Quench front propagation during bottom reflooding of a heated annular channel

A time-dependent two-dimensional conduction model of quenching, via bottom reflood, of hot, dry cylindrical tube of finite length in an annulus is presented. The method of finite integral transform is used in the solution. Multiregion heat transfer is considered at the heater surface, and no a priori assumption is made in regard to the quench front velocity. A possible mode of variation of the quench temperature with coolant inlet velocity is proposed. Predictions of the quench front velocity are compared with a few experimental data for Zircaloy-2 tube available in the literature.     

Theory of pump-induced pulsating coolant pressure in pressurized water reactors

The theory of pump-induced pulsating pressure distributions in a PWR coolant annulus is developed. The calculated pressure distribution can then be applied to predict the dynamic responses of the reactor internals. The mathematical analysis is formulated in accordance with the linearized Navier-Stokes' equations by assuming a compressible, inviscid liquid. These equations are combined to form a single equation in terms of the unknown pressure distribution. The boundary conditions are two concentric rigid walls in the radial direction and any combination of closed, open, and piston-spring supported end conditions in the axial direction. The pulsating pump pressure which induces the pressure fluctuation in the annulus is prescribed at a small opening of the outer cylindrical wall (pump inlet of the reactor).An approximate solution is obtained by introducing the concept of time-dependent body force in the governing differential equations. With this conceptual substitution for the actual loading, the time-dependent, mixed boundary value problem can be represented as a forced vibration problem with homogeneous boundary conditions. This problem can then be solved by the method of normal modes. Numerical examples are provided which give the pressure distribution in the axial and circumferential directions of the annulus for various configurations of one and/or several pumps.     

Rigorous solution to multiregion problems of monoenergetic neutron transport theory

A rigorous semianalytical algorithm is used, in the frame of the integral form of the transport equation, for the solution of some basic multilayer problems of monoenergetic neutron transport theory. The critical problem for a three region reactor is explicitly worked out, and numerical results are presented, in comparison with F_N calculations.     

On the use of conformal transformations to calculate the eigenvalues of the Helmholtz equation for cylinders with polygonal cross section

Solutions of the Helmholtz equation, as it arises in reactor criticality calculations, have been obtained in cylinders of polygonal cross section. The method used employs the Schwarz–Christoffel transformation which maps the vertices of the polygon onto the circumference of the unit circle. Thus the polygonal geometry is converted to r–θ geometry thereby allowing a solution to be constructed which satisfies the symmetry and boundary conditions of zero flux exactly. Three cases have been studied numerically; the equilateral triangle, the square and the hexagon. The square is particularly useful because an exact solution can be obtained by separation of variables in x–y geometry and hence allows the accuracy of the conformal mapping procedure to be assessed. Numerical results are obtained in the r–θ domain using a variational method and appropriate trial functions. The accuracy of the eigenvalues is very high and by comparison with the exact value in the square it is shown that only one term in the trial function leads to an accuracy in the eigenvalue of 0.1%. The lowest order approximation to the current at the square centre edge gives an accuracy of 4%. Details of convergence for all cases are given, as well as some useful approximate analytical formulae for eigenvalues and edge current. ©     

解瞬态热传导问题的边界元法

本文提出了一个用边界元法分析瞬态热传导问题的方法。采用稳态问题的基本群,并通过分离变量,导出了边界积分方程,叙述了数值求解的处理过程。对二维瞬态热传导的实例进行了计算,结果与解析解相比较,证明方法是方便有效的。     

Transient Temperature Response of an Annular Flow with Step Change in Heat Generating Rod

An analysis is made on the transient temperature response of fluid and of heat generating rod in the case of annular flow. Special consideration is given to the effects of the rod heat capacity and radial thermal conduction in the rod during the transient.

The analysis is based on the following assumptions: (1) Initially the system has a uniform temperature distribution, and the rod undergoes a step change in heat generation; (2) the temperature of the fluid entering the annular space is kept constant; (3) the velocity distribution of fluid is uniform; (4) all physical properties remain unchanged during the transient; (5) axial thermal conduction in the rod is neglected in comparison with that in the radial direction; (6) axial thermal conduction in the fluid is neglected in comparison with axial thermal convection.

The present result is compared with values by quasi-static solution for various heat transfer parameters.     

Transient heat conduction problems in inhomogeneous media discretized by means of boundary-volume element

An integral equation formulation is presented for the transient heat conduction problems in inhomogeneous media. The material constants are assumed to be prescribed as arbitrary, continuous and differentiable functions of position vector. The governing integral equations are derived from the weighted residual statement of the problems in which the fundamental solution to the corresponding heat conduction problems in homogeneous media is used as the weight function. The whole domain of interest is discretized into a series of boundary-volume-time elements, and then a set of linear simultaneous equations are obtained. Their solutions yield the temperature in the whole domain as well as the heat flux on the boundary.     

Time-dependent boundary source term for advanced nodal diffusion theory

An extension to nodal diffusion theory was developed to deal with time-dependent boundary source terms. It is shown that this extension is easily introduced in advanced, full-functional diffusion theory by means of Dirac's delta spatially-dependent functions at the nodal boundaries, and allows the evaluation of reactor transients showing the propagation of neutron waves or the effects of pulsed neutron sources. A model problem with exact solutions for both the diffusion and the P₁ (telegrapher's) equations was developed to test the capabilities of the theoretical extension. The larger discrepancies occur at the earliest times computed showing, at t=50 μs, mean-square deviations between the exact diffusion solution and the numerical approximations of 2.95, 1.72, 0.32, and 0.16% for the fourth, sixth, eighth, and tenth polynomial expansion order, respectively. The availability of the exact telegrapher's solution, however, demonstrates that improved accuracy is meaningless since the mean-square deviation between the exact diffusion and exact telegrapher's solution is very similar to the mean-square deviation between the exact diffusion and the poorer (fourth-order) diffusion approximation.     

Heat conduction in nuclear fuel rods

The numerical solution of the transient heat conduction in the fuel rods of light water reactors is studied in this paper, for the purpose of selecting an efficient, economic and accurate solution procedure to be used in fast running codes. The radial heat conduction equations are presented in terms of ordinary time dependent differential equations that are subsequently integrated. Comparisons were carried out using the lumped capacitance method, the method of lines, and the integral method with parabolic temperature profiles.The temperature-time history of the fuel pellet and cladding, predicted by the various methods, is shown for slow and fast transients representing the uncovery and reflood phases of a loss of coolant accident. It was found that the integral method, with parabolic temperature profiles in both the fuel pellet and cladding, was the most economic and gave sufficiently accurate predictions.     

Modeling of heat deposition on the W/Cu movable limiter in HT-7

The thermal behavior of a directly water-cooled W/Cu movable poloidal limiter was investigated in HT-7, a medium-sized superconducting tokamak with limiter configuration, major radius R = 1.22 m, and minor radius a = 0.27 m. The W/Cu movable limiter (ML) was exposed to the plasma at various radial positions at r < a. The surface and bulk temperatures were monitored by an IR-camera and the thermocouples, respectively. The heat flux deposited on the limiter was evaluated by an ANSYS code using the measured surface temperatures as boundary conditions. It was found that the maximum heat flux incident on the ML was less than 1 MW/m² in the Ohmic discharges, but reached up to 5-7 MW/m² in the discharges with lower hybrid current drive (LHCD). A simple model was developed to understand heat transport to the W/Cu ML, taking into account the “funnel effect”.     

Heat Transfer in Turbulent Flow with Internal Heat Generation in Concentric Annulus*,(I)

This paper deals with the problem of heat transfer in fully developed turbulent flow with uniform internal heat generation in a concentric annulus under conditions of uniform but different heat fluxes at the two wall surfaces. The Nusselt numbers at each wall surface are calculated by analyzing the radial distribution of the fluid temperature. We further derive approximate equations, which are found to represent very well the analytical results.

The Nusselt numbers vary with the ratios of the radii of annular space as well as of the wall heat fluxes, as also with internal that generation. These predictions agree well with available experimental data for various fluids.     

A method for solving a stochastic eigenvalue problem applied to criticality

We present a novel approach to calculate stochastic eigenvalues of differential and integral equations using polynomial chaos theory. The method is applied to a criticality problem using the diffusion equation. This technique has the advantage of avoiding the non-linear terms in the conventional method of stochastic eigenvalue calculation but it does require an additional, ‘pseudo-time’, independent variable t.     

Application of conformal mapping to the determination of unsteady temperature distribution in thermally orthotropic plates

Very few papers and reports are available on the solution of heat conduction problems in anisotropic solids. This situation is of particular interest in some nuclear engineering applications (for instance, fuel elements possess, in general, anisotropic characteristics). The present study deals with the solution of an unsteady heat conduction problem in domains of complicated boundary shape, considering a particular case of anisotropy: a thermally orthotropic material. It is shown that the conformal mapping technique coupled with the Ritz method leads to a unified solution of the title problem for an arbitrary orientation for the axes of orthotropy with respect to the directions of the sides of the plates.     

Investigation of radial heat conduction with 1D self-consistent model in helicon plasmas

A 1D radially self-consistent model in helicon plasmas has been established to investigate the influence of radial heat conduction on plasma transport and wave propagation.Two kinds of 1D radial fluid models,with and without considering heat conduction,have been developed to couple the 1D plasma-wave interaction model,and self-consistent solutions have been obtained.It is concluded that in the low magnetic field range the radial heat conduction plays a moderate role in the transport of helicon p...     

On the Initial-Value Problem of the Transport Equation for a Moderator Slab

A mathematical analysis of the time-dependent neutron transport equation is presented for the case of a moderator slab. The scattering of neutrons is treated on the basis of the Van Hove theory and is assumed to be isotropic.

First, the spectrum of the corresponding transport operator is investigated and it is shown that the complex λ-plane is decomposed by the operator into a band spectrum Re(λ) ≤ — (v∑_t)min, a discrete spectrum on the real axis λ≤— (v∑_t), _min a resolvent set Re(λ)> — (v∑_t)_min deleted by the discrete spectrum.

The discrete spectrum is carefully studied and is found to consist of isolated eigenvalues or otherwise empty. For all categories of moderator, there exists an upper limit to the thickness of a slab having empty discrete spectrum. A slab exceeding this limit has a finite number of eigenvalues if the moderator material is gas or solid. This is also true for a certain class of liquid moderators. For other liquids, there is another critical thickness for the slab thickness such that if it is exceeded, the set of eigenvalues turns out to be denumerable with an accumulation point at — (v∑_t)_min. Each eigenvalue possesses a finite multiplicity, and the index is one.

Finally the related initial value problem is considered, and it proves to have a unique solution upon application of the Hille-Yosida theorem. The contribution of the band spectrum to the solution is evaluated and is shown to decay faster than exp{ — (v∑_t)_mint}.     

Degradation of elastomer by heat and/or radiation

This article studied various problems on the degradation of elastomers by heat and/or radiation. Three kinds of elastomers were irradiated and evaluated by the radiation resistant property using the measurement of tensile test. The fluorine containing elastomer, which has excellent heat resistant properties, was found to be less durable for irradiation than ethylene-propylene-diene (EPDM) elastomer. Ten kinds of different compounding formulas of EPDM were prepared to investigate whether the compounding for heat resistant has durability for irradiation. The thermal exposure was performed in an air oven. The duration of thermal exposure at 140 °C was 384 h. The irradiation condition was 5.0 kGy/h at 70 °C, and the total dose was 0.9 MGy. Elongation retained was taken for the evaluation of the stability. It was found that the formulas for improving the thermal stability did not bring radiation resistant of samples in the experiment.The rate constant of the increase in CO concentration by heat and radiation was measured and defined as k_c(h) and k_c(r), respectively. The rate constant of that under the combined addition of the heat and the radiation is expressed as k_c(h + r). Eq. (1) was obtained by the experiment and it was found that there is a synergistic relationship between heat and radiation on the increase in CO concentration

(1)     

SCF Hartree-Fock results for elements with two open shells and the elements francium to nobelium

Hartree-Fock calculations of the electronic structure and energies of the elements from hydrogen to lawrencium were made. Exchange terms were included without approximation. The calculations were done with the configuration average energy stationary to first order. Results are presented here for the ground state configurations of elements Cr, Cu, Nb, Mo, Ru, Rh, Pd, Ag, La, Ce, Gd, Pt, and Au (all but Pd have two open shells), and for the elements Z = 87 to Z = 102; also included are results for neon, argon, krypton, xenon, radon, and mercury. The results consist of total average energy; one-electron eigenvalues and I(nl) energies; two-electron interaction energy integrals F^k, G^k; radial expectation values, the radii of orbital principal maxima, the radial wavefunctions, and a constant suitable for obtaining wavefunctions at very small radii.     

Cellular neural networks (CNN) simulation for the TN approximation of the time dependent neutron transport equation in slab geometry

This paper describes the application of a multilayer cellular neural network (CNN) to model and solve the time dependent one-speed neutron transport equation in slab geometry. We use a neutron angular flux in terms of the Chebyshev polynomials (T_N) of the first kind and then we attempt to implement the equations in an equivalent electrical circuit. We apply this equivalent circuit to analyze the T_N moments equation in a uniform finite slab using Marshak type vacuum boundary condition. The validity of the CNN results is evaluated with numerical solution of the steady state T_N moments equations by MATLAB. Steady state, as well as transient simulations, shows a very good comparison between the two methods. We used our CNN model to simulate space–time response of total flux and its moments for various c (where c is the mean number of secondary neutrons per collision).     

Existence theory and simulations for one-dimensional radiative flows

We consider the coupling of the radiative heat transfer equations and the energy equation for the temperature T of a compressible fluid within the finite segment [0, L]. Using the technique of upper and lower sequences associated to integro-parabolic equations, we establish the existence and uniqueness of a classical solution T, 0 ? Λ₋ ? T(x, t) ? Λ₊ < ∞ with corresponding radiative intensity I(x, Ω, ν, t). The boundary is considered to be semi-reflexive with reflection coefficient ρ, 0 ? ρ(μ) ? 1. The existence of the solution for the coupled system does not depend on any additional hypotheses besides that the total absorption coefficient is bounded and that the ratio between the coefficients of scattering and total absorption is uniformly bounded. As well we present numerical results for the coupled evolutive problem. Using the operational representation encountered in the course of establishing the existence theory, we derive vector Green’s functions for the transport equation which allow us to solve numerically the coupled system.     

Legendre expansion of neutron flux over entire multilayer slab geometry

The monoenergetic integral transport equation for a multilayer slab geometry has been solved by Legendre expansion method. The method utilizes an expansion of the neutron flux over entire multilayer slab geometry in Legendre polynomials of the position co-ordinate (Single Expansion Method—SEM). This formulation is an extension of Carlvik's (1968) method for a homogeneous slab. Earlier (Raghav, 1984) the expansion of the neutron flux was done in each layer in Legendre polynomials of the position co-ordinate (Multi Expansion Method—MEM). The aim in this paper is to compare both the approaches of SEM and MEM. A few multilayer slab systems with vacuum boundary conditions have been selected for this purpose and K_eff, the effective multiplication factor of the system, has been compared.SEM requires the evaluation of the integrals where the limits are not −1 to +1 (as they are in MEM and where analytical expressions can be derived), in these cases we have derived recurrence relations (which are described in the Appendix) to evaluate such integrals.