Proceedings of the 15th International Heat Transfer Conference, IHTC-15August 10-15, 2014, Kyoto, Japan

IHTC15-9914

PORE-SCALE EXPERIMENTAL STUDY OF BOILING IN POROUS MEDIA

Paul Sapin,1,∗ Paul Duru,1 Florian Fichot,2 Marc Prat,1,3 Michel Quintard1,3

1Universite de Toulouse; INPT, UPS; IMFT (Institut de Mecanique des Fluides de Toulouse); Allee

Camille Soula, F-31400 Toulouse, France2Institut de Radioprotection et de Surete Nucleaire (IRSN), Division of Major Accidents Prevention,

BP3 13115 St Paul lez Durance, France3CNRS ; IMFT ; F-31400 Toulouse, France

ABSTRACT

Following a long-lasting failure in the cooling system of a water reactor, the superheated core can be cooled

down efficiently by reflooding. At the French Institute of Radioprotection and Nuclear Safety (IRSN), the

macroscopic model used to simulate this process is based on strong assumptions on the local flow patterns.

This paper presents an experimental study of boiling in porous media with focus on different pore-scale boil-

ing regimes. The experimental setup is a ”two-dimensional” porous medium made of 2mm diameter cylinders

randomly placed between two ceramic plates. The 145 heating cylinders are RTD probes (Resistance Temper-

ature Detectors), used to give pore-scale thermal measurements as well as heat generation. The operating fluid

is HFE-7000, a Fluorinert liquid with a low saturation temperature (Tsat= 34◦C at the atmospheric pressure),

which allows all the boiling regimes to be observed with moderate thermal constraints. This paper presents

results for forced convection boiling on a single horizontal cylinder embedded in a porous medium, the influ-

ence of a heating porous medium on the different boiling regimes and the reflooding of a superheated model

porous medium.

KEY WORDS: Boiling and evaporation, Porous media, Nuclear energy, Measurement and instrumentation.

1. INTRODUCTION

A wealth of industrial processes are related to the phase change phenomena within porous media, examples of

which are: drying problems [1], geothermal energy production [2], hydrocarbon exploitation, heat exchangers

design [3], and nuclear safety analysis [4] [5]. The latter example is the main motivation of this paper.

Pressurized water reactors (PWRs) and boiling water reactors (BWRs) constitute a large majority of nuclear

power plants and are based on the same operating principle: the thermal energy generated by the fission of

atoms is recovered by the water flowing in the primary circuit. In case of a long-lasting failure or leak in this

cooling system, the control rods are lowered in order to stop the chain reaction. The decay heat of fission prod-

ucts keeps heating up the system, which induces water evaporation and can lead to the drying and degradation

of the fuel rods. The nuclear accidents which occurred in one PWR at the Three Mile Island nuclear power plant

in 1979 and more recently in three BWRs at the Fukushima Daiichi power station have shown that a LOCA

scenario (Loss of Coolant Accident) can result in a core meltdown and in the formation of a hot particulate

debris bed in the reactor vessel, comparable to a heat-generating porous medium. Results from the Phebus PF

experiment (1:5000 scale) [6] and the observations on the TMI-2 reactor characterized the successive stages of

∗Corresponding Paul Sapin: psapin@imft.fr

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a LOCA accident and the resulting debris bed. If a water source is available, the decay heat can be efficiently

removed by reflooding. In order to assess the coolability of this particle bed, the intense boiling mechanisms

encountered must be modeled properly. The thermohydraulics of heated porous media has been studied both

experimentally and theoretically so as to estimate the increase of pressure in the primary cooling circuit, the

hydrogen production and the release of fission products.

In the PRELUDE experimental setup (Fig. 1(a)) at IRSN, the heating debris bed is represented by a bed of

steel balls (from 2 to 8mm diameter) heated by induction. The reflooding of this hot porous medium has

been studied with water to validate a local thermal non-equilibrium model. The macroscopic properties of

this model, required to simulate properly the available experiments, depend on the boiling regimes and flow

patterns at the pore scale.

An experimental setup (Fig. 1(b)) consisting of a ”two-dimensional” porous medium made of heating cylinders

has thus been developed to observe the encountered microscopic phenomena. This paper presents experimental

results for boiling and for the reflooding process.

(a) (b)

Fig. 1 (a) PRELUDE experimental setup; (b) ”Two-dimensional” model porous medium.

2. PREVIOUS STUDIES

The modelization of the reflood of a heat-generating porous medium is highly complex. In order to take account

of the encountered phenomena at the microscopic scale, macroscopic models can be obtained by up-scaling

the problem as was done with the method of volume averaging in [7].

For the safety analysis of PWRs, a macroscopic model [5, 8, 9] has been developed at IRSN using the volume

averaging method. With the assumption that the gas-liquid interface velocity can be neglected in the up-scaling

process, which is called the quasi-static hypothesis, the momentum balance equations are generalized Darcy’s

laws for each fluid phase, with Forchheimer terms accounting for inertial effects (Eqn. (1) and (2)):

αρg(∂vg∂t

+ vg · ∇vg) = −α∇pg + αρgg − εα2(μg

Kkrgvg + εα

ρg

ηηrgvg|vg|) (1)

(1− α)ρl(∂vl∂t

+ vl · ∇vl) = −(1− α)∇pl + (1− α)ρlg − ε(1− α)2(μl

Kkrlvl + ε(1− α)

ρlηηrl

vl|vl|) (2)

In these equations, the macroscopic pressure, density and the intrinsic velocity of the β-phase (β=g,l for gas

and liquid) are respectively pβ, ρβ and vβ. The void fraction α represents the fraction of a fluid volume

occupied by the gas phase. For spherical particle beds, as used in the PRELUDE experimental setup, the

intrinsic permeability K and passability η are respectively calculated thanks to the Carman-Kozeny relation

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[10] and the Ergun’s law [11]. The relative permeability krβ and passability ηrβ are obtained with the Brooks-

Corey relations [12].

Saturation is an important parameter for the description of a gas-liquid two-phase flow in a rigid porous

medium. Consider the total volume VT of a representative elementary volume of such a medium, that can

be expressed as: VT = Vl + Vg + Vs. The saturation S represents the fraction of the pore space occupied

by the liquid phase: S = Vl / (Vl + Vg ). The capillary pressure, modeled as a function of saturation, is also

introduced in the equations to account for the effects of pressure between the wetting and the non-wetting

phases.

As for the heat and mass transfer problem, the three phases (solid, liquid, gas) are considered to be in local

thermal non-equilibrium. The macro-scale enthalpy and temperature of the β-phase (β=g,l,s for gas, liquid

and solid) are respectively represented by hβ and Tβ. The macro-scale energy balance equations relative to

each phase (Eqn. (3),(4),(5)) are obtained thanks to the up-scaling of the micro-scale mass and energy balance

equations:∂(αερghg)

∂t+∇(αερgvghg) = ∇(K∗

g · ∇Tg) + mghsatg + φsg − φgi (3)

∂((1− α)ερlhl

)∂t

+∇((1− α)ερlvlhl) = ∇(K∗l · ∇Tl) + mlh

satl + φsl − φli (4)

∂((1− ε)ρshs)

∂t= ∇(K∗

s · ∇Ts)− φsl − φsg − φsi +�s (5)

The macroscopic coefficients involved in these equations are the effective thermal dispersion tensors K∗β and

the thermal coefficients used in the definitions of thermal exchanges between phases φsβ and φβi, involving the

temperature differences and the spatial gradients of these differences. Indeed, the up-scaling of the microscopic

problem provides equations involving the macroscopic temperatures as well as their local deviations. Solving

the mixed problem with the two types of variables would be obviously too complex; this is why the deviations

are expressed as functions of the averaged temperatures and their spatial gradients.

The macroscopic coefficients may be calculated by solving the closure problems thus obtained. Duval et al.[13] have determined analytically these coefficients for simple configurations : Solid-Liquid-Gas (SLG) and

Solig-Gas-Liquid (SGL) stratified flows. Since the gas-liquid interface is considered stationary, only the heat

exchanges for evaporation or condensation of a fluid can be described properly. In other terms, the nucleation

phenomenon remains unmodeled. The heat exchanges for the nucleate, transition and film boiling regimes,

encountered during a reflooding process, are thus underestimated.

To improve this model, Bachrata et al. [14] proposed to use correlations from the literature to estimate the

contribution of boiling to the local heat exchanges. A modified version of the Berenson’s correlation [15]

is used to estimate the heat transfer coefficient for the film boiling regime. The critical heat flux (CHF) is

calculated with the CATHARE correlation [16] based on the Groeneveld table [17]. The wall-to-fluid heat flux

is then expressed as a function of the CHF for the transition boiling regime. In the nucleate boiling regime,

the forced convection hcv and nucleate boiling hnb heat transfer coefficients are obtained respectively with

the Dittus-Boelter correlation [18] and the Thom’s correlation [19]. For the local wall-to-fluid heat transfer

coefficient hwall calculation, these contributions are weighted by a factor depending on void fraction:

hwall = (1− α)hnb + ((1− α)hcv,l + αhcv,g) (6)

where hcv,l and hcv,g are respectively obtained from the SLG and SGL configurations.

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Boiling heat transfer has been extensively studied since Nukiyama first characterized it in 1934 [20]. In convec-

tive flows, the correlations for boiling heat transfer are essentially of three types: the summation models [21],

the asymptotic models [22] and models based on flow patterns [23]. The heuristic model to estimate forced

convective nucleate boiling proposed in equation (6) is similar to a summation model. This improvement seems

well under way since no correlation is available to evaluate boiling heat transfer in porous media. Besides, the

improved model is in good agreement with the PRELUDE experimental results [24].

However, regardless of the chosen macro-to-micro criterion [25] and according to the characteristics of the

observed debris beds, microscopic boiling takes place in the considered porous media during a reflooding,

while the correlations considered above were developed for macroscopic boiling. Furthermore, the interaction

between bubbles generated by close heating elements in a porous medium might in particular influence the

predicted CHF value, since the coalescence phenomenon is facilitated. Indeed, improvement of the CHF value

by using porous-layer coatings has been shown both theoretically and experimentally by Liter and Kaviany

[3].

The experimental setup designed for this study aims to provide local thermal measurements (heat flux and

temperature) for boiling in porous media, with the objective of obtaining indications for the type of correlations

that must be used to estimate the effective parameters of the thermal non-equilibrium macroscopic model. The

assumptions on the local flow patterns made to build this model can also be questioned thanks to the pore-scale

visualizations.

3. EXPERIMENTAL SETUP

The experimental setup aims to study forced convective boiling in a nearly two-dimensional porous medium

(Fig. 3), made of cylinders randomly placed between two ceramic plates; the test section is thus inserted in

a closed loop circuit described in the Figure 2. The fluid, circulated by a gear pump (flow rate from 2 to

Fig. 2 Experimental setup: Closed fluid circuit.

100mL /min), is brought close to the saturation temperature thanks to a thermostat before entering the test

section. An ice bath is used to recondense the vapor generated by the boiling processes. The function of the

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tank is to trap non condensed vapor and the possibly remaining non condensible gases (air principally). A valve

on the top of this plenum chamber allows these gases to be evacuated and to release the pressure in the loop.

So the setup can be used itself to degas the fluid.

The 3MTM NovecTM Engineered Fluid HFE-7000 was chosen as the operating fluid for many reasons. First,

all the boiling regimes can be observed with moderate thermal constraints because its saturation temperature is

very low: Tsat = 34◦C at the atmospheric pressure. Besides, safety is ensured during the manipulations since it

is a low-toxicity, non-flammable and dielectric fluid. The HFE is, though, highly volatile and with an important

global warming potential (370 for a 100-year integrated time horizon). The fluid circuit is filled by vacuum

with degassed HFE, in order to avoid the presence of air, which would significantly change the saturation

temperature. For both the boiling phenomenon and the study of flows in porous media, the hydrophobicity

is a matter of importance. The HFE is highly hydrophilic (contact angle close to 0◦ with ceramics), as is

water, which is necessary since the experiments have to be conducted in similar conditions as the PRELUDE

experiment. As for the capillary effects, the ratio between the HFE and the water surface tension is about 4.7,

so the HFE capillary length Lcap (defined in equation (7)) is 2.6 times as high as water’s.

Lcap =

√σ

ρg(7)

The test section is a ”two-dimensional” porous medium made of 145 heating cylinders randomly placed be-

tween two ceramic plates, spaced from one another by 3mm (Fig. 3). To obtain visualizations at the pore scale,

one of the plates is transparent. The diameter of the steel balls constituting the debris bed in the PRELUDE

experimental setup range between 2 and 8mm. The diameter of the heating cylinders has thus been chosen

to be 2mm, so that the capillary effects in this experimental setup are representative of those in the previous

experiments.

Fig. 3 Experimental setup: Test section (Dimensions in mm).

The constitutive elements of the model porous medium are cylindrical Resistance Temperature Detectors

(RTDs). They are used not only as heaters, thanks to the Joule effect, but also as heat flux and temperature

sensors. In a RTD probe, a fine coiled platine wire is wrapped around an insulating core and sealed in a ce-

ramic coating (Fig. 4). The temperature of such an element can be measured by measuring the platine wire

resistance (100Ω at 0◦C for the used PT100 probes); indeed, the platine resistance changes with temperature

according to a simple and well-known law. Because of this dependency of the resistance upon the temperature,

it is necessary to control the heat flux transmitted to each cylinder (Joule effect depending on R(T )). A micro-

electronic servo-control system, running at 1000Hz, is thus connected to each probe, in order to measure and

control both the heat flux and the temperature of the element. To avoid having too many control modules, most

of the elements are controlled by groups of four or five. In these groups, the RTDs are mounted in parallel, to

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Fig. 4 RTD ELEMENT

maintain them in thermal equilibrium: if an element heats up more than the others, its resistance increases so

that the current passing through it -and hence the Joule effect- decreases. Twenty elements distributed in the

test section are controlled individually (Fig. 5), to provide precise measurements near the CHF or the Leiden-

frost temperature. This way the different boiling regimes can be characterized, with the aim in mind to obtain

Nukiyama’s curves for porous media. A simple heat conduction analysis of the ceramic coating evaluates the

Fig. 5 Individually controlled elements

response time of the cylinders at 0.4 s. This time delay is non-negligible in comparison to the timing of tran-

sition between film and nucleate boiling regimes observed in the reflooding tests presented later in this paper.

However, providing Nukiyama’s curves for porous medium requires established regimes, so this time delay

has no impact for these measurements. The thermal control command is given to each module via a computer.

The servo-control systems are then autonomous, only the desired probes are linked to the computer to acquire

thermal measurements in real-time (Fig. 6).

Fig. 6 Data acquisition

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4. RESULTS

As a first step, the thermal resistance of the probe ceramic coating has to be determined (Fig. 4). For the

temperature measurement with RTD probes, an usual problem is to avoid self-heating, because this leads to

a temperature difference between the cylinder wall temperature and the platine wire temperature effectively

measured. Since self-heating is used in our experiment as the power source, this temperature difference must

be known at every time. That is why ceramic coating resistances have been measured thanks to infrared ther-

mography for different thermal equilibriums.

4.1 Forced convection boiling on a single horizontal cylinder embedded in a porous medium

Once this correction applied, forced convection boiling on a cylinder, selected from the bundle of cylinders

presented in Figure 5, has been studied. The chosen probe is the number 11 (numbered according to Figure 5)

because it is located in the middle of the test section. By controlling the heat flux imposed to the single element,

the main boiling regimes have been observed and a Nukiyama’s curve (Fig. 8) has been obtained. The assembly

of the heating elements between the two ceramic plates has been simply modelized in MATLAB. Thanks to

a heat conduction model, with natural convection in air on external boundaries and forced convection boiling

with HFE on inside boundaries, the heat losses and the effectively transmitted heat fluxes have been estimated.

The fluid circulates at a flow rate of 21.4mL /min. The temperature of the liquid impacting the studied element

(Tl = 24.4◦C) is also evaluated by this model, from the measurement of the liquid temperature at the inlet of

the test section. At the cylinder level, pressure is 1.18 bar, which corresponds to a saturation temperature of

39.9◦C. Subcooling effects are thus important.

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qwall

(W/cm²�

������ ���������ΔTsat

= Twall

- Tsat

(°C)

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Fig. 7 Onset of nucleate boiling on a single element amongst a bundle of horizontal cylinders in a crossward

flow

By increasing step-by-step the power applied to the cylinder, onsets of nucleate boiling are observed. Under the

conditions described earlier, the first nucleation is observed for a wall superheat of 10.5◦C (Fig. 7). The wall

superheat ΔTsat is defined as the difference of temperature between the outer wall of the cylinder and the fluid

saturation temperature (39.9◦C): ΔTsat = Twall − Tsat. In the partial nucleate boiling regime (Fig. 9(a)),

isolated vapor bubbles are generated at the different nucleation sites. The intense boiling regime, observed for

higher heat fluxes, is characterized by the formation of vapor pockets (Fig. 9(b)). Below the critical heat flux

(qchf = 23.7W / cm2), the liquid remains in contact with the heating cylinder wall despite the coalescence

of bubbles. The heat exchange reaches a maximum just before the burnout point, at which a vapor film is

formed at the heater surface. The so-called boiling crisis is violent and leads to an increase in wall superheat

and an important reduction in the heat transfer rate. The film boiling regime (Fig. 9(c)) is maintained while

decreasing the heat flux: the vapor film thickness decreases until the heat flux reaches a minimum at the so-

called Leidenfrost point, for which ΔTsat = 110.9◦C and qmin = 12.5W / cm2. Thanks to the control

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�

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��

��

��

��

��� �� �� ��� ���

������������

qwall

(W/cm²)

������ ���������ΔTsat

= Twall

- Tsat

(°C)

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Fig. 8 Nukiyama’s curve for subcooled forced convective boiling a single horizontal cylinder embedded in a

porous medium

(a) (b) (c)

Fig. 9 (a) Partial nucleate boiling; (b) Intense nucleate boiling; (c) Film boiling.

of temperature, the transition boiling regime can also be maintained. However, the large fluctuations of the

recorded temperatures (about 20◦C for each point) make these measurements unreliable.

The generation of vapor slugs in the film boiling regime is well brought into focus in Figure 9(c). The behavior

of these slugs in a heated porous medium is hardly predictable (coalescence with other slugs or vapor films,

simple Taylor bubbles. . . ) for a large number of pores. It will thus be very instructive to observe how the

confinement affects the boiling and convective heat exchanges.

4.2 Forced convection boiling in a heating porous medium

The purpose of this experiment is to obtain Nukiyama’s curves for an element within a heating porous medium.

This means that all the cylinders are set to apply a constant heat flux, while a single element in the very middle

of the others is heated up individually to observe all the boiling regimes, with the aim of measuring an influence

of the local void fraction on the boiling phenomena.

The heat flux imposed to the surrounding cylinders Qs varies from 0mW (results presented in the previous

section) to 350mW. The flow rate is still set to 21.4mL /min which corresponds to a filtration speed of

3.4mm / s in the test section. The influence on the nucleate boiling and the film boiling regimes are respectively

presented in Figures 10(a) and 10(b).

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��� ��� � �� �� �� �� �� � �

������������

qwall

(W/cm²)

������ ���������ΔTsat

= Twall

- Tsat

(°C)

Qs = 20mW

Qs = 50 mW

Qs = 100 mW

Qs= 150 mW

(a)

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(W/cm²)

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= Twall

- Tsat

(°C)

Qs = 20 mW

Qs = 50 mW

Qs = 100 mW

Qs = 150 mW

(b)

Fig. 10 Influence of a heating porous medium on the (a) Nucleate boiling regime and the (b) Film boiling

regime.

As expected, the vapor generated by the porous medium mainly influences the nucleate boiling regime (Fig. 10(a))

by changing the value of the critical heat flux: the increase in local void fraction helps the formation of a va-

por film around the studied cylinder. The CHF has also been measured for different imposed fluxes Qs, the

results are presented in Figure 11. As explained earlier, the case in which no surrounding cylinder is heated

(Qs= 0mW) is subject to subcooling effects. The difference with the other curves is thus not only due to the

presence of vapor but also to the local fluid temperature.

As for the film boiling regimes observed (Fig. 10(b)), the higher the amount of vapor, the lower the heat-

exchange rate. Indeed, in this regime, the quality of the heat transfer depends strongly on the thickness of the

vapor film. Besides, the presence of gas helps the film to be maintained for smaller fluxes, which explains the

differences in the measured minimum heat fluxes qmin.

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��������������

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qchf

(W/cm²)

�������� ������ ��������������������������Qs ��!

Fig. 11 Influence of a surrounding heat-generating porous medium on the critical heat flux

4.3 Bottom reflooding of a heat-generating porous medium

The aim of this section is to actually simulate a reflooding process. The porous medium is thus heated with a

homogeneous and constant heat flux up to an equilibrium temperature and is perfectly dried out: only HFE va-

por remains in the test section. The quenching of this hot and heat-generating porous medium from the bottom

can then be achieved with the liquid at a temperature close to the saturation. With a flow rate of 21.4mL /mindelivered by the gear pump, the velocity of the quenching front in the porous medium is 3.4mm / s without

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phase change. The valve at the outlet of the test section is open to prevent an important increase of pressure

due to intense boiling. So, the saturation is reached for a temperature of 34◦C. For the first experiment, the test

section is heated with a power of 80mW per element. The equilibrium temperatures range between 70 and

90◦C, depending on the position of the cylinder. The temperature evolution of the four in-line elements 13-16

(numbered according to the Figure 12(b)) during the reflooding is presented in Figure 12(a).

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Fig. 12 (a) Reflooding of four in-line elements represented in (b)

The initial temperature of the probes located in the middle of the test section (cylinders 14 and 15) is higher

than that of the probes located on the edges (cylinders 13 and 16). This is due to the important thermal mass

of the ceramic plate on the borders of the bundle of probes (Fig. 3). The quenching begins approximately 5 safter the beginning of recording; at this time, one can observe a slow cooling of the recorded elements, which

is due to the vapor flow generated below. For the central elements, the liquid is obviously put in contact with

these superheated cylinders at approximately t= 15 s. However, it appears that the cylinders on the edges

first enter in contact with the quenching front well in advance. Here, the visualization of the reflooding test

provides us precious indications: the by-pass effect due to the staggered arrangement of the bundle leads to the

development of instabilities on the edges. The first temperature drop is in fact due to liquid jets.

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Fig. 13 Evolution of the temperature of cylinders at different heights in the middle of the test section

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Figure 13(a) shows the temperature evolution during the reflooding of the cylinders 2-6-10-14 and 18 (Fig. 13(b)),

located at different heights in the middle of the test section. The differences in the initial temperature of the

cylinders 2-6-10-14 result from the natural convection taking place inside and outside the tube bundle. As

regards the temperature of element 18, its low initial value is also due to an important non-heated thermal

mass of ceramic above this last line. By detecting the moment when the quenching front touches each probe,

it is possible to determine its position and thus its velocity. This has been carried out for this test and for the

reflooding of bundles heated with power sources of 100, 120, 140 and 160mW. The results are presented in

Figure 14(a) for the cylinders located at the middle of the test section.

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Fig. 14 (a) Tracking of the quenching front; (b) Quenching front velocity for different heating powers.

The obtained average quenching front velocities are presented in the Figure 14(b). As expected, the liquid front

velocity decreases with the increasing power source imposed to the porous medium.

5. CONCLUSIONS

The presented experiments have validated many experimental settings and have shown the feasibility of our

project to visualize and quantify pore-scale boiling mechanisms. The results already obtained give us precious

informations for boiling of HFE-7000 in our specific configuration. They also confirm the need for a specific

heat-transfer model like the one proposed by Bachrata et al. taking into account various pore-scale boiling

regimes. The influence of the local void fraction on the characteristics of the different boiling regimes, espe-

cially on the film boiling regime, has been put into focus. This regime is indeed particularly influent during the

reflooding of a nuclear reactor because the initial temperatures are significantly higher than the water Leiden-

frost temperature. Therefore, the further analysis of the experiments needs to focus on a local measurement of

the void fraction inside the two-dimensional porous medium, thanks to the high speed visualization. Control-

ling the cylinders with different electronic mountings will also allow us to reach higher heat fluxes for every

point in the test section, the aim being to obtain Nukiyama’s curves for an uniformly heated medium. The

possibility of heating only part of the cylinders can also provide interesting results. Indeed, by heating only a

determined ratio of the cylinders, the situation in a reactor core, where only the fuel fragments release thermal

energy, is more faithfully reproduced.

ACKNOWLEDGMENTS

This work is supported by the French Institute of Radioprotection and Nuclear Safety (IRSN) and Electricite

de France (EDF). Thanks also go to H. Ayroles and R. Soeparno for all the time they spent on the experimental

setup development.

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NOMENCLATUREΔTsat Wall superheat: Tw − Tsat, K

m Evaporation rate, kg /m3 / s

g Standard gravity, m2 / s

K∗β Effective thermal dispersion tensor, W /m /K

vβ Averaged speed in the β phase, m / s

�s Volumetric thermal source, W /m3

h Volumetric heat transfer coefficient, W /m3 /K

hβ Averaged specific enthalpy of the β phase, J / kg

hsatβ Specific enthalpy of saturated β phase, J / kg

I Electric intensity, A

K Macroscopic permeability , m2

krβ Relative permeability of the β phase

Lcap Capillary length, m

pβ Averaged pressure of the β phase, Pa

Q Heat flux , W

q Surface heat flux , W /m2

R Electric resistance, Ω

S Saturation

T Temperature, K

Tβ Averaged temperature of the β phase, K

Subscripts

β Phase index: β = g, l, s

cv Convective exchange

g Gas phase

i Interface

inj Injection (inlet of the test section)

l Liquid phase

nb Nucleate boiling

s Solid phase

sat Saturation

Greek lettersα Void fraction or gas saturation

η Macroscopic passability, m

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ηrβ Relative passability of the β phase

μβ Viscosity of the β phase, Pa · sφ Volumetric heat flux , W /m3

ρ Density, kg /m3

σ Surface tension, N /m

ε Porosity

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