Experiments on flows, boiling and heat transfer in porous media: Emphasis on bottom injection

Nuclear Engineering and Design 236 (2006) 2084–2103

Experiments on flows, boiling and heat transfer in porous media: Emphasis on bottom injection M. Miscevic a,∗ , O. Rahli b , L. Tadrist b , F. Topin b a

b

Laboratoire d’Energ´etique, Universit´e Paul Sabatier, 31062 Toulouse Cedex 09, France Polytech Marseille, Laboratoire IUSTI, UMR CNRS 6595, Technopˆole de Chˆateau Gombert, 5 rue Enrico Fermi, 13453 Marseille Cedex 13, France

Received 1 January 2004; received in revised form 15 March 2006; accepted 20 March 2006

Abstract This paper deals with basic experiments conducted to analyse the effect of the particles’ shape and size distribution on intrinsic properties of porous beds as well as two-phase flow and heat transfer in these porous media. Structural, transport properties, flow laws and heat transfer with phase-change phenomena in several kinds of porous media are presented and discussed. The porosity of stacks constituted by spheres of various sizes is analysed. A variation law of the porosity as a function of the standard deviation of the particle size distribution is proposed. The porosity, tortuosity, permeability and inertial coefficient of the flow law in randomly stacked fibres are established experimentally and theoretically. The porosity of such media is found to vary from 0.35 to 0.92 according to the fibre aspect ratio. Tortuosity and Kozeny–Carman parameters are determined by both electric and hydrodynamic methods. These parameters are found to vary with the porosity of the fibrous bed. New relations of permeability and inertial coefficient are derived from experimental results. Finally, a pressure drop relation is proposed for fibrous beds. Convective boiling phenomena, with emphasis to application on bottom injection, are experimentally determined for fibrous porous media. Temperature field determination evidences the formation of three distinct zones in the porous medium: a liquid zone, a two-phase zone and a superheated zone. For higher heat flux density, a fourth zone is found in which vapour and liquid are in thermal non-equilibrium. A onedimensional analytical model of pressure drop in two-phase configuration has been performed. Comparisons with experimental data are found in good agreement with the results of this model for moderate heat fluxes. For higher heat flux values, discrepancies are found. These cases correspond to the appearance and the evolution of the thermal non-equilibrium two-phase zone. Heat transfer characteristics at the heated walls are analysed. Formation of vapour in the neighbourhood of the heated walls has a strong influence on the heat transfer coefficient. This behaviour may be related to the critical heat flux phenomenon encountered in usual ducts. © 2006 Elsevier B.V. All rights reserved.

1. Introduction There exist numerous experimental and theoretical studies about cooling of debris’ bed in nuclear reactors. Most of these studies concern boiling and dryout in a homogeneous bed. For these cases, the particles’ diameter distribution is replaced by a single diameter. In spite of this simplification, the boiling phenomenon analyses remain complex due to the several space scales that have to be considered. Heterogeneous structures and the appearance of liquid clusters characterise the phase change

∗

Corresponding author. E-mail addresses: [email protected] (M. Miscevic), [email protected] (L. Tadrist). 0029-5493/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2006.03.034

phenomenon. It is thus possible to observe structures of twophase flow similar to those met in channels (bubbly flow, slug flow and plug flow). On the contrary, specific two-phase flow regimes may take place. In this case, the effect of confinement plays an important role in the phase change and fluid motions. This may be partly estimated by comparing the capillary length with the characteristic pore size. Viscous, inertial, gravity and capillary effects control the two-phase flow when boiling occurs in the porous medium. The complex phenomena occurring in the porous matrix make difficult an accurate prediction of the transport phenomena. For these reasons the main studies of two-phase flow are often experimental. In porous media, the fluid motion is due to natural convection coupled with phase change phenomena. Experimental studies are generally realised in small vessels. A great number of these

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Nomenclature A ap d Dh E f F g G kk K L Lv m n P q r R Re S T un U V x X z

cross-section area (m2 ) specific surface area (m2 /m3 ) particle or fibre diameter (m) hydraulic diameter (m) electric potential (V) non-dimensional inertial coefficient formation factor (ratio of electrical resistances) gravity acceleration (m s−2 ) mass flux (kg m−2 s−1 ) Kozeny–Carman parameter permeability (m2 ) porous bed length (m) latent heat of vaporisation (J kg−1 ) number of layers or mass (kg) number of particles pressure (Pa) heat flux (W m−2 ) ratio between small and large sphere diameters or fibre aspect ratio electrical resistance (
) Reynolds number saturation temperature (K) fluid velocity component in the n direction (m s−1 ) superficial velocity (m s−1 ) volume (m3 ) quality or coordinate (m) particles volume fraction coordinate (m)

Greek symbols β inertial coefficient (m−1 ) γ sphericity factor ε porosity λ thermal conductivity (W m−1 K−1 ) µ dynamic viscosity (Pa s) ρ density (kg m−3 ) σ specific electric conductivity of the fluid τ tortuosity φ compactness (1 − ε) ϕ piezometric high (m) Subscripts e sample in inlet l, liq liquid m mixture p pore sat saturation v, vap vapour

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works concern the determination of the critical heat flux, corresponding to the appearance of dried zones inside the porous structure. Dhir and Catton (1975) were the pioneers of these studies that continued over several years (Dhir and Catton, 1977; Naik and Dhir, 1982). Lipinski (1982) made a detailed review of the previous works conducted in this field. Other works have been realised in order to determine the phenomenology of the boiling in porous media (Sondergeld and Turcott, 1977; Udell, 1983; Torrance, 1983; Stemmelen, 1991). These authors have considered the development of a liquid–vapour two-phase zone under thermal equilibrium conditions at saturation temperature of the fluid. Basic experiments are often carried out in porous media constituted by spherical particles. Several heating configurations are used: volumetric (induction) or on the walls of the porous medium with a fluid at rest (Udell, 1983; Torrance, 1983) or crossed by a fluid flow (Cioulachtjian et al., 1989; Rahli et al., 1996). Most of the works dealing with boiling porous media highlight the fundamental role of a biphasic zone, which appears and evolves in the medium, due to the heat and mass transfer. Relative intensity and orientation of heat and mass fluxes, as well as the structure of the solid matrix, govern the precise behaviour of the phenomena occurring in the porous medium. The complexity of coupled heat and mass transport phenomena with phase change associated to the experimental difficulties show the need of complementary numerical tools that would allow to precise both description and understanding of these phenomena. Different methods and tools such as homogenisation, averaging, percolation and fractals have been used to predict the macroscopic transport properties from the microstructure of the medium. The definitions of a continuous medium equivalent to the real porous structure, as well as the definition of the applicability level of the macroscopic model, constituted a tricky problem that has long been debated (see Marle, 1982; Baveye and Sposito, 1984; Auriault and Caillerie, 1989; Quintard and Whitaker, 1991 for example). The forms of the used phenomenological laws have been obtained from correlations resulting from experiments and not from physical laws. The valuable expressions are much diversified and the choice for one or the other is only justified when compared with specific experiments. Numerical models are, thus, essentially tools to test the impact of such laws on the system behaviour (Schmidt, 2000) and assess (and maybe identify) valuable expressions for the closure relations. Many approaches have been developed in order to model boiling phenomena in porous media at macroscopic level (i.e. the porous medium is assimilated as a fictitious, continuous and homogeneous medium) according to the application or field of research such as geothermal science, nuclear engineering, drying, etc. (Udell, 1985; Moyne, 1987; Tung and Dhir, 1990; Perre et al., 1993; Topin et al., 1997; Wang, 1997; Petit, 1998; Ghafir, 2000; D´ecossin, 2000; Najjari and BenNassrallah, 2002). The physical description is based on the same set of balance equations (Whitaker, 1977) and the used models differ mainly by additional hypotheses and used closure relation sets. Three families of models could be roughly distinguished according to

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the treatment of the energy equation: • Local thermal non-equilibrium (two, three temperatures model). • Local thermal equilibrium (one temperature model). • Imposed saturation temperature from system pressure independent of local pressure. Some distinction could be added depending on the treatment of the momentum equation: Darcy’s flow, introduction of an inertial term (Forchheimer), use of liquid-gas coupling terms (e.g. see Tung and Dhir (1988) or Grall (2001) who gives a comparison of momentum equation treatment). Moreover, one could also distinguish models where the phase change rate is imposed (e. g. Tung and Dhir, 1990), models where the phase change rate is deduced from closure relations (see Petit, 1998; D´ecossin, 2000) and models where the phase change rate is deduced from the balance equations. Another important aspect of these works is the applicability to experimental configurations, i.e. the possibility of simulating “real conditions”, which generally imply the coupling of the porous domain with other domains (solid, fluid). Additional distinctions could be added considering the numerical treatment of the equations and thus the induced limitations/capabilities of the developed tools: boundary conditions (fixed, free and evolving), transient regime, multidimensional or 1D. For example, Moyne (1987) uses finite elements applied to 1D drying case; Perre et al. (1993) treats the case of high temperature drying with finite volume (without specifying other possible applications). The works done in the field of nuclear engineering deal often with thermal nonequilibrium cases (WABE-2D, TRIO-REPOS, etc.) and thus use more complex procedure to determine saturation temperature and phase change rate (Petit, 1998; D´ecossin, 2000; Duval, 2002). On the other hand, most of the authors dealing with convective boiling in porous media suppose constant fluid saturation conditions (Udell, 1985; Stemmelen, 1991; Wang, 1997; Ghafir, 2000; Najjari and BenNassrallah, 2002). In all cases, the agreement between model and experiment is only obtained by adjusting various parameters (physical properties, boundary or initial conditions), see for example Schmidt et al. (2000), Quintard and Puiggali (1986) and Ferguson and Turner (1995). These tools enable access to variables difficult to measure such as the local phase change rate and local phase velocities as well as local heat exchange coefficient. On the other hand, the transport properties of porous media are still not well known and the used phenomenological laws have been obtained from correlations resulting from experiments (often in conditions quite different from the boiling one) and not from physical laws. Applications of such numerical models must be associated to basic experimental works in order to get assured quantitative results. In summary, the numerical tools are, now, efficient for getting qualitative description of the phase change phenomena and associated two-phase flow in porous media. The limitations of such tools are mainly linked to the lack of knowledge of appropriate closure laws such as flow properties, capillary pressure, interfacial heat transfer coefficient, etc.

In this paper, basic experiments are conducted to analyse the effect of the particles’ shape and size distribution on intrinsic properties of porous beds. Structural and transport properties, flow laws and heat transfer with phase-change phenomena in several kinds of porous media are presented and discussed. 2. Phenomenological behaviour In order to evaluate the local characteristic parameters of the two-phase flow (pressure, saturation, velocity of each phase), a numerical model of heat and mass transfers in a porous medium that takes into account the pressure distribution in the gaseous phase for variations of the boiling temperature, the capillary effects as well as the phase change and gravity effects has been developed for the transient case in 2D geometry (Daurelle et al., 1998; Topin et al., 2002). The choice of the phenomenological laws was by their simplicity and to limit the number of not wellknown closure relations. We assume that the porous medium stays in thermal equilibrium. We use the Galerkin type finite element method that is well adapted to our configuration, which requires a wide range of geometries and boundary conditions. The Gaussian numerical integration enables to compute easily the complex transport coefficients, which are strongly dependent on unknown variables. In order to illustrate multi-dimensional effects that may appear during the cooling of debris beds, this model is applied to a porous bed held inside a rectangular channel, heated on both lateral sides and crossed by a forced ascending fluid flow (bronze balls Φ = 150 ␮m, bed dimensions 1 cm × 5 cm × 20 cm, fluid n-pentane). The external sides of the lateral surfaces are heated uniformly. Figs. 1–3 present examples of results from initial time to steady state. Fig. 1 shows the temporal evolution of temperature T (a), the so-called saturation S (b), i.e. the liquid volume fraction of fluid, and (c) pressure P profiles along the centreline of the channel. Fig. 2 gives the vapour quality evolution at different points of the outlet section. Fig. 3 presents the evolutions of each phases flow rate on the outlet of the channel. These results highlight three stages in the behaviour of the transient evolution. In a first stage, the flow is mainly monophasic (Fig. 3). Temperature increases regularly at each point (Fig. 1a), small variations of gaseous pressure are observable (Fig. 1c). These latter are due to temperature and saturation gradients that contribute to the flow motion and reduce the pressure gradient. A second stage occurs when boiling starts as the temperature reaches the saturation value. Temperature reaches a plateau stage; the saturation decreases sharply (Fig. 1a and b). The vapour production (fluid volume expansion) expulses fluid and raises the total flow rate at the channel outlet that is higher than the inlet during this period (Fig. 3). The lowering of the saturation value as well as the increase of the flow rate above the boiling zone increase the pressure drop in the channel (Fig. 1c). This stage starts with a rapid expansion upstream along the wall of the channel of the boiling zone followed by a smooth evolution of temperature, pressure and saturation up to steady stage. As the outlet pressure value is fixed, the pressure increases in

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Fig. 1. Temporal evolution of temperature (a), saturation (b) and gaseous pressure (c) along the centreline of the channel (z upwards). ηin = 0.45 kg m−2 s−1 ; wall imposed heat flux 1500 W m−2 ; Tin = 30 ◦ C.

the channel and the local saturation temperature increases which tends to limit the phase change. During the third stage a steady state is reached. Vapour and liquid flow rates are constant and the total flow rate has decreased to the inlet value (Fig. 3). The flow is stratified at the outlet of the channel. At the channel centre, the fluid is mostly liquid (Fig. 2,

vapour quality <10%), while near the wall, the vapour phase is dominant (vapour quality >50%). Fig. 4 illustrates a slip (vvap − vliq ) velocity field in a rectangular channel heated on a part of the side as marked in Fig. 4. One can easily see that the liquid and vapour velocities are quite different in magnitude and direction. This is emphasised on the

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Fig. 2. Local vapour quality temporal evolution on the section z = 20 cm; ηin = 0.45 kg m−2 s; wall imposed heat flux 1500 W m−2 ; Tin = 30 ◦ C.

Fig. 3. Temporal evolution of liquid (ηl ), vapour (ηv ) and total flow rate (ηt ) across the section z = 20 cm; ηin = 0.45 kg m−2 s−1 ; wall imposed heat flux 1500 W m−2 ; Tin = 30 ◦ C.

“slip velocity” figure that shows up the multidimensional nature of such a complex flow structure, even for this simplified case. Fig. 5 gives the computed local heat transfer coefficient along the bronze plate boundary of the channel after stationary stage is reached for three thermo-hydraulic conditions with the same heat flux to flow rate ratio. One can distinguish two zones corresponding to the phase state of the fluid. In the lower zone (z < 5 cm), the heat exchange coefficient decreases sharply (entrance effect), then remains constant. This behaviour corresponds to the transfer between liquid and wall. The obtained values are similar to those usually proposed for these media (h ∼ 600 W m−2 K−1 ). In the boiling zone (z > 5 cm), the heat coefficient raises dramatically along the channel to reach a maximum near the outlet (5–7 × 105 W m−2 K−1 ). The three curves differ only in the transition zone between monophasic and boil-

ing regions. The higher values of the inlet flow tend to stretch out the transition zone. This behaviour is linked to the convective effect. Above this transition zone (z > 8 cm) the heat exchange coefficient is the same for all configurations. The phenomenological description of the dynamic behaviour is coherent with available experimental results (Rahli et al., 1996). Nevertheless, in order to obtain a quantitative description of these phenomena, it is necessary to understand the structure of such porous media as well as their impact on flow and transport laws. 3. Structural properties Various models of debris beds show a very important sensitivity to particle size distribution and to bed porosity: small

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Table 1 Porosities of the randomly stacked spheres (close packing) obtained experimentally by various authors Reference

Material

Porosity

Westmann and Hugill (1930)

Lead Steel

0.369 0.392

Steel Steel Lead

0.371 0.363–0.366 0.380 0.359

Rice (1944) Scott (1960) Berenson (1960) Smith et al. (1929)

Fig. 4. Liquid–vapour slip velocity (vvap − vliq ) fields in the vicinity of a heater in a rectangular channel crossed by a low velocity fluid flow. Bottom injection (grey rectangle: heated zone).

changes of these parameters could lead to easily coolable or non-coolable situations. In these conditions, the determination of a relation between particle size distribution and bed porosity appears necessary to be developed. Such a relation should allow to first evaluating the bed porosity and then to derive relations giving transport properties and to deduce flow and heat transfer laws. The aim of this section is to propose an approach in order

Fig. 5. Local heat exchange coefficient for three thermo-hydraulic conditions. Steady state results (Tin = 30 ◦ C).

to model the porosity of a bed constituted by a stack of various sizes of particles (polydisperse stack). A first approach consists in considering spherical particles. Then, particles having a cylindrical shape are considered, in order to illustrate the influence of deviation to sphericity on intrinsic properties of the porous bed. For disordered stacked spheres, the experiment remains the only source of reliable information for the porous structure. Indeed, in the real porous media, arrangements are much more complex due to several sphere sizes. Two states of packing exist in principle for a disordered bed of uniform spheres: close and loose packing. Numerous authors have realised porosity measurements of porous beds obtained with identical spheres poured into cells of variable shapes and dimensions (Table 1). The porosity depends strongly on the way of packing. However, this porosity does not exceed 0.42. The compact disordered stack seems to correspond to a fixed porosity value. Several authors such as Westmann and Hugill (1930) and Scott (1960) found porosity values ranging from 0.36 to 0.38 for close packing. Scott (1960) showed that the porosity limit, far from the walls, does not depend on the shape of the cell. It does not depend either on the friction coefficient of the spheres surface. In the case of binary mixtures, porosity has been measured by different authors (Westmann and Hugill, 1930; McGeary, 1961; Ben Aim, 1970; Jeschar, 1975). Such mixtures represent the first step modelling a real bed containing particles of several sizes. Let us consider a binary mixture constituted by large and small particles. If there are only small (or large) particles, the porosity of each class is equal to ε0 . A mixture constituted with large and small particles has a porosity ε. The porosity variation according to the volume fraction of the large particles is reported in Fig. 6. This figure shows that the porosity of a binary mixture is always smaller than the monodisperse bed porosity. A minimum value occurs for a volume fraction of large particles between 0.6 and 0.8. This minimum becomes smaller as the diameter ratio increases. Based on experimental results, Yu and Standish (1987) proposed a linear model to determine the porosity of a porous bed containing spheres with various sizes, based on a binary mixture model. The packing fraction may be deduced assuming a linear variation with the volume fraction as detailed in Yu and Standish (1987) and Rahli (1997): φiT =




φi

1− 1−

φi φij



Xi Xij

.

(1)

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Fig. 8. Porosity variation vs. relative standard deviation of spheres size distribution.

Fig. 6. Porosity variation vs. the fractional solid volume of small, related to large particles for relative diameter ratio df /dg varying from 0.0129 to 0.5.

For a ternary mixture, the packing of the bed is expressed as φiT =




1− 1−

φi φij



φi Xi Xij


− 1−

φi φik



Xk Xik

.

(2)

This relation may be extended to mixtures composed of N particle sizes: φi  . (3) φiT = N
1 − j=1 1 − φφiji XXiji The compactness (1 − ε) depends on the particle size distribution of the mixture as schematically illustrated in Fig. 7; it increases with the relative standard deviation built on the mean particle diameter σd¯ . For example, Fig. 8 represents the variations of the porosity for a powder constituted with several particle sizes (Rahli and Tadrist, 1992). Supposing that the porosity of the stack of monodisperse particles is equal to 0.40, the poros-

Fig. 7. Diagram illustrating the random packing structure of spheres.

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ity variation according to the relative standard deviation of the size distribution of particles is approximated by a second-degree polynomial law. This law gives the tendency of variation of the porosity of a polydisperse stack of particles for a relative
 standard deviation σd¯ varying from 0 to 0.7. This law can be derived using a model based on the mixing laws of binary mixture (McGeary, 1961; Rahli, 1997). So, in the case where particles can be considered as approximately spherical, the influence of the size distribution on the compactness (and so porosity) of the stack can be estimated by using relation (3). In real debris beds, the shape of the particles is irregular. In order to illustrate the influence of this shape on the bed porosity, a porous medium constituted by disordered stacked monodisperse fibres is now investigated. As indicated above, studies concerning the stacks of both monodisperse and polydisperse spherical particles are numerous in the literature. For media constituted of disordered stacks of spherical particles, the porosity as well as pore structures can be determined using the above approaches. For randomly stacked fibres, limited published works exist. Milewski (1978, 1986) and Nardin et al. (1985) investigated experimentally fibre stacks. Lee (1987) studied these stacks from a theoretical point of view. A methodology yielding laws for the porosity of randomly stacked monodisperse fibres was proposed by Rahli (1997). Scheidegger (1974) measured the porosity of media constituted by stacks of particles by using several techniques. These porosity values given by Kaviany (1985) for various porous media are grouped in Table 2. The use of fibres (Fig. 9) allows obtaining porous media having a wide range of porosities depending on the fibres aspect ratio. This porosity variation as a function of the aspect ratio of the fibres is presented in Fig. 10 for the close packing case. Similar trends for porosity are found

Table 2 Experimental porosity values of various porous media (Kaviany, 1985) Substance

Porosity

Metallic foam Fiberglas Berl saddles Wire crimps Silica grains Black slate powder Raschig rings Leather Catalyst Granular crushed rock Soil Sand Silica powder Packing of spheres Cigarette filters Brick Hot-compacted copper powder Sandstone Limestone, dolomite Coal Concrete

0.98 0.88–0.93 0.68–0.83 0.68–0.76 0.65 0.57–0.66 0.56–0.65 0.56–0.59 0.45 0.44–0.45 0.43–0.54 0.37–0.50 0.37–0.49 0.36–0.43 0.17–0.49 0.12–0.34 0.09–0.34 0.08–0.38 0.04–0.10 0.02–0.12 0.02–0.07

Fig. 9. Several used fibres and sample of randomly stacked bronze fibres (diameter = 150 ␮m and aspect ratio r = 25).

Fig. 10. Variation of the porosity of randomly packed fibres vs. the fibre aspect ratios (diameter = 150 ␮m and 4.5 ≤ r ≤ 66.7).

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Table 3 Porosity values obtained experimentally for various types of packing of fibres

4. Flow laws

Aspect ratio, r = L/D

Hydrodynamic effects play an important role for cooling of debris beds. Since the structure of the solid matrix is rather complex and not precisely known, a good understanding of the impact of the solid matrix on flow laws for basic configurations should be achieved in order to evaluate, with acceptable precision, the flow behaviour in the real beds. In this section, we evaluate the impact of the matrix structure on the fluid flow behaviour: first on viscous effects, and then we extend this approach to the inertial ones.

4.5 7.3 10.7 14.7 18.0 18.0 20.2 20.2 25.2 31.2 33.3 42.7 50.0 61.0 66.7

Porosity, ε Close packing

Loose packing

0.36 0.44 0.52 0.60 0.65 0.65 0.70 0.70 0.72 0.77 0.80 0.84 0.85 0.86 0.89

0.38 0.48 0.62 0.68 0.71 0.71 0.77 0.77 0.80 0.85 0.87 0.90 0.92

Nature of the fibres Copper Copper Copper Copper Copper Nylon Copper Bronze Bronze Bronze Bronze Bronze Bronze Bronze Bronze

for both close and loose packing, the close packing case having lower porosity values (Table 3). The porosity increases continuously with the fibre aspect ratio r. An asymptotic tendency is found for high aspect ratio. An empirical law of the porosity as a function of the fibre aspect ratio r is proposed ε=1−

1 . a + br

(4)

a and b are constants, respectively, equal to 1 and 0.1. Rahli (1997) derived a new expression of the porosity for randomly stacked fibres based on the excluded volume proposed by Onsager (1948): ε=1−

m . f Vexcl

(5)

f is a non-dimensional excluded volume; it represents the Vexcl averaged volume around a particle where the centre of one other f may be written particle cannot enter. For two rigid cylinders Vexcl as (Rahli, 1997): f Vexcl

Vexcl π = = + 6 + 2r. Vfibre 2r

(6)

From the experimental data, the value of parameter m is found equal to 11. A good agreement with experimental results is observed. Thus, the porosity of two kinds of porous media (randomly stacked fibres or polydisperse spheres) has been determined as a function of the geometry characteristics and/or size distribution of the particles. In both cases, the porosity may be controlled, allowing studying its influence on transport properties. Such media may be considered as basic models of debris beds, and so can be used in order to validate theoretical approaches. To model the porosity of more realistic porous beds, an equivalent approach has to be developed in order to take into account the specificities of the shape and size distribution of the particles.

4.1. Permeability The flow law in porous media depends on an important property of the bed: the intrinsic permeability. This property depends on several parameters which appear in the Kozeny–Carman relation. An approach to evaluate these different parameters in order to determine the permeability of the considered porous medium (with spherical or cylindrical particles shape) is described here. The tortuosity τ, as proposed by Carman (1937) characterises the trajectory of the fluid in a porous medium. The real length covered by the fluid to cross the medium is then τ times greater than the straight length of the porous structure. In the case of porous media constituted of monodisperse spheres, the tortuosity is equal to π/2. For other porous structures, numerous authors used the model of Kozeny–Carman to determine this parameter. Empirical approaches for real porous media, numerical or theoretical for simple or orderly geometries, were developed. In the case of disordered stacked fibres, a dependence of the Kozeny–Carman parameter on the porosity of the fibrous medium has been found using a hydrodynamic method (Rahli et al., 1996). This parameter may also be determined by electric methods (Guyon and Troadec, 1994; Rahli, 1997). For this, the forming factor F is introduced. It represents the ratio between the electric resistance of the porous matrix, saturated with an electrolyte, and the electric resistance of the electrolyte: F=

ρe Le Rp = AL = Re ρe Ae

Le L Ae A

=

τ Ae A

=

τ . ε

(7)

This relation is obtained assuming that the solid matrix is an insulating electrical material. Archie (1942) proposed the following relation for the forming factor F: a F = m, (8) ε where m and a are the parameters depending on the porous structure. The fitting of our experimental points by Eq. (8) gives the following relation: F=

1 . ε1.53

(9)

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Table 4 Experimental results of the tortuosity measurements and the corresponding kk parameter values according to the porosity of the medium (spheres) Aspect ratio, r = L/D

Porosity, ε

Tortuosity (Le /L)

Parameter, kk

1.00 7.00 10.25 12.00 14.75 20.40 25.50

0.40 0.52 0.57 0.59 0.62 0.65 0.71

1.52 1.40 1.34 1.33 1.30 1.27 1.23

5 7.43 6.51 6.19 5.75 5.13 4.70

Combining relations (7) and (9), the tortuosity is τ=

1 . ε0.53

(10)

The Kozeny–Carman parameter is expressed as (Rahli, 1997; Rahli et al., 1997): kk = a γτ,

(11)

where γ is the sphericity factor, calculated from the shape of fibres as √ 2/3 (3r 2) . (12) γ= 2r + 1 Relation (11) allows deducing the Kozeny–Carman parameter from the experimental results of the tortuosity measured by the electrical method (Table 4) and thus identifying also the parameter a . The variations of kk parameter according to the porosity of the medium are represented in Fig. 11. The values are continuously decreasing with an asymptotic tendency. The experimental results obtained by the electric method are well fitted by a hyperbolic function: kk = aε with a = 3.61. This value is almost equal

Fig. 11. Kozeny–Carman parameter kk measured by electric method vs. fibrous medium porosity.

Fig. 12. Test cell for the measurement of pressure drop laws.

to the value deduced from the hydrodynamic method (a = 3.60) (Rahli et al., 1996). In order to further validate this model, the permeability of randomly stacked fibres has been determined using the hydrodynamic method. Ergun’s law describes the pressure drop generated by a fluid flow through a packed bed made with monodisperse spheres. In the case of randomly stacked fibres, a different flow law must be taken into account. For this aim, a specific experimental device has been realised (Fig. 12). The variation in pressure drop P L is determined systematically according to the superficial velocity in the case of aspect ratios ranging from 4 to 70 (Fig. 13). The variation of the pressure drop versus the velocity is apparently almost linear in the range of superficial velocities explored. Nevertheless, as illustrated in Fig. 14 the inertial component is rather important if the superficial velocity is high enough (10 mm s−1 for this example).

Fig. 13. Pressure drop variations vs. liquid superficial velocity for several aspect ratios of fibres (4.5 ≤ r ≤ 66.7).

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following correlation: K π ε5.1 = 0.0606 2 d 4 (1 − ε)

(0.4 ≤ ε ≤ 0.8).

(13)

The variation law for permeability versus porosity, deduced from the measurements of Rahli et al. (1995), is as follows K 62.5ε6 = . d2 (1 − ε)2 (3.6 + 56.4ε)2

Fig. 14. Pressure drop variations vs. liquid superficial velocity. Comparison with linear behaviour: emphasis on non-Darcean effects. Aspect ratio of fibres: 25.2.

The experimental permeability evolves according to the bed porosity (Fig. 15). The permeability values deduced from models are also shown on this figure. This comparison is limited to the applicable porosity range for each model. The permeability varies from 20 to 20 000 Darcy, for porosity ranging between 0.35 and 0.92. Happel and Brenner (1986) developed analytical solutions for parallel and normal flows around a cylinder of a given diameter and proposed a linear combination of the permeability calculated for parallel and perpendicular flows. Numerical solutions of Navier–Stokes equations have been proposed by Sahraoui and Kaviany (1992) using a finite-difference method for twodimensional flows through an array of cylinders and lead to the

(14)

For porosity values greater than 0.7, the permeability values deduced from the model of Jackson and James (1986) are smaller than those obtained experimentally by Rahli et al. (1995). The variation laws proposed by several authors differ considerably from the experimental results if all porosity values are considered. The law deduced from the model of Happel and Brenner (1986) is similar in behaviour to the experimental results, but gives over- or underestimation of the permeability depending on the porosity. For porosity values lower than 0.6, the model of Happel tends to overestimate the permeability. When the permeability is greater than 0.7, the results tend to be lower than the experimental values. The comparisons indicate that models based on fibres arranged in simple, ordered patterns do not “accurately” predict the permeability of the randomly packed monodisperse rigid fibres. Such models do not take into account the influence of the fibre aspect ratio r and the contribution of the fibre-extremity surface. This contribution falls below 2% for r values greater than 25, but it is quite significant for low aspect ratios. As we obtained expressions describing relationships between porosity, tortuosity and aspect ratio, we could propose a relation giving the dependence of permeability with aspect ratio and fibre’s diameter for randomly stacked fibres of arbitrary length and diameter. This expression is deduced from Eqs. (5), (6), (10)–(12) along with the Kozeny expression and the
 geometri2+4r cal definition of fibres specific surface Sp = dr : π 3.53 d 2 r 4/3 2r + 2r − 5 ε3 d 2 K= =  1.53 . π kk (1 − ε)2 Sp 30483(1 + 2r) 6 + 2r + 2r

(15)

√ 2/3 Note that the value 30483 is, in fact, given by 484(3 2) a , with a the constant of expression (11) which has been identified from experimental data (here a = 24). 4.2. Inertial effects (passability) In the case of higher flow rate values, inertial effects are increased and the pressure drop law is given by the Forchheimer relation: Fig. 15. Permeability variation K vs. fibrous medium porosity and fibre’s aspect ratio: (
) experimental results (Rahli et al., 1995); (1) model of Jackson and James (1986); (2) model (flow perpendicular to the cylinders) of Happel and Brenner (1986); (3) model (flow parallel to the cylinders) of Happel and Brenner (1986); (4) model of Kyan et al. (1970); (5) model of Sahraoui and Kaviany (1992).

P µ = U + βρU 2 , L K

(16)

where the permeability K and the inertial coefficient β are intrinsic parameters of the considered porous matrix. Several authors proposed the use of a non-dimensional inertial coefficient (Angirasa, 2002; Bhattacharya et al., 2002),

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expressed as √ f = β K.

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(17)

For stacks of balls, Ergun (1952) proposed the following relation for the inertial coefficient β: 1−ε , with a = 1.75. (18) ε3 d This leads to the following expression for the fibres using the relationship between aspect ratio and porosity (Eqs. (5) and (6) with m = 11):

β=a

β=

22ar(π + 4r(3 + r))2 . d(π + 2r(2r − 5))3

(19)

The Ergun expression is the most often used to evaluate the pressure drop in any kind of porous medium. Nevertheless, this expression is valid only for porous media constituted by nearly monodisperse roughly spherical particles. As for the Kozeny–Carman constant appearing in the viscous term, the constant of the inertial coefficient is a function of the porous medium structure. In the case of a monodisperse fibre stack, it is possible to deduce the inertial coefficient value from the experimental results. For that purpose, each experimental pressure drop law is fitted by a second-degree polynomial equation (Figs. 13 and 14). The coefficient of the quadratic term corresponds to the value of βρ, from which the constant a of the inertial term is deduced. We notice a variation of this parameter as a function of fibre’s aspect ratio (Fig. 16). All these points are well fitted with a power law: 5532 . (20) r2 This coefficient varies by four orders of magnitude for porosity ranging from 0.35 to 0.92. This variation has to be connected to the strong modification of the texture of the porous medium with the porosity value. Indeed, an increase of porosity leads simultaneously to a reduction of the solid volume fraction and of singularities (constrictions, obstacles, etc.).

Fig. 17. Dimensionless inertial coefficient f vs. stacked fibres porosity: () experimental results; (grey triangles) values deduced from Eq. (21); (line) correlation from Rahli et al. (1996).

By using the expression of K from Eq. (15), the nondimensional inertial coefficient becomes     √ √ 3 6 2  r 4/3 π + 2r − 5 3.53 461 2 3(π + 4r(3 + r))  2r f = π 1.53 . r 0.83 (π + 2r(2r − 5))3 (1 + 2r) 2r + 2r + 6 (21) In Fig. 17, this expression is compared to experimental values and the semi-empirical expression given by Rahli et al. (1996). Both expressions give reasonable agreement with experimental data.

a=

Fig. 16. Variation of the inertial parameter a of Ergun’s law vs. fibres aspect ratio: () experimental results.

4.3. Pressure drop A new expression of the pressure drop in the case of a fibrous disordered medium may be derived from the previous sections (Eqs. (15) and (21)). This expression is valid for a disordered stack of monodisperse fibres. It takes into account the effect of the porosity variation in a wide range (0.35 ≤ ε ≤ 0.92). This law was verified experimentally for a fibre diameter value equal to 150 ␮m. In order to generalise this expression, its validity has to be confirmed for various fibre diameters. These results show all the complexity of the fluid flow through porous media of various textures. The flow laws established here show the necessity to use specific expressions for each type of porous media structure. The use of a law established for a different texture than the porous medium under study may lead to erroneous estimations of pressure losses. The viscous effects appear important for porous media having small hydraulic diameter while the inertial effects become dominant for high porosities and hydraulic diameters. In relation to the debris bed, the second behaviour is mostly encountered. Nevertheless the two effects may exist in such media because of the heterogeneity’s of the debris bed structure. In summary, for the flow laws in the debris bed, the Forchheimer law remains valid. To describe precisely the flows in such porous media, the main effort must be done to evaluate the

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coefficients which describe the porous structure, the fluid trajectories in the bed and flow regimes. We proposed in this paper a new approach to determine the coefficient which is characteristic of the viscous effects. Inertial effects have been experimentally investigated for randomly stacked fibres (150 ␮m diameter) for several fibres lengths. More investigations are needed in order to derive a general expression of pressure losses in such fibrous materials.

acteristics of these media). Experiments involving boiling have been conducted for different operating conditions (Miscevic et al., 1998, 2002). Even if such a medium could not be considered as representative of a debris bed, this study constituted a basic experiment in order to show up the influence of the morphology of a bed on the thermo-hydraulic behaviour under boiling conditions. 5.1. Experimental setup

5. Convective boiling in porous media: two-phase flow and heat transfer When boiling occurs in debris beds, heat transfer phenomena and fluid motion are complex because of the solid phase texture and the heterogeneous distribution of the heat source generated by radioactive decay. The modelling of these phenomena requires stronger simplifying hypotheses. As a result, the existing models may only describe the qualitative behaviour. The complexity of convective boiling configurations generally implies to develop experimental approaches in order to obtain a better understanding of the phase-change phenomena and twophase flows occurring in the porous matrix. Inside porous media, nucleation takes place over wide surfaces whose geometrical singularities greatly enhance the phenomenon. The liquid–vapour phase change occurs on several scales: • On the pore scale: this is the level where nucleation occurs. There are three coexisting phases (liquid, vapour and solid) separated by a triple line. As a result, the interfaces are highly complex and characterised by intense motions. • On the porous reservoir scale: at this level, there are largescale convective motions of the two-phase flow. • On intermediate scales: these latter are characteristic of the relative motion of the phases (liquid–vapour) and structure heterogeneity.

The porous medium is composed of bronze fibres (diameter df = 150 ␮m, length to diameter ratio = 14 ± 0.3, porosity 59 ± 0.8%) randomly stacked in a low thickness-to-width ratio channel of dimension 1 cm × 5 cm × 10 cm. The fibre medium is sintered simultaneously with the boundary two bronze plates of 5 cm × 10 cm. This eliminates the thermal contact resistances between the solid walls and the porous matrix. In these conditions, high thermal conductivity of the porous medium (λeff ∼ 11 W m−1 K−1 ) is obtained (Miscevic, 1997). The sintered system is held inside a fluoride plastic box (PVDF) that ensures both tightness and thermal insulation. The liquid (i.e. n-pentane) flows through the sintered porous media vertically from bottom to top. The bronze plates are heated by 10 embedded electrical resistors. The average surface heat fluxes at the plate surface could be varied from 0 kW m−2 up to 15 kW m−2 . The energy balance is checked for each experiment and the overall losses were found to be negligible (<3%). Fig. 18 shows the schematic diagram of the experimental apparatus consisting of seven main components: the box containing the porous channel, a cyclone, a condenser, a weighted tank, a buffer tank, a pump, a cryostat. The n-pentane contained in the storage vessel is circulated by a peristaltic pump. In this setup configuration, the mass flux is imposed and could be adjusted in the range 0–12 kg m−2 s−1 . Down this device, a vibration absorber, mounted as a Helmholtz resonator, is inserted in the loop. It allows the damping of the

At all scales, the motions are conditioned by the phase-change intensity, the vapour volume fraction, and the porous medium structure. It is thus possible to observe two-phase flow structures similar to those observed in channels (bubbles, slug, plug flows). Nevertheless, specific flow structures may appear, due to the great influence of the confinement effect on the two-phase flow dynamics. The distribution of phases during boiling in the porous medium is conditioned by viscous, inertial, gravity and capillary effects (Topin et al., 2002). In this part, experimental results about convective boiling in a sintered fibrous medium are presented (see Table 5 for charTable 5 Characteristics of the studied porous media Bronze balls Thermal conductivity (W m−1 K−1 ) Diameter (␮m) Porosity Permeability (×10−12 m2 )

180 150–180 0.40 24

Glass balls 1.1 140–160 0.37 16

Bronze fibers 180 150 0.69 250

Fig. 18. Schematic view of the experimental apparatus. The main components of the loop are represented.

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pressure fluctuations created by the pump. The n-pentane passes in a heat exchanger immersed in a cryostat whose temperature is adjusted to obtain the chosen fluid temperature at the inlet of the porous channel. The n-pentane flows into the heated porous channel and may vaporise. At the test section exit, the liquid flows to the weighted tank. The generated vapour flows to the channel exit, passes through a cyclone separator that collects residual liquid drops. The vapour passes then through a condenser where it is liquefied. The liquid is finally collected in the buffer tank. K type thermocouples are placed along three vertical lines (in the median plane, in the vicinity of the wall and at an intermediate position) at a depth of 2.5 cm (Miscevic et al., 1998). Twelve pressure sensors are placed along the median plane, at the same locations as the thermocouples. 5.2. Temperature profiles and phase distribution In boiling conditions, different zones, distinguished by their temperature profiles, are observed inside the sintered fibrous medium. From these results, a typical distribution of zones in the test section is proposed. Four zones are defined according to the thermodynamic state of the fluid. From the inlet to the channel outlet, one may observe a liquid zone Zl , a two-phase zone in thermal equilibrium Zd , a superheated two-phase zone out of equilibrium Ze and a dry vapour zone Zv (this latter zone was not observed in these experiments). Fig. 19 shows an example of the temperature profiles along all three measuring axes for a stationary regime as well as the axial pressure profile. In the lower part of the fibrous medium, the temperatures remain below saturation temperature: the fluid is thus in a liquid state (liquid zone Zl ). In this zone, the pressure evolves in a linear manner. A zone with nearly constant temperatures is then observed, corresponding to the boiling of the fluid (two-phase zone in thermal equilibrium Zd ). The pressure values, however, evolve in a

Fig. 19. Example of experimental temperature and pressure profiles in stationary regime (heat flux ≈ 6.6 W cm−2 . Inlet pentane mass flux G ≈ 2.8 kg m−2 s): ( ) pressure—y = 0; ( ) temperature—y = 5 mm (wall); () temperature—y = 2.5 mm; ( ) temperature—y = 0 (centre).

Fig. 20. Temperature variation in the boiling zone vs. the z-axis: ( ) measured temperature; ( ) saturation temperature deduced from pressure measurements.

non-linear manner. The pressure gradient increases as a result of the acceleration of the liquid–vapour mixture, and this acceleration results from the phase change in the fluid (expansion rate: ρl /ρv ≈ 200). In this boiling zone, there is a slight decrease in the temperature profile. Fig. 20 shows this drop along the central axis. This phenomenon is attributed to the pressure losses (Topin et al., 1996), and it is often disregarded (Vasil’ev and Maiorov, 1979; Udell, 1985; Stemmelen, 1991; Wang, 1997; Ghafir, 2000; Najjari and BenNassrallah, 2002). Nevertheless, concerning the flow properties of the medium and flow rate, this temperature change could be of importance. Partially, pressure and temperature gradients may also be correlated to the instabilities that are observed for some specific operating conditions. The profile of saturation temperature along the central axis in the boiling zone in Fig. 20 is determined from the measured pressures. The saturation temperatures are found to be identical to the temperature readings at the same location within the experimental uncertainties (i.e. maximum deviation: 0.1 ◦ C). This indicates that, in this zone, the fluid remains saturated and that both fluid phases are in thermal equilibrium. Using the temperature profile in the porous medium, we can delimit the different zones and particularly the limit between the Zd zone in thermal equilibrium and the Ze zone in nonequilibrium. This limit corresponds to the location where the measured temperature starts to become higher than the saturation value (Fig. 21, different case from that in Fig. 19). As we measure simultaneously the outlet quality of the two-phase flow (X < 100% in these experiments), we could verify that indeed this zone is not a superheated vapour one. One can see in this figure that the Ze zone starts at the same height (z ∼ 45 mm) for the two profiles (y = 0 and 2.5 mm; y-axis is oriented in the width direction, and y = 0 is the median plan of the fibrous channel). The change of slope of the wall temperature indicates that the Ze zone probably starts at z ∼ 25 mm in the vicinity of the wall. In order to reconstruct the pressure profile in the whole channel, we calculated it in each zone according to the state of the fluid: liquid, two-phase and vapour.

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Fig. 21. Temperature profiles along the three vertical axes. Inlet pentane mass flux: G = 6.7 kg m2 s−1 ; heat flux: q = 15 104 W m−2 (each point is a mean value of the temporal evolutions of the temperature).

In the liquid zone, pressure does not depend on the temperature. If the flow is supposed one-dimensional, the pressure profile may be deduced from Forchheimer law, using the flow rate and the pressure at the inlet. Although the temperature field is finely known in the two-phase zone, we cannot reach directly the other characteristic parameters of the two-phase flow. 5.3. Flow laws in boiling conditions: two-phase zone pressure profiles

dP 2 = αµm Um + βρm Um + ρm g, dz

configuration from expressions (15) and (19), respectively: α=

1 m−2 250 × 10−12

(22)

with Um is the equivalent mixture velocity, α and β the constants that characterise, respectively, viscous and inertial effects and are characteristic of the porous medium. These latter parameter values are deduced from the flow law obtained in single phase configuration and are determined for the present experimental

and

β = 48 710 m−1 .

(23)

In addition, the heat flux in the boiling zone is assumed to be constant and uniform. The energy conservation equation leads to x(z) =

In this zone, both heat transfer and fluid flow are supposed one-dimensional. The two-phase flow was described by an equivalent homogeneous fluid which also follows the Forchheimer law. Taking into account the full complexity of the two-phase flow structure leads to complex treatments (as illustrated previously) and is not attempted here because of the numerous uncertainties in the closure laws involved (slip velocities, capillarity, etc.). For the present purpose, we adopted here the simplest model. We neglected kinetic terms and effects of flashing and compressibility (although these assumptions are rigorously valid only at high pressure and low velocity when the overall pressure drop is nil compared to the absolute pressure). Moreover, we assumed that the slip velocity is nil. For the case under consideration here, with co-current flow of liquid and vapour, this is considered to be justified as first approach. With these simplifying assumptions, the law for pressure loss is expressed as follows −

Fig. 22. Pressure variation in the boiling zone vs. the z-axis (heat flux ≈ 3.77 W cm−2 ; G ≈ 5.3 kg m2 s−1 ; U ≈ 8.8 mm s−1 ): experimental results: ( ) measured pressure; ( ) saturation pressure deduced from the temperature measurements. Results of the model with the mixture viscosity relation of: (1) Mac Adams et al. (1942); (2) Beattie and Whalley (1981); (3) Duckler et al. (1964).

ql(z − zd ) m ˙ 0 Lv

if z > zd ,

(24)

where x is the local quality, q the heat flux in the boiling zone, l the width of the channel, Lv the latent heat of vaporisation and m ˙ 0 is the inlet flow rate. The liquid saturation can then be expressed as Sliq =

1 1+

x ρliq 1−x ρvap

.

(25)

Fig. 22 shows an example of the pressure variation obtained by this approach for a moderate heat flux. The three curves are obtained using the different laws for the mixture viscosity proposed by Mac Adams (1942), Duckler et al. (1964) or Beattie and Whalley (1981). The different laws yield the same shape. Although the different viscosity laws contain significant differences, they produce only slight discrepancies in the pressure loss results. This is primarily due to the importance of inertial effects over viscous ones. Given the assumptions upon which this work was based, the model results correspond quite closely to those obtained from the experiments. This agreement is found to be lost when the heat flux is increased as shown in Fig. 23. This discrepancy between model and experimental results is observed when a superheated zone is present in the medium. Given the weak thermal diffusion of the vapour, the temperature gradients in this zone are quite high and the temperature fields then present a two-dimensional pattern. For sintered fibrous media presenting a superheated vapour zone, a new model has to

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Fig. 23. Pressure variation in the boiling zone vs. the z-axis (heat flux ≈ 13.66 W cm−2 ; G ≈ 5.3 kg m2 s−1 ; U ≈ 8.8 mm s−1 ): experimental results: ( ) measured pressure; ( ) saturation pressure deduced from the temperature measurements. Results of the model with the mixture viscosity relation of: (1) Mac Adams et al. (1942); (2) Beattie and Whalley (1981); (3) Duckler et al. (1964).

be developed taking into account these two-dimensional effects to correctly describe the pressure variations in the boiling zone. Even more, in the case of a high vapour quality flow, a considerable temperature difference between the fluid phases could occur. This leads to suppose that the two-phase flow is not homogeneous in these conditions: its structure is probably an inverted annular flow generally observed in post-dryout conditions in tubes (vapour near the wall with a two phase core). Thus, the two phases flow globally at different velocities and slip phenomena become preeminent and generate a quite different pressure behaviour. 5.4. Non-stationary two-phase flow and non-equilibrium phenomena

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behaviours are highlighted for a given inlet velocity. At low heat flux, very small fluctuations (less than 0.5 ◦ C, Fig. 24, curve a) are observed. Above a critical heat flux, fluctuations amplitude increases sharply (Fig. 24, curve b). These fluctuations are typically of 10 ◦ C amplitude but could even reach 30 ◦ C for some operating conditions. These temperature fluctuations are correlated to the appearance of a superheated zone in the upper part of the channel (Fig. 21). They are not observed on readings taken in both liquid and boiling zones for all operating conditions. They may indicate that the sensor is alternatively in contact with liquid, two-phase mixture or superheated vapour. When this superheated zone exists, both measurements of outlet vapour quality (less than 100%) and enthalpy balance indicate that only a partial vaporisation of the liquid occurs in the channel. As we measure separately vapour and liquid temperature in a cyclone placed at the channel outlet, we verify that the liquid always exits at saturation temperature Tsat (p), even when the vapour exhibits high temperature superheat. This behaviour is distinctive of thermal non-equilibrium between the two fluid phases. The transition between Zd and Ze zones appears for a critical value of operating parameters (outlet quality, wall heat flux, etc.). We determined the “critical conditions” corresponding to the apparition of this thermal non-equilibrium behaviour and try to correlate them to the main operating parameters of these experiments. We use variations of vapour quality versus temperature difference Tvap as shown in Fig. 25 to determine the critical condition. Tvap represents the temperature difference between fluid temperature at the outlet of the channel and the saturation temperature at the same location. All curves present the same behaviour: a first part where the quality evolves at nil superheat (thermal equilibrium) and

Temporal evolutions of temperature for different heat fluxes, with a given inlet liquid mass flux, are reported in Fig. 24. These readings are taken close to the fibrous channel outlet; two

Fig. 24. Temperature variations in the porous channel at y = 0, z = 86 mm. Mass flux G = 6.7 kg m−2 s−1 ; (a) heat flux q = 2.4 × 104 W m−2 ; (b) heat flux q = 15 × 104 W m−2 .

Fig. 25. Variation of the outlet vapour quality vs. the temperature difference Tvap (outlet fluid temperature minus the saturation temperature) for several mass fluxes ranging from 0.5 to 10.1 kg m−2 s−1 . Heat flux is increased in different ranges (see Fig. 27) to yield a certain curve for a certain mass flux.

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Fig. 26. Critical vapour quality vs. the mass flux (these Xc values correspond to the appearance of the Ze zone).

a second part where the quality increases with the superheat Tvap . This behaviour corresponds to a thermal non-equilibrium between the liquid and its vapour. The critical quality, for each flow rate, is defined as the highest quality at which Tvap remains equal to zero. Thus, the critical conditions occur for the highest heat flux (for a given flow rate) at which the superheat remains nil. From these values, all parameters (temperature, pressure, quality, etc.) are deduced from experimental readings. In Fig. 26, we report critical outlet vapour qualities versus mass flow rates. The “critical” quality decreases continuously with increasing the mass flux G indicating that the thermal nonequilibrium phenomena are stimulated not only by the higher wall heat flux, but also by the increase of the mass flow rate. These experimental results lead us to assume that the thermohydraulic instabilities may result from a competition between two effects. The first corresponds to the phase change phenomenon in the porous matrix and the second is linked to the matrix structure (permeability, etc.). In fact, when the heat flux is small enough, the fluid flow may occur without great disturbances (homogeneous two-phase flow). For higher heat flux (high vapour hold up), the fluid flow is disturbed (its structure is no more homogeneous in the section: inverted annular configuration) and generates the thermal instabilities. The influence of these two effects may be seen more clearly in Fig. 27, representing the heat fluxes as a function of the mass fluxes, For a given heat flux, a variation of G leads to the transition from equilibrium to non-equilibrium. Similarly, for a given mass flux G, varying the heat flux leads also to the transition. The wall temperature increases sharply in the vicinity of the boundary between Zd and Ze zones. This shows that a local dryout near the wall probably appears. Several authors have studied the boiling phenomena in rectangular ducts without porous matrix (Collier and Thome, 1994). Katto (1981) proposed an empirical correlation for the critical heat flux (CHF) in the case of a uniformly heated rectangular channel. The comparison of this correlation with the experimental results shows a good agreement (Fig. 27). Nevertheless, it seems that the Katto model overestimates the experimental

Fig. 27. Wall heat flux in the two phase zone (Zd + Ze ) vs. the mass flux G: experimental results in a thermal equilibrium conditions ( ) and non-thermal equilibrium conditions (). Comparison of experimental CHF (transition between thermal equilibrium and non-equilibrium regions) with Katto correlation: experimental CHF (䊉); CHF from Katto correlation (1981) ( ).

transition for the higher mass fluxes. More experimental data are needed to confirm this trend. The CHF values obtained with or without porous medium are similar. This indicates that in the Zd zone, most of the heat exchanges between the fluid and the solid occurs in the vicinity of the wall. The porous structure does not play a significant role for the heat transfer process in this zone. On the other hand, the fibres probably increase the number of nucleation sites on the wall and thus lead to a very high heat exchange coefficient. 5.5. Heat transfer Two distinct convective boiling regimes were observed: thermal equilibrium and thermal non-equilibrium. The transition is found to behave similarly to the dryout heat flux according to the model of Katto. These two regimes lead to different heat transfer characteristic as illustrated in Figs. 27 and 28. The boiling characteristics are determined for each experiment. The heat flux is defined as the ratio between the transferred heat per time and the wall channel area where the boiling phenomenon occurs (Zd + Ze ). The temperature difference is defined as the mean difference between the wall and the fluid saturation temperatures in a cross-section of the channel. The derived boiling curve is reported in Fig. 28. In the thermal equilibrium case, there is only a weak influence of the mass flux on the heat transfer characteristics. This indicates that the phase change effect is dominant in comparison to the convective one. The onset of boiling occurs at very low temperature differences between the wall and the fluid (T ∼ 1 ◦ C), while on a smooth surface this temperature difference is about 10 ◦ C. This result has also been reported by Fukusako et al. (1986). These differences are attributed to the increase of the exchange surface. This leads to a sharp augmentation of the thermal exchange between fluid and solid phases, and of the number of nucleation sites.

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Fig. 28. Boiling curve, heat flux vs. mean superheat value. (*) Tcrit deduced from experimental data in CHF conditions; (grey zone) thermal equilibrium results.

In the thermal non-equilibrium case, the boiling curve depends clearly on the mass flux. For a given temperature difference, the heat flux increases with the mass flux. In fact, we may assume in this case the appearance of a dried zone in the vicinity of the heating walls. The convective effects play an important role and the heat transfer depends on the mass flow rate. By analogy, one can say that the Ze zone is similar to the dryout phenomenon observed in pool boiling within a bed of particles, or to the mist flow regime observed in the case of convective boiling in tubes. Groeneveld and Delorme (1976) have reported a study of a two-phase zone in non-equilibrium for a mist flow configuration. They proposed a relation for the exchange coefficient and the wall temperature. Their correlation, applied to our case, leads to abnormal wall temperature values (>1000 ◦ C). This result highlights the important role played by the porous matrix in the Ze zone. The fibres participate very efficiently to the heat transfer process in this zone and allow the wall temperature to remain moderate even for high flux in post-CHF configuration. In case of convective boiling in tubes (without porous medium), as soon as a vapour film appears near the wall the thermal resistance (and consequently the wall temperature) increase sharply. The solid surface is separated from the liquid (or twophase) core of the flow and nucleation is suppressed. The wall heat flux is transferred first to the vapour (monophasic convection) and then eventually to the liquid which still vaporises at the liquid–vapour interface. Thus, the global fluid-wall exchange coefficient is reduced in comparison with the nucleate boiling. On the other hand, with a channel filled with a highly conductive porous medium, such as sintered fibres, when a vapour zone appears near the wall, the heat flux can be transferred by conduction in the solid matrix across the vapour film to the core of the channel. Then, nucleation can occur on the fibres surface in the bulk of the channel. Thus, the fibrous matrix has a strong influence on the behaviour of the Ze zone: it allows the heat to

be transferred across a vapour film and they provide solid surface for the nucleation process in the bulk of the channel. We do not find experimental data from other authors that could be fully compared to the case presented in the present study. 6. Conclusions The heterogeneity of the solid matrix makes the transport phenomena complex. The porous structure may have significant differences depending on the form of the basic elements (spheres, fibres). As a result the fluid flowing through the pores exhibits a characteristic behaviour depending on these structural parameters. To improve the usual models used for predicting fluid flow, heat transfer and CHF, these have to be tested using porous media of different and well-controlled structural properties. We have applied this method for polydisperse spheres and stacked fibres. Classical relations may be applied to derive pressure drop and heat transfer of a fluid flow in a porous medium constituted with stacked spheres. For other porous structures, these relations remain only qualitative. A new pressure drop relation is proposed for stacked fibres whatever is the fibre aspect ratio and porosity of the stack. For convective boiling in porous media, typical zones are found: liquid, two-phase and vapour. In the two-phase zone, two regimes depending on the heat flux and the liquid superficial velocity are evidenced. For small heat fluxes, a liquid–vapour thermal equilibrium zone is found. A one-dimensional model is successfully used to derive the pressure drop in the two-phase zone. For high heat fluxes densities, a non-thermal equilibrium zone is observed. A deviation between the pressure drop model and the experiments is found. This deviation is enhanced when the heat flux is increased. This discrepancy is probably linked to the “heterogeneity” of phase distribution, heat transfer and fluid flow.

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The heat transfer behaviour is found to be quite different in each regime: in the non-equilibrium regime, the heat transfer coefficient depends clearly on the fluid velocity and is reduced compared to the thermal equilibrium regime coefficient. The transition conditions between these two regimes have been characterised and correspond roughly to the CHF model of Katto. This result has to be confirmed in others configurations.

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