- Email: [email protected]
The effects of debris bed's prototypical characteristics on corium coolability in a LWR severe accident
Nuclear Engineering and Design 240 (2010) 598–608
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
The effects of debris bed’s prototypical characteristics on corium coolability in a LWR severe accident Weimin Ma ∗ , Truc-Nam Dinh 1 Division of Nuclear Power Safety, Royal Institute of Technology (KTH), Roslagstullsbacken 21, 106 91 Stockholm, Sweden
a r t i c l e
i n f o
Article history: Received 25 June 2009 Received in revised form 16 October 2009 Accepted 30 October 2009
a b s t r a c t This paper is concerned with coolability assessment of a debris bed formed in fuel coolant interactions (FCIs) during a hypothetical severe accident in a light water reactor (LWR). The focus is placed the potential effect of the bed’s prototypical characteristics on its coolability, in terms of (i) porosity range, (ii) multi-dimensionality, (iii) inhomogeneity, (iv) particle morphology, and (v) heat generation method (e.g. volumetric heating vs. local heaters). The analysis results indicate availability of substantial coolability margins compared to previous assessments based on models and experiments using an idealized bed configuration (e.g. 1D homogenous debris layer). Notably, high porosity (up to 70%) of debris beds, obtained in experiments and expected to be the case of prototypical debris beds, could increase the dryout heat flux by 100% and more, depending on particle size, compared with the dryout heat flux predicted for debris beds with traditionally assumed porosity of approximately 40%. Bed inhomogeneity represented by micro-channels in a mini bed is predicted to enhance the dryout heat flux by up to ∼50%, even if the micro-channels occupy only a small volume fraction (e.g., less than 4%) of the bed. The effect of coolant side ingress into a multidimensional bed is predicted to enhance the dryout heat flux by up to 40% for the beds analyzed. © 2009 Elsevier B.V. All rights reserved.
1. Introduction In a hypothetical severe core-melt accident of a nuclear reactor, a debris bed may be formed in the vessel lower head (in-vessel) or in the reactor cavity (ex-vessel), provided that the molten corium penetrates into a water pool, fragmenting and forming debris particles which will then settle down on the pool bottom. Compared with a molten corium pool, the particulate corium (debris) bed is easier to be cooled, i.e., having more chance to remove the decay heat, for the inside of the bed is accessible for coolant through the internal pores of the bed. The debris bed coolability is therefore of paramount importance to termination and stabilization of the severe accident. As a result, the study on debris bed coolability has been a hot topic in assessment of melt risk of light water reactors (LWR). The quantification of ex-vessel debris bed coolability is of particular interest to the Swedish BWR plants which employ lower drywell flooding as a severe accident management strategy, with a hope that a coolable debris bed will be formed in a deep water pool of the wet cavity. To pursue a quantitative understanding of the coolability of both the in-vessel and ex-vessel debris beds, numerous experi-
∗ Corresponding author. Tel.: +46 8 5537 8821; fax: +46 8 5537 8830. E-mail address: [email protected] (W.M. Ma). 1 Present address: Idaho National Laboratory, Idaho Falls, ID-83415, USA. 0029-5493/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2009.10.026
ments and analyses have been conducted to study two-phase flow and heat transfer in such particulate debris beds. The debris beds employed in the past studies were characterized with an averaged particle size or with a size distribution of the debris obtained from a few fuel–coolant interaction (FCI) experiments (Spencer et al., 1994; Magallon, 2006; Huhtiniemi et al., 1997; Song et al., 2002; Kato et al., 1999; Dinh et al., 1999) using sieving technique. Moreover, a bed-averaged porosity (normally 40%) was assumed based on the free particles packing theory under cold conditions. Subsequently, the so-measured mean particle size and bed porosity serve as a basis to design the debris bed coolability experiments. Notably, neither the prototypic bed’s porosity nor bed’s interior structure (pore size distribution) are reproduced in such experimental beds (Hofmann, 1984; Schmidt, 2004; Konovalikhin, 2001; Bang et al., 2005; Hu and Theofanous, 1991). Furthermore, the coolability tests were conducted primarily for determining dryout heat fluxes in one-dimensional bed configuration with top flooding. In other words, the study of debris bed coolability was focused on top-flooding scenarios, which manifest situations when side and bottom coolant injection are assumed to be negligible or absent, for example, when corium debris is spread evenly over the cavity floor and coolant is added to top. For the top-flooding bed packed by spherical particles, the dryout heat flux (DHF) can be determined by counter-current flow limit (CCFL) and predicted, with a fair accuracy, by analytical models (Hu and Theofanous, 1991; ˝ Lipinski, 1984; Reed et al., 1985; Schulenberg and Muller, 1987;
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
Nomenclature C d Dh F g j K Kr L p s z
Cozeny–Karman constant particle diameter (m) hydraulic diameter (m) volumetric drag force (N/m3 ) gravitational acceleration (m/s2 ) superficial velocity (m/s) permeability relative permeability height of bed (m) pressure (Pa) liquid saturation vertical coordinate (m)
Greek symbols ˛ void fraction vapor production rate (kg/m3 /s) ε porosity passability r relative passability dynamic viscosity (kg/m/s) density (kg/m3 ) Subscripts g gas, steam i interface l liquid p particle
599
scheme circumvents the CCFL limit and consequently enhances the coolability. While such installations were proposed for some new reactor designs, their application in the existing nuclear plants is not straightforward, due to a cumbersome retrofit and impact on plant operation. Thus, instead of engineering solutions to enhance debris bed coolability, the present study explores inherent features of a debris bed formed from FCIs. The motivation is to quantify those characteristics which were ignored in the past studies but are judged to have potential impact on the bed’s coolability. For instance, in a prototypical accident scenario, the debris beds are expected to have a heap-like shape, with the peripheral region being more coolable; a porous decay-heat-free bottom layer may be formed as a result of earlier discharge of metallic melt (e.g. Zr an stainless steel); channels (micro/macro inhomogeneity) may be formed in the bed due to intensive boiling and flow; etc. On the other hand, in many experiments performed for investigation of debris bed coolability, ohmic heaters were used for heat source, which is distinct from the volumetrically uniform heating of the corium decay heat. While characterization of prototypical debris beds formed from FCIs is presently undertaken in the DEFOR program and discussed elsewhere (Karbojian et al., 2009; Kudinov et al., 2007), the present study is to provide a first-cut quantitative analysis of the impact of such prototypicalities on the debris bed coolability. 2. Identification of non-prototypicalities in debris bed models
Tung and Dhir, 1988). Given accident scenarios with formation of deep beds or dense beds with either small particles or low porosity, the models predict that the top-flooding is insufficient to remove decay heat released in such debris beds. This perception has motivated further search for additional means to enhance debris bed coolability, thereby benefiting reactor safety performance. One of the engineering solutions is to bring coolant down to the bottom of the bed through a device such as a downcomer (Konovalikhin, 2001) or a distributor embedded on the cavity floor. The bottom-fed
The debris beds formed in a severe accident are the outcome of molten corium breakup and subsequent solidification and settlement of the debris in a water pool (cf. Fig. 1, a test debris bed formation test (Karbojian et al., 2009)). Due to the particular process, one may expect characteristics of the so-formed debris beds to be distinct from the packed beds that were formed under cold conditions and employed as default in the past studies on debris coolability. It is the distinction of debris bed formation that determines the prototypicalities (characteristics of the bed) which is contrary to the non-prototypicalities (characteristics of an experimental bed). Although it is still difficult to identify all non-prototypicalities because of the complexity of debris bed formation process as
Fig. 1. Three-phase flow and heat transfer during FCI and debris bed formation.
600
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
Table 1 Identification of the characteristics of a debris bed formed from FCIs. Configuration
Inhomogeneity
Porosity
Particle morphology
Heat generation
Bed used in coolability experiments
1D
Homogeneous
From particle packing theory (ε = ∼0.4)
Electric heaters (local heating)
Bed formed in FCI experiments
2D/3D
Heterogeneous at pore scale
Governed by intense boiling and particle morphology (ε = ∼0.6–0.7)
Solid particles, regular shapes and smooth surface Some hollow particles, irregular shapes, roughened surface
g g dpg j + jg |jg | = g g + KKr,g l r,g dz
Decay heat (volumetric and uniform heating)
revealed in the relevant experiments and analyses (Spencer et al., 1994; Magallon, 2006; Karbojian et al., 2009; Kudinov et al., 2007), several characteristics of debris beds have been identified and considered as importance to corium coolability. The present study places focus on five aspects of the prototypicality that are believed to have potential influence on debris bed coolability, namely (i) porosity range, (ii) multi-dimensionality, (iii) inhomogeneity, (iv) particle morphology, and (v) heat generation method (e.g. volumetric heating vs. local heaters). Table 1 compares the prototypical beds with the one-dimensional homogeneous top-flooding bed used in previous studies. To enable analysis of separate effect, further modeling assumptions are made as discussed in the following sections. It should be noted that other prototypicalities such as the debris surface properties (related to chemical composition and fragmentation) and the ratio of particle surface to volume also affect the friction laws of two-phase flow in debris beds, and will be addressed in follow-on work.
−
3. Technical approach
There are several models for Kr and r , including the well-known Lipinski’s model (Lipinski, 1984) used to estimate the dryout heat flux. Reed’s model (m = 3, n = 5) is nearly identical to the Lipinski model (m = 3, n = 3), except for the use of a higher relative passability. The Reed’s correlation is used here as it has undergone most extensive validation against the existing experimental database on debris coolability.
Coolability of debris bed submerged in a water pool is assessed by means of the maximum heat flux (per unit area of the bed floor surface) that is generated in the bed and can be removed without dryout incipience, or so-called dryout heat flux. To predict the bed’s dryout heat flux, two approaches are employed in this study—the Reed’s analytical model (Reed et al., 1985) developed and validated for one-dimensional top-flooding homogeneous beds and the WABE-2D code (Schmidt, 2004) developed at IKE-Stuttgart University and validated for calculation of two-phase thermal-hydraulics in two-dimensional debris beds. While limitations (e.g., in friction laws) of these two stateof-the-art methods for debris coolability analysis are recognized and under improvement, the objective of the present study is to provide best estimate of the effect of the above-mentioned non-prototypicality so as to aid identification and first-cut quantification of major uncertainty in ex-vessel melt risk safety assessment and guide future experimentation. 3.1. Reed’s model At the basis of Reed’s and similar models of dryout heat flux in volumetrically heated debris beds is the notion that the coolability of a debris bed is restricted by hydrodynamic (counter-current flow) limitations due to the friction of two-phase flow through the debris bed. When the amount of steam generated in the bed is sufficient to prevent the coolant from reaching the bed’s bottom, dryout occurs there and the bed is considered as uncoolable. Thus, for the dryout models, the key in modeling was to provide the formulation of the friction laws for momentum equations. The well-known Lipinski’s model for the prediction of dryout heat flux in a debris bed is based on the Ergun’s equation (Ergun, 1952) for single phase flow in porous media, but adding relative permeability Kr and relative passability r to account for the effect of two-phase existence: −
dpl l l = l g + j + j |j | KKr,l l r,l l l dz
(1a)
(1b)
where jl and jg are the superficial velocities of fluids, and K and are the permeability and passability, respectively. For uniform spherical particles bed, they are usually expressed as K=
ε3 d 2 C(1 − ε)
2
;
=
ε3 d 1.75(1 − ε)
where C is called Cozeny–Karman constant varying from 150 to 180 and d and ε are the particle diameter and porosity of the debris bed, respectively. The relative permeability Kr and relative passability r are obtained via experiments, and considered as the functions of void fraction. Kr,l = (1 − ˛)m , r,l = (1 − ˛)n ,
Kr,g = ˛m r,g = ˛n
3.2. The WABE-2D code WABE-2D code (Schmidt, 2004) was developed at IKE-Stuttgart University, Germany for simulating transient behavior of debris bed formed in severe nuclear reactor accidents of LWRs. The debris bed is modeled in two dimensions with cylindrical or Cartesian geometry using a quasi-continuum approach. Three separate phases for solid particles, water and vapor are considered. The solid matrix is assumed to be fixed. The mass conservation is governed by the equations: ∂ (ε˛g ) + ∇ gjg = ∂t
(2)
∂ (ε˛l ) + ∇ ljl = − ∂t
(3)
for vapor and liquid, respectively, where ˛ is the void fraction, jg and jl are the superficial velocity vectors of vapor and liquid, and is the net mass conversion rate (vaporization or condensation) between liquid and vapor. In the energy conservation equations of vapor and liquid, the mechanical work due to friction and pressure forces is generally neglected. Radiation heat transfer is implicitly considered in the solid energy conservation equation via an effective thermal conductivity. The momentum conservation equations are simplified by an assumption that the temporal and spatial derivatives of the velocities are neglected, because the dominant force is the friction between the particles and the fluids, and velocity field can be simultaneously adjusted to the pressure field. Especially for quasi
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
601
Table 2 Dryout heat flux (DHF) of debris beds with high porosity. Bed
Porosity
Effective particle size (mm)
DHF (MW/m2 )
A B C
40% 70% 70%
3 3 1.5
0.9 3.37 2.28
in Table 2. The results depict a significant, and yet quite expected effect of the bed’s high porosity on its coolability. Another implication of the results is that the effect of the porosity on coolability should be more significant than the particle size. This can be explained by the flow resistance through the debris bed. From Ergun’s equation (Ergun, 1952), the frictional resistance of single phase flow through a debris bed can be predicted by
Fig. 2. Variation of dryout heat flux with porosity (Reeds’ model (Reed et al., 1985)).
steady-state processes, such an assumption is acceptable. As a result, the momentum conservation yields −∇ pg = g g + −∇ pl = l g +
Fpg F + i ε˛ ε˛ Fpl
ε(1 − ˛)
−
(4) Fi ε(1 − ˛)
(5)
for vapor and liquid, respectively. Here, Fpg is drag forces between particles and vapor, Fpl is drag forces between particles and liquid, and Fi is the interfacial drag between liquid and vapor. The drag forces are derived from Eq. (1) and experimental data (Schmidt, 2004). Capillary force is considered in the momentum equations through the relation pc = pg − pl , which is a function of surface tension , contact angle , porosity ε and saturation s [approximate to (1 − ˛)]. Details of constitutive laws and correlations to close the conservation equations, including flow heat and heat transfer, can be found in Schmidt (2004), Bürger et al. (2006) and Ma et al. (2007) which also provided validation of the WABE-2D code against relevant experimental data. 4. Case study and discussion 4.1. Effect of prototypical range of debris bed porosity From particle packing theory, the porosity of a bed packed by regular particles is known to vary in the range from 26% to ∼46%, and the average porosity with size distribution is around 40% (Schmidt, 2004). This value has become the reference porosity for debris bed configurations used in debris coolability studies. However, in both recent DEFOR scoping experiments (Karbojian et al., 2009) and earlier CCM tests (Spencer et al., 1994), it was found that porosity of the FCI-generated debris beds varied from 50% to 70%, with most cases resulted in porosity of 60% and higher. This implies that the porosity of a prototypical debris bed may be substantially higher than the value predicted by the particle packing theory. Fig. 2 shows the variation of dryout heat flux with porosity of the debris beds for top-flooding homogeneous debris beds. The coolant is saturated at atmospheric pressure. If the particle size is 3 mm and the porosity is 70% in the bed, the dryout heat flux (DHF) is 3.37 MW/m2 predicted by Reed’s model, which is 3.7 times of the DHF of the reference bed with 3 mm particles and 40% porosity. If the particle size was reduced to half of that of the reference bed while keeping 70% porosity, the DHF is 2.28 MW/m2 which is 2.53 times of that of the reference bed. The comparison is listed
Pf = 2f
L Dh
j 2 ε
(6)
with Dh (2ε/(3(1 − ε)))d, f = (1/3)(100/Re + 1.75) and Re = (Dh j/ε). For low flowrate (laminar flow), f = (100/3Re). Thus, the pressure drop is almost proportional to the reciprocal of the fourth power to the porosity, while it is nearly proportional to the reciprocal of the second power to the particle size. For two-phase flow, the principle of the friction law is the same (cf. Eq. (1)). It should be noted that the Ergun’s equation is obtained from flows through beds packed with regular particles. Moreover, the Reed’s model (derived from Ergun’s equation) was developed based on experimental data of dryout heat flux for porous beds with regular particles and 30–40% porosity. The applicability and accuracy of the model in a high-porosity range remain uncertain and should be the subject of future investigation. 4.2. Effect of the debris bed’s multi-dimensionality With few exceptions, debris beds used for the DHF study were characteristically homogenous and one dimensional, the debris bed formed in a reactor accident must be multi-dimensional in nature. Such multi-dimensionality (2D/3D) of debris bed has been recognized and studied by a few investigators (Bürger et al., 2006; Fichot et al., 2006; Chung and Catton, 1991). One of the most important debris bed configurations of multi-dimensionality is the heap-like debris beds (cf. Fig. 3) which does not uniformly spread on the pool floor as previously assumed. Thermal-hydraulics of such debris beds submerged in a water pool is characterized by coolant ingression from the sides and bottom of the bed as well as from the top. In these cases, a downcomer-like channel is formed surrounding the bed; cf. Fig. 4a. At atmospheric pressure, the dryout heat flux is predicted to be 325 W/m2 for the configuration (particle diameter is 1 mm and porosity is 0.36) as shown in Fig. 4b, which is higher than that of 1D top-flooding bed (Ma et al., 2007). A similar DHF enhancement was predicted by Bürger et al. (2006) for an in-vessel debris bed in the reactor pressure vessel lower head. 4.3. Effect of micro-inhomogeneity in a porous debris bed The multi-dimensionality may result from a macro inhomogeneity such as a radially stratified debris bed (Nayak et al., 2006). Since the macro inhomogeneity has received some attention, here the study on bed’s inhomogeneity will only focus on the effect of the micro-spatial structure of the bed on its coolability. To perform the study on the effect of micro-inhomogeneity on coolability, a unit volume was chosen as a mini bed as shown in Fig. 5. The micro-inhomogeneity was represented by a high-porosity zone incorporated in the middle of the unit volume. The WABE-2D code is used to analyze the thermal-hydraulics of such beds. The results showed that the inner zone (high-porosity) served as a ‘down-
602
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
Fig. 3. Debris beds from DEFOR tests (Karbojian et al., 2009).
comer’ which transports water to the bottom of the main bed, and thus enhanced coolability which can be observed in Table 3, for the coolant condition at atmospheric pressure and saturated temperature. More detailed discussion of the results can be found in Ma et al. (2007). The implication of the predicted results on debris bed coolability is straightforward, for the prototypical debris beds are likely heterogeneous and composed of many such mini beds. Such inhomogeneity surpasses the limitation of counter-current flow in homogeneous beds. However, the effect of the microinhomogeneity can not be reflected in the average approach where the beds are represented by a single mean porosity and particle size, since the high-porosity zone only occupies a very small portion in the bed (here the volume ratio of the inner zone to the whole bed
is less than 4%), and its contribution to the average porosity and particle size diminishes under averaging. This analysis also reveals a fundamental limit of the treatment of a bed thermal hydraulics using mean porosity and particle size. 4.4. Effect of particle morphology In previous studies on debris bed coolability (Hofmann, 1984; Schmidt, 2004; Konovalikhin, 2001; Bang et al., 2005; Hu and Theofanous, 1991; Lipinski, 1984; Reed et al., 1985; Schulenberg ˝ and Muller, 1987; Tung and Dhir, 1988), the particulate debris was assumed to comprise of solid particles with regular shapes and macroscopically smooth surfaces. There are only few studies which pay attention on the morphology of prototypical debris,
Fig. 4. Heap-like debris bed in the analysis and post-dryout profiles of temperature and velocity.
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
603
Table 3 Dryout heat flux of debris beds with micro-inhomogeneity. Case
1 2 3 4 a
Bed
Downcomer
dp (mm)
ε
Db (mm)
Hb (mm)
dp (mm)
ε
Ddc (mm)
1.0 1.0 1.0 1.0
0.36 0.36 0.36 0.36
50 50 50 50
450 450 450 450
3 1 2 1
0.6 0.6 0.72 0.6
10 6 6 10
DHF (kW/m2 )
DHF/DHFtop a
338 242 276 276
148% 106% 121% 121%
DHFtop is 228 kW/m2 for the top-flooding bed. Table 4 Comparison of DHF of beds with hollow spheres and solid spheres.
Bed 1 Bed 2 Bed 3
Fig. 5. Debris bed with micro-inhomogeneity.
and the effect of particle shape and size distributions. Recently in the DEFOR experiments conducted at KTH (Karbojian et al., 2009; Kudinov et al., 2007), it was found that the particle surface is generally roughened (cf. Fig. 6a). More importantly, debris fragments are also found to have encapsulated cavities or pores on the surfaces (cf. Fig. 6b). Such cavities (or internal pores), irregular shape and roughness were indeed observed in the debris fragments collected in the tests on prototypic corium melt–water interactions (Song et al., 2006). It should be noted that the internal pores were not separately counted for, when the porosity of the debris bed was determined in post-test measurements. In other words, most debris beds were assumed as Fig. 7a in the existing coolability studies, where pores
dp (mm)
ε
εi
εe
DHF (MW/m2 )
3.0 3.0 3.0
0.4 0.7 0.7
0.0 0.1 0.0
0.0 0.6 0.0
0.9 2.26 3.37
between the particles serve as channels of coolant flow, and the “external” porosity is used as a sole parameter in coolability analysis. However, for a prototypic debris bed, internal pores may contribute to the total porosity, reducing the effective “external” porosity and hence influencing the bed coolability. If the pores are encapsulated as illustrated in Fig. 7b, the coolant could not get access to them at all; if the pores are half-enclosed as in Fig. 7c, a degraded coolability due to such pore space can be expected. To evaluate the effect of such internal pores on coolability, in the present study we examined the coolability of a bed with the configuration of Fig. 6b, i.e. we assume that the debris bed is composed of spheres with some of them being hollow. Under this situation, the effective porosity εe equals to the nominal porosity ε minus the internal porosity εi (cf. Fig. 8). Furthermore, we assume that the internal porosity is uniformly distributed in all particles, and the beds are flooded by coolant at atmospheric pressure and saturated temperature. In Table 4 and Fig. 8, Bed 1 is the reference bed with 40% porosity and packed by 3 mm solid particles. Bed 2 is supposed a prototypical debris bed with high porosity (70%), but 10% being the contribution of the encapsulated pores, while Bed 3 is without the encapsulated pores. Comparing Bed 2 with Bed 3, we can see that the dryout heat flux was reduced by 33% (from 3.37 MW/m2 to 2.26 MW/m2 ) due to the internal porosity in Bed 2. This result implies the importance of determining the particle morphology and pore distribution in a prototypical bed. We refer to discussion in Section 3.1 on the cause of the remarkable impact of the porosity variation on coolability. A prototypical debris bed formed in a severe accident is expected to be inclusive of particle morphology such as those shown in Fig. 7b and c and even more complex. Coolability of such beds
Fig. 6. Debris from DEFOR experiments (Karbojian et al., 2009).
604
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
Fig. 7. Different configurations of debris beds.
Fig. 8. Dryout heat flux for debris beds with hollow and solid debris.
cannot be reliably predicted using the existing models and computational tools, which were developed and validated for the bed configuration of Fig. 7a. The fundamental question here pertains to the two-phase friction laws in a bed with enhanced surface-tovolume ratio for debris fragments (complex particle morphology). The situation suggests an important gap in contemporary knowledge of severe accident phenomena for which new experimental and analytical investigations are needed to reduce uncertainty in coolability assessment. Generally speaking, the particle size distribution and morphology affects the effective hydraulic diameter. On the one hand, the particle’s complex shapes result in an enhanced surface-tovolume ratio that leads to a decreased local heat flux through the particle surface, and therefore higher resilient to dryout. On the other hand, the particle surface roughness and enhanced surface area lead to increased flow resistance, with subsequent reduction of DHF.
4.5. Effect of heat generation method The decay heat in a prototypic corium debris bed provides volumetric heating, which has been postulated to be uniform in all the previous coolability assessment. To simulate the volumetric heating, coolability experiments in the past employed inductive heating (Schmidt, 2004) of metal spheres. This method of inductive heating is however not applicable to nonmetal materials (e.g. oxides). The inductive heating is also not suitable for a large-size bed, because of the induction’s skin effect and low efficiency. To provide a proper heat source, two other options were proposed and implemented in the coolability experiments: inductively heated metallic spheres are mixed with nonmetal debris (cf. Fig. 9a); and electric heaters [resistance wires, as shown in Fig. 9b] are embedded in the experimental debris beds. Naturally, such simulation of decay heating introduces additional non-prototypicality which has to be taken into account when one intends to apply the data obtained in such
Fig. 9. Local heating methods in debris beds.
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
605
Fig. 10. Configuration of beds in Table 5.
Table 5 Calculation matrix for non-uniform heating. Case
Non-uniformity
DHF (kW/m2 )
Beda
A B
See Fig. 10a, layer thickness = 10 mm See Fig. 10b, column thickness = 8 mm
∼230 ∼200
Height = 450 mm; diameter = 350 mm Porosity = 0.36; particle size = 1 mm
a
DHF of uniform heating of the bed is 228 kW/m2 .
experiments to reactor safety analysis. In the present study, we examine the influence of the local heating on dryout heat flux to quantify the effect of heating methods on the experimental findings. To enable implementation of the current analysis using the WABE-2D code, we assumed that the non-uniform heating was represented by stratification of heat generation (decay heat); the beds were composed of interlaced zones with and without heat generation, as shown in Fig. 10. For Case A in Table 5, the debris bed is axially stratified (cf. Fig. 10a), with interlaced layers with the thickness of heating layer being half of that of un-heating layer. With increasing heat flux, the dryout occurs first at the bottom of the bed, as illustrated in Fig. 11. The dryout heat flux is approximately 230 kW/m2 at atmospheric pressure, which is almost the same as that of the uniformly heating bed. For some variations of Case A (i.e. making the interval between the heating layers by half or double), a similar dryout heat flux was obtained. This indicates that the non-uniform heating along the
axial direction has little effect on coolability limitation. This can be explained by that the dryout heat flux is determined by countercurrent flow limit (CCFL), and for such a one-dimensional flow only the vapor generated in the bed matters. When the heat source is radially stratified (cf. Case B in Table 5 and Fig. 10), the picture is different. Fig. 12 shows the post-dryout temperature profile and water inflow directions in this case, for which the average dryout heat flux is around 200 kW/m2 (according to the calculation). Obviously, the coolability is deteriorated by the radial stratification of heating. It should be noted that the dryout heat flux considered here is an average one, which is related to the whole surface of a bed, and employed in experimental studies. However, the local dryout heat flux may be higher than the average one, since the water-rich and heat-source-free columns transport coolant to the water-starving columns with heat generation (cf. Fig. 12). The insights from Case B are also applicable to the induction-heating experiments where the power density in the center is lower than the periphery due to skin effect.
Fig. 11. Temperature and liquid velocity of the Case A (235 kW/m2 ).
Fig. 12. Temperature and liquid velocity of the Case B (207 kW/m2 ).
606
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
Fig. 13. Configurations of an ex-vessel debris bed.
Although the above analysis reveals the trend by which the use of local heaters may affect DHF, the absolute value of the effect depends on real heater and bed geometries. In the experiments, the heat-generating electric wires are typically much smaller in size than the representation in the current calculations, suggesting even higher conservatism of experimentally measured DHF. Furthermore, the wire heated surface and debris non-heated surfaces can have different wettability, rendering possibly different twophase flow patterns in microchannels, and therefore different flow friction. 4.6. Reactor application For prototypical-scale debris beds in reactor application, here we investigate the effect of debris bed configurations on coolability. The coolability analysis is performed for an ex-vessel debris bed formed in hypothetical severe accident scenarios of a boiling BWR which has thermal power of 2500 MW and the cavity diameter of 9 m (see Fig. 13). The maximum mass of the ex-vessel corium debris bed is about 190 tonnes, having the solid volume of around 21.5 m3 . Normally, the decay heat is 1–2% thermal power of the reactor power, which is 25–50 MW or 132–264 W/kg in term of the corium mass. The pressure in the containment is 1 bar in the calculation. The configuration (a) in Fig. 13 is representative of a debris bed formed by uniform spreading of particles over the cavity floor. If the mean diameter of the particulate debris is assumed as 3 mm and the bed’s porosity as 0.4, the dryout heat flux of such a topflooding configuration is 335 W/kg by the prediction of the WABE code, which is equivalent to 1 MW/m2 at the top surface, and comparable to that of a small-scale debris bed predicted by WABE or Reed’s model. The configuration (b) in Fig. 13 represents a scenario
where the debris does not occupy the entire floor of the cavity, leading to an annular gap between the bed and the cavity wall. This is more realistic than the uniform spreading assumption, due to the large area of the cavity floor. At the same time, a non-heated bottom layer is formed at the bottom, say, as a result of earlier discharge of metallic melt (e.g. stainless steel). This is very possible for severe accident scenario of a boiling water reactor, since it has a large amount of stainless steel and zircaloy, and stratified melt pool may be formed in the lower plenum. The vessel may fail first at the metallic layer due to the well-known focus effect, resulting in the earlier discharge of metallic melt. In this case, the dryout heat flux is 475 W/kg, 42% higher than that of the configuration (a). Besides, the dryout location for the configurations (b) is quite different from that of the configuration (a), as illustrated in Fig. 14b and a, respectively. 5. Discussion on modeling uncertainties Degree of confidence in computational results and findings presented in the preceding section depends on numerical solution, modeling assumptions and accuracy of correlations used for model closure. Performance of the simulation tool (i.e., WABE code) is affected by numerical uncertainty and physical fidelity of the models. The numerical uncertainty can be qualified by changing the mesh scheme. For a debris bed as Case 1 in Table 3, if the computational grid is refined twice, both the resulting heat flux and the location for dryout remain unchanged; see Fig. 15, suggesting minute sensitivity of the solution to the numerical discretization. For the physical model, the main source of uncertainty is in the closure correlations. Some uncertainties in this respect have been mentioned in Section 4 where concerned. As the debris coolabil-
Fig. 14. Dryout and liquid velocity for the respective configurations in Fig. 13.
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
607
Fig. 15. Effect of mesh refinement on coolability prediction.
ity is not restricted by heat transfer between the particles and coolant, but by hydrodynamic limitations due to the friction of two-phase flow through the debris bed (Schmidt, 2004), modeling uncertainty boils down to the friction laws in momentum conservation equations. In the WABE code, the friction laws are based on the Ergun’s equation (Ergun, 1952) which is derived and used broadly for porous media, and the closure correlations (e.g., relative permeability and passability) are deduced from dryout studies of debris beds, mostly with 1D configuration. There is a need to validate the model against data from experiments performed on multi-dimensional configurations. Such data are lacking for the time being. With the ongoing studies at KTH and elsewhere, we expect to fill in the data.
trends predicted in this first-cut evaluation appear significant for the quantification of coolability risk in severe accidents in LWRs. Notably, the results indicate the availability of substantial coolability margins, compared to that of the previous coolability assessments which considered idealized bed configurations (e.g. one-dimensional, homogenous, free-packing spheres). The present study also suggests the need for mechanisms that govern formation of and two-phase thermal-hydraulics in prototypic debris beds. They are investigated in the continuing DEFOR (Debris FORmation) and POMECO (Porous Media COolability) experimental projects currently underway at the Royal Institute of Technology (KTH), Stockholm. Acknowledgements
6. Concluding remarks In the present study, we identify and examine the potential effect of non-prototypical factors of the experimental configurations and characteristics of debris beds used in the past studies of corium coolability. The main technical findings are as follows. • The high porosity (up to 70%) of debris beds from prototypical debris beds is predicted to dramatically enhance the dryout heat flux (double or triple, depending on particle size), compared with that of the debris bed with a traditionally assumed porosity of 40%. • Bed inhomogeneity represented by micro-channels (i.e. locally higher porosity or particle size zone) in a mini bed is predicted to cause the dryout heat flux enhancement by up to ∼50%, even if the micro-channels occupies only an insignificant fraction of the bed (mere 4% of the bed total volume in the case study). • The multidimensional effect of coolant side ingress for the beds analyzed is predicted to increase the dryout heat flux by more than ∼40%. • The local heating of electric heaters embedded in a debris bed is predicted to reduce the dryout heat flux, in comparison with a uniformly and volumetrically heated bed. The effect depends on the heater’ size and distribution. The simulation suggests that the non-prototypical effect can be reduced by using fine wire-heaters distributed more uniformly in the bed’s radial direction. It is instructive to note that the present findings are made possible by using the analysis tools which in themselves are subject to modeling assumptions and uncertainty. Nonetheless, the
This study benefits from insights developed in DEFOR and POMECO projects and discussion with the DEFOR and POMECO team members at KTH. The support from SSM, EU, NKS and HSK under the APRI and SARNET programs is gratefully acknowledged. The authors thank Drs Michael Buck and Manfred Bürger at IKE of Stuttgart University who encouraged and enabled the application of the WABE code in the present study. References Bang, K.H., Kim, J.M., et al., 2005. Enhancement of dryout heat flux in a debris bed by forced coolability flow from below. In: Proceedings of NURETH-11, Aavignon, France, October 2–6. Bürger, M., Buck, M., Schmidt, W., Widmann, W., 2006. Validation and application of the WABE code: investigations of constitutive laws and 2D effects on debris coolability. Nuclear Engineering and Design 236, 2164–2188. Chung, M., Catton, I., 1991. Post-dryout heat transfer in multidimensional porous bed. Nuclear Engineering and Design 128, 289–304. Dinh, T.N., Bui, V.A., et al., 1999. Experimental and analytical studies of melt jetcoolant interactions: a synthesis. Nuclear Engineering and Design 189, 29–327. Ergun, S., 1952. Fluid flow through packed columns. Chemical Engineering Progress 48 (2), 89–94. Fichot, F., Duval, F., Trégourès, N., Béchaud, C., Quintard, M., 2006. The impact of thermal non-equilibrium and large-scale 2D/3D effects on debris bed reflooding and coolability. Nuclear Engineering and Design 236, 2144–2163. Hofmann, G., 1984. On the location and mechanisms of dryout in top-fed and bottom-fed particulate beds. Nuclear Technology 65. Hu, K., Theofanous, T.G., 1991. On the measurement and mechanism of dryout in volumetrically heated coarse particle beds. International Journal of Multiphase Flow 17 (4), 519–532. Huhtiniemi, I., Magallon, D., Hohmann, H., 1997. Results of recent KROTOS FCI tests: alumina vs. corium melts. In: Proceedings of OECD/CSNI Specialists Meeting on Fuel–Coolant Interactions, Tokai-Mura, Japan, May 19–21. Karbojian, A., Ma, W.M., Kudinov, P., Dinh, T.N., 2009. A scoping study of debris bed formation in the DEFOR test facility. Nuclear Engineering and Design 239 (9), 1653–1659.
608
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
Kato, M., Nagasaka, H., et al., 1999. Fuel coolant interaction tests under ex-vessel conditions. In: Proceedings of the OECD workshop on ex-vessel debris coolabilitty, Karlsruhe, Germany. Konovalikhin, M.J., 2001. Investigations on melt spreading and coolability in a LWR severe accident. Ph.D. Thesis of Royal Institute of Technology, Stockholm, November. Kudinov, P., Karbojian, A., Ma, W.M., Davydov, M., Dinh, T.N., 2007. A study of ex-vessel debris formation in a LWR severe accident. In: 2007 International Congress on Advanced Nuclear Power Plants, Nice, France, May 13–18. Lipinski, R.J., 1984. A coolability model for post accident nuclear reactor debris. Nuclear Technology 65, 53–66. Ma, W.M., Buck, M., Bürger, M., Dinh, T.N., 2007. Analysis of the effect of bed inhomogeneity on debris coolability. In: Proceedings of the 15th International Conference on Nuclear Engineering, ICONE15-10472, Nagoya, Japan, April 22–26. Magallon, D., 2006. Characteristics of corium debris bed generated in large-scale fuel–coolant interaction experiments. Nuclear Engineering and Design 236, 1998–2009.
Nayak, A.K., Sehgal, B.R., Stepanyan, A.V., 2006. An experimental study on quenching of a radially stratified heated porous bed. Nuclear Engineering and Design 236, 2189–2198. Reed, A.W., Bergeron, E.D., et al., 1985. Coolability of UO2 debris bed in pressurized water pools DCC-1 and DCC-2 experiment results. Nuclear Engineering and Design 97, 81–88. Schmidt, W., 2004. Influence of multidimentionality and interfacial friction on the coolability of fragmented corium. Ph.D. Thesis of University of Stuttgart, May. ˝ Schulenberg, T., Muller, U., 1987. An improved model for two-phase flow through beds of coarse particles. International Journal of Multiphase flow 13 (1), 87–97. Song, J.H., Park, I.K., et al., 2002. Experiments on the interactions of molten ZrO2 with water using TROI facility. Nuclear Engineering and Design 213, 97–110. Song, J.H., et al., 2006. The effect of corium composition and interaction vessel geometry on the prototypic steam explosion. Annuals of Nuclear Energy 33, 1437–1451. Spencer, B.W., Wang, K., et al., 1994. Fragmentation and quench behaviour of corium melt streams in water. NUREG/CR-6133, ANL-93/32, February. Tung, V.X., Dhir, V.K., 1988. A hydrodynamic model for two-phase flow through porous media. International Journal of Multiphase Flow 14 (1), 47–65.
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
The effects of debris bed’s prototypical characteristics on corium coolability in a LWR severe accident Weimin Ma ∗ , Truc-Nam Dinh 1 Division of Nuclear Power Safety, Royal Institute of Technology (KTH), Roslagstullsbacken 21, 106 91 Stockholm, Sweden
a r t i c l e
i n f o
Article history: Received 25 June 2009 Received in revised form 16 October 2009 Accepted 30 October 2009
a b s t r a c t This paper is concerned with coolability assessment of a debris bed formed in fuel coolant interactions (FCIs) during a hypothetical severe accident in a light water reactor (LWR). The focus is placed the potential effect of the bed’s prototypical characteristics on its coolability, in terms of (i) porosity range, (ii) multi-dimensionality, (iii) inhomogeneity, (iv) particle morphology, and (v) heat generation method (e.g. volumetric heating vs. local heaters). The analysis results indicate availability of substantial coolability margins compared to previous assessments based on models and experiments using an idealized bed configuration (e.g. 1D homogenous debris layer). Notably, high porosity (up to 70%) of debris beds, obtained in experiments and expected to be the case of prototypical debris beds, could increase the dryout heat flux by 100% and more, depending on particle size, compared with the dryout heat flux predicted for debris beds with traditionally assumed porosity of approximately 40%. Bed inhomogeneity represented by micro-channels in a mini bed is predicted to enhance the dryout heat flux by up to ∼50%, even if the micro-channels occupy only a small volume fraction (e.g., less than 4%) of the bed. The effect of coolant side ingress into a multidimensional bed is predicted to enhance the dryout heat flux by up to 40% for the beds analyzed. © 2009 Elsevier B.V. All rights reserved.
1. Introduction In a hypothetical severe core-melt accident of a nuclear reactor, a debris bed may be formed in the vessel lower head (in-vessel) or in the reactor cavity (ex-vessel), provided that the molten corium penetrates into a water pool, fragmenting and forming debris particles which will then settle down on the pool bottom. Compared with a molten corium pool, the particulate corium (debris) bed is easier to be cooled, i.e., having more chance to remove the decay heat, for the inside of the bed is accessible for coolant through the internal pores of the bed. The debris bed coolability is therefore of paramount importance to termination and stabilization of the severe accident. As a result, the study on debris bed coolability has been a hot topic in assessment of melt risk of light water reactors (LWR). The quantification of ex-vessel debris bed coolability is of particular interest to the Swedish BWR plants which employ lower drywell flooding as a severe accident management strategy, with a hope that a coolable debris bed will be formed in a deep water pool of the wet cavity. To pursue a quantitative understanding of the coolability of both the in-vessel and ex-vessel debris beds, numerous experi-
∗ Corresponding author. Tel.: +46 8 5537 8821; fax: +46 8 5537 8830. E-mail address: [email protected] (W.M. Ma). 1 Present address: Idaho National Laboratory, Idaho Falls, ID-83415, USA. 0029-5493/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nucengdes.2009.10.026
ments and analyses have been conducted to study two-phase flow and heat transfer in such particulate debris beds. The debris beds employed in the past studies were characterized with an averaged particle size or with a size distribution of the debris obtained from a few fuel–coolant interaction (FCI) experiments (Spencer et al., 1994; Magallon, 2006; Huhtiniemi et al., 1997; Song et al., 2002; Kato et al., 1999; Dinh et al., 1999) using sieving technique. Moreover, a bed-averaged porosity (normally 40%) was assumed based on the free particles packing theory under cold conditions. Subsequently, the so-measured mean particle size and bed porosity serve as a basis to design the debris bed coolability experiments. Notably, neither the prototypic bed’s porosity nor bed’s interior structure (pore size distribution) are reproduced in such experimental beds (Hofmann, 1984; Schmidt, 2004; Konovalikhin, 2001; Bang et al., 2005; Hu and Theofanous, 1991). Furthermore, the coolability tests were conducted primarily for determining dryout heat fluxes in one-dimensional bed configuration with top flooding. In other words, the study of debris bed coolability was focused on top-flooding scenarios, which manifest situations when side and bottom coolant injection are assumed to be negligible or absent, for example, when corium debris is spread evenly over the cavity floor and coolant is added to top. For the top-flooding bed packed by spherical particles, the dryout heat flux (DHF) can be determined by counter-current flow limit (CCFL) and predicted, with a fair accuracy, by analytical models (Hu and Theofanous, 1991; ˝ Lipinski, 1984; Reed et al., 1985; Schulenberg and Muller, 1987;
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
Nomenclature C d Dh F g j K Kr L p s z
Cozeny–Karman constant particle diameter (m) hydraulic diameter (m) volumetric drag force (N/m3 ) gravitational acceleration (m/s2 ) superficial velocity (m/s) permeability relative permeability height of bed (m) pressure (Pa) liquid saturation vertical coordinate (m)
Greek symbols ˛ void fraction vapor production rate (kg/m3 /s) ε porosity passability r relative passability dynamic viscosity (kg/m/s) density (kg/m3 ) Subscripts g gas, steam i interface l liquid p particle
599
scheme circumvents the CCFL limit and consequently enhances the coolability. While such installations were proposed for some new reactor designs, their application in the existing nuclear plants is not straightforward, due to a cumbersome retrofit and impact on plant operation. Thus, instead of engineering solutions to enhance debris bed coolability, the present study explores inherent features of a debris bed formed from FCIs. The motivation is to quantify those characteristics which were ignored in the past studies but are judged to have potential impact on the bed’s coolability. For instance, in a prototypical accident scenario, the debris beds are expected to have a heap-like shape, with the peripheral region being more coolable; a porous decay-heat-free bottom layer may be formed as a result of earlier discharge of metallic melt (e.g. Zr an stainless steel); channels (micro/macro inhomogeneity) may be formed in the bed due to intensive boiling and flow; etc. On the other hand, in many experiments performed for investigation of debris bed coolability, ohmic heaters were used for heat source, which is distinct from the volumetrically uniform heating of the corium decay heat. While characterization of prototypical debris beds formed from FCIs is presently undertaken in the DEFOR program and discussed elsewhere (Karbojian et al., 2009; Kudinov et al., 2007), the present study is to provide a first-cut quantitative analysis of the impact of such prototypicalities on the debris bed coolability. 2. Identification of non-prototypicalities in debris bed models
Tung and Dhir, 1988). Given accident scenarios with formation of deep beds or dense beds with either small particles or low porosity, the models predict that the top-flooding is insufficient to remove decay heat released in such debris beds. This perception has motivated further search for additional means to enhance debris bed coolability, thereby benefiting reactor safety performance. One of the engineering solutions is to bring coolant down to the bottom of the bed through a device such as a downcomer (Konovalikhin, 2001) or a distributor embedded on the cavity floor. The bottom-fed
The debris beds formed in a severe accident are the outcome of molten corium breakup and subsequent solidification and settlement of the debris in a water pool (cf. Fig. 1, a test debris bed formation test (Karbojian et al., 2009)). Due to the particular process, one may expect characteristics of the so-formed debris beds to be distinct from the packed beds that were formed under cold conditions and employed as default in the past studies on debris coolability. It is the distinction of debris bed formation that determines the prototypicalities (characteristics of the bed) which is contrary to the non-prototypicalities (characteristics of an experimental bed). Although it is still difficult to identify all non-prototypicalities because of the complexity of debris bed formation process as
Fig. 1. Three-phase flow and heat transfer during FCI and debris bed formation.
600
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
Table 1 Identification of the characteristics of a debris bed formed from FCIs. Configuration
Inhomogeneity
Porosity
Particle morphology
Heat generation
Bed used in coolability experiments
1D
Homogeneous
From particle packing theory (ε = ∼0.4)
Electric heaters (local heating)
Bed formed in FCI experiments
2D/3D
Heterogeneous at pore scale
Governed by intense boiling and particle morphology (ε = ∼0.6–0.7)
Solid particles, regular shapes and smooth surface Some hollow particles, irregular shapes, roughened surface
g g dpg j + jg |jg | = g g + KKr,g l r,g dz
Decay heat (volumetric and uniform heating)
revealed in the relevant experiments and analyses (Spencer et al., 1994; Magallon, 2006; Karbojian et al., 2009; Kudinov et al., 2007), several characteristics of debris beds have been identified and considered as importance to corium coolability. The present study places focus on five aspects of the prototypicality that are believed to have potential influence on debris bed coolability, namely (i) porosity range, (ii) multi-dimensionality, (iii) inhomogeneity, (iv) particle morphology, and (v) heat generation method (e.g. volumetric heating vs. local heaters). Table 1 compares the prototypical beds with the one-dimensional homogeneous top-flooding bed used in previous studies. To enable analysis of separate effect, further modeling assumptions are made as discussed in the following sections. It should be noted that other prototypicalities such as the debris surface properties (related to chemical composition and fragmentation) and the ratio of particle surface to volume also affect the friction laws of two-phase flow in debris beds, and will be addressed in follow-on work.
−
3. Technical approach
There are several models for Kr and r , including the well-known Lipinski’s model (Lipinski, 1984) used to estimate the dryout heat flux. Reed’s model (m = 3, n = 5) is nearly identical to the Lipinski model (m = 3, n = 3), except for the use of a higher relative passability. The Reed’s correlation is used here as it has undergone most extensive validation against the existing experimental database on debris coolability.
Coolability of debris bed submerged in a water pool is assessed by means of the maximum heat flux (per unit area of the bed floor surface) that is generated in the bed and can be removed without dryout incipience, or so-called dryout heat flux. To predict the bed’s dryout heat flux, two approaches are employed in this study—the Reed’s analytical model (Reed et al., 1985) developed and validated for one-dimensional top-flooding homogeneous beds and the WABE-2D code (Schmidt, 2004) developed at IKE-Stuttgart University and validated for calculation of two-phase thermal-hydraulics in two-dimensional debris beds. While limitations (e.g., in friction laws) of these two stateof-the-art methods for debris coolability analysis are recognized and under improvement, the objective of the present study is to provide best estimate of the effect of the above-mentioned non-prototypicality so as to aid identification and first-cut quantification of major uncertainty in ex-vessel melt risk safety assessment and guide future experimentation. 3.1. Reed’s model At the basis of Reed’s and similar models of dryout heat flux in volumetrically heated debris beds is the notion that the coolability of a debris bed is restricted by hydrodynamic (counter-current flow) limitations due to the friction of two-phase flow through the debris bed. When the amount of steam generated in the bed is sufficient to prevent the coolant from reaching the bed’s bottom, dryout occurs there and the bed is considered as uncoolable. Thus, for the dryout models, the key in modeling was to provide the formulation of the friction laws for momentum equations. The well-known Lipinski’s model for the prediction of dryout heat flux in a debris bed is based on the Ergun’s equation (Ergun, 1952) for single phase flow in porous media, but adding relative permeability Kr and relative passability r to account for the effect of two-phase existence: −
dpl l l = l g + j + j |j | KKr,l l r,l l l dz
(1a)
(1b)
where jl and jg are the superficial velocities of fluids, and K and are the permeability and passability, respectively. For uniform spherical particles bed, they are usually expressed as K=
ε3 d 2 C(1 − ε)
2
;
=
ε3 d 1.75(1 − ε)
where C is called Cozeny–Karman constant varying from 150 to 180 and d and ε are the particle diameter and porosity of the debris bed, respectively. The relative permeability Kr and relative passability r are obtained via experiments, and considered as the functions of void fraction. Kr,l = (1 − ˛)m , r,l = (1 − ˛)n ,
Kr,g = ˛m r,g = ˛n
3.2. The WABE-2D code WABE-2D code (Schmidt, 2004) was developed at IKE-Stuttgart University, Germany for simulating transient behavior of debris bed formed in severe nuclear reactor accidents of LWRs. The debris bed is modeled in two dimensions with cylindrical or Cartesian geometry using a quasi-continuum approach. Three separate phases for solid particles, water and vapor are considered. The solid matrix is assumed to be fixed. The mass conservation is governed by the equations: ∂ (ε˛g ) + ∇ gjg = ∂t
(2)
∂ (ε˛l ) + ∇ ljl = − ∂t
(3)
for vapor and liquid, respectively, where ˛ is the void fraction, jg and jl are the superficial velocity vectors of vapor and liquid, and is the net mass conversion rate (vaporization or condensation) between liquid and vapor. In the energy conservation equations of vapor and liquid, the mechanical work due to friction and pressure forces is generally neglected. Radiation heat transfer is implicitly considered in the solid energy conservation equation via an effective thermal conductivity. The momentum conservation equations are simplified by an assumption that the temporal and spatial derivatives of the velocities are neglected, because the dominant force is the friction between the particles and the fluids, and velocity field can be simultaneously adjusted to the pressure field. Especially for quasi
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
601
Table 2 Dryout heat flux (DHF) of debris beds with high porosity. Bed
Porosity
Effective particle size (mm)
DHF (MW/m2 )
A B C
40% 70% 70%
3 3 1.5
0.9 3.37 2.28
in Table 2. The results depict a significant, and yet quite expected effect of the bed’s high porosity on its coolability. Another implication of the results is that the effect of the porosity on coolability should be more significant than the particle size. This can be explained by the flow resistance through the debris bed. From Ergun’s equation (Ergun, 1952), the frictional resistance of single phase flow through a debris bed can be predicted by
Fig. 2. Variation of dryout heat flux with porosity (Reeds’ model (Reed et al., 1985)).
steady-state processes, such an assumption is acceptable. As a result, the momentum conservation yields −∇ pg = g g + −∇ pl = l g +
Fpg F + i ε˛ ε˛ Fpl
ε(1 − ˛)
−
(4) Fi ε(1 − ˛)
(5)
for vapor and liquid, respectively. Here, Fpg is drag forces between particles and vapor, Fpl is drag forces between particles and liquid, and Fi is the interfacial drag between liquid and vapor. The drag forces are derived from Eq. (1) and experimental data (Schmidt, 2004). Capillary force is considered in the momentum equations through the relation pc = pg − pl , which is a function of surface tension , contact angle , porosity ε and saturation s [approximate to (1 − ˛)]. Details of constitutive laws and correlations to close the conservation equations, including flow heat and heat transfer, can be found in Schmidt (2004), Bürger et al. (2006) and Ma et al. (2007) which also provided validation of the WABE-2D code against relevant experimental data. 4. Case study and discussion 4.1. Effect of prototypical range of debris bed porosity From particle packing theory, the porosity of a bed packed by regular particles is known to vary in the range from 26% to ∼46%, and the average porosity with size distribution is around 40% (Schmidt, 2004). This value has become the reference porosity for debris bed configurations used in debris coolability studies. However, in both recent DEFOR scoping experiments (Karbojian et al., 2009) and earlier CCM tests (Spencer et al., 1994), it was found that porosity of the FCI-generated debris beds varied from 50% to 70%, with most cases resulted in porosity of 60% and higher. This implies that the porosity of a prototypical debris bed may be substantially higher than the value predicted by the particle packing theory. Fig. 2 shows the variation of dryout heat flux with porosity of the debris beds for top-flooding homogeneous debris beds. The coolant is saturated at atmospheric pressure. If the particle size is 3 mm and the porosity is 70% in the bed, the dryout heat flux (DHF) is 3.37 MW/m2 predicted by Reed’s model, which is 3.7 times of the DHF of the reference bed with 3 mm particles and 40% porosity. If the particle size was reduced to half of that of the reference bed while keeping 70% porosity, the DHF is 2.28 MW/m2 which is 2.53 times of that of the reference bed. The comparison is listed
Pf = 2f
L Dh
j 2 ε
(6)
with Dh (2ε/(3(1 − ε)))d, f = (1/3)(100/Re + 1.75) and Re = (Dh j/ε). For low flowrate (laminar flow), f = (100/3Re). Thus, the pressure drop is almost proportional to the reciprocal of the fourth power to the porosity, while it is nearly proportional to the reciprocal of the second power to the particle size. For two-phase flow, the principle of the friction law is the same (cf. Eq. (1)). It should be noted that the Ergun’s equation is obtained from flows through beds packed with regular particles. Moreover, the Reed’s model (derived from Ergun’s equation) was developed based on experimental data of dryout heat flux for porous beds with regular particles and 30–40% porosity. The applicability and accuracy of the model in a high-porosity range remain uncertain and should be the subject of future investigation. 4.2. Effect of the debris bed’s multi-dimensionality With few exceptions, debris beds used for the DHF study were characteristically homogenous and one dimensional, the debris bed formed in a reactor accident must be multi-dimensional in nature. Such multi-dimensionality (2D/3D) of debris bed has been recognized and studied by a few investigators (Bürger et al., 2006; Fichot et al., 2006; Chung and Catton, 1991). One of the most important debris bed configurations of multi-dimensionality is the heap-like debris beds (cf. Fig. 3) which does not uniformly spread on the pool floor as previously assumed. Thermal-hydraulics of such debris beds submerged in a water pool is characterized by coolant ingression from the sides and bottom of the bed as well as from the top. In these cases, a downcomer-like channel is formed surrounding the bed; cf. Fig. 4a. At atmospheric pressure, the dryout heat flux is predicted to be 325 W/m2 for the configuration (particle diameter is 1 mm and porosity is 0.36) as shown in Fig. 4b, which is higher than that of 1D top-flooding bed (Ma et al., 2007). A similar DHF enhancement was predicted by Bürger et al. (2006) for an in-vessel debris bed in the reactor pressure vessel lower head. 4.3. Effect of micro-inhomogeneity in a porous debris bed The multi-dimensionality may result from a macro inhomogeneity such as a radially stratified debris bed (Nayak et al., 2006). Since the macro inhomogeneity has received some attention, here the study on bed’s inhomogeneity will only focus on the effect of the micro-spatial structure of the bed on its coolability. To perform the study on the effect of micro-inhomogeneity on coolability, a unit volume was chosen as a mini bed as shown in Fig. 5. The micro-inhomogeneity was represented by a high-porosity zone incorporated in the middle of the unit volume. The WABE-2D code is used to analyze the thermal-hydraulics of such beds. The results showed that the inner zone (high-porosity) served as a ‘down-
602
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
Fig. 3. Debris beds from DEFOR tests (Karbojian et al., 2009).
comer’ which transports water to the bottom of the main bed, and thus enhanced coolability which can be observed in Table 3, for the coolant condition at atmospheric pressure and saturated temperature. More detailed discussion of the results can be found in Ma et al. (2007). The implication of the predicted results on debris bed coolability is straightforward, for the prototypical debris beds are likely heterogeneous and composed of many such mini beds. Such inhomogeneity surpasses the limitation of counter-current flow in homogeneous beds. However, the effect of the microinhomogeneity can not be reflected in the average approach where the beds are represented by a single mean porosity and particle size, since the high-porosity zone only occupies a very small portion in the bed (here the volume ratio of the inner zone to the whole bed
is less than 4%), and its contribution to the average porosity and particle size diminishes under averaging. This analysis also reveals a fundamental limit of the treatment of a bed thermal hydraulics using mean porosity and particle size. 4.4. Effect of particle morphology In previous studies on debris bed coolability (Hofmann, 1984; Schmidt, 2004; Konovalikhin, 2001; Bang et al., 2005; Hu and Theofanous, 1991; Lipinski, 1984; Reed et al., 1985; Schulenberg ˝ and Muller, 1987; Tung and Dhir, 1988), the particulate debris was assumed to comprise of solid particles with regular shapes and macroscopically smooth surfaces. There are only few studies which pay attention on the morphology of prototypical debris,
Fig. 4. Heap-like debris bed in the analysis and post-dryout profiles of temperature and velocity.
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
603
Table 3 Dryout heat flux of debris beds with micro-inhomogeneity. Case
1 2 3 4 a
Bed
Downcomer
dp (mm)
ε
Db (mm)
Hb (mm)
dp (mm)
ε
Ddc (mm)
1.0 1.0 1.0 1.0
0.36 0.36 0.36 0.36
50 50 50 50
450 450 450 450
3 1 2 1
0.6 0.6 0.72 0.6
10 6 6 10
DHF (kW/m2 )
DHF/DHFtop a
338 242 276 276
148% 106% 121% 121%
DHFtop is 228 kW/m2 for the top-flooding bed. Table 4 Comparison of DHF of beds with hollow spheres and solid spheres.
Bed 1 Bed 2 Bed 3
Fig. 5. Debris bed with micro-inhomogeneity.
and the effect of particle shape and size distributions. Recently in the DEFOR experiments conducted at KTH (Karbojian et al., 2009; Kudinov et al., 2007), it was found that the particle surface is generally roughened (cf. Fig. 6a). More importantly, debris fragments are also found to have encapsulated cavities or pores on the surfaces (cf. Fig. 6b). Such cavities (or internal pores), irregular shape and roughness were indeed observed in the debris fragments collected in the tests on prototypic corium melt–water interactions (Song et al., 2006). It should be noted that the internal pores were not separately counted for, when the porosity of the debris bed was determined in post-test measurements. In other words, most debris beds were assumed as Fig. 7a in the existing coolability studies, where pores
dp (mm)
ε
εi
εe
DHF (MW/m2 )
3.0 3.0 3.0
0.4 0.7 0.7
0.0 0.1 0.0
0.0 0.6 0.0
0.9 2.26 3.37
between the particles serve as channels of coolant flow, and the “external” porosity is used as a sole parameter in coolability analysis. However, for a prototypic debris bed, internal pores may contribute to the total porosity, reducing the effective “external” porosity and hence influencing the bed coolability. If the pores are encapsulated as illustrated in Fig. 7b, the coolant could not get access to them at all; if the pores are half-enclosed as in Fig. 7c, a degraded coolability due to such pore space can be expected. To evaluate the effect of such internal pores on coolability, in the present study we examined the coolability of a bed with the configuration of Fig. 6b, i.e. we assume that the debris bed is composed of spheres with some of them being hollow. Under this situation, the effective porosity εe equals to the nominal porosity ε minus the internal porosity εi (cf. Fig. 8). Furthermore, we assume that the internal porosity is uniformly distributed in all particles, and the beds are flooded by coolant at atmospheric pressure and saturated temperature. In Table 4 and Fig. 8, Bed 1 is the reference bed with 40% porosity and packed by 3 mm solid particles. Bed 2 is supposed a prototypical debris bed with high porosity (70%), but 10% being the contribution of the encapsulated pores, while Bed 3 is without the encapsulated pores. Comparing Bed 2 with Bed 3, we can see that the dryout heat flux was reduced by 33% (from 3.37 MW/m2 to 2.26 MW/m2 ) due to the internal porosity in Bed 2. This result implies the importance of determining the particle morphology and pore distribution in a prototypical bed. We refer to discussion in Section 3.1 on the cause of the remarkable impact of the porosity variation on coolability. A prototypical debris bed formed in a severe accident is expected to be inclusive of particle morphology such as those shown in Fig. 7b and c and even more complex. Coolability of such beds
Fig. 6. Debris from DEFOR experiments (Karbojian et al., 2009).
604
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
Fig. 7. Different configurations of debris beds.
Fig. 8. Dryout heat flux for debris beds with hollow and solid debris.
cannot be reliably predicted using the existing models and computational tools, which were developed and validated for the bed configuration of Fig. 7a. The fundamental question here pertains to the two-phase friction laws in a bed with enhanced surface-tovolume ratio for debris fragments (complex particle morphology). The situation suggests an important gap in contemporary knowledge of severe accident phenomena for which new experimental and analytical investigations are needed to reduce uncertainty in coolability assessment. Generally speaking, the particle size distribution and morphology affects the effective hydraulic diameter. On the one hand, the particle’s complex shapes result in an enhanced surface-tovolume ratio that leads to a decreased local heat flux through the particle surface, and therefore higher resilient to dryout. On the other hand, the particle surface roughness and enhanced surface area lead to increased flow resistance, with subsequent reduction of DHF.
4.5. Effect of heat generation method The decay heat in a prototypic corium debris bed provides volumetric heating, which has been postulated to be uniform in all the previous coolability assessment. To simulate the volumetric heating, coolability experiments in the past employed inductive heating (Schmidt, 2004) of metal spheres. This method of inductive heating is however not applicable to nonmetal materials (e.g. oxides). The inductive heating is also not suitable for a large-size bed, because of the induction’s skin effect and low efficiency. To provide a proper heat source, two other options were proposed and implemented in the coolability experiments: inductively heated metallic spheres are mixed with nonmetal debris (cf. Fig. 9a); and electric heaters [resistance wires, as shown in Fig. 9b] are embedded in the experimental debris beds. Naturally, such simulation of decay heating introduces additional non-prototypicality which has to be taken into account when one intends to apply the data obtained in such
Fig. 9. Local heating methods in debris beds.
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
605
Fig. 10. Configuration of beds in Table 5.
Table 5 Calculation matrix for non-uniform heating. Case
Non-uniformity
DHF (kW/m2 )
Beda
A B
See Fig. 10a, layer thickness = 10 mm See Fig. 10b, column thickness = 8 mm
∼230 ∼200
Height = 450 mm; diameter = 350 mm Porosity = 0.36; particle size = 1 mm
a
DHF of uniform heating of the bed is 228 kW/m2 .
experiments to reactor safety analysis. In the present study, we examine the influence of the local heating on dryout heat flux to quantify the effect of heating methods on the experimental findings. To enable implementation of the current analysis using the WABE-2D code, we assumed that the non-uniform heating was represented by stratification of heat generation (decay heat); the beds were composed of interlaced zones with and without heat generation, as shown in Fig. 10. For Case A in Table 5, the debris bed is axially stratified (cf. Fig. 10a), with interlaced layers with the thickness of heating layer being half of that of un-heating layer. With increasing heat flux, the dryout occurs first at the bottom of the bed, as illustrated in Fig. 11. The dryout heat flux is approximately 230 kW/m2 at atmospheric pressure, which is almost the same as that of the uniformly heating bed. For some variations of Case A (i.e. making the interval between the heating layers by half or double), a similar dryout heat flux was obtained. This indicates that the non-uniform heating along the
axial direction has little effect on coolability limitation. This can be explained by that the dryout heat flux is determined by countercurrent flow limit (CCFL), and for such a one-dimensional flow only the vapor generated in the bed matters. When the heat source is radially stratified (cf. Case B in Table 5 and Fig. 10), the picture is different. Fig. 12 shows the post-dryout temperature profile and water inflow directions in this case, for which the average dryout heat flux is around 200 kW/m2 (according to the calculation). Obviously, the coolability is deteriorated by the radial stratification of heating. It should be noted that the dryout heat flux considered here is an average one, which is related to the whole surface of a bed, and employed in experimental studies. However, the local dryout heat flux may be higher than the average one, since the water-rich and heat-source-free columns transport coolant to the water-starving columns with heat generation (cf. Fig. 12). The insights from Case B are also applicable to the induction-heating experiments where the power density in the center is lower than the periphery due to skin effect.
Fig. 11. Temperature and liquid velocity of the Case A (235 kW/m2 ).
Fig. 12. Temperature and liquid velocity of the Case B (207 kW/m2 ).
606
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
Fig. 13. Configurations of an ex-vessel debris bed.
Although the above analysis reveals the trend by which the use of local heaters may affect DHF, the absolute value of the effect depends on real heater and bed geometries. In the experiments, the heat-generating electric wires are typically much smaller in size than the representation in the current calculations, suggesting even higher conservatism of experimentally measured DHF. Furthermore, the wire heated surface and debris non-heated surfaces can have different wettability, rendering possibly different twophase flow patterns in microchannels, and therefore different flow friction. 4.6. Reactor application For prototypical-scale debris beds in reactor application, here we investigate the effect of debris bed configurations on coolability. The coolability analysis is performed for an ex-vessel debris bed formed in hypothetical severe accident scenarios of a boiling BWR which has thermal power of 2500 MW and the cavity diameter of 9 m (see Fig. 13). The maximum mass of the ex-vessel corium debris bed is about 190 tonnes, having the solid volume of around 21.5 m3 . Normally, the decay heat is 1–2% thermal power of the reactor power, which is 25–50 MW or 132–264 W/kg in term of the corium mass. The pressure in the containment is 1 bar in the calculation. The configuration (a) in Fig. 13 is representative of a debris bed formed by uniform spreading of particles over the cavity floor. If the mean diameter of the particulate debris is assumed as 3 mm and the bed’s porosity as 0.4, the dryout heat flux of such a topflooding configuration is 335 W/kg by the prediction of the WABE code, which is equivalent to 1 MW/m2 at the top surface, and comparable to that of a small-scale debris bed predicted by WABE or Reed’s model. The configuration (b) in Fig. 13 represents a scenario
where the debris does not occupy the entire floor of the cavity, leading to an annular gap between the bed and the cavity wall. This is more realistic than the uniform spreading assumption, due to the large area of the cavity floor. At the same time, a non-heated bottom layer is formed at the bottom, say, as a result of earlier discharge of metallic melt (e.g. stainless steel). This is very possible for severe accident scenario of a boiling water reactor, since it has a large amount of stainless steel and zircaloy, and stratified melt pool may be formed in the lower plenum. The vessel may fail first at the metallic layer due to the well-known focus effect, resulting in the earlier discharge of metallic melt. In this case, the dryout heat flux is 475 W/kg, 42% higher than that of the configuration (a). Besides, the dryout location for the configurations (b) is quite different from that of the configuration (a), as illustrated in Fig. 14b and a, respectively. 5. Discussion on modeling uncertainties Degree of confidence in computational results and findings presented in the preceding section depends on numerical solution, modeling assumptions and accuracy of correlations used for model closure. Performance of the simulation tool (i.e., WABE code) is affected by numerical uncertainty and physical fidelity of the models. The numerical uncertainty can be qualified by changing the mesh scheme. For a debris bed as Case 1 in Table 3, if the computational grid is refined twice, both the resulting heat flux and the location for dryout remain unchanged; see Fig. 15, suggesting minute sensitivity of the solution to the numerical discretization. For the physical model, the main source of uncertainty is in the closure correlations. Some uncertainties in this respect have been mentioned in Section 4 where concerned. As the debris coolabil-
Fig. 14. Dryout and liquid velocity for the respective configurations in Fig. 13.
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
607
Fig. 15. Effect of mesh refinement on coolability prediction.
ity is not restricted by heat transfer between the particles and coolant, but by hydrodynamic limitations due to the friction of two-phase flow through the debris bed (Schmidt, 2004), modeling uncertainty boils down to the friction laws in momentum conservation equations. In the WABE code, the friction laws are based on the Ergun’s equation (Ergun, 1952) which is derived and used broadly for porous media, and the closure correlations (e.g., relative permeability and passability) are deduced from dryout studies of debris beds, mostly with 1D configuration. There is a need to validate the model against data from experiments performed on multi-dimensional configurations. Such data are lacking for the time being. With the ongoing studies at KTH and elsewhere, we expect to fill in the data.
trends predicted in this first-cut evaluation appear significant for the quantification of coolability risk in severe accidents in LWRs. Notably, the results indicate the availability of substantial coolability margins, compared to that of the previous coolability assessments which considered idealized bed configurations (e.g. one-dimensional, homogenous, free-packing spheres). The present study also suggests the need for mechanisms that govern formation of and two-phase thermal-hydraulics in prototypic debris beds. They are investigated in the continuing DEFOR (Debris FORmation) and POMECO (Porous Media COolability) experimental projects currently underway at the Royal Institute of Technology (KTH), Stockholm. Acknowledgements
6. Concluding remarks In the present study, we identify and examine the potential effect of non-prototypical factors of the experimental configurations and characteristics of debris beds used in the past studies of corium coolability. The main technical findings are as follows. • The high porosity (up to 70%) of debris beds from prototypical debris beds is predicted to dramatically enhance the dryout heat flux (double or triple, depending on particle size), compared with that of the debris bed with a traditionally assumed porosity of 40%. • Bed inhomogeneity represented by micro-channels (i.e. locally higher porosity or particle size zone) in a mini bed is predicted to cause the dryout heat flux enhancement by up to ∼50%, even if the micro-channels occupies only an insignificant fraction of the bed (mere 4% of the bed total volume in the case study). • The multidimensional effect of coolant side ingress for the beds analyzed is predicted to increase the dryout heat flux by more than ∼40%. • The local heating of electric heaters embedded in a debris bed is predicted to reduce the dryout heat flux, in comparison with a uniformly and volumetrically heated bed. The effect depends on the heater’ size and distribution. The simulation suggests that the non-prototypical effect can be reduced by using fine wire-heaters distributed more uniformly in the bed’s radial direction. It is instructive to note that the present findings are made possible by using the analysis tools which in themselves are subject to modeling assumptions and uncertainty. Nonetheless, the
This study benefits from insights developed in DEFOR and POMECO projects and discussion with the DEFOR and POMECO team members at KTH. The support from SSM, EU, NKS and HSK under the APRI and SARNET programs is gratefully acknowledged. The authors thank Drs Michael Buck and Manfred Bürger at IKE of Stuttgart University who encouraged and enabled the application of the WABE code in the present study. References Bang, K.H., Kim, J.M., et al., 2005. Enhancement of dryout heat flux in a debris bed by forced coolability flow from below. In: Proceedings of NURETH-11, Aavignon, France, October 2–6. Bürger, M., Buck, M., Schmidt, W., Widmann, W., 2006. Validation and application of the WABE code: investigations of constitutive laws and 2D effects on debris coolability. Nuclear Engineering and Design 236, 2164–2188. Chung, M., Catton, I., 1991. Post-dryout heat transfer in multidimensional porous bed. Nuclear Engineering and Design 128, 289–304. Dinh, T.N., Bui, V.A., et al., 1999. Experimental and analytical studies of melt jetcoolant interactions: a synthesis. Nuclear Engineering and Design 189, 29–327. Ergun, S., 1952. Fluid flow through packed columns. Chemical Engineering Progress 48 (2), 89–94. Fichot, F., Duval, F., Trégourès, N., Béchaud, C., Quintard, M., 2006. The impact of thermal non-equilibrium and large-scale 2D/3D effects on debris bed reflooding and coolability. Nuclear Engineering and Design 236, 2144–2163. Hofmann, G., 1984. On the location and mechanisms of dryout in top-fed and bottom-fed particulate beds. Nuclear Technology 65. Hu, K., Theofanous, T.G., 1991. On the measurement and mechanism of dryout in volumetrically heated coarse particle beds. International Journal of Multiphase Flow 17 (4), 519–532. Huhtiniemi, I., Magallon, D., Hohmann, H., 1997. Results of recent KROTOS FCI tests: alumina vs. corium melts. In: Proceedings of OECD/CSNI Specialists Meeting on Fuel–Coolant Interactions, Tokai-Mura, Japan, May 19–21. Karbojian, A., Ma, W.M., Kudinov, P., Dinh, T.N., 2009. A scoping study of debris bed formation in the DEFOR test facility. Nuclear Engineering and Design 239 (9), 1653–1659.
608
W. Ma, T.-N. Dinh / Nuclear Engineering and Design 240 (2010) 598–608
Kato, M., Nagasaka, H., et al., 1999. Fuel coolant interaction tests under ex-vessel conditions. In: Proceedings of the OECD workshop on ex-vessel debris coolabilitty, Karlsruhe, Germany. Konovalikhin, M.J., 2001. Investigations on melt spreading and coolability in a LWR severe accident. Ph.D. Thesis of Royal Institute of Technology, Stockholm, November. Kudinov, P., Karbojian, A., Ma, W.M., Davydov, M., Dinh, T.N., 2007. A study of ex-vessel debris formation in a LWR severe accident. In: 2007 International Congress on Advanced Nuclear Power Plants, Nice, France, May 13–18. Lipinski, R.J., 1984. A coolability model for post accident nuclear reactor debris. Nuclear Technology 65, 53–66. Ma, W.M., Buck, M., Bürger, M., Dinh, T.N., 2007. Analysis of the effect of bed inhomogeneity on debris coolability. In: Proceedings of the 15th International Conference on Nuclear Engineering, ICONE15-10472, Nagoya, Japan, April 22–26. Magallon, D., 2006. Characteristics of corium debris bed generated in large-scale fuel–coolant interaction experiments. Nuclear Engineering and Design 236, 1998–2009.
Nayak, A.K., Sehgal, B.R., Stepanyan, A.V., 2006. An experimental study on quenching of a radially stratified heated porous bed. Nuclear Engineering and Design 236, 2189–2198. Reed, A.W., Bergeron, E.D., et al., 1985. Coolability of UO2 debris bed in pressurized water pools DCC-1 and DCC-2 experiment results. Nuclear Engineering and Design 97, 81–88. Schmidt, W., 2004. Influence of multidimentionality and interfacial friction on the coolability of fragmented corium. Ph.D. Thesis of University of Stuttgart, May. ˝ Schulenberg, T., Muller, U., 1987. An improved model for two-phase flow through beds of coarse particles. International Journal of Multiphase flow 13 (1), 87–97. Song, J.H., Park, I.K., et al., 2002. Experiments on the interactions of molten ZrO2 with water using TROI facility. Nuclear Engineering and Design 213, 97–110. Song, J.H., et al., 2006. The effect of corium composition and interaction vessel geometry on the prototypic steam explosion. Annuals of Nuclear Energy 33, 1437–1451. Spencer, B.W., Wang, K., et al., 1994. Fragmentation and quench behaviour of corium melt streams in water. NUREG/CR-6133, ANL-93/32, February. Tung, V.X., Dhir, V.K., 1988. A hydrodynamic model for two-phase flow through porous media. International Journal of Multiphase Flow 14 (1), 47–65.