A. V. Ugarov,
Professor of the Tomsk Technological Institute of Emperor Nicholas II.

A Machine with Intermediate Steam Extraction

The Operating Regime of the Machine,

as the Basis for Its Calculation.

With 17 figures in the text and 6 tables of drawings.

Tomsk,
Typo-Lithography of the Siberian Association of Printing.
1915.

Noted Misprints

Page	Line	Printed	Should be
5	10 from bottom	$x_3$	$x$
7	11 from bottom	horizontal	horizontal
9	7 from top	$O$	$O_1$
10	3 from bottom	cylinder	cylinder
11	1 from top	in	on
11	2 from bottom	drawings	drawing
13	7 from bottom	cylinder	cylinder
13	4 from bottom	compression	compression
14	11 from top	deviation	cut-off
17	2 from bottom	K. Vodolinskii	Kh. Vodolinskii
24	18 from bottom	omitted:	low pressure
27	21 from bottom	$l_3 l_4$	$l_3 l_4'$
28	7 from bottom	algebraic	algebraic
30	7 from bottom	0.4	0.5
34	10 from bottom	analytic	analytically
34	6 from bottom	was equal	was equal
39	8 from bottom	7.5	7.6
39	6 from bottom	$O_2 t''$	$O_2 t_2$
43	15 from top	investigations	investigations
50	15 from top	complete,	complete
59	1 from top	broke,	broke-
60	11 from top	diagrams	diagrams --
65	5 from top	(Table V)	(drawing, Table V)
65	6 from top	Table VI	drawing, Table VI
71	12 from top	considered	considered
76	3 from top	15 mm	14 mm
77	11 from top	$Q$	$Q_i$
77	13 from top	$Q$	$Q_i$
77	table	$Q$	$Q_i$
78	table	$Q$	$Q_i$
85	4 from bottom	2.153	2.513
86	13 from bottom	2.153	2.513
86	8--9 from bottom	extracted	extracted
Throughout the text		ballance	balance

Introduction

The work set forth below has as its subject the investigation of the phenomena occurring in a machine with steam extraction when the magnitudes both of the extraction itself and of the machine load are changing.

Understanding by the working process of a steam engine, in its established state, the entire aggregate of periodically recurring phenomena, in an unchanging sequence, undergone by a certain quantity of steam admitted into the machine during the whole time of its passage through the machine, the author has set himself the aim: to clarify, on the basis of a study of the working process of engines with steam extraction, the conditions for calculating the operation and designing machines of this kind.

Theoretical investigations of an analogous character are usually conducted by one of the following methods. One method, followed by Reutlinger *), consists in assuming a machine operating without losses, and thereby considering the unnecessary drawing of indicator diagrams of the machine under study; the entire calculation is carried out with respect to the quantity of heat contained in the steam passing through the machine in a definite time.

The other method, adhered to by L. Schneider **), is based on the approximate drawing of indicator diagrams for each of the cylinders of the machine; at the same time, as far as possible, on the basis of experimental data from various experimenters, all losses occurring in the machine are taken into account, and, on the basis of the diagrams obtained by such construction, the conditions of operation of the machine when steam is extracted from it are derived.

L. Schneider constructs in his investigations a series of Rankine diagrams. Diagrams of this kind, making it possible to judge the quantity and character of the work delivered by each of the cylin-

*) E. Reutlinger—Die Zwischendampfverwertung in Entwicklung, Theorie und Wirtschaftlichkeit. Berlin—1912.

**) L. Schneider—Über die Verwertung des Zwischendampfes und des Abdampfes der Dampfmaschinen zu Heizzwecken. Berlin—1910.
L. Schneider—Die Abwärmeverwertung im Kraftmaschinenbetrieb. Berlin—1912.

the engines separately, at the same time they do not possess the clarity of an image of the separate parts of the working process from the point of view of the simultaneous action of the steam-distribution organs of each of its cylinders and of the phenomena occurring in the receiver, which alternately communicates with all the cavities of each of the cylinders.

The most graphic and convenient—in the indicated sense—is the method, proposed by Schröter, of jointly plotting the indicator diagrams of both cylinders.

In the work set forth below the author has decided, in his investigation, to follow an intermediate course between the two methods indicated above; namely, the author assumes a machine without losses arising from heat exchange between the steam and the cylinder walls, without losses due to leakage of steam through imperfect tightness of the machine, without radiation losses into space, without losses due to throttling of the steam produced by the distributing devices, and without friction losses. In a word, the presence of ideal conditions is assumed, under which it is possible to admit the constant existence, in all cavities of the machine, of dry saturated steam obeying Mariotte’s law \(pv=c\).

Since, under the indicated conditions, the quantity of steam entering the machine will depend only on the periods of filling and compression in the high-pressure cylinder, the determination of the steam consumption per unit power can, in and of itself, refer only to the so-called useful consumption of steam.

Thus, for such a machine, relying—as will be indicated in due course—on the introduction into its working process of a fictitious extraction cylinder, the author constructs Schröter diagrams for various operating conditions of the machine, and from them studies the dependence between the changes revealed in the various periods and factors of the working process.

As a consequence of the assumptions made, it is accepted that the mixing of vapors obeys Dalton’s law, according to which the pressure of the mixture is

\[ p_m=\frac{p_1v_1+p_2v_2+p_3v_3+\cdots}{v_1+v_2+v_3+\cdots}. \]

For the sake of clarity in plotting the separate phases of the working process, it is assumed that the pressure of the mixture of steam in two or several volumes communicating with one another is established instantaneously. In various cases the pressure of the mixture is determined either analytically or graphically by Mönch’s method (provided that the final volume of the mixture does not change).

In the investigation of the working process of steam engines it is sometimes necessary to find the pressure of a mixture also under the condition that the mixing volumes change their magnitude. The working spaces of the various cylinders of an engine, communicating with one another during the motion of their pistons, give us precisely an example of such conditions.

The determination of the pressure of a mixture in such cases is most simply reduced to finding the point of intersection of equilateral hyperbolas, which graphically express the changes in the state of the steam for each of the quantities being mixed.

In the problem of finding the point of intersection of two hyperbolas there may be two cases: a) the centers of both hyperbolas lie on different sides of the point of their meeting (see Fig. 1), and b) these centers are on one side of the sought point (Fig. 2).

For the case of a mixture of two quantities of steam at constant volume, we have that the pressure of the mixture is

\[ p_x=\frac{p_1v_1+p_2v_2}{v_1+v_2}. \]

For the case of changing volumes, the resultant volume \(v\), corresponding to a definite moment of the working process and occupied by the mixed quantities of steam, will be equal to:

\[ v=v_1+v_2\pm v_3. \]

Then, for the pressure of the mixture, we obtain the following analytical expression:

\[ p'_x=\frac{p_1v_1+p_2v_2}{v_1+v_2\pm v_3}, \]

whence, knowing the constants of the corresponding hyperbolas \(p_1v_1=c_1\) and \(p_2v_2=c_2\), we have

\[ p'_x=\frac{c_1+c_2}{v}, \]

i.e., the pressure of the mixture, with changing volumes, is equal to the sum of the corresponding constants divided by the final volume.

The indicated question is also easily solved graphically.*)

Since, by constructing individual points of an equilateral hyperbola, we can always find the volumes occupied by the steam at a definite pressure, we shall adopt the condition, always easily introduced: the initial pressures of the mixing volumes are identical.

Thus, we have the following data:

\[ p_1v_1=c_1,\quad p_2v_2=c_2\quad \text{and}\quad p_1=p_2. \]

) A. Ugarov. From the practice of the drafting room. Journal of the Society of Siberian Engineers*, 1915, No. 2.

Fig. 1.

Fig. 2.

According to the notation adopted in Fig. 1, we have:

\[ p_x(v_1+x)=p_1v_1, \]

\[ p_x(v_3-x+v_2)=p_2v_2. \]

Dividing these equalities term by term and reducing, we obtain the proportion:

\[ \frac{v_1+x}{v_3-x+v_2}=\frac{v_1}{v_2}, \]

whence, relying on the property of the difference of the antecedents and consequents, we have:

\[ \frac{x}{v_3-x}=\frac{v_1}{v_2}. \]

Consequently, in order to find the point of intersection of the line \(O_1O_2\)—the axis of volumes—with the vertical passing through the desired point of intersection of the hyperbolas, one must divide, internally, the segment \(AB=v_3\) into parts proportional to the given initial volumes \(v_1\) and \(v_2\).

For this purpose we lay off, on opposite sides of the line \(O_1O_2\), the segments \(AC=AO_1=v_1\) and \(BD=BO_2=v_2\). The straight line joining the points \(C\) and \(D\) gives the point \(E\) of the vertical \(EG\). Having this vertical, by the construction customary for equilateral hyperbolas, we find the required point \(F\) common to both curves.

For the case when the point of intersection of the hyperbolas lies on the same side of both origins of coordinates, with the same data as before, we have, according to the notation of Fig. 2:

\[ p_x(v_1+x)=p_1v_1 \]

and

\[ p_x(x_3-v_3+v_2)=p_2v_2, \]

whence, by the property of proportions, we obtain:

\[ \frac{x}{x-v_3}=\frac{v_1}{v_2}, \]

i.e., in order to find the position of the ordinate of the desired point of intersection, one must divide the segment \(AB=v_3\), externally, into parts proportional to the given volumes.

Laying off, on the same side of the line \(O_1O_2\), on the corresponding verticals, the segments \(AC=AO_1=v_1\) and \(BD=BO_2=v_2\), we draw the straight line \(CD\), which gives us the point \(E\). On the vertical \(EG\), in the usual manner, we find the point \(F\) of intersection of the hyperbolas.

In both examples analyzed, the point \(F\) determines the desired mixture pressure \(p_x\).

The mixture pressure for several communicating volumes may be found by the same method, applied successively.

In order to reveal the difference between the working processes of a Tandem machine of the ordinary type and a Tandem machine with steam extraction, and also to clarify better the individual periods of the process of the latter, the author has considered it necessary to examine, also for the first of the indicated machines, its working process in all its separate phases.

Thus, the exposition of the investigation undertaken falls of itself into the following parts:

a) a graphical representation of the working process of a Tandem machine of the ordinary type;

b) a graphical representation of the working process of a Tandem machine with intermediate steam extraction;

c) an approximate calculation of a Tandem machine without cooling, and determination of the limits of possible intermediate steam extraction;

d) an approximate calculation of a Tandem machine with cooling (under a low vacuum) and determination of the limits of possible extraction;

e) investigation of the operating conditions of the machine with variable extraction of steam at constant pressure;

f) investigation of the operating conditions of the machine with variable extraction of steam at variable pressure;

g) establishment of the aggregate of the various operating conditions of the machine under consideration, as the basis for designing machines of this kind;

h) conclusion; general deductions.

To avoid ambiguities, it will be appropriate to state here that in what follows, by the right and left working cavities, the right and left dead positions are meant those corresponding to their arrangement in the respective drawings.

By the final pressures of expansion—\((p_e)\)—and of compression—\((p_c)\)—are understood the pressures corresponding to the dead positions of the piston, in the absence of the periods of pre-release and admission. The steam pressures at the end of the periods of expansion and compression, otherwise, at the moments of pre-release and admission, are denoted: for the first—\((p_3)\), and for the second—\((p_1)\). The designations of quantities relating to the high-pressure cylinder are accompanied by one prime \((’)\); those of the low-pressure cylinder are marked by two primes \((’’)\).

CHAPTER ONE.

Graphical representation of the working process of a Tandem machine of the ordinary type.

Let us suppose that the following quantities are known to us:

\(p_c\)—the pressure, in \(kg\) per \(1\ cm^2\), of the steam admitted into the machine,

\(p_a\)—the pressure of the steam leaving the machine,

\(S_1\)—the volume of the high-pressure cylinder,

\(m_1\)—the volume of its clearance space,

\(S_2\)—the volume of the low-pressure cylinder,

\(m_2\)—the volume of its clearance space,

\(R\)—the volume of the receiver,

\(S_0\)—the degree of filling, in percent, of the volume of the high-pressure cylinder,

\(p_c\) and \(p_c''\)—the final compression pressures in both cylinders.

According to Schröter’s method, we assume all volumes to be referred to the area of the piston of the large cylinder; thus the quantities \(S_1\), \(S_0\), \(m_1\), \(m_2\), and \(R\) are expressed in the corresponding parts of \(S_2\)—the common stroke of the machine pistons.

On the upper horizontal line we lay off, to some scale, successively the quantities \(S_1\), \(m_1\), \(R\), \(m_2\), and \(S_2\) (drawing 1, table I), and through all the points bounding the segments laid off we draw vertical lines. At an arbitrary distance from the upper horizontal line, sufficient, however, for placing the volume and indicator diagrams, we draw the lower horizontal line, which is the base of the drawing and at the same time the line of absolute zero pressure. On this latter line we mark the points \(O_1\) and \(O_2\), obtained by the intersection of the verticals bounding the volume of the receiver.

In order to be able to take into account the volumes described by the pistons of the machine, we construct, under the segments \(S_1\) and \(S_2\), the corresponding volume diagrams. The construction of these diagrams is carried out in the most convenient, quite exact, and simple manner, giving volume diagrams independent of the length of the connecting rod, which does not prevent, when necessary, taking into account also the influence of the finite length of this latter. The method employed

The method is based on the fact that, on the verticals bounding the volume diagram, segments are laid off that are proportional to the parts of the piston stroke which are functions of the angles of rotation of the crank.

Under this condition, the volumes described by the piston, being directly proportional to its corresponding strokes, are expressed graphically by straight lines.

Since in Tandem machines both pistons move together and describe the full volumes at one and the same time, it is evident that, in order for the volume diagrams for the two cylinders to be mutually related in time, they must be constructed so that the dead positions of the pistons (in the volume diagrams) lie on the same horizontals. In Fig. 1 (Table 1), the volume diagram \((1—2—3)\) of the high-pressure cylinder corresponds to the volume diagram \((5—6—7)\) of the low-pressure cylinder; the pairs of points \((1—5)\), \((2—6)\), \((3—7)\) lie on common horizontals. With this mutual arrangement of the volume diagrams, the volumes simultaneously described by the pistons of both cylinders are easily found by drawing a horizontal through the point corresponding, on any one of the diagrams, to the moment of the working process under consideration.

Having drawn the volume diagrams, we shall proceed to the construction of the indicator diagrams of the work of the steam in each of the cylinders.

We shall begin with the high-pressure cylinder. From the point \(O_1\), on the vertical passing through it, we lay off, on the chosen scale, the admission pressure \(p_e\) into the machine and draw a horizontal line. From the dead position of the piston on this line we mark off \(ab = S_0\)—the degree of filling of the high-pressure cylinder. Dropping the point \(b\) (the moment of cut-off of the high-pressure cylinder) onto the volume diagram, we see that during the period of filling the volumes of steam entering the cylinder change as the horizontal segments between \((1—B)\) and the vertical through the point \(O_1\).

After cut-off, the steam enclosed in the cylinder must fill the volumes freed by the further motion of the piston, and its pressure must fall. Graphically this change of pressure, expressing Mariotte’s law \(pv = c\), will be represented by an equilateral hyperbola referred to its asymptotes as to coordinate axes. These axes in the drawing will be the axes \(p\) and \(v\). The line \(O_0O_4\), as was mentioned earlier, being the line of zero pressure, will at the same time serve as the axis of volumes for the indicator diagrams of both cylinders. To draw the Mariotte curve passing through the point \(b\), we must find the origin of coordinates. For this purpose...

the origin of coordinates corresponds to the condition: \(p=0,\ v=0\). Consequently, the \(p\)-axis for the period of steam under consideration (expansion in the high-pressure cylinder) will be the line corresponding to zero volume of the steam enclosed in the cylinder. Such a zero volume is obtained if we mentally advance the piston to the end of the clearance space. From this we conclude that the \(p\)-axis must be a vertical line passing through the point \(O\).

Constructing, by one of the methods known in geometry, an equilateral hyperbola passing through the point \(b\), using the point \(O_1\) as its center, we obtain a graphical representation of the change in steam pressure during the period of expansion in the small cylinder.

Let us specify the instant of advance release of steam from the high-pressure cylinder into the receiver. We shall mark the corresponding points \(C\) and \(c\) on the volume and indicator diagrams.

At the instant of advance release, the steam passes from the cylinder into the receiver, which in turn—when the machine is in steady operation—is filled with steam. Since the receiver communicates now with one, now with both cylinders, the pressure of the steam contained in it is a variable quantity. By the receiver pressure of the steam we shall everywhere in what follows mean the pressure of the steam in the receiver corresponding to the instant of advance release from the high-pressure cylinder.

To improve the conditions for the exit of steam from the small cylinder, a pressure drop is arranged when the cylinder communicates with the receiver. This is achieved by an appropriate choice of the receiver pressure \(p_r\).

In constructing the working process, we shall determine graphically the pressure of the mixture—of different quantities of steam at different pressures—using Mönch’s method.

Let us specify some drop in steam pressure at the instant of advance release, equal on the pressure scale to the segment \(cd^*\). Then the pressure of the mixture—the steam contained before this instant in the small cylinder, together with the steam filling the receiver—will be determined by the ordinate of the point \(d\), measured from the line \(O_0O_1\). By Mönch’s method, this pressure lies on the perpendicular restored to the boundary of the consecutively plotted mixing volumes. Such a perpendicular is the vertical passing through the point \(O_1\). Indeed, to the left of it there is already plotted the volume of steam corresponding to the point \(C\), while to the right there is the volume of the receiver.

* See fig. 2, table I, where the corresponding parts of the indicator diagrams of both cylinders are shown on an enlarged scale.

Thus, in order to find the receiver pressure required for a given pressure drop, we carry, along horizontals, point $d$ to $d_1$ on the vertical $O_1$ and point $c$ to $c_1$ on the vertical $O_2$. Connecting the points $c_1$ and $d_1$ by a straight line, we obtain the desired receiver pressure on the ordinate $C$ in the form of the segment $fk$.

After the common pressure has been instantaneously established in the small cylinder and the receiver, with the further motion of the piston toward the left dead position there occurs an expansion of the steam in the two volumes communicating with one another. Repeating the considerations set out above, we come to the conclusion that the origin of the coordinates corresponding to the Mariotte curve is the point $O_2$, which represents the zero volume for the period of the working process under consideration.

The period of steam expansion along the curve constructed from the center $O_2$ will continue until the moment of pre-admission of steam into the low-pressure cylinder occurs. Pre-admission, as is known, signifies the end of the corresponding compression period.

Having fixed, depending on the steam-distribution system of the large cylinder, the moment of pre-admission, we draw the corresponding vertical (in the drawing: $I - n_2$). Then, on the left dead position of the piston of the low-pressure cylinder, we lay off from the line $O_0O_4$ the final compression pressure $p_c''$ and construct the compression curve, using for this purpose the center $O_2$, which represents the zero volume of the period being analyzed. The curve obtained, at its intersection with the ordinate of pre-admission, gives the point $n''$, which determines the pressure and volume of the quantity of steam present in the large cylinder at the moment of its communication with the receiver and the small cylinder. In the general case this pressure is less than the steam pressure then existing in the volumes communicating with the large cylinder, since the final compression pressure $p_c''$ is chosen only so as to ensure smooth, shockless running of the machine.

Thus we see that the factor which has come into action entails a new pressure drop of the steam in the small cylinder and receiver, and a corresponding rise in pressure in the large cylinder. To determine the change in steam pressure that takes place, using again the method of Mönch, we lay off on the ordinate $(I - n_2)$ the pressure existing at the instant under consideration in the small cylinder and receiver, and on the ordinate of point $e$—the corresponding pressure $p_1'' - n_2n''$. Connecting the resulting points by a straight line, we determine the desired pressure—

of the mixture along the vertical \(O_2\), which in the present case is the line delimiting the volumes*). Having plotted the resulting pressure on the corresponding ordinates of the indicator diagrams of both cylinders, we may proceed further.

After the moment of premature admission of steam into the large cylinder, both pistons continue to move toward their left dead centers. The period that has begun is characterized by communication between the right working space of the small cylinder, the receiver, and the left working space of the large cylinder. Depending on the motion of the pistons, the steam pressure in all these volumes will change simultaneously and equally. It is obvious that—when the pistons move jointly—the volume described by the piston of the small cylinder will be less than the volume correspondingly described by the piston of the large cylinder.

As a result, the piston of the large cylinder, approaching its left dead center, diminishes the total volume of steam more considerably than this volume is increased by the corresponding stroke of the piston of the small cylinder. It is clear that in the period under consideration the steam will be compressed and its pressure will rise.

When the pistons reach their left dead centers and begin to move back, the following will occur: the steam expelled from the high-pressure cylinder into the receiver passes from the latter into the low-pressure cylinder. The volumes freed by the piston of the large cylinder are greater than those described correspondingly by the piston of the small cylinder.

The total volume of the spaces communicating with one another increases; the steam contained in the volume under consideration expands, and its pressure falls. The period we are considering is essentially the period of admission of the large cylinder. Thus, during the admission of the large cylinder, expansion of the steam occurs in all the interconnected spaces of the engine. Graphically, this expansion must again be expressed by an equilateral hyperbola. Since admission of the large cylinder corresponds to exhaust from the small cylinder, it is evident that one and the same change of pressure of the steam contained in the common volume must correspond, in the representation of the working process, to two ordinates, each separately in the indicator diagram of each of the cylinders. The change of pressure under consideration will be represented in the drawings by two equilateral hyperbolas expressing one and the same change of state of the steam,

*) See points \(g'\) and \(g''\) in drawing 2, table I.

plotted on the corresponding strokes of both pistons. In essence, these curves therefore represent one and the same Mariotte curve, drawn twice with a different scale of abscissae (volumes) for each case.

It is clear that the abscissae of both hyperbolas must be related to one another as the corresponding volumes of the cylinders, i.e. as \(S_1:S_2\); and the magnitude of the abscissae will become known when the origin of coordinates—the zero volume—has been determined, at the same time corresponding to both curves.

Since the axis of volumes of these two hyperbolas, which express, as already indicated, one and the same change of state of the steam, is the line of zero pressures \(O_0O_4\), common to the indicator diagrams, and since, on the other hand, both curves, in accordance with the course of the working process, have their convexity turned in the same direction, the origin of coordinates of these curves must also be common. We shall find this origin of coordinates on the continuation of the line \(O_1O_2\), which, as already said, is one of the asymptotes—the axis of volumes—for the indicator diagrams. To the origin of coordinates there corresponds zero volume for the period of the working process under consideration.

The volumes of steam for this period vary proportionally to the segments of the horizontals enclosed between the straight lines \((2—3)\) and \((6—7)\). Zero volume will correspond to a segment equal to zero, determined, obviously, by the point \(Q_1\) of intersection of the extended lines \((2—3)\) and \((6—7)\).

If the point \(Q_1\) lies outside the limits of the drawing, then for the zero volume one may use the point \(Q\) of intersection of the lines \((1—2)\) and \((5—6)\), symmetrical to the former, as was done in Fig. 1 (Table I).

The zero volume is characterized by the second asymptote, as a perpendicular to the line \(O_1O_2\), passing through the point \(Q\). Thus the required origin of coordinates \(O_0\) for the hyperbolas corresponding to the period of change of state of the steam under consideration has been found by us; the beginning of this entire period, as we indicated above, is the moment preceding admission into the large cylinder. The corresponding initial state of the steam, each in its own scale of abscissae, is marked in both indicator diagrams by the points \(g'\) and \(g''\). Using the points \(g'\) and \(g''\), we shall draw through them two equilateral hyperbolas from the center \(O_0\), where the parts of the hyperbolas lying above the points \(g'\) and \(g''\) will correspond to compression of the steam during the motion of both pistons toward the left dead position; during the reverse motion of the pistons these same curves as a whole will express the expansion of the steam. Thus we have outlined the graphical course of the exhaust of steam from

of the small cylinder and its admission into the large one. Determining the scales of the abscissas is superfluous, since the point \(O_0\) serves only as an auxiliary means for constructing the hyperbolas. During its expansion the steam occupies volumes that vary as the segments (on the scale of the drawing) of the horizontals between the straight lines \((2—3)\) and \((6—7)\). These segments, as is not difficult to see, are in turn proportional to the abscissas of both hyperbolas. Indeed, from Fig. 1 we have:

\[ \frac{\overline{2—6}}{\gamma\delta} = \frac{\overline{a—2}}{\beta\gamma} = \frac{\overline{a—6}}{\beta\delta} = \frac{Q_1\alpha}{Q_1\beta}. \]

In drawing the part of the working process that has been analyzed, one may confine oneself to the construction of a hyperbola passing through any one of the points \(g'\) or \(g''\), transferring the pressures determined by the hyperbola thus obtained, by means of volume diagrams, to the corresponding positions of the piston of the other cylinder.

As we see from all that has been set forth above, during the period of admission into the large cylinder the steam pressure gradually decreases; the receiver is connected with both cylinders and, consequently, a similar phenomenon will occur in it. For the proper course of the working process, it is necessary to prepare the receiver for communication with the left working space of the small cylinder at the moment of the corresponding pre-release.

Consequently, by this moment the pressure in the receiver must be brought to the receiver pressure \(p_r\) specified earlier. This can be done only by cutting off the large cylinder and compressing the remaining steam in the receiver and in the right space of the small cylinder communicating with it.

As soon as the pressure in the receiver rises to the specified value, it is necessary to cut off the receiver from the right space of the small cylinder, leaving it isolated from both cylinders until the moment of communication with the left working space of the high-pressure cylinder. This condition determines: the moment of cut-off in the left space of the large cylinder and the moment, which must occur, of disconnection of the small cylinder from the receiver—the moment of the beginning of compression in the right space of the high-pressure cylinder.

The final compression pressure \(p_c'\) in the small cylinder is known to us. The origin of coordinates for the corresponding Mariotte curve is, obviously, the point \(O_1\). Using the indicated point as center, we draw the compression hyperbola to its intersection at the point \(s\) with the horizontal line drawn from the line \(O_0O_4\) at a distance corresponding to the receiver pressure determined earlier. The point \(s\) determines the moment of disconnection of the small cylinder from the receiv—

the reservoir. Before the occurrence of this moment, as was indicated above, in the reservoir \(t\), communicating with the small cylinder, under the action of the piston of the latter, there took place a gradual rise of the steam pressure to the required value. It is clear that point \(s\) characterizes the end of the said process and, consequently, belongs to the hyperbola of steam compression in the reservoir and the small cylinder. The origin of the coordinates of this hyperbola is, obviously, at point \(O_2\). Drawing through point \(s\) a hyperbola from center \(O_2\) to its intersection with the previously drawn hyperbola of the general expansion of the steam in the small cylinder, the reservoir, and the large cylinder, we find point \(t'\), which determines the moment of separation of the small cylinder and the reservoir from the large cylinder, or the sought moment of cut-off of the latter.

Having specified, depending on the system of steam-distribution organs, the moment of pre-admission of the inlet (the moment of the end of the period of compression) into the small cylinder, we thereby determine point \(n'\) on the corresponding compression curve. By connecting point \(n'\) with point \(a\) by a straight line, we close the indicator diagram of the high-pressure cylinder.

For the further construction of the indicator diagram of the low-pressure cylinder, we shall transfer point \(t'\) onto the filling line at \(t''\), and through it draw a hyperbola corresponding to the period of expansion of the steam only in the large cylinder. The origin of the coordinates of this hyperbola is at point \(O_2\). Completing the indicator diagram of the large cylinder by plotting, according to the chosen moment of pre-release, the exhaust line, and then the release line drawn from the line \(O_0 O_4\) at a distance equal to \(pa\), and giving, at the point of intersection with the previously drawn hyperbola of steam compression in the large cylinder, the moment of the beginning of compression, we close the indicator diagram of the large cylinder as well.

Relying on the given numbers of revolutions and the indicated power of the machine, and determining from the drawn diagrams the calculated mean indicated pressure, we thereby obtain the possibility of calculating the principal dimensions of the machine and of proceeding, if necessary, to the calculation and design of its parts.

In Fig. 1 (Table I), for a clearer separation of the individual periods of the working process that has been analyzed, all the volumes successively occupied by the portion of steam from the moment of its admission into the small cylinder until its release from the large cylinder are shown, by means of volumetric diagrams, with various kinds of hatching.

After the character of the individual periods of the working process of the Tandem machine has thus been clarified and analyzed, it becomes clear and

and the necessary method of checking correctness, both of the design of the process and of its outline.

Indeed, in a multiple-expansion engine without losses there must exist, and be fulfilled, one simple condition: in the steady state of the engine, through each of its cylinders, in one stroke of the piston, there must pass the same quantity of steam as enters, in the same time, the high-pressure cylinder.

We know that, on an indicator diagram, the volume of steam which has entered the engine in one stroke of the piston is expressed by the segment of the horizontal steam-admission line between the continued compression curve of the steam and the moment of cut-off.

Generalizing this proposition, we may say: the volume of steam passing, in one stroke of the piston, through the cylinder of the engine for each given arbitrary pressure is expressed, in the indicator diagram, by the segment of the corresponding horizontal line between the expansion and compression hyperbolas.

Consequently, drawing at some distance (Fig. 1, Table I) a line \(p'q''\) parallel to the axis \(O_c O_4\), we obtain that—for the corresponding pressure \(p_x\)—the volume of steam which has entered the small cylinder will be expressed by the segment \(p'q' = v_1 - v_2\), where \(v_1\) and \(v_2\) are the volumes of the expanding and compressed steam. On this same straight line are found the points of its intersection with the corresponding hyperbolas of expansion and compression of the steam in the low-pressure cylinder, which also determine the magnitude of the segment \(p''q'' = v_3 - v_4\).

With the correct construction of the working process, on the basis of what has been said above, there must be:

\[ v_1 - v_2 = v_3 - v_4. \]

Multiplying this expression term by term by \(p_x\), and knowing the constants of the corresponding Mariotte curves, we obtain the following relation:

\[ c_1 - c_2 = c_3 - c_4. \]

This relation must be satisfied in all double-expansion steam engines. For a Tandem) engine there must exist, in addition to the one indicated, still another specific relation characterizing the working process analyzed. This relation is determined by the character of the exhaust line* of the high-pressure cylinder.

Turning to this latter line, we see that it consists of two parts. The first part, as was indicated above, represents—

*) Also for Woolf engines similar in working process.

constitutes the flow of steam through the receiver into the low-pressure cylinder, accompanied by expansion during the entire filling period of the large cylinder; the second part of the exhaust is the compression of the steam in the receiver and in the high-pressure cylinder.

Thus, the moment of cut-off of the low-pressure cylinder is marked on the exhaust line of the small cylinder by the special point \(t'\). At this latter point we have a division between two states of one definite quantity of steam, i.e. we have before us a case opposite to the mixing of vapors and therefore subject to Dalton’s law. Denoting the steam pressure at the point \(t'\) by the letter \(p_t\), and relying on the fact that immediately before the very moment of cut-off the steam occupied the volume \(\gamma\delta\), while at the moment of cut-off the volume \(\varepsilon\delta\) was separated from this entire volume, we may write:

\[ \gamma\delta=\gamma\varepsilon+\varepsilon\delta; \]

multiplying everything by \(p_t\) and, knowing the constants of the corresponding hyperbolas, we have the expression of Dalton’s law for the case under consideration in the following form:

\[ c_5=c_6+c_3. \]

This condition will be the second sought-after check. The two indicated conditions make it possible to verify the graphical method analytically. For this purpose it is necessary to multiply the volumes obtained by Schröter’s method, occupied by the steam in the various periods of the working process, by the corresponding pressures, and to compose from the products the equations of the steam balance of the Tandem machine just derived.

CHAPTER TWO.

Graphical Representation of the Working Process of a Machine with Intermediate Steam Extraction.

Although the literature on the question of intermediate steam extraction comprises a fairly extensive series of articles, a simple method, permitting one to construct the working process of the machine being designed on the basis of Schröter’s diagram, which is convenient by virtue of its clarity, has not yet been proposed*).

In view of this, in presenting to students the corresponding chapter of the course on steam engines, in order to elucidate all the details of the working process, the author makes use of a device, the description of which is set forth in the present chapter and which has been made the basis of the entire investigation carried out.

Let us suppose that there is a Tandem machine, from the receiver of which—in a given period of time—a definite quantity of steam is extracted. The phenomenon of extraction, evidently, consists in this: of the steam of a definite volume entering during one stroke of the piston into the high-pressure cylinder, only a certain part passes through the receiver into the low-pressure cylinder; the other part does not perform work in the low-pressure cylinder, because at each piston stroke it is correspondingly taken from the machine for the purposes of some production process, entirely analogously to the way in which exhaust steam is taken from a machine for heating purposes.

It is evident that the receiver of machines of this kind must be in constant communication with the pipeline extracting the steam.

The working process of a machine with steam extraction will correspond to the process of an ordinary Tandem machine only in those periods of the steam’s action in each of the cylinders when the latter are disconnected from the receiver, i.e.—for the high-pressure cylinder—from the moment compression begins to the moment admission is anticipated; for the low-pressure cylinder—from the moment cut-off occurs to the moment admission is anticipated.

*) The method described by K. Vodzinsky (Bulletin of the Moscow Polytechnic Society, 1909, No. 1) is distinguished by considerable complexity of construction.

Consequently, the indicated parts of the working process must be constructed in the usual way.

As for the graphical representation of the changes in steam pressure that occur together with the changes in the volumes of the spaces consequently occupied by it in the machine (in which chiefly consists the clarity of constructing the Schröter diagram), in those periods of the working process when the cylinders of the machine communicate with the receiver and with one another, here, obviously, one must take into account the influence of the volume of steam leaving the receiver upon the change in the pressures acting on the cylinder pistons.

We may identify the volumetric influence of the extraction of steam on the receiver and on the machine spaces connected with it with the presence in the machine of a certain fictitious cylinder, communicating alternately and continuously by its spaces with the receiver and with the extraction pipeline; this cylinder, being as it were a double-acting pump, pumping steam alternately with each of its spaces out of the receiver, must have a volume corresponding exactly to that part of the entire volume of steam flowing into the high-pressure cylinder during the admission period which expresses the relative magnitude of the extraction existing at the given time.

On the basis of all that has been said, the fictitious cylinder must possess the following properties:

a) its volume changes together with the change in the extraction of steam;

b) its volume depends on the volume of steam entering the high-pressure cylinder during the admission period;

c) the indicated work of the fictitious cylinder is equal to zero;

d) in it, as is also assumed for the whole machine under investigation, there are no steam losses whatsoever.

Having introduced the concept of the fictitious cylinder-pump (hereafter we shall also call it the extraction cylinder), we obtain the general diagram of the machine under study, as it is shown in Fig. 1, Plate II.

The extraction cylinder \(F\) is placed between the high- and low-pressure cylinders \(H\) and \(N\) in such a way that the pistons of all three cylinders are mounted on a common rod. Both spaces of the high-pressure cylinder are connected in the usual way with the steam-supply pipe \(A\) and the receiver \(R\); both spaces of the low-pressure cylinder communicate with the receiver (receiving from it steam for operation) and with the exhaust pipe \(B\) of the machine.

The extraction cylinder \(F\), having a separate exhaust (delivery) pipe \(C\), passing directly into the extraction pipeline, is connected by its spaces to the receiver as well.

Communication and separation among the various cavities of the machine, including the imaginary extraction cylinder, is effected by the corresponding steam-distribution organs.

In Fig. 1 (Plate II) is shown the moment when live steam enters the high-pressure cylinder into the cavity on the crank side; further, the arrows show the path along which the steam is directed in the remaining cavities of the machine.

Having the scheme of the machine, we may proceed to the construction of Schröter’s diagram, assuming the following data to be known:

\(p_c\)—the pressure of the steam admitted into the machine, in \(kg\) per \(1\ cm^2\),
\(p_r\)—the receiver pressure,
\(p_a\)—the pressure of the steam leaving the machine,
\(S_1\)—the volume of the high-pressure cylinder,
\(m_1\)—the volume of its clearance space,
\(S_2\)—and \(m_2\)—the corresponding volumes for the low-pressure cylinder,
\(R\)—the volume of the receiver,
\(S_0\)—the normal degree of admission, in % of the volume of the high-pressure cylinder,
\(p_c'\) and \(p_c''\)—the final compression pressures in the corresponding cylinders.

As for an ordinary-type Tandem machine, we assume, according to Schröter’s method, all volumes to be referred to the piston area of the large cylinder; thus the quantities \(S_0\), \(S_1\), \(m_1\), \(m_2\), and \(R\) are expressed in the corresponding parts of \(S_2\)—the total stroke of all the pistons of the machine.

On the basis of the data given, we may pass to carrying out the graphical representation of the working process of a Tandem machine connected with a steam-extraction pump. Let the horizontal straight line \(O_4O_6\) (Plate II, Fig. 2) represent the line of absolute zero pressure. On it let us set off successively, on some scale, the quantities \(S_1\), \(m_1\), \(R\), \(m_2\), and \(S_2\); then draw a series of corresponding verticals and mark the points \(O_1\) and \(O_2\), which bound the volume of the receiver.

In order to be able to take account of the volumes described by the working pistons of the machine, below the line \(O_4O_6\) under the segments \(S_1\) and \(S_2\) we construct volume diagrams in the form of straight lines, coordinated with one another in time. In Fig. 2 the volume diagram of the high-pressure cylinder \((1—2—3)\) corresponds to the volume diagram \((4—5—6)\) of the low-pressure cylinder.

Let us proceed to the construction of the indicator diagrams for each of the working cylinders of the machine.

Let us begin with the high-pressure cylinder. From the point \(O_1\) on the vertical passing through it, lay off upward, on the chosen scale, the pressure \(p_c\); draw a horizontal line and on it, measuring from the vertical corresponding to the right dead position of the piston, lay off the segment \(ab = S_0\)—the degree of admission. The hyperbolas of expansion and compression of steam in the machine under consideration, for the high-pressure cylinder, must be constructed as in the ordinary Tandem machine, using the point \(O_1\) as the center; having constructed the hyperbola passing through the point \(b\), we find the point \(c'\)—the final pressure of expansion.

Then lay off from the line \(O_4O_6\), on the vertical corresponding to the right dead position of the piston, the final compression pressure \(p_c'\), and through the point so obtained draw the hyperbola of steam compression in the small cylinder. We find the points \(m'\) and \(n'\) of intersection of this hyperbola with the horizontals passing through the points \(c'\) and \(b\). It is clear that the segment \(bm'\) represents the volume corresponding—at the pressure \(p_c\)—to the quantity of fresh steam admitted into the machine during one stroke of the piston. This quantity—under the absence of any losses in the machine assumed in our investigation—must be the measure of its total (useful) steam consumption. The segment \(c'm'\) will obviously represent the volume occupied by the same quantity of steam at the final expansion pressure \(p_e'\).

Let us turn temporarily to the low-pressure cylinder.

Draw, at a distance equal to the pressure \(p_a\) from the horizontal \(O_4O_6\), the line of steam exhaust from the machine. Then, laying off on the vertical corresponding to the left dead position of the piston the final compression pressure \(p_c''\), we are able, using the point \(O_2\), to construct the hyperbola of steam compression in the low-pressure cylinder. Having constructed this curve, we find the point \(z\) of its intersection with the exhaust line; the point \(z\) determines the beginning of the compression period.

Next find the intersection of this same hyperbola at the point \(c''\) with the extended horizontal \(c'm'\). If we now lay off \(c''m'' = c'm'\) and draw from the center \(O_2\) the expansion hyperbola \(m''r\), we shall have before us an example of an ordinary Tandem machine, in whose cylinders equal quantities of steam are working. We already know that in the machine under consideration precisely the opposite case occurs, since a certain portion of the steam is taken from the receiver. Suppose that the extraction is equal to \(40\%\) of all the steam entering the machine at the given admission of the high-pressure cylinder. Then, laying off to the left from the point \(m''\) a segment equal to \(0.4\,c'm'\), we obtain the point \(n''\). The segment \(c''n''\) represents the volume of steam entering during each

the stroke of the piston in the large cylinder at the pressure corresponding to the final pressure of expansion of the steam in the small cylinder.

Through the point \(n''\), using the point \(O_2\) as center, we draw the hyperbola of the expansion of the steam in the low-pressure cylinder which exists when extraction is present; the point \(q\) of this hyperbola determines the corresponding final pressure of expansion—\(p_2''\).

We have now constructed all those parts of the working process which are determined, as was mentioned above, by the work of cylinders connected neither with the receiver nor with one another. In order to be able further to construct the representation of the working process, we must determine and take into account the volume \(Sf\) of the extraction cylinder, referred to the area of the piston of the low-pressure cylinder.

This volume has already been determined by us.

Indeed, if it is required that, when the piston of the large cylinder leaves the left dead position, only the part \(c''n''\) of the whole volume \(c''m''\), enclosed in the same dead position in the high-pressure cylinder, should enter this latter cylinder, then, obviously, in this same dead position the extraction cylinder must contain within itself the volume of steam expressed by the segment \(n''m''\).

Thus, for constructing the volume diagram of the extraction cylinder, we already have one datum. Laying off from the vertical \(O_2\) to the right*) on the line \((2--7)\) a segment equal to \(n''m''\), we obtain the final point \(c_3\) of the volume line.

We shall then transfer the point \(b\) to the volume diagram of the high-pressure cylinder at the point \(b_1\), and through this latter draw a horizontal to its intersection at the point \(b_2\) with the vertical \(O_2\). If, at the moment of cutoff of the high-pressure cylinder, we were to create the possibility of communication of this cylinder with the extraction cylinder, then, according to everything said above, the volume of the cavity of this latter cylinder, which can be made available for the extraction of steam at pressure \(p_e\), must be equal to \(40\%\) of the segment \(bn'\). Laying off from the vertical \(O_2\), on the continued horizontal \(b_1b_2\), a segment equal to \(0.4\,bn'\), we obtain the point \(b_3\)—the second point of the volume diagram of the extraction cylinder. Joining the points \(b_3\) and \(c_3\) by a straight line, we obtain the required volume diagram for one piston stroke.

The point of intersection of the straight lines \((1--4)\) and \(b_3c_3\) characterizes, as it were, the mean clearance space—\(mf\)—of the extraction cylinder. It is not difficult to see

*) To the right in order that the volumes of the extraction cylinder should always be summed with the volume of the receiver; thus the point \(c_3\) corresponds to the left dead position of the piston of the extraction cylinder.

on the other hand, that the point \(\beta\), the intersection of the straight line \(b_3c_3\) with the vertical \(O_2\), lies on the horizontal passing through the point \(\alpha\), the intersection of the straight line \((1—2)\) with the vertical \(O_1\).

Indeed, the zero volume occupied by the steam of the high-pressure cylinder in the given position (the piston of the high-pressure cylinder having, as it were, advanced to the right to the end of the clearance space) must correspond to the same zero volume of the extraction cylinder.

It is further clear that, together with the change in the volume of the extracted steam, the magnitude of the segment “\(n''m''\)” changes as well, chiefly characterizing the volume of the extraction cylinder and its clearance space; thus we see that, if the easily found point \(\beta\) is available, the straight line \(\beta c_3\) can be drawn without determining the point \(b_3\).

Drawing the straight line \(\gamma c_3\) symmetrically to \(\beta c_3\) with respect to the line \((2—\gamma)\), we obtain the complete volume diagram of the extraction cylinder for one revolution of the main shaft of the machine.

In plotting the working process we must take into account the whole—gradually increasing during one revolution of the machine shaft—volume removed from the portions of steam admitted, for the position of the machine under consideration, into the right-hand working cavity of the high-pressure cylinder.

Since, with the arrival of the piston of the extraction cylinder at the left dead-center position—\(c_3\)—and the subsequent change in the direction of motion of the piston, the extraction of steam continues into the left-hand cavity, it is evident that the new volumes must be added to the volumes of the extraction cylinder previously described by the piston. Thus, for one revolution of the machine shaft, the law of variation of the extracted volumes will be expressed by the straight line \(\beta c_3\) and its continuation—the resulting straight line \(c_3\gamma_1\).

After all that has been said, we can return to the construction of the indicator diagram of the high-pressure cylinder.

For a better changeover of the pressures acting on the piston, it is known that advance exhaust is necessary. Let the advance exhaust occur at the piston position corresponding on the diagram to the point \(d\); at this moment the high-pressure cylinder communicates with the receiver, and through it also with the extraction cylinder.

In these last two volumes there prevails a certain common steam pressure, constantly varying according to the change in the volumes of the cavities with which the receiver communicates alternately; the pressure corresponding to the instant of advance exhaust from the small cylinder is the receiver pressure \(p_r\) known to us from the data.

Assuming, for the sake of generalizing the question, that the receiver pressure \(p_r\) is not equal to the pressure of the expanding steam in the piston position corresponding to point \(d\), we shall obtain that, at the moment when the release is advanced, a drop will occur in the high-pressure cylinder, and, in the receiver and in the extraction cylinder, an increase in the steam pressure; moreover, the pressures in all communicating spaces are equalized and—according to our assumption—the mixture pressure is established instantaneously. Using Mönch’s method, we can determine by construction the pressure of the mixture, if we know separately the pressures and volumes of the quantities of steam being mixed. The mixture pressure, as was already indicated earlier, is found on the perpendicular erected at the boundary of the successively added volumes being mixed. Such a perpendicular, for the moment of the working process under consideration, is the vertical \(O_1\), since to the left of it is the volume corresponding to point \(d\) (\(d_1d_2\) is equal to the volume of the right-hand working space of the high-pressure cylinder, including the clearance space); while on the right is the volume \(d_2d_3\), equal to the volume of the receiver added to the corresponding part of the volume of the right-hand space of the extraction cylinder. Both this latter space and the receiver are filled—at the established state of the engine and of the cylinder-pump—with steam that entered them before the moment under consideration from the left-hand space of the high-pressure cylinder.

Transferring, by Mönch’s method, the steam pressure at point \(d\) to the ordinate corresponding to point \(d_3\), and, conversely, the pressure \(p_r\)—of the steam in the volume \(d_2d_3\)—to the ordinate passing through point \(d\), we find on the vertical \(O_1\) the pressure of the mixture, whence, further, we find the pressure drop \(de\).

With the subsequent motion of the engine pistons toward the left dead position, expansion of the steam must occur in all three volumes communicating with one another.

In order to construct the corresponding hyperbola of expansion, we must—according to what preceded—find the origin of coordinates of this hyperbola on the line \(O_4O_6\). We already know that this origin of coordinates must be characterized by the condition \(v=0\); on the other hand, we know that, for the change of state of the steam under consideration, the volumes are measured as segments of horizontals between the straight lines \((1—2)\) and \(3c_3\).

To find the zero volume, let us mentally advance the pistons of the high-pressure cylinder and of the extraction cylinder (assuming such independent motion of them with respect to one another to be possible) in the direction in which the volumes, expressed by the corresponding segments of horizontals, while gradually decreasing, would be reduced to zero. Carrying out

said graphically, we must continue the lines \((1—2)\) and \(\beta c_3\) to their intersection at the point \(A\). The vertical through the point \(A\), being the desired asymptote—the \(p\)-axis—gives us the origin of coordinates \(O_3\). Using the point \(O_3\) as the center, we construct the hyperbola passing through the point \(e^*\)).

The expansion of the steam under consideration will proceed along the constructed hyperbola until such time as a new factor comes into operation, namely the advance admission into the low-pressure cylinder. Let us see what will occur in the engine.

Let the advance admission begin when the piston of the low-pressure cylinder is in the position determined by the ordinate \(f'\). The point of intersection \(f''\) of this ordinate with the previously constructed compression hyperbola \(c''z\) gives us the pressure existing at the moment under consideration in the large cylinder. In the general case this pressure must differ from the pressure existing at the same moment in the engine spaces connected to the low-pressure cylinder. As a result of the communication of the spaces, a new mixing of the steam will occur. In order to find the pressure of the mixture, we must again determine the volumes and pressures of the quantities of steam being mixed.

Using the volume diagrams \((4—5)\) and \((1—2)\), we find, by drawing verticals and horizontals \((f''f_2—f_2, f_1—f_1 f')\), on the indicator diagram of the high-pressure cylinder the point \(f\), and on the volume diagram of the cylinder the point \(f_3\).

Thus we obtain the following conditions of mixing: the volume \(f_1 f_3\) with pressure \(f_0 f\), and the volume \(f_2 \delta\) with pressure \(f'f''\). Laying off on the extended horizontal \(f_1 f_3\) the point \(f_4\) so that \(f_3 f_4 = f_2 \delta\), we can, by the same method of Mönck, find on the ordinate passing through the point \(f_3\) the pressure of the mixture**), and from this find the points \(g'\) and \(g''\) on the indicator diagrams of both cylinders (see Table II, drawing 3-a).

In order to learn what will occur in the engine after the moment considered, it is necessary to investigate how, with the further motion of all the pistons toward the left dead position, the volume of the four engine spaces communicated with one another will change (the high-pressure cylinder, receiver, extraction cylinder, and low-pressure cylinder). For this purpose we shall construct the summary volume diagram of the last two cylinders. This diagram is constructed very easily; it is necessary

*) See Table II, drawing 3-a, where the corresponding parts of the working process for both cylinders are shown in enlarged form.

**) In drawing 2 of Table II the points \(d_3\), \(f_3\), and \(c_3\) are arranged in consecutive order near one another.

add the volumes described simultaneously by the pistons of the low-pressure cylinder and the extraction cylinder, i.e. to the segments of the horizontals between the lines \((4—5)\) and the vertical \(O_2\) add the corresponding ones between the line \(\beta c_3\) (on the same horizontals) and the same vertical \(O_2\). The resulting volume diagram, being a straight line, will be obtained most easily if we connect the already existing point \(f_4\) with the point \(\beta_1\), corresponding on the line \((4—5)\) to the zero segment of the volume diagram of the extraction cylinder.

Thus we have that, in the period of the working process under consideration, the volumes occupied by the steam during the motion of all the pistons of the machine vary as the segments of the horizontals between the straight lines \((1—2)\) and \(\beta_1 f_4\). We see that—under the given conditions—these volumes tend to decrease. In order to find the origin of coordinates for constructing the corresponding compression hyperbola, similarly to what was said earlier for the period \(ef\), we continue the lines \((1—2)\) and \(\beta_1 f_4\) to their intersection at the point \(B\), whose projection on the line \(O_4 O_6\) of zero pressures gives the origin of coordinates at the point \(O_4\). Using this point, we draw up to the left dead position the compression hyperbola \(g'h'\) and—since we are now constructing the parts of the working process belonging to the indicator diagrams of both cylinders—respectively \(g''h''\).

Subsequently the pistons of the machine begin to move from the left dead position to the right; as before, all four cavities are in communication, but the general law of variation of the volumes becomes different.

Indeed, the volumes described by the pistons of the working cylinders are characterized by the straight lines \((2—3)\) and \((5—6)\), while the volumes of the extraction cylinder—as indicated above—must be taken into account by the resulting straight line \(c_3\gamma_1\). In order to determine how all the volumes as a whole will vary, we must construct a new line of the summed volumes of the low-pressure cylinder and the extraction cylinder. The summed volume diagram, similarly to what was said earlier, will be found if, on the extended horizontal \(\gamma\gamma_1\), we set off from the point \(\gamma_2\) to the right the point \(\gamma_3\), taking \(\gamma_2\gamma_3=\gamma\gamma_1=2(S_f+m_f)\). The straight line \(c_4\gamma_3\) represents the required diagram.

Considering the change of the segments of the horizontals between the straight lines \((2—3)\) and \(c_4\gamma_3\), we see that the volumes occupied by the steam when the pistons of the machine move to the right increase—expansion takes place in all the mutually communicating cavities, characterizing the period of admission in the low-pressure cylinder. In order

constructing the corresponding hyperbolas of the indicator diagrams of both cylinders, we shall find, as set forth in the preceding chapter, their common origin of coordinates. For this purpose we find the point \(C\) of intersection of the straight lines \((2—3)\) and \(c_4^7 b\), and from it the required \(O_5\).

We carry the hyperbola of steam expansion during the period of admission in the low-pressure cylinder to the point \(k''\)—the intersection with the hyperbola \(n''q\) drawn earlier. The point \(k''\) determines the moment of cut-off. By plotting on the hyperbola \(k''q\) the point \(u\), corresponding to the chosen moment of pre-release, we complete the construction of the graphical representation of the working process of the low-pressure cylinder.

Let us now turn to the indicator diagram of the high-pressure cylinder. We transfer the point \(k''\) to the corresponding hyperbola at the point \(k'\).

It is evident that the low-pressure cylinder, after cut-off at \(k''\), has been disconnected from all the remaining volumes. The subsequent change in the volumes of steam contained in the right-hand cavity of the high-pressure cylinder, the receiver, and the left-hand cavity of the extraction cylinder is characterized by the change in the segments of horizontals between the straight lines \((2—3)\) and \(c_3 l_1\). As before, having found the point \(D\) of intersection of these straight lines, we determine \(O_6\)—the origin of coordinates for constructing the corresponding hyperbola of compression in all three volumes.

The intersection of this hyperbola at the point \(l\) with the previously constructed hyperbola \(m'n'\) determines the moment of the beginning of compression in the high-pressure cylinder. Taking the necessary pre-admission as given, we determine, by the intersection of the corresponding ordinate with the hyperbola \(m'n'\), the point \(t\), which defines the steam pressure existing at that moment. Connecting, as is usually done, the point \(a\) by a straight line with the point \(t\), we close the indicator diagram of the high-pressure cylinder.

It now remains for us to verify the correctness of the quantities taken as the basis of the construction—the data adopted by us—the principal one being \(p_r\), the receiver pressure.

For this it is necessary to investigate what occurs in the receiver after the beginning of the compression period in the high-pressure cylinder.

We already know that at the point \((k',k'')\) the receiver became disconnected from the low-pressure cylinder; at the point \(l\) the receiver also became disconnected from the high-pressure cylinder, remaining in communication only with the extraction cylinder. Consequently, with the further motion of the pistons of the machine toward the right dead position, steam is extracted exclusively from the receiver, expressed by the expansion of the steam contained in both mutually communicating cavities.

If the data we have adopted are correct, this expansion must be carried out in such a way that, at the moment when release from the left-hand side of the high-pressure cylinder begins, the receiver and the extraction cylinder contain the pressure \(p_r\) we have chosen; in other words, so that the pressure drop in the left working space, corresponding to the moment under consideration, is again equal to the value \(de\) (of course, on the condition that equal quantities of steam are working in both spaces of the high-pressure cylinder, and that the separate principal moments of steam distribution are also equal, which is quite attainable with the valve gears ordinarily used for machines of this kind).

By successively finding the points \(l_1\), \(l_2\), and \(l_3\), we obtain the possibility of drawing, through the last of them, the hyperbola \(l_3 l_4'\) of the expansion under consideration, using the point \(O_1\) as center. Transferring this curve by means of the volume diagrams to the indicator diagram of the high-pressure cylinder, we obtain the hyperbola \(l l_4\); at the point \(\lambda'\)—its intersection with the ordinate of the preliminary release from the left-hand side—we should, if all the data have been correctly chosen, obtain the pressure \(p_r\) that we have adopted.

The same verification can be made with the aid of the volume diagrams also on the hyperbola \(l_3 l_4\), by determining on it the point \(\lambda'\).

With the completion of the indicated verification we obtain a finished picture of the separate moments and periods of the working process characterizing a Tandem machine with intermediate steam extraction.

For completeness of the investigation, on the volume diagrams the separate periods of the working process are shown by different hatching; from these diagrams it is easy to see the volumes occupied by the steam in the various spaces of the machine at each given moment. For the low-pressure cylinder the periods of expansion and compression of the steam are distinguished.

Here it will be appropriate to point out further details that appear in such an investigation of the working process being studied.

Whereas—when the extraction is changed—the point \(A\), the intersection of the lines \((1—2)\) and \(\beta c_3\), always remains between the verticals \(O_1\) and \(O_2\), and consequently the period \(ef\) is always an expansion of the steam, the point \(B\), the intersection of the lines \((1—2)\) and \(\beta_1 f_4\), does not possess this property.

Indeed, together with a change in the extraction, the inclination of the lines \(\beta c_3\) and \(\beta_1 c_4\) changes. Consequently, depending on the ratio of the volumes of the working cylinders—which determines the mutual inclination of the lines \((1—2)\) and \((4—5)\)—and on the volume of the extraction cylinder, there may be (as shown in Fig. 3, Pl. II) three cases:

a) the lines \((I—2)\) and \(\beta_1 c_4\), when prolonged downward, intersect—the case of compression \((g'h' - g''h'')\) considered above;

b) these lines are parallel—when the pistons move, the volumes occupied by the steam do not change—the pressure remains constant; the lines \(g'h'\), \(g''h''\) are parallel to \(O_4O_6\);

c) the lines \((1—2)\) and \(\beta_1 c_4\), when prolonged upward, intersect—the period \(g'h'\) \((g''h'')\) is an expansion.

In Fig. 3—for all the cases a, b, c—the same designations are retained for the separate periods of the diagrams of both cylinders.

Since from the point \(h''\), during the return stroke of the pistons, there always proceeds the expansion curve \(h''k''\), we conclude that, whatever the magnitude of the steam extraction may be, on the indicator diagram of the low-pressure cylinder near the dead position there is always obtained a loop.

We indicated above that the indicated work of the extraction cylinder must be equal to zero.

For the theoretically admissible case of unchanging steam pressure in the receiver this is self-evident. Indeed, the indicator diagram in such a case (Table II, Fig. 4-a) will be expressed by the straight line \(AB\), along which the pencil of the indicator will pass twice during one revolution of the engine shaft; the positive work of the extraction cylinder during the forward stroke of the piston is equal to the same negative work during the return stroke, and the sum of the works for one revolution of the shaft is equal to zero.

In reality, as we know, the steam pressure in the receiver varies together with the change in the volumes of the working cavities of the cylinders communicating at different moments with the receiver. The indicator diagram of the extraction cylinder in this case will take the form shown on an enlarged pressure scale in Fig. 4-b, where all the designations correspond to those of Fig. 2 (Table II).

It is easy to see that—on the one hand—the indicator diagram is expressed by the area of the hatched figure, and—on the other hand—that the algebraic sum of the works for both strokes of the piston is equal to zero.

Being able to depict graphically the working process of a steam engine with steam extraction, we must have at our disposal a means enabling us easily and quickly to verify the correctness both of the design and of the drawing of this process.

Such a means, as also for the usual type of Tandem engine, is the preparation of the steam balance.

If the constants of the corresponding hyperbolas are denoted, as shown in Fig. 2 (Table II), then, obviously, for the machine under consideration the following two conditions must hold simultaneously:

\[ c_5 = c_6 + c_3, \]

\[ \frac{(c_1 - c_2) - (c_3 - c_4)}{c_1 - c_2} = k, \]

where \(k\) is the relative steam extraction existing at the given time, i.e. expressed in fractions of all the steam flowing into the machine; \(c_5\) and \(c_6\) are the constants of the hyperbolas obtained when accounting for the corresponding volume of the extraction cylinder.

CHAPTER THREE

Approximate calculation of a Tandem machine without condensation, with the use of steam for purposes of technical heating.

In order, in the proposed investigation, to illuminate as far as possible all the conditions that must be taken into account in designing machines with steam extraction, it is most convenient to turn to a definite numerical example.

So we shall do.

Let us suppose that we are required to design a machine of 1000 indicated horsepower, intended to drive a number of machine tools serving the given production, and that the machine—by the very conditions of the production—must satisfy the requirements of a wide range of variation both in the load of the machine and in the use of the steam passing through it.

The number of revolutions of the main shaft of the machine per minute is 120. The admission pressure of the steam is equal to 11 Atm. abs.

Let us take, as usual, the exhaust pressure of the steam as equal to 1.1 Atm. abs., i.e. we shall build the machine without condensation, seeking as far as possible to utilize the heat of the steam passing through the machine and assuming that the exhaust steam of the machine is to be used—as is done in rationally conducted plants—for heating purposes.

To determine the ratio of the cylinder volumes of the machine being designed, we construct (Table III, drawing 1) the theoretical indicator diagram of a single-cylinder machine of the power given to us.

We shall take the clearance space of the machine as equal to 8% of the cylinder volume; the base of the diagram equal to 100 mm, and the pressure scale as 1 Atm = 15 mm.

Let us set the final expansion pressure \(p_e\) to exceed the exhaust pressure by approximately 0.4 Atm., which corresponds on the scale to 24.5 mm, and through the point \(E\) thus obtained draw the steam expansion curve \(DE\).

In distributing the work produced by the machine between the cylinders, we shall take as a basis, as is usually done for Tandem machines, an approximately equal drop in the steam temperature in each of the cylinders.

Using the tables of saturated-steam temperatures, we find for the admission temperature \(t_e^\circ = 183.05\) and for the exhaust \(t_a^\circ = 101.76\), whence we obtain the corresponding mean temperature \(t_m^\circ = 142.40\), on the basis of which we take the receiver pressure \(p_r = 4\ \mathrm{Atm.}\) \((t_r^\circ = 142.82)\).

Since by receiver pressure we have agreed to understand the pressure corresponding to the instant just before the release of steam from the small cylinder, it is evident that, in the case under consideration, the point \(B\) on the steam-expansion curve determines this indicated instant of steam distribution (under the initial assumption that there is no fall in steam pressure at the instant of communication with the receiver).

Taking the period of pre-release as equal to \(9\%\) of the corresponding piston stroke \(S_1\), and the clearance space as \(8\%\), we have that the segment \(CB\) is expressed in fractions of the stroke of the piston of the small cylinder as follows:

\[ CB = (100 - 9 + 8)\cdot S_1 = 0.99\,S_1. \]

From the drawing we have \(CB = 44\ \mathrm{mm.}\), whence

\[ S_1 = \frac{100\,CB}{99} = 44.44\ldots \]

Since we have taken \(S_2\) equal to \(100\ \mathrm{mm.}\), the sought ratio of the cylinder volumes is

\[ \frac{S_1}{S_2} = \frac{44.44}{100} \approx \frac{1}{2.25}. \]

Such a ratio of volumes is found in engines with steam extraction*), and therefore we shall stop at it.

Continuing the calculation begun, we then determine the length of the line \(FG\), equal to the piston stroke \(S_1\) plus the clearance space \(m_1\), from the condition:

\[ FG = (100 + 8)\cdot S_1 = 48.04\ \mathrm{mm.} \]

Having \(m_1 = 0.08\,S_1 = 3.6\ \mathrm{mm.}\), we draw the line \(LH\), bounding the clearance space of the indicator diagram of the small cylinder.

The segment \(LD\) of the horizontal straight line drawn from the axis of abscissas at a distance of \(165\ \mathrm{mm.} = 11\ \mathrm{Atm.}\), gives us the necessary degree of filling of the small cylinder,

\[ \varepsilon = \frac{S_0}{S_1}. \]

In order to eliminate the presence of steam extraction only from the receiver in the interval of time between the instant of the beginning of compression in one cavity

*) L. Schneider, Die Abdampfverwertung. S. 17.

of the high-pressure cylinder and the moments of pre-release of the exhaust from the other side of the same cylinder, and thereby—on the one hand—ensure a constant inflow of fresh steam into the receiver, and—on the other hand—simplify the course of the working process by removing from it the intrinsic period of expansion of steam in the receiver and the fictitious cylinder of extraction (see the preceding chapter), we make this period of time equal to zero, i.e., we take the periods of compression and pre-release in both chambers of the small cylinder to be equal to one another.

Setting off on the line \(CB\) the point \(A\), corresponding to the beginning of compression in the high-pressure cylinder, under the condition that \(HA = 0.09 \cdot S_1\), we obtain the possibility of drawing the curve of steam compression in the theoretical indicator diagram of a single-cylinder machine.

Finding the point \(N\) of intersection of this hyperbola with the line \(LH\), we find the final compression pressure \(p_c\) in the small cylinder of the machine being designed to be equal to \(8.44\ \mathrm{Atm}\), which, amounting to about \(0.77p\), the admission pressure, is permissible.

Considering the diagram constructed, we see that during one stroke of the piston there enters the machine an amount of fresh steam occupying—at a receiver pressure equal to \(4\ \mathrm{Atm}\)—a volume proportional to the segment \(AB\).

We shall take the period of steam compression in the low-pressure cylinder as equal to \(12\%\), which corresponds to a final compression pressure in the large cylinder \(p_c'' = 2.75\ \mathrm{Atm}\), i.e. about \(0.7p_r\).

Extending this last compression hyperbola to its intersection at the point \(A_1\) with the line \(CB\) and laying off on this latter \(A_1B_1 = AB\), by drawing through the point \(B_1\) the corresponding hyperbola of steam expansion in the low-pressure cylinder, we find the final pressure of steam expansion in the large cylinder \(p_s'' = 1.55\ \mathrm{Atm}\).

Thus we now have all the necessary data for drawing, by Schröter’s method, the representation of the working process of the machine being designed.

Since the periods of pre-release of admission and exhaust have as their purpose to prepare in each of the cylinders the corresponding steam pressure that is to act on the piston at its dead position, etc.—and, on the other hand, we have assumed for the machine under investigation that the steam pressures in the communicating chambers are established instantaneously—then, in drawing the working process, we shall take into account only the moment of communication of the small cylinder with the receiver as the moment of the beginning of extraction; the end of the corresponding

of extraction, on the basis of what has been set forth above, is the beginning of compression, because simultaneously with it begins the extraction from the second cavity of the small cylinder.

Having constructed, by the method indicated in the first chapter, the Schröter diagram for the machine being designed (Table III, drawing 2) and planimetered the resulting indicator diagrams of both cylinders, we find the calculated mean indicator pressure (referred to the piston area of the large cylinder) equal to:

\[ p_i = 3.225\ \text{klg.} \]

To obtain possibly more accurate results in constructing the working process, along with the graphical method of determining the values of the steam pressure at various moments of time, these same values were also determined analytically from the corresponding expressions \(p_x v_x = c_x\). The position of the point \(O_3\), the abscissa of the point \(A\) of intersection of the extended volume diagrams of both cylinders, needed for constructing the curve of steam filling in the large cylinder, was also checked analytically.

We shall dwell on this last check somewhat more fully. Taking point \((5)\) as the origin of the rectangular coordinate system, we easily write the equations of the lines of the volume diagrams in the following form:

for the line \((1—2)\)

\[ a x + S_1 y - a(S_1 + m_1 + R + m_2)=0, \]

for the line \((4—5)\)

\[ a x + S_2 y = 0. \]

Solving these equations simultaneously with respect to \(x\), we obtain:

\[ O_3 O_4 = x = \frac{S_2(S_1 + m_1 + R + m_2)}{S_2 - S_1}. \]

Subtracting from the value obtained the segment \(O_5 O_4 = (S_1 + m_1 + R + m_2)\), we determine the length of the segment \(O_3 O_5 = x'\) in the following form:

\[ x' = x - (S_1 + m_1 + R + m_2), \]

\[ x' = \frac{S_1(S_1 + m_1 + R + m_2)}{S_2 - S_1}. \]

From the ratios of these two quantities \((x, x')\) we find:

\[ O_3 O_5 : O_3 O_4 = S_1 : S_2. \]

The derived condition makes it possible to verify the position of the point \(O_3\) in the following manner: on the verticals passing through the dead positions of both pistons of the machine, lay off the corresponding strokes \(S_1\) and \(S_2\); the straight line joining these points passes through the point \(O_3\).

The indicated verification has been reproduced in Fig. 2 (Table III).

This same condition also confirms and proves the proposition stated in the first chapter a priori, namely that the scales of the abscissae of the curve of steam flow from the small cylinder into the large one, represented on the indicator diagrams of both cylinders by two hyperbolas drawn from the common center \(O_3\), must be related as the corresponding cylinder volumes (the piston strokes \(S_1\) and \(S_2\) on Schröter’s diagram).

Let us return to the machine we are designing.

When the machine is operating at reduced load, steam of low pressure enters the large cylinder, as a result of which loops may form on the corresponding indicator diagram. These loops are located at the ends of the diagram—at the beginning of steam admission into the cylinder and at the end of the period of steam expansion. The first loops indicate an existing delay of the valve gear, leading to a disturbance of the regularity of its action; the second indicate the suction into the cylinder of a mixture of air and steam from the exhaust pipe, which contributes to fouling of the cylinder, wear of its walls, and entails damage to the machine.

Both kinds of loops are therefore also undesirable because they indicate useless negative work of the steam in the machine.

In view of all that has been set forth, by the minimum power of the machine we shall henceforth understand such a power below which loops already form on the indicator diagram of the large cylinder.

By means of the corresponding calculations, based on the definite analytical position of the point \(O_3\), the volumetric diagrams of the machine being designed, and the exhaust pressure \(p_a\) from the large cylinder, it is possible to determine the beginning of the period and the final pressure of compression in such a way that, at the minimum power of the machine, the pressure \(p_e''\) of steam admission into the large cylinder would be equal to the pressure \(p_c''\), and the final pressure of expansion \(p_\varepsilon''\)—in turn—to the exhaust pressure \(p_a\).

Under the indicated conditions, with a further decrease in the power of the machine, loops form simultaneously at both ends of the indicator diagram of the large cylinder.

It is clear that the power of the low-pressure cylinder obtained under such conditions, and consequently also that of the entire machine, will be the least possible power of the machine without the formation of loops.

On the basis of the considerations given above, we must take the period of steam compression in the low-pressure cylinder as equal to 12% and the final compression pressure as \(p_c'' = 2.75 \, klg\).

Since the fluctuations in power of Tandem machines of the usual type are effected exclusively by decreasing or increasing the degree of cut-off in the high-pressure cylinder, while all the other elements of steam distribution undergo no changes whatever, then, relying on the minimum work of the large cylinder, it is not difficult to determine, by inverse graphical construction or by calculation, the steam pressures at the various moments of the working process; in other words—from the given Schröter diagram for the normal operation of the machine it is not difficult to construct the same diagram for the minimum, or any other, power of the machine.

Having constructed the indicator diagrams (Table III, drawing 2) for the minimum work of the machine and planimetered them, we obtain the minimum calculated mean indicated pressure

\[ {}^{\min} p_i = 2.25 \, klg. \]

We determine the maximum power of the machine from the same condition of the impossibility of loop formation—this time on the indicator diagram of the high-pressure cylinder.

With an increase of cut-off in the small cylinder, the receiver pressure also increases; together with it, evidently, the final compression pressure \(p_c'\) of the high-pressure cylinder also increases.

Any increase in cut-off occurring after the final compression pressure \(p_c'\) has become equal to the admission pressure \(p_e'\) causes the formation of a loop at the corresponding end of the diagram. Once again, the throttling of the steam-admission organs and negative work are present.

Thus it is clear that the maximum work of the machine is determined by the condition:

\[ p_c' = p_e'. \]

Proceeding from the indicated condition, we can, just as for the minimum work, by graphical construction, checked by analytical calculation, obtain on the same drawing 2 (Table III) the indicator diagrams of both cylinders at the maximum load permissible without the formation of loops.

Planimetering these diagrams, we obtain, for the corresponding indicated pressure:

\[ \max p_i = 4.05 \ \mathrm{kg}. \]

Since the indicated power of the engine is directly proportional to the mean indicated pressure, taking the normal power of the engine as unity, we have the corresponding powers of the engine being designed expressed by the following numbers:

\[ 1.256 \;—\; 1 \;—\; 0.698. \]

Hence we see that, without the formation of loops, i.e., under the condition of smooth operation of the steam-distribution organs, the load of the engine we have designed may fluctuate within the limits from 1260 to 700 indicated horsepower—approximately by 30% on either side of the normal.

If the maximum load is sufficient, then the minimum must be regarded as still very high.

Let us now determine the conditions for possible steam extraction.

The amount of extraction must vary from zero to some maximum, above which the working process becomes improper, as is indicated by uselessly performed negative work, accompanied by the appearance of loops in the indicator diagrams.

It is clear that the greatest possible amount of extraction is expressed by the difference between the quantities of steam entering the engine at its maximum and minimum duties.

Indeed, the limit of maximum duty is restricted by the properties of the indicator diagram of the small cylinder; the limit of minimum duty depends on the indicator diagram of the large cylinder.

By making the engine operate in such a way that each of its cylinders performs work corresponding to its limiting diagram, we thereby obtain the possibility, in this extreme case, of drawing off from the receiver the greatest quantity of steam.

In order to express the extraction conditions numerically, let us reduce the quantities of steam passing through the engine at its various loads to a certain common pressure (in our case this pressure is taken as \(p_r = 4 \ \mathrm{Atm.}\)) and determine the volumes corresponding to the indicated quantities.

The reduction to a common pressure has been carried out on the indicator diagrams of the low-pressure cylinder (Fig. 2, Table III), where the values obtained make it possible to judge clearly the possible extraction.

The segments \(\overline{\alpha\beta}\), \(\overline{\alpha\gamma}\), and \(\overline{\alpha\delta}\) of one common horizontal line, being proportional to the quantities of steam passing through the machine at its minimum, normal, and maximum loads, indicate by their difference

\[ \overline{\beta\delta}=\overline{\alpha\delta}-\overline{\alpha\beta} \]

a value proportional to the possible maximum extraction.

The relative maximum extraction \(k_{mx}\) will be equal to:

\[ k_{mx}=\frac{\overline{\beta\delta}}{\overline{\alpha\delta}}. \]

Determining from the drawing and checking by calculation the named segments, we find for them, at the scale of volumes adopted in constructing the diagrams, the following values:

\[ \overline{\alpha\delta}=49{,}16\ \mathrm{mm}. \]

\[ \overline{\alpha\gamma}=36{,}44\quad \text{''} \]

\[ \overline{\alpha\beta}=24{,}2\quad \text{''}. \]

From a comparison of the values given, we conclude that the maximum extraction, referred to the normal quantity of steam passing through the machine, will be expressed thus:

\[ {}^{n}k_{mx}=\frac{49{,}16-24{,}2}{36{,}44}=68{,}56\sim 70\%; \]

while, referred to the greatest possible quantity of steam passing through the machine, it will be equal to:

\[ k_{mx}=\frac{49{,}16-24{,}2}{49{,}16}=49{,}2\sim 50\%. \]

In the interests of further investigation, let us determine certain dimensions of the machine, namely: the useful piston area of the low-pressure cylinder and the piston stroke of the machine corresponding to a specified mean speed.

Since, under the conditions adopted by us, there are no losses in the machine, we shall not introduce the coefficient of incompleteness of indicator diagrams that is usually adopted in calculations.

From the basic formula for calculating steam engines

\[ N_i=\frac{O\cdot p_i\cdot c_m}{75} \]

we have:

\[ O=\frac{75\cdot N_i}{p_i\cdot c_m}, \]

where

\[ \begin{aligned} N_i&\text{—the given number of indicated horsepowers,}\\ c_m&\text{—the mean piston speed, } mtr/sec,\\ O&\text{—the useful piston area in } cm^2. \end{aligned} \]

Let us assume a mean piston speed \(c_m=3.1\ mtr/sec.\)

Then, taking into account the mean indicated pressure corresponding to normal operation of the engine, \(n p_i=3.225\), we have:

\[ O=\frac{75\cdot 1000}{3.225\cdot 3.1}. \]

Since the denominator of this expression may, without great error, be taken as equal to \(10\) (in reality—\(9.9975\)), the required piston area is

\[ O=7500\ cm^2. \]

From the mean piston speed \(c_m\) and from the given number of revolutions \(n\), we determine \(S_2\)—the stroke of the piston of our engine.

\[ c_m=\frac{S_2 n}{30}, \]

whence:

\[ S_2=\frac{3.1\cdot 30}{120}=0.775\ mtr. \]

Let us return to the clarification of the conditions accompanying the extraction of steam from the engine under investigation.

In order to form an approximately correct idea of how the greatest extraction will affect the amount of work delivered by the engine, we shall calculate, on the basis of the planimetrations made earlier, the computed mean indicated pressure, determined by the ultimate indicator diagrams of each of its cylinders.

Having done this, we find:

\[ k p_i=2.413\ klg. \]

With respect to the mean indicated pressure \(npi = 3.225\), corresponding to the operation of the machine taken as unity, we have:

\[ \frac{3.124}{3.225}=0.748, \]

i.e., with the greatest extraction the machine being designed can develop only slightly more than its minimum power, expressed in relation to unity by the number \(0.698\).

Considering that, on the one hand, the machine must operate well and steadily under possibly wide fluctuations of its load, and that, on the other hand, the greatest steam extraction may be required precisely when the machine must overcome the greatest load, we are compelled to acknowledge that the working process designed for the machine intended for investigation will not fully satisfy all the requirements imposed upon it.

In order to confirm the correctness of the conclusions drawn on the basis of the diagrams constructed for various loads of the machine, we shall compile, for each case, a steam balance.

For simplicity, we shall operate with quantities expressed in \(mm\) and representing, on the corresponding scales, the volumes and pressures of the steam.

The quantities needed by us are taken directly from the drawing with an accuracy of up to \(1/4\ mm\).

We turn first to the diagrams corresponding to the maximum power of the machine.

For point \(b\) we shall have volume \(=21\ mm.\), and pressure \(165\ mm.\), therefore:

\[ c_1 = 21.165 = 3465; \]

for point \(d\) we have volume \(7.5\ mm.\), pressure \(78.25\ mm\), whence

\[ c_2 = 586.875; \]

for point \(t''\)—respectively \(65\) and \(O_2t'' = 50.75\), and

\[ c_3 = 3298.75; \]

point \(n''\), with volume \(m_2=8\) and pressure \(41.25\), gives

\[ c_4 = 330; \]

for point \(t'\) we have \(p=65\) and \(v=t't''=179.75\), whence

\[ c_5 = 11683.75, \]

the same point \(t'\), when referred to the volume \(v - t_1 O_2 = 129\), determines

\[ c_6 = 8385. \]

Let us set up the balance:

\[ \begin{array}{r} 3465\\ -\;586{,}875\\ \hline c_1 - c_2 = 2878{,}125 \end{array} \qquad \begin{array}{r} 3298{,}75\\ -\;330\\ \hline c_3 - c_4 = 2968{,}75 \end{array} \]

Discrepancy:

\[ \begin{array}{r} 2968{,}75\\ -\;2878{,}125\\ \hline 90{,}625 \approx 3\%. \end{array} \]

Let us turn to the line of exhaust from the high-pressure cylinder and to the expansion curve of the large cylinder:

\[ \begin{array}{r} 3298{,}75\\ +\;8385{,}00\\ \hline c_3 + c_6 = 11683{,}75 \end{array} \]

\[ c_5 = 11683{,}75. \]

\[ \text{Discrepancy} = 0. \]

From the example given for calculating the steam balance, it is evident that the discrepancy may reach the comparatively large value of \(3\%\). This is readily explained by the inaccuracy of measurements from the drawing of the corresponding data. With the pressure scale we have adopted \((1\ \mathrm{Atm} = 15\ \mathrm{mm})\), an error of \(0{,}25\ \mathrm{mm}\) is equal to \(1/60\) of an atmosphere—a very small quantity. As for the measurement of volumes, however, an error of \(0{,}25\ \mathrm{mm}\) already gives significant results.

Above we determined the useful area and the piston stroke of the low-pressure cylinder:

\[ O = 0{,}75\ \mathrm{mtr}^2 \quad \text{and} \quad S_2 = 0{,}775\ \mathrm{mtr}, \]

whence the useful volume of the large cylinder is

\[ Q = 0{,}58125\ \mathrm{mtr}^3. \]

This volume on Schröter’s diagram is taken by us as equal to \(100\ \mathrm{mm}\), from which we see that \(0{,}25\ \mathrm{mm}\) represents...

volume equal to $1453.125\ \mathrm{cm}^3$, which is a very considerable quantity.

Thus we conclude that use of Schröter’s diagram, while possessing the merits of clarity and convenience of drawing, must—in order to avoid errors—be accompanied by calculation and by verification of at least some characteristic points of the indicator diagrams.

Owing to the fact that, in our case, the volume diagrams are expressed by straight lines, it is extraordinarily easy, by simple calculation, to obtain the volumes occupied by the steam for each of the principal moments of steam distribution.

In what follows, through the entire investigation, a similar graphico-analytical method of constructing the working process is carried out.

Let us return to drawing up the balance for the case of maximum filling of the high-pressure cylinder, and here we shall use the values of volumes and pressures obtained by calculation.

The whole corresponding working process is obtained by us (for the given unchanging moments of steam distribution) from the condition

\[ p_c' = p_c', \]

therefore all the points of all the curves can be determined very easily.

The values of the volumes and pressures shown by the corresponding segments will then be expressed as follows:

point—$b$:

\[ p = 165 \qquad v = 21.47 \]

\[ c_1 = 3542.55 \]

point—$d$:

\[ p = 78.16 \qquad v = 7.6 \]

\[ c_2 = 594.0 \]

point—$t''$:

\[ p = 65.13 \qquad v = 50.35 \]

\[ c_3 = 3279.2955. \]

point—$n''$:

\[ p = 41.25 \qquad v = 8. \]

\[ c_4 = 330 \]

Balance:

\[ \begin{array}{r@{\ }r@{\qquad\qquad}r@{\ }r} & 3542.55 && 3279.2955\\ - & 594.00 & - & 330.00\\ \hline c_1 - c_3 = & 2946.55 & c_2 - c_4 = & 2949.2955 \end{array} \]

Discrepancy:

\[ \begin{array}{r} 2949.296\\ -\,2946.550\\ \hline 2.746 < 0.1\% \end{array} \]

As may be seen from the last example, the balance is obtained more accurately than it was when the corresponding quantities were taken from the drawing, even with an accuracy of up to \(0.25\ mm\).

As we have seen above, the steam balance of the machine under study is based, among other quantities, on the constant \(C_3\) of the hyperbola of steam expansion in the low-pressure cylinder.

The value of this constant, in turn, is determined by the cut-off point of the large cylinder. We know that this latter point is characterized on the indicator diagram of the high-pressure cylinder by the point of intersection of two equilateral hyperbolas drawn from the centers \(O_2\) and \(O_3\).

The graphical and analytical methods for finding such points were indicated in the “Introduction.”

In Fig. 2, Table III, the position of the ordinate of the point under consideration is found by a construction based on the proportional division, internally, of the corresponding segment of the line of volumes \(O_3O_2\).

CHAPTER FOUR.

Approximate calculation of a tandem machine with cooling (at low vacuum). Determination of the limits of possible extraction.

In Chapter Three we saw that a machine without cooling is poorly adapted to operation under conditions of wide variation both of load and of extraction.

To build a machine with cooling, striving to obtain, at high vacuum, as small as possible a steam consumption per indicated horsepower per hour, is irrational from the standpoint of utilization of the hot water leaving the machine, since at high vacuum, corresponding in the newest machines to the pressure of the steam discharged from the machine \(p_a=(0.05—0.06)\ \mathrm{Atm}\), the water leaving the condenser of the machine will have a low temperature (about \(30^\circ—35^\circ C\)).

Therefore we shall take the pressure \(p_a\) of the steam discharge from the machine, as L. Schneider*) also does in his study, equal to \(0.55\ \mathrm{Atm}\), assuming that the exhaust steam in the machine is used, in accordance with production needs, for heating the corresponding quantity of water to a temperature of about \(80^\circ C\).

To make it possible to compare the working process of both machines considered, we shall leave unchanged the ratio of cylinder volumes, the receiver volume, and all the steam-distribution moments.

We shall take the final expansion pressure \(p_e''\) in the large cylinder equal to \(p_a\) plus \(0.45\ \mathrm{Atm}\), whence

\[ p_e'' = 1\ \mathrm{Atm}. \]

Applying, as before, the grapho-analytical method, we construct (drawing, Plate IV) the working process of the machine with a small vacuum for the normal and possible (without the formation of loops on the indicator diagrams) maximum and minimum loads of the machine. In the drawing of Plate IV the corresponding indicator diagrams are selected from the entire series of diagrams prepared for the study; in addition, individual moments of the working process of the machine at normal load are marked with circles.

By planimetering the diagrams of normal operation of the machine, we find the calculated mean indicated pressure

) Die Abwärmeverwertung.* S. 16.

\[ {}^{n}p_i = 2.627 \ \mathrm{klg}. \]

The mean indicated pressure of the machine without cooling we had earlier

\[ {}^{n}p_i = 3.225 \ \mathrm{klg}. \]

Thus we see that, for the former number of revolutions and the former piston speed

\[ c_m = 3.1 \ \mathrm{mtr/sec}. \]

we must have, for the specified normal power of the machine \(N_i = 1000\), the piston area \(O\) of the low-pressure cylinder increased by

\[ \frac{3.225}{2.627} = 1.228 \]

times.

Whence we determine the piston area

\[ O = 9210 \ \mathrm{cm}^2. \]

Consequently, we must note that the scale of volumes for Schröter’s diagram has increased for us by the same number of times.

Let us now determine the useful steam consumption by the machine, expressed, as we already know, by a certain volume for one piston stroke. Referring this volume to the former pressure of \(4 \ \mathrm{Atm.}\), we obtain (by simultaneous graphical and analytical calculation) that the useful steam consumption at the normal load of the machine (drawing, table IV) is expressed by the segment

\[ \alpha \gamma = 24.25 \ \mathrm{mm}. \]

The corresponding steam consumption in the first machine was expressed (see drawing 2, table III) by the segment

\[ \alpha \gamma = 36.44 \ \mathrm{mm}. \]

In order to compare the indicated steam consumptions with one another, it is obviously necessary to reduce these segments to one scale of volumes.

Doing this, we obtain that the steam consumption of the machine with low vacuum will be expressed, on the former scale, by the segment

\[ \alpha' \gamma' = 24.25 \cdot 1.228 = 29.78, \]

i.e. it amounts to \(0.817\) of the consumption of the machine without cooling.

Planimetering successively the diagrams of maximum and minimum load of the machine, we find respectively:

\[ {}^{max}p_i = 4.73 \ \mathrm{klg}, \qquad {}^{min}p_i = 1.267 \ \mathrm{klg}, \]

whence, taking as before \(np_i = 2.627\,klg.\) per unit, we find the following permissible fluctuations in the load of the machine:

\[ \begin{array}{ccc} \max. & \norm. & \min.\\[4pt] \dfrac{4.73}{2.627}=1.8005 & 1 & \dfrac{1.267}{2.627}=0.4823. \end{array} \]

In other words, the machine, without undergoing any disturbances in the action of its steam-distributing apparatus, can work at half load (\(N_i = 482\)) and can admit a very high overload, reaching up to \(80\%\) of its normal power.

This latter quality is especially valuable for us. From the conditions of the working process of machines with steam extraction we know that the deficiency of work delivered by the low-pressure cylinder must be made up by a corresponding increase in the power delivered by the small cylinder.

Consequently, the greater the permissible maximum filling of the high-pressure cylinder, the more considerable, one may say a priori, will be the magnitude of the possible extraction.

Relying on the planimetering already carried out, let us make a calculation of the power of the machine due to the presence of limiting indicator diagrams of each of its cylinders, thereby assuming the presence of maximum extraction.

Carrying out what has been said, we have:

\[ kp_i = 2.133, \]

or, in relation to the normal power, at the greatest possible extraction the machine develops

\[ \frac{2.133}{2.627}=0.812, \]

i.e. \(19\%\) less than normal and \(1.685\) times greater than its minimum work.

Let us now determine the maximum permissible extraction of steam. Reducing the useful steam consumption by the machine—at the maximum and minimum fillings of the high-pressure cylinder—to the volumes correspondingly represented (drawing, table IV) on the indicator diagram of the large cylinder by the segments \(\overline{\alpha\beta}\) and \(\overline{\alpha\delta}\), we obtain the greatest extraction in the form of the segment:

\[ \overline{\beta\delta}=\overline{\alpha\delta}-\overline{\alpha\beta}. \]

Determining by calculation the magnitudes of the segments characterizing the volume of steam passing through the engine, we find:

\[ \overline{a\delta}=51{,}864, \]

\[ \alpha\gamma=24{,}25, \]

\[ \alpha\beta=12{,}10, \]

whence we determine the maximum extraction:

\[ \overline{\beta\delta}=39{,}76. \]

For comparison, we must reduce this value to the scale of the volumes of the engine without cooling. Carrying this out, we have:

\[ 39{,}76 \cdot 1{,}228 = 48{,}83, \]

which, in comparison with the value of the maximum extraction given in Chapter Three \((24{,}96)\), gives an increase in the amount of possible extraction by

\[ \frac{48{,}84}{24{,}96}=1{,}957, \]

i.e. we have an increase of the extraction almost twofold.

Let us now determine the relative value of the maximum extraction in comparison with the quantity of steam passing through the high-pressure cylinder, at normal and maximum admission.

Then we have:

\[ n'_{max}=\frac{39{,}77}{24{,}25}=1{,}6387 \sim 164\%, \]

\[ k_{max}=\frac{39{,}77}{51{,}864}=0{,}7668 \sim 77\%. \]

From all that has been set forth it is clear that, by introducing into the working process of the engine a moderate vacuum, we have considerably improved all its qualities from the standpoint of intermediate steam extraction and, moreover, have obtained the possibility of proper operation of the engine when its load varies from \(482\) to \(1800{,}5\) horsepower.

We shall stop at this engine, selecting it as the object of the proposed investigation.

In order to confirm the conclusions made above, based on comparison of the plotted indicator diagrams, let us verify the cor—

the correctness of constructing these latter by compiling the steam balance for the three examined loads of the machine, while, for simplicity, taking the volumes and pressures in the form of the corresponding segments expressed in mm. (see the Appendix, Tables II and III).

A) Maximum load of the machine.

Data table for the balance.

Diagram points	Corresponding	Corresponding	Corresponding	Corresponding
	Volume	Pressure	Hyperbola constant	Hyperbola constant
$a_{mx}$	22,46	165,00	$c_1$	3705,90
$g'_{mx}$	7,60	78,16	$c_2$	594,016
$g''_{mx}$	108,00	30,34	$c_3$	3276,720
$h$	20,00	8,25	$c_4$	165,00
$e'_{mx}$	156,04	74.9	$c_5$	11687,396
$g'_{mx}$	107,60	78,16	$c_6$	8410,016

Balance.

\[ \begin{aligned} &\frac{\begin{array}{r} 3705{,}90\\[-2pt] -\,594{,}016 \end{array}}{c_1-c_2=3111{,}884} \qquad \frac{\begin{array}{r} 3276{,}720\\[-2pt] -\,165{,}00 \end{array}}{c_3-c_4=3111{,}720} \end{aligned} \]

\[ \frac{\begin{array}{r} 3276{,}720\\[-2pt] +\,8410{,}016 \end{array}}{c_3+c_6=11686{,}736} \qquad c_5=11687{,}396 \]

The diagrams have been drawn correctly.

B) Normal load of the machine.

Data table for the balance.

Diagram points	Corresponding	Corresponding	Corresponding	Corresponding
	Volume	Pressure	Hyperbola constant	Hyperbola constant
$a_n$	10,6	165,00	$c_1$	1749,00
$f'_n$	7,6	38,70	$c_2$	294,12
$g''_n$	108,00	15,00	$c_3$	1620,00
$h$	20,00	8,25	$c_4$	165,00
$e'_n$	156,04	37,05	$c_5$	5781,28
$g'_n$	107,60	38,70	$c_6$	416

Balance.

\[ \begin{array}{r} 1749{,}00\\ -\ \ 294{,}12\\ \hline c_1-c_2=1454{,}88 \end{array} \qquad \begin{array}{r} 1620{,}00\\ -\ \ 165{,}00\\ \hline c_3-c_4=1455{,}00 \end{array} \]

\[ \begin{array}{r} 1620{,}00\\ +\ 4164{,}12\\ \hline c_3+c_6=5784{,}12 \end{array} \qquad c_5=5773{,}48 \]

The discrepancy is less than \(0{,}2\%\).

The diagrams have been drawn correctly.

C) Lowest loads of the machine.

Table of data for the balance.

Points of the diagram	Corresponding	Corresponding	Corresponding	Corresponding
	Volume	Pressure	Constant hyperbola	Constant hyperbola
\(d_{\min}\)	5,38	165,00	\(c_1\)	887,70
\(f'_{\min}\)	7,60	21,30	\(c_2\)	161,88
\(g''_{\min}\)	108,00	8,25	\(c_3\)	891,00
\(h\)	20,00	8,25	\(c_4\)	165,00
\(c'_{\min}\)	156,04	20,40	\(c_5\)	3183,216
\(g'_{\min}\)	107,60	21,30	\(c_6\)	2291,88

Balance.

\[ \begin{array}{r} 887{,}70\\ -\ \ 161{,}88\\ \hline c_1-c_2=725{,}82 \end{array} \qquad \begin{array}{r} 891{,}00\\ -\ \ 165{,}00\\ \hline c_3-c_4=726{,}00 \end{array} \]

\[ \begin{array}{r} 891{,}00\\ +\ 2291{,}88\\ \hline c_3+c_6=3182{,}88 \end{array} \qquad c_5=3183{,}216 \]

The diagrams have been drawn correctly.

CHAPTER FIVE.

Investigation of the Tandem engine under variable load, without extracting steam from it.

In order to be able to trace the influence of extraction on the consumption of steam per one indicated horsepower of the engine per hour—on the distribution of the work delivered by the engine between the two cylinders, and also to compare the results obtained with an ordinary-type Tandem engine, we shall investigate—when extraction is absent—the operating conditions of the engine, the principal dimensions and working process of which were determined by us in the preceding chapter.

We shall conduct the investigation in the direction of changing the load of the engine from the greatest to the smallest possible limits. For each case we shall determine the indicated power of the whole engine and the distribution of this power between the cylinders; in addition, we shall find the corresponding useful consumption of steam.

Let us divide the greatest load of the engine into successive stages. We saw above that the greatest permissible steam extraction, being referred to a pressure of $4$ Atm., is expressed volumetrically by the segment $36 = 39.76\ mm.$ (drawing, plate IV). We shall divide this same segment into $9$ parts so that one of the divisions coincides with the amount of steam corresponding to the normal load of the engine, while the remaining divisions are spaced from one another by approximately $4\ mm.$ Under these conditions, assuming, as already stated, that there is no extraction of steam from the engine, we can construct, for each of the cylinders of the engine, a total of $11$ diagrams corresponding to eleven different volumetric quantities of steam passing through the cylinders of the engine at its various loads, from the greatest to the smallest.

For each load, by the adopted grapho-analytical method, the corresponding indicator diagrams were constructed (drawing, plate IV), the correctness of the construction being checked by compiling steam balances.

By planimetering the constructed diagrams, we obtain the possibility of determining the power of the engine for each case and the other data characterizing the engine under investigation in the absence of extraction.

Calling the volume of steam passing through the machine in one stroke of the piston by the letter \(q\), we have, on the scale of volumes (at a pressure of 4 atm.), a series of segments corresponding to the various degrees of load of the machine*).

Knowing the area of the piston \(O_2\) in \(mt^2\), its stroke—\(S_2\), the number of revolutions of the machine—\(n\) per minute, and the weight of a unit volume of steam at a pressure of 4 atm., equal to \(2.147\ kg\), we can easily compute \(G\)—the useful hourly consumption of steam in \(kg\)—by the following formula:

\[ G=\frac{S_2\cdot O_2}{100}\cdot q\cdot 2n\cdot 60\cdot 2.147. \]

Collecting all the constant quantities into one coefficient, we give this formula the simpler form:

\[ G=220.67q \]

and from it determine the hourly steam consumption at the various loads of the machine.

Knowing the total hourly steam consumption and the corresponding load of the machine, we determine \(Q_i\)—the steam consumption per one indicated horsepower per hour.

All the obtained quantities are collected in one table.

Table of quantities characterizing the operating conditions of the machine under investigation in the absence of extraction.

Volumetric quantities of steam, expressed in mm.	Volumetric quantities of steam, expressed in mm.	Indicated power of the small cylinder	Indicated power of the large cylinder	Indicated power of the whole machine	Total hourly steam consumption \(G\)—in kg.	Steam consumption per horsepower-hour \(Q_i\)—kg.
\(\min q_1\)	12,1	340,15	142,12	482,3	2670,20	5,54
\(q_2\)	16	421,29	246,03	667,3	3530,80	5,29
\(q_3\)	20	489,81	350,3	840,1	4413,50	5,25
\(q_4\)	24,25	550,72	449,28	1000	5351,40	5,35
\(q_5\)	28	588,79	548,17	1137	6178,90	5,43
\(q_6\)	32	611,63	636,9	1248,5	7061,70	5,66
\(q_7\)	36	642,08	755,9	1398	7944,40	5,68
\(q_8\)	40	644,62	852,7	1497,3	8827,10	5,90
\(q_9\)	44	647,16	948,8	1596	9709,80	6,08
\(q_{10}\)	48	659,94	1076,1	1736	10592,50	6,10
\(\max q_{11}\)	51,864	659,94	1140,6	1800,5	11445,20	6,36

Considering this table, we see that the smallest steam consumption per horsepower-hour is observed at about the third degree of load of the machine, from which, on both sides, it increases. The work delivered—

* See Appendix.—Table 1.

…delivered by the high-pressure cylinder at different loads fluctuates within considerably narrower limits (340.15–659.94) than the corresponding work of the large cylinder (142.12–1140.6).

All the analyzed properties of the machine appear especially clearly in Fig. 3, where the quantities \(q\) are plotted along the abscissa axis, and along the ordinate axis the work delivered by the high-pressure cylinder (curve I), by the low-pressure cylinder (curve II), and by the whole machine (curve III).

Fig. 3.

Examining these curves, we see that the work delivered by the larger cylinder changes almost directly in proportion to the quantity of steam passing through the machine.

In Fig. 4 the curve represents the variation of steam consumption per indicated horsepower per hour as a function of the load of the machine. We see that, within the limits between the smallest and the greatest loads, the steam consumption changes only slightly overall. Referring to the table, we may calculate that, when the total power of the machine is reduced from 840 horsepower to 482, i.e. by approximately 57%, the steam consumption increases by 4.73%; whereas, when the power is increased from 840

Fig. 4.

from horsepower to the greatest value (1800 horsepower), which corresponds to an increase in power of \(\sim 114\%\), the steam consumption increases by only \(20\%\) (both of these changes being measured relative to the minimum consumption).

CHAPTER SIX.

Investigation of the operating conditions of the machine when different quantities of steam at constant pressure are extracted from it.

We shall carry out the further investigation of the machine under study in the direction of varying the quantity extracted from the machine at constant steam pressure.

This condition is not formulated quite correctly. In Chapter Two we saw that the steam pressure in the receiver and in the cylinder (the extraction pipeline), under the influence of changes in the volumes of the working spaces of the machine cylinders that communicate in succession with the receiver, varies continuously.

Therefore—strictly speaking—there can be no extraction of steam at constant pressure. We, however, in order to distinguish the different conditions of extraction, shall hereafter speak of steam extraction at constant receiver pressure, understanding by this latter, as was said at the very beginning of our exposition, the pressure in the receiver, with the machine in a steady state, at the moment preceding release from the small cylinder.

Thus the indicated condition of extraction is essentially reduced to a variable quantity of steam being extracted from the machine, while at the same time one and the same quantity of steam enters the small cylinder for each stroke of the piston.

The latter condition is fulfilled when the degree of filling of the high-pressure cylinder remains constant.

Thus the general picture of the process under study is clear: the small cylinder performs a definite amount of work at the expense of all the steam entering it; the large cylinder performs work at the expense of the steam remaining in the receiver after a certain quantity of steam has been extracted. With a change in extraction, the work of the low-pressure cylinder changes, and along with it the work delivered by the entire machine.

Upon more detailed study we shall see that—with a change in extraction—the corresponding work of the small cylinder also changes, although only insignificantly.

In the preceding chapter we carried out an investigation of the working process of the machine, taking as our basis the various degrees of its load, corresponding to the different quantities of steam entering the machine. We shall make use of these same quantities of steam in the present investigation.

We shall pass through the small cylinder the greatest quantity of steam, corresponding, evidently, to the limiting indicator diagram; from this quantity, as it passes through the receiver, we shall extract steam, beginning with the greatest possible extraction, determined by the limiting indicator diagram of the large cylinder; then, by stages, we shall reduce the extraction until—with extraction equal to zero—we again obtain the indicator diagrams of an ordinary Tandem machine, corresponding to the greatest permissible load.

Carrying out what has been said for 11 stages of load of the machine, we obtain 10 stages of extraction, for which the corresponding indicator diagrams were constructed.

Since, according to Chapter II, the origins of coordinates for the hyperbolas characterizing one or another part of the working process are found as projections of the points of intersection of the straight lines of the specified volumetric diagrams, so also for the investigation described in the present chapter all constructions of the working process were carried out by a grapho-analytical method, i.e. the origins of coordinates for the hyperbolas were found by solving the corresponding equations of straight lines*), while the individual points of the indicator diagrams were found from the equations relating to these points of the equilateral hyperbolas \(p.v=c\). For a final verification of the results obtained, a steam balance was compiled in each separate case.

In carrying out the investigation it was necessary, for each separate quantity of extracted steam, to take into account the corresponding volumetric influence of the extraction on the steam pressures represented in the indicator diagrams of both working cylinders of the machine.

The volumetric influence of the extraction, as was indicated in the second chapter, is determined as a function of the volume corresponding to the final expansion pressure \(p_{\varepsilon}\), of the entire quantity of steam passing through the small cylinder of the machine during one stroke of the piston.

Since the degree of filling of the high-pressure cylinder—for the conditions of the extraction under study—remains constant, the moment at which the ex-

*) See Appendix—Table XI.

the period of compression does not change at all, then, in the established condition of the engine, the quantity of steam passing through it during one stroke of the piston remains constant.

In drawing 1 (Table V) are given all the indicator diagrams obtained for various quantities of steam extracted. The volume diagrams themselves, however—characteristic of the process—and their points of intersection are shown only for the case of extraction in which the quantity of steam entering the low-pressure cylinder corresponds to the normal operation of the engine (in this operation, as we already know, of the 1000 indicated horsepower delivered by the engine, 449.28 fall to the share of the large cylinder).

In order to obtain a representation of the working process corresponding to each case under consideration, it is necessary, according to Chapter Two, to determine the volume of the extraction cylinder.

In studying the engine we introduced the condition that the moment at which compression begins in one working space of the small cylinder coincides with the moment of pre-release of steam from the other space of the same cylinder.

It follows from this that the required quantity of steam must be extracted not during one revolution of the engine shaft, but only during the interval of time corresponding to the two above-mentioned moments of steam distribution in the small cylinder—so that extraction from one space proceeds without disturbing the regularity of extraction from the other.

Thus the total extraction of steam from the engine is the sum of two identical independent extractions from each space of its small cylinder, following one another continuously; the receiver is constantly in communication with the high-pressure cylinder (in turn with each of its working spaces), and expansion of steam into the receiver, unaccompanied by useful work of the steam in the engine, does not occur.

In accordance with the condition set forth—the absence of influence of each of the extracted quantities of steam on the one following it in time—the volume diagram of the extraction cylinder (drawing 2, Table II) must be correspondingly changed.

Indeed, the working process discussed in the second chapter relates to the presence in the engine of a pump-cylinder which alternately fills each of its spaces with steam taken from the two spaces of the small cylinder, while in the extraction cylinder these quantities of steam must mix. Taking this phenomenon into account, we said at the proper time that, at the moment of pre-release from the right-hand space of the small ...

of the small cylinder a receiver is connected to it, and through this latter the take-off belt and cylinder, partly occupied by the steam that entered it before the moment under consideration from the left-hand cavity of the high-pressure cylinder (see p. 23).

In order—when drawing the working process—to make the take-off from each cavity of the small cylinder independent, it is necessary to take into account the volumetric influence of the take-off cylinder only insofar as the increase of volume, resulting from the motion of the piston of the take-off cylinder, affects the change in steam pressure in the cavity communicating with this cylinder, without at all taking into account the quantity of steam present in the take-off cylinder that entered it earlier from the other working cavity of the small cylinder.

To accomplish this, from all the volumes represented by the resultant diagram \(\beta'\gamma_1\) (drawing 1, plate V), one must subtract the volume—equal to the segment \(b_2 b_3\)—which is precisely the volume of steam that penetrated, during the piston stroke preceding the one under consideration, from the left-hand cavity of the small cylinder.

The result of this subtraction will be the line \(b_2 \delta_1\), parallel to the line \(\beta'\gamma_1\) and drawn from it at a distance equal, horizontally, to the segment \(b_2 b_3\).

The straight line \(b_2 \delta_1\) will thus represent the corresponding line of the volumetric influence of the take-off.

In the case where the moments of the beginning of compression and of pre-release for the cavities of the small cylinder coincide, it is obvious that the segment \(\beta_1 \delta_1\) of the horizontal line must express the relative take-off existing at the given time.

Thus the line of the volumetric influence of the take-off for each separate case is constructed very simply: on the horizontal corresponding to the moment of the beginning of compression in the small cylinder, we set off from the vertical \(O_2\) the segment \(\beta_1 \delta_1\), expressing the volume of the available take-off corresponding to the final pressure of steam expansion in this cylinder, and connect the obtained point \(\delta_1\) by a straight line with the point \(b_2\), corresponding on the vertical \(O_2\) to the beginning of the take-off, i.e., to the moment of pre-release of steam from the small cylinder.

Since the subsequent character of the changes in the volumes successively occupied by the steam in the machine is in complete agreement with the exposition given in the second chapter, the procedure for drawing the working process also remains the same as before.

On the basis of the foregoing, in the drawing (plate V) we plotted only the corresponding lines \(b_2 \delta_1\) of the volumetric influence of the take-off, without using

...with diagrams of the fictitious pump-cylinder volumes, now superfluous and formerly useful—in their own time and place—for explaining to us all the details of the general picture of extraction.

Let us proceed to a more detailed description of the investigation carried out and of the results obtained.

According to the eleven stages of machine load, or more correctly—the quantities of steam passing through the small cylinder during one piston stroke at eleven different loads—we obtained ten stages of extraction.

Let us express the extraction \(k_x\)—for each arbitrary stage—by the difference:

\[ k_x = \max q_{11} - q_x, \]

where \(\max q_{11}\) is the maximum quantity of steam entering the small cylinder, expressed in mm according to the adopted scale of volumes, and \(q_x\), correspondingly, is the amount remaining after extraction and entering the large cylinder for further work.

Let us also determine the extraction as a percentage of the total quantity of steam entering the high-pressure cylinder*).

The quantities obtained are set out in the following table.

Table of quantities characterizing steam extraction.

Quantity of steam entering the large cylinder.	Quantity of steam entering the large cylinder.	Corresponding differences \(\max q_{11} - q_x\)	Corresponding differences \(\max q_{11} - q_x\)	Extraction in % of \(\max q_{11}\)	Values \(k_x\), reduced to a pressure of \(5.14\) atm.
\(\max q_{11}\)	51,864	\(k_{11}\)	0	0	0
\(q_{10}\)	48	\(k_{10}\)	3,864	7,45	3,00
\(q_9\)	44	\(k_9\)	7,864	15,16	6,12
\(q_8\)	40	\(k_8\)	11,864	22,88	9,23
\(q_7\)	36	\(k_7\)	15,864	30,59	12,34
\(q_6\)	32	\(k_6\)	19,864	38,30	15,45
\(q_5\)	28	\(k_5\)	23,864	46,01	18,56
\(n q_4\)	24,25	\(n k_4\)	27,614	53,24	21,48
\(q_3\)	20	\(k_3\)	31,864	61,44	24,78
\(q_2\)	16	\(k_2\)	35,864	69,15	27,90
\(\min q_1\)	12,1	\(\max k_1\)	39,764	76,67	30,93

*) See Appendix—Tables I, V and VI.

Examining the table, we see that, in accordance with the stages we have outlined, the extraction changes for each subsequent stage by approximately 8% of the entire quantity of steam passing through the small cylinder.

In the table, the data relating to the mutual arrangement of all the lines of the volume diagrams shown in the drawing are marked in italics.

To compare with one another the volume quantities of steam—and we must do this throughout the entire investigation—is possible only at identical pressures. In our case all the volumes have been reduced to a pressure equal to 4 atm. With respect to this pressure all the quantities \(q\) and \(k\), placed in the corresponding tables, were calculated.

For constructing the line of the volumetric effect of extraction, as we already know, it is necessary to reduce the quantities of steam passing through the small cylinder to the volume corresponding to the final pressure of expansion \(p_3\).

Since the volumes of dry saturated steam—obeying Mariotte’s law—are inversely proportional to the pressure, all the quantities \(k\) needed for the construction must be recalculated in accordance with the final pressure of expansion existing in the small cylinder.

For the extraction conditions considered in the present chapter, the final pressure of expansion remains unchanged (\(5.14\) atm.), and therefore all the values \(kx\) placed in the table must be multiplied by one and the same quantity \(\left(\frac{4}{5.14}=0.778\right)\). The values obtained are shown in the last column of the table*).

As was already said earlier, for each stage of extraction the indicator diagrams of both cylinders were drawn.

Let us analyze and describe here in detail one of the cases of extraction, namely, \(-kk_4\), corresponding to the normal operation of the large cylinder.

We shall begin with the indicator diagram of the high-pressure cylinder (see Drawing 1, Table V).

From the moment compression begins until the moment exhaust is admitted, the indicator diagram of the small cylinder remains the same for all cases of extraction, since throughout this entire part of the machine’s working process the right-hand cavity of the high-pressure cylinder under investigation by us is not connected with the receiver.

*) See also Appendix—Table 17, quantity \(z\).

This communication begins at point \(b\), and, since we are analyzing the case of steam extraction at a constant receiver pressure, evidently the pressure drop \(bc\) remains the same for all cases of extraction.

Starting from point \(c\), as we know from the second chapter, the general expansion of the steam proceeds in the high-pressure cylinder, in the receiver, and in the extraction cylinder. The origin of the coordinates of the corresponding hyperbola is found as the projection of point \(A\)—the intersection of the extended straight lines \((1—2)\) and \((b_2—\delta_1)\)—onto the zero-pressure line \(O_1O_5\).

Since, with a change in extraction, the length of the corresponding segment \(\beta_1\delta_1\) will also change, and with it the inclination of the line \(b_2\delta_1\) to the horizontal, it is evident that point \(O_3\)—the origin of the coordinates just described—will change its position on the line \(O_1O_2\).

Consequently, the expansion hyperbolas, being constructed for each of the cases of extraction analyzed from the corresponding center \(O_3\), must occupy on the drawing of Table V a consecutive series of different positions.

On the indicator diagram of the high-pressure cylinder, for all cases of extraction we have drawn only one curve \(cd\), since—owing to the short duration of the expansion process under consideration—the steam pressures toward the end of this period differ from one another extremely little. Indeed, the pressures in the dead position of the piston of the small cylinder, having been determined by calculation*) for the limiting values of extraction \(mxk_1=76.67\%\) and \(k_0=0\), were respectively \(5.09\) atm. and \(5.19\) atm., which—at the pressure scale adopted by us \((1\ \mathrm{Atm.}=15\ \mathrm{mm}.)\)—gives, for eleven successively drawn hyperbolas, a displacement of point \(d\) of only \(1.5\ \mathrm{mm}\).

Let us compute the greatest error obtained by us under the indicated assumption.

Since the distance between the ordinates passing through points \(c\) and \(d\) is equal to \(4\ \mathrm{mm}\), the assumed increase, contrary to the true value, of the area of the indicator diagram will be expressed by the area of the corresponding triangle \(cd_1d\), equal, as above, to

\[ \frac{4\cdot 1.5}{2}=3\ \mathrm{mm}^2. \]

Referring this area to the stroke \(S_2\) of the piston of the large cylinder, we shall obtain the corresponding change in the calculated mean indicator—

*) See Appendix—Table VI.

of pressure equal in kg:

\[ \frac{0.03}{15} = 0.002\ \text{kg}, \]

which is an insignificant quantity.

Let us proceed to the further exposition.

Since in the machine under consideration, according to the adopted condition, there is no advance admission of steam into the large cylinder, the expansion of the steam \(cd\) will continue up to the dead position of the piston, where the large cylinder enters into the process, the clearance space of which is filled with steam at pressure \(p_c''\)—the final compression pressure.

In all four cavities there is now established a common steam pressure, determined on the indicator diagrams by the method of Mönch, or analytically—by the points \(e'\) and \(e''\). These points, like point \(d\), are different for each case of extraction, but the difference between their positions on the corresponding verticals is even less perceptible, since by calculation we obtain*) for maximum extraction \(m x h_{10}\)—the pressure at the point \(e'(e'')\)—\(4.91\) Atm., and for zero extraction \(k_0 — e'(e'') — 4.99\) Atm. This difference, on the adopted scale, will be expressed as \(1.2\) mm, corresponding to eleven different positions of the point \(e'(e'')\).

For this reason, for all cases of the extraction under consideration, we have adopted, for the exhaust line from the small cylinder and the admission line into the large cylinder, as the initial points \(e'\) and \(e''\) corresponding to zero extraction; and therefore all the successively obtained curves \(e'f'\) are, for all cases of extraction, continuations of one another.

In reality such a phenomenon does not occur, and each degree of extraction is characterized by its own line of filling of the large cylinder.

Using the grapho-analytical method, and having the necessary centers \(O_1, O_2 \ldots O_5\), it is not difficult to draw, for each cylinder separately and at any scale, indicator diagrams corresponding to an arbitrarily chosen individual case of extraction.

For us, in the investigation undertaken, it is desirable to obtain, especially from the standpoint of the gradual change in the cut-off moment in the large cylinder, a series of indicator diagrams drawn one upon another in the order of the sequence of the change in extraction.

The error that we make in the indicator diagram of the high-pressure cylinder, as we have already seen, is reflected in the calculated mean indicator pressure to an extremely small extent. For the cylinder

*) See Appendix—Table VI.

low-pressure error arising from the assumption adopted is likewise so small; the following calculation convinces us of this.

If, for the sake of simplicity, we assume that—for the smallest adopted degree of extraction \(k_{10}\)—the filling of the large cylinder proceeds along the straight line \(e''f''\), instead of along the curve \(e''f''\) shown, then, by planimetering, with the corresponding determination of the mean indicated pressure, the error obtained is expressed by a value equal to \(0.04\ \mathrm{Atm}\). In reality the error that has occurred is considerably smaller, since the filling of the large cylinder always proceeds along a curve.*)

Thus, having accepted the assumptions made, let us return to the further description of the construction of the working process for the case of extraction under consideration.

In order to determine the corresponding moment of cut-off in the large cylinder of the machine, it is necessary to find the point \(f''\) of intersection of the hyperbola \(f''g''\) with the filling line \(e''f''\). This intersection may be found by graphical construction and checked analytically, as was indicated in the introduction.

Using the volume diagrams of the working cylinders of the machine, we may refer the curve \(f''g''\) to the stroke of the piston of the small cylinder and superpose it on the indicator diagram of this latter cylinder (the dotted curve \(f_3 g_3\)), whereby the origin of coordinates of the similarly transformed curve will pass to the point \(O_6\); then, in order to find the coordinates of the point \(f''\) in all cases of extraction, we obtain three combined conditions characterizing the intersection of three hyperbolas at one point.

These conditions, in accordance with what was said in the introduction, may be expressed as follows:

a) the intersection of the hyperbolas \(c_5\) and \(c_6\) gives, for the corresponding pressure \(p_x\), the expression:

\[ p_x=\frac{c_5+c_6}{O_4O_5}; \]

b) the hyperbolas \(c_7\) and \(c_6\) give, for determining \(p_x\), the value:

\[ p_x=\frac{c_6+c_7}{O_6O_5}; \]

c) for the hyperbolas \(c_5\) and \(c_7\) we obtain in the same way:

\[ p_x=\frac{c_5-c_7}{O_4O_6}. \]

*) The author’s attempt to draw jointly the corresponding parts of both indicator diagrams for all cases of the extraction under consideration, in full agreement with the values obtained by calculation, at the adopted scale \(1\ \mathrm{Atm.}=150\ \mathrm{mm}\), did not increase the accuracy of the results obtained.

By these three conditions the determination was made of the cut-off moment of the low-pressure cylinder for all the investigated cases of steam extraction.

After the cut-off moment of the large cylinder, independent expansion of the part of the steam that has fallen to its share proceeds in this latter cylinder.

The line \(f'g'\) of exhaust from the small cylinder must represent the compression of the steam, taking place in the high-pressure cylinder and the receiver simultaneously with the continuing extraction.

As we already know, the origin of coordinates for the corresponding hyperbola will be the point \(O_3\)—the projection on the line \(O_1O_2\) of the point \(C\)—the intersection of the volume straight lines \((2—3')\) and \(b_5\delta_1\).

The point \(g'\) characterizes the beginning of the compression period in the small cylinder.

Let us compile, for the analyzed case of extraction, the steam balance on the basis of the conditions derived in Chapter Two and the data given in Tables IV and VI (see the Appendices).

Table of data.

Points of the diagram	Corresponding volume	Corresponding pressure	Corresponding constant of the hyperbola	Corresponding constant of the hyperbola
\(a\)	22,46	165,0	\(c_1\)	3705,90
\(g'\)	7,60	78,16	\(c_2\)	594,02
\(f''\)	23,50	68,9	\(c_3\)	1619,15
\(r\)	8	20,63	\(c_4\)	165,0
\(c'\)	157,97	74,12	\(c_5\)	11708,74
\(g'\)	129,08	78,16	\(c_6\)	10088,89

Balance:

\[ \begin{aligned} &\begin{array}{r} 10088,89\\ +\,1619,15\\ \hline \end{array} && c_6 + c_3 = 11708,04 && c_5 = 11708,74 \end{aligned} \]

\[ \begin{aligned} &\begin{array}{r} 3705,90\\ -\,594,02\\ \hline \end{array} && c_1 - c_2 = 3111,88 \end{aligned} \qquad \begin{aligned} &\begin{array}{r} 1619,15\\ -\,165,00\\ \hline \end{array} && c_3 - c_4 = 1454,15 \end{aligned} \]

\[ \frac{(c_1-c_2)-(c_3-c_4)}{c_1-c_2} = \frac{3111,88-1454,15}{3111,88}. \]

The last expression must give the corresponding amount of extraction as a percentage of the total quantity of steam entering the machine.

Carrying out the indicated operations, we find:

\[ \frac{(c_1-c_2)-(c_3-c_4)}{c_1-c_2}=0{,}5327. \]

In the table in the text *), where all stages of extraction are indicated, opposite the corresponding value \(m_i\) we have \(53{,}24\%\).

The balance agrees. Consequently, the construction of the indicator diagrams was performed correctly.

Under the extraction conditions considered, as we know, one and the same quantity of steam always enters the small cylinder of the machine, corresponding to the maximum load of the machine, the magnitude of which was determined by us in the preceding chapter as equal to \(11445{,}20\ kg\) for 1 hour of operation **). We know that the stated hourly consumption corresponds to a power developed by the machine of \(1800{,}5\) indicated horsepower.

Knowing the magnitude of the extraction for each stage, as a percentage of the total (gross) quantity of steam entering the high-pressure cylinder, we can easily compute the quantity of steam that has passed through both cylinders of the machine and has been expended exclusively on overcoming the useful resistances represented by the machine-tools of the given production.

The steam extracted from the machine for the various heating apparatuses of the same production, while expanding in the small cylinder from the exhaust pressure \((p_e=11\ Atm.)\) to the pressure in the receiver \((p_r=5{,}21\ Atm.)\), also performs a certain definite amount of work in the machine.

As a result—for each stage of extraction we have at our disposal the total power of the machine, corresponding to a quite definite quantity of steam that has passed through the machine exclusively for the delivery of useful work.

We shall planimeter all the obtained indicator diagrams and determine the work delivered by each of the cylinders separately and by the whole machine in the aggregate.

Then we shall be able to determine how many \(kg\) of steam are consumed per unit—of the power delivered by the whole machine—under the various conditions of extraction.

) See p. 57.
*) See the table on p. 50.

All the quantities obtained from these determinations shall be collected into one table; as before, the volumes will be expressed in $\mathrm{mm}^3$.*)

Table of quantities characterizing the various operating conditions of the machine.

Stages of extraction in %	Stages of extraction in %	Volumes of steam admitted to the large cylinder	Volumes of steam admitted to the large cylinder	Indicated power: small cylinder	Indicated power: large cylinder	Indicated power: whole machine	Hourly steam consumption in kg: total	Hourly steam consumption in kg: per indicated horsepower
$k_0$	0	$q_{11}$	51.864	659.94	1140.60	1800.54	11445.20	6.36
$k_{10}$	7.45	$q_{10}$	48	665.02	1093.64	1758.66	10592.50	6.02
$k_9$	15.16	$q_9$	44	662.35	1032.74	1695.09	9709.80	5.73
$k_8$	22.88	$q_8$	40	657.40	971.83	1629.23	8827.10	5.42
$k_7$	30.59	$q_7$	36	649.79	910.93	1560.72	7944.40	5.09
$k_6$	38.30	$q_6$	32	644.46	850.02	1494.48	7061.70	4.73
$k_5$	46.01	$q_5$	28	642.18	756.38	1398.56	6178.90	4.42
$m k_4$	53.24	$q_4$	24.25	636.85	682.53	1319.38	5351.40	4.06
$k_3$	61.44	$q_3$	20	629.24	586.22	1215.46	4413.50	3.63
$k_2$	69.15	$q_2$	16	624.29	476.97	1101.26	3530.80	3.21
$k_1$	76.67	$q_1$	12	614.00	342.60	956.60	2670.20	2.79

Considering this table, we can note certain features of the operation of the machine.

Whereas, as the quantity of steam extracted from the machine increases, the total power delivered by the machine decreases, the power of the small cylinder changes only slightly. At low degrees of extraction ($k_{10}, k_9$), the power of the high-pressure cylinder even slightly exceeds the limiting power of the same cylinder in the absence of extraction. In contrast, the power of the low-pressure cylinder varies within very wide limits.

The least work that can be obtained from the machine at the highest degree of extraction ($k_1 = 76.67\%$) is equal to 956.60 indicated horsepower, i.e. only 4.34% less than the normal power of the machine.

In Chapter Four, relying on the planimetering of the limiting diagrams for each cylinder, we preliminarily determined the power of the machine at maximum extraction to be equal to 812 indicated horsepower.

) See Appendix*—Tables I, V, and VI.

A detailed study of the machine’s operating conditions shows us that the preliminary calculation could not give a complete picture of the phenomena taking place.

Considering the preliminary diagram of the low-pressure cylinder (Table V), we notice that its area, in comparison with that shown in Table IV, has increased considerably.

If we make a more detailed calculation of the change in the machine’s operation, we shall see that, at maximum extraction, the work corresponding to the preliminary indicator diagram of the small cylinder is only slightly less (\(\infty 7\%\)) than the work of the same cylinder in the absence of extraction; whereas the work of the large cylinder, expressed by the preliminary indicator diagram with extraction, is \(141\%\) greater than that corresponding to the preliminary diagram of the same cylinder without extraction.

For clarity, let us depict the results of the investigation in the form of certain curves.

Fig. 5.

Plotting along the abscissa axis the degrees of steam extraction in % of the total quantity of steam entering the machine (see Fig. 5), and along the ordinate axis the corresponding power, we obtain the general picture of the change in operating conditions in the form of three curves.

Curve I corresponds to the work delivered by the small cylinder. We see that, with an increase in extraction, it only approaches the abscissa axis insignificantly.

Curve II represents the changes in the power delivered by the large cylinder.

We see that changes in extraction are reflected very strongly in the magnitudes of the ordinates of this curve.

Curve III represents the total power of both cylinders and shows us that only at extractions above 74% can we not obtain from the machine its full normal power (1000 forces).

In the same Fig. 5, along the ordinate axis, on a different scale, is plotted the corresponding steam consumption per indicated horsepower per hour at various degrees of extraction.

The resulting curve IV shows that the change in steam consumption proceeds almost in the same way as the change in the total power of the machine.

Fig. 6.

Indeed, plotting on the abscissa axis (Fig. 6) the power developed by the machine at various degrees of extraction, and on the ordinate axis, on a scale of \(1\ \mathrm{kg}=1\ \mathrm{mm}\), the corresponding steam consumptions, we see that curve I deviates only very slightly from a straight line. Therefore we may state the proposition: in a steam engine with steam extraction at constant pressure, the steam consumption is approximately directly proportional to the power developed.

In the same Fig. 6, for comparison, curve II is placed for the steam consumption by the machine without extraction, at the same indicated work delivered.

We have considered the case of steam extraction, as it is called, at constant pressure. In reality we know that this constancy of pressure amounts to the fact that at the moment of pre-release from the small cylinder into the receiver—at all degrees of extraction—

selection—the steam pressure in the receiver must be the same. Owing to the presence of this condition, at the moment when the small cylinder is put in communication with the receiver, one and the same pressure is established in the latter for all stages of selection, which is the maximum receiver pressure.

As for the least pressure of the steam present in the receiver and passing into the selection pipeline, this pressure must obviously depend on the magnitude of the selection.

The least steam pressure in the receiver is determined by the moment of cutoff into the low-pressure cylinder.

The corresponding points have been obtained by us for all stages of selection. Plotting on the axis of abscissas, in Fig. 7, the selection in %, and on the axis of ordinates, on a large scale, the least receiver pressure (the axis of abscissas corresponds to a pressure of 4 atm.), we obtain curve I, which shows us that with an increase in selection the deviations of the pressure in the receiver from the greatest value (line II, parallel to the axis of abscissas) decrease.

Fig. 7.

This position, strange at first sight, is fully confirmed by the working process of the engine as studied by us under the given selection conditions.

Turning to the diagram of Table V, we see that with an increase in selection the degree of filling of the low-pressure cylinder decreases, and thereby the influence of the volume of this cylinder on the change in the steam pressure present in the receiver ceases earlier; this influence,

as we know from the study of the working process of a Tandem machine of the usual type, is expressed precisely by a decrease in the steam pressure.

The earlier the moment of cut-off occurs in the large cylinder, the longer the process of steam compression will take place before the piston of the small cylinder and in the receiver. It is this process of steam compression that moderates the effect of high extraction on the change in pressure of the steam contained in the receiver.

Thus we may state the proposition: with large extractions, fluctuations of the steam pressure in the receiver are insignificant.

In Fig. 7 there is a dotted line—parallel to the axis of abscissae—line III, showing the pressure in the receiver corresponding to the moment of its communication with the small cylinder.

For each degree of extraction we have that, at the moment when exhaust from the small cylinder into the receiver is interrupted, the pressure rises to line II, then gradually falls (along the ordinate corresponding to the same extraction) to curve I, in order to rise again to the pressure represented by straight line III.

In the same Fig. 7, curve IV is shown, representing, on another scale of ordinates, the law of variation of the degree of filling of the large cylinder as a function of the available extraction. We see that the degree of filling varies along a straight line, and therefore, as the result of the investigation carried out, there is one more proposition: the degree of filling of the large cylinder decreases with increasing extraction according to the law of a straight line connecting the corresponding points of zero and maximum extractions.

CHAPTER SEVEN

Study of the operation of a machine when various quantities of steam of variable pressure are extracted from it.

In every production process the amount of energy required to set the working machines in motion fluctuates depending on the course of the production itself.

A steam engine used as a prime mover, adapting itself to the varying magnitudes of the useful resistances, changes its power by a corresponding change in the degree of admission of the high-pressure cylinder.

A change in the degree of admission—with the other phases of steam distribution remaining unchanged (as is usually the case)—entails a change in receiver pressure, by which, as before, we mean the pressure at the moment of communication with the small cylinder.

As a result, the condition of operation of the machine is obtained, which may be formulated as follows: into the receiver of the machine, for each stroke of its piston, there enters, at different admissions, a different quantity of steam of different pressure.

From each of these quantities—depending on the requirements of the heating apparatus of the works—one or another portion of steam is extracted; thus, in contrast with the conditions of the preceding chapter, a quantity of steam enters the low-pressure cylinder that depends both on the amount of extraction and on the load of the machine.

From each quantity of steam, corresponding to a definite degree of admission of the small cylinder, as it passes through the receiver, various quantities may be extracted, varying from zero to a certain maximum determined by the operating conditions of the machine.

Obviously, this maximum must be such that the machine is provided with the limiting indicator diagram of the low-pressure cylinder.

Consequently, the constructive conditions for regulating the operation of the machine must be as follows: the degree of admission of the small cylinder must be increased only insofar as, with the existing extraction, a limiting diagram can be formed in the large cylinder.

Thus, the most important question in investigating the various operating conditions of a machine with extraction of steam at variable receiver pressure is the study of the conditions of maximum extraction for each stage of the quantity of steam entering the high-pressure cylinder.

These stages have already been examined by us in Chapter Five, when we studied the operation of the machine in the absence of extraction from it. The problem now before us is to find the operating conditions of the machine under extraction corresponding to the difference between the variable quantity of steam passing through the small cylinder and the constant quantity entering the large cylinder.

Relying on the various stages of the quantities of steam passing through the machine and analyzed in Chapter Five*), we can compile a table of the limiting cases of extraction under investigation. With the admission into the low-pressure cylinder always of one and the same volume, expressed to scale by the segment 12.1 mm. (corresponding to a pressure of 4 Atm.), the possible cases of extraction are characterized by the following quantities:

Volume of steam passing through the small cylinder	Volume of steam passing through the small cylinder	Possible limiting extraction	Possible limiting extraction: extracted volume, in mm.	Possible limiting extraction: same, in % of the volume of the l.p.c.	Possible limiting extraction: same, in 1 part klg.
$m q_1$	12,1	$k_{11}$	0	0	0
$q_2$	16	$k_{10}$	3,9	24,38	860,64
$q_3$	20	$k_9$	7,9	39,50	1743,35
$n q_4$	24,25	$n k_8$	12,5	50,10	2681,23
$q_5$	28	$k_7$	15,9	56,79	3508,77
$q_6$	32	$k_6$	19,9	62,19	4391,48
$q_7$	36	$k_5$	23,9	66,39	5274,19
$q_8$	40	$k_4$	27,9	69,75	6156,90
$q_9$	44	$k_3$	31,9	72,50	7039,61
$q_{10}$	48	$k_2$	35,9	74,79	7922,32
$q_{11}$	51,864	$m n k_1$	39,764	76,67	8775,00

Since we are now examining the case of various extractions expressed in % of the various initial quantities (which have passed through the small

*) See also the Appendix—Tables I, VIII and IX.

cylinder) quantity of steam, then, in order to make it possible to compare the results of the investigation, the last column of the table gives the hourly quantity in kg of steam (reduced, as at admission, to a pressure of 4 atm.) corresponding to each individual case of extraction.

The case \(max\, k_1\) (extraction equal to \(76.67\%\)—the possible maximum extraction) was considered by us in Chapter Six.

Consequently, we have to investigate only the remaining nine cases, which, as has already been said, are the limiting cases for the nine different degrees of filling of the small cylinder.

Let us proceed to set forth the results of the investigation of the machine.

In the drawing of Table VI the indicator diagrams of both cylinders are constructed together for all the cases of extraction under consideration.

As before, we shall describe only one case, \(n h_8\), corresponding to the normal degree of filling of the high-pressure cylinder.

The indicator diagram of the small cylinder, from the moment compression begins (point \(g'\)) to the moment of preliminary release (point \(b\)), is constructed in the usual way. In the usual way also are constructed the curve of compression of the steam, the exhaust line, and the line of expansion of the steam in the low-pressure cylinder; the last line is constructed starting from the point \(g''\),—this same line, transformed, is transferred to the corresponding strokes of the small cylinder, the origin of coordinates thereby passing to the point \(O_6\).

Below the line \(O_1O_2\) the usual volumetric diagrams of both cylinders are laid out. On them we mark the points corresponding to the already known moments of steam distribution—chiefly the moment of preliminary release (point \(b\)) and the beginning of compression \(g'\) in the high-pressure cylinder.

Let us determine, on the chosen scale, the volume, entering the small cylinder during one stroke of the piston, of the quantity of steam corresponding to its final expansion pressure \(p_e'\), and take a part of this volume equal to the percentage extraction existing for the case under consideration. Then, setting off on the vertical \(O_2\) of the points \(b_2\) and \(\delta_1\), and on the horizontal through this latter point the segment \(\beta_1\delta_1\), representing the existing extraction, we obtain the possibility of drawing on the diagram the straight line \(b_2\delta_1\) of the volumetric influence of the extraction; using it, it is easy to find graphically, and to verify analytically, the positions of the points \(O_3, O_4\), and \(O_5\), corresponding to the given extraction.

We shall also carry out all the further construction of the working process by the grapho-analytical method.

Let us return to the high-pressure cylinder and state, in a few words, everything that occurs in the machine.

At the moment of pre-exhaust release, upon communication with the receiver and the fictitious extraction cylinder, there will occur on the indicator diagram of the small cylinder a pressure drop \(bc\); then, up to the left dead position of the piston, the steam will expand along a hyperbola from the center \(O_3\), determined by the point \(A\) of intersection of the straight lines \((1—2)\) and \(b_2\delta_1\).

In the dead position there will occur a drop in steam pressure \(de'\), caused by the connection of the large cylinder to the former volumes. Then the general expansion of the steam will take place in all the mutually communicating cavities of the engine, both the working ones and the extraction cavity. The corresponding expansion hyperbola \(e'f'\) must be constructed from the center \(O_4\), determined by the point \(B\) of intersection of the lines \((2—3)\) and the line of the total volumes \(\delta_2 f_4\). The point \(f'\) of intersection of this hyperbola with the transformed hyperbola of steam expansion in the large cylinder determines the moment of cut-off \(f''\) in this latter. The hyperbola \(e''f''\) of expansion during the filling period (from the center \(O_4\)) completes the indicator diagram of the low-pressure cylinder.

In the high-pressure cylinder, from the point \(f'\) to the point \(g'\), there proceeds the compression of steam, already known to us, before the piston of the small cylinder, in the receiver, and in the extraction cylinder. The corresponding hyperbola is constructed from the center \(O_5\), determined by the point \(C\) of intersection of the volume lines \((2—3)\) and \(b_2\delta_1\). In this way the indicator diagram of the high-pressure cylinder is also closed. In the drawing, Table VI, the volumes successively occupied by the steam in the various cavities of the engine for the considered case of extraction are indicated by different kinds of hatching.

The points \(f'(f'')\) for all cases of extraction were determined analytically from the conditions that the three corresponding hyperbolas intersect at one point.

Let us draw up, as before, the steam balance for the described case of extraction.

Table of data for the balance).

Points of the diagram	Corresponding volume	Corresponding pressure	Corresponding hyperbola	Corresponding hyperbola
\(a\)	10,60	165,00	\(c_1\)	1749,0
\(g'\)	7,60	38,70	\(c_2\)	294,12
\(g''\)	108,0	8,25	\(c_3\)	891,0
\(r\)	8	20,62	\(c_4\)	165
\(e''\)	157,84	36,93	\(c_5\)	5830,03
\(g'\)	127,60	38,70	\(c_6\)	4938,12

*) See Appendix—Tables X and III.

Balance.

$$ \begin{array}{r} 4938{,}12\\ +\,891{,}00\\ \hline c_6+c_3=5829{,}12 \end{array} \qquad c_5=5822{,}72 $$

$$ \begin{array}{r} 1749{,}0\\ -\,294{,}12\\ \hline c_1-c_2=1454{,}88 \end{array} \qquad \begin{array}{r} 891{,}0\\ -\,165{,}0\\ \hline c_3-c_4=726{,}0 \end{array} $$

$$ \begin{array}{r} 1454{,}88\\ -\,726{,}00\\ \hline (c_1-c_2)-(c_3-c_4)=728{,}88 \end{array} $$

$$ \frac{(c_1-c_2)-(c_3-c_4)}{c_1-c_2} = \frac{728{,}88}{1454{,}88} = 0{,}501, $$

whence we see that the extraction, in %, must be equal to

$$ k=50{,}1\%, $$

which fully corresponds to the value \(m_k\) given in the table for the case under consideration. Consequently, the indicator diagrams have been drawn correctly.

Let us now turn to all the indicator diagrams of the extraction cases under consideration. We shall planimeter them, determine the power delivered by each of the cylinders of the machine and by the whole machine in aggregate. Since, for all extraction cases, a constant hourly quantity of steam passes through the low-pressure cylinder, corresponding to the limiting indicator diagram of this cylinder, already determined by us in Chapter Five as \(2670{,}20\ kg\), it will not be difficult for us to calculate also the steam consumption, in \(kg\), per one indicated horsepower-hour. Having made all the necessary calculations, we shall collect the values found in a table characterizing the operation of the machine under the investigated extraction conditions*).

*) See also Appendix—Table IX.

Hourly steam extraction, in kg	Hourly steam extraction, in kg	Indicated power	Indicated power	Indicated power	Steam consumption, in kg, per one horsepower
		of the small cylinder	of the large cylinder	of the whole machine
\(0^{[[unclear]]}_{t1}\)	0	340,15	142,12	82,27	5,54
\(k_{10}\)	860,64	408,45	193,00	601,45	4,44
\(k_9\)	1743,35	476,97	220,78	697,75	3,83
\(\eta k_8\)	2681,23	530,26	248,57	778,83	3,43
\(k_7\)	3508,77	573,66	276,74	850,40	3,14
\(k_6\)	4391,48	604,11	291,97	896,08	2,98
\(k_5\)	5274,19	621,62	304,53	926,15	2,88
\(k_4\)	6156,90	624,29	314,81	939,10	2,84
\(k_3\)	7039,61	634,56	327,37	961,93	2,78
\(k_2\)	7922,32	626,95	334,98	961,93	2,78
\(k_1\)	8775,00	614,00	342,60	956,60	2,79

The first and last rows of the table are, for the sake of completeness, taken from the preceding chapters.

To obtain an overall picture of the changes in the operating conditions, let us represent graphically the data given in the table.

Fig. 8.

Plotting on the axis of abscissas (Fig. 8) the hourly quantity of steam extracted, in kg, and on the axis of ordinates the corresponding power of the machine, we obtain three curves.

Curve \(I\) corresponds to the power delivered by the high-pressure cylinder. We see that an increase in extraction is possible only with an increase in the admission in the small cylinder, entailing ...

up to a certain limit, an increase of power; from the table we find that, after extraction \(k_3\), the work delivered by the high-pressure cylinder begins to decrease.

Curve \(II\) shows us that the power delivered by the low-pressure cylinder at first increases together with the power of the small cylinder; then this increase becomes less significant, and the curve almost passes into a straight line parallel to the axis of abscissas.

Curve \(III\) shows that an increase of extraction at variable receiver pressure is possible only with an increase of the total power delivered by the engine—again only up to a certain limit; at the highest degrees of extraction the power of the engine begins, although only slightly, to decrease.

Fig. 9.

In Fig. 9, curve \(I\) represents the steam consumption in \(klg.\) per one indicated horsepower per hour. We see that, with an increase in the power of the engine, accompanied by an increase in extraction, the steam consumption decreases considerably. Curve \(II\) in the same Fig. 9 shows the steam consumption per one indicated horsepower at the corresponding power of the same engine, but in the absence of extraction.

The fluctuations of pressure in the receiver are shown in Fig. 10, where line \(I\) corresponds to the pressures at the moment of cut-off of the large cylinder; line \(II\)—to the pressures established in the receiver after communication with the small cylinder; and line \(III\)—to the pressures in the receiver at the moment of the beginning of compression in the small cylinder. The character of these three lines indicates an increase of the corresponding pressures together with an increase of extraction.

In addition, we see that the pressures in the receiver for each given extraction generally vary insignificantly (pressure scale: 1 atm. = 15 mm).

Fig. 10.

Curve IV (Fig. 10) shows the change in the degree of filling of the low-pressure cylinder as a function of the change in the hourly quantity of steam extracted.

We see that, at small extractions, the degree of filling changes considerably more than at high extractions.

CHAPTER EIGHT.

Comparison of the obtained results of the investigation.

In order to make it possible to compare with one another the various cases of changes in the operating conditions of the machine, it is necessary to bring together all the quantities characterizing the amounts of steam—both that entering the high-pressure cylinder and that passing through the low-pressure cylinder.

For the operation of the machine without steam extraction and with extraction at constant receiver pressure, we obtain a table showing, for identical hourly quantities of steam \(G\) entering the small cylinder, the corresponding steam consumption \(Q\), in \(klg.\), falling per one indicated horsepower developed by the machine.*)

\(G\), \(klg.\)	Without extraction \(N_i\)	Without extraction \(q\)	With extraction \(N_i\)	With extraction \(q\)
2670,20	482,3	5,54	956,60	2,79
3530,80	667,3	5,29	1101,26	3,21
4413,50	840,1	5,25	1215,46	3,63
5351,40	1000,0	5,35	1319,38	4,06
6178,95	1137,0	5,43	1398,56	4,42
7061,70	1248,5	5,66	1494,48	4,73
7944,40	1398,0	5,68	1560,72	5,09
8827,10	1497,3	5,90	1629,23	5,42
9709,80	1596,0	6,08	1695,09	5,73
10592,50	1736,0	6,10	1758,66	6,02
11445,20	1800,5	6,36	1800,5	6,36

A table for comparing the operating conditions of the machine with steam extraction, based on the quantity \(Q\)—of steam that has passed through the large cylinder—must characterize the operation of the machine at various quantities \(P\ klg.\) extracted in part.

*) See Appendix.—Table I.

Machine operates with steam extraction	Machine operates with steam extraction	Machine operates with steam extraction	Machine operates with steam extraction	Machine operates with steam extraction	Machine operates with steam extraction
constant pressure.	constant pressure.	constant pressure.	variable pressure.	variable pressure.	variable pressure.
$P$	$N_i$	$Q$	$P$	$N_i$	$Q$
0	1800,5	6,36	0	482,27	5,54
852,69	1758,66	6,02	860,64	601,45	4,44
1735,40	1695,09	5,73	1743,35	697,75	3,83
2618,11	1629,23	5,42	2681,23	778,83	3,43
3500,82	1560,72	5,09	3508,77	850,40	3,14
4333,53	1494,48	4,73	4391,48	806,08	2,98
5266,24	1398,56	4,42	5274,19	926,15	2,83
6093,78	1319,33	4,06	6156,90	939,10	2,84
7031,66	1215,46	3,63	7039,61	961,93	2,78
7914,37	1101,26	3,21	7922,32	961,93	2,78
8775,01	956,60	2,79	8775,01	956,60	2,79

Considering this table, we see that the quantities of steam extracted at the various degrees of extraction differ little from one another for the one case and the other. This is understandable, since the degrees of extraction have been taken as almost identical; these degrees have been chosen by us in accordance with the degrees of the quantities of steam entering the machine without extraction at its various loads.

Let us represent graphically the dependence between the steam consumption and the load of the machine under various conditions of its operation.

[In the graph: vertical axis—“hourly steam consumption in kg per 1 h.p.-hour”; horizontal axis—“indicated power of the machine,” $N_i$; points marked $A$, $B$, $C$.]

Fig. 11.

Let us plot (Fig. 11) on the axis of abscissas, taking as the origin of coordinates the point corresponding to 400 indicated horsepower, on a definite scale all the points obtained in investigating the various loads of the machine; on the corresponding ordinates let us plot quantities proportional to the steam consumption per horsepower-hour.

The curve $AB$ will represent the steam consumption by the machine when operating without extraction, as was shown in Fig. 4.

The curve $AC$—just as in Fig. 9—shows the decrease in steam consumption with an increase in the power of the machine when steam of variable pressure is extracted.

The curve $BC$—repeating the curve of Fig. 6—represents the decrease in steam consumption that appears when steam of constant pressure is extracted together with a decrease in the power of the machine.

Thus, the closed curve $ABC$ shows the following quality of the designed machine: with the greatest hourly extraction, expressed as 8775 kg of steam, the machine can develop 956.60 indicated horsepower, expending per horsepower-hour the least amount of steam, equal to 2.79 kg (point $C$).

For each power developed by the machine, the segment of the ordinate enclosed between the separate branches of the curve shows the economy in steam consumption that can be obtained by applying, for the given load, the corresponding maximum extraction.

In order to obtain an idea of the magnitude of the possible extraction, when using the curve $ABC$ it is necessary to calculate—from the correspondingly taken ordinate—the least required total hourly steam consumption; then the greatest possible quantity of steam that can enter the high-pressure cylinder; the difference between these two quantities will correspond to the extraction.

Thus we see that the curve $ABC$, while characterizing in general the benefit of extraction in the sense of reducing the steam consumption per horsepower-hour, does not give a clear picture outlining the totality of the extraction conditions and of the total power delivered by the machine. Therefore, relying on the same data as before, we shall construct a new curve.

We shall plot (Fig. 12) on the axis of abscissas the magnitudes of the extraction, beginning with zero and ending with the greatest possible; on the ordinates we shall plot the corresponding power developed by the machine.

Fig. 12.

Scale of abscissas: \(75\ \mathrm{kg} = 1\ \mathrm{mm}\); of ordinates: \(20\) indicator horsepower \(= 1\ \mathrm{mm}\).

Plotting the quantities characterizing extraction at constant pressure, we obtain the descending curve \(KL\), showing the decrease in the power of the machine caused by an increase in extraction.

Then, plotting in succession the quantities characterizing the possible—at each given load—maximum extraction at variable pressure, we obtain the ascending curve \(ML\), showing the possibility of increasing the extraction, due to the increase in the power delivered by the machine.

It is clear that both these curves must meet at one common point, characterizing the limiting value of the possible extraction; for the machine investigated we obtained this value equal to \(8775\ \mathrm{kg}\) of steam per hour.

The closed figure \(KLM\) gives us a detailed picture of those varied conditions under which the machine may be placed, corresponding to changing requirements of load and extraction.

Indeed, a segment of any ordinate between the branches of the curve shows us within what limits the power of the engine may vary, in accordance with the existing extraction, expressed by the corresponding abscissa; while the part of the curved figure $KLM$ lying between any two ordinates shows within what limits the power of the engine will vary in accordance with changes in the extraction.

In Fig. 12, for completeness of the picture, a horizontal line has been drawn corresponding to the normal operation of the engine.

Considering the curve $KLM$, we see that, in the absence of extraction, the engine can develop its maximum power (1800 indicated horsepower); gradually, as the quantity of steam extracted at constant pressure is increased, the power of the engine falls, and with an extraction equal to $\sim 8475\ kg.$ of steam per hour (point $N$) the engine supplies the normal amount of work (1000 indicated horsepower); with a further increase of extraction the total power of the engine falls below normal, and finally the limiting maximum extraction is reached, with the corresponding power equal to $956.6$ indicated horsepower. We have reached the greatest power at the greatest extraction.

On the other hand, in the absence of extraction, as the same curve $KLM$ shows us, the engine can operate with a minimum load equal to $482.3$ indicated horsepower.

In order that, without harm to the steam-distributing organs of the low-pressure cylinder, it be possible to extract steam from the engine, it is necessary to increase the load of the engine. Then, with the corresponding change in the degree of filling of the small cylinder, the receiver pressure also changes, as we know.

With a further increase in the extraction of steam of variable pressure, the power of the engine must rise correspondingly, and at an extraction of about $7000\ kg.$ of steam per hour the power of the engine reaches its maximum ($962$ indicated horsepower); then the power falls, reaching at point $L$ again the greatest possible value ($956.6$ hp) at the greatest possible extraction ($8775\ kg.$).

Thus we see that the curve $KLM$ embraces all the variety of operating conditions of the engine possible with proper operation of both cylinders, characterizes the whole permissible regime of the engine, and therefore may be called the curve of the engine’s operating regime under various conditions of steam extraction, or, more briefly, the regime curve.

The point \(L\)—which characterizes the limiting compatible values of power and extraction, which cannot be changed—must be called the limiting point of the operating curve.

The product of the quantities defining the coordinates of the point \(L\) obviously constitutes a certain characteristic number.

Summarizing all that has been investigated, we may draw the following conclusion: the operating curve of the machine makes it possible to establish, depending on the requirements of production, the schedule of the machine’s operation; guided by this schedule, one can bring the machine into full conformity with the conditions imposed on it in every phase of production.

If the demand for the extracted steam exceeds the amount possible at the given load, it is evidently necessary to add to the extraction pipeline, through a throttling valve, steam directly from the boiler.

The curves of variation in the degree of admission of the large cylinder and the curves showing the fluctuations of receiver pressure (see Figs. 7 and 10) provide guiding indications in designing the steam-distribution organs of the low-pressure cylinder and the regulator controlling the admission of steam into this latter cylinder. Practice has shown that varying the pressure in the receiver in accordance with the changing demands of extraction is the most convenient force for acting upon the regulator of the low-pressure cylinder of machines with steam extraction.

CHAPTER NINE.

Review of the investigation carried out.

As the result of our investigation of a definite machine with steam extraction, we obtained the operating curve of the machine.

Let us repeat in a few words the description of the path by which we arrived at this curve.

Relying on the selected ratio of the cylinder volumes, on their clearance spaces, on the volume of the receiver, and on the steam pressures at admission into and exhaust from the machine, we constructed a series of indicator diagrams for various quantities of steam entering the small cylinder of the machine and extracted from the receiver.

In accordance with the dimensions of the machine, determined from the indicator diagrams for its normal operation, we calculated the power delivered by the machine in each separate case of extraction.

The indicator diagram of the normal operation of the machine was constructed by us with a somewhat arbitrarily chosen final expansion pressure in the large cylinder; in exactly the same way, in determining the dimensions of the machine, in general a rather arbitrary mean piston speed \(c_m\) was chosen.

For the choice of the latter quantity, as is known, there exists a series of empirical formulae, which give results that are not wholly consistent.

Let us see whether—in designing a machine with steam extraction—it is possible to avoid this arbitrariness and thereby to set definite limits to the conditions that may be imposed upon the machine.

Relying on the data given above, we can—without determining the dimensions of the machine—construct, by Schröter’s method, the indicator diagrams of the greatest and the least power delivered by the machine in the absence of extraction.

As we saw in Chapter Three, the difference in the quantities of steam consumed—corresponding to the two indicated limiting conditions of operation of the machine—determines the possible greatest extraction.

The quantities of steam, as was indicated in due course, are expressed on the indicator diagrams, for each pressure, by the segments of horizontals between the hyperbolas of expansion and compression.

Having constructed, for each of the machine’s cylinders, its limiting indicator diagram, we can, again without determining the dimensions of the machine, obtain in percent the relative magnitude of the greatest extraction.

Dividing this relative extraction into a series of intervals, we can, relying on the method of accounting for the volumetric effect of the extraction, construct the corresponding indicator diagrams.

By planimetering all these diagrams, we shall obtain the calculated mean indicator pressures, which determine the power of the machine when its dimensions are known, and which are proportional to this power when the dimensions are unknown.

Referring the percentage magnitude of each stage of extraction to a unit volume described by the piston of the low-pressure cylinder of the machine, we obtain, without determining the dimensions of this latter cylinder, a series of quantities proportional to the steam extraction possible—at each stage of load.

In short, without knowing the dimensions of the machine—or rather, knowing its relative dimensions—we are able to construct the regulation curve of the machine being designed, with all its accompanying curves, obtained by us in the investigation carried out.

In all these curves only the pressure scale will be known to us; the volume scale remains arbitrarily chosen—undetermined.

Let us suppose that all that we have just said has been done in designing the machine assigned to us.

Then the indicator diagrams constructed for the normal operation of the machine will represent one of the intermediate stages of extraction investigated.

As before, we draw (Fig. 13) the regulation curve of the machine, plotting on the axis of abscissas the relative magnitudes of the steam extraction at variable and constant pressure, and on the axis of ordinates the magnitudes proportional to the mean indicator pressure, referred, as usual, to a unit area of the piston.

The regulation curve $KLM$ will have the limiting point $L$, characterizing the power of the machine at the greatest extraction, corresponding to the full intensive course of production.

Let us suppose that the study of the requirements of the given production (such a study must always precede the determination of the machine’s operating conditions) has shown that, at the greatest extraction—corresponding to $12000\ klg.$ of steam per hour—the necessary ...

Fig. 13.

for production—the machine tools—an indicated power equal to 1500 horsepower is required, whereas the normal power of the machine, with all the preceding data, must be equal to 1000 horsepower.

We see from what has been said that the requirements of production determine for us the coordinates of the limiting point \(L\) of the operating curve.

If the true magnitudes of the coordinates are known, the scale of the abscissae of the operating curve is known, which determines the volumes described by the piston.

Let us determine the principal dimensions of the machine and the magnitude of the maximum and minimum power delivered by it.

The calculated mean indicated pressure corresponding to the limiting point is determined directly by planimetering the indicator diagrams—in the present case it is equal to \(2.153\ klg.\)*)

We calculate the volume of steam corresponding to the given hourly extraction at a specified pressure (as everywhere in our case—we take \(4\ Atm.\)).

*) See Appendix—Table X.

From the tables, for dry saturated steam we find the specific volume to be \(1\ mtr.^3 = 2.147\ klg.\). Then the hourly volume of the extracted steam will be equal to \(5589.1942\ mtr.^3\).

If the effective piston area of the large cylinder is equal to \(O\ cm^2\), with piston stroke \(S\ mtr.\), then the volume \(Q\) in \(mtr.^3\), swept by the piston in one stroke, will be expressed as follows:

\[ Q=\frac{O\cdot S}{10000}. \]

Since the mean piston speed is

\[ c_m=\frac{S\cdot n}{30}, \]

we can transform the formula for the indicated work of the machine

\[ N_i=\frac{O\cdot p_i\cdot c_m}{75}, \]

giving it the form:

\[ N_i=\frac{p_i\cdot n\cdot S\cdot O}{75\cdot 30\cdot 10000}, \]

or:

\[ N_i=\frac{p_i\cdot n\cdot Q}{75\cdot 30}. \]

Substituting the values known to us, \(N_i=1300\), \(n=120\), and the value \(p_i=2.153\), we obtain the effective volume of the low-pressure cylinder

\[ Q=0.9700\ mtr.^3. \]

On the other hand, we know the hourly volume of the extracted steam, whence, by a simple calculation, we can determine the volume of steam extracted during a certain part of one piston stroke of the large cylinder. Denoting this volume by the letter \(q\), we obtain:

\[ q=\frac{5589.1942}{2\cdot 120\cdot 60}=0.3881\ mtr.^3. \]

We had earlier (see Appendix—Table V) that, with respect to the piston stroke of the large cylinder, taken in Schröter’s diagram as equal to \(100\ mm.\), the greatest possible extraction at a pressure of \(4\ Atm.\) is expressed, with the adopted scale of volumes, by a segment of \(39.764\ mm.\), or, with respect to the volume of the low-pressure cylinder,

\[ q=0.39764Q. \]

Thus, from the steam-extraction conditions we obtain the equality:

\[ 0{,}39764 \cdot Q = 0{,}3881 \, mtr^3., \]

whence

\[ Q = 0{,}9761 \, mtr^3. \]

Comparing the two values found for the useful volume of the large cylinder, we see that they differ very little from one another. Indeed:

\[ \frac{0{,}9761}{0{,}9700} = 1{,}0063. \]

The latter value shows us that, with the selected ratio of the volumes of the cylinders, receiver, and clearance spaces, the required maximum extraction must correspond to a power of the engine of \(1308{,}2\) indicated horsepower, which is an almost exact fulfillment of the assignment.

In the event of obtaining a considerable difference in the determinations of the volume of the large cylinder, these conditions must be recognized as incompatible with the accepted relative volumes of the working cavities of the engine; and, after changing these ratios, indicator diagrams should be constructed that give, for the limiting extraction, the necessary mean indicated pressure. Having the indicator diagram for the limiting extraction, it is easy to construct the corresponding diagrams of the least and greatest power and, from them, to establish the entire operating regime of the engine.

Let us return, however, to the engine we are designing.

To obtain the principal dimensions of the engine—given the volume of the large cylinder—it is necessary to set the mean piston speed \(c_m\).

We shall choose this value on the basis of the consideration that, at the least work of the small cylinder, the correspondingly reduced final compression pressure \(p_c'\) should be sufficient to eliminate the harmful influence of inertial forces causing knocking in the clearances of the engine joints.

From the indicator diagram of the least work of the small cylinder (drawing, table IV) we find the driving working pressure (the difference of pressures on both sides of the piston) in the dead position to be equal to \(1{,}634 \, klg\).

The inertial pressure \(p_b\) in the dead position of the piston has the greatest value, expressed as follows:

\[ p_b = \frac{G}{O \cdot g} \cdot \frac{v^2}{R} (1 + \mu), \]

where

\[ \frac{G}{O} \]

is the weight of all reciprocating parts of the machine, referred to a unit area of the piston;

\(g\) is the acceleration of gravity;

\(\mu = 1/5\) is the ratio of the crank radius \(R\) to the length of the connecting rod;

\[ v = \frac{2\pi R n}{60} \]

is the mean circumferential velocity of the crank pin.

According to Radinger we shall take \(\frac{G}{O} = 0,3\ klg.\), which is a considerable weight; we shall further assume that the active working pressure of the small cylinder is equal to the corresponding inertial pressure (as a reserve we still have the active working pressure of the large cylinder). Express the inertial pressure through the crank radius:

\[ p_i = \frac{G}{O \cdot g}\cdot \frac{4\pi^2 n^2 R}{3600}\cdot \frac{6}{5}. \]

Taking, in view of the slight difference of the quantities,

\[ \pi^2 = 9, \]

then, substituting the numerical values, we have:

\[ 1,634 = \frac{0,3 \cdot 4 \cdot 14400 \cdot R}{3600}\cdot \frac{6}{5}. \]

Whence:

\[ R = 0,284\ mtr., \]

the piston stroke:

\[ S = 2R = 0,568\ mtr., \]

the mean speed:

\[ c_m = \frac{S \cdot n}{30} = 2,272\ mtr. \]

Having the piston stroke, from the previously calculated volume of the large cylinder it is not difficult to determine also the useful piston area \(O\).

\[ O\ mtr^2 = \frac{0,9761}{0,568} = 1,71848. \]

The piston area has turned out, in comparison with that previously calculated for the same machine, somewhat large, owing to the fact that the mean speed, instead of the former value \(3,1\ mtr.\), has been taken reduced to \(2,272\ mtr.\).

To determine the principal dimensions of the machine, we can, relying on the operating curve, calculate both limiting loads of the machine when it operates without steam extraction.

Carrying this out, we obtain, for the lowest power of the machine, a value of 659.53 indicated horsepower; the greatest permissible load, however, takes the value 2461.9 indicated horsepower, i.e., in the absence of extraction the machine can permit an overload of almost 150%.

Operation of the machine with such an overload is excessive and, of course, not very advantageous from the point of view of steam consumption. The designer of the machine, in any case, must know the upper possible limit of the load, in order to coordinate with it the determination of the strength dimensions of the machine parts.

Let us draw in Fig. 13 the horizontal line \(NP\), corresponding—on a scale already known to us—to the normal operation of the machine at 1000 indicated horsepower. Through point \(A\)—the intersection of this straight line with the descending branch of the operating curve \(KLM\)—let us draw the ordinate. Since the given limiting extraction determines the scale of the abscissas, we see that, under normal operation of the machine, 2820.4 kg of steam per hour can be extracted from it; moreover, without decreasing this extraction, we can give the machine a load of 2289.2 indicated horsepower.

The operating conditions of the designed machine are thus clarified in their main features.

If we were to set ourselves the condition that, at the maximum extraction, equal to 12000 kg per hour, the normal power of the machine should be sufficient, then the line \(NP\) would shift on the operating curve to the position \(LS\). It is clear that both the dimensions of the machine and its general operating conditions would accordingly be different.

CHAPTER TEN.

Addendum. Conclusion. General conclusions.

Addendum to Chapters One and Two. If it is desired to obtain a graphical representation of the working process of a machine served by superheated steam, it is necessary to construct the curves of change of state of the steam, relying on the corresponding polytropic indices; moreover, the coordinate origins for constructing these latter curves will be the same as for the equilateral hyperbolas that represent the expansion and compression curves of the working process of machines with dry saturated steam.

Addendum to Chapter Two. Steam working in the various cavities of the machine cannot be recognized as dry saturated, with steam content \(x = 1\), even though it enters the high-pressure cylinder superheated. This phenomenon is noted especially for the low-pressure cylinder of machines with steam extraction from the receiver.

In order to take into account the reduced steam content when constructing indicator diagrams, the corresponding curves of change of state of dry saturated steam must be referred to a new increased volume, so as to obtain, in this new volume of dry saturated steam, the same pressure as in the existing volume occupied by steam of reduced steam content.

Thus it is necessary to multiply all volumes described by the piston by the quantity \((1 : x)\).

For volumetric diagrams expressed by straight lines, the indicated transformation of volumes is performed very easily.

Suppose that we have (Fig. 14) the volumetric diagram \((\alpha—\beta—\gamma)\) of the low-pressure cylinder with the lines of harmful spaces belonging to it. Let us examine the influence of reduced steam content (let \(x = 0.75\)) on the change of the volumes described by the piston in the direction of the line \((\beta—\gamma)\), during the period of filling of the left cavity.

Increasing the volume \((B—\gamma)\) in the ratio \((1 : 0.75)\), we obtain the segment \((B—C) = {}^{4}/_{3}(B—\gamma)\); lay off this quantity from point \(B\) on the horizontal \((B—\gamma)\). Then extend the line \((\beta—\gamma)\) to its intersection with the vertical bounding the harmful space of the cylinder; the point of intersection \(A\) will denote, on the line \((\beta—\gamma)\), the zero volume for any steam content.

Fig. 14.

The straight line \((A-C)\) gives the sought new line of volumes of the low-pressure cylinder, which, as before, by its continuation until it meets the lines of the volume diagrams of the other cavities of the engine—being summed with them when necessary—will determine the origins of the coordinates of the corresponding curves, showing the change in the state of the steam in the engine when allowance is made for the reduced steam content in the large cylinder.

It is clear that, if the magnitudes of the mean steam content are known—in the general case different—for each of the separate cavities of the engine, then, carrying out what has just been said with respect to the volume diagrams of the working cylinders of the engine, including the clearance spaces and the volume of the receiver, as well as with respect to the volume of the fictitious extraction cylinder, we can obtain a picture of the working process that is considerably closer to the actual one.

Supplement to Chapter Nine. Let us take successively from Fig. 12 all the points found on the regime curve \(KLM\) and place them, together with the corresponding grid, on the drawing in Fig. 15. Join pairwise by straight lines the three principal points \(K\), \(L\), and \(M\). Find the center \(O\) of the circle circumscribed about the triangle \(KLM\). Dropping from the point \(O\) perpendiculars to the lines \(KL\) and \(LM\), prolong them toward the concavity of the curves. Then from the vertex of the angle \(B\) of the grid of the regime curve lay off to the left the segment \(AB = LN\), and draw downward a vertical line until it meets one of the perpendiculars at the point \(G\). Subtract from the segment \(AG\) the quantity \(GD = OC\), and through the point \(D\) draw a horizontal line until it meets the other perpendicular at the point \(E\).

If now, from the points \(G\) and \(E\) found in this way, as centers, circles are drawn passing through the points

\((K\ \text{and}\ L)\) and \((L\ \text{and}\ M)\), it will turn out that these circles pass through a series of points obtained earlier in the study of the various stages of extraction.

Thus there emerges the possibility of obtaining—by means of three principal points—the operating curve of the machine, without resorting to the construction of indicator diagrams corresponding to intermediate loads of the machine.

A series of operating curves, when constructed for definite values of the greatest extraction and of the corresponding power, for different relative dimensions of the chambers of the machine, can indicate the significance of the volume of the individual chambers of the machine and their connection with the general conditions of its operation.

A simplification of the method of finding individual points of the operating curve would greatly facilitate the generally laborious investigation of the operation of a machine with steam extraction for each separate case.

Refraining from final conclusions that might be based on the drawing, Fig. 15, the author leaves the questions of elucidating, by means of the operating curve, the influence of the volumes of the various chambers of the machine on the conditions of its operation with steam extraction, as well as the study of further properties of the operating curve and possible methods of its simplified construction, as material for further investigations; and with this he concludes the description of the work undertaken by him and of the results obtained*).

In conclusion, the author considers it necessary to present the principal general conclusions—most of which may be applied to machines without losses—that can be drawn on the basis of the work performed.

1) The use of straight lines as volume diagrams in the study of a Tandem machine and of machines similar to it in working process makes it possible easily and accurately—graphically and analytically—to determine the centers of the curves corresponding to different changes in the state of the steam in the various chambers of the machine.

2) Taking account—in the form of the segments of horizontals between the corresponding curves of change of state—of the quantities of steam passing through the cylinders of a multiple-expansion machine, and also calculating the constant quantities characterizing these curves—with the subsequent compilation of the steam balance—makes it possible to verify

*) The idea of constructing operating curves by means of circles occurred to the author just before the submission of the whole work to the press.

\[ AB = BF = LN \]

\[ AD = HG = RL \]

\[ DG = DS = AP \]

\[ AP = FR = OC \]

Fig. 15.

the correctness of constructing indicator diagrams for all the working cylinders of the machine.

3) In constructing the Schröter diagram for machines with intermediate steam extraction, the introduction into the working process of a fictitious extraction cylinder, as of a kind of double-acting pump, makes it possible—by analogy—to examine all the details of the working process for an arbitrary possible extraction.

4) Considering the action of the extraction cylinder on the quantity of steam that has entered only one of the cavities of the high-pressure cylinder, we obtain the possibility, by analogy, of passing to an account of the actual volumetric influence corresponding to the steam extraction existing in the machine.

5) A Tandem machine operating with steam exhaust at atmospheric pressure does not permit considerable fluctuations of load and does not make it possible to vary, within broad limits, the quantity of steam extracted from it.

6) A Tandem machine with a high vacuum is of little suitability, from the point of view of utilizing—the heat of the steam not converted by the low-pressure cylinder into work—since the water leaving the condenser of the machine has a low temperature.

7) A machine with a small back pressure (about \(0.5\ \mathrm{Atm}\)), yielding from the condenser hot water (about \(80^\circ C\)) suitable for many technical purposes, at the same time permits sufficiently wide fluctuations of load and considerable intermediate extraction of steam.

8) Limiting indicator diagrams for each of the working cylinders of the machine, while ensuring calm operation of the steam-distributing organs, at the same time determine the possible maximum extraction of steam.

9) Maximum extraction is carried out at an unchanged receiver pressure.

10) Steam extraction below the maximum may be carried out both at constant and at variable receiver pressure.

11) To each quantity of extracted steam there correspond definite possible fluctuations of the machine load.

12) The construction of the indicator diagrams of both cylinders of the machine for each individual case of extraction makes it possible to determine the influence of extraction on the steam consumption per one indicated horsepower-hour.

13) The construction of indicator diagrams for both cylinders, for each case of extraction, makes it possible to determine the design conditions for the steam-distribution organs of the machine.

14) The study of fluctuations of receiver pressure under various cases of extraction, in connection with the necessary changes in the degree of filling of the large cylinder, makes it possible to determine the design conditions for the regulator serving the steam-admission ports of the low-pressure cylinder.

15) The mutual relation between steam extraction and the work delivered by the machine is clearly represented by the operating curve of the machine.

16) The operating curve has three principal points, characterizing the greatest and the least power of the machine in the absence of extraction, and the greatest power corresponding to the greatest extraction.

17) The segments of the ordinates enclosed between the branches of the operating curve show the possible—at unchanged extraction—fluctuations in the load of the machine. The branches of the operating curve correspond to two qualitative conditions of extraction.

18) The portion of the operating curve enclosed between its two branches and any two ordinates shows the possible changes in the power of the machine when the extraction changes (qualitatively and quantitatively) within the limits corresponding to the ordinates taken.

19) The horizontal line of the normal operation of the machine, by its intersection with the operating curve, establishes the limits of the correspondingly possible extraction.

20) The operating curve, when plotted for averaged relative dimensions of the separate cavities of the machine, makes it possible—given the requirements of production: maximum extraction and the corresponding necessary power of the machine—to determine its principal dimensions.

21) A series of operating curves, obtained for one definite specification of the operating conditions of the machine, when constructed for various relative dimensions of the machine cavities, can give the designer guidance as to the most expedient choice of the receiver volume and of the volumes of the machine cylinders.

22) The indicated possibility of a simplified construction of the operating curve, by eliminating the necessity of drawing a large number of indicator diagrams, will thereby facilitate the study of various operating conditions of machines with intermediate steam extraction.

Brief index of periodical literature of the last five years on the question of piston steam engines with intermediate steam extraction.

Bulletins of the Polytechnic Society.

1. Kh. Vodoginskii.—Construction of the theoretical indicator diagram of a tandem engine using steam from the receiver.

The years 1909, 1910, and 1912 contain a number of translated and compiled articles with indications of foreign sources.

Journal of the Society of Siberian Engineers.

1. A. Ugarov.—Graphical representation of the working process of an engine with intermediate steam extraction.

Mémoires et compte rendu des travaux de la Société des ingénieurs civils.

1. Beauviscime.—Production simultanée d’énergie électrique et de chaleur.

Revue de mécanique.

1. Lecuir.—Note sur le chauffage à l’aide de la vapeur de décharge des machines monocylindriques ou le vapeur prise au receiver des machines compounds.

Engineering News.

1. Peek.—Bleeding steam from receivers of triple expansion engines for heating feed water.

Gesundheits-Ingenieur.

1. Brabbée.—Forschungsarbeiten der Prüfungsanstalt für Heizungs- und Lüftungseinrichtungen.

Zeitschrift des Bayerischen Revisions-Vereins.

1. Eberle.—Einfluss des Gegendruckes und der Zwischendampfentnahme auf den Dampfverbrauch von Kolbenmaschinen.

2. Kammerei.—Einige Untersuchungs-Ergebnisse von Maschinen und Turbinen mit Gegendruck und Zwischendampfentnahme.

Zeitschrift des Vereines deutscher Ingenieure.

1. Hottinger.—Some steam-power plants with utilization of waste heat.
2. Pfleiderer.—The single-cylinder engine with intermediate steam extraction.

Detailed references to the literature on the use, for purposes of technical heating, of exhaust and intermediate steam from steam engines are available in the following reference publications:

Fortschritte der Technik,
Repertorium der technischen Journal-Litteratur,
Revue de l’Ingénieur et index technique.

Appendix

Data characterizing the operating conditions and principal dimensions of the machine investigated.

$N_i$ — number of indicated horsepower $=1000$,

$n$ — “ revolutions of the machine shaft per minute $=120$,

$p_e$ — pressure of the steam admitted into the machine $=11$ Atm. abs.,

$p_a$ — “ pressure of the steam leaving the machine $=0.55$ Atm. abs.,

$c_m$ — mean piston speed per second $=3.1$ mtr.,

$S$ — piston stroke $=0.775$ mtr.,

$O_2$ — piston area of the low-pressure cylinder $0.921$ mtr.$^2$,

$O_2 \times S$ — useful volume of the low-pressure cylinder $=$
$=0.713775$ mtr.$^3$,

$O_1 \times S$ — useful volume of the high-pressure cylinder $=$
$=0.317230$ mtr.$^3$

$\varphi$ — ratio of the cylinder volumes $O_2 : O_1 = 2.25$,

$R$ — receiver volume $=0.713775$ mtr.$^3$,

$m_1, m_2$ — volumes of the clearance spaces of the corresponding
cylinders $=8\%$.

$\mu$ — scale of volumes for Schröter’s diagrams in the drawings
(Tables IV, V, VI) $1$ mm. $=7137.75$ cm.$^3$

The segment $O_1O_3$ (drawing, Tables III and IV) is equal to $172.84$ mm.

The volume of steam entering, during one piston stroke, the small cylinder, at the greatest power developed by the machine $(N_i = 1800.5)$, reduced to a pressure of $4$ Atm., is equal to $0.370192$ mtr.$^3$; under normal operation of the machine $(N_i = 1000)$ this volume becomes equal to $0.173090$ mtr.$^3$ and decreases, at the smallest output $(N_i = 482.27)$, to $0.086367$ mtr.$^3$. Thus the greatest possible extraction during one piston stroke is $0.370192 - 0.086367 = 0.283825$ mtr.$^3$. Since $1$ mtr.$^3$ of dry saturated steam at $4$ Atm. weighs $2.147$ klg., the greatest hourly steam extraction in klg. is:

$$ G_{max}=0.283825.2.120.60.2.147=8775\ \text{klg}. $$

Fig. 16.

Fig. 17.

Amounts of steam entering the engine—without extraction—at its various loads, calculated in kg per hour ($G$) and in volumes per stroke—($Q\ mtr.^3$), reduced to a pressure of 4 atm.

Volumes are expressed according to the scale of Schröter’s diagram in mm.

Table I.

Designation of stages	$G\ kg.$ per hour	$Q$—per one stroke, $mtr^3$	$Q$—per one stroke, $mm.$
$min\, q_1$	2670,2	0,086367	12,1
$q_2$	3530,8	0,114204	16
$q_3$	4413,5	0,142755	20
$nrm\, q_4$	5351,4	0,173090	24,25
$q_5$	6178,9	0,199857	28
$q_6$	7061,7	0,228408	32
$q_7$	7944,4	0,256959	36
$q_8$	8827,1	0,285510	40
$q_9$	9709,8	0,314061	44
$q_{10}$	10592,5	0,342612	48
$max\, q_{11}$	11445,2	0,370192	51,86

Values relating to the indicator diagrams of the engine without steam extraction (drawing, Table IV), at its various loads.

Designation of the points determining the volumes and pressures, according to the sketch in Fig. 16.

Table II.

Constant values.

Pressures	$Atm.$	On the scale	Volumes in mm of the diagram	Volumes in mm of the diagram	Volumes in mm of the diagram	Volumes in mm of the diagram
$p_2$	11	$165,00\ mm.$	$S_1$	44,44	$b_1O_1$	44,04
$p_u$	0,55	$8,25\ \text{”}$	$m_1$	3,6	$g_1O_1$	7,6
$p_c''$	1,375	$20,63\ \text{”}$	$S_2$	100	$f_2O_2$	50,35
			$m_2$	8	$f_1f_2$	179,60
			$R$	100	$f_1O_2$	129,24

Table III.
Quantities used in constructing the machine.

Stage of steam quantity	\(p' = g'\), atm.	\(p' = g'\), mm	\(p'_c\), atm.	\(p'_c\), mm	\(a\), volume in mm	\(b\), atm.	\(b\), mm	\(c\), atm.	\(c\), mm	\(d\), atm.	\(d\), mm	\(e' = e''\), atm.	\(e' = e''\), mm	\(f' = f''\), atm.	\(f' = f''\), mm	\(g'\), atm.	\(g'\), mm	Calculated mean indicator pressure, \(p_i\)
min. \(q_1\)	1,42	21,30	3,0	45	5,38	1,34	20,10	1,4	21,0	1,36	20,40	1,36	20,40	1,18	17,70	0,55	8,25	1,267
\(q_2\)	1,79	26,85	3,78	56,70	7,05	1,76	26,40	1,78	26,70	1,73	25,95	1,71	25,65	1,49	22,35	0,69	10,35	1,753
\(q_3\)	2,17	32,55	4,58	68,70	8,77	2,19	32,85	2,18	32,70	2,12	31,80	2,08	31,20	1,81	27,15	0,84	12,60	2,207
norm. \(q_4\)	2,58	38,70	5,45	81,75	10,60	2,65	39,75	2,53	37,95	2,55	37,05	2,47	37,05	2,14	32,10	1,0	15,0	2,617
\(q_5\)	2,93	43,95	6,19	92,85	12,21	3,05	45,75	2,97	44,55	2,89	43,35	2,81	42,15	2,44	36,60	1,14	17,10	2,987
\(q_6\)	3,32	49,80	6,99	104,85	13,92	3,48	52,20	3,37	50,55	3,27	49,05	3,18	47,70	2,76	41,40	1,29	19,35	3,280
\(q_7\)	3,70	55,50	7,84	117,60	15,64	3,91	58,65	3,76	56,40	3,66	54,90	3,54	53,10	3,08	46,20	1,44	21,60	3,673
\(q_8\)	4,08	61,20	8,61	129,15	17,36	4,34	65,10	4,16	62,4	4,0	60,75	3,91	58,65	3,40	51,0	1,58	23,70	3,930
\(q_9\)	4,46	66,90	9,41	141,15	19,09	4,77	71,55	4,55	68,25	4,43	66,45	4,27	64,05	3,71	55,65	1,73	25,95	4,193
\(q_{10}\)	4,84	72,60	10,22	153,30	20,80	5,20	78,0	4,95	74,25	4,82	72,30	4,64	69,60	4,03	60,45	1,88	28,2	4,560
max. \(q_{11}\)	5,21	78,158	11,0	165,0	22,46	5,61	84,15	5,33	79,85	5,19	77,95	4,99	74,85	4,34	65,10	2,02	30,30	4,730

Data relating to the indicator diagrams (drawing, Table V) of a machine with intermediate steam extraction at constant receiver pressure.

Names of the points defining the volumes and pressures, according to the sketch in Fig. 17.

Table IV

Quantities remaining unknown.

Pressure	Pressure, atm.	Pressure, mm.	Volume in mm, diagrams	Volume in mm, diagrams
$p_e$	11,0	165,0	$a'O_1$	22,46
$p_a$	0,55	8,25	$a\delta$	51,86
$p_c'=p_e$	11,0	165,0	$g'O_2$	107,6
$p_r$	5,21	78,15	$b'O_1$	44,04
$p_c''$	1,375	20,63

The stages of steam extraction are determined as the difference between $maxq_{11}$ and each smaller quantity of steam entering the large cylinder of the machine.

Table V

Stages of steam extraction at constant receiver pressure.

(Volumes for one piston stroke are referred to 4 atm.).

Name of stages	Name of stages	Extraction volume for one stroke, in mm.	Extraction in % of $maxq_{11}$
$max\ k_1$	$q_{11}-q_1$	39,764	76,67
$k_2$	$q_{11}-q_2$	35,834	69,15
$k_3$	$q_{11}-q_3$	31,864	61,44
const. $k_4$	$q_{11}-q_4$	27,614	53,24
$k_5$	$q_{11}-q_5$	23,864	46,01
$k_6$	$q_{11}-q_6$	19,864	38,30
$k_7$	$q_{11}-q_7$	15,864	30,59
$k_8$	$q_{11}-q_8$	11,864	22,88
$k_9$	$q_{11}-q_9$	7,864	15,16
$k_{10}$	$q_{11}-q_{10}$	3,864	7,45
${}^{0}k_{11}$	$q_{11}-q_{11}$	0	0

Table VI.

Values used in compiling the selection.

Extraction stage	Volume of diagrams, mm: \(\beta\)	Volume of diagrams, mm: \(d_1d_2\)	Volume of diagrams, mm: \(f_2f_3\)	Volume of diagrams, mm: \(f_1f_4\)	Volume of diagrams, mm: \(g_1\delta_1\)	Points of the saturation diagram: \(d\), Atm.	Points of the saturation diagram: \(d\), mm.	Points of the saturation diagram: \(c'=c''\), Atm.	Points of the saturation diagram: \(c'=c''\), mm.	Points of the saturation diagram: \(f'=f''\), Atm.	Points of the saturation diagram: \(f'=f''\), mm.	Points of the saturation diagram: \(g''\), Atm.	Points of the saturation diagram: \(g''\), mm.	Calculated mean indicated pressure, \(p_i\)—kg.
\(h_1\)	30,93	150,82	12,38	162,53	138,63	5,09	76,35	4,910	73,65	4,80	71,97	0,55	8,25	2,513
\(h_2\)	27,90	150,55	15,79	165,07	135,50	5,10	76,50	4,914	73,71	4,72	70,80	0,69	10,35	2,893
\(h_3\)	24,78	150,27	19,46	167,46	132,38	5,11	76,65	4,933	74,0	4,66	69,90	0,84	12,60	3,193
\(h_4\)	21,48	149,97	23,51	169,94	129,08	5,12	76,80	4,941	74,12	4,59	68,85	1,00	15,00	3,466
\(h_5\)	18,56	149,71	27,10	171,88	126,16	5,13	76,95	4,948	74,22	4,54	68,10	1,14	17,10	3,674
\(h_6\)	15,45	149,43	31,02	173,72	123,05	5,14	77,10	4,956	74,34	4,49	67,35	1,29	19,35	3,923
\(h_7\)	12,34	149,15	34,97	175,33	119,94	5,15	77,25	4,963	74,45	4,45	66,75	1,44	21,60	4,10
\(h_8\)	9,23	148,87	38,67	176,71	116,83	5,16	77,40	4,971	74,57	4,41	66,15	1,58	23,70	4,280
\(h_9\)	6,12	148,59	42,66	177,99	113,72	5,17	77,55	4,978	74,67	4,38	65,70	1,73	25,95	4,453
\(h_{10}\)	3,00	148,31	46,62	178,93	110,6	5,18	77,70	4,986	74,79	4,36	65,40	1,88	28,20	4,620
\(o_k t\)	0	148,04	50,48	179,75	107,6	5,19	77,85	4,990	74,85	4,35	65,25	2,02	30,35	4,733

To construct the lines of volumetric influence of the extraction, it is necessary to know the volume \(\alpha\beta\) passing, during one stroke of the piston, through the small cylinder of steam at its final expansion pressure \(p_\varepsilon'\). For the extraction under consideration—at a constant degree of admission \(\varepsilon\) of the small cylinder—\(p_\varepsilon'=5.14\) atm. Consequently, the values of \(k\) from Table V must be changed in the ratio \((4:5.14)\) (see Fig. 17); doing this, we obtain the numerical values of the extraction quantities \(\beta\delta=a\delta-\alpha\beta\). (Table VI).

Data relating to the indicator diagrams (drawing, Table VI) of a machine with steam extraction at variable receiver pressure.

Names of the points defining the volumes and pressures, according to the sketch in Fig. 17.

Table VII.

Quantities remaining unchanged.

Pressures	Pressures	Pressures	Volumes in mm. of the diagram (at 4 atm.)	Volumes in mm. of the diagram (at 4 atm.)
—	Atm.	mm.
\(p_\varepsilon\)	11,0	165,00	\(\gamma_\mu\)	12,1
\(p_a\)	0,55	8,25	\(y_1 O_2\)	107,6
\(p_\varepsilon''\)	1,375	20,63	\(b' O_1\)	44,04
\(p_\varepsilon''=p_a\)	0,55	8,25	—	—

The degrees of extraction are determined as the differences between \(qx\)—the quantities of steam entering, during one stroke of the piston, the small cylinder—and \(\min q_1\), the quantity entering unchanged into the large cylinder of the machine. As in all preceding tables, unless otherwise stated, the quantities of steam are expressed in mm. of the scale of the Schröter diagram and are referred to a pressure of 4 atm.

Table VIII.

Degrees of steam extraction at variable receiver pressure.

Degrees	Difference between quantities of steam	Volumes in mm.
\(\max k_1\)	\(q_{11}-q_1\)	39,76
\(k_2\)	\(q_{10}-q_1\)	35,90
\(k_3\)	\(q_9-q_1\)	31,90
\(k_4\)	\(q_8-q_1\)	27,90
\(k_5\)	\(q_7-q_1\)	23,90
\(k_6\)	\(q_6-q_1\)	19,90
\(k_7\)	\(q_5-q_1\)	15,90
\(k_8\)	\(q_4-q_1\)	12,15
\(k_9\)	\(q_3-q_1\)	7,90
\(k_{10}\)	\(q_2-q_1\)	3,90
\(k_{11}\)	\(q_1-q_1\)	0

Table IX.

Extraction quantities for each stage: \(Q\), expressed in % of the quantity of steam entering the small cylinder during one piston stroke, and \(G\)—in kg for one hour of operation of the machine.

Extraction stage	\(Q=\%\)	\(G=kg.\)
\(k_1\)	76,67	8775,0
\(k_2\)	74,79	7922,32
\(k_3\)	72,50	7029,61
\(k_4\)	69,75	6156,90
\(k_5\)	66,39	5274,13
\(k_6\)	62,19	4391,48
\(k_7\)	56,79	3508,77
\(k_8\)	50,10	2681,23
\(k_9\)	39,50	1743,35
\(k_{10}\)	24,38	860,64

For constructing on Schröter’s diagram the lines of the volumetric influence of extraction, it is necessary, corresponding to each extraction stage, to determine the volume \(a\beta\) (sketch, Fig. 17) of the quantity of steam passing through the small cylinder during one piston stroke at its final expansion pressure \(p'_e\). Since the degree of filling of the high-pressure cylinder varies together with the extraction stages, in order to obtain the numerical value of the quantities \(\beta\delta\), it is necessary to calculate the corresponding values of \(p'_e\), so as then to change the quantities \(kx\) and \(a\beta\), expressed in mm of the diagram scale, in the ratio \((4 : p'_e)\).

Table X.

Quantities varying with the stages of extraction of superheated steam.
Initial state according to the sketch in Fig. 17.

Extraction stage	$n'_e$, atm.	$\dfrac{\Delta}{p'_e}$	$\delta$	$\beta$	$d_1 d_2$	$f_2/f_3$	$f_3/f_4$	$f/f_4$	$g/\delta$	$c$, $\Delta tm.$	$c$, mm	$d$, $\Delta tm.$	$d$, mm	$e'=e''$, $\Delta tm.$	$e'=e''$, mm	$f'=f''$, $\Delta tm.$	$f'=f''$, mm	$g'-p_e$, $\Delta tm.$	$g'-p_e$, mm	$n$	$p'_e$, mm	Calculated mean pressure, $P'_i = L_i/V_0$
$h_1$	5,14	0,778	40,34	30,93	150,82	12,38	162,52	138,53	5,33	79,95	5,09	76,35	4,910	73,65	4,80	71,87	5,21	78,15	11,00	165,00	2,513
$h_2$	4,76	0,840	40,32	30,16	150,75	13,35	163,18	137,76	4,95	74,25	4,73	70,95	4,574	68,61	4,45	66,75	4,84	72,60	10,22	153,30	2,527
$h_3$	4,37	0,915	40,26	29,19	150,67	14,56	164,11	136,79	4,55	68,25	4,35	65,25	4,219	63,29	4,08	61,20	4,46	66,90	9,41	141,15	2,527
$h_4$	3,96	1,005	40,24	23,07	150,57	16,01	165,21	135,67	4,16	62,40	3,98	59,70	3,865	57,98	3,71	55,65	4,08	61,20	8,61	129,15	2,467
$h_5$	3,58	1,117	40,22	26,70	150,44	17,78	166,56	134,30	3,76	56,40	3,60	54,00	3,511	52,67	3,34	50,10	3,70	55,50	7,84	117,60	2,433
$h_6$	3,19	1,254	40,12	24,95	150,29	20,00	168,17	132,55	3,37	50,55	3,23	48,45	3,155	47,33	2,97	44,55	3,32	49,80	6,99	104,85	2,354
$h_7$	2,80	1,429	40,01	22,72	150,09	22,85	169,71	130,32	2,97	44,55	2,85	42,75	2,791	41,87	2,60	39,00	2,93	43,95	6,19	92,85	2,234
$h_8$	2,43	1,646	39,92	20,00	149,84	26,23	171,98	127,60	2,53	37,95	2,50	37,50	2,462	36,93	2,26	33,90	2,58	38,70	5,45	8,75	2,046
$h_9$	2,01	1,990	39,80	15,72	149,46	31,60	173,94	123,32	2,18	32,70	2,10	31,50	2,077	31,15	1,88	28,20	2,17	32,55	4,58	68,70	1,833
$h_{10}$	1,62	2,469	39,50	9,63	148,91	39,10	177,81	117,23	1,78	26,70	1,72	25,80	1,719	25,68	1,52	22,78	1,79	26,85	3,78	56,70	1,580

Many of the analytical quantities given, in correspondence with the hyperbolas \(O_4\), \(O_3\), and \(O_2O_5\) (Fig. VI), characterizing in particular the coordinates, transform into a specially selected state of the steam, for each stage of extraction considered.

Table XI.

Quantities, in mm, determining the origin of the coordinates of the hyperbolas for various cases of steam extraction.

\[ \left( \begin{array}{l} n_c \text{—extraction at constant pressure}\\ n_p \text{—extraction at variable pressure} \end{array} \right) \]

Extraction stage	Extraction	\(k_1\)	\(k_1\)	\(k_2\)	\(k_2\)	\(k_3\)	\(k_3\)	\(k_4\)	\(k_4\)	\(k_5\)	\(k_5\)	\(k_6\)	\(k_6\)	\(k_7\)	\(k_7\)	\(k_8\)	\(k_8\)	\(k_9\)	\(k_9\)	\(k_{10}\)
Extraction stage	Extraction	\(n_c\)	\(n_p\)	\(n_c\)	\(n_p\)	\(n_c\)	\(n_p\)	\(n_c\)	\(n_p\)	\(n_c\)	\(n_p\)	\(n_c\)	\(n_p\)	\(n_c\)	\(n_p\)	\(n_c\)	\(n_p\)	\(n_c\)	\(n_p\)	\(n_c\)
\(O_4\)		129,65	130,35	132,47	131,22	135,59	132,30	139,16	133,66	142,60	135,41	146,57	137,75	150,89	140,84	155,63	146,21	160,86	154,96	166,66
\(O_3\)		40,90	41,78	44,47	42,88	48,44	44,25	53,07	45,95	57,57	48,21	62,85	51,23	68,70	55,26	75,24	62,37	82,53	74,30	90,90
\(O_2O_5\)		348,08	320,88	236,22	291,60	194,60	260,71	142,24	228,36	109,04	194,69	81,03	159,57	58,45	121,73	39,86	83,21	24,28	42,21	10,91

Graphical representation of the working process of a tandem machine of the ordinary type

[A large schematic diagram occupies the page. Visible labels in the diagram include: \(Q_1\), \(Q\), \(O_0\), \(\alpha\), \(\beta\), \(O_3\), \(O_1\), \(S_1\), \(S_0\), \(R\), \(m_1\), \(m_2\), \(q x^0\), \(a\), \(b\), \(c\), \(d\), \(d_1\), \(e\), \(f\), \(g\), \(g'\), \(g''\), \(i'\), \(s\), \(C\), \(C_5\), \(C_6\), \(v_1\), \(v_2\), \(p'_3\), \(p'_c\), \(p_e\), \(p'_1\), \(p_x\), \(p_t\), \(p'_t\), \(p''_c\), and points numbered \(1\), \(2\), \(3\), \(4\).]

Table I.

Drawing 1.

Drawing 2.

In relation to drawing 1, the pressure scale is enlarged; the dimensions of the volumes are distorted non-proportionally.

Graphical representation of the working process of a machine with intermediate steam extraction

Fig. 1.

Fig. 4b.

Fig. 4a.

Fig. 3a.

Plate II.

Diagram 2.

Diagram 3ᵇ.

Diagram 3ᵛ.

Fig. 1.

Pressure scale:

\(1\ \mathrm{atm.} = 15\ \mathrm{mm}.\)

Visible diagram labels include: \(S_0\), \(S_1\), \(S_2\), \(p_c'\), \(p_c''\), \(p_e\), \(p_a = 1.1\ \mathrm{atm.}\), \(p_e'' = 1.55\ \mathrm{atm.}\), \(p_e = 4\ \mathrm{atm.}\), \(m_1\), \(m_2 = 8\%\), \(O\), \(O_3\), \(O_5\), \(A\), \(x'\), \(x\), \(a_1\), \(t\), \(t_1\), \(u_1\), \(u_n\), \(c_5\).

Working process of a tandem-compound machine without cooling—at three loads.

Plate III.

Drawing 2.

Calculated mean indicated pressure in kg, \(p_i\).	Calculated mean indicated pressure in kg, \(p_i\).	Calculated mean indicated pressure in kg, \(p_i\).	Calculated mean indicated pressure in kg, \(p_i\).
Load of the machine	Small cylinder	Large cylinder	\(p_i=p_i' + p_i''\)
Lowest	\(\min p_i' = 1{,}440\)	\(\min p_i'' = 0{,}810\)	\(\min p_i = 2{,}250\)
Normal	\(n p_i' = 1{,}625\)	\(n p_i'' = 1{,}600\)	\(n p_i = 3{,}225\)
Highest	\(\max p_i' = 1{,}600\)	\(\max p_i'' = 2{,}450\)	\(\max p_i = 4{,}050\)

Working Process of a Tandem Machine — at Various Loads

Pressure scale:
\(1\ \mathrm{atm} = 15\ \mathrm{mm}\).

Visible diagram annotations include:
\(p_e = 11\ \mathrm{atm}\); \(a_{\max}\), \(a_n\), \(a_{\min}\); \(C_1\), \(C_2\), \(C_5\), \(C_6\); \(e'_{\max}\), \(e'_n\), \(e'_{\min}\); \(g_{\max}\), \(g'_n\), \(g'_{\min}\); \(O_1\), \(O_3\), \(A\); points \(1\), \(2\), \(3\), \(4\); \((21{,}30)\).

with a low vacuum—
exhaust.

Table IV.

Machine load	Small cylinder	Large cylinder	\(p_i' + p_i''\)
Minimum.	\(\min p_i' = 0.867\)	\(\min p_i'' = 0.400\)	\(\min p_i = 1.267\)
Normal.	\(n p_i' = 1.447\)	\(n p_i'' = 1.180\)	\(n p_i = 2.627\)
Maximum.	\(\max p_i' = 1.733\)	\(\max p_i'' = 2.997\)	\(\max p_i = 4.730\)

Calculated mean indicated pressure in kg/cm².

[Diagram with visible labels: \((78.16\ \mathrm{mm})\), \(p = 4\ \mathrm{atm.}\ (60\ \mathrm{mm})\), \(3.70\ \mathrm{mm}\), \(O_2\), \(m_2\), \(12\%\), \(e''\), \(a\), \(b\), \(d\), \(f\), \(g\), \(C_3\), \(C_4\), \(h\), \(p_c = 0.55\ \mathrm{atm.}\), \(g_{\max}''\), \(g_m''\), \(g_{\min}''\), \(p_c = 1\ \mathrm{atm.}\), \(4\), \(5\), \(6\).]

Operating process of the Bandel machine with extraction from the nozzle of a variable quantity of steam at constant pressure.

[Diagram labels visible in the figure include: \(M_e\), Machine \(10\) atm., \(p_c' = p_e = 10\) atm., \(p_e' = 5.14\) atm., \(p_c = 5.21\) atm., and point/curve labels \(a, b, c, d, e, f, g, C_1, C_2, C_6, C_7, O_1, O_4, O_6\), etc.]

Diagram 2.
Corresponding indicator diagram of the extraction cylinder.

Plate V.

Drawing 1.

main pressure

\(n = 15\ \mathrm{mm}\).

[Diagram with labels: \(p_k = 4\ \mathrm{atm.}\), \(p_a = 0.55\ \mathrm{atm.}\), \(p''_c\), \(C_3\), \(C_4\), and points marked \(a, \beta, d, e'', f'', g'', \gamma, O_2, O_5, \beta', \beta_1, \beta_2, \beta_3, \beta_0, a_1, d_1, d_2, f_2, f_3, f_4, 4, 5, 6\).]

[Diagram only; no caption visible. Visible annotations include: “Working,” “by,” \(p_e = 14\ \mathrm{atm.}\), and point/curve labels \(a\), \(c_1\), \(c_2\), \(c_5\), \(c_6\), \(d\), \(e\), \(g\), \(g'\), \(g_1\), \(n\), \(O_1\), \(O_2\), \(O_6\), \(b\), \(\beta\), \(B\), \(1\), \(2\), \(3\), \(4\), \(\gamma_3\).]

Table VI.

Process of a Tandem machine with extraction from the receiver of an economical quantity of steam at variable pressure.

Pressure scale:

\(1\ \text{atm.} = 15\ \text{mm}\).

[Diagram with points and curves labeled \(A\), \(O_3\), \(O_a\), \(O_b\), \(C_3\), \(C_4\), \(e''\), \(f''\), \(g''\), \(p_a = 0.55\ \text{atm.}\), \(4\), \(5\), \(6\), \(f_1\), \(f_2\), \(f_3\), \(\beta_1\), \(\beta_2\), \(\delta_1\), \(\delta_2\), \(C\), and other construction lines.]