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dredth of a volt between its ends, or ten volts per thousand feet, which is by no means insignificant. Bars of this size or larger are often used in practice carrying correspondingly heavy currents; hence it is not safe to ignore resistance, even in the case of very large conductors.
Loss of Energy. — The second objectionable effect which resistance produces in electrical distribution is the loss of energy which it occasions. This loss is absolute, and must always occur whenever a current flows through a resistance. The exact value of this loss is given by the expressions :
in which I is the current in amperes, R is the resistance in ohms, and E is the drop or lost pressure in volts, being applicable either to the whole circuit or to any part of it. From one of these expressions the loss of energy can always be ascertained, provided any two of the three quantities are known. These equations give the loss which occurs continuously so long as the current flows; that is, the rate of dissipation of energy or the power wasted. For a given time t in seconds —
Loss of energy in joules or watt-seconds) = 12 Rt= E Ir=
To find the loss of energy in heat units, any of the above values may be multiplied by .24 for calories (gram-degree cent.), or by .00095 to obtain British thermal units (pound-degree Fahr.).
This loss of energy, while quite considerable in almost every electrical system, usually amounting to from 5 to 25 per cent, is rarely the controlling consideration in electric lighting. The drop, which has already been considered, and the heating limit, which will be discussed later, are usually of more consequence than the mere waste of a small fraction of the total energy, the success or failure of an electric lighting plant being dependent upon keeping them within certain limits.
Economy in Design of Conductors. — In many cases, particularly for long-distance transmission in contradistinction to local distribution, the relation between the first cost of the conductors and the energy lost in them may be a matter of prime importance. This subject was first attacked in 1881 by Lord Kelvin, then Sir William Thomson, who read before the British Association a paper on “ The Economy of Metal Conductors of Electricity,” in which he attempted to give a general solution of the problem. The conclusion reached by him, and now known as “Kelvin's Law," may be stated in the following language: The most economical size of conductor is that for which the annual interest on capital outlay equals the annual cost of energy wasted. In other words, the total annual expenditure for interest on the investment and energy lost on the line is a minimum when these two items are equal to each other.
The importance of this law has usually been greatly overestimated, but gradually its limitations have been brought out. In 1886 Professors Ayrton and Perry showed, in papers before the Society of Telegraph Engineers and Electricians, that Kelvin's Law applies only in certain cases; and they gave various modifications and extensions of it. Professor George Forbes has also contributed to this subject in his Cantor Lectures of 1885,* in which he showed that the portion of the investment which is not proportional to the cross-section of the conductor should be kept separate, so that the amended law becomes : The most economical area of conductor is that for which the annual cost of energy wasted is equal to the annual interest on that portion of the capital outlay which is proportional to the area or weight of metal used. Professor William A. Anthony, in an article on “Economy in Conductors, and the Limitations in the Applicability of Kelvin's Law,” † demonstrates that in some cases Kelvin's Law gives absurd results, and may, for example, require that all of the energy should be wasted in order to secure the highest economy. This is due to the fact that the minimum expense of operation is considered, and the energy delivered at the end of the line, which is still more important, is entirely ignored. In fact, a great many laws of this kind can be deduced according to what factors are considered.
Kilgour f and Abbott & give 15 possible combinations of the six variable factors involved in the problem, but state that only 11 of these are likely to be of any practical importance. The six factors and the 11 cases are as follows:
* London Electrician, vols. xv. and xvi. † Electrical Engineer (N. Y.), Oct. 31, 1894. | Electrical Distribution, London, 1893, p. 115. § Electrical Transmission of Energy, N. Y., 1895, p. 457.
V = the pressure at the receiving end of the conductor ;
the power given to the receiving end of the conductor ;
This is a far more complete treatment of the question than that originally given, Kelvin's Law being only one (No. 5) of these 11 different cases. But any of these solutions of the problem is of somewhat doubtful practical value ; and it is probably true that Kelvin's Law, or any modification or extension of it that has yet been brought out, has done more harm than good in electrical engineering. It gives a false confidence in the results of calculations which may be totally at variance with real commercial economy. The reason for this difficulty lies chiefly in the fact that the actual costs of some of the items cannot be expressed, even approximately, as mathematical functions. Furthermore, various incidental factors and particular conditions arise, such as the available sizes of machines, which render a general solution of this problem of questionable value in the actual cases which are found in practical work.
It is a common mistake to forget that the interest and depreciation on the investment is a fixed and irretrievable expense, while the energy lost on the conductor depends upon the power transmitted. When the plant is lightly loaded, or is shut down entirely, owing to hard times, strikes, etc., the fixed charges run on as usual, but the energy loss is greatly reduced, or stopped altogether. Hence it is not wise to lay the full amount of copper corresponding to the maximum or even ordinary demands, as there is no control over the investment after it is once made, whereas the energy loss adjusts itself to the working conditions.
Probably the safest, as well as the quickest, method to arrive at a correct result would be to obtain a general solution of the problem by means of some form of Kelvin's Law; then this result should be carefully checked by assuming a larger and also a smaller wire, and estimating the economy that would be secured if they were substituted for the size of wire obtained by the calculation. The difficulties of determining the various items of expense are greatly reduced by assuming a certain size of wire, and the several factors that are almost impossible to cover by a general formula, immediately become definite. Scientific and rational methods should always be preferred to empirical ones; but every experienced engineer will admit that when complicated questions of cost arise it is unwise to rely entirely upon general formulæ, which are almost necessarily abstract and incomplete. The attempt to force science beyond its legitimate limits has done great injury to many industrial enterprises as well as to science itself.
Specific examples of this problem will be considered later in the case of constant-current arc-lighting circuits and feeders for constant potential systems.
Current-Carrying Capacity of Conductors. — The third objectionable effect of resistance in electrical distribution is the heating which it causes. The production of heat in an electrical conductor has already been stated in terms of the various quantities involved. This heat is an absolutely definite and unavoidable result of the flow of the current. Its effect is to raise the temperature of the conductor, and this rise continues until the rate at which heat is lost equals the rate at which it is generated ; then the temperature becomes constant. It is obvious, therefore, that any electrical conductor is only capable of carrying a certain current with a given elevation of temperature, and in practical work the allowable temperature is limited by considerations of injury to insulation, danger of fire, etc. No exact general rule for current capacity can be given, as much depends upon the conditions in each case. But, since a wide margin must be allowed between the danger point and the permissible. current capacity, it is possible to establish rules which are somewhat arbitrary, but sufficiently safe in almost any case. This is practicably the basis upon which tables are made giving the current that it is allowable for any size of wire to carry. These tables are partly based upon general experience, and partly the results of experiment and calculation.
The first rule of this kind originated with Lord Kelvin, and
was adopted by the Board of Trade (London). It stated that the current density in copper conductors should not exceed 1,000 amperes per square inch of cross section.
Professor George Forbes discussed this problem in a paper read before the Institution of Electrical Engineers (London) in March, 1881, and showed that the Board of Trade rule was hardly safe for very large conductors, and gave an unnecessarily large margin for small wires. This fact is very evident when it is considered that the current at a given density and also the heating increase in proportion to the square of the diameter of a wire, while the heat-dissipating surface only increases as the diameter.
Dr. A. E. Kennelly has given the results of his investigations in two papers before the Association of Edison Illuminating Companies, Aug. 13, 1889, and Aug. 11, 1893.* He determined by calculation and experiment the heating of conductors submerged in water, buried in the earth, inclosed in wooden molding, and suspended in air.
It is found that there is not such a great difference between the heating effects under these various conditions. An insulated cable in water is the simplest case ; since the rise in temperature of the conductor depends merely upon the thermal resistance of the insulation, the outer surface (or sheathing) of the latter being kept at a constant temperature by the water. An underground conductor only differs from the foregoing in the fact that its sheathing may rise in temperature because heat is not taken from it rapidly enough by the surrounding soil or conduit. In other words, the thermal resistance of the conduit and soil is added to that of the insulating covering. For underground conductors in iron pipe conduits laid in cement, the temperature elevation due to this cause would be small, probably not more than 10 or 20 per cent greater than that of the same cables submerged in water. The heating of conductors in wooden or even earthenware conduits would be considerably greater, and in the case of the former might be considered to be the same as for those placed in wooden panels or molding, the rules for which will be given later. Insulated wires suspended in air are more highly heated than similar submarine or most underground conductors, for the reason that the thermal losses by radiation and convection through the air are
* Electrical World, Nov. 23 and 30, 1889 and Sept. 2 and 9, 1893.