China
Africa
Explore chapters and articles related to this topic
Three-Phase Transformer Models
Published in William H. Kersting, Robert J. Kerestes, Distribution System Modeling and Analysis with MATLAB® and WindMil®, 2022
William H. Kersting, Robert J. Kerestes
Because the matrix [Di] is singular, it is not possible to use Equation 8.56 to develop an equation relating the wye-side line currents at Node-n to the delta-side line currents at Node-m. To develop the necessary matrix equation, three independent equations must be written. Two independent KCL equations at the vertices of the delta can be used. Because there is no path for the high-side currents to flow to ground, they must sum to zero and, therefore, so must the delta currents in the transformer secondary sum to zero. This provides the third independent equation. The resulting three independent equations in matrix form are given by IaIb0=10−1−110111⋅IbaIcbIac
Fundamental Relationships for Flow and Transport
Published in James L. Martin, Steven C. McCutcheon, Robert W. Schottman, Hydrodynamics and Transport for Water Quality Modeling, 2018
James L. Martin, Steven C. McCutcheon, Robert W. Schottman
This expression contains terms for the new time (t + Δt) at more than one point along the x-axis. Therefore, this expression cannot be solved using the simple time-marching technique used for the explicit scheme. More than one independent equation is required. Instead, the equations must be solved simultaneously for all of the points along the x-axis. This implicit scheme, while computationally more complex, has numerous advantages that often outweigh the method’s increased complexity. For example, implicit schemes are generally not limited to small time steps by the Courant condition. Implicit schemes will be discussed in greater detail in Part II.
Three-Phase Transformer Models
Published in William H. Kersting, Distribution System Modeling and Analysis, 2017
Because the matrix [Di] is singular, it is not possible to use Equation 8.56 to develop an equation relating the wye-side line currents at node n to the delta-side line currents at node m. In order to develop the necessary matrix equation, three independent equations must be written. Two independent KCL equations at the vertices of the delta can be used. Because there is no path for the high-side currents to flow to the ground, they must sum to zero and, therefore, so must the delta currents in the transformer secondary sum to zero. This provides the third independent equation. The resulting three independent equations in matrix form are given by:
On the equations of open channel flow
Published in Journal of Hydraulic Research, 2023
William Guerin Gray, Cass Timothy Miller
It is useful to recognize that the mechanical energy balance is not an equation that is an additional independent conservation equation. Although named mechanical “energy”, it is actually derived directly from the momentum equation. The total energy equation, which accounts for both mechanical and thermal effects, is an independent equation. It is an important equation for use in entropy analyses or when there are temperature gradients. It is not provided here as we wish to concentrate on flow. The mechanical energy equation is used as an alternative to the component of the momentum equation in the direction of flow. When the mechanical energy equation is used, the full set of governing equations consists of mass conservation, Eq. (18), the mechanical energy equation to be derived here, and the two components of momentum conservation in the directions tangent to the cross-section of integration, Eqs (31) and (32). These latter two equations are typically not employed but are replaced implicitly with the assumption that and may be considered negligible and the pressure is hydrostatic. For cases when these assumptions might not hold, the lateral momentum equations can be employed. Analytically, the momentum and mechanical energy equations should lead to identical solutions. However, in some cases, the assumptions that may be applied to simplify or close the equation forms may be more readily applied to one of the two forms.
Critical point calculations by numerical inversion of functions
Published in Chemical Engineering Communications, 2021
C. N. Parajara, G. M. Platt, F. D. Moura Neto, M. Escobar, G. B. Libotte
In the simulations, we consider mixtures with two components, c = 2, therefore this setup can be simplified a little bit. In this case, Equation (5c) has only one independent equation. Using the first one, and letting we get i.e., Substituting these values in Equation (5b), and defining function F by the left hand side of Equations (5a) and (5b), the equations of the critical thermodynamic points are written simply as which is interpreted as computing the pre-image of a point, (0,0), by a nonlinear map from the plane to the plane, F.
Reflection of magneto-photothermal plasma waves in a diffusion semiconductor in two-temperature with multi-phase-lag thermoelasticity
Published in Mechanics Based Design of Structures and Machines, 2020
The solution of Eqs. (19)–(23) are sought in following form: where, and are the constants, is phase velocity and k, is wavenumber. Using Equation (24) into Eqs. (19)–(23) we get one independent equation in and four homogeneous equations in and for the non-trivial solution of these four equations require where,