Proceedings

of the Tomsk Technological Institute

of Emperor Nicholas II,

Vol. 9, 1908, No. 1.

III.

A. M. Krylov.

THEORY AND CALCULATION OF THE INJECTOR.

Chapters I, II, III, with an appendix of 6 plates of drawings, I–VI, 1–68.

Preface.

Despite the wide use of the injector for feeding factory boilers, it has firmly retained among engineers and technicians the reputation of a device that is highly capricious, unreliable, and therefore of little suitability for this purpose. The latter is made dependent on the impossibility of feeding a boiler with it regularly and in accordance with the current steam consumption. On the contrary, to the fullest extent the qualities of reliability, constancy of action, and complete regularity of feed are ascribed to pumps.

It seems to me that this comes from the fact that the working process of a pump, resting on the simple and obvious principle of equality between the working pressure and the existing resistances, is easy to understand, whereas of the working process of an injector, based on the direct conversion of thermal energy into kinetic energy and of the latter into potential energy, there exists a very indefinite conception.

Regularity in feeding a boiler can also be achieved with an injector, because in many of them the limits of maximum and minimum delivery may be significantly separated from one another. One need only be able to determine the conditions under which this can be achieved in each individual case.

Our law on the installation of steam boilers does not regulate in detail what the feeding appliances at a boiler must be; it says only that there must be two of them and that both must operate independently of one another.

Practice has developed the rule that injectors should be installed for double, or even triple, the amount of water consumed by the boiler. With such an injector, of course, it may be impossible to achieve continuous feeding, especially in the case of a lightly loaded boiler; but the price of the device is so small, and its installation requires so little space, that one can always have a third injector, whose capacity corresponds precisely to the normal working conditions of the boiler, and in this case the delivery of water can always be brought to the desired degree of regularity.

II.

As for the first two shortcomings, in the majority of cases they can more properly be attributed to the caprice of the personnel entrusted with tending the injector than to the injector itself. If the necessary requirements for the possibility of the working process are not met in the operating conditions, not only the injector but also the pump and any mechanism will become capricious; and if this is not observed in reality, it is because these processes take place externally, and are therefore more obvious and comprehensible.

To clarify the working process of the injector and, on this basis, to establish the conditions under which it can operate flawlessly and quite reliably—this was the aim I pursued in compiling this book.

I am far from thinking that it is entirely free of shortcomings; on the contrary, the need to rework certain parts, and the need to have more experimental data on which one could rely with confidence, are quite evident.

Nevertheless, I believe that even in its present form it may be of use to those who have to deal with injectors, especially since in the Russian language, apart from Mr. Mitte’s brochure “Steam-Jet Apparatus” and a few pages devoted to injectors in such large courses as “Steam Boilers” by Messrs. Depp and Predtechensky and “Locomotives” by Mr. Mukhachev, I have encountered nothing on the subject of injectors. I take the liberty of expressing the hope that those interested in injectors and in steam-jet apparatus in general will not refuse, in their comments, to communicate the practical data they possess for the necessary changes and additions.

In addition to the sources indicated in the text, I also had at my disposal the following manuals and journal articles.

Berthot, Traité de l’élévation des eaux.

Herrmann, E. Theorie des Injectors. Zeitschr. d. Oesterreich. Ing. und Archit. Ver., XXX, S. 114.

Richard. La chaudière locomotive et son outillage.

Transactions of the American Society of Mechanical Engineers, vol. IX, p. 505. The best form of nozzles.

Pullen. Experimental Engineering.

Theorie d. Injectors, Cario. Zeitschr. für Dampfkessel und Maschinenbetrieb, 1904, S. 333.

Contents

	Pg.
Introduction. General theory of the sucking action of jets	1–8
Steam nozzle. Outflow of steam from nozzles in the general case and as applied to the injector in particular. Velocity of steam outflow with a converging conical and a diverging nozzle. Determination of the nozzle diameter analytically and graphically by means of the temperature–entropy diagram. Velocity of outflow of wet steam	9–45
Condensing nozzle. General conditions for condensation of steam by water. Quantity of water required for condensation. The process of steam condensation in the nozzle and its influence on the operation of the injector. Temperature and density of the delivery water	46–68
Receiving nozzle. The role of the nozzle in the operation of the injector. The law of transformation of kinetic energy into potential energy. Nozzle profile under the condition of uniformly retarded motion of the jet within it. Losses of energy in the nozzle	69–82
Theory of action. Fundamental equations in the theory of the injector. Calculation of nozzles with the impact coefficient. Their calculation when a definite working back pressure is specified. Greatest and least power of the injector. Influence of the initial temperature of the water and of the suction height. Mechanical and thermal efficiency	83–106
Types of injectors. Structural difference between suction and non-suction injectors	107–128
Non-suction injectors of Schau, Friedman, Nathan, Webb, Rue
Non-suction injectors operating with wet steam. Hanrez, Davies, Metcalfe; Schaeffer and Budenberg; Holden & Brooke
Suction injectors. Giffard’s original. Metropolitan, Sellers; Schaeffer and Budenberg restarting injector, Holden & Brooke, Sirius, Siemens & Halske, Gresham & Craven, Friedman, Monitor, Penberthy

IV.

	Page
Double, or compound, injectors operating with live and exhaust steam. Davies & Metcalfe, Holden & Brooke.
Körting double injector, Schütte, Metropolitan, Centralheizungs-und Apparate-Bau-Anstalt, Belfield, Buffalo, Albion.
Materials for injectors; machining of their parts; repair; wear; rules for installation and maintenance	128–133
Testing of injectors. Purpose of the test. Graphical representation of test results. Arrangements for installing an injector for testing	134–155

Printing Errors Noted

Page	Line	Printed	Should read
5	20 from bottom	flowing out	flowing out
7	16 from top	weight of steam \(m\) to the weight of water \(m_1\)	weight of water \(m\) to the weight of steam \(m_1\)
8	11 from top	degrees	degree
9	5 from top	aperture	opening
11	5 from bottom	to the volume of the weight of a unit	to the volume of a unit
13	13 from top	Poche	Pochet
17	19 from top	ratio	relation
18	7 from top	known to the limit	to the known
18	11 from bottom	\(\delta \dfrac{dG}{dp} = -u \dfrac{d\delta}{dp}\)	\(\delta \dfrac{du}{dp} = -u \dfrac{d\delta}{dp}\)
19	1 from top	Printed: \(\displaystyle pv=\frac{2}{k-1}p_1v_1-\frac{2}{k-1}p_1v_1-\left(\frac{p_2}{p_1}\right)^{\frac{k-1}{k}}\) Should read: \(\displaystyle pv=\frac{2}{k-1}p_1v_1-\frac{2}{k-1}p_1v_1\cdot\left(\frac{p_2}{p_1}\right)^{\frac{k-1}{k}}\)	Printed: \(\displaystyle pv=\frac{2}{k-1}p_1v_1-\frac{2}{k-1}p_1v_1-\left(\frac{p_2}{p_1}\right)^{\frac{k-1}{k}}\) Should read: \(\displaystyle pv=\frac{2}{k-1}p_1v_1-\frac{2}{k-1}p_1v_1\cdot\left(\frac{p_2}{p_1}\right)^{\frac{k-1}{k}}\)
20	13 from top	at the inlet	at the outlet
25	3 from top	according to the setting	setting
25	12 from bottom	not overheated	overheated
27		The asterisks in the footnotes are shown incorrectly.	The asterisks in the footnotes are shown incorrectly.
32	6 from top	for the time being	while
39	8 from top	identical	identical
39	10 from bottom	\(\displaystyle w^2=\frac{2g(\lambda_0-\lambda_1)}{A}\)	\(\displaystyle w^2=\frac{2g(\lambda_0-\lambda')}{A}\)
56	13 from top	chamber serves	chamber \(f\) serves
57	5 from top	waters	water
73	3 from bottom	\(\displaystyle \frac{du}{dt}=\frac{g}{\gamma}\cdot\frac{dp}{dl}\)	\(\displaystyle \frac{du}{dt}=-\frac{g}{\gamma}\cdot\frac{dp}{dl}\)
75	15 from bottom	inside	within
78	5 from bottom	\(Z=20d_0\)	\(L=20d_0\)
85	8 from top	\(p'\)	\(p_1\)
85	3 from bottom	\(p'\)	\(p_1\)
94	16 from top	\(\displaystyle \left(\frac{d_0}{d_s}\right)^2=\frac{\pi^{\prime s}\ldots}{2(\ldots)}\)	\(\displaystyle \left(\frac{d_0}{d_s}\right)^2=\frac{\pi^{\prime s}\ldots}{2(\ldots)}\)
98	10 from top	\((p+1)\) kg	\((P+1)\) kg
100	formula (26a)	\(Q=0.04\,d_s\,s\sqrt{2p}\)	\(Q=0.04\,d_s\zeta_s\sqrt{2p}\)
102	formula (28b)	denominator \(t-t_a\)	\(t_e-t_a\)
113	5 from top	suction	suction

Introduction

The injector belongs to the large and continuously growing, in its variety, group of jet apparatuses. The operation of such an apparatus, so long as we are dealing, as the working body, with a jet of liquid, must be based on the mechanical theory of impact; the use of steam or of some gas-like substance as the working fluid introduces into the theory of the action of the apparatus a thermodynamic principle; finally, in the case where the liquid being moved, before receiving in the apparatus the living force necessary for motion and for overcoming resistances, must be raised to some height, a third principle is joined to the theory of action—the suction action of the flowing jet. Whatever the working liquid and the liquid receiving from it the impact may be, the principle of action of a jet apparatus remains unchanged: it is the impact of two bodies and the subsequent joint motion of both liquids.* Of special interest, of course, is an apparatus which not only forces or moves to a considerable height some liquid supplied to it under a weak head, but first sucks it up to a known height and then imparts to it a store of energy necessary and sufficient to overcome the existing resistances. Therefore, before proceeding to the study of the action of injectors, we shall consider, in the general case, the suction action of a flowing jet.

Suppose that through pipe \(a\) (Fig. 1) some liquid flows, being under a pressure \(p \frac{\mathrm{kgr}}{\mathrm{mtr}^2}\); the pipe ends in a conical converging nozzle, whose outlet area is \(\omega\), the cross-sectional area of the pipe is \(\omega_1\). Flowing out of the orifice, the liquid enters pipe \(b\), whose cross-section is \(\omega_2\); the pressure at the inlet opening of pipe \(b\) we shall call \(p_1 \frac{\mathrm{kgr}}{\mathrm{mtr}^2}\).

Pipe \(b\) is connected with a hermetically closed vessel \(c\); pipe \(c\), with cock \(d\), connects the interior of this vessel with a water reservoir. In vessel \(c\) and in the pipes adjoining it before the start—

* If we somewhat broaden the concept of liquids, then loose bodies as well may be included here, such as sand, flour, dust, and the like.

of the jet of working liquid through pipe \(a\) there is air, whose pressure is, of course, equal to \(p_1\).

Let us now suppose that liquid has begun to flow out of nozzle \(a\) with a definite and sufficiently great velocity so that, without discharging into vessel \(c\), it enters directly into pipe \(b\). At some distance from the inlet opening the liquid will fill the entire cross-section of pipe \(b\) and will push out the air lying before it; owing to the disturbance of equilibrium, the air contained in vessel \(c\) will begin to flow toward the inlet opening of pipe \(b\); its particles, partly under the impacts of the liquid particles and partly by friction against the surface of the jet, will be carried along by it, and through pipe \(b\), together with the liquid, will pass out into the open space. After a certain interval of outflow, part of the air will be removed from vessel \(c\), and a rarefaction will occur in it; if the cock on pipe \(d\) is now opened, suction of water from the reservoir will begin.

Let us consider the steady process: let \(p_x \, \frac{\mathrm{kgr}}{\mathrm{mtr}^2}\) be the pressure which is established in vessel \(c\) after a certain part of the air has been removed from it. In addition, denote:

\(\gamma\)—the density of the issuing liquid, which we assume constant throughout the entire time of outflow;

\(v\)—the velocity of the jet in mtr. per sec. at the outlet from the nozzle;

\(v_1\)—the velocity of the liquid in pipe \(b\);

\(\zeta\)—the velocity hydraulic coefficient;

\(\psi\)—the coefficient of contraction of the jet.

At the beginning of the outflow, the pressure in vessel \(c\) is equal to \(p_1 \, \frac{\mathrm{kgr}}{\mathrm{mtr}^2}\); we may write the following equality:

\[ \frac{1}{2g}\left(\frac{v}{\zeta}\right)^2 = \frac{p-p_1}{\gamma} + \frac{1}{2g}\left(\frac{v}{\zeta}\right)^2 \left(\frac{\psi \omega}{\omega_1}\right)^2 \quad \text{or} \]

\[ v=\zeta \sqrt{ 2g\frac{p-p_1}{\gamma} \cdot \frac{1}{1-\left(\frac{\psi\omega}{\omega_1}\right)^2} } \ldots\ldots (1). \]

For simplicity of consideration let us suppose that \(\omega_1\) is so much greater than \(\omega\) that, with \(\psi < 1\), the whole fraction \(\left(\frac{\psi\omega}{\omega_1}\right)^2\) is very small in comparison with unity; neglecting it, we shall have, instead of the preceding equality,

\[ v=\zeta \sqrt{2g\frac{p-p_1}{\gamma}} \ldots\ldots (1a). \]

When in vessel \(c\) the pressure \(p_x \, \frac{\mathrm{kgr}}{\mathrm{mtr}^2}\) is established, the velocity of outflow will be (analogously to 1a)

\[ v=\zeta \sqrt{2g\frac{p-p_x}{\gamma}} \ldots\ldots (1b). \]

In tube \(b\) the liquid passes almost instantaneously from velocity \(v\) to velocity \(v_1\); an impact occurs and, associated with it, a loss of energy, for which, referring it to a unit of weight, we obtain the expression

\[ \frac{1}{2g}(v-v_1)^2. \]

Let the length of tube \(b\) be equal to \(l_2\), and its diameter \(d_2\); flowing through it, the liquid experiences friction against the walls, and if \(\xi\) is the coefficient of friction, then the loss of energy depending on it may be estimated by the expression:

\[ \xi \frac{v_1^2}{2g}\frac{l_2}{d_2}. \]

Thus, on leaving the nozzle, the jet possesses the energy \(\dfrac{v^2}{2g}\), if we refer it to a unit of weight, while in chamber \(c\) it is under the pressure \(p_x\); the pressure at the outlet opening of tube \(b\) is equal to \(p_1\); let the density of the liquid in tube \(b\) be \(\gamma_1\).

By the law of conservation of energy we may write the following equality:

\[ \frac{v^2}{2g}+\frac{p_x-p_1}{\gamma_1} = \frac{v_1^2}{2g} + \xi \frac{v_1^2}{2g}\frac{l_2}{d_2} + \frac{1}{2g}(v-v_1)^2 \quad \text{or} \]

after transformations

\[ \frac{p_x-p_1}{\gamma_1} = \frac{1}{2g}\left[ v_1^2\left(2+\xi\frac{l_2}{d_2}\right)-vv_1 \right]\ldots\ldots\ldots(2). \]

For steady motion in tube \(b\) with constant density \(\gamma_1\), we must have:

\[ \omega v=\omega_2 v_1 \quad \text{or} \quad v_1=\frac{\omega}{\omega_2}v=\alpha v. \]

Since, by (1a),

\[ v=\varphi \sqrt{2g\frac{p_x-p_1}{\gamma}}, \]

then

\[ v_1=\alpha\varphi \sqrt{2g\frac{p_x-p_1}{\gamma}}. \]

Making the substitution in equality (2), we find

\[ \frac{p_x-p_1}{\gamma_1} = \frac{\alpha v^2}{2g} \left[ \alpha\left(2+\xi\frac{l_2}{d_2}\right)-2 \right]. \]

Since the length of nozzles in jet apparatus is very small, the term expressing the loss due to friction may be omitted, and we obtain

\[ p_x-p_1=\frac{\alpha v^2\gamma_1}{g}(\alpha-1)\ldots\ldots\ldots(3) \]

Here \( \alpha = \dfrac{\omega}{\omega_2} \), and since \( \omega_2 > \omega \), \( \alpha \) is less than unity.

The right-hand side of the equation is therefore negative, and the equality is possible only when its left-hand side is also negative. This leads us to the conclusion that \( p_x < p_1 \), i.e. that in vessel \( c \) there is indeed a rarefaction relative to the external pressure at the outlet opening of pipe \( b \).

Let us assume that \( p_1 \) is atmospheric pressure; then, upon opening cock \( d \), if the height of the pipe above the water level in the reservoir does not exceed the value \( (10000 - p_x) : 1000 \), and in the complete absence of any losses whatever, water from the reservoir will be sucked into vessel \( c \).

In equation (3) let us replace the velocity \( v \) from (1b); then we have

\[ p_1 - p_x = 2 \alpha \varphi^2 \gamma_1 (1-\alpha)\, \frac{p-p_x}{\gamma} \ldots\ldots\ldots (4) \]

Let us note that \( p-p_x = p-p_1 + p_1-p_x \); if, instead of the difference \( p-p_x \), we substitute the indicated quantity into the right-hand side, we shall find the final expression

\[ p_1 - p_x = \frac{ 2 \alpha \varphi^2 \gamma_1 (1-\alpha)(p-p_1) }{ \gamma - 2 \alpha \varphi^2 \gamma_1 (1-\alpha) } \ldots\ldots\ldots (5) \]

Equation (5) gives us the magnitude of the rarefaction in reservoir \( c \); at the same time it characterizes the suction capacity of the jet apparatus. We see that the suction height is directly proportional to the pressure difference \( (p-p_1) \); if \( \alpha = 1 \), then rarefaction in vessel \( c \), and consequently the suction of water from the reservoir, becomes impossible; depending on \( \alpha \), i.e. on the ratio \( \dfrac{\omega}{\omega_2} \), the suction capacity of the jet reaches a maximum at \( \alpha = \tfrac{1}{2} \), i.e. at \( \omega_2 = 2\omega \)—a value determined by means of the derivative of the function; moreover, the suction capacity is the greater, the greater \( \gamma_1 \), the density of the jet in pipe \( b \).

If now, in the left-hand side of equation (4), instead of the difference \( p_1-p_x \) equal to \( (p-p_x)-(p-p_1) \), we substitute it and perform the transformations, then we obtain the following expression:

\[ p-p_x = \frac{(p-p_1)\gamma}{ \gamma - 2 \alpha \varphi^2 \gamma_1 (1-\alpha) } \ldots\ldots\ldots (6). \]

By this equation is determined the pressure difference under the influence of which the flow into reservoir \( c \) takes place. It characterizes for us the amount of kinetic energy acquired by the jet; since

\[ v = \varphi \sqrt{2g\,\frac{p-p_x}{\gamma}}, \]

the kinetic energy of the jet will be

\[ \frac{v^2}{2g} = \frac{\varphi^2 (p-p_1)}{ \gamma - 2 \alpha \varphi^2 \gamma_1 (1-\alpha) } \ldots\ldots\ldots (7). \]

It is evident that the kinetic energy of the jet is directly proportional to the same difference of pressures \((p - p_1)\). It is not difficult to show that, when \(\alpha = 1/2\), the kinetic energy of the jet attains a greater value than in the cases \(\alpha \lessgtr 1/2\), all other conditions remaining unchanged.

Thus, this is the general principle on the basis of which the construction of suction jet apparatus becomes possible.

Steam, water, or air may serve as the working fluid; the fluid to be sucked in may be the same substances, any gas, and certain granular bodies. But for the construction of a jet apparatus that forces fluid, where the pressure of the reservoir into which the delivery is made is equal to or greater than the pressure of the vessel from which we take the working fluid, one further condition is necessary. In what follows we shall be concerned exclusively with the theory of the injector, the study of the process and of the operations occurring in it, and the determination of the most correct outlines and shape of its nozzles and of their most advantageous position relative to one another.

Here, however, having noted that the above theory of the suction action of a jet long served as the basis for calculations of the nozzles of an injector working with an elastic fluid, we shall familiarize ourselves with the apparatus itself, the manner of its action, and the terminology in use. In Fig. (2) Giffard’s injector is shown schematically. Through pipe a steam flows from the boiler; by means of spindle g one may regulate the quantity of steam issuing, or completely stop its outflow from the conical steam nozzle. Entering at high velocity into chamber H and expanding freely in it, whereby the jet takes the form of a diverging cone, the steam produces a rarefaction there, and is removed, together with the air, through pipe d. Water is sucked from the lower reservoir through pipe B into chamber H and condenses the steam; at the same time it receives an impulse from the steam jet and, at considerable speed, mixing with the steam that has not yet condensed, leaves the condensing nozzle b; by its outlet orifice it is directed into the receiving nozzle c, whence, overcoming the pressure on the valve from the side of the vessel into which delivery is being made, it is directed through the pipe to the latter.

These three nozzles—the steam, condensing, and receiving nozzles—are the essentially important parts of the apparatus.

In what follows we shall examine in greater detail the role of each of them, and indicate the influence of its size and shape on the operation of the injector, on its capacity and economy of action; here we shall merely mention that a given form of the steam nozzle will be the more advantageous the less steam it consumes; in this case the resistance, the quantity of water delivered by the apparatus, and the temperature of the water being sucked in must remain unchanged.

The better the given form of the condensation nozzle, the more perfectly the condensation of the steam will take place in it; depending on this, the greater will be the density of the mixture at the exit from this nozzle, and the greater the reserve of kinetic energy possessed by the jet entering the receiving nozzle.

Finally, the greater the area of the section of the narrowest part of the receiving nozzle, the greater the quantity of water the injector can deliver at the same velocity; the more correct the form of the nozzle profile in longitudinal section, the greater the quantity of kinetic energy of the jet that will be converted into useful work of the jet.

Giffard and his first followers selected the proper forms of the nozzle profiles and the dimensions of their cross-sections by experiment. Thus, for example, with one and the same outline for an entire series of nozzles differing from one another only in the dimensions of their cross-sections, tables were compiled in which the steam consumption and the greatest and smallest quantities of water delivered by the injector were indicated. A comparison among the data contained in such tables, for nozzles of one and the same size but with different profile outlines, and for nozzles differing only in the dimensions of the cross-section, indicated which of them was the most expedient and advantageous.

On the basis of data obtained through numerous experiments, Giffard proposed determining the velocity of the jet in the narrowest section of the receiving nozzle, at its mouth, by the following formula:

\[ \text{velocity}=\sqrt{2ghk}. \]

Here \(g\)—the acceleration of gravity—\(9.81\ \mathrm{mtr.}\text{--}\mathrm{sec};\)

\(h\)—the height of a column of water in mtr., corresponding to the pressure of the steam in the boiler, and

\(k\)—a coefficient introduced to allow for the resistances encountered by the jet on its way to the boiler, equal to \(1.7\text{--}2.0\).

Giffard further indicates that this coefficient \(k\) should represent the ratio of the area of the outlet opening of the steam nozzle to the area of the mouth of the receiving nozzle; such should be the ratio for the case when the steam issuing from the nozzle has the same density as in the boiler, and when it is all condensed by the supplied water. In reality, neither the one nor the other condition is fulfilled exactly; therefore the coefficient \(k\) is determined by a more complicated formula, depending on the steam pressures in the boiler and in the condensation nozzle, and on the densities of the steam in the boiler and at the exit from the steam nozzle.

The mechanical theory of the injector is based on the theorem of conservation of the quantities of motion of two impacting bodies before and after impact.

Neglecting the small magnitude of the velocity of entry of the water into the condensing nozzle, and denoting by \(w_1\) the velocity of exit of the steam from the steam nozzle, and by \(m_1\) its weight per unit time; by \(u_1\) the velocity of the jet upon leaving the condensing nozzle, and the weight of the mixture by \(m+m_1\), where \(m\) is the weight of water supplied to the injector corresponding to the weight of steam \(m_1\), according to this theorem we shall have the equality

\[ w_1 m_1=(m+m_1)u_1 \]

or

\[ u_1=w_1 \frac{1}{1+\frac{m}{m_1}} . \]

In order for the injector to force water into the boiler, it is necessary that this velocity \(u_1\) be equal to or greater than the velocity with which the water would flow out through the receiving nozzle under the head corresponding to the pressure in the boiler. Thus, the necessary condition is \(u_1 \geq u_2\), or

\[ w_1 \frac{1}{1+\frac{m}{m_1}} \geq u_2 \ldots\ldots (8) \]

As has been said, the velocity \(u_2\) is determined by the pressure of the boiler or reservoir into which the injector delivers water. It is clear that the ratio of the weight of steam \(m\) to the weight of water \(m_1\) must be the smaller, the higher the pressure against which the jet works; and conversely, the lower this pressure, the greater this ratio may be for one and the same velocity \(w_1\) in both cases.

This is in fact observed for the majority of injectors in use in practice, whose steam nozzle is a converging cone.

Without yet entering into consideration of the question of the velocity of outflow of steam from nozzles, let us merely note that the generally accepted formula for determining the velocity of outflow of steam from a conical converging nozzle gives, at various pressures from 2 to 14 atm., values differing very little. The kinetic energy of the jet remains almost unchanged, since it depends on the velocity; but at the same time the quantity of steam flowing out per unit time increases; therefore, naturally, with an increase of pressure, and consequently also of the velocity \(u_2\), it is necessary only that the ratio

\[ \frac{m}{m_1}. \]

decrease.

The very slight increase in the kinetic energy of the jet with increasing pressure clearly indicates the irrationality of using conical converging nozzles. By direct experiment it is not difficult to be convinced that the jet of steam issuing, after leaving the orifice, at a distance of 3–4 diameters from it becomes visible and assumes

in turn, the form of a cone with its apex turned toward the orifice. The jet assumes such a form solely because of the expansion of the steam taking place in directions perpendicular to its motion. Instantaneous photographic images of such a jet also indicate the presence of vortical motion of the particles within it; (see Figs. 3 and 4)*) by this, of course, the concentration of the impact is disturbed and a loss appears.

But if the steam, like an elastic fluid, is allowed to expand freely in the direction of motion, then on issuing from the orifice the jet has the appearance shown in Fig. 5. The impact of such a jet has a greater effect, and the kinetic energy of the jet increases to a very high degree.

If in (8) the equality sign is used, then in order to keep the ratio \(\frac{m}{m_1}\) unchanged when \(u_2\) is increased, it is evidently necessary correspondingly to increase \(w_1\), the velocity of outflow of the steam; since the action of the apparatus is based on the impact of two bodies, a loss of kinetic energy is inevitable; in this lies the reason why, from the mechanical point of view, the apparatus is very imperfect. Retaining the previous notation, we can estimate the loss of energy by the formula

\[ \frac{1}{2g}(w_1-u_1)^2 \cdot \frac{m \cdot m_1}{m+m_1}; \]

we see that it will be the greater, the greater the velocity \(w_1\). With regard to the injector, however, this is not of great importance. The point is that, first, we can attain an increase in the velocity of outflow of the steam only at the expense of its expansion in the nozzle, without increasing its quantity per unit time; secondly, because the kinetic energy lost in the impact in the form of heat will be carried back with the water into the boiler.

Thus, the arrangement of a nozzle of another form, in which expansion of the steam is possible, will enable the injector to force water in the same quantity against a greater back pressure, or in a greater quantity against the same back pressure; if both remain unchanged, the same results may be attained with a smaller expenditure of steam.

) On the appearance of the jet see Miette, Steam-jet apparatus*.

Steam Nozzle.

Let us now turn to the question of the discharge of steam from nozzles. Let \(x\) characterize the degree of dryness of the steam (the specific steam content, i.e., the weight of dry steam contained in a unit weight of wet steam); the steam pressure is given. Suppose now that we have a reservoir of large capacity, from which steam flows out, through a well-rounded orifice and through a pipe of small diameter, into some medium (Fig. 6). Let us assume that the walls of the reservoir have a temperature equal to that of the steam and that there is no heat exchange between them and the latter. In order for the discharge of steam to be possible, its pressure in the reservoir must be greater than the pressure of the medium into which the discharge takes place; in order that the process of discharge should have the character of a continuous phenomenon, and the motion of the steam be steady, it is necessary to assume that the difference between the pressures of the reservoir and of the medium is a finite quantity. Since this is so, the process of discharge will be an irreversible process, and therefore we can apply only the first principle of thermodynamics, whose algebraic expression is as follows:

\[ dQ = A(dE + dW + dK)\ldots\ldots(1) \]

Here, \(dQ\) is an infinitely small quantity of heat;

\(dE\)—the increment of internal energy;

\(dW\)—external work;

\(dK\)—the increment of kinetic energy;

\(A\)—the thermal equivalent of work.

Equality (1) expresses the general law of conservation of energy. Let us denote by \(E_1\) and \(E_2\) the internal energy of the steam in the reservoir, where its velocity shall be \(w_1\), and in some section of the pipe, where its velocity shall be \(w_2\); let the state of the steam in the reservoir be characterized by the quantities \(p_1\) and \(v_1\), and in the section where the velocity is \(w_2\)—by the quantities \(p_2\) and \(v_2\); \(p\) and \(v\) are the pressure and volume of a unit weight of steam.

The internal energy of the steam in its state \((p_1, v_1)\) will be

\[ -\frac{q_1+\rho_1 x_1}{A}; \quad \text{and in the state } (p_2, v_2) - \frac{q_2+\rho_2 x_2}{A}; \]

here \(q_1, \rho_1\) and \(q_2, \rho_2\) are found from the tables for water vapors.

corresponding to these, \(p_1\) and \(p_2\), \(x_1\) and \(x_2\) are determined experimentally, or suitable values are chosen for them.

Flowing from the reservoir into the pipe, the steam will expand; its internal energy will decrease, and this decrease may be represented by the difference \(E_1 - E_2\), i.e.

\[ \frac{q_1 + p_1 x_1}{A} - \frac{q_2 + p_2 x_2}{A_2}. \]

The change of internal energy, under the assumption we have made of an adiabatic process, may also be represented in the following form:

\[ dE = \int_{v_1}^{v_2} p\,dv. \]

(We refer the phenomenon to a unit weight per unit time.) The increment of kinetic energy may be represented by the difference

\[ \frac{w_2^2}{2g} - \frac{w_1^2}{2g}. \]

It remains for us still to determine the increment of the work of the external forces. Of these, first of all, attention must be paid to the work of gravity; its influence, however, in an injector, owing to the small dimensions of the nozzle, is so slight that it may be neglected without great error in the final result.

Let us divide the jet of steam flowing in the pipe into a number of streams of small cross-section; each such stream experiences on its periphery, from those adjacent to it, some pressure, but since this pressure is directed perpendicular to the axis, i.e. to the direction of displacement of the steam particles, the work of this pressure is equal to \(0\).

Let us now divide the jet into elements by sections perpendicular to the axis, drawn infinitely close to one another; owing to the expansion of the steam, the pressures on the end planes of an element will be different; with respect to this element, the pressures of the adjacent elements in front of and behind it will be external forces. We must also determine the sum of the works of such elementary forces, which gives us the desired increment of the work of the external forces. Denote the cross-sectional area of the large reservoir by \(F_1\), the cross-sectional area of the pipe by \(F_2\); let us single out the volume of steam \(F_1 ds_1\), the pressure of which is \(p_1\); suppose that under the influence of the indicated forces its displacements will successively be \(ds_1'\), \(ds_1''\), \(ds_1'''\), etc., and the pressures corresponding to its determined positions \(p_1'\), \(p_1''\), \(p_1'''\), etc.; for the pipe at the place where we took the velocity of the steam to be \(w_2\), its pressure is equal to \(p_2\); let the pressures in the preceding positions of the element \(F_2 ds_2\) be \(p_2'\), \(p_2''\), \(p_2'''\), etc., and its displacements before them respectively \(ds_2'\), \(ds_2''\), \(ds_2'''\), etc.

On the front face of the first element the pressure upon it will be \(F_1p_1\); if its displacement in a unit of time is \(ds_1\), then the work of this pressure on the element will be \(F_1p_1ds_1\); the work of the front pressure on the element at its next displacement will be \(F_1p_1'ds_1'\), and so on; the work of the pressure from the side of the preceding element before the section where the pressure is \(p_2\) will be \(F_2p_2\,ds_2\); the sum of these works is:

\[ F_1(p_1ds_1+p_1'ds_1'+p_1''ds_1''+\ldots)+F_2(\ldots+p_2''ds_2''+p_2'ds_2'+p_2ds_2). \]

The sum of such elementary works, produced by the rear pressures with respect to the element and its position, will be:

\[ F_1(p_1'ds_1'+p_1''ds_1''+\ldots)+F_2(\ldots+p_2''ds_2''+p_2'ds_2'+p_2ds_2). \]

Since the pressures \(p_1', p_1'' \ldots\) and \(p_2'', p_2'\) and \(p_2\) are back pressures, i.e. the action of the forces conditioned by them is directed in the direction opposite to the motion, the entire second sum must be taken with a negative sign; this is the negative work of the external forces. Subtracting it from the first, we find the required increment of the work of the external forces produced by them upon the jet. Thus, we shall have:

\[ dW=F_1p_1ds_1-F_2p_2ds_2. \]

If we assume that the displacements \(ds_1\) and \(ds_2\) take place in a finite interval of time, for example in one second, then obviously

\[ ds_1=w_1 \quad \text{and} \quad ds_2=w_2, \]

and, consequently,

\[ dW=F_1w_1p_1-F_2w_2p_2. \]

If the motion has become steady, then for the continuity of the jets we must have

\[ Fw_1=v_1 \quad \text{and} \quad F_2w_2=v_2; \]

in this case the areas and velocities must be such that in one second one unit by weight of steam flows through. Making the substitution, we find that

\[ dW=p_1v_1-p_2v_2. \]

For clarification of the question, the phenomenon may also be represented as occurring in the following manner. Suppose that a piston moving without friction is inserted into a large reservoir; let the outflow of steam begin; the pressure in the reservoir falls. In order to maintain it constant, it will be necessary to compress steam into the reservoir. If the outflow of one unit by weight of steam, as we are considering the phenomenon, corresponds to a displacement of the piston \(S_1\), then the work of the external force which must be applied to compress the steam will be \(F_1p_1S_1\); but \(F_1S_1\) is the volume described by the piston during the time in which a unit by weight of steam flows out into the medium, i.e. \(F_1S_1=v_1\), the volume of a unit weight of steam. Thus, the positive external work will be \(p_1v_1\). In reality, instead of the performance of this external work, there occurs a communication of heat to the boiler and evaporation, at the expense of it, of a known quantity of water into steam.

Let us now imagine that such a frictionless moving piston is inserted into the pipe at the very place where we assume the pressure \(p_2\); when the piston in the reservoir is displaced through the length \(S_1\), this second piston, overcoming the pressure \(p_2\), will be displaced through the length \(S_2\); the work of resistance will be \(p_2 v_2\), and the actual external work expended in the outflow of unit weight of steam will be

\[ p_1v_1-p_2v_2. \]

Returning now to the algebraic expression of the first law of thermodynamics and taking \(dQ=0\), i.e. assuming the process to be adiabatic, we shall find:

\[ \frac{w_2^2}{2g}+p_2v_2+\frac{q_2+\rho_2x_2}{A} = \frac{w_1^2}{2g}+p_1v_1+\frac{q_1+\rho_1x_1}{A}\ldots\ldots(2) \]

In reality, the process cannot be adiabatic, since there will always be an exchange of heat between the reservoir and the surrounding medium and, in addition, the metallic parts, which conduct heat well, will transmit it to the particles of steam at their lower temperature, taking it from the reservoir, in which the steam is at the higher temperature.

We can always choose the reservoir so large in volume that the velocity \(w_1\) will be small, and the term of the sum depending on it may be neglected. Solving then the equality thus obtained with respect to \(w_2\), we shall have:

\[ \frac{w_2^2}{2g} = \frac{q_1+\rho_1x_1}{A} - \frac{q_2+\rho_2x_2}{A} +p_1v_1-p_2v_2\ldots\ldots(3) \]

Let us now suppose that the volume \(v_1\) consists of \(x_1u_1\)—the volume occupied by dry steam, and \(\sigma_1\)—the volume occupied by water, which determines the wetness of the steam, i.e. \(v_1=x_1u_1+\sigma_1\); correspondingly \(v_2=x_2u_2+\sigma_2\). Since the change in the volume of water as a function of temperature is very small, one may put \(\sigma=\sigma_1=\sigma_2=0.001\). Substituting in equality (3), instead of \(v_1\) and \(v_2\), the indicated expressions, we obtain:

\[ A\frac{w_2^2}{2g} = q_1+\rho_1x_1-q_2-\rho_2x_2 +Ap_1u_1x_1-Ap_2u_2x_2 -A\sigma(p_1-p_2). \]

Since \(Apu+\rho=r\), it follows that

\[ \frac{Aw_2^2}{2g} = q_1-q_2+r_1x_1-r_2x_2+A\sigma(p_1-p_2),\ldots\ldots(4). \]

The last term on the right-hand side is very small: \(\sigma=0.001\) and \(A=\dfrac{1}{427}\); discarding it, we obtain, for determining \(w_2\), the formula:

\[ w_2= \sqrt{\frac{2g}{A}\left(q_1-q_2+r_1x_1-r_2x_2\right)}\ldots\ldots(5). \]

If the pressures \(p_1\) and \(p_2\) are known, then \(q_1,\ q_2,\ r_1\) and \(r_2\) can be found from the tables; \(x_1\) and \(x_2\) must be determined in each case directly or estimated approximately. If \(x_1\) is known (in most cases it may be taken equal to 0.98), then \(x_2\) can be found, first, from the equation representing the state of the substance in an adiabatic process:

\[ \frac{x_1 r_1}{T_1}+\Phi_1=\frac{x_2 r_2}{T_2}+\Phi_2 \ldots \ldots \ldots (6), \]

where \(\Phi_1\) and \(\Phi_2\) are the entropies of the steam in its two states, and \(T_1\) and \(T_2\) are the absolute temperatures;

2) approximately, from Clausius’s equation:

\[ x_2=\frac{T_2}{r_2}\left(\frac{x_1 r_1}{T_1}+c\lg\frac{T_1}{T_2}\right)\ldots \ldots (6a), \]

where \(c\) is the mean heat capacity of water;

3) finally, from Roche’s empirical formula:

\[ x_2=0.50+(x_1-0.50)\frac{147+t_2}{147+t_1}\ldots \ldots (6c). \]

This formula gives sufficiently accurate results for \(x_1>0.6\), which is the case for an injector operating with live steam.

Let us also note that \(q_1\) and \(q_2\)—the quantities of heat required to heat the water to the corresponding temperatures—may be taken equal to these latter temperatures, since \(q\) in general increases only slightly with a change in \(t\).

Making this substitution in formula (3) and calculating \(\sqrt{\frac{2g}{A}}\), we obtain

\[ w_2=91.53\sqrt{t_1-t_2+r_1x_1-r_2x_2}\ldots \ldots (5a). \]

For a tube area \(F_2\), the quantity of steam flowing out per second will be:

\[ \begin{aligned} G&=\frac{F_2w_2}{x_2u_2+\sigma_2}\\ \text{or approximately}\qquad G&=\frac{F_2w_2}{x_2u_2} \end{aligned} \left\}\ldots \ldots 7.\right. \]

Turning now to the consideration of the formula for the velocity (5a), we observe that, for a given initial state of the steam, \(w_2\) depends on its final state.

Let us write the same formula in the following form:

\[ w_2=91.53\sqrt{(t_1+x_1r_1)-(t_2+x_2r_2)}. \]

For all cases when \(p>p_1\), the quantity under the radical for \(x_1=x_2\) is positive; but since the adiabatic process is accompanied by condensation of steam, \(x_2\) will be less than \(x_1\), i.e. the numerical value of the difference increases. Consequently, \(w_2\) will be the greater, the lower the temperature \(t_2\) or the corresponding pressure \(p_2\).

As is seen from the preceding, the quantities \(p_2\), \(F_2\), etc., have been taken by us for some arbitrary cross-section of the pipe. The question now arises whether the formulas derived are applicable to the case when we wish to compute the velocity of outflow of steam and its quantity in the case of its discharge into a medium with constant and definite pressure \(p\).

In solving problems of this kind it was usually assumed that the steam pressure at the outlet opening is exactly equal to the pressure of the medium; under such an assumption, in order to determine \(w_2\) and \(G\) it would only be necessary to take from the tables the corresponding quantities for the letters in formulas (5a) and (7), and to determine \(x_2\) by one or another of the indicated methods.

Experience, however, shows that the quantity of steam computed by the formulas derived above and that determined directly by weighing after condensation differ greatly from one another, even allowing for possible inaccuracies; at the same time it was also observed that both results, at the given pressure \(p_1\), agree very closely with one another only up to a certain definite pressure of the medium into which the outflow takes place. Only when the pressure of the medium becomes lower than a certain limit does a discrepancy at once begin between the results given by theory and by experiment. This latter circumstance necessarily had to lead to the conclusion that the supposition that the steam pressure at its exit from the tube or nozzle is equal to the pressure of the medium is incorrect. We shall now show this theoretically.

We had (formula 3)

\[ \frac{w_2^2}{2g} = \frac{q_1+\rho_1 x_1}{A} - \frac{q_2+\rho_2 x_2}{A} + p_1 v_1 - p_2 v_2 \]

The difference

\[ \frac{1}{A}\left[(q_1+\rho_1 x_1)-(q_2+\rho_2 x_2)\right] \]

indicates to us what part of the internal energy of the steam has been transformed, in the phenomenon under consideration, into kinetic energy. To replace this difference by an expression that would depend on the pressures and volumes characterizing the steam in its initial and final states would mean that all the kinetic energy of the jet is acquired by it at the expense of external work. But since we are considering a process taking place without exchange of heat, i.e. adiabatic, and since in such a process the external work performed by the steam and equivalent ...

change of its internal energy \(E_1-E_2\), is equal to

\[ E_1-E_2=\int_{v_1}^{v_2} p\,dv, \]

then, instead of the equality indicated above, we may write:

\[ \frac{w_2^2}{2g}=p_1v_1-p_2v_2+\int_{v_1}^{v_2}p\,dv\ldots\ldots (8) \]

The law of variation of pressure and volume is then expressed by the relation:

\[ pv^k=p_1v_1^k=p_2v_2^k=\mathrm{const}; \]

if the steam before expansion is dry saturated, then \(k\) is equal to \(1.135\); if, however, it is wet and its dryness fraction is \(x\), then \(k=1.035+0.1x\) (for \(x\) within the limits from \(0.7\) to \(1.0\)).

Using this law, determine \(\int_{v_1}^{v_2} p\,dv\).

\[ \int_{v_1}^{v_2} p\,dv = p_1v_1^k\int_{v_1}^{v_2}\frac{dv}{v^k} = \frac{p_1v_1}{k-1} \left[ 1-\left(\frac{v_1}{v_2}\right)^{k-1} \right]. \]

Since

\[ \left(\frac{v_1}{v_2}\right)^{k-1} = \frac{p_2v_2}{p_1v_1} = \left(\frac{p_2}{p_1}\right)^{\frac{k-1}{k}}, \]

then in equality (8) the difference \((p_1v_1-p_2v_2)\) may be replaced by the corresponding expression:

\[ p_1v_1-p_2v_2 = p_1v_1 \left[ 1-\left(\frac{v_1}{v_2}\right)^{k-1} \right]; \]

we shall have:

\[ \frac{w_2^2}{2g} = p_1v_1 \left[ 1-\left(\frac{v_1}{v_2}\right)^{k-1} \right] + \frac{p_1v_1}{k-1} \left[ 1-\left(\frac{v_1}{v_2}\right)^{k-1} \right]. \]

Hence, after transformations and substitution, instead of \(\left(\frac{v_1}{v_2}\right)^{k-1}\), of the ratio equal to it, \(\left(\frac{p_2}{p_1}\right)^{\frac{k-1}{k}}\), for the velocity \(w_2\) we obtain the following expression:

\[ w_2 = \sqrt{ 2g\,\frac{k}{k-1}\,p_1v_1 \left[ 1-\left(\frac{p_2}{p_1}\right)^{\frac{k-1}{k}} \right] } \ldots\ldots (9). \]

Let us determine by weight the quantity of steam flowing through an arbitrary cross-section, whose area is \(F_2\), and the pressure in which is \(p_2\). Let the density of the steam be \(\delta_2=\frac{1}{v_2}\), and the weight of steam \(G\); then

\[ \frac{G}{\delta_2}=F_2w_2 \quad\text{or}\quad G=\frac{F_2w_2}{v_2}. \]

Since

\[ \frac{1}{v_2}=\frac{1}{v_1}\left(\frac{p_2}{p_1}\right)^{\frac{1}{k}}, \]

then

\[ G=\frac{F_2 w_2}{v_1}\left(\frac{p_2}{p_1}\right)^{\frac{1}{k}} = F_2 \sqrt{ 2g\,\frac{k}{k-1}\,\frac{p_1}{v_1}\cdot \sqrt{ \left(\frac{p_2}{p_1}\right)^{\frac{2}{k}} - \left(\frac{p_2}{p_1}\right)^{\frac{k+1}{k}} } }. \tag{10} \]

Let us apply this formula to the outlet opening of the pipe; let us suppose that the steam leaves the pipe precisely with the pressure under which the medium into which the outflow occurs is found. Suppose now that the pressure of the medium is equal to the initial pressure of the steam, i.e. \(p_2=p_1\); we find that \(G\)—the weight of the steam—is equal to 0. This is clear: if there is no difference of pressures, there can be no outflow. Suppose that the outflow takes place into an absolutely rarefied space; the pressure \(p_2\) must be equal to 0.

For the weight of steam \(G\) we again obtain the value 0. This result is plainly impossible, since in the given case there is the greatest difference of pressures, which would seem to have to condition the greatest weight of the issuing steam. We must conclude either that the formula itself is incorrect, or that the assumption made by us—that the pressure of the steam at the outlet and the pressure of the medium are equal to one another—is incorrect. We shall show below that precisely the latter is incorrect; for the present let us note that, for the weight \(G\), depending on the particular values of the letters entering into the second root, a maximum discharge is possible for the given area \(F_2\).

Denote by \(Z\) the difference

\[ \left(\frac{p_2}{p_1}\right)^{\frac{2}{k}} - \left(\frac{p_2}{p_1}\right)^{\frac{k+1}{k}}, \]

and, taking \(p_2\) as the independent variable, find that value \(p_2'\) at which \(Z\) assumes its greatest value. Differentiating*) and setting the first derivative equal to 0, we find that

\[ p_2'=p_1\left[\frac{2}{k+1}\right]^{\frac{k}{k-1}}. \]

If we put \(k=1.135\), i.e. assume the steam to be dry throughout the entire time of outflow, then the calculation gives us \(p_2'=0.5744p_1\).

*)

\[ Z=\frac{1}{p_1^{\frac{2}{k}}} \left[ p_2^{\frac{2}{k}} - p_1^{\frac{1-k}{k}}\cdot p_2^{\frac{k+1}{k}} \right]; \]

\[ \frac{dZ}{dp_2} = \frac{1}{p_1^{\frac{2}{k}}} \left[ \frac{2}{k}p_2^{\frac{2-k}{k}} - \frac{k+1}{k}p_1^{\frac{1-k}{k}}p_2^{\frac{1}{k}} \right] = \frac{p_2^{\frac{1}{k}}}{k p_1^{\frac{2}{k}}} \left[ 2p_2^{\frac{1-k}{k}} - (k+1)p_1^{\frac{1-k}{k}} \right] =0, \]

i.e.

\[ 2p_2'^{\frac{1-k}{k}}=(k+1)p_1^{\frac{1-k}{k}}; \]

whence

\[ \frac{p_2'}{p_1} = \left(\frac{2}{k+1}\right)^{\frac{k}{k-1}}. \]

The contradiction indicated above, consisting in the fact that for \(p_2 = 0\) the formula for the weight of steam \(G\) gives the value 0, while at the same time there should be a maximum of outflow, was first noted by Saint-Venant and Wantzel in 1839. To explain this, they put forward the supposition that the quantity of air flowing out (their work concerns the study of the outflow of the latter) does not depend on the pressure before the orifice within a certain interval of variation of the ratio \(\dfrac{p_2}{p_1}\), namely

when \(\dfrac{p_2}{p_1}\) changes from \(\dfrac{p_2}{p_1}=0\) to \(\dfrac{p_2}{p_1}=p_2'\). Fliegner’s experiments on the outflow of air confirmed the correctness of the supposition made by them. He found that until the air pressure in the reservoir exceeds twice the atmospheric pressure into which the outflow occurs, the pressure at the orifice is \(0.5767\) of the pressure in the reservoir.

This means that, provided the condition \(p_1 = 2p\) is satisfied, where \(p\) is the pressure of the medium, the air always leaves the orifice at a pressure greater than \(p\), independent of it and determined only by the indicated ratio of it to \(p_1\).

The first experiments concerning the outflow of steam were made by Napier. He found for steam as well exactly the same relation between the initial pressure and the pressure at the outlet orifice. Analyzing his experiments, Rankine came to the conclusion that the pressure in the orifice can never be lower than that which, under the given conditions, corresponds to the greatest discharge of steam. Napier’s experiments were repeated by Kunhardt in the laboratory of the Massachusetts Institute of Technology. The steam flowed out through short nozzles, \(0.25\) inch in diameter and leng-

That, at this value \(p_2 = p_2'\), a true maximum for \(Z\) occurs is shown by the following calculation:

\[ \frac{d_2 Z}{d(p_2)^2} = \frac{k+1}{k^2}\cdot \frac{ p_2^{\frac{2(1-k)}{k}} }{ p_1^{\frac{2}{k}} } \left[ \frac{2(2-k)}{k+1} - \frac{ p_1^{\frac{1-k}{k}} }{ p_2^{\frac{1-k}{k}} } \right] \quad \text{or} \]

\[ \frac{d_2 Z}{d(p_2)^2} = \frac{k+1}{k^2}\cdot \frac{ p_2^{\frac{2(1-k)}{k}} }{ p_1^{\frac{2}{k}} } \left[ \frac{2(2-k)}{k+1} - \left(\frac{p_2}{p_1}\right)^{\frac{k-1}{k}} \right]. \]

The sign of the second derivative depends on the expression in brackets; we determine it, replacing \(p_2\) by the value found above; we have:

\[ \frac{2(2-k)}{k+1} - \frac{2}{k+1} = \frac{2}{k+1}(1-k). \]

Since \(K = 1.135\) and in general is greater than unity, it follows that the second derivative is less than 0; i.e., at this particular value of \(p_2\), a maximum for \(Z\) must indeed occur.

... with diameters of 1.5, 0.5, and 0.25 inches. Direct observation established that the ratio between the pressure at the outlet orifice and the initial pressure was respectively 0.634, 0.615, and 0.583. The experiments of Blaess and Gutermuth, carried out in 1902 and 1904, confirmed the validity of Saint-Venant’s hypothesis for water vapor as well; the quantity of steam flowing out, when the ratio \(\frac{p}{p_1}\), where \(p\) is the pressure of the medium, decreases from unity, gradually increases up to a certain limit, upon reaching which it remains constant, even if \(\frac{p}{p_1}\) were brought down to 0, i.e. the outflow occurred into an absolute vacuum. In order to show theoretically the independence, under certain conditions, of the quantity of steam flowing out from the pressure of the medium, let us analyze the phenomenon in the following manner. Let \(G\) be the weight of steam flowing out per unit time through an orifice of given size; let \(p\) be the pressure of the medium into which the outflow takes place, \(\delta\) the density of the steam, and \(u\) its velocity at the instant of exit. If \(F_2\) is the area of the orifice, then

\[ G = F_2 u \delta . \]

Let us take the derivative with respect to \(p\); the maximum steam discharge will occur in the case when \(\frac{dG}{dp}=0\). Since \(F_2\) is constant, then

\[ \frac{dG}{dp} = F_2 \delta \frac{du}{dp} + F_2 u \frac{d\delta}{dp}; \]

hence, for \(\frac{dG}{dp}=0\), we have:

\[ \delta \frac{dG}{dp} = -u \frac{d\delta}{dp} \quad \text{or} \]

\[ \frac{du}{dp} = -\frac{u}{\delta}\frac{d\delta}{dp} \tag{11} \]

Replacing partial differentials by total differentials, we obtain

\[ \frac{du}{u} = -\frac{d\delta}{\delta}, \]

whence we find:

\[ \lg u = c - \lg \delta \quad \text{or} \]

\[ \lg (u \cdot \delta) = c, \]

i.e.

\[ u = \frac{1}{\delta} e^{c}, \]

where \(e\) is the base of Napierian logarithms.

Since \(e\) and \(c\) are constants, the velocity \(u\), for a given initial state of the steam, depends only on its density \(\delta\) at the instant of exit.

Theory and Calculation of the Injector

has openings. The latter, however, is determined by the final state of the steam, but not by the pressure of the medium.

Let us nevertheless assume that both these pressures, i.e. those of the medium and of the steam at the outlet, are equal.

The velocity \(u\), according to (9), will be

\[ u^2=\frac{2gk}{k-1}\,p_1v_1\left[1-\left(\frac{p}{p_1}\right)^{\frac{k-1}{k}}\right]. \]

Let us find the derivative with respect to \(p\).

\[ \frac{du}{dp}=-\frac{gv_1}{u}\cdot\left(\frac{p_1}{p}\right)^{\frac{1}{k}}. \]

Since \(pv^k=p_1v_1^k\), then \(\left(\dfrac{p_1}{p}\right)^{\frac{1}{k}}=\dfrac{v}{v_1}\), and

\[ \frac{du}{dp}=-\frac{gv}{u}. \]

The same relation \(pv^k\) may also be written thus:

\[ \frac{p}{\delta^k}=\frac{p_1}{\delta_1^k},\quad \text{or}\quad \delta=\frac{p}{p_1}\,\delta_1^k. \]

The derivative with respect to \(p\) will be

\[ \frac{d\delta}{dp}=\frac{1}{kpv}. \]

Substituting, in place of the derivatives, the values found into equation (11), we obtain:

\[ -\frac{gv}{u}=-\frac{u}{\delta}\cdot\frac{1}{kpv}. \]

Since \(\delta v=1\), then

\[ u^2=kgpv \]

and

\[ u=\sqrt{kgpv}\ldots\ldots(12). \]

Equation (12) determines for us that velocity of the steam at which the maximum discharge of it through a given pipe opening will take place. In appearance, the expression for the velocity is wholly identical with Laplace’s formula for the velocity of propagation of sound in a gaseous medium. But if \(u\) corresponds to this latter, then it is at once clear that the pressure of the steam at the outlet cannot be equal to the pressure of the medium, since the velocity of propagation of sound depends on the state of the gas or steam at the moment of outflow. Let us assume that the velocity \(u\), determined by (12), and \(w_2\), determined from (9), are equal to one another; we shall have:

\[ pv=\frac{2}{k-1}p_1v_1-\frac{2}{k-1}p_1v_1\left(\frac{p_2}{p_1}\right)^{\frac{k-1}{k}}. \]

Since \(p_2 v_2^{\,k}=p_1 v_1^{\,k}\), then

\[ \left(\frac{p_2}{p_1}\right)^{\frac{k-1}{k}} =\frac{p_2 v_2}{p_1 v_1}, \]

we obtain

\[ pv=\frac{2}{k-1}\left(p_1v_1-p_2v_2\right). \]

Let us further suppose that the state of the steam characterized by the quantities \(p_2, v_2\) is identical with its state determined by the quantities \(p, v\); doing this, we obtain

\[ pv+\frac{2}{k-1}pv=\frac{2}{k-1}p_1v_1. \]

and, after reduction,

\[ pv=\frac{2}{k+1}p_1v_1. \]

Using the same equality \(p_1v_1^{\,k}=pv^k\), we obtain

\[ p=\left(\frac{2}{k+1}\right)^{\frac{k}{k-1}}p_1, \]

i.e., for \(p\) the same value is obtained as before for \(p'_2\), the final steam pressure at which the greatest steam discharge takes place.

Thus, on the basis of all that has been said above, we arrive at the conclusion that the phenomenon of steam outflow under variable pressure at its beginning divides into two periods. So long as the steam pressure \(p_1\) remains less than \(\frac{p}{0.6}\), or, if \(p\) is equal to atmospheric pressure, so long as \(p_1\) is less than \(1.66\) atm., the quantity of steam discharged depends on the difference \(p_1-p\), since at the inlet opening the steam has this pressure \(p\). But as soon as this limit is reached and the pressure \(p_1\) becomes greater than \(\frac{p}{0.6}\), the quantity of steam discharged, for the same area, depends only on its initial state. Indeed, substituting in (10), instead of \(p_2\), its value

\[ \left(\frac{2}{k+1}\right)^{\frac{k}{k-1}}p_1, \]

we arrive at the following formula for the velocity:

\[ w_2^{\,2}=\frac{2gk}{k+1}p_1v_1 \ldots .(13). \]

Above, methods have already been indicated for determining the steam content at the moment of the steam’s exit. The following indication may also serve the same purpose.

Let the volume of dry saturated steam be \(u\); with steam content \(x\), the volume of a unit of the mixture of steam and water will be:

\[ v=xu+\sigma(1-x). \]

Since \(\sigma = 0.001\) is small, and \(x\) differs hardly at all from unity, it may, with sufficient accuracy, be assumed that

\[ v = xu . \]

Let us denote the initial steam content in the boiler, corresponding to the pressure \(p_1\), by \(x_1\); then

\[ v_1 = x_1 u_1 . \]

Dividing term by term, we obtain:

\[ \frac{x}{x_1} = \frac{v}{v_1}\cdot\frac{u_1}{u}\ldots\ldots\ldots (14). \]

The dependence between the pressure and the volume of dry saturated steam is given by the equation:

\[ p_1 u_1^{\,n} = p u^{\,n} = D, \]

where \(n = 1.0646\), and \(D\) is a constant quantity, equal to \(1.7617\), if the pressure is expressed in \(\mathrm{kg}/\mathrm{cm}^2\); since for a mixture of steam and water the dependence holds

\[ p v^{\,k} = p_1 v_1^{\,k}, \]

then (14) may be represented as follows:

\[ \frac{x}{x_1} = \left(\frac{p}{p_1}\right)^{\frac{k-n}{nk}} \ldots\ldots\ldots (15). \]

Once \(p_1\) and \(x_1\) are known, for each pressure the value of \(x\) may be found in the corresponding section of the tube.

If the outflow occurs into a medium whose pressure is \(< 0.5744\,p_1\), then the pressure of the steam at the outlet, as we have seen, is determined by the equation

\[ p = \left(\frac{2}{k+1}\right)^{\frac{k}{k-1}} p_1 . \]

Substituting into (15) the ratio \(\dfrac{p}{p_1}\) obtained from this, we find:

\[ \frac{x}{x_1} = \left(\frac{2}{k+1}\right)^{\frac{k-n}{k(n-1)}} . \]

Since \(n\) is exactly equal to \(\dfrac{33}{31}\), and \(k = 1.035 + 0.1x_1\), then, for a given \(x_1\), \(x\) can always be determined exactly.

If we assume that \(x_1 = 1\), then for all pressures \(p_1\) the steam content at the outlet remains unchanged and equal to

\[ x = 0.9685 . \]

If we do this, taking \(x_1=1\), the calculation in formulas (9) and (10) will take the following form:

\[ w_2 = 323 \sqrt{p_1 v_1}\ldots\ldots\ldots(16) \]

\[ \frac{G}{F} = 199 \sqrt{\frac{p_1}{v_1}}\ldots\ldots\ldots(17) \]

With the aid of these expressions, Table A has been calculated.

TABLE A.

Initial pressure in the boiler, kg/cm²	Pressure at the outlet orifice \(p=0.5744\,p_1\).	Velocity of outflow \(w_2\), in mtr.	Kinetic energy of the jet \(k=\dfrac{w_2^2}{2g}\)	Quantity of steam per 1 unit of cross-sectional area \(\dfrac{G}{F_2}\)
5	2,887	442,4	9977 kgr. mtr.	727
6	3,465	444,9	10088 „ „	867
7	4,042	447,0	10182 „ „	1007
8	4,619	448,8	10265 „ „	1146
9	5,197	450,4	10339 „ „	1285
10	5,774	451,8	10405 „ „	1423
11	6,352	453,1	10466 „ „	1561
12	6,929	454,3	10521 „ „	1698

The table shows us how little the velocity of outflow increases with an increase in the initial pressure. The kinetic energy of the jet also remains almost constant and, at pressures from 5 to 12 atm., is approximately 10300 kgr. mtr.

From this it is clear why the operation of the injector becomes less productive at higher steam pressures. If the quantity of water flowing into the condensation chamber remained unchanged, then the jet of steam, even at a higher pressure, could impart to it only a slightly increasing velocity, while at the same time the resistance to driving it into the boiler increases directly in proportion to the pressure. In order for operation to be possible, it is necessary to reduce the quantity of water; its velocity will be greater, but at the same time its temperature rises, and if the injector is working with an open overflow, then the action of the apparatus may soon be completely disturbed.

Table B, containing the results of experiments with the Giffard injector, performed by M. Deloy, well illustrates the above.

TABLE B.

Temperature of feed water.	Temperature of delivered water.	Steam pressure in the boiler, in atmospheres.	kg of water per 1 kg of steam expended.
9.3° C.	38° C.	2.25	20.88
"	47	5	15.94
"	52.5	5.5	14.30
"	70	6.5	9.63
"	59.5	7.75	11.80
"	60.5	8	11.63

TABLE C.

Temperature of feed water.	Temperature of delivered water.	Steam pressure in the boiler, in atm.	kg of water per 1 kg of steam.	Greatest pressure that the injector can overcome.
38.00 C.	65° C.	2	21.5	2.15
38.75	68	2.5	19.8	2.75
38.0	70	3	18.0	3.40
39.75	74.5	3.5	16.5	4.0
40.0	80	4	15.0	4.75
41.75	83	4.5	13.8	5.35

Table C contains the results of Villiers’s experiments, which, among other things, Grasof used to determine the density of the jet of the mixture. Here also one clearly sees the decrease in the quantity of water supplied with increasing pressure. The figures in the last column indicate those limiting pressures that can exist in the reservoir at the given working steam pressure; but the injector works against them only in the case when the inflow of water to it is reduced against that indicated in the table. As can be seen, the temperature of the delivered water in this case

the water is already sufficiently high; at a feed-water temperature of 45° C., the injector no longer worked. It is also interesting to note that the velocity of the steam at 4.5 atm., calculated by Villiers as equal to 550 mtr., is less than that which, according to the formulas given above, it cannot exceed with the usual conically converging steam nozzle of the first and many later injectors, namely 450 mtr., even at twice the steam pressure. At the same time, as we have seen, the greatest consumption of steam also occurs. All this obviously indicates the irrationality of using a conically converging nozzle as the steam nozzle of an injector.

But its deficiencies are not limited only to the circumstances indicated: it not only utilizes poorly the thermal energy of the steam, converting it into kinetic energy, but also makes the jet itself less capable of doing the greatest possible work. A sufficient indication of this may be the already noted coincidence of the outlet velocity with the velocity of propagation of sound: in the issuing jet of steam there must arise wave-like motions along the axis. This, indeed, is also confirmed by experiment. For a jet of air this was first shown by means of photography by Mach and Emden); in the photograph of the jet there are visible regularly following one another dark and light lines. The presence of similar waves in the issuing jet of steam, noticed earlier by Parent, was clearly proved by Stodola’s experiments*) (drawing 7) on the discharge of steam from nozzles.

Stodola established that there exists one counter-pressure at which these oscillations of the steam particles in the jet disappear, but the slightest deflection to one side or the other immediately causes the appearance of considerable waves, in the form of symmetrically arranged cruciform lines, observable even with the naked eye. The flow of steam from such a nozzle is, moreover, accompanied by a transverse expansion of the jet; on the resulting diverging cone, whose form the jet assumes, there are sometimes noticeable contractions of the transverse section and subsequent renewed expansions. Under such conditions it is impossible to use advantageously the impact force of the jet: besides losses of kinetic energy from the impact itself, in this case additional losses are also inevitable, depending on the formation of vortex motions, on the non-centrality of the impact, etc.

It should be noted that the use of a converging conical nozzle for steam is explained by insufficient knowledge of the laws governing the outflow of elastic fluids at the time of the appearance of the injector. To

) Wiedemanns Annalen. 1890, Bd. 41 and 1899, Bd. 69.
*) The presence of waves was also observed by Gutermuth, Z. d. V. d. Ing. 1904. Stodola. Dampfturbinen.

To page 24.

$p = 2.3$ atm.

Conical nozzle with diameter 2.65 mm; outflow pressure $p = 2.9$ atm.

$p = 3.85$ atm.

$p = 5$ atm.

$p = 2.6$ atm.

Conical nozzle with diameter 3.63 mm; $p = 3.3$ atm.

$p = 5$ atm.

$p = 6$ atm.

Outflow of carbon dioxide from the same nozzle; $p = 5$ atm.

Cylindrical nozzle $d = 2.72$ mm; $p = 4.1$ atm.

Conical nozzle with diameter 0.3 mm; $p = 7$ atm.

Photographic images of Enden’s outflowing air jets from nozzles.

outflow of steam, forgetting its ability to expand, the conclusions obtained experimentally from observations of the outflow of liquid were adopted directly. For water, for example, according to St. Venant, diverging nozzles entail an increase in the flow rate $o_1$ and a decrease in the final velocity. Both the one and the other are undesirable for an injector, and therefore the steam nozzle was given the form of a converging cone, which, it seemed, was to ensure the smallest flow rate at the greatest exit velocity. In reality, however, the matter stands otherwise: such a nozzle causes the energy of the jet to remain almost unchanged with an increase in the steam pressure and a greatly increasing consumption of it. Therefore, in addition to the reason already indicated above, namely that when the pressure is increased one must be content with a smaller quantity of supplied water, the strongly increasing quantity of heat brought by the steam, as a result of its greater consumption, caused a considerable increase in the temperature of the jet, and thereby narrowed still more the limits within which operation of the apparatus was possible. A diverging nozzle, while keeping the steam consumption almost unchanged, will make it possible for the steam to expand in the direction of motion of the jet, and this will lead to a considerable increase in the final velocity.

But besides this, as observation indicates, the jet in this case may be more efficient, owing to the absence of the eddy-like motions noted above. Figs. 4 and 5 show the appearance of the jet upon exit in the first case from a nozzle with a rectilinear profile, and in the second—from a nozzle outlined along a curve. The direction of motion of the steam particles is almost parallel to the axis of the nozzle; in the first case, however, within the jet there is nevertheless noticeable a small reverse cone, formed by particles of wetter steam, whereas the outer ones, owing to friction against the walls, consist of drier and even slightly superheated steam. Hence the advantage is clear of using a diverging nozzle in which free expansion of the steam could occur up to a definite final pressure. If this pressure is $p$, then it is necessary only to insert in formula (5a) the values taken from the tables for $t_1$, $t_2$, $r_1$ and $r_2$, and to determine $x_2$ by one of the preceding formulae, in order to obtain the value for the exit velocity $w$. Further, since the quantity of steam remains the same, the area of the exit section can at once be found.

If the area of the cross-section in the narrowest part of the nozzle is $F_1$, then $G$ is equal to:

\[ G = \frac{F_1 w_1}{x_1 u_1}, \]

where $w_1$, $x_1$ and $u_1$ must be determined from that pressure which is esta-

is established in this narrow section, i.e. \(p_2' = 0.5744\,p_1\). Let the outlet section be \(F\); then, to determine it, we have the relation

\[ G=\frac{F_1 w_1}{x_1 u_1}=\frac{F\cdot w}{x\cdot u} \quad\text{and} \]

\[ F=F_1\frac{u x w_1}{u_1 x_1 w}. \]

If we wished to determine the theoretically correct profile of the nozzle from its smallest section to the outlet, we would have, after specifying the length of this part, to draw the curve of adiabatic expansion; then in each section the steam pressure would be determined, and thereby the corresponding \(x_1\), \(u\), and \(w\), and the dimensions of the nozzle sections. The extent to which, with such a nozzle, the velocity of outflow and the kinetic energy of the jet increase is shown in Table D.

TABLE D.

Ratio of initial pressure in the boiler to that after expansion \(p_1/p\)	Ratio of exit velocity to velocity at the nozzle mouth \(w/w_2\)	Ratio of areas of the outlet and the mouth \(F/F_2\)	Ratio of diameters for circular nozzles \(d/d_2\)	Specific steam content at the outlet \(x\)
100	2,583	13,802	3,715	0,765
90	2,560	12,690	3,562	0,769
80	2,535	11,535	3,399	0,775
70	2,505	10,395	3,224	0,781
60	2,469	9,163	3,027	0,788
50	2,426	7,980	2,825	0,796
20	2,177	3,966	1,991	0,840
10	1,924	2,436	1,561	0,874
8	1,861	2,069	1,438	0,886
6	1,742	1,716	1,310	0,901
4	1,550	1,349	1,161	0,922
2	1,119	1,015	1,007	0,960
1,7318 or \((p=0.57p_1)\).	1	1	1	0,968

From the table it is seen that, if the initial steam pressure is 10 atm. and the expansion at the end is carried to 1 atm., then the exit velocity will be \(w = 1.924 w_2\), and since, according to Table A, \(w_2 = 451.8\), then \(w\) will be equal to 869.26 m.

In Fig. 8 is shown the theoretically correct nozzle, as it is obtained according to the above-indicated method of determining its various cross-sections. It should also be noted that a further expansion of the nozzle entails a diminution of velocity. This is understandable, since, after passing the section corresponding to the final pressure, equal to the pressure of the medium, the steam does not expand, but only a change of its state occurs at constant pressure \(p\).* It could be shown that, owing to the diminution of kinetic energy and to friction, the steam content of the jet will begin to increase; if the nozzle is sufficiently long, then at the outlet the steam may even be superheated.

The above-mentioned form of nozzle is difficult to manufacture; especially careful workmanship and grinding of the surfaces are needed in order, as far as possible, to eliminate an increase of the coefficient of friction. Below we shall indicate what the loss from friction is; a rough or grooved surface, at a high velocity of the jet, would increase this loss still more.

The expansion of steam in a nozzle was first applied by Laval in the steam turbine, where, as in the injector, it is important to have as high an exit velocity as possible. His nozzle is shown in Fig. 9; the part of it up to the smallest section, or throat, is very short, so as not to lengthen the path of the steam and thereby to reduce friction losses. In the expanding part the nozzle has the form of a cone with a straight-line generatrix and an angle of taper of about \(10^\circ\); with a larger angle, lagging of the steam from the walls becomes probable and possible: the jet will not fill the entire section of the nozzle. The angle of taper, in general, depends on the initial pressure of the steam: the higher it is, the smaller the angle; likewise, the expansion of the jet observed with the naked eye, when it issues into the atmosphere, indicates that the outline of the generatrix of the cone is actually curvilinear, and that the curvature is the greater, the greater the steam pressure in the boiler. In the following Table E are given various angles of taper for steam pressures from 8 to 1 atm., determined by observation of those traces of the profile which the jet leaves on a sheet of hygroscopic paper.** The length of such a cone varies from 70 to 35 diameters of the discharge opening, decreasing with the lowering of the elasticity of the steam.

* Mitte, Steam-jet apparatus.

* This is confirmed by Büchner’s experiments: at small steam pressures in the boiler, a vacuum appeared in the nozzle, the velocity decreased, and the steam content increased. See Z. d. V. d. Ing.*, 1904, p. 1098, Tables 8a and 8b.

TABLE E.

Steam pressure, atm.	8	7	6	5	4	3	2	1
Magnitude of the angle of conicity	6°	7°	8°	10°	12°	16°	19°	28°

The data of this table indicate that a conicity angle of \(10^\circ\) is an average value for steam pressures from 3 to 8 atmospheres.

Recently very many experiments have been carried out on the outflow of steam from nozzles. Stodola, in studying this phenomenon, used a nozzle whose mouth diameter was 7 mm and whose outlet-opening diameter was 12 mm; the final steam pressure was \(1.05\ \mathrm{kg}/\mathrm{cm}^2\). The pressure of the reservoir into which the steam issued could be varied. With a vacuum in this reservoir, the expansion of the steam proceeded correctly to the end of the nozzle, and then a wave-like motion of the steam particles began, as indicated by curve A in Fig. 10; when the back pressure in the reservoir was equal to the final pressure of the jet, the motion of the steam particles was more regular, as is seen from curve C, which gives only very small deviations from a straight line; but as soon as the back pressure was increased, large oscillatory motions appeared in the jet, indicated by curve D.

In a second experiment the steam pressure at the outlet opening was \(0.7\ \mathrm{kg}/\mathrm{cm}^2\); when issuing into a vacuum the same thing was observed as in the first case, curve A, Fig. 11. With a back pressure in the reservoir of \(0.7\ \mathrm{kg}/\mathrm{cm}^2\), the outflow of steam occurred without any trace of oscillatory motion, and the steam particles moved along an exactly straight line. When the back pressure was increased, oscillatory motions of the steam particles again arose, and at the same time the steam pressure increased in the nozzle itself before its exit from it.

Curves D in Fig. 10 and C and D in Fig. 11 indicate that when steam issues into a space where the pressure is greater than the final pressure of the steam after its expansion, a steam shock occurs, causing its compaction and an increase of pressure before its exit from the nozzle. The same results were obtained by Büchner as well, when he observed the dependence of the outflow of steam from a Laval nozzle on increased back pressure*). In Fig. 12 are shown the steam pressures observed by him in various sections of the nozzle at various back pressures at the outlet; Fig. 13 shows the nozzle itself, its outline, and the position of the holes in its wall through which the pressure observations were made. As

) Zeit. d. V. d. Ing.*, 1904.

as is evident from the diagram, in the different sections of the nozzle, depending on the back pressure, the curve of steam expansion, upon reaching atmospheric back pressure, rises; that is, before leaving, the steam becomes more elastic. This phenomenon is accompanied by a loss of kinetic energy, since the velocity of efflux decreases; at the same time the steam content increases. In Table F*) are given the results of observations and calculations relating to the same nozzle; the marks indicate the position of the sections to which the velocities and the values for the dryness of the steam refer.

TABLE F.

$p\ \dfrac{\mathrm{kg}}{\mathrm{cm}^2}$	$w_1$	$w_2$	$w_3$	$w_4$	$w_5$	$x_1$	$x_2$	$x_3$	$x_4$	$x_5$	$p_b$ back pressure
10,3	449	585	668	711	760	96,5	94,9	93,0	92,0	91,0	2,09
10,33	449	585	668	711	346	96,5	94,9	—	92,0	99,6	4,22
10,38	449	585	538	292	210	96,5	—	96,0	99,6	—	6,46
10,35	449	464	263	202	165	96,5	96,7	99,4	99,9	99,9	8,18
10,34	334	253	176	157	131	98,2	99,0	99,5	99,6	99,7	9,44

For the purposes of the present chapter on the steam nozzle of the injector, Büchner’s experimental data on the efflux of steam into the atmosphere (i.e., at atmospheric pressure) are of special interest.

In Table G are presented his observations of the steam pressure in different sections of the nozzle.

In Fig. 14 the curves of steam expansion in the nozzle are shown. In external appearance they recall the ordinary expansion curve of steam engines, and differ from it only by a more rapid fall of pressure. This is quite understandable, since in a steam engine, for equal portions of the stroke, the volumes occupied by the steam increase uniformly; whereas in a diverging nozzle, owing to its conicity, the volumes between equally spaced sections cannot be equal to one another. In view of the more rapid expansion of the steam in the nozzle, it is possible that the quantity of steam which should condense into water during adiabatic expansion does not have time to pass into the liquid state. In this, perhaps, lies the reason why, according to Büchner’s data, independently of the initial steam pressure, the steam content at the end of the expansion—

*) The table gives actual velocities, not theoretical ones.

...of expansion remains unchanged and independent of the degree of expansion. This is all the more probable, since greater degrees of expansion correspond—

TABLE 6.

\(P_0\) pressure in the boiler	\(p_1\)	\(\dfrac{p_1}{p_0}\)	\(p_2\)	\(p_3\)	\(p_4\)	\(p_5\)	\(p_6\) back-pressure at the outlet section
12,80	7,34	0,573	4,44	3,03	2,31	1,78	0,96
11,23	6,50	0,577	3,93	2,68	2,05	1,57	0,96
10,36	5,84	0,564	3,61	2,46	1,87	1,46	0,97
9,24	5,28	0,572	3,22	2,20	1,67	1,30	—
8,24	4,75	0,577	2,89	1,97	1,49	1,17	0,97
7,14	4,08	0,572	2,50	1,70	1,30	1,03	0,99
6,10	3,47	0,569	2,13	1,45	1,11	0,87	0,99
5,08	2,89	0,569	1,80	1,22	0,92	0,72	1,00
4,08	2,30	0,564	1,42	0,97	0,76	0,60	1,02
3,13	1,75	0,559	1,12	0,77	0,61	0,87	1,02
2,07	1,15	0,553	0,70	0,54	0,72	0,94	1,02

—also to a greater velocity of flow of the jet, and, in connection with this, to a greater loss to the work of friction, which, being converted into heat, goes to superheating the steam.

Further, from these diagrams we see that, despite a considerable difference of pressures in the boiler, the pressure at the outlet differs very little from one case to another, by only hundredths of an atmosphere. Hence there follows a very small fluctuation also in the outlet velocity of the jet within considerable limits of fluctuation of the initial pressure. In Fig. 15 curves of velocities are plotted for the same sections. We see that the outlet velocity of the jet from one and the same nozzle, as the initial pressure varies from 12,8 atm. to 5,08 atm., changes very little, and only at still lower pressures does a sharp decrease begin in the outlet velocity of the jet and in the velocity of its motion in the nozzle itself*).

*) This may explain why, for many injectors made for operation with high-pressure steam and having several divergent cones, the initial working pressure is specified as 3 atm. If the velocity is too small, the injector cannot produce the required rarefaction and suck in water.

If we turn to the preceding diagrams, which indicate the process of expansion of the steam in the nozzle, we shall note that it proceeds quite uniformly and regularly only for pressures from 12.8 atm. to 7.14; with a further lowering of the pressure in the boiler, the pressure at the outlet opening of the nozzle begins to increase, and this increase occurs the earlier, the lower the initial pressure. Corresponding to this increase of pressure inside the nozzle, in a certain section of it, there also occurs a retardation in the motion of the jet.

The nature of the change of the steam in the nozzle and of its kinetic energy, as both follow from the experimental data, fully answers the preceding theoretical consideration. The magnitude of the back pressure at these pores, so long as it is less than the pressure of the steam at the end of the expansion, has no effect on the process of the latter, and only in the case when it is greater does compression of the steam begin before the outlet opening from the nozzle, and a retarded motion of the jet.

We can now, relying on the conclusions of the preceding consideration and on a comparison of the experimental data with one another, say how the installation of a steam nozzle that diverges will affect the operation of the injector.

With a conical converging nozzle the pressure at the outlet opening is equal to 0.57 of the initial pressure, and the velocity of outflow is almost independent of the initial pressure, varying very little for different steam pressures (the difference is expressed in units of meters); the quantity of steam flowing out is always a maximum for the given pressure.

With a conical diverging nozzle the velocity of outflow of the steam increases almost twofold, and within the limits of variation of the initial steam pressure by 50–60% it remains almost one and the same; consequently, an injector with such a nozzle can theoretically deliver almost twice as much water.

As regards injectors for low steam pressure, according to Büchner’s experimental data, for them it is advantageous to install conical converging nozzles, or else shorter diverging ones. Thus, for example, instead of a nozzle 50 m/m long, counting from the inlet opening for the steam to the outlet from the nozzle, which is suitable for pressures from 12.8 atm. to 5.08 atm., one must install a nozzle 36.5 m/m long at a pressure of 4.08 atm., and 27 m/m long at a pressure of 2.07 atm.

These latter conclusions, however, do not have absolute significance. A fall of the steam pressure during expansion in the nozzle below the back pressure prevailing at the outlet opening, and the compression of the steam caused by this before the outlet, may result from a discrepancy between the angle of conicity of the nozzle and this initial pressure, i.e. the jet could not fill its entire section.

Finding the most advantageous angle of conicity, given prescribed limits of the upper and lower initial steam pressures at which the capacity of the injector should vary as little as possible, must be made the designer’s task. At present, however, there are not yet sufficient data for resolving the question.

Likewise, we still cannot say anything definite about the process of expansion in the nozzle, since in an injector it depends on the contact of its walls with the inflowing cold water.

It may only be supposed that, owing to the small length of the nozzle and the rapidity of the expansion process, this influence should not produce any considerable deviation of the expansion curve from the form it has when steam discharges into the atmosphere. Only at the exit of the steam from the nozzle, when its direct contact with water and its rapid condensation begin, must the reduction of pressure make itself sharply felt.

If the results of Stodola’s experiments are valid—namely, that only when the steam pressure at the outlet orifice and the back pressure are equal does the motion of the steam particles proceed in a straight line—then we must expect the occurrence of a wave-like motion of the particles of the mixture and, consequently, the loss of kinetic energy associated with this.

If this is so, then the steam nozzle of a suction injector should allow the steam to expand to a pressure below atmospheric, i.e. it should be longer than in an injector working with water supplied to it under pressure.

The questions outlined here concerning the process of steam expansion in the injector nozzle in the presence of cold water can, of course, be resolved only experimentally.

Up to this point we have considered the motion of steam jets as occurring without resistances; in reality, however, owing to the great velocity of motion, this resistance must considerably alter the store of kinetic energy of the jet in comparison with that computed from the formulae.

Stodola, on the basis of his experiments, assumes that for nozzles up to \(50^{m}/_{m}\) in length, with throat diameters from \(6\) to \(10^{m}/_{m}\) and with a small conicity, this loss of kinetic energy due to friction may be taken as \(10\text{–}15\%\) of its total amount.* According to the experiments of the Laval turbine-building works in Stockholm, the loss of energy is expressed by the same number; the decrease in velocity, however, should be taken as \(5\text{–}8\%\) of that theoretically calculated. The results of Büchner’s tests give this loss likewise as \(7\text{–}8\%\), so long as it depends only on friction; when the jet is retarded owing to back pressure from the outlet side, the loss in velocity amounts to \(35\text{–}49\%\).

) Lewicki’s experiments on the discharge of slightly superheated steam make it possible to conclude that the loss is \(8\%\) (Z. d. V. d. Ing.*, 1903).

Let us now consider the influence of the moisture content of the steam on its final velocity. For the velocity we take formula (5)

\[ w=91{,}53\sqrt{q_1-q_2+r_1x_1-r_2x_2}. \]

We assumed that the steam content in the initial state is \(x_1=1\).

If the steam is supplied wet, then we thereby reduce the numerical value of the term of the sum \(r_1x_1\); the steam content \(x_2\) at the end of the expansion will also decrease, but since \(r_2\) increases greatly, the radical quantity decreases.

Let, for example, the initial steam pressure be 10 atm. and \(x_1=1\); let the expansion be carried to 1 atm.

From the tables for saturated water vapor we find

\[ \begin{aligned} p_1&=10\ \text{atm.}; & q_1&=181{,}243; & r_1&=479{,}817; & x_1&=1.\\ p_2&=1\ \text{atm.}; & q_2&=99{,}576; & r_2&=537{,}146; & x_2&=0{,}873 \end{aligned} \]

Here \(x_2\) is determined from the formula

\[ \frac{x_1r_1}{T_1}+\Phi_1=\frac{x_2r_2}{T_2}+\Phi_2. \]

From the tables, \(p_1=10\) atm., \(\Phi_1=0{,}50986\); \(\dfrac{r_1}{T_1}=1{,}0618\) and \(x_1=1\).

\[ p_2=1\ \text{atm.},\qquad \Phi_2=0{,}31108;\qquad \frac{r_2}{T_2}=1{,}4436\ \text{and}\ x_2=? \]

The calculation gives

\[ 1{,}0618+0{,}19878=1{,}4436x_2 \quad \text{and} \quad x_2=0{,}873. \]

Substituting further in the formula for the velocity, we find the value \(w\).

\[ w=91{,}53\sqrt{81{,}667+10{,}889}=880{,}5\ \text{mtr.} \]

Let us now suppose that \(x_1=0{,}95\); for \(x_2\) we find the value \(0{,}836\), and correspondingly for the velocity we shall have

\[ w=91{,}53\sqrt{81{,}667+6{,}772}=91{,}53\sqrt{88{,}439}=860{,}9\ \text{mtr.}; \]

the velocity has decreased by 19 mtr., which amounts to about \(2{,}27\%\).

In 1873 Allaire took out a patent for an injector with superheated steam; he had in view chiefly the attainment of considerable heating of the water; it was also assumed that in this case hotter water could be supplied to the injector. However, his attempt was not successful.

From the calculation of the velocity with wet steam it is evident that, for the injector, the steam should be taken at the place where its dryness is greatest.

The presence of moisture in the steam jet also has an unfavorable effect on the service of the nozzle itself. They are cast of bronze, and especially careful [[unclear: continuation cut off]]

Careful molding and casting are required in order to obtain, at all points of the nozzle, material uniform in alloy and in surface quality. If softer spots appear on the surface of the nozzle, they will be rubbed away by the water passing over them at high velocity. How important it is to have a smooth, polished surface inside the nozzle not only at the beginning of its service but also after the lapse of a certain period may be judged from diagram 16 *). Curves B—E were obtained with a nozzle in which, at the corresponding point, a porous surface was found; the same porosity, scratches, or a poorly polished surface after turning at the entrance to the nozzle will cause compression of the steam jet; on the expansion curve a downward-pointing notch is obtained. Compression of the jet at the entrance entails a decrease in the quantity of issuing steam and a lowering of the pressure in all sections of the nozzle.

The expansion of steam in a nozzle was first applied to an injector by Schau in 1869 in Germany; however, such nozzles were most widely used in America. They had been introduced there as early as 1860, that is, earlier than Schau, by Will. Sellers. Through broadly conducted experimental investigations he succeeded in improving Giffard’s apparatus to such an extent that at the present time, on many railways of France, the inventor’s homeland, Sellers injectors are regarded as the only permissible ones for installation on locomotives.

In the following Table H are given the results of tests of a Sellers injector, type of 1876, carried out in 1879 in Boston.

TABLE H.

Steam pressure in the boiler	Water delivery, liters per hour — Maximum	Water delivery, liters per hour — Minimum	Ratio minim./maxim.	Temperature — Feed water	Temperature — Delivered, at maxim.	Temperature — Delivered, at minim.	Steam pressures required for operation against the back pressure indicated in column I
2.10 kg/cm²	2400	1500	0.625	20° C	50° C	51° C	1.10 kg/cm²
4.20 kg/cm²	3230	1900	0.559	—	54° C	57° C	2.30 kg/cm²
6.30 kg/cm²	3880	1870	0.479	—	59° C	67° C	3.50 kg/cm²
8.40 kg/cm²	4270	2210	0.516	—	65° C	69° C	4.90 kg/cm²
10.50 kg/cm²	4270	2490	0.586	—	71° C	77° C	6.80 kg/cm²

*) Stodola, Dampfturbinen.

M. K. N. Buell; injector No. 6, suction height 1.5 mtr., delivery closed.*)

The highest limiting temperatures for the feed water are shown for various pressures in the following Table J.

TABLE J.

Steam pressure	1,4	2,8	4,2	5,6	7,0	8,4	9,8	10,5
Temperature	59°	58°	55°	55°	56°	56°	53°	53°

If we compare the data of these tables with the results obtained by Villiers in testing Giffard’s injector, we shall have Table K.

TABLE K.

Giffard injector. \(p\) Working steam pressure.	Giffard injector. \(p_1\) Possible counter-pressure.	Giffard injector. Ratio \(p_1 : p\)	Sellers injector. Working steam pressure \(p\).	Sellers injector. Possible counter-pressure \(p_1\).	Sellers injector. Ratio \(p_1 : p\).
2	2,15	1,075	1,10	2,10	1,9
2,5	2,75	1,1	2,3	4,20	1,8
3	3,40	1,13	3,5	6,30	1,8
3,5	4,0	1,14	4,9	8,40	1,7
4	4,75	1,18	6,8	10,5	1,5
4,5	5,35	1,19	—	—	—

At a working steam pressure of 3.5 atm., Sellers’s injector can operate against a pressure \( \frac{6,3}{4} = 1,57 \) times greater than Giffard’s injector.

In the theory of the injector, as it is set forth in many manuals, with a converging conical nozzle a very prominent role is played by the ratio of the outlet area of the steam nozzle to the smallest section of the receiving nozzle. As Giffard established, it should lie within the limits 1.7–2.

*) Procee d. of Civil Engin., XXIV.

By changing this ratio, Rue*) constructed an injector which, at a working steam pressure of 5.5–6 atm., delivers water against a back pressure of 28–30 atm. This is possible, of course, with a large reduction in the quantity of water delivered by it. The advantage of one injector system over another, however, does not consist in this. What is important is that the injector, as a feed apparatus for a boiler, should make it possible to vary sufficiently widely the quantity of water delivered by it, in accordance with the evaporative capacity of the boiler; it is also important that the ratio between the limiting quantities of water delivered by it should vary as little as possible with fluctuations in the steam pressure. Table II shows how good the results were which it was possible to obtain in this respect with the apparatus.

Let us now pass to the determination of the dimensions of the steam nozzle. If the nozzle is a conical converging cone, then the final steam pressure on leaving it will be equal to 0.57 of the initial pressure. By one of the formulae given in the preceding section, we calculate the velocity of discharge of the steam.

Let us suppose that each kgr. of steam delivers \(y\) kgr. of water; let the total quantity of water which must be delivered per second be \(P\) kgr.; consequently, we shall have to expend \(\dfrac{P}{y}\) kgr. of steam per second.

The specific volume of the steam at discharge is

\[ v = xu + \sigma. \]

The volume of the issuing steam will be

\[ \frac{P}{y}\, v. \]

Since the velocity of discharge of the steam is \(w\), the area of the steam nozzle, having the form of a converging cone, will be

\[ \omega = \frac{P}{y}\,\frac{v}{w}. \]

Let us suppose that there is steam of 8 atm. pressure by the pressure gauge. In the outlet section of the nozzle its pressure will be equal to \(0.57 \cdot 8 = 4.56\) atm. If the initial steam content \(x = 1\), then at discharge \(x_2 = 0.9685\).

The velocity of discharge will be 448.8 mtr.

Let us assume that each kgr. of steam delivers 12 kgr. of water; the quantity of water can also be determined as a function of its initial and final temperatures. Let the initial temperature of the water be \(15^\circ\) C., and let it be desired to deliver the water to the boiler heated to \(70^\circ\) C.

) Handbook of the Locomotive*, 7 ed., p. 230.

1 kgr. of steam at the exit from the nozzle contains

$$ 150,918 + 0,9685 \cdot 501,170 = 636,3 \text{ calories}; $$

4.71 calories are contained in the water mixed with it.
Total 641 calories.

We have the equality:

$$ 571 = y(70 - 15) \quad \text{and} \quad y = 10,4 \text{ kgr.} $$

The specific volume of the steam is \(v = 0,3765\).

If the boiler consumes 1200 kgr. of water per hour, then 0.33 kgr. must be supplied per second. *).

Consequently, the area of the outlet opening of the nozzle will be

$$ \omega = \frac{0,33 \cdot 0,3765}{10,4 \cdot 448,8} \text{ sq. meters}, $$

$$ \omega = 0,000026619 \text{ sq. m.} \quad \text{or} \quad 26,62 \text{ mm}^2. $$

The diameter of the outlet opening must be \(\sim 5,82 \dfrac{\text{m}}{\text{m}}\).

If now the nozzle is expanded from the smallest section and the steam is allowed to expand freely to the pressure in the condensation nozzle, which we take as equal to 0.3 atm. abs. **), then the exit velocity in this case will be:

$$ w = 91,53 \sqrt{171,493 - 68,93 + 489,692 - 558,535x_2}. $$

To determine \(x_2\), we use the formula

$$ \frac{r_1 x_1}{T_1} + \Phi_1 = \frac{r_2 x_2}{T_2} + \Phi_2 $$

and, with \(x_1 = 1\),

$$ \Phi_1 = 0,488; \quad \frac{r_1}{T_1} = 1,1 $$

$$ \Phi_2 = 0,2253; \quad \frac{r_2}{T_2} = 1,635. $$

We compute \(x_2\),

$$ x_2 = 0,833. $$

Making the substitution and carrying out the indicated operations, we find the velocity \(w'\) equal to 1018.7 mtr.

The specific volume of the steam is \(0,833 \cdot 5,3 + 0,001 = 4,42\).

*) In practice, an injector is usually taken that supplies 2–3 times the quantity of water.

**) Almost the limiting value for injectors drawing in cold water. See the chapter “Testing of Injectors.”

To determine the outlet area, let us note that it must vary directly as the volumes and inversely as the velocities, i.e.

\[ \frac{\omega'}{\omega}=\frac{v'}{v}\cdot\frac{w}{w'} \]

or

\[ \omega'=\frac{0.000027\cdot 4.42\cdot 448.8}{0.3765\cdot 1018.7} =0.000124\ \text{sq. meter, or }124\ \text{mm}^2. \]

Hence the diameter of the widened nozzle at its end must be equal to 12.6 mm, so that the ratio

\[ \frac{d_1}{d}=\frac{12.6}{5.8}\sim 2.02. \]

We have made the calculation without taking into account the loss of velocity due to friction. Owing to this loss, in order not to retard the motion of the steam, the outlet section of the nozzle must be somewhat increased, depending on the coefficient representing the ratio of the actual velocity to the theoretically calculated one.

In particular, in the case we are considering—the application of a nozzle to an injector—the question remains entirely open as to how to estimate this coefficient and, consequently, by how much to increase the area of the outlet opening. The uncertainty is introduced precisely by the possible drop of pressure in the nozzle, owing to the contact of its walls with cold water, and by the resulting condensation of steam in the nozzle; the data of some experiments indicate that this condensation is considerable and has a noticeable effect on the total steam consumption.*)

Thus, for example, Deloy, in testing an injector operating with steam at 7 atm. pressure, determined the steam consumption to be 0.205 kg per second; when the steam discharged into a reservoir in which it was then condensed, the steam consumption through the same injector was determined (by the weight of the condensate) to be 0.120 kg per second; thus the influence of the water was expressed in an increase of the steam consumption by a factor of 1.7.

An increase in the quantity of outflowing steam was also observed by Zeuner.

All questions concerning the steam nozzle can be solved simply and clearly with the aid of an entropy diagram. We have six quantities, some of which are given and others must be found; these are the steam pressure, its volume, temperature, entropy, velocity, and the area of that section of the nozzle to which they refer.

*) This, however, agrees poorly with the observed rise in temperature of the feed water that has passed through the injector, and with the heating of it calculated from the quantity of steam determined by the formulas of outflow. The difference between the temperatures by observation and by calculation is very small; according to Deloy’s experiments one would have to expect a large discrepancy. In view of this, the reference made to these experiments has no absolute significance and serves only as an illustration of the insufficient investigation of questions connected with the operation of the injector.

Suppose we have steam whose initial pressure is \(p \frac{\mathrm{kgr}}{\mathrm{cm}_2}\); from the tables for aqueous vapors its temperature and entropy are known to us. In a system of rectangular coordinate axes, in which the abscissas are entropies and the ordinates temperatures, we construct the heat diagram (Fig. 17). The process of outflow, as before, we consider adiabatic; for \(x=1\), the adiabat of expansion of the steam in the heat diagram is represented by the straight line \(ab\); let the velocity of the steam at point \(a\) be equal to \(w_0\), and at point \(b\)—to \(w_1\).

In order to obtain from water at \(0^\circ\) C steam whose state is identical with its state at point \(a\) of our diagram, we must expend \(\lambda_0\) units of heat; it is equal to the area \(odcabe\); after expansion of the steam and the conversion of part of its heat energy into kinetic energy, it contains \(\lambda_1\) thermal units; if \(bd\) corresponds to the temperature of the steam, and, consequently, to the pressure to which the expansion is carried, then the heat \(\lambda_1\) contained in the steam will be given by the area \(odbe\). We have:

\[ \frac{A}{2g}(w_1 - w_0)^2 = \lambda_0 - \lambda_1 . \]

Since the initial velocity of the steam is equal to 0, then

\[ w_1^2 = \frac{2g(\lambda_0 - \lambda_1)}{A} \quad \text{or} \]

\[ w_1^2 = \frac{2g}{A} \times \text{area } abdc . \]

With the aid of the same diagram we can find any intermediate velocity.

For example, we want to know the velocity at the moment when the state of the steam corresponds to point \(k\) of the adiabat \(ab\); the heat of evaporation is equal to \(\lambda'\) and is equal to the area \(omke\).

The velocity \(w'\) at point \(k\) is equal to

\[ w'^2 = \frac{2g(\lambda_0 - \lambda_1)}{A} = \frac{2g}{A} = \text{area } acmk . \]

The heats of evaporation \(\lambda_0, \lambda_1 \ldots\), etc., knowing the scale, we can replace by measuring the corresponding area with a planimeter, or else take them from tables for saturated water vapor*).

If the process takes place not adiabatically, but along some curve \(ag\), then in this case also a simple solution is obtained for finding the velocity of the steam at point \(g\).

*) The question of determining the velocity, heat of evaporation, steam content, etc., is solved very simply by I. S. Mollier’s diagram (Z. d. V. d. Ing. 1904, p. 272).

The heat of evaporation for the initial state is given by the area \(ocae\), and at the end of expansion by the area \(odgh\); the velocity of efflux, therefore, will be found from the formula:

\[ w_1^2=\frac{2g}{A}\times(\text{area }abdc-\text{area }ebgh). \]

The velocity of efflux in this case will be smaller; the area \(ebgh\) represents the loss of heat owing to the non-adiabatic character of the process.

From the heat diagram it is clearly seen that, in order to obtain the greatest velocity, we must take the steam as dry as possible.

The closer the expansion curve comes to the limiting curve \(x=1\), the smaller will be the quantity of heat transformed into kinetic energy.

Assuming, as before, the process to be adiabatic, we can calculate the velocity by prescribing a known percentage of energy loss. Let us suppose that the loss is \(18\%\); then on the line \(db\), the line of the steam temperature to which we bring the expansion, we set off from point \(b\) a segment \(bg'\) of such magnitude that the area \(bg'eh'\) amounts to \(0.18\) of the area \(abdc\). In the actual case, in the non-adiabatic expansion of the steam along the curve \(ag'\), its velocity is determined by the difference of the areas \(abdc\) and \(bg'eh'\); since \(bg'eh'=0.18\,abdc\), the kinetic energy of the steam (velocity \(w_2\)) will be

\[ A\frac{w_2^2}{2g}=0.82\text{ of the area }abdc. \]

Thus, with the aid of the heat diagram, we shall find the velocity of the steam in any of its sections. Let us see how to determine the volume of the steam in any section where its velocity is known, or, what is the same thing, for that state of it which is given by some point on the heat diagram.

We construct, as before, the heat diagram in rectangular coordinate axes; along the axis of abscissas to the left of point \(O\), on some scale, we set off the steam pressures expressed in \(\mathrm{kg}/\mathrm{cm}^2\); since the ordinate axis upward from point \(O\) represents temperatures, in the upper left-hand quadrant we obtain a curve characterizing for us the variation of the steam pressure as a function of its temperature \([p=f(t)]\). (Fig. 18).

Downward along the ordinate axis we set off, to scale, the volumes of steam corresponding to its pressures \(p_1, p', p''\), etc., and construct the curve \(MN\), representing the law of variation of volumes with pressure at constant steam content \(x=1\).

Let the given state of the steam be indicated to us on the heat diagram by point \(a\); we wish to find the volume of the steam.

We draw a straight line parallel to the axis of abscissas until it intersects the curve \(p=f(t)\); from the point of intersection we draw a vertical to the intersec-

... with the curve \(v=f_1(p)_{x=1}\), and again parallel to the axis of abscissas through the point thus found of intersection of the vertical with the curve \(v=f_1(p)\).

From the point of intersection of the straight line passing through \(a\) parallel to the axis of abscissas, and the boundary curve \(x=1\), we drop a vertical and seek the point of its meeting with the preceding straight line; from the point of intersection of that same straight line through \(a\) with the boundary curve \(x=0\), we drop a perpendicular to the axis of abscissas; we join the points \(d\) and \(c\); drawing from \(a\) a vertical downward, and from the point where it meets the straight line \(cd\) a horizontal \(a_1a'\), we find the point \(a'\) as its intersection with the vertical corresponding to the given steam pressure at the point \(a\). The latter is thus transferred from the heat diagram to the diagram of \(p\) and \(v\).

The drawing shows a second construction for point \(b\).

The basis of this construction is the relation between \(p\), \(v\), and the entropy, given by Clapeyron’s equation:

\[ \frac{r}{u}=AT\frac{dp}{dt}\quad \text{or} \]

\[ Au=\frac{r}{T}\cdot\frac{dt}{dp}. \]

The segments of the horizontals between the two boundary curves in the heat diagram give, in a certain scale, the quotient \(\frac{r}{T}\); \(\frac{dp}{dt}\) is the tangent of the angle of the tangent to the curve \(p=f(t)\), which it makes with the axis \(t\) on the side of positive \(p\) (drawing 19), i.e. \(\frac{dp}{dt}=\tg\beta\).

Construct at the point \(a\) the angle \(\alpha=90-\beta\), and extend the side of the constructed angle to its intersection with the adiabat \(x=1\) at the point \(c\). We shall have

\[ \overline{bc}=\frac{r}{T}\tg\alpha=\frac{r}{T}\tg(90-\beta)=\frac{r}{T}\cdot\frac{1}{\tg\beta}. \]

But

\[ \tg\beta=\frac{dp}{dt}; \]

therefore,

\[ \overline{bc}=\frac{r}{T}\cdot\frac{dt}{dp}, \]

and the segment \(\overline{bc}\) gives, in scale, \(Au\).

We project the point \(a\) onto the axis of abscissas and construct at \(a'\) the same angle \(\alpha\) and the segment \(b'c_1'\); we project it onto the vertical corresponding to the pressure \(p\) at \(b_1c_1\); the point \(c_1\) belongs to the curve \(pu\).

It is known that \(v=u+\sigma x\); since \(x=1\), then

\[ v=u+\sigma. \]

We see that the axis of the diagram \(pv\) lies by \(\sigma\) higher than our horizontal axis, and, owing to the smallness of the quantity \(\sigma\), coincides with it; that is, \(c_1\) is also the point belonging to the diagram \(pv\).

Thus one may give the path for constructing the saturation curve in the same lower quadrant [i.e., by means of tangents to the curve \(p=f(t)\)]; but it can be constructed more quickly by using tables for water vapor; the left-hand boundary curve for \(x=0\) in the \(pv\) diagram will obviously be represented by a straight line coinciding with the axis of abscissas, since we neglect the quantity \(\sigma\).

Thus, any point of the heat diagram we shall transfer, by graphical construction, to the \(pv\) diagram, and at every instant of the expansion of the steam we shall find the volume occupied by it.

Let us now turn to Drawing 20.

From the initial pressure the steam expands to the pressure \(p'\) with a loss of energy equal to \(15\%\) (area \(b_1cfe=0.15\,abb_1d\)); the curve \(p=f(t)\) and \(p=\varphi(v)\) are constructed; with the aid of the graphical method indicated above, the curve of steam expansion is transferred to the left lower quadrant. For any point we find the numerical value of the velocity and the volume of the steam \(v\); laying off horizontally, along the line passing through \(v_k\), this velocity on a known scale, we obtain the curve of velocities, which will pass through the point \(O\), the origin of coordinates. One point of this curve, namely the point of its intersection with the curve \(p=\varphi(v)\), is determined directly as the critical velocity, depending on the pressure \(0.57\) of the initial pressure. The volume of steam determined by this point corresponds to the orifice of the steam nozzle. If the quantity of steam issuing per second is \(1\ \mathrm{kgr.}\), then the equality holds

\[ F \cdot w = Gv = v, \]

since \(G=1\).

Hence follows a simple method for finding graphically the area of the nozzle \(F\) at the place where the steam has the velocity \(w\). The straight line \(AB\) is drawn vertically downward at a distance from the axis of ordinates which, on the scale, gives \(1\ \mathrm{kgr.}\) of steam.

Join the point \(n\) (velocity \(w\)) with the origin of coordinates and extend the straight line \(On\) to its intersection with \(AB\); we have the point \(q\). The segment \(Aq\) gives the number of square units of area; indeed, from the similarity of the triangles \(Ons\) and \(OAq\) it follows:

\[ ns : Os = AO : Aq \]

or

\[ Os \cdot AO = ns \cdot Aq \]

or

\[ v \cdot 1 = w \cdot Aq = wF. \]

Further, at the point \(q\) we construct the angle \(\alpha=38^\circ 4'\) and extend its side to its intersection with \(AO\) at the point \(z\); the segment \(qz\) is equal to \(d^2\); indeed, we have:

\[ Aq = qz \cdot \cos\alpha. \]

\[ F = 0.784 \cdot d^2. \]

Laying off from the straight line \(AB\), to the right and to the left, \(\dfrac{d}{2}\) along the horizontals for the corresponding velocities \(w\), we find the profile of the nozzle through which one kilogram of steam passes per second; let the area of the inlet be \(\omega\) and that of the outlet \(\rho\); the nozzle must be made for a steam consumption of \(D\) kg; obviously, its inlet area must then be \(\omega D\) and its outlet area \(\rho D\). In this way we shall find the profile of the nozzle in which the expansion of the steam occurs with the loss of energy adopted by us, and through which a fully definite quantity of steam, \(D\) kg per second, flows.

We have already indicated above a peculiarity of the injector, first noted by Giffard and then by Deloy ), namely that it can deliver water against a back pressure greater than the pressure of the steam working in it; but this was not duly appreciated by them. Deloy, for example, observed that the ratio between the back pressure and the pressure of the working steam is equal to the ratio of the area of the outlet opening of the steam nozzle to the area of the inlet ). Only Hamer and Davies **) paid attention to this peculiarity from the practical side, by designing an injector for operation with exhaust steam.

We saw above that the kinetic energy of the jet is obtained at the expense of the heat of the steam; but it is known how little heat need be imparted to steam in order to raise its pressure from one atmosphere; for example, to obtain steam at a pressure of one atmosphere, 1 kg of water must be supplied with 637 calories, while to raise its pressure to 7 atm., it is enough to add only 20 calories in all; in this, indeed, lies chiefly the uneconomical nature of steam engines; even with condensation of the steam, when the pressure in her cylinder falls to 0.2 atm., the steam contains on average up to 80% of the heat required for its production and not used for mechanical work. Hence the low efficiency of the steam engine as a heat engine.

How great the energy of the steam exhausted in the cylinders of the engine, i.e. of the so-called exhaust steam, may be is seen from a numerical example.

Let us suppose that a simple-expansion engine works with an initial steam pressure of 7 atm.; the steam content \(x = 1\). When the steam expands to 1.2 atm., part of it condenses, but at the beginning of the compression period the quantity of water contained by one kg of steam will not be the same as at the end of expansion; what it actually is cannot be determined exactly. Hirn, for engines with condensation, considers that \(x\)

) Annal. des Mines*, 1860, XVII.

**) See above, the Rue injector, constructed in departure from the usual rule and forcing water against a very great back pressure.

*) Revue Industrielle, 1883.

will be unity; Zeuner, assuming \(x=1\) at the beginning of the operation and the steam not superheated, considers the steam content \(x=0.808\) and even lower.*)

We shall take \(x=0.8\).

When wet steam issues from a converging conical nozzle into the condensing chamber of the injector, the pressure in which may be taken as \(0.4 \dfrac{\mathrm{kgr}}{\mathrm{cm}^{2}}\), the pressure at the outlet orifice will be \(0.57 \cdot 1.1\) (we take the pressure of the steam entering the nozzle as \(1.1 \dfrac{\mathrm{kgr}}{\mathrm{cm}^{2}}\)); consequently, the pressure at the outlet is

\[ 0.627 \dfrac{\mathrm{kgr}}{\mathrm{cm}^{2}}. \]

We determine \(x_2\) from the formula

\[ \frac{r_1 x_1}{T_1}+\Phi_1=\frac{r_2 x_2}{T_2}+\Phi_2. \]

From the tables, for

\[ p=1.1,\qquad q=102.281;\qquad \frac{r}{T}=1.4283;\qquad \Phi_1=0.31833;\qquad x=0.8;\qquad r=535.255 \]

\[ p=0.6,\qquad q=85.818;\qquad \frac{r}{T}=1.5252;\qquad \Phi_2=0.27341;\qquad x=?;\qquad r=546.754. \]

\[ 1.4283 \cdot 0.8+0.31833=0.27341+1.5252x_2; \]

\[ x_2=0.78. \]

The velocity of efflux is

\[ w=91.53 \sqrt{q_1-q_2+r_1x_1-r_2x_2} \]

\[ w=390\ \mathrm{metr}. \]

We see that it differs little from the velocity of efflux of high-pressure steam through the same nozzle. We shall suppose that for \(1\ \mathrm{kgr}\) of steam there flows in \(18\ \mathrm{kgr}\) of water having a velocity of \(10\) meters; to such a velocity there corresponds a head of \(5.1\ \mathrm{mtr.}\), i.e. the rarefaction in the chamber must be about \(1/2\) atmosphere, which is quite possible. By the theorem on quantities of motion we have:

\[ 390+18\cdot 10=19v. \]

and

\[ v=30\ \mathrm{mtr}. \]

Such a velocity corresponds to a head of \(45.8\ \mathrm{mtr.}\); if the height of a water column for one atmosphere is taken as \(10\ \mathrm{mtr.}\), then the jet can overcome, neglecting harmful resistances, a pressure on the feed valve from the boiler side of \(4.58\) atm. This agrees with practical data. According to Sellers’s experiments,**) with wet steam it is possible to force water into a boiler with a pressure up to \(5\) atm., while the back pressure on the piston of the engine does not increase at all; with a back pressure from \(5\) to \(12\)

*) Zeuner, Technische Thermodynamik, S. 396 et seq.
**) American Mashinist, 1882.

atm. one may operate with the same injector, attaching to it a nozzle for live steam, which gives the final strong push to the water.

Figs. 21 and 22 show indicator diagrams of a steam engine, taken with Richard’s indicator for cases when the injector was working and was not working. In the first case the pressure on the back side of the machine piston was \(0.14 \dfrac{\mathrm{kg}}{\mathrm{cm}^2}\) less than in the second *).

The injector consumes about \(6.5\%\) of the steam produced by the boiler; if the steam-producing capacity of the boiler is \(2000\ \mathrm{kg}\), then the use of exhaust steam, when it is not consumed for other purposes, e.g. heating, preheating water, etc., can save these \(130\ \mathrm{kg}\) of steam per hour; the water will then enter the boiler at a temperature of \(70\text{--}90^\circ\).

The wide spread of such injectors is hindered, however, by the circumstance that, together with the exhaust steam, the oil contained in it enters the boiler; in the best case this oil will merely cause foaming of the water in the boiler and thereby hinder the production of dry steam. To avoid contaminating the boiler with oil, it is necessary to install a filter for the exhaust steam, which first separates the emulsion of water with oil and then filters the steam itself. In steam-turbine installations, the use of an injector operating with exhaust steam should find wider application.

In view of the considerably larger volume of steam having an elasticity of \(1.1\) atm., the dimensions of its nozzle are more considerable. Their determination may be made according to the formulas indicated above.

) Pompes et machines à élever les eaux*, Poillon.

Condensing Nozzle

Below, in the chapter on the receiving nozzle, we shall see that the area of its orifice and the form of the nozzle following it serve as the chief decisive factors in the operation of the injector as a feeding apparatus. But the role of the condensing nozzle is hardly less important: only thanks to the condensation of the steam does the action of the apparatus become possible, and only with those gaseous bodies which are capable of condensing can apparatuses of this kind be obtained. M. Truchot*) succeeded, namely, in reproducing the action of the injector by replacing water vapor with a jet of ammonia gas; although here there is not the phenomenon which we call condensation, there is instead the phenomenon of rapid absorption and dissolution of the gas jet by water. Nevertheless, the influence of the dimensions of the nozzle on the rapidity and completeness of condensation remains to this day little studied and clarified, and it is precisely in this part of the apparatus that, perhaps, many improvements can still be introduced which will contribute to the general perfection of the apparatus. The condensing nozzle, says Giffard, constitutes that part of the apparatus which must mainly be determined by experiment, both in its appearance and in its dimensions.**) All manufacturers of injectors still adhere to this rule, giving the nozzle that form which, according to the designer’s views, should best satisfy its purpose. For each injector it may indeed be the best, but only for certain, quite definite conditions; and as soon as the circumstances under which the injector has to operate change and no longer correspond to these conditions, the apparatus at once becomes little able to satisfy the existing requirements. The process taking place in this nozzle is, however, so complex that to express and determine it exactly and rigorously by mathematical formulas appears completely impossible. One can only indicate the general phenomenon of steam condensation as it occurs, determine the conditions which favor it, cite indications obtained experimentally, and already on this basis draw possible and useful conclusions.

*) La Nature, 1879.
**) Ann. des Mines, XVII.

If a jet of air, flowing out at high velocity, strikes the surface of water, we observe how, from the place of impact, the liquid is driven away, and its particles are set into a more or less strong oscillatory motion. If the velocity is very great and the water at the place of impact is not deep, then even the splashing of the liquid and its scattering to the sides is noticeable.

When steam strikes the surface of water, the same thing occurs, but in addition its condensation also appears: part of the steam condenses at the place of impact, while another part of it departs from this place; the steam spreads over the surface of the water, condensing in other places. From this follow the following two requirements with respect to the condensing nozzle.

The jet of steam flowing out of the steam nozzle has, as we have seen, a very high velocity, while the water entering the condensing nozzle proceeds in a comparatively thin layer. The shape of the condensing nozzle must be such that the jet of water is not broken up and does not splash at the moment when the steam meets the water, but would fully preserve its continuity.

The second requirement is that the form and length of the nozzle must be such that condensation is actually possible in it; if the nozzle is too wide or too short toward its outlet end, then perfect condensation cannot occur.

Whether complete condensation can be achieved in any injector whatever is difficult to say; the state of the mixture at the moment of its passage through the mouth of the receiving nozzle is very difficult to determine, and one has to judge it by the external appearance of the jet in the interval between the condensing and receiving nozzles. According to observations, the jet has a milky-white appearance, is opaque, and resembles a foamed sea wave. It is possible that this appearance is imparted to it by particles of steam contained in the water which have not thickened; but the jet can have this appearance also in the case when the water is strongly saturated with air, or when vortical motions take place inside the liquid. Zeuner*) observed exactly the same appearance of the jet in water-jet apparatuses, if one jet possessed considerable velocity.

Water always contains air; when it enters with it into the condensing chamber, where the pressure is slight and where rapid heating of the water occurs, this air is released. Its amount may then be considerably greater than the usual content in water, owing to the looseness and lack of tightness of the suction pipe and its connections. Minary et Resal**) determined, for example, its

*) Technische Thermodynamik.

*) Ann. des Mines.* 1862, 1, 6th series.

quantity in 1 liter per 1 kg of water. We shall speak further below about how incomplete condensation of the steam and an admixture of air affect the operation of the injector.

If we know the pressure and the specific volume of the steam \((v=ux+\sigma\), i.e., chiefly we need to know \(x\)—the specific steam content), then, specifying the initial and final temperatures of the water, we can at once determine the quantity of it which must enter simultaneously with it into the condensing nozzle. Suppose the pressure \(p,x\) are given; from tables for water vapor we find the external and internal heats of the steam. Let \(D\) kg of steam flow per second through the steam nozzle; let \(t_a\) denote the initial temperature of the water; the final temperature must be \(t_c\); determine \(P\)—the quantity of water, in kg, necessary for complete condensation of the steam.

In \(D\) kg of mixture there will be \(Dx\) kg of steam and \(D(1-x)\) kg of water; the heat of the steam will be \(Dx(q+\rho+Ap u)\), or, since \(\rho+Ap u=r\), \(Dx(q+r)\); the heat of the water contained in the steam is \(D(1-x)q\). After condensation there is formed \((D+P)\) kg of water, which at \(t_c\) will contain \((D+P)t_c\) units of heat; the heat of the cold water is \(Pt_a\).

We have the equality, neglecting the insignificant losses from radiation and transport by the air,

\[ Dx(q+r)+D(1-x)q+Pt_a=(D+P)t_c . \]

Hence we find \(P\).

\[ P=\frac{D(q+rx-t_c)}{t_c-t_a}\ldots\ldots\ldots(1). \]

Dividing by \(D\), we obtain

\[ \frac{P}{D}=\frac{q+rx-t_c}{t_c-t_a}\ldots\ldots\ldots(2) \]

an expression which enables us to draw certain conclusions as to how the quantity of water referred to 1 kg of steam changes depending on its final state and on the initial and final temperatures of the water. In connection with the equality obtained from the theorem of conservation of momentum, it is the fundamental equation in the theory of the injector.

It should be noted, however, that it determines exactly only the quantity of water which must be supplied for condensation of the steam; if it is not desired that the actual delivery temperature be higher than \(t_c\), an additional quantity of water must be introduced, which must absorb the heat obtained from the conversion, upon impact, of kinetic energy back into heat.

From (1) it is seen that \(\frac{P}{D}\) must decrease if \(t_c\) is to increase; the lower \(t_a\) is, the less water may be introduced per 1 kg of steam consumed for its condensation; the greater the steam content \(x\), the greater the quantity of water that must be supplied for condensation. If the escaping steam is always dry, so that \(x = 1\), then the quantity of water introduced depends almost not at all on the sum \((q + r)\), since with a change in pressure it changes insignificantly.

Experience indicates that suction injectors can operate with a suction lift of up to 7 m, i.e. the vacuum in the condensation chamber reaches 0.3 atm. If it is assumed that, on average, the vacuum in this nozzle reaches 0.45, then the total heat of dry saturated steam will be 630 calories. For this number the following table has been calculated, indicating how many kg of water are needed at various \(t_a\) and \(t_c\) for the complete condensation of one kg of steam.

This table gives a clear representation of how \(\frac{P}{D}\) changes depending on the initial and final temperatures of the water. It may serve for an approximate assessment of the working capacity of the injector and of its perfection with respect to completeness of condensation.

The condensation of steam does not occur instantaneously; only the steam, upon leaving the steam nozzle, enters the condensation chamber; for this it is necessary to allow it a certain time. The duration of the process varies depending on the surfaces of contact of the steam with the water; for the same weight quantities of steam and water, the time required for condensation varies inversely with these surfaces; for one and the same contact surface it will vary proportionally to the weight of the steam being condensed.

In the injector, water is supplied through the annular space formed by the steam and condensation nozzles; in the nozzle itself the cone of water envelops the jet of steam from the outside. With an insignificant nozzle length and sufficient water velocity (up to 10 m), it may be assumed that in its motion the water is directed only by the walls of the nozzle and is not acted upon by, for example, gravity. Since water is a poor conductor of heat, the transfer of heat from the steam to the water can occur only along the surfaces of their immediate contact; as it moves toward the outlet end of the nozzle, the water jet will be compressed more and more by its walls, enveloping the jet of condensing steam. It is possible that, before the very exit from the nozzle, or even from the moment of meeting of the steam and water, heat transfer and direct mixing appear; probably, after the impact, the water jet proceeds as a thinner layer than it enters the nozzle, and there occurs, so to speak, a stretching out of the layer.

otherwise the transition from the inlet velocity to a velocity 2–4 times greater would be associated with rupture of the jet; the action of the steam on the water would in this case consist of a series of rapidly succeeding shocks and would be unstable; this is evident, for example, from the fact that some injectors cease operating in the event of shocks or shaking of the suction pipe.

Initial temperature $t_a$	20°	25°	30°	35°	40°	45°	50°	55°	60°	65°	70°	75°	80°	85°	90°
Kgr. of water per 1 kgr. of steam at final temperature $t_c$.
5°	44,3	30	23,8	19,7	16,7	14,5	12,7	11,4	10,3	9,4	8,6	7,93	7,33	6,8	6,35
6°	45,2	31,5	24,7	20,5	17,2	14,9	13	11,6	10,5	9,6	8,75	8,0	7,43	6,9	6,43
7°	46,5	33,3	25,6	21,3	17,8	15,2	13,3	11,8	10,7	9,74	8,88	8,16	7,53	7,0	6,5
8°	50,5	35,3	27	22	18,3	15,7	13,7	12,13	10,9	9,9	9,03	8,28	7,64	7,1	6,6
9°	55	37,5	28,3	23	18,9	16,1	14	12,4	11,1	10,1	9,18	8,4	7,73	7,17	6,66
10°	60,5	40,0	29,3	24	19,6	16,4	14,4	12,7	11,3	10,3	9,33	8,54	7,85	7,25	6,8
11°	66,2	42,9	31,3	24,6	20	17,1	14,8	13	11,5	10,5	9,5	8,67	7,97	7,36	6,83
12°	75,6	46,2	33	25,6	20,9	17,6	15,1	13,25	11,8	10,66	9,65	8,8	8,1	7,46	6,9
13°	86,4	50	35	26,5	21,5	18,1	15,4	13,6	12	10,86	9,8	8,95	8,2	7,57	7,0
14°	101	55	37,2	28,1	22,5	19	16	14	12,3	11,1	10	9,1	8,3	7,67	7,1
15°	121	60	39,6	29,5	23,4	19,7	16,4	14,25	12,6	11,3	10,2	9,25	8,5	7,8	7,2
16°	152	66	42,5	31,1	24,1	20	16,9	14,6	12,83	11,5	10,4	9,4	8,6	7,9	7,3
17°	202	75	45,6	33	25,4	20,7	17,4	15	13,15	11,75	10,6	9,57	8,7	8,0	7,4
18°	303	86	49,6	34,5	26,6	21,5	18	15,4	13,4	12	10,8	9,74	8,9	8,13	7,5
19°	—	100	54,1	36,5	27,8	22,3	18,5	16	13,8	12,3	11,0	9,9	9	8,25	7,6
20°	—	120	59,5	39,5	29,3	23,2	19,1	16,3	14,1	12,5	11,2	10,1	9,2	8,4	7,7
21°	—	150	65,0	42,1	30,8	24,1	19,8	17	14,5	12,84	11,4	10,3	9,3	8,5	7,8
22°	—	200	74,4	45,4	32,4	25,1	20,6	17,3	14,8	13,1	11,65	10,5	9,5	8,65	7,94
23°	—	—	84,4	49,5	34,4	26,4	21,3	17,8	15,3	13,4	11,9	10,7	9,65	8,8	8,0
24°	—	—	99,2	53,6	36,5	27,6	22,1	18,4	15,7	13,8	12,2	10,9	9,8	8,93	8,18
25°	—	—	119	59	38,5	29,3	23	19	16	14,1	12,4	11,1	10	9,1	8,3
26°	—	—	149	65,6	42	30,5	23,9	19,6	16,4	14,5	12,7	11,3	10,2	9,24	8,43
27°	—	—	—	74,3	45	32,2	25	20,5	17,1	14,9	13,0	11,6	10,4	9,4	8,57
28°	—	—	—	84,3	49	34,1	26,14	20,7	17,7	15,3	13,3	11,8	10,6	9,6	8,7
29°	—	—	—	98,3	53,2	36,2	27,4	21,5	18,2	15,7	13,6	12,0	10,8	9,73	8,85
30°	—	—	—	147	58,5	38,6	28,75	22,4	19,2	16,1	14,0	12,3	11	9,9	9,0
31°	—	—	—	197	65	41,4	30,3	23,3	19,5	16,6	14,4	12,6	11,2	10,1	9,1
32°	—	—	—	—	73	44,6	32	24,1	20,2	17,0	14,7	13	11,5	10,3	9,3
33°	—	—	—	—	97,5	48,3	33,8	25,4	20,5	17,6	15,1	13,2	11,7	10,5	9,5
34°	—	—	—	—	117	53	35,9	26,7	21,7	18,2	15,5	13,5	11,9	10,7	9,64
35°	—	—	—	—	149	58	38,3	28	22,6	18,8	16	13,87	12,2	10,9	9,8
36°	—	—	—	—	—	—	41	29,4	23,5	19,5	16,5	14,2	12,5	11,1	10,0
37°	—	—	—	—	—	—	44,2	31,1	24,6	20,2	17,0	14,6	12,8	11,3	10,2
38°	—	—	—	—	—	—	48	33	25,7	20,9	17,5	15,0	13,1	11,6	10,4
39°	—	—	—	—	—	—	52,5	35	27	21,7	18,0	15,4	13,4	11,85	10,6
40°	—	—	—	—	—	—	57,5	37,3	28,3	22,6	18,6	15,9	13,75	12,1	10,8

Hence it is evident that the condensation nozzle, in its dimensions at the outlet end, must in every case correspond to a certain...

conditions of the condensation of the steam occurring in it, i.e. the state of the mixture; it is also clear that the dimensions of the nozzle therefore cannot be determined by any, even approximate, formulas, and only by experiment can one determine whether it is suitable for the given conditions or not.

The question may arise whether it is not better to bring the steam and water into the apparatus in the reverse position relative to one another, i.e. so that the steam jet surrounds the water jet. Engineer M. Audenet*) in France carried out experiments with Morton’s ejector condenser, intended to clarify in which case a greater vacuum is attained in the condenser and, in connection with this, the least back pressure on the piston of the steam engine. It turned out that the best results from a series of experiments with nozzles of various sizes were obtained in the case when the water jet surrounds the steam jet from outside. Consequently, in the injector as well, the arrangement of the steam and water jets as it is carried out is entirely correct.

Thus, for rapid condensation it is necessary to observe the following conditions:

1) The surface of the water must be as large as possible. Since, for a given volume, the surface will be the greater the thinner the layer of water, the water must be brought in, as far as possible, in a thin layer. Moreover, in order that there be enough water under these conditions, it must have as high an entrance velocity as possible. In a suction injector this velocity is produced only by a head equal to the difference between atmospheric pressure and that in the condensing nozzle. If this difference is a quite definite quantity, then the action of the apparatus should be better the smaller the suction height, since then there will be fewer harmful resistances reducing the velocity and requiring an increase in the thickness of the layer. At the same time the mechanical action of the injector will also improve, since the loss upon impact will be the smaller, the smaller the difference between the velocities of the colliding bodies. The following table gives the ratios of the surface in mtr.\(^2\) to the volume in mtr.\(^3\) for various thicknesses of the water layer in m/m for 1 kgr. of water.

Thickness of the water layer	1	2	3	4	5	6	7	8	9	10	11	12
Ratio \(\dfrac{s}{v}\)	1	0.5	0.33	0.25	0.2	0.1667	0.1429	0.125	0.111	0.1	0.09	0.083

It is evident how rapidly this ratio decreases with an increase in the thickness of the layer.

) Ledieu, Les nouvelles machines marines*.

The correctness of the stated proposition is confirmed by the results of Audenet’s experiments. With a water-layer thickness of 1 mm, the greatest vacuum was obtained; with layer thicknesses of 2 mm and 0.5 mm it was already smaller, and in the latter case, probably because the continuity of the jet was disturbed, the surface was torn apart by the steam.

2) The surface of the water must change as rapidly as possible.

This will be accomplished with a sufficient velocity of the water. Hence the requirement is clear that the greatest possible vacuum should always exist in the condensing nozzle.

3) Sufficient time must be allowed for condensation.

This requirement will be satisfied with a sufficient length of the nozzle. The greater the velocity with which the jet passes through the nozzle, the longer the nozzle must be. If the injector works well at low steam pressures, then at high pressures its operation will be less satisfactory; the percentage of uncondensed steam must be greater, the greater the velocity; the vacuum in the nozzle decreases and the apparatus finally ceases to act.

4) All joints of the suction pipe and of the injector connected with it must be absolutely airtight.

The penetration of air into the condensing nozzle will affect the degree of vacuum; but, in addition, its presence is harmful to the process of condensation, since, by virtue of its poor heat conductivity, it will impede the transfer of heat from the steam to the water, forming air layers in the latter.

Of these conditions, the first two must be regarded as especially important for the operation of the injector.

The coefficient of heat transfer of water is \(\lambda = 0.155\), i.e., through a sq. metre of the surface of the water, with a layer thickness of 1 mm and a temperature difference of the layer surfaces of \(1^\circ\) C., 0.155 calories pass in the course of one second.

If \(H\) is the surface of the water in square metres, \(z\) the time in seconds during which condensation takes place, \(\eta\) the thickness of the layer of water, and \(\vartheta\) the temperature difference on the surfaces of the layer, then the quantity of heat transmitted into the interior of the layer will be:

\[ Q = \frac{\lambda H z \vartheta}{\eta} \ldots\ldots (3) \]

Let the initial temperature of the water be \(15^\circ\); the final temperature of the steam, on its expansion to 0.45 atm., \(78^\circ\); then, for \(H = z = \eta = 1\),

\[ Q = 9.765 \text{ calories.} \]

Still less will be transmitted to the second layer, since \(\eta\) increases and \(\vartheta\) decreases.

On the other hand, the coefficient of external thermal conductivity, or the coefficient characterizing the ability of water to absorb heat, according to Péclet, may be taken as 3.2 calories per 1 sq. meter per second for a temperature difference of 1°.*) With the same temperature difference, \(78-15=63^\circ\), 1 sq. m per second can take 201.6 calories from the steam, i.e. almost 20 times more. It is clear how important it is to introduce the water in a thin layer at the greatest possible velocity, so that its surface is rapidly renewed.

Thus, for the process of condensation, the size of the inlet area for the water is of great importance; this was correctly appreciated by Giffard and by the first experimenters, who made the nozzle movable in order to regulate the quantity of incoming water, establishing, for a given velocity, its inflow in the thinnest possible layer.

As for the third condition, notwithstanding all its importance, its exact observance must almost be abandoned, since otherwise this would be done to the detriment of the compactness of the apparatus. The length of the nozzle in injectors of various types, in relation to the diameter of the mouth of the receiving nozzle, varies depending on the steam pressures and the temperature of the feed water. The higher the one and the other, the longer the nozzle must be. By observation Giffard established that, with the diameter of the mouth taken as 1, the length of its rear part should be

    at low pressures up to 3.5 atm. ........ 6.
    at medium and high pressures up to 10 atm. ............... 10.
    at very high pressures ....................... 14.
    the length of the nozzle in the front part for all cases ... 4.

It turns out, however, that even such a change of length from 6 to 10 is small: analyzing the results of tests, Giffard came to the conclusion that the density of the jet as it passes through the mouth of the receiving nozzle is 0.66; since for water it is 1, it is therefore evident that condensation is far from perfect. Villiers’s experiments**) indicated that only 90% of the steam is condensed, while 10% remains uncondensed upon entering the receiving nozzle; on this basis, for pressures up to 8 atm. the length of the nozzle should be made still somewhat greater, bringing it to 14, and to 18 at higher pressures.

Let us note that the interval between the condensing and receiving nozzles has no significance for the operation of the injector; without it, the forcing

*) This coefficient depends on the velocities with which the surfaces of the heated and cooled liquids move relative to one another. The indicated coefficient applies to the transfer of heat through a copper thin-walled tube; with direct contact of steam and water, it is probably considerably greater.

**) Ann. des Mines, 1860, XVII.

would occur just as successfully as with it. It is made only so that one may directly observe whether the injector is working, or why its action has ceased.

The necessity of regulating the quantity of incoming water may be seen from the following consideration. Suppose that the injector delivers the greatest possible quantity of water, after first sucking it up to a height of 1 meter, at a certain steam pressure.

Let us imagine that the steam pressure has increased; whatever the form of the nozzle, the quantity of steam flowing out will immediately increase; for the same successful condensation as before, it is necessary also to increase the quantity of water introduced. This may be done either by increasing the velocity, and consequently the head, or by increasing the inlet area. If the expansion of the steam is carried to the mean pressure in the condensing nozzle, the rarefaction in it can vary only very slightly, and an increase of velocity can be attained only by arranging the inflow under pressure from an upper tank; but an increase in the quantity of water may be attained, at the same inlet velocity, by increasing the annular clearance between the nozzles, by moving the condensing nozzle away from the receiving nozzle.

In Fig. 23 it is shown what must take place when there is a complete absence of water inflow: all the steam escapes through the overflow opening or the overflow pipe, doing no work. With an insufficient inflow of water, it will be broken up by the steam into small particles, some of which will enter the receiving nozzle; but since a continuous motion of the jet will not be established in the latter, it cannot lift the valve of the feed pipe, and the water will not go into the boiler.

If now, conversely, we imagine that the steam pressure is decreasing, then correspondingly the quantity of steam and its velocity also decrease. Water will flow in in excessive quantity, and although the condensation will be more complete, the steam will not be able to impart to it the proper velocity, and part of the water will be lost through the overflow pipe. If the steam pressure falls still lower, the water will have so little kinetic energy that it will be impossible for it to overcome the back pressure at the valve, and in large quantity it will begin to drain through the overflow pipe. Fig. 24 shows just this case; so it will be when the dimensions of the overflow opening are sufficient; but if all the water is not able to leave through the overflow pipe, braking will occur: the steam will push the water out of the suction pipe, thereby for a time freeing the overflow chamber; the injector will again take up water, and the same thing will again be repeated, as soon as the water level in the overflow chamber covers the outlet opening of the condensing nozzle.

In Fig. 25 there is shown schematically the condensation nozzle of the Rue injector, in which its position can be changed by increasing or decreasing the annular clearance; the dotted line indicates its extreme forward position. With a steam pressure of 6 atm., the injector delivered the greatest quantity of water without loss through the overflow; the distance of the end of the condensation nozzle from the line \(ab\) was found to be equal to \(15\ \mathrm{m/m}\); the suction lift was \(0.3\ \mathrm{mtr}\). When the steam pressure was raised to 8.5 atm., the temperature of the delivered water rose from 46 to \(74^\circ\) C.; since this temperature indicated an insufficiency of the water supplied, the nozzle was moved back another \(10\ \mathrm{m/m}\); working without overflow, the injector now delivered water at a temperature of \(64.5^\circ\). With the same initial steam pressures and nozzle position, the water level in the reservoir was lowered, so that the injector had to operate with a suction lift of \(1.8\ \mathrm{mtr}\); the quantity of water supplied per \(1\ \mathrm{kgr}\) of steam decreased, and the temperature rose from \(46^\circ\) to \(52.5^\circ\). The condensation nozzle was again moved back; the injector began to deliver the normal quantity of water for it at the same delivery temperature of \(46^\circ\).

These experimental data show that an injector with a fixed nozzle, when the steam pressures or the suction lift change, unless it attains the possible limiting values, will not be able to operate with equal economy. How many variations there may be in the position of the steam and condensation nozzles relative to one another is indicated by the following table*).

Suction lift.	Steam pressure.	Distance of the front end of the condensation nozzle from the extreme position. At the greatest delivery.	Distance of the front end of the condensation nozzle from the extreme position. At the least delivery.
\(0.3\ \mathrm{mt.}\)	\(8.5\ \mathrm{kg}/\mathrm{cm}^2\)	25 millimeters	3.5 millimeters
"	6 "	15 millimeters	1.0 millimeters
"	4 "	7 millimeters	0.5 millimeters
"	2 "	1.5 millimeters	0.5 millimeters

From the table it is evident that, at a steam pressure of 6 atm. and an incorrect position of the nozzle, the injector will deliver almost as much water as at a pressure of 2 atm.

Hence, among other things, the following rule may be derived for injectors with a fixed nozzle: its position at the most—

) Kneass, Injectors*.

less than the steam pressure at which the injector is to operate, must correspond to the greatest quantity of water delivered; in this case, at the higher steam pressure the injector will give the possible minimum of delivery. If the converse were done, i.e. if the position of the condensing nozzle corresponded to the higher steam pressure, then, when the steam pressure is lowered, the quantity of entering water would be so great that the injector would cease to operate at a pressure considerably greater than its lower limit. With a fixed nozzle, therefore, the operation of the injector at different steam pressures can in no case be equally economical.

Although the smaller quantity of water delivered at high steam pressures must in part be ascribed to the greatly increasing consumption of the latter, which diminishes the ratio

\[ \frac{P}{D}, \]

nevertheless the position of the nozzle itself, which limits the quantity of incoming water, will play an equally important role. It is clear that in the present case preference must be given to a nozzle that can be adjusted.

Manual adjustment, however, complicates the attendance of the apparatus, and therefore an injector with a self-adjusting nozzle, such as that made by Sellers (Self-adjusting Injector), is in this respect the most perfect apparatus. In Fig. 26 the nozzle of this injector is shown*). Here \(a\) is the steam nozzle, \(e\) the condensing nozzle, \(d\) the receiving nozzle, \(b\) represents the end of a spindle turned down inside the cylinder, through which a small jet of steam passes before the injector is put into operation; through pipe \(w\) water enters, and through \(D\) steam enters. The head of the condensing nozzle \(e\) has the form of a piston; piston springs are placed in the grooves. This is sufficient for chamber \(o\) to be almost hermetically separated from the water chamber \(g\). Nozzles \(e\) and \(d\) are connected with one another by a screw thread; the small chamber serves as an entrance into the receiving nozzle and, moreover, communicates by apertures with chamber \(o\). Below, behind the sleeve \(h\), which guides the motion of the receiving nozzle, there is a cock through which the water is drained before its delivery into the boiler begins.

The jet of steam entering along the spindle produces suction of the water; as soon as the latter appears through the cock, the spindle is withdrawn from the nozzle, and the steam jet begins to deliver water; this is indicated by the increased discharge, both in velocity and in volume, through the cock. The cock is then closed, and the energy of the jet overcomes the pressure on the feed valve, and the water enters the boiler. If the action of the injector should for any reason cease, this is indicated

*) Proceed. of inst. of Mechan. Eng., 1866.

the discharge of steam through the opening \(k\), in which there is a spring valve that opens when the pressure in the apparatus rises to a certain limit; during operation of the injector, it is pressed tightly against its seat by atmospheric pressure and by the force of a small spring.

The idea of self-regulation consists in the following: if too much water is flowing in, then its excess, through the slots of chamber \(f\), drains into chamber \(o\); a small excess of pressure in the latter, as compared with the pressure in the water chamber \(g\), is sufficient for the whole nozzle to move toward the latter; the annular clearance \(a\) decreases, and, consequently, so does the quantity of water entering; the nozzle will stop precisely in the position in which the quantity of water entering corresponds to the velocity of the steam and to its weight. If now the steam pressure changes—let us suppose it increases—then, with the former position of the nozzle, too little water will enter, in correspondence with the increased quantity of steam and velocity of the jet; since the receiving nozzle has a suction capacity, at the higher velocity it will take part of the water from chamber \(o\). The pressure in the latter will become less than in the water chamber \(g\); acting on the piston head, this excess pressure will force the nozzle to move back and thereby open greater access for the water; the nozzle will again assume the position that corresponds to the changed working conditions. It is not difficult to see that the nozzle will likewise be automatically reset in the event of a decrease in steam pressure, or of an increase or decrease in the suction height, i.e. when the water level in the reservoir from which the water is drawn is lowered or raised. The degree of perfection of an injector with such a nozzle may be seen from the following table, where results obtained experimentally with No. 6 are given,*) and, beside them, data determined for the same number of Giffard injector by his formula for the maximum capacity: \(E = 28 d^2 \sqrt{p}\), where \(d\) is the diameter of the mouth of the receiving nozzle and \(p\) is the steam pressure in the boiler.

The data for the Sellers injector were obtained in tests at the inventor’s works; the pressures and temperatures of the feed water required for operation of the injector, at which the injector ceases to operate, are indicated as averages from a series of experiments; the deviations are very small. The result of installing a self-regulating inlet nozzle is expressed in a considerable increase in the quantity of water delivered; the minimum of the Sellers injector, almost up to \(3.5\) atm., is equal to the maximum consumption of the Giffard injector.

It should further be noted how greatly the quantity of water delivered can vary; for example, at \(10.5\) atm. the discharge can be varied within the limits of \(58.6\%\) of the maximum. The Sellers injector was—

*) Peabody, Thermodynamics of the Steam Engine.

owing to its great elasticity in delivery when the pressure and the suction height change, is especially suitable for locomotives, where the steam pressure is subject to constant fluctuations depending on the profile of the track, and also where the height of the water in the tender varies.

Steam pressure in the boiler, kg/cm²	Diff.-farb. Maximum \(E = 28d^2 \sqrt{p}\)	Sellers with automatic adjustment, condensing nozzle: delivery, max.	Sellers with automatic adjustment, condensing nozzle: delivery, min.	Sellers with automatic adjustment, condensing nozzle: relative minimum to maximum	Temperature: feed water	Temperature: at maximum	Temperature: at minimum	Steam working pressure required	Highest temperature of feed water
0,7	845	2132	1800	0,845	19°	38°	34,5°	0,21	55,5°
1,4	1192	2333	1733	0,743	—	42,5	40,0	0,63	57,0
2,1	1461	2668	1600	0,600	—	45,5	47,0	1,1	57,0
2,8	1688	2835	1700	0,599	—	49,0	50,5	1,54	55,5
3,5	1885	3057	1832	0,597	—	51,0	51,5	1,89	55,0
4,2	2066	3300	1800	0,546	—	53,0	56,0	2,4	54,5
4,9	2228	3534	1800	0,510	19,5	54,5	61,0	2,8	54,5
5,6	2379	3766	1900	0,505	19,0	56,5	62,0	3,2	55,0
6,3	2520	4000	1968	0,492	19,5	58,0	64,5	3,6	55,5
7,0	2661	4168	1834	0,456	19,0	60,0	70,5	4,1	55,5
7,7	2792	4332	1900	0,439	19,5	62,0	72,0	4,4	—
8,4	2921	4435	2067	0,466	19,5	64,5	72,0	4,8	57,0
9,1	3040	4565	2100	0,46	19,0	65,5	74,0	5,3	54,5
9,8	3155	4690	2234	0,476	19,0	67,0	74,5	5,7	52
10,5	3266	4834	2000	0,414	19,0	69,5	75	6,2	49,5

Every injector can, within certain limits, vary the quantity of water delivered. The greatest output of the injector depends on the final velocity of the jet as it leaves the condensing nozzle; the temperature of the delivered water will be low, since in this case more water can flow into the nozzle than is essentially necessary for the process of condensation.

The smallest output of the injector is determined, on the contrary, by the temperature of the delivered water, which cannot be higher than a certain

limiting. Both the velocity and the limiting temperature depend on the rarefaction that takes place in the condensation nozzle, i.e., they depend on the rapidity and completeness of the condensation. Let us consider separately two processes occurring in the nozzle: the first—the imparting of velocity to the incoming water; the second—the condensation of the steam. Let \(g_1\) and \(g_2\) be the weights of steam and water entering the condensation nozzle per unit time, \(u_1\) and \(u_2\) their velocities; and \(u\) the final velocity of the mixture. By the theorem of quantities of motion we have

\[ g_1u_1+g_2u_2=g_1u+g_2u \]

\[ g_1(u_1-u)=g_2(u-u_2), \]

and

\[ \frac{g_2}{g_1}=\frac{u_1-u}{u-u_2}\ldots\ldots(4). \]

If the back pressure which the jet must overcome in order to enter the reservoir is known and constant, then the final velocity of the jet \(u\) is thereby determined. With this back pressure, the greater \(\frac{g_2}{g_1}\) will be, the greater are \(u_1\) and \(u_2\).

But \(u_1\) depends on the pressure in the nozzle to which the steam may have expanded, just as \(u_2\) depends on whether the water is sucked in or enters under head; in the first case, if the pressure in the nozzle is \(p'\), and the atmospheric pressure is \(p\), (in kgr. per sq. mt.),

\[ u_2=\sqrt{\frac{p-p'}{\gamma}\cdot 2g}; \]

in the second,

\[ u_2=\sqrt{2g\left(h+\frac{p-p'}{\gamma}\right)}; \]

here \(h\) is the head in mtr., i.e., the height of the water level in the reservoir above the axis of the condensation nozzle, and \(\gamma\) is the density of water. Thus the greatest delivery is determined by the rarefaction that occurs in the condensation nozzle; however, this is not the only condition: it is necessary that at the same time the condensation be as complete as possible, and this, as we have seen, besides the quantity of water entering for the whole amount of steam, depends on the time during which condensation can take place, or on the length of the nozzle. Suppose that water enters in a quantity in excess, yet sufficient for \(u\) to acquire the required magnitude; suppose that some part of the steam nevertheless remains uncondensed; its presence in the jet will be expressed by an increase in the volume of the mixture and, consequently, by a decrease in density; the velocity \(u\), however, already has a limiting value. If the area of the mouth of the receiving nozzle is \(F\), then in the first case the equality will hold:

\[ Fu\gamma=g_1+g_2,\ldots\ldots(5) \]

and in the second case there must be

\[ Fu\gamma_1=g_1+g_2\ldots\ldots(6). \]

Since \(\gamma_1<\gamma\), it is obvious that the second part of the equality in the second case cannot be equal to the same part in the first; part of the mixture will not enter the receiving nozzle and will drain through the standpipe, and thereby the actually possible ratio

\[ \frac{g_2}{g_1}. \]

will decrease.

Let us now suppose that the water is supplied in an amount just sufficient for condensation, and that the latter takes place completely and fully. In this case the velocity of the mixture \(u\) will have a different value, namely \(u'>u\). Part of the energy of the jet will, for the given back pressure, remain unused, and the jet could overcome a pressure greater than that against which it is working. At the same time, in this case there is created, so to speak, a certain limit for imperfect condensation, at which the amount of water delivered \(\frac{g_2}{g_1}\) remains unchanged. Let \(H\) denote the head in meters corresponding to the pressure in the delivery reservoir. The velocity \(u\) is then determined by it as

\[ \frac{u^2}{2g}=H \ldots\ldots (7). \]

But in reality the velocity \(u'\) is greater than \(u\); it corresponds to the head \(H'\), connected by the relation

\[ \frac{u'^2}{2g}=H' \ldots\ldots (8). \]

Let in the first case the density of the jet be \(\gamma\), and in the second \(\gamma'\); taking \(\gamma\) as unity, we find that \(H'\) must be equal to \(\frac{H}{\gamma'}\); the two preceding equalities give us

\[ \frac{u^2}{2g}=\frac{u'^2}{2g}\gamma' \]

or

\[ \gamma'=\frac{u^2}{u'^2}\ldots\ldots (9). \]

This is precisely that limiting value for \(\gamma'\) at which \(\frac{g_2}{g_1}\) will remain constant; suppose that, corresponding to the boiler pressure, the velocity \(u\) should be 30 m, while in reality it is found to be \(32\ \mathrm{m}=u'\).

If complete condensation takes place in the nozzle, then at the mouth of the receiving nozzle not the entire cross-section will be filled by the jet; with the standpipe open, air will be sucked in; with the drain closed,

one can arrange an additional inflow of water, or increase the inflow of water directly into the condensation chamber. If not all the steam is condensed in the nozzle, then the remaining quantity of it must give the mixture such a volume that the density be

\[ \gamma'=\frac{30^2}{32^2}=0.88. \]

If the condensation is still more imperfect, i.e. \(\gamma'\) obtains a value less than \(0.88\), then again the excess of water will be discharged through the overflow pipe. The loss of water through the overflow, both in the first and in the second case, will depend not on a deficiency of water, but on insufficient length of the nozzle or insufficient time for condensation. If it is undesirable, then it will not be possible to eliminate it by reducing the quantity of water: it is necessary either simultaneously to regulate the quantity of steam and of water, but in that case the possible capacity of the apparatus will not be utilized; or to replace the nozzle by a longer one, to supply colder water, or to introduce it under pressure in the same quantity, but in a thinner layer.

From the preceding it is clear that an injector for which the final density of the mixture at its entrance into the mouth of the receiving nozzle is less than unity delivers less water than it could. Indeed, if the area of the mouth is \(F\), then at \(\gamma=1\), and velocity \(u\), the weight \(Q_1\) which can be delivered is

\[ Q=Fu\gamma=Fu. \]

With imperfect condensation the density will be \(\gamma'\) and the velocity \(u'\); the weight \(Q_2\) will be

\[ Q_2=Fu'\gamma' \]

The ratio of the actually delivered weight to the possible or theoretical \(Q_1\) will be

\[ \frac{Q_2}{Q_1}=\frac{Fu'\gamma'}{Fu}=\frac{u'\gamma'}{u'\sqrt{\gamma'}}=\sqrt{\gamma'}\ldots\ldots (10) \]

For the preceding example, \(Q_2\) will amount to \(0.94\,Q_1\).

The presence of air in excessive quantity will affect the operation of the injector in the same direction. Some quantity of it, corresponding to the normal content in the water, is inevitable; but everything that contributes to increasing it in the jet must be eliminated; such are, above all, all leaks in the suction pipe and at the flanges.

If we know the vacuum in the condensing nozzle and it has been established that the quantity of water entering it is sufficient for complete condensation, then the value \(\sqrt{\gamma'}\) may serve as a characteristic of the perfection of this nozzle under the given conditions, in comparison with another injector operating under the same conditions. To determine \(\gamma'\) directly is difficult; the temperature of the mixture at the moment it passes the mouth of the receiving nozzle will differ from the temperature in the delivery pi—

it; for an exact determination one must know also its pressure. For this purpose one may choose two indirect methods. If the pressure of the reservoir into which the delivery is to occur is given, then we can know the velocity with which the water tends to issue from it. Let the pressure of the reservoir be \(p\), and before the orifice \(p'\); then

\[ u'=\sqrt{2g\frac{p-p'}{\gamma}}, \]

where \(\gamma\) corresponds to the temperature at pressure \(p\). The velocity of the jet entering the receiving nozzle must in any case be equal to or somewhat greater than \(u'\), i.e.

\[ u \geq u'. \]

By this the weight of water \(Q_1\) that can be delivered is determined; in reality, measurement has found that \(Q_2\) is delivered; then

\[ \frac{Q_2}{Q_1}=\sqrt{\gamma'}, \]

\[ \gamma'=\left(\frac{Q_2}{Q_1}\right)^2. \]

To determine \(\gamma'\) by the second method it is necessary to know the rarefaction in the condensing nozzle. For this one may recommend the following method, indicated by Zeuner*). Water is passed through the injector under a definite head (Fig. 27); in doing so it is necessary to see that the valve on the delivery pipe is opened for its full travel; in the figure only the steam and condensing nozzles are shown; if, by its construction, the injector does not allow this, then the water may be allowed to discharge through the overflow pipe, but it is necessary to make sure that all of it actually issues; for this the steam pipe must be removed and it must be observed whether the steam nozzle is being filled with water; if this occurs, it means that the overflow orifice is too small; it will be necessary either to enlarge the latter or to reduce the head. Determining, by weighing, the weight of all the water that has flowed out, and applying the laws of hydraulics, the coefficient of resistance is found. Let the mean head be \(h\); then

\[ h=(1+\xi)\frac{w^2}{2g}\quad \text{and} \quad w=\sqrt{\frac{2gh}{1+\xi}},\ldots (11) \]

If the area of the annular clearance \(F\) is in sq. m, then

\[ F \cdot w \cdot \gamma = \text{weight of the issuing water}=Q. \]

Hence it is determined that

\[ \xi=2gh\left(\frac{F\gamma}{Q}\right)^2-1.\ldots (12) \]

) Technische Thermodynamik*, Zeuner.

Install the steam pipe, if it has been removed, and put the injector into operation; in doing so, the valve on the delivery pipe must be in the same position; the operation of the apparatus should be regulated by the steam valve so that there is no loss through the overflow; the quantity of water taken must be measured; let it be \(Q'\); then \(Q' \cdot F\gamma=w'\).

If we denote by \(p\) the pressure of the atmosphere in \(\mathrm{kg}/\mathrm{mt}^{2}\), and by \(p'\) the pressure in the condensation nozzle, then at the mean head \(h\) the equality must hold:

\[ h+\frac{p-p'}{\gamma}=(1+\xi)\frac{w'^2}{2g}; \]

from which we find

\[ \frac{p-p'}{\gamma}=(1+\xi)\frac{w'^2}{2g}-h\ldots\ldots(13) \]

In one of the experiments \(\xi\) was found by Zeuner to be equal to \(1.277\); since \(\xi\) itself depends on the velocity, the additional head \(\frac{p-p'}{\gamma}\) cannot be determined quite exactly. In approximate calculations one may confine oneself merely to the following determination: since \(F\) and \(\gamma\) are constant, then

\[ \frac{Q}{Q'}=\sqrt{\frac{h}{h'}} \quad\text{and}\quad h'=\left(\frac{Q'}{Q}\right)^2 h\ldots\ldots(14) \]

Let, for example, \(Q'=1800\), \(Q=900\ \mathrm{kgr.}\), \(h=2\ \mathrm{mtr.}\); we determine \(h'\)

\[ h'=\left(\frac{1800}{900}\right)^2\cdot2=8\ \mathrm{mtr.} \]

The additional head due to rarefaction is \(6\ \mathrm{mtr.}\); let the height of the barometer be \(735.5\ \mathrm{m}/\mathrm{m}\), then \(p=10000\ \mathrm{kg}/\mathrm{mt}^{2}\), and we find \(p'\) equal to

\[ p'=4000\,\frac{\mathrm{kg}}{\mathrm{mt}^{2}} \quad\text{or}\quad 0.4\ \text{atm. abs.} \]

Having determined \(p'\), we find the velocities \(u_1\) and \(u_2\); for the given reservoir pressure, the velocity \(u\) is known.

We compute

\[ \frac{g_2}{g_1}=\frac{u_1-u}{u-u_0}. \]

Let \(\frac{g_2'}{g_1'}\) be determined directly by experiment; if \(\frac{g_2'}{g_1'}=\frac{g_2}{g_1}\), then complete condensation of the steam takes place; but if \(\frac{g_2'}{g_1'}<\frac{g_2}{g_1}\), then part of the steam remains uncondensed.

For the first case we have

\[ Q_1 = Fu\gamma, \]

for the second

\[ Q_2 = Fu'\gamma'. \]

The ratio is

\[ \frac{Q_1}{Q_2}=\frac{u\gamma}{u'\gamma'}, \]

since \(u=u'\sqrt{\gamma'}\), then

\[ \frac{Q_2}{Q_1}=\frac{u'\gamma'}{\gamma u'\sqrt{\gamma'}} \]

and, when \(\gamma=1\),

\[ \gamma'=\left(\frac{Q_2}{Q_1}\right)^2. \]

Thus, the greatest delivery of the injector is determined by the final velocity of the jet and by the perfection of condensation, since the latter affects this velocity. For every injector, under given working conditions, there exists a definite limit; let this be \(P\) kgr. of water per one kgr. of steam.

If one wished to increase \(P\) under the same conditions, then the excess would at once begin to discharge through the overflow pipe; with an increase of \(P\), \(u\) must necessarily decrease, and all the water would not be able to enter the mouth of the receiving nozzle. In exactly the same way there exists a limiting pressure at which the injector can work; if in this case \(P\) were decreased, then, although the velocity would become greater, the density of the jet would be diminished, and therefore a smaller potential energy would be obtained in the receiving nozzle than with the normal weight of water; with an increase of \(P\), the velocity would decrease, and, in dependence on it, the potential energy likewise.

The smallest delivery of the injector is determined by the temperature which the mixture has in the condensing chamber. When the quantity of water is decreased, it will be heated more strongly; in this case it is necessary that its temperature be lower than the temperature corresponding to the pressure in the nozzle. Thus, for example, at a pressure in the nozzle of \(0.4\) atm., the temperature of the mixture must be below \(75^\circ\), approximately \(65^\circ\)—\(70^\circ\), in order that the rarefaction in the nozzle should not be diminished by reverse evaporation.

If the pressure in the nozzle is \(p'\), the atmospheric pressure \(p\), and the pressure corresponding to the temperature of the mixture \(p_1\), then, at the suction height \(h_1\), the relation must hold:

\[ \frac{p}{\gamma}-h_1>\frac{p'}{\gamma}>\frac{p_1}{\gamma}\ldots\ldots\ldots (15). \]

Let us suppose that the temperature of the mixture has risen so much that \(p_1\) has become greater than \(p'\); if the suction height remains the same, the injector will refuse to work. The same will happen if we begin to preheat the feed water; therefore, for each suction height, for a given injector there is a definite limit for the initial temperature of the water; as soon as it is reached, suction will cease. The injector will begin to work again if the water is brought to it under pressure. But with an open overflow a certain limit is also established here: the temperature of the mixture as it passes through its interval between the nozzles must not be higher than \(85^\circ\) (Deloy’s experiments) and, as an extreme limit, \(90^\circ\).

As soon as steam begins to be discharged through the overflow pipe of a working injector, this indicates the imminent possibility that its operation will cease. The reason for this is that the jet begins to evaporate in the interval; its volume increases, and it is unable to enter the receiving nozzle. The essence of the action of the apparatus consists precisely in the fact that the large volume of steam issuing from the steam nozzle is, by condensation, reduced to so small a volume, possessing a great velocity, that a considerable quantity of water can be added to it, to which it can impart the velocity necessary for passing the mouth of the receiving nozzle and sufficient kinetic energy to overcome a definite counterpressure. The hotter the water, the greater the quantity of it required for condensing the steam; therefore, for heated water, the change in the injector delivery will be less than for cold water; at the same time the power of the injector will also decrease. In injectors with a closed overflow the final temperature of the mixture may be higher, sometimes reaching \(105^\circ\); double injectors, of which we shall speak below, can deliver hot water up to \(125^\circ\).

Thus, the determining quantities for the amount of water delivered by a suction injector are the velocity of the steam jet and the temperature of the mixture in the condensing nozzle; for a non-suction injector, instead of the temperature in the condensing nozzle, the temperature of the mixture in the interval between it and the receiving nozzle plays a role.

Above we gave a formula determining the quantity of water necessary for condensing the steam, and made the reservation that in the actual case somewhat more water must be supplied, so that the final temperature \(t_e\) should have a definite value. Let us now give a more general expression for \(P\). Let us suppose that the velocity of the steam at the instant of its meeting with the water is \(u_1\), the velocity of the water is \(u_2\), and the velocity of the mixture after impact is \(u\); let \(P\) denote the number of kg of water falling on 1 kg of effluent steam. The kinetic energy of the steam and water before impact is equal to

\[ \frac{u_1^2 + u_2^2 \cdot P}{2g}, \]

after impact

\[ \frac{u^2}{2g}(1+P); \]

their difference repres—

constitutes a loss due to impact; express it in heat units, and we shall have:

\[ \frac{A u_1^{\,2}}{2g}+\frac{A P u_2^{\,2}}{2g}-\frac{A(P+1)u^{2}}{2g}\ldots\ldots\ldots(16). \]

This lost kinetic energy, upon impact, is converted back into heat; in addition, part of the energy will be lost to friction in the nozzles during the motion in them of the steam, the water, and the mixture, but we may omit these losses from the calculation, since, occurring inside the apparatus, they will be returned to the mixture in the form of heat; the velocities \(u_1\), \(u_2\), and \(u\) are quantities determined theoretically by the formulas without corrections for losses. If the above-mentioned difference of energies, in heat units, is divided by the weight of water \(P\), taking its specific heat as constant and equal to unity, then we obtain the temperature increase of the water \(\Delta_1 t\); we may therefore write:

\[ \frac{A u_1^{\,2}}{2g}-\frac{A P u_2^{\,2}}{2g}-\frac{A(P+1)u^{2}}{2g}=P\Delta_1 t\ldots(17). \]

Let the final state of the steam at the moment of its exit from the steam nozzle be characterized by the quantities \(p_2,v_2,x_2\); the quantity of heat contained in \(1\ \mathrm{kg}\) of steam will be

\[ q_2+r_2x_2; \]

if the final temperature of the mixture is \(t_e\), then during condensation there must be removed \(q_2+r_2x_2-t_e\) units of heat, which will give the water an increase of its temperature \(\Delta_2 t\); we have

\[ q_2+r_2x_2-t_e=\Delta_2 t\cdot P\ldots\ldots(18). \]

The final temperature of the mixture, with initial temperature \(t_a\), will be

\[ t_e=t_a+\Delta_1 t+\Delta_2 t\ldots\ldots(19). \]

After substituting into this equality, in place of \(\Delta_1 t\) and \(\Delta_2 t\), the corresponding expressions, we obtain

\[ t_e=\frac{q_2+r_2x_2-t_e}{P}+\frac{A(u_1^{\,2}-u^{2})}{2gP}+\frac{A(u^{2}-u_2^{\,2})}{2g}+t_a\ldots(20) \]

Solving this equation with respect to \(P\), we find

\[ P= \frac{ q_2+r_2x_2-t_e+\dfrac{A}{2g}(u_1^{\,2}-u^{2}) }{ t_e-t_a+\dfrac{A}{2g}(u^{2}-u_2^{\,2}) } \ldots(21) \]

This is the more general expression determining the weight of water that must be brought to the injector per \(1\ \mathrm{kg}\) of steam.

It differs from that indicated above, which determines the entire weight of water required only for condensing the steam, by terms containing the factor \(A\). As for the term \(\dfrac{A}{2g}(u^2-u_2^2)\) in the denominator, its influence on the result is very small; thus, for \(u=50\) mt. and \(u_2=10\) mt., it amounts to only \(0.3\) calories; but the magnitude of the term in the numerator depends on the form of the steam nozzle. With a converging conical nozzle, as we have seen, the energy of the jet remains almost constant and may be taken on average as equal to \(10{,}300\ \mathrm{kgr./mtr.}\), i.e. \(\dfrac{u_1^2}{2g}=10{,}300\) up to pressures of 10 atm.; from the formula it is clearly seen that, with an increase in \(u\), the quantity of water delivered per unit weight of steam must decrease; since, with increasing pressure, the quantity of issuing steam greatly increases, the capacity of the injector increases, but the efficiency decreases. If, instead of \(\dfrac{A}{2g}u_1^2\), we put the equal quantity \((q_1+r_1x_1-q_2-r_2x_2)\), then we obtain

\[ P= \frac{ q_1+r_1x_1-t_e-A\dfrac{u^2}{2g} }{ t_e-t_a+\dfrac{Au^2}{2g}-\dfrac{Au_2^2}{2g} } \ldots\ldots (22) \]

Let \(p\) in \(\mathrm{kgr./mtr.^2}\) denote the pressure against which the injector must work; \(p_1\) the pressure in the condensing nozzle, which we shall then take as equal to the pressure in the receiving mouth; \(p_2\) the atmospheric pressure \(=10{,}000\ \mathrm{kgr./mtr.^2}\); and \(h\) and \(h_1\) the heads corresponding to the first two. The water tends to issue from the boiler with velocity

\[ u'=\sqrt{2g\,\frac{p-p'}{\gamma}}, \]

and, in the limiting case, with the almost complete absence of resistances, must be equal to \(u'\); hence,

\[ \frac{u^2}{2g}=\frac{p-p_1}{\gamma}=h-h_1,\ldots\ldots (23) \]

let \(h_2\) be the suction height; then the velocity \(u_2\) is determined from the equality

\[ u_2=\sqrt{\,2g\left(\frac{p_2-p_1}{\gamma}-h_2\right)} =\sqrt{2g\,(10-h_1-h_2)}\ldots\ldots (24) \]

Replacing the terms representing kinetic energy by the heads corresponding to them, we obtain for \(P\) the equality

\[ P= \frac{ q_1+r_1x_1-t_e-A(h-h_1) }{ t_e-t_a+A(h-10+h_2) } \ldots\ldots (25) \]

As we see, this expression for \(P\) includes a whole series of quantities that determine the weight of water per \(1\ \mathrm{kgr.}\) of steam; hence it is clear that, when comparing the operation of injectors, care must be taken to ensure that a whole series of conditions are fulfilled, and that these must remain unchanged in testing each of them. If we know the temperature of the water, \(t_e\), at which the injector operates without loss through the overflow opening, then this formula may be used to determine the minimum amount of water possible under the given other conditions. The maximum delivery must be determined by the formula derived on the basis of the theorem on equality of quantities of motion, i.e.

\[ P=\frac{u_1-u}{u-u_2}. \]

Below we shall discuss how correct it will be to use this formula for the case of an injector without introducing a corrective coefficient. Here, however, we shall point out that in deriving this relation the density of the mixture is taken as equal to the density of water, i.e. we apply the theorem to the case in which complete condensation of the steam takes place; in reality, however, it can be attained only under the particularly favorable conditions indicated earlier; the presence of steam and air in the jet will manifest itself in an increase of the final velocity and a decrease of \(P\). Likewise, the magnitude of \(P\) may be affected by greater or lesser condensation of steam in the steam nozzle, washed externally by the cold water flowing past, since the velocity of the steam is reduced; Giffard assumed that the density of the steam issuing from the nozzle is only \(0.66\) of its density in the boiler. The more strongly the condensation of steam due to cooling by water is expressed, the more strongly the effect of this circumstance will be reflected in the ability of the injector to take hot water, since, with the increased discharge of steam, the temperature of the mixture must nevertheless remain below that corresponding to the pressure in the condensation chamber; that is, the initial temperature in this case must be correspondingly lower than if the steam were dry.

Fig. 1.

Visible labels in the diagram: \(p_a\), \(c\), \(k\), \(d\), \(p_1\).

Fig. 2.

Visible labels in the diagram: \(a\), \(g\), \(n\), \(b\), \(k\), \(c\), \(d\), \(e\), \(f\).

Fig. 3.

Fig. 4.

Fig. 5.

Fig. 6.

Labels visible in the figure: \(p_1\), \(p'_1\), \(p''_1\), \(p'''_1\), \(p^{IV}_1\), \(p^V_1\); \(dS_1\), \(dS'_1\), \(dS''_1\), \(dS'''_1\), \(dS^{IV}_1\); \(dS'''_2\), \(dS''_2\), \(dS'_2\), \(dS_2\); \(p'''_2\), \(p''_2\), \(p'_2\), \(p_2\).

Fig. 7.

Pressure curve during the slamming shut of the mouths.

Labels visible in the graph: \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(C\); “jump of the jet”; “flow of steam”; bottom scale in seconds, from 20 to 10; right-hand scale from 0 to 7.

Fig. 8.

Labels visible in the figure: \(p_0\), \(w_0\), \(F_0\); \(p_m\), \(w_m\), \(F_m\); \(p_w\), \(w\), \(F\); \(p_{w2}\), \(w_2\), \(F_2\).

Fig. 9.

Labels visible in the figure: \(p_1\), \(p_m\), \(p_k\), \(p_2\); \(\delta_m\), \(\delta_c\), \(\delta_2\).

Fig. 10.

Labels visible in the graph: vertical scale \(\dfrac{k_0}{c_\delta}\), from 0 to 1.7; horizontal scale from \(-20\) to \(60\), \(m/m\); “end of the nozzle.” Curves marked \(A\), \(B\), \(C\), \(D\).

Fig. 11.

Graph scale: vertical axis, \(\mathrm{kg}/\mathrm{cm}^2\); horizontal axis, distance “from the end of the nozzle,” in mm. Curves marked \(a\), \(b\), \(c\), \(d\).

Fig. 12.

Graph scale: \(0, 1, 2, 3\ \mathrm{kg}/\mathrm{cm}^2\). Marked sections \(1, 2, 3, 4, 5\); dimensions \(9.1\) and \(40.2\).

Fig. 13.

Nozzle diagram. Marked sections \(1, 2, 3, 4, 5\); dimensions \(9.1\), \(40.9\), \(4.2\), \(4.65\).

Fig. 14.

Graph scale: \(0, 1, 2, 3\ \mathrm{kg}/\mathrm{cm}^2\). Marked sections \(1, 2, 3, 4, 5\); dimensions \(9.1\) and \(40.9\). Curve labels include \(7.34\), \(4.08\), \(1.15\), \(3.03\), \(1.73\).

Dimensions of the nozzle, Fig. 13.

Quantity	Value
Overall length	50.0
Length of the first section	9.1
Length of the annular channel	40.9
Diameter of the first section	4.01
Greatest diameter of the channel	6.05
Diameter of the last section	1.20

Quantity	No. 1	No. 2	No. 3	No. 4	No. 5
Distance of the openings from the plane of the first section	41.7	32.6	23.0	13.5	4.2
Diameters of the openings	12.6	15.3	18.9	22.7	26.8

Fig. 15.

Pressures corresponding to the curves.

1	2	3	4	5
12.8	11.28	10.36	9.24	8.24

6	7	8	9	10	11
7.14	6.10	5.08	4.02	3.15	2.02

Fig. 16.

Fig. 17.

Vapor pressure \(p\); \(t^\circ\text{C}\).

Superheated steam \(O - p\).

Supersaturated steam \(x - p\).

Fig. 18.

Fig. 19.

Visible labels: \(a\), \(a'\), \(b\), \(b'\), \(c\), \(c'\), \(d\), \(d'\), \(e\), \(\varphi\), \(\gamma/\delta\).

Fig. 20.

Visible labels: \(A\), \(B\), \(Z\), \(m\), \(o\), \(\nu\), \(S\), \(q\), \(x\), \(w\), \(a\), \(a_1\), \(b\), \(b_1\), \(c\), \(f\), \(\alpha\), \(p=f(t)\), \(p=\varphi(x)\), \(p'\).

Curve of consumption.

\(1\) kg of steam.

Fig. 21.

Fig. 22.

Fig. 23.

With excess steam.

Fig. 24.

With excess water.

Fig. 25.

Fig. 26.

Fig. 27.

Fig. 28.

Fig. 29.

Fig. 30.

\[ p_0 \omega_0 u_0 \]

\[ l;\quad \frac{dl}{du\,dp} \]

\[ \omega(p+dp) \]

\[ \omega p \]

\[ \omega,u,p \]

\[ u_a \]

Fig. 31.

critical pressures

critical velocities

Fig. 32.

Diagram labels: \(x\), \(y\), \(o\), \(a\), \(b\), \(x\), \(y\), \(L = [[unclear]]\).

Fig. 33.

Text within the graph: “Theoretical profile of the receiving nozzle with a gradual acceleration of the jet by \(1\) [[unclear]].”

Graph labels and numbers visible: \(1{,}00\), \(1{,}026\), \(1{,}0574\), “axis of the nozzle,” \(975{,}19\).

Fig. 34.

Diagram labels: \(O\), \(O'\), \(B\), \(B_1\), \(D\), \(D\), \(\omega\), \(\alpha\).