Response function of the convection-dispersion equations describing radionuclide migration in a finite medium

Analytical expressions are derived for the response functions for the set of n coupled convection-dispersion equations in 1-D with constant coefficients. The equations are solved for a finite medium, and for an n-member radioactive decay chain with radionuclide-specific retardation factors. An exponential transformation and the Laplace transformation are applied to this set of partial differential equations to obtain a general set of transformed solutions. These are then inverted by two procedures to produce complementary forms of the response functions. One form converges quickly for small values of dimensionless time while the other converges quickly for large values. These response functions are used to model radionuclide transport in an underground nuclear waste disposal vault.     

Analytic solutions of the multigroup space-time reactor kinetics equations—I: 1-D multiregion slab and spherical geometry

The development of analytical and numerical solutions to the reactor kinetics equations is reviewed. Analytic solutions of the multigroup space-time reactor kinetics equations are developed for bare and reflected slabs and spherical reactors for zero flux, zero current and extrapolated endpoint boundary conditions. The material properties of the reactors are assumed constant in space and time, but spatially-dependent source terms and initial conditions are investigated. The system of partial differential equations is reduced to a set of linear ordinary differential equations by the Laplace transform method. These equations are solved by matrix Green's functions yielding a general matrix solution for the neutron flux and precursor concentration in the Laplace transform space. The detailed pole structure of the Laplace transform matrix solutions is investigated. The temporally- and spatially-dependent solutions are determined from the inverse Laplace transform using the Cauchy residue theorem, the theorem of Frobenius, a knowledge of the detailed pole structure and matrix operators.     

Generalization of New Finite Element Method Using ‘Imaginary’ Nodal Points

A method for improving the accuracy of finite element solutions to diffusion equations has been developed. The author previously suggested a method for improving the accuracy of finite element solutions to neutron diffusion equations, a kind of Helmholtz equations, within a short computing time. The method has been generalized so that it can be applied to problems described by the Laplace equation, too, such as temperature distributions and electric fields. In this generalized method, 3 ‘imaginary’ nodal points are added at the midsides of each data-given triangular element and the element is subdivided into 4 triangular subelements of the same dimension to improve accuracy. Then, approximate expressions, which express solutions at the ‘imaginary’ nodal points using those at ‘real’ nodal points, are derived by Jacobi's iteration method. These approximate expressions are used to reduce the number of unknowns in the final linear equations. The computing time required for the method described here is much shorter than that required for the straightforward method of increasing the number of elements 4 times under the same accuracy.     

Transmission Line Models for Use with Circuit/System Analysis Programs

Techniques for modeling multiconductor transmission lines for use with the SCEPTRE computer program are presented. The transmission line models developed can be modified for compatibility with other circuit/ system transient analysis programs and are amenable to modification to include nuclear weapon effects. The general modeling approach has been to develop computationally efficient and accurate terminal models which can be arbitrarily loaded at the source and load ends and which can be used in conjunction with nonlinear electronic circuit models using either simplified or discrete modeling techniques. The concept of the method is to derive a set of transfer functions in the Laplace domain relating forward and backward traveling waves on the line to voltages and currents at the source and load ends of the line, approximate the transfer functions with a set of orthonormal polynomials, and represent the resulting rational polynomials in the time domain with state variable differential equations. For the multiconductor case, the orthogonal characteristics of wave propagation are used to decouple the modes of propagation except at the source and load boundary condition circuits.     

Reactivity calculation with reduction of the nuclear power fluctuations

A new formulation is presented in this paper for the calculation of reactivity, which is simpler than the formulation that uses the Laplace and Z transforms. A treatment is also made to reduce the intensity of the noise found in the nuclear power signal used in the calculation of reactivity. Two classes of different filters are used for that. This treatment is based on the fact that the reactivity can be written by using the compose Simpson’s rule resulting in a sum of two convolution terms with response to the impulse that is characteristic of a linear system. The linear part is calculated by using the filter named finite impulse response filter (FIR). The non-linear part is calculated using the filter exponentially adjusted by the least squares method, which does not cause attenuation in the reactivity calculation.     

Three-dimensional transport theory: An analytical solution of an internal beam searchlight problem-I

We describe a number of methods for obtaining analytical solutions and numerical results for three-dimensional one-speed neutron transport problems in a half-space containing a variety of source shapes which emit neutrons mono-directionally. For example, we consider an off-centre point source, a ring source and a disk source, or any combination of these, and calculate the surface scalar flux as a function of the radial and angular co-ordinates. Fourier transforms in the transverse directions are used and a Laplace transform in the axial direction. This enables the Wiener–Hopf method to be employed, followed by an inverse Fourier–Hankel transform. Some additional transformations are introduced which enable the inverse Hankel transforms involving Bessel functions to be evaluated numerically more efficiently. A hybrid diffusion theory method is also described which is shown to be a useful guide to the general behaviour of the solutions of the transport equation.     

The integral-transform method for solving neutron transport problems with general anisotropic scattering in a cylinder of finite height

In this paper, we present a mathematical technique for solving the integral transport equation for the criticality of a homogeneous cylinder of finite height. The purpose of the present paper is two-fold : firstly, to show that our earlier formalism can be generalized to any order of anisotropy, and secondly to generate the numerical results, which could serve as benchmarks when scattering is linearly anisotropic. We expand the scattering function in spherical harmonics to retain the Lth order of anisotropy. Thereafter, we write the integral transport equations for the Fourier-transformed spherical harmonic moments of the angular flux. In conformity with the integral-transform method for multidimensional geometry, the kernels of these integral equations are represented in their respective factorized form, which consists of a series of products of suitable spherical Bessel functions. The Fourier-transformed spherical harmonic moments are also represented in their separable form by expanding them in a series of products of spherical Bessel functions, commensurate with the symmetry of finite cylindrical geometry. The criticality problem for the cylinder of finite height is then posed as a matrix eigen value problem whose eigen vector is composed of the expansion coefficients mentioned above. The general matrix element is expressed as a product of certain integrals of Bessel functions, which can be evaluated by recursion relations derived in this paper. Finally, a comparison between the present benchmark results and S_N results (twotran) in (r–z) goemetry is presented when scattering is linearly anisotropic.     

Application of frequency domain analysis to transient response of nuclear containment structures

A combination of frequency domain and time domain analyses is proposed to obtain the dynamic responses of nuclear power plant containment structures. A soil-structure model of a boiling water reactor containment subjected to an assumed safety relief valve blowdown load is used as illustration. Linear time-invariant systems are analyzed using input forcing functions with varying frequency contents. Time domain analysis is performed using a synthesized input forcing function. The system characteristic function is generated in the frequency domain through Fourier transforms of the response time history and the synthesized input time history. The frequency response due to any other forcing function is obtained in frequency domain by using the system characteristic function, and the response time history is obtained by inverse Fourier transforms of the frequency response. The results obtained by the proposed method are in close agreement with the conventional time domain dynamic finite element analysis.     

An analytical solution for the general perturbative diffusion equation by integral transform techniques

In this work, we present analytical solutions for the eigenvalue problem of a neutron flux in a rectangular two dimensional geometry by a two step integral transform procedure. For a given effective multiplication factor _Keff$K_{\text{eff}}$ we consider a homogeneous problem for two energy groups, i.e. fast and thermal neutrons, respectively, where the problem is setup by two coupled bi-dimensional diffusion equations in agreement with general perturbation theory (GPT). These are solved in a two-fold way by integral transforms, in the sequence Laplace transform followed by GITT and vice versa. Although, the functional base and the employed integral transforms are the same for both sequences, the procedures differ. We verify the efficiency of the sequence on the solutions of such problems, further the results are compared to the solution obtained by the finite difference method.     

An analytical solution to time-dependent fission-product diffusion in an HTGR core

An analytical time-dependent fission-product diffusion model is solved for the fuel-moderator regions of a high temperature gas-cooled reactor (HTGR) during a hypothetical loss of forced circulation (LOFC) accident. A conservative approximate 1-D model is developed for the fuel and moderator regions, represented in cylindrical and slab geometries, from consideration of the hexagonal fuel-element symmetry. Transport is assumed along the shortest diffusion path and the concentration change across the fuel-moderator interface is approximated by a jump condition. The model is solved by construction of the Green's functions for the Laplace-transformed equations and identification of the pole structure. The concentration and current inverse Laplace transforms are obtained by the Cauchy residue theorem in each region for cubic piecewise polynomial initial conditions. A computer program was developed and validated to evaluate the solution, serve as a benchmark for more sophisticated numerical models and to investigate ⁹⁰Sr diffusion during a hypothetical LOFC.     

Development of two-energy group,two-dimensional,frequency dependent detector adjoint function based on the nodal method

A nodalization technique has been demonstrated to calculate the response of a detector to a vibrating absorber in a reactor core using a concept of local/global components, based on the frequency dependent detector adjoint function. The technique was developed for two-energy group one-dimensional or one-energy group two-dimensional reactor core geometry. The purpose of this research was to expand the applicability of a nodalization model technique to calculate the real and the imaginary parts of the detector adjoint function for two-energy group two-dimensional reactor geometry. The frequency dependent detector adjoint functions presented by complex equations were expanded into real and imaginary parts. In the nodalization technique, the flux or detector adjoint function is expanded into polynomials about the center point of each node. A computer code was developed to calculate static flux for two-energy group, two-dimensional reactor geometry. The eigen value (k_eff) and static flux were calculated for the Iowa State University UTR-10 reactor and the results were compared against the values calculated using the computer code exterminator. The eigen values were within less than 0.1% agreement. The phase angle and the detector adjoint function for the frequency of 10 rad/s were calculated for a detector located in the center of a 60×60 cm reactor. The phase angle calculated by the nodalization model technique varied from 0.2° near the source to 0.4° away from the source. These values are well within the range of the phase angle value of 0.2° calculated using the zero power transfer function. The thermal detector adjoint function peaked in the center as expected. The discontinuity in the current of the real thermal detector adjoint function at the detector position was observed as expected. The average current based on the polynomials on the left node of the interface and the right node of the interface matched within 1% of the average value at the interface. The current of the imaginary fast and thermal detector adjoint function on both sides of the interface varied ±2% from the average value at the interface. No discontinuity in the current was observed in the case of the fast real and imaginary and thermal imaginary components of the detector adjoint function at the detector location.     

Three dimensional analysis of thermally loaded thick plates

A mixed three dimensional formulation is presented for the analysis of thermally loaded thick plates. The method developed follows the guidelines furnished by the Vlasov-Iyengar technique applicable to the analysis of mechanically loaded thick plates, in which the three dimensional governing equations are derived from the elasticity equations by using a MacLaurin series approach. The expressions for the components of displacement, stress, and heat flux as well as the temperature are obtained in a series form in terms of the linear partial differential operators operating on a set of initial functions, which are the solutions to the governing equations. The procedure to be followed in arriving at the solution is illustrated by solving the problem of the thermal bending of a clamped-supported thick square plate; it is found that the present solution deviates significantly from those of the lower order theories for large values of the thickness ratio.     

Study and Design of a High-Frequency Interaction Cavity for a 140 GHz Megawatts Gyrotron

The gyrotron is one of the most promising high-power millimeter-wave sources for electron cyclotron resonance heating(ECRH) in controlled thermal nuclear fusion experiments.In this paper,the design of a high-frequency interaction cavity of a 1 MW/140 GHz gyrotron is described in detail.The cavity is designed by using eigen mode analysis and radio frequency(RF) behavior calculation.Rounded transitions at the input and output tapers are designed for reducing mode conversion.With the obtained cavity structure,non-linear self-consistent equations are adopted to calculate its output power and efficiency.A particle-in-cell(PIC) method is used to simulate the beam-wave interaction process for obtaining the resonant frequency and output power of the cavity.The PIC simulation results match considerably well with the results obtained by the non-linear self-consistent calculation.The cavity is currently under construction and will be integrated with other components for overall testing.     

Investigation of the Plasma Behavior in Filippov-Type Plasma Focus Using the State Space of Sing Lee’s Model

The state space of Lee’s model (SSL model) is a model developed for plasma behavior in Filippov-type plasma focus facilities which has been described and used. This model is attractive because it provides a practical approach for analysis of a plasma focus device. In this article, we turn to an alternative method of system analysis using time-domain methods. We will reconsider the differential equations describing the Filippov-type plasma focus device and select a certain form of differential equations. We will use a set of variables that can be used to establish a set of first-order differential equations. Using matrix methods, we will be able to determine the transient response of the Filippov-type plasma focus device and examine the performance of this system. This model is a derivation of modified Lee’s model and is based on the so-called slug model. Using the SSL model, the discharged current and its derivative as a function of time, pinch time, and maximum discharge current; as functions of pressure, have been predicted. The experimental data obtained by using the UIPFF1 facility with a maximum energy of 20 kJ is compared with the simulated data obtained through SSL model. This investigation shows that the SSL model is capable of predicting the plasma behavior in the Filippov type plasma focus.     

Influence of thermal gradient on natural frequency of rectangular plate vibration

Dynamic free response of thin rectangular plates subjected to steady state one dimensional and two dimensional temperature distributions satisfying the Laplace equation is analysed in this paper by using the finite difference method and finite element method. The governing equations of motion derived by the finite difference method are solved by a simultaneous iteration technique to obtain eigenvalues and eigenvectors. The results of both the methods compare well with those of classical methods in some typical cases. An attempt is made to correlate the non-dimensional frequency parameter and the temperature parameter. Plates of different boundary conditions with at least one edge simply supported are studied.     

Asymmetrical transient thermoelastic problems in a composite hollow circular cylinder

The present paper is concerned with the transient thermal stresses in a bonded composite hollow circular cylinder under an arbitrary asymmetrical heat supply. Numerical work is carried out using Laplace transforms and is given for the composite cylinder made of the different materials under three, band heat sources on its outer surface.     

Transient analysis of thick-walled piping under polynomial thermal loading

The transient response of a thick-walled pipe subjected to a generalized excitation of temperature on the internal surface was derived using Duhamel’s relationship. Generalization of the temperature excitation was achieved by using a polynomial composed of integral- and half-order terms. In order to avoid the evaluation of recurring functions in the complex domain, Laplace transformation and a 10-term Gaver–Stehfest inversion formula were used to perform part of the necessary integrations. Excellent agreement between the derived relationships and existing analytical and finite-element solutions was seen for a thick-walled cylinder subjected to an asymptotic temperature rise on the exposed surface. As intended, the use of a smoothed polynomial allows the incorporation of empirical date not easily represented by standard functions. Moreover, the resulting relationships are easily programmed and can be used for a wide range of nuclear, piping, and cylindrical vessel applications.     

Calculation of reactivity using a finite impulse response filter

A new formulation is presented in this paper to solve the inverse kinetics equation. This method is based on the Laplace transform of the point kinetics equations, resulting in an expression equivalent to the inverse kinetics equation as a function of the power history. Reactivity can be written in terms of the summation of convolution with response to impulse, characteristic of a linear system. For its digital form the Z-transform is used, which is the discrete version of the Laplace transform. This new method of reactivity calculation has very special features, amongst which it can be pointed out that the linear part is characterized by a filter named finite impulse response (FIR). The FIR filter will always be, stable and non-varying in time, and, apart from this, it can be implemented in the non-recursive form. This type of implementation does not require feedback, allowing the calculation of reactivity in a continuous way.