459 CHAPTER XXXIII. THEORETICAL INDICATOR DIAGRAMS OF STAGE To determine the relative proportions of the cylinders of stage expansion engines, and the points of cut-off in each to produce the most uniform strains on the shafting, and enable the work due to the expansion to be most fully realised, theoretical indicator diagrams showing the action of the steam in the several cylinders are very useful. For simplicity, we will neglect the effects of the release before the end of the stroke, the compression, the resistance of the passages between the cylinders, &c., simplicity being of more importance than extreme accuracy, especially as allowance can easily be made for the compression, early release, &c., after the diagrams have been drawn, if the principles involved are clearly understood. Let V represent the volume of the large cylinder; v the volume of the small cylinder; U the volume of the intermediate reservoir; R = total rate of expansion; Rate of expansion = r in high-pressure cylinder, and Ρ in low v-pressure; A ratio of cylinders; so that R=rλ; = ratio of reservoir to h p. cylinder; so that U=4v ; P1 = initial absolute pressure in the high-pressure cylinder. บ Since a volume of steam entering the high-pressure cylinder at pressure P, we shall have v V r R' P1, occupies finally a volume V at pressure R F C B C D a C d R F A Compound engines with cranks at 0 deg. or 180 deg. apart, but without an intermediate reservoir.-In Fig. 386, O B represents the initial absolute pressure of the steam on its admission to the high-pressure cylinder, and BC is the line of pressure during admission. At C the steam is cut off and expands in the small cylinder to D, the end of the stroke, when the communication is opened to the large cylinder, and the steam exerts a forward pressure on the large piston and a back pressure on the small piston. This part of the action of the steam is represented by the two curves, DA and E F, the ordinates of DA representing the back pressures on the small piston and the corresponding ordinates of E F the forward pressures on the large piston. OP is V, the volume of the large cylinder, and O N=v, the volume of the small cylinder. At the end of the stroke of the large piston the communication is opened to the condenser, and the pressure falls to PG, the constant condenser pressure. These diagrams inay be combined as follows:-Draw any straight line, a b c d, parallel to POQ, and intersecting the two diagrams, and lay off on it c d = a b, O M would be formed if the steam had been expanded in Er drawing a suiticieni zomber is distances on 15€. ALT DEmber it pimte no le di be reasoned about as of th, wh only. The pressure at any pot in the forward strat back stroke of the stall pena met brazed return stroke of the stall paste the pad via the steam whenits Clara which A cylinder only. nilaying the proper iwi the diagram can ise in one cylinder the large piston, and At any point M in the betspied by the steam is Compound engines with cranks at 0 deg or 180 deg apart, but with an intermediate reservoir.-This is a case that scllim as a practice unless the reservoir be used for the purpose Let p = pressure in the reR Server immediately before the high-pressure exänder exhausts intoil. 04. Fig.387 = p. = initial absolute pressure of the steam in the high-pressure cylinder. At B the steam is cut off and expan is to C. the end of the stroke of the high-pressure cylinder. At this point the communication is opened to the reservoir, and a volume, r, of steam at pressure E: is admitted to the reservoir; consequently the pres r T sure ND will be This is, of course, equal to the initial pressure OG in the low-pressure cylinder. The steam now acts on the lowpressure piston until 1th of the stroke of the low-pressure piston has been performed, when the admission to the large cylinder is cut off. At this point the steam occupies the volume ( 1 − 1) = and its pressure is therefore T P This part of the action of the steam is represented by the curve G H in the low-pressure diagram, and DE in the high-pressure diagram. After the steam is cut off, it expands in the cylinder to the final pressure, while R in the reservoir it is compressed to the pressure p,, represented by MR = OF. It only remains to determine p, in order that the diagrams may be completely drawn. We can easily find p, from the fact that a volume of steam V Ꭱ . The diagram can now be completely drawn. If the reservoir pressure p, be not so great as the pressure of release in the high-pressure cylinder, P1, there will be a fall of pressure on the admission to the reservoir, and the work due to expansion will be partly lost. If these pressures be equal we have: = λ + 1. 2 From this equation in any given case p can be determined, so that there shall be no loss on admission to the reservoir. When = o, p 1; this is the case previously discussed. Taking = 1, that is, taking the volume of the reservoir equal to that of the high-pressure cylinder, we have p In this case, if λ be greater than 3, p will be greater than 2, consequently arrangements should be fitted to cause the cut-off in the low-pressure cylinder to be before half-stroke if the work due to the expansion is to be fully realised. The following table gives a few values of rates of expansion necessary in the low-pressure cylinders of compound engines of this type when there is no fall of pressure on the admission to the reservoir. We now pass on to consider the type of compound engine most generally used, viz. engines with two cylinders, side by side, acting on cranks at right angles to each other, and having an intermediate reservoir. The cases in which the cut-off in the low-pressure cylinder is after half-stroke, and before half-stroke respectively, must be discussed separately. Two cylinder compound engines with cranks at right angles to each other, having an intermediate reservoir, the cut-off in the low-pressure cylinder being after halfstroke. It will be necessary in the first place to find an expression for the distance of the high-pressure piston from the end of its stroke, when the steam is cut off in the low-pressure cylinder. Let OA (Fig. 388) be the position of the crank of the low-pressure cylinder when the steam is cut off; OB the corresponding position of the crank of the high-pressure cylinder. 1- sin @ pressure piston, when steam is cut off in the low-pressure cylinder 2 In the following investigation we will denote this by m; consequently the fraction of the small cylinder that is occupied by the steam that acts on the low-pressure piston at the point of cut-off is (1 - m). The following table gives some values of (1-m) for different values = In Fig. 389, OA represents the initial pressure of steam in the high-pressure cylinder. At B the steam is cut off and expands to C, the end of the stroke of the high-pressure piston, when the communication is opened to the reservoir and the pressure falls to R D. After this the steam expands in the reservoir and low-pressure cylinder until it is cut off in the latter. This part of the action of the steam is represented by the curve DE, QE being the pressure at the point of cut-off. From this point the steam is compressed behind the high-pressure piston until it has completed half its return stroke, when its pressure is represented by P F. At this point the admission to the low-pressure cylinder commences, and the steam expands in the low-pressure cylinder until the end of the return stroke of the high-pressure cylinder, when its pressure is O G. The low-pressure diagram is easily deduced from this. The initial pressure OH is, of course, equal to the back pressure PF at the middle of the return stroke of the high-pressure piston. The steam expands in the low-pressure cylinder until half-stroke, when its pressure, SK, is obviously equal to OG. At this point the high-pressure cylinder, containing steam at the pressure RC, opens to the reservoir, and the pressure rises to SL, SL being equal to RD. From L the steam expands in the reservoir and low-pressure cylinder to W, the point of cut-off, TW being equal to QE. From W the steam in the cylinder expands to the final pressure N M, while that in the reservoir is compressed to V, NV being equal to the initial pressure in the low-pressure cylinder. At M the communication to the condenser is opened, and the pressure falls to N N',-the constant condenser pressure. We will now give the algebraical expressions for the pressures at the different points, in order that the diagrams may be drawn in any given case. Since the total rate of expansion is R, the final pressure, N M, in the lowpressure cylinder is The final pressure, R C, in the high-pressure P1 The steam in the low-pressure cylinder is expanded p times: consequently at the point of cut-off, the pressure, T W, is = PIP. This is also the pressure, QE, in the reservoir at the point of cut-off, and we have, therefore, steam at the pressure PIP occupying a volume U + v (1 - m). This steam is com R pressed behind the high-pressure piston until the beginning of the next stroke of the low-pressure piston, when its volume has been reduced to U + and its pressure has been increased to v 2 which is the initial pressure OH (= PF = N V) in the low-pressure cylinder. This steam is driven before the high-pressure piston, and drives the lowpressure piston before it till half-stroke, when its volume is U + V, and the pressure SK is, therefore, But at this point the high-pressure cylinder, containing a volume v of steam at pressure P1 V R v , opens to the reservoir, and the pressure becomes |