Patent Number: 
Section: description

The fluid flow in a boiling water reactor (BWR) will be generally described with reference to FIG. 1. Feed-water is admitted into a reactor pressure vessel (RPV) 10 via a feed-water inlet 12 and a feed-water sparger 14, which is a ring-shaped pipe having suitable apertures for circumferentially distributing the feed-water inside the RPV. A core spray inlet 11 supplies water to a core spray sparger 15 via core spray line 13. The feed-water from feed-water sparger 14 flows downwardly through the downcomer annulus 16, which is an annular region between RPV 10 and core shroud 18. Core shroud 18 is a stainless steel cylinder which surrounds the core 20 comprising numerous fuel assemblies 22 (only two 2xc3x972 arrays of which are depicted in FIG. 1). Each fuel assembly is supported at the top by top guide 19 and at the bottom by core plate 21. Water flowing through downcomer annulus 16 then flows to the core lower plenum 24. The water subsequently enters the fuel assemblies 22 disposed within core 20, wherein a boiling boundary layer (not shown) is established. A mixture of water and steam enters core upper plenum 26 under shroud head 28. Core upper plenum 26 provides standoff between the steam-water mixture exiting core 20 and entering vertical standpipes 30, which are disposed atop shroud head 28 and in fluid communication with core upper plenum 26. The steam-water mixture flows through standpipes 30 and enters steam separators 32, which are of the axial-flow centrifugal type. The separated liquid water then mixes with feed-water in the mixing plenum 33, which mixture then returns to the core via the downcomer annulus. The steam passes through steam dryers 34 and enters steam dome 36. The steam is withdrawn from the RPV via steam outlet 38. The BWR also includes a coolant recirculation system which provides the forced convection flow through the core necessary to attain the required power density. A portion of the water is sucked from the lower end of the downcomer annulus 16 via recirculation water outlet 43 and forced by a centrifugal recirculation pump (not shown) into jet pump assemblies 42 (only one of which is shown) via recirculation water inlets 45. The BWR has two recirculation pumps, each of which provides the driving flow for a plurality of jet pump assemblies. The pressurized driving water is supplied to each jet pump nozzle 44 via an inlet riser 47, an elbow 48 and an inlet mixer 46 in flow sequence. A typical BWR may have sixteen to twenty-four inlet mixers. During an in situ noble metal application process for a BWR, it is perhaps most useful to know of the degree of noble metal loading as a function of location along water flow paths within the BWR. Consequently, in the method of the present invention, distinct regions of water flow are subdivided into multiple cells for modeling and analysis. In addition, since mass balances must be maintained wherever parallel flow regions merge despite unequal flow residence times within the region, the modeling routine of the present invention selects cells having equal flow time. Referring now to FIG. 2, an example portable computing system 100 for utilizing the modeling software of present invention is shown along with a block diagram below it illustrating the general input/output processing flow of information. For example, text files 101 containing initialization parameters and other data may be created (or input from an alternate source) on a portable processor 102 using conventional programming applications such as Excel.(trademark) After the noble metal loading simulation/modeling program of the present invention is run, an ASCI output file 103 containing computed model data is produced for displaying charts/graphs 104 of the modeled noble metal loading. FIG. 3 shows a flow diagram 110 that illustrates exemplary steps the present method for evaluating and maintaining the proper noble metal loading during an in situ noble metal application process such as mentioned above. Each box of the diagram of FIG. 3 contains a concise explanation of a particular step for this example embodiment of the method of the present invention. As indicated at block 112, the initial state of reactor water chemistry and initial operating conditions are input by a user on a processing system (100) containing the modeling program of the present invention. The modeling program is then run which computes, at block 114, noble metal loading throughout the flow circuit, specifications for conductivity and other parameters based on the input parameters. The user then proceeds to obtain samples of noble metal concentrations at various locations throughout the water-flow circuit within the reactor, as indicated at block 116. The actual measured concentrations from the acquired samples can then be compared to the computed model results for corresponding locations and may be used to alter particular rate-constants employed by the modeling program, as indicated at block 118, until the modeling program output agrees with the actual measured concentrations. Once the modeling program is effectively xe2x80x9ccalibratedxe2x80x9d in this manner, it may then may be run again at various times during the application process, as indicated at block 120, to provide an immediate best estimate of noble metal loading for any desired location within the reactor water-flow circuit. If the computed best estimate of noble metal loading is consistent with predetermined target goals for the noble metal application process (block 122), further samples may be taken at later times (block 124) to continue the evaluation process or, alternatively, the evaluation may be terminated (block 128). After the above xe2x80x9ccalibrationxe2x80x9d of the modeling program (block 118) has been achieved, if a subsequent computed best estimate of noble metal loading (block 120) is not consistent with the targeted goals of the NobleChem(trademark) process for the reactor (block 122), then such divergence(s) may serve as an indication that the operating conditions of the reactor may need to be at least temporarily altered (block 126). For example, the noble metals in the water flow circuit can be altered by producing a change in the rate-constant effected by changing the current operational criteria/conditions of the reactor. FIG. 4 covers some example general procedural steps which may be followed for using the simulation/modeling software of the present invention. Referring now to FIG. 4, steps 202 through 210 illustrate example steps performed before the noble metal deposition process simulation/modeling routine is launched. For example, the user/operator may begin by opening the workbook application for running the modeling program installed on a portable (or other) computer (block 202). The user/operator then inputs simulation parameters for the process including, for example, initial concentrations, reactor operating parameters, and chemical parameters and rate constants for the particular reactor and the noble metal application process (block 204). Next, the user/operator runs a set-up macro for the noble metal loading simulation/modeling program (block 206). This is a simple set-up macro that has been created to produce a text file of predetermined format containing labels and values for various input parameters and reactor specific geometric parameters, which may be provided, for example, from reactor-specific spreadsheet data stored within the same computer (block 208). The macro then launches the executable modeling program (block 210) stored on the same computer for performing the noble metal loading computations and analysis, the results of which may be locally displayed for immediate review. Basically, the modeling method implemented by the noble metal application process simulation program of the present invention integrates kinetic flow equations in a non-steady state system comprised of individual fluid-slug flow regions and sub-regions within the reactor. Some of these flow regions are considered as coupled and flowing in series, and some in parallel. This approach can be applied using conventional kinetic equations, because it basically models a set of mass balances in seriesxe2x80x94although a somewhat more basic linear modeling embodiment is preferred as a reasonable and easily implemented approximation to actual reactor conditions. If desired, more detailed and non-linear fluid kinetics may be assumed and used to more accurately model conditions within particular reactors. The specific rate constants referred to in the following discussion of applicable kinetic data may also be altered accordingly, as determined by empirical data. Kinetic Equations The basic equations for determining loading and decomposition are as follows: d(c)/dt=xe2x88x92(ks+kt+kz+kr)c (rate of destruction of noble metal compound) d(s)/dt=ksc(Dh/4) (rate of surface loading of noble metal compound) d(d)/dt=xe2x88x92(kt+kz+kr)c (rate of creation of inactive noble metal species) where: c=concentration of noble metal for the injected compound, g (or g moles) metal/unit volume d=concentration of metal for the deactivated compound, g (or g-moles) metal/unit volume s=surface concentration of noble metal, g (or g-moles) metal/unit area As knowledge of the noble metal kinetics improves, one of ordinary skill will appreciate that it may become necessary to make this example model more complex. For example, although it is not commonly known whether deactivation is actually heterogeneous (occurring at surfaces) or homogeneous, it is assumed to be homogeneous in the example embodiment disclosed herein. Particles of deactivated noble metal may collide with the surface, and deposit some otherwise deactivated noble metal. Particles of deactivated noble metal or crud may also be responsible for deactivation. If so, then one must account for the particle size distribution. In any event, the framework of the model presented herein should work with any kinetics. In the example embodiment herein, it is assumed that all reaction rate constants may be represented by the Arrhenius formula as follows: ks=As(4/Dh)exp[xe2x88x92Bs/RT] surface kt=At exp[xe2x88x92Bt/RT] thermal kz=AzZn exp[xe2x88x92Bz/RT] zinc xe2x80x83kr=ArG exp[xe2x88x92Br/RT] radiation where: Dh is the hydraulic diameter Zn is the zinc concentration G is the gamma dose rate. The overall rate constant is defined by: ko=(As(4/Dh)exp[xe2x88x92Bs/RT]+At exp[xe2x88x92Bt/RT]+AzZn exp[xe2x88x92Bz/RT]+ArG exp[xe2x88x92Br/RT]) so that d(c)/dt=xe2x88x92koc Example General Modeling Approach The approach of the present invention in applying the above equations is to divide each flow region and sub-region of the reactor into sub-lengths such that the residence time of every sub-length is xcex94t. This makes it straight-forward to analyze any fluid element using a total time derivative. In this way, it is possible to follow individual xe2x80x9cslugsxe2x80x9d of water as they flow from one sub-length to the next. Also, using this approach, it is possible to add parallel streams together and ensure that a true mass balance is maintained. Example regions and sub-regions of the reactor are defined a list provided at the end of this discussion. For this example, xcex94t should equal 1 second, although the user can elect to select a different value. However, selecting a much larger value for xcex94t would result in bypassing the sub-regions of shortest residence time, which should be avoided. Each sub-region of the primary system is characterized by the following primary quantities: Length (L) Surface area (S) Volume (V) Flow rate (Q) The water flow velocity in a sub-region is given by: v=QL/xcfx81V In a given region, the sub-length is given by: xcex94L=xcex94tv=xcex94tQLxcfx81V The number of sub-lengths in a region is given by: n=L/xcex94L In practice, n is rounded up to the next nearest integer. Because of this rounding, the formulation only represents the geometry approximately. However, this should only have a small effect on the results. The effect is minor compared with some of the assumptions in the model, such as the use of simple rate expressions and kinetics, the neglect of any influence of crud, and the use of plug flow in all regions. At time mxcex94t, each sub-length in each region is characterized by values of c, d, and s. In most cases, these will be set equal to 0 when time equals 0. However, non-zero values may also be used at time 0, depending on the application. Suppose the sub-lengths of a region are numbered from 1 to n. For the general interior sub-length i, with concentrations c(i, mxcex94t), d(i, mxcex94t), and s(i, mxcex94t) at time mxcex94t, the concentrations at time (m+1)xcex94t are given by c(i, (m+1)xcex94t)=c(ixe2x88x921, mxcex94t)xe2x88x92koc(ixe2x88x921, mxcex94t)xcex94t d(i, (m+1)xcex94t)=d(ixe2x88x921, mxcex94t)+(kt+kz+kr)c(ixe2x88x921, mxcex94t)xcex94t s(i, (m+1)xcex94t)=s(i, mxcex94t)+(Dh/4)ksc(i, mxcex94t) An equation similar to the equation for c can be used for other species in the water, with appropriate modification of the rate constants to account for stoichiometry. For example, OH+ ions are generated by the decomposition of Pt(OH)6xe2x88x92 ions. The equation of change of OHxe2x88x92 is: OHxe2x88x92(i, (m+1)xcex94t))=OHxe2x88x92(ixe2x88x921, mxcex94t)+6fOHxe2x88x92ko,PtcPtxe2x80x94complex(ixe2x88x921, mxcex94t)xcex94t where c is on a molar basis, and fOHxe2x88x92 is the fraction of the OH groups converted to ions. Similarly, the equation for change of NO2xe2x88x92 due to the decomposition of Rh(NO2)6 +++ is NO2xe2x88x92(i, (m+1)xcex94t))=NO2xe2x88x92(ixe2x88x921, mxcex94t)+6ko,RhcRhxe2x80x94complex(ixe2x88x921, mxcex94t)xcex94t where the process is assumed to be 100% efficient. Region Entrance Conditions In addition to the general interior sub-lengths, the inlet and outlet conditions for each region, corresponding to the first and last sub-lengths of a region, are specified. For simplicity, any inlet flow or outlet flows will occur between the last sub-length of a sub-region and the first sub-length of the next sub-region. Consider the beginning of a sub-region receiving flows Q1, Q2, and Q3, with active concentrations c1, c2, and c3 and deactivated concentrations d1, d2, and d from three separate sources (this is the maximum I believe required for this system). The flows may either be from other sub-regions leading into the sub-region or inlet flows to the system. For example, at the entrance to the upper plenum, Q1 is the fuel channel flow, Q2 is the core bypass flow, and Q3 is the outer bypass flow. The flow rate for the new sub-region is Q=Q1+Q2+Q3 the active concentration is xe2x80x83c=(Q1c1+Q2c2+Q3c3)/Q and the inactive concentration is d=(Q1d1+Q2d2+Q3d3)/Q The same set of equations may be used if there is an outlet stream. For the outlet stream, the sign of Q is negative. Example Specifications for Pt and Rh Process Example species that are tracked in the flow water are shown below in Table 1. In the present example, in addition to the above listed components, surface-loaded concentrations of Pt and Rh are also tracked. The labels used for such are PT_SURF and RH_SURF, respectively. The concentrations of chemical species are calculated by mass balance throughout the reactor. The only species of interest that is not calculated by mass balance is zinc. This is because: 1) zinc appears to reach an equilibrium value, unaffected by the parameters of the model; 2) we do not currently have a method to calculate zinc concentrations; and 3) zinc is only used in order to predict the conductivity. Initially, the zinc concentration will only be measured and input (as listed in Table 6). If there is supporting data, the zinc concentration may later be used in the calculation of kz. Table 2 shown immediately below lists example chemical parameters used by the modeling program of the present invention. These are parameters used to 1) calculate the rate of change of the active chemicals, 2) calculate conductivity, 3) calculate pH. Numerical values for flow lengths, volumes, and surface areas are provided by the user for each of the reactor regions. Table 3 below provides a list of labels used to define these quantities in the example modeling program. The corresponding values are provided by the user (e.g., determined externally and provided to the modeling program). In this example, provisions are made for using up to three segments per sample line with differing geometry, but constant mass flow rate. In addition to the primary circuit, two sample lines are included. For the present example, Table 4 immediately below illustrates some additional reactor design parameters which the user may need to provide. Table 5 below shows example definitions and entrance conditions used for each sub-region. For each sub-region, the volume, length, and surface-to-volume ratio (which equals 4/Dh) are provided by the user. In, for example, the NobleChem(trademark) metal application process, the streams entering the loop (chemical injection, feedwater cleanup, control rod) and exiting the loop (sample line, recirculation to RWCU, drain line to RWCU) are extremely small compared with the core flow. None of these flows will significantly affect the velocity. Only the injection flow will significantly affect any local concentrations. The effects of the other streams on concentration, while ultimately important, are relatively slow and non-localized. Therefore, they can be lumped together. This is a useful simplification, in that it prevents the accumulation of numerical round-off errors. It is assumed that there are only two inlet flows (chemical injection and feedwater cleanup) and three outlets (RWCU from recirculation loop, and two sample lines). The control rod drive flow will be assumed to equal 0, and the feedwater cleanup flow rate assumed to equal the sum of the RWCU and sample line flow rates minus the injection stream flow rate. The total flow (TF) in the primary system varies as water enters and exits. TF equals CF from the middle of the mixing plenum until the next entrance or exit stream. TF decreases by QSAMPA at the recirculation discharge. TF increases by QIS, the injection stream flow rate, at the recirculation header. TF decreases by QRWCU at the RWCU take-off point, assumed to be the bottom of the lower plenum. TF decreases by QSAMPB at core plate. Finally, TF increases by the amount (QRWCU+QSAMPA+QSAMPBxe2x88x92QIS) in the middle of the mixing plenum. Table 6 shown immediately below lists example parameters to be input by the user. These values may be changed by the user. For example, the user may elect to change various chemical parameters in response to sample measurements taken in order to improve the match of the computed model to the actual NobleChem(trademark) process. The operating parameters may be entered in steps. For this example, at least fifty of such steps are permitted. Table 7 shown immediately below summarizes some example intermediate parameters. These are values that are used by the model, and which will be not altered by the user. The average core dose rate is calculated as a function of the thermal power at shutdown and the time between shutdown and the beginning of the simulation. The water density is calculated as a function of temperature, and will change from its initial value if the temperature changes. Example Simulation Program FIGS. 5A-5E are flow diagrams illustrating example functional steps performed by the executable noble metal application modeling routine of the present invention. One of ordinary skill will appreciate that such steps may be implemented on any particular computer by utilizing conventional programming techniques and programming tools well known in the art. Referring now to FIG. 5A, blocks 310 to 320 illustrate example steps for initializing the modeling computations. For example, a text file previously created by the user (see FIG. 4) defines the noble metal application process variables to be analyzed (block 310). Input values are obtained from the text file (blocks 312 and 314) and converted to model units (block 316). Intermediate parameters are then computed (block 318) and set up to prepare for the modeling/simulation run (320) as further described with respect to FIG. 5B. Referring now to FIG. 5B, blocks 410 through 434 illustrate example steps for preparing and computing internal parameters used in the process modeling/simulation. For example, times and core dose rates (e.g., gamma vs. time for base case reactor) for standard shutdown condition are defined (block 410). Relative power density at shutdown (compared to base case) is determined (block 412). Minimum core flow for all input steps is computed to determine the required number of cells (block 414). Flow rates (of all regions) at minimum core flow rate are computed to determine the number of cells (block 416). Region flows during operating step of minimum flow are determined (block 418). The cell Delta Length is computed for each region (block 420). The number of cells in each region is computed (block 422). Indexes for the first cell in each sub-region are determined (block 424) and the total number of cells is then determined (block 426). Next, indexes for inlet streams are defined (block 428) and arrays for tracking concentrations are created and initialized (block 430), converting values from ppb (as input) to moles/liter. Next, the concentrations of unidentified ions are computed based on initial measured pH and noble metals (block 432) and labels and indexes are set up for output (434). Referring next to FIG. 5C, blocks 436 through 456 illustrate example steps for performing computations for determining concentrations of chemicals and noble metal loading rates. For example, computations are performed for determining variable/parameter values; and variables representing time, Pt and Rh material balance components are initialized (block 436). At this point, operating variables (e.g., flow rates, temperatures, RWCU efficiency, injection rates) may be updated if the next operating time interval is reached (block 438). Next, operating variables are converted to MKS units (block 440), region flows are determined (in m3/s) for the current operating time interval (block 442), the values of the time step and number of time steps in the current operating interval are determined (block 446), and feedwater inlet concentrations based on RWCU efficiency are computed (block 448). The individual operating time interval computations are then performed, for example, using a For/next loop (block 450). The current time is updated (block 452) and results for each sub-region during the current time-step are computed (block 454). Next, a sub-region is selected (block 456). Referring now to FIG. 5D, entrance flow rates and labels of entrance streams are defined (block 458). The number of cells, first index, velocity, hydraulic diameter, surface-to-volume ratio, volume and temperature are also defined (block 458). Next, the rate constants foe all bulk and surface chemical reactions are updated based on current local gamma dose rate, geometry, velocity and temperature (block 460). Entrance concentrations for the sub-region are determined based on inlet streams concentrations in the previous operating time interval (block 462). Next, new concentrations and loadings of the first cell of the sub-region are computed using chemical kinetic equations based on local rate constants and bulk concentrations at the entrance (block 464). For the remaining cell concentrations in the sub-region, temperature in the cell is determined if the temperature varies within the region (block 468) and if local temperature is re-determined, the rate constants are also re-determined (block 470), and new concentrations and surface loadings in the cell are computed using chemical kinetic equations based on local rate constants and concentrations of the previous cell in the previous operating time interval (block 472). At this point, the steps in blocks 468 through 472 are performed again for each cell in the sub-region currently being evaluated until all cells in the sub-region have been evaluated (block 474). Next, the concentration arrays for the operating time interval for each cell of each sub-region and the material balances for Pt and Rh are updated (block 476 and block 478). Referring now to FIG. 5E, blocks 480 through 508 illustrate example steps for assembling selected computed modeling/simulation results for output. Next, the current time is checked to determine if an output should be generated (block 480) in accordance with some user-selected predetermined output interval. For example, printing or displaying model results may be scheduled for every x minutes. If the current time is greater than the next scheduled output time, then for selected locations (block 482) the following steps are performed (blocks 484 through 492): H+ and OHxe2x88x92 concentrations are selected based either on: 1) charge balance with other ions and initial unidentified ion concentration or 2) H+ and OHxe2x88x92 concentrations are determined via chemical reactions (block 484); the pH is computed using H+ (block 486); the water conductivity is then computed based on all ions and their respective conductance (block 488); the state of the system (i.e., ion concentrations, loading, pH, conductivity, etc.) is output for the selected locations (block 490); and the next print time is then updated (block 492), after which processing continues with determining if more time steps remain in the current operating interval (block 494). Referring back to block 480 in FIG. 5E, if the current time is earlier than the next scheduled output time, then processing continues with determining if more time steps remain in the current operating interval (block 494). If there more time steps remain within the current operating interval, the current time is updated (block 452 in FIG. 5C) and the modeling/simulation continues for another sub-region (block 454 in FIG. 5C). If there are no more time steps within the current operating interval (block 494), it is then determined if further operating time intervals remain to be evaluated (block 496). If further operating intervals are to be modeled, the processing continues with updating the operating variables for the next operating time interval (operating step), as indicated at block 438 (FIG. 5C). If no further operating intervals are to be processed, the simulation is essentially complete (block 498) and the final state data (i.e., ion concentrations, loadings, pH, and conductivity for all cells) may be output via printer or display (500). In the above described example, non-steady state calculations are preferably performed for each second after the start of the noble metal application process. The output of the simulation model may be stored and plotted, for example, for each five minute interval in a simulated forty-eight hour application process period. A forty-eight hour simulation may be based on a set of step-wise inputs, as mentioned above. The user may select or change the time, xcex94t, between inputting sample data in order to check convergence. With the present arrangement, the user is provided easy access to the kinetic parameters used to calculate rate constants, equivalent conductance constants, and all other chemical parameters, so that they can be changed from a set of default values. For example, individual default parameters may be selectively displayed prominently on the output display device so that the user can easily restore them to the default values. The input file and simulation results (tables and charts) may also be copied, for example, to a separate Excel(trademark) workbook. Standard statistical analysis charts may be input by the user, for example, in an Excel(trademark) workbook, along with the stored numeric results. Such standard charts may include information specifying, for example: 1. noble metal loading (1 five-curve chart for Pt and 1 for Rh versus time at two sample locations, bottom head, inner shroud, outer shroud). 2. Active noble metal concentrations (1 chart for Pt and Rh versus time at two sample locations). 3. Inactive noble metal concentrations (1 chart for Pt and Rh versus time at two sample locations). 4. Concentrations of ions (two charts for all ion concentrations versus time, one per sample location). 5. Conductivity (1 chart of conductivity versus time at two sample locations). 6. pH at 25 C. (1 chart of pH versus time at two sample locations). 7. Mass balance (2 charts for the sample locations, with curves of Rh loss, Pt loss, and %s unaccounted for based on other species concentrations). 8. Ionic balance (1 chart for net charge versus time at the sample locations) The user may also make additional charts, for example, from the spreadsheets. A small window(s) in the computer display or printed output may be used to provide current values of critical results, e.g., gmsPt/min, loading rates, % unaccounted for, conductivity, hrs to maximum conductivity, hrs to loading targets (Pt and Rh), etc. Such window contents may be defined as desired by the user. For the present example embodiment, the two sides of the reactor are assumed to be identical and a new forty-eight hour run is computed and stored whenever the user changes the inputs. The program code for the modeling/simulation routine of the present invention as illustrated in FIGS. 5A through 5E, may be written, for example, in a Visual Basic module in an Excel workbook. Sample data may be collected in the field and may be input to the modeling/simulation program from a spreadsheet. Such data may include measurements, for example, from GFAA (soluble Pt and Rh), zinc, IC ions, conductivity, pH, temperature, etc. The spreadsheet may be formatted so that its contents can easily be added to selected graphs generated as output by the modeling/simulation program. The following list provides a non-comprehensive listing of example regions and sub-regions of a reactor that may be modeled using the present invention: The modeling program/routine of the present invention may also be used to perform non-steady state evaluations of water chemistry transients in BWRs. For example, concentrations of water impurities in a BWR due to leaking fuel rods, corroding components or other intrusions can be easily modeled by including xe2x80x9csourcexe2x80x9d terms in the modeling routine to represent an impurities at probable locations that might account for their appearance. Likewise, the disappearance of various impurities, for example, due to incorporation into crud or radioactive decay, can be accounted for by including representative xe2x80x9csinkxe2x80x9d terms in the modeling routine. In this manner, the non-steady state concentration of radioactive isotopes, corrosion products and water impurities could be determined for the entire water flow circuit(s) throughout the reactor and in the steam for performing, for example, analysis on fuel leaks and corrosion. The foregoing method and apparatus has been disclosed for the purpose of illustration. As should be obvious to one of ordinary skill in the art, the noble metal application process modeling program routines of the present invention could be adapted, with only slight modifications, to model other non-steady state fluid systems having flow loops that contain multiple regions of disparate geometry and parallel flow paths. Variations and modifications of the disclosed invention will be readily apparent to computer programming practitioners of ordinary skill or those skilled in the arts of boiling-water nuclear reactor operation and/or other non-steady state physical chemistry systems. While the invention has been described in connection with what is presently considered to be the most practical and preferred embodiment, it is to be understood that the invention is not to be limited to the disclosed embodiment, but on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims.