Page:Encyclopædia Britannica, Ninth Edition, v. 22.djvu/510

 486 STEAM-ENGINE [rKOPERTIES OP STEAM. EiBjieucy of an engiiie u.sing steam non- ex pan - sively. the performance of actual engines, and as setting forth the advan- tage of high-pressure steam from the thermodyuamic point of view. 70. As a contrast to the ideally perfect steam-engine of 68 we may next consider a cyclic action such as occurred in the early engines of Newcomen or Leupold, when steam was used noii- expausively, or rather, such an action as would have occurred in engines of this type had the cylinder been a perfect non-conductor of heat. Let the cycle of operations be this : (1) Apply A and evaporate the water as before at Pj. Heat taken in = L|. (2) Remove A and apply C. This at once condenses a part of the steam, and reduces the pressure to ?._,. (3) Compress at P 2, in contact with C, till condensation is complete, and water at r. 2 is left. (4) Remove B and apply A. This heats the water again to T I and completes the cycle. Heat taken in = h^ - h. 2. The indicator diagram for this series of operations is shown in fig. 15. Here the action is not reversible, have Fig. 15. To calculate the efficiency, we Work done = (P^ - P 3 XVi-Q-017) Heat taken in J^ + /^ - A.,) The values of this will be found to range from 0'067 to 0'072 for the values of J > 1 which are stated in 69, when the temperature of condensation is 60 F. 71. In the ideal engine represented in fig. 14 the functions of boiler, cylinder, and condenser are combined in a single vessel ; but after what has been said in chap. II. it is scarcely necessary to re- mark that, 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 which performs the cycle of 70 as consisting of a boiler A (fig. 16) kept at TJ, a non-conducting cylin- der 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. Then for every pound of steam supplied and used non-expansively as in 70, we have work done on the piston = ( P - P 2 ) V x ; but an amount of work has to be ex- pended in driving thefeed- pump =(Pj-P 2 ) 0-017. Deducting this, the net work done per Ib of steam is the same as before, and the heat taken in is also FIG. 16. Organs of a Steam-Engine. the same. An indicator diagram taken from the cylinder would give the area cfcjh (fig. 17), where oe = P 1, cf=y i} o/i = P 2 ; an indi- cator diagram taken from the pump would give the negative area Jijie, where ei is the volume of the feed-water, or 0'017 cub. ft. The difference, namely, the shaded area, is the diagram of the complete cycle gone through by each pound of the working^* substance. In experimental measurements of the work done in steam-engines, only *' 17 - the action which occurs within the cylinder is shown on the indi- cator diagram. From this the work spent on the feed-pump is to be subtracted in any accurate determination of the thermodynaniic efficiency. If the feed-water is at any temperature r n other than that of the condenser as assumed in 70, it is clear that the heat taken in is H a - 7;. instead of Hj - h a . How 72. We have now to inquire how nearly, with the engine of fig. nearly 16 (that is to say, with an engine in which the boiler and condenser may the are separate from the cylinder), we can approach the reversible process cycle of 68. The first stage of that cycle corresponds to the be re- admission of steam from the boiler into the cylinder. Then the veraibie ? point known as the point of cut-off is reached, at which admis- sion ceases, and the steam already in the cylinder is allowed to expand, exerting a diminishing pressure 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, com- munication is opened with the condenser. Then the return stroke begins, and a period termed the exJutust occurs, that is to say, steam passes out of the cylinder, into the condenser, where it is condensed at pressure P 2, 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 com- pleted. 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 com- Efficiency of engiiie working expan- sivelv. plete, and then return the condensed water to the boiler by means of the feed-pump. It is true that we may, and in actual practice do, stop the exhaust before the return stroke is complete, and compress that portion of the steam which remains below the piston, but this does not materially affect the thermodynaniic efficiency ; it is done partly for mechanical reasons, and partly to avoid loss of power through clearance (see chap. IV. ). In the present instance it is supposed that there is no clearance, in which case this compression is out of the question. The indicator diagram given by a cylinder in which steam goes through the action de- scribed above is shown to scale in fig. 18 for a par- ticular example, in which it is supposed that 1 cubic foot of dry saturated steam is admitted at an absolute pressure of 90 Ib per square _ inch, and is expanded twelve r, u ;r times, or down to a pressure of 5'4 tt> per square inch, at FIG. 18. Ideal indicator Diagram for Steam i v. -4. v used expansively. which pressure it is dis- charged to the condenser. As we have assumed the cylinder to be non-conducting, and the steam to be initially dry, the expansion follows the law PV 1J35 = constant. The advantage of expansion is obvious, that part of the diagram which lies under the curve being so much clear gain. 73. To calculate the efficiency, we have Work done per Ib during admission = PiY, ; P V - P.,? Y ,, ,, during expansion to volume rV 1 = - J i (by 36), = (PjVj - Po'/'YJ/O "105 ; Work spent during return stroke = IVY] ; ,, on the feed-pump=(P 1 -"P 2 )0-Ol7; Heat taken in = H l - A . 74. These expressions refer to complete expansion. When the Incoin- expansion is incomplete, as it generally is, the expression given plete ex- above for the work done during expansion still applies if we take pausiou. P 2 to be the pressure at the end of expansion, while the work spent on the steam during the back-stroke is PtrVj and that spent on the feed-pump is (Pj Pt)0'017, P& being the back pressure. Incomplete expansion is illustrated by the dotted line in tig. 18. It is easy, by the aid of 64 and 67 to extend these calcula- tions to cases where the steam, instead of being initially dry, is supposed to have any assigned degree of wetness. The efficiency which is calculated in this way, which for the present purpose may be called the theoretical efficiency corresponding to the assumed conditions of working, is always much less than the ideal efficiency of a perfect endue, since the cycle we are now dealing with is not reversible. But even this theoretical efficiency, short as it falls of the ideal of a perfect engine, is far greater than can be realized in practice when the same boiler and condenser tem- peratures are used, and the same ratio of expansion. The reasons for this will be briefly considered in the next chapter ; at present the fact is mentioned to guard the reader from supposing that the results which the above formulas give apply to actual engines. 75. The results of 68 have been turned to account by Rankine Calcula- and Clausius for the purpose of deducing the density of steam tion of from other properties which admit of more exact direct measure- density of nieut. Let the perfect steam-engine there described work through saturated a very small interval of temperature AT between two temperatures r steam. and T - AT. The efficiency is AT/T, and the work done (in foot-lbs. ) is JLAT/T. The indicator diagram is now reduced to a long narrow strip, whose length is V-0'017 and its breadth AP, the difference in pressure between steam at tern peratures T and T - AT. Hence the work done is also AP(V - 0'017), and therefore V-0-017 =. ^. T Ar Here p, or (in the limit) -^ > is the rate of increase of tempera- ture with increase of pressure in saturated steam at the particular temperature T. It may be found roughly from Table II., p. 484, or more exactly by differentiating the equation given in 57. L is also known, and hence the value of V corresponding to any assigned temperature may be calculated with a degree of accuracy which it would be difficult to reach in direct experiment. The volumes given in the Table are determined in this way. 1 1 The result of 75 may be applied as follows to give the formula of G7 for the adiabatic expansion of wet steam. For brevity we may write V 017= u. In adiabatic expansion the work done is equal to the loss of internal energy, or P <%)= Jdl= Jd(h+qp). Since dh = dr, and p = L Pu/J , this may be written J By 5 75. dP= dT. hence 1 + -7- (L) - =0; T dr r and by integration, Iog 6 T+ "7 V T = constant = log e r t + jjLj which is the equation of G7.