PROCEEDINGS

of the Tomsk Technological Institute

of Emperor Nicholas II

Vol. 18. 1910. No. 2.

III.

V. G. Karpenko.

Graphical Construction of Thermal Diagrams of Internal Combustion Engines.

With 2 plates of drawings.

1–15.

Graphical Construction of Thermal Diagrams of Internal-Combustion Engines.

V. G. Karpenko.

Before proceeding to the exposition of my work, I must point out that it is a direct continuation and development of the work of Prof. V. L. Maleev, printed under the same title in 1908 in vol. XII of the Proceedings of the Tomsk Technological Institute. In order not to repeat what has already been analyzed by my predecessor, in the further exposition of my work I shall confine myself to references to the corresponding places, without dwelling on proofs of the propositions considered there. I also assume that persons reading my article are already familiar with the principle of constructing thermal diagrams according to the work of Prof. V. L. Maleev.

If we have some change of state of a gas, then for the initial state we may set the value of the entropy equal to zero. In fact, the initial state is characterized by the fact that no heat has yet been imparted to the gas, i.e. \(dQ = 0\); then the entropy is also zero with respect to the given change.

Thus, for the initial state one may take ¹)

\[ dS = \frac{dQ}{T_0} = 0, \tag{38} \]

where \(T_0\) is the temperature of the initial state of the gas.

Equation (38) represents the basic proposition for our further deductions. Since such a proposition exists, we have a particular value of the function when we can determine the constant \(C\). This condition is necessary for us in order to establish the beginning of the change of state of the gas relative to definite coordinates. As coordinates for the investigations we shall take the coordinates \(TS\). Every process of change of state

¹) We have adopted the numbering of equations and drawings in the order of the numbering in the work of V. L. Maleev: “Graphical Construction of Thermal Diagrams of Internal-Combustion Engines.”

gas, expressed in these coordinates, is characterized by how heat is supplied to the gas, and how much heat has been supplied to it. If there is a series of changes in the state of the gas, and the laws of the change are indicated, then, owing to the properties of gases, each process may be considered quite independently, since every new change depends not on the preceding one, but only on its own initial state. This proposition is important to us because it makes it possible each time to determine the constant in the expression for the entropy of the gas as soon as we pass to a new process of changes, and to consider the process with respect to new coordinates.

We shall confine ourselves to the consideration of only one entropy equation, namely equation \((12)\)²), which expresses \(S\) as a function of \(T\) and \(p\), \(S=f(T,p)\). It is more convenient than the other equations for the graphical construction of diagrams³).

Let us take the entropy equation

\[ S=c_p\lg_n T-AR\lg_n p+C; \tag{12} \]

substitute, in place of \(T\) and \(p\), the values of the initial state of the gas \(T_0\) and \(p_0\), and set it equal to zero

\[ S_0=c_p\lg_n T_0-AR\lg_n p_0+C=0, \]

whence

\[ C=-c_p\lg_n T_0+AR\lg_n p_0. \]

If we now substitute the value found for \(C\) in equation \((12)\), then for any state of the gas within the limits of the process under consideration we obtain for \(S\)

\[ S=c_p\lg_n T-c_p\lg_n T_0-AR\lg_n p+AR\lg_n p_0. \tag{39} \]

Such a change of \(S\) will take place only so long as \(c_p\) and \(R\) remain unchanged; but as soon as these quantities undergo a change, equation \((39)\) is no longer applicable, since it is not sufficient merely to replace \(c_p\) and \(R\) in equation \((39)\) by their new values; at the same time changes occur in \(T\) and \(p\), which are connected with \(c_p\) by the equations

\[ Pv=RT,\qquad c_p-c_v=AR. \]

This change, of course, could be expressed, but we would then lose the conception of the constant part of the integral, which can only be determined from a particular value of the function. We know the value of the entropy only in the initial state of the gas, when we take it to be equal to zero. This is what compels us to consider

²) The numbering of the formulas follows the work of Prof. V. L. Malyshev.
³) See on this ibid., p. 9.

a new process in new coordinates, which, as we shall see further on, not only does not hinder, but even simplifies the graphical construction of diagrams.

The initial state of the gas in the new process must be known; for this, in our case, from the available indicator diagram we must know the pressure and temperature, determining it from the characteristic equation

\[ Pv = RT. \]

When \(R\) changes, the temperature also changes; therefore the initial temperature of any new process in which we have a new gas constant \(R\) will differ from the temperature of the final state of the preceding process. In constructing thermal diagrams of internal-combustion engines, this change must be taken into account beforehand. This can be achieved by constructing the curve of temperatures as a function of the change of \(R\). Namely, for a series of values \(p\) of the indicator diagram, find the values \(T\) which correspond to \(R\) for the given composition of the gas. All the points obtained in this way are connected by a smooth curve, which indicates the change of temperature not only from the change of pressure and volume, but also from the change of \(R\). Therefore, dividing the cycle of the engine into a series of elementary processes, in each such process we have an initial temperature equal to the final temperature of the preceding process.

Thus, the new process of change of state of the gas, for which we have the heat capacity \(c_{p_1}\) and the gas constant \(R_1\), may be considered quite independently in the new coordinates. Entirely analogously to the preceding case, we substitute into equation (12) the values \(T_1\) and \(p_1\), as well as \(c_{p_1}\) and \(R_1\), and set equal to zero

\[ S_1 = c_{p_1}\lg_n T_1 - AR_1\lg_n p_1 + C = 0, \]

whence

\[ C = -c_{p_1}\lg_n T_1 + AR_1\lg_n p_1. \]

After substitution we obtain

\[ S = c_{p_1}\lg_n T - c_{p_1}\lg_n T_1 - AR_1\lg_n p + AR_1\lg_n p_1. \tag{40} \]

By this equation we determine the magnitude \(S\) for any state of the gas within the limits of the new process under consideration.

With a new change of \(c_p\) and \(R\), we again pass to new coordinates, i.e. we again consider the process independently; and so on.

If we begin to construct graphically these processes of change of the gas in the coordinates \(TS\), we shall obtain several areas \(Q_1, Q_2, Q_3,\ldots\), express-

constitute the heat imparted to or taken from the gas in the corresponding process. The sum of these areas represents the total quantity of heat that has participated in all the processes. If, at the end of the series of processes, the gas returns to the initial state \(p_0, v_0\), and \(T_0\), then the algebraic sum of all the areas will represent the heat converted into work. In order to add these areas graphically, it is most convenient to place them together so that the ordinate of the initial state of the gas of the second process coincides with the ordinate of the final state of the first process, and so on; that is, to obtain the total area of all the areas in Drawing 14 in the form of the area in Drawing 15.

Drawing 14.

Drawing 15.

In internal-combustion engines we encounter precisely the circumstance that, from a certain moment, \(c_p\) and \(R\) begin to vary, since during compression a combustible mixture is present, then combustion proceeds with a change in the composition of the gas and, finally, there are combustion products. If we wished to carry out the construction of the thermal diagram of an engine by the method indicated by Prof. V. L. Malev in (p. 7, Drawing 4), then we would first have to construct logarithmic curves for the change of state of the gas with the initial gas constant \(R\) and \(c\), \(ab\) and \(a'b'\) (Drawing 16), find from them the curve of change of state \(1—2\), then, with the change of \(c_p\) and \(R\), construct new logarithmic lines \(cd\) and \(c'd'\), taking as

Drawing 16.

from them the curve of change of state \(1—2\), then, with the change of \(c_p\) and \(R\), construct new logarithmic lines \(cd\) and \(c'd'\), taking as

take, as the beginning of the construction, the points \(c\) and \(c'\), and from them construct the curve of the new change of the gas, etc.

Since combustion of the mixture begins after compression, during which the composition of the gas is changing all the time, and consequently \(c_p\) and \(R\) as well, it will be necessary to construct not one but several logarithmic lines, dividing the combustion process into several elementary processes. The last logarithmic line will serve for constructing the curve of the change of the gas during expansion and exhaust.

We see that, in order to construct the thermal diagram, it is necessary each time to construct several logarithmic lines, which will require much time and labor. But this can be avoided if one has a table of logarithmic lines constructed in advance. Then, while observing the order of construction described above, one may, with the aid of tracing paper, transfer any logarithmic line from the tables to the drawing. To simplify the construction still further, one need not transfer the curve to the drawing, but may obtain the corresponding entropy values directly from the tables; but for this one must first study the properties of the tables.

We have constructed two tables: I is constructed according to the equation

\[ S_p = c_p \lg_n T - c_p \lg_n T_0, \tag{41} \]

and II according to the equation

\[ S_t = AR \lg_n p - AR \lg_n p, \tag{42} \]

Let us examine them in succession.

Table of logarithmic lines \(S_p\)). These logarithmic lines are constructed in the coordinates \(T S_p\). The origin of coordinates is taken as \(T = O\) and \(S_p = c_p \lg_n T_0\), where \(T = 288^\circ\) abs. or \(15^\circ\)C. Therefore the quantities \(S_p\), for any heat capacities whatever, at \(T = T_0\) become zero, or, in other words, intersect at one point on the axis of ordinates. This point will be \(T_0 = 288^\circ\) abs. From the point of intersection all the curves corresponding to different heat capacities diverge. The distance between two points corresponding to one and the same temperature, but lying on any two curves, we shall call the divergence of the curves and denote by \(\Phi_p\).

The divergence is equal to the difference of the values of \(S_p\) for different \(c_p\) and is a function of \(T\). This dependence may be obtained as follows: take the equation for some logarithmic curve

\[ S'_p = c_{p_n} \lg_n T - c_{p_n} \lg_n T_0, \]

then, for any other,

\[ S''_p = c_{p_m} \lg_n T - c_{p_m} \lg_n T_0 \]

*) See Table I, table of logarithmic lines I.

and take the difference in any way we like; let us put

\[ \Phi_p=S_p''-S_p=(c_{p_m}-c_{p_n})(\lg_n T-\lg_n T_0). \tag{43} \]

From this we see that the divergence of the curves is a function of \(T\) and of the difference of the heat capacities. We shall denote the difference of heat capacities in general by \(\Delta c_p\). Let us take \(\Delta c_p'\) and compose, as above, the equation for \(\Phi_p'\),

\[ \Phi_p'=\Delta c_p'(\lg_n T-\lg_n T_0), \]

and another equation, with \(\Delta c_p''\), for \(\Phi_p''\),

\[ \Phi_p''=\Delta c_p''(\lg_n T-\lg_n T_0) \]

and divide one equation by the other,

\[ \frac{\Phi_p'}{\Phi_p''} = \frac{\Delta c_p'(\lg_n T-\lg_n T_0)} {\Delta c_p''(\lg_n T-\lg_n T_0)} = \frac{\Delta c_p'}{\Delta c_p''}, \tag{44} \]

i.e. the divergences are related to one another as the differences of the heat capacities. This proposition makes it possible not to compute all the curves of the table, but to compute only two and construct the rest on the basis of equation (44). Since equal differences of heat capacities have been taken in the table for the logarithmic curves, the divergences of all the curves of the table at one and the same temperature will be equal.

When we begin to construct the curve of change of state of the gas for some process, we are not hindered by the fact that the origin of the coordinates of our process does not coincide with the origin of the coordinates of the logarithmic curves of the given table. Since this table is constructed on the assumption of investigating processes of internal-combustion engines, it should be expected that all processes in them will have a temperature higher than that at the beginning of these curves. Suppose now that the initial temperature of the process under investigation is equal to some \(T_0'\), and that the change proceeds along an isobar. According to our basic proposition, we must carry out the construction by setting the initial value of the entropy equal to zero, i.e. according to the equation

\[ S_1=c_p\lg_n T-c_p\lg_n T_0', \]

whereas, having taken the value of the entropy from the table, we shall have

\[ S_0=c_p\lg_n T-c_p\lg_n T_0; \]

but since the difference

\[ S_0-S_1=c_p\lg_n T_0'-c_p\lg_n T_0 \]

remains constant for every value of \(T\), this will not affect the curve of change of state of the gas.

If, from a certain moment, the heat capacity changes for us, then, using the table of logarithmic lines, we must shift the corresponding auxiliary logarithmic curve parallel to itself, so that it passes through the preceding auxiliary curve at the point by which the final change in the state of the gas was determined. For this it is necessary to determine the magnitude of the divergence of the curves at the given temperature in our table and, when plotting all subsequent \(S_p\), subtract it if the shift is made to the left, or add it if the shift is made to the right. Consequently, if we change the value of the heat capacity in equation (41) and add or subtract the shift, then we shall consider the new process with respect to a new auxiliary logarithmic curve passing through the final point of the preceding change of state. We must then plot all values of \(S_p\) from the former coordinates, whereby we shall at once carry out the summation of the heat areas indicated in Fig. 15. Thus the general form of the equation for \(S_p\), when our tables are used, will be

\[ S_p = c_p \lg_n T - c_p \lg_n T_0 \pm X_p, \tag{45} \]

where \(X_p\) is equal to the divergence of the curves at the given temperature.

The matter is much more complicated in determining the magnitude of the shift under a continuous change of \(c_p\), which we encounter in the combustion process. Since the temperature of the gases depends on the amount of heat released during the combustion of the working mixture, the changes of \(c_p\) may be made dependent on the temperature, since thereby we shall make \(c_p\) vary proportionally to the change in the composition of the gases. Since in advance we do not know whether combustion occupied only a strictly definite interval of time, or passed into the expansion line, we assume that the change in the composition of the gases, and consequently also in \(c_p\), occurred along the combustion line. This assumption is possible, since the change in composition during the afterburning of the mixture is comparatively negligible, and all the more negligible will be the change in \(c_p\). Accepting this, we divide the whole combustion process into a series of elementary processes and assume that, within the limits of each such process, the heat capacity does not change, but changes with the transition to the new process in accordance with the temperature. It is impossible to construct the combustion curve directly according to the final \(c_{p_{\mathrm{pr}}}\) of the combustion products, since the shift of the curves is not the same at different temperatures and depends on the magnitude of the divergence of the curves.

Let \(X_p\) be the required shift of the curves, \(\Delta c_p\) the difference between the heat capacities of the mixture \(c_{p_{\mathrm{sm}}}\) and of the products \(c_{p_{\mathrm{pr}}}\), and let \(n\) be the number into which we divide the whole combustion process.

V. G. Karpenko.

Let us consider the first elementary process in the \(1/n\)-th part of the entire combustion process. However small this elementary process may be, there nevertheless exists a difference in the divergence of the auxiliary logarithmic curves for the initial and final states of the gas. The displacement of the curves may, with sufficient accuracy, be considered equal to some mean value of the divergence over this interval of change,

\[ X_{p_1}=\Phi_{p_1}=\frac{\Phi'_{p}+\Phi''_{p}}{2}, \tag{46} \]

corresponding to some temperature \(T_1\), but not to the arithmetic mean temperature of the initial and final states of the gas during the elementary process.

Drawing 17

Let us now write the equation expressing the dependence of \(\Phi_{p_1}\) on \(T_1\) according to equation (43):

\[ \Phi_{p_1}=(c_{p_2}-c_{p_1})(\lg_n T_1-\lg_n T_0); \]

this expression may be rewritten in another form if we take

\[ c_{p_2}-c_{p_1}=\frac{c_{p_{\mathrm{cm}}}-c_{p_{\mathrm{np}}}}{n}=\frac{1}{n}\Delta c_p; \]

then we have

\[ \Phi_{p_1}=\frac{1}{n}\Delta c_p(\lg_n T_1-\lg_n T_0). \tag{47} \]

This value \(\Phi_{p_1}\) will also be the first displacement \(X_{p_1}\).

The graphical magnitude of the second displacement in Drawing 18 is expressed by the straight-line segment \(2-2''\), since the curve \(O1_3\) must be shifted parallel to itself into the equidistant position \(2-3\). We can divide the entire displacement into the magnitude \(2-2'\), which is equal to the first displacement owing to the equidistance of \(1-2\) and \(1'-2'\), and into \(2'-2''\), which represents the divergence \(\Phi_{p_2}\) at \(T_2\). Let us express the new divergence \(\Phi_{p_2}\) as the sum of the divergence \(\Phi_{p_1}\) and the difference of divergences \(\Phi_{p_2}-\Phi_{p_1}\). Graphically this is obtained if one draws the curve \(1''-2''\) equidistant to \(O1_2\); then \(2''-2' = 1'-1'' + 1-1'\), while \(2''-2'''\) is the difference of the divergences.

Drawing 18.

Analytically this is represented as:

\[ \Phi_{p_2}=(c_{p_2}-c_{p_1})(\lg_n T_1-\lg_n T_0)+(c_{p_3}-c_{p_2})(\lg_n T_2-\lg_n T_1); \]

this equation, knowing that

\[ c_{p_2}-c_{p_1}=c_{p_3}-c_{p_2}=\frac{1}{n}\Delta c_p, \]

can be rewritten as

\[ \Phi_{p_2}=\frac{1}{n}\Delta c_p(\lg_n T_1-\lg_n T_0)+\frac{1}{n}\Delta c_p(\lg_n T_2-\lg_n T_1) \]

or, what is the same,

\[ \Phi_{p_2}=\frac{1}{n}\Delta c_p(\lg_n T_2-\lg_n T_0). \]

Consequently, the second displacement is equal to

\[ X_{p_2}=\Phi_{p_1}+\Phi_{p_2}. \tag{48} \]

The third displacement will be greater than the preceding one by the amount of the divergence \(\Phi_{p_3}\) at \(T_3\), which is equal to

\[ \Phi_{p_3}=\frac{1}{n}\Delta c_p(\lg_n T_2-\lg_n T_0)+\frac{1}{n}\Delta c_p(\lg_n T_3-\lg_n T_2) \]

or

\[ \Phi_{p_2}=\frac{1}{n}\Delta c_p(\lg_n T_3-\lg_n T_0). \]

The third displacement will be equal to

\[ X_{p_3}=\Phi_{p_1}+\Phi_{p_2}+\Phi_{p_3}, \tag{49} \]

and so on; and, finally, \(\Phi_{p_n}\) at \(T_n\) will be expressed as

\[ \Phi_{p_n}=\frac{1}{n}\Delta c_p(\lg_n T_n-\lg_n T_0); \]

and the displacement, expressed by the sum of all \(\Phi_{p_i}\), will take the form

\[ X_{p_n}=\sum \Phi_{p_i}=\frac{1}{n}\Delta c_p \lg_n(T_1\cdot T_2\cdot T_3\ldots T_n)-\Delta c_p\lg_n T_0, \tag{50} \]

which can be rewritten as

\[ X_{p_n}= \frac{\Delta c_p(\lg_n T_1-\lg_n T_0)+\Delta c_p(\lg_n T_2-\lg_n T_0)+}{n} \]

\[ +\ldots+\frac{\Delta c_p(\lg_n T_n-\lg_n T_0)}{n}. \tag{50a} \]

Consequently, the final displacement along the \(S_p\) curves is equal to the arithmetic mean of all the divergences, within the limits of temperature variation, of the initial and final logarithmic curves.

Indeed—the equation

\[ \Phi_{p_i} = \Delta c_p (\lg_n T_i - \lg_n T_0) = (c_{p_{\mathrm{cm}}} - c_{p_{\mathrm{pr}}})(\lg_n T_i - \lg_n T_0) \]

expresses the divergence, for any \(T_i\), between the curve of the mixture and that of the products of combustion.

All equations from (47) to (50) make it possible to compute the magnitude of the displacement for each elementary process. Since in constructing the diagram we determine all displacements from the table graphically, it is important for us only to verify analytically the final displacement by equation (50).

Since the assumption made in composing equation (46) is given in the form of the supposition that the divergences in the elementary process are small, the accuracy of the curve of change of state of the gas during combustion will be the greater, the larger the number \(n\) of elementary processes. In practice the construction is sufficiently accurate for \(n = 4\), since a further increase of \(n\) changes the magnitude of the displacement so little that graphically it is imperceptible. When substituting the values of \(T\) into equation (50), we must first compute \(T_1\) from equation (46), while the remaining values of \(T\) may be obtained if we add to \(T_1\) the temperature difference between the initial and final values for the elementary process, which will be equal in all processes.

Table of logarithmic lines \(S_t\)). These logarithmic lines are constructed in the coordinates \(P S_t\). As the origin of coordinates, \(p = 0\) and \(S_t = A R \lg_n p_0 = 0\) are adopted, where \(p_0 = 1\). All values of \(S_t\), for any gas constants \(R\), will intersect at one point on the axis of ordinates, once \(p\) is made equal to \(p_0\). The point of intersection will be \(p_0 = 1\). From this point all curves diverge, and the divergence, which we shall denote by \(\Phi_t\), will depend on \(p\) and \(R\). Let us take the equation of some logarithmic line

\[ S_t' = A R_n \lg_n p - A R_n \lg_n p_0, \]

where \(A R_n \lg_n p_0 = 0\); therefore

\[ S_t' = A R_n \lg_n p; \]

then let us take the equation of some other logarithmic line

\[ S_t'' = A R_m \lg_n p; \]

*) See Table I, logarithmic lines, II.

the difference of these two equations will be the divergence at \(p\)

\[ \Phi_t = S_t'' - S_t' = (R_m - R_n) A \lg_n p; \tag{51} \]

denote the difference of the gas constants by \(\Delta R\); then equation (51) may be rewritten as

\[ \Phi_t = \Delta R A \lg_n p . \tag{51a} \]

Let us take

\[ \Phi_t' = \Delta R' A \lg_n p,\quad \Phi_t'' = \Delta R'' A \lg_n p \]

and divide one by the other

\[ \frac{\Phi_t'}{\Phi_t''} = \frac{\Delta R' A \lg_n p}{\Delta R'' A \lg_n p} = \frac{\Delta R'}{\Delta R''} \tag{52} \]

Equations (51) and (52) give the dependence of the magnitude of the divergence \(\Phi_t\) on \(p\) and \(R\).

Comparing equations (51) and (52) with equations (43) and (44), we see that the properties of the curves \(S_t\) are the same as the properties of the curves \(S_p\). Therefore the construction of the curves \(S_t\) has been carried out in the same manner as that in which the curves \(S_p\) were constructed. Moreover, we see that all further propositions will remain the same for determining \(S_t\) in constructing thermal diagrams as were derived for the curves \(S_p\).

These propositions are as follows:

a) the course of the curve of change of state of a gas in a thermal diagram is not affected by whether the initial pressure of the process under investigation coincides with \(p_0\) or not;

b) with a change in \(R\) we must shift the corresponding curve into a position equidistant from itself by the magnitude of the divergence of the curves at the given \(p\);

c) the plotting of the values \(S_t\) of a new process must be carried out from the former coordinates, if \(S_t\) is calculated from the equation

\[ S_t = AR \lg_n p \pm X_t, \tag{53} \]

where \(X_t\) is equal to the divergence of the curves at the given \(p\);

d) with a continuous change in the composition of the gases and with a continuous change in \(R\), the process must be regarded as consisting of a series of elementary processes, for which the shift is to be determined successively as a function of the preceding shifts.

The first shift \(X_{t_1}\) will be obtained analogously to the shift \(X_{p_1}\). It will be equal to the difference of the divergences and will be expressed as

\[ X_{t_1} = \Phi_{t_1} = \frac{1}{n}\Delta R A \lg_n p_1, \tag{54} \]

where \(\Delta R\) is the difference \(R_{cm} - R_{np}\).

Second shift

\[ X_{t_2}=\Phi_{t_1}+\Phi_{t_2}, \tag{55} \]

where \(\Phi_{t_2}\) is the discrepancy at \(p_2\) and is equal to

\[ \Phi_{t_2}=\frac{1}{n}\triangle R A \lg_n p_2 . \]

Third shift

\[ X_{t_3}=\Phi_{t_1}+\Phi_{t_2}+\Phi_{t_3}, \tag{56} \]

where \(\Phi_{t_3}\) is the discrepancy at \(p_3\) and is equal to

\[ \Phi_{t_3}=\frac{1}{n}\triangle R A \lg_n p_3, \]

and so on; and, finally, the \(n\)-th shift,

\[ X_{t_n}=\Phi_{t_1}+\Phi_{t_2}+\Phi_{t_3}+\ldots+\Phi_{t_n}. \tag{57} \]

This equation may be written in the following form:

\[ X_{t_n}=\sum \Phi_{t_i}=\frac{1}{n}\triangle R A \lg_n(p_1\cdot p_2\cdot p_3\ldots p_n) \tag{57a} \]

Consequently, the final shift along the curve \(St\) is equal to the arithmetic mean of all the discrepancies within the limits of variation of the pressures of the initial and final logarithmic curves.

Knowing how to find the values \(Sp\) and \(St\) from the corresponding tables, we can also find the magnitude of the entropy \(S\) from the difference of these quantities. From equations (45) and (53) one can compose an equation for \(S\), when plotting the entropy values from the same coordinates. This equation is the following:

\[ S=c_p \lg_n T-c_p \lg_n T_0+(\pm X_p)-AR\lg_n p-(\pm X_t), \tag{58} \]

where \(X_p\) and \(X_t\) are determined each time in passing to new logarithmic curves. The sign used is that which will express a shift to the right or to the left.

Example. Construct a thermal diagram for the working process of a machine operating according to the Diesel cycle.

Let us take the mean compression ratio \(\varepsilon=15.5\), the initial compression pressure \(p_a=0.9\), the initial compression temperature \(T_a=335^\circ\) abs., and, on the basis of the ratios of the quantities of oil and air, compute the values \(c_{p\,c.m}=0.24\), \(R_{c.m}=29.4\), \(c_{p\,p.r}=0.27\), and \(R_{p.r}=31.4\).

First we construct graphically the indicator diagram and, from it, the temperature curve. In constructing the temperature curve for the process—

for combustion, we must take into account the circumstance that the gas constant \(R\) changes all the time. For this we connect the points \(r_1\) and \(r_2\) (see the Diesel diagram), by an inclined line and determine the temperatures on it. We divide the whole combustion process into four elementary processes and, correspondingly, find the values of the temperatures and pressures for four points of the initial state of the gas. Before constructing the thermal diagram, from some vertical line we draw a series of horizontal lines corresponding to the found values of the temperatures).

The construction of the thermal diagram is usually begun with the beginning of compression and continued in the order in which the whole cycle of changes proceeds. As was indicated earlier, we do not need to shift the first auxiliary logarithmic curve, since the difference between the tabular entropy and that required for the process will, each time it is set off, enter as a constant quantity and will not affect the curve of change of state of the gas. Therefore, for compression the quantities in equation (58), \(X_p\) and \(X_t\), will be taken equal to zero. Following equation (58), we must take \(S_t\) from Table II along the logarithmic curve for \(R_{cm}=29.4\) and subtract it from the quantity \(S_p\), taken from Table I along the curve for \(c_{p,cm}=0.24\). Since \(S_t\) in Table II is obtained as a negative quantity at \(p_a=0.9\), in equation (58) it enters with a plus sign, and we shall have to add its linear magnitude to the linear magnitude \(S_p\). It is not difficult to see that all these operations can be determined with the aid of dividers. The entropy value thus obtained we set off in Fig. 19 from the axis \(OT\). The next point for us will have \(T=360^\circ\) abs. and \(p=1.1\). Here the quantity \(S_t\) from Table II is obtained positive, and we must subtract it from \(S_p\), i.e. it will enter equation (58) with a minus sign. In this way we arrive at the final point of compression. With the beginning of combustion we shall have to change \(c_p\) and \(R\). For this we divide the difference \(c_{p,np}-c_{p,cm}\) and \(R_{np}-R_{cm}\) into four parts, depending on the four found temperature values, and obtain the corresponding values of \(c_p\) and \(R\)

It is convenient to have such a table before one during the construction. Now, when passing to the new auxiliary curves, we need each time to find the magnitude of the shift. In the present case—

*) Table II.

this amount of displacement will enter equation (58) with a minus sign, since in constructing we must shift all subsequent curves to the left. Consequently, before finding the entropy value for the point \(p=35.2\) and \(T=1100^\circ\) abs., we find \(X_p\) and \(X_t\), which are equal to the arithmetic mean of the divergences of the curves at \(T=780\) and \(p=35.6\), and \(T=1100^\circ\) and \(p=35.2\). When the first displacement has been found, we find, as before, from Table II, \(S_t\) along the curve \(R=30.0\), subtract from this value the displacement \(X_t\), subtract the difference from \(S_p\), found along the curve \(c_p=0.248\), and from the new difference subtract \(X_p\). With the transition to new curves we again find the displacement and proceed to determine the entropy as before. Since the displacement is always equal to the sum of the new and preceding divergences of the curves (equations 50 and 57a), all the values of the divergences must be laid off somewhere on a straight line, so that they can be summed with the aid of compasses.

We construct expansion and exhaust by taking the final displacement along the curves \(c_{p\,pr}=0.27\) and \(R_{pr}=31.4\).

We see that the thermal diagram of the Diesel did not close; this occurred because the products of combustion differ in heat capacity \(c_p\) and gas constant \(R\) from the compressed and working mixture. This same circumstance also affected the fact that the obtained area \(abcde a'\) proved to be \(4\%\) smaller than the area \(a'b'c'd'e'a'\), constructed with constant \(c_{p\,pr}\) and \(R_{pr}\). This is as it should be, because with an increase in the heat capacity and the constant \(R\) of the products of combustion, the amount of heat carried away with the exhaust gases will also increase.

In order to determine the amount of heat converted into work, we must determine with a planimeter the area \(kbcdek\) and divide by the scale of the unit of heat.

The heat scale, if the process is considered for one kg of mixture, is equal to:

\[ m_q=m_s\cdot m_t=500\cdot 0.05=25\ \text{mm.}=1\ \text{T. unit}. \]

If the area \(kbcdek\) is divided by \(m_q\), we obtain the number of T. units expressed by this area:

\[ 4220:25=168.5\ \text{T. units}. \]

We take the calorific value of petroleum to be 10800 cal. per 1 kg. Since petroleum enters the mixture with air by weight as \(1:21.5\), the calorific value of one kg of mixture will be:

\[ H_u=\frac{10800}{22.5}=480\ \text{T. units}. \]

Thus, the thermal indicator efficiency will be equal to:

\[ 168{,}5 : 480 = 0{,}352. \]

From a comparison of the two areas \(kbcdek\) and \(a'b'e'd'e'a'\), we see that the difference obtained is not great, whether or not we take into account the changes in \(c_p\) and \(R\) during the process. Therefore, in those cases where these changes are still smaller than in the example given here, one may disregard changes in heat capacity and in the gas constant and construct the thermal diagram of the engine, considering \(R\) and \(c_p\) to be constant.

Tomsk, November 28, 1909.

Diagram II.

For the article by Eng. Vl. Karpenko, “Graphical Construction

[Plate: a ruled graphical chart with oblique families of lines labeled with decimal values such as 0.23–0.40, 0.45–0.49, and 0.30–0.40.]

Plate 1.

“Tables of logarithmic lines”

Scales, entropies: \(1 \text{ sq. dec.} = 500 \text{ mm}\).

“temperatures: \(1^\circ \mathrm{C} = 1/5 \text{ mm}\).

“pressures: \(1 \text{ atm.} = 10 \text{ mm}\).

Table I.

Proceedings of the St. Petersburg Institute of Railway Engineers.
V. G. Karpenko. “Graphical Construction of Diagrams of Internal-Combustion Engines.”
Plate II.

Drawing 19?

Thermal indicator diagram of a Diesel engine.

Visible handwritten annotations:

\[ m_v = 3 \ \mathrm{m/m} = 1 \ \mathrm{kg/cm^2}; \qquad n_v = 11 \ \mathrm{m/m}; \]

\[ \varepsilon = 15.5; \qquad \alpha_1 = 2.9; \qquad \alpha_2 = 3.4; \]

\[ m_v = 9.05 \ \mathrm{m/m} = 1 \ \mathrm{kg/cm^2}; \qquad m_v = 500 \ \mathrm{m/m} = 1 \ \mathrm{m^3/kg}; \]

\[ m_v = 9.5 \ \mathrm{m/m} = 1 \ \mathrm{kg/cm^2}; \qquad \alpha_1 = 0.62; \qquad \alpha_2 = 0.27. \]

Diagram labels visible in the plate include points and curves marked:

\[ a,\ a',\ a'',\ a''',\ b,\ b',\ c,\ c',\ d,\ d',\ e,\ e',\ z_1,\ z_2 \]

and temperature/scale markings such as:

\[ 355,\ 390,\ 430,\ 460,\ 500,\ 570,\ 640,\ 710,\ 780,\ 920,\ 1000,\ 1130,\ 1310,\ 1370,\ 1400,\ 1450,\ 1465,\ 1780,\ 1800. \]