Axial and radial void fraction measurements in convective boiling flows

Chemical Engineering Science ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Axial and radial void fraction measurements in convective boiling flows Ashutosh Yadav, Shantanu Roy n Department of Chemical Engineering, Indian Institute of Technology – Delhi, New Delhi 110016, India

H I G H L I G H T S


Axial and radial void fraction measurement for vertical forced convective boiling flow.
γ- ray densitometry has been used for measuring void fraction distribution profiles.
Drift flux model description used to rationalize the experimental observations.

art ic l e i nf o Article history: Received 15 September 2015 Received in revised form 10 April 2016 Accepted 17 April 2016 Keywords: Convective boiling flows Densitometry Sub-cooled boiling Saturated boiling Modified Fourier–Hankel method

a b s t r a c t This work is inspired by the need to have measurements and predictive capability on vertical saturated boiling flows, which are of importance in boiling water nuclear reactors. In these systems, sub-cooled water flows upwards in vertical tubes which contain a multiplicity of nuclear fuel rods, and the heat is taken away from these rods by natural convection boiling. A close analog of this situation exists when the liquid is in forced flow, and the heat flux from the vertical rods is quenched by forced convection boiling of the upward liquid water flux. In this work, we present experimental data has for axial, and radial vapor void fraction distributions in an annular boiling channel for low mass-flux forced the flow of water at high inlet sub-cooling. A single centrally placed electrical rod, designed to mimic the nuclear fuel rod has been used. The void fraction measurement is made using gamma ray densitometry technique. Axial and radial vapor void fractions have been reported, as a function of inlet liquid flux and inlet liquid temperature. The experimental data has been rationalized using a simple one-dimensional drift flux model adapted to the conditions of the experiment. & 2016 Elsevier Ltd. All rights reserved.

1. Introduction Convective boiling flows are of great importance to nuclear reactor systems and have been the subject of numerous theoretical and experimental investigations (Todreas and Kazimi, 2012). Boiling occurs at a planar interface in contact with liquid when the temperature of the liquid is raised sufficiently beyond the saturation temperature at that pressure. Boiling may occur under a quiescent fluid condition, which is referred to as pool boiling; or under forced flow conditions, which is referred to as forced convective boiling (Collier and Thome, 1994). When boiling flow occurs under forced flow condition, heat transfer includes a contribution from both convective as well as from nucleate boiling (Collier and Thome, 1994). The process of flow boiling is most commonly affected inside vertical tubes, in horizontal tubes, in annuli, and on the outside of horizontal tube bundles. The local flow boiling n

Corresponding author. E-mail address: [email protected] (S. Roy).

heat transfer coefficient is primarily a function of vapor void fraction, mass flux, heat flux, flow channel geometry and orientation, twophase flow pattern, and fluid properties. The particular relevance of convective boiling to the nuclear industry is for applications in thermal hydraulics in boiling water reactors. In these applications, liquid phase water is brought in typically under sub-cooled conditions and made to flow around nuclear fuel pins (vertical rods), held concentrically within a large vertical cylindrical containing vessel. The fuel pins serve as the principal source of heat, usually with very high energy fluxes. The liquid water undergoes phase change in the vicinity of the heated fuel rods, even when other parts of the column may continue to be under sub-cooled or saturated liquid conditions. This kind of configuration leads to a differential distribution of vapor and liquid phases, with the vapor tending to segregate both radially and axially, as it forms along the vertical height of the vessel. Indeed, in turn this segregated vapor drives the liquid circulation, and in natural circulation boiling water reactors, is the sole cause of the flow of the two-phase mixture to occur. On the other

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Please cite this article as: Yadav, A., Roy, S., Axial and radial void fraction measurements in convective boiling flows. Chem. Eng. Sci. (2016), http://dx.doi.org/10.1016/j.ces.2016.04.038i

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hand, in forced circulation boiling, in which there is a default flow profile even when the energy flux from the heater rods is zero, this local segregation of vapor and liquid is significantly modulated both by the energy flux from the rods as well as the flow patterns as a result of the inlet hydrodynamic conditions as well as the geometry available for the flow and heat transfer to occur. One of the main challenges in operating this kind of a reactor system are in the complexities of two-phase flow around the rods driven by a vertically distributed heat flux in the rods (Todreas and Kazimi, 2012). This is because the void fraction (vapor fraction) distribution significantly affects the reactor power and is one of the important parameters that determine the heat transfer capability and the possible occurrence of critical heat flux (Collier and Thome, 1994). The complex phenomenon of convective flow boiling can be divided into several regimes on a qualitative basis, depending on the local flow conditions, namely bubbly, slug, churn, annular, wispy annular and mist flow (Todreas and Kazimi, 2012). Bubbly flow regime is characterized by the vapor bubbles that are dispersed in the form of discrete bubbles in the continuous liquid phase. The shapes and sizes of the bubbles may vary widely, but they are notably smaller than the pipe diameter. Moving up the vertical tube, the vapor fraction increases, and slug flow regime is observed. In this regime, vapor bubbles collide and coalesce to form larger bubbles similar in size to the pipe diameter. Further moving up the containing tube, churn flow is observed, the flow becomes unstable, and the liquid travels up and down in a chaotic manner, although the net flow is directed upwards. Annular flow regime follows churn flow regime, where the bulk of the liquid flows as a thin film on the wall with the vapor as the continuous phase flowing up the center of the tube, forming a liquid annulus with a vapor core whose interface is disturbed by both large-magnitude waves and chaotic ripples. Annular flow regime is followed by wispy annular and mist flow, where the entrained droplets congregate to form large lumps or wisps of liquid in the central vapor core with a very disturbed annular liquid film. In mist flow regime, the annular liquid film becomes very thin, such that the shear of the vapor core on the interface is able to entrain all the liquid as droplets in the continuous vapor phase (this regime is inverse of the bubbly flow regime) (Todreas and Kazimi, 2012). It is important to note that the above description is largely qualitative and based on photographic visualization of the flow fields. In general, the underlying flow physics, involving the nucleation of bubbles, their growth and departure, and possible coalescence as they meet other bubbles in the neighborhood, as well as re-condensation as they move to regions of relatively cooler liquid, is not totally understood and even more difficult to model from first principles. One of the important impediments to developing reliable models also has to do with the inability to have reliable quantitative experimental observations of flow variables such as local void fraction, local liquid velocity, and local temperature. The present contribution relates to the measurement of local void fraction in such systems. The vapor void fraction inside a vertically heated tube varies axially as well as radially. The void fraction distribution in turn affects the liquid velocity distribution and hence is a characteristic feature of the prevalent flow regime. Void fraction distribution is dependent on the mass flux of liquid, inlet sub-cooling and heat flux of the heater. Even if the heat flux is independent of the elevation, as the liquid progressively vaporizes, the flow develops along the height of the boiling tube. Thus, it is pertinent to measure void fraction both along the height and along radial location in such a column. This information is crucial for providing validation data for thermal–hydraulic CFD codes, as well as for the design of nuclear safety systems based on more conventional methods. Extensive research has been done in the area of sub-cooled boiling flow and comprehensive review of these works has been reported by Lee and Bankoff (1998) and Bartel et al. (2001). These reviews suggest that several researchers have attempted to measure void fraction in

sub-cooled boiling flows; however all of them fall short on some aspect or the other. Roy et al. (1994) measured void fraction, gas velocity and bubble diameter in R-113 (refrigerant) boiling flows using dual sensor fiber optic probe, while Lee et al. (2002) measured void fraction distribution for water boiling flows using an intrusive conductivity probe. However, all these measurements were performed at certain axial positions only and hence no data on the axial development of local flow parameters have been reported. Situ et al. (2004) also measured void fraction distribution, interfacial area concentration, and interfacial liquid velocities using double sensor conductivity probe. Their experiments were conducted on sub-cooled flow boiling water and reported data on the axial development of void fraction, gas velocity, interfacial area concentration and interfacial velocity. They also validated constitutive equations of distribution parameters, drift velocity, and bubble Sauter mean diameter using their experimental data. From the literature review, it become clear that most of the past work is limited to sub-cooled boiling and limited experimental conditions (there is almost no data on saturated boiling conditions, which are of greater relevance to the more modern nuclear reactor technologies). Further, it is limited to void fraction measurement at only a few axial locations, and that too with intrusive probes (which could potentially alter the flow significantly in these highly unstable boiling flows, hence making the measurements inaccurate). Additionally, the experimental results available in literature are mostly in the sub-cooled flow regime using invasive void fraction measurement techniques. Whereas, in present case the void fraction measurements are done at axial locations which are in subcooled as well as in saturated regime using densitometry, which is clearly non-invasive in nature. Additionally, the experimental conditions presented are for low flow rate forced convective boiling flows, whereas the results reported in literature are predominantly for high flow rate conditions. In this contribution, measurements for radial and axial void fraction distribution are reported for vertical up-flowing boiling flows in a single cylindrical flow channel with a centrally located heating rod. All cases presented are for forced convection boiling, i.e., the liquid water is pumped into the system and flows independent of the boiling heat flux. For making the void fraction measurements, the gamma ray densitometry (GDT) technique has been used. GDT is a well-suited technique for measurement of void fraction non-invasively and has been extensively used for measuring void fractions in many nonboiling gas–liquid flows (Yadav et al., 2016), and in some flow boiling systems (Kok et al., 2001). The experimental data has been rationalized using the one-dimensional drift flux model.

2. Experimental 2.1. Boiling flow setup The schematic diagram of the experimental setup is shown in Fig. 1. It consists of a tank for holding the water, preheater having a 1 kW heater for heating the liquid water to desired temperature (in the sub-boiling range), and a pump to deliver the water from tank to the inlet of glass column. The experiments were carried out in the vertical, concentric annular test section shown in Fig. 1. The outer tube of this section is a 75 mm inner diameter glass tube that allows for visual observation. The inner tube is a heating rod, which has an outside diameter of 40 mm with a heating length of 690 mm. The entire inner heater tube was connected to a 3 kW power supply. The inlet water temperature could be varied by changing the power of the heater fixed in the pre-heater tank. The applied heat flux was assumed to be constant and was calculated based on the electric power consumed by the heater rod and was verified by performing a heat balance across the test section for the case in which heat flux and inlet temperature were such that no bubbles were being formed at the

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Fig. 1. Schematic diagram of boiling flow experimental setup.

heater surface (single phase heating). Heat flux calculated from the heat balance across the test section was 12% less than that calculated from the electric input power. This difference is attributed to the heat losses due to convection outside the test section. The temperature at the heater surface was also measured at five axial locations for the boiling flow case and were found to be nearly constant which also ascertains the fact that heat flux is constant at different axial locations. The power of the 3 kW heater could be regulated using a variable rheostat (Variac). Accordingly, the heat was generated uniformly in the heated length where the convective boiling flow takes place in the annulus. Five K-type thermocouples were placed at equal distances in glass column to measure the liquid temperature at different axial locations. A similar thermocouple had also been placed in the preheater tank to monitor the temperature of inlet water. De-mineralized water was used as the working fluid in all experiments. Using the Gamma-ray densitometry (GDT) technique (detailed described below), seven chordal scans were performed at a different location in annulus region to get radial (scans at same plane) and axial (scans at different planes) void fraction distribution. The distance between two consecutive chordal scans is 2 mm. Chordal time averaged vapor void fractions were calculated using the formula given by Eq. (5). 2.2. Gamma-ray densitometry (GDT) technique Gamma ray densitometry (GDT) has been in use as a non-invasive technique for monitoring two-phase/three-phase flows for several decades. In this technique, a γ-ray emitting radioactive source (such as Cs-137), and a scintillation detector that is sensitive to γ-ray photons, is placed on a horizontal plane and positioned precisely perpendicular to the vertically erected boiling flow column. A scintillation detector with an encased γ-ray sensitive NaI/Tl photo-sensor along with a hermetically sealed photomultiplier tube (PMT) is arranged collinearly with the source. To create a well-defined narrow beam, a lead collimator was placed in front of the source, which was already encased in a lead chamber. The collinearly arranged source–detector assembly is placed on a horseshoe shaped gantry, which can be translated in the horizontal plane to scan different chords in the flow boiling column. For scanning different axial planes, the gantry can be raised up and down using a hydraulic lift, and once a level of scanning was fixed, the horizontal movement of source–detector assembly was enabled. Schematic of the

densitometry setup used in present work are shown in Fig. 2. Densitometry is a transmission tomography technique that works on the principle of mass attenuation of radiation. When a material with a linear attenuation coefficient, μ, is placed between the source and the detector, the count rate (I) measured by the detector can be written as:

I = I0 e−μd

(1)

where I0 is the count rate measured in vacuum, and d is the distance traversed by radiation passing through the material. If the beam passes through a multiphase system (for example, in the present case, it would pass through liquid and vapor phase), the Eq. (1) can be modified as:

ITP = I0 e−μm dm e (−μliq dliq − μvap dvap )

(2)

where μliq is the attenuation coefficient of water, μvap is the attenuation coefficient of vapor and ITP is the count rate for two phase. In case of single-phase flow (either liquid phase or vapor phase), the count rate of single phase can be defined as:

Iliq = I0 e−μm dm e−μliq dliq

(3)

Ivap = I0 e−μm dm e−μvap dvap

(4)

The density of water vapor is very close to the density of air. Hence, both water vapor and air have nearly same attenuation property for γ -rays. Owing, to this similarity, the measurement of count rate in column filled with water vapor will be practically same as that of empty column. In the present contribution, the measurement was done for empty column and the count rate obtained is equal to count rate for vapor phase. Combining Eqs. (2)–(4), the following relationship is obtained for the chordal time averaged void fraction of vapor phase:

ln (ITP ) − ln (Iliq ) dvap = =ψ ln (Ivap ) − ln (Iliq ) d

(5)

where ψ is a discrete time-averaged chordal void fraction, Iliq is the count rate for pure liquid phase and Ivap is the count rate for pure vapor phase. The significance of Eq. (5) is that the time averaged chordal void fraction determined in this way is independent of I0 (which depends on the source strength, the distance between the

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Fig. 2. Schematic diagram of gamma ray densitometry.

source and detector, and the detector efficiency), independent of μ liq and μvap (which depend on the pressure and temperature), and also independent of the attenuation in structural material (glass column). This technique has been employed extensively for measuring void fraction in systems like bubble columns (Veera and Joshi, 1999), fluidized bed (Mudde et al., 2005), and boiling flows (Zeitoun and Shoukri, 1997,; Kok et al., 2001). Gamma ray densitometry is simple and relatively very cheap compared to other non-intrusive measurements techniques like X-ray tomography (Kok et al., 2001). The key element of using gamma ray densitometry in this work is to keep mechanical aspects simple and use of proper reconstruction algorithm for converting chordal time averaged void fraction discrete data to radial time-averaged data. When one performs a GDT experiment and back-calculates the void fraction using Eq. (5), what one obtains is a time-averaged chordal void fraction along a projection. The measurement is “chordal” because the movement of the source–detector assembly on the horseshoe-shaped gantry (as shown in Fig. 2) is along different chords of the circular cross-section of the column. However, for validating CFD results, or for feeding into design equations, or for calculation of area-averaged void fraction, the desirable void fraction profile should have a radial (and not chordal) variation. For effecting this conversion, we invoke the concept of the Abel transform, which involves the line integral of the radial axisymmetric function (ε(r)) along a chord, which also becomes the line of projection for source–detector assembly (Yadav et al., 2016). Under the assumption axisymmetric radial function (ε(r)) described above, the Abel transform can be defined as: Forward (relating the chordal profile to the radial profile):

φ (x) = 2

∫x

R

r 2 − x2

(6)

∫r

R

⎛ (dφ (x)/dx) ⎞ ⎞ ⎜ ⎟ dx⎟⎟ ⎝ x2 − r 2 ⎠ ⎠

(7)

The measured projection chord is just the line integral of ε and is related to ψ through the following relation:

ψ (x) =

ε ( ri ) =

φ (x) 2 R2 − x2

(8)

However, mechanical limitations posed by densitometry leads to a finite number of experimentally measured data ψ (x ) (and not a

n

1 2

2π [[2n + 1] Δx]

∑

n

ϕ ( xj )

j =−n

⎛

jk ⎞ ⎛ ik ⎞ ⎟ ⎟J ⎜ ⎝ 2n + 1 ⎠ 0 ⎝ 2n + 1 ⎠

∑ k cos ⎜ k =0

(9)

It suffices to state here that this allows the conversion of experimentally obtained chordal averaged void fraction profiles to radial void fraction profiles, to arbitrary accuracy. Details of the MFH method are given in Appendix A.

3. Theory Two-phase boiling flows in vertical channels may be, in a onedimensional sense, viewed in the perspective of the classical drift flux model. The first step in the analysis is the calculation of thermal equilibrium quality using a heat balance. The next step involves calculation of flow quality from thermal equilibrium quality. These two quantities are equal when flow is at thermodynamic equilibrium. Thermal equilibrium quality is an important parameter to discuss the thermal effect on the flow parameters. If no heat loss from the test tube and axial heat conduction is assumed, xeq, is estimated by (Kok et al., 2001):

ϵ (r ) rdr

Inverse (relating the radial profile to the chordal profile):

1⎛ ϵ (r ) = − ⎜⎜ π⎝

“continuous” function ψ ), and hence direct inversion of Eq. (8) using finite difference method becomes an ill-posed problem. In the present contribution, experimentally obtained time averaged chordal void fraction discrete data was converted to radial void fraction profiles using Modified Fourier–Hankel method (MFH) equation:

x eq (z ) =

Cpf ΔTin qzξ + ρf A c vf , in hfg hfg

(10)

where Cpf, ΔTin , hfg, z, ξ , ρf , and Ac are the liquid specific heat, the inlet sub-cooling, the latent heat, the heated length, the heated perimeter, the liquid density, and the flow channel cross-section area, respectively. The negative value of thermal equilibrium quality means liquid is in sub-cooled condition while a positive value of thermal equilibrium quality corresponds to the saturated liquid condition. The determination of flow quality, x, is trivial in all regions except in the subcooled region (Kok et al., 2001). The model of Lahey and Moody (1993) provides a functional relationship of flow quality with equilibrium quality at the bubble departure point:

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x (z ) = x eq (z ) − x eq, d e( ( xeq (z )− xeq, d )/ xeq, d )

5

(11)

where xeq, d is the equilibrium quality at the bubble departure point. Since, the value of xeq, d is negative ( x > xeq, d ), the correlation for determining equilibrium quality at the departure point is represented by following relation (Saha and Zuber, 1974):

x eq, d = − 0.0022

q′′De Cp kl hfg

for

NPe < 7 X 104

(12)

where q′ , is the wall heat flux, De is the equivalent diameter and kl is the liquid thermal conductivity. The calculation of void fraction, α, is done from quality using the one-dimensional drift flux model:

α=

x

(x +

ρg ρl

(1 − x)

)( C

0

+

vgj j

)

(13)

The one-dimensional drift flux model was first envisaged by Zuber and Findlay (1965) and later developed for different conditions by Ishii (1977). This is clearly an approximate formulation as compared to detailed two-fluid flow model. The appeal of one dimensional drift flux model lies in its flexibility and obvious simplicity over a complex two-fluid model, such as brought about by computational fluid dynamic (CFD) model involving fluid mechanics, heat transfer and phase change. Thus, such a model has immense utility in the scaling of the designs and rationalization of experimental data, such as in this work. The one-dimensional drift flux model is given by following equations (Zuber and Findlay, 1965):

vg

=

αvg α

=

jg α

=

αvgi αj j + α j α

= C0 j + Vgj

(14)

where C0 is the distribution parameter, Vgj is the void fraction weighted mean drift velocity and vgi is the drift velocity of the gas phase defined as gas phase velocity with respect to mixture volume center. C0 and Vgj are defined as:

C0 ≡

αj α j

and

Vgj =

αvgj α

(15)

The schematic representation of the drift flux model in shown in Fig. 3. In this work the value of distribution parameter used is given by (Edelman and Elias, 1981):

C0 = 1.2

(16)

The constitutive equation for Vgj is given for bubbly flow regime is given by following relation (Dix, 1971):

⎛ ⎞1/4 gσ Δρ ⎟ Vgj = 2.9 ⎜⎜ 2 ⎟ ⎝ ρf ⎠

(17)

where Δρ is the density difference between liquid and vapor, σ is the surface tension and ρf is the liquid density. As will be clear in the discussion that follows, this rather simple model for boiling two-phase flow in a vertical tube can rationalize well the experimental observations reported in this work.

4. Results and discussion As stated earlier, the void fraction measurement in this work has been performed using gamma ray densitometry (GDT), and the experimentally measured time-averaged chordal void fraction profiles have been converted to corresponding radial void fraction profiles using Modified Fourier–Hankel (MFH) method discussed

Fig. 3. Schematic representation of the drift flux model.

in Appendix A. At each radial location the time of acquisition is fixed at a long enough time: typical acquisition time for these experiments is 30 min for each chordal scan to collect sufficient experimental statistics. The experiments at each chordal location is repeated for five times. The error bars for each chordal location is added in Fig. 5(a). Since, radial void fraction profiles are reconstructed based on chordal data, same error bar is also reported for corresponding radial location. The error in terms of standard deviation for area averaged void fraction at each axial location is 0.0065. The experimental conditions are tabulated in Table 1. The preheater arrangement is shown in Fig. 1 gives us the freedom to fix the inlet liquid temperature at desired level of sub-cooling. The experimental conditions are chosen in such a way that conditions of sub-cooled as well as saturated boiling are observed in the test section. The void fraction measurements were performed at seven different axial levels, i.e. at z¼ 26, 32, 38, 44, 50, 58 and 64 cm, respectively. At each axial location, the radial void fraction measurements were performed at varying radial locations. The experimental conditions mentioned in Table 1 are essentially in laminar flow regime if one were to base the Reynolds number on the inlet liquid velocity. Fig. 4 shows photographs of the influence of different mass flux conditions for a fixed inlet liquid temperature on convective boiling flow. The photographs clearly depict that, at low mass flux the boiling is vigorous and net vapor generation starts at lower end of the heater. Also, the amount of vapor generated at any fixed axial location is more in the case of lower inlet mass flux as compared to high mass flux condition. It can be clearly seen that fraction of total axial length in which vapors are observed varies as mass flux changes, it is nearly nearly half for G three-fourth for G¼2.89 kg m  2 s  1, ¼8.67 kg m  2 s  1 and nearly one-fourth for G¼ 12.1 kg m  2 s  1, respectively. Clearly, when the inlet mass flux is low, the residence time of the fluid in the vessel is lower, so that more vaporization occurs, and the resultant vapor flux at the exit of the containing tube is higher. The experimentally measured time averaged chordal void fraction distribution is converted time-averaged void fraction using MFH method discussed in Appendix A. One such conversion of chordal to radial void fraction data is shown in Fig. 5. In Fig. 5(a), the abscissa corresponds to the dimensionless “chordal” distance in the annulus, i.e., the (x/(R − R0 )) value, while in Fig. 5(b) it corresponds to the dimensional “radial” distance in the annulus,

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Table 1 Experimental conditions. Tin (°C)

q″ (kW/m2)

G (kg/m2 s)

45 50 60 45 50 60 45 50 60 45 50 60

22.7 22.7 22.7 22.7 22.7 22.7 22.7 22.7 22.7 22.7 22.7 22.7

2.89 2.89 2.89 6.94 6.94 6.94 8.67 8.67 8.67 12.1 12.1 12.1

i.e., the (r /(R − R0 )) value. The value for both the former and latter dimensionless positions is zero for heater wall and unity for glass wall, respectively. One notes that while the two curves have similar shapes, their curvatures and quantitative values of the ordinate for a given abscissa are different, as a result of the MFH transformation. Indeed, it is the figure in Fig. 5(b) which is more relevant for design purposes, as well as serving as crucial validation data for possible CFD models of the system. The vapor void fraction profiles obtained experimentally show circular symmetry. Thus, one is justified in using the MFH transformation and reporting all results as radial profiles. In what follows, all profiles presented are the radial profiles and not the raw chordal profiles. The effect of temperature and mass flux on the radial void fraction distribution is shown in Fig. 6. The effect of varying the inlet liquid temperature is shown in Fig. 6(a). The cross-sectional mean void fraction increases with the increase in the inlet liquid temperature, as seen in Fig. 6(a). The radial void fraction profiles have vapor void fraction maximum in the region close to heater wall while the void fraction is minimum at the glass wall. The void fraction profile follows a uniform decreasing trend from heater wall to the glass wall, with the gradient in the void fraction being most significant at lower inlet liquid temperature. As the inlet subcooling decreases (i.e., the liquid enters the column closer to the

boiling point), the change in a void fraction at a fixed location near the heater wall is more than the change at a fixed location in the bulk liquid (near wall) region. The difference between radial profiles is observed more in case of 50–60 °C as compared to 45–50 °C. At lower inlet velocities of liquid, the convective heat transfer is low; hence bubbles generated at the heater wall do not condense much as they move radially outwards towards the glass wall. This effect is indirectly seen in Fig. 6(b). Therefore, the observed void fraction values obtained at the heater wall is not substantially high as compared to void fraction values at the glass wall. In fact, as the inlet velocity increases, both the average void fraction comes down, and the radial profiles also become flatter. It is prudent to examine the axial development of area-averaged void fraction. The significance of the axial variation of the void fraction comes from the fact that in forced boiling flows all the multiphase (two-phase) flow regimes can be observed in a single test section. The area-averaged void fraction is obtained by averaging the radial void fraction distribution at any given axial location is calculated using following equation: R

2π

1 α= A

∬A εdA =

∫0 ∫0 ε (r ) rdrdθ 2π

R

∫0 ∫0 rdrdθ

(18)

The evolution of area-averaged void fraction profiles at the different axial locations as a function of different inlet liquid temperature, and inlet mass flux is shown in Fig. 7. It is observed that with an increase in inlet liquid mass flux, the point of net vapor generation is shifting upwards along the direction of flow. It is observed that at minimum inlet liquid mass flux (Fig. 7(a)), the void fraction is started to be observed at axial position of 20 cm whereas for maximum mass flux (Fig. 7(d)) the void fraction started to be observed at axial position of 50 cm. The increment in inlet liquid mass flux leads to a reduction in the area-averaged void fraction at all axial locations. Increasing mass flux suppresses the activation of nucleation sites and also increases the bubble condensation rate. Hence, it leads to the production of fewer vapor bubbles, which in turn decreases the area-averaged void fraction along the length of the heater. The area-averaged void fraction at all axial locations for

Fig. 4. Photograph of boiling in experimental setup for inlet mass flux of (a) G ¼ 2.89 kg/(m2s); (b) G ¼ 8.67 kg/(m2s); (c) G ¼12.1 kg/(m2s).

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Fig. 5. Time-averaged (a) chordal void fraction profile (b) Time-averaged radial void fraction profile (dimensionless radial distance of 0 corresponds to heater wall, and dimensionless radial distance of 1 corresponds to glass wall) for q" ¼22.7 kW/m2, G ¼8.67 kg/(m2s) and inlet liquid temperature 60 °C.

Fig. 6. Variation of radial void fraction for q" ¼22.7 kW/m2 as a parameter of: (a) inlet liquid temperatures for G ¼ 8.67 kg/(m2s) and at measurement height ¼64 cm; (b) inlet liquid mass flux for inlet liquid temperature of 60 °C and at measurement height¼ 64 cm (dimensionless radial distance of 0 corresponds to heater wall, and dimensionless radial distance of 1 corresponds to glass wall).

G¼12.1 kg m  2 s  1 (Fig. 7(c)) is very low as compared to G¼2.89 kg m  2 s  1 (Fig. 7(a)). The thermal equilibrium quality at every measurement location for G¼2.89 kg m  2 s  1 is in saturated condition while that for G¼12.1 kg m  2 s  1 is in sub-cooled condition. Owing, to this fact the measured void fraction for these two mass flux condition differ by an order of magnitude. For a fixed inlet mass flux, the area-averaged void fraction increases with increase in inlet liquid temperature. The increase in the void fraction is steeper near the outlet of the test section as compared to the region near the inlet of the test section (i.e., at higher axial levels, as compared to lower axial levels, respectively). The difference between area-averaged void fraction profiles is not significant between inlet liquid temperature of 45 °C and 50 °C for all mass flux conditions. The large inlet sub-cooling at these inlet liquid temperatures makes time-averaged void fraction difference minimal (even though the transient visualization movies reveal significantly different bubbling behavior (submitted as supplementary material to this paper)). The shape of the void fraction profiles changes with the increase in the mass flux. The curve corresponding lowest mass flux (Fig. 7(a)) condition is concave downwards, till measurement location of z¼44 cm, and changes to concave upwards for axial locations thereon, while the void fraction curves corresponding to mass flux of G¼6.94 kg m  2 s  1 (Fig. 7(b)) and others (Fig. 7(c) and (d)) are concave upwards. The condition presented in the Fig. 7(a) is for

lowest mass flux and highest inlet water temperature (60 °C) among all conditions and thermal equilibrium quality is positive at all axial locations for this conditions. The regime is bubbly flow till z¼ 38 cm, hence void fraction increases from z¼ 26 cm to z¼38 cm. From z¼ 38 cm to z¼44 cm the regime is slug flow and void fraction is nearly constant between these two axial locations. As we go past z¼ 44 cm the regime changes to churn flow, owing to that the void fraction increases in concave upwards manner. The axial development of area-averaged void fraction with equilibrium quality is shown in Fig. 8. The condition with G ¼2.89 kg m  2 s  1 lies in the saturated boiling regime at all measurement locations whereas the flow condition of G ¼8.67 kg m  2 s  1 is sub-cooled at the inlet and saturated at the outlet. For mass flux of G ¼2.89 kg m  2 s  1, the area-averaged void fraction increases with increase in the thermal equilibrium quality and the void fraction reach a value of 0.52, for thermal equilibrium quality of 0.17. For mass flux of G ¼8.67 kg m  2 s  1, the area-averaged void fraction reaches zero for thermal equilibrium quality of 0.0132 and goes up to an area-averaged void fraction of 0.2 for thermal equilibrium quality of 0.018. The areaaveraged void fraction increases linearly with the increase in the thermal equilibrium quality for the case of G ¼2.89 kg m  2 s  1 whereas for G ¼8.67 kg m  2 s  1 the area-averaged void fraction follows nearly an exponential increase.

Please cite this article as: Yadav, A., Roy, S., Axial and radial void fraction measurements in convective boiling flows. Chem. Eng. Sci. (2016), http://dx.doi.org/10.1016/j.ces.2016.04.038i

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8

Fig. 7. Axial development of void fraction for q" ¼22.7 kW/m2 as function of different inlet liquid temperatures and inlet liquid mass flux of: (a) G ¼2.89 kg/(m2s), (b) G ¼ 6.94 kg/(m2s), (c) G ¼ 8.67 kg/(m2s) and (d) G ¼ 12.1 kg/(m2s).

To provide a fundamental perspective to the above experimental observations, the classical drift flux model (as discussed above) was applied for prediction of void fraction variation with axial location. The calculated void fraction using drift flux model is compared with measured experimental results in Figs. 9 and 10. The results are shown in Fig. 9 corresponds comparison of experimental data with drift flux model at different axial locations for the inlet liquid mass flux of G ¼ 2.89 kg m  2 s  1. Fig. 10 shows a comparison between experiments and drift flux model results at a different thermal equilibrium quality for G ¼ 6.94 kg m  2 s  1. The continuous line is the prediction based on the drift flux model whereas dots represent experimental data. The prediction of the drift flux model seems to be in fair agreement with the experimental data. In Fig. 9(a), the drift flux model predicts well the experimental data in upper and lower part but not in the middle part of the test section. The model prediction is quite acceptable for inlet liquid temperature of 60 °C as compared to that of 45 °C. Prediction error of drift flux model is defined as:

Prediction error for α ≡

αmea . − αcal . αmea .

(19)

The prediction errors for cases presented in Fig. 9(a) and (b) are 19.2% and 9.9% respectively, whereas for Fig. 10(a) and (b), they are 25.1% and 24.7% respectively. The drift flux model predictions in Fig. 10 seems to work very well in regions where thermal equilibrium quality is positive, but

Fig. 8. Variation of area-averaged void fraction with thermal equilibrium quality for q′′ ¼22.7 kW/m2.

not comparably well the sub-cooled regime. The reason for this discrepancy can be seen on closer inspection of the model of Lahey and Moody (1993). The model assumes that void fraction is zero before xeq, d , which is the equilibrium quality at the point of net vapor generation. Owing to this fact the drift flux model predicts

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9

Fig. 9. Comparison of drift flux model with experimental data for q′′ ¼22.7 kW/m2, G ¼2.89 kg/(m2s) and inlet liquid temperature of: (a) 45 °C and (b) 60 °C.

Fig. 10. Comparison of drift flux model with experimental data for q′′ ¼ 22.7 kW/m2, G ¼ 6.94 kg/(m2s) and inlet liquid temperature of: (a) 45 °C and (b) 60 °C.

Table 2 Known chordal and corresponding radial functions. Chordal function (1) (2)

I (x ) =

I (x ) =

16 (1 − 105 4 −(πx )2 e 3

x2)5/2 (19 + 72x2)

Radial function

ϵ (r ) = (1 − r 2)2 (1 + 12r 2) ϵ (r ) =

4 3

2

π e−(πr )

zero void fractions for high sub-cooling regions. However, void fraction is observed in the regions where thermal equilibrium quality less than quality corresponding to net vapor generation. The prediction of drift flux model combined with Dix's correlation (Dix, 1971) gives a poor match with experimental data in the region of a high degree of sub-cooling. In the present contribution, a constant value of distribution parameter, C0, has been found to be satisfactory for predicting void fraction at different axial locations and different thermal equilibrium quality conditions. The drift flux model with parameters used in this study over-predict the area-averaged void fractions in the saturated region, and under predict the void fractions for conditions which lies in the sub-cooled region. The phenomenon of sub-cooled boiling is very complex, and the parameters used in the drift flux model are not sensitive enough to capture the underlying physics. Also, for the saturated conditions the trend of the predictions is not exactly matching the experimental measurements. Since the experimental conditions in the present case are limited to fixed heat flux, high sub-cooling and laminar flow

conditions, no general correlation for the distribution parameter, C0, as a function of different underlying physical parameters influencing boiling flow can be recommended as of now. However, work on this is currently in progress. In general we feel that the value of distribution parameter, C0 should change gradually along the test section length in a more appropriate model. Indeed, this variation may be abrupt (steep) or gradual. Indeed, a better prediction by the drift flux model in the subcooled region can be made possible by having a model that accurately predicts the bubble detachment process and location in boiling test section. The saturated boiling phenomena start with bubbly flow regime and subsequently progresses to different flow regimes based on the imposed inlet conditions. The distribution parameter, C0 and Vgj should also accommodate the flow regime variation. The experimental data obtained in the present experiments points to the fact that area-averaged void fraction is a function of the axial location. Hence, relative vapor velocity with respect to the moving frame of reference, Vgj should have a component that assimilate the axial variation of void fraction. It has been experimentally observed that the vapor bubble sizes vary along the heater length. Hence, vapor bubbles have different rise velocities depending on their size at detachment. The vapor bubble velocity corresponding to different flow regimes should also be included in the constitutive relationship of the Vgj for accurate prediction of the void fraction using drift flux model.

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5. Conclusions Local measurements of void fraction have been performed for vertical upwards forced boiling flow in an internally heated annulus. The internally heated annulus consists of outer glass pipe of internal diameter 75 mm and inner heater rod of the diameter of 40 mm. The void fraction measurements were performed at seven axial locations and each axial location, and radial void fraction measurements were also done. The observed radial profiles of vapor void fraction are flat with a peak at heater surface. The axial void fraction starts with a lower value at the start of the heater, and the high void fraction is at the outlet. The trend of the axial void fraction profiles changes with a change in the inlet liquid flow conditions; effects that have been discussed in some detail in the manuscript. The existing drift flux model results were compared with the area-averaged void fraction experimental data. A good agreement of the drift flux model prediction with experimental data is found. The model predictions are poor in the highly sub-cooled region. However, our experimental data is also indicative of the need for modifications in the existing closures for drift flux models to increase its prediction accuracy.

FT (φ (x)) = φ˜(ω) =

ϵ (r ) =

1 2π

J0 (ωr ) =

The presence of singularity in the integral at the lower limit makes Abel inversion (Eq. (7)) an ill-posed problem. The experimental data are inherently noisy, and derivative of the projection greatly amplifies the noise. The discrete form of the experimental data coupled with difficulties discussed above makes the application of Abel inversion difficult. The Fourier–Hankel method avoids the singularity of Eq. (7) using the Fourier transform, by changing the variable x to its Fourier conjugate, ω . The Fourier transform of ϕ (x ) is given by the following relation:

∫0 2 π

∞

φ˜(ω) ωJ0 (ωr ) dω

(A2)

∞

∫r ( x2 − r2)−1/2 sin (ωr ) dx

(A3)

where J0 is the zero order Bessel function. The Eq. (A3) does not exhibit any discontinuity; hence it can be discretized as follows: ε ( ri ) =

n

1 2π [[2n + 1] Δx]2

∑

n

φ ( xj )

j =−n

⎛

jk ⎞ ⎛ ik ⎞ ⎟ ⎟J ⎜ ⎝ 2n + 1 ⎠ 0 ⎝ 2n + 1 ⎠

∑ k cos ⎜ k =0

(A4)

where it is assumed that I (xj ) is real and axisymmetric, the discrete Fourier transform of I (xj ) is:

⎛ αjkπ ⎞ ⎟ φ (xj ) cos ⎜ ⎝ n ⎠ j =−n n− 1

Acknowledgments

Appendix A

(A1)

Substituting the value of Eq. (6) in Eq. (A1) and changing the variable of the integration to polar coordinates, the Fourier transform of φ (x ) becomes equal to zero-order Hankel transform of ε (r ). The inverse Hankel transform leads to following Eq. (Ma et al., 2008):

φ˜(ωk ) =

The authors wish to acknowledge the financial support of the Board for Research in Nuclear Sciences (BRNS), Department of Atomic Energy (DAE), Government of India vide project no. 2012/ 35/13/BRNS. Technical and logistical support of Dr. H. J. Pant and Dr. A. K. Nayak of the Bhabha Atomic Research Centre (BARC), Mumbai is gratefully acknowledged.

∞

∫−∞ φ˜(x) e( i˜xω) dx

∑

(A5)

where n is the number of experimental data, r i=iΔr (i = 0, 1, ... n) , xj = jΔx (j = 0, 1, .... , n) and Δr = Δx = R/n, Δr and Δx denote spacing of data. Modified version of the Fourier– Hankel method has the following discrete form (Ma et al., 2008):

ε (ri ) =

α2 2nR

n− 1

∑ j =−n

⌊n / α⌋

I (xj )

∑ k=1

⎛ αjkπ ⎞ ⎛ αikπ ⎞ ⎟ ⎟ cos ⎜ kJ0 ⎜ ⎝ n ⎠ ⎝ n ⎠

(A6)

where ⌊y ⌋ denotes the nearest integer ≤ y , J0 is the zero order Bessel function. The introduction of parameter α (0 < α ≤ 1) in Eq. (A1) becomes necessary due to two reasons (Ma et al., 2008): (i) to minimize the discretization errors and (ii) to improve noise filtering capabilities. The validity of this method is tested using two chordal functions whose corresponding radial functions are known. These functions are tabulated in Table 2, and comparison between results obtained using MFH method and using radial function is shown in Fig. A1. The predictions of MFH method are exactly matching the values obtained using known radial functions. The unique quality of this method is that it is applicable even when the number of experimental data points is small. Having proved the theoretical efficacy of this method, the MFH

Fig. A1. Comparison of Abel inversion using MFH method with known radial functions (Table 1) (a) function 1 and (b) function 2.

Please cite this article as: Yadav, A., Roy, S., Axial and radial void fraction measurements in convective boiling flows. Chem. Eng. Sci. (2016), http://dx.doi.org/10.1016/j.ces.2016.04.038i

A. Yadav, S. Roy / Chemical Engineering Science ∎ (∎∎∎∎) ∎∎∎–∎∎∎

method was finally applied for converting chordal void fraction data obtained during convective boiling flow experiments into radial void fraction profiles. The following protocol is applied for converting chordal void fraction data into radial void fraction data: (i) The experimentally obtained time averaged chordal data ( ψ (x )) is converted into φ (x ) using Eq. (8). (ii) Obtained value of φ (x ) is then converted to radial holdup profiles using Eq. (A6). Results of this protocol are already demonstrated in Fig. 5.

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Please cite this article as: Yadav, A., Roy, S., Axial and radial void fraction measurements in convective boiling flows. Chem. Eng. Sci. (2016), http://dx.doi.org/10.1016/j.ces.2016.04.038i