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us the point E on the chart as representing the condition of the steam as supplied to the third stage. Proceeding in this way, a series of " state points " can be marked on the chart, each of which represents the condition of the steam as supplied to the next ele- mentary turbine of the series.

So long as the steam is superheated or supersaturated its volume can be determined, when the pressure and total heat are known, by Callendar's equation

V= 2-2436

- + 0-0123.

The relation between the volume, pressure and temperature under the same condition is

i -0706 T /373-

(V-o-oi6)= 0-4213 I JJT-

in which T denotes the absolute temperature on the centigrade scale.

With wet steam expanding in a condition of thermal equilibrium the volume of the steam is equal to the volume of dry saturated steam at tHe same pressure, multiplied by the dryness fraction as read from the chart. Since the steam in passing through a turbine never does expand in a condition of thermal equilibrium, this case is of no practical importance.

If MI denotes the adiabatic heat drop for the first stage of the series, MJ that for the second stage, and so on, then the aggregate of these values of u for the whole series will be greater, the greater the number of stages into which the whole turbine is divided. The ratio of the aggregate to the value of u obtained when the whole of the expansion is effected in a single stage, is known as the " re- heat factor " R. In the case of a reaction turbine the number of stages is so great that the expansion may, for practical purposes, be considered as effected continuously instead of in a series of steps. In this case the reheat factor for superheated or supersat- urated steam can be read off from the diagram fig. 14, which is reproduced from Martin's New Theory of the Steam Turbine.

most trustworthy experimental data. Callendar's formula for thi adiabatic expansion of superheated or supersaturated steam is

_I3

p 3 T = constant where T denotes the absolute temperature.

In a continuous expansion of superheated or supersaturated stean effected with a hydraulic efficiency i), the relation between volume and pressure during the expansion is represented accurately by thj expression

f V 0-016 j = constant-

-(2)

where r- =

1-0-230773.

A closely approximate expression has been given by Callendar ir the form

(H 464) p S3 =constant-

-(3)-

F I C. 14

The " efficiency ratio " of a turbine is denoted by, and is denned as the ratio which the useful work W actually done by the steam bears to that which would be performed by a turbine of unit effi- ciency, so that W = e. The hydraulic efficiency, denoted by T), is defined as the ratio of the work done to the total effective thermo- dynamic head, which head, as pointed out above, is always greater than u in the case of a multistage turbine, as it is the sum of the values of u for each stage. We thus have

W = 7,U = i;Rtt, so that R = -. 1

The hydraulic efficiency 77 of a turbine is a much more fundamental property than- the efficiency ratio, and remains unaltered what- ever the number of elementary turbines or stages, into which the whole turbine is divided, or whatever be the total ratio of expansion. In the ideal limiting case in which the expansion is carried down to zero pressure the efficiency ratio is always unity, whatever the hydraulic efficiency may be.

Where the heat drop per stage of a turbine is small, it cannot be measured with accuracy from a chart but must be calculated from formulas or derived from steam tables, of which Callendar's are the most reliable and self-consistent, and accord best with the

In practice in equation (2) may be taken as unity withou

involving serious error; and since, along the saturation line, the rela tion between pressure and volume is represented very approxim, by the equation

0-9406 log p + log (V-o-oi6) =2-5252, the point at which the saturation line is crossed in a continuous expansion, effected with an efficiency n, can be found approximate!) by combining this equation with (2), which gives:

j^-log p + log (V-o-oi6)=-^log p a + log (V<.-o-oi6).

The pressure thus obtained can be plotted on the steam chart as at M (fig. 13). A single additional point representing the state ol the steam at some intermediate pressure gives the "condition line " in the superheated field with sufficient accuracy as the curvature of this line is always very slight. The condition line for wet steam expanding in thermal equilibrium i> obtained from the chart. To this end a horizontal line is drawn from M to cut the exhaust pressure line at S. Th< length MS then represents, on the scale of total heats, tin adiabatic heat drop for an expansion from M in a condition of thermal equilibrium. Denoting this by u, the corrcs] ing useful work done is eu, and the heat wasted in friction is (l )..

If we add this wasted energy to the total heat corresponding to the point S we get ] as the state point representtn condition of the steam as finally discharged. A similar pro- cedure gives us the state point K at some intermediate pn-s- sure, and the three points M, K, J suffice to fix with practical accuracy the condition line for wet steam expanding from M to S in thermaj equilibrium.

From a condition line the total heat of the steam corres-l ponding to any pressure can be read off, and the correspoi volume then obtained as already described. The condition line for steam expanding beyond the saturation line in a w- dition of thermal equilibrium, has, as already mentioned, no practical significance in steam turbine work. Once tin uratipn line is passed the expansion never proceeds in tin equilibrium. This discovery renders obsolete the theory of the steam turbine working with non-superheated steam, as un- derstood up to the end of 1912, at which time attention was directed anew to certain remarkable anomalies observed in experiments on the discharge of non-superheated steam from nozzles. Numerous careful experiments had shown that the weight discharged was often in excess of what the then ac-' cepted theory declared to be possible. In discussing these re- sults in Engineering, Jan. 10 1913, Martin pointed out thatthe experiments of Aitken and Wilson on the sudden expansion of dust free vapour afforded conclusive evidence that in expanding through. a nozzle, the steam must be in the supersaturated condition and not in thermal equilibrium, so that the accepted theory was based on a fundamental error. Stodola succeeded in confirming this com ln- sion by direct experiment. He studied, under very strong illumination, the appearance of jets of steam discharged from a nozzle and found that the steam exhibited no signs of condensation occurring until the pressure had been reduced far below the saturation point. Finally, in 1915, Callendar, in a paper published in the Proceedings of the I nst. Mech. Engineers, gave an exhaustive study of the whole question and showed that the anomalies observed in nozzle experiments entirely disappeared if the steam were considered to remain in a supersaturated condition up to a point beyond the throat of the nozzles. Moreover, under such an assumption, the computed fric- tional losses became in good accord with those observed in experi- ments with air. There is however, of course, a point beyond which steam cannot be expanded without condensation occurring, from experiments of C. T. R. Wilson, H. M. Martin calculated the follow- ing table giving the properties of steam at the supersaturation limit, or the " Wilson line " as he called it (" A New Theory of the Steam Turbine," Engineering 1913) :