Experiments in Fluids

, 45:187

Analysis of heat and mass transfers in two-phase flow by coupling optical diagnostic techniques

Authors

    • Institut de Radioprotection et de Sûreté Nucléaire
  • E. Porcheron
    • Institut de Radioprotection et de Sûreté Nucléaire
Review Article

DOI: 10.1007/s00348-008-0519-z

Cite this article as:
Lemaitre, P. & Porcheron, E. Exp Fluids (2008) 45: 187. doi:10.1007/s00348-008-0519-z

Abstract

During the course of a hypothetical accident in a nuclear power plant, spraying might be actuated to reduce static pressure in the containment. To acquire a better understanding of the heat and mass transfers between a spray and the surrounding confined gas, non-intrusive optical measurements have to be carried out simultaneously on both phases. The coupling of global rainbow refractometry with out-of-focus imaging and spontaneous Raman scattering spectroscopy allows us to calculate the local Spalding parameter BM, which is useful in describing heat transfer associated with two-phase flow.

List of symbols

Latin symbols

BM

Spalding parameter

Cp

heat capacity

d

diameter

D

diffusion coefficient

k

conductivity

L

latent heat

M

molar mass

\( \dot m \)

mass flow rate at the droplet interface

n

number of moles

P

TOSQAN absolute pressure

RTOSQAN

TOSQAN radius

s

local saturation ratio

\( \overline S \)

average saturation ratio

t

Time

\( \overline T _{{}}^{{_{{}}^{{}} }} \)

mean temperature

T

temperature

Vsump

sump volume

VTOSQAN

TOSQAN volume

\( \bar V \)

mean radial velocity of gas

X

volume fraction

Y

mass fraction

Greek symbols

τphase

phase-change characteristic time

τentr

entrainment characteristic time

τconv

convective heat transfer characteristic time

ρ

density

List of subscripts

air

relative to dry air

drop

relative to the droplet

int

relative to the droplet interface

gas

relative to the gas mixture

sat

relative to the saturation state

steam

relative to steam

sump

relative to the sump zone

TOSQAN

relative to the TOSQAN entire volume

Water

relative to water in liquid phase

List of dimensionless numbers

Sc

Schmidt number

Sh

Sherwood number

Nu

Nusselt number

Pr

Prandlt number

Re

Reynolds number

List of acronyms

GRR

global rainbow refractometry

MS

mass spectroscopy

PDA

phase Doppler anemometry

PIV

particle imaging velocimetry

SRSS

spontaneous Raman scattering spectroscopy

1 Introduction

The aim of this article is to study the heat and mass exchanges between spray droplets and the surrounding gas with thermal–hydraulic conditions representative of a hypothetical severe accident in a nuclear pressurized water reactor (PWR). In fact, a breach in the primary cooling circuit during a hypothetical severe accident in a nuclear power plant could lead to pressurization of the containment with steam. To maintain the integrity of the containment enclosure, spraying systems are used to reduce pressure and temperature, by means of steam condensation on the droplets.

There is an abundant literature of different models for taking account of the heat and mass transfers in multiphase flows; Sirignano (1993) drew up a comprehensive review of modelling approaches to provide an interfacial transfer expression that could be directly implemented in transport equations. Unfortunately, there is a lack of accurate measurements to validate these models for PWR applications. In fact, these models are established and validated in the fields of aerospace and combustion, and thus applied to the case of droplet vaporization (Ranz and Marshall 1952; Spalding 1952; Abramzon and Sirignano 1987). Thus, we still require a validation in the symmetrical case of steam condensation on droplets.

Moreover, the emergence of optical diagnostics over the last twenty years has enabled increasingly accurate non-intrusive measurements of parameters such as droplet temperature (van Beeck et al. 1999) and size (Glover et al. 1995) as well as gas composition (Cohen-Tanoundji et al. 1986); these measurements give a higher level of qualification of the previously established models.

Because of the typical size of containment buildings (60,000 m3) experiments are often designed with scale different from prototypical. A variety of similarity and scaling methods is available to enable transferring such experimental results to real reactor conditions. The strategy adopted to study severe accidents is thus to perform experiments at different scales and to increase the level of instrumentation and its reliability with the decrease of the experimental scale.

In view of this state of the art, the TOSQAN experiment was created to simulate typical accidental thermal–hydraulic flow conditions in reactor containment. TOSQAN is a French acronym that means TONUS analytical qualification; TONUS is a CFD code developed by CEA and IRSN and dedicated to simulate severe accident in nuclear power plants. This experiment is small compared to containment buildings, but it is large compared to its level of instrumentation. A special scaling effort has been achieved to design the test presented in this article, in order to be representative to spray release during a severe accident (Fischer et al. 2002).

In the first part of this article, we describe the TOSQAN experiment and the associated optical instrumentation. The second part presents the implemented test and its associated measurements. Finally, we analyze the heat and mass exchanges on both phases while giving a detailed description.

2 Description of the facility and its instrumentation

2.1 TOSQAN facility

The IRSN has developed the TOSQAN facility to characterize the heat and mass transfers between spray droplets and the surrounding gas under thermal–hydraulic conditions representative of a severe accident in a PWR. This experimental facility consists of a large stainless steel cylindrical vessel (4.8 m high with a 1.5 m internal diameter) with double walls, with a coolant circulating in the annulus to regulate the wall temperature between 60 and 160°C (Fig. 1). Superheated steam can be injected at different flow rates via an injection pipe located in the central part of the vessel.
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Fig. 1

Schematic view of the facility

A single nozzle is used to spray injection into the upper part of the vessel. A special system was designed to allow vertical movements over a height interval of 90 cm, so as to carry out measurements at different distances from the nozzle but through the same porthole window. In addition, the nozzle is off-centred by 40 cm in relation to the axis of the enclosure.

The gas temperature and pressure are measured using a set of more than 150 thermocouples and two pressure sensors. Deflectors are fitted above the thermocouples to prevent the impact of droplets on the sensor, thus ensuring that the measurements are only related to the gas temperature.

The thermocouples of the experiment are stacked vertically in radial rods at the locations indicated on Fig. 2.
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Fig. 2

Location of the thermocouples, nozzle and porthole viewing windows for laser measurements

For this particular test, the nozzle is off-centred by 40 cm so the different measurements—particularly the droplet temperature—can be carried out in the axis of the nozzle.

The optical measurements are carried out through 14 porthole viewing windows distributed at four levels. Each porthole window is composed of a double-glass screen, with hot air circulation in between to avoid steam condensation. To characterize the two-phase flow, we rely on two kinds of instrumentation.

On the one hand, we use conventional devices such as thermocouples, pressure sensors and flow meters, as well as commercially available techniques such as out-of-focus imaging for particle sizing (Lemaitre et al. 2006a, b) and particle imaging velocimetry (Porcheron et al. (2003)) to measure the droplet velocity field. In addition, sampling is performed to determine the gas composition by mass spectrometry. On the other hand, we developed two optical benches, dedicated for non-intrusive measurement of the local gas volume fraction (spontaneous Raman scattering spectroscopy) and droplet temperature (global rainbow refractometry). The principles of these two techniques are briefly presented below.

2.2 Global rainbow refractometry (GRR)

This technique was introduced by van Beeck et al. (1999) to measure the mean droplet temperature on a set of droplets located in a small volume of measurement. This optical diagnostic tool is based on the analysis of oscillations generated by the interference between the optical rays refracted by the droplets. This allows a determination of the refractive index, and, hence, the temperature of the droplets. However, this technique is hindered by problems linked to droplet non-sphericity that can produce a shift in the temperature measurements (van Beeck and Riethmuller 1995).

A special calibration has been established by Lemaitre et al. (2006a, b) to avoid these disturbances. This procedure is implemented in the TOSQAN experiment after each test with GRR measurements; it consists in injecting a spray at the same mass flow rate as during the test (30 g s−1), and at exactly the same temperature as the gas when initially saturated with steam. Under these conditions, there is neither heat nor mass transfer between the droplets and the gas mixture; as a consequence, the average measured droplet temperature is equal to its injection temperature as well as the gas temperature. This allows us to correct the global rainbow temperature measurement and take into account the angular shift induced by the non-sphericity of the droplets.

Once the calibration is complete, we then need to validate the measurement, using exactly the same procedure as the calibration. A spray is injected into a gas mixture having the same temperature and saturated with steam, to check whether that the GRR measurement is close to its injection temperature. This method is reproduced at different temperatures close to the value expected to be measured in the test considered here.

As the difference between the GRR measurement and the gas temperature never exceeds 1°C, this method allows us to validate the GRR measurement obtained on the TOSQAN experiment (Lemaitre et al. 2006a, b).

2.3 Spontaneous Raman scattering spectroscopy (SRSS)

This non-intrusive technique was developed on the TOSQAN facility to characterize the gas composition. Spontaneous Raman scattering is an inelastic scattering process involving the interaction of a photon with a specific vibration-rotational state of a molecule. This scattering results from a process in which energy exchange occurs between the incident photons with frequency (υ0) and a given molecule. The generated energy flux undergoes a frequency shift (±υ) that is characteristic of each polyatomic molecule depending on the occurrence of a Stokes (+υ) or anti-Stokes (−υ) transition. A full description of the Raman effect is presented by Cohen-Tanoundji et al. (1986).

Thus, if all the constituents of the gas mixture are active in Raman scattering, then the molar fractions of each constituent can be calculated using the Raman power scattered by each molecule. Porcheron et al. (2003) have presented a full description of the principle of this technique, along with the optical set up and its validation on the TOSQAN facility in the presence of droplets.

2.4 Evaluation of measurement uncertainties

Since various techniques are coupled in this study, we need to evaluate the uncertainties of all the measurements obtained in the TOSQAN experiment. Table 1 summarizes the accuracy of all the measurements carried out.
Table 1

Instrumental accuracy in the TOSQAN experiment

 

Measurement technique

Uncertainty

Pressure

Pressure sensor

0.01 bar

Droplet temperature

GRR

1°C

Droplet and gas velocity

PIV

<5%

Gas temperature

Thermocouples

1°C

Droplet size

Out-of-focus imaging

<5%

Gas composition

SRSS

<2%

3 Description of the test

Prior to the beginning of the actual test, while the gas in the TOSQAN enclosure is composed of dry air at 1 bar, the walls of the experiment are regulated at 120°C by circulating coolant fluid inside the double wall.

Then, the vessel is pressurized with 1.5 bar of dry steam, added to the 1 bar of dry air, by means of a steam injection pipe (Fig. 1).

The steam injection ceases when the pressure reaches 2.5 bar, whereas sprinkling starts with a water flow rate of 30 g s−1 at a temperature of 23°C. For this test, the spray injection is off-centred by 0.4 m. During the test, the water is drained accordingly to prevent water accumulation in the sump. To reproduce exactly the same initial conditions, dry air circulation is imposed before each test inside the vessel for 20 h, at a mass flow rate of 5 g s−1, which represents a ventilation rate of two air changes per hour, and consequently a total of 40 air changes before a new test.

The spray is generated with a TG-3.5 nozzle (spraying systems). This starts a transient state of depressurization in the vessel, which is continued until an overall steady state is obtained where the pressure and the average temperature of the gas mixture present in the vessel remain stable. Throughout the test, the temperature of the vessel walls is regulated at 120°C, owing to the circulation of coolant fluid inside the double wall.

Figure 3 shows that the spray release is characterized, as expected, by a sharp decrease in mean temperature \( \left( {\bar T_{{{\text{TOSQAN}}}} } \right) \) and an overall depressurization. The mean gas temperature is computed by averaging the temperature measured by 42 thermocouples uniformly distributed over the entire vessel volume (Fig. 2).
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Fig. 3

Variation of mean temperature and relative pressure with tim, in the TOSQAN vessel throughout the test

To perform the measurements for characterization of both phases (droplet size, velocity and temperature, as well as gas volume fraction), this test must be reproduced identically several times.

In Fig. 4, we can observe that, as long as the temperature of the injected water is exactly the same, the test is perfectly reproduced.
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Fig. 4

Reproducibility of the spray test

The test analysis consists of two parts. First, we carry out a global analysis of the heat and mass transfers, followed by a more detailed and local study of the exchanges between the droplets and the gas.

4 Heat and mass transfer analysis

4.1 Global analysis

For a more in-depth description of the test, we require thermodynamic parameters that fully describe the time course of the test. These parameters are the total number of moles inside the vessel nTotal, the average saturation ratio \( \overline S (t) \) and the average gas temperature \( \overline T _{{{\text{TOSQAN}}}} (t). \)

The total number of moles inside the vessel is computed according to equation (1) i.e., assuming an ideal gas.

To compute this number, the volume of the dispersed phase is ignored. This hypothesis is justified since the spray mass flow rate is 30 g s−1, the experimental enclosure is 4.8 m high, and the droplet velocity is more than 1 m s−1. As a consequence, the residence time of the droplets in the gas is less than 5 s, leading to a liquid volume in suspension smaller than 1.5 × 10−4 m3, which is negligible compared to the entire TOSQAN volume (7 m3)
$$ n_{{{\text{total}}}} = \frac{{P_{{{\text{TOSQAN}}}}V_{{{\text{TOSQAN}}}} }}{{R\bar T_{{{\text{TOSQAN}}}} }} .$$
(1)
Then, according to equation (2), we compute the global saturation ratio. In this equation, the steam partial pressure \( \left( {P_{{{\text{steam}}}} } \right) \) is derived from the pressure and the initial air pressure before the steam injection (all values at time t, unless otherwise specified)
$$ \bar S(t) = \frac{{P_{{{\text{steam}}}} (t)}}{{P_{{{\text{sat}}}} \left( {\bar T_{{{\text{TOSQAN}}}} (t)} \right)}} = \frac{{P_{{{\text{TOSQAN}}}} (t) - P_{{{\text{air}}}} (t)}}{{P_{{{\text{sat}}}} \left( {\bar T_{{{\text{TOSQAN}}}} (t)} \right)}} = \frac{{P_{{{\text{TOSQAN}}}} (t) - P_{{{\text{air}}}} (t_{0} ) \cdot \frac{{\bar T_{{{\text{TOSQAN}}}} (t)}}{{\bar T_{{{\text{TOSQAN}}}} (t_{0} )}}}}{{P_{{{\text{sat}}}} \left( {\bar T_{{{\text{TOSQAN}}}} (t)} \right)}}. $$
(2)
The time-evolution of the total number of moles of gas in the vessel, the global saturation ratio and the mean gas temperature are presented together on Fig. 5. In this figure, the mean gas temperature (°C) is multiplied by four for scaling reasons, to represent the three curves on the same graph.
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Fig. 5

Global characterization of the test

Basing our analysis on the evolution of these three parameters, we can identify three characteristic times (t1, t2 and t3) from the test result.

For times after t2, the uncertainty on the saturation ratio is smaller than 0.03, which represents 3% of the average saturation ratio. As a consequence, in cases where the values of average saturation ratio \( \left( {\overline S } \right) \) are sometimes slightly higher than unity (Fig. 5), this is attributed to the measurement uncertainty.

Throughout this study, the uncertainties on the different calculated parameters are evaluated using the guide to the expression of uncertainty in measurement (ISO 1993).

Therefore, we observe that, until time t1, the total number of moles increases in the containment, which is correlated with a sharp decrease in temperature. Nevertheless, as the leakflow rate is negligible, the change in total number of moles corresponds to variations in the number of moles of vapor in the containment. The second characteristic time t2 corresponds to the time when the gas becomes totally saturated inside the vessel. Finally, time t3 corresponds to the time at which all the parameters of our analysis become stable.

These three characteristic times lead us to subdivide the test into four phases denoted A, B, C and D in Fig. 5.

Phase A of the test is a very short (100 s), corresponding to a strong droplet vaporization regime. This first phase is due to a low initial saturation ratio and an initial mass stratification, particularly in the bottom of the vessel (sump, Fig. 1). This last point is discussed in the section treating the local analysis of heat and mass transfers.

The variation in total number of moles highlights the fact that this second phase of the test (B phase) corresponds globally to condensation. The gas temperature decreases due to convective heat transfer between the gas and the droplets injected at 23°C. At the end of this phase, the atmosphere is globally saturated in the containment.

The convective heat transfer between the droplets and the gas drives this third phase (C phase). This heat transfer still cools the gas and, as a consequence, steam condenses on the droplets at a rate that keeps the vessel globally saturated.

The final phase of the test (D phase) corresponds to an equilibrium regime. During this phase, all the studied parameters remain constant. Throughout this period, the thermal power removed from the gas due to convective heat transfer with the droplets is exactly compensated by the heat flux transferred to the gas from the heated walls. As mentioned above, this test is repeated many times, in order to acquire all the local measurements needed to characterize the heat and mass transfers. The temperature of the injected water throughout these tests was not always exactly the same as during our measurement campaign. Thus, we examine the influence of this temperature on the evolution of the test, and particularly on the total depressurization of the containment (Fig. 6).
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Fig. 6

Influence of droplet temperature at the injection nozzle on the total number of moles condensed during the test

On this graph, we observe that the total number of moles of condensed steam increases with decreasing temperature of the injected water. Hence, more steam can condense on colder droplets before the system reaches its equilibrium temperature: lowering the temperature of spray droplets is a very efficient means of condensing steam in the containment atmosphere.

Consequently, the local measurements presented in the next section correspond to tests performed with an injection temperature of 23 ± 0.5°C (uncertainty of thermocouple measurements).

4.2 Local measurements

Different local measurements were carried out during this test. As mentioned above, not all these measurements were obtained on the same day, but they were always carried out under exactly the same thermodynamic conditions. The results of these measurements are presented either as variations with time at a given distance from the nozzle, or as fields or vertical profiles during the equilibrium phase.

4.2.1 Transient measurements

The time-evolution of the steam volume fraction (Xsteam) was recorded on the axis of the spray and at 25 cm from the nozzle orifice, using the SRSS technique (Fig. 7).
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Fig. 7

Time-evolution of steam volume fraction at 25 cm below the injection nozzle

We can subdivide this time-evolution into three phases for this particular measurement. The first phase corresponds to the first 60 s of the test, and is associated with an increase in steam volume fraction. The second phase corresponds to a strong decrease of the steam volume fraction, and continues until its stabilization (third phase) after 3,500 s of spraying.

4.2.2 Steady-state measurements

In this part of the paper, we present all the measurements needed for describing the test.

4.2.3 Gas temperature measurement

The most interesting zone for our study is the near field of the spray injection. Thus, we show the gas temperature during the steady state of the test along a radial profile 25 cm from the nozzle (Fig. 8) and on a vertical profile in the axis of the nozzle (Fig. 9).
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Fig. 8

Radial profile of gas temperature at 25 cm from the nozzle

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Fig. 9

Vertical profile of gas temperature in the axis of the nozzle

As one of the thermocouples of the highest rod is in the axis of the nozzle orifice (Fig. 2), a gas temperature vertical profile is measured in the axis of the spray. To acquire a vertical profile of gas temperature, the nozzle is displaced during the equilibrium phase; as a consequence, each data point plotted on Fig. 9 corresponds to the temperature recorded by the thermocouple in the axis of the spray when the measurement is stabilized (600 s after the nozzle displacement). It is important to point out that displacement of the nozzle does not modify the equilibrium state; this is because the displacement (250 mm) is an order of magnitude lower than the height of the TOSQAN experiment (4,800 mm).

The gas cooling induced by the spray is observed on Fig. 8, with a minimum temperature in the axis of the nozzle. Moreover, this profile is dissymmetric with regard to the nozzle axis, because the nozzle is off-centred; as a consequence, the temperature gradient is steeper on the side towards the off-centred nozzle, due to the proximity of the heated wall.

Figure 9 shows that the gas temperature increases by 1°C when the distance from the nozzle increases by 250 mm. This measurement is useful in the following sections of this article (Table 2).
Table 2

Inputs of the simulation

Inputs of the simulation

Measurement techniques

Measurement results

Uncertainty

Droplet initial temperature

GRR measurement at 5 cm from the nozzle orifice

First measurement point of Fig. 10

1°C

Droplet velocity vertical profile

PIV measurements

Fig. 13

<5%

Gas temperature vertical profile

Thermocouple measurements

Fig. 9

1°C

Droplet initial size

Out-of-focus imaging measurements

dGRR computed from Figure 11

<5%

Gas saturation ratio

SRSS associated with thermocouple measurements

97%

<2%

4.2.4 Droplet temperature measurement

Figure 10 presents a vertical profile of droplet temperature at different distances from the nozzle, measured using the GRR technique. In the figure, the measurement points are acquired on different tests by translating the nozzle vertically. Using this procedure, we ensure that the nozzle displacement does not modify the reproducibility of the test (for the global parameters \( P_{{{\text{TOSQAN}}}} \,{\text{and}}\,\bar T_{{{\text{TOSQAN}}}} \)).
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Fig. 10

Droplet temperature at different distances from the nozzle

4.2.5 Droplet size measurement

Figure 11 presents the size distribution of the droplets measured during the test steady state, at 1 m from the nozzle (on the second level of porthole windows, Fig. 1). Measurements cannot be performed closer to the nozzle with the out-of-focus imaging technique, due to overlapping of the interference fringes (Lemaitre et al. 2006a, b). Nevertheless, measurements can be performed closer to the nozzle, but outside the TOSQAN vessel, using another technique based on phase doppler anemometry. Unfortunately, this technique cannot be implemented in our experiments, due to the configuration of our optical accesses.
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Fig. 11

Droplet size distribution at 1 m from the nozzle during the steady-state phase

The spray size distribution can be satisfactorily fitted by a log-normal distribution with a median diameter of 100 μm and a geometrical standard deviation of 1.65. The computed arithmetic mean diameter and the Sauter mean diameter are equal to 141 and 207 μm, respectively \( \left( {d_{{10}} = 141 \pm 7\,\upmu{\text{m}},d_{{32}} = 207 \pm 10\,\upmu{\text{m}}} \right). \) Measurements of droplet size at injection were performed at the same distance from the injection nozzle but at low gas temperature to prevent droplet vaporization, and with addition of air in order to reproduce the same gas density. As a consequence, we reproduced the same aerodynamic effects on the droplets. The resulting measurements match very closely with the result presented in Fig. 11.

4.2.6 Measurement of droplet velocity

The droplet velocity fields are measured throughout the spray test using the PIV technique. Our PIV set up is a DANTEC commercial system composed of a double-cavity Nd YAG laser doubled in frequency to provide a laser beam at a wavelength of 532 nm. The flow is seeded with the spray droplets, whose displacement is measured between the two pulses of the laser. This displacement is measured with a Hisense camera (1280 × 1024), equipped with a 60 mm lens and an aperture of f22. Various tests were performed to determine the optimum setting for laser intensity and inter-pulse times (which vary with the distance from the nozzle due to droplet deceleration) and, finally, the interrogation size area and overlap. We chose an interrogation size of 32 × 32 for the images, without overlap. More details on the use of the PIV technique to measure droplet velocity are provided by Comer et al. (2001) and Porcheron et al. (2003) for the TOSQAN application.

Figure 12 shows the droplet velocity field measured in the near field of the nozzle, presented as iso-levels of vertical velocity, during the D phase of the test. A strong gradient of the axial droplet velocity is observed at the edges of the spray, linked to a strong shear velocity component between the droplets and the gas at that location, which induces a strong momentum transfer between the two phases.
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Fig. 12

Droplet velocity during the equilibrium phase (m s−1)

Different velocity fields were measured on different spray tests at different distances from the nozzle, by vertical displacement of the nozzle. The data points are connected up and plotted in Fig. 13 (different colors indicate iso-levels of vertical velocity component), which shows the variation of droplet axial velocity as a function of the distance to the orifice of the nozzle along the axis.
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Fig. 13

Vertical profile of the droplet vertical velocity

We can observe a perfect continuity of the vertical component of the droplet velocity measured in the axis of the nozzle, which reflects the deceleration of the droplets as a function of the distance to the nozzle, due to the transfer of momentum to the gas. This observation is useful in the following (Table 2), for validation of the two-boundary-layer model.

4.3 Local analysis

In Fig. 14, the time-evolution of steam volume fraction is plotted along with the gas temperature measured at the same point, in order to compute the local saturation ratio. These measurements are acquired in the axis of the spray and at 25 cm from the nozzle. The parameters presented in this figure (Tgas, Psteam, s) are analogous to those of the global analysis \( \left( {\overline T _{{{\text{TOSQAN}}}} ,n_{{{\text{total}}}} ,\overline {S(t} )} \right), \) and thus allow us to highlight the similarities and differences between the local and global dynamics of the test.
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Fig. 14

Local characterisation of test results

Using this figure, we can compute the local saturation ratio using equation (3), while the associated uncertainty is evaluated according to the GUM
$$ s = \frac{{P_{{{\text{steam}}}} }}{{P_{{{\text{sat}}}} (T_{{{\text{gas}}}} )}} = \frac{{X_{{{\text{steam}}}} P_{{{\text{TOSQAN}}}} }}{{P_{{{\text{sat}}}} (T_{{{\text{gas}}}} )}}. $$
(3)

First of all, we observe that, on the local scale, the initial droplet vaporization phase lasts only 60 s \( \left( {t_{1}^{{{\text{local}}}} = 60 \pm 3\,{\text{s}}} \right), \) whereas it corresponds to a 100 s phase on the global scale (of the whole vessel). This is due to the fact that, during the first 100 s of the test, the initial vaporization phase propagates from the near field of the nozzle to the far field. Moreover, we observe that, locally, the droplet vaporization phase ends when the gas nearly reaches saturation \( \left( {s = 0.97 \pm 0.03\quad {\text{for}}\,t > t_{1}^{{{\text{local}}}} } \right). \)

However, we observe that, globally, the first vaporization phase ends before the gas reaches overall saturation. This is because the spray does not extend throughout the entire containment volume. As a result, even if the spray zone is nearly saturated, the outer part of the spray zone is still far from saturation (but remains nearly at its initial water vapor content). Thus, we infer that the phase between times t1 and t2 corresponds to a phase of steam transfer from the spray zone to the non-spray zone.

To ensure that these phenomena occur successively, we compute the characteristic times associated with each of them. The entrainment characteristic time is evaluated using the radial gas velocity measured by the PIV technique (Fig. 15). To measure the gas velocity by this technique, the gas has to be seeded with particles before steam injection. This is carried out by introducing; silicon carbide particles (SiC) with a geometrical mean diameter of 1 μm. As the spray envelope is known from previous measurements (Fig. 12), we can obtain the gas velocity in the non-sprayed zone alone.
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Fig. 15

Gas velocity measurement

We measure an average radial gas velocity \( \left( {\overline V } \right) \) of 0.2 m s−1. Thus, using equation (4), we can derive the entrainment characteristic time (τentr) from this measurement
$$ \tau _{{{\text{entr}}}} = \frac{{R_{{{\text{TOSQAN}}}} }}{{\overline V }} \approx 3.8\,{\text{s}}. $$
(4)
The phase-change characteristic time (τphase) is computed in the following from equation (17) using the droplet temperature measurement. We observe that the phase-change characteristic time is several orders of magnitude lower than the entrainment characteristic time (see Fig. 18)
$$ \tau _{{{\text{entr}}}} \approx 5\,{\text{s}} \gg \tau _{{{\text{phase}}}} \approx 0.01\,{\text{s}}. $$
(5)

This supports our hypothesis that the phenomena are not simultaneous. Vaporization processes drive the first phase and gas entrainment drives the second phase, called here respectively, A and B.

Measurements of gas and droplet temperature are performed during the equilibrium phase of the test by vertically displacing the nozzle (this change of nozzle position has no influence on the equilibrium phase during the test). Gas temperatures are measured by thermocouples (Fig. 9) and droplet temperatures by GRR (Fig. 10).

By coupling these measurements with steam volume fraction measured using SRSS (Fig. 7) at the same location, we can compute the Spalding parameter (BM). Spalding (1952) showed that this latter parameter (Eq. 7) is needed to calculate the steam mass flow rate in equation (6) on the droplet.

In these equations, the director vectors are taken orthogonally to the interfaces and directed toward the outside of the droplet. Consequently, a positive Spalding parameter corresponds to droplet vaporization and a negative Spalding parameter to steam condensation on the droplet
$$ \dot m = \pi d_{{{\text{drop}}}} \rho _{{{\text{steam}}}} D_{{{\text{steam,air}}}} Sh_{{{\text{gas}}}} \ln \left( {1 + B_{{\text{M}}} } \right) $$
(6)
$$ B_{{\text{M}}} = \frac{{Y_{{{\text{steam}},\text{int} }} - Y_{{{\text{steam}},\infty }} }}{{1 - Y_{{{\text{steam,}}\text{int} }} }}. $$
(7)
In equation (7), the steam mass fraction (Ysteam,∞) is directly derived from SRSS measurements, whereas the steam mass fraction at the interface (Ysteam,int) is computed assuming the fluid is at thermodynamic equilibrium, and therefore saturated with steam according to equation (8)
$$ Y_{{{\text{steam,}}\text{int} }} = \frac{{P_{{{\text{steam}},\text{int} }} M_{{{\text{H}}_{ 2} {\text{O}}}} }}{{\rho _{{{\text{gas}},\text{int} }} T_{{\text{int} }} R}} $$
(8)
$$ \rho _{{{\text{gas,}}\text{int} }} = \frac{{\left( {P_{{{\text{TOSQAN}}}} - P_{{{\text{sat}}}} (T_{{\text{int} }} )} \right)M_{{{\text{air}}}} + P_{{{\text{sat}}}} (T_{{\text{int} }} )M_{{{\text{steam}}}} }}{{RT_{{\text{int} }} }}. $$
(9)
According to Abramzon and Sirignano (1987), the most appropriate results are obtained by computing the gas density (ρsteam) and the steam diffusion coefficient in air (Dsteam,air) at the mean temperature within the gaseous boundary layer around the droplet (Tref), which is computed using equation (10)
$$ T_{{{\text{ref}}}} = \frac{{T_{{{\text{gas}}}} + 2T_{{\text{int} }} }}{3}. $$
(10)
The droplet temperature at its interface is derived from the gas and droplet temperature measured using thermocouples and GRR, respectively, and a double boundary layer model (Fig. 16).
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-008-0519-z/MediaObjects/348_2008_519_Fig16_HTML.gif
Fig. 16

Double boundary layer model

This model developed by Lemaitre (2005) assumes a thermal boundary layer in the liquid phase and a non-isothermal diffusion boundary layer in the gaseous phase.

The heat transfer coefficient inside the liquid boundary layer is modeled using empirical correlations that depend on the flow regime inside the droplet as determined by Hendou (1992).

No flow inside the droplet
$$ Nu_{{{\text{boundary - layer}}}}^{{{\text{liquid}}}} = 0.994 + 0.848\text{Re} _{{{\text{drop}}}}^{{1/2}} {\Pr} _{{{\text{drop}}}}^{{1/3}} $$
(11)
Droplet with laminar recirculation
$$ Nu_{{{\text{boundary - layer}}}}^{{{\text{liquid}}}} = 2.567 + 0.794\text{Re} _{{{\text{drop}}}}^{{1/2}} {\Pr}_{{{\text{drop}}}}^{{1/3}} $$
(12)
Droplet with turbulent recirculation
$$ Nu_{{{\text{boundary - layer}}}}^{{{\text{liquid}}}} = 0.351 + 0.381\text{Re} _{{{\text{drop}}}}^{{1/2}} {\Pr} _{{{\text{drop}}}}^{{1/3}} $$
(13)
The heat and mass transfers inside the gas boundary layer are modeled using empirical correlations due to Ranz and Marshall (1952).
$$ Nu_{{{\text{boundary - layer}}}}^{{{\text{gas}}}} = 2 + 0.6\text{Re} _{{{\text{drop}}}}^{{1/2}} {\Pr} _{{{\text{gas}}}}^{{1/3}} $$
(14)
$$ Sh_{{{\text{boundary - layer}}}}^{{{\text{gas}}}} = 2 + 0.6\text{Re} _{{{\text{drop}}}}^{{1/2}} Sc_{{{\text{gas}}}}^{{1/3}} $$
(15)

In these empirical correlations, the Reynolds numbers are computed using droplet velocity, which is derived from PIV measurements. Hence, we compute the heat and mass transfers at the interface in order to obtain its temperature (Tint).

We show that, for our experimental conditions, the temperature gradients inside the droplets are so weak that we can consider the interface and droplet temperatures as equal. This is mainly due to the high saturation ratio in the spray zone during the D phase. Studies were previously performed by Lemaitre (2005) to ensure that temperature gradients inside the droplet do not disturb the droplet temperature measurements performed with the GRR technique.

Figure 17 presents a vertical profile of the computed experimental Spalding parameter. Using this dimensionless parameter, we can distinguish zones where steam is condensing on the droplets (BM < 0, equation 4) from zones where the droplets are vaporizing (BM > 0).
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-008-0519-z/MediaObjects/348_2008_519_Fig17_HTML.gif
Fig. 17

Computation of the experimental spalding parameter

The uncertainty on the Spalding parameter, which is also evaluated according to the GUM, varies according to the distance from the injection pipe, from 2 × 10−2 at 5 cm to 3 × 10−2 at 25 cm.

As a consequence, we can identify four zones that are designated as α, β, δ and γ in Fig. 17. To understand the phenomena occurring in each of these zones, we need to compute characteristic times for convective heat transfer (τconv) and phase change (τphase).

The expressions of these characteristic times are derived from a dimensional analysis applied to the energy balance equation for the droplet
$$ \left\{ {\begin{array}{*{20}c} {m_{{{\text{drop}}}} Cp_{{{\text{drop}}}} \frac{{{\text{d}}T_{{{\text{drop}}}} }}{{{\text{d}}t}} = \frac{{Nu_{{{\text{boundary - layer}}}}^{{{\text{gas}}}} k_{{{\text{boundary - layer}}}}^{{{\text{gas}}}} }}{{d_{{{\text{drop}}}} }}\pi d_{{{\text{drop}}}}^{ 2} (T_{{{\text{gas}}}} - T_{{{\text{drop}}}} ) + \dot m\left( {Cp_{{{\text{Steam}}}} (T_{{{\text{gas}}}} - T_{{\text{int} }} ) + L + Cp_{{{\text{water}}}} (T_{{\text{int} }} - T_{{{\text{drop}}}} )} \right)} \\ {\dot m = \pi d_{{{\text{drop}}}} \rho _{{{\text{steam}}}} D_{{{\text{steam,air}}}} Sh_{{{\text{boundary - layer}}}}^{{{\text{gas}}}} \ln (1 + B_{{\text{M}}} )} \\ \end{array} } \right. $$
(16)
$$ \tau _{{phase}} = \frac{{\rho _{{drop}} d_{{drop}}^{2} Cp_{{drop}} T_{{drop}} }}{{6LD_{{steam,air}} Sh_{{gas}} \rho _{{steam}} \ln (1 + B_{M} )}} $$
(17)
$$ \tau _{{{\text{conv}}}} = \frac{{\rho _{{{\text{drop}}}} d_{{{\text{drop}}}}^{2} Cp_{{{\text{drop}}}} }}{{6Nu_{{{\text{gas}}}} k_{{{\text{gas}}}} }}. $$
(18)
As the expression for the experimental Spalding parameter is known, the value of these two characteristic times can be computed as a function of the distance from the nozzle. In equations (16)–(18), the mean droplet diameter taken into account is dGRR (equation 22), as it is representative of GRR measurements. Moreover, it is close to the mean Sauter diameter (Fig. 11), which is a meaningful measurement of particle size since it represents the size of a droplet with the same surface-area to volume ratio as the entire spray. In addition, the Sauter diameter is known to characterize the heat and mass transfer of the droplets better than the arithmetic mean diameter (Lefebvre (1989)). These characteristic times are presented in Fig. 18.
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-008-0519-z/MediaObjects/348_2008_519_Fig18_HTML.gif
Fig. 18

Calculation of characteristic times

Since the phase-change, characteristic time in the α zone is an order of magnitude greater than the characteristic time for convective heat transfer and, moreover, because the Spalding parameter is negative, we infer that the main process leading to droplet heating is steam condensation on the latter.

In the β zone, the Spalding parameter changes from a negative to a positive value, characterizing a transition from a condensing to a vaporizing zone. Nevertheless, since the Spalding parameter is very close to zero, the characteristic time for the phase change becomes orders of magnitude greater than for the convective heat transfer. Consequently, as the gas remains hotter than the droplets, the latter continue to heat up due to the heat transfer.

In the δ zone, we observe that the phase-change characteristic time becomes smaller than the convective heat transfer time. Thus, as the Spalding parameter is positive, the droplets start to vaporize in order to dissipate the energy surplus accumulated in the β zone.

Finally, in the γ phase, the characteristic times have nearly all the same order of magnitude, thus at any single moment, the convective heat flux collected by the droplet is compensated by the vaporization.

4.4 Sump singularity

The lower part of the experimental vessel is occupied by the sump zone (Fig. 1), which displays a particular type of behavior, especially during the A phase. As shown previously, phase A of the test corresponds to vaporization, due to the fact that the initial saturation ratio is low inside the experimental vessel. This singularity of the sump zone is linked to the fact that the injected steam is homogeneously distributed inside the whole vessel, except for the sump. Indeed, we observe a very low steam volume fraction in the sump zone (Fig. 19).
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-008-0519-z/MediaObjects/348_2008_519_Fig19_HTML.gif
Fig. 19

Steam distribution inside the TOSQAN experiment measured by mass spectroscopy

This observation is attributed to the fact that the convective recirculation induced by the steam jet does not extend down into the sump zone. Thus, the steam is conveyed to the sump due to molecular diffusion processes alone, which are very slow compared to convective mixing. This hypothesis is confirmed by the temperature field inside the experimental vessel during steam injection (Fig. 20).
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-008-0519-z/MediaObjects/348_2008_519_Fig20_HTML.gif
Fig. 20

Temperature field during steam injection (°C)

Thus, the spray release will induce both gas mixing and strong droplet vaporization. As mentioned above, vaporization is predominant compared to mixing processes, since the characteristic times differ by several orders of magnitude.

To estimate the number of moles of steam vaporized in the sump during phase A, we apply the ideal gas law in this zone of the vessel between times t0 and t1
$$ \left\{ {\begin{array}{*{20}c} {P_{{{\text{steam}}}}^{{t_{0} }} V_{{{\text{sump}}}} = n_{{{\text{steam - sump}}}}^{{t_{0} }} R\bar T_{{{\text{sump}}}}^{{t_{0} }} } \\ {P_{{{\text{steam}}}}^{{t_{1} }} V_{{{\text{sump}}}} = n_{{{\text{steam - sump}}}}^{{t_{1} }} R\bar T_{{{\text{sump}}}}^{{t_{1} }} } \\ \end{array} } \right. $$
(19)
therefore, we can write:
$$ P_{{{\text{steam}}}} = X_{{{\text{steam}}}} P_{{{\text{TOSQAN}}}}. $$
(20)
Thus, by substituting equation (20) into system (19), we obtain the total number of moles of steam Δnsteam-sump vaporized in the sump
$$ \Updelta n_{{{\text{steam - sump}}}} = n_{{{\text{steam - sump}}}}^{{t_{1} }} - n_{{{\text{steam - sump}}}}^{{t_{0} }} = \frac{{V_{{{\text{sump}}}} }}{R}\left[ {\frac{{X_{{{\text{steam}}}}^{{t_{1} }} P_{{{\text{TOSQAN}}}}^{{t_{1} }} }}{{\overline T _{{{\text{sump}}}}^{{t_{1} }} }} - \frac{{X_{{{\text{steam}}}}^{{t_{0} }} P_{{{\text{TOSQAN}}}}^{{t_{ 0} }} }}{{\overline T _{{{\text{sump}}}}^{{t_{0} }} }}} \right]. $$
(21)

The average sump temperature is computed from the measurements of 18 thermocouples located throughout the sump volume.

Thus, we calculate that 23 mol are vaporized over the whole vessel (Fig. 5), whereas 9.37 mol of steam are vaporized in the sump zone, which represents only 3.5% of the total volume of the experimental vessel. This highlights the density of mass exchanges in the sump zone.

4.5 Validation of two-boundary-layer model

Finally, we perform a validation of the two boundary layer model. For this simulation, the heat and mass transfer coefficients are computed using empirical correlations based on the Nusselt and Sherwood numbers, respectively. The mass flow rate at the interface is computed using a model developed by Spalding (1952).

The droplet temperature obtained from the GRR measurements is averaged over a large set of droplets. The GRR technique is more sensitive to large droplets, because these generate most of the scattered intensity (with a ratio d7/3, van de Hulst 1957). Therefore, it is important to determine an average diameter dGRR that can be used as input for the droplet heating simulation, so we can make a pertinent comparison with the GRR measurement. Lemaitre (2005) used numerical simulations to show that the average droplet size representative of the GRR measurement (dGRR) can be computed from equation (22). Global rainbow simulations are performed using the Airy theory: in the first type of simulation, we assume a correlation between droplet diameter and refractive index (m = f(d)), while the second simulation takes all the droplets as having the same refractive index <mGRR>, which is computed with equation (23)
$$ d_{{GRR}} = \frac{{\int\nolimits_{{d = 0}}^{\infty } {d.d^{{7/3}} f(d){\text{d}}d} }}{{\int\nolimits_{{d = 0}}^{\infty } {d^{{7/3}} f(d){\text{d}}d} }} = \frac{{\int\nolimits_{{d = 0}}^{\infty } {d^{{10/3}} f(d){\text{d}}d} }}{{\int\nolimits_{{d = 0}}^{\infty } {d^{{7/3}} f(d){\text{d}}d} }} $$
(22)
$$ < m_{{{\text{GRR}}}} > = f(d_{{{\text{GRR}}}} ). $$
(23)

By applying the second type of simulation, we obtain a perfect superimposition with the global rainbow refactrometry measurements. This shows that the mean diameter representative of the mean refractive index, as derived from GRR measurements, is equal to the dGRR value computed from equation (23). Fortunately, this average diameter is very close to the Sauter mean diameter, which is very suitable for studying heat and mass transfer phenomena (Lefebvre (1989)).

Figure 21 reports the result of this calculation.
https://static-content.springer.com/image/art%3A10.1007%2Fs00348-008-0519-z/MediaObjects/348_2008_519_Fig21_HTML.gif
Fig. 21

Comparison between simulation and global rainbow refractometry measurements

In this figure, we observe a good consistency between the experimental results and the simulation. Thus, we conclude that the double boundary layer model is suitable for modeling the condensation on droplets under thermal–hydraulic conditions representative of a hypothetical nuclear accident.

Despite the satisfactory overall performance, we also observe that our model does not adequately describe the β and γ zones. Therefore, the model should be refined to achieve a better simulation of these previously defined zones.

In addition, supplementary measurements could be performed to improve the representativity of the simulation. Indeed, the simulation does not take into account any correlation between droplet size and velocity. Such a correlation could be investigated by coupling the out-of-focus imaging technique with a particle tracking velocimetry algorithm (Madsen et al. 2003).

However, in the case of our experimental arrangement with high droplet density, the out-of-focus imaging technique is hampered by overlapping interference fringes. These difficulties might be overcome using an advanced set-up developed by Kawaguchi et al. (2002), which involves an optical compression of the image. These measurements could be performed on smaller experiments with less uncertainties.

5 Conclusion

In this study, we present an analysis of heat and mass transfers between droplets and the surrounding gas during a test representative of thermal–hydraulic conditions during a hypothetical severe accident in a nuclear power plant. To carry out this study on the TOSQAN experiment, we coupled many innovative optical diagnostic techniques such as global rainbow refractometry, spontaneous Raman spectroscopy and out-of-focus imaging. An analysis of the local measurements enables us to explain the overall behavior of the test in terms of dimensionless numbers (Spalding parameter) and characteristic times. These measurements also highlight the efficiency of the Abramzon–Sirignano model in dealing with steam condensation on droplets, even though this model was initially established for the study of droplet vaporization.

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© Springer-Verlag 2008