Determination of Cell Averaged Diffusion Constants Based on Transport/Diffusion Perturbation Theory

A unified definition for cell-averaged diffusion constants has been established on the basis of transport and/or diffusion perturbatlon theories. The diffusion constants are derived by equating each component of reactivities obtained by transport theory for a heterogeneous cell to a corresponding component of reactivities by diffusion theory for a homogenized cell. The derived diffusion constants are applied to infinite uniform lattices and heterogeneous lattices composed of different cells. In infinite lattices the present diffusion constants are compared to the conventional flux-weighted cross sections and Benoist's anisotropic diffusion coefficients. In heterogeneous lattices they are compared with Rowlands-Eaton cross sections and the extended Benoist's diffusion coefficient. The present formula is applied to a heterogeneous slab lattice. It reveals that the present method gives a more accurate result for k _eff compared to the case with the conventional diffusion coefficient defined by D= 1/3Σ_tr or the extended Benoist's diffusion coefficient together with the conventional flux-weighted cross sections.     

Improvement of Coarse Mesh Difference Diffusion Scheme about Control Rods

A simple method is derived to improve the accuracy of the predicted neutron flux in and around control rods by coarse mesh difference diffusion scheme in fast reactor criticality calculations. In a two-region slab lattice, a simple polynomial function is used to approximate the flux in the integral transport model, and the coefficients in the polynomial are determined by the variational method. From this approximate transport solution, the terms of net current into the control rods are corrected in the difference diffusion scheme before treatment by iteration in the criticality calculation. The results represent an appreciable improvement over those obtained with the conventional diffusion scheme, without any increase of computer time in the iteration. This improvement is confirmed in comparison with the results obtained with the usual fine mesh transport scheme as the standard reference.     

An analytic benchmark solution to the linear,time-dependent isothermal laminar flow fission-product plateout problem

The time-dependent convective-diffusion equation with radioactive decay is solved analytically in axisymmetric cylindrical geometry for laminar and slug velocity profiles under isothermal conditions. Concentration-dependent diffusion is neglected. The laminar flow solution is derived using the method of separation of variables and Frobenius' technique for constructing a series expansion about a regular singular point. These solutions, which describe the transport of fission products in a flowing stream, are then used to determine the pointwise and integrated concentrations of radioactive material deposited on a conduit wall using a standard mass-transfer model.Extensive parametric investigations have been conducted by varying the wall mass-transfer coefficient, diffusion coefficient, flow velocity, pipe radius and species half-life in the deposition models. The computational results indicate that the plateout estimates for the slug flow model are typically 5–100% greater than for the laminar model. The effect of axial diffusion is necessarily negligible for Péclet numbers greater than 100. Little increased plateout is observed for Péclet numbers less than 100; an additional 8% is predicted for a Péclet number of 20 if axial diffusion is included. Characteristic stream, wall and integrated deposition profiles are shown.Representative results from the analysis of fission-product deposition measurements for diffusion tubes in the Fort St Vrain high-temperature gas-cooled reactor plateout probe are presented. Using single-region slug and laminar models, the wall mass-transfer coefficients, diffusion coefficients and inlet concentrations are determined using least-squares analysis. The diffusion coefficients and inlet concentrations are consistent between tubes. The computed diffusion coefficients and wall mass-transfer coefficients are in agreement with known literature values. The analysis indicates that little difference can be discerned between computed laminar and slub flow-model parameters unless accurate data is obtained under strictly regulated conditions.     

Anisotropic Thermal Expansion of Uranium-Refractory Metal-Carbides: UWC2 and UMoC2

The anisotropic diffusion coefficient has been calculated in a cylindrical cell with use made of the integral transport theory. The previous method⁽¹⁾ of calculating the diffusion coefficient requires much computer time to evaluate the generalized first-flight collision probabilities between two mesh points for a square cell. To circumvent this drawback, we introduce new calculation methods for determining the anisotropic diffusion coefficient in a cylindrical cell. Two methods are proposed–one uses the diffusion theory in the outermost moderator region of a cell; the other adopts the integral transport theory in that region as well. For the latter method, a new boundary condition is introduced for the cell surface, which, in the present problem, supersedes the usual isotropic return boundary condition. Using these two methods, the anisotropic diffusion coefficient can be evaluated with very short computer time. Moreover, an analytic expression is obtained for the special case where a cell is composed of a fuel and a moderator.     

Discretizing the diffusion equation on unstructured polygonal meshes in two dimensions

We derive a discretization of the two-dimensional diffusion equation for use with unstructured meshes of polygons. The scheme is presented in r–z geometry, but can easily be applied to x–y geometry. The method is “node”- or “point”-based and is constructed using a finite volume approach. The scheme is designed to have several important properties, including second-order accuracy, convergence to the exact result as the mesh is refined (regardless of the smoothness of the grid), and preservation of the homogeneous linear solution. Its principle disadvantage is that, in general, it generates an asymmetric coefficient matrix, and therefore requires more storage and the use of non-traditional, and sometimes slowly-converging, iterative linear solvers. On an unstructured triangular grid in x–y geometry, the scheme is equivalent to the linear continuous finite element method with “mass-matrix lumping”. We give computational examples that demonstrate the accuracy and convergence properties of the new scheme relative to other schemes.     

Measurement of Thermal Neutron Cross Section of 137Cs(n, γ)138Cs Reaction

A calculational model for a modified diffusion coefficient has been developed to incorporate the neutron streaming effect in heterogeneous low-density channels accurately into diffusion theory calculations. The model uses a supercell, and the axial and radial diffusion coefficients of the heterogeneous inner cell are so defined that they can reproduce Benoist's axial and radial diffusion coefficients of the supercell when the diffusion coefficient of the outer cell is given as 1/3 Σ_tr . In the case of the axial diffusion coefficient, the axial buckling effect is taken into account by modifying the neutron path length within the streaming channel in calculating the collision probabilities. This model has been applied to an RZ fast reactor Core model with a gas expansion module (GEM). By using the axial diffusion coefficient obtained with the presented model, calculational error of GEM worth was reduced to less than 1/7 compared to the formula of Rowlands and Eaton.     

Isotopic Approximation to Thermal Diffusion Coefficients in Multi-component Mixture

An approximate expression of thermal diffusion coefficient in a multi-component mixture was derived by using isotopic approximations to ordinary diffusion coefficients and thermal diffusion factors. The result is the same as an expression of the thermal diffusion coefficient, of which validity was ensured numerically before, in terms of binary thermal diffusion factors and binary diffusion coefficients.     

Three-dimensional transport coefficient model and prediction-correction numerical method for thermal margin analysis of PWR cores

Combustion Engineering Inc. designs its modern PWR reactor cores using open-core thermal-hydraulic methods where the mass, momentum and energy equations are solved in three dimensions (one axial and two lateral directions). The resultant fluid properties are used to compute the minimum Departure from Nucleate Boiling Ratio (DNBR) which utlimately sets the power capability of the core. The on-line digital monitoring and protection systems require a small fast-running algorithm of the design code. This paper presents two techniques used in the development of the on-line DNB algorithmFirst, a three-dimensional transport coefficient model is introduced to radially group the flow subchannel into channels for the thermal-hydraulic fluid properties calculation. Conservation equations of mass, momentum and energy for these channels are derived using transport coefficients to modify the calculation of the radial transport of enthalpy and momentum.Second, a simplified, non-iterative numerical method, called the prediction-correction method, is applied together with the transport coefficient model to reduce the computer execution time in the determination of fluid properties.Comparison of the algorithm and the design thermal-hydraulic code shows agreement to within 0.65% equivalent power at a 95/95 confidence/probability level for all normal operating conditions of the PWR core. This algorithm accuracy is achieved with 1/800th of the computer processing time of its parent design code.     

Diffusion Calculation Model for Streaming Effects in Low Density Channels in LMFBRs

A study is made of the use of modified diffusion theory to calculate the negative reactivity worth of recently proposed flow activated reactor shutdown devices, in which sodium is voided from purely sodium filled channels located at the boundary between the core and the radial blanket in a Liquid Metal cooled Fast Breeder Reactor (LMFBR). Three-dimensional diffusion theory calculations using various definitions of modified diffusion coefficients in the channels are compared with each other and with three-dimensional transport theory computations. While normal diffusion coefficient is found to be inadequate for these reactivity worth calculations, the various modified diffusion coefficients appear adequate for the considered case. Modelling method of the small thickness of steel wall and interstitial sodium layer of the modules are also studied.     

An AMR Capable Finite Element Diffusion Solver for ALE Hydrocodes

We present a novel method for the solution of the diffusion equation on a composite AMR mesh.This approach is suitable for including diffusion based physics modules to hydrocodes that support ALE and AMR capabilities.To illustrate,we proffer our implementations of diffusion based radiation transport and heat conduction in a hydrocode called ALE-AMR.Numerical experiments conducted with the diffusion solver and associated physics packages yield 2nd order convergence in the L₂ norm.     

Effect of Porous Surface Layer on Leaching Rate of Radionuclide from Cementitious Waste Form

Leach tests based on IAEA methods were carried out using three types of cementitious specimens containing ¹³⁴Cs and ¹²⁵I and an evaluation method of apparent diffusion coefficients from the leaching curves was discussed. Since the gradient of the leaching curve, i.e. the relationship between the leaching ratio and the square root of time, changes during the leach tests, it has been difficult to estimate the diffusion coefficient. The difference between the leaching rate in the early period of the leach test and that in the later period was attributed to the porous microstructure of the surface layer in the cementitious specimens. The duplicate layer model, which expressed differences in the properties between the surface layer and an inner region by different diffusion coefficients, was proposed. Its numerical solutions had good agreement with the experimental leaching curves.     

On the diffusion coefficient calculation in two-step light water reactor core analysis

This paper presents consistent and rigorous accuracy assessments of various methods for calculating the diffusion coefficients in a two-step reactor core analysis of light water reactors (LWRs). The diffusion coefficients are significantly affected by the transport correction and critical spectrum calculations. There are various methods for the transport corrections (inflow/outflow/hybrid corrections) and critical spectrum calculations (B₁/P₁/CASMO-4E methods) so that it is necessary to decide the best combination to achieve a high accuracy in the transport/diffusion two-step analysis. Numerical tests are performed step-by-step to search for the best combination of the methods by comparing each other the transport one-step results, transport/diffusion two-step results, and Monte Carlo results. Numerical test results with a large and a small LWR core show that the combination of inflow transport correction and CASMO-4E critical spectrum calculation is most accurate than the other combinations in terms of eigenvalues and assembly power distributions.     

Axial SPN and Radial MOC Coupled Whole Core Transport Calculation

The Simplified P_N (SP_N) method is applied to the axial solution of the two-dimensional (2-D) method of characteristics (MOC) solution based whole core transport calculation. A sub-plane scheme and the nodal expansion method (NEM) are employed for the solution of the one-dimensional (1-D) SP_N equations involving a radial transverse leakage. The SP_N solver replaces the axial diffusion solver of the DeCART direct whole core transport code to provide more accurate, transport theory based axial solutions. In the sub-plane scheme, the radial equivalent homogenization parameters generated by the local MOC for a thick plane are assigned to the multiple finer planes in the subsequent global three-dimensional (3-D) coarse mesh finite difference (CMFD) calculation in which the NEM is employed for the axial solution. The sub-plane scheme induces a much less nodal error while having little impact on the axial leakage representation of the radial MOC calculation. The performance of the sub-plane scheme and SP_N nodal transport solver is examined by solving a set of demonstrative problems and the C5G7MOX 3-D extension benchmark problems. It is shown in the demonstrative problems that the nodal error reaching upto 1,400 pcm in a rodded case is reduced to 10pcm by introducing 10 sub-planes per MOC plane and the transport error is reduced from about 150pcm to 10pcm by using SP₃. Also it is observed, in the C5G7MOX rodded configuration B problem, that the eigenvalues and pin power errors of 180 pcm and 2.2% of the 10 sub-planes diffusion case are reduced to 40 pcm and 1.4%, respectively, for SP₃ with only about a 15% increase in the computing time. It is shown that the SP₅ case gives very similar results to the SP₃ case.     

Surface effects on the diffusion of tritium in 304-stainless steel and zircaloy-2

Tritium diffusion measurements following tritium recoil injection into austenitic stainless steels and Zircaloy-2 exhibit slower hydrogen transport in the surface regions than in the bulk. Kinetics of tritium release from these materials indicate that the release is controlled by a diffusion coefficient that is two to three orders of magnitude lower than the bulk diffusion coefficient for tritium in stainless steel and seven to eight orders of magnitude lower in Zircaloy-2. A two-region classical diffusion model with different diffusion coefficients in each region has been developed which appears to adequately represent the surface data for short heating times. Release rates at long heating times are apparently influenced by trapping of hydrogen in surface films. The surface effects are shown not to be due to the helium which is injected into the specimens along with the tritium.     

Ag diffusion in cubic silicon carbide

The diffusion of Ag impurities in bulk 3C-SiC is studied using ab initio methods based on density functional theory. This work is motivated by the desire to reduce transport of radioactive Ag isotopes through the SiC boundary layer in the Tristructural-Isotropic (TRISO) fuel pellet, which is a significant concern for the Very High Temperature Reactor (VHTR) nuclear reactor concept. The structure and stability of charged Ag and Ag-vacancy clusters in SiC are calculated. Relevant intrinsic SiC defect energies are also determined. The most stable state for the Ag impurity in SiC is found to be a Ag atom substituting on the Si sub-lattice and bound to a C vacancy. Bulk diffusion coefficients are estimated for different impurity states and values are all found to have very high activation energy. The impurity state with the lowest activation energy for diffusion is found to be the Ag interstitial, with an activation energy of approximately 7.9 eV. The high activation energies for Ag diffusion in bulk 3C-SiC cause Ag transport to be very slow in the bulk and suggests that observed Ag transport in this material is due to an alternative mechanism (e.g., grain boundary diffusion).     

Study of the neutron-noise transmission characteristics of non-multiplying media in liquid-metal fast breeder reactors using transport theory

The neutron-noise transmission characteristics of non-multiplying media typical to LMFBRs are studied using transport theory and the results are compared with the corresponding results for diffusion theory. In transport theory suitable expressions are derived for the discrete (asymptotic) and continuum (transient) solutions for the frequency-dependent neutron flux. Further, it is examined how the break frequency of the neutron flux APSD, calculated using transport theory and diffusion theory, changes with distance from the neutron-noise source for materials like graphite, stainless steel and borated concrete. The 1-D, isotropic, monoenergetic neutron transport equation for an infinite medium is used.     

Efficient Method for Calculating Axial Diffusion Coefficients in Two-Dimensional Low Density Channel

An efficient calculation method of an axial diffusion coefficient which is applicable to multi-region and two-dimensional geometries has been developed. This diffusion coefficient which requires biquadratic numerical integration is quickly calculated by using a Good Lattice Point Method. Total computational time is about one-thirty hundredth that of Romberg's Method. The geometry dependency of axial diffusion coefficient is investigated. As a result the axial diffusion coefficient of a rectangular cross section of low density channel is larger than that of a square one, under the constraint that the channel volume is conserved.

Analysis of the reactivity worth measured for a square void channel in the Tank-type Critical Assembly (TCA) was performed using this diffusion coefficient. The calculated value agrees well with the experimental value, i.e. ratio of the calculated to experimental value (C/E) is 0.997.