Experimental study on bubble sliding for upward subcooled flow boiling in a narrow rectangular channel

International Journal of Heat and Mass Transfer 152 (2020) 119489

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Experimental study on bubble sliding for upward subcooled flow boiling in a narrow rectangular channel Tingting Ren a, Zhiqiang Zhu b, Jiangwu Shi a, Changqi Yan a, Rui Zhang a,∗ a b

Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, Harbin Engineering University, Harbin, 150001, China Wuhan Second Ship Design and Research Institute, Wuhan, Hubei, 430205, China

a r t i c l e

i n f o

Article history: Received 9 November 2019 Revised 19 January 2020 Accepted 8 February 2020

Keywords: Narrow rectangular channel Subcooled flow boiling Bubble growth rate Bubble sliding velocity

a b s t r a c t In order to analyze the flow boiling heat transfer in narrow channels, visualization experiments were carried out and bubble sliding behaviors were observed for upward subcooled flow boiling in a narrow rectangular channel. The effects of wall heat flux, mass flow rate and inlet subcooling on sliding bubble growth rate and velocity were studied. Based on experimental results, a bubble generating at the nucleation site on the heating surface experiences a rapid growth process in a short time at first and then the growth rate slows down, which continues before the bubble slides out of the observation area. The wall heat flux promotes the initial bubble growth. Additionally, the bubble growth rate increases with the decreasing mass flow rate and decreasing inlet subcooling. Compared with the heat flux, the mainstream temperature influences bubble growth more significantly in the narrow channel. Bubble sliding velocity increases rapidly at the initial stage and tends to stabilize when approaching the mainstream value. Since the heat flux and inlet subcooling affect the bubble size and the forces acting on the bubble, bubble sliding velocity increases as heat flux increases and inlet subcooling decreases. The dependence of bubble velocity on liquid velocity indicates that the drag force caused by velocity difference between bubble and mainstream is the main force driving the bubble sliding motion. New models are established and can well predict the sliding bubble diameter and velocity. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction With the application of heat exchange equipment in nuclear energy fields, boiling research in narrow rectangular channels has attracted more attention. The mechanism of boiling heat transfer in narrow channels with large length-width ratio is different from that in conventional channels. From the perspective of microscopic heat transfer, bubble dynamics is the basis of boiling heat transfer research. In order to better understand the subcooled boiling process and heat transfer mechanism, it is significant to study the bubble dynamics in narrow channels. Large efforts have been paid to bubble nucleation, growth, departure and lift-off. However, after leaving the nucleation site, a bubble slides along the heating surface for some distance before moving into the mainstream. Studies have shown that such bubble motion affects the heat transfer characteristic in boiling flow. Cornwell [1] found that the sliding bubbles significantly promoted the heat transfer in bubbly flow. Thorncroft [2] experimentally ∗

Corresponding author. E-mail addresses: [email protected] (T. Ren), [email protected] (R. Zhang). https://doi.org/10.1016/j.ijheatmasstransfer.2020.119489 0017-9310/© 2020 Elsevier Ltd. All rights reserved.

studied the bubble growth and departure in vertical up-flow and down-flow forced convection boiling. The results show that in upward flow, the bubble detaching from the nucleation site almost slides along the heating wall without lifting off, while in downward flows, the bubble lifts off directly from the nucleation site or after sliding. Under the same experimental conditions, the heat transfer coefficient for upward flow is larger due to the bubble sliding behavior. Thorncroft and Klausner [3] further studied the effect of bubble sliding on flow boiling heat transfer. They found that the turbulence enhancement caused by bubble sliding motion mainly contributed to heat transfer in forced convection boiling, accounting for approximately 52% of the total heat transfer capacity. A similar study was done by Houston and Cornwell [4]. They suggested that the effect of increased turbulence due to bubble sliding was greater in narrow channels than in normal-sized channels. Qiu and Dhir [5] obtained the temperature field in the liquid by using the holographic interferometry and confirmed the contribution of bubble sliding motion to heat transfer. Okawa [6] pointed out that bubble sliding along the heating wall after departure from a nucleation site played an important role in heat transfer, changing the cross-sectional void fraction as well. Basu [7], Sateesh [8] and others developed advanced wall boiling models, which

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Nomenclature General symbols qw wall heat flux (kW•m − 2 ) M mass flux (kg•s − 1 ) D bubble diameter (m) h enthalpy (J•kg−1 ) Q heating power (kW) W width of the channel (m) L length (m) T temperature ( °C) k heat conductivity coefficient (W•m − V volume (m3 ) A area (m2 ) qev evaporation heat flux (kW•m − 2 ) qc condensation heat flux (kW•m − 2 ) m fraction b growth constant P pressure (MPa) t time (s) y y-coordinate (m) G mass flow rate (kg•m − 2 •s − 1 ) u velocity (m•s − 1 ) T w wall superheat ( °C) Tsub fluid subcooling ( °C) N dimensionless number

1 •K − 1 )

Greek letters α thermal diffusivity δ Thickness of heating plate (m) ρ density (kg•m − 3 ) Superscript and Subscripts w wall in inlet out outlet l liquid g gas b bubble

considering bubble evaporation and transient heat conduction at the nucleation site, meanwhile, considering microlayer evaporation and transient heat conduction during bubble sliding process. Since bubble growth rate and sliding velocity are important input parameters for establishing reliable heat transfer mechanism and numerical simulation, both experimental and theoretical studies are necessary. For a bubble, the initial growth is controlled by inertia force and then by different heat transfer mechanisms. Fig. 1 shows several possible heat transfer mechanisms during bubble growth near the heating surface, including microlayer evaporation underneath a bubble, evaporation of superheated liquid layer surrounding a bubble and condensation on bubble cap. A number of models have been proposed based on one or several bubble growth heat transfer mechanisms. Among them, an analytical model proposed by Zuber [9] has been widely applied due to the simple form. The model is first proposed for a growing bubble in a uniform superheated liquid and assumes that bubble growth is merely dominated by transient heat conduction. In order to describe the growth and collapse of bubbles under subcooled boiling conditions, Zuber extended the application to the nonuniform temperature field. The model contains an empirical constant called the “growth constant”. Many researchers used their experimental data to obtain different growth constants to apply the Zuber bubble growth model [2,10].

Fig. 1. Heat transfer mechanisms during bubble growth.

Similarly, the model proposed by Plesset and Zwick [11] is based on the thermal diffusion of the superheated layer around the bubble. Cooper and Lloyd [12] believed that microlayer evaporation was the main source for bubble growth. Yun [13] introduced the Ranz and Marshall [14] correlation to estimate the condensation effect on the bubble top in contact with subcooled liquid. The model proposed by Hoang [15] also takes the evaporation of superheated liquid layer and the bubble condensation into account. The condensation heat transfer coefficient is calculated by Levenspiel [16] correlation. Recently, the bubble growth model proposed by Colombo and Fairweather [17] considers the microlayer evaporation, superheated liquid evaporation and condensation simultaneously. Yoo [18] later used a similar approach. Many researchers have studied the bubble departure and liftoff based on the forces analysis. Zeng [10] assumed that bubble sliding velocity was equivalent to the local liquid velocity in predicting bubble liftoff diameter by ignoring the shear lift force. Situ [19] assumed that the local liquid velocity was twice the bubble sliding velocity when establishing the bubble force balance. However, Chu [20] found that Situ’s model predictions were much larger than those of other researchers. They suspected that the result may be caused by an incorrect prediction of bubble sliding velocity. In addition to experimental and theoretical research on bubble characteristics, the development of computer technology and computational fluid dynamics (CFD) has gradually made numerical simulation an important research method. Many researchers studied bubble growth motion in nucleate boiling and bubble transport in the bubble column using the mathematical models. Genske and Stephan [21] simulated the growth of a single bubble in pool boiling and obtained parameters such as the velocity field, temperature field around the bubble, bubble shape and detachment diameter. M. van Sint Annaland [22] carried out numerical simulation on gas bubbles rising motion in quiescent liquids using a three-dimensional volume of fluid (VOF) method. Wei [23] performed a numerical investigation on the behaviors of bubble coalescence, sliding, detachment in subcooled flow boiling considering energy and mass transfer during phase change based on the VOF method. Based on Euler-Euler large eddy simulation (EELES), Liu [24] performed CFD simulations of gas-liquid flow in the bubble column to investigate the scale-adaptive of EELES and the sensitivity of different turbulence models. Lau [25] proposed a coalescence and break-up model on the basis of Euler-Lagrangian framework to predict the bubble size distribution (BSD) in a bubble column. Jain [26] combined the VOF model and the discrete bubble model (DBM) to study the coalescence and breakup in a bubble column. There are many recent studies on bubble sliding characteristics in narrow channels. Xu [27] found a sharp increase in bubble

T. Ren, Z. Zhu and J. Shi et al. / International Journal of Heat and Mass Transfer 152 (2020) 119489

sliding velocity at the initial moment, while the trend of increase decreased gradually and reached to a constant. Yuan [28] pointed out that the sliding bubble growth rate significantly affected the bubble sliding velocity and distance. Li [29] found that the sliding bubble velocity approximately followed normal distribution by statistical methods. In addition, Xu [30] obtained the “growth constant” according to experimental data and used the modified Zuber model to predict sliding bubble diameter in a narrow channel. The above literature review indicates that the bubble sliding characteristics are still unclear especially in narrow channels even though there are a few studies on bubble behaviors. It is valuable to understand the bubble sliding characteristics due to its important role in the flow boiling heat transfer. In this paper, visualization experiments were carried out and bubble sliding behaviors were observed for upward subcooled flow boiling in a narrow rectangular channel. The effects of wall heat flux, mass flow rate and inlet subcooling on bubble sliding parameters were analyzed. Meanwhile, the prediction models for sliding bubble growth rate and velocity were established.

2. Experimental apparatus 2.1. Experimental loop The circulation loop is shown in Fig. 2. The test section is a visualized single-side heated narrow rectangular channel. Deionized water is heated to a desired temperature by adjusting the power of the preheater and flows into the test section. The fluid is heated further to generate boiling and then flows into the condenser through the riser. The condensed water flows back to the preheater through the downcomer, thus a whole circulation is finished. The system pressure is maintained by a pressurizer. The con-

3

denser is connected to a water tank and an air cooling tower for taking away the heat from the major loop. 2.2. Test section Fig. 3 shows the test section details. A flow channel of 700 × 40 × 2mm3 is formed by a stainless steel heating plate and a quartz glass plate, which is sealed by an O-type ring. The plate is heated by a DC power with a capacity of 50 V/20 0 0A to provide a wide range of heat fluxes. The uniformity of material and thickness of the electric heating plate ensure the uniformity of heating power along the plate. In order to measure the external wall temperature, 33 K-type sheathed thermocouples are welded on the back of the heating plate in different axial positions. The distribution of thermocouples and pressure measuring points are shown in Fig. 4. Additionally, two N-type sheathed thermocouples are respectively arranged at the inlet and outlet of the test section to obtain fluid temperature. The bubble behaviors are captured by a high speed camera with a Sigma 105 mm F 2.8 macro lens, which is placed on a twodimensional guide rail that can move parallel or normal to the visualization window. Clear bubble images are obtained by setting camera frame rate and resolution to 4300 frame per second and 1024 × 1024 pixels respectively. The corresponding time interval between two adjacent frames is approximately 0.23 ms and the actual viewing area is 10.28 × 10.28 mm2 . Four N-type thermocouples are arranged at the inlets and outlets of the preheater and condenser. An electromagnetic flowmeter is equipped to measure the circulation flow rate. In order to avoid the influence on bubble motion, the solubility of noncondensable gas is reduced before formal experiment by heating water up to saturated boiling so that most non-condensable gas is discharged from the exhaust valve. The working conditions are listed in Table 1.

Fig. 2. Schematic diagram of the experimental loop.

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Fig. 3. Configuration of the visualization test section.

Fig. 4. Schematic diagram of the thermocouple position.

2.3. Data processing The total electric heating power Q is directly measured in the experiment. In order to reduce the heat loss, the whole test section is covered with insulating materials except for the visualization window. The heat loss caused by this part can be ignored due to the small thermal conductivity of quartz glass. The thermal efficiency η for heating section is calculated by the ratio of the enthalpy rise to the electric power. Since the void fraction is tiny within the range of experiments, the thermal efficiency for

Table 1 Working conditions. Parameters

Range

Operating pressure (MPa) Heat flux in test section (kW•m − 2 ) Inlet fluid subcooling ( °C) Mass flow rate (kg•m − 2 •s − 1 )

0.2–0.3 109–387 30–50 311–1654

two-phase flow is replaced by the corresponding single-phase flow thermal efficiency and is always larger than 95%. Therefore, wall

T. Ren, Z. Zhu and J. Shi et al. / International Journal of Heat and Mass Transfer 152 (2020) 119489

heat flux qw and thermal efficiency are expressed as:

Q W L0 M (hout − hin ) = Q

troid in two sequential images, the velocity is calculated by:

y j+1 − y j t

qw =

(1)

ub =

η

(2)

2.4. Uncertainty analysis

where W is the width of the heating plate, L0 is the heating length, M is the mass flux, hout and hin are the inlet and outlet enthalpies, respectively. The outer wall temperature Tw,out is directly measured by the thermocouples in the experiment The inner wall temperature Tw,in is calculated by the one-dimensional heat conduction equation.

Tw,in =Tw,out −

qw 2 δ 2k

(3)

where k is the thermal conductivity of the stainless steel heating plate and δ is the thickness of the heating plate. The inlet and outlet water temperature of the experimental section is measured by N-type thermocouples. The water temperature is considered varying approximately linearly along the axial position in view of the tiny void fraction. Therefore, the fluid temperature Tf,x at different position is calculated by

Tl,x = Tl,in +

 Lx  Tl,out − Tl,in L

(4)

where Tl,x is the liquid temperature at a certain location, Lx is the distance between this position and inlet, L is the test section length, Tl,in and Tl,out are the inlet and outlet liquid temperatures, respectively. Considering the effects of light and flow heat transfer, bubbles may appear in a variety of shapes. Therefore, Feret diameter is used to describe the bubble diameter. For a single bubble, its equivalent diameter is defined as the average Feret diameter in all directions through the centroid (interval 2°, 90 groups in total).

D=

90 

DiF /90

5

(5)

i=1

As shown in Fig. 5(b), bubbles in the flow channel are mainly affected by the mainstream, which is manifested as the movement along the flow direction. The lateral motion is extremely small. Therefore, bubble sliding velocity in this paper is the velocity along the flow direction. According to the y-coordinate of bubble cen-

(6)

The K-type and N-type sheathed thermocouples respectively have accuracies of ±1 °C and ±0.5 °C. The heating power has an accuracy of 0.55%. The accuracies for bubble diameter and position are ±2 pixel. Based on the Kline and McClintock [31] uncertainties analysis method, under the experimental conditions in this paper, the relative uncertainty range for heat flux and inner wall temperature are respectively 2.1–5.2% and ±1°C. The relative uncertainty for bubble diameter and bubble sliding velocity are respectively ±0.02 mm and ±0.08 m/s.



UF = ±

n 



i=1

∂F u ∂ pi pi

2 (7)

where F is a function of variables p1 , p2 , … pn ; UF is the uncertainty for F and upi is the uncertainty for the variable pi . The relative uncertainty for the friction factor λ can be calculated as



Uλ

λ

=±

n   up i=1

2 i

pi

(8)

3. Results and discussions 3.1. Bubble sliding characteristics Typical bubble sequence images observed in the experiments are shown in Fig. 6, including the bubble which generates at the nucleation site in the visualization window and the bubble which slides from the upstream. The growth curve and relative positions of bubbles generated by nucleation are shown in Fig. 7(a). It can be seen that the bubble departs from the nucleation site after undergoing a very short growth process (less than 0.4 ms) at the nucleation point. As sliding along the flow direction on the heating surface, a large proportion of bubble growth occurs during the sliding process. A bubble experiences two growth stages: first, in the inertial growth stage, the bubble grows fast in a short time (approximately 3 ms); afterwards, bubble growth rate slows down in

Fig. 5. Schematic diagrams for performing measurements.

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Fig. 6. Bubble sequence images.

the thermal diffusion control stage and bubble diameter almost increases linearly, which continues before the bubble slides out of the observation window. For this type of bubble, because of its small size, the bubble is still in the superheated liquid layer near the heating surface, continuously absorbing energy and growing.

However, for the bubbles sliding from the upstream, the size is obviously larger due to the longer growth time. This type of bubble is affected by fluid perturbation and condensation at the top of the bubble, thus there are many variation tendencies in bubbles diameter, including increasing all the time, decreasing first and then

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Fig. 7. Bubble diameter, relative position and velocity.

increasing or remaining almost unchanged. Nevertheless, the overall trend is increasing. The axial height increases with the increase of bubble sliding distance. The wall temperature and mainstream temperature increase accordingly, which is beneficial for bubble growth. Bubble collapse is rarely observed in the limited visualization area. As depicted in Fig. 7, the bubble sliding velocity increases rapidly at the initial growth process. The bubble sliding velocity tends to stabilize and fluctuate within a small range after approaching a specific value (approximately equal to mainstream velocity) over time. According to the study of Xu [32], the forces controlling the bubble sliding motion are mainly the buoyancy, quasisteady drag force and added-mass force. In the initial stage, bubble sliding velocity is less than the local mainstream velocity. The velocity difference between the bubble and the liquid is relatively larger. Therefore, the quasi-steady drag force and added-mass force driving the bubble sliding motion are large as well, which promotes a quick increase in bubble sliding velocity. As the bubble

sliding velocity increases, the drag force and added-mass force decrease. When it exceeds the liquid velocity, drag force becomes the resistance. Finally, as the resultant force on the bubble approaches zero, the bubble slides uniformly.

3.2. Effects of working conditions on bubble sliding parameters 3.2.1. Sliding bubble growth rate Fig. 8 shows the experimental results to investigate the effects of wall heat flux, mass flow rate and inlet subcooling on nucleated bubble growth behavior. Partial bubble diameter data under corresponding conditions are summarized in Tables 2 and 3. It can be seen from Fig. 8(a) that, for given mass flow rate and inlet subcooling, the bubble growth rate and diameter increase significantly in the early stage with the increasing wall heat flux, while in the later stage the bubble growth rate is nearly unchanged under different wall heat flux conditions. This is different from Thorncroft

Table 2 Experimental conditions. Exp.

Operating pressure P/(MPa)

Heat flux qw /(kW•m

A B C D E

0.3 0.3 0.3 0.3 0.3

212.0 233.4 235.4 167.5 166.1

− 2

)

Mass flow rate G/(kg•m 696.9 696.5 957.4 673.8 673.9

− 2•

s

− 1

)

Inlet subcooling Tsub,in /( °C) 30.2 30.4 30.3 31.3 23.7

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Fig. 8. (a) Effect of wall heat flux on bubble growth characteristic. (b) Effect of mass flow rate on bubble growth characteristic. (c) Effect of inlet subcooling on bubble growth characteristic.

T. Ren, Z. Zhu and J. Shi et al. / International Journal of Heat and Mass Transfer 152 (2020) 119489 Table 3 Bubble diameter data.

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mainstream temperature has a greater influence on bubble growth in the narrow channel.

Time (ms)

A

B

C

D

E

0.23 0.47 0.70 0.93 1.16 1.40 1.63 1.86 2.09 2.33 2.56 2.79 3.02 3.26 3.49 3.72 3.95 4.19 4.42 4.65 4.88 5.12 5.35 5.58 5.81 6.05 6.28 6.51 6.74 6.98 7.21 7.44 7.67 7.91 8.14 8.37 8.60 8.84 9.07 9.30 9.53

0.07 0.07 0.07 0.07 0.09 0.11 0.11 0.11 0.13 0.14 0.15 0.17 0.17 0.18 0.19 0.20 0.20 0.20 0.20 0.19 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.19 0.19 0.20 0.20 0.21 0.21 0.22 0.22 0.21 0.22 0.22 0.21 0.22 0.23

0.10 0.12 0.14 0.16 0.18 0.18 0.20 0.22 0.24 0.25 0.26 0.26 0.27 0.28 0.29 0.31 0.32 0.32 0.33 0.34 0.35 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.37 0.36 0.36 0.37 0.37 0.36 0.37 0.38 0.38 0.38 0.38 0.39 0.38

0.04 0.04 0.05 0.05 0.06 0.06 0.06 0.06 0.07 0.07 0.07 0.07 0.07 0.07 0.08 0.09 0.09 0.09 0.10 0.09 0.08 0.07 0.07 0.07 0.07 0.07 0.09 0.08 0.09 0.09 0.10 0.09 0.09 0.10 0.10 0.11 0.12 0.12 0.11 0.12

0.04 0.05 0.06 0.06 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.09 0.08 0.08 0.09 0.08 0.09

0.08 0.09 0.10 0.11 0.12 0.14 0.14 0.15 0.16 0.16 0.17 0.17 0.18 0.18 0.19 0.19 0.20 0.20 0.21 0.21 0.21 0.22 0.23 0.23 0.24 0.24 0.26 0.25 0.26 0.26 0.26 0.26 0.27 0.27 0.26 0.28 0.28 0.27 0.27 0.27 0.27

[2] and Kandlikar’s [33] results. Thorncroft demonstrated that bubble growth increased with increasing wall superheat in vertical upflow and downflow forced convection boiling and Kandlikar reported a similar result for subcooled horizontal flow boiling. The reason is that when the bubble diameter increases to a certain value, the top of the bubble may contact the subcooled fluid and condensation occurs. In Fig. 8(a), the mainstream water temperatures are not completely the same in two working conditions with the same heat flux. Under the combined action of evaporation and condensation, wall heat flux merely influences the bubble growth rate. In addition, other nucleation sites around the target bubbles are activated when the wall heat flux increases. The nucleation and growth of surrounding bubbles disturb the thermal boundary layer, which inhibits the target bubbles growth. The influence of mass flow rate on bubble growth is more obvious at lower heat fluxes than at higher heat fluxes, as shown in Fig. 8(b). However, bubble growth rate decreases with the increasing mass flow rate generally. Meanwhile, experimental results indicate that the inertia control stage is longer when mass flow rate is relatively larger. Bubble growth is dominated by energy when mass flow rate is small. Fig. 8(c) shows the effect of inlet subcooling on bubble growth. Inlet subcooling mainly affects the mainstream temperature and thus affects the bubble condensation. When the inlet subcooling decreases by 10 °C, the growth rate of bubble increases approximately 2.5 times. Compared with the heat flux, the

3.2.2. Bubble sliding velocity Based on the analysis in Section 3.1, the bubble basically slides at a uniform speed after a short period of accelerated motion. According to Li [29], the bubble average velocity at a certain moment approximately follows a normal distribution. Therefore, the average bubble sliding velocity over a period under corresponding operation condition is studied in this paper. Fig. 9(a) and (b) show the effect of heat flux and inlet subcooling on average bubble sliding velocity at mainstream velocities of 0.32 m/s and 0.66 m/s. For a given liquid velocity, bubble sliding velocity increases with the increasing heat flux and decreases with the increasing inlet subcooling. According to Section 3.2.1, bubble diameter is lager under high heat flux than low heat flux conditions. Bubble growth rate may increase by reducing the inlet subcooling and thus bubble size increases. Driven by buoyancy force, bubble sliding velocity increases. Yoo [34] reached the same conclusion in the study of subcooled flow boiling. In the study of Li [29], the influence of heat flux on sliding velocity is various. The velocity increases with the increasing heat flux at first, whereas decreases with the increasing heat flux when heat flux increases to a certain value. They considered that the tension between the top of bubble and the glass hinders the bubble movement since the bubble diameter is larger than the width of narrow channel when heat flux is high. However, for experiments in this paper, the maximum bubble diameter is 0.4 mm due to the lager mass flow rate, much smaller than the width of the narrow channel. This kind of phenomena does not exist. Fig. 9(c) shows the effect of fluid velocity on bubble sliding velocity. As liquid velocity increases, bubble sliding velocity increases significantly. The relative velocity of the mainstream and bubble increases when liquid velocity increases, thus increasing the drag force. On the contrary, bubble size decreases with the increasing fluid velocity, thus reducing the buoyancy force. Under the combined action of increasing drag force and decreasing buoyancy force, bubble sliding velocity increases, which indicates that the drag force caused by the velocity difference between bubble and mainstream is the main force driving the bubble sliding motion. Moreover, the mainstream velocity is shown in the figure. It is clear that bubble sliding velocity is higher than liquid velocity only at the liquid velocity of 0.32 m/s, while bubble sliding velocity is lower when liquid velocity exceeds 0.32 m/s. 3.3. Development of prediction models As shown in Fig. 1, several heat transfer mechanisms may exist in bubble growth process, such as the microlayer evaporation underneath the bubble, the superheated liquid layer evaporation surrounding the bubble and the condensation at the top of the bubble. However, Demiray [35] pointed out that only 12.5% of the energy required for bubble growth came from the microlayer evaporation. Kim [36] believed that most of the energy needed for bubble growth came from superheated layer and microlayer evaporation accounted for less than 25% of the entire bubble heat transfer. Moreover, Yoo [18] indicated that the microlayer is exhausted before sliding away from the nucleation site. Therefore, the latter two transfer mechanisms are considered in this paper. The energy balance equation for a single sliding bubble is expressed as

ρg hlg

dVb = (1 − m )qev,s Ab − mqc Ab dt

(9)

where Vb is the volume of the bubble (according to Xu [27], the bubble is approximately spherical during sliding, Vb = π D3 /6), Ab is the surface area of the bubble (Ab = π D2 ), m is the fraction of the bubble surface in contact with subcooled liquid layer. qev,s and

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Fig. 9. Effects of thermal parameters on bubble sliding velocity.

qc are superheated liquid layer evaporation and condensation heat flux, respectively. Zuber model is used to calculate the superheated liquid layer evaporation.

bk Tw qev,sli = √l π αt

(10)

where kl is the liquid thermal conductivity, α is the thermal diffusivity. Growth constant b is an empirical constant. Different growth constant values are obtained by many researchers using their own experimental data to apply Zuber bubble growth model. In this study, b is fixed as one without considering the flow boiling conditions. Condensation heat flux is expressed as

qc =hc Tsub

(11)

where hc is the condensation heat transfer coefficient. Based on the approach taken by Ünal [37] and Levenspiel [16], hc is expressed as

C ϕ hlg D hc = 2 ( 1 / ρg − 1 / ρl )

(12)

where the empirical constant C and ϕ represent the effect of system pressure P and liquid bulk velocity ul , respectively.



C=





65 − 5.60 × 10−5 P − 105 0.1 ≤ P ≤ 1 0.25 × 1010 P −1.418 1

 ϕ = max 1, (ul /0.61)0.47

(13) (14)

Substituting Eqs. (10) - (14) into Eq. (9) and simplifying Eq. (1) reduces to

dD 2(1 − m )kl bTw −0.5 mC ϕ Tsub = t − D=At −0.5 − BD √ dt 1 − ρg / ρl ρg hlg π α

(15)

where

A=

2(1 − m )kl bTw mC ϕ Tsub , B= √ 1 − ρg / ρl ρg hlg π α

(16)

Eq. (15) is similar to the model of Ünal and has an analytic solution, which gives bubble diameter as a function of time.

D(t ) =

2At 0.5 (1 + Bt /3 ) 1 + Bt

(17)

In order to predict the bubble growth diameter, the value of m needs to be determined. Some researchers suggested m = 0.5.

T. Ren, Z. Zhu and J. Shi et al. / International Journal of Heat and Mass Transfer 152 (2020) 119489

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Fig. 10. Sliding bubble diameter prediction compared with experimental data.

However, noted that m is simplified as a constant for the bubble growing at a nucleation site. Actually, for a sliding bubble in subcooled boiling flow, the fraction of the bubble surface in contact with subcooled liquid layer is related to the bubble size and the thickness of thermal boundary layer, which has a complex relation with the heat flux, bulk subcooling and mass flow rate. Few

mechanism model is found to provide such information at present. Therefore, referring to Hoang and Ünal’s model, m is an adjusting parameter in the present model. Fig. 10 shows the comparison between the predicted results calculated by Eq. (17) and the experimental data. Most experimental data are basically within 20%, indicating the model has a good

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T. Ren, Z. Zhu and J. Shi et al. / International Journal of Heat and Mass Transfer 152 (2020) 119489

The wall heat flux promotes the initial bubble growth. Additionally, the bubble growth rate increases with the decreasing mass flow rate and the decreasing inlet subcooling. Compared with the heat flux, the mainstream temperature significantly influences the bubble growth in the narrow channel. The heat flux and inlet subcooling affect the bubble size and thus affect the forces acting on the bubble, therefore, the bubble sliding velocity increases with the increasing wall heat flux and the decreasing inlet subcooling. The dependence on liquid velocity indicates that the drag force caused by the velocity difference between bubbles and the mainstream is the main force driving the bubble sliding motion. A semi-theoretical model is established to predict the bubble diameter by considering the combined effect of superheated liquid layer evaporation and condensation on bubble cap. Meanwhile, the liquid velocity and dimensionless number N are adopted to predict the bubble sliding velocity. The predicted values by the model agree well with the experimental data. Fig. 11. Bubble sliding velocity prediction compared with experimental data.

prediction on experimental data. Fig. 10(a), (b) and (c) show the comparison of experimental data and model prediction under different heat flux, mass flow rate and inlet subcooling, respectively. It is clear that the present model can well reflect the dependence of bubble growth on thermal conditions. The value range of m is 0.3 – 0.7. Although the predicted value is slightly different from experimental result in the initial stage of bubble growth, it agrees well with the experimental results in the later stage. It is particularly evident in Fig. 10(a). The alternation of the two different mechanism (early inertial control and later energy diffusion) results in a small peak under low mass flow rate and high heat flux conditions since the growth inertia is greater under those conditions. Yoo [34] obtained the same experimental results. Hence, the accuracy of the model needs to be improved through considering the mechanism of growth inertia in the future work. According to Section 3.2.2, a bubble slides along the heating surface under the action of various forces. The buoyancy force is proportional to bubble volume. The quasi-steady drag force and added-mass force are affected by the wall superheat and mainstream subcooling. Therefore, the liquid velocity and dimensionless number N are adopted to predict the bubble sliding velocity, which is expressed as Eqs. (18) – (19). Comparison between model predictions and experimental data is shown in Fig. 11. Almost all the experimental data are within ± 20% and the mean relative error is 9%.

Tw Tsub ub =(0.78ul + 0.11)N 0.09 N=

(18) (19)

4. Conclusions Visualization experiments were carried out and bubble sliding behaviors were observed for upward subcooled flow boiling in a narrow rectangular channel. The effects of wall heat flux, mass flow rate and inlet subcooling on sliding bubble growth rate and velocity were analyzed. The corresponding prediction models were established as well. A bubble generated by nucleation on the heating surface first experiences rapid growth for a short time, then the growth rate slows down and the bubble slides along the heating wall for a long time. Similarly, at the stage of rapid growth, the bubble performs an accelerated sliding motion. Then the bubble sliding velocity tends to stabilize and fluctuate within a small range when approaching the mainstream value.

Declaration of Competing Interest None. Acknowledgments The authors greatly appreciate support from the Natural Science Foundation of China (Grant No. 11675045), support from the State Key Laboratory of Nuclear Power Safety Monitoring Technology and Equipment (K-A2019.414) and support from the Natural Science Foundation of Heilongjiang Province (LH2019A009). Additionally, the authors are thankful for support from the Fundamental Science on Nuclear Safety and Simulation Technology Laboratory, Harbin Engineering University, China. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijheatmasstransfer. 2020.119489. References [1] K. Cornwell, The influence of bubbly flow on boiling from a tube in a bundle[J], Int. J. Heat. Mass Transf. 33 (12) (1990) 2579–2584. [2] G.E. Thorncroft, J.F. Klausner, R. Mei, An experimental investigation of bubble growth and detachment in vertical upflow and downflow boiling[J], Int. J. Heat. Mass. Transf. 41 (23) (1998) 3857–3871. [3] G.E. Thorncroft, J.F. Klausner, The influence of vapor bubble sliding on forced convection boiling heat transfer[J], J. Heat Transfer. 121 (1) (1999) 73–79. [4] S.D. Houston, K. Cornwell, Heat transfer to sliding bubbles on a tube under evaporating and non-evaporating conditions[J], Int. J. Heat Mass Transf. 39 (1) (1996) 211–214. [5] D. Qiu, V.K. Dhir, Experimental study of flow pattern and heat transfer associated with a bubble sliding on downward facing inclined surfaces[J], Exp. Therm. Fluid Sci. 26 (6–7) (2002) 605–616. [6] T. Okawa, T. Ishida, I. Kataoka, et al., An experimental study on bubble rise path after the departure from a nucleation site in vertical upflow boiling[J], Exp. Therm. Fluid Sci. 29 (3) (2005) 287–294. [7] N. Basu, G.R. Warrier, V.K. Dhir, Wall heat flux partitioning during subcooled flow boiling: part 1—model development[J], J. Heat Transfer. 127 (2) (2005) 131–140. [8] G. Sateesh, S.K. Das, A.R. Balakrishnan, Analysis of pool boiling heat transfer: effect of bubbles sliding on the heating surface[J], Int. J. Heat. Mass Transf. 48 (8) (2005) 1543–1553. [9] N. Zuber, The dynamics of vapor bubbles in nonuniform temperature fields[J], Int. J. Heat. Mass Transf. 2 (1–2) (1961) 83–98. [10] L.Z. Zeng, J.F. Klausner, D.M. Bernhard, et al., A unified model for the prediction of bubble detachment diameters in boiling systems—II, Flow boiling[J]. International journal of heat and mass transfer 36 (9) (1993) 2271–2279. [11] M.S. Plesset, S.A. Zwick, The growth of vapor bubbles in superheated liquids[J], J. Appl. Phys. 25 (4) (1954) 493–500. [12] M.G. Cooper, A. Lloyd, The microlayer in nucleate pool boiling[J], Int. J. Heat Mass Transf. 12 (8) (1969) 895–913.

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