Patent Number: 047708433
Section: description

DESCRIPTION OF THE PREFERRED EMBODIMENTS FIG. 1 illustrtes a typical boiling water reactor (BWR) 1 which comprises a reactor core 3 housed inside of a pressure vessel 5. The core 3 includes an array of elongated rectangular fuel assemblies or bundles 7 (only one shown) each of which includes a number fuel rods containing fissionable material. Reactor coolant 9 in the form of light water is circulated through the core 3 with the assistance of jet pumps 11. The jet pumps 11 in turn are driven by coolant recirculated through recirculating piping 13 by a recirculating pump 15. The water circulated through the core 3 is converted to steam by the heat generated by the fission reactions. The steam is utilized to drive a turbine generator 17 which produces electric power. Steam exhausted from the turbine generated 17 is condensed in condenser 19 and returned to the pressure vessel by a feedwater pump 21. The power generated by the reactor 1 is controlled in a well known manner through regulation of the flow of reactor coolant 9 through the core 3. This flow is controlled, in turn, by regulation of the speed of the recirculation pump 15. In practice, several jet pumps 11 are provided, but only two are shown in FIG. 1 to explain the principles of operation of the reactor 1. The thermal output of the reactor is also controlled by control rods 23 (again only 1 shown for clarity) made of neutron absorbing material which are inserted into and withdrawn from the reactor core by a rod control drive 25. A combination of reactor coolant flow and control rod position in used to operate the reactor at a desired power level, while maintaining prescribed limits on power distribution, under the control of a computer 27. The computer 27 monitors measurements of various parameters for use in controlling the reactor, including: feedwater flow as measured by flow meter 29, rod position as reported by rod position indicators 31, dome pressure as measured by pressure transducer 33, core inlet temperature as measured by temperature detectors 35, neutron flux as recorded by a plurality of strings of local power range monitors (LPRMs), or detectors, 37 distributed across the core (only one string shown), and other parameters not pertinent to this invention. FIG. 2 illustrates in plan view an advanced type of BWR water-cross fuel assembly 7 in which four minibundles of 4.times.4 arrays of fuel rods 33 communicate with one another laterally through ventilation holes 41 in a cruciform partition 43. The present invention takes into account such cross flow in determining the stability of the fuel assemblies 7. Normal BWR fuel assemblies are of the open lattice type without the partitioning as shown in FIG. 2, assembly 7. The present invention accommodates both designs. In a boiling system, such as BWR fuel assembly, the vapor voids act as compressible volumes that can expand and collapse. These bubbles are in constant motion, expanding and contracting. These oscillations are dampened by the forces of friction, et cetera. If the energy being pumped into such a boiling system is not dissipated/dampened the oscillations in the two-phase flow and nuclear power increase in amplitude and eventually undergo self sustained oscillations, of a specific magnitude dependent on certain key parameters such as power level, power shape, orificing/internal damping, nuclear feedback, pressure et cetera. Such a sustained oscillation is referred to as unstable operation, and must be carefully guarded against. Sustained unstable operation can cause fuel failure, safety violations, and also mechanical damage to core components. The present invention provides an on-line system for not only determining the stability of the individual fuel assemblies in a BWR, with or without cross-flow, but also for determining a course of action, which can be implemented automatically or by an operator, to steer the reactor to a condition wherein the stability margin of all the fuel assemblies is within prescribed limits. In order to determine the quantitative value of the stability of the individual fuel bundles in a BWR in real-time, on-line, it is necessary to have a fast and reliable algorithm for modeling the system. The mathematical model used in the invention is described at length in an article entitled "A Stability Analysis of Ventilated Boiling Channels", by R. P. Taleyarkham, M. Z. Podowski and R. T. Lahey, Jr. published in Nuclear Engineering and Design, 93, (May 1986) 39-50, Elsevier Science Publisher B. V. (North-Holland Physics Publishing Division), Amsterdam, Holland. This is a nodal model which accounts for subcooled boiling, an arbitrary heat flux distribution, distributed and local hydraulic losses, heated wall dynamics, void slip flow, turbulant mixing and arbitrary flow paths for transverse ventilation. The model is represented by a system of differential equations which can be written in matrix form. This matrix is reduced using a special matrix reduction scheme described in the above article. Further experience with the physical nature of the cross-connected system has indicated a further means of optimizing the matrix reduction process. It was found that cross-flow inertia in the axial direction has a negligible impact on stability. This aspect allows for matrix size reduction and thereafter for substantial savings in computation time (approximately 20%) - further speeding up the process. The resultant relationship derived by the model for analyzing channel stability is: EQU .delta..DELTA.P.sub.t =G.sub.1 (S).delta.W.sub.T (Equation 1) where: .DELTA.P.sub.t =the total pressure drop from the inlet to the outlet of the channel; PA1 W.sub.t =the total flow through the channel; and PA1 G.sub.1 (S)=a transfer function. PA1 .gamma.=is a time varying component of internal heat/power generation; PA1 G.sub.2 (S)=a second transfer function; and the remaining terms are as described above. PA1 .alpha..sub.i =perturbed void fraction for ith node PA1 h.sub.in =inlet enthalpy PA1 .DELTA.z.sub.i =nodal length PA1 k.sub.o =reactivity Equation 1 is a scalar equation in which the transfer function, G.sub.1 (S), is an element of the matrix. While this equation takes into account the hydrodynamic effects on channel stability, it has been found that considering only such effects can lead to sizeable inaccuracies in quantifying stability of a "Boiling Water Nuclear Reactor". Accordingly, equation (1) should be modified to take into account nuclear feedback effects. The resultant solution for the generalized matrix, using the matrix reduction scheme, produces the following equation: EQU .delta..DELTA.P.sub.T =G.sub.1 (S).delta.W.sub.T +G.sub.2 (S).delta..gamma.Equation (2) where: In order to examine the variation attributes of W.sub.T for stability margin evaluation, .delta..gamma. must be expressed in terms of .delta.W.sub.T. This involves incorporating the effects of nuclear reactivity feedback. This is accomplished as follows: Step 1 A relationship is obtained relating the void fraction perturbation for each node in terms of the nodal flow and enthalpy for the radially averaged system. Step 2 Next, expressions are obtained through perturbed mass and energy conservation equations, such as those given in the reference paper, for the perturbed flow and enthalpy for each node in terms of the total perturbed inlet flow rate/enthalpy entering the channels, and .DELTA..gamma., using the special matrix reduction scheme. Step 3 The equations for the perturbed nodal void fractions (from step 1) are now transformed so that the perturbed void fraction for each node can be expressed in terms of the channel inlet flow rate/enthalpy perturbation. That is, we now obtain for each node i: EQU .delta..alpha..sub.i =f(.delta.W.sub.t,.delta.h.sub.in,.delta..gamma.) Equation (3) where: The overall averaged void fraction, .delta..alpha. perturbation can now be obtained from: ##EQU1## where: L.sub.H =heated length or EQU .delta..alpha.=T.sub.1 (S).delta.W.sub.T +T.sub.2 (S).delta..gamma.+T.sub.3 (S).delta.h.sub.in Equation (5) The expression for .delta..alpha. is next related to the nuclear reactivity kinetics and void feedback to obtain, assuming constant inlet enthalpy: EQU .delta..gamma.=R(S).delta.W.sub.T Equation (6) where: ##EQU2## .phi.=Kinetics transfer function C.sub..alpha. =void reactivity coefficient Equation 6 is now implemented into equation (2) to obtain EQU .delta..DELTA.P.sub.T =G(S).delta.W.sub.T (Equation) (7) where: EQU G(S)=G.sub.1 (S)+G.sub.2 (S)R(S) Equation (8) Equation (7), as formulated, represents the overall characteristic equation to be submitted to nuclear - hydraulic stability margin analysis. The transfer function G (S) is a complex polynomial obtained from the matrix. The eigenvalue of G(S) having the most positive real part in the frequency range of interest to BWR operation is determined, such as by using complex mapping theory, and is a measure of the stability of the associated assembly. A suitable stability index, preferably the decay ratio, is derived from this dominate eigen value of G(S). As is well known, a decay ratio, DR, of a value of one indicates the onset of instability, with increasing values above 1 indicating increased instability, and decreasing values below one indicating increased stability. A system for implementing the invention utilizing the above model is illustrated in block diagram form in FIG. 3. The software, which may be executed in the plant process computer 27, includes a module 45 for updating the necessary steady-state parameters, a scanning routine 47, for selecting the fuel bundles most likely to experience instability, and a module 49 for carrying out the stability margin calculations. The module 45 uses on-line measurements of total flow, pressure, inlet temperature and control rod position to generate real time distributed values for a number of parameters with a radial mesh of one node per flow channel and an axial mesh of at least about 30 nodes for acceptable results, and preferably about 50 nodes. Each fuel assembly can be treated as a flow channel for stability calculation purposes, or if desired, sub channels within a fuel assembly can each be considered a separate flow channel. The distributed parameters calculated by the module 45 comprise: power level and shape, flow and enthalpy, void drift or slip, inlet subcooling, and reactivity constants. Computer codes for generating these distributed parameters are well known in the field. The scanning routine 47 is an important part of the invention which reduces the number of fuel assemblies or flow channels for which stability calculations need to be made. It is based upon the fact that the stability characteristics of a nuclear assembly are very sensitive to the axial power distribution as well as the power level. Calculations show that assemblies with the highest power levels also take in the least amount of flow. This further aggravates the bundle stability characteristics. From the standpoint of monitoring the BWR core for the prevention of the onset to instability, it is necessary, and sufficient, to only look at those assemblies which exhibit certain axial power distributions and high power levels. That is, analyze for stability, only those assemblies most susceptible. This avoids a brute force approach of evaluating all assembly locations for instability, provides accurate results with a minimum of computational effort on demand within seconds while simultaneously enveloping the remainder of the core, and provides a basis for additional calculations to guide the operator on how much and in what direction to change power/flow/control rod patterns. A flow chart of the scanning routine 47 is shown in FIG. 4. As indicated at 51, the real time steady state axial and radial power distributions generated by the module 45 are read. The axial power distributions for all of the assemblies are evaluated at 53 to determine the average axial power distribution profile and the axial location of power peaking. Next the radial power distribution is analyzed at 55 to determine the radial power fraction for each fuel assembly in accordance with the relationship. ##EQU3## and to determine the average power per fuel assembly. Those assemblies with the location of the peak of the axial power distribution below the average axial lcoation for all assemblies are are identified as assemblies with inlet peaking power distributions at 57. Similarly, those assemblies with a radial power fraction R.sub.p, which is greater than the average for all assemblies (i.e. R.sub.p &gt;1.00), are identified at 59. Finally, the assemblies which are identified both at 57 as having inlet peaking axial power distributions, and at 59 as having above average power levels, are selected at 61 for stability analysis. While a typical BWR would have 700-800 fuel assemblies, each of which would be analyzed for stability under current practice, the scanning routine 47 reduces the number of assemblies to be analyzed in a typical situation to about 5 to 10. Analysis of the fuel assemblies most susceptible to instability, as identified by the scanning routine 47 is carried out by the stability margin calculation routine 49, which is diagrammed in FIG. 5. The stability of each of the assemblies, k, selected by the scanning routine 47 is analyzed by reading the appropriate steady state constants for the assembly generated by the module 45 as indicated in block 63. Using these data the transfer function G(S).sub.k, for assembly k is determined at 65 using the techniques discussed above. The complex polynomial, G(S).sub.k, is then evaluated 67 to determine the characteristic root with the most positive real part, or most dominant eigen value, and a stability index, such as the decay ratio, DR, is calculated at 69. If the decay ratio is more than 1, indicating an unstable condition 71, a report is made immediately to the user 73. This procedure is repeated 75 for each of the K selected assemblies. The least stable assembly is then identified 77, 79 as the assembly, k, with the highest decay ratio. If the highest decay ratio identified at 79 is greater than a preselected limit, nominally 1, as determined at 81, an incremental change is assumed for the least stable fuel assembly in rod position or flow at 83, as selected by the user, and a new G(S).sub.k is calculated at 65 for evaluating stability at 67 and generating a new decay ratio at 69. The selected parameter, rod position or flow, is iteratively adjusted in this manner until the decay ratio is within the prescribed limits in block 81. Since the least stable assembly has been identified at this point, the iterative process bypasses the determination of the least stable assembly in 79 through the branching provided at 77. The decay ratios for the k assemblies, and where appropriate, an output signal representative of the suggested change required in rod position of flow to bring the stability of the assemblies within limits, are reported out to the user at 85. If the user selects rod position as the parameter to be adjusted, the axial location of power peaking for the assembly being evaluated, is iteratively, incrementally shifted upward to determine what axial power profile will lead to a decay ratio of less than 1.0. The operator is then informed at 85 of what adjustments are necessary in the position of the control rod banks in order to obtain the necessary axial power profile. If the user options for suggested changes in flow to reduce instability, the assembly with the highest decay ratio is iteratively analyzed at assumed flow rates increased by increments selected by the operator, or assumed as a preset value, such as say five percent. The power level at which these calculations are carried out is also increased at a rate corresponding to the established operating line on the power-flow map predetermined by the module 45. The combination of power and flow which brings the assembly decay ratio to less than 1.0 at 81 is reported to the operator at 85. Returning to FIG. 3, the output of the stability margin calculation routine is transmitted to an output device 83 which can be a monitor which generates a visual representation for the user of the identification of the assemblies analyzed, their stability margins, and any recommended action. The output device can also, or alternatively, comprise devices for making hard copies such a printers, and recorders or other storage devices. The output device may also comrise or include an interface for passing a control signal to automatic controls 85 for automatic implementation of the recommended control action to improve core stability. The user can initiate a stability evaluation, indicate the parameter to be used in determining the amount of control action needed to increase stability, manually make the adjustment in flow or rod position suggested by the digital computer and otherwise control the program through input device 87. The input device 87 can be a keyboard, a touchscreen on the output monitor or any other suitable device. The input device also includes devices for automatically inputting the required data on the plant parameters used by the program. While the invention is illustrated in the exemplary embodiment as being implemented using the plane process computer 27, the program runs fast enough, and are compact enough, that they could also be implemented on a dedicated minicomputer or personal computer. The described system provides an on-line effective expert system that is expected to take only a few seconds to execute. It is based upon accurate, physically based calculations which add structure to an otherwise cumbersome "black art" of trying to supress instabilities in a boiling water reactor. The benefits are economic savings from reduced down time, and improved flexibility and reactor control. While specific embodiments of the invention have been described in detail, it will be appreciated by those skilled in the art that various modifications and alternatives to those details could be developed in light of the overall teachings of the disclosure. Accordingly, the particular arrangements disclosed are meant to be illustrative only and not limiting as to the scope of the invention which is to be given to full breadth of the appended claims and any and all equivalents thereof.