Prof. G. V. Trapeznikov

GRAPHICAL METHODS

for dividing the area of the indicator diagram of a double-expansion steam engine according to the principle of equality of initial forces on the pistons

Tomsk

1929

Prof. G. V. Trapeznikov.

Graphical Methods for Dividing the Area of the Indicator Diagram of a Double-Expansion Steam Engine According to the Principle of Equality of the Initial Forces on the Pistons

Of the three methods for dividing the area of the indicator diagram of a double-expansion steam engine: 1) according to equality of work in both cylinders, 2) according to equality of the temperature drops in them, and 3) according to equality of the initial forces on the pistons, the last, so far as is known to the author of this article, has no more or less exact analytical or graphical solution. The author proposes three new methods for such a division.

The problem is as follows: it is required to divide the area of the theoretical indicator diagram so that:

\[ P_{\mathrm{в}} = P_{\mathrm{н}} \ldots \ldots \ldots \ldots \ldots (1) \]

where \(P_{\mathrm{в}}\) is the initial force on the piston in the H.P.C.,
and \(P_{\mathrm{н}}\) is the same in the L.P.C.

Let (Fig. 1), on the available indicator diagram, \(p_x\) be the exhaust pressure in the H.P.C.; it is also the admission pressure in the L.P.C. If, at the same time, condition (1) is satisfied, then this will also be the pressure on the line dividing the diagram.

Fig. 1.

Calling:

\[ \begin{aligned} p_e&\text{— admission pressure} &&\left.\begin{array}{l}\\[-1.1em]\end{array}\right\}\ \text{in the H.P.C.}\\ O_v&\text{— piston area} &&\\ p_a&\text{— exhaust pressure} &&\left.\begin{array}{l}\\[-1.1em]\end{array}\right\}\ \text{in the L.P.C.}\\ O_n&\text{— piston area} && \end{aligned} \]

we have:

\[ P_v=P_v=(p_e-p_x)O_v=(p_x-p_a)O_n; \]

since, in constructing the indicator diagram, it is assumed:

\[ O_v:S_v=O_n:S_n, \]

where \(S_v\) and \(S_n\) are the strokes of the pistons of the H.P.C. and L.P.C.,

then:

\[ (p_e-p_x)S_v=(p_x-p_a)S_n \]

or:

\[ \frac{P_e-P_x}{P_x-P_a}=\frac{S_n}{S_v} \]

This proportion permits the construction of two similar triangles: \(ACG\) and \(FBD\),

where:

\[ \frac{CG}{FD}=\frac{AG}{FB} \]

the values of these segments are marked on the drawing.

Thus the straight line \(AB\), which divides the area of the diagram according to the condition of the problem, proves to be the line connecting the points of intersection: of the straight line from the center \(C\) with the compression curve, and of another straight line, parallel to the first and drawn from the center \(D\), with the expansion curve. Of all the straight lines that may be obtained as a result of drawing arbitrary parallels from \(C\) and \(D\), only one satisfies our condition, namely the one parallel to the axis of abscissas (volume, strokes). It will evidently be tangent at the maximum (or minimum) of the curve enveloping all the straight lines obtained by the above-described method. Since the analytical expression of the curve is sufficiently complicated, it may be constructed by fitting a curve to several tangents (Fig. 2). The tangent drawn parallel to the axis of abscissas will give

Fig. 2.

...division line. In practice there is no need to construct a large number of tangents. It is sufficient to take two straight lines, close to being parallel to the axis of abscissas (by eye), as for example (Fig. 2) \(AB\) and \(CD\), and to draw the division line through their point of intersection \(G\).

The second proposed method consists in constructing two integral curves, which in their intersection give a point whose ordinate is equal to the division pressure—\(P_x\).

Fig. 3.

Let us divide the area of the indicator diagram into a number of elements of equal height (Fig. 3); let us draw the straight line \(A_1B_1\), parallel to the axis of ordinates; it will serve as the zero line for the integral curves. From it, along the corresponding extended straight lines dividing the diagram, we shall lay off, on some scale, the quantities:

\[ (p_e - p)S. \]

Here \(p\) and \(S\) are, respectively, the exhaust pressure and the piston stroke of a certain fictitious single-cylinder machine which performs the work represented by that part of the area of the diagram which lies above the pressure line \(p\). In Fig. 3 such a part is hatched.

Since, in constructing the integral curve, one has to add, going downward (decreasing \(p\)), elements of area of the same height, the differences \(p_e - p\) will increase, with the inclusion of each new element, by the constant value of the segment between the straight division lines.

Taking this segment as a conventional unit, we shall have, for the abscissas of the integral curve, counting from above:

\[ \begin{aligned} &\text{for the 1st section of the area}\ .\ .\ .\ (p_e-p_b)b = 1b,\\ &\text{“ the 2nd ” “}\ .\ .\ .\ (p_e-p_c)c = 2c,\\ &\text{“ the 2nd ” “}\ .\ .\ .\ (p_e-p_d)d = 3d, \end{aligned} \]

and so on. In the drawing the scale of the integral curve is taken smaller than in the diagram—by a factor of 3.

For constructing the second integral curve, we take, in the same sequence, the elements, beginning from below.

The abscissas of this curve will express, on the former scale,

\[ (p-p_a)S, \]

where \(p\) and \(S\) are the admission pressure and the piston stroke of a fictitious single-cylinder engine performing the work corresponding to the part of the area of our diagram between the pressures \(p\) and \(p_a\).

The abscissas of the curve being constructed will have magnitudes, successively, counting from below upward from the line \(ED\): \(v\), \(2u\), \(3t\), etc.

The upper and lower integral curves will intersect at a certain point \(C\). Drawing through this point a line parallel to the axis of abscissas, we shall divide the area of the indicator diagram in accordance with the condition of the problem.

The third method of dividing the area of the diagram consists in the anamorphic transformation of the integral curves and in the use of the expansion and compression curves as anamorphoses of the upper and lower integral curves.

Let (Fig. 3) the expansion curve of the indicator diagram be the anamorphosis of the upper integral curve. This means that each abscissa of the expansion curve, on a nonconstant scale in general, will, for the same ordinate, represent the abscissa of the upper curve (for brevity, we shall henceforth omit the word “integral”).

If we measure the abscissas of the expansion line from the line of the left clearance space—\(AB\), then, counting from above, we shall have, for the elements, the abscissas:

\[ \begin{array}{c} 1-b\\ 2-c\\ 3-d \end{array} \qquad \text{and so on.} \]

These abscissas must, on different scales, respectively represent the abscissas of the upper curve:

\[ \begin{array}{c} 1-b\\ 2-2c\\ 3-3d \end{array} \qquad \text{and so on.} \]

Consequently, the scale for representing the abscissas of the upper curve in the anamorphosis, when moving from top to bottom from the filling line, C.V.D., decreases linearly.

We shall regard the compression curve as the anamorphosis of the lower curve; in this case the abscissas must be measured from the line of the right clearance space—\(ED\). Comparing the compression line with the lower curve, we see that here too the scale of the anamorphic transformation changes linearly, decreasing from the exhaust line in C.N.D., when moving upward.

If now we reconstruct one of the anamorphoses in such a way that, for equal ordinates with the other anamorphosis, its scale of distortion of the integral curve is the same as that of the latter, then the point

the intersection of the anamorphosis of the transformed and untransformed curves gives an ordinate that gives the division pressure of the area of the diagram.

For graphical construction it is most convenient to leave the expansion curve unchanged, and to reduce the abscissas of the compression curve, reckoning them from \(ED\), to the scales of the corresponding abscissas of the expansion curve (reckoning from \(AB\)).

If the height of the area of the diagram is divided into \(m\) equal parts, then the height of each element will be:

\[ h=\frac{P_c-P_a}{m}. \]

Let us take a line parallel to the axis of abscissas and lying at \(n\) divisions from the admission line in the H.P.C.

The scale in which the abscissa \(EM\) (Fig. 4) of the expansion line represents the corresponding abscissa of the upper curve will be:

\[ \mathfrak{M}_{\mathrm{v}}=\frac{S_{\mathrm{v}}}{h n S_{\mathrm{v}}}=\frac{1}{n h}, \]

where \(S_{\mathrm{v}}\) is the piston stroke of the fictitious engine performing work corresponding to the area of the \(n\) upper elements of the diagram.

Fig. 4.

The scale of the abscissas of the compression line, \(FK\), is

\[ \mathfrak{M}_{\mathrm{n}}=\frac{S_{\mathrm{n}}}{h(m-n)S_{\mathrm{n}}}=\frac{1}{(m-n)h}, \]

where \(S_{\mathrm{n}}\) is the piston stroke of the fictitious engine performing work over the \(m-n\) lower elements of the diagram.

The ratio of the scales will indicate in what ratio the abscissa of compression \(FK\) must be changed so that, being an anamorphosis of the lower curve, it may have the same scale as the abscissa of the expansion line (as an anamorphosis as well):

\[ a=\frac{\mathfrak{M}_{\mathrm{в}}}{\mathfrak{M}_{\mathrm{н}}} =\frac{m-n}{n}. \]

When reduced to the common scale, the abscissa of the anamorphosis of the lower curve will have the value:

\[ x=a\,FK=\frac{m-n}{n}\,FK. \]

This abscissa must now be laid off from the line \(AB\), as the zero line for the abscissas of the expansion curve.

If we connect the center \(C\) with the point \(F\) by a straight line, and from the center \(A\) draw a straight line parallel to the first until it intersects the line \(EK\), then, from the similarity of the triangles \(ECK\) and \(EGA\), we obtain:

\[ \frac{EG}{FK}=\frac{EA}{CK} \]

whence

\[ EG=\frac{EA}{CK}\,FK \]

but

\[ EA=h(m-n) \]

\[ CK=hn \]

i.e.

\[ EG=\frac{m-n}{n}\,EK=x. \]

Thus the construction of the abscissas of the anamorphosis of the lower curve is reduced to finding a line segment as the fourth proportional to three.

There is no need to construct the entire anamorphosis of the lower curve; a small part of it near the point \(L\) of intersection with the expansion line is sufficient.

In Fig. 4 the diagram with a large filling is shown by a dotted line. We see that the anamorphosis of the lower curve is not changed by this.

Comparing the three methods proposed by the author, it should be noted that the first is the simplest, but less exact, although the accuracy of the construction is quite sufficient for practical purposes; the second is more exact, but has a certain drawback in the cumbersomeness of constructing the integral curves; the third surpasses the first two in accuracy (the first analytically and the second in a purely drafting sense), is simple, gives a compact drawing and the pressures of division for any varying filling of the engine.