Describing limits of integrable functions as grid functions of nonstandard analysis

In functional analysis, there are different notions of limit for a bounded sequence of $$L^1$$ L 1 functions. Besides the pointwise limit, that does not always exist, the behaviour of a bounded sequence of $$L^1$$ L 1 functions can be described in terms of its weak-$$\star $$ ⋆ limit or by introducing a measure-valued notion of limit in the sense of Young measures. Working in Robinson’s nonstandard analysis, we show that for every bounded sequence $$\{z_n\}_{n \in \mathbb {N}}$$ { z n } n ∈ N of $$L^1$$ L 1 functions there exists a function of a hyperfinite domain (i.e. a grid function) that represents both the weak-$$\star $$ ⋆ and the Young measure limits of the sequence. This result has relevant applications to the study of nonlinear PDEs. We discuss the example of an ill-posed forward–backward parabolic equation.

Virtualization Management Concept for Flexible and Fault-Tolerant Smart Grid Service Provision

In modern power systems, reliable provision of grid services (e.g., primary and ancillary services) are highly dependent on automation systems in order to have monitoring, processing, decision making and communication capabilities. The operational flexibility of automation systems is essential for the reliable operation of power systems during and after disruptive events. However, this is restricted by integrated hardware-software platforms. Therefore, it will be difficult to reconfigure control strategies during run time. This paper presents the concept of Grid Function Virtualization (GFV) as a potential approach to improve the operational flexibility of grid automation systems. GFV has been proposed to offer a new way to deploy and manage grid services by leveraging virtualization technology. The main idea of GFV is to run grid services (i.e., software implementation of services) independently from underlying hardware. To realize the important design considerations, the GFV architecture and its building blocks is elaborated in details. To this end, an exhaustive review of applications of virtualization in several domains is provided to show the importance of virtualization in improving flexibility and resource utilization. Finally, the advantages of the proposed concept to deal with disruptions in power systems is demonstrated in a proof of concept based on a CIGRE MV benchmark grid.

Experimental Investigation on Heat Transfer of Supercritical Water Flowing in the Subchannel with Grid Spacer in Supercritical Water-Cooled Reactor

Experimental investigations on the heat transfer performance of supercritical water flowing in the subchannel of supercritical water-cooled reactor (SCWR) simulated by a triangular channel were conducted at pressures of 23–28 MPa, mass flow rates of 700–1300 kg·m−2·s−1, and inner wall surface heat fluxes of 200–600 kW·m−2. An 8 mm diameter fuel rod with a 1.4 pitch to diameter ratio was used. The effects of pressure, mass flow rate, and heat flux on the heat transfer performance under the resistance of a standard grid spacer were analyzed. Experimental results showed the significant positive influence of the grid spacer on the supercritical water in the subchannel. Moreover, in the presence of the grid spacer, the parameters influenced the heat transfer with different degrees of strengthening reaction. In view of the phenomenon in the tests, the rule of the supercritical heat transfer was further revealed by the comparison between empirical formulas and experimental data. This paper mainly studied the positioning grid function and the fluid flow characteristics downstream of the subchannel under the influence of the standard grid spacer and the impact mechanism of each parameter on the whole heat transfer process coefficient.

NUMERICAL SIMULATION OF THE PROPAGATION OF A GAS SHOCK WAVE IN ELECTRICALLY CHARGED AND NEUTRAL GAS SUSPENSION IN A FLAT CHANNEL

In this paper, we consider the propagation of a shock wave from a pure gas into a heterogeneous mixture consisting of solid particles suspended in a gas and having an electric charge. The applied mathematical model takes into account the speed and thermal interaction of the carrier and dispersed components of the mixture. The force interaction of particles and gas was described by the Stokes force. The carrier medium was described as a viscous compressible heat–conducting gas. The equations of the mathematical model were solved by the explicit finite–difference method of the second order of accuracy, using the non–linear correction of the grid function. The system of equations of the mathematical model was supplemented by boundary and initial conditions for the desired functions. As a result of numerical simulation, it was found that in an electrically charged gas suspension there is a difference in gas pressure and velocity, “average density” and velocity of the dispersed component, compared with similar values in a gas suspension with an electrically neutral dispersed component. The revealed differences in the dynamics of neutral and electrically charged dusty media can be explained by the fact that the dispersed component of an electrically charged gas suspension is affected by both aerodynamic drag forces and Coulomb forces. Due to interfacial interaction, the dynamics of the carrier medium changes.

Numerical study of velocity slip of phases during the passage of a shock wave of low intensity from a pure gas to a dusty medium

In this paper, the process of the movement of a direct shock wave from a pure gas into a dusty medium is numerically modeled. The mathematical model took into account the viscosity, compressibility and thermal conductivity of the carrier phase. Also, the modeling technique made it possible to describe the interphase force interaction, which included the Stokes force, the dynamic force of Archimedes, the strength of the attached masses. In addition, interfacial interaction included heat transfer between the carrier and dispersed phases. The numerical solution was carried out using the explicit finite-difference method, with the subsequent application of the nonlinear correction scheme for the grid function. As a result of numerical calculations, it was revealed that with an increase in the linear particle size of the gas suspension, the velocity slip between the carrier and dispersed phases increases. Numerical modeling also showed that the absolute value of the difference between the velocities of the carrier and the dispersed phase reaches the largest value at the leading edge of the compression wave. The revealed regularities can be explained by the fact that the particles of the dispersed phase are assumed to be spherical in shape. Due to this, a multiple increase in particle size leads to a three-fold increase in their mass, a twofold increase in the area of one particle and a three-fold decrease in the number of particles. Thus, an increase in particle size leads to a decrease in the area of interfacial contact and an increase in the inertia of the particles, which in turn affects the interfacial velocity slip.