Gregory M. Lewis

Configuration determination for k-out-of-n partially redundant systems

The k-out-of-n configuration is a widely adopted structure for partially redundant safety systems. This configuration ensures a high level of reliability and safety with limited financial and space resources. It also facilitates on-line Test and Maintenance (T&M) without having to shut the system down. One question a decision maker needs to answer when adopting k-out-of-n systems is: what is the best configuration for the application, i.e. how many channels in total need to be utilized and among these channels, how many channels need to function simultaneously in order for the system to function. There are various factors to consider in order to make this decision. This paper looks at this problem from a reliability engineers point of view. A quantitative analysis is performed for both unavailability and probability of spurious operation due to independent failure. In particular, the relative gain and/or loss of these quantities that occurs due to changing from one configuration to another are compared through rigorous mathematical analysis. The results provide important information that can be used when choosing system configurations to meet regulatory requirements and financial constraints. The two different configurations for shutdown systems in Nuclear Power Plants, the 2-out-of-3 system and the 2-out-of-4 system, are utilized as an example to illustrate the theoretical results.

Linear stability analysis for the differentially heated rotating annulus

We use linear stability analysis to approximate the axisymmetric to nonaxisymmetric transition in the differentially heated rotating annulus. We study an accurate mathematical model that uses the Navier–Stokes equations in the Boussinesq approximation. The steady axisymmetric solution satisfies a two-dimensional partial differential boundary value problem. It is not possible to compute the solution analytically, and thus, numerical methods are used. The eigenvalues are also given by a two-dimensional partial differential problem, and are approximated using the matrix eigenvalue problem that results from discretizing the linear part of the appropriate equations. A comparison is made with experimental results. It is shown that the predictions using linear stability analysis accurately reproduce many of the experimental observations. Of particular interest is that the analysis predicts cusping of the axisymmetric to nonaxisymmetric transition curve at wave number transitions, and the wave number maximum along the lower part of the axisymmetric to nonaxisymmetric transition curve is accurately determined. The correspondence between theoretical and experimental results validates the numerical approximations as well as the application of linear stability analysis. A linear stability analysis is also performed with the effects of centrifugal buoyancy neglected. Along the lower part of the transition curve, the results are significantly qualitatively and quantitatively different than when the centrifugal effects are considered. In particular, the results indicate that the centrifugal buoyancy is the cause of the observation of a wave number maximum along the transition curve, and is the cause of a change in concavity of the transition curve.

DOUBLE HOPF BIFURCATIONS IN THE DIFFERENTIALLY HEATED ROTATING ANNULUS

We study a mathematical model of the differentially heated rotating fluid annulus experiment. In particular, we analyze the double Hopf bifurcations that occur along the transition between axisymmetric steady solutions and nonaxisymmetric rotating waves. The model uses the Navier--Stokes equations in the Boussinesq approximation. At the bifurcation points, center manifold reduction and normal form theory are used to deduce the local behavior of the full system of partial differential equations from a low-dimensional system of ordinary differential equations.It is not possible to compute the relevant eigenvalues and eigenfunctions analytically. Therefore, the linear part of the equations is discretized, and the eigenvalues and eigenfunctions are approximated from the resulting matrix eigenvalue problem. However, the projection onto the center manifold and reduction to normal form can be done analytically. Thus, a combination of analytical and numerical methods is used to obtain numerical approximations of ...

Hysteresis in a Rotating Differentially Heated Spherical Shell of Boussinesq Fluid

A mathematical model of convection of a Boussinesq fluid in a rotating spherical shell is analyzed using numerical computations guided by bifurcation theory. The fluid is differentially heated on its inner spherical surface, with the temperature increasing from both poles to a maximum at the equator. The model is assumed to be both rotationally symmetric about the polar axis and reflectionally symmetric across the equator. This work is an extension to spherical geometry of previous work on the differentially heated rotating annulus. The spherical geometry is motivated by applications to planetary atmospheres. As the temperature gradient increases from zero, large Hadley cells extending from equator to poles form immediately. For larger temperature differences, two or three convection cells appear in each hemisphere. An organizing center is shown to exist, at which two saddle-node bifurcations come together in a codimension-2 hysteresis bifurcation (or cusp) point, providing a mechanism for hysteretic tran...

The primary flow transition in a differentially heated rotating channel of fluid with O(2) symmetry

Abstract The primary flow transition in a periodic differentially heated rotating channel of fluid with O(2) symmetry is studied. This transition occurs when a time-independent flow that is uniform along the channel bifurcates to a stationary wave flow. The fluid is modelled using the Navier–Stokes equations in the Boussinesq approximation and the flow transition is found using linear stability analysis. The computation of the flow transition curve is performed efficiently by replacing the relevant eigenvalue problem with an equivalent bordered linear system, and by implementing a pseudoarclength continuation strategy that is appropriate for large-scale systems. The dynamics of the fluid near the transition are deduced by applying centre manifold reduction and normal form theory. The reduction produces analytical expressions for the normal form coefficients in terms of functions that must be computed numerically. The results indicate that the transition to stationary wave flow occurs via a supercritical pitchfork bifurcation to a group orbit, sometimes referred to as a pitchfork of revolution. Furthermore, at several points along the transition curve two such pitchfork bifurcations occur simultaneously, which physically corresponds to the interaction of two stationary wave modes. An analysis of the normal form equations that are associated with the steady-state mode interactions shows the possibility of bistability and hysteresis of the stationary waves. Many of the results obtained in the channel model show a remarkable quantitative similarity to those of theoretical and experimental studies of analogous experiments using a cylindrical annulus, even though the difference in the symmetries of the systems ensure certain qualitative differences. This suggests that the dynamics of the fluid are dominated by the differential heating and the rotation, and not by the curvature of the system.

Spatiotemporal Model for Depth Perception in Electric Sensing

Electric sensing involves measuring the voltage changes in an actively generated electric field, enabling an environment to be characterized by its electrical properties. It has been applied in a variety of contexts, from geophysics to biomedical imaging. Some species of fish also use an active electric sense to explore their environment in the dark. One of the primary challenges in such electric sensing involves mapping an environment in three-dimensions using voltage measurements that are limited to a two-dimensional sensor array (i.e. a two-dimensional electric image). In some special cases, the distance of simple objects from the sensor array can be estimated by combining properties of the electric image. Here, we describe a novel algorithm for distance estimation based on a single property of the electric image. Our algorithm can be implemented in two simple ways, involving either different electric field strengths or different sensor thresholds, and is robust to changes in object properties and noise.