A THERMAL-HYDRAULICS MODEL NUMERICAL ANALYSIS OF BWR INSTABILITY PREDICTIONS USING THE DYNOBOSS CODE

Mauricio A. Pinheiro Rosa* and Michael Z. Podowski**

*Instituto de Estudos Avançados – IEAv Centro Técnico Aeroespacial – CTA 12931-970 Sâo José dos Campos, SP, Brasil

**Rensselaer Polytechnic Institute – RPI 12181 Troy, New York, USA

ABSTRACT

This paper is concerned with the analysis of the effects of various modeling and computational concepts applied to the main system components of a BWR loop on the accuracy of the predicted system response for natural circulation operating conditions near the marginal stability boundary. Both stable and unstable loop operating conditions have been considered in the analysis using the DYNOBOSS (DYnamics of NOnlinear BOiling SystemS) computer code. The overall loop response has shown to be more sensitive to the numerics of the reactor core and downcomer than the others components of the loop.

Keywords: boiling reactor, numerical methods, two-phase flow, reactor dynamics, reactor instabilities.

I. INTRODUCTION A brief description of the models and numerical methods in DYNOBOSS is given below, followed by a The dynamics of boiling water nuclear reactors sensitivity study of this code`s predictions. (BWR) is governed by several complex physical phenomena, such as coolant thermal-hydraulics, transfer in reactor fuel element and core neutronics. The II. OVERVIEW OF THE DYNOBOSS CODE nonlinear nature of most phenomena involved and the coupling effects between them make the overall predictions DYNOBOSS (DYnamics of NOnliner BOiling of BWR stability very sensitive to the modeling SystemS) is a computer code for the analysis of transient assumptions and numerical methodologies used in the and instabilities in boiling systems in general and in boiling analysis. water nuclear reactors (BWR), in particular. The code has The objective of this paper is to study the effects of two options: a system parallel boiling channels, and a selected modeling and computational concepts applied to a boiling loop model. a BWR recirculation loop, in particular the reactor core and The BWR model in DYNOBOSS accounts for all downcomer, on the accuracy of the predicted system major components of the BWR nuclear supply system response for natural circulation operating conditions near as shown in Figs. 1 and 2. These components are the reactor the marginal stability boundary. The DYNOBOSS core, upper plenum, steam riser, steam separator and steam model/computer code [1-4] has been used in this analysis. dome in the boiling (two-phase) region of the loop and the An important feature of DYNOBOSS is the inclusion of downcomers and lower plenum in the nonboiling (single- various two-phase flow modeling assumptions and phase) part of the loop. The reactor thermal-hydraulics is numerical parameters in the integration scheme which based on a one-dimensional (1-D) modeling framework. In readily allow for numerical testing of the various this approach, both kinematics (phasic slip) and components of the loop and validation of the overall BWR thermodynamic (subcooled boilng) nonequilibrium are loop model against measured data such as of the LaSalle-2 accounted for using a four-equation model of two-phase BWR instability event [3]. flow. The equations governing two-phase/single phase flows in the various components of the loop are:

Volumetric Flux Equation parameters, and either position-and-time-dependent or average parameters.

¶ ¶ r j g = Gv f g - vg a (1) ¶ z ¶ t

STEAM DOME Void Propagation Equation STEAM STEAM DRYER ¶ a ¶ jg ¶r g + = vg (G - a ) (2) ¶ t ¶ z ¶ t STEAM SEPARATOR L SW FEEDWATER with G , the volumetric evaporation rate, given by RISER DOWM- COMER dh dh f g dp UPPER PLENUM , [1 - r (1 - a) - r a ] q f dp g dp dt G = + (3) h AXS h fg fg CORE JET PUMP Mixture Energy Conservation Equation LOWER PLENUM ¶ ¶ [r f hl (1 - a) + r h a] + [r j h + r j h ] = ¶ t g g ¶ z f l l g g g

q, ¶ p = + , (4) A ¶ t Figure 1. Schematic of a BWR Nuclear Steam Supply xs System.

Mixture Momentum Conservation Equation

r j 2 2 ¶ G ¶ f f r g jg ¶ p + [ + ]+ + F + F = 0 , (5) ¶ t ¶ z 1-a a ¶ z f g

, where G is the mass flux, q is the linear heat rate and

F f and Fg are, respectively, the wall friction and gravity forces per unit area. All flow parameters are averaged over the flow cross sectional area, and the remaining notation in Eqs (1) – (5) is as in Lahey & Moody [5]. The governing equations for single-phase liquid flow can be derived from the two-phase flow equations, Eqs. (1)

- (5), by setting a , G and jg to zero. In this case, Eqs. (1) and (2) reduce to the condition j = jl (z, t) = jin (t) . These equations can be used in the calculations for the single- phase region of the channels as well as for the single-phase part of the recirculation loop (lower and upper downcomers and lower plenum). An important feature of the model is that it includes two options regarding the subcooled boiling phenomena: a profile-fit model [6] and a mechanistic model [5]. Another option is also available in which subcooled boiling is ignored. Similarly, the effects of phasic slip can be Figure 2. Reactor Pressure Vessel Components and Coolant accounted for using different modeling assumptions, such as Flow Diagram Modeled in the DYNOBOSS Code. the EPRI [7] correlation or user specified drift-flux The numerical method of solution of the governing Alternatively, Eq. (10) can be integrated along the equations is based on a two-parameter approach which characteristic, d y / d t = jin (t) , between instants tn-1 and allows for testing and quantifying the effects of both tn to yield temporal and spatial implicitness on the results of calculations. The governing equations are first discretized in t space: n y(zn , tn ) = y(zn-1, tn-1)+ S(t) d t (11) - ò d yi R( yi ) R( yi-1) t + = ci (6) n-1 d t D z where where the node-average variable, yi , is expressed as a zn = zn-1 + jin (tn ) D t (12) weighted average of the values at node boundaries

By applying Eq. (11) to evaluate the energy of fluid yi = (1 - s ) yi-1 + s yi (7) particles entering the channel at time instants, tn , and

tracking down their positions using Eq. (12), the time- where 0 < s £ 1. dependent fluid energy at the channel exit can be calculated. Taking Eq. (6) over all the nodes, after some Because the method of characteristics is based on an manipulations one obtains analytical solution, it does not suffer the problem of

numerical dispersion and diffusion. d y = H y + d = g (8) Certain components of the BWR loop, typically d t those characterized by a low L / A ratio, can be modeled using a simplified lumped-parameter approach. Specifically, The, the finite-difference approximation of Eq. (8) is integrating Eq. (10) between the inlet (zin ) and the exit given as ( zex ) of a given control volume, yields

y n - y n-1 = (1-q ) y n-1 +q y n D t (9) [ ] d y jin (t) + [yex (t) - yin (t)] = S(t) (13) d t L where 0 < q £ 1 is a measure of temporal implicitness of the scheme. z Depending on the values of q and s , various 1 ex where L = zex - zin and y(t) = y(z, t) d z . numerical schemes are obtained, from fully explicit for L ò zin q = 0 , through semi-implicit (Crank-Nicholson) for Assuming y = y in Eq. (13) yields a perfect q = 0.5 to fully implicit q = 1. ex The method described above has also been used for mixing model, the nonboiling portion of the BWR recirculation loop. In this case, however, another approach has also been d yex jin (t) jin (t) + yex (t) = S(t) + yin (t) (14) developed based on the method of characteristics. The d t L L advantage of applying this method to single-phase liquid flows (where the effects of local compressibility are Eq. (14) is solved for the variable y at the exit of negligible) is that it does not require spatial discretization the control volume using implicit/semi-implicit methods of and thus, is more accurate and faster than nodal methods. integration for ordinary differential equations. For a constant-area section of a nonboiling channel, Table 1 summarizes the methods and approaches the propagation of the thermal waves can be evaluated by used in the modeling and respective numerical solution of rewriting Eq. (4) as the main components of a BWR loop implemented in the

DYNOBOSS code. ¶ y(z, t) ¶ + [ jin (t) y(z,t)] = S(z, t) (10) ¶ t ¶ z III. NUMERICAL ANALYSIS where y = r f hl . When Eq. (10) is applied to external As indicated in the previous section, the portions of the BWR loop, such as the downcomers, the DYNOBOSS code has been implemented in a way which source term, S , mainly accounts for with the readily allows for testing the accuracy of predictions by reactor vessel wall and is normally small. comparing the effects of various modeling assumptions and Similarly as for the two-phase flow equations, the parameters in the solution scheme on the results of same finite difference approximation given by Eq. (9) which calculations. Several results of DYNOBOSS testing and uses the two-parameter approach can be applied to Eq. (10). validations have been reported before [1-4] against both experimental [8] and other models computer code [9].

TABLE 1. Methods and Approaches

METHODS AND COMPONENTS OF APPROACHES THE LOOP Finite-difference method Reactor core, Steam with two interpolation riser and upper and parameters lower downcomers Lumped-parameter approach Upper and lower with implicit/semi-implicit plena, steam methods for ODEs separator and upper downcomer Method of characteristics Upper and lower (MOC) downcomers

Figure 3. The response of an unstable BWR loop to a The objective of the present analysis was to external reactivity perturbation: a) the core inlet flow rate investigate the effects of selected modeling assumptions and limit-cycle response; b) the envelope of the limit-cycle computational methodologies of a BWR loop on the response for different core nodalizations (s = q = 0.5 ). accuracy of predicted system response for the reactor operating in natural circulation mode and at, or close to, marginally stability conditions, i.e. about 50% rated power and 30% rated flow. In particular, the importance of proper modeling of the nonboiling portion of the loop, i.e. the downcomer, on the onset-of-stability conditions has been investigated. The tests included the effect of integration method used for the downcomer and lower plenum, and the sensitivity of the results to the downcomer nodalization and to time step of integration. The analysis is separated in two parts: first, the components of the two-phase region of the loop are analyzed, then the most suitable numerical methods for this region are used to analyze the numerical methods for the components of the single-phase part of the loop. Fig. 3 shows the response of a unstable BWR loop Figure 4. The decay ratio of the core inlet flow rate (limit-cycle solution) for different nodalizations and Fig. 4 response of a stable BWR loop for different nodalizations shows the decay ratio of the dumped oscilating solution and interpolation parameters in the reactor core. (stable loop conditions) calculated for different nodalizations of the reactor core and for two combinations component, two methods of solution of the energy equation of the interpolation parameters. As can be seen in these have been tested: an implicit/semi-implicit finite difference figures, the best combination is for s = q = 0.5 and less method and the method of characteristics (MOC). The than 100 nodes are needed for the two-phase reactor core upper downcomer which has a relatively small length-to- model to reach satisfactory accuracy for stable and unstable area ratio (0.25 m-1) consequently, a fairly low fluid loop conditions velocity, mainly when the reactor is operating in the natural The governing equation integration methods for the circulation mode, several methods of solution of the energy single-phase part of the loop, i.e., in the downcomers and equation have been considered, including: the method of lower plenum have been extensively analyzed. characteristics (MOC), implicit/semi-implicit finite The lower downcomer, which includes the jet difference methods and a implicit/semi-implicit method for pumps, has a relatively large length-to-area ratio (4.5 m-1) a lumped-parameter model. First, a series of calculations so that a wave front takes a very short time to travel through was performed for the purpose of stablishing a reference for because of the high speed of the liquid. In this further testing. In all the calculations, the time step used in the other loop components was 0.01 seconds which is the largest permissible value to assure accuracy of the solution for this specific operating condition as shown in Fig. 5. Using the method of characteristics (MOC), the effect of the time step of integration was checked on the reactor response downcomer is relatively short, lumped parameter model is to a temporary perturbation in the external core reactivity. unable to properly predict the magnitude of oscillations. The results are shown in Figure 6 and 7. As can be seen, The next test involved a comparison between the method of whereas there is a small phase shift in the self-sustained characteristics and various nodalization and numerical oscillations in the core inlet flow rate, these oscillations schemes. The results are in Figure 9. As can be seen, the remain basically unchanged even if the integration time step model with s = q = 0.5 yields the best convergence, for the upper downcomer is increased to 0.24 seconds. For followed by the model in which q = 1 and s = 0.5 . The the test of the lower downcomer, the impact of the last case, with s = q = 1, behaves in a manner similar to the integration time step is more significant and the needed time perfect mixing model even if a very large number of nodes step is close to the overall loop integration time step of 0.01 (240) is used. seconds.

Figure 5. The responses of a stable BWR loop to a perturbation in the core external reactivity for various loop overall model time steps.

Figure 7. The responses of an unstable BWR loop for different time steps of the MOC method in the lower downcomer: a) the core inlet flow rate; b) the core mass flux envelope.

IV. CONCLUSION

A sensitivity analysis has been performed to quantify both modeling and computational effects of the major components of BWR loop on the predictions of the system dynamics and stability. It has been shown that semi-implicit schemes with the center-of-node approximation of time derivatives yields fast convergence and best accuracy, and require lowest numbers of nodes (typically, less than 100 axial nodes in all major components of the BWR loop). This is only valid for a BWR operating in natural circulation mode at, or close to, the marginally stability condition. Since for other operating conditions the results Figure 6. The responses of an unstable BWR loop for can be quite different, for instance, the system operating different time steps of the MOC method in the upper point can stable or unstable, or have considerable different downcomer: a) the core inlet flow rate; b) the core mass flow velocities, and so on, new numerical testing regarding flux envelope. the numerical methods and associated timestep and nodalization is recommended. Having stablished the time step requirements, the The importance of modeling and computational spatial modeling effects in the upper downcomer were accuracy of reactor system thermal-hydraulics is augmented investigated. A comparison between the results for a perfect by the neutronics coupling due to the combined mixing model and the method of characteristics is shown in and void reactivity effects. Differences such as those in the Fig. 8. As can be seen, although this section of the calculated core flow rate observed in Figs. 8 and 9 directly affect the core void fraction, then the reactor thermal power REFERENCES and, finally, the overall response and the predicted stability characteristics of the reactor system. Indeed, the ability to [1] Rosa, M. P. and Podowski, M. Z., Nonlinear Effect in predict the onset-of-instability conditions and the transient Two-Phase Flow Dynamics, and response of an unstable reactor can be used as a measure of Design, vol. 146, p 277-288, 1994. accuracy and correctness of any given computational model. [2] Rosa, M. P. and Podowski, M. Z., Neutronically Coupled Two-Phase Flow Modeling and Numerical Solution for Application in Boiling Water Nuclear Reactos, Proceedings of the Second International Conference in Nuclear Reactor Engineering, ICONE-2, San Francisco, 1993.

{3] Rosa, M. P. and Podowski, M. Z., Simulation and Analysis of the LaSalle-2 BWR Power Plant Insttability Event with the DYNOBOSS Code, VI Congresso Geral de Energia Nuclear (VI CGEN), Rio de Janeiro, 1996.

[4] Rosa, M. P. and Podowski, M. Z., DYNOBOSS: A Computer Code for the Nonlinear Analysis of Boiling Water Nuclear Reactors, Proceedings of the Fifth International Topical Meeting on Nuclear Thermal- hydraulics, Operations & Safety, NUTHOS-5, Beijing, China, 1997.

[5] Lahey, Jr., R. T. and Moody F., The Thermal- Figure 8. The responses of an unstable BWR loop using two Hydraulics of Boiling Water Nuclear Reactors, different integration methods for the energy equation in the Published by the American Nuclear Society, 1977. upper downcomer: the MOC method and the lumped parameter approach. a) the core inlet flow rate, b) the core [6] Levy, S., Forced Subcooled Boiling mass flux envelope. Prediction of Vapor Volumetric Fraction, GEAP-5157,

General Electric Company, 1996.

[7] Chexal B. and Lellouche, G., A Full Range Drift-Flux Correlation for Vertical Flows, ANS Proceedings of the 1985 National Heat Transfer Conference, Denver, 1985.

[8] Saha, P., Thermally Induced Two-Phase Flow Instabilities, Including the Effect of Thermal Nonequilibrium Between Phases, Ph.D. Thesis, Georgia Tech. Institute, 1974.

[9] Peng, S. J., Podowski, M. Z., Lahey, Jr., R. T. and Becker M., NUFREQ-NP: A Computer Code for the Stability Analysis of Boiling Water Nuclear Reactors, Nuclear Science and Engineering., 88, 3, 1984.

Figure 9. The responses of an unstable BWR loop using two different integration methods for the energy equation in the upper downcomer: the MOC method and the finite difference method with different combinations of the interpolation parameters. a) the core inlet flow rate, b) the core mass flux envelope.