Prof. I. I. Butakov.

Changes in the Power of Steam Engines

The power of a steam engine, as is known, is expressed in h.p. in the form: \(N_e = \eta \cdot \dfrac{l \cdot n}{30.75} O p_i\). With \(O\)—the useful area of the piston in sq. cm, and \(l\)—the length of the piston stroke in m, unchanged in a given engine, the effective power \(N_e\) could change with a change in \(n\)—the number of revolutions per minute—and \(p_i\)—the mean indicated pressure, since \(\eta\)—the mechanical efficiency—is practically a constant quantity. But a change in \(n\), for given production conditions, does not appear convenient, since the driven mechanisms set in motion by the given steam engine have definite most advantageous speeds of motion, violation of which leads to a lowering of the qualitative and quantitative aspects of the product output. On the other hand, precisely this number of revolutions \(n\) is taken into account in designing the steam engine itself, in the part relating to its dynamics. Thus, in essence, there remains only one possibility for changing the power \(N_e\) of the given steam engine—namely, changing \(p_i\), while keeping \(n\) constant.

The mean indicated pressure of any steam engine may be represented as follows: \(p_i = \sigma (k p - k_e p_e)\) . . . . . (1), where \(p\) and \(p_e\) are the admission and exhaust pressures in abs. atm.; \(k\) and \(k_e\) are the averaging coefficients of the front and back pressures; and \(\sigma\) is the coefficient of reliability, characterizing the losses in the actual indicator diagram due to steam distribution and heat exchange with the cylinder walls; \(\sigma = 0.90—0.95\).

From formula (1) for \(p_i\) it is clear that changes in \(p_i\) may be obtained by changing the magnitude \(p\), which occurs in actual steam engines when they are regulated by a throttle valve. In this case the change of \(p_i\) as a function of \(p\) will take place according to the law of a straight line drawn from the origin at a distance \(-\sigma k_e p_e\) at an angle to the \(p\)-axis whose tangent is equal to \(\sigma k\). In Fig. 1 this law is shown for a particular example: \(p = 6.15\); \(p_e = 1.1\); \(k = 0.58\); \(k_e = 1.13\) and \(\sigma = 0.93\), whereby for \(p = 6.15\) atm. \(p_i = 2.2\) atm., and for \(p = 4\) atm. \(p_i = 1.05\) atm.

The method of changing the power of an engine by means of “throttling the steam” is fairly widespread in small steam engines. It has, however, the following generally known shortcomings: 1) let, in the theoretical indicator diagram (Fig. 2), the initial pressure \(AC\) be lowered to \(AC^1\), while the degree of filling \(\dfrac{CD}{AB} = \dfrac{C^1D^1}{AB}\) is left as before. We obtain the diagram \(C^1D^1E^1GKC^1\). Extending the expansion line \(E^1D^1\) until it meets the straight line \(CD\), we obtain the theoretical diagram \(KCFE^1GK\) at the initial pressure \(AC\) and with the admission ratio \(\dfrac{CF}{AB} < \dfrac{CD}{AB}\). The steam consumption by weight is the same in both of the latter diagrams, while in the second there is an excess of work, represented by the area \(CFC^1D^1\). And since the additional heat consumption for producing steam of higher pressure is negligible, it is clear that it is more advantageous to reduce the power of an engine by earlier cut-off than by throttling the steam, which is confirmed by experience.

2) There exist steam engines in which the transition of the equilibrating horizontal forces through zero occurs after the dead position. With throttling valves, an increase of \(p\) in this case causes the zero of the pressures to approach the dead point, up to possible coincidence with it, as a result of which impacts may occur in the joints of the machine.

3) When the inertia forces of the mechanism are balanced by sufficient compression, the pressure of the latter at the end, \(p_c\), limits a further lowering of \(p\), since for \(p < p_c\) a loop will appear at the end of compression, and a distortion of the form of the indicator diagram will occur, whereby the limits of regulation are narrowed; or else, in the case of a large reduction of power, one must put up with losses in the indicator diagram.

It is therefore natural that the method of changing the power \(N_e\) by changing the degree of filling \(\varepsilon\), i.e. by changing \(k\) in formula (1) for \(p_i\), is the most rational.

The coefficient of mean forward pressure \(k\) is expressed through the degree of filling \(\varepsilon\) in the following form:
\[ k = \varepsilon + (\varepsilon + m)\ln \frac{1+m}{\varepsilon+m}, \]
where \(m\) is the magnitude of the clearance space, in \(\%/\%\) of the working volume of the steam cylinder. Putting \(x = \varepsilon + m\), we transform:
\[ k = \alpha x - xm + \beta x\ln(1+m) - \beta x\ln x. \]
To get rid of \(\ln x\), we resort to expanding it in a series, and, for the possibility of ignoring all terms of the expansion beginning with the second, multiply \(x\) under the sign \(\ln\) by a certain factor \(\gamma > 1\), at the same time adding the quantity \(+\beta x\ln\gamma\). Then:
\[ k = x[\alpha+\beta\ln(1+m)+\beta\ln\gamma] - \alpha m - 2\beta x\frac{\gamma x-1}{\gamma x+1}, \]
whence:
\[ \gamma[\alpha+\beta\ln(1+m)+\beta\ln\gamma-2\beta]x^2 -\gamma kx-k+[\alpha+\beta\ln(1+m)+\beta\ln\gamma-\alpha m+2\beta]x-\alpha m=0. \]

Thus we have an equation of a curve of the second order of the form:
\[ Ax^2 - Bxy + Dx - y - F = 0, \]
where
\[ AC-B^2 = -(-\gamma)^2 < 0, \]
i.e. the curve is a hyperbola. For the choice of the value of \(\gamma\), let us note that the second term of the series expansion
\[ \ln \gamma x \]
is equal to
\[ \frac{2}{3}\beta x\left(\frac{\gamma x-1}{\gamma x+1}\right)^3 \]
and vanishes when \(\gamma=\frac{1}{x}\); for this reason it would be necessary to give variable values of \(\gamma\) for every \(x\), which is obviously impossible. In a single-cylinder machine, when working on the condenser, the usual fluctuations for \(x=0.1\text{--}0.5\), and when working to the atmosphere \(x=0.2\text{--}0.7\), correspond to values of \(\gamma\) in the first case \(\gamma=10\text{--}2\), and in the second \(\gamma=5\text{--}1.43\); therefore, keeping closer to the normal degrees of filling, we choose for the first case \(\gamma=5\), and for the second, moreover with the calculation that the coefficient \(A\) at \(x^2\) be made zero in order to simplify the problem. If we take \(\beta=0.85\), \(a=0.92\), \(m=0.08\), we obtain \(\gamma=2.3\), when \(A=0\). Denote, for the latter case:
\[ \psi = D - By \]
and
\[ \tau = -Bx - 1. \]
Then:
\[ \psi\tau = (D-By)(-Bx-1) = -B(Dx - Bxy - y) - D = -BF - D = \text{const.} \]
The equation of the asymptotes of the hyperbola is
\[ \psi\tau = 0, \]
i.e.
\[ y=\frac{D}{B} \quad\text{and}\quad x=-\frac{1}{B}. \]
We have the numerical values for the above particular example:

Coefficients	Their letter expressions	Their numerical values
\(A\).	\(\gamma[\alpha+\beta\ln(1+m)+\beta\ln\gamma-2\beta]\)	\(0\)
\(B\).	\(\gamma\)	\(2.3\)
\(D\).	\(\alpha+\beta\ln(1+m)+\beta\ln\gamma-\alpha m+2\beta\)	\(3.23\)
\(F\).	\(\alpha m\)	\(0.074\)

Thus, for the asymptotes we have \(k=-\dfrac{3.23}{2.3}=-1.4\) and \(x=-\dfrac{1}{2.3}=-0.43\).

The point necessary for constructing the hyperbola is found by putting \(x=0\), whence \(k=-F=-0.074\). The construction of the hyperbola is carried out in Fig. 3.

Passing to the second case of operation with the refrigerator, let us note that \(A\) is not equal to zero. Then the equation of our hyperbola may be represented in the form

\[ \left[0.5By-\left(0.5D-\frac{A}{B}\right)\right]^2 - \left[Ax-0.5(By-D)\right]^2 +0.25D^2+AF - \left(0.5D-\frac{A}{B}\right)^2 =0, \]

so that the equation for the asymptotes is

\[ \left[0.5By-\left(0.5D-\frac{A}{B}\right)\right]^2 - \left[Ax-0.5(By-D)\right]^2 =0, \]

whence, after decomposing the difference of squares into the product of their sum and their difference, we have:

\[ x=-\frac{1}{B}, \]

and the equation of the other asymptote:

\[ By-D+\frac{A}{B}-Ax=0, \]

which, for \(x=0\), gives

\[ y=\frac{D}{B}-\frac{A}{B^2}, \]

and for \(x=1\) gives

\[ y=\frac{D}{B}-\frac{A}{B^2}+\frac{A}{B}. \]

The coefficients in the numerical values of our particular example will be, for the given case:

Coefficients	Their literal expressions	Their numerical values
\(A.\)	\(\gamma\{\alpha+\beta\ln(1+m)+\beta\ln\gamma-2\beta\}\)	\(3.3\)
\(B.\)	\(\gamma\)	\(5.0\)
\(D.\)	\(\alpha+\beta\ln(1+m)+\beta\ln\gamma-\alpha m\gamma+2\beta\)	\(3.7\)
\(F.\)	\(\alpha m\)	\(0.074\)

For both,

\[ x=-\frac{1}{B}=-\frac{1}{5}=-0.20; \]

for the other asymptote, at \(x=0\) we have

\[ y=\frac{D}{B}-\frac{A}{B^2} =\frac{3.7}{5}-\frac{3.3}{25} =0.61, \]

and at \(x=1\) we have

\[ y=\frac{D}{B}-\frac{A}{B^2}+\frac{A}{B} =\frac{3.7}{5}-\frac{3.3}{25}+\frac{3.3}{5} =1.27. \]

The point needed for constructing the hyperbola is obtained at \(x=0\), when \(k=-F=-0.074\). The hyperbola is constructed in Fig. 4 by the method of secants.

The inaccuracies from ignoring the second term of the expansion

\[ \frac{2}{3}\beta x\left(\frac{\gamma x-1}{\gamma x+1}\right)^3 \]

are expressed, for the limiting values of \(x\), as:

\(x\)	\(\gamma=5\)	\(\gamma=2.3\)
\(0.1\)	\(0.002\)	—
\(0.2\)	—	\(0.006\)
\(0.5\)	\(0.02\)	—
\(0.7\)	—	\(0.004\)

The change of \(p_i\) in the basic equation (1) as a function of \(k\) will be represented in the form of a straight line,

\[ p_i=\sigma p\,k-\sigma k_e p_o, \]

passing from the origin at a distance \(\sigma k_o p_o\) and making an angle with the \(k\)-axis whose tangent is \(\sigma p\). For the above particular example of the machine,

\[ p_i=-0.93\cdot 6.15\,k-0.93\cdot 1.13\cdot 1.1; \]

for \(k=0\) we have

\[ p_i=-0.93\cdot 1.13\cdot 1.1=-1.15; \]

for \(k=1.0\) we have

\[ p_i=5.7-1.15=4.55. \]

This straight line is shown in Fig. 5. From the found \(k\), for any \(p_i\) it is easy to find \(x=\varepsilon+m\), using one or another hyperbola for \(k\) (Figs. 3–4), depending on the circumstances. Thus, for \(p_i=2.2\) atm.

we find \(k=0.57\), to which in Fig. 3 there corresponds \(x=0.33\), and hence \(\varepsilon=0.25\). The inverse problem is also easily solved: for a given \(\varepsilon=x-m\), find \(p_i\) by means of the hyperbola \(k=f(x)\) and the straight line \(p_i=f(k)\).

If it were desirable to establish a direct dependence between \(p_i\) and \(x=\varepsilon+m\), then it is not difficult to verify, proceeding from the basic equation (1),

\[ p_i=\sigma kp-\sigma k_e p_e, \]

that for any \(x\) the values obtained for \(p_i\) are precisely the differences between the ordinates of one or another hyperbola \(k\) (Figs. 3–4) and the ordinates of a straight line drawn parallel to the \(x\)-axis at a distance from it equal to \(\sigma k_e p_e\), the scale for these and the other ordinates being taken, in comparison with Figs. 3–4, \(\sigma p\) times smaller. Thus, for the particular example mentioned we have the value of \(p_i\) for \(x=0.5\) when exhausting to the atmosphere (Fig. 3):

\[ p_i=\frac{6.15\cdot0.93}{100}(51.5-20)=2.95\ \text{atm.} \]

Thus the direct dependence \(p_i=f(x)\) is hyperbolic.

The basic equation (1),

\[ p_i=\sigma (kp-k_e p_e), \]

shows the possibility of varying \(p_i\) by varying \(k_e\). The coefficient of back pressure \(k_e\) is expressed through \(c\)—the compression ratio—in the form:

\[ k_e=1-c+\varphi(c+m)\ln\frac{c+m}{m} =1-Z(1+\varphi\ln m)+m+\varphi Z\ln Z, \]

if we denote \(Z=c+m\), and through \(\varphi\)—the coefficient characterizing the softness of the steam at the moment compression begins—\(=1.1\). Knowing that \(Z\) is usually within the limits \(Z=0.1\text{—}0.5\), and that, consequently, the multiplier at \(Z\) under the sign \(\ln\), when expanded in a series with neglect of the second term of the expansion, may be \(\gamma_1=10-2\), we may take \(\gamma_1=5\). Then we have:

\[ \gamma_1[2\varphi-1-\varphi\ln(m\gamma_1)]Z^2 -\gamma_1 k_e Z+ [\gamma_1(1+m)-2\varphi-1-\varphi\ln(m\gamma_1)]Z -k_e+1+m=0, \]

i.e. again for the dependence between \(k_e\) and \(Z\) we have a hyperbola, for

\[ AC-B^2=-(-\gamma_1)^2<0. \]

The coordinates of the center of the hyperbola will be:

\[ z=-\frac{1}{\gamma_1}=-0.2 \]

and

\[ k_e= -\frac{2\gamma_1[2\varphi-1-\varphi\ln(m\gamma_1)]}{\gamma_1^2} + \frac{\gamma_1(1+m)-2\varphi-1-\varphi\ln(m\gamma_1)}{\gamma_1} = -\frac{2\cdot11}{25}+\frac{3.2}{5} =-0.24, \]

whereby the equation of the second asymptote is

\[ k_e=-0.24+[2\varphi-1-\varphi\ln(m\gamma_1)]Z, \]

which for \(Z=1\) gives \(k_e'=2.4\). The point needed for constructing the hyperbola is found by putting \(Z=0\), so that \(k_e=1+m=1.08\). The hyperbola has been constructed by the method of secants in Fig. 6. Let us note that the relative error from neglecting the second term of the expansion here will be less than in the case of \(k\), since \(k_e\) is always \(>1\).

The use of variations of \(k_e\) for changing the power of steam engines is not usually applied in practice, for it has the following disadvantages: 1) the pressure \(p_c\) at the end of compression is commonly chosen sufficient to balance the inertia forces of parts with reciprocating motion; this ensures smooth running. When \(p_c\) is decreased, impacts may easily arise in the joints of the engine; 2) when \(p_c\) is increased, the work of back pressure increases, while the increase of \(p_c\) from compression of the steam is obtained by an unfavorable path, as the conversion of mechanical work into heat, and in the case of considerable clearance spaces even preliminary condensation may occur. The thermally most advantageous pressure at the end of compression should not exceed the mean temperature of the walls of the cylinder ends: departure from this leads to an increase in steam consumption; 3) the limits of power regulation are much wider in the case of changing the degree of admission or using a throttle valve. Although it is not customary in practice to resort to changing the degree of compression as an independent means of changing power, nevertheless in engines with a slide-valve steam distribution, and also in cases of flat regulators, together with a change in the degree of admission the degree of compression also changes in the opposite direction, i.e. an increase, for example, in the degree of admission entails a decrease in the degree of compression, and conversely. Thus the change in power occurs—

comes, in these cases, from both sides simultaneously, whereby the power of the engine changes more rapidly. In order to be able to determine the law of variation of \(p_i\) as a function of \(\varepsilon\) (or \(x=\varepsilon+m\)), it is necessary, in each such separate case, to know the dependence \(c=f(\varepsilon)\). If one confines oneself to a roughly approximate solution of the problem, it will turn out that sometimes this dependence can be expressed by an extremely simple linear equation of the form: \(c=a-b\varepsilon\). Such a dependence occurs, for example, in certain types of locomotives.¹ Thus, for a locomotive of series K, in numerical values it will be: \(c=0.38-0.42\varepsilon\). Expressing through \(x\) and \(Z\), we obtain: \(Z=a+m+bm-bx\) and \(Z=0.49-0.42x\), assuming \(m=0.08\). Using this dependence, we find the graph of the dependence of \(p_i\) on \(x\) for this particular case as the difference of the ordinates of the curve \(k=f(x)\) with a scale reduced by \(\sigma p\) times (Fig. 3), since in locomotives the segments before \(x=0.2\) are not used, and of the ordinates of the curve \(k_e=f(Z)\) (Fig. 6) with the scale reduced by \(p/p_e\) times, bearing in mind that the scales of Figs. 3–6 are the same; moreover, the ordinates of the latter curve (Fig. 6) are taken for those values of \(Z\) that correspond to the corresponding values of \(x\) from the linear graph (Fig. 7): \(Z=0.49-0.42x\). The required dependence \(p_i=f^1(x)\) is plotted in Fig. 8, where for \(x=0.5\) we have \(p_i\), as the difference of the ordinates \(ab-ad\):

\[ =(71-10.5)\frac{0.93\cdot 12\cdot 5}{100}\simeq 6.9\ \text{atm.}, \]

with \(p=12.5\) atm. and \(p_e=1.2\) atm.

It goes without saying that the influence of the speed of motion of the locomotive on \(p_i\) is not taken into account here.

Finally, from the basic equation (1), \(p_i=\sigma(kp-k_e p_e)\), it follows that \(p_i\) can be varied by changing \(p_e\). The disadvantages of regulating power by changing \(p_e\) are as follows: 1) the pressure at the end of compression changes, and consequently the possibility of shocks is not eliminated; 2) when \(p_e\) is increased there is a danger of loops forming on the indicator diagram at the end of expansion and compression; 3) when \(p_e\) is changed the engine is uneconomical, since for each degree of cutoff there corresponds only one most economical exhaust pressure \(p_e\), depending on depreciation and operating costs and on the cost of fuel. As is known, when selecting the most advantageous condenser pressure it is necessary to calculate all costs for a specified \(p_i\) at different \(p_e\), whereby, in order to determine the useful steam consumption, the corresponding degree of cutoff \(\varepsilon\) is determined for each \(p_e\). Thus the problem, in the case of changing \(p_e\) in the basic equation (1), \(p_i=\sigma(kp-k_e p_e)\), is modified: one has to find \(x=\varepsilon+m\) as \(x=f(p_e)\) for \(p_i=\text{const.}\) An analogous problem also arises in another practical case, when it is necessary to find the degree of cutoff when the back pressure is changed while maintaining the power, which occurs in cases of converting an installation that worked on a condenser to work on heating, and in the reverse conversion.

It is not difficult to see that

\[ \frac{p_i}{\sigma p}+\frac{k_e}{p}p_e=k, \]

i.e., \(k\) varies as \(k=f(p_e)\), according to the law of a straight line drawn from the origin at a distance \(\dfrac{p_i}{\sigma p}\) at an angle to the \(p_e\) axis whose tangent \(=\dfrac{k_e}{p}\). For our particular example, at \(p_e=0\) we have \(k=p_i/\sigma p=0.38\), and for \(p_e=1\) we have \(k=0.56\), which is also plotted in Fig. 9. Given \(p_e\), we easily find \(k\), and, using the curves (Figs. 3–4) \(k=f(x)\), we simply determine the corresponding \(x\). But one can have a direct representation of the dependence

\[ p_e=f(x)=\frac{p}{k_e}K-\frac{p_i}{\sigma k_e}, \]

i.e., this dependence is of hyperbolic form.

¹ Locomotive passports: 1-5-0 Ef (Petrograd, 1917, p. 14); 1-4-0 Shchp (Petrograd, 1915, p. 7); 1-3-1 S (Petrograd, 1915, p. 7); 2-3-0 Ku (Petrograd, 1915, p. 7); 0-5-0 E (Moscow, 1918, p. 7).

Each value of \(p_e\) is obtained as the difference between the ordinates of the curve \(k=f(x)\) (Figs. 3–4), on a scale reduced by a factor of \(\dfrac{p}{k_e}\), and the ordinates of the straight line \(\dfrac{p_1}{\sigma k_e}\) parallel to the \(x\)-axis on the same scale \(k\). Thus, for example, for \(p_e=0.3\) atm. we have in Fig. 10:

\[ p_e=\frac{0.3}{6.15}\times 1.13 \times 100=5.5\ \text{mm}, \]

so that \(x=0.23\), whence \(\varepsilon=0.15\).

All that has been said above applies to engines operating with saturated steam. For superheated-steam engines the coefficient of initial pressure has the form:

\[ k_n=\alpha \varepsilon+\frac{\beta}{n-1}(\varepsilon+m) \left[ 1-\left(\frac{\varepsilon+m}{m+1}\right)^{n-1} \right], \]

which may be represented, putting \(x=\varepsilon+m\), in the form:

\[ k_n=\left[ \alpha+\frac{\beta}{n-1} - \frac{x\beta\cdot n^{-1}}{(n-1)(m+1)^{\,n-1}} \right]x-\alpha m . \]

A complicating circumstance here is that the exponent \(n\) of the polytrope, being dependent on the temperature of the superheated steam, at the same time also depends on the degree of cut-off \(\varepsilon=x-m\).

For characterizing the latter dependence one may use the curves of Fig. 11, relating to the experiments of Prof. Schröter with compound-expansion engines; these curves give the dependence of \(n\) on \(\varepsilon\) in the HP cylinder. Since in modern engines the temperature of superheated steam is usually \(300\)—\(350^\circ\) C., then, assuming an approximate linear dependence between \(n\) and \(x=\varepsilon-m\) within the limits \(x=0.1\), when \(n=1.0\), and \(x=0.4\), when \(n=1.24\), we obtain:

\[ n=0.8x+0.92, \]

where \(m\) is taken as \(0.08\), as is often the value encountered for valve and slide-valve steam distribution with cylindrical slide valves, used with superheated steam. Very often, in choosing the value of the exponent \(n\) for pressures¹ \(p\geq 9\) kg/sq. cm, it is recommended to use the following table of Berner:

Mean exponents of expansion for the HP cylinder	Mean exponents of expansion for the HP cylinder	Mean exponents of expansion for the HP cylinder	Mean exponents of expansion for the HP cylinder	Mean exponents of expansion for the HP cylinder
Steam temperature in °C	200	250	300	350
Low load	1.00	1.05	1.10	1.14
Normal load	1.05	1.10	1.14	1.17
Heavy load	1.10	1.15	1.18	1.20

But here, in the first place, the dependence on the degree of cut-off exists only in a concealed and, moreover, indefinite form (low, normal, heavy load), and, secondly, for large engines it is recommended to raise the values of \(n\) by \(0.02\)—\(0.04\), and together this indicates that, in practice, even larger deviations from the actual values are usually of little significance. In general there is some uncertainty in the choice of \(n\), and it must be thought that the use of the approximate formula

\[ n=0.8x+0.92 \]

within the indicated limits cannot produce gross errors. Multiplying and dividing the subtracted term inside the square brackets of the expression for \(k_n\) by \(1.5^{\,n-1}\) and expanding into a series the numerator of the subtrahend \(\beta(1.5x)^{n-1}\), and in its denominator \((n-1)[1.5(1+m)]^{n-1}\), we then obtain:

\[ k_n= \left\{ \alpha+\frac{\beta}{n-1} - \frac{\beta\left[1+(n-1)\ln 1.5x\right]} {(n-1)\left[1+(n-1)\ln 1.5(1+m)\right]} \right\} (x-\alpha m). \]

The second terms of the expansion may be neglected, since for \(x=0.1\), \(n-1\) becomes zero, while for \(x=0.4\) the second terms of the expansion will be:

\[ 0.9\cdot 0.24^2\cdot \frac{(\ln 0.6)^2}{2}=0.002 \]

and

\[ 0.24\cdot 0.24^2\cdot \frac{(\ln 0.6)^2}{2}=0.001 . \]

We shall also expand in a series \(\beta(n-1)x\ln 1.5x\). The second term of the expansion, for \(x=0.4\), is equal to

\[ \frac{2}{3}\left(\frac{1.5x-1}{1.5x+1}\right)^3 (n-1)\beta x = 0.0008, \]

which value may likewise be ignored; consequently, instead of \(\ln 1.5\) it is necessary to substitute only

\[ 2\left(\frac{1.5x-1}{1.5x+1}\right). \]

We now reduce the right-hand side of the expression for \(k_n\) to a single denominator, substituting for \(n\) its approximate value \(=0.8x+0.92\), and carry out the actual division of the numerator polynomial by the denominator polynomial, after which we have:

\[ \begin{aligned} k_n={}&\Bigl\{1.2\alpha\ln[1.5(1+m)]x^3 +\{1.5\alpha-3\beta+[0.68\alpha+1.5\beta-1.2\alpha m]\ln[1.5(1+m)]\}x^2 \\ &+\{\alpha+2\beta-1.5\alpha m+[\beta-0.08\alpha-0.04\alpha m]\ln[1.5(1+m)]\}x \\ &+0.08\alpha m\ln[1.5(1+m)]-\alpha m\Bigr\}:\Bigl\{1.2\ln[1.5(1+m)]x^2 \\ &+\{1.5+0.68\ln[1.5(1+m)]\}x+1-0.08\ln[1.5(1+m)]\Bigr\} \\ ={}&\frac{0.55x^3-0.35x^2+3.04x-0.07}{0.57x^2+1.83x+0.96} =0.97x-3.72+\frac{8.81x+3.49}{0.57x^2+1.83x+0.96} =y_1+y_2 . \end{aligned} \]

Thus \(k_n\) for each \(x\) is obtained as the sum of the ordinates \(y_1\) and \(y_2\). But

\[ y_1=0.97x-3.72 \]

is the equation of a straight line, while

\[ y_2=\frac{8.81x+3.49}{0.57x^2+1.83x+0.94} \]

is obtained as the quotient from dividing the ordinates of the numerator straight line by the corresponding ordinates of the parabola*) of the denominator for the same abscissae \(x\). The whole matter is therefore reduced to the construction of two straight lines and one parabola. It is interesting to note that the denominator parabola is entirely independent of \(\alpha\) and \(\beta\), i.e., it is not connected with the character of the steam distribution and can be constructed once and for all, both for valve and for slide-valve steam distribution. The construction for \(y_1\) and \(y_2\) is carried out in Fig. 12, where the curve \(k_n\) is also plotted as a function of \(x\); the scale for the abscissae is \(1=100\) mm, and for the ordinates of the straight lines \(1=10\) mm, in order to make it possible to place them within the limits of the drawing, while for the curve \(k_n\) itself again \(1=100\) mm.

Beyond the cut-off limit, when \(x \ge 0.4\), the polytropic exponent \(n\) can apparently be taken as constant \(=1.24\). Therefore, for this case we have:

\[ \begin{aligned} k_n={}&\Bigl(\{1.5\alpha-3\beta+[\alpha(n-1)+\beta]1.5\ln[1.5(1+m)]\}x^2 \\ &+\{\alpha+2\beta-1.5\alpha m+[\alpha(n-1)+\beta-1.5\alpha m(n-1)]\ln[(1.5(1+m)]\}x \\ &-\alpha m-\alpha m(n-1)\ln[1.5(1+m)]\Bigr) : \Bigl\{1.5\{1+(n-1)\ln[1.5(1-m)]\}x \\ &+1+(n-1)\ln[1.5(1+m)]\Bigr\} \\ ={}&\frac{-0.47x^2+3.16x-0.08}{1.67x+1.11} =-0.28x+2.08-\frac{2.38}{1.67x+1.11}, \end{aligned} \]

i.e., in this case the matter is reduced to the construction of three straight lines, which is also carried out in Fig. 13. The combined curve of the variations of \(k_n\) with \(x\) within the limits from \(x=0.1\) to \(x=0.7\) is plotted in the same Fig. 13. This same curve can also represent the law of variation of the mean front pressure \(k_n\cdot p\) for any \(p\), as a function of \(x\), with the scale of the drawing, of course, reduced by \(p\) times.

*) For \(AC-B^2=0\).

Let us now pass to double-expansion engines. The aggregate mean indicated pressure \(p_i\) is represented here in the form of the sum of the partial such pressures in each of the cylinders, with \(p'_i\) in the HP cylinder having to be referred, of course, to the area of the LP piston, i.e.

\[ p_i=\frac{p'_i}{\delta}+p''_i, \]

where \(\delta\) is the ratio of the cylinder volumes. We may write:

\[ p'_i=\sigma(kp-k_r\rho p_r), \]

where \(p_r\) is the mean receiver pressure, and

\[ p''_i=\sigma(k_r-k_c p_c), \]

whereby by \(\rho\) is estimated the difference between the back pressure on the HP piston and the pressure on the LP piston. Usually \(\rho\) is taken as \(0.1p_r\), i.e. \(\rho=1.1\). Thus \(p_i\), being dependent on \(k\), as in a single-cylinder engine, when the degree of admission \(\varepsilon\) in the HP cylinder or \(E\) in the LP cylinder changes, depends at the same time also on \(p_r\), which changes together with \(\varepsilon\) and \(E\). For the weight quantity of steam passing through the steam engine, we have:

\[ \gamma F_1(m_0+\varepsilon)=\gamma_r F_2(M_0+E). \]

Taking approximately:

\[ \gamma=\mu p,\qquad \gamma_r=\mu p_r,\qquad m_0=m-\frac{\rho p_r(c+m)}{p} \]

and

\[ M_0=M-\frac{p_c(C+M)}{p_r}\,^{1)}, \]

and also putting

\[ \delta=\frac{F_2}{F_1}, \]

we obtain:

\[ p_r=\frac{p}{\delta M+\delta E+\rho(c+m)}\,\varepsilon +\frac{pm+\delta(C+M)p_c}{\delta M+\delta E+\rho(c+m)}, \]

i.e. the dependence of the mean receiver pressure on the variable degree \(\varepsilon\) of admission of the HP cylinder is expressed by the law of a straight line, on the assumption that at the same time the compression ratio of the HP cylinder \(c\) remains constant, as does the degree of admission \(E\) and the compression \(C\) in the LP cylinder. Put \(p_r\) as a function of \(x=\varepsilon+m\). Then:

\[ p_r=\frac{p}{\delta M+\delta E+\rho(c+m)}\,x +\frac{\delta(C+M)p_c}{\delta M+\delta E+\rho(c+m)}. \]

We take the particular example of a compound engine: \(\delta=2.5\); \(p=7\); \(p_c=0.2\); \(m=M=0.08\); \(c=0.08\); \(C=0.12\), and \(E=0.6\). We have:

\[ p_r=3.8x+0.055. \]

The construction is made in Fig. 14, with the scale for ordinates \(1=10\) mm, where, for example, for \(x=0.5\) we obtain the ordinate value 20.5 mm, which gives

\[ p_r=\frac{20.5}{10}=2.05\ \text{atm}. \]

We turn first to finding the dependence of the mean indicated pressure \(p''_i\) in the LP cylinder on \(x\). Substituting in the general equation \(p''_i=\sigma(k_r-k_c p_c)\) its expression through \(x\) in place of \(p_r\), we obtain:

\[ p''_i= \frac{\sigma k p}{\delta M+\delta E+\rho(C+m)}\,x +\frac{\delta(C+M)p_c\cdot\sigma k}{\delta M+\delta E+\rho(C+m)} -\sigma k_c p_c. \]

Thus the dependence for the LP cylinder between its mean indicated pressure \(p''_i\) and \(x\) is also of linear form.

Let us express this dependence in numerical values for the above example of a compound engine, taking \(\beta=0.85\), \(\alpha=0.93\), and \(\sigma=0.93\). Then:

\[ p''_i=2.9x-0.17. \]

In Fig. 15 we have, for example, for \(x=0.5\), the value

\[ p''_i=\frac{13.5}{10}=1.35\ \text{atm}. \]

In the HP cylinder the change of \(p'_i\) is governed by the change of two factors \(k\) and \(p_r\). Using Fig. 4, we easily find the curve for \(\sigma kp\) in Fig. 16, on which, at its scale, we also plot the straight line \(\sigma k_r\rho p_r\), applying Fig. 14. The difference of the ordinates of these two lines gives the value of \(p'_i\) for any \(x\). Thus, for example, for \(x=0.5\) we have

\[ p'_i=(74-33.5)\cdot\frac{0.93\cdot7}{100}=2.76\ \text{atm}, \]

so that the aggregate

\[ p_i=1.35+\frac{2.76}{2.5}=2.45\ \text{atm}. \]

\(^{1)}\) The volume of the clearance spaces filled with steam at admission pressure.

It follows from Fig. 16 that, with changes in \(\varepsilon\), the power of the HPC changes little, beginning with \(x \ge 0.3\).

Let us note, incidentally, that Fig. 14 makes it possible to establish the maximum cut-off in the HPC that we may allow without distorting the form of the indicator diagram in the HPC. The maximum pressure at the end of compression \(p_c\) must not exceed the admission pressure \(p\), in order to avoid the formation of a loop, i.e., approximately we have:

\[ p_c \cdot m = p_r(c+m), \quad \text{whence} \quad p_r=\frac{p_c m}{c+m}. \]

From the value of \(p_r\) found in this way and Fig. 14 we determine \(x\), and consequently also

\[ \varepsilon_{\max}=x-m. \]

If, with all other conditions of steam distribution unchanged, only the degree of filling \(E\) in the LPC is varied, then, proceeding again from the constancy of the weight quantity of steam, we obtain:

\[ \delta p_r X+\rho(c+m)p_r-p(m+\varepsilon)-\delta p_e(C+M)=0, \]

where \(X=E+M\). Thus, between the receiver pressure \(p_r\) and \(X\) there exists a hyperbolic dependence, since

\[ AC-B^2=-\delta^2<0. \]

Let

\[ \tau=\delta p_r \]

and

\[ \psi=\delta X+\rho(c+m). \]

Then

\[ \tau\psi=\delta\{\delta Xp_r+\rho(c+m)p_r\} =\delta\{p(m+\varepsilon)+\delta p_e(C+M)\}=\text{const}. \]

For the asymptotes we have \(\tau\psi=0\), so that

\[ \delta p_r=0, \quad \text{i.e.} \quad p_r=0 \]

and

\[ \psi=\delta X+\rho(c+m)=0, \quad \text{i.e.} \quad X=-\frac{\rho(c+m)}{\delta}. \]

Take as an example:

\[ \varepsilon=0.27;\quad m=M=0.08;\quad p=11;\quad p_e=1.1;\quad \delta=2.25;\quad c=0.09 \]

and

\[ C=0.12. \]

For the asymptote we have

\[ X=-\frac{1.1(0.09+0.08)}{2.25}=-0.083. \]

The point necessary for constructing the hyperbola will be found by putting \(X=1\), so that

\[ p_r=\frac{p(m+\varepsilon)+\delta p_e(C+M)}{\delta X+\rho(c+m)}=1.78. \]

The construction is carried out in Fig. 17.

Let us next proceed to finding the dependence of the mean indicated pressure \(p'_i\) in the HPC on \(X\). Substituting into the equation

\[ p'_i=\sigma(kp-\rho k_r p_r) \]

for \(p_r\) its value from the preceding expression, we obtain:

\[ p'_i=\sigma\left\{kp-k_r\cdot \frac{\rho[p(m+\varepsilon)+\delta p_e(C+M)]}{\rho(c+m)+\delta X}\right\}, \]

which is reduced to the form:

\[ \rho(c+m)p'_i+\delta p'_iX-\sigma\delta kpX-S=0, \]

i.e., the dependence of \(p'_i\) on \(X\) is also hyperbolic. We have the equations of the asymptotes:

\[ \tau=\delta p'_i-\sigma\delta kp=0 \]

and

\[ \psi=\delta X+\rho(c+m)=0, \]

whence:

\[ p_i=\frac{\sigma\delta kp}{\delta}=\sigma pk \]

and

\[ X=-\frac{\rho(c+m)}{\delta}, \]

which gives, in the numerical values of our particular example:

\[ p'_i=5.9 \]

and

\[ X=0.083. \]

The required point is obtained for \(X=1.0\), which gives

\[ p'_i=4.6 \text{ atm.} \]

A graphical representation of this law is given in Fig. 18.

As for the mean indicated pressure of the LPC:

\[ p''_i=\sigma(kp_r-k_e p_e), \]

then, as is evident from the formula, when \(E\) changes we shall have simultaneous changes in \(k\) and \(p_r\); moreover, for the dependence of \(k\) on \(X\) one must use the hyperbola of Fig. 3, since in the LPC \(X=0.3\text{--}0.8\). Multiplying the ordinates of this hyperbola by the corresponding ordinates of the hyperbola

\[ p_r=f(X) \]

of Fig. 17 for the same abscissae, and plotting the products (Fig. 19) for the same abscissae in the form of new ordinates on the scale

\[ \frac{\text{Fig. }3 \times \text{Fig. }17}{\text{Fig. }3}, \]

and subtracting from the latter the constant ordinates of the straight line \(-k_e p_e\), plotted on the same scale, we obtain, in the form of the difference of these ordinates, the values of \(p''_i\) between the lines \(ab\) and \(vg\), if the scale for measurement is the scale of Fig. 17 \(\cdot \sigma\). Thus, for

\[ x=0.5 \]

we have

\[ p''_i=\frac{11.5}{10\cdot 0.93}=1.2 \text{ atm.} \]

Finally, let us construct in Fig. 20 the curve of variation of the total

\[ p_i=\frac{p'_i}{\delta}+p''_i \]

as a function of \(X\), by adding the ordinates of Fig. 18 to the difference segments between \(ab\) and \(vg\) of Fig. 19.

We see that the maximum power is obtained at \(X = 0.5\). A further increase of \(X\) leads to a decrease in the power of the engine, which is explained by the growth of losses from the pressure drop when the H.P.C. communicates with the receiver, and also by the possibility of a loop forming at the end of compression in the L.P.C. A decrease of \(X < 0.5\) also lowers the power of the engine, since the receiver pressure begins to rise, and consequently also the counterpressure in the H.P.C.; as a result of this, the pressure at the end of compression in the H.P.C. increases, giving rise to the danger of the formation here of loops at the end of compression, as well as loops at the end of expansion.

Incidentally, one cannot fail to note one interesting consequence following from this, namely the possibility of using a graph of the type of Fig. 20 for the purpose of distributing the power between the cylinders equally in compound engines. Here two cases must be distinguished: the case of slide-valve steam distribution in the H.P.C. and valve distribution. Slide-valve steam distribution makes it necessary to obtain a certain definite degree of admission in the H.P.C., depending on the type of slide valves. Thus, for example, for simple box slide valves, degrees of admission of less than \(50\%\) of the piston stroke are disadvantageous, since they worsen the conditions of exhaust and compression and require large eccentricities and, consequently, considerable friction work; for Trick-Weyss slide valves one may take \(\varepsilon \geq 0.35\), while for double slide valves one usually takes \(\varepsilon = 0.25\), etc. These degrees of admission are, as it were, specified in advance. Let the admission pressure in the H.P.C. be \(p = 8\) atm. For this pressure, and for operation with a condenser, the most advantageous degree of admission, referred to the L.P.C., say, is \(\varepsilon_0 = 0.07\text{--}0.06\). Then, in order to make it possible to obtain it, for example with a double slide valve in the H.P.C., one should choose the ratio of cylinder volumes

\[ \frac{0.25}{0.07} \simeq 3.5. \]

For economical slide-valve engines and constant load, larger volume ratios, \(3\text{--}5\), are customary in America.^1) Having assigned the degrees of compression \(c\) and \(C\) from the conditions of slide-valve steam distribution and smooth running (the possibility of unloading the inertia forces by sufficient pressure at the end of compression), we obtain all the data, knowing \(\varepsilon\) and \(p\) and considering \(X = E + M\) as the varying quantity, to construct the curve \(p_r = f(X)\), as in Fig. 17, the curve \(p'_i = f'(X)\), and the curve \(p''_i = f''(X)\), after the manner of Figs. 18 and 19. By constructing the curve of the total \(p_i\) according to Fig. 20, we determine the \(X\) at which \(p'_i = p''_i\). This \(X = E + M\) is then adopted as the degree of admission of the L.P.C., ensuring an equal distribution of the steam work between the cylinders in a compound engine. In Fig. 20 this occurs at \(X = 0.43\). It is necessary to take the degree of admission \(X = E + M\) rather than \(E\), since the finite volume of the receiver causes a decrease of the area of the indicator diagram of the H.P.C. in a compound engine and an increase of the same in the L.P.C. by approximately up to \(10\%\).^2) In the case of a valve engine, besides the stated degree of admission \(\varepsilon_0\), what is usually specified is the volume ratio \(\delta\) of the cylinders, which can be determined from the theoretical diagram of the corresponding single-cylinder engine by dividing its area in half, at least by the method of the integral curve, or with the aid of the tables of Hrabak or Haeder. Knowing \(\delta\), we find \(\varepsilon = \varepsilon_0 \cdot \delta\), after which the problem of finding \(X = E + M\) for uniform distribution of the work between the cylinders is solved as before.

A special case is represented by engines in which, simultaneously with a change in the degrees of admission in both cylinders, the degrees of compression also change; this occurs, for example, in engines with link motions, or in engines with flat governors, where, when the degree of admission of the H.P.C. changes, there simultaneously changes—

^1) Levenson. Modern American Steam Engines. St. Petersburg, 1912, p. 6.
^2) Dubbel. Design and Calculation of Steam Engines. 1907. St. Petersburg, p. 27.

here and the degree of compression is also varied. In these cases it is first of all necessary to establish the law of dependence between \(\varepsilon\) and \(c\) in the HP cylinder and between \(E\) and \(C\) in the LP cylinder, and also the dependence between \(\varepsilon\) and \(E\), i.e. to what extent changes in \(\varepsilon\) cause changes in \(E\). As a particular example, indicating the convenience of solving such problems by the methods set forth here, let us take the passenger locomotive, type n/t, issue of 1901. As is seen from Fig. 21, the dependence between \(\varepsilon\) and \(c\), on the one hand, and \(E\) and \(C\), on the other, approaches a linear type: \(Z=a-bx\), where \(Z=c+m\), and the values of the constants in the HP cylinder \(a\) and \(b\), as \(d\) and \(t\) in the LP cylinder, are indicated in Fig. 21. As for the dependence between \(\varepsilon\) and \(E\), for this type of locomotive one may approximately take: \(E=\varepsilon+0.69\). Starting from the constancy of the steam flow through the locomotive: \(\gamma F_1(m_0+\varepsilon)=\gamma_r F_2(M_0+E)\), and using the above-indicated expressions for \(m_0\), \(M_0\), and the just-indicated dependences of \(E\) on \(\varepsilon\) and \(C\) on \(E\), after transformations we reduce the equation of steam flow to the form: \(Bx\,p_r' + Dx - Fp_r' - N = 0\), which defines a hyperbola, the coefficients having the following values:

Coefficients.	Their literal expressions.	Their numerical values.
\(B.\)	\(\dfrac{F_1}{F_2}\cdot \rho \cdot b - 1\)	\(-0.82\)
\(D.\)	\(p\,\dfrac{F_1}{F_2} - p_e \cdot f\)	\(+5.3\)
\(F.\)	\(\rho \cdot a \cdot \dfrac{F_1}{F_2} + i\)	\(+0.31\)
\(N.\)	\(d - f \cdot i\)	\(+0.55\)

Here \(\dfrac{F_1}{F_2}=\dfrac{1}{\delta}=0.47\) is the ratio of the cylinder volumes of the n/t locomotives; \(m=N=0.08\); \(p=12.5\); \(p_e=1.25\); \(i=E-\varepsilon=0.09\); \(\rho=1.1\); \(d\) and \(f\) in the LP cylinder are the same as \(a\) and \(b\) in the HP cylinder, i.e. the coefficients in the dependence between \(C\) and \(E\). It must be borne in mind that, as the speed of motion of the locomotive increases, there will occur a distortion in the form of the indicator diagrams in both cylinders, owing to changes in \(\rho\), \(\alpha\), \(\beta\), \(\varphi\), and \(p_e\); the tracking of this is not part of the problem of the present work.

The hyperbola for the adopted data is plotted in Fig. 22, and for its asymptotes we have: \(p_r'=-\dfrac{D}{B}=+6.47\) and \(x=\dfrac{F}{B}=-0.38\); and for the necessary point at \(x=0\) we have: \(p_r'=1.78\).—

Multiplying the ordinates of the latter hyperbola \(p_r'\) by the corresponding ordinates of the hyperbola \(K_r\) (Fig. 6) for the same abscissae \(x\), and by \(\sigma\rho\), gives the curve of change of the back pressure \(\sigma\rho K_r p_r'\); when using the hyperbola \(K_r\) (Fig. 6), \(Z\) is determined beforehand from \(x\) from the above-mentioned linear equation (Fig. 21). This curve is plotted in Fig. 23, where above it is drawn the curve of the forward pressure for the same abscissae \(x\), as the product of the ordinates of the hyperbola \(K\) (Fig. 3) by \(\sigma p\), for the cut-off in a locomotive usually does not go below \(x=0.3\). The difference of the ordinates of the curves in Fig. 23 \((\sigma pK-\sigma\rho K_r p_r')\) for any value of \(x\) gives the value of the mean indicated pressure \(p_i'\) in the HP cylinder. Thus, for \(x=0.6\) we have \(p_i'=\dfrac{40}{100}\cdot 0.93\cdot 12.5 = 4.65\) atm. In a completely analogous way we carry out the construction for the curves

of the front and back pressures in the LPC, which is also done in Fig. 24, using the linear dependence between \(E\) and \(C\) in the LPC: \(Z=0.48+0.4X\). In Fig. 25 a diagram is constructed of the change in the combined pressure; \(p_i=\dfrac{p'_i}{\delta}+p''_i\), where, in constructing this diagram, it was assumed that the degree of filling \(\varepsilon\) in the HPC corresponds in the LPC to \(E=\varepsilon+0.09\), in accordance with which the ordinates from Fig. 24 were taken for transfer to Fig. 25.—For example, for \(x=0.6\) we have

\[ p_i=\frac{36}{100}\cdot 12.5\cdot 0.93 \simeq 4.2\ \text{atm.} \]

Let us also examine the case of a change in the back pressure in the LPC while preserving the total power of the steam engine, in view of the practical significance of this case, as was noted above in the corresponding part concerning the single-cylinder engine. From the equation for the LPC:

\[ p''_i=\sigma( k p_r-k_e p_e) \]

we have:

\[ p_r=\frac{p''_i}{\sigma k}+\frac{k_e}{k}p_e, \]

i.e., with \(p''_i=\mathrm{const.}\) preserved, we have a change of \(p_r\) as a function of \(p_e\) according to the law of a straight line. Substituting into the equation of constancy of the weight quantity of steam passing through the steam engine,

\[ (\varepsilon+m_0)p\cdot F_1=(E+M_0)\,p_r\cdot F_2, \]

instead of \(m_0\), \(M_0\), and \(p_r\) their values from the preceding, we obtain:

\[ \varepsilon= \left\{ \left[(E+M)\delta+\rho(c+m)\right]\frac{k_e}{k} -(C+M)\delta \right\}\frac{p_e}{p} -m+ \left[(E+M)\delta+\rho(c+m)\right]\frac{p''_i}{\delta k p}, \]

i.e., the change in filling \(\varepsilon\) in the HPC when \(p_e\) changes also occurs according to the law of a straight line drawn from the origin at a distance

\[ \left[(E+M)\delta+\rho(c+m)\right]\frac{p''_i}{\delta k p}-m \]

and at an angle of inclination to the \(p_e\) axis whose tangent is

\[ \frac{1}{p}\left\{\left[(E+M)\delta+\rho(c+m)\right]\frac{k_e}{k} -(C+M)\delta\right\}, \]

where it is assumed that all steam-distribution conditions, such as \(c\), \(C\), and \(E\), remain unchanged. The straight line is shown in Fig. 26 for our particular example: \(p=7\); \(c=0.08\); \(C=0.12\); \(E=0.6\); \(p''_i=0.9\); \(m=M=0.08\); \(\delta=2.5\). Then

\[ \varepsilon=0.275\,p_e+0.23, \]

so that, for example, for \(p_e=0.2\) atm., \(\varepsilon=0.27\). The power of the HPC, as we saw above (Fig. 16), changes little when \(\varepsilon\geq 0.25\).

The method of regulation by steam throttling in double-expansion engines is at present rarely used, but in locomotives, when running “on a small valve,” this method (manually) is employed. Likewise in stationary engines there are always possible cases of operation of the engine not at full initial pressure, either for permanent reasons (structural weakness of boilers, etc.) or temporary ones (poor fuel, inexperience of stokers, etc.). Proceeding from the condition of constancy of the steam flow by weight, we have:

\[ p(\varepsilon+m_0)=\delta\,p_r(E+M_0), \]

whence, after substituting instead of \(M_0\) and \(m_0\) their expressions from the preceding, we obtain:

\[ p_r= \frac{\varepsilon+m}{\rho(c+m)+\delta(E+M)}\,p - \frac{p_e(C+M)\cdot\delta}{\delta(c+m)+\delta(E+M)}, \]

i.e., the change in \(p_r\) with changes in \(p\) occurs according to the law of a straight line, as does \(p'_i\) as a function of \(p\):

\[ p'_i= \frac{\sigma k(\varepsilon+m)}{\rho(c+m)+\delta(E+M)}\,p - \frac{p_e(C+M)\cdot\sigma k\cdot\delta}{\rho(c+m)-\delta(E+M)} -\sigma k_e p_e. \]

For our particular example of a compound engine we have, at \(\varepsilon=0.25\):

\[ p_r=0.175\,p-0.054 \]

and

\[ p'_i=0.135\,p-0.23, \]

which straight lines are shown in Figs. 27 and 28. For \(p'_i\) we have:

\[ p'_i=\sigma k p-\sigma k_r \rho p_r, \]

i.e., \(p'_i\) is obtained, when \(p\) changes, as the difference of the ordinates of the straight line \(y_1=\sigma k p\), passing through the origin, and the straight line

\[ y_2=\sigma k_r\rho p_r= \frac{\sigma k_r(\varepsilon+m)\cdot\rho}{\rho(c+m)+\delta(E+M)}\,p- \]

$$ +\frac{p e (C+M)\delta \sigma k_r \cdot \rho}{p(c+m)+\delta(E+M)}. $$

For our numerical example: \(y_1=0.93 \cdot 0.56 \cdot p=0.52p\) and \(y_2=0.186p-0.057\). The construction is carried out in Fig. 29. In Fig. 28 the upper straight line represents the law of variation of the total

$$ p_i=\frac{p'}{\delta}+p'_i $$

with variation of \(p\).

Now, in the adopted order of exposition, we must deal with compound-expansion engines with superheated steam. It is usually assumed that steam enters the LPC already saturated or only slightly superheated; therefore the expansion period here, as well as the compression periods in both cylinders, proceeds in the same way as in saturated-steam engines. Thus, in the equation of constancy of the weight quantity of steam passing through the steam engine, the changes will concern only the HPC. Denoting by \(\gamma_n\) the specific weight of superheated steam, we obtain:

\[ \gamma_n F_1(\varepsilon+m_0)=\gamma_r F_2(E+M_0). \]

The specific volume of superheated steam, as is known, is expressed by

\[ V_n=\frac{1}{\gamma_n}=\frac{47.1\,T_n}{p}-0.016, \]

where \(T_n\) is the absolute temperature of the superheated steam, and \(p\) is its pressure in kg/sq. m. But approximately one may assume that the specific volumes of superheated steam vary as for gases, i.e.

\[ \frac{1}{\gamma_n}:\frac{1}{\gamma}=T_n:T, \]

where \(\gamma\), as before, is the specific weight of dry saturated steam at the same pressure, and \(T\) its absolute temperature. One may therefore write:

\[ p \cdot \frac{T}{T_n}(\varepsilon+m_0)=\delta p_r(E+M_0). \]

Taking the degree of filling \(\varepsilon\) in the HPC to be unchanged, as are \(c\) and \(C\), and only the degree of filling \(E\) in the LPC to vary, substituting instead of \(m_0\) and \(M_0\) their expressions from the preceding, we have:

\[ \delta p_r X+\frac{T}{T_n}\rho(c+m)p_r-\frac{T}{T_n}p(m+\varepsilon)-\delta p_e(C+M)=0, \]

i.e., as for saturated steam, the relation between \(p_r\) and \(X=E+M\) is hyperbolic. The point needed for constructing the hyperbola at \(X=1\) will be:

\[ p_r= \frac{\frac{T}{T_n}p(m+\varepsilon)+\delta p_e(C+M)} {\delta X+\frac{T}{T_n}\rho(c+m)}, \]

which, in the numerical values of the example in Fig. 17, for superheating to \(325^\circ\) C, gives \(p_r=1.43\).

For the asymptotes we have:

\[ p_r=0 \]

and

\[ X=-\frac{\frac{T}{T_n}\rho(c+m)}{\delta}=-0.06. \]

The hyperbola is constructed in Fig. 30. For the HPC we have:

\[ p'_i=\sigma\left[ k_n p - k_r\cdot \rho\cdot \frac{\frac{T}{T_n}p(m+\varepsilon)+\delta p_e(C+M)} {\delta X+\frac{T}{T_n}\rho(c+m)} \right], \]

which gives:

\[ \delta p'_i X+\frac{T}{T_n}\rho(c+m)p'_i-\sigma k_n p\,\delta X+Q=0, \]

i.e. \(p'_i=\varphi(X)\) is again a hyperbola. The asymptotes are

\[ p'_i=\sigma k_n p=5.9 \]

and

\[ X=-\frac{\frac{T}{T_n}\rho(c+m)}{\delta}=-0.06. \]

The necessary point at \(X=1\) will be \(p'_i=4.22\). The hyperbola is constructed in Fig. 31. The construction of the graph \(p'_i=\varphi(X)\) for a superheated-steam engine will differ from that for a saturated-steam engine (Fig. 19) only in that the ordinates of the hyperbola \(k\) (Fig. 3) must in the present case be multiplied by the ordinates of the hyperbola \(p_r\) in Fig. 30, and not in Fig. 17. The graph is constructed in Fig. 32. The combined

the graph for \(p_i=\dfrac{p'_i}{\delta}+p''_i\) is given in Fig. 33, from which it is seen that, in the case of superheated steam, equality of the works in the cylinders will occur at \(x=0.38\).

In the case where only the degree \(\varepsilon\) of cut-off in the HPC changes, from the equality \(p \cdot T/T_n\,(\varepsilon+m_0)=\delta p_r(E+M_0)\) we obtain:

\[ p_r= \frac{p\cdot T/T_n}{\delta(E+M)+T/T_n\,\rho(c+m)}\,x + \frac{\delta p_e(C+M)}{\delta(E+M)+T/T_n\,\rho(c+m)} \quad \text{and} \]

\[ p''_i= \frac{\sigma k\,p\,T/T_n}{\delta(E+M)+T/T_n\,\rho(c+m)}\,x + \frac{\delta p_e(C+M)\,\sigma k}{\delta(E+M)+T/T_n\,\rho(c+m)} - \sigma k_e p_e . \]

Thus, conversely, \(p_r\) and \(p''_i\), as functions of \(x\), vary according to the law of a straight line. These straight lines are plotted in Fig. 34 and Fig. 35 for the particular example of a compound engine (Fig. 14), assuming superheating to \(300^\circ\) C. After substituting the numerical values, we obtain: \(p_r=2.9x+0.055\) and \(p''_i=2.24x-0.16\). For the HPC, using Fig. 13, we easily find the curve \(\sigma k_n p\) in Fig. 36; on the same scale we also plot the straight line \(\sigma k_r \rho p_r\), applying Fig. 34. The difference of the ordinates of these two lines gives the values of \(p'_i\). The combined diagram of the variations \(p_i=p'_i/\delta+p''_i\) as a function of \(x\) is presented in Fig. 35, where it is seen that for all \(x\) the work of the HPC is predominant. Equalization of the work can be achieved by reducing the degree of cut-off of the LPC.

These particular cases may also be sufficient to illustrate the application, to double-expansion engines using superheated steam, of the equality

\[ p\cdot T/T_n\,(\varepsilon+m_0)=\delta\,p_r(E+M_0) \]

and of the hyperbola \(k_n\) (Fig. 13); for these cases, being typical, in connection with what has been set forth concerning saturated-steam engines, can sufficiently serve as a guide in the numerous and varied situations of practice.

In conclusion it is necessary to note that the dependences considered above for the coefficients of the forward \(k\) and back \(k_e\) pressures, which have been adopted as fundamental, were derived in accordance with existing practice under the assumption that the coefficients \(\alpha\), \(\beta\), \(\varphi\) remain constant when the degrees of cut-off \(\varepsilon\) and compression \(c\) vary. Meanwhile there is reason to suppose that these latter coefficients, in turn, vary when \(\varepsilon\) and \(c\) vary, and are different for different types of steam distribution. There is little experimental material in the literature for judging the laws of such variation, and therefore one of the forthcoming works in the steam-engine laboratory of the Tomsk Technological Institute is planned to establish such dependences for \(\alpha\) and \(\beta\) as \(f(\varepsilon)\), and for \(\varphi\) as \(f(c)\).

Prof. I. N. Butakov. “Changes in the Power of Steam Engines.”

Fig. 1.

Visible labels and markings: \(p\); \(1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 6.1, 10.0\); \(0, 10, 20, 30, 40\).

Fig. 2.

Visible labels and markings: \(A, B, C, D, E, F\); \(p_1\); \(f\); \(h\); Fig. 2.

Fig. 3.

Visible labels and markings: \(x\), \(y\), \(0\), \(-0.05\), \(-0.445\), \(-0.24\), \(-0.30\), \(-0.33\), \(-0.424\), \(-0.50\), \(-0.98\), \(-1.31\), \(8.0\), \(8.32\), \(8.53\), \(8.62\); \(h\); \(\varphi\); \(k\); Fig. 3.

Fig. 4.

Visible labels and markings: \(K\), \(0\), \(0.1\), \(0.2\), \(0.3\), \(0.4\), \(0.5\), \(0.6\), \(0.7\), \(0.8\), \(0.9\); \(+0.01\), \(-0.001\), \(-0.001\); Fig. 4.

Fig. 5.

Visible labels and markings: \(K\), \(Z\), \(0\), \(0.1\), \(0.2\), \(0.24\), \(0.26\), \(0.281\), \(0.3\), \(0.4\), \(0.5\); \(1.0\), \(1.1\), \(1.2\), \(1.3\), \(1.4\), \(1.5\), \(1.6\), \(1.7\); \(-0.30\), \(-0.24\), \(+2.40\); Fig. 5.

Fig. 7.

Visible labels and markings: \(P_i\); \(0\); \(1.0\); \(2.0\); \(3.0\); \(4.0\); \(0.1\); \(0.2\); \(0.3\); \(0.4\); \(0.5\); \(0.57\); \(0.6\); \(0.9\); \(1.0\); \(2.2\).

Fig. 7.

Visible labels and markings: \(Z\); \(0\); \(0.1\); \(0.2\); \(0.3\); \(0.4\); \(0.5\); \(0.6\); \(0.7\); \(0.8\); \(0.9\); \(1.0\); \(0.15\); \(0.2\); \(0.25\); \(0.28\); \(0.32\); \(0.37\); \(0.4\); \(0.5\).

Fig. 8.

Visible labels and markings: \(P_i\); \(0\); \(0.3\); \(0.4\); \(m\).

Fig. 10.

Visible labels and markings: \(P_e\); \(0\); \(0.1\); \(0.2\); \(0.25\); \(0.3\); \(0.4\); \(0.5\); \(p_e\), kg/cm\(^2\).

Fig. 12.

Visible labels and inscriptions:

• \(k_m\)
• \(k_n\)
• \(Y\)
• \(X\)
• \(X_1\)
• \(0.33x + 1.83 + 0.96 = Y\)
• \(0.87x + 0.8 = Y_0\)
• \(0.038\, [[unclear: formula fragment]] = X_1\)

Fig. 13.

Visible labels and inscriptions:

• \(k_m\)
• \(k_n\)
• \(X\)
• \(Y\)
• \(Y = 1.6x + 1.11\)
• \(Y = -0.28x + 2.68\)

Fig. 14.

Visible labels:

• \(X\)
• \(Y\)

Fig. 15.

Visible labels:

• \(X\)
• \(Y\)

Fig. 16.

Visible labels and inscriptions:

• \(Y\)
• \(X\)
• \(50\%\)
• \(40\%\)
• \(30\%\)
• \(20\%\)
• \(15\%\)

Fig. 17.

Visible labels and inscriptions:

• \(n\)
• \(t_{\mathrm{avg.}\, [[unclear: Russian abbreviation]]}\)
• \(20\%\)
• \(30\%\)

Fig. 18.

Visible caption:

• Fig. 18.

Visible labels in the diagrams:

• \(p_1\)
• \(p_0\)
• \(0{,}4\)
• \(0{,}8\)
• \(90\)
• \(80\)
• \(70\)
• \(60\)
• \(50\)
• \(40\)
• \(30\)
• \(20\)
• \(1{,}8\ \mathrm{rpm}\)
• \(19\ \mathrm{rpm}\)
• \(2{,}1\ \mathrm{rpm}\)
• \(x,\ \mathrm{frac.}\)
• \(X\)
• \(82{,}6\)
• \(x\)
• \(x_0\)
• \(0{,}1\)
• \(0{,}2\)
• \(0{,}3\)
• \(0{,}4\)
• \(0{,}5\)
• \(0{,}6\)
• \(0{,}7\)
• \(0{,}8\)
• \(0{,}9\)
• \(1{,}0\)
• \([[unclear: formula beginning with X]]\)
• \([[unclear: formula beginning with x]]\)

Prof. I. I. Vukalov. Changes in the Power of Steam Engines.

Fig. 21.

Visible labels: \(p_i\), \(p_{\mathrm{fr}}\), \(p'\), \(X\).

Axis markings: \(0\), \(0.3\), \(0.4\), \(0.5\), \(0.6\), \(0.7\), \(0.8\).

Fig. 22.

Visible labels: \(p_i\), \(p'\), \(p_{\mathrm{fr}}\), \(X\).

Axis markings: \(0\), \(1.0\), \(2.0\), \(3.0\), \(4.0\); \(0.3\), \(0.4\), \(0.5\), \(0.6\), \(0.7\), \(0.8\).

Fig. 23.

Visible labels: \(P_i\), \(X\), \(0.47\), \(1.76\), \(-0.58\).

Axis markings: \(0\), \(0.3\), \(0.4\), \(0.5\), \(0.6\), \(0.7\).

Fig. 24.

Visible labels: \(p_i\), \(p'\), \(p_{\mathrm{fr}}\).

Axis markings: \(0\), \(0.0\), \(0.1\), \(0.3\), \(0.6\), \(0.7\), \(0.8\).

Fig. 25.

Visible labels: \(P_i\), \(n\).

Axis markings: \(0\), \(1\), \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(9\), \(10\).

Fig. 26.

Visible labels: \(\dfrac{P_i}{P_i'}\), \(p_i\), \(p_i'\).

Axis markings: \(0\), \(1\), \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(9\), \(10\).

Fig. 27.

Visible labels: \(p_i'\), \(p_{\mathrm{fr}}\), \(p'\).

Axis markings: \(0\), \(0.1\), \(0.2\), \(0.3\), \(0.4\), \(0.5\), \(0.6\), \(1.0\).

Fig. 32

Visible labels: \(0{,}1\), \(0{,}2\), \(0{,}3\), \(0{,}4\), \(0{,}5\), \(0{,}6\), \(0{,}7\), \(0{,}8\), \(0{,}9\); \(d\); \(d_1/d\); [[unclear: small curve label]].

Fig. 33

Visible labels: \(0{,}1\), \(0{,}2\), \(0{,}3\), \(0{,}4\), \(0{,}5\), \(0{,}6\), \(0{,}7\), \(0{,}8\), \(0{,}9\); [[unclear: axis/curve labels]].

Fig. 34

Visible labels: \(0{,}1\), \(0{,}2\), \(0{,}3\), \(0{,}4\), \(0{,}5\), \(0{,}6\), \(0{,}7\), \(0{,}8\), \(0{,}9\); [[unclear: axis/curve labels]].

Fig. 35

Visible labels: \(0{,}1\), \(0{,}2\), \(0{,}3\), \(0{,}4\), \(0{,}5\), \(0{,}6\), \(0{,}7\), \(0{,}8\), \(0{,}9\); [[unclear: axis/curve labels]].

Fig. 36

Visible labels: \(0{,}1\), \(0{,}2\), \(0{,}3\), \(0{,}4\), \(0{,}5\), \(0{,}6\), \(0{,}7\), \(0{,}8\), \(0{,}9\); [[unclear: axis/curve labels]].

Fig. 37

Visible labels: \(0{,}1\), \(0{,}2\), \(0{,}3\), \(0{,}4\), \(0{,}5\), \(0{,}6\), \(0{,}7\), \(0{,}8\), \(0{,}9\); [[unclear: axis/curve labels]].

Fig. 38

Visible labels: \(0{,}1\), \(0{,}2\), \(0{,}3\), \(0{,}4\), \(0{,}5\), \(0{,}6\), \(0{,}7\), \(0{,}8\), \(0{,}9\); [[unclear: axis/curve labels]].

Fig. 39

Visible labels: \(0{,}1\), \(0{,}2\), \(0{,}3\), \(0{,}4\), \(0{,}5\), \(0{,}6\), \(0{,}7\), \(0{,}8\), \(0{,}9\); [[unclear: axis/curve labels]].

1. Berner. The Application of Superheated Steam to the Piston Steam Engine. Moscow, 1908, p. 83. ↩