On a technique for flow splitting in the method of expected flow paths
Abstract Generated abstract
This paper presents a graph-analytical technique for splitting field-vector flow in the method of expected flow paths, addressing cases where symmetry-based division is insufficient for systems of bodies with specified potential differences or isolated bodies of complex form. The proposed procedure begins with a sketch of the field pattern, approximates each flow part by simple geometric bodies to compute conductance or capacitance, and then redistributes terminal areas iteratively until the total conductance reaches a maximum. The method is illustrated through calculation of current spreading conductance from a grounding pipe ending in a hemisphere, including formulas for resistance under assumed laws of flow cross-section variation. It offers a practical approximate approach for estimating conductance, current density, and field strength when exact analytical solutions or physical modeling are inconvenient.
Full Text
UDC 621.319.7.001.24
ON ONE TECHNIQUE FOR FLOW SPLITTING IN THE METHOD OF EXPECTED FLOW PATHS
P. P. Galinskii
(Submitted by the scientific seminar of the Department of TOE)
In engineering practice there often arises the need to investigate the field of a system of bodies for which an exact analytical calculation, even if possible, is very cumbersome. For approximate calculations that determine the direction of the search or the range of possible values of the parameters of devices being designed, the necessary initial data can be obtained by field modeling. However, the construction of a modeling device and the preparation of the experiment require much time. The graphical method of constructing a field pattern is likewise not sufficiently convenient for these purposes. The most suitable method for a number of such cases is evidently the graph-analytical method of expected flow paths [1].
Usually, in the method of expected flow paths, the flow of the field vector is divided into parts on the basis of symmetry considerations. This principle of flow division does not always lead to the goal in the case of a system of two or more bodies with a specified potential difference between the bodies, and in the case of an isolated body whose shape makes it possible to divide the flow of the vector issuing from this body into parts.
In the variant of the graph-analytical method set forth here, the flows are first divided into parts by means of a sketch construction of the field pattern. For separate parts, reducible to a set of the simplest bodies (parts of cylinders, tori, spherical sectors, etc.), the conductance (capacitance) is calculated in the first approximation. Then a certain redistribution is made of the areas at the end of the flow of the field vector among the selected areas of the source of the field vector. If this produces an increase in the sum of conductances for the flow, then a further redistribution of the areas is made in the same direction (an increase, for example, of the 1st area in comparison with the 2nd).
Such a change is made until the conductance begins to decrease. The maximum conductance will correspond to the true distribution of the flow. The calculation can be refined if the number of parts into which the flow is divided is increased. However, in this case there evidently exists an optimal limit to the number of parts of the flow division, which can be estimated from the relative change in the total conductance for the flow of the field vector. Increasing the number of parts of the flow is not expedient if the percentage change in the maximum conductance becomes less than the specified error.
The described technique for splitting the flow is most expedient when the space along the flow path expands monotonically. If the space of the flow path has a limited number of expansions, contractions, and constant cross sections, then, after the flow path has been divided into parts at the points of contraction, the results of calculations for fields close in configuration and having an analytical solution can be used.
As an example, let us calculate the conductance \(G\) of current spreading from a grounding pipe of length \(l\), ending in a hemisphere of the same radius as the pipe. The medium surrounding the pipe has a constant specific conductivity \(\gamma\). We shall take the contact resistance between the pipe and the medium to be zero. The pipe is the first electrode; the second electrode is a sphere of infinite radius.
We divide the surface of the first electrode—the ground electrode—into two parts: the cylindrical part and the hemisphere. The infinite surface of the second electrode must be divided into the corresponding parts, but this cannot be done graphically without mapping; therefore it should be assumed that, at an observable distance, for example at a distance of 4–5 maximum dimensions of the first electrode, an equipotential surface repeats the surface of the second electrode, i.e., in the present case is a hemisphere. The surface of this hemisphere is divided in proportion to the parts of the surface of the first electrode, and in this way the boundary of the flow parts is determined in the first approximation. The line of the conductivity vector is drawn perpendicular to the equipotential surface of the first electrode and of the second electrode. The beginning and end of the boundary line are joined by a smooth line in accordance with the known rules for constructing the field pattern.
After constructing the boundary surfaces of the flow division, equipotentials are drawn, which make it possible to specify the law of variation of the flow cross section as a function of the distance from one of the electrodes, or from the axis of rotation (for the field of bodies of revolution), along the mean line of the flow. This law is specified in a form convenient for integration in order to determine the conductance or resistance. In the case of a linear law, the flow cross section is
\[ S(x)=S_1+\frac{S_2-S_1}{d}\,x, \tag{1} \]
where
\[ \begin{aligned} &S_1,\ S_2 &&\text{are the minimum and maximum cross sections of the flow part (at the electrodes),}\\ &d &&\text{is the maximum value of the distance } x \text{ of the flow cross section from}\\ &&&\text{the first electrode along the mean line of the flow.} \end{aligned} \]
The differential resistance along the path of a uniformly expanding flow is
\[ dR=\frac{dx}{\gamma S(x)}. \]
After integration and substitution of the limits, the total resistance for the flow part is
\[ R=\frac{d}{\gamma\,(S_2-S_1)}\ln\frac{S_2}{S_1}. \tag{2} \]
If, in expression (1), the ratio \(\frac{x}{d}\) is taken to the second power, then instead of (2) we obtain
\[ R=\frac{d}{\gamma S_1}\cdot\frac{\operatorname{arctg} m}{m},\qquad m=\left(\frac{S_2}{S_1}-1\right)^{0.5}. \tag{2a} \]
Dependence (1) may be replaced by a piecewise-linear or other function in accordance with the degree of expansion of the space along the path of the flow (a field close to uniform, cylindrical, spherical, etc.). In specific cases, the form of the dependence of the cross section on the coordinate and the expression for the differential resistance can be refined.
Having determined the resistances and conductances of both parts of the flow, we carry out a certain small redistribution of the surface of the second electrode between the selected parts of the surface of the first electrode, repeat the calculation of the total conductance, and compare it with the conductance of the first approximation. If it has increased, then we intensify the redistribution in the same direction and continue doing this until we find the maximum conductance, which should be taken as the true one.
Multiplying the potential difference between the electrodes by the conductance of the resulting current tubes, one can determine the current in each tube; then, by dividing the current by the cross section of the tube at characteristic points, one can find the mean current density in these cross sections and the mean field strength.
REFERENCES
- V. A. Govorkov, Electric and Magnetic Fields. GEI, 1960.