evaluated (6)]. With the use of all the data (i.e., at superheats, liquids, and surface materials), for the nucleation site density was obtained N A

218.8(Pr)1.63 ( 1 ) 0

later [see Eq. different wall a correlation as

0.4(AT)3 '

(3)

Nucleation Site Density in Pool Boiling 39 where Pr is the Prandlt number and is defined as

• 0 . 0 2 / z m < R a < 1.17 ~ m

• 5K
Cp/x

Pr

(4)

K

• 13 x 10 - 3 N / m

and accounts for the physical properties of the liquid being boiled. The surface-liquid interaction parameter, 7, is defined by

KwPwCpw 3' =

During bubble growth, there is convective flow of the liquid away from a nucleation site. When a bubble has departed, there is flow toward the site--envisaged as "source" and "wake" flows by Zuber [35]. Therefore, although there is no net flow, the viscosity of the boiled liquid plays an important role in pool boiling heat transfer. Han and Griffith [40] concluded that the viscosity of the liquid determines the contact angle and the bubble departure diameter. Moreover, the Prandtl number (which takes the viscosity into account) was used by Rohsenow [1] to develop his widely used correlation. He concluded that, although there is no flow during pool boiling, the Prandtl number is significant. Furthermore, Nishikawa and Fujita [41] also used the Prandtl number while developing their bubble agitation and model for pool boiling. They justified this approach by pointing out that nucleate pool boiling is similar to natural convection. In the present study, the Prandtl number of the liquid also has been used as a correlating parameter. To characterize the surface microroughness and its effect on nucleation, the profilometer data alone are not sufficient, but the surface and liquid properties (i.e., a "smooth" surface for one liquid can serve as a "rough" surface for another) have to be taken into account as well. This was also the conclusion of Ramilson et al. [42]. They hypothesized that "smooth" and "rough" surfaces can be discussed only in connection with a given surface-liquid

(5)

and 0 is given by

) + 0.4

N/m

DISCUSSION

KI P l C p l

0 = 14.5 - 4.5~

< o- < 5 9 × 10 - 3

• 2.2 < 0 < 14

(6)

The wall superheat, AT, is defined as TW - Ts. The correlation has been further validated by using the data of Griffith and Wallis [5], Kurihara and Myers [6], Gaertner and Westwater [8], Gaertner [12], Wang and Dhir [23], and Zuber [35]. Only in Wang and Dhir's [23] work was the R a value explicitly stated, whereas the others have stated only the grade of emery paper used in polishing the surface. Hence, in the present study, the surfaces used by them were polished with the appropriate grades of emery paper and the R, value was measured by using the profilometer. The performance of the proposed correlation is shown in Fig. 6. 90% of the data points lie within + 35% and 30% of the correlation. The various surfaces, surface finishes, R a values, and liquids used to validate the correlation proposed are given in Table 1. The range of parameters covered in developing and validating the correlation are: • 1.7 < Pr < 5 • 4.7 < 3 ' < 9 3

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Figure 6. Performance of the nucleation site density correlation.

40

R.J. Benjamin and A. R. Balakrishnan

combination. Therefore, in the present study, a surfaceliquid interaction parameter, 3' (the ratio of the thermal conductivity, density, and specific heat of the solid to the liquid), was used in conjunction with the profilometer data in developing the nucleation site density correlation. The surface-liquid interaction parameter 3' has been used in the literature by Kant and Weber [39], Sernas and Hopper [43], and Unal [44]. It is the product of the ratio of the thermal diffusivity of the solid to the liquid and the heat content of the solid to the liquid. That is, o-w p w C p . 3" =

KI PlCpt

=

A vapor bubble cannot spontaneously originate in a saturated liquid. This would imply that latent heat is taken from the liquid, which would in turn cool, violating the second law of thermodynamics. If a vapor is to exist, an interface between the vapor and the liquid is necessary, and this will be curved in the case of a bubble. But the liquid will resist the formation of such a curved surface by virtue of is surface tension. W. F. Thomson (Lord Kelvin) has shown that the vapor pressure on a curved surface, P .... on a concave liquid surface of radius r is smaller than on a plane surface, Pv, at the same temperature according to the equation

(7)

plCp I

2o- Pv r Pl .

Pvr, = ev

The temperature of the heating surface during bubble growth is a function of 3'. The surface-liquid interaction parameter 3' also indirectly affects the bubble growth time [39]. The variation of nucleation site density with the wall superheat for the four surface finishes for acetone, hexane, carbon tetrachloride, and distilled water is shown in Fig. 4. It can be seen that, for all four liquids, as the roughness increases, the nucleation site density decreases and then increases. The dependence of heat flux on the wall superheat for each of the four liquids shows the same trend, as can be seen in Fig. 5 (i.e., as the roughness increases, the heat flux decreases and then increases). This can also be seen in the set of four photographs shown in Fig. 3 but can best be seen in Fig. 7, where ( N / A ) / ( A T ) 3 is plotted against R a. These curves are generated from the correlation developed and are not experimental data points; they are shown in this form to clearly show the dependence of N / A o n R a. Conventional wisdom would lead one to expect higher nucleation site densities and hence higher heat fluxes with increasing R a values--that is, rougher surface give rise to higher heat fluxes. To examine why this minimum occurs, it is necessary to ask how a vapor bubble is formed in a liquid. This has been described in detail in the classical monograph in heat transfer by Jakob [45].

(8)

Equation (8) can be derived easily by considering the hydrostatic forces in the rise of a liquid in a capillary column. It can also be expressed in terms of the temperature driving force, as given by Webb [46], who defined as necessary condition for boiling nucleation on a surface as qr c 2 o-l Tw - T~ = ~ + m----~c,

(9)

where r c is the cavity mouth radius and m is the slope of the vapor pressure curve. Assuming the cavity mouth radius as approximately the R a value, we can rewrite Eq. (9) as

Tw - T ~ =

qR a 2o'1 kl + mR----~a.

(10)

For increasing Ra, 2 o - / R a becomes less significant. The first term of Eq. (10) is not affected very much by the R a increase because q is very large. Therefore, increasing R a decreases the available superheat, leading to lower nucleation site densities. Beyond the minimum, probably one or more individual bubbles are nucleated in the same cavity, so N / A starts increasing again. This also explains the choice of the parameters in Eq. (2) for defining 0. In addition, inclusion of the surface tension also incorporates the effect of the contact angle noticed by earlier investiga-

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3 on Ra; curves generated for the four liquids from the correlation.

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0.8 Ro (IJ.m|

1.2

Nucleation Site Density in Pool Boiling 41 tors [24, 25]. Table 1 gives the R a values obtained with different grades of emery paper on different materials. When the N / A data obtained on an aluminum surface are compared with those obtained on a copper surface [6], it is found that, for a given surface finish, liquid, and wall superheat, the N / A of copper is lower than that of aluminum. This indicates that there is a residual dependence of N / A on the material properties of the heating surface and its interaction with the liquid being boiled. This is taken care of in the correlation by including the surface-liquid interaction parameter, 3'. The surface-liquid interaction parameter, % is much higher for copper than for aluminum. By inclusion of both the roughness parameter, 0, and the surface-liquid interaction parameter, % in the correlation to evaluate N / A , the empirical constant Csf, which has to be determined for each surface (and finish)-liquid pair and tabulated to use the classical Rohsenow correlation [1], has been quantified on the basis of the physical and metrological properties of the surface and the thermophysical properties of the liquid. EXPERIMENTAL UNCERTAINTY The resolution in the measurement of temperature is 0.1 C °, and the resolution in the measurement of the R a value is 0.01 ~ m (instrument specifications). Over the temperature range encountered in this study, the average uncertainty in the measurement of temperature (based on C °) is estimated to be +0.2%. Hence the uncertainty in estimating the temperature of the heating surface is + 0.4%. The uncertainty in estimating the wall superheat and the heat flUX is + 0.8%. The uncertainty in estimating 0 [Eq. (6)] is + 5 % . The physical properties of the heating surface and the liquids used (which are functions of temperature) were obtained from property tables given by Piret and Isbin [47]. Gallant [48], and Liley et al. [49]. The uncertainty in estimating the Prandtl number and 3, is + 2% and + 4 % , respectively. The relative uncertainty is estimated by using the method of Moffat [50] and is found to be 6.03%. Therefore for a 95% confidence interval, the relative uncertainty is 12.06%. S I G N I F I C A N C E A N D USEFULNESS The boiling heat flUX during nucleate boiling depends on the nucleation site density to a very large extent. The correlation developed in this study facilitates a quick estimate of the nucleation site density from the heating surface properties, the surface microroughness, the liquid properties, and the wall superheat. Further, it can be used to optimize the surface finish for a given surface-liquid combination, which in turn will increase the heat transfer. Moreover, many correlations and models in the literature require an estimate of the nucleation site density, which can be obtained from the correlation proposed in the present study. CONCLUSIONS A correlation for the nucleation site density in terms of the thermophysical properties of the heating surface and the liquid and the metrological properties of the surface has been developed. The N / A value decreases and then increases with R a. This has been explained in terms of the

bubble radius, the wall superheat, and the ability of the trapped vapor in the cavities to grow at a given wall superheat. The correlation fits the present experimental data and the data from the literature. This study was funded by a research grant from the Department of Atomic Energy, Government of India, through the Board of Research in Nuclear Sciences, Bhabha Atomic Research Centre, Mumbai, India. NOMENCLATURE A area of heat transfer surface, m 2 Cp specific heat at constant pressure, J / ( k g K) Csf surface-liquid interaction constant in the Rohsenow [1] correlation, dimensionless k thermal conductivity, W / ( m K) m slope of the vapor pressure curve, N / ( m 2 K) N number of nucleation sites, dimensionless P external pressure, N / m 2 q heat flUX, W / m 2 R a arithemetic average roughness, m (or /zm where specified) R z averaged roughness depth, m (or/xm where specified) r radius, m T temperature, K AT wall superheat, K

Greek Symbols a y

thermal diffusivity, m 2 / s surface-liquid interaction parameter, defined by Eq. (5), dimensionless 0 roughness parameter, defined by Eq. (6), dimensionless h latent heat of vaporation, J / k g /x dynamic viscosity, (N s ) / m 2 p density, k g / m 3 cr surface tension, N / m

Subscripts c w 1 v r s

cavity surface liquid vapor curved surface saturation REFERENCES

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Received December 5, 1995; accepted October 28, 1996