- Email: [email protected]
One-dimensional model for containment in AP1000 nuclear power plant based on thermal stratification
Applied Thermal Engineering 70 (2014) 25e32
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
One-dimensional model for containment in AP1000 nuclear power plant based on thermal stratification Yu Yu, Fenglei Niu*, Shengfei Wang, Yingqiu Hu School of Nuclear Science and Technology, North China Electric Power University, No.2 Beinong Road, Huilongguan, Changping District, Beijing 102206, China
h i g h l i g h t s 1-D model for AP1000 containment has been established. Thermal stratification and circulation exist simultaneously in the containment. Pressure positive gradient affords the force of the circulation. Steam concentration at the top enhances the heat and mass transfer. The pressure in the containment can be kept at the safe level.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 October 2013 Accepted 27 April 2014 Available online 9 May 2014
Passive containment cooling system operating based on natural circulation is innovatively used in AP1000 nuclear power plant to improve safety. However, the overall heat and mass transfer will decrease and the heat transfer through the containment shell will be slow due to stable stratification which will occur if the forced convection mixing is not sufficiently strong enough to disrupt the stable fluid layers, so it is important for system design and accident analysis to evaluate whether the circulation can establish or not in the containment accurately and efficiently. Many researches indicate that the gradients of such parameters in horizontal direction are so small that can be ignored, in this paper, onedimensional model is developed for AP1000 containment based on thermal stratification theory, natural circulations in and outside the containment are both included. Based on the results, the thermal stratification and circulation exist simultaneously, the pressure in the containment can be kept at the safe level, and positive pressure gradient in the vertical direction affords the force of circulation. The steam and air with higher temperature concentrating at the top will strengthen the heat and mass transfer because the cooling capacity of top head is higher than of other parts. Ó 2014 Elsevier Ltd. All rights reserved.
Keywords: Passive containment cooling system Natural circulation Thermal stratification One-dimensional model
1. Introduction Passive safety systems [1,2] are widely used in new generation reactor design to improve nuclear power plant safety especially under external disasters such as earthquake, tsunami, since they are independent of emergency AC power or offset power [2]. Passive containment cooling system [1] is one of the important safety systems in AP1000 nuclear power plant [2] by which the heat produced in the containment can be transferred to the atmosphere based on natural circulation [3]. However, stratification [1,5e7] will
* Corresponding author. E-mail addresses: [email protected] (Y. Yu), [email protected] (F. Niu), [email protected] (S. Wang). http://dx.doi.org/10.1016/j.applthermaleng.2014.04.070 1359-4311/Ó 2014 Elsevier Ltd. All rights reserved.
occur if the density of the layers decreases in the upward vertical direction and if the forced convection mixing is not strong enough to disrupt the stable fluid layers. As a result of the stable stratification, the overall heat and mass transfer decrease and the heat transfer through the containment shell is slow, which may induce adverse effect on the containment and should be considered in the system thermal-hydraulic analysis. So it is important for system design and accident analysis to simulate thermal-hydraulic performance, to recognize influence of the factors such as temperature distribution, fluid velocity on system operation [4], to predict distributions of pressure, temperature, and steam concentration, and to evaluate whether the circulation can establish in the containment accurately and efficiently. Many thermal-hydraulic analyses have been done mainly in two ways: lumped parameter [8e10] model and multi-dimensional
26
Y. Yu et al. / Applied Thermal Engineering 70 (2014) 25e32
model [11e14]. For AP1000 containment design and system reliability evaluation it is important to describe distributions of the physical parameters such as temperature, pressure and density, at the same time high calculation speed is needed for accident analysis especially under abnormal condition. Since the gradients of physical parameters in the horizontal direction are so small [11,14] that can be ignored, in this paper it is proposed that onedimensional model [15e19] based on thermal stratification theory can be used to simulate the thermal-hydraulic performance in the containment after being modified according to the accident conditions: Firstly, the original model [15,16] is mainly developed for a large volume with a free outlet always at the upper part, so the thermal stratification can be simulated yet the circulation is too weak to be considered. While for the volume with outlet at the bottom, the fluid is supposed to flow down along the wall when arriving at the top and wall jet is used to simulate the down coming fluid [19]. Nevertheless, for fluid including air and steam in the containment after accident, we consider that the hot fluid at the top will come down through the whole volume induced by the pressure difference and be cooled by the cold wall, so both of the stratification and circulation should be simulated. Secondly, the original model [15,16] is developed mainly for the volume having cold wall with constant temperature or constant heat flux density, while for the accident analysis in the containment, the heat transfer process is more complicated, nether the wall temperature nor the heat flux density is constant, they are determined by the heat transfer processes both inside and outside the containment. In this paper, a new model including stratification and circulation is developed based on one-dimensional model, in which the modified momentum equation is used to replace the wall jet model and the energy equation is also modified to reflect the coupling heat transfer process. The temperature and pressure distributions are gained, the influences of convection and stratification are analyzed, flow in the air channel outside the containment and the couple heat transfer between the fluids in and outside the containment are simulated.
The local momentum flux M and the volume flux Qbj are: 2
4
M ¼ km B3 z3 1
(2)
5
Qbj ¼ km B3 z3
(3)
Here, z is the height, and constants km and km are approximately 0.35 and 0.15. The characteristic plume dimension dbj and local 0 are given by following equations volumetric entrainment Qbj respectively:
dbj ¼
Qbj 1
M2
¼
km 1
z
(4)
k2m
5km 1 2 B3 z3 3
0 Qbj ¼
(5)
Then the characteristic plume entrainment velocity can be calculated as:
ue ¼
0 Qbj
(6)
pdbj
Here, for thermal-hydraulic performance analysis in the containment after accident, Q0 is the volume flow rate and r0 is the density of the steam rising from the break, ra is the density of the fluid in the containment. 2.2. Model for fluid in the containment 2.2.1. One-dimensional model introduction The governing equations [15,16] for the stratified ambient fluid are:
AðzÞ
vG vF þ ¼ S vt vz
(7)
A(z) is the horizontal cross section area of the volume at elevation z,
2
2. One-dimensional model Since the physical parameters in the horizontal direction are similar [1,11], the one-dimensional model is used for the fluid in the large volume, the conservative equations are modified to be applied to circulation and coupling heat transfer simulation, and the steam condensation is also considered. Under the accidents of loss of coolant accident (LOCA) [6,9] and main steam line break (MSLB), the fluid injecting to the containment is steam, so the plume jet model is used to simulate the steam jet. 2.1. Steam jet model Since the stratification is easier to form under plume jet [15,16] and the jet becomes weaker and weaker after accident happens, in this paper attention is paid to long term cooling phase and buoyant plume jet model is used for steam jet, and the following empirical treatments for entrainment into turbulent buoyant plumes are used [15,16]: The specific buoyancy flux B is related to the densities of the injected fluid and ambient fluid, r0 and ra, and the volume flow rate Q0 at the source,
B ¼ g
ðra r0 Þ
ra
Q0
(1)
2
r
3
6 0 7 7 G ¼ 6 4 ri 5
rcj
3 rQsf 6 7 6 7 P 6 7 6 7 vT 6 F ¼ 6 riQ Ak sf 7 7 sf 6 vz 7 6 7 4 vcj 5 rcj Qsf rAD vz
2
3
rQ 0 6 rg 7 7 s ¼ 6 4 riQ 0 5 rcj Q 0
(8)
The subscripts sf represents ambient fluid and j presents constituents respectively.
2.2.2. One-dimensional model for accident analysis After accident such as LOCA, MSLB etc, steam injected to the containment should be condensed and then be collected in the containment sump, and the air will be heated by sensible and latent heat. We develop the momentum and energy equations based on Formula (8) to simulate the thermal-hydraulic performance in the containment. In Equation (8), the momentum equation is
vP ¼ rg vZ
(9)
Since the top of the containment is not free surface and circulation exists, the pressure difference is the motivation of the circulation which should be considered in the model, and the
Y. Yu et al. / Applied Thermal Engineering 70 (2014) 25e32
viscous force can be ignored, so Equation (9) should be modified as:
r
vusf vu þ usf sf vt vz
vP ¼ rg vz
(10)
Here, usf is the velocity of the fluid in the containment,
Qsf ðzÞ ¼ AðzÞ$usf ðzÞ
kG RTpBM L DV P
(18)
Tfilm þ Tbulk 2
(19)
pAG pAi PpAi ln Pp AG
(20)
Sh ¼ Here,
T ¼ (11)
Thus, the circulation can be simulated in the model. The heat transfer between the fluid in the containment and the internal surface should be added to the energy equation which is modified as:
pBM ¼
vTsf vri v riQsf Ak vz 0 þ ¼ riQ þ ql AðzÞ vt vz
2.5. Model assumption
(12)
So in this paper, the one-dimensional model developed for accident analysis is
vr vrQsf 0 ¼ rQ AðzÞ þ vt vz vu vu vP r þu ¼ rg vt vz vz vT v riQsf Ak sf 0 vri vz þ ¼ riQ þ ql AðzÞ vt vz vcj v rcj Qsf rAD vrcj 0 vz þ ¼ rcj Q ; AðzÞ vt vz
3. Small-scale system analysis 3.1. System description
j ¼ 1; 2
2.3. Model for the air outside containment The performance of the fluid in the air channel outside the containment is also described by one-dimensional model and the nodes are according with those in the containment, the governing equations are:
rout
vuout vuout vPout þ uout ¼ rout g vt vz vz
Aout
0 vrout iout vðrout iout uout Þ ¼ ql þ vz vt
(14)
(15)
(16)
2.4. Heat and mass transfer model In the containment, the steam is condensed when arriving at the surface of the containment, and cooling water is sprayed to the outside surface of the containment to enhance the heat transfer, some of which will be vaporize, the condensation and evaporation amount can be calculated based on the correlation [19]:
G ¼ kG MA ðpAi pAG Þ
Since steam injected to the containment will be collected in the containment sump after condensed, then be sent back to the reactor and be heated in the core, which is another process excluding in our analysis. So in our model, we suppose the condensed water will be entirely sent back to the core, that is, the total amount of steam injected to the containment is equal to the sum amount of condensed water and steam retaining in the containment.
(13)
In this paper, only one steam jet is considered and in the containment there are two constituents e air and steam, so subscript j only has two values: 1 presents air and 2 presents steam.
vrout vðrout uout Þ ¼ 0 þ vz vt
27
(17)
The mass transfer coefficient kG can be predicted using empirical correlation:
In order to verify the reliability of the new model, a small-scale system of AP1000 containment which has been simulated by COMMIX and whose result is verified to be reasonable [20] is used as an example. COMMIX [13,20] is a general-purpose, multidimensional computer code for thermal hydraulic analysis of single or multi component engineering systems, in order to apply it to the passive containment cooling system analysis, tracking models, pertinent heat and mass transfer models are developed and implemented to simulate the heat and mass transfer process at both sides of the steel vessel, and the results are validated with Westinghouse AP-600 PCCS small-scale test data [13]. Furthermore, we also do the simplified simulation by CFD which is widely used in thermal-hydraulic analysis. The small-scale system [10] is shown in Fig. 1, containing a steel vessel which consists of top head, bottom head and a cylinder. The long and short semi-axes of the heads are 1.5 m and 0.8 m respectively, and the height of the cylinder is 3 m. The air channel outside the steel is annular with the width of 0.2 m. The initial status in the steel is temperature of 25 C and pressure of 0.1 MPa. The steam with temperature of 140 C is injected into the vessel from the center of the bottom with the mass flow rate of 0.1 kg/s, at the same time the cooling water with temperature of 25 C is sprayed onto the surface of the steel with the mass flow rate of 0.5 kg/s. The cooling air flow into the channel outside the steel from the bottom, at the inlet the flow rate is 2 m/s and the temperature is 25 C, then the air is heated by the steel vessel surface and flow out through the outlet at the top whose radius is 0.65 m. 3.2. Results In order to validate our model, we compare our result with those of COMMIX and CFD. There are 298 computational cells in the COMMIX model [20], including horizontal and vertical partitions. Since the steel vessel is an axial symmetry structure, we use 2-D model in our simulation by CFD and the total number of computational cells is 11,576.
28
Y. Yu et al. / Applied Thermal Engineering 70 (2014) 25e32
Fig. 3. Comparison of temperature distributions.
Fig. 1. Small-scale system of AP1000 containment.
The one-dimensional model is used to simulate the thermalhydraulic performance of system in Fig. 1, the conservative Equation (13) describing the fluid performance in the containment and Equations (14)e(16) describing the fluid performance in the air channel can be solved by Finite Differential Method (FDM), in our analysis there are only 10 computational cells in the vertical directions. The empirical Formulas (1)e(6) are used to describe the steam jet and Formulas (19) and (20) are used to describe the heat and mass transfer process. For the cooling water sprayed onto the surface from the top, the simplified model is used: the evaporation capacity in each node is also calculated based on the correlation (17), if the total amount of the cooling water is evaporated, in the nodes bellowing there is no water film on the surface. The transient temperature distribution in the steel vessel is shown in Fig. 2, the ordinate represents temperature of fluid in steel
vessel, and the abscissa is the time after accident occurring, from the result it can be seen that the temperature distribution tends to be steady after 2000 s. Influenced by the steam jet source, the temperature of the bottom node is a little higher than of the middle node in a short period. Since we care for the long-term phase after accident, the result of COMMIX [10,20] is steady one, we do the steady analysis by CFD and use the result at 3600 s of our model as the steady one, then compare the results of the three codes. Curves of temperature variations with the height from three models are shown in Fig. 3, here the mean values of horizontal cross-sections are used for COMMIX [10,20] and CFD results, due to the temperature gradient is mainly in the vertical direction. In the figure the ordinate represents temperature of fluid in steel vessel, and the abscissa is the height of steel vessel. It can be seen that the temperature distributions of the three curves are almost in accordance, varies from about 60 C at the bottom to 110 C at the top with positive gradient in the vertical direction. Since COMMIX is a multi-dimensional code, the influence of source can be simulated more detailed, and we use mean values of the cross-sections to compare, the
0.25
0.2
usf(m/s)
0.15
0.1
0.05
0 0.5
Fig. 2. Transient temperature distributions in the steel vessel.
1
1.5
2
2.5 H (m)
3
3.5
Fig. 4. Fluid velocity distribution in the steel vessel.
4
4.5
Y. Yu et al. / Applied Thermal Engineering 70 (2014) 25e32
29
0.7
0.24 0.23
0.6
0.22
0.5 steam portion
P (MPa)
0.21 0.2 0.19 0.18
0.3 0.2
0.17 0.16 0.15 0.5
0.4
0.1 1
1.5
2
2.5 H (m)
3
3.5
4
4.5
0 0.5
1
1.5
2
2.5 H (m)
3
3.5
4
4.5
Fig. 5. Total pressure distribution in the steel vessel. Fig. 7. Steam portion distribution in the steel vessel.
temperature of COMMIX result is a little higher than of our result at the bottom. In CFD simulation we simply suppose the wall temperature is a constant value of 25 C since we care for the thermal stratification and circulation in the containment and there are cooling water sprayed to the steel surface to enhance the heat transfer between wall and atmosphere, so the whole temperature is a little lower than the results of the other two models. Furthermore since the wall temperature is constant, the heat transfer amount is more even along the height, and the temperature distribution curve is a little smoother. Fig. 4 shows the velocity of fluid including air and steam in the containment, it can be seen that the circulation establishes e the steam jet and air entrained into the jet upward, while air with steam diffused in the volume downward e though the thermal stratification exists. Fig. 5 shows the distribution of total pressure and Figs. 6 and 7 show the steam partial pressure and steam portion. The total pressure is a little higher at the top than at the bottom, which is the causation of the circulation. The steam partial pressure and steam portion is much higher at the top than at the bottom, since the steam density is lower than the air density, steam
is gathering at the top and then cooled at the surface of the steel vessel. 4. AP1000 containment system analysis 4.1. AP1000 containment system Passive containment cooling system [1,3] is innovatively used in AP1000 nuclear power plant [2] to improve the safety. The containment is a steel vessel, the long axes and short axes of top and bottom heads are 43 m and 13 m respectively, and the height of the vessel is about 73 m. Since the free volume is above the operating platform, we care for the volume in the top head and cylinder in the analysis as shown in Fig. 8. In this paper, MSLB accident is analyzed since it is one of the accidents challenge the containment pressure limitation. As
0.14
0.12
Psteam (MPa)
0.1
0.08
0.06
0.04
0.02
0 0.5
1
1.5
2
2.5 H (m)
3
3.5
4
Fig. 6. Steam partial pressure distribution in the steel vessel.
4.5
Fig. 8. AP1000 containment structure.
Y. Yu et al. / Applied Thermal Engineering 70 (2014) 25e32
described in Section 2.1, here we care for whether the circulation can establish or not in the long-term cooling phase, the temperature and mass flow of steam jet are 163 C and 43 kg/s respectively, and the cooling water is sprayed to the containment surface. Moreover, since the pressure in the containment is crucial to safety, we also analyze another typical mass flow of steam jet 218 kg/s under MSLB accident to validate whether the pressure in the containment can be kept at the safe level.
4.2. Results The temperature and fluid velocity distributions in the containment are shown in Figs. 9 and 10, total pressure, steam partial pressure and steam portion are shown in Figs. 11e13 respectively. From the results, it can be seen that the temperature stratification and circulation also exist synchronously, both the gradients of temperature and pressure in the vertical direction are positive, and the pressure gradient can afford the force of the circulation. It can be also seen that the steam is concentrated in the top of the containment, the steam partial pressure and the steam portion in the top head is higher than in the bottom volume. For another typical steam mass flow rate at 218 kg/s after MSLB accident, the pressure distribution direction is similar to curve in Fig. 11, and the peak value of the pressure is 0.40 MPa which is under the safety threshold of 0.5 MPa.
0.7
0.6
0.5
usf (m/s)
30
0.4
0.3
0.2
0.1
0
5
10
15
20
25
30 H (m)
35
40
45
50
55
Fig. 10. Fluid velocity distribution in the containment.
After an accident, the heat produced in the containment should be transferred to the steel vessel by natural circulation, while the circulation will be weakened or even disrupted if stable stratified layers exist due to the positive density gradient in the vertical direction and then the pressure will be building-up. So our attention is paid to the thermal stratification and circulation, the model is applied to simulate the distributions of temperature, pressure, concentration, velocity, etc., after a certain accident. Based on our analysis in this paper, after MSLB the circulation can be established even though the thermal stratification exists, so the pressure can be retained at the safe level.
Since the density of steam is lighter than density of air in the volume, so the steam jet will go into the top of the containment, and the characteristic diameter of the steam jet is much shorter than the diameter of the cylinder, the bulk of the steam will be condensed when arrive at the internal surface of the vessel in the top head. Latent heat transferring to air in the containment is the mainly heat source, so the air temperature in the top head is higher than in lower parts. Along with the temperature of air in the top increasing, the pressure increases simultaneously, the air mingled with steam will flow down, and the temperature and pressure in the whole containment will increase later. WGOTHIC [1] is a special computer code developed to simulate containment performance in AP1000 after accidents based on lumped parameter method. Compared with the results of WGOTHIC, the average temperature in the containment is lower about 10%, and the average pressure in the containment is lower about 7% as shown in Figs. 9 and 11, the reason can be analyzed as following:
Fig. 9. Temperature distribution in the containment.
Fig. 11. Pressure distribution in the containment.
5. Discuss
Y. Yu et al. / Applied Thermal Engineering 70 (2014) 25e32
31
0.11 0.1 0.09
Psteam (MPa)
0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
5
10
15
20
25
30 H (m)
35
40
45
50
55
Fig. 12. Steam partial pressure distribution in the containment.
Fig. 14. Heat transfer coefficient in the steel vessel (small-scale system).
Firstly, the steam in the vessel will concentrate in the top and the cooling capacity of the top head is higher than lower parts, the steam concentrating in the top will strengthen the mass transfer, and air with higher temperature in the top will enhance the heat transfer. Furthermore, heat transfer coefficient increases along with the height increasing, which can be seen in Fig. 14 (for small-scale system) and Fig. 15 (for containment). From the results of smallscale system and containment, Gr/Re2 >> 1, the heat and mass transfer depend on the natural circulation, Nusselt number mainly depends on Grashof number which is:
Gr ¼
gðr rs ÞL3
rn2
Here, rs is the density of the steam jet and r is the density of the fluid in the volume, L is the characteristic length which is height here, and n is the kinematic viscosity. The variation of density difference is much smaller than the height, so the Nusselt number increases along with the height, which is benefit for the heat and
mass transfer for AP1000 containment design since steam and hot air will concentrate in the upper part. 6. Conclusions Since the gradients of the physical parameters in the horizontal direction is much smaller than in the vertical direction, the onedimensional model is suitable for the thermal-hydraulic performance analysis in the containment, that is, the flow along the diameter direction can be ignored, which has little influence on the heat and mass transfer process in the containment. The temperature stratification exists in the containment, meanwhile the circulation also exists at the same time because of the density difference of the air and the steam, the positive pressure gradient of the fluid in the containment affords the force of the circulation, so the pressure in the containment can be kept at the safe level. For AP1000 containment design, since the cooling surface of the top head is much higher than lower parts, the steam and air with higher temperature and higher Nusselt number concentrating at the top will enhance the heat and mass transfer.
0.8
0.7
Steam Portion
0.6
0.5
0.4
0.3
0.2
0.1
5
10
15
20
25
30 H (m)
35
40
45
Fig. 13. Steam portion distribution in the containment.
50
55
Fig. 15. Heat transfer coefficient in the containment.
32
Y. Yu et al. / Applied Thermal Engineering 70 (2014) 25e32
Acknowledgements This work is supported by “The National Natural Science Foundation of China (51206042), (91326108)” and “the Fundamental Research Funds for the Central Universities (12ZX05)”. References [1] J. Woodcock, T.S. Andreychek, L. Conway, et al., WGOTHIC Application to AP600 and AP1000 (WCAP-15862, Class3), Westinghouse Electric Company LLC, PA, 2004. [2] T.L. Schulz, Westinghouse AP1000 advanced passive plant, Nucl. Eng. Des. 236 (2006) 1547e1557. [3] Y.D. Hwang, B.D. Chung, B.H. Cho, et al., PCCS analysis model for the passively cooled steel containment, J. Korean Nucl. Soc. 30 (1998) 26e39. [4] J. Oh, G.W. Michael, Methods for comparative assessment of active and passive safety systems with respect to reliability, uncertainty, economy, and flexibility, in: Proceedings of the 9th Probabilistic Assessment Analysis and Management, Hongkong, China, 2008. [5] Y.S. Kim, Y.J. Youn, Experimental study of turbulent jet induced by steam jet condensation through a hole in a water tank, Int. Commun. Heat Mass Transfer 35 (2008) 21e29. [6] M.S. Shin, H.S. Kim, D.S. Jang, et al., Numerical and experimental study on the design of a stratified thermal storage system, Appl. Therm. Eng. 24 (2007) 17e 27. [7] F.L. Niu, H.H. Zhao, P.F. Peterson, et al., Investigation of mixed convection in a large rectangular enclosure, Nucl. Eng. Des. 237 (2007) 1025e1032. [8] F.C. Rahim, M. Rahgoshay, S.K. Mousavian, A study of large break LOCA in the AP1000 reactor containment, Prog. Nucl. Energy 54 (2012) 132e137. [9] F.C. Rahim, P. Yousefi, E. Aliakbari, Simulation of the AP1000 reactor containment pressurization during loss of coolant accident, Prog. Nucl. Energy 60 (2012) 129e134. [10] R.P. Ofstun, J.H. Scobel, Westinghouse Containment Analysis Methodology (WCAP-16608-NP, class3), Westinghouse Electric Company LLC, PA, 2006. [11] B.S. Jia, J.Y. Yu, J.Y. Shi, AC600 passive containment cooling system performance research, in: Proceedings of the 5th International Topical Meeting on Nuclear Thermal-Hydraulics, Operations and Safety, Beijing, China, 1997. [12] Jimmy F.C. Chang, T.H. Chien, J.M. Ding, et al., COMMIX analysis of AP-600 passive containment cooling system, in: Proceedings of the 20th Water Reactor Safety Meeting, Bethesda, Maryland, 1992. [13] J.G. Sun, T.H. Chien, J. Ding, et al., Validation of COMMIX with Westinghouse AP-600 PCCS test data, in: Proceedings of the 21st Water Reactor Safety Information Meeting, Bethesda, MD, 1993. [14] Y.Q. Hu, Y. Yu, F.L. Niu, et al., Analysis of the thermal hydraulic phenomena caused by steam jets in AP1000 containment, in: Proceedings of the 21st International Conference of Nuclear Engineering, Chengdu, China, 2013. [15] P.F. Peterson, Scaling and analysis of mixing in large stratified volumes, Int. J. Heat Mass Transfer 37 (1994) 97e106. [16] P.F. Peterson, V.E. Schrock, R. Greif, Scaling for integral simulation of mixing in large, stratified volumes, Nucl. Eng. Des. 186 (1998) 213e224. [17] H.H. Zhao, P.F. Peterson, One-dimensional analysis of thermal stratification in the AHTR coolant pool, Nucl. Eng. Technol. 41 (2009) 953e968. [18] H.H. Zhao, P.F. Peterson, An overview of modeling methods for thermal mixing and stratification in large enclosures for reactor safety analysis, in:
Proceedings of the 8th International Topical Meeting on Nuclear ThermalHydraulic, Operation and Safety, Shanghai, China, 2010. [19] A.L. Hines, Mass Transfer and Applications, Englewood Clifs: Prentice-Hall, Inc., 1985. [20] J.G. Sun, W.T. Sha, Y.S. Chen, Development of Liquid-film Tracking Models for Analysis of AP-600 Passive Containment Cooling System, 1993.
Nomenclature
Symbols dbj: characteristic plume dimension, m g: acceleration of gravity, m/s2 i: enthalpy of fluid in the containment, J/kg iout: enthalpy of the air in the air tunnel, J/kg k: thermal conductivity, W/(m K) kG: mass transfer coefficient, mol/(m2 s MPa) usf: velocity of the fluid in the containment, m/s ue: characteristic plume entrainment velocity, m/s uout: velocity of the air in the air tunnel, m/s pAi: partial pressure of gas A at the interface, MPa pAG: partial pressure of gas A at the bulk gas mixture, MPa pBM: log mean partial pressure, MPa ql: heat flux in the unit length, W/m A: cross section area, m2 B: specific buoyancy flux, m4/s3 D: mass diffusion coefficient, m2/s DV: diffusion coefficient, m2/s G: condensing or evaporating mass flux, kg/(m2 s) Gr: Grashof number L: characteristic length, m M: momentum flux, kg/s2 MA: molecular weight of gas A, kg/mol Nu: Nusselt number P: total pressure, MPa Q0: volume flow rate at the source, m3/s Qbj: volume flow rate of the jet, m3/s Qsf: volume flow rate in the containment, m3/s Q0 : volumetric entrainment, m2/s R: universal gas constant, (m3 MPa)/(mol K) Re: Reynolds number Sh: Sherwood number T: boundary layer temperature, K Tbulk: temperature of fluid in the volume, K Tfilm: film temperature, K Tsf: temperature in the containment, K r: density of the ambient fluid, kg/m3 r0: density of the injected fluid at the source, kg/m3 ra: density of the ambient fluid at the source, kg/m3 rout: density of the air in the air tunnel, kg/m3 rs: density of the injected fluid, kg/m3 c: mass fraction n: kinematic viscosity, m2/s
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
One-dimensional model for containment in AP1000 nuclear power plant based on thermal stratification Yu Yu, Fenglei Niu*, Shengfei Wang, Yingqiu Hu School of Nuclear Science and Technology, North China Electric Power University, No.2 Beinong Road, Huilongguan, Changping District, Beijing 102206, China
h i g h l i g h t s 1-D model for AP1000 containment has been established. Thermal stratification and circulation exist simultaneously in the containment. Pressure positive gradient affords the force of the circulation. Steam concentration at the top enhances the heat and mass transfer. The pressure in the containment can be kept at the safe level.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 October 2013 Accepted 27 April 2014 Available online 9 May 2014
Passive containment cooling system operating based on natural circulation is innovatively used in AP1000 nuclear power plant to improve safety. However, the overall heat and mass transfer will decrease and the heat transfer through the containment shell will be slow due to stable stratification which will occur if the forced convection mixing is not sufficiently strong enough to disrupt the stable fluid layers, so it is important for system design and accident analysis to evaluate whether the circulation can establish or not in the containment accurately and efficiently. Many researches indicate that the gradients of such parameters in horizontal direction are so small that can be ignored, in this paper, onedimensional model is developed for AP1000 containment based on thermal stratification theory, natural circulations in and outside the containment are both included. Based on the results, the thermal stratification and circulation exist simultaneously, the pressure in the containment can be kept at the safe level, and positive pressure gradient in the vertical direction affords the force of circulation. The steam and air with higher temperature concentrating at the top will strengthen the heat and mass transfer because the cooling capacity of top head is higher than of other parts. Ó 2014 Elsevier Ltd. All rights reserved.
Keywords: Passive containment cooling system Natural circulation Thermal stratification One-dimensional model
1. Introduction Passive safety systems [1,2] are widely used in new generation reactor design to improve nuclear power plant safety especially under external disasters such as earthquake, tsunami, since they are independent of emergency AC power or offset power [2]. Passive containment cooling system [1] is one of the important safety systems in AP1000 nuclear power plant [2] by which the heat produced in the containment can be transferred to the atmosphere based on natural circulation [3]. However, stratification [1,5e7] will
* Corresponding author. E-mail addresses: [email protected] (Y. Yu), [email protected] (F. Niu), [email protected] (S. Wang). http://dx.doi.org/10.1016/j.applthermaleng.2014.04.070 1359-4311/Ó 2014 Elsevier Ltd. All rights reserved.
occur if the density of the layers decreases in the upward vertical direction and if the forced convection mixing is not strong enough to disrupt the stable fluid layers. As a result of the stable stratification, the overall heat and mass transfer decrease and the heat transfer through the containment shell is slow, which may induce adverse effect on the containment and should be considered in the system thermal-hydraulic analysis. So it is important for system design and accident analysis to simulate thermal-hydraulic performance, to recognize influence of the factors such as temperature distribution, fluid velocity on system operation [4], to predict distributions of pressure, temperature, and steam concentration, and to evaluate whether the circulation can establish in the containment accurately and efficiently. Many thermal-hydraulic analyses have been done mainly in two ways: lumped parameter [8e10] model and multi-dimensional
26
Y. Yu et al. / Applied Thermal Engineering 70 (2014) 25e32
model [11e14]. For AP1000 containment design and system reliability evaluation it is important to describe distributions of the physical parameters such as temperature, pressure and density, at the same time high calculation speed is needed for accident analysis especially under abnormal condition. Since the gradients of physical parameters in the horizontal direction are so small [11,14] that can be ignored, in this paper it is proposed that onedimensional model [15e19] based on thermal stratification theory can be used to simulate the thermal-hydraulic performance in the containment after being modified according to the accident conditions: Firstly, the original model [15,16] is mainly developed for a large volume with a free outlet always at the upper part, so the thermal stratification can be simulated yet the circulation is too weak to be considered. While for the volume with outlet at the bottom, the fluid is supposed to flow down along the wall when arriving at the top and wall jet is used to simulate the down coming fluid [19]. Nevertheless, for fluid including air and steam in the containment after accident, we consider that the hot fluid at the top will come down through the whole volume induced by the pressure difference and be cooled by the cold wall, so both of the stratification and circulation should be simulated. Secondly, the original model [15,16] is developed mainly for the volume having cold wall with constant temperature or constant heat flux density, while for the accident analysis in the containment, the heat transfer process is more complicated, nether the wall temperature nor the heat flux density is constant, they are determined by the heat transfer processes both inside and outside the containment. In this paper, a new model including stratification and circulation is developed based on one-dimensional model, in which the modified momentum equation is used to replace the wall jet model and the energy equation is also modified to reflect the coupling heat transfer process. The temperature and pressure distributions are gained, the influences of convection and stratification are analyzed, flow in the air channel outside the containment and the couple heat transfer between the fluids in and outside the containment are simulated.
The local momentum flux M and the volume flux Qbj are: 2
4
M ¼ km B3 z3 1
(2)
5
Qbj ¼ km B3 z3
(3)
Here, z is the height, and constants km and km are approximately 0.35 and 0.15. The characteristic plume dimension dbj and local 0 are given by following equations volumetric entrainment Qbj respectively:
dbj ¼
Qbj 1
M2
¼
km 1
z
(4)
k2m
5km 1 2 B3 z3 3
0 Qbj ¼
(5)
Then the characteristic plume entrainment velocity can be calculated as:
ue ¼
0 Qbj
(6)
pdbj
Here, for thermal-hydraulic performance analysis in the containment after accident, Q0 is the volume flow rate and r0 is the density of the steam rising from the break, ra is the density of the fluid in the containment. 2.2. Model for fluid in the containment 2.2.1. One-dimensional model introduction The governing equations [15,16] for the stratified ambient fluid are:
AðzÞ
vG vF þ ¼ S vt vz
(7)
A(z) is the horizontal cross section area of the volume at elevation z,
2
2. One-dimensional model Since the physical parameters in the horizontal direction are similar [1,11], the one-dimensional model is used for the fluid in the large volume, the conservative equations are modified to be applied to circulation and coupling heat transfer simulation, and the steam condensation is also considered. Under the accidents of loss of coolant accident (LOCA) [6,9] and main steam line break (MSLB), the fluid injecting to the containment is steam, so the plume jet model is used to simulate the steam jet. 2.1. Steam jet model Since the stratification is easier to form under plume jet [15,16] and the jet becomes weaker and weaker after accident happens, in this paper attention is paid to long term cooling phase and buoyant plume jet model is used for steam jet, and the following empirical treatments for entrainment into turbulent buoyant plumes are used [15,16]: The specific buoyancy flux B is related to the densities of the injected fluid and ambient fluid, r0 and ra, and the volume flow rate Q0 at the source,
B ¼ g
ðra r0 Þ
ra
Q0
(1)
2
r
3
6 0 7 7 G ¼ 6 4 ri 5
rcj
3 rQsf 6 7 6 7 P 6 7 6 7 vT 6 F ¼ 6 riQ Ak sf 7 7 sf 6 vz 7 6 7 4 vcj 5 rcj Qsf rAD vz
2
3
rQ 0 6 rg 7 7 s ¼ 6 4 riQ 0 5 rcj Q 0
(8)
The subscripts sf represents ambient fluid and j presents constituents respectively.
2.2.2. One-dimensional model for accident analysis After accident such as LOCA, MSLB etc, steam injected to the containment should be condensed and then be collected in the containment sump, and the air will be heated by sensible and latent heat. We develop the momentum and energy equations based on Formula (8) to simulate the thermal-hydraulic performance in the containment. In Equation (8), the momentum equation is
vP ¼ rg vZ
(9)
Since the top of the containment is not free surface and circulation exists, the pressure difference is the motivation of the circulation which should be considered in the model, and the
Y. Yu et al. / Applied Thermal Engineering 70 (2014) 25e32
viscous force can be ignored, so Equation (9) should be modified as:
r
vusf vu þ usf sf vt vz
vP ¼ rg vz
(10)
Here, usf is the velocity of the fluid in the containment,
Qsf ðzÞ ¼ AðzÞ$usf ðzÞ
kG RTpBM L DV P
(18)
Tfilm þ Tbulk 2
(19)
pAG pAi PpAi ln Pp AG
(20)
Sh ¼ Here,
T ¼ (11)
Thus, the circulation can be simulated in the model. The heat transfer between the fluid in the containment and the internal surface should be added to the energy equation which is modified as:
pBM ¼
vTsf vri v riQsf Ak vz 0 þ ¼ riQ þ ql AðzÞ vt vz
2.5. Model assumption
(12)
So in this paper, the one-dimensional model developed for accident analysis is
vr vrQsf 0 ¼ rQ AðzÞ þ vt vz vu vu vP r þu ¼ rg vt vz vz vT v riQsf Ak sf 0 vri vz þ ¼ riQ þ ql AðzÞ vt vz vcj v rcj Qsf rAD vrcj 0 vz þ ¼ rcj Q ; AðzÞ vt vz
3. Small-scale system analysis 3.1. System description
j ¼ 1; 2
2.3. Model for the air outside containment The performance of the fluid in the air channel outside the containment is also described by one-dimensional model and the nodes are according with those in the containment, the governing equations are:
rout
vuout vuout vPout þ uout ¼ rout g vt vz vz
Aout
0 vrout iout vðrout iout uout Þ ¼ ql þ vz vt
(14)
(15)
(16)
2.4. Heat and mass transfer model In the containment, the steam is condensed when arriving at the surface of the containment, and cooling water is sprayed to the outside surface of the containment to enhance the heat transfer, some of which will be vaporize, the condensation and evaporation amount can be calculated based on the correlation [19]:
G ¼ kG MA ðpAi pAG Þ
Since steam injected to the containment will be collected in the containment sump after condensed, then be sent back to the reactor and be heated in the core, which is another process excluding in our analysis. So in our model, we suppose the condensed water will be entirely sent back to the core, that is, the total amount of steam injected to the containment is equal to the sum amount of condensed water and steam retaining in the containment.
(13)
In this paper, only one steam jet is considered and in the containment there are two constituents e air and steam, so subscript j only has two values: 1 presents air and 2 presents steam.
vrout vðrout uout Þ ¼ 0 þ vz vt
27
(17)
The mass transfer coefficient kG can be predicted using empirical correlation:
In order to verify the reliability of the new model, a small-scale system of AP1000 containment which has been simulated by COMMIX and whose result is verified to be reasonable [20] is used as an example. COMMIX [13,20] is a general-purpose, multidimensional computer code for thermal hydraulic analysis of single or multi component engineering systems, in order to apply it to the passive containment cooling system analysis, tracking models, pertinent heat and mass transfer models are developed and implemented to simulate the heat and mass transfer process at both sides of the steel vessel, and the results are validated with Westinghouse AP-600 PCCS small-scale test data [13]. Furthermore, we also do the simplified simulation by CFD which is widely used in thermal-hydraulic analysis. The small-scale system [10] is shown in Fig. 1, containing a steel vessel which consists of top head, bottom head and a cylinder. The long and short semi-axes of the heads are 1.5 m and 0.8 m respectively, and the height of the cylinder is 3 m. The air channel outside the steel is annular with the width of 0.2 m. The initial status in the steel is temperature of 25 C and pressure of 0.1 MPa. The steam with temperature of 140 C is injected into the vessel from the center of the bottom with the mass flow rate of 0.1 kg/s, at the same time the cooling water with temperature of 25 C is sprayed onto the surface of the steel with the mass flow rate of 0.5 kg/s. The cooling air flow into the channel outside the steel from the bottom, at the inlet the flow rate is 2 m/s and the temperature is 25 C, then the air is heated by the steel vessel surface and flow out through the outlet at the top whose radius is 0.65 m. 3.2. Results In order to validate our model, we compare our result with those of COMMIX and CFD. There are 298 computational cells in the COMMIX model [20], including horizontal and vertical partitions. Since the steel vessel is an axial symmetry structure, we use 2-D model in our simulation by CFD and the total number of computational cells is 11,576.
28
Y. Yu et al. / Applied Thermal Engineering 70 (2014) 25e32
Fig. 3. Comparison of temperature distributions.
Fig. 1. Small-scale system of AP1000 containment.
The one-dimensional model is used to simulate the thermalhydraulic performance of system in Fig. 1, the conservative Equation (13) describing the fluid performance in the containment and Equations (14)e(16) describing the fluid performance in the air channel can be solved by Finite Differential Method (FDM), in our analysis there are only 10 computational cells in the vertical directions. The empirical Formulas (1)e(6) are used to describe the steam jet and Formulas (19) and (20) are used to describe the heat and mass transfer process. For the cooling water sprayed onto the surface from the top, the simplified model is used: the evaporation capacity in each node is also calculated based on the correlation (17), if the total amount of the cooling water is evaporated, in the nodes bellowing there is no water film on the surface. The transient temperature distribution in the steel vessel is shown in Fig. 2, the ordinate represents temperature of fluid in steel
vessel, and the abscissa is the time after accident occurring, from the result it can be seen that the temperature distribution tends to be steady after 2000 s. Influenced by the steam jet source, the temperature of the bottom node is a little higher than of the middle node in a short period. Since we care for the long-term phase after accident, the result of COMMIX [10,20] is steady one, we do the steady analysis by CFD and use the result at 3600 s of our model as the steady one, then compare the results of the three codes. Curves of temperature variations with the height from three models are shown in Fig. 3, here the mean values of horizontal cross-sections are used for COMMIX [10,20] and CFD results, due to the temperature gradient is mainly in the vertical direction. In the figure the ordinate represents temperature of fluid in steel vessel, and the abscissa is the height of steel vessel. It can be seen that the temperature distributions of the three curves are almost in accordance, varies from about 60 C at the bottom to 110 C at the top with positive gradient in the vertical direction. Since COMMIX is a multi-dimensional code, the influence of source can be simulated more detailed, and we use mean values of the cross-sections to compare, the
0.25
0.2
usf(m/s)
0.15
0.1
0.05
0 0.5
Fig. 2. Transient temperature distributions in the steel vessel.
1
1.5
2
2.5 H (m)
3
3.5
Fig. 4. Fluid velocity distribution in the steel vessel.
4
4.5
Y. Yu et al. / Applied Thermal Engineering 70 (2014) 25e32
29
0.7
0.24 0.23
0.6
0.22
0.5 steam portion
P (MPa)
0.21 0.2 0.19 0.18
0.3 0.2
0.17 0.16 0.15 0.5
0.4
0.1 1
1.5
2
2.5 H (m)
3
3.5
4
4.5
0 0.5
1
1.5
2
2.5 H (m)
3
3.5
4
4.5
Fig. 5. Total pressure distribution in the steel vessel. Fig. 7. Steam portion distribution in the steel vessel.
temperature of COMMIX result is a little higher than of our result at the bottom. In CFD simulation we simply suppose the wall temperature is a constant value of 25 C since we care for the thermal stratification and circulation in the containment and there are cooling water sprayed to the steel surface to enhance the heat transfer between wall and atmosphere, so the whole temperature is a little lower than the results of the other two models. Furthermore since the wall temperature is constant, the heat transfer amount is more even along the height, and the temperature distribution curve is a little smoother. Fig. 4 shows the velocity of fluid including air and steam in the containment, it can be seen that the circulation establishes e the steam jet and air entrained into the jet upward, while air with steam diffused in the volume downward e though the thermal stratification exists. Fig. 5 shows the distribution of total pressure and Figs. 6 and 7 show the steam partial pressure and steam portion. The total pressure is a little higher at the top than at the bottom, which is the causation of the circulation. The steam partial pressure and steam portion is much higher at the top than at the bottom, since the steam density is lower than the air density, steam
is gathering at the top and then cooled at the surface of the steel vessel. 4. AP1000 containment system analysis 4.1. AP1000 containment system Passive containment cooling system [1,3] is innovatively used in AP1000 nuclear power plant [2] to improve the safety. The containment is a steel vessel, the long axes and short axes of top and bottom heads are 43 m and 13 m respectively, and the height of the vessel is about 73 m. Since the free volume is above the operating platform, we care for the volume in the top head and cylinder in the analysis as shown in Fig. 8. In this paper, MSLB accident is analyzed since it is one of the accidents challenge the containment pressure limitation. As
0.14
0.12
Psteam (MPa)
0.1
0.08
0.06
0.04
0.02
0 0.5
1
1.5
2
2.5 H (m)
3
3.5
4
Fig. 6. Steam partial pressure distribution in the steel vessel.
4.5
Fig. 8. AP1000 containment structure.
Y. Yu et al. / Applied Thermal Engineering 70 (2014) 25e32
described in Section 2.1, here we care for whether the circulation can establish or not in the long-term cooling phase, the temperature and mass flow of steam jet are 163 C and 43 kg/s respectively, and the cooling water is sprayed to the containment surface. Moreover, since the pressure in the containment is crucial to safety, we also analyze another typical mass flow of steam jet 218 kg/s under MSLB accident to validate whether the pressure in the containment can be kept at the safe level.
4.2. Results The temperature and fluid velocity distributions in the containment are shown in Figs. 9 and 10, total pressure, steam partial pressure and steam portion are shown in Figs. 11e13 respectively. From the results, it can be seen that the temperature stratification and circulation also exist synchronously, both the gradients of temperature and pressure in the vertical direction are positive, and the pressure gradient can afford the force of the circulation. It can be also seen that the steam is concentrated in the top of the containment, the steam partial pressure and the steam portion in the top head is higher than in the bottom volume. For another typical steam mass flow rate at 218 kg/s after MSLB accident, the pressure distribution direction is similar to curve in Fig. 11, and the peak value of the pressure is 0.40 MPa which is under the safety threshold of 0.5 MPa.
0.7
0.6
0.5
usf (m/s)
30
0.4
0.3
0.2
0.1
0
5
10
15
20
25
30 H (m)
35
40
45
50
55
Fig. 10. Fluid velocity distribution in the containment.
After an accident, the heat produced in the containment should be transferred to the steel vessel by natural circulation, while the circulation will be weakened or even disrupted if stable stratified layers exist due to the positive density gradient in the vertical direction and then the pressure will be building-up. So our attention is paid to the thermal stratification and circulation, the model is applied to simulate the distributions of temperature, pressure, concentration, velocity, etc., after a certain accident. Based on our analysis in this paper, after MSLB the circulation can be established even though the thermal stratification exists, so the pressure can be retained at the safe level.
Since the density of steam is lighter than density of air in the volume, so the steam jet will go into the top of the containment, and the characteristic diameter of the steam jet is much shorter than the diameter of the cylinder, the bulk of the steam will be condensed when arrive at the internal surface of the vessel in the top head. Latent heat transferring to air in the containment is the mainly heat source, so the air temperature in the top head is higher than in lower parts. Along with the temperature of air in the top increasing, the pressure increases simultaneously, the air mingled with steam will flow down, and the temperature and pressure in the whole containment will increase later. WGOTHIC [1] is a special computer code developed to simulate containment performance in AP1000 after accidents based on lumped parameter method. Compared with the results of WGOTHIC, the average temperature in the containment is lower about 10%, and the average pressure in the containment is lower about 7% as shown in Figs. 9 and 11, the reason can be analyzed as following:
Fig. 9. Temperature distribution in the containment.
Fig. 11. Pressure distribution in the containment.
5. Discuss
Y. Yu et al. / Applied Thermal Engineering 70 (2014) 25e32
31
0.11 0.1 0.09
Psteam (MPa)
0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01
5
10
15
20
25
30 H (m)
35
40
45
50
55
Fig. 12. Steam partial pressure distribution in the containment.
Fig. 14. Heat transfer coefficient in the steel vessel (small-scale system).
Firstly, the steam in the vessel will concentrate in the top and the cooling capacity of the top head is higher than lower parts, the steam concentrating in the top will strengthen the mass transfer, and air with higher temperature in the top will enhance the heat transfer. Furthermore, heat transfer coefficient increases along with the height increasing, which can be seen in Fig. 14 (for small-scale system) and Fig. 15 (for containment). From the results of smallscale system and containment, Gr/Re2 >> 1, the heat and mass transfer depend on the natural circulation, Nusselt number mainly depends on Grashof number which is:
Gr ¼
gðr rs ÞL3
rn2
Here, rs is the density of the steam jet and r is the density of the fluid in the volume, L is the characteristic length which is height here, and n is the kinematic viscosity. The variation of density difference is much smaller than the height, so the Nusselt number increases along with the height, which is benefit for the heat and
mass transfer for AP1000 containment design since steam and hot air will concentrate in the upper part. 6. Conclusions Since the gradients of the physical parameters in the horizontal direction is much smaller than in the vertical direction, the onedimensional model is suitable for the thermal-hydraulic performance analysis in the containment, that is, the flow along the diameter direction can be ignored, which has little influence on the heat and mass transfer process in the containment. The temperature stratification exists in the containment, meanwhile the circulation also exists at the same time because of the density difference of the air and the steam, the positive pressure gradient of the fluid in the containment affords the force of the circulation, so the pressure in the containment can be kept at the safe level. For AP1000 containment design, since the cooling surface of the top head is much higher than lower parts, the steam and air with higher temperature and higher Nusselt number concentrating at the top will enhance the heat and mass transfer.
0.8
0.7
Steam Portion
0.6
0.5
0.4
0.3
0.2
0.1
5
10
15
20
25
30 H (m)
35
40
45
Fig. 13. Steam portion distribution in the containment.
50
55
Fig. 15. Heat transfer coefficient in the containment.
32
Y. Yu et al. / Applied Thermal Engineering 70 (2014) 25e32
Acknowledgements This work is supported by “The National Natural Science Foundation of China (51206042), (91326108)” and “the Fundamental Research Funds for the Central Universities (12ZX05)”. References [1] J. Woodcock, T.S. Andreychek, L. Conway, et al., WGOTHIC Application to AP600 and AP1000 (WCAP-15862, Class3), Westinghouse Electric Company LLC, PA, 2004. [2] T.L. Schulz, Westinghouse AP1000 advanced passive plant, Nucl. Eng. Des. 236 (2006) 1547e1557. [3] Y.D. Hwang, B.D. Chung, B.H. Cho, et al., PCCS analysis model for the passively cooled steel containment, J. Korean Nucl. Soc. 30 (1998) 26e39. [4] J. Oh, G.W. Michael, Methods for comparative assessment of active and passive safety systems with respect to reliability, uncertainty, economy, and flexibility, in: Proceedings of the 9th Probabilistic Assessment Analysis and Management, Hongkong, China, 2008. [5] Y.S. Kim, Y.J. Youn, Experimental study of turbulent jet induced by steam jet condensation through a hole in a water tank, Int. Commun. Heat Mass Transfer 35 (2008) 21e29. [6] M.S. Shin, H.S. Kim, D.S. Jang, et al., Numerical and experimental study on the design of a stratified thermal storage system, Appl. Therm. Eng. 24 (2007) 17e 27. [7] F.L. Niu, H.H. Zhao, P.F. Peterson, et al., Investigation of mixed convection in a large rectangular enclosure, Nucl. Eng. Des. 237 (2007) 1025e1032. [8] F.C. Rahim, M. Rahgoshay, S.K. Mousavian, A study of large break LOCA in the AP1000 reactor containment, Prog. Nucl. Energy 54 (2012) 132e137. [9] F.C. Rahim, P. Yousefi, E. Aliakbari, Simulation of the AP1000 reactor containment pressurization during loss of coolant accident, Prog. Nucl. Energy 60 (2012) 129e134. [10] R.P. Ofstun, J.H. Scobel, Westinghouse Containment Analysis Methodology (WCAP-16608-NP, class3), Westinghouse Electric Company LLC, PA, 2006. [11] B.S. Jia, J.Y. Yu, J.Y. Shi, AC600 passive containment cooling system performance research, in: Proceedings of the 5th International Topical Meeting on Nuclear Thermal-Hydraulics, Operations and Safety, Beijing, China, 1997. [12] Jimmy F.C. Chang, T.H. Chien, J.M. Ding, et al., COMMIX analysis of AP-600 passive containment cooling system, in: Proceedings of the 20th Water Reactor Safety Meeting, Bethesda, Maryland, 1992. [13] J.G. Sun, T.H. Chien, J. Ding, et al., Validation of COMMIX with Westinghouse AP-600 PCCS test data, in: Proceedings of the 21st Water Reactor Safety Information Meeting, Bethesda, MD, 1993. [14] Y.Q. Hu, Y. Yu, F.L. Niu, et al., Analysis of the thermal hydraulic phenomena caused by steam jets in AP1000 containment, in: Proceedings of the 21st International Conference of Nuclear Engineering, Chengdu, China, 2013. [15] P.F. Peterson, Scaling and analysis of mixing in large stratified volumes, Int. J. Heat Mass Transfer 37 (1994) 97e106. [16] P.F. Peterson, V.E. Schrock, R. Greif, Scaling for integral simulation of mixing in large, stratified volumes, Nucl. Eng. Des. 186 (1998) 213e224. [17] H.H. Zhao, P.F. Peterson, One-dimensional analysis of thermal stratification in the AHTR coolant pool, Nucl. Eng. Technol. 41 (2009) 953e968. [18] H.H. Zhao, P.F. Peterson, An overview of modeling methods for thermal mixing and stratification in large enclosures for reactor safety analysis, in:
Proceedings of the 8th International Topical Meeting on Nuclear ThermalHydraulic, Operation and Safety, Shanghai, China, 2010. [19] A.L. Hines, Mass Transfer and Applications, Englewood Clifs: Prentice-Hall, Inc., 1985. [20] J.G. Sun, W.T. Sha, Y.S. Chen, Development of Liquid-film Tracking Models for Analysis of AP-600 Passive Containment Cooling System, 1993.
Nomenclature
Symbols dbj: characteristic plume dimension, m g: acceleration of gravity, m/s2 i: enthalpy of fluid in the containment, J/kg iout: enthalpy of the air in the air tunnel, J/kg k: thermal conductivity, W/(m K) kG: mass transfer coefficient, mol/(m2 s MPa) usf: velocity of the fluid in the containment, m/s ue: characteristic plume entrainment velocity, m/s uout: velocity of the air in the air tunnel, m/s pAi: partial pressure of gas A at the interface, MPa pAG: partial pressure of gas A at the bulk gas mixture, MPa pBM: log mean partial pressure, MPa ql: heat flux in the unit length, W/m A: cross section area, m2 B: specific buoyancy flux, m4/s3 D: mass diffusion coefficient, m2/s DV: diffusion coefficient, m2/s G: condensing or evaporating mass flux, kg/(m2 s) Gr: Grashof number L: characteristic length, m M: momentum flux, kg/s2 MA: molecular weight of gas A, kg/mol Nu: Nusselt number P: total pressure, MPa Q0: volume flow rate at the source, m3/s Qbj: volume flow rate of the jet, m3/s Qsf: volume flow rate in the containment, m3/s Q0 : volumetric entrainment, m2/s R: universal gas constant, (m3 MPa)/(mol K) Re: Reynolds number Sh: Sherwood number T: boundary layer temperature, K Tbulk: temperature of fluid in the volume, K Tfilm: film temperature, K Tsf: temperature in the containment, K r: density of the ambient fluid, kg/m3 r0: density of the injected fluid at the source, kg/m3 ra: density of the ambient fluid at the source, kg/m3 rout: density of the air in the air tunnel, kg/m3 rs: density of the injected fluid, kg/m3 c: mass fraction n: kinematic viscosity, m2/s