Experimental investigations on bubble departure diameter and frequency of methane saturated nucleate pool boiling at four different pressures

International Journal of Heat and Mass Transfer 112 (2017) 662–675

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Experimental investigations on bubble departure diameter and frequency of methane saturated nucleate pool boiling at four different pressures Hanzhi Chen a,b, Gaofei Chen a, Xin Zou a, Yuan Yao a,b, Maoqiong Gong a,⇑ a b

CAS Key Laboratory of Cryogenics, Technical Institute of Physics and Chemistry, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049, China

a r t i c l e

i n f o

Article history: Received 7 December 2016 Received in revised form 3 May 2017 Accepted 8 May 2017

Keywords: Nucleate pool boiling Methane Bubble departure diameter Bubble departure frequency

a b s t r a c t In this work, bubble departure diameters and bubble departure frequencies on saturated nucleate pool boiling of methane were studied. The experiments were conducted on the upper surface of a smooth vertical copper cylinder, at pressures of 0.15 MPa, 0.2 MPa, 0.3 MPa and 0.4 MPa with heat fluxes varying from 10.64 kW m
2 to 79.25 kW m
2. Bubble departure diameters were measured from the images captured by a high-speed digital camera at lower heat fluxes less than 79.25 kW m
2, at which isolated bubbles were obtained. Bubble departure frequencies were calculated by counting the numbers of the detachment bubbles and the corresponding time intervals. Their relationship with Jacob number (Ja) was analyzed. With an increase in Ja at a given pressure, bubble departure diameter increases while bubble departure frequency tends to decrease. After the comparisons with six most used correlations for bubble departure diameter, a new correlation was developed within ±20% deviation from most of the experimental data. Additionally, a new correlation for the relationship between bubble departure diameter and departure frequency was also proposed within ±20% deviation from most of the experimental fD2d. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Nucleate boiling has raised wide attention in the world because of its considerable capacity to carry heat from a heater to a liquid, which also makes it an important process in refrigeration and other industrial applications. During nucleate boiling, a bubble nucleates and grows from a single site on the heated surface. When it grows to a certain size, it will depart from the surface into the liquid as a result of the joint effects of various forces, such as surface tension force, buoyancy force, inertia force and so on. It is supposed that bubble detachment on the heated surface is directly related to the boiling heat transfer. Due to the importance of the bubble departure process, bubble departure diameter has been shown to be one of the most important parameters in the heat transfer analysis. Bubble departure frequency, as another important parameter, usually correlates with bubble departure diameter. Extensive research has been conducted for nearly eighty years, and the most used correlations for them are summarized in Tables 1 and 2. ⇑ Corresponding author. E-mail address: [email protected] (M. Gong). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.05.031 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

For bubble departure diameter, the correlations can be grouped into two types. One incorporates the term of {r/[g(ql
qv)]}1/2, the other doesn’t. The correlations of the first type have tried to relate Bond number to the effects of liquid properties, pressure, superheat or Ja number, heat flux, as well as surface properties (Fritz [1], Borinshansky and Fokin [2], Ruckenstein [3], Cole and Shulman [4], Cole [5], Cole and Rohsenow [6], Kutateladze and Gogonin [7], Jensen and Memmel [8], Kim and Kim [9], Fazel and Shafaee [10], Hamzekhani et al. [11]). The correlations of the second type usually incorporate the bubble growth rate (Golorin et al. [12], Zeng et al. [13], Yang et al. [14]). In Golorin et al. [12], bd is a term related to the bubble growth rate, and bd = 6.0 for water, alcohol and benzene. Analytical models about bubble growth (Zuber [15], Cooper and Lloyd [16], Mikic et al. [17]) have been developed. Recently, experimental and numerical investigations (Mukherjee and Dhir [18], Siedel [19]) were also conducted on the bubble growth, as well as the interaction and coalescence of adjacent bubbles. Besides all these correlations, other efforts have been done to explore bubble departure diameters. Gravity effects on bubble departure diameter were studied. Kim [20] made good reviews

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663

Nomenclature a A AAD AD Ar a(t) b Bo c C Ca Cd Cf cp d d1, d2 dw d⁄ Dd Dd,m DF f g h hlv Ja Jac K K1 L m n N

the length of the horizontal axis of a departure bubble (m) amplitude, a parameter in Gaussamp function the average absolute deviation (%) the average deviation (%) Archimedes number, Ar = [gql(ql
qv)/l2l ] [r/g(ql
qv)]3/2 bubble growth rate (m s
1) the length of the vertical axis of a departure bubble (m) Bond number, Bo = gD2d(ql
qv)/r the length of the vertical axis of the upper part of a departure bubble (m) a parameter given by Borinshansky and Fokin [2] or Cole and Rohsenow [6] capillary number a bubble drag coefficient in Cole [43] drag coefficient specific heat capacity (J kg
1 K
1) the length of the vertical axis of the lower part of a departure bubble (m) a parameter in Golorin et al. [12] contact diameter (m) the gap between roughness elements with the same order of magnitude as the roughness height (mm) bubble departure diameter (m) mean bubble departure diameter (m) diameter with Fritz [1] bubble departure frequency (Hz) gravitational acceleration (m s
2) heat transfer coefficient (kW m
2 K
1) latent heat (J kg
1) Jacob number, Ja = qlcp4T/(qvhlv) modified Jacob number in Cole and Rohsenow [6], Ja = qlcplTc/(qvhlv) an empirical value in bubble growth rate in Zeng et al. [13] a parameter defined by Jensen and Memmel [8], Kl=(Ja/ Prl)2(Ar)
1 the calibrated length of the ruler (m) an empirical exponent an empirical value in bubble growth rate the number of bubbles

on reduced gravity boiling heat transfer, which showed that gravity level was one of the important parameters in analyzing bubble sizes. Numerical simulation, as the advances in computing power and modeling techniques, has also been a successful approach for modeling bubble dynamics. Dhir [21] presented a review of numerical simulation of pool boiling. The predictions of bubble departure diameter were included as well as other influencing parameters. For bubble departure frequency, most of the correlations were before 1970. It usually correlates with bubble departure diameter and can be written in this form: m

fDd ¼ f ðql ; qv ; g; r; al ; JaÞ

ð1Þ

where exponent m is an empirical value. fDdm is fitted as a function of liquid and vapor density, gravity level, thermal diffusivity and so 2 on. From Table 2, three products, fD1/2 d , fDd, fDd, are respectively considered to find their relationship with other influencing parameters. Recently, Kim et al. [22] used wire heaters immersed in FC-72 coolant and water for investigations of pool boiling. Regardless of heat

P pressure (MPa) Pr Prandtl number, Pr = ??cp/?? q heat flux (kW m
2) R Radius in Eq. (19) s1, s2 fitting parameters in Eqs. (29) and (30) t time (s) T temperature (K) Tc critical temperature (K) tG growth time (s) tW waiting time (s) 4T surface superheat (K) u velocity (m s
1) v1, v2 fitting parameters in Eq. (41) V volume in Eq. (2) w width, a parameter in Gaussamp function x a parameter in Eq. (39) X1, X2, X3 pixels of L, a and b respectively (pixel) y the relative frequency of bubble departure diameter y0 offset, a parameter in Gaussamp function Greek letters a thermal diffusivity (m2 s
1), a = k/(qcp) b the mean linear thermal expansion coefficient for stainless steel relative to 299.15 K bd a coefficient related to the growth of vapor bubbles the proportionality factor b⁄ d thermal layer thickness e a factor that represents the effect of superheat in Yang et al. [14] ?? percentage of data predicted within ±30% deviation (%) h contact angle, deg ?? thermal conductivity (W m
1 K
1) l dynamic viscosity (Pa s) q density (kg m
3) r surface tension (N m
1) w a modified factor in Yang et al. [14] Subscripts l liquid phase s saturation state v vapor phase w wall

flux, the relationship between f and Dd was found to be written as fD4.85 = 7.2  10
8. From the pentane pool boiling on artificial d nucleation sites, Siedel et al. [19] found that the bubble frequency was proportional to the wall superheat, and fDd was also proportional to the wall superheat. The complexity of predicting the relationship between the bubble departure diameter and bubble departure frequency is not only due to the effects of thermodynamic properties but also the influence of the heated surface roughness. As the experiments shown in McHale and Garimella [23], the surface roughness was one of the important affecting parameters. As mentioned above, many efforts have been devoted to investigating bubble departure mechanisms. However, owing to the complexity of boiling phenomena, correlations to predict bubble departure diameter or bubble departure frequency remain a hotspot. Moreover, methane as the working fluid in pool boiling experiments, especially its departure behavior, can hardly be found in the published literature. Methane, as the predominant component of the liquefied natural gas (LNG), is important for the sustainable growth of economy and society all over the world. Due to its much larger density than the gaseous state, it can be stored

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Table 1 Correlations for bubble departure diameter.

Table 2 Correlations for the relationship between bubble departure diameter and bubble departure frequency.

Reference

Correlation

Fritz [1]

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dd ¼ 0:0208h gðq r
q Þ l

Borinshansky and Fokin [2]

Ruckenstein [3]

Cole and Shulman [4] Cole [5] Cole and Rohsenow [6]

Golorin et al. [12]

Kutateladze and Gogonin [7]

Jensen and Memmel [8] Zeng et al. [13]

Yang et al. [14]

Jamialahmadi et al. [40] Kim and Kim [9] Fazel and Shafaee [10] Hamzekhani et al. [11]

v

where h is the contact angle in degrees sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C2 þ1 D2F where DF is the diameter with Fritz [1]  0:4     6 ql qv h C¼ g ql
qv ql qv hlv " #1=3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3p2 q2l a2l g 1=2 ðql
qv Þ1=2 r Ja4=3 Dd ¼ 3=2 gðql
qv Þ r rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1000 r Dd ¼ p gðql
qv Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r Dd ¼ 0:04Ja gðql
qv Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 5=4 r Jac Dd ¼ C gðql
qv Þ where C = 1.5  10
4 for water, C = 4.65  10
4 for fluids other than water Dd d2 ¼1þ d1 d1  1:65d r d1 ¼ gðql
qv Þ     15:6ql 1=3 bd kl ðT w
T s Þ 2=3 d2 ¼ gðql
qv Þ qv hlv where kl is the liquid conductivity, d⁄ = 6.0  10
3 mm, bd = 6.0 for water, alcohol and benzene rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 r Dd ¼ 0:25ð1 þ 105 K 1 Þ gðql
qv Þ  2 Ja ðArÞ
1 For K1 < 0.06, where K 1 ¼ Pr l rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=3 r Dd ¼ 0:19ð1:8 þ 105 K 1 Þ gðql
qv Þ ( )n=ð2
nÞ 3 K 2=n 3 Dd ¼ 2 ½ Cs n2 þ nðn
1Þ 4 g 2 Dd C ¼
þ DF DF

bubble growth rateaðtÞ ¼ Ktn pffiffiffiffiffiffiffiffiffiffiffiffi 1 q C pl T s aPr3 Dd ¼ 3:0557  103 l qv hlv e

e ¼ wJa0:3 where w is a modified factor 1 0:01425q ¼ 96:75 þ Dd ln q where, q is in W m
2, Dd is in m rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r Dd ¼ 0:1649Ja0:7 gðql
qv Þ 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 lv q r Þ Dd ¼ 40ð ql hlv r cos h gðql
qv Þ Bo ¼ Ca0:25 Ja0:75 Ar 0:05 lv u Ca ¼ r cos h u ¼ q qh

l lv

and transported over long distances. In the process of liquefaction and gasification, extensive heat exchangers are used. Therefore, bubble departure characteristics for methane are necessary for the optimization and manufacture of LNG facilities, which makes pool boiling experiments of methane significant. Furthermore, methane is also one of the hydrocarbons that can be environmentally friendly refrigerants. It is a common component in mixed refrigerants for the low-temperature Joule-Thomson refrigerator. To provide more basic data for future research, this work highlights concern on bubble departure diameters and bubble departure frequencies of methane in nucleate pool boiling. The experiments were conducted on the upper surface of a smooth vertical copper cylinder. The corresponding heat fluxes, superheats and heat transfer coefficients were presented. Meanwhile, the results were compared with some existed correlations and new correlations were developed.

Reference

Correlation

Jakob and Fritz [41] Peebles and Garber [42]

fDd ¼ 0:078

Cole [43]

McFadden and Grassmann [37] Zuber [38]

#1=4  " tG rgðql
qv Þ fDd ¼ 1:18 t W þ tG q2l  12 1 4gð q
q Þ l v fD2d ¼ 3C d ql where Cd is a bubble drag coefficient, Cd = 1 for water at 1 atm pffiffiffi 1=2 fDd ¼ 0:56 g " fDd ¼ 0:59

Ivey [44]

rgðql
qv Þ q2l

#1=4

1

For dynamically controlled growth: fD2d ¼ constant 2

Mikic and Rohsenow [39]

For thermally controlled growth: fDd ¼ constant  1=2 1=2 4 pffiffiffiffiffiffiffiffiffiffiffi tG tG 1=2 f Dd ¼ Ja 3pal fð Þ þ 1þ
1g p tG þ tW tG þ tW tG For 0:15 < tG þtW < 0:8

Kim et al. [22]

fDd

2

fDd ¼ 0:6889Ja2 pal 4:85

¼ 7:2  10
8

2. Experiments 2.1. Test facility Fig. 1 shows a schematic diagram of the experimental apparatus. It is mainly composed of four parts, namely, boiling vessel, heating system, refrigeration system and data acquisition system. Compared to our previous work (Zhao et al. [24]), the experimental apparatus has a ruler inside the pool as the reference length to measure the bubble size. As shown in Fig. 2, the ruler is made of stainless steel, and its calibrated length is 2.007 mm with an uncertainty ±0.003 mm. The calibrated length in the image is 68 pixels, which leads to 0.030 mm per pixel. The maximum uncertainty of the length measurement is 2 pixels. Therefore, the maximum uncertainty for length measurement is 0.060 mm. The experiments were conducted on the upper surface of a smooth vertical copper cylinder. The measured roughness of the boiling surface was 68.1 nm (root mean square average of the height deviations) and 50.7 nm (the arithmetic average of the absolute values of the surface height deviations). In order to capture both bubble behavior and clear images of the ruler inside the pool, two cold light sources were placed on both sides of the boiling vessel. The images of bubble behavior were recorded by the Phantom high-speed digital camera, with a frame rate of 2619 fps and a full resolution of 960  720 pixels. Accurate time intervals of images were obtained from the Phantom video player. Sizes of bubbles were measured from the images. Bubble departure frequencies were calculated by counting the numbers of the detachment bubbles and the corresponding time intervals. 2.2. Experimental procedures and uncertainties In this work, nucleate pool boiling experiments of methane were performed at saturation. For the purpose of taking clear images of the departure bubbles, the experiments were carried out at low heat fluxes in the range from 10.64 kW m
2 to 79.25 kW m
2. The data were recorded in the descending heat fluxes from the largest to the lowest, in which way the hysteresis effects were minimized.

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Fig. 1. Schematic diagram of experimental apparatus: 1-Visualization boiling vessel; 2-Condenser; 3-Auxiliary electric heater; 4-Main electric heater; 5-Copper cylinder; 6Visual windows; 7-Cryocooler; 8-Refrigerant tank; 9-Illuminant; 10-High speed camera; 11-Ruler.

Fig. 2. The calibrated length of the ruler.

Before the experiments, the boiling vessel and the condenser were initially well cleaned by filling with acetone for half an hour and then swept away with pure nitrogen gas. Next, the system was filled with Helium gas. After the bottom of the copper cylinder immersed in the liquid nitrogen for five minutes, the Helium Mass Spectrometer Leak Detector was used to check the airtightness of the system under low-temperature conditions. Following this, the system was also kept at the pressure of about 1 MPa for 24 h. In addition, the experimental system was purged with methane for three times to ensure no contamination inside it. Finally, the system was fitted with a vacuum insulation vessel. In the experiments, the vacuum pump and the vacuum gauge were first turned on. When the vacuum pressure was under 10 Pa, the cryocooler system began to work and the methane was gradually condensed into liquid. Meanwhile, Keithley 2700 was monitoring the experimental process. The valve of the refrigerant tank was closed when the methane reached the needed amount. In the next step, the auxiliary electric heater and main electric heater worked together with the cryocooler to keep the

required pressure in the boiling vessel constant. The data were recorded when the system was stable at the corresponding heat flux. The experimental measurements were even repeated three times to ensure the repeatability and reliability. The calculation of the uncertainties, as well as the determination of the surface temperature and the heat flux along the copper cylinder, was described in our previous work (Gong et al. [25]). Different uncertainties are listed in Table 3. Table 3 Measurement and calculation of the uncertainties. Parameters

Instruments

Temperature PT100 thermometer Pressure Pressure transducer Heat flux Superheat Heat transfer coefficient Calibrated length Ruler

Range

Uncertainties

55–300 K 0–4 MPa 10.64–79.25 kW m
2 4.80–11.08 K 2.21–9.13 kW m
2 K
1

±0.1 K ±0.04% 4.6% 6.2% 9.4%

0–2.007 mm

±0.060 mm

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2.3. Equivalent bubble diameter and its uncertainty The equivalent bubble diameter was defined as the diameter of a sphere with the same volume. Kim et al. [26] assumed that the departure bubbles were composed of two rotation semiellipsoids with the vertical rotation axis. As can be seen in the images captured, when a bubble departs, its shape is like that in Fig. 3. The lengths of the horizontal and vertical axis in the bubble (a and b) were measured by an image software. The equivalent bubble diameter can be derived as below.

V total ¼ V 1 þ V 2 ¼

 3 4 D p d 3 2

ð2Þ

2 a2 p c 3 2

ð3Þ

1=3

ð6Þ

Other different equations were used to calculate the bubble departure diameters. Han and Griffith [27] assumed that the equivalent bubble departure diameter was equal to the geometric mean of a and b (Eq. (7)), while Siegel and Keshock [28] assumed that it was equal to the arithmetic mean of a and b (Eq. (8)). Cole and Shulman [29] took the bubbles as rotation ellipsoids with the horizontal axis as rotation axis (Eq. (9)), while Van Wijk and Van Stralen [30] used the vertical axis as rotation axis, whose equation was the same as Eq. (6). Approximately 6500 bubble departure diameters at four pressures were measured, and the calculated results with Eqs. (7)–(9) were compared with Eq. (6) in this work. The deviations and the percent of bubble numbers are shown in Fig. 4. For Eqs. (7) and (8), about 95% percent of the bubbles are within ±4% deviations. For Eq. (9), about 95% of bubbles are within ±10% deviations. The bigger deviations are caused by the very different assumption of the bubble geometry. In this work, the rotation axis of a departure bubble is obviously the vertical axis instead of horizontal axis. Comparisons with another three equations indicate that the calculation of the bubble departure diameters in this work is convincible.

Fig. 3. The shape of a departure bubble.

1=2

ð7Þ

aþb 2

ð8Þ

21.17

Equation 8

18.14 15.12 12.10 9.07 6.05 3.02 0.00 -10

-8

-6

-4

-2

0

2

4

6

Percent of the bubbles (%)

b

a Percent of the bubbles (%)

ð5Þ

Dd ¼ ða2 bÞ

Dd ¼

24.19

Equation 9

21.17 18.14 15.12 12.10 9.07 6.05 3.02 0.00

-6

21.21

Equation 10

18.18 15.15 12.12 9.09 6.06 3.03 0.00

-4

-2

0

Deviation (%)

Deviation (%) c

ð4Þ

cþd¼b

Dd ¼ ðabÞ

Percent of the bubbles (%)

V1 ¼

2 a2 p d 3 2

V2 ¼

-16-14-12-10 -8 -6 -4 -2 0 2 4 6 8 10 12

Deviation (%) Fig. 4. The calculation deviations with Eq. (6).

2

4

6

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H. Chen et al. / International Journal of Heat and Mass Transfer 112 (2017) 662–675 2 1=3

Dd ¼ ðab Þ

ð9Þ

As can be seen from the images, bubbles and the ruler might not be accurately in the same plane, which leads to measurement uncertainty. The uncertainty can be analyzed as below in Fig. 5. Three pictures of the ruler were taken in the following steps. Firstly, the camera was placed in the same position as that in the experiments with the same lens and aperture, and a very clear picture was taken with 65 pixels for the calibrated length L. Secondly, move the camera backward and another picture was taken, which shows the range from 63 to 67 pixels. Thirdly, keep moving the camera backward and another blurred picture was taken, which shows the range from 61 to 69 pixels. The pictures indicate that the maximum uncertainty between the clear ruler and the blurred ruler is ±4 pixels. If the bubbles were clear while the ruler was blurred, we could assume that a very clear ruler with the same calibrated length was put in the same plane with the bubbles. Therefore, the maximum uncertainty for the calibrated length L can be obtained: u(L)/L = ±4/65 = ±6.2%. X1, X2, X3 are the pixels for the calibrated length L, the length of a, the length of b, respectively. b is the mean linear thermal expansion coefficient for stainless steel relative to 299.15 K, and it was precisely measured by Research Centre of Cryogenic Materials and Cryogenic Technology, Technical Institute of Physics and Chemistry, Chinese Academy of Sciences. Its relationship with temperature is shown in Fig. 6. In this work, the values of b at the four increasing pressures are
1.52726  10
3,
1.50197  10
3,
1.46732  10
3,
1.43080  10
3, respectively. With the method of uncertainty analysis described in [31], the maximum uncertainty of bubble departure diameter can be derived as below.

X2 X3 Lð1
bÞ; b ¼ Lð1
bÞ X1 X1 

Dd ¼

1

X 32 X 33 Lð1
bÞ X1

uðDd Þ ¼ Dd

ð13Þ The pixels for a and b are more than 20 pixels, for calibrated length L are 68 pixels, with the maximum measurement uncertainty ±2 pixels. The uncertainty of b was so small that it has little influence in bubble departure diameter, which was ignored here.

uðX 2 Þ 2 uðX 3 Þ 2 uðX 1 Þ 2 uðLÞ ; ; ; ¼ 6:2% ¼ ¼ ¼ X2 20 X 3 20 X 1 68 L

ð14Þ

uðDd Þ ¼ 10:3% Dd

ð15Þ

Therefore, the maximum uncertainty of bubble departure diameter is 10.8%. 3. Results and discussion 3.1. Heat transfer Fig. 7 and Fig. 8 illustrate the experimental ranges. The surface superheat is in the range from 4.8 K to 11.1 K. The heat transfer coefficients are in the range from 2.21 kW m
2 K
1 to 9.13 kW m
2 K
1. For a given pressure, heat flux increases with the increasing surface superheat, and heat transfer coefficient increases with the

ð10Þ

23  13 X2 X3 Lð1
bÞ Lð1
bÞ X1 X1

ð12Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2  2  2 2 uðX 2 Þ 1 uðX 3 Þ uðX 1 Þ uðLÞ uðbÞ þ þ þ þ 3 X2 3 X3 X1 L 1
b

80

ð11Þ

q (kW m-2)

a¼

2

Dd ¼

60

40 0.15 MPa 0.2 MPa

20

0.3 MPa 0.4 MPa

4

8

12

ΔT (K) Fig. 5. Analysis of the uncertainty caused by that the ruler and bubbles are not in the same plane.

9

Mean linear thermal expansion coefficient

-2 -1 h (kW m K )

β ×106

0

Fig. 7. Nucleate boiling curves.

-600

-1200

6

0.15 MPa

3

0.2 MPa 0.3 MPa

-1800 80

160

240

320

Temperature (K) Fig. 6. Mean linear thermal expansion coefficient for stainless steel ruler relative to 299.15 K.

0.4 MPa 25

50 -2

q (kW m ) Fig. 8. Nucleate pool boiling heat transfer data.

75

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Fig. 10 illustrates the bubble sizes at different heat fluxes. At a given pressure, it’s obvious that with the increase in the heat flux, the bubble sizes increase, the number of bubbles increases as well. If heat fluxes become bigger than the maximum heat fluxes the bubble diameters were measured, bubbles begin to coalesce into bigger bubbles, which makes them difficult to distinguish. At a heat flux, big and small bubbles exist. Very small bubbles appear at the periphery of the copper rod, especially at higher pressures. The periphery of the copper rod is not the investigated area and thus the very small bubbles were ignored. It can also be seen that bubbles are ellipsoids rather than spheres, which leads to the assumption of Fig. 3 for the bubbles at a departure. The quantitative evaluation of bubble departure diameters at different heat fluxes was carried out. Due to the wide range of bubble departure diameters at a given heat flux, the distribution histograms of bubble departure diameters are displayed for analysis. Fig. 11 is the distribution histograms corresponding to bubble images in Fig. 10(c). The histogram portion of the graphs represents the actual data in experiments. It can be seen that the bubble departure diameters are mainly distributed in the center of the dataset. In the meanwhile, smaller and bigger diameters also exist, which is in good agreement with the visual observation. Some adjacent bubbles coalesce before their departure from the heated surface, which leads to bigger bubbles, and smaller bubble usually departs after the bigger one on the same nucleation site. The same phenomenon was also found by Kulenovic et al. [33] who perform experiments of pool boiling from tubular heat transfer surfaces with propane as the working fluid. The curves drawn in the graphs are fitted gaussamp function which is defined as:

Dd
Dd;m 2 y ¼ y0 þ Ae
0:5 w

Fig. 9. Single bubbles chosen to measure bubble departure diameters.

increasing heat flux. The trends in this work agree well with Gong et al. [25] which performed pool boiling of methane at saturation pressure of 0.13 MPa, and both together made a contribution to the research data. Additionally, for a given heat flux, lower superheat can be found at higher pressure, thus, higher heat transfer coefficient is presented. Higher pressure improved the performance of nucleate pool boiling, as explained by Nishikawa et al. [32], was due to that the active nucleation site density increased with increasing pressure at a given surface superheat. 3.2. Bubble departure diameter 3.2.1. The statistical analysis of bubble departure diameter For the purpose of obtaining clear individual bubbles, the experiments were conducted in the regime of isolated bubbles at low heat fluxes. When the system reached a steady state at a given heat flux, the data such as temperature and pressure were recorded, and the bubble images were also captured by the high-speed camera. Among about 2000 captured photos, the sizes of more than 200 departure bubbles were measured. As shown in Fig. 9, the pictures in the horizontal direction are two successive frames in the video. The bubbles in the red1 rectangles on the left are those chosen to measure bubble departure diameters at different sites. 1 For interpretation of color in Fig. 9, the reader is referred to the web version of this article.

ð16Þ

where y is the relative frequency of bubble departure diameters, y0 is offset, A is amplitude, Dd is bubble departure diameter, Dd,m is mean bubble departure diameter, w is width. The parameters of the fitted gaussamp function in Eq. (16) is shown in Table 4. The standard errors of all the fitted values are within 3%, with adjusted R-square (Adj. R-square) from 0.88 to 0.99. Adj. R-square shows the goodness of fit. The closer it is to 1, the closer the fit is to the data points. The gaussamp function describes normal distribution in statistics. Therefore, it comes to a conclusion that the distribution of bubble departure diameters is very similar to a normal distribution at a given condition, which can also prove the reliability of the measured data. Paul and Abdel-Khalik [34] made a statistical analysis of bubble departure diameters and bubble departure frequencies for the saturated nucleate boiling of water on a heated platinum wire. As a result, he found that the distributions of bubble departure diameters and bubble departure frequencies are both normal distributions. In Ramaswamy et al. [35], the bubble departure diameters were also found to fit normal distributions. Therefore, the mean bubble departure diameters Dd,m can be taken as the bubble departure diameter at a given heat flux. 3.2.2. Comparisons with six most used correlations for bubble departure diameters As shown in Fig. 12, bubble departure diameter increases with the increasing Ja at a given pressure. However, it’s not easy to find its relationship with Ja at different pressures. The reason is that at lower pressure, the onset of nucleate boiling needs a higher superheat, higher Ja is presented as well. Therefore, even if the Ja at 0.15 MPa is higher than that at other pressures, its departure diameter might not be bigger. Bubble departure diameter is a coupling result of so many parameters that more analyses are needed.

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q=32.89 kW m-2 q=19.36 kW m-2 q=49.99 kW m-2 (a) Bubble images at different heat fluxes at 0.15 MPa.

q=24.18 kW m-2 q=10.85 kW m-2 q=43.35 kW m-2 (b) Bubble images at different heat fluxes at 0.2 MPa.

q=24.19 kW m-2 q=12.69 kW m-2 q=40.42 kW m-2 (c) Bubble images at different heat fluxes at 0.3 MPa.

q=31.03 kW m-2 q=15.99 kW m-2 q=48.38 kW m-2 (d) Bubble images at different heat fluxes at 0.4 MPa. Fig. 10. Bubble images at different heat fluxes at different pressures.

Six most used correlations were chosen to compare with the experimental data in this work. The correlations were evaluated by three criteria, which are respectively the average deviation (AD), the average absolute deviation (AAD), and the percent of data, ??, predicted within ±30% deviation. equations of AD and AAD are defined as:

AD ¼

N 1X predicted value
experimental value  100 N 1 experimental value

AAD ¼

 N   1X predicted value
experimental value  100   N 1 experimental value

ð17Þ

ð18Þ

The statistical results are listed in Table 5. In the meanwhile, comparisons of experimental and predicted bubble departure diameters are shown in Fig. 13.

Kim and Kim [9] predicted best among the six correlations. However, it could only agree well within ±30% deviation at 0.15 MPa and 0.2 MPa. Its deviation was close to
30% at 0.3 MPa and beyond
45% at 0.4 MPa. The predicted results are especially due to the influence of system pressures. After the quantitative analysis of bubble departure radius and time by using the dimensionless scales of radius and time, Kim and Kim [9] proposed the correlation by fitting the data at atmospheric and subatmospheric pressures, which was limited at higher pressures. Good results were predicted at lower pressures. The predicted values were bigger than the experimental values at 0.15 MPa. As the increase in pressure, the predicted value became much smaller than the experimental value. In order to involve the significant influence of system pressure, Cole and Shulman [4] proposed a correlation, in which an inverse relationship between the bubble departure diameter and system

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b 0.2

q=24.19 kW m-2

q=40.42 kW m-2

Relative frequency

Relative frequency

a

0.1

0.0

0.7

0.8

0.9

1.0

1.1

1.2

0.2

0.1

0.0 0.7

1.3

Bubble departure diameter (mm)

Relative frequency

c

0.3

0.8

0.9

1.0

1.1

Bubble departure diameter (mm)

q=12.69 kW m-2

0.2

0.1

0.0 0.6

0.7

0.8

0.9

1.0

Bubble departure diameter (mm) Fig. 11. Distribution of bubble departure diameters at different fluxes at 0.3 MPa.

Table 4 Parameters of the fitted gaussamp function in Eq. (16). Parameters Group

1 2 3

q

y0

Dd,m

w

A

Adj. R-square

40.38 24.19 12.69

0.00491 0.03609 0.00324

0.98806 0.91744 0.81122

0.09801 0.06537 0.077

0.19069 0.21759 0.25462

0.93069 0.93656 0.90055

Dd (mm)

1.6

0.8

0.15 MPa 0.2 MPa 0.3 MPa 0.4 MPa 12

6

Ja Fig. 12. Bubble departure diameters at different Ja.

pressure was presented. 1000/P was taken as a combined dimensionless number affected by pressures. Cole and Shulman [4] also agreed well within ±30% at two pressures (0.2 MPa and 0.3 MPa).

As the increase in pressure, the predicted value became much smaller than the experimental value, which was similar to Kim and Kim [9]. However, bubble departure diameters are affected by so many factors besides pressures, more influencing factors should be considered. Two correlations, Kutateladze and Gogonin [7] and Jensen and Memmel [8], both predicted the same trends. Most of their predicted values were within ±30% at 0.15 MPa and beyond ±30% at other three pressures. The expression of Jensen and Memmel [8] is similar to Kutateladze and Gogonin [7]. They both involved the dimensionless number Kl, which is composed of three dimensionless numbers Ja, Pr and Ar. Kutateladze and Gogonin [7] proposed the correlation based on the balance of forces at the moment of bubble separation, good derivation of the equation was presented. However, the correlation was fitted with the data under free convection in pool boiling, which can be different in nucleate pool boiling for methane. By changing some coefficients and the exponent, Jensen and Memmel [8] developed a correlation similar to Kutateladze and Gogonin [7] by comparing twelve correlations using a large data bank. The correlation somehow predicted better data for the data bank, but it showed big deviations due to the lack of mechanism model.

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H. Chen et al. / International Journal of Heat and Mass Transfer 112 (2017) 662–675 Table 5 Statistical analysis of the bubble departure diameter at different pressures. Pressures (MPa)

Correlations

AD (%)

AAD (%)

?? (%)

0.1

Cole and Shulman [29] Cole [5] Cole and Rosehnow [6] Kutateladze and Gogonin [7] Jensen and Memmel [8] Kim and Kim [9]

48.04
45.46
36.91
23.26
19.83 18.86

48.04 45.46 36.91 23.26 19.83 18.86

25 0 12.5 100 100 100

0.2

Cole and Shulman [4] Cole [5] Cole and Rosehnow [6] Kutateladze and Gogonin [7] Jensen and Memmel [8] Kim and Kim [9]

5.34
67.03
56.42
46.47
42.66
17.44

5.81 67.03 56.42 46.47 42.66 17.44

100 0 0 0 0 100

0.3

Cole and Shulman [4] Cole [5] Cole and Rosehnow [6] Kutateladze and Gogonin [7] Jensen and Memmel [8] Kim and Kim [9]


22.07
77.00
68.96
49.02
44.47
33.73

22.07 77.00 68.96 49.02 44.47 33.73

100 0 0 0 0 0

0.4

Cole and Shulman [4] Cole [5] Cole and Rosehnow [6] Kutateladze and Gogonin [7] Jensen and Memmel [8] Kim and Kim [9]


40.10
82.84
76.69
50.83
45.85
45.64

40.10 82.84 76.69 50.83 45.85 45.64

12.5 0 0 0 0 0

1.6

0.2 MPa

0.15 MPa +30% -30%

0.8

0.0 0.0

0.8

Predicted Dd (mm)

Predicted Dd (mm)

1.0

+30% 0.5

-30%

0.0 0.0

1.6

0.3 MPa

1.0

Predicted Dd (mm)

Predicted Dd (mm)

1.0

+30% 0.5

-30% 0.0 0.0

0.4

0.8

0.8

1.2

0.4 MPa

0.5

+30% -30%

0.0 0.0

Experimental Dd (mm)

0.4

0.8

Experimental Dd (mm)

Fig. 13. Comparison of experimental and predicted bubble departure diameters: j, Cole and Shulman [4]; [7]; , Jensen and Memmel [8]; , Kim and Kim [9].

Cole [5] and Cole and Rohsenow [6] didn’t predict well at four pressures. As can be seen in figures and tables, big deviations were displayed. Cole [5] derived a correlation based on Cole and Shulman [4], in which the effect of system pressure was accounted

0.4

Experimental Dd (mm)

Experimental Dd (mm)

, Cole [5];

, Cole and Rohsenow [6];

, Kutaledze and Gogonin

for through the vapor density term in Ja number. The data fit into it were at sub-atmospheric pressures, which limited the use in wide range of pressure conditions. Cole and Rohsenow [6] proposed a correlation in order to solve the proportionality

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contradictions between bubble departure diameter and the wall superheat. The correlation showed the same bubble departure diameter at the same pressure, which deviated from experimental results even if it somehow showed the trends. As mentioned above, none of these correlations can predict well for the bubble departure diameters at all four pressures in this work. An attempt should be made to find a new correlation. 3.2.3. New correlation At the moment of a bubble’s detachment from the wall, the balance of the forces, including buoyancy force, surface tension force, hydrodynamic resistance to the bubble growth on the wall and inertial force, were considered in Kutateladze and Gogonin [7]. With the expression of surface tension force in Jensen and Memmel [8], the equation can be rewritten as below,

p 6

D3d gðql
qv Þ ¼ pdw r sin h þ þ

p 6

 D3d

p 8

C f D2d ql

 2 dR dt

D2d gðql
qv Þ

r

¼

6dw sin h 3C f Dd ql þ Dd 4r

ð19Þ

dR 2 dt

þ



2 D2d qv þ 12 ql dd2Rt

r

ð20Þ

Kutateladze and Gogonin [7] assumed that the thickness of the liquid layer beneath the bubble was proportional to its radius. Considering conduction heat transfer and latent heat transfer, the bubble growth rate can be written in this form,

dR k DT ¼ b dt Rqv hlv 2

d R 2

d t

¼ b

ð21Þ

 2   kDT 1 dR 1 dR ¼

2  dt R dt qv hlv R

ð22Þ

Eq. (20) can be rewritten as

D2d gðql
qv Þ

r

¼

2 6dw sin h 3C f Dd ql dR dt þ Dd 4r   2

D2d qv þ 12 ql 
D2 dR dt d þ

ð23Þ

r

qv  ql, so qv + ql/2 can be approximated to ql/2, D2d gðql
qv Þ

r

 1 3 Bo2 ¼ s2 K 1 þ s1

ð29Þ

Therefore, Bo1/2 can be written as a function of Kl. 1

1 2

Bo ¼ ðs1 þ s2 K 1 Þ3

ð30Þ

Based on the experimental data, the fitting parameters s1 and s2 can be obtained. The relationship between Bo and Kl is presented as 1

1

Bo2 ¼ ð0:2 þ 5046:6K 1 Þ3

ð31Þ

As illustrated in Fig. 14, the new correlation can predict well for the bubble departure diameter. Most of the predicted departure diameters were within ±20% deviation, which showed great improvement to the prediction of methane departure diameters. 3.3. Bubble departure frequency and its new correlation

 2 1 d R qv þ ql 2 2 d t

Both sides are divided by pr6Dd ,

(Bo1/2)3 shows a linear dependence on Kl. In order to fit the experimental data better, the linear dependence can be written as

2   Dd ql dR 6dw sin h 3C f dt ¼ þ
1 Dd 4 r

ð24Þ

Hutter et al. [36] investigated pool boiling of FC-72 with cavities on silicon, and bubble departure frequency was the average value of five successive bubbles. In this work, as the first bubble detached, more than five successive bubbles were counted and the corresponding time interval for these bubbles was gained from the Phantom video player. The ratio of bubble number to the time interval was equal to the average bubble departure frequency.

f ¼

N bubble t total

ð32Þ

The time interval of consecutive frames is 381 or 382 ls. In the calculation of bubble departure frequency, the maximum uncertainty of total time is 764 ls, so the uncertainty in bubble departure frequency is up to 1.2 Hz. At a given heat flux, more than twenty bubble departure frequencies were measured. The mean value and deviation were also calculated. Bubble departure frequencies are shown in Fig. 15. The results show that bubble departure frequencies range from 34 Hz to 94 Hz. At a given pressure, bubble departure frequency fluctuates but tends to decrease with the increasing Ja. For the pressure of 0.15 MPa, bubble departure frequency varies with a wide range. The reason is that as Ja increases, bubble departure diameter becomes bigger, which needs more time to grow. Even at the same nucleation site, bubble departure frequency is not the same. Using the measured mean departure diameters and mean bub2 ble departure frequencies, fD1/2 d , fDd, fDd, the three products were calculated and compared with some correlations in Table 2.

Substitute Bo and Eq. (21) into it,

6dw sin h 1 ql k2 DT 2 ffi þ ð3C f b2
4b2 Þ  1  qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Dd Bo2 r gðq r
q Þ q2v hlv

ð25Þ

6dw sin h 1 þ ð3C f b2
4b2 Þ  1  K 1 Dd Bo2

ð26Þ

l

Bo ¼

v

The surface tension force was neglected by Zeng et al. [13] at the point of departure and a thorough discussion was provided. In this equation, the contact diameter is very small compared to the bubble departure diameter, so the limit of dw/Dd is equal to zero. Therefore, the term 6dwsinh/Dd could be neglected.

Bo ¼ ð3C f b2
4b2 Þ 

1

 K1 1 Bo2

 1 3 Bo2 ¼ ð3C f b2
4b2 ÞK 1

ð27Þ

ð28Þ

1.3

Predicted D d (mm)

Bo ¼

20% -20%

0.0 0.0

0.15 MPa 0.2 MPa 0.3 MPa 0.4 MPa

1.3

Experimental D d (mm) Fig. 14. Comparison of experimental and new predicted bubble departure diameters.

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H. Chen et al. / International Journal of Heat and Mass Transfer 112 (2017) 662–675

1 tG þ tW

d2

ð38Þ

pal

Assume that d = xR,

tW ¼

x2 R 2

ð39Þ

pal

Substitute Eqs. (36) and (39) into Eq. (33),

f ¼

2Ja  al pb R2 ðp þ 2x2 b  JaÞ

8Ja  al pb

ð40Þ

D2d ðp þ 2x2 b  JaÞ

0.15 MPa 0.2 MPa 0.3 MPa 0.4 MPa 30%

-30%

0

0

3

Experimental fDd

1/2

(m

1/2

-1

s )

(a) Comparison of experimental fDd1/2 with McFadden and Grassmann (1962) [37].

ð33Þ 0.08

Due to: 

dR k DT b Ja  al ¼ b ¼ dt Rqv hlv R

ð34Þ

Evaluate the integrals on both sides,

1 2 R ¼ b Ja  al tG 2

ð35Þ

R2 tG ¼  2b Ja  al

0.15 MPa 0.2 MPa 0.3 MPa 0.4 MPa

ð36Þ

30%

-30%

0.00 0.00

pffiffiffiffiffiffiffiffiffiffiffiffiffi pal tW

(b) Comparison of experimental fDd with Zuber (1963) [38].

ð37Þ

f (Hz)

60

Predicted fDd2 (m2 s-1)

9.0x10-5

0.15 MPa 0.2 MPa 0.3 MPa 0.4 MPa

30

0.08

Experimental fDd (m s-1)

The thermal layer thickness was derived by Han and Griffith [27],

d¼

¼

3

Predicted fDd (m s-1)

f ¼

tW ¼

Predicted fDd1/2 (m1/2 s-1)

Fig. 16 shows the comparisons of experimental data with pre2 dicted fD1/2 d , fDd, fDd. For fD1/2 , McFadden and Grassmann [37] agreed well within d ±30% deviation at 0.15 MPa and 0.2 MPa while beyond ±30% deviation at other two pressures. The predicted values were constant at four pressures while the experimental values were not. McFadden and Grassmann [37] proposed the correlation under the assumption that f2Dd is proportional to the acceleration of a departure bubble, but neglecting the analysis of the waiting time. Waiting time is a major factor affecting bubble departure frequency, and more analysis should be considered. For fDd, Zuber [38] agreed well within ±30% deviation only at 0.15 MPa. The predicted values were also constant at a given pressure while the experimental values were not. Zuber (1963) derived the correlation based on the assumption that growth time was equal to waiting time. However, as observed in the experiments, the relationship between growth time and waiting time is complicated and cannot be simplified to be equal. For fD2d, Mikic and Rohsenow [39] predicted beyond ±60% deviation, however, the equation showed the trend that fD2d varied with Ja number, which will also be discussed in this work. Mikic and Rohsenow [39] proposed the correlation based on their model of heat diffusion controlled growth of a bubble. Both waiting time and growth time were included. Due to the limited quantity of data fit into it, the prediction accuracy in this work is not satisfactory. The comparisons indicate that the existed correlations don’t fit well with the experimental values of methane. Therefore, a new correlation should be developed. The bubble departure frequency can be defined as

0.15 MPa 0.2 MPa 0.3 MPa 0.4 MPa 60%

-60%

0.0 0.0

9.0x10-5

Experimental fDd (m s ) 2

6

12

Ja Fig. 15. Bubble departure frequencies at different Ja.

2

-1

(c) Comparison of experimental fDd2 with Mikic and Rohsenow (1969) [39]. Fig. 16. Comparison of experimental values with the predicted values for fDm d.

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References

Predicted fDd2 (m2 s-1)

9.0x10-5

20% 0.15 MPa 0.2 MPa 0.3 MPa 0.4 MPa

-20% 0.0 0.0

Experimental fDd (m s ) 2

2

9.0x10-5

-1

Fig. 17. Comparison of experimental data with new predicted fD2d.

fDd 8pb v1 ¼ ¼ Ja  al ðp þ 2x2 b  JaÞ v 2 þ Ja 2

ð41Þ

Based on the experimental data, the fitting parameters v1 and v2 can be obtained. The relationship between fD2d/(Jaa) and Ja is presented as 2

fDd 31:2623 ¼ Ja  al 0:2297 þ Ja

ð42Þ

The dimensionless parameter, fD2d/(Jaal), is important in analyzing bubble departure frequency. By considering its relationship with Ja, a better model for methane bubble departure frequency was proposed. As illustrated in Fig. 17, most of the predicted fD2d were within ±20% deviation.

4. Conclusion Saturated nucleate pool boiling experiments of methane were carried out at pressures of 0.15 MPa, 0.2 MPa, 0.3 MPa and 0.4 MPa with heat fluxes varying from 10.64 kW m
2 to 79.25 kW m
2. The bubble departure images were captured by a high-speed digital camera, and bubble departure diameters and bubble departure frequencies were measured and analyzed. The conclusions can be drawn as follows. At a given pressure, bubble departure diameter increases with the increasing Ja while bubble departure frequency tends to decrease. Six most used correlations for bubble departure diameter were compared with the experimental results. Kim and Kim [9] and Cole and Shulman [4] were found well fitted at two pressures. Additionally, a new correlation for bubble departure diameter was developed. Most of the predicted departure diameters were within ±20% deviation. Three correlations for the relationship between bubble departure diameter and departure frequency were compared with the experimental data. However, none of them obtained a satisfactory result. A new correlation was also proposed within ±20% deviation from most of the experimental fD2d.

Acknowledgment This work was supported by the National Natural Science Foundation of China under the contract number of 51506210 and 51625603.

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