Quantity

Summary

Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit of measurement. Mass, time, distance, heat, and angle are among the familiar examples of quantitative properties. Quantity is among the basic classes of things along with quality, substance, change, and relation. Some quantities are such by their inner nature (as number), while others function as states (properties, dimensions, attributes) of things such as heavy and light, long and short, broad and narrow, small and great, or much and little. Under the name of multitude comes what is discontinuous and discrete and divisible ultimately into indivisibles, such as: army, fleet, flock, government, company, party, people, mess (military), chorus, crowd, and number; all which are cases of collective nouns. Under the name of magnitude comes what is c

Official source

https://en.wikipedia.org/wiki/Quantity

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Aluminium is a metal sought in the industry because of its various physical properties. It is produced by an electrolysis reduction process in large cells. In these cells, a large electric current goes through the electrolytic bath and the liquid aluminium. This electric current generates electromagnetic forces that set the bath and the aluminium into motion. Moreover, large quantities of carbon dioxide gas are produced through chemical reactions in the electrolytic bath: the presence of these gases alleviates the density of the liquid bath and changes the dynamics of the flow. Accurate knowledge of this fluid flow is essential to improve the efficiency of the whole process.The purpose of this thesis is to study and approximate the interaction of carbon dioxide with the fluid flow in the aluminium electrolysis process.In the first chapter of this work, a mixture-averaged model is developed for mixtures of gas and liquid. The model is based on the conservation of mass and momentum equations of the two phases, liquid and gas. By combining these equations, a system is established that takes into account the velocity of the liquid-gas mixture, the pressure, the gas velocity and the local gas concentration as unknowns.In the second chapter, a simplified problem is studied theoretically. It is shown that under the assumption that the gas concentration is small, the problem is well-posed. Moreover, we prove a priori error estimates of a finite element approximation of this problem.In the third chapter, we compare this liquid-gas model with a water column reactor experiment. Finally, the last chapter shows that the fluid flow is changed in aluminium electrolysis cells when we take into account the density of the bath reduced by carbon dioxide. These changes are quantified as being of the order of 30% and explain partially the differences between previous models and observations from Rio Tinto Aluminium engineers.

EPFL2022

Linear Recursive State Estimation of Hybrid and Unbalanced AC/DC Micro-Grids using Synchronized Measurements

Willem Lambrichts, Mario Paolone

In this paper, we present an exact (i.e. non-approximated) and linear measurement model for hybrid AC/DC micro-grids for recursive state estimation (SE). More specifically, an exact linear model of a voltage source converter (VSC) is proposed. It relies on the complex VSC modulation index to relate the quantities at the converters DC side to the phasors at the AC side. The VSC model is derived from a transformer-like representation and accounts for the VSC conduction and switching losses. In the case of three-phase unbalanced grids, the measurement model is extended using the symmetrical component decomposition where each sequence individually affects the DC quantities. Synchronized measurements are provided by phasor measurement units and DC measurement units in the DC system. To make the SE more resilient to vive step changes in the grid states, an adaptive Kalman Filter that uses an approximation of the prediction-error covariance estimation method is proposed. This approximation reduces the computational speed significantly with only a limited reduction in the SE performance. The hybrid SE is validated in an EMTP-RV time-domain simulation of the CIGRE AC benchmark micro-grid that is connected to a DC grid using 4 VSCs. Bad data detection and identification using the largest normalised residual is assessed with respect to such a system. Furthermore, the proposed method is compared with a non-linear weighted least squares SE in terms of accuracy and computational time.

2022

Cartographie et spectroscopie des propriétés mécaniques à l'échelle du nanomètre par spectrométrie acoustique locale à fréquence variable

This work aims at completing the development of local mechanical spectroscopy as initiated by F. Oulevey during his thesis. More precisely, the measurement of the amplitude and phase lag of the strain as a function of the applied stress frequency bas to be settled. Parallel to the development of the technical aspects making such an acquisition possible, the model used to analyze the measured quantities is modified in order to describe the studied system more realistically. The local mechanical spectrometer is based on an Atomic Force Microscope (AFM) using an optical method to detect the deflection of the lever. The stress is applied locally on the sample by a transducer fixed at the base of the cantilever. The bandwidth of the microscope' s electronics does not allow one to detect the cantilever movement at frequencies higher than the MHz. Two different approaches are followed to overcome this limit. The first approach exploits the nonlinearity of the contact to down-convert, via mechanical mixing, the high-frequency signals into low-frequency ones. This work demonstrates the feasibility of this technique, as well as its limits, mainly due to the difficulty to analyze confidently the obtained results. The second approach takes advantage of the stroboscopic detection of the movement of the cantilever. The intensity of the laser used to detect the deflection of the cantilever is modulated at a frequency close to the applied stress frequency. The amplitude of the signal recorded at the difference frequency corresponds then to the high-frequency amplitude of the cantilever movement. The frequency range attainable with this method is only limited by the bandwidth of the transducer providing the excitation. The model connecting the measured quantities (strain amplitude and phase lag) to the desired mechanical properties (elasticity and damping) consists in a beam clamped at one end, with a tip attached at some point along its length. The tip itself is connected to the sample via a spring and a dashpot in parallel, representing the normal interaction, and a spring and a dashpot in parallel representing the interaction in the plane of the sample. Combined with the stroboscopic detection, this model enables one to quantitatively determine the elastic modulus of stiff samples. Many results obtained on a wide range of samples exhibiting very diverse mechanical properties are presented. They demonstrate the validity of the stroboscopic approach, which combines simplicity of use and efficiency.

EPFL2000

Cartographie et spectroscopie des propriétés mécaniques à l'échelle du nanomètre par spectrométrie acoustique locale à fréquence variable

EPFL2000