V '

The submitted manuscript has been authored by a contractor of the U. S. Government

contract No. W-31-104ENG-38. 1 E % r d i n g l v , the U. S. Government retains a

-. : \ " " 4 i

I $ ; J '%; k=a L - ' Principal Author: Hussein S . Khalil

Argonne National Laboratory A 9 9 1 4 4 r: *FL'iq kak!"J & J Iudr.27 Building 208 s*q- 1 Reactor Analysis Division

9700 South Cass Avenue Argonne, Illinois 60439

U. S. Government purposes.

Phone: Fax: Email:

(630) 252-7266 (630) 252-4500 khalil @ra.anl.gov

Invited Paper

Coupled Reactor Physics and Thermal-Hydraulics Computations'

by

Hussein S. Khalil

For Presentation At

Saratoga '97 Joint International Conference on Mathematical Methods

and Supercomputing for Nuclear Applications

Intended Session: Coupled Reactor Physics and Thermal-Hydraulics Calculations

Q, 'Work supported by the U. S. Department of Energy under Contract W-31-109-ENG-38.

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, make any warranty, express or implied, or assumes any legal liabili- ty or responsibility for the accuracy, completeness, or usefulness of any information, appa- ratus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily comtitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessar- ily state or reflect those of the United States Government or any agency thereof.

COUPLED REACTOR PHYSICS AND THERMAL-HYDRAULICS COMPUTATIONS

Hussein S. Khalil Reactor Analysis Division

Argonne National Laboratory

ABSTRACT

This paper provides an overview of coupled reactor physics and thermal-hydraulics computations with the SAS- DIF3DK computer code. The assumptions and capabilities of the DIF3D-K nodal diffusion theory spatial kinetics model are presented. Results from DIF3D-K benchmark verification calculations are reported. SAS thermal-hydraulics model assumptions and features are described, and results from a prototypic application are shown. The details of data flow in the coupling of spatial kinetics with thermal-hydraulics are reviewed, and coupled applications are described. Future model developments are listed.

I. INTRODUCTION

Two factors have contributed to continuing developments in the area of coupled reactor physics and thermal-hydraulics computations. First, safety-related analysis requirements for research, production, and power reactors have grown in response to increased public awareness and heightened regulatory scrutiny. Second, the widespread availability of microcomputers and workstations with memory sizes and computational speeds approaching mainframe capabilities has prompted model developers to implement ever more complex analysis algorithms. The union of the DIF3D-K spatial kinetics model’ with the SAS thermal-hydraulics model’ is an example of such a developmental algorithm. The computer code at Argonne National Laboratory that couples these models has been given the working title of “SAS-DIF3DK.” This paper provides an overview of spatial kinetics and thermal-hydraulics models in SAS-DIF3DK, presents results from benchmark and testing calculations, summarizes the coupling methodology, and indicates future developmental needs. The details of the spatial kinetics model,’ the thermal-hydraulics model? and the cross section methodologf employed in their coupling are presented in separate papers in these proceedings.

11. REACTOR PHYSICS METHODS

The DIF3D-K’ computer code embodies the basic reactor physics component for coupled calculations at ANL by providing solutions of the energy, space, and time-dependent neutron diffusion equation. In DIF3D-K, the energy dependence of the neutron flux is represented with the standard multigroup approximation, and nodal diffusion methods are employed to discretize the multidimensional spatial domain. Two methods are available in DIF3D-K to solve for the time dependence of the neutron flux, the variable implicit (theta) method and the factorization method. DIF3D-K has been designed to be executed either in a “stand-alone” mode, or in the coupled mode described here.

Two nodal spatial discretization options are provided in DIF3D-K for solution of multidimensional problems, hexagonal-z or Cartesian. The hexagonal-z mesh option4 was developed onginally as a solution technique for the stationary eigenvalue equation applied to analysis of liquid-metal-cooled reactors having hexagonal fuel subassemblies. The nodal equations are derived using higher-order polynomial approximations to the spatial dependence of the flux within the hexagonal-z node. The final equations, which are cast in the form of inhomogeneous response mamx equations for each energy group, involve spatial moments of the node-interior flux distribution plus surface-averaged partial currents across the faces of the node. These equations are solved using conventional fission source iteration accelerated by coarse- mesh rebalance and asymptotic source extrapolation.

Applications‘ of the hexagonal-z nodal mesh option to models of heterogeneous-core liquid-metal-reactor designs showed that the accuracy of the nodal scheme was supenor to that of the standard DIF3D finite difference optionS with

2

six triangular mesh cells per hesrtgonal f w l assembly. The higher-order axial approximation in the nodal scheme permitted the use of an axial mesh ivhich was at least four times coarser than a typical finite difference mesh. Particular improvement was seen in the average fluxes in the internal blanket regions and in the computed values for the eigenvalue. leading to more accurate predictions of internal blanket burnups, breeding ratios, and burnup reactivity swings. This enhanced accuracy was obtained with a potential order-of-magnitude reduction in the Computational cost of a three- dimensional calculation.

The computational efficiency of the nodal solution techniques prompted the development of the first version of the DIF3D-K spatial kinetics code.6 This code employed the hexagonal-z nodal diffusion theory approximation for the spatial variables, and provided two options for the temporal solution, the theta method and the factorization method. The theta method is a variable time integration scheme that permits the resulting time-differenced equations to range from fully explicit ( 0 4 ) to fully implicit (O=l>. The factorization method is based on a formal separation of the neutron flux time dependence into a rapidly varying function of time only (the flux amplitude), and a relatively slowly varying function of time that also depends on neutron enerey and spatial position (the flux shape). The options available in the factorization method include the improved quasistatic option, the adiabatic option, and the conventional point kinetics option. In the limit of a flux shape recalculation each time step, the improved quasistatic method is equivalent to the fully implicit theta method, based on observed performance characteristics. In the limit of no shape recalculation, the factorization method is formally equivalent to point kinetics. Besides the opportunity for easily assessing the accuracy of spatial, adiabatic, and point kinetics approximations. the factorization method produces superior computational efficiency for the same accuracy as the fully implicit theta method for most reactor transient analyses. The DIF3D-K hexagonal-z nodal spatial kinetics model was successfully employed in safety analyses of severe accidents in a large heavy-water reactor design.'

To extend the applicability of the DUF3D-K code, a Cartesian mesh geometry option' has been added to the nodal space-time solution. As was the case in the conversion of the static hexagonal-z nodal code to a space-time capability, the Cartesian nodal spatial solution involves time-dependent polynomial approximations to the intranodal flux distribution. The Cartesian geometry option in DIF3D-K has been verified by analysis of a wide variety of benchmark problems and by comparison to analysis results produced by other state-of-the-art spatial kinetics codes. Reference 1 provides details of the Cartesian geometry option and its application to analysis of an NEACRP benchmark problem and the Chernobyl accident.

Testing and verification of the DIF3D-K Cartesian geometry option were performed by execution of a variety of benchmark problems and comparisons with other spatial kinetics codes. Results from three of these analyses are summarized here:

Three-Dimensional Analytical Benchmark. This two-neutron-group, three-dimensional benchmark was developed and solved analytically by Dias! The problem involves the perturbation of a fictitious homogeneous reactor from an initial condition by a step reduction of the thermal group absorption cross section by about 0.6%. The perturbation results in a delayed supercritical transient that causes the core power to increase by a factor of about 5.25 over 6.0 seconds. The results provided in Ref. 8 include the core power evolution over this time period, and the kinetics parameters (p and A) at 6.0 seconds. The DIF3D-K results employing the various solution approaches are compared to the reference analytical results in Table 1. As is evident from Table 1, the core power and kinetics parameters are predicted very accurately by the DIF3D-K solution options. The conventional point kinetics scheme provides a very accurate solution to this transient problem because the flux shape is fairly constant, even though an adjustment of the fast-to-slow flux ratio occurs after the perturbation. Two theta method solutions are presented in Table 1 to demonstrate the compatibility of fully- and semi-implicit solutions.

Two-Dimensional TWIGL Ramp Transient Benchmark. This benchmark problem and its accompanying solutions were obtained from Ref. 9. The problem is a two-dimensional model of a 160 cm square unreflected seed-blanket reactor. This problem is modeled with two group cross sections and one delayed neutron precursor family. The transient is initiated by a 2.3% ramp reduction of the thermal capture cross section in the corner seed assemblies. The duration of the ramp is 0.2 seconds. Table 2 is a summary of results obtained from DIF3D-K (theta method and improved quasistatic scheme), along with results from Ref. 9 obtained from the analytical nodal code QUANDRY,

3

Time (s) TWIGL Reference

0.0 1 .OOo

1.307

1.957

2.074

2.096

2.109

-1 0.91419

Table 1. 3-D Analytical Benchmark

DIF3D-K DIF3D-K IQS 2DTD QUANDRY 0=1.0 (Default At=0.005 s At=0.005 s

At=0.005 s Parameters)

1 .om 1 .om 1 .OOo 1 .Ooo

1.309 1.306 1.305 1.305

1.961 1.949 1.95 1 1.954

2.074 2.097 2.064 2.074

2.092 2. loo 2.08 1 2.092

2.109 2.111 2.098 2.109

0.9 1323 ~~

0.9 13 12 0.9 1312 0.9 1368

Table 2. TWIGL Ramp Transient Results

4

Time (s)

0.0

5.0

10.0

20.0

30.0

40.0

50.0

60.0

1.

the nodal code 2DTD. and the (reference) finite difference code TWIGL. Again. the DIF3D-K solution options provide very accurate results. The xcuracy of the DIF3D-K code is comparable to that of QUXVDRY, ind appears to be better than that of ZDTD. The improved quasistatic result can be made more accurate by reducing the time between shape calculations. and b> forcing a shape calculation at 0.2 seconds (when the perturbation ended).

Core Mean Power Density (wkc)

CUBBOX QUANDRY QUANDRY DIF3D-K DIF3D-K (Reference) A td .25 s A t=0.50 s Auto-step IQS

Cusping No Cusping A td .25 s Correction Correction

150.0 150.0 150.0 150.0 150.0

169.4 169.1 167.1 168.7 168.8

202.0 202.0 196.9 200.4 200.5

260.5 262.2 250.3 254.2 254.6

209.9 210.8 20 1 .o 203.5 203.8

123.9 123.0 119.0 120.2 120.6

76.5 75.7 74.1 74.7 71.9

58.6 57.9 56.9 57.4 57.5

0.99971 0.99974 0.99959 0.99959

The 3-D LMW Light Water Reactor Transient Without Feedback. This benchmark problem from Ref. 9 is a three-dimensional model of a light water reactor operational transient. The problem is modeled with two neutron energy groups and six delayed neutron precursor families. Feedback effects are not modeled. The transient is initiated by withdrawing a bank of partially-inserted rods at a rate of 3 crn per second. Subsequently, a bank of rods initially parked in the upper reflector is inserted at the same rate. Results from the DIF3D-K code, the QUANDRY code, and the CUBBOX code are presented in Table 3. Following Ref. 9, the CUBBOX code is used as the reference result. The DIF3D-K solution options provide acceptable results. However, maximum differences of about 3% in the core mean power density are observed. This is due mainly to a feature in CUBBOX and QUANDRY that provides corrections for simulation of control rods traversing the large axial mesh intervals used in nodal codes. DIF3D-K does not currently contain this correction, and thus exhibits the so-called rod cusping effect. Two QUANDRY results, with and without rod cusping correction, are presented to indicate the magnitude of this effect.

Table 3. 3-D LMW LWR Transient Without Feedback

In the transient simulation, DIF3D-K calculates the space-time dependent flux distribution. The corresponding transient reactor power distribution is employed in the coupled calculation to drive the thermal-hydraulic calculation.

HI. THERMAL-HYDRAULICS METHODS

The SAS"' family of computer codes provides the basis for the thermal-hydraulic computational component coupled to DIF3D-K. This union was first applied to safety analysis of a large heavy-water reactor design.' The thermal-hydraulic model has since been significantly extended to treat high-pressure, high-temperature water applications. The details of this

5

current thermal-hydr3.ulics model are given in Ref. 2.

The SAS thermal-hydraulic model geometry assumes a configuration in which the reactor core is represented by a large number of coolant channels, each of which is modeled hydraulically with a one-dimensional coolant flow path. Multiple fuel pins communicate thermally with a common coolant channel through their cladding, based on a conduction heat transfer calculation solved on a detailed two-dimensional (WZ) spatial mesh for each fuel pin. In a typical reactor model, there are normally from ten to several hundred coolant channels, with several unique fuel pins in each channel, and each unique fuel pin standing for one or more physical pins. Each fuel pin may have different dimensions and operating conditions compared to the other fuel pins in their common coolant channel, and each coolant channel may have geometry and flow conditions different from other channels. At present, a coolant channel does not communicate thermally or hydraulically with other coolant channels except at common channel inlet and outlet junctions. Explicit geometric and heat transfer models are also provided for non-fuel members such as assembly wrapper cans, structural rods, pressure tubes, or moderator regions separate from the coolant, as may be found in W E R or RBMK designs.

Figure 1 provides a schematic presentation for a SAS thermal-hydraulic model of an RBMK-1500 fuel channel. This design features a circular pressure tube positioned vertically along the centerline of a stack of 25 cm square graphite blocks. Within the tube is suspended a fuel subassembly, consisting of two pin bundles supported one above the other on a structural central rod. A gap exists at the core nudplane between the two pin bundles, each of which holds eighteen pins. Grid spacers position six fuel pins on the circumference of an inner circle, and twelve on an outer circle. The axial mesh employed in the SAS thermal-hydraulic model of this fuel channel is indicated in Fig. 1. Each of the axial zones shown in the figure is characterized by a unique coolant flow area and combination of material solids, which may be specified from a number of available solid types.

The equation set employed in the SAS coolant thermal-hydraulic model consists of separate energy conservation equations for the liquid and vapor, separate mass conservation equations for the vapor and the mixture, and a momentum conservation equation for the mixture. The vapor and liquid are allowed to be in thermodynamic nonequilibrium, so the set may be described as five-equation, homogeneous, nonequilibrium model. This model is solved on an axial spatial mesh that need not coincide with the neutronics mesh, and normally has many more axial intervals than the neutronics mesh. Initial conditions for the transient simulation are obtained by a direct solution of the stationary equations, given a channel axial power distribution, coolant flow, and outlet pressure. Steady-state pressure drops across all channels are adjusted to obtain a common pressure drop by introducing an inlet orifice coefficient in all channels except the peak pressure drop channel. In the transient, common inlet and outlet pressure boundary conditions are applied to all channels, and the individual channel flows respond to the boundary pressures. All solid and liquid thermophysical and transport properties are formally temperature dependent, and steam properties are dependent on pressure, temperature, and quality. Convective heat transfer from the cladding to pure liquid, two-phase, and pure vapor regions is modeled, with coolant boiling in subcooled or saturated regimes.

Figure 2 provides a graphical depiction of the calculated temperatures and coolant density conditions for the channel geometry shown in Fig. 1 for steady state, nominal power, fresh fuel channel conditions. For the axial plots in Fig. 2, the axial location is measured from the bottom of the fuel.

IV. COUPLING OF REACTOR PHYSICS AND THERMAL-HYDRAULICS

For coupled calculations in which a thermal-hydraulics computation is providing predictions of changes in reactor material temperatures and densities due to changes in the reactor power amplitude and distribution, it is necessary to model transient behavior of the he multigroup cross sections used in the space-time neutronics solution. The MACOEF method" was developed for coupling DIF3D-K with SAS thermal hydraulics and first applied in the analysis reported in Ref. 7. The MACOEF method is essentially a phenomenologically-based correlation technique in which changes in macroscopic, few-group cross sections are expressed as mathematical functions of changes in reactor material properties (temperatures, densities, etc.). A database of macroscopic few-group cross sections is generated using a lattice cell physics computer code. where the problems posed to the lattice physics code reflect assumed perturbations in reactor material properties. The nature and degree of the assumed perturbations to the reactor material properties are determined by the analyst based on

6

AXIAL MESH

2 x 190 cm - - - - I

1 x 100 crn

1 x 193 crn

- - - -

I - - - -

4 x l 5 c m

l x 4 c K I 1 1 I

1 x l 9 c m

- - - - I

2 6 x l 3 c m

I

26 x 13 crn

I

l x 4 c r r i I I I

1 x lOcrn

I I 3 x l 5 c r n

1 x222cm - - - - I - - - - I

SHIELD 1 SMEARED GRID SPACERS

/

t

END CAPS

GAP

t >OOLANT FLOW

PRESSURE TUBE / (NOT TO SCALE)

u1 k r a, a [I (3

Fig. 1. SAS Thermal-Hydraulics Modeling Assumptions and Axial Mesh for RBMK Application

7

570 . . I

2 i .si I i O i I I I I

Saturation Temperature v1

7 ccn -I

f a, Saturated Boiling 550 - I- -

6 530 1 I I I

0 1 2 3 4 5

Axial Localion (melers) 6 7

Fig. 2. SAS Thermal-Hydraulics Model Results for RBMK-1500 Application

known reactor operating conditions and changes to those conditions expected during the transients to be analyzed. The MACOEF method requires that the analyst must first identify the neutronically significant phenomena for the problem at hand, and then perform lattice physics calculations as appropriate to produce a cross section database with sufficient generality to envelope all relevant phenomenological perturbations. Nonlinear dependencies of macroscopic cross sections on interactions among multiple material perturbations are formally accommodated in the MACOEF methodology, but must be recognized by the analyst and reflected in the fitting of the correlation to the database. An example of the application of the MACOEF method is given in Ref. 3.

The spatially-dependent material mass and temperature fields computed on the SAS thermal-hydraulics axiakhannel mesh are mapped to the DIF3D-K three-dimensional reactor mesh according to a channel-to-assembly position correspondence specified by the analyst. The MACOEF correlation is employed to compute a full set of macroscopic multigroup cross sections for each homogeneous node in the DIF3D-K solution. The reverse spatial mapping is performed to transfer the reactor power distribution from the DIF3D-K mesh to the SAS axialkhannel mesh. All mapping operations are designed to preserve mass, energy, and power.

At the initial, steady state, multiple iterations are performed, each of which consists of thermal-hydraulics and neutronics solutions for the reactor, until two subsequent iterations produce neutronics equation eigenvalues converged to within a user-specified criterion.

In the transient calculation, multiple time steps are taken to advance the transient solution. On each time step, the thermal-hydraulics solution is advanced using an extrapolated reactor power amplitude and the power shape determined by DIF3D-K for the time for the beginning of the time step. Once the thermal-hydraulics solution has been advanced to the end of the time step, end-of-step cross sections are computed and the DIF3D-K transient solution is advanced to the end of the time step. The newly-computed total reactor power is then used in an heuristic algorithm to determine the length of the next time step, and in the next reactor power amplitude extrapolation. However, the transient coupling between SAS

8

and DIF3D-K on a given time step is non-iterative. and accumulated errors are controlled by the time step selection algorithm.

The transient time step linking SAS and DIF3D-K is at the top level of a hierarchy of time steps. On the SAS side, the top level step is subdivided into coolant loop time steps common to all thermal-hydraulic channels, and each coolant loop time step is further divided into heat transfer and coolant dynamics time steps within individual channels. On the DIF3D-K side, the top level time step is further divided in to multiple flux shape recalculation time steps, and for the factorization methods, the shape time steps are divided into point kinetics solution time steps. The architecture of this time step hierarchy and the non-iterative coupling of SAS thermal-hydraulics with DIF3D-K spatial kinetics are the result of an evolutionary learning experience.

V. APPLICATIONS

Coupling of reactor physics solutions with SAS thermal hydraulics began during the development of the SAS family of codes" and their application to analysis of liquid-metal-cooled reactors. More recent efforts to couple SAS with DIETD- K are most notably exemplified by application to large heavy water moderated reactors.' Most recently, the extension of the SAS thermal-hydraulics model to high pressure, high temperature waterlsteam conditions? and the addition of Cartesian geometry to DIF3D-K,' have permitted applications to light water reactors, including graphite-moderated Russian RBMK reactors.

The SAS-DIF3DK code is in a developmental stage, and its current applications include test and verification analyses consisting of benchmark calculations and simulations of actual reactor transients reported at this In the near future, applications to known reactor transients will continue, and the experience gained from those applications will be used to focus the development process. The ultimate application goals for SAS-DIF3DK development are 1) a high accuracy capability for analysis of design basis transients (safety analysis report applications), 2 ) a high performance capability for real time simulation of operational and off-normal transients (operator training applications), and 3) a high reliability capability for safety research transient analyses (severe accident analysis and accident management studies).

VI. FUTURE DEVELOPMENTS

Recent applications of SAS-DIF3DK have provided new physical insights and understandings of the safety and design implications of existing reactor features, and have indicated areas for optimization and improvement of reactor designs. For example, the calculations reported in Ref. 1 tend to verify the effectiveness of the design changes made to RBMK reactors following the Chernobyl accident. At the same time, these applications have shown that improvements to SAS- DIF3DK would increase its usefulness and open new areas of applications.

The very large spatial mesh sizes characteristic of nodal neutronics methods promotes computational efficiency, but their use requires nodally-homogenized cross sections that accurately preserve reaction rates. In calculations with material relocation (e.g. control rod motion, coolant void propagation, or fuel relocation), it has been found that while global quantities (e.g. reactivity) are computed with good accuracy, local quantities (e.g. flux, or power) often contain an unacceptable level of error. One example of such error is the control rod cusping effect mentioned earlier in connection with the DIF3D-K benchmark solutions. Another is the error associated with using a nodal-average power in a thermal- hydraulic calculation computing material temperature and density changes on a much finer spatial mesh. For both of these examples, transient (perhaps unforeseen) material relocations evolve with physical dimensions smaller than the original neutronics mesh. To correct these errors in a fixed-mesh computation, it is necessary to implement 1) an intranodal heterogeneity model that, given significant changes in material distributions within a node, would produce the appropriate homogeneous nodal cross section for the flux solution. and 2) an intranodal reconstruction method that would produce flux transients within the node from the nodal flux information and the material distributions. An alternative would be to implement an adaptive mesh algorithm. that would produce refined mesh layouts to resolve transient material relocations. The intranodal heterogeneity and reconstruction technique would require increased computational effort for calculation of the homogenized cross sections and detailed power distribution, while the mesh refinement technique might require increased flux solution computation times.

9

Distinct from the mesh effect. there also exists another area for improvement of nodal neutronics accuracy, namely extension of nodal techniques to higher-order transport solutions. The development of nodal transport techniques provides the prospect for dealing with abrupt density changes (control rods, voids, or fuel compactions) that are poorly treated with diffusion theory. The utility of nodal transport techniques will depend on the application at hand, and the availability of computational resources.

The SAS thermal-hydraulic model has been developed with capabilities equivalent to other state-of-the-art fluid dynamics models, while performing with a computational efficiency that permits real-time simulation on current-generation 32-bit workstation computers. This goal has been achieved within the context of a homogeneous flow model, and the next step is to extend the model’s utility by implementation of a separated flow model while maintaining computational efficiency.

To compliment the detailed neutronics and thermal-hydraulics models, it is necessary to incorporate a detailed pin- cladding mechanics model for prediction of fuel element performance in design basis transients and accidents. T h i s model will include fission gas generation and release effects, irradiation effects on properties of cladding and structural materials.

At present, the SAS thermal-hydraulics model is structured to represent the reactor core, or the manifold-to-drum section of a channel reactor system. It is envisioned that the current model may be coupled with models of ex-vessel systems, as has been the case for other codes in the SAS family, thus extending overall capability toward the ultimate modeling goal of whole-plant simulation.

Finally, because the ultimate performance goal for SAS-DIF3DK has been to execute at real-time or faster for simulation applications, the ability to execute in a parallel computational environment is a continuing code design objective. The target environment for whole-plant simulation is a network of high-end personal computers or 32-bit workstations, but data management and transfer algorithms have been designed to accommodate either a network architecture or a single- machine, multiple processor architecture.

ACKNOWLEDGMENTS

This paper was compiled by J. E. Cahalan from contributions by F. E. Dunn, E. E. Moms, T. A. Taiwo, and R. B. Turski. The work described here was supported by the U. S. Department of Energy under contract W-31-109-Eng-38.

REFERENCES

1. T. A. Taiwo, “SAS-DIF3DK Spatial Kinetics Capability for Thermal Reactor Systems,” these proceedings.

2. F. E. Dunn, “Compiltationally Efficient Thermal Hydraulics Calculations in the SAS-DIF3DK Coupled Reactor Physics and Thermal Hydraulics Code,” these proceedings.

3. R. B. Turski, “hhcroscopic Cross Section Generation and Application for Coupled Spatial Kinetics and Thermal Hydraulics Analysis in the SAS-DIF3DK Code,” these proceedings.

4. R. D. Lawrence, “The DIF3D Nodal Neutronics Option for Two- and Three-Dimensional Diffusion Theory Calculations in Hcxagonal Geometry,” ANL-83-1, Argonne National Laboratory, March 1983.

5. K. L. Dentine, “DIF3D: A Code to Solve One-, Two, and Three-Dimensional Finite-Difference Diffusion Theory Problems.” ANL-82-64, Argonne National Laboratory, April, 1984.

6 . T. A. Taiwo and H. S. Khalil, “The DIF3D Nodal Kinetics Capability in Hexagonal-2 Geometry: Formulation and Preliminary Tests,” International ToDical Meetinn on Advances in Mathematics. ComDutations, and Reactor Phvsics, April 28-May 2. 1991, p. 23.2 2-1, Pittsburgh, American Nuclear Society, 1991.

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7. E. E. Moms, “Impact of Spatial Kinetics in Severe Accident Analysis for a Large HWR.” Proceedinas of the International Topical h4eerin.e on Advanced Reactors Safety, Vol . 1, pp. 290-297, Pittsburgh, American Nuclear Society, 1994.

8. A. F. V. Dias, “Systematic Derivation, from 3-D Nodal Equations, of Simpler Models for Describing Reactor Transients,” Ph. D. Thesis, Massachusetts Institute of Technology, May 1987.

K. S. Smith, “An Analytical Nodal Method for Solving the Two-Group, Multidimensional, Static and Transient Neutron Diffusion Equations,” M.S. and N.E. Thesis, Massachusetts Institute of Technology, March 1979.

9.

10. J. E. Cahalan, et al., “Advanced LMR Safety Analysis Capabilities in the SASSYS-1 and SAS4A Computer Codes,” Proceedings o f the International Touical Meetinn on Advanced Reactors Safetv, Vol. 2, pp. 1038-1045, Pittsburgh, American Nuclear Society, 1994.

11. W. S. Yang, Argonne National Laboratory, Private Communication, 1992.

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