Relationship of mode transitions and standing waves in helicon plasmas

Helicon wave plasma sources have the well-known advantages of high efficiency and high plasma density, with broad applications in many areas. The crucial mechanism lies with mode transitions, which has been an outstanding issue for years. We have built a fluid simulation model and further developed the Peking University Helicon Discharge code. The mode transitions, also known as density jumps, of a single-loop antenna discharge are reproduced in simulations for the first time. It is found that large-amplitude standing helicon waves (SHWs) are responsible for the mode transitions, similar to those of a resonant cavity for laser generation. This paper intends to give a complete and quantitative SHW resonance theory to explain the relationship of the mode transitions and the SHWs. The SHW resonance theory reasonably explains several key questions in helicon plasmas, such as mode transition and efficient power absorption, and helps to improve future plasma generation methods.     

Reversal of radial glow distribution in helicon plasma induced by reversed magnetic field

In this work,the reversal of radial glow distribution induced by reversed magnetic field is reported.Based on the Boswell antenna which is symmetric and insensitive to the magnetic field direction,it seems such a phenomenon in theory appears impossible.However,according to the diagnostic of the helicon waves by magnetic probe,it is found that the direction of magnetic field significantly affects the propagation characteristic of helicon waves,i.e.,the interchange of the helicon waves at the upper and the lower half of tube was caused by reversing the direction of magnetic field.It is suggested that the variation of helicon wave against the direction of magnetic field causes the reversed radial glow distribution.The appearance of the traveling wave does not only improve the discharge strength,but also determines the transition of the discharge mode.     

Fuel rod model based on Non-Fourier heat conduction equation

In this paper we explore the applicability of a fuel rod mathematical model based on Non-Fourier transient heat conduction as constitutive law for the Light Water Reactors transient analysis (LWRs). In the classical theory of diffusion, Fourier law of heat conduction is used to describe the relation between the heat flux vector and the temperature gradient assuming that the heat propagation speeds are infinite. The motivation for this research was to eliminate the paradox of an infinite thermal wave speed. The time-dependent heat sources were considered in the fuel rod heat transfer model. The close of the Main Steam Isolated Valves (MSIV) transient in a Boiling Water Reactor (BWR) was analyzed by different relaxation times. The results show that for long-times the heat fluxes on the clad surface under Non-Fourier approach can be important, while for short-times and from the engineering point of view the changes are very small. Some results from transient calculations are examined.     

Steady-state heat conduction in slabs, cylindrical and spherical shells with non-uniform heat generation

A comparison is made between temperature drops in cylindrical and spherical shells and in slabs in the cases of uniform and non-uniform internal heat generation with the same total amount of heat. One boundary is assumed to. be insulated; the rate of internal heat generation may be an arbitrary function of the radius. The (variational) ratio between the relative difference in temperature drops (in the non-uniform and uniform cases) and a (dimensionless) measure of the non-uniformity in heat generation is found to vary between narrow bounds which depend only upon the geometry. When the power density is better known, the bounds on the variational ratio get closer together. Similar derivations are made for the average temperature when one boundary is insulated or when both boundaries are at the same temperature. Kernels allowing the exact computation of temperatures of interest are listed in tables for various geometries and boundary conditions.     

Geodesic acoustic modes in tokamak plasmas with anisotropic distribution and a radial equilibrium electric field

The dispersion relation of standard geodesic acoustic modes in tokamak plasmas with anisotropic distribution and a radial equilibrium electric field is derived and analyzed. Both frequencies and damping rates increase with respect to the poloidal Mach number which indicates the strength of the radial electric field. The strength of anisotropy is denoted by the ratio of the parallel temperature(T_‖) to the perpendicular temperature(T_⊥). It is shown that, when the parallel temperature is lower than the perpendicular temperature, the enhanced anisotropy tends to enlarge the real frequency but reduces the damping rate, and when the parallel temperature is higher than the perpendicular temperature, the effect is opposite. The radial equilibrium electric field has stronger effect on the frequency and damping rate for the case with higher parallel temperature than the case with higher perpendicular temperature.     

Electron population properties with different energies in a helicon plasma source

The characteristics of electrons play a dominant role in determining the ionization and acceleration processes of plasmas. Compared with electrostatic diagnostics, the optical method is independent of the radio frequency(RF) noise, magnetic field, and electric field. In this paper, an optical emission spectroscope was used to determine the plasma emission spectra, electron excitation energy population distributions(EEEPDs), growth rates of low-energy and highenergy electrons, and their intensity jumps with input powers. The 56 emission lines with the highest signal-to-noise ratio and their corresponding electron excitation energy were used for the translation of the spectrum into EEEPD. One discrete EEEPD has two clear different regions,namely the low-energy electron excitation region(neutral lines with threshold energy of13–15 eV) and the high-energy electron excitation region(ionic lines with threshold energy?19 e V). The EEEPD variations with different diameters of discharge tubes(20 mm, 40 mm,and 60 mm) and different input RF powers(200–1800 W) were investigated. By normalized intensity comparison of the ionic and neutral lines, the growth rate of the ionic population was higher than the neutral one, especially when the tube diameter was less than 40 mm and the input power was higher than 1000 W. Moreover, we found that the intensities of low-energy electrons and high-energy electrons jump at different input powers from inductively coupled(H) mode to helicon(W) mode; therefore, the determination of W mode needs to be carefully considered.     

Observation of low-frequency oscillation in argon helicon discharge

We present here a kind of low-frequency oscillation in argon helicon discharge with a half helical antenna. This time-dependent instability shows a global quasi-periodic oscillation of plasma density and electron temperature, with a typical frequency of a few tens of Hz which increases with external magnetic field as well as radiofrequency(RF) power. The relative oscillation amplitude decreases with magnetic field and RF power, but the rising time and pulse width do not change significantly unde...     

Parametric study on rewetting velocities obtained with a two-dimensional heat conduction code

A comprehensive parametric study has been performed to quantify the effect of different variables on the rewetting velocity in a light water reactor following a loss-of-coolant accident. To this purpose, a numerical solution of the general two-dimensional (axial and radial) heat conduction equation in cylindrical geometry has been obtained. The method used is the alternating-direction implicit procedure developed by Peaceman and Rachford. The model accounts for decay heat generation in the fuel, coolant subcooling, different wall temperatures and different heat transfer coefficients across the gap and at the clad surface. The two-dimensional model can be reduced to a one-dimensional model by setting the heat conduction in either the radial or axial direction to zero. Results with the new model agree with previous models and with experimental data.The variables studied were: axial and/or radial heat conduction, clad temperature, quench temperature, coolant temperature, temperature for the onset of nucleate boiling, heat transfer coefficients, stored and decay heats, clad material and clad thickness. The critical thickness (clad thickness for which the calculated rewetting velocity remains constant) was also determined and found to be larger than the clad thickness of light water reactor fuel pins under usual reflood conditions. According to these calculations, the stored and decay heats affect the rewetting velocity significantly.     

A comparison of analytical and numerical solutions in heat conduction problems

This paper deals with a comparison of analytical and finite element results in two types of steady state diffusion problems: (a) determination of solutions in domains of complicated boundary shape, and (b) analysis diffusion-type problems in nonhomogeneous media.     

Assessment of the CATHARE 1D pump model

The development of an advanced model to determine the dynamic pump performance under two-phase flow conditions is presented. This model is included in CATHARE 2, version V1.3. It is based on the two-fluid six-equation CATHARE model which describes the mechanical and thermal non-equilibria.In a previous review (P. Van den Hove and G. Geffraye, The CATHARE code— one-dimension pump model, Fifth Int. Topical Meet. on Nuclear Reactor Thermal Hydraulics (NURETH-5), Salt Lake City, USA, September, 1992), various calculations were presented concerning Eva single-phase and two-phase steam–water test results in the first three quadrants. Here, the range of assessment of the first quadrant is enlarged with Eva air–water tests and Bethsy pump steam–water tests. Both pumps are mixed flow pumps, the Bethsy one being radial at the impeller outlet.Some improvements suggested in the above cited paper are tested against all single-phase liquid, single-phase vapor, two-phase steam–water, and two-phase air–water data in the first quadrant. They concern a new deviation model and head losses model, and the model of mechanical interaction between phases.     

Analysis of transient heat conduction with phase changes by a boundary integral equation method

A combined method of a latent heat model and a boundary integral equation is presented for solving transient heat conduction problems with solidification and melting. Some typical problems are solved in order to confirm the validity of the proposed method. The present results show good agreement with results obtained by using finite difference, finite element and boundary element methods. It is shown that the method admits the use of rather large time steps and a relatively coarse mesh. The computational property of the function which is introduced in deriving the boundary integral equation is discussed in connection with a condition for smooth motion of the interface.     

Feedback model of secondary electron emission in DC gas discharge plasmas

Feedback is said to exist in any amplifier when the fraction of output power in fed back as an input.Similarly,in gaseous discharge ions that incident on the cathode act as a natural feedback element to stabilize and self sustain the discharge.The present investigation is intended to emphasize the feedback nature of ions that emits secondary electrons(SEs)from the cathode surface in DC gas discharges.The average number of SEs emitted per incident ion and non ionic species(energetic neutrals,metastables and photons)which results from ion is defined as effective secondary electronemission coefficient(ESEEC,_Eg).In this study,we derive an analytic expression that corroborates the relation between_Eg and power influx by ion to the cathode based on the feedback theory of an amplifier.In addition,experimentally,we confirmed the typical positive feedback nature of SEEfrom the cathode in argon DC glow discharges.The experiment is done for three different cathode material of same dimension(tungsten(W),copper(Cu)and brass)under identical discharge conditions(pressure:0.45 mbar,cathode bias:-600 V,discharge gab:15 cm and operating gas:argon).Further,we found that the_Eg value of these cathode material controls the amount of feedback power given by ions.The difference in feedback leads different final output i.e the power carried by ion at cathode(P_i _C￠∣).The experimentally obtained value of P_i _C￠∣is 4.28 W,6.87 W and9.26 W respectively for W,Cu and brass.In addition,the present investigation reveals that the amount of feedback power in a DC gas discharges not only affect the fraction of power fed back to the cathode but also the entire characteristics of the discharge.     

Molecular dynamics calculations of heat conduction in actinide oxides under thermal gradient

Thermal conductivities of UO₂, PuO₂ and (U_0.8,Pu_0.2)O₂ have been investigated by non-equilibrium molecular dynamics (NEMD) simulation between 300 K and 2000 K. The thermal conductivity was directly calculated by the temperature gradient on the system according to Fourier's law in NEMD simulation. The thermal conductivity obtained from the NEMD simulation decreases with a decrease of the supercell size, which means the phonon scattering occurs at the system boundaries in the microsystem. In addition, the present NEMD simulation, as well as previous EMD simulation studies, clearly shows that the Umklapp process causes the decrease of thermal conductivity at high temperatures. When comparison is made with literature data, the calculated results obtained from the relatively small supercell are in good agreement with the measured ones for the above actinide dioxides.     

Some steady and unsteady heat conduction solutions in orthotropic circular cylinders

Simple, approximate solutions are obtained in the case of prescribed axially symmetric boundary conditions. The results are compared in one example with values obtained using a finite element code and good agreement is shown to exist. The problem is of interest in several fields of engineering and applied science, for instance when performing thermal analysis of nuclear fuel elements which in some instances exhibit thermally orthotropic characteristics.     

Real time application of filtering theory to one-dimensional heat conduction

This paper applies filtering of a distributed parameter system to one-dimensional heat conduction, to obtain information on the real time application of the estimation theory. This involves a trial to study a simple algorithm for estimating states of power reactors limited to a small number of sensors. The authors could estimate the transient axial temperature profile in a copper rod in real time using a small computer, NOVA . The filter and covariance equations studied were derived from the approximation of coupled ordinary differential equations obtained by the orthogonal collocation method with eigenfunction expansion. It was also a problem here to examine the feasibility of introducing the steady state covariance value to the equations which provided for the simplified design of the filter. The filter tested was found to converge on real time and produced reasonable estimates of the transient temperature profile in the copper rod even in the presence of contamination by a rather high level of measurement noise.     

Measurements of critical heat flux in CANDU 37-element bundle with a steep variation in radial power profile

An experiment has been performed to obtain dryout power measurements with a 37-element bundle string simulating radial power profiles of high-enriched fuel (which contains slightly higher enrichment than the CANDU fuel and is not the same as the traditional highly enriched uranium) and natural-uranium fuel. The electrically heated bundle string was cooled with Refrigerant-134a and installed vertically inside the test station. Occurrences of CHF were detected using sliding thermocouples installed inside each element. Measurements showed that the dryout power for the high-enriched fuel bundle is lower than that for the natural-uranium fuel bundle by 26%, on average. Similarly, the corresponding critical heat flux (CHF) values for the high-enriched fuel power profile are lower than those for the natural-uranium fuel power profile by about 48%. A correlation previously proposed for the effect of radial power profile on CHF underpredicts considerably the CHF for the high-enriched fuel power profile. The correlation has been revised to improve the prediction accuracy. The revised correlation represents closely the available database. It exhibits a correct asymptotic trend and hence can be extended to bundles with more severe variation in radial power profile than those covered in the current experiment.     

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.     

Finite integral transform method to solve asymmetric heat conduction in a multilayer annulus with time-dependent boundary conditions

The separation of variables (SOV) method has recently been applied to solve time-dependent heat conduction problem in a multilayer annulus. It is observed that the transverse (radial) eigenvalues for the solution in polar (r-θ) coordinate system are always real numbers (unlike in the case of similar multidimensional Cartesian problems where they may be imaginary) allowing one to obtain the solution analytically. However, the SOV method cannot be applied when the boundary conditions and/or the source terms are time-dependent, for example, in a nuclear fuel rod subjected to time-dependent boundaries or heat sources. In this paper, we present an alternative approach using the finite integral transform method to solve the asymmetric heat conduction problem in a multilayer annulus with time-dependent boundary conditions and/or heat sources. An eigenfunction expansion approach satisfying periodic boundary condition in the angular direction is used. After integral transformation and subsequent weighted summation over the radial layers, partial derivative with respect to r-variable is eliminated and, first order ordinary differential equations (ODEs) are formed for the transformed temperatures. Solutions of ODEs are then inverted to obtain the temperature distribution in each layer. Since the proposed solution requires the same eigenfunctions as those in the similar problem with time-independent sources and/or boundary conditions—a problem solved using the SOV method—it is also “free” from imaginary eigenvalues.     

Spontaneous magnetic field multipolar structure in toroidal plasmas based on 2D equilibrium

The multipolar magnetic field structure is investigated by the momentum conservation equation with self-consistent 3D sheared flows during transition of plasma properties from local paramagnetic to diamagnetic fields. Numerical results show that the traditional poloidal magnetic field (B_P) is one part of equilibrium magnetic fields. The non-zero-order quantities are originated from the higher-order terms of 2D equilibrium treatment based on a Fourier expansion of ψ (r, θ). The distributions of magnetic field vectors of the order of 1, 2, and 3 terms are presented respectively in two, four, and six polar fields with the local vortex structures (spontaneous magnetic connection). The excitation mechanisms of the magnetic vortices are the coupling effects of the magnetic fluid structure pattern and the toroidal effects. These results can help us understand the physical mechanism of the interaction between the external perturbation fields and control tearing modes, as well as the radial plasma flow and magnetic vortices.     

An axisymmetric rectangular element used in the stress analysis and heat conduction problem

The application of an axisymmetric element of rectangular cross section (AXSHER) to stress analysis and the steady-state heat conduction problem is considered. The shape functions are continuous at the inter-element boundaries up to the first derivatives. Several examples are presented to demonstrate the efficiency of the element for both thick and thin axisymmetric structures. The comparison between the results of different axisymmetric elements and the exact results confirms the accuracy and convergence of the element. Economic limitation in data preparation and computing time is achieved by the used formulation.