Progress in understanding and modelling of annular two-phase flows with heat transfer

Nuclear Engineering and Design 345 (2019) 166–182

Contents lists available at ScienceDirect

Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes

Progress in understanding and modelling of annular two-phase flows with heat transfer

T

Henryk Anglart Dept. of Physics, School of Engineering Sciences, KTH Royal Institute of Technology, Roslagstullsbacken 21, 106 91 Stockholm, Sweden

A R T I C LE I N FO

A B S T R A C T

Keywords: Annular flow Liquid film Droplets Dryout Post-dryout

Annular two-phase flows with heat transfer play important role in many industrial applications. In particular, thermal margins of Boiling Water Reactors (BWR) are entirely determined by this type of flow and heat transfer conditions. To avoid dryout, a liquid film must be present on heated rods of BWR fuel assemblies during normal operation. The present paper describes the recent progress in understanding and modelling of the governing phenomena of annular two-phase flow and heat transfer. A special attention has been devoted to experimental observations that have the most significant influence on the adopted modelling approach. The primary goal is to pave a path to mechanistic modelling of dryout and post-dryout heat transfer applicable to nuclear fuel assemblies. Current Computational Fluid Dynamics (CFD) approaches to model the governing phenomena are presented and their further improvements are suggested.

1. Introduction Multiphase flows exhibit a complex dynamical behavior resulting in several different possible flow regimes, which can be in general divided into separated and mixed flows. One of the examples of a separate flow is the annular two-phase flow, which distinguishes itself with presence of a liquid film, flowing along channel walls, and a gas core, occupying central regions of the channel. The gas core contains droplets, which are participating in mass, momentum and energy transfer with the liquid film through main two mechanisms: (a) entrainment of droplets from the liquid film; (b) deposition of droplets from the gas core to the liquid film. The liquid film itself may contain gas bubbles, which can be entrained from the gas core or which can be created within the film due to nucleate boiling. This effect is particularly relevant for thick liquid films. In the present paper annular flows with thin liquid films will be considered, where presence of bubbles can be neglected. One of the fundamental aspects in annular two-phase flow analyses is to determine the liquid film mass flow rate and its thickness. This is because the presence of the film on walls has a significant influence on the heat transfer coefficient and the local pressure gradient. Thus, annular flow models have to include calculation of the liquid film flow rate in order to capture these fundamental phenomena. This requirement asks for a high accuracy in prediction of mass transfer rates on the liquid surface, such as due to drop entrainment and deposition. When channel is heated, the liquid film is depleted by evaporation as well and this effect must be included.

There are various levels of approximation to model annular twophase flows (Levy, 1966; Wallis, 1970; Milashenko et al., 1989; Hewitt and Govan, 1990; Sugawara and Miyamoto, 1990; Anglart, 1994; Anglart et al., 1997; Anglart, 1999; Anglart and Adamsson, 2002; Anglart and Adamsson, 2003; Hemlin and Anglart, 2004; Adamsson and Anglart, 2004; Krejci et al., 2005; Caraghiaur and Anglart, 2006; Adamsson and Anglart, 2006a; Caraghiaur and Anglart, 2006; Anglart, 2006; Adamsson and Anglart, 2007; Caraghiaur and Anglart, 2009; Caraghiaur et al., 2009; Adamsson and Anglart, 2010; Caraghiaur and Anglart, 2010; Caraghiaur et al., 2013; Caraghiaur et al., 2013; Caraghiaur and Anglart, 2013; Anglart, 2014a; Anglart, 2015; Li and Anglart, 2015, 2016a–c; Rodriguez, 2009). In the simplest approach only a mass conservation equation for the liquid film is considered and the liquid film mass flow rate is calculated from a first-order nonlinear differential equation (Levy, 1966; Wallis, 1970; Milashenko et al., 1989; Hewitt and Govan, 1990). The advantage of the method is its robustness due to the simplicity. The disadvantage stems mainly from unclear boundary conditions at the onset of annular point, but also from neglect of pressure change along the channel and inherent inability to account for the spatial three-dimensional effects, which are important in complex geometries such as rod bundles. To remedy the above-mentioned disadvantages, the phenomenological film model has been implemented into sub-channel codes (Sugawara and Miyamoto, 1990) and CFD codes (Anglart et al., 1997; Anglart, 1999; Li and Anglart, 2015, 2016a–c), where an increasing level of detail has been applied. However, in all these models certain

E-mail address: [email protected]. https://doi.org/10.1016/j.nucengdes.2019.02.007 Received 31 July 2018; Received in revised form 3 January 2019; Accepted 12 February 2019 0029-5493/ © 2019 Elsevier B.V. All rights reserved.

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

u+ ur v We x y y+

Nomenclature a A C C Ca Cf Cfi d d32 de dpq D D Dh Dt Eo f f F Fo g g gz G h hfg j k k′ M M n On p P q Q q″ R Re Sδ Sh SU Str t u U u*

coefficient in Eq. (33) channel cross-section area disturbance wave velocity concentration of entrained liquid in gas core capillary number Fanning friction factor (= 2τw / ρU 2 ) interfacial friction factor (Eq. (15)) drop diameter Sauter mean diameter equivalent drop diameter equivalent drop diameter (Eq. (6)) pipe diameter deposition rate hydraulic diameter tube diameter Eötvös number (Eq. (1)) disturbance wave frequency probability distribution function of drop diameter (Eq. (4)) force vector Froud number gravity acceleration vector of gravity acceleration gravity acceleration vector projected on the flow direction mass flux specific enthalpy latent heat superficial velocity deposition coefficient fraction of liquid deposited to film Morton number (Eq. (2)) vector of interfacial momentum transfer normal vector Onhenzoge number pressure perimeter heat volumetric film flow rate per unit wetted perimeter heat flux vector radius Reynolds number (Eq. (3)) source term in film mass balance equation source term in film energy balance equation source term in film momentum balance equation Strouhal number time velocity mean film velocity vector friction velocity (= τw / ρ )

non-dimensional velocity in wall region (= u/ u∗) relative velocity velocity vector Weber number position vector distance from the wall non-dimensional distance in wall region (= yu∗/ ν )

Greek

α α β Γ ΓF δ ε μ ν ρ Δρ σ τ

coefficient in Eq. (41) volume fraction coefficient in Eq. (41) mass source per unit volume film mass flow rate per unit perimeter liquid film thickness wall roughness dynamic viscosity kinematic viscosity density liquid-gas density difference (= ρg − ρl ) surface tension shear stress

Subscripts c C d e F g i k l le w

critical gas core droplet, deposition entrainment liquid film gas interfacial phase index liquid entrained liquid wall

Acronyms BWR CFD DNS FT ITM LFD LPT RANS VOF SMD SST

Boiling Water Reactor Computational Fluid Dynamics Direct Numerical Simulation Front Tracking Interfacial Tracking Method Laser Focus Displacement Lagrangian Particle Tracking Reynolds-Averaged Navier Stokes Volume of Fluid Sauter Mean Diameter Shear Stress Transport

mechanistic principles. It is convenient to divide the fluid mechanics phenomena in the following research areas:

assumptions on location and shape of the film-gas interface have to be adopted. In the most rigorous approach, the film-gas interface can be resolved using the Interface Tracking Methods (ITM) such as the Volume of Fluid (VOF), Front Tracking (FT) and Level Set methods. Such methods are free from any assumptions on the interface shape and dynamics, however, they are still too demanding in terms of the computational effort to be used for industrial applications (Rodriguez, 2009). Thus, substantial amount of modelling is required to successfully predict annular two-phase flow with heat transfer of industrial relevance. To this end it is necessary to investigate and discuss the governing phenomena in detail and to build the required models using

• gas core flow, • liquid film flow, • interfacial wave propagation, • shear stress and pressure drop, • droplet deposition and entrainment. The relevant heat transfer phenomena are as follows,

• pre-dryout heat transfer, 167

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

• onset of dryout, • post-dryout heat transfer.

Eo =

Above-mentioned phenomena are summarized in Sections 2 and 3, whereas Section 4 contains a description of a CFD model of annular two-phase flow with heat transfer and Section 5 presents model’s applications.

M=

g Δρde2 (E\"o tv\"o s number), σ

gμg4 Δρ

Re =

2. Fluid mechanics of annular two-phase flows

ρg2 σ 3

(Morton number), (2)

ρg de ur μg

(1)

(Reynolds number), (3)

where de – equivalent drop diameter (volume-equivalent spherical diameter), g – gravity acceleration, μg – gas viscosity, ρg – gas density, σ – surface tension, ur – relative drop velocity and Δρ = ρl − ρg , where ρl – liquid density. It can be noted that the Morton number is purely a property group (sometime referenced to as M-group), whereas Eo depends on the drop size, and Re depends both on the drop size and on its relative velocity in the carrier fluid. The Eötvös number is a measure of the importance of the gravitational force related to the surface tension force. For small values of the Eötvös number the surface tension force is prevailing and drops tend to be spherical in shape. For large Eötvös number values the drop shapes can vary from spherical through ellipsoidal to spherical-cap, depending on the Reynolds number values. Non-dimensional analysis shows that where the gravity force is dominant (low Froude number, Fo = ur / gde ), the appropriate parameter to describe the drop deformation is the Eötvös number. A similar analysis shows that when viscous forces are dominant (high capillary number Ca = μg ur / σ ) and inertial effects are negligible (low Reynolds number), the relevant parameter to be considered in a study of drop deformation is the capillary number. By nature, drop swarms in annular flows of industrial interest have various diameters ranging from microns to millimeters. For accurate modelling of interfacial transfer processes it is crucial to adequately represent the drop size distribution and to determine a characteristic drop size. A wide variety of drop-size distribution functions has been proposed. The drop size is affected by many processes, such as: atomization of liquid on the crests of the disturbance waves, splashing of drops on the liquid film surface and on walls, coalescence, breakup, evaporation and condensation. All these processes result in a swarm of drops where the sizes are distributed between some non-zero minimum diameter and a finite maximum diameter. A finite maximum diameter is caused by aerodynamic forces acting on large drops and leading to their breakup into smaller ones. The drops cannot breakup indefinitely to zero-size drops, since the cohesive surface tension force increases with decreasing drop diameter and the available aerodynamic force cannot overcome the surface tension force. Anyway, for a mathematical convenience, most drop size distribution functions assume that the drop size varies from zero to infinity. However, there are constraints on such functions, that their values are approaching zero for the drop size approaching either zero or infinity. With these constraints, a probability density function for a drop diameter distribution f can be assumed, when the following obvious properties are satisfied,

Annular flow of industrial interest typically contains liquid film flowing along channel walls and gas core flowing in the channel central region. The interface between the liquid film and the gas core is wavy and unstable, and locally affected by the mass and momentum transfer, partly due to a shear stress between the phases and partly due to droplet deposition and entrainment phenomena. The gas core consists of a continuous gas phase which is carrying liquid drops. For given values of mass flow rates of both phases at a certain cross section of a channel, there is a unique set of such dependent parameters as liquid film thickness, liquid film flow rate, droplet size, deposition rate, entrainment rate, interfacial shear stress and total axial pressure gradient. The main purpose of a consistent and accurate model of annular two-phase flow is to determine all these dependent parameters in a good agreement with experimental data. 2.1. Gas core flow Particle-laden flow prevails in the gas core, in which the gas phase (sometime referred to as the carrier phase) is continuous and the liquid phase exists as drops (referred to as the dispersed phase as well). The boundaries of the gas core flow are not known, since they are set by the core-film interface, which is one of the unknown dependent parameters of the annular two-phase flow. The gas core flow has been intensively investigated both experimentally and numerically (Gill et al., 1963, 1964; Gill and Hewitt, 1968; Jepson et al., 1989; Azzopardi et al., 1991; Bates and Sheriff, 1992; Azzopardi and Teixeira, 1994a,b; Kocamustafaogullari et al., 1994; Fore and Dukler, 1995; Soldati and Andreussi, 1996; Azzopardi, 1997; Zaichik and Alipchenkov, 2001; Babinsky and Sojka, 2002; Fore et al., 2002; van’t Westende, 2007; Alamu and Azzopardi, 2011; Le Corre et al., 2017; Wang et al., 2017). In numerical models, the gas core flow can be described in many different ways, beginning from a onedimensional homogeneous two-phase flow formulation and ending with a Direct Numerical Simulation (DNS) of the free surface and interfacial flow (Rodriguez, 2009; Rodriguez et al., 2013). It should be mentioned, however, that DNS approach can be only applied to resolve a few drops with relatively low Reynolds and Weber numbers. Thus, such phenomena as droplet breakup in industrial gas core flows cannot be addressed in full scale yet. Two dominant CFD approaches which are applicable to gas core flows are based either on a two-fluid (or Eulerian-Eulerian) framework or on a Lagrangian Particle Tracking (LPT) formulation (Li and Anglart, 2016a–c).

f (ς ) ≥ 0,

2.1.1. Dispersed liquid phase flow Dispersed phase flow in the gas core can be characterized with distribution of drop sizes and velocities. These parameters determine the drop dynamics and are very important for mass transfer rates of the liquid phase between the core and the liquid film.

0 < ς < ∞,

∫0

∞

f (ς )dς = 1,

(4)

that is, the distribution must be positive everywhere and normalized. Moments about the origin of a drop size distribution can be then calculated as,

Mn = 2.1.1.1. Drop size. Usually a characteristic size of a drop cannot be determined uniquely due to its changing shape. Various shape regimes can be observed depending on the prevailing flow conditions. A graphical representation of these regimes was provided by, e.g. Clift et al. (1978), using the following three characteristic non-dimensional numbers,

∫0

∞

ς nf (ς )dς .

(5)

The mean value of a drop size distribution is then equal to the first moment, dm = M1. In statistics, three additional central moments (that is moments calculated about the mean value) are considered: the variance (n = 2), the coefficient of skewness (n = 3) and the kurtosis (n = 4). 168

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

mass flux of entrained liquid. As can be noted, the above equation indicates that the drop size depends on the square root of the tube diameter. This is not in agreement with experimental results obtained in large diameter tubes. Azzopardi (1985) argued that this was due to curvatures other than that of tube wall being more important, e.g. that due to the wave. An alternative length scale was proposed λT = σ / ρl g , which resulted in:

None of the above-mentioned higher moment definitions is used in a derivation of an equivalent drop diameter for a polydisperse drop flow, however. For that purpose the following equation is used, ∞

dpq

1

p−q p ⎡ ∫ ς f (ς )dς ⎤ = ⎢ 0∞ q , ⎥ ς f (ς )dς ⎣ ∫0 ⎦

(6)

where p and q are positive integers. The following diameters are commonly used: d10 (which is the arithmetic-mean diameter, equivalent to M1), d30 (the volume-mean diameter), d32 (the Sauter Mean Diameter – SMD) and d43 (the de Brouckere mean diameter). Currently three methods have been used to model drop size distribution: the empirical method, the maximum entropy method and the discrete probability function method (Babibsky and Sojka, 2002). In empirical approaches, the characteristic drop diameter is correlated to the main flow parameters such as the channel-mean Weber number and the channel-mean Reynolds number. The drop flow is then treated as a monodispersed flow, in which all drops have the same diameter obtained from an empirical correlation. Azzopardi et al. (1991) measured drop size for annular two-phase flow in a 20 mm diameter vertical tube. They confirmed the drop size dependence on tube diameter and showed that measured drop diameter fell between those from earlier work, obtained in 10 mm and 32 mm internal diameter tubes. The influence of gas and liquid flow rate on the measured film flow rate shows trends similar to those seen in data from the 10 mm and 32 mm diameter tubes. Azzopardi and Teixeira (1994a,b) performed detailed measurements of vertical annular two-phase flow. For air/water flow in a vertical tube with 32 mm inner diameter, the measured drop diameter was in a range from 89 μm to 209 μm. Gas mass flux was between 24.5 and 55.6 kg/m2s and liquid mass flux from 15.9 to 47.6 kg/m2s. Fore and Dukler (1995) performed an experimental study of the drop size and velocity distribution in gas-liquid annular upflow in a 50.8 mm internal diameter vertical tube. They observed bimodal drop size distributions, with increasing mean drop sizes for increasing liquid flow and viscosity. They reported that on average, a drop at the tube centerline travels at 80% of the local gas velocity. For liquid Reynolds number in a range from 140 to 560, the gas superficial velocity from 16.9 to 33 m/s and the liquid viscosity from 1 cP to 6 cP, the SMD was in a range from 339 to 1151 μm. They observed nearly single dependence for SMD versus the drop concentration, which was consistent with Azzopardi’s (1985) argument that increasing the concentration should cause drops to coalesce, resulting in larger sizes. Soldati and Andreussi (1996) performed a theoretical assessment of the effect of collisions and coalescence on the motion of large drops. According to this study the drop coalescence plays an important role in lengthening of the residence time of a drop in the gas core and in decreasing of the deposition constant. The model supported the earlier observations of the effects of drop concentration on its size. Several semi-empirical equations have been proposed to predict the drop size. Tatterson et al. (1977) obtained the following equation:

d32 = λT

where We =

Gle , ρl jg (9)

(10)

where correlations for the film thickness (δ ) and the interfacial friction factor (Cfi), are provided separately. Caraghiaur and Anglart (2013) proposed a correlation that is accurate within ± 40% with most of the data found in the literature. The equation for the Sauter mean diameter is as follows, 0.15 ⎡ Reg ⎞0.85 ⎛ μg ⎞ d32 ⎛G ρ μ ⎞ + ⎜ le l l ⎟ = 10−3 ⎢ ⎛ ⎜ ⎟ Dh We ⎠ ⎝ μl ⎠ ρj ρ μ ⎝ ⎢ ⎝ l g g g⎠ ⎣ ⎜

⎟

0.75

⎤ ⎥. ⎥ ⎦

(11)

The drop entrainment from a liquid film is a major source of the entrained liquid phase in the gas core of annular two-phase flow. The drop initial velocity vector and size are two important parameters that determine the path along which it travels in the gas core. Measurements made by the shadowgraph technique showed that the initial transversal velocity and the direction of the velocity vector of entrained drop do not depend on the drop size. Using this observation, Andreussi and Azzopardi (1983) suggested the following relationship between the drop initial transverse mean velocity udn0 and the gas friction velocity, ug* = τi/ ρg ,

udn0 = 12ug*

ρg ρl

. (12)

The drop mean velocity tangential to the film interface can be assumed to be equal to the celerity of the disturbance wave at the location where the drop is created. 2.1.1.2. Drop velocity. Drop velocity distribution shows similarities to gas velocity distribution, but in general, the velocity of drops is lower than the velocity of gas at the same location. Azzopardi and Teixeira (1994a,b) measured drop size and velocity distributions in annular airwater flow in a vertical pipe with 32 mm inner diameter. They concluded that drop velocities were 20 percent below the corresponding local velocity of the gas, whereas the standard deviations of the drop velocities normalized by the mean velocity were 10–65 percent higher than those of the gas and their values were increasing once moving from the channel centerline toward the interface. A trend was observed for smaller drops to be travelling at higher velocities, with wider range of velocities. This effect is due to a stronger influence of gas turbulence on small drops than on large ones. It is of interest to note that a completely different trend was observed by Lopes and Dukler (1987), who showed that larger velocities were obtained by the bigger drops. However, as pointed out by Azzopardi and Teixeira, the conditions in the latter experiments corresponded to churn-annular flow, whereas annular flow was prevailing in experiments performed by them. Fig. 1(a–c) shows profiles of the drop velocity and the Sauter mean diameter of drops measured for various flow conditions. The error bars shown for the velocity values correspond to the standard deviation of velocity fluctuations and thus are a measure of the local turbulence

(7)

where – gas friction velocity, d – drop diameter, δ – liquid film thickness, We – Weber number based on the gas friction velocity and the liquid film thickness, and A is a constant. Other correlations for drop size were proposed by Andreussi et al. (1978) and Ueda (1979). Azzopardi et al. (1980) accounted for the effect of liquid flow rate on the drop size due to coalescence (second term) and proposed the following equation,

ρg jg2 Dt / σ ,

⎛ Weλ ⎞ ⎝ ρg ⎠

+ 3.5

0.83 d32 σ G D 99 ⎞ ⎛ ⎞ ⎛ ρg ⎞ exp ⎜⎛0.6 le t + = 22 ⎜ ⎜ ⎟ ⎟ δ ρl ug d32 We ⎠ ρg Cfi ug2 δ ⎟ ⎝ ρl ⎠ ⎝ ⎝ ⎠

ug∗

0.6 Re0.1 d32 G g ⎛ ρg ⎞ = 1.91 + 0.4 le , ⎜ ⎟ We0.6 ⎝ ρl ⎠ Dt ρl jg

0.58

where Weλ = ρg jg2 λT / σ . Ambrosini et al. (1991) proposed the following equation,

0.5

∗2 d ⎛ ρg ug δ ⎞ = A We0.5 , m = A⎜ δ σ ⎟ ⎝ ⎠

15.4 ρl

(8)

Reg = ρg jg Dt / μg , jg – gas superficial velocity, Gle – 169

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

a pipe measured by Azzopardi and Teixeira (1994a,b). It can be seen that, to some extent, the measured data follow the 1/7-th shape valid for single phase flows in circular pipes: 1

y n u =⎛ ⎞ , uM ⎝R⎠

where n = 7 and y is the distance from the wall: y = R − r. For air-water two-phase flow the measured gas velocity profiles do not follow the same shape as those for single phase flow. Fig. 4 shows the velocity profile data obtained by Azzopardi and Teixeira (1994a,b) for various air-water flow conditions. It can be seen that the measured velocity values at r/R = 0.5 are significantly under the 1/7-th profile. The influence of the entrained drop phase on the turbulence intensity in the gas phase can be seen in Figs. 5 and 6. Fig. 5 shows typical axial velocity fluctuations observed in single-phase flows. The relative axial velocity fluctuations u′/u vary between 3 ÷ 5 percent at the centerline and they increase with the radius to about 6 ÷ 9 percent at r/ R = 0.69. The local values decrease with increasing gas Reynolds number. Fig. 6 shows velocity fluctuation intensities for annular two-phase flow. They vary from 6 ÷ 12 percent at the pipe centerline and increase with increasing radius to 10 ÷ 17 percent at radial distance r/R = 0.5. Fig. 6 indicates that the intensity of the axial velocity fluctuation is significantly increased for two-phase flow when compared to singlephase flow. Azzopardi and Teixeira concluded that the mean velocity and the Reynolds’ stresses in the gas appear to be similar to those in corresponding rough pipes. Thus single-phase and two-phase quantities should collapse to similar curves when properly normalized using corresponding friction velocities. However, turbulence intensities are higher than the values corresponding to flow over a wall with a roughness equivalent to the film interface. This disproportionately high increase of turbulence intensity must be attributed to presence of droplets. Indeed, it can be shown that the percent change in turbulence intensity over standard values (that is values characteristic to the single phase flow) increases with increasing rate of drop entrainment per unit area of film surface (Azzopardi and Teixeira, 1994b). New born drops at the film surface have relatively low axial velocity thus the relative velocity between gas and drops is large. When in addition the drop size is large enough, the drop Reynolds number will be high enough for vortex shedding from the drop to occur, which, in turn, will increase local turbulence intensity in the gas phase. Interactions between a particle and a fluid depend on the particle size in comparison to a most energetic eddy size in the fluid. When the particle is much smaller than the corresponding eddy length scale in the fluid, the particle will follow the eddy and effectively contribute to turbulence attenuation. On the contrary, larger particles will augment the turbulence. For solid particles the threshold value of the particle-to-

Fig. 1. Profiles of the droplet velocity and the Sauter mean diameter in annular air-water two phase flow in a vertical pipe with inner diameter D = 32 mm: (a) Gg = 24.5 kg/m2s, Gl = 15.9 kg/m2s; (b) Gg = 31.8 kg/m2s, Gl = 15.9 kg/m2s; (c) Gg = 31.8 kg/m2s, Gl = 31.7 kg/m2s (Azzopardi and Teixeira, 1994a,b).

intensity. A normalized standard deviation of velocity of each phase is shown in Fig. 2. The values of the standard deviation were calculated as,

σk =

[∑ (uk, j − u¯ k )2]/ n u¯ k

,

(14)

(13)

where k = (g,d) is the phase index for either gas or drop, u¯ k is the local mean velocity of phase k, uk,j is the velocity of the individual measurement j of the velocity and n is the number of measurements. As can be seen, the standard deviations decrease with the increasing gas superficial velocity and the standard deviation of drop velocity remains higher than the corresponding value of the gas phase in the whole range of investigated flow conditions. This type of behavior requires such modelling of the turbulence in the gas core that the observed trends are properly captured. 2.1.2. Continuous gas phase flow The parameters of interest for the continuous phase flow include gas velocity distribution, turbulence and interfacial interactions. These parameters have been investigated by several researchers, e.g. Gill et al. (1964), Gill and Hewitt (1968) and Azzopardi and Teixeira (1994a,b). The gas velocity profile in annular two-phase flow differs from that of the single-phase flow. Fig. 3 shows the velocity profiles for air flow in

Fig. 2. Variation of standard deviations of drop and gas velocity with the gas superficial velocity (Azzopardi and Teixeira, 1994a,b). 170

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

Fig. 3. Gas velocity in single-phase flow versus radial distance for various gas Reynolds numbers (Azzopardi and Teixeira, 1994a,b).

Fig. 6. Gas velocity fluctuations in two-phase annular flow versus radial distance for various flow conditions (Azzopardi and Teixeira, 1994a,b); C1: Gg = 24.5 kg/m2s, Gl = 15.9 kg/m2s; C2: Gg = 31.8 kg/m2s, Gl = 15.9 kg/m2s; C3: Gg = 31.8 kg/m2s, Gl = 31.7 kg/m2s; C4: Gg = 31.8 kg/m2s, Gl = 47.6 kg/ m2s; C5: Gg = 43.7 kg/m2s, Gl = 15.9 kg/m2s; C6: Gg = 55.6 kg/m2s, Gl = 15.9 kg/m2s.

Okawa et al., 2010; Yu and Wang, 2012; Ashwood et al., 2014; Key et al., 2014; Ju et al., 2015; Ito et al., 2016). The simplest model of annular two phase flow assumes existence of a liquid film and a gas core separated from each-other with a smooth interface. No mass transfer through the interface due to entrainment and deposition is considered. Such models can be solved analytically. In more practical cases the interface is wavy and substantial mass transfer through the interface takes place. Levy (1966) developed one of the first annular two-phase flow models applicable to such conditions. He considered a turbulent flow system in which the local shear stress is affected by the fact that the two-phase mixture density, once crossing the interface, is not constant. He showed that a non-dimensional parameter F ′ ∝ (τi/ ρg ug2 )1/2 correlates very well with a non-dimensional liquid film thickness δ / D . Wallis (1969) obtained a similar result and indicated that the F′ parameter derived in the Levy theory is equivalent to (Cfi/8)1/2 . Here Cfi is an interfacial friction factor defined as,

Fig. 4. Gas velocity in two-phase annular flow versus radial distance for various flow conditions (Azzopardi and Teixeira, 1994a,b): C1 – Gg = 24.5 kg/m2s, Gl = 15.9 kg/m2s; C2 – Gg = 31.8 kg/m2s, Gl = 15.9 kg/m2s; C3 – Gg = 31.8 kg/m2s, Gl = 31.7 kg/m2s; C4 – Gg = 31.8 kg/m2s, Gl = 47.6 kg/m2s; C5 – Gg = 43.7 kg/m2s, Gl = 15.9 kg/m2s.

Cfi =

2τi , ρg ug2

(15)

and τi is the interfacial shear stress. A correct estimation of this shear stress and thus of the interfacial friction factor has important implications on the liquid film flow. One of the important aspects of liquid film flow analysis is the relationship between the liquid film thickness and the liquid film flow rate. The two parameters are related to each-other with the following equation, representing the conservation of mass in the liquid film: Fig. 5. Gas velocity fluctuations for single-phase flow versus radial distance at various gas Reynolds numbers (Azzopardi and Teixeira, 1994a,b).

ΓF = ρl

∫0

δ

uF (y ) dy,

(16)

where ΓF is the film mass flowrate per unit length of channel perimeter, uF(y) is the velocity distribution in the film, y is a distance from the wall and δ is the local film thickness. The equation shows that if the velocity distribution in the film is known, the mass flow rate and the film thickness are uniquely related to each-other. The velocity distribution in the film can be obtained, assuming fullydeveloped flow conditions in the film with no mass transfer at the interface, from the following momentum conservation equation,

eddy size ratio is 0.1 (Gore and Crowe, 1989; Crowe, 2000); whereas for drops this threshold value is less than 0.1 (Azzopardi and Teixeira, 1994a,b). 2.2. Liquid film flow In annular flow, the liquid phase is split and flows partly as a film along walls, and partly as drops in the gas core. The distribution and interchange of the liquid between the film and the drops are important features of annular flow and determine both fluid mechanics and heat transfer phenomena. The phenomena related to the liquid film flow have been studied both experimentally and numerically (Coney, 1973; Thwaites et al., 1976; Mudawar and Houpt, 1993; Rosskamp et al., 1998; Moran et al., 2002; Adamsson and Anglart, 2005, 2006b; Gatapova and Kabov, 2008; Hazuku et al., 2008; Desoutter et al., 2009;

−

∂pF d ⎛ e duF ⎞ + μ + ρF gz = 0, ∂z dy ⎝ F dy ⎠ ⎜

⎟

(17)

where μFe is the effective dynamic viscosity of liquid in the film, pF is the pressure in the film, ρF is the density in the film and gz is the gravitational acceleration projected on the flow direction. The boundary conditions are as follows, 171

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

uF = 0 at y = 0,

duF = τi at y = δ . dy

μFe

(18)

Assuming laminar flow of liquid only in the film ( μFe = μl , ρF = ρl ), the velocity distribution is obtained as,

(− u (y ) = −

∂pF ∂z

F

)

+ ρl gz δ 2 2μl

2 ⎡ ⎛ y ⎞ − 2 ⎛ y ⎞ ⎤ + τi y. ⎢ δ μl ⎝ δ ⎠⎥ ⎣⎝ ⎠ ⎦

(19)

As expected, for laminar flow, the parabolic velocity profile exists in the liquid film, with zero velocity value at the wall and the maximum velocity value at the interface,

(− u = u (δ ) = i

F

∂pF ∂z

)

+ ρl gz δ 2 2μl

+

τi δ. μl

(20)

This equation gives a simple condition to be satisfied for a co-current annular two-phase flow, in which both ui > 0 and τi > 0 must be satisfied in a coordinate system oriented positively with the gas flow direction,

τi > −

1 ⎛ ∂pF − + ρl gz ⎞ δ . 2 ⎝ ∂z ⎠ ⎜

Fig. 7. Liquid film structure (after Hazuku et al. (2008)).

development of liquid film in a vertical upward annular two-phase flow. They used the laser focus displacement (LFD) meter to perform a detailed analysis of liquid film surface behavior under various flow conditions, including gas Reynolds number ranging from 31,800 to 98,300 and liquid Reynolds number ranging from 1050 to 9430. Measurements were performed for upward flow of air and water in a 3 m long pipe with 11 mm internal diameter, at atmospheric pressure and temperature 25 °C. To be able to discern the disturbance wave effect on the liquid film thickness, Hazuku et al. introduced several definitions of the liquid film thickness, as illustrated in Fig. 7. Based on this, a disturbance wave is defined as a wave whose thickness at both ends is smaller than the average film thickness and the maximum thickness is greater than the average thickness of the disturbance wave layer. Also, the minimum and maximum film thicknesses were calculated to represent 1% and 99% probability level, respectively. Hazuku et al. proposed the following correlation for the minimum film thickness,

⎟

(21)

The above equation indicates that for a fixed (and negative) value of −∂pF / ∂z + ρl gz the minimum interfacial shear stress is linearly increasing with the film thickness. A simplified force balance in the core region (here again the interfacial mass transfer is neglected) gives,

−

∂pC P = C τi − ρC gz . AC ∂z

(22)

Here pC is the pressure in the gas core, ρC is the density in the gas core, AC is the gas core flow area and PC is the gas core friction perimeter. Assuming annular flow in a circular pipe, we have,

PC / AC ≈ 4/(D − 2δ ).

(23)

Further, assuming pC = pF = p and ρC = ρg , we can derive a relationship between the liquid film mass flow rate, the interfacial shear stress and the film thickness as follows,

ΓF = −

ρl (ρl − ρg ) gz δ 3 3μl

4δ 3 δ 2 ⎤ ρl τi +⎡ + . ⎢ 3( D − 2 δ ) 2⎥ ⎣ ⎦ μl

1

∗ δmin = 0.977Rel0.143 τi∗− 0.117, τi∗ =

(24)

τi ⎛ g ⎞ 3 ∗ ⎛ g ⎟⎞. ⎜ ⎟ , δmin = δ min ⎜ 2 ρl g ⎝ νl2 ⎠ ⎝ νl ⎠

(25)

The correlation agreed with their data within ± 5% and for Reg = 31800 ÷ 98300, Rel = 1050 ÷ 9430 and z/D = 50 ÷ 250. Ju et al. (2015) used experimental data obtained by Sawant (2008), Whalley et al. (1973) and Fukano and Furukawa (1998) to derive a correlation for a liquid film thickness in vertical upward co-current adiabatic annular flow in a pipe. Using the transition criteria to annular flow for the derivation of the maximum liquid film thickness, they arrived at the following expression for the liquid film thickness,

This equation shows that for a given film thickness the film mass flow rate is still not determined uniquely and it is a function of the interfacial shear stress. For example, it is possible to conceive such a situation that two liquid films of the same thickness will have different mass flow rates because of different shear stress values. Such behavior is particularly important in channels with complex shapes, where local shear stress varies and influences the relationship between the film flow rate and the film thickness. A similar analysis can be performed for turbulent flow in the film; however, for that case no general closedform relationship can be obtained due to unknown velocity distribution in the film. Nevertheless, both for laminar and for turbulent follows, it is important to independently calculate the local interfacial shear stress in order to close the relationship between the film mass flow rate and the film thickness. In addition, for turbulent flow in the liquid film, the relationship between the mass flow rate and the film thickness is affected by the turbulence intensity. Some experimental results on liquid film flows are presented in the following sections.

δ = tanh(14.22Wel0.24 We″g − 0.47 N 0.21 μl ), δmax

(26)

where non-dimensional numbers are defined as,

Wel =

ρl jl2 D σ

1

, We″g =

ρg jg2 D ⎛ Δρ ⎞ 4 , Nμl = σ ⎜ ρg ⎟ ⎝ ⎠

μl ρl σ

σ g Δρ

. (27)

Based on the experimental data, the maximum film thickness is taken as δmax = 0.071D , where D is the pipe diameter. Ito et al. (2016) used sensors based on the electrical conductance method to measure liquid film behavior on walls in square-lattice and tight-lattice subchannels. The liquid film sensor had a network of electrodes on the surface and the electrical conductance between transmitter and receiver electrodes was detected by using a high-speed wire-mesh measuring system. The method allowed for observing the flow structure of the liquid film, including the spatial distribution of the

2.2.1. Liquid film thickness An average liquid film thickness has to be determined in a mechanistic annular two-phase flow model. Its value is important for such phenomena as the onset of dryout. For this reason, liquid film thickness has been intensively investigated both experimentally and analytically. Hazuku et al. (2008) performed an experimental study on axial 172

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

film thickness. It was observed that under all investigated flow conditions the time-averaged liquid film thickness was non-uniformly distributed along a rod circumference and consistently the thinnest film was observed towards open subchannel region. The film was increasing and getting a local maximum once moving towards the narrow gap between the neighboring rods. Shedd and Newell (2004) measured liquid film thickness for air and water annular two-phase flow through horizontal round, square and triangle tubes. They observed that in the square and triangle tubes the liquid film was thinned in the corners, indicating an effect of turbulent secondary flows in these geometries. They noted that the film tended to follow curves of constant velocity (isotachs) in the gas, for which secondary flows are responsible. However, the whole picture is not that clear since the liquid film roughness may influence the secondary flows in two-phase annular flow and in this way it may indirectly influence the liquid film thickness as well.

Here y+ = yul∗/ νl and ul∗ = τw / ρl . It has been shown by many researchers (Vassallo, 1999; Ashwood et al., 2014) that single-phase theory does not apply to liquid films. Even though similar three regions can be confirmed, the velocity profile seems to be altered by the action of disturbance waves and ripples at the interface. In case of boiling heat transfer, presence of bubbles is affecting the velocity profile as well. Based on their own data, Ashwood et al. (2014) proposed the following equations to calculate the velocity distribution in a liquid film,

u+

+ y+ < 5 for ⎧y + = 7.2 ln y − 6.6 for 5 ≤ y+ ≤ 30 , ⎨ + 30 < y+ ⎩ 7.38 ln y − 7.1 for

(29)

where u+ = ul / ul∗. 2.3. Interfacial wave propagation

2.2.2. Liquid film flow rate As discussed above, liquid film flow rate is intimately coupled to the liquid film thickness and both quantities are often investigated together. The relationship between the two is not unique, however, since it is influenced by the interfacial shear stress and the velocity distribution within the film. Several investigations have been performed in tubes and annuli. It has been observed that in concentric annuli the liquid film flow rate per unit perimeter is much higher for the tube wall than for the rod wall. This was observed by Moeck (1970) for both diabatic and adiabatic cases. Similar asymmetry was observed by Mannov (1973), Jensen and Mannov (1974) and Würtz (1978). For eccentric annuli the situation is somewhat unclear: Schraub et al. (1969) and Andersen et al. (1974) observed a minimum film flow rate at the minimum gap. In contrast, Butterworth (1968) measured uniform film flow distribution in an eccentric annulus and found the thickest film at the narrow gap. These results are in general agreement with those reported by Ito et al. (2016), who measured an increasing film thickness towards the narrow gaps in square and triangular rod lattices. However, since interfacial shear stress is not uniform in such irregular channels, the local film thickness and the corresponding film flow rate can have different profiles from each other. Experiments performed by Shedd and Newell (2004) in comparison with previous results obtained by Fukano et al. (1984) shed some light on the relation between the film thickness and the film flow rate in noncircular tubes. The non-uniform liquid film flow rate distribution observed by Fukano et al. was confirmed by a qualitatively similar liquid film thickness distribution obtained by Shedd and Newell. Quantitative comparison is not possible, since both geometries were not exactly the same. Nevertheless, Shedd and Newall suggest that the secondary flow in the gas carries axial momentum from the center of the tube into the corners. This increased momentum will contribute to increased interfacial shear in the corner regions versus central regions at the same distance from the wall, which could act to thin the liquid film in the corners. As the film thins in the corner with respect to the center, the interfacial shear will tend to approach a constant value across the side.

Waves appearing on the liquid film surface play important role in mass, momentum and energy heat transfer between the film and the gas core. Their nature has been investigated by many researchers both experimentally and numerically (Hall Taylor et al., 1963, 2014; Nedderman and Shearer, 1963; Hall Taylor and Nedderman, 1968; Azzopardi, 1986; Wolf et al., 1996; Inada et al., 2004; Sawant et al., 2008; Belt et al., 2010; Schubring et al., 2010; Zhao et al., 2013; Alekseenko et al., 2014, 2015; Pan et al., 2015; Yang et al., 2016). A wave that appears to roll over the surface of a liquid is called a “roll wave”. It is specifically a type of surface wave that occurs as one of a series of such waves separated by calm liquid and moving faster than the bulk liquid. Roll waves are quasi-periodic waves of finite amplitude generated in a channel due to instability of the uniform steady-state flow. They typically develop in inclined channels, when the angle of inclination exceeds a certain critical value. In annular vertical flow interfacial waves are always present. For low liquid flow rates only small ripple waves are observed. With increasing liquid flow rate larger disturbance waves appear. For a given gas mass flow rate it is possible to observe a specific value of the liquid flow rate, when transition from the ripple to disturbance waves occurs. This transition (sometimes called disturbance wave inception) is, however, more sensitive to the gas flow rate than to the liquid flow rate. With increasing liquid flow rates, the transition depends on the gas flow rate, but is insensitive to the liquid flow rate. The transition manifests itself with appearance of a “pulse” region (Hall Taylor et al., 1963), in which disturbance waves do occur but do not form continuously and appear in groups or “pulses”. Sekoguchi et al. (1985a,b) and Sekoguchi and Takeishi (1989) reported another type of waves, with amplitudes comparable to the disturbance waves, but lacking axial coherence, which is characteristic for the disturbance waves. These are the so-called ephemeral waves and were identified by Wolf et al. (1996) as developing disturbance waves. Hall Taylor and Nedderman (1968) investigated the influence of several parameters on the disturbance wave inception. Their and others results indicate that the pipe diameter has very small (if any) influence on the inception of disturbance waves. That conclusion is drawn based on air-water annular flows at atmospheric pressure in pipes with diameter in a range from 15.8 to 31.8 mm. The same authors also investigated the influence of liquid viscosity and surface tension on the wave inception. Only viscosity showed a strong influence. It can be observed that with increasing viscosity, the liquid Reynolds number at inception clearly decreases, once keeping the gas Reynolds number constant. Thus highly viscous liquids tend to easier develop disturbance waves, when their liquid Reynolds number (defined as Re = Q/ν, where Q – volumetric flow rate per unit wetted perimeter, ν – kinematic viscosity) is still low. Azzopardi et al. (1983) suggested that the appropriate dimensionless groups to represent data on the disturbance wave inception are the

2.2.3. Velocity profile and turbulence A unique relationship between the liquid film thickness and the liquid film flow rate can be obtained only when the velocity distribution in the film is known. It is common to apply the law-of-the wall formulation for single phase flows to predict velocity distribution in the film. For wall bounded single-phase flow three regions are usually distinguished,

viscous region: y+ < 5, transition region: 5 ≤ log region:y+

> 30.

y+

(28a)

≤ 30,

(28b) (28c) 173

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

Sekoguchi et al. (1985a,b) proposed the following correlation to calculate the disturbance wave frequency,

liquid Reynolds number Rel = Gl Dh / μl , the Onhenzoge number On = μl / σρl Dh , the dimensionless gas velocity, and a ratio of the Weber number and gas Reynolds number: We/Reg = μg jg / σ . Here Gl is the liquid mass flux, Dh is the hydraulic (or pipe) diameter, jg – the superficial gas velocity, μl , μg – the liquid and gas viscosities respectively, ρl – the liquid density and σ – the surface tension. There are various theories to explain the disturbance wave inception:

Str ≡ fDW D / jg =

•

and Gromles (1975) suggested that the phenomenon responsible for the start of entrainment is the penetration of the boundary layer of the gas and there are two asymptotic limits of gas and liquid flow rates: o minimum limit gas flow rate when the liquid film is increasing to infinity; o minimum limit liquid flow rate when the gas velocity is increasing to infinity. Abolfadl (1984) suggested that the onset of entrainment is linked to the onset of turbulence in the film, which occurs when the film Reynolds number exceeds 268. Asali et al. (1985) analyzed the stability of disturbance waves on the interface to conclude that the critical liquid Reynolds number for their inception depended on the group (μl / μg ) ρg / ρl .

C=

u¯ i + u¯ g ρg / ρl 1+

ρg / ρl

K jg + jl 1+K

,

(32)

where 1

1

ρg 2 Re 4 K = a ⎜⎛ ⎟⎞ ⎜⎛ l ⎟⎞ . ⎝ ρl ⎠ ⎝ Reg ⎠

The frequency and velocity of disturbance waves can be correlated to some flow parameters. Based on experimental observations (Hall Taylor and Nedderman, 1968), velocities of individual waves are found normally distributed about the mean with a standard deviation that is independent of the mean velocity. The rate of change of wave frequency with distance along the tube is predicted from the velocity distribution. Azzopardi (1986) studied disturbance wave frequencies, velocities and spacing in vertical annular two-phase flow. He employed a more objective method to determine wave frequency based on the power spectral density function of the film thickness recorded. If the disturbance waves are the most dominant feature of the flow then the disturbance wave frequency manifests itself by the peak in the power spectral density function. The measured disturbance wave frequency was shown to agree reasonably well with results obtained by other workers, who used similar (objective) methods. However, results obtained by workers who used manual method to determine frequencies were approximately higher by 50%. The general observed trend was that for a given constant gas mass flux, the frequency increases with increasing liquid mass flux, getting some asymptotic limit value for high liquid mass fluxes. For the cases studied (air-water at atmospheric pressure, Gg = 39.7 kg/m2s, Gl = 10–150 kg/m2s and D = 38.1 mm) the frequencies were between 5 and 10 Hz. It was also shown that the velocity of disturbance waves increase with both gas and liquid mass flux. The agreement with results obtained by other workers (who were using objective methods) was good. It was shown that a correlation suggested by Pearce (1979) is well representing the obtained results:

C=

(31)

where Eo = gD 2 (ρl − ρg )/ σ and Fr = jg / gD . Sawant et al. (2008) investigated properties of disturbance waves in vertical annular two-phase flow. Air-water experiments have been conducted at three different pressures (1.2, 4.0 and 5.8 bars) in a tube with 9.4 mm inner diameter. The liquid film thickness was measured from which the disturbance wave velocity, frequency, amplitude and wavelength could be calculated. It was concluded that two non-dimensional numbers: the liquid Reynolds number and the Weber number are enough to satisfactorily predict the dependence of the disturbance wave velocity and the amplitude on pressure, gas phase flow rate and liquid phase flow rate. The measured disturbance wave velocity could be reasonably well predicted with the Kumar et al. (2002) equation,

• Ishii

•

Re2.5 0.5 ln Eo − 0.47 ⎡ ⎤ 0.0076 ln ⎛⎜ l ⎞⎟ − 0.051⎥, ⎢ 0.5 Eo Fr g ⎝ ⎠ ⎣ ⎦

(33)

Sawant et al. suggested that coefficient a should be 9.0 in order to fit their own experimental data. This value differs from the one given by Kumar et al. (2002) and equal to 5.5. The disturbance wave frequency was proposed to be calculated as, −0.64

⎛ρ ⎞ Str = 0.086Rel0.27 ⎜ l ⎟ ρ ⎝ g⎠

, (34)

where the Strouhal number is defined as Str = fDt / jg and f is the disturbance wave frequency. 2.4. Shear stress and pressure drop Shear stresses on wall and on liquid film interface, as well as pressure gradients in the channel are influencing other annular two-phase flow parameters, such as the liquid film thickness. Clearly, these parameters play important role in over-all annular flow behavior and due to that they attracted attention of many researchers (Shearer and Nedderman, 1965; Wallis, 1969; Chisholm, 1973; Govan et al., 1989; Fukano and Furukawa, 1998; Fore et al., 2000; Pan et al., 2015). A short description of some findings is provided in the following subsections. 2.4.1. Wall shear stress Experimental observations of annular two-phase flow indicate that the wall friction factor is almost the same function of the liquid Reynolds number as it is for single phase flow. The liquid Reynolds number is defined as:

, (30)

where C is the disturbance wave velocity, u¯ i is the film-core interface velocity at mean conditions, u¯ g is the mean gas velocity and ρg , ρl are gas and liquid densities, respectively. Mean wave spacing, defined as the ratio of the wave velocity and the frequency, is shown to decrease with increasing liquid mass flux, when the gas mass flux is kept constant. This behavior can be explained in terms of the coalescence of waves. Wave spacing has been related to the mean film thickness. It is shown that the wave spacing increases with the mean film thickness. This increase is quite rapid for air-water cases, but less rapid for steam-water at 70 bars. Azzopardi (1986) concludes that the relationship between wave spacing and film thickness is not unique.

Rel =

ρl jl D , μl

(35)

and thus the Fanning friction factor is given as:

Cfw =

16 , Rel

(36)

for laminar flow, and as:

Cfw =

0.079 , Re1/4 l

(37)

for turbulent flow, when using the Blasius correlation. It should be noted that the definition of the liquid Reynolds number 174

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

is valid for one-dimensional flow, where superficial liquid velocity includes both the liquid film and the entrained liquid. If the entrained fraction is negligible, the liquid Reynolds number is equal to the Reynolds number of the liquid film,

4Γ ReF = F , μl

2011; Dasgupta et al., 2015; Erkan et al., 2015; Aliyu et al., 2017; Liu and Bai, 2017). The most important findings are reported below. 2.5.1. Drop deposition The entrained liquid is carried by the turbulent gas stream as droplets, which will deposit on bounding walls. The mass rate of deposition per unit area, D, called further the deposition rate, is one of the important quantities in annular two-phase flows, which needs to be measured and predicted. Throughout the years three types of experimental techniques have been developed to investigate deposition rates:

(38)

where ΓF is the film flow rate per unit perimeter. 2.4.2. Interfacial shear stress One can expect that the interfacial shear stress in annular two-phase flow will depend on a difference between the gas velocity and the liquid film velocity, taking their characteristic mean values. Since in most practical cases the gas velocity is much larger than the liquid velocity, the latter can be neglected. In that way the interfacial shear stress can be expressed in terms of the gas velocity only, which greatly simplifies the analysis. Adopting this assumption, the interfacial shear stress is as follows,

τi = Cfi

ρg u¯ g2 2

.

• Injection of droplets into a gas stream and studying their deposition on the wall. • Introduction of liquid as an annulus through a porous section of the wall of a duct. • Injection of dye into the wall layer and measuring the change of the dye concentration due to deposition of droplets.

Azzopardi (1997) analyzed these various methods and concluded that the tracer interchange technique provides best measurements of deposition. Two general types of expressions have been used to correlate the deposition rate. The most common procedure is to use the concentration of the entrained liquid in the unit of mass/volume, C, as the driving force:

(39)

Here Cfi is the Fanning friction factor, which was shown (Wallis, 1969) to be a linear function of the non-dimensional film thickness,

δ Cfi = Cfw ⎛1 + 300 ⎞, D⎠ ⎝

(40)

where Cfw is the wall shear stress (note that Cfi → Cfw ) and coefficient

D = kC.

δ→0

300 results from fitting of the above equation to experimental data. An interesting point is that this equation is equivalent to Nikuradse’s and Moody’s equations to account for the influence of sand roughness on the friction factor, assuming that the sand roughness is equivalent to four times the film thickness (Wallis, 1969).

Here k is a deposition coefficient that needs to be determined. Another approach, less frequently employed, treats the deposition rate as a fraction of the entrained liquid mass flow rate per unit channel perimeter P:

D = k′

2.4.3. Pressure drop Experiments indicate that a pressure gradient in annular two-phase flows increases with both gas and liquid mass flow rates. Shearer and Nedderman (1965) measured pressure gradient and liquid film thickness in co-current upwards annular flow. They concluded that there is a universal relationship between the film thickness and the apparent wall roughness,

δ+ = α

ε + β, D

Wd , P

(43)

where k′ has an interpretation as the fraction of the liquid deposited in a unit length of pipe. Both coefficients k and k′ are related to each other, but they are different. One should note that k′ is dimensionless (as a fraction), whereas k has a dimension of m/s. 2.5.2. Drop entrainment Droplets in annular flow are created from the wall film by the action of the gas flowing over it. The droplets are not created over the entire film interface but rather arise from the disturbance waves. Azzopardi and Whalley (1980) investigated annular flow in which the film flow rate was just below a threshold that is required for disturbance wave creation and suddenly injected a small volume of liquid into the film, such that a small number of waves were created. The flow at the top was observed by means of a high-speed camera. They observed no droplets before the waves were created. However, as soon as liquid was injected to film, droplets could be seen and their number was increasing when the waves approached the end of the pipe. Once the waves had passed out of the pipe, no drops were observed. A mechanistic model of entrainment rate has to take this physical picture into account (Liu and Bai, 2017).

(41)

= τi/ ρl is the is a dimensionless film thickness, where δ + = friction velocity based on interfacial shear stress τi , ε is the wall roughness, α is a function of the gas mass flow rate and liquid viscosity, and β is a function of tube diameter and surface tension. The liquid film in annular flow acts as a rough wall and thus is affecting the frictional component of the pressure drop. The roughness depends on the flow rate of the liquid in the film. Apart from this, there are contributions to pressure drop from the acceleration of droplets from their initial low velocities, close to the liquid film velocity, towards that of the gas velocity. In addition, there exists a contribution from the momentum transferred when drops deposit.

δui∗/ νl

(42)

ui∗

2.5. Deposition and entrainment rates 3. Heat transfer to annular two-phase flow Deposition and entrainment rates play important role for prediction of the liquid mass flow rate and thus are important for an accurate prediction of the onset of dryout. Both experimental and analytical works have been performed in this area (Andersson and Mantzouranis, 1960; Ishii and Grolmes, 1975; McCoy and Hanratty, 1977; Andreussi and Azzopardi, 1983; Andreussi, 1983; Lee et al., 1989; Binder and Hanratty, 1991; Lopez de Bertodano et al., 1997; Assad et al., 1998; Samenfink et al., 1999; Holowach et al., 2002, 2003; Okawa et al., 2002; Nikolopoulos et al., 2005; Okawa et al., 2005; Ryu and Park,

Heat transfer in annular two-phase flow is changing dramatically at locations where the liquid film breaks up and the heated wall is no longer covered with the continuous liquid layer. The phenomenon of a liquid film break-up and a sudden heat transfer mode change from essentially boiling heat transfer to liquid film to boiling heat transfer to vapor-droplet mixture is called dryout. Thus, a model of heat transfer to annular two-phase flow has to address all these complex phenomena. The pre-dryout heat transfer, onset of dryout and post-dryout heat 175

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

of mass, momentum and energy for phase k,

transfer have been investigated by many researchers (Xie et al., 2017; Hartley and Murgatroyd, 1964; Hewitt and Lacey, 1965; Hewitt et al., 1965; Whalley, 1977; Fujita and Ueda, 1978a,b, 1978; France et al., 1982; Ryley and Xu, 1983; Milashenko et al., 1989; Azzopardi, 1996; Carvalho et al., 2002; Fukano et al., 2002, 2003; Okawa et al., 2009; Wu et al., 2009; Gong et al., 2011; Konishi et al., 2013; Adamsson and Le Corre, 2014), providing great insight into the heat transfer mechanisms in annular flow.

∂ (αk ρk ) + ∇ (αk ρk vk) = Γk ∂t

(45)

∂ (αk ρk vk ) + ∇ ·(αk ρk vk vk) = −αk ∇p + ∇ ·(αk τ ) + αk ρk g + Γk vki + Mki ∂t (46)

3.1. Pre-dryout heat transfer

∂ (αk ρk hk ) + ∇ (αk ρk hk vk) = −∇ ·(αk q″) + Γk hki + qki ∂t

In pre-dryout heat transfer regime wall temperature can be determined from standard law-of-the-wall equations implemented in industrial CFD codes (Menter and Esch, 2001). However, a modified treatment of heat transfer has to be adopted for post-dryout heat transfer, as described in the following sections.

where the subscript k represents a phase index, which can be either the gas or the liquid, αk is the volume fraction of phase k, Γk is the mass source gained by phase k, vki is the interfacial velocity, Mki is the interfacial momentum transfer, hki is the interfacial enthalpy, and qki is the interfacial heat transfer.

3.2. Onset of dryout

4.1.1.2. Particle tracking model (Eulerian-Lagrangian approach). A unified annular flow model with Eulerian-Lagrangian approach to model the gas core has been developed by Li and Anglart (2015). In this approach the gas phase is treated as a continuum and RANS (Reynolds-Averaged Navier-Stokes) equations are solved to find the velocity and pressure field in the core, while the drops are solved by the Lagrangian Particle Tracking (LPT) method. The drops can exchange mass, momentum and energy with the gas phase. A two-way coupling between gas and drops is employed, thus the influence of drops on the gas velocity distribution is considered. Currently drop-drop interactions are not taken into account, thus the coalescence and breakup of drops is not included. The governing equations for the continuous (gas) phase are as follows:

If a stable dry patch forms on a heated wall, the onset of dryout takes place. There are variety of approaches to determine the stable dry patch conditions, based either on the minimum total energy of the liquid film (Hartley and Murgatroyd, 1964; Ryley and Xu, 1983) or on a force balance on the leading edge of the film (Anglart, 2014b). From these models, the minimum (or “critical”) liquid film thickness can be determined. Thus, the local dryout criterion is as follows,

δ < δc ,

(47)

(44)

where δc is the critical film thickness. A more detailed description of the dryout criterion used in the current model is given by Li and Anglart (2016b).

∂ρg

+ ∇ ·(ρg vg) = Γg ,

3.3. Post-dryout heat transfer

∂t

In post-dryout heat transfer regime, liquid film does not exist any longer on the heated wall. In the current model it is assumed that the heat flux is partitioned into several contributions, representing the wallgas convective heat transfer, wall-droplet heat transfer and thermal radiation heat transfer to the gas and droplets (Li and Anglart, 2016b).

∂ (ρg vg )

4. CFD model formulation

where Γg is the mass source gained by the gas phase from evaporation of the liquid film, Uf is the thickness-averaged liquid film velocity, τ is the effective stress due to both molecular and turbulent effects, Md is the momentum source from the gas-drop interaction due to the two-way coupling, q″ is the effective heat flux including both molecular and turbulent effects, hf is the thickness-averaged enthalpy of the liquid film, and qd is the heat source in the gas phase resulting from the gasdrop heat transfer. The effective stress tensor τ consists of two components, the molecular component and the turbulent shear stress, which is based on the Boussinesq approximation. The turbulent viscosity is determined from a suitable model, such as either standard k-ε model (Jones and Launder, 1972) or the SST turbulence model (Menter, 1993). The droplet motion is tracked individually according to the motion equation as:

∂t

∂ (ρg hg ) ∂t

The CFD model is formulated separately for the gas core and the liquid film, with proper boundary conditions at the interface for the conservation of mass, momentum and energy. The gas core is treated as fully three-dimensional domain, whereas the conservation equations in the liquid film domain are averaged in a layer distributed along the channel walls. 4.1. Governing equations Governing equations are formulated separately for each of the domains. A two-phase flow is assumed to prevail in the gas core, whereas single-phase liquid flow is considered for the liquid film domain. 4.1.1. Gas core Two different formulations are available for the gas core. The first one is based on ensemble averaging of the flow field, resulting with two sets of mass, momentum and energy conservation equation for each of the phases. The gas phase is continuous, whereas the liquid phase is mono-dispersed and is characterized by a specific mean drop size, where the drop size is calculated from equations presented in Section 2.1.1.

(48)

+ ∇ ·(ρg vg vg) = −∇p + ∇ ·τ + ρg g + Γg Uf + Md ,

(49)

+ ∇ ·(ρg hg vg) = −∇ ·q″ + Γg hf + qd ,

(50)

dxd = vd , dt

(51)

where x d is the droplet position, and vd is the droplet velocity, which is calculated based on the droplet momentum equation as:

md

d vd = F = FD + FL + md g dt

(52)

Here F is the total force acting between gas and a drop, md is the drop mass and g is the gravity acceleration vector. Force F is often decomposed into two components: the drag force FD and the lift force FL.

4.1.1.1. Two-fluid model for gas core (Eulerian-Eulerian approach). Li and Anglart (2015) used the following set of equations for conservation 176

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

4.1.2. Liquid film The liquid film, especially that in the upstream of the dryout point, is sufficiently thin to safely make the following major thin-film assumptions:

Sδ, ent = −ke ρl

ϕdy,

kd

∂ (ρl δ U) + ∇s (ρl δ UU) = −∇s p + SU , ∂t

(55)

∂ (ρl δh) + ∇s (ρl δhU) = Sh, ∂t

(56)

ρg Dh σ

−0.5

⎛C ⎞ = 0.0632 ⎜ ⎟ ρ ⎝ g⎠

. (59)

Finally, the total mass source term for the liquid film is given as,

Sδ = Sδ, evp + Sδ, ent + Sδ, dep.

(60)

4.2.2. Mass transfer at the droplet-gas interface When dispersed drops are surrounded by a gas at temperature higher than the saturation temperature, evaporation of drops takes place. Assuming saturated conditions for a drop, the mass transfer rate can be found from the heat transfer rate at the drop surface. The RanzMarshall model of evaporation from drops is frequently used in calculations (Ranz and Marshall, 1952). However, to account for the effect of the evaporation on heat transfer, the model proposed by Renksizbulut and Yuen (1983) is used in the present formulation.

where δ is the film thickness, ϕ is any liquid film property variable, and y is the coordinate for wall normal direction. For simplicity, the bar is omitted for all the depth-averaged liquid film properties used in the following description. Then the mass, momentum, and energy equations are integrated in the wall normal direction as (Li and Anglart, 2015):

(54)

(58)

where C is the droplet concentration in the gas and kd is the deposition coefficient, calculated as,

(53)

∂ (ρl δ ) + ∇s (ρl δ U) = Sδ , ∂t

(57)

Sδ, dep = kd C,

These assumptions imply that the advection can be treated in the wall tangential direction and diffusion in the wall normal direction. As a result, the transport equations for the liquid film can be integrated in the wall normal direction to obtain the two-dimensional equations. All the liquid film properties, which vary across the film thickness, appear as depth-averaged quantities and are in general defined as:

∫0

,

where jg is the gas volumetric flux, ke = 4.79 ∙ 10 m/s and Cfi is the interfacial friction factor. The mass transfer due to deposition can be calculated from the equation given by Okawa et al. (2003) as follows,

wall surface are negligible compared to those in the wall normal direction.

1 ϕ¯ = δ

0.111

−4

• the flow in the wall normal direction can be reasonably assumed to be negligible, the • spatial gradients of the dependent variables tangential to the

δ

Cfi ρg jg2 δ ⎛ ρl ⎞ ⎜ρ ⎟ σ ⎝ g⎠

4.2.3. Momentum transfer at the film-gas interface Momentum transfer term at the film-gas interface is split into the normal-to-wall and the tangential-to-wall components. The normal component results from several sources, such as the hydrostatic pressure, capillary pressure, vapor recoil pressure, drop deposition and drop entrainment (Li and Anglart, 2015). For the tangential component, the following contributions are included: gravity force, shear stress, thermocapillary force, contact angle force, drop deposition and drop entrainment. The hydrostatic pressure is given as,

where U is the mean film velocity, h is the mean film enthalpy, ∇s is the nabla operator tangential to the surface, ρ is the density, p is the total pressure, and Sδ, SU and Sh are the source terms of mass, momentum and energy, respectively. The corresponding source terms could originate from the interface between the gas core and the liquid film, e.g., the interfacial shear stress, and from the wall surface, e.g., the wall shear stress. This means that all the source terms are considered in the boundary cells facing the wall and the liquid film surface. It is noted that the advection for all the equations is explicitly described, however, the diffusion and the external sources are modeled as source terms. The liquid film has complex interaction with the gas core flow, which means that corresponding models should be included as source terms to consider all the phenomena of concern.

pδ = −ρl δn ·g ,

(61)

where n is the surface-normal vector, g is gravity vector and δ is the film thickness. The capillary pressure results from the interface curvature and can be found as,

pσ = −σ ∇2s δ,

(62)

where σ is the surface tension and is an approximation of the local curvature of the liquid film. The vapor recoil pressure is calculated as,

∇2s δ

pevp = 4.2. Closure relationships

Γ2evp 2ρg

, (63)

where Γevp is the evaporation rate of the liquid film. The deposition and entrainment pressure effects are as follows, respectively:

To be solved, the governing equations presented in the previous sections require additional closure relationships for source terms. These closure relationships are discussed in the following sections.

pdep = Γdep (vd ·n), pent = Γent Ufn,

(64)

where vd is the droplet velocity and Ufn is the entrained droplet velocity normal to the surface. The wall-parallel component of the gravity force is given as,

4.2.1. Mass transfer at the film-gas interface There are three main mass transfer terms that have to be considered: the evaporation of the liquid film, the entrainment of drops from the crests of disturbance waves and the deposition of drops. The evaporation rate Sδ, evp is calculated from the energy balance for the liquid film assuming a local thermodynamic equilibrium, whereas the deposition and entrainment rates can be found from proper correlations. For example, using the Okawa et al. (2003) model, the entrainment rate is given as,

SU , δ = ρl δ gt .

(65)

The shear stress contains two contributions: one due to the wall friction and one due to the interfacial friction. The latter is given as,

SU , τi = 177

1 Cfi ρg |vg − Uf |(vg − Uf ), 2

(66)

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

Three curves are shown in the figure: a horizontal line, which shows the evaporation rate of the liquid film, a monotonically decreasing curve which shows the calculated entrainment rate using the EulerianEulerian approach, and fluctuating line representing the calculated instantaneous deposition rate using the LPT method and the droplet size found from a correlation proposed by Caraghiaur and Anglart (2013). With dots, a deposition rate obtained from the Okawa correlation is shown for comparison. A good agreement between the deposition rate obtained from the LPT calculations and the correlation can be noticed. This confirms that the implemented deposition model produces reasonable results. The high value of the entrainment rate at the pipe inlet results from inlet conditions, which are far from the developed flow conditions.

where Cfi is the interfacial friction factor. The shear stress caused by drop deposition and entrainment is found as, respectively,

SU , dep = Γdep vdt , SU , ent = Γent Uft ,

(67)

where vdt is the droplet velocity vector tangential to the surface, and Uft is the entrained liquid film velocity vector tangential to the surface. 4.2.4. Energy transfer at the film-gas interface It is assumed that the heat flux applied to a wall covered with a liquid film is causing the film evaporation. For saturated conditions, the relationship between the mass transfer and the energy transfer at the film interface is as follows:

Sh, vap = Sδ, vap hfg ,

(68)

5.2. Comparison with measurements

where hfg is the latent heat. In most cases the wall heat flux q″w is known and thus:

Sh, vap = q″w .

Fig. 11 shows the measured and calculated film flow rate for watersteam annular two-phase flow in a pipe with 14 mm internal diameter. The operational conditions are as follows: mass flux G = 750 kg/m2s, system pressure at exit p = 7 MPa, inlet subcooling 10 K, exit thermodynamic quality 0.75 and uniformly distributed heat flux in a 3.65 m long pipe. The figure reveals that the current model is not very sensitive to the choice of closure relationship for the interfacial mass transfer on the film surface. Both the Okawa et al. (2003) and the Hewitt and Govan (1990) model give a reasonably good agreement with experimental data. The model developed by Li and Anglart (2015, 2016a–c) is applicable to prediction of the onset of dryout and post-dryout heat transfer. Fig. 12 shows a comparison of predicted and measured wall temperature in a 14.9 mm inner diameter pipe. The measured data shown with dots were obtained by Becker et al. (1983) and the three curves were obtained with the current model, using three different closure-relationship options for the interfacial mass transfer rates, but applying the same dryout criterion based on the critical film thickness, as described in Section 3.2. Unlike for the liquid film flow rate, the location of the dryout point is quite sensitive to the choice of the interfacial mass transfer model. The results presented in the figure suggest that for the tested models for deposition and entrainment rates, the calculated dryout point moves up to 0.5 m upstream of the measured dryout point. A probable reason for the observed sensitivity is that the investigated channel is quite long (the test section length in experiments was 7 m) and a relatively small discrepancy in the local mass transfer rate may have significant impact on liquid film flow rate over a long distance. These results indicate a possible future development of the model and, in particular, suggest that more accurate, preferably locally-based closure relationships for the interfacial mass transfer terms on the film surface are required. The experimental temperature distribution downstream of the dryout point has been correctly captured by the model as far as its overall level is concerned. However, the predicted temperature shape is not following the trends observed in the experiment. The model clearly over-estimates the temperature just downstream of the dryout point.

(69)

Combining the above two equations, the interfacial mass transfer can be found. The energy transfer due to entrainment and deposition rates can be found as follows:

Sh, dep = Sδ, dep hd ,

(70)

where Sδ, dep is the deposition rate and hd is the drop specific enthalpy. For the energy transfer due to entrainment rate we have,

Sh, ent = Sδ, ent hf .

(71)

Here Sδ, ent is the entrainment rate and hf is the thickness-averaged liquid film specific enthalpy. 5. CFD model predictions The model described in the previous section has been implemented into OpenFOAM suit of CFD solvers and applied to predictions of annular two-phase flow in various channels (Li and Anglart, 2015, 2016a–c). Fig. 8 shows a typical computational mesh which has been used for calculations performed in round tubes. Multi-block hexagonal mesh is used with 544 cells in the cross section, for which y+ varies in a range from 30 to 150 for various cases analyzed. Mesh-independence studies have been performed for the gascore predictions for both the Eulerian-Eulerian and Eulerian-Lagrangian approach (Li and Anglart, 2016a,b). 5.1. Verification of results Fig. 9 shows drop concentration distribution in a gas core for watersteam flow in a pipe with 14 mm internal diameter, calculated with two different methods. The operational conditions are as follows: the total mass flux G = 750 kg/m2s, the system pressure at exit p = 7 MPa, the inlet subcooling 10 K, the exit thermodynamic quality 0.75 and a uniformly distributed heat flux in a 3.65 m long pipe. Two curves are shown in the figure: one obtained from the Eulerian-Eulerian calculations and one obtained from the LPT calculation. Interestingly, both curves indicate that the droplet concentration has its maximum close to the wall, where a liquid film is present. It should be noted that the Eulerian-Eulerian solution represents the ensemble-averaged distribution of the drop concentration in the pipe, whereas the LPT result shows the instantaneous drop concentration. Fig. 10 shows mass transfer rates between a gas core and a liquid film for water-steam flow in a pipe with 14 mm internal diameter, calculated with two different approaches. The operational conditions are as follows: mass flux G = 750 kg/m2s, system pressure at exit p = 7 MPa, inlet subcooling 10 K, exit thermodynamic quality 0.75 and uniformly distributed heat flux in a 3.65 m long pipe.

Fig. 8. Computational domain and mesh for annular flow calculations in round tubes (Li and Anglart, 2016b). 178

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

Fig. 11. Film mass flow rate vs distance for annular two-phase flow of boiling water in pipe with 14 mm inner diameter: G = 750 kg/m2 s, p = 7 MPa; experimental data from Adamsson and Anglart (2006a,b) and Li and Anglart (2015).

Fig. 9. Profile of the droplet concentration at the outlet of a test section (Li and Anglart, 2016b).

Fig. 10. Mass transfer rates between gas core and liquid film for annular twophase flow of boiling water in pipe with 14 mm inner diameter: G = 750 kg/ m2 s, p = 7 MPa (Li and Anglart, 2015).

Fig. 12. Wall temperature vs distance for annular two-phase flow of boiling water in pipe with 14.9 mm inner diameter: G = 1000.9 kg/m2s, p = 7.01 MPa, inlet subcooling 11.7 K and uniform wall heat flux 0.765 MW/m2; experimental data from Becker et al. (1983) and Li and Anglart (2016b).

Two possible reasons of this discrepancy are due to the neglect of the axial heat conduction in the wall and also due to a sharp transition from the pre-dryout to the post-dryout heat transfer regime at the dryout point. Clearly more work is needed to improve the model accuracy in the near-dryout region.

two-dimensional approach is used to resolve the liquid film moving along the channel walls. Such approach combines the ability of CFD to mechanistically treat the phasic conservation equations in three-dimensional space with the simplicity and robustness of presently available closure relationships applicable to annular flows. It has been demonstrated that with a careful choice of closure laws, which adequately represent the governing phenomena, the predictions of main features of annular flows are feasible. Nevertheless, it has been also demonstrated that additional development in the field is needed. In particular, more accurate, locally based closure relationships for the interfacial mass, momentum and energy transfer rates at the liquid film interface are required. Currently available correlations for the liquid film thickness, the interfacial wave propagation, the deposition rates and the entrained rates are mainly derived from air-water data obtained at ambient conditions. Moreover, they are obtained in simple, effectively two-dimensional geometries such as, e.g., circular tubes. Their relevance and accuracy, when applying to high-pressure boiling systems in complex rod-bundle configurations, should be carefully investigated in the

6. Conclusions Significant progress has been achieved in prediction of annular twophase flows with heat transfer within recent years. On the one hand more detailed experimental data are currently available to develop proper closure relationships for the governing equations of mass, momentum and energy transfer. Such recent and most important experimental results have been extensively discussed in the paper. On the other hand, more powerful computers and efficient numerical methods have become available, allowing for detailed, steady-state and transient solutions of annular flows. In this paper a computational framework is discussed that allows prediction of annular flows with heat transfer of industrial interest. The proposed model is employing CFD approach to resolve the geometry effects in the gas core flow, whereas a simplified 179

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

future work. To this end, more detailed experimental data on the film and the gas core interface behavior at elevated pressures, preferentially supported with DNS results, will be necessary.

Reactor Thermal Hydraulics, NURETH-8, Kyoto, Japan, September 30–October 4, 1997, pp. 1647–1655. Asali, J.C., Hanratty, T.J., Andreussi, P., 1985. Interfacial drag and film height for vertical annular flow. AIChE J. 31, 895–902. Ashwood, A.C., Vanden Hogen, S.J., Rodarte, M.A., Kopplin, C.R., Rodriguez, D.J., Hurlburt, E.T., 2014. Reprint of: a multiphase, micro-scale PIV measurement technique for liquid film velocity measurements in annular two-phase flow. Int. J. Heat Mass Transfer 67, 200–212. Assad, A., Jan, C., Lopez de Bertodano, M.A., Beus, S.G., 1998. Scaled entrainment measurements in ripple-annular flow in a small tube. Nucl. Eng. Des. 184, 437–447. Azzopardi, B.J., 1985. Drop sizes in annular two-phase flow. Exp. Fluids 3, 53–59. Azzopardi, B.J., 1986. Disturbance wave frequencies, velocities and spacing in vertical annular two-phase flow. Nucl. Eng. Des. 92, 121–133. Azzopardi, B.J., 1996. Prediction of dryout and post-burnout heat transfer with axially non-uniform heat input by means of an annular flow model. Nucl. Eng. Des. 163, 51–57. Azzopardi, B.J., 1997. Drops in annular two-phase flow. Int. J. Multiphase Flow 23, 1–53. Azzopardi, B.J., Freeman, G. and King, D.J., Drop sizes and deposition in annular twophase flow, UKAEA Report AERE-R9634, 1980. Azzopardi, B.J., Taylor, S., Gibbons, D.B., 1983. Annular Two-Phase Flow in a Large Diameter Tube. Int. Conf. Physical Modelling of Multiphase Flow, Coventry. Azzopardi, B.J., Whalley, P.B., 1980. Artificial Waves in Annular Two-Phase Flow. ASME Winter Annual Meeting, Chicago. Azzopardi, B.J., Piearcey, A., Jepson, D.M., 1991. Drop size measurements for annular two-phase flow in a 20 mm diameter vertical tube. Exp. Fluids 11, 191–197. Azzopardi, B.J., Teixeira, J.C.F., 1994a. Detailed measurements of vertical annular twophase flow – Part II: gas core turbulence. J. Fluids Eng. Trans. ASME 116, 796–1795. Azzopardi, B.J., Teixeira, J.C.F., 1994a. Detailed measurements of vertical annular twophase flow – Part I: drop velocities and sizes. J. Fluids Eng. Trans. ASME 116, 792–795. Babibsky, E., Sojka, P.E., 2002. Modeling drop size distribution. Progr. Energy Combust. Sci. 28, 303–329. Babinsky, E., Sojka, P.E., 2002. Modeling drop size distributions. Prog. Energy Combust. Sci. 28, 303–329. Bates, C.J., Sheriff, J.M., 1992. High data rate measurements of droplet dynamics in a vertical gas-liquid annular flow. Flow Meas. Instrum. 3 (4), 247–256. Becker, K.M., Ling, C.H., Hedberg, S., Strand, G., 1983. An Experimental Investigation of Post Dryout Heat Transfer, Report KTH-NEL-33. Royal Institute of Technology, Sweden. Belt, R.J., Van’t Westende, J.M.C., Prasser, H.M., Portela, L.M., 2010. Time and spatially resolved measurements of interfacial waves in vertical annular flow. Int. J. Multiphase Flow 36, 570–587. Binder, J.L., Hanratty, T.J., 1991. A diffusion model for droplet deposition in gas/liquid annular flow. Int. J. Multiphase Flow 17 (1), 1–11. Butterworth, D., 1968. Air-water climbing film flow in an eccentric annulus. Paper B2. Int. Symposium on Research in Co-current Gas-Liquid Flow. University of Waterloo, Ontario. Caraghiaur, D., Adamsson, C., Anglart, H., 2009. Verification of the applicability of random walk models to calculate drop deposition in nuclear fuel assemblies. Proceedings of 13th Int. Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-13), Kanazawa, Japan, September 27–October 2. Caraghiaur, D., Adamsson, C., Anglart, H., 2013. A model for inertial drop deposition suitable to predict obstacle effect. Nucl. Eng. Des. 260, 121–133. Caraghiaur, D., Anglart, H., 2006. Numerical study of deposition rates in annular dispersed flows in non-circular channels with obstacles. Proceedings of Int. Workshop on Modeling and Measurements of Two-Phase Flows and Heat Transfer in Nuclear Fuel Assemblies, Stockholm, Sweden, October 11–12. Caraghiaur, D., Anglart, H., 2006. Development of drop deposition model for annular flows in fuel assemblies with spacers using particle tracking approach. Proceedings of 14th Int. Conference on Nuclear Engineering, ICONE-14, Miami, FL, USA, July 17–20. Caraghiaur, D., Adamsson, C., Anglart, H., 2013. Deposition of inertial drops in Eulerian formulation. Proceedings of 15th Int. Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-15), Pisa, Italy, May 12–17. Caraghiaur, D., Anglart, H., 2009. Lagrangian particle tracking as a tool for deposition modeling in annular flow. Proceedings of 17th Int. Conf. on Nucl. Engineering, ICONE17, Brussels, Belgium, July 12–16. Caraghiaur, D., Anglart, H., 2010. Eulerian description model for deposition of drops in annular flow regime. Proceedings of 8th Int. Top. Meeting on Nuclear ThermalHydraulics, Operation and Safety NUTHOS-8, October 10–14. Caraghiaur, D., Anglart, H., 2013. Drop deposition in annular two-phase flow calculated with Lagrangian particle tracking. Nucl. Eng. Des. 265, 856–866. Carvalho, I.S., Heitroyr, M.V., Santos, D., 2002. Liquid film disintegration regimes and proposed correlations. Int. J. Multiphase Flow 28, 773–789. Chisholm, D., 1973. Pressure gradients due to friction during the flow of evaporating twophase mixtures in smooth tubes and channels. Int. J. Multiphase Flow 16, 347–1158. Clift, R., Grace, J.R., Weber, M.E., 1978. Bubbles, Drops and Particles. Dover Publications Inc., Mineola, New York. Coney, M.W.E., 1973. The theory and application of conductance probes for the measurement of liquid film thickness in two-phase flow. J. Phys. E: Sci. Instrum. 6, 903–910. Crowe, C.T., 2000. On models for turbulence modulation in fluid-particle flows. Int. J. Multiphase Flow 26, 719–727. Dasgupta, A., Chandraker, D.K., Vishnoi, A.K., Vijayan, P.K., 2015. A new methodology for estimation of initial entrainment fraction in annular flow for improved dryout prediction. Ann. Nucl. Energy 75, 323–330.

Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.nucengdes.2019.02.007. References Abolfadl, M.A., 1984. Analysis of Annular Two-phase Flow with Liquid Entrainment. [Ph.D. Thesis]. Thayler School of Engineering, Dartmouth College, USA. Adamsson, C., Anglart, H., 2004. Dryout predictions in BWR Fuel assemblies with spacers. Proc. 6th International Conference on Nuclear Thermal Hydraulics, Operation and Safety (NUTHOS-6), Nara, Japan, October 4–8. Adamsson, C., Anglart, H., 2005. Experimental investigation of the liquid film for annular flow in a tube with various axial power distributions. Proc. 11th Int. Topical Meeting on Nuclear Reactor Thermal Hydraulics, NURETH-11, Avignon, France, October 2–6. Adamsson, C., Anglart, H., 2006a. Lagrangian modeling of dispersion and deposition of drops in annular flow. Proceedings of Int. Workshop on Modeling and Measurements of Two-Phase Flows and Heat Transfer in Nuclear Fuel Assemblies, Stockholm, Sweden, October 11–12. Adamsson, C., Anglart, H., 2006b. Film flow measurements in high pressure diabatic annular flow in tubes with various axial power distributions. Nucl. Eng. Des. 236, 2485–2493. Adamsson, C., Anglart, H., 2007. An investigation of cross-section geometry effects on the deposition rate in annular two-phase flows with a lagrangian model. Proceedings of 12th Int. Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-12), Pittsburgh, PA, USA, September 30–October 4. Adamsson, C., Anglart, H., 2010. Influence of axial power distribution on dryout: film flow models and experiments. Nucl. Eng. Des. 240, 1495–1505. Adamsson, C., Le Corre, J.M., 2014. Transient dryout prediction using a computationally efficient method for sub-channel film-flow analysis. Nucl. Eng. Des. 280, 316–325. Alamu, M.B., Azzopardi, B.J., 2011. Wave and drop periodicity in transient annular flow. Nucl. Eng. Des. 241, 5079–5092. Alekseenko, S.V., Cherdantsev, A.V., Heinz, O.M., Kharlamov, S.M., 2014. Analysis of spatial and temporal evolution of disturbance waves and ripples in annular gas-liquid flow. Int. J. Multiphase Flow 67, 122–134. Alekseenko, S.V., Cherdantsev, A.V., Cherdantsev, M.V., Isaenkov, S.V., Markovich, D.M., 2015. Study of formation and development of disturbance waves in annular gas-liquid flow. Int. J. Multiphase Flow 77, 65–75. Aliyu, A.M., Almabrok, A.A., Baba, Y.D., Archibong, A.-E., Lao, L., Yeung, H., Kim, K.C., 2017. Prediction of entrained droplet fraction in co-current annular gas-liquid flow in vertical pipes. Exp. Therm. Fluid Sci. 85, 287–304. Ambrosini, W., Andreussi, P., Azzopardi, B.J., 1991. A physically based correlation for drop size in annular flow. Int. J. Multiphase Flow 17, 497–507. Andersen, P.S., Jensen, A., Mannov, G., Olsen, A., 1974. Burn-out, circumferential film flow distribution and pressure drop for an eccentric annulus with heated rod. Int. J. Multiphase Flow 1, 585–600. Andersson, G.H., Mantzouranis, B.G., 1960. Two-phase (gas/liquid) flow phenomena – II Liquid entrainment. Chem. Eng. Sci. 12, 233–242. Andreussi, P., 1983. Droplet transfer in annular two-phase flow. Int. J. Multiphase Flow 9, 697–713. Andreussi, P., Azzopardi, B.J., 1983. Droplet deposition and interchange in annular twophase flow. Int. J. Multiphase Flow 9, 681–695. Andreussi, P., Romano, G., Zanelli, S., 1978. Drop Size Distribution in Annular Mist Flows. 1st Int Conf. on Liquid Atomization and Spray Systems, Tokyo. Anglart, H., 1994. An annular flow model for predicting film thickness and pressure drop in vertical heated channels. Proc. 2nd CFDS Int. User Conference, Pittsburgh, USA, 7–9 December. Anglart, H., 1999. ABC – advanced bundle code for thermal-hydraulics predictions in LWR fuel assemblies. In: Proceedings of Mathematics and Computation, Reactor Physics and Environmental Analysis in Nuclear Applications, Madrid, Spain, September, 1999, pp. 200–209. Anglart, H., 2006. Thermal-hydraulic design of nuclear fuel assemblies – current needs and challenges. Proceedings of Int. Workshop on Modeling and Measurements of Two-Phase Flows and Heat Transfer in Nuclear Fuel Assemblies, Stockholm, Sweden, October 11–12. Anglart, H., 2014b. Dry patch formation in diabatic annular two-phase flows. J. Power Technol. 94, 85–95. Anglart, H., 2014a. Modelling of liquid film flow in annuli. J. Power Technol. 94, 8–15. Anglart, H., 2015. Current trends in CHF predictions: from correlations to simulations. Proceedings of Int. Seminar on Subchannel Analysis (ISACC-2015). Anglart, H., Adamsson, C., 2002. Numerical study of the effect of the axial power distribution on dryout conditions in an annulus. Presented at European Two-Phase Flow Group Meeting, Stockholm, June 10–13. Anglart, H., Adamsson, C., 2003. Modeling of dryout in channels with various spatial power distributions. Proceedings of 10th Int. Topical Meeting on Nuclear Reactor Thermal Hydraulics, NURETH-10, Seoul, South Korea, October 5–10. Anglart, H., Ji, W., Gu, C.-Y., 1997. Numerical simulation of multidimensional two-phase boiling flow in rod bundles. In: Proceedings of 8th Int. Topical Meeting on Nuclear

180

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

Jones, W.P., Launder, B.E., 1972. The prediction of laminarization with a two-equation model of turbulence. Int. J. Heat and Mass Transfer 15, 301–314. Ju, P., Brooks, C.S., Ishii, M., Liu, Y., Hibiki, T., 2015. Film thickness of vertical upward co-current adiabatic flow in pipes. Int. J. Heat Mass Transfer 89, 985–995. Key, E.D., Hibberd, S., Power, H., 2014. A depth-averaged model for non-isothermal thinfilm rimming flow. Int. J. Heat Mass Transfer 70, 1003–1015. Kocamustafaogullari, G., Smith, S.R., Razi, J., 1994. Maximum and mean droplet sizes in annular two-phase flow. Int. J. Heat Mass Transfer 37 (6), 955–965. Konishi, Ch., Mudawar, I., Hasan, M.M., 2013. Investigation of localized dryout versus CHF in saturated flow boiling. Int. J. Heat Mass Transfer 67, 131–146. Krejci, M., Adamsson, C., Anglart, H., 2005. Development of a model for prediction of annular flow and dryout in BWR fuel assemblies with spacers. Proceedings of 11th Int. Topical Meeting on Nuclear Reactor Thermal Hydraulics, NURETH-11, Avignon, France, October 2–6. Kumar, R., Gottmann, M., Sridhar, K.R., 2002. Film thickness and wave velocity measurements in a vertical duct. Trans. ASME J. Fluids Eng. 124, 634–642. Le Corre, J.M., Bergman, U.C., Hallehn, A., Tejne, H., Waldermarsson, F., Morrenius, B., Baghai, R., 2017. Measurement of local two-phase flow parameters in fuel bundle under BWR operating conditions. Nucl. Eng. Des. 336, 15–23. Lee, M.M., Hanratty, T.J., Adrian, R.J., 1989. The interpretation of droplet deposition measurements with diffusion model. Int. J. Multiphase Flow 15 (3), 459–469. Levy, S., 1966. Prediction of two-phase annular flow with liquid entrainment. Int. J. Heat Mass Transfer 9, 171–188. Li, H., Anglart, H., 2015. CFD model of diabatic annular two-phase flow using the Eulerian-Lagrangian approach. Ann. Nucl. Energy 77, 415–424. Li, H., Anglart, H., 2016c. CFD prediction of droplet deposition in steam-water annular flow with flow obstacle effects. Nucl. Eng. Des. 321, 173–179. Li, H., Anglart, H., 2016a. Modeling of annular two-phase flow using a unified CFD approach. Nucl. Eng. Des. 303, 17–24. Li, H., Anglart, H., 2016b. Prediction of dryout and post-dryout heat transfer using a twophase CFD model. Int. J. Heat Mass Transfer 99, 839–850. Liu, L., Bai, B., 2017. Generalization of droplet entrainment rate correlation for annular flow considering disturbance wave properties. Chem. Eng. Sci. 164, 279–291. Lopes, J.C.B., Dukler, A.E., 1987. Droplets dynamics in vertical gas-liquid annular flow. AIChE J. 33 (6), 1013–1024. Lopez de Bertodano, M.A., Jan, C.-S., Beus, S.G., 1997. Annular flow entrainment rate experiment in a small vertical pipe. Nucl. Eng. Des. 178, 61–70. Mannov, G., 1973. Film Flow Measurements in Concentric Annulus 3500 × 27.2 × 17 mm with Heated and Unheated Rod. Risö Internal Report SDS-65. McCoy, D.D., Hanratty, T.J., 1977. Rate of deposition of droplets in annular two-phase flow. Int. J. Multiphase Flow 1, 319–331. Menter, F.R., 1993. Multiscale model for turbulent flows. In: Proc. 24th AIAA Fluid Dynamics Conf., Orlando, Fla, USA,, pp. 1311–1320. Menter, F., Esch, T., 2001. Elements of industrial heat transfer predictions. In: 16th Brazilian Congress of Mechanical Engineering, pp. 117–127. Milashenko, V.I., Nigmatulin, B.I., Petukhov, V.V., Trubkin, N.I., 1989. Burnout and distribution of liquid in evaporative channels of various lengths. Int. J. Multiphase Flow 15 (3), 393–401. Milashenko, V.I., Nigmatulin, B.I., Petukhov, V.V., Trubkin, N.I., 1989. Burnout and distribution of liquid in evaporative channels of various lengths. Int. J. Multiphase Flow 15 (3), 393–401. Moeck, E.O., 1970. Annular-dispersed Two-phase Flow and Critical Heat Flux. AECL3656. Moran, K., Inumaru, J., Kawaji, M., 2002. Instantaneous hydrodynamics of a laminar wavy liquid film. Int. J. Multiphase Flow 28, 731–755. Mudawar, I., Houpt, R.A., 1993. Mass and momentum transport in smooth falling liquid films laminarized at relatively high Reynolds numbers. Int. J. Heat Mass Transfer 36 (14), 3437–3448. Nedderman, R.M., Shearer, C.J., 1963. The motion and frequency of large disturbance waves in annular two-phase flow of air-water mixtures. Chem. Eng. Sci. 18, 661–670. Nikolopoulos, N., Theodorakakos, A., Bergles, G., 2005. Normal impingement of a droplet onto a wall film: a numerical investigation. Int. J. Heat Fluid Flow 26, 119–132. Okawa, T., Kitahara, T., Yoshida, K., Matsumoto, T., Kataoka, I., 2002. New entrainment rate correlation in annular two-phase flow applicable to wide range of flow conditions. Int. J. Heat Mass Transfer 45, 87–98. Okawa, T., Kotani, A., Kataoka, I., Naito, M., 2003. Prediction of critical heat flux in annular flow using a film flow model. J. Nucl. Sci. Technol. 40, 388–396. Okawa, T., Kotani, A., Kataoka, I., 2005. Experiments for liquid phase mass transfer rate in annular regime for a small vertical tube. Int. J. Heat Mass Transfer 48, 585–598. Okawa, T., Goto, T., Minamitani, J., Yamagoe, Y., 2009. Liquid film dryout in a boiling channel under flow oscillation conditions. Int. J. Heat Mass Transfer 52, 3665–3675. Okawa, T., Goto, T., Yamagoe, Y., 2010. Liquid film behavior in annular two-phase flow under flow oscillation conditions. Int. J. Heat Mass Transfer 53, 962–971. Pan, L.-M., He, H., Ju, P., Hibiki, T., Ishii, M., 2015. The influences of gas-liquid interfacial properties on interfacial shear stress for vertical annular flow. Int. J. Multiphase Flow 89, 1172–1183. Pan, L.-M., He, H., Ju, P., Hibiki, T., Ishii, M., 2015. Experimental study and modeling of disturbance wave height of vertical annular flow. Int. J. Multiphase Flow 89, 165–175. Pearce, D.L., 1979. Film Waves in Horizontal Annular Flow: Space-time Correlator Experiments. CERL Note RD/L/N111/79. Ranz, W.E., Marshall, W.R.J., 1952. Evaporation from drops, Part I and Part II. Chem. Eng. Prog. 48, 173–180. Renksizbulut, M., Yuen, M.C., 1983. Experimental study of droplet evaporation in a hightemperature air stream. Trans. ASME 105, 384–388. Rodriguez, J.M., 2009. Numerical Modeling of Annular Two-Phase Flow. [Ph.D. Thesis].

Desoutter, G., Habchi, C., Cuenot, B., Poinsot, T., 2009. DNS and modelling of the turbulent boundary layer over an evaporating liquid film. Int. J. Heat Mass Transfer 52, 6028–6041. Erkan, N., Kawakami, T., Madokoro, H., Chai, P., Ishiwatari, Y., Okamoto, K., 2015. Numerical simulation of droplet deposition onto a liquid film by VOF-MPS hybrid method. J. Vis. 18, 381–391. Fore, L.B., Dukler, A.E., 1995. The distribution of drop size and velocity in gas-liquid annular flow. Int. J. Multiphase Flow 21 (2), 137–149. Fore, L.B., Beus, S.G., Bauer, R.C., 2000. Interfacial friction in gas-liquid annular flow: analogies to full and transition roughness. Int. J. Multiphase Flow 26, 1755–1769. Fore, L.B., Ibrahim, B.B., Beus, S.G., 2002. Visual measurements of droplet size in gasliquid annular flow. Int. J. Multiphase Flow 28, 1895–1910. France, D.M., Chiang, T., Carlson, R.D., Priemer, R., 1982. Experimental evidence supporting two-mechanism critical heat flux. Int. J. Heat Mass Transfer 25, 691–698. Fujita, T., Ueda, T., 1978a. Heat transfer to falling liquid films and film breakdown – I Subcooled liquid films. Int. J. Heat Mass Transfer 21, 97–108. Fujita, T., Ueda, T., 1978b. Heat transfer to falling liquid films and film breakdown – II Saturated liquid films with nucleate boiling. Int. J. Heat Mass Transfer 21, 109–118. Fukano, T., Furukawa, T., 1998. Prediction of the effects of liquid viscosity on interfacial shear stress and frictional pressure drop in vertical upward gas-liquid annular flow. Int. J. Multiphase Flow 24, 587–603. Fukano, T., Akenaga, H., Ikeda, M., Itoh, A., Kuriwaki, T., Takamatsu, Y., 1984. Liquid films flowing concurrently with air in horizontal duct (4th report, effect of the geometry of duct cross-section on liquid film flow). Bull. JSME 27 (230), 1644–1651. Fukano, T., Mori, S., Akamatsu, S., Baba, A., 2002. Relation between temperature fluctuation of a heating surface and generation of drypatch caused by a cylindrical spacer in a vertical boiling two-phase upward flow in a narrow annular channel. Nucl. Eng. Des. 217, 81–90. Fukano, T., Mori, S., Nakagawa, T., 2003. Fluctuation characteristics of heating surface temperature near obstacle in transient boiling two-phase flow in a vertical annular channel. Nucl. Eng. Des. 219, 47–60. Gatapova, E.Ya, Kabov, O.A., 2008. Shear-driven flows of locally heated liquid film. Int. J. Heat Mass Transfer 51, 4797–4810. Gill, L.E., Hewitt, G.F., Hitchon, J.W., Lacey, P.M.C., 1963. Sampling probe studies of the gas core in annular two-phase flow – I The effect of length on phase and velocity distribution. Chem. Eng. Sci. 18, 525–535. Gill, E., Hewitt, G.F., Lacey, P.M.C., 1964. Sampling probe studies of the gas core in annular two-phase flow – II Studies of the effect of phase flow rates on phase and velocity distribution. Chem. Eng. Sci. 19, 665–682. Gill, L.E., Hewitt, G.F., 1968. Sampling probe studies of the gas core in annular two-phase flow – III Distribution of velocity and droplet flowrate after injection through and axial jet. Chem. Eng. Sci. 23, 677–686. Gong, S., Ma, W., Dinh, T.-N., 2011. An experimental study of rupture dynamics of evaporating liquid films on different heater surfaces. Int. J. Heat Mass Transfer 54, 1538–1547. Gore, R.A., Crowe, C.T., 1989. The effect of particle size on modulating turbulent intensity. Int. J. Multiphase Flow 15, 279–285. Govan, A.H., Hewitt, G.F., Owen, D.G., Burnett, G., 1989. Wall shear stress measurements in vertical air-water annular two-phase flow. Int. J. Multiphase Flow 15 (3), 307–325. Hall Taylor, N., Hewitt, G.F., Lacey, P.M.C., 1963. The motion and frequency of large disturbance waves in annular two-phase flow of air-water mixtures. Chem. Eng. Sci. 18, 537–552. Hall Taylor, N.S., Hewitt, I.J., Ockendon, J.R., Witelski, T.P., 2014. A new model for disturbance waves. Int. J. Multiphase Flow 66, 38–45. Hall Taylor, N.S., Nedderman, R.M., 1968. The coalescence of disturbance waves in annular two-phase flow. Chem. Eng. Sci. 23, 551–564. Hartley, D.E., Murgatroyd, W., 1964. Criteria for the break-up of thin liquid layers flowing isothermally over solid surfaces. Int. J. Heat Mass Transfer 7, 1003–1015. Hazuku, T., Takamasa, T., Matsumoto, Y., 2008. Experimental study on axial development of liquid film in vertical upward annular two-phase flow. Int. J. Multiphase Flow 34, 111–127. Hemlin, M., Anglart, H., 2004. Modeling of entrainment rates for annular flow in vertical annuli. Proc. of the 6th International Conference on Nuclear Thermal Hydraulics, Operation and Safety (NUTHOS-6), Nara, Japan, October 4–8. Hewitt, G.F., Govan, A.H., 1990. Phenomenological modelling of non-equilibrium flows with phase change. Int. Heat Mass Transfer 33, 229–242. Hewitt, G.F., Lacey, P.M.C., 1965. The breakdown of the liquid film in annular two-phase flow. Int. J. Heat Mass Transfer 8, 781–791. Hewitt, G.F., Kearsey, H.A., Lacey, P.M.C., Pulling, D.J., 1965. Burnout and nucleation in climbing film flow. Int. J. Heat Mass Transfer 8, 793–814. Holowach, M.J., Hochreier, L.E., Cheung, F.B., 2002. A model for droplet entrainment in heated annular flow. Int. J. Heat Fluid Flow 23, 807–822. Holowach, M.J., Hochreiter, L.E., Mahaffy, J.H., Cheung, F.B., 2003. Modeling of droplet entrainment phenomena at a quench front. Int. J. Heat Fluid Flow 24, 902–918. Inada, F., Drew, D.A., Lahey Jr., R.T., 2004. An analytical study on interfacial wave structure between the liquid film and gas core in a vertical tube. Int. J. Multiphase Flow 30, 827–851. Ishii, M., Grolmes, M.A., 1975. Inception criteria for droplet entrainment in two-phase concurrent film flow. AIChE J. 21 (2), 308–318. Ito, D., Papadopoulos, P., Prasser, H.-M., 2016. Liquid film dynamics of two-phase annular flow in square and light lattice subchannels. Nucl. Eng. Des. 300, 467–474. Jensen, A., Mannov, G., 1974. Measurements of Burnout, Film Flow, Film Thickness and Pressure Drop in a Concentric Annulus 3500 × 26 × 17 mm with Heated Rod and Tube. Risö Internal Report SDS-77. Jepson, D.M., Azzopardi, B.J., Whalley, P.B., 1989. The effect of gas properties on drops in annular flow. Int. J. Multiphase Flow 15 (3), 327–339.

181

Nuclear Engineering and Design 345 (2019) 166–182

H. Anglart

Tatterson, D.F., Dallman, J.C., Hanratty, T.J., 1977. Drop sizes in annular gas-liquid flows. AIChE J. 23, 68–76. Thwaites, G.R., Kulov, N.N., Nedderman, R.M., 1976. Liquid film properties in two-phase annular flow. Chem. Eng. Sci. 31 (3), 481–486. Ueda, T., 1979. Entrainment rate and size of entrained droplets in annular two-phase flow. Bull. JSME 22, 1258–1265. van’t Westende, J.M.C., 2007. Droplets in Annular-dispersed Gas-liquid Pipe-flows. [Ph.D. Thesis]. TU Delft. Vassallo, P., 1999. Near wall structure in vertical air-water annular flows. Int. J. Multiphase Flow 25, 459–476. Wallis, G.B., 1969. One-dimensional two-phase flow, McGraw. Hill Inc. Wallis, G.B., 1970. Annular two phase flow, Part I: A simple theory; Part II: Additional effects. J. Basic Eng. 92, 59–82. Wang, K., Ye, J., Bai, B., 2017. Entrained droplets in two-phase churn flow. Chem. Eng. Sci. 164, 270–278. Whalley, P.B., 1977. The calculation of dryout in a rod bundle. Int. J. Multiphase Flow 3, 501–515. Whalley, P.B., Hewitt, G.F. and Hutchinson, P., Experimental wave and entrainment measurements in vertical annular two-phase flow, UKAEA Report AERE-R7521, 1973. Wolf, A., Jayanti, S., Hewitt, G.F., 1996. On the nature of ephemeral waves in vertical annular flow. Int. J. Multiphase Flow 22, 325–333. Wu, Y.W., Su, G.H., Qiu, S.Z., Hu, B.X., 2009. Experimental study on critical heat flux in bilaterally heated narrow annuli. Int. J. Multiphase Flow 35, 977–986. Würtz, J., 1978. An Experimental and Theoretical Investigation of Annular Steam-water Flow in Tube and Annuli at 30–90 bar, Technical Report 372. Risø National Laboratory. Xie, Z., Hewitt, G.F., Pavlidis, D., Salinas, P., Pain, Ch.C., Matar, O.K., 2017. Numerical study of three-dimensional droplet impact on a flowing liquid film in annular twophase flow. Chem. Eng. Sci. 166, 303–312. Yang, J., Narayanan, C., Lakehal, D., 2016. Large Eddy & interface simulation (LEIS) of disturbance waves and heat transfer in annular flows. Nucl. Eng. Des. 321, 190–198. Yu, J., Wang, H., 2012. A molecular dynamics investigation on evaporation of thin liquid films. Int. J. Heat Mass Transfer 55, 1218–1225. Zaichik, L.I., Alipchenkov, V.M., 2001. A statistical model for transport and deposition of high-inertia cooling particles in turbulent flow. Int. J. Heat Fluid Flow 22, 365–371. Zhao, Y., Markides, ChN., Matar, O.K., Hewitt, G.F., 2013. Disturbance wave development in two-phase gas-liquid upwards vertical annular flow. Int. J. Multiphase Flow 55, 111–129.

RPI. Rodriguez, J.M., Sahni, O., Lahey Jr., R.T., Jansen, K.E., 2013. A parallel adaptive mesh method for the numerical simulation of multiphase flows. Comput. Fluids 87, 115–131. Rosskamp, H., Willmann, M., Wittig, S., 1998. Heat up and evaporation of shear driven liquid wall films in hot turbulent air flow. Int. J. Heat Fluid Flow 19, 167–172. Ryley, D.J., Xu, T., 1983. Free energy loss during the breakdown of liquid films. Int. J. Heat Mass Transfer 26, 185–196. Ryu, S.-H., Park, G.-C., 2011. A droplet entrainment model based on the force balance of an interfacial wave in two-phase annular flow. Nucl. Eng. Des. 241, 3890–3897. Samenfink, W., Elsäßer, A., Dullenkopf, K., Wittig, S., 1999. Droplet interaction with shear-driven liquid films: analysis of deposition and secondary droplet characteristics. Int. J. Heat Fluid Flow 20, 462–469. Sawant, P., 2008. Dynamics of Annular Two-Phase Flow. Purdue University, West Lafayette, USA PhD thesis. Sawant, P., Ishii, M., Hazuku, T., Takamasa, T., Mori, M., 2008. Properties of disturbance waves in vertical annular two-phase flow. Nucl. Eng. Des. 238, 3528–3541. Schraub, F.A., Simpson, R.L., Janssen, E., 1969. Two-phase Flow and Heat Transfer in Multirod Geometries. Air-water Flow Structure Data for a Round Tube Concentric and Eccentric Annulus, and Nine-rod Bundle. GEAP-5739. Schubring, D., Shedd, T.A., Hurlburt, E.T., 2010. Studying disturbance waves in vertical annular flow with high-speed video. Int. J. Multiphase Flow 36, 385–396. Sekoguchi, K., Ueno, T., Tanaka, O., 1985b. Flow characteristics of air-water annular flow (2nd report: correlation for flow parameter). Trans. Jpn. Soc. Eng. 51-466 (B), 1798–1806. Sekoguchi, K., Takeishi, M., Ishimatsu, T., 1985a. Interfacial structure in vertical upward annular flow. Physicochem. Hydrodyn. 6, 239–255. Sekoguchi, K., Takeishi, M., 1989. Interfacial structures in upward huge wave flow and annular flow regimes. Int. J. Multiphase Flow 15, 295–305. Shearer, C.J., Nedderman, R.M., 1965. Pressure gradient and liquid film thickness in cocurrent upwards flow of gas/liquid mixtures: application to film-cooler design. Chem. Eng. Sci. 20, 671–683. Shedd, T.A., Newell, T.A., 2004. Characteristics of the liquid film and pressure drop in horizontal, annular, two-phase flow through round, square and triangular tubes. J. Fluids Eng. 126, 807–817. Soldati, A., Andreussi, P., 1996. The influence of coalescence on droplet transfer in vertical annular flow. Chem. Eng. Sci. 51 (3), 353–363. Sugawara, S., Miyamoto, Y., 1990. FIDAS: detailed subchannel analysis code based on the three-fluid and three-field model. Nucl. Eng. Des. 120, 147–161.

182