On the Question of Selecting the Cylinder Volume Ratio of Double-Expansion Steamship Engines

I. N. Butakov and E. N. Shadrin

In steam engines of the river fleet, in selecting the periods of steam action in the cylinders, we are constrained by the conditions of the steam-distribution gear. Thus, for example, a box-type slide valve can usually provide a degree of admission of not less than 50--60% of the piston stroke; with Trick-Weyss slide valves, the degree of admission is usually not less than 35% of the piston stroke, and so on. These conditions predetermine the choice of the cylinder volume ratio of compound engines, being connected with the requirement to distribute the work, as far as possible, equally between the cylinders.

Proceeding from equality of the weight quantity of saturated steam passing through the engine, and assuming approximately that the clearance steam consumption will be the same for both cylinders, we have

\[ \gamma_i O_{\mathrm{HPC}}(m_0+\varepsilon)=\gamma_r O_{\mathrm{LPC}}(M_0+E). \]

Let \(\varepsilon\) and \(E\) be the degrees of admission, in fractions of the piston stroke, in the HPC and LPC; \(c\) and \(C\) the degrees of compression; \(m\) and \(M\) the clearance volumes, in fractions of the working volume in the cylinders; \(P_1\) and \(P_2\) the pressures of steam admission into the HPC and exhaust from the LPC into the condenser, in ata; \(P_r\) the pressure, in ata, at admission into the LPC; \(\rho=1.05\) a coefficient characterizing the increase of back pressure in the HPC relative to \(P_r\). Then

\[ m_0=m-\frac{\rho P_r(c+m)}{P_1}, \qquad M_0=M-\frac{P_2(C+M)}{P_r}. \]

Assuming approximately \(\gamma_i=\mu P_1\) and \(\gamma_r=\mu P_r\), and also calling

\[ \delta=\frac{O_{\mathrm{LPC}}}{O_{\mathrm{HPC}}} \]

the cylinder volume ratio, we obtain

\[ \delta P_r(E+M)+\rho(c+m)P_r-\delta P_2(C+M)-P_1(m+\varepsilon)=0 \tag{1} \]

As can be seen, the dependence between \(P_r\) and \(\delta\) is hyperbolic. From equation (1) we determine

\[ \delta= \frac{P_1(m+\varepsilon)-\rho P_r(c+m)} {P_r(E+M)-P_2(C+M)} = -\frac{\rho(c+m)}{E+M} + \frac{ P_1(m+\varepsilon)-\dfrac{\rho P_2(C+M)(c+m)}{E+M} } {P_r(E+M)-P_2(C+M)}. \]

If \(P_r=\infty\), then

\[ \delta=-\frac{\rho(c+m)}{E+M}. \]

If \(\delta=\infty\), then

\[ P_r=\frac{P_2(C+M)}{E+M}. \]

The point needed for constructing the hyperbola is found by setting \(\delta=0\). Then

\[ P_r=\frac{P_1(m+\varepsilon)}{\rho(c+m)}. \]

The hyperbola is constructed in Fig. 1 for the particular example given below.

For the mean indicated pressure of the high-pressure cylinder we write

\[ P_i'=\sigma_{\mathrm{hpc}}\left(k'P_1-k_2\rho P_r\right)= \sigma_{\mathrm{hpc}}\left\{ \left[ \alpha_{\mathrm{hpc}}\varepsilon+ \beta_{\mathrm{hpc}}(\varepsilon+m)\ln\frac{1+m}{\varepsilon+m} \right]P_1-\right. \]

\[ \left. -\rho\, \frac{P_1(m+\varepsilon)+\delta P_2(C+M)} {\delta(E+M)+\rho(c+m)} \cdot \left[ 1-c+(c+m)\ln\frac{m+c}{m} \right] \right\}. \]

Fig. 1

which is transformed as follows:

\[ (E+M)\delta P_i' + \rho(c+m)P_i' - F\delta - Q = 0, \tag{2} \]

where

\[ F=\sigma_{\mathrm{hpc}}(E+M) \left[ \alpha_{\mathrm{hpc}}\varepsilon+ \beta_{\mathrm{hpc}}(\varepsilon+m)\ln\frac{1+m}{\varepsilon+m} \right]P_1- \]

\[ -\sigma_{\mathrm{hpc}}\rho P_2(C+M) \left[ 1-c+(c+m)\ln\frac{m+c}{m} \right], \]

and

\[ Q=\rho\sigma_{\mathrm{hpc}}(c+m) \left[ \alpha_{\mathrm{hpc}}\varepsilon+ \beta_{\mathrm{hpc}}(\varepsilon+m)\ln\frac{1+m}{\varepsilon+m} \right]P_1- \]

\[ -\rho P_1(m+\varepsilon)\sigma_{\mathrm{hpc}} \left[ 1-c+(c+m)\ln\frac{m+c}{m} \right]. \]

Equation (2) again characterizes a hyperbola. We determine

\[ \delta= \frac{Q-\rho(c+m)P_i'}{(E+M)P_i'-F} = -\frac{\rho(c+m)}{E+M} + \frac{Q-\dfrac{F\rho(c+m)}{E+M}} {(E+M)P_i'-F}. \]

If \(\delta=\infty\), then \(P_i'=\dfrac{F}{E+M}\). If \(P_i'=\infty\), then \(\delta=-\dfrac{\rho(c+m)}{E+M}\). The point necessary for constructing the hyperbola is found by setting \(P_i'=0\). Then

\[ \delta = -\frac{Q}{F} \]

Finally, for the mean indicated pressure of the LPC we obtain

\[ P_i''=\sigma_{\text{LPC}}\left(k''P_r-k_2P_2\right)= \]

\[ =\sigma_{\text{LPC}}\left\{\left[\alpha_{\text{LPC}}E+\beta_{\text{LPC}}(E+M)\ln\frac{1+M}{E+M}\right] \left[\frac{P_1(m+e)+\delta P_2(c+m)}{\delta(E+M)+\rho(c+m)}\right]\right. \]

\[ \left.-P_2\left[1-C+(C+M)\ln\frac{M+C}{M}\right]\right\}, \]

whence

\[ (E+M)\delta\cdot P_i''+\rho(c+m)P_i''-P\delta-S=0, \tag{3} \]

i.e., here too the relation between \(P_i''\) and \(\delta\) is hyperbolic, with

\[ P=\sigma_{\text{LPC}}P_2(C+M)\left[\alpha_{\text{LPC}}E+\beta_{\text{LPC}}(E+M)\ln\frac{1+M}{E+M}\right]- \]

\[ -\sigma_{\text{LPC}}P_2(E+M)\left[1-C+(C+M)\ln\frac{M+C}{M}\right] \]

and

\[ S=\sigma_{\text{LPC}}P_1(m+e)\left[\alpha_{\text{LPC}}E+\beta_{\text{LPC}}(E+M)\ln\frac{1+M}{E+M}\right]- \]

\[ -\sigma_{\text{LPC}}P_2\rho(c+m)\left[1-C+(C+M)\ln\frac{M+C}{M}\right]. \]

To find the asymptotes we have

\[ \delta=\frac{S-\rho(c+m)P_i''}{(E+M)P_i''-P} = -\frac{\rho(c+m)}{E+M} - \frac{\dfrac{\rho(c+m)P}{E+M}-S}{(E+M)P_i''-P}. \]

If \(\delta=\infty\), then \(P_i''=\dfrac{P}{E+M}\). If \(P_i''=\infty\), then \(\delta=-\dfrac{\rho(c+m)}{E+M}\). The point required for constructing the latter hyperbola, in order to reduce the scale of the drawing, should expediently be determined from the condition \(P_i'=P_i''\), which determines the equality of \(\delta\), i.e.,

\[ \frac{Q-\rho(c+m)P_i'}{(E+M)P_i'-F} = \frac{S-\rho(c+m)P_i''}{(E+M)P_i''-P}. \]

Replacing \(P_i''\) on the right-hand side by the equal quantity \(P_i'\), we obtain

\[ P_i'=\frac{QP-SF}{(E+M)(Q-S)+\rho(c+m)(P-F)}, \]

which corresponds to a definite point on the hyperbola \(P_i'\), and which we take as the desired point for constructing the hyperbola \(P_i''\).

Having constructed the hyperbolas for \(P_i'\) and \(P_i''\) as functions of \(\delta\), and reducing the ordinates of the hyperbola \(P_i'\) by \(\delta\) times, we find the point of intersection of these hyperbolas; its abscissa gives the required volume ratio \(\delta\), ensuring equality of work in both cylinders.

In Fig. 2 the hyperbolas \(P_i'\) and \(P_i''\) are constructed for a particular example:

\[ e=E=0.6;\quad c=0.08;\quad C=0.15;\quad \beta_{\text{HPC}}=0.85;\quad \beta_{\text{LPC}}=0.80;\quad \rho=1.05;\quad m=0.12; \]

\[ M=0.08;\quad P_1=10\ \text{ata};\quad P_2=0.2\ \text{ata};\quad \sigma_{\text{HPC}}=0.96;\quad \sigma_{\text{LPC}}=0.93;\quad \alpha_{\text{HPC}}=0.93; \]

\(\delta_{\text{ind}} = 0.90\). The intersection of the hyperbolas is obtained at \(\delta \simeq 6.3\), when \(P_i'/\delta = P_i''\). In the practice of designing river steam engines, most often \(\delta = 3\text{—}4.5\) is adopted. As is seen from Fig. 2, under these conditions \(P_i'' > \dfrac{P_i'}{\delta}\), which occurs at the degree of admission \(\varepsilon = E = 0.6\). In the case of a decrease in the degree of admission in the H.P. cylinder, and hence also in the L.P. cylinder, the pressure in the H.P. cylinder \(P_r\) will fall (Fig. 1), in accordance with which \(P_i''\) will decrease, so that an equalization of the powers in both cylinders will take place.

Fig. 2

From the numerical data of the particular example it is seen that in the L.P. cylinder smaller values are adopted: \(\delta_{\text{ind}} = 0.93\), \(\beta_{\text{ind}} = 0.8\), and the quantity \(\alpha_{\text{ind}}\), which depends on \(\beta_{\text{ind}}\). With respect to \(\delta_{\text{ind}}\), this is explained by the fact that the magnitude of the advance release in the L.P. cylinder is greater than in the H.P. cylinder, and also by the fact that in river engines the vacuum is usually not established immediately (at the beginning of the run it is worse). As for the smaller value of \(\beta_{\text{ind}}\), it is due to the preceding expansion of the steam in the receiver and in the L.P. cylinder, which creates a lowering of pressure, especially noticeable with small receiver dimensions before the instant of steam cut-off in the L.P. cylinder.

Turning now to superheated-steam engines, we may write the equation of constancy of the weight quantity of steam passing through the engines in the following form

\[ \gamma_{\text{sup}} O_{\text{hpc}}(\varepsilon + m_0) = \gamma_r O_{\text{lpc}}(E + M_0). \]

Since it may be assumed approximately that the specific volumes of superheated steam vary as for gases, i.e. \(\dfrac{1}{\gamma_{\text{sup}}} : \dfrac{1}{\gamma_1} \simeq T_{\text{sup}} : T_1\), where \(\gamma_1\) and \(T_1\) are the specific weight and absolute temperature of saturated steam of the same pressure, then

\[ p_1 \cdot \frac{T_1}{T_{\text{sup}}}(\varepsilon + m_0) = \delta \cdot p_r(E + M). \]

It is usually assumed that saturated steam already operates in the L.P. cylinder. After introducing corrections for superheated steam, from equation (1) we obtain the asymptotes: \(P_r = \dfrac{P_2(C + M)}{E + M}\) (remains

(without change) and

\[ \delta=-\frac{T_1/T_{\mathrm{per}}\cdot \rho(c+m)}{E+M}. \]

The asymptotes of hyperbola (2) take the form

\[ P_i'=\frac{F}{E+M} \]

(remains unchanged) and

\[ \delta=-\frac{T_1/T_{\mathrm{per}}\cdot \rho(c+m)}{E+M}, \]

where the point required for constructing the hyperbola is

\[ \delta=-\frac{T_1/T_{\mathrm{per}}\cdot Q}{F}. \]

Finally, for the asymptotes of hyperbola (3) we have, in the case of superheated steam,

\[ P_i''=\frac{P}{E+M} \]

(remains unchanged) and

\[ \delta=-\frac{T_1/T_{\mathrm{per}}\cdot \rho(c+m)}{E+M}. \]

The point of the hyperbola remains the same as for saturated steam. The hyperbolas in the case of superheated steam are constructed in the same Fig. 2 for the same particular example as for saturated steam, but are drawn with a dotted line. From Fig. 2 it is seen that \(P_i'/\delta\) turns out to be already higher than \(P_i''\). This is why the indication is entirely correct that, in the case of superheated steam, in order to equalize the power in the cylinders it is necessary to have a degree of filling in the low-pressure cylinder greater than in the high-pressure cylinder, since in this case \(P_r\) will increase (the dotted hyperbola in Fig. 1).