Difference equation approach to two-thermocouple sensor characterisation in constant velocity flow environments

Thermocouples are one of the most popular devices for temperature measurement due to their robustness, ease of manufacture and installation, and low cost. However, when used in certain harsh environments, for example, in combustion systems and engine exhausts, large wire diameters are required, and consequently the measurement bandwidth is reduced. This article discusses a software compensation technique to address the loss of high frequency fluctuations based on measurements from two thermocouples. In particular, a difference equation sDEd approach is proposed and compared with existing methods both in simulation and on experimental test rig data with constant flow velocity. It is found that the DE algorithm, combined with the use of generalized total least squares for parameter identification, provides better performance in terms of time constant estimation without any a priori assumption on the time constant ratios of the thermocouples.

Exploiting a priori time constant ratio information in difference equation two-thermocouple sensor characterisation

The characterization of thermocouple sensors for temperature measurement in varying-flow environments is a challenging problem. Recently, the authors introduced novel difference-equation-based algorithms that allow in situ characterization of temperature measurement probes consisting of two-thermocouple sensors with differing time constants. In particular, a linear least squares (LS) lambda formulation of the characterization problem, which yields unbiased estimates when identified using generalized total LS, was introduced. These algorithms assume that time constants do not change during operation and are, therefore, appropriate for temperature measurement in homogenous constant-velocity liquid or gas flows. This paper develops an alternative ß-formulation of the characterization problem that has the major advantage of allowing exploitation of a priori knowledge of the ratio of the sensor time constants, thereby facilitating the implementation of computationally efficient algorithms that are less sensitive to measurement noise. A number of variants of the ß-formulation are developed, and appropriate unbiased estimators are identified. Monte Carlo simulation results are used to support the analysis.

Solving the advection-dispersion equation using the discrete puff particle method

A flexible, mass-conservative numerical technique for solving the advection-dispersion equation for miscible contaminant transport is presented. The method combines features of puff transport models from air pollution studies with features from the random walk particle method used in water resources studies, providing a deterministic time-marching algorithm which is independent of the grid Peclet number and scales from one to higher dimensions simply. The concentration field is discretised into a number of particles, each of which is treated as a point release which advects and disperses over the time interval. The dispersed puff is itself discretised into a spatial distribution of particles whose masses can be pre-calculated. Concentration within the simulation domain is then calculated from the mass distribution as an average over some small volume. Comparison with analytical solutions for a one-dimensional fixed-duration concentration pulse and for two-dimensional transport in an axisymmetric flow field indicate that the algorithm performs well. For a given level of accuracy the new method has lower computation times than the random walk particle method.

A Van der Pol-Mathieu equation for the dynamics of dust grain charge in dusty plasmas

The chaotic profile of dust grain dynamics associated with dust-acoustic oscillations in a dusty plasma is considered. The collective behaviour of the dust plasma component is described via a multi-fluid model, comprising Boltzmann distributed electrons and ions, as well as an equation of continuity possessing a source term for the dust grains, the dust momentum and Poisson's equations. A Van der Pol–Mathieu-type nonlinear ordinary differential equation for the dust grain density dynamics is derived. The dynamical system is cast into an autonomous form by employing an averaging method. Critical stability boundaries for a particular trivial solution of the governing equation with varying parameters are specified. The equation is analysed to determine the resonance region, and finally numerically solved by using a fourth-order Runge–Kutta method. The presence of chaotic limit cycles is pointed out.

Difference equation sensor characterization algorithms for two-thermocouple probes

The characterization of thermocouple sensors for temperature measurement in variable flow environments is a challenging problem. In this paper, novel difference equation-based algorithms are presented that allow in situ characterization of temperature measurement probes consisting of two-thermocouple sensors with differing time constants. Linear and non-linear least squares formulations of the characterization problem are introduced and compared in terms of their computational complexity, robustness to noise and statistical properties. With the aid of this analysis, least squares optimization procedures that yield unbiased estimates are identified. The main contribution of the paper is the development of a linear two-parameter generalized total least squares formulation of the sensor characterization problem. Monte-Carlo simulation results are used to support the analysis.

Combined R-matrix eigenstate basis set and finite-difference propagation method for the time-dependent Schrodinger equation: The one-electron case

In this work we present the theoretical framework for the solution of the time-dependent Schrödinger equation (TDSE) of atomic and molecular systems under strong electromagnetic fields with the configuration space of the electron’s coordinates separated over two regions; that is, regions I and II. In region I the solution of the TDSE is obtained by an R-matrix basis set representation of the time-dependent wave function. In region II a grid representation of the wave function is considered and propagation in space and time is obtained through the finite-difference method. With this, a combination of basis set and grid methods is put forward for tackling multiregion time-dependent problems. In both regions, a high-order explicit scheme is employed for the time propagation. While, in a purely hydrogenic system no approximation is involved due to this separation, in multielectron systems the validity and the usefulness of the present method relies on the basic assumption of R-matrix theory, namely, that beyond a certain distance (encompassing region I) a single ejected electron is distinguishable from the other electrons of the multielectron system and evolves there (region II) effectively as a one-electron system. The method is developed in detail for single active electron systems and applied to the exemplar case of the hydrogen atom in an intense laser field.

Fokker-Planck equation for a test-particle weakly coupled to a magnetized plasma

We consider the derivation of a kinetic equation for a charged test particle weakly interacting with an electrostatic plasma in thermal equilibrium, subject to a uniform external magnetic field. The Liouville equation leads to a generalized master equation to second order in the `weak' interaction; a Fokker-Planck-type equation then follows as a `Markovian' approximation. It is shown that such an equation does not preserve the positivity of the distribution function f(x,v;t). By applying techniques developed in the theory of open systems, a correct Fokker-Planck equation is derived. Explicit expressions for the diffusion and drift coefficients, depending on the magnetic field, are obtained.

A comparison of nonstaggered compact FDTD schemes for the 3D wave equation

This paper aims at providing a better insight into the 3D approximations of the wave equation using compact finite-difference time-domain (FDTD) schemes in the context of room acoustic simulations. A general family of 3D compact explicit and implicit schemes based on a nonstaggered rectilinear grid is analyzed in terms of stability, numerical error, and accuracy. Various special cases are compared and the most accurate explicit and implicit schemes are identified. Further considerations presented in the paper include the direct relationship with other numerical approaches found in the literature on room acoustic modeling such as the 3D digital waveguide mesh and Yee's staggered grid technique.

Berry phase in open quantum systems: a quantum Langevin equation approach

The evolution of a two level system with a slowly varying Hamiltonian, modeled as a spin 1/2 in a slowly varying magnetic field, and interacting with a quantum environment, modeled as a bath of harmonic oscillators is analyzed using a quantum Langevin approach. This allows to easily obtain the dissipation time and the correction to the Berry phase in the case of an adiabatic cyclic evolution.

CRITICAL PROCESSES, LANGEVIN EQUATION AND UNIVERSALITY

In nature there are ubiquitous systems that can naturally approach critical states, The Langevin equation in the discrete version can be used to describe a class of critical processes, which are characterized by power-law behaviors and scaling relations. As an example, we present a simple model for a clinical thermometer, whose reading cannot fall even when its temperature decreases. The fibers bundle model and the spring-block model are also shown to belong to such a class.