Abstract

Saturated water at one-atmosphere pressure was boiled on horizontal flat copper disks of diameters 1.0, 1.5, and 2.0 cm, respectively. The contact angle was varied from about 10 to 80 deg by controlling thermal oxidation of the disks, while the surrounding vessel size was changed by placing glass tubes of different inner diameters around the disks. Nucleate boiling heat transfer data were obtained up to critical heat flux (CHF), where vapor removal pattern was photographed. Vapor jet diameter and the dominant wavelength at water–steam interface were measured from the photographs for the well wetted disks. For well wetted surfaces, the magnitude of CHF increased when the heater size was reduced from 2.0 to 1.0 cm. Improving the wettability enhanced the CHF substantially, whereas the increased size of the liquid-holding vessel had a smaller effect. The highest measured CHF is 233 W/cm2 or 2.11 times Zuber's CHF prediction for infinite horizontal flat plates. It was obtained on a 1.0-cm-diameter disk of contact angle about 10 deg surrounded by a large vessel. The CHF for this surface was increased from 203 to 233 W/cm2 when the ratio of heater size to surrounding vessel size was decreased from 1 to about 0.

1 Introduction

Boiling heat transfer has been increasingly applied for efficient heat dissipation on small scales required by the continuous power density increase and miniaturization of electronic devices. The ongoing breakthroughs achieved in the semiconductor industry, e.g., shrinking transistor dimensions and three-dimensional packaging, enhance electronic performance. However, at the same time, the associated higher power density poses a significant challenge to thermal management. For example, the power dissipation from a microprocessor chip has been predicted to exceed 800 W by 2026 from 270 W in 2015 [1]. Assuming the die to be 2 cm × 2 cm in size, the average heat flux would increase to 200 W/cm2. Besides, local hot spots with heat fluxes exceeding 1 kW/cm2 are very common and can potentially degrade the chip performance [2]. Furthermore, in other small electronic devices like insulated-gate bipolar transistors used in hybrid electric vehicles, heat fluxes up to 500 W/cm2 are expected in the next generation as the operating voltage, current, and frequency increase [3]. Numerous studies have been performed in applying and improving boiling for heat dissipation in various configurations, such as microchannel flow boiling, flash evaporation, two-phase jet impingement, and spray cooling to meet different expectations of electronic cooling [47]. However, some fundamentals, such as the characteristics of boiling on small surfaces, still lack understanding. Since the bubble dynamics and vapor removal pattern can be quite different from those on large scales, the pool boiling critical heat flux (CHF) on chip-scale surfaces is worth a systematic study for the scientific design of electronic cooling systems.

This study was carried out to experimentally investigate the individual and interacting effects of heater size, contact angle, and surrounding vessel size on CHF for pool boiling on small horizontal flat heaters. As shown in Fig. 1, the pool boiling heat transfer occurring near critical condition on horizontal flat heaters that are well wetted can be classified into three categories depending on the vapor removal pattern. Figure 1(a) illustrates the vapor removal pattern at CHF for pool boiling on infinite horizontal flat plates as proposed by Zuber [8]. In Zuber's model, near critical condition, the vapor generated on horizontal flat heaters escapes in the form of jets. These vapor jets were assumed to locate on a square grid with spacings of two-dimensional Taylor unstable wavelength. Zuber could not provide a basis for selecting either the critical Taylor wavelength
λc=2πσ(ρlρv)g
(1)
or the “most dangerous” Taylor wavelength
λd=2π3σ(ρlρv)g
(2)
where ρl and ρv are the densities of liquid and vapor, respectively, σ is the surface tension of the liquid, and g is the gravitational acceleration. For saturated water at one-atmosphere pressure, λc = 1.57 cm and λd = 2.72 cm. The diameters of the vapor jets were assumed to be equal to half of their spacings. It was proposed that at CHF, the liquid–vapor interface becomes unstable when the relative vapor velocity reaches a critical value determined by the Helmholtz unstable wavelength λH. Thus, the instability of the vapor jets represents the maximum possible rate at which vapor can be removed under surface tension and buoyancy. Taking the Helmholtz unstable wavelength λH to be equal to the circumference of the vapor jet and using a Taylor unstable wavelength between λc and λd, Zuber [8,9] obtained the corresponding CHF expression as
qCHF,Z=π24ρvhfg[σ(ρlρv)gρv2]1/4
(3)

where hfg is the latent heat of vaporization of the liquid, e.g., for saturated water at one-atmosphere pressure, Eq. (3) gives qCHF,Z = 110 W/cm2. Equation (3) applies as long as the system pressure is much less than the critical pressure. Zuber's formulation benefited from earlier works of Kutateladze [1012]. Later, Lienhard and Dhir [13,14] proposed that the leading constant should be 0.149 instead of π/24 for large horizontal flat plate heaters by arguing that the liquid–vapor interface would become unstable at a shorter available Taylor wavelength, λd, where the spacings of the vapor jets were equal to λd as well. With the modified constant, the ratio qCHF/qCHF,Z is 1.14. They also showed that a heater could not be called an infinite flat plate when its size was less than 3λd [15]. Figures 1(b) and 1(c) show the postulated vapor removal pattern at CHF for pool boiling on small horizontal flat plate heaters of sizes between λd and 3λd and those of diameters less than λd surrounded by enclosing sidewalls, respectively.

Fig. 1
Vapor removal pattern at CHF for pool boiling on (a) infinite horizontal flat plates, (b) small horizontal flat plate heaters of sizes between λd and 3λd, and (c) those of diameters less than λd surrounded by enclosing sidewalls
Fig. 1
Vapor removal pattern at CHF for pool boiling on (a) infinite horizontal flat plates, (b) small horizontal flat plate heaters of sizes between λd and 3λd, and (c) those of diameters less than λd surrounded by enclosing sidewalls
Lienhard et al. [15] investigated the heater size effect on CHF as most heaters in earlier experimental studies could not be called infinite flat plates. Gogonin and Kutateladze [16] boiled ethanol on horizontal flat ribbons of length 150 mm and widths varying from 5 to 50 mm under different pressures. In their study, no size effect was observed on the CHF for the upward-facing pool boiling in a large chamber. In the first-of-its-kind study performed by Lienhard et al. [15], a series of experiments were conducted by boiling various liquids on horizontal square and circular flat plate heaters of different sizes. In their study, all the heater surfaces were fully wetted and surrounded by enclosing sidewalls. No prominent heater size effect was found until the characteristic lengths Lc of the heaters became less than 3λd. The characteristic length Lc is the diameter of a circular heater or the width of a square heater. CHFs approximately 2.0–2.5 times Zuber's CHF prediction were obtained from the heaters when their characteristic lengths Lc decreased to 0.5λd. They attributed this outcome to the change in the fraction of heater surface area occupied by vapor jet. They argued that the fraction depended on the number of vapor jets that could be accommodated on a heater when its characteristic length Lc was less than 3λd. The vapor jet diameter was still assumed to be 0.5λd. As such, CHF should then be determined by the actual number of vapor jets Nj that could be accommodated on the heater of surface area A [15], i.e.,
qCHF,smallqCHF,Z=1.14NjA/λd2
(4)

Similar dependences of CHF on heater size have also been found by Saylor [17] and Rainey and You [18] who boiled FC-72 on horizontal flat heaters of different sizes. Equation (4) could effectively explain the heater size effect on CHF within the range λdLc3λd, while it could not describe the trend of CHF against heater size when the heater size becomes less than λd.

Many experimental studies have been performed to investigate the surface wettability effect on CHF. Generally, the contact angle is employed as an indicator of surface wettability. Berenson [19,20] investigated the contact angle effect on CHF by boiling n-pentane on a horizontal flat 5.1-cm-diameter copper surface surrounded by a brass tube of inner diameter 5.8 cm. By adding a slight amount of oleic acid into the liquid, the contact angle decreased from 10 to 0 deg. No obvious contact angle effect was found on the CHF within this narrow contact angle range. Several other studies had shown reductions in CHF with increasing contact angle. Roy Chowdhury and Winterton [21] performed a transient boiling experiment to study the contact angle effect by quenching vertical cylindrical heaters of diameter 18 mm and length 40 mm in water and methanol, respectively. Aluminum and copper specimens with various surface treatments were used. They found that increasing the contact angle led to decrease in the CHF. The contact angle varied from 102 to 0 deg in their study, and the variations in contact angle were achieved by etching or anodizing the specimens and aging the surfaces via quenching repeatedly. Later, Liaw and Dhir [22,23] thoroughly studied the contact angle effect by boiling water on a vertical flat copper surface of width 6.3 cm and height 10.3 cm. The CHF was found to decrease with increasing contact angle based on their steady-state and transient cooling and heating measurements. In their study, the contact angle was decreased by thermally oxidizing the copper surface. A contact angle above 90 deg was obtained by depositing a thin coating of fluorosilicone sealant on the surface. The measured CHF at a contact angle of 107 deg was about half of that measured at a contact angle of 14 deg. Meanwhile, by performing transient boiling of water on a horizontal flat 26.6-mm-diameter copper surface confined by a glass tube, Maracy and Winterton [24] found that the CHF increased with decreasing contact angle for both the heating and cooling runs. In their study, the contact angle was decreased from 75 to 0 deg through surface aging by increasing the number of experimental runs. Dhir and Liaw [25] argued that while the hydrodynamic theory is based on the maximum vapor removal rate possible from well wetted surfaces under available buoyancy, CHF on partially wetted surfaces is limited by the maximum vapor generation rate that can be achieved on the surface. With increasing contact angle, there is less area available for vaporization of micro/macrolayers surrounding the vapor stems. Thus, the maximum rate at which vapor can be generated before the surface is substantially covered with vapor sets the upper limit of nucleate boiling heat flux on partially wetted surfaces. In contrast, for well wetted surfaces, the wall void fraction is much smaller, and a large fraction of the surface is covered with liquid. In such cases, the upper limit of nucleate boiling heat flux is set by the maximum rate at which vapor generated on the surface can be removed. The vapor removal limit is larger than the vapor production limit on partially wetted surfaces. Kandlikar [26] postulated that CHF is reached when the force due to the change of evaporation momentum equals the sum of the forces due to surface tension and hydrostatic pressure at the contact line of a bubble on the surface. Based on that hypothesis, he developed a CHF correlation with dynamic receding contact angle θrec and surface inclination angle from horizontal ϕ taken into account, i.e.,
qCHF=(1+cosθrec16)[2π+π4(1+cosθrec)cosϕ]1/2ρvhfg[σ(ρlρv)gρv2]1/4
(5)

It also shows that CHF decreases as the contact angle increases with fixed surface orientation. Similar dependences of CHF on contact angle have also been shown in experimental studies by Chen et al. [27], Kwark et al. [28], Ahn et al. [29], O'Hanley et al. [30], and Girard et al. [31]. Even though some of these studies mainly focused on pool boiling on structured surfaces, it can still be concluded that CHF decreases with increasing contact angle by comparing the reported CHFs solely for the plain surfaces of different contact angles.

Much less attention has been given to the surrounding sidewall effect on CHF. Costello et al. [32] found that a flat ribbon heater, mounted on a slightly wider block, induced strong side flow in their pool boiling experiment. When the side flow was blocked by the sidewall, the CHF was found to be much lower than it was when the side flow was allowed. Moreover, they observed that the CHF increased when the ribbon width was reduced and the side flow was permitted. Lienhard and Keeling [33] boiled various liquids on horizontal nichrome ribbons of length 10.2 cm and widths ranging from 0.1 to 2.5 cm in a chamber of length 17.8 cm, width 9.5 cm, and height 7.6 cm. The CHF was found to increase with decreasing ribbon width. They attributed this enhancement to the stronger induced convection when the heater size was reduced with fixed surrounding sidewall. The surrounding sidewall effect on CHF was further studied by Elkassabgi and Lienhard [34]. They boiled methanol on a horizontal nichrome wire of diameter 0.813 mm and revealed that enlarging the spacing of the parallel sidewalls from 2 to 12 mm produced an enhancement up to 50% in the CHF. Later, Bockwoldt et al. [35] used a horizontal flat nickel-plated copper surface of diameter 1.5 cm to boil water. The CHF was enhanced by 25% and reached 227 W/cm2 by increasing the inner diameter of the surrounding tube from 1.6 to 14.4 cm. Taken together, it is believed that surrounding sidewall spacing can influence CHF for finite heaters. However, it is worth pointing out that the surrounding sidewall effect has been ignored in several earlier and some recent experimental studies [3640]. Failing to keep the surrounding sidewall spacing or ratio of heater size to surrounding sidewall spacing unchanged among the experiments makes the comparison of results improper.

Thus, several disagreements in the results of different experimental studies may be attributed to a lack of specification or control of various experimental parameters. In this work, a systematic study investigating the effects of heater size, contact angle, and surrounding vessel size on pool boiling CHF from horizontal surfaces was carried out using disk heaters of diameters 0.37λd, 0.55λd, and 0.74λd.

2 Experiments

Each boiling experiment was configured based on its heater size and surrounding vessel size. The method of preparing test surfaces of different contact angles is described in Sec. 2.1.

2.1 Preparation of Test Surfaces.

Copper disks (110 Cu) of thickness d = 3 mm and diameters D = 1.0, 1.5, and 2.0 cm were used as horizontal flat heaters in the boiling experiments. Wetting characteristics of the surfaces were identified through the use of static contact angles. Advancing or receding contact angles were not used. These dynamic angles change during the boiling process and depend on the liquid–vapor interface velocity which in turn would be affected by the surface superheat [41,42]. Before conducting any treatment on the disks, a polishing routine was performed to ensure the disk top surface was smooth without any bumps or scratches. For each disk, the top surface was first rubbed with a 1200-grit sanding cloth (400 MX, Micro-Mesh) in a certain direction 200 times and rubbed another 200 times after being rotated 90 deg. This step was then repeated using a 1500-grit sanding cloth (4000 AO, Micro-Mesh). The final polishing was performed using a 3000-grit sanding cloth (12000 AO, Micro-Mesh) until a uniform mirror finish was reached. The prepared surface was then cleaned with a disposable ethyl alcohol cotton swab and dried spontaneously in air. With all the steps above completed, the disk was ready for subsequent use.

In this study, to get different contact angles of the disks without altering their surface topographies, the disks were oxidized in air following a predeveloped method [43,44]. For this purpose, an experiment was performed on a representative disk of thickness d = 3 mm and diameter D = 1.5 cm to determine the dependence of contact angle on heating temperature Th and heating time th. In present experiments, the polished disks were placed on a digital hotplate (SP131635Q, Thermo Scientific) to be heated in air at temperature Th. The heating temperature Th was precisely controlled by rotating the knob of the hotplate and was directly displayed on its digital screen. After heating time th, the power was turned off. The disk subsequently cooled down along with the hotplate to room temperature. Thereafter, a local contact angle was measured from a photograph of a 0.01 mL deionized water droplet placed on the disk top surface. This measurement was repeated three times by placing droplets at different locations on the surface, and the average of the three values was used to represent the contact angle θ of the disk.

Figure 2(a) shows the contact angle θ as a function of heating temperature Th and heating time th for the representative disk. Contact angles ranging from 90 to 9 deg were obtained following this procedure, as shown in Fig. 2(b). The largest contact angle 90 deg was obtained on the unoxidized disk, and the smallest contact angle 9 deg was obtained on the disk when it had been heated at 275 °C for 30 min. When the disk had been heated to 300 °C for 30 min, the oxide layer forming on its top surface was found to be unstable and peeled off during its cooldown period, as the photographs in Fig. 2(a) show. Thus, it was not feasible to obtain reliable contact angles below 9 deg using this approach. At lower temperatures, the oxide layer was chemically stable and physically robust on the disk top surface, as has also been noted in Ref. [43].

Fig. 2
(a) Contact angle θ as a function of heating temperature Th and heating time th for the representative copper disk. (b) Photographs of the unoxidized and oxidized representative copper disks after being heated at different temperatures Th for different times th and then cooling down to room temperature and corresponding pictures of droplets on the surface.
Fig. 2
(a) Contact angle θ as a function of heating temperature Th and heating time th for the representative copper disk. (b) Photographs of the unoxidized and oxidized representative copper disks after being heated at different temperatures Th for different times th and then cooling down to room temperature and corresponding pictures of droplets on the surface.

2.2 Experimental Apparatus.

Figure 3(a) shows a schematic of the experimental apparatus assembled to measure saturated boiling heat transfer data of water at normal earth gravity and one-atmosphere pressure for horizontal flat plate heaters. The heating section consisted of the sample disk, a copper block (110 Cu), inner four cartridge heaters (CIR-30224, Omega Engineering), and an outer ceramic fiber pedestal. The disk was bonded on the block top by graphite adhesive (0.2 mm thickness, 931C, Resbond) and was surrounded by a PTFE ring (upper, 1 mm thickness) and a fiberglass ring (lower, 2 mm thickness) to insulate the outer edge of the disk. The upper cylindrical part (3.0 cm diameter, 2.5 cm height) of the copper block was used for conducting heat to the disk, whereas the four cartridge heaters (1.3 cm diameter, 5.7 cm length) with a maximum power of 500 W each were embedded in the lower cuboidal part (3.5 cm × 6.4 cm × 3.5 cm) of the copper block. The clearances between the copper block and cartridge heaters were filled with thermal cement (OB-600, Omega Engineering) to ensure they were completely contacted. Ceramic fiber boards (2732F, BXI) were fabricated into the pedestal to insulate the copper block from its surroundings, and the remaining space between the pedestal and copper block was filled up with fragments of ceramic fiber cloth. A glass chamber (20.0 cm inner diameter, 30.0 cm height), mounted over the heating section, was set on an aluminum plate and covered by another aluminum plate on its top. Two immersion heaters rated at 300 W each were submerged in stored water. Their on/off switch was controlled by a digital thermostat (DTC101, Digiten), which monitored the water temperature and formed a feedback loop with the immersion heaters to ensure the water was saturated. Besides, a ceramic fiber blanket of thickness 2 cm was wrapped around the glass chamber for reducing heat dissipation while leaving an uncovered window for simultaneous observation using a high-speed camera (IL4, Fastec Imaging). The four cartridge heaters embedded in the copper block were connected to an adjustable voltage transformer (TDGC2-5, YaeCCC). By increasing the supplied voltage from 0 to 120 V, the heating section could provide heating power from 0 to 2000 W. To change the surrounding vessel size, glass tubes of different inner diameters DT were vertically placed around the disk.

Fig. 3
(a) Schematic of the experimental apparatus. (b) Sectional view of the cylindrical part of the copper block with a 1.5-cm-diameter disk bonded on top with the graphite adhesive. The outer surface is thermally insulated except for the top and bottom.
Fig. 3
(a) Schematic of the experimental apparatus. (b) Sectional view of the cylindrical part of the copper block with a 1.5-cm-diameter disk bonded on top with the graphite adhesive. The outer surface is thermally insulated except for the top and bottom.

Temperatures at the centerline of the cylindrical part of the copper block were measured with three type-K thermocouples. Their measuring junctions were of diameter 0.5 mm, and their lengths embedded in the cylinder were 1.5 cm. The thermocouples, denoted by T1, T2, and T3, were placed along the vertical axis downwards in order with a spacing of 5 mm between each, as shown in Fig. 3(b). The thermocouple T1 was right at the cylinder top and under the graphite adhesive that bonded the disk. The holes of diameter 1 mm, through which the thermocouples were inserted, were filled with thermal cement (OB-600, Omega Engineering) to minimize their effect on the temperature distribution within the cylinder and maintain the thermocouples in their desired positions. This cement is thermally conductive but electrically insulated, thus insulating the thermocouple wires from the surroundings in addition to their PFA insulation skin until their junctions. All the thermocouples were connected to a digital thermometer (HH501DK, Omega Engineering) with a certificate of NIST traceable calibration. Before use, they were calibrated again using a high-precision platinum RTD thermometer (6412, Control Company) of uncertainty ±0.05 °C for temperatures ranging from 100 to 300 °C in a sand bath. The uncertainty in measuring temperatures with the thermocouples was less than ±0.2 °C for temperatures ranging from 100 to 250 °C and less than ±0.4 °C for temperatures ranging from 250 to 300 °C.

2.3 Experimental Procedure.

Deionized water was deaerated by boiling in an electric kettle and then poured into the container. The atmospheric pressure in the laboratory was measured to be 101.3 kPa using a built-in barometric pressure sensor in a smartphone. As monitored by the digital thermostat, water within the pool was maintained at 99 ± 1 °C. The voltage transformer was adjusted by about 2 V each time to change the heating power of the cartridge heaters. When a steady-state was reached, the supplied voltage and temperatures T1, T2, and T3 were recorded. The steady-state condition was determined based on a criterion that the temperatures measured with the thermocouples changed less than 0.2 °C in 1 min. It typically took 15–20 min after changing the supplied voltage to obtain steady-state temperatures. Meanwhile, the high-speed camera recorded the corresponding vapor removal pattern on the disk. The input power was increased in steps and immediately cut off when a sudden increase in T1 appeared, which indicated a shift to transition boiling. Thus, the boiling heat transfer data were recorded up to CHF. In each boiling experiment, the liquid height was maintained at about 20 cm by regularly adding deaerated water.

Due to the possibility of surface aging and fouling during the boiling experiments, the contact angles could change [21,24,45]. Therefore, the contact angle was measured before and after each boiling experiment, and a mean value of the pre- and postboiling contact angles was used to correlate the CHF data. Table 1 lists the pre-, postboiling, and mean contact angles of the boiling cases on 1.5-cm-diameter disks. Generally, on a copper surface with water as the test liquid, the contact angle decreases with time due to surface oxidation. In this work, no attempt was made to correlate the decrease in contact angle with boiling time. It should also be noted that with the oxidation of an initially highly oxidized copper surface, the oxide layer may become thick enough to peel off. This, in turn, can lead to an increase in the postboiling contact angle, as given in Table 1.

Table 1

Pre-, post‐boiling, and mean contact angles of the boiling cases on 1.5-cm-diameter disks

Case numberPreboiling contact angle (deg)Postboiling contact angle (deg)Mean contact angle, θ (deg)
1 (unoxidized)90 ± 269 ± 480
274 ± 251 ± 663
359 ± 346 ± 553
445 ± 329 ± 437
528 ± 425 ± 427
68 ± 315 ± 512
Case numberPreboiling contact angle (deg)Postboiling contact angle (deg)Mean contact angle, θ (deg)
1 (unoxidized)90 ± 269 ± 480
274 ± 251 ± 663
359 ± 346 ± 553
445 ± 329 ± 437
528 ± 425 ± 427
68 ± 315 ± 512

2.4 Data Reduction.

Assuming that no heat is dissipated through the ceramic fiber insulation, the heat conduction domain can be regarded as a collection of the cylindrical part of the copper block, graphite adhesive, and disk, as shown in Fig. 3(b). Due to the irregular geometry of the heating section, the boiling heat flux q and surface temperature Ts at each steady-state were determined numerically. For this purpose, a computational model built in comsol was used. Unlike a classic heat transfer problem, namely, solving spatial temperature distribution using the governing equations along with the specified boundary conditions, the temperatures T1, T2, and T3 were assumed to be known here while the heat input at the bottom of the cylinder and the temperature at the disk top surface Ts were the unknowns to be solved. Thus, a trial-and-error algorithm was applied: the spatial temperature distribution is solved once the heat input and surface temperature Ts are set in the comsol model, the heat input and Ts should be adjusted until the calculated T1 and T3 match the measured T1 and T3. The measured T2 is not involved in the algorithm but compared with the calculated T2 for validation. The boiling heat flux q and surface temperature Ts were thus determined with the boundary conditions identified. An energy balance was also performed between the heat generation from the cartridge heaters and heat lost at the boiling and ceramic fiber surfaces to validate the calculated heat flux q.

The heat flux q is considered to be proportional to the temperature difference (T3T1), as discussed in detail in Ref. [46], i.e.,
qkc(T3T1)dt
(6)
where kc = 386 ± 5 W/K m is the thermal conductivity of 110 Cu, and dt = 10.0 ± 0.5 mm is the distance between the thermocouples T1 and T3. As all the temperatures recorded by the thermocouples T1, T2, and T3 in this study are less than 250 °C, the temperature uncertainties are ±0.2 °C. With given boundary conditions and thermal conductivities of the materials, the proportionality constant depends on the geometry of the heating section and thus differs for the disks of different diameters. The proportionality constant was determined by fitting the numerically calculated heat flux q against kc(T3T1)/dt and features uncertainty less than 0.3% for all the disks in this study. Thus, with the uncertainty in the proportionality constant reasonably small, the uncertainty of the heat flux q was then determined by
δqq=(δkckc)2+(δdtdt)2+[δ(T3T1)T3T1]2
(7)

The heat flux uncertainty was calculated using Eq. (7), e.g., for a case of qCHF = 77 W/cm2, the heat flux uncertainty was ±4 W/cm2. Overall, the uncertainties of the measured CHFs in this study were within ±6.3%. Any two of the temperatures T1, T2, and T3 can be used to calculate the heat flux q and surface temperature Ts. The temperatures T1 and T3 were chosen as the smallest heat flux uncertainty resulted by their largest distance, as described in Eq. (7).

Table 2 summarizes the parameters of all the experiments performed to study the effects of heater size, contact angle, and surrounding vessel size on pool boiling CHF from horizontal surfaces. The experiments, calculation of CHF, and uncertainty analysis are further detailed in Ref. [46].

Table 2

Experimental parameters: disk diameter D, contact angle θ, and surrounding vessel diameter DT

D (cm)D/λdθ (deg)DT (cm)D/DT
1.00.3710, 27, 43, 54, 65, 801.01
2.00.5
5.00.2
20.00.05
1.50.5512, 27, 37, 53, 63, 801.51
3.00.5
7.50.2
20.00.075
2.00.7412, 24, 39, 51, 60, 802.01
4.00.5
10.00.2
20.00.1
D (cm)D/λdθ (deg)DT (cm)D/DT
1.00.3710, 27, 43, 54, 65, 801.01
2.00.5
5.00.2
20.00.05
1.50.5512, 27, 37, 53, 63, 801.51
3.00.5
7.50.2
20.00.075
2.00.7412, 24, 39, 51, 60, 802.01
4.00.5
10.00.2
20.00.1

3 Results and Discussion

CHF for the disks of diameters D = 1.0, 1.5, and 2.0 cm (0.37λd, 0.55λd, and 0.74λd) was measured for different contact angles θ and surrounding vessel diameters DT.

3.1 Heater Size Effect.

Figure 4 shows the dimensionless CHF measured for the disks of contact angle θ10deg as a function of heater size nondimensionalized with the two-dimensional “most dangerous” wavelength Lc/λd when the surrounding vessel diameter DT was equal to the disk diameter D, i.e., D/DT=1. As the disk diameter D decreases from 2.0 to 1.5 and to 1.0 cm, the CHF increases from 143 to 181 and to 203 W/cm2, and the dimensionless CHF, accordingly, increases from 1.30 to 1.64 and to 1.85. For comparison, dimensionless CHFs of various liquids reported by Costello et al. [32], Lienhard et al. [15], and Maracy and Winterton [24] for fully wetted (θ=0deg) horizontal heaters of different sizes are also given in Fig. 4. As shown in Fig. 4, the dependence of dimensionless CHF on heater size in wavelengths Lc/λd in this study is comparable to that of cited CHF data. The consistent trend formed by the CHF data collectively confirms that CHF increases with decreasing heater size and is higher than that for an infinite flat plate when the heater size is less than λd, i.e., Lc/λd<1. Prediction from Eq. (4) proposed by Lienhard et al. [15] is also shown as a dotted line in Fig. 4 with Nj = 1. The enhancement in CHF resulting from decreasing heater size when Lc/λd1 is captured by Eq. (4). However, when Lc/λd<1, the observed CHFs are much less than the prediction with the one-jet model. As the heater size continues to decrease, the fraction of heater surface area occupied by vapor jet increases, allowing a higher fractional vapor removal rate, thereby enhancing the CHF.

Fig. 4
Dimensionless CHF as a function of heater size in wavelengths Lc/λd for the well wetted disks
Fig. 4
Dimensionless CHF as a function of heater size in wavelengths Lc/λd for the well wetted disks

Figure 5 shows the sequential photographs of boiling near CHF for the disks of diameters (a) D = 1.0 cm, (b) D = 1.5 cm, and (c) D = 2.0 cm and contact angle θ10deg to visualize the corresponding vapor removal pattern. In the past, some had argued about the validity of the assumption of the existence of vapor jets in Zuber's model. Here, a single vapor jet is clearly seen on the disk surface near CHF. Each row of the sequential photographs depicts a complete cycle of one vapor jet forming on the disk surface. From 0 to 100 ms, with a time interval of 20 ms, six photographs in order show the development of the vapor jet. The sequential photographs in Fig. 5 visually validate the argument by Lienhard et al. [15], that is, unlike the vapor removal pattern at CHF on large horizontal flat plate heaters depicted in Zuber's hydrodynamic theory, only one vapor jet exists on heaters of sizes less than the “most dangerous” Taylor wavelength λd. Furthermore, it is also observed that the vapor jet size, confined by the surrounding vessel, decreases with decreasing heater size.

Fig. 5
Sequential photographs of vapor removal pattern near CHF for the well wetted disks of diameters (a) D = 1.0 cm, (b) D = 1.5 cm, and (c) D = 2.0 cm. For all the cases, the ratio of heater size to surrounding vessel size D/DT is 1.
Fig. 5
Sequential photographs of vapor removal pattern near CHF for the well wetted disks of diameters (a) D = 1.0 cm, (b) D = 1.5 cm, and (c) D = 2.0 cm. For all the cases, the ratio of heater size to surrounding vessel size D/DT is 1.
The dimensionless CHF for each disk can be calculated based on the hydrodynamic theory using the actual vapor jet diameter that was directly measured from the photographs taken when the vapor jet was fully developed. For the disks of diameters D = 1.0, 1.5, and 2.0 cm, the measured jet diameters ranged from 0.65 to 0.87 cm, 0.87 to 1.37 cm, and 1.05 to 1.55 cm, respectively, the corresponding average jet diameters Dv were 0.76, 1.12, and 1.30 cm. It is worth noting that the vapor jet diameter Dv = 1.30 cm observed on the 2.0-cm-diameter disk of D/λd=0.74 is still about half of the two-dimensional “most dangerous” Taylor wavelength λd = 2.72 cm. The critical relative velocity for a plane interface of inviscid liquid and vapor streams flowing parallel to each other is given by Kelvin–Helmholtz instability, i.e.,
(UvUl)crit=2πσ(ρl+ρv)λHρlρv
(8)
where Ul and Uv are the velocities of the liquid and vapor, respectively, in the direction of the vapor stream. An effort was also made to measure the Helmholtz unstable wavelength at water–steam interface from the photographs taken, as shown in Fig. 6. There is significant uncertainty in the measurement because of the challenges of distinguishing the peak and valley of a wave and differentiating the wave from breakups of the jet. For the disks of diameters D = 1.0, 1.5, and 2.0 cm, the measured wavelengths ranged from 1.5 to 2.3 cm, 1.6 to 2.6 cm, and 1.3 to 2.6 cm, respectively, the corresponding average wavelengths λH were 1.9, 2.1, and 2.0 cm. Figure 6 shows one of the clearest photographs of wavelengths measured at water-steam interface for the disks of diameters (a) D = 1.0 cm and (b) D = 1.5 cm, and (c) half wavelengths measured for the 2.0-cm-diameter disk. It is of great interest to note that the measured Helmholtz wavelengths λH, varying from 1.3 to 2.6 cm, are comparable to the two-dimensional critical and “most dangerous” Taylor wavelengths of 1.57 and 2.72 cm, respectively. The velocities of the liquid Ul and vapor Uv are also governed by an equation of mass balance performed at a horizontal cross section of the vapor stream, i.e.,
ρvAvUv+ρl(AAv)Ul=0
(9)
where Av is the cross-sectional area of the vapor jet, and A is the area of a unit “cell” on which the vapor jet is located. As there is only one vapor jet existing on the disk surface in this study, A is the top surface area of the disk. The critical vapor velocity Uv,crit can be obtained by solving Eqs. (8) and (9) and expressed as
Uv,crit=2πσ(ρl+ρv)λHρlρv[1+ρvAvρl(AAv)]1
(10)
Fig. 6
Wavelengths at water–steam interface for the well wetted disks of diameters (a) D = 1.0 cm and (b) D = 1.5 cm. (c) Half wavelengths at water-steam interface for the well wetted 2.0-cm-diameter disk.
Fig. 6
Wavelengths at water–steam interface for the well wetted disks of diameters (a) D = 1.0 cm and (b) D = 1.5 cm. (c) Half wavelengths at water-steam interface for the well wetted 2.0-cm-diameter disk.
The CHF can be expressed in the form of energy balance as
qCHF=ρvhfgUv,critAvA
(11)
By substituting the critical vapor velocity Uv,crit with Eq. (10) and recognizing (ρl+ρv)/ρl1 as long as the system pressure is enough less than the critical pressure, the CHF expression can be written as
qCHF=ρvhfg2πσλHρv[1+ρvAvρl(AAv)]1AvA
(12)
Replacing Av/A by r and dividing Eq. (12) by Eq. (3) give the dimensionless CHF expression
qCHFqCHF,Z=19.15λH(1r+ρv/ρl1r)1[σ(ρlρv)g]1/4
(13)

According to Eq. (13), the dimensionless CHF should be a function of the Helmholtz unstable wavelength λH and fraction of heater surface area occupied by vapor jet r for a given liquid under a certain condition. When r is relatively small, the term (ρv/ρl)/(1r) at low pressures is much less than the term 1/r in Eq. (13) and so can be reasonably omitted. For large horizontal flat plate heaters, λH=λd=2.72cm,r=π/16, and the dimensionless CHF is calculated to be 1.14 [13,14]. In this study, however, the value of r might be large for the small disks. Thus, the term (ρv/ρl)/(1r) in Eq. (13) was included in the following calculations.

By substituting the properties of saturated water and steam at one-atmosphere pressure, i.e., σ = 0.0589 N/m, ρl = 957.9 kg/m3, and ρv = 0.596 kg/m3, normal earth gravitational acceleration g = 9.8 m/s2, as well as the average measured vapor jet diameter Dv (r=(Dv/D)2) and the average measured Helmholtz unstable wavelength λH for each disk into Eq. (13), the corresponding dimensionless CHF was calculated to be 4.02, 3.69, and 2.86 for the disks of diameters D = 1.0, 1.5, and 2.0 cm, respectively. Figure 7 shows the comparison of the measured dimensionless CHFs and those calculated from Eq. (13). The bounds on the calculated average values correspond to the cases when maximum and minimum values of Dv were combined with maximum and minimum values of λH, respectively. It is not a surprise that the predicted CHFs are higher than the measured CHFs. The two-phase flow is speculated to be more complicated than just parallel flows of vapor and liquid traveling in opposite directions. The vapor can drag part of the liquid nearby to move upward. This effect will not be significant until the vapor jet is confined by the surrounding vessel. Besides, the route of the liquid flowing back is squeezed into a thin film where the viscosity dominates. These mechanisms not considered in the hydrodynamic theory dictate CHF when the heater size becomes less than the two-dimensional “most dangerous” Taylor wavelength λd.

Fig. 7
Comparison of the measured dimensionless CHF and that calculated from Eq. (13) for the well wetted disks of diameters D = 1.0, 1.5, and 2.0 cm
Fig. 7
Comparison of the measured dimensionless CHF and that calculated from Eq. (13) for the well wetted disks of diameters D = 1.0, 1.5, and 2.0 cm

3.2 Contact Angle Effect.

Figure 8 shows the dimensionless CHF measured for the disks as a function of contact angle θ when the surrounding vessel diameter DT was equal to the disk diameter D, i.e., D/DT=1. For the disks of diameters D = 1.0, 1.5, and 2.0 cm, as the contact angle increases from about 10 to 80 deg, the CHF decreases from 203 to 76 W/cm2, 181 to 74 W/cm2, and 143 to 77 W/cm2, respectively, and the dimensionless CHF decreases from 1.85 to 0.69, 1.64 to 0.68, and 1.30 to 0.70, accordingly. CHFs reported by Maracy and Winterton [24] for a 2.66-cm-diameter surface and Girard et al. [31] for a 1 cm × 1 cm surface of different contact angles are also added for comparison. It should be pointed out that Maracy and Winterton conducted transient boiling and the surface used by them was confined by a glass tube of inner diameter that was equal to the surface diameter, namely, D/DT=1. Girard et al. performed steady-state boiling without surrounding vessel size reported. The corresponding D/DT was assumed to be close to 0 by observing the schematic of their experimental apparatus. As shown in Fig. 8, the trend of CHF against contact angle θ for the 1.0-cm-diameter disk from this study is in good agreement with that for the 1 cm × 1 cm surface reported by Girard et al. [31]. The CHF data in Fig. 8 collectively show that CHF decreases with increasing contact angle regardless of the heater size. In addition, Fig. 8 shows that CHF increases with decreasing heater size only for low contact angles (or well wetted surfaces), which is attributed to the increase of vapor removal limit, as discussed in the earlier sections. However, for high contact angles (or partially wetted surfaces), it appears that it is not the vapor removal limit that determines the maximum heat flux. Even the cited CHF data show some variances at high contact angles, the heater size effect clearly diminishes as the contact angle increases by observing the CHFs from this study. It is in good agreement with the argument by Dhir and Liaw [25] that CHF for partially wetted surfaces is set by the vapor production limit, which is independent of the heater size as long as there are many active nucleation cavities on the surface. The difference in CHFs for high contact angles reported in Refs. [23,24,31] might be attributed to experimental uncertainties, different types of boiling (steady-state or transient), different methods or criteria of determining contact angle, different surface orientations (horizontal or vertical), and other uncontrolled parameters, such as surrounding vessel size and liquid height.

Fig. 8
Dimensionless CHF as a function of contact angle θ for the disks of diameters D = 1.0, 1.5, and 2.0 cm
Fig. 8
Dimensionless CHF as a function of contact angle θ for the disks of diameters D = 1.0, 1.5, and 2.0 cm

Figure 9 shows the sequential photographs of boiling near CHF on the 1.5-cm-diameter disk to visualize the corresponding vapor removal pattern when the disk featured contact angles (a) θ = 12 deg, (b) θ = 37 deg, and (c) θ = 63 deg. When the disk features different contact angles θ, the vapor generated always tends to fill the surrounding vessel and shows no obvious difference in the vapor jet size. Even so, as the contact angle θ increases from 12 to 63 deg, the vapor jet appears to take less space and leave more space for liquid flowing back toward the disk surface, as shown in the photographs taken at 60 ms in Fig. 9, which is possibly attributed to the reduced vapor generation rate with increasing contact angle θ.

Fig. 9
Sequential photographs of vapor removal pattern near CHF for the 1.5-cm-diameter disk when it featured contact angles (a) θ = 12 deg, (b) θ = 37 deg, and (c) θ = 63 deg. For all the cases, the ratio of heater size to surrounding vessel size D/DT is 1.
Fig. 9
Sequential photographs of vapor removal pattern near CHF for the 1.5-cm-diameter disk when it featured contact angles (a) θ = 12 deg, (b) θ = 37 deg, and (c) θ = 63 deg. For all the cases, the ratio of heater size to surrounding vessel size D/DT is 1.

Both Dhir and Liaw's vapor generation model [25] and Kandlikar's force balance model [26] can explain the decrease in CHF with worsening surface wettability. Meanwhile, the former can also capture the interacting effect of heater size and contact angle on CHF. Figure 8 shows no apparent heater size effect for the partially wetted disks as the CHF is set by the maximum vapor generation rate, which is independent of the heater size. However, for the well wetted disks, the CHF is set by the maximum vapor removal rate as assumed in the hydrodynamic theory.

3.3 Surrounding Vessel Size Effect.

Figure 10 shows the dimensionless CHF measured for the disks of contact angle θ10deg as a function of the ratio of heater size to surrounding vessel size D/DT. For the disks of diameters D = 1.0, 1.5, and 2.0 cm, as the vessel diameter increases from the disk diameter to 20.0 cm, the CHF increases from 203 to 233 W/cm2, 181 to 218 W/cm2, and 143 to 150 W/cm2, respectively, and the dimensionless CHF increases from 1.85 to 2.11, 1.64 to 1.98, and 1.30 to 1.36, accordingly. CHFs reported by Bockwoldt et al. [35] for a horizontal 1.5-cm-diameter surface are also added for comparison. They did not report contact angle but stated that CHF was measured for the nickel-plated copper surface that was aged until the CHF became stable. Thus, it is assumed that the surface used by them was well wetted. Besides, as they investigated the effects of surrounding vessel size and liquid height, only the CHFs measured for liquid heights of about 20 cm are plotted in Fig. 10 to eliminate any effect of changed liquid height. It can be seen from Fig. 10 that the trend of CHF against the ratio of heater size to surrounding vessel size D/DT for the 1.5-cm-diameter disk from this study agrees well with that for the 1.5-cm-diameter surface reported by Bockwoldt et al. [35]. Figure 10 establishes that CHF increases as the surrounding vessel size increases. Besides, the CHF data also point out that the magnitude of enhancement in CHF decreases as the heater size increases. For the 2.0-cm-diameter disk, the effect of surrounding vessel size is within the uncertainty of the CHF data.

Fig. 10
Dimensionless CHF as a function of the ratio of heater size to surrounding vessel size D/DT for the well wetted disks of diameters D = 1.0, 1.5, and 2.0 cm
Fig. 10
Dimensionless CHF as a function of the ratio of heater size to surrounding vessel size D/DT for the well wetted disks of diameters D = 1.0, 1.5, and 2.0 cm

Figure 11 shows the sequential photographs of boiling near CHF for the 1.5-cm-diameter disk of contact angle θ = 12 deg to visualize the corresponding vapor removal pattern when the disk was surrounded by vessels of diameters (a) DT = 1.5 cm, (b) DT = 3.0 cm, (c) DT = 7.5 cm, and (d) DT = 20.0 cm. As seen in Fig. 11(a), when the vessel diameter DT is 1.5 cm, the generated vapor develops vertically into a continuous vapor jet. However, when the vessel diameter DT increases to 3.0, to 7.5, and to 20.0 cm, as shown in Figs. 11(b)11(d), the removal of generated vapor in a continuous columnar structure appears to break down, and large vapor slugs, which appear to be discontinuous, now leave the disk surface.

Fig. 11
Sequential photographs of vapor removal pattern near CHF for the 1.5-cm-diameter disk of contact angle θ = 12 deg when it was surrounded by vessels of diameters (a) DT = 1.5 cm, (b) DT = 3.0 cm, (c) DT = 7.5 cm, and (d) DT = 20.0 cm
Fig. 11
Sequential photographs of vapor removal pattern near CHF for the 1.5-cm-diameter disk of contact angle θ = 12 deg when it was surrounded by vessels of diameters (a) DT = 1.5 cm, (b) DT = 3.0 cm, (c) DT = 7.5 cm, and (d) DT = 20.0 cm

The enhancement in CHF by enlarging the surrounding vessel can be attributed to induced convection flow as well as improved stability of vapor columns. As the surrounding vessel size increases, the induced convection flow increases convective heat transfer contribution and affects the formation and stability of vapor jets. As can be deduced from Eq. (8), the upward flow of liquid adjacent to the vapor jet will increase the critical vapor velocity for instability to occur, thus increasing the CHF. However, for larger heaters, when the vapor jet diameter is much smaller than the heater size, the convective contribution will increase. In the absence of measurement of the flow field in the vicinity of the vapor jets, it is difficult to separate the contribution of each.

4 Conclusions

By boiling saturated water at one-atmosphere pressure on horizontal flat heaters of sizes less than the two-dimensional “most dangerous” Taylor wavelength, of different contact angles, and surrounded by vessels of different sizes, the following conclusions are arrived at.

  1. For well wetted surfaces, CHF increases with decreasing heater size when the heater size Lc is less than the two-dimensional “most dangerous” Taylor wavelength λd and appears to reach an asymptotic value at Lc/λd0.3.

  2. CHF decreases with increasing contact angle regardless of the heater size.

  3. The heater size effect on CHF is not pronounced for partially wetted surfaces where the CHF is limited by the maximum vapor generation rate, which is independent of the heater size when many active nucleation sites are present.

  4. CHF increases with increasing surrounding vessel size. However, this enhancement degrades with increasing heater size.

  5. The highest measured CHF of 233 W/cm2 or 2.11 times Zuber's CHF prediction was obtained on a 1.0-cm-diameter disk of contact angle about 10 deg surrounded by a large vessel.

Acknowledgment

The Boiling and Phase Change Heat Transfer Laboratory at the University of California, Los Angeles (UCLA) acknowledges the support of various donors that has made its operations possible.

Nomenclature

     
  • d =

    thickness of disk, mm or m

  •  
  • g =

    gravitational acceleration, m/s2

  •  
  • q =

    heat flux, W/cm2 or W/m2

  •  
  • r =

    fraction of heater surface area occupied by vapor jet

  •  
  • A =

    surface area, cm2 or m2

  •  
  • U =

    velocity, m/s

  •  
  • D =

    diameter of disk, cm or m

  •  
  • dt =

    distance between thermocouples T1 and T3, mm or m

  •  
  • hfg =

    latent heat of vaporization, J/kg

  •  
  • kc =

    thermal conductivity of 110 Cu, W/m K

  •  
  • th =

    heating time of thermal oxidation, min

  •  
  • Av =

    cross-sectional area of vapor jet, cm2 or m2

  •  
  • DT =

    inner diameter of vessel, cm or m

  •  
  • Dv =

    diameter of vapor jet, cm or m

  •  
  • Lc =

    characteristic length, cm or m

  •  
  • Nj =

    number of vapor jets on heater surface

  •  
  • Th =

    heating temperature of thermal oxidation, ° C

  •  
  • Ts =

    surface temperature, ° C

Greek Symbols

    Greek Symbols
     
  • θ =

    contact angle, deg

  •  
  • λc =

    critical Taylor wavelength, cm or m

  •  
  • λd =

    “most dangerous” Taylor wavelength, cm or m

  •  
  • λH =

    Helmholtz unstable wavelength, cm or m

  •  
  • ρ =

    density, kg/m3

  •  
  • σ =

    surface tension, N/m

  •  
  • ϕ =

    inclination angle from horizontal, deg

Subscripts

    Subscripts
     
  • j =

    vapor jet

  •  
  • l =

    liquid

  •  
  • v =

    vapor

  •  
  • Z =

    Zuber's critical heat flux prediction

  •  
  • crit =

    critical condition

  •  
  • rec =

    dynamic receding contact angle

  •  
  • CHF =

    critical heat flux

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