For very long distances the term under the square root approaches unity, and the most economical current the value 2/ ; from which it follows that under no circumstances will it be economical to lose more than half the total power in the line.
Sprague has developed some very interesting formulm, from which he deduces the follow ing laws: With fixed conditions of cost and efficiency of apparatus, the number of volts fall to get the minimum, cost of the plant is a function of distance alone, and is independent of the electro motive force used at the motor.
1Vith any fixed couple and commercial efficiency, the cost of the wire bears a definite and fixed ratio to the cost of the generating plant.
The cost of the were varies directly with the cost of the generating plant.
If we do not limit ourselves in the electromotive force used, the cost per horse-power deliv ered exclusive of line erection is, for least cost and for a given commercial efficiency, absolutely independent of the distance.
By the aid of these laws and Sprague's formulae, and assuming— K = Cost in cents of bare copper wire per lb. delivered at the poles = 25 a = Commercial efficiency of motor = .90 b = Commercial efficiency of generator G = Cost in dollars of generator set up, per electric horse-power delivered at its ter minals = 45 P = Cost in dollars of power (water) set up per mechanical horse-power delivered at generator pulleys.. = 25 —the accompanying diagram, Fig. 1, has been constructed, which shows the commercial efficiency at various distances and voltages for a minimum total initial cost of a transmis sion plant.
As with fixed conditions of cost and efficiency of apparatus, the number of volts to get the minimum cost of plant is a function of the distance alone, and is independent of the electromotive force used at the motor, Table IV. can be calculated. The values for K, a, b, G, and P, are assumed as before.
Badt (Electric Trawanission Hand-Book) expresses the principles governing the minimum cost of a transmission plant, in the following rale : For miNitM117} initial cost of plaid, and assuming certain prices per horse-power of' motors, generators, and power plant (all erected and reaay for operation), and assuming a certain price per pound of copper (delivered at the poles), Mc total cost of the plant, excluding line con struction, is a- constant for a certain efficiency of the electric system-, no matter what the elec tromotive force of the motor and the distance may be.
At a given efficiency of the electric system, the cleetromotive force of the motor and distance will increase and decrease in the same ratio.
This rifle is embodied in Table V., from which it kill be seen, for instance, that the cost of plant per horse-power delivered by motor at 1,000 volts and 25.000 ft. distance, and at an efficiency of 56.4 per cent., is $205.82. It will also be seen that the cost is the same at 4,000 volts, 100,000 ft. distance, and the sante efficieney of 504 per cent. While the cost and efficiency in both cases are the same, with an electromotive force lour times greater we can reach four times the distance.
The annexed comparative table shows the commercial efficiency of four different systems of transmission. See Table VI.
It will be seen that for distances less than 5 kilometres (about three miles) transmission by wire rope is more economical than that by any other system. For distances greater than
5 kilometres the electric transmission is most economical. As regards capital outlay, the wire-rope system is also for short distances more advantageous than electric transmission, the limit being at about 3 kilometres (a little under two miles). Beyond that the electrical system is the cheapest, as will be seen from the annexed Table VII.
The table shows that for short distances the cost of electric transmission is very consider able as compared to that of the other systems. The reason for this is that the price of dynamos and motors have been rather overestimated in the above table. For long distances this is not so noticeable, as the conductor forms the more important item, and especially since an electric wire is cheaper than an equivalent hydraulic or pneumatic tube. If we compare the conductors only, we find that for the transmission of 10 horse-power, a copper wire of 127 mils diameter [No. 10/ B. W. G.] is equivalent to a water-pipe of 3i in. diameter, or to an air-pipe of 3A in. diameter, or to a wire rope of in. diameter. The proportion between the cost of these conductors calculated for equal distances is as 1 -4 : 27.8 :1. The conductor with hydraulic transmission costs, therefore, twenty-five times as much, and with pneumatic transmission it costs nearly twenty times as much as with electric trans mission. These figures prove that as far as capital outlay is concerned, the electric system has the greatest advantage where the conductor is long, that is, where the energy has to be transmitted over a long distance. It would, however, not be correct to compare the four systems on this basis alone. The comparison must he made on the question of capital outlay combined with efficiency ; in other words, the figure of merit for each system is the price which has to be paid for 1 horse-power-hour at the receiving station. The smaller this price. the better the system. A glance at the annexed table (see Table VIII.) will show that the cost of 1 horse-power-hour increases in all systems with the distance, but with electric trans mission the increase is not so rapid as with the other systems. The table also shows that up to a distance of 1,000 meters [five-eighths of a mile], wire-rope transmission is better than electric transmission, but above that limit the electrical system is better. Hydraulic and pneumatic transmission are in some few cases better than electric transmission, but then the wire rope is again better than either, so that there does not seem to be a field for the applica tion of the hydraulic or pneumatic system, except in eases where the other two systems are for some local reason inadmissible, or where the water and air may be of further use after the power has been obtained from them. This, for instance, is the case with the pneumatic transmission employed in the building of tunnels. Here it is an absolute necessity to force air to the end of the workings for ventilating purposes, and pneumatic transmission is adopted in preference to any other system which would require some special ventilating plant being erected.