- Email: [email protected]
Hydrodynamic formation of a microlayer underneath a boiling bubble
International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Hydrodynamic formation of a microlayer underneath a boiling bubble Satbyoul Jung, Hyungdae Kim ⇑ Department of Nuclear Engineering, Kyung Hee University, Yongin, Republic of Korea
a r t i c l e
i n f o
Article history: Received 21 August 2017 Received in revised form 2 November 2017 Accepted 21 December 2017 Available online 4 January 2018 Keywords: Bubble growth Laser interferometry Microlayer formation Theoretical model Single bubble
a b s t r a c t The hydrodynamic formation of a microlayer at the base of a vapor bubble growing on a heated wall was experimentally and theoretically studied. Single bubble nucleate boiling experiments were conducted in a pool of saturated water under atmospheric pressure. A complete picture of the bubble geometry was obtained, including the three-phase contact line, microlayer and macroscopic bubble. This was performed using integrated laser interferometry, infrared thermometry and shadowgraph techniques. Existing models of the initial microlayer thickness that use an idealized hemispherical bubble shape and neglect surface tension force significantly overestimate microlayer thickness measured in experiment. The visualization results revealed that the non-hemispherical shape and surface tension of the bubble play a critical role in determining initial microlayer thickness. Theoretical analysis also indicated that the short-lived residual flow could play a key role in microlayer formation. Finally, a sophisticated model of initial microlayer thickness, developed to take into account the three identified mechanisms, provides predictions in agreement with experiments for different fluids. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Nucleate boiling is a very effective heat transfer mode that is widely used in industry where removal of high heat fluxes is required, such as in high-power-density electronics cooling, material processing, and nuclear power plants. Numerous mechanisms have been proposed to explain the effective heat transfer characteristics of nucleate boiling in connection with nucleation, growth and detachment behavior of a vapor bubble on a heated wall [1–5]. In particular, the evaporation of a thin liquid layer formed underneath a rapidly growing bubble, the so-called microlayer (Fig. 1), has been proposed as a key candidate mechanism. Hence, the initial profile of the microlayer is of great importance to evaluate and predict the contribution of the microlayer to bubble growth. Theoretical models of the initial microlayer thickness were proposed by Cooper and Lloyd [4] and Ouwerkerk [5] based on approximate boundary layer analysis, and by Smirnov [6], who used equations of the continuity and motion of a microlayer. In parallel with the analytical approaches, there has been considerable effort to measure the initial microlayer thickness experimentally using laser interferometry [7–9], microelectromechanical system (MEMS) sensors [10] and the laser extinction method [11]. Several empirical models of the initial thickness of a micro-
⇑ Corresponding author. E-mail address: [email protected] (H. Kim). https://doi.org/10.1016/j.ijheatmasstransfer.2017.12.098 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
layer have also been developed by approximating the measured data [9–11]. These models have been used to describe the initial condition of the microlayer in the numerical analysis of boiling heat transfer. Sato and Niceno [12] adopted the pure empirical model proposed by Utaka et al. [11] to describe the initial profile of a microlayer. Lee and Nydahl [13] and Guion et al. [14] used the theoretical model by Cooper and Lloyd [4] but the proportional constant value in their models was estimated by fitting other experimental data [9,15]. Dhir [16], Lee and Son [17], and Kunkelmann and Stephan [18] assumed that the microlayer has an initial thickness of half of the height of the cell used for the numerical simulation at the radial location of the micro-macro boundary. As such, in most cases the initial condition of a microlayer in numerical analysis depends on an empirical basis or includes a non-physical assumption. Even though there are a few prediction models based on theoretical analysis, systematical comparison and evaluation with experimental data has rarely been conducted. This seems to explain why theoretical prediction models have not been adopted in advanced numerical simulations of nucleate boiling heat transfer [5,7]. Recently, high resolution space and time measurements of microlayer formation and bubble growth behavior have been developed [10,11,19–21]. Hence, existing prediction models for initial microlayer thickness can be assessed and improved upon, based on these better observation of microlayer formation.
1230
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
Nomenclature a C C0 k m n q’’ r T u t
ratio of microlayer radius to bubble radius constant in the equation R = Ctn constant in the equation d0 = C0(mt)1/2 thermal conductivity (W m1 K1) fringe order refractive index or constant in the equation R = Ctn local wall heat flux (W m2) radius (m) temperature (K) velocity (m/s) time (s)
Greek symbols d microlayer thickness (m) initial microlayer thickness (m) d0 k laser wavelength (m)
m q r h
kinematic viscosity (m2/s) density (kg m-3) surface tension (N/m) angle of refraction (°)
Subscripts b bubble g bubble growth int vapor-liquid interface l liquid m microlayer r residual flow w wall
However, previous research on microlayer has focused on change in microlayer thickness due to its evaporation or investigated initial microlayer thickness, excluding the effect of bubble dynamics on microlayer formation process. The characteristics of microlayer formation considering bubble dynamics have rarely been studied. The main objective of this paper is to conduct a new experiment to improve our understanding of hydrodynamic phenomena during the formation of a microlayer underneath a boiling bubble, and to develop an improved theoretical model of the initial microlayer thickness that agrees with experimental data. Detailed descriptions on the experimental methods and observations are presented in Section 2, while the shortcomings of existing theoretical models are highlighted through comparisons with experimental results in Section 3. Lastly, a modified theoretical model compensating for the detected shortcomings is proposed and validated based on comparison with various experimental data (Section 4).
Fig. 2 shows a schematic diagram of the pool boiling experimental apparatus. This consists of a boiling chamber, a test sample and optical devices. The boiling chamber was composed of an immersion heater to maintain the liquid temperature, a reflux condenser to maintain the liquid level in the pool, and a T-type thermocouple to measure the pool temperature. The test sample was a 10 mm thick calcium fluoride (CaF2) plate coated with a 700nm-thick indium tin oxide (ITO) film electric heater (15 8 mm2). The electro-conductive ITO film had a sheet resistance of 10 X/sq. Optical devices included an infrared (IR) camera, two high-speed cameras, and a He–Ne laser. A high-speed camera was used to measure the bubble growth behavior from the side of the pool, and an IR camera was used to measure the wall
Fig. 1. Microlayer formation underneath a growing bubble.
Fig. 2. Schematic diagram of the pool boiling experimental apparatus.
2. Experiment 2.1. Pool boiling apparatus
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
temperature distribution. The other devices, including an additional high-speed camera, were used for measurement of the microlayer thickness. The measurement details of bubble growth and microlayer thickness are given in the following section. The experiment for a single bubble in a pool of saturated water was conducted under atmospheric pressure. Data on microlayer thickness and bubble growth were collected at an averaged wall superheat of about 9, 10 and 12 °C, corresponding to an average heat flux of 114, 144 and 209 kW/m2, respectively. The average wall temperature was obtained by spatially averaging the wall temperature distribution measured with the IR camera just before bubble nucleation. It was used as the initial value of wall temperature. The average heat flux was calculated from the electric power supplied by the DC power supply divided by the heater area.
1231
the liquid, k is the wavelength of laser light, and m is the fringe order. From the Eq. (2), the slope that means a unit change in the microlayer thickness at two adjacent fringe positions was measured, and the microlayer thickness with the radius was calculated by integrating all the unit changes in microlayer thickness from the center to the point of interest. To demonstrate the measurement of microlayer thickness using the laser interferometry technique, a convex lens with similar scale and geometry to the microlayer was used. A curved thin liquid film was made between the substrate
2.2. Measurement methods 2.2.1. Bubble growth The bubble growth behavior, from bubble nucleation to departure, was captured using the high-speed camera, as seen in Fig. 3 (a). The captured two-dimensional images show a bubble with symmetric shape. Thus, the equivalent bubble radius with time was calculated based on the assumption that the cross-sectional area of a bubble has an axial symmetry. The bubble growth was expressed by the conventional model of Mikic et al. [22],
R ¼ Ct n
ð1Þ
where R is the equivalent bubble radius, t is the time and C is a constant related to the fluid properties and the wall superheat, and n is a constant that is associated with the bubble growth phases (i.e. inertia and thermal diffusion). The recording frequency of the high-speed camera was 50 kHz and the spatial resolution was 25 mm per pixel. The uncertainty resulting from the spatial resolution of measurement was estimated to be less than 25 mm. 2.2.2. Microlayer thickness The microlayer thickness was measured using a laser interferometry method. The interference fringe distribution was observed, as shown in Fig. 3(b). The optical components of a He-Ne laser, reflection mirrors, prisms, an interference filter and a high-speed camera, which is similar to that used in previous paper [21], were used to perform this measurement. The fringe spacing between the adjacent light and dark rings can be determined by the following equation:
dmþ1 dm ¼
k 2nl cos hl
ð2Þ
where d is the thickness of the microlayer, nl is the refraction index of the film layer, hl is the angle of refraction from the substrate into
Fig. 4. (a) Experimental setup to demonstrate the measurement of curved liquid film using a convex lens and (b) comparison between the measured and actual curvature profiles of a convex lens.
Fig. 3. High-speed camera images of bubble behavior and corresponding interference fringe due to microlayer.
1232
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
and the convex lens, having a curvature radius of 387.6 mm within a tolerance of ±1%, as shown in Fig. 4(a). The optical setup of this measurement is consistent with the boiling experiment. The measured thickness of the curved liquid film was in good agreement with the curvature profile of the convex lens. The results are shown in Fig. 4(b). The measurement error was less than 10% at thicknesses above 0.5 lm. In the region thinner than 0.5 lm, the maximum error was about 30%. The recording frequency for the fringe pattern of a microlayer was 50 kHz. During the initial stage when the dry spot is not apparent (<2.0 ms), detection of fringe patterns near the bubble nucleation site is difficult because the spacing of adjacent fringe patterns is smaller than the spatial resolution of the measurement setup, as seen in Fig. 3 (t = 0.5 ms). After the dry spot appears due to the microlayer evaporation, the first fringe pattern near the triple contact line can be detected clearly (t = 2.0 ms). In such a case, the order of the fringe patterns before the appearance of the dry spot can be inferred from patterns when the dry spot does exist. Fig. 5 shows the time evolution of the laser intensity profile across the microlayer. As the microlayer evaporates, the dry spot appears, and then the triple contact line moves outward from the bubble nucleation site. Backwards through time from (t + Dt) to t, the first fringe pattern moves inward, as seen in Fig. 5(a), but it is not detected
at (t Dt) in Fig. 5(b). In this case, the fringe order at (t Dt) can be determined by tracking the fringe movement from (t + Dt) to t. 2.3. Results 2.3.1. Bubble growth and microlayer formation Fig. 6 shows the equivalent bubble radius for a cycle from bubble nucleation to detachment with a wall superheat of 9 °C. The measured data was compared with Mikic’s model, both for the inertial- and diffusion-controlled bubble growth, as noted above. The average temperature of the heat transfer surface was measured using an IR camera, which was used to calculate the Mikic’s model for the initial wall temperature. The measured data show good agreement with the prediction curve for the diffusion stage, except at the very early stage (<0.05 ms) and the final stage (>3 ms). It is expected that the early stage corresponds to the transition region from the inertia to the diffusion stage. Mikic et al. assumed an infinite superheated liquid around the growing bubble, which leads to a prediction over-estimate as the bubble grows, as seen in the final stage of Fig. 6. Therefore, the bubble growth trend follows the form of t0.5 until 3 ms, and the proportional constant has a range of 0.025–0.035 within the wall superheat of 9–12 °C in this study.
Fig. 6. Time evolution of equivalent bubble radius for an isolated bubble.
Fig. 5. Change in laser intensity profile according to the microlayer evaporation; (a) from t to t + Dt and (b) from t Dt to t.
Fig. 7. Time evolution of microlayer profile.
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
1233
Fig. 7 shows the instantaneous microlayer profiles corresponding to the bubble growth in Fig. 6 until just before complete depletion of the microlayer. The error bars represent the maximum measurement error in each data point. The error increases with the thickness, and the maximum error is less than 0.2 lm at the thickness of 3.5 lm. The data point where the thickness is zero indicates the triple contact line and the outermost data point corresponds to the outer edge of the microlayer at each time step. The instantaneous microlayer thickness at a particular radial position decreases, and the triple contact line moves outward from the bubble nucleation site due to the microlayer evaporation. It should be noted that the radial extent of the microlayer lasts until 5.6 ms. This indicates that the microlayer maintains its formation while undergoing evaporation during the diffusion-controlled stage of bubble growth. While the bubble radius complies the form of R t1 at the very early stage of the bubble growth (<0.05 ms), it grows as obeying the diffusion-controlled growth equation of R t0.5 approximately after 0.1 ms, as shown in Fig. 6. It was commonly reported in previous experimental observations [4] and theoretical analysis [4,5] that the bubble growth behavior during microlayer formation can be approximated with the functional form of the diffusion-controlled growth stage, R t0.5. Thus, to focus the discussion on the microlayer formation and corresponding bubble growth, in this paper the presented data will be confined to the period of microlayer formation, which corresponds to the diffusion-controlled stage of bubble growth. To ensure detection of the outermost boundary of the microlayer using the laser interferometry method, it was cross-checked by calculating the microlayer thickness from the local wall heat flux distribution. The local wall heat flux was obtained by solving the three-dimensional transient heat conduction using the measured time-varying temperature distribution data [21], which were obtained using the IR camera. The local wall temperature and heat flux distributions vary temporally and spatially due to the microlayer formation and evaporation. Such data were used to calculate the thickness profile of microlayer using one-dimensional steadystate conduction equation, which is
q00w ¼
kl ðT w T int Þ d
ð3Þ
00
where qw is local wall heat flux, kl is the thermal conductivity of the liquid, Tw is the local wall temperature and Tint is the temperature at the liquid-vapor interface. The interface temperature can be assumed to be the same as the saturated temperature in the microlayer region, except in the vicinity of the triple contact line due to evaporative resistance [23]. The microlayer thickness calculated by the two different methods shows reasonable consistency in the outermost boundary of the microlayer, as shown in Fig. 8. The outermost boundary obtained by the fringe analysis reveals accurate detection of the transition point of the liquid-vapor interface profile derived from Eq. (3), as the fringe patterns cannot appear when the sharp change in the slope of the liquid-vapor interface occurs. In addition, the profile at the transition region between the microlayer and macroscopic liquid can be roughly estimated. It is noted that the thickness at the transition region may be slightly over-estimated because it was derived with the assumption of onedimensional conduction heat transfer from the wall to the liquidvapor interface. The complete geometry of the boiling bubble can be determined by combining the macroscopic bubble shown in Fig. 3(a) with the corresponding profile of the microlayer and transition region shown in Fig. 8. Fig. 9 shows the bubble growth history during the microlayer formation. It is clearly shown that the microlayer
Fig. 8. Comparison between the microlayer thickness obtained by the fringe analysis and the interface profile calculated by Eq. (3): (a) 1.6 ms, (b) 3.6 ms, (c) 5.6 ms.
1234
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
thickness is of the order of 1 lm and the macroscopic bubble has an order of 1 mm. There is a short transition region between the two, with a thickness of approximately 50 lm. (A more detailed analysis of this transition region taking into account surface tension effects [6], is a possible subject for future investigation.) In addition, contrary to the general expectation that the bubble has a hemispherical shape during microlayer formation, the bubble has an oblate shape with an apparent contact angle less than 50°.
2.3.2. Initial microlayer thickness In several experimental studies [10,11], initial microlayer thickness at a point radially away from a nucleation site was defined as thickness of microlayer initially observed at the specific point while a rapidly growing bubble forms microlayer on the heater surface. The initial microlayer points determined according to the above-mentioned definition were marked in Fig. 10. The initial microlayer thickness increased up to 3.7 lm as the bubble expanded to the radial point of 1.2 mm, but then thinned during further expansion. As mentioned in the previous section, the microlayer formation occurs simultaneous with its evaporation. Therefore, the initial microlayer thickness in Fig. 10 incorporates not only radial expansion induced by microlayer formation but also the thinning effect induced by microlayer evaporation; this is why initial microlayer thickness gets thinned at the later part of the microlayer formation, as shown in Fig. 10. To determine the isothermal initial microlayer thickness, that is, the microlayer thickness initially formed on the wall during bubble growth in the isothermal field without microlayer evaporation, the initial microlayer thickness in Fig. 10 was utilized. Without microlayer evaporation, the microlayer thickness at a specific radial position does not change. In other words, the thickness reduction can be caused by the evaporation. Thus, the isothermal thickness can be reconstructed by tracking the reduced thickness at the same radial position for a time step. Here, the microlayer thickness at 0.2 ms is assumed to be the initial microlayer thickness without the evaporation, because the thickness reduction due to the microlayer evaporation is negligibly small at the early stage of microlayer formation, as seen in Fig. 7. Fig. 11 shows the reconstruction results for the isothermal initial microlayer thickness. It was approximately 5 lm at 1.5 mm of the radial position.
Fig. 9. Complete bubble geometries including the microlayer, transition region and macroscopic bubble shape for bubble growing: (a) 1.6 ms, (b) 3.6 ms, (c) 5.6 ms.
Fig. 10. Initial microlayer thickness.
1235
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
Fig. 11. Isothermal initial microlayer thickness.
There have been several experiments measuring the initial microlayer thickness, such as those by Koffman [9], Yabuki and Nakabeppu [10], and Utaka et al. [11]. The experimental condition, measurement method and empirical model for initial microlayer thickness in the previous studies are summarized in Table 1. The experiments were conducted with the same fluid and under similar liquid temperature and pressure conditions, but measurements were done using different methods. A comparison of the results for the initial microlayer thickness is shown in Fig. 12. The filled symbols show the isothermal initial thickness obtained by the reconstruction while the values predicted by the empirical models [9– 11] represent the initial microlayer thicknesses that necessarily include the thinning effect due to evaporation as illustrated in Fig. 10. However, the resulting difference is not significant as seen in Fig. 12. This is because only data obtained at the very beginning of microlayer formation with negligible evaporation were used for the development of the previous empirical models [9–11]. Therefore, the comparison with the prediction by the empirical models is plausible. Utaka’s result shows a linear shape of the initial microlayer thickness while the others show that the shape is close to convex. Nevertheless, the results for the initial microlayer thickness show very good agreement in order of magnitude. Therefore, it can be concluded that the present experimental data are generally representative of the initial microlayer thickness.
Fig. 12. Comparison between the initial microlayer thickness and the measured data of other experiments for saturated water under atmospheric pressure conditions.
Stralen et al. [25] derived a model based on approximate boundary layer analysis, in which the initial microlayer thickness is equal to the displacement thickness of the hydrodynamic boundary layer outside the bubble. Assuming R / t0.5, the expression derived for the initial microlayer thickness was
d0 ¼ C 0
pffiffiffiffiffi vt
ð4Þ
where C0 is a constant, m is the kinematic viscosity and t denotes the bubble growth time. Cooper and Lloyd [4] suggested a value of 0.8 for C0, which has an empirical basis, Ouwerkerk [5] derived the value 1.26 for C0 using a self-similarity solution, and Olander and Watts [24] derived the value 0.88 for C0 by neglecting the convective term in boundary layer equation. Van Stalen et al. [25] derived C0 as a function of liquid Prandtl number using Pohlhausen solution for laminar boundary layer.
3. Assessment of existing theoretical models 3.1. Review of previous theoretical models In addition to the preceding empirical models, some models based on theoretical analysis have been developed. Cooper and Lloyd [4], Van Ouwerkerk [5], Olander and Watts [24] and Van
Fig. 13. Schematic of the formation of initial microlayer thickness.
Table 1 Experimental conditions of previous studies and the present study for measurement of the initial microlayer thickness using different measurement methods.
Present study Koffman [9] Yabuki and Nakabeppu [10] Utaka et al. [11]
Experiment condition
Measurement method
Empirical model
Saturated water at atmospheric pressure Subcooled water at atmospheric pressure
Laser interferometry
–
Laser interferometry
Saturated water at atmospheric pressure Saturated water at atmospheric pressure
MEMS sensor
d0 = 0.00188r0.6, (r: cm, d0: cm) d0 = 4.34r0.69 (r: mm, d0: lm) d0 = 4.46r, (r: mm, d0: lm)
Laser extinction
1236
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
Smirnov [6] attempted to calculate the initial microlayer thickness by solving the continuity and linear momentum equations for the hydrodynamics of one-dimensional radial flow during formation of the microlayer. Since this model uses a generalized form of bubble growth behavior compared with other models, it was used as a reference model for improving the prediction performance in the later section. Hence, we briefly review Smirnov’s modelling of the initial microlayer thickness (it is helpful to see Fig. 13):
@um @um 1 @P um þ m r2 u m 2 ; þ um ¼ @t @r q @r r
ð5Þ
@ ðrum Þ ¼ 0: @t
ð6Þ
Using the assumption of a hemispherical bubble, the velocity at the microlayer boundary, um(r, d0), was taken to be equal to that of the expanding bubble interface:
um ðR; d0 Þ ¼
3.2. Comparison between measurement data and extant models A comparison of the initial microlayer thickness between the theoretical models and the experimental measurement data obtained in this study is given in Fig. 14. The measured values of C and n for bubble growth, used to calculate the initial microlayer thickness using the theoretical model, were 0.035 and 0.5, respectively. It was found that there were significant differences between the experimental data and the predictions in terms of both the order of magnitude and the shape, even though the predicted values for each model are almost consistent. This may be because the theoretical model for the microlayer thickness is not realistic for bubbles affected by widely varying parameters, such as inertia and surface tension forces, fluid properties and so on. In this regard, the shortcomings of the existing models, with respect to the accuracy of predictions of microlayer formation, need to be identified through systematic comparison with the detailed experimental observations done in the present study. 3.3. Identification of shortcomings
dR : dt
ð7Þ
Smirnov’s model also shows the same form of d0 = C0(mt)1/2 as obtained by Ref. [4,5,24,25]. When the exponent of the bubble growth is chosen as n = 0.5, the constant C0 is 1.05, which is consistent with other models.
Several issues regarding the existing theoretical model were identified based on the present and previous studies [27–30]. The first issue is the assumption of neglecting the surface tension force acting on the bubble growth in the theoretical model. Our experiment provided the decisive insight that surface tension acts on the growing bubble during the microlayer formation. When a bubble grows quickly, the inertial force dominates and the resulting shape is hemispherical, whereas for slow bubble growth, the surface tension dominates so that a spherical bubble forms. In our experiment, the bubble shape was oblate during the whole process of microlayer formation (Fig. 9). This indicates that both the inertial and surface tension forces had an effect on bubble growth. The second issue is the assumption in classical models of a hemispherical bubble. In the classical models, the boiling bubble is approximated to be hemispherical and thus the microlayer formation velocity is the same as the bubble growth velocity. However, in the experiment in this study, an oblate bubble including transition region between microlayer and macroscopic bubble was observed and the interface velocity sweeping liquid to form the microlayer was considerably slower than the bubble growth velocity, as seen in Fig. 15. The final issue concerns the residual flow inside the microlayer. The theoretical analysis considers the initial microlayer thickness
Fig. 14. Comparison of the initial microlayer thickness among the theoretical prediction models for saturated water under atmospheric pressure conditions.
Fig. 15. Radii of bubble and microlayer with time during the microlayer formation.
Then, the following equation is obtained:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ," u 2 # 2 u 2 dP d R 2 dR 1 : d0 ¼ t2mR þ q dR dt2 3 dt R
ð8Þ
The value of dP/dR can be determined using the Rayleigh-Plesset equation [26]: 3
3
dP @ 2 R Rðd R=dt Þ 2r ¼4 2 þ 2 : dR dR=dt @t qR ðdR=dtÞ
ð9Þ
If the surface force for the rapid growth stage of the bubble is negligible, and the bubble growth follows the growth law in the form R = Ctn, then Eq. (8) becomes:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2v t d0 ¼ 9ð1 nÞ þ 2 1n 1 ðn 2Þ þ 0:66n
ð10Þ
1237
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
with no flow inside the microlayer, because viscous dissipation of the kinetic energy of residual flow in the microlayer occurs suddenly over a time interval that is much smaller compared to the microlayer formation time. However, it is found that the residual flow exists once the microlayer has formed, and during that short time it can affect the initial microlayer thickness, making it thinner [29,30]. The residual flow can affect the microlayer thickness. 4. Development of a modified theoretical model With reference to the issues discussed in the previous section, we attempt to show the effect of the surface tension force, the non-hemispherical bubble shape and the residual flow on the microlayer formation using the generalized model of Smirnov [6]. Detailed discussion on each aspect are presented. 4.1. Surface tension effect The surface tension term in Eq. (9) is neglected in the classical model, while the oblate bubble shape in the experiment indicates
that its effect is considerable. Thus, the effect of the surface tension force on microlayer formation should be considered. The effect of surface tension can be included by retaining the surface tension term when solving the Rayleigh-Plesset equation. For the dynamics of a bubble in Eq. (9), the following microlayer thickness equation including surface tension is obtained:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2v t d0 ¼ 9ð1 nÞ þ 2 1n 1 ðn 2Þ þ 0:66n þ qC 4 n42rt4n3
ð11Þ
4.2. Effect of non-hemispherical bubble shape As noted above, the bubble shape is not hemispherical, but oblate during the whole period of microlayer formation in our experiment. This indicates a considerable difference between the radius of the bubble and the outer edge of the microlayer, with a ratio of approximately Rb/Rm 0.6, as seen in Fig. 15. The interface velocity sweeping liquid to form the microlayer was about half the bubble growth velocity. The effect of the difference in radius between the bubble and the microlayer, Rm/Rb = a (a < 1), can be included by using um = dRm/dt = a(dRb/dt) as the boundary condition at the outer edge of the microlayer in Eq. (7). The following initial microlayer thickness equation is obtained:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2v t d0 ¼ 8 2 1 þ 1 ð1 nÞ þ 1 ðn 2Þ þ a2 0:66n þ a3 qC 44nr2 t4n3 3 3 a a n ð12Þ
Fig. 16. Comparison of the initial microlayer thickness measured in the experiment and calculated by the modified prediction model.
This analysis, by incorporating the effects of surface tension and the bubble-microlayer radius difference predicts a sequential thinning of the microlayer, as shown in Fig. 16, because both lead to a decrease in sweep velocity and shear stress for microlayer formation. In addition, the initial microlayer thickness profile changes from linear to convex due to the increasing influence of surface tension in the denominator of Eqs. (11) and (12), which is consistent with the experimental observations (Fig. 12). As a result of these two effects, the experimental data are more closely predicted by Eq. (12); however, a distinct discrepancy still remains.
Fig. 17. Effect of residual flow on microlayer formation: (a) Residual flow velocity degradation by viscous dissipation. (b) Moving distance of residual flow in the radial direction from the bubble center outwards. (c) Conceptual picture of the reduced thickness of the microlayer according to the residual flow.
1238
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
4.3. Effect of short-lived residual flow inside the microlayer There is one further physical aspect of microlayer formation that has not yet been considered, which may have a significant effect on the initial microlayer thickness. In the preceding analysis, residual flow in the microlayer was neglected because the characteristic timescale related to viscous friction in a thin liquid film, tr d02/m, is very small (e.g., 103–101 ms for a water liquid layer with thickness of 1–10 lm). As it is almost impossible to experimentally observe the short-lived residual flow, its influence has not been taken into account. However, some studies using numerical simulations argue that the residual flow may have a considerable impact on the initial microlayer profile over such a short time interval [29,30]. To give some indication of the importance of residual flow, we have attempted to estimate the residual flow effect on the initial microlayer thickness analytically. When stagnant liquid on a flat substrate is swept by a rapidly expanding bubble, the residual flow may further push the trapped liquid along the direction of bubble expansion, and the microlayer thickness will be reduced by the amount of liquid pumped out by the residual flow. The equivalent amount of pumped liquid can be estimated by integrating the volumetric flow rate of the residual flow during its characteristic time so that the reduced thickness can be predicted. A conceptual picture of the thickness thinning effect induced by the residual flow is presented in Fig. 17. If we approximate the microlayer as an axisymmetric wedge, the resultant thickness (dr) incorporating the effect of residual flow can be given by:
dr ¼
3 ! 3 ur 1 ur d0 1 þ 2 um 2 um
ð13Þ
where d0 is the initial microlayer thickness without residual flow, r is the average velocum is the microlayer formation velocity and u ity of the residual flow. The velocity evolution of the residual flow can be presumed to simply follow the viscous diffusion equation, q(our/ot) l(o2ur/oy2), which gives an approximate solution ur(t) = um exp(lt/qd2), as seen in Fig. 17(a). During the effective residual flow dissipation time (tr), which means a period in which the residual flow moves 95% of the terminal moving distance of the r/um is about 0.32. The terminal velocity solution (lr = 0.95 l1), u moving distance can be estimated by integration of the residual flow velocity over time. Finally, the initial microlayer thickness
Fig. 18. Effect of residual flow inside the microlayer on the initial microlayer thickness.
Fig. 19. Comparison between experimental data for initial microlayer thickness and theoretical models: (a) saturated methanol at 58.5 kPa [7], (b) saturated water at 101.3 kPa [10] and (c) saturated toluene at 13.9 kPa [4].
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
equation, which includes the effects of surface tension force, bubble shape and residual flow, is obtained:
d0 ¼ 0:53 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2v t 8 2 1 þ 1 ð1 nÞ þ 1 ðn 2Þ þ a2 0:66n þ a3 qC 44nr2 t4n3 3 3 n a a ð14Þ The proposed model (solid line) shows remarkable agreement with the experimental data measured in the present study, as seen in Fig. 18. Note that the contribution of residual flow to the radial heat conduction in the microlayer can be negligible due to the very short timescale of the residual flow. 4.4. Assessment of the modified model The model was further validated by comparison with experimental data from other studies [4,7,10], which provide quantitative data for bubble growth history as well as microlayer thickness. The values of d0 in [4] and [10] were obtained by calculating conductive heat transfer based on the measured wall temperature, while the value in [7] was measured using the same laser interferometry technique as used in our experiment. The comparison of the prediction of the existing models and the modified model with the measured data for methanol, water and toluene is shown in Fig. 19. The constant C in the equation R = Ct0.5 is determined to best fit the bubble growth presented in each study, which is presented as an inset. The dotted lines present the prediction by existing models including Cooper and Lloyd [4], Van Ouwerkerk [5], and Smirnov [6]. These significantly over-predict the measured data. Based on the Smirnov’s model, it overpredicts the thickness to be 2 to 2.5 times. The solid line presents the prediction by the proposed model. It was found that the proposed model predicts the experimental results well, regardless of the liquid (water, toluene or methanol) or pressure conditions. It is noted in Fig. 19(c) that the prediction by the proposed model slightly underestimated the measured thickness. It can be interpreted as a result of the use of the inaccurate value of ‘a’, which is a parameter associated with the bubble shape. Indeed, the constant C in the equation R = Ct0.5 for toluene is 0.084, which is relatively high. This tends to make the bubble shape more hemispherical [29]. In other word, the ‘a’ value may be greater than 0.6, which is a suggested value in this study. If the value greater than 0.6 was used for the prediction by the proposed model, the microlayer thickness would have been predicted to be thicker, which could result in a more consistent agreement between the prediction and the measured data. The value of 0.6 was suggested based on the observation in our experimental condition. Therefore, some further investigation of the value, which is related to the bubble shape as well as the transition region between the microlayer and macroscopic liquid, is required to generalize the proposed model in future. 5. Conclusion Hydrodynamic formation of a microlayer at the base of a vapor bubble growing on a heated wall was studied. Single bubble nucleate boiling experiments were conducted in a pool of saturated water under atmospheric pressure. A complete bubble geometry, including the microlayer and macroscopic bubble, was obtained using the integrated optical measurement method. Using experimental and theoretical insights, we developed an improved theoretical model of initial microlayer thickness that is in agreement with experimental data. Major findings from this study can be summarized as follows:
1239
– High-resolved measurement data on the initial microlayer thickness were obtained and compared with existing theoretical models. The models contain several idealizations (hemispherical bubble shape and negligible surface tension effects) and significantly overestimate the initial microlayer thickness relative to experimental results. – Precise observation of the growing bubble geometry shows that the bubble is non-hemispherical, which indicates that both inertia and surface tension forces have an effect on bubble growth during microlayer formation. In addition, the critical role played by the short-lived residual flow in microlayer formation was demonstrated by analytical estimates of the residual flow. – Finally, an improved theoretical model of initial microlayer thickness, which incorporates the effects of surface tension, non-hemispherical bubble shape and residual flow, is proposed. The proposed model is in good agreement with the experimental results for other fluids and pressure conditions. The proposed model was developed based on the observation in pool boiling of saturated water on an upward-facing horizontal heater wall under atmospheric condition. However, various parameters such as flow condition, liquid subcooling, fluid, working pressure and wall inclination can affect microlayer formation as well as bubble growth characteristics (i.e. growth rate and bubble shape, especially the bubble shape is strongly related to ‘a’ value which is a ratio of microlayer radius to bubble radius). Therefore, further investigations about effects of the above-mentioned parameters would be next subjects for improving the predictive performance and extending its applications. Conflicts of interest Authors declare that there is no conflict of interest. Acknowledgements This work was supported by National R&D Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2015M1A7A1A01002428). This research was also supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP: Ministry of Science, ICT and Future Planning) (No. 2017M2A8A4015283). References [1] J. Kim, Review of nucleate pool boiling bubble heat transfer mechanisms, Int. J. Multiphase Flow 35 (2009) 1067–1076. [2] S. Moghaddam, K. Kiger, Physical mechanism of heat transfer during single bubble nucleate boiling of FC-72 under saturation conditions I. Experimental investigation, Int. J. Heat Mass Transfer 52 (2009) 1284–1294. [3] F. Demiray, J. Kim, Microscale heat transfer measurements during pool boiling of FC-72: effect of subcooling, Int. J. Heat Mass Transfer 47 (2004) 3257–3268. [4] M.G. Cooper, A.J.P. Lloyd, The microlayer in nucleate pool boiling, Int. J. Heat Mass Transfer 12 (1969) 895–9133. [5] H.J. Van Ouwerkerk, The rapid growth of a vapour bubble at a liquid-solid interface, Int. J. Heat Mass Transfer 14 (1971) 1415–1431. [6] G.F. Smirnov, Calculation of the initial thickness of the microlayer during bubble boiling, J. Eng. Phys. 28 (1975) 369–374. [7] H.G. MacGregor, H.H. Jawurek, High speed cine laser interferometry technique for microlayer studies in boiling, N&O Joernaal 4 (1992) 1–7. [8] M. Gao, L. Zhang, P. Cheng, X. Quan, An investigation of microlayer beneath nucleation bubble by laser interferometric method, Int. J. Heat Mass Transfer 57 (2012) 183–189. [9] L.D. Koffman, Microlayer evaporation in subcooled nucleate boiling, in: Proceeding of 3rd Multi-Phase Flow Heat Transfer Symposium – Workshop, April 18–20, Miami Beach, USA, 1983. [10] T. Yabuki, O. Nakabeppu, Heat transfer mechanisms in isolated bubble boiling of water observed with MEMS sensor, Int. J. Heat Mass Transfer 76 (2014) 286– 297.
1240
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
[11] Y. Utaka, Y. Kashiwabara, M. Ozaki, Microlayer structure in nucleate boiling of water and ethanol at atmospheric pressure, Int. J. Heat Mass Transfer 57 (2013) 222–230. [12] Y. Sato, B. Niceno, A depletable micro-layer model for nucleate pool boiling, J. Comput. Phys. 300 (2015) 20–52. [13] R.C. Lee, J.E. Nydahl, Numerical calculation of bubble growth in nucleate boiling from inception through departure, J. Heat Transfer – Trans. ASME 111 (1989) 474–479. [14] A. Guion, D. Langewisch, J. Buongiorno, On the extended liquid microlayer underneath a vapor bubble growing at a heated wall, in: Proceeding of 8th International Conference on Multiphase Flow, May 26–31, Jeju, Korea, 2013. [15] B.E. Staniszewski, Nucleate boiling bubble growth and departure, DSR project No. 7-7673, Technical Report No. 16, 1959. [16] V.K. Dhir, Mechanistic prediction of nucleate boiling heat transfer – achievable or a hopeless task?, J Heat Transfer 128 (2006) 1–12. [17] W. Lee, G. Son, Bubble dynamics and heat transfer during nucleate boiling in a microchannel, Numer. Heat Transfer 53 (2008) 1074–1090. [18] C. Kunkelmann, P. Stephan, CFD simulation of boiling flows using the volumeof-fluid method within OpenFOAM, Numer. Heat Transfer 56 (2009) 631–646. [19] C. Gerardi, J. Buongiorno, L.W. Hu, T. McKrell, Study of bubble growth in water pool boiling through synchronized, infrared thermometry and high-speed video, Int. J. Heat Mass Transfer 53 (2010) 4185–4192. [20] I.C. Chu, H.C. No, C.H. Song, Visualization of boiling structure and critical heat flux phenomenon for a narrow heating surface in a horizontal pool of saturated water, Int. J. Heat Mass Transfer 62 (2013) 142–152.
[21] S. Jung, H. Kim, An experimental method to simultaneously measure the dynamics and heat transfer associated with a single bubble during nucleate boiling on a horizontal surface, Int. J. Heat Mass Transfer 73 (2014) 365–375. [22] B.B. Mikic, W.M. Rohsenow, P. Griffith, On bubble growth rates, Int. J. Heat Mass Transfer 13 (1970) 657–666. [23] G. Giustini, S. Jung, H. Kim, S.P. Walker, Evaporative thermal resistance and its influence on microscopic bubble growth, Int. J. Heat Mass Transfer 101 (2016) 733–741. [24] R.R. Olander, R.G. Watts, An analytical expression of microlayer thickness in nucleate boiling, J. Heat Transfer 91 (1969) 178–180. [25] S.J.D. Van Stralen, M.S. Sohal, R. Cole, W.M. Sluyter, Bubble growth rates in pure and binary systems: combined effect of relaxation and evaporation microlayers, Int. J. Heat Mass Transfer 18 (1975) 453–467. [26] M.S. Plesset, The dynamics of cavitation bubbles, J. Appl. Mech. 16 (1949) 277– 282. [27] Y. Zhang, Y. Utaka, Characteristics of a liquid microlayer formed by a confined vapor bubble in micro gap boiling between two parallel plates, Int. J. Heat Mass Transfer 84 (2015) 475–485. [28] A. Guion, J. Buongiorno, S. Zaleski, S. Afkhami, C. Narayanan, Numerical simulation of nucleate boiling using dynamic models of microlayer formation and evaporation, in: Proceeding of CFD4NRS, Cambridge, MA-USA, 2016. [29] S. Hansch, S. Walker, The hydrodynamics of microlayer formation beneath vapour bubbles, Int. J. Heat Mass Transfer 102 (2016) 1282–1292. [30] H.S. Fath, Microlayer formation, evaporation and bubble growth in nucleate boiling (Ph. D. dissertation), McMaster University, 1981.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Hydrodynamic formation of a microlayer underneath a boiling bubble Satbyoul Jung, Hyungdae Kim ⇑ Department of Nuclear Engineering, Kyung Hee University, Yongin, Republic of Korea
a r t i c l e
i n f o
Article history: Received 21 August 2017 Received in revised form 2 November 2017 Accepted 21 December 2017 Available online 4 January 2018 Keywords: Bubble growth Laser interferometry Microlayer formation Theoretical model Single bubble
a b s t r a c t The hydrodynamic formation of a microlayer at the base of a vapor bubble growing on a heated wall was experimentally and theoretically studied. Single bubble nucleate boiling experiments were conducted in a pool of saturated water under atmospheric pressure. A complete picture of the bubble geometry was obtained, including the three-phase contact line, microlayer and macroscopic bubble. This was performed using integrated laser interferometry, infrared thermometry and shadowgraph techniques. Existing models of the initial microlayer thickness that use an idealized hemispherical bubble shape and neglect surface tension force significantly overestimate microlayer thickness measured in experiment. The visualization results revealed that the non-hemispherical shape and surface tension of the bubble play a critical role in determining initial microlayer thickness. Theoretical analysis also indicated that the short-lived residual flow could play a key role in microlayer formation. Finally, a sophisticated model of initial microlayer thickness, developed to take into account the three identified mechanisms, provides predictions in agreement with experiments for different fluids. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Nucleate boiling is a very effective heat transfer mode that is widely used in industry where removal of high heat fluxes is required, such as in high-power-density electronics cooling, material processing, and nuclear power plants. Numerous mechanisms have been proposed to explain the effective heat transfer characteristics of nucleate boiling in connection with nucleation, growth and detachment behavior of a vapor bubble on a heated wall [1–5]. In particular, the evaporation of a thin liquid layer formed underneath a rapidly growing bubble, the so-called microlayer (Fig. 1), has been proposed as a key candidate mechanism. Hence, the initial profile of the microlayer is of great importance to evaluate and predict the contribution of the microlayer to bubble growth. Theoretical models of the initial microlayer thickness were proposed by Cooper and Lloyd [4] and Ouwerkerk [5] based on approximate boundary layer analysis, and by Smirnov [6], who used equations of the continuity and motion of a microlayer. In parallel with the analytical approaches, there has been considerable effort to measure the initial microlayer thickness experimentally using laser interferometry [7–9], microelectromechanical system (MEMS) sensors [10] and the laser extinction method [11]. Several empirical models of the initial thickness of a micro-
⇑ Corresponding author. E-mail address: [email protected] (H. Kim). https://doi.org/10.1016/j.ijheatmasstransfer.2017.12.098 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
layer have also been developed by approximating the measured data [9–11]. These models have been used to describe the initial condition of the microlayer in the numerical analysis of boiling heat transfer. Sato and Niceno [12] adopted the pure empirical model proposed by Utaka et al. [11] to describe the initial profile of a microlayer. Lee and Nydahl [13] and Guion et al. [14] used the theoretical model by Cooper and Lloyd [4] but the proportional constant value in their models was estimated by fitting other experimental data [9,15]. Dhir [16], Lee and Son [17], and Kunkelmann and Stephan [18] assumed that the microlayer has an initial thickness of half of the height of the cell used for the numerical simulation at the radial location of the micro-macro boundary. As such, in most cases the initial condition of a microlayer in numerical analysis depends on an empirical basis or includes a non-physical assumption. Even though there are a few prediction models based on theoretical analysis, systematical comparison and evaluation with experimental data has rarely been conducted. This seems to explain why theoretical prediction models have not been adopted in advanced numerical simulations of nucleate boiling heat transfer [5,7]. Recently, high resolution space and time measurements of microlayer formation and bubble growth behavior have been developed [10,11,19–21]. Hence, existing prediction models for initial microlayer thickness can be assessed and improved upon, based on these better observation of microlayer formation.
1230
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
Nomenclature a C C0 k m n q’’ r T u t
ratio of microlayer radius to bubble radius constant in the equation R = Ctn constant in the equation d0 = C0(mt)1/2 thermal conductivity (W m1 K1) fringe order refractive index or constant in the equation R = Ctn local wall heat flux (W m2) radius (m) temperature (K) velocity (m/s) time (s)
Greek symbols d microlayer thickness (m) initial microlayer thickness (m) d0 k laser wavelength (m)
m q r h
kinematic viscosity (m2/s) density (kg m-3) surface tension (N/m) angle of refraction (°)
Subscripts b bubble g bubble growth int vapor-liquid interface l liquid m microlayer r residual flow w wall
However, previous research on microlayer has focused on change in microlayer thickness due to its evaporation or investigated initial microlayer thickness, excluding the effect of bubble dynamics on microlayer formation process. The characteristics of microlayer formation considering bubble dynamics have rarely been studied. The main objective of this paper is to conduct a new experiment to improve our understanding of hydrodynamic phenomena during the formation of a microlayer underneath a boiling bubble, and to develop an improved theoretical model of the initial microlayer thickness that agrees with experimental data. Detailed descriptions on the experimental methods and observations are presented in Section 2, while the shortcomings of existing theoretical models are highlighted through comparisons with experimental results in Section 3. Lastly, a modified theoretical model compensating for the detected shortcomings is proposed and validated based on comparison with various experimental data (Section 4).
Fig. 2 shows a schematic diagram of the pool boiling experimental apparatus. This consists of a boiling chamber, a test sample and optical devices. The boiling chamber was composed of an immersion heater to maintain the liquid temperature, a reflux condenser to maintain the liquid level in the pool, and a T-type thermocouple to measure the pool temperature. The test sample was a 10 mm thick calcium fluoride (CaF2) plate coated with a 700nm-thick indium tin oxide (ITO) film electric heater (15 8 mm2). The electro-conductive ITO film had a sheet resistance of 10 X/sq. Optical devices included an infrared (IR) camera, two high-speed cameras, and a He–Ne laser. A high-speed camera was used to measure the bubble growth behavior from the side of the pool, and an IR camera was used to measure the wall
Fig. 1. Microlayer formation underneath a growing bubble.
Fig. 2. Schematic diagram of the pool boiling experimental apparatus.
2. Experiment 2.1. Pool boiling apparatus
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
temperature distribution. The other devices, including an additional high-speed camera, were used for measurement of the microlayer thickness. The measurement details of bubble growth and microlayer thickness are given in the following section. The experiment for a single bubble in a pool of saturated water was conducted under atmospheric pressure. Data on microlayer thickness and bubble growth were collected at an averaged wall superheat of about 9, 10 and 12 °C, corresponding to an average heat flux of 114, 144 and 209 kW/m2, respectively. The average wall temperature was obtained by spatially averaging the wall temperature distribution measured with the IR camera just before bubble nucleation. It was used as the initial value of wall temperature. The average heat flux was calculated from the electric power supplied by the DC power supply divided by the heater area.
1231
the liquid, k is the wavelength of laser light, and m is the fringe order. From the Eq. (2), the slope that means a unit change in the microlayer thickness at two adjacent fringe positions was measured, and the microlayer thickness with the radius was calculated by integrating all the unit changes in microlayer thickness from the center to the point of interest. To demonstrate the measurement of microlayer thickness using the laser interferometry technique, a convex lens with similar scale and geometry to the microlayer was used. A curved thin liquid film was made between the substrate
2.2. Measurement methods 2.2.1. Bubble growth The bubble growth behavior, from bubble nucleation to departure, was captured using the high-speed camera, as seen in Fig. 3 (a). The captured two-dimensional images show a bubble with symmetric shape. Thus, the equivalent bubble radius with time was calculated based on the assumption that the cross-sectional area of a bubble has an axial symmetry. The bubble growth was expressed by the conventional model of Mikic et al. [22],
R ¼ Ct n
ð1Þ
where R is the equivalent bubble radius, t is the time and C is a constant related to the fluid properties and the wall superheat, and n is a constant that is associated with the bubble growth phases (i.e. inertia and thermal diffusion). The recording frequency of the high-speed camera was 50 kHz and the spatial resolution was 25 mm per pixel. The uncertainty resulting from the spatial resolution of measurement was estimated to be less than 25 mm. 2.2.2. Microlayer thickness The microlayer thickness was measured using a laser interferometry method. The interference fringe distribution was observed, as shown in Fig. 3(b). The optical components of a He-Ne laser, reflection mirrors, prisms, an interference filter and a high-speed camera, which is similar to that used in previous paper [21], were used to perform this measurement. The fringe spacing between the adjacent light and dark rings can be determined by the following equation:
dmþ1 dm ¼
k 2nl cos hl
ð2Þ
where d is the thickness of the microlayer, nl is the refraction index of the film layer, hl is the angle of refraction from the substrate into
Fig. 4. (a) Experimental setup to demonstrate the measurement of curved liquid film using a convex lens and (b) comparison between the measured and actual curvature profiles of a convex lens.
Fig. 3. High-speed camera images of bubble behavior and corresponding interference fringe due to microlayer.
1232
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
and the convex lens, having a curvature radius of 387.6 mm within a tolerance of ±1%, as shown in Fig. 4(a). The optical setup of this measurement is consistent with the boiling experiment. The measured thickness of the curved liquid film was in good agreement with the curvature profile of the convex lens. The results are shown in Fig. 4(b). The measurement error was less than 10% at thicknesses above 0.5 lm. In the region thinner than 0.5 lm, the maximum error was about 30%. The recording frequency for the fringe pattern of a microlayer was 50 kHz. During the initial stage when the dry spot is not apparent (<2.0 ms), detection of fringe patterns near the bubble nucleation site is difficult because the spacing of adjacent fringe patterns is smaller than the spatial resolution of the measurement setup, as seen in Fig. 3 (t = 0.5 ms). After the dry spot appears due to the microlayer evaporation, the first fringe pattern near the triple contact line can be detected clearly (t = 2.0 ms). In such a case, the order of the fringe patterns before the appearance of the dry spot can be inferred from patterns when the dry spot does exist. Fig. 5 shows the time evolution of the laser intensity profile across the microlayer. As the microlayer evaporates, the dry spot appears, and then the triple contact line moves outward from the bubble nucleation site. Backwards through time from (t + Dt) to t, the first fringe pattern moves inward, as seen in Fig. 5(a), but it is not detected
at (t Dt) in Fig. 5(b). In this case, the fringe order at (t Dt) can be determined by tracking the fringe movement from (t + Dt) to t. 2.3. Results 2.3.1. Bubble growth and microlayer formation Fig. 6 shows the equivalent bubble radius for a cycle from bubble nucleation to detachment with a wall superheat of 9 °C. The measured data was compared with Mikic’s model, both for the inertial- and diffusion-controlled bubble growth, as noted above. The average temperature of the heat transfer surface was measured using an IR camera, which was used to calculate the Mikic’s model for the initial wall temperature. The measured data show good agreement with the prediction curve for the diffusion stage, except at the very early stage (<0.05 ms) and the final stage (>3 ms). It is expected that the early stage corresponds to the transition region from the inertia to the diffusion stage. Mikic et al. assumed an infinite superheated liquid around the growing bubble, which leads to a prediction over-estimate as the bubble grows, as seen in the final stage of Fig. 6. Therefore, the bubble growth trend follows the form of t0.5 until 3 ms, and the proportional constant has a range of 0.025–0.035 within the wall superheat of 9–12 °C in this study.
Fig. 6. Time evolution of equivalent bubble radius for an isolated bubble.
Fig. 5. Change in laser intensity profile according to the microlayer evaporation; (a) from t to t + Dt and (b) from t Dt to t.
Fig. 7. Time evolution of microlayer profile.
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
1233
Fig. 7 shows the instantaneous microlayer profiles corresponding to the bubble growth in Fig. 6 until just before complete depletion of the microlayer. The error bars represent the maximum measurement error in each data point. The error increases with the thickness, and the maximum error is less than 0.2 lm at the thickness of 3.5 lm. The data point where the thickness is zero indicates the triple contact line and the outermost data point corresponds to the outer edge of the microlayer at each time step. The instantaneous microlayer thickness at a particular radial position decreases, and the triple contact line moves outward from the bubble nucleation site due to the microlayer evaporation. It should be noted that the radial extent of the microlayer lasts until 5.6 ms. This indicates that the microlayer maintains its formation while undergoing evaporation during the diffusion-controlled stage of bubble growth. While the bubble radius complies the form of R t1 at the very early stage of the bubble growth (<0.05 ms), it grows as obeying the diffusion-controlled growth equation of R t0.5 approximately after 0.1 ms, as shown in Fig. 6. It was commonly reported in previous experimental observations [4] and theoretical analysis [4,5] that the bubble growth behavior during microlayer formation can be approximated with the functional form of the diffusion-controlled growth stage, R t0.5. Thus, to focus the discussion on the microlayer formation and corresponding bubble growth, in this paper the presented data will be confined to the period of microlayer formation, which corresponds to the diffusion-controlled stage of bubble growth. To ensure detection of the outermost boundary of the microlayer using the laser interferometry method, it was cross-checked by calculating the microlayer thickness from the local wall heat flux distribution. The local wall heat flux was obtained by solving the three-dimensional transient heat conduction using the measured time-varying temperature distribution data [21], which were obtained using the IR camera. The local wall temperature and heat flux distributions vary temporally and spatially due to the microlayer formation and evaporation. Such data were used to calculate the thickness profile of microlayer using one-dimensional steadystate conduction equation, which is
q00w ¼
kl ðT w T int Þ d
ð3Þ
00
where qw is local wall heat flux, kl is the thermal conductivity of the liquid, Tw is the local wall temperature and Tint is the temperature at the liquid-vapor interface. The interface temperature can be assumed to be the same as the saturated temperature in the microlayer region, except in the vicinity of the triple contact line due to evaporative resistance [23]. The microlayer thickness calculated by the two different methods shows reasonable consistency in the outermost boundary of the microlayer, as shown in Fig. 8. The outermost boundary obtained by the fringe analysis reveals accurate detection of the transition point of the liquid-vapor interface profile derived from Eq. (3), as the fringe patterns cannot appear when the sharp change in the slope of the liquid-vapor interface occurs. In addition, the profile at the transition region between the microlayer and macroscopic liquid can be roughly estimated. It is noted that the thickness at the transition region may be slightly over-estimated because it was derived with the assumption of onedimensional conduction heat transfer from the wall to the liquidvapor interface. The complete geometry of the boiling bubble can be determined by combining the macroscopic bubble shown in Fig. 3(a) with the corresponding profile of the microlayer and transition region shown in Fig. 8. Fig. 9 shows the bubble growth history during the microlayer formation. It is clearly shown that the microlayer
Fig. 8. Comparison between the microlayer thickness obtained by the fringe analysis and the interface profile calculated by Eq. (3): (a) 1.6 ms, (b) 3.6 ms, (c) 5.6 ms.
1234
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
thickness is of the order of 1 lm and the macroscopic bubble has an order of 1 mm. There is a short transition region between the two, with a thickness of approximately 50 lm. (A more detailed analysis of this transition region taking into account surface tension effects [6], is a possible subject for future investigation.) In addition, contrary to the general expectation that the bubble has a hemispherical shape during microlayer formation, the bubble has an oblate shape with an apparent contact angle less than 50°.
2.3.2. Initial microlayer thickness In several experimental studies [10,11], initial microlayer thickness at a point radially away from a nucleation site was defined as thickness of microlayer initially observed at the specific point while a rapidly growing bubble forms microlayer on the heater surface. The initial microlayer points determined according to the above-mentioned definition were marked in Fig. 10. The initial microlayer thickness increased up to 3.7 lm as the bubble expanded to the radial point of 1.2 mm, but then thinned during further expansion. As mentioned in the previous section, the microlayer formation occurs simultaneous with its evaporation. Therefore, the initial microlayer thickness in Fig. 10 incorporates not only radial expansion induced by microlayer formation but also the thinning effect induced by microlayer evaporation; this is why initial microlayer thickness gets thinned at the later part of the microlayer formation, as shown in Fig. 10. To determine the isothermal initial microlayer thickness, that is, the microlayer thickness initially formed on the wall during bubble growth in the isothermal field without microlayer evaporation, the initial microlayer thickness in Fig. 10 was utilized. Without microlayer evaporation, the microlayer thickness at a specific radial position does not change. In other words, the thickness reduction can be caused by the evaporation. Thus, the isothermal thickness can be reconstructed by tracking the reduced thickness at the same radial position for a time step. Here, the microlayer thickness at 0.2 ms is assumed to be the initial microlayer thickness without the evaporation, because the thickness reduction due to the microlayer evaporation is negligibly small at the early stage of microlayer formation, as seen in Fig. 7. Fig. 11 shows the reconstruction results for the isothermal initial microlayer thickness. It was approximately 5 lm at 1.5 mm of the radial position.
Fig. 9. Complete bubble geometries including the microlayer, transition region and macroscopic bubble shape for bubble growing: (a) 1.6 ms, (b) 3.6 ms, (c) 5.6 ms.
Fig. 10. Initial microlayer thickness.
1235
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
Fig. 11. Isothermal initial microlayer thickness.
There have been several experiments measuring the initial microlayer thickness, such as those by Koffman [9], Yabuki and Nakabeppu [10], and Utaka et al. [11]. The experimental condition, measurement method and empirical model for initial microlayer thickness in the previous studies are summarized in Table 1. The experiments were conducted with the same fluid and under similar liquid temperature and pressure conditions, but measurements were done using different methods. A comparison of the results for the initial microlayer thickness is shown in Fig. 12. The filled symbols show the isothermal initial thickness obtained by the reconstruction while the values predicted by the empirical models [9– 11] represent the initial microlayer thicknesses that necessarily include the thinning effect due to evaporation as illustrated in Fig. 10. However, the resulting difference is not significant as seen in Fig. 12. This is because only data obtained at the very beginning of microlayer formation with negligible evaporation were used for the development of the previous empirical models [9–11]. Therefore, the comparison with the prediction by the empirical models is plausible. Utaka’s result shows a linear shape of the initial microlayer thickness while the others show that the shape is close to convex. Nevertheless, the results for the initial microlayer thickness show very good agreement in order of magnitude. Therefore, it can be concluded that the present experimental data are generally representative of the initial microlayer thickness.
Fig. 12. Comparison between the initial microlayer thickness and the measured data of other experiments for saturated water under atmospheric pressure conditions.
Stralen et al. [25] derived a model based on approximate boundary layer analysis, in which the initial microlayer thickness is equal to the displacement thickness of the hydrodynamic boundary layer outside the bubble. Assuming R / t0.5, the expression derived for the initial microlayer thickness was
d0 ¼ C 0
pffiffiffiffiffi vt
ð4Þ
where C0 is a constant, m is the kinematic viscosity and t denotes the bubble growth time. Cooper and Lloyd [4] suggested a value of 0.8 for C0, which has an empirical basis, Ouwerkerk [5] derived the value 1.26 for C0 using a self-similarity solution, and Olander and Watts [24] derived the value 0.88 for C0 by neglecting the convective term in boundary layer equation. Van Stalen et al. [25] derived C0 as a function of liquid Prandtl number using Pohlhausen solution for laminar boundary layer.
3. Assessment of existing theoretical models 3.1. Review of previous theoretical models In addition to the preceding empirical models, some models based on theoretical analysis have been developed. Cooper and Lloyd [4], Van Ouwerkerk [5], Olander and Watts [24] and Van
Fig. 13. Schematic of the formation of initial microlayer thickness.
Table 1 Experimental conditions of previous studies and the present study for measurement of the initial microlayer thickness using different measurement methods.
Present study Koffman [9] Yabuki and Nakabeppu [10] Utaka et al. [11]
Experiment condition
Measurement method
Empirical model
Saturated water at atmospheric pressure Subcooled water at atmospheric pressure
Laser interferometry
–
Laser interferometry
Saturated water at atmospheric pressure Saturated water at atmospheric pressure
MEMS sensor
d0 = 0.00188r0.6, (r: cm, d0: cm) d0 = 4.34r0.69 (r: mm, d0: lm) d0 = 4.46r, (r: mm, d0: lm)
Laser extinction
1236
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
Smirnov [6] attempted to calculate the initial microlayer thickness by solving the continuity and linear momentum equations for the hydrodynamics of one-dimensional radial flow during formation of the microlayer. Since this model uses a generalized form of bubble growth behavior compared with other models, it was used as a reference model for improving the prediction performance in the later section. Hence, we briefly review Smirnov’s modelling of the initial microlayer thickness (it is helpful to see Fig. 13):
@um @um 1 @P um þ m r2 u m 2 ; þ um ¼ @t @r q @r r
ð5Þ
@ ðrum Þ ¼ 0: @t
ð6Þ
Using the assumption of a hemispherical bubble, the velocity at the microlayer boundary, um(r, d0), was taken to be equal to that of the expanding bubble interface:
um ðR; d0 Þ ¼
3.2. Comparison between measurement data and extant models A comparison of the initial microlayer thickness between the theoretical models and the experimental measurement data obtained in this study is given in Fig. 14. The measured values of C and n for bubble growth, used to calculate the initial microlayer thickness using the theoretical model, were 0.035 and 0.5, respectively. It was found that there were significant differences between the experimental data and the predictions in terms of both the order of magnitude and the shape, even though the predicted values for each model are almost consistent. This may be because the theoretical model for the microlayer thickness is not realistic for bubbles affected by widely varying parameters, such as inertia and surface tension forces, fluid properties and so on. In this regard, the shortcomings of the existing models, with respect to the accuracy of predictions of microlayer formation, need to be identified through systematic comparison with the detailed experimental observations done in the present study. 3.3. Identification of shortcomings
dR : dt
ð7Þ
Smirnov’s model also shows the same form of d0 = C0(mt)1/2 as obtained by Ref. [4,5,24,25]. When the exponent of the bubble growth is chosen as n = 0.5, the constant C0 is 1.05, which is consistent with other models.
Several issues regarding the existing theoretical model were identified based on the present and previous studies [27–30]. The first issue is the assumption of neglecting the surface tension force acting on the bubble growth in the theoretical model. Our experiment provided the decisive insight that surface tension acts on the growing bubble during the microlayer formation. When a bubble grows quickly, the inertial force dominates and the resulting shape is hemispherical, whereas for slow bubble growth, the surface tension dominates so that a spherical bubble forms. In our experiment, the bubble shape was oblate during the whole process of microlayer formation (Fig. 9). This indicates that both the inertial and surface tension forces had an effect on bubble growth. The second issue is the assumption in classical models of a hemispherical bubble. In the classical models, the boiling bubble is approximated to be hemispherical and thus the microlayer formation velocity is the same as the bubble growth velocity. However, in the experiment in this study, an oblate bubble including transition region between microlayer and macroscopic bubble was observed and the interface velocity sweeping liquid to form the microlayer was considerably slower than the bubble growth velocity, as seen in Fig. 15. The final issue concerns the residual flow inside the microlayer. The theoretical analysis considers the initial microlayer thickness
Fig. 14. Comparison of the initial microlayer thickness among the theoretical prediction models for saturated water under atmospheric pressure conditions.
Fig. 15. Radii of bubble and microlayer with time during the microlayer formation.
Then, the following equation is obtained:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ," u 2 # 2 u 2 dP d R 2 dR 1 : d0 ¼ t2mR þ q dR dt2 3 dt R
ð8Þ
The value of dP/dR can be determined using the Rayleigh-Plesset equation [26]: 3
3
dP @ 2 R Rðd R=dt Þ 2r ¼4 2 þ 2 : dR dR=dt @t qR ðdR=dtÞ
ð9Þ
If the surface force for the rapid growth stage of the bubble is negligible, and the bubble growth follows the growth law in the form R = Ctn, then Eq. (8) becomes:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2v t d0 ¼ 9ð1 nÞ þ 2 1n 1 ðn 2Þ þ 0:66n
ð10Þ
1237
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
with no flow inside the microlayer, because viscous dissipation of the kinetic energy of residual flow in the microlayer occurs suddenly over a time interval that is much smaller compared to the microlayer formation time. However, it is found that the residual flow exists once the microlayer has formed, and during that short time it can affect the initial microlayer thickness, making it thinner [29,30]. The residual flow can affect the microlayer thickness. 4. Development of a modified theoretical model With reference to the issues discussed in the previous section, we attempt to show the effect of the surface tension force, the non-hemispherical bubble shape and the residual flow on the microlayer formation using the generalized model of Smirnov [6]. Detailed discussion on each aspect are presented. 4.1. Surface tension effect The surface tension term in Eq. (9) is neglected in the classical model, while the oblate bubble shape in the experiment indicates
that its effect is considerable. Thus, the effect of the surface tension force on microlayer formation should be considered. The effect of surface tension can be included by retaining the surface tension term when solving the Rayleigh-Plesset equation. For the dynamics of a bubble in Eq. (9), the following microlayer thickness equation including surface tension is obtained:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2v t d0 ¼ 9ð1 nÞ þ 2 1n 1 ðn 2Þ þ 0:66n þ qC 4 n42rt4n3
ð11Þ
4.2. Effect of non-hemispherical bubble shape As noted above, the bubble shape is not hemispherical, but oblate during the whole period of microlayer formation in our experiment. This indicates a considerable difference between the radius of the bubble and the outer edge of the microlayer, with a ratio of approximately Rb/Rm 0.6, as seen in Fig. 15. The interface velocity sweeping liquid to form the microlayer was about half the bubble growth velocity. The effect of the difference in radius between the bubble and the microlayer, Rm/Rb = a (a < 1), can be included by using um = dRm/dt = a(dRb/dt) as the boundary condition at the outer edge of the microlayer in Eq. (7). The following initial microlayer thickness equation is obtained:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2v t d0 ¼ 8 2 1 þ 1 ð1 nÞ þ 1 ðn 2Þ þ a2 0:66n þ a3 qC 44nr2 t4n3 3 3 a a n ð12Þ
Fig. 16. Comparison of the initial microlayer thickness measured in the experiment and calculated by the modified prediction model.
This analysis, by incorporating the effects of surface tension and the bubble-microlayer radius difference predicts a sequential thinning of the microlayer, as shown in Fig. 16, because both lead to a decrease in sweep velocity and shear stress for microlayer formation. In addition, the initial microlayer thickness profile changes from linear to convex due to the increasing influence of surface tension in the denominator of Eqs. (11) and (12), which is consistent with the experimental observations (Fig. 12). As a result of these two effects, the experimental data are more closely predicted by Eq. (12); however, a distinct discrepancy still remains.
Fig. 17. Effect of residual flow on microlayer formation: (a) Residual flow velocity degradation by viscous dissipation. (b) Moving distance of residual flow in the radial direction from the bubble center outwards. (c) Conceptual picture of the reduced thickness of the microlayer according to the residual flow.
1238
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
4.3. Effect of short-lived residual flow inside the microlayer There is one further physical aspect of microlayer formation that has not yet been considered, which may have a significant effect on the initial microlayer thickness. In the preceding analysis, residual flow in the microlayer was neglected because the characteristic timescale related to viscous friction in a thin liquid film, tr d02/m, is very small (e.g., 103–101 ms for a water liquid layer with thickness of 1–10 lm). As it is almost impossible to experimentally observe the short-lived residual flow, its influence has not been taken into account. However, some studies using numerical simulations argue that the residual flow may have a considerable impact on the initial microlayer profile over such a short time interval [29,30]. To give some indication of the importance of residual flow, we have attempted to estimate the residual flow effect on the initial microlayer thickness analytically. When stagnant liquid on a flat substrate is swept by a rapidly expanding bubble, the residual flow may further push the trapped liquid along the direction of bubble expansion, and the microlayer thickness will be reduced by the amount of liquid pumped out by the residual flow. The equivalent amount of pumped liquid can be estimated by integrating the volumetric flow rate of the residual flow during its characteristic time so that the reduced thickness can be predicted. A conceptual picture of the thickness thinning effect induced by the residual flow is presented in Fig. 17. If we approximate the microlayer as an axisymmetric wedge, the resultant thickness (dr) incorporating the effect of residual flow can be given by:
dr ¼
3 ! 3 ur 1 ur d0 1 þ 2 um 2 um
ð13Þ
where d0 is the initial microlayer thickness without residual flow, r is the average velocum is the microlayer formation velocity and u ity of the residual flow. The velocity evolution of the residual flow can be presumed to simply follow the viscous diffusion equation, q(our/ot) l(o2ur/oy2), which gives an approximate solution ur(t) = um exp(lt/qd2), as seen in Fig. 17(a). During the effective residual flow dissipation time (tr), which means a period in which the residual flow moves 95% of the terminal moving distance of the r/um is about 0.32. The terminal velocity solution (lr = 0.95 l1), u moving distance can be estimated by integration of the residual flow velocity over time. Finally, the initial microlayer thickness
Fig. 18. Effect of residual flow inside the microlayer on the initial microlayer thickness.
Fig. 19. Comparison between experimental data for initial microlayer thickness and theoretical models: (a) saturated methanol at 58.5 kPa [7], (b) saturated water at 101.3 kPa [10] and (c) saturated toluene at 13.9 kPa [4].
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
equation, which includes the effects of surface tension force, bubble shape and residual flow, is obtained:
d0 ¼ 0:53 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2v t 8 2 1 þ 1 ð1 nÞ þ 1 ðn 2Þ þ a2 0:66n þ a3 qC 44nr2 t4n3 3 3 n a a ð14Þ The proposed model (solid line) shows remarkable agreement with the experimental data measured in the present study, as seen in Fig. 18. Note that the contribution of residual flow to the radial heat conduction in the microlayer can be negligible due to the very short timescale of the residual flow. 4.4. Assessment of the modified model The model was further validated by comparison with experimental data from other studies [4,7,10], which provide quantitative data for bubble growth history as well as microlayer thickness. The values of d0 in [4] and [10] were obtained by calculating conductive heat transfer based on the measured wall temperature, while the value in [7] was measured using the same laser interferometry technique as used in our experiment. The comparison of the prediction of the existing models and the modified model with the measured data for methanol, water and toluene is shown in Fig. 19. The constant C in the equation R = Ct0.5 is determined to best fit the bubble growth presented in each study, which is presented as an inset. The dotted lines present the prediction by existing models including Cooper and Lloyd [4], Van Ouwerkerk [5], and Smirnov [6]. These significantly over-predict the measured data. Based on the Smirnov’s model, it overpredicts the thickness to be 2 to 2.5 times. The solid line presents the prediction by the proposed model. It was found that the proposed model predicts the experimental results well, regardless of the liquid (water, toluene or methanol) or pressure conditions. It is noted in Fig. 19(c) that the prediction by the proposed model slightly underestimated the measured thickness. It can be interpreted as a result of the use of the inaccurate value of ‘a’, which is a parameter associated with the bubble shape. Indeed, the constant C in the equation R = Ct0.5 for toluene is 0.084, which is relatively high. This tends to make the bubble shape more hemispherical [29]. In other word, the ‘a’ value may be greater than 0.6, which is a suggested value in this study. If the value greater than 0.6 was used for the prediction by the proposed model, the microlayer thickness would have been predicted to be thicker, which could result in a more consistent agreement between the prediction and the measured data. The value of 0.6 was suggested based on the observation in our experimental condition. Therefore, some further investigation of the value, which is related to the bubble shape as well as the transition region between the microlayer and macroscopic liquid, is required to generalize the proposed model in future. 5. Conclusion Hydrodynamic formation of a microlayer at the base of a vapor bubble growing on a heated wall was studied. Single bubble nucleate boiling experiments were conducted in a pool of saturated water under atmospheric pressure. A complete bubble geometry, including the microlayer and macroscopic bubble, was obtained using the integrated optical measurement method. Using experimental and theoretical insights, we developed an improved theoretical model of initial microlayer thickness that is in agreement with experimental data. Major findings from this study can be summarized as follows:
1239
– High-resolved measurement data on the initial microlayer thickness were obtained and compared with existing theoretical models. The models contain several idealizations (hemispherical bubble shape and negligible surface tension effects) and significantly overestimate the initial microlayer thickness relative to experimental results. – Precise observation of the growing bubble geometry shows that the bubble is non-hemispherical, which indicates that both inertia and surface tension forces have an effect on bubble growth during microlayer formation. In addition, the critical role played by the short-lived residual flow in microlayer formation was demonstrated by analytical estimates of the residual flow. – Finally, an improved theoretical model of initial microlayer thickness, which incorporates the effects of surface tension, non-hemispherical bubble shape and residual flow, is proposed. The proposed model is in good agreement with the experimental results for other fluids and pressure conditions. The proposed model was developed based on the observation in pool boiling of saturated water on an upward-facing horizontal heater wall under atmospheric condition. However, various parameters such as flow condition, liquid subcooling, fluid, working pressure and wall inclination can affect microlayer formation as well as bubble growth characteristics (i.e. growth rate and bubble shape, especially the bubble shape is strongly related to ‘a’ value which is a ratio of microlayer radius to bubble radius). Therefore, further investigations about effects of the above-mentioned parameters would be next subjects for improving the predictive performance and extending its applications. Conflicts of interest Authors declare that there is no conflict of interest. Acknowledgements This work was supported by National R&D Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2015M1A7A1A01002428). This research was also supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP: Ministry of Science, ICT and Future Planning) (No. 2017M2A8A4015283). References [1] J. Kim, Review of nucleate pool boiling bubble heat transfer mechanisms, Int. J. Multiphase Flow 35 (2009) 1067–1076. [2] S. Moghaddam, K. Kiger, Physical mechanism of heat transfer during single bubble nucleate boiling of FC-72 under saturation conditions I. Experimental investigation, Int. J. Heat Mass Transfer 52 (2009) 1284–1294. [3] F. Demiray, J. Kim, Microscale heat transfer measurements during pool boiling of FC-72: effect of subcooling, Int. J. Heat Mass Transfer 47 (2004) 3257–3268. [4] M.G. Cooper, A.J.P. Lloyd, The microlayer in nucleate pool boiling, Int. J. Heat Mass Transfer 12 (1969) 895–9133. [5] H.J. Van Ouwerkerk, The rapid growth of a vapour bubble at a liquid-solid interface, Int. J. Heat Mass Transfer 14 (1971) 1415–1431. [6] G.F. Smirnov, Calculation of the initial thickness of the microlayer during bubble boiling, J. Eng. Phys. 28 (1975) 369–374. [7] H.G. MacGregor, H.H. Jawurek, High speed cine laser interferometry technique for microlayer studies in boiling, N&O Joernaal 4 (1992) 1–7. [8] M. Gao, L. Zhang, P. Cheng, X. Quan, An investigation of microlayer beneath nucleation bubble by laser interferometric method, Int. J. Heat Mass Transfer 57 (2012) 183–189. [9] L.D. Koffman, Microlayer evaporation in subcooled nucleate boiling, in: Proceeding of 3rd Multi-Phase Flow Heat Transfer Symposium – Workshop, April 18–20, Miami Beach, USA, 1983. [10] T. Yabuki, O. Nakabeppu, Heat transfer mechanisms in isolated bubble boiling of water observed with MEMS sensor, Int. J. Heat Mass Transfer 76 (2014) 286– 297.
1240
S. Jung, H. Kim / International Journal of Heat and Mass Transfer 120 (2018) 1229–1240
[11] Y. Utaka, Y. Kashiwabara, M. Ozaki, Microlayer structure in nucleate boiling of water and ethanol at atmospheric pressure, Int. J. Heat Mass Transfer 57 (2013) 222–230. [12] Y. Sato, B. Niceno, A depletable micro-layer model for nucleate pool boiling, J. Comput. Phys. 300 (2015) 20–52. [13] R.C. Lee, J.E. Nydahl, Numerical calculation of bubble growth in nucleate boiling from inception through departure, J. Heat Transfer – Trans. ASME 111 (1989) 474–479. [14] A. Guion, D. Langewisch, J. Buongiorno, On the extended liquid microlayer underneath a vapor bubble growing at a heated wall, in: Proceeding of 8th International Conference on Multiphase Flow, May 26–31, Jeju, Korea, 2013. [15] B.E. Staniszewski, Nucleate boiling bubble growth and departure, DSR project No. 7-7673, Technical Report No. 16, 1959. [16] V.K. Dhir, Mechanistic prediction of nucleate boiling heat transfer – achievable or a hopeless task?, J Heat Transfer 128 (2006) 1–12. [17] W. Lee, G. Son, Bubble dynamics and heat transfer during nucleate boiling in a microchannel, Numer. Heat Transfer 53 (2008) 1074–1090. [18] C. Kunkelmann, P. Stephan, CFD simulation of boiling flows using the volumeof-fluid method within OpenFOAM, Numer. Heat Transfer 56 (2009) 631–646. [19] C. Gerardi, J. Buongiorno, L.W. Hu, T. McKrell, Study of bubble growth in water pool boiling through synchronized, infrared thermometry and high-speed video, Int. J. Heat Mass Transfer 53 (2010) 4185–4192. [20] I.C. Chu, H.C. No, C.H. Song, Visualization of boiling structure and critical heat flux phenomenon for a narrow heating surface in a horizontal pool of saturated water, Int. J. Heat Mass Transfer 62 (2013) 142–152.
[21] S. Jung, H. Kim, An experimental method to simultaneously measure the dynamics and heat transfer associated with a single bubble during nucleate boiling on a horizontal surface, Int. J. Heat Mass Transfer 73 (2014) 365–375. [22] B.B. Mikic, W.M. Rohsenow, P. Griffith, On bubble growth rates, Int. J. Heat Mass Transfer 13 (1970) 657–666. [23] G. Giustini, S. Jung, H. Kim, S.P. Walker, Evaporative thermal resistance and its influence on microscopic bubble growth, Int. J. Heat Mass Transfer 101 (2016) 733–741. [24] R.R. Olander, R.G. Watts, An analytical expression of microlayer thickness in nucleate boiling, J. Heat Transfer 91 (1969) 178–180. [25] S.J.D. Van Stralen, M.S. Sohal, R. Cole, W.M. Sluyter, Bubble growth rates in pure and binary systems: combined effect of relaxation and evaporation microlayers, Int. J. Heat Mass Transfer 18 (1975) 453–467. [26] M.S. Plesset, The dynamics of cavitation bubbles, J. Appl. Mech. 16 (1949) 277– 282. [27] Y. Zhang, Y. Utaka, Characteristics of a liquid microlayer formed by a confined vapor bubble in micro gap boiling between two parallel plates, Int. J. Heat Mass Transfer 84 (2015) 475–485. [28] A. Guion, J. Buongiorno, S. Zaleski, S. Afkhami, C. Narayanan, Numerical simulation of nucleate boiling using dynamic models of microlayer formation and evaporation, in: Proceeding of CFD4NRS, Cambridge, MA-USA, 2016. [29] S. Hansch, S. Walker, The hydrodynamics of microlayer formation beneath vapour bubbles, Int. J. Heat Mass Transfer 102 (2016) 1282–1292. [30] H.S. Fath, Microlayer formation, evaporation and bubble growth in nucleate boiling (Ph. D. dissertation), McMaster University, 1981.