CHANGE IN THE POWER OF A COMPOUND STEAMSHIP ENGINE WITH CHANGES IN THE DEGREE OF ADMISSION

I. N. BUTAKOV and E. N. SHADRIN

The mean indicated pressure, as is known, may be expressed for the HP cylinder in the form

\[ P_i'=\delta_{\mathrm{cyl}}\cdot\left(K_1P_1-K_r\rho P_r\right), \]

where \(K_1\) is the averaging coefficient of the initial pressure, \(K_r\) is the same for the back pressure, which is \(\rho P_r\), with \(P_r\) being the admission pressure in the LP cylinder, and \(\rho \simeq 1.05\) is a coefficient characterizing the pressure drop during the flow of steam from the HP cylinder. When the degree of admission of the HP cylinder changes, \(K_1\), \(K_r\), and \(P_r\) will change. For \(K_1\), in the case of saturated steam, we have

\[ K_1=\alpha_{\mathrm{cyl}}\cdot\varepsilon+\beta_{\mathrm{cyl}}\cdot(\varepsilon+m)\cdot\ln\frac{1+m}{\varepsilon+m}, \]

where \(m\) is the clearance volume of the HP cylinder in fractions of the working volume of the cylinder, \(\alpha_{\mathrm{cyl}}\simeq0.93\) is the averaging coefficient of admission, and \(\beta_{\mathrm{cyl}}=0.85\) is the coefficient characterizing the pressure drop at the moment of cut-off. Putting \(x=\varepsilon+m\), we transform

\[ K_1=\alpha_{\mathrm{cyl}}\cdot x-\alpha_{\mathrm{cyl}}\cdot m+\beta_{\mathrm{cyl}}\cdot x\cdot\ln(1+m)-\beta_{\mathrm{cyl}}\cdot x\cdot\ln x . \]

In order to get rid of \(\ln x\), we resort to expansion in a series; moreover, in order to be able to ignore all terms of the expansion beginning with the second, we multiply \(x\) under the sign of the logarithm by a certain factor \(\psi>1\), while at the same time adding the quantity \(\beta_{\mathrm{cyl}}\cdot x\cdot\ln\psi\). Then

\[ K_1=x\left[\alpha_{\mathrm{cyl}}+\beta_{\mathrm{cyl}}\cdot\ln(1+m)+\beta_{\mathrm{cyl}}\cdot\ln\psi\right] -\alpha_{\mathrm{cyl}}\cdot m-2\beta_{\mathrm{cyl}}\cdot x\cdot\frac{\psi x-1}{\psi x+1} \]

or

\[ \psi\left[\alpha_{\mathrm{cyl}}+\beta_{\mathrm{cyl}}\cdot\ln(1+m)+\beta_{\mathrm{cyl}}\cdot\ln\psi-2\beta_{\mathrm{cyl}}\right]x^2+ \]

\[ +\left[\alpha_{\mathrm{cyl}}+\beta_{\mathrm{cyl}}\cdot\ln(1+m)+\beta_{\mathrm{cyl}}\cdot\ln\psi-\alpha_{\mathrm{cyl}}\cdot m\cdot\psi+2\beta_{\mathrm{cyl}}-\psi K_1\right]x- \]

\[ -K_1-\alpha_{\mathrm{cyl}}\cdot m=0. \tag{1} \]

This is the equation of a hyperbola. Considering that in steamship engines the usual fluctuations are \(x=0.3\div0.8\), in order to be able to ignore the second term

\[ \frac{2}{3}\,\beta_{\mathrm{cyl}}\cdot x\left(\frac{\psi x-1}{\psi x+1}\right)^3 \]

of the series expansion of \(\ln\psi x\), we take

\[ \psi=\frac{1}{x}=3.5-1.25, \]

keeping closer to the mean values \(\psi=2\div2.5\), and in such a way that the multiplier at \(x^2\) of equation (1) becomes zero, for the purpose of simplifying the problem.

If we take \(\beta_{\mathrm{tsvo}}=0.85;\ \alpha_{\mathrm{tsvo}}=0.93;\ m=0.08\), we obtain \(\psi=2.3\), when the indicated factor is made equal to zero. Then, for determining the asymptotes, we have

\[ K_1= \frac{ [\alpha_{\mathrm{tsvo}}+\beta_{\mathrm{tsvo}}\ln(1+m)+\beta_{\mathrm{tsvo}}\ln\psi-\alpha_{\mathrm{tsvo}}m\psi+2\beta_{\mathrm{tsvo}}]-\alpha_{\mathrm{tsvo}}m }{ \psi x+1 } = \]

\[ = \frac{ [\alpha_{\mathrm{tsvo}}+\beta_{\mathrm{tsvo}}\ln(1+m)-\beta_{\mathrm{tsvo}}\ln\psi-\alpha_{\mathrm{tsvo}}m\psi+2\beta_{\mathrm{tsvo}}] }{ \psi } - \]

\[ - \frac{ \frac{ [\alpha_{\mathrm{tsvo}}+\beta_{\mathrm{tsvo}}\ln(1+m)+\beta_{\mathrm{tsvo}}\ln\psi-\alpha_{\mathrm{tsvo}}m\psi+2\beta_{\mathrm{tsvo}}] }{ \psi } -\alpha_{\mathrm{tsvo}}m }{ \psi x+1 }. \]

If \(K_1=\infty\), then \(\psi x+1=0\), or \(x=-\dfrac{1}{\psi}\). If \(x=\infty\), then

\[ K_1=-\alpha_{\mathrm{tsvo}}m+\frac{4\beta_{\mathrm{tsvo}}}{\psi}. \]

For the numerical values of the particular example indicated below, we have \(K_1=1.4\) and \(x=-0.43\). The necessary point for constructing the hyperbola

Fig. 1

will be found by putting \(x=0\), when \(K_1=-0.074\). The hyperbola \(K_1\) is plotted in Fig. 1 as a function of \(x\).

As for the back-pressure coefficient \(K_r\), for it we have

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

if we denote \(z=c+m\), and by \(\varphi\) the coefficient characterizing the crushing of the steam by the moment at which compression begins, \(\varphi=1.1\). Knowing that \(z\) usually is found—

vary within the limits \(z=0.1 \div 0.4\), and, consequently, the multiplier at \(z\) under the sign \(\ln\), when expanded in a series with the second term of the expansion neglected, may be \(\psi_1=10 \div 2.5\); it is permissible to take \(\psi_1=5\), taking into account that \(K_r>1\), for which reason the inaccuracy due to omitting the second term of the expansion will be negligible. We then obtain

\[ \psi[2\varphi-1-\varphi\ln(m\psi_1)]z^2+ [\psi_1(1+m)-2\varphi-1-\varphi\ln(m\psi)]z- \]

\[ -\psi_1K_rz-K_r+1+m=0, \tag{2} \]

i.e., again the dependence \(K_r=f(z)\) is hyperbolic.
Putting \(z=0\), we find the point of the hyperbola necessary for its construction, when \(K_r=1+m=1.08\). To determine the asymptotes we have

\[ K_r=\frac{Az^2+\beta z+1+m}{\psi_1 z+1} =\frac{Az}{\psi_1} +\frac{\beta-\dfrac{A}{\psi_1}}{\psi_1} -\frac{\dfrac{\beta-\dfrac{A}{\psi_1}}{\psi_1}-1-m}{\psi_1 z+1}. \]

For \(K_r=\infty\) we obtain

\[ z=-\frac{1}{\psi_1}=-0.2. \]

For the second asymptote we have the equation of the straight line

\[ K_r=\frac{Az}{\psi_1}+\frac{\beta-\dfrac{A}{\psi_1}}{\psi_1} =[2\varphi-1-\varphi\ln(m\psi_1)]z+1+m-\frac{4\varphi}{\psi_1}. \]

The hyperbola is constructed in the same Fig. 1, with its position in the upper left corner of the rectangular coordinates (the second quadrant), for the reason evident from what follows.

Between the degree of admission \(\varepsilon\) and the degree of compression \(c\) there exists, in every steamship engine, its own definite dependence. The latter, for link motions that ensure equality of the linear lead of admission under changes of the degree of admission \(\varepsilon\) (Joy, Hackworth, Klug, etc.), is close to a straight line of the form \(c \cong a_1-b_1\varepsilon\), where \(a_1\) and \(b_1\) are certain constants less than unity, different for different engines. In Fig. 2, for example, such a dependence is shown for the HPC, of the form \(c=0.31-0.35\varepsilon\), and consequently,

\[ z=c+m=a_1+m+b_1m-b_1x \cong u_1-b_1x=0.42-0.35x. \]

When \(x\) is laid off along the abscissa axis in the first quadrant (Fig. 1), the values of \(z\) are obtained on the ordinate axis of this same quadrant, so that the values \(K_r\) for the values of \(z\), and consequently also \(x\), are conveniently placed on the abscissa axis of the second quadrant.

Turning now to changes of \(P_r\) as a function of \(x\), we write the equation expressing constancy of the weight quantity of steam passing through the engine,

\[ \gamma_1F_1(m_0+\varepsilon)=\gamma_rF_2(M_0+E), \tag{3} \]

where

\[ m_0=m-\frac{\rho P_r(c+m)}{P_1}, \qquad M_0=M-\frac{P_2(C+M)}{P_r}, \qquad \gamma_1=\zeta\cdot P_1, \qquad \gamma_r=\zeta\cdot P \]

and

\[ E=q+i\varepsilon, \]

where, in the case \(q=0\) and \(i=1\), we have \(E=\varepsilon\), as is recommended, for example, by P. G. Rufanov (Increasing the Efficiency of River Steam Engines, 1946, p. 51). Taking

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

to be the ratio of the cylinder volumes, we obtain from [3] the following equation:

\[ (\rho b_1-\delta i)P_r\cdot x+(\delta i m-\rho u_1-\delta M-\delta q)P_r+(P_1-i\delta P_2 b_2)x+ \]

\[ +P_2\delta(u_2-qb_2-Mb_2+imb_2)=0 \tag{4} \]

Here \(u_2\) and \(b_2\) in the LPC are the same as \(u_1\) and \(b_1\) in the HPC, i.e., in the LPC

Fig. 2

\[ Z=u_2-b_2X,\quad \text{where } X=E+M,\quad \text{and } Z=C+M. \]

Equation (4) is a hyperbola. To determine its asymptotes we have

\[ P_r=-\frac{F_1x+K_1}{A_1x+B_1} = -\frac{F_1}{A_1} - \frac{K_1-\dfrac{B_1F_1}{A_1}}{A_1x+B_1}. \]

For \(P_r=\infty\), \(x=-\dfrac{B_1}{A_1}\). For \(x=\infty\), \(P_r=-\dfrac{F_1}{A_1}\). The required point is found by setting \(x=0\), when \(P_r=-\dfrac{K_1}{B_1}\). In Fig. 3 the hyperbola is shown, with the following values adopted for a particular example:

\[ \rho=1.05;\quad b_1=0.35;\quad i=1;\quad M=m=0.08;\quad u_1=0.42;\quad q=0;\quad P_1=10\ \text{ata};\quad P_2=0.2\ \text{ata}; \]

\[ b_2=0.35;\quad u_2=0.42;\quad \delta=4. \]

The curve of the front pressure of the HPC, \(\sigma_{\text{hpc}}\cdot K_1P_1\), as a function of \(x\), may be the same hyperbola \(K_1=f_1(x)\), if the scale of the drawing is taken to be the scale of Fig. 1, reduced by a factor of \(\sigma_{\text{hpc}}\cdot P_1\). Multiplying the ordinates of the hyperbola \(P_r=f_2(x)\) by the corresponding ordinates of the hyperbola \(K_r=f(z)\) for the same abscissas \(x\) (Fig. 1) gives the curve of variation of the back pressure in the HPC, \(\sigma_{\text{hpc}}\cdot \rho\cdot K_rP_r\), as a function of \(x\), and the ordinates of the latter curve must-

must be plotted on the same scale as the curve \(\sigma_{\mathrm{ivo}}\cdot K_{1}\cdot P_{1}\). The difference between the ordinates of the curves

\[ \left(\sigma_{\mathrm{ivo}}\cdot K_{1}\cdot P_{1}-\sigma_{\mathrm{ivo}}\cdot K_{r}\cdot \rho\cdot P_{r}\right) \]

in Fig. 4, for any value of \(x\), determines the magnitude of the mean indicated pressure \(P_i'\) in the low-pressure cylinder, as a function of \(x\).

Fig. 3

Similarly we carry out the construction for the curves of the forward and back pressures in the high-pressure cylinder, starting from the basic equation \(P_i''=\sigma_{\mathrm{ind}}(K_{1}P_{r}-K_{2}P_{2})\). Here \(K_{1}=f_{4}(X)\) is the very same hyperbola \(K_{1}=f_{1}(x)\) as in Fig. 1, using

Fig. 4

which, however, should be used so that, for the assigned values

\[ X=E+M \]

on the negative ordinate axis (fourth quadrant), the corresponding values of \(x\) are found on the positive abscissa axis (first quadrant), using the straight line

\[ X=E+M=q+M-im+ix=Q+ix, \]

constructed in the fourth quadrant; and, from the values of \(x\), the corresponding ordinate values (in the first quadrant) are found from the hyperbola \(K_1\) in Fig. 1. Multiplying these latter ordinates by the ordinates of the curve \(P_r=f_2(x)\) in Fig. 3 for the same values of \(x\), and laying off the products to the scale of Fig. 4, we obtain the curve of variation of the front pressure in the LPC.

Fig. 5

\(\sigma_{ind}\cdot K_1\cdot P_r\), as functions of \(x\) (Fig. 5). Here we also construct the curve \(\sigma_{ind}\cdot K_2\cdot P_2\), as the product \(\sigma_{ind}\cdot P_2\) by \(K_2=f_5(x)\), for which we use Fig. 1, where, for the given value \(X=E+M\), we find \(x\), and through this latter quantity we determine \(Z\), whose values correspond to certain values \(K_2=K_r\). The ordinates of the curve \(\sigma_{ind}\cdot K_2\cdot P_2=f_6(x)\) are laid off in Fig. 5 to the scale of Fig. 4. The difference of the ordinates \(\sigma_{ind}\cdot K_1\cdot P_r-\sigma_{ind}\cdot K_2\cdot P_r\) gives \(P_i''\) as a function of \(x\). In Fig. 6 the curve of variation of the total mean indicated pressure is constructed,

\[ P_i'=\frac{P_i'}{\delta}+P_i'' \]

as a function of \(x\).

The example analyzed should be regarded as a special case illustrating the general method for solving such a rather complex problem of the variable operating regime of a steamship engine. It should not be overlooked here that the construction of the hyperbolas of the coefficients of front and back pressures can be performed once and for all for different values of the harmful spaces. What are specific, however, are the curves of dependence between the degrees of admission and compression of the HPC and LPC, and also, in connection with this, the curves of mean receiver pressure as a function of \(x\) in the HPC.

Fig. 6

Fig. 7

In the case of the use of superheated steam, the averaging coefficient of the mean pressure has the form

\[ K_n=\alpha_{\mathrm{hpc}}\cdot \varepsilon+ \frac{\beta_{\mathrm{hpc}}}{n-1}(\varepsilon+m) \left[1-\left(\frac{\varepsilon+m}{1+m}\right)^{n-1}\right]. \]

Here \(n\) is the polytropic exponent, depending on the pressure and temperature of the steam. Let us call

\[ y=\left(\frac{\varepsilon+m}{1+m}\right)^{n-1}. \]

Then

\[ \ln y=(n-1)\ln x-(n-1)\ln(1+m). \]

Introducing the coefficient \(\psi_n \ge 2\) and expanding \(\ln \psi_n x\) in a series, after transformations we obtain

\[ \ln y=(n-1)\left[2\left(1-\frac{2}{\psi_n x+1}\right)-\ln \psi_n(1+m)\right]. \]

The asymptote of the hyperbola for \(\ln y=\infty\) will be

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

and for \(x=\infty\)

\[ \ln y=(n-1)[2-\ln \psi_n(1+m)]. \]

If \(x=0\), then

\[ \ln y=(n-1)[-2-\ln \psi_n(1+m)]. \]

On the ordinate axis, opposite the values of \(\ln y\), their numbers must be placed and the dependence \(y=f(x)\) constructed. As a result we obtain

\[ K_n=x\left(\alpha_{\mathrm{hpc}}+\frac{\beta_{\mathrm{hpc}}}{n-1}\right) -\alpha_{\mathrm{hpc}}\cdot m -\frac{\beta_{\mathrm{hpc}}}{n-1}xy =Z_1-Z_2, \]

where

\[ Z_1=x\left(\alpha_{\mathrm{hpc}}+\frac{\beta_{\mathrm{hpc}}}{n-1}\right) -\alpha_{\mathrm{hpc}}\cdot m \quad \text{is the equation of a straight line.} \]

The curves

\[ Z_2=-\frac{\beta_{\mathrm{hpc}}}{n-1}xy \]

as functions of \(x\) can be constructed for different values of \(n\). Then the values for \(K_n\) are obtained as indicated in Fig. 7.

Equation (3), for constancy of the weight quantity of steam passing through the engine, in the case of superheated steam, is rewritten in the form

\[ \frac{T_1}{T_{\mathrm{sup}}}\cdot \gamma_1 F(m_0+\varepsilon) = \gamma_2 F_2(M_0+E), \]

where \(T_1\) and \(T_{\mathrm{sup}}\) are the absolute temperatures of saturated and superheated steam at the initial pressure in the high-pressure cylinder. In accordance with this, the hyperbola \(P_r=f_2(x)\) in Fig. 3 will take another position, and changes in the position of the curves \(\sigma_{\mathrm{hpc}}\), \(\rho\), \(K_r\), \(P_r\), and \(P_i'\) in Fig. 4 will result.

In conclusion, it is necessary to note that, with changes in the degrees of admission \(\varepsilon\) and \(E\) in the high-pressure cylinder and in the low-pressure cylinder, the coefficients \(\alpha\), \(\beta\), and \(\sigma\) will change to some extent; this can be determined only experimentally. In accordance with the generalization of experimental data, it will probably be necessary to introduce certain correction coefficients to the values \(P_i'\) and \(P_i''\) obtained on the basis of the considerations set forth above.

Noted Misprints