J. Dorning

Some nonlinear dynamics of a heated channel

Linear and nonlinear mathematical stability analyses of parallel channel density wave oscillations are reported. The two phase flow is represented by the drift flux model. A constant characteristic velocity v0∗ is used to make the set of equations dimensionless to ensure the mutual independence of the dimensionless variables and parameters: the steady-state inlet velocity v, the inlet subcooling number Nsub and the phase change number Npch. The exact equation for the total channel pressure drop is perturbed about the steady-state for the linear and nonlinear analyses. The surface defining the marginal stability boundary (MSB) is determined in the three-dimensional equilibrium-solution/operating-parameter space v − Nsub − Npch. The effects of the void distribution parameter C0 and the drift velocity Vgj on the MSB are examined. The MSB is shown to be sensitive to the value of C0 and comparison with experimental data shows that the drift flux model with C0 > 1 predicts the experimental MSB and the neighboring region of stable oscillations (limit cycles) considerably better than do the homogeneous equilibrium model (C0 = 1, Vgj = 0) or a slip flow model. The nonlinear analysis shows that supercritical Hopf bifurcation occurs for the regions of parameter space studied; hence stable oscillatory solutions exist in the linearly unstable region in the vicinity of the MSB. That is, the stable fixed point v becomes unstable and bifurcates to a stable limit cycle as the MSB is crossed by varying Nsub and/or Npch.

Stability analysis of BWR nuclear-coupled thermal-hydraulics using a simple model

Abstract A simple mathematical model is developed to describe the dynamics of the nuclear-coupled thermal-hydraulics in a boiling water reactor (BWR) core. The model, which incorporates the essential features of neutron kinetics and single-phase and two-phase thermal-hydraulics, leads to a simple dynamical system comprised of a set of nonlinear ordinary differential equations (ODEs). The stability boundary is determined and plotted in the inlet-subcooling-number (enthalpy)/external-reactivity operating parameter plane. The eigenvalues of the Jacobian matrix of the dynamical system also are calculated at various steady-states (fixed points); the results are consistent with those of the direct stability analysis and indicate that a Hopf bifurcation occurs as the stability boundary in the operating parameter plane is crossed. Numerical simulations of the time-dependent, nonlinear ODEs are carried out for selected points in the operating parameter plane to obtain the actual damped and growing oscillations in the neutron number density, the channel inlet flow velocity, and the other phase variables. These indicate that the Hopf bifurcation is subcritical, hence, density wave oscillations with growing amplitude could result from a finite perturbation of the system even when it is being operated in the parameter region thought to be safe, i.e. where the steady-state is stable. Finally, the power-flow map, frequently used by reactor operators during start-up and shut-down operation of a BWR, is mapped to the inlet-subcooling-number/neutron-density (operating-parameter/phase-variable) plane, and then related to the stability boundaries for different fixed inlet velocities corresponding to selected points on the flow-control line. Also, the stability boundaries for different fixed inlet subcooling numbers corresponding to those selected points, are plotted in the neutron-density/inlet-velocity phase variable plane and then the points on the flow-control line are related to their respective stability boundaries in this plane. The relationship of the operating points on the flow-control line to their respective stability boundaries in these two planes provides insight into the instability observed in BWRs during low-flow/high-power operating conditions. It also shows that the normal operating point of a BWR is very stable in comparison with other possible operating points on the power-flow map.

Systematic Homogenization and Self-Consistent Flux and Pin Power Reconstruction for Nodal Diffusion Methods —I: Diffusion Equation-Based Theory

A diffusion equation-based systematic homogenization theory and a self-consistent dehomogenization theory for fuel assemblies have been developed for use with coarse-mesh nodal diffusion calculations of light water reactors. The theoretical development is based on a multiple-scales asymptotic expansion carried out through second order in a small parameter, the ratio of the average diffusion length to the reactor characteristic dimension. By starting from the neutron diffusion equation for a three-dimensional heterogeneous medium and introducing two spatial scales, the development systematically yields an assembly-homogenized global diffusion equation with self-consistent expressions for the assembly-homogenized diffusion tensor elements and cross sections and assembly-surface-flux discontinuity factors. The rector eigenvalue 1/k{sub eff} is shown to be obtained to the second order in the small parameter, and the heterogeneous diffusion theory flux is shown to be obtained to leading order in that parameter. The latter of these two results provides a natural procedure for the reconstruction of the local fluxes and the determination of pin powers, even though homogenized assemblies are used in the global nodal diffusion calculation.

Studies on Nodal Integral Methods for the Convection-Diffusion Equation

Abstract The computational efficiencies of two nodal integral methods for the numerical solution of linear convection-diffusion equations are studied. Although the first, which leads to a second-order spatial truncation error, has been reported earlier, it is reviewed in order to lead logically to the development here of the second, which has a third-order error. This third-order nodal integral method is developed by introducing an upwind approximation for the linear terms in the “pseudo-sources” that appear in the transverse-averaged equations introduced in the formulation of nodal integral methods. This upwind approximation obviates the need to develop and solve additional equations for the transverse-averaged first moments of the unknown, as would have to be done in a more straightforwardly developed higher-order nodal integral method. The computational efficiencies of the second-order nodal method and the third-order nodal method—of which there are two versions: one, a full third-order method and the other, which uses simpler second-order equations near the boundaries—are compared with those of both a very traditional method and a recently developed state-of-the-art method. Based on the comparisons reported here for a challenging recirculating flow benchmark problem it appears that, among the five methods studied, the second-order nodal integral method has the highest computational efficiency (the lowest CPU computing times for the same accuracy requirements) in the practical 1% error regime.

Undamped longitudinal plasma waves

Abstract We have derived necessary and sufficient conditions for the existence of undamped nonlinear plasma waves in one dimension. The analysis establishes the existence of exact nonlinear plasma waves which are predicted to Landau damp according to the standard linear theory, but which in fact do not damp.

Time-asymptotic traveling-wave solutions to the nonlinear Vlasov-Poisson-Ampere equations

We consider the Vlasov–Poisson–Ampere system of equations, and we seek solutions for the electric field E(x,t) that are periodic in space and asymptotically almost periodic in time. Introducing the representation E(x,t)=T(x,t)+A(x,t) (where T and A are, respectively, the transient and time-asymptotic parts of E) enables us to decompose the nonlinear Poisson equation into a transient equation and a time-asymptotic equation. We then study the latter in isolation as a bifurcation problem for A with the initial condition and T as parameters. We show that the Frechet derivative at a generic bifurcation point has a nontrivial null space determined by the roots of a Vlasov dispersion relation. Hence, the bifurcation analysis leads to a general solution for A given (at leading order) by a discrete superposition of traveling-wave modes, whose frequencies and wave numbers satisfy the Vlasov dispersion relation, and whose amplitudes satisfy a system of nonlinear algebraic equations. In applications, there is usually...

Nonlinear but Small Amplitude Longitudinal Plasma Waves

This paper describes new and rigorous results on small amplitude spatially periodic traveling wave solutions of the kinetic equations that govern the evolution of collisionless plasmas in one dimension. The exact nonlinear analysis described here shows that it is in fact possible for electrostatic waves to travel through a spatially uniform background without decaying, even in the absence of a particle beam or “bump-on-the-tail” of the equilibrium particle distribution function. The results include the derivation of conditions necessary and sufficient for the existence of traveling wave solutions which are small perturbations on a spatially uniform background plasma; they also include the construction of approximations to these solutions. This nonlinear analysis of small amplitude waves shows that, in contrast to the standard linear analysis, spatially periodic traveling waves are quite common perturbations to all spatially uniform plasma equilibria. The damping of the electric field commonly predicted by the linear theory need not always occur. The conclusions of the linear analysis do not apply to the undamped waves reported here because the term neglected in the linear analysis is not small, even though the electric field is small and the particle distribution functions that describe the waves, which always include trapped particles, represent only a minor rearrangement of the particles in the background distribution.

Systematic homogenization and self-consistent flux and pin power reconstruction for nodal diffusion methods part II: Transport equation based theory

Abstract Starting from the transport equation, a systematic homogenization theory and a self-consistent de-homogenization theory for fuel assemblies have been developed using a multiple-scales asymptotic expansion method. The resulting theory provides a framework for coarse-mesh nodal diffusion calculations of light water reactors. The theoretical development is carried out through second order in a small parameter—the ratio of the neutron mean free path to the reactor characteristic dimension. Introducing two spatial scales—a fast scale for the rapid variation of the flux over a fuel assembly and a slow scale for the slow variation of the flux over the whole core—into the neutron transport equation for a three-dimensional heterogeneous medium, the development systematically yields: an assembly-homogenized global diffusion equation with self-consistent expressions for the assembly-homogenized diffusion tensor elements and cross sections, and assembly-surface flux discontinuity factors. The analysis shows ...

Nonlinear waves in collisionless plasmas

Abstract We report results showing that arbitrarily close to any spatially uniform equilibrium of the one-dimensional Vlasov-Maxwell equations for collisionless plasmas there exist exact traveling solitary wave and traveling periodic wave solutions which, by virtue of particle trapping, neither damp nor grow even when their amplitude is arbitrarily small.

Thermal-Neutron Decay in Small Systems

Experiments of critical heat flux(CHF) under natural circulation condition in narrow rectangular channel were conducted in Natural Circulation Laboratory of North China Electric Power University.The phenomena of flow stagnation and heat transfer deterioration were observed during the experiments.The developing mechanism of flow stagnation/ heat transfer deterioration in narrow rectangular channel under saturation condition was proposed.Flow pattern instability would appear when the flow excursion occurred,thus leading to the continuous flow oscillation and flow stagnation.Then the liquid layer near the outlet would completely evaporate under a certain heat condition,and the CHF occurred.While in narrow rectangular channel,the vapor stream could produce extrusion action to the liquid layer on the heating wall due to the limitation caused by gap size,which cause the liquid to become very thin and heat transfer deterioration could happen under a much lower heat flux condition.Based on the mechanistic analysis,a corresponding calculation model was provided.A dimensionless number of constraints and a natural circulation characteristic factor were brought in to optimize the proposed model for the purpose of considering the effects of gap size and circulation mode in the model respectively.The multiple regressions were applied to the calculation model according to the experimental results,and the accuracy of which was validated.The comparison between experimental results and model calculation values indicated that CHF would increase with the inlet velocity and system pressure increased,and would decrease with the outlet quality increased.