It must not be supposed that the efficiency On â€” rj)/^ is actually attained, or is even attainable. Many causes conspire to prevent steam engines from being thermodynamically perfect, and some of the causes of imperfection cannot be removed.

40. Engine with Separate Organs. â€” In the ideal engine represented in fig. 10 the functions of boiler, cylinder and condenser are com- bined in a single vessel; but, provided the working substance passes through the same cycle of operations, it is indifferent whether these are performed in several vessels or in one. To approach a little more closely the conditions that hold in practice, we may think of

the engine as consisting of a boiler A (fig. 10) kept at tj, a non-conducting cylinder and piston B, a surface con- denser C kept at t 2 , and a feed-pump D which restores the condensed water to the boiler. When the several organs of the engine are separated in this way we can still carry out the first three stages of the cyclic process described in Â§ 38. The first stage of that cycle

Flo. xo.-Organs of a Steam Engine. STSSSff^tfifSSS into the cylinder. Then the point known as the point of cut-off is reached, at which admission ceases, and the steam already in the cylinder is allowed to expand, exerting a diminishing pres- sure on the piston. This is the second stage, or the stage of expansion. The process of expansion may be carried on until the pressure falls to that of the condenser, in which case the expansion is said to be complete. .At the end of the expansion release takes place, that is to say, communication is opened with the con- denser. Then the return stroke begins, and a period termed the exhaust occurs, that is to say, steam passes out of the cylinder, into the condenser, where it is condensed at the pressure in the condenser, which is felt as a back pressure opposing the return of the piston. So far, all has been essentially reversible and identical with the corresponding parts of Carnot's cycle.

But we cannot complete the cycle as Carnot's cycle was completed. The existence of a separate condenser makes the fourth stage, that of adiabatic compression, impracticable, and the best we can do is to continue the exhaust until condensation is complete, and then return the condensed water to the boiler.

41. Rankine Cycle. â€” It follows that the ideal cycle of Garnot is not an appropriate standard with which to compare the action of a real steam engine. Instead of it we have, in the engine with separated organs, a cycle which is commonly called the Rankine cycle, which differs from the Carnot cycle only in this, that the stage of adiabatic compression, is wanting and its place is taken by a direct return of the condensed water to the boiler, a process which makes the water receive heat at various temperatures, ranging from the temperature of the condenser up to that of the boiler. The chief part of the heat which the working substance receives is still taken in at the upper limit of temperature, during the process of changing from water to steam. But a small part is taken in at lower tempera- tures, namely, in the heating of the feed water in its transfer to the boiler. Any heat so taken in has less availability ior conversion into work than if it were taken in at the top of the range, and conse- quently the ideal efficiency of the cycle falls somewhat short of this ideal reached in the cycle of Carnot.

But the principle still applies that with respect to each portion of the heat that is taken in, the fraction convertible into work under ideally favourable conditions is measured by (t â€” ti)/t, where t is the absolute temperature at which that portion of heat is received, and t 2 is the temperature at which heat is rejected. Accordingly, we may investigate as follows the ideal performance of an engine following the Rankine cycle. Let &Q represent that portion of the whole heat which is taken in at any temperature t. Then the greatest amount of work obtainable from that portion of heat is ?Q(râ€” t 2 )/t, and the whole amount of work ideally obtainable in the complete process is found by calculating, 25Q(t â€” t2)/t where the summation includes all the heat that is taken in. In a steam engine using saturated steam the principal item in this sum is the latent heat Li, which is taken in at constant temperature n, during the change of state from water to steam. But there is, in addition, the heat taken in by the feed-water before it reaches the temperature at which steam is formed, and this may be represented as the sum of a series of elements ahr taken in at varying temperatures t, where a is the specific heat of water. Thus if W represents the thermal equivalent of the work theoretically obtainable per ft of steam, under ideally favourable conditions

determined by Regnault, between the limits of ri and tj. basis

W= t r(r 1 -T. 2 )- J T2log^+ Ll(T '~ T2 ) -

T 2 Ti

It is usual to take a as practically equal to 1 , which makes W =

827

On this

= (r

\ Tl/ T 2

This expresses the greatest amount of work which each pound of steam can yield when the temperature ti at which it reaches the engine and the temperature t 2 at which it leaves the engine are assigned. It consequently serves as a standard with which the actual per- formance may usefully be compared. The actual yield per lb of steam is always considerably less, chiefly because the ideal condition of adiabatic expansion from the higher to the lower extreme of temperature is never satisfied.

A more simple expression for the work theoretically obtainable per lb of steam when expanded adiabatically under the conditions of the Rankine cycle, is

I1-I2, where Ii is the total heat of the working substance in the initial state, before the adiabatic expansion, and I 2 is its total heat after that expansion. For it may readily be proved that, in an adiabatic process,

y,r_ ; , q-Â»T(Tâ€” T;) _J_ Llfriâ€” T 2 ) T Ti

The experiments of Regnault show that a, within the limits of temperature that obtain in boilers, is a nearly constant quantity, and no serious error will be introduced in this integration by treating it as a constant, with a value equal to the mean value, as

l'-I.=j/Â£Â«fc.

and this integral is the area of the indicator diagram when the substance is taken in at pi, expanded to pi and discharged at pi.

This expression applies whether the steam is initially superheated or not. I 2 will in general be the total heat of a wet mixture, and to calculate it we must know the condition as to wetness which results from the expansion. This is most easily found, especially when there has been initial superheat, by making use of the entropy- temperature diagram to be presently described, or by other graphic methods, for an account of which the reader should refer to the paper by Mollier already cited, or to J. A. Ewing's The Steam Engine and other Heat Engines (3rd ed.).

42. Entropy. â€” The study of steam-engine problems is greatly assisted by introducing the idea of entropy and making use of dia- grams in which the two co-ordinates are entropy and temperature. Entropy is a condition of the working substance defined by the statement that when any quantity of heat SQ is received by, or generated in, or rejected by the substance, when its absolute tem- perature is t, the substance gains or loses entropy by the amount SQ/t. Thus 2SQ/r measures the whole change of entropy in a process which involves the taking in or rejection of heat at more than one temperature. We shall denote entropy by tf>, and consider it as reckoned per unit of mass of the substance. Since by definition of entropy SQâ€”SQ/t, t5<Â£ = oQ, and hence if a curve be drawn with t and <f> for ordinates to exhibit the action of a working substance, the area under the curve, or jrd<j>, being equal to S5Q, measures the heat which the substance has received or rejected during the operation which the curve represents.

In a reversible cycle of operations Carnot's principle shows that S5Q/t = o, and it is obvious in such a case that the entropy returns at the end of the cycle to its primitive value. The same result may be extended to a cycle which includes any non-reversible step, by taking account of the heat generated within the substance by such a step, as if it were heat communicated from outside, in the reckoning of entropy. Thus, for example, if at one stage in the cycle the sub- stance passes through a throttle-valve, which lowers its pressure without letting it do work, the action is equivalent in effect to an adiabatic expansion, together with the communication to the sub- stance, as heat, of the work which is lost in consequence of the irreversible expansion through the throttle-valve taking the place of adiabatic expansion against a piston. If this heat be included in the reckoning 25Q/r = o for the complete cycle.

The entropy-temperature diagram for any complete cyclic process is a closed curve, and the area it encloses, being the excess of the heat received over the heat rejected, measures the work done. The entropy-temperature diagram shares this useful characteristic with the pressure-volume diagram, and in addition it shows directly the heat received and the heat rejected by the areas under the forward and backward limbs of the curve. To draw the entropy-tempera- ture diagram for the ideal steam engine (namely, the engine following the Rankine cycle), we have to reckon first the entropy which water acquires in being heated, and next the entropy Li/n which is acquired when the conversion into steam has taken place. Reckoning from any standard temperature tâ€ž, in the heating of the feed-water up to any temperature r, the entropy acquired is

. (~t adr <*V = J T o â€” ' and taking a as sensibly constant,

4>â€ž, = ff(logeT â€” logero)-

During evaporation at n a quantity of heat Li is taken in at temperature n, and hence the entropy of the steam

<t>, = *Â«, +L1/T1 = O- (lOgeTj - logÂ«T ) +Li/ti>