Numerical Analysis on Pressure Propagation in Pressure Suppression System Due to Steam Bubble Collapse

Transcription

1 Journal of NUCLEAR SCIENCE and TECHNOLOGY, 21[4] pp.279~287 (April 1984). 279 Numerical Analysis on Pressure Propagation in Pressure Suppression System Due to Steam Bubble Collapse Motoaki UTAMURA, Energy Research Laboratory, Hitachi Ltd.* Kumiaki MORIYA, Nuclear Power Plant Engineering Department, Nuclear Power Generation Division, Hitachi Ltd.** Hiroto UOZUMI Nuclear Power Plant Design Department, Hitachi Works, Hitachi Ltd.** Received June 18, 1983 Revised January 5, 1984 In order to provide a numerical tool for the analysis of pressure response to steam bubble collapse in a BWR pressure suppression system, a computer program has been developed. The program is featured in the capability of handling three-dimensional fluid structure interaction based on finite element method. Numerical results are compared with experiments in both frequency and time domains. Natural frequencies of the fluid structure system are well predicted. Fluid damping associated with the bubble collapse and pressure wave travelling needs further investigation. KEYWORDS: numerical method, finite element method, fluid structure interaction, dynamic load, chugging, primary containment vessel, BWR type reactors, computer codes I. INTRODUCTION For the purpose of limiting the maximum pressure in the primary containment vessel (PCV) during a postulated loss-of-coolant accident (LOCA), a boiling water reactor is equipped with a pressure suppression system. In the LOCA, the reactor coolant is discharged into drywell, passes through vent pipes and condenses in the water of a suppression chamber. It has been pointed out that steam condensation in subcooled water occurs in unsteady manner, called condensation oscillation or chugging depending on its condensation manner. The former is considered to occur under the condition of high steam mass flux in the vent pipe and the latter at low steam mass flux(1)~(3). Both phenomena are of interest in view of hydrodynamic load against structure of the suppression chamber and its internals. The hydrodynamic load is brought by combined responses of vent acoustics, pool a- coustics and fluid structure interactions to the steam condensation. To clarify the characteristics of those mechanisms, a large scale Mark-II Containment Response Test (JAERI CRT) * Moriyama-cho, Hitachi-shi 316. ** Saiwai-cho, Hitachi-shi

2 280 J. Nucl. Sci. Technol., has been performed by Japan Atomic Energy Research Institute(4). The JAERI CRT results have shown many important features of condensation phenomena under full scale test conditions. One of the most important features is that the dominant frequency components of induced dynamic pressure in the water are corresponding to the vent acoustics(5). Fluid structure interactions and pool acoustics were also observed and numerical treatments of these responses were attempted(6)(7). Thus, the hydrodynamic load is strongly coupled with those system responses. This paper describes a numerical simulation of pressure wave propagation in the pressure suppression system due to the steam condensation. Numerical model is primarily based on the finite element method being capable of the analysis for three-dimensional fluid structure interactions. In this formulation, forcing function originated from the steam condensation is represented by two sided triangular pressure pulse. Numerical results have been compared with JAERI CRT data during chugging phase of steam condensation and system effects have been discussed. II. PHENOMENOLOGICAL BASIS When flowing steam in a pipe condenses in water at pipe exit, steam bubble is formed at the pipe exit at high steam flow rate. As steam mass flux is decreased, the steam bubble no longer maintains its shape due to the unbalance between the rate of steam supply to the bubble and the condensation rate of the steam in water. This stage of condensation called chugging is featured in intermittent steam bubble formation and pool water entry to the pipe as is illustrated in Fig. 1. Typical pressure transient inside bubble usually starts with a negative spike followed by sinusoidal oscillation with its decreasing magnitude. The negative spike comes from low pressure development due to steam condensation being accelerated by increase in heat transfer area during bubble growth. The rest of the pressure trace is a result from propagation and reflexion of pressure wave travelling along the pipe. Therefore, primary part of pressure history is the first nagative and positive peaks from the standpoint of triggering the chugging. Physical process during bubble formation and collapse is highly complex depending on local thermal-hydraulic conditions on bubble surface. As a result, measured pressure is not necessarily reproducible. Therefore, the triggering of the chugging is represented by a pair of triangular pulses in the present numerical model. Fig. 1 Description of chugging phenomenon Figure 2 illustrates mechanisms of pressure suppression system response to the chugging phenomenon. Liquid acceleration during the bubble collapsing phase produces pressure spike in pool being exciting various oscillation modes. These are excited in (1) steam column inside vent pipe, (2) pool water and (3) pool boundary structure. The modes of items (2) and (3) are strongly coupled together (fluid structure interaction) and should be 34

3 Vol. 21, No. 4 (Apr. 1984) 281 treated simultaneously. To the contrary, coupling of acoustic media (1) and (2) may be considered to be weak because of much difference in their bulk moduli and the pressure on the steam water interface is regarded as dominated by the vent acoustic phenomenon. Fig. 2 Mechanism of hydrodynamic loading II NUMERICAL MODEL To evaluate hydrodynamic load due to the chugging, a computer code has been developed. This code accounts for the foregoing mechanisms among the entire system effects. Numerical scheme of model is depicted in Fig. 3. The whole model is subdivided into (1) vent drywell model, (2) pool acoustic model and (3) structure vibration model respectively. Given pointwise source with dual triangle shape in time, the code calculates resultant pressure distribution along vent pipe and on pool boundary, and stress of structure. Details of each submodel is described below. 1. Vent Drywell Model Vent drywell model is illustrated in Fig. 4. The model involves drywell thermodynamics and vent acoustics. No bubble model is incorporated for simplicity. Fig. 3 Scheme of numerical model Under the condition that no heat transfer exists on the drywell wall, mass and energy conservation for steam in the drywell gives pressure gradient with respect to time as M(ah/ap where PD: Drywell pressure (Pa) p)v Vo M: Mass of steam in drywell (kg) V0: Drywell volume (m3) 0: Specific enthalpy of hinlet steam (stagnant) (J/kg) W0: Rate of inlet steam flow (kg/s) W1 : Rate of outlet steam flow (kg/s) v : Specific volume of steam (m3/kg) h: Specific enthalpy in drywell steam (J/kg) : Total derivative with respect to time. (1) Mass and momentum conservation for onedimensional vent flow gives Fig. 4 Modeling of vent acoustics 35

4 282 J. Nucl. Sci. Technol., Mass (2) Momentum (3) where pg: Steam density (kg/m3), u: Steam velocity (m/s) p: Vent pressure (Pa), x: Friction coefficient d: Diameter of vent tube (m), x: Coordinate (m). It should be noted that positive directions of velocity and coordinate are opposite. Let no, condensation or evaporation of steam occur during pressure propagation in the vent tube, mass conservation (Eq. (2)) may be rewritten as where cg: Velocity of sound in steam (=r(pp/ppg)s). Equations (3) and (4) are fundamental equations for vent model. Method of characteristics was incorporated to discretize partial differential equations. The feature of the method is capable of handling nonlinear momentum advection. Hence pressure wave propagation superposing steady state steam velocity can be analyzed. Resultant equations follow: Equation (5) shows two characteristic lines which mean backward wave path (C-) in the upper signature and forward (C+) in the lower signature respectively. Calculation procedure using method of characteristics is given in Fig. 4. Total time derivatives of pressure and steam velocity p and u respectively are defined along characteristic lines. Each of momentum equations holds on the corresponding characteristic line. Then, at the time-space grid point A where both lines intersect, each of the momentum equation holds simultaneously and both pressure and velocity at the point A can be determined provided their last values are known along vent length at the time t. Orifice model was applied to describe vent inlet flow condition. Rate of vent inlet flow may be expressed as which is related to steam velocity at vent inlet as Wi=u*A/v,,(8) where x: Constant, p*: Vent pressure at vent inlet (Pa) K : Specific heat ratio, A: Flow area (m2). Given drywell pressure at the current time step using Eq. (1), then unknowns W1, p* and u* can be determined by solving Eqs. (7), (8) and (6) on the C+ characteristic line simultaneously using iteration procedure. It is very difficult to track location and velocity of steam water interface after reentry of water to the vent pipe. However, as shown in Fig. 1, the interface is localized 6 (4) (5) (6) (7)

5 Vol. 21, No. 4 (Apr. 1984) 283 near the vent exit with 1.5 m above the exit at most compared with the total length of 13.6 m of the vent pipe. Furthermore, maximum velocity of the interface is estimated around 2 m/s which is much smaller than vent acoustic speed of around 500 m/s and thus the interface may be considered to be stationary. Therefore, in the present model null velocity condition was imposed at the vent exit as boundary conditions in the rest of calculation period after duration of pressure input. In this way, pressure time history at the vent exit can be obtained, which was transferred to the pool acoustics model to give one of its boundary conditions. Calculation procedure follows: stationary flow calculation precedes transient calculation given rate of inlet steam flow W0 which is equal to W1 in this part. Vent exit pressure was specified to be equal to static pressure equivalent to vent submergence. Resultant pressure and velocity distributions along vent pipe were set forth as initial values for following transient calculation. 2. Pool Structure Model Based on finite element method, a computer program has been developed to treat fluid structure interaction in three dimensions. Figure 5 shows modeling of fluid structure interaction. With the assumptions of ideal fluid and linear elasticity of structure, governing equations and boundary conditions follow: For fluid (9) where boundary conditions are given as, Fig. 5 Modeling of fluid structure interaction pp/pxini=-pfp2ui/pt2ni on F. S. I. boundary (Gi), (10) on rigid boundary (G6), (11) pp/pxini=0 P=P0 on source boundary (G2), (12) P=0 on free surface (G3), (13) For structure where boundary conditions are given as, PSp2i/pt2psij/pxi0,,(14) =Pni on sijnj F. S. I. boundary (Gt), (15) =0 on free boundary (G4), (16) Ui=0 on fixed boundary (GP5), (17) where G: Boundary,P: Fluid pressure Ui: i-th component of structure displacement c: Velocity of sound in fluid ni: i-th component of outward normal vector : Stress tensor, p: Density. sij 37 sijnj

6 284 J. Nucl. Sci. Technol., Summation convention was used in Eqs. (10), (11), (14), (15) and (16). Equation (10) shows equation of motion for fluid particle attached on the interface and Eq. (15) means that surface force can be equated to fluid pressure. Fluid pressure was specified as P0 on the inner surface, which is forcing function to fluid structure system. Actually, P0 is vent exit pressure calculated by the vent drywell model. Finite element method was adoptedto approximate Eqs. (9)~(17). Space concerned was subdivided into elements and spatial distribution of dependent variables (i.e. pressure and displacement) were interpolated element by element(8), that is P and Ui were expressed by where m: Number of nodes in an element Nj,Nj: Interpolation function related to j-th node (equal to unity at j-th node) Pj,Uji: Nodal parameters (unknowns). Substitution of Eq. (18) for Eqs. (9)~(17) and performing integration in space, matrix form of linear equations is obtained as (18) (19) where submatrices are given as, Mass matrices (20) (21) Stiffness matrices (22) (23) (24) (25) 38

7 Vol. 21, No. 4 (Apr. 1984) 285 where n: Poisson ratio, E: Young's modulus k : Shear correction factor (=1.2). Coupling matrix Load vector S=pfSGNTinTNjdG, nt=(n1, n2, n3), (26) (27) Submatrices and 1/PfST link fluid to structure. Subspace method was used to extract eigen-frequencies and modes and modal superposition technique was adopted to get solution in time domain.. IVRESULTS AND DISCUSSIONS The JAERI CRT full scale experiments(1) were analyzed to examine the numerical model. Outline of CRT test facility and its finite element representation is given in Fig. 6. With consideration of geometrical symmetry, half of the test facility was modeled using forty eight volumetric elements for fluid region and forty shell elements for structural region, which corresponds to the upper half pool and shell of section A-A in Fig. 6. Twenty noded isoparametric element and eight noded isoparametric element were used for fluid and shell respectively, i.e. m=20 for fluid and m=8 for shell structure. On the symmetry plane and surfaces Fig. 6 Modeling of test facility contacting concrete filler and bottom plate, rigid boundary condition was applied to fluid nodes. For nodes on the edge within symmetry plane, tangential shell displacements were fixed to zero. On the other edges of shell surface, all of nodal displacements were constrained to be zero. Vent pipes of CRT were not modeled in geometry but were represented by point source located at grid point corresponding to each vent exit. Pool acoustic velocity was chosen to be 1,500 m/s. Figure 7 shows calculated results on natural frequencies of the test facility compared with the experiment. The pool water level was set to normal value. Precision of calculation has been within 8% discrepancy at the highest, which shows that fluid structure interaction modeling is basically satisfactory. Following conditions were imposed on vent flow calculation. Drywell inlet flow rate W0 was chosen to be Fig. 7 Measured and predicted natural frequencies 39

8 286 J. Nucl. Sci. Technol., 2.8 kg/s. It is equivalent to vent mass flux of 10 kg/m2,s where typical chugging is supposed to occur. Vent acoustic velocity and other steam properties were evaluated at the saturated state corresponding to time averaged vent exit pressure observed in the experiment. In the case of TEST 3102, the pressure was 325 kpa and the acoustic velocity was evaluated to be 460 m/s. Friction coefficient was evaluated to be 0.01 based on Blasius formula and orifice constant was chosen to be typical value of 1.0. Drywell volume V0 was determined to be the one seventh of the actual volume. Vent pipe was subdivided into twenty five nodes and time step was 1 ms. A pressure time history at the vent exit is given in Fig. 8 in comparison with the experiment (TEST 3102). The pressure in the indicated part of time domain was simplified by a pair of triangle and was input to the model. Good agreement was obtained with respect to frequency and damping. Pool boundary pressure was predicted through fluid structure interaction model. The actual phenomenon at the vent exit varies from vent to vent and is very complex. However, in this part of calculation, the same shape of forcing function was applied to each of four vents assuming simultaneous collapsing of steam bubbles among vents for simplicity. Result is given in Fig. 9, which shows that the first negative peak was overestimated and higher mode of oscillation was significant more than in the experiment. Power spectral density functions of the pool boundary pressure are shown in Fig. 10. The first mode of vent acoustic was found to have most significant effect on the pool boundary load. The other acoustic media (fluid structure interaction and pool acoustic) were not likely to be excited in the experiment although calculation exhibited their effects. Damping mechanism in fluid, especially in the higher frequency, is not understood clearly at this stage but may come from desyncronization of bubble collapses among vents. Fig. 8 Pressure transient at vent exit Fig. 9 Pressure transient on pool boundary Fig. 10 PSD of pool boundary pressure V. CONCLUSIONS A computer program has been developed to simulate pressure wave propagation in a flexible container. 40

9 Vol. 21, No. 4 (Apr. 1984) 287 Numerical predictions were compared with JAERI CRT experiments, which showed (1) Frequency response of the fluid structure system was well predicted. (2) Transmission of vent acoustic effect on pool boundary was well simulated in time domain. (3) Both effects of fluid structure interaction and pool acoustics were overpredicted in their magnitude. (4) Fluid damping associated with steam bubble collapse and desyncronization effect in multi-vent system need precise modeling. ACKNOWLEDGMENT Authors would like to express their thanks to Mr. T. Horiuchi, senior engineer of Nuclear Generation Department, Hitachi Ltd. for his valuable discussions. They are also indebted to Dr. K. Taniguchi, general manager of Energy Research Laboratory, Hitachi Ltd., for his encouragements. R EFERENCES (1) KUKITA, Y., et al.: Results of pressure supression tests for Mark II containment, Preprint 1980 Annu. Mtg. of AESJ, A13, (1980). (2) NARIAI, H., AYA, I.: Pressure oscillations in vent tubes induced by steam condensation in pressure suppression containment, Proc. ANS/ASME 2nd Int. Topical Mtg. on Nuclear Reactor Thermal-Hydraulics, (1983). (3) AYA, I., NARIAI, H.: Frequency of condensation oscillation during steam condensation in water, Proc. 20th Symp. on Heat Transfer, pp.283~285 (1983). (4) KUKITA, Y., et al.: Statistical evaluation of steam condensation loads in pressure suppression pool (1), JAERI-M 9665, (1981). (5) HATTORI, T., et al.: Evaluation of LOCA hydrodynamic loads for BWR Mark-II containments caused by steam condensation, Ref. (2). (6) UTAMURA, M., et al.: Evaluation of chugging load coupled with system response, Trans. Am. Nucl. Soc., pp.694~695 (1982). (7) UTAMURA, M., et al.: Development of load evaluation methodology for steam condensation in consideration of system response, Preprint 1982 Annu. Mtg. of AESJ, A39, (1982). (8) ZIENKIEWICZ, O.C.: "The Finite Element Method", (3rd Ed.), (1977), McGraw Hill. 41