scholarly journals Geometric Algebra Framework Applied to Symmetrical Balanced Three-Phase Systems for Sinusoidal and Non-Sinusoidal Voltage Supply

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1259
Author(s):  
Francisco G. Montoya ◽  
Raúl Baños ◽  
Alfredo Alcayde ◽  
Francisco Manuel Arrabal-Campos ◽  
Javier Roldán Roldán Pérez
Keyword(s):  
Geometric Algebra ◽  
Dimensional Euclidean Space ◽  
Active Components ◽  
Electrical Circuits ◽  
Current Harmonics ◽  
Operation Conditions ◽  
Multiple Phase ◽  
Three Phase ◽  
Phase Systems ◽  
Definition Of

This paper presents a new framework based on geometric algebra (GA) to solve and analyse three-phase balanced electrical circuits under sinusoidal and non-sinusoidal conditions. The proposed approach is an exploratory application of the geometric algebra power theory (GAPoT) to multiple-phase systems. A definition of geometric apparent power for three-phase systems, that complies with the energy conservation principle, is also introduced. Power calculations are performed in a multi-dimensional Euclidean space where cross effects between voltage and current harmonics are taken into consideration. By using the proposed framework, the current can be easily geometrically decomposed into active- and non-active components for current compensation purposes. The paper includes detailed examples in which electrical circuits are solved and the results are analysed. This work is a first step towards a more advanced polyphase proposal that can be applied to systems under real operation conditions, where unbalance and asymmetry is considered.

The Effect of Several Polymers on the Phase Behavior of Micellar Fluids

1982 ◽  
Vol 22 (06) ◽  
pp. 816-830 ◽  
Author(s):  
Gary A. Pope ◽  
Kerming Tsaur ◽  
Robert S. Schechter ◽  
Ben Wang
Keyword(s):  
Phase Behavior ◽  
Nonionic Surfactants ◽  
Polymer Concentration ◽  
Salinity Range ◽  
Lower Phase ◽  
Optimal Salinity ◽  
Rich Phase ◽  
Three Phase ◽  
Phase Systems ◽  
Increasing Temperature

Abstract We made static measurements of the phase volumes of mixtures of surfactant, polymer, alcohol, water, oil, sodium chloride, and in some cases polymer additives. We also made a limited number of viscosity, phase concentration, and interfacial tension (IFT) measurements. The purpose was to determine systematically the effect of various polymers on the phase behavior of various surfactant formulations. We made measurements with and without oil (n-octane and n-octane/benzene mixtures) across a range of salinity appropriate to the particular surfactant at temperatures between 24 and 75 degrees C. Introduction The oil-free (i.e., no added oil) solutions showed a characteristic phase separation into an aqueous surfactant-rich phase and an aqueous polymer-rich phase at some sufficiently high salinity (NaCl concentration), which we call the critical electrolyte concentration (CEC). The CEC was found to be a characteristic of a given surfactant/alcohol combination that shifts with the solubility of the surfactant qualitatively the same way as does the optimal salinity: but the CEC was found independent of the polymer type, polymer concentration (between the 100- and 1,000-ppm limits investigated), and surfactant concentration. The CEC increases with increasing temperature for the anionic surfactants and decreases with increasing temperature for the nonionic surfactants. When oil was added to the mixtures, an entirely different pattern of phase behavior was observed. As salinity increases, the particular formulations form the typical sequence of lower-phase microemulsion and excess oil, middle-phase microemulsion. excess oil, and excess brine: and upper-phase microemulsion and excess brine. The sequence with polymer was precisely the same over most of the salinity range but deviated over a limited range of salinity; the three-phase region simply shifted a small distance to the left on the salinity scale. Also, and probably more significantly, some of the aqueous phases in the critical region of the shift (which is also just above oil-free CEC salinity) were found to be gel-like in nature. These apparently occur under conditions such that the polymer concentration in the excess brine of the three-phase systems becomes very high because almost all the polymer is always in the brine phase, even when the brine phase is very small. Thus an overall 1,000 ppm of polymer easily can be concentrated to 10,000 ppm or more. One of the most remarkable aspects of the phase behavior of the surfactant/polymer systems is that the same patterns are observed for all combinations of anionic and nonionic surfactants and polymers. Also, little difference was observed in the IFT values with and without polymer. The three-phase systems still exhibited ultralow IFT values. Obviously, significant differences did occur in the brine viscosities when polymer was added. The polymer-free mixtures were themselves quite viscous, however, and the viscosity of the oil-free surfactant-rich phases (above the CEC) was significantly higher when the phases were in equilibrium with a polymer-rich aqueous phase, even though they apparently contained almost no polymer. We found that the polymer-rich phases had normal viscosities, as judged by the same polymer in the same brine at the expected concentration, assuming all the polymer was in the polymer-rich phase. The effect of polymer on the systems with oil was to increase the viscosity of the water-rich phase only, with little effect on the microemulsion phase unless it was the water-rich phase. SPEJ P. 816^

Calculation of the Phase-Angle-Jump for Voltage Dips in Three-Phase Systems

2015 ◽  
Vol 30 (1) ◽  
pp. 480-487 ◽  
Author(s):  
Ying Wang ◽  
Math H. J. Bollen ◽  
Xian-Yong Xiao
Keyword(s):  
Phase Angle ◽  
Three Phase ◽  
Voltage Dips ◽  
Phase Systems
Sign in / Sign up

Export Citation Format

Share Document