Wall nucleation modeling in subcooled boiling flow

International Journal of Heat and Mass Transfer 86 (2015) 183–196

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Wall nucleation modeling in subcooled boiling flow Caleb S. Brooks a,⇑, Takashi Hibiki b a b

Department of Nuclear, Plasma, and Radiological Engineering, University of Illinois, Urbana, IL 61801, USA School of Nuclear Engineering, Purdue University, West Lafayette, IN 47907-1290, USA

a r t i c l e

i n f o

Article history: Received 30 December 2014 Received in revised form 2 March 2015 Accepted 2 March 2015

Keywords: Departure diameter Departure frequency Subcooled boiling Wall nucleation Interfacial area concentration Refrigerants Mini-channel

a b s t r a c t The bubble departure characteristics are studied in flow boiling considering the available experimental data in the literature. The current bubble departure diameter and departure frequency models are reviewed and compared with the collected database, revealing the current modeling shortcomings in forced convective flows. Based on an energy balance approach at the heated surface, the important dimensionless groups are identified and correlated for the bubble departure diameter and departure frequency. These dimensionless groups are the Jakob number, Boiling number, density ratio, and Prandtl number. The bubble departure diameter data also suggests some effect of the flow geometry in minichannels. The newly proposed semi-empirical models for the bubble departure diameter and departure frequency are shown based on the available database to be accurate to within ±22% and ±35% respectively. This work also identifies important future experimental considerations for the bubble departure characteristics including data at elevated pressures, larger Prandtl numbers, and around the boundary between mini and conventional channels. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Fundamentally, wall nucleation is the generation of the vapor phase from a liquid state by super-heating the liquid in small cavities on the heater surface. Microscopic roughness on a heated surface traps gas forming nucleation sites, locations where the required energy to generate a vapor bubble is possible. Therefore the initiation of the bubble growth is dependent on the local heat transfer process occurring in the very small cavity. Once the gas phase is generated, the bubble grows very rapidly to a size greater or equal to the critical bubble size related to the pressure jump due to surface tension at the vapor–liquid interface. From this state, the bubble may continue to grow due to the superheated liquid layer near the heated surface, depart from the nucleation site, or collapse as it condenses back into the bulk fluid. Wall nucleation characteristics, including bubble growth and departure, is a classic heat transfer problem still requiring study for a wide range of advanced applications. Understanding and predicting the boiling flow dynamics at the heated surface is of great importance in boiling heat transfer systems. Nucleate boiling provides an efficient method to remove heat from surfaces and is common in electronic cooling applications as well as large scale nuclear reactors [1–5]. Furthermore, with the ⇑ Corresponding author. Tel.: +1 217 265 0519; fax: +1 217 333 2295. E-mail address: [email protected] (C.S. Brooks). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.03.005 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

increase in recent computational power and interest in high resolution two-phase flow fluid dynamics simulation, bubbles generated on the heater surface in subcooled boiling flow is of particular importance as nucleating bubbles represent the gas phase boundary condition. Without a clear understanding of gas generated from the heated boundary in computational fluid dynamics simulation, the gas phase distribution within the flow cross-section cannot be accurately predicted and stands as the limiting factor in current subcooled boiling simulation capability [6]. This limitation based on the nucleation characteristics is due to the dependence of the interfacial area concentration in the twofluid model (i.e. Eulerian–Eulerian modeling). The two-fluid model treats each phase separately with its own set of balance equations for mass, momentum and energy. Therefore, the interaction between phases is accounted for in the six balance equations through interfacial transfer terms (e.g. interfacial drag force, interfacial shear, interfacial heat transfer, interfacial mass transfer, etc.). These interfacial transfer terms are modeled based on the interfacial area concentration and the appropriate driving forces, defining the degree of coupling between the phases [7]. Wall nucleation produces a lot of very small bubbles and therefore is the most important source of bubble number density and interfacial area concentration in subcooled boiling flows [1]. This strong dependence of wall nucleation on the total interfacial area concentration is reflected by the importance of the wall nucleation source term in the interfacial area transport equation which dynamically predicts

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Nomenclature a b C c d m n constants [–] Ac flow cross-sectional area [m2] interfacial area generation rate per active nucleation A_ i;WN site [m2/s] Bo Boiling number [–] cp specific heat capacity [J/(kg K)] Dd departure diameter [m] Dh hydraulic diameter [m] fd departure frequency [1/s] G mass flux [kg/(m2 s)] g gravitational acceleration [m/s2] h heat transfer coefficient [W/(m2 K)] hfg latent heat of vaporization [J/kg] Ja Jacob number [–] jf,in average liquid velocity at the inlet [m/s] k thermal conductivity [W/(m K)] Lc characteristic length scale [m] Lo Laplace length [m] Nn active nucleation site density [1/m2] NqNB non-dimensional nucleate boiling heat flux [–] NT non-dimensional temperature [–] Nu Nusselt number [–] Pr Prandtl number [–] P pressure [Pa] q00NB nucleate boiling partition of the wall heat flux [W/m2] 00 qw wall heat flux [W/m2] Re Reynolds number [–] T temperature [K] tG growth time [s] tW wait time [s] td delay time [s] Vc characteristic velocity scale [m/s]

the evolution of the interfacial area concentration throughout any two-phase flow structure [7]. In the interfacial area transport equation [7], the wall nucleation source term, /WN , and wall nucleation volume source term, gWN , are given as a function of three important nucleation terms: active nucleation site density, Nn, bubble departure diameter, Dd, and bubble departure frequency, fd,

/WN ¼

gWN ¼

pNn f d D2d nh Ac

pNn f d D3d nh 6Ac

;

ð1Þ

;

ð2Þ

where the remaining terms, Ac and nh , are the flow cross sectional area and heated perimeter respectively. The active nucleation site density represents the number of locations that are actively nucleating bubbles and has been given significant attention by many researchers [8–14], including Hibiki and Ishii [15] who developed a comprehensive model that predicts a large database collected from literature. However, there currently exists a major shortcoming in the prediction of bubble departure diameter, which represents the average bubble size at the departure from the active nucleation sites and departure frequency, which gives the average rate of bubbles departing from the active nucleation sites [1,4,16]. In this work, the classical heat transfer phenomenon of wall nucleation is studied in forced convective flows for application in the interfacial area transport equation. After reviewing the available modeling and experimental data for bubble departure diameter and departure frequency, a new modeling approach is proposed and supported with available data. The experimental data included

V_ WN y

volume generation rate per active nucleation site [m3/s] distance from the wall [m]

Greek symbols a thermal diffusivity [m2/s] Dq density difference [kg/m3] DT temperature difference [K] gWN volume source term from wall nucleation [1/s] nh heated perimeter [m] q density [kg/m3] ⁄ q density ratio [–] r surface tension [N/m] l dynamic viscosity [kg/(m s)] sw wall shear stress [kg/(m s2)] m kinematic viscosity [m2/s] /WN interfacial area concentration source term from wall nucleation [1/(m s)] Subscripts Calc calculated value d departure Exp experimental value f liquid g gas sat saturation sub subcooling w wall Superscripts ⁄ non-dimensional parameter + non-dimensionalized by the wall length scale

in this work is limited to forced convective systems and does not consider the special case of pool boiling. The impact of convective effects in flow boiling are significant as the liquid flow adds additional forces on the nucleating bubble [3]. Furthermore, a nucleating bubble in a flow boiling system is required to grow through a much larger temperature gradient in the near-wall region [12] which can affect the bubble size, growth rate, and wait time. Therefore, pool boiling data and empirical models based on pool boiling data are not considered and the analysis is focused on the modeling departure characteristics in forced convective flows. A good review of modeling and bubble departure diameter data in pool boiling systems is given by Jensen and Memmel [17]. 2. Existing work 2.1. Modeling of wall nucleation characteristics 2.1.1. Bubble departure diameter There have been three major approaches to modeling the characteristic nucleation bubble diameter: energy balance approach, force balance approach, and correlation approach. The energy balance approach was performed by Unal [18] by considering energy input through the liquid film between the bubble and the heater wall, output through convection to the bulk fluid, and latent heat. The bubble departure is assumed to occur at the maximum bubble size. The model was compared with limited data; however, showed reasonable agreement over a large pressure range. A force balance approach on the departing bubbles was first suggested by Fritz [19] for application in pool boiling. Fritz [19] model considered surface versus buoyancy forces and was later

C.S. Brooks, T. Hibiki / International Journal of Heat and Mass Transfer 86 (2015) 183–196

modified by Kocamustafaogullari and Ishii [12] to account for pressure dependence using the density ratio. In forced convective flows, Chang [20] includes a dynamic force similar to a drag force, acting in both the normal and tangential directions. At bubble departure, the bubble is assumed to be under hydrodynamic equilibrium. Under this condition the governing equation is given by the magnitude of the external forces (buoyance force and drag force) equal to the magnitude of the surface force. However this model is proposed without comparison with experimental data. A simple force balance model is also proposed by Levy [21] which considers surface tension, friction, and buoyancy forces where the buoyancy force is neglected for high flow rate applications. Levy [21] adopts the same buoyancy and surface tension forces proposed by Chang [20] however considers a frictional force based on the wall shear stress. For upward flow boiling on a vertical surface the frictional and buoyancy forces act to remove the bubble from the surface while the surface force resists the bubble motion. Similar to the other early modeling attempts, experimental data was unavailable for benchmarking the model and determining the constants. Conflicting with this early model of Levy [21], the buoyancy is cited by Klausner et al. [3] as playing a significant role regardless of flow rate and Harada et al. [22] later neglects the frictional term by Levy [21] rather than the buoyancy term to model their experimental data. Furthermore, the surface tension force is said to be insufficient to hold bubbles at the surface and therefore Klausner et al. [3] introduces an unsteady drag force from asymmetric bubble growth force which helps hold the bubble on the site and also acts opposite to the fluid motion. A sensitivity study of the forces found that the dominant forces are quasi-steady drag and unsteady drag forces in the flow direction and surface tension, unsteady drag, and shear lift force in the transverse direction. The model showed good agreement with the dataset taken at relatively low heat flux and mass flux of refrigerant R113. The study of Koumoutsos et al. [23] suggests that the drag force is too small compared with the surface tension force and buoyancy force to account for the velocity dependence on departure size shown by visual observation. This was also observed by Kandlikar and Stumn [24] prompting the investigation of bubble contact angle effect. The forces proposed by Alhayes and Winterton [25], which first considers the effect of different upstream and down-stream contact angles, were adopted by Kandlikar and Stumn [24] in a control volume method. Bubble departure is said to be initiated due to a sweep removal at the upstream edge of the bubble from insufficient surface tension on the leading edge to support the pressure force and surface tension of the downstream edge. This perspective leads the model by Kandlikar and Stumn [24] to have a strong dependence on upsteam and downstream contact angles. More recently, Situ et al. [26] proposes a departure diameter model which utilizes the modeling foundation set by Klausner et al. [3]. The growth force is modified by using the bubble growth rate of Zuber [27] and employs pressure, gravity, and the quasisteady drag force. Here, the departure condition assumed by Situ et al. [26] is that the surface force is zero. The model by Situ et al. [26] is not validated against experimental data however a similar approach was used by Situ et al. [28] to model the bubble lift-off diameter. At lift-off, Situ et al. [28] considers the shear lift force suggested by Klausner et al. [3] and growth force modified by the Zuber [27] growth rate. Prodanovic et al. [29] correlated experimental data for the maximum bubble diameter during growth on a nucleation site and the bubble ejection diameter. These diameters were non-dimensionalized with the surface tension, liquid density and liquid thermal diffusivity, then correlated against the density ratio, a dimensionless temperature, Boiling number, and Jakob number.

185

With these non-dimensional parameters and five adjustable constants Prodanovic et al. [29] showed good agreement with experimental data. Recently, Brooks [30] studied wall nucleation of water in vertical upward flow systems. The model was developed and benchmarked for boiling of water and given as a function of density ratio, Jakob number, and Boiling number. Brooks [30] also identified and discussed the large difference in bubble departure diameter between flow in conventional and mini-channels. 2.1.2. Bubble departure frequency The departure frequency can be described by the bubble wait time, tW, and bubble growth time, tG, simply by

fd ¼

1 ; tW þ tG

ð3Þ

where the wait time represents the time from bubble departure until the next bubble is first initiated in the nucleation site cavity, and the growth time is the time from the initiation until the bubble departs the nucleation site. In forced convective boiling, the nucleation phenomenon is exposed to a much higher temperature gradient of the liquid phase. This may have a large effect on the wait time and growth time due to large temperature fluctuations carried by the turbulence in the flow field. Most departure frequency modeling considers an energy balance approach; however, a force balance approach and raw correlation approach are also attempted. An early correlation for bubble departure frequency in pool boiling was formulated by Cole [31]. In this study, the author considers the conditions approaching critical heat flux on a horizontal heated surface and proposed a force balance between drag resisting the bubble rise and buoyancy driving the departure where the drag coefficient is assumed to be unity and the characteristic velocity is given by the product of the bubble diameter and departure frequency. In vertical flow boiling this modeling is not physically consistent as the drag at the bubble interface and the buoyancy both act to remove the bubble from the nucleation cite. Ivey [32] correlated the bubble departure frequency in pool boiling as a function of bubble departure diameter and gravity. Three regimes are identified and labeled as hydrodynamic which is controlled by drag and buoyancy forces, transition region where the surface tension, drag and buoyancy are comparable, and the thermodynamic region which is controlled by the bubble growth conditions. These correlations were confirmed by experimental data of pool boiling available in literature. Expressions for wait time and growth time are analytically derived by Podowski et al. [33] considering the heat conduction through the heating surface and from the surface to the bulk flow. In the formulation for the bubble wait time, Podowski et al. [33] considers the wall surface temperature as a time dependent, fluctuating parameter which is solved for by a one-dimensional heat conduction equation through the wall. The growth time is modeled by assuming that the local time dependent surface heat flux is used entirely for the bubble. The instantaneous wall temperature at the nucleation site cannot be measured and is typically taken to be the average wall temperature. Thorncroft et al. [2] studied nucleation in upward and downward flows and found that the bubble diameter growth could be correlated well with a simple power law curve and therefore the bubble growth time can be found from the departure diameter. Basu et al. [34] also correlated the bubble growth time, as well as wait time, and found that the wait time was a strong function of wall super heat, while the growth time could be expressed by the Jakob number considering both the subcooling and superheat. Interestingly, the wait time is proposed as independent of bulk liquid subcooling.

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In the recent study of Situ et al. [26], the bubble departure frequency is correlated with the nucleate boiling heat flux proposed by Chen [35], 

f d ¼ 10:7N0:634 q;NB ;

ð4Þ  f d,

where the non-dimensional bubble departure frequency, and non-dimensional nucleate boiling heat flux, Nq,NB, are given by 

fd 

f d D2d

af

ð5Þ

;

and

Nq;NB 

q00NB Dd

af qg hfg

ð6Þ

;

respectively. In this relation, af is the liquid thermal diffusivity, qg is the density of the gas phase, hfg is the latent heat, and q00NB is the nucleated boiling component of the Chen [35] boiling heat flux partition. The empirical constants were later adjusted by Euh et al. [36] which publish experimental data of departure frequency at slightly higher pressure. Brooks [30] proposes a modification to the model of Situ et al. [26] by considering the nucleate boiling component of boiling heat flux partition as proposed by Liu and Winterton [37]. The energy balance proposed by Brooks [30] considers the average energy to all nucleating bubbles and therefore includes a dependence on the active nucleation site density. It is important to note that all of the available models for bubble departure frequency require accurate closure of the bubble departure diameter. It is clear that without an accurate model of bubble departure diameter, the performance of the bubble departure frequency prediction will be significantly affected. 2.2. Existing experimental data of bubble departure characteristics and evaluation of current modeling Reliable experimental data is essential for model development and benchmark. In boiling two-phase flows, experimental data providing the detail required in this study has only been possible in recent decades. The available data for bubble departure diameter and departure frequency in flow boiling systems is very limited. As discussed previously, the bubble departure frequency has been modeled as a function of bubble departure diameter and shown to be a strong function of the departure size. Therefore the development and benchmark of the bubble departure frequency database requires experimental data of the departure diameter to isolate the

effect of flow conditions on frequency. The datasets of Basu [38], Murshed et al. [39], and Brooks et al. [16] provide the required information necessary to describe the bubble departure frequency phenomena. From an extensive literature review, six datasets shown in Table 1 have been selected for the study of the bubble departure diameter in forced convective flows. The datasets of Yuan et al. [40], Basu [38], Murshed et al. [39], Klausner et al. [3], Thorncroft et al. [2], and Brooks et al. [16] are considered due to the completeness of the dataset and range of conditions covered. The details of each experimental work are reviewed in Table 1. An addition nine data points were also taken in the test section of Brooks et al. [16] with an identical procedure at a pressure of approximately 450 kPa and are tabulated in Appendix A. The third pressure condition is essential for determining the pressure effect on the bubble departure diameter and therefore is reported in this work. The evaluation of the current modeling of the bubble departure diameter is given in Table 2 with select models displayed in Fig. 1. The bubble departure diameter models have large error when compared against the available database, in many cases resulting in a significant over prediction in the departure diameter. The model of Prodanovic et al. [29], while still resulting in large error relative to the database, is most able to capture the trends in the data for the data with water as the working fluid. The large under prediction by Prodanovic et al. [29] for the non-water data of Murshed et al. [39] (R134a), Klausner et al. [3] (R113), and Thorncroft et al. [2] (FC87) suggest that the model is missing an important dimensionless group to account for different fluid properties. Therefore a new model must be derived with theoretical justification in order to determine the correct dimensionless groups. The relative successes of the basic models by Fritz [19] and Kocamustafaogullari and Ishii [12] suggest the ability of the Laplace length to scale the departure diameter. The bubble departure frequency models are benchmarked against the available experimental data with errors tabulated in Table 3 and comparison presented in Fig. 2. The model by Basu et al. [34] works well for data at similar conditions to the data of Basu [38], which was used to develop the correlation. Overall the early model of Cole [31] and more recent model of Situ et al. [28] show the best agreement with the experimental data. The success in the model of Situ et al. [28] suggests that the dimensionless frequency based on the thermal diffusivity may be a good way to correlate the data. It is important to note that the frequency comparison in Table 3 and Fig. 2 was performed using the measured value of bubble departure diameter. Based on the large

Table 1 Summary of available bubble departure data in flow boiling.

a b c d e f

Study

Basu [38]

Murshed et al. [39]

Yuan et al. [40]

Brooks et al. [16]

Klausner et al. [3]

Thorncroft et al. [2]

Measurement conditions Orientation Fluid Heated surface Geometry Hydr. Diameter [mm] Contact Angle [deg.] Pressure [kPa] Heat flux [kW/m2] Mass flux [kg/m2 s] Subcooling [°C]

Dd, fd 19 Vertical Water Copper Square 39.2 30 103 210–950 235–685 7–46

Dd, fd 3 Vertical R134a Stainless steel Rectangular 5.56 NA 690, 758, 827 130 1206 4, 7, 9.5b

Dd 21 Vertical Water Stainless Steel Rectangular 3.85 57c 121–1040 71–334 76–603 20–36

Dd, fd 92a Vertical Water Stainless Steel Annulus 19 57 150, 300, 450a 100–492 235–986 5–40

Dd 35 Horizontal R113 Nichrome Square 25 40.5d 131–212 11–26 113–287 1–19e

Dd 19f Vertical FC87 Nichrome Square 12.7 NA 142–155 1.32–14.6 192–666 1.96–4.91

The 9 conditions taken at 450 kPa are tabulated in Appendix A. Listed with respect to the pressure conditions. Contact angle not reported but assumed based on similarity of working fluid and heated surface to Brooks et al. [16]. Contact angle is assumed to be an average value between the reported advancing and receding contact angle. Bulk liquid temperature calculated from Liu and Winterton [37]. 9 conditions in upward flow, 10 conditions in downward flow.

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C.S. Brooks, T. Hibiki / International Journal of Heat and Mass Transfer 86 (2015) 183–196 Table 2 Comparison of available departure diameter models with experimental data. Mean absolute relative error [%]a

Model

Situ et al. [26] Harada et al. [22] Prodanovic et al. [29] Unal [18] Kocamustafaogullari and Ishii [12] Fritz [19] Brooks [30] New Model

Klausner et al. [3]

Thorncroft et al. [2]

Murshed et al. [39]

Yuan et al. [40]

Brooks et al. [16]

All Data

608 213 48.1 413 168 198 26.3 25.4

61.0 144 91.4 1201 68.3 215 98.0 23.0

52.4 249 93.8 235 NAb NAb 98.8 21.2

97.3 123 97.8 156 NAb NAb 99.7 44.0

57.1 377 65.6 54.9 80.8 764 87.5 28.1

853 1210 173 279 879 2280 19.9 18.3

501 706 124.2 433.2 527.8 1419.7 51.7 21.7

Mean Absolute Relative Error = 100  |Dd,Exp  Dd,Calc|/Dd,Exp. No reported contact angle.

10

4

+50%

10 10 10

2

1

Brooks et al. (2014) Data Basu (2003) Data Klausner et al. (1993) Data Yuan et al. (2011) Data

10 0 10

10

1

10

2

10

3

Exp. Dep. Diameter, D

d,Exp

10

+50%

3

2

Brooks et al. (2014) Data Basu (2003) Data Klausner et al. (1993) Data Yuan et al. (2011) Data

0

10

1

10

2

10

3

10 d,Exp

4

10

10 10

10

10 10 10 10

Brooks et al. (2014) Data Basu (2003) Data Klausner et al. (1993) Data Thorncroft et al. (1998) Data Yuan et al. (2011) Data Murshed et al. (2010) Data

1

10

1

10

2

10

3

10

4

10

5

[μm]

d,Exp

5

Model by Prodanovic et al. (2002) Brooks et al. (2014) Data Basu (2003) Data Klausner et al. (1993) Data Thorncroft et al. (1998) Data Yuan et al. (2011) Data Murshed et al. (2010) Data

4

3

+50% -50%

2

1

[μm]

10

Model by Situ et al. (2008) 4

+50%

3

2

Brooks et al. (2014) Data Basu (2003) Data Klausner et al. (1993) Data Thorncroft et al. (1998) Data Yuan et al. (2011) Data Murshed et al. (2010) Data

1

0

10

1

10

2

10

3

Exp. Dep. Diameter, D

10 d,Exp

4

10

1

10

2

10

3

Exp. Dep. Diameter, D

5

10 0 10

2

10 0 10

5

-50%

10

10

0

10

Exp. Dep. Diameter, D

10

10

10

Model by Kocamustafaogullari and Ishii (1983)

1

-50%

3

Exp. Dep. Diameter, D

Calc. Dep. Diameter, D

10

+50%

[μm]

-50%

10

Model by Unal (1976) 4

10 0 10

5

[μm]

4

5

0

10

5

10 0 10

[μm]

4

d,Calc

10

10

10

[μm]

Calc. Dep. Diameter, D

d,Calc

[μm]

10

d,Calc

Calc. Dep. Diameter, D

-50%

3

0

Calc. Dep. Diameter, D

[μm]

Model by Fritz (1935)

d,Calc

10

5

d,Calc

Calc. Dep. Diameter, D

d,Calc

[μm]

10

Calc. Dep. Diameter, D

a b

Basu [38]

10

10 10 10 10

10 d,Exp

4

10

5

[μm]

5

Model by Brooks (2014) 4

3

Brooks et al. (2014) Data Basu (2003) Data Klausner et al. (1993) Data Thorncroft et al. (1998) Data Yuan et al. (2011) Data Murshed et al. (2010) Data

+50% -50%

2

1

0

5

[μm]

10 0 10

10

1

10

2

10

3

Exp. Dep. Diameter, D

10 d,Exp

4

10

5

[μm]

Fig. 1. Comparison of select bubble departure diameter models from literature.

prediction error of the bubble departure diameter models in Table 2, clearly the bubble departure frequency will have very

larger errors if the correlation is calculated using a previously discussed departure diameter model.

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Table 3 Comparison of available departure frequency models with experimental data. Model

Cole [31] Basu et al. [34] Situ et al. [26] Euh et al. [36] Podowski et al. [33] Brooks [30] New Model a b

qf V c Lc ; lf cp;f lf ; kf

ð9Þ

Mean absolute relative error [%]a Basu [38]

Murshed et al. [39]

Brooks et al. [16]

All Data

Pr 

77.6 22.6 55.9 45.6 439 175 32.0

54.7 70.9 80.5 89.1 46.6 NAb 39.8

48.9 54.9 54.0 79.2 96.5 39.0 34.3

53.9 49.9 55.0 73.8 152 62.2 34.1

where h, kf, cp,f, lf , qf , Lc, and Vc are the heat transfer coefficient, liquid thermal conductivity, liquid specific heat capacity, liquid viscosity, liquid density, characteristic length, and characteristic velocity respectively. The characteristic velocity, Vc, in the Reynolds number may be given by the agitation of the liquid as proposed by Rohsenow [41]. In boiling this agitation of the liquid comes from the bubble departure at the heated surface. The characteristic length of this liquid agitation is the distance the liquid travels in order to replace the departing bubble (proportional to the bubble size), and the characteristic time is given by the bubble departure frequency,

Mean Absolute Relative Error = 100  |fd,Exp  fd,Calc|/fd,Exp. No reported contact angle.

3. Theoretical considerations for departure characteristics The bubble departure characteristics have been modeled with several approaches (i.e. correlation, force balance, energy balance) as discussed in the previous section. New models for bubble departure diameter and frequency are derived here using an energy balance approach. The Nusselt number which describes the ratio of convective to conductive heat transfer is an important non-dimensional number in heat transfer analysis and can be given in terms of the Reynolds number, Re, and Prandtl number, Pr, as,

Nu ¼ CRemfc Prnfc ;

ð7Þ

where C is a constant, and mfc and nfc are the power dependence in forced convective flow. The non-dimensional groups are defined as,

hLc ; kf

1 fd

¼ td /

[Hz] Calc

+50% -50%

10 10 10

Calc. Dep. Frequency, f

[Hz] Calc

Calc. Dep. Frequency, f

10

10

Brooks et al. (2014) Data Basu (2003) Data Murshed et al. (2010) Data

4

3

2

1

10

1

10

2

10

3

10

Exp. Dep. Frequency, f

Exp

4

10

10 10 10

10 10 10 10

Model by Basu et al. (2005) Brooks et al. (2014) Data Basu (2003) Data Murshed et al. (2010) Data

4

+50% -50%

3

2

1

10 [Hz]

Brooks et al. (2014) Data Basu (2003) Data Murshed et al. (2010) Data

,Calc

+50% -50%

3

2

1

0

10 0 10

ð13Þ

5

[Hz]

Model by Situ et al. (2008)

10

ð12Þ

;

10

1

10

2

10

3

10

Exp. Dep. Frequency, f

5

4

q00w D2d

q00w qf Dd : qg hfg lf

10 0 10

5

Calc. Dep. Frequency, fd

Calc

[Hz]

10

!

0

0

10 0 10

qg hfg D3d

where q00w is the wall heat flux. Therefore the resulting bubble departure Reynolds number used to describe the boiling flow Nusselt number is given as,

5

Model by Cole (1960)

ð11Þ

Rohsenow [41] gives the time between bubble departures, the inverse of frequency, as the energy required to form a nucleating bubble divided by the heat transfer rate to the nucleating bubble [42],

ð8Þ

10

ð10Þ

V c / Dd f d :

Red 

Calc. Dep. Frequency, f

Nu 

Re 

10

4

10

5

[Hz]

Exp

5

Model by Brooks (2014) 4

Brooks et al. (2014) Data Basu (2003)

+50% -50%

10 10 10

3

2

1

0

10

1

10

2

10

3

Exp. Dep. Frequency, f

10 Exp

4

[Hz]

10

5

10 0 10

10

1

10

2

10

3

Exp. Dep. Frequency, fd

10 ,Exp

Fig. 2. Comparison of select bubble departure frequency models from literature.

4

[Hz]

10

5

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The heat transfer coefficient can be given by Newton’s law of cooling considering the total temperature difference between the wall, Tw, and liquid, Tf,

q00w h¼ : ðT w  T f Þ

ð14Þ

Bubble departure in flow boiling, particularly at low pressure, can be a strong function of bulk liquid subcooling [16] and therefore the total temperature difference between the wall and bulk fluid is considered. The Nusselt number expression, Eq. (7), can be rearranged, and by introducing non-dimensional groups, the nondimensional bubble departure diameter can be given as, m0

Dd ¼ CJaT fc q

mfc m0 fc

1 Dh Re1 h Bo Pr

n0fc m0fc

;

ð15Þ

Dd ; Lo

Dh 

Dh ; Lo

Lo 

ð16Þ

and

q 

cp;f lf qg q00 GDh ; Bo  w ; Reh  ; Pr  : qf Ghfg lf kf

ð17Þ

Here, Lo, JaT, Bo, r, g, and G are the Laplace length, total Jakob number, Boiling number, surface tension, gravitational constant, and mass flux respectively. The equation was non-dimensionalized considering the potential shown by the groups proposed by Prodanovic et al. [29]. The Boiling number describes the ratio of heat flux to mass flux, while the form of the Jakob number in Eq. (16) describes the total available sensible heat available in the system to the latent heat. This derivation shows that there may be other important dimensionless groups to the bubble departure diameter not included by Prodanovic et al. [29], including Reynolds number, Prandtl number, and channel size. The bubble departure diameter is non-dimensionalized based on the Laplace length. The Laplace length is a common length scale in two-phase flow as it describes the balance of surface and buoyancy forces. The Laplace length is a convenient and appropriate choice for the bubble length scale in two-phase flows. The use of the Laplace length to non-dimensionalize the departure diameter is also supported by the relative success of the models by Fritz [19] and Kocamustafaogullari and Ishii [12] which correlate the departure diameter to the Laplace length. Based on Eq. (15) and this discussion, the bubble departure diameter may be correlated based on the following dimensionless groups,

Dd 

Dd ¼ f ðJaT ; q ; Bo; Pr; Reh ; Dh Þ: Lo

ð18Þ

This overall approach is similar to the approach by Rohsenow [41] to derive his widely used nucleate boiling heat flux model. In the case of Rohsenow [41] nucleate boiling heat flux, the Laplace length is used for bubble departure diameter and the equations are solved for the wall heat flux. The bubble departure frequency can be derived based on the same approach as the bubble departure diameter. Combining Eqs. (7)–(11) the governing equation for the bubble departure frequency may be given as,

qf f d Dd Dd q00w Dd ¼C DTkf lf

!mfc   cp;f lf nfc : kf

af

 1=m n00  1Reh ¼ C BoJa1 : T Pr Dd Dh

ð20Þ

By non-dimensionalizing the frequency based on the liquid thermal diffusivity, the conduction and heat capacity in the liquid at the nucleation site during the wait time, and through the thin layer of liquid between the wall and bubble during the growth time are considered. Based on the final function form of the bubble departure diameter, Eq. (15), the bubble departure frequency reduces to,  fd

¼C

0

000 Jam T

qg qf

!m000 000

Bo2=m Prn :

ð21Þ

Therefore the bubble departure diameter may be correlated based on the following dimensionless groups, 

rffiffiffiffiffiffiffiffiffiffi

cp;f ðT w  T f Þ r ; JaT  g Dq hfg

f d D2d

f d ¼ f ðJaT ; q ; Bo; PrÞ:

where the new groups are defined as,

Dd 



fd 

ð19Þ

Rearranging terms, considering the non-dimensional frequency as proposed by Situ et al. [28], Eq. (5), the non-dimensional frequency may be written as,

ð22Þ

The identified important dimensionless groups of Eqs. (18) and (22) are an important contribution to understanding the wall nucleation phenomenon. Deriving theoretically justifiable non-dimensional groups are important to understanding important mechanisms, designing experiments, and indispensable for working with scaling fluids. 4. Results and discussion 4.1. Bubble departure diameter 4.1.1. Correlating departure diameter Based on the available experimental data shown in Table 1, the bubble departure diameter can be correlated based on the parameters derived in the previous section and given in Eq. (18). The channel Reynolds number was shown by the experimental data to have an insignificant contribution to the bubble departure diameter, as the Boiling number is sufficient to account for changes in the mass flux. Based on the available experimental data, the bubble departure diameter can be given as 1:72 Dd  C Dd Ja0:49 q0:78 Bo0:44 Prsat ; T

ð23Þ 3

2

where the coefficient, CDd, is 2.11  10 and 1.36  10 for conventional and mini-channels respectively. Classification of conventional and mini-channels will be discussed later in this section. The mean absolute error is given in Table 2 for each dataset and the final model comparison is given in Fig. 3. With exception to the limited data points of Murshed et al. [38], the resulting model is able to collapse each bubble departure diameter dataset to within ±30%. With an error of less than ±22% for the departure diameter database as a whole, this new bubble departure diameter model represents a significant improvement as compared to the current modeling prediction shown in Fig. 1. It is important to note that the wall temperature was calculated from the model by Liu and Winterton [37] for the database of Brooks et al. [16], as well as, the bulk liquid temperature for the dataset of Klausner et al. [3], and therefore the adjustable constants have some inherent influence from that model. The resulting trend in the bubble departure diameter with each important dimensionless group is discussed as follows with their relationship presented in Fig. 4. 4.1.2. Jakob number dependence The Jakob number given in Eq. (18) is modified from the typical Jakob numbers based on wall superheat or bulk liquid subcooling. This Jakob number considers the total temperature difference between the wall and bulk liquid. In forced convective boiling, a departing bubble may extend through the laminar sublayer into the transition zone to the turbulent core. Therefore the departing

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Calc. Dep. Diameter, D

d,Calc

[μm]

10 10

4

New Model in Conventional Channel

+50% -50%

10 10 10

3

2

Brooks et al. (2014) Data

1

Basu (2003) Data Klausner et al. (1993) Data Thorncroft et al. (1998) Data

0

10 0 10

10

1

10

2

10

3

Exp. Dep. Diameter, D

d,Calc

[μm]

10

Calc. Dep. Diameter, D

bubble is exposed to convective effects of the bulk subcooling rather than steady saturated liquid as in the case of pool boiling. This Jakob number, which represents the total available sensible heat to latent heat, correlates better with the experimental data than the wall or subcooling Jakob numbers, particularly at low pressure where the departure diameter is largest. Fig. 4 shows the trend in Jacob number for the available departure diameter database to decrease the bubble departure diameter with increase Jakob number, particularly for the low pressure water data. This total Jakob number may be decomposed into the wall Jakob number and dimensionless temperature which describes the subcooling effect,

5

10

10 d,Exp

4

10

5

JaT ¼ JaW NT ;

[μm]

where the non-dimensional temperature, NT, is defined as,

5

4

New Model in Mini-channels

ð24Þ

NT 

+50%

Tw  Tf : T w  T sat

ð25Þ

-50%

10 10 10

Therefore, this total Jakob number accounts for the effect of increasing the bulk subcooling, which is shown to reduce the bubble departure diameter by many experimental and visual studies [16,29,43–46]. The elevated pressure data of Brooks et al. [16] is less sensitive to the Jakob number as the bubble departure size is small compared to the distance from the wall to the turbulent core. In the Jakob number plot of Fig. 4, the lower blue circles that do not follow the trend negative one-half slope corresponded to the elevated pressure cases and it is theorized that the bubble departure under these conditions are less sensitive to subcooling because their departure is influenced less by the turbulent subcooled bulk flow. A conservative demonstration of this turbulent effect is shown in Fig. 5 using the simple single phase velocity profile of Martinelli [47] based on Prandtl’s classic mixing length model which is well

3

2

1 Yuan et al. (2011) Data Murshed et al. (2010) Data

0

10 0 10

10

1

10

2

10

3

Exp. Dep. Diameter, D

10 d,Exp

4

10

5

[μm]

Fig. 3. Comparison of the newly developed bubble departure diameter model with experimental data.

1

10

10

10

0

-1

-2

10 -2 10

10

10

-1

Jakob Number, JaT [-]

10

10

10

0

-1

10 -4 10

0

1

10 Brooks et al. (2014) Basu (2003) Klausner et al. (1993) Thorcroft et al. (1998) Yuan et al. (2011) Murshed et al. (2010)

0

10

-3

-2

10

10

10

-2

10

Density Ratio, ρ* [-]

-1

1 Brooks et al. (2014) Basu (2003) Klausner et al. (1993) Thorcroft et al. (1998) Yuan et al. (2011) Murshed et al. (2010)

*

-1

10 -5 10

Brooks et al. (2014) Basu (2003) Klausner et al. (1993) Thorcroft et al. (1998) Yuan et al. (2011) Murshed et al. (2010)

-2

10

Non-D Departure Diameter, Dd [-]

Non-D Departure Diameter, D*d [-]

10

1

*

Non-D Departure Diameter, Dd [-]

Brooks et al. (2014) Basu (2003) Klausner et al. (1993) Thorcroft et al. (1998) Yuan et al. (2011) Murshed et al. (2010)

*

Non-D Departure Diameter, Dd [-]

10

0

-1

-2

10

-4

10

-3

Boiling Number, Bo [-]

10

-2

10 -1 10

10

0

10

Prandtl Number, Pr

1

sat

[-]

Fig. 4. Important dimensionless groups for correlating the bubble departure diameter.

10

2

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observed to increase the bubble departure diameter [3,50]. An increase in heat flux and a decrease in the mass flux both result in an increase in Boiling number, and in turn, an increase in the bubble departure diameter. However, the low pressure water data and the mini channel data are not sensitive to changes in the Boiling number as these datasets are mostly described by the thermal condition accounted for by the Jakob number.

P~450 kPa P~300 kPa P~150 kPa

70 60 50 40

y+=30

30 20 10

y+=5

0 0

2 4 6 8 Liquid Reynolds Number, Ref [-]

10 4

x 10

Fig. 5. Bubble departure position relative to single phase liquid velocity profile, highlighting a conservative effect of turbulence mixing of the subcooled bulk liquid on bubble departure.

summarized by Todreas and Kazimi [48]. Here the superscript ‘+’ represents a length non-dimensionalized by the wall length scale,

Dþd ¼ Dd

qffiffiffiffiffiffiffiffiffiffiffiffiffi sw =qf

mf

;

yþ ¼ y

qffiffiffiffiffiffiffiffiffiffiffiffiffi sw =qf

mf

;

ð26Þ

where sw is the wall shear stress, y is the distance from the wall, and mf is the kinematic liquid viscosity. The non-dimensional wall length, y+, of 5 is the boundary of the laminar sub-layer in single phase flow, y+ of 5 to 30 represents the transition region, and y+ greater than 30 is the turbulent core region. In subcooling boiling flow with bubbles nucleating on the heated surface the velocity profile is likely to transition to turbulent closer to the surface, however Fig. 5 gives a conservative effect of turbulence on the departing bubble at the three pressures considered by the data of Brooks et al. [16]. At low pressure the departing bubbles extend further into the bulk fluid and are exposed to more turbulent effects, and in turn more affected by the bulk liquid subcooling. 4.1.3. Density ratio dependence A clear trend exists in the experimental data of decreasing departure diameter with increasing density ratio as shown in Fig. 4. This trend is a consequence of increasing pressure which increases the density ratio and decreases the departure diameter. Decreasing bubble departure size with increasing pressure is a trend that is well documented in literature [16,29,49]. The datasets of Klausner et al. [3], Thorncroft et al. [2], and Basu [37] do not consider a range of pressure and are only acquired near atmospheric pressure. The applicability of the departure diameter model to higher system pressure is likely to be dependent on the correct determination of the density ratio dependence. More experimental data at higher system pressures is important to confirm the correct trend in the departure diameter with increasing density ratio. 4.1.4. Boiling number dependence The Boiling number accounts for the effect of changes in the wall heat flux as well as changes in the mass flux. The combined result is an increase in the bubble departure diameter with increases in the Boiling number. This trend is shown by the elevated pressure data of Brooks et al. [16] as well as the non-water datasets of Thorncroft et al. [2] and Klausner et al. [3] in Fig 4. Experimental data and nucleation trends reported in literature describe the departure diameter as decreasing due to increasing the mass flux [3,23,29,38,43–45,50]. The increasing mass flux increases the drag on the nucleating bubble causing a sweep removal from the site before the bubble reaches sizes found in flows of smaller mass flux. An increase in the heat flux is also

4.1.5. Prandtl number dependence Considering the available data which spans approximately one order of magnitude in Prandtl number, the bubble departure diameter is shown in Fig. 4 to increase with increasing Prandtl number. Here the Prandtl number is used to consider the effect of the different fluid properties. Therefore the Prandtl number is calculated from the saturation properties, which results in the Prandtl number being a function of pressure and working fluid. Considering that the density ratio accounts for the effects of system pressure, together the density ratio and Prandtl number should collapse the experimental data of different pressures and fluid properties, as shown in Fig. 6. As the Prandtl number increases the thermal boundary layer grows relative to the velocity boundary layer which may suggest that the departing bubbles have more room to grow in the superheated layer. Currently there are no available studies in literature which explore the effect of different working fluids on the bubble departure characteristics. Future experimental work can improve the model capability by extending the available experimental data to a wider range of Prandtl numbers. This influence of the Prandtl number should be considered in experiments involving scaling fluids to properly account for differences in fluid properties. 4.1.6. Channel size dependence Based on the experimental data of Yuan et al. [40], and Murshed et al. [39], there is a clear effect of the channel size on the bubble departure diameter. The gap size of Yuan et al. [40], 2 mm, is on the same order of magnitude as the measured bubble departure diameter at low pressure which results in much different thermal and mechanical conditions at the nucleation site than seen in the conventional flow geometries. Intuitively, it is clear that if the channel thickness is reduced beyond a certain critical distance, the bubble size must be influenced by the adjacent surface. For large channels where the hydraulic diameter is orders of magnitude larger than the departing bubble size, beyond some critical threshold, the effect of channel size is expected to saturate for the highly localized and microscopic wall nucleation phenomenon. Given the limited experimental data in these narrow channels, the channel size dependence within this critical region cannot be

10 Non-D Departure Diameter, D*d [-]

d

Non-Dimensional Departure Diameter, D+ [-]

80

10

10

1 Brooks et al. (2014) Basu (2003) Klausner et al. (1993) Thorcroft et al. (1998) Yuan et al. (2011) Murshed et al. (2010)

0

-1

-2

10 1 10

10

2

10

3

10

4

Prandtl & Density Ratio Dep., Pr1.72 ρ*-0.78 [-] sat

Fig. 6. Prandtl number and density ratio dependence on bubble departure diameter, accounting for the effect of fluid properties.

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completely determined. However this channel size effect is analogous to a similar criterion applied to the Rohsenow nucleate boiling heat flux. Pioro [51] reports that significant changes to the Rohsenow model [41], occur in pool boiling of liquids with less than 2.4 mm of fluid level. The resulting influence of a reduced fluid height on the heat transfer coefficient in pool boiling is likely due to a similar impact seen in the mini channel effect on departure diameter (i.e. in both cases the nucleating bubbles on the heat surface are of similar size as the medium in which they depart). Many other studies analyzing the two-phase flow behavior in mini-channels describe differences from conventional channels in important two-phase flow phenomena [5,52–54]. The mini-channel definition is often considered to be channels with less than approximately 3 mm gap thickness [5,53] which would include the mini-channels identified in this study of Mushed et al. [39] and Yuan et al. [40]. More data on wall nucleation in these very narrow channels are necessary to understand this channel size effect. 4.2. Bubble departure frequency 4.2.1. Correlating departure frequency An important consideration in correlating the bubble departure frequency based on the important dimensionless groups determined in the previous section is the limited range of Prandtl number in the available data. Therefore we first successfully correlate the experimental data consider the following functional form based on Eq. (20), 

 f d ¼ C fd Ja2:27 W q

0:26

Bo0:68 D1:37 ; d

ð27Þ

where the dependence of the Reynolds number was shown by the experimental data to be negligible and the dependence of the dimensionless hydraulic diameter indeterminate based on limited data. The strong function of the non-dimensional departure frequency with non-dimensional departure diameter is shown in Fig. 7. In Eq. (27) the wall Jakob number was most successful in correlating the data as compared to the total or subcooling Jakob number. Based on the final proposed model for dimensionless bubble departure diameter, Eq. (23), the dimensionless frequency may be given as, 

b

f d ¼ C fd JaaW q JacT Prdsat ;

ð28Þ

where the Prandtl number dependence, d, is 2.36. The Prandtl number dependence will be discussed in more detail later in this section. Based on the available experimental data, this new expression of bubble departure frequency is correlated as,

0:93



1:46  f d ¼ C fd Ja0:82 q W NT

10

10

2

1 Brooks et al. (2014) Data Basu (2003) Data Murshed et al. (2010) Data

0

10 -3 10

10

-2

10

-1

10

Non-D Departure Diameter,

0

D* d

10

Pr2:36 sat ;

ð30Þ

4.2.3. Density ratio dependence The decrease in the non-dimensional bubble departure frequency with increasing density ratio as shown in Fig. 9 is likely a

[Hz]

10

0:93

4.2.2. Jakob number dependence The strong dependence of the wall Jakob number in Eq. (30), and shown in Fig. 9, accounts for the large increase in departure frequency with increase in the wall superheat. This is also reflected by the experimental observation of a sharp increase in frequency with increase wall heat flux [16] for a given flow rate, pressure, and bulk liquid subcooling. The increase in the departure frequency with increase in wall Jakob number can be attributed to the decrease in bubble wait time and growth time due to more energy input form the heated surface [2,36]. The total and wall Jakob numbers in Eq. (29) are simplified to the wall Jakob number and non-dimensional temperature in Eq. (30). The Non-dimensional temperature, Eq. (25), increases with increasing bulk liquid subcooling. Increasing the bulk liquid subcooling has been shown to decrease the bubble departure frequency [16,36,45,46] which is consistent with the negative power dependence of the non-dimensional temperature in the newly proposed model. An increase in the bulk liquid subcooling may increase the wait time between nucleations due to local mixing caused by the departure of the previous bubble [36], or increase the growth time as the bubble expands through the thermal boundary layer [45].

4

3

ð29Þ

where one leading constant, Cfd, is given as 5.5. While the strong function of departure diameter on departure frequency would suggest separate constants for conventional and mini-channels, the limited data of frequency in mini-channels does not allow for identification of separate constants. The channel size effect will be discussed in detail later in this section. Comparison of this bubble departure frequency model with the experimental data is given in Fig. 8 with tabulated errors in Table 3. Here the measured bubble departure diameter is used to calculate the departure frequency in order to isolate the errors of the frequency model. This newly proposed bubble departure frequency model results in an average error of less than ±35% considering the entire frequency database. The important dimensionless groups for the non-dimensional bubble departure frequency are presented in Fig. 9 and discussed as follows.

Calc

10

2:36 Ja1:46 Pr sat ; T

or

Calc. Dep. Frequency, f

Non-D Departure Frequency, f*d [-]

10



 f d ¼ C fd Ja2:28 W q

10

5

4

+50% -50%

10 10 10

3

2

1 Brooks et al. (2014) Data Basu (2003) Data Murshed et al. (2010) Data

0

1

[-]

Fig. 7. Relationship between non-dimensional bubble departure frequency and non-dimensional departure diameter.

10 0 10

10

1

10

2

10

3

Exp. Dep. Frequency, f

10 Exp

4

10

5

[Hz]

Fig. 8. Comparison of the newly developed bubble departure frequency model with experimental data.

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10

10

10

10

4

*

Non-D Departure Frequency, f [-]

Brooks et al. (2014) Data Basu (2003) Data Murshed et al. (2010) Data

*

Non-D Departure Frequency, fd [-]

10

3

2

1

10 10 10

3

2

1

0

10 -3 10

0

10 -4 10

10

-3

10

-2

10

-1

10

0

10

10 10 10

3

2

1

0

10 0 10

-2

10

-1

10

0

4 Brooks et al. (2014) Data Basu (2003) Data Murshed et al. (2010) Data

*

Non-D Departure Frequency, fd [-]

*

Non-D Departure Frequency, f [-]

4 Brooks et al. (2014) Data Basu (2003) Data Murshed et al. (2010) Data

10

Jakob Number, JaW [-]

Density Ratio, ρ* [-]

10

Brooks et al. (2014) Data Basu (2003) Data Murshed et al. (2010) Data

10

10

10

3

2

1

0

10

1

10

2

Non-dimensional Temperature, NT [-]

10 -1 10

10

0

10

1

Prandtl Number, Prsat [-]

Fig. 9. Important dimensionless groups for correlating the bubble departure frequency.

10 Non-D Departure Frequency, f d* [-]

result of the non-dimensional frequency definition, Eq. (5), being a function of departure diameter squared. There is a significant effect of pressure on the bubble departure size as discussed previously and shown by the bubble departure diameter model, Eq. (23). No direct impact of pressure on the bubble departure frequency could be observed by Brooks et al. [16]. Therefore given the correct trend in the pressure dependence of the bubble departure diameter, the dependence on the non-dimensional frequency is confirmed, which is shown to be accurate for the data of Yuan et al. [40] spanning atmospheric to over 1 MPa. Therefore, this departure frequency model may work well at these elevated pressures.

4 Brooks et al. (2014) Data Basu (2003) Data Murshed et al. (2010) Data

10

10

10

3

2

1

0

4.2.4. Prandtl number dependence Currently the Prandtl number dependence is observed from the effect of Prandtl number on departure diameter. Given the very limited range of the experimental data for different Prandtl numbers, more experimental data is necessary to confirm this trend. As the Prandtl number increases, the thermal boundary layer grows relative to the velocity boundary layer which may suggest larger frequencies due to a reduced wait time from reduced mixing effects of the subcooled bulk fluid following a bubble departure. Fig. 10 shows the trend of the proposed Prandtl and density ratio dependences on the bubble departure frequency. The combination of density, which accounts for pressure, and Prandtl number calculated based on the liquid saturation properties, which is a function of pressure and working fluid, should collapse the data of different working fluids. Fig. 10 suggests that the Prandlt number dependence may be too small; however, there is very little data available. 4.2.5. Channel size dependence The large difference between bubble departure diameter in conventional versus mini-channels suggests there could be some

10 1 10

10

2

10

3

10

4

10

5

Prandtl & Density Ratio Dep., Pr2.36 ρ*-0.93 [-] sat

Fig. 10. Prandtl number and density ratio dependence on bubble departure frequency, accounting for the effect of fluid properties.

difference in the non-dimensional frequency with channel size. However, only three data points of Murshed et al. [39] are available for mini-channel bubble departure frequency. Much more data is necessary in mini-channels to identify the potential channel size effect on the departure frequency. 4.3. Application of departure modeling The range of important parameters used in the development of the new departure modeling is given in Table 4. Use of the models outside the range of data should be done so with caution and more data extending the range of these parameters would be beneficial for confirming the correct dependencies. Furthermore, it is important to note that the current database (shown in Table 1) considers

i,WN,Calc

3.85–39.2 0.98–7.76 7.6  104–1.2  101 1.0–99 6.4  104–3.4  102 7.3  105–1.0  103

5.56–39.2 1.16–3.41 9.0  103–9.2  102 1.3–5.4 6.4  104–3.4  102 NA

a relatively small range of contact angle. Given the significance of the contact angle in the force balance approach (stemming from the surface force which acts to hold the bubble at the nucleation site), there is some question of the models application to fluid-surface combinations with drastically different contact angles. Model agreement for bubble departure on hydrophobic surfaces and super-hydrophilic surfaces should be considered with new experimental data in the future. The effect of the buoyancy force on bubble departure, often considered in the force balance approaches, is a fundamental difference between vertical and horizontal configurations as the force direction changes relative to the flow direction. The energy balance approach does not consider a separate bubble departure model for horizontal flows. The agreement of the departure diameter model to the horizontal data of Klausner et al. [3], suggests the independence of the model to flow orientation, however more experimental data would help to confirm this application of the model.

.

Calc. Interfacial Area Gen., A

Dh [mm] Prsat [–] JaW [–] NT [–] q⁄ [–] Bo [–]

Departure frequency

10 10 10 10

-2

+50%

-3

-50%

-4

-5 Brooks et al. (2014) Data Basu (2003) Data Murshed et al. (2010) Data

-6

10 -6 10

10

-5

10

-4

. 10 Exp. Interfacial Area Gen., A

-3

.

10

-2

2

[m /s]

-6 +50%

10 10 10 10 10

-7 -50%

-8

-9

-10 Brooks et al. (2014) Data Basu (2003) Data Murshed et al. (2010) Data

-11

10

-11

10

-10

10

-9

10 .

Exp. Volume Gen., V

-8

10

WN,Exp

-7

3

10

-6

[m /s]

Fig. 11. Agreement of the newly proposed bubble departure models for use in the interfacial area transport equation.

4.4. Interfacial area concentration The rate of interfacial area generation rate per active nucleation site, A_ i;WN , is of particular importance to the interfacial area transport equation wall nucleation source term, Eq. (1),

A_ i;WN ¼ pf d D2d ;

10

i,WN,Exp

[m3/s]

Departure diameter

WN,Calc

Table 4 Range of important parameters considered in the development of the new bubble departure modeling.

[m2/s]

C.S. Brooks, T. Hibiki / International Journal of Heat and Mass Transfer 86 (2015) 183–196

Calc. Volume Gen., V

194

relative to the total net vapor generation term after the point of net vapor generation [7].

ð31Þ 5. Conclusions

as well as the volume generation rate per active nucleation site, V_ WN , in the volume source term, Eq. (2) ,

p V_ WN ¼ f d D3d : 6

ð32Þ

The combined error of the bubble departure diameter and departure frequency models for the calculation of important interfacial area transport equation parameters are compared with the experimental data in Fig. 11. The interfacial area generation rate can be calculated from the non-dimensional frequency expression, Eq. (30), and results in a ±35% mean absolute relative error when compared with the available data. The volume generation rate employs both the frequency and departure diameter models, and results in a ±53% mean absolute relative error when compared with the available data. These uncertainties are important in considering the uncertainty in the interfacial area transport equation wall nucleation terms. The active nucleation site density model of Hibiki and Ishii [15] is reported to be within ±29% when compared with the available active nucleation site data with measured wall temperature. Therefore, the wall nucleation interfacial area source term may be considered accurate to within approximately ±65% when used in conditions considered by each model. Similarly, the volume source term from wall nucleation may be considered accurate to within approximately ±90%. While the new bubble departure modeling has drastically improved these predictions in the interfacial area concentration and volume source terms, a continued improvement is clearly necessary. The volume source term error is large however the volume produced by wall nucleation is very small

The bubble departure diameter and departure frequency are studied in great detail by this work. Past work is reviewed and the available databases have been collected from literature to present new semi-empirical models for departure diameter and frequency. The bubble departure diameter includes datasets of vertical upward flow, vertical downward flow, and horizontal flow, including four different working fluids and a wide range of flow conditions. The major findings of this work can be summarized as follows,  The available models for bubble departure diameter and bubble departure frequency show very large error when benchmarked against the available data from literature.  Important dimensionless groups for modeling bubble departure diameter and departure frequency are derived considering a heat transfer balance at the heated surface. The effects of each important dimensionless group are evaluated with the available database. The non-dimensional bubble departure diameter is shown to be a strong function of Jakob number, density ratio, Boiling number, and Prandtl number. The non-dimensional bubble departure frequency is given as a strong function of Jakob number, density ratio, Prandtl number, and dimensionless temperature.  The experimental data suggests that the bubble departure diameter is affected by the narrow gap of mini channels, leading to larger bubble departure diameters than in conventional channels.

C.S. Brooks, T. Hibiki / International Journal of Heat and Mass Transfer 86 (2015) 183–196

 With exception to the limited data points of Murshed et al. [39], the new bubble departure diameter model can predict each available dataset to within ±30%, and the new bubbled departure frequency model within ±35%. Considering the departure diameter database as a whole the bubble departure diameter model has an average error within ±22%. Both models result in substantial improvement in prediction accuracy over the available modeling.  The current bubble departure diameter and departure frequency database is shown to be sorely lacking in the range of Prandtl number. Also experimental data at elevated pressures is necessary to confirm the density ratio dependence. More attention is required in mini-channels and the boundary between conventional and mini-channel to understand the geometry effect on departure characteristics.  The interfacial area concentration and volume source terms were evaluated based on the new modeling of bubble departure diameter and frequency. From this analysis, assuming the use of Hibiki and Ishii [15] active nucleation site density model, the prediction uncertainty in the sources terms are ±65% and ±90% for interfacial area concentration and volume respectively. Therefore more work is necessary to further reduce these uncertainties. Conflict of interest None declared. Acknowledgment The lead author would like to thank Prof. Mamoru Ishii for his suggestions in the early developmental stages of this work.

Appendix A Additional data was taken in the same facility, with identical test procedure, instrumentation, and measurement as the data of Brooks et al. [16]. In line with the averaging procedure of the data by Brooks et al. [16] at least 100 successive bubble departures were identified to calculate the bubble departure frequency and average bubble departure diameter. By averaging over 100 bubbles, the experimental error in departure diameter from the measurement technique is less than 2% of the measured value. The statistical accuracy of the data is more difficult to determine since a very large number of bubbles cannot be measured for each condition. This is an issue with all past datasets and is discussed in detail for the dataset of Brooks et al. [16] by Martinez-Cuenca et al. [55]. This additional data was taken in order to better determine the effect of pressure on the bubble departure characteristics. The data is tabulated here in Table A1.

Table A1 Additional data corresponding to the tests of Brooks et al. [16]. Run

jf,in [m/s]

q00w [kW/m2]

P [kPa]

Tf [°C]

Dd [lm]

fd [Hz]

1 2 3 4 5 6 7 8 9

0.48 0.50 0.50 0.50 0.50 1.0 1.0 0.25 0.25

304 304 304 352 352 352 304 304 304

453 453 450 449 450 450 452 452 450

129 134 123 124 129 122 128 123 128

81 75 86 88 81 52 46 108 107

788 928 754 697 568 1340 1150 469 551

195

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