Compound Expansion
111. In the original form of compound engine, invented by Hornblower and revived by Woolf, steam passed directly from the first to the second cylinder; the exhaust from the first and admission to the second went on together throughout the whole of the back stroke. This arrangement is possible only when the high and low pressure pistons begin and end their strokes together, that is to say, when their movements either coincides in phase of differ by half a revolution. Engines of the "tandem" type satisfy this condition—engines, namely, whose high and low pressure cylinders are in one line, with one piston-rod common to both pistons. Engines which the high and low pressure cylinders are placed side by side, and act either on the same crank or on cranks set at 180&Mac251; apart, may also discharge steam directly from one to the other cylinder; the same remark applies to beam engines, whether of the class in which both pistons act on one end of the beam, or of the class introduced by M’Naught, in which the high and low pressure cylinders stand on opposite sides of the centre. By a convenient usage which is now pretty general the name "Woolf engine" is restricted to those compound engines which discharge steam directly from the high to the low pressure cylinders without the use of an intermediate receiver.
112. An intermediate receiver becomes necessary when the places of the pistons in a compound engine do not agree. With two cranks at right angles, for example, a portion of the discharge from the high-pressure cylinder occurs at a
time when the low-pressure cylinder cannot properly receive steam. The receiver is to some cases an entirely independent vessel connected to the cylinders by pipes; very often, however, a sufficient amount of receiver volume is afforded by the valve casings and the steam-pipe which connects the cylinders. The receiver, when it is a distinct vessel, is frequently jacketed.
The use of a receiver is of course not restricted to engines in which the "Woolf" system of compound working is impracticable. On the contrary, it is frequently applied with advantage to beam and tandem compound engines. Communication need not then be maintained between the high and low-pressure cylinders during the whole of the stroke; admission to the low-pressure cylinder is stopped before the stroke is completed; the steam already admitted is allowed to expand independently; and the remainder of the discharge from the high-pressure cylinder is compressed into the intermediate receiver. Each cylinder has then a definite point of cut-off, and by varying these points the distribution of work between the two cylinders may be adjusted at will. In general it is desirable to make both cylinders of a compound engine contribute equal quantities of work. If they act on separate cranks this has the effect of giving the same value to the mean twisting moment on both cranks.
113. Wherever a receiver is used, care should be taken that there is no unresisted expansion into it; in other words, the pressure in the receiver should be equal to that in the high-pressure cylinder at the moment of release. If the receiver pressure is less than this there will be what is termed a "drop" in the steam pressure between the high-pressure cylinder and the receiver, which will show itself in an indicator diagram by a sudden fall at the end of the high-pressure expansion. The "drop" is, from the thermodynamic point of view, irreversible, and therefore wasteful. It can be avoided by selecting a proper point of cut-off in the low-pressure cylinder. When there is no "drop" the expansion that occurs in a compound engine has precisely the same effect in doing work as the same amount of expansion in a simple engine would have, provided the law of expansion be the same in both and the waste of energy which occurs by the friction of ports and passages in the transfer of steam from one to the other cylinder be negligible. The work done in either case depends merely on the relation of pressure to volume throughout the process; and so long as that relation is unchanged it is a matter of indifference whether the expansion be performed in one vessel or in more than one. It has, however, been fully pointed out in chap. IV. that is general a compound engine has a thermodynamic advantage over a simple engine using the same pressure and the same expansion, inasmuch as it reduces the exchange of heat between the working substance and the cylinder walls and so makes the process of expansion more nearly adiabatic. The compound engine has also a mechanical advantage which will be presently described. The ultimate ratio of expansion in any compound engine is the ratio of the volume of the low-pressure cylinder to the volume of steam admitted to the high-pressure cylinder. Fig. 29 illustrates the combined action of the two cylinders in a hypothetical compound engine of the Woolf type, in which for simplicity the effect of clearance is neglected and also the loss of pressure which the steam undergoes in transfer from one to the other cylinder. ABCD is the indicator diagram of high-pressure cylinder. The exhaust line CD shows a falling pressure in consequence of the increase of volume which the steam is then undergoing through the advantage of the low-pressure piston. EFGH is the diagram of the low-pressure cylinder drawn alongside of the other for convenience in the construction which follows. It has no point of cut-off; its admission line is the continuous curve of expansion EF, which is the same as the high-pressure exhaust line CD, but drawn to a different scale of volumes. At any point K, the actual volume of the steam is KL + MN. By drawing OP equal to KL + MN, so that OP represents the whole volume, and repeating the same construction at other points of the diagram, we may set out the curve QPR, the upper part of which is identical with BC, and so complete a single diagram which exhibits the equivalent expansion in a single cylinder.
In a tandem compound engine of the receiver type the diagrams resemble those shown in fig. 30. During CD (which corresponds to FG) expansion is taking place into the large of low pressure cylinder. D and G mark the point of cut-off in the large cylinder, after which GH shows the independent expansion of the steam now shut within the large cylinder, and DE shows the compression of steam by continued discharge from the small cylinder into the receiver. At the end of the stroke the receiver pressure is OE, and this must be the same as the pressure at C, if there is to be no "drop." Diagrams of a similar kind may be sketched without difficulty for the case of a receiver engine with any assigned phase relation between the pistons.1
114. By making the cut-off take place earlier in the large cylinder we increase the mean pressure in the receiver; the work done in the small cylinder has consequently diminished. The work done in the large cylinder is correspondingly increased, for the total work (depending as it does on the initial pressure and the total ratio of expansion) is unaffected by the change. The same adjustment serves, in case there is "drop," to remove it. By selecting a suitable ratio of cylinder volumes to one another and to be volume of the receiver, and also by choosing a proper point for the low pressure cut-off, it is possible to secure absence of drop along with equality in the division of the work between the two cylinders.
To determine that point of cut-off in the low-pressure cylinder which will prevent drop when the ratio of cylinder and receiver volumes in assigned is a problem most easily solved by a graphic process. The process consists in drawing the curve of pressure during admission to the low-pressure cylinder until it meets the curve of expansion which is common to both cylinders.2 Thus in fig 31 (where for the sake of simplicity the effects of clearance are neglected) AB represents the admission line and BC the expansion line in the small cylinder. Release occurs at C, and from C to D steam is being is being taken by the large cylinder. D corresponds to the cut-off in the large cylinder, which is the point to be found. From D to E steam is being compressed into the receiver. To avoid drop the receiver pressure at E is to be the same as the pressure at C. E is therefore known, and may be employed as the starting-point in drawing a curve EF which is the admission line of the low-pressure diagram EFGHI. This line is drawn by considering at each point in the low-pressure piston’s stroke what the whole volume of the steam is then. The place at which EF intersects the continuous expansion curve BCG determines the proper point of cut-off. The sketch (fig. 31) refers to the case of a tandem receiver engine; but the process may also be a applied to an engine with any assumed phase relation between the cranks. Fig. 32 shows a pair of theoretical indicator diagrams determined in the same way for an engine with cranks at right angles, the high-pressure crank leading. In using the graphic method any form may be assigned to the curve of expansion. Generally this curve may be treated without serious inaccuracy as a common hyperbola, in which the pressure varies inversely as the volume.
115. If this simple relation between pressure and volume be assumed, it is practicable to find algebraically the low-pressure cut-off which will give no drop, with assigned ratios of cylinder and receiver volumes. Taking the simplest case—that of a tandem engine or of an engine with parallel cylinders whose pistons move together or in opposition—we may proceed thus. Since the point of cut-off to be determined depends on volume ratios we may for brevity treat the volume of the small cylinder as unity. Let R be the ratio to it of the receiver’s volume, and L that of the low-pressure cylinder. Let x be the required fraction of the stroke at which cut-off is to occur in the large cylinder; and let p be the pressure at release from the small cylinder. As there is to be no drop, p is also the pressure in the receiver at the beginning of admission to the large cylinder. During that admission the volume changes from 1 + R to 1 – x + R +xL, and the pressure at cut-off is
Therefore p(1+R) . The steam that remains is now compressed
1 – x + R + xL
Into the receiver, from volume 1 – x + R to volume R. its pressure therefore rises to p(1+R) . (1 – x + R) , and this, by assumption, is to be equal to
1 – x + R + xL R
p. We therefore have (1 + R)( 1 – x + R) = R(1 – x + R +xL) ,
whence x = (R+1)/(RL+1)
Thus, with R=1 and L=3, cut-off should occur in the large cylinder at half-stroke; with a greater cylinder ratio the cut-off should earlier.
A similar calculation1 for a compound engine whose cranks are at right angles, and in which cut-off occurs in the large cylinder before half-stroke, shows that the condition of no drop is secured when___
2R(xL – 1) = 2&Mac195;x(1-x).
In some compound engines a pair of high-pressure cylinders discharge into a common receiver; in some a pair of low-pressure cylinders are fed from a receiver which takes steam from one high-pressure cylinder, or in some instances from two. With these arrangements the pressure in the receiver may be kept much more nearly constant than is possible with the ordinary two-cylinder type.
116. An important mechanical advantage belongs to the compound engine in the fact that it avoids the extreme thrust and pulls which would have to be borne by the piston-rod of a single-cylinder engine working at the same power with the same initial pressure and the same ratio of expansion. If all expansion took place in the low –pressure cylinder, the piston at the beginning of the stroke would be exposed to a thrust much greater than the sum of the thrusts on the tow pistons of a compound engine in which a fair proportion of the expansion is performed in the small cylinder. Thus in the tandem engine of fig. 29 the greatest sum of the thrusts will found to amount to less than two-thirds of the thrust which the large piston would be subjected to if the engine were simple. The mean thrust throughout the stroke is of course not affected by compounding; only the range of variation in the thrust is reduced. The effort on the crank-pin is consequently made more uniform, the strength of the parts may be reduced, and the friction at slides and journals is lessened. The advantage in this respect is obviously much greater when the cylinders are placed side by side, instead of tandem, and work on cranks at right angles. As set-off to its advantage in giving a more uniform effort, the compound engine has the drawback of requiring more working parts than a simple engine with one cylinder. But in many instances—as
in marine engines—two cranks and two cylinders are almost indispensable, to give a tolerably uniform effort, and to get over the dead points; and the comparison should then be made between a pair of simple cylinders and a pair of compounded cylinders. Another point in favour of the compound engine is that, although the whole ratio of expansion is great, their need not to be a very early cut-off in either cylinder; hence the common slide-valve, which is unsuited to give an early cut-off, may be used in place of a more complex arrangement. The mechanical advantage of the compound engine has long been recognized, and had much to do with its adoption in the nearly days of high-pressure steam.2 Its subsequent development has been due in part to this, but probably in much greater part to the thermodynamic advantage which has been discussed above (§ 93).
117. Indicator diagrams taken form compound engines show that the transfer of steam from one cylinder to another is never, under the most favourable conditions, performed without loss of energy. Fig. 33 shows a pair of diagrams from the two cylinders of a tandem Woolf engine, in which the steam passed as directly as possible from the small to large cylinder. The diagrams are drawn to the same scale of stroke and therefore to different scale of volume, and the low-pressure diagram is turned round so that it may fit into the high-pressure diagram. There is some drop at the high-pressure release, and, apart from this, there is a loss through friction of the passages, which shows itself by the admission line to the large cylinder lying below the exhaust line form the small one.
118. Fig. 34. Is a pair of diagrams taken from a compound tandem receiver engine running at 50 revolutions per minute, with cylinders 30 inches and 52 inches in diameter, and with a 6-feet stroke. The ratio of cylinder volumes is therefore 3 to 1. The capacity of the receiver is nearly 1_ times that of the cylinder. There is a comparatively early cut-off in both cylinders, and nearly complete absence of drop. The small cylinder, however, doe more work than the large one, in the ratio of nearly 3 to 2.
Fig. 35 shows the same of pair of diagrams combined by drawing both to the same scale of volume and of pressure, and by setting out each by an amount equal to the clearance space from the line of no volume. This makes the expansion curve in each diagram present correctly the relation of the pressure to the absolute volume of the expanding steam. The broken line is a continuous curve of adiabatic expansion, drawn from the point of high-pressure cut-off, on the assumption that the steam then contained about 25 per cent. of condensed water. If the expansion were actually adiabatic, and if there were no loss in the transfer of the steam, the expansion curves for both cylinders would fall into this line.
119. Fig. 36 exhibits, in the same manner as fig. 35, a set of diagram taken by Mr Kirk from the triple expansion engines of the S.S. "Aberdeen." Each diagram is set out form the line of no volume by a distance which represents the clearance in the corresponding cylinder. The boiler pressure is 125_ per square inch. The cylinders are 32 inches, and 70 inches in diameter, and the stroke is 4_ feet. The cranks make 120&Mac251; with each other. The means of the diagrams for the ends of each cylinder have been used in drawing this and the next figure, a practice which should be followed in drawing combined diagrams of the kind here exemplified.
120. Fig. 37 shows in the same way a set of diagrams taken by Mr. Brock from the quadruple expansion engines of the S.S. "Lohara" (by Messrs Denny & Co.). Here the boiler pressure was 154 _ by gauge, or 169 _ absolute, the cylinders were 24 inches, 34 inches, 48 inches, and 68 inches in diameter, the stroke was 4 feet, and the number of revolutions 65 per minute.
121. In all of these cases continuous curve, shown by a broken line, has been drawn to represent result of adiabatic expansion, on the same assumption as before—that the steam contains about 25 per cent. of water a the point of cut-off in the first cylinder. The equation of the curve may then be taken as
PV10/9 = constant (§ 67). In the absence of data regarding the wetness of the steam this assumption may be considered fair.
122. Lastly, fig. 38 shows a pair of diagrams, treated in the same manner, for a two-cylinder compound engine with cranks at right angles each other, the high-pressure crank being 20&Mac251; in advance. During the back stroke of the high-pressure piston there is at first compression into the receiver until the large cylinder opens; the high-pressure diagram consequently takes a peculiar form, which should be compared with the diagram already given for a tandem engine (§ 118). In this example there is a considerable amount of drop and also of loss between the two cylinders.
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