Jerzy Gołębiowski

The paper has analyzed a stationary thermal field in a direct current gas-insulated line (DC GIL ). After assuming some simplifications, a mathematical model for the system is the boundary problem of the Poisson equation. The conditions of continuity of temperature and heat flux increments were assumed at the boundaries of material layers. The cooling of the system was modelled using Hankel’s condition (of the 3rd kind) with the total heat transfer coefficient (i.e. with the sum of the convective and the radiation coefficients ). The problem was solved analytically. After tabulating the formulas, the respective distributions were shown graphically. Additionally, the steady-state current rating was determined. The obtained results have a very good physical interpretation.

The passage of current generates heat and increases the temperature of electrical components, which affects the environment, support insulators and contacts. Knowledge of the temperature allows for the determination of important operational parameters. Time-varying currents result in a nonuniform current density distribution due to the skin and proximity effects. As a result, temperature and energy losses are increased compared to the uniform DC current density case. In this paper, these effects are considered for three-phase systems with round conductors in flat and trefoil arrangements. In the first step, the analytical expressions for current distributions are determined and used to construct the heat source density. Then, a suitable Green’s function, which allows for obtaining temperature distribution in analytical form, is used to evaluate temperature at any point throughout the conductors. The temperature differences throughout individual wires are usually negligible, whereas noticeable differences can be observed between the wires. The impact of various parameters is examined, and an approximate closed formula is derived to assess the influence of the skin and proximity effects. When the skin depth is not smaller than the wire radius, the skin effect enlarges the temperature increase by around 2% compared to the DC case. As for the proximity effect, the additional increase can be neglected if the distance is above around 10 wire radii, but for closely spaced wires, it can reach up to around 17%, depending on the arrangement and the distance between the wires. Such an additional increase may result in exceeding the permissible temperatures, which damages particular components of the system; therefore, it is important to take it into account at the design stage.

An analytical–numerical method for determining the two-dimensional (2D) thermal field in a layer-inhomogeneous elliptic cylinder (elliptical roller) was developed in the article. A mathematical model was formulated in the form of a boundary problem for Poisson equations with an external boundary condition of the third kind (Hankel’s). The conditions of continuity of temperature and heat flux increment were assumed at the inner boundaries of material layers. The eigenfunctions of the boundary problem were determined analytically. Hankel’s condition was subjected to appropriate mathematical transformations. As a result, a system of algebraic equations with respect to the unknown coefficients of the eigenfunctions was obtained. The above-mentioned system of equations was solved numerically (iteratively). As an example of an application of the aforementioned method, an analysis of the thermal field in an elliptical electric wire was presented. The system consists of an aluminum core and two layers of insulation (PVC and rubber). In addition to the field distribution, the steady-state current rating was also determined. The thermal conductivities of PVC and rubber are very similar to each other. For this reason, apart from the real model, a test system was also considered. Significantly different values of thermal conductivity were assumed in individual layers of the test model. The temperature distributions were presented graphically. The graphs showed that the temperature drop is almost linear in the insulation of an electrical conductor. On the other hand, in the analogous area of the test model, a broken line was observed. It was also found that the elliptical layer boundaries are not isothermal. The results obtained by the method presented in this paper were verified numerically.

The paper determines the stationary thermal field in an elliptical cross-section electric conductor coated with insulation. Heat is generated by the flow of alternating current (AC) through the conducting core, and then dissipated from the insulation surface via convection and radiation. The authors have developed an original method for hybrid (analytical–numerical) modeling of a field. This method has been used to solve the relevant boundary problem of Poisson’s equation. While the eigenfunctions of the Laplace operator were determined analytically, the coefficients of the eigenfunctions were calculated by iteratively solving an appropriate system of algebraic equations. The proposed method enables the analysis of systems with an elliptical geometry and a heterogeneous layered structure (e.g., air, aluminum alloy, PCV), and does not require area discretization (grid). The developed analytical–numerical (AN) method has been positively verified using finite elements (FEs). The determined thermal field is illustrated graphically. The obtained solution has a good physical interpretation.

The paper investigates the dynamics of the thermal field of the ACCC (aluminum conductor composite core) line. The system was heated by solar radiation and current flow. Conductor cooling was modeled using the total heat transfer coefficient as the sum of convective and radiative components. The temperature increase generated by the current is described by a system of parabolic differential equations with an appropriate set of boundary, initial and continuity condition. The mentioned boundary-initial problem was solved by a modified Green’s method, adapted to the layered structure of the system. For this purpose, Green’s functions, as the kernels of integral operators inverse to differential ones, were determined. Aluminum resistivity and heat transfer coefficient change significantly with temperature. For this reason, the solution to the problem is presented in the form of a lower and upper estimation of the heating curve and local time constant. A steady-state current rating was also determined. The results are presented graphically and verified by other methods (power balance and finite element). The physical interpretation of the presented solution is also given.

In the paper temperature-time profiles generated in a tubular bus under short-circuit conditions were determined. The initial-boundary parabolic problem was the mathematical model. Adiabatic boundaries of the system and field uniformity at the initial moment of shorting were assumed and motivated. The two models of the bus-bar electric resistivity were considered: the temperature dependent (variable) and averaged one (constant). The problem was solved by means of Green's function. The one-second shorting current and its multi-second equivalents were determined based on the above. The analytical results were verified numerically by the finite element method.

Purpose: The purpose of this paper is to analyse the dynamics of a thermal field generated in a tubular bus with rated current by using two models of electrical resistivity of copper. Design/methodology/approach: The boundary-initial problem of the modified heat equation was formulated for the tubular bus. Analytical solutions were obtained by means of Green’s functions as the kernels of the integral operator inverse to the corresponding differential operator. The results were presented graphically and verified using the finite element method. The calculations were made by considering the example of the Storm Power Components tubular bus (USA). Findings: Analytical field models were used to determine time- and space-variable heating curves, time constants and steady-state current ratings. Research limitations/implications: This paper is related to the structure of a hollow cylinder. Other bus sections can be taken into account by using the coordinate systems of different curvilinear orthogonal symmetry. Originality/value: Using the analytical method, the influence of the variable (temperature dependent) electrical resistivity on some important parameters and characteristics of the tubular bus was investigated. The system was considered as an element with distributed parameters.

Purpose: The purpose of this article is investigating the impact of the spatially variable heat transfer coefficient on the thermal field in the double insulated wire. Design/methodology/approach: The effect of the air boundary layer was modelled by means of changing the total heat transfer coefficient on the external perimeter of the wire. This leads to an elliptical boundary problem with Hankel’s condition dependent on the angular coordinate. The eigenfunctions of the problem were determined analytically. On the other hand, the unknown coefficients of eigenfunctions and the constants were calculated numerically by solving a respective system of algebraic equations. The steady state current rating was determined with an iterative method. Findings: By means of the presented method, the thermal field distribution deprived of axial symmetry in the double insulated wire was determined. The obtained results have good physical interpretation and were verified with the finite element method (by means of NISA v. 16 software). The determined values of the steady-state current rating were compared with those calculated by means of the equivalent heat transfer coefficient method and the International Electrotechnical Commission (IEC) standard. Research limitations/implications: The method is applied to analyse scalar fields in layered cylindrical structures. This could be expanded to the case of a wire of any number of insulation layers. What is more, one could also consider heat sources without axial symmetry and located within the external area. Originality/value: The analytical method of determining a thermal field deprived of axial symmetry in heterogeneous cylindrical system (the wire composed of three different materials) was developed.

This article presents a way of calibration of an unconventional two-current circuit, named 2J+2R, which consists of two current sources and two referential resistors connected to the circuit mass. This bridge was used to measure the beam deflection and the temperature increase simultaneously with the use of a pair of metal strain gauges. This paper contains theoretical and corrected (after calibration) processing characteristics of the measurement circuit. Calibration coefficients of both inputs, responsible for measurement of the measured values in the places where the strain gauges are attached, were calculated. Moreover, the standard combined and expanded uncertainties of both calibration coefficients were calculated and an uncertainty budget was made.