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Electrification is one of the key factors to be considered in the design of power transformers utilizing dielectric liquid as a coolant. Compared with enormous quantity of experimental and analytical studies on electrification, numerical simulations are very few. This paper describes essential elements of numerical solution methods for the charge transport equations in a space between concentric cylinders. It is found that maintaining the conservation property of the convective terms in the governing equations is of the uttermost importance for numerical accuracy, in particular at low reaction rates. Parametric study on the charge transport on the axial plane of the annular space with a predetermined velocity shows that when the convection effect is weak the solutions tend to a one-dimensional nature, where diffusion is simply balanced by conduction. As the convection effect is increased the contours of charge distribution approach the fluid streamlines. Thus, when the conduction effect is weak, charge distribution tends to be uniform and the role of the convection effect becomes insignificant. At an increased conduction effect, on the other hand, the fluid motion transports the charge within the electric double layers toward the top and bottom boundaries leading to an increased amount of total charge in the domain.

When a dielectric liquid containing impurities is in contact with a solid surface, a certain physicochemical process occurs at the interface yielding free ions near the surface of the liquid. Usually negative ions are adsorbed to the solid surface and the positive ions are diffused away forming the electric double layer (to be referred to as EDL). Since the positive ions are mobile, they are convected by the fluid flow giving rise to the streaming current, which is called flow electrification. Problem occurs when they are accumulated in a certain location downstream resulting in locally high electric-field intensity which can cause electrical discharge, breakdown, and local failure of the device employing the liquid transport.

Electrification becomes one of the key factors to be considered in the design of electrical devices utilizing dielectric liquid (mineral or ester oil) as a coolant, such as power transformers. Demand for increased capacity from the users of power transformers tempts designers to increase the oil flow rate for increased cooling capacity, which, however, brings increased electrification and makes the device more susceptible to the electrical failure.

Studies on electrification and discharge with full-scale transformers were carried out by Higaki et al. [

In order to perform more fundamental studies, researchers have considered simple experimental apparatus other than actual transformers, which is easier to build and easier to measure data with, such as electrical charge tendency (ECT). In addition, how to interpret the measured data in relation to the actual transformers is also an important issue in selecting suitable geometries for study. Most of the initial studies focused on the flow between parallel plates and circular pipes [

With the simple flow apparatus in hand, researchers can perform investigations on the effect of various factors on the electrification independently. There are many factors influencing electrification or ECT. They can be categorized into two kinds, fluid/flow properties and electrical properties. Included in the former are flow rate, geometrical features determined by the fluid path, and fluid viscosity, while in the latter electrical conductivity and permittivity are the key elements; the operating temperature and material degradation may influence many of these properties, such as viscosity, conductivity and permittivity. Properties of the pressboard such as chemical composition of the material and surface roughness may also influence the ECT. General understanding of the effect of various parameters on ECT and design aspect for avoiding discharge in transformers was given in [

Touchard [

Most studies on electrification even with simple geometry have been performed experimentally and/or analytically, and numerical studies, at least in a two-dimensional space, are very few. In [

The main purpose of the present study is to develop a two-dimensional numerical code and perform simulations for charge transport in a confined space under a various range of parameters. In particular, we select as the computational domain the annulus between concentric cylinders, following [

We consider transport of a space charge density distributed in an annulus space between two concentric circular cylinders of radii

where

It is assumed that metals or pressboards in contact with dielectric liquid create charges (or they may be adsorbed) by certain chemical reaction, and we simply employ the model used in [

where

The fluid flow within the annulus can be assumed to be created by two kinds of forcing; one is by the rotation of the inner cylinder and the other by the so-called induced charge electroosmotic effect. While the former is driven by the boundary condition for the Navier-Stokes equation (

As a first step in our series of studies on the charge transport within an annulus, we in this paper focus on steady and axisymmetric solutions. Then the azimuthal component of the fluid velocity does not contribute to the charge transport. Thus, even the circular Couette flow driven by the inner cylinder’s rotation has no effect on the charge transport when it is stable, which is relevant at low Reynolds numbers exhibiting only the primary azimuthal flow (referred to as steady circular Couette flow; see, e.g., Liao et al. [

In the present study, convection due to the induced charge electroosmotic flow effect is assumed to be negligible compared with the effect of the secondary flow caused by the flow instability mentioned above following El-Adawy et al. [

Based on the above reasoning, we can write the governing equations for the dimensionless charge density

Here,

Boundary conditions for

where

As for the conditions on the upper and lower boundaries we apply zero gradient for

The fact that

When the convection effect is neglected, we may well assume, in view of the boundary conditions, that solutions are independent of

The general solution to (

For the case with

where

When the velocity field is arbitrarily imposed, we must use numerical methods to solve the 2D charge transport problem governed by (

We briefly address first the numerical method employed in the in-house code. Although the steady-state solutions are our primary concern, we add the transient term to the left-hand side of (

Since

Both

Notation for grid sizes, coordinates, and points in the variable grid system.

The two algebraic systems of equations constructed in this way are solved by using the SOR (successive-over relaxation) method in a coupled manner. Relaxation parameter for (

In the use of the commercial software COMSOL, we employ two models, “transport of a diluted species” and “electrostatics.” The original form of the model however leads to numerical instability due to the fact that the conductivity is set to be proportional to the charge density in the original model, whereas in this study the conductivity is set to be constant. So, we modified the model in such a way that the conduction term is excluded from the charge flux

The velocity field we are interested in is the secondary Taylor-vortex flow observed in the axial plane caused by hydrodynamic instability. Instead of using the exact solution of the secondary flow given from the numerical simulation of the Navier-Stokes equations or the experimental measurement, we set the flow in an arbitrary manner but with physical relevance if possible. For this, we assume that the axial plane between the coaxial cylinders is occupied by the series of spatially periodic flow cells. The velocity components

Typical profiles of the velocity component

We have also prepared 1D code applicable to the case where no fluid motion exists so that the convection terms vanish. The numerical schemes are identical to those employed in the 2D code except that the variables’ dependence on

The standard parameter set is given as follows [^{2}/s] in Washabaugh [^{2}/s]. So, in this study we set ^{2}/s] as the standard diffusivity. Increasing the diffusivity is equivalent to decreasing the geometric scale as can be seen from the definition of the two main dimensionless parameters,

Figure

Dependence of

The fact that the numerical solutions are sensitively dependent on the form of the convection or conduction (source) term in the governing equations implies that a small error in the equations can yield a significantly different solution. In order to explore the reason, we perform a simple analysis with the one-dimensional equation without convection effect:

where the prime denotes differentiation with respect to the new variable

Here, ^{2}/s]. Then, after some algebraic work we derive

On the other hand, the numerical data of

Figure

Numerical results of 1D simulation with

Two important dimensionless parameters explicitly appearing in the governing equations (

Figure

Numerical results of 2D simulation at

Passage of the charge transmission from the inner cylindrical wall at

At a reference velocity 10 times higher,

Numerical results of 2D simulation at

Further increase of

Numerical results of 2D simulation at

When

As

In order to confirm the above reasoning, we calculate the Lagrangian variation of

Lagrangian variation of the charge density versus the normalized distance travelling from the initial point of a fluid particle while flowing along the streamline, numerically given at the four reference velocities indicated and at

Now we investigate the effect of

Numerical results of 2D simulation at

As

Numerical results of 2D simulation at

We can estimate the dependence of the charge density distribution on

Lagrangian variation of the charge density versus the normalized distance travelling from the initial point of a fluid particle while flowing along the streamline, numerically given at the three indicated values of

A typical solution structure of the charge transport equations with

Typical solution structure of the system of charge transport equations given numerically at

Electrification is known to be directly related to the amount of charge accumulated in the bulk, which in this study is quantified by the averaged charge density

Volume-averaged charge density within the annulus given numerically at (a)

We studied the physics of charge transport in an annulus between concentric circular cylinders from theoretical and numerical analysis by using a commercial software COMSOL and 2D in-house code.

We have found that the conservation property of the convective terms in the charge transport equation affects numerical accuracy significantly. In both the COMSOL and in-house code simulations, keeping the convective terms in conservative form is essential in maintaining the numerical accuracy. In COMSOL, the conductive terms being treated as sources must also be written in the gradient-of-field form, not in the form of charge so as not to deteriorate the numerical accuracy. Such sensitive dependence of the numerical solutions’ accuracy on a small error in the governing equations can be explained in terms of 1D simplified equations.

In the absence of the convection effect, the analytical and numerical solutions of the 1D equations show that the diffusive charge flux is balanced by the conductive flux and the sum of the two fluxes yields the total flux which remains much smaller than the two fluxes for small values of

The effect of two dimensionless parameters,

Increase of

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean government (MSIP) (no. 2009-0083510). This work was also supported by the Human Resources Development Program (no. 20114030200030) of the Korean Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korean government Ministry of Trade, Industry and Energy. This paper has been read by Professor M. Duffy.

_{60}as flow electrification inhibitor in mineral insulating oil