- Email: [email protected]
Model development for fragment-size distribution based on upper-limit log-normal distribution
Nuclear Engineering and Design 349 (2019) 86–91
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Model development for fragment-size distribution based on upper-limit lognormal distribution J. Kim, H.C. NO
T
⁎
Department of Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology, Yuseong-Gu, Daejeon 34141, Republic of Korea
A R T I C LE I N FO
A B S T R A C T
Keywords: FCI Corium Fragmentation Severe accident Breakup
As molten corium is ejected from a reactor pressure vessel (RPV), the melt undergoes fuel-coolant interaction (FCI). Through the FCI molten corium is fragmented into small particles, and the fragments will form a debris bed stacking on the bottom of the cavity. Heat removal performance from both fragments and the debris bed is highly related to the size of fragments. Also, as FCI is a thermal-hydraulic interaction on the interface between the corium and the coolant, the estimation of the Sauter mean diameter is required. Thus, in this study, we develop a simplified strategy to predict the Sauter mean diameter with particle size distribution. The model is proposed to calculate the particle size distribution based on upper-limit log-normal distribution and the critical Weber number criterion. The calculation results are compared with 9 cases of FARO experiments and 7 cases of TROI experiments. The particle size distribution calculated with the suggested method well follows the trend of measured distribution from experiments. The estimated Sauter mean diameter varies from 1.58 mm to 3.13 mm, while the results for the ratio of the mass mean diameter to the Sauter mean diameter is almost constant to 1.55. It turns out that the suggested model gives good predictions of the particle size distribution and the Sauter mean diameter without large computational effort.
1. Introduction
1.1. Literature
As a severe accident with a core meltdown occurs, the molten corium may discharge from the reactor pressure vessel (RPV). Safety systems such as a core catcher and a pre-flooded cavity are introduced and designed to trap the discharged molten corium in the power plant. As molten corium release to a pre-flooded cavity, the corium will be fragmented under fuel-coolant interaction (FCI). In the falling stage, the fragmented molten corium loses heat and is solidified by thermal interaction with the coolant. Among the fragments, small size particle more likely to be solidified in the falling stage, while a large size particle may not be. The solidified corium particles are stacked on the bottom of the pre-flooded cavity and form loose debris bed, while unsolidified corium particles would form hard debris or cake. As solidified particles size are smaller, the porosity of the formed loosed debris bed will be lower. Since the heat removal performance of the debris bed is highly related to the porosity of the bed, the fragmented particle size effects to heat removal performance in not only the falling stage but also the debris bed. Therefore, the fragmented particle size comes up for the most influential parameters of the corium fragmentation.
As FCI analysis requires the fragmented droplet diameter, several models have been developed to predict the particle size.
⁎
1.1.1. MC3D MC3D, which is the multicomponent three-dimensional FCI code, modeled the local jet breakup based on Kelvin-Helmholtz instability. The model calculates the diameter of droplets d p with the wavelength λ of instability (Leskovar and Ursic, 2016):
d p = Nd λ, λ =
2π k max
(1)
where Nd and k max are the droplet diameter parameter and the wave number, respectively. The wave number, k max , used in Eq. (1) is defined as follows:
k max =
2 ρj ρamb (vj − vamb )2 3 ρj + ρamb σj
(2)
where ρ, v , and σ are density, velocity, and surface tension of the fluid, individually. The subscript j and amb stand for the jet and the ambient
Corresponding author. E-mail addresses: [email protected] (J. Kim), [email protected] (H.C. NO).
https://doi.org/10.1016/j.nucengdes.2019.04.029 Received 17 April 2018; Received in revised form 28 February 2019; Accepted 19 April 2019 Available online 24 April 2019 0029-5493/ © 2019 Elsevier B.V. All rights reserved.
Nuclear Engineering and Design 349 (2019) 86–91
J. Kim and H.C. NO
fluid, respectively.
the mass mean diameter information. Thus, converting the mass mean diameter into the Sauter mean diameter requires the distribution of the fragmented particle size. In this study, we aim to develop a simplified strategy to generate the distribution of the fragmented particle size based on upper-limit lognormal distribution and critical Weber number criterion so that we may predict the Sauter mean diameter of the fragments.
1.1.2. Namiech correlation Namiech et al. suggested an empirical correlation to approximate the droplet diameter (Nameich et al., 2004): −0.958
P d p = 33.272Dj ⎛ ⎞ P ⎝ 0⎠ ⎜
⎟
−0.035
⎛ v M0 Dj ⎞ ⎟ ⎝ υ vj ⎠
⎜
0.484
⎡1 − Ts − TL ⎤ ⎢ Tj − Ts ⎥ ⎣ ⎦
⎛ σ ⎞ ⎜ρ D v2 ⎟ ⎝ j j M0 ⎠
−1.485
⎛ vj ⎞ ⎟ ⎝ vj0 ⎠
⎜
1.591 ⎛ vj ⎞ ⎝ v M0 ⎠
⎜
⎟
2. Particle size distributions
0.065
As mentioned in the introduction section, the size distribution is required to get the desired representative particle size. Mugele et al. has conducted comparison studies between the drop size distribution equations in the spray system (Rosin-Rammler, Nukiyama-Tanasawa, the log-probability, and the upper-limit log-normal distribution), and found that the upper-limit log normal distribution well follows the trend in all cases, and gives a good fit quantitatively (Mugele and Evans, 1961).
(3)
where D, P, T, and υ are diameter, pressure, temperature, and kinematic viscosity, respectively. The subscript p, s, L, and M stand for the particles, surrounds, liquid and vapor, respectively. The reference values with a subscript 0 are defined as follows:
P0 = 1 bar, Uj0 = 5 m/s, UM0 =
ρL gDj ρvj
(4)
2.1. Upper-limit log-normal distribution 1.1.3. MUDROPS Escobar et al. suggested the MUDROPS model implemented in the MC3D code in order to obtain the particle size distribution with a lognormal distribution function (Escobar et al., 2015). The MUDROP calculates the droplet Sauter mean diameter (SMD) using the maximum wavelength which is defined as Eq. (1) as same as MC3D does. However, the MUDROP uses the log-normal distribution and MC3D code to identify the droplet characteristics.
The basic form of the upper-limit log-normal distribution is as follows: 2
f3 (d p) =
0.394
δ= log
Since particles are not fragmented into a single diameter, we need to specify the most representative particle size. As the FCI is thermal-hydraulic behavior between the coolant and the corium particle to remove the heat generated from the corium particle, we need to consider the most representative effective diameter. Chikhi et al. conducted experiments (Chikhi et al., 2014) using the different sizes of particles with a different shape in order to propose an most representative concept of the effective diameter in the debris bed (Chikhi et al., 2014). They reported that the Sauter diameter gave the best agreement with a measured pressure drop compared with other effective diameter definitions such as mass mean diameter. Therefore, in our study, the volume to the surface area ratio is considered as the most representative size, which is the Sauter mean diameter, defined as follows (Pacek et al., 1998):
α=
{
d90 / (dmax − d90) d50 / (dmax − d50)
}
d max −1 d50
(7)
(8)
where d max , d50 , and d 90 is the maximum diameter, the mass mean diameter, and the diameter which 90% of the mass is involved. 2.2. Maximum diameter Fig. 1 shows the visualization of TROI-38, which is a corium melt ejection experiment (Song et al., 2006). As we can observe from the very left picture of the figure, the size of the primarily fragmented mass is comparable to the jet diameter. The size is too large to be considered that the chunk is the final size of the fragmented particle which composes the debris bed. Thus, there will be secondary fragmentation that the massive chunk breakup into small-size particles by hydrodynamic interaction with the coolant. To sum up, the fragmentation process is divided into two phases, primary and secondary fragmentation. In the primary fragmentation,
max
∫min d3f (d)dd max ∫min d 2f (d)dd
(6)
where δ , and α are the dimensionless distribution parameter and defined as follows:
1.2. Objectives
dSMD =
αd p ⎞ ⎤ ⎫ ⎧ δd max ⎡ exp −δ 2 ⎢ln ⎛⎜ ⎟ ⎨ d π d p (d max − d p) − dp ⎠ ⎥ ⎬ max ⎣ ⎝ ⎦⎭ ⎩
(5)
On the other hand, since experimental studies measure the particle diameter with sieving the fragments, most of the experiment can give us
Fig. 1. Visualization of corium ejection experiment (TROI-38), Song et al. (2006). 87
Nuclear Engineering and Design 349 (2019) 86–91
J. Kim and H.C. NO
Table 1 FARO experiment conditions. Test
L-06
L-08
L-11
L-14
L-19
L-20
L-24
L-28
L-31
Average
System Pressure, (MPa) Melt temperature, (K) Subcooled degree, (K) Mean particle diameter, (mm)
5 2923 0 4.5
5.8 3023 12 3.8
5 2823 2 3.5
5 3123 0 4.8
5 3073 1 3.7
2 3173 0 4.4
0.5 3023 0 2.6
0.5 3052 1 3.0
0.2 2990 104 ?
2.6 3037 15.4 3.63
Fig. 2. Calculated particle size distribution results using the KAIST model for each case of FARO experiments.
particle is calculated with the following relation:
the large mass of corium is detached from the jet column. The detached mass interacts with the coolant under the local hydrodynamic condition and undergoes the secondary fragmentation. Therefore, we can say that the final fragmented particle size is a result of the secondary fragmentation. A falling-down particle generated by the secondary fragmentation in the water will quickly reach its terminal velocity, and its velocity is calculated as follows:
up =
Cd = 0.2 + 10.4d p + 0.00218ΔTsub
The relationship is fitted to experimental data performed by Gylys et al. (2015). They measured the falling velocity of the spherical body with a high-temperature (650 °C) metallic specimen with various diameters (10–35 mm). Since the temperature condition used in the experiment is far from the typical accident condition (over 3000 K), the applicability of Eq. (10) is arguable. Yoon simulated FARO cases, which are large-scale corium experiments conducted under high temperature (above 2800 K), using Eq. (10) as a drag coefficient (Yoon, 2018). The simulation is performed under 7 FARO cases, and gives good agreement with experimental data in terms of pressure and released energy.
8m p g πd p2 ρc Cd
(10)
(9)
where m p, ρc , and Cd are the mass of a single particle, corium density, and the drag coefficient, respectively. The drag coefficient of the 88
Nuclear Engineering and Design 349 (2019) 86–91
J. Kim and H.C. NO
the particle one for the upper-limit log-normal distribution. 2.3. Calculation process To sum up, the strategies explained in the above section, the following overall calculation process will be taken to estimate the fragmented particle size distribution: 2.3.1. Set the input parameters In the maximum diameter calculation with the critical We, we need mean diameter information. FCI experiments such as FARO and TROI show that the mean diameter of the fragmented particles is around 3.4 mm, and most of the mean diameter is in the range of 2.5–4.8 mm. Also, system pressure, coolant subcooled degree, and corium temperature is required for the calculation. 2.3.2. Determine empirical parameters in ULLN distribution function The basic form of ULLN distribution function contains three experimental parameters (d max , δ , and α ). The maximum diameter is calculated with the critical We. The skewness factor, α , could be calculated with the given mean diameter and the calculated maximum diameter. Finally, since δ , which is an unknown distribution parameter, has little sensitivity, it is assumed as 0.75 which is experimentally averaged value. (Hubbard and Dukler, 1966; Cousins and Hewitt, 1968; Tatterson, 1975).
Fig. 3. comparison between average FARO results and calculated particle size distribution using the KAIST model with averaged experiment parameters.
Table 2 Calculated the maximum diameter, the Sauter mean diameter and the ratio of the mean diameter to SMD for each case of FARO.
L-06 L-08 L-11 L-14 L-19 L-20 L-24 L-28 L-31 Average
Calculated dmax [mm]
Sauter Mean Diameter [mm]
d50
78.4 76.2 92.9 72.9 78.4 138.0 111.1 139.9 183.9 108.0
2.93 2.47 2.26 3.13 2.40 2.84 1.67 1.92 2.18 2.42
1.54 1.54 1.55 1.53 1.54 1.55 1.56 1.56 1.56 1.55
SMD
[–]
2.3.3. Obtain the satuer mean diameter by applying the parameters to ULLN distribution function From the above process, every required parameter for the distribution function is given or calculated. Therefore, we could get the distribution using the upper-limit log-normal distribution function. Finally, we calculate the Sauter mean diameter with Eq. (5). 3. Comparison with experiments For the validation of suggested logic, the particle size distribution is calculated based on experiment conditions
Therefore, it is a reasonable assumption that applying the drag coefficient correlation to the high-temperature condition. From the calculated velocity information, we can also calculate the Weber number defined as follows:
We =
3.1. FARO experiment Table 1 shows the FARO experiment conditions which are required to set the input parameters (Magallon, 2006). Among the FARO experiments, L-27 and L-29 are not involved because there is no mean diameter information in the report. Also, since L33 is the case that a steam explosion is observed, it is not included in the calculation to avoid steam explosion effect.
ρc u p2 d p (11)
σm
where σm is surface tension of the particle. If the falling particle has higher than the We of 12 (Critical We) due to such as initial large particle sizes and coalescences among relatively large particles, it breaks up into smaller particles (Pilch and Erdman, 1987). It also means, if a particle has smaller We than 12, it will be in the stable stage, and no breakup will occur. Therefore, the maximum diameter is determined when the particle has 12 of We as follows:
d max =
12σm ρc u p2
3.1.1. Results Fig. 2 is the comparison between the FARO experiment results and calculation results with the KAIST model. Besides, Fig. 3 shows the comparison between the average FARO experiment results and KAIST model results with averaged experiment parameters for cumulative mass fraction. As shown in Fig. 2 and Fig. 3, the KAIST model not only follows the trend of the curve but also well fits to experiment results. The Sauter mean diameter results calculated using the KAIST model is listed in Table 2.
(12)
We propose this maximum possible diameter as the upper limit of Table 3 TROI Experiment conditions. Test
11
19
23
28
29
32
38
Average
System Pressure, (MPa) Melt temperature, (K) Subcooled degree, (K) Mean particle diameter, (mm)
0.111 4150 77 3.73
0.118 3200 78 3.38
0.11 3600 80 3.29
0.105 3500 89 2.71
0.11 3450 86 2.46
0.113 3530 83 3.21
0.105 3000 85 2.27
0.110 3490 82.6 3.01
89
Nuclear Engineering and Design 349 (2019) 86–91
J. Kim and H.C. NO
Fig. 4. Calculated particle size distribution results using the KAIST model for each case of TROI experiments. Table 4 Calculated the maximum diameter, the Sauter mean diameter and the ratio of the mean diameter to SMD for each case of TROI.
TROI 11 TROI 19 TROI 23 TROI 28 TROI 29 TROI 32 TROI 38 Average
Calculated dmax [mm]
Sauter Mean Diameter [mm]
d50
210.2 183.3 175.3 126.1 105.7 168.5 91.3 151.5
2.39 2.17 2.11 1.74 1.58 2.06 1.46 1.93
1.56 1.56 1.56 1.56 1.56 1.56 1.56 1.56
SMD
[−]
The KAIST model gives 2.42 mm of average the Sauter mean diameter and 1.55 of the ratio of mean diameter to SMD. 3.2. TROI experiment Table 3 shows the TROI experiment conditions which are required to set the input parameters (Korea Atomic Energy Research Institute, 2006). As compared with the FARO experiment, the TROI experiment
Fig. 5. comparison between average TROI results and calculated particle size distribution using the KAIST model with averaged experiment parameters.
90
Nuclear Engineering and Design 349 (2019) 86–91
J. Kim and H.C. NO
Although the KAIST model gives good agreement with corium experiment results, we are undergoing improvement to extend the application of the model to the real accident condition. The KAIST model requires measured mass mean diameter information, which is not given for the real accident situation, for the prediction. Therefore, in accident condition simulation, the averaged mass mean diameter value from a large-scale experiment such as FARO should be adopted. In addition, as a next step it can be essential to compare the current approach with other ones in the future.
conditions have a higher subcooled degree. The measured particle diameters of TROI are in a similar range with that of FARO but have relatively smaller values than those of FARO. It is not clear why the particle diameters of TROI are lower values than those of FARO. 3.2.1. Results Fig. 4 shows the comparison between the measured particle diameter distribution in the TROI experiment and the calculation results of the KAIST model. Fig. 5 shows the comparison between the average TROI experiment results and KAIST model results with averaged experiment parameters for cumulative mass fraction. Except for one case (TROI 11), the KAIST model shows the good prediction of size distribution. The deviation in TROI 11 may come from the extremely high temperature of corium (4150 K). As shown in Table 4, for the TROI experiment, the KAIST model gives 1.93 mm of the Sauter mean diameter and 1.56 of the ratio of mean diameter to SMD. The results of the TROI case show smaller SMD than the results for FARO cases, but the ratios of mean diameter to SMD are almost the same as ones for the FARO experiments. The comparison with experimental data shows that the KAIST model can predict the size distribution for fragmented particles. Also, the results for the ratio of mean diameter to SMD are almost constant for all cases, while SMD results show some deviation. The consistent calculation results for the proportion of mean diameter to SMD may indicate that the KAIST model, which is a simplified strategy, could be applied to the real scale accident analysis. Also, since the suggested method adopts the simplified calculation method to get the particle size distribution, it requires not much computational effort. Therefore, applying the proposed method to the severe accident analysis will significantly lessen a computational load.
Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (Ministry of Science and ICT) (No. NRF-2017M2B2A9072062). References Chikhi, N., Coindreau, O., Li, L.X., Ma, W.M., Taivassalo, V., Takasuo, E., Leininger, S., Kulenovic, R., Laurien, E., 2014. Evaluation of an effective diameter to study quenching and dry-out of complex debris bed. Ann. Nucl. Energy 74, 24–41. Cousins, L.B., Hewitt, G.F., 1968. Liquid phase mass transfer in annular two phase flow: droplet deposition and liquid entrainment. UKAEA Report, AERE-R5657; 1968. Escobar, S.C., Meignen, R., Picchi, S., Rimbert, N., Gradeck, M., 2015. A two-scale approach for modeling the corium melt fragmentation during fuel-coolant interaction. NURETH-16, Chicago, IL, pp. 6543–6556. Gylys, J., Skvorcinskiene, R., Paukstaitis, L., Gylys, M., Adomavicius, A., 2015. Water temperature influence on the spherical body’s falling velocity. Int. J. Heat Mass Transfer 89, 913–919. Hubbard, M.G., Dukler, A.E., 1966. The characterization of flow regimes for horizontal two-phase flow. In: Proceeding of the 1966 Heat Transfer and Fluid Mechanics Institute, pp. 100–121. Korea Atomic Energy Research Institute, 2006. Fuel-coolant interaction experiments in the TROI facility. J. Kim, B. Min, S. Hong, S. Hong, I. Park, H. Kim, J. Song, and H. Kim, KAERI/TR-3171/2006. Leskovar, M., Ursic, M., 2016. Ex-vessel steam explosion analysis for Pressurized Water Reactor and Boiling Water Reactor. Nucl. Eng. Technol. 48, 72–86. Magallon, D., 2006. Characteristics of corium debris bed generated in large-scale fuelcoolant interaction experiments. Nucl. Eng. Des. 236, 1998–2009. Mugele, R.A., Evans, H.D., 1961. Droplet size distribution in sprays. Ind. Eng. Chem. 43, 1317–1324. Nameich, J., Berthoud, G., Coutris, N., 2004. Fragmentation of a molten corium jet falling into water. Nucl. Eng. Des. 229, 265–287. Pacek, A.W., Man, C.C., Nienow, A.W., 1998. On the Sauter mean diameter and size distributions in turbulent liquid/liquid dispersions in a stirred vessel. Chem. Eng. Sci. 53 (11), 2005–2011. Pilch, M., Erdman, C.A., 1987. Use of breakup time data and velocity history data to predict the maximum size of stable fragments for acceleration-induced breakup of a liquid drop. Int. J. Multiphase Flow 13, 741–757. Song, J.H., Kim, J.H., Hong, S.W., Min, B.T., Kim, H.D., 2006. The effect of corium composition and interaction vessel geometry on the prototypic steam explosion. Ann. Nucl. Energy 33, 1437–1451. Tatterson, D.F., 1975. Rates of atomization and drop size in annular two-phase flow (Ph.D. thesis). Univ. of Illinois, Urbana. Yoon, S.H., 2018. Experimental investigations of film boiling heat transfer on a sphere for evaluating thermal-hydraulic behavior of corium particle during fuel-coolant interaction (Ph.D. thesis). KAIST.
4. Conclusions This study aims to develop a simplified strategy to get the fragments size distribution and to predict the Sauter mean diameter of fragments. The upper-limit log-normal distribution equation is applied to obtain the particle size distribution. The critical We criterion is used to get the maximum particle size distribution with a spherical shape of particles. The developed KAIST strategy is compared with 9 cases of FARO experiments and 7 cases of TROI experiments which are large-scale corium ejection experiments. We can make the following conclusions:
• The KAIST model well predicts the trend of the fragmented particle size distribution without significant deviation. • The calculated the Sauter mean diameter varies from 1.58 mm to •
3.13 mm, but the ratio of mean diameter to SMD shows to be nearly constant (1.55 of the rate). Since the suggested method does not require considerable computational effort, it would lessen a computational load of severe accident analysis.
91
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
Model development for fragment-size distribution based on upper-limit lognormal distribution J. Kim, H.C. NO
T
⁎
Department of Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology, Yuseong-Gu, Daejeon 34141, Republic of Korea
A R T I C LE I N FO
A B S T R A C T
Keywords: FCI Corium Fragmentation Severe accident Breakup
As molten corium is ejected from a reactor pressure vessel (RPV), the melt undergoes fuel-coolant interaction (FCI). Through the FCI molten corium is fragmented into small particles, and the fragments will form a debris bed stacking on the bottom of the cavity. Heat removal performance from both fragments and the debris bed is highly related to the size of fragments. Also, as FCI is a thermal-hydraulic interaction on the interface between the corium and the coolant, the estimation of the Sauter mean diameter is required. Thus, in this study, we develop a simplified strategy to predict the Sauter mean diameter with particle size distribution. The model is proposed to calculate the particle size distribution based on upper-limit log-normal distribution and the critical Weber number criterion. The calculation results are compared with 9 cases of FARO experiments and 7 cases of TROI experiments. The particle size distribution calculated with the suggested method well follows the trend of measured distribution from experiments. The estimated Sauter mean diameter varies from 1.58 mm to 3.13 mm, while the results for the ratio of the mass mean diameter to the Sauter mean diameter is almost constant to 1.55. It turns out that the suggested model gives good predictions of the particle size distribution and the Sauter mean diameter without large computational effort.
1. Introduction
1.1. Literature
As a severe accident with a core meltdown occurs, the molten corium may discharge from the reactor pressure vessel (RPV). Safety systems such as a core catcher and a pre-flooded cavity are introduced and designed to trap the discharged molten corium in the power plant. As molten corium release to a pre-flooded cavity, the corium will be fragmented under fuel-coolant interaction (FCI). In the falling stage, the fragmented molten corium loses heat and is solidified by thermal interaction with the coolant. Among the fragments, small size particle more likely to be solidified in the falling stage, while a large size particle may not be. The solidified corium particles are stacked on the bottom of the pre-flooded cavity and form loose debris bed, while unsolidified corium particles would form hard debris or cake. As solidified particles size are smaller, the porosity of the formed loosed debris bed will be lower. Since the heat removal performance of the debris bed is highly related to the porosity of the bed, the fragmented particle size effects to heat removal performance in not only the falling stage but also the debris bed. Therefore, the fragmented particle size comes up for the most influential parameters of the corium fragmentation.
As FCI analysis requires the fragmented droplet diameter, several models have been developed to predict the particle size.
⁎
1.1.1. MC3D MC3D, which is the multicomponent three-dimensional FCI code, modeled the local jet breakup based on Kelvin-Helmholtz instability. The model calculates the diameter of droplets d p with the wavelength λ of instability (Leskovar and Ursic, 2016):
d p = Nd λ, λ =
2π k max
(1)
where Nd and k max are the droplet diameter parameter and the wave number, respectively. The wave number, k max , used in Eq. (1) is defined as follows:
k max =
2 ρj ρamb (vj − vamb )2 3 ρj + ρamb σj
(2)
where ρ, v , and σ are density, velocity, and surface tension of the fluid, individually. The subscript j and amb stand for the jet and the ambient
Corresponding author. E-mail addresses: [email protected] (J. Kim), [email protected] (H.C. NO).
https://doi.org/10.1016/j.nucengdes.2019.04.029 Received 17 April 2018; Received in revised form 28 February 2019; Accepted 19 April 2019 Available online 24 April 2019 0029-5493/ © 2019 Elsevier B.V. All rights reserved.
Nuclear Engineering and Design 349 (2019) 86–91
J. Kim and H.C. NO
fluid, respectively.
the mass mean diameter information. Thus, converting the mass mean diameter into the Sauter mean diameter requires the distribution of the fragmented particle size. In this study, we aim to develop a simplified strategy to generate the distribution of the fragmented particle size based on upper-limit lognormal distribution and critical Weber number criterion so that we may predict the Sauter mean diameter of the fragments.
1.1.2. Namiech correlation Namiech et al. suggested an empirical correlation to approximate the droplet diameter (Nameich et al., 2004): −0.958
P d p = 33.272Dj ⎛ ⎞ P ⎝ 0⎠ ⎜
⎟
−0.035
⎛ v M0 Dj ⎞ ⎟ ⎝ υ vj ⎠
⎜
0.484
⎡1 − Ts − TL ⎤ ⎢ Tj − Ts ⎥ ⎣ ⎦
⎛ σ ⎞ ⎜ρ D v2 ⎟ ⎝ j j M0 ⎠
−1.485
⎛ vj ⎞ ⎟ ⎝ vj0 ⎠
⎜
1.591 ⎛ vj ⎞ ⎝ v M0 ⎠
⎜
⎟
2. Particle size distributions
0.065
As mentioned in the introduction section, the size distribution is required to get the desired representative particle size. Mugele et al. has conducted comparison studies between the drop size distribution equations in the spray system (Rosin-Rammler, Nukiyama-Tanasawa, the log-probability, and the upper-limit log-normal distribution), and found that the upper-limit log normal distribution well follows the trend in all cases, and gives a good fit quantitatively (Mugele and Evans, 1961).
(3)
where D, P, T, and υ are diameter, pressure, temperature, and kinematic viscosity, respectively. The subscript p, s, L, and M stand for the particles, surrounds, liquid and vapor, respectively. The reference values with a subscript 0 are defined as follows:
P0 = 1 bar, Uj0 = 5 m/s, UM0 =
ρL gDj ρvj
(4)
2.1. Upper-limit log-normal distribution 1.1.3. MUDROPS Escobar et al. suggested the MUDROPS model implemented in the MC3D code in order to obtain the particle size distribution with a lognormal distribution function (Escobar et al., 2015). The MUDROP calculates the droplet Sauter mean diameter (SMD) using the maximum wavelength which is defined as Eq. (1) as same as MC3D does. However, the MUDROP uses the log-normal distribution and MC3D code to identify the droplet characteristics.
The basic form of the upper-limit log-normal distribution is as follows: 2
f3 (d p) =
0.394
δ= log
Since particles are not fragmented into a single diameter, we need to specify the most representative particle size. As the FCI is thermal-hydraulic behavior between the coolant and the corium particle to remove the heat generated from the corium particle, we need to consider the most representative effective diameter. Chikhi et al. conducted experiments (Chikhi et al., 2014) using the different sizes of particles with a different shape in order to propose an most representative concept of the effective diameter in the debris bed (Chikhi et al., 2014). They reported that the Sauter diameter gave the best agreement with a measured pressure drop compared with other effective diameter definitions such as mass mean diameter. Therefore, in our study, the volume to the surface area ratio is considered as the most representative size, which is the Sauter mean diameter, defined as follows (Pacek et al., 1998):
α=
{
d90 / (dmax − d90) d50 / (dmax − d50)
}
d max −1 d50
(7)
(8)
where d max , d50 , and d 90 is the maximum diameter, the mass mean diameter, and the diameter which 90% of the mass is involved. 2.2. Maximum diameter Fig. 1 shows the visualization of TROI-38, which is a corium melt ejection experiment (Song et al., 2006). As we can observe from the very left picture of the figure, the size of the primarily fragmented mass is comparable to the jet diameter. The size is too large to be considered that the chunk is the final size of the fragmented particle which composes the debris bed. Thus, there will be secondary fragmentation that the massive chunk breakup into small-size particles by hydrodynamic interaction with the coolant. To sum up, the fragmentation process is divided into two phases, primary and secondary fragmentation. In the primary fragmentation,
max
∫min d3f (d)dd max ∫min d 2f (d)dd
(6)
where δ , and α are the dimensionless distribution parameter and defined as follows:
1.2. Objectives
dSMD =
αd p ⎞ ⎤ ⎫ ⎧ δd max ⎡ exp −δ 2 ⎢ln ⎛⎜ ⎟ ⎨ d π d p (d max − d p) − dp ⎠ ⎥ ⎬ max ⎣ ⎝ ⎦⎭ ⎩
(5)
On the other hand, since experimental studies measure the particle diameter with sieving the fragments, most of the experiment can give us
Fig. 1. Visualization of corium ejection experiment (TROI-38), Song et al. (2006). 87
Nuclear Engineering and Design 349 (2019) 86–91
J. Kim and H.C. NO
Table 1 FARO experiment conditions. Test
L-06
L-08
L-11
L-14
L-19
L-20
L-24
L-28
L-31
Average
System Pressure, (MPa) Melt temperature, (K) Subcooled degree, (K) Mean particle diameter, (mm)
5 2923 0 4.5
5.8 3023 12 3.8
5 2823 2 3.5
5 3123 0 4.8
5 3073 1 3.7
2 3173 0 4.4
0.5 3023 0 2.6
0.5 3052 1 3.0
0.2 2990 104 ?
2.6 3037 15.4 3.63
Fig. 2. Calculated particle size distribution results using the KAIST model for each case of FARO experiments.
particle is calculated with the following relation:
the large mass of corium is detached from the jet column. The detached mass interacts with the coolant under the local hydrodynamic condition and undergoes the secondary fragmentation. Therefore, we can say that the final fragmented particle size is a result of the secondary fragmentation. A falling-down particle generated by the secondary fragmentation in the water will quickly reach its terminal velocity, and its velocity is calculated as follows:
up =
Cd = 0.2 + 10.4d p + 0.00218ΔTsub
The relationship is fitted to experimental data performed by Gylys et al. (2015). They measured the falling velocity of the spherical body with a high-temperature (650 °C) metallic specimen with various diameters (10–35 mm). Since the temperature condition used in the experiment is far from the typical accident condition (over 3000 K), the applicability of Eq. (10) is arguable. Yoon simulated FARO cases, which are large-scale corium experiments conducted under high temperature (above 2800 K), using Eq. (10) as a drag coefficient (Yoon, 2018). The simulation is performed under 7 FARO cases, and gives good agreement with experimental data in terms of pressure and released energy.
8m p g πd p2 ρc Cd
(10)
(9)
where m p, ρc , and Cd are the mass of a single particle, corium density, and the drag coefficient, respectively. The drag coefficient of the 88
Nuclear Engineering and Design 349 (2019) 86–91
J. Kim and H.C. NO
the particle one for the upper-limit log-normal distribution. 2.3. Calculation process To sum up, the strategies explained in the above section, the following overall calculation process will be taken to estimate the fragmented particle size distribution: 2.3.1. Set the input parameters In the maximum diameter calculation with the critical We, we need mean diameter information. FCI experiments such as FARO and TROI show that the mean diameter of the fragmented particles is around 3.4 mm, and most of the mean diameter is in the range of 2.5–4.8 mm. Also, system pressure, coolant subcooled degree, and corium temperature is required for the calculation. 2.3.2. Determine empirical parameters in ULLN distribution function The basic form of ULLN distribution function contains three experimental parameters (d max , δ , and α ). The maximum diameter is calculated with the critical We. The skewness factor, α , could be calculated with the given mean diameter and the calculated maximum diameter. Finally, since δ , which is an unknown distribution parameter, has little sensitivity, it is assumed as 0.75 which is experimentally averaged value. (Hubbard and Dukler, 1966; Cousins and Hewitt, 1968; Tatterson, 1975).
Fig. 3. comparison between average FARO results and calculated particle size distribution using the KAIST model with averaged experiment parameters.
Table 2 Calculated the maximum diameter, the Sauter mean diameter and the ratio of the mean diameter to SMD for each case of FARO.
L-06 L-08 L-11 L-14 L-19 L-20 L-24 L-28 L-31 Average
Calculated dmax [mm]
Sauter Mean Diameter [mm]
d50
78.4 76.2 92.9 72.9 78.4 138.0 111.1 139.9 183.9 108.0
2.93 2.47 2.26 3.13 2.40 2.84 1.67 1.92 2.18 2.42
1.54 1.54 1.55 1.53 1.54 1.55 1.56 1.56 1.56 1.55
SMD
[–]
2.3.3. Obtain the satuer mean diameter by applying the parameters to ULLN distribution function From the above process, every required parameter for the distribution function is given or calculated. Therefore, we could get the distribution using the upper-limit log-normal distribution function. Finally, we calculate the Sauter mean diameter with Eq. (5). 3. Comparison with experiments For the validation of suggested logic, the particle size distribution is calculated based on experiment conditions
Therefore, it is a reasonable assumption that applying the drag coefficient correlation to the high-temperature condition. From the calculated velocity information, we can also calculate the Weber number defined as follows:
We =
3.1. FARO experiment Table 1 shows the FARO experiment conditions which are required to set the input parameters (Magallon, 2006). Among the FARO experiments, L-27 and L-29 are not involved because there is no mean diameter information in the report. Also, since L33 is the case that a steam explosion is observed, it is not included in the calculation to avoid steam explosion effect.
ρc u p2 d p (11)
σm
where σm is surface tension of the particle. If the falling particle has higher than the We of 12 (Critical We) due to such as initial large particle sizes and coalescences among relatively large particles, it breaks up into smaller particles (Pilch and Erdman, 1987). It also means, if a particle has smaller We than 12, it will be in the stable stage, and no breakup will occur. Therefore, the maximum diameter is determined when the particle has 12 of We as follows:
d max =
12σm ρc u p2
3.1.1. Results Fig. 2 is the comparison between the FARO experiment results and calculation results with the KAIST model. Besides, Fig. 3 shows the comparison between the average FARO experiment results and KAIST model results with averaged experiment parameters for cumulative mass fraction. As shown in Fig. 2 and Fig. 3, the KAIST model not only follows the trend of the curve but also well fits to experiment results. The Sauter mean diameter results calculated using the KAIST model is listed in Table 2.
(12)
We propose this maximum possible diameter as the upper limit of Table 3 TROI Experiment conditions. Test
11
19
23
28
29
32
38
Average
System Pressure, (MPa) Melt temperature, (K) Subcooled degree, (K) Mean particle diameter, (mm)
0.111 4150 77 3.73
0.118 3200 78 3.38
0.11 3600 80 3.29
0.105 3500 89 2.71
0.11 3450 86 2.46
0.113 3530 83 3.21
0.105 3000 85 2.27
0.110 3490 82.6 3.01
89
Nuclear Engineering and Design 349 (2019) 86–91
J. Kim and H.C. NO
Fig. 4. Calculated particle size distribution results using the KAIST model for each case of TROI experiments. Table 4 Calculated the maximum diameter, the Sauter mean diameter and the ratio of the mean diameter to SMD for each case of TROI.
TROI 11 TROI 19 TROI 23 TROI 28 TROI 29 TROI 32 TROI 38 Average
Calculated dmax [mm]
Sauter Mean Diameter [mm]
d50
210.2 183.3 175.3 126.1 105.7 168.5 91.3 151.5
2.39 2.17 2.11 1.74 1.58 2.06 1.46 1.93
1.56 1.56 1.56 1.56 1.56 1.56 1.56 1.56
SMD
[−]
The KAIST model gives 2.42 mm of average the Sauter mean diameter and 1.55 of the ratio of mean diameter to SMD. 3.2. TROI experiment Table 3 shows the TROI experiment conditions which are required to set the input parameters (Korea Atomic Energy Research Institute, 2006). As compared with the FARO experiment, the TROI experiment
Fig. 5. comparison between average TROI results and calculated particle size distribution using the KAIST model with averaged experiment parameters.
90
Nuclear Engineering and Design 349 (2019) 86–91
J. Kim and H.C. NO
Although the KAIST model gives good agreement with corium experiment results, we are undergoing improvement to extend the application of the model to the real accident condition. The KAIST model requires measured mass mean diameter information, which is not given for the real accident situation, for the prediction. Therefore, in accident condition simulation, the averaged mass mean diameter value from a large-scale experiment such as FARO should be adopted. In addition, as a next step it can be essential to compare the current approach with other ones in the future.
conditions have a higher subcooled degree. The measured particle diameters of TROI are in a similar range with that of FARO but have relatively smaller values than those of FARO. It is not clear why the particle diameters of TROI are lower values than those of FARO. 3.2.1. Results Fig. 4 shows the comparison between the measured particle diameter distribution in the TROI experiment and the calculation results of the KAIST model. Fig. 5 shows the comparison between the average TROI experiment results and KAIST model results with averaged experiment parameters for cumulative mass fraction. Except for one case (TROI 11), the KAIST model shows the good prediction of size distribution. The deviation in TROI 11 may come from the extremely high temperature of corium (4150 K). As shown in Table 4, for the TROI experiment, the KAIST model gives 1.93 mm of the Sauter mean diameter and 1.56 of the ratio of mean diameter to SMD. The results of the TROI case show smaller SMD than the results for FARO cases, but the ratios of mean diameter to SMD are almost the same as ones for the FARO experiments. The comparison with experimental data shows that the KAIST model can predict the size distribution for fragmented particles. Also, the results for the ratio of mean diameter to SMD are almost constant for all cases, while SMD results show some deviation. The consistent calculation results for the proportion of mean diameter to SMD may indicate that the KAIST model, which is a simplified strategy, could be applied to the real scale accident analysis. Also, since the suggested method adopts the simplified calculation method to get the particle size distribution, it requires not much computational effort. Therefore, applying the proposed method to the severe accident analysis will significantly lessen a computational load.
Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (Ministry of Science and ICT) (No. NRF-2017M2B2A9072062). References Chikhi, N., Coindreau, O., Li, L.X., Ma, W.M., Taivassalo, V., Takasuo, E., Leininger, S., Kulenovic, R., Laurien, E., 2014. Evaluation of an effective diameter to study quenching and dry-out of complex debris bed. Ann. Nucl. Energy 74, 24–41. Cousins, L.B., Hewitt, G.F., 1968. Liquid phase mass transfer in annular two phase flow: droplet deposition and liquid entrainment. UKAEA Report, AERE-R5657; 1968. Escobar, S.C., Meignen, R., Picchi, S., Rimbert, N., Gradeck, M., 2015. A two-scale approach for modeling the corium melt fragmentation during fuel-coolant interaction. NURETH-16, Chicago, IL, pp. 6543–6556. Gylys, J., Skvorcinskiene, R., Paukstaitis, L., Gylys, M., Adomavicius, A., 2015. Water temperature influence on the spherical body’s falling velocity. Int. J. Heat Mass Transfer 89, 913–919. Hubbard, M.G., Dukler, A.E., 1966. The characterization of flow regimes for horizontal two-phase flow. In: Proceeding of the 1966 Heat Transfer and Fluid Mechanics Institute, pp. 100–121. Korea Atomic Energy Research Institute, 2006. Fuel-coolant interaction experiments in the TROI facility. J. Kim, B. Min, S. Hong, S. Hong, I. Park, H. Kim, J. Song, and H. Kim, KAERI/TR-3171/2006. Leskovar, M., Ursic, M., 2016. Ex-vessel steam explosion analysis for Pressurized Water Reactor and Boiling Water Reactor. Nucl. Eng. Technol. 48, 72–86. Magallon, D., 2006. Characteristics of corium debris bed generated in large-scale fuelcoolant interaction experiments. Nucl. Eng. Des. 236, 1998–2009. Mugele, R.A., Evans, H.D., 1961. Droplet size distribution in sprays. Ind. Eng. Chem. 43, 1317–1324. Nameich, J., Berthoud, G., Coutris, N., 2004. Fragmentation of a molten corium jet falling into water. Nucl. Eng. Des. 229, 265–287. Pacek, A.W., Man, C.C., Nienow, A.W., 1998. On the Sauter mean diameter and size distributions in turbulent liquid/liquid dispersions in a stirred vessel. Chem. Eng. Sci. 53 (11), 2005–2011. Pilch, M., Erdman, C.A., 1987. Use of breakup time data and velocity history data to predict the maximum size of stable fragments for acceleration-induced breakup of a liquid drop. Int. J. Multiphase Flow 13, 741–757. Song, J.H., Kim, J.H., Hong, S.W., Min, B.T., Kim, H.D., 2006. The effect of corium composition and interaction vessel geometry on the prototypic steam explosion. Ann. Nucl. Energy 33, 1437–1451. Tatterson, D.F., 1975. Rates of atomization and drop size in annular two-phase flow (Ph.D. thesis). Univ. of Illinois, Urbana. Yoon, S.H., 2018. Experimental investigations of film boiling heat transfer on a sphere for evaluating thermal-hydraulic behavior of corium particle during fuel-coolant interaction (Ph.D. thesis). KAIST.
4. Conclusions This study aims to develop a simplified strategy to get the fragments size distribution and to predict the Sauter mean diameter of fragments. The upper-limit log-normal distribution equation is applied to obtain the particle size distribution. The critical We criterion is used to get the maximum particle size distribution with a spherical shape of particles. The developed KAIST strategy is compared with 9 cases of FARO experiments and 7 cases of TROI experiments which are large-scale corium ejection experiments. We can make the following conclusions:
• The KAIST model well predicts the trend of the fragmented particle size distribution without significant deviation. • The calculated the Sauter mean diameter varies from 1.58 mm to •
3.13 mm, but the ratio of mean diameter to SMD shows to be nearly constant (1.55 of the rate). Since the suggested method does not require considerable computational effort, it would lessen a computational load of severe accident analysis.
91