. Let p2 be the value to which the pressure is reduced by the loss of pressure at the end of the conduit, and v2 the volume which the air occupies at this pressure and at the same temperature; the force stored up in the air at the end of its course through the conduit is p2 v2 log(p2/p0); consequently, the efficiency of the conduit is
, a fraction that may be reduced to the simple form
, if there is no leakage during the passage of the air, because in that cause p2 v2 = p1 v1. Lastly, if W1 is the net disposable force on the shaft of the compressed air motor, the efficiency of this engine will be,
and the product of these three partial efficiencies is equal to W1/W, the general efficiency of the transmission.
III. Transmission by Pressure Water.--As transmission of power by compressed air has been specially applied to the driving of tunnels, so transmission by pressure water has been specially resorted to for lifting heavy loads, or for work of a similar nature, such as the operations connected with the manufacture of Bessemer steel or of cast-iron pipes. The author does not propose to treat of transmissions established for this special purpose, and depending on the use of accumulators at high pressure, as he has no fresh matter to impart on this subject, and as he believes that the remarkable invention of Sir William Armstrong was described for the first time, in the "Proceedings of the Institution of Mechanical Engineers." His object is to refer to transmissions applicable to general purposes.
The transmission of power by water may occur in another form. The motive force to be transmitted may be employed for working pumps which raise the water, not to a fictitious height in an accumulator, but to a real height in a reservoir, with a channel from this reservoir to distribute the water so raised among several motors arranged for utilizing the pressure. The author is not aware that works have been carried out for this purpose. However, in many towns a part of the water from the public mains serves to supply small motors--consequently, if the water, instead of being brought by a natural fall, has been previously lifted artificially, it might be said that a transmission of power is here grafted on to the ordinary distribution of water.
Unless a positive or negative force of gravity is introduced into the problem, independently of the force to be transmitted, the receivers of the water pressure must be assumed to be at the same level as the forcing pumps, or more correctly, the water discharged from the receivers to be at the same level as the surface of the water from which the pumps draw their supply. In this case the general efficiency of transmission is the product of three partial efficiencies, which correspond exactly to those mentioned with regard to compressed air. The height of lift, contained in the numerator of the fraction which expresses the efficiency of the pumps, is not to be taken as the difference in level between the surface of the water in the reservoir and the surface of the water whence the pumps draw their supply; but as this difference in level, plus the loss of pressure in the suction pipe, which is usually very short, and plus the loss in the channel to the reservoir, which may be very long. A similar loss of initial pressure affects the efficiency of the discharge channel. The reservoir, if of sufficient capacity, may become an important store of power, while the compressed air reservoir can only do so to a very limited extent.