Effect of subatmospheric pressures on heat transfer, vapor bubbles and dry spots evolution during water boiling

Experimental Thermal and Fluid Science 112 (2020) 109974

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Effect of subatmospheric pressures on heat transfer, vapor bubbles and dry spots evolution during water boiling

T

⁎

Anton Surtaeva,b, , Vladimir Serdyukova,b,c, Ivan Malakhova,b a

Novosibirsk State University, Pirogov str. 1, Novosibirsk, Russia Kutateladze Institute of Thermophysics SB RAS, Lavrentiev ave. 1, Novosibirsk, Russia c Chinakal Institute of Mining SB RAS, Krasny ave. 54, Novosibirsk, Russia b

A R T I C LE I N FO

A B S T R A C T

Keywords: Boiling Subatmospheric pressure Dry spot evolution Bubble dynamics Multiscale heat transfer

The present paper reports the results of the comprehensive experimental investigation of an influence of subatmospheric pressures on multiscale heat transfer characteristics during liquid pool boiling. Experiments were carried out in the pressure range of 8.8–103 kPa at saturated water boiling using high-speed IR thermography, high-speed visualization from different sides and the specially designed transparent ITO heater. This made it possible to obtain simultaneously extensive data set on the effect of reduced pressure on main characteristics of boiling, including heat transfer coefficients, nucleation site density, growth rate and departure diameter of vapor bubbles. High-speed visualization from a bottom side of transparent heater allowed to investigate an evolution of dry spots bounded by triple contact line depending on pressure for the first time. It was demonstrated that the growth rate of dry spots is constant in time and has a non-monotonic dependence on pressure.

1. Introduction Being one of the most effective heat transfer regimes boiling is quite often used in practice. But despite numerous studies there are still questions related to the description of dynamics of two-phase flows, the theory of heat transfer and crisis phenomena development during nucleate boiling [1,2]. Commonly, dimensionless correlations presented in the literature were obtained for certain fluids and are valid only in certain pressure range. For example, at pressures range of p/ pcr < 0.002 the well-known hydrodynamic theory of pool boiling crisis [3,4] shows significantly overestimated results than the experiments [5,6]. The complexity of the theoretical description of the boiling process is primarily due to the fact that this is conjugate task, which requires taking into account the influence of the physical and chemical surface properties, including its geometry, morphology, wetting properties, etc. Secondly, boiling is a multiscale non-stationary process and for its description it is necessary to consider the effects that occur on different spatial and temporal scales. These features of the boiling also create additional complexity for the experimental study of this process. It is well known that the system pressure is one of the most important parameters which has the complex effect on the nucleation, the heat transfer rate and critical heat fluxes at nucleate boiling. In the second half of the last century various authors [7–15] showed that with pressure reduction, the sharp decrease in the density of nucleation sites ⁎

and the emission frequency of vapor bubbles, as well as the increase in the growth rate and departure diameters of bubbles are observed. This reflects the fact, that with pressure reduction vapor density and surface tension dramatically change, which leads to the increase in the critical radius of the vapor bubble and wall superheating corresponding to boiling incipience, and as a result to the increase in the Jakob number. The change in the nucleation site density and the emission frequency of vapor bubbles leads to significant surface temperature fluctuations. A significant change in the local boiling characteristics and in the dynamics of two-phase flows near a heated wall at subatmospheric pressures has a negative effect on the intensity of heat transfer and the value of the critical heat flux. In the literature, a lot of attention is paid to an investigation of the dynamics of vapor bubbles during boiling of various liquids at subatmospheric pressures. In particular, authors of [7,8,13,16–20] analyzed a growth rate of vapor bubbles at pool boiling down to p = 1 kPa with the use of high-speed video recording from the side of heating surface. It was shown that a growth rate of vapor bubbles at subatmospheric pressures boiling cannot be described in frame of heat diffusion-controlled scheme of bubble growth, at which the interfacial heat transfer is the only limiting factor. Bubble growth curves obtained for different reduced pressures are characterized by different exponents n in power law Req(t) ~ tn, which demonstrates the manifestation of different mechanisms of bubble growth with pressure change [21]. The

Corresponding author. E-mail address: [email protected] (A. Surtaev).

https://doi.org/10.1016/j.expthermflusci.2019.109974 Received 5 July 2019; Received in revised form 29 September 2019; Accepted 27 October 2019 Available online 30 October 2019 0894-1777/ © 2019 Elsevier Inc. All rights reserved.

Experimental Thermal and Fluid Science 112 (2020) 109974

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ΔT

Nomenclature A a, C C0, c Cp Ddep Deq Dh Db g H h hfg HSV I IR K n NSD p q Ra Rb Rds Req T t V

wall superheat (K)

Greek letters

area (m2) empirical constants empirical constants of initial microlayer thickness heat capacity (J/(kg⋅K)) bubble departure diameter (m) equivalent bubble diameter (m) bubble vertical size (m) bubble outer diameter (m) gravitational acceleration (m/s2) liquid level height (m) heat transfer coefficient (W/(m2∙K)) latent heat of vaporization (J/kg) high-speed visualization current (A) infrared parameter defined by Kutateladze and Gogonin [49] bubble growth exponent nucleation site density (1/m2) pressure (Pa) heat flux density (W/m2) surface roughness (m) bubble outer radius (m) dry spot radius (m) equivalent bubble radius (m) temperature (°C) time (s) voltage (V)

α δ δ0 Λ λ μ ρ σ υ

thermal diffusivity (m2/s) thickness (m) initial microlayer thickness (m) capillary length (m) thermal conductivity (W/(m⋅K)) dynamic viscosity (Pa⋅s) density (kg/m3) surface tension (N/m) kinematic viscosity (m2/s)

Non-dimensional groups Ar Ja Pr

Archimedes number Jakob number Prandtl number

Subscripts bi cr g l s sat v

boiling incipience critical growth liquid surface saturation vapor

to the formation of so-called liquid microlayer under the vapor bubble. Its small thickness of several microns provides the extremely high values of heat transfer coefficient in this region. The presence of a rapidly evaporating thin liquid film at the base of vapor bubble was supposed back in the 1960s [40–42]. However, only the development of highspeed laser interferometry techniques has made it possible to investigate in recent years the evolution of the microlayer region and its thickness with high temporal and spatial resolutions [28,34–37]. An equally important local characteristic of boiling is the evolution of dry spots under vapor bubbles, which is closely related to the microlayer evaporation rate. The description of the dry spots dynamics is important not only to determine local heat transfer rate in the area of a single nucleation site during the vapor bubble growth and departure, but also to describe the crisis phenomena development at boiling, especially at subatmospheric pressures. Recently Surtaev et al. [31] have demonstrated that the high-speed video recording from the bottom side of a transparent heater allows to analyse in detail the evolution of microlayer region and dry spots under vapor bubbles. In particular, it was shown, that at water and ethanol pool boiling the dry spots growth rate is constant in time in a wide range of heat fluxes. Despite the fact that the usage of above described modern experimental techniques with high temporal and spatial resolutions allows to obtain fundamentally new information on the boiling process, the vast majority of experimental data, including local heat transfer in the region of the nucleation site, the evolution of liquid microlayer and dry spots were obtained at water boiling only at atmospheric pressure. Of course, this fact does not allow to fully test existing theoretical models to describe local boiling characteristics, and also limits the possibility of creating new correlations that would be applicable to describe multiscale heat transfer characteristics and crisis phenomena during boiling of liquids in a wide range of pressures. The aim of this work is the experimental study of heat transfer, nucleation site density, the evolution of vapor bubbles, as well as dry spots during water boiling at subatmospheric pressures using both high-speed visualization and

pressure reduction also has a significant effect on the bubbles shape - in particular, massive vapor bubbles formed at very low pressures have specific, so-called “mushroom” shape. The departure of such bubbles is accompanied by the appearance of high-velocity liquid jet, penetrated into its flattened base. Also recently, Rullière et al. [17] noted the “cyclic boiling regime” at water boiling at 1.2 kPa, which is characterized by the appearance of numerous small bubbles on a surface after massive bubble liftoff. The nucleation cycle can restart as soon as this “bubble crisis” ends. A detailed review of the experimental studies devoted to the analysis of vapor bubbles dynamics at subatmospheric pressures boiling is presented in the recent paper of Rullière et al. [20]. However, despite the fact that evolution of vapor bubbles at subatmospheric pressures boiling has been studied in detail, nowadays this direction of researches continues to actively develop [16–25]. Primarily this is due to the expansion of the field of application of boiling regimes at reduced pressures. For example, subatmospheric pressures boiling is realized in liquid desiccant dehumidification systems and absorption chillers, which are used to remove excess heat and to maintain an optimal thermal regime during operation of various types of equipment. In addition, boiling at low pressures is a promising method for cooling microelectronic devices that require maintaining a given low operating temperature [22,23], including space applications. The development of new experimental techniques in the last two decades, including high-speed IR thermography [26–33], laser interferometry [28,34–37], rainbow schlieren technique [38,39] and so forth, has made it possible today to obtain fundamentally new information on the bubble dynamics and multiscale characteristics of boiling including local heat transfer in the area of nucleation site, the thickness and evaporation rate of liquid microlayer, the evolution of dry spots formed under vapor bubbles, etc. In particular, Jung and Kim [28] and Serdyukov et al. [32] based on the data of high-speed IR thermography showed that the local heat flux density at the moment of bubble appearance and at the initial stage of its growth during water boiling at atmospheric pressure can be as high as 1 MW/m2. This is due 2

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2.2. Measurements techniques

infrared thermography and specially designed transparent ITO heater.

To study the effect of pressure on the vapor bubbles dynamics and triple contact line evolution at pool boiling the high-speed video recording with «Vision Research» Phantom v.7.0 camera was performed from the bottom side of the heater, as shown in Fig. 1. The maximum video recording frame rate of 20,000 fps has allowed to achieve high temporal resolution to study in detail all stages of vapor bubble lifecycle. To increase the spatial resolution, «Nikon» 105 mm f/2.8G macro lens was used and, as a result, the maximum spatial resolution of the video recording in experiments was 35 μm/pixel. In addition to boiling visualization from the bottom side of the heater, also traditional format of video recording from its side through windows was performed. As it will be shown below, this made it possible to measure the equivalent and departure diameters of vapor bubbles in a wide pressure range. High-speed infrared camera «FLIR» Titanium HD 570 M with the spectral response of 3.7–4.8 μm and NETD (noise equivalent differential temperature) value of 18 mK was used to measure the non-stationary temperature field of the heating surface. As configured for this study, the thermographic camera had a frame rate of 1000 fps and maximum spatial resolution of 150 μm/pixel. The uncertainty analysis of surface temperature measurements is described in following subsection.

2. Experimental facility and measuring techniques 2.1. Experimental setup and test section The scheme of experimental setup for the investigation of local and integral characteristics of heat transfer during pool boiling at subatmospheric pressures is shown in Fig. 1a. The setup consists of two sealed cylindrical stainless steel vessels inserted one inside an other. To observe boiling in the inner chamber, the setup has two sealed windows located on one optical axis. The setup is vacuumed according to DIN 28400-3:1992-06 standard. Deionized water (MiliQ by «Merck») on the saturation line for a given system pressure ps is used as working fluid. The thermophysical properties of water for studied pressures are presented in Table 1. To maintain a constant temperature of the working liquid, the boiling chamber is mounted in the isothermal bath with two tubular electric pre-heaters with total power of 2.4 kW. The temperature of water in the isothermal bath is controlled using an electronic temperature regulator. The working volume of boiling chamber is 2.5 × 10−3 m3. The temperature in the inner reservoir and isothermal bath (T1 and T2 in Fig. 1a) is measured during experiments using «Honeywell» HEL 700 platinum resistance temperature detector (T1) and chromel-alumel thermocouple (T2). The inner chamber was evacuated through a valve connected to the «MEZ» vacuum pump at its top. To eliminate the effect of pressure increasing during boiling, the working volume was equipped with water-cooled condenser. The system pressure was controlled using the “Manometer” BO1227 vacuum gauge. As it was highlighted by Rullière et al. [17] the boiling environment at subatmospheric pressure is particularly non-homogeneous. This is due to the fact that for subatmospheric conditions the pressure head due to the height of liquid level over the heating surface may be of the same order of magnitude as the vapor pressure. This leads to a non-negligible variation of the local saturation temperature inside the liquid bulk with the depth. Therefore for correct measurement of pressure at the heated surface level (ps) it is also necessary to take into account the hydrostatic pressure of liquid column over the heater - ρlgH. For this purpose before each experiment liquid level over the heater surface H was measured. The value of H in experiments was 12–15 cm. As a result, the experiments were carried out in the pressure range of ps = 8.8–103 kPa (corresponding Tsat range is 43.3–100.4 °C). In the experiments sapphire substrate 60 mm in diameter, on the bottom side of which electro conductive indium – tin oxide (ITO) film was deposited by ion-plasma sputtering, was used as a heating element (Fig. 1b). The thicknesses of the sapphire substrate and ITO film heater are 3 mm and 1 μm respectively. As noted in a number of papers [26,28,29,31,32], the advantage of the usage ITO as heater material in experiments devoted to the investigation of local and integral characteristics of heat transfer at nucleate boiling is its transparency in the visible wavelength range (380–750 nm) and opacity in the IR range (3–5 μm). At the same time, sapphire transmission in the wavelength range of 0.3–5 μm exceeds 80%. The combination of these properties makes it possible to measure non-stationary temperature field of the ITO film surface by infrared camera and to observe vapor bubbles evolution directly on the sapphire substrate by high-speed video camera [26,31]. Fabricated samples had the electrical resistance of 8.6 Ohm and the heating area of 28 × 30 mm2. According to the manufacturer («RostoxN», Russia), the surface roughness of the sapphire substrate was Ra ≈ 8 nm, what classifies it as “ultra-smooth” surface. Samples were heated by Joule effect using DC power supply «Elektro Automatik» PS 8080-60 DT with a maximum output power of 1.5 kW via thin silver electrodes vacuum deposited onto the ITO film.

Fig. 1. Scheme of the experimental setup (a) and test section (b) for visual studies of boiling at subatmospheric pressures. 3

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Table 1 Thermophysical properties of water at measured pressures. Items

Unit

ps = 8.8 kPa

ps = 22 kPa

ps = 42 kPa

ps = 73 kPa

ps = 103 kPa

Tsat ρl ρv Cpl hfg λ σ μl

°C kg/m3 kg/m3 J/(kg⋅°C) J/kg W/(m⋅°C) N/m Pa⋅sec

43.33 990.9 0.060 4178.8 2395.6⋅103 0.635 0.069 0.614⋅10−3

62.13 982.1 0.143 4183.9 2352.4⋅103 0.656 0.0659 0.452⋅10−3

77.03 973.6 0.262 4193.2 2315.3⋅103 0.668 0.0632 0.368⋅10−3

91.04 964.6 0.440 4206.2 2279.8⋅103 0.676 0.0606 0.311⋅10−3

100.4 958.0 0.606 4217.2 2255.4⋅103 0.679 0.0588 0.281⋅10−3

averaged temperature of the heating surface was calculated by averaging the temperature field over the heater area and time for 10 s, taking into account the temperature drop across the sapphire substrate in the stationary approximation. As can be seen in the figure, with the pressure reduction the heat transfer rate decreases for the given heat fluxes. Obtained result agrees with the results of previous studies [6,8,11,22] and, as it will be shown below, is associated with a significant decrease in the density of nucleation sites, increase in surface superheating, corresponding to the onset of nucleate boiling and decrease in the bubble emission frequency. The analysis of evolution of temperature field shows that for subatmospheric pressures at ps ≤ 20 kPa, the dependence of integral surface temperature (averaged over the heater area at each time step) on time is cyclic and clearly reflects different stages of bubble lifecycle during boiling. This is due to the fact that at pressures ps ≤ 20 kPa, as it will be shown in Section 3.2, as a rule, a single bubble occurs on a heating surface, which departure diameter is comparable to or even exceeds a linear size of the used heater. A decrease in the nucleation site density and an increase in the bubble departure diameters lead to the fact that at indicated pressures the integral temperature fluctuations are similar to those, which were observed in previous experimental studies for evolution of local temperature of the thin-film and thin-walled heating surfaces under a single nucleation site at atmospheric pressure boiling [26–28,31,32]. It is generally accepted to characterize the liquid superheating above a saturation temperature at nucleate boiling by the Jakob number (Ja). Since the integral temperature of a heater surface at subatmospheric pressures varies with time and is cyclic, to estimate the Jakob number it is necessary to use the following expression:

2.3. Uncertainty analysis The measurement error of the pressure p using the precision pressure gauge is ± 0.25 kPa. The error in measuring a liquid level over the heater surface H is ± 2 mm. The measurement error of the input heat flux density q is composed of the errors of measure the current through the heater I, the voltage drop V on it and the area of the heating surface A: q = V∙I/A. Thus, the relative error in measuring the value of q in the study was less than 5%. Furthermore, the heat losses due to thermal conductivity through a heater also should be taken into account. To estimate these losses, test experiments were carried out with saturated water under atmospheric pressure at the heat flux densities corresponding to the conditions of convective heat transfer. Comparison with the McAdams' model [43] showed that a discrepancy between experimental and calculated data is within 10%, which means that the main amount of heat from the heater goes directly into a working liquid. For correct measurement of the heaters temperature field by infrared camera, preliminary calibration procedure was performed before each series of experiments. The boiling chamber was filled with deionized water at room temperature and then it was heated to approximately 100 °C by outer isothermal bath. The temperature of the heater surface was monitored using «Honeywell» HEL 700 platinum resistance temperature detector, which was placed directly on the ITO film. The temperature field also was monitored using the infrared camera from the bottom side of the heater. Based on the results of the experiments, the calibration dependencies of the camera digital level on the surface temperature were plotted and then used in data analysis. To estimate corresponding wall superheat ΔT = Ts − Tsat for given heat flux density q and pressure ps, the temperature of the sapphire substrate surface was recalculated in the stationary approximation based on the data on the temperature field of the ITO film obtained by IR camera. Due to the relatively high thermal conductivity of sapphire (λ = 27 W/(m⋅K)) and small thickness of the ITO film, the temperature difference between the ITO and boiling surfaces was no more than 3 K. The geometry parameters of vapor bubbles and dry spots were measured by counting pixels in a captured frame of video recording. Therefore the measurement error is determined by the spatial resolution of the used video camera and it was about ± 35 μm. The determination of the vertical size of vapor bubble based on side view HSV video data was performed taking into account the camera tilt to a heater surface. Thus, the relative error in measuring the Dh value was 6%. To analyze such values, as departure diameters, growth rates of dry spots, etc., statistical analysis of the ensemble of 10–20 bubbles was performed.

Ja =

ρl Cp ρv hfg

(Tbi − Tsat ),

where Tbi is the surface temperature at the moment of boiling incipience

3. Experimental results 3.1. Heat transfer rate at subatmospheric boiling This subsection presents the results obtained from a data analysis of high-speed infrared thermography. In Fig. 2 boiling curves for saturated water obtained in the experiments at different pressures are shown. The

Fig. 2. Boiling curves of saturated water at different pressure. 4

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and the evolution of dry spots bounded by triple contact line at their base. In particular, it can be seen that for all measured pressures, after the appearance of a vapor bubble, dry spot is formed. Over time, the dry spot area expands due to the intense liquid evaporation in a region of microlayer. The departure of vapor bubble is accompanied by collapse of the triple contact line and dry spot rewetting. In addition, with the use of obtained data, individual nucleation sites can be identified with much higher accuracy in comparison with conventional visualization from the heater side in a wide range of heat fluxes up to CHF. Below the results of quantitative analysis of obtained HSV data are discussed. First of all, the analysis of experimental data on the effect of reduced pressure on a nucleation site density (NSD) was performed. The estimation of the number of active nucleation sites was carried out based on the data of high-speed visualization from the bottom heater side. Fig. 6 shows the dependence of the NSD value on the heat flux density for different pressures. From the figure it can be seen, that the pressure reduction at given heat fluxes leads to the decreasing of the number of active nucleation sites on the heating surface. Secondly, one can see that for pressures range from 42 to 103 kPa, the NSD values increase with heat flux increasing. At the same time, for pressures ps = 8.8 kPa and ps = 22 kPa, this value remains constant for all measured heat fluxes, which is due to the fact that as a rule only one vapor bubble appears on a heating surface. At the next stage, the analysis of the bubbles growth rates for various pressures was carried out. Fig. 7 shows the evolution of outer

for given nucleation site. This temperature corresponds to the maximum wall superheat before the vapor bubble appearance [32]. The Tbi value was determined based on the analysis of IR thermography data taking into account the temperature drop through the sapphire substrate. The analysis of IR thermography data was carried out at decreasing heat flux density. The dependence of the Jakob number on pressure is shown in Fig. 3. An analysis of obtained curve shows that with pressure reduction, the Jakob number increases significantly up to Ja ≈ 440, which is primarily associated with a significant decrease in vapor density. Also, some contribution is made by an increase in wall superheat ΔΤ at the boiling incipience with pressure reduction. This fact agrees with the generally accepted theory of nucleation and is explained by an increase in the critical radius of viable vapor nuclei with pressure reduction. It is also seen from the figure that the data obtained in this study are in good agreement with data of previous studies [8,13,19], which confirms the correctness of the made temperature measurements and its subsequent processing. The usage of IR thermography in experiments, when the boiling process takes place directly on a thin-film heater, would significantly simplify the processing of measurements of non-stationary temperature field, as well as improve a temperature measurement accuracy. Moreover, carrying out of these experiments would allow to analyze an evolution of local heat transfer in a region of active nucleation site with high accuracy, as it was done previously at water boiling at atmospheric pressure [32]. Therefore, perform of such experiments is the task of future research.

1

diameters Db, together with equivalent diameters Deq = (Dh Db2 ) 3 of vapor bubbles for various pressures. The data analysis shows that with the pressure reduction and increase in Jakob number, the bubble growth rate, the time of its contact with the solid wall and the departure diameter increase. Moreover with pressure reduction the discrepancy between Db and Deq values is growing, which is a direct result of the significant change in the bubble shape during its growth at subatmospheric pressures boiling. As can be seen in the frames of high-speed visualization (Fig. 4), fast-growing bubbles have a flattened shape, which is due to the subcooling gradient in the bulk liquid [17]. The next graph presents the comparison of the results obtained for bubble growth rates for various pressures with the known power laws Req(t) ~ tn describing bubble growth (Fig. 8). According to Labuntsov approach [44,45], there are several schemes of bubble growth, including an inertia-controlled and a heat diffusion-controlled regimes, which are basic in the overwhelming majority of cases. For example, in the case of water boiling at atmospheric pressure the heat diffusioncontrolled regime is realized for a considerable time on the bubble growth stage. For this regime the growth of the vapor bubble is limited by heat transfer rate on the interface. At the same time, at the initial

3.2. Analysis of high speed visualization This subsection presents the data analysis of high-speed visualization of the evolution of vapor bubbles at different pressures at two video camera positions: from the side and from the bottom of the transparent heater. In Fig. 4 and Fig. 5 the frames of high-speed video recording (HSV) of the water boiling at different pressures (ps = 8.8, 42, 103 kPa, corresponding Tsat = 43.3, 77, 100.4 °C), obtained from the side (Fig. 4) and from the bottom of transparent heater (Fig. 5) are shown. From HSV frames of side view it can be seen that the boiling behavior significantly changes with pressure reduction at the given heat flux density. First of all, the decrease in ps value leads to the noticeable decrease in the nucleation site density and to the increase of vapor bubbles sizes. For atmospheric pressure at q = 40 kW/m2 on the heating surface there are about 5 bubbles with a transverse size of no more than 5 mm, while for the lowest pressure (ps = 8.8 kPa) the surface is occupied by one, “massive” bubble, which size during its growth reaches 50 mm, which exceeds the size of the heater. In addition, the pressure reduction from 103 to 8.8 kPa results both in higher growth rates and departure diameters of vapor bubbles. An increase in the bubble growth time and an increase in the duration of the stage corresponding to transient heating of the liquid until the next bubble appears leads to a significant decrease in the bubble emission frequency with pressure decrease. The shape of vapor bubbles also dramatically changes with a pressure decrease. In particular, during boiling under atmospheric pressure, vapor bubbles throughout their growth have a quasi-spherical shape, while boiling at subatmospheric pressures is characterized by “flattening” of bubbles. In addition, as can be seen in the frames at boiling at the lowest pressure, a vapor bubble at the stage of departure from a surface has “mushroom” shape with pronounced vapor stem at its base. It should be noted, that presented results of high-speed visualization from the side of the heater are consistent with the previous experimental observations [7,8,13,16–20]. Fundamentally new information on the effect of pressure on the multiscale nucleate boiling characteristics could be obtained by highspeed recording from the bottom side of the transparent heating surface (Fig. 5). From the presented frames it is clearly seen that such recording type allows to trace in detail both the outer size of the vapor bubbles

Fig. 3. The influence of pressure reduction on the Jakob number at water boiling. 5

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Fig. 4. Frames of side view high-speed video recording of bubble dynamics at different pressure at water boiling (q = 40 kW/m2).

range of Jakob numbers the heat diffusion-controlled regime is realized for a long stage. However, when the pressure is much lower than atmospheric pressure (ps = 8.8 kPa), it can be seen that the bubble growth curve can be divided into 3 following stages. At the first stage (I), as has already been noted above, the inertia-controlled regime takes place. The next stage (II) is intermediate, at which it is necessary to take into account both the heat diffusion and inertial effects [45]. As it is illustrated in Fig. 8 at this quite long stage, the experimental data can be approximated by the dependence Req(t) ~ t0.75, which is an intermediate law of bubble growth between inertia-controlled and heat diffusion-controlled regimes. It is worth mentioning here the results of Yagov [47], who based on the joint solution of the Rayleigh equation, the energy flow equation and the ideal gas equation of state has obtained the analytic solution for bubble growth rate at high Jakob numbers (Ja > 300), which also presents the dependence Req(t) ~ t0.75.

stage, when the bubble size is rather small and the growth rate is maximal, the increase in the bubble volume is limited by the inertia force of the surrounding liquid. As can be seen in the Fig. 8, for two reported pressures the linear bubble growth law Req(t) ~ t, which corresponds to an inertia-controlled regime, is observed. At this stage, the bubble growth rate is described by the Rayleigh equation. At the same time, the duration of the initial stage, at which inertial forces of a fluid play the crucial role, increases significantly with decreasing pressure. For atmospheric pressure the initial stage does not exceed 0.6 ms, whereas for pressure ps = 8.8 kPa the duration of the initial stage (or in other words, Rayleigh stage) is already about 2 ms. It is also seen from the figure that over time the growth rate of bubbles at atmospheric pressure decreases noticeably and can be described by the Labuntsov-Yagov model [46], which has relationship Req(t) ~ t0.5, which confirms that for a given

6

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Fig. 5. Frames of bottom view high-speed video recording of bubble dynamics at different pressure at water boiling (q = 40 kW/m2).

However, at the final stage of bubble evolution (III), when its growth rate and pressure in the bubble decrease significantly, experimental data can be approximated by the dependence Req(t) ~ t0.5, which corresponds to the heat diffusion-controlled regime. Also, based on HSV data, the bubbles departure diameters were measured. In Fig. 9 the dependence of average departure diameters normalized by the capillary length (Λ) on pressure is presented. In this case values of bubble departure size were calculated from the data on the equivalent diameters (as the maximum value of Deq). It can be seen that with decreasing pressure, the dimensionless diameter increases significantly. This indirectly indicates that the bubble departure diameter value at boiling is not determined only by the balance between surface tension and buoyancy forces (Fritz correlation [48]). Gao et al. [19] examined the balance of forces acting on the vapor bubble and showed that during boiling at subatmospheric pressures, along with the

buoyancy force, the inertia force is decisive. Also in Fig. 9 the Kutateladze-Gogonin correlation [49] for bubble departure diameter, based on the criterial approach of considering the balance of the forces acting on the vapor bubble, is presented:

1 Ja 2 ∼ ⎛ ⎞ , Ddep = a (1 + 105K ), K = Ar ⎝ Pr ⎠

(1)

where a = 0.31 – is empirical constant. As illustrated in the figure, experimental data on the bubbles departure diameters for water are consistent with Kutateladze-Gogonin correlation in wide pressure range. 3.3. Evolution of dry spots Bottom side HSV allowed us to investigate the evolution of dry spots 7

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∼ Fig. 9. Dimensionless departure diameter Ddep at nucleate boiling in dependence on pressure.

Fig. 6. The influence of pressure reduction on the nucleation site density at water boiling.

presented. The data analysis shows, that the growth rate of dry spots and their maximum size significantly depend on the system pressure. It is also shown that the radius of a dry spot grows linearly with time, and therefore the expansion rate of unwetted areas under the bubbles is constant. This result is consistent with the previous experimental results on the dry spots evolution at water boiling at atmospheric pressure [31]. At the same time, Zhao et al. [50] after analyzing the growth of dry spot under the bubble have proposed the following dependence: 1 2

2 3 ⎡ 8C pl λl ΔT (t − tg ) ⎤ Rds (t ) = ⎢ 3 ⎥ c 2αhfg ρv2 ⎣ ⎦

(2)

where tg – is the time while the liquid microlayer is just formed. From this relation, it follows that the growth rate of a dry spot is not constant in time and its radius has the dependence Rds(t) ~ t0.5, which does not agree with presented experimental data. Therefore, further development of theoretical models is needed to describe the dynamics of dry spots and for that the experimental data obtained in this research will be useful. Based on the experimental results on dry spots evolution, the dependence of dry spots growth rate on pressure at the given heat flux q = 40 kW/m2 was plotted (Fig. 11). For the analysis, the average values of dry spots growth rates for the bubbles ensemble were calculated

Fig. 7. Evolution of vapor bubbles outer (Db) and equivalent (Deq) diameters at boiling at two subatmospheric pressures (q = 40 kW/m2).

Fig. 8. Vapor bubble growth curves for different pressures (q = 40 kW/m2).

(Fig. 5) which appear under vapor bubbles at water boiling in the pressures range of 8.8–103 kPa for the first time. In Fig. 10 the dependences of dry spots radius on time for various pressures are

Fig. 10. Dependence of the dry spot radius on time for different pressures at water boiling (q = 40 kW/m2). 8

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(usually, for calculating the microlayer thickness the inertia-controlled stage of the bubble lifecycle is not taken into account), at subatmospheric pressures boiling to describe the initial microlayer thickness it is necessary to take into account the initial stage of vapor bubble growth, which, as was shown above, represents linear growth in time. For example, for the lowest pressure (ps = 8.8 kPa) in experiments, the analysis of the data presented in Section 3.2 shows that the duration of the initial bubble growth stage is about 2 ms, over which the bubble reaches size of Db ~ 6 mm, which is commensurate with the bubble departure diameter at atmospheric pressure boiling. Furthermore, it is also necessary to take into account the intermediate stage at which the combined heat diffusion and inertial effects are observed. For a rough estimate of the microlayer thickness at subatmospheric pressures boiling, it is possible to accept the bubble growth law in the form of Rb(t) = Ct0.75. However, the estimations show that in this case the microlayer thickness slightly varies compared with result when we take n = 0.5 in the bubble growth law. Moreover, with the increase in n from 0.75 to 1, the microlayer thickness will only increase. Therefore, on the basis of the above conclusions, the changes in the liquid viscosity and in the bubble growth rate will lead only to an increase in the microlayer thickness with pressure reduction and, consequently, to the decrease in the dry spots growth rate (if not taking into account the influence of other factors, for example, evaporation rate). Therefore, a decrease in the dry spots growth rate with pressure reduction from 103 kPa to 42 kPa is most likely associated with an increase in a liquid viscosity with saturation temperature decreasing, as well as with an increase in the vapor bubbles growth rate. At the same time, the influence of above factors cannot explain the observed increase in the dry spots growth rate with pressure reduction from 42 kPa to 8.8 kPa. It is obvious that the enthalpy of superheated liquid also affects the evaporation rate of liquid in the microlayer region, which, for example, follows from dependence (2). The increase in liquid superheating before the bubble appearance will lead to the increase in the microlayer evaporation rate and, consequently, to the increase in the dry spots growth rate. In Section 3.1, we noted the fact that pressure reduction results in noticeable increase in wall superheat, corresponding to the activation of nucleation sites. It is possible that this fact leads to the increase in the dry spots growth rate with pressure reduction from 42 kPa to 8.8 kPa at the given heat flux. Therefore, the analysis of the results presented in this subsection shows that the theoretical description of the dry spots evolution and characteristics of the liquid microlayer under vapor bubbles at boiling is not a trivial task and still unsolved problem, which needs more detailed theoretical analysis. Such analysis requires consideration of many factors, including changing of the microlayer characteristics due to changes in the viscosity and bubbles growth rate, as well as the effect of wall superheat at vapor bubble appearance and the possible manifestation of kinetic effects associated with intense evaporation of the liquid in the microlayer region. Authors of the present study hope that obtained results will be useful to deepen understanding of the physical processes of vapor bubbles and dry spots evolution and for development new theoretical correlations for describing multiscale boiling characteristics.

Fig. 11. Dependence of dry spot growth rate on pressure at water boiling (q = 40 kW/m2).

for the given pressure. It is seen that the dry spots growth rate has not monotonous dependence on pressure. For presented heat flux density, as the pressure reduces from 103 kPa to 42 kPa, the dry spots growth rate decreases by almost 2 times. However, with a further pressure reducing from 42 kPa to 8.8 kPa, the growth rate increases by almost 2 times. For the lowest pressure (ps = 8.8 kPa), the data on the growth rate practically coincide with the data obtained for atmospheric pressure boiling. Thus, the lower extremum of dry spots growth rate is observed in Fig. 11. It is obvious that the dry spots growth rate is directly related to the evaporation of liquid microlayer under a vapor bubble at its growth stage. Therefore, one of the most important characteristics for describing the dynamics of microlayer evaporation is its initial thickness. In most semi-empirical models, for example, proposed by Cooper and Lloyd [42], Van Stralen et al. [13], Van Ouwerkerk [51], based on the approximate boundary layer analysis the following correlation for the microlayer initial thickness was obtained: (3)

δ0 = C0 νt

From this expression it can be seen that the initial microlayer thickness significantly depends on both the liquid viscosity ν and its growth time. Since at pressure reduction, the saturation temperature decreases and, consequently, the viscosity of water increases, therefore according to (3), the microlayer thickness increases. An increase in the microlayer thickness means an increase in the volume of liquid in its region. Thus, assuming that the evaporation rate from the interface of a liquid microlayer does not change with pressure (in reality, this is most likely not happen), an increase in a liquid volume due to an increase in a microlayer thickness with pressure reduction will lead to an increase in the time of full evaporation of a liquid in a microlayer region. This, in turn, should lead to a decrease in the dry spots growth rate. More generalized equation for the initial microlayer thickness, taking into account the bubble growth law Rb(t) = Ctn, was obtained in [52,53] by solving the continuity and linear momentum equations for the hydrodynamics of one-dimensional radial flow during formation of the microlayer:

4. Conclusions In the present study an experimental data on the main characteristics of water boiling depending on pressure in the range of 8.8–103 kPa were obtained using the complex of experimental techniques, including high-speed IR thermography and high-speed visualization. It was shown, that:

2νt

δ0 = 9(1 − n) + 2

(

1 n

)

− 1 (n − 2) + 0.66n

(4)

From this relationship follows that the initial microlayer thickness depends not only on the viscosity coefficient, but also on the law of vapor bubble growth by means of growth exponent n. However, if in the case of boiling at atmospheric pressure, the bubble growth rate for a long period of microlayer formation can be described as Rb(t) ~ t0.5

- Heat transfer coefficient, as well as the nucleation site density and emission bubble frequency decreases with pressure reduction; - The vapor bubbles growth rate and departure diameter of vapor bubbles increase with pressure reduction; 9

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- At low subatmospheric pressures bubble growth can be divided into three stages – inertia-controlled stage (Req(t) ~ t), the stage at which both the heat diffusion and inertial effects take place (Req(t) ~ t0.75) and heat diffusion-controlled stage (Req(t) ~ t0.5). At the same time, the duration of the inertial stage increases significantly with decreasing pressure; - The bubble departure diameter at subatmospheric pressures for water can be described by the Kutateladze – Gogonin dependence [49].

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High-speed visualization from the bottom side of the transparent heating surface made it possible to investigate the effect of pressure reduction on the evolution of dry spots under bubbles for the first time. It is shown that the growth rate of dry spots is constant over time and also has a non-monotonic dependence on pressure for a given heat flux. Meanwhile the minimum value of the growth rate of dry spots under vapor bubbles at the heat flux density of 40 kW/m2 is observed at pressure of ps = 42 kPa. The obtained experimental data can be used to verify existing and to construct new theoretical approaches to describe the multiscale boiling characteristics. At the same time, further experimental studies on the evolution of local heat transfer during bubble lifecycle and on the liquid microlayer characteristics in a wide pressure range will facilitate the theoretical description of boiling process. Declaration of Competing Interest We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. Acknowledgments The reported study was funded by the Russian Science Foundation (Project № 18-79-00078). References [1] L.S. Tong, Boiling heat transfer and two-phase flow, Routledge, 2018. [2] Y. Koizumi, M. Shoji, M. Monde, Y. Takata, N. Nagai, Boiling: Research and Advances, Elsevier, 2017. [3] S.S. Kutateladze, Hydrodynamic model of heat transfer crisis in free-convection boiling, J. Tech. Phys. 20 (11) (1950) 1389–1392. [4] N. Zuber, On the stability of boiling heat transfer, Trans. Am. Soc. Mech. Engrs. 80 (1958). [5] A.B. Ponter, C.P. Haigh, The boiling crisis in saturated and subcooled pool boiling at reduced pressures, Int. J. Heat Mass Transf. 12 (4) (1969) 429–437. [6] G.I. Samokhin, V.V. Yagov, Heat transfer and critical thermal loads under liquid boiling in the range of low reduced pressures, Teploehnergetika (1988) 72–74. [7] M.A. Johnson Jr, J. De La Peña, R.B. Mesler, Bubble shapes in nucleate boiling, AIChE J. 12 (2) (1966) 344–348. [8] N.N. Mamontova, Boiling of certain liquids at reduced pressures, J. Appl. Mech. Tech. Phys. 7 (3) (1966) 94–98. [9] R. Cole, Bubble frequencies and departure volumes at subatmospheric pressures, Am. Inst. Chem. Eng. J. 13 (4) (1967) 779–783. [10] M. Akiyama, F. Tachibana, N. Ogawa, Effect of pressure on bubble growth in pool boiling, Bull. Jpn. Soc. Mech. Eng. 12 (53) (1969) 1121–1128. [11] V.V. Yagov, A.K. Gorodov, D.A. Labuntsov, Experimental study of heat transfer in the boiling of liquids at low pressures under conditions of free motion, J. Eng. Phys. 18 (4) (1970) 421–425. [12] J.K. Stewart, R. Cole, Bubble growth rates during nucleate boiling at high Jakob numbers, Int. J. Heat Mass Transf. 15 (4) (1972) 655–664. [13] S.J.D. Van Stralen, R. Cole, W.M. Sluyter, M.S. Sohal, Bubble growth rates in nucleate boiling of water at subatmospheric pressures, Int. J. Heat Mass Transf. 18 (5) (1975) 655–669. [14] S.J.D. Van Stralen, W. Zijl, D.A. De Vries, The behaviour of vapour bubbles during growth at subatmospheric pressures, Chem. Eng. Sci. 32 (10) (1977) 1189–1195. [15] K.A. Joudi, D.D. James, Incipient boiling characteristics at atmospheric and subatmospheric pressures, J. Heat Transf. 99 (3) (1977) 398–403. [16] J. Kim, C. Huh, M.H. Kim, On the growth behavior of bubbles during saturated nucleate pool boiling at sub-atmospheric pressure, Int. J. Heat Mass Transf. 50 (17–18) (2007) 3695–3699. [17] F. Giraud, R. Rullière, C. Toublanc, M. Clausse, J. Bonjour, Experimental evidence of a new regime for boiling of water at subatmospheric pressure, Exp. Therm. Fluid

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